LMFDB - Embedded newform 784.2.w.d.619.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,2,Mod(19,784)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 9, 10]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("784.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	784.w (of order $12$, degree $4$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.26027151847$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			619.2
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	784.619
Dual form			784.2.w.d.19.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.36603 + 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(3.34607 + 0.896575i) q^{5} +(2.00000 + 2.00000i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})$	\(q+(1.36603 + 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(3.34607 + 0.896575i) q^{5} +(2.00000 + 2.00000i) q^{8} +(2.59808 - 1.50000i) q^{9} +(4.24264 + 2.44949i) q^{10} +(0.366025 + 1.36603i) q^{11} +(-2.44949 - 2.44949i) q^{13} +(2.00000 + 3.46410i) q^{16} +(-4.24264 - 2.44949i) q^{17} +(4.09808 - 1.09808i) q^{18} +(-6.69213 - 1.79315i) q^{19} +(4.89898 + 4.89898i) q^{20} +2.00000i q^{22} +(-2.00000 - 3.46410i) q^{23} +(6.06218 + 3.50000i) q^{25} +(-2.44949 - 4.24264i) q^{26} +(-3.00000 + 3.00000i) q^{29} +(-2.44949 + 4.24264i) q^{31} +(1.46410 + 5.46410i) q^{32} +(-4.89898 - 4.89898i) q^{34} +6.00000 q^{36} +(1.83013 - 6.83013i) q^{37} +(-8.48528 - 4.89898i) q^{38} +(4.89898 + 8.48528i) q^{40} +4.89898 q^{41} +(-5.00000 + 5.00000i) q^{43} +(-0.732051 + 2.73205i) q^{44} +(10.0382 - 2.68973i) q^{45} +(-1.46410 - 5.46410i) q^{46} +(2.44949 + 4.24264i) q^{47} +(7.00000 + 7.00000i) q^{50} +(-1.79315 - 6.69213i) q^{52} +(1.36603 - 0.366025i) q^{53} +4.89898i q^{55} +(-5.19615 + 3.00000i) q^{58} +(6.69213 - 1.79315i) q^{59} +(0.896575 - 3.34607i) q^{61} +(-4.89898 + 4.89898i) q^{62} +8.00000i q^{64} +(-6.00000 - 10.3923i) q^{65} +(-6.83013 + 1.83013i) q^{67} +(-4.89898 - 8.48528i) q^{68} +2.00000 q^{71} +(8.19615 + 2.19615i) q^{72} +(-4.89898 + 8.48528i) q^{73} +(5.00000 - 8.66025i) q^{74} +(-9.79796 - 9.79796i) q^{76} +(-3.46410 + 2.00000i) q^{79} +(3.58630 + 13.3843i) q^{80} +(4.50000 - 7.79423i) q^{81} +(6.69213 + 1.79315i) q^{82} +(-12.0000 - 12.0000i) q^{85} +(-8.66025 + 5.00000i) q^{86} +(-2.00000 + 3.46410i) q^{88} +14.6969 q^{90} -8.00000i q^{92} +(1.79315 + 6.69213i) q^{94} +(-20.7846 - 12.0000i) q^{95} +4.89898i q^{97} +(3.00000 + 3.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 16 q^{8}+O(q^{10}) $ 8 * q + 4 * q^2 + 16 * q^8	$ 8 q + 4 q^{2} + 16 q^{8} - 4 q^{11} + 16 q^{16} + 12 q^{18} - 16 q^{23} - 24 q^{29} - 16 q^{32} + 48 q^{36} - 20 q^{37} - 40 q^{43} + 8 q^{44} + 16 q^{46} + 56 q^{50} + 4 q^{53} - 48 q^{65} - 20 q^{67} + 16 q^{71} + 24 q^{72} + 40 q^{74} + 36 q^{81} - 96 q^{85} - 16 q^{88} + 24 q^{99}+O(q^{100}) $ 8 * q + 4 * q^2 + 16 * q^8 - 4 * q^11 + 16 * q^16 + 12 * q^18 - 16 * q^23 - 24 * q^29 - 16 * q^32 + 48 * q^36 - 20 * q^37 - 40 * q^43 + 8 * q^44 + 16 * q^46 + 56 * q^50 + 4 * q^53 - 48 * q^65 - 20 * q^67 + 16 * q^71 + 24 * q^72 + 40 * q^74 + 36 * q^81 - 96 * q^85 - 16 * q^88 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.36603	+	0.366025i	0.965926	+	0.258819i
$2$	1.36603	+	0.366025i	0.965926	+	0.258819i
$3$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$3$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$4$	1.73205	+	1.00000i	0.866025	+	0.500000i
$5$	3.34607	+	0.896575i	1.49641	+	0.400961i	0.911894	−	0.410425i	$-0.134620\pi$
$5$	3.34607	+	0.896575i	1.49641	+	0.400961i	0.584511	+	0.811386i	$0.301286\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	2.00000	+	2.00000i	0.707107	+	0.707107i
$9$	2.59808	−	1.50000i	0.866025	−	0.500000i
$10$	4.24264	+	2.44949i	1.34164	+	0.774597i
$11$	0.366025	+	1.36603i	0.110361	+	0.411872i	0.998898	−	0.0469323i	$-0.0149445\pi$
$11$	0.366025	+	1.36603i	0.110361	+	0.411872i	−0.888537	+	0.458804i	$0.848278\pi$
$12$		0			0
$13$	−2.44949	−	2.44949i	−0.679366	−	0.679366i	0.280491	−	0.959857i	$-0.409503\pi$
$13$	−2.44949	−	2.44949i	−0.679366	−	0.679366i	−0.959857	+	0.280491i	$0.909503\pi$
$14$		0			0
$15$		0			0
$16$	2.00000	+	3.46410i	0.500000	+	0.866025i
$17$	−4.24264	−	2.44949i	−1.02899	−	0.594089i	−0.112296	−	0.993675i	$-0.535820\pi$
$17$	−4.24264	−	2.44949i	−1.02899	−	0.594089i	−0.916696	+	0.399586i	$0.869154\pi$
$18$	4.09808	−	1.09808i	0.965926	−	0.258819i
$19$	−6.69213	−	1.79315i	−1.53528	−	0.411377i	−0.610543	−	0.791983i	$-0.709049\pi$
$19$	−6.69213	−	1.79315i	−1.53528	−	0.411377i	−0.924737	+	0.380606i	$0.875715\pi$
$20$	4.89898	+	4.89898i	1.09545	+	1.09545i
$21$		0			0
$22$			2.00000i			0.426401i
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	0.578610	−	0.815604i	$-0.303595\pi$
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	−0.995639	+	0.0932891i	$0.970262\pi$
$24$		0			0
$25$	6.06218	+	3.50000i	1.21244	+	0.700000i
$26$	−2.44949	−	4.24264i	−0.480384	−	0.832050i
$27$		0			0
$28$		0			0
$29$	−3.00000	+	3.00000i	−0.557086	+	0.557086i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−3.00000	+	3.00000i	−0.557086	+	0.557086i	0.371391	+	0.928477i	$0.378881\pi$
$30$		0			0
$31$	−2.44949	+	4.24264i	−0.439941	+	0.762001i	−0.997684	−	0.0680129i	$-0.978334\pi$
$31$	−2.44949	+	4.24264i	−0.439941	+	0.762001i	0.557743	+	0.830014i	$0.311667\pi$
$32$	1.46410	+	5.46410i	0.258819	+	0.965926i
$33$		0			0
$34$	−4.89898	−	4.89898i	−0.840168	−	0.840168i
$35$		0			0
$36$	6.00000			1.00000
$37$	1.83013	−	6.83013i	0.300871	−	1.12287i	−0.635571	−	0.772043i	$-0.719235\pi$
$37$	1.83013	−	6.83013i	0.300871	−	1.12287i	0.936442	−	0.350823i	$-0.114098\pi$
$38$	−8.48528	−	4.89898i	−1.37649	−	0.794719i
$39$		0			0
$40$	4.89898	+	8.48528i	0.774597	+	1.34164i
$41$	4.89898			0.765092			0.382546	−	0.923936i	$-0.375047\pi$
$41$	4.89898			0.765092			0.382546	+	0.923936i	$0.375047\pi$
$42$		0			0
$43$	−5.00000	+	5.00000i	−0.762493	+	0.762493i	−0.976772	−	0.214280i	$-0.931260\pi$
$43$	−5.00000	+	5.00000i	−0.762493	+	0.762493i	0.214280	+	0.976772i	$0.431260\pi$
$44$	−0.732051	+	2.73205i	−0.110361	+	0.411872i
$45$	10.0382	−	2.68973i	1.49641	−	0.400961i
$46$	−1.46410	−	5.46410i	−0.215870	−	0.805638i
$47$	2.44949	+	4.24264i	0.357295	+	0.618853i	0.987508	−	0.157569i	$-0.0503658\pi$
$47$	2.44949	+	4.24264i	0.357295	+	0.618853i	−0.630213	+	0.776422i	$0.717032\pi$
$48$		0			0
$49$		0			0
$50$	7.00000	+	7.00000i	0.989949	+	0.989949i
$51$		0			0
$52$	−1.79315	−	6.69213i	−0.248665	−	0.928032i
$53$	1.36603	−	0.366025i	0.187638	−	0.0502775i	−0.163776	−	0.986498i	$-0.552368\pi$
$53$	1.36603	−	0.366025i	0.187638	−	0.0502775i	0.351414	+	0.936220i	$0.385701\pi$
$54$		0			0
$55$			4.89898i			0.660578i
$56$		0			0
$57$		0			0
$58$	−5.19615	+	3.00000i	−0.682288	+	0.393919i
$59$	6.69213	−	1.79315i	0.871241	−	0.233448i	0.204617	−	0.978842i	$-0.434405\pi$
$59$	6.69213	−	1.79315i	0.871241	−	0.233448i	0.666624	+	0.745394i	$0.267739\pi$
$60$		0			0
$61$	0.896575	−	3.34607i	0.114795	−	0.428420i	−0.884477	−	0.466584i	$-0.845484\pi$
$61$	0.896575	−	3.34607i	0.114795	−	0.428420i	0.999271	+	0.0381645i	$0.0121511\pi$
$62$	−4.89898	+	4.89898i	−0.622171	+	0.622171i
$63$		0			0
$64$			8.00000i			1.00000i
$65$	−6.00000	−	10.3923i	−0.744208	−	1.28901i
$66$		0			0
$67$	−6.83013	+	1.83013i	−0.834433	+	0.223586i	−0.650647	−	0.759381i	$-0.725502\pi$
$67$	−6.83013	+	1.83013i	−0.834433	+	0.223586i	−0.183786	+	0.982966i	$0.558835\pi$
$68$	−4.89898	−	8.48528i	−0.594089	−	1.02899i
$69$		0			0
$70$		0			0
$71$	2.00000			0.237356			0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000			0.237356			0.118678	+	0.992933i	$0.462134\pi$
$72$	8.19615	+	2.19615i	0.965926	+	0.258819i
$73$	−4.89898	+	8.48528i	−0.573382	+	0.993127i	0.422833	+	0.906208i	$0.361036\pi$
$73$	−4.89898	+	8.48528i	−0.573382	+	0.993127i	−0.996215	+	0.0869195i	$0.972298\pi$
$74$	5.00000	−	8.66025i	0.581238	−	1.00673i
$75$		0			0
$76$	−9.79796	−	9.79796i	−1.12390	−	1.12390i
$77$		0			0
$78$		0			0
$79$	−3.46410	+	2.00000i	−0.389742	+	0.225018i	−0.682048	−	0.731307i	$-0.738911\pi$
$79$	−3.46410	+	2.00000i	−0.389742	+	0.225018i	0.292306	+	0.956325i	$0.405577\pi$
$80$	3.58630	+	13.3843i	0.400961	+	1.49641i
$81$	4.50000	−	7.79423i	0.500000	−	0.866025i
$82$	6.69213	+	1.79315i	0.739022	+	0.198020i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	−12.0000	−	12.0000i	−1.30158	−	1.30158i
$86$	−8.66025	+	5.00000i	−0.933859	+	0.539164i
$87$		0			0
$88$	−2.00000	+	3.46410i	−0.213201	+	0.369274i
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$	14.6969			1.54919
$91$		0			0
$92$		−	8.00000i		−	0.834058i
$93$		0			0
$94$	1.79315	+	6.69213i	0.184949	+	0.690241i
$95$	−20.7846	−	12.0000i	−2.13246	−	1.23117i
$96$		0			0
$97$			4.89898i			0.497416i	0.968579	+	0.248708i	$0.0800060\pi$
$97$			4.89898i			0.497416i	−0.968579	+	0.248708i	$0.919994\pi$
$98$		0			0
$99$	3.00000	+	3.00000i	0.301511	+	0.301511i
$100$	7.00000	+	12.1244i	0.700000	+	1.21244i
$101$	0.896575	+	3.34607i	0.0892126	+	0.332946i	0.996079	−	0.0884733i	$-0.0281988\pi$
$101$	0.896575	+	3.34607i	0.0892126	+	0.332946i	−0.906866	+	0.421419i	$0.861532\pi$
$102$		0			0
$103$	8.48528	−	4.89898i	0.836080	−	0.482711i	−0.0198501	−	0.999803i	$-0.506319\pi$
$103$	8.48528	−	4.89898i	0.836080	−	0.482711i	0.855930	+	0.517092i	$0.172986\pi$
$104$		−	9.79796i		−	0.960769i
$105$		0			0
$106$	2.00000			0.194257
$107$	−4.09808	−	1.09808i	−0.396176	−	0.106155i	0.0552301	−	0.998474i	$-0.482411\pi$
$107$	−4.09808	−	1.09808i	−0.396176	−	0.106155i	−0.451406	+	0.892319i	$0.649077\pi$
$108$		0			0
$109$	2.56218	+	9.56218i	0.245412	+	0.915891i	0.973176	+	0.230063i	$0.0738931\pi$
$109$	2.56218	+	9.56218i	0.245412	+	0.915891i	−0.727764	+	0.685828i	$0.759440\pi$
$110$	−1.79315	+	6.69213i	−0.170970	+	0.638070i
$111$		0			0
$112$		0			0
$113$	4.00000			0.376288			0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000			0.376288			0.188144	+	0.982141i	$0.439753\pi$
$114$		0			0
$115$	−3.58630	−	13.3843i	−0.334424	−	1.24809i
$116$	−8.19615	+	2.19615i	−0.760994	+	0.203908i
$117$	−10.0382	−	2.68973i	−0.928032	−	0.248665i
$118$	9.79796			0.901975
$119$		0			0
$120$		0			0
$121$	7.79423	−	4.50000i	0.708566	−	0.409091i
$122$	2.44949	−	4.24264i	0.221766	−	0.384111i
$123$		0			0
$124$	−8.48528	+	4.89898i	−0.762001	+	0.439941i
$125$	4.89898	+	4.89898i	0.438178	+	0.438178i
$126$		0			0
$127$		−	18.0000i		−	1.59724i	−0.601834	−	0.798621i	$-0.705563\pi$
$127$		−	18.0000i		−	1.59724i	0.601834	−	0.798621i	$-0.294437\pi$
$128$	−2.92820	+	10.9282i	−0.258819	+	0.965926i
$129$		0			0
$130$	−4.39230	−	16.3923i	−0.385231	−	1.43770i
$131$	13.3843	+	3.58630i	1.16939	+	0.313337i	0.790710	−	0.612192i	$-0.209712\pi$
$131$	13.3843	+	3.58630i	1.16939	+	0.313337i	0.378679	+	0.925528i	$0.376379\pi$
$132$		0			0
$133$		0			0
$134$	−10.0000			−0.863868
$135$		0			0
$136$	−3.58630	−	13.3843i	−0.307523	−	1.14769i
$137$	6.92820	+	4.00000i	0.591916	+	0.341743i	0.765855	−	0.643013i	$-0.222316\pi$
$137$	6.92820	+	4.00000i	0.591916	+	0.341743i	−0.173939	+	0.984757i	$0.555649\pi$
$138$		0			0
$139$	−4.89898	−	4.89898i	−0.415526	−	0.415526i	0.468132	−	0.883658i	$-0.344927\pi$
$139$	−4.89898	−	4.89898i	−0.415526	−	0.415526i	−0.883658	+	0.468132i	$0.844927\pi$
$140$		0			0
$141$		0			0
$142$	2.73205	+	0.732051i	0.229269	+	0.0614323i
$143$	2.44949	−	4.24264i	0.204837	−	0.354787i
$144$	10.3923	+	6.00000i	0.866025	+	0.500000i
$145$	−12.7279	+	7.34847i	−1.05700	+	0.610257i
$146$	−9.79796	+	9.79796i	−0.810885	+	0.810885i
$147$		0			0
$148$	10.0000	−	10.0000i	0.821995	−	0.821995i
$149$	1.09808	−	4.09808i	0.0899579	−	0.335727i	−0.906249	−	0.422744i	$-0.861067\pi$
$149$	1.09808	−	4.09808i	0.0899579	−	0.335727i	0.996207	+	0.0870170i	$0.0277334\pi$
$150$		0			0
$151$	−6.00000	+	10.3923i	−0.488273	+	0.845714i	−0.999909	−	0.0134886i	$-0.995706\pi$
$151$	−6.00000	+	10.3923i	−0.488273	+	0.845714i	0.511636	+	0.859202i	$0.329040\pi$
$152$	−9.79796	−	16.9706i	−0.794719	−	1.37649i
$153$	−14.6969			−1.18818
$154$		0			0
$155$	−12.0000	+	12.0000i	−0.963863	+	0.963863i
$156$		0			0
$157$	−16.7303	+	4.48288i	−1.33523	+	0.357773i	−0.854661	−	0.519187i	$-0.826235\pi$
$157$	−16.7303	+	4.48288i	−1.33523	+	0.357773i	−0.480565	+	0.876959i	$0.659568\pi$
$158$	−5.46410	+	1.46410i	−0.434701	+	0.116478i
$159$		0			0
$160$			19.5959i			1.54919i
$161$		0			0
$162$	9.00000	−	9.00000i	0.707107	−	0.707107i
$163$	−0.366025	+	1.36603i	−0.0286693	+	0.106995i	−0.978778	−	0.204924i	$-0.934305\pi$
$163$	−0.366025	+	1.36603i	−0.0286693	+	0.106995i	0.950109	+	0.311919i	$0.100972\pi$
$164$	8.48528	+	4.89898i	0.662589	+	0.382546i
$165$		0			0
$166$		0			0
$167$		−	19.5959i		−	1.51638i	−0.652035	−	0.758189i	$-0.726085\pi$
$167$		−	19.5959i		−	1.51638i	0.652035	−	0.758189i	$-0.273915\pi$
$168$		0			0
$169$		−	1.00000i		−	0.0769231i
$170$	−12.0000	−	20.7846i	−0.920358	−	1.59411i
$171$	−20.0764	+	5.37945i	−1.53528	+	0.411377i
$172$	−13.6603	+	3.66025i	−1.04158	+	0.279092i
$173$	−0.896575	+	3.34607i	−0.0681654	+	0.254397i	−0.991597	−	0.129367i	$-0.958706\pi$
$173$	−0.896575	+	3.34607i	−0.0681654	+	0.254397i	0.923431	+	0.383763i	$0.125372\pi$
$174$		0			0
$175$		0			0
$176$	−4.00000	+	4.00000i	−0.301511	+	0.301511i
$177$		0			0
$178$		0			0
$179$	−4.09808	+	1.09808i	−0.306305	+	0.0820741i	−0.408697	−	0.912670i	$-0.634017\pi$
$179$	−4.09808	+	1.09808i	−0.306305	+	0.0820741i	0.102393	+	0.994744i	$0.467350\pi$
$180$	20.0764	+	5.37945i	1.49641	+	0.400961i
$181$	2.44949	−	2.44949i	0.182069	−	0.182069i	−0.610188	−	0.792257i	$-0.708906\pi$
$181$	2.44949	−	2.44949i	0.182069	−	0.182069i	0.792257	+	0.610188i	$0.208906\pi$
$182$		0			0
$183$		0			0
$184$	2.92820	−	10.9282i	0.215870	−	0.805638i
$185$	12.2474	−	21.2132i	0.900450	−	1.55963i
$186$		0			0
$187$	1.79315	−	6.69213i	0.131128	−	0.489377i
$188$			9.79796i			0.714590i
$189$		0			0
$190$	−24.0000	−	24.0000i	−1.74114	−	1.74114i
$191$	−17.3205	+	10.0000i	−1.25327	+	0.723575i	−0.971757	−	0.235983i	$-0.924169\pi$
$191$	−17.3205	+	10.0000i	−1.25327	+	0.723575i	−0.281511	+	0.959558i	$0.590836\pi$
$192$		0			0
$193$	8.00000	−	13.8564i	0.575853	−	0.997406i	−0.420096	−	0.907480i	$-0.638004\pi$
$193$	8.00000	−	13.8564i	0.575853	−	0.997406i	0.995948	−	0.0899262i	$-0.0286631\pi$
$194$	−1.79315	+	6.69213i	−0.128741	+	0.480467i
$195$		0			0
$196$		0			0
$197$	−15.0000	−	15.0000i	−1.06871	−	1.06871i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	−15.0000	−	15.0000i	−1.06871	−	1.06871i	−0.0712470	−	0.997459i	$-0.522698\pi$
$198$	3.00000	+	5.19615i	0.213201	+	0.369274i
$199$	8.48528	+	4.89898i	0.601506	+	0.347279i	0.769634	−	0.638486i	$-0.220439\pi$
$199$	8.48528	+	4.89898i	0.601506	+	0.347279i	−0.168128	+	0.985765i	$0.553772\pi$
$200$	5.12436	+	19.1244i	0.362347	+	1.35230i
$201$		0			0
$202$			4.89898i			0.344691i
$203$		0			0
$204$		0			0
$205$	16.3923	+	4.39230i	1.14489	+	0.306772i
$206$	13.3843	−	3.58630i	0.932526	−	0.249869i
$207$	−10.3923	−	6.00000i	−0.722315	−	0.417029i
$208$	3.58630	−	13.3843i	0.248665	−	0.928032i
$209$		−	9.79796i		−	0.677739i
$210$		0			0
$211$	1.00000	+	1.00000i	0.0688428	+	0.0688428i	0.740690	−	0.671847i	$-0.234499\pi$
$211$	1.00000	+	1.00000i	0.0688428	+	0.0688428i	−0.671847	+	0.740690i	$0.734499\pi$
$212$	2.73205	+	0.732051i	0.187638	+	0.0502775i
$213$		0			0
$214$	−5.19615	−	3.00000i	−0.355202	−	0.205076i
$215$	−21.2132	+	12.2474i	−1.44673	+	0.835269i
$216$		0			0
$217$		0			0
$218$			14.0000i			0.948200i
$219$		0			0
$220$	−4.89898	+	8.48528i	−0.330289	+	0.572078i
$221$	4.39230	+	16.3923i	0.295458	+	1.10267i
$222$		0			0
$223$	9.79796			0.656120			0.328060	−	0.944657i	$-0.393605\pi$
$223$	9.79796			0.656120			0.328060	+	0.944657i	$0.393605\pi$
$224$		0			0
$225$	21.0000			1.40000
$226$	5.46410	+	1.46410i	0.363467	+	0.0973906i
$227$	7.17260	+	26.7685i	0.476062	+	1.77669i	0.617316	+	0.786716i	$0.288220\pi$
$227$	7.17260	+	26.7685i	0.476062	+	1.77669i	−0.141253	+	0.989974i	$0.545113\pi$
$228$		0			0
$229$	10.0382	+	2.68973i	0.663343	+	0.177742i	0.574754	−	0.818326i	$-0.305098\pi$
$229$	10.0382	+	2.68973i	0.663343	+	0.177742i	0.0885886	+	0.996068i	$0.471764\pi$
$230$		−	19.5959i		−	1.29212i
$231$		0			0
$232$	−12.0000			−0.787839
$233$	3.46410	−	2.00000i	0.226941	−	0.131024i	−0.382219	−	0.924072i	$-0.624840\pi$
$233$	3.46410	−	2.00000i	0.226941	−	0.131024i	0.609160	+	0.793047i	$0.291507\pi$
$234$	−12.7279	−	7.34847i	−0.832050	−	0.480384i
$235$	4.39230	+	16.3923i	0.286522	+	1.06932i
$236$	13.3843	+	3.58630i	0.871241	+	0.233448i
$237$		0			0
$238$		0			0
$239$		−	14.0000i		−	0.905585i	−0.891616	−	0.452792i	$-0.850428\pi$
$239$		−	14.0000i		−	0.905585i	0.891616	−	0.452792i	$-0.149572\pi$
$240$		0			0
$241$	21.2132	+	12.2474i	1.36646	+	0.788928i	0.990474	−	0.137697i	$-0.0439700\pi$
$241$	21.2132	+	12.2474i	1.36646	+	0.788928i	0.375988	+	0.926624i	$0.377303\pi$
$242$	12.2942	−	3.29423i	0.790303	−	0.211761i
$243$		0			0
$244$	4.89898	−	4.89898i	0.313625	−	0.313625i
$245$		0			0
$246$		0			0
$247$	12.0000	+	20.7846i	0.763542	+	1.32249i
$248$	−13.3843	+	3.58630i	−0.849901	+	0.227730i
$249$		0			0
$250$	4.89898	+	8.48528i	0.309839	+	0.536656i
$251$	14.6969	+	14.6969i	0.927663	+	0.927663i	0.997555	−	0.0698920i	$-0.0222655\pi$
$251$	14.6969	+	14.6969i	0.927663	+	0.927663i	−0.0698920	+	0.997555i	$0.522265\pi$
$252$		0			0
$253$	4.00000	−	4.00000i	0.251478	−	0.251478i
$254$	6.58846	−	24.5885i	0.413397	−	1.54282i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	25.4558	−	14.6969i	1.58789	−	0.916770i	0.594238	−	0.804289i	$-0.297454\pi$
$257$	25.4558	−	14.6969i	1.58789	−	0.916770i	0.993654	−	0.112481i	$-0.0358796\pi$
$258$		0			0
$259$		0			0
$260$		−	24.0000i		−	1.48842i
$261$	−3.29423	+	12.2942i	−0.203908	+	0.760994i
$262$	16.9706	+	9.79796i	1.04844	+	0.605320i
$263$	13.0000	−	22.5167i	0.801614	−	1.38844i	−0.116939	−	0.993139i	$-0.537308\pi$
$263$	13.0000	−	22.5167i	0.801614	−	1.38844i	0.918553	−	0.395298i	$-0.129359\pi$
$264$		0			0
$265$	4.89898			0.300942
$266$		0			0
$267$		0			0
$268$	−13.6603	−	3.66025i	−0.834433	−	0.223586i
$269$	−10.0382	+	2.68973i	−0.612040	+	0.163996i	−0.551506	−	0.834171i	$-0.685947\pi$
$269$	−10.0382	+	2.68973i	−0.612040	+	0.163996i	−0.0605332	+	0.998166i	$0.519280\pi$
$270$		0			0
$271$	−14.6969	−	25.4558i	−0.892775	−	1.54633i	−0.836534	−	0.547915i	$-0.815422\pi$
$271$	−14.6969	−	25.4558i	−0.892775	−	1.54633i	−0.0562416	−	0.998417i	$-0.517912\pi$
$272$		−	19.5959i		−	1.18818i
$273$		0			0
$274$	8.00000	+	8.00000i	0.483298	+	0.483298i
$275$	−2.56218	+	9.56218i	−0.154505	+	0.576621i
$276$		0			0
$277$	−6.83013	+	1.83013i	−0.410383	+	0.109962i	−0.458103	−	0.888899i	$-0.651471\pi$
$277$	−6.83013	+	1.83013i	−0.410383	+	0.109962i	0.0477206	+	0.998861i	$0.484804\pi$
$278$	−4.89898	−	8.48528i	−0.293821	−	0.508913i
$279$			14.6969i			0.879883i
$280$		0			0
$281$			20.0000i			1.19310i	0.802576	+	0.596550i	$0.203462\pi$
$281$			20.0000i			1.19310i	−0.802576	+	0.596550i	$0.796538\pi$
$282$		0			0
$283$	20.0764	−	5.37945i	1.19342	−	0.319775i	0.393182	−	0.919461i	$-0.371374\pi$
$283$	20.0764	−	5.37945i	1.19342	−	0.319775i	0.800236	+	0.599685i	$0.204708\pi$
$284$	3.46410	+	2.00000i	0.205557	+	0.118678i
$285$		0			0
$286$	4.89898	−	4.89898i	0.289683	−	0.289683i
$287$		0			0
$288$	12.0000	+	12.0000i	0.707107	+	0.707107i
$289$	3.50000	+	6.06218i	0.205882	+	0.356599i
$290$	−20.0764	+	5.37945i	−1.17893	+	0.315892i
$291$		0			0
$292$	−16.9706	+	9.79796i	−0.993127	+	0.573382i
$293$	−12.2474	+	12.2474i	−0.715504	+	0.715504i	−0.967681	−	0.252177i	$-0.918853\pi$
$293$	−12.2474	+	12.2474i	−0.715504	+	0.715504i	0.252177	+	0.967681i	$0.418853\pi$
$294$		0			0
$295$	24.0000			1.39733
$296$	17.3205	−	10.0000i	1.00673	−	0.581238i
$297$		0			0
$298$	3.00000	−	5.19615i	0.173785	−	0.301005i
$299$	−3.58630	+	13.3843i	−0.207401	+	0.774032i
$300$		0			0
$301$		0			0
$302$	−12.0000	+	12.0000i	−0.690522	+	0.690522i
$303$		0			0
$304$	−7.17260	−	26.7685i	−0.411377	−	1.53528i
$305$	6.00000	−	10.3923i	0.343559	−	0.595062i
$306$	−20.0764	−	5.37945i	−1.14769	−	0.307523i
$307$	−4.89898	+	4.89898i	−0.279600	+	0.279600i	−0.832949	−	0.553350i	$-0.813349\pi$
$307$	−4.89898	+	4.89898i	−0.279600	+	0.279600i	0.553350	+	0.832949i	$0.313349\pi$
$308$		0			0
$309$		0			0
$310$	−20.7846	+	12.0000i	−1.18049	+	0.681554i
$311$	−21.2132	−	12.2474i	−1.20289	−	0.694489i	−0.241694	−	0.970353i	$-0.577703\pi$
$311$	−21.2132	−	12.2474i	−1.20289	−	0.694489i	−0.961197	+	0.275864i	$0.911036\pi$
$312$		0			0
$313$	7.34847	+	12.7279i	0.415360	+	0.719425i	0.995466	−	0.0951162i	$-0.0303223\pi$
$313$	7.34847	+	12.7279i	0.415360	+	0.719425i	−0.580106	+	0.814541i	$0.696989\pi$
$314$	−24.4949			−1.38233
$315$		0			0
$316$	−8.00000			−0.450035
$317$	−17.7583	−	4.75833i	−0.997407	−	0.267254i	−0.277048	−	0.960856i	$-0.589356\pi$
$317$	−17.7583	−	4.75833i	−0.997407	−	0.267254i	−0.720359	+	0.693602i	$0.756023\pi$
$318$		0			0
$319$	−5.19615	−	3.00000i	−0.290929	−	0.167968i
$320$	−7.17260	+	26.7685i	−0.400961	+	1.49641i
$321$		0			0
$322$		0			0
$323$	24.0000	+	24.0000i	1.33540	+	1.33540i
$324$	15.5885	−	9.00000i	0.866025	−	0.500000i
$325$	−6.27603	−	23.4225i	−0.348131	−	1.29924i
$326$	−1.00000	+	1.73205i	−0.0553849	+	0.0959294i
$327$		0			0
$328$	9.79796	+	9.79796i	0.541002	+	0.541002i
$329$		0			0
$330$		0			0
$331$	25.9545	+	6.95448i	1.42659	+	0.382253i	0.887816	−	0.460199i	$-0.152222\pi$
$331$	25.9545	+	6.95448i	1.42659	+	0.382253i	0.538772	+	0.842452i	$0.318889\pi$
$332$		0			0
$333$	−5.49038	−	20.4904i	−0.300871	−	1.12287i
$334$	7.17260	−	26.7685i	0.392467	−	1.46471i
$335$	−24.4949			−1.33830
$336$		0			0
$337$	28.0000			1.52526			0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000			1.52526			0.762629	+	0.646837i	$0.223908\pi$
$338$	0.366025	−	1.36603i	0.0199092	−	0.0743020i
$339$		0			0
$340$	−8.78461	−	32.7846i	−0.476412	−	1.77800i
$341$	−6.69213	−	1.79315i	−0.362399	−	0.0971046i
$342$	−29.3939			−1.58944
$343$		0			0
$344$	−20.0000			−1.07833
$345$		0			0
$346$	−2.44949	+	4.24264i	−0.131685	+	0.228086i
$347$	1.09808	+	4.09808i	0.0589478	+	0.219996i	0.989116	−	0.147137i	$-0.0470059\pi$
$347$	1.09808	+	4.09808i	0.0589478	+	0.219996i	−0.930168	+	0.367133i	$0.880339\pi$
$348$		0			0
$349$	7.34847	+	7.34847i	0.393355	+	0.393355i	0.875881	−	0.482527i	$-0.160281\pi$
$349$	7.34847	+	7.34847i	0.393355	+	0.393355i	−0.482527	+	0.875881i	$0.660281\pi$
$350$		0			0
$351$		0			0
$352$	−6.92820	+	4.00000i	−0.369274	+	0.213201i
$353$	−8.48528	−	4.89898i	−0.451626	−	0.260746i	0.256891	−	0.966440i	$-0.417302\pi$
$353$	−8.48528	−	4.89898i	−0.451626	−	0.260746i	−0.708517	+	0.705694i	$0.750635\pi$
$354$		0			0
$355$	6.69213	+	1.79315i	0.355181	+	0.0951706i
$356$		0			0
$357$		0			0
$358$	−6.00000			−0.317110
$359$	−10.0000	−	17.3205i	−0.527780	−	0.914141i	−0.999476	−	0.0323801i	$-0.989691\pi$
$359$	−10.0000	−	17.3205i	−0.527780	−	0.914141i	0.471696	−	0.881761i	$-0.343642\pi$
$360$	25.4558	+	14.6969i	1.34164	+	0.774597i
$361$	25.1147	+	14.5000i	1.32183	+	0.763158i
$362$	4.24264	−	2.44949i	0.222988	−	0.128742i
$363$		0			0
$364$		0			0
$365$	−24.0000	+	24.0000i	−1.25622	+	1.25622i
$366$		0			0
$367$	14.6969	−	25.4558i	0.767174	−	1.32878i	−0.171916	−	0.985112i	$-0.554996\pi$
$367$	14.6969	−	25.4558i	0.767174	−	1.32878i	0.939090	−	0.343673i	$-0.111671\pi$
$368$	8.00000	−	13.8564i	0.417029	−	0.722315i
$369$	12.7279	−	7.34847i	0.662589	−	0.382546i
$370$	24.4949	−	24.4949i	1.27343	−	1.27343i
$371$		0			0
$372$		0			0
$373$	−4.02628	+	15.0263i	−0.208473	+	0.778031i	0.779890	+	0.625917i	$0.215275\pi$
$373$	−4.02628	+	15.0263i	−0.208473	+	0.778031i	−0.988363	+	0.152115i	$0.951392\pi$
$374$	4.89898	−	8.48528i	0.253320	−	0.438763i
$375$		0			0
$376$	−3.58630	+	13.3843i	−0.184949	+	0.690241i
$377$	14.6969			0.756931
$378$		0			0
$379$	−23.0000	+	23.0000i	−1.18143	+	1.18143i	−0.202057	+	0.979374i	$0.564763\pi$
$379$	−23.0000	+	23.0000i	−1.18143	+	1.18143i	−0.979374	+	0.202057i	$0.935237\pi$
$380$	−24.0000	−	41.5692i	−1.23117	−	2.13246i
$381$		0			0
$382$	−27.3205	+	7.32051i	−1.39784	+	0.374550i
$383$	7.34847	+	12.7279i	0.375489	+	0.650366i	0.990400	−	0.138230i	$-0.0441414\pi$
$383$	7.34847	+	12.7279i	0.375489	+	0.650366i	−0.614911	+	0.788597i	$0.710808\pi$
$384$		0			0
$385$		0			0
$386$	16.0000	−	16.0000i	0.814379	−	0.814379i
$387$	−5.49038	+	20.4904i	−0.279092	+	1.04158i
$388$	−4.89898	+	8.48528i	−0.248708	+	0.430775i
$389$	−17.7583	+	4.75833i	−0.900383	+	0.241257i	−0.679181	−	0.733971i	$-0.737665\pi$
$389$	−17.7583	+	4.75833i	−0.900383	+	0.241257i	−0.221202	+	0.975228i	$0.570998\pi$
$390$		0			0
$391$			19.5959i			0.991008i
$392$		0			0
$393$		0			0
$394$	−15.0000	−	25.9808i	−0.755689	−	1.30889i
$395$	−13.3843	+	3.58630i	−0.673435	+	0.180446i
$396$	2.19615	+	8.19615i	0.110361	+	0.411872i
$397$	−4.48288	+	16.7303i	−0.224989	+	0.839671i	0.757420	+	0.652928i	$0.226460\pi$
$397$	−4.48288	+	16.7303i	−0.224989	+	0.839671i	−0.982409	+	0.186743i	$0.940207\pi$
$398$	9.79796	+	9.79796i	0.491127	+	0.491127i
$399$		0			0
$400$			28.0000i			1.40000i
$401$	4.00000	+	6.92820i	0.199750	+	0.345978i	0.948447	−	0.316934i	$-0.102654\pi$
$401$	4.00000	+	6.92820i	0.199750	+	0.345978i	−0.748697	+	0.662912i	$0.769320\pi$
$402$		0			0
$403$	16.3923	−	4.39230i	0.816559	−	0.218796i
$404$	−1.79315	+	6.69213i	−0.0892126	+	0.332946i
$405$	22.0454	−	22.0454i	1.09545	−	1.09545i
$406$		0			0
$407$	10.0000			0.495682
$408$		0			0
$409$	−12.2474	+	21.2132i	−0.605597	+	1.04893i	0.386359	+	0.922348i	$0.373732\pi$
$409$	−12.2474	+	21.2132i	−0.605597	+	1.04893i	−0.991957	+	0.126577i	$0.959601\pi$
$410$	20.7846	+	12.0000i	1.02648	+	0.592638i
$411$		0			0
$412$	19.5959			0.965422
$413$		0			0
$414$	−12.0000	−	12.0000i	−0.589768	−	0.589768i
$415$		0			0
$416$	9.79796	−	16.9706i	0.480384	−	0.832050i
$417$		0			0
$418$	3.58630	−	13.3843i	0.175412	−	0.654646i
$419$	4.89898	−	4.89898i	0.239331	−	0.239331i	−0.577242	−	0.816573i	$-0.695871\pi$
$419$	4.89898	−	4.89898i	0.239331	−	0.239331i	0.816573	+	0.577242i	$0.195871\pi$
$420$		0			0
$421$	11.0000	+	11.0000i	0.536107	+	0.536107i	0.922383	−	0.386276i	$-0.126239\pi$
$421$	11.0000	+	11.0000i	0.536107	+	0.536107i	−0.386276	+	0.922383i	$0.626239\pi$
$422$	1.00000	+	1.73205i	0.0486792	+	0.0843149i
$423$	12.7279	+	7.34847i	0.618853	+	0.357295i
$424$	3.46410	+	2.00000i	0.168232	+	0.0971286i
$425$	−17.1464	−	29.6985i	−0.831724	−	1.44059i
$426$		0			0
$427$		0			0
$428$	−6.00000	−	6.00000i	−0.290021	−	0.290021i
$429$		0			0
$430$	−33.4607	+	8.96575i	−1.61362	+	0.432367i
$431$	8.66025	+	5.00000i	0.417150	+	0.240842i	0.693857	−	0.720113i	$-0.255910\pi$
$431$	8.66025	+	5.00000i	0.417150	+	0.240842i	−0.276707	+	0.960954i	$0.589243\pi$
$432$		0			0
$433$		−	14.6969i		−	0.706290i	−0.935569	−	0.353145i	$-0.885112\pi$
$433$		−	14.6969i		−	0.706290i	0.935569	−	0.353145i	$-0.114888\pi$
$434$		0			0
$435$		0			0
$436$	−5.12436	+	19.1244i	−0.245412	+	0.915891i
$437$	7.17260	+	26.7685i	0.343112	+	1.28051i
$438$		0			0
$439$	12.7279	−	7.34847i	0.607471	−	0.350723i	−0.164504	−	0.986376i	$-0.552602\pi$
$439$	12.7279	−	7.34847i	0.607471	−	0.350723i	0.771975	+	0.635653i	$0.219269\pi$
$440$	−9.79796	+	9.79796i	−0.467099	+	0.467099i
$441$		0			0
$442$			24.0000i			1.14156i
$443$	−6.83013	−	1.83013i	−0.324509	−	0.0869520i	0.0928868	−	0.995677i	$-0.470391\pi$
$443$	−6.83013	−	1.83013i	−0.324509	−	0.0869520i	−0.417396	+	0.908725i	$0.637057\pi$
$444$		0			0
$445$		0			0
$446$	13.3843	+	3.58630i	0.633763	+	0.169816i
$447$		0			0
$448$		0			0
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$	28.6865	+	7.68653i	1.35230	+	0.362347i
$451$	1.79315	+	6.69213i	0.0844362	+	0.315120i
$452$	6.92820	+	4.00000i	0.325875	+	0.188144i
$453$		0			0
$454$			39.1918i			1.83936i
$455$		0			0
$456$		0			0
$457$	−15.5885	+	9.00000i	−0.729197	+	0.421002i	−0.818128	−	0.575036i	$-0.804988\pi$
$457$	−15.5885	+	9.00000i	−0.729197	+	0.421002i	0.0889312	+	0.996038i	$0.471655\pi$
$458$	12.7279	+	7.34847i	0.594737	+	0.343371i
$459$		0			0
$460$	7.17260	−	26.7685i	0.334424	−	1.24809i
$461$	−22.0454	−	22.0454i	−1.02676	−	1.02676i	−0.999632	−	0.0271249i	$-0.991365\pi$
$461$	−22.0454	−	22.0454i	−1.02676	−	1.02676i	−0.0271249	−	0.999632i	$-0.508635\pi$
$462$		0			0
$463$			4.00000i			0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$			4.00000i			0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	−16.3923	−	4.39230i	−0.760994	−	0.203908i
$465$		0			0
$466$	5.46410	−	1.46410i	0.253120	−	0.0678232i
$467$	−26.7685	−	7.17260i	−1.23870	−	0.331909i	−0.420738	−	0.907182i	$-0.638229\pi$
$467$	−26.7685	−	7.17260i	−1.23870	−	0.331909i	−0.817961	+	0.575274i	$0.804896\pi$
$468$	−14.6969	−	14.6969i	−0.679366	−	0.679366i
$469$		0			0
$470$			24.0000i			1.10704i
$471$		0			0
$472$	16.9706	+	9.79796i	0.781133	+	0.450988i
$473$	−8.66025	−	5.00000i	−0.398199	−	0.229900i
$474$		0			0
$475$	−34.2929	−	34.2929i	−1.57346	−	1.57346i
$476$		0			0
$477$	3.00000	−	3.00000i	0.137361	−	0.137361i
$478$	5.12436	−	19.1244i	0.234383	−	0.874728i
$479$	−12.2474	+	21.2132i	−0.559600	+	0.969256i	0.437929	+	0.899009i	$0.355712\pi$
$479$	−12.2474	+	21.2132i	−0.559600	+	0.969256i	−0.997530	+	0.0702467i	$0.977621\pi$
$480$		0			0
$481$	−21.2132	+	12.2474i	−0.967239	+	0.558436i
$482$	24.4949	+	24.4949i	1.11571	+	1.11571i
$483$		0			0
$484$	18.0000			0.818182
$485$	−4.39230	+	16.3923i	−0.199444	+	0.744336i
$486$		0			0
$487$	6.00000	−	10.3923i	0.271886	−	0.470920i	−0.697459	−	0.716625i	$-0.745686\pi$
$487$	6.00000	−	10.3923i	0.271886	−	0.470920i	0.969345	+	0.245705i	$0.0790193\pi$
$488$	8.48528	−	4.89898i	0.384111	−	0.221766i
$489$		0			0
$490$		0			0
$491$	11.0000	−	11.0000i	0.496423	−	0.496423i	−0.413900	−	0.910323i	$-0.635834\pi$
$491$	11.0000	−	11.0000i	0.496423	−	0.496423i	0.910323	+	0.413900i	$0.135834\pi$
$492$		0			0
$493$	20.0764	−	5.37945i	0.904195	−	0.242278i
$494$	8.78461	+	32.7846i	0.395238	+	1.47505i
$495$	7.34847	+	12.7279i	0.330289	+	0.572078i
$496$	−19.5959			−0.879883
$497$		0			0
$498$		0			0
$499$	8.41858	−	31.4186i	0.376868	−	1.40649i	−0.473729	−	0.880671i	$-0.657092\pi$
$499$	8.41858	−	31.4186i	0.376868	−	1.40649i	0.850597	−	0.525818i	$-0.176241\pi$
$500$	3.58630	+	13.3843i	0.160384	+	0.598562i
$501$		0			0
$502$	14.6969	+	25.4558i	0.655956	+	1.13615i
$503$		−	14.6969i		−	0.655304i	−0.944798	−	0.327652i	$-0.893743\pi$
$503$		−	14.6969i		−	0.655304i	0.944798	−	0.327652i	$-0.106257\pi$
$504$		0			0
$505$			12.0000i			0.533993i
$506$	6.92820	−	4.00000i	0.307996	−	0.177822i
$507$		0			0
$508$	18.0000	−	31.1769i	0.798621	−	1.38325i
$509$	2.68973	−	10.0382i	0.119220	−	0.444935i	−0.880348	−	0.474329i	$-0.842691\pi$
$509$	2.68973	−	10.0382i	0.119220	−	0.444935i	0.999568	+	0.0293934i	$0.00935755\pi$
$510$		0			0
$511$		0			0
$512$	−16.0000	+	16.0000i	−0.707107	+	0.707107i
$513$		0			0
$514$	40.1528	−	10.7589i	1.77106	−	0.474555i
$515$	32.7846	−	8.78461i	1.44466	−	0.387096i
$516$		0			0
$517$	−4.89898	+	4.89898i	−0.215457	+	0.215457i
$518$		0			0
$519$		0			0
$520$	8.78461	−	32.7846i	0.385231	−	1.43770i
$521$	−2.44949	+	4.24264i	−0.107314	+	0.185873i	−0.914681	−	0.404176i	$-0.867558\pi$
$521$	−2.44949	+	4.24264i	−0.107314	+	0.185873i	0.807367	+	0.590049i	$0.200892\pi$
$522$	−9.00000	+	15.5885i	−0.393919	+	0.682288i
$523$	−5.37945	+	20.0764i	−0.235227	+	0.877879i	0.742819	+	0.669492i	$0.233488\pi$
$523$	−5.37945	+	20.0764i	−0.235227	+	0.877879i	−0.978046	+	0.208387i	$0.933179\pi$
$524$	19.5959	+	19.5959i	0.856052	+	0.856052i
$525$		0			0
$526$	26.0000	−	26.0000i	1.13365	−	1.13365i
$527$	20.7846	−	12.0000i	0.905392	−	0.522728i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$	6.69213	+	1.79315i	0.290688	+	0.0778895i
$531$	14.6969	−	14.6969i	0.637793	−	0.637793i
$532$		0			0
$533$	−12.0000	−	12.0000i	−0.519778	−	0.519778i
$534$		0			0
$535$	−12.7279	−	7.34847i	−0.550276	−	0.317702i
$536$	−17.3205	−	10.0000i	−0.748132	−	0.431934i
$537$		0			0
$538$	−14.6969			−0.633630
$539$		0			0
$540$		0			0
$541$	39.6147	+	10.6147i	1.70317	+	0.456363i	0.973735	−	0.227686i	$-0.0731160\pi$
$541$	39.6147	+	10.6147i	1.70317	+	0.456363i	0.729436	+	0.684049i	$0.239783\pi$
$542$	−10.7589	−	40.1528i	−0.462135	−	1.72471i
$543$		0			0
$544$	7.17260	−	26.7685i	0.307523	−	1.14769i
$545$			34.2929i			1.46894i
$546$		0			0
$547$	−5.00000	−	5.00000i	−0.213785	−	0.213785i	0.592088	−	0.805873i	$-0.298304\pi$
$547$	−5.00000	−	5.00000i	−0.213785	−	0.213785i	−0.805873	+	0.592088i	$0.798304\pi$
$548$	8.00000	+	13.8564i	0.341743	+	0.591916i
$549$	−2.68973	−	10.0382i	−0.114795	−	0.428420i
$550$	−7.00000	+	12.1244i	−0.298481	+	0.516984i
$551$	25.4558	−	14.6969i	1.08446	−	0.626111i
$552$		0			0
$553$		0			0
$554$	−10.0000			−0.424859
$555$		0			0
$556$	−3.58630	−	13.3843i	−0.152093	−	0.567619i
$557$	1.09808	+	4.09808i	0.0465270	+	0.173641i	0.985280	−	0.170951i	$-0.0546839\pi$
$557$	1.09808	+	4.09808i	0.0465270	+	0.173641i	−0.938753	+	0.344592i	$0.888017\pi$
$558$	−5.37945	+	20.0764i	−0.227730	+	0.849901i
$559$	24.4949			1.03602
$560$		0			0
$561$		0			0
$562$	−7.32051	+	27.3205i	−0.308797	+	1.15245i
$563$	−8.96575	−	33.4607i	−0.377862	−	1.41020i	−0.849118	−	0.528202i	$-0.822866\pi$
$563$	−8.96575	−	33.4607i	−0.377862	−	1.41020i	0.471257	−	0.881996i	$-0.343800\pi$
$564$		0			0
$565$	13.3843	+	3.58630i	0.563080	+	0.150877i
$566$	29.3939			1.23552
$567$		0			0
$568$	4.00000	+	4.00000i	0.167836	+	0.167836i
$569$	−12.1244	+	7.00000i	−0.508279	+	0.293455i	−0.732126	−	0.681169i	$-0.761472\pi$
$569$	−12.1244	+	7.00000i	−0.508279	+	0.293455i	0.223847	+	0.974624i	$0.428139\pi$
$570$		0			0
$571$	11.3468	+	42.3468i	0.474848	+	1.77216i	0.621972	+	0.783040i	$0.286332\pi$
$571$	11.3468	+	42.3468i	0.474848	+	1.77216i	−0.147123	+	0.989118i	$0.547001\pi$
$572$	8.48528	−	4.89898i	0.354787	−	0.204837i
$573$		0			0
$574$		0			0
$575$		−	28.0000i		−	1.16768i
$576$	12.0000	+	20.7846i	0.500000	+	0.866025i
$577$	−25.4558	−	14.6969i	−1.05974	−	0.611842i	−0.134380	−	0.990930i	$-0.542904\pi$
$577$	−25.4558	−	14.6969i	−1.05974	−	0.611842i	−0.925361	+	0.379088i	$0.876238\pi$
$578$	2.56218	+	9.56218i	0.106573	+	0.397734i
$579$		0			0
$580$	−29.3939			−1.22051
$581$		0			0
$582$		0			0
$583$	1.00000	+	1.73205i	0.0414158	+	0.0717342i
$584$	−26.7685	+	7.17260i	−1.10769	+	0.296804i
$585$	−31.1769	−	18.0000i	−1.28901	−	0.744208i
$586$	−21.2132	+	12.2474i	−0.876309	+	0.505937i
$587$	24.4949	+	24.4949i	1.01101	+	1.01101i	0.999939	+	0.0110739i	$0.00352501\pi$
$587$	24.4949	+	24.4949i	1.01101	+	1.01101i	0.0110739	+	0.999939i	$0.496475\pi$
$588$		0			0
$589$	24.0000	−	24.0000i	0.988903	−	0.988903i
$590$	32.7846	+	8.78461i	1.34972	+	0.361657i
$591$		0			0
$592$	27.3205	−	7.32051i	1.12287	−	0.300871i
$593$	−33.9411	+	19.5959i	−1.39379	+	0.804708i	−0.993733	−	0.111780i	$-0.964345\pi$
$593$	−33.9411	+	19.5959i	−1.39379	+	0.804708i	−0.400062	+	0.916488i	$0.631011\pi$
$594$		0			0
$595$		0			0
$596$	6.00000	−	6.00000i	0.245770	−	0.245770i
$597$		0			0
$598$	−9.79796	+	16.9706i	−0.400668	+	0.693978i
$599$	−5.00000	+	8.66025i	−0.204294	+	0.353848i	−0.949908	−	0.312531i	$-0.898823\pi$
$599$	−5.00000	+	8.66025i	−0.204294	+	0.353848i	0.745613	+	0.666379i	$0.232157\pi$
$600$		0			0
$601$	−19.5959			−0.799334			−0.399667	−	0.916660i	$-0.630874\pi$
$601$	−19.5959			−0.799334			−0.399667	+	0.916660i	$0.630874\pi$
$602$		0			0
$603$	−15.0000	+	15.0000i	−0.610847	+	0.610847i
$604$	−20.7846	+	12.0000i	−0.845714	+	0.488273i
$605$	30.1146	−	8.06918i	1.22433	−	0.328059i
$606$		0			0
$607$	14.6969	+	25.4558i	0.596530	+	1.03322i	0.993329	+	0.115315i	$0.0367878\pi$
$607$	14.6969	+	25.4558i	0.596530	+	1.03322i	−0.396799	+	0.917906i	$0.629879\pi$
$608$		−	39.1918i		−	1.58944i
$609$		0			0
$610$	12.0000	−	12.0000i	0.485866	−	0.485866i
$611$	4.39230	−	16.3923i	0.177694	−	0.663162i
$612$	−25.4558	−	14.6969i	−1.02899	−	0.594089i
$613$	−25.9545	+	6.95448i	−1.04829	+	0.280889i	−0.741546	−	0.670902i	$-0.765907\pi$
$613$	−25.9545	+	6.95448i	−1.04829	+	0.280889i	−0.306746	+	0.951791i	$0.599240\pi$
$614$	−8.48528	+	4.89898i	−0.342438	+	0.197707i
$615$		0			0
$616$		0			0
$617$			2.00000i			0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$			2.00000i			0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$		0			0
$619$	6.69213	−	1.79315i	0.268979	−	0.0720728i	−0.121808	−	0.992554i	$-0.538869\pi$
$619$	6.69213	−	1.79315i	0.268979	−	0.0720728i	0.390788	+	0.920481i	$0.372203\pi$
$620$	−32.7846	+	8.78461i	−1.31666	+	0.352798i
$621$		0			0
$622$	−24.4949	−	24.4949i	−0.982156	−	0.982156i
$623$		0			0
$624$		0			0
$625$	−5.50000	−	9.52628i	−0.220000	−	0.381051i
$626$	5.37945	+	20.0764i	0.215006	+	0.802414i
$627$		0			0
$628$	−33.4607	−	8.96575i	−1.33523	−	0.357773i
$629$	−24.4949	+	24.4949i	−0.976676	+	0.976676i
$630$		0			0
$631$	−18.0000			−0.716569			−0.358284	−	0.933613i	$-0.616638\pi$
$631$	−18.0000			−0.716569			−0.358284	+	0.933613i	$0.616638\pi$
$632$	−10.9282	−	2.92820i	−0.434701	−	0.116478i
$633$		0			0
$634$	−22.5167	−	13.0000i	−0.894251	−	0.516296i
$635$	16.1384	−	60.2292i	0.640431	−	2.39012i
$636$		0			0
$637$		0			0
$638$	−6.00000	−	6.00000i	−0.237542	−	0.237542i
$639$	5.19615	−	3.00000i	0.205557	−	0.118678i
$640$	−19.5959	+	33.9411i	−0.774597	+	1.34164i
$641$	−16.0000	+	27.7128i	−0.631962	+	1.09459i	0.355188	+	0.934795i	$0.384417\pi$
$641$	−16.0000	+	27.7128i	−0.631962	+	1.09459i	−0.987150	+	0.159795i	$0.948917\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$	24.0000	+	41.5692i	0.944267	+	1.63552i
$647$	16.9706	+	9.79796i	0.667182	+	0.385198i	0.795008	−	0.606599i	$-0.207467\pi$
$647$	16.9706	+	9.79796i	0.667182	+	0.385198i	−0.127826	+	0.991797i	$0.540800\pi$
$648$	24.5885	−	6.58846i	0.965926	−	0.258819i
$649$	4.89898	+	8.48528i	0.192302	+	0.333076i
$650$		−	34.2929i		−	1.34508i
$651$		0			0
$652$	−2.00000	+	2.00000i	−0.0783260	+	0.0783260i
$653$	−20.4904	−	5.49038i	−0.801851	−	0.214855i	−0.165454	−	0.986217i	$-0.552909\pi$
$653$	−20.4904	−	5.49038i	−0.801851	−	0.214855i	−0.636396	+	0.771362i	$0.719576\pi$
$654$		0			0
$655$	41.5692	+	24.0000i	1.62424	+	0.937758i
$656$	9.79796	+	16.9706i	0.382546	+	0.662589i
$657$			29.3939i			1.14676i
$658$		0			0
$659$	−27.0000	−	27.0000i	−1.05177	−	1.05177i	−0.998585	−	0.0531861i	$-0.983062\pi$
$659$	−27.0000	−	27.0000i	−1.05177	−	1.05177i	−0.0531861	−	0.998585i	$-0.516938\pi$
$660$		0			0
$661$	0.896575	+	3.34607i	0.0348727	+	0.130147i	0.981168	−	0.193158i	$-0.0618729\pi$
$661$	0.896575	+	3.34607i	0.0348727	+	0.130147i	−0.946295	+	0.323304i	$0.895206\pi$
$662$	32.9090	+	19.0000i	1.27904	+	0.738456i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		−	30.0000i		−	1.16248i
$667$	16.3923	+	4.39230i	0.634713	+	0.170071i
$668$	19.5959	−	33.9411i	0.758189	−	1.31322i
$669$		0			0
$670$	−33.4607	−	8.96575i	−1.29270	−	0.346377i
$671$	4.89898			0.189123
$672$		0			0
$673$	−6.00000			−0.231283			−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000			−0.231283			−0.115642	+	0.993291i	$0.536892\pi$
$674$	38.2487	+	10.2487i	1.47329	+	0.394766i
$675$		0			0
$676$	1.00000	−	1.73205i	0.0384615	−	0.0666173i
$677$	−10.0382	−	2.68973i	−0.385799	−	0.103375i	0.0607058	−	0.998156i	$-0.480665\pi$
$677$	−10.0382	−	2.68973i	−0.385799	−	0.103375i	−0.446505	+	0.894781i	$0.647332\pi$
$678$		0			0
$679$		0			0
$680$		−	48.0000i		−	1.84072i
$681$		0			0
$682$	−8.48528	−	4.89898i	−0.324918	−	0.187592i
$683$	−9.15064	−	34.1506i	−0.350139	−	1.30674i	−0.886492	−	0.462744i	$-0.846865\pi$
$683$	−9.15064	−	34.1506i	−0.350139	−	1.30674i	0.536353	−	0.843994i	$-0.319802\pi$
$684$	−40.1528	−	10.7589i	−1.53528	−	0.411377i
$685$	19.5959	+	19.5959i	0.748722	+	0.748722i
$686$		0			0
$687$		0			0
$688$	−27.3205	−	7.32051i	−1.04158	−	0.279092i
$689$	−4.24264	−	2.44949i	−0.161632	−	0.0933181i
$690$		0			0
$691$	13.3843	+	3.58630i	0.509161	+	0.136429i	0.504249	−	0.863559i	$-0.331770\pi$
$691$	13.3843	+	3.58630i	0.509161	+	0.136429i	0.00491275	+	0.999988i	$0.498436\pi$
$692$	−4.89898	+	4.89898i	−0.186231	+	0.186231i
$693$		0			0
$694$			6.00000i			0.227757i
$695$	−12.0000	−	20.7846i	−0.455186	−	0.788405i
$696$		0			0
$697$	−20.7846	−	12.0000i	−0.787273	−	0.454532i
$698$	7.34847	+	12.7279i	0.278144	+	0.481759i
$699$		0			0
$700$		0			0
$701$	−29.0000	+	29.0000i	−1.09531	+	1.09531i	−0.100364	+	0.994951i	$0.532001\pi$
$701$	−29.0000	+	29.0000i	−1.09531	+	1.09531i	−0.994951	+	0.100364i	$0.967999\pi$
$702$		0			0
$703$	−24.4949	+	42.4264i	−0.923843	+	1.60014i
$704$	−10.9282	+	2.92820i	−0.411872	+	0.110361i
$705$		0			0
$706$	−9.79796	−	9.79796i	−0.368751	−	0.368751i
$707$		0			0
$708$		0			0
$709$	8.41858	−	31.4186i	0.316167	−	1.17995i	−0.606732	−	0.794906i	$-0.707520\pi$
$709$	8.41858	−	31.4186i	0.316167	−	1.17995i	0.922899	−	0.385043i	$-0.125813\pi$
$710$	8.48528	+	4.89898i	0.318447	+	0.183855i
$711$	−6.00000	+	10.3923i	−0.225018	+	0.389742i
$712$		0			0
$713$	19.5959			0.733873
$714$		0			0
$715$	12.0000	−	12.0000i	0.448775	−	0.448775i
$716$	−8.19615	−	2.19615i	−0.306305	−	0.0820741i
$717$		0			0
$718$	−7.32051	−	27.3205i	−0.273199	−	1.01959i
$719$	−12.2474	−	21.2132i	−0.456753	−	0.791119i	0.542034	−	0.840356i	$-0.317654\pi$
$719$	−12.2474	−	21.2132i	−0.456753	−	0.791119i	−0.998787	+	0.0492373i	$0.984321\pi$
$720$	29.3939	+	29.3939i	1.09545	+	1.09545i
$721$		0			0
$722$	29.0000	+	29.0000i	1.07927	+	1.07927i
$723$		0			0
$724$	6.69213	−	1.79315i	0.248711	−	0.0666419i
$725$	−28.6865	+	7.68653i	−1.06539	+	0.285471i
$726$		0			0
$727$		−	19.5959i		−	0.726772i	−0.931639	−	0.363386i	$-0.881621\pi$
$727$		−	19.5959i		−	0.726772i	0.931639	−	0.363386i	$-0.118379\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$	−41.5692	+	24.0000i	−1.53855	+	0.888280i
$731$	33.4607	−	8.96575i	1.23759	−	0.331610i
$732$		0			0
$733$	8.06918	−	30.1146i	0.298042	−	1.11231i	−0.640729	−	0.767767i	$-0.721368\pi$
$733$	8.06918	−	30.1146i	0.298042	−	1.11231i	0.938771	−	0.344541i	$-0.111965\pi$
$734$	29.3939	−	29.3939i	1.08495	−	1.08495i
$735$		0			0
$736$	16.0000	−	16.0000i	0.589768	−	0.589768i
$737$	−5.00000	−	8.66025i	−0.184177	−	0.319005i
$738$	20.0764	−	5.37945i	0.739022	−	0.198020i
$739$	23.2224	−	6.22243i	0.854251	−	0.228896i	0.194986	−	0.980806i	$-0.437534\pi$
$739$	23.2224	−	6.22243i	0.854251	−	0.228896i	0.659265	+	0.751910i	$0.270867\pi$
$740$	42.4264	−	24.4949i	1.55963	−	0.900450i
$741$		0			0
$742$		0			0
$743$	4.00000			0.146746			0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000			0.146746			0.0733729	+	0.997305i	$0.476624\pi$
$744$		0			0
$745$	7.34847	−	12.7279i	0.269227	−	0.466315i
$746$	−11.0000	+	19.0526i	−0.402739	+	0.697564i
$747$		0			0
$748$	9.79796	−	9.79796i	0.358249	−	0.358249i
$749$		0			0
$750$		0			0
$751$	8.66025	−	5.00000i	0.316017	−	0.182453i	−0.333599	−	0.942715i	$-0.608263\pi$
$751$	8.66025	−	5.00000i	0.316017	−	0.182453i	0.649616	+	0.760263i	$0.274930\pi$
$752$	−9.79796	+	16.9706i	−0.357295	+	0.618853i
$753$		0			0
$754$	20.0764	+	5.37945i	0.731139	+	0.195908i
$755$	−29.3939	+	29.3939i	−1.06975	+	1.06975i
$756$		0			0
$757$	−5.00000	−	5.00000i	−0.181728	−	0.181728i	0.610380	−	0.792108i	$-0.291017\pi$
$757$	−5.00000	−	5.00000i	−0.181728	−	0.181728i	−0.792108	+	0.610380i	$0.791017\pi$
$758$	−39.8372	+	23.0000i	−1.44695	+	0.835398i
$759$		0			0
$760$	−17.5692	−	65.5692i	−0.637303	−	2.37845i
$761$	22.0454	+	38.1838i	0.799145	+	1.38416i	0.920173	+	0.391511i	$0.128048\pi$
$761$	22.0454	+	38.1838i	0.799145	+	1.38416i	−0.121028	+	0.992649i	$0.538619\pi$
$762$		0			0
$763$		0			0
$764$	−40.0000			−1.44715
$765$	−49.1769	−	13.1769i	−1.77800	−	0.476412i
$766$	5.37945	+	20.0764i	0.194368	+	0.725390i
$767$	−20.7846	−	12.0000i	−0.750489	−	0.433295i
$768$		0			0
$769$		−	34.2929i		−	1.23663i	−0.785930	−	0.618316i	$-0.787815\pi$
$769$		−	34.2929i		−	1.23663i	0.785930	−	0.618316i	$-0.212185\pi$
$770$		0			0
$771$		0			0
$772$	27.7128	−	16.0000i	0.997406	−	0.575853i
$773$	4.48288	+	16.7303i	0.161238	+	0.601748i	0.998490	+	0.0549312i	$0.0174939\pi$
$773$	4.48288	+	16.7303i	0.161238	+	0.601748i	−0.837252	+	0.546817i	$0.815839\pi$
$774$	−15.0000	+	25.9808i	−0.539164	+	0.933859i
$775$	−29.6985	+	17.1464i	−1.06680	+	0.615918i
$776$	−9.79796	+	9.79796i	−0.351726	+	0.351726i
$777$		0			0
$778$	−26.0000			−0.932145
$779$	−32.7846	−	8.78461i	−1.17463	−	0.314741i
$780$		0			0
$781$	0.732051	+	2.73205i	0.0261948	+	0.0977605i
$782$	−7.17260	+	26.7685i	−0.256492	+	0.957240i
$783$		0			0
$784$		0			0
$785$	−60.0000			−2.14149
$786$		0			0
$787$	−1.79315	−	6.69213i	−0.0639189	−	0.238549i	0.926574	−	0.376113i	$-0.122739\pi$
$787$	−1.79315	−	6.69213i	−0.0639189	−	0.238549i	−0.990493	+	0.137564i	$0.956073\pi$
$788$	−10.9808	−	40.9808i	−0.391173	−	1.45988i
$789$		0			0
$790$	−19.5959			−0.697191
$791$		0			0
$792$			12.0000i			0.426401i
$793$	−10.3923	+	6.00000i	−0.369042	+	0.213066i
$794$	−12.2474	+	21.2132i	−0.434646	+	0.752828i
$795$		0			0
$796$	9.79796	+	16.9706i	0.347279	+	0.601506i
$797$	−12.2474	−	12.2474i	−0.433827	−	0.433827i	0.456101	−	0.889928i	$-0.349246\pi$
$797$	−12.2474	−	12.2474i	−0.433827	−	0.433827i	−0.889928	+	0.456101i	$0.849246\pi$
$798$		0			0
$799$		−	24.0000i		−	0.849059i
$800$	−10.2487	+	38.2487i	−0.362347	+	1.35230i
$801$		0			0
$802$	2.92820	+	10.9282i	0.103398	+	0.385888i
$803$	−13.3843	−	3.58630i	−0.472320	−	0.126558i
$804$		0			0
$805$		0			0
$806$	24.0000			0.845364
$807$		0			0
$808$	−4.89898	+	8.48528i	−0.172345	+	0.298511i
$809$	−39.8372	−	23.0000i	−1.40060	−	0.808637i	−0.406146	−	0.913808i	$-0.633128\pi$
$809$	−39.8372	−	23.0000i	−1.40060	−	0.808637i	−0.994454	+	0.105171i	$0.966461\pi$
$810$	38.1838	−	22.0454i	1.34164	−	0.774597i
$811$	14.6969	+	14.6969i	0.516079	+	0.516079i	0.916383	−	0.400303i	$-0.131095\pi$
$811$	14.6969	+	14.6969i	0.516079	+	0.516079i	−0.400303	+	0.916383i	$0.631095\pi$
$812$		0			0
$813$		0			0
$814$	13.6603	+	3.66025i	0.478792	+	0.128292i
$815$	−2.44949	+	4.24264i	−0.0858019	+	0.148613i
$816$		0			0
$817$	42.4264	−	24.4949i	1.48431	−	0.856968i
$818$	−24.4949	+	24.4949i	−0.856444	+	0.856444i
$819$		0			0
$820$	24.0000	+	24.0000i	0.838116	+	0.838116i
$821$	−6.95448	+	25.9545i	−0.242713	+	0.905818i	0.731806	+	0.681513i	$0.238678\pi$
$821$	−6.95448	+	25.9545i	−0.242713	+	0.905818i	−0.974519	+	0.224305i	$0.927989\pi$
$822$		0			0
$823$	3.00000	−	5.19615i	0.104573	−	0.181126i	−0.808990	−	0.587822i	$-0.799986\pi$
$823$	3.00000	−	5.19615i	0.104573	−	0.181126i	0.913564	+	0.406695i	$0.133319\pi$
$824$	26.7685	+	7.17260i	0.932526	+	0.249869i
$825$		0			0
$826$		0			0
$827$	33.0000	−	33.0000i	1.14752	−	1.14752i	0.160484	−	0.987038i	$-0.448695\pi$
$827$	33.0000	−	33.0000i	1.14752	−	1.14752i	0.987038	−	0.160484i	$-0.0513055\pi$
$828$	−12.0000	−	20.7846i	−0.417029	−	0.722315i
$829$	−10.0382	+	2.68973i	−0.348641	+	0.0934181i	−0.428890	−	0.903357i	$-0.641095\pi$
$829$	−10.0382	+	2.68973i	−0.348641	+	0.0934181i	0.0802489	+	0.996775i	$0.474428\pi$
$830$		0			0
$831$		0			0
$832$	19.5959	−	19.5959i	0.679366	−	0.679366i
$833$		0			0
$834$		0			0
$835$	17.5692	−	65.5692i	0.608008	−	2.26912i
$836$	9.79796	−	16.9706i	0.338869	−	0.586939i
$837$		0			0
$838$	8.48528	−	4.89898i	0.293119	−	0.169232i
$839$		−	9.79796i		−	0.338263i	−0.985593	−	0.169132i	$-0.945904\pi$
$839$		−	9.79796i		−	0.338263i	0.985593	−	0.169132i	$-0.0540963\pi$
$840$		0			0
$841$			11.0000i			0.379310i
$842$	11.0000	+	19.0526i	0.379085	+	0.656595i
$843$		0			0
$844$	0.732051	+	2.73205i	0.0251982	+	0.0940411i
$845$	0.896575	−	3.34607i	0.0308431	−	0.115108i
$846$	14.6969	+	14.6969i	0.505291	+	0.505291i
$847$		0			0
$848$	4.00000	+	4.00000i	0.137361	+	0.137361i
$849$		0			0
$850$	−12.5521	−	46.8449i	−0.430532	−	1.60677i
$851$	−27.3205	+	7.32051i	−0.936535	+	0.250944i
$852$		0			0
$853$	12.2474	−	12.2474i	0.419345	−	0.419345i	−0.465633	−	0.884978i	$-0.654173\pi$
$853$	12.2474	−	12.2474i	0.419345	−	0.419345i	0.884978	+	0.465633i	$0.154173\pi$
$854$		0			0
$855$	−72.0000			−2.46235
$856$	−6.00000	−	10.3923i	−0.205076	−	0.355202i
$857$	2.44949	−	4.24264i	0.0836730	−	0.144926i	−0.821152	−	0.570710i	$-0.806668\pi$
$857$	2.44949	−	4.24264i	0.0836730	−	0.144926i	0.904825	+	0.425784i	$0.140002\pi$
$858$		0			0
$859$	−10.7589	+	40.1528i	−0.367089	+	1.37000i	0.497477	+	0.867477i	$0.334260\pi$
$859$	−10.7589	+	40.1528i	−0.367089	+	1.37000i	−0.864567	+	0.502518i	$0.832407\pi$
$860$	−48.9898			−1.67054
$861$		0			0
$862$	10.0000	+	10.0000i	0.340601	+	0.340601i
$863$	3.46410	−	2.00000i	0.117919	−	0.0680808i	−0.439880	−	0.898056i	$-0.644979\pi$
$863$	3.46410	−	2.00000i	0.117919	−	0.0680808i	0.557800	+	0.829976i	$0.311646\pi$
$864$		0			0
$865$	−6.00000	+	10.3923i	−0.204006	+	0.353349i
$866$	5.37945	−	20.0764i	0.182801	−	0.682224i
$867$		0			0
$868$		0			0
$869$	−4.00000	−	4.00000i	−0.135691	−	0.135691i
$870$		0			0
$871$	21.2132	+	12.2474i	0.718782	+	0.414989i
$872$	−14.0000	+	24.2487i	−0.474100	+	0.821165i
$873$	7.34847	+	12.7279i	0.248708	+	0.430775i
$874$			39.1918i			1.32568i
$875$		0			0
$876$		0			0
$877$	−17.7583	−	4.75833i	−0.599656	−	0.160677i	−0.0537936	−	0.998552i	$-0.517131\pi$
$877$	−17.7583	−	4.75833i	−0.599656	−	0.160677i	−0.545863	+	0.837875i	$0.683798\pi$
$878$	20.0764	−	5.37945i	0.677545	−	0.181548i
$879$		0			0
$880$	−16.9706	+	9.79796i	−0.572078	+	0.330289i
$881$		−	48.9898i		−	1.65051i	−0.564762	−	0.825254i	$-0.691032\pi$
$881$		−	48.9898i		−	1.65051i	0.564762	−	0.825254i	$-0.308968\pi$
$882$		0			0
$883$	−31.0000	−	31.0000i	−1.04323	−	1.04323i	−0.999022	−	0.0442108i	$-0.985923\pi$
$883$	−31.0000	−	31.0000i	−1.04323	−	1.04323i	−0.0442108	−	0.999022i	$-0.514077\pi$
$884$	−8.78461	+	32.7846i	−0.295458	+	1.10267i
$885$		0			0
$886$	−8.66025	−	5.00000i	−0.290947	−	0.167978i
$887$	−16.9706	+	9.79796i	−0.569816	+	0.328983i	−0.757076	−	0.653327i	$-0.773373\pi$
$887$	−16.9706	+	9.79796i	−0.569816	+	0.328983i	0.187260	+	0.982310i	$0.440039\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	12.2942	+	3.29423i	0.411872	+	0.110361i
$892$	16.9706	+	9.79796i	0.568216	+	0.328060i
$893$	−8.78461	−	32.7846i	−0.293966	−	1.09710i
$894$		0			0
$895$	−14.6969			−0.491264
$896$		0			0
$897$		0			0
$898$	13.6603	+	3.66025i	0.455849	+	0.122144i
$899$	−5.37945	−	20.0764i	−0.179415	−	0.669585i
$900$	36.3731	+	21.0000i	1.21244	+	0.700000i
$901$	−6.69213	−	1.79315i	−0.222947	−	0.0597385i
$902$			9.79796i			0.326236i
$903$		0			0
$904$	8.00000	+	8.00000i	0.266076	+	0.266076i
$905$	10.3923	−	6.00000i	0.345452	−	0.199447i
$906$		0			0
$907$	−2.56218	−	9.56218i	−0.0850757	−	0.317507i	0.910253	−	0.414053i	$-0.135887\pi$
$907$	−2.56218	−	9.56218i	−0.0850757	−	0.317507i	−0.995329	+	0.0965460i	$0.969220\pi$
$908$	−14.3452	+	53.5370i	−0.476062	+	1.77669i
$909$	7.34847	+	7.34847i	0.243733	+	0.243733i
$910$		0			0
$911$			50.0000i			1.65657i	0.560304	+	0.828287i	$0.310684\pi$
$911$			50.0000i			1.65657i	−0.560304	+	0.828287i	$0.689316\pi$
$912$		0			0
$913$		0			0
$914$	−24.5885	+	6.58846i	−0.813314	+	0.217927i
$915$		0			0
$916$	14.6969	+	14.6969i	0.485601	+	0.485601i
$917$		0			0
$918$		0			0
$919$	5.00000	+	8.66025i	0.164935	+	0.285675i	0.936632	−	0.350315i	$-0.113925\pi$
$919$	5.00000	+	8.66025i	0.164935	+	0.285675i	−0.771697	+	0.635990i	$0.780592\pi$
$920$	19.5959	−	33.9411i	0.646058	−	1.11901i
$921$		0			0
$922$	−22.0454	−	38.1838i	−0.726027	−	1.25752i
$923$	−4.89898	−	4.89898i	−0.161252	−	0.161252i
$924$		0			0
$925$	35.0000	−	35.0000i	1.15079	−	1.15079i
$926$	−1.46410	+	5.46410i	−0.0481134	+	0.179562i
$927$	14.6969	−	25.4558i	0.482711	−	0.836080i
$928$	−20.7846	−	12.0000i	−0.682288	−	0.393919i
$929$	−29.6985	+	17.1464i	−0.974376	+	0.562556i	−0.900567	−	0.434716i	$-0.856849\pi$
$929$	−29.6985	+	17.1464i	−0.974376	+	0.562556i	−0.0738083	+	0.997272i	$0.523515\pi$
$930$		0			0
$931$		0			0
$932$	8.00000			0.262049
$933$		0			0
$934$	−33.9411	−	19.5959i	−1.11059	−	0.641198i
$935$	12.0000	−	20.7846i	0.392442	−	0.679729i
$936$	−14.6969	−	25.4558i	−0.480384	−	0.832050i
$937$	−29.3939			−0.960256			−0.480128	−	0.877198i	$-0.659410\pi$
$937$	−29.3939			−0.960256			−0.480128	+	0.877198i	$0.659410\pi$
$938$		0			0
$939$		0			0
$940$	−8.78461	+	32.7846i	−0.286522	+	1.06932i
$941$	−36.8067	+	9.86233i	−1.19986	+	0.321503i	−0.802778	−	0.596278i	$-0.796646\pi$
$941$	−36.8067	+	9.86233i	−1.19986	+	0.321503i	−0.397087	+	0.917781i	$0.629979\pi$
$942$		0			0
$943$	−9.79796	−	16.9706i	−0.319065	−	0.552638i
$944$	19.5959	+	19.5959i	0.637793	+	0.637793i
$945$		0			0
$946$	−10.0000	−	10.0000i	−0.325128	−	0.325128i
$947$	−5.49038	+	20.4904i	−0.178413	+	0.665848i	0.817532	+	0.575884i	$0.195342\pi$
$947$	−5.49038	+	20.4904i	−0.178413	+	0.665848i	−0.995945	+	0.0899642i	$0.971325\pi$
$948$		0			0
$949$	32.7846	−	8.78461i	1.06423	−	0.285160i
$950$	−34.2929	−	59.3970i	−1.11261	−	1.92709i
$951$		0			0
$952$		0			0
$953$		−	16.0000i		−	0.518291i	−0.965838	−	0.259145i	$-0.916559\pi$
$953$		−	16.0000i		−	0.518291i	0.965838	−	0.259145i	$-0.0834409\pi$
$954$	5.19615	−	3.00000i	0.168232	−	0.0971286i
$955$	−66.9213	+	17.9315i	−2.16552	+	0.580250i
$956$	14.0000	−	24.2487i	0.452792	−	0.784259i
$957$		0			0
$958$	−24.4949	+	24.4949i	−0.791394	+	0.791394i
$959$		0			0
$960$		0			0
$961$	3.50000	+	6.06218i	0.112903	+	0.195554i
$962$	−33.4607	+	8.96575i	−1.07881	+	0.289068i
$963$	−12.2942	+	3.29423i	−0.396176	+	0.106155i
$964$	24.4949	+	42.4264i	0.788928	+	1.36646i
$965$	39.1918	−	39.1918i	1.26163	−	1.26163i
$966$		0			0
$967$	28.0000			0.900419			0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000			0.900419			0.450210	+	0.892923i	$0.351349\pi$
$968$	24.5885	+	6.58846i	0.790303	+	0.211761i
$969$		0			0
$970$	−12.0000	+	20.7846i	−0.385297	+	0.667354i
$971$	−3.58630	+	13.3843i	−0.115090	+	0.429521i	−0.999294	−	0.0375801i	$-0.988035\pi$
$971$	−3.58630	+	13.3843i	−0.115090	+	0.429521i	0.884204	+	0.467101i	$0.154702\pi$
$972$		0			0
$973$		0			0
$974$	12.0000	−	12.0000i	0.384505	−	0.384505i
$975$		0			0
$976$	13.3843	−	3.58630i	0.428420	−	0.114795i
$977$	−19.0000	+	32.9090i	−0.607864	+	1.05285i	0.383728	+	0.923446i	$0.374640\pi$
$977$	−19.0000	+	32.9090i	−0.607864	+	1.05285i	−0.991592	+	0.129405i	$0.958693\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	21.0000	+	21.0000i	0.670478	+	0.670478i
$982$	19.0526	−	11.0000i	0.607992	−	0.351024i
$983$	−8.48528	−	4.89898i	−0.270638	−	0.156253i	0.358539	−	0.933515i	$-0.383275\pi$
$983$	−8.48528	−	4.89898i	−0.270638	−	0.156253i	−0.629178	+	0.777261i	$0.716608\pi$
$984$		0			0
$985$	−36.7423	−	63.6396i	−1.17071	−	2.02773i
$986$	29.3939			0.936092
$987$		0			0
$988$			48.0000i			1.52708i
$989$	27.3205	+	7.32051i	0.868742	+	0.232779i
$990$	5.37945	+	20.0764i	0.170970	+	0.638070i
$991$	51.9615	+	30.0000i	1.65061	+	0.952981i	0.976820	+	0.214060i	$0.0686688\pi$
$991$	51.9615	+	30.0000i	1.65061	+	0.952981i	0.673792	+	0.738921i	$0.264664\pi$
$992$	−26.7685	−	7.17260i	−0.849901	−	0.227730i
$993$		0			0
$994$		0			0
$995$	24.0000	+	24.0000i	0.760851	+	0.760851i
$996$		0			0
$997$	2.68973	+	10.0382i	0.0851845	+	0.317913i	0.995349	−	0.0963340i	$-0.0307117\pi$
$997$	2.68973	+	10.0382i	0.0851845	+	0.317913i	−0.910165	+	0.414247i	$0.864045\pi$
$998$	23.0000	−	39.8372i	0.728052	−	1.26102i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.w.d.619.2		8
7.2	even	3	inner	784.2.w.d.411.1		8
7.3	odd	6		112.2.j.a.27.2	yes	4
7.4	even	3		112.2.j.a.27.1	✓	4
7.5	odd	6	inner	784.2.w.d.411.2		8
7.6	odd	2	inner	784.2.w.d.619.1		8
16.3	odd	4	inner	784.2.w.d.227.2		8
28.3	even	6		448.2.j.b.335.2		4
28.11	odd	6		448.2.j.b.335.1		4
56.3	even	6		896.2.j.e.671.1		4
56.11	odd	6		896.2.j.e.671.2		4
56.45	odd	6		896.2.j.b.671.1		4
56.53	even	6		896.2.j.b.671.2		4
112.3	even	12		112.2.j.a.83.1	yes	4
112.11	odd	12		896.2.j.b.223.1		4
112.19	even	12	inner	784.2.w.d.19.2		8
112.45	odd	12		448.2.j.b.111.1		4
112.51	odd	12	inner	784.2.w.d.19.1		8
112.53	even	12		896.2.j.e.223.1		4
112.59	even	12		896.2.j.b.223.2		4
112.67	odd	12		112.2.j.a.83.2	yes	4
112.83	even	4	inner	784.2.w.d.227.1		8
112.101	odd	12		896.2.j.e.223.2		4
112.109	even	12		448.2.j.b.111.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.j.a.27.1	✓	4	7.4	even	3
112.2.j.a.27.2	yes	4	7.3	odd	6
112.2.j.a.83.1	yes	4	112.3	even	12
112.2.j.a.83.2	yes	4	112.67	odd	12
448.2.j.b.111.1		4	112.45	odd	12
448.2.j.b.111.2		4	112.109	even	12
448.2.j.b.335.1		4	28.11	odd	6
448.2.j.b.335.2		4	28.3	even	6
784.2.w.d.19.1		8	112.51	odd	12	inner
784.2.w.d.19.2		8	112.19	even	12	inner
784.2.w.d.227.1		8	112.83	even	4	inner
784.2.w.d.227.2		8	16.3	odd	4	inner
784.2.w.d.411.1		8	7.2	even	3	inner
784.2.w.d.411.2		8	7.5	odd	6	inner
784.2.w.d.619.1		8	7.6	odd	2	inner
784.2.w.d.619.2		8	1.1	even	1	trivial
896.2.j.b.223.1		4	112.11	odd	12
896.2.j.b.223.2		4	112.59	even	12
896.2.j.b.671.1		4	56.45	odd	6
896.2.j.b.671.2		4	56.53	even	6
896.2.j.e.223.1		4	112.53	even	12
896.2.j.e.223.2		4	112.101	odd	12
896.2.j.e.671.1		4	56.3	even	6
896.2.j.e.671.2		4	56.11	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.w (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.36603 + 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(3.34607 + 0.896575i) q^{5} +(2.00000 + 2.00000i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)	\(q+(1.36603 + 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(3.34607 + 0.896575i) q^{5} +(2.00000 + 2.00000i) q^{8} +(2.59808 - 1.50000i) q^{9} +(4.24264 + 2.44949i) q^{10} +(0.366025 + 1.36603i) q^{11} +(-2.44949 - 2.44949i) q^{13} +(2.00000 + 3.46410i) q^{16} +(-4.24264 - 2.44949i) q^{17} +(4.09808 - 1.09808i) q^{18} +(-6.69213 - 1.79315i) q^{19} +(4.89898 + 4.89898i) q^{20} +2.00000i q^{22} +(-2.00000 - 3.46410i) q^{23} +(6.06218 + 3.50000i) q^{25} +(-2.44949 - 4.24264i) q^{26} +(-3.00000 + 3.00000i) q^{29} +(-2.44949 + 4.24264i) q^{31} +(1.46410 + 5.46410i) q^{32} +(-4.89898 - 4.89898i) q^{34} +6.00000 q^{36} +(1.83013 - 6.83013i) q^{37} +(-8.48528 - 4.89898i) q^{38} +(4.89898 + 8.48528i) q^{40} +4.89898 q^{41} +(-5.00000 + 5.00000i) q^{43} +(-0.732051 + 2.73205i) q^{44} +(10.0382 - 2.68973i) q^{45} +(-1.46410 - 5.46410i) q^{46} +(2.44949 + 4.24264i) q^{47} +(7.00000 + 7.00000i) q^{50} +(-1.79315 - 6.69213i) q^{52} +(1.36603 - 0.366025i) q^{53} +4.89898i q^{55} +(-5.19615 + 3.00000i) q^{58} +(6.69213 - 1.79315i) q^{59} +(0.896575 - 3.34607i) q^{61} +(-4.89898 + 4.89898i) q^{62} +8.00000i q^{64} +(-6.00000 - 10.3923i) q^{65} +(-6.83013 + 1.83013i) q^{67} +(-4.89898 - 8.48528i) q^{68} +2.00000 q^{71} +(8.19615 + 2.19615i) q^{72} +(-4.89898 + 8.48528i) q^{73} +(5.00000 - 8.66025i) q^{74} +(-9.79796 - 9.79796i) q^{76} +(-3.46410 + 2.00000i) q^{79} +(3.58630 + 13.3843i) q^{80} +(4.50000 - 7.79423i) q^{81} +(6.69213 + 1.79315i) q^{82} +(-12.0000 - 12.0000i) q^{85} +(-8.66025 + 5.00000i) q^{86} +(-2.00000 + 3.46410i) q^{88} +14.6969 q^{90} -8.00000i q^{92} +(1.79315 + 6.69213i) q^{94} +(-20.7846 - 12.0000i) q^{95} +4.89898i q^{97} +(3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 16 q^{8}+O(q^{10}) \) 8 * q + 4 * q^2 + 16 * q^8	\( 8 q + 4 q^{2} + 16 q^{8} - 4 q^{11} + 16 q^{16} + 12 q^{18} - 16 q^{23} - 24 q^{29} - 16 q^{32} + 48 q^{36} - 20 q^{37} - 40 q^{43} + 8 q^{44} + 16 q^{46} + 56 q^{50} + 4 q^{53} - 48 q^{65} - 20 q^{67} + 16 q^{71} + 24 q^{72} + 40 q^{74} + 36 q^{81} - 96 q^{85} - 16 q^{88} + 24 q^{99}+O(q^{100}) \) 8 * q + 4 * q^2 + 16 * q^8 - 4 * q^11 + 16 * q^16 + 12 * q^18 - 16 * q^23 - 24 * q^29 - 16 * q^32 + 48 * q^36 - 20 * q^37 - 40 * q^43 + 8 * q^44 + 16 * q^46 + 56 * q^50 + 4 * q^53 - 48 * q^65 - 20 * q^67 + 16 * q^71 + 24 * q^72 + 40 * q^74 + 36 * q^81 - 96 * q^85 - 16 * q^88 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more