LMFDB - Embedded newform 784.2.i.n.753.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,2,Mod(177,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("784.177");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	784.i (of order $3$, degree $2$, 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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 392)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			753.1
Root			$-0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	784.753
Dual form			784.2.i.n.177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 1.22474i) q^{3} +(1.41421 + 2.44949i) q^{5} +(0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.707107 + 1.22474i) q^{3} +(1.41421 + 2.44949i) q^{5} +(0.500000 + 0.866025i) q^{9} +(3.00000 - 5.19615i) q^{11} +5.65685 q^{13} -4.00000 q^{15} +(0.707107 - 1.22474i) q^{17} +(2.12132 + 3.67423i) q^{19} +(2.00000 + 3.46410i) q^{23} +(-1.50000 + 2.59808i) q^{25} -5.65685 q^{27} -6.00000 q^{29} +(-1.41421 + 2.44949i) q^{31} +(4.24264 + 7.34847i) q^{33} +(-1.00000 - 1.73205i) q^{37} +(-4.00000 + 6.92820i) q^{39} +1.41421 q^{41} -10.0000 q^{43} +(-1.41421 + 2.44949i) q^{45} +(1.41421 + 2.44949i) q^{47} +(1.00000 + 1.73205i) q^{51} +(1.00000 - 1.73205i) q^{53} +16.9706 q^{55} -6.00000 q^{57} +(-0.707107 + 1.22474i) q^{59} +(-4.24264 - 7.34847i) q^{61} +(8.00000 + 13.8564i) q^{65} +(2.00000 - 3.46410i) q^{67} -5.65685 q^{69} +12.0000 q^{71} +(-4.94975 + 8.57321i) q^{73} +(-2.12132 - 3.67423i) q^{75} +(-2.00000 - 3.46410i) q^{79} +(2.50000 - 4.33013i) q^{81} +1.41421 q^{83} +4.00000 q^{85} +(4.24264 - 7.34847i) q^{87} +(-2.12132 - 3.67423i) q^{89} +(-2.00000 - 3.46410i) q^{93} +(-6.00000 + 10.3923i) q^{95} -12.7279 q^{97} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^9	$ 4 q + 2 q^{9} + 12 q^{11} - 16 q^{15} + 8 q^{23} - 6 q^{25} - 24 q^{29} - 4 q^{37} - 16 q^{39} - 40 q^{43} + 4 q^{51} + 4 q^{53} - 24 q^{57} + 32 q^{65} + 8 q^{67} + 48 q^{71} - 8 q^{79} + 10 q^{81} + 16 q^{85} - 8 q^{93} - 24 q^{95} + 24 q^{99}+O(q^{100}) $ 4 * q + 2 * q^9 + 12 * q^11 - 16 * q^15 + 8 * q^23 - 6 * q^25 - 24 * q^29 - 4 * q^37 - 16 * q^39 - 40 * q^43 + 4 * q^51 + 4 * q^53 - 24 * q^57 + 32 * q^65 + 8 * q^67 + 48 * q^71 - 8 * q^79 + 10 * q^81 + 16 * q^85 - 8 * q^93 - 24 * q^95 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	−0.707107	+	1.22474i	−0.408248	+	0.707107i	−0.994694	−	0.102882i	$-0.967194\pi$
$3$	−0.707107	+	1.22474i	−0.408248	+	0.707107i	0.586445	+	0.809989i	$0.300527\pi$
$4$		0			0
$5$	1.41421	+	2.44949i	0.632456	+	1.09545i	0.987048	+	0.160424i	$0.0512862\pi$
$5$	1.41421	+	2.44949i	0.632456	+	1.09545i	−0.354593	+	0.935021i	$0.615380\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	0.500000	+	0.866025i	0.166667	+	0.288675i
$10$		0			0
$11$	3.00000	−	5.19615i	0.904534	−	1.56670i	0.0829925	−	0.996550i	$-0.473552\pi$
$11$	3.00000	−	5.19615i	0.904534	−	1.56670i	0.821541	−	0.570149i	$-0.193114\pi$
$12$		0			0
$13$	5.65685			1.56893			0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685			1.56893			0.784465	+	0.620174i	$0.212938\pi$
$14$		0			0
$15$	−4.00000			−1.03280
$16$		0			0
$17$	0.707107	−	1.22474i	0.171499	−	0.297044i	−0.767445	−	0.641114i	$-0.778472\pi$
$17$	0.707107	−	1.22474i	0.171499	−	0.297044i	0.938944	+	0.344070i	$0.111806\pi$
$18$		0			0
$19$	2.12132	+	3.67423i	0.486664	+	0.842927i	0.999882	−	0.0153309i	$-0.00488018\pi$
$19$	2.12132	+	3.67423i	0.486664	+	0.842927i	−0.513218	+	0.858258i	$0.671547\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	−1.50000	+	2.59808i	−0.300000	+	0.519615i
$26$		0			0
$27$	−5.65685			−1.08866
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	−1.41421	+	2.44949i	−0.254000	+	0.439941i	−0.964623	−	0.263631i	$-0.915080\pi$
$31$	−1.41421	+	2.44949i	−0.254000	+	0.439941i	0.710623	+	0.703573i	$0.248413\pi$
$32$		0			0
$33$	4.24264	+	7.34847i	0.738549	+	1.27920i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.00000	−	1.73205i	−0.164399	−	0.284747i	0.772043	−	0.635571i	$-0.219235\pi$
$37$	−1.00000	−	1.73205i	−0.164399	−	0.284747i	−0.936442	+	0.350823i	$0.885902\pi$
$38$		0			0
$39$	−4.00000	+	6.92820i	−0.640513	+	1.10940i
$40$		0			0
$41$	1.41421			0.220863			0.110432	−	0.993884i	$-0.464777\pi$
$41$	1.41421			0.220863			0.110432	+	0.993884i	$0.464777\pi$
$42$		0			0
$43$	−10.0000			−1.52499			−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000			−1.52499			−0.762493	+	0.646997i	$0.776025\pi$
$44$		0			0
$45$	−1.41421	+	2.44949i	−0.210819	+	0.365148i
$46$		0			0
$47$	1.41421	+	2.44949i	0.206284	+	0.357295i	0.950541	−	0.310599i	$-0.100530\pi$
$47$	1.41421	+	2.44949i	0.206284	+	0.357295i	−0.744257	+	0.667893i	$0.767196\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	1.00000	+	1.73205i	0.140028	+	0.242536i
$52$		0			0
$53$	1.00000	−	1.73205i	0.137361	−	0.237915i	−0.789136	−	0.614218i	$-0.789471\pi$
$53$	1.00000	−	1.73205i	0.137361	−	0.237915i	0.926497	+	0.376303i	$0.122805\pi$
$54$		0			0
$55$	16.9706			2.28831
$56$		0			0
$57$	−6.00000			−0.794719
$58$		0			0
$59$	−0.707107	+	1.22474i	−0.0920575	+	0.159448i	−0.908377	−	0.418153i	$-0.862678\pi$
$59$	−0.707107	+	1.22474i	−0.0920575	+	0.159448i	0.816319	+	0.577601i	$0.196011\pi$
$60$		0			0
$61$	−4.24264	−	7.34847i	−0.543214	−	0.940875i	−0.998717	−	0.0506406i	$-0.983874\pi$
$61$	−4.24264	−	7.34847i	−0.543214	−	0.940875i	0.455502	−	0.890235i	$-0.349460\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	8.00000	+	13.8564i	0.992278	+	1.71868i
$66$		0			0
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	−0.717607	−	0.696449i	$-0.754762\pi$
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	0.961946	+	0.273241i	$0.0880957\pi$
$68$		0			0
$69$	−5.65685			−0.681005
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	−4.94975	+	8.57321i	−0.579324	+	1.00342i	0.416233	+	0.909258i	$0.363350\pi$
$73$	−4.94975	+	8.57321i	−0.579324	+	1.00342i	−0.995557	+	0.0941608i	$0.969983\pi$
$74$		0			0
$75$	−2.12132	−	3.67423i	−0.244949	−	0.424264i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.00000	−	3.46410i	−0.225018	−	0.389742i	0.731307	−	0.682048i	$-0.238911\pi$
$79$	−2.00000	−	3.46410i	−0.225018	−	0.389742i	−0.956325	+	0.292306i	$0.905577\pi$
$80$		0			0
$81$	2.50000	−	4.33013i	0.277778	−	0.481125i
$82$		0			0
$83$	1.41421			0.155230			0.0776151	−	0.996983i	$-0.475269\pi$
$83$	1.41421			0.155230			0.0776151	+	0.996983i	$0.475269\pi$
$84$		0			0
$85$	4.00000			0.433861
$86$		0			0
$87$	4.24264	−	7.34847i	0.454859	−	0.787839i
$88$		0			0
$89$	−2.12132	−	3.67423i	−0.224860	−	0.389468i	0.731418	−	0.681930i	$-0.238859\pi$
$89$	−2.12132	−	3.67423i	−0.224860	−	0.389468i	−0.956277	+	0.292462i	$0.905526\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−2.00000	−	3.46410i	−0.207390	−	0.359211i
$94$		0			0
$95$	−6.00000	+	10.3923i	−0.615587	+	1.06623i
$96$		0			0
$97$	−12.7279			−1.29232			−0.646162	−	0.763200i	$-0.723627\pi$
$97$	−12.7279			−1.29232			−0.646162	+	0.763200i	$0.723627\pi$
$98$		0			0
$99$	6.00000			0.603023
$100$		0			0
$101$	−1.41421	+	2.44949i	−0.140720	+	0.243733i	−0.927768	−	0.373158i	$-0.878275\pi$
$101$	−1.41421	+	2.44949i	−0.140720	+	0.243733i	0.787048	+	0.616891i	$0.211608\pi$
$102$		0			0
$103$	7.07107	+	12.2474i	0.696733	+	1.20678i	0.969593	+	0.244723i	$0.0786971\pi$
$103$	7.07107	+	12.2474i	0.696733	+	1.20678i	−0.272860	+	0.962054i	$0.587970\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.00000	+	3.46410i	0.193347	+	0.334887i	0.946357	−	0.323122i	$-0.104732\pi$
$107$	2.00000	+	3.46410i	0.193347	+	0.334887i	−0.753010	+	0.658009i	$0.771399\pi$
$108$		0			0
$109$	5.00000	−	8.66025i	0.478913	−	0.829502i	−0.520794	−	0.853682i	$-0.674364\pi$
$109$	5.00000	−	8.66025i	0.478913	−	0.829502i	0.999708	+	0.0241802i	$0.00769755\pi$
$110$		0			0
$111$	2.82843			0.268462
$112$		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
$115$	−5.65685	+	9.79796i	−0.527504	+	0.913664i
$116$		0			0
$117$	2.82843	+	4.89898i	0.261488	+	0.452911i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−12.5000	−	21.6506i	−1.13636	−	1.96824i
$122$		0			0
$123$	−1.00000	+	1.73205i	−0.0901670	+	0.156174i
$124$		0			0
$125$	5.65685			0.505964
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$	7.07107	−	12.2474i	0.622573	−	1.07833i
$130$		0			0
$131$	−10.6066	−	18.3712i	−0.926703	−	1.60510i	−0.788799	−	0.614651i	$-0.789297\pi$
$131$	−10.6066	−	18.3712i	−0.926703	−	1.60510i	−0.137904	−	0.990446i	$-0.544037\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−8.00000	−	13.8564i	−0.688530	−	1.19257i
$136$		0			0
$137$	2.00000	−	3.46410i	0.170872	−	0.295958i	−0.767853	−	0.640626i	$-0.778675\pi$
$137$	2.00000	−	3.46410i	0.170872	−	0.295958i	0.938725	+	0.344668i	$0.112008\pi$
$138$		0			0
$139$	−21.2132			−1.79928			−0.899640	−	0.436632i	$-0.856171\pi$
$139$	−21.2132			−1.79928			−0.899640	+	0.436632i	$0.856171\pi$
$140$		0			0
$141$	−4.00000			−0.336861
$142$		0			0
$143$	16.9706	−	29.3939i	1.41915	−	2.45804i
$144$		0			0
$145$	−8.48528	−	14.6969i	−0.704664	−	1.22051i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	0.962348	−	0.271821i	$-0.0876260\pi$
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	−0.716578	+	0.697507i	$0.754293\pi$
$150$		0			0
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	−0.656101	−	0.754673i	$-0.727796\pi$
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	0.981617	+	0.190864i	$0.0611289\pi$
$152$		0			0
$153$	1.41421			0.114332
$154$		0			0
$155$	−8.00000			−0.642575
$156$		0			0
$157$	8.48528	−	14.6969i	0.677199	−	1.17294i	−0.298622	−	0.954372i	$-0.596527\pi$
$157$	8.48528	−	14.6969i	0.677199	−	1.17294i	0.975821	−	0.218572i	$-0.0701398\pi$
$158$		0			0
$159$	1.41421	+	2.44949i	0.112154	+	0.194257i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.00000	+	1.73205i	0.0783260	+	0.135665i	0.902528	−	0.430632i	$-0.141709\pi$
$163$	1.00000	+	1.73205i	0.0783260	+	0.135665i	−0.824202	+	0.566296i	$0.808376\pi$
$164$		0			0
$165$	−12.0000	+	20.7846i	−0.934199	+	1.61808i
$166$		0			0
$167$	−2.82843			−0.218870			−0.109435	−	0.993994i	$-0.534904\pi$
$167$	−2.82843			−0.218870			−0.109435	+	0.993994i	$0.534904\pi$
$168$		0			0
$169$	19.0000			1.46154
$170$		0			0
$171$	−2.12132	+	3.67423i	−0.162221	+	0.280976i
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.00000	−	1.73205i	−0.0751646	−	0.130189i
$178$		0			0
$179$	10.0000	−	17.3205i	0.747435	−	1.29460i	−0.201613	−	0.979465i	$-0.564618\pi$
$179$	10.0000	−	17.3205i	0.747435	−	1.29460i	0.949048	−	0.315130i	$-0.102048\pi$
$180$		0			0
$181$	−16.9706			−1.26141			−0.630706	−	0.776022i	$-0.717235\pi$
$181$	−16.9706			−1.26141			−0.630706	+	0.776022i	$0.717235\pi$
$182$		0			0
$183$	12.0000			0.887066
$184$		0			0
$185$	2.82843	−	4.89898i	0.207950	−	0.360180i
$186$		0			0
$187$	−4.24264	−	7.34847i	−0.310253	−	0.537373i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.00000	−	3.46410i	−0.144715	−	0.250654i	0.784552	−	0.620063i	$-0.212893\pi$
$191$	−2.00000	−	3.46410i	−0.144715	−	0.250654i	−0.929267	+	0.369410i	$0.879560\pi$
$192$		0			0
$193$	−8.00000	+	13.8564i	−0.575853	+	0.997406i	0.420096	+	0.907480i	$0.361996\pi$
$193$	−8.00000	+	13.8564i	−0.575853	+	0.997406i	−0.995948	+	0.0899262i	$0.971337\pi$
$194$		0			0
$195$	−22.6274			−1.62038
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	−12.7279	+	22.0454i	−0.902258	+	1.56276i	−0.0777029	+	0.996977i	$0.524759\pi$
$199$	−12.7279	+	22.0454i	−0.902258	+	1.56276i	−0.824556	+	0.565781i	$0.808575\pi$
$200$		0			0
$201$	2.82843	+	4.89898i	0.199502	+	0.345547i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	2.00000	+	3.46410i	0.139686	+	0.241943i
$206$		0			0
$207$	−2.00000	+	3.46410i	−0.139010	+	0.240772i
$208$		0			0
$209$	25.4558			1.76082
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		0			0
$213$	−8.48528	+	14.6969i	−0.581402	+	1.00702i
$214$		0			0
$215$	−14.1421	−	24.4949i	−0.964486	−	1.67054i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−7.00000	−	12.1244i	−0.473016	−	0.819288i
$220$		0			0
$221$	4.00000	−	6.92820i	0.269069	−	0.466041i
$222$		0			0
$223$	11.3137			0.757622			0.378811	−	0.925474i	$-0.376333\pi$
$223$	11.3137			0.757622			0.378811	+	0.925474i	$0.376333\pi$
$224$		0			0
$225$	−3.00000			−0.200000
$226$		0			0
$227$	−4.94975	+	8.57321i	−0.328526	+	0.569024i	−0.982220	−	0.187735i	$-0.939885\pi$
$227$	−4.94975	+	8.57321i	−0.328526	+	0.569024i	0.653693	+	0.756760i	$0.273219\pi$
$228$		0			0
$229$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$229$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.00000	−	6.92820i	−0.262049	−	0.453882i	0.704737	−	0.709468i	$-0.251065\pi$
$233$	−4.00000	−	6.92820i	−0.262049	−	0.453882i	−0.966786	+	0.255586i	$0.917731\pi$
$234$		0			0
$235$	−4.00000	+	6.92820i	−0.260931	+	0.451946i
$236$		0			0
$237$	5.65685			0.367452
$238$		0			0
$239$	4.00000			0.258738			0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000			0.258738			0.129369	+	0.991596i	$0.458705\pi$
$240$		0			0
$241$	−0.707107	+	1.22474i	−0.0455488	+	0.0788928i	−0.887901	−	0.460035i	$-0.847837\pi$
$241$	−0.707107	+	1.22474i	−0.0455488	+	0.0788928i	0.842352	+	0.538927i	$0.181170\pi$
$242$		0			0
$243$	−4.94975	−	8.57321i	−0.317526	−	0.549972i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	12.0000	+	20.7846i	0.763542	+	1.32249i
$248$		0			0
$249$	−1.00000	+	1.73205i	−0.0633724	+	0.109764i
$250$		0			0
$251$	1.41421			0.0892644			0.0446322	−	0.999003i	$-0.485788\pi$
$251$	1.41421			0.0892644			0.0446322	+	0.999003i	$0.485788\pi$
$252$		0			0
$253$	24.0000			1.50887
$254$		0			0
$255$	−2.82843	+	4.89898i	−0.177123	+	0.306786i
$256$		0			0
$257$	−6.36396	−	11.0227i	−0.396973	−	0.687577i	0.596378	−	0.802704i	$-0.296606\pi$
$257$	−6.36396	−	11.0227i	−0.396973	−	0.687577i	−0.993351	+	0.115126i	$0.963273\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−3.00000	−	5.19615i	−0.185695	−	0.321634i
$262$		0			0
$263$	−2.00000	+	3.46410i	−0.123325	+	0.213606i	−0.921077	−	0.389380i	$-0.872689\pi$
$263$	−2.00000	+	3.46410i	−0.123325	+	0.213606i	0.797752	+	0.602986i	$0.206023\pi$
$264$		0			0
$265$	5.65685			0.347498
$266$		0			0
$267$	6.00000			0.367194
$268$		0			0
$269$	14.1421	−	24.4949i	0.862261	−	1.49348i	−0.00747990	−	0.999972i	$-0.502381\pi$
$269$	14.1421	−	24.4949i	0.862261	−	1.49348i	0.869741	−	0.493508i	$-0.164286\pi$
$270$		0			0
$271$	5.65685	+	9.79796i	0.343629	+	0.595184i	0.985104	−	0.171961i	$-0.0550103\pi$
$271$	5.65685	+	9.79796i	0.343629	+	0.595184i	−0.641474	+	0.767144i	$0.721677\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	9.00000	+	15.5885i	0.542720	+	0.940019i
$276$		0			0
$277$	−7.00000	+	12.1244i	−0.420589	+	0.728482i	−0.995997	−	0.0893846i	$-0.971510\pi$
$277$	−7.00000	+	12.1244i	−0.420589	+	0.728482i	0.575408	+	0.817867i	$0.304843\pi$
$278$		0			0
$279$	−2.82843			−0.169334
$280$		0			0
$281$	−32.0000			−1.90896			−0.954480	−	0.298275i	$-0.903589\pi$
$281$	−32.0000			−1.90896			−0.954480	+	0.298275i	$0.903589\pi$
$282$		0			0
$283$	10.6066	−	18.3712i	0.630497	−	1.09205i	−0.356953	−	0.934122i	$-0.616184\pi$
$283$	10.6066	−	18.3712i	0.630497	−	1.09205i	0.987450	−	0.157931i	$-0.0504822\pi$
$284$		0			0
$285$	−8.48528	−	14.6969i	−0.502625	−	0.870572i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	7.50000	+	12.9904i	0.441176	+	0.764140i
$290$		0			0
$291$	9.00000	−	15.5885i	0.527589	−	0.913812i
$292$		0			0
$293$	−2.82843			−0.165238			−0.0826192	−	0.996581i	$-0.526329\pi$
$293$	−2.82843			−0.165238			−0.0826192	+	0.996581i	$0.526329\pi$
$294$		0			0
$295$	−4.00000			−0.232889
$296$		0			0
$297$	−16.9706	+	29.3939i	−0.984732	+	1.70561i
$298$		0			0
$299$	11.3137	+	19.5959i	0.654289	+	1.13326i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−2.00000	−	3.46410i	−0.114897	−	0.199007i
$304$		0			0
$305$	12.0000	−	20.7846i	0.687118	−	1.19012i
$306$		0			0
$307$	−1.41421			−0.0807134			−0.0403567	−	0.999185i	$-0.512849\pi$
$307$	−1.41421			−0.0807134			−0.0403567	+	0.999185i	$0.512849\pi$
$308$		0			0
$309$	−20.0000			−1.13776
$310$		0			0
$311$	5.65685	−	9.79796i	0.320771	−	0.555591i	−0.659877	−	0.751374i	$-0.729391\pi$
$311$	5.65685	−	9.79796i	0.320771	−	0.555591i	0.980647	+	0.195783i	$0.0627248\pi$
$312$		0			0
$313$	10.6066	+	18.3712i	0.599521	+	1.03840i	0.992892	+	0.119020i	$0.0379754\pi$
$313$	10.6066	+	18.3712i	0.599521	+	1.03840i	−0.393371	+	0.919380i	$0.628691\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	11.0000	+	19.0526i	0.617822	+	1.07010i	0.989882	+	0.141890i	$0.0453179\pi$
$317$	11.0000	+	19.0526i	0.617822	+	1.07010i	−0.372061	+	0.928208i	$0.621349\pi$
$318$		0			0
$319$	−18.0000	+	31.1769i	−1.00781	+	1.74557i
$320$		0			0
$321$	−5.65685			−0.315735
$322$		0			0
$323$	6.00000			0.333849
$324$		0			0
$325$	−8.48528	+	14.6969i	−0.470679	+	0.815239i
$326$		0			0
$327$	7.07107	+	12.2474i	0.391031	+	0.677285i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−7.00000	−	12.1244i	−0.384755	−	0.666415i	0.606980	−	0.794717i	$-0.292381\pi$
$331$	−7.00000	−	12.1244i	−0.384755	−	0.666415i	−0.991735	+	0.128302i	$0.959047\pi$
$332$		0			0
$333$	1.00000	−	1.73205i	0.0547997	−	0.0949158i
$334$		0			0
$335$	11.3137			0.618134
$336$		0			0
$337$	−14.0000			−0.762629			−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000			−0.762629			−0.381314	+	0.924445i	$0.624528\pi$
$338$		0			0
$339$	−2.82843	+	4.89898i	−0.153619	+	0.266076i
$340$		0			0
$341$	8.48528	+	14.6969i	0.459504	+	0.795884i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−8.00000	−	13.8564i	−0.430706	−	0.746004i
$346$		0			0
$347$	5.00000	−	8.66025i	0.268414	−	0.464907i	−0.700038	−	0.714105i	$-0.746834\pi$
$347$	5.00000	−	8.66025i	0.268414	−	0.464907i	0.968452	+	0.249198i	$0.0801671\pi$
$348$		0			0
$349$	−16.9706			−0.908413			−0.454207	−	0.890896i	$-0.650077\pi$
$349$	−16.9706			−0.908413			−0.454207	+	0.890896i	$0.650077\pi$
$350$		0			0
$351$	−32.0000			−1.70803
$352$		0			0
$353$	12.0208	−	20.8207i	0.639803	−	1.10817i	−0.345672	−	0.938355i	$-0.612349\pi$
$353$	12.0208	−	20.8207i	0.639803	−	1.10817i	0.985476	−	0.169817i	$-0.0543176\pi$
$354$		0			0
$355$	16.9706	+	29.3939i	0.900704	+	1.56007i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.00000	−	6.92820i	−0.211112	−	0.365657i	0.740951	−	0.671559i	$-0.234375\pi$
$359$	−4.00000	−	6.92820i	−0.211112	−	0.365657i	−0.952063	+	0.305903i	$0.901042\pi$
$360$		0			0
$361$	0.500000	−	0.866025i	0.0263158	−	0.0455803i
$362$		0			0
$363$	35.3553			1.85567
$364$		0			0
$365$	−28.0000			−1.46559
$366$		0			0
$367$	−2.82843	+	4.89898i	−0.147643	+	0.255725i	−0.930356	−	0.366658i	$-0.880502\pi$
$367$	−2.82843	+	4.89898i	−0.147643	+	0.255725i	0.782713	+	0.622383i	$0.213835\pi$
$368$		0			0
$369$	0.707107	+	1.22474i	0.0368105	+	0.0637577i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−13.0000	−	22.5167i	−0.673114	−	1.16587i	−0.977016	−	0.213165i	$-0.931623\pi$
$373$	−13.0000	−	22.5167i	−0.673114	−	1.16587i	0.303902	−	0.952703i	$-0.401711\pi$
$374$		0			0
$375$	−4.00000	+	6.92820i	−0.206559	+	0.357771i
$376$		0			0
$377$	−33.9411			−1.74806
$378$		0			0
$379$	2.00000			0.102733			0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000			0.102733			0.0513665	+	0.998680i	$0.483642\pi$
$380$		0			0
$381$	−5.65685	+	9.79796i	−0.289809	+	0.501965i
$382$		0			0
$383$	−7.07107	−	12.2474i	−0.361315	−	0.625815i	0.626863	−	0.779130i	$-0.284339\pi$
$383$	−7.07107	−	12.2474i	−0.361315	−	0.625815i	−0.988177	+	0.153314i	$0.951005\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−5.00000	−	8.66025i	−0.254164	−	0.440225i
$388$		0			0
$389$	15.0000	−	25.9808i	0.760530	−	1.31728i	−0.182047	−	0.983290i	$-0.558272\pi$
$389$	15.0000	−	25.9808i	0.760530	−	1.31728i	0.942578	−	0.333987i	$-0.108394\pi$
$390$		0			0
$391$	5.65685			0.286079
$392$		0			0
$393$	30.0000			1.51330
$394$		0			0
$395$	5.65685	−	9.79796i	0.284627	−	0.492989i
$396$		0			0
$397$	14.1421	+	24.4949i	0.709773	+	1.22936i	0.964941	+	0.262467i	$0.0845360\pi$
$397$	14.1421	+	24.4949i	0.709773	+	1.22936i	−0.255168	+	0.966897i	$0.582131\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	0.998350	−	0.0574304i	$-0.0182907\pi$
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	−0.548911	+	0.835881i	$0.684957\pi$
$402$		0			0
$403$	−8.00000	+	13.8564i	−0.398508	+	0.690237i
$404$		0			0
$405$	14.1421			0.702728
$406$		0			0
$407$	−12.0000			−0.594818
$408$		0			0
$409$	−2.12132	+	3.67423i	−0.104893	+	0.181679i	−0.913694	−	0.406402i	$-0.866783\pi$
$409$	−2.12132	+	3.67423i	−0.104893	+	0.181679i	0.808802	+	0.588081i	$0.200117\pi$
$410$		0			0
$411$	2.82843	+	4.89898i	0.139516	+	0.241649i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	2.00000	+	3.46410i	0.0981761	+	0.170046i
$416$		0			0
$417$	15.0000	−	25.9808i	0.734553	−	1.27228i
$418$		0			0
$419$	32.5269			1.58904			0.794522	−	0.607236i	$-0.207722\pi$
$419$	32.5269			1.58904			0.794522	+	0.607236i	$0.207722\pi$
$420$		0			0
$421$	−18.0000			−0.877266			−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000			−0.877266			−0.438633	+	0.898666i	$0.644537\pi$
$422$		0			0
$423$	−1.41421	+	2.44949i	−0.0687614	+	0.119098i
$424$		0			0
$425$	2.12132	+	3.67423i	0.102899	+	0.178227i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	24.0000	+	41.5692i	1.15873	+	2.00698i
$430$		0			0
$431$	18.0000	−	31.1769i	0.867029	−	1.50174i	0.00201168	−	0.999998i	$-0.499360\pi$
$431$	18.0000	−	31.1769i	0.867029	−	1.50174i	0.865018	−	0.501741i	$-0.167307\pi$
$432$		0			0
$433$	15.5563			0.747590			0.373795	−	0.927511i	$-0.378056\pi$
$433$	15.5563			0.747590			0.373795	+	0.927511i	$0.378056\pi$
$434$		0			0
$435$	24.0000			1.15071
$436$		0			0
$437$	−8.48528	+	14.6969i	−0.405906	+	0.703050i
$438$		0			0
$439$	8.48528	+	14.6969i	0.404980	+	0.701447i	0.994319	−	0.106439i	$-0.0339450\pi$
$439$	8.48528	+	14.6969i	0.404980	+	0.701447i	−0.589339	+	0.807886i	$0.700612\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	18.0000	+	31.1769i	0.855206	+	1.48126i	0.876454	+	0.481486i	$0.159903\pi$
$443$	18.0000	+	31.1769i	0.855206	+	1.48126i	−0.0212481	+	0.999774i	$0.506764\pi$
$444$		0			0
$445$	6.00000	−	10.3923i	0.284427	−	0.492642i
$446$		0			0
$447$	−8.48528			−0.401340
$448$		0			0
$449$	−2.00000			−0.0943858			−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000			−0.0943858			−0.0471929	+	0.998886i	$0.515028\pi$
$450$		0			0
$451$	4.24264	−	7.34847i	0.199778	−	0.346026i
$452$		0			0
$453$	5.65685	+	9.79796i	0.265782	+	0.460348i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	12.0000	+	20.7846i	0.561336	+	0.972263i	0.997380	+	0.0723376i	$0.0230459\pi$
$457$	12.0000	+	20.7846i	0.561336	+	0.972263i	−0.436044	+	0.899925i	$0.643621\pi$
$458$		0			0
$459$	−4.00000	+	6.92820i	−0.186704	+	0.323381i
$460$		0			0
$461$	22.6274			1.05386			0.526932	−	0.849907i	$-0.323342\pi$
$461$	22.6274			1.05386			0.526932	+	0.849907i	$0.323342\pi$
$462$		0			0
$463$	16.0000			0.743583			0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000			0.743583			0.371792	+	0.928316i	$0.378744\pi$
$464$		0			0
$465$	5.65685	−	9.79796i	0.262330	−	0.454369i
$466$		0			0
$467$	10.6066	+	18.3712i	0.490815	+	0.850117i	0.999944	−	0.0105737i	$-0.00336576\pi$
$467$	10.6066	+	18.3712i	0.490815	+	0.850117i	−0.509129	+	0.860690i	$0.670032\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	12.0000	+	20.7846i	0.552931	+	0.957704i
$472$		0			0
$473$	−30.0000	+	51.9615i	−1.37940	+	2.38919i
$474$		0			0
$475$	−12.7279			−0.583997
$476$		0			0
$477$	2.00000			0.0915737
$478$		0			0
$479$	−9.89949	+	17.1464i	−0.452319	+	0.783440i	−0.998530	−	0.0542078i	$-0.982737\pi$
$479$	−9.89949	+	17.1464i	−0.452319	+	0.783440i	0.546210	+	0.837648i	$0.316070\pi$
$480$		0			0
$481$	−5.65685	−	9.79796i	−0.257930	−	0.446748i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−18.0000	−	31.1769i	−0.817338	−	1.41567i
$486$		0			0
$487$	18.0000	−	31.1769i	0.815658	−	1.41276i	−0.0931967	−	0.995648i	$-0.529709\pi$
$487$	18.0000	−	31.1769i	0.815658	−	1.41276i	0.908855	−	0.417113i	$-0.136958\pi$
$488$		0			0
$489$	−2.82843			−0.127906
$490$		0			0
$491$	−28.0000			−1.26362			−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000			−1.26362			−0.631811	+	0.775122i	$0.717688\pi$
$492$		0			0
$493$	−4.24264	+	7.34847i	−0.191079	+	0.330958i
$494$		0			0
$495$	8.48528	+	14.6969i	0.381385	+	0.660578i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−14.0000	−	24.2487i	−0.626726	−	1.08552i	−0.988204	−	0.153141i	$-0.951061\pi$
$499$	−14.0000	−	24.2487i	−0.626726	−	1.08552i	0.361478	−	0.932381i	$-0.382272\pi$
$500$		0			0
$501$	2.00000	−	3.46410i	0.0893534	−	0.154765i
$502$		0			0
$503$	−5.65685			−0.252227			−0.126113	−	0.992016i	$-0.540250\pi$
$503$	−5.65685			−0.252227			−0.126113	+	0.992016i	$0.540250\pi$
$504$		0			0
$505$	−8.00000			−0.355995
$506$		0			0
$507$	−13.4350	+	23.2702i	−0.596671	+	1.03346i
$508$		0			0
$509$	−19.7990	−	34.2929i	−0.877575	−	1.52000i	−0.853994	−	0.520282i	$-0.825827\pi$
$509$	−19.7990	−	34.2929i	−0.877575	−	1.52000i	−0.0235804	−	0.999722i	$-0.507507\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−12.0000	−	20.7846i	−0.529813	−	0.917663i
$514$		0			0
$515$	−20.0000	+	34.6410i	−0.881305	+	1.52647i
$516$		0			0
$517$	16.9706			0.746364
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−16.2635	+	28.1691i	−0.712515	+	1.23411i	0.251395	+	0.967885i	$0.419111\pi$
$521$	−16.2635	+	28.1691i	−0.712515	+	1.23411i	−0.963910	+	0.266228i	$0.914223\pi$
$522$		0			0
$523$	−16.2635	−	28.1691i	−0.711151	−	1.23175i	−0.964425	−	0.264356i	$-0.914841\pi$
$523$	−16.2635	−	28.1691i	−0.711151	−	1.23175i	0.253274	−	0.967395i	$-0.418493\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.00000	+	3.46410i	0.0871214	+	0.150899i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$	−1.41421			−0.0613716
$532$		0			0
$533$	8.00000			0.346518
$534$		0			0
$535$	−5.65685	+	9.79796i	−0.244567	+	0.423603i
$536$		0			0
$537$	14.1421	+	24.4949i	0.610278	+	1.05703i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−5.00000	−	8.66025i	−0.214967	−	0.372333i	0.738296	−	0.674477i	$-0.235631\pi$
$541$	−5.00000	−	8.66025i	−0.214967	−	0.372333i	−0.953262	+	0.302144i	$0.902298\pi$
$542$		0			0
$543$	12.0000	−	20.7846i	0.514969	−	0.891953i
$544$		0			0
$545$	28.2843			1.21157
$546$		0			0
$547$	−14.0000			−0.598597			−0.299298	−	0.954160i	$-0.596753\pi$
$547$	−14.0000			−0.598597			−0.299298	+	0.954160i	$0.596753\pi$
$548$		0			0
$549$	4.24264	−	7.34847i	0.181071	−	0.313625i
$550$		0			0
$551$	−12.7279	−	22.0454i	−0.542228	−	0.939166i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	4.00000	+	6.92820i	0.169791	+	0.294086i
$556$		0			0
$557$	−9.00000	+	15.5885i	−0.381342	+	0.660504i	−0.991254	−	0.131965i	$-0.957871\pi$
$557$	−9.00000	+	15.5885i	−0.381342	+	0.660504i	0.609912	+	0.792469i	$0.291205\pi$
$558$		0			0
$559$	−56.5685			−2.39259
$560$		0			0
$561$	12.0000			0.506640
$562$		0			0
$563$	10.6066	−	18.3712i	0.447015	−	0.774253i	−0.551175	−	0.834390i	$-0.685820\pi$
$563$	10.6066	−	18.3712i	0.447015	−	0.774253i	0.998190	+	0.0601369i	$0.0191537\pi$
$564$		0			0
$565$	5.65685	+	9.79796i	0.237986	+	0.412203i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	11.0000	+	19.0526i	0.461144	+	0.798725i	0.999018	−	0.0443003i	$-0.0141058\pi$
$569$	11.0000	+	19.0526i	0.461144	+	0.798725i	−0.537874	+	0.843025i	$0.680772\pi$
$570$		0			0
$571$	−5.00000	+	8.66025i	−0.209243	+	0.362420i	−0.951476	−	0.307722i	$-0.900433\pi$
$571$	−5.00000	+	8.66025i	−0.209243	+	0.362420i	0.742233	+	0.670142i	$0.233767\pi$
$572$		0			0
$573$	5.65685			0.236318
$574$		0			0
$575$	−12.0000			−0.500435
$576$		0			0
$577$	10.6066	−	18.3712i	0.441559	−	0.764802i	−0.556247	−	0.831017i	$-0.687759\pi$
$577$	10.6066	−	18.3712i	0.441559	−	0.764802i	0.997805	+	0.0662152i	$0.0210924\pi$
$578$		0			0
$579$	−11.3137	−	19.5959i	−0.470182	−	0.814379i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−6.00000	−	10.3923i	−0.248495	−	0.430405i
$584$		0			0
$585$	−8.00000	+	13.8564i	−0.330759	+	0.572892i
$586$		0			0
$587$	−4.24264			−0.175113			−0.0875563	−	0.996160i	$-0.527906\pi$
$587$	−4.24264			−0.175113			−0.0875563	+	0.996160i	$0.527906\pi$
$588$		0			0
$589$	−12.0000			−0.494451
$590$		0			0
$591$	−12.7279	+	22.0454i	−0.523557	+	0.906827i
$592$		0			0
$593$	3.53553	+	6.12372i	0.145187	+	0.251471i	0.929443	−	0.368967i	$-0.120288\pi$
$593$	3.53553	+	6.12372i	0.145187	+	0.251471i	−0.784256	+	0.620438i	$0.786955\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−18.0000	−	31.1769i	−0.736691	−	1.27599i
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	−26.8701			−1.09605			−0.548026	−	0.836461i	$-0.684621\pi$
$601$	−26.8701			−1.09605			−0.548026	+	0.836461i	$0.684621\pi$
$602$		0			0
$603$	4.00000			0.162893
$604$		0			0
$605$	35.3553	−	61.2372i	1.43740	−	2.48965i
$606$		0			0
$607$	−19.7990	−	34.2929i	−0.803616	−	1.39190i	−0.917221	−	0.398378i	$-0.869573\pi$
$607$	−19.7990	−	34.2929i	−0.803616	−	1.39190i	0.113605	−	0.993526i	$-0.463760\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.00000	+	13.8564i	0.323645	+	0.560570i
$612$		0			0
$613$	−21.0000	+	36.3731i	−0.848182	+	1.46909i	0.0346469	+	0.999400i	$0.488969\pi$
$613$	−21.0000	+	36.3731i	−0.848182	+	1.46909i	−0.882829	+	0.469695i	$0.844364\pi$
$614$		0			0
$615$	−5.65685			−0.228106
$616$		0			0
$617$	22.0000			0.885687			0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000			0.885687			0.442843	+	0.896599i	$0.353970\pi$
$618$		0			0
$619$	−2.12132	+	3.67423i	−0.0852631	+	0.147680i	−0.905503	−	0.424339i	$-0.860506\pi$
$619$	−2.12132	+	3.67423i	−0.0852631	+	0.147680i	0.820240	+	0.572019i	$0.193840\pi$
$620$		0			0
$621$	−11.3137	−	19.5959i	−0.454003	−	0.786357i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	15.5000	+	26.8468i	0.620000	+	1.07387i
$626$		0			0
$627$	−18.0000	+	31.1769i	−0.718851	+	1.24509i
$628$		0			0
$629$	−2.82843			−0.112777
$630$		0			0
$631$	−28.0000			−1.11466			−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000			−1.11466			−0.557331	+	0.830290i	$0.688175\pi$
$632$		0			0
$633$	8.48528	−	14.6969i	0.337260	−	0.584151i
$634$		0			0
$635$	11.3137	+	19.5959i	0.448971	+	0.777640i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	6.00000	+	10.3923i	0.237356	+	0.411113i
$640$		0			0
$641$	3.00000	−	5.19615i	0.118493	−	0.205236i	−0.800678	−	0.599095i	$-0.795527\pi$
$641$	3.00000	−	5.19615i	0.118493	−	0.205236i	0.919171	+	0.393860i	$0.128860\pi$
$642$		0			0
$643$	35.3553			1.39428			0.697139	−	0.716936i	$-0.254456\pi$
$643$	35.3553			1.39428			0.697139	+	0.716936i	$0.254456\pi$
$644$		0			0
$645$	40.0000			1.57500
$646$		0			0
$647$	21.2132	−	36.7423i	0.833977	−	1.44449i	−0.0608835	−	0.998145i	$-0.519392\pi$
$647$	21.2132	−	36.7423i	0.833977	−	1.44449i	0.894861	−	0.446346i	$-0.147275\pi$
$648$		0			0
$649$	4.24264	+	7.34847i	0.166538	+	0.288453i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13.0000	+	22.5167i	0.508729	+	0.881145i	0.999949	+	0.0101092i	$0.00321793\pi$
$653$	13.0000	+	22.5167i	0.508729	+	0.881145i	−0.491220	+	0.871036i	$0.663449\pi$
$654$		0			0
$655$	30.0000	−	51.9615i	1.17220	−	2.03030i
$656$		0			0
$657$	−9.89949			−0.386216
$658$		0			0
$659$	−6.00000			−0.233727			−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000			−0.233727			−0.116863	+	0.993148i	$0.537284\pi$
$660$		0			0
$661$	−18.3848	+	31.8434i	−0.715085	+	1.23856i	0.247842	+	0.968801i	$0.420279\pi$
$661$	−18.3848	+	31.8434i	−0.715085	+	1.23856i	−0.962927	+	0.269763i	$0.913055\pi$
$662$		0			0
$663$	5.65685	+	9.79796i	0.219694	+	0.380521i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−12.0000	−	20.7846i	−0.464642	−	0.804783i
$668$		0			0
$669$	−8.00000	+	13.8564i	−0.309298	+	0.535720i
$670$		0			0
$671$	−50.9117			−1.96542
$672$		0			0
$673$	−44.0000			−1.69608			−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000			−1.69608			−0.848038	+	0.529936i	$0.822216\pi$
$674$		0			0
$675$	8.48528	−	14.6969i	0.326599	−	0.565685i
$676$		0			0
$677$	−16.9706	−	29.3939i	−0.652232	−	1.12970i	−0.982580	−	0.185839i	$-0.940500\pi$
$677$	−16.9706	−	29.3939i	−0.652232	−	1.12970i	0.330348	−	0.943859i	$-0.392834\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−7.00000	−	12.1244i	−0.268241	−	0.464606i
$682$		0			0
$683$	2.00000	−	3.46410i	0.0765279	−	0.132550i	−0.825222	−	0.564809i	$-0.808950\pi$
$683$	2.00000	−	3.46410i	0.0765279	−	0.132550i	0.901750	+	0.432259i	$0.142283\pi$
$684$		0			0
$685$	11.3137			0.432275
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5.65685	−	9.79796i	0.215509	−	0.373273i
$690$		0			0
$691$	−21.9203	−	37.9671i	−0.833888	−	1.44434i	−0.894933	−	0.446201i	$-0.852777\pi$
$691$	−21.9203	−	37.9671i	−0.833888	−	1.44434i	0.0610448	−	0.998135i	$-0.480557\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−30.0000	−	51.9615i	−1.13796	−	1.97101i
$696$		0			0
$697$	1.00000	−	1.73205i	0.0378777	−	0.0656061i
$698$		0			0
$699$	11.3137			0.427924
$700$		0			0
$701$	38.0000			1.43524			0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000			1.43524			0.717620	+	0.696435i	$0.245231\pi$
$702$		0			0
$703$	4.24264	−	7.34847i	0.160014	−	0.277153i
$704$		0			0
$705$	−5.65685	−	9.79796i	−0.213049	−	0.369012i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−17.0000	−	29.4449i	−0.638448	−	1.10583i	−0.985773	−	0.168080i	$-0.946243\pi$
$709$	−17.0000	−	29.4449i	−0.638448	−	1.10583i	0.347325	−	0.937745i	$-0.387090\pi$
$710$		0			0
$711$	2.00000	−	3.46410i	0.0750059	−	0.129914i
$712$		0			0
$713$	−11.3137			−0.423702
$714$		0			0
$715$	96.0000			3.59020
$716$		0			0
$717$	−2.82843	+	4.89898i	−0.105630	+	0.182956i
$718$		0			0
$719$	1.41421	+	2.44949i	0.0527413	+	0.0913506i	0.891191	−	0.453629i	$-0.149871\pi$
$719$	1.41421	+	2.44949i	0.0527413	+	0.0913506i	−0.838449	+	0.544979i	$0.816537\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−1.00000	−	1.73205i	−0.0371904	−	0.0644157i
$724$		0			0
$725$	9.00000	−	15.5885i	0.334252	−	0.578941i
$726$		0			0
$727$	−25.4558			−0.944105			−0.472052	−	0.881570i	$-0.656487\pi$
$727$	−25.4558			−0.944105			−0.472052	+	0.881570i	$0.656487\pi$
$728$		0			0
$729$	29.0000			1.07407
$730$		0			0
$731$	−7.07107	+	12.2474i	−0.261533	+	0.452988i
$732$		0			0
$733$	4.24264	+	7.34847i	0.156706	+	0.271422i	0.933679	−	0.358112i	$-0.116579\pi$
$733$	4.24264	+	7.34847i	0.156706	+	0.271422i	−0.776973	+	0.629534i	$0.783246\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−12.0000	−	20.7846i	−0.442026	−	0.765611i
$738$		0			0
$739$	−3.00000	+	5.19615i	−0.110357	+	0.191144i	−0.915914	−	0.401374i	$-0.868533\pi$
$739$	−3.00000	+	5.19615i	−0.110357	+	0.191144i	0.805557	+	0.592518i	$0.201866\pi$
$740$		0			0
$741$	−33.9411			−1.24686
$742$		0			0
$743$	24.0000			0.880475			0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000			0.880475			0.440237	+	0.897881i	$0.354894\pi$
$744$		0			0
$745$	−8.48528	+	14.6969i	−0.310877	+	0.538454i
$746$		0			0
$747$	0.707107	+	1.22474i	0.0258717	+	0.0448111i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−6.00000	−	10.3923i	−0.218943	−	0.379221i	0.735542	−	0.677479i	$-0.236928\pi$
$751$	−6.00000	−	10.3923i	−0.218943	−	0.379221i	−0.954485	+	0.298259i	$0.903594\pi$
$752$		0			0
$753$	−1.00000	+	1.73205i	−0.0364420	+	0.0631194i
$754$		0			0
$755$	22.6274			0.823496
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$		0			0
$759$	−16.9706	+	29.3939i	−0.615992	+	1.06693i
$760$		0			0
$761$	−13.4350	−	23.2702i	−0.487019	−	0.843542i	0.512869	−	0.858467i	$-0.328583\pi$
$761$	−13.4350	−	23.2702i	−0.487019	−	0.843542i	−0.999889	+	0.0149244i	$0.995249\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	2.00000	+	3.46410i	0.0723102	+	0.125245i
$766$		0			0
$767$	−4.00000	+	6.92820i	−0.144432	+	0.250163i
$768$		0			0
$769$	29.6985			1.07095			0.535477	−	0.844550i	$-0.320132\pi$
$769$	29.6985			1.07095			0.535477	+	0.844550i	$0.320132\pi$
$770$		0			0
$771$	18.0000			0.648254
$772$		0			0
$773$	−9.89949	+	17.1464i	−0.356060	+	0.616714i	−0.987299	−	0.158874i	$-0.949213\pi$
$773$	−9.89949	+	17.1464i	−0.356060	+	0.616714i	0.631239	+	0.775589i	$0.282547\pi$
$774$		0			0
$775$	−4.24264	−	7.34847i	−0.152400	−	0.263965i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.00000	+	5.19615i	0.107486	+	0.186171i
$780$		0			0
$781$	36.0000	−	62.3538i	1.28818	−	2.23120i
$782$		0			0
$783$	33.9411			1.21296
$784$		0			0
$785$	48.0000			1.71319
$786$		0			0
$787$	16.2635	−	28.1691i	0.579730	−	1.00412i	−0.415780	−	0.909465i	$-0.636492\pi$
$787$	16.2635	−	28.1691i	0.579730	−	1.00412i	0.995510	−	0.0946561i	$-0.0301751\pi$
$788$		0			0
$789$	−2.82843	−	4.89898i	−0.100695	−	0.174408i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−24.0000	−	41.5692i	−0.852265	−	1.47617i
$794$		0			0
$795$	−4.00000	+	6.92820i	−0.141865	+	0.245718i
$796$		0			0
$797$	−39.5980			−1.40263			−0.701316	−	0.712850i	$-0.747404\pi$
$797$	−39.5980			−1.40263			−0.701316	+	0.712850i	$0.747404\pi$
$798$		0			0
$799$	4.00000			0.141510
$800$		0			0
$801$	2.12132	−	3.67423i	0.0749532	−	0.129823i
$802$		0			0
$803$	29.6985	+	51.4393i	1.04804	+	1.81525i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	20.0000	+	34.6410i	0.704033	+	1.21942i
$808$		0			0
$809$	16.0000	−	27.7128i	0.562530	−	0.974331i	−0.434745	−	0.900554i	$-0.643161\pi$
$809$	16.0000	−	27.7128i	0.562530	−	0.974331i	0.997275	−	0.0737769i	$-0.0235053\pi$
$810$		0			0
$811$	7.07107			0.248299			0.124149	−	0.992264i	$-0.460380\pi$
$811$	7.07107			0.248299			0.124149	+	0.992264i	$0.460380\pi$
$812$		0			0
$813$	−16.0000			−0.561144
$814$		0			0
$815$	−2.82843	+	4.89898i	−0.0990755	+	0.171604i
$816$		0			0
$817$	−21.2132	−	36.7423i	−0.742156	−	1.28545i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	9.00000	+	15.5885i	0.314102	+	0.544041i	0.979246	−	0.202674i	$-0.0649632\pi$
$821$	9.00000	+	15.5885i	0.314102	+	0.544041i	−0.665144	+	0.746715i	$0.731630\pi$
$822$		0			0
$823$	−28.0000	+	48.4974i	−0.976019	+	1.69051i	−0.299487	+	0.954100i	$0.596815\pi$
$823$	−28.0000	+	48.4974i	−0.976019	+	1.69051i	−0.676532	+	0.736413i	$0.736518\pi$
$824$		0			0
$825$	−25.4558			−0.886259
$826$		0			0
$827$	4.00000			0.139094			0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000			0.139094			0.0695468	+	0.997579i	$0.477845\pi$
$828$		0			0
$829$	12.7279	−	22.0454i	0.442059	−	0.765669i	−0.555783	−	0.831327i	$-0.687582\pi$
$829$	12.7279	−	22.0454i	0.442059	−	0.765669i	0.997842	+	0.0656587i	$0.0209148\pi$
$830$		0			0
$831$	−9.89949	−	17.1464i	−0.343410	−	0.594803i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−4.00000	−	6.92820i	−0.138426	−	0.239760i
$836$		0			0
$837$	8.00000	−	13.8564i	0.276520	−	0.478947i
$838$		0			0
$839$	−31.1127			−1.07413			−0.537065	−	0.843541i	$-0.680467\pi$
$839$	−31.1127			−1.07413			−0.537065	+	0.843541i	$0.680467\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	22.6274	−	39.1918i	0.779330	−	1.34984i
$844$		0			0
$845$	26.8701	+	46.5403i	0.924358	+	1.60104i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	15.0000	+	25.9808i	0.514799	+	0.891657i
$850$		0			0
$851$	4.00000	−	6.92820i	0.137118	−	0.237496i
$852$		0			0
$853$	16.9706			0.581061			0.290531	−	0.956866i	$-0.406168\pi$
$853$	16.9706			0.581061			0.290531	+	0.956866i	$0.406168\pi$
$854$		0			0
$855$	−12.0000			−0.410391
$856$		0			0
$857$	−3.53553	+	6.12372i	−0.120772	+	0.209182i	−0.920072	−	0.391749i	$-0.871870\pi$
$857$	−3.53553	+	6.12372i	−0.120772	+	0.209182i	0.799301	+	0.600931i	$0.205204\pi$
$858$		0			0
$859$	−7.77817	−	13.4722i	−0.265388	−	0.459665i	0.702277	−	0.711904i	$-0.252167\pi$
$859$	−7.77817	−	13.4722i	−0.265388	−	0.459665i	−0.967665	+	0.252238i	$0.918833\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	22.0000	+	38.1051i	0.748889	+	1.29711i	0.948356	+	0.317209i	$0.102746\pi$
$863$	22.0000	+	38.1051i	0.748889	+	1.29711i	−0.199467	+	0.979905i	$0.563921\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−21.2132			−0.720438
$868$		0			0
$869$	−24.0000			−0.814144
$870$		0			0
$871$	11.3137	−	19.5959i	0.383350	−	0.663982i
$872$		0			0
$873$	−6.36396	−	11.0227i	−0.215387	−	0.373062i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	19.0000	+	32.9090i	0.641584	+	1.11126i	0.985079	+	0.172102i	$0.0550559\pi$
$877$	19.0000	+	32.9090i	0.641584	+	1.11126i	−0.343495	+	0.939155i	$0.611611\pi$
$878$		0			0
$879$	2.00000	−	3.46410i	0.0674583	−	0.116841i
$880$		0			0
$881$	−38.1838			−1.28644			−0.643222	−	0.765680i	$-0.722403\pi$
$881$	−38.1838			−1.28644			−0.643222	+	0.765680i	$0.722403\pi$
$882$		0			0
$883$	28.0000			0.942275			0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000			0.942275			0.471138	+	0.882060i	$0.343844\pi$
$884$		0			0
$885$	2.82843	−	4.89898i	0.0950765	−	0.164677i
$886$		0			0
$887$	−18.3848	−	31.8434i	−0.617300	−	1.06920i	−0.989976	−	0.141234i	$-0.954893\pi$
$887$	−18.3848	−	31.8434i	−0.617300	−	1.06920i	0.372676	−	0.927962i	$-0.378440\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−15.0000	−	25.9808i	−0.502519	−	0.870388i
$892$		0			0
$893$	−6.00000	+	10.3923i	−0.200782	+	0.347765i
$894$		0			0
$895$	56.5685			1.89088
$896$		0			0
$897$	−32.0000			−1.06845
$898$		0			0
$899$	8.48528	−	14.6969i	0.283000	−	0.490170i
$900$		0			0
$901$	−1.41421	−	2.44949i	−0.0471143	−	0.0816043i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−24.0000	−	41.5692i	−0.797787	−	1.38181i
$906$		0			0
$907$	−10.0000	+	17.3205i	−0.332045	+	0.575118i	−0.982913	−	0.184073i	$-0.941072\pi$
$907$	−10.0000	+	17.3205i	−0.332045	+	0.575118i	0.650868	+	0.759191i	$0.274405\pi$
$908$		0			0
$909$	−2.82843			−0.0938130
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$	4.24264	−	7.34847i	0.140411	−	0.243199i
$914$		0			0
$915$	16.9706	+	29.3939i	0.561029	+	0.971732i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−16.0000	−	27.7128i	−0.527791	−	0.914161i	−0.999475	−	0.0323936i	$-0.989687\pi$
$919$	−16.0000	−	27.7128i	−0.527791	−	0.914161i	0.471684	−	0.881768i	$-0.343646\pi$
$920$		0			0
$921$	1.00000	−	1.73205i	0.0329511	−	0.0570730i
$922$		0			0
$923$	67.8823			2.23437
$924$		0			0
$925$	6.00000			0.197279
$926$		0			0
$927$	−7.07107	+	12.2474i	−0.232244	+	0.402259i
$928$		0			0
$929$	−16.2635	−	28.1691i	−0.533587	−	0.924199i	−0.999230	−	0.0392269i	$-0.987510\pi$
$929$	−16.2635	−	28.1691i	−0.533587	−	0.924199i	0.465644	−	0.884972i	$-0.345823\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	8.00000	+	13.8564i	0.261908	+	0.453638i
$934$		0			0
$935$	12.0000	−	20.7846i	0.392442	−	0.679729i
$936$		0			0
$937$	21.2132			0.693005			0.346503	−	0.938049i	$-0.387369\pi$
$937$	21.2132			0.693005			0.346503	+	0.938049i	$0.387369\pi$
$938$		0			0
$939$	−30.0000			−0.979013
$940$		0			0
$941$	−15.5563	+	26.9444i	−0.507122	+	0.878362i	0.492844	+	0.870118i	$0.335957\pi$
$941$	−15.5563	+	26.9444i	−0.507122	+	0.878362i	−0.999966	+	0.00824396i	$0.997376\pi$
$942$		0			0
$943$	2.82843	+	4.89898i	0.0921063	+	0.159533i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−21.0000	−	36.3731i	−0.682408	−	1.18197i	−0.974244	−	0.225497i	$-0.927599\pi$
$947$	−21.0000	−	36.3731i	−0.682408	−	1.18197i	0.291835	−	0.956469i	$-0.405734\pi$
$948$		0			0
$949$	−28.0000	+	48.4974i	−0.908918	+	1.57429i
$950$		0			0
$951$	−31.1127			−1.00890
$952$		0			0
$953$	−26.0000			−0.842223			−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000			−0.842223			−0.421111	+	0.907009i	$0.638360\pi$
$954$		0			0
$955$	5.65685	−	9.79796i	0.183052	−	0.317055i
$956$		0			0
$957$	−25.4558	−	44.0908i	−0.822871	−	1.42525i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	11.5000	+	19.9186i	0.370968	+	0.642535i
$962$		0			0
$963$	−2.00000	+	3.46410i	−0.0644491	+	0.111629i
$964$		0			0
$965$	−45.2548			−1.45680
$966$		0			0
$967$	−60.0000			−1.92947			−0.964735	−	0.263223i	$-0.915214\pi$
$967$	−60.0000			−1.92947			−0.964735	+	0.263223i	$0.915214\pi$
$968$		0			0
$969$	−4.24264	+	7.34847i	−0.136293	+	0.236067i
$970$		0			0
$971$	16.2635	+	28.1691i	0.521919	+	0.903990i	0.999675	+	0.0254978i	$0.00811707\pi$
$971$	16.2635	+	28.1691i	0.521919	+	0.903990i	−0.477756	+	0.878493i	$0.658550\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−12.0000	−	20.7846i	−0.384308	−	0.665640i
$976$		0			0
$977$	−6.00000	+	10.3923i	−0.191957	+	0.332479i	−0.945899	−	0.324462i	$-0.894817\pi$
$977$	−6.00000	+	10.3923i	−0.191957	+	0.332479i	0.753942	+	0.656941i	$0.228150\pi$
$978$		0			0
$979$	−25.4558			−0.813572
$980$		0			0
$981$	10.0000			0.319275
$982$		0			0
$983$	−26.8701	+	46.5403i	−0.857022	+	1.48441i	0.0177349	+	0.999843i	$0.494355\pi$
$983$	−26.8701	+	46.5403i	−0.857022	+	1.48441i	−0.874757	+	0.484562i	$0.838979\pi$
$984$		0			0
$985$	25.4558	+	44.0908i	0.811091	+	1.40485i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−20.0000	−	34.6410i	−0.635963	−	1.10152i
$990$		0			0
$991$	−16.0000	+	27.7128i	−0.508257	+	0.880327i	0.491698	+	0.870766i	$0.336377\pi$
$991$	−16.0000	+	27.7128i	−0.508257	+	0.880327i	−0.999954	+	0.00956046i	$0.996957\pi$
$992$		0			0
$993$	19.7990			0.628302
$994$		0			0
$995$	−72.0000			−2.28255
$996$		0			0
$997$	−4.24264	+	7.34847i	−0.134366	+	0.232728i	−0.925355	−	0.379102i	$-0.876233\pi$
$997$	−4.24264	+	7.34847i	−0.134366	+	0.232728i	0.790989	+	0.611830i	$0.209566\pi$
$998$		0			0
$999$	5.65685	+	9.79796i	0.178975	+	0.309994i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.i.n.753.1		4
4.3	odd	2		392.2.i.h.361.2		4
7.2	even	3	inner	784.2.i.n.177.1		4
7.3	odd	6		784.2.a.k.1.1		2
7.4	even	3		784.2.a.k.1.2		2
7.5	odd	6	inner	784.2.i.n.177.2		4
7.6	odd	2	inner	784.2.i.n.753.2		4
12.11	even	2		3528.2.s.bj.361.1		4
21.11	odd	6		7056.2.a.ct.1.2		2
21.17	even	6		7056.2.a.ct.1.1		2
28.3	even	6		392.2.a.g.1.2	yes	2
28.11	odd	6		392.2.a.g.1.1	✓	2
28.19	even	6		392.2.i.h.177.1		4
28.23	odd	6		392.2.i.h.177.2		4
28.27	even	2		392.2.i.h.361.1		4
56.3	even	6		3136.2.a.bk.1.1		2
56.11	odd	6		3136.2.a.bk.1.2		2
56.45	odd	6		3136.2.a.bp.1.2		2
56.53	even	6		3136.2.a.bp.1.1		2
84.11	even	6		3528.2.a.be.1.2		2
84.23	even	6		3528.2.s.bj.3313.1		4
84.47	odd	6		3528.2.s.bj.3313.2		4
84.59	odd	6		3528.2.a.be.1.1		2
84.83	odd	2		3528.2.s.bj.361.2		4
140.39	odd	6		9800.2.a.bv.1.2		2
140.59	even	6		9800.2.a.bv.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.a.g.1.1	✓	2	28.11	odd	6
392.2.a.g.1.2	yes	2	28.3	even	6
392.2.i.h.177.1		4	28.19	even	6
392.2.i.h.177.2		4	28.23	odd	6
392.2.i.h.361.1		4	28.27	even	2
392.2.i.h.361.2		4	4.3	odd	2
784.2.a.k.1.1		2	7.3	odd	6
784.2.a.k.1.2		2	7.4	even	3
784.2.i.n.177.1		4	7.2	even	3	inner
784.2.i.n.177.2		4	7.5	odd	6	inner
784.2.i.n.753.1		4	1.1	even	1	trivial
784.2.i.n.753.2		4	7.6	odd	2	inner
3136.2.a.bk.1.1		2	56.3	even	6
3136.2.a.bk.1.2		2	56.11	odd	6
3136.2.a.bp.1.1		2	56.53	even	6
3136.2.a.bp.1.2		2	56.45	odd	6
3528.2.a.be.1.1		2	84.59	odd	6
3528.2.a.be.1.2		2	84.11	even	6
3528.2.s.bj.361.1		4	12.11	even	2
3528.2.s.bj.361.2		4	84.83	odd	2
3528.2.s.bj.3313.1		4	84.23	even	6
3528.2.s.bj.3313.2		4	84.47	odd	6
7056.2.a.ct.1.1		2	21.17	even	6
7056.2.a.ct.1.2		2	21.11	odd	6
9800.2.a.bv.1.1		2	140.59	even	6
9800.2.a.bv.1.2		2	140.39	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.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 1.22474i) q^{3} +(1.41421 + 2.44949i) q^{5} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.707107 + 1.22474i) q^{3} +(1.41421 + 2.44949i) q^{5} +(0.500000 + 0.866025i) q^{9} +(3.00000 - 5.19615i) q^{11} +5.65685 q^{13} -4.00000 q^{15} +(0.707107 - 1.22474i) q^{17} +(2.12132 + 3.67423i) q^{19} +(2.00000 + 3.46410i) q^{23} +(-1.50000 + 2.59808i) q^{25} -5.65685 q^{27} -6.00000 q^{29} +(-1.41421 + 2.44949i) q^{31} +(4.24264 + 7.34847i) q^{33} +(-1.00000 - 1.73205i) q^{37} +(-4.00000 + 6.92820i) q^{39} +1.41421 q^{41} -10.0000 q^{43} +(-1.41421 + 2.44949i) q^{45} +(1.41421 + 2.44949i) q^{47} +(1.00000 + 1.73205i) q^{51} +(1.00000 - 1.73205i) q^{53} +16.9706 q^{55} -6.00000 q^{57} +(-0.707107 + 1.22474i) q^{59} +(-4.24264 - 7.34847i) q^{61} +(8.00000 + 13.8564i) q^{65} +(2.00000 - 3.46410i) q^{67} -5.65685 q^{69} +12.0000 q^{71} +(-4.94975 + 8.57321i) q^{73} +(-2.12132 - 3.67423i) q^{75} +(-2.00000 - 3.46410i) q^{79} +(2.50000 - 4.33013i) q^{81} +1.41421 q^{83} +4.00000 q^{85} +(4.24264 - 7.34847i) q^{87} +(-2.12132 - 3.67423i) q^{89} +(-2.00000 - 3.46410i) q^{93} +(-6.00000 + 10.3923i) q^{95} -12.7279 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^9	\( 4 q + 2 q^{9} + 12 q^{11} - 16 q^{15} + 8 q^{23} - 6 q^{25} - 24 q^{29} - 4 q^{37} - 16 q^{39} - 40 q^{43} + 4 q^{51} + 4 q^{53} - 24 q^{57} + 32 q^{65} + 8 q^{67} + 48 q^{71} - 8 q^{79} + 10 q^{81} + 16 q^{85} - 8 q^{93} - 24 q^{95} + 24 q^{99}+O(q^{100}) \) 4 * q + 2 * q^9 + 12 * q^11 - 16 * q^15 + 8 * q^23 - 6 * q^25 - 24 * q^29 - 4 * q^37 - 16 * q^39 - 40 * q^43 + 4 * q^51 + 4 * q^53 - 24 * q^57 + 32 * q^65 + 8 * q^67 + 48 * q^71 - 8 * q^79 + 10 * q^81 + 16 * q^85 - 8 * q^93 - 24 * q^95 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more