LMFDB - Embedded newform 416.2.w.c.257.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(225,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("416.225");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.w (of order $6$, degree $2$, 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:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.56070144.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 63x^4 - 74x^3 + 70x^2 - 38*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			257.1
Root			$0.500000 + 1.56488i$ of defining polynomial
Character	$\chi$	$=$	416.257
Dual form			416.2.w.c.225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.45466 - 2.51954i) q^{3} +2.00000i q^{5} +(-0.675108 - 0.389774i) q^{7} +(-2.73205 + 4.73205i) q^{9} +O(q^{10})$	\(q+(-1.45466 - 2.51954i) q^{3} +2.00000i q^{5} +(-0.675108 - 0.389774i) q^{7} +(-2.73205 + 4.73205i) q^{9} +(-4.36397 + 2.51954i) q^{11} +(-1.00000 - 3.46410i) q^{13} +(5.03908 - 2.90931i) q^{15} +(-3.23205 + 5.59808i) q^{17} +(-0.675108 - 0.389774i) q^{19} +2.26795i q^{21} +(3.58442 + 6.20840i) q^{23} +1.00000 q^{25} +7.16884 q^{27} +(-1.50000 - 2.59808i) q^{29} +1.55910i q^{31} +(12.6962 + 7.33013i) q^{33} +(0.779548 - 1.35022i) q^{35} +(-3.69615 + 2.13397i) q^{37} +(-7.27328 + 7.55862i) q^{39} +(-2.76795 + 1.59808i) q^{41} +(-4.36397 + 7.55862i) q^{43} +(-9.46410 - 5.46410i) q^{45} -7.37772i q^{47} +(-3.19615 - 5.53590i) q^{49} +18.8061 q^{51} +9.46410 q^{53} +(-5.03908 - 8.72794i) q^{55} +2.26795i q^{57} +(-10.7533 - 6.20840i) q^{59} +(-1.50000 + 2.59808i) q^{61} +(3.68886 - 2.12976i) q^{63} +(6.92820 - 2.00000i) q^{65} +(0.675108 - 0.389774i) q^{67} +(10.4282 - 18.0622i) q^{69} +(-13.0919 - 7.55862i) q^{71} -6.00000i q^{73} +(-1.45466 - 2.51954i) q^{75} +3.92820 q^{77} -10.0782 q^{79} +(-2.23205 - 3.86603i) q^{81} +10.0782i q^{83} +(-11.1962 - 6.46410i) q^{85} +(-4.36397 + 7.55862i) q^{87} +(-7.96410 + 4.59808i) q^{89} +(-0.675108 + 2.72842i) q^{91} +(3.92820 - 2.26795i) q^{93} +(0.779548 - 1.35022i) q^{95} +(-8.89230 - 5.13397i) q^{97} -27.5340i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 8 q^{13} - 12 q^{17} + 8 q^{25} - 12 q^{29} + 60 q^{33} + 12 q^{37} - 36 q^{41} - 48 q^{45} + 16 q^{49} + 48 q^{53} - 12 q^{61} + 28 q^{69} - 24 q^{77} - 4 q^{81} - 48 q^{85} - 36 q^{89} - 24 q^{93} + 12 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 - 8 * q^13 - 12 * q^17 + 8 * q^25 - 12 * q^29 + 60 * q^33 + 12 * q^37 - 36 * q^41 - 48 * q^45 + 16 * q^49 + 48 * q^53 - 12 * q^61 + 28 * q^69 - 24 * q^77 - 4 * q^81 - 48 * q^85 - 36 * q^89 - 24 * q^93 + 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	−1.45466	−	2.51954i	−0.839846	−	1.45466i	−0.890023	−	0.455916i	$-0.849312\pi$
$3$	−1.45466	−	2.51954i	−0.839846	−	1.45466i	0.0501765	−	0.998740i	$-0.484022\pi$
$4$		0			0
$5$			2.00000i			0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$			2.00000i			0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		0			0
$7$	−0.675108	−	0.389774i	−0.255167	−	0.147321i	0.366961	−	0.930236i	$-0.380398\pi$
$7$	−0.675108	−	0.389774i	−0.255167	−	0.147321i	−0.622128	+	0.782916i	$0.713732\pi$
$8$		0			0
$9$	−2.73205	+	4.73205i	−0.910684	+	1.57735i
$10$		0			0
$11$	−4.36397	+	2.51954i	−1.31579	+	0.759670i	−0.983048	−	0.183350i	$-0.941306\pi$
$11$	−4.36397	+	2.51954i	−1.31579	+	0.759670i	−0.332738	+	0.943019i	$0.607973\pi$
$12$		0			0
$13$	−1.00000	−	3.46410i	−0.277350	−	0.960769i
$13$	−1.00000	−	3.46410i	−0.277350	−	0.960769i
$14$		0			0
$15$	5.03908	−	2.90931i	1.30108	−	0.751181i
$16$		0			0
$17$	−3.23205	+	5.59808i	−0.783887	+	1.35773i	0.145774	+	0.989318i	$0.453433\pi$
$17$	−3.23205	+	5.59808i	−0.783887	+	1.35773i	−0.929661	+	0.368415i	$0.879901\pi$
$18$		0			0
$19$	−0.675108	−	0.389774i	−0.154880	−	0.0894203i	0.420557	−	0.907266i	$-0.361835\pi$
$19$	−0.675108	−	0.389774i	−0.154880	−	0.0894203i	−0.575437	+	0.817846i	$0.695168\pi$
$20$		0			0
$21$			2.26795i			0.494907i
$22$		0			0
$23$	3.58442	+	6.20840i	0.747404	+	1.29454i	0.949063	+	0.315085i	$0.102033\pi$
$23$	3.58442	+	6.20840i	0.747404	+	1.29454i	−0.201660	+	0.979456i	$0.564634\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	7.16884			1.37964
$28$		0			0
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	0.692480	−	0.721437i	$-0.256518\pi$
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	−0.971023	+	0.238987i	$0.923185\pi$
$30$		0			0
$31$			1.55910i			0.280022i	0.990150	+	0.140011i	$0.0447138\pi$
$31$			1.55910i			0.280022i	−0.990150	+	0.140011i	$0.955286\pi$
$32$		0			0
$33$	12.6962	+	7.33013i	2.21012	+	1.27601i
$34$		0			0
$35$	0.779548	−	1.35022i	0.131768	−	0.228228i
$36$		0			0
$37$	−3.69615	+	2.13397i	−0.607644	+	0.350823i	−0.772043	−	0.635571i	$-0.780765\pi$
$37$	−3.69615	+	2.13397i	−0.607644	+	0.350823i	0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$	−7.27328	+	7.55862i	−1.16466	+	1.21035i
$40$		0			0
$41$	−2.76795	+	1.59808i	−0.432281	+	0.249578i	−0.700318	−	0.713831i	$-0.746958\pi$
$41$	−2.76795	+	1.59808i	−0.432281	+	0.249578i	0.268037	+	0.963409i	$0.413625\pi$
$42$		0			0
$43$	−4.36397	+	7.55862i	−0.665499	+	1.15268i	0.313651	+	0.949538i	$0.398448\pi$
$43$	−4.36397	+	7.55862i	−0.665499	+	1.15268i	−0.979150	+	0.203140i	$0.934885\pi$
$44$		0			0
$45$	−9.46410	−	5.46410i	−1.41082	−	0.814540i
$46$		0			0
$47$		−	7.37772i		−	1.07615i	−0.842897	−	0.538076i	$-0.819151\pi$
$47$		−	7.37772i		−	1.07615i	0.842897	−	0.538076i	$-0.180849\pi$
$48$		0			0
$49$	−3.19615	−	5.53590i	−0.456593	−	0.790843i
$50$		0			0
$51$	18.8061			2.63338
$52$		0			0
$53$	9.46410			1.29999			0.649997	−	0.759937i	$-0.274770\pi$
$53$	9.46410			1.29999			0.649997	+	0.759937i	$0.274770\pi$
$54$		0			0
$55$	−5.03908	−	8.72794i	−0.679469	−	1.17688i
$56$		0			0
$57$			2.26795i			0.300397i
$58$		0			0
$59$	−10.7533	−	6.20840i	−1.39996	−	0.808265i	−0.405568	−	0.914065i	$-0.632926\pi$
$59$	−10.7533	−	6.20840i	−1.39996	−	0.808265i	−0.994387	+	0.105800i	$0.966260\pi$
$60$		0			0
$61$	−1.50000	+	2.59808i	−0.192055	+	0.332650i	−0.945931	−	0.324367i	$-0.894849\pi$
$61$	−1.50000	+	2.59808i	−0.192055	+	0.332650i	0.753876	+	0.657017i	$0.228182\pi$
$62$		0			0
$63$	3.68886	−	2.12976i	0.464753	−	0.268325i
$64$		0			0
$65$	6.92820	−	2.00000i	0.859338	−	0.248069i
$66$		0			0
$67$	0.675108	−	0.389774i	0.0824776	−	0.0476185i	−0.458194	−	0.888852i	$-0.651504\pi$
$67$	0.675108	−	0.389774i	0.0824776	−	0.0476185i	0.540672	+	0.841234i	$0.318170\pi$
$68$		0			0
$69$	10.4282	−	18.0622i	1.25541	−	2.17443i
$70$		0			0
$71$	−13.0919	−	7.55862i	−1.55372	−	0.897043i	−0.997834	−	0.0657869i	$-0.979044\pi$
$71$	−13.0919	−	7.55862i	−1.55372	−	0.897043i	−0.555890	−	0.831256i	$-0.687622\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$		0			0
$75$	−1.45466	−	2.51954i	−0.167969	−	0.290931i
$76$		0			0
$77$	3.92820			0.447660
$78$		0			0
$79$	−10.0782			−1.13388			−0.566941	−	0.823759i	$-0.691873\pi$
$79$	−10.0782			−1.13388			−0.566941	+	0.823759i	$0.691873\pi$
$80$		0			0
$81$	−2.23205	−	3.86603i	−0.248006	−	0.429558i
$82$		0			0
$83$			10.0782i			1.10622i	0.833108	+	0.553111i	$0.186559\pi$
$83$			10.0782i			1.10622i	−0.833108	+	0.553111i	$0.813441\pi$
$84$		0			0
$85$	−11.1962	−	6.46410i	−1.21439	−	0.701130i
$86$		0			0
$87$	−4.36397	+	7.55862i	−0.467867	+	0.810369i
$88$		0			0
$89$	−7.96410	+	4.59808i	−0.844193	+	0.487395i	−0.858687	−	0.512500i	$-0.828720\pi$
$89$	−7.96410	+	4.59808i	−0.844193	+	0.487395i	0.0144942	+	0.999895i	$0.495386\pi$
$90$		0			0
$91$	−0.675108	+	2.72842i	−0.0707706	+	0.286016i
$92$		0			0
$93$	3.92820	−	2.26795i	0.407336	−	0.235175i
$94$		0			0
$95$	0.779548	−	1.35022i	0.0799799	−	0.138529i
$96$		0			0
$97$	−8.89230	−	5.13397i	−0.902877	−	0.521276i	−0.0247444	−	0.999694i	$-0.507877\pi$
$97$	−8.89230	−	5.13397i	−0.902877	−	0.521276i	−0.878132	+	0.478418i	$0.841211\pi$
$98$		0			0
$99$		−	27.5340i		−	2.76727i
$100$		0			0
$101$	−0.696152	−	1.20577i	−0.0692698	−	0.119979i	0.829310	−	0.558788i	$-0.188734\pi$
$101$	−0.696152	−	1.20577i	−0.0692698	−	0.119979i	−0.898580	+	0.438810i	$0.855400\pi$
$102$		0			0
$103$	20.1563			1.98606			0.993030	−	0.117860i	$-0.0376035\pi$
$103$	20.1563			1.98606			0.993030	+	0.117860i	$0.0376035\pi$
$104$		0			0
$105$	−4.53590			−0.442658
$106$		0			0
$107$	4.36397	+	7.55862i	0.421881	+	0.730719i	0.996123	−	0.0879661i	$-0.0280367\pi$
$107$	4.36397	+	7.55862i	0.421881	+	0.730719i	−0.574243	+	0.818685i	$0.694703\pi$
$108$		0			0
$109$		−	6.00000i		−	0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$		−	6.00000i		−	0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$		0			0
$111$	10.7533	+	6.20840i	1.02065	+	0.589275i
$112$		0			0
$113$	2.76795	−	4.79423i	0.260387	−	0.451003i	−0.705958	−	0.708254i	$-0.749483\pi$
$113$	2.76795	−	4.79423i	0.260387	−	0.451003i	0.966345	+	0.257251i	$0.0828166\pi$
$114$		0			0
$115$	−12.4168	+	7.16884i	−1.15787	+	0.668498i
$116$		0			0
$117$	19.1244	+	4.73205i	1.76805	+	0.437478i
$118$		0			0
$119$	4.36397	−	2.51954i	0.400044	−	0.230966i
$120$		0			0
$121$	7.19615	−	12.4641i	0.654196	−	1.13310i
$122$		0			0
$123$	8.05283	+	4.64930i	0.726099	+	0.419214i
$124$		0			0
$125$			12.0000i			1.07331i
$126$		0			0
$127$	4.36397	+	7.55862i	0.387240	+	0.670719i	0.992077	−	0.125630i	$-0.0400953\pi$
$127$	4.36397	+	7.55862i	0.387240	+	0.670719i	−0.604837	+	0.796349i	$0.706762\pi$
$128$		0			0
$129$	25.3923			2.23567
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	0.303848	+	0.526279i	0.0263469	+	0.0456342i
$134$		0			0
$135$			14.3377i			1.23399i
$136$		0			0
$137$	2.42820	+	1.40192i	0.207455	+	0.119774i	0.600128	−	0.799904i	$-0.295116\pi$
$137$	2.42820	+	1.40192i	0.207455	+	0.119774i	−0.392673	+	0.919678i	$0.628450\pi$
$138$		0			0
$139$	0.675108	−	1.16932i	0.0572619	−	0.0991806i	−0.835973	−	0.548770i	$-0.815096\pi$
$139$	0.675108	−	1.16932i	0.0572619	−	0.0991806i	0.893235	+	0.449589i	$0.148430\pi$
$140$		0			0
$141$	−18.5885	+	10.7321i	−1.56543	+	0.903802i
$142$		0			0
$143$	13.0919	+	12.5977i	1.09480	+	1.05347i
$144$		0			0
$145$	5.19615	−	3.00000i	0.431517	−	0.249136i
$146$		0			0
$147$	−9.29861	+	16.1057i	−0.766936	+	1.32837i
$148$		0			0
$149$	16.9641	+	9.79423i	1.38975	+	0.802374i	0.993287	−	0.115675i	$-0.0369031\pi$
$149$	16.9641	+	9.79423i	1.38975	+	0.802374i	0.396466	+	0.918049i	$0.370236\pi$
$150$		0			0
$151$			1.55910i			0.126877i	0.997986	+	0.0634387i	$0.0202067\pi$
$151$			1.55910i			0.126877i	−0.997986	+	0.0634387i	$0.979793\pi$
$152$		0			0
$153$	−17.6603	−	30.5885i	−1.42775	−	2.47293i
$154$		0			0
$155$	−3.11819			−0.250459
$156$		0			0
$157$	−16.3923			−1.30825			−0.654124	−	0.756387i	$-0.726963\pi$
$157$	−16.3923			−1.30825			−0.654124	+	0.756387i	$0.726963\pi$
$158$		0			0
$159$	−13.7670	−	23.8452i	−1.09180	−	1.89105i
$160$		0			0
$161$		−	5.58846i		−	0.440432i
$162$		0			0
$163$	−10.3915	−	5.99952i	−0.813923	−	0.469919i	0.0343933	−	0.999408i	$-0.489050\pi$
$163$	−10.3915	−	5.99952i	−0.813923	−	0.469919i	−0.848316	+	0.529490i	$0.822383\pi$
$164$		0			0
$165$	−14.6603	+	25.3923i	−1.14130	+	1.97679i
$166$		0			0
$167$	15.4306	−	8.90883i	1.19405	−	0.689386i	0.234829	−	0.972037i	$-0.424547\pi$
$167$	15.4306	−	8.90883i	1.19405	−	0.689386i	0.959223	+	0.282650i	$0.0912136\pi$
$168$		0			0
$169$	−11.0000	+	6.92820i	−0.846154	+	0.532939i
$170$		0			0
$171$	3.68886	−	2.12976i	0.282094	−	0.162867i
$172$		0			0
$173$	1.96410	−	3.40192i	0.149328	−	0.258643i	−0.781651	−	0.623716i	$-0.785622\pi$
$173$	1.96410	−	3.40192i	0.149328	−	0.258643i	0.930979	+	0.365072i	$0.118956\pi$
$174$		0			0
$175$	−0.675108	−	0.389774i	−0.0510334	−	0.0294641i
$176$		0			0
$177$			36.1244i			2.71527i
$178$		0			0
$179$	4.36397	+	7.55862i	0.326178	+	0.564958i	0.981750	−	0.190175i	$-0.0609057\pi$
$179$	4.36397	+	7.55862i	0.326178	+	0.564958i	−0.655572	+	0.755133i	$0.727572\pi$
$180$		0			0
$181$	4.39230			0.326477			0.163239	−	0.986587i	$-0.447806\pi$
$181$	4.39230			0.326477			0.163239	+	0.986587i	$0.447806\pi$
$182$		0			0
$183$	8.72794			0.645188
$184$		0			0
$185$	−4.26795	−	7.39230i	−0.313786	−	0.543493i
$186$		0			0
$187$		−	32.5731i		−	2.38198i
$188$		0			0
$189$	−4.83975	−	2.79423i	−0.352040	−	0.203250i
$190$		0			0
$191$	−5.14352	+	8.90883i	−0.372172	+	0.644621i	−0.989899	−	0.141772i	$-0.954720\pi$
$191$	−5.14352	+	8.90883i	−0.372172	+	0.644621i	0.617728	+	0.786392i	$0.288053\pi$
$192$		0			0
$193$	−14.0885	+	8.13397i	−1.01411	+	0.585496i	−0.912392	−	0.409317i	$-0.865767\pi$
$193$	−14.0885	+	8.13397i	−1.01411	+	0.585496i	−0.101717	+	0.994813i	$0.532434\pi$
$194$		0			0
$195$	−15.1172	−	14.5466i	−1.08257	−	1.04170i
$196$		0			0
$197$	6.57180	−	3.79423i	0.468221	−	0.270328i	−0.247274	−	0.968946i	$-0.579535\pi$
$197$	6.57180	−	3.79423i	0.468221	−	0.270328i	0.715495	+	0.698618i	$0.246201\pi$
$198$		0			0
$199$	5.71419	−	9.89726i	0.405068	−	0.701598i	−0.589261	−	0.807942i	$-0.700581\pi$
$199$	5.71419	−	9.89726i	0.405068	−	0.701598i	0.994329	+	0.106344i	$0.0339145\pi$
$200$		0			0
$201$	−1.96410	−	1.13397i	−0.138537	−	0.0799844i
$202$		0			0
$203$			2.33864i			0.164141i
$204$		0			0
$205$	−3.19615	−	5.53590i	−0.223229	−	0.386644i
$206$		0			0
$207$	−39.1713			−2.72259
$208$		0			0
$209$	3.92820			0.271719
$210$		0			0
$211$	11.7417	+	20.3372i	0.808331	+	1.40007i	0.914019	+	0.405672i	$0.132962\pi$
$211$	11.7417	+	20.3372i	0.808331	+	1.40007i	−0.105688	+	0.994399i	$0.533704\pi$
$212$		0			0
$213$			43.9808i			3.01351i
$214$		0			0
$215$	−15.1172	−	8.72794i	−1.03099	−	0.595240i
$216$		0			0
$217$	0.607695	−	1.05256i	0.0412530	−	0.0714524i
$218$		0			0
$219$	−15.1172	+	8.72794i	−1.02153	+	0.589779i
$220$		0			0
$221$	22.6244	+	5.59808i	1.52188	+	0.376567i
$222$		0			0
$223$	0.675108	−	0.389774i	0.0452086	−	0.0261012i	−0.477225	−	0.878781i	$-0.658357\pi$
$223$	0.675108	−	0.389774i	0.0452086	−	0.0261012i	0.522434	+	0.852680i	$0.325024\pi$
$224$		0			0
$225$	−2.73205	+	4.73205i	−0.182137	+	0.315470i
$226$		0			0
$227$	2.02533	+	1.16932i	0.134426	+	0.0776106i	0.565705	−	0.824608i	$-0.308604\pi$
$227$	2.02533	+	1.16932i	0.134426	+	0.0776106i	−0.431279	+	0.902219i	$0.641938\pi$
$228$		0			0
$229$		−	12.9282i		−	0.854320i	−0.904176	−	0.427160i	$-0.859514\pi$
$229$		−	12.9282i		−	0.854320i	0.904176	−	0.427160i	$-0.140486\pi$
$230$		0			0
$231$	−5.71419	−	9.89726i	−0.375966	−	0.651192i
$232$		0			0
$233$	−21.4641			−1.40616			−0.703080	−	0.711111i	$-0.748192\pi$
$233$	−21.4641			−1.40616			−0.703080	+	0.711111i	$0.748192\pi$
$234$		0			0
$235$	14.7554			0.962539
$236$		0			0
$237$	14.6603	+	25.3923i	0.952286	+	1.64941i
$238$		0			0
$239$		−	10.0782i		−	0.651902i	−0.945387	−	0.325951i	$-0.894316\pi$
$239$		−	10.0782i		−	0.651902i	0.945387	−	0.325951i	$-0.105684\pi$
$240$		0			0
$241$	8.30385	+	4.79423i	0.534898	+	0.308823i	0.743009	−	0.669282i	$-0.233398\pi$
$241$	8.30385	+	4.79423i	0.534898	+	0.308823i	−0.208111	+	0.978105i	$0.566731\pi$
$242$		0			0
$243$	4.25953	−	7.37772i	0.273249	−	0.473281i
$244$		0			0
$245$	11.0718	−	6.39230i	0.707351	−	0.408389i
$246$		0			0
$247$	−0.675108	+	2.72842i	−0.0429561	+	0.173605i
$248$		0			0
$249$	25.3923	−	14.6603i	1.60917	−	0.929056i
$250$		0			0
$251$	9.97372	−	17.2750i	0.629535	−	1.09039i	−0.358110	−	0.933680i	$-0.616579\pi$
$251$	9.97372	−	17.2750i	0.629535	−	1.09039i	0.987645	−	0.156708i	$-0.0500881\pi$
$252$		0			0
$253$	−31.2846	−	18.0622i	−1.96685	−	1.13556i
$254$		0			0
$255$			37.6122i			2.35537i
$256$		0			0
$257$	−3.23205	−	5.59808i	−0.201610	−	0.349198i	0.747437	−	0.664332i	$-0.231284\pi$
$257$	−3.23205	−	5.59808i	−0.201610	−	0.349198i	−0.949047	+	0.315134i	$0.897951\pi$
$258$		0			0
$259$	3.32707			0.206734
$260$		0			0
$261$	16.3923			1.01466
$262$		0			0
$263$	2.02533	+	3.50797i	0.124887	+	0.216310i	0.921689	−	0.387930i	$-0.126810\pi$
$263$	2.02533	+	3.50797i	0.124887	+	0.216310i	−0.796802	+	0.604241i	$0.793477\pi$
$264$		0			0
$265$			18.9282i			1.16275i
$266$		0			0
$267$	23.1701	+	13.3772i	1.41798	+	0.818674i
$268$		0			0
$269$	−12.6962	+	21.9904i	−0.774098	+	1.34078i	0.161202	+	0.986921i	$0.448463\pi$
$269$	−12.6962	+	21.9904i	−0.774098	+	1.34078i	−0.935300	+	0.353856i	$0.884870\pi$
$270$		0			0
$271$	3.37554	−	1.94887i	0.205050	−	0.118385i	−0.393959	−	0.919128i	$-0.628895\pi$
$271$	3.37554	−	1.94887i	0.205050	−	0.118385i	0.599009	+	0.800743i	$0.295561\pi$
$272$		0			0
$273$	7.85641	−	2.26795i	0.475491	−	0.137263i
$274$		0			0
$275$	−4.36397	+	2.51954i	−0.263157	+	0.151934i
$276$		0			0
$277$	−8.69615	+	15.0622i	−0.522501	+	0.904999i	0.477156	+	0.878819i	$0.341668\pi$
$277$	−8.69615	+	15.0622i	−0.522501	+	0.904999i	−0.999657	+	0.0261800i	$0.991666\pi$
$278$		0			0
$279$	−7.37772	−	4.25953i	−0.441693	−	0.255011i
$280$		0			0
$281$			10.0000i			0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$			10.0000i			0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$		0			0
$283$	−3.01375	−	5.21997i	−0.179149	−	0.310295i	0.762440	−	0.647059i	$-0.224001\pi$
$283$	−3.01375	−	5.21997i	−0.179149	−	0.310295i	−0.941589	+	0.336763i	$0.890668\pi$
$284$		0			0
$285$	−4.53590			−0.268683
$286$		0			0
$287$	2.49155			0.147072
$288$		0			0
$289$	−12.3923	−	21.4641i	−0.728959	−	1.26259i
$290$		0			0
$291$			29.8727i			1.75117i
$292$		0			0
$293$	12.8205	+	7.40192i	0.748982	+	0.432425i	0.825326	−	0.564657i	$-0.190991\pi$
$293$	12.8205	+	7.40192i	0.748982	+	0.432425i	−0.0763439	+	0.997082i	$0.524325\pi$
$294$		0			0
$295$	12.4168	−	21.5065i	0.722934	−	1.25216i
$296$		0			0
$297$	−31.2846	+	18.0622i	−1.81532	+	1.04807i
$298$		0			0
$299$	17.9221	−	18.6252i	1.03646	−	1.07712i
$300$		0			0
$301$	5.89230	−	3.40192i	0.339627	−	0.196084i
$302$		0			0
$303$	−2.02533	+	3.50797i	−0.116352	+	0.201527i
$304$		0			0
$305$	−5.19615	−	3.00000i	−0.297531	−	0.171780i
$306$		0			0
$307$			33.3527i			1.90354i	0.306817	+	0.951768i	$0.400736\pi$
$307$			33.3527i			1.90354i	−0.306817	+	0.951768i	$0.599264\pi$
$308$		0			0
$309$	−29.3205	−	50.7846i	−1.66799	−	2.88904i
$310$		0			0
$311$	3.11819			0.176816			0.0884082	−	0.996084i	$-0.471822\pi$
$311$	3.11819			0.176816			0.0884082	+	0.996084i	$0.471822\pi$
$312$		0			0
$313$	12.3923			0.700454			0.350227	−	0.936665i	$-0.386104\pi$
$313$	12.3923			0.700454			0.350227	+	0.936665i	$0.386104\pi$
$314$		0			0
$315$	4.25953	+	7.37772i	0.239997	+	0.415688i
$316$		0			0
$317$			4.00000i			0.224662i	0.993671	+	0.112331i	$0.0358318\pi$
$317$			4.00000i			0.224662i	−0.993671	+	0.112331i	$0.964168\pi$
$318$		0			0
$319$	13.0919	+	7.55862i	0.733006	+	0.423201i
$320$		0			0
$321$	12.6962	−	21.9904i	0.708630	−	1.22738i
$322$		0			0
$323$	4.36397	−	2.51954i	0.242818	−	0.140191i
$324$		0			0
$325$	−1.00000	−	3.46410i	−0.0554700	−	0.192154i
$326$		0			0
$327$	−15.1172	+	8.72794i	−0.835985	+	0.482656i
$328$		0			0
$329$	−2.87564	+	4.98076i	−0.158539	+	0.274598i
$330$		0			0
$331$	14.4421	+	8.33816i	0.793811	+	0.458307i	0.841302	−	0.540565i	$-0.181789\pi$
$331$	14.4421	+	8.33816i	0.793811	+	0.458307i	−0.0474915	+	0.998872i	$0.515123\pi$
$332$		0			0
$333$		−	23.3205i		−	1.27796i
$334$		0			0
$335$	0.779548	+	1.35022i	0.0425913	+	0.0737702i
$336$		0			0
$337$	20.3923			1.11084			0.555420	−	0.831570i	$-0.312558\pi$
$337$	20.3923			1.11084			0.555420	+	0.831570i	$0.312558\pi$
$338$		0			0
$339$	−16.1057			−0.874739
$340$		0			0
$341$	−3.92820	−	6.80385i	−0.212724	−	0.368449i
$342$		0			0
$343$			10.4399i			0.563704i
$344$		0			0
$345$	36.1244	+	20.8564i	1.94487	+	1.12287i
$346$		0			0
$347$	−2.80487	+	4.85818i	−0.150573	+	0.260801i	−0.931438	−	0.363899i	$-0.881445\pi$
$347$	−2.80487	+	4.85818i	−0.150573	+	0.260801i	0.780865	+	0.624700i	$0.214779\pi$
$348$		0			0
$349$	18.6962	−	10.7942i	1.00078	−	0.577802i	0.0923025	−	0.995731i	$-0.470577\pi$
$349$	18.6962	−	10.7942i	1.00078	−	0.577802i	0.908480	+	0.417929i	$0.137244\pi$
$350$		0			0
$351$	−7.16884	−	24.8336i	−0.382645	−	1.32552i
$352$		0			0
$353$	−1.03590	+	0.598076i	−0.0551353	+	0.0318324i	−0.527314	−	0.849670i	$-0.676801\pi$
$353$	−1.03590	+	0.598076i	−0.0551353	+	0.0318324i	0.472179	+	0.881503i	$0.343468\pi$
$354$		0			0
$355$	15.1172	−	26.1838i	0.802339	−	1.38969i
$356$		0			0
$357$	−12.6962	−	7.33013i	−0.671952	−	0.387951i
$358$		0			0
$359$			10.0782i			0.531905i	0.963986	+	0.265952i	$0.0856864\pi$
$359$			10.0782i			0.531905i	−0.963986	+	0.265952i	$0.914314\pi$
$360$		0			0
$361$	−9.19615	−	15.9282i	−0.484008	−	0.838326i
$362$		0			0
$363$	−41.8717			−2.19770
$364$		0			0
$365$	12.0000			0.628109
$366$		0			0
$367$	9.40305	+	16.2866i	0.490835	+	0.850151i	0.999944	−	0.0105506i	$-0.00335843\pi$
$367$	9.40305	+	16.2866i	0.490835	+	0.850151i	−0.509109	+	0.860702i	$0.670025\pi$
$368$		0			0
$369$		−	17.4641i		−	0.909145i
$370$		0			0
$371$	−6.38929	−	3.68886i	−0.331716	−	0.191516i
$372$		0			0
$373$	5.69615	−	9.86603i	0.294936	−	0.510843i	−0.680034	−	0.733180i	$-0.738035\pi$
$373$	5.69615	−	9.86603i	0.294936	−	0.510843i	0.974970	+	0.222337i	$0.0713685\pi$
$374$		0			0
$375$	30.2345	−	17.4559i	1.56130	−	0.901418i
$376$		0			0
$377$	−7.50000	+	7.79423i	−0.386270	+	0.401423i
$378$		0			0
$379$	−14.4421	+	8.33816i	−0.741842	+	0.428303i	−0.822739	−	0.568420i	$-0.807555\pi$
$379$	−14.4421	+	8.33816i	−0.741842	+	0.428303i	0.0808966	+	0.996723i	$0.474222\pi$
$380$		0			0
$381$	12.6962	−	21.9904i	0.650444	−	1.12660i
$382$		0			0
$383$	24.1585	+	13.9479i	1.23444	+	0.712705i	0.967953	−	0.251133i	$-0.0808032\pi$
$383$	24.1585	+	13.9479i	1.23444	+	0.712705i	0.266489	+	0.963838i	$0.414137\pi$
$384$		0			0
$385$			7.85641i			0.400400i
$386$		0			0
$387$	−23.8452	−	41.3010i	−1.21212	−	2.09945i
$388$		0			0
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		0			0
$391$	−46.3401			−2.34352
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	20.1563i		−	1.01417i
$396$		0			0
$397$	−11.3038	−	6.52628i	−0.567324	−	0.327545i	0.188756	−	0.982024i	$-0.439554\pi$
$397$	−11.3038	−	6.52628i	−0.567324	−	0.327545i	−0.756080	+	0.654479i	$0.772888\pi$
$398$		0			0
$399$	0.883988	−	1.53111i	0.0442547	−	0.0766515i
$400$		0			0
$401$	−13.8397	+	7.99038i	−0.691124	+	0.399021i	−0.804033	−	0.594585i	$-0.797316\pi$
$401$	−13.8397	+	7.99038i	−0.691124	+	0.399021i	0.112909	+	0.993605i	$0.463983\pi$
$402$		0			0
$403$	5.40087	−	1.55910i	0.269036	−	0.0776641i
$404$		0			0
$405$	7.73205	−	4.46410i	0.384209	−	0.221823i
$406$		0			0
$407$	10.7533	−	18.6252i	0.533020	−	0.923217i
$408$		0			0
$409$	14.3038	+	8.25833i	0.707280	+	0.408348i	0.810053	−	0.586357i	$-0.199438\pi$
$409$	14.3038	+	8.25833i	0.707280	+	0.408348i	−0.102773	+	0.994705i	$0.532772\pi$
$410$		0			0
$411$		−	8.15727i		−	0.402368i
$412$		0			0
$413$	4.83975	+	8.38269i	0.238148	+	0.412485i
$414$		0			0
$415$	−20.1563			−0.989434
$416$		0			0
$417$	−3.92820			−0.192365
$418$		0			0
$419$	2.02533	+	3.50797i	0.0989436	+	0.171375i	0.911248	−	0.411859i	$-0.135120\pi$
$419$	2.02533	+	3.50797i	0.0989436	+	0.171375i	−0.812304	+	0.583234i	$0.801787\pi$
$420$		0			0
$421$			12.9282i			0.630082i	0.949078	+	0.315041i	$0.102018\pi$
$421$			12.9282i			0.630082i	−0.949078	+	0.315041i	$0.897982\pi$
$422$		0			0
$423$	34.9118	+	20.1563i	1.69747	+	0.980033i
$424$		0			0
$425$	−3.23205	+	5.59808i	−0.156777	+	0.271547i
$426$		0			0
$427$	2.02533	−	1.16932i	0.0980124	−	0.0565875i
$428$		0			0
$429$	12.6962	−	51.3109i	0.612976	−	2.47731i
$430$		0			0
$431$	−17.1426	+	9.89726i	−0.825728	+	0.476734i	−0.852388	−	0.522910i	$-0.824846\pi$
$431$	−17.1426	+	9.89726i	−0.825728	+	0.476734i	0.0266597	+	0.999645i	$0.491513\pi$
$432$		0			0
$433$	3.69615	−	6.40192i	0.177626	−	0.307657i	−0.763441	−	0.645877i	$-0.776492\pi$
$433$	3.69615	−	6.40192i	0.177626	−	0.307657i	0.941067	+	0.338221i	$0.109825\pi$
$434$		0			0
$435$	−15.1172	−	8.72794i	−0.724816	−	0.418473i
$436$		0			0
$437$		−	5.58846i		−	0.267332i
$438$		0			0
$439$	−3.01375	−	5.21997i	−0.143839	−	0.249136i	0.785100	−	0.619368i	$-0.212611\pi$
$439$	−3.01375	−	5.21997i	−0.143839	−	0.249136i	−0.928939	+	0.370233i	$0.879278\pi$
$440$		0			0
$441$	34.9282			1.66325
$442$		0			0
$443$	−25.5572			−1.21426			−0.607129	−	0.794603i	$-0.707679\pi$
$443$	−25.5572			−1.21426			−0.607129	+	0.794603i	$0.707679\pi$
$444$		0			0
$445$	−9.19615	−	15.9282i	−0.435939	−	0.755069i
$446$		0			0
$447$		−	56.9890i		−	2.69548i
$448$		0			0
$449$	−5.55256	−	3.20577i	−0.262041	−	0.151290i	0.363224	−	0.931702i	$-0.381676\pi$
$449$	−5.55256	−	3.20577i	−0.262041	−	0.151290i	−0.625265	+	0.780412i	$0.715009\pi$
$450$		0			0
$451$	8.05283	−	13.9479i	0.379193	−	0.656781i
$452$		0			0
$453$	3.92820	−	2.26795i	0.184563	−	0.106558i
$454$		0			0
$455$	−5.45684	−	1.35022i	−0.255820	−	0.0632991i
$456$		0			0
$457$	−0.107695	+	0.0621778i	−0.00503777	+	0.00290856i	−0.502517	−	0.864567i	$-0.667592\pi$
$457$	−0.107695	+	0.0621778i	−0.00503777	+	0.00290856i	0.497479	+	0.867476i	$0.334259\pi$
$458$		0			0
$459$	−23.1701	+	40.1317i	−1.08149	+	1.87319i
$460$		0			0
$461$	−6.23205	−	3.59808i	−0.290256	−	0.167579i	0.347802	−	0.937568i	$-0.386928\pi$
$461$	−6.23205	−	3.59808i	−0.290256	−	0.167579i	−0.638057	+	0.769989i	$0.720262\pi$
$462$		0			0
$463$		−	27.1163i		−	1.26020i	−0.776514	−	0.630100i	$-0.783014\pi$
$463$		−	27.1163i		−	1.26020i	0.776514	−	0.630100i	$-0.216986\pi$
$464$		0			0
$465$	4.53590	+	7.85641i	0.210347	+	0.364332i
$466$		0			0
$467$	1.55910			0.0721464			0.0360732	−	0.999349i	$-0.488515\pi$
$467$	1.55910			0.0721464			0.0360732	+	0.999349i	$0.488515\pi$
$468$		0			0
$469$	−0.607695			−0.0280608
$470$		0			0
$471$	23.8452	+	41.3010i	1.09873	+	1.90305i
$472$		0			0
$473$		−	43.9808i		−	2.02224i
$474$		0			0
$475$	−0.675108	−	0.389774i	−0.0309761	−	0.0178841i
$476$		0			0
$477$	−25.8564	+	44.7846i	−1.18388	+	2.05055i
$478$		0			0
$479$	−21.8198	+	12.5977i	−0.996974	+	0.575603i	−0.907352	−	0.420372i	$-0.861899\pi$
$479$	−21.8198	+	12.5977i	−0.996974	+	0.575603i	−0.0896226	+	0.995976i	$0.528566\pi$
$480$		0			0
$481$	11.0885	+	10.6699i	0.505590	+	0.486504i
$482$		0			0
$483$	−14.0803	+	8.12929i	−0.640677	+	0.369895i
$484$		0			0
$485$	10.2679	−	17.7846i	0.466244	−	0.807558i
$486$		0			0
$487$	14.4421	+	8.33816i	0.654435	+	0.377838i	0.790153	−	0.612909i	$-0.210001\pi$
$487$	14.4421	+	8.33816i	0.654435	+	0.377838i	−0.135718	+	0.990747i	$0.543334\pi$
$488$		0			0
$489$			34.9090i			1.57864i
$490$		0			0
$491$	5.14352	+	8.90883i	0.232124	+	0.402050i	0.958433	−	0.285318i	$-0.0920993\pi$
$491$	5.14352	+	8.90883i	0.232124	+	0.402050i	−0.726309	+	0.687368i	$0.758766\pi$
$492$		0			0
$493$	19.3923			0.873385
$494$		0			0
$495$	55.0681			2.47513
$496$		0			0
$497$	5.89230	+	10.2058i	0.264306	+	0.457791i
$498$		0			0
$499$			1.55910i			0.0697947i	0.999391	+	0.0348974i	$0.0111104\pi$
$499$			1.55910i			0.0697947i	−0.999391	+	0.0348974i	$0.988890\pi$
$500$		0			0
$501$	−44.8923	−	25.9186i	−2.00564	−	1.15796i
$502$		0			0
$503$	−6.70261	+	11.6093i	−0.298855	+	0.517632i	−0.975874	−	0.218333i	$-0.929938\pi$
$503$	−6.70261	+	11.6093i	−0.298855	+	0.517632i	0.677019	+	0.735965i	$0.263271\pi$
$504$		0			0
$505$	2.41154	−	1.39230i	0.107312	−	0.0619568i
$506$		0			0
$507$	33.4571	+	17.6368i	1.48588	+	0.783277i
$508$		0			0
$509$	14.5526	−	8.40192i	0.645031	−	0.372409i	−0.141519	−	0.989936i	$-0.545199\pi$
$509$	14.5526	−	8.40192i	0.645031	−	0.372409i	0.786550	+	0.617527i	$0.211865\pi$
$510$		0			0
$511$	−2.33864	+	4.05065i	−0.103456	+	0.179190i
$512$		0			0
$513$	−4.83975	−	2.79423i	−0.213680	−	0.123368i
$514$		0			0
$515$			40.3126i			1.77639i
$516$		0			0
$517$	18.5885	+	32.1962i	0.817519	+	1.41599i
$518$		0			0
$519$	−11.4284			−0.501650
$520$		0			0
$521$	−42.2487			−1.85095			−0.925475	−	0.378809i	$-0.876334\pi$
$521$	−42.2487			−1.85095			−0.925475	+	0.378809i	$0.876334\pi$
$522$		0			0
$523$	−18.1310	−	31.4038i	−0.792813	−	1.37319i	−0.924219	−	0.381863i	$-0.875283\pi$
$523$	−18.1310	−	31.4038i	−0.792813	−	1.37319i	0.131406	−	0.991329i	$-0.458051\pi$
$524$		0			0
$525$			2.26795i			0.0989814i
$526$		0			0
$527$	−8.72794	−	5.03908i	−0.380195	−	0.219506i
$528$		0			0
$529$	−14.1962	+	24.5885i	−0.617224	+	1.06906i
$530$		0			0
$531$	58.7569	−	33.9233i	2.54983	−	1.47215i
$532$		0			0
$533$	8.30385	+	7.99038i	0.359680	+	0.346102i
$534$		0			0
$535$	−15.1172	+	8.72794i	−0.653575	+	0.377342i
$536$		0			0
$537$	12.6962	−	21.9904i	0.547879	−	0.948955i
$538$		0			0
$539$	27.8958	+	16.1057i	1.20156	+	0.693720i
$540$		0			0
$541$		−	6.92820i		−	0.297867i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$		−	6.92820i		−	0.297867i	0.988847	−	0.148933i	$-0.0475840\pi$
$542$		0			0
$543$	−6.38929	−	11.0666i	−0.274191	−	0.474913i
$544$		0			0
$545$	12.0000			0.514024
$546$		0			0
$547$	20.1563			0.861822			0.430911	−	0.902395i	$-0.358192\pi$
$547$	20.1563			0.861822			0.430911	+	0.902395i	$0.358192\pi$
$548$		0			0
$549$	−8.19615	−	14.1962i	−0.349803	−	0.605877i
$550$		0			0
$551$			2.33864i			0.0996296i
$552$		0			0
$553$	6.80385	+	3.92820i	0.289329	+	0.167044i
$554$		0			0
$555$	−12.4168	+	21.5065i	−0.527064	+	0.912901i
$556$		0			0
$557$	−28.7487	+	16.5981i	−1.21812	+	0.703283i	−0.964516	−	0.264025i	$-0.914950\pi$
$557$	−28.7487	+	16.5981i	−1.21812	+	0.703283i	−0.253606	+	0.967308i	$0.581617\pi$
$558$		0			0
$559$	30.5478	+	7.55862i	1.29203	+	0.319695i
$560$		0			0
$561$	−82.0692	+	47.3827i	−3.46497	+	2.00050i
$562$		0			0
$563$	15.4306	−	26.7265i	0.650320	−	1.12639i	−0.332725	−	0.943024i	$-0.607968\pi$
$563$	15.4306	−	26.7265i	0.650320	−	1.12639i	0.983045	−	0.183364i	$-0.0586986\pi$
$564$		0			0
$565$	9.58846	+	5.53590i	0.403389	+	0.232897i
$566$		0			0
$567$			3.47998i			0.146145i
$568$		0			0
$569$	1.96410	+	3.40192i	0.0823394	+	0.142616i	0.904254	−	0.426994i	$-0.140428\pi$
$569$	1.96410	+	3.40192i	0.0823394	+	0.142616i	−0.821915	+	0.569610i	$0.807094\pi$
$570$		0			0
$571$	−14.7554			−0.617496			−0.308748	−	0.951144i	$-0.599910\pi$
$571$	−14.7554			−0.617496			−0.308748	+	0.951144i	$0.599910\pi$
$572$		0			0
$573$	29.9282			1.25027
$574$		0			0
$575$	3.58442	+	6.20840i	0.149481	+	0.258908i
$576$		0			0
$577$			16.1436i			0.672067i	0.941850	+	0.336033i	$0.109085\pi$
$577$			16.1436i			0.672067i	−0.941850	+	0.336033i	$0.890915\pi$
$578$		0			0
$579$	40.9877	+	23.6643i	1.70339	+	0.983454i
$580$		0			0
$581$	3.92820	−	6.80385i	0.162969	−	0.282271i
$582$		0			0
$583$	−41.3010	+	23.8452i	−1.71051	+	0.987566i
$584$		0			0
$585$	−9.46410	+	38.2487i	−0.391292	+	1.58139i
$586$		0			0
$587$	15.4306	−	8.90883i	0.636887	−	0.367707i	−0.146527	−	0.989207i	$-0.546810\pi$
$587$	15.4306	−	8.90883i	0.636887	−	0.367707i	0.783414	+	0.621500i	$0.213476\pi$
$588$		0			0
$589$	0.607695	−	1.05256i	0.0250396	−	0.0433699i
$590$		0			0
$591$	−19.1194	−	11.0386i	−0.786468	−	0.454067i
$592$		0			0
$593$		−	18.7846i		−	0.771391i	−0.922626	−	0.385696i	$-0.873961\pi$
$593$		−	18.7846i		−	0.771391i	0.922626	−	0.385696i	$-0.126039\pi$
$594$		0			0
$595$	5.03908	+	8.72794i	0.206582	+	0.357811i
$596$		0			0
$597$	−33.2487			−1.36078
$598$		0			0
$599$	−30.2345			−1.23535			−0.617673	−	0.786435i	$-0.711925\pi$
$599$	−30.2345			−1.23535			−0.617673	+	0.786435i	$0.711925\pi$
$600$		0			0
$601$	7.69615	+	13.3301i	0.313933	+	0.543747i	0.979210	−	0.202849i	$-0.0650201\pi$
$601$	7.69615	+	13.3301i	0.313933	+	0.543747i	−0.665277	+	0.746596i	$0.731687\pi$
$602$		0			0
$603$			4.25953i			0.173461i
$604$		0			0
$605$	24.9282	+	14.3923i	1.01348	+	0.585130i
$606$		0			0
$607$	−14.4421	+	25.0145i	−0.586188	+	1.01531i	0.408538	+	0.912741i	$0.366039\pi$
$607$	−14.4421	+	25.0145i	−0.586188	+	1.01531i	−0.994726	+	0.102566i	$0.967295\pi$
$608$		0			0
$609$	5.89230	−	3.40192i	0.238768	−	0.137853i
$610$		0			0
$611$	−25.5572	+	7.37772i	−1.03393	+	0.298471i
$612$		0			0
$613$	−9.69615	+	5.59808i	−0.391624	+	0.226104i	−0.682864	−	0.730546i	$-0.739266\pi$
$613$	−9.69615	+	5.59808i	−0.391624	+	0.226104i	0.291240	+	0.956650i	$0.405932\pi$
$614$		0			0
$615$	−9.29861	+	16.1057i	−0.374956	+	0.649443i
$616$		0			0
$617$	16.9641	+	9.79423i	0.682949	+	0.394301i	0.800965	−	0.598711i	$-0.204320\pi$
$617$	16.9641	+	9.79423i	0.682949	+	0.394301i	−0.118016	+	0.993012i	$0.537653\pi$
$618$		0			0
$619$			1.55910i			0.0626654i	0.999509	+	0.0313327i	$0.00997513\pi$
$619$			1.55910i			0.0626654i	−0.999509	+	0.0313327i	$0.990025\pi$
$620$		0			0
$621$	25.6962	+	44.5070i	1.03115	+	1.78601i
$622$		0			0
$623$	7.16884			0.287214
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$	−5.71419	−	9.89726i	−0.228203	−	0.395259i
$628$		0			0
$629$		−	27.5885i		−	1.10002i
$630$		0			0
$631$	−37.9255	−	21.8963i	−1.50979	−	0.871678i	−0.999935	−	0.0114181i	$-0.996365\pi$
$631$	−37.9255	−	21.8963i	−1.50979	−	0.871678i	−0.509856	−	0.860260i	$-0.670301\pi$
$632$		0			0
$633$	34.1603	−	59.1673i	1.35775	−	2.35169i
$634$		0			0
$635$	−15.1172	+	8.72794i	−0.599909	+	0.346358i
$636$		0			0
$637$	−15.9808	+	16.6077i	−0.633181	+	0.658021i
$638$		0			0
$639$	71.5355	−	41.3010i	2.82990	−	1.63384i
$640$		0			0
$641$	−0.696152	+	1.20577i	−0.0274964	+	0.0476251i	−0.879446	−	0.475998i	$-0.842087\pi$
$641$	−0.696152	+	1.20577i	−0.0274964	+	0.0476251i	0.851950	+	0.523624i	$0.175420\pi$
$642$		0			0
$643$	−18.4928	−	10.6768i	−0.729284	−	0.421052i	0.0888763	−	0.996043i	$-0.471672\pi$
$643$	−18.4928	−	10.6768i	−0.729284	−	0.421052i	−0.818160	+	0.574990i	$0.805006\pi$
$644$		0			0
$645$			50.7846i			1.99964i
$646$		0			0
$647$	−13.0919	−	22.6758i	−0.514696	−	0.891480i	−0.999855	−	0.0170535i	$-0.994571\pi$
$647$	−13.0919	−	22.6758i	−0.514696	−	0.891480i	0.485159	−	0.874426i	$-0.338762\pi$
$648$		0			0
$649$	62.5692			2.45606
$650$		0			0
$651$	−3.53595			−0.138585
$652$		0			0
$653$	16.6244	+	28.7942i	0.650561	+	1.12681i	0.982987	+	0.183676i	$0.0587996\pi$
$653$	16.6244	+	28.7942i	0.650561	+	1.12681i	−0.332426	+	0.943129i	$0.607867\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	28.3923	+	16.3923i	1.10769	+	0.639525i
$658$		0			0
$659$	11.5328	−	19.9754i	0.449255	−	0.778132i	−0.549083	−	0.835768i	$-0.685023\pi$
$659$	11.5328	−	19.9754i	0.449255	−	0.778132i	0.998338	+	0.0576360i	$0.0183563\pi$
$660$		0			0
$661$	−21.6962	+	12.5263i	−0.843883	+	0.487216i	−0.858582	−	0.512676i	$-0.828654\pi$
$661$	−21.6962	+	12.5263i	−0.843883	+	0.487216i	0.0146995	+	0.999892i	$0.495321\pi$
$662$		0			0
$663$	−18.8061	−	65.1462i	−0.730368	−	2.53007i
$664$		0			0
$665$	−1.05256	+	0.607695i	−0.0408165	+	0.0235654i
$666$		0			0
$667$	10.7533	−	18.6252i	0.416368	−	0.721171i
$668$		0			0
$669$	−1.96410	−	1.13397i	−0.0759366	−	0.0438420i
$670$		0			0
$671$		−	15.1172i		−	0.583594i
$672$		0			0
$673$	8.89230	+	15.4019i	0.342773	+	0.593701i	0.984947	−	0.172859i	$-0.0553004\pi$
$673$	8.89230	+	15.4019i	0.342773	+	0.593701i	−0.642173	+	0.766559i	$0.721967\pi$
$674$		0			0
$675$	7.16884			0.275929
$676$		0			0
$677$	2.53590			0.0974625			0.0487312	−	0.998812i	$-0.484482\pi$
$677$	2.53590			0.0974625			0.0487312	+	0.998812i	$0.484482\pi$
$678$		0			0
$679$	4.00218	+	6.93198i	0.153590	+	0.266025i
$680$		0			0
$681$		−	6.80385i		−	0.260724i
$682$		0			0
$683$	−10.7533	−	6.20840i	−0.411462	−	0.237558i	0.279956	−	0.960013i	$-0.409680\pi$
$683$	−10.7533	−	6.20840i	−0.411462	−	0.237558i	−0.691418	+	0.722455i	$0.743014\pi$
$684$		0			0
$685$	−2.80385	+	4.85641i	−0.107130	+	0.185554i
$686$		0			0
$687$	−32.5731	+	18.8061i	−1.24274	+	0.717497i
$688$		0			0
$689$	−9.46410	−	32.7846i	−0.360554	−	1.24899i
$690$		0			0
$691$	−19.8430	+	11.4564i	−0.754863	+	0.435820i	−0.827448	−	0.561542i	$-0.810208\pi$
$691$	−19.8430	+	11.4564i	−0.754863	+	0.435820i	0.0725853	+	0.997362i	$0.476875\pi$
$692$		0			0
$693$	−10.7321	+	18.5885i	−0.407677	+	0.706117i
$694$		0			0
$695$	2.33864	+	1.35022i	0.0887098	+	0.0512166i
$696$		0			0
$697$		−	20.6603i		−	0.782563i
$698$		0			0
$699$	31.2229	+	54.0796i	1.18096	+	2.04548i
$700$		0			0
$701$	−44.1051			−1.66583			−0.832914	−	0.553403i	$-0.813329\pi$
$701$	−44.1051			−1.66583			−0.832914	+	0.553403i	$0.813329\pi$
$702$		0			0
$703$	3.32707			0.125483
$704$		0			0
$705$	−21.4641	−	37.1769i	−0.808385	−	1.40016i
$706$		0			0
$707$			1.08537i			0.0408195i
$708$		0			0
$709$	−33.6962	−	19.4545i	−1.26549	−	0.730628i	−0.291355	−	0.956615i	$-0.594106\pi$
$709$	−33.6962	−	19.4545i	−1.26549	−	0.730628i	−0.974130	+	0.225987i	$0.927439\pi$
$710$		0			0
$711$	27.5340	−	47.6903i	1.03261	−	1.78853i
$712$		0			0
$713$	−9.67949	+	5.58846i	−0.362500	+	0.209289i
$714$		0			0
$715$	−25.1954	+	26.1838i	−0.942254	+	0.979219i
$716$		0			0
$717$	−25.3923	+	14.6603i	−0.948293	+	0.547497i
$718$		0			0
$719$	−5.14352	+	8.90883i	−0.191821	+	0.332243i	−0.945854	−	0.324593i	$-0.894773\pi$
$719$	−5.14352	+	8.90883i	−0.191821	+	0.332243i	0.754033	+	0.656837i	$0.228106\pi$
$720$		0			0
$721$	−13.6077	−	7.85641i	−0.506777	−	0.292588i
$722$		0			0
$723$		−	27.8958i		−	1.03746i
$724$		0			0
$725$	−1.50000	−	2.59808i	−0.0557086	−	0.0964901i
$726$		0			0
$727$	40.3126			1.49511			0.747556	−	0.664199i	$-0.231227\pi$
$727$	40.3126			1.49511			0.747556	+	0.664199i	$0.231227\pi$
$728$		0			0
$729$	−38.1769			−1.41396
$730$		0			0
$731$	−28.2091	−	48.8597i	−1.04335	−	1.80714i
$732$		0			0
$733$			39.7128i			1.46683i	0.679783	+	0.733413i	$0.262074\pi$
$733$			39.7128i			1.46683i	−0.679783	+	0.733413i	$0.737926\pi$
$734$		0			0
$735$	−32.2113	−	18.5972i	−1.18813	−	0.685969i
$736$		0			0
$737$	−1.96410	+	3.40192i	−0.0723486	+	0.125311i
$738$		0			0
$739$	22.8083	−	13.1684i	0.839016	−	0.484406i	−0.0179137	−	0.999840i	$-0.505702\pi$
$739$	22.8083	−	13.1684i	0.839016	−	0.484406i	0.856930	+	0.515433i	$0.172369\pi$
$740$		0			0
$741$	7.85641	−	2.26795i	0.288612	−	0.0833152i
$742$		0			0
$743$	−26.4971	+	15.2981i	−0.972086	+	0.561234i	−0.899871	−	0.436155i	$-0.856340\pi$
$743$	−26.4971	+	15.2981i	−0.972086	+	0.561234i	−0.0722142	+	0.997389i	$0.523007\pi$
$744$		0			0
$745$	−19.5885	+	33.9282i	−0.717666	+	1.24303i
$746$		0			0
$747$	−47.6903	−	27.5340i	−1.74490	−	1.00742i
$748$		0			0
$749$		−	6.80385i		−	0.248607i
$750$		0			0
$751$	9.40305	+	16.2866i	0.343122	+	0.594305i	0.985011	−	0.172493i	$-0.0551822\pi$
$751$	9.40305	+	16.2866i	0.343122	+	0.594305i	−0.641889	+	0.766798i	$0.721849\pi$
$752$		0			0
$753$	−58.0333			−2.11485
$754$		0			0
$755$	−3.11819			−0.113483
$756$		0			0
$757$	21.6962	+	37.5788i	0.788560	+	1.36583i	0.926849	+	0.375434i	$0.122506\pi$
$757$	21.6962	+	37.5788i	0.788560	+	1.36583i	−0.138289	+	0.990392i	$0.544160\pi$
$758$		0			0
$759$			105.097i			3.81478i
$760$		0			0
$761$	−6.23205	−	3.59808i	−0.225912	−	0.130430i	0.382773	−	0.923842i	$-0.374969\pi$
$761$	−6.23205	−	3.59808i	−0.225912	−	0.130430i	−0.608685	+	0.793412i	$0.708303\pi$
$762$		0			0
$763$	−2.33864	+	4.05065i	−0.0846646	+	0.146643i
$764$		0			0
$765$	61.1769	−	35.3205i	2.21186	−	1.27702i
$766$		0			0
$767$	−10.7533	+	43.4588i	−0.388278	+	1.56921i
$768$		0			0
$769$	15.4808	−	8.93782i	0.558251	−	0.322306i	−0.194192	−	0.980963i	$-0.562209\pi$
$769$	15.4808	−	8.93782i	0.558251	−	0.322306i	0.752443	+	0.658657i	$0.228875\pi$
$770$		0			0
$771$	−9.40305	+	16.2866i	−0.338642	+	0.586546i
$772$		0			0
$773$	−23.5526	−	13.5981i	−0.847127	−	0.489089i	0.0125537	−	0.999921i	$-0.496004\pi$
$773$	−23.5526	−	13.5981i	−0.847127	−	0.489089i	−0.859680	+	0.510832i	$0.829337\pi$
$774$		0			0
$775$			1.55910i			0.0560044i
$776$		0			0
$777$	−4.83975	−	8.38269i	−0.173625	−	0.300727i
$778$		0			0
$779$	2.49155			0.0892692
$780$		0			0
$781$	76.1769			2.72582
$782$		0			0
$783$	−10.7533	−	18.6252i	−0.384290	−	0.665610i
$784$		0			0
$785$		−	32.7846i		−	1.17013i
$786$		0			0
$787$	−46.0268	−	26.5736i	−1.64068	−	0.947246i	−0.980593	−	0.196056i	$-0.937187\pi$
$787$	−46.0268	−	26.5736i	−1.64068	−	0.947246i	−0.660086	−	0.751190i	$-0.729480\pi$
$788$		0			0
$789$	5.89230	−	10.2058i	0.209772	−	0.363335i
$790$		0

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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.w (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.45466 - 2.51954i) q^{3} +2.00000i q^{5} +(-0.675108 - 0.389774i) q^{7} +(-2.73205 + 4.73205i) q^{9} +O(q^{10})\)	\(q+(-1.45466 - 2.51954i) q^{3} +2.00000i q^{5} +(-0.675108 - 0.389774i) q^{7} +(-2.73205 + 4.73205i) q^{9} +(-4.36397 + 2.51954i) q^{11} +(-1.00000 - 3.46410i) q^{13} +(5.03908 - 2.90931i) q^{15} +(-3.23205 + 5.59808i) q^{17} +(-0.675108 - 0.389774i) q^{19} +2.26795i q^{21} +(3.58442 + 6.20840i) q^{23} +1.00000 q^{25} +7.16884 q^{27} +(-1.50000 - 2.59808i) q^{29} +1.55910i q^{31} +(12.6962 + 7.33013i) q^{33} +(0.779548 - 1.35022i) q^{35} +(-3.69615 + 2.13397i) q^{37} +(-7.27328 + 7.55862i) q^{39} +(-2.76795 + 1.59808i) q^{41} +(-4.36397 + 7.55862i) q^{43} +(-9.46410 - 5.46410i) q^{45} -7.37772i q^{47} +(-3.19615 - 5.53590i) q^{49} +18.8061 q^{51} +9.46410 q^{53} +(-5.03908 - 8.72794i) q^{55} +2.26795i q^{57} +(-10.7533 - 6.20840i) q^{59} +(-1.50000 + 2.59808i) q^{61} +(3.68886 - 2.12976i) q^{63} +(6.92820 - 2.00000i) q^{65} +(0.675108 - 0.389774i) q^{67} +(10.4282 - 18.0622i) q^{69} +(-13.0919 - 7.55862i) q^{71} -6.00000i q^{73} +(-1.45466 - 2.51954i) q^{75} +3.92820 q^{77} -10.0782 q^{79} +(-2.23205 - 3.86603i) q^{81} +10.0782i q^{83} +(-11.1962 - 6.46410i) q^{85} +(-4.36397 + 7.55862i) q^{87} +(-7.96410 + 4.59808i) q^{89} +(-0.675108 + 2.72842i) q^{91} +(3.92820 - 2.26795i) q^{93} +(0.779548 - 1.35022i) q^{95} +(-8.89230 - 5.13397i) q^{97} -27.5340i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 8 q^{13} - 12 q^{17} + 8 q^{25} - 12 q^{29} + 60 q^{33} + 12 q^{37} - 36 q^{41} - 48 q^{45} + 16 q^{49} + 48 q^{53} - 12 q^{61} + 28 q^{69} - 24 q^{77} - 4 q^{81} - 48 q^{85} - 36 q^{89} - 24 q^{93} + 12 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 - 8 * q^13 - 12 * q^17 + 8 * q^25 - 12 * q^29 + 60 * q^33 + 12 * q^37 - 36 * q^41 - 48 * q^45 + 16 * q^49 + 48 * q^53 - 12 * q^61 + 28 * q^69 - 24 * q^77 - 4 * q^81 - 48 * q^85 - 36 * q^89 - 24 * q^93 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more