LMFDB - Embedded newform 476.2.i.b.137.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [476,2,Mod(137,476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(476, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("476.137");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 476 = 2^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	476.i (of order $3$, 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:	$3.80087913621$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			137.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	476.137
Dual form			476.2.i.b.205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{3} +(2.00000 - 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{9} +O(q^{10})$	$q+(0.500000 - 0.866025i) q^{3} +(2.00000 - 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{9} +(2.50000 - 4.33013i) q^{11} -5.00000 q^{13} +(0.500000 - 0.866025i) q^{17} +(3.00000 + 5.19615i) q^{19} +(-0.500000 - 2.59808i) q^{21} +(-2.00000 - 3.46410i) q^{23} +(2.50000 - 4.33013i) q^{25} +5.00000 q^{27} +4.00000 q^{29} +(-2.50000 - 4.33013i) q^{33} +(-4.00000 - 6.92820i) q^{37} +(-2.50000 + 4.33013i) q^{39} +4.00000 q^{41} -6.00000 q^{43} +(3.00000 + 5.19615i) q^{47} +(1.00000 - 6.92820i) q^{49} +(-0.500000 - 0.866025i) q^{51} +(-5.50000 + 9.52628i) q^{53} +6.00000 q^{57} +(-5.00000 + 8.66025i) q^{59} +(5.00000 + 1.73205i) q^{63} +(-5.00000 + 8.66025i) q^{67} -4.00000 q^{69} +9.00000 q^{71} +(-2.00000 + 3.46410i) q^{73} +(-2.50000 - 4.33013i) q^{75} +(-2.50000 - 12.9904i) q^{77} +(-4.50000 - 7.79423i) q^{79} +(-0.500000 + 0.866025i) q^{81} +4.00000 q^{83} +(2.00000 - 3.46410i) q^{87} +(7.50000 + 12.9904i) q^{89} +(-10.0000 + 8.66025i) q^{91} -18.0000 q^{97} +10.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + q^3 + 4 * q^7 + 2 * q^9	$ 2 q + q^{3} + 4 q^{7} + 2 q^{9} + 5 q^{11} - 10 q^{13} + q^{17} + 6 q^{19} - q^{21} - 4 q^{23} + 5 q^{25} + 10 q^{27} + 8 q^{29} - 5 q^{33} - 8 q^{37} - 5 q^{39} + 8 q^{41} - 12 q^{43} + 6 q^{47} + 2 q^{49} - q^{51} - 11 q^{53} + 12 q^{57} - 10 q^{59} + 10 q^{63} - 10 q^{67} - 8 q^{69} + 18 q^{71} - 4 q^{73} - 5 q^{75} - 5 q^{77} - 9 q^{79} - q^{81} + 8 q^{83} + 4 q^{87} + 15 q^{89} - 20 q^{91} - 36 q^{97} + 20 q^{99}+O(q^{100}) $ 2 * q + q^3 + 4 * q^7 + 2 * q^9 + 5 * q^11 - 10 * q^13 + q^17 + 6 * q^19 - q^21 - 4 * q^23 + 5 * q^25 + 10 * q^27 + 8 * q^29 - 5 * q^33 - 8 * q^37 - 5 * q^39 + 8 * q^41 - 12 * q^43 + 6 * q^47 + 2 * q^49 - q^51 - 11 * q^53 + 12 * q^57 - 10 * q^59 + 10 * q^63 - 10 * q^67 - 8 * q^69 + 18 * q^71 - 4 * q^73 - 5 * q^75 - 5 * q^77 - 9 * q^79 - q^81 + 8 * q^83 + 4 * q^87 + 15 * q^89 - 20 * q^91 - 36 * q^97 + 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/476\mathbb{Z}\right)^\times$.

$n$	$239$	$309$	$409$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i	−0.684819	−	0.728714i	$-0.740119\pi$
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i	0.973494	+	0.228714i	$0.0734519\pi$
$4$		0			0
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$		0			0
$9$	1.00000	+	1.73205i	0.333333	+	0.577350i
$10$		0			0
$11$	2.50000	−	4.33013i	0.753778	−	1.30558i	−0.192201	−	0.981356i	$-0.561563\pi$
$11$	2.50000	−	4.33013i	0.753778	−	1.30558i	0.945979	−	0.324227i	$-0.105104\pi$
$12$		0			0
$13$	−5.00000			−1.38675			−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000			−1.38675			−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.500000	−	0.866025i	0.121268	−	0.210042i
$17$	0.500000	−	0.866025i	0.121268	−	0.210042i
$18$		0			0
$19$	3.00000	+	5.19615i	0.688247	+	1.19208i	0.972404	+	0.233301i	$0.0749529\pi$
$19$	3.00000	+	5.19615i	0.688247	+	1.19208i	−0.284157	+	0.958778i	$0.591714\pi$
$20$		0			0
$21$	−0.500000	−	2.59808i	−0.109109	−	0.566947i
$22$		0			0
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	0.578610	−	0.815604i	$-0.303595\pi$
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	−0.995639	+	0.0932891i	$0.970262\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$	5.00000			0.962250
$28$		0			0
$29$	4.00000			0.742781			0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000			0.742781			0.371391	+	0.928477i	$0.378881\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$		0			0
$33$	−2.50000	−	4.33013i	−0.435194	−	0.753778i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	−0.981236	−	0.192809i	$-0.938240\pi$
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	0.323640	−	0.946180i	$-0.395093\pi$
$38$		0			0
$39$	−2.50000	+	4.33013i	−0.400320	+	0.693375i
$40$		0			0
$41$	4.00000			0.624695			0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000			0.624695			0.312348	+	0.949968i	$0.398885\pi$
$42$		0			0
$43$	−6.00000			−0.914991			−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000			−0.914991			−0.457496	+	0.889212i	$0.651253\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	0.997503	−	0.0706177i	$-0.0224970\pi$
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	−0.559908	+	0.828554i	$0.689164\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$	−0.500000	−	0.866025i	−0.0700140	−	0.121268i
$52$		0			0
$53$	−5.50000	+	9.52628i	−0.755483	+	1.30854i	0.189651	+	0.981852i	$0.439264\pi$
$53$	−5.50000	+	9.52628i	−0.755483	+	1.30854i	−0.945134	+	0.326683i	$0.894069\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.00000			0.794719
$58$		0			0
$59$	−5.00000	+	8.66025i	−0.650945	+	1.12747i	0.331949	+	0.943297i	$0.392294\pi$
$59$	−5.00000	+	8.66025i	−0.650945	+	1.12747i	−0.982894	+	0.184172i	$0.941040\pi$
$60$		0			0
$61$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$61$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$62$		0			0
$63$	5.00000	+	1.73205i	0.629941	+	0.218218i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−5.00000	+	8.66025i	−0.610847	+	1.05802i	0.380251	+	0.924883i	$0.375838\pi$
$67$	−5.00000	+	8.66025i	−0.610847	+	1.05802i	−0.991098	+	0.133135i	$0.957496\pi$
$68$		0			0
$69$	−4.00000			−0.481543
$70$		0			0
$71$	9.00000			1.06810			0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000			1.06810			0.534052	+	0.845452i	$0.320669\pi$
$72$		0			0
$73$	−2.00000	+	3.46410i	−0.234082	+	0.405442i	−0.959006	−	0.283387i	$-0.908542\pi$
$73$	−2.00000	+	3.46410i	−0.234082	+	0.405442i	0.724923	+	0.688830i	$0.241875\pi$
$74$		0			0
$75$	−2.50000	−	4.33013i	−0.288675	−	0.500000i
$76$		0			0
$77$	−2.50000	−	12.9904i	−0.284901	−	1.48039i
$78$		0			0
$79$	−4.50000	−	7.79423i	−0.506290	−	0.876919i	−0.999974	−	0.00727784i	$-0.997683\pi$
$79$	−4.50000	−	7.79423i	−0.506290	−	0.876919i	0.493684	−	0.869641i	$-0.335650\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	4.00000			0.439057			0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000			0.439057			0.219529	+	0.975606i	$0.429548\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.00000	−	3.46410i	0.214423	−	0.371391i
$88$		0			0
$89$	7.50000	+	12.9904i	0.794998	+	1.37698i	0.922840	+	0.385183i	$0.125862\pi$
$89$	7.50000	+	12.9904i	0.794998	+	1.37698i	−0.127842	+	0.991795i	$0.540805\pi$
$90$		0			0
$91$	−10.0000	+	8.66025i	−1.04828	+	0.907841i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−18.0000			−1.82762			−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000			−1.82762			−0.913812	+	0.406138i	$0.866875\pi$
$98$		0			0
$99$	10.0000			1.00504
$100$		0			0
$101$	−1.00000	+	1.73205i	−0.0995037	+	0.172345i	−0.911479	−	0.411346i	$-0.865059\pi$
$101$	−1.00000	+	1.73205i	−0.0995037	+	0.172345i	0.811976	+	0.583691i	$0.198392\pi$
$102$		0			0
$103$	8.00000	+	13.8564i	0.788263	+	1.36531i	0.927030	+	0.374987i	$0.122353\pi$
$103$	8.00000	+	13.8564i	0.788263	+	1.36531i	−0.138767	+	0.990325i	$0.544314\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.50000	+	14.7224i	0.821726	+	1.42327i	0.904396	+	0.426694i	$0.140322\pi$
$107$	8.50000	+	14.7224i	0.821726	+	1.42327i	−0.0826699	+	0.996577i	$0.526345\pi$
$108$		0			0
$109$	−8.00000	+	13.8564i	−0.766261	+	1.32720i	0.173316	+	0.984866i	$0.444552\pi$
$109$	−8.00000	+	13.8564i	−0.766261	+	1.32720i	−0.939577	+	0.342337i	$0.888782\pi$
$110$		0			0
$111$	−8.00000			−0.759326
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−5.00000	−	8.66025i	−0.462250	−	0.800641i
$118$		0			0
$119$	−0.500000	−	2.59808i	−0.0458349	−	0.238165i
$120$		0			0
$121$	−7.00000	−	12.1244i	−0.636364	−	1.10221i
$122$		0			0
$123$	2.00000	−	3.46410i	0.180334	−	0.312348i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−4.00000			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$		0			0
$129$	−3.00000	+	5.19615i	−0.264135	+	0.457496i
$130$		0			0
$131$	2.00000	+	3.46410i	0.174741	+	0.302660i	0.940072	−	0.340977i	$-0.110758\pi$
$131$	2.00000	+	3.46410i	0.174741	+	0.302660i	−0.765331	+	0.643637i	$0.777425\pi$
$132$		0			0
$133$	15.0000	+	5.19615i	1.30066	+	0.450564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.50000	−	12.9904i	0.640768	−	1.10984i	−0.344493	−	0.938789i	$-0.611949\pi$
$137$	7.50000	−	12.9904i	0.640768	−	1.10984i	0.985262	−	0.171054i	$-0.0547174\pi$
$138$		0			0
$139$	−7.00000			−0.593732			−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000			−0.593732			−0.296866	+	0.954919i	$0.595942\pi$
$140$		0			0
$141$	6.00000			0.505291
$142$		0			0
$143$	−12.5000	+	21.6506i	−1.04530	+	1.81052i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−5.50000	−	4.33013i	−0.453632	−	0.357143i
$148$		0			0
$149$	−10.5000	−	18.1865i	−0.860194	−	1.48990i	−0.871742	−	0.489966i	$-0.837009\pi$
$149$	−10.5000	−	18.1865i	−0.860194	−	1.48990i	0.0115483	−	0.999933i	$-0.496324\pi$
$150$		0			0
$151$	1.00000	−	1.73205i	0.0813788	−	0.140952i	−0.822464	−	0.568818i	$-0.807401\pi$
$151$	1.00000	−	1.73205i	0.0813788	−	0.140952i	0.903842	+	0.427865i	$0.140734\pi$
$152$		0			0
$153$	2.00000			0.161690
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−8.50000	+	14.7224i	−0.678374	+	1.17498i	0.297097	+	0.954847i	$0.403982\pi$
$157$	−8.50000	+	14.7224i	−0.678374	+	1.17498i	−0.975470	+	0.220131i	$0.929352\pi$
$158$		0			0
$159$	5.50000	+	9.52628i	0.436178	+	0.755483i
$160$		0			0
$161$	−10.0000	−	3.46410i	−0.788110	−	0.273009i
$162$		0			0
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	0.777007	−	0.629492i	$-0.216737\pi$
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	−0.933659	+	0.358162i	$0.883403\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−21.0000			−1.62503			−0.812514	−	0.582941i	$-0.801902\pi$
$167$	−21.0000			−1.62503			−0.812514	+	0.582941i	$0.801902\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$	−6.00000	+	10.3923i	−0.458831	+	0.794719i
$172$		0			0
$173$	−9.00000	−	15.5885i	−0.684257	−	1.18517i	−0.973670	−	0.227964i	$-0.926793\pi$
$173$	−9.00000	−	15.5885i	−0.684257	−	1.18517i	0.289412	−	0.957205i	$-0.406540\pi$
$174$		0			0
$175$	−2.50000	−	12.9904i	−0.188982	−	0.981981i
$176$		0			0
$177$	5.00000	+	8.66025i	0.375823	+	0.650945i
$178$		0			0
$179$	12.0000	−	20.7846i	0.896922	−	1.55351i	0.0655145	−	0.997852i	$-0.479131\pi$
$179$	12.0000	−	20.7846i	0.896922	−	1.55351i	0.831408	−	0.555663i	$-0.187536\pi$
$180$		0			0
$181$	6.00000			0.445976			0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000			0.445976			0.222988	+	0.974821i	$0.428419\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.50000	−	4.33013i	−0.182818	−	0.316650i
$188$		0			0
$189$	10.0000	−	8.66025i	0.727393	−	0.629941i
$190$		0			0
$191$	6.00000	+	10.3923i	0.434145	+	0.751961i	0.997225	−	0.0744412i	$-0.0237173\pi$
$191$	6.00000	+	10.3923i	0.434145	+	0.751961i	−0.563081	+	0.826402i	$0.690384\pi$
$192$		0			0
$193$	−5.00000	+	8.66025i	−0.359908	+	0.623379i	−0.987945	−	0.154805i	$-0.950525\pi$
$193$	−5.00000	+	8.66025i	−0.359908	+	0.623379i	0.628037	+	0.778183i	$0.283859\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$	0.500000	−	0.866025i	0.0354441	−	0.0613909i	−0.847759	−	0.530381i	$-0.822049\pi$
$199$	0.500000	−	0.866025i	0.0354441	−	0.0613909i	0.883203	+	0.468990i	$0.155382\pi$
$200$		0			0
$201$	5.00000	+	8.66025i	0.352673	+	0.610847i
$202$		0			0
$203$	8.00000	−	6.92820i	0.561490	−	0.486265i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	4.00000	−	6.92820i	0.278019	−	0.481543i
$208$		0			0
$209$	30.0000			2.07514
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$	4.50000	−	7.79423i	0.308335	−	0.534052i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	2.00000	+	3.46410i	0.135147	+	0.234082i
$220$		0			0
$221$	−2.50000	+	4.33013i	−0.168168	+	0.291276i
$222$		0			0
$223$	18.0000			1.20537			0.602685	−	0.797980i	$-0.294098\pi$
$223$	18.0000			1.20537			0.602685	+	0.797980i	$0.294098\pi$
$224$		0			0
$225$	10.0000			0.666667
$226$		0			0
$227$	4.50000	−	7.79423i	0.298675	−	0.517321i	−0.677158	−	0.735838i	$-0.736789\pi$
$227$	4.50000	−	7.79423i	0.298675	−	0.517321i	0.975833	+	0.218517i	$0.0701218\pi$
$228$		0			0
$229$	−11.0000	−	19.0526i	−0.726900	−	1.25903i	−0.958187	−	0.286143i	$-0.907627\pi$
$229$	−11.0000	−	19.0526i	−0.726900	−	1.25903i	0.231287	−	0.972886i	$-0.425707\pi$
$230$		0			0
$231$	−12.5000	−	4.33013i	−0.822440	−	0.284901i
$232$		0			0
$233$	6.00000	+	10.3923i	0.393073	+	0.680823i	0.992853	−	0.119342i	$-0.0380786\pi$
$233$	6.00000	+	10.3923i	0.393073	+	0.680823i	−0.599780	+	0.800165i	$0.704745\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−9.00000			−0.584613
$238$		0			0
$239$	18.0000			1.16432			0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000			1.16432			0.582162	+	0.813073i	$0.302207\pi$
$240$		0			0
$241$	10.0000	−	17.3205i	0.644157	−	1.11571i	−0.340339	−	0.940303i	$-0.610542\pi$
$241$	10.0000	−	17.3205i	0.644157	−	1.11571i	0.984496	−	0.175409i	$-0.0561248\pi$
$242$		0			0
$243$	8.00000	+	13.8564i	0.513200	+	0.888889i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−15.0000	−	25.9808i	−0.954427	−	1.65312i
$248$		0			0
$249$	2.00000	−	3.46410i	0.126745	−	0.219529i
$250$		0			0
$251$	−4.00000			−0.252478			−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000			−0.252478			−0.126239	+	0.992000i	$0.540291\pi$
$252$		0			0
$253$	−20.0000			−1.25739
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.50000	−	2.59808i	−0.0935674	−	0.162064i	0.815442	−	0.578838i	$-0.196494\pi$
$257$	−1.50000	−	2.59808i	−0.0935674	−	0.162064i	−0.909010	+	0.416775i	$0.863160\pi$
$258$		0			0
$259$	−20.0000	−	6.92820i	−1.24274	−	0.430498i
$260$		0			0
$261$	4.00000	+	6.92820i	0.247594	+	0.428845i
$262$		0			0
$263$	4.00000	−	6.92820i	0.246651	−	0.427211i	−0.715944	−	0.698158i	$-0.754003\pi$
$263$	4.00000	−	6.92820i	0.246651	−	0.427211i	0.962594	+	0.270947i	$0.0873367\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	15.0000			0.917985
$268$		0			0
$269$	−3.00000	+	5.19615i	−0.182913	+	0.316815i	−0.942871	−	0.333157i	$-0.891886\pi$
$269$	−3.00000	+	5.19615i	−0.182913	+	0.316815i	0.759958	+	0.649972i	$0.225219\pi$
$270$		0			0
$271$	−12.0000	−	20.7846i	−0.728948	−	1.26258i	−0.957328	−	0.289003i	$-0.906676\pi$
$271$	−12.0000	−	20.7846i	−0.728948	−	1.26258i	0.228380	−	0.973572i	$-0.426657\pi$
$272$		0			0
$273$	2.50000	+	12.9904i	0.151307	+	0.786214i
$274$		0			0
$275$	−12.5000	−	21.6506i	−0.753778	−	1.30558i
$276$		0			0
$277$	−12.0000	+	20.7846i	−0.721010	+	1.24883i	0.239585	+	0.970875i	$0.422989\pi$
$277$	−12.0000	+	20.7846i	−0.721010	+	1.24883i	−0.960595	+	0.277951i	$0.910345\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.00000			0.298275			0.149137	−	0.988816i	$-0.452350\pi$
$281$	5.00000			0.298275			0.149137	+	0.988816i	$0.452350\pi$
$282$		0			0
$283$	−4.50000	+	7.79423i	−0.267497	+	0.463319i	−0.968215	−	0.250120i	$-0.919530\pi$
$283$	−4.50000	+	7.79423i	−0.267497	+	0.463319i	0.700718	+	0.713439i	$0.252863\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.00000	−	6.92820i	0.472225	−	0.408959i
$288$		0			0
$289$	−0.500000	−	0.866025i	−0.0294118	−	0.0509427i
$290$		0			0
$291$	−9.00000	+	15.5885i	−0.527589	+	0.913812i
$292$		0			0
$293$	9.00000			0.525786			0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000			0.525786			0.262893	+	0.964825i	$0.415323\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	12.5000	−	21.6506i	0.725324	−	1.25630i
$298$		0			0
$299$	10.0000	+	17.3205i	0.578315	+	1.00167i
$300$		0			0
$301$	−12.0000	+	10.3923i	−0.691669	+	0.599002i
$302$		0			0
$303$	1.00000	+	1.73205i	0.0574485	+	0.0995037i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	16.0000			0.913168			0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000			0.913168			0.456584	+	0.889680i	$0.349073\pi$
$308$		0			0
$309$	16.0000			0.910208
$310$		0			0
$311$	1.50000	−	2.59808i	0.0850572	−	0.147323i	−0.820358	−	0.571850i	$-0.806226\pi$
$311$	1.50000	−	2.59808i	0.0850572	−	0.147323i	0.905416	+	0.424526i	$0.139559\pi$
$312$		0			0
$313$	−3.00000	−	5.19615i	−0.169570	−	0.293704i	0.768699	−	0.639611i	$-0.220905\pi$
$313$	−3.00000	−	5.19615i	−0.169570	−	0.293704i	−0.938269	+	0.345907i	$0.887571\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	−0.999980	−	0.00635137i	$-0.997978\pi$
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	0.494489	−	0.869184i	$-0.335355\pi$
$318$		0			0
$319$	10.0000	−	17.3205i	0.559893	−	0.969762i
$320$		0			0
$321$	17.0000			0.948847
$322$		0			0
$323$	6.00000			0.333849
$324$		0			0
$325$	−12.5000	+	21.6506i	−0.693375	+	1.20096i
$326$		0			0
$327$	8.00000	+	13.8564i	0.442401	+	0.766261i
$328$		0			0
$329$	15.0000	+	5.19615i	0.826977	+	0.286473i
$330$		0			0
$331$	−5.00000	−	8.66025i	−0.274825	−	0.476011i	0.695266	−	0.718752i	$-0.255287\pi$
$331$	−5.00000	−	8.66025i	−0.274825	−	0.476011i	−0.970091	+	0.242742i	$0.921953\pi$
$332$		0			0
$333$	8.00000	−	13.8564i	0.438397	−	0.759326i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	8.00000			0.435788			0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000			0.435788			0.217894	+	0.975972i	$0.430081\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	−0.658824	−	0.752297i	$-0.728946\pi$
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	0.980921	+	0.194409i	$0.0622790\pi$
$348$		0			0
$349$	−34.0000			−1.81998			−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000			−1.81998			−0.909989	+	0.414632i	$0.863910\pi$
$350$		0			0
$351$	−25.0000			−1.33440
$352$		0			0
$353$	−5.50000	+	9.52628i	−0.292735	+	0.507033i	−0.974456	−	0.224580i	$-0.927899\pi$
$353$	−5.50000	+	9.52628i	−0.292735	+	0.507033i	0.681720	+	0.731613i	$0.261232\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−2.50000	−	0.866025i	−0.132314	−	0.0458349i
$358$		0			0
$359$	13.0000	+	22.5167i	0.686114	+	1.18838i	0.973085	+	0.230445i	$0.0740181\pi$
$359$	13.0000	+	22.5167i	0.686114	+	1.18838i	−0.286972	+	0.957939i	$0.592649\pi$
$360$		0			0
$361$	−8.50000	+	14.7224i	−0.447368	+	0.774865i
$362$		0			0
$363$	−14.0000			−0.734809
$364$		0			0
$365$		0			0
$366$		0			0
$367$	7.50000	−	12.9904i	0.391497	−	0.678092i	−0.601150	−	0.799136i	$-0.705291\pi$
$367$	7.50000	−	12.9904i	0.391497	−	0.678092i	0.992647	+	0.121044i	$0.0386241\pi$
$368$		0			0
$369$	4.00000	+	6.92820i	0.208232	+	0.360668i
$370$		0			0
$371$	5.50000	+	28.5788i	0.285546	+	1.48374i
$372$		0			0
$373$	0.500000	+	0.866025i	0.0258890	+	0.0448411i	0.878680	−	0.477412i	$-0.158425\pi$
$373$	0.500000	+	0.866025i	0.0258890	+	0.0448411i	−0.852791	+	0.522253i	$0.825092\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−20.0000			−1.03005
$378$		0			0
$379$	−25.0000			−1.28416			−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000			−1.28416			−0.642082	+	0.766636i	$0.721929\pi$
$380$		0			0
$381$	−2.00000	+	3.46410i	−0.102463	+	0.177471i
$382$		0			0
$383$	−9.00000	−	15.5885i	−0.459879	−	0.796533i	0.539076	−	0.842257i	$-0.318774\pi$
$383$	−9.00000	−	15.5885i	−0.459879	−	0.796533i	−0.998954	+	0.0457244i	$0.985440\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−6.00000	−	10.3923i	−0.304997	−	0.528271i
$388$		0			0
$389$	−10.5000	+	18.1865i	−0.532371	+	0.922094i	0.466915	+	0.884302i	$0.345366\pi$
$389$	−10.5000	+	18.1865i	−0.532371	+	0.922094i	−0.999286	+	0.0377914i	$0.987968\pi$
$390$		0			0
$391$	−4.00000			−0.202289
$392$		0			0
$393$	4.00000			0.201773
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−7.00000	−	12.1244i	−0.351320	−	0.608504i	0.635161	−	0.772380i	$-0.280934\pi$
$397$	−7.00000	−	12.1244i	−0.351320	−	0.608504i	−0.986481	+	0.163876i	$0.947600\pi$
$398$		0			0
$399$	12.0000	−	10.3923i	0.600751	−	0.520266i
$400$		0			0
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	0.948272	+	0.317460i	$0.102830\pi$
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	−0.199207	+	0.979957i	$0.563837\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−40.0000			−1.98273
$408$		0			0
$409$	−17.5000	+	30.3109i	−0.865319	+	1.49878i	0.00141047	+	0.999999i	$0.499551\pi$
$409$	−17.5000	+	30.3109i	−0.865319	+	1.49878i	−0.866730	+	0.498778i	$0.833782\pi$
$410$		0			0
$411$	−7.50000	−	12.9904i	−0.369948	−	0.640768i
$412$		0			0
$413$	5.00000	+	25.9808i	0.246034	+	1.27843i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3.50000	+	6.06218i	−0.171396	+	0.296866i
$418$		0			0
$419$	−5.00000			−0.244266			−0.122133	−	0.992514i	$-0.538973\pi$
$419$	−5.00000			−0.244266			−0.122133	+	0.992514i	$0.538973\pi$
$420$		0			0
$421$	6.00000			0.292422			0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000			0.292422			0.146211	+	0.989253i	$0.453292\pi$
$422$		0			0
$423$	−6.00000	+	10.3923i	−0.291730	+	0.505291i
$424$		0			0
$425$	−2.50000	−	4.33013i	−0.121268	−	0.210042i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	12.5000	+	21.6506i	0.603506	+	1.04530i
$430$		0			0
$431$	8.50000	−	14.7224i	0.409431	−	0.709155i	−0.585395	−	0.810748i	$-0.699061\pi$
$431$	8.50000	−	14.7224i	0.409431	−	0.709155i	0.994826	+	0.101593i	$0.0323941\pi$
$432$		0			0
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.0000	−	20.7846i	0.574038	−	0.994263i
$438$		0			0
$439$	11.5000	+	19.9186i	0.548865	+	0.950662i	0.998353	+	0.0573756i	$0.0182733\pi$
$439$	11.5000	+	19.9186i	0.548865	+	0.950662i	−0.449488	+	0.893287i	$0.648393\pi$
$440$		0			0
$441$	13.0000	−	5.19615i	0.619048	−	0.247436i
$442$		0			0
$443$	−1.00000	−	1.73205i	−0.0475114	−	0.0822922i	0.841292	−	0.540581i	$-0.181796\pi$
$443$	−1.00000	−	1.73205i	−0.0475114	−	0.0822922i	−0.888803	+	0.458289i	$0.848462\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−21.0000			−0.993266
$448$		0			0
$449$	14.0000			0.660701			0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000			0.660701			0.330350	+	0.943858i	$0.392833\pi$
$450$		0			0
$451$	10.0000	−	17.3205i	0.470882	−	0.815591i
$452$		0			0
$453$	−1.00000	−	1.73205i	−0.0469841	−	0.0813788i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−7.00000	−	12.1244i	−0.327446	−	0.567153i	0.654558	−	0.756012i	$-0.272855\pi$
$457$	−7.00000	−	12.1244i	−0.327446	−	0.567153i	−0.982004	+	0.188858i	$0.939521\pi$
$458$		0			0
$459$	2.50000	−	4.33013i	0.116690	−	0.202113i
$460$		0			0
$461$	−25.0000			−1.16437			−0.582183	−	0.813058i	$-0.697801\pi$
$461$	−25.0000			−1.16437			−0.582183	+	0.813058i	$0.697801\pi$
$462$		0			0
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.00000	−	13.8564i	−0.370196	−	0.641198i	0.619400	−	0.785076i	$-0.287376\pi$
$467$	−8.00000	−	13.8564i	−0.370196	−	0.641198i	−0.989595	+	0.143878i	$0.954043\pi$
$468$		0			0
$469$	5.00000	+	25.9808i	0.230879	+	1.19968i
$470$		0			0
$471$	8.50000	+	14.7224i	0.391659	+	0.678374i
$472$		0			0
$473$	−15.0000	+	25.9808i	−0.689701	+	1.19460i
$474$		0			0
$475$	30.0000			1.37649
$476$		0			0
$477$	−22.0000			−1.00731
$478$		0			0
$479$	−12.0000	+	20.7846i	−0.548294	+	0.949673i	0.450098	+	0.892979i	$0.351389\pi$
$479$	−12.0000	+	20.7846i	−0.548294	+	0.949673i	−0.998392	+	0.0566937i	$0.981944\pi$
$480$		0			0
$481$	20.0000	+	34.6410i	0.911922	+	1.57949i
$482$		0			0
$483$	−8.00000	+	6.92820i	−0.364013	+	0.315244i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	9.50000	−	16.4545i	0.430486	−	0.745624i	−0.566429	−	0.824110i	$-0.691675\pi$
$487$	9.50000	−	16.4545i	0.430486	−	0.745624i	0.996915	+	0.0784867i	$0.0250088\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$	−22.0000			−0.992846			−0.496423	−	0.868081i	$-0.665354\pi$
$491$	−22.0000			−0.992846			−0.496423	+	0.868081i	$0.665354\pi$
$492$		0			0
$493$	2.00000	−	3.46410i	0.0900755	−	0.156015i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	18.0000	−	15.5885i	0.807410	−	0.699238i
$498$		0			0
$499$	7.50000	+	12.9904i	0.335746	+	0.581529i	0.983628	−	0.180212i	$-0.0576783\pi$
$499$	7.50000	+	12.9904i	0.335746	+	0.581529i	−0.647882	+	0.761741i	$0.724345\pi$
$500$		0			0
$501$	−10.5000	+	18.1865i	−0.469105	+	0.812514i
$502$		0			0
$503$	−13.0000			−0.579641			−0.289821	−	0.957081i	$-0.593596\pi$
$503$	−13.0000			−0.579641			−0.289821	+	0.957081i	$0.593596\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	6.00000	−	10.3923i	0.266469	−	0.461538i
$508$		0			0
$509$	13.0000	+	22.5167i	0.576215	+	0.998033i	0.995908	+	0.0903676i	$0.0288042\pi$
$509$	13.0000	+	22.5167i	0.576215	+	0.998033i	−0.419694	+	0.907666i	$0.637862\pi$
$510$		0			0
$511$	2.00000	+	10.3923i	0.0884748	+	0.459728i
$512$		0			0
$513$	15.0000	+	25.9808i	0.662266	+	1.14708i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	30.0000			1.31940
$518$		0			0
$519$	−18.0000			−0.790112
$520$		0			0
$521$	14.0000	−	24.2487i	0.613351	−	1.06236i	−0.377320	−	0.926083i	$-0.623154\pi$
$521$	14.0000	−	24.2487i	0.613351	−	1.06236i	0.990671	−	0.136272i	$-0.0435123\pi$
$522$		0			0
$523$	−3.00000	−	5.19615i	−0.131181	−	0.227212i	0.792951	−	0.609285i	$-0.208544\pi$
$523$	−3.00000	−	5.19615i	−0.131181	−	0.227212i	−0.924132	+	0.382073i	$0.875210\pi$
$524$		0			0
$525$	−12.5000	−	4.33013i	−0.545545	−	0.188982i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$	−20.0000			−0.867926
$532$		0			0
$533$	−20.0000			−0.866296
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−12.0000	−	20.7846i	−0.517838	−	0.896922i
$538$		0			0
$539$	−27.5000	−	21.6506i	−1.18451	−	0.932559i
$540$		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
$543$	3.00000	−	5.19615i	0.128742	−	0.222988i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−43.0000			−1.83855			−0.919274	−	0.393619i	$-0.871223\pi$
$547$	−43.0000			−1.83855			−0.919274	+	0.393619i	$0.871223\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.0000	+	20.7846i	0.511217	+	0.885454i
$552$		0			0
$553$	−22.5000	−	7.79423i	−0.956797	−	0.331444i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8.50000	−	14.7224i	0.360157	−	0.623809i	−0.627830	−	0.778351i	$-0.716057\pi$
$557$	8.50000	−	14.7224i	0.360157	−	0.623809i	0.987986	+	0.154541i	$0.0493899\pi$
$558$		0			0
$559$	30.0000			1.26886
$560$		0			0
$561$	−5.00000			−0.211100
$562$		0			0
$563$	13.0000	−	22.5167i	0.547885	−	0.948964i	−0.450535	−	0.892759i	$-0.648767\pi$
$563$	13.0000	−	22.5167i	0.547885	−	0.948964i	0.998419	−	0.0562051i	$-0.0179001\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.500000	+	2.59808i	0.0209980	+	0.109109i
$568$		0			0
$569$	−0.500000	−	0.866025i	−0.0209611	−	0.0363057i	0.855355	−	0.518043i	$-0.173339\pi$
$569$	−0.500000	−	0.866025i	−0.0209611	−	0.0363057i	−0.876316	+	0.481737i	$0.840006\pi$
$570$		0			0
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	−0.821138	−	0.570730i	$-0.806660\pi$
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	0.904835	+	0.425762i	$0.139994\pi$
$572$		0			0
$573$	12.0000			0.501307
$574$		0			0
$575$	−20.0000			−0.834058
$576$		0			0
$577$	9.50000	−	16.4545i	0.395490	−	0.685009i	−0.597673	−	0.801740i	$-0.703908\pi$
$577$	9.50000	−	16.4545i	0.395490	−	0.685009i	0.993164	+	0.116731i	$0.0372414\pi$
$578$		0			0
$579$	5.00000	+	8.66025i	0.207793	+	0.359908i
$580$		0			0
$581$	8.00000	−	6.92820i	0.331896	−	0.287430i
$582$		0			0
$583$	27.5000	+	47.6314i	1.13893	+	1.97269i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	34.0000			1.40333			0.701665	−	0.712507i	$-0.252440\pi$
$587$	34.0000			1.40333			0.701665	+	0.712507i	$0.252440\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	1.00000	−	1.73205i	0.0411345	−	0.0712470i
$592$		0			0
$593$	−22.5000	−	38.9711i	−0.923964	−	1.60035i	−0.793219	−	0.608937i	$-0.791596\pi$
$593$	−22.5000	−	38.9711i	−0.923964	−	1.60035i	−0.130746	−	0.991416i	$-0.541737\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.500000	−	0.866025i	−0.0204636	−	0.0354441i
$598$		0			0
$599$	−15.0000	+	25.9808i	−0.612883	+	1.06155i	0.377869	+	0.925859i	$0.376657\pi$
$599$	−15.0000	+	25.9808i	−0.612883	+	1.06155i	−0.990752	+	0.135686i	$0.956676\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$	−20.0000			−0.814463
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	0.786212	−	0.617957i	$-0.212039\pi$
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	−0.928272	+	0.371901i	$0.878706\pi$
$608$		0			0
$609$	−2.00000	−	10.3923i	−0.0810441	−	0.421117i
$610$		0			0
$611$	−15.0000	−	25.9808i	−0.606835	−	1.05107i
$612$		0			0
$613$	−8.50000	+	14.7224i	−0.343312	+	0.594633i	−0.985046	−	0.172294i	$-0.944882\pi$
$613$	−8.50000	+	14.7224i	−0.343312	+	0.594633i	0.641734	+	0.766927i	$0.278215\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	16.0000			0.644136			0.322068	−	0.946717i	$-0.395622\pi$
$617$	16.0000			0.644136			0.322068	+	0.946717i	$0.395622\pi$
$618$		0			0
$619$	−8.50000	+	14.7224i	−0.341644	+	0.591744i	−0.984738	−	0.174042i	$-0.944317\pi$
$619$	−8.50000	+	14.7224i	−0.341644	+	0.591744i	0.643094	+	0.765787i	$0.277650\pi$
$620$		0			0
$621$	−10.0000	−	17.3205i	−0.401286	−	0.695048i
$622$		0			0
$623$	37.5000	+	12.9904i	1.50241	+	0.520449i
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$	15.0000	−	25.9808i	0.599042	−	1.03757i
$628$		0			0
$629$	−8.00000			−0.318981
$630$		0			0
$631$	18.0000			0.716569			0.358284	−	0.933613i	$-0.383362\pi$
$631$	18.0000			0.716569			0.358284	+	0.933613i	$0.383362\pi$
$632$		0			0
$633$	10.0000	−	17.3205i	0.397464	−	0.688428i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−5.00000	+	34.6410i	−0.198107	+	1.37253i
$638$		0			0
$639$	9.00000	+	15.5885i	0.356034	+	0.616670i
$640$		0			0
$641$	−11.0000	+	19.0526i	−0.434474	+	0.752531i	−0.997253	−	0.0740768i	$-0.976399\pi$
$641$	−11.0000	+	19.0526i	−0.434474	+	0.752531i	0.562779	+	0.826608i	$0.309732\pi$
$642$		0			0
$643$	41.0000			1.61688			0.808441	−	0.588577i	$-0.200312\pi$
$643$	41.0000			1.61688			0.808441	+	0.588577i	$0.200312\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−14.0000	+	24.2487i	−0.550397	+	0.953315i	0.447849	+	0.894109i	$0.352190\pi$
$647$	−14.0000	+	24.2487i	−0.550397	+	0.953315i	−0.998246	+	0.0592060i	$0.981143\pi$
$648$		0			0
$649$	25.0000	+	43.3013i	0.981336	+	1.69972i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−21.0000	−	36.3731i	−0.821794	−	1.42339i	−0.904345	−	0.426801i	$-0.859640\pi$
$653$	−21.0000	−	36.3731i	−0.821794	−	1.42339i	0.0825519	−	0.996587i	$-0.473693\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−8.00000			−0.312110
$658$		0			0
$659$	−30.0000			−1.16863			−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000			−1.16863			−0.584317	+	0.811525i	$0.698638\pi$
$660$		0			0
$661$	11.0000	−	19.0526i	0.427850	−	0.741059i	−0.568831	−	0.822454i	$-0.692604\pi$
$661$	11.0000	−	19.0526i	0.427850	−	0.741059i	0.996682	+	0.0813955i	$0.0259377\pi$
$662$		0			0
$663$	2.50000	+	4.33013i	0.0970920	+	0.168168i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−8.00000	−	13.8564i	−0.309761	−	0.536522i
$668$		0			0
$669$	9.00000	−	15.5885i	0.347960	−	0.602685i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−24.0000			−0.925132			−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000			−0.925132			−0.462566	+	0.886585i	$0.653071\pi$
$674$		0			0
$675$	12.5000	−	21.6506i	0.481125	−	0.833333i
$676$		0			0
$677$	−6.00000	−	10.3923i	−0.230599	−	0.399409i	0.727386	−	0.686229i	$-0.240735\pi$
$677$	−6.00000	−	10.3923i	−0.230599	−	0.399409i	−0.957984	+	0.286820i	$0.907402\pi$
$678$		0			0
$679$	−36.0000	+	31.1769i	−1.38155	+	1.19646i
$680$		0			0
$681$	−4.50000	−	7.79423i	−0.172440	−	0.298675i
$682$		0			0
$683$	−6.50000	+	11.2583i	−0.248716	+	0.430788i	−0.963170	−	0.268894i	$-0.913342\pi$
$683$	−6.50000	+	11.2583i	−0.248716	+	0.430788i	0.714454	+	0.699682i	$0.246675\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−22.0000			−0.839352
$688$		0			0
$689$	27.5000	−	47.6314i	1.04767	−	1.81461i
$690$		0			0
$691$	−12.0000	−	20.7846i	−0.456502	−	0.790684i	0.542272	−	0.840203i	$-0.317564\pi$
$691$	−12.0000	−	20.7846i	−0.456502	−	0.790684i	−0.998773	+	0.0495194i	$0.984231\pi$
$692$		0			0
$693$	20.0000	−	17.3205i	0.759737	−	0.657952i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.00000	−	3.46410i	0.0757554	−	0.131212i
$698$		0			0
$699$	12.0000			0.453882
$700$		0			0
$701$	23.0000			0.868698			0.434349	−	0.900745i	$-0.356978\pi$
$701$	23.0000			0.868698			0.434349	+	0.900745i	$0.356978\pi$
$702$		0			0
$703$	24.0000	−	41.5692i	0.905177	−	1.56781i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.00000	+	5.19615i	0.0376089	+	0.195421i
$708$		0			0
$709$	5.00000	+	8.66025i	0.187779	+	0.325243i	0.944509	−	0.328484i	$-0.106538\pi$
$709$	5.00000	+	8.66025i	0.187779	+	0.325243i	−0.756730	+	0.653727i	$0.773204\pi$
$710$		0			0
$711$	9.00000	−	15.5885i	0.337526	−	0.584613i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	9.00000	−	15.5885i	0.336111	−	0.582162i
$718$		0			0
$719$	−17.5000	−	30.3109i	−0.652640	−	1.13041i	−0.982480	−	0.186369i	$-0.940328\pi$
$719$	−17.5000	−	30.3109i	−0.652640	−	1.13041i	0.329840	−	0.944037i	$-0.393005\pi$
$720$		0			0
$721$	40.0000	+	13.8564i	1.48968	+	0.516040i
$722$		0			0
$723$	−10.0000	−	17.3205i	−0.371904	−	0.644157i
$724$		0			0
$725$	10.0000	−	17.3205i	0.371391	−	0.643268i
$726$		0			0
$727$	26.0000			0.964287			0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000			0.964287			0.482143	+	0.876092i	$0.339858\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	−3.00000	+	5.19615i	−0.110959	+	0.192187i
$732$		0			0
$733$	2.50000	+	4.33013i	0.0923396	+	0.159937i	0.908495	−	0.417895i	$-0.137232\pi$
$733$	2.50000	+	4.33013i	0.0923396	+	0.159937i	−0.816156	+	0.577832i	$0.803899\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	25.0000	+	43.3013i	0.920887	+	1.59502i
$738$		0			0
$739$	9.00000	−	15.5885i	0.331070	−	0.573431i	−0.651652	−	0.758518i	$-0.725924\pi$
$739$	9.00000	−	15.5885i	0.331070	−	0.573431i	0.982722	+	0.185088i	$0.0592569\pi$
$740$		0			0
$741$	−30.0000			−1.10208
$742$		0			0
$743$	7.00000			0.256805			0.128403	−	0.991722i	$-0.459015\pi$
$743$	7.00000			0.256805			0.128403	+	0.991722i	$0.459015\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	4.00000	+	6.92820i	0.146352	+	0.253490i
$748$		0			0
$749$	42.5000	+	14.7224i	1.55292	+	0.537946i
$750$		0			0
$751$	−10.5000	−	18.1865i	−0.383150	−	0.663636i	0.608360	−	0.793661i	$-0.291828\pi$
$751$	−10.5000	−	18.1865i	−0.383150	−	0.663636i	−0.991511	+	0.130025i	$0.958494\pi$
$752$		0			0
$753$	−2.00000	+	3.46410i	−0.0728841	+	0.126239i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	33.0000			1.19941			0.599703	−	0.800223i	$-0.295286\pi$
$757$	33.0000			1.19941			0.599703	+	0.800223i	$0.295286\pi$
$758$		0			0
$759$	−10.0000	+	17.3205i	−0.362977	+	0.628695i
$760$		0			0
$761$	−6.50000	−	11.2583i	−0.235625	−	0.408114i	0.723829	−	0.689979i	$-0.242380\pi$
$761$	−6.50000	−	11.2583i	−0.235625	−	0.408114i	−0.959454	+	0.281865i	$0.909047\pi$
$762$		0			0
$763$	8.00000	+	41.5692i	0.289619	+	1.50491i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	25.0000	−	43.3013i	0.902698	−	1.56352i
$768$		0			0
$769$	−7.00000			−0.252426			−0.126213	−	0.992003i	$-0.540282\pi$
$769$	−7.00000			−0.252426			−0.126213	+	0.992003i	$0.540282\pi$
$770$		0			0
$771$	−3.00000			−0.108042
$772$		0			0
$773$	−0.500000	+	0.866025i	−0.0179838	+	0.0311488i	−0.874877	−	0.484345i	$-0.839058\pi$
$773$	−0.500000	+	0.866025i	−0.0179838	+	0.0311488i	0.856893	+	0.515494i	$0.172391\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−16.0000	+	13.8564i	−0.573997	+	0.497096i
$778$		0			0
$779$	12.0000	+	20.7846i	0.429945	+	0.744686i
$780$		0			0
$781$	22.5000	−	38.9711i	0.805113	−	1.39450i
$782$		0			0
$783$	20.0000			0.714742
$784$		0			0
$785$		0			0
$786$		0			0
$787$	22.0000	−	38.1051i	0.784215	−	1.35830i	−0.145251	−	0.989395i	$-0.546399\pi$
$787$	22.0000	−	38.1051i	0.784215	−	1.35830i	0.929467	−	0.368906i	$-0.120268\pi$
$788$		0			0
$789$	−4.00000	−	6.92820i	−0.142404	−	0.246651i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$

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:	\( N \)	\(=\)	\( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	476.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} +(2.00000 - 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{3} +(2.00000 - 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{9} +(2.50000 - 4.33013i) q^{11} -5.00000 q^{13} +(0.500000 - 0.866025i) q^{17} +(3.00000 + 5.19615i) q^{19} +(-0.500000 - 2.59808i) q^{21} +(-2.00000 - 3.46410i) q^{23} +(2.50000 - 4.33013i) q^{25} +5.00000 q^{27} +4.00000 q^{29} +(-2.50000 - 4.33013i) q^{33} +(-4.00000 - 6.92820i) q^{37} +(-2.50000 + 4.33013i) q^{39} +4.00000 q^{41} -6.00000 q^{43} +(3.00000 + 5.19615i) q^{47} +(1.00000 - 6.92820i) q^{49} +(-0.500000 - 0.866025i) q^{51} +(-5.50000 + 9.52628i) q^{53} +6.00000 q^{57} +(-5.00000 + 8.66025i) q^{59} +(5.00000 + 1.73205i) q^{63} +(-5.00000 + 8.66025i) q^{67} -4.00000 q^{69} +9.00000 q^{71} +(-2.00000 + 3.46410i) q^{73} +(-2.50000 - 4.33013i) q^{75} +(-2.50000 - 12.9904i) q^{77} +(-4.50000 - 7.79423i) q^{79} +(-0.500000 + 0.866025i) q^{81} +4.00000 q^{83} +(2.00000 - 3.46410i) q^{87} +(7.50000 + 12.9904i) q^{89} +(-10.0000 + 8.66025i) q^{91} -18.0000 q^{97} +10.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + q^3 + 4 * q^7 + 2 * q^9	\( 2 q + q^{3} + 4 q^{7} + 2 q^{9} + 5 q^{11} - 10 q^{13} + q^{17} + 6 q^{19} - q^{21} - 4 q^{23} + 5 q^{25} + 10 q^{27} + 8 q^{29} - 5 q^{33} - 8 q^{37} - 5 q^{39} + 8 q^{41} - 12 q^{43} + 6 q^{47} + 2 q^{49} - q^{51} - 11 q^{53} + 12 q^{57} - 10 q^{59} + 10 q^{63} - 10 q^{67} - 8 q^{69} + 18 q^{71} - 4 q^{73} - 5 q^{75} - 5 q^{77} - 9 q^{79} - q^{81} + 8 q^{83} + 4 q^{87} + 15 q^{89} - 20 q^{91} - 36 q^{97} + 20 q^{99}+O(q^{100}) \) 2 * q + q^3 + 4 * q^7 + 2 * q^9 + 5 * q^11 - 10 * q^13 + q^17 + 6 * q^19 - q^21 - 4 * q^23 + 5 * q^25 + 10 * q^27 + 8 * q^29 - 5 * q^33 - 8 * q^37 - 5 * q^39 + 8 * q^41 - 12 * q^43 + 6 * q^47 + 2 * q^49 - q^51 - 11 * q^53 + 12 * q^57 - 10 * q^59 + 10 * q^63 - 10 * q^67 - 8 * q^69 + 18 * q^71 - 4 * q^73 - 5 * q^75 - 5 * q^77 - 9 * q^79 - q^81 + 8 * q^83 + 4 * q^87 + 15 * q^89 - 20 * q^91 - 36 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

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