LMFDB - Embedded newform 476.2.i.b.205.1

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			205.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	476.205
Dual form			476.2.i.b.137.1

$q$-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 $\left(\mathbb{Z}/476\mathbb{Z}\right)^\times$.

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

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	0.973494	−	0.228714i	$-0.0734519\pi$
$3$	0.500000	+	0.866025i	0.288675	+	0.500000i	−0.684819	+	0.728714i	$0.740119\pi$
$4$		0			0
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		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	0.945979	+	0.324227i	$0.105104\pi$
$11$	2.50000	+	4.33013i	0.753778	+	1.30558i	−0.192201	+	0.981356i	$0.561563\pi$
$12$		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	−0.284157	−	0.958778i	$-0.591714\pi$
$19$	3.00000	−	5.19615i	0.688247	−	1.19208i	0.972404	−	0.233301i	$-0.0749529\pi$
$20$		0			0
$21$	−0.500000	+	2.59808i	−0.109109	+	0.566947i
$22$		0			0
$23$	−2.00000	+	3.46410i	−0.417029	+	0.722315i	−0.995639	−	0.0932891i	$-0.970262\pi$
$23$	−2.00000	+	3.46410i	−0.417029	+	0.722315i	0.578610	+	0.815604i	$0.303595\pi$
$24$		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	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$		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	0.323640	+	0.946180i	$0.395093\pi$
$37$	−4.00000	+	6.92820i	−0.657596	+	1.13899i	−0.981236	+	0.192809i	$0.938240\pi$
$38$		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	−0.559908	−	0.828554i	$-0.689164\pi$
$47$	3.00000	−	5.19615i	0.437595	−	0.757937i	0.997503	+	0.0706177i	$0.0224970\pi$
$48$		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	−0.945134	−	0.326683i	$-0.894069\pi$
$53$	−5.50000	−	9.52628i	−0.755483	−	1.30854i	0.189651	−	0.981852i	$-0.439264\pi$
$54$		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	−0.982894	−	0.184172i	$-0.941040\pi$
$59$	−5.00000	−	8.66025i	−0.650945	−	1.12747i	0.331949	−	0.943297i	$-0.392294\pi$
$60$		0			0
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		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	−0.991098	−	0.133135i	$-0.957496\pi$
$67$	−5.00000	−	8.66025i	−0.610847	−	1.05802i	0.380251	−	0.924883i	$-0.375838\pi$
$68$		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	0.724923	−	0.688830i	$-0.241875\pi$
$73$	−2.00000	−	3.46410i	−0.234082	−	0.405442i	−0.959006	+	0.283387i	$0.908542\pi$
$74$		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	0.493684	+	0.869641i	$0.335650\pi$
$79$	−4.50000	+	7.79423i	−0.506290	+	0.876919i	−0.999974	+	0.00727784i	$0.997683\pi$
$80$		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	−0.127842	−	0.991795i	$-0.540805\pi$
$89$	7.50000	−	12.9904i	0.794998	−	1.37698i	0.922840	−	0.385183i	$-0.125862\pi$
$90$		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	0.811976	−	0.583691i	$-0.198392\pi$
$101$	−1.00000	−	1.73205i	−0.0995037	−	0.172345i	−0.911479	+	0.411346i	$0.865059\pi$
$102$		0			0
$103$	8.00000	−	13.8564i	0.788263	−	1.36531i	−0.138767	−	0.990325i	$-0.544314\pi$
$103$	8.00000	−	13.8564i	0.788263	−	1.36531i	0.927030	−	0.374987i	$-0.122353\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.50000	−	14.7224i	0.821726	−	1.42327i	−0.0826699	−	0.996577i	$-0.526345\pi$
$107$	8.50000	−	14.7224i	0.821726	−	1.42327i	0.904396	−	0.426694i	$-0.140322\pi$
$108$		0			0
$109$	−8.00000	−	13.8564i	−0.766261	−	1.32720i	−0.939577	−	0.342337i	$-0.888782\pi$
$109$	−8.00000	−	13.8564i	−0.766261	−	1.32720i	0.173316	−	0.984866i	$-0.444552\pi$
$110$		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	−0.765331	−	0.643637i	$-0.777425\pi$
$131$	2.00000	−	3.46410i	0.174741	−	0.302660i	0.940072	+	0.340977i	$0.110758\pi$
$132$		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	0.985262	+	0.171054i	$0.0547174\pi$
$137$	7.50000	+	12.9904i	0.640768	+	1.10984i	−0.344493	+	0.938789i	$0.611949\pi$
$138$		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	0.0115483	+	0.999933i	$0.496324\pi$
$149$	−10.5000	+	18.1865i	−0.860194	+	1.48990i	−0.871742	+	0.489966i	$0.837009\pi$
$150$		0			0
$151$	1.00000	+	1.73205i	0.0813788	+	0.140952i	0.903842	−	0.427865i	$-0.140734\pi$
$151$	1.00000	+	1.73205i	0.0813788	+	0.140952i	−0.822464	+	0.568818i	$0.807401\pi$
$152$		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	−0.975470	−	0.220131i	$-0.929352\pi$
$157$	−8.50000	−	14.7224i	−0.678374	−	1.17498i	0.297097	−	0.954847i	$-0.403982\pi$
$158$		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	−0.933659	−	0.358162i	$-0.883403\pi$
$163$	−2.00000	+	3.46410i	−0.156652	+	0.271329i	0.777007	+	0.629492i	$0.216737\pi$
$164$		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	0.289412	+	0.957205i	$0.406540\pi$
$173$	−9.00000	+	15.5885i	−0.684257	+	1.18517i	−0.973670	+	0.227964i	$0.926793\pi$
$174$		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	0.831408	+	0.555663i	$0.187536\pi$
$179$	12.0000	+	20.7846i	0.896922	+	1.55351i	0.0655145	+	0.997852i	$0.479131\pi$
$180$		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	−0.563081	−	0.826402i	$-0.690384\pi$
$191$	6.00000	−	10.3923i	0.434145	−	0.751961i	0.997225	+	0.0744412i	$0.0237173\pi$
$192$		0			0
$193$	−5.00000	−	8.66025i	−0.359908	−	0.623379i	0.628037	−	0.778183i	$-0.283859\pi$
$193$	−5.00000	−	8.66025i	−0.359908	−	0.623379i	−0.987945	+	0.154805i	$0.950525\pi$
$194$		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	0.883203	−	0.468990i	$-0.155382\pi$
$199$	0.500000	+	0.866025i	0.0354441	+	0.0613909i	−0.847759	+	0.530381i	$0.822049\pi$
$200$		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	0.975833	−	0.218517i	$-0.0701218\pi$
$227$	4.50000	+	7.79423i	0.298675	+	0.517321i	−0.677158	+	0.735838i	$0.736789\pi$
$228$		0			0
$229$	−11.0000	+	19.0526i	−0.726900	+	1.25903i	0.231287	+	0.972886i	$0.425707\pi$
$229$	−11.0000	+	19.0526i	−0.726900	+	1.25903i	−0.958187	+	0.286143i	$0.907627\pi$
$230$		0			0
$231$	−12.5000	+	4.33013i	−0.822440	+	0.284901i
$232$		0			0
$233$	6.00000	−	10.3923i	0.393073	−	0.680823i	−0.599780	−	0.800165i	$-0.704745\pi$
$233$	6.00000	−	10.3923i	0.393073	−	0.680823i	0.992853	+	0.119342i	$0.0380786\pi$
$234$		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	0.984496	+	0.175409i	$0.0561248\pi$
$241$	10.0000	+	17.3205i	0.644157	+	1.11571i	−0.340339	+	0.940303i	$0.610542\pi$
$242$		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	−0.909010	−	0.416775i	$-0.863160\pi$
$257$	−1.50000	+	2.59808i	−0.0935674	+	0.162064i	0.815442	+	0.578838i	$0.196494\pi$
$258$		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	0.962594	−	0.270947i	$-0.0873367\pi$
$263$	4.00000	+	6.92820i	0.246651	+	0.427211i	−0.715944	+	0.698158i	$0.754003\pi$
$264$		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	0.759958	−	0.649972i	$-0.225219\pi$
$269$	−3.00000	−	5.19615i	−0.182913	−	0.316815i	−0.942871	+	0.333157i	$0.891886\pi$
$270$		0			0
$271$	−12.0000	+	20.7846i	−0.728948	+	1.26258i	0.228380	+	0.973572i	$0.426657\pi$
$271$	−12.0000	+	20.7846i	−0.728948	+	1.26258i	−0.957328	+	0.289003i	$0.906676\pi$
$272$		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	−0.960595	−	0.277951i	$-0.910345\pi$
$277$	−12.0000	−	20.7846i	−0.721010	−	1.24883i	0.239585	−	0.970875i	$-0.422989\pi$
$278$		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	0.700718	−	0.713439i	$-0.252863\pi$
$283$	−4.50000	−	7.79423i	−0.267497	−	0.463319i	−0.968215	+	0.250120i	$0.919530\pi$
$284$		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	0.905416	−	0.424526i	$-0.139559\pi$
$311$	1.50000	+	2.59808i	0.0850572	+	0.147323i	−0.820358	+	0.571850i	$0.806226\pi$
$312$		0			0
$313$	−3.00000	+	5.19615i	−0.169570	+	0.293704i	−0.938269	−	0.345907i	$-0.887571\pi$
$313$	−3.00000	+	5.19615i	−0.169570	+	0.293704i	0.768699	+	0.639611i	$0.220905\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	0.494489	+	0.869184i	$0.335355\pi$
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	−0.999980	+	0.00635137i	$0.997978\pi$
$318$		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	−0.970091	−	0.242742i	$-0.921953\pi$
$331$	−5.00000	+	8.66025i	−0.274825	+	0.476011i	0.695266	+	0.718752i	$0.255287\pi$
$332$		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	0.980921	−	0.194409i	$-0.0622790\pi$
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	−0.658824	+	0.752297i	$0.728946\pi$
$348$		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	0.681720	−	0.731613i	$-0.261232\pi$
$353$	−5.50000	−	9.52628i	−0.292735	−	0.507033i	−0.974456	+	0.224580i	$0.927899\pi$
$354$		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	−0.286972	−	0.957939i	$-0.592649\pi$
$359$	13.0000	−	22.5167i	0.686114	−	1.18838i	0.973085	−	0.230445i	$-0.0740181\pi$
$360$		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	0.992647	−	0.121044i	$-0.0386241\pi$
$367$	7.50000	+	12.9904i	0.391497	+	0.678092i	−0.601150	+	0.799136i	$0.705291\pi$
$368$		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	−0.852791	−	0.522253i	$-0.825092\pi$
$373$	0.500000	−	0.866025i	0.0258890	−	0.0448411i	0.878680	+	0.477412i	$0.158425\pi$
$374$		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	−0.998954	−	0.0457244i	$-0.985440\pi$
$383$	−9.00000	+	15.5885i	−0.459879	+	0.796533i	0.539076	+	0.842257i	$0.318774\pi$
$384$		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	−0.999286	−	0.0377914i	$-0.987968\pi$
$389$	−10.5000	−	18.1865i	−0.532371	−	0.922094i	0.466915	−	0.884302i	$-0.345366\pi$
$390$		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	−0.986481	−	0.163876i	$-0.947600\pi$
$397$	−7.00000	+	12.1244i	−0.351320	+	0.608504i	0.635161	+	0.772380i	$0.280934\pi$
$398$		0			0
$399$	12.0000	+	10.3923i	0.600751	+	0.520266i
$400$		0			0
$401$	15.0000	−	25.9808i	0.749064	−	1.29742i	−0.199207	−	0.979957i	$-0.563837\pi$
$401$	15.0000	−	25.9808i	0.749064	−	1.29742i	0.948272	−	0.317460i	$-0.102830\pi$
$402$		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	−0.866730	−	0.498778i	$-0.833782\pi$
$409$	−17.5000	−	30.3109i	−0.865319	−	1.49878i	0.00141047	−	0.999999i	$-0.499551\pi$
$410$		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	0.994826	−	0.101593i	$-0.0323941\pi$
$431$	8.50000	+	14.7224i	0.409431	+	0.709155i	−0.585395	+	0.810748i	$0.699061\pi$
$432$		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	−0.449488	−	0.893287i	$-0.648393\pi$
$439$	11.5000	−	19.9186i	0.548865	−	0.950662i	0.998353	−	0.0573756i	$-0.0182733\pi$
$440$		0			0
$441$	13.0000	+	5.19615i	0.619048	+	0.247436i
$442$		0			0
$443$	−1.00000	+	1.73205i	−0.0475114	+	0.0822922i	−0.888803	−	0.458289i	$-0.848462\pi$
$443$	−1.00000	+	1.73205i	−0.0475114	+	0.0822922i	0.841292	+	0.540581i	$0.181796\pi$
$444$		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	−0.982004	−	0.188858i	$-0.939521\pi$
$457$	−7.00000	+	12.1244i	−0.327446	+	0.567153i	0.654558	+	0.756012i	$0.272855\pi$
$458$		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	−0.989595	−	0.143878i	$-0.954043\pi$
$467$	−8.00000	+	13.8564i	−0.370196	+	0.641198i	0.619400	+	0.785076i	$0.287376\pi$
$468$		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	−0.998392	−	0.0566937i	$-0.981944\pi$
$479$	−12.0000	−	20.7846i	−0.548294	−	0.949673i	0.450098	−	0.892979i	$-0.351389\pi$
$480$		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	0.996915	−	0.0784867i	$-0.0250088\pi$
$487$	9.50000	+	16.4545i	0.430486	+	0.745624i	−0.566429	+	0.824110i	$0.691675\pi$
$488$		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	−0.647882	−	0.761741i	$-0.724345\pi$
$499$	7.50000	−	12.9904i	0.335746	−	0.581529i	0.983628	+	0.180212i	$0.0576783\pi$
$500$		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	−0.419694	−	0.907666i	$-0.637862\pi$
$509$	13.0000	−	22.5167i	0.576215	−	0.998033i	0.995908	−	0.0903676i	$-0.0288042\pi$
$510$		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	0.990671	+	0.136272i	$0.0435123\pi$
$521$	14.0000	+	24.2487i	0.613351	+	1.06236i	−0.377320	+	0.926083i	$0.623154\pi$
$522$		0			0
$523$	−3.00000	+	5.19615i	−0.131181	+	0.227212i	−0.924132	−	0.382073i	$-0.875210\pi$
$523$	−3.00000	+	5.19615i	−0.131181	+	0.227212i	0.792951	+	0.609285i	$0.208544\pi$
$524$		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	−0.953262	−	0.302144i	$-0.902298\pi$
$541$	−5.00000	+	8.66025i	−0.214967	+	0.372333i	0.738296	+	0.674477i	$0.235631\pi$
$542$		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	0.987986	−	0.154541i	$-0.0493899\pi$
$557$	8.50000	+	14.7224i	0.360157	+	0.623809i	−0.627830	+	0.778351i	$0.716057\pi$
$558$		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	0.998419	+	0.0562051i	$0.0179001\pi$
$563$	13.0000	+	22.5167i	0.547885	+	0.948964i	−0.450535	+	0.892759i	$0.648767\pi$
$564$		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	−0.876316	−	0.481737i	$-0.840006\pi$
$569$	−0.500000	+	0.866025i	−0.0209611	+	0.0363057i	0.855355	+	0.518043i	$0.173339\pi$
$570$		0			0
$571$	2.00000	+	3.46410i	0.0836974	+	0.144968i	0.904835	−	0.425762i	$-0.139994\pi$
$571$	2.00000	+	3.46410i	0.0836974	+	0.144968i	−0.821138	+	0.570730i	$0.806660\pi$
$572$		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	0.993164	−	0.116731i	$-0.0372414\pi$
$577$	9.50000	+	16.4545i	0.395490	+	0.685009i	−0.597673	+	0.801740i	$0.703908\pi$
$578$		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	−0.130746	+	0.991416i	$0.541737\pi$
$593$	−22.5000	+	38.9711i	−0.923964	+	1.60035i	−0.793219	+	0.608937i	$0.791596\pi$
$594$		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	−0.990752	−	0.135686i	$-0.956676\pi$
$599$	−15.0000	−	25.9808i	−0.612883	−	1.06155i	0.377869	−	0.925859i	$-0.376657\pi$
$600$		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	−0.928272	−	0.371901i	$-0.878706\pi$
$607$	−3.50000	+	6.06218i	−0.142061	+	0.246056i	0.786212	+	0.617957i	$0.212039\pi$
$608$		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	0.641734	−	0.766927i	$-0.278215\pi$
$613$	−8.50000	−	14.7224i	−0.343312	−	0.594633i	−0.985046	+	0.172294i	$0.944882\pi$
$614$		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	0.643094	−	0.765787i	$-0.277650\pi$
$619$	−8.50000	−	14.7224i	−0.341644	−	0.591744i	−0.984738	+	0.174042i	$0.944317\pi$
$620$		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	0.562779	−	0.826608i	$-0.309732\pi$
$641$	−11.0000	−	19.0526i	−0.434474	−	0.752531i	−0.997253	+	0.0740768i	$0.976399\pi$
$642$		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	−0.998246	−	0.0592060i	$-0.981143\pi$
$647$	−14.0000	−	24.2487i	−0.550397	−	0.953315i	0.447849	−	0.894109i	$-0.352190\pi$
$648$		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	0.0825519	+	0.996587i	$0.473693\pi$
$653$	−21.0000	+	36.3731i	−0.821794	+	1.42339i	−0.904345	+	0.426801i	$0.859640\pi$
$654$		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	0.996682	−	0.0813955i	$-0.0259377\pi$
$661$	11.0000	+	19.0526i	0.427850	+	0.741059i	−0.568831	+	0.822454i	$0.692604\pi$
$662$		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	−0.957984	−	0.286820i	$-0.907402\pi$
$677$	−6.00000	+	10.3923i	−0.230599	+	0.399409i	0.727386	+	0.686229i	$0.240735\pi$
$678$		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	0.714454	−	0.699682i	$-0.246675\pi$
$683$	−6.50000	−	11.2583i	−0.248716	−	0.430788i	−0.963170	+	0.268894i	$0.913342\pi$
$684$		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	−0.998773	−	0.0495194i	$-0.984231\pi$
$691$	−12.0000	+	20.7846i	−0.456502	+	0.790684i	0.542272	+	0.840203i	$0.317564\pi$
$692$		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	−0.756730	−	0.653727i	$-0.773204\pi$
$709$	5.00000	−	8.66025i	0.187779	−	0.325243i	0.944509	+	0.328484i	$0.106538\pi$
$710$		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	0.329840	+	0.944037i	$0.393005\pi$
$719$	−17.5000	+	30.3109i	−0.652640	+	1.13041i	−0.982480	+	0.186369i	$0.940328\pi$
$720$		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	−0.816156	−	0.577832i	$-0.803899\pi$
$733$	2.50000	−	4.33013i	0.0923396	−	0.159937i	0.908495	+	0.417895i	$0.137232\pi$
$734$		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	0.982722	−	0.185088i	$-0.0592569\pi$
$739$	9.00000	+	15.5885i	0.331070	+	0.573431i	−0.651652	+	0.758518i	$0.725924\pi$
$740$		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	−0.991511	−	0.130025i	$-0.958494\pi$
$751$	−10.5000	+	18.1865i	−0.383150	+	0.663636i	0.608360	+	0.793661i	$0.291828\pi$
$752$		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	−0.959454	−	0.281865i	$-0.909047\pi$
$761$	−6.50000	+	11.2583i	−0.235625	+	0.408114i	0.723829	+	0.689979i	$0.242380\pi$
$762$		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	0.856893	−	0.515494i	$-0.172391\pi$
$773$	−0.500000	−	0.866025i	−0.0179838	−	0.0311488i	−0.874877	+	0.484345i	$0.839058\pi$
$774$		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	0.929467	+	0.368906i	$0.120268\pi$
$787$	22.0000	+	38.1051i	0.784215	+	1.35830i	−0.145251	+	0.989395i	$0.546399\pi$
$788$		0			0
$789$	−4.00000	+	6.92820i	−0.142404	+	0.246651i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		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:	\( N \)	\(=\)	\( 476 = 2^{2} \cdot 7 \cdot 17 \)
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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