LMFDB - Embedded newform 2600.2.f.a.649.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2600,2,Mod(649,2600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2600.649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2600 = 2^{3} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2600.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$20.7611045255$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	2600.649
Dual form			2600.2.f.a.649.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155i q^{3} -4.56155 q^{7} -3.56155 q^{9} +3.12311i q^{11} +(0.561553 - 3.56155i) q^{13} -0.561553i q^{17} +2.00000i q^{19} +11.6847i q^{21} +5.12311i q^{23} +1.43845i q^{27} -3.12311 q^{29} -3.12311i q^{31} +8.00000 q^{33} -5.43845 q^{37} +(-9.12311 - 1.43845i) q^{39} +8.00000i q^{41} +5.43845i q^{43} +3.43845 q^{47} +13.8078 q^{49} -1.43845 q^{51} -4.24621i q^{53} +5.12311 q^{57} +10.0000i q^{59} +11.1231 q^{61} +16.2462 q^{63} +0.876894 q^{67} +13.1231 q^{69} +5.68466i q^{71} +15.3693 q^{73} -14.2462i q^{77} +2.87689 q^{79} -7.00000 q^{81} -8.24621 q^{83} +8.00000i q^{87} +5.12311i q^{89} +(-2.56155 + 16.2462i) q^{91} -8.00000 q^{93} +10.2462 q^{97} -11.1231i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{7} - 6 q^{9} - 6 q^{13} + 4 q^{29} + 32 q^{33} - 30 q^{37} - 20 q^{39} + 22 q^{47} + 14 q^{49} - 14 q^{51} + 4 q^{57} + 28 q^{61} + 32 q^{63} + 20 q^{67} + 36 q^{69} + 12 q^{73} + 28 q^{79}+ \cdots + 8 q^{97}+O(q^{100}) $ 4 * q - 10 * q^7 - 6 * q^9 - 6 * q^13 + 4 * q^29 + 32 * q^33 - 30 * q^37 - 20 * q^39 + 22 * q^47 + 14 * q^49 - 14 * q^51 + 4 * q^57 + 28 * q^61 + 32 * q^63 + 20 * q^67 + 36 * q^69 + 12 * q^73 + 28 * q^79 - 28 * q^81 - 2 * q^91 - 32 * q^93 + 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times$.

$n$	$1301$	$1601$	$1951$	$1977$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$		−	2.56155i		−	1.47891i	−0.673204	−	0.739457i	$-0.735083\pi$
$3$		−	2.56155i		−	1.47891i	0.673204	−	0.739457i	$-0.264917\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−4.56155			−1.72410			−0.862052	−	0.506819i	$-0.830821\pi$
$7$	−4.56155			−1.72410			−0.862052	+	0.506819i	$0.830821\pi$
$8$		0			0
$9$	−3.56155			−1.18718
$10$		0			0
$11$			3.12311i			0.941652i	0.882226	+	0.470826i	$0.156044\pi$
$11$			3.12311i			0.941652i	−0.882226	+	0.470826i	$0.843956\pi$
$12$		0			0
$13$	0.561553	−	3.56155i	0.155747	−	0.987797i
$13$	0.561553	−	3.56155i	0.155747	−	0.987797i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	0.561553i		−	0.136197i	−0.997679	−	0.0680983i	$-0.978307\pi$
$17$		−	0.561553i		−	0.136197i	0.997679	−	0.0680983i	$-0.0216931\pi$
$18$		0			0
$19$			2.00000i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$			2.00000i			0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$		0			0
$21$			11.6847i			2.54980i
$22$		0			0
$23$			5.12311i			1.06824i	0.845408	+	0.534121i	$0.179357\pi$
$23$			5.12311i			1.06824i	−0.845408	+	0.534121i	$0.820643\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			1.43845i			0.276829i
$28$		0			0
$29$	−3.12311			−0.579946			−0.289973	−	0.957035i	$-0.593646\pi$
$29$	−3.12311			−0.579946			−0.289973	+	0.957035i	$0.593646\pi$
$30$		0			0
$31$		−	3.12311i		−	0.560926i	−0.959865	−	0.280463i	$-0.909512\pi$
$31$		−	3.12311i		−	0.560926i	0.959865	−	0.280463i	$-0.0904881\pi$
$32$		0			0
$33$	8.00000			1.39262
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.43845			−0.894075			−0.447038	−	0.894515i	$-0.647521\pi$
$37$	−5.43845			−0.894075			−0.447038	+	0.894515i	$0.647521\pi$
$38$		0			0
$39$	−9.12311	−	1.43845i	−1.46087	−	0.230336i
$40$		0			0
$41$			8.00000i			1.24939i	0.780869	+	0.624695i	$0.214777\pi$
$41$			8.00000i			1.24939i	−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$			5.43845i			0.829355i	0.909968	+	0.414678i	$0.136106\pi$
$43$			5.43845i			0.829355i	−0.909968	+	0.414678i	$0.863894\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.43845			0.501549			0.250775	−	0.968046i	$-0.419315\pi$
$47$	3.43845			0.501549			0.250775	+	0.968046i	$0.419315\pi$
$48$		0			0
$49$	13.8078			1.97254
$50$		0			0
$51$	−1.43845			−0.201423
$52$		0			0
$53$		−	4.24621i		−	0.583262i	−0.956531	−	0.291631i	$-0.905802\pi$
$53$		−	4.24621i		−	0.583262i	0.956531	−	0.291631i	$-0.0941979\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	5.12311			0.678572
$58$		0			0
$59$			10.0000i			1.30189i	0.759125	+	0.650945i	$0.225627\pi$
$59$			10.0000i			1.30189i	−0.759125	+	0.650945i	$0.774373\pi$
$60$		0			0
$61$	11.1231			1.42417			0.712084	−	0.702094i	$-0.247752\pi$
$61$	11.1231			1.42417			0.712084	+	0.702094i	$0.247752\pi$
$62$		0			0
$63$	16.2462			2.04683
$64$		0			0
$65$		0			0
$66$		0			0
$67$	0.876894			0.107130			0.0535648	−	0.998564i	$-0.482942\pi$
$67$	0.876894			0.107130			0.0535648	+	0.998564i	$0.482942\pi$
$68$		0			0
$69$	13.1231			1.57984
$70$		0			0
$71$			5.68466i			0.674645i	0.941389	+	0.337322i	$0.109521\pi$
$71$			5.68466i			0.674645i	−0.941389	+	0.337322i	$0.890479\pi$
$72$		0			0
$73$	15.3693			1.79884			0.899421	−	0.437083i	$-0.143988\pi$
$73$	15.3693			1.79884			0.899421	+	0.437083i	$0.143988\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	14.2462i		−	1.62351i
$78$		0			0
$79$	2.87689			0.323676			0.161838	−	0.986817i	$-0.448258\pi$
$79$	2.87689			0.323676			0.161838	+	0.986817i	$0.448258\pi$
$80$		0			0
$81$	−7.00000			−0.777778
$82$		0			0
$83$	−8.24621			−0.905139			−0.452570	−	0.891729i	$-0.649493\pi$
$83$	−8.24621			−0.905139			−0.452570	+	0.891729i	$0.649493\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			8.00000i			0.857690i
$88$		0			0
$89$			5.12311i			0.543048i	0.962432	+	0.271524i	$0.0875277\pi$
$89$			5.12311i			0.543048i	−0.962432	+	0.271524i	$0.912472\pi$
$90$		0			0
$91$	−2.56155	+	16.2462i	−0.268524	+	1.70307i
$92$		0			0
$93$	−8.00000			−0.829561
$94$		0			0
$95$		0			0
$96$		0			0
$97$	10.2462			1.04035			0.520173	−	0.854061i	$-0.325868\pi$
$97$	10.2462			1.04035			0.520173	+	0.854061i	$0.325868\pi$
$98$		0			0
$99$		−	11.1231i		−	1.11791i
$100$		0			0
$101$	13.3693			1.33030			0.665148	−	0.746711i	$-0.268368\pi$
$101$	13.3693			1.33030			0.665148	+	0.746711i	$0.268368\pi$
$102$		0			0
$103$		−	5.12311i		−	0.504795i	−0.967624	−	0.252397i	$-0.918781\pi$
$103$		−	5.12311i		−	0.504795i	0.967624	−	0.252397i	$-0.0812190\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			4.00000i			0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$			4.00000i			0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$		0			0
$109$		−	15.6847i		−	1.50232i	−0.660121	−	0.751159i	$-0.729495\pi$
$109$		−	15.6847i		−	1.50232i	0.660121	−	0.751159i	$-0.270505\pi$
$110$		0			0
$111$			13.9309i			1.32226i
$112$		0			0
$113$			10.0000i			0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$			10.0000i			0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.00000	+	12.6847i	−0.184900	+	1.17270i
$118$		0			0
$119$			2.56155i			0.234817i
$120$		0			0
$121$	1.24621			0.113292
$122$		0			0
$123$	20.4924			1.84774
$124$		0			0
$125$		0			0
$126$		0			0
$127$			13.1231i			1.16449i	0.813014	+	0.582244i	$0.197825\pi$
$127$			13.1231i			1.16449i	−0.813014	+	0.582244i	$0.802175\pi$
$128$		0			0
$129$	13.9309			1.22654
$130$		0			0
$131$	−15.6847			−1.37037			−0.685187	−	0.728367i	$-0.740280\pi$
$131$	−15.6847			−1.37037			−0.685187	+	0.728367i	$0.740280\pi$
$132$		0			0
$133$		−	9.12311i		−	0.791074i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.12311			−0.437696			−0.218848	−	0.975759i	$-0.570230\pi$
$137$	−5.12311			−0.437696			−0.218848	+	0.975759i	$0.570230\pi$
$138$		0			0
$139$	−12.8078			−1.08634			−0.543170	−	0.839623i	$-0.682776\pi$
$139$	−12.8078			−1.08634			−0.543170	+	0.839623i	$0.682776\pi$
$140$		0			0
$141$		−	8.80776i		−	0.741748i
$142$		0			0
$143$	11.1231	+	1.75379i	0.930161	+	0.146659i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	35.3693i		−	2.91721i
$148$		0			0
$149$		−	1.12311i		−	0.0920084i	−0.998941	−	0.0460042i	$-0.985351\pi$
$149$		−	1.12311i		−	0.0920084i	0.998941	−	0.0460042i	$-0.0146488\pi$
$150$		0			0
$151$		−	7.43845i		−	0.605332i	−0.953097	−	0.302666i	$-0.902123\pi$
$151$		−	7.43845i		−	0.605332i	0.953097	−	0.302666i	$-0.0978767\pi$
$152$		0			0
$153$			2.00000i			0.161690i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			4.24621i			0.338885i	0.985540	+	0.169442i	$0.0541966\pi$
$157$			4.24621i			0.338885i	−0.985540	+	0.169442i	$0.945803\pi$
$158$		0			0
$159$	−10.8769			−0.862594
$160$		0			0
$161$		−	23.3693i		−	1.84176i
$162$		0			0
$163$	−3.75379			−0.294019			−0.147010	−	0.989135i	$-0.546965\pi$
$163$	−3.75379			−0.294019			−0.147010	+	0.989135i	$0.546965\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	15.1231			1.17026			0.585131	−	0.810939i	$-0.301043\pi$
$167$	15.1231			1.17026			0.585131	+	0.810939i	$0.301043\pi$
$168$		0			0
$169$	−12.3693	−	4.00000i	−0.951486	−	0.307692i
$170$		0			0
$171$		−	7.12311i		−	0.544718i
$172$		0			0
$173$		−	12.2462i		−	0.931062i	−0.885032	−	0.465531i	$-0.845863\pi$
$173$		−	12.2462i		−	0.931062i	0.885032	−	0.465531i	$-0.154137\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	25.6155			1.92538
$178$		0			0
$179$	−13.4384			−1.00444			−0.502218	−	0.864741i	$-0.667483\pi$
$179$	−13.4384			−1.00444			−0.502218	+	0.864741i	$0.667483\pi$
$180$		0			0
$181$	8.24621			0.612936			0.306468	−	0.951881i	$-0.400853\pi$
$181$	8.24621			0.612936			0.306468	+	0.951881i	$0.400853\pi$
$182$		0			0
$183$		−	28.4924i		−	2.10622i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	1.75379			0.128250
$188$		0			0
$189$		−	6.56155i		−	0.477283i
$190$		0			0
$191$	−5.75379			−0.416330			−0.208165	−	0.978094i	$-0.566749\pi$
$191$	−5.75379			−0.416330			−0.208165	+	0.978094i	$0.566749\pi$
$192$		0			0
$193$	−17.6155			−1.26799			−0.633997	−	0.773336i	$-0.718587\pi$
$193$	−17.6155			−1.26799			−0.633997	+	0.773336i	$0.718587\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.43845			−0.387473			−0.193737	−	0.981054i	$-0.562061\pi$
$197$	−5.43845			−0.387473			−0.193737	+	0.981054i	$0.562061\pi$
$198$		0			0
$199$	−2.24621			−0.159230			−0.0796148	−	0.996826i	$-0.525369\pi$
$199$	−2.24621			−0.159230			−0.0796148	+	0.996826i	$0.525369\pi$
$200$		0			0
$201$		−	2.24621i		−	0.158436i
$202$		0			0
$203$	14.2462			0.999888
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	18.2462i		−	1.26820i
$208$		0			0
$209$	−6.24621			−0.432059
$210$		0			0
$211$	15.0540			1.03636			0.518179	−	0.855272i	$-0.326610\pi$
$211$	15.0540			1.03636			0.518179	+	0.855272i	$0.326610\pi$
$212$		0			0
$213$	14.5616			0.997741
$214$		0			0
$215$		0			0
$216$		0			0
$217$			14.2462i			0.967096i
$218$		0			0
$219$		−	39.3693i		−	2.66033i
$220$		0			0
$221$	−2.00000	−	0.315342i	−0.134535	−	0.0212122i
$222$		0			0
$223$	−18.1771			−1.21723			−0.608614	−	0.793467i	$-0.708274\pi$
$223$	−18.1771			−1.21723			−0.608614	+	0.793467i	$0.708274\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	21.3693			1.41833			0.709166	−	0.705042i	$-0.249072\pi$
$227$	21.3693			1.41833			0.709166	+	0.705042i	$0.249072\pi$
$228$		0			0
$229$			8.31534i			0.549493i	0.961517	+	0.274747i	$0.0885940\pi$
$229$			8.31534i			0.549493i	−0.961517	+	0.274747i	$0.911406\pi$
$230$		0			0
$231$	−36.4924			−2.40103
$232$		0			0
$233$			0.561553i			0.0367885i	0.999831	+	0.0183943i	$0.00585541\pi$
$233$			0.561553i			0.0367885i	−0.999831	+	0.0183943i	$0.994145\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	7.36932i		−	0.478689i
$238$		0			0
$239$			19.9309i			1.28922i	0.764511	+	0.644610i	$0.222980\pi$
$239$			19.9309i			1.28922i	−0.764511	+	0.644610i	$0.777020\pi$
$240$		0			0
$241$			18.8769i			1.21597i	0.793949	+	0.607984i	$0.208021\pi$
$241$			18.8769i			1.21597i	−0.793949	+	0.607984i	$0.791979\pi$
$242$		0			0
$243$			22.2462i			1.42710i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	7.12311	+	1.12311i	0.453232	+	0.0714615i
$248$		0			0
$249$			21.1231i			1.33862i
$250$		0			0
$251$	−8.49242			−0.536037			−0.268018	−	0.963414i	$-0.586369\pi$
$251$	−8.49242			−0.536037			−0.268018	+	0.963414i	$0.586369\pi$
$252$		0			0
$253$	−16.0000			−1.00591
$254$		0			0
$255$		0			0
$256$		0			0
$257$			27.9309i			1.74228i	0.491035	+	0.871140i	$0.336619\pi$
$257$			27.9309i			1.74228i	−0.491035	+	0.871140i	$0.663381\pi$
$258$		0			0
$259$	24.8078			1.54148
$260$		0			0
$261$	11.1231			0.688503
$262$		0			0
$263$			28.4924i			1.75692i	0.477818	+	0.878459i	$0.341428\pi$
$263$			28.4924i			1.75692i	−0.477818	+	0.878459i	$0.658572\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	13.1231			0.803121
$268$		0			0
$269$	−19.1231			−1.16596			−0.582978	−	0.812488i	$-0.698113\pi$
$269$	−19.1231			−1.16596			−0.582978	+	0.812488i	$0.698113\pi$
$270$		0			0
$271$		−	2.31534i		−	0.140647i	−0.997524	−	0.0703235i	$-0.977597\pi$
$271$		−	2.31534i		−	0.140647i	0.997524	−	0.0703235i	$-0.0224032\pi$
$272$		0			0
$273$	41.6155	+	6.56155i	2.51869	+	0.397123i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			17.3693i			1.04362i	0.853061	+	0.521811i	$0.174743\pi$
$277$			17.3693i			1.04362i	−0.853061	+	0.521811i	$0.825257\pi$
$278$		0			0
$279$			11.1231i			0.665923i
$280$		0			0
$281$			15.3693i			0.916857i	0.888731	+	0.458428i	$0.151587\pi$
$281$			15.3693i			0.916857i	−0.888731	+	0.458428i	$0.848413\pi$
$282$		0			0
$283$			12.0000i			0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$			12.0000i			0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	36.4924i		−	2.15408i
$288$		0			0
$289$	16.6847			0.981450
$290$		0			0
$291$		−	26.2462i		−	1.53858i
$292$		0			0
$293$	−0.315342			−0.0184225			−0.00921123	−	0.999958i	$-0.502932\pi$
$293$	−0.315342			−0.0184225			−0.00921123	+	0.999958i	$0.502932\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−4.49242			−0.260677
$298$		0			0
$299$	18.2462	+	2.87689i	1.05521	+	0.166375i
$300$		0			0
$301$		−	24.8078i		−	1.42990i
$302$		0			0
$303$		−	34.2462i		−	1.96739i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−18.0000			−1.02731			−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000			−1.02731			−0.513657	+	0.857996i	$0.671710\pi$
$308$		0			0
$309$	−13.1231			−0.746547
$310$		0			0
$311$	10.8769			0.616772			0.308386	−	0.951261i	$-0.400211\pi$
$311$	10.8769			0.616772			0.308386	+	0.951261i	$0.400211\pi$
$312$		0			0
$313$			8.56155i			0.483928i	0.970285	+	0.241964i	$0.0777915\pi$
$313$			8.56155i			0.483928i	−0.970285	+	0.241964i	$0.922208\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.87689			−0.386245			−0.193122	−	0.981175i	$-0.561861\pi$
$317$	−6.87689			−0.386245			−0.193122	+	0.981175i	$0.561861\pi$
$318$		0			0
$319$		−	9.75379i		−	0.546107i
$320$		0			0
$321$	10.2462			0.571888
$322$		0			0
$323$	1.12311			0.0624913
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−40.1771			−2.22180
$328$		0			0
$329$	−15.6847			−0.864723
$330$		0			0
$331$			16.2462i			0.892973i	0.894790	+	0.446486i	$0.147325\pi$
$331$			16.2462i			0.892973i	−0.894790	+	0.446486i	$0.852675\pi$
$332$		0			0
$333$	19.3693			1.06143
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	13.6847i		−	0.745451i	−0.927942	−	0.372725i	$-0.878423\pi$
$337$		−	13.6847i		−	0.745451i	0.927942	−	0.372725i	$-0.121577\pi$
$338$		0			0
$339$	25.6155			1.39124
$340$		0			0
$341$	9.75379			0.528197
$342$		0			0
$343$	−31.0540			−1.67676
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	2.56155i		−	0.137511i	−0.997634	−	0.0687557i	$-0.978097\pi$
$347$		−	2.56155i		−	0.137511i	0.997634	−	0.0687557i	$-0.0219029\pi$
$348$		0			0
$349$			17.9309i			0.959817i	0.877318	+	0.479909i	$0.159330\pi$
$349$			17.9309i			0.959817i	−0.877318	+	0.479909i	$0.840670\pi$
$350$		0			0
$351$	5.12311	+	0.807764i	0.273451	+	0.0431153i
$352$		0			0
$353$	−33.6155			−1.78917			−0.894587	−	0.446894i	$-0.852530\pi$
$353$	−33.6155			−1.78917			−0.894587	+	0.446894i	$0.852530\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	6.56155			0.347274
$358$		0			0
$359$		−	4.87689i		−	0.257393i	−0.991684	−	0.128696i	$-0.958921\pi$
$359$		−	4.87689i		−	0.257393i	0.991684	−	0.128696i	$-0.0410792\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$		−	3.19224i		−	0.167549i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			4.49242i			0.234503i	0.993102	+	0.117251i	$0.0374083\pi$
$367$			4.49242i			0.234503i	−0.993102	+	0.117251i	$0.962592\pi$
$368$		0			0
$369$		−	28.4924i		−	1.48326i
$370$		0			0
$371$			19.3693i			1.00560i
$372$		0			0
$373$		−	10.6307i		−	0.550436i	−0.961382	−	0.275218i	$-0.911250\pi$
$373$		−	10.6307i		−	0.550436i	0.961382	−	0.275218i	$-0.0887500\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.75379	+	11.1231i	−0.0903247	+	0.572869i
$378$		0			0
$379$			26.0000i			1.33553i	0.744372	+	0.667765i	$0.232749\pi$
$379$			26.0000i			1.33553i	−0.744372	+	0.667765i	$0.767251\pi$
$380$		0			0
$381$	33.6155			1.72218
$382$		0			0
$383$	−9.19224			−0.469701			−0.234851	−	0.972031i	$-0.575460\pi$
$383$	−9.19224			−0.469701			−0.234851	+	0.972031i	$0.575460\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	19.3693i		−	0.984598i
$388$		0			0
$389$	28.2462			1.43214			0.716070	−	0.698029i	$-0.245939\pi$
$389$	28.2462			1.43214			0.716070	+	0.698029i	$0.245939\pi$
$390$		0			0
$391$	2.87689			0.145491
$392$		0			0
$393$			40.1771i			2.02667i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	9.12311			0.457876			0.228938	−	0.973441i	$-0.426475\pi$
$397$	9.12311			0.457876			0.228938	+	0.973441i	$0.426475\pi$
$398$		0			0
$399$	−23.3693			−1.16993
$400$		0			0
$401$		−	7.36932i		−	0.368006i	−0.982926	−	0.184003i	$-0.941094\pi$
$401$		−	7.36932i		−	0.368006i	0.982926	−	0.184003i	$-0.0589056\pi$
$402$		0			0
$403$	−11.1231	−	1.75379i	−0.554081	−	0.0873624i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	16.9848i		−	0.841908i
$408$		0			0
$409$			0.630683i			0.0311853i	0.999878	+	0.0155926i	$0.00496349\pi$
$409$			0.630683i			0.0311853i	−0.999878	+	0.0155926i	$0.995037\pi$
$410$		0			0
$411$			13.1231i			0.647315i
$412$		0			0
$413$		−	45.6155i		−	2.24459i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			32.8078i			1.60660i
$418$		0			0
$419$	34.5616			1.68844			0.844221	−	0.535995i	$-0.180063\pi$
$419$	34.5616			1.68844			0.844221	+	0.535995i	$0.180063\pi$
$420$		0			0
$421$			3.19224i			0.155580i	0.996970	+	0.0777900i	$0.0247864\pi$
$421$			3.19224i			0.155580i	−0.996970	+	0.0777900i	$0.975214\pi$
$422$		0			0
$423$	−12.2462			−0.595431
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−50.7386			−2.45541
$428$		0			0
$429$	4.49242	−	28.4924i	0.216896	−	1.37563i
$430$		0			0
$431$		−	24.4233i		−	1.17643i	−0.808705	−	0.588214i	$-0.799831\pi$
$431$		−	24.4233i		−	1.17643i	0.808705	−	0.588214i	$-0.200169\pi$
$432$		0			0
$433$			21.6847i			1.04210i	0.853527	+	0.521049i	$0.174459\pi$
$433$			21.6847i			1.04210i	−0.853527	+	0.521049i	$0.825541\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.2462			−0.490143
$438$		0			0
$439$	−19.8617			−0.947949			−0.473975	−	0.880539i	$-0.657181\pi$
$439$	−19.8617			−0.947949			−0.473975	+	0.880539i	$0.657181\pi$
$440$		0			0
$441$	−49.1771			−2.34177
$442$		0			0
$443$			36.1771i			1.71882i	0.511283	+	0.859412i	$0.329170\pi$
$443$			36.1771i			1.71882i	−0.511283	+	0.859412i	$0.670830\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−2.87689			−0.136072
$448$		0			0
$449$			16.0000i			0.755087i	0.925992	+	0.377543i	$0.123231\pi$
$449$			16.0000i			0.755087i	−0.925992	+	0.377543i	$0.876769\pi$
$450$		0			0
$451$	−24.9848			−1.17649
$452$		0			0
$453$	−19.0540			−0.895234
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−25.6155			−1.19824			−0.599122	−	0.800658i	$-0.704484\pi$
$457$	−25.6155			−1.19824			−0.599122	+	0.800658i	$0.704484\pi$
$458$		0			0
$459$	0.807764			0.0377032
$460$		0			0
$461$			36.1771i			1.68493i	0.538747	+	0.842467i	$0.318898\pi$
$461$			36.1771i			1.68493i	−0.538747	+	0.842467i	$0.681102\pi$
$462$		0			0
$463$	39.6155			1.84109			0.920545	−	0.390637i	$-0.127745\pi$
$463$	39.6155			1.84109			0.920545	+	0.390637i	$0.127745\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			16.4924i			0.763178i	0.924332	+	0.381589i	$0.124623\pi$
$467$			16.4924i			0.763178i	−0.924332	+	0.381589i	$0.875377\pi$
$468$		0			0
$469$	−4.00000			−0.184703
$470$		0			0
$471$	10.8769			0.501181
$472$		0			0
$473$	−16.9848			−0.780964
$474$		0			0
$475$		0			0
$476$		0			0
$477$			15.1231i			0.692439i
$478$		0			0
$479$		−	9.19224i		−	0.420004i	−0.977701	−	0.210002i	$-0.932653\pi$
$479$		−	9.19224i		−	0.420004i	0.977701	−	0.210002i	$-0.0673470\pi$
$480$		0			0
$481$	−3.05398	+	19.3693i	−0.139249	+	0.883165i
$482$		0			0
$483$	−59.8617			−2.72380
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−41.8617			−1.89694			−0.948468	−	0.316872i	$-0.897367\pi$
$487$	−41.8617			−1.89694			−0.948468	+	0.316872i	$0.897367\pi$
$488$		0			0
$489$			9.61553i			0.434829i
$490$		0			0
$491$	11.1922			0.505099			0.252549	−	0.967584i	$-0.418731\pi$
$491$	11.1922			0.505099			0.252549	+	0.967584i	$0.418731\pi$
$492$		0			0
$493$			1.75379i			0.0789867i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	25.9309i		−	1.16316i
$498$		0			0
$499$		−	16.8769i		−	0.755514i	−0.925905	−	0.377757i	$-0.876696\pi$
$499$		−	16.8769i		−	0.755514i	0.925905	−	0.377757i	$-0.123304\pi$
$500$		0			0
$501$		−	38.7386i		−	1.73071i
$502$		0			0
$503$		−	15.3693i		−	0.685284i	−0.939466	−	0.342642i	$-0.888678\pi$
$503$		−	15.3693i		−	0.685284i	0.939466	−	0.342642i	$-0.111322\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−10.2462	+	31.6847i	−0.455050	+	1.40717i
$508$		0			0
$509$			27.3693i			1.21312i	0.795036	+	0.606562i	$0.207452\pi$
$509$			27.3693i			1.21312i	−0.795036	+	0.606562i	$0.792548\pi$
$510$		0			0
$511$	−70.1080			−3.10139
$512$		0			0
$513$	−2.87689			−0.127018
$514$		0			0
$515$		0			0
$516$		0			0
$517$			10.7386i			0.472285i
$518$		0			0
$519$	−31.3693			−1.37696
$520$		0			0
$521$	−27.9309			−1.22367			−0.611837	−	0.790984i	$-0.709569\pi$
$521$	−27.9309			−1.22367			−0.611837	+	0.790984i	$0.709569\pi$
$522$		0			0
$523$		−	30.2462i		−	1.32257i	−0.750133	−	0.661287i	$-0.770010\pi$
$523$		−	30.2462i		−	1.32257i	0.750133	−	0.661287i	$-0.229990\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.75379			−0.0763962
$528$		0			0
$529$	−3.24621			−0.141140
$530$		0			0
$531$		−	35.6155i		−	1.54558i
$532$		0			0
$533$	28.4924	+	4.49242i	1.23414	+	0.194588i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			34.4233i			1.48547i
$538$		0			0
$539$			43.1231i			1.85744i
$540$		0			0
$541$			20.1771i			0.867480i	0.901038	+	0.433740i	$0.142806\pi$
$541$			20.1771i			0.867480i	−0.901038	+	0.433740i	$0.857194\pi$
$542$		0			0
$543$		−	21.1231i		−	0.906479i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	3.19224i		−	0.136490i	−0.997669	−	0.0682451i	$-0.978260\pi$
$547$		−	3.19224i		−	0.136490i	0.997669	−	0.0682451i	$-0.0217400\pi$
$548$		0			0
$549$	−39.6155			−1.69075
$550$		0			0
$551$		−	6.24621i		−	0.266098i
$552$		0			0
$553$	−13.1231			−0.558051
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.93087			0.420784			0.210392	−	0.977617i	$-0.432526\pi$
$557$	9.93087			0.420784			0.210392	+	0.977617i	$0.432526\pi$
$558$		0			0
$559$	19.3693	+	3.05398i	0.819235	+	0.129169i
$560$		0			0
$561$		−	4.49242i		−	0.189670i
$562$		0			0
$563$			1.93087i			0.0813765i	0.999172	+	0.0406882i	$0.0129550\pi$
$563$			1.93087i			0.0813765i	−0.999172	+	0.0406882i	$0.987045\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	31.9309			1.34097
$568$		0			0
$569$	22.8078			0.956151			0.478076	−	0.878319i	$-0.341334\pi$
$569$	22.8078			0.956151			0.478076	+	0.878319i	$0.341334\pi$
$570$		0			0
$571$	23.0540			0.964779			0.482389	−	0.875957i	$-0.339769\pi$
$571$	23.0540			0.964779			0.482389	+	0.875957i	$0.339769\pi$
$572$		0			0
$573$			14.7386i			0.615715i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−27.8617			−1.15990			−0.579950	−	0.814652i	$-0.696928\pi$
$577$	−27.8617			−1.15990			−0.579950	+	0.814652i	$0.696928\pi$
$578$		0			0
$579$			45.1231i			1.87525i
$580$		0			0
$581$	37.6155			1.56056
$582$		0			0
$583$	13.2614			0.549230
$584$		0			0
$585$		0			0
$586$		0			0
$587$	42.4924			1.75385			0.876925	−	0.480627i	$-0.159591\pi$
$587$	42.4924			1.75385			0.876925	+	0.480627i	$0.159591\pi$
$588$		0			0
$589$	6.24621			0.257371
$590$		0			0
$591$			13.9309i			0.573039i
$592$		0			0
$593$	−42.2462			−1.73484			−0.867422	−	0.497573i	$-0.834225\pi$
$593$	−42.2462			−1.73484			−0.867422	+	0.497573i	$0.834225\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			5.75379i			0.235487i
$598$		0			0
$599$	−16.6307			−0.679511			−0.339756	−	0.940514i	$-0.610344\pi$
$599$	−16.6307			−0.679511			−0.339756	+	0.940514i	$0.610344\pi$
$600$		0			0
$601$	1.19224			0.0486323			0.0243162	−	0.999704i	$-0.492259\pi$
$601$	1.19224			0.0486323			0.0243162	+	0.999704i	$0.492259\pi$
$602$		0			0
$603$	−3.12311			−0.127183
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	33.6155i		−	1.36441i	−0.731160	−	0.682206i	$-0.761021\pi$
$607$		−	33.6155i		−	1.36441i	0.731160	−	0.682206i	$-0.238979\pi$
$608$		0			0
$609$		−	36.4924i		−	1.47875i
$610$		0			0
$611$	1.93087	−	12.2462i	0.0781146	−	0.495429i
$612$		0			0
$613$	−43.3693			−1.75167			−0.875835	−	0.482610i	$-0.839689\pi$
$613$	−43.3693			−1.75167			−0.875835	+	0.482610i	$0.839689\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.2462			0.734565			0.367282	−	0.930109i	$-0.380288\pi$
$617$	18.2462			0.734565			0.367282	+	0.930109i	$0.380288\pi$
$618$		0			0
$619$		−	23.6155i		−	0.949188i	−0.880205	−	0.474594i	$-0.842595\pi$
$619$		−	23.6155i		−	0.949188i	0.880205	−	0.474594i	$-0.157405\pi$
$620$		0			0
$621$	−7.36932			−0.295720
$622$		0			0
$623$		−	23.3693i		−	0.936272i
$624$		0			0
$625$		0			0
$626$		0			0
$627$			16.0000i			0.638978i
$628$		0			0
$629$			3.05398i			0.121770i
$630$		0			0
$631$		−	7.43845i		−	0.296120i	−0.988978	−	0.148060i	$-0.952697\pi$
$631$		−	7.43845i		−	0.296120i	0.988978	−	0.148060i	$-0.0473029\pi$
$632$		0			0
$633$		−	38.5616i		−	1.53268i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	7.75379	−	49.1771i	0.307216	−	1.94847i
$638$		0			0
$639$		−	20.2462i		−	0.800928i
$640$		0			0
$641$	20.2462			0.799677			0.399839	−	0.916586i	$-0.369066\pi$
$641$	20.2462			0.799677			0.399839	+	0.916586i	$0.369066\pi$
$642$		0			0
$643$	−16.8769			−0.665560			−0.332780	−	0.943005i	$-0.607987\pi$
$643$	−16.8769			−0.665560			−0.332780	+	0.943005i	$0.607987\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			5.12311i			0.201410i	0.994916	+	0.100705i	$0.0321098\pi$
$647$			5.12311i			0.201410i	−0.994916	+	0.100705i	$0.967890\pi$
$648$		0			0
$649$	−31.2311			−1.22593
$650$		0			0
$651$	36.4924			1.43025
$652$		0			0
$653$		−	27.6155i		−	1.08068i	−0.841447	−	0.540340i	$-0.818296\pi$
$653$		−	27.6155i		−	1.08068i	0.841447	−	0.540340i	$-0.181704\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−54.7386			−2.13556
$658$		0			0
$659$	42.7386			1.66486			0.832430	−	0.554130i	$-0.186949\pi$
$659$	42.7386			1.66486			0.832430	+	0.554130i	$0.186949\pi$
$660$		0			0
$661$		−	41.1231i		−	1.59950i	−0.600331	−	0.799752i	$-0.704964\pi$
$661$		−	41.1231i		−	1.59950i	0.600331	−	0.799752i	$-0.295036\pi$
$662$		0			0
$663$	−0.807764	+	5.12311i	−0.0313710	+	0.198965i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	16.0000i		−	0.619522i
$668$		0			0
$669$			46.5616i			1.80017i
$670$		0			0
$671$			34.7386i			1.34107i
$672$		0			0
$673$			31.3002i			1.20653i	0.797539	+	0.603267i	$0.206135\pi$
$673$			31.3002i			1.20653i	−0.797539	+	0.603267i	$0.793865\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	20.7386i		−	0.797050i	−0.917157	−	0.398525i	$-0.869522\pi$
$677$		−	20.7386i		−	0.797050i	0.917157	−	0.398525i	$-0.130478\pi$
$678$		0			0
$679$	−46.7386			−1.79366
$680$		0			0
$681$		−	54.7386i		−	2.09759i
$682$		0			0
$683$	4.24621			0.162477			0.0812384	−	0.996695i	$-0.474112\pi$
$683$	4.24621			0.162477			0.0812384	+	0.996695i	$0.474112\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	21.3002			0.812653
$688$		0			0
$689$	−15.1231	−	2.38447i	−0.576144	−	0.0908411i
$690$		0			0
$691$		−	29.8617i		−	1.13599i	−0.823031	−	0.567997i	$-0.807718\pi$
$691$		−	29.8617i		−	1.13599i	0.823031	−	0.567997i	$-0.192282\pi$
$692$		0			0
$693$			50.7386i			1.92740i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	4.49242			0.170163
$698$		0			0
$699$	1.43845			0.0544071
$700$		0			0
$701$	−45.2311			−1.70835			−0.854177	−	0.519983i	$-0.825938\pi$
$701$	−45.2311			−1.70835			−0.854177	+	0.519983i	$0.825938\pi$
$702$		0			0
$703$		−	10.8769i		−	0.410230i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−60.9848			−2.29357
$708$		0			0
$709$		−	1.12311i		−	0.0421791i	−0.999778	−	0.0210896i	$-0.993286\pi$
$709$		−	1.12311i		−	0.0421791i	0.999778	−	0.0210896i	$-0.00671351\pi$
$710$		0			0
$711$	−10.2462			−0.384263
$712$		0			0
$713$	16.0000			0.599205
$714$		0			0
$715$		0			0
$716$		0			0
$717$	51.0540			1.90665
$718$		0			0
$719$	16.0000			0.596699			0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000			0.596699			0.298350	+	0.954457i	$0.403564\pi$
$720$		0			0
$721$			23.3693i			0.870319i
$722$		0			0
$723$	48.3542			1.79831
$724$		0			0
$725$		0			0
$726$		0			0
$727$			43.2311i			1.60335i	0.597759	+	0.801676i	$0.296058\pi$
$727$			43.2311i			1.60335i	−0.597759	+	0.801676i	$0.703942\pi$
$728$		0			0
$729$	35.9848			1.33277
$730$		0			0
$731$	3.05398			0.112955
$732$		0			0
$733$	−49.3002			−1.82094			−0.910472	−	0.413571i	$-0.864281\pi$
$733$	−49.3002			−1.82094			−0.910472	+	0.413571i	$0.864281\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			2.73863i			0.100879i
$738$		0			0
$739$		−	41.8617i		−	1.53991i	−0.638099	−	0.769954i	$-0.720279\pi$
$739$		−	41.8617i		−	1.53991i	0.638099	−	0.769954i	$-0.279721\pi$
$740$		0			0
$741$	2.87689	−	18.2462i	0.105685	−	0.670291i
$742$		0			0
$743$	−45.0540			−1.65287			−0.826435	−	0.563032i	$-0.809635\pi$
$743$	−45.0540			−1.65287			−0.826435	+	0.563032i	$0.809635\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	29.3693			1.07457
$748$		0			0
$749$		−	18.2462i		−	0.666702i
$750$		0			0
$751$	−36.4924			−1.33163			−0.665814	−	0.746118i	$-0.731915\pi$
$751$	−36.4924			−1.33163			−0.665814	+	0.746118i	$0.731915\pi$
$752$		0			0
$753$			21.7538i			0.792752i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	22.0000i		−	0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$		−	22.0000i		−	0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$		0			0
$759$			40.9848i			1.48766i
$760$		0			0
$761$		−	53.1231i		−	1.92571i	−0.270017	−	0.962856i	$-0.587029\pi$
$761$		−	53.1231i		−	1.92571i	0.270017	−	0.962856i	$-0.412971\pi$
$762$		0			0
$763$			71.5464i			2.59015i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	35.6155	+	5.61553i	1.28600	+	0.202765i
$768$		0			0
$769$		−	4.49242i		−	0.162001i	−0.996714	−	0.0810004i	$-0.974188\pi$
$769$		−	4.49242i		−	0.162001i	0.996714	−	0.0810004i	$-0.0258115\pi$
$770$		0			0
$771$	71.5464			2.57668
$772$		0			0
$773$	23.0540			0.829194			0.414597	−	0.910005i	$-0.363923\pi$
$773$	23.0540			0.829194			0.414597	+	0.910005i	$0.363923\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	63.5464i		−	2.27971i
$778$		0			0
$779$	−16.0000			−0.573259
$780$		0			0
$781$	−17.7538			−0.635281
$782$		0			0
$783$		−	4.49242i		−	0.160546i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	21.3693			0.761734			0.380867	−	0.924630i	$-0.375626\pi$
$787$	21.3693			0.761734			0.380867	+	0.924630i	$0.375626\pi$
$788$		0			0
$789$	72.9848			2.59833
$790$		0			0
$791$		−	45.6155i		−	1.62190i
$792$		0			0
$793$	6.24621	−	39.6155i	0.221809	−	1.40679i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			4.24621i			0.150409i	0.997168	+	0.0752043i	$0.0239609\pi$
$797$			4.24621i			0.150409i	−0.997168	+	0.0752043i	$0.976039\pi$
$798$		0			0
$799$		−	1.93087i		−	0.0683093i
$800$		0			0
$801$		−	18.2462i		−	0.644698i
$802$		0			0
$803$			48.0000i			1.69388i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			48.9848i			1.72435i
$808$		0			0
$809$	−30.3153			−1.06583			−0.532915	−	0.846169i	$-0.678904\pi$
$809$	−30.3153			−1.06583			−0.532915	+	0.846169i	$0.678904\pi$
$810$		0			0
$811$		−	1.36932i		−	0.0480832i	−0.999711	−	0.0240416i	$-0.992347\pi$
$811$		−	1.36932i		−	0.0480832i	0.999711	−	0.0240416i	$-0.00765342\pi$
$812$		0			0
$813$	−5.93087			−0.208005
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−10.8769			−0.380534
$818$		0			0
$819$	9.12311	−	57.8617i	0.318787	−	2.02185i
$820$		0			0
$821$		−	27.5464i		−	0.961376i	−0.876892	−	0.480688i	$-0.840387\pi$
$821$		−	27.5464i		−	0.961376i	0.876892	−	0.480688i	$-0.159613\pi$
$822$		0			0
$823$			40.3542i			1.40666i	0.710865	+	0.703329i	$0.248304\pi$
$823$			40.3542i			1.40666i	−0.710865	+	0.703329i	$0.751696\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	22.0000			0.765015			0.382507	−	0.923952i	$-0.375061\pi$
$827$	22.0000			0.765015			0.382507	+	0.923952i	$0.375061\pi$
$828$		0			0
$829$	15.1231			0.525247			0.262624	−	0.964898i	$-0.415412\pi$
$829$	15.1231			0.525247			0.262624	+	0.964898i	$0.415412\pi$
$830$		0			0
$831$	44.4924			1.54343
$832$		0			0
$833$		−	7.75379i		−	0.268653i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	4.49242			0.155281
$838$		0			0
$839$		−	13.8617i		−	0.478560i	−0.970951	−	0.239280i	$-0.923089\pi$
$839$		−	13.8617i		−	0.478560i	0.970951	−	0.239280i	$-0.0769114\pi$
$840$		0			0
$841$	−19.2462			−0.663662
$842$		0			0
$843$	39.3693			1.35595
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−5.68466			−0.195327
$848$		0			0
$849$	30.7386			1.05495
$850$		0			0
$851$		−	27.8617i		−	0.955088i
$852$		0			0
$853$	−7.68466			−0.263118			−0.131559	−	0.991308i	$-0.541998\pi$
$853$	−7.68466			−0.263118			−0.131559	+	0.991308i	$0.541998\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			17.5076i			0.598047i	0.954246	+	0.299024i	$0.0966610\pi$
$857$			17.5076i			0.598047i	−0.954246	+	0.299024i	$0.903339\pi$
$858$		0			0
$859$	−6.24621			−0.213118			−0.106559	−	0.994306i	$-0.533983\pi$
$859$	−6.24621			−0.213118			−0.106559	+	0.994306i	$0.533983\pi$
$860$		0			0
$861$	−93.4773			−3.18570
$862$		0			0
$863$	10.9460			0.372607			0.186304	−	0.982492i	$-0.440349\pi$
$863$	10.9460			0.372607			0.186304	+	0.982492i	$0.440349\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	42.7386i		−	1.45148i
$868$		0			0
$869$			8.98485i			0.304790i
$870$		0			0
$871$	0.492423	−	3.12311i	0.0166851	−	0.105822i
$872$		0			0
$873$	−36.4924			−1.23508
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−54.4233			−1.83774			−0.918872	−	0.394556i	$-0.870898\pi$
$877$	−54.4233			−1.83774			−0.918872	+	0.394556i	$0.870898\pi$
$878$		0			0
$879$			0.807764i			0.0272452i
$880$		0			0
$881$	43.7926			1.47541			0.737705	−	0.675123i	$-0.235909\pi$
$881$	43.7926			1.47541			0.737705	+	0.675123i	$0.235909\pi$
$882$		0			0
$883$		−	11.1922i		−	0.376649i	−0.982107	−	0.188324i	$-0.939694\pi$
$883$		−	11.1922i		−	0.376649i	0.982107	−	0.188324i	$-0.0603056\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			54.7386i			1.83794i	0.394323	+	0.918972i	$0.370979\pi$
$887$			54.7386i			1.83794i	−0.394323	+	0.918972i	$0.629021\pi$
$888$		0			0
$889$		−	59.8617i		−	2.00770i
$890$		0			0
$891$		−	21.8617i		−	0.732396i
$892$		0			0
$893$			6.87689i			0.230126i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	7.36932	−	46.7386i	0.246054	−	1.56056i
$898$		0			0
$899$			9.75379i			0.325307i
$900$		0			0
$901$	−2.38447			−0.0794383
$902$		0			0
$903$	−63.5464			−2.11469
$904$		0			0
$905$		0			0
$906$		0			0
$907$			26.5616i			0.881962i	0.897516	+	0.440981i	$0.145369\pi$
$907$			26.5616i			0.881962i	−0.897516	+	0.440981i	$0.854631\pi$
$908$		0			0
$909$	−47.6155			−1.57931
$910$		0			0
$911$	42.2462			1.39968			0.699840	−	0.714300i	$-0.253255\pi$
$911$	42.2462			1.39968			0.699840	+	0.714300i	$0.253255\pi$
$912$		0			0
$913$		−	25.7538i		−	0.852326i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	71.5464			2.36267
$918$		0			0
$919$	−18.2462			−0.601887			−0.300943	−	0.953642i	$-0.597302\pi$
$919$	−18.2462			−0.601887			−0.300943	+	0.953642i	$0.597302\pi$
$920$		0			0
$921$			46.1080i			1.51931i
$922$		0			0
$923$	20.2462	+	3.19224i	0.666412	+	0.105074i
$924$		0			0
$925$		0			0
$926$		0			0
$927$			18.2462i			0.599284i
$928$		0			0
$929$			30.7386i			1.00850i	0.863557	+	0.504251i	$0.168231\pi$
$929$			30.7386i			1.00850i	−0.863557	+	0.504251i	$0.831769\pi$
$930$		0			0
$931$			27.6155i			0.905062i
$932$		0			0
$933$		−	27.8617i		−	0.912152i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	36.2462i		−	1.18411i	−0.805897	−	0.592056i	$-0.798316\pi$
$937$		−	36.2462i		−	1.18411i	0.805897	−	0.592056i	$-0.201684\pi$
$938$		0			0
$939$	21.9309			0.715687
$940$		0			0
$941$			27.5464i			0.897987i	0.893535	+	0.448993i	$0.148217\pi$
$941$			27.5464i			0.897987i	−0.893535	+	0.448993i	$0.851783\pi$
$942$		0			0
$943$	−40.9848			−1.33465
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.36932			−0.304462			−0.152231	−	0.988345i	$-0.548646\pi$
$947$	−9.36932			−0.304462			−0.152231	+	0.988345i	$0.548646\pi$
$948$		0			0
$949$	8.63068	−	54.7386i	0.280164	−	1.77689i
$950$		0			0
$951$			17.6155i			0.571223i
$952$		0			0
$953$		−	17.0540i		−	0.552432i	−0.961096	−	0.276216i	$-0.910919\pi$
$953$		−	17.0540i		−	0.552432i	0.961096	−	0.276216i	$-0.0890806\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−24.9848			−0.807645
$958$		0			0
$959$	23.3693			0.754635
$960$		0			0
$961$	21.2462			0.685362
$962$		0			0
$963$		−	14.2462i		−	0.459078i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	11.4384			0.367836			0.183918	−	0.982942i	$-0.441122\pi$
$967$	11.4384			0.367836			0.183918	+	0.982942i	$0.441122\pi$
$968$		0			0
$969$		−	2.87689i		−	0.0924192i
$970$		0			0
$971$	−11.8229			−0.379416			−0.189708	−	0.981841i	$-0.560754\pi$
$971$	−11.8229			−0.379416			−0.189708	+	0.981841i	$0.560754\pi$
$972$		0			0
$973$	58.4233			1.87296
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−35.2311			−1.12714			−0.563571	−	0.826068i	$-0.690573\pi$
$977$	−35.2311			−1.12714			−0.563571	+	0.826068i	$0.690573\pi$
$978$		0			0
$979$	−16.0000			−0.511362
$980$		0			0
$981$			55.8617i			1.78353i
$982$		0			0
$983$	30.8078			0.982615			0.491308	−	0.870986i	$-0.336519\pi$
$983$	30.8078			0.982615			0.491308	+	0.870986i	$0.336519\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			40.1771i			1.27885i
$988$		0			0
$989$	−27.8617			−0.885952
$990$		0			0
$991$	33.6155			1.06783			0.533916	−	0.845537i	$-0.320720\pi$
$991$	33.6155			1.06783			0.533916	+	0.845537i	$0.320720\pi$
$992$		0			0
$993$	41.6155			1.32063
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	39.6155i		−	1.25464i	−0.778763	−	0.627318i	$-0.784153\pi$
$997$		−	39.6155i		−	1.25464i	0.778763	−	0.627318i	$-0.215847\pi$
$998$		0			0
$999$		−	7.82292i		−	0.247506i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2600.2.f.a.649.1		4
5.2	odd	4		104.2.f.a.25.1	✓	4
5.3	odd	4		2600.2.k.a.2001.4		4
5.4	even	2		2600.2.f.b.649.4		4
13.12	even	2		2600.2.f.b.649.1		4
15.2	even	4		936.2.c.d.649.4		4
20.7	even	4		208.2.f.b.129.3		4
40.27	even	4		832.2.f.f.129.2		4
40.37	odd	4		832.2.f.i.129.4		4
60.47	odd	4		1872.2.c.j.1585.4		4
65.2	even	12		1352.2.i.g.529.2		4
65.7	even	12		1352.2.i.h.1329.2		4
65.12	odd	4		104.2.f.a.25.2	yes	4
65.17	odd	12		1352.2.o.e.361.4		8
65.22	odd	12		1352.2.o.e.361.3		8
65.32	even	12		1352.2.i.g.1329.2		4
65.37	even	12		1352.2.i.h.529.2		4
65.38	odd	4		2600.2.k.a.2001.3		4
65.42	odd	12		1352.2.o.e.1161.3		8
65.47	even	4		1352.2.a.e.1.1		2
65.57	even	4		1352.2.a.d.1.1		2
65.62	odd	12		1352.2.o.e.1161.4		8
65.64	even	2	inner	2600.2.f.a.649.4		4
195.77	even	4		936.2.c.d.649.1		4
260.47	odd	4		2704.2.a.t.1.2		2
260.187	odd	4		2704.2.a.s.1.2		2
260.207	even	4		208.2.f.b.129.4		4
520.77	odd	4		832.2.f.i.129.3		4
520.467	even	4		832.2.f.f.129.1		4
780.467	odd	4		1872.2.c.j.1585.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.f.a.25.1	✓	4	5.2	odd	4
104.2.f.a.25.2	yes	4	65.12	odd	4
208.2.f.b.129.3		4	20.7	even	4
208.2.f.b.129.4		4	260.207	even	4
832.2.f.f.129.1		4	520.467	even	4
832.2.f.f.129.2		4	40.27	even	4
832.2.f.i.129.3		4	520.77	odd	4
832.2.f.i.129.4		4	40.37	odd	4
936.2.c.d.649.1		4	195.77	even	4
936.2.c.d.649.4		4	15.2	even	4
1352.2.a.d.1.1		2	65.57	even	4
1352.2.a.e.1.1		2	65.47	even	4
1352.2.i.g.529.2		4	65.2	even	12
1352.2.i.g.1329.2		4	65.32	even	12
1352.2.i.h.529.2		4	65.37	even	12
1352.2.i.h.1329.2		4	65.7	even	12
1352.2.o.e.361.3		8	65.22	odd	12
1352.2.o.e.361.4		8	65.17	odd	12
1352.2.o.e.1161.3		8	65.42	odd	12
1352.2.o.e.1161.4		8	65.62	odd	12
1872.2.c.j.1585.1		4	780.467	odd	4
1872.2.c.j.1585.4		4	60.47	odd	4
2600.2.f.a.649.1		4	1.1	even	1	trivial
2600.2.f.a.649.4		4	65.64	even	2	inner
2600.2.f.b.649.1		4	13.12	even	2
2600.2.f.b.649.4		4	5.4	even	2
2600.2.k.a.2001.3		4	65.38	odd	4
2600.2.k.a.2001.4		4	5.3	odd	4
2704.2.a.s.1.2		2	260.187	odd	4
2704.2.a.t.1.2		2	260.47	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2600.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.56155i q^{3} -4.56155 q^{7} -3.56155 q^{9} +3.12311i q^{11} +(0.561553 - 3.56155i) q^{13} -0.561553i q^{17} +2.00000i q^{19} +11.6847i q^{21} +5.12311i q^{23} +1.43845i q^{27} -3.12311 q^{29} -3.12311i q^{31} +8.00000 q^{33} -5.43845 q^{37} +(-9.12311 - 1.43845i) q^{39} +8.00000i q^{41} +5.43845i q^{43} +3.43845 q^{47} +13.8078 q^{49} -1.43845 q^{51} -4.24621i q^{53} +5.12311 q^{57} +10.0000i q^{59} +11.1231 q^{61} +16.2462 q^{63} +0.876894 q^{67} +13.1231 q^{69} +5.68466i q^{71} +15.3693 q^{73} -14.2462i q^{77} +2.87689 q^{79} -7.00000 q^{81} -8.24621 q^{83} +8.00000i q^{87} +5.12311i q^{89} +(-2.56155 + 16.2462i) q^{91} -8.00000 q^{93} +10.2462 q^{97} -11.1231i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{7} - 6 q^{9} - 6 q^{13} + 4 q^{29} + 32 q^{33} - 30 q^{37} - 20 q^{39} + 22 q^{47} + 14 q^{49} - 14 q^{51} + 4 q^{57} + 28 q^{61} + 32 q^{63} + 20 q^{67} + 36 q^{69} + 12 q^{73} + 28 q^{79}+ \cdots + 8 q^{97}+O(q^{100}) \) 4 * q - 10 * q^7 - 6 * q^9 - 6 * q^13 + 4 * q^29 + 32 * q^33 - 30 * q^37 - 20 * q^39 + 22 * q^47 + 14 * q^49 - 14 * q^51 + 4 * q^57 + 28 * q^61 + 32 * q^63 + 20 * q^67 + 36 * q^69 + 12 * q^73 + 28 * q^79 - 28 * q^81 - 2 * q^91 - 32 * q^93 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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