LMFDB - Embedded newform 2912.2.i.a.337.74

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2912,2,Mod(337,2912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2912.337"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.i (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.74
Character	$\chi$	$=$	2912.337
Dual form			2912.2.i.a.337.73

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.58847i q^{3} +3.12702 q^{5} +1.00000i q^{7} +0.476774 q^{9} +5.71492 q^{11} +(0.942265 - 3.48025i) q^{13} +4.96717i q^{15} -2.06335 q^{17} -7.11418 q^{19} -1.58847 q^{21} +7.24092 q^{23} +4.77826 q^{25} +5.52274i q^{27} -1.66947i q^{29} +5.84705i q^{31} +9.07796i q^{33} +3.12702i q^{35} +9.75151 q^{37} +(5.52826 + 1.49676i) q^{39} -1.47733i q^{41} -2.72493i q^{43} +1.49088 q^{45} -13.5277i q^{47} -1.00000 q^{49} -3.27757i q^{51} +2.27021i q^{53} +17.8707 q^{55} -11.3006i q^{57} -8.38121 q^{59} +8.37594i q^{61} +0.476774i q^{63} +(2.94648 - 10.8828i) q^{65} -0.139962 q^{67} +11.5020i q^{69} -6.92315i q^{71} +2.08543i q^{73} +7.59011i q^{75} +5.71492i q^{77} +4.96160 q^{79} -7.34237 q^{81} -10.3016 q^{83} -6.45215 q^{85} +2.65190 q^{87} -9.56091i q^{89} +(3.48025 + 0.942265i) q^{91} -9.28785 q^{93} -22.2462 q^{95} +1.34231i q^{97} +2.72472 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 84 q^{9} + 8 q^{17} + 24 q^{23} + 92 q^{25} + 24 q^{39} - 84 q^{49} - 32 q^{55} - 24 q^{65} + 40 q^{79} + 84 q^{81} + 48 q^{87} + 16 q^{95}+O(q^{100}) $ 84 * q - 84 * q^9 + 8 * q^17 + 24 * q^23 + 92 * q^25 + 24 * q^39 - 84 * q^49 - 32 * q^55 - 24 * q^65 + 40 * q^79 + 84 * q^81 + 48 * q^87 + 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$			1.58847i			0.917102i	0.888668	+	0.458551i	$0.151631\pi$
$3$			1.58847i			0.917102i	−0.888668	+	0.458551i	$0.848369\pi$
$4$		0			0
$5$	3.12702			1.39845			0.699223	−	0.714903i	$-0.253529\pi$
$5$	3.12702			1.39845			0.699223	+	0.714903i	$0.253529\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	0.476774			0.158925
$10$		0			0
$11$	5.71492			1.72311			0.861556	−	0.507662i	$-0.169490\pi$
$11$	5.71492			1.72311			0.861556	+	0.507662i	$0.169490\pi$
$12$		0			0
$13$	0.942265	−	3.48025i	0.261337	−	0.965248i
$13$	0.942265	−	3.48025i	0.261337	−	0.965248i
$14$		0			0
$15$			4.96717i			1.28252i
$16$		0			0
$17$	−2.06335			−0.500437			−0.250218	−	0.968189i	$-0.580502\pi$
$17$	−2.06335			−0.500437			−0.250218	+	0.968189i	$0.580502\pi$
$18$		0			0
$19$	−7.11418			−1.63210			−0.816052	−	0.577978i	$-0.803842\pi$
$19$	−7.11418			−1.63210			−0.816052	+	0.577978i	$0.803842\pi$
$20$		0			0
$21$	−1.58847			−0.346632
$22$		0			0
$23$	7.24092			1.50984			0.754918	−	0.655819i	$-0.227676\pi$
$23$	7.24092			1.50984			0.754918	+	0.655819i	$0.227676\pi$
$24$		0			0
$25$	4.77826			0.955652
$26$		0			0
$27$			5.52274i			1.06285i
$28$		0			0
$29$		−	1.66947i		−	0.310013i	−0.987913	−	0.155006i	$-0.950460\pi$
$29$		−	1.66947i		−	0.310013i	0.987913	−	0.155006i	$-0.0495398\pi$
$30$		0			0
$31$			5.84705i			1.05016i	0.851052	+	0.525081i	$0.175965\pi$
$31$			5.84705i			1.05016i	−0.851052	+	0.525081i	$0.824035\pi$
$32$		0			0
$33$			9.07796i			1.58027i
$34$		0			0
$35$			3.12702i			0.528563i
$36$		0			0
$37$	9.75151			1.60314			0.801569	−	0.597902i	$-0.203999\pi$
$37$	9.75151			1.60314			0.801569	+	0.597902i	$0.203999\pi$
$38$		0			0
$39$	5.52826	+	1.49676i	0.885230	+	0.239673i
$40$		0			0
$41$		−	1.47733i		−	0.230720i	−0.993324	−	0.115360i	$-0.963198\pi$
$41$		−	1.47733i		−	0.230720i	0.993324	−	0.115360i	$-0.0368022\pi$
$42$		0			0
$43$		−	2.72493i		−	0.415548i	−0.978177	−	0.207774i	$-0.933378\pi$
$43$		−	2.72493i		−	0.415548i	0.978177	−	0.207774i	$-0.0666218\pi$
$44$		0			0
$45$	1.49088			0.222248
$46$		0			0
$47$		−	13.5277i		−	1.97321i	−0.163116	−	0.986607i	$-0.552154\pi$
$47$		−	13.5277i		−	1.97321i	0.163116	−	0.986607i	$-0.447846\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		−	3.27757i		−	0.458951i
$52$		0			0
$53$			2.27021i			0.311838i	0.987770	+	0.155919i	$0.0498339\pi$
$53$			2.27021i			0.311838i	−0.987770	+	0.155919i	$0.950166\pi$
$54$		0			0
$55$	17.8707			2.40968
$56$		0			0
$57$		−	11.3006i		−	1.49681i
$58$		0			0
$59$	−8.38121			−1.09114			−0.545570	−	0.838065i	$-0.683687\pi$
$59$	−8.38121			−1.09114			−0.545570	+	0.838065i	$0.683687\pi$
$60$		0			0
$61$			8.37594i			1.07243i	0.844082	+	0.536215i	$0.180146\pi$
$61$			8.37594i			1.07243i	−0.844082	+	0.536215i	$0.819854\pi$
$62$		0			0
$63$			0.476774i			0.0600679i
$64$		0			0
$65$	2.94648	−	10.8828i	0.365466	−	1.34985i
$66$		0			0
$67$	−0.139962			−0.0170991			−0.00854956	−	0.999963i	$-0.502721\pi$
$67$	−0.139962			−0.0170991			−0.00854956	+	0.999963i	$0.502721\pi$
$68$		0			0
$69$			11.5020i			1.38467i
$70$		0			0
$71$		−	6.92315i		−	0.821627i	−0.911719	−	0.410814i	$-0.865245\pi$
$71$		−	6.92315i		−	0.821627i	0.911719	−	0.410814i	$-0.134755\pi$
$72$		0			0
$73$			2.08543i			0.244081i	0.992525	+	0.122040i	$0.0389438\pi$
$73$			2.08543i			0.244081i	−0.992525	+	0.122040i	$0.961056\pi$
$74$		0			0
$75$			7.59011i			0.876430i
$76$		0			0
$77$			5.71492i			0.651275i
$78$		0			0
$79$	4.96160			0.558223			0.279112	−	0.960259i	$-0.409960\pi$
$79$	4.96160			0.558223			0.279112	+	0.960259i	$0.409960\pi$
$80$		0			0
$81$	−7.34237			−0.815818
$82$		0			0
$83$	−10.3016			−1.13075			−0.565374	−	0.824835i	$-0.691268\pi$
$83$	−10.3016			−1.13075			−0.565374	+	0.824835i	$0.691268\pi$
$84$		0			0
$85$	−6.45215			−0.699834
$86$		0			0
$87$	2.65190			0.284313
$88$		0			0
$89$		−	9.56091i		−	1.01345i	−0.862107	−	0.506727i	$-0.830855\pi$
$89$		−	9.56091i		−	1.01345i	0.862107	−	0.506727i	$-0.169145\pi$
$90$		0			0
$91$	3.48025	+	0.942265i	0.364829	+	0.0987762i
$92$		0			0
$93$	−9.28785			−0.963105
$94$		0			0
$95$	−22.2462			−2.28241
$96$		0			0
$97$			1.34231i			0.136291i	0.997675	+	0.0681455i	$0.0217082\pi$
$97$			1.34231i			0.136291i	−0.997675	+	0.0681455i	$0.978292\pi$
$98$		0			0
$99$	2.72472			0.273845
$100$		0			0
$101$		−	7.97650i		−	0.793692i	−0.917885	−	0.396846i	$-0.870105\pi$
$101$		−	7.97650i		−	0.793692i	0.917885	−	0.396846i	$-0.129895\pi$
$102$		0			0
$103$	−4.05005			−0.399063			−0.199531	−	0.979891i	$-0.563942\pi$
$103$	−4.05005			−0.399063			−0.199531	+	0.979891i	$0.563942\pi$
$104$		0			0
$105$	−4.96717			−0.484746
$106$		0			0
$107$			11.3278i			1.09510i	0.836774	+	0.547548i	$0.184439\pi$
$107$			11.3278i			1.09510i	−0.836774	+	0.547548i	$0.815561\pi$
$108$		0			0
$109$	10.2168			0.978588			0.489294	−	0.872119i	$-0.337254\pi$
$109$	10.2168			0.978588			0.489294	+	0.872119i	$0.337254\pi$
$110$		0			0
$111$			15.4900i			1.47024i
$112$		0			0
$113$	5.69472			0.535714			0.267857	−	0.963459i	$-0.413684\pi$
$113$	5.69472			0.535714			0.267857	+	0.963459i	$0.413684\pi$
$114$		0			0
$115$	22.6425			2.11143
$116$		0			0
$117$	0.449247	−	1.65929i	0.0415329	−	0.153402i
$118$		0			0
$119$		−	2.06335i		−	0.189147i
$120$		0			0
$121$	21.6603			1.96912
$122$		0			0
$123$	2.34669			0.211594
$124$		0			0
$125$	−0.693378			−0.0620176
$126$		0			0
$127$	−15.2373			−1.35209			−0.676047	−	0.736859i	$-0.736308\pi$
$127$	−15.2373			−1.35209			−0.676047	+	0.736859i	$0.736308\pi$
$128$		0			0
$129$	4.32846			0.381099
$130$		0			0
$131$			9.35861i			0.817666i	0.912609	+	0.408833i	$0.134064\pi$
$131$			9.35861i			0.817666i	−0.912609	+	0.408833i	$0.865936\pi$
$132$		0			0
$133$		−	7.11418i		−	0.616878i
$134$		0			0
$135$			17.2697i			1.48634i
$136$		0			0
$137$			1.38971i			0.118731i	0.998236	+	0.0593655i	$0.0189077\pi$
$137$			1.38971i			0.118731i	−0.998236	+	0.0593655i	$0.981092\pi$
$138$		0			0
$139$		−	3.04493i		−	0.258268i	−0.991627	−	0.129134i	$-0.958780\pi$
$139$		−	3.04493i		−	0.258268i	0.991627	−	0.129134i	$-0.0412197\pi$
$140$		0			0
$141$	21.4883			1.80964
$142$		0			0
$143$	5.38497	−	19.8893i	0.450314	−	1.66323i
$144$		0			0
$145$		−	5.22047i		−	0.433536i
$146$		0			0
$147$		−	1.58847i		−	0.131015i
$148$		0			0
$149$	−3.72346			−0.305038			−0.152519	−	0.988301i	$-0.548738\pi$
$149$	−3.72346			−0.305038			−0.152519	+	0.988301i	$0.548738\pi$
$150$		0			0
$151$			18.1030i			1.47320i	0.676328	+	0.736600i	$0.263570\pi$
$151$			18.1030i			1.47320i	−0.676328	+	0.736600i	$0.736430\pi$
$152$		0			0
$153$	−0.983753			−0.0795317
$154$		0			0
$155$			18.2839i			1.46859i
$156$		0			0
$157$			11.6727i			0.931583i	0.884894	+	0.465792i	$0.154230\pi$
$157$			11.6727i			0.931583i	−0.884894	+	0.465792i	$0.845770\pi$
$158$		0			0
$159$	−3.60616			−0.285987
$160$		0			0
$161$			7.24092i			0.570665i
$162$		0			0
$163$	−17.1521			−1.34346			−0.671730	−	0.740796i	$-0.734448\pi$
$163$	−17.1521			−1.34346			−0.671730	+	0.740796i	$0.734448\pi$
$164$		0			0
$165$			28.3870i			2.20992i
$166$		0			0
$167$		−	13.2013i		−	1.02154i	−0.859716	−	0.510772i	$-0.829359\pi$
$167$		−	13.2013i		−	1.02154i	0.859716	−	0.510772i	$-0.170641\pi$
$168$		0			0
$169$	−11.2243	−	6.55864i	−0.863406	−	0.504510i
$170$		0			0
$171$	−3.39186			−0.259382
$172$		0			0
$173$			11.8696i			0.902430i	0.892415	+	0.451215i	$0.149009\pi$
$173$			11.8696i			0.902430i	−0.892415	+	0.451215i	$0.850991\pi$
$174$		0			0
$175$			4.77826i			0.361203i
$176$		0			0
$177$		−	13.3133i		−	1.00069i
$178$		0			0
$179$			3.07406i			0.229766i	0.993379	+	0.114883i	$0.0366493\pi$
$179$			3.07406i			0.229766i	−0.993379	+	0.114883i	$0.963351\pi$
$180$		0			0
$181$			0.896220i			0.0666155i	0.999445	+	0.0333078i	$0.0106041\pi$
$181$			0.896220i			0.0666155i	−0.999445	+	0.0333078i	$0.989396\pi$
$182$		0			0
$183$	−13.3049			−0.983527
$184$		0			0
$185$	30.4932			2.24190
$186$		0			0
$187$	−11.7919			−0.862309
$188$		0			0
$189$	−5.52274			−0.401720
$190$		0			0
$191$	9.06136			0.655657			0.327828	−	0.944737i	$-0.393683\pi$
$191$	9.06136			0.655657			0.327828	+	0.944737i	$0.393683\pi$
$192$		0			0
$193$			16.9037i			1.21676i	0.793647	+	0.608379i	$0.208180\pi$
$193$			16.9037i			1.21676i	−0.793647	+	0.608379i	$0.791820\pi$
$194$		0			0
$195$	17.2870	+	4.68039i	1.23795	+	0.335170i
$196$		0			0
$197$	−15.2870			−1.08915			−0.544575	−	0.838712i	$-0.683309\pi$
$197$	−15.2870			−1.08915			−0.544575	+	0.838712i	$0.683309\pi$
$198$		0			0
$199$	18.3528			1.30100			0.650498	−	0.759508i	$-0.274560\pi$
$199$	18.3528			1.30100			0.650498	+	0.759508i	$0.274560\pi$
$200$		0			0
$201$		−	0.222325i		−	0.0156816i
$202$		0			0
$203$	1.66947			0.117174
$204$		0			0
$205$		−	4.61964i		−	0.322650i
$206$		0			0
$207$	3.45228			0.239950
$208$		0			0
$209$	−40.6570			−2.81230
$210$		0			0
$211$			18.2550i			1.25672i	0.777921	+	0.628362i	$0.216274\pi$
$211$			18.2550i			1.25672i	−0.777921	+	0.628362i	$0.783726\pi$
$212$		0			0
$213$	10.9972			0.753516
$214$		0			0
$215$		−	8.52091i		−	0.581121i
$216$		0			0
$217$	−5.84705			−0.396924
$218$		0			0
$219$	−3.31263			−0.223847
$220$		0			0
$221$	−1.94423	+	7.18098i	−0.130783	+	0.483045i
$222$		0			0
$223$		−	7.07767i		−	0.473956i	−0.971515	−	0.236978i	$-0.923843\pi$
$223$		−	7.07767i		−	0.473956i	0.971515	−	0.236978i	$-0.0761570\pi$
$224$		0			0
$225$	2.27815			0.151877
$226$		0			0
$227$	8.47553			0.562541			0.281270	−	0.959629i	$-0.409244\pi$
$227$	8.47553			0.562541			0.281270	+	0.959629i	$0.409244\pi$
$228$		0			0
$229$	−29.3790			−1.94142			−0.970709	−	0.240260i	$-0.922767\pi$
$229$	−29.3790			−1.94142			−0.970709	+	0.240260i	$0.922767\pi$
$230$		0			0
$231$	−9.07796			−0.597286
$232$		0			0
$233$	0.232073			0.0152036			0.00760180	−	0.999971i	$-0.497580\pi$
$233$	0.232073			0.0152036			0.00760180	+	0.999971i	$0.497580\pi$
$234$		0			0
$235$		−	42.3013i		−	2.75943i
$236$		0			0
$237$			7.88133i			0.511947i
$238$		0			0
$239$		−	22.7221i		−	1.46977i	−0.678193	−	0.734884i	$-0.737237\pi$
$239$		−	22.7221i		−	1.46977i	0.678193	−	0.734884i	$-0.262763\pi$
$240$		0			0
$241$		−	6.78095i		−	0.436800i	−0.975859	−	0.218400i	$-0.929916\pi$
$241$		−	6.78095i		−	0.436800i	0.975859	−	0.218400i	$-0.0700837\pi$
$242$		0			0
$243$			4.90512i			0.314663i
$244$		0			0
$245$	−3.12702			−0.199778
$246$		0			0
$247$	−6.70344	+	24.7591i	−0.426530	+	1.57539i
$248$		0			0
$249$		−	16.3638i		−	1.03701i
$250$		0			0
$251$		−	3.20949i		−	0.202582i	−0.994857	−	0.101291i	$-0.967703\pi$
$251$		−	3.20949i		−	0.202582i	0.994857	−	0.101291i	$-0.0322972\pi$
$252$		0			0
$253$	41.3813			2.60162
$254$		0			0
$255$		−	10.2490i		−	0.641819i
$256$		0			0
$257$	13.4708			0.840288			0.420144	−	0.907457i	$-0.361980\pi$
$257$	13.4708			0.840288			0.420144	+	0.907457i	$0.361980\pi$
$258$		0			0
$259$			9.75151i			0.605929i
$260$		0			0
$261$		−	0.795960i		−	0.0492687i
$262$		0			0
$263$	−17.7133			−1.09225			−0.546125	−	0.837704i	$-0.683897\pi$
$263$	−17.7133			−1.09225			−0.546125	+	0.837704i	$0.683897\pi$
$264$		0			0
$265$			7.09901i			0.436088i
$266$		0			0
$267$	15.1872			0.929440
$268$		0			0
$269$		−	4.99183i		−	0.304357i	−0.988353	−	0.152178i	$-0.951371\pi$
$269$		−	4.99183i		−	0.304357i	0.988353	−	0.152178i	$-0.0486289\pi$
$270$		0			0
$271$		−	21.5931i		−	1.31169i	−0.754896	−	0.655845i	$-0.772313\pi$
$271$		−	21.5931i		−	1.31169i	0.754896	−	0.655845i	$-0.227687\pi$
$272$		0			0
$273$	−1.49676	+	5.52826i	−0.0905878	+	0.334586i
$274$		0			0
$275$	27.3074			1.64670
$276$		0			0
$277$			23.4358i			1.40812i	0.710139	+	0.704061i	$0.248632\pi$
$277$			23.4358i			1.40812i	−0.710139	+	0.704061i	$0.751368\pi$
$278$		0			0
$279$			2.78772i			0.166897i
$280$		0			0
$281$			16.9262i			1.00973i	0.863197	+	0.504867i	$0.168459\pi$
$281$			16.9262i			1.00973i	−0.863197	+	0.504867i	$0.831541\pi$
$282$		0			0
$283$		−	8.33061i		−	0.495203i	−0.968862	−	0.247602i	$-0.920358\pi$
$283$		−	8.33061i		−	0.495203i	0.968862	−	0.247602i	$-0.0796425\pi$
$284$		0			0
$285$		−	35.3373i		−	2.09320i
$286$		0			0
$287$	1.47733			0.0872041
$288$		0			0
$289$	−12.7426			−0.749563
$290$		0			0
$291$	−2.13222			−0.124993
$292$		0			0
$293$	−1.81142			−0.105824			−0.0529122	−	0.998599i	$-0.516850\pi$
$293$	−1.81142			−0.105824			−0.0529122	+	0.998599i	$0.516850\pi$
$294$		0			0
$295$	−26.2082			−1.52590
$296$		0			0
$297$			31.5620i			1.83141i
$298$		0			0
$299$	6.82287	−	25.2002i	0.394577	−	1.45737i
$300$		0			0
$301$	2.72493			0.157062
$302$		0			0
$303$	12.6704			0.727896
$304$		0			0
$305$			26.1917i			1.49973i
$306$		0			0
$307$	−33.0031			−1.88359			−0.941794	−	0.336192i	$-0.890861\pi$
$307$	−33.0031			−1.88359			−0.941794	+	0.336192i	$0.890861\pi$
$308$		0			0
$309$		−	6.43336i		−	0.365981i
$310$		0			0
$311$	9.23137			0.523463			0.261732	−	0.965141i	$-0.415707\pi$
$311$	9.23137			0.523463			0.261732	+	0.965141i	$0.415707\pi$
$312$		0			0
$313$	20.1921			1.14133			0.570664	−	0.821184i	$-0.306686\pi$
$313$	20.1921			1.14133			0.570664	+	0.821184i	$0.306686\pi$
$314$		0			0
$315$			1.49088i			0.0840017i
$316$		0			0
$317$	−4.03071			−0.226387			−0.113194	−	0.993573i	$-0.536108\pi$
$317$	−4.03071			−0.226387			−0.113194	+	0.993573i	$0.536108\pi$
$318$		0			0
$319$		−	9.54089i		−	0.534187i
$320$		0			0
$321$	−17.9938			−1.00431
$322$		0			0
$323$	14.6791			0.816765
$324$		0			0
$325$	4.50239	−	16.6295i	0.249748	−	0.922441i
$326$		0			0
$327$			16.2290i			0.897465i
$328$		0			0
$329$	13.5277			0.745805
$330$		0			0
$331$	−11.5976			−0.637463			−0.318731	−	0.947845i	$-0.603257\pi$
$331$	−11.5976			−0.637463			−0.318731	+	0.947845i	$0.603257\pi$
$332$		0			0
$333$	4.64927			0.254778
$334$		0			0
$335$	−0.437665			−0.0239122
$336$		0			0
$337$	−24.0032			−1.30754			−0.653769	−	0.756695i	$-0.726813\pi$
$337$	−24.0032			−1.30754			−0.653769	+	0.756695i	$0.726813\pi$
$338$		0			0
$339$			9.04588i			0.491305i
$340$		0			0
$341$			33.4154i			1.80955i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			35.9669i			1.93639i
$346$		0			0
$347$			33.8649i			1.81796i	0.416836	+	0.908982i	$0.363139\pi$
$347$			33.8649i			1.81796i	−0.416836	+	0.908982i	$0.636861\pi$
$348$		0			0
$349$	−8.65554			−0.463320			−0.231660	−	0.972797i	$-0.574416\pi$
$349$	−8.65554			−0.463320			−0.231660	+	0.972797i	$0.574416\pi$
$350$		0			0
$351$	19.2205	+	5.20388i	1.02591	+	0.277763i
$352$		0			0
$353$			18.6871i			0.994615i	0.867574	+	0.497308i	$0.165678\pi$
$353$			18.6871i			0.994615i	−0.867574	+	0.497308i	$0.834322\pi$
$354$		0			0
$355$		−	21.6489i		−	1.14900i
$356$		0			0
$357$	3.27757			0.173467
$358$		0			0
$359$			11.5291i			0.608485i	0.952595	+	0.304242i	$0.0984032\pi$
$359$			11.5291i			0.608485i	−0.952595	+	0.304242i	$0.901597\pi$
$360$		0			0
$361$	31.6116			1.66377
$362$		0			0
$363$			34.4066i			1.80588i
$364$		0			0
$365$			6.52118i			0.341334i
$366$		0			0
$367$	23.7235			1.23836			0.619178	−	0.785250i	$-0.287466\pi$
$367$	23.7235			1.23836			0.619178	+	0.785250i	$0.287466\pi$
$368$		0			0
$369$		−	0.704353i		−	0.0366671i
$370$		0			0
$371$	−2.27021			−0.117864
$372$		0			0
$373$			0.110535i			0.00572327i	0.999996	+	0.00286163i	$0.000910888\pi$
$373$			0.110535i			0.00572327i	−0.999996	+	0.00286163i	$0.999089\pi$
$374$		0			0
$375$		−	1.10141i		−	0.0568765i
$376$		0			0
$377$	−5.81017	−	1.57308i	−0.299239	−	0.0810180i
$378$		0			0
$379$	22.8948			1.17603			0.588014	−	0.808851i	$-0.299910\pi$
$379$	22.8948			1.17603			0.588014	+	0.808851i	$0.299910\pi$
$380$		0			0
$381$		−	24.2040i		−	1.24001i
$382$		0			0
$383$			5.53185i			0.282664i	0.989962	+	0.141332i	$0.0451386\pi$
$383$			5.53185i			0.282664i	−0.989962	+	0.141332i	$0.954861\pi$
$384$		0			0
$385$			17.8707i			0.910774i
$386$		0			0
$387$		−	1.29917i		−	0.0660408i
$388$		0			0
$389$		−	17.9481i		−	0.910005i	−0.890490	−	0.455003i	$-0.849638\pi$
$389$		−	17.9481i		−	0.910005i	0.890490	−	0.455003i	$-0.150362\pi$
$390$		0			0
$391$	−14.9406			−0.755578
$392$		0			0
$393$	−14.8658			−0.749883
$394$		0			0
$395$	15.5150			0.780645
$396$		0			0
$397$	−27.2050			−1.36538			−0.682690	−	0.730708i	$-0.739190\pi$
$397$	−27.2050			−1.36538			−0.682690	+	0.730708i	$0.739190\pi$
$398$		0			0
$399$	11.3006			0.565739
$400$		0			0
$401$			0.884157i			0.0441527i	0.999756	+	0.0220764i	$0.00702769\pi$
$401$			0.884157i			0.0441527i	−0.999756	+	0.0220764i	$0.992972\pi$
$402$		0			0
$403$	20.3492	+	5.50947i	1.01367	+	0.274446i
$404$		0			0
$405$	−22.9597			−1.14088
$406$		0			0
$407$	55.7291			2.76239
$408$		0			0
$409$		−	32.4864i		−	1.60635i	−0.595744	−	0.803174i	$-0.703143\pi$
$409$		−	32.4864i		−	1.60635i	0.595744	−	0.803174i	$-0.296857\pi$
$410$		0			0
$411$	−2.20751			−0.108888
$412$		0			0
$413$		−	8.38121i		−	0.412412i
$414$		0			0
$415$	−32.2133			−1.58129
$416$		0			0
$417$	4.83678			0.236858
$418$		0			0
$419$		−	3.36137i		−	0.164214i	−0.996624	−	0.0821068i	$-0.973835\pi$
$419$		−	3.36137i		−	0.164214i	0.996624	−	0.0821068i	$-0.0261649\pi$
$420$		0			0
$421$	−1.18230			−0.0576220			−0.0288110	−	0.999585i	$-0.509172\pi$
$421$	−1.18230			−0.0576220			−0.0288110	+	0.999585i	$0.509172\pi$
$422$		0			0
$423$		−	6.44964i		−	0.313592i
$424$		0			0
$425$	−9.85924			−0.478244
$426$		0			0
$427$	−8.37594			−0.405340
$428$		0			0
$429$	31.5936	+	8.55384i	1.52535	+	0.412983i
$430$		0			0
$431$			6.88144i			0.331467i	0.986171	+	0.165734i	$0.0529992\pi$
$431$			6.88144i			0.331467i	−0.986171	+	0.165734i	$0.947001\pi$
$432$		0			0
$433$	13.0548			0.627371			0.313686	−	0.949527i	$-0.398436\pi$
$433$	13.0548			0.627371			0.313686	+	0.949527i	$0.398436\pi$
$434$		0			0
$435$	8.29254			0.397597
$436$		0			0
$437$	−51.5132			−2.46421
$438$		0			0
$439$	−11.1615			−0.532712			−0.266356	−	0.963875i	$-0.585820\pi$
$439$	−11.1615			−0.532712			−0.266356	+	0.963875i	$0.585820\pi$
$440$		0			0
$441$	−0.476774			−0.0227035
$442$		0			0
$443$		−	37.5895i		−	1.78593i	−0.450123	−	0.892966i	$-0.648620\pi$
$443$		−	37.5895i		−	1.78593i	0.450123	−	0.892966i	$-0.351380\pi$
$444$		0			0
$445$		−	29.8972i		−	1.41726i
$446$		0			0
$447$		−	5.91459i		−	0.279751i
$448$		0			0
$449$		−	22.7894i		−	1.07550i	−0.843105	−	0.537749i	$-0.819275\pi$
$449$		−	22.7894i		−	1.07550i	0.843105	−	0.537749i	$-0.180725\pi$
$450$		0			0
$451$		−	8.44282i		−	0.397557i
$452$		0			0
$453$	−28.7560			−1.35107
$454$		0			0
$455$	10.8828	+	2.94648i	0.510194	+	0.138133i
$456$		0			0
$457$			35.4078i			1.65630i	0.560503	+	0.828152i	$0.310608\pi$
$457$			35.4078i			1.65630i	−0.560503	+	0.828152i	$0.689392\pi$
$458$		0			0
$459$		−	11.3954i		−	0.531890i
$460$		0			0
$461$	−6.70049			−0.312073			−0.156036	−	0.987751i	$-0.549872\pi$
$461$	−6.70049			−0.312073			−0.156036	+	0.987751i	$0.549872\pi$
$462$		0			0
$463$			13.1048i			0.609033i	0.952507	+	0.304517i	$0.0984949\pi$
$463$			13.1048i			0.609033i	−0.952507	+	0.304517i	$0.901505\pi$
$464$		0			0
$465$	−29.0433			−1.34685
$466$		0			0
$467$		−	36.0468i		−	1.66805i	−0.551730	−	0.834023i	$-0.686032\pi$
$467$		−	36.0468i		−	1.66805i	0.551730	−	0.834023i	$-0.313968\pi$
$468$		0			0
$469$		−	0.139962i		−	0.00646286i
$470$		0			0
$471$	−18.5417			−0.854356
$472$		0			0
$473$		−	15.5727i		−	0.716035i
$474$		0			0
$475$	−33.9934			−1.55972
$476$		0			0
$477$			1.08238i			0.0495587i
$478$		0			0
$479$		−	21.1536i		−	0.966532i	−0.875474	−	0.483266i	$-0.839450\pi$
$479$		−	21.1536i		−	0.966532i	0.875474	−	0.483266i	$-0.160550\pi$
$480$		0			0
$481$	9.18851	−	33.9377i	0.418960	−	1.54743i
$482$		0			0
$483$	−11.5020			−0.523358
$484$		0			0
$485$			4.19744i			0.190596i
$486$		0			0
$487$		−	35.8876i		−	1.62622i	−0.582110	−	0.813110i	$-0.697773\pi$
$487$		−	35.8876i		−	1.62622i	0.582110	−	0.813110i	$-0.302227\pi$
$488$		0			0
$489$		−	27.2456i		−	1.23209i
$490$		0			0
$491$		−	29.6539i		−	1.33826i	−0.743145	−	0.669130i	$-0.766667\pi$
$491$		−	29.6539i		−	1.33826i	0.743145	−	0.669130i	$-0.233333\pi$
$492$		0			0
$493$			3.44471i			0.155142i
$494$		0			0
$495$	8.52027			0.382958
$496$		0			0
$497$	6.92315			0.310546
$498$		0			0
$499$	−10.4844			−0.469346			−0.234673	−	0.972074i	$-0.575402\pi$
$499$	−10.4844			−0.469346			−0.234673	+	0.972074i	$0.575402\pi$
$500$		0			0
$501$	20.9698			0.936860
$502$		0			0
$503$	−0.831971			−0.0370958			−0.0185479	−	0.999828i	$-0.505904\pi$
$503$	−0.831971			−0.0370958			−0.0185479	+	0.999828i	$0.505904\pi$
$504$		0			0
$505$		−	24.9427i		−	1.10994i
$506$		0			0
$507$	10.4182	−	17.8294i	0.462687	−	0.791831i
$508$		0			0
$509$	−6.83338			−0.302884			−0.151442	−	0.988466i	$-0.548392\pi$
$509$	−6.83338			−0.302884			−0.151442	+	0.988466i	$0.548392\pi$
$510$		0			0
$511$	−2.08543			−0.0922539
$512$		0			0
$513$		−	39.2898i		−	1.73469i
$514$		0			0
$515$	−12.6646			−0.558068
$516$		0			0
$517$		−	77.3095i		−	3.40007i
$518$		0			0
$519$	−18.8545			−0.827620
$520$		0			0
$521$	−15.0251			−0.658260			−0.329130	−	0.944285i	$-0.606755\pi$
$521$	−15.0251			−0.658260			−0.329130	+	0.944285i	$0.606755\pi$
$522$		0			0
$523$			39.8747i			1.74360i	0.489861	+	0.871800i	$0.337047\pi$
$523$			39.8747i			1.74360i	−0.489861	+	0.871800i	$0.662953\pi$
$524$		0			0
$525$	−7.59011			−0.331260
$526$		0			0
$527$		−	12.0645i		−	0.525539i
$528$		0			0
$529$	29.4310			1.27961
$530$		0			0
$531$	−3.99594			−0.173409
$532$		0			0
$533$	−5.14148	−	1.39204i	−0.222702	−	0.0602958i
$534$		0			0
$535$			35.4222i			1.53143i
$536$		0			0
$537$	−4.88304			−0.210719
$538$		0			0
$539$	−5.71492			−0.246159
$540$		0			0
$541$	−20.3026			−0.872879			−0.436439	−	0.899734i	$-0.643761\pi$
$541$	−20.3026			−0.872879			−0.436439	+	0.899734i	$0.643761\pi$
$542$		0			0
$543$	−1.42362			−0.0610932
$544$		0			0
$545$	31.9480			1.36850
$546$		0			0
$547$		−	13.5144i		−	0.577833i	−0.957354	−	0.288916i	$-0.906705\pi$
$547$		−	13.5144i		−	0.577833i	0.957354	−	0.288916i	$-0.0932949\pi$
$548$		0			0
$549$			3.99343i			0.170435i
$550$		0			0
$551$			11.8769i			0.505974i
$552$		0			0
$553$			4.96160i			0.210989i
$554$		0			0
$555$			48.4374i			2.05605i
$556$		0			0
$557$	27.9302			1.18344			0.591720	−	0.806144i	$-0.298449\pi$
$557$	27.9302			1.18344			0.591720	+	0.806144i	$0.298449\pi$
$558$		0			0
$559$	−9.48343	−	2.56761i	−0.401106	−	0.108598i
$560$		0			0
$561$		−	18.7310i		−	0.790825i
$562$		0			0
$563$			17.8687i			0.753076i	0.926401	+	0.376538i	$0.122886\pi$
$563$			17.8687i			0.753076i	−0.926401	+	0.376538i	$0.877114\pi$
$564$		0			0
$565$	17.8075			0.749168
$566$		0			0
$567$		−	7.34237i		−	0.308350i
$568$		0			0
$569$	−7.97207			−0.334207			−0.167103	−	0.985939i	$-0.553441\pi$
$569$	−7.97207			−0.334207			−0.167103	+	0.985939i	$0.553441\pi$
$570$		0			0
$571$		−	35.3020i		−	1.47734i	−0.674065	−	0.738672i	$-0.735453\pi$
$571$		−	35.3020i		−	1.47734i	0.674065	−	0.738672i	$-0.264547\pi$
$572$		0			0
$573$			14.3937i			0.601304i
$574$		0			0
$575$	34.5990			1.44288
$576$		0			0
$577$		−	3.82990i		−	0.159441i	−0.996817	−	0.0797204i	$-0.974597\pi$
$577$		−	3.82990i		−	0.159441i	0.996817	−	0.0797204i	$-0.0254027\pi$
$578$		0			0
$579$	−26.8510			−1.11589
$580$		0			0
$581$		−	10.3016i		−	0.427382i
$582$		0			0
$583$			12.9741i			0.537332i
$584$		0			0
$585$	1.40481	−	5.18864i	0.0580816	−	0.214524i
$586$		0			0
$587$	32.9822			1.36132			0.680661	−	0.732599i	$-0.261693\pi$
$587$	32.9822			1.36132			0.680661	+	0.732599i	$0.261693\pi$
$588$		0			0
$589$		−	41.5970i		−	1.71397i
$590$		0			0
$591$		−	24.2828i		−	0.998861i
$592$		0			0
$593$			9.15921i			0.376124i	0.982157	+	0.188062i	$0.0602205\pi$
$593$			9.15921i			0.376124i	−0.982157	+	0.188062i	$0.939779\pi$
$594$		0			0
$595$		−	6.45215i		−	0.264512i
$596$		0			0
$597$			29.1528i			1.19315i
$598$		0			0
$599$	−28.6067			−1.16884			−0.584419	−	0.811452i	$-0.698678\pi$
$599$	−28.6067			−1.16884			−0.584419	+	0.811452i	$0.698678\pi$
$600$		0			0
$601$	23.6133			0.963205			0.481603	−	0.876390i	$-0.340055\pi$
$601$	23.6133			0.963205			0.481603	+	0.876390i	$0.340055\pi$
$602$		0			0
$603$	−0.0667304			−0.00271747
$604$		0			0
$605$	67.7322			2.75370
$606$		0			0
$607$	−1.91438			−0.0777025			−0.0388512	−	0.999245i	$-0.512370\pi$
$607$	−1.91438			−0.0777025			−0.0388512	+	0.999245i	$0.512370\pi$
$608$		0			0
$609$			2.65190i			0.107460i
$610$		0			0
$611$	−47.0797	−	12.7467i	−1.90464	−	0.515674i
$612$		0			0
$613$	−5.37235			−0.216987			−0.108494	−	0.994097i	$-0.534603\pi$
$613$	−5.37235			−0.216987			−0.108494	+	0.994097i	$0.534603\pi$
$614$		0			0
$615$	7.33815			0.295903
$616$		0			0
$617$		−	8.91871i		−	0.359054i	−0.983753	−	0.179527i	$-0.942543\pi$
$617$		−	8.91871i		−	0.359054i	0.983753	−	0.179527i	$-0.0574567\pi$
$618$		0			0
$619$	27.5206			1.10614			0.553072	−	0.833133i	$-0.313455\pi$
$619$	27.5206			1.10614			0.553072	+	0.833133i	$0.313455\pi$
$620$		0			0
$621$			39.9897i			1.60473i
$622$		0			0
$623$	9.56091			0.383050
$624$		0			0
$625$	−26.0595			−1.04238
$626$		0			0
$627$		−	64.5822i		−	2.57917i
$628$		0			0
$629$	−20.1208			−0.802269
$630$		0			0
$631$		−	28.3172i		−	1.12729i	−0.826017	−	0.563645i	$-0.809399\pi$
$631$		−	28.3172i		−	1.12729i	0.826017	−	0.563645i	$-0.190601\pi$
$632$		0			0
$633$	−28.9974			−1.15254
$634$		0			0
$635$	−47.6474			−1.89083
$636$		0			0
$637$	−0.942265	+	3.48025i	−0.0373339	+	0.137893i
$638$		0			0
$639$		−	3.30078i		−	0.130577i
$640$		0			0
$641$	−9.42420			−0.372234			−0.186117	−	0.982528i	$-0.559590\pi$
$641$	−9.42420			−0.372234			−0.186117	+	0.982528i	$0.559590\pi$
$642$		0			0
$643$	7.56475			0.298325			0.149162	−	0.988813i	$-0.452342\pi$
$643$	7.56475			0.298325			0.149162	+	0.988813i	$0.452342\pi$
$644$		0			0
$645$	13.5352			0.532947
$646$		0			0
$647$	3.76754			0.148117			0.0740586	−	0.997254i	$-0.476405\pi$
$647$	3.76754			0.148117			0.0740586	+	0.997254i	$0.476405\pi$
$648$		0			0
$649$	−47.8979			−1.88016
$650$		0			0
$651$		−	9.28785i		−	0.364019i
$652$		0			0
$653$		−	13.6124i		−	0.532694i	−0.963877	−	0.266347i	$-0.914183\pi$
$653$		−	13.6124i		−	0.532694i	0.963877	−	0.266347i	$-0.0858166\pi$
$654$		0			0
$655$			29.2646i			1.14346i
$656$		0			0
$657$			0.994278i			0.0387905i
$658$		0			0
$659$		−	24.9897i		−	0.973461i	−0.873552	−	0.486731i	$-0.838189\pi$
$659$		−	24.9897i		−	0.973461i	0.873552	−	0.486731i	$-0.161811\pi$
$660$		0			0
$661$	−3.69054			−0.143545			−0.0717726	−	0.997421i	$-0.522866\pi$
$661$	−3.69054			−0.143545			−0.0717726	+	0.997421i	$0.522866\pi$
$662$		0			0
$663$	−11.4068	−	3.08834i	−0.443002	−	0.119941i
$664$		0			0
$665$		−	22.2462i		−	0.862670i
$666$		0			0
$667$		−	12.0885i		−	0.468069i
$668$		0			0
$669$	11.2426			0.434666
$670$		0			0
$671$			47.8678i			1.84792i
$672$		0			0
$673$	−22.8132			−0.879385			−0.439693	−	0.898148i	$-0.644913\pi$
$673$	−22.8132			−0.879385			−0.439693	+	0.898148i	$0.644913\pi$
$674$		0			0
$675$			26.3891i			1.01572i
$676$		0			0
$677$			28.7882i			1.10642i	0.833042	+	0.553210i	$0.186597\pi$
$677$			28.7882i			1.10642i	−0.833042	+	0.553210i	$0.813403\pi$
$678$		0			0
$679$	−1.34231			−0.0515132
$680$		0			0
$681$			13.4631i			0.515907i
$682$		0			0
$683$	−13.8519			−0.530030			−0.265015	−	0.964244i	$-0.585377\pi$
$683$	−13.8519			−0.530030			−0.265015	+	0.964244i	$0.585377\pi$
$684$		0			0
$685$			4.34565i			0.166039i
$686$		0			0
$687$		−	46.6675i		−	1.78048i
$688$		0			0
$689$	7.90091	+	2.13914i	0.301001	+	0.0814949i
$690$		0			0
$691$	2.49342			0.0948544			0.0474272	−	0.998875i	$-0.484898\pi$
$691$	2.49342			0.0948544			0.0474272	+	0.998875i	$0.484898\pi$
$692$		0			0
$693$			2.72472i			0.103504i
$694$		0			0
$695$		−	9.52158i		−	0.361174i
$696$		0			0
$697$			3.04825i			0.115461i
$698$		0			0
$699$			0.368640i			0.0139432i
$700$		0			0
$701$		−	27.1472i		−	1.02534i	−0.858587	−	0.512668i	$-0.828657\pi$
$701$		−	27.1472i		−	1.02534i	0.858587	−	0.512668i	$-0.171343\pi$
$702$		0			0
$703$	−69.3740			−2.61649
$704$		0			0
$705$	67.1942			2.53068
$706$		0			0
$707$	7.97650			0.299987
$708$		0			0
$709$	−17.4367			−0.654848			−0.327424	−	0.944878i	$-0.606180\pi$
$709$	−17.4367			−0.654848			−0.327424	+	0.944878i	$0.606180\pi$
$710$		0			0
$711$	2.36556			0.0887154
$712$		0			0
$713$			42.3380i			1.58557i
$714$		0			0
$715$	16.8389	−	62.1944i	0.629740	−	2.32594i
$716$		0			0
$717$	36.0932			1.34793
$718$		0			0
$719$	−8.99987			−0.335639			−0.167819	−	0.985818i	$-0.553673\pi$
$719$	−8.99987			−0.335639			−0.167819	+	0.985818i	$0.553673\pi$
$720$		0			0
$721$		−	4.05005i		−	0.150832i
$722$		0			0
$723$	10.7713			0.400590
$724$		0			0
$725$		−	7.97717i		−	0.296265i
$726$		0			0
$727$	17.6410			0.654269			0.327135	−	0.944978i	$-0.393917\pi$
$727$	17.6410			0.654269			0.327135	+	0.944978i	$0.393917\pi$
$728$		0			0
$729$	−29.8187			−1.10440
$730$		0			0
$731$			5.62249i			0.207955i
$732$		0			0
$733$	18.9500			0.699933			0.349967	−	0.936762i	$-0.386193\pi$
$733$	18.9500			0.699933			0.349967	+	0.936762i	$0.386193\pi$
$734$		0			0
$735$		−	4.96717i		−	0.183217i
$736$		0			0
$737$	−0.799873			−0.0294637
$738$		0			0
$739$	−45.9766			−1.69128			−0.845638	−	0.533756i	$-0.820780\pi$
$739$	−45.9766			−1.69128			−0.845638	+	0.533756i	$0.820780\pi$
$740$		0			0
$741$	−39.3290	−	10.6482i	−1.44479	−	0.391171i
$742$		0			0
$743$		−	19.1217i		−	0.701508i	−0.936468	−	0.350754i	$-0.885925\pi$
$743$		−	19.1217i		−	0.701508i	0.936468	−	0.350754i	$-0.114075\pi$
$744$		0			0
$745$	−11.6433			−0.426579
$746$		0			0
$747$	−4.91153			−0.179704
$748$		0			0
$749$	−11.3278			−0.413908
$750$		0			0
$751$	−18.2696			−0.666667			−0.333333	−	0.942809i	$-0.608174\pi$
$751$	−18.2696			−0.666667			−0.333333	+	0.942809i	$0.608174\pi$
$752$		0			0
$753$	5.09818			0.185788
$754$		0			0
$755$			56.6084i			2.06019i
$756$		0			0
$757$		−	20.9226i		−	0.760446i	−0.924895	−	0.380223i	$-0.875847\pi$
$757$		−	20.9226i		−	0.760446i	0.924895	−	0.380223i	$-0.124153\pi$
$758$		0			0
$759$			65.7328i			2.38595i
$760$		0			0
$761$		−	11.7957i		−	0.427595i	−0.976878	−	0.213798i	$-0.931417\pi$
$761$		−	11.7957i		−	0.427595i	0.976878	−	0.213798i	$-0.0685833\pi$
$762$		0			0
$763$			10.2168i			0.369872i
$764$		0			0
$765$	−3.07622			−0.111221
$766$		0			0
$767$	−7.89732	+	29.1687i	−0.285156	+	1.05322i
$768$		0			0
$769$		−	20.0712i		−	0.723785i	−0.932220	−	0.361892i	$-0.882131\pi$
$769$		−	20.0712i		−	0.723785i	0.932220	−	0.361892i	$-0.117869\pi$
$770$		0			0
$771$			21.3980i			0.770630i
$772$		0			0
$773$	2.21427			0.0796419			0.0398210	−	0.999207i	$-0.487321\pi$
$773$	2.21427			0.0796419			0.0398210	+	0.999207i	$0.487321\pi$
$774$		0			0
$775$			27.9387i			1.00359i
$776$		0			0
$777$	−15.4900			−0.555699
$778$		0			0
$779$			10.5100i			0.376560i
$780$		0			0
$781$		−	39.5653i		−	1.41576i
$782$		0			0
$783$	9.22005			0.329498
$784$		0			0
$785$			36.5008i			1.30277i
$786$		0			0
$787$	−20.3879			−0.726751			−0.363376	−	0.931643i	$-0.618376\pi$
$787$	−20.3879			−0.726751			−0.363376	+	0.931643i	$0.618376\pi$
$788$		0			0
$789$		−	28.1370i		−	1.00170i
$790$		0			0
$791$			5.69472i			0.202481i
$792$		0			0
$793$	29.1504	+	7.89236i	1.03516	+	0.280266i
$794$		0			0
$795$	−11.2765			−0.399937
$796$		0			0
$797$		−	16.3765i		−	0.580085i	−0.957014	−	0.290043i	$-0.906331\pi$
$797$		−	16.3765i		−	0.580085i	0.957014	−	0.290043i	$-0.0936695\pi$
$798$		0			0
$799$			27.9124i			0.987469i
$800$		0			0
$801$		−	4.55839i		−	0.161063i
$802$		0			0
$803$			11.9180i			0.420579i
$804$		0			0
$805$			22.6425i			0.798044i
$806$		0			0
$807$	7.92935			0.279126
$808$		0			0
$809$	25.7490			0.905286			0.452643	−	0.891692i	$-0.350481\pi$
$809$	25.7490			0.905286			0.452643	+	0.891692i	$0.350481\pi$
$810$		0			0
$811$	−16.1352			−0.566583			−0.283292	−	0.959034i	$-0.591426\pi$
$811$	−16.1352			−0.566583			−0.283292	+	0.959034i	$0.591426\pi$
$812$		0			0
$813$	34.3000			1.20295
$814$		0			0
$815$	−53.6351			−1.87876
$816$		0			0
$817$			19.3856i			0.678217i
$818$		0			0
$819$	1.65929	+	0.449247i	0.0579804	+	0.0156980i
$820$		0			0
$821$	−2.63810			−0.0920704			−0.0460352	−	0.998940i	$-0.514659\pi$
$821$	−2.63810			−0.0920704			−0.0460352	+	0.998940i	$0.514659\pi$
$822$		0			0
$823$	50.7066			1.76752			0.883760	−	0.467940i	$-0.155004\pi$
$823$	50.7066			1.76752			0.883760	+	0.467940i	$0.155004\pi$
$824$		0			0
$825$			43.3769i			1.51019i
$826$		0			0
$827$	14.2904			0.496924			0.248462	−	0.968642i	$-0.420075\pi$
$827$	14.2904			0.496924			0.248462	+	0.968642i	$0.420075\pi$
$828$		0			0
$829$		−	45.1542i		−	1.56827i	−0.620591	−	0.784134i	$-0.713107\pi$
$829$		−	45.1542i		−	1.56827i	0.620591	−	0.784134i	$-0.286893\pi$
$830$		0			0
$831$	−37.2270			−1.29139
$832$		0			0
$833$	2.06335			0.0714910
$834$		0			0
$835$		−	41.2806i		−	1.42858i
$836$		0			0
$837$	−32.2917			−1.11617
$838$		0			0
$839$			23.2329i			0.802090i	0.916058	+	0.401045i	$0.131353\pi$
$839$			23.2329i			0.802090i	−0.916058	+	0.401045i	$0.868647\pi$
$840$		0			0
$841$	26.2129			0.903892
$842$		0			0
$843$	−26.8868			−0.926029
$844$		0			0
$845$	−35.0985	−	20.5090i	−1.20743	−	0.705531i
$846$		0			0
$847$			21.6603i			0.744256i
$848$		0			0
$849$	13.2329			0.454152
$850$		0			0
$851$	70.6099			2.42048
$852$		0			0
$853$	31.4442			1.07663			0.538315	−	0.842744i	$-0.319061\pi$
$853$	31.4442			1.07663			0.538315	+	0.842744i	$0.319061\pi$
$854$		0			0
$855$	−10.6064			−0.362731
$856$		0			0
$857$	−31.3725			−1.07166			−0.535832	−	0.844324i	$-0.680002\pi$
$857$	−31.3725			−1.07166			−0.535832	+	0.844324i	$0.680002\pi$
$858$		0			0
$859$		−	3.80519i		−	0.129831i	−0.997891	−	0.0649157i	$-0.979322\pi$
$859$		−	3.80519i		−	0.129831i	0.997891	−	0.0649157i	$-0.0206778\pi$
$860$		0			0
$861$			2.34669i			0.0799750i
$862$		0			0
$863$		−	23.0017i		−	0.782987i	−0.920181	−	0.391494i	$-0.871958\pi$
$863$		−	23.0017i		−	0.782987i	0.920181	−	0.391494i	$-0.128042\pi$
$864$		0			0
$865$			37.1165i			1.26200i
$866$		0			0
$867$		−	20.2412i		−	0.687426i
$868$		0			0
$869$	28.3551			0.961881
$870$		0			0
$871$	−0.131882	+	0.487104i	−0.00446864	+	0.0165049i
$872$		0			0
$873$			0.639979i			0.0216600i
$874$		0			0
$875$		−	0.693378i		−	0.0234405i
$876$		0			0
$877$	18.4131			0.621766			0.310883	−	0.950448i	$-0.399375\pi$
$877$	18.4131			0.621766			0.310883	+	0.950448i	$0.399375\pi$
$878$		0			0
$879$		−	2.87738i		−	0.0970518i
$880$		0			0
$881$	50.2962			1.69452			0.847261	−	0.531177i	$-0.178250\pi$
$881$	50.2962			1.69452			0.847261	+	0.531177i	$0.178250\pi$
$882$		0			0
$883$			6.05996i			0.203934i	0.994788	+	0.101967i	$0.0325136\pi$
$883$			6.05996i			0.203934i	−0.994788	+	0.101967i	$0.967486\pi$
$884$		0			0
$885$		−	41.6309i		−	1.39941i
$886$		0			0
$887$	8.11875			0.272601			0.136301	−	0.990668i	$-0.456479\pi$
$887$	8.11875			0.272601			0.136301	+	0.990668i	$0.456479\pi$
$888$		0			0
$889$		−	15.2373i		−	0.511043i
$890$		0			0
$891$	−41.9610			−1.40575
$892$		0			0
$893$			96.2383i			3.22049i
$894$		0			0
$895$			9.61264i			0.321315i
$896$		0			0
$897$	40.0297	+	10.8379i	1.33655	+	0.361867i
$898$		0			0
$899$	9.76148			0.325564
$900$		0			0
$901$		−	4.68425i		−	0.156055i
$902$		0			0
$903$			4.32846i			0.144042i
$904$		0			0
$905$			2.80250i			0.0931582i
$906$		0			0
$907$			47.5887i			1.58016i	0.613006	+	0.790078i	$0.289960\pi$
$907$			47.5887i			1.58016i	−0.613006	+	0.790078i	$0.710040\pi$
$908$		0			0
$909$		−	3.80299i		−	0.126137i
$910$		0			0
$911$	40.4216			1.33923			0.669613	−	0.742710i	$-0.266460\pi$
$911$	40.4216			1.33923			0.669613	+	0.742710i	$0.266460\pi$
$912$		0			0
$913$	−58.8728			−1.94841
$914$		0			0
$915$	−41.6047			−1.37541
$916$		0			0
$917$	−9.35861			−0.309049
$918$		0			0
$919$	5.01405			0.165398			0.0826991	−	0.996575i	$-0.473646\pi$
$919$	5.01405			0.165398			0.0826991	+	0.996575i	$0.473646\pi$
$920$		0			0
$921$		−	52.4243i		−	1.72744i
$922$		0			0
$923$	−24.0943	−	6.52345i	−0.793074	−	0.214722i
$924$		0			0
$925$	46.5953			1.53204
$926$		0			0
$927$	−1.93096			−0.0634209
$928$		0			0
$929$			56.9403i			1.86815i	0.357075	+	0.934076i	$0.383774\pi$
$929$			56.9403i			1.86815i	−0.357075	+	0.934076i	$0.616226\pi$
$930$		0			0
$931$	7.11418			0.233158
$932$		0			0
$933$			14.6637i			0.480069i
$934$		0			0
$935$	−36.8735			−1.20589
$936$		0			0
$937$	−14.0722			−0.459718			−0.229859	−	0.973224i	$-0.573826\pi$
$937$	−14.0722			−0.459718			−0.229859	+	0.973224i	$0.573826\pi$
$938$		0			0
$939$			32.0745i			1.04671i
$940$		0			0
$941$	36.9952			1.20601			0.603005	−	0.797737i	$-0.293970\pi$
$941$	36.9952			1.20601			0.603005	+	0.797737i	$0.293970\pi$
$942$		0			0
$943$		−	10.6972i		−	0.348350i
$944$		0			0
$945$	−17.2697			−0.561784
$946$		0			0
$947$	34.5738			1.12350			0.561748	−	0.827308i	$-0.310129\pi$
$947$	34.5738			1.12350			0.561748	+	0.827308i	$0.310129\pi$
$948$		0			0
$949$	7.25781	+	1.96503i	0.235598	+	0.0637874i
$950$		0			0
$951$		−	6.40264i		−	0.207620i
$952$		0			0
$953$	−55.5958			−1.80093			−0.900463	−	0.434933i	$-0.856772\pi$
$953$	−55.5958			−1.80093			−0.900463	+	0.434933i	$0.856772\pi$
$954$		0			0
$955$	28.3351			0.916901
$956$		0			0
$957$	15.1554			0.489904
$958$		0			0
$959$	−1.38971			−0.0448761
$960$		0			0
$961$	−3.18801			−0.102839
$962$		0			0
$963$			5.40078i			0.174038i
$964$		0			0
$965$			52.8584i			1.70157i
$966$		0			0
$967$		−	33.2862i		−	1.07041i	−0.844722	−	0.535206i	$-0.820234\pi$
$967$		−	33.2862i		−	1.07041i	0.844722	−	0.535206i	$-0.179766\pi$
$968$		0			0
$969$			23.3172i			0.749057i
$970$		0			0
$971$			36.4853i			1.17087i	0.810719	+	0.585435i	$0.199076\pi$
$971$			36.4853i			1.17087i	−0.810719	+	0.585435i	$0.800924\pi$
$972$		0			0
$973$	3.04493			0.0976161
$974$		0			0
$975$	26.4155	+	7.15190i	0.845972	+	0.229044i
$976$		0			0
$977$		−	38.7164i		−	1.23865i	−0.785136	−	0.619323i	$-0.787407\pi$
$977$		−	38.7164i		−	1.23865i	0.785136	−	0.619323i	$-0.212593\pi$
$978$		0			0
$979$		−	54.6398i		−	1.74630i
$980$		0			0
$981$	4.87109			0.155522
$982$		0			0
$983$		−	12.2739i		−	0.391478i	−0.980656	−	0.195739i	$-0.937289\pi$
$983$		−	12.2739i		−	0.391478i	0.980656	−	0.195739i	$-0.0627105\pi$
$984$		0			0
$985$	−47.8026			−1.52312
$986$		0			0
$987$			21.4883i			0.683979i
$988$		0			0
$989$		−	19.7310i		−	0.627409i
$990$		0			0
$991$	6.99246			0.222123			0.111061	−	0.993814i	$-0.464575\pi$
$991$	6.99246			0.222123			0.111061	+	0.993814i	$0.464575\pi$
$992$		0			0
$993$		−	18.4224i		−	0.584618i
$994$		0			0
$995$	57.3896			1.81937
$996$		0			0
$997$			24.7971i			0.785333i	0.919681	+	0.392666i	$0.128447\pi$
$997$			24.7971i			0.785333i	−0.919681	+	0.392666i	$0.871553\pi$
$998$		0			0
$999$			53.8551i			1.70390i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.i.a.337.74		84
4.3	odd	2		728.2.i.a.701.49	yes	84
8.3	odd	2		728.2.i.a.701.35	✓	84
8.5	even	2	inner	2912.2.i.a.337.11		84
13.12	even	2	inner	2912.2.i.a.337.12		84
52.51	odd	2		728.2.i.a.701.36	yes	84
104.51	odd	2		728.2.i.a.701.50	yes	84
104.77	even	2	inner	2912.2.i.a.337.73		84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.i.a.701.35	✓	84	8.3	odd	2
728.2.i.a.701.36	yes	84	52.51	odd	2
728.2.i.a.701.49	yes	84	4.3	odd	2
728.2.i.a.701.50	yes	84	104.51	odd	2
2912.2.i.a.337.11		84	8.5	even	2	inner
2912.2.i.a.337.12		84	13.12	even	2	inner
2912.2.i.a.337.73		84	104.77	even	2	inner
2912.2.i.a.337.74		84	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.58847i q^{3} +3.12702 q^{5} +1.00000i q^{7} +0.476774 q^{9} +5.71492 q^{11} +(0.942265 - 3.48025i) q^{13} +4.96717i q^{15} -2.06335 q^{17} -7.11418 q^{19} -1.58847 q^{21} +7.24092 q^{23} +4.77826 q^{25} +5.52274i q^{27} -1.66947i q^{29} +5.84705i q^{31} +9.07796i q^{33} +3.12702i q^{35} +9.75151 q^{37} +(5.52826 + 1.49676i) q^{39} -1.47733i q^{41} -2.72493i q^{43} +1.49088 q^{45} -13.5277i q^{47} -1.00000 q^{49} -3.27757i q^{51} +2.27021i q^{53} +17.8707 q^{55} -11.3006i q^{57} -8.38121 q^{59} +8.37594i q^{61} +0.476774i q^{63} +(2.94648 - 10.8828i) q^{65} -0.139962 q^{67} +11.5020i q^{69} -6.92315i q^{71} +2.08543i q^{73} +7.59011i q^{75} +5.71492i q^{77} +4.96160 q^{79} -7.34237 q^{81} -10.3016 q^{83} -6.45215 q^{85} +2.65190 q^{87} -9.56091i q^{89} +(3.48025 + 0.942265i) q^{91} -9.28785 q^{93} -22.2462 q^{95} +1.34231i q^{97} +2.72472 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 84 q^{9} + 8 q^{17} + 24 q^{23} + 92 q^{25} + 24 q^{39} - 84 q^{49} - 32 q^{55} - 24 q^{65} + 40 q^{79} + 84 q^{81} + 48 q^{87} + 16 q^{95}+O(q^{100}) \) 84 * q - 84 * q^9 + 8 * q^17 + 24 * q^23 + 92 * q^25 + 24 * q^39 - 84 * q^49 - 32 * q^55 - 24 * q^65 + 40 * q^79 + 84 * q^81 + 48 * q^87 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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