LMFDB - Embedded newform 9280.2.a.bg.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9280,2,Mod(1,9280)] 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("9280.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9280 = 2^{6} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9280.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,3,0,7,0,2,0,2,0,9,0,-3,0,1,0,6,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	yes
Analytic conductor:	$74.1011730757$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4640)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	9280.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.11491 q^{3} +1.00000 q^{5} +4.75698 q^{7} +6.70265 q^{9} -2.94567 q^{11} +0.885092 q^{13} -3.11491 q^{15} -1.47283 q^{17} -2.22982 q^{19} -14.8176 q^{21} -3.11491 q^{23} +1.00000 q^{25} -11.5334 q^{27} -1.00000 q^{29} -4.18869 q^{31} +9.17548 q^{33} +4.75698 q^{35} -7.51396 q^{37} -2.75698 q^{39} -0.945668 q^{41} -3.70265 q^{43} +6.70265 q^{45} -3.17548 q^{47} +15.6289 q^{49} +4.58774 q^{51} +13.9325 q^{53} -2.94567 q^{55} +6.94567 q^{57} +13.2361 q^{59} -4.62887 q^{61} +31.8844 q^{63} +0.885092 q^{65} -0.824517 q^{67} +9.70265 q^{69} -9.89134 q^{71} -2.90454 q^{73} -3.11491 q^{75} -14.0125 q^{77} +6.54661 q^{79} +15.8176 q^{81} -4.94567 q^{83} -1.47283 q^{85} +3.11491 q^{87} +13.6615 q^{89} +4.21037 q^{91} +13.0474 q^{93} -2.22982 q^{95} +11.5745 q^{97} -19.7438 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} + 7 q^{7} + 2 q^{9} + 2 q^{11} + 9 q^{13} - 3 q^{15} + q^{17} + 6 q^{19} - 20 q^{21} - 3 q^{23} + 3 q^{25} - 12 q^{27} - 3 q^{29} - 9 q^{31} + 4 q^{33} + 7 q^{35} - 8 q^{37} - q^{39}+ \cdots - 32 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 + 7 * q^7 + 2 * q^9 + 2 * q^11 + 9 * q^13 - 3 * q^15 + q^17 + 6 * q^19 - 20 * q^21 - 3 * q^23 + 3 * q^25 - 12 * q^27 - 3 * q^29 - 9 * q^31 + 4 * q^33 + 7 * q^35 - 8 * q^37 - q^39 + 8 * q^41 + 7 * q^43 + 2 * q^45 + 14 * q^47 + 26 * q^49 + 2 * q^51 + 11 * q^53 + 2 * q^55 + 10 * q^57 - q^59 + 7 * q^61 + 26 * q^63 + 9 * q^65 - 26 * q^67 + 11 * q^69 - 8 * q^71 - 7 * q^73 - 3 * q^75 + 14 * q^77 + 17 * q^79 + 23 * q^81 - 4 * q^83 + q^85 + 3 * q^87 + 32 * q^89 + 8 * q^91 + 2 * q^93 + 6 * q^95 + 3 * q^97 - 32 * q^99

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
$3$	−3.11491	−1.79839	−0.899196	−	0.437545i	$-0.855848\pi$
$3$	−3.11491	−1.79839	−0.899196	+	0.437545i	$0.855848\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.75698	1.79797	0.898985	−	0.437980i	$-0.144306\pi$
$7$	4.75698	1.79797	0.898985	+	0.437980i	$0.144306\pi$
$8$	0	0
$9$	6.70265	2.23422
$10$	0	0
$11$	−2.94567	−0.888152	−0.444076	−	0.895989i	$-0.646468\pi$
$11$	−2.94567	−0.888152	−0.444076	+	0.895989i	$0.646468\pi$
$12$	0	0
$13$	0.885092	0.245480	0.122740	−	0.992439i	$-0.460832\pi$
$13$	0.885092	0.245480	0.122740	+	0.992439i	$0.460832\pi$
$14$	0	0
$15$	−3.11491	−0.804266
$16$	0	0
$17$	−1.47283	−0.357215	−0.178607	−	0.983920i	$-0.557159\pi$
$17$	−1.47283	−0.357215	−0.178607	+	0.983920i	$0.557159\pi$
$18$	0	0
$19$	−2.22982	−0.511555	−0.255777	−	0.966736i	$-0.582331\pi$
$19$	−2.22982	−0.511555	−0.255777	+	0.966736i	$0.582331\pi$
$20$	0	0
$21$	−14.8176	−3.23346
$22$	0	0
$23$	−3.11491	−0.649503	−0.324752	−	0.945799i	$-0.605281\pi$
$23$	−3.11491	−0.649503	−0.324752	+	0.945799i	$0.605281\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−11.5334	−2.21961
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−4.18869	−0.752310	−0.376155	−	0.926557i	$-0.622754\pi$
$31$	−4.18869	−0.752310	−0.376155	+	0.926557i	$0.622754\pi$
$32$	0	0
$33$	9.17548	1.59725
$34$	0	0
$35$	4.75698	0.804077
$36$	0	0
$37$	−7.51396	−1.23529	−0.617644	−	0.786458i	$-0.711913\pi$
$37$	−7.51396	−1.23529	−0.617644	+	0.786458i	$0.711913\pi$
$38$	0	0
$39$	−2.75698	−0.441470
$40$	0	0
$41$	−0.945668	−0.147689	−0.0738443	−	0.997270i	$-0.523527\pi$
$41$	−0.945668	−0.147689	−0.0738443	+	0.997270i	$0.523527\pi$
$42$	0	0
$43$	−3.70265	−0.564649	−0.282324	−	0.959319i	$-0.591105\pi$
$43$	−3.70265	−0.564649	−0.282324	+	0.959319i	$0.591105\pi$
$44$	0	0
$45$	6.70265	0.999172
$46$	0	0
$47$	−3.17548	−0.463192	−0.231596	−	0.972812i	$-0.574395\pi$
$47$	−3.17548	−0.463192	−0.231596	+	0.972812i	$0.574395\pi$
$48$	0	0
$49$	15.6289	2.23270
$50$	0	0
$51$	4.58774	0.642412
$52$	0	0
$53$	13.9325	1.91377	0.956886	−	0.290465i	$-0.0938100\pi$
$53$	13.9325	1.91377	0.956886	+	0.290465i	$0.0938100\pi$
$54$	0	0
$55$	−2.94567	−0.397194
$56$	0	0
$57$	6.94567	0.919976
$58$	0	0
$59$	13.2361	1.72319	0.861594	−	0.507598i	$-0.169466\pi$
$59$	13.2361	1.72319	0.861594	+	0.507598i	$0.169466\pi$
$60$	0	0
$61$	−4.62887	−0.592666	−0.296333	−	0.955085i	$-0.595764\pi$
$61$	−4.62887	−0.592666	−0.296333	+	0.955085i	$0.595764\pi$
$62$	0	0
$63$	31.8844	4.01705
$64$	0	0
$65$	0.885092	0.109782
$66$	0	0
$67$	−0.824517	−0.100731	−0.0503654	−	0.998731i	$-0.516039\pi$
$67$	−0.824517	−0.100731	−0.0503654	+	0.998731i	$0.516039\pi$
$68$	0	0
$69$	9.70265	1.16806
$70$	0	0
$71$	−9.89134	−1.17389	−0.586943	−	0.809628i	$-0.699669\pi$
$71$	−9.89134	−1.17389	−0.586943	+	0.809628i	$0.699669\pi$
$72$	0	0
$73$	−2.90454	−0.339951	−0.169975	−	0.985448i	$-0.554369\pi$
$73$	−2.90454	−0.339951	−0.169975	+	0.985448i	$0.554369\pi$
$74$	0	0
$75$	−3.11491	−0.359679
$76$	0	0
$77$	−14.0125	−1.59687
$78$	0	0
$79$	6.54661	0.736552	0.368276	−	0.929717i	$-0.379948\pi$
$79$	6.54661	0.736552	0.368276	+	0.929717i	$0.379948\pi$
$80$	0	0
$81$	15.8176	1.75751
$82$	0	0
$83$	−4.94567	−0.542858	−0.271429	−	0.962459i	$-0.587496\pi$
$83$	−4.94567	−0.542858	−0.271429	+	0.962459i	$0.587496\pi$
$84$	0	0
$85$	−1.47283	−0.159751
$86$	0	0
$87$	3.11491	0.333953
$88$	0	0
$89$	13.6615	1.44812	0.724059	−	0.689738i	$-0.242274\pi$
$89$	13.6615	1.44812	0.724059	+	0.689738i	$0.242274\pi$
$90$	0	0
$91$	4.21037	0.441367
$92$	0	0
$93$	13.0474	1.35295
$94$	0	0
$95$	−2.22982	−0.228774
$96$	0	0
$97$	11.5745	1.17522	0.587608	−	0.809146i	$-0.300070\pi$
$97$	11.5745	1.17522	0.587608	+	0.809146i	$0.300070\pi$
$98$	0	0
$99$	−19.7438	−1.98432
$100$	0	0
$101$	7.49228	0.745510	0.372755	−	0.927930i	$-0.378413\pi$
$101$	7.49228	0.745510	0.372755	+	0.927930i	$0.378413\pi$
$102$	0	0
$103$	−7.17548	−0.707021	−0.353511	−	0.935430i	$-0.615012\pi$
$103$	−7.17548	−0.707021	−0.353511	+	0.935430i	$0.615012\pi$
$104$	0	0
$105$	−14.8176	−1.44605
$106$	0	0
$107$	9.40530	0.909244	0.454622	−	0.890684i	$-0.349774\pi$
$107$	9.40530	0.909244	0.454622	+	0.890684i	$0.349774\pi$
$108$	0	0
$109$	18.8370	1.80426	0.902129	−	0.431467i	$-0.142004\pi$
$109$	18.8370	1.80426	0.902129	+	0.431467i	$0.142004\pi$
$110$	0	0
$111$	23.4053	2.22153
$112$	0	0
$113$	18.8976	1.77773	0.888867	−	0.458165i	$-0.151493\pi$
$113$	18.8976	1.77773	0.888867	+	0.458165i	$0.151493\pi$
$114$	0	0
$115$	−3.11491	−0.290467
$116$	0	0
$117$	5.93246	0.548456
$118$	0	0
$119$	−7.00624	−0.642261
$120$	0	0
$121$	−2.32304	−0.211186
$122$	0	0
$123$	2.94567	0.265602
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.74378	−0.154735	−0.0773676	−	0.997003i	$-0.524652\pi$
$127$	−1.74378	−0.154735	−0.0773676	+	0.997003i	$0.524652\pi$
$128$	0	0
$129$	11.5334	1.01546
$130$	0	0
$131$	13.1755	1.15115	0.575574	−	0.817750i	$-0.304779\pi$
$131$	13.1755	1.15115	0.575574	+	0.817750i	$0.304779\pi$
$132$	0	0
$133$	−10.6072	−0.919760
$134$	0	0
$135$	−11.5334	−0.992638
$136$	0	0
$137$	−11.9130	−1.01780	−0.508899	−	0.860826i	$-0.669947\pi$
$137$	−11.9130	−1.01780	−0.508899	+	0.860826i	$0.669947\pi$
$138$	0	0
$139$	10.1623	0.861953	0.430977	−	0.902363i	$-0.358169\pi$
$139$	10.1623	0.861953	0.430977	+	0.902363i	$0.358169\pi$
$140$	0	0
$141$	9.89134	0.833001
$142$	0	0
$143$	−2.60719	−0.218024
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−48.6825	−4.01526
$148$	0	0
$149$	−2.12115	−0.173771	−0.0868857	−	0.996218i	$-0.527691\pi$
$149$	−2.12115	−0.173771	−0.0868857	+	0.996218i	$0.527691\pi$
$150$	0	0
$151$	−6.22982	−0.506975	−0.253488	−	0.967339i	$-0.581578\pi$
$151$	−6.22982	−0.506975	−0.253488	+	0.967339i	$0.581578\pi$
$152$	0	0
$153$	−9.87189	−0.798095
$154$	0	0
$155$	−4.18869	−0.336443
$156$	0	0
$157$	18.4596	1.47324	0.736619	−	0.676307i	$-0.236421\pi$
$157$	18.4596	1.47324	0.736619	+	0.676307i	$0.236421\pi$
$158$	0	0
$159$	−43.3983	−3.44171
$160$	0	0
$161$	−14.8176	−1.16779
$162$	0	0
$163$	−9.28415	−0.727191	−0.363595	−	0.931557i	$-0.618451\pi$
$163$	−9.28415	−0.727191	−0.363595	+	0.931557i	$0.618451\pi$
$164$	0	0
$165$	9.17548	0.714310
$166$	0	0
$167$	16.5397	1.27988	0.639938	−	0.768426i	$-0.278960\pi$
$167$	16.5397	1.27988	0.639938	+	0.768426i	$0.278960\pi$
$168$	0	0
$169$	−12.2166	−0.939739
$170$	0	0
$171$	−14.9457	−1.14292
$172$	0	0
$173$	−2.77643	−0.211088	−0.105544	−	0.994415i	$-0.533658\pi$
$173$	−2.77643	−0.211088	−0.105544	+	0.994415i	$0.533658\pi$
$174$	0	0
$175$	4.75698	0.359594
$176$	0	0
$177$	−41.2291	−3.09897
$178$	0	0
$179$	−26.3726	−1.97118	−0.985592	−	0.169140i	$-0.945901\pi$
$179$	−26.3726	−1.97118	−0.985592	+	0.169140i	$0.945901\pi$
$180$	0	0
$181$	−18.8370	−1.40014	−0.700072	−	0.714073i	$-0.746849\pi$
$181$	−18.8370	−1.40014	−0.700072	+	0.714073i	$0.746849\pi$
$182$	0	0
$183$	14.4185	1.06585
$184$	0	0
$185$	−7.51396	−0.552437
$186$	0	0
$187$	4.33848	0.317261
$188$	0	0
$189$	−54.8642	−3.99078
$190$	0	0
$191$	13.1079	0.948458	0.474229	−	0.880402i	$-0.342727\pi$
$191$	13.1079	0.948458	0.474229	+	0.880402i	$0.342727\pi$
$192$	0	0
$193$	16.6678	1.19977	0.599886	−	0.800086i	$-0.295213\pi$
$193$	16.6678	1.19977	0.599886	+	0.800086i	$0.295213\pi$
$194$	0	0
$195$	−2.75698	−0.197432
$196$	0	0
$197$	21.9325	1.56262	0.781312	−	0.624141i	$-0.214551\pi$
$197$	21.9325	1.56262	0.781312	+	0.624141i	$0.214551\pi$
$198$	0	0
$199$	−13.5140	−0.957979	−0.478990	−	0.877821i	$-0.658997\pi$
$199$	−13.5140	−0.957979	−0.478990	+	0.877821i	$0.658997\pi$
$200$	0	0
$201$	2.56829	0.181154
$202$	0	0
$203$	−4.75698	−0.333875
$204$	0	0
$205$	−0.945668	−0.0660483
$206$	0	0
$207$	−20.8781	−1.45113
$208$	0	0
$209$	6.56829	0.454338
$210$	0	0
$211$	10.2687	0.706927	0.353464	−	0.935448i	$-0.385004\pi$
$211$	10.2687	0.706927	0.353464	+	0.935448i	$0.385004\pi$
$212$	0	0
$213$	30.8106	2.11111
$214$	0	0
$215$	−3.70265	−0.252519
$216$	0	0
$217$	−19.9255	−1.35263
$218$	0	0
$219$	9.04737	0.611365
$220$	0	0
$221$	−1.30359	−0.0876892
$222$	0	0
$223$	−6.65528	−0.445670	−0.222835	−	0.974856i	$-0.571531\pi$
$223$	−6.65528	−0.445670	−0.222835	+	0.974856i	$0.571531\pi$
$224$	0	0
$225$	6.70265	0.446843
$226$	0	0
$227$	3.89134	0.258277	0.129139	−	0.991627i	$-0.458779\pi$
$227$	3.89134	0.258277	0.129139	+	0.991627i	$0.458779\pi$
$228$	0	0
$229$	−25.2166	−1.66636	−0.833180	−	0.553001i	$-0.813482\pi$
$229$	−25.2166	−1.66636	−0.833180	+	0.553001i	$0.813482\pi$
$230$	0	0
$231$	43.6476	2.87180
$232$	0	0
$233$	−4.56829	−0.299279	−0.149639	−	0.988741i	$-0.547811\pi$
$233$	−4.56829	−0.299279	−0.149639	+	0.988741i	$0.547811\pi$
$234$	0	0
$235$	−3.17548	−0.207146
$236$	0	0
$237$	−20.3921	−1.32461
$238$	0	0
$239$	12.0389	0.778731	0.389366	−	0.921083i	$-0.372694\pi$
$239$	12.0389	0.778731	0.389366	+	0.921083i	$0.372694\pi$
$240$	0	0
$241$	−11.9519	−0.769890	−0.384945	−	0.922939i	$-0.625780\pi$
$241$	−11.9519	−0.769890	−0.384945	+	0.922939i	$0.625780\pi$
$242$	0	0
$243$	−14.6700	−0.941081
$244$	0	0
$245$	15.6289	0.998492
$246$	0	0
$247$	−1.97359	−0.125577
$248$	0	0
$249$	15.4053	0.976271
$250$	0	0
$251$	6.90677	0.435952	0.217976	−	0.975954i	$-0.430055\pi$
$251$	6.90677	0.435952	0.217976	+	0.975954i	$0.430055\pi$
$252$	0	0
$253$	9.17548	0.576858
$254$	0	0
$255$	4.58774	0.287296
$256$	0	0
$257$	−14.5419	−0.907098	−0.453549	−	0.891231i	$-0.649842\pi$
$257$	−14.5419	−0.907098	−0.453549	+	0.891231i	$0.649842\pi$
$258$	0	0
$259$	−35.7438	−2.22101
$260$	0	0
$261$	−6.70265	−0.414884
$262$	0	0
$263$	26.4985	1.63397	0.816984	−	0.576660i	$-0.195644\pi$
$263$	26.4985	1.63397	0.816984	+	0.576660i	$0.195644\pi$
$264$	0	0
$265$	13.9325	0.855864
$266$	0	0
$267$	−42.5544	−2.60429
$268$	0	0
$269$	−23.5676	−1.43694	−0.718470	−	0.695558i	$-0.755157\pi$
$269$	−23.5676	−1.43694	−0.718470	+	0.695558i	$0.755157\pi$
$270$	0	0
$271$	7.54037	0.458045	0.229022	−	0.973421i	$-0.426447\pi$
$271$	7.54037	0.458045	0.229022	+	0.973421i	$0.426447\pi$
$272$	0	0
$273$	−13.1149	−0.793750
$274$	0	0
$275$	−2.94567	−0.177630
$276$	0	0
$277$	33.2702	1.99901	0.999507	−	0.0313944i	$-0.00999479\pi$
$277$	33.2702	1.99901	0.999507	+	0.0313944i	$0.00999479\pi$
$278$	0	0
$279$	−28.0753	−1.68082
$280$	0	0
$281$	16.2974	0.972218	0.486109	−	0.873898i	$-0.338416\pi$
$281$	16.2974	0.972218	0.486109	+	0.873898i	$0.338416\pi$
$282$	0	0
$283$	−0.486038	−0.0288919	−0.0144460	−	0.999896i	$-0.504598\pi$
$283$	−0.486038	−0.0288919	−0.0144460	+	0.999896i	$0.504598\pi$
$284$	0	0
$285$	6.94567	0.411426
$286$	0	0
$287$	−4.49852	−0.265539
$288$	0	0
$289$	−14.8308	−0.872398
$290$	0	0
$291$	−36.0536	−2.11350
$292$	0	0
$293$	6.37737	0.372570	0.186285	−	0.982496i	$-0.440355\pi$
$293$	6.37737	0.372570	0.186285	+	0.982496i	$0.440355\pi$
$294$	0	0
$295$	13.2361	0.770633
$296$	0	0
$297$	33.9736	1.97135
$298$	0	0
$299$	−2.75698	−0.159440
$300$	0	0
$301$	−17.6134	−1.01522
$302$	0	0
$303$	−23.3378	−1.34072
$304$	0	0
$305$	−4.62887	−0.265048
$306$	0	0
$307$	−10.4207	−0.594743	−0.297371	−	0.954762i	$-0.596110\pi$
$307$	−10.4207	−0.594743	−0.297371	+	0.954762i	$0.596110\pi$
$308$	0	0
$309$	22.3510	1.27150
$310$	0	0
$311$	22.3726	1.26864	0.634318	−	0.773072i	$-0.281281\pi$
$311$	22.3726	1.26864	0.634318	+	0.773072i	$0.281281\pi$
$312$	0	0
$313$	30.4596	1.72168	0.860840	−	0.508876i	$-0.169939\pi$
$313$	30.4596	1.72168	0.860840	+	0.508876i	$0.169939\pi$
$314$	0	0
$315$	31.8844	1.79648
$316$	0	0
$317$	25.6615	1.44129	0.720647	−	0.693302i	$-0.243845\pi$
$317$	25.6615	1.44129	0.720647	+	0.693302i	$0.243845\pi$
$318$	0	0
$319$	2.94567	0.164926
$320$	0	0
$321$	−29.2966	−1.63518
$322$	0	0
$323$	3.28415	0.182735
$324$	0	0
$325$	0.885092	0.0490961
$326$	0	0
$327$	−58.6755	−3.24476
$328$	0	0
$329$	−15.1057	−0.832805
$330$	0	0
$331$	−19.8649	−1.09188	−0.545938	−	0.837826i	$-0.683826\pi$
$331$	−19.8649	−1.09188	−0.545938	+	0.837826i	$0.683826\pi$
$332$	0	0
$333$	−50.3635	−2.75990
$334$	0	0
$335$	−0.824517	−0.0450482
$336$	0	0
$337$	19.3642	1.05483	0.527417	−	0.849607i	$-0.323161\pi$
$337$	19.3642	1.05483	0.527417	+	0.849607i	$0.323161\pi$
$338$	0	0
$339$	−58.8642	−3.19707
$340$	0	0
$341$	12.3385	0.668166
$342$	0	0
$343$	41.0474	2.21635
$344$	0	0
$345$	9.70265	0.522373
$346$	0	0
$347$	−0.945668	−0.0507661	−0.0253831	−	0.999678i	$-0.508081\pi$
$347$	−0.945668	−0.0507661	−0.0253831	+	0.999678i	$0.508081\pi$
$348$	0	0
$349$	10.4985	0.561973	0.280987	−	0.959712i	$-0.409338\pi$
$349$	10.4985	0.561973	0.280987	+	0.959712i	$0.409338\pi$
$350$	0	0
$351$	−10.2081	−0.544870
$352$	0	0
$353$	22.8804	1.21780	0.608900	−	0.793247i	$-0.291611\pi$
$353$	22.8804	1.21780	0.608900	+	0.793247i	$0.291611\pi$
$354$	0	0
$355$	−9.89134	−0.524978
$356$	0	0
$357$	21.8238	1.15504
$358$	0	0
$359$	26.8851	1.41894	0.709470	−	0.704735i	$-0.248934\pi$
$359$	26.8851	1.41894	0.709470	+	0.704735i	$0.248934\pi$
$360$	0	0
$361$	−14.0279	−0.738312
$362$	0	0
$363$	7.23606	0.379795
$364$	0	0
$365$	−2.90454	−0.152031
$366$	0	0
$367$	7.05433	0.368233	0.184117	−	0.982904i	$-0.441058\pi$
$367$	7.05433	0.368233	0.184117	+	0.982904i	$0.441058\pi$
$368$	0	0
$369$	−6.33848	−0.329968
$370$	0	0
$371$	66.2765	3.44090
$372$	0	0
$373$	15.3642	0.795527	0.397763	−	0.917488i	$-0.369787\pi$
$373$	15.3642	0.795527	0.397763	+	0.917488i	$0.369787\pi$
$374$	0	0
$375$	−3.11491	−0.160853
$376$	0	0
$377$	−0.885092	−0.0455846
$378$	0	0
$379$	3.40530	0.174918	0.0874592	−	0.996168i	$-0.472125\pi$
$379$	3.40530	0.174918	0.0874592	+	0.996168i	$0.472125\pi$
$380$	0	0
$381$	5.43171	0.278275
$382$	0	0
$383$	−16.1623	−0.825854	−0.412927	−	0.910764i	$-0.635494\pi$
$383$	−16.1623	−0.825854	−0.412927	+	0.910764i	$0.635494\pi$
$384$	0	0
$385$	−14.0125	−0.714142
$386$	0	0
$387$	−24.8176	−1.26155
$388$	0	0
$389$	−15.2144	−0.771400	−0.385700	−	0.922624i	$-0.626040\pi$
$389$	−15.2144	−0.771400	−0.385700	+	0.922624i	$0.626040\pi$
$390$	0	0
$391$	4.58774	0.232012
$392$	0	0
$393$	−41.0404	−2.07021
$394$	0	0
$395$	6.54661	0.329396
$396$	0	0
$397$	−0.500759	−0.0251324	−0.0125662	−	0.999921i	$-0.504000\pi$
$397$	−0.500759	−0.0251324	−0.0125662	+	0.999921i	$0.504000\pi$
$398$	0	0
$399$	33.0404	1.65409
$400$	0	0
$401$	15.6134	0.779698	0.389849	−	0.920879i	$-0.372527\pi$
$401$	15.6134	0.779698	0.389849	+	0.920879i	$0.372527\pi$
$402$	0	0
$403$	−3.70737	−0.184678
$404$	0	0
$405$	15.8176	0.785981
$406$	0	0
$407$	22.1336	1.09712
$408$	0	0
$409$	−19.4178	−0.960148	−0.480074	−	0.877228i	$-0.659390\pi$
$409$	−19.4178	−0.960148	−0.480074	+	0.877228i	$0.659390\pi$
$410$	0	0
$411$	37.1079	1.83040
$412$	0	0
$413$	62.9637	3.09824
$414$	0	0
$415$	−4.94567	−0.242773
$416$	0	0
$417$	−31.6546	−1.55013
$418$	0	0
$419$	1.62039	0.0791613	0.0395807	−	0.999216i	$-0.487398\pi$
$419$	1.62039	0.0791613	0.0395807	+	0.999216i	$0.487398\pi$
$420$	0	0
$421$	−23.3789	−1.13942	−0.569709	−	0.821847i	$-0.692944\pi$
$421$	−23.3789	−1.13942	−0.569709	+	0.821847i	$0.692944\pi$
$422$	0	0
$423$	−21.2841	−1.03487
$424$	0	0
$425$	−1.47283	−0.0714429
$426$	0	0
$427$	−22.0194	−1.06560
$428$	0	0
$429$	8.12115	0.392093
$430$	0	0
$431$	−25.7702	−1.24131	−0.620653	−	0.784085i	$-0.713132\pi$
$431$	−25.7702	−1.24131	−0.620653	+	0.784085i	$0.713132\pi$
$432$	0	0
$433$	9.32304	0.448037	0.224018	−	0.974585i	$-0.428082\pi$
$433$	9.32304	0.448037	0.224018	+	0.974585i	$0.428082\pi$
$434$	0	0
$435$	3.11491	0.149348
$436$	0	0
$437$	6.94567	0.332256
$438$	0	0
$439$	25.3400	1.20941	0.604706	−	0.796449i	$-0.293291\pi$
$439$	25.3400	1.20941	0.604706	+	0.796449i	$0.293291\pi$
$440$	0	0
$441$	104.755	4.98833
$442$	0	0
$443$	38.8448	1.84557	0.922785	−	0.385315i	$-0.125907\pi$
$443$	38.8448	1.84557	0.922785	+	0.385315i	$0.125907\pi$
$444$	0	0
$445$	13.6615	0.647618
$446$	0	0
$447$	6.60719	0.312509
$448$	0	0
$449$	5.86493	0.276783	0.138392	−	0.990378i	$-0.455807\pi$
$449$	5.86493	0.276783	0.138392	+	0.990378i	$0.455807\pi$
$450$	0	0
$451$	2.78562	0.131170
$452$	0	0
$453$	19.4053	0.911740
$454$	0	0
$455$	4.21037	0.197385
$456$	0	0
$457$	19.4317	0.908977	0.454488	−	0.890753i	$-0.349822\pi$
$457$	19.4317	0.908977	0.454488	+	0.890753i	$0.349822\pi$
$458$	0	0
$459$	16.9868	0.792876
$460$	0	0
$461$	40.8642	1.90324	0.951618	−	0.307283i	$-0.0994200\pi$
$461$	40.8642	1.90324	0.951618	+	0.307283i	$0.0994200\pi$
$462$	0	0
$463$	−11.4706	−0.533084	−0.266542	−	0.963823i	$-0.585881\pi$
$463$	−11.4706	−0.533084	−0.266542	+	0.963823i	$0.585881\pi$
$464$	0	0
$465$	13.0474	0.605057
$466$	0	0
$467$	22.3532	1.03438	0.517191	−	0.855870i	$-0.326978\pi$
$467$	22.3532	1.03438	0.517191	+	0.855870i	$0.326978\pi$
$468$	0	0
$469$	−3.92221	−0.181111
$470$	0	0
$471$	−57.5000	−2.64946
$472$	0	0
$473$	10.9068	0.501494
$474$	0	0
$475$	−2.22982	−0.102311
$476$	0	0
$477$	93.3844	4.27578
$478$	0	0
$479$	−13.7026	−0.626090	−0.313045	−	0.949738i	$-0.601349\pi$
$479$	−13.7026	−0.626090	−0.313045	+	0.949738i	$0.601349\pi$
$480$	0	0
$481$	−6.65055	−0.303239
$482$	0	0
$483$	46.1553	2.10014
$484$	0	0
$485$	11.5745	0.525573
$486$	0	0
$487$	−29.1274	−1.31989	−0.659944	−	0.751315i	$-0.729420\pi$
$487$	−29.1274	−1.31989	−0.659944	+	0.751315i	$0.729420\pi$
$488$	0	0
$489$	28.9193	1.30777
$490$	0	0
$491$	24.4596	1.10385	0.551924	−	0.833895i	$-0.313894\pi$
$491$	24.4596	1.10385	0.551924	+	0.833895i	$0.313894\pi$
$492$	0	0
$493$	1.47283	0.0663331
$494$	0	0
$495$	−19.7438	−0.887417
$496$	0	0
$497$	−47.0529	−2.11061
$498$	0	0
$499$	−20.9938	−0.939810	−0.469905	−	0.882717i	$-0.655712\pi$
$499$	−20.9938	−0.939810	−0.469905	+	0.882717i	$0.655712\pi$
$500$	0	0
$501$	−51.5195	−2.30172
$502$	0	0
$503$	−15.9302	−0.710294	−0.355147	−	0.934810i	$-0.615569\pi$
$503$	−15.9302	−0.710294	−0.355147	+	0.934810i	$0.615569\pi$
$504$	0	0
$505$	7.49228	0.333402
$506$	0	0
$507$	38.0536	1.69002
$508$	0	0
$509$	−12.2687	−0.543801	−0.271900	−	0.962325i	$-0.587652\pi$
$509$	−12.2687	−0.543801	−0.271900	+	0.962325i	$0.587652\pi$
$510$	0	0
$511$	−13.8168	−0.611221
$512$	0	0
$513$	25.7174	1.13545
$514$	0	0
$515$	−7.17548	−0.316190
$516$	0	0
$517$	9.35392	0.411385
$518$	0	0
$519$	8.64832	0.379619
$520$	0	0
$521$	−40.3268	−1.76675	−0.883374	−	0.468668i	$-0.844734\pi$
$521$	−40.3268	−1.76675	−0.883374	+	0.468668i	$0.844734\pi$
$522$	0	0
$523$	20.7283	0.906387	0.453193	−	0.891412i	$-0.350285\pi$
$523$	20.7283	0.906387	0.453193	+	0.891412i	$0.350285\pi$
$524$	0	0
$525$	−14.8176	−0.646691
$526$	0	0
$527$	6.16924	0.268736
$528$	0	0
$529$	−13.2974	−0.578146
$530$	0	0
$531$	88.7167	3.84997
$532$	0	0
$533$	−0.837003	−0.0362546
$534$	0	0
$535$	9.40530	0.406626
$536$	0	0
$537$	82.1484	3.54496
$538$	0	0
$539$	−46.0375	−1.98297
$540$	0	0
$541$	42.2181	1.81510	0.907550	−	0.419945i	$-0.137951\pi$
$541$	42.2181	1.81510	0.907550	+	0.419945i	$0.137951\pi$
$542$	0	0
$543$	58.6755	2.51801
$544$	0	0
$545$	18.8370	0.806889
$546$	0	0
$547$	−32.4332	−1.38674	−0.693372	−	0.720580i	$-0.743876\pi$
$547$	−32.4332	−1.38674	−0.693372	+	0.720580i	$0.743876\pi$
$548$	0	0
$549$	−31.0257	−1.32414
$550$	0	0
$551$	2.22982	0.0949933
$552$	0	0
$553$	31.1421	1.32430
$554$	0	0
$555$	23.4053	0.993500
$556$	0	0
$557$	9.68320	0.410290	0.205145	−	0.978732i	$-0.434233\pi$
$557$	9.68320	0.410290	0.205145	+	0.978732i	$0.434233\pi$
$558$	0	0
$559$	−3.27719	−0.138610
$560$	0	0
$561$	−13.5140	−0.570560
$562$	0	0
$563$	39.4006	1.66054	0.830268	−	0.557364i	$-0.188187\pi$
$563$	39.4006	1.66054	0.830268	+	0.557364i	$0.188187\pi$
$564$	0	0
$565$	18.8976	0.795027
$566$	0	0
$567$	75.2438	3.15994
$568$	0	0
$569$	25.3619	1.06323	0.531614	−	0.846987i	$-0.321586\pi$
$569$	25.3619	1.06323	0.531614	+	0.846987i	$0.321586\pi$
$570$	0	0
$571$	21.5745	0.902866	0.451433	−	0.892305i	$-0.350913\pi$
$571$	21.5745	0.902866	0.451433	+	0.892305i	$0.350913\pi$
$572$	0	0
$573$	−40.8300	−1.70570
$574$	0	0
$575$	−3.11491	−0.129901
$576$	0	0
$577$	0.800344	0.0333188	0.0166594	−	0.999861i	$-0.494697\pi$
$577$	0.800344	0.0333188	0.0166594	+	0.999861i	$0.494697\pi$
$578$	0	0
$579$	−51.9185	−2.15766
$580$	0	0
$581$	−23.5264	−0.976042
$582$	0	0
$583$	−41.0404	−1.69972
$584$	0	0
$585$	5.93246	0.245277
$586$	0	0
$587$	28.1336	1.16120	0.580600	−	0.814189i	$-0.302818\pi$
$587$	28.1336	1.16120	0.580600	+	0.814189i	$0.302818\pi$
$588$	0	0
$589$	9.34000	0.384848
$590$	0	0
$591$	−68.3176	−2.81021
$592$	0	0
$593$	0.447144	0.0183620	0.00918100	−	0.999958i	$-0.497078\pi$
$593$	0.447144	0.0183620	0.00918100	+	0.999958i	$0.497078\pi$
$594$	0	0
$595$	−7.00624	−0.287228
$596$	0	0
$597$	42.0947	1.72282
$598$	0	0
$599$	−27.9714	−1.14288	−0.571439	−	0.820644i	$-0.693615\pi$
$599$	−27.9714	−1.14288	−0.571439	+	0.820644i	$0.693615\pi$
$600$	0	0
$601$	−15.7702	−0.643279	−0.321640	−	0.946862i	$-0.604234\pi$
$601$	−15.7702	−0.643279	−0.321640	+	0.946862i	$0.604234\pi$
$602$	0	0
$603$	−5.52645	−0.225054
$604$	0	0
$605$	−2.32304	−0.0944451
$606$	0	0
$607$	34.2812	1.39143	0.695715	−	0.718318i	$-0.255087\pi$
$607$	34.2812	1.39143	0.695715	+	0.718318i	$0.255087\pi$
$608$	0	0
$609$	14.8176	0.600438
$610$	0	0
$611$	−2.81060	−0.113705
$612$	0	0
$613$	−33.1204	−1.33772	−0.668861	−	0.743388i	$-0.733218\pi$
$613$	−33.1204	−1.33772	−0.668861	+	0.743388i	$0.733218\pi$
$614$	0	0
$615$	2.94567	0.118781
$616$	0	0
$617$	−21.7974	−0.877530	−0.438765	−	0.898602i	$-0.644584\pi$
$617$	−21.7974	−0.877530	−0.438765	+	0.898602i	$0.644584\pi$
$618$	0	0
$619$	−45.6351	−1.83423	−0.917115	−	0.398623i	$-0.869488\pi$
$619$	−45.6351	−1.83423	−0.917115	+	0.398623i	$0.869488\pi$
$620$	0	0
$621$	35.9255	1.44164
$622$	0	0
$623$	64.9876	2.60367
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−20.4596	−0.817079
$628$	0	0
$629$	11.0668	0.441263
$630$	0	0
$631$	30.0947	1.19805	0.599026	−	0.800729i	$-0.295554\pi$
$631$	30.0947	1.19805	0.599026	+	0.800729i	$0.295554\pi$
$632$	0	0
$633$	−31.9861	−1.27133
$634$	0	0
$635$	−1.74378	−0.0691997
$636$	0	0
$637$	13.8330	0.548083
$638$	0	0
$639$	−66.2982	−2.62271
$640$	0	0
$641$	−20.4860	−0.809150	−0.404575	−	0.914505i	$-0.632581\pi$
$641$	−20.4860	−0.809150	−0.404575	+	0.914505i	$0.632581\pi$
$642$	0	0
$643$	−36.9317	−1.45645	−0.728223	−	0.685340i	$-0.759653\pi$
$643$	−36.9317	−1.45645	−0.728223	+	0.685340i	$0.759653\pi$
$644$	0	0
$645$	11.5334	0.454128
$646$	0	0
$647$	32.7717	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$647$	32.7717	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$648$	0	0
$649$	−38.9890	−1.53045
$650$	0	0
$651$	62.0661	2.43256
$652$	0	0
$653$	−14.5030	−0.567546	−0.283773	−	0.958891i	$-0.591586\pi$
$653$	−14.5030	−0.567546	−0.283773	+	0.958891i	$0.591586\pi$
$654$	0	0
$655$	13.1755	0.514809
$656$	0	0
$657$	−19.4681	−0.759523
$658$	0	0
$659$	0.135072	0.00526165	0.00263083	−	0.999997i	$-0.499163\pi$
$659$	0.135072	0.00526165	0.00263083	+	0.999997i	$0.499163\pi$
$660$	0	0
$661$	39.5529	1.53843	0.769214	−	0.638991i	$-0.220648\pi$
$661$	39.5529	1.53843	0.769214	+	0.638991i	$0.220648\pi$
$662$	0	0
$663$	4.06058	0.157700
$664$	0	0
$665$	−10.6072	−0.411329
$666$	0	0
$667$	3.11491	0.120610
$668$	0	0
$669$	20.7306	0.801490
$670$	0	0
$671$	13.6351	0.526378
$672$	0	0
$673$	11.9736	0.461548	0.230774	−	0.973007i	$-0.425874\pi$
$673$	11.9736	0.461548	0.230774	+	0.973007i	$0.425874\pi$
$674$	0	0
$675$	−11.5334	−0.443921
$676$	0	0
$677$	−14.9582	−0.574889	−0.287444	−	0.957797i	$-0.592806\pi$
$677$	−14.9582	−0.574889	−0.287444	+	0.957797i	$0.592806\pi$
$678$	0	0
$679$	55.0599	2.11300
$680$	0	0
$681$	−12.1212	−0.464484
$682$	0	0
$683$	12.7283	0.487036	0.243518	−	0.969896i	$-0.421698\pi$
$683$	12.7283	0.487036	0.243518	+	0.969896i	$0.421698\pi$
$684$	0	0
$685$	−11.9130	−0.455173
$686$	0	0
$687$	78.5474	2.99677
$688$	0	0
$689$	12.3315	0.469793
$690$	0	0
$691$	25.1902	0.958281	0.479140	−	0.877738i	$-0.340949\pi$
$691$	25.1902	0.958281	0.479140	+	0.877738i	$0.340949\pi$
$692$	0	0
$693$	−93.9208	−3.56776
$694$	0	0
$695$	10.1623	0.385477
$696$	0	0
$697$	1.39281	0.0527565
$698$	0	0
$699$	14.2298	0.538221
$700$	0	0
$701$	−43.7174	−1.65118	−0.825591	−	0.564269i	$-0.809158\pi$
$701$	−43.7174	−1.65118	−0.825591	+	0.564269i	$0.809158\pi$
$702$	0	0
$703$	16.7547	0.631917
$704$	0	0
$705$	9.89134	0.372529
$706$	0	0
$707$	35.6406	1.34040
$708$	0	0
$709$	−41.1491	−1.54539	−0.772693	−	0.634780i	$-0.781091\pi$
$709$	−41.1491	−1.54539	−0.772693	+	0.634780i	$0.781091\pi$
$710$	0	0
$711$	43.8796	1.64562
$712$	0	0
$713$	13.0474	0.488628
$714$	0	0
$715$	−2.60719	−0.0975033
$716$	0	0
$717$	−37.5000	−1.40046
$718$	0	0
$719$	23.4442	0.874321	0.437160	−	0.899384i	$-0.355984\pi$
$719$	23.4442	0.874321	0.437160	+	0.899384i	$0.355984\pi$
$720$	0	0
$721$	−34.1336	−1.27120
$722$	0	0
$723$	37.2291	1.38457
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−16.6894	−0.618977	−0.309489	−	0.950903i	$-0.600158\pi$
$727$	−16.6894	−0.618977	−0.309489	+	0.950903i	$0.600158\pi$
$728$	0	0
$729$	−1.75698	−0.0650734
$730$	0	0
$731$	5.45339	0.201701
$732$	0	0
$733$	−23.8774	−0.881932	−0.440966	−	0.897524i	$-0.645364\pi$
$733$	−23.8774	−0.881932	−0.440966	+	0.897524i	$0.645364\pi$
$734$	0	0
$735$	−48.6825	−1.79568
$736$	0	0
$737$	2.42875	0.0894643
$738$	0	0
$739$	34.3510	1.26362	0.631810	−	0.775123i	$-0.282312\pi$
$739$	34.3510	1.26362	0.631810	+	0.775123i	$0.282312\pi$
$740$	0	0
$741$	6.14756	0.225836
$742$	0	0
$743$	36.6072	1.34299	0.671494	−	0.741010i	$-0.265653\pi$
$743$	36.6072	1.34299	0.671494	+	0.741010i	$0.265653\pi$
$744$	0	0
$745$	−2.12115	−0.0777129
$746$	0	0
$747$	−33.1491	−1.21286
$748$	0	0
$749$	44.7408	1.63479
$750$	0	0
$751$	26.0558	0.950791	0.475396	−	0.879772i	$-0.342305\pi$
$751$	26.0558	0.950791	0.475396	+	0.879772i	$0.342305\pi$
$752$	0	0
$753$	−21.5140	−0.784012
$754$	0	0
$755$	−6.22982	−0.226726
$756$	0	0
$757$	−18.1351	−0.659130	−0.329565	−	0.944133i	$-0.606902\pi$
$757$	−18.1351	−0.659130	−0.329565	+	0.944133i	$0.606902\pi$
$758$	0	0
$759$	−28.5808	−1.03742
$760$	0	0
$761$	8.37961	0.303761	0.151880	−	0.988399i	$-0.451467\pi$
$761$	8.37961	0.303761	0.151880	+	0.988399i	$0.451467\pi$
$762$	0	0
$763$	89.6073	3.24400
$764$	0	0
$765$	−9.87189	−0.356919
$766$	0	0
$767$	11.7151	0.423009
$768$	0	0
$769$	−26.5808	−0.958527	−0.479264	−	0.877671i	$-0.659096\pi$
$769$	−26.5808	−0.958527	−0.479264	+	0.877671i	$0.659096\pi$
$770$	0	0
$771$	45.2966	1.63132
$772$	0	0
$773$	−41.0529	−1.47657	−0.738285	−	0.674489i	$-0.764364\pi$
$773$	−41.0529	−1.47657	−0.738285	+	0.674489i	$0.764364\pi$
$774$	0	0
$775$	−4.18869	−0.150462
$776$	0	0
$777$	111.339	3.99425
$778$	0	0
$779$	2.10866	0.0755507
$780$	0	0
$781$	29.1366	1.04259
$782$	0	0
$783$	11.5334	0.412170
$784$	0	0
$785$	18.4596	0.658852
$786$	0	0
$787$	−42.6197	−1.51923	−0.759614	−	0.650375i	$-0.774612\pi$
$787$	−42.6197	−1.51923	−0.759614	+	0.650375i	$0.774612\pi$
$788$	0	0
$789$	−82.5405	−2.93852
$790$	0	0
$791$	89.8954	3.19631
$792$	0	0
$793$	−4.09698	−0.145488
$794$	0	0
$795$	−43.3983	−1.53918
$796$	0	0
$797$	43.6351	1.54564	0.772818	−	0.634628i	$-0.218847\pi$
$797$	43.6351	1.54564	0.772818	+	0.634628i	$0.218847\pi$
$798$	0	0
$799$	4.67696	0.165459
$800$	0	0
$801$	91.5684	3.23541
$802$	0	0
$803$	8.55581	0.301928
$804$	0	0
$805$	−14.8176	−0.522250
$806$	0	0
$807$	73.4108	2.58418
$808$	0	0
$809$	12.7717	0.449029	0.224515	−	0.974471i	$-0.427920\pi$
$809$	12.7717	0.449029	0.224515	+	0.974471i	$0.427920\pi$
$810$	0	0
$811$	26.0800	0.915793	0.457897	−	0.889005i	$-0.348603\pi$
$811$	26.0800	0.915793	0.457897	+	0.889005i	$0.348603\pi$
$812$	0	0
$813$	−23.4876	−0.823745
$814$	0	0
$815$	−9.28415	−0.325209
$816$	0	0
$817$	8.25622	0.288849
$818$	0	0
$819$	28.2206	0.986108
$820$	0	0
$821$	34.0683	1.18899	0.594497	−	0.804098i	$-0.297351\pi$
$821$	34.0683	1.18899	0.594497	+	0.804098i	$0.297351\pi$
$822$	0	0
$823$	−8.10866	−0.282650	−0.141325	−	0.989963i	$-0.545136\pi$
$823$	−8.10866	−0.282650	−0.141325	+	0.989963i	$0.545136\pi$
$824$	0	0
$825$	9.17548	0.319449
$826$	0	0
$827$	41.7585	1.45209	0.726043	−	0.687650i	$-0.241357\pi$
$827$	41.7585	1.45209	0.726043	+	0.687650i	$0.241357\pi$
$828$	0	0
$829$	24.9170	0.865404	0.432702	−	0.901537i	$-0.357560\pi$
$829$	24.9170	0.865404	0.432702	+	0.901537i	$0.357560\pi$
$830$	0	0
$831$	−103.634	−3.59501
$832$	0	0
$833$	−23.0187	−0.797552
$834$	0	0
$835$	16.5397	0.572378
$836$	0	0
$837$	48.3098	1.66983
$838$	0	0
$839$	−12.9721	−0.447846	−0.223923	−	0.974607i	$-0.571886\pi$
$839$	−12.9721	−0.447846	−0.223923	+	0.974607i	$0.571886\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−50.7647	−1.74843
$844$	0	0
$845$	−12.2166	−0.420264
$846$	0	0
$847$	−11.0507	−0.379705
$848$	0	0
$849$	1.51396	0.0519591
$850$	0	0
$851$	23.4053	0.802323
$852$	0	0
$853$	4.43322	0.151791	0.0758953	−	0.997116i	$-0.475819\pi$
$853$	4.43322	0.151791	0.0758953	+	0.997116i	$0.475819\pi$
$854$	0	0
$855$	−14.9457	−0.511131
$856$	0	0
$857$	23.7702	0.811974	0.405987	−	0.913879i	$-0.366928\pi$
$857$	23.7702	0.811974	0.405987	+	0.913879i	$0.366928\pi$
$858$	0	0
$859$	53.5000	1.82540	0.912699	−	0.408633i	$-0.133994\pi$
$859$	53.5000	1.82540	0.912699	+	0.408633i	$0.133994\pi$
$860$	0	0
$861$	14.0125	0.477544
$862$	0	0
$863$	−10.5591	−0.359436	−0.179718	−	0.983718i	$-0.557519\pi$
$863$	−10.5591	−0.359436	−0.179718	+	0.983718i	$0.557519\pi$
$864$	0	0
$865$	−2.77643	−0.0944014
$866$	0	0
$867$	46.1964	1.56891
$868$	0	0
$869$	−19.2841	−0.654170
$870$	0	0
$871$	−0.729774	−0.0247274
$872$	0	0
$873$	77.5801	2.62569
$874$	0	0
$875$	4.75698	0.160815
$876$	0	0
$877$	−45.2919	−1.52940	−0.764700	−	0.644387i	$-0.777113\pi$
$877$	−45.2919	−1.52940	−0.764700	+	0.644387i	$0.777113\pi$
$878$	0	0
$879$	−19.8649	−0.670027
$880$	0	0
$881$	−40.6755	−1.37039	−0.685197	−	0.728358i	$-0.740284\pi$
$881$	−40.6755	−1.37039	−0.685197	+	0.728358i	$0.740284\pi$
$882$	0	0
$883$	14.5947	0.491151	0.245575	−	0.969377i	$-0.421023\pi$
$883$	14.5947	0.491151	0.245575	+	0.969377i	$0.421023\pi$
$884$	0	0
$885$	−41.2291	−1.38590
$886$	0	0
$887$	−39.8774	−1.33895	−0.669476	−	0.742833i	$-0.733481\pi$
$887$	−39.8774	−1.33895	−0.669476	+	0.742833i	$0.733481\pi$
$888$	0	0
$889$	−8.29512	−0.278209
$890$	0	0
$891$	−46.5933	−1.56093
$892$	0	0
$893$	7.08074	0.236948
$894$	0	0
$895$	−26.3726	−0.881540
$896$	0	0
$897$	8.58774	0.286736
$898$	0	0
$899$	4.18869	0.139701
$900$	0	0
$901$	−20.5202	−0.683627
$902$	0	0
$903$	54.8642	1.82577
$904$	0	0
$905$	−18.8370	−0.626163
$906$	0	0
$907$	21.5412	0.715263	0.357631	−	0.933863i	$-0.383584\pi$
$907$	21.5412	0.715263	0.357631	+	0.933863i	$0.383584\pi$
$908$	0	0
$909$	50.2181	1.66563
$910$	0	0
$911$	−35.9783	−1.19201	−0.596007	−	0.802979i	$-0.703247\pi$
$911$	−35.9783	−1.19201	−0.596007	+	0.802979i	$0.703247\pi$
$912$	0	0
$913$	14.5683	0.482140
$914$	0	0
$915$	14.4185	0.476661
$916$	0	0
$917$	62.6755	2.06973
$918$	0	0
$919$	44.4596	1.46659	0.733294	−	0.679912i	$-0.237982\pi$
$919$	44.4596	1.46659	0.733294	+	0.679912i	$0.237982\pi$
$920$	0	0
$921$	32.4596	1.06958
$922$	0	0
$923$	−8.75475	−0.288166
$924$	0	0
$925$	−7.51396	−0.247058
$926$	0	0
$927$	−48.0947	−1.57964
$928$	0	0
$929$	−2.94343	−0.0965709	−0.0482855	−	0.998834i	$-0.515376\pi$
$929$	−2.94343	−0.0965709	−0.0482855	+	0.998834i	$0.515376\pi$
$930$	0	0
$931$	−34.8495	−1.14215
$932$	0	0
$933$	−69.6887	−2.28151
$934$	0	0
$935$	4.33848	0.141883
$936$	0	0
$937$	−47.9736	−1.56723	−0.783614	−	0.621247i	$-0.786626\pi$
$937$	−47.9736	−1.56723	−0.783614	+	0.621247i	$0.786626\pi$
$938$	0	0
$939$	−94.8789	−3.09626
$940$	0	0
$941$	−43.8774	−1.43036	−0.715181	−	0.698939i	$-0.753656\pi$
$941$	−43.8774	−1.43036	−0.715181	+	0.698939i	$0.753656\pi$
$942$	0	0
$943$	2.94567	0.0959241
$944$	0	0
$945$	−54.8642	−1.78473
$946$	0	0
$947$	25.8044	0.838529	0.419264	−	0.907864i	$-0.362288\pi$
$947$	25.8044	0.838529	0.419264	+	0.907864i	$0.362288\pi$
$948$	0	0
$949$	−2.57079	−0.0834512
$950$	0	0
$951$	−79.9333	−2.59201
$952$	0	0
$953$	−15.6740	−0.507731	−0.253865	−	0.967240i	$-0.581702\pi$
$953$	−15.6740	−0.507731	−0.253865	+	0.967240i	$0.581702\pi$
$954$	0	0
$955$	13.1079	0.424163
$956$	0	0
$957$	−9.17548	−0.296601
$958$	0	0
$959$	−56.6700	−1.82997
$960$	0	0
$961$	−13.4549	−0.434029
$962$	0	0
$963$	63.0404	2.03145
$964$	0	0
$965$	16.6678	0.536554
$966$	0	0
$967$	23.5529	0.757409	0.378704	−	0.925518i	$-0.376370\pi$
$967$	23.5529	0.757409	0.378704	+	0.925518i	$0.376370\pi$
$968$	0	0
$969$	−10.2298	−0.328629
$970$	0	0
$971$	−40.7453	−1.30758	−0.653789	−	0.756677i	$-0.726822\pi$
$971$	−40.7453	−1.30758	−0.653789	+	0.756677i	$0.726822\pi$
$972$	0	0
$973$	48.3418	1.54977
$974$	0	0
$975$	−2.75698	−0.0882941
$976$	0	0
$977$	−12.0125	−0.384313	−0.192157	−	0.981364i	$-0.561548\pi$
$977$	−12.0125	−0.384313	−0.192157	+	0.981364i	$0.561548\pi$
$978$	0	0
$979$	−40.2423	−1.28615
$980$	0	0
$981$	126.258	4.03110
$982$	0	0
$983$	−42.5374	−1.35673	−0.678366	−	0.734724i	$-0.737312\pi$
$983$	−42.5374	−1.35673	−0.678366	+	0.734724i	$0.737312\pi$
$984$	0	0
$985$	21.9325	0.698826
$986$	0	0
$987$	47.0529	1.49771
$988$	0	0
$989$	11.5334	0.366741
$990$	0	0
$991$	−11.6226	−0.369205	−0.184602	−	0.982813i	$-0.559100\pi$
$991$	−11.6226	−0.369205	−0.184602	+	0.982813i	$0.559100\pi$
$992$	0	0
$993$	61.8774	1.96362
$994$	0	0
$995$	−13.5140	−0.428421
$996$	0	0
$997$	−4.22982	−0.133960	−0.0669798	−	0.997754i	$-0.521336\pi$
$997$	−4.22982	−0.133960	−0.0669798	+	0.997754i	$0.521336\pi$
$998$	0	0
$999$	86.6616	2.74185

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9280.2.a.bg.1.1		3
4.3	odd	2		9280.2.a.bx.1.3		3
8.3	odd	2		4640.2.a.j.1.1	✓	3
8.5	even	2		4640.2.a.k.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4640.2.a.j.1.1	✓	3	8.3	odd	2
4640.2.a.k.1.3	yes	3	8.5	even	2
9280.2.a.bg.1.1		3	1.1	even	1	trivial
9280.2.a.bx.1.3		3	4.3	odd	2

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:	\( N \)	\(=\)	\( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9280.a (trivial)

\(f(q)\)	\(=\)	\(q-3.11491 q^{3} +1.00000 q^{5} +4.75698 q^{7} +6.70265 q^{9} -2.94567 q^{11} +0.885092 q^{13} -3.11491 q^{15} -1.47283 q^{17} -2.22982 q^{19} -14.8176 q^{21} -3.11491 q^{23} +1.00000 q^{25} -11.5334 q^{27} -1.00000 q^{29} -4.18869 q^{31} +9.17548 q^{33} +4.75698 q^{35} -7.51396 q^{37} -2.75698 q^{39} -0.945668 q^{41} -3.70265 q^{43} +6.70265 q^{45} -3.17548 q^{47} +15.6289 q^{49} +4.58774 q^{51} +13.9325 q^{53} -2.94567 q^{55} +6.94567 q^{57} +13.2361 q^{59} -4.62887 q^{61} +31.8844 q^{63} +0.885092 q^{65} -0.824517 q^{67} +9.70265 q^{69} -9.89134 q^{71} -2.90454 q^{73} -3.11491 q^{75} -14.0125 q^{77} +6.54661 q^{79} +15.8176 q^{81} -4.94567 q^{83} -1.47283 q^{85} +3.11491 q^{87} +13.6615 q^{89} +4.21037 q^{91} +13.0474 q^{93} -2.22982 q^{95} +11.5745 q^{97} -19.7438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} + 7 q^{7} + 2 q^{9} + 2 q^{11} + 9 q^{13} - 3 q^{15} + q^{17} + 6 q^{19} - 20 q^{21} - 3 q^{23} + 3 q^{25} - 12 q^{27} - 3 q^{29} - 9 q^{31} + 4 q^{33} + 7 q^{35} - 8 q^{37} - q^{39}+ \cdots - 32 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 + 7 * q^7 + 2 * q^9 + 2 * q^11 + 9 * q^13 - 3 * q^15 + q^17 + 6 * q^19 - 20 * q^21 - 3 * q^23 + 3 * q^25 - 12 * q^27 - 3 * q^29 - 9 * q^31 + 4 * q^33 + 7 * q^35 - 8 * q^37 - q^39 + 8 * q^41 + 7 * q^43 + 2 * q^45 + 14 * q^47 + 26 * q^49 + 2 * q^51 + 11 * q^53 + 2 * q^55 + 10 * q^57 - q^59 + 7 * q^61 + 26 * q^63 + 9 * q^65 - 26 * q^67 + 11 * q^69 - 8 * q^71 - 7 * q^73 - 3 * q^75 + 14 * q^77 + 17 * q^79 + 23 * q^81 - 4 * q^83 + q^85 + 3 * q^87 + 32 * q^89 + 8 * q^91 + 2 * q^93 + 6 * q^95 + 3 * q^97 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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