LMFDB - Embedded newform 9280.2.a.bu.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 = [3,0,2,0,3,0,-2,0,3,0,-8,0,6,0,2,0,0,0,0,0,0] 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:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ 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)

$f(q)$	$=$	$q-1.70928 q^{3} +1.00000 q^{5} +0.630898 q^{7} -0.0783777 q^{9} -0.290725 q^{11} +0.921622 q^{13} -1.70928 q^{15} +4.97107 q^{17} +6.04945 q^{19} -1.07838 q^{21} +2.29072 q^{23} +1.00000 q^{25} +5.26180 q^{27} -1.00000 q^{29} +10.0494 q^{31} +0.496928 q^{33} +0.630898 q^{35} -1.55252 q^{37} -1.57531 q^{39} +0.340173 q^{41} +5.70928 q^{43} -0.0783777 q^{45} -1.12783 q^{47} -6.60197 q^{49} -8.49693 q^{51} +0.340173 q^{53} -0.290725 q^{55} -10.3402 q^{57} -9.75872 q^{59} -3.07838 q^{61} -0.0494483 q^{63} +0.921622 q^{65} +5.70928 q^{67} -3.91548 q^{69} +9.07838 q^{71} -6.94441 q^{73} -1.70928 q^{75} -0.183417 q^{77} +12.3896 q^{79} -8.75872 q^{81} -2.78765 q^{83} +4.97107 q^{85} +1.70928 q^{87} +4.73820 q^{89} +0.581449 q^{91} -17.1773 q^{93} +6.04945 q^{95} -15.8927 q^{97} +0.0227863 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 8 q^{11} + 6 q^{13} + 2 q^{15} + 14 q^{23} + 3 q^{25} + 8 q^{27} - 3 q^{29} + 12 q^{31} - 16 q^{33} - 2 q^{35} - 4 q^{37} + 16 q^{39} - 10 q^{41} + 10 q^{43}+ \cdots - 20 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 8 * q^11 + 6 * q^13 + 2 * q^15 + 14 * q^23 + 3 * q^25 + 8 * q^27 - 3 * q^29 + 12 * q^31 - 16 * q^33 - 2 * q^35 - 4 * q^37 + 16 * q^39 - 10 * q^41 + 10 * q^43 + 3 * q^45 + 18 * q^47 - q^49 - 8 * q^51 - 10 * q^53 - 8 * q^55 - 20 * q^57 - 4 * q^59 - 6 * q^61 + 18 * q^63 + 6 * q^65 + 10 * q^67 + 20 * q^69 + 24 * q^71 - 4 * q^73 + 2 * q^75 + 4 * q^77 + 8 * q^79 - q^81 + 2 * q^83 - 2 * q^87 + 22 * q^89 + 16 * q^91 - 12 * q^93 - 36 * q^97 - 20 * 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
$3$	−1.70928	−0.986851	−0.493425	−	0.869788i	$-0.664255\pi$
$3$	−1.70928	−0.986851	−0.493425	+	0.869788i	$0.664255\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.630898	0.238457	0.119228	−	0.992867i	$-0.461958\pi$
$7$	0.630898	0.238457	0.119228	+	0.992867i	$0.461958\pi$
$8$	0	0
$9$	−0.0783777	−0.0261259
$10$	0	0
$11$	−0.290725	−0.0876568	−0.0438284	−	0.999039i	$-0.513955\pi$
$11$	−0.290725	−0.0876568	−0.0438284	+	0.999039i	$0.513955\pi$
$12$	0	0
$13$	0.921622	0.255612	0.127806	−	0.991799i	$-0.459207\pi$
$13$	0.921622	0.255612	0.127806	+	0.991799i	$0.459207\pi$
$14$	0	0
$15$	−1.70928	−0.441333
$16$	0	0
$17$	4.97107	1.20566	0.602831	−	0.797869i	$-0.294039\pi$
$17$	4.97107	1.20566	0.602831	+	0.797869i	$0.294039\pi$
$18$	0	0
$19$	6.04945	1.38784	0.693919	−	0.720053i	$-0.255882\pi$
$19$	6.04945	1.38784	0.693919	+	0.720053i	$0.255882\pi$
$20$	0	0
$21$	−1.07838	−0.235321
$22$	0	0
$23$	2.29072	0.477649	0.238825	−	0.971063i	$-0.423238\pi$
$23$	2.29072	0.477649	0.238825	+	0.971063i	$0.423238\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.26180	1.01263
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	10.0494	1.80493	0.902467	−	0.430759i	$-0.141754\pi$
$31$	10.0494	1.80493	0.902467	+	0.430759i	$0.141754\pi$
$32$	0	0
$33$	0.496928	0.0865041
$34$	0	0
$35$	0.630898	0.106641
$36$	0	0
$37$	−1.55252	−0.255233	−0.127616	−	0.991824i	$-0.540733\pi$
$37$	−1.55252	−0.255233	−0.127616	+	0.991824i	$0.540733\pi$
$38$	0	0
$39$	−1.57531	−0.252251
$40$	0	0
$41$	0.340173	0.0531261	0.0265630	−	0.999647i	$-0.491544\pi$
$41$	0.340173	0.0531261	0.0265630	+	0.999647i	$0.491544\pi$
$42$	0	0
$43$	5.70928	0.870656	0.435328	−	0.900272i	$-0.356632\pi$
$43$	5.70928	0.870656	0.435328	+	0.900272i	$0.356632\pi$
$44$	0	0
$45$	−0.0783777	−0.0116839
$46$	0	0
$47$	−1.12783	−0.164510	−0.0822552	−	0.996611i	$-0.526212\pi$
$47$	−1.12783	−0.164510	−0.0822552	+	0.996611i	$0.526212\pi$
$48$	0	0
$49$	−6.60197	−0.943138
$50$	0	0
$51$	−8.49693	−1.18981
$52$	0	0
$53$	0.340173	0.0467264	0.0233632	−	0.999727i	$-0.492563\pi$
$53$	0.340173	0.0467264	0.0233632	+	0.999727i	$0.492563\pi$
$54$	0	0
$55$	−0.290725	−0.0392013
$56$	0	0
$57$	−10.3402	−1.36959
$58$	0	0
$59$	−9.75872	−1.27048	−0.635239	−	0.772316i	$-0.719098\pi$
$59$	−9.75872	−1.27048	−0.635239	+	0.772316i	$0.719098\pi$
$60$	0	0
$61$	−3.07838	−0.394146	−0.197073	−	0.980389i	$-0.563144\pi$
$61$	−3.07838	−0.394146	−0.197073	+	0.980389i	$0.563144\pi$
$62$	0	0
$63$	−0.0494483	−0.00622990
$64$	0	0
$65$	0.921622	0.114313
$66$	0	0
$67$	5.70928	0.697499	0.348749	−	0.937216i	$-0.386606\pi$
$67$	5.70928	0.697499	0.348749	+	0.937216i	$0.386606\pi$
$68$	0	0
$69$	−3.91548	−0.471368
$70$	0	0
$71$	9.07838	1.07741	0.538703	−	0.842496i	$-0.318915\pi$
$71$	9.07838	1.07741	0.538703	+	0.842496i	$0.318915\pi$
$72$	0	0
$73$	−6.94441	−0.812782	−0.406391	−	0.913699i	$-0.633213\pi$
$73$	−6.94441	−0.812782	−0.406391	+	0.913699i	$0.633213\pi$
$74$	0	0
$75$	−1.70928	−0.197370
$76$	0	0
$77$	−0.183417	−0.0209024
$78$	0	0
$79$	12.3896	1.39394	0.696971	−	0.717100i	$-0.254531\pi$
$79$	12.3896	1.39394	0.696971	+	0.717100i	$0.254531\pi$
$80$	0	0
$81$	−8.75872	−0.973192
$82$	0	0
$83$	−2.78765	−0.305985	−0.152992	−	0.988227i	$-0.548891\pi$
$83$	−2.78765	−0.305985	−0.152992	+	0.988227i	$0.548891\pi$
$84$	0	0
$85$	4.97107	0.539188
$86$	0	0
$87$	1.70928	0.183254
$88$	0	0
$89$	4.73820	0.502249	0.251124	−	0.967955i	$-0.419200\pi$
$89$	4.73820	0.502249	0.251124	+	0.967955i	$0.419200\pi$
$90$	0	0
$91$	0.581449	0.0609524
$92$	0	0
$93$	−17.1773	−1.78120
$94$	0	0
$95$	6.04945	0.620660
$96$	0	0
$97$	−15.8927	−1.61366	−0.806829	−	0.590785i	$-0.798818\pi$
$97$	−15.8927	−1.61366	−0.806829	+	0.590785i	$0.798818\pi$
$98$	0	0
$99$	0.0227863	0.00229011
$100$	0	0
$101$	12.2557	1.21948	0.609741	−	0.792600i	$-0.291273\pi$
$101$	12.2557	1.21948	0.609741	+	0.792600i	$0.291273\pi$
$102$	0	0
$103$	−7.86603	−0.775063	−0.387532	−	0.921856i	$-0.626672\pi$
$103$	−7.86603	−0.775063	−0.387532	+	0.921856i	$0.626672\pi$
$104$	0	0
$105$	−1.07838	−0.105239
$106$	0	0
$107$	−12.7298	−1.23064	−0.615318	−	0.788279i	$-0.710972\pi$
$107$	−12.7298	−1.23064	−0.615318	+	0.788279i	$0.710972\pi$
$108$	0	0
$109$	−12.4391	−1.19145	−0.595723	−	0.803190i	$-0.703135\pi$
$109$	−12.4391	−1.19145	−0.595723	+	0.803190i	$0.703135\pi$
$110$	0	0
$111$	2.65368	0.251877
$112$	0	0
$113$	12.5730	1.18277	0.591386	−	0.806389i	$-0.298581\pi$
$113$	12.5730	1.18277	0.591386	+	0.806389i	$0.298581\pi$
$114$	0	0
$115$	2.29072	0.213611
$116$	0	0
$117$	−0.0722347	−0.00667810
$118$	0	0
$119$	3.13624	0.287498
$120$	0	0
$121$	−10.9155	−0.992316
$122$	0	0
$123$	−0.581449	−0.0524275
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−20.9132	−1.85575	−0.927874	−	0.372895i	$-0.878365\pi$
$127$	−20.9132	−1.85575	−0.927874	+	0.372895i	$0.878365\pi$
$128$	0	0
$129$	−9.75872	−0.859208
$130$	0	0
$131$	13.4680	1.17670	0.588352	−	0.808605i	$-0.299777\pi$
$131$	13.4680	1.17670	0.588352	+	0.808605i	$0.299777\pi$
$132$	0	0
$133$	3.81658	0.330940
$134$	0	0
$135$	5.26180	0.452863
$136$	0	0
$137$	−13.5525	−1.15787	−0.578935	−	0.815374i	$-0.696531\pi$
$137$	−13.5525	−1.15787	−0.578935	+	0.815374i	$0.696531\pi$
$138$	0	0
$139$	4.89496	0.415185	0.207593	−	0.978215i	$-0.433437\pi$
$139$	4.89496	0.415185	0.207593	+	0.978215i	$0.433437\pi$
$140$	0	0
$141$	1.92777	0.162347
$142$	0	0
$143$	−0.267938	−0.0224061
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	11.2846	0.930737
$148$	0	0
$149$	12.5236	1.02597	0.512986	−	0.858397i	$-0.328539\pi$
$149$	12.5236	1.02597	0.512986	+	0.858397i	$0.328539\pi$
$150$	0	0
$151$	7.60197	0.618639	0.309320	−	0.950958i	$-0.399899\pi$
$151$	7.60197	0.618639	0.309320	+	0.950958i	$0.399899\pi$
$152$	0	0
$153$	−0.389621	−0.0314990
$154$	0	0
$155$	10.0494	0.807191
$156$	0	0
$157$	24.8865	1.98616	0.993081	−	0.117428i	$-0.0374648\pi$
$157$	24.8865	1.98616	0.993081	+	0.117428i	$0.0374648\pi$
$158$	0	0
$159$	−0.581449	−0.0461119
$160$	0	0
$161$	1.44521	0.113899
$162$	0	0
$163$	−0.447480	−0.0350493	−0.0175247	−	0.999846i	$-0.505579\pi$
$163$	−0.447480	−0.0350493	−0.0175247	+	0.999846i	$0.505579\pi$
$164$	0	0
$165$	0.496928	0.0386858
$166$	0	0
$167$	19.8660	1.53728	0.768640	−	0.639682i	$-0.220934\pi$
$167$	19.8660	1.53728	0.768640	+	0.639682i	$0.220934\pi$
$168$	0	0
$169$	−12.1506	−0.934662
$170$	0	0
$171$	−0.474142	−0.0362586
$172$	0	0
$173$	25.4329	1.93363	0.966815	−	0.255478i	$-0.0822329\pi$
$173$	25.4329	1.93363	0.966815	+	0.255478i	$0.0822329\pi$
$174$	0	0
$175$	0.630898	0.0476914
$176$	0	0
$177$	16.6803	1.25377
$178$	0	0
$179$	−14.8371	−1.10898	−0.554489	−	0.832191i	$-0.687086\pi$
$179$	−14.8371	−1.10898	−0.554489	+	0.832191i	$0.687086\pi$
$180$	0	0
$181$	5.91548	0.439694	0.219847	−	0.975534i	$-0.429444\pi$
$181$	5.91548	0.439694	0.219847	+	0.975534i	$0.429444\pi$
$182$	0	0
$183$	5.26180	0.388963
$184$	0	0
$185$	−1.55252	−0.114144
$186$	0	0
$187$	−1.44521	−0.105684
$188$	0	0
$189$	3.31965	0.241469
$190$	0	0
$191$	7.02893	0.508595	0.254298	−	0.967126i	$-0.418156\pi$
$191$	7.02893	0.508595	0.254298	+	0.967126i	$0.418156\pi$
$192$	0	0
$193$	17.8660	1.28603	0.643013	−	0.765856i	$-0.277684\pi$
$193$	17.8660	1.28603	0.643013	+	0.765856i	$0.277684\pi$
$194$	0	0
$195$	−1.57531	−0.112810
$196$	0	0
$197$	−6.09890	−0.434528	−0.217264	−	0.976113i	$-0.569713\pi$
$197$	−6.09890	−0.434528	−0.217264	+	0.976113i	$0.569713\pi$
$198$	0	0
$199$	−9.75872	−0.691778	−0.345889	−	0.938276i	$-0.612423\pi$
$199$	−9.75872	−0.691778	−0.345889	+	0.938276i	$0.612423\pi$
$200$	0	0
$201$	−9.75872	−0.688327
$202$	0	0
$203$	−0.630898	−0.0442803
$204$	0	0
$205$	0.340173	0.0237587
$206$	0	0
$207$	−0.179542	−0.0124790
$208$	0	0
$209$	−1.75872	−0.121653
$210$	0	0
$211$	−9.86603	−0.679206	−0.339603	−	0.940569i	$-0.610293\pi$
$211$	−9.86603	−0.679206	−0.339603	+	0.940569i	$0.610293\pi$
$212$	0	0
$213$	−15.5174	−1.06324
$214$	0	0
$215$	5.70928	0.389369
$216$	0	0
$217$	6.34017	0.430399
$218$	0	0
$219$	11.8699	0.802094
$220$	0	0
$221$	4.58145	0.308182
$222$	0	0
$223$	10.9711	0.734677	0.367339	−	0.930087i	$-0.380269\pi$
$223$	10.9711	0.734677	0.367339	+	0.930087i	$0.380269\pi$
$224$	0	0
$225$	−0.0783777	−0.00522518
$226$	0	0
$227$	12.5464	0.832732	0.416366	−	0.909197i	$-0.363303\pi$
$227$	12.5464	0.832732	0.416366	+	0.909197i	$0.363303\pi$
$228$	0	0
$229$	−23.3607	−1.54372	−0.771859	−	0.635794i	$-0.780673\pi$
$229$	−23.3607	−1.54372	−0.771859	+	0.635794i	$0.780673\pi$
$230$	0	0
$231$	0.313511	0.0206275
$232$	0	0
$233$	12.4703	0.816954	0.408477	−	0.912769i	$-0.366060\pi$
$233$	12.4703	0.816954	0.408477	+	0.912769i	$0.366060\pi$
$234$	0	0
$235$	−1.12783	−0.0735713
$236$	0	0
$237$	−21.1773	−1.37561
$238$	0	0
$239$	13.7587	0.889978	0.444989	−	0.895536i	$-0.353208\pi$
$239$	13.7587	0.889978	0.444989	+	0.895536i	$0.353208\pi$
$240$	0	0
$241$	−14.6803	−0.945644	−0.472822	−	0.881158i	$-0.656765\pi$
$241$	−14.6803	−0.945644	−0.472822	+	0.881158i	$0.656765\pi$
$242$	0	0
$243$	−0.814315	−0.0522383
$244$	0	0
$245$	−6.60197	−0.421784
$246$	0	0
$247$	5.57531	0.354748
$248$	0	0
$249$	4.76487	0.301961
$250$	0	0
$251$	−15.4413	−0.974649	−0.487324	−	0.873221i	$-0.662027\pi$
$251$	−15.4413	−0.974649	−0.487324	+	0.873221i	$0.662027\pi$
$252$	0	0
$253$	−0.665970	−0.0418692
$254$	0	0
$255$	−8.49693	−0.532098
$256$	0	0
$257$	−6.28231	−0.391880	−0.195940	−	0.980616i	$-0.562776\pi$
$257$	−6.28231	−0.391880	−0.195940	+	0.980616i	$0.562776\pi$
$258$	0	0
$259$	−0.979481	−0.0608620
$260$	0	0
$261$	0.0783777	0.00485146
$262$	0	0
$263$	10.0761	0.621320	0.310660	−	0.950521i	$-0.399450\pi$
$263$	10.0761	0.621320	0.310660	+	0.950521i	$0.399450\pi$
$264$	0	0
$265$	0.340173	0.0208967
$266$	0	0
$267$	−8.09890	−0.495644
$268$	0	0
$269$	−28.1711	−1.71762	−0.858812	−	0.512291i	$-0.828797\pi$
$269$	−28.1711	−1.71762	−0.858812	+	0.512291i	$0.828797\pi$
$270$	0	0
$271$	28.8020	1.74960	0.874799	−	0.484485i	$-0.160993\pi$
$271$	28.8020	1.74960	0.874799	+	0.484485i	$0.160993\pi$
$272$	0	0
$273$	−0.993857	−0.0601510
$274$	0	0
$275$	−0.290725	−0.0175314
$276$	0	0
$277$	−0.0266620	−0.00160196	−0.000800982	−	1.00000i	$-0.500255\pi$
$277$	−0.0266620	−0.00160196	−0.000800982		1.00000i	$0.500255\pi$
$278$	0	0
$279$	−0.787653	−0.0471556
$280$	0	0
$281$	−28.0722	−1.67465	−0.837325	−	0.546706i	$-0.815881\pi$
$281$	−28.0722	−1.67465	−0.837325	+	0.546706i	$0.815881\pi$
$282$	0	0
$283$	20.8143	1.23728	0.618641	−	0.785674i	$-0.287683\pi$
$283$	20.8143	1.23728	0.618641	+	0.785674i	$0.287683\pi$
$284$	0	0
$285$	−10.3402	−0.612499
$286$	0	0
$287$	0.214614	0.0126683
$288$	0	0
$289$	7.71154	0.453620
$290$	0	0
$291$	27.1650	1.59244
$292$	0	0
$293$	15.4101	0.900270	0.450135	−	0.892961i	$-0.351376\pi$
$293$	15.4101	0.900270	0.450135	+	0.892961i	$0.351376\pi$
$294$	0	0
$295$	−9.75872	−0.568175
$296$	0	0
$297$	−1.52973	−0.0887641
$298$	0	0
$299$	2.11118	0.122093
$300$	0	0
$301$	3.60197	0.207614
$302$	0	0
$303$	−20.9483	−1.20345
$304$	0	0
$305$	−3.07838	−0.176267
$306$	0	0
$307$	28.4307	1.62262	0.811312	−	0.584614i	$-0.198754\pi$
$307$	28.4307	1.62262	0.811312	+	0.584614i	$0.198754\pi$
$308$	0	0
$309$	13.4452	0.764871
$310$	0	0
$311$	−19.6248	−1.11282	−0.556409	−	0.830909i	$-0.687821\pi$
$311$	−19.6248	−1.11282	−0.556409	+	0.830909i	$0.687821\pi$
$312$	0	0
$313$	22.9093	1.29491	0.647456	−	0.762103i	$-0.275833\pi$
$313$	22.9093	1.29491	0.647456	+	0.762103i	$0.275833\pi$
$314$	0	0
$315$	−0.0494483	−0.00278610
$316$	0	0
$317$	−22.8599	−1.28394	−0.641970	−	0.766730i	$-0.721882\pi$
$317$	−22.8599	−1.28394	−0.641970	+	0.766730i	$0.721882\pi$
$318$	0	0
$319$	0.290725	0.0162775
$320$	0	0
$321$	21.7587	1.21445
$322$	0	0
$323$	30.0722	1.67326
$324$	0	0
$325$	0.921622	0.0511224
$326$	0	0
$327$	21.2618	1.17578
$328$	0	0
$329$	−0.711543	−0.0392286
$330$	0	0
$331$	−24.0905	−1.32413	−0.662066	−	0.749445i	$-0.730320\pi$
$331$	−24.0905	−1.32413	−0.662066	+	0.749445i	$0.730320\pi$
$332$	0	0
$333$	0.121683	0.00666819
$334$	0	0
$335$	5.70928	0.311931
$336$	0	0
$337$	12.7877	0.696588	0.348294	−	0.937385i	$-0.386761\pi$
$337$	12.7877	0.696588	0.348294	+	0.937385i	$0.386761\pi$
$338$	0	0
$339$	−21.4908	−1.16722
$340$	0	0
$341$	−2.92162	−0.158215
$342$	0	0
$343$	−8.58145	−0.463355
$344$	0	0
$345$	−3.91548	−0.210802
$346$	0	0
$347$	−8.41628	−0.451810	−0.225905	−	0.974149i	$-0.572534\pi$
$347$	−8.41628	−0.451810	−0.225905	+	0.974149i	$0.572534\pi$
$348$	0	0
$349$	−22.1978	−1.18822	−0.594110	−	0.804384i	$-0.702496\pi$
$349$	−22.1978	−1.18822	−0.594110	+	0.804384i	$0.702496\pi$
$350$	0	0
$351$	4.84939	0.258841
$352$	0	0
$353$	−6.18342	−0.329110	−0.164555	−	0.986368i	$-0.552619\pi$
$353$	−6.18342	−0.329110	−0.164555	+	0.986368i	$0.552619\pi$
$354$	0	0
$355$	9.07838	0.481830
$356$	0	0
$357$	−5.36069	−0.283718
$358$	0	0
$359$	5.05559	0.266824	0.133412	−	0.991061i	$-0.457407\pi$
$359$	5.05559	0.266824	0.133412	+	0.991061i	$0.457407\pi$
$360$	0	0
$361$	17.5958	0.926096
$362$	0	0
$363$	18.6576	0.979268
$364$	0	0
$365$	−6.94441	−0.363487
$366$	0	0
$367$	29.5402	1.54199	0.770994	−	0.636843i	$-0.219760\pi$
$367$	29.5402	1.54199	0.770994	+	0.636843i	$0.219760\pi$
$368$	0	0
$369$	−0.0266620	−0.00138797
$370$	0	0
$371$	0.214614	0.0111422
$372$	0	0
$373$	−14.4124	−0.746246	−0.373123	−	0.927782i	$-0.621713\pi$
$373$	−14.4124	−0.746246	−0.373123	+	0.927782i	$0.621713\pi$
$374$	0	0
$375$	−1.70928	−0.0882666
$376$	0	0
$377$	−0.921622	−0.0474660
$378$	0	0
$379$	−14.1340	−0.726013	−0.363007	−	0.931787i	$-0.618250\pi$
$379$	−14.1340	−0.726013	−0.363007	+	0.931787i	$0.618250\pi$
$380$	0	0
$381$	35.7464	1.83135
$382$	0	0
$383$	−15.7815	−0.806397	−0.403199	−	0.915112i	$-0.632102\pi$
$383$	−15.7815	−0.806397	−0.403199	+	0.915112i	$0.632102\pi$
$384$	0	0
$385$	−0.183417	−0.00934782
$386$	0	0
$387$	−0.447480	−0.0227467
$388$	0	0
$389$	13.8166	0.700529	0.350264	−	0.936651i	$-0.386092\pi$
$389$	13.8166	0.700529	0.350264	+	0.936651i	$0.386092\pi$
$390$	0	0
$391$	11.3874	0.575883
$392$	0	0
$393$	−23.0205	−1.16123
$394$	0	0
$395$	12.3896	0.623390
$396$	0	0
$397$	−9.05172	−0.454293	−0.227146	−	0.973861i	$-0.572940\pi$
$397$	−9.05172	−0.454293	−0.227146	+	0.973861i	$0.572940\pi$
$398$	0	0
$399$	−6.52359	−0.326588
$400$	0	0
$401$	19.7587	0.986704	0.493352	−	0.869830i	$-0.335772\pi$
$401$	19.7587	0.986704	0.493352	+	0.869830i	$0.335772\pi$
$402$	0	0
$403$	9.26180	0.461363
$404$	0	0
$405$	−8.75872	−0.435224
$406$	0	0
$407$	0.451356	0.0223729
$408$	0	0
$409$	−1.71769	−0.0849341	−0.0424670	−	0.999098i	$-0.513522\pi$
$409$	−1.71769	−0.0849341	−0.0424670	+	0.999098i	$0.513522\pi$
$410$	0	0
$411$	23.1650	1.14264
$412$	0	0
$413$	−6.15676	−0.302954
$414$	0	0
$415$	−2.78765	−0.136841
$416$	0	0
$417$	−8.36683	−0.409726
$418$	0	0
$419$	35.5318	1.73584	0.867922	−	0.496701i	$-0.165456\pi$
$419$	35.5318	1.73584	0.867922	+	0.496701i	$0.165456\pi$
$420$	0	0
$421$	12.0722	0.588365	0.294182	−	0.955749i	$-0.404953\pi$
$421$	12.0722	0.588365	0.294182	+	0.955749i	$0.404953\pi$
$422$	0	0
$423$	0.0883965	0.00429798
$424$	0	0
$425$	4.97107	0.241132
$426$	0	0
$427$	−1.94214	−0.0939868
$428$	0	0
$429$	0.457980	0.0221115
$430$	0	0
$431$	19.8310	0.955224	0.477612	−	0.878571i	$-0.341503\pi$
$431$	19.8310	0.955224	0.477612	+	0.878571i	$0.341503\pi$
$432$	0	0
$433$	14.8143	0.711931	0.355965	−	0.934499i	$-0.384152\pi$
$433$	14.8143	0.711931	0.355965	+	0.934499i	$0.384152\pi$
$434$	0	0
$435$	1.70928	0.0819535
$436$	0	0
$437$	13.8576	0.662900
$438$	0	0
$439$	−17.8576	−0.852298	−0.426149	−	0.904653i	$-0.640130\pi$
$439$	−17.8576	−0.852298	−0.426149	+	0.904653i	$0.640130\pi$
$440$	0	0
$441$	0.517447	0.0246404
$442$	0	0
$443$	−33.5936	−1.59608	−0.798039	−	0.602606i	$-0.794129\pi$
$443$	−33.5936	−1.59608	−0.798039	+	0.602606i	$0.794129\pi$
$444$	0	0
$445$	4.73820	0.224612
$446$	0	0
$447$	−21.4063	−1.01248
$448$	0	0
$449$	7.07838	0.334049	0.167025	−	0.985953i	$-0.446584\pi$
$449$	7.07838	0.334049	0.167025	+	0.985953i	$0.446584\pi$
$450$	0	0
$451$	−0.0988967	−0.00465686
$452$	0	0
$453$	−12.9939	−0.610505
$454$	0	0
$455$	0.581449	0.0272588
$456$	0	0
$457$	−5.81658	−0.272088	−0.136044	−	0.990703i	$-0.543439\pi$
$457$	−5.81658	−0.272088	−0.136044	+	0.990703i	$0.543439\pi$
$458$	0	0
$459$	26.1568	1.22089
$460$	0	0
$461$	32.3090	1.50478	0.752390	−	0.658718i	$-0.228901\pi$
$461$	32.3090	1.50478	0.752390	+	0.658718i	$0.228901\pi$
$462$	0	0
$463$	1.44134	0.0669846	0.0334923	−	0.999439i	$-0.489337\pi$
$463$	1.44134	0.0669846	0.0334923	+	0.999439i	$0.489337\pi$
$464$	0	0
$465$	−17.1773	−0.796577
$466$	0	0
$467$	11.7503	0.543740	0.271870	−	0.962334i	$-0.412358\pi$
$467$	11.7503	0.543740	0.271870	+	0.962334i	$0.412358\pi$
$468$	0	0
$469$	3.60197	0.166323
$470$	0	0
$471$	−42.5380	−1.96005
$472$	0	0
$473$	−1.65983	−0.0763189
$474$	0	0
$475$	6.04945	0.277568
$476$	0	0
$477$	−0.0266620	−0.00122077
$478$	0	0
$479$	17.1689	0.784465	0.392233	−	0.919866i	$-0.371703\pi$
$479$	17.1689	0.784465	0.392233	+	0.919866i	$0.371703\pi$
$480$	0	0
$481$	−1.43084	−0.0652405
$482$	0	0
$483$	−2.47027	−0.112401
$484$	0	0
$485$	−15.8927	−0.721650
$486$	0	0
$487$	−4.10277	−0.185914	−0.0929572	−	0.995670i	$-0.529632\pi$
$487$	−4.10277	−0.185914	−0.0929572	+	0.995670i	$0.529632\pi$
$488$	0	0
$489$	0.764867	0.0345885
$490$	0	0
$491$	−40.7708	−1.83996	−0.919981	−	0.391963i	$-0.871796\pi$
$491$	−40.7708	−1.83996	−0.919981	+	0.391963i	$0.871796\pi$
$492$	0	0
$493$	−4.97107	−0.223886
$494$	0	0
$495$	0.0227863	0.00102417
$496$	0	0
$497$	5.72753	0.256915
$498$	0	0
$499$	18.4703	0.826843	0.413421	−	0.910540i	$-0.364334\pi$
$499$	18.4703	0.826843	0.413421	+	0.910540i	$0.364334\pi$
$500$	0	0
$501$	−33.9565	−1.51707
$502$	0	0
$503$	−21.4947	−0.958400	−0.479200	−	0.877706i	$-0.659073\pi$
$503$	−21.4947	−0.958400	−0.479200	+	0.877706i	$0.659073\pi$
$504$	0	0
$505$	12.2557	0.545369
$506$	0	0
$507$	20.7687	0.922372
$508$	0	0
$509$	−3.75872	−0.166602	−0.0833012	−	0.996524i	$-0.526546\pi$
$509$	−3.75872	−0.166602	−0.0833012	+	0.996524i	$0.526546\pi$
$510$	0	0
$511$	−4.38121	−0.193813
$512$	0	0
$513$	31.8310	1.40537
$514$	0	0
$515$	−7.86603	−0.346619
$516$	0	0
$517$	0.327887	0.0144204
$518$	0	0
$519$	−43.4719	−1.90820
$520$	0	0
$521$	12.8059	0.561037	0.280518	−	0.959849i	$-0.409494\pi$
$521$	12.8059	0.561037	0.280518	+	0.959849i	$0.409494\pi$
$522$	0	0
$523$	21.1278	0.923855	0.461928	−	0.886918i	$-0.347158\pi$
$523$	21.1278	0.923855	0.461928	+	0.886918i	$0.347158\pi$
$524$	0	0
$525$	−1.07838	−0.0470643
$526$	0	0
$527$	49.9565	2.17614
$528$	0	0
$529$	−17.7526	−0.771851
$530$	0	0
$531$	0.764867	0.0331924
$532$	0	0
$533$	0.313511	0.0135797
$534$	0	0
$535$	−12.7298	−0.550357
$536$	0	0
$537$	25.3607	1.09439
$538$	0	0
$539$	1.91935	0.0826725
$540$	0	0
$541$	−32.7382	−1.40753	−0.703763	−	0.710435i	$-0.748498\pi$
$541$	−32.7382	−1.40753	−0.703763	+	0.710435i	$0.748498\pi$
$542$	0	0
$543$	−10.1112	−0.433912
$544$	0	0
$545$	−12.4391	−0.532831
$546$	0	0
$547$	22.1073	0.945240	0.472620	−	0.881266i	$-0.343308\pi$
$547$	22.1073	0.945240	0.472620	+	0.881266i	$0.343308\pi$
$548$	0	0
$549$	0.241276	0.0102974
$550$	0	0
$551$	−6.04945	−0.257715
$552$	0	0
$553$	7.81658	0.332395
$554$	0	0
$555$	2.65368	0.112643
$556$	0	0
$557$	39.8720	1.68943	0.844715	−	0.535216i	$-0.179770\pi$
$557$	39.8720	1.68943	0.844715	+	0.535216i	$0.179770\pi$
$558$	0	0
$559$	5.26180	0.222550
$560$	0	0
$561$	2.47027	0.104295
$562$	0	0
$563$	−10.1217	−0.426578	−0.213289	−	0.976989i	$-0.568418\pi$
$563$	−10.1217	−0.426578	−0.213289	+	0.976989i	$0.568418\pi$
$564$	0	0
$565$	12.5730	0.528952
$566$	0	0
$567$	−5.52586	−0.232064
$568$	0	0
$569$	−24.4391	−1.02454	−0.512270	−	0.858825i	$-0.671195\pi$
$569$	−24.4391	−1.02454	−0.512270	+	0.858825i	$0.671195\pi$
$570$	0	0
$571$	28.2511	1.18227	0.591136	−	0.806572i	$-0.298680\pi$
$571$	28.2511	1.18227	0.591136	+	0.806572i	$0.298680\pi$
$572$	0	0
$573$	−12.0144	−0.501908
$574$	0	0
$575$	2.29072	0.0955298
$576$	0	0
$577$	46.1171	1.91988	0.959941	−	0.280202i	$-0.0904015\pi$
$577$	46.1171	1.91988	0.959941	+	0.280202i	$0.0904015\pi$
$578$	0	0
$579$	−30.5380	−1.26911
$580$	0	0
$581$	−1.75872	−0.0729642
$582$	0	0
$583$	−0.0988967	−0.00409588
$584$	0	0
$585$	−0.0722347	−0.00298654
$586$	0	0
$587$	0.715418	0.0295285	0.0147642	−	0.999891i	$-0.495300\pi$
$587$	0.715418	0.0295285	0.0147642	+	0.999891i	$0.495300\pi$
$588$	0	0
$589$	60.7936	2.50496
$590$	0	0
$591$	10.4247	0.428815
$592$	0	0
$593$	15.5441	0.638320	0.319160	−	0.947701i	$-0.396599\pi$
$593$	15.5441	0.638320	0.319160	+	0.947701i	$0.396599\pi$
$594$	0	0
$595$	3.13624	0.128573
$596$	0	0
$597$	16.6803	0.682681
$598$	0	0
$599$	−9.59809	−0.392167	−0.196084	−	0.980587i	$-0.562822\pi$
$599$	−9.59809	−0.392167	−0.196084	+	0.980587i	$0.562822\pi$
$600$	0	0
$601$	6.81044	0.277804	0.138902	−	0.990306i	$-0.455643\pi$
$601$	6.81044	0.277804	0.138902	+	0.990306i	$0.455643\pi$
$602$	0	0
$603$	−0.447480	−0.0182228
$604$	0	0
$605$	−10.9155	−0.443777
$606$	0	0
$607$	31.6970	1.28654	0.643271	−	0.765639i	$-0.277577\pi$
$607$	31.6970	1.28654	0.643271	+	0.765639i	$0.277577\pi$
$608$	0	0
$609$	1.07838	0.0436981
$610$	0	0
$611$	−1.03943	−0.0420508
$612$	0	0
$613$	−1.20394	−0.0486265	−0.0243133	−	0.999704i	$-0.507740\pi$
$613$	−1.20394	−0.0486265	−0.0243133	+	0.999704i	$0.507740\pi$
$614$	0	0
$615$	−0.581449	−0.0234463
$616$	0	0
$617$	−37.9337	−1.52715	−0.763577	−	0.645716i	$-0.776559\pi$
$617$	−37.9337	−1.52715	−0.763577	+	0.645716i	$0.776559\pi$
$618$	0	0
$619$	4.60424	0.185060	0.0925299	−	0.995710i	$-0.470505\pi$
$619$	4.60424	0.185060	0.0925299	+	0.995710i	$0.470505\pi$
$620$	0	0
$621$	12.0533	0.483683
$622$	0	0
$623$	2.98932	0.119765
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.00614	0.120054
$628$	0	0
$629$	−7.71769	−0.307724
$630$	0	0
$631$	−8.41241	−0.334893	−0.167446	−	0.985881i	$-0.553552\pi$
$631$	−8.41241	−0.334893	−0.167446	+	0.985881i	$0.553552\pi$
$632$	0	0
$633$	16.8638	0.670274
$634$	0	0
$635$	−20.9132	−0.829915
$636$	0	0
$637$	−6.08452	−0.241077
$638$	0	0
$639$	−0.711543	−0.0281482
$640$	0	0
$641$	32.5380	1.28517	0.642586	−	0.766213i	$-0.277861\pi$
$641$	32.5380	1.28517	0.642586	+	0.766213i	$0.277861\pi$
$642$	0	0
$643$	2.09293	0.0825372	0.0412686	−	0.999148i	$-0.486860\pi$
$643$	2.09293	0.0825372	0.0412686	+	0.999148i	$0.486860\pi$
$644$	0	0
$645$	−9.75872	−0.384249
$646$	0	0
$647$	45.1955	1.77682	0.888410	−	0.459051i	$-0.151811\pi$
$647$	45.1955	1.77682	0.888410	+	0.459051i	$0.151811\pi$
$648$	0	0
$649$	2.83710	0.111366
$650$	0	0
$651$	−10.8371	−0.424739
$652$	0	0
$653$	2.14834	0.0840712	0.0420356	−	0.999116i	$-0.486616\pi$
$653$	2.14834	0.0840712	0.0420356	+	0.999116i	$0.486616\pi$
$654$	0	0
$655$	13.4680	0.526238
$656$	0	0
$657$	0.544287	0.0212347
$658$	0	0
$659$	45.0843	1.75624	0.878118	−	0.478444i	$-0.158799\pi$
$659$	45.0843	1.75624	0.878118	+	0.478444i	$0.158799\pi$
$660$	0	0
$661$	36.3234	1.41281	0.706407	−	0.707806i	$-0.250315\pi$
$661$	36.3234	1.41281	0.706407	+	0.707806i	$0.250315\pi$
$662$	0	0
$663$	−7.83096	−0.304129
$664$	0	0
$665$	3.81658	0.148001
$666$	0	0
$667$	−2.29072	−0.0886972
$668$	0	0
$669$	−18.7526	−0.725017
$670$	0	0
$671$	0.894960	0.0345496
$672$	0	0
$673$	−17.4719	−0.673491	−0.336746	−	0.941596i	$-0.609326\pi$
$673$	−17.4719	−0.673491	−0.336746	+	0.941596i	$0.609326\pi$
$674$	0	0
$675$	5.26180	0.202527
$676$	0	0
$677$	40.0372	1.53875	0.769377	−	0.638796i	$-0.220567\pi$
$677$	40.0372	1.53875	0.769377	+	0.638796i	$0.220567\pi$
$678$	0	0
$679$	−10.0267	−0.384788
$680$	0	0
$681$	−21.4452	−0.821782
$682$	0	0
$683$	−2.07611	−0.0794402	−0.0397201	−	0.999211i	$-0.512647\pi$
$683$	−2.07611	−0.0794402	−0.0397201	+	0.999211i	$0.512647\pi$
$684$	0	0
$685$	−13.5525	−0.517815
$686$	0	0
$687$	39.9299	1.52342
$688$	0	0
$689$	0.313511	0.0119438
$690$	0	0
$691$	−26.7070	−1.01598	−0.507991	−	0.861362i	$-0.669612\pi$
$691$	−26.7070	−1.01598	−0.507991	+	0.861362i	$0.669612\pi$
$692$	0	0
$693$	0.0143758	0.000546093 0
$694$	0	0
$695$	4.89496	0.185676
$696$	0	0
$697$	1.69102	0.0640521
$698$	0	0
$699$	−21.3151	−0.806212
$700$	0	0
$701$	21.9155	0.827736	0.413868	−	0.910337i	$-0.364177\pi$
$701$	21.9155	0.827736	0.413868	+	0.910337i	$0.364177\pi$
$702$	0	0
$703$	−9.39189	−0.354222
$704$	0	0
$705$	1.92777	0.0726038
$706$	0	0
$707$	7.73206	0.290794
$708$	0	0
$709$	4.60811	0.173061	0.0865306	−	0.996249i	$-0.472422\pi$
$709$	4.60811	0.173061	0.0865306	+	0.996249i	$0.472422\pi$
$710$	0	0
$711$	−0.971071	−0.0364180
$712$	0	0
$713$	23.0205	0.862125
$714$	0	0
$715$	−0.267938	−0.0100203
$716$	0	0
$717$	−23.5174	−0.878275
$718$	0	0
$719$	6.80590	0.253817	0.126909	−	0.991914i	$-0.459494\pi$
$719$	6.80590	0.253817	0.126909	+	0.991914i	$0.459494\pi$
$720$	0	0
$721$	−4.96266	−0.184819
$722$	0	0
$723$	25.0928	0.933210
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−26.9711	−1.00030	−0.500151	−	0.865938i	$-0.666722\pi$
$727$	−26.9711	−1.00030	−0.500151	+	0.865938i	$0.666722\pi$
$728$	0	0
$729$	27.6681	1.02474
$730$	0	0
$731$	28.3812	1.04972
$732$	0	0
$733$	30.0638	1.11043	0.555216	−	0.831706i	$-0.312635\pi$
$733$	30.0638	1.11043	0.555216	+	0.831706i	$0.312635\pi$
$734$	0	0
$735$	11.2846	0.416238
$736$	0	0
$737$	−1.65983	−0.0611405
$738$	0	0
$739$	−51.1422	−1.88130	−0.940648	−	0.339383i	$-0.889782\pi$
$739$	−51.1422	−1.88130	−0.940648	+	0.339383i	$0.889782\pi$
$740$	0	0
$741$	−9.52973	−0.350084
$742$	0	0
$743$	11.1857	0.410363	0.205181	−	0.978724i	$-0.434222\pi$
$743$	11.1857	0.410363	0.205181	+	0.978724i	$0.434222\pi$
$744$	0	0
$745$	12.5236	0.458829
$746$	0	0
$747$	0.218490	0.00799413
$748$	0	0
$749$	−8.03120	−0.293454
$750$	0	0
$751$	−18.3630	−0.670074	−0.335037	−	0.942205i	$-0.608749\pi$
$751$	−18.3630	−0.670074	−0.335037	+	0.942205i	$0.608749\pi$
$752$	0	0
$753$	26.3935	0.961832
$754$	0	0
$755$	7.60197	0.276664
$756$	0	0
$757$	−15.8927	−0.577630	−0.288815	−	0.957385i	$-0.593261\pi$
$757$	−15.8927	−0.577630	−0.288815	+	0.957385i	$0.593261\pi$
$758$	0	0
$759$	1.13833	0.0413186
$760$	0	0
$761$	13.8843	0.503305	0.251652	−	0.967818i	$-0.419026\pi$
$761$	13.8843	0.503305	0.251652	+	0.967818i	$0.419026\pi$
$762$	0	0
$763$	−7.84778	−0.284109
$764$	0	0
$765$	−0.389621	−0.0140868
$766$	0	0
$767$	−8.99386	−0.324749
$768$	0	0
$769$	−35.4063	−1.27678	−0.638391	−	0.769712i	$-0.720400\pi$
$769$	−35.4063	−1.27678	−0.638391	+	0.769712i	$0.720400\pi$
$770$	0	0
$771$	10.7382	0.386727
$772$	0	0
$773$	0.488518	0.0175708	0.00878539	−	0.999961i	$-0.497203\pi$
$773$	0.488518	0.0175708	0.00878539	+	0.999961i	$0.497203\pi$
$774$	0	0
$775$	10.0494	0.360987
$776$	0	0
$777$	1.67420	0.0600617
$778$	0	0
$779$	2.05786	0.0737304
$780$	0	0
$781$	−2.63931	−0.0944419
$782$	0	0
$783$	−5.26180	−0.188041
$784$	0	0
$785$	24.8865	0.888239
$786$	0	0
$787$	1.99159	0.0709925	0.0354962	−	0.999370i	$-0.488699\pi$
$787$	1.99159	0.0709925	0.0354962	+	0.999370i	$0.488699\pi$
$788$	0	0
$789$	−17.2228	−0.613150
$790$	0	0
$791$	7.93230	0.282040
$792$	0	0
$793$	−2.83710	−0.100748
$794$	0	0
$795$	−0.581449	−0.0206219
$796$	0	0
$797$	17.2702	0.611742	0.305871	−	0.952073i	$-0.401052\pi$
$797$	17.2702	0.611742	0.305871	+	0.952073i	$0.401052\pi$
$798$	0	0
$799$	−5.60650	−0.198344
$800$	0	0
$801$	−0.371370	−0.0131217
$802$	0	0
$803$	2.01891	0.0712458
$804$	0	0
$805$	1.44521	0.0509371
$806$	0	0
$807$	48.1522	1.69504
$808$	0	0
$809$	−56.5068	−1.98667	−0.993336	−	0.115254i	$-0.963232\pi$
$809$	−56.5068	−1.98667	−0.993336	+	0.115254i	$0.963232\pi$
$810$	0	0
$811$	8.77924	0.308281	0.154140	−	0.988049i	$-0.450739\pi$
$811$	8.77924	0.308281	0.154140	+	0.988049i	$0.450739\pi$
$812$	0	0
$813$	−49.2306	−1.72659
$814$	0	0
$815$	−0.447480	−0.0156745
$816$	0	0
$817$	34.5380	1.20833
$818$	0	0
$819$	−0.0455727	−0.00159244
$820$	0	0
$821$	28.1568	0.982678	0.491339	−	0.870969i	$-0.336508\pi$
$821$	28.1568	0.982678	0.491339	+	0.870969i	$0.336508\pi$
$822$	0	0
$823$	20.7442	0.723096	0.361548	−	0.932353i	$-0.382248\pi$
$823$	20.7442	0.723096	0.361548	+	0.932353i	$0.382248\pi$
$824$	0	0
$825$	0.496928	0.0173008
$826$	0	0
$827$	9.12783	0.317406	0.158703	−	0.987326i	$-0.449269\pi$
$827$	9.12783	0.317406	0.158703	+	0.987326i	$0.449269\pi$
$828$	0	0
$829$	−31.8576	−1.10646	−0.553230	−	0.833028i	$-0.686605\pi$
$829$	−31.8576	−1.10646	−0.553230	+	0.833028i	$0.686605\pi$
$830$	0	0
$831$	0.0455727	0.00158090
$832$	0	0
$833$	−32.8188	−1.13711
$834$	0	0
$835$	19.8660	0.687492
$836$	0	0
$837$	52.8781	1.82774
$838$	0	0
$839$	27.4413	0.947380	0.473690	−	0.880692i	$-0.342922\pi$
$839$	27.4413	0.947380	0.473690	+	0.880692i	$0.342922\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	47.9832	1.65263
$844$	0	0
$845$	−12.1506	−0.417994
$846$	0	0
$847$	−6.88655	−0.236625
$848$	0	0
$849$	−35.5774	−1.22101
$850$	0	0
$851$	−3.55640	−0.121912
$852$	0	0
$853$	56.0515	1.91917	0.959584	−	0.281422i	$-0.0908061\pi$
$853$	56.0515	1.91917	0.959584	+	0.281422i	$0.0908061\pi$
$854$	0	0
$855$	−0.474142	−0.0162153
$856$	0	0
$857$	−6.08452	−0.207843	−0.103922	−	0.994585i	$-0.533139\pi$
$857$	−6.08452	−0.207843	−0.103922	+	0.994585i	$0.533139\pi$
$858$	0	0
$859$	−35.5936	−1.21444	−0.607218	−	0.794535i	$-0.707715\pi$
$859$	−35.5936	−1.21444	−0.607218	+	0.794535i	$0.707715\pi$
$860$	0	0
$861$	−0.366835	−0.0125017
$862$	0	0
$863$	−12.8287	−0.436694	−0.218347	−	0.975871i	$-0.570066\pi$
$863$	−12.8287	−0.436694	−0.218347	+	0.975871i	$0.570066\pi$
$864$	0	0
$865$	25.4329	0.864745
$866$	0	0
$867$	−13.1812	−0.447655
$868$	0	0
$869$	−3.60197	−0.122188
$870$	0	0
$871$	5.26180	0.178289
$872$	0	0
$873$	1.24563	0.0421583
$874$	0	0
$875$	0.630898	0.0213282
$876$	0	0
$877$	1.50307	0.0507551	0.0253776	−	0.999678i	$-0.491921\pi$
$877$	1.50307	0.0507551	0.0253776	+	0.999678i	$0.491921\pi$
$878$	0	0
$879$	−26.3402	−0.888432
$880$	0	0
$881$	23.4908	0.791425	0.395712	−	0.918375i	$-0.370498\pi$
$881$	23.4908	0.791425	0.395712	+	0.918375i	$0.370498\pi$
$882$	0	0
$883$	29.0433	0.977385	0.488693	−	0.872456i	$-0.337474\pi$
$883$	29.0433	0.977385	0.488693	+	0.872456i	$0.337474\pi$
$884$	0	0
$885$	16.6803	0.560704
$886$	0	0
$887$	−19.0700	−0.640307	−0.320153	−	0.947366i	$-0.603734\pi$
$887$	−19.0700	−0.640307	−0.320153	+	0.947366i	$0.603734\pi$
$888$	0	0
$889$	−13.1941	−0.442516
$890$	0	0
$891$	2.54638	0.0853068
$892$	0	0
$893$	−6.82273	−0.228314
$894$	0	0
$895$	−14.8371	−0.495950
$896$	0	0
$897$	−3.60859	−0.120487
$898$	0	0
$899$	−10.0494	−0.335168
$900$	0	0
$901$	1.69102	0.0563362
$902$	0	0
$903$	−6.15676	−0.204884
$904$	0	0
$905$	5.91548	0.196637
$906$	0	0
$907$	−5.54023	−0.183960	−0.0919802	−	0.995761i	$-0.529320\pi$
$907$	−5.54023	−0.183960	−0.0919802	+	0.995761i	$0.529320\pi$
$908$	0	0
$909$	−0.960570	−0.0318601
$910$	0	0
$911$	53.2990	1.76587	0.882937	−	0.469492i	$-0.155563\pi$
$911$	53.2990	1.76587	0.882937	+	0.469492i	$0.155563\pi$
$912$	0	0
$913$	0.810439	0.0268216
$914$	0	0
$915$	5.26180	0.173950
$916$	0	0
$917$	8.49693	0.280593
$918$	0	0
$919$	37.5897	1.23997	0.619985	−	0.784614i	$-0.287139\pi$
$919$	37.5897	1.23997	0.619985	+	0.784614i	$0.287139\pi$
$920$	0	0
$921$	−48.5958	−1.60129
$922$	0	0
$923$	8.36683	0.275398
$924$	0	0
$925$	−1.55252	−0.0510465
$926$	0	0
$927$	0.616522	0.0202492
$928$	0	0
$929$	37.3197	1.22442	0.612209	−	0.790696i	$-0.290281\pi$
$929$	37.3197	1.22442	0.612209	+	0.790696i	$0.290281\pi$
$930$	0	0
$931$	−39.9383	−1.30892
$932$	0	0
$933$	33.5441	1.09818
$934$	0	0
$935$	−1.44521	−0.0472635
$936$	0	0
$937$	−22.8638	−0.746927	−0.373463	−	0.927645i	$-0.621830\pi$
$937$	−22.8638	−0.746927	−0.373463	+	0.927645i	$0.621830\pi$
$938$	0	0
$939$	−39.1584	−1.27788
$940$	0	0
$941$	−0.523590	−0.0170686	−0.00853428	−	0.999964i	$-0.502717\pi$
$941$	−0.523590	−0.0170686	−0.00853428	+	0.999964i	$0.502717\pi$
$942$	0	0
$943$	0.779243	0.0253756
$944$	0	0
$945$	3.31965	0.107988
$946$	0	0
$947$	10.0228	0.325697	0.162848	−	0.986651i	$-0.447932\pi$
$947$	10.0228	0.325697	0.162848	+	0.986651i	$0.447932\pi$
$948$	0	0
$949$	−6.40012	−0.207757
$950$	0	0
$951$	39.0738	1.26706
$952$	0	0
$953$	−8.15676	−0.264223	−0.132112	−	0.991235i	$-0.542176\pi$
$953$	−8.15676	−0.264223	−0.132112	+	0.991235i	$0.542176\pi$
$954$	0	0
$955$	7.02893	0.227451
$956$	0	0
$957$	−0.496928	−0.0160634
$958$	0	0
$959$	−8.55025	−0.276102
$960$	0	0
$961$	69.9914	2.25779
$962$	0	0
$963$	0.997733	0.0321515
$964$	0	0
$965$	17.8660	0.575128
$966$	0	0
$967$	15.7671	0.507037	0.253518	−	0.967331i	$-0.418412\pi$
$967$	15.7671	0.507037	0.253518	+	0.967331i	$0.418412\pi$
$968$	0	0
$969$	−51.4017	−1.65126
$970$	0	0
$971$	17.8804	0.573810	0.286905	−	0.957959i	$-0.407374\pi$
$971$	17.8804	0.573810	0.286905	+	0.957959i	$0.407374\pi$
$972$	0	0
$973$	3.08822	0.0990037
$974$	0	0
$975$	−1.57531	−0.0504502
$976$	0	0
$977$	−55.1071	−1.76303	−0.881517	−	0.472153i	$-0.843477\pi$
$977$	−55.1071	−1.76303	−0.881517	+	0.472153i	$0.843477\pi$
$978$	0	0
$979$	−1.37751	−0.0440255
$980$	0	0
$981$	0.974946	0.0311276
$982$	0	0
$983$	−1.29687	−0.0413637	−0.0206818	−	0.999786i	$-0.506584\pi$
$983$	−1.29687	−0.0413637	−0.0206818	+	0.999786i	$0.506584\pi$
$984$	0	0
$985$	−6.09890	−0.194327
$986$	0	0
$987$	1.21622	0.0387128
$988$	0	0
$989$	13.0784	0.415868
$990$	0	0
$991$	−3.11942	−0.0990915	−0.0495458	−	0.998772i	$-0.515777\pi$
$991$	−3.11942	−0.0990915	−0.0495458	+	0.998772i	$0.515777\pi$
$992$	0	0
$993$	41.1773	1.30672
$994$	0	0
$995$	−9.75872	−0.309372
$996$	0	0
$997$	−30.2472	−0.957940	−0.478970	−	0.877831i	$-0.658990\pi$
$997$	−30.2472	−0.957940	−0.478970	+	0.877831i	$0.658990\pi$
$998$	0	0
$999$	−8.16904	−0.258457

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9280.2.a.bu.1.1		3
4.3	odd	2		9280.2.a.bm.1.3		3
8.3	odd	2		2320.2.a.s.1.1		3
8.5	even	2		145.2.a.d.1.2	✓	3
24.5	odd	2		1305.2.a.o.1.2		3
40.13	odd	4		725.2.b.d.349.2		6
40.29	even	2		725.2.a.d.1.2		3
40.37	odd	4		725.2.b.d.349.5		6
56.13	odd	2		7105.2.a.p.1.2		3
120.29	odd	2		6525.2.a.bh.1.2		3
232.173	even	2		4205.2.a.e.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.d.1.2	✓	3	8.5	even	2
725.2.a.d.1.2		3	40.29	even	2
725.2.b.d.349.2		6	40.13	odd	4
725.2.b.d.349.5		6	40.37	odd	4
1305.2.a.o.1.2		3	24.5	odd	2
2320.2.a.s.1.1		3	8.3	odd	2
4205.2.a.e.1.2		3	232.173	even	2
6525.2.a.bh.1.2		3	120.29	odd	2
7105.2.a.p.1.2		3	56.13	odd	2
9280.2.a.bm.1.3		3	4.3	odd	2
9280.2.a.bu.1.1		3	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:	\( N \)	\(=\)	\( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.70928 q^{3} +1.00000 q^{5} +0.630898 q^{7} -0.0783777 q^{9} -0.290725 q^{11} +0.921622 q^{13} -1.70928 q^{15} +4.97107 q^{17} +6.04945 q^{19} -1.07838 q^{21} +2.29072 q^{23} +1.00000 q^{25} +5.26180 q^{27} -1.00000 q^{29} +10.0494 q^{31} +0.496928 q^{33} +0.630898 q^{35} -1.55252 q^{37} -1.57531 q^{39} +0.340173 q^{41} +5.70928 q^{43} -0.0783777 q^{45} -1.12783 q^{47} -6.60197 q^{49} -8.49693 q^{51} +0.340173 q^{53} -0.290725 q^{55} -10.3402 q^{57} -9.75872 q^{59} -3.07838 q^{61} -0.0494483 q^{63} +0.921622 q^{65} +5.70928 q^{67} -3.91548 q^{69} +9.07838 q^{71} -6.94441 q^{73} -1.70928 q^{75} -0.183417 q^{77} +12.3896 q^{79} -8.75872 q^{81} -2.78765 q^{83} +4.97107 q^{85} +1.70928 q^{87} +4.73820 q^{89} +0.581449 q^{91} -17.1773 q^{93} +6.04945 q^{95} -15.8927 q^{97} +0.0227863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 8 q^{11} + 6 q^{13} + 2 q^{15} + 14 q^{23} + 3 q^{25} + 8 q^{27} - 3 q^{29} + 12 q^{31} - 16 q^{33} - 2 q^{35} - 4 q^{37} + 16 q^{39} - 10 q^{41} + 10 q^{43}+ \cdots - 20 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 8 * q^11 + 6 * q^13 + 2 * q^15 + 14 * q^23 + 3 * q^25 + 8 * q^27 - 3 * q^29 + 12 * q^31 - 16 * q^33 - 2 * q^35 - 4 * q^37 + 16 * q^39 - 10 * q^41 + 10 * q^43 + 3 * q^45 + 18 * q^47 - q^49 - 8 * q^51 - 10 * q^53 - 8 * q^55 - 20 * q^57 - 4 * q^59 - 6 * q^61 + 18 * q^63 + 6 * q^65 + 10 * q^67 + 20 * q^69 + 24 * q^71 - 4 * q^73 + 2 * q^75 + 4 * q^77 + 8 * q^79 - q^81 + 2 * q^83 - 2 * q^87 + 22 * q^89 + 16 * q^91 - 12 * q^93 - 36 * q^97 - 20 * q^99

Introduction

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