LMFDB - Embedded newform 5580.2.a.j.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5580,2,Mod(1,5580)] 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("5580.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5580, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5580 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5580.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,3,0,-2,0,0,0,-2,0,2,0,0,0,-10,0,0,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$44.5565243279$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1860)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.86620$ of defining polynomial
Character	$\chi$	$=$	5580.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +1.34889 q^{7} -4.69779 q^{11} -2.38350 q^{13} +1.21509 q^{17} +1.03461 q^{19} -6.24970 q^{23} +1.00000 q^{25} +5.59859 q^{29} +1.00000 q^{31} +1.34889 q^{35} +5.41811 q^{37} -1.30221 q^{41} +5.73240 q^{43} -3.48270 q^{47} -5.18048 q^{49} -2.51730 q^{53} -4.69779 q^{55} -7.86620 q^{59} -5.46479 q^{61} -2.38350 q^{65} -1.41811 q^{67} -3.79698 q^{71} -1.41811 q^{73} -6.33682 q^{77} -1.81952 q^{79} -13.2151 q^{83} +1.21509 q^{85} +2.20302 q^{89} -3.21509 q^{91} +1.03461 q^{95} +6.69779 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 2 q^{7} - 2 q^{11} + 2 q^{13} - 10 q^{17} - 2 q^{23} + 3 q^{25} - 6 q^{29} + 3 q^{31} - 2 q^{35} + 4 q^{37} - 16 q^{41} + 2 q^{43} - 12 q^{47} - 5 q^{49} - 6 q^{53} - 2 q^{55} - 16 q^{59}+ \cdots + 8 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 2 * q^7 - 2 * q^11 + 2 * q^13 - 10 * q^17 - 2 * q^23 + 3 * q^25 - 6 * q^29 + 3 * q^31 - 2 * q^35 + 4 * q^37 - 16 * q^41 + 2 * q^43 - 12 * q^47 - 5 * q^49 - 6 * q^53 - 2 * q^55 - 16 * q^59 + 14 * q^61 + 2 * q^65 + 8 * q^67 - 10 * q^71 + 8 * q^73 - 28 * q^77 - 16 * q^79 - 26 * q^83 - 10 * q^85 + 8 * q^89 + 4 * q^91 + 8 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.34889	0.509834	0.254917	−	0.966963i	$-0.417952\pi$
$7$	1.34889	0.509834	0.254917	+	0.966963i	$0.417952\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.69779	−1.41644	−0.708218	−	0.705994i	$-0.750501\pi$
$11$	−4.69779	−1.41644	−0.708218	+	0.705994i	$0.750501\pi$
$12$	0	0
$13$	−2.38350	−0.661065	−0.330532	−	0.943795i	$-0.607228\pi$
$13$	−2.38350	−0.661065	−0.330532	+	0.943795i	$0.607228\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.21509	0.294703	0.147352	−	0.989084i	$-0.452925\pi$
$17$	1.21509	0.294703	0.147352	+	0.989084i	$0.452925\pi$
$18$	0	0
$19$	1.03461	0.237355	0.118678	−	0.992933i	$-0.462134\pi$
$19$	1.03461	0.237355	0.118678	+	0.992933i	$0.462134\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.24970	−1.30315	−0.651576	−	0.758583i	$-0.725892\pi$
$23$	−6.24970	−1.30315	−0.651576	+	0.758583i	$0.725892\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.59859	1.03963	0.519816	−	0.854278i	$-0.326000\pi$
$29$	5.59859	1.03963	0.519816	+	0.854278i	$0.326000\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.34889	0.228005
$36$	0	0
$37$	5.41811	0.890732	0.445366	−	0.895349i	$-0.353074\pi$
$37$	5.41811	0.890732	0.445366	+	0.895349i	$0.353074\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.30221	−0.203371	−0.101686	−	0.994817i	$-0.532424\pi$
$41$	−1.30221	−0.203371	−0.101686	+	0.994817i	$0.532424\pi$
$42$	0	0
$43$	5.73240	0.874182	0.437091	−	0.899417i	$-0.356009\pi$
$43$	5.73240	0.874182	0.437091	+	0.899417i	$0.356009\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.48270	−0.508003	−0.254002	−	0.967204i	$-0.581747\pi$
$47$	−3.48270	−0.508003	−0.254002	+	0.967204i	$0.581747\pi$
$48$	0	0
$49$	−5.18048	−0.740069
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.51730	−0.345778	−0.172889	−	0.984941i	$-0.555310\pi$
$53$	−2.51730	−0.345778	−0.172889	+	0.984941i	$0.555310\pi$
$54$	0	0
$55$	−4.69779	−0.633450
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.86620	−1.02409	−0.512046	−	0.858958i	$-0.671112\pi$
$59$	−7.86620	−1.02409	−0.512046	+	0.858958i	$0.671112\pi$
$60$	0	0
$61$	−5.46479	−0.699695	−0.349848	−	0.936807i	$-0.613767\pi$
$61$	−5.46479	−0.699695	−0.349848	+	0.936807i	$0.613767\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.38350	−0.295637
$66$	0	0
$67$	−1.41811	−0.173250	−0.0866249	−	0.996241i	$-0.527608\pi$
$67$	−1.41811	−0.173250	−0.0866249	+	0.996241i	$0.527608\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.79698	−0.450619	−0.225309	−	0.974287i	$-0.572339\pi$
$71$	−3.79698	−0.450619	−0.225309	+	0.974287i	$0.572339\pi$
$72$	0	0
$73$	−1.41811	−0.165977	−0.0829886	−	0.996550i	$-0.526447\pi$
$73$	−1.41811	−0.165977	−0.0829886	+	0.996550i	$0.526447\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.33682	−0.722148
$78$	0	0
$79$	−1.81952	−0.204711	−0.102356	−	0.994748i	$-0.532638\pi$
$79$	−1.81952	−0.204711	−0.102356	+	0.994748i	$0.532638\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.2151	−1.45054	−0.725272	−	0.688462i	$-0.758286\pi$
$83$	−13.2151	−1.45054	−0.725272	+	0.688462i	$0.758286\pi$
$84$	0	0
$85$	1.21509	0.131795
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.20302	0.233519	0.116760	−	0.993160i	$-0.462749\pi$
$89$	2.20302	0.233519	0.116760	+	0.993160i	$0.462749\pi$
$90$	0	0
$91$	−3.21509	−0.337033
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.03461	0.106149
$96$	0	0
$97$	6.69779	0.680057	0.340029	−	0.940415i	$-0.389563\pi$
$97$	6.69779	0.680057	0.340029	+	0.940415i	$0.389563\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.1280	−1.30628	−0.653141	−	0.757236i	$-0.726549\pi$
$101$	−13.1280	−1.30628	−0.653141	+	0.757236i	$0.726549\pi$
$102$	0	0
$103$	−16.4769	−1.62351	−0.811757	−	0.583995i	$-0.801489\pi$
$103$	−16.4769	−1.62351	−0.811757	+	0.583995i	$0.801489\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.78491	−0.462574	−0.231287	−	0.972886i	$-0.574294\pi$
$107$	−4.78491	−0.462574	−0.231287	+	0.972886i	$0.574294\pi$
$108$	0	0
$109$	0.947489	0.0907530	0.0453765	−	0.998970i	$-0.485551\pi$
$109$	0.947489	0.0907530	0.0453765	+	0.998970i	$0.485551\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−17.1972	−1.61778	−0.808888	−	0.587963i	$-0.799930\pi$
$113$	−17.1972	−1.61778	−0.808888	+	0.587963i	$0.799930\pi$
$114$	0	0
$115$	−6.24970	−0.582788
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.63903	0.150250
$120$	0	0
$121$	11.0692	1.00629
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.16258	0.191898	0.0959490	−	0.995386i	$-0.469411\pi$
$127$	2.16258	0.191898	0.0959490	+	0.995386i	$0.469411\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−10.9700	−0.958455	−0.479228	−	0.877691i	$-0.659083\pi$
$131$	−10.9700	−0.958455	−0.479228	+	0.877691i	$0.659083\pi$
$132$	0	0
$133$	1.39558	0.121012
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.67989	0.741573	0.370786	−	0.928718i	$-0.379088\pi$
$137$	8.67989	0.741573	0.370786	+	0.928718i	$0.379088\pi$
$138$	0	0
$139$	4.76700	0.404332	0.202166	−	0.979351i	$-0.435202\pi$
$139$	4.76700	0.404332	0.202166	+	0.979351i	$0.435202\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	11.1972	0.936356
$144$	0	0
$145$	5.59859	0.464938
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.6978	0.876397	0.438198	−	0.898878i	$-0.355617\pi$
$149$	10.6978	0.876397	0.438198	+	0.898878i	$0.355617\pi$
$150$	0	0
$151$	5.01671	0.408254	0.204127	−	0.978944i	$-0.434565\pi$
$151$	5.01671	0.408254	0.204127	+	0.978944i	$0.434565\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−5.80161	−0.463019	−0.231510	−	0.972833i	$-0.574366\pi$
$157$	−5.80161	−0.463019	−0.231510	+	0.972833i	$0.574366\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.43018	−0.664392
$162$	0	0
$163$	−10.3143	−0.807877	−0.403939	−	0.914786i	$-0.632359\pi$
$163$	−10.3143	−0.807877	−0.403939	+	0.914786i	$0.632359\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.66318	0.438230	0.219115	−	0.975699i	$-0.429683\pi$
$167$	5.66318	0.438230	0.219115	+	0.975699i	$0.429683\pi$
$168$	0	0
$169$	−7.31892	−0.562994
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.6632	0.886735	0.443368	−	0.896340i	$-0.353784\pi$
$173$	11.6632	0.886735	0.443368	+	0.896340i	$0.353784\pi$
$174$	0	0
$175$	1.34889	0.101967
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.8258	0.883899	0.441949	−	0.897040i	$-0.354287\pi$
$179$	11.8258	0.883899	0.441949	+	0.897040i	$0.354287\pi$
$180$	0	0
$181$	−14.8604	−1.10456	−0.552281	−	0.833658i	$-0.686243\pi$
$181$	−14.8604	−1.10456	−0.552281	+	0.833658i	$0.686243\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.41811	0.398347
$186$	0	0
$187$	−5.70825	−0.417428
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.93541	0.429472	0.214736	−	0.976672i	$-0.431111\pi$
$191$	5.93541	0.429472	0.214736	+	0.976672i	$0.431111\pi$
$192$	0	0
$193$	−3.19719	−0.230139	−0.115069	−	0.993357i	$-0.536709\pi$
$193$	−3.19719	−0.230139	−0.115069	+	0.993357i	$0.536709\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.7491	−1.33582	−0.667909	−	0.744243i	$-0.732811\pi$
$197$	−18.7491	−1.33582	−0.667909	+	0.744243i	$0.732811\pi$
$198$	0	0
$199$	−21.9129	−1.55336	−0.776681	−	0.629894i	$-0.783099\pi$
$199$	−21.9129	−1.55336	−0.776681	+	0.629894i	$0.783099\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.55191	0.530040
$204$	0	0
$205$	−1.30221	−0.0909504
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.86037	−0.336199
$210$	0	0
$211$	1.66318	0.114498	0.0572490	−	0.998360i	$-0.481767\pi$
$211$	1.66318	0.114498	0.0572490	+	0.998360i	$0.481767\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.73240	0.390946
$216$	0	0
$217$	1.34889	0.0915689
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.89618	−0.194818
$222$	0	0
$223$	5.10382	0.341777	0.170889	−	0.985290i	$-0.445336\pi$
$223$	5.10382	0.341777	0.170889	+	0.985290i	$0.445336\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.85412	−0.189435	−0.0947174	−	0.995504i	$-0.530195\pi$
$227$	−2.85412	−0.189435	−0.0947174	+	0.995504i	$0.530195\pi$
$228$	0	0
$229$	−18.0934	−1.19564	−0.597822	−	0.801629i	$-0.703967\pi$
$229$	−18.0934	−1.19564	−0.597822	+	0.801629i	$0.703967\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.01671	0.590704	0.295352	−	0.955389i	$-0.404563\pi$
$233$	9.01671	0.590704	0.295352	+	0.955389i	$0.404563\pi$
$234$	0	0
$235$	−3.48270	−0.227186
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.9642	−0.903269	−0.451634	−	0.892203i	$-0.649159\pi$
$239$	−13.9642	−0.903269	−0.451634	+	0.892203i	$0.649159\pi$
$240$	0	0
$241$	13.1972	0.850106	0.425053	−	0.905169i	$-0.360256\pi$
$241$	13.1972	0.850106	0.425053	+	0.905169i	$0.360256\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.18048	−0.330969
$246$	0	0
$247$	−2.46599	−0.156907
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.36097	0.275262	0.137631	−	0.990484i	$-0.456051\pi$
$251$	4.36097	0.275262	0.137631	+	0.990484i	$0.456051\pi$
$252$	0	0
$253$	29.3598	1.84583
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.6211	−0.724906	−0.362453	−	0.932002i	$-0.618061\pi$
$257$	−11.6211	−0.724906	−0.362453	+	0.932002i	$0.618061\pi$
$258$	0	0
$259$	7.30846	0.454125
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.16258	0.380001	0.190001	−	0.981784i	$-0.439151\pi$
$263$	6.16258	0.380001	0.190001	+	0.981784i	$0.439151\pi$
$264$	0	0
$265$	−2.51730	−0.154637
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.4590	1.36935	0.684674	−	0.728850i	$-0.259945\pi$
$269$	22.4590	1.36935	0.684674	+	0.728850i	$0.259945\pi$
$270$	0	0
$271$	−22.8362	−1.38720	−0.693601	−	0.720360i	$-0.743977\pi$
$271$	−22.8362	−1.38720	−0.693601	+	0.720360i	$0.743977\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.69779	−0.283287
$276$	0	0
$277$	14.1851	0.852301	0.426150	−	0.904652i	$-0.359869\pi$
$277$	14.1851	0.852301	0.426150	+	0.904652i	$0.359869\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.8362	−0.765745	−0.382872	−	0.923801i	$-0.625065\pi$
$281$	−12.8362	−0.765745	−0.382872	+	0.923801i	$0.625065\pi$
$282$	0	0
$283$	5.70986	0.339416	0.169708	−	0.985494i	$-0.445718\pi$
$283$	5.70986	0.339416	0.169708	+	0.985494i	$0.445718\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.75655	−0.103686
$288$	0	0
$289$	−15.5236	−0.913150
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8.01790	−0.468411	−0.234205	−	0.972187i	$-0.575249\pi$
$293$	−8.01790	−0.468411	−0.234205	+	0.972187i	$0.575249\pi$
$294$	0	0
$295$	−7.86620	−0.457988
$296$	0	0
$297$	0	0
$298$	0	0
$299$	14.8962	0.861468
$300$	0	0
$301$	7.73240	0.445688
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5.46479	−0.312913
$306$	0	0
$307$	−18.7445	−1.06980	−0.534902	−	0.844914i	$-0.679651\pi$
$307$	−18.7445	−1.06980	−0.534902	+	0.844914i	$0.679651\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.2260	1.09020	0.545102	−	0.838370i	$-0.316491\pi$
$311$	19.2260	1.09020	0.545102	+	0.838370i	$0.316491\pi$
$312$	0	0
$313$	14.5219	0.820828	0.410414	−	0.911899i	$-0.365384\pi$
$313$	14.5219	0.820828	0.410414	+	0.911899i	$0.365384\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.378872	−0.0212796	−0.0106398	−	0.999943i	$-0.503387\pi$
$317$	−0.378872	−0.0212796	−0.0106398	+	0.999943i	$0.503387\pi$
$318$	0	0
$319$	−26.3010	−1.47257
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.25714	0.0699494
$324$	0	0
$325$	−2.38350	−0.132213
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.69779	−0.258997
$330$	0	0
$331$	17.1459	0.942423	0.471211	−	0.882020i	$-0.343817\pi$
$331$	17.1459	0.942423	0.471211	+	0.882020i	$0.343817\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.41811	−0.0774796
$336$	0	0
$337$	−2.04668	−0.111490	−0.0557450	−	0.998445i	$-0.517753\pi$
$337$	−2.04668	−0.111490	−0.0557450	+	0.998445i	$0.517753\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.69779	−0.254400
$342$	0	0
$343$	−16.4302	−0.887147
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.26641	0.497447	0.248723	−	0.968575i	$-0.419989\pi$
$347$	9.26641	0.497447	0.248723	+	0.968575i	$0.419989\pi$
$348$	0	0
$349$	−3.41348	−0.182719	−0.0913597	−	0.995818i	$-0.529121\pi$
$349$	−3.41348	−0.182719	−0.0913597	+	0.995818i	$0.529121\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.6799	1.10068	0.550340	−	0.834941i	$-0.314498\pi$
$353$	20.6799	1.10068	0.550340	+	0.834941i	$0.314498\pi$
$354$	0	0
$355$	−3.79698	−0.201523
$356$	0	0
$357$	0	0
$358$	0	0
$359$	33.6227	1.77454	0.887270	−	0.461250i	$-0.152599\pi$
$359$	33.6227	1.77454	0.887270	+	0.461250i	$0.152599\pi$
$360$	0	0
$361$	−17.9296	−0.943662
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.41811	−0.0742273
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.39558	−0.176290
$372$	0	0
$373$	35.2664	1.82603	0.913013	−	0.407931i	$-0.133750\pi$
$373$	35.2664	1.82603	0.913013	+	0.407931i	$0.133750\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13.3443	−0.687265
$378$	0	0
$379$	−19.1038	−0.981298	−0.490649	−	0.871357i	$-0.663240\pi$
$379$	−19.1038	−0.981298	−0.490649	+	0.871357i	$0.663240\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−20.0062	−1.02227	−0.511136	−	0.859500i	$-0.670775\pi$
$383$	−20.0062	−1.02227	−0.511136	+	0.859500i	$0.670775\pi$
$384$	0	0
$385$	−6.33682	−0.322954
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13.7036	0.694801	0.347400	−	0.937717i	$-0.387064\pi$
$389$	13.7036	0.694801	0.347400	+	0.937717i	$0.387064\pi$
$390$	0	0
$391$	−7.59396	−0.384043
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.81952	−0.0915498
$396$	0	0
$397$	−27.3598	−1.37315	−0.686574	−	0.727060i	$-0.740886\pi$
$397$	−27.3598	−1.37315	−0.686574	+	0.727060i	$0.740886\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.79698	0.489238	0.244619	−	0.969619i	$-0.421337\pi$
$401$	9.79698	0.489238	0.244619	+	0.969619i	$0.421337\pi$
$402$	0	0
$403$	−2.38350	−0.118731
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−25.4531	−1.26167
$408$	0	0
$409$	16.2559	0.803805	0.401902	−	0.915683i	$-0.368349\pi$
$409$	16.2559	0.803805	0.401902	+	0.915683i	$0.368349\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10.6107	−0.522117
$414$	0	0
$415$	−13.2151	−0.648703
$416$	0	0
$417$	0	0
$418$	0	0
$419$	25.6920	1.25513	0.627567	−	0.778562i	$-0.284051\pi$
$419$	25.6920	1.25513	0.627567	+	0.778562i	$0.284051\pi$
$420$	0	0
$421$	−12.8541	−0.626472	−0.313236	−	0.949675i	$-0.601413\pi$
$421$	−12.8541	−0.626472	−0.313236	+	0.949675i	$0.601413\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.21509	0.0589406
$426$	0	0
$427$	−7.37143	−0.356728
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−36.8891	−1.77689	−0.888444	−	0.458985i	$-0.848213\pi$
$431$	−36.8891	−1.77689	−0.888444	+	0.458985i	$0.848213\pi$
$432$	0	0
$433$	0.512673	0.0246375	0.0123188	−	0.999924i	$-0.496079\pi$
$433$	0.512673	0.0246375	0.0123188	+	0.999924i	$0.496079\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.46599	−0.309310
$438$	0	0
$439$	−24.7219	−1.17991	−0.589957	−	0.807435i	$-0.700855\pi$
$439$	−24.7219	−1.17991	−0.589957	+	0.807435i	$0.700855\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−35.4018	−1.68199	−0.840996	−	0.541042i	$-0.818030\pi$
$443$	−35.4018	−1.68199	−0.840996	+	0.541042i	$0.818030\pi$
$444$	0	0
$445$	2.20302	0.104433
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.53984	0.403020	0.201510	−	0.979486i	$-0.435415\pi$
$449$	8.53984	0.403020	0.201510	+	0.979486i	$0.435415\pi$
$450$	0	0
$451$	6.11751	0.288063
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.21509	−0.150726
$456$	0	0
$457$	−23.3131	−1.09054	−0.545270	−	0.838260i	$-0.683573\pi$
$457$	−23.3131	−1.09054	−0.545270	+	0.838260i	$0.683573\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	28.7958	1.34115	0.670577	−	0.741840i	$-0.266047\pi$
$461$	28.7958	1.34115	0.670577	+	0.741840i	$0.266047\pi$
$462$	0	0
$463$	−18.7670	−0.872177	−0.436088	−	0.899904i	$-0.643636\pi$
$463$	−18.7670	−0.872177	−0.436088	+	0.899904i	$0.643636\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.48270	0.161160	0.0805800	−	0.996748i	$-0.474323\pi$
$467$	3.48270	0.161160	0.0805800	+	0.996748i	$0.474323\pi$
$468$	0	0
$469$	−1.91288	−0.0883286
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−26.9296	−1.23822
$474$	0	0
$475$	1.03461	0.0474711
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.02998	0.321208	0.160604	−	0.987019i	$-0.448656\pi$
$479$	7.02998	0.321208	0.160604	+	0.987019i	$0.448656\pi$
$480$	0	0
$481$	−12.9141	−0.588831
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.69779	0.304131
$486$	0	0
$487$	−16.4543	−0.745617	−0.372809	−	0.927908i	$-0.621605\pi$
$487$	−16.4543	−0.745617	−0.372809	+	0.927908i	$0.621605\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	11.1972	0.505322	0.252661	−	0.967555i	$-0.418694\pi$
$491$	11.1972	0.505322	0.252661	+	0.967555i	$0.418694\pi$
$492$	0	0
$493$	6.80281	0.306383
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.12173	−0.229741
$498$	0	0
$499$	−2.74286	−0.122787	−0.0613935	−	0.998114i	$-0.519554\pi$
$499$	−2.74286	−0.122787	−0.0613935	+	0.998114i	$0.519554\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.9821	0.712606	0.356303	−	0.934370i	$-0.384037\pi$
$503$	15.9821	0.712606	0.356303	+	0.934370i	$0.384037\pi$
$504$	0	0
$505$	−13.1280	−0.584187
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.49357	−0.420795	−0.210398	−	0.977616i	$-0.567476\pi$
$509$	−9.49357	−0.420795	−0.210398	+	0.977616i	$0.567476\pi$
$510$	0	0
$511$	−1.91288	−0.0846209
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.4769	−0.726058
$516$	0	0
$517$	16.3610	0.719555
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.83622	−0.299500	−0.149750	−	0.988724i	$-0.547847\pi$
$521$	−6.83622	−0.299500	−0.149750	+	0.988724i	$0.547847\pi$
$522$	0	0
$523$	−27.1972	−1.18925	−0.594625	−	0.804003i	$-0.702699\pi$
$523$	−27.1972	−1.18925	−0.594625	+	0.804003i	$0.702699\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.21509	0.0529303
$528$	0	0
$529$	16.0588	0.698207
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.10382	0.134442
$534$	0	0
$535$	−4.78491	−0.206870
$536$	0	0
$537$	0	0
$538$	0	0
$539$	24.3368	1.04826
$540$	0	0
$541$	41.6787	1.79191	0.895953	−	0.444148i	$-0.146494\pi$
$541$	41.6787	1.79191	0.895953	+	0.444148i	$0.146494\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.947489	0.0405860
$546$	0	0
$547$	−19.0362	−0.813930	−0.406965	−	0.913444i	$-0.633413\pi$
$547$	−19.0362	−0.813930	−0.406965	+	0.913444i	$0.633413\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.79235	0.246762
$552$	0	0
$553$	−2.45433	−0.104369
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.2831	−1.53736	−0.768682	−	0.639631i	$-0.779087\pi$
$557$	−36.2831	−1.53736	−0.768682	+	0.639631i	$0.779087\pi$
$558$	0	0
$559$	−13.6632	−0.577891
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−24.0755	−1.01466	−0.507330	−	0.861752i	$-0.669367\pi$
$563$	−24.0755	−1.01466	−0.507330	+	0.861752i	$0.669367\pi$
$564$	0	0
$565$	−17.1972	−0.723491
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11.4360	0.479423	0.239711	−	0.970844i	$-0.422947\pi$
$569$	11.4360	0.479423	0.239711	+	0.970844i	$0.422947\pi$
$570$	0	0
$571$	24.2318	1.01407	0.507035	−	0.861926i	$-0.330742\pi$
$571$	24.2318	1.01407	0.507035	+	0.861926i	$0.330742\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.24970	−0.260631
$576$	0	0
$577$	1.53401	0.0638616	0.0319308	−	0.999490i	$-0.489834\pi$
$577$	1.53401	0.0638616	0.0319308	+	0.999490i	$0.489834\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−17.8258	−0.739537
$582$	0	0
$583$	11.8258	0.489773
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.1626	0.749650	0.374825	−	0.927096i	$-0.377703\pi$
$587$	18.1626	0.749650	0.374825	+	0.927096i	$0.377703\pi$
$588$	0	0
$589$	1.03461	0.0426303
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19.6274	0.806000	0.403000	−	0.915200i	$-0.367968\pi$
$593$	19.6274	0.806000	0.403000	+	0.915200i	$0.367968\pi$
$594$	0	0
$595$	1.63903	0.0671937
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0.609054	0.0248853	0.0124426	−	0.999923i	$-0.496039\pi$
$599$	0.609054	0.0248853	0.0124426	+	0.999923i	$0.496039\pi$
$600$	0	0
$601$	−23.6274	−0.963781	−0.481890	−	0.876232i	$-0.660050\pi$
$601$	−23.6274	−0.963781	−0.481890	+	0.876232i	$0.660050\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	11.0692	0.450028
$606$	0	0
$607$	−41.1505	−1.67025	−0.835124	−	0.550062i	$-0.814604\pi$
$607$	−41.1505	−1.67025	−0.835124	+	0.550062i	$0.814604\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.30101	0.335823
$612$	0	0
$613$	45.8966	1.85375	0.926873	−	0.375375i	$-0.122486\pi$
$613$	45.8966	1.85375	0.926873	+	0.375375i	$0.122486\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−46.8604	−1.88653	−0.943264	−	0.332044i	$-0.892262\pi$
$617$	−46.8604	−1.88653	−0.943264	+	0.332044i	$0.892262\pi$
$618$	0	0
$619$	11.8529	0.476409	0.238205	−	0.971215i	$-0.423441\pi$
$619$	11.8529	0.476409	0.238205	+	0.971215i	$0.423441\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.97164	0.119056
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.58351	0.262502
$630$	0	0
$631$	−17.2571	−0.686996	−0.343498	−	0.939153i	$-0.611612\pi$
$631$	−17.2571	−0.686996	−0.343498	+	0.939153i	$0.611612\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.16258	0.0858194
$636$	0	0
$637$	12.3477	0.489234
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.15795	0.322220	0.161110	−	0.986936i	$-0.448493\pi$
$641$	8.15795	0.322220	0.161110	+	0.986936i	$0.448493\pi$
$642$	0	0
$643$	7.19719	0.283829	0.141915	−	0.989879i	$-0.454674\pi$
$643$	7.19719	0.283829	0.141915	+	0.989879i	$0.454674\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.6873	0.616733	0.308366	−	0.951268i	$-0.400218\pi$
$647$	15.6873	0.616733	0.308366	+	0.951268i	$0.400218\pi$
$648$	0	0
$649$	36.9537	1.45056
$650$	0	0
$651$	0	0
$652$	0	0
$653$	37.8771	1.48224	0.741122	−	0.671370i	$-0.234294\pi$
$653$	37.8771	1.48224	0.741122	+	0.671370i	$0.234294\pi$
$654$	0	0
$655$	−10.9700	−0.428634
$656$	0	0
$657$	0	0
$658$	0	0
$659$	46.3298	1.80475	0.902376	−	0.430949i	$-0.141821\pi$
$659$	46.3298	1.80475	0.902376	+	0.430949i	$0.141821\pi$
$660$	0	0
$661$	−9.19719	−0.357729	−0.178865	−	0.983874i	$-0.557242\pi$
$661$	−9.19719	−0.357729	−0.178865	+	0.983874i	$0.557242\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.39558	0.0541181
$666$	0	0
$667$	−34.9895	−1.35480
$668$	0	0
$669$	0	0
$670$	0	0
$671$	25.6724	0.991074
$672$	0	0
$673$	−20.4169	−0.787014	−0.393507	−	0.919322i	$-0.628738\pi$
$673$	−20.4169	−0.787014	−0.393507	+	0.919322i	$0.628738\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15.7744	−0.606261	−0.303131	−	0.952949i	$-0.598032\pi$
$677$	−15.7744	−0.606261	−0.303131	+	0.952949i	$0.598032\pi$
$678$	0	0
$679$	9.03461	0.346716
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11.4920	−0.439728	−0.219864	−	0.975531i	$-0.570561\pi$
$683$	−11.4920	−0.439728	−0.219864	+	0.975531i	$0.570561\pi$
$684$	0	0
$685$	8.67989	0.331641
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.00000	0.228582
$690$	0	0
$691$	13.1638	0.500774	0.250387	−	0.968146i	$-0.419442\pi$
$691$	13.1638	0.500774	0.250387	+	0.968146i	$0.419442\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.76700	0.180823
$696$	0	0
$697$	−1.58231	−0.0599342
$698$	0	0
$699$	0	0
$700$	0	0
$701$	33.1372	1.25158	0.625788	−	0.779993i	$-0.284777\pi$
$701$	33.1372	1.25158	0.625788	+	0.779993i	$0.284777\pi$
$702$	0	0
$703$	5.60562	0.211420
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.7082	−0.665987
$708$	0	0
$709$	6.86037	0.257647	0.128823	−	0.991668i	$-0.458880\pi$
$709$	6.86037	0.257647	0.128823	+	0.991668i	$0.458880\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.24970	−0.234053
$714$	0	0
$715$	11.1972	0.418751
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.90663	0.369455	0.184728	−	0.982790i	$-0.440860\pi$
$719$	9.90663	0.369455	0.184728	+	0.982790i	$0.440860\pi$
$720$	0	0
$721$	−22.2256	−0.827723
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.59859	0.207927
$726$	0	0
$727$	15.4423	0.572722	0.286361	−	0.958122i	$-0.407554\pi$
$727$	15.4423	0.572722	0.286361	+	0.958122i	$0.407554\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.96539	0.257624
$732$	0	0
$733$	40.9296	1.51177	0.755884	−	0.654705i	$-0.227207\pi$
$733$	40.9296	1.51177	0.755884	+	0.654705i	$0.227207\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.66198	0.245397
$738$	0	0
$739$	50.4123	1.85445	0.927223	−	0.374510i	$-0.122189\pi$
$739$	50.4123	1.85445	0.927223	+	0.374510i	$0.122189\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21.3505	−0.783274	−0.391637	−	0.920120i	$-0.628091\pi$
$743$	−21.3505	−0.783274	−0.391637	+	0.920120i	$0.628091\pi$
$744$	0	0
$745$	10.6978	0.391937
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6.45433	−0.235836
$750$	0	0
$751$	32.1960	1.17485	0.587424	−	0.809279i	$-0.300142\pi$
$751$	32.1960	1.17485	0.587424	+	0.809279i	$0.300142\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.01671	0.182577
$756$	0	0
$757$	28.2334	1.02616	0.513080	−	0.858341i	$-0.328504\pi$
$757$	28.2334	1.02616	0.513080	+	0.858341i	$0.328504\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.9145	1.22940	0.614700	−	0.788761i	$-0.289277\pi$
$761$	33.9145	1.22940	0.614700	+	0.788761i	$0.289277\pi$
$762$	0	0
$763$	1.27806	0.0462690
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.7491	0.676991
$768$	0	0
$769$	36.5990	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$769$	36.5990	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	33.2634	1.19640	0.598201	−	0.801346i	$-0.295883\pi$
$773$	33.2634	1.19640	0.598201	+	0.801346i	$0.295883\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.34728	−0.0482713
$780$	0	0
$781$	17.8374	0.638273
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.80161	−0.207068
$786$	0	0
$787$	16.2559	0.579462	0.289731	−	0.957108i	$-0.406434\pi$
$787$	16.2559	0.579462	0.289731	+	0.957108i	$0.406434\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−23.1972	−0.824797
$792$	0	0
$793$	13.0253	0.462544
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24.7157	−0.875475	−0.437737	−	0.899103i	$-0.644220\pi$
$797$	−24.7157	−0.875475	−0.437737	+	0.899103i	$0.644220\pi$
$798$	0	0
$799$	−4.23180	−0.149710
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.66198	0.235096
$804$	0	0
$805$	−8.43018	−0.297125
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−19.8211	−0.696874	−0.348437	−	0.937332i	$-0.613287\pi$
$809$	−19.8211	−0.696874	−0.348437	+	0.937332i	$0.613287\pi$
$810$	0	0
$811$	11.2330	0.394444	0.197222	−	0.980359i	$-0.436808\pi$
$811$	11.2330	0.394444	0.197222	+	0.980359i	$0.436808\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−10.3143	−0.361294
$816$	0	0
$817$	5.93078	0.207492
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−22.9700	−0.801659	−0.400830	−	0.916153i	$-0.631278\pi$
$821$	−22.9700	−0.801659	−0.400830	+	0.916153i	$0.631278\pi$
$822$	0	0
$823$	0.697788	0.0243234	0.0121617	−	0.999926i	$-0.496129\pi$
$823$	0.697788	0.0243234	0.0121617	+	0.999926i	$0.496129\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30.0755	−1.04583	−0.522913	−	0.852386i	$-0.675155\pi$
$827$	−30.0755	−1.04583	−0.522913	+	0.852386i	$0.675155\pi$
$828$	0	0
$829$	−16.6620	−0.578695	−0.289347	−	0.957224i	$-0.593438\pi$
$829$	−16.6620	−0.578695	−0.289347	+	0.957224i	$0.593438\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.29477	−0.218101
$834$	0	0
$835$	5.66318	0.195982
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−16.9584	−0.585468	−0.292734	−	0.956194i	$-0.594565\pi$
$839$	−16.9584	−0.585468	−0.292734	+	0.956194i	$0.594565\pi$
$840$	0	0
$841$	2.34426	0.0808367
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7.31892	−0.251778
$846$	0	0
$847$	14.9312	0.513042
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−33.8616	−1.16076
$852$	0	0
$853$	−5.43138	−0.185967	−0.0929835	−	0.995668i	$-0.529640\pi$
$853$	−5.43138	−0.185967	−0.0929835	+	0.995668i	$0.529640\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.6632	0.398407	0.199203	−	0.979958i	$-0.436165\pi$
$857$	11.6632	0.398407	0.199203	+	0.979958i	$0.436165\pi$
$858$	0	0
$859$	54.9779	1.87582	0.937911	−	0.346877i	$-0.112758\pi$
$859$	54.9779	1.87582	0.937911	+	0.346877i	$0.112758\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−20.4636	−0.696589	−0.348294	−	0.937385i	$-0.613239\pi$
$863$	−20.4636	−0.696589	−0.348294	+	0.937385i	$0.613239\pi$
$864$	0	0
$865$	11.6632	0.396560
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.54770	0.289961
$870$	0	0
$871$	3.38007	0.114529
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.34889	0.0456009
$876$	0	0
$877$	−33.1855	−1.12060	−0.560298	−	0.828291i	$-0.689313\pi$
$877$	−33.1855	−1.12060	−0.560298	+	0.828291i	$0.689313\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5.30684	0.178792	0.0893960	−	0.995996i	$-0.471506\pi$
$881$	5.30684	0.178792	0.0893960	+	0.995996i	$0.471506\pi$
$882$	0	0
$883$	29.6032	0.996228	0.498114	−	0.867112i	$-0.334026\pi$
$883$	29.6032	0.996228	0.498114	+	0.867112i	$0.334026\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−47.2727	−1.58726	−0.793630	−	0.608401i	$-0.791811\pi$
$887$	−47.2727	−1.58726	−0.793630	+	0.608401i	$0.791811\pi$
$888$	0	0
$889$	2.91709	0.0978362
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.60323	−0.120577
$894$	0	0
$895$	11.8258	0.395292
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.59859	0.186724
$900$	0	0
$901$	−3.05876	−0.101902
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14.8604	−0.493975
$906$	0	0
$907$	−20.5010	−0.680725	−0.340363	−	0.940294i	$-0.610550\pi$
$907$	−20.5010	−0.680725	−0.340363	+	0.940294i	$0.610550\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−7.62737	−0.252706	−0.126353	−	0.991985i	$-0.540327\pi$
$911$	−7.62737	−0.252706	−0.126353	+	0.991985i	$0.540327\pi$
$912$	0	0
$913$	62.0817	2.05460
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14.7974	−0.488653
$918$	0	0
$919$	−6.92959	−0.228586	−0.114293	−	0.993447i	$-0.536460\pi$
$919$	−6.92959	−0.228586	−0.114293	+	0.993447i	$0.536460\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.05012	0.297888
$924$	0	0
$925$	5.41811	0.178146
$926$	0	0
$927$	0	0
$928$	0	0
$929$	39.6678	1.30146	0.650729	−	0.759310i	$-0.274463\pi$
$929$	39.6678	1.30146	0.650729	+	0.759310i	$0.274463\pi$
$930$	0	0
$931$	−5.35977	−0.175659
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.70825	−0.186680
$936$	0	0
$937$	31.6966	1.03548	0.517741	−	0.855537i	$-0.326773\pi$
$937$	31.6966	1.03548	0.517741	+	0.855537i	$0.326773\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	36.8557	1.20146	0.600731	−	0.799451i	$-0.294876\pi$
$941$	36.8557	1.20146	0.600731	+	0.799451i	$0.294876\pi$
$942$	0	0
$943$	8.13843	0.265024
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.6107	1.12470	0.562348	−	0.826901i	$-0.309898\pi$
$947$	34.6107	1.12470	0.562348	+	0.826901i	$0.309898\pi$
$948$	0	0
$949$	3.38007	0.109722
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.14468	−0.134259	−0.0671297	−	0.997744i	$-0.521384\pi$
$953$	−4.14468	−0.134259	−0.0671297	+	0.997744i	$0.521384\pi$
$954$	0	0
$955$	5.93541	0.192066
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.7082	0.378079
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3.19719	−0.102921
$966$	0	0
$967$	−24.7553	−0.796078	−0.398039	−	0.917368i	$-0.630309\pi$
$967$	−24.7553	−0.796078	−0.398039	+	0.917368i	$0.630309\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−39.9930	−1.28344	−0.641718	−	0.766941i	$-0.721778\pi$
$971$	−39.9930	−1.28344	−0.641718	+	0.766941i	$0.721778\pi$
$972$	0	0
$973$	6.43018	0.206142
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.8604	−0.347454	−0.173727	−	0.984794i	$-0.555581\pi$
$977$	−10.8604	−0.347454	−0.173727	+	0.984794i	$0.555581\pi$
$978$	0	0
$979$	−10.3493	−0.330765
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−5.03461	−0.160579	−0.0802895	−	0.996772i	$-0.525584\pi$
$983$	−5.03461	−0.160579	−0.0802895	+	0.996772i	$0.525584\pi$
$984$	0	0
$985$	−18.7491	−0.597396
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−35.8258	−1.13919
$990$	0	0
$991$	55.0230	1.74786	0.873931	−	0.486050i	$-0.161563\pi$
$991$	55.0230	1.74786	0.873931	+	0.486050i	$0.161563\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−21.9129	−0.694685
$996$	0	0
$997$	5.40723	0.171249	0.0856244	−	0.996327i	$-0.472711\pi$
$997$	5.40723	0.171249	0.0856244	+	0.996327i	$0.472711\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5580.2.a.j.1.3		3
3.2	odd	2		1860.2.a.g.1.3	✓	3
12.11	even	2		7440.2.a.bn.1.1		3
15.2	even	4		9300.2.g.q.3349.3		6
15.8	even	4		9300.2.g.q.3349.4		6
15.14	odd	2		9300.2.a.u.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.g.1.3	✓	3	3.2	odd	2
5580.2.a.j.1.3		3	1.1	even	1	trivial
7440.2.a.bn.1.1		3	12.11	even	2
9300.2.a.u.1.1		3	15.14	odd	2
9300.2.g.q.3349.3		6	15.2	even	4
9300.2.g.q.3349.4		6	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5580 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5580.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +1.34889 q^{7} -4.69779 q^{11} -2.38350 q^{13} +1.21509 q^{17} +1.03461 q^{19} -6.24970 q^{23} +1.00000 q^{25} +5.59859 q^{29} +1.00000 q^{31} +1.34889 q^{35} +5.41811 q^{37} -1.30221 q^{41} +5.73240 q^{43} -3.48270 q^{47} -5.18048 q^{49} -2.51730 q^{53} -4.69779 q^{55} -7.86620 q^{59} -5.46479 q^{61} -2.38350 q^{65} -1.41811 q^{67} -3.79698 q^{71} -1.41811 q^{73} -6.33682 q^{77} -1.81952 q^{79} -13.2151 q^{83} +1.21509 q^{85} +2.20302 q^{89} -3.21509 q^{91} +1.03461 q^{95} +6.69779 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 2 q^{7} - 2 q^{11} + 2 q^{13} - 10 q^{17} - 2 q^{23} + 3 q^{25} - 6 q^{29} + 3 q^{31} - 2 q^{35} + 4 q^{37} - 16 q^{41} + 2 q^{43} - 12 q^{47} - 5 q^{49} - 6 q^{53} - 2 q^{55} - 16 q^{59}+ \cdots + 8 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 2 * q^7 - 2 * q^11 + 2 * q^13 - 10 * q^17 - 2 * q^23 + 3 * q^25 - 6 * q^29 + 3 * q^31 - 2 * q^35 + 4 * q^37 - 16 * q^41 + 2 * q^43 - 12 * q^47 - 5 * q^49 - 6 * q^53 - 2 * q^55 - 16 * q^59 + 14 * q^61 + 2 * q^65 + 8 * q^67 - 10 * q^71 + 8 * q^73 - 28 * q^77 - 16 * q^79 - 26 * q^83 - 10 * q^85 + 8 * q^89 + 4 * q^91 + 8 * q^97

Introduction

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