LMFDB - Embedded newform 2376.2.a.h.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2376,2,Mod(1,2376)] 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("2376.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2376, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2376 = 2^{3} \cdot 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2376.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-2,0,0,0,2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.9724555203$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2376.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{5} -2.41421 q^{7} +1.00000 q^{11} +0.414214 q^{13} +2.24264 q^{17} +3.24264 q^{19} +5.65685 q^{23} -3.00000 q^{25} -3.41421 q^{29} -7.65685 q^{31} +3.41421 q^{35} +1.82843 q^{37} +2.82843 q^{41} -0.343146 q^{43} -0.242641 q^{47} -1.17157 q^{49} -13.8995 q^{53} -1.41421 q^{55} +5.41421 q^{59} -10.4142 q^{61} -0.585786 q^{65} -14.3137 q^{67} +4.00000 q^{71} -8.89949 q^{73} -2.41421 q^{77} -14.0711 q^{79} -1.65685 q^{83} -3.17157 q^{85} -8.24264 q^{89} -1.00000 q^{91} -4.58579 q^{95} -3.34315 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7} + 2 q^{11} - 2 q^{13} - 4 q^{17} - 2 q^{19} - 6 q^{25} - 4 q^{29} - 4 q^{31} + 4 q^{35} - 2 q^{37} - 12 q^{43} + 8 q^{47} - 8 q^{49} - 8 q^{53} + 8 q^{59} - 18 q^{61} - 4 q^{65} - 6 q^{67}+ \cdots - 18 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 + 2 * q^11 - 2 * q^13 - 4 * q^17 - 2 * q^19 - 6 * q^25 - 4 * q^29 - 4 * q^31 + 4 * q^35 - 2 * q^37 - 12 * q^43 + 8 * q^47 - 8 * q^49 - 8 * q^53 + 8 * q^59 - 18 * q^61 - 4 * q^65 - 6 * q^67 + 8 * q^71 + 2 * q^73 - 2 * q^77 - 14 * q^79 + 8 * q^83 - 12 * q^85 - 8 * q^89 - 2 * q^91 - 12 * q^95 - 18 * 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
$5$	−1.41421	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	0	0
$7$	−2.41421	−0.912487	−0.456243	−	0.889855i	$-0.650805\pi$
$7$	−2.41421	−0.912487	−0.456243	+	0.889855i	$0.650805\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.414214	0.114882	0.0574411	−	0.998349i	$-0.481706\pi$
$13$	0.414214	0.114882	0.0574411	+	0.998349i	$0.481706\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.24264	0.543920	0.271960	−	0.962309i	$-0.412328\pi$
$17$	2.24264	0.543920	0.271960	+	0.962309i	$0.412328\pi$
$18$	0	0
$19$	3.24264	0.743913	0.371956	−	0.928250i	$-0.378687\pi$
$19$	3.24264	0.743913	0.371956	+	0.928250i	$0.378687\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.65685	1.17954	0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685	1.17954	0.589768	+	0.807573i	$0.299219\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.41421	−0.634004	−0.317002	−	0.948425i	$-0.602676\pi$
$29$	−3.41421	−0.634004	−0.317002	+	0.948425i	$0.602676\pi$
$30$	0	0
$31$	−7.65685	−1.37521	−0.687606	−	0.726084i	$-0.741338\pi$
$31$	−7.65685	−1.37521	−0.687606	+	0.726084i	$0.741338\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.41421	0.577107
$36$	0	0
$37$	1.82843	0.300592	0.150296	−	0.988641i	$-0.451977\pi$
$37$	1.82843	0.300592	0.150296	+	0.988641i	$0.451977\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.82843	0.441726	0.220863	−	0.975305i	$-0.429113\pi$
$41$	2.82843	0.441726	0.220863	+	0.975305i	$0.429113\pi$
$42$	0	0
$43$	−0.343146	−0.0523292	−0.0261646	−	0.999658i	$-0.508329\pi$
$43$	−0.343146	−0.0523292	−0.0261646	+	0.999658i	$0.508329\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.242641	−0.0353928	−0.0176964	−	0.999843i	$-0.505633\pi$
$47$	−0.242641	−0.0353928	−0.0176964	+	0.999843i	$0.505633\pi$
$48$	0	0
$49$	−1.17157	−0.167368
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−13.8995	−1.90924	−0.954621	−	0.297823i	$-0.903740\pi$
$53$	−13.8995	−1.90924	−0.954621	+	0.297823i	$0.903740\pi$
$54$	0	0
$55$	−1.41421	−0.190693
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.41421	0.704871	0.352435	−	0.935836i	$-0.385354\pi$
$59$	5.41421	0.704871	0.352435	+	0.935836i	$0.385354\pi$
$60$	0	0
$61$	−10.4142	−1.33340	−0.666702	−	0.745325i	$-0.732294\pi$
$61$	−10.4142	−1.33340	−0.666702	+	0.745325i	$0.732294\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.585786	−0.0726579
$66$	0	0
$67$	−14.3137	−1.74870	−0.874349	−	0.485298i	$-0.838711\pi$
$67$	−14.3137	−1.74870	−0.874349	+	0.485298i	$0.838711\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	−8.89949	−1.04161	−0.520804	−	0.853677i	$-0.674368\pi$
$73$	−8.89949	−1.04161	−0.520804	+	0.853677i	$0.674368\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.41421	−0.275125
$78$	0	0
$79$	−14.0711	−1.58312	−0.791559	−	0.611092i	$-0.790730\pi$
$79$	−14.0711	−1.58312	−0.791559	+	0.611092i	$0.790730\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.65685	−0.181863	−0.0909317	−	0.995857i	$-0.528984\pi$
$83$	−1.65685	−0.181863	−0.0909317	+	0.995857i	$0.528984\pi$
$84$	0	0
$85$	−3.17157	−0.344005
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.24264	−0.873718	−0.436859	−	0.899530i	$-0.643909\pi$
$89$	−8.24264	−0.873718	−0.436859	+	0.899530i	$0.643909\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.58579	−0.470492
$96$	0	0
$97$	−3.34315	−0.339445	−0.169723	−	0.985492i	$-0.554287\pi$
$97$	−3.34315	−0.339445	−0.169723	+	0.985492i	$0.554287\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.75736	−0.572879	−0.286439	−	0.958098i	$-0.592472\pi$
$101$	−5.75736	−0.572879	−0.286439	+	0.958098i	$0.592472\pi$
$102$	0	0
$103$	5.48528	0.540481	0.270240	−	0.962793i	$-0.412897\pi$
$103$	5.48528	0.540481	0.270240	+	0.962793i	$0.412897\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.75736	0.556585	0.278292	−	0.960496i	$-0.410232\pi$
$107$	5.75736	0.556585	0.278292	+	0.960496i	$0.410232\pi$
$108$	0	0
$109$	3.65685	0.350263	0.175132	−	0.984545i	$-0.443965\pi$
$109$	3.65685	0.350263	0.175132	+	0.984545i	$0.443965\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.58579	0.243250	0.121625	−	0.992576i	$-0.461189\pi$
$113$	2.58579	0.243250	0.121625	+	0.992576i	$0.461189\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.41421	−0.496320
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.3137	1.01193
$126$	0	0
$127$	1.17157	0.103960	0.0519801	−	0.998648i	$-0.483447\pi$
$127$	1.17157	0.103960	0.0519801	+	0.998648i	$0.483447\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.24264	−0.195940	−0.0979702	−	0.995189i	$-0.531235\pi$
$131$	−2.24264	−0.195940	−0.0979702	+	0.995189i	$0.531235\pi$
$132$	0	0
$133$	−7.82843	−0.678811
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.82843	0.241649	0.120824	−	0.992674i	$-0.461446\pi$
$137$	2.82843	0.241649	0.120824	+	0.992674i	$0.461446\pi$
$138$	0	0
$139$	8.55635	0.725740	0.362870	−	0.931840i	$-0.381797\pi$
$139$	8.55635	0.725740	0.362870	+	0.931840i	$0.381797\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.414214	0.0346383
$144$	0	0
$145$	4.82843	0.400979
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.31371	0.271470	0.135735	−	0.990745i	$-0.456660\pi$
$149$	3.31371	0.271470	0.135735	+	0.990745i	$0.456660\pi$
$150$	0	0
$151$	−13.2426	−1.07767	−0.538835	−	0.842411i	$-0.681136\pi$
$151$	−13.2426	−1.07767	−0.538835	+	0.842411i	$0.681136\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.8284	0.869760
$156$	0	0
$157$	−7.31371	−0.583697	−0.291849	−	0.956464i	$-0.594270\pi$
$157$	−7.31371	−0.583697	−0.291849	+	0.956464i	$0.594270\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−13.6569	−1.07631
$162$	0	0
$163$	13.4853	1.05625	0.528124	−	0.849167i	$-0.322895\pi$
$163$	13.4853	1.05625	0.528124	+	0.849167i	$0.322895\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−23.2132	−1.79629	−0.898146	−	0.439698i	$-0.855086\pi$
$167$	−23.2132	−1.79629	−0.898146	+	0.439698i	$0.855086\pi$
$168$	0	0
$169$	−12.8284	−0.986802
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.1421	−1.37932	−0.689661	−	0.724133i	$-0.742240\pi$
$173$	−18.1421	−1.37932	−0.689661	+	0.724133i	$0.742240\pi$
$174$	0	0
$175$	7.24264	0.547492
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.5858	1.09019	0.545096	−	0.838373i	$-0.316493\pi$
$179$	14.5858	1.09019	0.545096	+	0.838373i	$0.316493\pi$
$180$	0	0
$181$	−13.9706	−1.03842	−0.519212	−	0.854646i	$-0.673774\pi$
$181$	−13.9706	−1.03842	−0.519212	+	0.854646i	$0.673774\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.58579	−0.190111
$186$	0	0
$187$	2.24264	0.163998
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.3137	1.39749	0.698745	−	0.715370i	$-0.253742\pi$
$191$	19.3137	1.39749	0.698745	+	0.715370i	$0.253742\pi$
$192$	0	0
$193$	−3.58579	−0.258111	−0.129055	−	0.991637i	$-0.541194\pi$
$193$	−3.58579	−0.258111	−0.129055	+	0.991637i	$0.541194\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.41421	−0.243253	−0.121626	−	0.992576i	$-0.538811\pi$
$197$	−3.41421	−0.243253	−0.121626	+	0.992576i	$0.538811\pi$
$198$	0	0
$199$	0.656854	0.0465632	0.0232816	−	0.999729i	$-0.492589\pi$
$199$	0.656854	0.0465632	0.0232816	+	0.999729i	$0.492589\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.24264	0.578520
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.24264	0.224298
$210$	0	0
$211$	−0.414214	−0.0285156	−0.0142578	−	0.999898i	$-0.504539\pi$
$211$	−0.414214	−0.0285156	−0.0142578	+	0.999898i	$0.504539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.485281	0.0330959
$216$	0	0
$217$	18.4853	1.25486
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.928932	0.0624867
$222$	0	0
$223$	−6.34315	−0.424768	−0.212384	−	0.977186i	$-0.568123\pi$
$223$	−6.34315	−0.424768	−0.212384	+	0.977186i	$0.568123\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.8995	1.58627	0.793133	−	0.609049i	$-0.208449\pi$
$227$	23.8995	1.58627	0.793133	+	0.609049i	$0.208449\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.75736	−0.639226	−0.319613	−	0.947548i	$-0.603553\pi$
$233$	−9.75736	−0.639226	−0.319613	+	0.947548i	$0.603553\pi$
$234$	0	0
$235$	0.343146	0.0223844
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.1421	0.914779	0.457389	−	0.889267i	$-0.348785\pi$
$239$	14.1421	0.914779	0.457389	+	0.889267i	$0.348785\pi$
$240$	0	0
$241$	−3.92893	−0.253085	−0.126542	−	0.991961i	$-0.540388\pi$
$241$	−3.92893	−0.253085	−0.126542	+	0.991961i	$0.540388\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.65685	0.105853
$246$	0	0
$247$	1.34315	0.0854623
$248$	0	0
$249$	0	0
$250$	0	0
$251$	11.5563	0.729430	0.364715	−	0.931119i	$-0.381166\pi$
$251$	11.5563	0.729430	0.364715	+	0.931119i	$0.381166\pi$
$252$	0	0
$253$	5.65685	0.355643
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.92893	0.556971	0.278486	−	0.960440i	$-0.410168\pi$
$257$	8.92893	0.556971	0.278486	+	0.960440i	$0.410168\pi$
$258$	0	0
$259$	−4.41421	−0.274286
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.55635	−0.589270	−0.294635	−	0.955610i	$-0.595198\pi$
$263$	−9.55635	−0.589270	−0.294635	+	0.955610i	$0.595198\pi$
$264$	0	0
$265$	19.6569	1.20751
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.00000	0.487769	0.243884	−	0.969804i	$-0.421578\pi$
$269$	8.00000	0.487769	0.243884	+	0.969804i	$0.421578\pi$
$270$	0	0
$271$	−4.07107	−0.247300	−0.123650	−	0.992326i	$-0.539460\pi$
$271$	−4.07107	−0.247300	−0.123650	+	0.992326i	$0.539460\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.00000	−0.180907
$276$	0	0
$277$	8.48528	0.509831	0.254916	−	0.966963i	$-0.417952\pi$
$277$	8.48528	0.509831	0.254916	+	0.966963i	$0.417952\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−25.6569	−1.53056	−0.765280	−	0.643698i	$-0.777399\pi$
$281$	−25.6569	−1.53056	−0.765280	+	0.643698i	$0.777399\pi$
$282$	0	0
$283$	−0.485281	−0.0288470	−0.0144235	−	0.999896i	$-0.504591\pi$
$283$	−0.485281	−0.0288470	−0.0144235	+	0.999896i	$0.504591\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.82843	−0.403069
$288$	0	0
$289$	−11.9706	−0.704151
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−13.7574	−0.803714	−0.401857	−	0.915703i	$-0.631635\pi$
$293$	−13.7574	−0.803714	−0.401857	+	0.915703i	$0.631635\pi$
$294$	0	0
$295$	−7.65685	−0.445799
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.34315	0.135508
$300$	0	0
$301$	0.828427	0.0477497
$302$	0	0
$303$	0	0
$304$	0	0
$305$	14.7279	0.843318
$306$	0	0
$307$	17.1716	0.980033	0.490017	−	0.871713i	$-0.336991\pi$
$307$	17.1716	0.980033	0.490017	+	0.871713i	$0.336991\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.89949	0.561349	0.280674	−	0.959803i	$-0.409442\pi$
$311$	9.89949	0.561349	0.280674	+	0.959803i	$0.409442\pi$
$312$	0	0
$313$	−26.7990	−1.51477	−0.757384	−	0.652969i	$-0.773523\pi$
$313$	−26.7990	−1.51477	−0.757384	+	0.652969i	$0.773523\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.5858	−1.49321	−0.746603	−	0.665270i	$-0.768317\pi$
$317$	−26.5858	−1.49321	−0.746603	+	0.665270i	$0.768317\pi$
$318$	0	0
$319$	−3.41421	−0.191159
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.27208	0.404629
$324$	0	0
$325$	−1.24264	−0.0689293
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.585786	0.0322955
$330$	0	0
$331$	−19.8284	−1.08987	−0.544934	−	0.838479i	$-0.683445\pi$
$331$	−19.8284	−1.08987	−0.544934	+	0.838479i	$0.683445\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	20.2426	1.10597
$336$	0	0
$337$	26.2132	1.42792	0.713962	−	0.700184i	$-0.246899\pi$
$337$	26.2132	1.42792	0.713962	+	0.700184i	$0.246899\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−7.65685	−0.414642
$342$	0	0
$343$	19.7279	1.06521
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.7279	−1.22010	−0.610049	−	0.792363i	$-0.708850\pi$
$347$	−22.7279	−1.22010	−0.610049	+	0.792363i	$0.708850\pi$
$348$	0	0
$349$	−4.41421	−0.236287	−0.118144	−	0.992997i	$-0.537694\pi$
$349$	−4.41421	−0.236287	−0.118144	+	0.992997i	$0.537694\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.2132	0.703268	0.351634	−	0.936138i	$-0.385626\pi$
$353$	13.2132	0.703268	0.351634	+	0.936138i	$0.385626\pi$
$354$	0	0
$355$	−5.65685	−0.300235
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.7990	−1.67829	−0.839143	−	0.543910i	$-0.816943\pi$
$359$	−31.7990	−1.67829	−0.839143	+	0.543910i	$0.816943\pi$
$360$	0	0
$361$	−8.48528	−0.446594
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.5858	0.658770
$366$	0	0
$367$	15.1421	0.790413	0.395207	−	0.918592i	$-0.370673\pi$
$367$	15.1421	0.790413	0.395207	+	0.918592i	$0.370673\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	33.5563	1.74216
$372$	0	0
$373$	−0.899495	−0.0465741	−0.0232870	−	0.999729i	$-0.507413\pi$
$373$	−0.899495	−0.0465741	−0.0232870	+	0.999729i	$0.507413\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.41421	−0.0728357
$378$	0	0
$379$	12.1716	0.625212	0.312606	−	0.949883i	$-0.398798\pi$
$379$	12.1716	0.625212	0.312606	+	0.949883i	$0.398798\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7.75736	−0.396383	−0.198191	−	0.980163i	$-0.563507\pi$
$383$	−7.75736	−0.396383	−0.198191	+	0.980163i	$0.563507\pi$
$384$	0	0
$385$	3.41421	0.174004
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9.17157	−0.465017	−0.232509	−	0.972594i	$-0.574693\pi$
$389$	−9.17157	−0.465017	−0.232509	+	0.972594i	$0.574693\pi$
$390$	0	0
$391$	12.6863	0.641573
$392$	0	0
$393$	0	0
$394$	0	0
$395$	19.8995	1.00125
$396$	0	0
$397$	35.3137	1.77234	0.886172	−	0.463356i	$-0.153355\pi$
$397$	35.3137	1.77234	0.886172	+	0.463356i	$0.153355\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.7990	−0.788964	−0.394482	−	0.918904i	$-0.629076\pi$
$401$	−15.7990	−0.788964	−0.394482	+	0.918904i	$0.629076\pi$
$402$	0	0
$403$	−3.17157	−0.157987
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.82843	0.0906318
$408$	0	0
$409$	−27.3848	−1.35409	−0.677045	−	0.735942i	$-0.736740\pi$
$409$	−27.3848	−1.35409	−0.677045	+	0.735942i	$0.736740\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−13.0711	−0.643185
$414$	0	0
$415$	2.34315	0.115021
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.6274	−0.910009	−0.455004	−	0.890489i	$-0.650362\pi$
$419$	−18.6274	−0.910009	−0.455004	+	0.890489i	$0.650362\pi$
$420$	0	0
$421$	14.7990	0.721259	0.360629	−	0.932709i	$-0.382562\pi$
$421$	14.7990	0.721259	0.360629	+	0.932709i	$0.382562\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.72792	−0.326352
$426$	0	0
$427$	25.1421	1.21671
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.8284	−0.521587	−0.260793	−	0.965395i	$-0.583984\pi$
$431$	−10.8284	−0.521587	−0.260793	+	0.965395i	$0.583984\pi$
$432$	0	0
$433$	24.0000	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.3431	0.877472
$438$	0	0
$439$	2.82843	0.134993	0.0674967	−	0.997719i	$-0.478499\pi$
$439$	2.82843	0.134993	0.0674967	+	0.997719i	$0.478499\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.2426	1.72194	0.860970	−	0.508656i	$-0.169857\pi$
$443$	36.2426	1.72194	0.860970	+	0.508656i	$0.169857\pi$
$444$	0	0
$445$	11.6569	0.552588
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−26.1421	−1.23372	−0.616862	−	0.787071i	$-0.711596\pi$
$449$	−26.1421	−1.23372	−0.616862	+	0.787071i	$0.711596\pi$
$450$	0	0
$451$	2.82843	0.133185
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.41421	0.0662994
$456$	0	0
$457$	4.34315	0.203164	0.101582	−	0.994827i	$-0.467610\pi$
$457$	4.34315	0.203164	0.101582	+	0.994827i	$0.467610\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4.58579	−0.213581	−0.106791	−	0.994282i	$-0.534057\pi$
$461$	−4.58579	−0.213581	−0.106791	+	0.994282i	$0.534057\pi$
$462$	0	0
$463$	−27.4853	−1.27735	−0.638675	−	0.769477i	$-0.720517\pi$
$463$	−27.4853	−1.27735	−0.638675	+	0.769477i	$0.720517\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	35.3137	1.63412	0.817062	−	0.576550i	$-0.195601\pi$
$467$	35.3137	1.63412	0.817062	+	0.576550i	$0.195601\pi$
$468$	0	0
$469$	34.5563	1.59566
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.343146	−0.0157779
$474$	0	0
$475$	−9.72792	−0.446348
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−21.7574	−0.994119	−0.497060	−	0.867716i	$-0.665587\pi$
$479$	−21.7574	−0.994119	−0.497060	+	0.867716i	$0.665587\pi$
$480$	0	0
$481$	0.757359	0.0345326
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.72792	0.214684
$486$	0	0
$487$	28.6569	1.29857	0.649283	−	0.760547i	$-0.275069\pi$
$487$	28.6569	1.29857	0.649283	+	0.760547i	$0.275069\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−37.5563	−1.69489	−0.847447	−	0.530880i	$-0.821862\pi$
$491$	−37.5563	−1.69489	−0.847447	+	0.530880i	$0.821862\pi$
$492$	0	0
$493$	−7.65685	−0.344847
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.65685	−0.433169
$498$	0	0
$499$	38.2843	1.71384	0.856920	−	0.515450i	$-0.172375\pi$
$499$	38.2843	1.71384	0.856920	+	0.515450i	$0.172375\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.31371	−0.147751	−0.0738755	−	0.997267i	$-0.523537\pi$
$503$	−3.31371	−0.147751	−0.0738755	+	0.997267i	$0.523537\pi$
$504$	0	0
$505$	8.14214	0.362320
$506$	0	0
$507$	0	0
$508$	0	0
$509$	24.9289	1.10496	0.552478	−	0.833528i	$-0.313682\pi$
$509$	24.9289	1.10496	0.552478	+	0.833528i	$0.313682\pi$
$510$	0	0
$511$	21.4853	0.950453
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−7.75736	−0.341830
$516$	0	0
$517$	−0.242641	−0.0106713
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−16.9706	−0.743494	−0.371747	−	0.928334i	$-0.621241\pi$
$521$	−16.9706	−0.743494	−0.371747	+	0.928334i	$0.621241\pi$
$522$	0	0
$523$	10.7574	0.470386	0.235193	−	0.971949i	$-0.424428\pi$
$523$	10.7574	0.470386	0.235193	+	0.971949i	$0.424428\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−17.1716	−0.748005
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.17157	0.0507465
$534$	0	0
$535$	−8.14214	−0.352015
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.17157	−0.0504632
$540$	0	0
$541$	15.9289	0.684838	0.342419	−	0.939547i	$-0.388754\pi$
$541$	15.9289	0.684838	0.342419	+	0.939547i	$0.388754\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.17157	−0.221526
$546$	0	0
$547$	9.58579	0.409859	0.204929	−	0.978777i	$-0.434304\pi$
$547$	9.58579	0.409859	0.204929	+	0.978777i	$0.434304\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11.0711	−0.471643
$552$	0	0
$553$	33.9706	1.44458
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.0416	1.27290	0.636452	−	0.771316i	$-0.280401\pi$
$557$	30.0416	1.27290	0.636452	+	0.771316i	$0.280401\pi$
$558$	0	0
$559$	−0.142136	−0.00601170
$560$	0	0
$561$	0	0
$562$	0	0
$563$	43.5980	1.83744	0.918718	−	0.394914i	$-0.129226\pi$
$563$	43.5980	1.83744	0.918718	+	0.394914i	$0.129226\pi$
$564$	0	0
$565$	−3.65685	−0.153845
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−15.7990	−0.662328	−0.331164	−	0.943573i	$-0.607441\pi$
$569$	−15.7990	−0.662328	−0.331164	+	0.943573i	$0.607441\pi$
$570$	0	0
$571$	−4.41421	−0.184729	−0.0923645	−	0.995725i	$-0.529443\pi$
$571$	−4.41421	−0.184729	−0.0923645	+	0.995725i	$0.529443\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−16.9706	−0.707721
$576$	0	0
$577$	28.1127	1.17035	0.585173	−	0.810908i	$-0.301026\pi$
$577$	28.1127	1.17035	0.585173	+	0.810908i	$0.301026\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	−13.8995	−0.575658
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.3848	−0.428626	−0.214313	−	0.976765i	$-0.568751\pi$
$587$	−10.3848	−0.428626	−0.214313	+	0.976765i	$0.568751\pi$
$588$	0	0
$589$	−24.8284	−1.02304
$590$	0	0
$591$	0	0
$592$	0	0
$593$	33.9411	1.39379	0.696897	−	0.717171i	$-0.254563\pi$
$593$	33.9411	1.39379	0.696897	+	0.717171i	$0.254563\pi$
$594$	0	0
$595$	7.65685	0.313900
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−23.5147	−0.960785	−0.480393	−	0.877054i	$-0.659506\pi$
$599$	−23.5147	−0.960785	−0.480393	+	0.877054i	$0.659506\pi$
$600$	0	0
$601$	−16.6274	−0.678246	−0.339123	−	0.940742i	$-0.610130\pi$
$601$	−16.6274	−0.678246	−0.339123	+	0.940742i	$0.610130\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.41421	−0.0574960
$606$	0	0
$607$	28.8995	1.17299	0.586497	−	0.809951i	$-0.300507\pi$
$607$	28.8995	1.17299	0.586497	+	0.809951i	$0.300507\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.100505	−0.00406600
$612$	0	0
$613$	33.5269	1.35414	0.677070	−	0.735919i	$-0.263250\pi$
$613$	33.5269	1.35414	0.677070	+	0.735919i	$0.263250\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.2843	1.29972	0.649858	−	0.760056i	$-0.274828\pi$
$617$	32.2843	1.29972	0.649858	+	0.760056i	$0.274828\pi$
$618$	0	0
$619$	−22.7990	−0.916369	−0.458184	−	0.888857i	$-0.651500\pi$
$619$	−22.7990	−0.916369	−0.458184	+	0.888857i	$0.651500\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	19.8995	0.797256
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.10051	0.163498
$630$	0	0
$631$	−29.9706	−1.19311	−0.596555	−	0.802572i	$-0.703464\pi$
$631$	−29.9706	−1.19311	−0.596555	+	0.802572i	$0.703464\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.65685	−0.0657503
$636$	0	0
$637$	−0.485281	−0.0192275
$638$	0	0
$639$	0	0
$640$	0	0
$641$	37.4142	1.47777	0.738886	−	0.673830i	$-0.235352\pi$
$641$	37.4142	1.47777	0.738886	+	0.673830i	$0.235352\pi$
$642$	0	0
$643$	−22.2843	−0.878806	−0.439403	−	0.898290i	$-0.644810\pi$
$643$	−22.2843	−0.878806	−0.439403	+	0.898290i	$0.644810\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.1716	−0.989597	−0.494798	−	0.869008i	$-0.664758\pi$
$647$	−25.1716	−0.989597	−0.494798	+	0.869008i	$0.664758\pi$
$648$	0	0
$649$	5.41421	0.212526
$650$	0	0
$651$	0	0
$652$	0	0
$653$	42.1421	1.64915	0.824575	−	0.565753i	$-0.191415\pi$
$653$	42.1421	1.64915	0.824575	+	0.565753i	$0.191415\pi$
$654$	0	0
$655$	3.17157	0.123924
$656$	0	0
$657$	0	0
$658$	0	0
$659$	10.9289	0.425731	0.212865	−	0.977082i	$-0.431720\pi$
$659$	10.9289	0.425731	0.212865	+	0.977082i	$0.431720\pi$
$660$	0	0
$661$	12.8579	0.500113	0.250056	−	0.968231i	$-0.419551\pi$
$661$	12.8579	0.500113	0.250056	+	0.968231i	$0.419551\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	11.0711	0.429318
$666$	0	0
$667$	−19.3137	−0.747830
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.4142	−0.402036
$672$	0	0
$673$	2.07107	0.0798338	0.0399169	−	0.999203i	$-0.487291\pi$
$673$	2.07107	0.0798338	0.0399169	+	0.999203i	$0.487291\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−48.4853	−1.86344	−0.931720	−	0.363176i	$-0.881692\pi$
$677$	−48.4853	−1.86344	−0.931720	+	0.363176i	$0.881692\pi$
$678$	0	0
$679$	8.07107	0.309739
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−21.2132	−0.811701	−0.405850	−	0.913940i	$-0.633025\pi$
$683$	−21.2132	−0.811701	−0.405850	+	0.913940i	$0.633025\pi$
$684$	0	0
$685$	−4.00000	−0.152832
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.75736	−0.219338
$690$	0	0
$691$	16.9706	0.645591	0.322795	−	0.946469i	$-0.395377\pi$
$691$	16.9706	0.645591	0.322795	+	0.946469i	$0.395377\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.1005	−0.458998
$696$	0	0
$697$	6.34315	0.240264
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.9289	0.714936	0.357468	−	0.933925i	$-0.383640\pi$
$701$	18.9289	0.714936	0.357468	+	0.933925i	$0.383640\pi$
$702$	0	0
$703$	5.92893	0.223614
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.8995	0.522744
$708$	0	0
$709$	12.1127	0.454902	0.227451	−	0.973790i	$-0.426961\pi$
$709$	12.1127	0.454902	0.227451	+	0.973790i	$0.426961\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−43.3137	−1.62211
$714$	0	0
$715$	−0.585786	−0.0219072
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0.727922	0.0271469	0.0135735	−	0.999908i	$-0.495679\pi$
$719$	0.727922	0.0271469	0.0135735	+	0.999908i	$0.495679\pi$
$720$	0	0
$721$	−13.2426	−0.493182
$722$	0	0
$723$	0	0
$724$	0	0
$725$	10.2426	0.380402
$726$	0	0
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.769553	−0.0284629
$732$	0	0
$733$	−33.5980	−1.24097	−0.620485	−	0.784218i	$-0.713064\pi$
$733$	−33.5980	−1.24097	−0.620485	+	0.784218i	$0.713064\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14.3137	−0.527252
$738$	0	0
$739$	−32.4853	−1.19499	−0.597495	−	0.801872i	$-0.703837\pi$
$739$	−32.4853	−1.19499	−0.597495	+	0.801872i	$0.703837\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−50.2426	−1.84322	−0.921612	−	0.388113i	$-0.873127\pi$
$743$	−50.2426	−1.84322	−0.921612	+	0.388113i	$0.873127\pi$
$744$	0	0
$745$	−4.68629	−0.171692
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−13.8995	−0.507876
$750$	0	0
$751$	26.5980	0.970574	0.485287	−	0.874355i	$-0.338715\pi$
$751$	26.5980	0.970574	0.485287	+	0.874355i	$0.338715\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	18.7279	0.681579
$756$	0	0
$757$	−22.1127	−0.803700	−0.401850	−	0.915706i	$-0.631633\pi$
$757$	−22.1127	−0.803700	−0.401850	+	0.915706i	$0.631633\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34.1421	1.23765	0.618826	−	0.785528i	$-0.287609\pi$
$761$	34.1421	1.23765	0.618826	+	0.785528i	$0.287609\pi$
$762$	0	0
$763$	−8.82843	−0.319611
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.24264	0.0809771
$768$	0	0
$769$	2.89949	0.104558	0.0522792	−	0.998633i	$-0.483351\pi$
$769$	2.89949	0.104558	0.0522792	+	0.998633i	$0.483351\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	24.2426	0.871947	0.435974	−	0.899959i	$-0.356404\pi$
$773$	24.2426	0.871947	0.435974	+	0.899959i	$0.356404\pi$
$774$	0	0
$775$	22.9706	0.825127
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.17157	0.328606
$780$	0	0
$781$	4.00000	0.143131
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.3431	0.369163
$786$	0	0
$787$	27.8701	0.993460	0.496730	−	0.867905i	$-0.334534\pi$
$787$	27.8701	0.993460	0.496730	+	0.867905i	$0.334534\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.24264	−0.221963
$792$	0	0
$793$	−4.31371	−0.153184
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.6274	1.50994	0.754970	−	0.655759i	$-0.227651\pi$
$797$	42.6274	1.50994	0.754970	+	0.655759i	$0.227651\pi$
$798$	0	0
$799$	−0.544156	−0.0192509
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8.89949	−0.314056
$804$	0	0
$805$	19.3137	0.680719
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−24.5858	−0.864390	−0.432195	−	0.901780i	$-0.642261\pi$
$809$	−24.5858	−0.864390	−0.432195	+	0.901780i	$0.642261\pi$
$810$	0	0
$811$	14.1421	0.496598	0.248299	−	0.968683i	$-0.420129\pi$
$811$	14.1421	0.496598	0.248299	+	0.968683i	$0.420129\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−19.0711	−0.668030
$816$	0	0
$817$	−1.11270	−0.0389284
$818$	0	0
$819$	0	0
$820$	0	0
$821$	23.1127	0.806639	0.403319	−	0.915059i	$-0.367856\pi$
$821$	23.1127	0.806639	0.403319	+	0.915059i	$0.367856\pi$
$822$	0	0
$823$	5.14214	0.179244	0.0896218	−	0.995976i	$-0.471434\pi$
$823$	5.14214	0.179244	0.0896218	+	0.995976i	$0.471434\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−6.24264	−0.217078	−0.108539	−	0.994092i	$-0.534617\pi$
$827$	−6.24264	−0.217078	−0.108539	+	0.994092i	$0.534617\pi$
$828$	0	0
$829$	−56.5980	−1.96573	−0.982865	−	0.184329i	$-0.940989\pi$
$829$	−56.5980	−1.96573	−0.982865	+	0.184329i	$0.940989\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.62742	−0.0910346
$834$	0	0
$835$	32.8284	1.13607
$836$	0	0
$837$	0	0
$838$	0	0
$839$	44.7696	1.54562	0.772808	−	0.634640i	$-0.218851\pi$
$839$	44.7696	1.54562	0.772808	+	0.634640i	$0.218851\pi$
$840$	0	0
$841$	−17.3431	−0.598040
$842$	0	0
$843$	0	0
$844$	0	0
$845$	18.1421	0.624108
$846$	0	0
$847$	−2.41421	−0.0829534
$848$	0	0
$849$	0	0
$850$	0	0
$851$	10.3431	0.354558
$852$	0	0
$853$	−0.129942	−0.00444914	−0.00222457	−	0.999998i	$-0.500708\pi$
$853$	−0.129942	−0.00444914	−0.00222457	+	0.999998i	$0.500708\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43.2132	1.47613	0.738067	−	0.674727i	$-0.235739\pi$
$857$	43.2132	1.47613	0.738067	+	0.674727i	$0.235739\pi$
$858$	0	0
$859$	−31.4853	−1.07426	−0.537132	−	0.843498i	$-0.680492\pi$
$859$	−31.4853	−1.07426	−0.537132	+	0.843498i	$0.680492\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.58579	0.224183	0.112091	−	0.993698i	$-0.464245\pi$
$863$	6.58579	0.224183	0.112091	+	0.993698i	$0.464245\pi$
$864$	0	0
$865$	25.6569	0.872359
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−14.0711	−0.477328
$870$	0	0
$871$	−5.92893	−0.200894
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−27.3137	−0.923372
$876$	0	0
$877$	−51.6690	−1.74474	−0.872370	−	0.488846i	$-0.837418\pi$
$877$	−51.6690	−1.74474	−0.872370	+	0.488846i	$0.837418\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−26.1421	−0.880751	−0.440375	−	0.897814i	$-0.645155\pi$
$881$	−26.1421	−0.880751	−0.440375	+	0.897814i	$0.645155\pi$
$882$	0	0
$883$	−33.6274	−1.13165	−0.565826	−	0.824524i	$-0.691443\pi$
$883$	−33.6274	−1.13165	−0.565826	+	0.824524i	$0.691443\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−42.9289	−1.44141	−0.720706	−	0.693241i	$-0.756182\pi$
$887$	−42.9289	−1.44141	−0.720706	+	0.693241i	$0.756182\pi$
$888$	0	0
$889$	−2.82843	−0.0948624
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.786797	−0.0263291
$894$	0	0
$895$	−20.6274	−0.689499
$896$	0	0
$897$	0	0
$898$	0	0
$899$	26.1421	0.871889
$900$	0	0
$901$	−31.1716	−1.03848
$902$	0	0
$903$	0	0
$904$	0	0
$905$	19.7574	0.656757
$906$	0	0
$907$	6.85786	0.227712	0.113856	−	0.993497i	$-0.463680\pi$
$907$	6.85786	0.227712	0.113856	+	0.993497i	$0.463680\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48.4853	−1.60639	−0.803195	−	0.595717i	$-0.796868\pi$
$911$	−48.4853	−1.60639	−0.803195	+	0.595717i	$0.796868\pi$
$912$	0	0
$913$	−1.65685	−0.0548339
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.41421	0.178793
$918$	0	0
$919$	29.4558	0.971659	0.485829	−	0.874054i	$-0.338518\pi$
$919$	29.4558	0.971659	0.485829	+	0.874054i	$0.338518\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.65685	0.0545360
$924$	0	0
$925$	−5.48528	−0.180355
$926$	0	0
$927$	0	0
$928$	0	0
$929$	40.9289	1.34283	0.671417	−	0.741079i	$-0.265686\pi$
$929$	40.9289	1.34283	0.671417	+	0.741079i	$0.265686\pi$
$930$	0	0
$931$	−3.79899	−0.124507
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.17157	−0.103722
$936$	0	0
$937$	13.7279	0.448472	0.224236	−	0.974535i	$-0.428011\pi$
$937$	13.7279	0.448472	0.224236	+	0.974535i	$0.428011\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−14.8284	−0.483393	−0.241696	−	0.970352i	$-0.577704\pi$
$941$	−14.8284	−0.483393	−0.241696	+	0.970352i	$0.577704\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.6985	0.835089	0.417544	−	0.908657i	$-0.362891\pi$
$947$	25.6985	0.835089	0.417544	+	0.908657i	$0.362891\pi$
$948$	0	0
$949$	−3.68629	−0.119662
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−55.5980	−1.80100	−0.900498	−	0.434861i	$-0.856798\pi$
$953$	−55.5980	−1.80100	−0.900498	+	0.434861i	$0.856798\pi$
$954$	0	0
$955$	−27.3137	−0.883851
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.82843	−0.220501
$960$	0	0
$961$	27.6274	0.891207
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.07107	0.163243
$966$	0	0
$967$	−34.2132	−1.10022	−0.550111	−	0.835091i	$-0.685415\pi$
$967$	−34.2132	−1.10022	−0.550111	+	0.835091i	$0.685415\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	14.5858	0.468080	0.234040	−	0.972227i	$-0.424805\pi$
$971$	14.5858	0.468080	0.234040	+	0.972227i	$0.424805\pi$
$972$	0	0
$973$	−20.6569	−0.662228
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.9289	0.541605	0.270802	−	0.962635i	$-0.412711\pi$
$977$	16.9289	0.541605	0.270802	+	0.962635i	$0.412711\pi$
$978$	0	0
$979$	−8.24264	−0.263436
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−30.3848	−0.969124	−0.484562	−	0.874757i	$-0.661021\pi$
$983$	−30.3848	−0.969124	−0.484562	+	0.874757i	$0.661021\pi$
$984$	0	0
$985$	4.82843	0.153846
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.94113	−0.0617242
$990$	0	0
$991$	−4.37258	−0.138900	−0.0694498	−	0.997585i	$-0.522124\pi$
$991$	−4.37258	−0.138900	−0.0694498	+	0.997585i	$0.522124\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−0.928932	−0.0294491
$996$	0	0
$997$	60.6274	1.92009	0.960045	−	0.279846i	$-0.0902835\pi$
$997$	60.6274	1.92009	0.960045	+	0.279846i	$0.0902835\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2376.2.a.h.1.1	yes	2
3.2	odd	2		2376.2.a.f.1.2	✓	2
4.3	odd	2		4752.2.a.ba.1.1		2
12.11	even	2		4752.2.a.bc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2376.2.a.f.1.2	✓	2	3.2	odd	2
2376.2.a.h.1.1	yes	2	1.1	even	1	trivial
4752.2.a.ba.1.1		2	4.3	odd	2
4752.2.a.bc.1.2		2	12.11	even	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:	\( N \)	\(=\)	\( 2376 = 2^{3} \cdot 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2376.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{5} -2.41421 q^{7} +1.00000 q^{11} +0.414214 q^{13} +2.24264 q^{17} +3.24264 q^{19} +5.65685 q^{23} -3.00000 q^{25} -3.41421 q^{29} -7.65685 q^{31} +3.41421 q^{35} +1.82843 q^{37} +2.82843 q^{41} -0.343146 q^{43} -0.242641 q^{47} -1.17157 q^{49} -13.8995 q^{53} -1.41421 q^{55} +5.41421 q^{59} -10.4142 q^{61} -0.585786 q^{65} -14.3137 q^{67} +4.00000 q^{71} -8.89949 q^{73} -2.41421 q^{77} -14.0711 q^{79} -1.65685 q^{83} -3.17157 q^{85} -8.24264 q^{89} -1.00000 q^{91} -4.58579 q^{95} -3.34315 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} + 2 q^{11} - 2 q^{13} - 4 q^{17} - 2 q^{19} - 6 q^{25} - 4 q^{29} - 4 q^{31} + 4 q^{35} - 2 q^{37} - 12 q^{43} + 8 q^{47} - 8 q^{49} - 8 q^{53} + 8 q^{59} - 18 q^{61} - 4 q^{65} - 6 q^{67}+ \cdots - 18 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 + 2 * q^11 - 2 * q^13 - 4 * q^17 - 2 * q^19 - 6 * q^25 - 4 * q^29 - 4 * q^31 + 4 * q^35 - 2 * q^37 - 12 * q^43 + 8 * q^47 - 8 * q^49 - 8 * q^53 + 8 * q^59 - 18 * q^61 - 4 * q^65 - 6 * q^67 + 8 * q^71 + 2 * q^73 - 2 * q^77 - 14 * q^79 + 8 * q^83 - 12 * q^85 - 8 * q^89 - 2 * q^91 - 12 * q^95 - 18 * q^97

Introduction

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