LMFDB - Embedded newform 2376.2.a.l.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 = [3,0,0,0,-2,0,1,0,0,0,-3,0,0] 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:	$3$
Coefficient field:	3.3.788.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.476452$ 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)

$f(q)$	$=$	$q-3.82009 q^{5} -0.476452 q^{7} -1.00000 q^{11} +4.77299 q^{13} +2.29654 q^{17} +0.867185 q^{19} +0.820089 q^{23} +9.59308 q^{25} -1.52355 q^{29} +1.82009 q^{31} +1.82009 q^{35} -3.77299 q^{37} -10.1166 q^{41} -2.39073 q^{43} +5.59308 q^{47} -6.77299 q^{49} -6.95290 q^{53} +3.82009 q^{55} -13.5074 q^{59} +9.82009 q^{61} -18.2333 q^{65} -9.82009 q^{67} +12.3275 q^{71} -10.2333 q^{73} +0.476452 q^{77} -6.98382 q^{79} -10.6788 q^{83} -8.77299 q^{85} +11.1862 q^{89} -2.27410 q^{91} -3.31273 q^{95} +5.17991 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{5} + q^{7} - 3 q^{11} - 5 q^{17} - 2 q^{19} - 7 q^{23} + 5 q^{25} - 7 q^{29} - 4 q^{31} - 4 q^{35} + 3 q^{37} - 9 q^{41} - 5 q^{43} - 7 q^{47} - 6 q^{49} - 16 q^{53} + 2 q^{55} - 17 q^{59} + 20 q^{61}+ \cdots + 25 q^{97}+O(q^{100}) $ 3 * q - 2 * q^5 + q^7 - 3 * q^11 - 5 * q^17 - 2 * q^19 - 7 * q^23 + 5 * q^25 - 7 * q^29 - 4 * q^31 - 4 * q^35 + 3 * q^37 - 9 * q^41 - 5 * q^43 - 7 * q^47 - 6 * q^49 - 16 * q^53 + 2 * q^55 - 17 * q^59 + 20 * q^61 - 12 * q^65 - 20 * q^67 + 4 * q^71 + 12 * q^73 - q^77 + 5 * q^79 - 8 * q^83 - 12 * q^85 - 14 * q^89 - 26 * q^91 - 24 * q^95 + 25 * 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
$5$	−3.82009	−1.70840	−0.854198	−	0.519948i	$-0.825951\pi$
$5$	−3.82009	−1.70840	−0.854198	+	0.519948i	$0.825951\pi$
$6$	0	0
$7$	−0.476452	−0.180082	−0.0900410	−	0.995938i	$-0.528700\pi$
$7$	−0.476452	−0.180082	−0.0900410	+	0.995938i	$0.528700\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.77299	1.32379	0.661895	−	0.749596i	$-0.269752\pi$
$13$	4.77299	1.32379	0.661895	+	0.749596i	$0.269752\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.29654	0.556993	0.278497	−	0.960437i	$-0.410164\pi$
$17$	2.29654	0.556993	0.278497	+	0.960437i	$0.410164\pi$
$18$	0	0
$19$	0.867185	0.198946	0.0994730	−	0.995040i	$-0.468284\pi$
$19$	0.867185	0.198946	0.0994730	+	0.995040i	$0.468284\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.820089	0.171000	0.0855002	−	0.996338i	$-0.472751\pi$
$23$	0.820089	0.171000	0.0855002	+	0.996338i	$0.472751\pi$
$24$	0	0
$25$	9.59308	1.91862
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.52355	−0.282916	−0.141458	−	0.989944i	$-0.545179\pi$
$29$	−1.52355	−0.282916	−0.141458	+	0.989944i	$0.545179\pi$
$30$	0	0
$31$	1.82009	0.326898	0.163449	−	0.986552i	$-0.447738\pi$
$31$	1.82009	0.326898	0.163449	+	0.986552i	$0.447738\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.82009	0.307651
$36$	0	0
$37$	−3.77299	−0.620276	−0.310138	−	0.950691i	$-0.600375\pi$
$37$	−3.77299	−0.620276	−0.310138	+	0.950691i	$0.600375\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.1166	−1.57995	−0.789976	−	0.613138i	$-0.789907\pi$
$41$	−10.1166	−1.57995	−0.789976	+	0.613138i	$0.789907\pi$
$42$	0	0
$43$	−2.39073	−0.364583	−0.182292	−	0.983244i	$-0.558352\pi$
$43$	−2.39073	−0.364583	−0.182292	+	0.983244i	$0.558352\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.59308	0.815835	0.407917	−	0.913019i	$-0.366255\pi$
$47$	5.59308	0.815835	0.407917	+	0.913019i	$0.366255\pi$
$48$	0	0
$49$	−6.77299	−0.967570
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.95290	−0.955055	−0.477527	−	0.878617i	$-0.658467\pi$
$53$	−6.95290	−0.955055	−0.477527	+	0.878617i	$0.658467\pi$
$54$	0	0
$55$	3.82009	0.515101
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−13.5074	−1.75851	−0.879255	−	0.476352i	$-0.841959\pi$
$59$	−13.5074	−1.75851	−0.879255	+	0.476352i	$0.841959\pi$
$60$	0	0
$61$	9.82009	1.25733	0.628667	−	0.777675i	$-0.283601\pi$
$61$	9.82009	1.25733	0.628667	+	0.777675i	$0.283601\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−18.2333	−2.26156
$66$	0	0
$67$	−9.82009	−1.19971	−0.599857	−	0.800107i	$-0.704776\pi$
$67$	−9.82009	−1.19971	−0.599857	+	0.800107i	$0.704776\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.3275	1.46300	0.731500	−	0.681842i	$-0.238821\pi$
$71$	12.3275	1.46300	0.731500	+	0.681842i	$0.238821\pi$
$72$	0	0
$73$	−10.2333	−1.19771	−0.598856	−	0.800856i	$-0.704378\pi$
$73$	−10.2333	−1.19771	−0.598856	+	0.800856i	$0.704378\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.476452	0.0542967
$78$	0	0
$79$	−6.98382	−0.785741	−0.392870	−	0.919594i	$-0.628518\pi$
$79$	−6.98382	−0.785741	−0.392870	+	0.919594i	$0.628518\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.6788	−1.17215	−0.586075	−	0.810257i	$-0.699328\pi$
$83$	−10.6788	−1.17215	−0.586075	+	0.810257i	$0.699328\pi$
$84$	0	0
$85$	−8.77299	−0.951565
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.1862	1.18573	0.592866	−	0.805301i	$-0.297997\pi$
$89$	11.1862	1.18573	0.592866	+	0.805301i	$0.297997\pi$
$90$	0	0
$91$	−2.27410	−0.238391
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.31273	−0.339878
$96$	0	0
$97$	5.17991	0.525940	0.262970	−	0.964804i	$-0.415298\pi$
$97$	5.17991	0.525940	0.262970	+	0.964804i	$0.415298\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.29654	−0.825537	−0.412768	−	0.910836i	$-0.635438\pi$
$101$	−8.29654	−0.825537	−0.412768	+	0.910836i	$0.635438\pi$
$102$	0	0
$103$	−13.7259	−1.35245	−0.676226	−	0.736694i	$-0.736386\pi$
$103$	−13.7259	−1.35245	−0.676226	+	0.736694i	$0.736386\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.46027	−0.914559	−0.457279	−	0.889323i	$-0.651176\pi$
$107$	−9.46027	−0.914559	−0.457279	+	0.889323i	$0.651176\pi$
$108$	0	0
$109$	14.4989	1.38874	0.694371	−	0.719617i	$-0.255683\pi$
$109$	14.4989	1.38874	0.694371	+	0.719617i	$0.255683\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.6788	−1.00458	−0.502289	−	0.864700i	$-0.667508\pi$
$113$	−10.6788	−1.00458	−0.502289	+	0.864700i	$0.667508\pi$
$114$	0	0
$115$	−3.13281	−0.292136
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.09419	−0.100304
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−17.5460	−1.56936
$126$	0	0
$127$	−6.41317	−0.569077	−0.284539	−	0.958665i	$-0.591840\pi$
$127$	−6.41317	−0.569077	−0.284539	+	0.958665i	$0.591840\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.3275	−1.60128	−0.800639	−	0.599148i	$-0.795506\pi$
$131$	−18.3275	−1.60128	−0.800639	+	0.599148i	$0.795506\pi$
$132$	0	0
$133$	−0.413172	−0.0358266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.2333	1.21603	0.608015	−	0.793926i	$-0.291966\pi$
$137$	14.2333	1.21603	0.608015	+	0.793926i	$0.291966\pi$
$138$	0	0
$139$	−4.75055	−0.402937	−0.201468	−	0.979495i	$-0.564571\pi$
$139$	−4.75055	−0.402937	−0.201468	+	0.979495i	$0.564571\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.77299	−0.399138
$144$	0	0
$145$	5.82009	0.483332
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.9592	1.47127	0.735636	−	0.677377i	$-0.236884\pi$
$149$	17.9592	1.47127	0.735636	+	0.677377i	$0.236884\pi$
$150$	0	0
$151$	−11.4603	−0.932623	−0.466312	−	0.884620i	$-0.654417\pi$
$151$	−11.4603	−0.932623	−0.466312	+	0.884620i	$0.654417\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.95290	−0.558471
$156$	0	0
$157$	−13.6873	−1.09236	−0.546182	−	0.837667i	$-0.683919\pi$
$157$	−13.6873	−1.09236	−0.546182	+	0.837667i	$0.683919\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.390733	−0.0307941
$162$	0	0
$163$	15.1862	1.18947	0.594736	−	0.803921i	$-0.297257\pi$
$163$	15.1862	1.18947	0.594736	+	0.803921i	$0.297257\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.86719	−0.221869	−0.110935	−	0.993828i	$-0.535384\pi$
$167$	−2.86719	−0.221869	−0.110935	+	0.993828i	$0.535384\pi$
$168$	0	0
$169$	9.78147	0.752421
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.36608	0.560032	0.280016	−	0.959995i	$-0.409660\pi$
$173$	7.36608	0.560032	0.280016	+	0.959995i	$0.409660\pi$
$174$	0	0
$175$	−4.57064	−0.345508
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.49889	0.709980	0.354990	−	0.934870i	$-0.384484\pi$
$179$	9.49889	0.709980	0.354990	+	0.934870i	$0.384484\pi$
$180$	0	0
$181$	3.00000	0.222988	0.111494	−	0.993765i	$-0.464436\pi$
$181$	3.00000	0.222988	0.111494	+	0.993765i	$0.464436\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	14.4132	1.05968
$186$	0	0
$187$	−2.29654	−0.167940
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.68727	−0.556232	−0.278116	−	0.960548i	$-0.589710\pi$
$191$	−7.68727	−0.556232	−0.278116	+	0.960548i	$0.589710\pi$
$192$	0	0
$193$	3.91428	0.281756	0.140878	−	0.990027i	$-0.455007\pi$
$193$	3.91428	0.281756	0.140878	+	0.990027i	$0.455007\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.8510	−1.27183	−0.635916	−	0.771759i	$-0.719377\pi$
$197$	−17.8510	−1.27183	−0.635916	+	0.771759i	$0.719377\pi$
$198$	0	0
$199$	17.1004	1.21222	0.606109	−	0.795381i	$-0.292729\pi$
$199$	17.1004	1.21222	0.606109	+	0.795381i	$0.292729\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.725898	0.0509480
$204$	0	0
$205$	38.6464	2.69918
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.867185	−0.0599845
$210$	0	0
$211$	−23.0695	−1.58817	−0.794086	−	0.607805i	$-0.792050\pi$
$211$	−23.0695	−1.58817	−0.794086	+	0.607805i	$0.792050\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.13281	0.622853
$216$	0	0
$217$	−0.867185	−0.0588684
$218$	0	0
$219$	0	0
$220$	0	0
$221$	10.9614	0.737342
$222$	0	0
$223$	−0.0857188	−0.00574015	−0.00287008	−	0.999996i	$-0.500914\pi$
$223$	−0.0857188	−0.00574015	−0.00287008	+	0.999996i	$0.500914\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.5074	−0.962888	−0.481444	−	0.876477i	$-0.659887\pi$
$227$	−14.5074	−0.962888	−0.481444	+	0.876477i	$0.659887\pi$
$228$	0	0
$229$	−8.19464	−0.541517	−0.270758	−	0.962647i	$-0.587274\pi$
$229$	−8.19464	−0.541517	−0.270758	+	0.962647i	$0.587274\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.46027	0.0956653	0.0478327	−	0.998855i	$-0.484769\pi$
$233$	1.46027	0.0956653	0.0478327	+	0.998855i	$0.484769\pi$
$234$	0	0
$235$	−21.3661	−1.39377
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.55446	0.100550	0.0502748	−	0.998735i	$-0.483990\pi$
$239$	1.55446	0.100550	0.0502748	+	0.998735i	$0.483990\pi$
$240$	0	0
$241$	23.0147	1.48251	0.741254	−	0.671224i	$-0.234231\pi$
$241$	23.0147	1.48251	0.741254	+	0.671224i	$0.234231\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	25.8734	1.65299
$246$	0	0
$247$	4.13907	0.263363
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−23.7322	−1.49796	−0.748980	−	0.662592i	$-0.769456\pi$
$251$	−23.7322	−1.49796	−0.748980	+	0.662592i	$0.769456\pi$
$252$	0	0
$253$	−0.820089	−0.0515586
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.36608	−0.0852135	−0.0426067	−	0.999092i	$-0.513566\pi$
$257$	−1.36608	−0.0852135	−0.0426067	+	0.999092i	$0.513566\pi$
$258$	0	0
$259$	1.79765	0.111701
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.0920	−1.05394	−0.526968	−	0.849885i	$-0.676671\pi$
$263$	−17.0920	−1.05394	−0.526968	+	0.849885i	$0.676671\pi$
$264$	0	0
$265$	26.5607	1.63161
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.99153	−0.365310	−0.182655	−	0.983177i	$-0.558469\pi$
$269$	−5.99153	−0.365310	−0.182655	+	0.983177i	$0.558469\pi$
$270$	0	0
$271$	3.01618	0.183220	0.0916101	−	0.995795i	$-0.470799\pi$
$271$	3.01618	0.183220	0.0916101	+	0.995795i	$0.470799\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−9.59308	−0.578485
$276$	0	0
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0.484925	0.0289282	0.0144641	−	0.999895i	$-0.495396\pi$
$281$	0.484925	0.0289282	0.0144641	+	0.999895i	$0.495396\pi$
$282$	0	0
$283$	8.64643	0.513977	0.256989	−	0.966414i	$-0.417270\pi$
$283$	8.64643	0.513977	0.256989	+	0.966414i	$0.417270\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.82009	0.284521
$288$	0	0
$289$	−11.7259	−0.689759
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.31898	−0.135476	−0.0677381	−	0.997703i	$-0.521578\pi$
$293$	−2.31898	−0.135476	−0.0677381	+	0.997703i	$0.521578\pi$
$294$	0	0
$295$	51.5993	3.00423
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.91428	0.226369
$300$	0	0
$301$	1.13907	0.0656549
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−37.5136	−2.14802
$306$	0	0
$307$	−28.9753	−1.65371	−0.826855	−	0.562415i	$-0.809872\pi$
$307$	−28.9753	−1.65371	−0.826855	+	0.562415i	$0.809872\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	11.1475	0.632119	0.316060	−	0.948739i	$-0.397640\pi$
$311$	11.1475	0.632119	0.316060	+	0.948739i	$0.397640\pi$
$312$	0	0
$313$	2.17144	0.122737	0.0613685	−	0.998115i	$-0.480454\pi$
$313$	2.17144	0.122737	0.0613685	+	0.998115i	$0.480454\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.95290	0.502845	0.251423	−	0.967877i	$-0.419102\pi$
$317$	8.95290	0.502845	0.251423	+	0.967877i	$0.419102\pi$
$318$	0	0
$319$	1.52355	0.0853023
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.99153	0.110812
$324$	0	0
$325$	45.7877	2.53985
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.66484	−0.146917
$330$	0	0
$331$	−28.7406	−1.57973	−0.789864	−	0.613282i	$-0.789849\pi$
$331$	−28.7406	−1.57973	−0.789864	+	0.613282i	$0.789849\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	37.5136	2.04959
$336$	0	0
$337$	−9.46027	−0.515334	−0.257667	−	0.966234i	$-0.582954\pi$
$337$	−9.46027	−0.515334	−0.257667	+	0.966234i	$0.582954\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.82009	−0.0985634
$342$	0	0
$343$	6.56217	0.354324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	33.5136	1.79911	0.899553	−	0.436812i	$-0.143893\pi$
$347$	33.5136	1.79911	0.899553	+	0.436812i	$0.143893\pi$
$348$	0	0
$349$	−36.0210	−1.92816	−0.964080	−	0.265614i	$-0.914425\pi$
$349$	−36.0210	−1.92816	−0.964080	+	0.265614i	$0.914425\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	22.8587	1.21665	0.608323	−	0.793689i	$-0.291842\pi$
$353$	22.8587	1.21665	0.608323	+	0.793689i	$0.291842\pi$
$354$	0	0
$355$	−47.0920	−2.49938
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.10045	−0.0580794	−0.0290397	−	0.999578i	$-0.509245\pi$
$359$	−1.10045	−0.0580794	−0.0290397	+	0.999578i	$0.509245\pi$
$360$	0	0
$361$	−18.2480	−0.960421
$362$	0	0
$363$	0	0
$364$	0	0
$365$	39.0920	2.04617
$366$	0	0
$367$	31.4194	1.64008	0.820040	−	0.572306i	$-0.193951\pi$
$367$	31.4194	1.64008	0.820040	+	0.572306i	$0.193951\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.31273	0.171988
$372$	0	0
$373$	−18.2248	−0.943644	−0.471822	−	0.881694i	$-0.656403\pi$
$373$	−18.2248	−0.943644	−0.471822	+	0.881694i	$0.656403\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.27188	−0.374521
$378$	0	0
$379$	−22.4132	−1.15129	−0.575644	−	0.817701i	$-0.695248\pi$
$379$	−22.4132	−1.15129	−0.575644	+	0.817701i	$0.695248\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.32745	−0.221123	−0.110561	−	0.993869i	$-0.535265\pi$
$383$	−4.32745	−0.221123	−0.110561	+	0.993869i	$0.535265\pi$
$384$	0	0
$385$	−1.82009	−0.0927603
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−34.4665	−1.74752	−0.873761	−	0.486355i	$-0.838326\pi$
$389$	−34.4665	−1.74752	−0.873761	+	0.486355i	$0.838326\pi$
$390$	0	0
$391$	1.88337	0.0952461
$392$	0	0
$393$	0	0
$394$	0	0
$395$	26.6788	1.34236
$396$	0	0
$397$	19.0147	0.954322	0.477161	−	0.878816i	$-0.341666\pi$
$397$	19.0147	0.954322	0.477161	+	0.878816i	$0.341666\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	37.0596	1.85067	0.925334	−	0.379152i	$-0.123784\pi$
$401$	37.0596	1.85067	0.925334	+	0.379152i	$0.123784\pi$
$402$	0	0
$403$	8.68727	0.432744
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.77299	0.187020
$408$	0	0
$409$	18.7322	0.926245	0.463123	−	0.886294i	$-0.346729\pi$
$409$	18.7322	0.926245	0.463123	+	0.886294i	$0.346729\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.43561	0.316676
$414$	0	0
$415$	40.7940	2.00250
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.51584	0.220613	0.110307	−	0.993898i	$-0.464817\pi$
$419$	4.51584	0.220613	0.110307	+	0.993898i	$0.464817\pi$
$420$	0	0
$421$	−25.2333	−1.22979	−0.614897	−	0.788607i	$-0.710802\pi$
$421$	−25.2333	−1.22979	−0.614897	+	0.788607i	$0.710802\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	22.0309	1.06866
$426$	0	0
$427$	−4.67880	−0.226423
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−0.772993	−0.0372338	−0.0186169	−	0.999827i	$-0.505926\pi$
$431$	−0.772993	−0.0372338	−0.0186169	+	0.999827i	$0.505926\pi$
$432$	0	0
$433$	−41.1152	−1.97587	−0.987935	−	0.154872i	$-0.950504\pi$
$433$	−41.1152	−1.97587	−0.987935	+	0.154872i	$0.950504\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.711169	0.0340198
$438$	0	0
$439$	−25.4503	−1.21468	−0.607339	−	0.794443i	$-0.707763\pi$
$439$	−25.4503	−1.21468	−0.607339	+	0.794443i	$0.707763\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18.9205	−0.898942	−0.449471	−	0.893295i	$-0.648387\pi$
$443$	−18.9205	−0.898942	−0.449471	+	0.893295i	$0.648387\pi$
$444$	0	0
$445$	−42.7322	−2.02570
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−15.2270	−0.718607	−0.359303	−	0.933221i	$-0.616986\pi$
$449$	−15.2270	−0.718607	−0.359303	+	0.933221i	$0.616986\pi$
$450$	0	0
$451$	10.1166	0.476374
$452$	0	0
$453$	0	0
$454$	0	0
$455$	8.68727	0.407266
$456$	0	0
$457$	28.0857	1.31379	0.656897	−	0.753980i	$-0.271869\pi$
$457$	28.0857	1.31379	0.656897	+	0.753980i	$0.271869\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5.63171	−0.262295	−0.131147	−	0.991363i	$-0.541866\pi$
$461$	−5.63171	−0.262295	−0.131147	+	0.991363i	$0.541866\pi$
$462$	0	0
$463$	−40.1067	−1.86392	−0.931958	−	0.362566i	$-0.881901\pi$
$463$	−40.1067	−1.86392	−0.931958	+	0.362566i	$0.881901\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	35.1799	1.62793	0.813966	−	0.580912i	$-0.197304\pi$
$467$	35.1799	1.62793	0.813966	+	0.580912i	$0.197304\pi$
$468$	0	0
$469$	4.67880	0.216047
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.39073	0.109926
$474$	0	0
$475$	8.31898	0.381701
$476$	0	0
$477$	0	0
$478$	0	0
$479$	23.4518	1.07154	0.535770	−	0.844364i	$-0.320021\pi$
$479$	23.4518	1.07154	0.535770	+	0.844364i	$0.320021\pi$
$480$	0	0
$481$	−18.0085	−0.821116
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−19.7877	−0.898514
$486$	0	0
$487$	−29.0147	−1.31478	−0.657391	−	0.753549i	$-0.728340\pi$
$487$	−29.0147	−1.31478	−0.657391	+	0.753549i	$0.728340\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17.3127	−0.781312	−0.390656	−	0.920537i	$-0.627752\pi$
$491$	−17.3127	−0.781312	−0.390656	+	0.920537i	$0.627752\pi$
$492$	0	0
$493$	−3.49889	−0.157582
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.87344	−0.263460
$498$	0	0
$499$	−20.5074	−0.918036	−0.459018	−	0.888427i	$-0.651799\pi$
$499$	−20.5074	−0.918036	−0.459018	+	0.888427i	$0.651799\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	28.4216	1.26726	0.633629	−	0.773637i	$-0.281564\pi$
$503$	28.4216	1.26726	0.633629	+	0.773637i	$0.281564\pi$
$504$	0	0
$505$	31.6935	1.41034
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.6788	1.27117	0.635583	−	0.772033i	$-0.280760\pi$
$509$	28.6788	1.27117	0.635583	+	0.772033i	$0.280760\pi$
$510$	0	0
$511$	4.87566	0.215686
$512$	0	0
$513$	0	0
$514$	0	0
$515$	52.4342	2.31052
$516$	0	0
$517$	−5.59308	−0.245984
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.86719	−0.213235	−0.106618	−	0.994300i	$-0.534002\pi$
$521$	−4.86719	−0.213235	−0.106618	+	0.994300i	$0.534002\pi$
$522$	0	0
$523$	−7.76528	−0.339552	−0.169776	−	0.985483i	$-0.554304\pi$
$523$	−7.76528	−0.339552	−0.169776	+	0.985483i	$0.554304\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.17991	0.182080
$528$	0	0
$529$	−22.3275	−0.970759
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−48.2866	−2.09153
$534$	0	0
$535$	36.1391	1.56243
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.77299	0.291733
$540$	0	0
$541$	33.4364	1.43754	0.718771	−	0.695247i	$-0.244705\pi$
$541$	33.4364	1.43754	0.718771	+	0.695247i	$0.244705\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−55.3871	−2.37252
$546$	0	0
$547$	−27.0780	−1.15777	−0.578886	−	0.815408i	$-0.696512\pi$
$547$	−27.0780	−1.15777	−0.578886	+	0.815408i	$0.696512\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.32120	−0.0562849
$552$	0	0
$553$	3.32745	0.141498
$554$	0	0
$555$	0	0
$556$	0	0
$557$	41.6711	1.76566	0.882830	−	0.469692i	$-0.155635\pi$
$557$	41.6711	1.76566	0.882830	+	0.469692i	$0.155635\pi$
$558$	0	0
$559$	−11.4110	−0.482632
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.6851	−1.16679	−0.583393	−	0.812190i	$-0.698275\pi$
$563$	−27.6851	−1.16679	−0.583393	+	0.812190i	$0.698275\pi$
$564$	0	0
$565$	40.7940	1.71622
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.7406	1.20487	0.602435	−	0.798168i	$-0.294197\pi$
$569$	28.7406	1.20487	0.602435	+	0.798168i	$0.294197\pi$
$570$	0	0
$571$	−30.8039	−1.28910	−0.644552	−	0.764561i	$-0.722956\pi$
$571$	−30.8039	−1.28910	−0.644552	+	0.764561i	$0.722956\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.86719	0.328084
$576$	0	0
$577$	−19.3598	−0.805960	−0.402980	−	0.915209i	$-0.632026\pi$
$577$	−19.3598	−0.805960	−0.402980	+	0.915209i	$0.632026\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.08794	0.211083
$582$	0	0
$583$	6.95290	0.287960
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.3661	1.17079	0.585397	−	0.810747i	$-0.300939\pi$
$587$	28.3661	1.17079	0.585397	+	0.810747i	$0.300939\pi$
$588$	0	0
$589$	1.57835	0.0650350
$590$	0	0
$591$	0	0
$592$	0	0
$593$	23.2171	0.953411	0.476706	−	0.879063i	$-0.341831\pi$
$593$	23.2171	0.953411	0.476706	+	0.879063i	$0.341831\pi$
$594$	0	0
$595$	4.17991	0.171360
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−45.1152	−1.84336	−0.921678	−	0.387956i	$-0.873181\pi$
$599$	−45.1152	−1.84336	−0.921678	+	0.387956i	$0.873181\pi$
$600$	0	0
$601$	24.6788	1.00667	0.503335	−	0.864092i	$-0.332106\pi$
$601$	24.6788	1.00667	0.503335	+	0.864092i	$0.332106\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.82009	−0.155309
$606$	0	0
$607$	15.3212	0.621868	0.310934	−	0.950431i	$-0.399358\pi$
$607$	15.3212	0.621868	0.310934	+	0.950431i	$0.399358\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	26.6957	1.07999
$612$	0	0
$613$	29.0681	1.17405	0.587024	−	0.809569i	$-0.300299\pi$
$613$	29.0681	1.17405	0.587024	+	0.809569i	$0.300299\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.6317	1.51500	0.757498	−	0.652838i	$-0.226422\pi$
$617$	37.6317	1.51500	0.757498	+	0.652838i	$0.226422\pi$
$618$	0	0
$619$	42.6141	1.71280	0.856402	−	0.516310i	$-0.172695\pi$
$619$	42.6141	1.71280	0.856402	+	0.516310i	$0.172695\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.32967	−0.213529
$624$	0	0
$625$	19.0618	0.762473
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−8.66484	−0.345490
$630$	0	0
$631$	−6.60156	−0.262804	−0.131402	−	0.991329i	$-0.541948\pi$
$631$	−6.60156	−0.262804	−0.131402	+	0.991329i	$0.541948\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	24.4989	0.972209
$636$	0	0
$637$	−32.3275	−1.28086
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.10045	−0.280451	−0.140225	−	0.990120i	$-0.544783\pi$
$641$	−7.10045	−0.280451	−0.140225	+	0.990120i	$0.544783\pi$
$642$	0	0
$643$	11.6851	0.460814	0.230407	−	0.973094i	$-0.425994\pi$
$643$	11.6851	0.460814	0.230407	+	0.973094i	$0.425994\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.8263	0.976024	0.488012	−	0.872837i	$-0.337722\pi$
$647$	24.8263	0.976024	0.488012	+	0.872837i	$0.337722\pi$
$648$	0	0
$649$	13.5074	0.530210
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.2804	1.38063	0.690314	−	0.723510i	$-0.257473\pi$
$653$	35.2804	1.38063	0.690314	+	0.723510i	$0.257473\pi$
$654$	0	0
$655$	70.0125	2.73562
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−12.5074	−0.487218	−0.243609	−	0.969874i	$-0.578331\pi$
$659$	−12.5074	−0.487218	−0.243609	+	0.969874i	$0.578331\pi$
$660$	0	0
$661$	1.95290	0.0759592	0.0379796	−	0.999279i	$-0.487908\pi$
$661$	1.95290	0.0759592	0.0379796	+	0.999279i	$0.487908\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.57835	0.0612060
$666$	0	0
$667$	−1.24945	−0.0483787
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9.82009	−0.379100
$672$	0	0
$673$	−22.2866	−0.859086	−0.429543	−	0.903046i	$-0.641325\pi$
$673$	−22.2866	−0.859086	−0.429543	+	0.903046i	$0.641325\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.2881	−0.472269	−0.236134	−	0.971720i	$-0.575881\pi$
$677$	−12.2881	−0.472269	−0.236134	+	0.971720i	$0.575881\pi$
$678$	0	0
$679$	−2.46798	−0.0947123
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−4.67255	−0.178790	−0.0893950	−	0.995996i	$-0.528493\pi$
$683$	−4.67255	−0.178790	−0.0893950	+	0.995996i	$0.528493\pi$
$684$	0	0
$685$	−54.3723	−2.07746
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−33.1862	−1.26429
$690$	0	0
$691$	48.9824	1.86338	0.931688	−	0.363258i	$-0.118336\pi$
$691$	48.9824	1.86338	0.931688	+	0.363258i	$0.118336\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	18.1475	0.688376
$696$	0	0
$697$	−23.2333	−0.880023
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.4974	0.698638	0.349319	−	0.937004i	$-0.386413\pi$
$701$	18.4974	0.698638	0.349319	+	0.937004i	$0.386413\pi$
$702$	0	0
$703$	−3.27188	−0.123401
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.95290	0.148664
$708$	0	0
$709$	−4.22701	−0.158749	−0.0793743	−	0.996845i	$-0.525292\pi$
$709$	−4.22701	−0.158749	−0.0793743	+	0.996845i	$0.525292\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.49264	0.0558997
$714$	0	0
$715$	18.2333	0.681885
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−11.1243	−0.414868	−0.207434	−	0.978249i	$-0.566511\pi$
$719$	−11.1243	−0.414868	−0.207434	+	0.978249i	$0.566511\pi$
$720$	0	0
$721$	6.53973	0.243552
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−14.6155	−0.542807
$726$	0	0
$727$	4.80536	0.178221	0.0891105	−	0.996022i	$-0.471598\pi$
$727$	4.80536	0.178221	0.0891105	+	0.996022i	$0.471598\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.49042	−0.203070
$732$	0	0
$733$	−4.16296	−0.153763	−0.0768813	−	0.997040i	$-0.524496\pi$
$733$	−4.16296	−0.153763	−0.0768813	+	0.997040i	$0.524496\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.82009	0.361728
$738$	0	0
$739$	3.15525	0.116068	0.0580339	−	0.998315i	$-0.481517\pi$
$739$	3.15525	0.116068	0.0580339	+	0.998315i	$0.481517\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.2031	−0.484375	−0.242188	−	0.970229i	$-0.577865\pi$
$743$	−13.2031	−0.484375	−0.242188	+	0.970229i	$0.577865\pi$
$744$	0	0
$745$	−68.6056	−2.51351
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.50736	0.164695
$750$	0	0
$751$	11.9592	0.436396	0.218198	−	0.975905i	$-0.429982\pi$
$751$	11.9592	0.436396	0.218198	+	0.975905i	$0.429982\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	43.7792	1.59329
$756$	0	0
$757$	13.3598	0.485571	0.242785	−	0.970080i	$-0.421939\pi$
$757$	13.3598	0.485571	0.242785	+	0.970080i	$0.421939\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.28883	0.336720	0.168360	−	0.985726i	$-0.446153\pi$
$761$	9.28883	0.336720	0.168360	+	0.985726i	$0.446153\pi$
$762$	0	0
$763$	−6.90803	−0.250087
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−64.4706	−2.32790
$768$	0	0
$769$	36.0125	1.29864	0.649322	−	0.760513i	$-0.275053\pi$
$769$	36.0125	1.29864	0.649322	+	0.760513i	$0.275053\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	12.9121	0.464415	0.232207	−	0.972666i	$-0.425405\pi$
$773$	12.9121	0.464415	0.232207	+	0.972666i	$0.425405\pi$
$774$	0	0
$775$	17.4603	0.627191
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8.77299	−0.314325
$780$	0	0
$781$	−12.3275	−0.441111
$782$	0	0
$783$	0	0
$784$	0	0
$785$	52.2866	1.86619
$786$	0	0
$787$	34.2248	1.21998	0.609991	−	0.792408i	$-0.291173\pi$
$787$	34.2248	1.21998	0.609991	+	0.792408i	$0.291173\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.08794	0.180906
$792$	0	0
$793$	46.8712	1.66445
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.8819	0.562566	0.281283	−	0.959625i	$-0.409240\pi$
$797$	15.8819	0.562566	0.281283	+	0.959625i	$0.409240\pi$
$798$	0	0
$799$	12.8447	0.454414
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10.2333	0.361124
$804$	0	0
$805$	1.49264	0.0526085
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−18.1939	−0.639663	−0.319831	−	0.947475i	$-0.603626\pi$
$809$	−18.1939	−0.639663	−0.319831	+	0.947475i	$0.603626\pi$
$810$	0	0
$811$	0.210823	0.00740298	0.00370149	−	0.999993i	$-0.498822\pi$
$811$	0.210823	0.00740298	0.00370149	+	0.999993i	$0.498822\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−58.0125	−2.03209
$816$	0	0
$817$	−2.07321	−0.0725324
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33.0287	1.15271	0.576355	−	0.817200i	$-0.304475\pi$
$821$	33.0287	1.15271	0.576355	+	0.817200i	$0.304475\pi$
$822$	0	0
$823$	50.2333	1.75102	0.875511	−	0.483199i	$-0.160525\pi$
$823$	50.2333	1.75102	0.875511	+	0.483199i	$0.160525\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.5051	1.02599	0.512997	−	0.858390i	$-0.328535\pi$
$827$	29.5051	1.02599	0.512997	+	0.858390i	$0.328535\pi$
$828$	0	0
$829$	−5.40066	−0.187573	−0.0937864	−	0.995592i	$-0.529897\pi$
$829$	−5.40066	−0.187573	−0.0937864	+	0.995592i	$0.529897\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−15.5545	−0.538930
$834$	0	0
$835$	10.9529	0.379041
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−50.3337	−1.73771	−0.868856	−	0.495064i	$-0.835144\pi$
$839$	−50.3337	−1.73771	−0.868856	+	0.495064i	$0.835144\pi$
$840$	0	0
$841$	−26.6788	−0.919959
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−37.3661	−1.28543
$846$	0	0
$847$	−0.476452	−0.0163711
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.09419	−0.106068
$852$	0	0
$853$	13.0471	0.446724	0.223362	−	0.974736i	$-0.428297\pi$
$853$	13.0471	0.446724	0.223362	+	0.974736i	$0.428297\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	33.5361	1.14557	0.572785	−	0.819706i	$-0.305863\pi$
$857$	33.5361	1.14557	0.572785	+	0.819706i	$0.305863\pi$
$858$	0	0
$859$	34.6464	1.18212	0.591061	−	0.806627i	$-0.298709\pi$
$859$	34.6464	1.18212	0.591061	+	0.806627i	$0.298709\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.11809	0.0380601	0.0190301	−	0.999819i	$-0.493942\pi$
$863$	1.11809	0.0380601	0.0190301	+	0.999819i	$0.493942\pi$
$864$	0	0
$865$	−28.1391	−0.956757
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.98382	0.236910
$870$	0	0
$871$	−46.8712	−1.58817
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.35982	0.282614
$876$	0	0
$877$	−10.8587	−0.366673	−0.183336	−	0.983050i	$-0.558690\pi$
$877$	−10.8587	−0.366673	−0.183336	+	0.983050i	$0.558690\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.5074	0.421384	0.210692	−	0.977553i	$-0.432428\pi$
$881$	12.5074	0.421384	0.210692	+	0.977553i	$0.432428\pi$
$882$	0	0
$883$	−52.2333	−1.75779	−0.878895	−	0.477016i	$-0.841718\pi$
$883$	−52.2333	−1.75779	−0.878895	+	0.477016i	$0.841718\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38.4132	−1.28979	−0.644894	−	0.764272i	$-0.723099\pi$
$887$	−38.4132	−1.28979	−0.644894	+	0.764272i	$0.723099\pi$
$888$	0	0
$889$	3.05557	0.102480
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.85024	0.162307
$894$	0	0
$895$	−36.2866	−1.21293
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.77299	−0.0924845
$900$	0	0
$901$	−15.9676	−0.531959
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11.4603	−0.380952
$906$	0	0
$907$	−36.5074	−1.21221	−0.606104	−	0.795386i	$-0.707268\pi$
$907$	−36.5074	−1.21221	−0.606104	+	0.795386i	$0.707268\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−38.8411	−1.28686	−0.643431	−	0.765504i	$-0.722490\pi$
$911$	−38.8411	−1.28686	−0.643431	+	0.765504i	$0.722490\pi$
$912$	0	0
$913$	10.6788	0.353417
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.73215	0.288361
$918$	0	0
$919$	47.4294	1.56455	0.782275	−	0.622933i	$-0.214059\pi$
$919$	47.4294	1.56455	0.782275	+	0.622933i	$0.214059\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	58.8389	1.93670
$924$	0	0
$925$	−36.1946	−1.19007
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−49.1538	−1.61268	−0.806342	−	0.591450i	$-0.798556\pi$
$929$	−49.1538	−1.61268	−0.806342	+	0.591450i	$0.798556\pi$
$930$	0	0
$931$	−5.87344	−0.192494
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.77299	0.286908
$936$	0	0
$937$	−35.5051	−1.15990	−0.579951	−	0.814651i	$-0.696928\pi$
$937$	−35.5051	−1.15990	−0.579951	+	0.814651i	$0.696928\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−41.1314	−1.34084	−0.670422	−	0.741980i	$-0.733887\pi$
$941$	−41.1314	−1.34084	−0.670422	+	0.741980i	$0.733887\pi$
$942$	0	0
$943$	−8.29654	−0.270173
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34.0618	−1.10686	−0.553430	−	0.832896i	$-0.686681\pi$
$947$	−34.0618	−1.10686	−0.553430	+	0.832896i	$0.686681\pi$
$948$	0	0
$949$	−48.8433	−1.58552
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12.2193	0.395822	0.197911	−	0.980220i	$-0.436584\pi$
$953$	12.2193	0.395822	0.197911	+	0.980220i	$0.436584\pi$
$954$	0	0
$955$	29.3661	0.950264
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.78147	−0.218985
$960$	0	0
$961$	−27.6873	−0.893138
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−14.9529	−0.481351
$966$	0	0
$967$	42.3414	1.36161	0.680804	−	0.732466i	$-0.261630\pi$
$967$	42.3414	1.36161	0.680804	+	0.732466i	$0.261630\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−36.9978	−1.18732	−0.593658	−	0.804718i	$-0.702317\pi$
$971$	−36.9978	−1.18732	−0.593658	+	0.804718i	$0.702317\pi$
$972$	0	0
$973$	2.26341	0.0725616
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.5460	0.369389	0.184694	−	0.982796i	$-0.440870\pi$
$977$	11.5460	0.369389	0.184694	+	0.982796i	$0.440870\pi$
$978$	0	0
$979$	−11.1862	−0.357511
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.5290	−0.814250	−0.407125	−	0.913372i	$-0.633469\pi$
$983$	−25.5290	−0.814250	−0.407125	+	0.913372i	$0.633469\pi$
$984$	0	0
$985$	68.1924	2.17279
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.96061	−0.0623439
$990$	0	0
$991$	−3.28036	−0.104204	−0.0521020	−	0.998642i	$-0.516592\pi$
$991$	−3.28036	−0.104204	−0.0521020	+	0.998642i	$0.516592\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−65.3252	−2.07095
$996$	0	0
$997$	28.5158	0.903106	0.451553	−	0.892244i	$-0.350870\pi$
$997$	28.5158	0.903106	0.451553	+	0.892244i	$0.350870\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2376.2.a.l.1.1	✓	3
3.2	odd	2		2376.2.a.q.1.3	yes	3
4.3	odd	2		4752.2.a.bh.1.1		3
12.11	even	2		4752.2.a.bp.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2376.2.a.l.1.1	✓	3	1.1	even	1	trivial
2376.2.a.q.1.3	yes	3	3.2	odd	2
4752.2.a.bh.1.1		3	4.3	odd	2
4752.2.a.bp.1.3		3	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)

\(f(q)\)	\(=\)	\(q-3.82009 q^{5} -0.476452 q^{7} -1.00000 q^{11} +4.77299 q^{13} +2.29654 q^{17} +0.867185 q^{19} +0.820089 q^{23} +9.59308 q^{25} -1.52355 q^{29} +1.82009 q^{31} +1.82009 q^{35} -3.77299 q^{37} -10.1166 q^{41} -2.39073 q^{43} +5.59308 q^{47} -6.77299 q^{49} -6.95290 q^{53} +3.82009 q^{55} -13.5074 q^{59} +9.82009 q^{61} -18.2333 q^{65} -9.82009 q^{67} +12.3275 q^{71} -10.2333 q^{73} +0.476452 q^{77} -6.98382 q^{79} -10.6788 q^{83} -8.77299 q^{85} +11.1862 q^{89} -2.27410 q^{91} -3.31273 q^{95} +5.17991 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{5} + q^{7} - 3 q^{11} - 5 q^{17} - 2 q^{19} - 7 q^{23} + 5 q^{25} - 7 q^{29} - 4 q^{31} - 4 q^{35} + 3 q^{37} - 9 q^{41} - 5 q^{43} - 7 q^{47} - 6 q^{49} - 16 q^{53} + 2 q^{55} - 17 q^{59} + 20 q^{61}+ \cdots + 25 q^{97}+O(q^{100}) \) 3 * q - 2 * q^5 + q^7 - 3 * q^11 - 5 * q^17 - 2 * q^19 - 7 * q^23 + 5 * q^25 - 7 * q^29 - 4 * q^31 - 4 * q^35 + 3 * q^37 - 9 * q^41 - 5 * q^43 - 7 * q^47 - 6 * q^49 - 16 * q^53 + 2 * q^55 - 17 * q^59 + 20 * q^61 - 12 * q^65 - 20 * q^67 + 4 * q^71 + 12 * q^73 - q^77 + 5 * q^79 - 8 * q^83 - 12 * q^85 - 14 * q^89 - 26 * q^91 - 24 * q^95 + 25 * q^97

Introduction

L-functions

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