LMFDB - Embedded newform 396.4.a.h.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [396,4,Mod(1,396)] 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("396.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(396, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 396 = 2^{2} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	396.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-12,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$23.3647563623$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{31}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 31 $ x^2 - 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.56776$ of defining polynomial
Character	$\chi$	$=$	396.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.13553 q^{5} -0.864471 q^{7} +11.0000 q^{11} +51.6776 q^{13} -52.8132 q^{17} -56.2711 q^{19} +211.949 q^{23} -98.6263 q^{25} +174.542 q^{29} +211.897 q^{31} -4.43952 q^{35} -86.9816 q^{37} +151.084 q^{41} -74.0000 q^{43} +163.509 q^{47} -342.253 q^{49} +539.099 q^{53} +56.4908 q^{55} +365.187 q^{59} +499.912 q^{61} +265.392 q^{65} +189.421 q^{67} +751.744 q^{71} +352.374 q^{73} -9.50918 q^{77} -319.678 q^{79} +811.018 q^{83} -271.224 q^{85} -1103.18 q^{89} -44.6738 q^{91} -288.982 q^{95} -551.758 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{5} - 24 q^{7} + 22 q^{11} - 8 q^{13} + 28 q^{17} - 68 q^{19} + 268 q^{23} + 70 q^{25} + 260 q^{29} + 112 q^{31} + 392 q^{35} + 316 q^{37} + 124 q^{41} - 148 q^{43} + 572 q^{47} - 150 q^{49}+ \cdots + 500 q^{97}+O(q^{100}) $ 2 * q - 12 * q^5 - 24 * q^7 + 22 * q^11 - 8 * q^13 + 28 * q^17 - 68 * q^19 + 268 * q^23 + 70 * q^25 + 260 * q^29 + 112 * q^31 + 392 * q^35 + 316 * q^37 + 124 * q^41 - 148 * q^43 + 572 * q^47 - 150 * q^49 + 76 * q^53 - 132 * q^55 + 864 * q^59 - 136 * q^61 + 1288 * q^65 - 512 * q^67 + 724 * q^71 + 972 * q^73 - 264 * q^77 - 528 * q^79 + 2112 * q^83 - 1656 * q^85 + 288 * q^89 + 1336 * q^91 - 88 * q^95 + 500 * 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
$5$	5.13553	0.459336	0.229668	−	0.973269i	$-0.426236\pi$
$5$	5.13553	0.459336	0.229668	+	0.973269i	$0.426236\pi$
$6$	0	0
$7$	−0.864471	−0.0466771	−0.0233385	−	0.999728i	$-0.507430\pi$
$7$	−0.864471	−0.0466771	−0.0233385	+	0.999728i	$0.507430\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	51.6776	1.10252	0.551262	−	0.834333i	$-0.314147\pi$
$13$	51.6776	1.10252	0.551262	+	0.834333i	$0.314147\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−52.8132	−0.753475	−0.376738	−	0.926320i	$-0.622954\pi$
$17$	−52.8132	−0.753475	−0.376738	+	0.926320i	$0.622954\pi$
$18$	0	0
$19$	−56.2711	−0.679446	−0.339723	−	0.940526i	$-0.610333\pi$
$19$	−56.2711	−0.679446	−0.339723	+	0.940526i	$0.610333\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	211.949	1.92149	0.960747	−	0.277426i	$-0.0894814\pi$
$23$	211.949	1.92149	0.960747	+	0.277426i	$0.0894814\pi$
$24$	0	0
$25$	−98.6263	−0.789011
$26$	0	0
$27$	0	0
$28$	0	0
$29$	174.542	1.11764	0.558822	−	0.829288i	$-0.311254\pi$
$29$	174.542	1.11764	0.558822	+	0.829288i	$0.311254\pi$
$30$	0	0
$31$	211.897	1.22767	0.613837	−	0.789433i	$-0.289625\pi$
$31$	211.897	1.22767	0.613837	+	0.789433i	$0.289625\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.43952	−0.0214404
$36$	0	0
$37$	−86.9816	−0.386478	−0.193239	−	0.981152i	$-0.561899\pi$
$37$	−86.9816	−0.386478	−0.193239	+	0.981152i	$0.561899\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	151.084	0.575497	0.287749	−	0.957706i	$-0.407093\pi$
$41$	151.084	0.575497	0.287749	+	0.957706i	$0.407093\pi$
$42$	0	0
$43$	−74.0000	−0.262439	−0.131220	−	0.991353i	$-0.541889\pi$
$43$	−74.0000	−0.262439	−0.131220	+	0.991353i	$0.541889\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	163.509	0.507452	0.253726	−	0.967276i	$-0.418344\pi$
$47$	163.509	0.507452	0.253726	+	0.967276i	$0.418344\pi$
$48$	0	0
$49$	−342.253	−0.997821
$50$	0	0
$51$	0	0
$52$	0	0
$53$	539.099	1.39719	0.698594	−	0.715519i	$-0.253810\pi$
$53$	539.099	1.39719	0.698594	+	0.715519i	$0.253810\pi$
$54$	0	0
$55$	56.4908	0.138495
$56$	0	0
$57$	0	0
$58$	0	0
$59$	365.187	0.805818	0.402909	−	0.915240i	$-0.367999\pi$
$59$	365.187	0.805818	0.402909	+	0.915240i	$0.367999\pi$
$60$	0	0
$61$	499.912	1.04930	0.524649	−	0.851319i	$-0.324197\pi$
$61$	499.912	1.04930	0.524649	+	0.851319i	$0.324197\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	265.392	0.506428
$66$	0	0
$67$	189.421	0.345395	0.172698	−	0.984975i	$-0.444752\pi$
$67$	189.421	0.345395	0.172698	+	0.984975i	$0.444752\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	751.744	1.25656	0.628278	−	0.777989i	$-0.283760\pi$
$71$	751.744	1.25656	0.628278	+	0.777989i	$0.283760\pi$
$72$	0	0
$73$	352.374	0.564962	0.282481	−	0.959273i	$-0.408843\pi$
$73$	352.374	0.564962	0.282481	+	0.959273i	$0.408843\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9.50918	−0.0140737
$78$	0	0
$79$	−319.678	−0.455273	−0.227636	−	0.973746i	$-0.573100\pi$
$79$	−319.678	−0.455273	−0.227636	+	0.973746i	$0.573100\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	811.018	1.07254	0.536270	−	0.844046i	$-0.319833\pi$
$83$	811.018	1.07254	0.536270	+	0.844046i	$0.319833\pi$
$84$	0	0
$85$	−271.224	−0.346098
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1103.18	−1.31390	−0.656948	−	0.753936i	$-0.728153\pi$
$89$	−1103.18	−1.31390	−0.656948	+	0.753936i	$0.728153\pi$
$90$	0	0
$91$	−44.6738	−0.0514625
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−288.982	−0.312094
$96$	0	0
$97$	−551.758	−0.577552	−0.288776	−	0.957397i	$-0.593248\pi$
$97$	−551.758	−0.577552	−0.288776	+	0.957397i	$0.593248\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	196.476	0.193566	0.0967828	−	0.995306i	$-0.469145\pi$
$101$	196.476	0.193566	0.0967828	+	0.995306i	$0.469145\pi$
$102$	0	0
$103$	261.832	0.250476	0.125238	−	0.992127i	$-0.460031\pi$
$103$	261.832	0.250476	0.125238	+	0.992127i	$0.460031\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1393.68	1.25918	0.629591	−	0.776926i	$-0.283222\pi$
$107$	1393.68	1.25918	0.629591	+	0.776926i	$0.283222\pi$
$108$	0	0
$109$	−190.491	−0.167392	−0.0836959	−	0.996491i	$-0.526672\pi$
$109$	−190.491	−0.167392	−0.0836959	+	0.996491i	$0.526672\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	226.095	0.188223	0.0941116	−	0.995562i	$-0.469999\pi$
$113$	226.095	0.188223	0.0941116	+	0.995562i	$0.469999\pi$
$114$	0	0
$115$	1088.47	0.882611
$116$	0	0
$117$	0	0
$118$	0	0
$119$	45.6555	0.0351700
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1148.44	−0.821756
$126$	0	0
$127$	−2585.70	−1.80664	−0.903322	−	0.428964i	$-0.858879\pi$
$127$	−2585.70	−1.80664	−0.903322	+	0.428964i	$0.858879\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1867.55	−1.24556	−0.622779	−	0.782398i	$-0.713997\pi$
$131$	−1867.55	−1.24556	−0.622779	+	0.782398i	$0.713997\pi$
$132$	0	0
$133$	48.6447	0.0317145
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1447.56	−0.902726	−0.451363	−	0.892340i	$-0.649062\pi$
$137$	−1447.56	−0.902726	−0.451363	+	0.892340i	$0.649062\pi$
$138$	0	0
$139$	298.513	0.182155	0.0910775	−	0.995844i	$-0.470969\pi$
$139$	298.513	0.182155	0.0910775	+	0.995844i	$0.470969\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	568.454	0.332423
$144$	0	0
$145$	896.366	0.513373
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3240.09	−1.78147	−0.890735	−	0.454524i	$-0.849809\pi$
$149$	−3240.09	−1.78147	−0.890735	+	0.454524i	$0.849809\pi$
$150$	0	0
$151$	169.875	0.0915513	0.0457757	−	0.998952i	$-0.485424\pi$
$151$	169.875	0.0915513	0.0457757	+	0.998952i	$0.485424\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1088.21	0.563914
$156$	0	0
$157$	2622.42	1.33307	0.666536	−	0.745473i	$-0.267776\pi$
$157$	2622.42	1.33307	0.666536	+	0.745473i	$0.267776\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−183.224	−0.0896897
$162$	0	0
$163$	−684.879	−0.329103	−0.164552	−	0.986368i	$-0.552618\pi$
$163$	−684.879	−0.329103	−0.164552	+	0.986368i	$0.552618\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1302.51	−0.603538	−0.301769	−	0.953381i	$-0.597577\pi$
$167$	−1302.51	−0.603538	−0.301769	+	0.953381i	$0.597577\pi$
$168$	0	0
$169$	473.579	0.215557
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3157.79	−1.38776	−0.693881	−	0.720090i	$-0.744100\pi$
$173$	−3157.79	−1.38776	−0.693881	+	0.720090i	$0.744100\pi$
$174$	0	0
$175$	85.2596	0.0368287
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−843.619	−0.352263	−0.176131	−	0.984367i	$-0.556358\pi$
$179$	−843.619	−0.352263	−0.176131	+	0.984367i	$0.556358\pi$
$180$	0	0
$181$	−2539.42	−1.04284	−0.521419	−	0.853301i	$-0.674597\pi$
$181$	−2539.42	−1.04284	−0.521419	+	0.853301i	$0.674597\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−446.697	−0.177523
$186$	0	0
$187$	−580.945	−0.227181
$188$	0	0
$189$	0	0
$190$	0	0
$191$	365.883	0.138609	0.0693046	−	0.997596i	$-0.477922\pi$
$191$	365.883	0.138609	0.0693046	+	0.997596i	$0.477922\pi$
$192$	0	0
$193$	−1566.10	−0.584096	−0.292048	−	0.956404i	$-0.594337\pi$
$193$	−1566.10	−0.584096	−0.292048	+	0.956404i	$0.594337\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2487.39	−0.899591	−0.449795	−	0.893132i	$-0.648503\pi$
$197$	−2487.39	−0.899591	−0.449795	+	0.893132i	$0.648503\pi$
$198$	0	0
$199$	4288.44	1.52764	0.763818	−	0.645432i	$-0.223323\pi$
$199$	4288.44	1.52764	0.763818	+	0.645432i	$0.223323\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−150.887	−0.0521683
$204$	0	0
$205$	775.897	0.264346
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−618.982	−0.204861
$210$	0	0
$211$	−1667.95	−0.544200	−0.272100	−	0.962269i	$-0.587718\pi$
$211$	−1667.95	−0.544200	−0.272100	+	0.962269i	$0.587718\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−380.029	−0.120548
$216$	0	0
$217$	−183.179	−0.0573042
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2729.26	−0.830724
$222$	0	0
$223$	−301.479	−0.0905316	−0.0452658	−	0.998975i	$-0.514413\pi$
$223$	−301.479	−0.0905316	−0.0452658	+	0.998975i	$0.514413\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1153.01	0.337128	0.168564	−	0.985691i	$-0.446087\pi$
$227$	1153.01	0.337128	0.168564	+	0.985691i	$0.446087\pi$
$228$	0	0
$229$	495.861	0.143089	0.0715445	−	0.997437i	$-0.477207\pi$
$229$	495.861	0.143089	0.0715445	+	0.997437i	$0.477207\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5009.50	−1.40851	−0.704256	−	0.709946i	$-0.748719\pi$
$233$	−5009.50	−1.40851	−0.704256	+	0.709946i	$0.748719\pi$
$234$	0	0
$235$	839.706	0.233091
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1971.52	0.533585	0.266792	−	0.963754i	$-0.414036\pi$
$239$	1971.52	0.533585	0.266792	+	0.963754i	$0.414036\pi$
$240$	0	0
$241$	−5622.20	−1.50273	−0.751365	−	0.659887i	$-0.770604\pi$
$241$	−5622.20	−1.50273	−0.751365	+	0.659887i	$0.770604\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1757.65	−0.458335
$246$	0	0
$247$	−2907.96	−0.749104
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2142.29	0.538725	0.269362	−	0.963039i	$-0.413187\pi$
$251$	2142.29	0.538725	0.269362	+	0.963039i	$0.413187\pi$
$252$	0	0
$253$	2331.44	0.579352
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1393.29	−0.338175	−0.169088	−	0.985601i	$-0.554082\pi$
$257$	−1393.29	−0.338175	−0.169088	+	0.985601i	$0.554082\pi$
$258$	0	0
$259$	75.1931	0.0180397
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7244.29	1.69849	0.849244	−	0.528001i	$-0.177058\pi$
$263$	7244.29	1.69849	0.849244	+	0.528001i	$0.177058\pi$
$264$	0	0
$265$	2768.56	0.641778
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3082.48	0.698670	0.349335	−	0.936998i	$-0.386407\pi$
$269$	3082.48	0.698670	0.349335	+	0.936998i	$0.386407\pi$
$270$	0	0
$271$	5463.79	1.22473	0.612365	−	0.790575i	$-0.290218\pi$
$271$	5463.79	1.22473	0.612365	+	0.790575i	$0.290218\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1084.89	−0.237896
$276$	0	0
$277$	2395.43	0.519593	0.259797	−	0.965663i	$-0.416344\pi$
$277$	2395.43	0.519593	0.259797	+	0.965663i	$0.416344\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6345.21	1.34706	0.673529	−	0.739161i	$-0.264778\pi$
$281$	6345.21	1.34706	0.673529	+	0.739161i	$0.264778\pi$
$282$	0	0
$283$	−6903.80	−1.45014	−0.725068	−	0.688677i	$-0.758192\pi$
$283$	−6903.80	−1.45014	−0.725068	+	0.688677i	$0.758192\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−130.608	−0.0268625
$288$	0	0
$289$	−2123.77	−0.432275
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7349.12	−1.46532	−0.732662	−	0.680592i	$-0.761723\pi$
$293$	−7349.12	−1.46532	−0.732662	+	0.680592i	$0.761723\pi$
$294$	0	0
$295$	1875.43	0.370141
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10953.0	2.11849
$300$	0	0
$301$	63.9709	0.0122499
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2567.31	0.481980
$306$	0	0
$307$	621.325	0.115508	0.0577539	−	0.998331i	$-0.481606\pi$
$307$	621.325	0.115508	0.0577539	+	0.998331i	$0.481606\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	674.748	0.123027	0.0615136	−	0.998106i	$-0.480407\pi$
$311$	674.748	0.123027	0.0615136	+	0.998106i	$0.480407\pi$
$312$	0	0
$313$	−367.510	−0.0663670	−0.0331835	−	0.999449i	$-0.510565\pi$
$313$	−367.510	−0.0663670	−0.0331835	+	0.999449i	$0.510565\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3027.35	−0.536381	−0.268190	−	0.963366i	$-0.586426\pi$
$317$	−3027.35	−0.536381	−0.268190	+	0.963366i	$0.586426\pi$
$318$	0	0
$319$	1919.96	0.336982
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2971.85	0.511945
$324$	0	0
$325$	−5096.78	−0.869903
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−141.349	−0.0236864
$330$	0	0
$331$	8180.62	1.35845	0.679226	−	0.733930i	$-0.262316\pi$
$331$	8180.62	1.35845	0.679226	+	0.733930i	$0.262316\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	972.778	0.158652
$336$	0	0
$337$	8582.67	1.38732	0.693662	−	0.720301i	$-0.255996\pi$
$337$	8582.67	1.38732	0.693662	+	0.720301i	$0.255996\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2330.87	0.370158
$342$	0	0
$343$	592.381	0.0932524
$344$	0	0
$345$	0	0
$346$	0	0
$347$	169.435	0.0262125	0.0131063	−	0.999914i	$-0.495828\pi$
$347$	169.435	0.0262125	0.0131063	+	0.999914i	$0.495828\pi$
$348$	0	0
$349$	−6107.13	−0.936697	−0.468349	−	0.883544i	$-0.655151\pi$
$349$	−6107.13	−0.936697	−0.468349	+	0.883544i	$0.655151\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10191.0	1.53658	0.768292	−	0.640100i	$-0.221107\pi$
$353$	10191.0	1.53658	0.768292	+	0.640100i	$0.221107\pi$
$354$	0	0
$355$	3860.60	0.577181
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13015.3	1.91343	0.956716	−	0.291025i	$-0.0939962\pi$
$359$	13015.3	1.91343	0.956716	+	0.291025i	$0.0939962\pi$
$360$	0	0
$361$	−3692.57	−0.538354
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1809.63	0.259507
$366$	0	0
$367$	−7541.19	−1.07261	−0.536303	−	0.844025i	$-0.680180\pi$
$367$	−7541.19	−1.07261	−0.536303	+	0.844025i	$0.680180\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−466.035	−0.0652166
$372$	0	0
$373$	−3186.02	−0.442268	−0.221134	−	0.975243i	$-0.570976\pi$
$373$	−3186.02	−0.442268	−0.221134	+	0.975243i	$0.570976\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9019.93	1.23223
$378$	0	0
$379$	−10641.9	−1.44232	−0.721159	−	0.692769i	$-0.756390\pi$
$379$	−10641.9	−1.44232	−0.721159	+	0.692769i	$0.756390\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1845.84	−0.246261	−0.123130	−	0.992391i	$-0.539293\pi$
$383$	−1845.84	−0.246261	−0.123130	+	0.992391i	$0.539293\pi$
$384$	0	0
$385$	−48.8347	−0.00646454
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4615.33	0.601560	0.300780	−	0.953694i	$-0.402753\pi$
$389$	4615.33	0.601560	0.300780	+	0.953694i	$0.402753\pi$
$390$	0	0
$391$	−11193.7	−1.44780
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1641.71	−0.209123
$396$	0	0
$397$	10168.0	1.28544	0.642719	−	0.766102i	$-0.277806\pi$
$397$	10168.0	1.28544	0.642719	+	0.766102i	$0.277806\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9577.61	−1.19273	−0.596363	−	0.802715i	$-0.703388\pi$
$401$	−9577.61	−1.19273	−0.596363	+	0.802715i	$0.703388\pi$
$402$	0	0
$403$	10950.4	1.35354
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−956.798	−0.116528
$408$	0	0
$409$	−3895.73	−0.470981	−0.235491	−	0.971877i	$-0.575670\pi$
$409$	−3895.73	−0.470981	−0.235491	+	0.971877i	$0.575670\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−315.694	−0.0376132
$414$	0	0
$415$	4165.01	0.492656
$416$	0	0
$417$	0	0
$418$	0	0
$419$	163.208	0.0190292	0.00951462	−	0.999955i	$-0.496971\pi$
$419$	163.208	0.0190292	0.00951462	+	0.999955i	$0.496971\pi$
$420$	0	0
$421$	13611.6	1.57574	0.787871	−	0.615841i	$-0.211183\pi$
$421$	13611.6	1.57574	0.787871	+	0.615841i	$0.211183\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5208.77	0.594500
$426$	0	0
$427$	−432.160	−0.0489781
$428$	0	0
$429$	0	0
$430$	0	0
$431$	309.524	0.0345922	0.0172961	−	0.999850i	$-0.494494\pi$
$431$	309.524	0.0345922	0.0172961	+	0.999850i	$0.494494\pi$
$432$	0	0
$433$	11406.1	1.26592	0.632960	−	0.774185i	$-0.281840\pi$
$433$	11406.1	1.26592	0.632960	+	0.774185i	$0.281840\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−11926.6	−1.30555
$438$	0	0
$439$	−16541.7	−1.79838	−0.899192	−	0.437554i	$-0.855845\pi$
$439$	−16541.7	−1.79838	−0.899192	+	0.437554i	$0.855845\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3728.94	0.399926	0.199963	−	0.979803i	$-0.435918\pi$
$443$	3728.94	0.399926	0.199963	+	0.979803i	$0.435918\pi$
$444$	0	0
$445$	−5665.41	−0.603519
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3344.45	−0.351524	−0.175762	−	0.984433i	$-0.556239\pi$
$449$	−3344.45	−0.351524	−0.175762	+	0.984433i	$0.556239\pi$
$450$	0	0
$451$	1661.93	0.173519
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−229.424	−0.0236386
$456$	0	0
$457$	−16692.1	−1.70858	−0.854291	−	0.519795i	$-0.826008\pi$
$457$	−16692.1	−1.70858	−0.854291	+	0.519795i	$0.826008\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8755.78	0.884593	0.442297	−	0.896869i	$-0.354164\pi$
$461$	8755.78	0.884593	0.442297	+	0.896869i	$0.354164\pi$
$462$	0	0
$463$	11017.1	1.10585	0.552924	−	0.833232i	$-0.313512\pi$
$463$	11017.1	1.10585	0.552924	+	0.833232i	$0.313512\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3146.12	0.311746	0.155873	−	0.987777i	$-0.450181\pi$
$467$	3146.12	0.311746	0.155873	+	0.987777i	$0.450181\pi$
$468$	0	0
$469$	−163.749	−0.0161220
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−814.000	−0.0791285
$474$	0	0
$475$	5549.81	0.536090
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1789.35	−0.170683	−0.0853417	−	0.996352i	$-0.527198\pi$
$479$	−1789.35	−0.170683	−0.0853417	+	0.996352i	$0.527198\pi$
$480$	0	0
$481$	−4495.01	−0.426101
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2833.57	−0.265290
$486$	0	0
$487$	7891.84	0.734319	0.367160	−	0.930158i	$-0.380330\pi$
$487$	7891.84	0.734319	0.367160	+	0.930158i	$0.380330\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−12340.1	−1.13422	−0.567111	−	0.823641i	$-0.691939\pi$
$491$	−12340.1	−1.13422	−0.567111	+	0.823641i	$0.691939\pi$
$492$	0	0
$493$	−9218.12	−0.842116
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−649.861	−0.0586524
$498$	0	0
$499$	13615.4	1.22146	0.610732	−	0.791837i	$-0.290875\pi$
$499$	13615.4	1.22146	0.610732	+	0.791837i	$0.290875\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4092.13	−0.362742	−0.181371	−	0.983415i	$-0.558053\pi$
$503$	−4092.13	−0.362742	−0.181371	+	0.983415i	$0.558053\pi$
$504$	0	0
$505$	1009.01	0.0889115
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−18821.2	−1.63897	−0.819483	−	0.573104i	$-0.805739\pi$
$509$	−18821.2	−1.63897	−0.819483	+	0.573104i	$0.805739\pi$
$510$	0	0
$511$	−304.617	−0.0263708
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1344.64	0.115053
$516$	0	0
$517$	1798.60	0.153003
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6851.87	−0.576172	−0.288086	−	0.957605i	$-0.593019\pi$
$521$	−6851.87	−0.576172	−0.288086	+	0.957605i	$0.593019\pi$
$522$	0	0
$523$	−20338.2	−1.70043	−0.850216	−	0.526433i	$-0.823529\pi$
$523$	−20338.2	−1.70043	−0.850216	+	0.526433i	$0.823529\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11191.0	−0.925022
$528$	0	0
$529$	32755.3	2.69214
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7807.68	0.634499
$534$	0	0
$535$	7157.31	0.578388
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3764.78	−0.300854
$540$	0	0
$541$	−6446.58	−0.512311	−0.256155	−	0.966636i	$-0.582456\pi$
$541$	−6446.58	−0.512311	−0.256155	+	0.966636i	$0.582456\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−978.271	−0.0768890
$546$	0	0
$547$	−20285.9	−1.58567	−0.792834	−	0.609438i	$-0.791395\pi$
$547$	−20285.9	−1.58567	−0.792834	+	0.609438i	$0.791395\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9821.67	−0.759378
$552$	0	0
$553$	276.352	0.0212508
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18441.2	−1.40284	−0.701418	−	0.712750i	$-0.747449\pi$
$557$	−18441.2	−1.40284	−0.701418	+	0.712750i	$0.747449\pi$
$558$	0	0
$559$	−3824.15	−0.289345
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20335.0	−1.52224	−0.761118	−	0.648614i	$-0.775349\pi$
$563$	−20335.0	−1.52224	−0.761118	+	0.648614i	$0.775349\pi$
$564$	0	0
$565$	1161.12	0.0864576
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22960.4	1.69165	0.845825	−	0.533460i	$-0.179109\pi$
$569$	22960.4	1.69165	0.845825	+	0.533460i	$0.179109\pi$
$570$	0	0
$571$	−13239.5	−0.970323	−0.485161	−	0.874425i	$-0.661239\pi$
$571$	−13239.5	−0.970323	−0.485161	+	0.874425i	$0.661239\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−20903.7	−1.51608
$576$	0	0
$577$	−11439.5	−0.825361	−0.412681	−	0.910876i	$-0.635407\pi$
$577$	−11439.5	−0.825361	−0.412681	+	0.910876i	$0.635407\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−701.102	−0.0500630
$582$	0	0
$583$	5930.09	0.421268
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8474.43	−0.595873	−0.297936	−	0.954586i	$-0.596298\pi$
$587$	−8474.43	−0.595873	−0.297936	+	0.954586i	$0.596298\pi$
$588$	0	0
$589$	−11923.7	−0.834138
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16321.7	1.13027	0.565136	−	0.824998i	$-0.308824\pi$
$593$	16321.7	1.13027	0.565136	+	0.824998i	$0.308824\pi$
$594$	0	0
$595$	234.465	0.0161548
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−11407.4	−0.778117	−0.389059	−	0.921213i	$-0.627200\pi$
$599$	−11407.4	−0.778117	−0.389059	+	0.921213i	$0.627200\pi$
$600$	0	0
$601$	22493.8	1.52669	0.763347	−	0.645989i	$-0.223555\pi$
$601$	22493.8	1.52669	0.763347	+	0.645989i	$0.223555\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	621.399	0.0417578
$606$	0	0
$607$	14720.9	0.984354	0.492177	−	0.870495i	$-0.336201\pi$
$607$	14720.9	0.984354	0.492177	+	0.870495i	$0.336201\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8449.77	0.559478
$612$	0	0
$613$	−10291.2	−0.678068	−0.339034	−	0.940774i	$-0.610100\pi$
$613$	−10291.2	−0.678068	−0.339034	+	0.940774i	$0.610100\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5682.09	−0.370750	−0.185375	−	0.982668i	$-0.559350\pi$
$617$	−5682.09	−0.370750	−0.185375	+	0.982668i	$0.559350\pi$
$618$	0	0
$619$	19433.1	1.26184	0.630922	−	0.775847i	$-0.282677\pi$
$619$	19433.1	1.26184	0.630922	+	0.775847i	$0.282677\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	953.667	0.0613288
$624$	0	0
$625$	6430.45	0.411549
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4593.78	0.291202
$630$	0	0
$631$	−15008.3	−0.946863	−0.473432	−	0.880831i	$-0.656985\pi$
$631$	−15008.3	−0.946863	−0.473432	+	0.880831i	$0.656985\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13278.9	−0.829856
$636$	0	0
$637$	−17686.8	−1.10012
$638$	0	0
$639$	0	0
$640$	0	0
$641$	26697.2	1.64505	0.822524	−	0.568731i	$-0.192565\pi$
$641$	26697.2	1.64505	0.822524	+	0.568731i	$0.192565\pi$
$642$	0	0
$643$	−17052.5	−1.04586	−0.522928	−	0.852377i	$-0.675161\pi$
$643$	−17052.5	−1.04586	−0.522928	+	0.852377i	$0.675161\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−19430.9	−1.18069	−0.590345	−	0.807151i	$-0.701008\pi$
$647$	−19430.9	−1.18069	−0.590345	+	0.807151i	$0.701008\pi$
$648$	0	0
$649$	4017.06	0.242963
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13484.8	−0.808116	−0.404058	−	0.914733i	$-0.632401\pi$
$653$	−13484.8	−0.808116	−0.404058	+	0.914733i	$0.632401\pi$
$654$	0	0
$655$	−9590.83	−0.572130
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−23752.4	−1.40404	−0.702020	−	0.712157i	$-0.747718\pi$
$659$	−23752.4	−1.40404	−0.702020	+	0.712157i	$0.747718\pi$
$660$	0	0
$661$	22078.0	1.29914	0.649571	−	0.760301i	$-0.274949\pi$
$661$	22078.0	1.29914	0.649571	+	0.760301i	$0.274949\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	249.816	0.0145676
$666$	0	0
$667$	36994.0	2.14754
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5499.03	0.316375
$672$	0	0
$673$	−20273.4	−1.16119	−0.580595	−	0.814192i	$-0.697180\pi$
$673$	−20273.4	−1.16119	−0.580595	+	0.814192i	$0.697180\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25947.6	−1.47304	−0.736519	−	0.676417i	$-0.763532\pi$
$677$	−25947.6	−1.47304	−0.736519	+	0.676417i	$0.763532\pi$
$678$	0	0
$679$	476.979	0.0269584
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9293.09	0.520630	0.260315	−	0.965524i	$-0.416174\pi$
$683$	9293.09	0.520630	0.260315	+	0.965524i	$0.416174\pi$
$684$	0	0
$685$	−7433.99	−0.414654
$686$	0	0
$687$	0	0
$688$	0	0
$689$	27859.4	1.54043
$690$	0	0
$691$	2879.37	0.158518	0.0792592	−	0.996854i	$-0.474745\pi$
$691$	2879.37	0.158518	0.0792592	+	0.996854i	$0.474745\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1533.02	0.0836703
$696$	0	0
$697$	−7979.24	−0.433623
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−14194.6	−0.764795	−0.382398	−	0.923998i	$-0.624902\pi$
$701$	−14194.6	−0.764795	−0.382398	+	0.923998i	$0.624902\pi$
$702$	0	0
$703$	4894.55	0.262591
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−169.848	−0.00903507
$708$	0	0
$709$	−19233.2	−1.01879	−0.509393	−	0.860534i	$-0.670130\pi$
$709$	−19233.2	−1.01879	−0.509393	+	0.860534i	$0.670130\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	44911.4	2.35897
$714$	0	0
$715$	2919.31	0.152694
$716$	0	0
$717$	0	0
$718$	0	0
$719$	32319.6	1.67638	0.838190	−	0.545378i	$-0.183614\pi$
$719$	32319.6	1.67638	0.838190	+	0.545378i	$0.183614\pi$
$720$	0	0
$721$	−226.346	−0.0116915
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−17214.5	−0.881833
$726$	0	0
$727$	26551.7	1.35454	0.677268	−	0.735737i	$-0.263164\pi$
$727$	26551.7	1.35454	0.677268	+	0.735737i	$0.263164\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3908.17	0.197742
$732$	0	0
$733$	31494.5	1.58701	0.793504	−	0.608565i	$-0.208255\pi$
$733$	31494.5	1.58701	0.793504	+	0.608565i	$0.208255\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2083.63	0.104141
$738$	0	0
$739$	14450.0	0.719283	0.359642	−	0.933091i	$-0.382899\pi$
$739$	14450.0	0.719283	0.359642	+	0.933091i	$0.382899\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−11693.6	−0.577383	−0.288692	−	0.957422i	$-0.593220\pi$
$743$	−11693.6	−0.577383	−0.288692	+	0.957422i	$0.593220\pi$
$744$	0	0
$745$	−16639.6	−0.818292
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1204.80	−0.0587750
$750$	0	0
$751$	−1082.31	−0.0525888	−0.0262944	−	0.999654i	$-0.508371\pi$
$751$	−1082.31	−0.0525888	−0.0262944	+	0.999654i	$0.508371\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	872.399	0.0420528
$756$	0	0
$757$	3150.38	0.151258	0.0756291	−	0.997136i	$-0.475904\pi$
$757$	3150.38	0.151258	0.0756291	+	0.997136i	$0.475904\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	16026.3	0.763406	0.381703	−	0.924285i	$-0.375338\pi$
$761$	16026.3	0.763406	0.381703	+	0.924285i	$0.375338\pi$
$762$	0	0
$763$	164.674	0.00781336
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18872.0	0.888433
$768$	0	0
$769$	−24674.3	−1.15706	−0.578528	−	0.815662i	$-0.696373\pi$
$769$	−24674.3	−1.15706	−0.578528	+	0.815662i	$0.696373\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28881.8	1.34386	0.671932	−	0.740613i	$-0.265465\pi$
$773$	28881.8	1.34386	0.671932	+	0.740613i	$0.265465\pi$
$774$	0	0
$775$	−20898.7	−0.968648
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8501.67	−0.391019
$780$	0	0
$781$	8269.18	0.378866
$782$	0	0
$783$	0	0
$784$	0	0
$785$	13467.5	0.612327
$786$	0	0
$787$	−24240.9	−1.09796	−0.548981	−	0.835835i	$-0.684984\pi$
$787$	−24240.9	−1.09796	−0.548981	+	0.835835i	$0.684984\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−195.453	−0.00878571
$792$	0	0
$793$	25834.3	1.15688
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1476.34	0.0656145	0.0328072	−	0.999462i	$-0.489555\pi$
$797$	1476.34	0.0656145	0.0328072	+	0.999462i	$0.489555\pi$
$798$	0	0
$799$	−8635.44	−0.382353
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3876.11	0.170342
$804$	0	0
$805$	−940.950	−0.0411977
$806$	0	0
$807$	0	0
$808$	0	0
$809$	36215.0	1.57386	0.786930	−	0.617042i	$-0.211669\pi$
$809$	36215.0	1.57386	0.786930	+	0.617042i	$0.211669\pi$
$810$	0	0
$811$	−36279.6	−1.57084	−0.785419	−	0.618964i	$-0.787553\pi$
$811$	−36279.6	−1.57084	−0.785419	+	0.618964i	$0.787553\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3517.22	−0.151169
$816$	0	0
$817$	4164.06	0.178313
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15206.6	0.646422	0.323211	−	0.946327i	$-0.395238\pi$
$821$	15206.6	0.646422	0.323211	+	0.946327i	$0.395238\pi$
$822$	0	0
$823$	−15023.1	−0.636296	−0.318148	−	0.948041i	$-0.603061\pi$
$823$	−15023.1	−0.636296	−0.318148	+	0.948041i	$0.603061\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1108.16	0.0465957	0.0232978	−	0.999729i	$-0.492583\pi$
$827$	1108.16	0.0465957	0.0232978	+	0.999729i	$0.492583\pi$
$828$	0	0
$829$	37519.7	1.57191	0.785955	−	0.618283i	$-0.212172\pi$
$829$	37519.7	1.57191	0.785955	+	0.618283i	$0.212172\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	18075.5	0.751833
$834$	0	0
$835$	−6689.05	−0.277227
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−112.813	−0.00464210	−0.00232105	−	0.999997i	$-0.500739\pi$
$839$	−112.813	−0.00464210	−0.00232105	+	0.999997i	$0.500739\pi$
$840$	0	0
$841$	6075.95	0.249127
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2432.08	0.0990130
$846$	0	0
$847$	−104.601	−0.00424337
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−18435.6	−0.742615
$852$	0	0
$853$	−15005.7	−0.602326	−0.301163	−	0.953573i	$-0.597375\pi$
$853$	−15005.7	−0.602326	−0.301163	+	0.953573i	$0.597375\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16517.0	0.658353	0.329177	−	0.944268i	$-0.393229\pi$
$857$	16517.0	0.658353	0.329177	+	0.944268i	$0.393229\pi$
$858$	0	0
$859$	21272.5	0.844945	0.422473	−	0.906376i	$-0.361162\pi$
$859$	21272.5	0.844945	0.422473	+	0.906376i	$0.361162\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1173.70	−0.0462959	−0.0231479	−	0.999732i	$-0.507369\pi$
$863$	−1173.70	−0.0462959	−0.0231479	+	0.999732i	$0.507369\pi$
$864$	0	0
$865$	−16216.9	−0.637448
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3516.45	−0.137270
$870$	0	0
$871$	9788.84	0.380806
$872$	0	0
$873$	0	0
$874$	0	0
$875$	992.793	0.0383572
$876$	0	0
$877$	5338.75	0.205561	0.102780	−	0.994704i	$-0.467226\pi$
$877$	5338.75	0.205561	0.102780	+	0.994704i	$0.467226\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	21936.8	0.838898	0.419449	−	0.907779i	$-0.362223\pi$
$881$	21936.8	0.838898	0.419449	+	0.907779i	$0.362223\pi$
$882$	0	0
$883$	30167.4	1.14973	0.574866	−	0.818247i	$-0.305054\pi$
$883$	30167.4	1.14973	0.574866	+	0.818247i	$0.305054\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22302.6	−0.844250	−0.422125	−	0.906538i	$-0.638716\pi$
$887$	−22302.6	−0.844250	−0.422125	+	0.906538i	$0.638716\pi$
$888$	0	0
$889$	2235.26	0.0843288
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9200.83	−0.344786
$894$	0	0
$895$	−4332.43	−0.161807
$896$	0	0
$897$	0	0
$898$	0	0
$899$	36985.0	1.37210
$900$	0	0
$901$	−28471.5	−1.05275
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13041.3	−0.479013
$906$	0	0
$907$	−1261.17	−0.0461704	−0.0230852	−	0.999734i	$-0.507349\pi$
$907$	−1261.17	−0.0461704	−0.0230852	+	0.999734i	$0.507349\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	33089.3	1.20340	0.601700	−	0.798722i	$-0.294490\pi$
$911$	33089.3	1.20340	0.601700	+	0.798722i	$0.294490\pi$
$912$	0	0
$913$	8921.20	0.323383
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1614.44	0.0581390
$918$	0	0
$919$	−43973.8	−1.57841	−0.789207	−	0.614128i	$-0.789508\pi$
$919$	−43973.8	−1.57841	−0.789207	+	0.614128i	$0.789508\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	38848.3	1.38538
$924$	0	0
$925$	8578.68	0.304935
$926$	0	0
$927$	0	0
$928$	0	0
$929$	51539.4	1.82019	0.910093	−	0.414403i	$-0.136010\pi$
$929$	51539.4	1.82019	0.910093	+	0.414403i	$0.136010\pi$
$930$	0	0
$931$	19258.9	0.677965
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2983.46	−0.104352
$936$	0	0
$937$	−4384.73	−0.152874	−0.0764369	−	0.997074i	$-0.524354\pi$
$937$	−4384.73	−0.152874	−0.0764369	+	0.997074i	$0.524354\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	23862.3	0.826663	0.413332	−	0.910581i	$-0.364365\pi$
$941$	23862.3	0.826663	0.413332	+	0.910581i	$0.364365\pi$
$942$	0	0
$943$	32022.1	1.10581
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−33377.7	−1.14533	−0.572666	−	0.819789i	$-0.694091\pi$
$947$	−33377.7	−1.14533	−0.572666	+	0.819789i	$0.694091\pi$
$948$	0	0
$949$	18209.8	0.622883
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−25097.9	−0.853095	−0.426547	−	0.904465i	$-0.640270\pi$
$953$	−25097.9	−0.853095	−0.426547	+	0.904465i	$0.640270\pi$
$954$	0	0
$955$	1879.00	0.0636681
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1251.37	0.0421366
$960$	0	0
$961$	15109.5	0.507184
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−8042.76	−0.268296
$966$	0	0
$967$	963.547	0.0320430	0.0160215	−	0.999872i	$-0.494900\pi$
$967$	963.547	0.0320430	0.0160215	+	0.999872i	$0.494900\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−28702.2	−0.948607	−0.474304	−	0.880361i	$-0.657300\pi$
$971$	−28702.2	−0.948607	−0.474304	+	0.880361i	$0.657300\pi$
$972$	0	0
$973$	−258.056	−0.00850246
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18696.2	−0.612226	−0.306113	−	0.951995i	$-0.599028\pi$
$977$	−18696.2	−0.612226	−0.306113	+	0.951995i	$0.599028\pi$
$978$	0	0
$979$	−12135.0	−0.396155
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−35820.2	−1.16224	−0.581122	−	0.813817i	$-0.697386\pi$
$983$	−35820.2	−1.16224	−0.581122	+	0.813817i	$0.697386\pi$
$984$	0	0
$985$	−12774.1	−0.413214
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−15684.2	−0.504276
$990$	0	0
$991$	32090.3	1.02864	0.514320	−	0.857598i	$-0.328044\pi$
$991$	32090.3	1.02864	0.514320	+	0.857598i	$0.328044\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	22023.4	0.701697
$996$	0	0
$997$	3963.89	0.125915	0.0629577	−	0.998016i	$-0.479947\pi$
$997$	3963.89	0.125915	0.0629577	+	0.998016i	$0.479947\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	396.4.a.h.1.2	✓	2
3.2	odd	2		396.4.a.j.1.1	yes	2
4.3	odd	2		1584.4.a.y.1.2		2
12.11	even	2		1584.4.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
396.4.a.h.1.2	✓	2	1.1	even	1	trivial
396.4.a.j.1.1	yes	2	3.2	odd	2
1584.4.a.y.1.2		2	4.3	odd	2
1584.4.a.bi.1.1		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:	\( N \)	\(=\)	\( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	396.a (trivial)

\(f(q)\)	\(=\)	\(q+5.13553 q^{5} -0.864471 q^{7} +11.0000 q^{11} +51.6776 q^{13} -52.8132 q^{17} -56.2711 q^{19} +211.949 q^{23} -98.6263 q^{25} +174.542 q^{29} +211.897 q^{31} -4.43952 q^{35} -86.9816 q^{37} +151.084 q^{41} -74.0000 q^{43} +163.509 q^{47} -342.253 q^{49} +539.099 q^{53} +56.4908 q^{55} +365.187 q^{59} +499.912 q^{61} +265.392 q^{65} +189.421 q^{67} +751.744 q^{71} +352.374 q^{73} -9.50918 q^{77} -319.678 q^{79} +811.018 q^{83} -271.224 q^{85} -1103.18 q^{89} -44.6738 q^{91} -288.982 q^{95} -551.758 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{5} - 24 q^{7} + 22 q^{11} - 8 q^{13} + 28 q^{17} - 68 q^{19} + 268 q^{23} + 70 q^{25} + 260 q^{29} + 112 q^{31} + 392 q^{35} + 316 q^{37} + 124 q^{41} - 148 q^{43} + 572 q^{47} - 150 q^{49}+ \cdots + 500 q^{97}+O(q^{100}) \) 2 * q - 12 * q^5 - 24 * q^7 + 22 * q^11 - 8 * q^13 + 28 * q^17 - 68 * q^19 + 268 * q^23 + 70 * q^25 + 260 * q^29 + 112 * q^31 + 392 * q^35 + 316 * q^37 + 124 * q^41 - 148 * q^43 + 572 * q^47 - 150 * q^49 + 76 * q^53 - 132 * q^55 + 864 * q^59 - 136 * q^61 + 1288 * q^65 - 512 * q^67 + 724 * q^71 + 972 * q^73 - 264 * q^77 - 528 * q^79 + 2112 * q^83 - 1656 * q^85 + 288 * q^89 + 1336 * q^91 - 88 * q^95 + 500 * q^97

Introduction

L-functions

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