LMFDB - Embedded newform 1368.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,2,Mod(1,1368)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$10.9235349965$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.961.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 10x + 8 $ x^3 - x^2 - 10*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.786802$ of defining polynomial
Character	$\chi$	$=$	1368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.786802 q^{5} +4.29707 q^{7} +1.21320 q^{11} +5.08387 q^{13} +2.29707 q^{17} -1.00000 q^{19} -7.67801 q^{23} -4.38094 q^{25} +0.489731 q^{29} -3.38094 q^{35} -2.00000 q^{37} +4.16774 q^{41} +12.9545 q^{43} +5.80734 q^{47} +11.4648 q^{49} -1.93667 q^{53} -0.954547 q^{55} +11.0839 q^{59} -5.38094 q^{61} -4.00000 q^{65} +2.48973 q^{67} +11.7413 q^{71} -8.46482 q^{73} +5.21320 q^{77} +1.83226 q^{79} +7.02054 q^{83} -1.80734 q^{85} -13.7413 q^{89} +21.8458 q^{91} +0.786802 q^{95} -3.57360 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{5} + 4 q^{7} + 5 q^{11} + 5 q^{13} - 2 q^{17} - 3 q^{19} + 5 q^{23} + 6 q^{25} + 9 q^{29} + 9 q^{35} - 6 q^{37} - 8 q^{41} + 17 q^{43} + q^{47} + 5 q^{49} - q^{53} + 19 q^{55} + 23 q^{59} + 3 q^{61}+ \cdots - 8 q^{97}+O(q^{100}) $ 3 * q - q^5 + 4 * q^7 + 5 * q^11 + 5 * q^13 - 2 * q^17 - 3 * q^19 + 5 * q^23 + 6 * q^25 + 9 * q^29 + 9 * q^35 - 6 * q^37 - 8 * q^41 + 17 * q^43 + q^47 + 5 * q^49 - q^53 + 19 * q^55 + 23 * q^59 + 3 * q^61 - 12 * q^65 + 15 * q^67 + 12 * q^71 + 4 * q^73 + 17 * q^77 + 26 * q^79 + 6 * q^83 + 11 * q^85 - 18 * q^89 + 17 * q^91 + q^95 - 8 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−0.786802	−0.351868	−0.175934	−	0.984402i	$-0.556295\pi$
$5$	−0.786802	−0.351868	−0.175934	+	0.984402i	$0.556295\pi$
$6$	0	0
$7$	4.29707	1.62414	0.812070	−	0.583560i	$-0.198341\pi$
$7$	4.29707	1.62414	0.812070	+	0.583560i	$0.198341\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.21320	0.365793	0.182897	−	0.983132i	$-0.441453\pi$
$11$	1.21320	0.365793	0.182897	+	0.983132i	$0.441453\pi$
$12$	0	0
$13$	5.08387	1.41001	0.705006	−	0.709201i	$-0.250944\pi$
$13$	5.08387	1.41001	0.705006	+	0.709201i	$0.250944\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.29707	0.557121	0.278561	−	0.960419i	$-0.410143\pi$
$17$	2.29707	0.557121	0.278561	+	0.960419i	$0.410143\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.67801	−1.60098	−0.800488	−	0.599348i	$-0.795426\pi$
$23$	−7.67801	−1.60098	−0.800488	+	0.599348i	$0.795426\pi$
$24$	0	0
$25$	−4.38094	−0.876189
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.489731	0.0909408	0.0454704	−	0.998966i	$-0.485521\pi$
$29$	0.489731	0.0909408	0.0454704	+	0.998966i	$0.485521\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.38094	−0.571484
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.16774	0.650892	0.325446	−	0.945561i	$-0.394486\pi$
$41$	4.16774	0.650892	0.325446	+	0.945561i	$0.394486\pi$
$42$	0	0
$43$	12.9545	1.97555	0.987775	−	0.155887i	$-0.0498235\pi$
$43$	12.9545	1.97555	0.987775	+	0.155887i	$0.0498235\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.80734	0.847087	0.423544	−	0.905876i	$-0.360786\pi$
$47$	5.80734	0.847087	0.423544	+	0.905876i	$0.360786\pi$
$48$	0	0
$49$	11.4648	1.63783
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.93667	−0.266021	−0.133011	−	0.991115i	$-0.542464\pi$
$53$	−1.93667	−0.266021	−0.133011	+	0.991115i	$0.542464\pi$
$54$	0	0
$55$	−0.954547	−0.128711
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.0839	1.44300	0.721499	−	0.692416i	$-0.243454\pi$
$59$	11.0839	1.44300	0.721499	+	0.692416i	$0.243454\pi$
$60$	0	0
$61$	−5.38094	−0.688959	−0.344480	−	0.938794i	$-0.611945\pi$
$61$	−5.38094	−0.688959	−0.344480	+	0.938794i	$0.611945\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	2.48973	0.304169	0.152085	−	0.988367i	$-0.451401\pi$
$67$	2.48973	0.304169	0.152085	+	0.988367i	$0.451401\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.7413	1.39344	0.696721	−	0.717342i	$-0.254642\pi$
$71$	11.7413	1.39344	0.696721	+	0.717342i	$0.254642\pi$
$72$	0	0
$73$	−8.46482	−0.990732	−0.495366	−	0.868684i	$-0.664966\pi$
$73$	−8.46482	−0.990732	−0.495366	+	0.868684i	$0.664966\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.21320	0.594099
$78$	0	0
$79$	1.83226	0.206145	0.103072	−	0.994674i	$-0.467133\pi$
$79$	1.83226	0.206145	0.103072	+	0.994674i	$0.467133\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.02054	0.770604	0.385302	−	0.922791i	$-0.374097\pi$
$83$	7.02054	0.770604	0.385302	+	0.922791i	$0.374097\pi$
$84$	0	0
$85$	−1.80734	−0.196033
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.7413	−1.45658	−0.728290	−	0.685269i	$-0.759685\pi$
$89$	−13.7413	−1.45658	−0.728290	+	0.685269i	$0.759685\pi$
$90$	0	0
$91$	21.8458	2.29006
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.786802	0.0807242
$96$	0	0
$97$	−3.57360	−0.362844	−0.181422	−	0.983405i	$-0.558070\pi$
$97$	−3.57360	−0.362844	−0.181422	+	0.983405i	$0.558070\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.33549	0.630405	0.315202	−	0.949024i	$-0.397928\pi$
$101$	6.33549	0.630405	0.315202	+	0.949024i	$0.397928\pi$
$102$	0	0
$103$	−5.44693	−0.536702	−0.268351	−	0.963321i	$-0.586479\pi$
$103$	−5.44693	−0.536702	−0.268351	+	0.963321i	$0.586479\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.08387	−0.298129	−0.149065	−	0.988827i	$-0.547626\pi$
$107$	−3.08387	−0.298129	−0.149065	+	0.988827i	$0.547626\pi$
$108$	0	0
$109$	8.10441	0.776262	0.388131	−	0.921604i	$-0.373121\pi$
$109$	8.10441	0.776262	0.388131	+	0.921604i	$0.373121\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.3355	−1.34857	−0.674285	−	0.738471i	$-0.735548\pi$
$113$	−14.3355	−1.34857	−0.674285	+	0.738471i	$0.735548\pi$
$114$	0	0
$115$	6.04107	0.563333
$116$	0	0
$117$	0	0
$118$	0	0
$119$	9.87067	0.904843
$120$	0	0
$121$	−9.52815	−0.866195
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.38094	0.660172
$126$	0	0
$127$	−3.74135	−0.331991	−0.165995	−	0.986127i	$-0.553084\pi$
$127$	−3.74135	−0.331991	−0.165995	+	0.986127i	$0.553084\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.9545	1.13184	0.565922	−	0.824459i	$-0.308520\pi$
$131$	12.9545	1.13184	0.565922	+	0.824459i	$0.308520\pi$
$132$	0	0
$133$	−4.29707	−0.372603
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.63256	0.566658	0.283329	−	0.959023i	$-0.408561\pi$
$137$	6.63256	0.566658	0.283329	+	0.959023i	$0.408561\pi$
$138$	0	0
$139$	−4.23374	−0.359101	−0.179550	−	0.983749i	$-0.557464\pi$
$139$	−4.23374	−0.359101	−0.179550	+	0.983749i	$0.557464\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.16774	0.515773
$144$	0	0
$145$	−0.385321	−0.0319992
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.63959	0.789706	0.394853	−	0.918744i	$-0.370795\pi$
$149$	9.63959	0.789706	0.394853	+	0.918744i	$0.370795\pi$
$150$	0	0
$151$	−5.44693	−0.443265	−0.221633	−	0.975130i	$-0.571139\pi$
$151$	−5.44693	−0.443265	−0.221633	+	0.975130i	$0.571139\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−32.9930	−2.60021
$162$	0	0
$163$	−16.3355	−1.27949	−0.639747	−	0.768585i	$-0.720961\pi$
$163$	−16.3355	−1.27949	−0.639747	+	0.768585i	$0.720961\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.5736	1.05036	0.525178	−	0.850992i	$-0.323999\pi$
$167$	13.5736	1.05036	0.525178	+	0.850992i	$0.323999\pi$
$168$	0	0
$169$	12.8458	0.988135
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.3355	−1.39402	−0.697011	−	0.717061i	$-0.745487\pi$
$173$	−18.3355	−1.39402	−0.697011	+	0.717061i	$0.745487\pi$
$174$	0	0
$175$	−18.8252	−1.42305
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−19.9502	−1.49115	−0.745573	−	0.666424i	$-0.767824\pi$
$179$	−19.9502	−1.49115	−0.745573	+	0.666424i	$0.767824\pi$
$180$	0	0
$181$	−6.85279	−0.509364	−0.254682	−	0.967025i	$-0.581971\pi$
$181$	−6.85279	−0.509364	−0.254682	+	0.967025i	$0.581971\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.57360	0.115694
$186$	0	0
$187$	2.78680	0.203791
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.7798	−1.43121	−0.715607	−	0.698503i	$-0.753850\pi$
$191$	−19.7798	−1.43121	−0.715607	+	0.698503i	$0.753850\pi$
$192$	0	0
$193$	−9.61468	−0.692080	−0.346040	−	0.938220i	$-0.612474\pi$
$193$	−9.61468	−0.692080	−0.346040	+	0.938220i	$0.612474\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.16774	−0.296940	−0.148470	−	0.988917i	$-0.547435\pi$
$197$	−4.16774	−0.296940	−0.148470	+	0.988917i	$0.547435\pi$
$198$	0	0
$199$	5.48535	0.388846	0.194423	−	0.980918i	$-0.437717\pi$
$199$	5.48535	0.388846	0.194423	+	0.980918i	$0.437717\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.10441	0.147701
$204$	0	0
$205$	−3.27919	−0.229029
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.21320	−0.0839187
$210$	0	0
$211$	−1.89559	−0.130498	−0.0652489	−	0.997869i	$-0.520784\pi$
$211$	−1.89559	−0.130498	−0.0652489	+	0.997869i	$0.520784\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.1927	−0.695134
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	11.6780	0.785548
$222$	0	0
$223$	3.74135	0.250539	0.125270	−	0.992123i	$-0.460020\pi$
$223$	3.74135	0.250539	0.125270	+	0.992123i	$0.460020\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−19.6780	−1.30608	−0.653038	−	0.757325i	$-0.726506\pi$
$227$	−19.6780	−1.30608	−0.653038	+	0.757325i	$0.726506\pi$
$228$	0	0
$229$	21.7164	1.43506	0.717531	−	0.696526i	$-0.245272\pi$
$229$	21.7164	1.43506	0.717531	+	0.696526i	$0.245272\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.5692	0.692413	0.346206	−	0.938158i	$-0.387470\pi$
$233$	10.5692	0.692413	0.346206	+	0.938158i	$0.387470\pi$
$234$	0	0
$235$	−4.56923	−0.298063
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.0384	1.03744	0.518720	−	0.854944i	$-0.326409\pi$
$239$	16.0384	1.03744	0.518720	+	0.854944i	$0.326409\pi$
$240$	0	0
$241$	−17.0205	−1.09639	−0.548195	−	0.836351i	$-0.684685\pi$
$241$	−17.0205	−1.09639	−0.548195	+	0.836351i	$0.684685\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−9.02054	−0.576301
$246$	0	0
$247$	−5.08387	−0.323479
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.1223	1.20699	0.603494	−	0.797367i	$-0.293775\pi$
$251$	19.1223	1.20699	0.603494	+	0.797367i	$0.293775\pi$
$252$	0	0
$253$	−9.31495	−0.585626
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.553066	0.0344993	0.0172497	−	0.999851i	$-0.494509\pi$
$257$	0.553066	0.0344993	0.0172497	+	0.999851i	$0.494509\pi$
$258$	0	0
$259$	−8.59414	−0.534014
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.04545	−0.434441	−0.217221	−	0.976123i	$-0.569699\pi$
$263$	−7.04545	−0.434441	−0.217221	+	0.976123i	$0.569699\pi$
$264$	0	0
$265$	1.52377	0.0936045
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−12.7619	−0.778106	−0.389053	−	0.921215i	$-0.627198\pi$
$269$	−12.7619	−0.778106	−0.389053	+	0.921215i	$0.627198\pi$
$270$	0	0
$271$	15.6780	0.952371	0.476186	−	0.879345i	$-0.342019\pi$
$271$	15.6780	0.952371	0.476186	+	0.879345i	$0.342019\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.31495	−0.320504
$276$	0	0
$277$	−5.97508	−0.359008	−0.179504	−	0.983757i	$-0.557449\pi$
$277$	−5.97508	−0.359008	−0.179504	+	0.983757i	$0.557449\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−30.2088	−1.80211	−0.901054	−	0.433708i	$-0.857205\pi$
$281$	−30.2088	−1.80211	−0.901054	+	0.433708i	$0.857205\pi$
$282$	0	0
$283$	−1.21320	−0.0721171	−0.0360586	−	0.999350i	$-0.511480\pi$
$283$	−1.21320	−0.0721171	−0.0360586	+	0.999350i	$0.511480\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	17.9091	1.05714
$288$	0	0
$289$	−11.7235	−0.689616
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−0.104410	−0.00609969	−0.00304984	−	0.999995i	$-0.500971\pi$
$293$	−0.104410	−0.00609969	−0.00304984	+	0.999995i	$0.500971\pi$
$294$	0	0
$295$	−8.72081	−0.507745
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−39.0340	−2.25740
$300$	0	0
$301$	55.6666	3.20857
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.23374	0.242423
$306$	0	0
$307$	0.852793	0.0486715	0.0243357	−	0.999704i	$-0.492253\pi$
$307$	0.852793	0.0486715	0.0243357	+	0.999704i	$0.492253\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.12933	0.347562	0.173781	−	0.984784i	$-0.444401\pi$
$311$	6.12933	0.347562	0.173781	+	0.984784i	$0.444401\pi$
$312$	0	0
$313$	−3.17478	−0.179449	−0.0897246	−	0.995967i	$-0.528599\pi$
$313$	−3.17478	−0.179449	−0.0897246	+	0.995967i	$0.528599\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.6986	1.16255	0.581273	−	0.813708i	$-0.302555\pi$
$317$	20.6986	1.16255	0.581273	+	0.813708i	$0.302555\pi$
$318$	0	0
$319$	0.594141	0.0332655
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.29707	−0.127812
$324$	0	0
$325$	−22.2722	−1.23544
$326$	0	0
$327$	0	0
$328$	0	0
$329$	24.9545	1.37579
$330$	0	0
$331$	9.84576	0.541172	0.270586	−	0.962696i	$-0.412783\pi$
$331$	9.84576	0.541172	0.270586	+	0.962696i	$0.412783\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.95893	−0.107028
$336$	0	0
$337$	−18.9296	−1.03116	−0.515581	−	0.856841i	$-0.672424\pi$
$337$	−18.9296	−1.03116	−0.515581	+	0.856841i	$0.672424\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.1856	1.03593
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.78680	0.364335	0.182167	−	0.983268i	$-0.441689\pi$
$347$	6.78680	0.364335	0.182167	+	0.983268i	$0.441689\pi$
$348$	0	0
$349$	6.23374	0.333684	0.166842	−	0.985984i	$-0.446643\pi$
$349$	6.23374	0.333684	0.166842	+	0.985984i	$0.446643\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−31.7191	−1.68824	−0.844118	−	0.536157i	$-0.819876\pi$
$353$	−31.7191	−1.68824	−0.844118	+	0.536157i	$0.819876\pi$
$354$	0	0
$355$	−9.23811	−0.490308
$356$	0	0
$357$	0	0
$358$	0	0
$359$	23.3944	1.23471	0.617356	−	0.786684i	$-0.288204\pi$
$359$	23.3944	1.23471	0.617356	+	0.786684i	$0.288204\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.66013	0.348607
$366$	0	0
$367$	−20.3355	−1.06150	−0.530752	−	0.847527i	$-0.678090\pi$
$367$	−20.3355	−1.06150	−0.530752	+	0.847527i	$0.678090\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.32199	−0.432056
$372$	0	0
$373$	−36.9021	−1.91072	−0.955358	−	0.295450i	$-0.904530\pi$
$373$	−36.9021	−1.91072	−0.955358	+	0.295450i	$0.904530\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.48973	0.128228
$378$	0	0
$379$	−15.9367	−0.818611	−0.409306	−	0.912397i	$-0.634229\pi$
$379$	−15.9367	−0.818611	−0.409306	+	0.912397i	$0.634229\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.0205	0.971904	0.485952	−	0.873985i	$-0.338473\pi$
$383$	19.0205	0.971904	0.485952	+	0.873985i	$0.338473\pi$
$384$	0	0
$385$	−4.10175	−0.209045
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.61906	−0.132791	−0.0663957	−	0.997793i	$-0.521150\pi$
$389$	−2.61906	−0.132791	−0.0663957	+	0.997793i	$0.521150\pi$
$390$	0	0
$391$	−17.6369	−0.891938
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.44162	−0.0725359
$396$	0	0
$397$	−31.1634	−1.56404	−0.782022	−	0.623251i	$-0.785812\pi$
$397$	−31.1634	−1.56404	−0.782022	+	0.623251i	$0.785812\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−21.7413	−1.08571	−0.542856	−	0.839826i	$-0.682657\pi$
$401$	−21.7413	−1.08571	−0.542856	+	0.839826i	$0.682657\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.42640	−0.120272
$408$	0	0
$409$	−23.2651	−1.15039	−0.575193	−	0.818018i	$-0.695073\pi$
$409$	−23.2651	−1.15039	−0.575193	+	0.818018i	$0.695073\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	47.6282	2.34363
$414$	0	0
$415$	−5.52377	−0.271151
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−8.33549	−0.407215	−0.203608	−	0.979053i	$-0.565267\pi$
$419$	−8.33549	−0.407215	−0.203608	+	0.979053i	$0.565267\pi$
$420$	0	0
$421$	0.748383	0.0364740	0.0182370	−	0.999834i	$-0.494195\pi$
$421$	0.748383	0.0364740	0.0182370	+	0.999834i	$0.494195\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−10.0633	−0.488143
$426$	0	0
$427$	−23.1223	−1.11897
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−31.3560	−1.51037	−0.755183	−	0.655514i	$-0.772452\pi$
$431$	−31.3560	−1.51037	−0.755183	+	0.655514i	$0.772452\pi$
$432$	0	0
$433$	9.48270	0.455709	0.227855	−	0.973695i	$-0.426829\pi$
$433$	9.48270	0.455709	0.227855	+	0.973695i	$0.426829\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.67801	0.367289
$438$	0	0
$439$	−31.9502	−1.52490	−0.762449	−	0.647048i	$-0.776003\pi$
$439$	−31.9502	−1.52490	−0.762449	+	0.647048i	$0.776003\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.91878	−0.138676	−0.0693378	−	0.997593i	$-0.522089\pi$
$443$	−2.91878	−0.138676	−0.0693378	+	0.997593i	$0.522089\pi$
$444$	0	0
$445$	10.8117	0.512525
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−0.0909069	−0.00429016	−0.00214508	−	0.999998i	$-0.500683\pi$
$449$	−0.0909069	−0.00429016	−0.00214508	+	0.999998i	$0.500683\pi$
$450$	0	0
$451$	5.05630	0.238092
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−17.1883	−0.805799
$456$	0	0
$457$	19.1499	0.895793	0.447896	−	0.894085i	$-0.352173\pi$
$457$	19.1499	0.895793	0.447896	+	0.894085i	$0.352173\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−33.9253	−1.58006	−0.790028	−	0.613070i	$-0.789934\pi$
$461$	−33.9253	−1.58006	−0.790028	+	0.613070i	$0.789934\pi$
$462$	0	0
$463$	27.3809	1.27250	0.636250	−	0.771483i	$-0.280485\pi$
$463$	27.3809	1.27250	0.636250	+	0.771483i	$0.280485\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.2132	−0.796532	−0.398266	−	0.917270i	$-0.630388\pi$
$467$	−17.2132	−0.796532	−0.398266	+	0.917270i	$0.630388\pi$
$468$	0	0
$469$	10.6986	0.494013
$470$	0	0
$471$	0	0
$472$	0	0
$473$	15.7164	0.722642
$474$	0	0
$475$	4.38094	0.201011
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.4827	−0.707422	−0.353711	−	0.935355i	$-0.615080\pi$
$479$	−15.4827	−0.707422	−0.353711	+	0.935355i	$0.615080\pi$
$480$	0	0
$481$	−10.1677	−0.463609
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.81172	0.127674
$486$	0	0
$487$	30.1179	1.36477	0.682386	−	0.730992i	$-0.260942\pi$
$487$	30.1179	1.36477	0.682386	+	0.730992i	$0.260942\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.2944	0.825615	0.412808	−	0.910818i	$-0.364548\pi$
$491$	18.2944	0.825615	0.412808	+	0.910818i	$0.364548\pi$
$492$	0	0
$493$	1.12495	0.0506651
$494$	0	0
$495$	0	0
$496$	0	0
$497$	50.4534	2.26314
$498$	0	0
$499$	10.5282	0.471305	0.235652	−	0.971837i	$-0.424277\pi$
$499$	10.5282	0.471305	0.235652	+	0.971837i	$0.424277\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−37.2018	−1.65875	−0.829373	−	0.558696i	$-0.811302\pi$
$503$	−37.2018	−1.65875	−0.829373	+	0.558696i	$0.811302\pi$
$504$	0	0
$505$	−4.98477	−0.221820
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20.4264	0.905384	0.452692	−	0.891667i	$-0.350464\pi$
$509$	20.4264	0.905384	0.452692	+	0.891667i	$0.350464\pi$
$510$	0	0
$511$	−36.3739	−1.60909
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.28566	0.188849
$516$	0	0
$517$	7.04545	0.309859
$518$	0	0
$519$	0	0
$520$	0	0
$521$	37.8235	1.65708	0.828539	−	0.559932i	$-0.189173\pi$
$521$	37.8235	1.65708	0.828539	+	0.559932i	$0.189173\pi$
$522$	0	0
$523$	13.0428	0.570322	0.285161	−	0.958480i	$-0.407953\pi$
$523$	13.0428	0.570322	0.285161	+	0.958480i	$0.407953\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	35.9519	1.56313
$530$	0	0
$531$	0	0
$532$	0	0
$533$	21.1883	0.917766
$534$	0	0
$535$	2.42640	0.104902
$536$	0	0
$537$	0	0
$538$	0	0
$539$	13.9091	0.599107
$540$	0	0
$541$	−27.6754	−1.18986	−0.594928	−	0.803779i	$-0.702820\pi$
$541$	−27.6754	−1.18986	−0.594928	+	0.803779i	$0.702820\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.37656	−0.273142
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.489731	−0.0208633
$552$	0	0
$553$	7.87333	0.334808
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.2695	1.53679	0.768394	−	0.639977i	$-0.221056\pi$
$557$	36.2695	1.53679	0.768394	+	0.639977i	$0.221056\pi$
$558$	0	0
$559$	65.8593	2.78555
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13.9091	−0.586198	−0.293099	−	0.956082i	$-0.594687\pi$
$563$	−13.9091	−0.586198	−0.293099	+	0.956082i	$0.594687\pi$
$564$	0	0
$565$	11.2792	0.474519
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−29.7413	−1.24682	−0.623411	−	0.781894i	$-0.714254\pi$
$569$	−29.7413	−1.24682	−0.623411	+	0.781894i	$0.714254\pi$
$570$	0	0
$571$	−36.6710	−1.53463	−0.767316	−	0.641269i	$-0.778408\pi$
$571$	−36.6710	−1.53463	−0.767316	+	0.641269i	$0.778408\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	33.6369	1.40276
$576$	0	0
$577$	20.4648	0.851961	0.425981	−	0.904732i	$-0.359929\pi$
$577$	20.4648	0.851961	0.425981	+	0.904732i	$0.359929\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	30.1677	1.25157
$582$	0	0
$583$	−2.34956	−0.0973088
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.2748	−0.754282	−0.377141	−	0.926156i	$-0.623093\pi$
$587$	−18.2748	−0.754282	−0.377141	+	0.926156i	$0.623093\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−31.5238	−1.29453	−0.647263	−	0.762267i	$-0.724086\pi$
$593$	−31.5238	−1.29453	−0.647263	+	0.762267i	$0.724086\pi$
$594$	0	0
$595$	−7.76626	−0.318386
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26.3766	1.07772	0.538859	−	0.842396i	$-0.318856\pi$
$599$	26.3766	1.07772	0.538859	+	0.842396i	$0.318856\pi$
$600$	0	0
$601$	−27.3971	−1.11755	−0.558776	−	0.829319i	$-0.688729\pi$
$601$	−27.3971	−1.11755	−0.558776	+	0.829319i	$0.688729\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.49677	0.304787
$606$	0	0
$607$	33.9859	1.37945	0.689723	−	0.724073i	$-0.257732\pi$
$607$	33.9859	1.37945	0.689723	+	0.724073i	$0.257732\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	29.5238	1.19440
$612$	0	0
$613$	3.08653	0.124664	0.0623319	−	0.998055i	$-0.480146\pi$
$613$	3.08653	0.124664	0.0623319	+	0.998055i	$0.480146\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.1018	1.05082	0.525409	−	0.850850i	$-0.323913\pi$
$617$	26.1018	1.05082	0.525409	+	0.850850i	$0.323913\pi$
$618$	0	0
$619$	29.0616	1.16808	0.584042	−	0.811723i	$-0.301470\pi$
$619$	29.0616	1.16808	0.584042	+	0.811723i	$0.301470\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−59.0475	−2.36569
$624$	0	0
$625$	16.0974	0.643895
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.59414	−0.183180
$630$	0	0
$631$	20.1018	0.800238	0.400119	−	0.916463i	$-0.368969\pi$
$631$	20.1018	0.800238	0.400119	+	0.916463i	$0.368969\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.94370	0.116817
$636$	0	0
$637$	58.2857	2.30936
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−24.7619	−0.978036	−0.489018	−	0.872274i	$-0.662645\pi$
$641$	−24.7619	−0.978036	−0.489018	+	0.872274i	$0.662645\pi$
$642$	0	0
$643$	−11.8431	−0.467046	−0.233523	−	0.972351i	$-0.575025\pi$
$643$	−11.8431	−0.467046	−0.233523	+	0.972351i	$0.575025\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−37.4854	−1.47370	−0.736851	−	0.676056i	$-0.763688\pi$
$647$	−37.4854	−1.47370	−0.736851	+	0.676056i	$0.763688\pi$
$648$	0	0
$649$	13.4469	0.527838
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.8279	1.36292	0.681460	−	0.731855i	$-0.261345\pi$
$653$	34.8279	1.36292	0.681460	+	0.731855i	$0.261345\pi$
$654$	0	0
$655$	−10.1927	−0.398260
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−28.2722	−1.10133	−0.550663	−	0.834727i	$-0.685625\pi$
$659$	−28.2722	−1.10133	−0.550663	+	0.834727i	$0.685625\pi$
$660$	0	0
$661$	2.58064	0.100375	0.0501876	−	0.998740i	$-0.484018\pi$
$661$	2.58064	0.100375	0.0501876	+	0.998740i	$0.484018\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.38094	0.131107
$666$	0	0
$667$	−3.76016	−0.145594
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.52815	−0.252016
$672$	0	0
$673$	50.5443	1.94834	0.974170	−	0.225816i	$-0.0725048\pi$
$673$	50.5443	1.94834	0.974170	+	0.225816i	$0.0725048\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−0.825221	−0.0317158	−0.0158579	−	0.999874i	$-0.505048\pi$
$677$	−0.825221	−0.0317158	−0.0158579	+	0.999874i	$0.505048\pi$
$678$	0	0
$679$	−15.3560	−0.589310
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	−5.21851	−0.199389
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−9.84576	−0.375094
$690$	0	0
$691$	0.0249160	0.000947850 0	0.000473925	−	1.00000i	$-0.499849\pi$
$691$	0.0249160	0.000947850 0	0.000473925		1.00000i	$0.499849\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.33111	0.126356
$696$	0	0
$697$	9.57360	0.362626
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−29.3560	−1.10876	−0.554381	−	0.832263i	$-0.687045\pi$
$701$	−29.3560	−1.10876	−0.554381	+	0.832263i	$0.687045\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	0	0
$706$	0	0
$707$	27.2240	1.02387
$708$	0	0
$709$	−17.4827	−0.656576	−0.328288	−	0.944578i	$-0.606472\pi$
$709$	−17.4827	−0.656576	−0.328288	+	0.944578i	$0.606472\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−4.85279	−0.181484
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−18.5915	−0.693345	−0.346673	−	0.937986i	$-0.612689\pi$
$719$	−18.5915	−0.693345	−0.346673	+	0.937986i	$0.612689\pi$
$720$	0	0
$721$	−23.4059	−0.871680
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.14548	−0.0796813
$726$	0	0
$727$	−8.50589	−0.315466	−0.157733	−	0.987482i	$-0.550419\pi$
$727$	−8.50589	−0.315466	−0.157733	+	0.987482i	$0.550419\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	29.7575	1.10062
$732$	0	0
$733$	24.5032	0.905048	0.452524	−	0.891752i	$-0.350524\pi$
$733$	24.5032	0.905048	0.452524	+	0.891752i	$0.350524\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.02054	0.111263
$738$	0	0
$739$	−20.9047	−0.768992	−0.384496	−	0.923127i	$-0.625625\pi$
$739$	−20.9047	−0.768992	−0.384496	+	0.923127i	$0.625625\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21.5238	−0.789631	−0.394815	−	0.918761i	$-0.629191\pi$
$743$	−21.5238	−0.789631	−0.394815	+	0.918761i	$0.629191\pi$
$744$	0	0
$745$	−7.58445	−0.277873
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−13.2516	−0.484204
$750$	0	0
$751$	2.37656	0.0867221	0.0433610	−	0.999059i	$-0.486193\pi$
$751$	2.37656	0.0867221	0.0433610	+	0.999059i	$0.486193\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.28566	0.155971
$756$	0	0
$757$	46.9545	1.70659	0.853296	−	0.521427i	$-0.174600\pi$
$757$	46.9545	1.70659	0.853296	+	0.521427i	$0.174600\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.44428	0.197355	0.0986775	−	0.995119i	$-0.468539\pi$
$761$	5.44428	0.197355	0.0986775	+	0.995119i	$0.468539\pi$
$762$	0	0
$763$	34.8252	1.26076
$764$	0	0
$765$	0	0
$766$	0	0
$767$	56.3490	2.03464
$768$	0	0
$769$	8.92697	0.321915	0.160957	−	0.986961i	$-0.448542\pi$
$769$	8.92697	0.321915	0.160957	+	0.986961i	$0.448542\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	17.8047	0.640390	0.320195	−	0.947352i	$-0.396252\pi$
$773$	17.8047	0.640390	0.320195	+	0.947352i	$0.396252\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.16774	−0.149325
$780$	0	0
$781$	14.2446	0.509711
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−11.0152	−0.393150
$786$	0	0
$787$	13.3836	0.477074	0.238537	−	0.971133i	$-0.423332\pi$
$787$	13.3836	0.477074	0.238537	+	0.971133i	$0.423332\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−61.6006	−2.19027
$792$	0	0
$793$	−27.3560	−0.971441
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28.5666	−1.01188	−0.505940	−	0.862569i	$-0.668854\pi$
$797$	−28.5666	−1.01188	−0.505940	+	0.862569i	$0.668854\pi$
$798$	0	0
$799$	13.3399	0.471931
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.2695	−0.362403
$804$	0	0
$805$	25.9589	0.914932
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−45.2625	−1.59134	−0.795672	−	0.605728i	$-0.792882\pi$
$809$	−45.2625	−1.59134	−0.795672	+	0.605728i	$0.792882\pi$
$810$	0	0
$811$	−49.4551	−1.73660	−0.868302	−	0.496036i	$-0.834789\pi$
$811$	−49.4551	−1.73660	−0.868302	+	0.496036i	$0.834789\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12.8528	0.450214
$816$	0	0
$817$	−12.9545	−0.453222
$818$	0	0
$819$	0	0
$820$	0	0
$821$	41.7933	1.45860	0.729298	−	0.684197i	$-0.239847\pi$
$821$	41.7933	1.45860	0.729298	+	0.684197i	$0.239847\pi$
$822$	0	0
$823$	−8.68239	−0.302649	−0.151325	−	0.988484i	$-0.548354\pi$
$823$	−8.68239	−0.302649	−0.151325	+	0.988484i	$0.548354\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13.1196	−0.456214	−0.228107	−	0.973636i	$-0.573254\pi$
$827$	−13.1196	−0.456214	−0.228107	+	0.973636i	$0.573254\pi$
$828$	0	0
$829$	−8.83053	−0.306697	−0.153349	−	0.988172i	$-0.549006\pi$
$829$	−8.83053	−0.306697	−0.153349	+	0.988172i	$0.549006\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	26.3355	0.912471
$834$	0	0
$835$	−10.6797	−0.369588
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4.80296	−0.165817	−0.0829083	−	0.996557i	$-0.526421\pi$
$839$	−4.80296	−0.165817	−0.0829083	+	0.996557i	$0.526421\pi$
$840$	0	0
$841$	−28.7602	−0.991730
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10.1071	−0.347694
$846$	0	0
$847$	−40.9431	−1.40682
$848$	0	0
$849$	0	0
$850$	0	0
$851$	15.3560	0.526398
$852$	0	0
$853$	−2.51730	−0.0861908	−0.0430954	−	0.999071i	$-0.513722\pi$
$853$	−2.51730	−0.0861908	−0.0430954	+	0.999071i	$0.513722\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37.3560	1.27606	0.638029	−	0.770013i	$-0.279750\pi$
$857$	37.3560	1.27606	0.638029	+	0.770013i	$0.279750\pi$
$858$	0	0
$859$	40.6461	1.38683	0.693413	−	0.720540i	$-0.256106\pi$
$859$	40.6461	1.38683	0.693413	+	0.720540i	$0.256106\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.40586	0.252098	0.126049	−	0.992024i	$-0.459770\pi$
$863$	7.40586	0.252098	0.126049	+	0.992024i	$0.459770\pi$
$864$	0	0
$865$	14.4264	0.490512
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.22289	0.0754063
$870$	0	0
$871$	12.6575	0.428882
$872$	0	0
$873$	0	0
$874$	0	0
$875$	31.7164	1.07221
$876$	0	0
$877$	50.5308	1.70630	0.853152	−	0.521662i	$-0.174688\pi$
$877$	50.5308	1.70630	0.853152	+	0.521662i	$0.174688\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−33.4631	−1.12740	−0.563700	−	0.825980i	$-0.690623\pi$
$881$	−33.4631	−1.12740	−0.563700	+	0.825980i	$0.690623\pi$
$882$	0	0
$883$	30.0162	1.01012	0.505062	−	0.863083i	$-0.331470\pi$
$883$	30.0162	1.01012	0.505062	+	0.863083i	$0.331470\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.0704	−0.640320	−0.320160	−	0.947363i	$-0.603737\pi$
$887$	−19.0704	−0.640320	−0.320160	+	0.947363i	$0.603737\pi$
$888$	0	0
$889$	−16.0768	−0.539200
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.80734	−0.194335
$894$	0	0
$895$	15.6968	0.524687
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−4.44866	−0.148206
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.39179	0.179229
$906$	0	0
$907$	−48.0135	−1.59426	−0.797131	−	0.603806i	$-0.793650\pi$
$907$	−48.0135	−1.59426	−0.797131	+	0.603806i	$0.793650\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	46.2446	1.53215	0.766076	−	0.642750i	$-0.222207\pi$
$911$	46.2446	1.53215	0.766076	+	0.642750i	$0.222207\pi$
$912$	0	0
$913$	8.51730	0.281882
$914$	0	0
$915$	0	0
$916$	0	0
$917$	55.6666	1.83827
$918$	0	0
$919$	41.5103	1.36930	0.684649	−	0.728873i	$-0.259956\pi$
$919$	41.5103	1.36930	0.684649	+	0.728873i	$0.259956\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	59.6915	1.96477
$924$	0	0
$925$	8.76189	0.288089
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−55.4551	−1.81942	−0.909712	−	0.415240i	$-0.863698\pi$
$929$	−55.4551	−1.81942	−0.909712	+	0.415240i	$0.863698\pi$
$930$	0	0
$931$	−11.4648	−0.375744
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.19266	−0.0717077
$936$	0	0
$937$	−6.08825	−0.198894	−0.0994472	−	0.995043i	$-0.531707\pi$
$937$	−6.08825	−0.198894	−0.0994472	+	0.995043i	$0.531707\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11.3836	0.371095	0.185547	−	0.982635i	$-0.440594\pi$
$941$	11.3836	0.371095	0.185547	+	0.982635i	$0.440594\pi$
$942$	0	0
$943$	−32.0000	−1.04206
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38.3766	1.24707	0.623535	−	0.781795i	$-0.285696\pi$
$947$	38.3766	1.24707	0.623535	+	0.781795i	$0.285696\pi$
$948$	0	0
$949$	−43.0340	−1.39694
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−30.2857	−0.981049	−0.490524	−	0.871427i	$-0.663195\pi$
$953$	−30.2857	−0.981049	−0.490524	+	0.871427i	$0.663195\pi$
$954$	0	0
$955$	15.5628	0.503599
$956$	0	0
$957$	0	0
$958$	0	0
$959$	28.5006	0.920332
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7.56485	0.243521
$966$	0	0
$967$	−24.6710	−0.793365	−0.396683	−	0.917956i	$-0.629839\pi$
$967$	−24.6710	−0.793365	−0.396683	+	0.917956i	$0.629839\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	−18.1927	−0.583230
$974$	0	0
$975$	0	0
$976$	0	0
$977$	21.4059	0.684834	0.342417	−	0.939548i	$-0.388754\pi$
$977$	21.4059	0.684834	0.342417	+	0.939548i	$0.388754\pi$
$978$	0	0
$979$	−16.6710	−0.532807
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−38.1677	−1.21736	−0.608681	−	0.793415i	$-0.708301\pi$
$983$	−38.1677	−1.21736	−0.608681	+	0.793415i	$0.708301\pi$
$984$	0	0
$985$	3.27919	0.104484
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−99.4652	−3.16281
$990$	0	0
$991$	16.4675	0.523106	0.261553	−	0.965189i	$-0.415765\pi$
$991$	16.4675	0.523106	0.261553	+	0.965189i	$0.415765\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4.31589	−0.136823
$996$	0	0
$997$	13.8431	0.438415	0.219208	−	0.975678i	$-0.429653\pi$
$997$	13.8431	0.438415	0.219208	+	0.975678i	$0.429653\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.a.n.1.2		3
3.2	odd	2		152.2.a.c.1.1	✓	3
4.3	odd	2		2736.2.a.bd.1.2		3
12.11	even	2		304.2.a.g.1.3		3
15.2	even	4		3800.2.d.j.3649.5		6
15.8	even	4		3800.2.d.j.3649.2		6
15.14	odd	2		3800.2.a.r.1.3		3
21.20	even	2		7448.2.a.bf.1.3		3
24.5	odd	2		1216.2.a.u.1.3		3
24.11	even	2		1216.2.a.v.1.1		3
57.56	even	2		2888.2.a.o.1.3		3
60.59	even	2		7600.2.a.bv.1.1		3
228.227	odd	2		5776.2.a.bp.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.a.c.1.1	✓	3	3.2	odd	2
304.2.a.g.1.3		3	12.11	even	2
1216.2.a.u.1.3		3	24.5	odd	2
1216.2.a.v.1.1		3	24.11	even	2
1368.2.a.n.1.2		3	1.1	even	1	trivial
2736.2.a.bd.1.2		3	4.3	odd	2
2888.2.a.o.1.3		3	57.56	even	2
3800.2.a.r.1.3		3	15.14	odd	2
3800.2.d.j.3649.2		6	15.8	even	4
3800.2.d.j.3649.5		6	15.2	even	4
5776.2.a.bp.1.1		3	228.227	odd	2
7448.2.a.bf.1.3		3	21.20	even	2
7600.2.a.bv.1.1		3	60.59	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.a (trivial)

\(f(q)\)	\(=\)	\(q-0.786802 q^{5} +4.29707 q^{7} +1.21320 q^{11} +5.08387 q^{13} +2.29707 q^{17} -1.00000 q^{19} -7.67801 q^{23} -4.38094 q^{25} +0.489731 q^{29} -3.38094 q^{35} -2.00000 q^{37} +4.16774 q^{41} +12.9545 q^{43} +5.80734 q^{47} +11.4648 q^{49} -1.93667 q^{53} -0.954547 q^{55} +11.0839 q^{59} -5.38094 q^{61} -4.00000 q^{65} +2.48973 q^{67} +11.7413 q^{71} -8.46482 q^{73} +5.21320 q^{77} +1.83226 q^{79} +7.02054 q^{83} -1.80734 q^{85} -13.7413 q^{89} +21.8458 q^{91} +0.786802 q^{95} -3.57360 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{5} + 4 q^{7} + 5 q^{11} + 5 q^{13} - 2 q^{17} - 3 q^{19} + 5 q^{23} + 6 q^{25} + 9 q^{29} + 9 q^{35} - 6 q^{37} - 8 q^{41} + 17 q^{43} + q^{47} + 5 q^{49} - q^{53} + 19 q^{55} + 23 q^{59} + 3 q^{61}+ \cdots - 8 q^{97}+O(q^{100}) \) 3 * q - q^5 + 4 * q^7 + 5 * q^11 + 5 * q^13 - 2 * q^17 - 3 * q^19 + 5 * q^23 + 6 * q^25 + 9 * q^29 + 9 * q^35 - 6 * q^37 - 8 * q^41 + 17 * q^43 + q^47 + 5 * q^49 - q^53 + 19 * q^55 + 23 * q^59 + 3 * q^61 - 12 * q^65 + 15 * q^67 + 12 * q^71 + 4 * q^73 + 17 * q^77 + 26 * q^79 + 6 * q^83 + 11 * q^85 - 18 * q^89 + 17 * q^91 + q^95 - 8 * q^97

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