LMFDB - Embedded newform 2268.4.f.a.1133.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,4,Mod(1133,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2268.1133");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2268.f (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$133.816331893$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1133.8
Character	$\chi$	$=$	2268.1133
Dual form			2268.4.f.a.1133.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.6490 q^{5} +(-3.87814 + 18.1097i) q^{7} +O(q^{10})$	$q-15.6490 q^{5} +(-3.87814 + 18.1097i) q^{7} -39.5146i q^{11} +64.1226i q^{13} +56.6997 q^{17} -117.055i q^{19} -7.61632i q^{23} +119.892 q^{25} -45.9831i q^{29} +290.942i q^{31} +(60.6891 - 283.399i) q^{35} -335.540 q^{37} -194.600 q^{41} -304.528 q^{43} -636.058 q^{47} +(-312.920 - 140.464i) q^{49} -274.953i q^{53} +618.365i q^{55} +517.741 q^{59} -164.360i q^{61} -1003.46i q^{65} -736.206 q^{67} +599.619i q^{71} -214.435i q^{73} +(715.595 + 153.243i) q^{77} +908.690 q^{79} +779.846 q^{83} -887.295 q^{85} -443.089 q^{89} +(-1161.24 - 248.676i) q^{91} +1831.79i q^{95} -389.951i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 12 q^{7}+O(q^{10}) $ 48 * q - 12 * q^7	$ 48 q - 12 q^{7} + 1200 q^{25} + 336 q^{37} - 168 q^{43} - 636 q^{49} + 1176 q^{67} - 408 q^{79} + 720 q^{85} - 1080 q^{91}+O(q^{100}) $ 48 * q - 12 * q^7 + 1200 * q^25 + 336 * q^37 - 168 * q^43 - 636 * q^49 + 1176 * q^67 - 408 * q^79 + 720 * q^85 - 1080 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	−15.6490			−1.39969			−0.699846	−	0.714294i	$-0.746748\pi$
$5$	−15.6490			−1.39969			−0.699846	+	0.714294i	$0.746748\pi$
$6$		0			0
$7$	−3.87814	+	18.1097i	−0.209400	+	0.977830i
$7$	−3.87814	+	18.1097i	−0.209400	+	0.977830i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	39.5146i		−	1.08310i	−0.840669	−	0.541549i	$-0.817838\pi$
$11$		−	39.5146i		−	1.08310i	0.840669	−	0.541549i	$-0.182162\pi$
$12$		0			0
$13$			64.1226i			1.36803i	0.729467	+	0.684015i	$0.239768\pi$
$13$			64.1226i			1.36803i	−0.729467	+	0.684015i	$0.760232\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	56.6997			0.808923			0.404462	−	0.914555i	$-0.367459\pi$
$17$	56.6997			0.808923			0.404462	+	0.914555i	$0.367459\pi$
$18$		0			0
$19$		−	117.055i		−	1.41338i	−0.707525	−	0.706689i	$-0.750188\pi$
$19$		−	117.055i		−	1.41338i	0.707525	−	0.706689i	$-0.249812\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.61632i		−	0.0690484i	−0.999404	−	0.0345242i	$-0.989008\pi$
$23$		−	7.61632i		−	0.0690484i	0.999404	−	0.0345242i	$-0.0109916\pi$
$24$		0			0
$25$	119.892			0.959138
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	45.9831i		−	0.294443i	−0.989104	−	0.147222i	$-0.952967\pi$
$29$		−	45.9831i		−	0.294443i	0.989104	−	0.147222i	$-0.0470330\pi$
$30$		0			0
$31$			290.942i			1.68564i	0.538198	+	0.842819i	$0.319105\pi$
$31$			290.942i			1.68564i	−0.538198	+	0.842819i	$0.680895\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	60.6891	−	283.399i	0.293095	−	1.36866i
$36$		0			0
$37$	−335.540			−1.49088			−0.745438	−	0.666575i	$-0.767759\pi$
$37$	−335.540			−1.49088			−0.745438	+	0.666575i	$0.767759\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−194.600			−0.741254			−0.370627	−	0.928782i	$-0.620857\pi$
$41$	−194.600			−0.741254			−0.370627	+	0.928782i	$0.620857\pi$
$42$		0			0
$43$	−304.528			−1.08000			−0.540000	−	0.841665i	$-0.681576\pi$
$43$	−304.528			−1.08000			−0.540000	+	0.841665i	$0.681576\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−636.058			−1.97401			−0.987006	−	0.160683i	$-0.948630\pi$
$47$	−636.058			−1.97401			−0.987006	+	0.160683i	$0.948630\pi$
$48$		0			0
$49$	−312.920	−	140.464i	−0.912304	−	0.409515i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	274.953i		−	0.712597i	−0.934372	−	0.356299i	$-0.884039\pi$
$53$		−	274.953i		−	0.712597i	0.934372	−	0.356299i	$-0.115961\pi$
$54$		0			0
$55$			618.365i			1.51600i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	517.741			1.14244			0.571222	−	0.820796i	$-0.306470\pi$
$59$	517.741			1.14244			0.571222	+	0.820796i	$0.306470\pi$
$60$		0			0
$61$		−	164.360i		−	0.344986i	−0.985011	−	0.172493i	$-0.944818\pi$
$61$		−	164.360i		−	0.344986i	0.985011	−	0.172493i	$-0.0551823\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	1003.46i		−	1.91482i
$66$		0			0
$67$	−736.206			−1.34242			−0.671208	−	0.741269i	$-0.734224\pi$
$67$	−736.206			−1.34242			−0.671208	+	0.741269i	$0.734224\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			599.619i			1.00228i	0.865367	+	0.501139i	$0.167085\pi$
$71$			599.619i			1.00228i	−0.865367	+	0.501139i	$0.832915\pi$
$72$		0			0
$73$		−	214.435i		−	0.343804i	−0.985114	−	0.171902i	$-0.945009\pi$
$73$		−	214.435i		−	0.343804i	0.985114	−	0.171902i	$-0.0549913\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	715.595	+	153.243i	1.05909	+	0.226800i
$78$		0			0
$79$	908.690			1.29412			0.647061	−	0.762438i	$-0.275998\pi$
$79$	908.690			1.29412			0.647061	+	0.762438i	$0.275998\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	779.846			1.03132			0.515658	−	0.856795i	$-0.327548\pi$
$83$	779.846			1.03132			0.515658	+	0.856795i	$0.327548\pi$
$84$		0			0
$85$	−887.295			−1.13224
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−443.089			−0.527723			−0.263862	−	0.964561i	$-0.584996\pi$
$89$	−443.089			−0.527723			−0.263862	+	0.964561i	$0.584996\pi$
$90$		0			0
$91$	−1161.24	−	248.676i	−1.33770	−	0.286465i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			1831.79i			1.97829i
$96$		0			0
$97$		−	389.951i		−	0.408180i	−0.978952	−	0.204090i	$-0.934576\pi$
$97$		−	389.951i		−	0.408180i	0.978952	−	0.204090i	$-0.0654236\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1402.54			1.38176			0.690881	−	0.722969i	$-0.257223\pi$
$101$	1402.54			1.38176			0.690881	+	0.722969i	$0.257223\pi$
$102$		0			0
$103$		−	1062.22i		−	1.01615i	−0.861312	−	0.508077i	$-0.830357\pi$
$103$		−	1062.22i		−	1.01615i	0.861312	−	0.508077i	$-0.169643\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	216.811i		−	0.195887i	−0.995192	−	0.0979433i	$-0.968774\pi$
$107$		−	216.811i		−	0.195887i	0.995192	−	0.0979433i	$-0.0312264\pi$
$108$		0			0
$109$	−947.744			−0.832820			−0.416410	−	0.909177i	$-0.636712\pi$
$109$	−947.744			−0.832820			−0.416410	+	0.909177i	$0.636712\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1407.68i			1.17188i	0.810353	+	0.585942i	$0.199276\pi$
$113$			1407.68i			1.17188i	−0.810353	+	0.585942i	$0.800724\pi$
$114$		0			0
$115$			119.188i			0.0966464i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−219.889	+	1026.81i	−0.169388	+	0.790989i
$120$		0			0
$121$	−230.400			−0.173103
$122$		0			0
$123$		0			0
$124$		0			0
$125$	79.9306			0.0571937
$126$		0			0
$127$	1271.11			0.888131			0.444066	−	0.895994i	$-0.353536\pi$
$127$	1271.11			0.888131			0.444066	+	0.895994i	$0.353536\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1074.40			−0.716572			−0.358286	−	0.933612i	$-0.616639\pi$
$131$	−1074.40			−0.716572			−0.358286	+	0.933612i	$0.616639\pi$
$132$		0			0
$133$	2119.82	+	453.954i	1.38204	+	0.295961i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1504.73i		−	0.938377i	−0.883098	−	0.469188i	$-0.844547\pi$
$137$		−	1504.73i		−	0.938377i	0.883098	−	0.469188i	$-0.155453\pi$
$138$		0			0
$139$			852.515i			0.520211i	0.965580	+	0.260106i	$0.0837574\pi$
$139$			852.515i			0.520211i	−0.965580	+	0.260106i	$0.916243\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2533.77			1.48171
$144$		0			0
$145$			719.592i			0.412130i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	773.536i		−	0.425305i	−0.977128	−	0.212653i	$-0.931790\pi$
$149$		−	773.536i		−	0.425305i	0.977128	−	0.212653i	$-0.0682103\pi$
$150$		0			0
$151$	2517.76			1.35690			0.678451	−	0.734646i	$-0.262652\pi$
$151$	2517.76			1.35690			0.678451	+	0.734646i	$0.262652\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	4552.96i		−	2.35937i
$156$		0			0
$157$		−	73.0447i		−	0.0371312i	−0.999828	−	0.0185656i	$-0.994090\pi$
$157$		−	73.0447i		−	0.0371312i	0.999828	−	0.0185656i	$-0.00590996\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	137.929	+	29.5371i	0.0675176	+	0.0144587i
$162$		0			0
$163$	1604.80			0.771151			0.385576	−	0.922676i	$-0.374003\pi$
$163$	1604.80			0.771151			0.385576	+	0.922676i	$0.374003\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1333.09			0.617712			0.308856	−	0.951109i	$-0.400054\pi$
$167$	1333.09			0.617712			0.308856	+	0.951109i	$0.400054\pi$
$168$		0			0
$169$	−1914.70			−0.871509
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2469.86			1.08543			0.542717	−	0.839915i	$-0.317395\pi$
$173$	2469.86			1.08543			0.542717	+	0.839915i	$0.317395\pi$
$174$		0			0
$175$	−464.959	+	2171.21i	−0.200843	+	0.937874i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	4348.42i		−	1.81573i	−0.419261	−	0.907866i	$-0.637711\pi$
$179$		−	4348.42i		−	1.81573i	0.419261	−	0.907866i	$-0.362289\pi$
$180$		0			0
$181$		−	1523.82i		−	0.625770i	−0.949791	−	0.312885i	$-0.898705\pi$
$181$		−	1523.82i		−	0.625770i	0.949791	−	0.312885i	$-0.101295\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5250.87			2.08677
$186$		0			0
$187$		−	2240.46i		−	0.876144i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			3892.28i			1.47453i	0.675603	+	0.737266i	$0.263883\pi$
$191$			3892.28i			1.47453i	−0.675603	+	0.737266i	$0.736117\pi$
$192$		0			0
$193$	379.617			0.141583			0.0707914	−	0.997491i	$-0.477448\pi$
$193$	379.617			0.141583			0.0707914	+	0.997491i	$0.477448\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2916.71i			1.05486i	0.849599	+	0.527429i	$0.176844\pi$
$197$			2916.71i			1.05486i	−0.849599	+	0.527429i	$0.823156\pi$
$198$		0			0
$199$			2913.59i			1.03789i	0.854809	+	0.518943i	$0.173674\pi$
$199$			2913.59i			1.03789i	−0.854809	+	0.518943i	$0.826326\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	832.739	+	178.329i	0.287915	+	0.0616563i
$204$		0			0
$205$	3045.30			1.03753
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4625.36			−1.53083
$210$		0			0
$211$	1174.26			0.383123			0.191562	−	0.981481i	$-0.438645\pi$
$211$	1174.26			0.383123			0.191562	+	0.981481i	$0.438645\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4765.56			1.51167
$216$		0			0
$217$	−5268.87	−	1128.31i	−1.64827	−	0.352972i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			3635.73i			1.10663i
$222$		0			0
$223$			1860.47i			0.558684i	0.960192	+	0.279342i	$0.0901163\pi$
$223$			1860.47i			0.558684i	−0.960192	+	0.279342i	$0.909884\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2480.42			0.725247			0.362624	−	0.931936i	$-0.381881\pi$
$227$	2480.42			0.725247			0.362624	+	0.931936i	$0.381881\pi$
$228$		0			0
$229$		−	3323.85i		−	0.959154i	−0.877500	−	0.479577i	$-0.840790\pi$
$229$		−	3323.85i		−	0.959154i	0.877500	−	0.479577i	$-0.159210\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			2378.25i			0.668689i	0.942451	+	0.334344i	$0.108515\pi$
$233$			2378.25i			0.668689i	−0.942451	+	0.334344i	$0.891485\pi$
$234$		0			0
$235$	9953.69			2.76301
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	6070.93i		−	1.64308i	−0.570152	−	0.821539i	$-0.693116\pi$
$239$		−	6070.93i		−	1.64308i	0.570152	−	0.821539i	$-0.306884\pi$
$240$		0			0
$241$			3585.36i			0.958313i	0.877730	+	0.479156i	$0.159057\pi$
$241$			3585.36i			0.958313i	−0.877730	+	0.479156i	$0.840943\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4896.90	+	2198.12i	1.27694	+	0.573194i
$246$		0			0
$247$	7505.84			1.93354
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4359.22			−1.09622			−0.548111	−	0.836406i	$-0.684653\pi$
$251$	−4359.22			−1.09622			−0.548111	+	0.836406i	$0.684653\pi$
$252$		0			0
$253$	−300.955			−0.0747862
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3242.80			0.787083			0.393541	−	0.919307i	$-0.371250\pi$
$257$	3242.80			0.787083			0.393541	+	0.919307i	$0.371250\pi$
$258$		0			0
$259$	1301.27	−	6076.51i	0.312189	−	1.45782i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			349.079i			0.0818447i	0.999162	+	0.0409223i	$0.0130296\pi$
$263$			349.079i			0.0818447i	−0.999162	+	0.0409223i	$0.986970\pi$
$264$		0			0
$265$			4302.74i			0.997417i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7924.66			1.79619			0.898095	−	0.439802i	$-0.144952\pi$
$269$	7924.66			1.79619			0.898095	+	0.439802i	$0.144952\pi$
$270$		0			0
$271$		−	6601.56i		−	1.47976i	−0.672737	−	0.739882i	$-0.734881\pi$
$271$		−	6601.56i		−	1.47976i	0.672737	−	0.739882i	$-0.265119\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	4737.49i		−	1.03884i
$276$		0			0
$277$	−5737.02			−1.24442			−0.622210	−	0.782850i	$-0.713765\pi$
$277$	−5737.02			−1.24442			−0.622210	+	0.782850i	$0.713765\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			739.657i			0.157026i	0.996913	+	0.0785128i	$0.0250172\pi$
$281$			739.657i			0.157026i	−0.996913	+	0.0785128i	$0.974983\pi$
$282$		0			0
$283$			1578.71i			0.331606i	0.986159	+	0.165803i	$0.0530216\pi$
$283$			1578.71i			0.331606i	−0.986159	+	0.165803i	$0.946978\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	754.685	−	3524.14i	0.155218	−	0.724820i
$288$		0			0
$289$	−1698.15			−0.345643
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1995.42			0.397862			0.198931	−	0.980014i	$-0.436253\pi$
$293$	1995.42			0.397862			0.198931	+	0.980014i	$0.436253\pi$
$294$		0			0
$295$	−8102.15			−1.59907
$296$		0			0
$297$		0			0
$298$		0			0
$299$	488.378			0.0944603
$300$		0			0
$301$	1181.00	−	5514.89i	0.226152	−	1.05606i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			2572.08i			0.482875i
$306$		0			0
$307$		−	1713.47i		−	0.318543i	−0.987235	−	0.159272i	$-0.949085\pi$
$307$		−	1713.47i		−	0.318543i	0.987235	−	0.159272i	$-0.0509146\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6529.50			−1.19053			−0.595263	−	0.803531i	$-0.702952\pi$
$311$	−6529.50			−1.19053			−0.595263	+	0.803531i	$0.702952\pi$
$312$		0			0
$313$		−	436.330i		−	0.0787950i	−0.999224	−	0.0393975i	$-0.987456\pi$
$313$		−	436.330i		−	0.0787950i	0.999224	−	0.0393975i	$-0.0125439\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	1386.97i		−	0.245742i	−0.992423	−	0.122871i	$-0.960790\pi$
$317$		−	1386.97i		−	0.245742i	0.992423	−	0.122871i	$-0.0392101\pi$
$318$		0			0
$319$	−1817.00			−0.318911
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	6636.96i		−	1.14331i
$324$		0			0
$325$			7687.80i			1.31213i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2466.72	−	11518.8i	0.413357	−	1.93025i
$330$		0			0
$331$	−466.235			−0.0774217			−0.0387108	−	0.999250i	$-0.512325\pi$
$331$	−466.235			−0.0774217			−0.0387108	+	0.999250i	$0.512325\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	11520.9			1.87897
$336$		0			0
$337$	−1568.47			−0.253531			−0.126766	−	0.991933i	$-0.540460\pi$
$337$	−1568.47			−0.253531			−0.126766	+	0.991933i	$0.540460\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	11496.4			1.82571
$342$		0			0
$343$	3757.29	−	5122.14i	0.591472	−	0.806326i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	5397.74i		−	0.835061i	−0.908663	−	0.417530i	$-0.862896\pi$
$347$		−	5397.74i		−	0.835061i	0.908663	−	0.417530i	$-0.137104\pi$
$348$		0			0
$349$			2595.28i			0.398057i	0.979994	+	0.199029i	$0.0637786\pi$
$349$			2595.28i			0.398057i	−0.979994	+	0.199029i	$0.936221\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4606.85			0.694612			0.347306	−	0.937752i	$-0.387097\pi$
$353$	4606.85			0.694612			0.347306	+	0.937752i	$0.387097\pi$
$354$		0			0
$355$		−	9383.46i		−	1.40288i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	2973.15i		−	0.437095i	−0.975826	−	0.218547i	$-0.929868\pi$
$359$		−	2973.15i		−	0.437095i	0.975826	−	0.218547i	$-0.0701319\pi$
$360$		0			0
$361$	−6842.78			−0.997635
$362$		0			0
$363$		0			0
$364$		0			0
$365$			3355.70i			0.481220i
$366$		0			0
$367$			9518.45i			1.35384i	0.736057	+	0.676919i	$0.236685\pi$
$367$			9518.45i			1.35384i	−0.736057	+	0.676919i	$0.763315\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4979.30	+	1066.30i	0.696799	+	0.149218i
$372$		0			0
$373$	−1944.90			−0.269981			−0.134991	−	0.990847i	$-0.543100\pi$
$373$	−1944.90			−0.269981			−0.134991	+	0.990847i	$0.543100\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2948.56			0.402807
$378$		0			0
$379$	4140.33			0.561147			0.280573	−	0.959833i	$-0.409475\pi$
$379$	4140.33			0.561147			0.280573	+	0.959833i	$0.409475\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1098.48			−0.146553			−0.0732767	−	0.997312i	$-0.523346\pi$
$383$	−1098.48			−0.146553			−0.0732767	+	0.997312i	$0.523346\pi$
$384$		0			0
$385$	−11198.4	−	2398.10i	−1.48240	−	0.317451i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	9204.27i		−	1.19968i	−0.800120	−	0.599839i	$-0.795231\pi$
$389$		−	9204.27i		−	1.19968i	0.800120	−	0.599839i	$-0.204769\pi$
$390$		0			0
$391$		−	431.843i		−	0.0558548i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−14220.1			−1.81137
$396$		0			0
$397$			13251.6i			1.67526i	0.546237	+	0.837630i	$0.316060\pi$
$397$			13251.6i			1.67526i	−0.546237	+	0.837630i	$0.683940\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	11673.4i		−	1.45372i	−0.686783	−	0.726862i	$-0.740978\pi$
$401$		−	11673.4i		−	1.45372i	0.686783	−	0.726862i	$-0.259022\pi$
$402$		0			0
$403$	−18656.0			−2.30600
$404$		0			0
$405$		0			0
$406$		0			0
$407$			13258.7i			1.61476i
$408$		0			0
$409$			11510.3i			1.39156i	0.718256	+	0.695779i	$0.244941\pi$
$409$			11510.3i			1.39156i	−0.718256	+	0.695779i	$0.755059\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2007.87	+	9376.12i	−0.239227	+	1.11712i
$414$		0			0
$415$	−12203.8			−1.44352
$416$		0			0
$417$		0			0
$418$		0			0
$419$	15629.8			1.82235			0.911174	−	0.412021i	$-0.135177\pi$
$419$	15629.8			1.82235			0.911174	+	0.412021i	$0.135177\pi$
$420$		0			0
$421$	7542.88			0.873201			0.436600	−	0.899656i	$-0.356182\pi$
$421$	7542.88			0.873201			0.436600	+	0.899656i	$0.356182\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6797.86			0.775869
$426$		0			0
$427$	2976.51	+	637.411i	0.337338	+	0.0722400i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			13115.9i			1.46582i	0.680324	+	0.732911i	$0.261839\pi$
$431$			13115.9i			1.46582i	−0.680324	+	0.732911i	$0.738161\pi$
$432$		0			0
$433$			1126.39i			0.125013i	0.998045	+	0.0625065i	$0.0199094\pi$
$433$			1126.39i			0.125013i	−0.998045	+	0.0625065i	$0.980091\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−891.525			−0.0975914
$438$		0			0
$439$			13489.6i			1.46657i	0.679922	+	0.733285i	$0.262014\pi$
$439$			13489.6i			1.46657i	−0.679922	+	0.733285i	$0.737986\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			8295.99i			0.889739i	0.895595	+	0.444869i	$0.146750\pi$
$443$			8295.99i			0.889739i	−0.895595	+	0.444869i	$0.853250\pi$
$444$		0			0
$445$	6933.92			0.738650
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	14603.8i		−	1.53496i	−0.641075	−	0.767478i	$-0.721511\pi$
$449$		−	14603.8i		−	1.53496i	0.641075	−	0.767478i	$-0.278489\pi$
$450$		0			0
$451$			7689.53i			0.802851i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	18172.3	+	3891.54i	1.87237	+	0.400963i
$456$		0			0
$457$	−655.066			−0.0670519			−0.0335259	−	0.999438i	$-0.510674\pi$
$457$	−655.066			−0.0670519			−0.0335259	+	0.999438i	$0.510674\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	12273.7			1.24000			0.620002	−	0.784600i	$-0.287132\pi$
$461$	12273.7			1.24000			0.620002	+	0.784600i	$0.287132\pi$
$462$		0			0
$463$	−9224.88			−0.925954			−0.462977	−	0.886370i	$-0.653219\pi$
$463$	−9224.88			−0.925954			−0.462977	+	0.886370i	$0.653219\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	14189.7			1.40605			0.703023	−	0.711167i	$-0.251833\pi$
$467$	14189.7			1.40605			0.703023	+	0.711167i	$0.251833\pi$
$468$		0			0
$469$	2855.11	−	13332.5i	0.281102	−	1.31266i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			12033.3i			1.16975i
$474$		0			0
$475$		−	14033.9i		−	1.35562i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10520.5			−1.00353			−0.501766	−	0.865003i	$-0.667316\pi$
$479$	−10520.5			−1.00353			−0.501766	+	0.865003i	$0.667316\pi$
$480$		0			0
$481$		−	21515.7i		−	2.03956i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			6102.35i			0.571327i
$486$		0			0
$487$	19692.0			1.83230			0.916149	−	0.400838i	$-0.131281\pi$
$487$	19692.0			1.83230			0.916149	+	0.400838i	$0.131281\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	676.336i		−	0.0621642i	−0.999517	−	0.0310821i	$-0.990105\pi$
$491$		−	676.336i		−	0.0621642i	0.999517	−	0.0310821i	$-0.00989533\pi$
$492$		0			0
$493$		−	2607.23i		−	0.238182i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−10858.9	−	2325.40i	−0.980057	−	0.209877i
$498$		0			0
$499$	−6191.62			−0.555461			−0.277730	−	0.960659i	$-0.589582\pi$
$499$	−6191.62			−0.555461			−0.277730	+	0.960659i	$0.589582\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	9490.52			0.841275			0.420638	−	0.907229i	$-0.361806\pi$
$503$	9490.52			0.841275			0.420638	+	0.907229i	$0.361806\pi$
$504$		0			0
$505$	−21948.4			−1.93404
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12969.7			−1.12942			−0.564709	−	0.825290i	$-0.691012\pi$
$509$	−12969.7			−1.12942			−0.564709	+	0.825290i	$0.691012\pi$
$510$		0			0
$511$	3883.34	+	831.608i	0.336182	+	0.0719925i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			16622.7i			1.42230i
$516$		0			0
$517$			25133.5i			2.13805i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	20707.4			1.74128			0.870642	−	0.491917i	$-0.163704\pi$
$521$	20707.4			1.74128			0.870642	+	0.491917i	$0.163704\pi$
$522$		0			0
$523$			10537.8i			0.881043i	0.897742	+	0.440522i	$0.145206\pi$
$523$			10537.8i			0.881043i	−0.897742	+	0.440522i	$0.854794\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			16496.3i			1.36355i
$528$		0			0
$529$	12109.0			0.995232
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	12478.3i		−	1.01406i
$534$		0			0
$535$			3392.88i			0.274181i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5550.35	+	12364.9i	−0.443545	+	0.988115i
$540$		0			0
$541$	11394.7			0.905542			0.452771	−	0.891627i	$-0.350436\pi$
$541$	11394.7			0.905542			0.452771	+	0.891627i	$0.350436\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14831.3			1.16569
$546$		0			0
$547$	22706.1			1.77485			0.887426	−	0.460951i	$-0.152492\pi$
$547$	22706.1			1.77485			0.887426	+	0.460951i	$0.152492\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−5382.54			−0.416159
$552$		0			0
$553$	−3524.02	+	16456.1i	−0.270989	+	1.26543i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	6229.16i		−	0.473857i	−0.971527	−	0.236928i	$-0.923859\pi$
$557$		−	6229.16i		−	0.473857i	0.971527	−	0.236928i	$-0.0761407\pi$
$558$		0			0
$559$		−	19527.1i		−	1.47747i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3117.62			−0.233379			−0.116689	−	0.993168i	$-0.537228\pi$
$563$	−3117.62			−0.233379			−0.116689	+	0.993168i	$0.537228\pi$
$564$		0			0
$565$		−	22028.8i		−	1.64028i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	12562.4i		−	0.925563i	−0.886472	−	0.462781i	$-0.846851\pi$
$569$		−	12562.4i		−	0.925563i	0.886472	−	0.462781i	$-0.153149\pi$
$570$		0			0
$571$	−7289.60			−0.534256			−0.267128	−	0.963661i	$-0.586075\pi$
$571$	−7289.60			−0.534256			−0.267128	+	0.963661i	$0.586075\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	913.138i		−	0.0662269i
$576$		0			0
$577$		−	5616.75i		−	0.405248i	−0.979257	−	0.202624i	$-0.935053\pi$
$577$		−	5616.75i		−	0.405248i	0.979257	−	0.202624i	$-0.0649470\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3024.35	+	14122.7i	−0.215957	+	1.00845i
$582$		0			0
$583$	−10864.6			−0.771813
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1794.10			0.126151			0.0630753	−	0.998009i	$-0.479909\pi$
$587$	1794.10			0.126151			0.0630753	+	0.998009i	$0.479909\pi$
$588$		0			0
$589$	34056.1			2.38244
$590$		0			0
$591$		0			0
$592$		0			0
$593$	16863.9			1.16782			0.583912	−	0.811817i	$-0.301521\pi$
$593$	16863.9			1.16782			0.583912	+	0.811817i	$0.301521\pi$
$594$		0			0
$595$	3441.05	−	16068.6i	0.237091	−	1.10714i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	14669.7i		−	1.00065i	−0.865839	−	0.500323i	$-0.833215\pi$
$599$		−	14669.7i		−	1.00065i	0.865839	−	0.500323i	$-0.166785\pi$
$600$		0			0
$601$			987.065i			0.0669937i	0.999439	+	0.0334969i	$0.0106644\pi$
$601$			987.065i			0.0669937i	−0.999439	+	0.0334969i	$0.989336\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3605.53			0.242291
$606$		0			0
$607$		−	14575.7i		−	0.974647i	−0.873222	−	0.487323i	$-0.837973\pi$
$607$		−	14575.7i		−	0.974647i	0.873222	−	0.487323i	$-0.162027\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	40785.7i		−	2.70051i
$612$		0			0
$613$	−20361.1			−1.34156			−0.670780	−	0.741656i	$-0.734041\pi$
$613$	−20361.1			−1.34156			−0.670780	+	0.741656i	$0.734041\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	4183.72i		−	0.272983i	−0.990641	−	0.136491i	$-0.956417\pi$
$617$		−	4183.72i		−	0.272983i	0.990641	−	0.136491i	$-0.0435826\pi$
$618$		0			0
$619$			27720.1i			1.79994i	0.435951	+	0.899970i	$0.356412\pi$
$619$			27720.1i			1.79994i	−0.435951	+	0.899970i	$0.643588\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1718.36	−	8024.20i	0.110505	−	0.516024i
$624$		0			0
$625$	−16237.4			−1.03919
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−19025.0			−1.20600
$630$		0			0
$631$	6437.94			0.406165			0.203083	−	0.979162i	$-0.434904\pi$
$631$	6437.94			0.406165			0.203083	+	0.979162i	$0.434904\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−19891.6			−1.24311
$636$		0			0
$637$	9006.88	−	20065.2i	0.560229	−	1.24806i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	29364.0i		−	1.80937i	−0.426076	−	0.904687i	$-0.640104\pi$
$641$		−	29364.0i		−	1.80937i	0.426076	−	0.904687i	$-0.359896\pi$
$642$		0			0
$643$			1098.23i			0.0673562i	0.999433	+	0.0336781i	$0.0107221\pi$
$643$			1098.23i			0.0673562i	−0.999433	+	0.0336781i	$0.989278\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8485.25			−0.515595			−0.257797	−	0.966199i	$-0.582997\pi$
$647$	−8485.25			−0.515595			−0.257797	+	0.966199i	$0.582997\pi$
$648$		0			0
$649$		−	20458.3i		−	1.23738i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			3167.42i			0.189817i	0.995486	+	0.0949085i	$0.0302559\pi$
$653$			3167.42i			0.189817i	−0.995486	+	0.0949085i	$0.969744\pi$
$654$		0			0
$655$	16813.3			1.00298
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	15539.8i		−	0.918582i	−0.888286	−	0.459291i	$-0.848103\pi$
$659$		−	15539.8i		−	0.918582i	0.888286	−	0.459291i	$-0.151897\pi$
$660$		0			0
$661$		−	10519.1i		−	0.618982i	−0.950902	−	0.309491i	$-0.899841\pi$
$661$		−	10519.1i		−	0.618982i	0.950902	−	0.309491i	$-0.100159\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−33173.1	−	7103.94i	−1.93443	−	0.414254i
$666$		0			0
$667$	−350.222			−0.0203308
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6494.62			−0.373654
$672$		0			0
$673$	29285.0			1.67735			0.838674	−	0.544634i	$-0.183331\pi$
$673$	29285.0			1.67735			0.838674	+	0.544634i	$0.183331\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5846.12			−0.331883			−0.165941	−	0.986136i	$-0.553066\pi$
$677$	−5846.12			−0.331883			−0.165941	+	0.986136i	$0.553066\pi$
$678$		0			0
$679$	7061.88	+	1512.28i	0.399131	+	0.0854728i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			7438.18i			0.416712i	0.978053	+	0.208356i	$0.0668112\pi$
$683$			7438.18i			0.416712i	−0.978053	+	0.208356i	$0.933189\pi$
$684$		0			0
$685$			23547.5i			1.31344i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	17630.7			0.974855
$690$		0			0
$691$		−	8101.50i		−	0.446014i	−0.974817	−	0.223007i	$-0.928413\pi$
$691$		−	8101.50i		−	0.446014i	0.974817	−	0.223007i	$-0.0715872\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	13341.0i		−	0.728136i
$696$		0			0
$697$	−11033.8			−0.599617
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	80.5954i		−	0.00434243i	−0.999998	−	0.00217122i	$-0.999309\pi$
$701$		−	80.5954i		−	0.00434243i	0.999998	−	0.00217122i	$-0.000691120\pi$
$702$		0			0
$703$			39276.5i			2.10717i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5439.24	+	25399.5i	−0.289340	+	1.35113i
$708$		0			0
$709$	20618.1			1.09214			0.546071	−	0.837739i	$-0.316123\pi$
$709$	20618.1			1.09214			0.546071	+	0.837739i	$0.316123\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2215.91			0.116390
$714$		0			0
$715$	−39651.1			−2.07394
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7395.39			0.383591			0.191795	−	0.981435i	$-0.438569\pi$
$719$	7395.39			0.383591			0.191795	+	0.981435i	$0.438569\pi$
$720$		0			0
$721$	19236.5	+	4119.44i	0.993626	+	0.212782i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	5513.02i		−	0.282412i
$726$		0			0
$727$			2697.28i			0.137602i	0.997630	+	0.0688008i	$0.0219173\pi$
$727$			2697.28i			0.137602i	−0.997630	+	0.0688008i	$0.978083\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−17266.6			−0.873637
$732$		0			0
$733$		−	39338.1i		−	1.98224i	−0.132952	−	0.991122i	$-0.542446\pi$
$733$		−	39338.1i		−	1.98224i	0.132952	−	0.991122i	$-0.457554\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			29090.9i			1.45397i
$738$		0			0
$739$	−16780.5			−0.835293			−0.417646	−	0.908610i	$-0.637145\pi$
$739$	−16780.5			−0.835293			−0.417646	+	0.908610i	$0.637145\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			12531.8i			0.618773i	0.950936	+	0.309386i	$0.100124\pi$
$743$			12531.8i			0.618773i	−0.950936	+	0.309386i	$0.899876\pi$
$744$		0			0
$745$			12105.1i			0.595297i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3926.37	+	840.821i	0.191544	+	0.0410186i
$750$		0			0
$751$	10627.3			0.516371			0.258186	−	0.966095i	$-0.416875\pi$
$751$	10627.3			0.516371			0.258186	+	0.966095i	$0.416875\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−39400.5			−1.89924
$756$		0			0
$757$	19420.8			0.932443			0.466221	−	0.884668i	$-0.345615\pi$
$757$	19420.8			0.932443			0.466221	+	0.884668i	$0.345615\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−2258.25			−0.107571			−0.0537854	−	0.998553i	$-0.517129\pi$
$761$	−2258.25			−0.107571			−0.0537854	+	0.998553i	$0.517129\pi$
$762$		0			0
$763$	3675.48	−	17163.3i	0.174392	−	0.814357i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			33198.9i			1.56290i
$768$		0			0
$769$			7682.01i			0.360234i	0.983645	+	0.180117i	$0.0576477\pi$
$769$			7682.01i			0.360234i	−0.983645	+	0.180117i	$0.942352\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	13041.2			0.606804			0.303402	−	0.952863i	$-0.401878\pi$
$773$	13041.2			0.606804			0.303402	+	0.952863i	$0.401878\pi$
$774$		0			0
$775$			34881.7i			1.61676i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			22778.8i			1.04767i
$780$		0			0
$781$	23693.7			1.08557
$782$		0			0
$783$		0			0
$784$		0			0
$785$			1143.08i			0.0519723i
$786$		0			0
$787$			9115.65i			0.412882i	0.978459	+	0.206441i	$0.0661881\pi$
$787$			9115.65i			0.412882i	−0.978459	+	0.206441i	$0.933812\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−25492.5	−	5459.16i	−1.14590	−	0.245392i
$792$		0			0
$793$	10539.2			0.471952
$794$		0			0
$795$		0			0
$796$		0			0
$797$	44493.3			1.97746			0.988729	−	0.149714i	$-0.0478352\pi$
$797$	44493.3			1.97746			0.988729	+	0.149714i	$0.0478352\pi$
$798$		0			0
$799$	−36064.3			−1.59682
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−8473.30			−0.372374
$804$		0			0
$805$	−2158.46	−	462.227i	−0.0945038	−	0.0202377i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	8457.35i		−	0.367546i	−0.982969	−	0.183773i	$-0.941169\pi$
$809$		−	8457.35i		−	0.367546i	0.982969	−	0.183773i	$-0.0588311\pi$
$810$		0			0
$811$		−	26796.4i		−	1.16023i	−0.814533	−	0.580117i	$-0.803007\pi$
$811$		−	26796.4i		−	1.16023i	0.814533	−	0.580117i	$-0.196993\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−25113.6			−1.07937
$816$		0			0
$817$			35646.3i			1.52645i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	28671.5i		−	1.21881i	−0.792859	−	0.609405i	$-0.791408\pi$
$821$		−	28671.5i		−	1.21881i	0.792859	−	0.609405i	$-0.208592\pi$
$822$		0			0
$823$	23207.1			0.982928			0.491464	−	0.870898i	$-0.336462\pi$
$823$	23207.1			0.982928			0.491464	+	0.870898i	$0.336462\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1777.49i		−	0.0747393i	−0.999302	−	0.0373696i	$-0.988102\pi$
$827$		−	1777.49i		−	0.0747393i	0.999302	−	0.0373696i	$-0.0118979\pi$
$828$		0			0
$829$			13834.0i			0.579585i	0.957090	+	0.289792i	$0.0935862\pi$
$829$			13834.0i			0.579585i	−0.957090	+	0.289792i	$0.906414\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−17742.5	−	7964.24i	−0.737983	−	0.331266i
$834$		0			0
$835$	−20861.6			−0.864606
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−24910.9			−1.02505			−0.512526	−	0.858672i	$-0.671290\pi$
$839$	−24910.9			−1.02505			−0.512526	+	0.858672i	$0.671290\pi$
$840$		0			0
$841$	22274.6			0.913303
$842$		0			0
$843$		0			0
$844$		0			0
$845$	29963.3			1.21984
$846$		0			0
$847$	893.522	−	4172.46i	0.0362477	−	0.169265i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			2555.58i			0.102942i
$852$		0			0
$853$			3512.35i			0.140986i	0.997512	+	0.0704928i	$0.0224572\pi$
$853$			3512.35i			0.140986i	−0.997512	+	0.0704928i	$0.977543\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	16423.3			0.654619			0.327310	−	0.944917i	$-0.393858\pi$
$857$	16423.3			0.654619			0.327310	+	0.944917i	$0.393858\pi$
$858$		0			0
$859$			1651.75i			0.0656078i	0.999462	+	0.0328039i	$0.0104437\pi$
$859$			1651.75i			0.0656078i	−0.999462	+	0.0328039i	$0.989556\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			20778.9i			0.819610i	0.912173	+	0.409805i	$0.134403\pi$
$863$			20778.9i			0.819610i	−0.912173	+	0.409805i	$0.865597\pi$
$864$		0			0
$865$	−38651.0			−1.51927
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	35906.5i		−	1.40166i
$870$		0			0
$871$		−	47207.4i		−	1.83647i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−309.982	+	1447.52i	−0.0119763	+	0.0559257i
$876$		0			0
$877$	−39579.8			−1.52396			−0.761981	−	0.647599i	$-0.775773\pi$
$877$	−39579.8			−1.52396			−0.761981	+	0.647599i	$0.775773\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−15881.0			−0.607314			−0.303657	−	0.952781i	$-0.598208\pi$
$881$	−15881.0			−0.607314			−0.303657	+	0.952781i	$0.598208\pi$
$882$		0			0
$883$	10903.6			0.415555			0.207778	−	0.978176i	$-0.433377\pi$
$883$	10903.6			0.415555			0.207778	+	0.978176i	$0.433377\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−47516.3			−1.79869			−0.899346	−	0.437237i	$-0.855957\pi$
$887$	−47516.3			−1.79869			−0.899346	+	0.437237i	$0.855957\pi$
$888$		0			0
$889$	−4929.53	+	23019.4i	−0.185974	+	0.868442i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			74453.5i			2.79002i
$894$		0			0
$895$			68048.5i			2.54146i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	13378.4			0.496324
$900$		0			0
$901$		−	15589.7i		−	0.576436i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			23846.2i			0.875885i
$906$		0			0
$907$	21213.0			0.776589			0.388294	−	0.921535i	$-0.373064\pi$
$907$	21213.0			0.776589			0.388294	+	0.921535i	$0.373064\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	16811.2i		−	0.611395i	−0.952129	−	0.305698i	$-0.901110\pi$
$911$		−	16811.2i		−	0.611395i	0.952129	−	0.305698i	$-0.0988897\pi$
$912$		0			0
$913$		−	30815.3i		−	1.11702i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4166.68	−	19457.1i	0.150050	−	0.700685i
$918$		0			0
$919$	−34695.2			−1.24536			−0.622682	−	0.782475i	$-0.713957\pi$
$919$	−34695.2			−1.24536			−0.622682	+	0.782475i	$0.713957\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−38449.1			−1.37115
$924$		0			0
$925$	−40228.6			−1.42996
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−44434.2			−1.56926			−0.784628	−	0.619966i	$-0.787146\pi$
$929$	−44434.2			−1.56926			−0.784628	+	0.619966i	$0.787146\pi$
$930$		0			0
$931$	−16441.9	+	36628.7i	−0.578799	+	1.28943i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			35061.1i			1.22633i
$936$		0			0
$937$		−	39975.1i		−	1.39374i	−0.717199	−	0.696868i	$-0.754576\pi$
$937$		−	39975.1i		−	1.39374i	0.717199	−	0.696868i	$-0.245424\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−30437.0			−1.05443			−0.527214	−	0.849733i	$-0.676763\pi$
$941$	−30437.0			−1.05443			−0.527214	+	0.849733i	$0.676763\pi$
$942$		0			0
$943$			1482.14i			0.0511824i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			15729.7i			0.539754i	0.962895	+	0.269877i	$0.0869830\pi$
$947$			15729.7i			0.539754i	−0.962895	+	0.269877i	$0.913017\pi$
$948$		0			0
$949$	13750.1			0.470335
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1741.59i			0.0591979i	0.999562	+	0.0295990i	$0.00942302\pi$
$953$			1741.59i			0.0591979i	−0.999562	+	0.0295990i	$0.990577\pi$
$954$		0			0
$955$		−	60910.4i		−	2.06389i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	27250.1	+	5835.54i	0.917573	+	0.196496i
$960$		0			0
$961$	−54856.3			−1.84137
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5940.65			−0.198172
$966$		0			0
$967$	−8193.06			−0.272462			−0.136231	−	0.990677i	$-0.543499\pi$
$967$	−8193.06			−0.272462			−0.136231	+	0.990677i	$0.543499\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−7844.92			−0.259275			−0.129637	−	0.991561i	$-0.541381\pi$
$971$	−7844.92			−0.259275			−0.129637	+	0.991561i	$0.541381\pi$
$972$		0			0
$973$	−15438.8	−	3306.17i	−0.508678	−	0.108932i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			50166.8i			1.64276i	0.570381	+	0.821380i	$0.306796\pi$
$977$			50166.8i			1.64276i	−0.570381	+	0.821380i	$0.693204\pi$
$978$		0			0
$979$			17508.5i			0.571576i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1307.85			−0.0424354			−0.0212177	−	0.999775i	$-0.506754\pi$
$983$	−1307.85			−0.0424354			−0.0212177	+	0.999775i	$0.506754\pi$
$984$		0			0
$985$		−	45643.7i		−	1.47648i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			2319.38i			0.0745722i
$990$		0			0
$991$	−43094.8			−1.38138			−0.690691	−	0.723150i	$-0.742694\pi$
$991$	−43094.8			−1.38138			−0.690691	+	0.723150i	$0.742694\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	45594.9i		−	1.45272i
$996$		0			0
$997$			8595.89i			0.273054i	0.990636	+	0.136527i	$0.0435940\pi$
$997$			8595.89i			0.273054i	−0.990636	+	0.136527i	$0.956406\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.4.f.a.1133.8		48
3.2	odd	2	inner	2268.4.f.a.1133.41		48
7.6	odd	2	inner	2268.4.f.a.1133.42		48
9.2	odd	6		252.4.x.a.41.12	✓	48
9.4	even	3		252.4.x.a.209.13	yes	48
9.5	odd	6		756.4.x.a.629.4		48
9.7	even	3		756.4.x.a.125.21		48
21.20	even	2	inner	2268.4.f.a.1133.7		48
63.13	odd	6		252.4.x.a.209.12	yes	48
63.20	even	6		252.4.x.a.41.13	yes	48
63.34	odd	6		756.4.x.a.125.4		48
63.41	even	6		756.4.x.a.629.21		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.x.a.41.12	✓	48	9.2	odd	6
252.4.x.a.41.13	yes	48	63.20	even	6
252.4.x.a.209.12	yes	48	63.13	odd	6
252.4.x.a.209.13	yes	48	9.4	even	3
756.4.x.a.125.4		48	63.34	odd	6
756.4.x.a.125.21		48	9.7	even	3
756.4.x.a.629.4		48	9.5	odd	6
756.4.x.a.629.21		48	63.41	even	6
2268.4.f.a.1133.7		48	21.20	even	2	inner
2268.4.f.a.1133.8		48	1.1	even	1	trivial
2268.4.f.a.1133.41		48	3.2	odd	2	inner
2268.4.f.a.1133.42		48	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2268.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-15.6490 q^{5} +(-3.87814 + 18.1097i) q^{7} +O(q^{10})\)	\(q-15.6490 q^{5} +(-3.87814 + 18.1097i) q^{7} -39.5146i q^{11} +64.1226i q^{13} +56.6997 q^{17} -117.055i q^{19} -7.61632i q^{23} +119.892 q^{25} -45.9831i q^{29} +290.942i q^{31} +(60.6891 - 283.399i) q^{35} -335.540 q^{37} -194.600 q^{41} -304.528 q^{43} -636.058 q^{47} +(-312.920 - 140.464i) q^{49} -274.953i q^{53} +618.365i q^{55} +517.741 q^{59} -164.360i q^{61} -1003.46i q^{65} -736.206 q^{67} +599.619i q^{71} -214.435i q^{73} +(715.595 + 153.243i) q^{77} +908.690 q^{79} +779.846 q^{83} -887.295 q^{85} -443.089 q^{89} +(-1161.24 - 248.676i) q^{91} +1831.79i q^{95} -389.951i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12 q^{7}+O(q^{10}) \) 48 * q - 12 * q^7	\( 48 q - 12 q^{7} + 1200 q^{25} + 336 q^{37} - 168 q^{43} - 636 q^{49} + 1176 q^{67} - 408 q^{79} + 720 q^{85} - 1080 q^{91}+O(q^{100}) \) 48 * q - 12 * q^7 + 1200 * q^25 + 336 * q^37 - 168 * q^43 - 636 * q^49 + 1176 * q^67 - 408 * q^79 + 720 * q^85 - 1080 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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