LMFDB - Embedded newform 4140.2.n.b.2069.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(2069,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4140.2069");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.n (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2069.2
Character	$\chi$	$=$	4140.2069
Dual form			4140.2.n.b.2069.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.23315 - 0.114288i) q^{5} +3.07318 q^{7} +O(q^{10})$	$q+(-2.23315 - 0.114288i) q^{5} +3.07318 q^{7} -1.12391 q^{11} +3.92844i q^{13} -0.286096i q^{17} -5.12159i q^{19} +(-4.17587 - 2.35841i) q^{23} +(4.97388 + 0.510445i) q^{25} -3.70765i q^{29} -5.14223 q^{31} +(-6.86287 - 0.351229i) q^{35} +1.46535 q^{37} -4.70374i q^{41} +4.70122 q^{43} -3.27234 q^{47} +2.44447 q^{49} +3.68936i q^{53} +(2.50986 + 0.128450i) q^{55} +1.30665i q^{59} +5.75653i q^{61} +(0.448975 - 8.77278i) q^{65} +7.13634 q^{67} +1.29736i q^{71} -9.83401i q^{73} -3.45399 q^{77} -0.954506i q^{79} -7.87601i q^{83} +(-0.0326975 + 0.638895i) q^{85} +12.6073 q^{89} +12.0728i q^{91} +(-0.585338 + 11.4372i) q^{95} +11.6079 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 56 q^{25} + 16 q^{31} - 96 q^{49} - 16 q^{55} - 40 q^{85}+O(q^{100}) $ 32 * q + 56 * q^25 + 16 * q^31 - 96 * q^49 - 16 * q^55 - 40 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$5$	−2.23315	−	0.114288i	−0.998693	−	0.0511113i
$5$	−2.23315	−	0.114288i	−0.998693	−	0.0511113i
$6$		0			0
$7$	3.07318			1.16155			0.580777	−	0.814062i	$-0.302749\pi$
$7$	3.07318			1.16155			0.580777	+	0.814062i	$0.302749\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.12391			−0.338872			−0.169436	−	0.985541i	$-0.554195\pi$
$11$	−1.12391			−0.338872			−0.169436	+	0.985541i	$0.554195\pi$
$12$		0			0
$13$			3.92844i			1.08955i	0.838581	+	0.544777i	$0.183386\pi$
$13$			3.92844i			1.08955i	−0.838581	+	0.544777i	$0.816614\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	0.286096i		−	0.0693886i	−0.999398	−	0.0346943i	$-0.988954\pi$
$17$		−	0.286096i		−	0.0693886i	0.999398	−	0.0346943i	$-0.0110458\pi$
$18$		0			0
$19$		−	5.12159i		−	1.17497i	−0.809234	−	0.587486i	$-0.800118\pi$
$19$		−	5.12159i		−	1.17497i	0.809234	−	0.587486i	$-0.199882\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.17587	−	2.35841i	−0.870729	−	0.491763i
$23$	−4.17587	−	2.35841i	−0.870729	−	0.491763i
$24$		0			0
$25$	4.97388	+	0.510445i	0.994775	+	0.102089i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	3.70765i		−	0.688494i	−0.938879	−	0.344247i	$-0.888134\pi$
$29$		−	3.70765i		−	0.688494i	0.938879	−	0.344247i	$-0.111866\pi$
$30$		0			0
$31$	−5.14223			−0.923572			−0.461786	−	0.886991i	$-0.652791\pi$
$31$	−5.14223			−0.923572			−0.461786	+	0.886991i	$0.652791\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.86287	−	0.351229i	−1.16004	−	0.0593686i
$36$		0			0
$37$	1.46535			0.240902			0.120451	−	0.992719i	$-0.461566\pi$
$37$	1.46535			0.240902			0.120451	+	0.992719i	$0.461566\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	4.70374i		−	0.734600i	−0.930102	−	0.367300i	$-0.880282\pi$
$41$		−	4.70374i		−	0.734600i	0.930102	−	0.367300i	$-0.119718\pi$
$42$		0			0
$43$	4.70122			0.716930			0.358465	−	0.933543i	$-0.383300\pi$
$43$	4.70122			0.716930			0.358465	+	0.933543i	$0.383300\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.27234			−0.477319			−0.238660	−	0.971103i	$-0.576708\pi$
$47$	−3.27234			−0.477319			−0.238660	+	0.971103i	$0.576708\pi$
$48$		0			0
$49$	2.44447			0.349209
$50$		0			0
$51$		0			0
$52$		0			0
$53$			3.68936i			0.506772i	0.967365	+	0.253386i	$0.0815443\pi$
$53$			3.68936i			0.506772i	−0.967365	+	0.253386i	$0.918456\pi$
$54$		0			0
$55$	2.50986	+	0.128450i	0.338430	+	0.0173202i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			1.30665i			0.170112i	0.996376	+	0.0850560i	$0.0271069\pi$
$59$			1.30665i			0.170112i	−0.996376	+	0.0850560i	$0.972893\pi$
$60$		0			0
$61$			5.75653i			0.737049i	0.929618	+	0.368524i	$0.120137\pi$
$61$			5.75653i			0.737049i	−0.929618	+	0.368524i	$0.879863\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.448975	−	8.77278i	0.0556885	−	1.08813i
$66$		0			0
$67$	7.13634			0.871843			0.435921	−	0.899985i	$-0.356423\pi$
$67$	7.13634			0.871843			0.435921	+	0.899985i	$0.356423\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			1.29736i			0.153968i	0.997032	+	0.0769840i	$0.0245290\pi$
$71$			1.29736i			0.153968i	−0.997032	+	0.0769840i	$0.975471\pi$
$72$		0			0
$73$		−	9.83401i		−	1.15098i	−0.817807	−	0.575492i	$-0.804811\pi$
$73$		−	9.83401i		−	1.15098i	0.817807	−	0.575492i	$-0.195189\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.45399			−0.393619
$78$		0			0
$79$		−	0.954506i		−	0.107390i	−0.998557	−	0.0536952i	$-0.982900\pi$
$79$		−	0.954506i		−	0.107390i	0.998557	−	0.0536952i	$-0.0170999\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	7.87601i		−	0.864504i	−0.901753	−	0.432252i	$-0.857719\pi$
$83$		−	7.87601i		−	0.864504i	0.901753	−	0.432252i	$-0.142281\pi$
$84$		0			0
$85$	−0.0326975	+	0.638895i	−0.00354654	+	0.0692979i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	12.6073			1.33637			0.668183	−	0.743997i	$-0.267072\pi$
$89$	12.6073			1.33637			0.668183	+	0.743997i	$0.267072\pi$
$90$		0			0
$91$			12.0728i			1.26558i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.585338	+	11.4372i	−0.0600544	+	1.17344i
$96$		0			0
$97$	11.6079			1.17861			0.589304	−	0.807911i	$-0.299402\pi$
$97$	11.6079			1.17861			0.589304	+	0.807911i	$0.299402\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			0.451800i			0.0449558i	0.999747	+	0.0224779i	$0.00715554\pi$
$101$			0.451800i			0.0449558i	−0.999747	+	0.0224779i	$0.992844\pi$
$102$		0			0
$103$	−5.45101			−0.537104			−0.268552	−	0.963265i	$-0.586545\pi$
$103$	−5.45101			−0.537104			−0.268552	+	0.963265i	$0.586545\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	15.4848i		−	1.49697i	−0.663150	−	0.748486i	$-0.730781\pi$
$107$		−	15.4848i		−	1.49697i	0.663150	−	0.748486i	$-0.269219\pi$
$108$		0			0
$109$		−	13.6623i		−	1.30861i	−0.756231	−	0.654304i	$-0.772962\pi$
$109$		−	13.6623i		−	1.30861i	0.756231	−	0.654304i	$-0.227038\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	15.0547i		−	1.41623i	−0.706098	−	0.708114i	$-0.749546\pi$
$113$		−	15.0547i		−	1.41623i	0.706098	−	0.708114i	$-0.250454\pi$
$114$		0			0
$115$	9.05579	+	5.74393i	0.844456	+	0.535624i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	0.879227i		−	0.0805986i
$120$		0			0
$121$	−9.73682			−0.885165
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.0491	−	1.70835i	−0.988257	−	0.152800i
$126$		0			0
$127$		−	9.57711i		−	0.849831i	−0.905233	−	0.424916i	$-0.860304\pi$
$127$		−	9.57711i		−	0.849831i	0.905233	−	0.424916i	$-0.139696\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			12.3790i			1.08156i	0.841165	+	0.540779i	$0.181871\pi$
$131$			12.3790i			1.08156i	−0.841165	+	0.540779i	$0.818129\pi$
$132$		0			0
$133$		−	15.7396i		−	1.36480i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	5.31609i		−	0.454184i	−0.973873	−	0.227092i	$-0.927078\pi$
$137$		−	5.31609i		−	0.454184i	0.973873	−	0.227092i	$-0.0729218\pi$
$138$		0			0
$139$	−20.8856			−1.77150			−0.885748	−	0.464167i	$-0.846354\pi$
$139$	−20.8856			−1.77150			−0.885748	+	0.464167i	$0.846354\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	4.41522i		−	0.369220i
$144$		0			0
$145$	−0.423742	+	8.27973i	−0.0351898	+	0.687594i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.34238			0.273818			0.136909	−	0.990584i	$-0.456283\pi$
$149$	3.34238			0.273818			0.136909	+	0.990584i	$0.456283\pi$
$150$		0			0
$151$	13.1258			1.06816			0.534080	−	0.845434i	$-0.320658\pi$
$151$	13.1258			1.06816			0.534080	+	0.845434i	$0.320658\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	11.4833	+	0.587697i	0.922365	+	0.0472049i
$156$		0			0
$157$	−5.91675			−0.472208			−0.236104	−	0.971728i	$-0.575871\pi$
$157$	−5.91675			−0.472208			−0.236104	+	0.971728i	$0.575871\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.8332	−	7.24784i	−1.01140	−	0.571210i
$162$		0			0
$163$			2.97220i			0.232800i	0.993202	+	0.116400i	$0.0371355\pi$
$163$			2.97220i			0.232800i	−0.993202	+	0.116400i	$0.962864\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.4424			−0.885441			−0.442721	−	0.896660i	$-0.645987\pi$
$167$	−11.4424			−0.885441			−0.442721	+	0.896660i	$0.645987\pi$
$168$		0			0
$169$	−2.43264			−0.187126
$170$		0			0
$171$		0			0
$172$		0			0
$173$	19.6447			1.49356			0.746780	−	0.665072i	$-0.231599\pi$
$173$	19.6447			1.49356			0.746780	+	0.665072i	$0.231599\pi$
$174$		0			0
$175$	15.2856	+	1.56869i	1.15549	+	0.118582i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			10.4449i			0.780687i	0.920669	+	0.390343i	$0.127644\pi$
$179$			10.4449i			0.780687i	−0.920669	+	0.390343i	$0.872356\pi$
$180$		0			0
$181$		−	17.8294i		−	1.32525i	−0.748953	−	0.662623i	$-0.769443\pi$
$181$		−	17.8294i		−	1.32525i	0.748953	−	0.662623i	$-0.230557\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.27234	−	0.167472i	−0.240587	−	0.0123128i
$186$		0			0
$187$			0.321548i			0.0235139i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.46898			−0.540437			−0.270218	−	0.962799i	$-0.587096\pi$
$191$	−7.46898			−0.540437			−0.270218	+	0.962799i	$0.587096\pi$
$192$		0			0
$193$		−	25.3167i		−	1.82234i	−0.412035	−	0.911168i	$-0.635182\pi$
$193$		−	25.3167i		−	1.82234i	0.412035	−	0.911168i	$-0.364818\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.1695			−1.08078			−0.540391	−	0.841414i	$-0.681724\pi$
$197$	−15.1695			−1.08078			−0.540391	+	0.841414i	$0.681724\pi$
$198$		0			0
$199$		−	5.68134i		−	0.402740i	−0.979515	−	0.201370i	$-0.935461\pi$
$199$		−	5.68134i		−	0.402740i	0.979515	−	0.201370i	$-0.0645393\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	11.3943i		−	0.799724i
$204$		0			0
$205$	−0.537582	+	10.5041i	−0.0375464	+	0.733640i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			5.75622i			0.398166i
$210$		0			0
$211$	−8.58802			−0.591223			−0.295612	−	0.955308i	$-0.595523\pi$
$211$	−8.58802			−0.591223			−0.295612	+	0.955308i	$0.595523\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−10.4985	−	0.537295i	−0.715993	−	0.0366432i
$216$		0			0
$217$	−15.8030			−1.07278
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.12391			0.0756026
$222$		0			0
$223$		−	16.4389i		−	1.10083i	−0.834891	−	0.550416i	$-0.814469\pi$
$223$		−	16.4389i		−	1.10083i	0.834891	−	0.550416i	$-0.185531\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	15.0106i		−	0.996291i	−0.867093	−	0.498146i	$-0.834014\pi$
$227$		−	15.0106i		−	0.996291i	0.867093	−	0.498146i	$-0.165986\pi$
$228$		0			0
$229$			4.72684i			0.312358i	0.987729	+	0.156179i	$0.0499177\pi$
$229$			4.72684i			0.312358i	−0.987729	+	0.156179i	$0.950082\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.48589			−0.621442			−0.310721	−	0.950501i	$-0.600570\pi$
$233$	−9.48589			−0.621442			−0.310721	+	0.950501i	$0.600570\pi$
$234$		0			0
$235$	7.30760	+	0.373990i	0.476695	+	0.0243964i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	27.1235i		−	1.75447i	−0.480060	−	0.877236i	$-0.659385\pi$
$239$		−	27.1235i		−	1.75447i	0.480060	−	0.877236i	$-0.340615\pi$
$240$		0			0
$241$		−	2.14921i		−	0.138443i	−0.997601	−	0.0692213i	$-0.977949\pi$
$241$		−	2.14921i		−	0.138443i	0.997601	−	0.0692213i	$-0.0220515\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5.45885	−	0.279374i	−0.348753	−	0.0178485i
$246$		0			0
$247$	20.1198			1.28020
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5.89314			0.371972			0.185986	−	0.982552i	$-0.440452\pi$
$251$	5.89314			0.371972			0.185986	+	0.982552i	$0.440452\pi$
$252$		0			0
$253$	4.69331	+	2.65065i	0.295066	+	0.166645i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.92000			0.119767			0.0598833	−	0.998205i	$-0.480927\pi$
$257$	1.92000			0.119767			0.0598833	+	0.998205i	$0.480927\pi$
$258$		0			0
$259$	4.50329			0.279821
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	25.4501i		−	1.56932i	−0.619925	−	0.784661i	$-0.712837\pi$
$263$		−	25.4501i		−	1.56932i	0.619925	−	0.784661i	$-0.287163\pi$
$264$		0			0
$265$	0.421651	−	8.23887i	0.0259018	−	0.506110i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			10.0949i			0.615494i	0.951468	+	0.307747i	$0.0995751\pi$
$269$			10.0949i			0.615494i	−0.951468	+	0.307747i	$0.900425\pi$
$270$		0			0
$271$	17.6422			1.07169			0.535844	−	0.844317i	$-0.319994\pi$
$271$	17.6422			1.07169			0.535844	+	0.844317i	$0.319994\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5.59020	−	0.573696i	−0.337102	−	0.0345951i
$276$		0			0
$277$			7.18336i			0.431606i	0.976437	+	0.215803i	$0.0692370\pi$
$277$			7.18336i			0.431606i	−0.976437	+	0.215803i	$0.930763\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.754865			−0.0450315			−0.0225157	−	0.999746i	$-0.507168\pi$
$281$	−0.754865			−0.0450315			−0.0225157	+	0.999746i	$0.507168\pi$
$282$		0			0
$283$	26.5310			1.57710			0.788552	−	0.614968i	$-0.210831\pi$
$283$	26.5310			1.57710			0.788552	+	0.614968i	$0.210831\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	14.4555i		−	0.853279i
$288$		0			0
$289$	16.9181			0.995185
$290$		0			0
$291$		0			0
$292$		0			0
$293$			1.76065i			0.102858i	0.998677	+	0.0514292i	$0.0163777\pi$
$293$			1.76065i			0.102858i	−0.998677	+	0.0514292i	$0.983622\pi$
$294$		0			0
$295$	0.149335	−	2.91795i	0.00869464	−	0.169890i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	9.26488	−	16.4047i	0.535802	−	0.948706i
$300$		0			0
$301$	14.4477			0.832753
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.657905	−	12.8552i	0.0376715	−	0.736085i
$306$		0			0
$307$		−	3.16445i		−	0.180605i	−0.995914	−	0.0903024i	$-0.971217\pi$
$307$		−	3.16445i		−	0.180605i	0.995914	−	0.0903024i	$-0.0287833\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	15.0364i		−	0.852638i	−0.904573	−	0.426319i	$-0.859810\pi$
$311$		−	15.0364i		−	0.852638i	0.904573	−	0.426319i	$-0.140190\pi$
$312$		0			0
$313$	−13.0302			−0.736512			−0.368256	−	0.929724i	$-0.620045\pi$
$313$	−13.0302			−0.736512			−0.368256	+	0.929724i	$0.620045\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.46468			0.475424			0.237712	−	0.971336i	$-0.423603\pi$
$317$	8.46468			0.475424			0.237712	+	0.971336i	$0.423603\pi$
$318$		0			0
$319$			4.16708i			0.233312i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.46527			−0.0815297
$324$		0			0
$325$	−2.00525	+	19.5396i	−0.111231	+	1.08386i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−10.0565			−0.554432
$330$		0			0
$331$	11.9251			0.655463			0.327732	−	0.944771i	$-0.393716\pi$
$331$	11.9251			0.655463			0.327732	+	0.944771i	$0.393716\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−15.9365	−	0.815601i	−0.870703	−	0.0445610i
$336$		0			0
$337$	−3.23760			−0.176363			−0.0881816	−	0.996104i	$-0.528106\pi$
$337$	−3.23760			−0.176363			−0.0881816	+	0.996104i	$0.528106\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5.77942			0.312973
$342$		0			0
$343$	−14.0000			−0.755929
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−26.1894			−1.40592			−0.702960	−	0.711230i	$-0.748139\pi$
$347$	−26.1894			−1.40592			−0.702960	+	0.711230i	$0.748139\pi$
$348$		0			0
$349$	−14.4855			−0.775392			−0.387696	−	0.921787i	$-0.626729\pi$
$349$	−14.4855			−0.775392			−0.387696	+	0.921787i	$0.626729\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	25.6229			1.36377			0.681885	−	0.731460i	$-0.261161\pi$
$353$	25.6229			1.36377			0.681885	+	0.731460i	$0.261161\pi$
$354$		0			0
$355$	0.148273	−	2.89719i	0.00786950	−	0.153767i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.16335			−0.219733			−0.109867	−	0.993946i	$-0.535042\pi$
$359$	−4.16335			−0.219733			−0.109867	+	0.993946i	$0.535042\pi$
$360$		0			0
$361$	−7.23065			−0.380561
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.12391	+	21.9608i	−0.0588283	+	1.14948i
$366$		0			0
$367$	17.6691			0.922321			0.461160	−	0.887317i	$-0.347433\pi$
$367$	17.6691			0.922321			0.461160	+	0.887317i	$0.347433\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			11.3381i			0.588644i
$372$		0			0
$373$	6.21136			0.321612			0.160806	−	0.986986i	$-0.448591\pi$
$373$	6.21136			0.321612			0.160806	+	0.986986i	$0.448591\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	14.5653			0.750151
$378$		0			0
$379$		−	0.912984i		−	0.0468968i	−0.999725	−	0.0234484i	$-0.992535\pi$
$379$		−	0.912984i		−	0.0468968i	0.999725	−	0.0234484i	$-0.00746455\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	5.00142i		−	0.255561i	−0.991803	−	0.127780i	$-0.959215\pi$
$383$		−	5.00142i		−	0.255561i	0.991803	−	0.127780i	$-0.0407852\pi$
$384$		0			0
$385$	7.71327	+	0.394751i	0.393104	+	0.0201184i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.5004			−0.938008			−0.469004	−	0.883196i	$-0.655387\pi$
$389$	−18.5004			−0.938008			−0.469004	+	0.883196i	$0.655387\pi$
$390$		0			0
$391$	−0.674733	+	1.19470i	−0.0341227	+	0.0604187i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.109089	+	2.13155i	−0.00548886	+	0.107250i
$396$		0			0
$397$			33.8083i			1.69679i	0.529365	+	0.848394i	$0.322430\pi$
$397$			33.8083i			1.69679i	−0.529365	+	0.848394i	$0.677570\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	26.1953			1.30813			0.654067	−	0.756437i	$-0.273062\pi$
$401$	26.1953			1.30813			0.654067	+	0.756437i	$0.273062\pi$
$402$		0			0
$403$		−	20.2009i		−	1.00628i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.64692			−0.0816350
$408$		0			0
$409$	−5.41817			−0.267911			−0.133956	−	0.990987i	$-0.542768\pi$
$409$	−5.41817			−0.267911			−0.133956	+	0.990987i	$0.542768\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			4.01559i			0.197594i
$414$		0			0
$415$	−0.900136	+	17.5883i	−0.0441859	+	0.863374i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.56084			0.271665			0.135832	−	0.990732i	$-0.456629\pi$
$419$	5.56084			0.271665			0.135832	+	0.990732i	$0.456629\pi$
$420$		0			0
$421$		−	11.2729i		−	0.549406i	−0.961529	−	0.274703i	$-0.911420\pi$
$421$		−	11.2729i		−	0.549406i	0.961529	−	0.274703i	$-0.0885796\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.146036	−	1.42301i	0.00708381	−	0.0690261i
$426$		0			0
$427$			17.6909i			0.856122i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5.55345			−0.267500			−0.133750	−	0.991015i	$-0.542702\pi$
$431$	−5.55345			−0.267500			−0.133750	+	0.991015i	$0.542702\pi$
$432$		0			0
$433$	−19.3797			−0.931329			−0.465664	−	0.884961i	$-0.654185\pi$
$433$	−19.3797			−0.931329			−0.465664	+	0.884961i	$0.654185\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.0788	+	21.3871i	−0.577808	+	1.02308i
$438$		0			0
$439$	−0.0344694			−0.00164513			−0.000822567	−	1.00000i	$-0.500262\pi$
$439$	−0.0344694			−0.00164513			−0.000822567		1.00000i	$0.500262\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−26.7570			−1.27127			−0.635633	−	0.771991i	$-0.719261\pi$
$443$	−26.7570			−1.27127			−0.635633	+	0.771991i	$0.719261\pi$
$444$		0			0
$445$	−28.1538	−	1.44086i	−1.33462	−	0.0683034i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	0.148879i		−	0.00702602i	−0.999994	−	0.00351301i	$-0.998882\pi$
$449$		−	0.148879i		−	0.00702602i	0.999994	−	0.00351301i	$-0.00111823\pi$
$450$		0			0
$451$			5.28659i			0.248936i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.37978	−	26.9604i	0.0646852	−	1.26392i
$456$		0			0
$457$	21.1785			0.990687			0.495343	−	0.868697i	$-0.335042\pi$
$457$	21.1785			0.990687			0.495343	+	0.868697i	$0.335042\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			24.2474i			1.12931i	0.825326	+	0.564657i	$0.190991\pi$
$461$			24.2474i			1.12931i	−0.825326	+	0.564657i	$0.809009\pi$
$462$		0			0
$463$		−	27.4603i		−	1.27619i	−0.769959	−	0.638093i	$-0.779723\pi$
$463$		−	27.4603i		−	1.27619i	0.769959	−	0.638093i	$-0.220277\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			12.8862i			0.596305i	0.954518	+	0.298152i	$0.0963703\pi$
$467$			12.8862i			0.596305i	−0.954518	+	0.298152i	$0.903630\pi$
$468$		0			0
$469$	21.9313			1.01269
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−5.28377			−0.242948
$474$		0			0
$475$	2.61429	−	25.4741i	0.119952	−	1.16883i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−27.6883			−1.26511			−0.632555	−	0.774515i	$-0.717994\pi$
$479$	−27.6883			−1.26511			−0.632555	+	0.774515i	$0.717994\pi$
$480$		0			0
$481$			5.75653i			0.262475i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−25.9222	−	1.32665i	−1.17707	−	0.0602402i
$486$		0			0
$487$		−	4.94933i		−	0.224276i	−0.993693	−	0.112138i	$-0.964230\pi$
$487$		−	4.94933i		−	0.224276i	0.993693	−	0.112138i	$-0.0357698\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			33.4915i			1.51145i	0.654888	+	0.755726i	$0.272716\pi$
$491$			33.4915i			1.51145i	−0.654888	+	0.755726i	$0.727284\pi$
$492$		0			0
$493$	−1.06075			−0.0477736
$494$		0			0
$495$		0			0
$496$		0			0
$497$			3.98702i			0.178842i
$498$		0			0
$499$	−14.0098			−0.627167			−0.313583	−	0.949561i	$-0.601529\pi$
$499$	−14.0098			−0.627167			−0.313583	+	0.949561i	$0.601529\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	4.68788i		−	0.209022i	−0.994524	−	0.104511i	$-0.966672\pi$
$503$		−	4.68788i		−	0.209022i	0.994524	−	0.104511i	$-0.0333277\pi$
$504$		0			0
$505$	0.0516355	−	1.00894i	0.00229775	−	0.0448971i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			31.1173i			1.37925i	0.724166	+	0.689625i	$0.242225\pi$
$509$			31.1173i			1.37925i	−0.724166	+	0.689625i	$0.757775\pi$
$510$		0			0
$511$		−	30.2217i		−	1.33693i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12.1729	+	0.622987i	0.536402	+	0.0274521i
$516$		0			0
$517$	3.67782			0.161750
$518$		0			0
$519$		0			0
$520$		0			0
$521$	25.8630			1.13308			0.566540	−	0.824034i	$-0.308282\pi$
$521$	25.8630			1.13308			0.566540	+	0.824034i	$0.308282\pi$
$522$		0			0
$523$	−7.97153			−0.348571			−0.174285	−	0.984695i	$-0.555762\pi$
$523$	−7.97153			−0.348571			−0.174285	+	0.984695i	$0.555762\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			1.47117i			0.0640853i
$528$		0			0
$529$	11.8758	+	19.6968i	0.516338	+	0.856385i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	18.4784			0.800386
$534$		0			0
$535$	−1.76973	+	34.5798i	−0.0765122	+	1.49502i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.74737			−0.118337
$540$		0			0
$541$	−12.5867			−0.541144			−0.270572	−	0.962700i	$-0.587213\pi$
$541$	−12.5867			−0.541144			−0.270572	+	0.962700i	$0.587213\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.56144	+	30.5098i	−0.0668847	+	1.30690i
$546$		0			0
$547$		−	23.7820i		−	1.01685i	−0.861108	−	0.508423i	$-0.830229\pi$
$547$		−	23.7820i		−	1.01685i	0.861108	−	0.508423i	$-0.169771\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−18.9891			−0.808962
$552$		0			0
$553$		−	2.93337i		−	0.124740i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			35.1988i			1.49142i	0.666269	+	0.745711i	$0.267890\pi$
$557$			35.1988i			1.49142i	−0.666269	+	0.745711i	$0.732110\pi$
$558$		0			0
$559$			18.4685i			0.781133i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	13.4381i		−	0.566346i	−0.959069	−	0.283173i	$-0.908613\pi$
$563$		−	13.4381i		−	0.566346i	0.959069	−	0.283173i	$-0.0913871\pi$
$564$		0			0
$565$	−1.72058	+	33.6194i	−0.0723853	+	1.41438i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	17.8945			0.750177			0.375089	−	0.926989i	$-0.377612\pi$
$569$	17.8945			0.750177			0.375089	+	0.926989i	$0.377612\pi$
$570$		0			0
$571$			44.2372i			1.85127i	0.378416	+	0.925635i	$0.376469\pi$
$571$			44.2372i			1.85127i	−0.378416	+	0.925635i	$0.623531\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−19.5664	−	13.8620i	−0.815976	−	0.578085i
$576$		0			0
$577$			41.4987i			1.72761i	0.503822	+	0.863807i	$0.331927\pi$
$577$			41.4987i			1.72761i	−0.503822	+	0.863807i	$0.668073\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	24.2044i		−	1.00417i
$582$		0			0
$583$		−	4.14652i		−	0.171731i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.46423			0.225533			0.112766	−	0.993622i	$-0.464029\pi$
$587$	5.46423			0.225533			0.112766	+	0.993622i	$0.464029\pi$
$588$		0			0
$589$			26.3364i			1.08517i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	19.6125			0.805389			0.402695	−	0.915334i	$-0.368074\pi$
$593$	19.6125			0.805389			0.402695	+	0.915334i	$0.368074\pi$
$594$		0			0
$595$	−0.100485	+	1.96344i	−0.00411950	+	0.0804933i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			16.9199i			0.691330i	0.938358	+	0.345665i	$0.112347\pi$
$599$			16.9199i			0.691330i	−0.938358	+	0.345665i	$0.887653\pi$
$600$		0			0
$601$	−36.0423			−1.47020			−0.735098	−	0.677960i	$-0.762864\pi$
$601$	−36.0423			−1.47020			−0.735098	+	0.677960i	$0.762864\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	21.7437	+	1.11280i	0.884009	+	0.0452420i
$606$		0			0
$607$		−	26.6591i		−	1.08206i	−0.841003	−	0.541030i	$-0.818034\pi$
$607$		−	26.6591i		−	1.08206i	0.841003	−	0.541030i	$-0.181966\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	12.8552i		−	0.520065i
$612$		0			0
$613$	−45.9355			−1.85532			−0.927658	−	0.373430i	$-0.878182\pi$
$613$	−45.9355			−1.85532			−0.927658	+	0.373430i	$0.878182\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			10.2025i			0.410736i	0.978685	+	0.205368i	$0.0658391\pi$
$617$			10.2025i			0.410736i	−0.978685	+	0.205368i	$0.934161\pi$
$618$		0			0
$619$		−	10.9196i		−	0.438898i	−0.975624	−	0.219449i	$-0.929574\pi$
$619$		−	10.9196i		−	0.438898i	0.975624	−	0.219449i	$-0.0704259\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	38.7444			1.55226
$624$		0			0
$625$	24.4789	+	5.07778i	0.979156	+	0.203111i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	0.419231i		−	0.0167158i
$630$		0			0
$631$			13.8168i			0.550039i	0.961439	+	0.275020i	$0.0886844\pi$
$631$			13.8168i			0.550039i	−0.961439	+	0.275020i	$0.911316\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.09455	+	21.3871i	−0.0434360	+	0.848720i
$636$		0			0
$637$			9.60294i			0.380482i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−17.0368			−0.672913			−0.336457	−	0.941699i	$-0.609229\pi$
$641$	−17.0368			−0.672913			−0.336457	+	0.941699i	$0.609229\pi$
$642$		0			0
$643$	−37.4467			−1.47675			−0.738377	−	0.674388i	$-0.764407\pi$
$643$	−37.4467			−1.47675			−0.738377	+	0.674388i	$0.764407\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	17.2712			0.678999			0.339500	−	0.940606i	$-0.389742\pi$
$647$	17.2712			0.678999			0.339500	+	0.940606i	$0.389742\pi$
$648$		0			0
$649$		−	1.46857i		−	0.0576462i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	22.3172			0.873338			0.436669	−	0.899622i	$-0.356158\pi$
$653$	22.3172			0.873338			0.436669	+	0.899622i	$0.356158\pi$
$654$		0			0
$655$	1.41478	−	27.6441i	0.0552799	−	1.08014i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	17.6392			0.687126			0.343563	−	0.939130i	$-0.388366\pi$
$659$	17.6392			0.687126			0.343563	+	0.939130i	$0.388366\pi$
$660$		0			0
$661$		−	30.7400i		−	1.19565i	−0.801628	−	0.597823i	$-0.796032\pi$
$661$		−	30.7400i		−	1.19565i	0.801628	−	0.597823i	$-0.203968\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.79885	+	35.1488i	−0.0697564	+	1.36301i
$666$		0			0
$667$	−8.74418	+	15.4827i	−0.338576	+	0.599492i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	6.46984i		−	0.249765i
$672$		0			0
$673$		−	30.7798i		−	1.18648i	−0.805027	−	0.593238i	$-0.797849\pi$
$673$		−	30.7798i		−	1.18648i	0.805027	−	0.593238i	$-0.202151\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			18.9532i			0.728431i	0.931315	+	0.364216i	$0.118663\pi$
$677$			18.9532i			0.728431i	−0.931315	+	0.364216i	$0.881337\pi$
$678$		0			0
$679$	35.6734			1.36902
$680$		0			0
$681$		0			0
$682$		0			0
$683$	45.2676			1.73212			0.866058	−	0.499943i	$-0.166646\pi$
$683$	45.2676			1.73212			0.866058	+	0.499943i	$0.166646\pi$
$684$		0			0
$685$	−0.607567	+	11.8716i	−0.0232139	+	0.453590i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−14.4934			−0.552155
$690$		0			0
$691$	−10.0739			−0.383230			−0.191615	−	0.981470i	$-0.561372\pi$
$691$	−10.0739			−0.383230			−0.191615	+	0.981470i	$0.561372\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	46.6406	+	2.38698i	1.76918	+	0.0905434i
$696$		0			0
$697$	−1.34572			−0.0509729
$698$		0			0
$699$		0			0
$700$		0			0
$701$	40.7475			1.53901			0.769506	−	0.638640i	$-0.220503\pi$
$701$	40.7475			1.53901			0.769506	+	0.638640i	$0.220503\pi$
$702$		0			0
$703$		−	7.50491i		−	0.283053i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1.38847i			0.0522187i
$708$		0			0
$709$			43.6958i			1.64103i	0.571625	+	0.820515i	$0.306313\pi$
$709$			43.6958i			1.64103i	−0.571625	+	0.820515i	$0.693687\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	21.4733	+	12.1275i	0.804181	+	0.454178i
$714$		0			0
$715$	−0.504609	+	9.85984i	−0.0188713	+	0.368737i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			20.6484i			0.770055i	0.922905	+	0.385028i	$0.125808\pi$
$719$			20.6484i			0.770055i	−0.922905	+	0.385028i	$0.874192\pi$
$720$		0			0
$721$	−16.7520			−0.623875
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.89255	−	18.4414i	0.0702877	−	0.684897i
$726$		0			0
$727$	18.5843			0.689252			0.344626	−	0.938740i	$-0.388006\pi$
$727$	18.5843			0.689252			0.344626	+	0.938740i	$0.388006\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	1.34500i		−	0.0497468i
$732$		0			0
$733$	−25.2963			−0.934339			−0.467169	−	0.884168i	$-0.654726\pi$
$733$	−25.2963			−0.934339			−0.467169	+	0.884168i	$0.654726\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.02063			−0.295444
$738$		0			0
$739$	−10.3380			−0.380290			−0.190145	−	0.981756i	$-0.560896\pi$
$739$	−10.3380			−0.380290			−0.190145	+	0.981756i	$0.560896\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	15.6250i		−	0.573227i	−0.958046	−	0.286613i	$-0.907470\pi$
$743$		−	15.6250i		−	0.573227i	0.958046	−	0.286613i	$-0.0925295\pi$
$744$		0			0
$745$	−7.46402	−	0.381995i	−0.273460	−	0.0139952i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	47.5877i		−	1.73882i
$750$		0			0
$751$		−	29.3192i		−	1.06987i	−0.844892	−	0.534936i	$-0.820336\pi$
$751$		−	29.3192i		−	1.06987i	0.844892	−	0.534936i	$-0.179664\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−29.3118	−	1.50012i	−1.06676	−	0.0545951i
$756$		0			0
$757$	−14.0615			−0.511076			−0.255538	−	0.966799i	$-0.582253\pi$
$757$	−14.0615			−0.511076			−0.255538	+	0.966799i	$0.582253\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	33.1705i		−	1.20243i	−0.799087	−	0.601215i	$-0.794684\pi$
$761$		−	33.1705i		−	1.20243i	0.799087	−	0.601215i	$-0.205316\pi$
$762$		0			0
$763$		−	41.9867i		−	1.52002i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.13311			−0.185346
$768$		0			0
$769$		−	12.1143i		−	0.436854i	−0.975853	−	0.218427i	$-0.929907\pi$
$769$		−	12.1143i		−	0.436854i	0.975853	−	0.218427i	$-0.0700926\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			15.7655i			0.567044i	0.958966	+	0.283522i	$0.0915029\pi$
$773$			15.7655i			0.567044i	−0.958966	+	0.283522i	$0.908497\pi$
$774$		0			0
$775$	−25.5768	−	2.62482i	−0.918746	−	0.0942865i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24.0906			−0.863135
$780$		0			0
$781$		−	1.45812i		−	0.0521755i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	13.2130	+	0.676216i	0.471591	+	0.0241352i
$786$		0			0
$787$	−32.4552			−1.15690			−0.578452	−	0.815716i	$-0.696343\pi$
$787$	−32.4552			−1.15690			−0.578452	+	0.815716i	$0.696343\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	46.2659i		−	1.64503i
$792$		0			0
$793$	−22.6142			−0.803054
$794$		0			0
$795$		0			0
$796$		0			0
$797$			43.1609i			1.52884i	0.644720	+	0.764419i	$0.276974\pi$
$797$			43.1609i			1.52884i	−0.644720	+	0.764419i	$0.723026\pi$
$798$		0			0
$799$			0.936204i			0.0331205i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			11.0526i			0.390037i
$804$		0			0
$805$	27.8301	+	17.6522i	0.980882	+	0.622157i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			40.6898i			1.43058i	0.698828	+	0.715289i	$0.253705\pi$
$809$			40.6898i			1.43058i	−0.698828	+	0.715289i	$0.746295\pi$
$810$		0			0
$811$	29.5830			1.03880			0.519400	−	0.854531i	$-0.326156\pi$
$811$	29.5830			1.03880			0.519400	+	0.854531i	$0.326156\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.339687	−	6.63735i	0.0118987	−	0.232496i
$816$		0			0
$817$		−	24.0777i		−	0.842373i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			15.9332i			0.556071i	0.960571	+	0.278036i	$0.0896833\pi$
$821$			15.9332i			0.556071i	−0.960571	+	0.278036i	$0.910317\pi$
$822$		0			0
$823$		−	33.5209i		−	1.16847i	−0.811586	−	0.584233i	$-0.801395\pi$
$823$		−	33.5209i		−	1.16847i	0.811586	−	0.584233i	$-0.198605\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	39.2783i		−	1.36584i	−0.730493	−	0.682920i	$-0.760710\pi$
$827$		−	39.2783i		−	1.36584i	0.730493	−	0.682920i	$-0.239290\pi$
$828$		0			0
$829$	−3.03816			−0.105520			−0.0527599	−	0.998607i	$-0.516802\pi$
$829$	−3.03816			−0.105520			−0.0527599	+	0.998607i	$0.516802\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	0.699353i		−	0.0242311i
$834$		0			0
$835$	25.5526	+	1.30774i	0.884284	+	0.0452560i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−10.9471			−0.377934			−0.188967	−	0.981983i	$-0.560514\pi$
$839$	−10.9471			−0.377934			−0.188967	+	0.981983i	$0.560514\pi$
$840$		0			0
$841$	15.2533			0.525976
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5.43244	+	0.278022i	0.186882	+	0.00956426i
$846$		0			0
$847$	−29.9230			−1.02817
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.11911	−	3.45590i	−0.209760	−	0.118467i
$852$		0			0
$853$			20.7582i			0.710746i	0.934725	+	0.355373i	$0.115646\pi$
$853$			20.7582i			0.710746i	−0.934725	+	0.355373i	$0.884354\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.48648			0.0849367			0.0424683	−	0.999098i	$-0.486478\pi$
$857$	2.48648			0.0849367			0.0424683	+	0.999098i	$0.486478\pi$
$858$		0			0
$859$	26.0114			0.887497			0.443749	−	0.896151i	$-0.353648\pi$
$859$	26.0114			0.887497			0.443749	+	0.896151i	$0.353648\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	47.9572			1.63248			0.816242	−	0.577710i	$-0.196054\pi$
$863$	47.9572			1.63248			0.816242	+	0.577710i	$0.196054\pi$
$864$		0			0
$865$	−43.8695	−	2.24516i	−1.49161	−	0.0763377i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1.07278i			0.0363916i
$870$		0			0
$871$			28.0347i			0.949919i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−33.9558	−	5.25009i	−1.14791	−	0.177485i
$876$		0			0
$877$			38.1487i			1.28819i	0.764945	+	0.644096i	$0.222766\pi$
$877$			38.1487i			1.28819i	−0.764945	+	0.644096i	$0.777234\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−38.5473			−1.29869			−0.649345	−	0.760494i	$-0.724957\pi$
$881$	−38.5473			−1.29869			−0.649345	+	0.760494i	$0.724957\pi$
$882$		0			0
$883$		−	41.4425i		−	1.39465i	−0.716755	−	0.697325i	$-0.754373\pi$
$883$		−	41.4425i		−	1.39465i	0.716755	−	0.697325i	$-0.245627\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9.48589			0.318505			0.159253	−	0.987238i	$-0.449092\pi$
$887$	9.48589			0.318505			0.159253	+	0.987238i	$0.449092\pi$
$888$		0			0
$889$		−	29.4322i		−	0.987126i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			16.7596i			0.560837i
$894$		0			0
$895$	1.19373	−	23.3249i	0.0399019	−	0.779667i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			19.0656i			0.635874i
$900$		0			0
$901$	1.05551			0.0351642
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.03769	+	39.8155i	−0.0677350	+	1.32351i
$906$		0			0
$907$	1.35629			0.0450350			0.0225175	−	0.999746i	$-0.492832\pi$
$907$	1.35629			0.0450350			0.0225175	+	0.999746i	$0.492832\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−47.1769			−1.56304			−0.781521	−	0.623879i	$-0.785556\pi$
$911$	−47.1769			−1.56304			−0.781521	+	0.623879i	$0.785556\pi$
$912$		0			0
$913$			8.85194i			0.292957i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			38.0430i			1.25629i
$918$		0			0
$919$			19.1139i			0.630508i	0.949007	+	0.315254i	$0.102090\pi$
$919$			19.1139i			0.630508i	−0.949007	+	0.315254i	$0.897910\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.09659			−0.167756
$924$		0			0
$925$	7.28846	+	0.747980i	0.239643	+	0.0245934i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	38.2935i		−	1.25637i	−0.778064	−	0.628185i	$-0.783798\pi$
$929$		−	38.2935i		−	1.25637i	0.778064	−	0.628185i	$-0.216202\pi$
$930$		0			0
$931$		−	12.5195i		−	0.410311i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.0367491	−	0.718062i	0.00120183	−	0.0234832i
$936$		0			0
$937$	−47.0133			−1.53586			−0.767929	−	0.640535i	$-0.778713\pi$
$937$	−47.0133			−1.53586			−0.767929	+	0.640535i	$0.778713\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4.35257			0.141890			0.0709448	−	0.997480i	$-0.477399\pi$
$941$	4.35257			0.141890			0.0709448	+	0.997480i	$0.477399\pi$
$942$		0			0
$943$	−11.0934	+	19.6422i	−0.361249	+	0.639638i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	19.6447			0.638367			0.319184	−	0.947693i	$-0.396591\pi$
$947$	19.6447			0.638367			0.319184	+	0.947693i	$0.396591\pi$
$948$		0			0
$949$	38.6323			1.25406
$950$		0			0
$951$		0			0
$952$		0			0
$953$			20.7662i			0.672683i	0.941740	+	0.336342i	$0.109190\pi$
$953$			20.7662i			0.672683i	−0.941740	+	0.336342i	$0.890810\pi$
$954$		0			0
$955$	16.6793	+	0.853617i	0.539730	+	0.0276224i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	16.3373i		−	0.527559i
$960$		0			0
$961$	−4.55748			−0.147015
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.89340	+	56.5359i	−0.0931419	+	1.81995i
$966$		0			0
$967$			19.4758i			0.626299i	0.949704	+	0.313149i	$0.101384\pi$
$967$			19.4758i			0.626299i	−0.949704	+	0.313149i	$0.898616\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−21.8354			−0.700731			−0.350366	−	0.936613i	$-0.613943\pi$
$971$	−21.8354			−0.700731			−0.350366	+	0.936613i	$0.613943\pi$
$972$		0			0
$973$	−64.1854			−2.05769
$974$		0			0
$975$		0			0
$976$		0			0
$977$			20.6620i			0.661035i	0.943800	+	0.330517i	$0.107223\pi$
$977$			20.6620i			0.661035i	−0.943800	+	0.330517i	$0.892777\pi$
$978$		0			0
$979$	−14.1695			−0.452858
$980$		0			0
$981$		0			0
$982$		0			0
$983$			24.3517i			0.776699i	0.921512	+	0.388349i	$0.126955\pi$
$983$			24.3517i			0.776699i	−0.921512	+	0.388349i	$0.873045\pi$
$984$		0			0
$985$	33.8757	+	1.73370i	1.07937	+	0.0552402i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−19.6317	−	11.0874i	−0.624252	−	0.352560i
$990$		0			0
$991$	−42.0032			−1.33428			−0.667138	−	0.744934i	$-0.732481\pi$
$991$	−42.0032			−1.33428			−0.667138	+	0.744934i	$0.732481\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.649311	+	12.6873i	−0.0205845	+	0.402213i
$996$		0			0
$997$			24.9001i			0.788593i	0.918983	+	0.394297i	$0.129012\pi$
$997$			24.9001i			0.788593i	−0.918983	+	0.394297i	$0.870988\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.n.b.2069.2	yes	32
3.2	odd	2	inner	4140.2.n.b.2069.32	yes	32
5.4	even	2	inner	4140.2.n.b.2069.3	yes	32
15.14	odd	2	inner	4140.2.n.b.2069.29	yes	32
23.22	odd	2	inner	4140.2.n.b.2069.31	yes	32
69.68	even	2	inner	4140.2.n.b.2069.1	✓	32
115.114	odd	2	inner	4140.2.n.b.2069.30	yes	32
345.344	even	2	inner	4140.2.n.b.2069.4	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.n.b.2069.1	✓	32	69.68	even	2	inner
4140.2.n.b.2069.2	yes	32	1.1	even	1	trivial
4140.2.n.b.2069.3	yes	32	5.4	even	2	inner
4140.2.n.b.2069.4	yes	32	345.344	even	2	inner
4140.2.n.b.2069.29	yes	32	15.14	odd	2	inner
4140.2.n.b.2069.30	yes	32	115.114	odd	2	inner
4140.2.n.b.2069.31	yes	32	23.22	odd	2	inner
4140.2.n.b.2069.32	yes	32	3.2	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.23315 - 0.114288i) q^{5} +3.07318 q^{7} +O(q^{10})\)	\(q+(-2.23315 - 0.114288i) q^{5} +3.07318 q^{7} -1.12391 q^{11} +3.92844i q^{13} -0.286096i q^{17} -5.12159i q^{19} +(-4.17587 - 2.35841i) q^{23} +(4.97388 + 0.510445i) q^{25} -3.70765i q^{29} -5.14223 q^{31} +(-6.86287 - 0.351229i) q^{35} +1.46535 q^{37} -4.70374i q^{41} +4.70122 q^{43} -3.27234 q^{47} +2.44447 q^{49} +3.68936i q^{53} +(2.50986 + 0.128450i) q^{55} +1.30665i q^{59} +5.75653i q^{61} +(0.448975 - 8.77278i) q^{65} +7.13634 q^{67} +1.29736i q^{71} -9.83401i q^{73} -3.45399 q^{77} -0.954506i q^{79} -7.87601i q^{83} +(-0.0326975 + 0.638895i) q^{85} +12.6073 q^{89} +12.0728i q^{91} +(-0.585338 + 11.4372i) q^{95} +11.6079 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 56 q^{25} + 16 q^{31} - 96 q^{49} - 16 q^{55} - 40 q^{85}+O(q^{100}) \) 32 * q + 56 * q^25 + 16 * q^31 - 96 * q^49 - 16 * q^55 - 40 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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