LMFDB - Embedded newform 4320.2.m.c.2159.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(2159,4320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("4320.2159");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.m (of order $2$, degree $1$, not 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:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2159.6
Character	$\chi$	$=$	4320.2159
Dual form			4320.2.m.c.2159.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.13531 - 0.663669i) q^{5} +4.39866 q^{7} +O(q^{10})$	$q+(-2.13531 - 0.663669i) q^{5} +4.39866 q^{7} -0.633050i q^{11} +3.40649 q^{13} +5.48913 q^{17} -0.323423 q^{19} +6.26839i q^{23} +(4.11909 + 2.83428i) q^{25} -3.77896 q^{29} -7.86723i q^{31} +(-9.39250 - 2.91926i) q^{35} +5.90433 q^{37} -0.877482i q^{41} -2.31397i q^{43} +8.67090i q^{47} +12.3482 q^{49} -7.09416i q^{53} +(-0.420136 + 1.35176i) q^{55} +10.5321i q^{59} +14.0673i q^{61} +(-7.27391 - 2.26078i) q^{65} +5.99382i q^{67} -10.7806 q^{71} -13.9654i q^{73} -2.78457i q^{77} +7.33728i q^{79} -1.90716 q^{83} +(-11.7210 - 3.64297i) q^{85} -8.16899i q^{89} +14.9840 q^{91} +(0.690609 + 0.214646i) q^{95} -6.27222i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 16 q^{19} + 48 q^{49}+O(q^{100}) $ 48 * q + 16 * q^19 + 48 * q^49

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times$.

$n$	$2081$	$2431$	$3457$	$3781$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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
$5$	−2.13531	−	0.663669i	−0.954939	−	0.296802i
$5$	−2.13531	−	0.663669i	−0.954939	−	0.296802i
$6$		0			0
$7$	4.39866			1.66254			0.831269	−	0.555871i	$-0.187615\pi$
$7$	4.39866			1.66254			0.831269	+	0.555871i	$0.187615\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.633050i		−	0.190872i	−0.995436	−	0.0954359i	$-0.969576\pi$
$11$		−	0.633050i		−	0.190872i	0.995436	−	0.0954359i	$-0.0304245\pi$
$12$		0			0
$13$	3.40649			0.944790			0.472395	−	0.881387i	$-0.343390\pi$
$13$	3.40649			0.944790			0.472395	+	0.881387i	$0.343390\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.48913			1.33131			0.665655	−	0.746260i	$-0.268152\pi$
$17$	5.48913			1.33131			0.665655	+	0.746260i	$0.268152\pi$
$18$		0			0
$19$	−0.323423			−0.0741984			−0.0370992	−	0.999312i	$-0.511812\pi$
$19$	−0.323423			−0.0741984			−0.0370992	+	0.999312i	$0.511812\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.26839i			1.30705i	0.756905	+	0.653525i	$0.226710\pi$
$23$			6.26839i			1.30705i	−0.756905	+	0.653525i	$0.773290\pi$
$24$		0			0
$25$	4.11909	+	2.83428i	0.823817	+	0.566856i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.77896			−0.701736			−0.350868	−	0.936425i	$-0.614113\pi$
$29$	−3.77896			−0.701736			−0.350868	+	0.936425i	$0.614113\pi$
$30$		0			0
$31$		−	7.86723i		−	1.41300i	−0.707715	−	0.706498i	$-0.750274\pi$
$31$		−	7.86723i		−	1.41300i	0.707715	−	0.706498i	$-0.249726\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9.39250	−	2.91926i	−1.58762	−	0.493444i
$36$		0			0
$37$	5.90433			0.970666			0.485333	−	0.874329i	$-0.338698\pi$
$37$	5.90433			0.970666			0.485333	+	0.874329i	$0.338698\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	0.877482i		−	0.137040i	−0.997650	−	0.0685198i	$-0.978172\pi$
$41$		−	0.877482i		−	0.137040i	0.997650	−	0.0685198i	$-0.0218276\pi$
$42$		0			0
$43$		−	2.31397i		−	0.352878i	−0.984312	−	0.176439i	$-0.943542\pi$
$43$		−	2.31397i		−	0.352878i	0.984312	−	0.176439i	$-0.0564578\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.67090i			1.26478i	0.774650	+	0.632391i	$0.217926\pi$
$47$			8.67090i			1.26478i	−0.774650	+	0.632391i	$0.782074\pi$
$48$		0			0
$49$	12.3482			1.76403
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	7.09416i		−	0.974457i	−0.873274	−	0.487229i	$-0.838008\pi$
$53$		−	7.09416i		−	0.974457i	0.873274	−	0.487229i	$-0.161992\pi$
$54$		0			0
$55$	−0.420136	+	1.35176i	−0.0566511	+	0.182271i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			10.5321i			1.37117i	0.727993	+	0.685584i	$0.240453\pi$
$59$			10.5321i			1.37117i	−0.727993	+	0.685584i	$0.759547\pi$
$60$		0			0
$61$			14.0673i			1.80113i	0.434717	+	0.900567i	$0.356848\pi$
$61$			14.0673i			1.80113i	−0.434717	+	0.900567i	$0.643152\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.27391	−	2.26078i	−0.902217	−	0.280416i
$66$		0			0
$67$			5.99382i			0.732262i	0.930563	+	0.366131i	$0.119318\pi$
$67$			5.99382i			0.732262i	−0.930563	+	0.366131i	$0.880682\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−10.7806			−1.27942			−0.639710	−	0.768617i	$-0.720945\pi$
$71$	−10.7806			−1.27942			−0.639710	+	0.768617i	$0.720945\pi$
$72$		0			0
$73$		−	13.9654i		−	1.63453i	−0.576264	−	0.817264i	$-0.695490\pi$
$73$		−	13.9654i		−	1.63453i	0.576264	−	0.817264i	$-0.304510\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.78457i		−	0.317331i
$78$		0			0
$79$			7.33728i			0.825508i	0.910843	+	0.412754i	$0.135433\pi$
$79$			7.33728i			0.825508i	−0.910843	+	0.412754i	$0.864567\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.90716			−0.209338			−0.104669	−	0.994507i	$-0.533378\pi$
$83$	−1.90716			−0.209338			−0.104669	+	0.994507i	$0.533378\pi$
$84$		0			0
$85$	−11.7210	−	3.64297i	−1.27132	−	0.395135i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	8.16899i		−	0.865911i	−0.901415	−	0.432956i	$-0.857471\pi$
$89$		−	8.16899i		−	0.865911i	0.901415	−	0.432956i	$-0.142529\pi$
$90$		0			0
$91$	14.9840			1.57075
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.690609	+	0.214646i	0.0708549	+	0.0220222i
$96$		0			0
$97$		−	6.27222i		−	0.636847i	−0.947949	−	0.318424i	$-0.896847\pi$
$97$		−	6.27222i		−	0.636847i	0.947949	−	0.318424i	$-0.103153\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.535487			−0.0532830			−0.0266415	−	0.999645i	$-0.508481\pi$
$101$	−0.535487			−0.0532830			−0.0266415	+	0.999645i	$0.508481\pi$
$102$		0			0
$103$	−12.5566			−1.23724			−0.618621	−	0.785689i	$-0.712308\pi$
$103$	−12.5566			−1.23724			−0.618621	+	0.785689i	$0.712308\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.4267			1.29801			0.649003	−	0.760786i	$-0.275186\pi$
$107$	13.4267			1.29801			0.649003	+	0.760786i	$0.275186\pi$
$108$		0			0
$109$			1.43706i			0.137646i	0.997629	+	0.0688228i	$0.0219243\pi$
$109$			1.43706i			0.137646i	−0.997629	+	0.0688228i	$0.978076\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.45479			0.607216			0.303608	−	0.952797i	$-0.401809\pi$
$113$	6.45479			0.607216			0.303608	+	0.952797i	$0.401809\pi$
$114$		0			0
$115$	4.16014	−	13.3849i	0.387935	−	1.24815i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	24.1448			2.21335
$120$		0			0
$121$	10.5992			0.963568
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−6.91450	−	8.78577i	−0.618451	−	0.785823i
$126$		0			0
$127$	−10.8023			−0.958550			−0.479275	−	0.877665i	$-0.659100\pi$
$127$	−10.8023			−0.958550			−0.479275	+	0.877665i	$0.659100\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	18.2846i		−	1.59754i	−0.601639	−	0.798768i	$-0.705485\pi$
$131$		−	18.2846i		−	1.59754i	0.601639	−	0.798768i	$-0.294515\pi$
$132$		0			0
$133$	−1.42263			−0.123358
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.9351			1.27599			0.637997	−	0.770039i	$-0.279763\pi$
$137$	14.9351			1.27599			0.637997	+	0.770039i	$0.279763\pi$
$138$		0			0
$139$	16.0327			1.35988			0.679939	−	0.733269i	$-0.262006\pi$
$139$	16.0327			1.35988			0.679939	+	0.733269i	$0.262006\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	2.15648i		−	0.180334i
$144$		0			0
$145$	8.06925	+	2.50798i	0.670115	+	0.208276i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	11.4476			0.937820			0.468910	−	0.883246i	$-0.344647\pi$
$149$	11.4476			0.937820			0.468910	+	0.883246i	$0.344647\pi$
$150$		0			0
$151$			12.0407i			0.979859i	0.871762	+	0.489930i	$0.162977\pi$
$151$			12.0407i			0.979859i	−0.871762	+	0.489930i	$0.837023\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.22124	+	16.7990i	−0.419380	+	1.34933i
$156$		0			0
$157$	−4.16913			−0.332732			−0.166366	−	0.986064i	$-0.553203\pi$
$157$	−4.16913			−0.332732			−0.166366	+	0.986064i	$0.553203\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			27.5725i			2.17302i
$162$		0			0
$163$		−	1.77553i		−	0.139070i	−0.997580	−	0.0695350i	$-0.977848\pi$
$163$		−	1.77553i		−	0.139070i	0.997580	−	0.0695350i	$-0.0221516\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2.93164i			0.226857i	0.993546	+	0.113429i	$0.0361833\pi$
$167$			2.93164i			0.226857i	−0.993546	+	0.113429i	$0.963817\pi$
$168$		0			0
$169$	−1.39582			−0.107371
$170$		0			0
$171$		0			0
$172$		0			0
$173$			5.27138i			0.400775i	0.979717	+	0.200388i	$0.0642202\pi$
$173$			5.27138i			0.400775i	−0.979717	+	0.200388i	$0.935780\pi$
$174$		0			0
$175$	18.1185	+	12.4670i	1.36963	+	0.942419i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	11.6443i		−	0.870340i	−0.900348	−	0.435170i	$-0.856688\pi$
$179$		−	11.6443i		−	0.870340i	0.900348	−	0.435170i	$-0.143312\pi$
$180$		0			0
$181$		−	13.6718i		−	1.01622i	−0.861293	−	0.508109i	$-0.830345\pi$
$181$		−	13.6718i		−	1.01622i	0.861293	−	0.508109i	$-0.169655\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−12.6076	−	3.91852i	−0.926927	−	0.288095i
$186$		0			0
$187$		−	3.47489i		−	0.254109i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.04522			0.437417			0.218709	−	0.975790i	$-0.429816\pi$
$191$	6.04522			0.437417			0.218709	+	0.975790i	$0.429816\pi$
$192$		0			0
$193$			25.9530i			1.86814i	0.357088	+	0.934071i	$0.383770\pi$
$193$			25.9530i			1.86814i	−0.357088	+	0.934071i	$0.616230\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	10.4843i		−	0.746973i	−0.927636	−	0.373487i	$-0.878162\pi$
$197$		−	10.4843i		−	0.746973i	0.927636	−	0.373487i	$-0.121838\pi$
$198$		0			0
$199$		−	3.19670i		−	0.226608i	−0.993560	−	0.113304i	$-0.963857\pi$
$199$		−	3.19670i		−	0.226608i	0.993560	−	0.113304i	$-0.0361434\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−16.6224			−1.16666
$204$		0			0
$205$	−0.582358	+	1.87370i	−0.0406736	+	0.130865i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			0.204743i			0.0141624i
$210$		0			0
$211$	17.2016			1.18421			0.592105	−	0.805861i	$-0.298297\pi$
$211$	17.2016			1.18421			0.592105	+	0.805861i	$0.298297\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.53571	+	4.94105i	−0.104735	+	0.336977i
$216$		0			0
$217$		−	34.6053i		−	2.34916i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	18.6987			1.25781
$222$		0			0
$223$	11.8087			0.790766			0.395383	−	0.918516i	$-0.370612\pi$
$223$	11.8087			0.790766			0.395383	+	0.918516i	$0.370612\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.14942			0.275407			0.137703	−	0.990474i	$-0.456028\pi$
$227$	4.14942			0.275407			0.137703	+	0.990474i	$0.456028\pi$
$228$		0			0
$229$			17.7520i			1.17308i	0.809919	+	0.586542i	$0.199511\pi$
$229$			17.7520i			1.17308i	−0.809919	+	0.586542i	$0.800489\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−7.55127			−0.494700			−0.247350	−	0.968926i	$-0.579560\pi$
$233$	−7.55127			−0.494700			−0.247350	+	0.968926i	$0.579560\pi$
$234$		0			0
$235$	5.75461	−	18.5151i	0.375389	−	1.20779i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.11479			0.0721100			0.0360550	−	0.999350i	$-0.488521\pi$
$239$	1.11479			0.0721100			0.0360550	+	0.999350i	$0.488521\pi$
$240$		0			0
$241$	23.4788			1.51240			0.756201	−	0.654339i	$-0.227053\pi$
$241$	23.4788			1.51240			0.756201	+	0.654339i	$0.227053\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−26.3673	−	8.19514i	−1.68454	−	0.523568i
$246$		0			0
$247$	−1.10174			−0.0701019
$248$		0			0
$249$		0			0
$250$		0			0
$251$			19.8709i			1.25424i	0.778923	+	0.627119i	$0.215766\pi$
$251$			19.8709i			1.25424i	−0.778923	+	0.627119i	$0.784234\pi$
$252$		0			0
$253$	3.96820			0.249479
$254$		0			0
$255$		0			0
$256$		0			0
$257$	16.1801			1.00929			0.504643	−	0.863328i	$-0.331624\pi$
$257$	16.1801			1.00929			0.504643	+	0.863328i	$0.331624\pi$
$258$		0			0
$259$	25.9711			1.61377
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	12.5368i		−	0.773051i	−0.922279	−	0.386525i	$-0.873675\pi$
$263$		−	12.5368i		−	0.773051i	0.922279	−	0.386525i	$-0.126325\pi$
$264$		0			0
$265$	−4.70817	+	15.1482i	−0.289221	+	0.930548i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	23.1238			1.40988			0.704941	−	0.709266i	$-0.250973\pi$
$269$	23.1238			1.40988			0.704941	+	0.709266i	$0.250973\pi$
$270$		0			0
$271$			10.1608i			0.617224i	0.951188	+	0.308612i	$0.0998645\pi$
$271$			10.1608i			0.617224i	−0.951188	+	0.308612i	$0.900136\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.79424	−	2.60759i	0.108197	−	0.157243i
$276$		0			0
$277$	5.64642			0.339260			0.169630	−	0.985508i	$-0.445743\pi$
$277$	5.64642			0.339260			0.169630	+	0.985508i	$0.445743\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	29.4258i		−	1.75540i	−0.479212	−	0.877699i	$-0.659077\pi$
$281$		−	29.4258i		−	1.75540i	0.479212	−	0.877699i	$-0.340923\pi$
$282$		0			0
$283$			2.34373i			0.139320i	0.997571	+	0.0696602i	$0.0221915\pi$
$283$			2.34373i			0.139320i	−0.997571	+	0.0696602i	$0.977809\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	3.85975i		−	0.227834i
$288$		0			0
$289$	13.1306			0.772387
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	23.4485i		−	1.36987i	−0.728603	−	0.684937i	$-0.759830\pi$
$293$		−	23.4485i		−	1.36987i	0.728603	−	0.684937i	$-0.240170\pi$
$294$		0			0
$295$	6.98986	−	22.4894i	0.406965	−	1.30938i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			21.3532i			1.23489i
$300$		0			0
$301$		−	10.1784i		−	0.586673i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	9.33604	−	30.0380i	0.534580	−	1.71997i
$306$		0			0
$307$			1.32995i			0.0759041i	0.999280	+	0.0379520i	$0.0120834\pi$
$307$			1.32995i			0.0759041i	−0.999280	+	0.0379520i	$0.987917\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−32.6861			−1.85346			−0.926729	−	0.375732i	$-0.877391\pi$
$311$	−32.6861			−1.85346			−0.926729	+	0.375732i	$0.877391\pi$
$312$		0			0
$313$		−	8.29685i		−	0.468966i	−0.972120	−	0.234483i	$-0.924660\pi$
$313$		−	8.29685i		−	0.468966i	0.972120	−	0.234483i	$-0.0753397\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			29.8855i			1.67854i	0.543718	+	0.839268i	$0.317016\pi$
$317$			29.8855i			1.67854i	−0.543718	+	0.839268i	$0.682984\pi$
$318$		0			0
$319$			2.39227i			0.133941i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.77531			−0.0987811
$324$		0			0
$325$	14.0316	+	9.65494i	0.778335	+	0.535560i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			38.1404i			2.10275i
$330$		0			0
$331$	−11.6308			−0.639289			−0.319645	−	0.947538i	$-0.603563\pi$
$331$	−11.6308			−0.639289			−0.319645	+	0.947538i	$0.603563\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.97791	−	12.7987i	0.217337	−	0.699265i
$336$		0			0
$337$			8.39265i			0.457177i	0.973523	+	0.228588i	$0.0734110\pi$
$337$			8.39265i			0.457177i	−0.973523	+	0.228588i	$0.926589\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.98035			−0.269701
$342$		0			0
$343$	23.5250			1.27023
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.3801			0.771965			0.385982	−	0.922506i	$-0.373863\pi$
$347$	14.3801			0.771965			0.385982	+	0.922506i	$0.373863\pi$
$348$		0			0
$349$		−	33.6982i		−	1.80382i	−0.431920	−	0.901912i	$-0.642164\pi$
$349$		−	33.6982i		−	1.80382i	0.431920	−	0.901912i	$-0.357836\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−26.3609			−1.40305			−0.701525	−	0.712645i	$-0.747497\pi$
$353$	−26.3609			−1.40305			−0.701525	+	0.712645i	$0.747497\pi$
$354$		0			0
$355$	23.0199	+	7.15474i	1.22177	+	0.379734i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	28.3508			1.49630			0.748149	−	0.663531i	$-0.230943\pi$
$359$	28.3508			1.49630			0.748149	+	0.663531i	$0.230943\pi$
$360$		0			0
$361$	−18.8954			−0.994495
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−9.26841	+	29.8205i	−0.485131	+	1.56087i
$366$		0			0
$367$	−18.2123			−0.950675			−0.475337	−	0.879804i	$-0.657674\pi$
$367$	−18.2123			−0.950675			−0.475337	+	0.879804i	$0.657674\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	31.2048i		−	1.62007i
$372$		0			0
$373$	7.21739			0.373702			0.186851	−	0.982388i	$-0.440172\pi$
$373$	7.21739			0.373702			0.186851	+	0.982388i	$0.440172\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.8730			−0.662993
$378$		0			0
$379$	−6.54353			−0.336118			−0.168059	−	0.985777i	$-0.553750\pi$
$379$	−6.54353			−0.336118			−0.168059	+	0.985777i	$0.553750\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			15.3236i			0.782999i	0.920178	+	0.391500i	$0.128044\pi$
$383$			15.3236i			0.782999i	−0.920178	+	0.391500i	$0.871956\pi$
$384$		0			0
$385$	−1.84803	+	5.94592i	−0.0941846	+	0.303032i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.599067			−0.0303739			−0.0151869	−	0.999885i	$-0.504834\pi$
$389$	−0.599067			−0.0303739			−0.0151869	+	0.999885i	$0.504834\pi$
$390$		0			0
$391$			34.4080i			1.74009i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.86953	−	15.6674i	0.245012	−	0.788310i
$396$		0			0
$397$	12.8269			0.643763			0.321881	−	0.946780i	$-0.395685\pi$
$397$	12.8269			0.643763			0.321881	+	0.946780i	$0.395685\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			5.19431i			0.259391i	0.991554	+	0.129696i	$0.0414000\pi$
$401$			5.19431i			0.259391i	−0.991554	+	0.129696i	$0.958600\pi$
$402$		0			0
$403$		−	26.7996i		−	1.33499i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	3.73774i		−	0.185273i
$408$		0			0
$409$	8.10266			0.400651			0.200325	−	0.979729i	$-0.435800\pi$
$409$	8.10266			0.400651			0.200325	+	0.979729i	$0.435800\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			46.3273i			2.27962i
$414$		0			0
$415$	4.07237	+	1.26572i	0.199905	+	0.0621318i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	3.51505i		−	0.171721i	−0.996307	−	0.0858607i	$-0.972636\pi$
$419$		−	3.51505i		−	0.171721i	0.996307	−	0.0858607i	$-0.0273640\pi$
$420$		0			0
$421$		−	5.14007i		−	0.250512i	−0.992124	−	0.125256i	$-0.960025\pi$
$421$		−	5.14007i		−	0.250512i	0.992124	−	0.125256i	$-0.0399751\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	22.6102	+	15.5577i	1.09676	+	0.754661i
$426$		0			0
$427$			61.8773i			2.99445i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−33.9083			−1.63330			−0.816652	−	0.577130i	$-0.804172\pi$
$431$	−33.9083			−1.63330			−0.816652	+	0.577130i	$0.804172\pi$
$432$		0			0
$433$		−	6.97168i		−	0.335038i	−0.985869	−	0.167519i	$-0.946425\pi$
$433$		−	6.97168i		−	0.335038i	0.985869	−	0.167519i	$-0.0535755\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	2.02734i		−	0.0969810i
$438$		0			0
$439$		−	1.99992i		−	0.0954509i	−0.998860	−	0.0477255i	$-0.984803\pi$
$439$		−	1.99992i		−	0.0954509i	0.998860	−	0.0477255i	$-0.0151973\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	30.0436			1.42741			0.713707	−	0.700445i	$-0.247015\pi$
$443$	30.0436			1.42741			0.713707	+	0.700445i	$0.247015\pi$
$444$		0			0
$445$	−5.42151	+	17.4433i	−0.257004	+	0.826893i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	10.1557i		−	0.479275i	−0.970862	−	0.239638i	$-0.922971\pi$
$449$		−	10.1557i		−	0.479275i	0.970862	−	0.239638i	$-0.0770286\pi$
$450$		0			0
$451$	−0.555490			−0.0261570
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−31.9955	−	9.94442i	−1.49997	−	0.466202i
$456$		0			0
$457$		−	37.9896i		−	1.77708i	−0.458800	−	0.888540i	$-0.651720\pi$
$457$		−	37.9896i		−	1.77708i	0.458800	−	0.888540i	$-0.348280\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−20.5143			−0.955445			−0.477722	−	0.878511i	$-0.658538\pi$
$461$	−20.5143			−0.955445			−0.477722	+	0.878511i	$0.658538\pi$
$462$		0			0
$463$	0.0629776			0.00292682			0.00146341	−	0.999999i	$-0.499534\pi$
$463$	0.0629776			0.00292682			0.00146341	+	0.999999i	$0.499534\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−21.9620			−1.01628			−0.508140	−	0.861274i	$-0.669667\pi$
$467$	−21.9620			−1.01628			−0.508140	+	0.861274i	$0.669667\pi$
$468$		0			0
$469$			26.3648i			1.21741i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.46486			−0.0673544
$474$		0			0
$475$	−1.33221	−	0.916671i	−0.0611259	−	0.0420598i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7.64777			0.349436			0.174718	−	0.984619i	$-0.444099\pi$
$479$	7.64777			0.349436			0.174718	+	0.984619i	$0.444099\pi$
$480$		0			0
$481$	20.1130			0.917076
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.16268	+	13.3931i	−0.189017	+	0.608150i
$486$		0			0
$487$	−2.23404			−0.101234			−0.0506171	−	0.998718i	$-0.516119\pi$
$487$	−2.23404			−0.101234			−0.0506171	+	0.998718i	$0.516119\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	22.4276i		−	1.01214i	−0.862491	−	0.506072i	$-0.831097\pi$
$491$		−	22.4276i		−	1.01214i	0.862491	−	0.506072i	$-0.168903\pi$
$492$		0			0
$493$	−20.7432			−0.934228
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−47.4201			−2.12708
$498$		0			0
$499$	−20.1585			−0.902420			−0.451210	−	0.892418i	$-0.649007\pi$
$499$	−20.1585			−0.902420			−0.451210	+	0.892418i	$0.649007\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	17.3479i		−	0.773506i	−0.922183	−	0.386753i	$-0.873597\pi$
$503$		−	17.3479i		−	0.773506i	0.922183	−	0.386753i	$-0.126403\pi$
$504$		0			0
$505$	1.14343	+	0.355386i	0.0508820	+	0.0158145i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	28.4216			1.25976			0.629882	−	0.776691i	$-0.283103\pi$
$509$	28.4216			1.25976			0.629882	+	0.776691i	$0.283103\pi$
$510$		0			0
$511$		−	61.4291i		−	2.71746i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	26.8123	+	8.33345i	1.18149	+	0.367216i
$516$		0			0
$517$	5.48911			0.241411
$518$		0			0
$519$		0			0
$520$		0			0
$521$			32.7743i			1.43587i	0.696111	+	0.717934i	$0.254912\pi$
$521$			32.7743i			1.43587i	−0.696111	+	0.717934i	$0.745088\pi$
$522$		0			0
$523$		−	43.3032i		−	1.89351i	−0.321948	−	0.946757i	$-0.604338\pi$
$523$		−	43.3032i		−	1.89351i	0.321948	−	0.946757i	$-0.395662\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	43.1843i		−	1.88114i
$528$		0			0
$529$	−16.2927			−0.708378
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	2.98913i		−	0.129474i
$534$		0			0
$535$	−28.6701	−	8.91088i	−1.23952	−	0.385251i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	7.81704i		−	0.336704i
$540$		0			0
$541$			8.86985i			0.381345i	0.981654	+	0.190672i	$0.0610668\pi$
$541$			8.86985i			0.381345i	−0.981654	+	0.190672i	$0.938933\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.953734	−	3.06857i	0.0408535	−	0.131443i
$546$		0			0
$547$			19.1126i			0.817197i	0.912714	+	0.408598i	$0.133982\pi$
$547$			19.1126i			0.817197i	−0.912714	+	0.408598i	$0.866018\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.22220			0.0520676
$552$		0			0
$553$			32.2742i			1.37244i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			18.4103i			0.780070i	0.920800	+	0.390035i	$0.127537\pi$
$557$			18.4103i			0.780070i	−0.920800	+	0.390035i	$0.872463\pi$
$558$		0			0
$559$		−	7.88253i		−	0.333396i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.23305			−0.0519669			−0.0259835	−	0.999662i	$-0.508272\pi$
$563$	−1.23305			−0.0519669			−0.0259835	+	0.999662i	$0.508272\pi$
$564$		0			0
$565$	−13.7830	−	4.28385i	−0.579854	−	0.180223i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	13.3647i		−	0.560279i	−0.959959	−	0.280140i	$-0.909619\pi$
$569$		−	13.3647i		−	0.560279i	0.959959	−	0.280140i	$-0.0903808\pi$
$570$		0			0
$571$	−20.7664			−0.869045			−0.434522	−	0.900661i	$-0.643083\pi$
$571$	−20.7664			−0.869045			−0.434522	+	0.900661i	$0.643083\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−17.7664	+	25.8200i	−0.740908	+	1.07677i
$576$		0			0
$577$		−	8.90816i		−	0.370852i	−0.982658	−	0.185426i	$-0.940634\pi$
$577$		−	8.90816i		−	0.370852i	0.982658	−	0.185426i	$-0.0593664\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−8.38894			−0.348032
$582$		0			0
$583$	−4.49096			−0.185996
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−33.0278			−1.36320			−0.681601	−	0.731724i	$-0.738716\pi$
$587$	−33.0278			−1.36320			−0.681601	+	0.731724i	$0.738716\pi$
$588$		0			0
$589$			2.54445i			0.104842i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.685529			−0.0281513			−0.0140757	−	0.999901i	$-0.504481\pi$
$593$	−0.685529			−0.0281513			−0.0140757	+	0.999901i	$0.504481\pi$
$594$		0			0
$595$	−51.5567	−	16.0242i	−2.11362	−	0.656928i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	23.7808			0.971656			0.485828	−	0.874055i	$-0.338518\pi$
$599$	23.7808			0.971656			0.485828	+	0.874055i	$0.338518\pi$
$600$		0			0
$601$	2.43712			0.0994123			0.0497062	−	0.998764i	$-0.484172\pi$
$601$	2.43712			0.0994123			0.0497062	+	0.998764i	$0.484172\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−22.6327	−	7.03440i	−0.920149	−	0.285989i
$606$		0			0
$607$	−28.3537			−1.15084			−0.575420	−	0.817858i	$-0.695161\pi$
$607$	−28.3537			−1.15084			−0.575420	+	0.817858i	$0.695161\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			29.5373i			1.19495i
$612$		0			0
$613$	−39.2558			−1.58553			−0.792763	−	0.609530i	$-0.791358\pi$
$613$	−39.2558			−1.58553			−0.792763	+	0.609530i	$0.791358\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−40.7603			−1.64095			−0.820473	−	0.571685i	$-0.806290\pi$
$617$	−40.7603			−1.64095			−0.820473	+	0.571685i	$0.806290\pi$
$618$		0			0
$619$	−11.3834			−0.457538			−0.228769	−	0.973481i	$-0.573470\pi$
$619$	−11.3834			−0.457538			−0.228769	+	0.973481i	$0.573470\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	35.9326i		−	1.43961i
$624$		0			0
$625$	8.93374	+	23.3493i	0.357350	+	0.933971i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	32.4096			1.29226
$630$		0			0
$631$			20.7246i			0.825035i	0.910950	+	0.412517i	$0.135350\pi$
$631$			20.7246i			0.825035i	−0.910950	+	0.412517i	$0.864650\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	23.0663	+	7.16916i	0.915357	+	0.284500i
$636$		0			0
$637$	42.0641			1.66664
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1.64857i			0.0651148i	0.999470	+	0.0325574i	$0.0103652\pi$
$641$			1.64857i			0.0651148i	−0.999470	+	0.0325574i	$0.989635\pi$
$642$		0			0
$643$			32.4116i			1.27819i	0.769128	+	0.639095i	$0.220691\pi$
$643$			32.4116i			1.27819i	−0.769128	+	0.639095i	$0.779309\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			3.92161i			0.154174i	0.997024	+	0.0770871i	$0.0245620\pi$
$647$			3.92161i			0.154174i	−0.997024	+	0.0770871i	$0.975438\pi$
$648$		0			0
$649$	6.66737			0.261717
$650$		0			0
$651$		0			0
$652$		0			0
$653$			16.8499i			0.659387i	0.944088	+	0.329693i	$0.106945\pi$
$653$			16.8499i			0.659387i	−0.944088	+	0.329693i	$0.893055\pi$
$654$		0			0
$655$	−12.1350	+	39.0433i	−0.474152	+	1.52555i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			31.2375i			1.21684i	0.793615	+	0.608420i	$0.208196\pi$
$659$			31.2375i			1.21684i	−0.793615	+	0.608420i	$0.791804\pi$
$660$		0			0
$661$			37.8548i			1.47238i	0.676774	+	0.736191i	$0.263377\pi$
$661$			37.8548i			1.47238i	−0.676774	+	0.736191i	$0.736623\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.03775	+	0.944155i	0.117799	+	0.0366128i
$666$		0			0
$667$		−	23.6880i		−	0.917203i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	8.90531			0.343786
$672$		0			0
$673$		−	6.36802i		−	0.245469i	−0.992440	−	0.122735i	$-0.960834\pi$
$673$		−	6.36802i		−	0.245469i	0.992440	−	0.122735i	$-0.0391664\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	16.6484i		−	0.639851i	−0.947443	−	0.319926i	$-0.896342\pi$
$677$		−	16.6484i		−	0.639851i	0.947443	−	0.319926i	$-0.103658\pi$
$678$		0			0
$679$		−	27.5894i		−	1.05878i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	29.5071			1.12906			0.564529	−	0.825413i	$-0.309058\pi$
$683$	29.5071			1.12906			0.564529	+	0.825413i	$0.309058\pi$
$684$		0			0
$685$	−31.8911	−	9.91199i	−1.21850	−	0.378718i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	24.1662i		−	0.920658i
$690$		0			0
$691$	−31.8365			−1.21112			−0.605559	−	0.795800i	$-0.707051\pi$
$691$	−31.8365			−1.21112			−0.605559	+	0.795800i	$0.707051\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−34.2348	−	10.6404i	−1.29860	−	0.403614i
$696$		0			0
$697$		−	4.81662i		−	0.182442i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.54314			0.171592			0.0857960	−	0.996313i	$-0.472657\pi$
$701$	4.54314			0.171592			0.0857960	+	0.996313i	$0.472657\pi$
$702$		0			0
$703$	−1.90960			−0.0720218
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.35543			−0.0885849
$708$		0			0
$709$			7.01898i			0.263603i	0.991276	+	0.131802i	$0.0420762\pi$
$709$			7.01898i			0.263603i	−0.991276	+	0.131802i	$0.957924\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	49.3149			1.84686
$714$		0			0
$715$	−1.43119	+	4.60475i	−0.0535234	+	0.172208i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	32.6924			1.21922			0.609610	−	0.792702i	$-0.291326\pi$
$719$	32.6924			1.21922			0.609610	+	0.792702i	$0.291326\pi$
$720$		0			0
$721$	−55.2324			−2.05696
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−15.5659	−	10.7106i	−0.578102	−	0.397783i
$726$		0			0
$727$	−15.4300			−0.572268			−0.286134	−	0.958190i	$-0.592370\pi$
$727$	−15.4300			−0.572268			−0.286134	+	0.958190i	$0.592370\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	12.7017i		−	0.469790i
$732$		0			0
$733$	−50.2010			−1.85421			−0.927107	−	0.374796i	$-0.877713\pi$
$733$	−50.2010			−1.85421			−0.927107	+	0.374796i	$0.877713\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.79439			0.139768
$738$		0			0
$739$	−36.6261			−1.34731			−0.673657	−	0.739044i	$-0.735277\pi$
$739$	−36.6261			−1.34731			−0.673657	+	0.739044i	$0.735277\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			35.5474i			1.30411i	0.758172	+	0.652055i	$0.226093\pi$
$743$			35.5474i			1.30411i	−0.758172	+	0.652055i	$0.773907\pi$
$744$		0			0
$745$	−24.4441	−	7.59739i	−0.895561	−	0.278347i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	59.0594			2.15798
$750$		0			0
$751$		−	41.1833i		−	1.50280i	−0.659848	−	0.751399i	$-0.729379\pi$
$751$		−	41.1833i		−	1.50280i	0.659848	−	0.751399i	$-0.270621\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.99105	−	25.7106i	0.290824	−	0.935706i
$756$		0			0
$757$	5.89555			0.214278			0.107139	−	0.994244i	$-0.465831\pi$
$757$	5.89555			0.214278			0.107139	+	0.994244i	$0.465831\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	41.5797i		−	1.50726i	−0.657297	−	0.753632i	$-0.728300\pi$
$761$		−	41.5797i		−	1.50726i	0.657297	−	0.753632i	$-0.271700\pi$
$762$		0			0
$763$			6.32115i			0.228841i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			35.8777i			1.29547i
$768$		0			0
$769$	−26.0235			−0.938432			−0.469216	−	0.883083i	$-0.655463\pi$
$769$	−26.0235			−0.938432			−0.469216	+	0.883083i	$0.655463\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	10.4488i		−	0.375819i	−0.982186	−	0.187909i	$-0.939829\pi$
$773$		−	10.4488i		−	0.375819i	0.982186	−	0.187909i	$-0.0601711\pi$
$774$		0			0
$775$	22.2979	−	32.4058i	0.800965	−	1.16405i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			0.283798i			0.0101681i
$780$		0			0
$781$			6.82465i			0.244205i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8.90237	+	2.76692i	0.317739	+	0.0987556i
$786$		0			0
$787$			35.5171i			1.26605i	0.774132	+	0.633024i	$0.218187\pi$
$787$			35.5171i			1.26605i	−0.774132	+	0.633024i	$0.781813\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	28.3924			1.00952
$792$		0			0
$793$			47.9202i			1.70169i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	20.0999i		−	0.711977i	−0.934490	−	0.355988i	$-0.884144\pi$
$797$		−	20.0999i		−	0.711977i	0.934490	−	0.355988i	$-0.115856\pi$
$798$		0			0
$799$			47.5957i			1.68382i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−8.84080			−0.311985
$804$		0			0
$805$	18.2990	−	58.8758i	0.644956	−	2.07510i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			32.5036i			1.14276i	0.820684	+	0.571382i	$0.193593\pi$
$809$			32.5036i			1.14276i	−0.820684	+	0.571382i	$0.806407\pi$
$810$		0			0
$811$	35.5091			1.24689			0.623447	−	0.781866i	$-0.285732\pi$
$811$	35.5091			1.24689			0.623447	+	0.781866i	$0.285732\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.17836	+	3.79130i	−0.0412763	+	0.132803i
$816$		0			0
$817$			0.748393i			0.0261830i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12.6596			−0.441821			−0.220911	−	0.975294i	$-0.570903\pi$
$821$	−12.6596			−0.441821			−0.220911	+	0.975294i	$0.570903\pi$
$822$		0			0
$823$	27.2106			0.948500			0.474250	−	0.880390i	$-0.342719\pi$
$823$	27.2106			0.948500			0.474250	+	0.880390i	$0.342719\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−37.0039			−1.28675			−0.643376	−	0.765551i	$-0.722467\pi$
$827$	−37.0039			−1.28675			−0.643376	+	0.765551i	$0.722467\pi$
$828$		0			0
$829$		−	33.1179i		−	1.15023i	−0.818072	−	0.575116i	$-0.804957\pi$
$829$		−	33.1179i		−	1.15023i	0.818072	−	0.575116i	$-0.195043\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	67.7810			2.34847
$834$		0			0
$835$	1.94564	−	6.25996i	0.0673317	−	0.216635i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	29.0655			1.00345			0.501726	−	0.865026i	$-0.332698\pi$
$839$	29.0655			1.00345			0.501726	+	0.865026i	$0.332698\pi$
$840$		0			0
$841$	−14.7194			−0.507567
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.98051	+	0.926364i	0.102533	+	0.0318679i
$846$		0			0
$847$	46.6225			1.60197
$848$		0			0
$849$		0			0
$850$		0			0
$851$			37.0106i			1.26871i
$852$		0			0
$853$	43.9301			1.50414			0.752069	−	0.659084i	$-0.229056\pi$
$853$	43.9301			1.50414			0.752069	+	0.659084i	$0.229056\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	8.72872			0.298167			0.149084	−	0.988825i	$-0.452368\pi$
$857$	8.72872			0.298167			0.149084	+	0.988825i	$0.452368\pi$
$858$		0			0
$859$	−7.59497			−0.259137			−0.129569	−	0.991570i	$-0.541359\pi$
$859$	−7.59497			−0.259137			−0.129569	+	0.991570i	$0.541359\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			11.9244i			0.405910i	0.979188	+	0.202955i	$0.0650546\pi$
$863$			11.9244i			0.405910i	−0.979188	+	0.202955i	$0.934945\pi$
$864$		0			0
$865$	3.49845	−	11.2560i	0.118951	−	0.382716i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.64486			0.157566
$870$		0			0
$871$			20.4179i			0.691834i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−30.4145	−	38.6456i	−1.02820	−	1.30646i
$876$		0			0
$877$	22.3668			0.755272			0.377636	−	0.925954i	$-0.376737\pi$
$877$	22.3668			0.755272			0.377636	+	0.925954i	$0.376737\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	18.7665i		−	0.632259i	−0.948716	−	0.316130i	$-0.897617\pi$
$881$		−	18.7665i		−	0.632259i	0.948716	−	0.316130i	$-0.102383\pi$
$882$		0			0
$883$		−	22.0307i		−	0.741393i	−0.928754	−	0.370697i	$-0.879119\pi$
$883$		−	22.0307i		−	0.741393i	0.928754	−	0.370697i	$-0.120881\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	39.7930i		−	1.33612i	−0.744108	−	0.668059i	$-0.767125\pi$
$887$		−	39.7930i		−	1.33612i	0.744108	−	0.668059i	$-0.232875\pi$
$888$		0			0
$889$	−47.5157			−1.59363
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	2.80437i		−	0.0938447i
$894$		0			0
$895$	−7.72800	+	24.8643i	−0.258318	+	0.831121i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			29.7300i			0.991550i
$900$		0			0
$901$		−	38.9408i		−	1.29731i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9.07357	+	29.1936i	−0.301616	+	0.970427i
$906$		0			0
$907$			24.7832i			0.822914i	0.911429	+	0.411457i	$0.134980\pi$
$907$			24.7832i			0.822914i	−0.911429	+	0.411457i	$0.865020\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−26.0401			−0.862747			−0.431374	−	0.902173i	$-0.641971\pi$
$911$	−26.0401			−0.862747			−0.431374	+	0.902173i	$0.641971\pi$
$912$		0			0
$913$			1.20733i			0.0399566i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	80.4279i		−	2.65596i
$918$		0			0
$919$		−	31.9640i		−	1.05439i	−0.849743	−	0.527197i	$-0.823243\pi$
$919$		−	31.9640i		−	1.05439i	0.849743	−	0.527197i	$-0.176757\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−36.7240			−1.20878
$924$		0			0
$925$	24.3204	+	16.7345i	0.799651	+	0.550227i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			55.1162i			1.80830i	0.427211	+	0.904152i	$0.359496\pi$
$929$			55.1162i			1.80830i	−0.427211	+	0.904152i	$0.640504\pi$
$930$		0			0
$931$	−3.99370			−0.130888
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.30618	+	7.41997i	−0.0754202	+	0.242659i
$936$		0			0
$937$		−	17.0794i		−	0.557960i	−0.960297	−	0.278980i	$-0.910004\pi$
$937$		−	17.0794i		−	0.557960i	0.960297	−	0.278980i	$-0.0899964\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−46.5090			−1.51615			−0.758075	−	0.652168i	$-0.773860\pi$
$941$	−46.5090			−1.51615			−0.758075	+	0.652168i	$0.773860\pi$
$942$		0			0
$943$	5.50040			0.179118
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.59658			−0.0843776			−0.0421888	−	0.999110i	$-0.513433\pi$
$947$	−2.59658			−0.0843776			−0.0421888	+	0.999110i	$0.513433\pi$
$948$		0			0
$949$		−	47.5730i		−	1.54429i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−34.8096			−1.12759			−0.563797	−	0.825914i	$-0.690660\pi$
$953$	−34.8096			−1.12759			−0.563797	+	0.825914i	$0.690660\pi$
$954$		0			0
$955$	−12.9084	−	4.01203i	−0.417707	−	0.129826i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	65.6946			2.12139
$960$		0			0
$961$	−30.8933			−0.996559
$962$		0			0
$963$		0			0
$964$		0			0
$965$	17.2242	−	55.4178i	0.554468	−	1.78396i
$966$		0			0
$967$	−6.55004			−0.210635			−0.105318	−	0.994439i	$-0.533586\pi$
$967$	−6.55004			−0.210635			−0.105318	+	0.994439i	$0.533586\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	27.4146i		−	0.879778i	−0.898052	−	0.439889i	$-0.855018\pi$
$971$		−	27.4146i		−	0.879778i	0.898052	−	0.439889i	$-0.144982\pi$
$972$		0			0
$973$	70.5225			2.26085
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−14.1922			−0.454048			−0.227024	−	0.973889i	$-0.572900\pi$
$977$	−14.1922			−0.454048			−0.227024	+	0.973889i	$0.572900\pi$
$978$		0			0
$979$	−5.17138			−0.165278
$980$		0			0
$981$		0			0
$982$		0			0
$983$			6.12231i			0.195272i	0.995222	+	0.0976358i	$0.0311280\pi$
$983$			6.12231i			0.195272i	−0.995222	+	0.0976358i	$0.968872\pi$
$984$		0			0
$985$	−6.95809	+	22.3871i	−0.221703	+	0.713314i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	14.5049			0.461229
$990$		0			0
$991$		−	36.8386i		−	1.17022i	−0.810955	−	0.585109i	$-0.801052\pi$
$991$		−	36.8386i		−	1.17022i	0.810955	−	0.585109i	$-0.198948\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2.12155	+	6.82594i	−0.0672577	+	0.216397i
$996$		0			0
$997$	−20.2385			−0.640960			−0.320480	−	0.947255i	$-0.603844\pi$
$997$	−20.2385			−0.640960			−0.320480	+	0.947255i	$0.603844\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.m.c.2159.6		48
3.2	odd	2	inner	4320.2.m.c.2159.44		48
4.3	odd	2		1080.2.m.c.539.27	yes	48
5.4	even	2	inner	4320.2.m.c.2159.7		48
8.3	odd	2	inner	4320.2.m.c.2159.43		48
8.5	even	2		1080.2.m.c.539.26	yes	48
12.11	even	2		1080.2.m.c.539.22	yes	48
15.14	odd	2	inner	4320.2.m.c.2159.41		48
20.19	odd	2		1080.2.m.c.539.21	✓	48
24.5	odd	2		1080.2.m.c.539.23	yes	48
24.11	even	2	inner	4320.2.m.c.2159.5		48
40.19	odd	2	inner	4320.2.m.c.2159.42		48
40.29	even	2		1080.2.m.c.539.24	yes	48
60.59	even	2		1080.2.m.c.539.28	yes	48
120.29	odd	2		1080.2.m.c.539.25	yes	48
120.59	even	2	inner	4320.2.m.c.2159.8		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.m.c.539.21	✓	48	20.19	odd	2
1080.2.m.c.539.22	yes	48	12.11	even	2
1080.2.m.c.539.23	yes	48	24.5	odd	2
1080.2.m.c.539.24	yes	48	40.29	even	2
1080.2.m.c.539.25	yes	48	120.29	odd	2
1080.2.m.c.539.26	yes	48	8.5	even	2
1080.2.m.c.539.27	yes	48	4.3	odd	2
1080.2.m.c.539.28	yes	48	60.59	even	2
4320.2.m.c.2159.5		48	24.11	even	2	inner
4320.2.m.c.2159.6		48	1.1	even	1	trivial
4320.2.m.c.2159.7		48	5.4	even	2	inner
4320.2.m.c.2159.8		48	120.59	even	2	inner
4320.2.m.c.2159.41		48	15.14	odd	2	inner
4320.2.m.c.2159.42		48	40.19	odd	2	inner
4320.2.m.c.2159.43		48	8.3	odd	2	inner
4320.2.m.c.2159.44		48	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:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.13531 - 0.663669i) q^{5} +4.39866 q^{7} +O(q^{10})\)	\(q+(-2.13531 - 0.663669i) q^{5} +4.39866 q^{7} -0.633050i q^{11} +3.40649 q^{13} +5.48913 q^{17} -0.323423 q^{19} +6.26839i q^{23} +(4.11909 + 2.83428i) q^{25} -3.77896 q^{29} -7.86723i q^{31} +(-9.39250 - 2.91926i) q^{35} +5.90433 q^{37} -0.877482i q^{41} -2.31397i q^{43} +8.67090i q^{47} +12.3482 q^{49} -7.09416i q^{53} +(-0.420136 + 1.35176i) q^{55} +10.5321i q^{59} +14.0673i q^{61} +(-7.27391 - 2.26078i) q^{65} +5.99382i q^{67} -10.7806 q^{71} -13.9654i q^{73} -2.78457i q^{77} +7.33728i q^{79} -1.90716 q^{83} +(-11.7210 - 3.64297i) q^{85} -8.16899i q^{89} +14.9840 q^{91} +(0.690609 + 0.214646i) q^{95} -6.27222i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 16 q^{19} + 48 q^{49}+O(q^{100}) \) 48 * q + 16 * q^19 + 48 * q^49

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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