LMFDB - Embedded newform 4320.2.m.b.2159.8

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:	$40$
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.94670 + 1.10016i) q^{5} +1.41383 q^{7} -1.85786i q^{11} -5.28485 q^{13} +3.39687 q^{17} -0.958964 q^{19} +1.26014i q^{23} +(2.57929 - 4.28337i) q^{25} +6.97811 q^{29} +8.83008i q^{31} +(-2.75230 + 1.55544i) q^{35} -7.01845 q^{37} -9.81991i q^{41} +7.74226i q^{43} -6.43323i q^{47} -5.00109 q^{49} -1.38393i q^{53} +(2.04394 + 3.61669i) q^{55} -1.37865i q^{59} -6.88857i q^{61} +(10.2880 - 5.81419i) q^{65} -6.08777i q^{67} -3.62010 q^{71} +9.30568i q^{73} -2.62669i q^{77} -7.07671i q^{79} +8.12610 q^{83} +(-6.61269 + 3.73711i) q^{85} -16.1673i q^{89} -7.47186 q^{91} +(1.86682 - 1.05502i) q^{95} -16.1841i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 24 q^{19} + 40 q^{25} + 104 q^{49}+O(q^{100}) $ 40 * q - 24 * q^19 + 40 * q^25 + 104 * q^49

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
$5$	−1.94670	+	1.10016i	−0.870591	+	0.492007i
$5$	−1.94670	+	1.10016i	−0.870591	+	0.492007i
$6$		0			0
$7$	1.41383			0.534377			0.267188	−	0.963644i	$-0.413905\pi$
$7$	1.41383			0.534377			0.267188	+	0.963644i	$0.413905\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.85786i		−	0.560165i	−0.959976	−	0.280082i	$-0.909638\pi$
$11$		−	1.85786i		−	0.560165i	0.959976	−	0.280082i	$-0.0903618\pi$
$12$		0			0
$13$	−5.28485			−1.46575			−0.732876	−	0.680362i	$-0.761823\pi$
$13$	−5.28485			−1.46575			−0.732876	+	0.680362i	$0.761823\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.39687			0.823862			0.411931	−	0.911215i	$-0.364855\pi$
$17$	3.39687			0.823862			0.411931	+	0.911215i	$0.364855\pi$
$18$		0			0
$19$	−0.958964			−0.220002			−0.110001	−	0.993932i	$-0.535085\pi$
$19$	−0.958964			−0.220002			−0.110001	+	0.993932i	$0.535085\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.26014i			0.262757i	0.991332	+	0.131378i	$0.0419403\pi$
$23$			1.26014i			0.262757i	−0.991332	+	0.131378i	$0.958060\pi$
$24$		0			0
$25$	2.57929	−	4.28337i	0.515858	−	0.856674i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.97811			1.29580			0.647901	−	0.761725i	$-0.275647\pi$
$29$	6.97811			1.29580			0.647901	+	0.761725i	$0.275647\pi$
$30$		0			0
$31$			8.83008i			1.58593i	0.609267	+	0.792965i	$0.291464\pi$
$31$			8.83008i			1.58593i	−0.609267	+	0.792965i	$0.708536\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.75230	+	1.55544i	−0.465224	+	0.262917i
$36$		0			0
$37$	−7.01845			−1.15383			−0.576913	−	0.816805i	$-0.695743\pi$
$37$	−7.01845			−1.15383			−0.576913	+	0.816805i	$0.695743\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	9.81991i		−	1.53361i	−0.641879	−	0.766806i	$-0.721845\pi$
$41$		−	9.81991i		−	1.53361i	0.641879	−	0.766806i	$-0.278155\pi$
$42$		0			0
$43$			7.74226i			1.18068i	0.807153	+	0.590342i	$0.201007\pi$
$43$			7.74226i			1.18068i	−0.807153	+	0.590342i	$0.798993\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.43323i		−	0.938383i	−0.883096	−	0.469192i	$-0.844545\pi$
$47$		−	6.43323i		−	0.938383i	0.883096	−	0.469192i	$-0.155455\pi$
$48$		0			0
$49$	−5.00109			−0.714442
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1.38393i		−	0.190098i	−0.995473	−	0.0950488i	$-0.969699\pi$
$53$		−	1.38393i		−	0.190098i	0.995473	−	0.0950488i	$-0.0303007\pi$
$54$		0			0
$55$	2.04394	+	3.61669i	0.275605	+	0.487675i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	1.37865i		−	0.179485i	−0.995965	−	0.0897426i	$-0.971396\pi$
$59$		−	1.37865i		−	0.179485i	0.995965	−	0.0897426i	$-0.0286044\pi$
$60$		0			0
$61$		−	6.88857i		−	0.881991i	−0.897509	−	0.440996i	$-0.854625\pi$
$61$		−	6.88857i		−	0.881991i	0.897509	−	0.440996i	$-0.145375\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.2880	−	5.81419i	1.27607	−	0.721161i
$66$		0			0
$67$		−	6.08777i		−	0.743740i	−0.928285	−	0.371870i	$-0.878717\pi$
$67$		−	6.08777i		−	0.743740i	0.928285	−	0.371870i	$-0.121283\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.62010			−0.429627			−0.214813	−	0.976655i	$-0.568914\pi$
$71$	−3.62010			−0.429627			−0.214813	+	0.976655i	$0.568914\pi$
$72$		0			0
$73$			9.30568i			1.08915i	0.838713	+	0.544573i	$0.183308\pi$
$73$			9.30568i			1.08915i	−0.838713	+	0.544573i	$0.816692\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.62669i		−	0.299339i
$78$		0			0
$79$		−	7.07671i		−	0.796192i	−0.917344	−	0.398096i	$-0.869671\pi$
$79$		−	7.07671i		−	0.796192i	0.917344	−	0.398096i	$-0.130329\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.12610			0.891956			0.445978	−	0.895044i	$-0.352856\pi$
$83$	8.12610			0.891956			0.445978	+	0.895044i	$0.352856\pi$
$84$		0			0
$85$	−6.61269	+	3.73711i	−0.717247	+	0.405346i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	16.1673i		−	1.71373i	−0.515541	−	0.856865i	$-0.672409\pi$
$89$		−	16.1673i		−	1.71373i	0.515541	−	0.856865i	$-0.327591\pi$
$90$		0			0
$91$	−7.47186			−0.783264
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.86682	−	1.05502i	0.191531	−	0.108242i
$96$		0			0
$97$		−	16.1841i		−	1.64325i	−0.570031	−	0.821623i	$-0.693069\pi$
$97$		−	16.1841i		−	1.64325i	0.570031	−	0.821623i	$-0.306931\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.26292			0.822191			0.411096	−	0.911592i	$-0.365146\pi$
$101$	8.26292			0.822191			0.411096	+	0.911592i	$0.365146\pi$
$102$		0			0
$103$	7.01845			0.691549			0.345774	−	0.938318i	$-0.387616\pi$
$103$	7.01845			0.691549			0.345774	+	0.938318i	$0.387616\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.6459			1.31920			0.659601	−	0.751616i	$-0.270725\pi$
$107$	13.6459			1.31920			0.659601	+	0.751616i	$0.270725\pi$
$108$		0			0
$109$		−	3.05122i		−	0.292254i	−0.989266	−	0.146127i	$-0.953319\pi$
$109$		−	3.05122i		−	0.292254i	0.989266	−	0.146127i	$-0.0466808\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.34103			0.314298			0.157149	−	0.987575i	$-0.449770\pi$
$113$	3.34103			0.314298			0.157149	+	0.987575i	$0.449770\pi$
$114$		0			0
$115$	−1.38636	−	2.45311i	−0.129278	−	0.228754i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.80259			0.440253
$120$		0			0
$121$	7.54837			0.686215
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.308701	+	11.1761i	−0.0276110	+	0.999619i
$126$		0			0
$127$	10.2948			0.913517			0.456758	−	0.889591i	$-0.349010\pi$
$127$	10.2948			0.913517			0.456758	+	0.889591i	$0.349010\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	11.1492i		−	0.974107i	−0.873372	−	0.487054i	$-0.838072\pi$
$131$		−	11.1492i		−	0.974107i	0.873372	−	0.487054i	$-0.161928\pi$
$132$		0			0
$133$	−1.35581			−0.117564
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.44890			−0.807274			−0.403637	−	0.914919i	$-0.632254\pi$
$137$	−9.44890			−0.807274			−0.403637	+	0.914919i	$0.632254\pi$
$138$		0			0
$139$	14.7721			1.25295			0.626477	−	0.779440i	$-0.284496\pi$
$139$	14.7721			1.25295			0.626477	+	0.779440i	$0.284496\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			9.81849i			0.821063i
$144$		0			0
$145$	−13.5843	+	7.67705i	−1.12811	+	0.637544i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−15.2225			−1.24708			−0.623540	−	0.781792i	$-0.714306\pi$
$149$	−15.2225			−1.24708			−0.623540	+	0.781792i	$0.714306\pi$
$150$		0			0
$151$			13.8353i			1.12590i	0.826491	+	0.562951i	$0.190334\pi$
$151$			13.8353i			1.12590i	−0.826491	+	0.562951i	$0.809666\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.71452	−	17.1895i	−0.780289	−	1.38070i
$156$		0			0
$157$	7.94053			0.633723			0.316862	−	0.948472i	$-0.397371\pi$
$157$	7.94053			0.633723			0.316862	+	0.948472i	$0.397371\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			1.78162i			0.140411i
$162$		0			0
$163$			9.26632i			0.725794i	0.931829	+	0.362897i	$0.118212\pi$
$163$			9.26632i			0.725794i	−0.931829	+	0.362897i	$0.881788\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			24.7574i			1.91578i	0.287129	+	0.957892i	$0.407299\pi$
$167$			24.7574i			1.91578i	−0.287129	+	0.957892i	$0.592701\pi$
$168$		0			0
$169$	14.9296			1.14843
$170$		0			0
$171$		0			0
$172$		0			0
$173$			2.42769i			0.184574i	0.995732	+	0.0922870i	$0.0294177\pi$
$173$			2.42769i			0.184574i	−0.995732	+	0.0922870i	$0.970582\pi$
$174$		0			0
$175$	3.64667	−	6.05595i	0.275662	−	0.457787i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	26.4664i		−	1.97819i	−0.147272	−	0.989096i	$-0.547049\pi$
$179$		−	26.4664i		−	1.97819i	0.147272	−	0.989096i	$-0.452951\pi$
$180$		0			0
$181$		−	16.4205i		−	1.22052i	−0.792199	−	0.610262i	$-0.791064\pi$
$181$		−	16.4205i		−	1.22052i	0.792199	−	0.610262i	$-0.208936\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	13.6628	−	7.72143i	1.00451	−	0.567691i
$186$		0			0
$187$		−	6.31090i		−	0.461498i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.528127			−0.0382140			−0.0191070	−	0.999817i	$-0.506082\pi$
$191$	−0.528127			−0.0382140			−0.0191070	+	0.999817i	$0.506082\pi$
$192$		0			0
$193$		−	3.21790i		−	0.231630i	−0.993271	−	0.115815i	$-0.963052\pi$
$193$		−	3.21790i		−	0.231630i	0.993271	−	0.115815i	$-0.0369479\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			1.79418i			0.127830i	0.997955	+	0.0639149i	$0.0203586\pi$
$197$			1.79418i			0.127830i	−0.997955	+	0.0639149i	$0.979641\pi$
$198$		0			0
$199$		−	16.5191i		−	1.17101i	−0.810669	−	0.585505i	$-0.800896\pi$
$199$		−	16.5191i		−	1.17101i	0.810669	−	0.585505i	$-0.199104\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.86584			0.692446
$204$		0			0
$205$	10.8035	+	19.1164i	0.754548	+	1.33515i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			1.78162i			0.123237i
$210$		0			0
$211$	−12.8886			−0.887286			−0.443643	−	0.896204i	$-0.646314\pi$
$211$	−12.8886			−0.887286			−0.443643	+	0.896204i	$0.646314\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.51774	−	15.0719i	−0.580905	−	1.02789i
$216$		0			0
$217$			12.4842i			0.847484i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−17.9519			−1.20758
$222$		0			0
$223$	7.01845			0.469990			0.234995	−	0.971997i	$-0.424493\pi$
$223$	7.01845			0.469990			0.234995	+	0.971997i	$0.424493\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	26.0959			1.73204			0.866022	−	0.500006i	$-0.166669\pi$
$227$	26.0959			1.73204			0.866022	+	0.500006i	$0.166669\pi$
$228$		0			0
$229$		−	27.3805i		−	1.80935i	−0.426098	−	0.904677i	$-0.640112\pi$
$229$		−	27.3805i		−	1.80935i	0.426098	−	0.904677i	$-0.359888\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.60062			0.628957			0.314479	−	0.949265i	$-0.398170\pi$
$233$	9.60062			0.628957			0.314479	+	0.949265i	$0.398170\pi$
$234$		0			0
$235$	7.07760	+	12.5236i	0.461691	+	0.816948i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	17.6898			1.14426			0.572130	−	0.820163i	$-0.306117\pi$
$239$	17.6898			1.14426			0.572130	+	0.820163i	$0.306117\pi$
$240$		0			0
$241$	3.99330			0.257231			0.128615	−	0.991695i	$-0.458947\pi$
$241$	3.99330			0.257231			0.128615	+	0.991695i	$0.458947\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.73563	−	5.50201i	0.621986	−	0.351510i
$246$		0			0
$247$	5.06798			0.322468
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	19.6398i		−	1.23965i	−0.784738	−	0.619827i	$-0.787203\pi$
$251$		−	19.6398i		−	1.23965i	0.784738	−	0.619827i	$-0.212797\pi$
$252$		0			0
$253$	2.34116			0.147187
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.736991			−0.0459723			−0.0229861	−	0.999736i	$-0.507317\pi$
$257$	−0.736991			−0.0459723			−0.0229861	+	0.999736i	$0.507317\pi$
$258$		0			0
$259$	−9.92289			−0.616578
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	10.6712i		−	0.658016i	−0.944327	−	0.329008i	$-0.893286\pi$
$263$		−	10.6712i		−	0.658016i	0.944327	−	0.329008i	$-0.106714\pi$
$264$		0			0
$265$	1.52255	+	2.69410i	0.0935294	+	0.165497i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	22.0409			1.34386			0.671928	−	0.740616i	$-0.265466\pi$
$269$	22.0409			1.34386			0.671928	+	0.740616i	$0.265466\pi$
$270$		0			0
$271$			14.4928i			0.880373i	0.897906	+	0.440186i	$0.145088\pi$
$271$			14.4928i			0.880373i	−0.897906	+	0.440186i	$0.854912\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.95789	−	4.79195i	−0.479879	−	0.288965i
$276$		0			0
$277$	−8.92588			−0.536304			−0.268152	−	0.963377i	$-0.586413\pi$
$277$	−8.92588			−0.536304			−0.268152	+	0.963377i	$0.586413\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	11.9703i		−	0.714091i	−0.934087	−	0.357046i	$-0.883784\pi$
$281$		−	11.9703i		−	0.714091i	0.934087	−	0.357046i	$-0.116216\pi$
$282$		0			0
$283$		−	17.0086i		−	1.01106i	−0.862810	−	0.505528i	$-0.831298\pi$
$283$		−	17.0086i		−	1.01106i	0.862810	−	0.505528i	$-0.168702\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	13.8837i		−	0.819526i
$288$		0			0
$289$	−5.46128			−0.321252
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	9.28221i		−	0.542272i	−0.962541	−	0.271136i	$-0.912601\pi$
$293$		−	9.28221i		−	0.542272i	0.962541	−	0.271136i	$-0.0873994\pi$
$294$		0			0
$295$	1.51674	+	2.68382i	0.0883080	+	0.156258i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	6.65964i		−	0.385137i
$300$		0			0
$301$			10.9462i			0.630930i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7.57854	+	13.4100i	0.433946	+	0.767854i
$306$		0			0
$307$			28.2257i			1.61093i	0.592646	+	0.805463i	$0.298083\pi$
$307$			28.2257i			1.61093i	−0.592646	+	0.805463i	$0.701917\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	27.3716			1.55210			0.776050	−	0.630671i	$-0.217220\pi$
$311$	27.3716			1.55210			0.776050	+	0.630671i	$0.217220\pi$
$312$		0			0
$313$		−	27.1781i		−	1.53620i	−0.640332	−	0.768098i	$-0.721203\pi$
$313$		−	27.1781i		−	1.53620i	0.640332	−	0.768098i	$-0.278797\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2.01745i		−	0.113311i	−0.998394	−	0.0566555i	$-0.981956\pi$
$317$		−	2.01745i		−	0.113311i	0.998394	−	0.0566555i	$-0.0180437\pi$
$318$		0			0
$319$		−	12.9643i		−	0.725863i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.25748			−0.181251
$324$		0			0
$325$	−13.6311	+	22.6370i	−0.756120	+	1.25567i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	9.09548i		−	0.501450i
$330$		0			0
$331$	25.5573			1.40475			0.702377	−	0.711805i	$-0.252122\pi$
$331$	25.5573			1.40475			0.702377	+	0.711805i	$0.252122\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.69754	+	11.8511i	0.365925	+	0.647493i
$336$		0			0
$337$		−	2.38790i		−	0.130077i	−0.997883	−	0.0650385i	$-0.979283\pi$
$337$		−	2.38790i		−	0.130077i	0.997883	−	0.0650385i	$-0.0207170\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16.4050			0.888382
$342$		0			0
$343$	−16.9675			−0.916158
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.6403			0.946980			0.473490	−	0.880799i	$-0.342994\pi$
$347$	17.6403			0.946980			0.473490	+	0.880799i	$0.342994\pi$
$348$		0			0
$349$		−	3.06501i		−	0.164066i	−0.996630	−	0.0820332i	$-0.973859\pi$
$349$		−	3.06501i		−	0.164066i	0.996630	−	0.0820332i	$-0.0261414\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.9754			0.584162			0.292081	−	0.956394i	$-0.405652\pi$
$353$	10.9754			0.584162			0.292081	+	0.956394i	$0.405652\pi$
$354$		0			0
$355$	7.04725	−	3.98269i	0.374029	−	0.211379i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−21.1071			−1.11399			−0.556996	−	0.830515i	$-0.688046\pi$
$359$	−21.1071			−1.11399			−0.556996	+	0.830515i	$0.688046\pi$
$360$		0			0
$361$	−18.0804			−0.951599
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−10.2377	−	18.1154i	−0.535868	−	0.948202i
$366$		0			0
$367$	−26.1942			−1.36733			−0.683664	−	0.729797i	$-0.739615\pi$
$367$	−26.1942			−1.36733			−0.683664	+	0.729797i	$0.739615\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	1.95664i		−	0.101584i
$372$		0			0
$373$	6.92866			0.358753			0.179376	−	0.983781i	$-0.442592\pi$
$373$	6.92866			0.358753			0.179376	+	0.983781i	$0.442592\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−36.8782			−1.89932
$378$		0			0
$379$	−23.3481			−1.19931			−0.599656	−	0.800258i	$-0.704696\pi$
$379$	−23.3481			−1.19931			−0.599656	+	0.800258i	$0.704696\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	10.6684i		−	0.545129i	−0.962138	−	0.272564i	$-0.912128\pi$
$383$		−	10.6684i		−	0.545129i	0.962138	−	0.272564i	$-0.0878718\pi$
$384$		0			0
$385$	2.88978	+	5.11338i	0.147277	+	0.260602i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.8584			−0.804054			−0.402027	−	0.915628i	$-0.631694\pi$
$389$	−15.8584			−0.804054			−0.402027	+	0.915628i	$0.631694\pi$
$390$		0			0
$391$			4.28052i			0.216475i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7.78553	+	13.7762i	0.391732	+	0.693158i
$396$		0			0
$397$	22.5850			1.13351			0.566754	−	0.823887i	$-0.308199\pi$
$397$	22.5850			1.13351			0.566754	+	0.823887i	$0.308199\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	15.2185i		−	0.759973i	−0.924992	−	0.379987i	$-0.875929\pi$
$401$		−	15.2185i		−	0.759973i	0.924992	−	0.379987i	$-0.124071\pi$
$402$		0			0
$403$		−	46.6656i		−	2.32458i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			13.0393i			0.646333i
$408$		0			0
$409$	−16.1485			−0.798494			−0.399247	−	0.916843i	$-0.630728\pi$
$409$	−16.1485			−0.798494			−0.399247	+	0.916843i	$0.630728\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	1.94918i		−	0.0959127i
$414$		0			0
$415$	−15.8191	+	8.94003i	−0.776529	+	0.438849i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			12.9802i			0.634123i	0.948405	+	0.317062i	$0.102696\pi$
$419$			12.9802i			0.634123i	−0.948405	+	0.317062i	$0.897304\pi$
$420$		0			0
$421$			3.77558i			0.184010i	0.995759	+	0.0920052i	$0.0293276\pi$
$421$			3.77558i			0.184010i	−0.995759	+	0.0920052i	$0.970672\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	8.76150	−	14.5501i	0.424995	−	0.705781i
$426$		0			0
$427$		−	9.73926i		−	0.471316i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9.99810			−0.481591			−0.240796	−	0.970576i	$-0.577408\pi$
$431$	−9.99810			−0.481591			−0.240796	+	0.970576i	$0.577408\pi$
$432$		0			0
$433$			18.9200i			0.909238i	0.890686	+	0.454619i	$0.150225\pi$
$433$			18.9200i			0.909238i	−0.890686	+	0.454619i	$0.849775\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	1.20843i		−	0.0578069i
$438$		0			0
$439$		−	35.3788i		−	1.68854i	−0.535920	−	0.844269i	$-0.680035\pi$
$439$		−	35.3788i		−	1.68854i	0.535920	−	0.844269i	$-0.319965\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−24.6061			−1.16907			−0.584536	−	0.811368i	$-0.698723\pi$
$443$	−24.6061			−1.16907			−0.584536	+	0.811368i	$0.698723\pi$
$444$		0			0
$445$	17.7866	+	31.4729i	0.843168	+	1.49196i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			4.53892i			0.214205i	0.994248	+	0.107102i	$0.0341573\pi$
$449$			4.53892i			0.214205i	−0.994248	+	0.107102i	$0.965843\pi$
$450$		0			0
$451$	−18.2440			−0.859076
$452$		0			0
$453$		0			0
$454$		0			0
$455$	14.5455	−	8.22026i	0.681903	−	0.385372i
$456$		0			0
$457$			8.34915i			0.390557i	0.980748	+	0.195278i	$0.0625610\pi$
$457$			8.34915i			0.390557i	−0.980748	+	0.195278i	$0.937439\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−10.5909			−0.493269			−0.246635	−	0.969109i	$-0.579325\pi$
$461$	−10.5909			−0.493269			−0.246635	+	0.969109i	$0.579325\pi$
$462$		0			0
$463$	−20.7805			−0.965751			−0.482875	−	0.875689i	$-0.660408\pi$
$463$	−20.7805			−0.965751			−0.482875	+	0.875689i	$0.660408\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	15.4779			0.716233			0.358116	−	0.933677i	$-0.383419\pi$
$467$	15.4779			0.716233			0.358116	+	0.933677i	$0.383419\pi$
$468$		0			0
$469$		−	8.60706i		−	0.397437i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	14.3840			0.661378
$474$		0			0
$475$	−2.47345	+	4.10760i	−0.113489	+	0.188470i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−38.1553			−1.74336			−0.871680	−	0.490075i	$-0.836969\pi$
$479$	−38.1553			−1.74336			−0.871680	+	0.490075i	$0.836969\pi$
$480$		0			0
$481$	37.0915			1.69122
$482$		0			0
$483$		0			0
$484$		0			0
$485$	17.8051	+	31.5056i	0.808489	+	1.43060i
$486$		0			0
$487$	−32.2405			−1.46096			−0.730478	−	0.682937i	$-0.760703\pi$
$487$	−32.2405			−1.46096			−0.730478	+	0.682937i	$0.760703\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			12.2064i			0.550865i	0.961320	+	0.275433i	$0.0888211\pi$
$491$			12.2064i			0.550865i	−0.961320	+	0.275433i	$0.911179\pi$
$492$		0			0
$493$	23.7037			1.06756
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5.11819			−0.229582
$498$		0			0
$499$	38.3721			1.71777			0.858886	−	0.512167i	$-0.171157\pi$
$499$	38.3721			1.71777			0.858886	+	0.512167i	$0.171157\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			10.4283i			0.464974i	0.972600	+	0.232487i	$0.0746863\pi$
$503$			10.4283i			0.464974i	−0.972600	+	0.232487i	$0.925314\pi$
$504$		0			0
$505$	−16.0854	+	9.09055i	−0.715792	+	0.404524i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.51620			0.421798			0.210899	−	0.977508i	$-0.432361\pi$
$509$	9.51620			0.421798			0.210899	+	0.977508i	$0.432361\pi$
$510$		0			0
$511$			13.1566i			0.582015i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−13.6628	+	7.72143i	−0.602056	+	0.340247i
$516$		0			0
$517$	−11.9520			−0.525649
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	12.3260i		−	0.540010i	−0.962859	−	0.270005i	$-0.912975\pi$
$521$		−	12.3260i		−	0.540010i	0.962859	−	0.270005i	$-0.0870253\pi$
$522$		0			0
$523$			26.5712i			1.16188i	0.813947	+	0.580939i	$0.197315\pi$
$523$			26.5712i			1.16188i	−0.813947	+	0.580939i	$0.802685\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			29.9946i			1.30659i
$528$		0			0
$529$	21.4121			0.930959
$530$		0			0
$531$		0			0
$532$		0			0
$533$			51.8967i			2.24790i
$534$		0			0
$535$	−26.5646	+	15.0127i	−1.14849	+	0.649057i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			9.29131i			0.400205i
$540$		0			0
$541$			36.6675i			1.57646i	0.615382	+	0.788229i	$0.289002\pi$
$541$			36.6675i			1.57646i	−0.615382	+	0.788229i	$0.710998\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.35684	+	5.93982i	0.143791	+	0.254434i
$546$		0			0
$547$		−	27.7063i		−	1.18463i	−0.805705	−	0.592317i	$-0.798213\pi$
$547$		−	27.7063i		−	1.18463i	0.805705	−	0.592317i	$-0.201787\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.69175			−0.285078
$552$		0			0
$553$		−	10.0052i		−	0.425466i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	17.7763i		−	0.753206i	−0.926375	−	0.376603i	$-0.877092\pi$
$557$		−	17.7763i		−	0.753206i	0.926375	−	0.376603i	$-0.122908\pi$
$558$		0			0
$559$		−	40.9167i		−	1.73059i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−36.7044			−1.54690			−0.773452	−	0.633854i	$-0.781472\pi$
$563$	−36.7044			−1.54690			−0.773452	+	0.633854i	$0.781472\pi$
$564$		0			0
$565$	−6.50399	+	3.67567i	−0.273625	+	0.154637i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	5.97856i		−	0.250634i	−0.992117	−	0.125317i	$-0.960005\pi$
$569$		−	5.97856i		−	0.250634i	0.992117	−	0.125317i	$-0.0399949\pi$
$570$		0			0
$571$	−26.6100			−1.11359			−0.556796	−	0.830649i	$-0.687970\pi$
$571$	−26.6100			−1.11359			−0.556796	+	0.830649i	$0.687970\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.39764	+	3.25026i	0.225097	+	0.135545i
$576$		0			0
$577$			47.2797i			1.96828i	0.177398	+	0.984139i	$0.443232\pi$
$577$			47.2797i			1.96828i	−0.177398	+	0.984139i	$0.556768\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	11.4889			0.476640
$582$		0			0
$583$	−2.57115			−0.106486
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9.59894			0.396191			0.198095	−	0.980183i	$-0.436524\pi$
$587$	9.59894			0.396191			0.198095	+	0.980183i	$0.436524\pi$
$588$		0			0
$589$		−	8.46774i		−	0.348907i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	34.4142			1.41322			0.706610	−	0.707603i	$-0.250223\pi$
$593$	34.4142			1.41322			0.706610	+	0.707603i	$0.250223\pi$
$594$		0			0
$595$	−9.34920	+	5.28362i	−0.383280	+	0.216607i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.71332			−0.151722			−0.0758611	−	0.997118i	$-0.524171\pi$
$599$	−3.71332			−0.151722			−0.0758611	+	0.997118i	$0.524171\pi$
$600$		0			0
$601$	−45.0267			−1.83668			−0.918338	−	0.395797i	$-0.870468\pi$
$601$	−45.0267			−1.83668			−0.918338	+	0.395797i	$0.870468\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−14.6944	+	8.30443i	−0.597413	+	0.337623i
$606$		0			0
$607$	4.98395			0.202292			0.101146	−	0.994872i	$-0.467749\pi$
$607$	4.98395			0.202292			0.101146	+	0.994872i	$0.467749\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			33.9986i			1.37544i
$612$		0			0
$613$	−14.1267			−0.570572			−0.285286	−	0.958442i	$-0.592089\pi$
$613$	−14.1267			−0.570572			−0.285286	+	0.958442i	$0.592089\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−34.3479			−1.38279			−0.691396	−	0.722476i	$-0.743004\pi$
$617$	−34.3479			−1.38279			−0.691396	+	0.722476i	$0.743004\pi$
$618$		0			0
$619$	35.3327			1.42014			0.710071	−	0.704130i	$-0.248663\pi$
$619$	35.3327			1.42014			0.710071	+	0.704130i	$0.248663\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	22.8578i		−	0.915777i
$624$		0			0
$625$	−11.6945	−	22.0961i	−0.467782	−	0.883844i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−23.8408			−0.950594
$630$		0			0
$631$			7.89084i			0.314129i	0.987588	+	0.157065i	$0.0502031\pi$
$631$			7.89084i			0.314129i	−0.987588	+	0.157065i	$0.949797\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−20.0409	+	11.3260i	−0.795299	+	0.449457i
$636$		0			0
$637$	26.4300			1.04719
$638$		0			0
$639$		0			0
$640$		0			0
$641$			48.6371i			1.92105i	0.278195	+	0.960525i	$0.410264\pi$
$641$			48.6371i			1.92105i	−0.278195	+	0.960525i	$0.589736\pi$
$642$		0			0
$643$			18.4399i			0.727200i	0.931555	+	0.363600i	$0.118453\pi$
$643$			18.4399i			0.727200i	−0.931555	+	0.363600i	$0.881547\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	36.7711i		−	1.44562i	−0.691046	−	0.722811i	$-0.742850\pi$
$647$		−	36.7711i		−	1.44562i	0.691046	−	0.722811i	$-0.257150\pi$
$648$		0			0
$649$	−2.56134			−0.100541
$650$		0			0
$651$		0			0
$652$		0			0
$653$			8.74915i			0.342381i	0.985238	+	0.171190i	$0.0547613\pi$
$653$			8.74915i			0.342381i	−0.985238	+	0.171190i	$0.945239\pi$
$654$		0			0
$655$	12.2659	+	21.7041i	0.479268	+	0.848049i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			13.4862i			0.525349i	0.964885	+	0.262674i	$0.0846045\pi$
$659$			13.4862i			0.525349i	−0.964885	+	0.262674i	$0.915395\pi$
$660$		0			0
$661$			11.8459i			0.460751i	0.973102	+	0.230375i	$0.0739954\pi$
$661$			11.8459i			0.460751i	−0.973102	+	0.230375i	$0.926005\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.63936	−	1.49161i	0.102350	−	0.0578422i
$666$		0			0
$667$			8.79338i			0.340481i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12.7980			−0.494061
$672$		0			0
$673$		−	3.43550i		−	0.132429i	−0.997805	−	0.0662144i	$-0.978908\pi$
$673$		−	3.43550i		−	0.132429i	0.997805	−	0.0662144i	$-0.0210921\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	40.8927i		−	1.57163i	−0.618459	−	0.785817i	$-0.712243\pi$
$677$		−	40.8927i		−	1.57163i	0.618459	−	0.785817i	$-0.287757\pi$
$678$		0			0
$679$		−	22.8815i		−	0.878112i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−44.9939			−1.72164			−0.860822	−	0.508906i	$-0.830050\pi$
$683$	−44.9939			−1.72164			−0.860822	+	0.508906i	$0.830050\pi$
$684$		0			0
$685$	18.3942	−	10.3953i	0.702806	−	0.397185i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			7.31386i			0.278636i
$690$		0			0
$691$	42.6781			1.62355			0.811776	−	0.583969i	$-0.198501\pi$
$691$	42.6781			1.62355			0.811776	+	0.583969i	$0.198501\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−28.7569	+	16.2517i	−1.09081	+	0.616463i
$696$		0			0
$697$		−	33.3569i		−	1.26348i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.25752			0.123035			0.0615174	−	0.998106i	$-0.480406\pi$
$701$	3.25752			0.123035			0.0615174	+	0.998106i	$0.480406\pi$
$702$		0			0
$703$	6.73045			0.253844
$704$		0			0
$705$		0			0
$706$		0			0
$707$	11.6823			0.439360
$708$		0			0
$709$		−	42.0351i		−	1.57866i	−0.613968	−	0.789331i	$-0.710428\pi$
$709$		−	42.0351i		−	1.57866i	0.613968	−	0.789331i	$-0.289572\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−11.1271			−0.416714
$714$		0			0
$715$	−10.8019	−	19.1137i	−0.403969	−	0.714810i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	42.2143			1.57433			0.787163	−	0.616745i	$-0.211549\pi$
$719$	42.2143			1.57433			0.787163	+	0.616745i	$0.211549\pi$
$720$		0			0
$721$	9.92289			0.369548
$722$		0			0
$723$		0			0
$724$		0			0
$725$	17.9985	−	29.8898i	0.668449	−	1.11008i
$726$		0			0
$727$	36.5526			1.35566			0.677830	−	0.735219i	$-0.262921\pi$
$727$	36.5526			1.35566			0.677830	+	0.735219i	$0.262921\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			26.2995i			0.972720i
$732$		0			0
$733$	27.9037			1.03065			0.515323	−	0.856996i	$-0.327672\pi$
$733$	27.9037			1.03065			0.515323	+	0.856996i	$0.327672\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.3102			−0.416617
$738$		0			0
$739$	−4.57103			−0.168148			−0.0840740	−	0.996460i	$-0.526793\pi$
$739$	−4.57103			−0.168148			−0.0840740	+	0.996460i	$0.526793\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	31.6650i		−	1.16168i	−0.814019	−	0.580839i	$-0.802725\pi$
$743$		−	31.6650i		−	1.16168i	0.814019	−	0.580839i	$-0.197275\pi$
$744$		0			0
$745$	29.6337	−	16.7473i	1.08570	−	0.613572i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	19.2930			0.704951
$750$		0			0
$751$		−	24.8129i		−	0.905434i	−0.891654	−	0.452717i	$-0.850455\pi$
$751$		−	24.8129i		−	0.905434i	0.891654	−	0.452717i	$-0.149545\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−15.2211	−	26.9332i	−0.553952	−	0.980200i
$756$		0			0
$757$	−25.9889			−0.944584			−0.472292	−	0.881442i	$-0.656573\pi$
$757$	−25.9889			−0.944584			−0.472292	+	0.881442i	$0.656573\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			33.5362i			1.21569i	0.794057	+	0.607843i	$0.207965\pi$
$761$			33.5362i			1.21569i	−0.794057	+	0.607843i	$0.792035\pi$
$762$		0			0
$763$		−	4.31390i		−	0.156174i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			7.28596i			0.263081i
$768$		0			0
$769$	6.84258			0.246750			0.123375	−	0.992360i	$-0.460628\pi$
$769$	6.84258			0.246750			0.123375	+	0.992360i	$0.460628\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	38.3944i		−	1.38095i	−0.723355	−	0.690476i	$-0.757401\pi$
$773$		−	38.3944i		−	1.38095i	0.723355	−	0.690476i	$-0.242599\pi$
$774$		0			0
$775$	37.8225	+	22.7753i	1.35863	+	0.818114i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			9.41694i			0.337397i
$780$		0			0
$781$			6.72562i			0.240662i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−15.4578	+	8.73587i	−0.551714	+	0.311796i
$786$		0			0
$787$			37.3993i			1.33314i	0.745442	+	0.666571i	$0.232239\pi$
$787$			37.3993i			1.33314i	−0.745442	+	0.666571i	$0.767761\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4.72364			0.167953
$792$		0			0
$793$			36.4051i			1.29278i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			29.6980i			1.05196i	0.850497	+	0.525979i	$0.176301\pi$
$797$			29.6980i			1.05196i	−0.850497	+	0.525979i	$0.823699\pi$
$798$		0			0
$799$		−	21.8528i		−	0.773098i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	17.2886			0.610102
$804$		0			0
$805$	−1.96007	−	3.46828i	−0.0690833	−	0.122241i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			41.0881i			1.44458i	0.691590	+	0.722290i	$0.256911\pi$
$809$			41.0881i			1.44458i	−0.691590	+	0.722290i	$0.743089\pi$
$810$		0			0
$811$	−30.7023			−1.07810			−0.539052	−	0.842272i	$-0.681217\pi$
$811$	−30.7023			−1.07810			−0.539052	+	0.842272i	$0.681217\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10.1944	−	18.0387i	−0.357096	−	0.631870i
$816$		0			0
$817$		−	7.42455i		−	0.259752i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13.3388			−0.465528			−0.232764	−	0.972533i	$-0.574777\pi$
$821$	−13.3388			−0.465528			−0.232764	+	0.972533i	$0.574777\pi$
$822$		0			0
$823$	8.08650			0.281877			0.140939	−	0.990018i	$-0.454988\pi$
$823$	8.08650			0.281877			0.140939	+	0.990018i	$0.454988\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	22.7070			0.789601			0.394800	−	0.918767i	$-0.370814\pi$
$827$	22.7070			0.789601			0.394800	+	0.918767i	$0.370814\pi$
$828$		0			0
$829$		−	46.8393i		−	1.62680i	−0.581708	−	0.813398i	$-0.697615\pi$
$829$		−	46.8393i		−	1.62680i	0.581708	−	0.813398i	$-0.302385\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−16.9881			−0.588601
$834$		0			0
$835$	−27.2371	−	48.1952i	−0.942580	−	1.66786i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	51.3436			1.77258			0.886289	−	0.463132i	$-0.153274\pi$
$839$	51.3436			1.77258			0.886289	+	0.463132i	$0.153274\pi$
$840$		0			0
$841$	19.6940			0.679102
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−29.0635	+	16.4250i	−0.999814	+	0.565036i
$846$		0			0
$847$	10.6721			0.366697
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	8.84422i		−	0.303176i
$852$		0			0
$853$	−48.8546			−1.67275			−0.836375	−	0.548158i	$-0.815329\pi$
$853$	−48.8546			−1.67275			−0.836375	+	0.548158i	$0.815329\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−23.8972			−0.816312			−0.408156	−	0.912912i	$-0.633828\pi$
$857$	−23.8972			−0.816312			−0.408156	+	0.912912i	$0.633828\pi$
$858$		0			0
$859$	−18.7193			−0.638696			−0.319348	−	0.947638i	$-0.603464\pi$
$859$	−18.7193			−0.638696			−0.319348	+	0.947638i	$0.603464\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			43.0074i			1.46399i	0.681311	+	0.731994i	$0.261410\pi$
$863$			43.0074i			1.46399i	−0.681311	+	0.731994i	$0.738590\pi$
$864$		0			0
$865$	−2.67085	−	4.72599i	−0.0908117	−	0.160688i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−13.1475			−0.445999
$870$		0			0
$871$			32.1729i			1.09014i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.436450	+	15.8010i	−0.0147547	+	0.534173i
$876$		0			0
$877$	21.5156			0.726531			0.363265	−	0.931686i	$-0.381662\pi$
$877$	21.5156			0.726531			0.363265	+	0.931686i	$0.381662\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			8.76271i			0.295223i	0.989045	+	0.147612i	$0.0471586\pi$
$881$			8.76271i			0.295223i	−0.989045	+	0.147612i	$0.952841\pi$
$882$		0			0
$883$			34.2208i			1.15162i	0.817583	+	0.575811i	$0.195313\pi$
$883$			34.2208i			1.15162i	−0.817583	+	0.575811i	$0.804687\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			3.61165i			0.121267i	0.998160	+	0.0606337i	$0.0193121\pi$
$887$			3.61165i			0.121267i	−0.998160	+	0.0606337i	$0.980688\pi$
$888$		0			0
$889$	14.5551			0.488162
$890$		0			0
$891$		0			0
$892$		0			0
$893$			6.16924i			0.206446i
$894$		0			0
$895$	29.1173	+	51.5222i	0.973285	+	1.72220i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			61.6173i			2.05505i
$900$		0			0
$901$		−	4.70103i		−	0.156614i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	18.0652	+	31.9658i	0.600507	+	1.06258i
$906$		0			0
$907$		−	0.296316i		−	0.00983900i	−0.999988	−	0.00491950i	$-0.998434\pi$
$907$		−	0.296316i		−	0.00983900i	0.999988	−	0.00491950i	$-0.00156593\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−38.2561			−1.26748			−0.633741	−	0.773545i	$-0.718482\pi$
$911$	−38.2561			−1.26748			−0.633741	+	0.773545i	$0.718482\pi$
$912$		0			0
$913$		−	15.0971i		−	0.499642i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	15.7630i		−	0.520540i
$918$		0			0
$919$		−	27.0814i		−	0.893332i	−0.894701	−	0.446666i	$-0.852611\pi$
$919$		−	27.0814i		−	0.893332i	0.894701	−	0.446666i	$-0.147389\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	19.1317			0.629726
$924$		0			0
$925$	−18.1026	+	30.0626i	−0.595210	+	0.988454i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			2.41533i			0.0792443i	0.999215	+	0.0396222i	$0.0126154\pi$
$929$			2.41533i			0.0792443i	−0.999215	+	0.0396222i	$0.987385\pi$
$930$		0			0
$931$	4.79587			0.157178
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.94301	+	12.2854i	0.227061	+	0.401776i
$936$		0			0
$937$		−	14.7062i		−	0.480432i	−0.970720	−	0.240216i	$-0.922782\pi$
$937$		−	14.7062i		−	0.480432i	0.970720	−	0.240216i	$-0.0772182\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	22.6540			0.738500			0.369250	−	0.929330i	$-0.379615\pi$
$941$	22.6540			0.738500			0.369250	+	0.929330i	$0.379615\pi$
$942$		0			0
$943$	12.3744			0.402967
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−39.0269			−1.26820			−0.634101	−	0.773250i	$-0.718630\pi$
$947$	−39.0269			−1.26820			−0.634101	+	0.773250i	$0.718630\pi$
$948$		0			0
$949$		−	49.1791i		−	1.59642i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−9.05724			−0.293393			−0.146696	−	0.989182i	$-0.546864\pi$
$953$	−9.05724			−0.293393			−0.146696	+	0.989182i	$0.546864\pi$
$954$		0			0
$955$	1.02811	−	0.581026i	0.0332687	−	0.0188015i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−13.3591			−0.431389
$960$		0			0
$961$	−46.9704			−1.51517
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3.54021	+	6.26429i	0.113963	+	0.201655i
$966$		0			0
$967$	0.330822			0.0106385			0.00531926	−	0.999986i	$-0.498307\pi$
$967$	0.330822			0.0106385			0.00531926	+	0.999986i	$0.498307\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			3.23651i			0.103865i	0.998651	+	0.0519323i	$0.0165380\pi$
$971$			3.23651i			0.103865i	−0.998651	+	0.0519323i	$0.983462\pi$
$972$		0			0
$973$	20.8852			0.669550
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−34.3972			−1.10046			−0.550231	−	0.835012i	$-0.685460\pi$
$977$	−34.3972			−1.10046			−0.550231	+	0.835012i	$0.685460\pi$
$978$		0			0
$979$	−30.0365			−0.959971
$980$		0			0
$981$		0			0
$982$		0			0
$983$			4.15590i			0.132553i	0.997801	+	0.0662763i	$0.0211119\pi$
$983$			4.15590i			0.132553i	−0.997801	+	0.0662763i	$0.978888\pi$
$984$		0			0
$985$	−1.97388	−	3.49272i	−0.0628932	−	0.111287i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−9.75632			−0.310233
$990$		0			0
$991$		−	21.0947i		−	0.670095i	−0.942201	−	0.335047i	$-0.891248\pi$
$991$		−	21.0947i		−	0.670095i	0.942201	−	0.335047i	$-0.108752\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	18.1737	+	32.1578i	0.576145	+	1.01947i
$996$		0			0
$997$	−27.0375			−0.856288			−0.428144	−	0.903711i	$-0.640832\pi$
$997$	−27.0375			−0.856288			−0.428144	+	0.903711i	$0.640832\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.m.b.2159.8		40
3.2	odd	2	inner	4320.2.m.b.2159.34		40
4.3	odd	2		1080.2.m.b.539.23	yes	40
5.4	even	2	inner	4320.2.m.b.2159.5		40
8.3	odd	2	inner	4320.2.m.b.2159.33		40
8.5	even	2		1080.2.m.b.539.22	yes	40
12.11	even	2		1080.2.m.b.539.18	yes	40
15.14	odd	2	inner	4320.2.m.b.2159.35		40
20.19	odd	2		1080.2.m.b.539.17	✓	40
24.5	odd	2		1080.2.m.b.539.19	yes	40
24.11	even	2	inner	4320.2.m.b.2159.7		40
40.19	odd	2	inner	4320.2.m.b.2159.36		40
40.29	even	2		1080.2.m.b.539.20	yes	40
60.59	even	2		1080.2.m.b.539.24	yes	40
120.29	odd	2		1080.2.m.b.539.21	yes	40
120.59	even	2	inner	4320.2.m.b.2159.6		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.m.b.539.17	✓	40	20.19	odd	2
1080.2.m.b.539.18	yes	40	12.11	even	2
1080.2.m.b.539.19	yes	40	24.5	odd	2
1080.2.m.b.539.20	yes	40	40.29	even	2
1080.2.m.b.539.21	yes	40	120.29	odd	2
1080.2.m.b.539.22	yes	40	8.5	even	2
1080.2.m.b.539.23	yes	40	4.3	odd	2
1080.2.m.b.539.24	yes	40	60.59	even	2
4320.2.m.b.2159.5		40	5.4	even	2	inner
4320.2.m.b.2159.6		40	120.59	even	2	inner
4320.2.m.b.2159.7		40	24.11	even	2	inner
4320.2.m.b.2159.8		40	1.1	even	1	trivial
4320.2.m.b.2159.33		40	8.3	odd	2	inner
4320.2.m.b.2159.34		40	3.2	odd	2	inner
4320.2.m.b.2159.35		40	15.14	odd	2	inner
4320.2.m.b.2159.36		40	40.19	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)

\(f(q)\)	\(=\)	\(q+(-1.94670 + 1.10016i) q^{5} +1.41383 q^{7} -1.85786i q^{11} -5.28485 q^{13} +3.39687 q^{17} -0.958964 q^{19} +1.26014i q^{23} +(2.57929 - 4.28337i) q^{25} +6.97811 q^{29} +8.83008i q^{31} +(-2.75230 + 1.55544i) q^{35} -7.01845 q^{37} -9.81991i q^{41} +7.74226i q^{43} -6.43323i q^{47} -5.00109 q^{49} -1.38393i q^{53} +(2.04394 + 3.61669i) q^{55} -1.37865i q^{59} -6.88857i q^{61} +(10.2880 - 5.81419i) q^{65} -6.08777i q^{67} -3.62010 q^{71} +9.30568i q^{73} -2.62669i q^{77} -7.07671i q^{79} +8.12610 q^{83} +(-6.61269 + 3.73711i) q^{85} -16.1673i q^{89} -7.47186 q^{91} +(1.86682 - 1.05502i) q^{95} -16.1841i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 24 q^{19} + 40 q^{25} + 104 q^{49}+O(q^{100}) \) 40 * q - 24 * q^19 + 40 * q^25 + 104 * q^49

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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