LMFDB - Embedded newform 4320.2.m.c.2159.15

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			2159.15
Character	$\chi$	$=$	4320.2159
Dual form			4320.2.m.c.2159.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.17345 + 1.90342i) q^{5} -0.450397 q^{7} -5.11202i q^{11} +1.15380 q^{13} +6.13070 q^{17} -2.16656 q^{19} -0.211029i q^{23} +(-2.24604 - 4.46713i) q^{25} -6.08377 q^{29} -0.417311i q^{31} +(0.528517 - 0.857296i) q^{35} -9.60013 q^{37} +9.71138i q^{41} -3.06432i q^{43} +2.65854i q^{47} -6.79714 q^{49} +8.81551i q^{53} +(9.73034 + 5.99869i) q^{55} +4.73857i q^{59} +7.64787i q^{61} +(-1.35392 + 2.19617i) q^{65} -6.49513i q^{67} +11.4430 q^{71} -13.8904i q^{73} +2.30244i q^{77} -13.7469i q^{79} -5.22116 q^{83} +(-7.19405 + 11.6693i) q^{85} -1.39700i q^{89} -0.519668 q^{91} +(2.54234 - 4.12388i) q^{95} -4.41107i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 16 q^{19} + 48 q^{49}+O(q^{100}) $ 48 * q + 16 * q^19 + 48 * 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
$5$	−1.17345	+	1.90342i	−0.524781	+	0.851237i
$5$	−1.17345	+	1.90342i	−0.524781	+	0.851237i
$6$		0			0
$7$	−0.450397			−0.170234			−0.0851170	−	0.996371i	$-0.527126\pi$
$7$	−0.450397			−0.170234			−0.0851170	+	0.996371i	$0.527126\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.11202i		−	1.54133i	−0.637239	−	0.770666i	$-0.719924\pi$
$11$		−	5.11202i		−	1.54133i	0.637239	−	0.770666i	$-0.280076\pi$
$12$		0			0
$13$	1.15380			0.320007			0.160003	−	0.987116i	$-0.448849\pi$
$13$	1.15380			0.320007			0.160003	+	0.987116i	$0.448849\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.13070			1.48691			0.743457	−	0.668784i	$-0.233185\pi$
$17$	6.13070			1.48691			0.743457	+	0.668784i	$0.233185\pi$
$18$		0			0
$19$	−2.16656			−0.497043			−0.248521	−	0.968626i	$-0.579945\pi$
$19$	−2.16656			−0.497043			−0.248521	+	0.968626i	$0.579945\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	0.211029i		−	0.0440025i	−0.999758	−	0.0220013i	$-0.992996\pi$
$23$		−	0.211029i		−	0.0440025i	0.999758	−	0.0220013i	$-0.00700379\pi$
$24$		0			0
$25$	−2.24604	−	4.46713i	−0.449209	−	0.893427i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.08377			−1.12973			−0.564864	−	0.825184i	$-0.691071\pi$
$29$	−6.08377			−1.12973			−0.564864	+	0.825184i	$0.691071\pi$
$30$		0			0
$31$		−	0.417311i		−	0.0749512i	−0.999298	−	0.0374756i	$-0.988068\pi$
$31$		−	0.417311i		−	0.0749512i	0.999298	−	0.0374756i	$-0.0119317\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.528517	−	0.857296i	0.0893356	−	0.144909i
$36$		0			0
$37$	−9.60013			−1.57825			−0.789126	−	0.614232i	$-0.789466\pi$
$37$	−9.60013			−1.57825			−0.789126	+	0.614232i	$0.789466\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			9.71138i			1.51666i	0.651869	+	0.758332i	$0.273985\pi$
$41$			9.71138i			1.51666i	−0.651869	+	0.758332i	$0.726015\pi$
$42$		0			0
$43$		−	3.06432i		−	0.467305i	−0.972320	−	0.233652i	$-0.924932\pi$
$43$		−	3.06432i		−	0.467305i	0.972320	−	0.233652i	$-0.0750678\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.65854i			0.387788i	0.981023	+	0.193894i	$0.0621117\pi$
$47$			2.65854i			0.387788i	−0.981023	+	0.193894i	$0.937888\pi$
$48$		0			0
$49$	−6.79714			−0.971020
$50$		0			0
$51$		0			0
$52$		0			0
$53$			8.81551i			1.21090i	0.795882	+	0.605452i	$0.207008\pi$
$53$			8.81551i			1.21090i	−0.795882	+	0.605452i	$0.792992\pi$
$54$		0			0
$55$	9.73034	+	5.99869i	1.31204	+	0.808863i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			4.73857i			0.616910i	0.951239	+	0.308455i	$0.0998119\pi$
$59$			4.73857i			0.616910i	−0.951239	+	0.308455i	$0.900188\pi$
$60$		0			0
$61$			7.64787i			0.979209i	0.871945	+	0.489605i	$0.162859\pi$
$61$			7.64787i			0.979209i	−0.871945	+	0.489605i	$0.837141\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.35392	+	2.19617i	−0.167934	+	0.272402i
$66$		0			0
$67$		−	6.49513i		−	0.793506i	−0.917925	−	0.396753i	$-0.870137\pi$
$67$		−	6.49513i		−	0.793506i	0.917925	−	0.396753i	$-0.129863\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	11.4430			1.35803			0.679016	−	0.734124i	$-0.262407\pi$
$71$	11.4430			1.35803			0.679016	+	0.734124i	$0.262407\pi$
$72$		0			0
$73$		−	13.8904i		−	1.62574i	−0.582443	−	0.812871i	$-0.697903\pi$
$73$		−	13.8904i		−	1.62574i	0.582443	−	0.812871i	$-0.302097\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.30244i			0.262387i
$78$		0			0
$79$		−	13.7469i		−	1.54664i	−0.634015	−	0.773321i	$-0.718594\pi$
$79$		−	13.7469i		−	1.54664i	0.634015	−	0.773321i	$-0.281406\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.22116			−0.573096			−0.286548	−	0.958066i	$-0.592508\pi$
$83$	−5.22116			−0.573096			−0.286548	+	0.958066i	$0.592508\pi$
$84$		0			0
$85$	−7.19405	+	11.6693i	−0.780305	+	1.26572i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1.39700i		−	0.148082i	−0.997255	−	0.0740408i	$-0.976411\pi$
$89$		−	1.39700i		−	0.148082i	0.997255	−	0.0740408i	$-0.0235895\pi$
$90$		0			0
$91$	−0.519668			−0.0544761
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.54234	−	4.12388i	0.260839	−	0.423101i
$96$		0			0
$97$		−	4.41107i		−	0.447876i	−0.974603	−	0.223938i	$-0.928109\pi$
$97$		−	4.41107i		−	0.447876i	0.974603	−	0.223938i	$-0.0718913\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.5630			−1.25006			−0.625031	−	0.780600i	$-0.714914\pi$
$101$	−12.5630			−1.25006			−0.625031	+	0.780600i	$0.714914\pi$
$102$		0			0
$103$	−5.95376			−0.586641			−0.293320	−	0.956014i	$-0.594760\pi$
$103$	−5.95376			−0.586641			−0.293320	+	0.956014i	$0.594760\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.98524			−0.868636			−0.434318	−	0.900760i	$-0.643011\pi$
$107$	−8.98524			−0.868636			−0.434318	+	0.900760i	$0.643011\pi$
$108$		0			0
$109$		−	11.7484i		−	1.12529i	−0.826697	−	0.562647i	$-0.809783\pi$
$109$		−	11.7484i		−	1.12529i	0.826697	−	0.562647i	$-0.190217\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.926078			−0.0871181			−0.0435590	−	0.999051i	$-0.513870\pi$
$113$	−0.926078			−0.0871181			−0.0435590	+	0.999051i	$0.513870\pi$
$114$		0			0
$115$	0.401677	+	0.247631i	0.0374566	+	0.0230917i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.76125			−0.253123
$120$		0			0
$121$	−15.1328			−1.37571
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.1385	+	0.966769i	0.996254	+	0.0864704i
$126$		0			0
$127$	−16.6083			−1.47375			−0.736873	−	0.676031i	$-0.763698\pi$
$127$	−16.6083			−1.47375			−0.736873	+	0.676031i	$0.763698\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	13.5426i		−	1.18323i	−0.806222	−	0.591613i	$-0.798491\pi$
$131$		−	13.5426i		−	1.18323i	0.806222	−	0.591613i	$-0.201509\pi$
$132$		0			0
$133$	0.975811			0.0846136
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.89160			−0.759661			−0.379830	−	0.925056i	$-0.624018\pi$
$137$	−8.89160			−0.759661			−0.379830	+	0.925056i	$0.624018\pi$
$138$		0			0
$139$	−14.2713			−1.21047			−0.605236	−	0.796046i	$-0.706921\pi$
$139$	−14.2713			−1.21047			−0.605236	+	0.796046i	$0.706921\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	5.89826i		−	0.493237i
$144$		0			0
$145$	7.13898	−	11.5800i	0.592860	−	0.961666i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.96593			−0.734518			−0.367259	−	0.930119i	$-0.619704\pi$
$149$	−8.96593			−0.734518			−0.367259	+	0.930119i	$0.619704\pi$
$150$		0			0
$151$			15.5560i			1.26593i	0.774180	+	0.632966i	$0.218163\pi$
$151$			15.5560i			1.26593i	−0.774180	+	0.632966i	$0.781837\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.794319	+	0.489692i	0.0638013	+	0.0393330i
$156$		0			0
$157$	13.2345			1.05623			0.528115	−	0.849173i	$-0.322899\pi$
$157$	13.2345			1.05623			0.528115	+	0.849173i	$0.322899\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			0.0950467i			0.00749073i
$162$		0			0
$163$		−	15.3659i		−	1.20355i	−0.798666	−	0.601774i	$-0.794461\pi$
$163$		−	15.3659i		−	1.20355i	0.798666	−	0.601774i	$-0.205539\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	17.9401i		−	1.38824i	−0.719857	−	0.694122i	$-0.755793\pi$
$167$		−	17.9401i		−	1.38824i	0.719857	−	0.694122i	$-0.244207\pi$
$168$		0			0
$169$	−11.6687			−0.897596
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	17.2609i		−	1.31232i	−0.754622	−	0.656160i	$-0.772180\pi$
$173$		−	17.2609i		−	1.31232i	0.754622	−	0.656160i	$-0.227820\pi$
$174$		0			0
$175$	1.01161	+	2.01198i	0.0764706	+	0.152092i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			0.229984i			0.0171898i	0.999963	+	0.00859490i	$0.00273588\pi$
$179$			0.229984i			0.0171898i	−0.999963	+	0.00859490i	$0.997264\pi$
$180$		0			0
$181$		−	2.26929i		−	0.168675i	−0.996437	−	0.0843377i	$-0.973123\pi$
$181$		−	2.26929i		−	0.168675i	0.996437	−	0.0843377i	$-0.0268774\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	11.2652	−	18.2731i	0.828237	−	1.34347i
$186$		0			0
$187$		−	31.3403i		−	2.29183i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	17.1688			1.24229			0.621145	−	0.783696i	$-0.286668\pi$
$191$	17.1688			1.24229			0.621145	+	0.783696i	$0.286668\pi$
$192$		0			0
$193$			0.900092i			0.0647901i	0.999475	+	0.0323950i	$0.0103135\pi$
$193$			0.900092i			0.0647901i	−0.999475	+	0.0323950i	$0.989687\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			1.61680i			0.115193i	0.998340	+	0.0575963i	$0.0183436\pi$
$197$			1.61680i			0.115193i	−0.998340	+	0.0575963i	$0.981656\pi$
$198$		0			0
$199$			13.9958i			0.992137i	0.868283	+	0.496069i	$0.165224\pi$
$199$			13.9958i			0.992137i	−0.868283	+	0.496069i	$0.834776\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.74011			0.192318
$204$		0			0
$205$	−18.4849	−	11.3958i	−1.29104	−	0.795917i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			11.0755i			0.766108i
$210$		0			0
$211$	−24.9023			−1.71435			−0.857174	−	0.515028i	$-0.827782\pi$
$211$	−24.9023			−1.71435			−0.857174	+	0.515028i	$0.827782\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.83270	+	3.59582i	0.397787	+	0.245233i
$216$		0			0
$217$			0.187956i			0.0127593i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.07361			0.475823
$222$		0			0
$223$	−19.2003			−1.28574			−0.642872	−	0.765974i	$-0.722257\pi$
$223$	−19.2003			−1.28574			−0.642872	+	0.765974i	$0.722257\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	28.6073			1.89874			0.949368	−	0.314167i	$-0.101725\pi$
$227$	28.6073			1.89874			0.949368	+	0.314167i	$0.101725\pi$
$228$		0			0
$229$			3.59647i			0.237662i	0.992915	+	0.118831i	$0.0379146\pi$
$229$			3.59647i			0.237662i	−0.992915	+	0.118831i	$0.962085\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.78036			0.640733			0.320366	−	0.947294i	$-0.396194\pi$
$233$	9.78036			0.640733			0.320366	+	0.947294i	$0.396194\pi$
$234$		0			0
$235$	−5.06033	−	3.11965i	−0.330099	−	0.203504i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−17.5597			−1.13584			−0.567921	−	0.823083i	$-0.692252\pi$
$239$	−17.5597			−1.13584			−0.567921	+	0.823083i	$0.692252\pi$
$240$		0			0
$241$	11.7884			0.759356			0.379678	−	0.925119i	$-0.376035\pi$
$241$	11.7884			0.759356			0.379678	+	0.925119i	$0.376035\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.97609	−	12.9378i	0.509573	−	0.826569i
$246$		0			0
$247$	−2.49978			−0.159057
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	12.8261i		−	0.809578i	−0.914410	−	0.404789i	$-0.867345\pi$
$251$		−	12.8261i		−	0.809578i	0.914410	−	0.404789i	$-0.132655\pi$
$252$		0			0
$253$	−1.07878			−0.0678226
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.70361			0.480538			0.240269	−	0.970706i	$-0.422764\pi$
$257$	7.70361			0.480538			0.240269	+	0.970706i	$0.422764\pi$
$258$		0			0
$259$	4.32387			0.268672
$260$		0			0
$261$		0			0
$262$		0			0
$263$			0.422058i			0.0260252i	0.999915	+	0.0130126i	$0.00414215\pi$
$263$			0.422058i			0.0260252i	−0.999915	+	0.0130126i	$0.995858\pi$
$264$		0			0
$265$	−16.7797	−	10.3445i	−1.03077	−	0.635460i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.49688			−0.0912663			−0.0456332	−	0.998958i	$-0.514531\pi$
$269$	−1.49688			−0.0912663			−0.0456332	+	0.998958i	$0.514531\pi$
$270$		0			0
$271$		−	13.1516i		−	0.798904i	−0.916754	−	0.399452i	$-0.869200\pi$
$271$		−	13.1516i		−	0.798904i	0.916754	−	0.399452i	$-0.130800\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−22.8361	+	11.4818i	−1.37707	+	0.692380i
$276$		0			0
$277$	28.3454			1.70311			0.851556	−	0.524263i	$-0.175659\pi$
$277$	28.3454			1.70311			0.851556	+	0.524263i	$0.175659\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			28.0322i			1.67226i	0.548529	+	0.836132i	$0.315188\pi$
$281$			28.0322i			1.67226i	−0.548529	+	0.836132i	$0.684812\pi$
$282$		0			0
$283$			4.04108i			0.240217i	0.992761	+	0.120109i	$0.0383243\pi$
$283$			4.04108i			0.240217i	−0.992761	+	0.120109i	$0.961676\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	4.37398i		−	0.258188i
$288$		0			0
$289$	20.5855			1.21091
$290$		0			0
$291$		0			0
$292$		0			0
$293$			9.69326i			0.566287i	0.959078	+	0.283143i	$0.0913772\pi$
$293$			9.69326i			0.566287i	−0.959078	+	0.283143i	$0.908623\pi$
$294$		0			0
$295$	−9.01951	−	5.56046i	−0.525136	−	0.323743i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	0.243485i		−	0.0140811i
$300$		0			0
$301$			1.38016i			0.0795512i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−14.5571	−	8.97437i	−0.833539	−	0.513871i
$306$		0			0
$307$		−	30.4121i		−	1.73571i	−0.496815	−	0.867856i	$-0.665497\pi$
$307$		−	30.4121i		−	1.73571i	0.496815	−	0.867856i	$-0.334503\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.93254			−0.109584			−0.0547921	−	0.998498i	$-0.517450\pi$
$311$	−1.93254			−0.109584			−0.0547921	+	0.998498i	$0.517450\pi$
$312$		0			0
$313$		−	22.8246i		−	1.29012i	−0.764130	−	0.645062i	$-0.776832\pi$
$313$		−	22.8246i		−	1.29012i	0.764130	−	0.645062i	$-0.223168\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	3.60376i		−	0.202407i	−0.994866	−	0.101204i	$-0.967731\pi$
$317$		−	3.60376i		−	0.202407i	0.994866	−	0.101204i	$-0.0322694\pi$
$318$		0			0
$319$			31.1004i			1.74129i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−13.2825			−0.739060
$324$		0			0
$325$	−2.59149	−	5.15418i	−0.143750	−	0.285903i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	1.19740i		−	0.0660147i
$330$		0			0
$331$	0.186551			0.0102538			0.00512689	−	0.999987i	$-0.498368\pi$
$331$	0.186551			0.0102538			0.00512689	+	0.999987i	$0.498368\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	12.3630	+	7.62169i	0.675462	+	0.416417i
$336$		0			0
$337$			15.2379i			0.830063i	0.909807	+	0.415032i	$0.136230\pi$
$337$			15.2379i			0.830063i	−0.909807	+	0.415032i	$0.863770\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.13330			−0.115525
$342$		0			0
$343$	6.21419			0.335535
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−26.1263			−1.40253			−0.701266	−	0.712900i	$-0.747382\pi$
$347$	−26.1263			−1.40253			−0.701266	+	0.712900i	$0.747382\pi$
$348$		0			0
$349$			36.0891i			1.93180i	0.258910	+	0.965901i	$0.416637\pi$
$349$			36.0891i			1.93180i	−0.258910	+	0.965901i	$0.583363\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.70998			−0.516810			−0.258405	−	0.966037i	$-0.583197\pi$
$353$	−9.70998			−0.516810			−0.258405	+	0.966037i	$0.583197\pi$
$354$		0			0
$355$	−13.4277	+	21.7808i	−0.712670	+	1.15601i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−33.8455			−1.78630			−0.893149	−	0.449761i	$-0.851509\pi$
$359$	−33.8455			−1.78630			−0.893149	+	0.449761i	$0.851509\pi$
$360$		0			0
$361$	−14.3060			−0.752949
$362$		0			0
$363$		0			0
$364$		0			0
$365$	26.4392	+	16.2996i	1.38389	+	0.853160i
$366$		0			0
$367$	2.14159			0.111790			0.0558949	−	0.998437i	$-0.482199\pi$
$367$	2.14159			0.111790			0.0558949	+	0.998437i	$0.482199\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	3.97048i		−	0.206137i
$372$		0			0
$373$	−14.6799			−0.760097			−0.380049	−	0.924967i	$-0.624093\pi$
$373$	−14.6799			−0.760097			−0.380049	+	0.924967i	$0.624093\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7.01946			−0.361521
$378$		0			0
$379$	4.44355			0.228250			0.114125	−	0.993466i	$-0.463594\pi$
$379$	4.44355			0.228250			0.114125	+	0.993466i	$0.463594\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			15.1342i			0.773324i	0.922222	+	0.386662i	$0.126372\pi$
$383$			15.1342i			0.773324i	−0.922222	+	0.386662i	$0.873628\pi$
$384$		0			0
$385$	−4.38252	−	2.70179i	−0.223354	−	0.137696i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−27.0036			−1.36914			−0.684570	−	0.728948i	$-0.740010\pi$
$389$	−27.0036			−1.36914			−0.684570	+	0.728948i	$0.740010\pi$
$390$		0			0
$391$		−	1.29375i		−	0.0654280i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	26.1661	+	16.1312i	1.31656	+	0.811649i
$396$		0			0
$397$	−3.18855			−0.160029			−0.0800143	−	0.996794i	$-0.525497\pi$
$397$	−3.18855			−0.160029			−0.0800143	+	0.996794i	$0.525497\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	25.1792i		−	1.25739i	−0.777652	−	0.628695i	$-0.783589\pi$
$401$		−	25.1792i		−	1.25739i	0.777652	−	0.628695i	$-0.216411\pi$
$402$		0			0
$403$		−	0.481494i		−	0.0239849i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			49.0761i			2.43261i
$408$		0			0
$409$	−27.5416			−1.36185			−0.680923	−	0.732355i	$-0.738421\pi$
$409$	−27.5416			−1.36185			−0.680923	+	0.732355i	$0.738421\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	2.13424i		−	0.105019i
$414$		0			0
$415$	6.12675	−	9.93808i	0.300750	−	0.487841i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			27.5259i			1.34473i	0.740221	+	0.672364i	$0.234721\pi$
$419$			27.5259i			1.34473i	−0.740221	+	0.672364i	$0.765279\pi$
$420$		0			0
$421$		−	29.6555i		−	1.44532i	−0.691203	−	0.722660i	$-0.742919\pi$
$421$		−	29.6555i		−	1.44532i	0.691203	−	0.722660i	$-0.257081\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−13.7698	−	27.3867i	−0.667935	−	1.32845i
$426$		0			0
$427$		−	3.44458i		−	0.166695i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−15.1134			−0.727986			−0.363993	−	0.931402i	$-0.618587\pi$
$431$	−15.1134			−0.727986			−0.363993	+	0.931402i	$0.618587\pi$
$432$		0			0
$433$		−	10.1697i		−	0.488726i	−0.969684	−	0.244363i	$-0.921421\pi$
$433$		−	10.1697i		−	0.488726i	0.969684	−	0.244363i	$-0.0785788\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			0.457206i			0.0218711i
$438$		0			0
$439$		−	17.3055i		−	0.825947i	−0.910743	−	0.412974i	$-0.864490\pi$
$439$		−	17.3055i		−	0.825947i	0.910743	−	0.412974i	$-0.135510\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	28.5925			1.35847			0.679235	−	0.733921i	$-0.262312\pi$
$443$	28.5925			1.35847			0.679235	+	0.733921i	$0.262312\pi$
$444$		0			0
$445$	2.65908	+	1.63930i	0.126053	+	0.0777104i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			3.59276i			0.169553i	0.996400	+	0.0847763i	$0.0270176\pi$
$449$			3.59276i			0.169553i	−0.996400	+	0.0847763i	$0.972982\pi$
$450$		0			0
$451$	49.6448			2.33768
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.609803	−	0.989149i	0.0285880	−	0.0463720i
$456$		0			0
$457$			11.2098i			0.524372i	0.965017	+	0.262186i	$0.0844434\pi$
$457$			11.2098i			0.524372i	−0.965017	+	0.262186i	$0.915557\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	29.8529			1.39039			0.695194	−	0.718823i	$-0.255319\pi$
$461$	29.8529			1.39039			0.695194	+	0.718823i	$0.255319\pi$
$462$		0			0
$463$	34.4700			1.60196			0.800978	−	0.598694i	$-0.204313\pi$
$463$	34.4700			1.60196			0.800978	+	0.598694i	$0.204313\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	15.7950			0.730907			0.365453	−	0.930830i	$-0.380914\pi$
$467$	15.7950			0.730907			0.365453	+	0.930830i	$0.380914\pi$
$468$		0			0
$469$			2.92539i			0.135082i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−15.6649			−0.720272
$474$		0			0
$475$	4.86619	+	9.67831i	0.223276	+	0.444071i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−32.1810			−1.47039			−0.735194	−	0.677857i	$-0.762909\pi$
$479$	−32.1810			−1.47039			−0.735194	+	0.677857i	$0.762909\pi$
$480$		0			0
$481$	−11.0766			−0.505051
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.39613	+	5.17615i	0.381249	+	0.235037i
$486$		0			0
$487$	−35.6312			−1.61460			−0.807302	−	0.590139i	$-0.799073\pi$
$487$	−35.6312			−1.61460			−0.807302	+	0.590139i	$0.799073\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			20.1909i			0.911201i	0.890184	+	0.455600i	$0.150575\pi$
$491$			20.1909i			0.911201i	−0.890184	+	0.455600i	$0.849425\pi$
$492$		0			0
$493$	−37.2978			−1.67981
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5.15388			−0.231183
$498$		0			0
$499$	−15.9850			−0.715588			−0.357794	−	0.933801i	$-0.616471\pi$
$499$	−15.9850			−0.715588			−0.357794	+	0.933801i	$0.616471\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			2.13750i			0.0953062i	0.998864	+	0.0476531i	$0.0151742\pi$
$503$			2.13750i			0.0953062i	−0.998864	+	0.0476531i	$0.984826\pi$
$504$		0			0
$505$	14.7420	−	23.9127i	0.656010	−	1.06410i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.67611			0.428886			0.214443	−	0.976736i	$-0.431206\pi$
$509$	9.67611			0.428886			0.214443	+	0.976736i	$0.431206\pi$
$510$		0			0
$511$			6.25617i			0.276757i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.98642	−	11.3325i	0.307858	−	0.499371i
$516$		0			0
$517$	13.5905			0.597710
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	3.11652i		−	0.136537i	−0.997667	−	0.0682687i	$-0.978252\pi$
$521$		−	3.11652i		−	0.136537i	0.997667	−	0.0682687i	$-0.0217475\pi$
$522$		0			0
$523$			23.5599i			1.03020i	0.857129	+	0.515102i	$0.172246\pi$
$523$			23.5599i			1.03020i	−0.857129	+	0.515102i	$0.827754\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	2.55841i		−	0.111446i
$528$		0			0
$529$	22.9555			0.998064
$530$		0			0
$531$		0			0
$532$		0			0
$533$			11.2050i			0.485343i
$534$		0			0
$535$	10.5437	−	17.1027i	0.455844	−	0.739415i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			34.7471i			1.49667i
$540$		0			0
$541$			21.4076i			0.920384i	0.887819	+	0.460192i	$0.152219\pi$
$541$			21.4076i			0.920384i	−0.887819	+	0.460192i	$0.847781\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	22.3622	+	13.7861i	0.957892	+	0.590533i
$546$		0			0
$547$			7.81382i			0.334095i	0.985949	+	0.167047i	$0.0534233\pi$
$547$			7.81382i			0.334095i	−0.985949	+	0.167047i	$0.946577\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	13.1808			0.561523
$552$		0			0
$553$			6.19154i			0.263291i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	26.4914i		−	1.12248i	−0.827655	−	0.561238i	$-0.810325\pi$
$557$		−	26.4914i		−	1.12248i	0.827655	−	0.561238i	$-0.189675\pi$
$558$		0			0
$559$		−	3.53562i		−	0.149541i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	40.0138			1.68638			0.843191	−	0.537614i	$-0.180674\pi$
$563$	40.0138			1.68638			0.843191	+	0.537614i	$0.180674\pi$
$564$		0			0
$565$	1.08670	−	1.76272i	0.0457179	−	0.0741581i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	17.5658i		−	0.736397i	−0.929747	−	0.368198i	$-0.879975\pi$
$569$		−	17.5658i		−	0.736397i	0.929747	−	0.368198i	$-0.120025\pi$
$570$		0			0
$571$	−31.8630			−1.33342			−0.666712	−	0.745315i	$-0.732299\pi$
$571$	−31.8630			−1.33342			−0.666712	+	0.745315i	$0.732299\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.942694	+	0.473980i	−0.0393130	+	0.0197663i
$576$		0			0
$577$			13.2498i			0.551596i	0.961216	+	0.275798i	$0.0889421\pi$
$577$			13.2498i			0.551596i	−0.961216	+	0.275798i	$0.911058\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.35159			0.0975605
$582$		0			0
$583$	45.0651			1.86641
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−21.1959			−0.874848			−0.437424	−	0.899255i	$-0.644109\pi$
$587$	−21.1959			−0.874848			−0.437424	+	0.899255i	$0.644109\pi$
$588$		0			0
$589$			0.904128i			0.0372540i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	24.0661			0.988275			0.494137	−	0.869384i	$-0.335484\pi$
$593$	24.0661			0.988275			0.494137	+	0.869384i	$0.335484\pi$
$594$		0			0
$595$	3.24018	−	5.25583i	0.132834	−	0.215468i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−37.1635			−1.51846			−0.759231	−	0.650822i	$-0.774425\pi$
$599$	−37.1635			−1.51846			−0.759231	+	0.650822i	$0.774425\pi$
$600$		0			0
$601$	23.4082			0.954841			0.477421	−	0.878675i	$-0.341572\pi$
$601$	23.4082			0.954841			0.477421	+	0.878675i	$0.341572\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	17.7575	−	28.8041i	0.721945	−	1.17105i
$606$		0			0
$607$	−11.7302			−0.476114			−0.238057	−	0.971251i	$-0.576510\pi$
$607$	−11.7302			−0.476114			−0.238057	+	0.971251i	$0.576510\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			3.06743i			0.124095i
$612$		0			0
$613$	−10.9631			−0.442797			−0.221399	−	0.975183i	$-0.571062\pi$
$613$	−10.9631			−0.442797			−0.221399	+	0.975183i	$0.571062\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	13.0317			0.524635			0.262318	−	0.964982i	$-0.415513\pi$
$617$	13.0317			0.524635			0.262318	+	0.964982i	$0.415513\pi$
$618$		0			0
$619$	36.3769			1.46211			0.731056	−	0.682317i	$-0.239028\pi$
$619$	36.3769			1.46211			0.731056	+	0.682317i	$0.239028\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			0.629204i			0.0252085i
$624$		0			0
$625$	−14.9106	+	20.0668i	−0.596423	+	0.802671i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−58.8555			−2.34672
$630$		0			0
$631$		−	42.0264i		−	1.67304i	−0.547933	−	0.836522i	$-0.684585\pi$
$631$		−	42.0264i		−	1.67304i	0.547933	−	0.836522i	$-0.315415\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	19.4889	−	31.6126i	0.773394	−	1.25451i
$636$		0			0
$637$	−7.84255			−0.310733
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	39.2028i		−	1.54842i	−0.632931	−	0.774208i	$-0.718148\pi$
$641$		−	39.2028i		−	1.54842i	0.632931	−	0.774208i	$-0.281852\pi$
$642$		0			0
$643$			40.0866i			1.58086i	0.612552	+	0.790431i	$0.290143\pi$
$643$			40.0866i			1.58086i	−0.612552	+	0.790431i	$0.709857\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	10.2214i		−	0.401844i	−0.979607	−	0.200922i	$-0.935606\pi$
$647$		−	10.2214i		−	0.401844i	0.979607	−	0.200922i	$-0.0643938\pi$
$648$		0			0
$649$	24.2237			0.950863
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	33.7435i		−	1.32049i	−0.751052	−	0.660243i	$-0.770453\pi$
$653$		−	33.7435i		−	1.32049i	0.751052	−	0.660243i	$-0.229547\pi$
$654$		0			0
$655$	25.7774	+	15.8916i	1.00721	+	0.620935i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			7.62965i			0.297209i	0.988897	+	0.148604i	$0.0474781\pi$
$659$			7.62965i			0.297209i	−0.988897	+	0.148604i	$0.952522\pi$
$660$		0			0
$661$			27.2772i			1.06096i	0.847698	+	0.530480i	$0.177988\pi$
$661$			27.2772i			1.06096i	−0.847698	+	0.530480i	$0.822012\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.14506	+	1.85738i	−0.0444036	+	0.0720262i
$666$		0			0
$667$			1.28385i			0.0497109i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	39.0961			1.50929
$672$		0			0
$673$			3.17561i			0.122411i	0.998125	+	0.0612055i	$0.0194945\pi$
$673$			3.17561i			0.122411i	−0.998125	+	0.0612055i	$0.980506\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			32.1530i			1.23574i	0.786280	+	0.617870i	$0.212004\pi$
$677$			32.1530i			1.23574i	−0.786280	+	0.617870i	$0.787996\pi$
$678$		0			0
$679$			1.98673i			0.0762437i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−25.5185			−0.976437			−0.488219	−	0.872721i	$-0.662353\pi$
$683$	−25.5185			−0.976437			−0.488219	+	0.872721i	$0.662353\pi$
$684$		0			0
$685$	10.4338	−	16.9245i	0.398656	−	0.646651i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			10.1714i			0.387498i
$690$		0			0
$691$	−32.6320			−1.24138			−0.620691	−	0.784056i	$-0.713148\pi$
$691$	−32.6320			−1.24138			−0.620691	+	0.784056i	$0.713148\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	16.7466	−	27.1643i	0.635234	−	1.03040i
$696$		0			0
$697$			59.5376i			2.25515i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	35.0817			1.32502			0.662509	−	0.749054i	$-0.269492\pi$
$701$	35.0817			1.32502			0.662509	+	0.749054i	$0.269492\pi$
$702$		0			0
$703$	20.7992			0.784458
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.65832			0.212803
$708$		0			0
$709$		−	17.6779i		−	0.663907i	−0.943296	−	0.331954i	$-0.892292\pi$
$709$		−	17.6779i		−	0.663907i	0.943296	−	0.331954i	$-0.107708\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.0880646			−0.00329805
$714$		0			0
$715$	11.2269	+	6.92129i	0.419862	+	0.258842i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.9011			−0.816773			−0.408386	−	0.912809i	$-0.633908\pi$
$719$	−21.9011			−0.816773			−0.408386	+	0.912809i	$0.633908\pi$
$720$		0			0
$721$	2.68155			0.0998662
$722$		0			0
$723$		0			0
$724$		0			0
$725$	13.6644	+	27.1770i	0.507484	+	1.00933i
$726$		0			0
$727$	−34.2800			−1.27138			−0.635688	−	0.771946i	$-0.719283\pi$
$727$	−34.2800			−1.27138			−0.635688	+	0.771946i	$0.719283\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	18.7864i		−	0.694842i
$732$		0			0
$733$	16.8179			0.621182			0.310591	−	0.950544i	$-0.399473\pi$
$733$	16.8179			0.621182			0.310591	+	0.950544i	$0.399473\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−33.2032			−1.22306
$738$		0			0
$739$	25.4981			0.937964			0.468982	−	0.883208i	$-0.344621\pi$
$739$	25.4981			0.937964			0.468982	+	0.883208i	$0.344621\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			2.19747i			0.0806172i	0.999187	+	0.0403086i	$0.0128341\pi$
$743$			2.19747i			0.0806172i	−0.999187	+	0.0403086i	$0.987166\pi$
$744$		0			0
$745$	10.5210	−	17.0660i	0.385461	−	0.625249i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.04692			0.147871
$750$		0			0
$751$			34.8488i			1.27165i	0.771832	+	0.635826i	$0.219340\pi$
$751$			34.8488i			1.27165i	−0.771832	+	0.635826i	$0.780660\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−29.6097	−	18.2542i	−1.07761	−	0.664338i
$756$		0			0
$757$	21.5026			0.781527			0.390763	−	0.920491i	$-0.372211\pi$
$757$	21.5026			0.781527			0.390763	+	0.920491i	$0.372211\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			6.95386i			0.252077i	0.992025	+	0.126039i	$0.0402263\pi$
$761$			6.95386i			0.252077i	−0.992025	+	0.126039i	$0.959774\pi$
$762$		0			0
$763$			5.29145i			0.191563i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			5.46737i			0.197415i
$768$		0			0
$769$	−15.1190			−0.545205			−0.272602	−	0.962127i	$-0.587884\pi$
$769$	−15.1190			−0.545205			−0.272602	+	0.962127i	$0.587884\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			13.0796i			0.470439i	0.971942	+	0.235220i	$0.0755810\pi$
$773$			13.0796i			0.470439i	−0.971942	+	0.235220i	$0.924419\pi$
$774$		0			0
$775$	−1.86418	+	0.937299i	−0.0669634	+	0.0336688i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	21.0403i		−	0.753846i
$780$		0			0
$781$		−	58.4968i		−	2.09318i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−15.5300	+	25.1909i	−0.554290	+	0.899102i
$786$		0			0
$787$			25.0403i			0.892591i	0.894886	+	0.446296i	$0.147257\pi$
$787$			25.0403i			0.892591i	−0.894886	+	0.446296i	$0.852743\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.417102			0.0148305
$792$		0			0
$793$			8.82412i			0.313354i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	29.0728i		−	1.02981i	−0.857247	−	0.514905i	$-0.827827\pi$
$797$		−	29.0728i		−	1.02981i	0.857247	−	0.514905i	$-0.172173\pi$
$798$		0			0
$799$			16.2987i			0.576607i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−71.0078			−2.50581
$804$		0			0
$805$	−0.180914	−	0.111532i	−0.00637639	−	0.00393100i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			17.8537i			0.627702i	0.949472	+	0.313851i	$0.101619\pi$
$809$			17.8537i			0.627702i	−0.949472	+	0.313851i	$0.898381\pi$
$810$		0			0
$811$	47.8173			1.67909			0.839546	−	0.543289i	$-0.182821\pi$
$811$	47.8173			1.67909			0.839546	+	0.543289i	$0.182821\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	29.2478	+	18.0310i	1.02450	+	0.631600i
$816$		0			0
$817$			6.63903i			0.232270i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.00469899			0.000163996			8.19979e−5	−	1.00000i	$-0.499974\pi$
$821$	0.00469899			0.000163996			8.19979e−5		1.00000i	$0.499974\pi$
$822$		0			0
$823$	−15.1981			−0.529773			−0.264887	−	0.964280i	$-0.585335\pi$
$823$	−15.1981			−0.529773			−0.264887	+	0.964280i	$0.585335\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.42753			−0.119187			−0.0595934	−	0.998223i	$-0.518980\pi$
$827$	−3.42753			−0.119187			−0.0595934	+	0.998223i	$0.518980\pi$
$828$		0			0
$829$		−	16.1960i		−	0.562511i	−0.959633	−	0.281256i	$-0.909249\pi$
$829$		−	16.1960i		−	0.562511i	0.959633	−	0.281256i	$-0.0907509\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−41.6713			−1.44382
$834$		0			0
$835$	34.1475	+	21.0517i	1.18172	+	0.728525i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	12.9846			0.448279			0.224139	−	0.974557i	$-0.428043\pi$
$839$	12.9846			0.448279			0.224139	+	0.974557i	$0.428043\pi$
$840$		0			0
$841$	8.01224			0.276284
$842$		0			0
$843$		0			0
$844$		0			0
$845$	13.6926	−	22.2106i	0.471041	−	0.764067i
$846$		0			0
$847$	6.81575			0.234192
$848$		0			0
$849$		0			0
$850$		0			0
$851$			2.02590i			0.0694471i
$852$		0			0
$853$	15.9478			0.546044			0.273022	−	0.962008i	$-0.411977\pi$
$853$	15.9478			0.546044			0.273022	+	0.962008i	$0.411977\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	36.3078			1.24025			0.620126	−	0.784502i	$-0.287081\pi$
$857$	36.3078			1.24025			0.620126	+	0.784502i	$0.287081\pi$
$858$		0			0
$859$	−16.7215			−0.570530			−0.285265	−	0.958449i	$-0.592082\pi$
$859$	−16.7215			−0.570530			−0.285265	+	0.958449i	$0.592082\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	32.0732i		−	1.09178i	−0.837856	−	0.545892i	$-0.816191\pi$
$863$		−	32.0732i		−	1.09178i	0.837856	−	0.545892i	$-0.183809\pi$
$864$		0			0
$865$	32.8548	+	20.2547i	1.11710	+	0.688681i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−70.2742			−2.38389
$870$		0			0
$871$		−	7.49409i		−	0.253928i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5.01673	−	0.435430i	−0.169596	−	0.0147202i
$876$		0			0
$877$	24.1662			0.816035			0.408018	−	0.912974i	$-0.366220\pi$
$877$	24.1662			0.816035			0.408018	+	0.912974i	$0.366220\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			16.3886i			0.552146i	0.961137	+	0.276073i	$0.0890332\pi$
$881$			16.3886i			0.552146i	−0.961137	+	0.276073i	$0.910967\pi$
$882$		0			0
$883$		−	17.3417i		−	0.583595i	−0.956480	−	0.291798i	$-0.905747\pi$
$883$		−	17.3417i		−	0.583595i	0.956480	−	0.291798i	$-0.0942534\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	47.8185i		−	1.60559i	−0.596256	−	0.802794i	$-0.703346\pi$
$887$		−	47.8185i		−	1.60559i	0.596256	−	0.802794i	$-0.296654\pi$
$888$		0			0
$889$	7.48031			0.250882
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	5.75988i		−	0.192747i
$894$		0			0
$895$	−0.437757	−	0.269874i	−0.0146326	−	0.00902089i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			2.53882i			0.0846745i
$900$		0			0
$901$			54.0453i			1.80051i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4.31943	+	2.66290i	0.143583	+	0.0885177i
$906$		0			0
$907$			8.22699i			0.273173i	0.990628	+	0.136586i	$0.0436131\pi$
$907$			8.22699i			0.273173i	−0.990628	+	0.136586i	$0.956387\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−22.9871			−0.761597			−0.380798	−	0.924658i	$-0.624351\pi$
$911$	−22.9871			−0.761597			−0.380798	+	0.924658i	$0.624351\pi$
$912$		0			0
$913$			26.6907i			0.883332i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			6.09957i			0.201425i
$918$		0			0
$919$		−	34.6477i		−	1.14292i	−0.820629	−	0.571461i	$-0.806377\pi$
$919$		−	34.6477i		−	1.14292i	0.820629	−	0.571461i	$-0.193623\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	13.2029			0.434579
$924$		0			0
$925$	21.5623	+	42.8851i	0.708965	+	1.41005i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			37.7103i			1.23723i	0.785692	+	0.618617i	$0.212307\pi$
$929$			37.7103i			1.23723i	−0.785692	+	0.618617i	$0.787693\pi$
$930$		0			0
$931$	14.7264			0.482639
$932$		0			0
$933$		0			0
$934$		0			0
$935$	59.6539	+	36.7762i	1.95089	+	1.20271i
$936$		0			0
$937$			27.9404i			0.912773i	0.889782	+	0.456386i	$0.150857\pi$
$937$			27.9404i			0.912773i	−0.889782	+	0.456386i	$0.849143\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	30.4349			0.992148			0.496074	−	0.868280i	$-0.334774\pi$
$941$	30.4349			0.992148			0.496074	+	0.868280i	$0.334774\pi$
$942$		0			0
$943$	2.04938			0.0667370
$944$		0			0
$945$		0			0
$946$		0			0
$947$	35.3004			1.14711			0.573554	−	0.819167i	$-0.305564\pi$
$947$	35.3004			1.14711			0.573554	+	0.819167i	$0.305564\pi$
$948$		0			0
$949$		−	16.0267i		−	0.520249i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	12.1033			0.392064			0.196032	−	0.980598i	$-0.437194\pi$
$953$	12.1033			0.392064			0.196032	+	0.980598i	$0.437194\pi$
$954$		0			0
$955$	−20.1467	+	32.6795i	−0.651930	+	1.05748i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.00475			0.129320
$960$		0			0
$961$	30.8259			0.994382
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.71326	−	1.05621i	−0.0551517	−	0.0340006i
$966$		0			0
$967$	30.6826			0.986687			0.493344	−	0.869834i	$-0.335774\pi$
$967$	30.6826			0.986687			0.493344	+	0.869834i	$0.335774\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	4.77033i		−	0.153087i	−0.997066	−	0.0765436i	$-0.975612\pi$
$971$		−	4.77033i		−	0.153087i	0.997066	−	0.0765436i	$-0.0243884\pi$
$972$		0			0
$973$	6.42773			0.206064
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−28.1550			−0.900760			−0.450380	−	0.892837i	$-0.648711\pi$
$977$	−28.1550			−0.900760			−0.450380	+	0.892837i	$0.648711\pi$
$978$		0			0
$979$	−7.14149			−0.228243
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	24.2134i		−	0.772288i	−0.922439	−	0.386144i	$-0.873807\pi$
$983$		−	24.2134i		−	0.772288i	0.922439	−	0.386144i	$-0.126193\pi$
$984$		0			0
$985$	−3.07746	−	1.89723i	−0.0980561	−	0.0604509i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.646660			−0.0205626
$990$		0			0
$991$		−	18.6181i		−	0.591424i	−0.955277	−	0.295712i	$-0.904443\pi$
$991$		−	18.6181i		−	0.591424i	0.955277	−	0.295712i	$-0.0955569\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−26.6400	−	16.4233i	−0.844544	−	0.520655i
$996$		0			0
$997$	−13.7405			−0.435167			−0.217584	−	0.976042i	$-0.569817\pi$
$997$	−13.7405			−0.435167			−0.217584	+	0.976042i	$0.569817\pi$
$998$		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.2159.15		48
3.2	odd	2	inner	4320.2.m.c.2159.33		48
4.3	odd	2		1080.2.m.c.539.19	yes	48
5.4	even	2	inner	4320.2.m.c.2159.14		48
8.3	odd	2	inner	4320.2.m.c.2159.34		48
8.5	even	2		1080.2.m.c.539.18	yes	48
12.11	even	2		1080.2.m.c.539.30	yes	48
15.14	odd	2	inner	4320.2.m.c.2159.36		48
20.19	odd	2		1080.2.m.c.539.29	yes	48
24.5	odd	2		1080.2.m.c.539.31	yes	48
24.11	even	2	inner	4320.2.m.c.2159.16		48
40.19	odd	2	inner	4320.2.m.c.2159.35		48
40.29	even	2		1080.2.m.c.539.32	yes	48
60.59	even	2		1080.2.m.c.539.20	yes	48
120.29	odd	2		1080.2.m.c.539.17	✓	48
120.59	even	2	inner	4320.2.m.c.2159.13		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.m.c.539.17	✓	48	120.29	odd	2
1080.2.m.c.539.18	yes	48	8.5	even	2
1080.2.m.c.539.19	yes	48	4.3	odd	2
1080.2.m.c.539.20	yes	48	60.59	even	2
1080.2.m.c.539.29	yes	48	20.19	odd	2
1080.2.m.c.539.30	yes	48	12.11	even	2
1080.2.m.c.539.31	yes	48	24.5	odd	2
1080.2.m.c.539.32	yes	48	40.29	even	2
4320.2.m.c.2159.13		48	120.59	even	2	inner
4320.2.m.c.2159.14		48	5.4	even	2	inner
4320.2.m.c.2159.15		48	1.1	even	1	trivial
4320.2.m.c.2159.16		48	24.11	even	2	inner
4320.2.m.c.2159.33		48	3.2	odd	2	inner
4320.2.m.c.2159.34		48	8.3	odd	2	inner
4320.2.m.c.2159.35		48	40.19	odd	2	inner
4320.2.m.c.2159.36		48	15.14	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)

\(f(q)\)	\(=\)	\(q+(-1.17345 + 1.90342i) q^{5} -0.450397 q^{7} -5.11202i q^{11} +1.15380 q^{13} +6.13070 q^{17} -2.16656 q^{19} -0.211029i q^{23} +(-2.24604 - 4.46713i) q^{25} -6.08377 q^{29} -0.417311i q^{31} +(0.528517 - 0.857296i) q^{35} -9.60013 q^{37} +9.71138i q^{41} -3.06432i q^{43} +2.65854i q^{47} -6.79714 q^{49} +8.81551i q^{53} +(9.73034 + 5.99869i) q^{55} +4.73857i q^{59} +7.64787i q^{61} +(-1.35392 + 2.19617i) q^{65} -6.49513i q^{67} +11.4430 q^{71} -13.8904i q^{73} +2.30244i q^{77} -13.7469i q^{79} -5.22116 q^{83} +(-7.19405 + 11.6693i) q^{85} -1.39700i q^{89} -0.519668 q^{91} +(2.54234 - 4.12388i) q^{95} -4.41107i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 16 q^{19} + 48 q^{49}+O(q^{100}) \) 48 * q + 16 * q^19 + 48 * q^49

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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