LMFDB - Embedded newform 2880.2.m.b.1439.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(1439,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, 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("2880.1439");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.m (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1439.7
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2880.1439
Dual form			2880.2.m.b.1439.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.41421 + 1.73205i) q^{5} -2.44949 q^{7} +O(q^{10})$	$q+(1.41421 + 1.73205i) q^{5} -2.44949 q^{7} -1.41421i q^{11} -2.44949 q^{13} -3.46410 q^{17} +6.00000 q^{19} -6.92820i q^{23} +(-1.00000 + 4.89898i) q^{25} -5.65685 q^{29} +2.00000i q^{31} +(-3.46410 - 4.24264i) q^{35} -7.34847 q^{37} +4.24264i q^{41} -4.89898i q^{43} +3.46410i q^{47} -1.00000 q^{49} +6.92820i q^{53} +(2.44949 - 2.00000i) q^{55} -1.41421i q^{59} -12.0000i q^{61} +(-3.46410 - 4.24264i) q^{65} -14.6969i q^{67} -8.48528 q^{71} -9.79796i q^{73} +3.46410i q^{77} -2.00000i q^{79} -13.8564 q^{83} +(-4.89898 - 6.00000i) q^{85} -4.24264i q^{89} +6.00000 q^{91} +(8.48528 + 10.3923i) q^{95} +4.89898i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 48 q^{19} - 8 q^{25} - 8 q^{49} + 48 q^{91}+O(q^{100}) $ 8 * q + 48 * q^19 - 8 * q^25 - 8 * q^49 + 48 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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
$5$	1.41421	+	1.73205i	0.632456	+	0.774597i
$5$	1.41421	+	1.73205i	0.632456	+	0.774597i
$6$		0			0
$7$	−2.44949			−0.925820			−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949			−0.925820			−0.462910	+	0.886405i	$0.653195\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.41421i		−	0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$		−	1.41421i		−	0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$		0			0
$13$	−2.44949			−0.679366			−0.339683	−	0.940540i	$-0.610320\pi$
$13$	−2.44949			−0.679366			−0.339683	+	0.940540i	$0.610320\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.46410			−0.840168			−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410			−0.840168			−0.420084	+	0.907485i	$0.637999\pi$
$18$		0			0
$19$	6.00000			1.37649			0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000			1.37649			0.688247	+	0.725476i	$0.258380\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	6.92820i		−	1.44463i	−0.691564	−	0.722315i	$-0.743078\pi$
$23$		−	6.92820i		−	1.44463i	0.691564	−	0.722315i	$-0.256922\pi$
$24$		0			0
$25$	−1.00000	+	4.89898i	−0.200000	+	0.979796i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.65685			−1.05045			−0.525226	−	0.850963i	$-0.676019\pi$
$29$	−5.65685			−1.05045			−0.525226	+	0.850963i	$0.676019\pi$
$30$		0			0
$31$			2.00000i			0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$			2.00000i			0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.46410	−	4.24264i	−0.585540	−	0.717137i
$36$		0			0
$37$	−7.34847			−1.20808			−0.604040	−	0.796954i	$-0.706443\pi$
$37$	−7.34847			−1.20808			−0.604040	+	0.796954i	$0.706443\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			4.24264i			0.662589i	0.943527	+	0.331295i	$0.107485\pi$
$41$			4.24264i			0.662589i	−0.943527	+	0.331295i	$0.892515\pi$
$42$		0			0
$43$		−	4.89898i		−	0.747087i	−0.927613	−	0.373544i	$-0.878143\pi$
$43$		−	4.89898i		−	0.747087i	0.927613	−	0.373544i	$-0.121857\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			3.46410i			0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$			3.46410i			0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.92820i			0.951662i	0.879537	+	0.475831i	$0.157853\pi$
$53$			6.92820i			0.951662i	−0.879537	+	0.475831i	$0.842147\pi$
$54$		0			0
$55$	2.44949	−	2.00000i	0.330289	−	0.269680i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	1.41421i		−	0.184115i	−0.995754	−	0.0920575i	$-0.970656\pi$
$59$		−	1.41421i		−	0.184115i	0.995754	−	0.0920575i	$-0.0293443\pi$
$60$		0			0
$61$		−	12.0000i		−	1.53644i	−0.640184	−	0.768221i	$-0.721142\pi$
$61$		−	12.0000i		−	1.53644i	0.640184	−	0.768221i	$-0.278858\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.46410	−	4.24264i	−0.429669	−	0.526235i
$66$		0			0
$67$		−	14.6969i		−	1.79552i	−0.440488	−	0.897758i	$-0.645195\pi$
$67$		−	14.6969i		−	1.79552i	0.440488	−	0.897758i	$-0.354805\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.48528			−1.00702			−0.503509	−	0.863990i	$-0.667958\pi$
$71$	−8.48528			−1.00702			−0.503509	+	0.863990i	$0.667958\pi$
$72$		0			0
$73$		−	9.79796i		−	1.14676i	−0.819288	−	0.573382i	$-0.805631\pi$
$73$		−	9.79796i		−	1.14676i	0.819288	−	0.573382i	$-0.194369\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			3.46410i			0.394771i
$78$		0			0
$79$		−	2.00000i		−	0.225018i	−0.993651	−	0.112509i	$-0.964111\pi$
$79$		−	2.00000i		−	0.225018i	0.993651	−	0.112509i	$-0.0358886\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−13.8564			−1.52094			−0.760469	−	0.649374i	$-0.775031\pi$
$83$	−13.8564			−1.52094			−0.760469	+	0.649374i	$0.775031\pi$
$84$		0			0
$85$	−4.89898	−	6.00000i	−0.531369	−	0.650791i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	4.24264i		−	0.449719i	−0.974391	−	0.224860i	$-0.927808\pi$
$89$		−	4.24264i		−	0.449719i	0.974391	−	0.224860i	$-0.0721923\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.48528	+	10.3923i	0.870572	+	1.06623i
$96$		0			0
$97$			4.89898i			0.497416i	0.968579	+	0.248708i	$0.0800060\pi$
$97$			4.89898i			0.497416i	−0.968579	+	0.248708i	$0.919994\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−14.1421			−1.40720			−0.703598	−	0.710599i	$-0.748424\pi$
$101$	−14.1421			−1.40720			−0.703598	+	0.710599i	$0.748424\pi$
$102$		0			0
$103$	−7.34847			−0.724066			−0.362033	−	0.932165i	$-0.617917\pi$
$103$	−7.34847			−0.724066			−0.362033	+	0.932165i	$0.617917\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.92820			−0.669775			−0.334887	−	0.942258i	$-0.608698\pi$
$107$	−6.92820			−0.669775			−0.334887	+	0.942258i	$0.608698\pi$
$108$		0			0
$109$		−	12.0000i		−	1.14939i	−0.818367	−	0.574696i	$-0.805120\pi$
$109$		−	12.0000i		−	1.14939i	0.818367	−	0.574696i	$-0.194880\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	17.3205			1.62938			0.814688	−	0.579899i	$-0.196908\pi$
$113$	17.3205			1.62938			0.814688	+	0.579899i	$0.196908\pi$
$114$		0			0
$115$	12.0000	−	9.79796i	1.11901	−	0.913664i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	8.48528			0.777844
$120$		0			0
$121$	9.00000			0.818182
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.89949	+	5.19615i	−0.885438	+	0.464758i
$126$		0			0
$127$	17.1464			1.52150			0.760750	−	0.649045i	$-0.224831\pi$
$127$	17.1464			1.52150			0.760750	+	0.649045i	$0.224831\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	7.07107i		−	0.617802i	−0.951094	−	0.308901i	$-0.900039\pi$
$131$		−	7.07107i		−	0.617802i	0.951094	−	0.308901i	$-0.0999612\pi$
$132$		0			0
$133$	−14.6969			−1.27439
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.3923			0.887875			0.443937	−	0.896058i	$-0.353581\pi$
$137$	10.3923			0.887875			0.443937	+	0.896058i	$0.353581\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			3.46410i			0.289683i
$144$		0			0
$145$	−8.00000	−	9.79796i	−0.664364	−	0.813676i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.65685			0.463428			0.231714	−	0.972784i	$-0.425567\pi$
$149$	5.65685			0.463428			0.231714	+	0.972784i	$0.425567\pi$
$150$		0			0
$151$			14.0000i			1.13930i	0.821886	+	0.569652i	$0.192922\pi$
$151$			14.0000i			1.13930i	−0.821886	+	0.569652i	$0.807078\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.46410	+	2.82843i	−0.278243	+	0.227185i
$156$		0			0
$157$	−12.2474			−0.977453			−0.488726	−	0.872437i	$-0.662538\pi$
$157$	−12.2474			−0.977453			−0.488726	+	0.872437i	$0.662538\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			16.9706i			1.33747i
$162$		0			0
$163$		−	19.5959i		−	1.53487i	−0.641126	−	0.767435i	$-0.721533\pi$
$163$		−	19.5959i		−	1.53487i	0.641126	−	0.767435i	$-0.278467\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−7.00000			−0.538462
$170$		0			0
$171$		0			0
$172$		0			0
$173$			6.92820i			0.526742i	0.964695	+	0.263371i	$0.0848343\pi$
$173$			6.92820i			0.526742i	−0.964695	+	0.263371i	$0.915166\pi$
$174$		0			0
$175$	2.44949	−	12.0000i	0.185164	−	0.907115i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	24.0416i		−	1.79696i	−0.439019	−	0.898478i	$-0.644674\pi$
$179$		−	24.0416i		−	1.79696i	0.439019	−	0.898478i	$-0.355326\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−10.3923	−	12.7279i	−0.764057	−	0.935775i
$186$		0			0
$187$			4.89898i			0.358249i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−16.9706			−1.22795			−0.613973	−	0.789327i	$-0.710430\pi$
$191$	−16.9706			−1.22795			−0.613973	+	0.789327i	$0.710430\pi$
$192$		0			0
$193$			4.89898i			0.352636i	0.984333	+	0.176318i	$0.0564187\pi$
$193$			4.89898i			0.352636i	−0.984333	+	0.176318i	$0.943581\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			17.3205i			1.23404i	0.786949	+	0.617018i	$0.211659\pi$
$197$			17.3205i			1.23404i	−0.786949	+	0.617018i	$0.788341\pi$
$198$		0			0
$199$		−	10.0000i		−	0.708881i	−0.935079	−	0.354441i	$-0.884671\pi$
$199$		−	10.0000i		−	0.708881i	0.935079	−	0.354441i	$-0.115329\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	13.8564			0.972529
$204$		0			0
$205$	−7.34847	+	6.00000i	−0.513239	+	0.419058i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	8.48528i		−	0.586939i
$210$		0			0
$211$	24.0000			1.65223			0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000			1.65223			0.826114	+	0.563503i	$0.190547\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.48528	−	6.92820i	0.578691	−	0.472500i
$216$		0			0
$217$		−	4.89898i		−	0.332564i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.48528			0.570782
$222$		0			0
$223$	−7.34847			−0.492090			−0.246045	−	0.969258i	$-0.579131\pi$
$223$	−7.34847			−0.492090			−0.246045	+	0.969258i	$0.579131\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−20.7846			−1.37952			−0.689761	−	0.724037i	$-0.742285\pi$
$227$	−20.7846			−1.37952			−0.689761	+	0.724037i	$0.742285\pi$
$228$		0			0
$229$		−	12.0000i		−	0.792982i	−0.918039	−	0.396491i	$-0.870228\pi$
$229$		−	12.0000i		−	0.792982i	0.918039	−	0.396491i	$-0.129772\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.3923			−0.680823			−0.340411	−	0.940277i	$-0.610566\pi$
$233$	−10.3923			−0.680823			−0.340411	+	0.940277i	$0.610566\pi$
$234$		0			0
$235$	−6.00000	+	4.89898i	−0.391397	+	0.319574i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	20.0000			1.28831			0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000			1.28831			0.644157	+	0.764894i	$0.277208\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.41421	−	1.73205i	−0.0903508	−	0.110657i
$246$		0			0
$247$	−14.6969			−0.935144
$248$		0			0
$249$		0			0
$250$		0			0
$251$			1.41421i			0.0892644i	0.999003	+	0.0446322i	$0.0142116\pi$
$251$			1.41421i			0.0892644i	−0.999003	+	0.0446322i	$0.985788\pi$
$252$		0			0
$253$	−9.79796			−0.615992
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−24.2487			−1.51259			−0.756297	−	0.654229i	$-0.772993\pi$
$257$	−24.2487			−1.51259			−0.756297	+	0.654229i	$0.772993\pi$
$258$		0			0
$259$	18.0000			1.11847
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	−12.0000	+	9.79796i	−0.737154	+	0.601884i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.3137			−0.689809			−0.344904	−	0.938638i	$-0.612089\pi$
$269$	−11.3137			−0.689809			−0.344904	+	0.938638i	$0.612089\pi$
$270$		0			0
$271$			14.0000i			0.850439i	0.905090	+	0.425220i	$0.139803\pi$
$271$			14.0000i			0.850439i	−0.905090	+	0.425220i	$0.860197\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.92820	+	1.41421i	0.417786	+	0.0852803i
$276$		0			0
$277$	7.34847			0.441527			0.220763	−	0.975327i	$-0.429145\pi$
$277$	7.34847			0.441527			0.220763	+	0.975327i	$0.429145\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			29.6985i			1.77166i	0.464007	+	0.885832i	$0.346411\pi$
$281$			29.6985i			1.77166i	−0.464007	+	0.885832i	$0.653589\pi$
$282$		0			0
$283$			14.6969i			0.873642i	0.899548	+	0.436821i	$0.143896\pi$
$283$			14.6969i			0.873642i	−0.899548	+	0.436821i	$0.856104\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	10.3923i		−	0.613438i
$288$		0			0
$289$	−5.00000			−0.294118
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	10.3923i		−	0.607125i	−0.952812	−	0.303562i	$-0.901824\pi$
$293$		−	10.3923i		−	0.607125i	0.952812	−	0.303562i	$-0.0981761\pi$
$294$		0			0
$295$	2.44949	−	2.00000i	0.142615	−	0.116445i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			16.9706i			0.981433i
$300$		0			0
$301$			12.0000i			0.691669i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	20.7846	−	16.9706i	1.19012	−	0.971732i
$306$		0			0
$307$		−	9.79796i		−	0.559199i	−0.960117	−	0.279600i	$-0.909798\pi$
$307$		−	9.79796i		−	0.559199i	0.960117	−	0.279600i	$-0.0902017\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−33.9411			−1.92462			−0.962312	−	0.271947i	$-0.912333\pi$
$311$	−33.9411			−1.92462			−0.962312	+	0.271947i	$0.912333\pi$
$312$		0			0
$313$			24.4949i			1.38453i	0.721642	+	0.692267i	$0.243388\pi$
$313$			24.4949i			1.38453i	−0.721642	+	0.692267i	$0.756612\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			3.46410i			0.194563i	0.995257	+	0.0972817i	$0.0310148\pi$
$317$			3.46410i			0.194563i	−0.995257	+	0.0972817i	$0.968985\pi$
$318$		0			0
$319$			8.00000i			0.447914i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−20.7846			−1.15649
$324$		0			0
$325$	2.44949	−	12.0000i	0.135873	−	0.665640i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	8.48528i		−	0.467809i
$330$		0			0
$331$	−30.0000			−1.64895			−0.824475	−	0.565899i	$-0.808529\pi$
$331$	−30.0000			−1.64895			−0.824475	+	0.565899i	$0.808529\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	25.4558	−	20.7846i	1.39080	−	1.13558i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.82843			0.153168
$342$		0			0
$343$	19.5959			1.05808
$344$		0			0
$345$		0			0
$346$		0			0
$347$	27.7128			1.48770			0.743851	−	0.668346i	$-0.232997\pi$
$347$	27.7128			1.48770			0.743851	+	0.668346i	$0.232997\pi$
$348$		0			0
$349$			12.0000i			0.642345i	0.947021	+	0.321173i	$0.104077\pi$
$349$			12.0000i			0.642345i	−0.947021	+	0.321173i	$0.895923\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−24.2487			−1.29063			−0.645314	−	0.763917i	$-0.723274\pi$
$353$	−24.2487			−1.29063			−0.645314	+	0.763917i	$0.723274\pi$
$354$		0			0
$355$	−12.0000	−	14.6969i	−0.636894	−	0.780033i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	25.4558			1.34351			0.671754	−	0.740774i	$-0.265541\pi$
$359$	25.4558			1.34351			0.671754	+	0.740774i	$0.265541\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		0			0
$364$		0			0
$365$	16.9706	−	13.8564i	0.888280	−	0.725277i
$366$		0			0
$367$	−31.8434			−1.66221			−0.831105	−	0.556115i	$-0.812291\pi$
$367$	−31.8434			−1.66221			−0.831105	+	0.556115i	$0.812291\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	16.9706i		−	0.881068i
$372$		0			0
$373$	26.9444			1.39513			0.697564	−	0.716523i	$-0.254267\pi$
$373$	26.9444			1.39513			0.697564	+	0.716523i	$0.254267\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	13.8564			0.713641
$378$		0			0
$379$	−24.0000			−1.23280			−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000			−1.23280			−0.616399	+	0.787434i	$0.711409\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	3.46410i		−	0.177007i	−0.996076	−	0.0885037i	$-0.971792\pi$
$383$		−	3.46410i		−	0.177007i	0.996076	−	0.0885037i	$-0.0282085\pi$
$384$		0			0
$385$	−6.00000	+	4.89898i	−0.305788	+	0.249675i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−14.1421			−0.717035			−0.358517	−	0.933523i	$-0.616718\pi$
$389$	−14.1421			−0.717035			−0.358517	+	0.933523i	$0.616718\pi$
$390$		0			0
$391$			24.0000i			1.21373i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.46410	−	2.82843i	0.174298	−	0.142314i
$396$		0			0
$397$	17.1464			0.860555			0.430277	−	0.902697i	$-0.358416\pi$
$397$	17.1464			0.860555			0.430277	+	0.902697i	$0.358416\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			29.6985i			1.48307i	0.670913	+	0.741536i	$0.265902\pi$
$401$			29.6985i			1.48307i	−0.670913	+	0.741536i	$0.734098\pi$
$402$		0			0
$403$		−	4.89898i		−	0.244036i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			10.3923i			0.515127i
$408$		0			0
$409$	−2.00000			−0.0988936			−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000			−0.0988936			−0.0494468	+	0.998777i	$0.515746\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			3.46410i			0.170457i
$414$		0			0
$415$	−19.5959	−	24.0000i	−0.961926	−	1.17811i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			35.3553i			1.72722i	0.504159	+	0.863611i	$0.331802\pi$
$419$			35.3553i			1.72722i	−0.504159	+	0.863611i	$0.668198\pi$
$420$		0			0
$421$		−	12.0000i		−	0.584844i	−0.956289	−	0.292422i	$-0.905539\pi$
$421$		−	12.0000i		−	0.584844i	0.956289	−	0.292422i	$-0.0944612\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.46410	−	16.9706i	0.168034	−	0.823193i
$426$		0			0
$427$			29.3939i			1.42247i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−25.4558			−1.22616			−0.613082	−	0.790019i	$-0.710071\pi$
$431$	−25.4558			−1.22616			−0.613082	+	0.790019i	$0.710071\pi$
$432$		0			0
$433$		−	19.5959i		−	0.941720i	−0.882208	−	0.470860i	$-0.843944\pi$
$433$		−	19.5959i		−	0.941720i	0.882208	−	0.470860i	$-0.156056\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	41.5692i		−	1.98853i
$438$		0			0
$439$		−	34.0000i		−	1.62273i	−0.584539	−	0.811366i	$-0.698725\pi$
$439$		−	34.0000i		−	1.62273i	0.584539	−	0.811366i	$-0.301275\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	20.7846			0.987507			0.493753	−	0.869602i	$-0.335625\pi$
$443$	20.7846			0.987507			0.493753	+	0.869602i	$0.335625\pi$
$444$		0			0
$445$	7.34847	−	6.00000i	0.348351	−	0.284427i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	12.7279i		−	0.600668i	−0.953834	−	0.300334i	$-0.902902\pi$
$449$		−	12.7279i		−	0.600668i	0.953834	−	0.300334i	$-0.0970981\pi$
$450$		0			0
$451$	6.00000			0.282529
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.48528	+	10.3923i	0.397796	+	0.487199i
$456$		0			0
$457$		−	24.4949i		−	1.14582i	−0.819617	−	0.572911i	$-0.805814\pi$
$457$		−	24.4949i		−	1.14582i	0.819617	−	0.572911i	$-0.194186\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5.65685			0.263466			0.131733	−	0.991285i	$-0.457946\pi$
$461$	5.65685			0.263466			0.131733	+	0.991285i	$0.457946\pi$
$462$		0			0
$463$	26.9444			1.25221			0.626106	−	0.779738i	$-0.284648\pi$
$463$	26.9444			1.25221			0.626106	+	0.779738i	$0.284648\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.92820			−0.320599			−0.160300	−	0.987068i	$-0.551246\pi$
$467$	−6.92820			−0.320599			−0.160300	+	0.987068i	$0.551246\pi$
$468$		0			0
$469$			36.0000i			1.66233i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6.92820			−0.318559
$474$		0			0
$475$	−6.00000	+	29.3939i	−0.275299	+	1.34868i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.48528			−0.387702			−0.193851	−	0.981031i	$-0.562098\pi$
$479$	−8.48528			−0.387702			−0.193851	+	0.981031i	$0.562098\pi$
$480$		0			0
$481$	18.0000			0.820729
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.48528	+	6.92820i	−0.385297	+	0.314594i
$486$		0			0
$487$	36.7423			1.66495			0.832477	−	0.554059i	$-0.186922\pi$
$487$	36.7423			1.66495			0.832477	+	0.554059i	$0.186922\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			41.0122i			1.85085i	0.378925	+	0.925427i	$0.376294\pi$
$491$			41.0122i			1.85085i	−0.378925	+	0.925427i	$0.623706\pi$
$492$		0			0
$493$	19.5959			0.882556
$494$		0			0
$495$		0			0
$496$		0			0
$497$	20.7846			0.932317
$498$		0			0
$499$	−6.00000			−0.268597			−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000			−0.268597			−0.134298	+	0.990941i	$0.542878\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			38.1051i			1.69902i	0.527570	+	0.849512i	$0.323103\pi$
$503$			38.1051i			1.69902i	−0.527570	+	0.849512i	$0.676897\pi$
$504$		0			0
$505$	−20.0000	−	24.4949i	−0.889988	−	1.09001i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	31.1127			1.37905			0.689523	−	0.724264i	$-0.257820\pi$
$509$	31.1127			1.37905			0.689523	+	0.724264i	$0.257820\pi$
$510$		0			0
$511$			24.0000i			1.06170i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−10.3923	−	12.7279i	−0.457940	−	0.560859i
$516$		0			0
$517$	4.89898			0.215457
$518$		0			0
$519$		0			0
$520$		0			0
$521$			21.2132i			0.929367i	0.885477	+	0.464684i	$0.153832\pi$
$521$			21.2132i			0.929367i	−0.885477	+	0.464684i	$0.846168\pi$
$522$		0			0
$523$			39.1918i			1.71374i	0.515533	+	0.856870i	$0.327594\pi$
$523$			39.1918i			1.71374i	−0.515533	+	0.856870i	$0.672406\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	6.92820i		−	0.301797i
$528$		0			0
$529$	−25.0000			−1.08696
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	10.3923i		−	0.450141i
$534$		0			0
$535$	−9.79796	−	12.0000i	−0.423603	−	0.518805i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			1.41421i			0.0609145i
$540$		0			0
$541$			24.0000i			1.03184i	0.856637	+	0.515920i	$0.172550\pi$
$541$			24.0000i			1.03184i	−0.856637	+	0.515920i	$0.827450\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	20.7846	−	16.9706i	0.890315	−	0.726939i
$546$		0			0
$547$		−	4.89898i		−	0.209465i	−0.994500	−	0.104733i	$-0.966601\pi$
$547$		−	4.89898i		−	0.209465i	0.994500	−	0.104733i	$-0.0333987\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−33.9411			−1.44594
$552$		0			0
$553$			4.89898i			0.208326i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	6.92820i		−	0.293557i	−0.989169	−	0.146779i	$-0.953109\pi$
$557$		−	6.92820i		−	0.293557i	0.989169	−	0.146779i	$-0.0468905\pi$
$558$		0			0
$559$			12.0000i			0.507546i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	27.7128			1.16796			0.583978	−	0.811770i	$-0.301495\pi$
$563$	27.7128			1.16796			0.583978	+	0.811770i	$0.301495\pi$
$564$		0			0
$565$	24.4949	+	30.0000i	1.03051	+	1.26211i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	12.7279i		−	0.533582i	−0.963754	−	0.266791i	$-0.914037\pi$
$569$		−	12.7279i		−	0.533582i	0.963754	−	0.266791i	$-0.0859634\pi$
$570$		0			0
$571$	−30.0000			−1.25546			−0.627730	−	0.778431i	$-0.716016\pi$
$571$	−30.0000			−1.25546			−0.627730	+	0.778431i	$0.716016\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	33.9411	+	6.92820i	1.41544	+	0.288926i
$576$		0			0
$577$			44.0908i			1.83552i	0.397130	+	0.917762i	$0.370006\pi$
$577$			44.0908i			1.83552i	−0.397130	+	0.917762i	$0.629994\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	33.9411			1.40812
$582$		0			0
$583$	9.79796			0.405790
$584$		0			0
$585$		0			0
$586$		0			0
$587$	34.6410			1.42979			0.714894	−	0.699233i	$-0.246475\pi$
$587$	34.6410			1.42979			0.714894	+	0.699233i	$0.246475\pi$
$588$		0			0
$589$			12.0000i			0.494451i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.46410			−0.142254			−0.0711268	−	0.997467i	$-0.522659\pi$
$593$	−3.46410			−0.142254			−0.0711268	+	0.997467i	$0.522659\pi$
$594$		0			0
$595$	12.0000	+	14.6969i	0.491952	+	0.602516i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	42.4264			1.73350			0.866748	−	0.498746i	$-0.166206\pi$
$599$	42.4264			1.73350			0.866748	+	0.498746i	$0.166206\pi$
$600$		0			0
$601$	32.0000			1.30531			0.652654	−	0.757656i	$-0.273656\pi$
$601$	32.0000			1.30531			0.652654	+	0.757656i	$0.273656\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	12.7279	+	15.5885i	0.517464	+	0.633761i
$606$		0			0
$607$	7.34847			0.298265			0.149133	−	0.988817i	$-0.452352\pi$
$607$	7.34847			0.298265			0.149133	+	0.988817i	$0.452352\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	8.48528i		−	0.343278i
$612$		0			0
$613$	−2.44949			−0.0989340			−0.0494670	−	0.998776i	$-0.515752\pi$
$613$	−2.44949			−0.0989340			−0.0494670	+	0.998776i	$0.515752\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	24.2487			0.976216			0.488108	−	0.872783i	$-0.337687\pi$
$617$	24.2487			0.976216			0.488108	+	0.872783i	$0.337687\pi$
$618$		0			0
$619$	−24.0000			−0.964641			−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000			−0.964641			−0.482321	+	0.875995i	$0.660206\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			10.3923i			0.416359i
$624$		0			0
$625$	−23.0000	−	9.79796i	−0.920000	−	0.391918i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	25.4558			1.01499
$630$		0			0
$631$			22.0000i			0.875806i	0.899022	+	0.437903i	$0.144279\pi$
$631$			22.0000i			0.875806i	−0.899022	+	0.437903i	$0.855721\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	24.2487	+	29.6985i	0.962281	+	1.17855i
$636$		0			0
$637$	2.44949			0.0970523
$638$		0			0
$639$		0			0
$640$		0			0
$641$			29.6985i			1.17302i	0.809942	+	0.586510i	$0.199498\pi$
$641$			29.6985i			1.17302i	−0.809942	+	0.586510i	$0.800502\pi$
$642$		0			0
$643$			9.79796i			0.386394i	0.981160	+	0.193197i	$0.0618856\pi$
$643$			9.79796i			0.386394i	−0.981160	+	0.193197i	$0.938114\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	10.3923i		−	0.408564i	−0.978912	−	0.204282i	$-0.934514\pi$
$647$		−	10.3923i		−	0.408564i	0.978912	−	0.204282i	$-0.0654859\pi$
$648$		0			0
$649$	−2.00000			−0.0785069
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	45.0333i		−	1.76229i	−0.472846	−	0.881145i	$-0.656773\pi$
$653$		−	45.0333i		−	1.76229i	0.472846	−	0.881145i	$-0.343227\pi$
$654$		0			0
$655$	12.2474	−	10.0000i	0.478547	−	0.390732i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	43.8406i		−	1.70779i	−0.520447	−	0.853894i	$-0.674235\pi$
$659$		−	43.8406i		−	1.70779i	0.520447	−	0.853894i	$-0.325765\pi$
$660$		0			0
$661$			24.0000i			0.933492i	0.884391	+	0.466746i	$0.154574\pi$
$661$			24.0000i			0.933492i	−0.884391	+	0.466746i	$0.845426\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−20.7846	−	25.4558i	−0.805993	−	0.987135i
$666$		0			0
$667$			39.1918i			1.51751i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−16.9706			−0.655141
$672$		0			0
$673$		−	9.79796i		−	0.377684i	−0.982008	−	0.188842i	$-0.939527\pi$
$673$		−	9.79796i		−	0.377684i	0.982008	−	0.188842i	$-0.0604733\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			10.3923i			0.399409i	0.979856	+	0.199704i	$0.0639982\pi$
$677$			10.3923i			0.399409i	−0.979856	+	0.199704i	$0.936002\pi$
$678$		0			0
$679$		−	12.0000i		−	0.460518i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$	14.6969	+	18.0000i	0.561541	+	0.687745i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	16.9706i		−	0.646527i
$690$		0			0
$691$	−6.00000			−0.228251			−0.114125	−	0.993466i	$-0.536407\pi$
$691$	−6.00000			−0.228251			−0.114125	+	0.993466i	$0.536407\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	14.6969i		−	0.556686i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−48.0833			−1.81608			−0.908040	−	0.418884i	$-0.862421\pi$
$701$	−48.0833			−1.81608			−0.908040	+	0.418884i	$0.862421\pi$
$702$		0			0
$703$	−44.0908			−1.66292
$704$		0			0
$705$		0			0
$706$		0			0
$707$	34.6410			1.30281
$708$		0			0
$709$		−	12.0000i		−	0.450669i	−0.974281	−	0.225335i	$-0.927652\pi$
$709$		−	12.0000i		−	0.450669i	0.974281	−	0.225335i	$-0.0723476\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	13.8564			0.518927
$714$		0			0
$715$	−6.00000	+	4.89898i	−0.224387	+	0.183211i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−25.4558			−0.949343			−0.474671	−	0.880163i	$-0.657433\pi$
$719$	−25.4558			−0.949343			−0.474671	+	0.880163i	$0.657433\pi$
$720$		0			0
$721$	18.0000			0.670355
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.65685	−	27.7128i	0.210090	−	1.02923i
$726$		0			0
$727$	2.44949			0.0908465			0.0454233	−	0.998968i	$-0.485536\pi$
$727$	2.44949			0.0908465			0.0454233	+	0.998968i	$0.485536\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			16.9706i			0.627679i
$732$		0			0
$733$	22.0454			0.814266			0.407133	−	0.913369i	$-0.366529\pi$
$733$	22.0454			0.814266			0.407133	+	0.913369i	$0.366529\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−20.7846			−0.765611
$738$		0			0
$739$	6.00000			0.220714			0.110357	−	0.993892i	$-0.464801\pi$
$739$	6.00000			0.220714			0.110357	+	0.993892i	$0.464801\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	3.46410i		−	0.127086i	−0.997979	−	0.0635428i	$-0.979760\pi$
$743$		−	3.46410i		−	0.127086i	0.997979	−	0.0635428i	$-0.0202399\pi$
$744$		0			0
$745$	8.00000	+	9.79796i	0.293097	+	0.358969i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	16.9706			0.620091
$750$		0			0
$751$			2.00000i			0.0729810i	0.999334	+	0.0364905i	$0.0116179\pi$
$751$			2.00000i			0.0729810i	−0.999334	+	0.0364905i	$0.988382\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−24.2487	+	19.7990i	−0.882501	+	0.720559i
$756$		0			0
$757$	26.9444			0.979310			0.489655	−	0.871916i	$-0.337123\pi$
$757$	26.9444			0.979310			0.489655	+	0.871916i	$0.337123\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			46.6690i			1.69175i	0.533380	+	0.845876i	$0.320922\pi$
$761$			46.6690i			1.69175i	−0.533380	+	0.845876i	$0.679078\pi$
$762$		0			0
$763$			29.3939i			1.06413i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			3.46410i			0.125081i
$768$		0			0
$769$	16.0000			0.576975			0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000			0.576975			0.288487	+	0.957484i	$0.406848\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			31.1769i			1.12136i	0.828034	+	0.560678i	$0.189459\pi$
$773$			31.1769i			1.12136i	−0.828034	+	0.560678i	$0.810541\pi$
$774$		0			0
$775$	−9.79796	−	2.00000i	−0.351953	−	0.0718421i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			25.4558i			0.912050i
$780$		0			0
$781$			12.0000i			0.429394i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−17.3205	−	21.2132i	−0.618195	−	0.757132i
$786$		0			0
$787$		−	9.79796i		−	0.349260i	−0.984634	−	0.174630i	$-0.944127\pi$
$787$		−	9.79796i		−	0.349260i	0.984634	−	0.174630i	$-0.0558729\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−42.4264			−1.50851
$792$		0			0
$793$			29.3939i			1.04381i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	48.4974i		−	1.71787i	−0.512087	−	0.858933i	$-0.671128\pi$
$797$		−	48.4974i		−	1.71787i	0.512087	−	0.858933i	$-0.328872\pi$
$798$		0			0
$799$		−	12.0000i		−	0.424529i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−13.8564			−0.488982
$804$		0			0
$805$	−29.3939	+	24.0000i	−1.03600	+	0.845889i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	4.24264i		−	0.149163i	−0.997215	−	0.0745817i	$-0.976238\pi$
$809$		−	4.24264i		−	0.149163i	0.997215	−	0.0745817i	$-0.0237621\pi$
$810$		0			0
$811$	−24.0000			−0.842754			−0.421377	−	0.906886i	$-0.638453\pi$
$811$	−24.0000			−0.842754			−0.421377	+	0.906886i	$0.638453\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	33.9411	−	27.7128i	1.18891	−	0.970737i
$816$		0			0
$817$		−	29.3939i		−	1.02836i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	28.2843			0.987128			0.493564	−	0.869710i	$-0.335694\pi$
$821$	28.2843			0.987128			0.493564	+	0.869710i	$0.335694\pi$
$822$		0			0
$823$	−26.9444			−0.939222			−0.469611	−	0.882873i	$-0.655606\pi$
$823$	−26.9444			−0.939222			−0.469611	+	0.882873i	$0.655606\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−20.7846			−0.722752			−0.361376	−	0.932420i	$-0.617693\pi$
$827$	−20.7846			−0.722752			−0.361376	+	0.932420i	$0.617693\pi$
$828$		0			0
$829$		−	24.0000i		−	0.833554i	−0.909009	−	0.416777i	$-0.863160\pi$
$829$		−	24.0000i		−	0.833554i	0.909009	−	0.416777i	$-0.136840\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.46410			0.120024
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	42.4264			1.46472			0.732361	−	0.680916i	$-0.238418\pi$
$839$	42.4264			1.46472			0.732361	+	0.680916i	$0.238418\pi$
$840$		0			0
$841$	3.00000			0.103448
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.89949	−	12.1244i	−0.340553	−	0.417091i
$846$		0			0
$847$	−22.0454			−0.757489
$848$		0			0
$849$		0			0
$850$		0			0
$851$			50.9117i			1.74523i
$852$		0			0
$853$	26.9444			0.922558			0.461279	−	0.887255i	$-0.347391\pi$
$853$	26.9444			0.922558			0.461279	+	0.887255i	$0.347391\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−51.9615			−1.77497			−0.887486	−	0.460835i	$-0.847550\pi$
$857$	−51.9615			−1.77497			−0.887486	+	0.460835i	$0.847550\pi$
$858$		0			0
$859$	−24.0000			−0.818869			−0.409435	−	0.912339i	$-0.634274\pi$
$859$	−24.0000			−0.818869			−0.409435	+	0.912339i	$0.634274\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	55.4256i		−	1.88671i	−0.331785	−	0.943355i	$-0.607651\pi$
$863$		−	55.4256i		−	1.88671i	0.331785	−	0.943355i	$-0.392349\pi$
$864$		0			0
$865$	−12.0000	+	9.79796i	−0.408012	+	0.333141i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.82843			−0.0959478
$870$		0			0
$871$			36.0000i			1.21981i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	24.2487	−	12.7279i	0.819756	−	0.430282i
$876$		0			0
$877$	31.8434			1.07527			0.537637	−	0.843176i	$-0.319317\pi$
$877$	31.8434			1.07527			0.537637	+	0.843176i	$0.319317\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			29.6985i			1.00057i	0.865862	+	0.500284i	$0.166771\pi$
$881$			29.6985i			1.00057i	−0.865862	+	0.500284i	$0.833229\pi$
$882$		0			0
$883$		−	29.3939i		−	0.989183i	−0.869126	−	0.494591i	$-0.835318\pi$
$883$		−	29.3939i		−	0.989183i	0.869126	−	0.494591i	$-0.164682\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	13.8564i		−	0.465253i	−0.972566	−	0.232626i	$-0.925268\pi$
$887$		−	13.8564i		−	0.465253i	0.972566	−	0.232626i	$-0.0747319\pi$
$888$		0			0
$889$	−42.0000			−1.40863
$890$		0			0
$891$		0			0
$892$		0			0
$893$			20.7846i			0.695530i
$894$		0			0
$895$	41.6413	−	34.0000i	1.39192	−	1.13649i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	11.3137i		−	0.377333i
$900$		0			0
$901$		−	24.0000i		−	0.799556i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$			4.89898i			0.162668i	0.996687	+	0.0813340i	$0.0259180\pi$
$907$			4.89898i			0.162668i	−0.996687	+	0.0813340i	$0.974082\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−16.9706			−0.562260			−0.281130	−	0.959670i	$-0.590709\pi$
$911$	−16.9706			−0.562260			−0.281130	+	0.959670i	$0.590709\pi$
$912$		0			0
$913$			19.5959i			0.648530i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			17.3205i			0.571974i
$918$		0			0
$919$		−	10.0000i		−	0.329870i	−0.986304	−	0.164935i	$-0.947259\pi$
$919$		−	10.0000i		−	0.329870i	0.986304	−	0.164935i	$-0.0527414\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	20.7846			0.684134
$924$		0			0
$925$	7.34847	−	36.0000i	0.241616	−	1.18367i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			4.24264i			0.139197i	0.997575	+	0.0695983i	$0.0221717\pi$
$929$			4.24264i			0.139197i	−0.997575	+	0.0695983i	$0.977828\pi$
$930$		0			0
$931$	−6.00000			−0.196642
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8.48528	+	6.92820i	−0.277498	+	0.226576i
$936$		0			0
$937$			39.1918i			1.28034i	0.768233	+	0.640171i	$0.221136\pi$
$937$			39.1918i			1.28034i	−0.768233	+	0.640171i	$0.778864\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19.7990			0.645429			0.322714	−	0.946496i	$-0.395405\pi$
$941$	19.7990			0.645429			0.322714	+	0.946496i	$0.395405\pi$
$942$		0			0
$943$	29.3939			0.957196
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−34.6410			−1.12568			−0.562841	−	0.826565i	$-0.690292\pi$
$947$	−34.6410			−1.12568			−0.562841	+	0.826565i	$0.690292\pi$
$948$		0			0
$949$			24.0000i			0.779073i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	24.2487			0.785493			0.392746	−	0.919647i	$-0.371525\pi$
$953$	24.2487			0.785493			0.392746	+	0.919647i	$0.371525\pi$
$954$		0			0
$955$	−24.0000	−	29.3939i	−0.776622	−	0.951164i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−25.4558			−0.822012
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8.48528	+	6.92820i	−0.273151	+	0.223027i
$966$		0			0
$967$	−26.9444			−0.866473			−0.433237	−	0.901280i	$-0.642629\pi$
$967$	−26.9444			−0.866473			−0.433237	+	0.901280i	$0.642629\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	41.0122i		−	1.31614i	−0.752955	−	0.658072i	$-0.771372\pi$
$971$		−	41.0122i		−	1.31614i	0.752955	−	0.658072i	$-0.228628\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−17.3205			−0.554132			−0.277066	−	0.960851i	$-0.589362\pi$
$977$	−17.3205			−0.554132			−0.277066	+	0.960851i	$0.589362\pi$
$978$		0			0
$979$	−6.00000			−0.191761
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	38.1051i		−	1.21536i	−0.794180	−	0.607682i	$-0.792099\pi$
$983$		−	38.1051i		−	1.21536i	0.794180	−	0.607682i	$-0.207901\pi$
$984$		0			0
$985$	−30.0000	+	24.4949i	−0.955879	+	0.780472i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−33.9411			−1.07927
$990$		0			0
$991$			10.0000i			0.317660i	0.987306	+	0.158830i	$0.0507723\pi$
$991$			10.0000i			0.317660i	−0.987306	+	0.158830i	$0.949228\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	17.3205	−	14.1421i	0.549097	−	0.448336i
$996$		0			0
$997$	−31.8434			−1.00849			−0.504245	−	0.863561i	$-0.668229\pi$
$997$	−31.8434			−1.00849			−0.504245	+	0.863561i	$0.668229\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.m.b.1439.7	yes	8
3.2	odd	2	inner	2880.2.m.b.1439.1	yes	8
4.3	odd	2		2880.2.m.a.1439.8	yes	8
5.4	even	2	inner	2880.2.m.b.1439.6	yes	8
8.3	odd	2	inner	2880.2.m.b.1439.2	yes	8
8.5	even	2		2880.2.m.a.1439.1	✓	8
12.11	even	2		2880.2.m.a.1439.2	yes	8
15.14	odd	2	inner	2880.2.m.b.1439.4	yes	8
20.19	odd	2		2880.2.m.a.1439.5	yes	8
24.5	odd	2		2880.2.m.a.1439.7	yes	8
24.11	even	2	inner	2880.2.m.b.1439.8	yes	8
40.19	odd	2	inner	2880.2.m.b.1439.3	yes	8
40.29	even	2		2880.2.m.a.1439.4	yes	8
60.59	even	2		2880.2.m.a.1439.3	yes	8
120.29	odd	2		2880.2.m.a.1439.6	yes	8
120.59	even	2	inner	2880.2.m.b.1439.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2880.2.m.a.1439.1	✓	8	8.5	even	2
2880.2.m.a.1439.2	yes	8	12.11	even	2
2880.2.m.a.1439.3	yes	8	60.59	even	2
2880.2.m.a.1439.4	yes	8	40.29	even	2
2880.2.m.a.1439.5	yes	8	20.19	odd	2
2880.2.m.a.1439.6	yes	8	120.29	odd	2
2880.2.m.a.1439.7	yes	8	24.5	odd	2
2880.2.m.a.1439.8	yes	8	4.3	odd	2
2880.2.m.b.1439.1	yes	8	3.2	odd	2	inner
2880.2.m.b.1439.2	yes	8	8.3	odd	2	inner
2880.2.m.b.1439.3	yes	8	40.19	odd	2	inner
2880.2.m.b.1439.4	yes	8	15.14	odd	2	inner
2880.2.m.b.1439.5	yes	8	120.59	even	2	inner
2880.2.m.b.1439.6	yes	8	5.4	even	2	inner
2880.2.m.b.1439.7	yes	8	1.1	even	1	trivial
2880.2.m.b.1439.8	yes	8	24.11	even	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:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.41421 + 1.73205i) q^{5} -2.44949 q^{7} +O(q^{10})\)	\(q+(1.41421 + 1.73205i) q^{5} -2.44949 q^{7} -1.41421i q^{11} -2.44949 q^{13} -3.46410 q^{17} +6.00000 q^{19} -6.92820i q^{23} +(-1.00000 + 4.89898i) q^{25} -5.65685 q^{29} +2.00000i q^{31} +(-3.46410 - 4.24264i) q^{35} -7.34847 q^{37} +4.24264i q^{41} -4.89898i q^{43} +3.46410i q^{47} -1.00000 q^{49} +6.92820i q^{53} +(2.44949 - 2.00000i) q^{55} -1.41421i q^{59} -12.0000i q^{61} +(-3.46410 - 4.24264i) q^{65} -14.6969i q^{67} -8.48528 q^{71} -9.79796i q^{73} +3.46410i q^{77} -2.00000i q^{79} -13.8564 q^{83} +(-4.89898 - 6.00000i) q^{85} -4.24264i q^{89} +6.00000 q^{91} +(8.48528 + 10.3923i) q^{95} +4.89898i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 48 q^{19} - 8 q^{25} - 8 q^{49} + 48 q^{91}+O(q^{100}) \) 8 * q + 48 * q^19 - 8 * q^25 - 8 * q^49 + 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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