LMFDB - Embedded newform 2880.2.d.g.289.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(289,2880)]

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([0, 1, 0, 1]))

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

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

chi := DirichletCharacter("2880.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.d (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:	no (minimal twist has level 320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.7
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2880.289
Dual form			2880.2.d.g.289.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.41421 + 1.73205i) q^{5} -1.41421i q^{7} +2.00000i q^{11} +5.65685 q^{13} -4.89898i q^{17} -6.00000i q^{19} -7.07107i q^{23} +(-1.00000 + 4.89898i) q^{25} -6.92820i q^{29} -6.92820 q^{31} +(2.44949 - 2.00000i) q^{35} -2.82843 q^{37} +4.00000 q^{41} +2.44949 q^{43} +4.24264i q^{47} +5.00000 q^{49} +(-3.46410 + 2.82843i) q^{55} -2.00000i q^{59} -3.46410i q^{61} +(8.00000 + 9.79796i) q^{65} -2.44949 q^{67} -6.92820 q^{71} +4.89898i q^{73} +2.82843 q^{77} +6.92820 q^{79} +12.2474 q^{83} +(8.48528 - 6.92820i) q^{85} +2.00000 q^{89} -8.00000i q^{91} +(10.3923 - 8.48528i) q^{95} +14.6969i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{25} + 32 q^{41} + 40 q^{49} + 64 q^{65} + 16 q^{89}+O(q^{100}) $ 8 * q - 8 * q^25 + 32 * q^41 + 40 * q^49 + 64 * q^65 + 16 * q^89

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
$7$		−	1.41421i		−	0.534522i	−0.963624	−	0.267261i	$-0.913881\pi$
$7$		−	1.41421i		−	0.534522i	0.963624	−	0.267261i	$-0.0861187\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			2.00000i			0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$			2.00000i			0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$		0			0
$13$	5.65685			1.56893			0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685			1.56893			0.784465	+	0.620174i	$0.212938\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.89898i		−	1.18818i	−0.804400	−	0.594089i	$-0.797513\pi$
$17$		−	4.89898i		−	1.18818i	0.804400	−	0.594089i	$-0.202487\pi$
$18$		0			0
$19$		−	6.00000i		−	1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.07107i		−	1.47442i	−0.675664	−	0.737210i	$-0.736143\pi$
$23$		−	7.07107i		−	1.47442i	0.675664	−	0.737210i	$-0.263857\pi$
$24$		0			0
$25$	−1.00000	+	4.89898i	−0.200000	+	0.979796i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	6.92820i		−	1.28654i	−0.765641	−	0.643268i	$-0.777578\pi$
$29$		−	6.92820i		−	1.28654i	0.765641	−	0.643268i	$-0.222422\pi$
$30$		0			0
$31$	−6.92820			−1.24434			−0.622171	−	0.782881i	$-0.713749\pi$
$31$	−6.92820			−1.24434			−0.622171	+	0.782881i	$0.713749\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.44949	−	2.00000i	0.414039	−	0.338062i
$36$		0			0
$37$	−2.82843			−0.464991			−0.232495	−	0.972598i	$-0.574689\pi$
$37$	−2.82843			−0.464991			−0.232495	+	0.972598i	$0.574689\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.00000			0.624695			0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000			0.624695			0.312348	+	0.949968i	$0.398885\pi$
$42$		0			0
$43$	2.44949			0.373544			0.186772	−	0.982403i	$-0.440197\pi$
$43$	2.44949			0.373544			0.186772	+	0.982403i	$0.440197\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			4.24264i			0.618853i	0.950923	+	0.309426i	$0.100137\pi$
$47$			4.24264i			0.618853i	−0.950923	+	0.309426i	$0.899863\pi$
$48$		0			0
$49$	5.00000			0.714286
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−3.46410	+	2.82843i	−0.467099	+	0.381385i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	2.00000i		−	0.260378i	−0.991489	−	0.130189i	$-0.958442\pi$
$59$		−	2.00000i		−	0.260378i	0.991489	−	0.130189i	$-0.0415584\pi$
$60$		0			0
$61$		−	3.46410i		−	0.443533i	−0.975100	−	0.221766i	$-0.928818\pi$
$61$		−	3.46410i		−	0.443533i	0.975100	−	0.221766i	$-0.0711822\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	8.00000	+	9.79796i	0.992278	+	1.21529i
$66$		0			0
$67$	−2.44949			−0.299253			−0.149626	−	0.988743i	$-0.547807\pi$
$67$	−2.44949			−0.299253			−0.149626	+	0.988743i	$0.547807\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.92820			−0.822226			−0.411113	−	0.911584i	$-0.634860\pi$
$71$	−6.92820			−0.822226			−0.411113	+	0.911584i	$0.634860\pi$
$72$		0			0
$73$			4.89898i			0.573382i	0.958023	+	0.286691i	$0.0925553\pi$
$73$			4.89898i			0.573382i	−0.958023	+	0.286691i	$0.907445\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.82843			0.322329
$78$		0			0
$79$	6.92820			0.779484			0.389742	−	0.920924i	$-0.372564\pi$
$79$	6.92820			0.779484			0.389742	+	0.920924i	$0.372564\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	12.2474			1.34433			0.672166	−	0.740400i	$-0.265364\pi$
$83$	12.2474			1.34433			0.672166	+	0.740400i	$0.265364\pi$
$84$		0			0
$85$	8.48528	−	6.92820i	0.920358	−	0.751469i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.00000			0.212000			0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000			0.212000			0.106000	+	0.994366i	$0.466196\pi$
$90$		0			0
$91$		−	8.00000i		−	0.838628i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	10.3923	−	8.48528i	1.06623	−	0.870572i
$96$		0			0
$97$			14.6969i			1.49225i	0.665807	+	0.746124i	$0.268087\pi$
$97$			14.6969i			1.49225i	−0.665807	+	0.746124i	$0.731913\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	6.92820i		−	0.689382i	−0.938716	−	0.344691i	$-0.887984\pi$
$101$		−	6.92820i		−	0.689382i	0.938716	−	0.344691i	$-0.112016\pi$
$102$		0			0
$103$		−	9.89949i		−	0.975426i	−0.873004	−	0.487713i	$-0.837831\pi$
$103$		−	9.89949i		−	0.975426i	0.873004	−	0.487713i	$-0.162169\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.44949			0.236801			0.118401	−	0.992966i	$-0.462223\pi$
$107$	2.44949			0.236801			0.118401	+	0.992966i	$0.462223\pi$
$108$		0			0
$109$			3.46410i			0.331801i	0.986143	+	0.165900i	$0.0530530\pi$
$109$			3.46410i			0.331801i	−0.986143	+	0.165900i	$0.946947\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	12.2474	−	10.0000i	1.14208	−	0.932505i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.92820			−0.635107
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.89949	+	5.19615i	−0.885438	+	0.464758i
$126$		0			0
$127$		−	7.07107i		−	0.627456i	−0.949513	−	0.313728i	$-0.898422\pi$
$127$		−	7.07107i		−	0.627456i	0.949513	−	0.313728i	$-0.101578\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	10.0000i		−	0.873704i	−0.899533	−	0.436852i	$-0.856093\pi$
$131$		−	10.0000i		−	0.873704i	0.899533	−	0.436852i	$-0.143907\pi$
$132$		0			0
$133$	−8.48528			−0.735767
$134$		0			0
$135$		0			0
$136$		0			0
$137$			9.79796i			0.837096i	0.908195	+	0.418548i	$0.137461\pi$
$137$			9.79796i			0.837096i	−0.908195	+	0.418548i	$0.862539\pi$
$138$		0			0
$139$		−	10.0000i		−	0.848189i	−0.905618	−	0.424094i	$-0.860592\pi$
$139$		−	10.0000i		−	0.848189i	0.905618	−	0.424094i	$-0.139408\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			11.3137i			0.946100i
$144$		0			0
$145$	12.0000	−	9.79796i	0.996546	−	0.813676i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			10.3923i			0.851371i	0.904871	+	0.425685i	$0.139967\pi$
$149$			10.3923i			0.851371i	−0.904871	+	0.425685i	$0.860033\pi$
$150$		0			0
$151$	−13.8564			−1.12762			−0.563809	−	0.825905i	$-0.690665\pi$
$151$	−13.8564			−1.12762			−0.563809	+	0.825905i	$0.690665\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.79796	−	12.0000i	−0.786991	−	0.963863i
$156$		0			0
$157$	2.82843			0.225733			0.112867	−	0.993610i	$-0.463997\pi$
$157$	2.82843			0.225733			0.112867	+	0.993610i	$0.463997\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−10.0000			−0.788110
$162$		0			0
$163$	22.0454			1.72673			0.863365	−	0.504580i	$-0.168353\pi$
$163$	22.0454			1.72673			0.863365	+	0.504580i	$0.168353\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			9.89949i			0.766046i	0.923739	+	0.383023i	$0.125117\pi$
$167$			9.89949i			0.766046i	−0.923739	+	0.383023i	$0.874883\pi$
$168$		0			0
$169$	19.0000			1.46154
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.48528			−0.645124			−0.322562	−	0.946548i	$-0.604544\pi$
$173$	−8.48528			−0.645124			−0.322562	+	0.946548i	$0.604544\pi$
$174$		0			0
$175$	6.92820	+	1.41421i	0.523723	+	0.106904i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			2.00000i			0.149487i	0.997203	+	0.0747435i	$0.0238138\pi$
$179$			2.00000i			0.149487i	−0.997203	+	0.0747435i	$0.976186\pi$
$180$		0			0
$181$		−	20.7846i		−	1.54491i	−0.635071	−	0.772454i	$-0.719029\pi$
$181$		−	20.7846i		−	1.54491i	0.635071	−	0.772454i	$-0.280971\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.00000	−	4.89898i	−0.294086	−	0.360180i
$186$		0			0
$187$	9.79796			0.716498
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−13.8564			−1.00261			−0.501307	−	0.865269i	$-0.667147\pi$
$191$	−13.8564			−1.00261			−0.501307	+	0.865269i	$0.667147\pi$
$192$		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
$197$	22.6274			1.61214			0.806068	−	0.591822i	$-0.201591\pi$
$197$	22.6274			1.61214			0.806068	+	0.591822i	$0.201591\pi$
$198$		0			0
$199$	−20.7846			−1.47338			−0.736691	−	0.676230i	$-0.763613\pi$
$199$	−20.7846			−1.47338			−0.736691	+	0.676230i	$0.763613\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−9.79796			−0.687682
$204$		0			0
$205$	5.65685	+	6.92820i	0.395092	+	0.483887i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$			10.0000i			0.688428i	0.938891	+	0.344214i	$0.111855\pi$
$211$			10.0000i			0.688428i	−0.938891	+	0.344214i	$0.888145\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.46410	+	4.24264i	0.236250	+	0.289346i
$216$		0			0
$217$			9.79796i			0.665129i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	27.7128i		−	1.86417i
$222$		0			0
$223$			26.8701i			1.79935i	0.436558	+	0.899676i	$0.356197\pi$
$223$			26.8701i			1.79935i	−0.436558	+	0.899676i	$0.643803\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	17.1464			1.13805			0.569024	−	0.822321i	$-0.307321\pi$
$227$	17.1464			1.13805			0.569024	+	0.822321i	$0.307321\pi$
$228$		0			0
$229$		−	6.92820i		−	0.457829i	−0.973447	−	0.228914i	$-0.926482\pi$
$229$		−	6.92820i		−	0.457829i	0.973447	−	0.228914i	$-0.0735176\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			14.6969i			0.962828i	0.876493	+	0.481414i	$0.159877\pi$
$233$			14.6969i			0.962828i	−0.876493	+	0.481414i	$0.840123\pi$
$234$		0			0
$235$	−7.34847	+	6.00000i	−0.479361	+	0.391397i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.92820			0.448148			0.224074	−	0.974572i	$-0.428064\pi$
$239$	6.92820			0.448148			0.224074	+	0.974572i	$0.428064\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.07107	+	8.66025i	0.451754	+	0.553283i
$246$		0			0
$247$		−	33.9411i		−	2.15962i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			26.0000i			1.64111i	0.571571	+	0.820553i	$0.306334\pi$
$251$			26.0000i			1.64111i	−0.571571	+	0.820553i	$0.693666\pi$
$252$		0			0
$253$	14.1421			0.889108
$254$		0			0
$255$		0			0
$256$		0			0
$257$			9.79796i			0.611180i	0.952163	+	0.305590i	$0.0988537\pi$
$257$			9.79796i			0.611180i	−0.952163	+	0.305590i	$0.901146\pi$
$258$		0			0
$259$			4.00000i			0.248548i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	26.8701i		−	1.65688i	−0.560079	−	0.828439i	$-0.689229\pi$
$263$		−	26.8701i		−	1.65688i	0.560079	−	0.828439i	$-0.310771\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			17.3205i			1.05605i	0.849229	+	0.528025i	$0.177067\pi$
$269$			17.3205i			1.05605i	−0.849229	+	0.528025i	$0.822933\pi$
$270$		0			0
$271$	20.7846			1.26258			0.631288	−	0.775549i	$-0.282527\pi$
$271$	20.7846			1.26258			0.631288	+	0.775549i	$0.282527\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−9.79796	−	2.00000i	−0.590839	−	0.120605i
$276$		0			0
$277$	−14.1421			−0.849719			−0.424859	−	0.905259i	$-0.639676\pi$
$277$	−14.1421			−0.849719			−0.424859	+	0.905259i	$0.639676\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.00000			−0.477240			−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000			−0.477240			−0.238620	+	0.971113i	$0.576695\pi$
$282$		0			0
$283$	−2.44949			−0.145607			−0.0728035	−	0.997346i	$-0.523195\pi$
$283$	−2.44949			−0.145607			−0.0728035	+	0.997346i	$0.523195\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	5.65685i		−	0.333914i
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.82843			0.165238			0.0826192	−	0.996581i	$-0.473671\pi$
$293$	2.82843			0.165238			0.0826192	+	0.996581i	$0.473671\pi$
$294$		0			0
$295$	3.46410	−	2.82843i	0.201688	−	0.164677i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	40.0000i		−	2.31326i
$300$		0			0
$301$		−	3.46410i		−	0.199667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.00000	−	4.89898i	0.343559	−	0.280515i
$306$		0			0
$307$	−2.44949			−0.139800			−0.0698999	−	0.997554i	$-0.522268\pi$
$307$	−2.44949			−0.139800			−0.0698999	+	0.997554i	$0.522268\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−27.7128			−1.57145			−0.785725	−	0.618576i	$-0.787710\pi$
$311$	−27.7128			−1.57145			−0.785725	+	0.618576i	$0.787710\pi$
$312$		0			0
$313$		−	29.3939i		−	1.66144i	−0.556690	−	0.830720i	$-0.687929\pi$
$313$		−	29.3939i		−	1.66144i	0.556690	−	0.830720i	$-0.312071\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	22.6274			1.27088			0.635441	−	0.772149i	$-0.280818\pi$
$317$	22.6274			1.27088			0.635441	+	0.772149i	$0.280818\pi$
$318$		0			0
$319$	13.8564			0.775810
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−29.3939			−1.63552
$324$		0			0
$325$	−5.65685	+	27.7128i	−0.313786	+	1.53723i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.00000			0.330791
$330$		0			0
$331$		−	6.00000i		−	0.329790i	−0.986311	−	0.164895i	$-0.947272\pi$
$331$		−	6.00000i		−	0.329790i	0.986311	−	0.164895i	$-0.0527285\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.46410	−	4.24264i	−0.189264	−	0.231800i
$336$		0			0
$337$		−	19.5959i		−	1.06746i	−0.845656	−	0.533729i	$-0.820790\pi$
$337$		−	19.5959i		−	1.06746i	0.845656	−	0.533729i	$-0.179210\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	13.8564i		−	0.750366i
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.1464			0.920468			0.460234	−	0.887798i	$-0.347765\pi$
$347$	17.1464			0.920468			0.460234	+	0.887798i	$0.347765\pi$
$348$		0			0
$349$		−	6.92820i		−	0.370858i	−0.982658	−	0.185429i	$-0.940632\pi$
$349$		−	6.92820i		−	0.370858i	0.982658	−	0.185429i	$-0.0593675\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	9.79796i		−	0.521493i	−0.965407	−	0.260746i	$-0.916031\pi$
$353$		−	9.79796i		−	0.521493i	0.965407	−	0.260746i	$-0.0839686\pi$
$354$		0			0
$355$	−9.79796	−	12.0000i	−0.520022	−	0.636894i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	27.7128			1.46263			0.731313	−	0.682042i	$-0.238908\pi$
$359$	27.7128			1.46263			0.731313	+	0.682042i	$0.238908\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.48528	+	6.92820i	−0.444140	+	0.362639i
$366$		0			0
$367$			9.89949i			0.516749i	0.966045	+	0.258375i	$0.0831869\pi$
$367$			9.89949i			0.516749i	−0.966045	+	0.258375i	$0.916813\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	19.7990			1.02515			0.512576	−	0.858642i	$-0.328691\pi$
$373$	19.7990			1.02515			0.512576	+	0.858642i	$0.328691\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	39.1918i		−	2.01848i
$378$		0			0
$379$			14.0000i			0.719132i	0.933120	+	0.359566i	$0.117075\pi$
$379$			14.0000i			0.719132i	−0.933120	+	0.359566i	$0.882925\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			12.7279i			0.650366i	0.945651	+	0.325183i	$0.105426\pi$
$383$			12.7279i			0.650366i	−0.945651	+	0.325183i	$0.894574\pi$
$384$		0			0
$385$	4.00000	+	4.89898i	0.203859	+	0.249675i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	3.46410i		−	0.175637i	−0.996136	−	0.0878185i	$-0.972010\pi$
$389$		−	3.46410i		−	0.175637i	0.996136	−	0.0878185i	$-0.0279895\pi$
$390$		0			0
$391$	−34.6410			−1.75187
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.79796	+	12.0000i	0.492989	+	0.603786i
$396$		0			0
$397$	−11.3137			−0.567819			−0.283909	−	0.958851i	$-0.591631\pi$
$397$	−11.3137			−0.567819			−0.283909	+	0.958851i	$0.591631\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−22.0000			−1.09863			−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000			−1.09863			−0.549314	+	0.835616i	$0.685111\pi$
$402$		0			0
$403$	−39.1918			−1.95228
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	5.65685i		−	0.280400i
$408$		0			0
$409$	−20.0000			−0.988936			−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000			−0.988936			−0.494468	+	0.869196i	$0.664637\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.82843			−0.139178
$414$		0			0
$415$	17.3205	+	21.2132i	0.850230	+	1.04132i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			26.0000i			1.27018i	0.772437	+	0.635092i	$0.219038\pi$
$419$			26.0000i			1.27018i	−0.772437	+	0.635092i	$0.780962\pi$
$420$		0			0
$421$			3.46410i			0.168830i	0.996431	+	0.0844150i	$0.0269021\pi$
$421$			3.46410i			0.168830i	−0.996431	+	0.0844150i	$0.973098\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	24.0000	+	4.89898i	1.16417	+	0.237635i
$426$		0			0
$427$	−4.89898			−0.237078
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.92820			−0.333720			−0.166860	−	0.985981i	$-0.553363\pi$
$431$	−6.92820			−0.333720			−0.166860	+	0.985981i	$0.553363\pi$
$432$		0			0
$433$			34.2929i			1.64801i	0.566583	+	0.824005i	$0.308265\pi$
$433$			34.2929i			1.64801i	−0.566583	+	0.824005i	$0.691735\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−42.4264			−2.02953
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	22.0454			1.04741			0.523704	−	0.851900i	$-0.324550\pi$
$443$	22.0454			1.04741			0.523704	+	0.851900i	$0.324550\pi$
$444$		0			0
$445$	2.82843	+	3.46410i	0.134080	+	0.164214i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−16.0000			−0.755087			−0.377543	−	0.925992i	$-0.623231\pi$
$449$	−16.0000			−0.755087			−0.377543	+	0.925992i	$0.623231\pi$
$450$		0			0
$451$			8.00000i			0.376705i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	13.8564	−	11.3137i	0.649598	−	0.530395i
$456$		0			0
$457$			19.5959i			0.916658i	0.888783	+	0.458329i	$0.151552\pi$
$457$			19.5959i			0.916658i	−0.888783	+	0.458329i	$0.848448\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	20.7846i		−	0.968036i	−0.875058	−	0.484018i	$-0.839177\pi$
$461$		−	20.7846i		−	0.968036i	0.875058	−	0.484018i	$-0.160823\pi$
$462$		0			0
$463$			1.41421i			0.0657241i	0.999460	+	0.0328620i	$0.0104622\pi$
$463$			1.41421i			0.0657241i	−0.999460	+	0.0328620i	$0.989538\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−31.8434			−1.47354			−0.736768	−	0.676146i	$-0.763649\pi$
$467$	−31.8434			−1.47354			−0.736768	+	0.676146i	$0.763649\pi$
$468$		0			0
$469$			3.46410i			0.159957i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			4.89898i			0.225255i
$474$		0			0
$475$	29.3939	+	6.00000i	1.34868	+	0.275299i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	41.5692			1.89935			0.949673	−	0.313243i	$-0.101415\pi$
$479$	41.5692			1.89935			0.949673	+	0.313243i	$0.101415\pi$
$480$		0			0
$481$	−16.0000			−0.729537
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−25.4558	+	20.7846i	−1.15589	+	0.943781i
$486$		0			0
$487$			26.8701i			1.21760i	0.793324	+	0.608799i	$0.208349\pi$
$487$			26.8701i			1.21760i	−0.793324	+	0.608799i	$0.791651\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			38.0000i			1.71492i	0.514554	+	0.857458i	$0.327958\pi$
$491$			38.0000i			1.71492i	−0.514554	+	0.857458i	$0.672042\pi$
$492$		0			0
$493$	−33.9411			−1.52863
$494$		0			0
$495$		0			0
$496$		0			0
$497$			9.79796i			0.439499i
$498$		0			0
$499$			6.00000i			0.268597i	0.990941	+	0.134298i	$0.0428781\pi$
$499$			6.00000i			0.268597i	−0.990941	+	0.134298i	$0.957122\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			21.2132i			0.945850i	0.881103	+	0.472925i	$0.156802\pi$
$503$			21.2132i			0.945850i	−0.881103	+	0.472925i	$0.843198\pi$
$504$		0			0
$505$	12.0000	−	9.79796i	0.533993	−	0.436003i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			20.7846i			0.921262i	0.887592	+	0.460631i	$0.152377\pi$
$509$			20.7846i			0.921262i	−0.887592	+	0.460631i	$0.847623\pi$
$510$		0			0
$511$	6.92820			0.306486
$512$		0			0
$513$		0			0
$514$		0			0
$515$	17.1464	−	14.0000i	0.755562	−	0.616914i
$516$		0			0
$517$	−8.48528			−0.373182
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−34.0000			−1.48957			−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000			−1.48957			−0.744784	+	0.667306i	$0.767447\pi$
$522$		0			0
$523$	−2.44949			−0.107109			−0.0535544	−	0.998565i	$-0.517055\pi$
$523$	−2.44949			−0.107109			−0.0535544	+	0.998565i	$0.517055\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			33.9411i			1.47850i
$528$		0			0
$529$	−27.0000			−1.17391
$530$		0			0
$531$		0			0
$532$		0			0
$533$	22.6274			0.980102
$534$		0			0
$535$	3.46410	+	4.24264i	0.149766	+	0.183425i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			10.0000i			0.430730i
$540$		0			0
$541$		−	6.92820i		−	0.297867i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$		−	6.92820i		−	0.297867i	0.988847	−	0.148933i	$-0.0475840\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.00000	+	4.89898i	−0.257012	+	0.209849i
$546$		0			0
$547$	−26.9444			−1.15206			−0.576029	−	0.817429i	$-0.695399\pi$
$547$	−26.9444			−1.15206			−0.576029	+	0.817429i	$0.695399\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−41.5692			−1.77091
$552$		0			0
$553$		−	9.79796i		−	0.416652i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−25.4558			−1.07860			−0.539299	−	0.842114i	$-0.681311\pi$
$557$	−25.4558			−1.07860			−0.539299	+	0.842114i	$0.681311\pi$
$558$		0			0
$559$	13.8564			0.586064
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.34847			0.309701			0.154851	−	0.987938i	$-0.450510\pi$
$563$	7.34847			0.309701			0.154851	+	0.987938i	$0.450510\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	28.0000			1.17382			0.586911	−	0.809652i	$-0.300344\pi$
$569$	28.0000			1.17382			0.586911	+	0.809652i	$0.300344\pi$
$570$		0			0
$571$			30.0000i			1.25546i	0.778431	+	0.627730i	$0.216016\pi$
$571$			30.0000i			1.25546i	−0.778431	+	0.627730i	$0.783984\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	34.6410	+	7.07107i	1.44463	+	0.294884i
$576$		0			0
$577$		−	29.3939i		−	1.22368i	−0.790980	−	0.611842i	$-0.790429\pi$
$577$		−	29.3939i		−	1.22368i	0.790980	−	0.611842i	$-0.209571\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	17.3205i		−	0.718576i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.2474			−0.505506			−0.252753	−	0.967531i	$-0.581336\pi$
$587$	−12.2474			−0.505506			−0.252753	+	0.967531i	$0.581336\pi$
$588$		0			0
$589$			41.5692i			1.71283i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	29.3939i		−	1.20706i	−0.797340	−	0.603531i	$-0.793760\pi$
$593$		−	29.3939i		−	1.20706i	0.797340	−	0.603531i	$-0.206240\pi$
$594$		0			0
$595$	−9.79796	−	12.0000i	−0.401677	−	0.491952i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−20.7846			−0.849236			−0.424618	−	0.905373i	$-0.639592\pi$
$599$	−20.7846			−0.849236			−0.424618	+	0.905373i	$0.639592\pi$
$600$		0			0
$601$	−24.0000			−0.978980			−0.489490	−	0.872009i	$-0.662817\pi$
$601$	−24.0000			−0.978980			−0.489490	+	0.872009i	$0.662817\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9.89949	+	12.1244i	0.402472	+	0.492925i
$606$		0			0
$607$			41.0122i			1.66463i	0.554300	+	0.832317i	$0.312986\pi$
$607$			41.0122i			1.66463i	−0.554300	+	0.832317i	$0.687014\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			24.0000i			0.970936i
$612$		0			0
$613$	5.65685			0.228478			0.114239	−	0.993453i	$-0.463557\pi$
$613$	5.65685			0.228478			0.114239	+	0.993453i	$0.463557\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	24.4949i		−	0.986127i	−0.869993	−	0.493064i	$-0.835877\pi$
$617$		−	24.4949i		−	0.986127i	0.869993	−	0.493064i	$-0.164123\pi$
$618$		0			0
$619$		−	14.0000i		−	0.562708i	−0.959604	−	0.281354i	$-0.909217\pi$
$619$		−	14.0000i		−	0.562708i	0.959604	−	0.281354i	$-0.0907834\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	2.82843i		−	0.113319i
$624$		0			0
$625$	−23.0000	−	9.79796i	−0.920000	−	0.391918i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			13.8564i			0.552491i
$630$		0			0
$631$	−13.8564			−0.551615			−0.275807	−	0.961213i	$-0.588945\pi$
$631$	−13.8564			−0.551615			−0.275807	+	0.961213i	$0.588945\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	12.2474	−	10.0000i	0.486025	−	0.396838i
$636$		0			0
$637$	28.2843			1.12066
$638$		0			0
$639$		0			0
$640$		0			0
$641$	40.0000			1.57991			0.789953	−	0.613168i	$-0.210105\pi$
$641$	40.0000			1.57991			0.789953	+	0.613168i	$0.210105\pi$
$642$		0			0
$643$	−2.44949			−0.0965984			−0.0482992	−	0.998833i	$-0.515380\pi$
$643$	−2.44949			−0.0965984			−0.0482992	+	0.998833i	$0.515380\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	4.24264i		−	0.166795i	−0.996516	−	0.0833977i	$-0.973423\pi$
$647$		−	4.24264i		−	0.166795i	0.996516	−	0.0833977i	$-0.0265772\pi$
$648$		0			0
$649$	4.00000			0.157014
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.65685			−0.221370			−0.110685	−	0.993856i	$-0.535304\pi$
$653$	−5.65685			−0.221370			−0.110685	+	0.993856i	$0.535304\pi$
$654$		0			0
$655$	17.3205	−	14.1421i	0.676768	−	0.552579i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			2.00000i			0.0779089i	0.999241	+	0.0389545i	$0.0124027\pi$
$659$			2.00000i			0.0779089i	−0.999241	+	0.0389545i	$0.987597\pi$
$660$		0			0
$661$			10.3923i			0.404214i	0.979363	+	0.202107i	$0.0647788\pi$
$661$			10.3923i			0.404214i	−0.979363	+	0.202107i	$0.935221\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−12.0000	−	14.6969i	−0.465340	−	0.569923i
$666$		0			0
$667$	−48.9898			−1.89689
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6.92820			0.267460
$672$		0			0
$673$		−	24.4949i		−	0.944209i	−0.881543	−	0.472104i	$-0.843495\pi$
$673$		−	24.4949i		−	0.944209i	0.881543	−	0.472104i	$-0.156505\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.65685			−0.217411			−0.108705	−	0.994074i	$-0.534670\pi$
$677$	−5.65685			−0.217411			−0.108705	+	0.994074i	$0.534670\pi$
$678$		0			0
$679$	20.7846			0.797640
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.44949			−0.0937271			−0.0468636	−	0.998901i	$-0.514923\pi$
$683$	−2.44949			−0.0937271			−0.0468636	+	0.998901i	$0.514923\pi$
$684$		0			0
$685$	−16.9706	+	13.8564i	−0.648412	+	0.529426i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$			42.0000i			1.59776i	0.601494	+	0.798878i	$0.294573\pi$
$691$			42.0000i			1.59776i	−0.601494	+	0.798878i	$0.705427\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	17.3205	−	14.1421i	0.657004	−	0.536442i
$696$		0			0
$697$		−	19.5959i		−	0.742248i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	3.46410i		−	0.130837i	−0.997858	−	0.0654187i	$-0.979162\pi$
$701$		−	3.46410i		−	0.130837i	0.997858	−	0.0654187i	$-0.0208383\pi$
$702$		0			0
$703$			16.9706i			0.640057i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−9.79796			−0.368490
$708$		0			0
$709$			34.6410i			1.30097i	0.759519	+	0.650485i	$0.225434\pi$
$709$			34.6410i			1.30097i	−0.759519	+	0.650485i	$0.774566\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			48.9898i			1.83468i
$714$		0			0
$715$	−19.5959	+	16.0000i	−0.732846	+	0.598366i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6.92820			−0.258378			−0.129189	−	0.991620i	$-0.541237\pi$
$719$	−6.92820			−0.258378			−0.129189	+	0.991620i	$0.541237\pi$
$720$		0			0
$721$	−14.0000			−0.521387
$722$		0			0
$723$		0			0
$724$		0			0
$725$	33.9411	+	6.92820i	1.26054	+	0.257307i
$726$		0			0
$727$		−	18.3848i		−	0.681854i	−0.940090	−	0.340927i	$-0.889259\pi$
$727$		−	18.3848i		−	0.681854i	0.940090	−	0.340927i	$-0.110741\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	12.0000i		−	0.443836i
$732$		0			0
$733$	−19.7990			−0.731292			−0.365646	−	0.930754i	$-0.619152\pi$
$733$	−19.7990			−0.731292			−0.365646	+	0.930754i	$0.619152\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	4.89898i		−	0.180456i
$738$		0			0
$739$			30.0000i			1.10357i	0.833987	+	0.551784i	$0.186053\pi$
$739$			30.0000i			1.10357i	−0.833987	+	0.551784i	$0.813947\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	4.24264i		−	0.155647i	−0.996967	−	0.0778237i	$-0.975203\pi$
$743$		−	4.24264i		−	0.155647i	0.996967	−	0.0778237i	$-0.0247971\pi$
$744$		0			0
$745$	−18.0000	+	14.6969i	−0.659469	+	0.538454i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	3.46410i		−	0.126576i
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−19.5959	−	24.0000i	−0.713168	−	0.873449i
$756$		0			0
$757$	2.82843			0.102801			0.0514005	−	0.998678i	$-0.483632\pi$
$757$	2.82843			0.102801			0.0514005	+	0.998678i	$0.483632\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−38.0000			−1.37750			−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000			−1.37750			−0.688749	+	0.724999i	$0.741840\pi$
$762$		0			0
$763$	4.89898			0.177355
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	11.3137i		−	0.408514i
$768$		0			0
$769$	42.0000			1.51456			0.757279	−	0.653091i	$-0.226528\pi$
$769$	42.0000			1.51456			0.757279	+	0.653091i	$0.226528\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−45.2548			−1.62770			−0.813852	−	0.581073i	$-0.802633\pi$
$773$	−45.2548			−1.62770			−0.813852	+	0.581073i	$0.802633\pi$
$774$		0			0
$775$	6.92820	−	33.9411i	0.248868	−	1.21920i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	24.0000i		−	0.859889i
$780$		0			0
$781$		−	13.8564i		−	0.495821i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.00000	+	4.89898i	0.142766	+	0.174852i
$786$		0			0
$787$	46.5403			1.65898			0.829491	−	0.558520i	$-0.188630\pi$
$787$	46.5403			1.65898			0.829491	+	0.558520i	$0.188630\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	19.5959i		−	0.695871i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	33.9411			1.20226			0.601128	−	0.799153i	$-0.294718\pi$
$797$	33.9411			1.20226			0.601128	+	0.799153i	$0.294718\pi$
$798$		0			0
$799$	20.7846			0.735307
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9.79796			−0.345762
$804$		0			0
$805$	−14.1421	−	17.3205i	−0.498445	−	0.610468i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−50.0000			−1.75791			−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000			−1.75791			−0.878953	+	0.476908i	$0.841757\pi$
$810$		0			0
$811$			22.0000i			0.772524i	0.922389	+	0.386262i	$0.126234\pi$
$811$			22.0000i			0.772524i	−0.922389	+	0.386262i	$0.873766\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	31.1769	+	38.1838i	1.09208	+	1.33752i
$816$		0			0
$817$		−	14.6969i		−	0.514181i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	51.9615i		−	1.81347i	−0.421701	−	0.906735i	$-0.638567\pi$
$821$		−	51.9615i		−	1.81347i	0.421701	−	0.906735i	$-0.361433\pi$
$822$		0			0
$823$			35.3553i			1.23241i	0.787586	+	0.616205i	$0.211331\pi$
$823$			35.3553i			1.23241i	−0.787586	+	0.616205i	$0.788669\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−31.8434			−1.10730			−0.553651	−	0.832749i	$-0.686766\pi$
$827$	−31.8434			−1.10730			−0.553651	+	0.832749i	$0.686766\pi$
$828$		0			0
$829$		−	45.0333i		−	1.56407i	−0.623233	−	0.782036i	$-0.714181\pi$
$829$		−	45.0333i		−	1.56407i	0.623233	−	0.782036i	$-0.285819\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	24.4949i		−	0.848698i
$834$		0			0
$835$	−17.1464	+	14.0000i	−0.593377	+	0.484490i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−41.5692			−1.43513			−0.717564	−	0.696492i	$-0.754743\pi$
$839$	−41.5692			−1.43513			−0.717564	+	0.696492i	$0.754743\pi$
$840$		0			0
$841$	−19.0000			−0.655172
$842$		0			0
$843$		0			0
$844$		0			0
$845$	26.8701	+	32.9090i	0.924358	+	1.13210i
$846$		0			0
$847$		−	9.89949i		−	0.340151i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			20.0000i			0.685591i
$852$		0			0
$853$	5.65685			0.193687			0.0968435	−	0.995300i	$-0.469125\pi$
$853$	5.65685			0.193687			0.0968435	+	0.995300i	$0.469125\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			39.1918i			1.33877i	0.742917	+	0.669384i	$0.233442\pi$
$857$			39.1918i			1.33877i	−0.742917	+	0.669384i	$0.766558\pi$
$858$		0			0
$859$		−	22.0000i		−	0.750630i	−0.926897	−	0.375315i	$-0.877534\pi$
$859$		−	22.0000i		−	0.750630i	0.926897	−	0.375315i	$-0.122466\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	24.0416i		−	0.818387i	−0.912448	−	0.409193i	$-0.865810\pi$
$863$		−	24.0416i		−	0.818387i	0.912448	−	0.409193i	$-0.134190\pi$
$864$		0			0
$865$	−12.0000	−	14.6969i	−0.408012	−	0.499711i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			13.8564i			0.470046i
$870$		0			0
$871$	−13.8564			−0.469506
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7.34847	+	14.0000i	0.248424	+	0.473286i
$876$		0			0
$877$	−36.7696			−1.24162			−0.620810	−	0.783961i	$-0.713196\pi$
$877$	−36.7696			−1.24162			−0.620810	+	0.783961i	$0.713196\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−52.0000			−1.75192			−0.875962	−	0.482380i	$-0.839773\pi$
$881$	−52.0000			−1.75192			−0.875962	+	0.482380i	$0.839773\pi$
$882$		0			0
$883$	12.2474			0.412159			0.206080	−	0.978535i	$-0.433929\pi$
$883$	12.2474			0.412159			0.206080	+	0.978535i	$0.433929\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	49.4975i		−	1.66196i	−0.556300	−	0.830981i	$-0.687780\pi$
$887$		−	49.4975i		−	1.66196i	0.556300	−	0.830981i	$-0.312220\pi$
$888$		0			0
$889$	−10.0000			−0.335389
$890$		0			0
$891$		0			0
$892$		0			0
$893$	25.4558			0.851847
$894$		0			0
$895$	−3.46410	+	2.82843i	−0.115792	+	0.0945439i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			48.0000i			1.60089i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	36.0000	−	29.3939i	1.19668	−	0.977086i
$906$		0			0
$907$	26.9444			0.894674			0.447337	−	0.894366i	$-0.352373\pi$
$907$	26.9444			0.894674			0.447337	+	0.894366i	$0.352373\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−48.4974			−1.60679			−0.803396	−	0.595446i	$-0.796976\pi$
$911$	−48.4974			−1.60679			−0.803396	+	0.595446i	$0.796976\pi$
$912$		0			0
$913$			24.4949i			0.810663i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−14.1421			−0.467014
$918$		0			0
$919$	−6.92820			−0.228540			−0.114270	−	0.993450i	$-0.536453\pi$
$919$	−6.92820			−0.228540			−0.114270	+	0.993450i	$0.536453\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−39.1918			−1.29001
$924$		0			0
$925$	2.82843	−	13.8564i	0.0929981	−	0.455596i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	8.00000			0.262471			0.131236	−	0.991351i	$-0.458106\pi$
$929$	8.00000			0.262471			0.131236	+	0.991351i	$0.458106\pi$
$930$		0			0
$931$		−	30.0000i		−	0.983210i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	13.8564	+	16.9706i	0.453153	+	0.554997i
$936$		0			0
$937$		−	24.4949i		−	0.800213i	−0.916469	−	0.400107i	$-0.868973\pi$
$937$		−	24.4949i		−	0.800213i	0.916469	−	0.400107i	$-0.131027\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			34.6410i			1.12926i	0.825342	+	0.564632i	$0.190982\pi$
$941$			34.6410i			1.12926i	−0.825342	+	0.564632i	$0.809018\pi$
$942$		0			0
$943$		−	28.2843i		−	0.921063i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	41.6413			1.35316			0.676581	−	0.736369i	$-0.263461\pi$
$947$	41.6413			1.35316			0.676581	+	0.736369i	$0.263461\pi$
$948$		0			0
$949$			27.7128i			0.899596i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	39.1918i		−	1.26955i	−0.772698	−	0.634774i	$-0.781093\pi$
$953$		−	39.1918i		−	1.26955i	0.772698	−	0.634774i	$-0.218907\pi$
$954$		0			0
$955$	−19.5959	−	24.0000i	−0.634109	−	0.776622i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	13.8564			0.447447
$960$		0			0
$961$	17.0000			0.548387
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8.48528	+	6.92820i	−0.273151	+	0.223027i
$966$		0			0
$967$		−	15.5563i		−	0.500258i	−0.968212	−	0.250129i	$-0.919527\pi$
$967$		−	15.5563i		−	0.500258i	0.968212	−	0.250129i	$-0.0804731\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			22.0000i			0.706014i	0.935621	+	0.353007i	$0.114841\pi$
$971$			22.0000i			0.706014i	−0.935621	+	0.353007i	$0.885159\pi$
$972$		0			0
$973$	−14.1421			−0.453376
$974$		0			0
$975$		0			0
$976$		0			0
$977$			4.89898i			0.156732i	0.996925	+	0.0783661i	$0.0249703\pi$
$977$			4.89898i			0.156732i	−0.996925	+	0.0783661i	$0.975030\pi$
$978$		0			0
$979$			4.00000i			0.127841i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			46.6690i			1.48851i	0.667895	+	0.744256i	$0.267196\pi$
$983$			46.6690i			1.48851i	−0.667895	+	0.744256i	$0.732804\pi$
$984$		0			0
$985$	32.0000	+	39.1918i	1.01960	+	1.24876i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	17.3205i		−	0.550760i
$990$		0			0
$991$	−27.7128			−0.880327			−0.440163	−	0.897918i	$-0.645079\pi$
$991$	−27.7128			−0.880327			−0.440163	+	0.897918i	$0.645079\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−29.3939	−	36.0000i	−0.931849	−	1.14128i
$996$		0			0
$997$	56.5685			1.79154			0.895772	−	0.444514i	$-0.146624\pi$
$997$	56.5685			1.79154			0.895772	+	0.444514i	$0.146624\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.d.g.289.7		8
3.2	odd	2		320.2.f.b.289.5	yes	8
4.3	odd	2	inner	2880.2.d.g.289.8		8
5.4	even	2	inner	2880.2.d.g.289.4		8
8.3	odd	2	inner	2880.2.d.g.289.2		8
8.5	even	2	inner	2880.2.d.g.289.1		8
12.11	even	2		320.2.f.b.289.1	✓	8
15.2	even	4		1600.2.d.i.801.3		8
15.8	even	4		1600.2.d.i.801.5		8
15.14	odd	2		320.2.f.b.289.3	yes	8
20.19	odd	2	inner	2880.2.d.g.289.3		8
24.5	odd	2		320.2.f.b.289.4	yes	8
24.11	even	2		320.2.f.b.289.8	yes	8
40.19	odd	2	inner	2880.2.d.g.289.5		8
40.29	even	2	inner	2880.2.d.g.289.6		8
48.5	odd	4		1280.2.c.h.769.2		4
48.11	even	4		1280.2.c.g.769.4		4
48.29	odd	4		1280.2.c.g.769.3		4
48.35	even	4		1280.2.c.h.769.1		4
60.23	odd	4		1600.2.d.i.801.4		8
60.47	odd	4		1600.2.d.i.801.6		8
60.59	even	2		320.2.f.b.289.7	yes	8
120.29	odd	2		320.2.f.b.289.6	yes	8
120.53	even	4		1600.2.d.i.801.2		8
120.59	even	2		320.2.f.b.289.2	yes	8
120.77	even	4		1600.2.d.i.801.8		8
120.83	odd	4		1600.2.d.i.801.7		8
120.107	odd	4		1600.2.d.i.801.1		8
240.29	odd	4		1280.2.c.g.769.1		4
240.53	even	4		6400.2.a.cu.1.4		4
240.59	even	4		1280.2.c.g.769.2		4
240.77	even	4		6400.2.a.ct.1.3		4
240.83	odd	4		6400.2.a.cu.1.3		4
240.107	odd	4		6400.2.a.ct.1.4		4
240.149	odd	4		1280.2.c.h.769.4		4
240.173	even	4		6400.2.a.ct.1.2		4
240.179	even	4		1280.2.c.h.769.3		4
240.197	even	4		6400.2.a.cu.1.1		4
240.203	odd	4		6400.2.a.ct.1.1		4
240.227	odd	4		6400.2.a.cu.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.f.b.289.1	✓	8	12.11	even	2
320.2.f.b.289.2	yes	8	120.59	even	2
320.2.f.b.289.3	yes	8	15.14	odd	2
320.2.f.b.289.4	yes	8	24.5	odd	2
320.2.f.b.289.5	yes	8	3.2	odd	2
320.2.f.b.289.6	yes	8	120.29	odd	2
320.2.f.b.289.7	yes	8	60.59	even	2
320.2.f.b.289.8	yes	8	24.11	even	2
1280.2.c.g.769.1		4	240.29	odd	4
1280.2.c.g.769.2		4	240.59	even	4
1280.2.c.g.769.3		4	48.29	odd	4
1280.2.c.g.769.4		4	48.11	even	4
1280.2.c.h.769.1		4	48.35	even	4
1280.2.c.h.769.2		4	48.5	odd	4
1280.2.c.h.769.3		4	240.179	even	4
1280.2.c.h.769.4		4	240.149	odd	4
1600.2.d.i.801.1		8	120.107	odd	4
1600.2.d.i.801.2		8	120.53	even	4
1600.2.d.i.801.3		8	15.2	even	4
1600.2.d.i.801.4		8	60.23	odd	4
1600.2.d.i.801.5		8	15.8	even	4
1600.2.d.i.801.6		8	60.47	odd	4
1600.2.d.i.801.7		8	120.83	odd	4
1600.2.d.i.801.8		8	120.77	even	4
2880.2.d.g.289.1		8	8.5	even	2	inner
2880.2.d.g.289.2		8	8.3	odd	2	inner
2880.2.d.g.289.3		8	20.19	odd	2	inner
2880.2.d.g.289.4		8	5.4	even	2	inner
2880.2.d.g.289.5		8	40.19	odd	2	inner
2880.2.d.g.289.6		8	40.29	even	2	inner
2880.2.d.g.289.7		8	1.1	even	1	trivial
2880.2.d.g.289.8		8	4.3	odd	2	inner
6400.2.a.ct.1.1		4	240.203	odd	4
6400.2.a.ct.1.2		4	240.173	even	4
6400.2.a.ct.1.3		4	240.77	even	4
6400.2.a.ct.1.4		4	240.107	odd	4
6400.2.a.cu.1.1		4	240.197	even	4
6400.2.a.cu.1.2		4	240.227	odd	4
6400.2.a.cu.1.3		4	240.83	odd	4
6400.2.a.cu.1.4		4	240.53	even	4

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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.41421 + 1.73205i) q^{5} -1.41421i q^{7} +2.00000i q^{11} +5.65685 q^{13} -4.89898i q^{17} -6.00000i q^{19} -7.07107i q^{23} +(-1.00000 + 4.89898i) q^{25} -6.92820i q^{29} -6.92820 q^{31} +(2.44949 - 2.00000i) q^{35} -2.82843 q^{37} +4.00000 q^{41} +2.44949 q^{43} +4.24264i q^{47} +5.00000 q^{49} +(-3.46410 + 2.82843i) q^{55} -2.00000i q^{59} -3.46410i q^{61} +(8.00000 + 9.79796i) q^{65} -2.44949 q^{67} -6.92820 q^{71} +4.89898i q^{73} +2.82843 q^{77} +6.92820 q^{79} +12.2474 q^{83} +(8.48528 - 6.92820i) q^{85} +2.00000 q^{89} -8.00000i q^{91} +(10.3923 - 8.48528i) q^{95} +14.6969i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{25} + 32 q^{41} + 40 q^{49} + 64 q^{65} + 16 q^{89}+O(q^{100}) \) 8 * q - 8 * q^25 + 32 * q^41 + 40 * q^49 + 64 * q^65 + 16 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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