LMFDB - Embedded newform 2880.2.d.h.289.1

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^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.1
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2880.289
Dual form			2880.2.d.h.289.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 - 1.73205i) q^{5} -2.82843i q^{7} +O(q^{10})$	$q+(-1.41421 - 1.73205i) q^{5} -2.82843i q^{7} +2.00000i q^{11} -2.82843 q^{13} +4.89898i q^{17} +5.65685i q^{23} +(-1.00000 + 4.89898i) q^{25} -3.46410i q^{29} -3.46410 q^{31} +(-4.89898 + 4.00000i) q^{35} -2.82843 q^{37} +8.00000 q^{41} -9.79796 q^{43} -1.00000 q^{49} +8.48528 q^{53} +(3.46410 - 2.82843i) q^{55} +10.0000i q^{59} +6.92820i q^{61} +(4.00000 + 4.89898i) q^{65} +9.79796 q^{67} +13.8564 q^{71} -9.79796i q^{73} +5.65685 q^{77} +3.46410 q^{79} -9.79796 q^{83} +(8.48528 - 6.92820i) q^{85} +16.0000 q^{89} +8.00000i q^{91} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{25} + 64 q^{41} - 8 q^{49} + 32 q^{65} + 128 q^{89}+O(q^{100}) $ 8 * q - 8 * q^25 + 64 * q^41 - 8 * q^49 + 32 * q^65 + 128 * q^89

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
$7$		−	2.82843i		−	1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$		−	2.82843i		−	1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$		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
$13$	−2.82843			−0.784465			−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843			−0.784465			−0.392232	+	0.919866i	$0.628297\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.89898i			1.18818i	0.804400	+	0.594089i	$0.202487\pi$
$17$			4.89898i			1.18818i	−0.804400	+	0.594089i	$0.797513\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.65685i			1.17954i	0.807573	+	0.589768i	$0.200781\pi$
$23$			5.65685i			1.17954i	−0.807573	+	0.589768i	$0.799219\pi$
$24$		0			0
$25$	−1.00000	+	4.89898i	−0.200000	+	0.979796i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	3.46410i		−	0.643268i	−0.946864	−	0.321634i	$-0.895768\pi$
$29$		−	3.46410i		−	0.643268i	0.946864	−	0.321634i	$-0.104232\pi$
$30$		0			0
$31$	−3.46410			−0.622171			−0.311086	−	0.950382i	$-0.600693\pi$
$31$	−3.46410			−0.622171			−0.311086	+	0.950382i	$0.600693\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.89898	+	4.00000i	−0.828079	+	0.676123i
$36$		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
$41$	8.00000			1.24939			0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000			1.24939			0.624695	+	0.780869i	$0.285223\pi$
$42$		0			0
$43$	−9.79796			−1.49417			−0.747087	−	0.664726i	$-0.768548\pi$
$43$	−9.79796			−1.49417			−0.747087	+	0.664726i	$0.768548\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8.48528			1.16554			0.582772	−	0.812636i	$-0.301968\pi$
$53$	8.48528			1.16554			0.582772	+	0.812636i	$0.301968\pi$
$54$		0			0
$55$	3.46410	−	2.82843i	0.467099	−	0.381385i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			10.0000i			1.30189i	0.759125	+	0.650945i	$0.225627\pi$
$59$			10.0000i			1.30189i	−0.759125	+	0.650945i	$0.774373\pi$
$60$		0			0
$61$			6.92820i			0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$			6.92820i			0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.00000	+	4.89898i	0.496139	+	0.607644i
$66$		0			0
$67$	9.79796			1.19701			0.598506	−	0.801119i	$-0.295761\pi$
$67$	9.79796			1.19701			0.598506	+	0.801119i	$0.295761\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	13.8564			1.64445			0.822226	−	0.569160i	$-0.192732\pi$
$71$	13.8564			1.64445			0.822226	+	0.569160i	$0.192732\pi$
$72$		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
$77$	5.65685			0.644658
$78$		0			0
$79$	3.46410			0.389742			0.194871	−	0.980829i	$-0.437571\pi$
$79$	3.46410			0.389742			0.194871	+	0.980829i	$0.437571\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−9.79796			−1.07547			−0.537733	−	0.843115i	$-0.680719\pi$
$83$	−9.79796			−1.07547			−0.537733	+	0.843115i	$0.680719\pi$
$84$		0			0
$85$	8.48528	−	6.92820i	0.920358	−	0.751469i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	16.0000			1.69600			0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000			1.69600			0.847998	+	0.529999i	$0.177808\pi$
$90$		0			0
$91$			8.00000i			0.838628i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	3.46410i		−	0.344691i	−0.985037	−	0.172345i	$-0.944865\pi$
$101$		−	3.46410i		−	0.344691i	0.985037	−	0.172345i	$-0.0551346\pi$
$102$		0			0
$103$			14.1421i			1.39347i	0.717331	+	0.696733i	$0.245364\pi$
$103$			14.1421i			1.39347i	−0.717331	+	0.696733i	$0.754636\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−19.5959			−1.89441			−0.947204	−	0.320630i	$-0.896105\pi$
$107$	−19.5959			−1.89441			−0.947204	+	0.320630i	$0.896105\pi$
$108$		0			0
$109$		−	6.92820i		−	0.663602i	−0.943349	−	0.331801i	$-0.892344\pi$
$109$		−	6.92820i		−	0.663602i	0.943349	−	0.331801i	$-0.107656\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			14.6969i			1.38257i	0.722581	+	0.691286i	$0.242955\pi$
$113$			14.6969i			1.38257i	−0.722581	+	0.691286i	$0.757045\pi$
$114$		0			0
$115$	9.79796	−	8.00000i	0.913664	−	0.746004i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	13.8564			1.27021
$120$		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
$127$			2.82843i			0.250982i	0.992095	+	0.125491i	$0.0400507\pi$
$127$			2.82843i			0.250982i	−0.992095	+	0.125491i	$0.959949\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			14.0000i			1.22319i	0.791173	+	0.611593i	$0.209471\pi$
$131$			14.0000i			1.22319i	−0.791173	+	0.611593i	$0.790529\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.89898i			0.418548i	0.977857	+	0.209274i	$0.0671101\pi$
$137$			4.89898i			0.418548i	−0.977857	+	0.209274i	$0.932890\pi$
$138$		0			0
$139$		−	8.00000i		−	0.678551i	−0.940687	−	0.339276i	$-0.889818\pi$
$139$		−	8.00000i		−	0.678551i	0.940687	−	0.339276i	$-0.110182\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	5.65685i		−	0.473050i
$144$		0			0
$145$	−6.00000	+	4.89898i	−0.498273	+	0.406838i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	10.3923i		−	0.851371i	−0.904871	−	0.425685i	$-0.860033\pi$
$149$		−	10.3923i		−	0.851371i	0.904871	−	0.425685i	$-0.139967\pi$
$150$		0			0
$151$	−17.3205			−1.40952			−0.704761	−	0.709444i	$-0.748946\pi$
$151$	−17.3205			−1.40952			−0.704761	+	0.709444i	$0.748946\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.89898	+	6.00000i	0.393496	+	0.481932i
$156$		0			0
$157$	19.7990			1.58013			0.790066	−	0.613022i	$-0.210046\pi$
$157$	19.7990			1.58013			0.790066	+	0.613022i	$0.210046\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	16.0000			1.26098
$162$		0			0
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			22.6274i			1.75096i	0.483252	+	0.875481i	$0.339455\pi$
$167$			22.6274i			1.75096i	−0.483252	+	0.875481i	$0.660545\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		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
$175$	13.8564	+	2.82843i	1.04745	+	0.213809i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			14.0000i			1.04641i	0.852207	+	0.523205i	$0.175264\pi$
$179$			14.0000i			1.04641i	−0.852207	+	0.523205i	$0.824736\pi$
$180$		0			0
$181$			20.7846i			1.54491i	0.635071	+	0.772454i	$0.280971\pi$
$181$			20.7846i			1.54491i	−0.635071	+	0.772454i	$0.719029\pi$
$182$		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
$193$		−	9.79796i		−	0.705273i	−0.935760	−	0.352636i	$-0.885285\pi$
$193$		−	9.79796i		−	0.705273i	0.935760	−	0.352636i	$-0.114715\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.82843			0.201517			0.100759	−	0.994911i	$-0.467873\pi$
$197$	2.82843			0.201517			0.100759	+	0.994911i	$0.467873\pi$
$198$		0			0
$199$	10.3923			0.736691			0.368345	−	0.929689i	$-0.379924\pi$
$199$	10.3923			0.736691			0.368345	+	0.929689i	$0.379924\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−9.79796			−0.687682
$204$		0			0
$205$	−11.3137	−	13.8564i	−0.790184	−	0.967773i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$			8.00000i			0.550743i	0.961338	+	0.275371i	$0.0888008\pi$
$211$			8.00000i			0.550743i	−0.961338	+	0.275371i	$0.911199\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	13.8564	+	16.9706i	0.944999	+	1.15738i
$216$		0			0
$217$			9.79796i			0.665129i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	13.8564i		−	0.932083i
$222$		0			0
$223$		−	14.1421i		−	0.947027i	−0.880786	−	0.473514i	$-0.842985\pi$
$223$		−	14.1421i		−	0.947027i	0.880786	−	0.473514i	$-0.157015\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.79796			0.650313			0.325157	−	0.945660i	$-0.394583\pi$
$227$	9.79796			0.650313			0.325157	+	0.945660i	$0.394583\pi$
$228$		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
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.8564			−0.896296			−0.448148	−	0.893959i	$-0.647916\pi$
$239$	−13.8564			−0.896296			−0.448148	+	0.893959i	$0.647916\pi$
$240$		0			0
$241$	18.0000			1.15948			0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000			1.15948			0.579741	+	0.814801i	$0.303154\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.41421	+	1.73205i	0.0903508	+	0.110657i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			14.0000i			0.883672i	0.897096	+	0.441836i	$0.145673\pi$
$251$			14.0000i			0.883672i	−0.897096	+	0.441836i	$0.854327\pi$
$252$		0			0
$253$	−11.3137			−0.711287
$254$		0			0
$255$		0			0
$256$		0			0
$257$			4.89898i			0.305590i	0.988258	+	0.152795i	$0.0488274\pi$
$257$			4.89898i			0.305590i	−0.988258	+	0.152795i	$0.951173\pi$
$258$		0			0
$259$			8.00000i			0.497096i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			11.3137i			0.697633i	0.937191	+	0.348817i	$0.113416\pi$
$263$			11.3137i			0.697633i	−0.937191	+	0.348817i	$0.886584\pi$
$264$		0			0
$265$	−12.0000	−	14.6969i	−0.737154	−	0.902826i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			3.46410i			0.211210i	0.994408	+	0.105605i	$0.0336779\pi$
$269$			3.46410i			0.211210i	−0.994408	+	0.105605i	$0.966322\pi$
$270$		0			0
$271$	−31.1769			−1.89386			−0.946931	−	0.321436i	$-0.895835\pi$
$271$	−31.1769			−1.89386			−0.946931	+	0.321436i	$0.895835\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−9.79796	−	2.00000i	−0.590839	−	0.120605i
$276$		0			0
$277$	−31.1127			−1.86938			−0.934690	−	0.355463i	$-0.884323\pi$
$277$	−31.1127			−1.86938			−0.934690	+	0.355463i	$0.884323\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−16.0000			−0.954480			−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000			−0.954480			−0.477240	+	0.878773i	$0.658363\pi$
$282$		0			0
$283$	9.79796			0.582428			0.291214	−	0.956658i	$-0.405941\pi$
$283$	9.79796			0.582428			0.291214	+	0.956658i	$0.405941\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	22.6274i		−	1.33565i
$288$		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	$-0.526329\pi$
$293$	−2.82843			−0.165238			−0.0826192	+	0.996581i	$0.526329\pi$
$294$		0			0
$295$	17.3205	−	14.1421i	1.00844	−	0.823387i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	16.0000i		−	0.925304i
$300$		0			0
$301$			27.7128i			1.59734i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	12.0000	−	9.79796i	0.687118	−	0.561029i
$306$		0			0
$307$	−19.5959			−1.11840			−0.559199	−	0.829033i	$-0.688891\pi$
$307$	−19.5959			−1.11840			−0.559199	+	0.829033i	$0.688891\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.8564			0.785725			0.392862	−	0.919597i	$-0.371485\pi$
$311$	13.8564			0.785725			0.392862	+	0.919597i	$0.371485\pi$
$312$		0			0
$313$			29.3939i			1.66144i	0.556690	+	0.830720i	$0.312071\pi$
$313$			29.3939i			1.66144i	−0.556690	+	0.830720i	$0.687929\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.82843			0.158860			0.0794301	−	0.996840i	$-0.474690\pi$
$317$	2.82843			0.158860			0.0794301	+	0.996840i	$0.474690\pi$
$318$		0			0
$319$	6.92820			0.387905
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	2.82843	−	13.8564i	0.156893	−	0.768615i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			24.0000i			1.31916i	0.751635	+	0.659580i	$0.229266\pi$
$331$			24.0000i			1.31916i	−0.751635	+	0.659580i	$0.770734\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−13.8564	−	16.9706i	−0.757056	−	0.927201i
$336$		0			0
$337$			9.79796i			0.533729i	0.963734	+	0.266864i	$0.0859876\pi$
$337$			9.79796i			0.533729i	−0.963734	+	0.266864i	$0.914012\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	6.92820i		−	0.375183i
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9.79796			0.525982			0.262991	−	0.964798i	$-0.415291\pi$
$347$	9.79796			0.525982			0.262991	+	0.964798i	$0.415291\pi$
$348$		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
$353$		−	34.2929i		−	1.82522i	−0.408826	−	0.912612i	$-0.634062\pi$
$353$		−	34.2929i		−	1.82522i	0.408826	−	0.912612i	$-0.365938\pi$
$354$		0			0
$355$	−19.5959	−	24.0000i	−1.04004	−	1.27379i
$356$		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
$361$	19.0000			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−16.9706	+	13.8564i	−0.888280	+	0.725277i
$366$		0			0
$367$			2.82843i			0.147643i	0.997271	+	0.0738213i	$0.0235195\pi$
$367$			2.82843i			0.147643i	−0.997271	+	0.0738213i	$0.976481\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	24.0000i		−	1.24602i
$372$		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
$377$			9.79796i			0.504621i
$378$		0			0
$379$		−	32.0000i		−	1.64373i	−0.569683	−	0.821865i	$-0.692934\pi$
$379$		−	32.0000i		−	1.64373i	0.569683	−	0.821865i	$-0.307066\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	16.9706i		−	0.867155i	−0.901116	−	0.433578i	$-0.857251\pi$
$383$		−	16.9706i		−	0.867155i	0.901116	−	0.433578i	$-0.142749\pi$
$384$		0			0
$385$	−8.00000	−	9.79796i	−0.407718	−	0.499350i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			24.2487i			1.22946i	0.788738	+	0.614729i	$0.210735\pi$
$389$			24.2487i			1.22946i	−0.788738	+	0.614729i	$0.789265\pi$
$390$		0			0
$391$	−27.7128			−1.40150
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.89898	−	6.00000i	−0.246494	−	0.301893i
$396$		0			0
$397$	14.1421			0.709773			0.354887	−	0.934909i	$-0.384519\pi$
$397$	14.1421			0.709773			0.354887	+	0.934909i	$0.384519\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−8.00000			−0.399501			−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000			−0.399501			−0.199750	+	0.979847i	$0.564013\pi$
$402$		0			0
$403$	9.79796			0.488071
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	5.65685i		−	0.280400i
$408$		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
$413$	28.2843			1.39178
$414$		0			0
$415$	13.8564	+	16.9706i	0.680184	+	0.833052i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	10.0000i		−	0.488532i	−0.969708	−	0.244266i	$-0.921453\pi$
$419$		−	10.0000i		−	0.488532i	0.969708	−	0.244266i	$-0.0785470\pi$
$420$		0			0
$421$			34.6410i			1.68830i	0.536107	+	0.844150i	$0.319894\pi$
$421$			34.6410i			1.68830i	−0.536107	+	0.844150i	$0.680106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−24.0000	−	4.89898i	−1.16417	−	0.237635i
$426$		0			0
$427$	19.5959			0.948313
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−27.7128			−1.33488			−0.667440	−	0.744664i	$-0.732610\pi$
$431$	−27.7128			−1.33488			−0.667440	+	0.744664i	$0.732610\pi$
$432$		0			0
$433$			19.5959i			0.941720i	0.882208	+	0.470860i	$0.156056\pi$
$433$			19.5959i			0.941720i	−0.882208	+	0.470860i	$0.843944\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	10.3923			0.495998			0.247999	−	0.968760i	$-0.420227\pi$
$439$	10.3923			0.495998			0.247999	+	0.968760i	$0.420227\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	29.3939			1.39655			0.698273	−	0.715832i	$-0.253952\pi$
$443$	29.3939			1.39655			0.698273	+	0.715832i	$0.253952\pi$
$444$		0			0
$445$	−22.6274	−	27.7128i	−1.07264	−	1.31371i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−8.00000			−0.377543			−0.188772	−	0.982021i	$-0.560451\pi$
$449$	−8.00000			−0.377543			−0.188772	+	0.982021i	$0.560451\pi$
$450$		0			0
$451$			16.0000i			0.753411i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	13.8564	−	11.3137i	0.649598	−	0.530395i
$456$		0			0
$457$		−	39.1918i		−	1.83332i	−0.399672	−	0.916658i	$-0.630876\pi$
$457$		−	39.1918i		−	1.83332i	0.399672	−	0.916658i	$-0.369124\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			31.1769i			1.45205i	0.687666	+	0.726027i	$0.258635\pi$
$461$			31.1769i			1.45205i	−0.687666	+	0.726027i	$0.741365\pi$
$462$		0			0
$463$			2.82843i			0.131448i	0.997838	+	0.0657241i	$0.0209357\pi$
$463$			2.82843i			0.131448i	−0.997838	+	0.0657241i	$0.979064\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9.79796			−0.453395			−0.226698	−	0.973965i	$-0.572793\pi$
$467$	−9.79796			−0.453395			−0.226698	+	0.973965i	$0.572793\pi$
$468$		0			0
$469$		−	27.7128i		−	1.27966i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	19.5959i		−	0.901021i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−41.5692			−1.89935			−0.949673	−	0.313243i	$-0.898585\pi$
$479$	−41.5692			−1.89935			−0.949673	+	0.313243i	$0.898585\pi$
$480$		0			0
$481$	8.00000			0.364769
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$			2.82843i			0.128168i	0.997944	+	0.0640841i	$0.0204126\pi$
$487$			2.82843i			0.128168i	−0.997944	+	0.0640841i	$0.979587\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	22.0000i		−	0.992846i	−0.868081	−	0.496423i	$-0.834646\pi$
$491$		−	22.0000i		−	0.992846i	0.868081	−	0.496423i	$-0.165354\pi$
$492$		0			0
$493$	16.9706			0.764316
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	39.1918i		−	1.75799i
$498$		0			0
$499$			24.0000i			1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$			24.0000i			1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			33.9411i			1.51336i	0.653785	+	0.756680i	$0.273180\pi$
$503$			33.9411i			1.51336i	−0.653785	+	0.756680i	$0.726820\pi$
$504$		0			0
$505$	−6.00000	+	4.89898i	−0.266996	+	0.218002i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	10.3923i		−	0.460631i	−0.973116	−	0.230315i	$-0.926024\pi$
$509$		−	10.3923i		−	0.460631i	0.973116	−	0.230315i	$-0.0739758\pi$
$510$		0			0
$511$	−27.7128			−1.22594
$512$		0			0
$513$		0			0
$514$		0			0
$515$	24.4949	−	20.0000i	1.07937	−	0.881305i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8.00000			−0.350486			−0.175243	−	0.984525i	$-0.556071\pi$
$521$	−8.00000			−0.350486			−0.175243	+	0.984525i	$0.556071\pi$
$522$		0			0
$523$	39.1918			1.71374			0.856870	−	0.515533i	$-0.172406\pi$
$523$	39.1918			1.71374			0.856870	+	0.515533i	$0.172406\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	16.9706i		−	0.739249i
$528$		0			0
$529$	−9.00000			−0.391304
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−22.6274			−0.980102
$534$		0			0
$535$	27.7128	+	33.9411i	1.19813	+	1.46740i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	2.00000i		−	0.0861461i
$540$		0			0
$541$			34.6410i			1.48933i	0.667436	+	0.744667i	$0.267392\pi$
$541$			34.6410i			1.48933i	−0.667436	+	0.744667i	$0.732608\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.0000	+	9.79796i	−0.514024	+	0.419698i
$546$		0			0
$547$	−9.79796			−0.418930			−0.209465	−	0.977816i	$-0.567172\pi$
$547$	−9.79796			−0.418930			−0.209465	+	0.977816i	$0.567172\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	9.79796i		−	0.416652i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−42.4264			−1.79766			−0.898832	−	0.438293i	$-0.855583\pi$
$557$	−42.4264			−1.79766			−0.898832	+	0.438293i	$0.855583\pi$
$558$		0			0
$559$	27.7128			1.17213
$560$		0			0
$561$		0			0
$562$		0			0
$563$	29.3939			1.23880			0.619402	−	0.785074i	$-0.287375\pi$
$563$	29.3939			1.23880			0.619402	+	0.785074i	$0.287375\pi$
$564$		0			0
$565$	25.4558	−	20.7846i	1.07094	−	0.874415i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−40.0000			−1.67689			−0.838444	−	0.544988i	$-0.816534\pi$
$569$	−40.0000			−1.67689			−0.838444	+	0.544988i	$0.816534\pi$
$570$		0			0
$571$			24.0000i			1.00437i	0.864761	+	0.502184i	$0.167470\pi$
$571$			24.0000i			1.00437i	−0.864761	+	0.502184i	$0.832530\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−27.7128	−	5.65685i	−1.15570	−	0.235907i
$576$		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
$581$			27.7128i			1.14972i
$582$		0			0
$583$			16.9706i			0.702849i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	39.1918			1.61762			0.808810	−	0.588070i	$-0.200112\pi$
$587$	39.1918			1.61762			0.808810	+	0.588070i	$0.200112\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			14.6969i			0.603531i	0.953382	+	0.301765i	$0.0975760\pi$
$593$			14.6969i			0.603531i	−0.953382	+	0.301765i	$0.902424\pi$
$594$		0			0
$595$	−19.5959	−	24.0000i	−0.803354	−	0.983904i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	18.0000			0.734235			0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000			0.734235			0.367118	+	0.930175i	$0.380345\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9.89949	−	12.1244i	−0.402472	−	0.492925i
$606$		0			0
$607$		−	2.82843i		−	0.114802i	−0.998351	−	0.0574012i	$-0.981719\pi$
$607$		−	2.82843i		−	0.114802i	0.998351	−	0.0574012i	$-0.0182814\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	14.1421			0.571195			0.285598	−	0.958350i	$-0.407808\pi$
$613$	14.1421			0.571195			0.285598	+	0.958350i	$0.407808\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	34.2929i		−	1.38058i	−0.723534	−	0.690289i	$-0.757483\pi$
$617$		−	34.2929i		−	1.38058i	0.723534	−	0.690289i	$-0.242517\pi$
$618$		0			0
$619$		−	40.0000i		−	1.60774i	−0.594808	−	0.803868i	$-0.702772\pi$
$619$		−	40.0000i		−	1.60774i	0.594808	−	0.803868i	$-0.297228\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	45.2548i		−	1.81310i
$624$		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
$631$	−38.1051			−1.51694			−0.758470	−	0.651707i	$-0.774053\pi$
$631$	−38.1051			−1.51694			−0.758470	+	0.651707i	$0.774053\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.89898	−	4.00000i	0.194410	−	0.158735i
$636$		0			0
$637$	2.82843			0.112066
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−40.0000			−1.57991			−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000			−1.57991			−0.789953	+	0.613168i	$0.789895\pi$
$642$		0			0
$643$	−19.5959			−0.772788			−0.386394	−	0.922334i	$-0.626279\pi$
$643$	−19.5959			−0.772788			−0.386394	+	0.922334i	$0.626279\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			33.9411i			1.33436i	0.744895	+	0.667182i	$0.232500\pi$
$647$			33.9411i			1.33436i	−0.744895	+	0.667182i	$0.767500\pi$
$648$		0			0
$649$	−20.0000			−0.785069
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−19.7990			−0.774794			−0.387397	−	0.921913i	$-0.626626\pi$
$653$	−19.7990			−0.774794			−0.387397	+	0.921913i	$0.626626\pi$
$654$		0			0
$655$	24.2487	−	19.7990i	0.947476	−	0.773611i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	46.0000i		−	1.79191i	−0.444149	−	0.895953i	$-0.646494\pi$
$659$		−	46.0000i		−	1.79191i	0.444149	−	0.895953i	$-0.353506\pi$
$660$		0			0
$661$			20.7846i			0.808428i	0.914665	+	0.404214i	$0.132455\pi$
$661$			20.7846i			0.808428i	−0.914665	+	0.404214i	$0.867545\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	19.5959			0.758757
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−13.8564			−0.534921
$672$		0			0
$673$			19.5959i			0.755367i	0.925935	+	0.377684i	$0.123279\pi$
$673$			19.5959i			0.755367i	−0.925935	+	0.377684i	$0.876721\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−19.7990			−0.760937			−0.380468	−	0.924794i	$-0.624237\pi$
$677$	−19.7990			−0.760937			−0.380468	+	0.924794i	$0.624237\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9.79796			−0.374908			−0.187454	−	0.982273i	$-0.560024\pi$
$683$	−9.79796			−0.374908			−0.187454	+	0.982273i	$0.560024\pi$
$684$		0			0
$685$	8.48528	−	6.92820i	0.324206	−	0.264713i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−24.0000			−0.914327
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−13.8564	+	11.3137i	−0.525603	+	0.429153i
$696$		0			0
$697$			39.1918i			1.48450i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	17.3205i		−	0.654187i	−0.944992	−	0.327093i	$-0.893931\pi$
$701$		−	17.3205i		−	0.654187i	0.944992	−	0.327093i	$-0.106069\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−9.79796			−0.368490
$708$		0			0
$709$		−	6.92820i		−	0.260194i	−0.991501	−	0.130097i	$-0.958471\pi$
$709$		−	6.92820i		−	0.260194i	0.991501	−	0.130097i	$-0.0415289\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	19.5959i		−	0.733873i
$714$		0			0
$715$	−9.79796	+	8.00000i	−0.366423	+	0.299183i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−27.7128			−1.03351			−0.516757	−	0.856132i	$-0.672861\pi$
$719$	−27.7128			−1.03351			−0.516757	+	0.856132i	$0.672861\pi$
$720$		0			0
$721$	40.0000			1.48968
$722$		0			0
$723$		0			0
$724$		0			0
$725$	16.9706	+	3.46410i	0.630271	+	0.128654i
$726$		0			0
$727$			14.1421i			0.524503i	0.965000	+	0.262251i	$0.0844650\pi$
$727$			14.1421i			0.524503i	−0.965000	+	0.262251i	$0.915535\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	48.0000i		−	1.77534i
$732$		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
$737$			19.5959i			0.721825i
$738$		0			0
$739$			24.0000i			0.882854i	0.897297	+	0.441427i	$0.145528\pi$
$739$			24.0000i			0.882854i	−0.897297	+	0.441427i	$0.854472\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	16.9706i		−	0.622590i	−0.950313	−	0.311295i	$-0.899237\pi$
$743$		−	16.9706i		−	0.622590i	0.950313	−	0.311295i	$-0.100763\pi$
$744$		0			0
$745$	−18.0000	+	14.6969i	−0.659469	+	0.538454i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			55.4256i			2.02521i
$750$		0			0
$751$	−31.1769			−1.13766			−0.568831	−	0.822455i	$-0.692604\pi$
$751$	−31.1769			−1.13766			−0.568831	+	0.822455i	$0.692604\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	24.4949	+	30.0000i	0.891461	+	1.09181i
$756$		0			0
$757$	−31.1127			−1.13081			−0.565405	−	0.824813i	$-0.691280\pi$
$757$	−31.1127			−1.13081			−0.565405	+	0.824813i	$0.691280\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	8.00000			0.290000			0.145000	−	0.989432i	$-0.453682\pi$
$761$	8.00000			0.290000			0.145000	+	0.989432i	$0.453682\pi$
$762$		0			0
$763$	−19.5959			−0.709420
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	28.2843i		−	1.02129i
$768$		0			0
$769$	6.00000			0.216366			0.108183	−	0.994131i	$-0.465497\pi$
$769$	6.00000			0.216366			0.108183	+	0.994131i	$0.465497\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−31.1127			−1.11905			−0.559523	−	0.828815i	$-0.689016\pi$
$773$	−31.1127			−1.11905			−0.559523	+	0.828815i	$0.689016\pi$
$774$		0			0
$775$	3.46410	−	16.9706i	0.124434	−	0.609601i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$			27.7128i			0.991642i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−28.0000	−	34.2929i	−0.999363	−	1.22396i
$786$		0			0
$787$	19.5959			0.698519			0.349260	−	0.937026i	$-0.386433\pi$
$787$	19.5959			0.698519			0.349260	+	0.937026i	$0.386433\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	41.5692			1.47803
$792$		0			0
$793$		−	19.5959i		−	0.695871i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	42.4264			1.50282			0.751410	−	0.659835i	$-0.229374\pi$
$797$	42.4264			1.50282			0.751410	+	0.659835i	$0.229374\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	19.5959			0.691525
$804$		0			0
$805$	−22.6274	−	27.7128i	−0.797512	−	0.976748i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	8.00000			0.281265			0.140633	−	0.990062i	$-0.455086\pi$
$809$	8.00000			0.281265			0.140633	+	0.990062i	$0.455086\pi$
$810$		0			0
$811$			32.0000i			1.12367i	0.827249	+	0.561836i	$0.189905\pi$
$811$			32.0000i			1.12367i	−0.827249	+	0.561836i	$0.810095\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$			10.3923i			0.362694i	0.983419	+	0.181347i	$0.0580457\pi$
$821$			10.3923i			0.362694i	−0.983419	+	0.181347i	$0.941954\pi$
$822$		0			0
$823$		−	48.0833i		−	1.67608i	−0.545611	−	0.838039i	$-0.683702\pi$
$823$		−	48.0833i		−	1.67608i	0.545611	−	0.838039i	$-0.316298\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	19.5959			0.681417			0.340708	−	0.940169i	$-0.389333\pi$
$827$	19.5959			0.681417			0.340708	+	0.940169i	$0.389333\pi$
$828$		0			0
$829$		−	34.6410i		−	1.20313i	−0.798823	−	0.601566i	$-0.794544\pi$
$829$		−	34.6410i		−	1.20313i	0.798823	−	0.601566i	$-0.205456\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	4.89898i		−	0.169740i
$834$		0			0
$835$	39.1918	−	32.0000i	1.35629	−	1.10741i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	17.0000			0.586207
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.07107	+	8.66025i	0.243252	+	0.297922i
$846$		0			0
$847$		−	19.7990i		−	0.680301i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	16.0000i		−	0.548473i
$852$		0			0
$853$	−19.7990			−0.677905			−0.338952	−	0.940804i	$-0.610073\pi$
$853$	−19.7990			−0.677905			−0.338952	+	0.940804i	$0.610073\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	24.4949i		−	0.836730i	−0.908279	−	0.418365i	$-0.862603\pi$
$857$		−	24.4949i		−	0.836730i	0.908279	−	0.418365i	$-0.137397\pi$
$858$		0			0
$859$		−	32.0000i		−	1.09183i	−0.837842	−	0.545913i	$-0.816183\pi$
$859$		−	32.0000i		−	1.09183i	0.837842	−	0.545913i	$-0.183817\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	11.3137i		−	0.385123i	−0.981285	−	0.192562i	$-0.938320\pi$
$863$		−	11.3137i		−	0.385123i	0.981285	−	0.192562i	$-0.0616795\pi$
$864$		0			0
$865$	12.0000	+	14.6969i	0.408012	+	0.499711i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			6.92820i			0.235023i
$870$		0			0
$871$	−27.7128			−0.939013
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−14.6969	−	28.0000i	−0.496847	−	0.946573i
$876$		0			0
$877$	−2.82843			−0.0955092			−0.0477546	−	0.998859i	$-0.515207\pi$
$877$	−2.82843			−0.0955092			−0.0477546	+	0.998859i	$0.515207\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	16.0000			0.539054			0.269527	−	0.962993i	$-0.413133\pi$
$881$	16.0000			0.539054			0.269527	+	0.962993i	$0.413133\pi$
$882$		0			0
$883$	−19.5959			−0.659455			−0.329728	−	0.944076i	$-0.606957\pi$
$883$	−19.5959			−0.659455			−0.329728	+	0.944076i	$0.606957\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	11.3137i		−	0.379877i	−0.981796	−	0.189939i	$-0.939171\pi$
$887$		−	11.3137i		−	0.379877i	0.981796	−	0.189939i	$-0.0608289\pi$
$888$		0			0
$889$	8.00000			0.268311
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	24.2487	−	19.7990i	0.810545	−	0.661807i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			12.0000i			0.400222i
$900$		0			0
$901$			41.5692i			1.38487i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	36.0000	−	29.3939i	1.19668	−	0.977086i
$906$		0			0
$907$	−48.9898			−1.62668			−0.813340	−	0.581789i	$-0.802353\pi$
$907$	−48.9898			−1.62668			−0.813340	+	0.581789i	$0.802353\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−27.7128			−0.918166			−0.459083	−	0.888393i	$-0.651822\pi$
$911$	−27.7128			−0.918166			−0.459083	+	0.888393i	$0.651822\pi$
$912$		0			0
$913$		−	19.5959i		−	0.648530i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	39.5980			1.30764
$918$		0			0
$919$	38.1051			1.25697			0.628486	−	0.777821i	$-0.283675\pi$
$919$	38.1051			1.25697			0.628486	+	0.777821i	$0.283675\pi$
$920$		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	$-0.541894\pi$
$929$	−8.00000			−0.262471			−0.131236	+	0.991351i	$0.541894\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	13.8564	+	16.9706i	0.453153	+	0.554997i
$936$		0			0
$937$			19.5959i			0.640171i	0.947389	+	0.320085i	$0.103712\pi$
$937$			19.5959i			0.640171i	−0.947389	+	0.320085i	$0.896288\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			38.1051i			1.24219i	0.783735	+	0.621096i	$0.213312\pi$
$941$			38.1051i			1.24219i	−0.783735	+	0.621096i	$0.786688\pi$
$942$		0			0
$943$			45.2548i			1.47370i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	19.5959			0.636782			0.318391	−	0.947960i	$-0.396858\pi$
$947$	19.5959			0.636782			0.318391	+	0.947960i	$0.396858\pi$
$948$		0			0
$949$			27.7128i			0.899596i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	4.89898i		−	0.158694i	−0.996847	−	0.0793468i	$-0.974717\pi$
$953$		−	4.89898i		−	0.158694i	0.996847	−	0.0793468i	$-0.0252834\pi$
$954$		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
$961$	−19.0000			−0.612903
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−16.9706	+	13.8564i	−0.546302	+	0.446054i
$966$		0			0
$967$		−	48.0833i		−	1.54625i	−0.634252	−	0.773127i	$-0.718692\pi$
$967$		−	48.0833i		−	1.54625i	0.634252	−	0.773127i	$-0.281308\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			46.0000i			1.47621i	0.674686	+	0.738105i	$0.264279\pi$
$971$			46.0000i			1.47621i	−0.674686	+	0.738105i	$0.735721\pi$
$972$		0			0
$973$	−22.6274			−0.725402
$974$		0			0
$975$		0			0
$976$		0			0
$977$			24.4949i			0.783661i	0.920037	+	0.391831i	$0.128158\pi$
$977$			24.4949i			0.783661i	−0.920037	+	0.391831i	$0.871842\pi$
$978$		0			0
$979$			32.0000i			1.02272i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	16.9706i		−	0.541277i	−0.962681	−	0.270638i	$-0.912765\pi$
$983$		−	16.9706i		−	0.541277i	0.962681	−	0.270638i	$-0.0872348\pi$
$984$		0			0
$985$	−4.00000	−	4.89898i	−0.127451	−	0.156094i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	55.4256i		−	1.76243i
$990$		0			0
$991$	−24.2487			−0.770286			−0.385143	−	0.922857i	$-0.625848\pi$
$991$	−24.2487			−0.770286			−0.385143	+	0.922857i	$0.625848\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−14.6969	−	18.0000i	−0.465924	−	0.570638i
$996$		0			0
$997$	48.0833			1.52281			0.761406	−	0.648275i	$-0.224509\pi$
$997$	48.0833			1.52281			0.761406	+	0.648275i	$0.224509\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.d.h.289.1	yes	8
3.2	odd	2		2880.2.d.f.289.7	yes	8
4.3	odd	2	inner	2880.2.d.h.289.2	yes	8
5.4	even	2	inner	2880.2.d.h.289.6	yes	8
8.3	odd	2	inner	2880.2.d.h.289.8	yes	8
8.5	even	2	inner	2880.2.d.h.289.7	yes	8
12.11	even	2		2880.2.d.f.289.8	yes	8
15.14	odd	2		2880.2.d.f.289.4	yes	8
20.19	odd	2	inner	2880.2.d.h.289.5	yes	8
24.5	odd	2		2880.2.d.f.289.1	✓	8
24.11	even	2		2880.2.d.f.289.2	yes	8
40.19	odd	2	inner	2880.2.d.h.289.3	yes	8
40.29	even	2	inner	2880.2.d.h.289.4	yes	8
60.59	even	2		2880.2.d.f.289.3	yes	8
120.29	odd	2		2880.2.d.f.289.6	yes	8
120.59	even	2		2880.2.d.f.289.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2880.2.d.f.289.1	✓	8	24.5	odd	2
2880.2.d.f.289.2	yes	8	24.11	even	2
2880.2.d.f.289.3	yes	8	60.59	even	2
2880.2.d.f.289.4	yes	8	15.14	odd	2
2880.2.d.f.289.5	yes	8	120.59	even	2
2880.2.d.f.289.6	yes	8	120.29	odd	2
2880.2.d.f.289.7	yes	8	3.2	odd	2
2880.2.d.f.289.8	yes	8	12.11	even	2
2880.2.d.h.289.1	yes	8	1.1	even	1	trivial
2880.2.d.h.289.2	yes	8	4.3	odd	2	inner
2880.2.d.h.289.3	yes	8	40.19	odd	2	inner
2880.2.d.h.289.4	yes	8	40.29	even	2	inner
2880.2.d.h.289.5	yes	8	20.19	odd	2	inner
2880.2.d.h.289.6	yes	8	5.4	even	2	inner
2880.2.d.h.289.7	yes	8	8.5	even	2	inner
2880.2.d.h.289.8	yes	8	8.3	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:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 - 1.73205i) q^{5} -2.82843i q^{7} +O(q^{10})\)	\(q+(-1.41421 - 1.73205i) q^{5} -2.82843i q^{7} +2.00000i q^{11} -2.82843 q^{13} +4.89898i q^{17} +5.65685i q^{23} +(-1.00000 + 4.89898i) q^{25} -3.46410i q^{29} -3.46410 q^{31} +(-4.89898 + 4.00000i) q^{35} -2.82843 q^{37} +8.00000 q^{41} -9.79796 q^{43} -1.00000 q^{49} +8.48528 q^{53} +(3.46410 - 2.82843i) q^{55} +10.0000i q^{59} +6.92820i q^{61} +(4.00000 + 4.89898i) q^{65} +9.79796 q^{67} +13.8564 q^{71} -9.79796i q^{73} +5.65685 q^{77} +3.46410 q^{79} -9.79796 q^{83} +(8.48528 - 6.92820i) q^{85} +16.0000 q^{89} +8.00000i q^{91} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{25} + 64 q^{41} - 8 q^{49} + 32 q^{65} + 128 q^{89}+O(q^{100}) \) 8 * q - 8 * q^25 + 64 * q^41 - 8 * q^49 + 32 * q^65 + 128 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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