LMFDB - Embedded newform 2880.2.w.n.2177.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(2177,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))

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

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

chi := DirichletCharacter("2880.2177");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.w (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 1440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2177.4
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2880.2177
Dual form			2880.2.w.n.2753.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.12132 + 0.707107i) q^{5} +(3.46410 + 3.46410i) q^{7} +O(q^{10})$	$q+(2.12132 + 0.707107i) q^{5} +(3.46410 + 3.46410i) q^{7} -4.89898i q^{11} +(-1.00000 + 1.00000i) q^{13} +(-2.82843 + 2.82843i) q^{17} +6.92820i q^{19} +(4.00000 + 3.00000i) q^{25} +7.07107 q^{29} -6.92820 q^{31} +(4.89898 + 9.79796i) q^{35} +(-1.00000 - 1.00000i) q^{37} +9.89949i q^{41} +(6.92820 - 6.92820i) q^{43} +(4.89898 - 4.89898i) q^{47} +17.0000i q^{49} +(2.82843 + 2.82843i) q^{53} +(3.46410 - 10.3923i) q^{55} -4.89898 q^{59} +(-2.82843 + 1.41421i) q^{65} +(-6.92820 - 6.92820i) q^{67} +(-7.00000 + 7.00000i) q^{73} +(16.9706 - 16.9706i) q^{77} +6.92820i q^{79} +(-4.89898 - 4.89898i) q^{83} +(-8.00000 + 4.00000i) q^{85} +9.89949 q^{89} -6.92820 q^{91} +(-4.89898 + 14.6969i) q^{95} +(5.00000 + 5.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{13} + 32 q^{25} - 8 q^{37} - 56 q^{73} - 64 q^{85} + 40 q^{97}+O(q^{100}) $ 8 * q - 8 * q^13 + 32 * q^25 - 8 * q^37 - 56 * q^73 - 64 * q^85 + 40 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$1$	$1$

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
$5$	2.12132	+	0.707107i	0.948683	+	0.316228i
$5$	2.12132	+	0.707107i	0.948683	+	0.316228i
$6$		0			0
$7$	3.46410	+	3.46410i	1.30931	+	1.30931i	0.921915	+	0.387392i	$0.126624\pi$
$7$	3.46410	+	3.46410i	1.30931	+	1.30931i	0.387392	+	0.921915i	$0.373376\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	4.89898i		−	1.47710i	−0.674200	−	0.738549i	$-0.735511\pi$
$11$		−	4.89898i		−	1.47710i	0.674200	−	0.738549i	$-0.264489\pi$
$12$		0			0
$13$	−1.00000	+	1.00000i	−0.277350	+	0.277350i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−1.00000	+	1.00000i	−0.277350	+	0.277350i	0.554700	+	0.832050i	$0.312833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.82843	+	2.82843i	−0.685994	+	0.685994i	−0.961344	−	0.275350i	$-0.911206\pi$
$17$	−2.82843	+	2.82843i	−0.685994	+	0.685994i	0.275350	+	0.961344i	$0.411206\pi$
$18$		0			0
$19$			6.92820i			1.58944i	0.606977	+	0.794719i	$0.292382\pi$
$19$			6.92820i			1.58944i	−0.606977	+	0.794719i	$0.707618\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$	4.00000	+	3.00000i	0.800000	+	0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.07107			1.31306			0.656532	−	0.754298i	$-0.272023\pi$
$29$	7.07107			1.31306			0.656532	+	0.754298i	$0.272023\pi$
$30$		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
$35$	4.89898	+	9.79796i	0.828079	+	1.65616i
$36$		0			0
$37$	−1.00000	−	1.00000i	−0.164399	−	0.164399i	0.620113	−	0.784512i	$-0.287087\pi$
$37$	−1.00000	−	1.00000i	−0.164399	−	0.164399i	−0.784512	+	0.620113i	$0.787087\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			9.89949i			1.54604i	0.634381	+	0.773021i	$0.281255\pi$
$41$			9.89949i			1.54604i	−0.634381	+	0.773021i	$0.718745\pi$
$42$		0			0
$43$	6.92820	−	6.92820i	1.05654	−	1.05654i	0.0582384	−	0.998303i	$-0.481452\pi$
$43$	6.92820	−	6.92820i	1.05654	−	1.05654i	0.998303	−	0.0582384i	$-0.0185483\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.89898	−	4.89898i	0.714590	−	0.714590i	−0.252902	−	0.967492i	$-0.581385\pi$
$47$	4.89898	−	4.89898i	0.714590	−	0.714590i	0.967492	+	0.252902i	$0.0813851\pi$
$48$		0			0
$49$			17.0000i			2.42857i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.82843	+	2.82843i	0.388514	+	0.388514i	0.874157	−	0.485643i	$-0.161414\pi$
$53$	2.82843	+	2.82843i	0.388514	+	0.388514i	−0.485643	+	0.874157i	$0.661414\pi$
$54$		0			0
$55$	3.46410	−	10.3923i	0.467099	−	1.40130i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.89898			−0.637793			−0.318896	−	0.947790i	$-0.603312\pi$
$59$	−4.89898			−0.637793			−0.318896	+	0.947790i	$0.603312\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.82843	+	1.41421i	−0.350823	+	0.175412i
$66$		0			0
$67$	−6.92820	−	6.92820i	−0.846415	−	0.846415i	0.143269	−	0.989684i	$-0.454239\pi$
$67$	−6.92820	−	6.92820i	−0.846415	−	0.846415i	−0.989684	+	0.143269i	$0.954239\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−7.00000	+	7.00000i	−0.819288	+	0.819288i	−0.986005	−	0.166717i	$-0.946683\pi$
$73$	−7.00000	+	7.00000i	−0.819288	+	0.819288i	0.166717	+	0.986005i	$0.446683\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	16.9706	−	16.9706i	1.93398	−	1.93398i
$78$		0			0
$79$			6.92820i			0.779484i	0.920924	+	0.389742i	$0.127436\pi$
$79$			6.92820i			0.779484i	−0.920924	+	0.389742i	$0.872564\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.89898	−	4.89898i	−0.537733	−	0.537733i	0.385130	−	0.922862i	$-0.374157\pi$
$83$	−4.89898	−	4.89898i	−0.537733	−	0.537733i	−0.922862	+	0.385130i	$0.874157\pi$
$84$		0			0
$85$	−8.00000	+	4.00000i	−0.867722	+	0.433861i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.89949			1.04934			0.524672	−	0.851304i	$-0.324188\pi$
$89$	9.89949			1.04934			0.524672	+	0.851304i	$0.324188\pi$
$90$		0			0
$91$	−6.92820			−0.726273
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.89898	+	14.6969i	−0.502625	+	1.50787i
$96$		0			0
$97$	5.00000	+	5.00000i	0.507673	+	0.507673i	0.913812	−	0.406138i	$-0.133125\pi$
$97$	5.00000	+	5.00000i	0.507673	+	0.507673i	−0.406138	+	0.913812i	$0.633125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	4.24264i		−	0.422159i	−0.977469	−	0.211079i	$-0.932302\pi$
$101$		−	4.24264i		−	0.422159i	0.977469	−	0.211079i	$-0.0676978\pi$
$102$		0			0
$103$	3.46410	−	3.46410i	0.341328	−	0.341328i	−0.515538	−	0.856866i	$-0.672408\pi$
$103$	3.46410	−	3.46410i	0.341328	−	0.341328i	0.856866	+	0.515538i	$0.172408\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.89898	−	4.89898i	0.473602	−	0.473602i	−0.429476	−	0.903078i	$-0.641302\pi$
$107$	4.89898	−	4.89898i	0.473602	−	0.473602i	0.903078	+	0.429476i	$0.141302\pi$
$108$		0			0
$109$		−	8.00000i		−	0.766261i	−0.923694	−	0.383131i	$-0.874846\pi$
$109$		−	8.00000i		−	0.766261i	0.923694	−	0.383131i	$-0.125154\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.07107	−	7.07107i	−0.665190	−	0.665190i	0.291409	−	0.956599i	$-0.405876\pi$
$113$	−7.07107	−	7.07107i	−0.665190	−	0.665190i	−0.956599	+	0.291409i	$0.905876\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−19.5959			−1.79635
$120$		0			0
$121$	−13.0000			−1.18182
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.36396	+	9.19239i	0.569210	+	0.822192i
$126$		0			0
$127$	−10.3923	−	10.3923i	−0.922168	−	0.922168i	0.0750145	−	0.997182i	$-0.476100\pi$
$127$	−10.3923	−	10.3923i	−0.922168	−	0.922168i	−0.997182	+	0.0750145i	$0.976100\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			4.89898i			0.428026i	0.976831	+	0.214013i	$0.0686535\pi$
$131$			4.89898i			0.428026i	−0.976831	+	0.214013i	$0.931347\pi$
$132$		0			0
$133$	−24.0000	+	24.0000i	−2.08106	+	2.08106i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.41421	−	1.41421i	0.120824	−	0.120824i	−0.644109	−	0.764934i	$-0.722772\pi$
$137$	1.41421	−	1.41421i	0.120824	−	0.120824i	0.764934	+	0.644109i	$0.222772\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.89898	+	4.89898i	0.409673	+	0.409673i
$144$		0			0
$145$	15.0000	+	5.00000i	1.24568	+	0.415227i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.24264			0.347571			0.173785	−	0.984784i	$-0.444400\pi$
$149$	4.24264			0.347571			0.173785	+	0.984784i	$0.444400\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−14.6969	−	4.89898i	−1.18049	−	0.393496i
$156$		0			0
$157$	13.0000	+	13.0000i	1.03751	+	1.03751i	0.999268	+	0.0382445i	$0.0121766\pi$
$157$	13.0000	+	13.0000i	1.03751	+	1.03751i	0.0382445	+	0.999268i	$0.487823\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−6.92820	+	6.92820i	−0.542659	+	0.542659i	−0.924307	−	0.381649i	$-0.875356\pi$
$163$	−6.92820	+	6.92820i	−0.542659	+	0.542659i	0.381649	+	0.924307i	$0.375356\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.79796	−	9.79796i	0.758189	−	0.758189i	−0.217804	−	0.975993i	$-0.569889\pi$
$167$	9.79796	−	9.79796i	0.758189	−	0.758189i	0.975993	+	0.217804i	$0.0698893\pi$
$168$		0			0
$169$			11.0000i			0.846154i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.41421	+	1.41421i	0.107521	+	0.107521i	0.758820	−	0.651300i	$-0.225776\pi$
$173$	1.41421	+	1.41421i	0.107521	+	0.107521i	−0.651300	+	0.758820i	$0.725776\pi$
$174$		0			0
$175$	3.46410	+	24.2487i	0.261861	+	1.83303i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.89898			−0.366167			−0.183083	−	0.983097i	$-0.558608\pi$
$179$	−4.89898			−0.366167			−0.183083	+	0.983097i	$0.558608\pi$
$180$		0			0
$181$	8.00000			0.594635			0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000			0.594635			0.297318	+	0.954779i	$0.403908\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.41421	−	2.82843i	−0.103975	−	0.207950i
$186$		0			0
$187$	13.8564	+	13.8564i	1.01328	+	1.01328i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	19.5959i		−	1.41791i	−0.705253	−	0.708955i	$-0.749167\pi$
$191$		−	19.5959i		−	1.41791i	0.705253	−	0.708955i	$-0.250833\pi$
$192$		0			0
$193$	−7.00000	+	7.00000i	−0.503871	+	0.503871i	−0.912639	−	0.408768i	$-0.865959\pi$
$193$	−7.00000	+	7.00000i	−0.503871	+	0.503871i	0.408768	+	0.912639i	$0.365959\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.41421	+	1.41421i	−0.100759	+	0.100759i	−0.755689	−	0.654931i	$-0.772698\pi$
$197$	−1.41421	+	1.41421i	−0.100759	+	0.100759i	0.654931	+	0.755689i	$0.272698\pi$
$198$		0			0
$199$			13.8564i			0.982255i	0.871088	+	0.491127i	$0.163415\pi$
$199$			13.8564i			0.982255i	−0.871088	+	0.491127i	$0.836585\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	24.4949	+	24.4949i	1.71920	+	1.71920i
$204$		0			0
$205$	−7.00000	+	21.0000i	−0.488901	+	1.46670i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	33.9411			2.34776
$210$		0			0
$211$	13.8564			0.953914			0.476957	−	0.878927i	$-0.341740\pi$
$211$	13.8564			0.953914			0.476957	+	0.878927i	$0.341740\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	19.5959	−	9.79796i	1.33643	−	0.668215i
$216$		0			0
$217$	−24.0000	−	24.0000i	−1.62923	−	1.62923i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	5.65685i		−	0.380521i
$222$		0			0
$223$	10.3923	−	10.3923i	0.695920	−	0.695920i	−0.267608	−	0.963528i	$-0.586233\pi$
$223$	10.3923	−	10.3923i	0.695920	−	0.695920i	0.963528	+	0.267608i	$0.0862331\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.79796	−	9.79796i	0.650313	−	0.650313i	−0.302755	−	0.953068i	$-0.597906\pi$
$227$	9.79796	−	9.79796i	0.650313	−	0.650313i	0.953068	+	0.302755i	$0.0979064\pi$
$228$		0			0
$229$		−	18.0000i		−	1.18947i	−0.803921	−	0.594737i	$-0.797256\pi$
$229$		−	18.0000i		−	1.18947i	0.803921	−	0.594737i	$-0.202744\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.82843	+	2.82843i	0.185296	+	0.185296i	0.793659	−	0.608363i	$-0.208173\pi$
$233$	2.82843	+	2.82843i	0.185296	+	0.185296i	−0.608363	+	0.793659i	$0.708173\pi$
$234$		0			0
$235$	13.8564	−	6.92820i	0.903892	−	0.451946i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	19.5959			1.26755			0.633777	−	0.773516i	$-0.281504\pi$
$239$	19.5959			1.26755			0.633777	+	0.773516i	$0.281504\pi$
$240$		0			0
$241$	6.00000			0.386494			0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000			0.386494			0.193247	+	0.981150i	$0.438098\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−12.0208	+	36.0624i	−0.767982	+	2.30395i
$246$		0			0
$247$	−6.92820	−	6.92820i	−0.440831	−	0.440831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	4.89898i		−	0.309221i	−0.987976	−	0.154610i	$-0.950588\pi$
$251$		−	4.89898i		−	0.309221i	0.987976	−	0.154610i	$-0.0494122\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	15.5563	−	15.5563i	0.970378	−	0.970378i	−0.0291953	−	0.999574i	$-0.509294\pi$
$257$	15.5563	−	15.5563i	0.970378	−	0.970378i	0.999574	+	0.0291953i	$0.00929448\pi$
$258$		0			0
$259$		−	6.92820i		−	0.430498i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$	4.00000	+	8.00000i	0.245718	+	0.491436i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−21.2132			−1.29339			−0.646696	−	0.762748i	$-0.723850\pi$
$269$	−21.2132			−1.29339			−0.646696	+	0.762748i	$0.723850\pi$
$270$		0			0
$271$	27.7128			1.68343			0.841717	−	0.539919i	$-0.181545\pi$
$271$	27.7128			1.68343			0.841717	+	0.539919i	$0.181545\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	14.6969	−	19.5959i	0.886259	−	1.18168i
$276$		0			0
$277$	11.0000	+	11.0000i	0.660926	+	0.660926i	0.955598	−	0.294672i	$-0.0952105\pi$
$277$	11.0000	+	11.0000i	0.660926	+	0.660926i	−0.294672	+	0.955598i	$0.595211\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			24.0416i			1.43420i	0.696969	+	0.717102i	$0.254532\pi$
$281$			24.0416i			1.43420i	−0.696969	+	0.717102i	$0.745468\pi$
$282$		0			0
$283$	−13.8564	+	13.8564i	−0.823678	+	0.823678i	−0.986633	−	0.162956i	$-0.947897\pi$
$283$	−13.8564	+	13.8564i	−0.823678	+	0.823678i	0.162956	+	0.986633i	$0.447897\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−34.2929	+	34.2929i	−2.02424	+	2.02424i
$288$		0			0
$289$			1.00000i			0.0588235i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.89949	−	9.89949i	−0.578335	−	0.578335i	0.356110	−	0.934444i	$-0.384103\pi$
$293$	−9.89949	−	9.89949i	−0.578335	−	0.578335i	−0.934444	+	0.356110i	$0.884103\pi$
$294$		0			0
$295$	−10.3923	−	3.46410i	−0.605063	−	0.201688i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	48.0000			2.76667
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−6.92820	−	6.92820i	−0.395413	−	0.395413i	0.481198	−	0.876612i	$-0.340202\pi$
$307$	−6.92820	−	6.92820i	−0.395413	−	0.395413i	−0.876612	+	0.481198i	$0.840202\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			29.3939i			1.66677i	0.552690	+	0.833387i	$0.313601\pi$
$311$			29.3939i			1.66677i	−0.552690	+	0.833387i	$0.686399\pi$
$312$		0			0
$313$	11.0000	−	11.0000i	0.621757	−	0.621757i	−0.324224	−	0.945980i	$-0.605103\pi$
$313$	11.0000	−	11.0000i	0.621757	−	0.621757i	0.945980	+	0.324224i	$0.105103\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.82843	−	2.82843i	0.158860	−	0.158860i	−0.623201	−	0.782062i	$-0.714168\pi$
$317$	2.82843	−	2.82843i	0.158860	−	0.158860i	0.782062	+	0.623201i	$0.214168\pi$
$318$		0			0
$319$		−	34.6410i		−	1.93952i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−19.5959	−	19.5959i	−1.09035	−	1.09035i
$324$		0			0
$325$	−7.00000	+	1.00000i	−0.388290	+	0.0554700i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	33.9411			1.87123
$330$		0			0
$331$	−34.6410			−1.90404			−0.952021	−	0.306032i	$-0.900999\pi$
$331$	−34.6410			−1.90404			−0.952021	+	0.306032i	$0.900999\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−9.79796	−	19.5959i	−0.535320	−	1.07064i
$336$		0			0
$337$	−13.0000	−	13.0000i	−0.708155	−	0.708155i	0.257992	−	0.966147i	$-0.416939\pi$
$337$	−13.0000	−	13.0000i	−0.708155	−	0.708155i	−0.966147	+	0.257992i	$0.916939\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			33.9411i			1.83801i
$342$		0			0
$343$	−34.6410	+	34.6410i	−1.87044	+	1.87044i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−19.5959	+	19.5959i	−1.05196	+	1.05196i	−0.0533903	+	0.998574i	$0.517003\pi$
$347$	−19.5959	+	19.5959i	−1.05196	+	1.05196i	−0.998574	+	0.0533903i	$0.982997\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−19.7990	−	19.7990i	−1.05379	−	1.05379i	−0.998468	−	0.0553255i	$-0.982380\pi$
$353$	−19.7990	−	19.7990i	−1.05379	−	1.05379i	−0.0553255	−	0.998468i	$-0.517620\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.79796			0.517116			0.258558	−	0.965996i	$-0.416753\pi$
$359$	9.79796			0.517116			0.258558	+	0.965996i	$0.416753\pi$
$360$		0			0
$361$	−29.0000			−1.52632
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−19.7990	+	9.89949i	−1.03633	+	0.518163i
$366$		0			0
$367$	10.3923	+	10.3923i	0.542474	+	0.542474i	0.924253	−	0.381780i	$-0.124689\pi$
$367$	10.3923	+	10.3923i	0.542474	+	0.542474i	−0.381780	+	0.924253i	$0.624689\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			19.5959i			1.01737i
$372$		0			0
$373$	−7.00000	+	7.00000i	−0.362446	+	0.362446i	−0.864713	−	0.502267i	$-0.832500\pi$
$373$	−7.00000	+	7.00000i	−0.362446	+	0.362446i	0.502267	+	0.864713i	$0.332500\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7.07107	+	7.07107i	−0.364179	+	0.364179i
$378$		0			0
$379$		−	27.7128i		−	1.42351i	−0.702427	−	0.711756i	$-0.747900\pi$
$379$		−	27.7128i		−	1.42351i	0.702427	−	0.711756i	$-0.252100\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−14.6969	−	14.6969i	−0.750978	−	0.750978i	0.223683	−	0.974662i	$-0.428192\pi$
$383$	−14.6969	−	14.6969i	−0.750978	−	0.750978i	−0.974662	+	0.223683i	$0.928192\pi$
$384$		0			0
$385$	48.0000	−	24.0000i	2.44631	−	1.22315i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.24264			−0.215110			−0.107555	−	0.994199i	$-0.534302\pi$
$389$	−4.24264			−0.215110			−0.107555	+	0.994199i	$0.534302\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.89898	+	14.6969i	−0.246494	+	0.739483i
$396$		0			0
$397$	5.00000	+	5.00000i	0.250943	+	0.250943i	0.821357	−	0.570414i	$-0.193217\pi$
$397$	5.00000	+	5.00000i	0.250943	+	0.250943i	−0.570414	+	0.821357i	$0.693217\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	21.2132i		−	1.05934i	−0.848205	−	0.529668i	$-0.822316\pi$
$401$		−	21.2132i		−	1.05934i	0.848205	−	0.529668i	$-0.177684\pi$
$402$		0			0
$403$	6.92820	−	6.92820i	0.345118	−	0.345118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.89898	+	4.89898i	−0.242833	+	0.242833i
$408$		0			0
$409$		−	8.00000i		−	0.395575i	−0.980245	−	0.197787i	$-0.936624\pi$
$409$		−	8.00000i		−	0.395575i	0.980245	−	0.197787i	$-0.0633755\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−16.9706	−	16.9706i	−0.835067	−	0.835067i
$414$		0			0
$415$	−6.92820	−	13.8564i	−0.340092	−	0.680184i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.89898			−0.239331			−0.119665	−	0.992814i	$-0.538182\pi$
$419$	−4.89898			−0.239331			−0.119665	+	0.992814i	$0.538182\pi$
$420$		0			0
$421$	−30.0000			−1.46211			−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000			−1.46211			−0.731055	+	0.682318i	$0.760972\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−19.7990	+	2.82843i	−0.960392	+	0.137199i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	19.5959i		−	0.943902i	−0.881625	−	0.471951i	$-0.843550\pi$
$431$		−	19.5959i		−	0.943902i	0.881625	−	0.471951i	$-0.156450\pi$
$432$		0			0
$433$	7.00000	−	7.00000i	0.336399	−	0.336399i	−0.518611	−	0.855010i	$-0.673551\pi$
$433$	7.00000	−	7.00000i	0.336399	−	0.336399i	0.855010	+	0.518611i	$0.173551\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	27.7128i		−	1.32266i	−0.750095	−	0.661330i	$-0.769992\pi$
$439$		−	27.7128i		−	1.32266i	0.750095	−	0.661330i	$-0.230008\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9.79796	−	9.79796i	−0.465515	−	0.465515i	0.434943	−	0.900458i	$-0.356769\pi$
$443$	−9.79796	−	9.79796i	−0.465515	−	0.465515i	−0.900458	+	0.434943i	$0.856769\pi$
$444$		0			0
$445$	21.0000	+	7.00000i	0.995495	+	0.331832i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	21.2132			1.00111			0.500556	−	0.865704i	$-0.333129\pi$
$449$	21.2132			1.00111			0.500556	+	0.865704i	$0.333129\pi$
$450$		0			0
$451$	48.4974			2.28365
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−14.6969	−	4.89898i	−0.689003	−	0.229668i
$456$		0			0
$457$	−23.0000	−	23.0000i	−1.07589	−	1.07589i	−0.996873	−	0.0790217i	$-0.974820\pi$
$457$	−23.0000	−	23.0000i	−1.07589	−	1.07589i	−0.0790217	−	0.996873i	$-0.525180\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	12.7279i		−	0.592798i	−0.955064	−	0.296399i	$-0.904214\pi$
$461$		−	12.7279i		−	0.592798i	0.955064	−	0.296399i	$-0.0957859\pi$
$462$		0			0
$463$	3.46410	−	3.46410i	0.160990	−	0.160990i	−0.622015	−	0.783005i	$-0.713686\pi$
$463$	3.46410	−	3.46410i	0.160990	−	0.160990i	0.783005	+	0.622015i	$0.213686\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−24.4949	+	24.4949i	−1.13349	+	1.13349i	−0.143896	+	0.989593i	$0.545963\pi$
$467$	−24.4949	+	24.4949i	−1.13349	+	1.13349i	−0.989593	+	0.143896i	$0.954037\pi$
$468$		0			0
$469$		−	48.0000i		−	2.21643i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−33.9411	−	33.9411i	−1.56061	−	1.56061i
$474$		0			0
$475$	−20.7846	+	27.7128i	−0.953663	+	1.27155i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	19.5959			0.895360			0.447680	−	0.894194i	$-0.352250\pi$
$479$	19.5959			0.895360			0.447680	+	0.894194i	$0.352250\pi$
$480$		0			0
$481$	2.00000			0.0911922
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7.07107	+	14.1421i	0.321081	+	0.642161i
$486$		0			0
$487$	17.3205	+	17.3205i	0.784867	+	0.784867i	0.980648	−	0.195781i	$-0.0627240\pi$
$487$	17.3205	+	17.3205i	0.784867	+	0.784867i	−0.195781	+	0.980648i	$0.562724\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	24.4949i		−	1.10544i	−0.833367	−	0.552720i	$-0.813590\pi$
$491$		−	24.4949i		−	1.10544i	0.833367	−	0.552720i	$-0.186410\pi$
$492$		0			0
$493$	−20.0000	+	20.0000i	−0.900755	+	0.900755i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			6.92820i			0.310149i	0.987903	+	0.155074i	$0.0495618\pi$
$499$			6.92820i			0.310149i	−0.987903	+	0.155074i	$0.950438\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4.89898	−	4.89898i	−0.218435	−	0.218435i	0.589404	−	0.807839i	$-0.299363\pi$
$503$	−4.89898	−	4.89898i	−0.218435	−	0.218435i	−0.807839	+	0.589404i	$0.799363\pi$
$504$		0			0
$505$	3.00000	−	9.00000i	0.133498	−	0.400495i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.07107			−0.313420			−0.156710	−	0.987645i	$-0.550089\pi$
$509$	−7.07107			−0.313420			−0.156710	+	0.987645i	$0.550089\pi$
$510$		0			0
$511$	−48.4974			−2.14540
$512$		0			0
$513$		0			0
$514$		0			0
$515$	9.79796	−	4.89898i	0.431750	−	0.215875i
$516$		0			0
$517$	−24.0000	−	24.0000i	−1.05552	−	1.05552i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			38.1838i			1.67286i	0.548073	+	0.836431i	$0.315362\pi$
$521$			38.1838i			1.67286i	−0.548073	+	0.836431i	$0.684638\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	19.5959	−	19.5959i	0.853612	−	0.853612i
$528$		0			0
$529$		−	23.0000i		−	1.00000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−9.89949	−	9.89949i	−0.428795	−	0.428795i
$534$		0			0
$535$	13.8564	−	6.92820i	0.599065	−	0.299532i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	83.2827			3.58724
$540$		0			0
$541$	40.0000			1.71973			0.859867	−	0.510518i	$-0.170546\pi$
$541$	40.0000			1.71973			0.859867	+	0.510518i	$0.170546\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.65685	−	16.9706i	0.242313	−	0.726939i
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			48.9898i			2.08704i
$552$		0			0
$553$	−24.0000	+	24.0000i	−1.02058	+	1.02058i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	31.1127	−	31.1127i	1.31829	−	1.31829i	0.403156	−	0.915131i	$-0.367913\pi$
$557$	31.1127	−	31.1127i	1.31829	−	1.31829i	0.915131	−	0.403156i	$-0.132087\pi$
$558$		0			0
$559$			13.8564i			0.586064i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$	−10.0000	−	20.0000i	−0.420703	−	0.841406i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−38.1838			−1.60075			−0.800373	−	0.599502i	$-0.795365\pi$
$569$	−38.1838			−1.60075			−0.800373	+	0.599502i	$0.795365\pi$
$570$		0			0
$571$	20.7846			0.869809			0.434904	−	0.900477i	$-0.356782\pi$
$571$	20.7846			0.869809			0.434904	+	0.900477i	$0.356782\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.00000	−	1.00000i	−0.0416305	−	0.0416305i	0.685985	−	0.727616i	$-0.259372\pi$
$577$	−1.00000	−	1.00000i	−0.0416305	−	0.0416305i	−0.727616	+	0.685985i	$0.759372\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	33.9411i		−	1.40812i
$582$		0			0
$583$	13.8564	−	13.8564i	0.573874	−	0.573874i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.89898	−	4.89898i	0.202203	−	0.202203i	−0.598741	−	0.800943i	$-0.704332\pi$
$587$	4.89898	−	4.89898i	0.202203	−	0.202203i	0.800943	+	0.598741i	$0.204332\pi$
$588$		0			0
$589$		−	48.0000i		−	1.97781i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−18.3848	−	18.3848i	−0.754972	−	0.754972i	0.220430	−	0.975403i	$-0.429254\pi$
$593$	−18.3848	−	18.3848i	−0.754972	−	0.754972i	−0.975403	+	0.220430i	$0.929254\pi$
$594$		0			0
$595$	−41.5692	−	13.8564i	−1.70417	−	0.568057i
$596$		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
$601$	24.0000			0.978980			0.489490	−	0.872009i	$-0.337183\pi$
$601$	24.0000			0.978980			0.489490	+	0.872009i	$0.337183\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−27.5772	−	9.19239i	−1.12117	−	0.373724i
$606$		0			0
$607$	31.1769	+	31.1769i	1.26543	+	1.26543i	0.948422	+	0.317010i	$0.102679\pi$
$607$	31.1769	+	31.1769i	1.26543	+	1.26543i	0.317010	+	0.948422i	$0.397321\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			9.79796i			0.396383i
$612$		0			0
$613$	7.00000	−	7.00000i	0.282727	−	0.282727i	−0.551468	−	0.834196i	$-0.685932\pi$
$613$	7.00000	−	7.00000i	0.282727	−	0.282727i	0.834196	+	0.551468i	$0.185932\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	14.1421	−	14.1421i	0.569341	−	0.569341i	−0.362603	−	0.931944i	$-0.618112\pi$
$617$	14.1421	−	14.1421i	0.569341	−	0.569341i	0.931944	+	0.362603i	$0.118112\pi$
$618$		0			0
$619$			13.8564i			0.556936i	0.960446	+	0.278468i	$0.0898266\pi$
$619$			13.8564i			0.556936i	−0.960446	+	0.278468i	$0.910173\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	34.2929	+	34.2929i	1.37391	+	1.37391i
$624$		0			0
$625$	7.00000	+	24.0000i	0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.65685			0.225554
$630$		0			0
$631$	34.6410			1.37904			0.689519	−	0.724268i	$-0.257822\pi$
$631$	34.6410			1.37904			0.689519	+	0.724268i	$0.257822\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−14.6969	−	29.3939i	−0.583230	−	1.16646i
$636$		0			0
$637$	−17.0000	−	17.0000i	−0.673565	−	0.673565i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			15.5563i			0.614439i	0.951639	+	0.307219i	$0.0993986\pi$
$641$			15.5563i			0.614439i	−0.951639	+	0.307219i	$0.900601\pi$
$642$		0			0
$643$	13.8564	−	13.8564i	0.546443	−	0.546443i	−0.378967	−	0.925410i	$-0.623721\pi$
$643$	13.8564	−	13.8564i	0.546443	−	0.546443i	0.925410	+	0.378967i	$0.123721\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	24.4949	−	24.4949i	0.962994	−	0.962994i	−0.0363455	−	0.999339i	$-0.511572\pi$
$647$	24.4949	−	24.4949i	0.962994	−	0.962994i	0.999339	+	0.0363455i	$0.0115717\pi$
$648$		0			0
$649$			24.0000i			0.942082i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−31.1127	−	31.1127i	−1.21753	−	1.21753i	−0.968493	−	0.249041i	$-0.919885\pi$
$653$	−31.1127	−	31.1127i	−1.21753	−	1.21753i	−0.249041	−	0.968493i	$-0.580115\pi$
$654$		0			0
$655$	−3.46410	+	10.3923i	−0.135354	+	0.406061i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.89898			0.190837			0.0954186	−	0.995437i	$-0.469581\pi$
$659$	4.89898			0.190837			0.0954186	+	0.995437i	$0.469581\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−67.8823	+	33.9411i	−2.63236	+	1.31618i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	25.0000	−	25.0000i	0.963679	−	0.963679i	−0.0356839	−	0.999363i	$-0.511361\pi$
$673$	25.0000	−	25.0000i	0.963679	−	0.963679i	0.999363	+	0.0356839i	$0.0113610\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.82843	+	2.82843i	−0.108705	+	0.108705i	−0.759367	−	0.650662i	$-0.774491\pi$
$677$	−2.82843	+	2.82843i	−0.108705	+	0.108705i	0.650662	+	0.759367i	$0.274491\pi$
$678$		0			0
$679$			34.6410i			1.32940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−24.4949	−	24.4949i	−0.937271	−	0.937271i	0.0608742	−	0.998145i	$-0.480611\pi$
$683$	−24.4949	−	24.4949i	−0.937271	−	0.937271i	−0.998145	+	0.0608742i	$0.980611\pi$
$684$		0			0
$685$	4.00000	−	2.00000i	0.152832	−	0.0764161i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.65685			−0.215509
$690$		0			0
$691$	−20.7846			−0.790684			−0.395342	−	0.918534i	$-0.629374\pi$
$691$	−20.7846			−0.790684			−0.395342	+	0.918534i	$0.629374\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−28.0000	−	28.0000i	−1.06058	−	1.06058i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			26.8701i			1.01487i	0.861691	+	0.507434i	$0.169406\pi$
$701$			26.8701i			1.01487i	−0.861691	+	0.507434i	$0.830594\pi$
$702$		0			0
$703$	6.92820	−	6.92820i	0.261302	−	0.261302i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	14.6969	−	14.6969i	0.552735	−	0.552735i
$708$		0			0
$709$		−	30.0000i		−	1.12667i	−0.826227	−	0.563337i	$-0.809517\pi$
$709$		−	30.0000i		−	1.12667i	0.826227	−	0.563337i	$-0.190483\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	6.92820	+	13.8564i	0.259100	+	0.518200i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−39.1918			−1.46161			−0.730804	−	0.682587i	$-0.760855\pi$
$719$	−39.1918			−1.46161			−0.730804	+	0.682587i	$0.760855\pi$
$720$		0			0
$721$	24.0000			0.893807
$722$		0			0
$723$		0			0
$724$		0			0
$725$	28.2843	+	21.2132i	1.05045	+	0.787839i
$726$		0			0
$727$	−17.3205	−	17.3205i	−0.642382	−	0.642382i	0.308758	−	0.951140i	$-0.400087\pi$
$727$	−17.3205	−	17.3205i	−0.642382	−	0.642382i	−0.951140	+	0.308758i	$0.900087\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			39.1918i			1.44956i
$732$		0			0
$733$	31.0000	−	31.0000i	1.14501	−	1.14501i	0.157491	−	0.987520i	$-0.449660\pi$
$733$	31.0000	−	31.0000i	1.14501	−	1.14501i	0.987520	−	0.157491i	$-0.0503404\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−33.9411	+	33.9411i	−1.25024	+	1.25024i
$738$		0			0
$739$			48.4974i			1.78401i	0.452029	+	0.892003i	$0.350700\pi$
$739$			48.4974i			1.78401i	−0.452029	+	0.892003i	$0.649300\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−34.2929	−	34.2929i	−1.25808	−	1.25808i	−0.952007	−	0.306076i	$-0.900984\pi$
$743$	−34.2929	−	34.2929i	−1.25808	−	1.25808i	−0.306076	−	0.952007i	$-0.599016\pi$
$744$		0			0
$745$	9.00000	+	3.00000i	0.329734	+	0.109911i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	33.9411			1.24018
$750$		0			0
$751$	−20.7846			−0.758441			−0.379221	−	0.925306i	$-0.623808\pi$
$751$	−20.7846			−0.758441			−0.379221	+	0.925306i	$0.623808\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−19.0000	−	19.0000i	−0.690567	−	0.690567i	0.271790	−	0.962357i	$-0.412384\pi$
$757$	−19.0000	−	19.0000i	−0.690567	−	0.690567i	−0.962357	+	0.271790i	$0.912384\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			4.24264i			0.153796i	0.997039	+	0.0768978i	$0.0245015\pi$
$761$			4.24264i			0.153796i	−0.997039	+	0.0768978i	$0.975498\pi$
$762$		0			0
$763$	27.7128	−	27.7128i	1.00327	−	1.00327i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.89898	−	4.89898i	0.176892	−	0.176892i
$768$		0			0
$769$		−	24.0000i		−	0.865462i	−0.901523	−	0.432731i	$-0.857550\pi$
$769$		−	24.0000i		−	0.865462i	0.901523	−	0.432731i	$-0.142450\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19.7990	−	19.7990i	−0.712120	−	0.712120i	0.254858	−	0.966978i	$-0.417971\pi$
$773$	−19.7990	−	19.7990i	−0.712120	−	0.712120i	−0.966978	+	0.254858i	$0.917971\pi$
$774$		0			0
$775$	−27.7128	−	20.7846i	−0.995474	−	0.746605i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−68.5857			−2.45734
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	18.3848	+	36.7696i	0.656181	+	1.31236i
$786$		0			0
$787$	20.7846	+	20.7846i	0.740891	+	0.740891i	0.972750	−	0.231858i	$-0.0744806\pi$
$787$	20.7846	+	20.7846i	0.740891	+	0.740891i	−0.231858	+	0.972750i	$0.574481\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	48.9898i		−	1.74188i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	35.3553	−	35.3553i	1.25235	−	1.25235i	0.297687	−	0.954664i	$-0.403785\pi$
$797$	35.3553	−	35.3553i	1.25235	−	1.25235i	0.954664	−	0.297687i	$-0.0962151\pi$
$798$		0			0
$799$			27.7128i			0.980409i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	34.2929	+	34.2929i	1.21017	+	1.21017i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−9.89949			−0.348048			−0.174024	−	0.984741i	$-0.555677\pi$
$809$	−9.89949			−0.348048			−0.174024	+	0.984741i	$0.555677\pi$
$810$		0			0
$811$	13.8564			0.486564			0.243282	−	0.969956i	$-0.421776\pi$
$811$	13.8564			0.486564			0.243282	+	0.969956i	$0.421776\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19.5959	+	9.79796i	−0.686415	+	0.343208i
$816$		0			0
$817$	48.0000	+	48.0000i	1.67931	+	1.67931i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			1.41421i			0.0493564i	0.999695	+	0.0246782i	$0.00785611\pi$
$821$			1.41421i			0.0493564i	−0.999695	+	0.0246782i	$0.992144\pi$
$822$		0			0
$823$	10.3923	−	10.3923i	0.362253	−	0.362253i	−0.502389	−	0.864642i	$-0.667545\pi$
$823$	10.3923	−	10.3923i	0.362253	−	0.362253i	0.864642	+	0.502389i	$0.167545\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	9.79796	−	9.79796i	0.340708	−	0.340708i	−0.515925	−	0.856634i	$-0.672552\pi$
$827$	9.79796	−	9.79796i	0.340708	−	0.340708i	0.856634	+	0.515925i	$0.172552\pi$
$828$		0			0
$829$			8.00000i			0.277851i	0.990303	+	0.138926i	$0.0443649\pi$
$829$			8.00000i			0.277851i	−0.990303	+	0.138926i	$0.955635\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−48.0833	−	48.0833i	−1.66599	−	1.66599i
$834$		0			0
$835$	27.7128	−	13.8564i	0.959041	−	0.479521i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	9.79796			0.338263			0.169132	−	0.985593i	$-0.445904\pi$
$839$	9.79796			0.338263			0.169132	+	0.985593i	$0.445904\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7.77817	+	23.3345i	−0.267577	+	0.802732i
$846$		0			0
$847$	−45.0333	−	45.0333i	−1.54736	−	1.54736i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−23.0000	+	23.0000i	−0.787505	+	0.787505i	−0.981085	−	0.193580i	$-0.937990\pi$
$853$	−23.0000	+	23.0000i	−0.787505	+	0.787505i	0.193580	+	0.981085i	$0.437990\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−14.1421	+	14.1421i	−0.483086	+	0.483086i	−0.906116	−	0.423030i	$-0.860967\pi$
$857$	−14.1421	+	14.1421i	−0.483086	+	0.483086i	0.423030	+	0.906116i	$0.360967\pi$
$858$		0			0
$859$		−	41.5692i		−	1.41832i	−0.705046	−	0.709162i	$-0.749074\pi$
$859$		−	41.5692i		−	1.41832i	0.705046	−	0.709162i	$-0.250926\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	9.79796	+	9.79796i	0.333526	+	0.333526i	0.853924	−	0.520398i	$-0.174216\pi$
$863$	9.79796	+	9.79796i	0.333526	+	0.333526i	−0.520398	+	0.853924i	$0.674216\pi$
$864$		0			0
$865$	2.00000	+	4.00000i	0.0680020	+	0.136004i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	33.9411			1.15137
$870$		0			0
$871$	13.8564			0.469506
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−9.79796	+	53.8888i	−0.331231	+	1.82177i
$876$		0			0
$877$	−5.00000	−	5.00000i	−0.168838	−	0.168838i	0.617630	−	0.786468i	$-0.288093\pi$
$877$	−5.00000	−	5.00000i	−0.168838	−	0.168838i	−0.786468	+	0.617630i	$0.788093\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	15.5563i		−	0.524107i	−0.965053	−	0.262053i	$-0.915600\pi$
$881$		−	15.5563i		−	0.524107i	0.965053	−	0.262053i	$-0.0843996\pi$
$882$		0			0
$883$	−34.6410	+	34.6410i	−1.16576	+	1.16576i	−0.182570	+	0.983193i	$0.558442\pi$
$883$	−34.6410	+	34.6410i	−1.16576	+	1.16576i	−0.983193	+	0.182570i	$0.941558\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−29.3939	+	29.3939i	−0.986950	+	0.986950i	−0.999916	−	0.0129661i	$-0.995873\pi$
$887$	−29.3939	+	29.3939i	−0.986950	+	0.986950i	0.0129661	+	0.999916i	$0.495873\pi$
$888$		0			0
$889$		−	72.0000i		−	2.41480i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	33.9411	+	33.9411i	1.13580	+	1.13580i
$894$		0			0
$895$	−10.3923	−	3.46410i	−0.347376	−	0.115792i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−48.9898			−1.63390
$900$		0			0
$901$	−16.0000			−0.533037
$902$		0			0
$903$		0			0
$904$		0			0
$905$	16.9706	+	5.65685i	0.564121	+	0.188040i
$906$		0			0
$907$	−20.7846	−	20.7846i	−0.690142	−	0.690142i	0.272121	−	0.962263i	$-0.412275\pi$
$907$	−20.7846	−	20.7846i	−0.690142	−	0.690142i	−0.962263	+	0.272121i	$0.912275\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			29.3939i			0.973863i	0.873440	+	0.486931i	$0.161884\pi$
$911$			29.3939i			0.973863i	−0.873440	+	0.486931i	$0.838116\pi$
$912$		0			0
$913$	−24.0000	+	24.0000i	−0.794284	+	0.794284i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−16.9706	+	16.9706i	−0.560417	+	0.560417i
$918$		0			0
$919$		−	20.7846i		−	0.685621i	−0.939405	−	0.342811i	$-0.888621\pi$
$919$		−	20.7846i		−	0.685621i	0.939405	−	0.342811i	$-0.111379\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.00000	−	7.00000i	−0.0328798	−	0.230159i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−12.7279			−0.417590			−0.208795	−	0.977959i	$-0.566954\pi$
$929$	−12.7279			−0.417590			−0.208795	+	0.977959i	$0.566954\pi$
$930$		0			0
$931$	−117.779			−3.86007
$932$		0			0
$933$		0			0
$934$		0			0
$935$	19.5959	+	39.1918i	0.640855	+	1.28171i
$936$		0			0
$937$	5.00000	+	5.00000i	0.163343	+	0.163343i	0.784046	−	0.620703i	$-0.213153\pi$
$937$	5.00000	+	5.00000i	0.163343	+	0.163343i	−0.620703	+	0.784046i	$0.713153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	21.2132i		−	0.691531i	−0.938321	−	0.345765i	$-0.887619\pi$
$941$		−	21.2132i		−	0.691531i	0.938321	−	0.345765i	$-0.112381\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	29.3939	−	29.3939i	0.955173	−	0.955173i	−0.0438648	−	0.999037i	$-0.513967\pi$
$947$	29.3939	−	29.3939i	0.955173	−	0.955173i	0.999037	+	0.0438648i	$0.0139671\pi$
$948$		0			0
$949$		−	14.0000i		−	0.454459i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	35.3553	+	35.3553i	1.14527	+	1.14527i	0.987471	+	0.157801i	$0.0504404\pi$
$953$	35.3553	+	35.3553i	1.14527	+	1.14527i	0.157801	+	0.987471i	$0.449560\pi$
$954$		0			0
$955$	13.8564	−	41.5692i	0.448383	−	1.34515i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	9.79796			0.316393
$960$		0			0
$961$	17.0000			0.548387
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−19.7990	+	9.89949i	−0.637352	+	0.318676i
$966$		0			0
$967$	−24.2487	−	24.2487i	−0.779786	−	0.779786i	0.200008	−	0.979794i	$-0.435903\pi$
$967$	−24.2487	−	24.2487i	−0.779786	−	0.779786i	−0.979794	+	0.200008i	$0.935903\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	44.0908i		−	1.41494i	−0.706743	−	0.707471i	$-0.749836\pi$
$971$		−	44.0908i		−	1.41494i	0.706743	−	0.707471i	$-0.250164\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19.7990	−	19.7990i	0.633426	−	0.633426i	−0.315500	−	0.948926i	$-0.602172\pi$
$977$	19.7990	−	19.7990i	0.633426	−	0.633426i	0.948926	+	0.315500i	$0.102172\pi$
$978$		0			0
$979$		−	48.4974i		−	1.54998i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−4.89898	−	4.89898i	−0.156253	−	0.156253i	0.624651	−	0.780904i	$-0.285241\pi$
$983$	−4.89898	−	4.89898i	−0.156253	−	0.156253i	−0.780904	+	0.624651i	$0.785241\pi$
$984$		0			0
$985$	−4.00000	+	2.00000i	−0.127451	+	0.0637253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	27.7128			0.880327			0.440163	−	0.897918i	$-0.354921\pi$
$991$	27.7128			0.880327			0.440163	+	0.897918i	$0.354921\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−9.79796	+	29.3939i	−0.310616	+	0.931849i
$996$		0			0
$997$	7.00000	+	7.00000i	0.221692	+	0.221692i	0.809211	−	0.587519i	$-0.199895\pi$
$997$	7.00000	+	7.00000i	0.221692	+	0.221692i	−0.587519	+	0.809211i	$0.699895\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.w.n.2177.4		8
3.2	odd	2	inner	2880.2.w.n.2177.2		8
4.3	odd	2	inner	2880.2.w.n.2177.3		8
5.3	odd	4	inner	2880.2.w.n.2753.2		8
8.3	odd	2		1440.2.w.e.737.1	✓	8
8.5	even	2		1440.2.w.e.737.2	yes	8
12.11	even	2	inner	2880.2.w.n.2177.1		8
15.8	even	4	inner	2880.2.w.n.2753.4		8
20.3	even	4	inner	2880.2.w.n.2753.1		8
24.5	odd	2		1440.2.w.e.737.4	yes	8
24.11	even	2		1440.2.w.e.737.3	yes	8
40.3	even	4		1440.2.w.e.1313.3	yes	8
40.13	odd	4		1440.2.w.e.1313.4	yes	8
60.23	odd	4	inner	2880.2.w.n.2753.3		8
120.53	even	4		1440.2.w.e.1313.2	yes	8
120.83	odd	4		1440.2.w.e.1313.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.w.e.737.1	✓	8	8.3	odd	2
1440.2.w.e.737.2	yes	8	8.5	even	2
1440.2.w.e.737.3	yes	8	24.11	even	2
1440.2.w.e.737.4	yes	8	24.5	odd	2
1440.2.w.e.1313.1	yes	8	120.83	odd	4
1440.2.w.e.1313.2	yes	8	120.53	even	4
1440.2.w.e.1313.3	yes	8	40.3	even	4
1440.2.w.e.1313.4	yes	8	40.13	odd	4
2880.2.w.n.2177.1		8	12.11	even	2	inner
2880.2.w.n.2177.2		8	3.2	odd	2	inner
2880.2.w.n.2177.3		8	4.3	odd	2	inner
2880.2.w.n.2177.4		8	1.1	even	1	trivial
2880.2.w.n.2753.1		8	20.3	even	4	inner
2880.2.w.n.2753.2		8	5.3	odd	4	inner
2880.2.w.n.2753.3		8	60.23	odd	4	inner
2880.2.w.n.2753.4		8	15.8	even	4	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.w (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.12132 + 0.707107i) q^{5} +(3.46410 + 3.46410i) q^{7} +O(q^{10})\)	\(q+(2.12132 + 0.707107i) q^{5} +(3.46410 + 3.46410i) q^{7} -4.89898i q^{11} +(-1.00000 + 1.00000i) q^{13} +(-2.82843 + 2.82843i) q^{17} +6.92820i q^{19} +(4.00000 + 3.00000i) q^{25} +7.07107 q^{29} -6.92820 q^{31} +(4.89898 + 9.79796i) q^{35} +(-1.00000 - 1.00000i) q^{37} +9.89949i q^{41} +(6.92820 - 6.92820i) q^{43} +(4.89898 - 4.89898i) q^{47} +17.0000i q^{49} +(2.82843 + 2.82843i) q^{53} +(3.46410 - 10.3923i) q^{55} -4.89898 q^{59} +(-2.82843 + 1.41421i) q^{65} +(-6.92820 - 6.92820i) q^{67} +(-7.00000 + 7.00000i) q^{73} +(16.9706 - 16.9706i) q^{77} +6.92820i q^{79} +(-4.89898 - 4.89898i) q^{83} +(-8.00000 + 4.00000i) q^{85} +9.89949 q^{89} -6.92820 q^{91} +(-4.89898 + 14.6969i) q^{95} +(5.00000 + 5.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{13} + 32 q^{25} - 8 q^{37} - 56 q^{73} - 64 q^{85} + 40 q^{97}+O(q^{100}) \) 8 * q - 8 * q^13 + 32 * q^25 - 8 * q^37 - 56 * q^73 - 64 * q^85 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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