LMFDB - Embedded newform 2880.2.t.e.721.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(721,2880)]

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, 3, 0, 0]))

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

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

chi := DirichletCharacter("2880.721");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.t (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:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			721.11
Character	$\chi$	$=$	2880.721
Dual form			2880.2.t.e.2161.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{5} -0.511707i q^{7} +O(q^{10})$	$q+(0.707107 - 0.707107i) q^{5} -0.511707i q^{7} +(-0.751529 + 0.751529i) q^{11} +(-4.22826 - 4.22826i) q^{13} +4.73728 q^{17} +(4.78289 + 4.78289i) q^{19} -0.0927028i q^{23} -1.00000i q^{25} +(-0.979239 - 0.979239i) q^{29} +3.03913 q^{31} +(-0.361831 - 0.361831i) q^{35} +(1.17130 - 1.17130i) q^{37} -11.9119i q^{41} +(-6.52641 + 6.52641i) q^{43} -5.91895 q^{47} +6.73816 q^{49} +(6.44018 - 6.44018i) q^{53} +1.06282i q^{55} +(7.60826 - 7.60826i) q^{59} +(-2.76756 - 2.76756i) q^{61} -5.97966 q^{65} +(-4.94156 - 4.94156i) q^{67} -9.80855i q^{71} -5.10381i q^{73} +(0.384562 + 0.384562i) q^{77} +6.92723 q^{79} +(-8.04926 - 8.04926i) q^{83} +(3.34976 - 3.34976i) q^{85} -2.41356i q^{89} +(-2.16363 + 2.16363i) q^{91} +6.76403 q^{95} -4.51711 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 8 q^{19} - 32 q^{37} - 16 q^{43} - 32 q^{49} - 16 q^{61} + 16 q^{67} + 16 q^{79} + 16 q^{85} + 80 q^{91}+O(q^{100}) $ 32 * q - 8 * q^19 - 32 * q^37 - 16 * q^43 - 32 * q^49 - 16 * q^61 + 16 * q^67 + 16 * q^79 + 16 * q^85 + 80 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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
$5$	0.707107	−	0.707107i	0.316228	−	0.316228i
$5$	0.707107	−	0.707107i	0.316228	−	0.316228i
$6$		0			0
$7$		−	0.511707i		−	0.193407i	−0.995313	−	0.0967035i	$-0.969170\pi$
$7$		−	0.511707i		−	0.193407i	0.995313	−	0.0967035i	$-0.0308299\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.751529	+	0.751529i	−0.226594	+	0.226594i	−0.811268	−	0.584674i	$-0.801222\pi$
$11$	−0.751529	+	0.751529i	−0.226594	+	0.226594i	0.584674	+	0.811268i	$0.301222\pi$
$12$		0			0
$13$	−4.22826	−	4.22826i	−1.17271	−	1.17271i	−0.981562	−	0.191146i	$-0.938780\pi$
$13$	−4.22826	−	4.22826i	−1.17271	−	1.17271i	−0.191146	−	0.981562i	$-0.561220\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.73728			1.14896			0.574479	−	0.818519i	$-0.305205\pi$
$17$	4.73728			1.14896			0.574479	+	0.818519i	$0.305205\pi$
$18$		0			0
$19$	4.78289	+	4.78289i	1.09727	+	1.09727i	0.994729	+	0.102542i	$0.0326976\pi$
$19$	4.78289	+	4.78289i	1.09727	+	1.09727i	0.102542	+	0.994729i	$0.467302\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	0.0927028i		−	0.0193299i	−0.999953	−	0.00966493i	$-0.996924\pi$
$23$		−	0.0927028i		−	0.0193299i	0.999953	−	0.00966493i	$-0.00307649\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.979239	−	0.979239i	−0.181840	−	0.181840i	0.610317	−	0.792157i	$-0.291042\pi$
$29$	−0.979239	−	0.979239i	−0.181840	−	0.181840i	−0.792157	+	0.610317i	$0.791042\pi$
$30$		0			0
$31$	3.03913			0.545845			0.272922	−	0.962036i	$-0.412010\pi$
$31$	3.03913			0.545845			0.272922	+	0.962036i	$0.412010\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.361831	−	0.361831i	−0.0611606	−	0.0611606i
$36$		0			0
$37$	1.17130	−	1.17130i	0.192561	−	0.192561i	−0.604241	−	0.796802i	$-0.706524\pi$
$37$	1.17130	−	1.17130i	0.192561	−	0.192561i	0.796802	+	0.604241i	$0.206524\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	11.9119i		−	1.86032i	−0.367150	−	0.930162i	$-0.619667\pi$
$41$		−	11.9119i		−	1.86032i	0.367150	−	0.930162i	$-0.380333\pi$
$42$		0			0
$43$	−6.52641	+	6.52641i	−0.995268	+	0.995268i	−0.999989	−	0.00472051i	$-0.998497\pi$
$43$	−6.52641	+	6.52641i	−0.995268	+	0.995268i	0.00472051	+	0.999989i	$0.498497\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.91895			−0.863367			−0.431684	−	0.902025i	$-0.642080\pi$
$47$	−5.91895			−0.863367			−0.431684	+	0.902025i	$0.642080\pi$
$48$		0			0
$49$	6.73816			0.962594
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.44018	−	6.44018i	0.884626	−	0.884626i	−0.109374	−	0.994001i	$-0.534885\pi$
$53$	6.44018	−	6.44018i	0.884626	−	0.884626i	0.994001	+	0.109374i	$0.0348847\pi$
$54$		0			0
$55$			1.06282i			0.143311i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.60826	−	7.60826i	0.990511	−	0.990511i	−0.00944406	−	0.999955i	$-0.503006\pi$
$59$	7.60826	−	7.60826i	0.990511	−	0.990511i	0.999955	+	0.00944406i	$0.00300618\pi$
$60$		0			0
$61$	−2.76756	−	2.76756i	−0.354349	−	0.354349i	0.507376	−	0.861725i	$-0.330616\pi$
$61$	−2.76756	−	2.76756i	−0.354349	−	0.354349i	−0.861725	+	0.507376i	$0.830616\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5.97966			−0.741685
$66$		0			0
$67$	−4.94156	−	4.94156i	−0.603708	−	0.603708i	0.337587	−	0.941294i	$-0.390389\pi$
$67$	−4.94156	−	4.94156i	−0.603708	−	0.603708i	−0.941294	+	0.337587i	$0.890389\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	9.80855i		−	1.16406i	−0.813167	−	0.582031i	$-0.802258\pi$
$71$		−	9.80855i		−	1.16406i	0.813167	−	0.582031i	$-0.197742\pi$
$72$		0			0
$73$		−	5.10381i		−	0.597356i	−0.954354	−	0.298678i	$-0.903454\pi$
$73$		−	5.10381i		−	0.597356i	0.954354	−	0.298678i	$-0.0965457\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.384562	+	0.384562i	0.0438249	+	0.0438249i
$78$		0			0
$79$	6.92723			0.779374			0.389687	−	0.920947i	$-0.372583\pi$
$79$	6.92723			0.779374			0.389687	+	0.920947i	$0.372583\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−8.04926	−	8.04926i	−0.883521	−	0.883521i	0.110370	−	0.993891i	$-0.464797\pi$
$83$	−8.04926	−	8.04926i	−0.883521	−	0.883521i	−0.993891	+	0.110370i	$0.964797\pi$
$84$		0			0
$85$	3.34976	−	3.34976i	0.363332	−	0.363332i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	2.41356i		−	0.255837i	−0.991785	−	0.127919i	$-0.959170\pi$
$89$		−	2.41356i		−	0.255837i	0.991785	−	0.127919i	$-0.0408296\pi$
$90$		0			0
$91$	−2.16363	+	2.16363i	−0.226810	+	0.226810i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.76403			0.693975
$96$		0			0
$97$	−4.51711			−0.458643			−0.229321	−	0.973351i	$-0.573651\pi$
$97$	−4.51711			−0.458643			−0.229321	+	0.973351i	$0.573651\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.90898	+	2.90898i	−0.289454	+	0.289454i	−0.836864	−	0.547410i	$-0.815614\pi$
$101$	−2.90898	+	2.90898i	−0.289454	+	0.289454i	0.547410	+	0.836864i	$0.315614\pi$
$102$		0			0
$103$			6.98357i			0.688111i	0.938949	+	0.344056i	$0.111801\pi$
$103$			6.98357i			0.688111i	−0.938949	+	0.344056i	$0.888199\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.31968	−	5.31968i	0.514273	−	0.514273i	−0.401560	−	0.915833i	$-0.631532\pi$
$107$	5.31968	−	5.31968i	0.514273	−	0.514273i	0.915833	+	0.401560i	$0.131532\pi$
$108$		0			0
$109$	−3.50835	−	3.50835i	−0.336039	−	0.336039i	0.518835	−	0.854874i	$-0.326366\pi$
$109$	−3.50835	−	3.50835i	−0.336039	−	0.336039i	−0.854874	+	0.518835i	$0.826366\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.57039			0.524018			0.262009	−	0.965065i	$-0.415615\pi$
$113$	5.57039			0.524018			0.262009	+	0.965065i	$0.415615\pi$
$114$		0			0
$115$	−0.0655508	−	0.0655508i	−0.00611264	−	0.00611264i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	2.42410i		−	0.222216i
$120$		0			0
$121$			9.87041i			0.897310i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	−	0.707107i	−0.0632456	−	0.0632456i
$126$		0			0
$127$	1.39206			0.123525			0.0617626	−	0.998091i	$-0.480328\pi$
$127$	1.39206			0.123525			0.0617626	+	0.998091i	$0.480328\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	14.4259	+	14.4259i	1.26039	+	1.26039i	0.950902	+	0.309493i	$0.100159\pi$
$131$	14.4259	+	14.4259i	1.26039	+	1.26039i	0.309493	+	0.950902i	$0.399841\pi$
$132$		0			0
$133$	2.44744	−	2.44744i	0.212220	−	0.212220i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	2.30854i		−	0.197232i	−0.995126	−	0.0986158i	$-0.968559\pi$
$137$		−	2.30854i		−	0.197232i	0.995126	−	0.0986158i	$-0.0314415\pi$
$138$		0			0
$139$	−12.4111	+	12.4111i	−1.05270	+	1.05270i	−0.0541659	+	0.998532i	$0.517250\pi$
$139$	−12.4111	+	12.4111i	−1.05270	+	1.05270i	−0.998532	+	0.0541659i	$0.982750\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.35531			0.531458
$144$		0			0
$145$	−1.38485			−0.115006
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.09810	−	5.09810i	0.417653	−	0.417653i	−0.466741	−	0.884394i	$-0.654572\pi$
$149$	5.09810	−	5.09810i	0.417653	−	0.417653i	0.884394	+	0.466741i	$0.154572\pi$
$150$		0			0
$151$		−	16.1276i		−	1.31245i	−0.754567	−	0.656223i	$-0.772153\pi$
$151$		−	16.1276i		−	1.31245i	0.754567	−	0.656223i	$-0.227847\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.14899	−	2.14899i	0.172611	−	0.172611i
$156$		0			0
$157$	−15.2703	−	15.2703i	−1.21871	−	1.21871i	−0.968085	−	0.250621i	$-0.919365\pi$
$157$	−15.2703	−	15.2703i	−1.21871	−	1.21871i	−0.250621	−	0.968085i	$-0.580635\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.0474366			−0.00373853
$162$		0			0
$163$	11.5036	+	11.5036i	0.901033	+	0.901033i	0.995526	−	0.0944929i	$-0.0301230\pi$
$163$	11.5036	+	11.5036i	0.901033	+	0.901033i	−0.0944929	+	0.995526i	$0.530123\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	21.7707i		−	1.68467i	−0.538958	−	0.842333i	$-0.681182\pi$
$167$		−	21.7707i		−	1.68467i	0.538958	−	0.842333i	$-0.318818\pi$
$168$		0			0
$169$			22.7563i			1.75049i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.88873	−	6.88873i	−0.523740	−	0.523740i	0.394959	−	0.918699i	$-0.370759\pi$
$173$	−6.88873	−	6.88873i	−0.523740	−	0.523740i	−0.918699	+	0.394959i	$0.870759\pi$
$174$		0			0
$175$	−0.511707			−0.0386814
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−13.7914	−	13.7914i	−1.03082	−	1.03082i	−0.999510	−	0.0313081i	$-0.990033\pi$
$179$	−13.7914	−	13.7914i	−1.03082	−	1.03082i	−0.0313081	−	0.999510i	$-0.509967\pi$
$180$		0			0
$181$	−8.89478	+	8.89478i	−0.661144	+	0.661144i	−0.955650	−	0.294506i	$-0.904845\pi$
$181$	−8.89478	+	8.89478i	−0.661144	+	0.661144i	0.294506	+	0.955650i	$0.404845\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	1.65647i		−	0.121786i
$186$		0			0
$187$	−3.56020	+	3.56020i	−0.260348	+	0.260348i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.8682			0.786392			0.393196	−	0.919455i	$-0.371369\pi$
$191$	10.8682			0.786392			0.393196	+	0.919455i	$0.371369\pi$
$192$		0			0
$193$	24.7380			1.78068			0.890339	−	0.455298i	$-0.150467\pi$
$193$	24.7380			1.78068			0.890339	+	0.455298i	$0.150467\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.70710	−	5.70710i	0.406614	−	0.406614i	−0.473942	−	0.880556i	$-0.657169\pi$
$197$	5.70710	−	5.70710i	0.406614	−	0.406614i	0.880556	+	0.473942i	$0.157169\pi$
$198$		0			0
$199$		−	8.60444i		−	0.609952i	−0.952360	−	0.304976i	$-0.901352\pi$
$199$		−	8.60444i		−	0.609952i	0.952360	−	0.304976i	$-0.0986485\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.501083	+	0.501083i	−0.0351692	+	0.0351692i
$204$		0			0
$205$	−8.42297	−	8.42297i	−0.588286	−	0.588286i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.18896			−0.497271
$210$		0			0
$211$	−3.85260	−	3.85260i	−0.265224	−	0.265224i	0.561949	−	0.827172i	$-0.310052\pi$
$211$	−3.85260	−	3.85260i	−0.265224	−	0.265224i	−0.827172	+	0.561949i	$0.810052\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			9.22974i			0.629463i
$216$		0			0
$217$		−	1.55515i		−	0.105570i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−20.0304	−	20.0304i	−1.34739	−	1.34739i
$222$		0			0
$223$	19.8126			1.32675			0.663374	−	0.748288i	$-0.269124\pi$
$223$	19.8126			1.32675			0.663374	+	0.748288i	$0.269124\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.4672	−	12.4672i	−0.827479	−	0.827479i	0.159689	−	0.987167i	$-0.448951\pi$
$227$	−12.4672	−	12.4672i	−0.827479	−	0.827479i	−0.987167	+	0.159689i	$0.948951\pi$
$228$		0			0
$229$	−5.02737	+	5.02737i	−0.332218	+	0.332218i	−0.853428	−	0.521210i	$-0.825481\pi$
$229$	−5.02737	+	5.02737i	−0.332218	+	0.332218i	0.521210	+	0.853428i	$0.325481\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			0.461523i			0.0302354i	0.999886	+	0.0151177i	$0.00481230\pi$
$233$			0.461523i			0.0302354i	−0.999886	+	0.0151177i	$0.995188\pi$
$234$		0			0
$235$	−4.18533	+	4.18533i	−0.273021	+	0.273021i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.95248			0.190980			0.0954900	−	0.995430i	$-0.469558\pi$
$239$	2.95248			0.190980			0.0954900	+	0.995430i	$0.469558\pi$
$240$		0			0
$241$	15.8276			1.01955			0.509773	−	0.860309i	$-0.329729\pi$
$241$	15.8276			1.01955			0.509773	+	0.860309i	$0.329729\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.76460	−	4.76460i	0.304399	−	0.304399i
$246$		0			0
$247$		−	40.4466i		−	2.57356i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.34784	−	9.34784i	0.590030	−	0.590030i	−0.347609	−	0.937639i	$-0.613006\pi$
$251$	9.34784	−	9.34784i	0.590030	−	0.590030i	0.937639	+	0.347609i	$0.113006\pi$
$252$		0			0
$253$	0.0696688	+	0.0696688i	0.00438004	+	0.00438004i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.3509			0.645671			0.322836	−	0.946455i	$-0.395364\pi$
$257$	10.3509			0.645671			0.322836	+	0.946455i	$0.395364\pi$
$258$		0			0
$259$	−0.599363	−	0.599363i	−0.0372426	−	0.0372426i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	21.3723i		−	1.31787i	−0.752198	−	0.658937i	$-0.771007\pi$
$263$		−	21.3723i		−	1.31787i	0.752198	−	0.658937i	$-0.228993\pi$
$264$		0			0
$265$		−	9.10779i		−	0.559487i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	14.7481	+	14.7481i	0.899208	+	0.899208i	0.995366	−	0.0961584i	$-0.0306555\pi$
$269$	14.7481	+	14.7481i	0.899208	+	0.899208i	−0.0961584	+	0.995366i	$0.530656\pi$
$270$		0			0
$271$	13.8661			0.842308			0.421154	−	0.906989i	$-0.361625\pi$
$271$	13.8661			0.842308			0.421154	+	0.906989i	$0.361625\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.751529	+	0.751529i	0.0453189	+	0.0453189i
$276$		0			0
$277$	−3.33259	+	3.33259i	−0.200236	+	0.200236i	−0.800101	−	0.599865i	$-0.795221\pi$
$277$	−3.33259	+	3.33259i	−0.200236	+	0.200236i	0.599865	+	0.800101i	$0.295221\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	0.301204i		−	0.0179683i	−0.999960	−	0.00898416i	$-0.997140\pi$
$281$		−	0.301204i		−	0.0179683i	0.999960	−	0.00898416i	$-0.00285978\pi$
$282$		0			0
$283$	−12.9592	+	12.9592i	−0.770347	+	0.770347i	−0.978167	−	0.207820i	$-0.933363\pi$
$283$	−12.9592	+	12.9592i	−0.770347	+	0.770347i	0.207820	+	0.978167i	$0.433363\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.09539			−0.359799
$288$		0			0
$289$	5.44178			0.320105
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−11.6268	+	11.6268i	−0.679244	+	0.679244i	−0.959829	−	0.280585i	$-0.909471\pi$
$293$	−11.6268	+	11.6268i	−0.679244	+	0.679244i	0.280585	+	0.959829i	$0.409471\pi$
$294$		0			0
$295$		−	10.7597i		−	0.626454i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.391971	+	0.391971i	−0.0226683	+	0.0226683i
$300$		0			0
$301$	3.33961	+	3.33961i	0.192492	+	0.192492i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.91392			−0.224110
$306$		0			0
$307$	−14.0025	−	14.0025i	−0.799164	−	0.799164i	0.183799	−	0.982964i	$-0.441160\pi$
$307$	−14.0025	−	14.0025i	−0.799164	−	0.799164i	−0.982964	+	0.183799i	$0.941160\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			27.3773i			1.55243i	0.630471	+	0.776213i	$0.282862\pi$
$311$			27.3773i			1.55243i	−0.630471	+	0.776213i	$0.717138\pi$
$312$		0			0
$313$			29.4232i			1.66310i	0.555453	+	0.831548i	$0.312545\pi$
$313$			29.4232i			1.66310i	−0.555453	+	0.831548i	$0.687455\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.2115	+	18.2115i	1.02286	+	1.02286i	0.999732	+	0.0231291i	$0.00736288\pi$
$317$	18.2115	+	18.2115i	1.02286	+	1.02286i	0.0231291	+	0.999732i	$0.492637\pi$
$318$		0			0
$319$	1.47185			0.0824080
$320$		0			0
$321$		0			0
$322$		0			0
$323$	22.6579	+	22.6579i	1.26072	+	1.26072i
$324$		0			0
$325$	−4.22826	+	4.22826i	−0.234542	+	0.234542i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			3.02877i			0.166981i
$330$		0			0
$331$	−1.61201	+	1.61201i	−0.0886040	+	0.0886040i	−0.750020	−	0.661416i	$-0.769956\pi$
$331$	−1.61201	+	1.61201i	−0.0886040	+	0.0886040i	0.661416	+	0.750020i	$0.269956\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.98842			−0.381818
$336$		0			0
$337$	16.9784			0.924874			0.462437	−	0.886652i	$-0.346975\pi$
$337$	16.9784			0.924874			0.462437	+	0.886652i	$0.346975\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.28400	+	2.28400i	−0.123685	+	0.123685i
$342$		0			0
$343$		−	7.02991i		−	0.379579i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−23.4829	+	23.4829i	−1.26063	+	1.26063i	−0.309839	+	0.950789i	$0.600275\pi$
$347$	−23.4829	+	23.4829i	−1.26063	+	1.26063i	−0.950789	+	0.309839i	$0.899725\pi$
$348$		0			0
$349$	14.0616	+	14.0616i	0.752699	+	0.752699i	0.974982	−	0.222283i	$-0.0713509\pi$
$349$	14.0616	+	14.0616i	0.752699	+	0.752699i	−0.222283	+	0.974982i	$0.571351\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	19.5402			1.04002			0.520009	−	0.854161i	$-0.325929\pi$
$353$	19.5402			1.04002			0.520009	+	0.854161i	$0.325929\pi$
$354$		0			0
$355$	−6.93569	−	6.93569i	−0.368108	−	0.368108i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			29.6910i			1.56703i	0.621372	+	0.783516i	$0.286576\pi$
$359$			29.6910i			1.56703i	−0.621372	+	0.783516i	$0.713424\pi$
$360$		0			0
$361$			26.7521i			1.40801i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.60894	−	3.60894i	−0.188901	−	0.188901i
$366$		0			0
$367$	0.925465			0.0483089			0.0241544	−	0.999708i	$-0.492311\pi$
$367$	0.925465			0.0483089			0.0241544	+	0.999708i	$0.492311\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.29548	−	3.29548i	−0.171093	−	0.171093i
$372$		0			0
$373$	4.29793	−	4.29793i	0.222538	−	0.222538i	−0.587028	−	0.809566i	$-0.699702\pi$
$373$	4.29793	−	4.29793i	0.222538	−	0.222538i	0.809566	+	0.587028i	$0.199702\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			8.28095i			0.426491i
$378$		0			0
$379$	9.20296	−	9.20296i	0.472724	−	0.472724i	−0.430071	−	0.902795i	$-0.641511\pi$
$379$	9.20296	−	9.20296i	0.472724	−	0.472724i	0.902795	+	0.430071i	$0.141511\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−17.2021			−0.878984			−0.439492	−	0.898247i	$-0.644841\pi$
$383$	−17.2021			−0.878984			−0.439492	+	0.898247i	$0.644841\pi$
$384$		0			0
$385$	0.543853			0.0277173
$386$		0			0
$387$		0			0
$388$		0			0
$389$	20.9608	−	20.9608i	1.06276	−	1.06276i	0.0648607	−	0.997894i	$-0.479340\pi$
$389$	20.9608	−	20.9608i	1.06276	−	1.06276i	0.997894	−	0.0648607i	$-0.0206603\pi$
$390$		0			0
$391$		−	0.439159i		−	0.0222092i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.89829	−	4.89829i	0.246460	−	0.246460i
$396$		0			0
$397$	−17.6135	−	17.6135i	−0.883995	−	0.883995i	0.109943	−	0.993938i	$-0.464933\pi$
$397$	−17.6135	−	17.6135i	−0.883995	−	0.883995i	−0.993938	+	0.109943i	$0.964933\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.4827			−0.573419			−0.286709	−	0.958018i	$-0.592561\pi$
$401$	−11.4827			−0.573419			−0.286709	+	0.958018i	$0.592561\pi$
$402$		0			0
$403$	−12.8502	−	12.8502i	−0.640116	−	0.640116i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			1.76053i			0.0872664i
$408$		0			0
$409$			14.0740i			0.695916i	0.937510	+	0.347958i	$0.113125\pi$
$409$			14.0740i			0.695916i	−0.937510	+	0.347958i	$0.886875\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.89320	−	3.89320i	−0.191572	−	0.191572i
$414$		0			0
$415$	−11.3834			−0.558788
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−14.6490	−	14.6490i	−0.715650	−	0.715650i	0.252062	−	0.967711i	$-0.418891\pi$
$419$	−14.6490	−	14.6490i	−0.715650	−	0.715650i	−0.967711	+	0.252062i	$0.918891\pi$
$420$		0			0
$421$	11.0244	−	11.0244i	0.537296	−	0.537296i	−0.385437	−	0.922734i	$-0.625949\pi$
$421$	11.0244	−	11.0244i	0.537296	−	0.537296i	0.922734	+	0.385437i	$0.125949\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	4.73728i		−	0.229792i
$426$		0			0
$427$	−1.41618	+	1.41618i	−0.0685336	+	0.0685336i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	16.9583			0.816851			0.408426	−	0.912792i	$-0.366078\pi$
$431$	16.9583			0.816851			0.408426	+	0.912792i	$0.366078\pi$
$432$		0			0
$433$	−21.0507			−1.01163			−0.505816	−	0.862642i	$-0.668808\pi$
$433$	−21.0507			−1.01163			−0.505816	+	0.862642i	$0.668808\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.443387	−	0.443387i	0.0212101	−	0.0212101i
$438$		0			0
$439$			40.2881i			1.92285i	0.275071	+	0.961424i	$0.411299\pi$
$439$			40.2881i			1.92285i	−0.275071	+	0.961424i	$0.588701\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.4658	+	17.4658i	−0.829827	+	0.829827i	−0.987493	−	0.157665i	$-0.949603\pi$
$443$	−17.4658	+	17.4658i	−0.829827	+	0.829827i	0.157665	+	0.987493i	$0.449603\pi$
$444$		0			0
$445$	−1.70665	−	1.70665i	−0.0809028	−	0.0809028i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9.56773			−0.451529			−0.225765	−	0.974182i	$-0.572488\pi$
$449$	−9.56773			−0.451529			−0.225765	+	0.974182i	$0.572488\pi$
$450$		0			0
$451$	8.95212	+	8.95212i	0.421539	+	0.421539i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			3.05983i			0.143447i
$456$		0			0
$457$		−	11.5509i		−	0.540330i	−0.962814	−	0.270165i	$-0.912922\pi$
$457$		−	11.5509i		−	0.540330i	0.962814	−	0.270165i	$-0.0870783\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−21.2403	−	21.2403i	−0.989260	−	0.989260i	0.0106834	−	0.999943i	$-0.496599\pi$
$461$	−21.2403	−	21.2403i	−0.989260	−	0.989260i	−0.999943	+	0.0106834i	$0.996599\pi$
$462$		0			0
$463$	−12.9880			−0.603605			−0.301802	−	0.953370i	$-0.597588\pi$
$463$	−12.9880			−0.603605			−0.301802	+	0.953370i	$0.597588\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.552173	+	0.552173i	0.0255515	+	0.0255515i	0.719767	−	0.694216i	$-0.244249\pi$
$467$	0.552173	+	0.552173i	0.0255515	+	0.0255515i	−0.694216	+	0.719767i	$0.744249\pi$
$468$		0			0
$469$	−2.52863	+	2.52863i	−0.116761	+	0.116761i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	9.80957i		−	0.451045i
$474$		0			0
$475$	4.78289	−	4.78289i	0.219454	−	0.219454i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−34.8968			−1.59447			−0.797237	−	0.603666i	$-0.793706\pi$
$479$	−34.8968			−1.59447			−0.797237	+	0.603666i	$0.793706\pi$
$480$		0			0
$481$	−9.90512			−0.451635
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.19408	+	3.19408i	−0.145036	+	0.145036i
$486$		0			0
$487$		−	28.2023i		−	1.27797i	−0.769220	−	0.638984i	$-0.779355\pi$
$487$		−	28.2023i		−	1.27797i	0.769220	−	0.638984i	$-0.220645\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−10.1166	+	10.1166i	−0.456556	+	0.456556i	−0.897523	−	0.440967i	$-0.854635\pi$
$491$	−10.1166	+	10.1166i	−0.456556	+	0.456556i	0.440967	+	0.897523i	$0.354635\pi$
$492$		0			0
$493$	−4.63893	−	4.63893i	−0.208927	−	0.208927i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5.01910			−0.225138
$498$		0			0
$499$	23.8551	+	23.8551i	1.06790	+	1.06790i	0.997520	+	0.0703790i	$0.0224209\pi$
$499$	23.8551	+	23.8551i	1.06790	+	1.06790i	0.0703790	+	0.997520i	$0.477579\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	26.2491i		−	1.17039i	−0.810892	−	0.585195i	$-0.801018\pi$
$503$		−	26.2491i		−	1.17039i	0.810892	−	0.585195i	$-0.198982\pi$
$504$		0			0
$505$			4.11391i			0.183067i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	29.0368	+	29.0368i	1.28704	+	1.28704i	0.936579	+	0.350456i	$0.113974\pi$
$509$	29.0368	+	29.0368i	1.28704	+	1.28704i	0.350456	+	0.936579i	$0.386026\pi$
$510$		0			0
$511$	−2.61165			−0.115533
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.93813	+	4.93813i	0.217600	+	0.217600i
$516$		0			0
$517$	4.44826	−	4.44826i	0.195634	−	0.195634i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			16.2624i			0.712468i	0.934397	+	0.356234i	$0.115939\pi$
$521$			16.2624i			0.712468i	−0.934397	+	0.356234i	$0.884061\pi$
$522$		0			0
$523$	10.5530	−	10.5530i	0.461451	−	0.461451i	−0.437680	−	0.899131i	$-0.644200\pi$
$523$	10.5530	−	10.5530i	0.461451	−	0.461451i	0.899131	+	0.437680i	$0.144200\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	14.3972			0.627153
$528$		0			0
$529$	22.9914			0.999626
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−50.3665	+	50.3665i	−2.18161	+	2.18161i
$534$		0			0
$535$		−	7.52316i		−	0.325255i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5.06392	+	5.06392i	−0.218118	+	0.218118i
$540$		0			0
$541$	−17.0580	−	17.0580i	−0.733380	−	0.733380i	0.237908	−	0.971288i	$-0.423538\pi$
$541$	−17.0580	−	17.0580i	−0.733380	−	0.733380i	−0.971288	+	0.237908i	$0.923538\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.96155			−0.212530
$546$		0			0
$547$	−12.4561	−	12.4561i	−0.532584	−	0.532584i	0.388757	−	0.921341i	$-0.372905\pi$
$547$	−12.4561	−	12.4561i	−0.532584	−	0.532584i	−0.921341	+	0.388757i	$0.872905\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	9.36719i		−	0.399056i
$552$		0			0
$553$		−	3.54471i		−	0.150736i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.43060	+	4.43060i	0.187731	+	0.187731i	0.794714	−	0.606984i	$-0.207621\pi$
$557$	4.43060	+	4.43060i	0.187731	+	0.187731i	−0.606984	+	0.794714i	$0.707621\pi$
$558$		0			0
$559$	55.1907			2.33432
$560$		0			0
$561$		0			0
$562$		0			0
$563$	25.8113	+	25.8113i	1.08782	+	1.08782i	0.995753	+	0.0920639i	$0.0293464\pi$
$563$	25.8113	+	25.8113i	1.08782	+	1.08782i	0.0920639	+	0.995753i	$0.470654\pi$
$564$		0			0
$565$	3.93886	−	3.93886i	0.165709	−	0.165709i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			24.0679i			1.00898i	0.863418	+	0.504489i	$0.168319\pi$
$569$			24.0679i			1.00898i	−0.863418	+	0.504489i	$0.831681\pi$
$570$		0			0
$571$	−19.4966	+	19.4966i	−0.815909	+	0.815909i	−0.985512	−	0.169603i	$-0.945751\pi$
$571$	−19.4966	+	19.4966i	−0.815909	+	0.815909i	0.169603	+	0.985512i	$0.445751\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.0927028			−0.00386597
$576$		0			0
$577$	−14.5789			−0.606929			−0.303464	−	0.952843i	$-0.598143\pi$
$577$	−14.5789			−0.606929			−0.303464	+	0.952843i	$0.598143\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.11886	+	4.11886i	−0.170879	+	0.170879i
$582$		0			0
$583$			9.67996i			0.400903i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	17.3865	−	17.3865i	0.717618	−	0.717618i	−0.250499	−	0.968117i	$-0.580595\pi$
$587$	17.3865	−	17.3865i	0.717618	−	0.717618i	0.968117	+	0.250499i	$0.0805947\pi$
$588$		0			0
$589$	14.5359	+	14.5359i	0.598939	+	0.598939i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	29.4761			1.21044			0.605220	−	0.796058i	$-0.293085\pi$
$593$	29.4761			1.21044			0.605220	+	0.796058i	$0.293085\pi$
$594$		0			0
$595$	−1.71409	−	1.71409i	−0.0702710	−	0.0702710i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			41.8704i			1.71078i	0.517985	+	0.855390i	$0.326682\pi$
$599$			41.8704i			1.71078i	−0.517985	+	0.855390i	$0.673318\pi$
$600$		0			0
$601$		−	35.2862i		−	1.43936i	−0.694308	−	0.719678i	$-0.744290\pi$
$601$		−	35.2862i		−	1.43936i	0.694308	−	0.719678i	$-0.255710\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.97943	+	6.97943i	0.283754	+	0.283754i
$606$		0			0
$607$	−35.7338			−1.45039			−0.725194	−	0.688544i	$-0.758250\pi$
$607$	−35.7338			−1.45039			−0.725194	+	0.688544i	$0.758250\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	25.0268	+	25.0268i	1.01248	+	1.01248i
$612$		0			0
$613$	−27.7737	+	27.7737i	−1.12177	+	1.12177i	−0.130295	+	0.991475i	$0.541592\pi$
$613$	−27.7737	+	27.7737i	−1.12177	+	1.12177i	−0.991475	+	0.130295i	$0.958408\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			41.2603i			1.66108i	0.556962	+	0.830538i	$0.311967\pi$
$617$			41.2603i			1.66108i	−0.556962	+	0.830538i	$0.688033\pi$
$618$		0			0
$619$	−28.8755	+	28.8755i	−1.16060	+	1.16060i	−0.176261	+	0.984344i	$0.556400\pi$
$619$	−28.8755	+	28.8755i	−1.16060	+	1.16060i	−0.984344	+	0.176261i	$0.943600\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.23504			−0.0494807
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.54878	−	5.54878i	0.221244	−	0.221244i
$630$		0			0
$631$			19.1784i			0.763481i	0.924270	+	0.381740i	$0.124675\pi$
$631$			19.1784i			0.763481i	−0.924270	+	0.381740i	$0.875325\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.984333	−	0.984333i	0.0390621	−	0.0390621i
$636$		0			0
$637$	−28.4907	−	28.4907i	−1.12884	−	1.12884i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−16.5636			−0.654223			−0.327111	−	0.944986i	$-0.606075\pi$
$641$	−16.5636			−0.654223			−0.327111	+	0.944986i	$0.606075\pi$
$642$		0			0
$643$	27.0589	+	27.0589i	1.06710	+	1.06710i	0.997581	+	0.0695192i	$0.0221465\pi$
$643$	27.0589	+	27.0589i	1.06710	+	1.06710i	0.0695192	+	0.997581i	$0.477854\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			29.7374i			1.16910i	0.811358	+	0.584550i	$0.198729\pi$
$647$			29.7374i			1.16910i	−0.811358	+	0.584550i	$0.801271\pi$
$648$		0			0
$649$			11.4357i			0.448889i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.504139	−	0.504139i	−0.0197285	−	0.0197285i	0.697174	−	0.716902i	$-0.254441\pi$
$653$	−0.504139	−	0.504139i	−0.0197285	−	0.0197285i	−0.716902	+	0.697174i	$0.754441\pi$
$654$		0			0
$655$	20.4013			0.797144
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−12.1441	−	12.1441i	−0.473066	−	0.473066i	0.429839	−	0.902905i	$-0.358570\pi$
$659$	−12.1441	−	12.1441i	−0.473066	−	0.473066i	−0.902905	+	0.429839i	$0.858570\pi$
$660$		0			0
$661$	−10.2032	+	10.2032i	−0.396859	+	0.396859i	−0.877124	−	0.480265i	$-0.840541\pi$
$661$	−10.2032	+	10.2032i	−0.396859	+	0.396859i	0.480265	+	0.877124i	$0.340541\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	3.46120i		−	0.134220i
$666$		0			0
$667$	−0.0907782	+	0.0907782i	−0.00351495	+	0.00351495i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.15980			0.160587
$672$		0			0
$673$	10.8274			0.417367			0.208683	−	0.977983i	$-0.433082\pi$
$673$	10.8274			0.417367			0.208683	+	0.977983i	$0.433082\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−22.8507	+	22.8507i	−0.878224	+	0.878224i	−0.993351	−	0.115127i	$-0.963272\pi$
$677$	−22.8507	+	22.8507i	−0.878224	+	0.878224i	0.115127	+	0.993351i	$0.463272\pi$
$678$		0			0
$679$			2.31143i			0.0887047i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.24450	+	3.24450i	−0.124147	+	0.124147i	−0.766451	−	0.642303i	$-0.777979\pi$
$683$	−3.24450	+	3.24450i	−0.124147	+	0.124147i	0.642303	+	0.766451i	$0.277979\pi$
$684$		0			0
$685$	−1.63238	−	1.63238i	−0.0623701	−	0.0623701i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−54.4615			−2.07482
$690$		0			0
$691$	−3.77564	−	3.77564i	−0.143632	−	0.143632i	0.631634	−	0.775266i	$-0.282384\pi$
$691$	−3.77564	−	3.77564i	−0.143632	−	0.143632i	−0.775266	+	0.631634i	$0.782384\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			17.5520i			0.665785i
$696$		0			0
$697$		−	56.4299i		−	2.13743i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	28.6912	+	28.6912i	1.08365	+	1.08365i	0.996166	+	0.0874865i	$0.0278835\pi$
$701$	28.6912	+	28.6912i	1.08365	+	1.08365i	0.0874865	+	0.996166i	$0.472117\pi$
$702$		0			0
$703$	11.2044			0.422582
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.48854	+	1.48854i	0.0559824	+	0.0559824i
$708$		0			0
$709$	6.56582	−	6.56582i	0.246585	−	0.246585i	−0.572983	−	0.819567i	$-0.694214\pi$
$709$	6.56582	−	6.56582i	0.246585	−	0.246585i	0.819567	+	0.572983i	$0.194214\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	0.281736i		−	0.0105511i
$714$		0			0
$715$	4.49389	−	4.49389i	0.168062	−	0.168062i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	39.8676			1.48681			0.743405	−	0.668842i	$-0.233210\pi$
$719$	39.8676			1.48681			0.743405	+	0.668842i	$0.233210\pi$
$720$		0			0
$721$	3.57354			0.133086
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.979239	+	0.979239i	−0.0363680	+	0.0363680i
$726$		0			0
$727$			18.7495i			0.695380i	0.937610	+	0.347690i	$0.113034\pi$
$727$			18.7495i			0.695380i	−0.937610	+	0.347690i	$0.886966\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−30.9174	+	30.9174i	−1.14352	+	1.14352i
$732$		0			0
$733$	33.3263	+	33.3263i	1.23093	+	1.23093i	0.963605	+	0.267329i	$0.0861410\pi$
$733$	33.3263	+	33.3263i	1.23093	+	1.23093i	0.267329	+	0.963605i	$0.413859\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.42745			0.273594
$738$		0			0
$739$	15.5069	+	15.5069i	0.570429	+	0.570429i	0.932248	−	0.361819i	$-0.117844\pi$
$739$	15.5069	+	15.5069i	0.570429	+	0.570429i	−0.361819	+	0.932248i	$0.617844\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	11.4896i		−	0.421514i	−0.977538	−	0.210757i	$-0.932407\pi$
$743$		−	11.4896i		−	0.421514i	0.977538	−	0.210757i	$-0.0675929\pi$
$744$		0			0
$745$		−	7.20980i		−	0.264147i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.72212	−	2.72212i	−0.0994640	−	0.0994640i
$750$		0			0
$751$	−33.4720			−1.22141			−0.610706	−	0.791857i	$-0.709114\pi$
$751$	−33.4720			−1.22141			−0.610706	+	0.791857i	$0.709114\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−11.4039	−	11.4039i	−0.415032	−	0.415032i
$756$		0			0
$757$	26.7408	−	26.7408i	0.971910	−	0.971910i	−0.0277060	−	0.999616i	$-0.508820\pi$
$757$	26.7408	−	26.7408i	0.971910	−	0.971910i	0.999616	+	0.0277060i	$0.00882023\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	6.83099i		−	0.247623i	−0.992306	−	0.123812i	$-0.960488\pi$
$761$		−	6.83099i		−	0.247623i	0.992306	−	0.123812i	$-0.0395118\pi$
$762$		0			0
$763$	−1.79525	+	1.79525i	−0.0649922	+	0.0649922i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−64.3394			−2.32316
$768$		0			0
$769$	23.5044			0.847590			0.423795	−	0.905758i	$-0.360698\pi$
$769$	23.5044			0.847590			0.423795	+	0.905758i	$0.360698\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.848545	−	0.848545i	0.0305200	−	0.0305200i	−0.691682	−	0.722202i	$-0.743130\pi$
$773$	0.848545	−	0.848545i	0.0305200	−	0.0305200i	0.722202	+	0.691682i	$0.243130\pi$
$774$		0			0
$775$		−	3.03913i		−	0.109169i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	56.9732	−	56.9732i	2.04128	−	2.04128i
$780$		0			0
$781$	7.37141	+	7.37141i	0.263770	+	0.263770i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−21.5955			−0.770778
$786$		0			0
$787$	−6.03949	−	6.03949i	−0.215285	−	0.215285i	0.591223	−	0.806508i	$-0.298645\pi$
$787$	−6.03949	−	6.03949i	−0.215285	−	0.215285i	−0.806508	+	0.591223i	$0.798645\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	2.85040i		−	0.101349i
$792$		0			0
$793$			23.4039i			0.831096i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−24.1586	−	24.1586i	−0.855742	−	0.855742i	0.135091	−	0.990833i	$-0.456867\pi$
$797$	−24.1586	−	24.1586i	−0.855742	−	0.855742i	−0.990833	+	0.135091i	$0.956867\pi$
$798$		0			0
$799$	−28.0397			−0.991973
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.83566	+	3.83566i	0.135358	+	0.135358i
$804$		0			0
$805$	−0.0335428	+	0.0335428i	−0.00118223	+	0.00118223i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1.35392i		−	0.0476012i	−0.999717	−	0.0238006i	$-0.992423\pi$
$809$		−	1.35392i		−	0.0476012i	0.999717	−	0.0238006i	$-0.00757668\pi$
$810$		0			0
$811$	6.36687	−	6.36687i	0.223571	−	0.223571i	−0.586429	−	0.810000i	$-0.699467\pi$
$811$	6.36687	−	6.36687i	0.223571	−	0.223571i	0.810000	+	0.586429i	$0.199467\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	16.2686			0.569863
$816$		0			0
$817$	−62.4302			−2.18416
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−17.6457	+	17.6457i	−0.615840	+	0.615840i	−0.944462	−	0.328622i	$-0.893416\pi$
$821$	−17.6457	+	17.6457i	−0.615840	+	0.615840i	0.328622	+	0.944462i	$0.393416\pi$
$822$		0			0
$823$		−	27.2150i		−	0.948654i	−0.880349	−	0.474327i	$-0.842691\pi$
$823$		−	27.2150i		−	0.948654i	0.880349	−	0.474327i	$-0.157309\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−11.4348	+	11.4348i	−0.397627	+	0.397627i	−0.877395	−	0.479768i	$-0.840721\pi$
$827$	−11.4348	+	11.4348i	−0.397627	+	0.397627i	0.479768	+	0.877395i	$0.340721\pi$
$828$		0			0
$829$	−0.300129	−	0.300129i	−0.0104239	−	0.0104239i	0.701876	−	0.712300i	$-0.252346\pi$
$829$	−0.300129	−	0.300129i	−0.0104239	−	0.0104239i	−0.712300	+	0.701876i	$0.752346\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	31.9205			1.10598
$834$		0			0
$835$	−15.3942	−	15.3942i	−0.532738	−	0.532738i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			11.0790i			0.382488i	0.981543	+	0.191244i	$0.0612522\pi$
$839$			11.0790i			0.382488i	−0.981543	+	0.191244i	$0.938748\pi$
$840$		0			0
$841$		−	27.0822i		−	0.933868i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	16.0911	+	16.0911i	0.553552	+	0.553552i
$846$		0			0
$847$	5.05075			0.173546
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.108583	−	0.108583i	−0.00372217	−	0.00372217i
$852$		0			0
$853$	20.3141	−	20.3141i	0.695541	−	0.695541i	−0.267904	−	0.963446i	$-0.586331\pi$
$853$	20.3141	−	20.3141i	0.695541	−	0.695541i	0.963446	+	0.267904i	$0.0863311\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	23.8031i		−	0.813099i	−0.913629	−	0.406549i	$-0.866732\pi$
$857$		−	23.8031i		−	0.813099i	0.913629	−	0.406549i	$-0.133268\pi$
$858$		0			0
$859$	22.8263	−	22.8263i	0.778822	−	0.778822i	−0.200808	−	0.979631i	$-0.564357\pi$
$859$	22.8263	−	22.8263i	0.778822	−	0.778822i	0.979631	+	0.200808i	$0.0643568\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−19.0336			−0.647911			−0.323956	−	0.946072i	$-0.605013\pi$
$863$	−19.0336			−0.647911			−0.323956	+	0.946072i	$0.605013\pi$
$864$		0			0
$865$	−9.74213			−0.331242
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−5.20601	+	5.20601i	−0.176602	+	0.176602i
$870$		0			0
$871$			41.7884i			1.41595i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.361831	+	0.361831i	−0.0122321	+	0.0122321i
$876$		0			0
$877$	6.77519	+	6.77519i	0.228782	+	0.228782i	0.812184	−	0.583402i	$-0.198279\pi$
$877$	6.77519	+	6.77519i	0.228782	+	0.228782i	−0.583402	+	0.812184i	$0.698279\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	34.1867			1.15178			0.575890	−	0.817527i	$-0.304656\pi$
$881$	34.1867			1.15178			0.575890	+	0.817527i	$0.304656\pi$
$882$		0			0
$883$	−6.99012	−	6.99012i	−0.235236	−	0.235236i	0.579638	−	0.814874i	$-0.303194\pi$
$883$	−6.99012	−	6.99012i	−0.235236	−	0.235236i	−0.814874	+	0.579638i	$0.803194\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	9.23335i		−	0.310026i	−0.987912	−	0.155013i	$-0.950458\pi$
$887$		−	9.23335i		−	0.310026i	0.987912	−	0.155013i	$-0.0495419\pi$
$888$		0			0
$889$		−	0.712325i		−	0.0238906i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−28.3097	−	28.3097i	−0.947348	−	0.947348i
$894$		0			0
$895$	−19.5040			−0.651946
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.97604	−	2.97604i	−0.0992565	−	0.0992565i
$900$		0			0
$901$	30.5089	−	30.5089i	1.01640	−	1.01640i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			12.5791i			0.418144i
$906$		0			0
$907$	27.5823	−	27.5823i	0.915854	−	0.915854i	−0.0808703	−	0.996725i	$-0.525770\pi$
$907$	27.5823	−	27.5823i	0.915854	−	0.915854i	0.996725	+	0.0808703i	$0.0257700\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−6.80204			−0.225362			−0.112681	−	0.993631i	$-0.535944\pi$
$911$	−6.80204			−0.225362			−0.112681	+	0.993631i	$0.535944\pi$
$912$		0			0
$913$	12.0985			0.400402
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.38182	−	7.38182i	0.243769	−	0.243769i
$918$		0			0
$919$			40.4075i			1.33292i	0.745541	+	0.666460i	$0.232191\pi$
$919$			40.4075i			1.33292i	−0.745541	+	0.666460i	$0.767809\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−41.4731	+	41.4731i	−1.36510	+	1.36510i
$924$		0			0
$925$	−1.17130	−	1.17130i	−0.0385121	−	0.0385121i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−49.3976			−1.62068			−0.810342	−	0.585957i	$-0.800719\pi$
$929$	−49.3976			−1.62068			−0.810342	+	0.585957i	$0.800719\pi$
$930$		0			0
$931$	32.2279	+	32.2279i	1.05623	+	1.05623i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			5.03488i			0.164658i
$936$		0			0
$937$		−	31.0115i		−	1.01310i	−0.862210	−	0.506551i	$-0.830920\pi$
$937$		−	31.0115i		−	1.01310i	0.862210	−	0.506551i	$-0.169080\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	13.2147	+	13.2147i	0.430786	+	0.430786i	0.888896	−	0.458110i	$-0.151473\pi$
$941$	13.2147	+	13.2147i	0.430786	+	0.430786i	−0.458110	+	0.888896i	$0.651473\pi$
$942$		0			0
$943$	−1.10426			−0.0359598
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.219269	−	0.219269i	−0.00712528	−	0.00712528i	0.703535	−	0.710660i	$-0.251604\pi$
$947$	−0.219269	−	0.219269i	−0.00712528	−	0.00712528i	−0.710660	+	0.703535i	$0.751604\pi$
$948$		0			0
$949$	−21.5802	+	21.5802i	−0.700524	+	0.700524i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			43.8339i			1.41992i	0.704242	+	0.709960i	$0.251287\pi$
$953$			43.8339i			1.41992i	−0.704242	+	0.709960i	$0.748713\pi$
$954$		0			0
$955$	7.68494	−	7.68494i	0.248679	−	0.248679i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.18129			−0.0381460
$960$		0			0
$961$	−21.7637			−0.702054
$962$		0			0
$963$		0			0
$964$		0			0
$965$	17.4924	−	17.4924i	0.563100	−	0.563100i
$966$		0			0
$967$			45.5600i			1.46511i	0.680707	+	0.732556i	$0.261673\pi$
$967$			45.5600i			1.46511i	−0.680707	+	0.732556i	$0.738327\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.79605	+	3.79605i	−0.121821	+	0.121821i	−0.765389	−	0.643568i	$-0.777453\pi$
$971$	−3.79605	+	3.79605i	−0.121821	+	0.121821i	0.643568	+	0.765389i	$0.277453\pi$
$972$		0			0
$973$	6.35086	+	6.35086i	0.203599	+	0.203599i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	46.9906			1.50336			0.751682	−	0.659526i	$-0.229243\pi$
$977$	46.9906			1.50336			0.751682	+	0.659526i	$0.229243\pi$
$978$		0			0
$979$	1.81386	+	1.81386i	0.0579713	+	0.0579713i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	5.05860i		−	0.161344i	−0.996741	−	0.0806722i	$-0.974293\pi$
$983$		−	5.05860i		−	0.161344i	0.996741	−	0.0806722i	$-0.0257067\pi$
$984$		0			0
$985$		−	8.07106i		−	0.257165i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.605017	+	0.605017i	0.0192384	+	0.0192384i
$990$		0			0
$991$	37.9925			1.20687			0.603436	−	0.797411i	$-0.293798\pi$
$991$	37.9925			1.20687			0.603436	+	0.797411i	$0.293798\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.08425	−	6.08425i	−0.192884	−	0.192884i
$996$		0			0
$997$	8.28839	−	8.28839i	0.262496	−	0.262496i	−0.563571	−	0.826067i	$-0.690573\pi$
$997$	8.28839	−	8.28839i	0.262496	−	0.262496i	0.826067	+	0.563571i	$0.190573\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.e.721.11		32
3.2	odd	2	inner	2880.2.t.e.721.4		32
4.3	odd	2		720.2.t.e.541.5	yes	32
12.11	even	2		720.2.t.e.541.12	yes	32
16.5	even	4	inner	2880.2.t.e.2161.11		32
16.11	odd	4		720.2.t.e.181.5	✓	32
48.5	odd	4	inner	2880.2.t.e.2161.4		32
48.11	even	4		720.2.t.e.181.12	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.t.e.181.5	✓	32	16.11	odd	4
720.2.t.e.181.12	yes	32	48.11	even	4
720.2.t.e.541.5	yes	32	4.3	odd	2
720.2.t.e.541.12	yes	32	12.11	even	2
2880.2.t.e.721.4		32	3.2	odd	2	inner
2880.2.t.e.721.11		32	1.1	even	1	trivial
2880.2.t.e.2161.4		32	48.5	odd	4	inner
2880.2.t.e.2161.11		32	16.5	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.t (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{5} -0.511707i q^{7} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{5} -0.511707i q^{7} +(-0.751529 + 0.751529i) q^{11} +(-4.22826 - 4.22826i) q^{13} +4.73728 q^{17} +(4.78289 + 4.78289i) q^{19} -0.0927028i q^{23} -1.00000i q^{25} +(-0.979239 - 0.979239i) q^{29} +3.03913 q^{31} +(-0.361831 - 0.361831i) q^{35} +(1.17130 - 1.17130i) q^{37} -11.9119i q^{41} +(-6.52641 + 6.52641i) q^{43} -5.91895 q^{47} +6.73816 q^{49} +(6.44018 - 6.44018i) q^{53} +1.06282i q^{55} +(7.60826 - 7.60826i) q^{59} +(-2.76756 - 2.76756i) q^{61} -5.97966 q^{65} +(-4.94156 - 4.94156i) q^{67} -9.80855i q^{71} -5.10381i q^{73} +(0.384562 + 0.384562i) q^{77} +6.92723 q^{79} +(-8.04926 - 8.04926i) q^{83} +(3.34976 - 3.34976i) q^{85} -2.41356i q^{89} +(-2.16363 + 2.16363i) q^{91} +6.76403 q^{95} -4.51711 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 8 q^{19} - 32 q^{37} - 16 q^{43} - 32 q^{49} - 16 q^{61} + 16 q^{67} + 16 q^{79} + 16 q^{85} + 80 q^{91}+O(q^{100}) \) 32 * q - 8 * q^19 - 32 * q^37 - 16 * q^43 - 32 * q^49 - 16 * q^61 + 16 * q^67 + 16 * q^79 + 16 * q^85 + 80 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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