LMFDB - Embedded newform 2880.2.bl.c.1871.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(431,2880)]

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([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("2880.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.bl (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			1871.3
Character	$\chi$	$=$	2880.1871
Dual form			2880.2.bl.c.431.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{5} -0.841567 q^{7} +O(q^{10})$	$q+(-0.707107 + 0.707107i) q^{5} -0.841567 q^{7} +(3.43874 + 3.43874i) q^{11} +(-0.690971 + 0.690971i) q^{13} +0.821521i q^{17} +(3.35742 + 3.35742i) q^{19} +0.473453i q^{23} -1.00000i q^{25} +(-0.820301 - 0.820301i) q^{29} -2.43031i q^{31} +(0.595078 - 0.595078i) q^{35} +(-6.22715 - 6.22715i) q^{37} -3.45570 q^{41} +(-3.26189 + 3.26189i) q^{43} +0.936881 q^{47} -6.29176 q^{49} +(6.73154 - 6.73154i) q^{53} -4.86311 q^{55} +(9.53265 + 9.53265i) q^{59} +(-6.32938 + 6.32938i) q^{61} -0.977181i q^{65} +(-2.77712 - 2.77712i) q^{67} +10.4924i q^{71} +6.65631i q^{73} +(-2.89393 - 2.89393i) q^{77} +15.9843i q^{79} +(-3.05663 + 3.05663i) q^{83} +(-0.580903 - 0.580903i) q^{85} +1.62307 q^{89} +(0.581499 - 0.581499i) q^{91} -4.74810 q^{95} -13.7207 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 8 q^{19} - 32 q^{49} + 16 q^{55} + 16 q^{61} - 16 q^{67} + 16 q^{85} + 16 q^{91}+O(q^{100}) $ 32 * q + 8 * q^19 - 32 * q^49 + 16 * q^55 + 16 * q^61 - 16 * q^67 + 16 * q^85 + 16 * q^91

Display 100 coefficients 10 coefficients

Character values

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

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

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
$7$	−0.841567			−0.318082			−0.159041	−	0.987272i	$-0.550840\pi$
$7$	−0.841567			−0.318082			−0.159041	+	0.987272i	$0.550840\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.43874	+	3.43874i	1.03682	+	1.03682i	0.999296	+	0.0375218i	$0.0119464\pi$
$11$	3.43874	+	3.43874i	1.03682	+	1.03682i	0.0375218	+	0.999296i	$0.488054\pi$
$12$		0			0
$13$	−0.690971	+	0.690971i	−0.191641	+	0.191641i	−0.796405	−	0.604764i	$-0.793267\pi$
$13$	−0.690971	+	0.690971i	−0.191641	+	0.191641i	0.604764	+	0.796405i	$0.293267\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.821521i			0.199248i	0.995025	+	0.0996240i	$0.0317640\pi$
$17$			0.821521i			0.199248i	−0.995025	+	0.0996240i	$0.968236\pi$
$18$		0			0
$19$	3.35742	+	3.35742i	0.770244	+	0.770244i	0.978149	−	0.207905i	$-0.0666645\pi$
$19$	3.35742	+	3.35742i	0.770244	+	0.770244i	−0.207905	+	0.978149i	$0.566665\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			0.473453i			0.0987218i	0.998781	+	0.0493609i	$0.0157185\pi$
$23$			0.473453i			0.0987218i	−0.998781	+	0.0493609i	$0.984282\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.820301	−	0.820301i	−0.152326	−	0.152326i	0.626830	−	0.779156i	$-0.284352\pi$
$29$	−0.820301	−	0.820301i	−0.152326	−	0.152326i	−0.779156	+	0.626830i	$0.784352\pi$
$30$		0			0
$31$		−	2.43031i		−	0.436496i	−0.975893	−	0.218248i	$-0.929966\pi$
$31$		−	2.43031i		−	0.436496i	0.975893	−	0.218248i	$-0.0700342\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.595078	−	0.595078i	0.100587	−	0.100587i
$36$		0			0
$37$	−6.22715	−	6.22715i	−1.02374	−	1.02374i	−0.999711	−	0.0240256i	$-0.992352\pi$
$37$	−6.22715	−	6.22715i	−1.02374	−	1.02374i	−0.0240256	−	0.999711i	$-0.507648\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.45570			−0.539689			−0.269845	−	0.962904i	$-0.586972\pi$
$41$	−3.45570			−0.539689			−0.269845	+	0.962904i	$0.586972\pi$
$42$		0			0
$43$	−3.26189	+	3.26189i	−0.497434	+	0.497434i	−0.910638	−	0.413205i	$-0.864409\pi$
$43$	−3.26189	+	3.26189i	−0.497434	+	0.497434i	0.413205	+	0.910638i	$0.364409\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.936881			0.136658			0.0683291	−	0.997663i	$-0.478233\pi$
$47$	0.936881			0.136658			0.0683291	+	0.997663i	$0.478233\pi$
$48$		0			0
$49$	−6.29176			−0.898824
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.73154	−	6.73154i	0.924649	−	0.924649i	−0.0727049	−	0.997353i	$-0.523163\pi$
$53$	6.73154	−	6.73154i	0.924649	−	0.924649i	0.997353	+	0.0727049i	$0.0231631\pi$
$54$		0			0
$55$	−4.86311			−0.655741
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.53265	+	9.53265i	1.24105	+	1.24105i	0.959568	+	0.281478i	$0.0908247\pi$
$59$	9.53265	+	9.53265i	1.24105	+	1.24105i	0.281478	+	0.959568i	$0.409175\pi$
$60$		0			0
$61$	−6.32938	+	6.32938i	−0.810394	+	0.810394i	−0.984693	−	0.174299i	$-0.944234\pi$
$61$	−6.32938	+	6.32938i	−0.810394	+	0.810394i	0.174299	+	0.984693i	$0.444234\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	0.977181i		−	0.121204i
$66$		0			0
$67$	−2.77712	−	2.77712i	−0.339279	−	0.339279i	0.516817	−	0.856096i	$-0.327117\pi$
$67$	−2.77712	−	2.77712i	−0.339279	−	0.339279i	−0.856096	+	0.516817i	$0.827117\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			10.4924i			1.24522i	0.782534	+	0.622608i	$0.213927\pi$
$71$			10.4924i			1.24522i	−0.782534	+	0.622608i	$0.786073\pi$
$72$		0			0
$73$			6.65631i			0.779062i	0.921013	+	0.389531i	$0.127363\pi$
$73$			6.65631i			0.779062i	−0.921013	+	0.389531i	$0.872637\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.89393	−	2.89393i	−0.329794	−	0.329794i
$78$		0			0
$79$			15.9843i			1.79837i	0.437570	+	0.899185i	$0.355839\pi$
$79$			15.9843i			1.79837i	−0.437570	+	0.899185i	$0.644161\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.05663	+	3.05663i	−0.335509	+	0.335509i	−0.854674	−	0.519165i	$-0.826243\pi$
$83$	−3.05663	+	3.05663i	−0.335509	+	0.335509i	0.519165	+	0.854674i	$0.326243\pi$
$84$		0			0
$85$	−0.580903	−	0.580903i	−0.0630078	−	0.0630078i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.62307			0.172045			0.0860225	−	0.996293i	$-0.472584\pi$
$89$	1.62307			0.172045			0.0860225	+	0.996293i	$0.472584\pi$
$90$		0			0
$91$	0.581499	−	0.581499i	0.0609576	−	0.0609576i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.74810			−0.487145
$96$		0			0
$97$	−13.7207			−1.39313			−0.696564	−	0.717494i	$-0.745289\pi$
$97$	−13.7207			−1.39313			−0.696564	+	0.717494i	$0.745289\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	10.8444	−	10.8444i	1.07905	−	1.07905i	0.0824589	−	0.996594i	$-0.473723\pi$
$101$	10.8444	−	10.8444i	1.07905	−	1.07905i	0.996594	−	0.0824589i	$-0.0262773\pi$
$102$		0			0
$103$	−1.94638			−0.191783			−0.0958913	−	0.995392i	$-0.530570\pi$
$103$	−1.94638			−0.191783			−0.0958913	+	0.995392i	$0.530570\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.32616	+	6.32616i	0.611573	+	0.611573i	0.943356	−	0.331783i	$-0.107650\pi$
$107$	6.32616	+	6.32616i	0.611573	+	0.611573i	−0.331783	+	0.943356i	$0.607650\pi$
$108$		0			0
$109$	0.780964	−	0.780964i	0.0748028	−	0.0748028i	−0.668716	−	0.743518i	$-0.733156\pi$
$109$	0.780964	−	0.780964i	0.0748028	−	0.0748028i	0.743518	+	0.668716i	$0.233156\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			21.1016i			1.98507i	0.121960	+	0.992535i	$0.461082\pi$
$113$			21.1016i			1.98507i	−0.121960	+	0.992535i	$0.538918\pi$
$114$		0			0
$115$	−0.334782	−	0.334782i	−0.0312186	−	0.0312186i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	0.691365i		−	0.0633773i
$120$		0			0
$121$			12.6498i			1.14998i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	+	0.707107i	0.0632456	+	0.0632456i
$126$		0			0
$127$			2.26053i			0.200590i	0.994958	+	0.100295i	$0.0319787\pi$
$127$			2.26053i			0.200590i	−0.994958	+	0.100295i	$0.968021\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−9.12372	+	9.12372i	−0.797143	+	0.797143i	−0.982644	−	0.185501i	$-0.940609\pi$
$131$	−9.12372	+	9.12372i	−0.797143	+	0.797143i	0.185501	+	0.982644i	$0.440609\pi$
$132$		0			0
$133$	−2.82549	−	2.82549i	−0.245001	−	0.245001i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.04620			−0.687433			−0.343717	−	0.939073i	$-0.611686\pi$
$137$	−8.04620			−0.687433			−0.343717	+	0.939073i	$0.611686\pi$
$138$		0			0
$139$	12.0739	−	12.0739i	1.02409	−	1.02409i	0.0243919	−	0.999702i	$-0.492235\pi$
$139$	12.0739	−	12.0739i	1.02409	−	1.02409i	0.999702	−	0.0243919i	$-0.00776495\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.75214			−0.397394
$144$		0			0
$145$	1.16008			0.0963395
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.2467	+	11.2467i	−0.921367	+	0.921367i	−0.997126	−	0.0757593i	$-0.975862\pi$
$149$	−11.2467	+	11.2467i	−0.921367	+	0.921367i	0.0757593	+	0.997126i	$0.475862\pi$
$150$		0			0
$151$	−9.01384			−0.733536			−0.366768	−	0.930313i	$-0.619536\pi$
$151$	−9.01384			−0.733536			−0.366768	+	0.930313i	$0.619536\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.71849	+	1.71849i	0.138032	+	0.138032i
$156$		0			0
$157$	−14.2480	+	14.2480i	−1.13711	+	1.13711i	−0.148149	+	0.988965i	$0.547332\pi$
$157$	−14.2480	+	14.2480i	−1.13711	+	1.13711i	−0.988965	+	0.148149i	$0.952668\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	0.398443i		−	0.0314017i
$162$		0			0
$163$	−4.00165	−	4.00165i	−0.313433	−	0.313433i	0.532805	−	0.846238i	$-0.321138\pi$
$163$	−4.00165	−	4.00165i	−0.313433	−	0.313433i	−0.846238	+	0.532805i	$0.821138\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	15.0907i		−	1.16775i	−0.811843	−	0.583876i	$-0.801536\pi$
$167$		−	15.0907i		−	1.16775i	0.811843	−	0.583876i	$-0.198464\pi$
$168$		0			0
$169$			12.0451i			0.926547i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	15.4229	+	15.4229i	1.17258	+	1.17258i	0.981592	+	0.190988i	$0.0611692\pi$
$173$	15.4229	+	15.4229i	1.17258	+	1.17258i	0.190988	+	0.981592i	$0.438831\pi$
$174$		0			0
$175$			0.841567i			0.0636165i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.528146	−	0.528146i	0.0394755	−	0.0394755i	−0.687093	−	0.726569i	$-0.741114\pi$
$179$	0.528146	−	0.528146i	0.0394755	−	0.0394755i	0.726569	+	0.687093i	$0.241114\pi$
$180$		0			0
$181$	4.34505	+	4.34505i	0.322965	+	0.322965i	0.849904	−	0.526938i	$-0.176660\pi$
$181$	4.34505	+	4.34505i	0.322965	+	0.322965i	−0.526938	+	0.849904i	$0.676660\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.80652			0.647468
$186$		0			0
$187$	−2.82499	+	2.82499i	−0.206584	+	0.206584i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.39873			−0.462996			−0.231498	−	0.972835i	$-0.574363\pi$
$191$	−6.39873			−0.462996			−0.231498	+	0.972835i	$0.574363\pi$
$192$		0			0
$193$	12.7016			0.914279			0.457140	−	0.889395i	$-0.348874\pi$
$193$	12.7016			0.914279			0.457140	+	0.889395i	$0.348874\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	14.3378	−	14.3378i	1.02152	−	1.02152i	0.0217606	−	0.999763i	$-0.493073\pi$
$197$	14.3378	−	14.3378i	1.02152	−	1.02152i	0.999763	−	0.0217606i	$-0.00692715\pi$
$198$		0			0
$199$	−17.4221			−1.23502			−0.617510	−	0.786563i	$-0.711859\pi$
$199$	−17.4221			−1.23502			−0.617510	+	0.786563i	$0.711859\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.690339	+	0.690339i	0.0484523	+	0.0484523i
$204$		0			0
$205$	2.44355	−	2.44355i	0.170665	−	0.170665i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			23.0905i			1.59720i
$210$		0			0
$211$	−15.1953	−	15.1953i	−1.04609	−	1.04609i	−0.998885	−	0.0472026i	$-0.984969\pi$
$211$	−15.1953	−	15.1953i	−1.04609	−	1.04609i	−0.0472026	−	0.998885i	$-0.515031\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	4.61301i		−	0.314605i
$216$		0			0
$217$			2.04527i			0.138842i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.567647	−	0.567647i	−0.0381841	−	0.0381841i
$222$		0			0
$223$			11.7905i			0.789552i	0.918777	+	0.394776i	$0.129178\pi$
$223$			11.7905i			0.789552i	−0.918777	+	0.394776i	$0.870822\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	19.0467	−	19.0467i	1.26418	−	1.26418i	0.315127	−	0.949050i	$-0.397953\pi$
$227$	19.0467	−	19.0467i	1.26418	−	1.26418i	0.949050	−	0.315127i	$-0.102047\pi$
$228$		0			0
$229$	−12.3779	−	12.3779i	−0.817958	−	0.817958i	0.167854	−	0.985812i	$-0.446316\pi$
$229$	−12.3779	−	12.3779i	−0.817958	−	0.817958i	−0.985812	+	0.167854i	$0.946316\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−15.6667			−1.02636			−0.513179	−	0.858282i	$-0.671532\pi$
$233$	−15.6667			−1.02636			−0.513179	+	0.858282i	$0.671532\pi$
$234$		0			0
$235$	−0.662475	+	0.662475i	−0.0432151	+	0.0432151i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−14.8657			−0.961585			−0.480792	−	0.876834i	$-0.659651\pi$
$239$	−14.8657			−0.961585			−0.480792	+	0.876834i	$0.659651\pi$
$240$		0			0
$241$	15.2617			0.983094			0.491547	−	0.870851i	$-0.336432\pi$
$241$	15.2617			0.983094			0.491547	+	0.870851i	$0.336432\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.44895	−	4.44895i	0.284233	−	0.284233i
$246$		0			0
$247$	−4.63976			−0.295221
$248$		0			0
$249$		0			0
$250$		0			0
$251$	17.7371	+	17.7371i	1.11955	+	1.11955i	0.991807	+	0.127747i	$0.0407747\pi$
$251$	17.7371	+	17.7371i	1.11955	+	1.11955i	0.127747	+	0.991807i	$0.459225\pi$
$252$		0			0
$253$	−1.62808	+	1.62808i	−0.102357	+	0.102357i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	4.38966i		−	0.273819i	−0.990584	−	0.136910i	$-0.956283\pi$
$257$		−	4.38966i		−	0.273819i	0.990584	−	0.136910i	$-0.0437170\pi$
$258$		0			0
$259$	5.24056	+	5.24056i	0.325633	+	0.325633i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	14.5203i		−	0.895363i	−0.894193	−	0.447681i	$-0.852250\pi$
$263$		−	14.5203i		−	0.895363i	0.894193	−	0.447681i	$-0.147750\pi$
$264$		0			0
$265$			9.51984i			0.584799i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.50032	+	6.50032i	0.396332	+	0.396332i	0.876937	−	0.480605i	$-0.159583\pi$
$269$	6.50032	+	6.50032i	0.396332	+	0.396332i	−0.480605	+	0.876937i	$0.659583\pi$
$270$		0			0
$271$			1.92906i			0.117182i	0.998282	+	0.0585910i	$0.0186608\pi$
$271$			1.92906i			0.117182i	−0.998282	+	0.0585910i	$0.981339\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.43874	−	3.43874i	0.207364	−	0.207364i
$276$		0			0
$277$	−22.6271	−	22.6271i	−1.35953	−	1.35953i	−0.874496	−	0.485033i	$-0.838808\pi$
$277$	−22.6271	−	22.6271i	−1.35953	−	1.35953i	−0.485033	−	0.874496i	$-0.661192\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−11.1584			−0.665657			−0.332828	−	0.942987i	$-0.608003\pi$
$281$	−11.1584			−0.665657			−0.332828	+	0.942987i	$0.608003\pi$
$282$		0			0
$283$	−7.00055	+	7.00055i	−0.416140	+	0.416140i	−0.883871	−	0.467731i	$-0.845072\pi$
$283$	−7.00055	+	7.00055i	−0.416140	+	0.416140i	0.467731	+	0.883871i	$0.345072\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.90820			0.171666
$288$		0			0
$289$	16.3251			0.960300
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.66244	+	2.66244i	−0.155541	+	0.155541i	−0.780588	−	0.625046i	$-0.785080\pi$
$293$	−2.66244	+	2.66244i	−0.155541	+	0.155541i	0.625046	+	0.780588i	$0.285080\pi$
$294$		0			0
$295$	−13.4812			−0.784906
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.327143	−	0.327143i	−0.0189191	−	0.0189191i
$300$		0			0
$301$	2.74510	−	2.74510i	0.158225	−	0.158225i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	8.95110i		−	0.512538i
$306$		0			0
$307$	16.4971	+	16.4971i	0.941540	+	0.941540i	0.998383	−	0.0568432i	$-0.0181035\pi$
$307$	16.4971	+	16.4971i	0.941540	+	0.941540i	−0.0568432	+	0.998383i	$0.518103\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	13.6245i		−	0.772577i	−0.922378	−	0.386288i	$-0.873757\pi$
$311$		−	13.6245i		−	0.772577i	0.922378	−	0.386288i	$-0.126243\pi$
$312$		0			0
$313$		−	29.1568i		−	1.64804i	−0.566561	−	0.824020i	$-0.691726\pi$
$313$		−	29.1568i		−	1.64804i	0.566561	−	0.824020i	$-0.308274\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.45854	+	6.45854i	0.362748	+	0.362748i	0.864824	−	0.502076i	$-0.167430\pi$
$317$	6.45854	+	6.45854i	0.362748	+	0.362748i	−0.502076	+	0.864824i	$0.667430\pi$
$318$		0			0
$319$		−	5.64160i		−	0.315869i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.75819	+	2.75819i	−0.153470	+	0.153470i
$324$		0			0
$325$	0.690971	+	0.690971i	0.0383282	+	0.0383282i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.788448			−0.0434686
$330$		0			0
$331$	0.0499803	−	0.0499803i	0.00274716	−	0.00274716i	−0.705732	−	0.708479i	$-0.749382\pi$
$331$	0.0499803	−	0.0499803i	0.00274716	−	0.00274716i	0.708479	+	0.705732i	$0.249382\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.92744			0.214579
$336$		0			0
$337$	11.5272			0.627929			0.313964	−	0.949435i	$-0.398343\pi$
$337$	11.5272			0.627929			0.313964	+	0.949435i	$0.398343\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8.35718	−	8.35718i	0.452567	−	0.452567i
$342$		0			0
$343$	11.1859			0.603982
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.992679	−	0.992679i	−0.0532898	−	0.0532898i	0.679960	−	0.733249i	$-0.261997\pi$
$347$	−0.992679	−	0.992679i	−0.0532898	−	0.0532898i	−0.733249	+	0.679960i	$0.761997\pi$
$348$		0			0
$349$	15.4778	−	15.4778i	0.828507	−	0.828507i	−0.158804	−	0.987310i	$-0.550764\pi$
$349$	15.4778	−	15.4778i	0.828507	−	0.828507i	0.987310	+	0.158804i	$0.0507637\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			18.6931i			0.994933i	0.867483	+	0.497466i	$0.165736\pi$
$353$			18.6931i			0.994933i	−0.867483	+	0.497466i	$0.834264\pi$
$354$		0			0
$355$	−7.41923	−	7.41923i	−0.393772	−	0.393772i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			28.2960i			1.49341i	0.665157	+	0.746704i	$0.268365\pi$
$359$			28.2960i			1.49341i	−0.665157	+	0.746704i	$0.731635\pi$
$360$		0			0
$361$			3.54448i			0.186551i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.70672	−	4.70672i	−0.246361	−	0.246361i
$366$		0			0
$367$			21.2027i			1.10677i	0.832924	+	0.553387i	$0.186665\pi$
$367$			21.2027i			1.10677i	−0.832924	+	0.553387i	$0.813335\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.66505	+	5.66505i	−0.294115	+	0.294115i
$372$		0			0
$373$	5.99827	+	5.99827i	0.310579	+	0.310579i	0.845134	−	0.534555i	$-0.179521\pi$
$373$	5.99827	+	5.99827i	0.310579	+	0.310579i	−0.534555	+	0.845134i	$0.679521\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.13361			0.0583839
$378$		0			0
$379$	13.8697	−	13.8697i	0.712437	−	0.712437i	−0.254607	−	0.967044i	$-0.581946\pi$
$379$	13.8697	−	13.8697i	0.712437	−	0.712437i	0.967044	+	0.254607i	$0.0819463\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−14.1151			−0.721249			−0.360624	−	0.932711i	$-0.617436\pi$
$383$	−14.1151			−0.721249			−0.360624	+	0.932711i	$0.617436\pi$
$384$		0			0
$385$	4.09263			0.208580
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.95205	+	3.95205i	−0.200377	+	0.200377i	−0.800161	−	0.599785i	$-0.795253\pi$
$389$	−3.95205	+	3.95205i	−0.200377	+	0.200377i	0.599785	+	0.800161i	$0.295253\pi$
$390$		0			0
$391$	−0.388952			−0.0196701
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11.3026	−	11.3026i	−0.568694	−	0.568694i
$396$		0			0
$397$	3.74520	−	3.74520i	0.187966	−	0.187966i	−0.606850	−	0.794816i	$-0.707567\pi$
$397$	3.74520	−	3.74520i	0.187966	−	0.187966i	0.794816	+	0.606850i	$0.207567\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			11.9464i			0.596575i	0.954476	+	0.298288i	$0.0964154\pi$
$401$			11.9464i			0.596575i	−0.954476	+	0.298288i	$0.903585\pi$
$402$		0			0
$403$	1.67927	+	1.67927i	0.0836505	+	0.0836505i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	42.8270i		−	2.12286i
$408$		0			0
$409$			13.9982i			0.692167i	0.938204	+	0.346083i	$0.112489\pi$
$409$			13.9982i			0.692167i	−0.938204	+	0.346083i	$0.887511\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8.02237	−	8.02237i	−0.394755	−	0.394755i
$414$		0			0
$415$		−	4.32273i		−	0.212195i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−9.62674	+	9.62674i	−0.470297	+	0.470297i	−0.902011	−	0.431714i	$-0.857909\pi$
$419$	−9.62674	+	9.62674i	−0.470297	+	0.470297i	0.431714	+	0.902011i	$0.357909\pi$
$420$		0			0
$421$	9.57727	+	9.57727i	0.466767	+	0.466767i	0.900866	−	0.434098i	$-0.142933\pi$
$421$	9.57727	+	9.57727i	0.466767	+	0.466767i	−0.434098	+	0.900866i	$0.642933\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.821521			0.0398496
$426$		0			0
$427$	5.32660	−	5.32660i	0.257772	−	0.257772i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.22660			−0.203588			−0.101794	−	0.994805i	$-0.532458\pi$
$431$	−4.22660			−0.203588			−0.101794	+	0.994805i	$0.532458\pi$
$432$		0			0
$433$	−5.38743			−0.258903			−0.129452	−	0.991586i	$-0.541322\pi$
$433$	−5.38743			−0.258903			−0.129452	+	0.991586i	$0.541322\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.58958	+	1.58958i	−0.0760399	+	0.0760399i
$438$		0			0
$439$	10.5436			0.503220			0.251610	−	0.967829i	$-0.419040\pi$
$439$	10.5436			0.503220			0.251610	+	0.967829i	$0.419040\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	26.2397	+	26.2397i	1.24669	+	1.24669i	0.957174	+	0.289513i	$0.0934935\pi$
$443$	26.2397	+	26.2397i	1.24669	+	1.24669i	0.289513	+	0.957174i	$0.406507\pi$
$444$		0			0
$445$	−1.14768	+	1.14768i	−0.0544054	+	0.0544054i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			25.3732i			1.19744i	0.800960	+	0.598718i	$0.204323\pi$
$449$			25.3732i			1.19744i	−0.800960	+	0.598718i	$0.795677\pi$
$450$		0			0
$451$	−11.8832	−	11.8832i	−0.559559	−	0.559559i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			0.822364i			0.0385530i
$456$		0			0
$457$		−	24.7233i		−	1.15651i	−0.815858	−	0.578253i	$-0.803735\pi$
$457$		−	24.7233i		−	1.15651i	0.815858	−	0.578253i	$-0.196265\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8.97177	−	8.97177i	−0.417857	−	0.417857i	0.466608	−	0.884464i	$-0.345476\pi$
$461$	−8.97177	−	8.97177i	−0.417857	−	0.417857i	−0.884464	+	0.466608i	$0.845476\pi$
$462$		0			0
$463$			17.6715i			0.821264i	0.911801	+	0.410632i	$0.134692\pi$
$463$			17.6715i			0.821264i	−0.911801	+	0.410632i	$0.865308\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	18.9454	−	18.9454i	0.876688	−	0.876688i	−0.116503	−	0.993190i	$-0.537168\pi$
$467$	18.9454	−	18.9454i	0.876688	−	0.876688i	0.993190	+	0.116503i	$0.0371684\pi$
$468$		0			0
$469$	2.33713	+	2.33713i	0.107919	+	0.107919i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−22.4336			−1.03150
$474$		0			0
$475$	3.35742	−	3.35742i	0.154049	−	0.154049i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.71437			0.169714			0.0848569	−	0.996393i	$-0.472957\pi$
$479$	3.71437			0.169714			0.0848569	+	0.996393i	$0.472957\pi$
$480$		0			0
$481$	8.60556			0.392380
$482$		0			0
$483$		0			0
$484$		0			0
$485$	9.70202	−	9.70202i	0.440546	−	0.440546i
$486$		0			0
$487$	−18.2673			−0.827769			−0.413885	−	0.910329i	$-0.635828\pi$
$487$	−18.2673			−0.827769			−0.413885	+	0.910329i	$0.635828\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.61901	−	6.61901i	−0.298712	−	0.298712i	0.541797	−	0.840509i	$-0.317744\pi$
$491$	−6.61901	−	6.61901i	−0.298712	−	0.298712i	−0.840509	+	0.541797i	$0.817744\pi$
$492$		0			0
$493$	0.673895	−	0.673895i	0.0303507	−	0.0303507i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	8.83004i		−	0.396081i
$498$		0			0
$499$	29.6389	+	29.6389i	1.32682	+	1.32682i	0.908129	+	0.418691i	$0.137511\pi$
$499$	29.6389	+	29.6389i	1.32682	+	1.32682i	0.418691	+	0.908129i	$0.362489\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	30.2450i		−	1.34856i	−0.738476	−	0.674280i	$-0.764454\pi$
$503$		−	30.2450i		−	1.34856i	0.738476	−	0.674280i	$-0.235546\pi$
$504$		0			0
$505$			15.3362i			0.682453i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12.9669	−	12.9669i	−0.574748	−	0.574748i	0.358703	−	0.933452i	$-0.383219\pi$
$509$	−12.9669	−	12.9669i	−0.574748	−	0.574748i	−0.933452	+	0.358703i	$0.883219\pi$
$510$		0			0
$511$		−	5.60173i		−	0.247806i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.37630	−	1.37630i	0.0606470	−	0.0606470i
$516$		0			0
$517$	3.22169	+	3.22169i	0.141690	+	0.141690i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	20.6410			0.904300			0.452150	−	0.891942i	$-0.350657\pi$
$521$	20.6410			0.904300			0.452150	+	0.891942i	$0.350657\pi$
$522$		0			0
$523$	16.0705	−	16.0705i	0.702715	−	0.702715i	−0.262278	−	0.964992i	$-0.584474\pi$
$523$	16.0705	−	16.0705i	0.702715	−	0.702715i	0.964992	+	0.262278i	$0.0844737\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.99655			0.0869710
$528$		0			0
$529$	22.7758			0.990254
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.38779	−	2.38779i	0.103427	−	0.103427i
$534$		0			0
$535$	−8.94654			−0.386793
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−21.6357	−	21.6357i	−0.931916	−	0.931916i
$540$		0			0
$541$	21.6485	−	21.6485i	0.930744	−	0.930744i	−0.0670086	−	0.997752i	$-0.521346\pi$
$541$	21.6485	−	21.6485i	0.930744	−	0.930744i	0.997752	+	0.0670086i	$0.0213455\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			1.10445i			0.0473094i
$546$		0			0
$547$	−22.9542	−	22.9542i	−0.981450	−	0.981450i	0.0183808	−	0.999831i	$-0.494149\pi$
$547$	−22.9542	−	22.9542i	−0.981450	−	0.981450i	−0.999831	+	0.0183808i	$0.994149\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	5.50818i		−	0.234657i
$552$		0			0
$553$		−	13.4518i		−	0.572030i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−21.3924	−	21.3924i	−0.906424	−	0.906424i	0.0895579	−	0.995982i	$-0.471455\pi$
$557$	−21.3924	−	21.3924i	−0.906424	−	0.906424i	−0.995982	+	0.0895579i	$0.971455\pi$
$558$		0			0
$559$		−	4.50775i		−	0.190657i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−11.6919	+	11.6919i	−0.492755	+	0.492755i	−0.909173	−	0.416418i	$-0.863285\pi$
$563$	−11.6919	+	11.6919i	−0.492755	+	0.492755i	0.416418	+	0.909173i	$0.363285\pi$
$564$		0			0
$565$	−14.9211	−	14.9211i	−0.627734	−	0.627734i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−13.1233			−0.550156			−0.275078	−	0.961422i	$-0.588704\pi$
$569$	−13.1233			−0.550156			−0.275078	+	0.961422i	$0.588704\pi$
$570$		0			0
$571$	−2.01503	+	2.01503i	−0.0843262	+	0.0843262i	−0.748012	−	0.663686i	$-0.768991\pi$
$571$	−2.01503	+	2.01503i	−0.0843262	+	0.0843262i	0.663686	+	0.748012i	$0.268991\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.473453			0.0197444
$576$		0			0
$577$	−42.3102			−1.76140			−0.880699	−	0.473677i	$-0.842926\pi$
$577$	−42.3102			−1.76140			−0.880699	+	0.473677i	$0.842926\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.57236	−	2.57236i	0.106720	−	0.106720i
$582$		0			0
$583$	46.2960			1.91738
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−22.8534	−	22.8534i	−0.943259	−	0.943259i	0.0552157	−	0.998474i	$-0.482415\pi$
$587$	−22.8534	−	22.8534i	−0.943259	−	0.943259i	−0.998474	+	0.0552157i	$0.982415\pi$
$588$		0			0
$589$	8.15955	−	8.15955i	0.336208	−	0.336208i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	46.8333i		−	1.92321i	−0.274431	−	0.961607i	$-0.588490\pi$
$593$		−	46.8333i		−	1.92321i	0.274431	−	0.961607i	$-0.411510\pi$
$594$		0			0
$595$	0.488869	+	0.488869i	0.0200417	+	0.0200417i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			34.7811i			1.42112i	0.703637	+	0.710559i	$0.251558\pi$
$599$			34.7811i			1.42112i	−0.703637	+	0.710559i	$0.748442\pi$
$600$		0			0
$601$		−	26.9860i		−	1.10078i	−0.834907	−	0.550392i	$-0.814478\pi$
$601$		−	26.9860i		−	1.10078i	0.834907	−	0.550392i	$-0.185522\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−8.94476	−	8.94476i	−0.363656	−	0.363656i
$606$		0			0
$607$		−	35.0694i		−	1.42342i	−0.702472	−	0.711712i	$-0.747920\pi$
$607$		−	35.0694i		−	1.42342i	0.702472	−	0.711712i	$-0.252080\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.647358	+	0.647358i	−0.0261893	+	0.0261893i
$612$		0			0
$613$	27.9876	+	27.9876i	1.13041	+	1.13041i	0.990109	+	0.140302i	$0.0448072\pi$
$613$	27.9876	+	27.9876i	1.13041	+	1.13041i	0.140302	+	0.990109i	$0.455193\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−18.7144			−0.753415			−0.376708	−	0.926332i	$-0.622944\pi$
$617$	−18.7144			−0.753415			−0.376708	+	0.926332i	$0.622944\pi$
$618$		0			0
$619$	19.7167	−	19.7167i	0.792483	−	0.792483i	−0.189414	−	0.981897i	$-0.560659\pi$
$619$	19.7167	−	19.7167i	0.792483	−	0.792483i	0.981897	+	0.189414i	$0.0606589\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.36592			−0.0547245
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.11573	−	5.11573i	0.203978	−	0.203978i
$630$		0			0
$631$	8.42943			0.335570			0.167785	−	0.985824i	$-0.446339\pi$
$631$	8.42943			0.335570			0.167785	+	0.985824i	$0.446339\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.59844	−	1.59844i	−0.0634321	−	0.0634321i
$636$		0			0
$637$	4.34743	−	4.34743i	0.172251	−	0.172251i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	31.5099i		−	1.24457i	−0.782793	−	0.622283i	$-0.786205\pi$
$641$		−	31.5099i		−	1.24457i	0.782793	−	0.622283i	$-0.213795\pi$
$642$		0			0
$643$	−8.32899	−	8.32899i	−0.328463	−	0.328463i	0.523539	−	0.852002i	$-0.324612\pi$
$643$	−8.32899	−	8.32899i	−0.328463	−	0.328463i	−0.852002	+	0.523539i	$0.824612\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	23.0627i		−	0.906690i	−0.891335	−	0.453345i	$-0.850231\pi$
$647$		−	23.0627i		−	0.906690i	0.891335	−	0.453345i	$-0.149769\pi$
$648$		0			0
$649$			65.5605i			2.57348i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.20182	+	5.20182i	0.203563	+	0.203563i	0.801525	−	0.597962i	$-0.204023\pi$
$653$	5.20182	+	5.20182i	0.203563	+	0.203563i	−0.597962	+	0.801525i	$0.704023\pi$
$654$		0			0
$655$		−	12.9029i		−	0.504157i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	13.6928	−	13.6928i	0.533394	−	0.533394i	−0.388187	−	0.921581i	$-0.626898\pi$
$659$	13.6928	−	13.6928i	0.533394	−	0.533394i	0.921581	+	0.388187i	$0.126898\pi$
$660$		0			0
$661$	18.5618	+	18.5618i	0.721971	+	0.721971i	0.969006	−	0.247035i	$-0.0794564\pi$
$661$	18.5618	+	18.5618i	0.721971	+	0.721971i	−0.247035	+	0.969006i	$0.579456\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.99585			0.154952
$666$		0			0
$667$	0.388374	−	0.388374i	0.0150379	−	0.0150379i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−43.5301			−1.68046
$672$		0			0
$673$	11.4250			0.440401			0.220200	−	0.975455i	$-0.429329\pi$
$673$	11.4250			0.440401			0.220200	+	0.975455i	$0.429329\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	21.2746	−	21.2746i	0.817648	−	0.817648i	−0.168119	−	0.985767i	$-0.553769\pi$
$677$	21.2746	−	21.2746i	0.817648	−	0.817648i	0.985767	+	0.168119i	$0.0537693\pi$
$678$		0			0
$679$	11.5469			0.443130
$680$		0			0
$681$		0			0
$682$		0			0
$683$	25.4104	+	25.4104i	0.972301	+	0.972301i	0.999627	−	0.0273256i	$-0.00869910\pi$
$683$	25.4104	+	25.4104i	0.972301	+	0.972301i	−0.0273256	+	0.999627i	$0.508699\pi$
$684$		0			0
$685$	5.68952	−	5.68952i	0.217385	−	0.217385i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			9.30261i			0.354401i
$690$		0			0
$691$	23.1775	+	23.1775i	0.881713	+	0.881713i	0.993709	−	0.111996i	$-0.0357244\pi$
$691$	23.1775	+	23.1775i	0.881713	+	0.881713i	−0.111996	+	0.993709i	$0.535724\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			17.0751i			0.647694i
$696$		0			0
$697$		−	2.83893i		−	0.107532i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	24.0995	+	24.0995i	0.910226	+	0.910226i	0.996290	−	0.0860634i	$-0.0274288\pi$
$701$	24.0995	+	24.0995i	0.910226	+	0.910226i	−0.0860634	+	0.996290i	$0.527429\pi$
$702$		0			0
$703$		−	41.8143i		−	1.57705i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−9.12625	+	9.12625i	−0.343228	+	0.343228i
$708$		0			0
$709$	−22.2081	−	22.2081i	−0.834043	−	0.834043i	0.154024	−	0.988067i	$-0.450777\pi$
$709$	−22.2081	−	22.2081i	−0.834043	−	0.834043i	−0.988067	+	0.154024i	$0.950777\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.15064			0.0430917
$714$		0			0
$715$	3.36027	−	3.36027i	0.125667	−	0.125667i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	10.9990			0.410194			0.205097	−	0.978742i	$-0.434249\pi$
$719$	10.9990			0.410194			0.205097	+	0.978742i	$0.434249\pi$
$720$		0			0
$721$	1.63801			0.0610027
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.820301	+	0.820301i	−0.0304652	+	0.0304652i
$726$		0			0
$727$	38.9343			1.44399			0.721996	−	0.691898i	$-0.243225\pi$
$727$	38.9343			1.44399			0.721996	+	0.691898i	$0.243225\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.67971	−	2.67971i	−0.0991127	−	0.0991127i
$732$		0			0
$733$	−24.8069	+	24.8069i	−0.916262	+	0.916262i	−0.996755	−	0.0804930i	$-0.974351\pi$
$733$	−24.8069	+	24.8069i	−0.916262	+	0.916262i	0.0804930	+	0.996755i	$0.474351\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	19.0996i		−	0.703541i
$738$		0			0
$739$	30.3710	+	30.3710i	1.11722	+	1.11722i	0.992148	+	0.125067i	$0.0399146\pi$
$739$	30.3710	+	30.3710i	1.11722	+	1.11722i	0.125067	+	0.992148i	$0.460085\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			41.6117i			1.52659i	0.646052	+	0.763293i	$0.276419\pi$
$743$			41.6117i			1.52659i	−0.646052	+	0.763293i	$0.723581\pi$
$744$		0			0
$745$		−	15.9053i		−	0.582724i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.32389	−	5.32389i	−0.194531	−	0.194531i
$750$		0			0
$751$			1.91507i			0.0698820i	0.999389	+	0.0349410i	$0.0111243\pi$
$751$			1.91507i			0.0698820i	−0.999389	+	0.0349410i	$0.988876\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.37375	−	6.37375i	0.231964	−	0.231964i
$756$		0			0
$757$	−17.7086	−	17.7086i	−0.643631	−	0.643631i	0.307816	−	0.951446i	$-0.400402\pi$
$757$	−17.7086	−	17.7086i	−0.643631	−	0.643631i	−0.951446	+	0.307816i	$0.900402\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−25.1308			−0.910989			−0.455495	−	0.890239i	$-0.650538\pi$
$761$	−25.1308			−0.910989			−0.455495	+	0.890239i	$0.650538\pi$
$762$		0			0
$763$	−0.657233	+	0.657233i	−0.0237934	+	0.0237934i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−13.1736			−0.475671
$768$		0			0
$769$	38.4717			1.38732			0.693662	−	0.720300i	$-0.255996\pi$
$769$	38.4717			1.38732			0.693662	+	0.720300i	$0.255996\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.33101	+	6.33101i	−0.227710	+	0.227710i	−0.811736	−	0.584025i	$-0.801477\pi$
$773$	−6.33101	+	6.33101i	−0.227710	+	0.227710i	0.584025	+	0.811736i	$0.301477\pi$
$774$		0			0
$775$	−2.43031			−0.0872992
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−11.6022	−	11.6022i	−0.415692	−	0.415692i
$780$		0			0
$781$	−36.0805	+	36.0805i	−1.29106	+	1.29106i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	20.1497i		−	0.719174i
$786$		0			0
$787$	−19.0873	−	19.0873i	−0.680388	−	0.680388i	0.279700	−	0.960088i	$-0.409765\pi$
$787$	−19.0873	−	19.0873i	−0.680388	−	0.680388i	−0.960088	+	0.279700i	$0.909765\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	17.7584i		−	0.631416i
$792$		0			0
$793$		−	8.74684i		−	0.310609i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−10.3591	−	10.3591i	−0.366936	−	0.366936i	0.499422	−	0.866359i	$-0.333546\pi$
$797$	−10.3591	−	10.3591i	−0.366936	−	0.366936i	−0.866359	+	0.499422i	$0.833546\pi$
$798$		0			0
$799$			0.769667i			0.0272289i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−22.8893	+	22.8893i	−0.807745	+	0.807745i
$804$		0			0
$805$	0.281742	+	0.281742i	0.00993008	+	0.00993008i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	50.0727			1.76046			0.880231	−	0.474545i	$-0.157387\pi$
$809$	50.0727			1.76046			0.880231	+	0.474545i	$0.157387\pi$
$810$		0			0
$811$	6.70984	−	6.70984i	0.235614	−	0.235614i	−0.579417	−	0.815031i	$-0.696720\pi$
$811$	6.70984	−	6.70984i	0.235614	−	0.235614i	0.815031	+	0.579417i	$0.196720\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5.65919			0.198233
$816$		0			0
$817$	−21.9030			−0.766290
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.00596	−	2.00596i	0.0700086	−	0.0700086i	−0.671236	−	0.741244i	$-0.734236\pi$
$821$	2.00596	−	2.00596i	0.0700086	−	0.0700086i	0.741244	+	0.671236i	$0.234236\pi$
$822$		0			0
$823$	54.2229			1.89009			0.945045	−	0.326939i	$-0.106017\pi$
$823$	54.2229			1.89009			0.945045	+	0.326939i	$0.106017\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	7.72144	+	7.72144i	0.268501	+	0.268501i	0.828496	−	0.559995i	$-0.189197\pi$
$827$	7.72144	+	7.72144i	0.268501	+	0.268501i	−0.559995	+	0.828496i	$0.689197\pi$
$828$		0			0
$829$	38.1216	−	38.1216i	1.32402	−	1.32402i	0.413528	−	0.910491i	$-0.364296\pi$
$829$	38.1216	−	38.1216i	1.32402	−	1.32402i	0.910491	−	0.413528i	$-0.135704\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	5.16882i		−	0.179089i
$834$		0			0
$835$	10.6707	+	10.6707i	0.369275	+	0.369275i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			43.5906i			1.50491i	0.658642	+	0.752457i	$0.271131\pi$
$839$			43.5906i			1.50491i	−0.658642	+	0.752457i	$0.728869\pi$
$840$		0			0
$841$		−	27.6542i		−	0.953593i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.51718	−	8.51718i	−0.293000	−	0.293000i
$846$		0			0
$847$		−	10.6457i		−	0.365789i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.94826	−	2.94826i	0.101065	−	0.101065i
$852$		0			0
$853$	21.4142	+	21.4142i	0.733207	+	0.733207i	0.971254	−	0.238047i	$-0.0765072\pi$
$853$	21.4142	+	21.4142i	0.733207	+	0.733207i	−0.238047	+	0.971254i	$0.576507\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	38.2983			1.30824			0.654122	−	0.756389i	$-0.273038\pi$
$857$	38.2983			1.30824			0.654122	+	0.756389i	$0.273038\pi$
$858$		0			0
$859$	−0.0830442	+	0.0830442i	−0.00283343	+	0.00283343i	−0.708522	−	0.705689i	$-0.750638\pi$
$859$	−0.0830442	+	0.0830442i	−0.00283343	+	0.00283343i	0.705689	+	0.708522i	$0.250638\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	14.3081			0.487053			0.243527	−	0.969894i	$-0.421696\pi$
$863$	14.3081			0.487053			0.243527	+	0.969894i	$0.421696\pi$
$864$		0			0
$865$	−21.8113			−0.741605
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−54.9656	+	54.9656i	−1.86458	+	1.86458i
$870$		0			0
$871$	3.83782			0.130040
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.595078	−	0.595078i	−0.0201173	−	0.0201173i
$876$		0			0
$877$	5.59288	−	5.59288i	0.188858	−	0.188858i	−0.606344	−	0.795202i	$-0.707365\pi$
$877$	5.59288	−	5.59288i	0.188858	−	0.188858i	0.795202	+	0.606344i	$0.207365\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	6.76857i		−	0.228039i	−0.993479	−	0.114019i	$-0.963627\pi$
$881$		−	6.76857i		−	0.228039i	0.993479	−	0.114019i	$-0.0363726\pi$
$882$		0			0
$883$	10.6497	+	10.6497i	0.358391	+	0.358391i	0.863220	−	0.504828i	$-0.168444\pi$
$883$	10.6497	+	10.6497i	0.358391	+	0.358391i	−0.504828	+	0.863220i	$0.668444\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			13.2188i			0.443843i	0.975065	+	0.221921i	$0.0712328\pi$
$887$			13.2188i			0.443843i	−0.975065	+	0.221921i	$0.928767\pi$
$888$		0			0
$889$		−	1.90239i		−	0.0638042i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.14550	+	3.14550i	0.105260	+	0.105260i
$894$		0			0
$895$			0.746912i			0.0249665i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.99358	+	1.99358i	−0.0664898	+	0.0664898i
$900$		0			0
$901$	5.53010	+	5.53010i	0.184234	+	0.184234i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.14484			−0.204261
$906$		0			0
$907$	−17.5104	+	17.5104i	−0.581422	+	0.581422i	−0.935294	−	0.353872i	$-0.884865\pi$
$907$	−17.5104	+	17.5104i	−0.581422	+	0.581422i	0.353872	+	0.935294i	$0.384865\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	22.9819			0.761424			0.380712	−	0.924694i	$-0.375679\pi$
$911$	22.9819			0.761424			0.380712	+	0.924694i	$0.375679\pi$
$912$		0			0
$913$	−21.0219			−0.695724
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.67822	−	7.67822i	0.253557	−	0.253557i
$918$		0			0
$919$	−26.2789			−0.866862			−0.433431	−	0.901187i	$-0.642697\pi$
$919$	−26.2789			−0.866862			−0.433431	+	0.901187i	$0.642697\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−7.24993	−	7.24993i	−0.238634	−	0.238634i
$924$		0			0
$925$	−6.22715	+	6.22715i	−0.204747	+	0.204747i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	28.5430i		−	0.936466i	−0.883605	−	0.468233i	$-0.844891\pi$
$929$		−	28.5430i		−	0.936466i	0.883605	−	0.468233i	$-0.155109\pi$
$930$		0			0
$931$	−21.1241	−	21.1241i	−0.692313	−	0.692313i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	3.99514i		−	0.130655i
$936$		0			0
$937$		−	32.8627i		−	1.07358i	−0.843717	−	0.536789i	$-0.819637\pi$
$937$		−	32.8627i		−	1.07358i	0.843717	−	0.536789i	$-0.180363\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4.97747	+	4.97747i	0.162261	+	0.162261i	0.783568	−	0.621307i	$-0.213398\pi$
$941$	4.97747	+	4.97747i	0.162261	+	0.162261i	−0.621307	+	0.783568i	$0.713398\pi$
$942$		0			0
$943$		−	1.63611i		−	0.0532791i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.28264	−	1.28264i	0.0416802	−	0.0416802i	−0.685959	−	0.727640i	$-0.740617\pi$
$947$	1.28264	−	1.28264i	0.0416802	−	0.0416802i	0.727640	+	0.685959i	$0.240617\pi$
$948$		0			0
$949$	−4.59932	−	4.59932i	−0.149300	−	0.149300i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−15.9255			−0.515877			−0.257938	−	0.966161i	$-0.583043\pi$
$953$	−15.9255			−0.515877			−0.257938	+	0.966161i	$0.583043\pi$
$954$		0			0
$955$	4.52459	−	4.52459i	0.146412	−	0.146412i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.77142			0.218660
$960$		0			0
$961$	25.0936			0.809471
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8.98137	+	8.98137i	−0.289121	+	0.289121i
$966$		0			0
$967$	−40.6717			−1.30791			−0.653956	−	0.756532i	$-0.726892\pi$
$967$	−40.6717			−1.30791			−0.653956	+	0.756532i	$0.726892\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−10.1841	−	10.1841i	−0.326823	−	0.326823i	0.524554	−	0.851377i	$-0.324232\pi$
$971$	−10.1841	−	10.1841i	−0.326823	−	0.326823i	−0.851377	+	0.524554i	$0.824232\pi$
$972$		0			0
$973$	−10.1610	+	10.1610i	−0.325746	+	0.325746i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	53.2395i		−	1.70328i	−0.524126	−	0.851641i	$-0.675608\pi$
$977$		−	53.2395i		−	1.70328i	0.524126	−	0.851641i	$-0.324392\pi$
$978$		0			0
$979$	5.58131	+	5.58131i	0.178379	+	0.178379i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	18.3154i		−	0.584170i	−0.956392	−	0.292085i	$-0.905651\pi$
$983$		−	18.3154i		−	0.584170i	0.956392	−	0.292085i	$-0.0943491\pi$
$984$		0			0
$985$			20.2767i			0.646068i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.54435	−	1.54435i	−0.0491076	−	0.0491076i
$990$		0			0
$991$		−	37.3151i		−	1.18535i	−0.805440	−	0.592677i	$-0.798071\pi$
$991$		−	37.3151i		−	1.18535i	0.805440	−	0.592677i	$-0.201929\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	12.3193	−	12.3193i	0.390548	−	0.390548i
$996$		0			0
$997$	19.6001	+	19.6001i	0.620741	+	0.620741i	0.945721	−	0.324980i	$-0.105358\pi$
$997$	19.6001	+	19.6001i	0.620741	+	0.620741i	−0.324980	+	0.945721i	$0.605358\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bl.c.1871.3		32
3.2	odd	2	inner	2880.2.bl.c.1871.10		32
4.3	odd	2		720.2.bl.c.251.1	✓	32
12.11	even	2		720.2.bl.c.251.16	yes	32
16.3	odd	4	inner	2880.2.bl.c.431.10		32
16.13	even	4		720.2.bl.c.611.16	yes	32
48.29	odd	4		720.2.bl.c.611.1	yes	32
48.35	even	4	inner	2880.2.bl.c.431.3		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bl.c.251.1	✓	32	4.3	odd	2
720.2.bl.c.251.16	yes	32	12.11	even	2
720.2.bl.c.611.1	yes	32	48.29	odd	4
720.2.bl.c.611.16	yes	32	16.13	even	4
2880.2.bl.c.431.3		32	48.35	even	4	inner
2880.2.bl.c.431.10		32	16.3	odd	4	inner
2880.2.bl.c.1871.3		32	1.1	even	1	trivial
2880.2.bl.c.1871.10		32	3.2	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.bl (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{5} -0.841567 q^{7} +O(q^{10})\)	\(q+(-0.707107 + 0.707107i) q^{5} -0.841567 q^{7} +(3.43874 + 3.43874i) q^{11} +(-0.690971 + 0.690971i) q^{13} +0.821521i q^{17} +(3.35742 + 3.35742i) q^{19} +0.473453i q^{23} -1.00000i q^{25} +(-0.820301 - 0.820301i) q^{29} -2.43031i q^{31} +(0.595078 - 0.595078i) q^{35} +(-6.22715 - 6.22715i) q^{37} -3.45570 q^{41} +(-3.26189 + 3.26189i) q^{43} +0.936881 q^{47} -6.29176 q^{49} +(6.73154 - 6.73154i) q^{53} -4.86311 q^{55} +(9.53265 + 9.53265i) q^{59} +(-6.32938 + 6.32938i) q^{61} -0.977181i q^{65} +(-2.77712 - 2.77712i) q^{67} +10.4924i q^{71} +6.65631i q^{73} +(-2.89393 - 2.89393i) q^{77} +15.9843i q^{79} +(-3.05663 + 3.05663i) q^{83} +(-0.580903 - 0.580903i) q^{85} +1.62307 q^{89} +(0.581499 - 0.581499i) q^{91} -4.74810 q^{95} -13.7207 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 8 q^{19} - 32 q^{49} + 16 q^{55} + 16 q^{61} - 16 q^{67} + 16 q^{85} + 16 q^{91}+O(q^{100}) \) 32 * q + 8 * q^19 - 32 * q^49 + 16 * q^55 + 16 * q^61 - 16 * q^67 + 16 * q^85 + 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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