LMFDB - Embedded newform 2880.2.bl.c.431.1

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			431.1
Character	$\chi$	$=$	2880.431
Dual form			2880.2.bl.c.1871.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{5} +3.01345 q^{7} +O(q^{10})$	$q+(-0.707107 - 0.707107i) q^{5} +3.01345 q^{7} +(2.52053 - 2.52053i) q^{11} +(0.848016 + 0.848016i) q^{13} +8.11753i q^{17} +(-1.76944 + 1.76944i) q^{19} +8.68585i q^{23} +1.00000i q^{25} +(-5.71652 + 5.71652i) q^{29} -1.57453i q^{31} +(-2.13083 - 2.13083i) q^{35} +(4.56150 - 4.56150i) q^{37} -11.3535 q^{41} +(4.33674 + 4.33674i) q^{43} +7.46215 q^{47} +2.08089 q^{49} +(-4.02550 - 4.02550i) q^{53} -3.56457 q^{55} +(2.67476 - 2.67476i) q^{59} +(5.84844 + 5.84844i) q^{61} -1.19928i q^{65} +(8.64918 - 8.64918i) q^{67} -6.90468i q^{71} -2.42961i q^{73} +(7.59549 - 7.59549i) q^{77} +12.0406i q^{79} +(4.26433 + 4.26433i) q^{83} +(5.73996 - 5.73996i) q^{85} +3.00434 q^{89} +(2.55545 + 2.55545i) q^{91} +2.50237 q^{95} -7.39856 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

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

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
$7$	3.01345			1.13898			0.569489	−	0.821999i	$-0.307141\pi$
$7$	3.01345			1.13898			0.569489	+	0.821999i	$0.307141\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.52053	−	2.52053i	0.759968	−	0.759968i	−0.216348	−	0.976316i	$-0.569415\pi$
$11$	2.52053	−	2.52053i	0.759968	−	0.759968i	0.976316	+	0.216348i	$0.0694146\pi$
$12$		0			0
$13$	0.848016	+	0.848016i	0.235197	+	0.235197i	0.814858	−	0.579661i	$-0.196815\pi$
$13$	0.848016	+	0.848016i	0.235197	+	0.235197i	−0.579661	+	0.814858i	$0.696815\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			8.11753i			1.96879i	0.175969	+	0.984396i	$0.443694\pi$
$17$			8.11753i			1.96879i	−0.175969	+	0.984396i	$0.556306\pi$
$18$		0			0
$19$	−1.76944	+	1.76944i	−0.405938	+	0.405938i	−0.880319	−	0.474382i	$-0.842672\pi$
$19$	−1.76944	+	1.76944i	−0.405938	+	0.405938i	0.474382	+	0.880319i	$0.342672\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			8.68585i			1.81112i	0.424213	+	0.905562i	$0.360551\pi$
$23$			8.68585i			1.81112i	−0.424213	+	0.905562i	$0.639449\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.71652	+	5.71652i	−1.06153	+	1.06153i	−0.0635516	+	0.997979i	$0.520243\pi$
$29$	−5.71652	+	5.71652i	−1.06153	+	1.06153i	−0.997979	+	0.0635516i	$0.979757\pi$
$30$		0			0
$31$		−	1.57453i		−	0.282794i	−0.989953	−	0.141397i	$-0.954841\pi$
$31$		−	1.57453i		−	0.282794i	0.989953	−	0.141397i	$-0.0451594\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.13083	−	2.13083i	−0.360176	−	0.360176i
$36$		0			0
$37$	4.56150	−	4.56150i	0.749907	−	0.749907i	−0.224555	−	0.974461i	$-0.572093\pi$
$37$	4.56150	−	4.56150i	0.749907	−	0.749907i	0.974461	+	0.224555i	$0.0720927\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−11.3535			−1.77312			−0.886561	−	0.462611i	$-0.846913\pi$
$41$	−11.3535			−1.77312			−0.886561	+	0.462611i	$0.846913\pi$
$42$		0			0
$43$	4.33674	+	4.33674i	0.661346	+	0.661346i	0.955697	−	0.294351i	$-0.0951036\pi$
$43$	4.33674	+	4.33674i	0.661346	+	0.661346i	−0.294351	+	0.955697i	$0.595104\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.46215			1.08847			0.544233	−	0.838934i	$-0.316821\pi$
$47$	7.46215			1.08847			0.544233	+	0.838934i	$0.316821\pi$
$48$		0			0
$49$	2.08089			0.297269
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.02550	−	4.02550i	−0.552944	−	0.552944i	0.374345	−	0.927289i	$-0.377868\pi$
$53$	−4.02550	−	4.02550i	−0.552944	−	0.552944i	−0.927289	+	0.374345i	$0.877868\pi$
$54$		0			0
$55$	−3.56457			−0.480646
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.67476	−	2.67476i	0.348224	−	0.348224i	−0.511224	−	0.859448i	$-0.670808\pi$
$59$	2.67476	−	2.67476i	0.348224	−	0.348224i	0.859448	+	0.511224i	$0.170808\pi$
$60$		0			0
$61$	5.84844	+	5.84844i	0.748816	+	0.748816i	0.974257	−	0.225441i	$-0.0723822\pi$
$61$	5.84844	+	5.84844i	0.748816	+	0.748816i	−0.225441	+	0.974257i	$0.572382\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	1.19928i		−	0.148752i
$66$		0			0
$67$	8.64918	−	8.64918i	1.05667	−	1.05667i	0.0583711	−	0.998295i	$-0.481409\pi$
$67$	8.64918	−	8.64918i	1.05667	−	1.05667i	0.998295	−	0.0583711i	$-0.0185907\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	6.90468i		−	0.819435i	−0.912212	−	0.409718i	$-0.865627\pi$
$71$		−	6.90468i		−	0.819435i	0.912212	−	0.409718i	$-0.134373\pi$
$72$		0			0
$73$		−	2.42961i		−	0.284365i	−0.989840	−	0.142182i	$-0.954588\pi$
$73$		−	2.42961i		−	0.284365i	0.989840	−	0.142182i	$-0.0454119\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.59549	−	7.59549i	0.865586	−	0.865586i
$78$		0			0
$79$			12.0406i			1.35467i	0.735674	+	0.677336i	$0.236866\pi$
$79$			12.0406i			1.35467i	−0.735674	+	0.677336i	$0.763134\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.26433	+	4.26433i	0.468071	+	0.468071i	0.901289	−	0.433218i	$-0.142622\pi$
$83$	4.26433	+	4.26433i	0.468071	+	0.468071i	−0.433218	+	0.901289i	$0.642622\pi$
$84$		0			0
$85$	5.73996	−	5.73996i	0.622586	−	0.622586i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.00434			0.318460			0.159230	−	0.987242i	$-0.449099\pi$
$89$	3.00434			0.318460			0.159230	+	0.987242i	$0.449099\pi$
$90$		0			0
$91$	2.55545	+	2.55545i	0.267884	+	0.267884i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.50237			0.256738
$96$		0			0
$97$	−7.39856			−0.751210			−0.375605	−	0.926780i	$-0.622565\pi$
$97$	−7.39856			−0.751210			−0.375605	+	0.926780i	$0.622565\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.99480	+	1.99480i	0.198490	+	0.198490i	0.799353	−	0.600862i	$-0.205176\pi$
$101$	1.99480	+	1.99480i	0.198490	+	0.198490i	−0.600862	+	0.799353i	$0.705176\pi$
$102$		0			0
$103$	2.24428			0.221135			0.110568	−	0.993869i	$-0.464733\pi$
$103$	2.24428			0.221135			0.110568	+	0.993869i	$0.464733\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.71105	+	8.71105i	−0.842129	+	0.842129i	−0.989136	−	0.147006i	$-0.953036\pi$
$107$	−8.71105	+	8.71105i	−0.842129	+	0.842129i	0.147006	+	0.989136i	$0.453036\pi$
$108$		0			0
$109$	4.87027	+	4.87027i	0.466488	+	0.466488i	0.900775	−	0.434287i	$-0.143000\pi$
$109$	4.87027	+	4.87027i	0.466488	+	0.466488i	−0.434287	+	0.900775i	$0.643000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			5.51465i			0.518775i	0.965773	+	0.259388i	$0.0835207\pi$
$113$			5.51465i			0.518775i	−0.965773	+	0.259388i	$0.916479\pi$
$114$		0			0
$115$	6.14182	−	6.14182i	0.572728	−	0.572728i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			24.4618i			2.24241i
$120$		0			0
$121$		−	1.70613i		−	0.155103i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	−	0.707107i	0.0632456	−	0.0632456i
$126$		0			0
$127$			16.4561i			1.46024i	0.683318	+	0.730121i	$0.260536\pi$
$127$			16.4561i			1.46024i	−0.683318	+	0.730121i	$0.739464\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.48031	−	7.48031i	−0.653558	−	0.653558i	0.300290	−	0.953848i	$-0.402916\pi$
$131$	−7.48031	−	7.48031i	−0.653558	−	0.653558i	−0.953848	+	0.300290i	$0.902916\pi$
$132$		0			0
$133$	−5.33212	+	5.33212i	−0.462354	+	0.462354i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−15.3388			−1.31048			−0.655241	−	0.755420i	$-0.727433\pi$
$137$	−15.3388			−1.31048			−0.655241	+	0.755420i	$0.727433\pi$
$138$		0			0
$139$	−0.237401	−	0.237401i	−0.0201361	−	0.0201361i	0.696967	−	0.717103i	$-0.254532\pi$
$139$	−0.237401	−	0.237401i	−0.0201361	−	0.0201361i	−0.717103	+	0.696967i	$0.754532\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.27490			0.357485
$144$		0			0
$145$	8.08437			0.671371
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.8045	+	13.8045i	1.13091	+	1.13091i	0.990026	+	0.140884i	$0.0449945\pi$
$149$	13.8045	+	13.8045i	1.13091	+	1.13091i	0.140884	+	0.990026i	$0.455006\pi$
$150$		0			0
$151$	11.7363			0.955086			0.477543	−	0.878608i	$-0.341527\pi$
$151$	11.7363			0.955086			0.477543	+	0.878608i	$0.341527\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.11336	+	1.11336i	−0.0894272	+	0.0894272i
$156$		0			0
$157$	−3.37861	−	3.37861i	−0.269642	−	0.269642i	0.559314	−	0.828956i	$-0.311065\pi$
$157$	−3.37861	−	3.37861i	−0.269642	−	0.269642i	−0.828956	+	0.559314i	$0.811065\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			26.1744i			2.06283i
$162$		0			0
$163$	−7.07092	+	7.07092i	−0.553837	+	0.553837i	−0.927546	−	0.373709i	$-0.878086\pi$
$163$	−7.07092	+	7.07092i	−0.553837	+	0.553837i	0.373709	+	0.927546i	$0.378086\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	0.385104i		−	0.0298002i	−0.999889	−	0.0149001i	$-0.995257\pi$
$167$		−	0.385104i		−	0.0298002i	0.999889	−	0.0149001i	$-0.00474303\pi$
$168$		0			0
$169$		−	11.5617i		−	0.889365i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	14.7701	−	14.7701i	1.12295	−	1.12295i	0.131655	−	0.991296i	$-0.457971\pi$
$173$	14.7701	−	14.7701i	1.12295	−	1.12295i	0.991296	−	0.131655i	$-0.0420291\pi$
$174$		0			0
$175$			3.01345i			0.227795i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.56679	−	5.56679i	−0.416081	−	0.416081i	0.467769	−	0.883851i	$-0.345058\pi$
$179$	−5.56679	−	5.56679i	−0.416081	−	0.416081i	−0.883851	+	0.467769i	$0.845058\pi$
$180$		0			0
$181$	4.50933	−	4.50933i	0.335176	−	0.335176i	−0.519372	−	0.854548i	$-0.673834\pi$
$181$	4.50933	−	4.50933i	0.335176	−	0.335176i	0.854548	+	0.519372i	$0.173834\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.45094			−0.474283
$186$		0			0
$187$	20.4605	+	20.4605i	1.49622	+	1.49622i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	15.9032			1.15071			0.575357	−	0.817902i	$-0.304863\pi$
$191$	15.9032			1.15071			0.575357	+	0.817902i	$0.304863\pi$
$192$		0			0
$193$	11.1744			0.804349			0.402175	−	0.915563i	$-0.368254\pi$
$193$	11.1744			0.804349			0.402175	+	0.915563i	$0.368254\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.95845	−	2.95845i	−0.210781	−	0.210781i	0.593818	−	0.804599i	$-0.297620\pi$
$197$	−2.95845	−	2.95845i	−0.210781	−	0.210781i	−0.804599	+	0.593818i	$0.797620\pi$
$198$		0			0
$199$	5.14905			0.365007			0.182503	−	0.983205i	$-0.441580\pi$
$199$	5.14905			0.365007			0.182503	+	0.983205i	$0.441580\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−17.2264	+	17.2264i	−1.20906	+	1.20906i
$204$		0			0
$205$	8.02816	+	8.02816i	0.560711	+	0.560711i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			8.91985i			0.616999i
$210$		0			0
$211$	11.5176	−	11.5176i	0.792902	−	0.792902i	−0.189063	−	0.981965i	$-0.560545\pi$
$211$	11.5176	−	11.5176i	0.792902	−	0.792902i	0.981965	+	0.189063i	$0.0605452\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	6.13307i		−	0.418272i
$216$		0			0
$217$		−	4.74476i		−	0.322096i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.88380	+	6.88380i	−0.463054	+	0.463054i
$222$		0			0
$223$		−	7.73829i		−	0.518194i	−0.965851	−	0.259097i	$-0.916575\pi$
$223$		−	7.73829i		−	0.518194i	0.965851	−	0.259097i	$-0.0834250\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.6412	−	12.6412i	−0.839024	−	0.839024i	0.149706	−	0.988731i	$-0.452167\pi$
$227$	−12.6412	−	12.6412i	−0.839024	−	0.839024i	−0.988731	+	0.149706i	$0.952167\pi$
$228$		0			0
$229$	−9.03817	+	9.03817i	−0.597259	+	0.597259i	−0.939582	−	0.342323i	$-0.888786\pi$
$229$	−9.03817	+	9.03817i	−0.597259	+	0.597259i	0.342323	+	0.939582i	$0.388786\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−8.49700			−0.556657			−0.278329	−	0.960486i	$-0.589780\pi$
$233$	−8.49700			−0.556657			−0.278329	+	0.960486i	$0.589780\pi$
$234$		0			0
$235$	−5.27654	−	5.27654i	−0.344203	−	0.344203i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.67636			0.625912			0.312956	−	0.949768i	$-0.398681\pi$
$239$	9.67636			0.625912			0.312956	+	0.949768i	$0.398681\pi$
$240$		0			0
$241$	9.79399			0.630887			0.315443	−	0.948944i	$-0.397847\pi$
$241$	9.79399			0.630887			0.315443	+	0.948944i	$0.397847\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.47141	−	1.47141i	−0.0940048	−	0.0940048i
$246$		0			0
$247$	−3.00103			−0.190951
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.969387	−	0.969387i	0.0611872	−	0.0611872i	−0.675851	−	0.737038i	$-0.736224\pi$
$251$	0.969387	−	0.969387i	0.0611872	−	0.0611872i	0.737038	+	0.675851i	$0.236224\pi$
$252$		0			0
$253$	21.8929	+	21.8929i	1.37640	+	1.37640i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	25.2455i		−	1.57477i	−0.616462	−	0.787384i	$-0.711435\pi$
$257$		−	25.2455i		−	1.57477i	0.616462	−	0.787384i	$-0.288565\pi$
$258$		0			0
$259$	13.7459	−	13.7459i	0.854127	−	0.854127i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			13.5917i			0.838103i	0.907963	+	0.419051i	$0.137637\pi$
$263$			13.5917i			0.838103i	−0.907963	+	0.419051i	$0.862363\pi$
$264$		0			0
$265$			5.69291i			0.349713i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.59732	−	3.59732i	0.219332	−	0.219332i	−0.588885	−	0.808217i	$-0.700433\pi$
$269$	3.59732	−	3.59732i	0.219332	−	0.219332i	0.808217	+	0.588885i	$0.200433\pi$
$270$		0			0
$271$		−	4.95267i		−	0.300853i	−0.988621	−	0.150427i	$-0.951935\pi$
$271$		−	4.95267i		−	0.300853i	0.988621	−	0.150427i	$-0.0480648\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.52053	+	2.52053i	0.151994	+	0.151994i
$276$		0			0
$277$	2.20929	−	2.20929i	0.132743	−	0.132743i	−0.637613	−	0.770357i	$-0.720078\pi$
$277$	2.20929	−	2.20929i	0.132743	−	0.132743i	0.770357	+	0.637613i	$0.220078\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	13.8056			0.823576			0.411788	−	0.911280i	$-0.364904\pi$
$281$	13.8056			0.823576			0.411788	+	0.911280i	$0.364904\pi$
$282$		0			0
$283$	−11.2766	−	11.2766i	−0.670322	−	0.670322i	0.287468	−	0.957790i	$-0.407186\pi$
$283$	−11.2766	−	11.2766i	−0.670322	−	0.670322i	−0.957790	+	0.287468i	$0.907186\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−34.2133			−2.01955
$288$		0			0
$289$	−48.8944			−2.87614
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.41817	+	6.41817i	0.374954	+	0.374954i	0.869278	−	0.494324i	$-0.164584\pi$
$293$	6.41817	+	6.41817i	0.374954	+	0.374954i	−0.494324	+	0.869278i	$0.664584\pi$
$294$		0			0
$295$	−3.78268			−0.220236
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.36574	+	7.36574i	−0.425972	+	0.425972i
$300$		0			0
$301$	13.0685	+	13.0685i	0.753258	+	0.753258i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	8.27095i		−	0.473593i
$306$		0			0
$307$	17.7483	−	17.7483i	1.01295	−	1.01295i	0.0130312	−	0.999915i	$-0.495852\pi$
$307$	17.7483	−	17.7483i	1.01295	−	1.01295i	0.999915	−	0.0130312i	$-0.00414808\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			17.2303i			0.977043i	0.872552	+	0.488521i	$0.162464\pi$
$311$			17.2303i			0.977043i	−0.872552	+	0.488521i	$0.837536\pi$
$312$		0			0
$313$		−	20.3557i		−	1.15057i	−0.817952	−	0.575287i	$-0.804890\pi$
$313$		−	20.3557i		−	1.15057i	0.817952	−	0.575287i	$-0.195110\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	15.6949	−	15.6949i	0.881511	−	0.881511i	−0.112178	−	0.993688i	$-0.535783\pi$
$317$	15.6949	−	15.6949i	0.881511	−	0.881511i	0.993688	+	0.112178i	$0.0357825\pi$
$318$		0			0
$319$			28.8173i			1.61346i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−14.3635	−	14.3635i	−0.799206	−	0.799206i
$324$		0			0
$325$	−0.848016	+	0.848016i	−0.0470394	+	0.0470394i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	22.4868			1.23974
$330$		0			0
$331$	−17.3241	−	17.3241i	−0.952221	−	0.952221i	0.0466885	−	0.998909i	$-0.485133\pi$
$331$	−17.3241	−	17.3241i	−0.952221	−	0.952221i	−0.998909	+	0.0466885i	$0.985133\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.2318			−0.668294
$336$		0			0
$337$	15.4336			0.840721			0.420360	−	0.907357i	$-0.361904\pi$
$337$	15.4336			0.840721			0.420360	+	0.907357i	$0.361904\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.96865	−	3.96865i	−0.214914	−	0.214914i
$342$		0			0
$343$	−14.8235			−0.800394
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.92959	−	8.92959i	0.479365	−	0.479365i	−0.425563	−	0.904929i	$-0.639924\pi$
$347$	8.92959	−	8.92959i	0.479365	−	0.479365i	0.904929	+	0.425563i	$0.139924\pi$
$348$		0			0
$349$	13.7519	+	13.7519i	0.736121	+	0.736121i	0.971825	−	0.235704i	$-0.0757395\pi$
$349$	13.7519	+	13.7519i	0.736121	+	0.736121i	−0.235704	+	0.971825i	$0.575740\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			34.2519i			1.82305i	0.411249	+	0.911523i	$0.365093\pi$
$353$			34.2519i			1.82305i	−0.411249	+	0.911523i	$0.634907\pi$
$354$		0			0
$355$	−4.88235	+	4.88235i	−0.259128	+	0.259128i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			14.0096i			0.739396i	0.929152	+	0.369698i	$0.120539\pi$
$359$			14.0096i			0.739396i	−0.929152	+	0.369698i	$0.879461\pi$
$360$		0			0
$361$			12.7382i			0.670429i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.71800	+	1.71800i	−0.0899240	+	0.0899240i
$366$		0			0
$367$		−	35.2824i		−	1.84173i	−0.389886	−	0.920863i	$-0.627486\pi$
$367$		−	35.2824i		−	1.84173i	0.389886	−	0.920863i	$-0.372514\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.1306	−	12.1306i	−0.629791	−	0.629791i
$372$		0			0
$373$	−4.79228	+	4.79228i	−0.248135	+	0.248135i	−0.820205	−	0.572070i	$-0.806141\pi$
$373$	−4.79228	+	4.79228i	−0.248135	+	0.248135i	0.572070	+	0.820205i	$0.306141\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−9.69539			−0.499338
$378$		0			0
$379$	−24.7085	−	24.7085i	−1.26919	−	1.26919i	−0.946506	−	0.322686i	$-0.895414\pi$
$379$	−24.7085	−	24.7085i	−1.26919	−	1.26919i	−0.322686	−	0.946506i	$-0.604586\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	31.7665			1.62319			0.811597	−	0.584217i	$-0.198598\pi$
$383$	31.7665			1.62319			0.811597	+	0.584217i	$0.198598\pi$
$384$		0			0
$385$	−10.7416			−0.547445
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.2982	−	10.2982i	−0.522138	−	0.522138i	0.396079	−	0.918217i	$-0.370371\pi$
$389$	−10.2982	−	10.2982i	−0.522138	−	0.522138i	−0.918217	+	0.396079i	$0.870371\pi$
$390$		0			0
$391$	−70.5077			−3.56573
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.51398	−	8.51398i	0.428385	−	0.428385i
$396$		0			0
$397$	−8.50627	−	8.50627i	−0.426917	−	0.426917i	0.460660	−	0.887577i	$-0.347613\pi$
$397$	−8.50627	−	8.50627i	−0.426917	−	0.426917i	−0.887577	+	0.460660i	$0.847613\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			21.7116i			1.08423i	0.840306	+	0.542113i	$0.182375\pi$
$401$			21.7116i			1.08423i	−0.840306	+	0.542113i	$0.817625\pi$
$402$		0			0
$403$	1.33523	−	1.33523i	0.0665123	−	0.0665123i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	22.9948i		−	1.13981i
$408$		0			0
$409$			10.3338i			0.510971i	0.966813	+	0.255486i	$0.0822354\pi$
$409$			10.3338i			0.510971i	−0.966813	+	0.255486i	$0.917765\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.06025	−	8.06025i	0.396619	−	0.396619i
$414$		0			0
$415$		−	6.03067i		−	0.296034i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	21.9568	+	21.9568i	1.07266	+	1.07266i	0.997145	+	0.0755134i	$0.0240596\pi$
$419$	21.9568	+	21.9568i	1.07266	+	1.07266i	0.0755134	+	0.997145i	$0.475940\pi$
$420$		0			0
$421$	−22.9282	+	22.9282i	−1.11745	+	1.11745i	−0.125336	+	0.992114i	$0.540001\pi$
$421$	−22.9282	+	22.9282i	−1.11745	+	1.11745i	−0.992114	+	0.125336i	$0.959999\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−8.11753			−0.393758
$426$		0			0
$427$	17.6240	+	17.6240i	0.852885	+	0.852885i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	11.1272			0.535978			0.267989	−	0.963422i	$-0.413641\pi$
$431$	11.1272			0.535978			0.267989	+	0.963422i	$0.413641\pi$
$432$		0			0
$433$	−6.03454			−0.290002			−0.145001	−	0.989432i	$-0.546318\pi$
$433$	−6.03454			−0.290002			−0.145001	+	0.989432i	$0.546318\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−15.3691	−	15.3691i	−0.735204	−	0.735204i
$438$		0			0
$439$	−4.42398			−0.211145			−0.105572	−	0.994412i	$-0.533667\pi$
$439$	−4.42398			−0.211145			−0.105572	+	0.994412i	$0.533667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.90545	−	6.90545i	0.328088	−	0.328088i	−0.523771	−	0.851859i	$-0.675475\pi$
$443$	6.90545	−	6.90545i	0.328088	−	0.328088i	0.851859	+	0.523771i	$0.175475\pi$
$444$		0			0
$445$	−2.12439	−	2.12439i	−0.100706	−	0.100706i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			1.94420i			0.0917522i	0.998947	+	0.0458761i	$0.0146079\pi$
$449$			1.94420i			0.0917522i	−0.998947	+	0.0458761i	$0.985392\pi$
$450$		0			0
$451$	−28.6169	+	28.6169i	−1.34752	+	1.34752i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	3.61396i		−	0.169425i
$456$		0			0
$457$			3.83227i			0.179266i	0.995975	+	0.0896329i	$0.0285694\pi$
$457$			3.83227i			0.179266i	−0.995975	+	0.0896329i	$0.971431\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.1424	−	15.1424i	0.705251	−	0.705251i	−0.260282	−	0.965533i	$-0.583815\pi$
$461$	15.1424	−	15.1424i	0.705251	−	0.705251i	0.965533	+	0.260282i	$0.0838154\pi$
$462$		0			0
$463$		−	2.33174i		−	0.108365i	−0.998531	−	0.0541826i	$-0.982745\pi$
$463$		−	2.33174i		−	0.108365i	0.998531	−	0.0541826i	$-0.0172553\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−29.0100	−	29.0100i	−1.34242	−	1.34242i	−0.893641	−	0.448783i	$-0.851858\pi$
$467$	−29.0100	−	29.0100i	−1.34242	−	1.34242i	−0.448783	−	0.893641i	$-0.648142\pi$
$468$		0			0
$469$	26.0639	−	26.0639i	1.20352	−	1.20352i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	21.8617			1.00520
$474$		0			0
$475$	−1.76944	−	1.76944i	−0.0811875	−	0.0811875i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.05620			−0.276715			−0.138357	−	0.990382i	$-0.544182\pi$
$479$	−6.05620			−0.276715			−0.138357	+	0.990382i	$0.544182\pi$
$480$		0			0
$481$	7.73646			0.352752
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.23157	+	5.23157i	0.237553	+	0.237553i
$486$		0			0
$487$	33.4429			1.51544			0.757720	−	0.652580i	$-0.226313\pi$
$487$	33.4429			1.51544			0.757720	+	0.652580i	$0.226313\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	17.7813	−	17.7813i	0.802460	−	0.802460i	−0.181020	−	0.983479i	$-0.557940\pi$
$491$	17.7813	−	17.7813i	0.802460	−	0.802460i	0.983479	+	0.181020i	$0.0579398\pi$
$492$		0			0
$493$	−46.4040	−	46.4040i	−2.08993	−	2.08993i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	20.8069i		−	0.933318i
$498$		0			0
$499$	4.19132	−	4.19132i	0.187629	−	0.187629i	−0.607041	−	0.794670i	$-0.707644\pi$
$499$	4.19132	−	4.19132i	0.187629	−	0.187629i	0.794670	+	0.607041i	$0.207644\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			5.40502i			0.240998i	0.992713	+	0.120499i	$0.0384495\pi$
$503$			5.40502i			0.240998i	−0.992713	+	0.120499i	$0.961551\pi$
$504$		0			0
$505$		−	2.82108i		−	0.125536i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	19.9696	−	19.9696i	0.885137	−	0.885137i	−0.108914	−	0.994051i	$-0.534737\pi$
$509$	19.9696	−	19.9696i	0.885137	−	0.885137i	0.994051	+	0.108914i	$0.0347374\pi$
$510$		0			0
$511$		−	7.32152i		−	0.323885i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.58694	−	1.58694i	−0.0699290	−	0.0699290i
$516$		0			0
$517$	18.8086	−	18.8086i	0.827200	−	0.827200i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−26.2170			−1.14859			−0.574295	−	0.818649i	$-0.694724\pi$
$521$	−26.2170			−1.14859			−0.574295	+	0.818649i	$0.694724\pi$
$522$		0			0
$523$	22.7344	+	22.7344i	0.994107	+	0.994107i	0.999983	−	0.00587606i	$-0.00187042\pi$
$523$	22.7344	+	22.7344i	0.994107	+	0.994107i	−0.00587606	+	0.999983i	$0.501870\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	12.7813			0.556762
$528$		0			0
$529$	−52.4440			−2.28017
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−9.62797	−	9.62797i	−0.417034	−	0.417034i
$534$		0			0
$535$	12.3193			0.532609
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.24493	−	5.24493i	0.225915	−	0.225915i
$540$		0			0
$541$	−6.13107	−	6.13107i	−0.263595	−	0.263595i	0.562918	−	0.826513i	$-0.309679\pi$
$541$	−6.13107	−	6.13107i	−0.263595	−	0.263595i	−0.826513	+	0.562918i	$0.809679\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	6.88761i		−	0.295033i
$546$		0			0
$547$	4.27936	−	4.27936i	0.182972	−	0.182972i	−0.609677	−	0.792650i	$-0.708701\pi$
$547$	4.27936	−	4.27936i	0.182972	−	0.182972i	0.792650	+	0.609677i	$0.208701\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	20.2301i		−	0.861830i
$552$		0			0
$553$			36.2837i			1.54294i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.08181	−	3.08181i	0.130580	−	0.130580i	−0.638796	−	0.769376i	$-0.720567\pi$
$557$	3.08181	−	3.08181i	0.130580	−	0.130580i	0.769376	+	0.638796i	$0.220567\pi$
$558$		0			0
$559$			7.35524i			0.311094i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−14.2684	−	14.2684i	−0.601340	−	0.601340i	0.339328	−	0.940668i	$-0.389800\pi$
$563$	−14.2684	−	14.2684i	−0.601340	−	0.601340i	−0.940668	+	0.339328i	$0.889800\pi$
$564$		0			0
$565$	3.89945	−	3.89945i	0.164051	−	0.164051i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−42.3124			−1.77383			−0.886913	−	0.461936i	$-0.847155\pi$
$569$	−42.3124			−1.77383			−0.886913	+	0.461936i	$0.847155\pi$
$570$		0			0
$571$	1.91588	+	1.91588i	0.0801771	+	0.0801771i	0.746058	−	0.665881i	$-0.231944\pi$
$571$	1.91588	+	1.91588i	0.0801771	+	0.0801771i	−0.665881	+	0.746058i	$0.731944\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.68585			−0.362225
$576$		0			0
$577$	10.1779			0.423710			0.211855	−	0.977301i	$-0.432049\pi$
$577$	10.1779			0.423710			0.211855	+	0.977301i	$0.432049\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12.8503	+	12.8503i	0.533122	+	0.533122i
$582$		0			0
$583$	−20.2928			−0.840440
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.66609	+	6.66609i	−0.275139	+	0.275139i	−0.831165	−	0.556026i	$-0.812326\pi$
$587$	−6.66609	+	6.66609i	−0.275139	+	0.275139i	0.556026	+	0.831165i	$0.312326\pi$
$588$		0			0
$589$	2.78604	+	2.78604i	0.114797	+	0.114797i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	8.61068i		−	0.353598i	−0.984247	−	0.176799i	$-0.943426\pi$
$593$		−	8.61068i		−	0.353598i	0.984247	−	0.176799i	$-0.0565743\pi$
$594$		0			0
$595$	17.2971	−	17.2971i	0.709112	−	0.709112i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			3.06132i			0.125082i	0.998042	+	0.0625412i	$0.0199205\pi$
$599$			3.06132i			0.125082i	−0.998042	+	0.0625412i	$0.980080\pi$
$600$		0			0
$601$			3.52721i			0.143878i	0.997409	+	0.0719389i	$0.0229187\pi$
$601$			3.52721i			0.143878i	−0.997409	+	0.0719389i	$0.977081\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.20642	+	1.20642i	−0.0490478	+	0.0490478i
$606$		0			0
$607$			16.3560i			0.663868i	0.943303	+	0.331934i	$0.107701\pi$
$607$			16.3560i			0.663868i	−0.943303	+	0.331934i	$0.892299\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.32802	+	6.32802i	0.256004	+	0.256004i
$612$		0			0
$613$	−12.3997	+	12.3997i	−0.500820	+	0.500820i	−0.911693	−	0.410873i	$-0.865224\pi$
$613$	−12.3997	+	12.3997i	−0.500820	+	0.500820i	0.410873	+	0.911693i	$0.365224\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	14.5593			0.586137			0.293068	−	0.956092i	$-0.405324\pi$
$617$	14.5593			0.586137			0.293068	+	0.956092i	$0.405324\pi$
$618$		0			0
$619$	10.4182	+	10.4182i	0.418742	+	0.418742i	0.884770	−	0.466028i	$-0.154315\pi$
$619$	10.4182	+	10.4182i	0.418742	+	0.418742i	−0.466028	+	0.884770i	$0.654315\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9.05344			0.362718
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	37.0282	+	37.0282i	1.47641	+	1.47641i
$630$		0			0
$631$	19.3547			0.770497			0.385248	−	0.922813i	$-0.374116\pi$
$631$	19.3547			0.770497			0.385248	+	0.922813i	$0.374116\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	11.6362	−	11.6362i	0.461769	−	0.461769i
$636$		0			0
$637$	1.76462	+	1.76462i	0.0699169	+	0.0699169i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	14.1094i		−	0.557287i	−0.960395	−	0.278643i	$-0.910115\pi$
$641$		−	14.1094i		−	0.557287i	0.960395	−	0.278643i	$-0.0898848\pi$
$642$		0			0
$643$	−19.7000	+	19.7000i	−0.776893	+	0.776893i	−0.979301	−	0.202408i	$-0.935123\pi$
$643$	−19.7000	+	19.7000i	−0.776893	+	0.776893i	0.202408	+	0.979301i	$0.435123\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	10.8503i		−	0.426568i	−0.976990	−	0.213284i	$-0.931584\pi$
$647$		−	10.8503i		−	0.426568i	0.976990	−	0.213284i	$-0.0684160\pi$
$648$		0			0
$649$		−	13.4836i		−	0.529278i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−16.6089	+	16.6089i	−0.649958	+	0.649958i	−0.952983	−	0.303024i	$-0.902004\pi$
$653$	−16.6089	+	16.6089i	−0.649958	+	0.649958i	0.303024	+	0.952983i	$0.402004\pi$
$654$		0			0
$655$			10.5788i			0.413346i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	33.6937	+	33.6937i	1.31252	+	1.31252i	0.919552	+	0.392968i	$0.128552\pi$
$659$	33.6937	+	33.6937i	1.31252	+	1.31252i	0.392968	+	0.919552i	$0.371448\pi$
$660$		0			0
$661$	−5.19105	+	5.19105i	−0.201908	+	0.201908i	−0.800817	−	0.598909i	$-0.795601\pi$
$661$	−5.19105	+	5.19105i	−0.201908	+	0.201908i	0.598909	+	0.800817i	$0.295601\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7.54076			0.292418
$666$		0			0
$667$	−49.6528	−	49.6528i	−1.92256	−	1.92256i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	29.4823			1.13815
$672$		0			0
$673$	3.81665			0.147121			0.0735605	−	0.997291i	$-0.476564\pi$
$673$	3.81665			0.147121			0.0735605	+	0.997291i	$0.476564\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−34.3078	−	34.3078i	−1.31855	−	1.31855i	−0.914920	−	0.403634i	$-0.867747\pi$
$677$	−34.3078	−	34.3078i	−1.31855	−	1.31855i	−0.403634	−	0.914920i	$-0.632253\pi$
$678$		0			0
$679$	−22.2952			−0.855611
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.920545	−	0.920545i	0.0352237	−	0.0352237i	−0.689276	−	0.724499i	$-0.742071\pi$
$683$	0.920545	−	0.920545i	0.0352237	−	0.0352237i	0.724499	+	0.689276i	$0.242071\pi$
$684$		0			0
$685$	10.8462	+	10.8462i	0.414411	+	0.414411i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	6.82737i		−	0.260102i
$690$		0			0
$691$	−4.68394	+	4.68394i	−0.178186	+	0.178186i	−0.790564	−	0.612379i	$-0.790213\pi$
$691$	−4.68394	+	4.68394i	−0.178186	+	0.178186i	0.612379	+	0.790564i	$0.290213\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			0.335736i			0.0127352i
$696$		0			0
$697$		−	92.1626i		−	3.49091i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−16.6667	+	16.6667i	−0.629491	+	0.629491i	−0.947940	−	0.318449i	$-0.896838\pi$
$701$	−16.6667	+	16.6667i	−0.629491	+	0.629491i	0.318449	+	0.947940i	$0.396838\pi$
$702$		0			0
$703$			16.1426i			0.608831i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.01124	+	6.01124i	0.226076	+	0.226076i
$708$		0			0
$709$	−18.5140	+	18.5140i	−0.695308	+	0.695308i	−0.963395	−	0.268087i	$-0.913609\pi$
$709$	−18.5140	+	18.5140i	−0.695308	+	0.695308i	0.268087	+	0.963395i	$0.413609\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	13.6761			0.512175
$714$		0			0
$715$	−3.02281	−	3.02281i	−0.113047	−	0.113047i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−36.8474			−1.37418			−0.687088	−	0.726575i	$-0.741111\pi$
$719$	−36.8474			−1.37418			−0.687088	+	0.726575i	$0.741111\pi$
$720$		0			0
$721$	6.76301			0.251868
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.71652	−	5.71652i	−0.212306	−	0.212306i
$726$		0			0
$727$	18.5492			0.687951			0.343975	−	0.938979i	$-0.388226\pi$
$727$	18.5492			0.687951			0.343975	+	0.938979i	$0.388226\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−35.2036	+	35.2036i	−1.30205	+	1.30205i
$732$		0			0
$733$	−37.0722	−	37.0722i	−1.36929	−	1.36929i	−0.861449	−	0.507844i	$-0.830443\pi$
$733$	−37.0722	−	37.0722i	−1.36929	−	1.36929i	−0.507844	−	0.861449i	$-0.669557\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	43.6010i		−	1.60606i
$738$		0			0
$739$	−28.0888	+	28.0888i	−1.03326	+	1.03326i	−0.0338349	+	0.999427i	$0.510772\pi$
$739$	−28.0888	+	28.0888i	−1.03326	+	1.03326i	−0.999427	+	0.0338349i	$0.989228\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	14.0700i		−	0.516177i	−0.966121	−	0.258088i	$-0.916907\pi$
$743$		−	14.0700i		−	0.516177i	0.966121	−	0.258088i	$-0.0830926\pi$
$744$		0			0
$745$		−	19.5225i		−	0.715250i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−26.2503	+	26.2503i	−0.959166	+	0.959166i
$750$		0			0
$751$		−	29.3907i		−	1.07248i	−0.844065	−	0.536242i	$-0.819844\pi$
$751$		−	29.3907i		−	1.07248i	0.844065	−	0.536242i	$-0.180156\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.29882	−	8.29882i	−0.302025	−	0.302025i
$756$		0			0
$757$	−17.8240	+	17.8240i	−0.647824	+	0.647824i	−0.952467	−	0.304643i	$-0.901463\pi$
$757$	−17.8240	+	17.8240i	−0.647824	+	0.647824i	0.304643	+	0.952467i	$0.401463\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	16.5211			0.598888			0.299444	−	0.954114i	$-0.403199\pi$
$761$	16.5211			0.598888			0.299444	+	0.954114i	$0.403199\pi$
$762$		0			0
$763$	14.6763	+	14.6763i	0.531319	+	0.531319i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.53647			0.163803
$768$		0			0
$769$	−4.76747			−0.171919			−0.0859596	−	0.996299i	$-0.527396\pi$
$769$	−4.76747			−0.171919			−0.0859596	+	0.996299i	$0.527396\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8.07952	−	8.07952i	−0.290600	−	0.290600i	0.546717	−	0.837317i	$-0.315877\pi$
$773$	−8.07952	−	8.07952i	−0.290600	−	0.290600i	−0.837317	+	0.546717i	$0.815877\pi$
$774$		0			0
$775$	1.57453			0.0565587
$776$		0			0
$777$		0			0
$778$		0			0
$779$	20.0894	−	20.0894i	0.719777	−	0.719777i
$780$		0			0
$781$	−17.4035	−	17.4035i	−0.622745	−	0.622745i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			4.77807i			0.170537i
$786$		0			0
$787$	−20.0847	+	20.0847i	−0.715941	+	0.715941i	−0.967771	−	0.251831i	$-0.918967\pi$
$787$	−20.0847	+	20.0847i	−0.715941	+	0.715941i	0.251831	+	0.967771i	$0.418967\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			16.6181i			0.590873i
$792$		0			0
$793$			9.91914i			0.352239i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	19.5861	−	19.5861i	0.693777	−	0.693777i	−0.269284	−	0.963061i	$-0.586787\pi$
$797$	19.5861	−	19.5861i	0.693777	−	0.693777i	0.963061	+	0.269284i	$0.0867871\pi$
$798$		0			0
$799$			60.5742i			2.14296i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.12391	−	6.12391i	−0.216108	−	0.216108i
$804$		0			0
$805$	18.5081	−	18.5081i	0.652324	−	0.652324i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	9.77227			0.343575			0.171787	−	0.985134i	$-0.445046\pi$
$809$	9.77227			0.343575			0.171787	+	0.985134i	$0.445046\pi$
$810$		0			0
$811$	−6.20987	−	6.20987i	−0.218058	−	0.218058i	0.589622	−	0.807680i	$-0.299277\pi$
$811$	−6.20987	−	6.20987i	−0.218058	−	0.218058i	−0.807680	+	0.589622i	$0.799277\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	9.99979			0.350278
$816$		0			0
$817$	−15.3472			−0.536930
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−11.7702	−	11.7702i	−0.410783	−	0.410783i	0.471228	−	0.882011i	$-0.343811\pi$
$821$	−11.7702	−	11.7702i	−0.410783	−	0.410783i	−0.882011	+	0.471228i	$0.843811\pi$
$822$		0			0
$823$	16.0990			0.561177			0.280588	−	0.959828i	$-0.409470\pi$
$823$	16.0990			0.561177			0.280588	+	0.959828i	$0.409470\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	24.0523	−	24.0523i	0.836382	−	0.836382i	−0.151999	−	0.988381i	$-0.548571\pi$
$827$	24.0523	−	24.0523i	0.836382	−	0.836382i	0.988381	+	0.151999i	$0.0485710\pi$
$828$		0			0
$829$	−17.9325	−	17.9325i	−0.622821	−	0.622821i	0.323431	−	0.946252i	$-0.395164\pi$
$829$	−17.9325	−	17.9325i	−0.622821	−	0.622821i	−0.946252	+	0.323431i	$0.895164\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			16.8917i			0.585261i
$834$		0			0
$835$	−0.272309	+	0.272309i	−0.00942366	+	0.00942366i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	34.8736i		−	1.20397i	−0.798508	−	0.601984i	$-0.794377\pi$
$839$		−	34.8736i		−	1.20397i	0.798508	−	0.601984i	$-0.205623\pi$
$840$		0			0
$841$		−	36.3571i		−	1.25369i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.17538	+	8.17538i	−0.281242	+	0.281242i
$846$		0			0
$847$		−	5.14134i		−	0.176659i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	39.6205	+	39.6205i	1.35817	+	1.35817i
$852$		0			0
$853$	−38.8650	+	38.8650i	−1.33071	+	1.33071i	−0.425982	+	0.904732i	$0.640071\pi$
$853$	−38.8650	+	38.8650i	−1.33071	+	1.33071i	−0.904732	+	0.425982i	$0.859929\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−10.4986			−0.358626			−0.179313	−	0.983792i	$-0.557387\pi$
$857$	−10.4986			−0.358626			−0.179313	+	0.983792i	$0.557387\pi$
$858$		0			0
$859$	−34.4230	−	34.4230i	−1.17450	−	1.17450i	−0.981126	−	0.193370i	$-0.938058\pi$
$859$	−34.4230	−	34.4230i	−1.17450	−	1.17450i	−0.193370	−	0.981126i	$-0.561942\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	27.5044			0.936260			0.468130	−	0.883660i	$-0.344928\pi$
$863$	27.5044			0.936260			0.468130	+	0.883660i	$0.344928\pi$
$864$		0			0
$865$	−20.8881			−0.710216
$866$		0			0
$867$		0			0
$868$		0			0
$869$	30.3487	+	30.3487i	1.02951	+	1.02951i
$870$		0			0
$871$	14.6693			0.497050
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.13083	−	2.13083i	0.0720352	−	0.0720352i
$876$		0			0
$877$	14.9071	+	14.9071i	0.503377	+	0.503377i	0.912486	−	0.409108i	$-0.134160\pi$
$877$	14.9071	+	14.9071i	0.503377	+	0.503377i	−0.409108	+	0.912486i	$0.634160\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	42.9744i		−	1.44785i	−0.689881	−	0.723923i	$-0.742337\pi$
$881$		−	42.9744i		−	1.44785i	0.689881	−	0.723923i	$-0.257663\pi$
$882$		0			0
$883$	−4.99084	+	4.99084i	−0.167955	+	0.167955i	−0.786080	−	0.618125i	$-0.787893\pi$
$883$	−4.99084	+	4.99084i	−0.167955	+	0.167955i	0.618125	+	0.786080i	$0.287893\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	2.39361i		−	0.0803695i	−0.999192	−	0.0401848i	$-0.987205\pi$
$887$		−	2.39361i		−	0.0803695i	0.999192	−	0.0401848i	$-0.0127947\pi$
$888$		0			0
$889$			49.5896i			1.66318i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−13.2038	+	13.2038i	−0.441849	+	0.441849i
$894$		0			0
$895$			7.87262i			0.263153i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	9.00082	+	9.00082i	0.300194	+	0.300194i
$900$		0			0
$901$	32.6771	−	32.6771i	1.08863	−	1.08863i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.37715			−0.211984
$906$		0			0
$907$	14.1617	+	14.1617i	0.470231	+	0.470231i	0.901989	−	0.431758i	$-0.142107\pi$
$907$	14.1617	+	14.1617i	0.470231	+	0.470231i	−0.431758	+	0.901989i	$0.642107\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−14.4762			−0.479618			−0.239809	−	0.970820i	$-0.577085\pi$
$911$	−14.4762			−0.479618			−0.239809	+	0.970820i	$0.577085\pi$
$912$		0			0
$913$	21.4967			0.711438
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−22.5415	−	22.5415i	−0.744387	−	0.744387i
$918$		0			0
$919$	−40.5751			−1.33845			−0.669224	−	0.743061i	$-0.733373\pi$
$919$	−40.5751			−1.33845			−0.669224	+	0.743061i	$0.733373\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.85528	−	5.85528i	0.192729	−	0.192729i
$924$		0			0
$925$	4.56150	+	4.56150i	0.149981	+	0.149981i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	51.5792i		−	1.69226i	−0.532977	−	0.846130i	$-0.678927\pi$
$929$		−	51.5792i		−	1.69226i	0.532977	−	0.846130i	$-0.321073\pi$
$930$		0			0
$931$	−3.68200	+	3.68200i	−0.120673	+	0.120673i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	28.9355i		−	0.946292i
$936$		0			0
$937$			39.7380i			1.29818i	0.760710	+	0.649092i	$0.224851\pi$
$937$			39.7380i			1.29818i	−0.760710	+	0.649092i	$0.775149\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	39.6225	−	39.6225i	1.29166	−	1.29166i	0.357894	−	0.933762i	$-0.383495\pi$
$941$	39.6225	−	39.6225i	1.29166	−	1.29166i	0.933762	−	0.357894i	$-0.116505\pi$
$942$		0			0
$943$		−	98.6150i		−	3.21135i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−25.4435	−	25.4435i	−0.826804	−	0.826804i	0.160270	−	0.987073i	$-0.448764\pi$
$947$	−25.4435	−	25.4435i	−0.826804	−	0.826804i	−0.987073	+	0.160270i	$0.948764\pi$
$948$		0			0
$949$	2.06035	−	2.06035i	0.0668818	−	0.0668818i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−44.7057			−1.44816			−0.724080	−	0.689716i	$-0.757735\pi$
$953$	−44.7057			−1.44816			−0.724080	+	0.689716i	$0.757735\pi$
$954$		0			0
$955$	−11.2453	−	11.2453i	−0.363888	−	0.363888i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−46.2227			−1.49261
$960$		0			0
$961$	28.5209			0.920028
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.90148	−	7.90148i	−0.254358	−	0.254358i
$966$		0			0
$967$	−17.1288			−0.550825			−0.275413	−	0.961326i	$-0.588814\pi$
$967$	−17.1288			−0.550825			−0.275413	+	0.961326i	$0.588814\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−25.7130	+	25.7130i	−0.825169	+	0.825169i	−0.986844	−	0.161675i	$-0.948310\pi$
$971$	−25.7130	+	25.7130i	−0.825169	+	0.825169i	0.161675	+	0.986844i	$0.448310\pi$
$972$		0			0
$973$	−0.715396	−	0.715396i	−0.0229345	−	0.0229345i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			43.9028i			1.40457i	0.711893	+	0.702287i	$0.247838\pi$
$977$			43.9028i			1.40457i	−0.711893	+	0.702287i	$0.752162\pi$
$978$		0			0
$979$	7.57253	−	7.57253i	0.242019	−	0.242019i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	30.4538i		−	0.971326i	−0.874146	−	0.485663i	$-0.838578\pi$
$983$		−	30.4538i		−	0.971326i	0.874146	−	0.485663i	$-0.161422\pi$
$984$		0			0
$985$			4.18388i			0.133310i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−37.6682	+	37.6682i	−1.19778	+	1.19778i
$990$		0			0
$991$		−	0.0569758i		−	0.00180990i	−1.00000		0.000904948i	$-0.999712\pi$
$991$		−	0.0569758i		−	0.00180990i	1.00000		0.000904948i	$-0.000288054\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.64093	−	3.64093i	−0.115425	−	0.115425i
$996$		0			0
$997$	11.4845	−	11.4845i	0.363719	−	0.363719i	−0.501461	−	0.865180i	$-0.667204\pi$
$997$	11.4845	−	11.4845i	0.363719	−	0.363719i	0.865180	+	0.501461i	$0.167204\pi$
$998$		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.431.1		32
3.2	odd	2	inner	2880.2.bl.c.431.13		32
4.3	odd	2		720.2.bl.c.611.12	yes	32
12.11	even	2		720.2.bl.c.611.5	yes	32
16.5	even	4		720.2.bl.c.251.5	✓	32
16.11	odd	4	inner	2880.2.bl.c.1871.14		32
48.5	odd	4		720.2.bl.c.251.12	yes	32
48.11	even	4	inner	2880.2.bl.c.1871.1		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bl.c.251.5	✓	32	16.5	even	4
720.2.bl.c.251.12	yes	32	48.5	odd	4
720.2.bl.c.611.5	yes	32	12.11	even	2
720.2.bl.c.611.12	yes	32	4.3	odd	2
2880.2.bl.c.431.1		32	1.1	even	1	trivial
2880.2.bl.c.431.13		32	3.2	odd	2	inner
2880.2.bl.c.1871.1		32	48.11	even	4	inner
2880.2.bl.c.1871.14		32	16.11	odd	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.bl (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{5} +3.01345 q^{7} +O(q^{10})\)	\(q+(-0.707107 - 0.707107i) q^{5} +3.01345 q^{7} +(2.52053 - 2.52053i) q^{11} +(0.848016 + 0.848016i) q^{13} +8.11753i q^{17} +(-1.76944 + 1.76944i) q^{19} +8.68585i q^{23} +1.00000i q^{25} +(-5.71652 + 5.71652i) q^{29} -1.57453i q^{31} +(-2.13083 - 2.13083i) q^{35} +(4.56150 - 4.56150i) q^{37} -11.3535 q^{41} +(4.33674 + 4.33674i) q^{43} +7.46215 q^{47} +2.08089 q^{49} +(-4.02550 - 4.02550i) q^{53} -3.56457 q^{55} +(2.67476 - 2.67476i) q^{59} +(5.84844 + 5.84844i) q^{61} -1.19928i q^{65} +(8.64918 - 8.64918i) q^{67} -6.90468i q^{71} -2.42961i q^{73} +(7.59549 - 7.59549i) q^{77} +12.0406i q^{79} +(4.26433 + 4.26433i) q^{83} +(5.73996 - 5.73996i) q^{85} +3.00434 q^{89} +(2.55545 + 2.55545i) q^{91} +2.50237 q^{95} -7.39856 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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