LMFDB - Embedded newform 2880.2.bl.b.431.10

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:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			431.10
Character	$\chi$	$=$	2880.431
Dual form			2880.2.bl.b.1871.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} -1.61527 q^{7} +O(q^{10})$	$q+(0.707107 + 0.707107i) q^{5} -1.61527 q^{7} +(-0.222034 + 0.222034i) q^{11} +(2.23818 + 2.23818i) q^{13} +2.72120i q^{17} +(-1.00945 + 1.00945i) q^{19} -1.97741i q^{23} +1.00000i q^{25} +(1.27708 - 1.27708i) q^{29} +2.63373i q^{31} +(-1.14217 - 1.14217i) q^{35} +(-1.18316 + 1.18316i) q^{37} -0.870130 q^{41} +(-3.10642 - 3.10642i) q^{43} -6.36609 q^{47} -4.39089 q^{49} +(0.945994 + 0.945994i) q^{53} -0.314004 q^{55} +(-8.08015 + 8.08015i) q^{59} +(10.5837 + 10.5837i) q^{61} +3.16526i q^{65} +(2.50886 - 2.50886i) q^{67} -3.39923i q^{71} +15.2186i q^{73} +(0.358646 - 0.358646i) q^{77} +8.65263i q^{79} +(-10.1425 - 10.1425i) q^{83} +(-1.92418 + 1.92418i) q^{85} +9.65640 q^{89} +(-3.61527 - 3.61527i) q^{91} -1.42757 q^{95} -6.56301 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 8 q^{7}+O(q^{10}) $ 24 * q + 8 * q^7	$ 24 q + 8 q^{7} + 24 q^{19} + 8 q^{37} - 48 q^{43} + 24 q^{49} - 24 q^{55} + 40 q^{61} + 40 q^{67} + 24 q^{85} - 40 q^{91} - 16 q^{97}+O(q^{100}) $ 24 * q + 8 * q^7 + 24 * q^19 + 8 * q^37 - 48 * q^43 + 24 * q^49 - 24 * q^55 + 40 * q^61 + 40 * q^67 + 24 * q^85 - 40 * q^91 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$7$	−1.61527			−0.610516			−0.305258	−	0.952270i	$-0.598743\pi$
$7$	−1.61527			−0.610516			−0.305258	+	0.952270i	$0.598743\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.222034	+	0.222034i	−0.0669458	+	0.0669458i	−0.739787	−	0.672841i	$-0.765074\pi$
$11$	−0.222034	+	0.222034i	−0.0669458	+	0.0669458i	0.672841	+	0.739787i	$0.265074\pi$
$12$		0			0
$13$	2.23818	+	2.23818i	0.620759	+	0.620759i	0.945726	−	0.324966i	$-0.105353\pi$
$13$	2.23818	+	2.23818i	0.620759	+	0.620759i	−0.324966	+	0.945726i	$0.605353\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.72120i			0.659987i	0.943983	+	0.329993i	$0.107047\pi$
$17$			2.72120i			0.659987i	−0.943983	+	0.329993i	$0.892953\pi$
$18$		0			0
$19$	−1.00945	+	1.00945i	−0.231583	+	0.231583i	−0.813353	−	0.581770i	$-0.802360\pi$
$19$	−1.00945	+	1.00945i	−0.231583	+	0.231583i	0.581770	+	0.813353i	$0.302360\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.97741i		−	0.412319i	−0.978518	−	0.206159i	$-0.933903\pi$
$23$		−	1.97741i		−	0.412319i	0.978518	−	0.206159i	$-0.0660965\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.27708	−	1.27708i	0.237148	−	0.237148i	−0.578520	−	0.815668i	$-0.696370\pi$
$29$	1.27708	−	1.27708i	0.237148	−	0.237148i	0.815668	+	0.578520i	$0.196370\pi$
$30$		0			0
$31$			2.63373i			0.473032i	0.971628	+	0.236516i	$0.0760056\pi$
$31$			2.63373i			0.473032i	−0.971628	+	0.236516i	$0.923994\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.14217	−	1.14217i	−0.193062	−	0.193062i
$36$		0			0
$37$	−1.18316	+	1.18316i	−0.194510	+	0.194510i	−0.797642	−	0.603132i	$-0.793919\pi$
$37$	−1.18316	+	1.18316i	−0.194510	+	0.194510i	0.603132	+	0.797642i	$0.293919\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.870130			−0.135891			−0.0679457	−	0.997689i	$-0.521644\pi$
$41$	−0.870130			−0.135891			−0.0679457	+	0.997689i	$0.521644\pi$
$42$		0			0
$43$	−3.10642	−	3.10642i	−0.473724	−	0.473724i	0.429393	−	0.903118i	$-0.358727\pi$
$43$	−3.10642	−	3.10642i	−0.473724	−	0.473724i	−0.903118	+	0.429393i	$0.858727\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.36609			−0.928590			−0.464295	−	0.885681i	$-0.653692\pi$
$47$	−6.36609			−0.928590			−0.464295	+	0.885681i	$0.653692\pi$
$48$		0			0
$49$	−4.39089			−0.627270
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.945994	+	0.945994i	0.129942	+	0.129942i	0.769087	−	0.639144i	$-0.220711\pi$
$53$	0.945994	+	0.945994i	0.129942	+	0.129942i	−0.639144	+	0.769087i	$0.720711\pi$
$54$		0			0
$55$	−0.314004			−0.0423402
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.08015	+	8.08015i	−1.05195	+	1.05195i	−0.0533714	+	0.998575i	$0.516997\pi$
$59$	−8.08015	+	8.08015i	−1.05195	+	1.05195i	−0.998575	+	0.0533714i	$0.983003\pi$
$60$		0			0
$61$	10.5837	+	10.5837i	1.35510	+	1.35510i	0.879848	+	0.475254i	$0.157644\pi$
$61$	10.5837	+	10.5837i	1.35510	+	1.35510i	0.475254	+	0.879848i	$0.342356\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			3.16526i			0.392603i
$66$		0			0
$67$	2.50886	−	2.50886i	0.306506	−	0.306506i	−0.537047	−	0.843553i	$-0.680460\pi$
$67$	2.50886	−	2.50886i	0.306506	−	0.306506i	0.843553	+	0.537047i	$0.180460\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.39923i		−	0.403415i	−0.979446	−	0.201707i	$-0.935351\pi$
$71$		−	3.39923i		−	0.403415i	0.979446	−	0.201707i	$-0.0646490\pi$
$72$		0			0
$73$			15.2186i			1.78120i	0.454785	+	0.890601i	$0.349716\pi$
$73$			15.2186i			1.78120i	−0.454785	+	0.890601i	$0.650284\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.358646	−	0.358646i	0.0408715	−	0.0408715i
$78$		0			0
$79$			8.65263i			0.973496i	0.873542	+	0.486748i	$0.161817\pi$
$79$			8.65263i			0.973496i	−0.873542	+	0.486748i	$0.838183\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.1425	−	10.1425i	−1.11328	−	1.11328i	−0.992704	−	0.120576i	$-0.961526\pi$
$83$	−10.1425	−	10.1425i	−1.11328	−	1.11328i	−0.120576	−	0.992704i	$-0.538474\pi$
$84$		0			0
$85$	−1.92418	+	1.92418i	−0.208706	+	0.208706i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.65640			1.02358			0.511788	−	0.859112i	$-0.328983\pi$
$89$	9.65640			1.02358			0.511788	+	0.859112i	$0.328983\pi$
$90$		0			0
$91$	−3.61527	−	3.61527i	−0.378984	−	0.378984i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.42757			−0.146466
$96$		0			0
$97$	−6.56301			−0.666373			−0.333186	−	0.942861i	$-0.608124\pi$
$97$	−6.56301			−0.666373			−0.333186	+	0.942861i	$0.608124\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.01588	−	9.01588i	−0.897113	−	0.897113i	0.0980665	−	0.995180i	$-0.468734\pi$
$101$	−9.01588	−	9.01588i	−0.897113	−	0.897113i	−0.995180	+	0.0980665i	$0.968734\pi$
$102$		0			0
$103$	13.7442			1.35426			0.677129	−	0.735864i	$-0.263224\pi$
$103$	13.7442			1.35426			0.677129	+	0.735864i	$0.263224\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.9131	+	13.9131i	−1.34503	+	1.34503i	−0.454054	+	0.890974i	$0.650023\pi$
$107$	−13.9131	+	13.9131i	−1.34503	+	1.34503i	−0.890974	+	0.454054i	$0.849977\pi$
$108$		0			0
$109$	5.43979	+	5.43979i	0.521038	+	0.521038i	0.917885	−	0.396847i	$-0.129896\pi$
$109$	5.43979	+	5.43979i	0.521038	+	0.521038i	−0.396847	+	0.917885i	$0.629896\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			9.32599i			0.877316i	0.898654	+	0.438658i	$0.144546\pi$
$113$			9.32599i			0.877316i	−0.898654	+	0.438658i	$0.855454\pi$
$114$		0			0
$115$	1.39824	−	1.39824i	0.130387	−	0.130387i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	4.39548i		−	0.402933i
$120$		0			0
$121$			10.9014i			0.991037i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$			16.3486i			1.45070i	0.688379	+	0.725351i	$0.258323\pi$
$127$			16.3486i			1.45070i	−0.688379	+	0.725351i	$0.741677\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.0522362	+	0.0522362i	0.00456390	+	0.00456390i	0.709385	−	0.704821i	$-0.248973\pi$
$131$	0.0522362	+	0.0522362i	0.00456390	+	0.00456390i	−0.704821	+	0.709385i	$0.748973\pi$
$132$		0			0
$133$	1.63053	−	1.63053i	0.141385	−	0.141385i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.96658			−0.851502			−0.425751	−	0.904840i	$-0.639990\pi$
$137$	−9.96658			−0.851502			−0.425751	+	0.904840i	$0.639990\pi$
$138$		0			0
$139$	12.7555	+	12.7555i	1.08191	+	1.08191i	0.996332	+	0.0855752i	$0.0272728\pi$
$139$	12.7555	+	12.7555i	1.08191	+	1.08191i	0.0855752	+	0.996332i	$0.472727\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.993904			−0.0831144
$144$		0			0
$145$	1.80606			0.149985
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.8030	−	11.8030i	−0.966939	−	0.966939i	0.0325320	−	0.999471i	$-0.489643\pi$
$149$	−11.8030	−	11.8030i	−0.966939	−	0.966939i	−0.999471	+	0.0325320i	$0.989643\pi$
$150$		0			0
$151$	−8.83695			−0.719141			−0.359570	−	0.933118i	$-0.617077\pi$
$151$	−8.83695			−0.719141			−0.359570	+	0.933118i	$0.617077\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.86233	+	1.86233i	−0.149586	+	0.149586i
$156$		0			0
$157$	−8.57404	−	8.57404i	−0.684283	−	0.684283i	0.276679	−	0.960962i	$-0.410766\pi$
$157$	−8.57404	−	8.57404i	−0.684283	−	0.684283i	−0.960962	+	0.276679i	$0.910766\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			3.19406i			0.251727i
$162$		0			0
$163$	0.614096	−	0.614096i	0.0480997	−	0.0480997i	−0.682648	−	0.730748i	$-0.739172\pi$
$163$	0.614096	−	0.614096i	0.0480997	−	0.0480997i	0.730748	+	0.682648i	$0.239172\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			18.1821i			1.40697i	0.710710	+	0.703485i	$0.248374\pi$
$167$			18.1821i			1.40697i	−0.710710	+	0.703485i	$0.751626\pi$
$168$		0			0
$169$		−	2.98111i		−	0.229316i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.64075	−	4.64075i	0.352830	−	0.352830i	−0.508332	−	0.861161i	$-0.669738\pi$
$173$	4.64075	−	4.64075i	0.352830	−	0.352830i	0.861161	+	0.508332i	$0.169738\pi$
$174$		0			0
$175$		−	1.61527i		−	0.122103i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.3762	−	10.3762i	−0.775553	−	0.775553i	0.203518	−	0.979071i	$-0.434762\pi$
$179$	−10.3762	−	10.3762i	−0.775553	−	0.775553i	−0.979071	+	0.203518i	$0.934762\pi$
$180$		0			0
$181$	6.84539	−	6.84539i	0.508813	−	0.508813i	−0.405349	−	0.914162i	$-0.632850\pi$
$181$	6.84539	−	6.84539i	0.508813	−	0.508813i	0.914162	+	0.405349i	$0.132850\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.67324			−0.123019
$186$		0			0
$187$	−0.604198	−	0.604198i	−0.0441833	−	0.0441833i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	22.9436			1.66014			0.830069	−	0.557661i	$-0.188301\pi$
$191$	22.9436			1.66014			0.830069	+	0.557661i	$0.188301\pi$
$192$		0			0
$193$	−19.5400			−1.40652			−0.703260	−	0.710933i	$-0.748273\pi$
$193$	−19.5400			−1.40652			−0.703260	+	0.710933i	$0.748273\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.1888	−	12.1888i	−0.868413	−	0.868413i	0.123884	−	0.992297i	$-0.460465\pi$
$197$	−12.1888	−	12.1888i	−0.868413	−	0.868413i	−0.992297	+	0.123884i	$0.960465\pi$
$198$		0			0
$199$	12.2270			0.866746			0.433373	−	0.901215i	$-0.357323\pi$
$199$	12.2270			0.866746			0.433373	+	0.901215i	$0.357323\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.06283	+	2.06283i	−0.144782	+	0.144782i
$204$		0			0
$205$	−0.615275	−	0.615275i	−0.0429726	−	0.0429726i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	0.448263i		−	0.0310070i
$210$		0			0
$211$	−4.23501	+	4.23501i	−0.291550	+	0.291550i	−0.837693	−	0.546142i	$-0.816096\pi$
$211$	−4.23501	+	4.23501i	−0.291550	+	0.291550i	0.546142	+	0.837693i	$0.316096\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	4.39314i		−	0.299609i
$216$		0			0
$217$		−	4.25420i		−	0.288794i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.09052	+	6.09052i	−0.409693	+	0.409693i
$222$		0			0
$223$			15.5126i			1.03880i	0.854532	+	0.519399i	$0.173844\pi$
$223$			15.5126i			1.03880i	−0.854532	+	0.519399i	$0.826156\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.22928	+	4.22928i	0.280707	+	0.280707i	0.833391	−	0.552684i	$-0.186396\pi$
$227$	4.22928	+	4.22928i	0.280707	+	0.280707i	−0.552684	+	0.833391i	$0.686396\pi$
$228$		0			0
$229$	−4.82199	+	4.82199i	−0.318646	+	0.318646i	−0.848247	−	0.529601i	$-0.822342\pi$
$229$	−4.82199	+	4.82199i	−0.318646	+	0.318646i	0.529601	+	0.848247i	$0.322342\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.58292			0.103700			0.0518502	−	0.998655i	$-0.483488\pi$
$233$	1.58292			0.103700			0.0518502	+	0.998655i	$0.483488\pi$
$234$		0			0
$235$	−4.50151	−	4.50151i	−0.293646	−	0.293646i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	24.9048			1.61096			0.805478	−	0.592626i	$-0.201909\pi$
$239$	24.9048			1.61096			0.805478	+	0.592626i	$0.201909\pi$
$240$		0			0
$241$	7.10161			0.457455			0.228727	−	0.973491i	$-0.426544\pi$
$241$	7.10161			0.457455			0.228727	+	0.973491i	$0.426544\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.10483	−	3.10483i	−0.198360	−	0.198360i
$246$		0			0
$247$	−4.51865			−0.287515
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.58500	−	2.58500i	0.163164	−	0.163164i	−0.620803	−	0.783967i	$-0.713193\pi$
$251$	2.58500	−	2.58500i	0.163164	−	0.163164i	0.783967	+	0.620803i	$0.213193\pi$
$252$		0			0
$253$	0.439052	+	0.439052i	0.0276030	+	0.0276030i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			21.8836i			1.36506i	0.730856	+	0.682531i	$0.239121\pi$
$257$			21.8836i			1.36506i	−0.730856	+	0.682531i	$0.760879\pi$
$258$		0			0
$259$	1.91112	−	1.91112i	0.118751	−	0.118751i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			0.412701i			0.0254482i	0.999919	+	0.0127241i	$0.00405032\pi$
$263$			0.412701i			0.0254482i	−0.999919	+	0.0127241i	$0.995950\pi$
$264$		0			0
$265$			1.33784i			0.0821827i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.5656	−	10.5656i	0.644197	−	0.644197i	−0.307387	−	0.951585i	$-0.599455\pi$
$269$	10.5656	−	10.5656i	0.644197	−	0.644197i	0.951585	+	0.307387i	$0.0994547\pi$
$270$		0			0
$271$			10.6881i			0.649253i	0.945842	+	0.324627i	$0.105239\pi$
$271$			10.6881i			0.649253i	−0.945842	+	0.324627i	$0.894761\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.222034	−	0.222034i	−0.0133892	−	0.0133892i
$276$		0			0
$277$	8.82048	−	8.82048i	0.529971	−	0.529971i	−0.390593	−	0.920564i	$-0.627730\pi$
$277$	8.82048	−	8.82048i	0.529971	−	0.529971i	0.920564	+	0.390593i	$0.127730\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−20.4617			−1.22064			−0.610321	−	0.792154i	$-0.708960\pi$
$281$	−20.4617			−1.22064			−0.610321	+	0.792154i	$0.708960\pi$
$282$		0			0
$283$	3.77081	+	3.77081i	0.224151	+	0.224151i	0.810244	−	0.586093i	$-0.199335\pi$
$283$	3.77081	+	3.77081i	0.224151	+	0.224151i	−0.586093	+	0.810244i	$0.699335\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.40550			0.0829640
$288$		0			0
$289$	9.59509			0.564417
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.48556	+	3.48556i	0.203628	+	0.203628i	0.801553	−	0.597924i	$-0.204008\pi$
$293$	3.48556	+	3.48556i	0.203628	+	0.203628i	−0.597924	+	0.801553i	$0.704008\pi$
$294$		0			0
$295$	−11.4271			−0.665309
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.42580	−	4.42580i	0.255951	−	0.255951i
$300$		0			0
$301$	5.01772	+	5.01772i	0.289216	+	0.289216i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			14.9676i			0.857042i
$306$		0			0
$307$	−4.85403	+	4.85403i	−0.277034	+	0.277034i	−0.831924	−	0.554890i	$-0.812760\pi$
$307$	−4.85403	+	4.85403i	−0.277034	+	0.277034i	0.554890	+	0.831924i	$0.312760\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			0.408055i			0.0231387i	0.999933	+	0.0115693i	$0.00368272\pi$
$311$			0.408055i			0.0231387i	−0.999933	+	0.0115693i	$0.996317\pi$
$312$		0			0
$313$			1.92296i			0.108692i	0.998522	+	0.0543462i	$0.0173075\pi$
$313$			1.92296i			0.108692i	−0.998522	+	0.0543462i	$0.982693\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.80304	+	2.80304i	−0.157435	+	0.157435i	−0.781429	−	0.623994i	$-0.785509\pi$
$317$	−2.80304	+	2.80304i	−0.157435	+	0.157435i	0.623994	+	0.781429i	$0.285509\pi$
$318$		0			0
$319$			0.567110i			0.0317521i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.74690	−	2.74690i	−0.152842	−	0.152842i
$324$		0			0
$325$	−2.23818	+	2.23818i	−0.124152	+	0.124152i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	10.2830			0.566919
$330$		0			0
$331$	−24.6354	−	24.6354i	−1.35409	−	1.35409i	−0.881035	−	0.473050i	$-0.843153\pi$
$331$	−24.6354	−	24.6354i	−1.35409	−	1.35409i	−0.473050	−	0.881035i	$-0.656847\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.54806			0.193851
$336$		0			0
$337$	−20.4305			−1.11292			−0.556461	−	0.830874i	$-0.687841\pi$
$337$	−20.4305			−1.11292			−0.556461	+	0.830874i	$0.687841\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.584778	−	0.584778i	−0.0316675	−	0.0316675i
$342$		0			0
$343$	18.3994			0.993475
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−18.6399	+	18.6399i	−1.00064	+	1.00064i	−0.000640862		1.00000i	$0.500204\pi$
$347$	−18.6399	+	18.6399i	−1.00064	+	1.00064i	−1.00000		0.000640862i	$0.999796\pi$
$348$		0			0
$349$	17.8823	+	17.8823i	0.957217	+	0.957217i	0.999122	−	0.0419049i	$-0.0133426\pi$
$349$	17.8823	+	17.8823i	0.957217	+	0.957217i	−0.0419049	+	0.999122i	$0.513343\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	3.09388i		−	0.164670i	−0.996605	−	0.0823352i	$-0.973762\pi$
$353$		−	3.09388i		−	0.164670i	0.996605	−	0.0823352i	$-0.0262378\pi$
$354$		0			0
$355$	2.40362	−	2.40362i	0.127571	−	0.127571i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	30.9370i		−	1.63279i	−0.577494	−	0.816395i	$-0.695969\pi$
$359$		−	30.9370i		−	1.63279i	0.577494	−	0.816395i	$-0.304031\pi$
$360$		0			0
$361$			16.9620i			0.892739i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−10.7612	+	10.7612i	−0.563266	+	0.563266i
$366$		0			0
$367$			11.7378i			0.612709i	0.951917	+	0.306355i	$0.0991094\pi$
$367$			11.7378i			0.612709i	−0.951917	+	0.306355i	$0.900891\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.52804	−	1.52804i	−0.0793319	−	0.0793319i
$372$		0			0
$373$	18.3260	−	18.3260i	0.948884	−	0.948884i	−0.0498713	−	0.998756i	$-0.515881\pi$
$373$	18.3260	−	18.3260i	0.948884	−	0.948884i	0.998756	+	0.0498713i	$0.0158811\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5.71666			0.294423
$378$		0			0
$379$	13.6400	+	13.6400i	0.700641	+	0.700641i	0.964548	−	0.263907i	$-0.0850112\pi$
$379$	13.6400	+	13.6400i	0.700641	+	0.700641i	−0.263907	+	0.964548i	$0.585011\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	17.4598			0.892156			0.446078	−	0.894994i	$-0.352820\pi$
$383$	17.4598			0.892156			0.446078	+	0.894994i	$0.352820\pi$
$384$		0			0
$385$	0.507202			0.0258494
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.5528	−	13.5528i	−0.687153	−	0.687153i	0.274449	−	0.961602i	$-0.411505\pi$
$389$	−13.5528	−	13.5528i	−0.687153	−	0.687153i	−0.961602	+	0.274449i	$0.911505\pi$
$390$		0			0
$391$	5.38092			0.272125
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.11833	+	6.11833i	−0.307847	+	0.307847i
$396$		0			0
$397$	−24.5611	−	24.5611i	−1.23269	−	1.23269i	−0.962927	−	0.269761i	$-0.913055\pi$
$397$	−24.5611	−	24.5611i	−1.23269	−	1.23269i	−0.269761	−	0.962927i	$-0.586945\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			25.6907i			1.28293i	0.767152	+	0.641465i	$0.221673\pi$
$401$			25.6907i			1.28293i	−0.767152	+	0.641465i	$0.778327\pi$
$402$		0			0
$403$	−5.89476	+	5.89476i	−0.293639	+	0.293639i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	0.525402i		−	0.0260432i
$408$		0			0
$409$		−	21.2837i		−	1.05241i	−0.850357	−	0.526206i	$-0.823614\pi$
$409$		−	21.2837i		−	1.05241i	0.850357	−	0.526206i	$-0.176386\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	13.0517	−	13.0517i	0.642230	−	0.642230i
$414$		0			0
$415$		−	14.3436i		−	0.704100i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.7685	+	12.7685i	0.623783	+	0.623783i	0.946497	−	0.322713i	$-0.104595\pi$
$419$	12.7685	+	12.7685i	0.623783	+	0.623783i	−0.322713	+	0.946497i	$0.604595\pi$
$420$		0			0
$421$	17.3514	−	17.3514i	0.845655	−	0.845655i	−0.143932	−	0.989588i	$-0.545975\pi$
$421$	17.3514	−	17.3514i	0.845655	−	0.845655i	0.989588	+	0.143932i	$0.0459747\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.72120			−0.131997
$426$		0			0
$427$	−17.0956	−	17.0956i	−0.827313	−	0.827313i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.0732942			−0.00353046			−0.00176523	−	0.999998i	$-0.500562\pi$
$431$	−0.0732942			−0.00353046			−0.00176523	+	0.999998i	$0.500562\pi$
$432$		0			0
$433$	−1.56003			−0.0749702			−0.0374851	−	0.999297i	$-0.511935\pi$
$433$	−1.56003			−0.0749702			−0.0374851	+	0.999297i	$0.511935\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.99609	+	1.99609i	0.0954860	+	0.0954860i
$438$		0			0
$439$	−31.2288			−1.49047			−0.745235	−	0.666802i	$-0.767663\pi$
$439$	−31.2288			−1.49047			−0.745235	+	0.666802i	$0.767663\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	14.0427	−	14.0427i	0.667187	−	0.667187i	−0.289877	−	0.957064i	$-0.593614\pi$
$443$	14.0427	−	14.0427i	0.667187	−	0.667187i	0.957064	+	0.289877i	$0.0936143\pi$
$444$		0			0
$445$	6.82811	+	6.82811i	0.323683	+	0.323683i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	6.14224i		−	0.289870i	−0.989441	−	0.144935i	$-0.953703\pi$
$449$		−	6.14224i		−	0.289870i	0.989441	−	0.144935i	$-0.0462973\pi$
$450$		0			0
$451$	0.193198	−	0.193198i	0.00909736	−	0.00909736i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	5.11277i		−	0.239690i
$456$		0			0
$457$		−	22.3962i		−	1.04765i	−0.851826	−	0.523825i	$-0.824505\pi$
$457$		−	22.3962i		−	1.04765i	0.851826	−	0.523825i	$-0.175495\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.15833	−	4.15833i	0.193673	−	0.193673i	−0.603608	−	0.797281i	$-0.706271\pi$
$461$	4.15833	−	4.15833i	0.193673	−	0.193673i	0.797281	+	0.603608i	$0.206271\pi$
$462$		0			0
$463$			20.5950i			0.957132i	0.878052	+	0.478566i	$0.158843\pi$
$463$			20.5950i			0.957132i	−0.878052	+	0.478566i	$0.841157\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9.95417	−	9.95417i	−0.460624	−	0.460624i	0.438236	−	0.898860i	$-0.355603\pi$
$467$	−9.95417	−	9.95417i	−0.460624	−	0.460624i	−0.898860	+	0.438236i	$0.855603\pi$
$468$		0			0
$469$	−4.05249	+	4.05249i	−0.187127	+	0.187127i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.37946			0.0634277
$474$		0			0
$475$	−1.00945	−	1.00945i	−0.0463166	−	0.0463166i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	33.6982			1.53971			0.769854	−	0.638220i	$-0.220329\pi$
$479$	33.6982			1.53971			0.769854	+	0.638220i	$0.220329\pi$
$480$		0			0
$481$	−5.29623			−0.241488
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.64075	−	4.64075i	−0.210726	−	0.210726i
$486$		0			0
$487$	27.0521			1.22585			0.612924	−	0.790142i	$-0.289993\pi$
$487$	27.0521			1.22585			0.612924	+	0.790142i	$0.289993\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10.9603	−	10.9603i	0.494631	−	0.494631i	−0.415131	−	0.909762i	$-0.636264\pi$
$491$	10.9603	−	10.9603i	0.494631	−	0.494631i	0.909762	+	0.415131i	$0.136264\pi$
$492$		0			0
$493$	3.47518	+	3.47518i	0.156514	+	0.156514i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			5.49069i			0.246291i
$498$		0			0
$499$	8.99453	−	8.99453i	0.402651	−	0.402651i	−0.476515	−	0.879166i	$-0.658100\pi$
$499$	8.99453	−	8.99453i	0.402651	−	0.402651i	0.879166	+	0.476515i	$0.158100\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	5.12105i		−	0.228337i	−0.993461	−	0.114168i	$-0.963580\pi$
$503$		−	5.12105i		−	0.228337i	0.993461	−	0.114168i	$-0.0364203\pi$
$504$		0			0
$505$		−	12.7504i		−	0.567384i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	11.3121	−	11.3121i	0.501400	−	0.501400i	−0.410473	−	0.911873i	$-0.634636\pi$
$509$	11.3121	−	11.3121i	0.501400	−	0.501400i	0.911873	+	0.410473i	$0.134636\pi$
$510$		0			0
$511$		−	24.5822i		−	1.08745i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	9.71863	+	9.71863i	0.428254	+	0.428254i
$516$		0			0
$517$	1.41349	−	1.41349i	0.0621652	−	0.0621652i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.487851			−0.0213731			−0.0106866	−	0.999943i	$-0.503402\pi$
$521$	−0.487851			−0.0213731			−0.0106866	+	0.999943i	$0.503402\pi$
$522$		0			0
$523$	−22.7818	−	22.7818i	−0.996178	−	0.996178i	0.00381502	−	0.999993i	$-0.498786\pi$
$523$	−22.7818	−	22.7818i	−0.996178	−	0.996178i	−0.999993	+	0.00381502i	$0.998786\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.16690			−0.312195
$528$		0			0
$529$	19.0898			0.829993
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.94751	−	1.94751i	−0.0843559	−	0.0843559i
$534$		0			0
$535$	−19.6761			−0.850670
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.974927	−	0.974927i	0.0419931	−	0.0419931i
$540$		0			0
$541$	−8.69246	−	8.69246i	−0.373718	−	0.373718i	0.495111	−	0.868829i	$-0.335127\pi$
$541$	−8.69246	−	8.69246i	−0.373718	−	0.373718i	−0.868829	+	0.495111i	$0.835127\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			7.69303i			0.329533i
$546$		0			0
$547$	8.43231	−	8.43231i	0.360539	−	0.360539i	−0.503472	−	0.864011i	$-0.667944\pi$
$547$	8.43231	−	8.43231i	0.360539	−	0.360539i	0.864011	+	0.503472i	$0.167944\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			2.57829i			0.109839i
$552$		0			0
$553$		−	13.9764i		−	0.594336i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.142402	−	0.142402i	0.00603375	−	0.00603375i	−0.704083	−	0.710117i	$-0.748642\pi$
$557$	0.142402	−	0.142402i	0.00603375	−	0.00603375i	0.710117	+	0.704083i	$0.248642\pi$
$558$		0			0
$559$		−	13.9054i		−	0.588137i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12.9174	+	12.9174i	0.544403	+	0.544403i	0.924816	−	0.380414i	$-0.124218\pi$
$563$	12.9174	+	12.9174i	0.544403	+	0.544403i	−0.380414	+	0.924816i	$0.624218\pi$
$564$		0			0
$565$	−6.59447	+	6.59447i	−0.277432	+	0.277432i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	38.1354			1.59872			0.799360	−	0.600852i	$-0.205172\pi$
$569$	38.1354			1.59872			0.799360	+	0.600852i	$0.205172\pi$
$570$		0			0
$571$	−1.44110	−	1.44110i	−0.0603084	−	0.0603084i	0.676309	−	0.736618i	$-0.263578\pi$
$571$	−1.44110	−	1.44110i	−0.0603084	−	0.0603084i	−0.736618	+	0.676309i	$0.763578\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.97741			0.0824637
$576$		0			0
$577$	−35.2224			−1.46633			−0.733164	−	0.680052i	$-0.761957\pi$
$577$	−35.2224			−1.46633			−0.733164	+	0.680052i	$0.761957\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	16.3829	+	16.3829i	0.679676	+	0.679676i
$582$		0			0
$583$	−0.420086			−0.0173982
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9.42412	+	9.42412i	−0.388975	+	0.388975i	−0.874322	−	0.485347i	$-0.838693\pi$
$587$	−9.42412	+	9.42412i	−0.388975	+	0.388975i	0.485347	+	0.874322i	$0.338693\pi$
$588$		0			0
$589$	−2.65861	−	2.65861i	−0.109546	−	0.109546i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	47.6546i		−	1.95694i	−0.206394	−	0.978469i	$-0.566173\pi$
$593$		−	47.6546i		−	1.95694i	0.206394	−	0.978469i	$-0.433827\pi$
$594$		0			0
$595$	3.10807	−	3.10807i	0.127419	−	0.127419i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	10.4812i		−	0.428250i	−0.976806	−	0.214125i	$-0.931310\pi$
$599$		−	10.4812i		−	0.428250i	0.976806	−	0.214125i	$-0.0686899\pi$
$600$		0			0
$601$		−	6.31247i		−	0.257491i	−0.991678	−	0.128746i	$-0.958905\pi$
$601$		−	6.31247i		−	0.257491i	0.991678	−	0.128746i	$-0.0410951\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.70846	+	7.70846i	−0.313393	+	0.313393i
$606$		0			0
$607$		−	42.1907i		−	1.71247i	−0.516589	−	0.856234i	$-0.672798\pi$
$607$		−	42.1907i		−	1.71247i	0.516589	−	0.856234i	$-0.327202\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−14.2485	−	14.2485i	−0.576431	−	0.576431i
$612$		0			0
$613$	16.3819	−	16.3819i	0.661659	−	0.661659i	−0.294112	−	0.955771i	$-0.595024\pi$
$613$	16.3819	−	16.3819i	0.661659	−	0.661659i	0.955771	+	0.294112i	$0.0950238\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.80136			−0.112778			−0.0563892	−	0.998409i	$-0.517959\pi$
$617$	−2.80136			−0.112778			−0.0563892	+	0.998409i	$0.517959\pi$
$618$		0			0
$619$	16.0566	+	16.0566i	0.645367	+	0.645367i	0.951870	−	0.306502i	$-0.0991588\pi$
$619$	16.0566	+	16.0566i	0.645367	+	0.645367i	−0.306502	+	0.951870i	$0.599159\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−15.5977			−0.624910
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.21960	−	3.21960i	−0.128374	−	0.128374i
$630$		0			0
$631$	19.5230			0.777199			0.388599	−	0.921407i	$-0.372959\pi$
$631$	19.5230			0.777199			0.388599	+	0.921407i	$0.372959\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11.5602	+	11.5602i	−0.458752	+	0.458752i
$636$		0			0
$637$	−9.82759	−	9.82759i	−0.389383	−	0.389383i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	22.2088i		−	0.877194i	−0.898684	−	0.438597i	$-0.855475\pi$
$641$		−	22.2088i		−	0.877194i	0.898684	−	0.438597i	$-0.144525\pi$
$642$		0			0
$643$	27.8111	−	27.8111i	1.09676	−	1.09676i	0.101977	−	0.994787i	$-0.467483\pi$
$643$	27.8111	−	27.8111i	1.09676	−	1.09676i	0.994787	−	0.101977i	$-0.0325169\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			28.3505i			1.11457i	0.830321	+	0.557286i	$0.188157\pi$
$647$			28.3505i			1.11457i	−0.830321	+	0.557286i	$0.811843\pi$
$648$		0			0
$649$		−	3.58814i		−	0.140847i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	8.74983	−	8.74983i	0.342407	−	0.342407i	−0.514864	−	0.857272i	$-0.672158\pi$
$653$	8.74983	−	8.74983i	0.342407	−	0.342407i	0.857272	+	0.514864i	$0.172158\pi$
$654$		0			0
$655$			0.0738731i			0.00288646i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.732843	+	0.732843i	0.0285475	+	0.0285475i	0.721236	−	0.692689i	$-0.243574\pi$
$659$	0.732843	+	0.732843i	0.0285475	+	0.0285475i	−0.692689	+	0.721236i	$0.743574\pi$
$660$		0			0
$661$	−8.00818	+	8.00818i	−0.311482	+	0.311482i	−0.845484	−	0.534001i	$-0.820688\pi$
$661$	−8.00818	+	8.00818i	−0.311482	+	0.311482i	0.534001	+	0.845484i	$0.320688\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.30592			0.0894199
$666$		0			0
$667$	−2.52531	−	2.52531i	−0.0977803	−	0.0977803i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.69988			−0.181437
$672$		0			0
$673$	9.28534			0.357924			0.178962	−	0.983856i	$-0.442726\pi$
$673$	9.28534			0.357924			0.178962	+	0.983856i	$0.442726\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	30.9535	+	30.9535i	1.18964	+	1.18964i	0.977167	+	0.212471i	$0.0681510\pi$
$677$	30.9535	+	30.9535i	1.18964	+	1.18964i	0.212471	+	0.977167i	$0.431849\pi$
$678$		0			0
$679$	10.6011			0.406831
$680$		0			0
$681$		0			0
$682$		0			0
$683$	21.1224	−	21.1224i	0.808226	−	0.808226i	−0.176139	−	0.984365i	$-0.556361\pi$
$683$	21.1224	−	21.1224i	0.808226	−	0.808226i	0.984365	+	0.176139i	$0.0563609\pi$
$684$		0			0
$685$	−7.04744	−	7.04744i	−0.269269	−	0.269269i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			4.23461i			0.161326i
$690$		0			0
$691$	14.0845	−	14.0845i	0.535801	−	0.535801i	−0.386492	−	0.922293i	$-0.626313\pi$
$691$	14.0845	−	14.0845i	0.535801	−	0.535801i	0.922293	+	0.386492i	$0.126313\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			18.0390i			0.684258i
$696$		0			0
$697$		−	2.36779i		−	0.0896866i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	17.3494	−	17.3494i	0.655279	−	0.655279i	−0.298980	−	0.954259i	$-0.596646\pi$
$701$	17.3494	−	17.3494i	0.655279	−	0.655279i	0.954259	+	0.298980i	$0.0966465\pi$
$702$		0			0
$703$		−	2.38867i		−	0.0900904i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	14.5631	+	14.5631i	0.547702	+	0.547702i
$708$		0			0
$709$	−23.5874	+	23.5874i	−0.885843	+	0.885843i	−0.994121	−	0.108278i	$-0.965466\pi$
$709$	−23.5874	+	23.5874i	−0.885843	+	0.885843i	0.108278	+	0.994121i	$0.465466\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5.20797			0.195040
$714$		0			0
$715$	−0.702796	−	0.702796i	−0.0262831	−	0.0262831i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−24.8339			−0.926148			−0.463074	−	0.886320i	$-0.653254\pi$
$719$	−24.8339			−0.926148			−0.463074	+	0.886320i	$0.653254\pi$
$720$		0			0
$721$	−22.2007			−0.826797
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.27708	+	1.27708i	0.0474295	+	0.0474295i
$726$		0			0
$727$	−30.3495			−1.12560			−0.562801	−	0.826592i	$-0.690276\pi$
$727$	−30.3495			−1.12560			−0.562801	+	0.826592i	$0.690276\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8.45317	−	8.45317i	0.312652	−	0.312652i
$732$		0			0
$733$	−5.52943	−	5.52943i	−0.204234	−	0.204234i	0.597577	−	0.801811i	$-0.296130\pi$
$733$	−5.52943	−	5.52943i	−0.204234	−	0.204234i	−0.801811	+	0.597577i	$0.796130\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1.11410i			0.0410385i
$738$		0			0
$739$	23.2339	−	23.2339i	0.854674	−	0.854674i	−0.136030	−	0.990705i	$-0.543435\pi$
$739$	23.2339	−	23.2339i	0.854674	−	0.854674i	0.990705	+	0.136030i	$0.0434345\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			31.6110i			1.15970i	0.814725	+	0.579848i	$0.196888\pi$
$743$			31.6110i			1.15970i	−0.814725	+	0.579848i	$0.803112\pi$
$744$		0			0
$745$		−	16.6919i		−	0.611546i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	22.4734	−	22.4734i	0.821162	−	0.821162i
$750$		0			0
$751$		−	2.13706i		−	0.0779823i	−0.999240	−	0.0389911i	$-0.987586\pi$
$751$		−	2.13706i		−	0.0779823i	0.999240	−	0.0389911i	$-0.0124144\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.24867	−	6.24867i	−0.227412	−	0.227412i
$756$		0			0
$757$	−0.568578	+	0.568578i	−0.0206653	+	0.0206653i	−0.717364	−	0.696699i	$-0.754651\pi$
$757$	−0.568578	+	0.568578i	−0.0206653	+	0.0206653i	0.696699	+	0.717364i	$0.254651\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	31.0124			1.12420			0.562099	−	0.827070i	$-0.309994\pi$
$761$	31.0124			1.12420			0.562099	+	0.827070i	$0.309994\pi$
$762$		0			0
$763$	−8.78676	−	8.78676i	−0.318102	−	0.318102i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−36.1697			−1.30601
$768$		0			0
$769$	10.4145			0.375555			0.187778	−	0.982212i	$-0.439872\pi$
$769$	10.4145			0.375555			0.187778	+	0.982212i	$0.439872\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4.79164	+	4.79164i	0.172343	+	0.172343i	0.788008	−	0.615665i	$-0.211112\pi$
$773$	4.79164	+	4.79164i	0.172343	+	0.172343i	−0.615665	+	0.788008i	$0.711112\pi$
$774$		0			0
$775$	−2.63373			−0.0946064
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.878350	−	0.878350i	0.0314702	−	0.0314702i
$780$		0			0
$781$	0.754745	+	0.754745i	0.0270069	+	0.0270069i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	12.1255i		−	0.432779i
$786$		0			0
$787$	−1.31041	+	1.31041i	−0.0467112	+	0.0467112i	−0.730077	−	0.683365i	$-0.760516\pi$
$787$	−1.31041	+	1.31041i	−0.0467112	+	0.0467112i	0.683365	+	0.730077i	$0.260516\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	15.0640i		−	0.535616i
$792$		0			0
$793$			47.3764i			1.68239i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−25.2095	+	25.2095i	−0.892965	+	0.892965i	−0.994801	−	0.101836i	$-0.967528\pi$
$797$	−25.2095	+	25.2095i	−0.892965	+	0.892965i	0.101836	+	0.994801i	$0.467528\pi$
$798$		0			0
$799$		−	17.3234i		−	0.612857i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.37905	−	3.37905i	−0.119244	−	0.119244i
$804$		0			0
$805$	−2.25854	+	2.25854i	−0.0796031	+	0.0796031i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	49.0623			1.72494			0.862469	−	0.506109i	$-0.168917\pi$
$809$	49.0623			1.72494			0.862469	+	0.506109i	$0.168917\pi$
$810$		0			0
$811$	−24.5170	−	24.5170i	−0.860908	−	0.860908i	0.130536	−	0.991444i	$-0.458330\pi$
$811$	−24.5170	−	24.5170i	−0.860908	−	0.860908i	−0.991444	+	0.130536i	$0.958330\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.868463			0.0304209
$816$		0			0
$817$	6.27153			0.219413
$818$		0			0
$819$		0			0
$820$		0			0
$821$	30.9736	+	30.9736i	1.08099	+	1.08099i	0.996418	+	0.0845701i	$0.0269517\pi$
$821$	30.9736	+	30.9736i	1.08099	+	1.08099i	0.0845701	+	0.996418i	$0.473048\pi$
$822$		0			0
$823$	7.87324			0.274444			0.137222	−	0.990540i	$-0.456183\pi$
$823$	7.87324			0.274444			0.137222	+	0.990540i	$0.456183\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.61513	−	1.61513i	0.0561635	−	0.0561635i	−0.678467	−	0.734631i	$-0.737355\pi$
$827$	1.61513	−	1.61513i	0.0561635	−	0.0561635i	0.734631	+	0.678467i	$0.237355\pi$
$828$		0			0
$829$	−11.9759	−	11.9759i	−0.415939	−	0.415939i	0.467863	−	0.883801i	$-0.345024\pi$
$829$	−11.9759	−	11.9759i	−0.415939	−	0.415939i	−0.883801	+	0.467863i	$0.845024\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	11.9485i		−	0.413990i
$834$		0			0
$835$	−12.8567	+	12.8567i	−0.444923	+	0.444923i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	44.0118i		−	1.51946i	−0.650240	−	0.759729i	$-0.725332\pi$
$839$		−	44.0118i		−	1.51946i	0.650240	−	0.759729i	$-0.274668\pi$
$840$		0			0
$841$			25.7381i			0.887522i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.10796	−	2.10796i	0.0725160	−	0.0725160i
$846$		0			0
$847$		−	17.6088i		−	0.605044i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.33959	+	2.33959i	0.0802000	+	0.0802000i
$852$		0			0
$853$	16.4933	−	16.4933i	0.564720	−	0.564720i	−0.365925	−	0.930645i	$-0.619247\pi$
$853$	16.4933	−	16.4933i	0.564720	−	0.564720i	0.930645	+	0.365925i	$0.119247\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.83536			−0.233491			−0.116746	−	0.993162i	$-0.537246\pi$
$857$	−6.83536			−0.233491			−0.116746	+	0.993162i	$0.537246\pi$
$858$		0			0
$859$	40.4961	+	40.4961i	1.38171	+	1.38171i	0.841587	+	0.540121i	$0.181622\pi$
$859$	40.4961	+	40.4961i	1.38171	+	1.38171i	0.540121	+	0.841587i	$0.318378\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	24.5094			0.834311			0.417156	−	0.908835i	$-0.363027\pi$
$863$	24.5094			0.834311			0.417156	+	0.908835i	$0.363027\pi$
$864$		0			0
$865$	6.56301			0.223149
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.92118	−	1.92118i	−0.0651715	−	0.0651715i
$870$		0			0
$871$	11.2305			0.380533
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.14217	−	1.14217i	0.0386125	−	0.0386125i
$876$		0			0
$877$	16.6770	+	16.6770i	0.563141	+	0.563141i	0.930198	−	0.367057i	$-0.119635\pi$
$877$	16.6770	+	16.6770i	0.563141	+	0.563141i	−0.367057	+	0.930198i	$0.619635\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	4.38490i		−	0.147731i	−0.997268	−	0.0738655i	$-0.976466\pi$
$881$		−	4.38490i		−	0.147731i	0.997268	−	0.0738655i	$-0.0235335\pi$
$882$		0			0
$883$	−38.6918	+	38.6918i	−1.30208	+	1.30208i	−0.375100	+	0.926984i	$0.622392\pi$
$883$	−38.6918	+	38.6918i	−1.30208	+	1.30208i	−0.926984	+	0.375100i	$0.877608\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			31.6694i			1.06335i	0.846947	+	0.531677i	$0.178438\pi$
$887$			31.6694i			1.06335i	−0.846947	+	0.531677i	$0.821562\pi$
$888$		0			0
$889$		−	26.4075i		−	0.885678i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6.42623	−	6.42623i	0.215046	−	0.215046i
$894$		0			0
$895$		−	14.6741i		−	0.490503i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.36348	+	3.36348i	0.112178	+	0.112178i
$900$		0			0
$901$	−2.57424	+	2.57424i	−0.0857602	+	0.0857602i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.68084			0.321802
$906$		0			0
$907$	−13.5135	−	13.5135i	−0.448707	−	0.448707i	0.446218	−	0.894925i	$-0.352771\pi$
$907$	−13.5135	−	13.5135i	−0.448707	−	0.448707i	−0.894925	+	0.446218i	$0.852771\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−36.9869			−1.22543			−0.612716	−	0.790303i	$-0.709923\pi$
$911$	−36.9869			−1.22543			−0.612716	+	0.790303i	$0.709923\pi$
$912$		0			0
$913$	4.50394			0.149059
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.0843758	−	0.0843758i	−0.00278633	−	0.00278633i
$918$		0			0
$919$	−31.7315			−1.04673			−0.523363	−	0.852110i	$-0.675323\pi$
$919$	−31.7315			−1.04673			−0.523363	+	0.852110i	$0.675323\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	7.60809	−	7.60809i	0.250423	−	0.250423i
$924$		0			0
$925$	−1.18316	−	1.18316i	−0.0389020	−	0.0389020i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1.71396i		−	0.0562333i	−0.999605	−	0.0281167i	$-0.991049\pi$
$929$		−	1.71396i		−	0.0562333i	0.999605	−	0.0281167i	$-0.00895099\pi$
$930$		0			0
$931$	4.43237	−	4.43237i	0.145265	−	0.145265i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	0.854465i		−	0.0279440i
$936$		0			0
$937$		−	7.46789i		−	0.243965i	−0.992532	−	0.121983i	$-0.961075\pi$
$937$		−	7.46789i		−	0.243965i	0.992532	−	0.121983i	$-0.0389252\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−21.8502	+	21.8502i	−0.712296	+	0.712296i	−0.967015	−	0.254719i	$-0.918017\pi$
$941$	−21.8502	+	21.8502i	−0.712296	+	0.712296i	0.254719	+	0.967015i	$0.418017\pi$
$942$		0			0
$943$			1.72060i			0.0560306i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−24.8599	−	24.8599i	−0.807837	−	0.807837i	0.176469	−	0.984306i	$-0.443532\pi$
$947$	−24.8599	−	24.8599i	−0.807837	−	0.807837i	−0.984306	+	0.176469i	$0.943532\pi$
$948$		0			0
$949$	−34.0620	+	34.0620i	−1.10570	+	1.10570i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−51.3477			−1.66332			−0.831658	−	0.555288i	$-0.812608\pi$
$953$	−51.3477			−1.66332			−0.831658	+	0.555288i	$0.812608\pi$
$954$		0			0
$955$	16.2235	+	16.2235i	0.524981	+	0.524981i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	16.0988			0.519856
$960$		0			0
$961$	24.0635			0.776241
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−13.8169	−	13.8169i	−0.444781	−	0.444781i
$966$		0			0
$967$	−32.0984			−1.03222			−0.516108	−	0.856524i	$-0.672620\pi$
$967$	−32.0984			−1.03222			−0.516108	+	0.856524i	$0.672620\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	27.8651	−	27.8651i	0.894234	−	0.894234i	−0.100685	−	0.994918i	$-0.532103\pi$
$971$	27.8651	−	27.8651i	0.894234	−	0.894234i	0.994918	+	0.100685i	$0.0321034\pi$
$972$		0			0
$973$	−20.6036	−	20.6036i	−0.660522	−	0.660522i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			8.60870i			0.275417i	0.990473	+	0.137708i	$0.0439737\pi$
$977$			8.60870i			0.275417i	−0.990473	+	0.137708i	$0.956026\pi$
$978$		0			0
$979$	−2.14405	+	2.14405i	−0.0685242	+	0.0685242i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			10.0033i			0.319057i	0.987193	+	0.159528i	$0.0509973\pi$
$983$			10.0033i			0.319057i	−0.987193	+	0.159528i	$0.949003\pi$
$984$		0			0
$985$		−	17.2375i		−	0.549232i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6.14266	+	6.14266i	−0.195325	+	0.195325i
$990$		0			0
$991$		−	12.6490i		−	0.401807i	−0.979611	−	0.200904i	$-0.935612\pi$
$991$		−	12.6490i		−	0.401807i	0.979611	−	0.200904i	$-0.0643878\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.64577	+	8.64577i	0.274089	+	0.274089i
$996$		0			0
$997$	−7.40190	+	7.40190i	−0.234420	+	0.234420i	−0.814535	−	0.580114i	$-0.803008\pi$
$997$	−7.40190	+	7.40190i	−0.234420	+	0.234420i	0.580114	+	0.814535i	$0.303008\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bl.b.431.10		24
3.2	odd	2	inner	2880.2.bl.b.431.4		24
4.3	odd	2		720.2.bl.b.611.1	yes	24
12.11	even	2		720.2.bl.b.611.12	yes	24
16.5	even	4		720.2.bl.b.251.12	yes	24
16.11	odd	4	inner	2880.2.bl.b.1871.4		24
48.5	odd	4		720.2.bl.b.251.1	✓	24
48.11	even	4	inner	2880.2.bl.b.1871.10		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bl.b.251.1	✓	24	48.5	odd	4
720.2.bl.b.251.12	yes	24	16.5	even	4
720.2.bl.b.611.1	yes	24	4.3	odd	2
720.2.bl.b.611.12	yes	24	12.11	even	2
2880.2.bl.b.431.4		24	3.2	odd	2	inner
2880.2.bl.b.431.10		24	1.1	even	1	trivial
2880.2.bl.b.1871.4		24	16.11	odd	4	inner
2880.2.bl.b.1871.10		24	48.11	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.bl (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} -1.61527 q^{7} +O(q^{10})\)	\(q+(0.707107 + 0.707107i) q^{5} -1.61527 q^{7} +(-0.222034 + 0.222034i) q^{11} +(2.23818 + 2.23818i) q^{13} +2.72120i q^{17} +(-1.00945 + 1.00945i) q^{19} -1.97741i q^{23} +1.00000i q^{25} +(1.27708 - 1.27708i) q^{29} +2.63373i q^{31} +(-1.14217 - 1.14217i) q^{35} +(-1.18316 + 1.18316i) q^{37} -0.870130 q^{41} +(-3.10642 - 3.10642i) q^{43} -6.36609 q^{47} -4.39089 q^{49} +(0.945994 + 0.945994i) q^{53} -0.314004 q^{55} +(-8.08015 + 8.08015i) q^{59} +(10.5837 + 10.5837i) q^{61} +3.16526i q^{65} +(2.50886 - 2.50886i) q^{67} -3.39923i q^{71} +15.2186i q^{73} +(0.358646 - 0.358646i) q^{77} +8.65263i q^{79} +(-10.1425 - 10.1425i) q^{83} +(-1.92418 + 1.92418i) q^{85} +9.65640 q^{89} +(-3.61527 - 3.61527i) q^{91} -1.42757 q^{95} -6.56301 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8 q^{7}+O(q^{10}) \) 24 * q + 8 * q^7	\( 24 q + 8 q^{7} + 24 q^{19} + 8 q^{37} - 48 q^{43} + 24 q^{49} - 24 q^{55} + 40 q^{61} + 40 q^{67} + 24 q^{85} - 40 q^{91} - 16 q^{97}+O(q^{100}) \) 24 * q + 8 * q^7 + 24 * q^19 + 8 * q^37 - 48 * q^43 + 24 * q^49 - 24 * q^55 + 40 * q^61 + 40 * q^67 + 24 * q^85 - 40 * q^91 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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