LMFDB - Embedded newform 2880.2.t.b.721.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))

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

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

chi := DirichletCharacter("2880.721");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.18939904.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 $ x^8 - 4x^7 + 14x^6 - 28x^5 + 43x^4 - 44x^3 + 30x^2 - 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			721.2
Root			$0.500000 - 1.44392i$ of defining polynomial
Character	$\chi$	$=$	2880.721
Dual form			2880.2.t.b.2161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{5} +1.41421i q^{7} +O(q^{10})$	$q+(-0.707107 + 0.707107i) q^{5} +1.41421i q^{7} +(4.22274 - 4.22274i) q^{11} +(1.33490 + 1.33490i) q^{13} +1.14343 q^{17} +(-1.05941 - 1.05941i) q^{19} -7.77568i q^{23} -1.00000i q^{25} +(-2.94059 - 2.94059i) q^{29} -0.389604 q^{31} +(-1.00000 - 1.00000i) q^{35} +(-4.28216 + 4.28216i) q^{37} +5.45844i q^{41} +(4.91245 - 4.91245i) q^{43} -10.7324 q^{47} +5.00000 q^{49} +(0.863230 - 0.863230i) q^{53} +5.97186i q^{55} +(8.50961 - 8.50961i) q^{59} +(2.22746 + 2.22746i) q^{61} -1.88784 q^{65} +(-2.94725 - 2.94725i) q^{67} +3.27391i q^{71} -1.84138i q^{73} +(5.97186 + 5.97186i) q^{77} +11.3861 q^{79} +(11.1153 + 11.1153i) q^{83} +(-0.808530 + 0.808530i) q^{85} -18.6533i q^{89} +(-1.88784 + 1.88784i) q^{91} +1.49824 q^{95} -5.44902 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{11} + 8 q^{13} - 8 q^{17} - 8 q^{19} - 24 q^{29} - 8 q^{31} - 8 q^{35} - 8 q^{37} + 40 q^{49} + 8 q^{59} - 16 q^{61} + 8 q^{65} + 8 q^{77} + 40 q^{79} + 32 q^{83} + 8 q^{85} + 8 q^{91} - 16 q^{95} - 48 q^{97}+O(q^{100}) $ 8 * q + 8 * q^11 + 8 * q^13 - 8 * q^17 - 8 * q^19 - 24 * q^29 - 8 * q^31 - 8 * q^35 - 8 * q^37 + 40 * q^49 + 8 * q^59 - 16 * q^61 + 8 * q^65 + 8 * q^77 + 40 * q^79 + 32 * q^83 + 8 * q^85 + 8 * q^91 - 16 * q^95 - 48 * 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
$7$			1.41421i			0.534522i	0.963624	+	0.267261i	$0.0861187\pi$
$7$			1.41421i			0.534522i	−0.963624	+	0.267261i	$0.913881\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.22274	−	4.22274i	1.27321	−	1.27321i	0.328808	−	0.944397i	$-0.393353\pi$
$11$	4.22274	−	4.22274i	1.27321	−	1.27321i	0.944397	−	0.328808i	$-0.106647\pi$
$12$		0			0
$13$	1.33490	+	1.33490i	0.370236	+	0.370236i	0.867563	−	0.497327i	$-0.165685\pi$
$13$	1.33490	+	1.33490i	0.370236	+	0.370236i	−0.497327	+	0.867563i	$0.665685\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.14343			0.277323			0.138662	−	0.990340i	$-0.455720\pi$
$17$	1.14343			0.277323			0.138662	+	0.990340i	$0.455720\pi$
$18$		0			0
$19$	−1.05941	−	1.05941i	−0.243046	−	0.243046i	0.575063	−	0.818109i	$-0.304977\pi$
$19$	−1.05941	−	1.05941i	−0.243046	−	0.243046i	−0.818109	+	0.575063i	$0.804977\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.77568i		−	1.62134i	−0.585503	−	0.810671i	$-0.699103\pi$
$23$		−	7.77568i		−	1.62134i	0.585503	−	0.810671i	$-0.300897\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.94059	−	2.94059i	−0.546053	−	0.546053i	0.379243	−	0.925297i	$-0.376184\pi$
$29$	−2.94059	−	2.94059i	−0.546053	−	0.546053i	−0.925297	+	0.379243i	$0.876184\pi$
$30$		0			0
$31$	−0.389604			−0.0699750			−0.0349875	−	0.999388i	$-0.511139\pi$
$31$	−0.389604			−0.0699750			−0.0349875	+	0.999388i	$0.511139\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.00000	−	1.00000i	−0.169031	−	0.169031i
$36$		0			0
$37$	−4.28216	+	4.28216i	−0.703982	+	0.703982i	−0.965263	−	0.261281i	$-0.915855\pi$
$37$	−4.28216	+	4.28216i	−0.703982	+	0.703982i	0.261281	+	0.965263i	$0.415855\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			5.45844i			0.852465i	0.904614	+	0.426233i	$0.140159\pi$
$41$			5.45844i			0.852465i	−0.904614	+	0.426233i	$0.859841\pi$
$42$		0			0
$43$	4.91245	−	4.91245i	0.749141	−	0.749141i	−0.225177	−	0.974318i	$-0.572296\pi$
$43$	4.91245	−	4.91245i	0.749141	−	0.749141i	0.974318	+	0.225177i	$0.0722959\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−10.7324			−1.56547			−0.782737	−	0.622352i	$-0.786177\pi$
$47$	−10.7324			−1.56547			−0.782737	+	0.622352i	$0.786177\pi$
$48$		0			0
$49$	5.00000			0.714286
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.863230	−	0.863230i	0.118574	−	0.118574i	−0.645330	−	0.763904i	$-0.723280\pi$
$53$	0.863230	−	0.863230i	0.118574	−	0.118574i	0.763904	+	0.645330i	$0.223280\pi$
$54$		0			0
$55$			5.97186i			0.805246i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.50961	−	8.50961i	1.10786	−	1.10786i	0.114425	−	0.993432i	$-0.463497\pi$
$59$	8.50961	−	8.50961i	1.10786	−	1.10786i	0.993432	−	0.114425i	$-0.0365026\pi$
$60$		0			0
$61$	2.22746	+	2.22746i	0.285196	+	0.285196i	0.835177	−	0.549981i	$-0.185365\pi$
$61$	2.22746	+	2.22746i	0.285196	+	0.285196i	−0.549981	+	0.835177i	$0.685365\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.88784			−0.234158
$66$		0			0
$67$	−2.94725	−	2.94725i	−0.360064	−	0.360064i	0.503772	−	0.863836i	$-0.331945\pi$
$67$	−2.94725	−	2.94725i	−0.360064	−	0.360064i	−0.863836	+	0.503772i	$0.831945\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.27391i			0.388542i	0.980948	+	0.194271i	$0.0622341\pi$
$71$			3.27391i			0.388542i	−0.980948	+	0.194271i	$0.937766\pi$
$72$		0			0
$73$		−	1.84138i		−	0.215517i	−0.994177	−	0.107759i	$-0.965633\pi$
$73$		−	1.84138i		−	0.215517i	0.994177	−	0.107759i	$-0.0343674\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.97186	+	5.97186i	0.680557	+	0.680557i
$78$		0			0
$79$	11.3861			1.28103			0.640517	−	0.767944i	$-0.278720\pi$
$79$	11.3861			1.28103			0.640517	+	0.767944i	$0.278720\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.1153	+	11.1153i	1.22006	+	1.22006i	0.967609	+	0.252453i	$0.0812373\pi$
$83$	11.1153	+	11.1153i	1.22006	+	1.22006i	0.252453	+	0.967609i	$0.418763\pi$
$84$		0			0
$85$	−0.808530	+	0.808530i	−0.0876974	+	0.0876974i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	18.6533i		−	1.97725i	−0.150407	−	0.988624i	$-0.548058\pi$
$89$		−	18.6533i		−	1.97725i	0.150407	−	0.988624i	$-0.451942\pi$
$90$		0			0
$91$	−1.88784	+	1.88784i	−0.197899	+	0.197899i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.49824			0.153716
$96$		0			0
$97$	−5.44902			−0.553264			−0.276632	−	0.960976i	$-0.589218\pi$
$97$	−5.44902			−0.553264			−0.276632	+	0.960976i	$0.589218\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.1899	−	11.1899i	1.11344	−	1.11344i	0.120753	−	0.992683i	$-0.461469\pi$
$101$	11.1899	−	11.1899i	1.11344	−	1.11344i	0.992683	−	0.120753i	$-0.0385310\pi$
$102$		0			0
$103$		−	18.2919i		−	1.80235i	−0.433455	−	0.901175i	$-0.642706\pi$
$103$		−	18.2919i		−	1.80235i	0.433455	−	0.901175i	$-0.357294\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.99647	−	6.99647i	0.676374	−	0.676374i	−0.282804	−	0.959178i	$-0.591264\pi$
$107$	6.99647	−	6.99647i	0.676374	−	0.676374i	0.959178	+	0.282804i	$0.0912644\pi$
$108$		0			0
$109$	12.6729	+	12.6729i	1.21385	+	1.21385i	0.969751	+	0.244097i	$0.0784915\pi$
$109$	12.6729	+	12.6729i	1.21385	+	1.21385i	0.244097	+	0.969751i	$0.421509\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.8101			−1.67543			−0.837716	−	0.546106i	$-0.816110\pi$
$113$	−17.8101			−1.67543			−0.837716	+	0.546106i	$0.816110\pi$
$114$		0			0
$115$	5.49824	+	5.49824i	0.512713	+	0.512713i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			1.61706i			0.148236i
$120$		0			0
$121$		−	24.6631i		−	2.24210i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	+	0.707107i	0.0632456	+	0.0632456i
$126$		0			0
$127$	10.0442			0.891281			0.445641	−	0.895212i	$-0.352976\pi$
$127$	10.0442			0.891281			0.445641	+	0.895212i	$0.352976\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.60215	+	7.60215i	0.664203	+	0.664203i	0.956368	−	0.292165i	$-0.0943756\pi$
$131$	7.60215	+	7.60215i	0.664203	+	0.664203i	−0.292165	+	0.956368i	$0.594376\pi$
$132$		0			0
$133$	1.49824	−	1.49824i	0.129913	−	0.129913i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			14.8003i			1.26447i	0.774775	+	0.632237i	$0.217863\pi$
$137$			14.8003i			1.26447i	−0.774775	+	0.632237i	$0.782137\pi$
$138$		0			0
$139$	8.43606	−	8.43606i	0.715537	−	0.715537i	−0.252151	−	0.967688i	$-0.581138\pi$
$139$	8.43606	−	8.43606i	0.715537	−	0.715537i	0.967688	+	0.252151i	$0.0811378\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	11.2739			0.942772
$144$		0			0
$145$	4.15862			0.345355
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.772545	+	0.772545i	−0.0632893	+	0.0632893i	−0.738043	−	0.674754i	$-0.764250\pi$
$149$	−0.772545	+	0.772545i	−0.0632893	+	0.0632893i	0.674754	+	0.738043i	$0.264250\pi$
$150$		0			0
$151$		−	2.90079i		−	0.236063i	−0.993010	−	0.118032i	$-0.962342\pi$
$151$		−	2.90079i		−	0.236063i	0.993010	−	0.118032i	$-0.0376584\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.275492	−	0.275492i	0.0221280	−	0.0221280i
$156$		0			0
$157$	2.60882	+	2.60882i	0.208206	+	0.208206i	0.803505	−	0.595298i	$-0.202966\pi$
$157$	2.60882	+	2.60882i	0.208206	+	0.208206i	−0.595298	+	0.803505i	$0.702966\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.9965			0.866643
$162$		0			0
$163$	1.42364	+	1.42364i	0.111508	+	0.111508i	0.760659	−	0.649151i	$-0.224876\pi$
$163$	1.42364	+	1.42364i	0.111508	+	0.111508i	−0.649151	+	0.760659i	$0.724876\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			4.22117i			0.326644i	0.986573	+	0.163322i	$0.0522209\pi$
$167$			4.22117i			0.326644i	−0.986573	+	0.163322i	$0.947779\pi$
$168$		0			0
$169$		−	9.43606i		−	0.725851i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	14.1582	+	14.1582i	1.07643	+	1.07643i	0.996827	+	0.0796031i	$0.0253653\pi$
$173$	14.1582	+	14.1582i	1.07643	+	1.07643i	0.0796031	+	0.996827i	$0.474635\pi$
$174$		0			0
$175$	1.41421			0.106904
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−11.8796	−	11.8796i	−0.887923	−	0.887923i	0.106401	−	0.994323i	$-0.466067\pi$
$179$	−11.8796	−	11.8796i	−0.887923	−	0.887923i	−0.994323	+	0.106401i	$0.966067\pi$
$180$		0			0
$181$	7.04646	−	7.04646i	0.523759	−	0.523759i	−0.394945	−	0.918705i	$-0.629237\pi$
$181$	7.04646	−	7.04646i	0.523759	−	0.523759i	0.918705	+	0.394945i	$0.129237\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	6.05588i		−	0.445237i
$186$		0			0
$187$	4.82843	−	4.82843i	0.353090	−	0.353090i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−15.3857			−1.11327			−0.556634	−	0.830758i	$-0.687908\pi$
$191$	−15.3857			−1.11327			−0.556634	+	0.830758i	$0.687908\pi$
$192$		0			0
$193$	7.52099			0.541372			0.270686	−	0.962668i	$-0.412749\pi$
$193$	7.52099			0.541372			0.270686	+	0.962668i	$0.412749\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−9.66352	+	9.66352i	−0.688497	+	0.688497i	−0.961900	−	0.273403i	$-0.911851\pi$
$197$	−9.66352	+	9.66352i	−0.688497	+	0.688497i	0.273403	+	0.961900i	$0.411851\pi$
$198$		0			0
$199$			9.35570i			0.663208i	0.943419	+	0.331604i	$0.107590\pi$
$199$			9.35570i			0.663208i	−0.943419	+	0.331604i	$0.892410\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.15862	−	4.15862i	0.291878	−	0.291878i
$204$		0			0
$205$	−3.85970	−	3.85970i	−0.269573	−	0.269573i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.94725			−0.618894
$210$		0			0
$211$	0.0625461	+	0.0625461i	0.00430585	+	0.00430585i	0.709256	−	0.704951i	$-0.249031\pi$
$211$	0.0625461	+	0.0625461i	0.00430585	+	0.00430585i	−0.704951	+	0.709256i	$0.749031\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			6.94725i			0.473799i
$216$		0			0
$217$		−	0.550984i		−	0.0374032i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.52637	+	1.52637i	0.102675	+	0.102675i
$222$		0			0
$223$	−13.1203			−0.878599			−0.439300	−	0.898341i	$-0.644773\pi$
$223$	−13.1203			−0.878599			−0.439300	+	0.898341i	$0.644773\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.76235	+	5.76235i	0.382461	+	0.382461i	0.871988	−	0.489527i	$-0.162831\pi$
$227$	5.76235	+	5.76235i	0.382461	+	0.382461i	−0.489527	+	0.871988i	$0.662831\pi$
$228$		0			0
$229$	5.12549	−	5.12549i	0.338702	−	0.338702i	−0.517177	−	0.855879i	$-0.673017\pi$
$229$	5.12549	−	5.12549i	0.338702	−	0.338702i	0.855879	+	0.517177i	$0.173017\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	13.2552i		−	0.868377i	−0.900822	−	0.434188i	$-0.857035\pi$
$233$		−	13.2552i		−	0.868377i	0.900822	−	0.434188i	$-0.142965\pi$
$234$		0			0
$235$	7.58892	−	7.58892i	0.495047	−	0.495047i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−18.8119			−1.21684			−0.608422	−	0.793614i	$-0.708197\pi$
$239$	−18.8119			−1.21684			−0.608422	+	0.793614i	$0.708197\pi$
$240$		0			0
$241$	−2.66665			−0.171774			−0.0858871	−	0.996305i	$-0.527372\pi$
$241$	−2.66665			−0.171774			−0.0858871	+	0.996305i	$0.527372\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.53553	+	3.53553i	−0.225877	+	0.225877i
$246$		0			0
$247$		−	2.82843i		−	0.179969i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7.77726	+	7.77726i	−0.490896	+	0.490896i	−0.908588	−	0.417692i	$-0.862839\pi$
$251$	−7.77726	+	7.77726i	−0.490896	+	0.490896i	0.417692	+	0.908588i	$0.362839\pi$
$252$		0			0
$253$	−32.8347	−	32.8347i	−2.06430	−	2.06430i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.83971			0.364271			0.182136	−	0.983273i	$-0.441699\pi$
$257$	5.83971			0.364271			0.182136	+	0.983273i	$0.441699\pi$
$258$		0			0
$259$	−6.05588	−	6.05588i	−0.376294	−	0.376294i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			10.0094i			0.617208i	0.951191	+	0.308604i	$0.0998617\pi$
$263$			10.0094i			0.617208i	−0.951191	+	0.308604i	$0.900138\pi$
$264$		0			0
$265$			1.22079i			0.0749926i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.82529	−	1.82529i	−0.111290	−	0.111290i	0.649269	−	0.760559i	$-0.275075\pi$
$269$	−1.82529	−	1.82529i	−0.111290	−	0.111290i	−0.760559	+	0.649269i	$0.775075\pi$
$270$		0			0
$271$	5.44862			0.330980			0.165490	−	0.986211i	$-0.447079\pi$
$271$	5.44862			0.330980			0.165490	+	0.986211i	$0.447079\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.22274	−	4.22274i	−0.254641	−	0.254641i
$276$		0			0
$277$	10.5690	−	10.5690i	0.635031	−	0.635031i	−0.314294	−	0.949326i	$-0.601768\pi$
$277$	10.5690	−	10.5690i	0.635031	−	0.635031i	0.949326	+	0.314294i	$0.101768\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	10.2376i		−	0.610727i	−0.952236	−	0.305363i	$-0.901222\pi$
$281$		−	10.2376i		−	0.610727i	0.952236	−	0.305363i	$-0.0987780\pi$
$282$		0			0
$283$	9.88744	−	9.88744i	0.587748	−	0.587748i	−0.349273	−	0.937021i	$-0.613572\pi$
$283$	9.88744	−	9.88744i	0.587748	−	0.587748i	0.937021	+	0.349273i	$0.113572\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−7.71940			−0.455662
$288$		0			0
$289$	−15.6926			−0.923092
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.05235	+	9.05235i	−0.528844	+	0.528844i	−0.920228	−	0.391384i	$-0.871997\pi$
$293$	−9.05235	+	9.05235i	−0.528844	+	0.528844i	0.391384	+	0.920228i	$0.371997\pi$
$294$		0			0
$295$			12.0344i			0.700670i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.3798	−	10.3798i	0.600278	−	0.600278i
$300$		0			0
$301$	6.94725	+	6.94725i	0.400433	+	0.400433i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.15010			−0.180374
$306$		0			0
$307$	−2.77215	−	2.77215i	−0.158215	−	0.158215i	0.623560	−	0.781775i	$-0.285686\pi$
$307$	−2.77215	−	2.77215i	−0.158215	−	0.158215i	−0.781775	+	0.623560i	$0.785686\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			20.9441i			1.18763i	0.804601	+	0.593815i	$0.202379\pi$
$311$			20.9441i			1.18763i	−0.804601	+	0.593815i	$0.797621\pi$
$312$		0			0
$313$			10.1814i			0.575485i	0.957708	+	0.287743i	$0.0929047\pi$
$313$			10.1814i			0.575485i	−0.957708	+	0.287743i	$0.907095\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	14.7655	+	14.7655i	0.829312	+	0.829312i	0.987422	−	0.158109i	$-0.0505398\pi$
$317$	14.7655	+	14.7655i	0.829312	+	0.829312i	−0.158109	+	0.987422i	$0.550540\pi$
$318$		0			0
$319$	−24.8347			−1.39048
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.21137	−	1.21137i	−0.0674023	−	0.0674023i
$324$		0			0
$325$	1.33490	−	1.33490i	0.0740472	−	0.0740472i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	15.1778i		−	0.836781i
$330$		0			0
$331$	−16.8815	+	16.8815i	−0.927894	+	0.927894i	−0.997570	−	0.0696758i	$-0.977804\pi$
$331$	−16.8815	+	16.8815i	−0.927894	+	0.927894i	0.0696758	+	0.997570i	$0.477804\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.16804			0.227725
$336$		0			0
$337$	23.8343			1.29834			0.649169	−	0.760644i	$-0.275117\pi$
$337$	23.8343			1.29834			0.649169	+	0.760644i	$0.275117\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.64520	+	1.64520i	−0.0890925	+	0.0890925i
$342$		0			0
$343$			16.9706i			0.916324i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4.38294	+	4.38294i	−0.235289	+	0.235289i	−0.814896	−	0.579607i	$-0.803206\pi$
$347$	−4.38294	+	4.38294i	−0.235289	+	0.235289i	0.579607	+	0.814896i	$0.303206\pi$
$348$		0			0
$349$	17.4357	+	17.4357i	0.933310	+	0.933310i	0.997911	−	0.0646013i	$-0.0205776\pi$
$349$	17.4357	+	17.4357i	0.933310	+	0.933310i	−0.0646013	+	0.997911i	$0.520578\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	31.5795			1.68081			0.840403	−	0.541961i	$-0.182318\pi$
$353$	31.5795			1.68081			0.840403	+	0.541961i	$0.182318\pi$
$354$		0			0
$355$	−2.31501	−	2.31501i	−0.122868	−	0.122868i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	25.5310i		−	1.34747i	−0.738972	−	0.673737i	$-0.764688\pi$
$359$		−	25.5310i		−	1.34747i	0.738972	−	0.673737i	$-0.235312\pi$
$360$		0			0
$361$		−	16.7553i		−	0.881857i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.30205	+	1.30205i	0.0681526	+	0.0681526i
$366$		0			0
$367$	−13.9084			−0.726010			−0.363005	−	0.931787i	$-0.618249\pi$
$367$	−13.9084			−0.726010			−0.363005	+	0.931787i	$0.618249\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.22079	+	1.22079i	0.0633803	+	0.0633803i
$372$		0			0
$373$	2.88942	−	2.88942i	0.149608	−	0.149608i	−0.628335	−	0.777943i	$-0.716263\pi$
$373$	2.88942	−	2.88942i	0.149608	−	0.149608i	0.777943	+	0.628335i	$0.216263\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	7.85080i		−	0.404337i
$378$		0			0
$379$	5.96687	−	5.96687i	0.306497	−	0.306497i	−0.537052	−	0.843549i	$-0.680462\pi$
$379$	5.96687	−	5.96687i	0.306497	−	0.306497i	0.843549	+	0.537052i	$0.180462\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	13.8382			0.707100			0.353550	−	0.935416i	$-0.384974\pi$
$383$	13.8382			0.707100			0.353550	+	0.935416i	$0.384974\pi$
$384$		0			0
$385$	−8.44549			−0.430422
$386$		0			0
$387$		0			0
$388$		0			0
$389$	20.2560	−	20.2560i	1.02702	−	1.02702i	0.0273936	−	0.999625i	$-0.491279\pi$
$389$	20.2560	−	20.2560i	1.02702	−	1.02702i	0.999625	−	0.0273936i	$-0.00872075\pi$
$390$		0			0
$391$		−	8.89097i		−	0.449636i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.05117	+	8.05117i	−0.405098	+	0.405098i
$396$		0			0
$397$	21.0941	+	21.0941i	1.05868	+	1.05868i	0.998167	+	0.0605152i	$0.0192744\pi$
$397$	21.0941	+	21.0941i	1.05868	+	1.05868i	0.0605152	+	0.998167i	$0.480726\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.98314			−0.248846			−0.124423	−	0.992229i	$-0.539708\pi$
$401$	−4.98314			−0.248846			−0.124423	+	0.992229i	$0.539708\pi$
$402$		0			0
$403$	−0.520084	−	0.520084i	−0.0259072	−	0.0259072i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			36.1649i			1.79263i
$408$		0			0
$409$			33.7686i			1.66975i	0.550439	+	0.834875i	$0.314460\pi$
$409$			33.7686i			1.66975i	−0.550439	+	0.834875i	$0.685540\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	12.0344	+	12.0344i	0.592174	+	0.592174i
$414$		0			0
$415$	−15.7194			−0.771635
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−15.9453	−	15.9453i	−0.778979	−	0.778979i	0.200678	−	0.979657i	$-0.435685\pi$
$419$	−15.9453	−	15.9453i	−0.778979	−	0.778979i	−0.979657	+	0.200678i	$0.935685\pi$
$420$		0			0
$421$	−2.33295	+	2.33295i	−0.113701	+	0.113701i	−0.761668	−	0.647967i	$-0.775619\pi$
$421$	−2.33295	+	2.33295i	−0.113701	+	0.113701i	0.647967	+	0.761668i	$0.275619\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	1.14343i		−	0.0554647i
$426$		0			0
$427$	−3.15010	+	3.15010i	−0.152444	+	0.152444i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−20.6400			−0.994194			−0.497097	−	0.867695i	$-0.665601\pi$
$431$	−20.6400			−0.994194			−0.497097	+	0.867695i	$0.665601\pi$
$432$		0			0
$433$	1.07550			0.0516852			0.0258426	−	0.999666i	$-0.491773\pi$
$433$	1.07550			0.0516852			0.0258426	+	0.999666i	$0.491773\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.23765	+	8.23765i	−0.394060	+	0.394060i
$438$		0			0
$439$			4.24041i			0.202384i	0.994867	+	0.101192i	$0.0322656\pi$
$439$			4.24041i			0.202384i	−0.994867	+	0.101192i	$0.967734\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−19.8969	+	19.8969i	−0.945329	+	0.945329i	−0.998581	−	0.0532524i	$-0.983041\pi$
$443$	−19.8969	+	19.8969i	−0.945329	+	0.945329i	0.0532524	+	0.998581i	$0.483041\pi$
$444$		0			0
$445$	13.1899	+	13.1899i	0.625261	+	0.625261i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−17.2441			−0.813800			−0.406900	−	0.913473i	$-0.633390\pi$
$449$	−17.2441			−0.813800			−0.406900	+	0.913473i	$0.633390\pi$
$450$		0			0
$451$	23.0496	+	23.0496i	1.08536	+	1.08536i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	2.66981i		−	0.125163i
$456$		0			0
$457$			32.3141i			1.51159i	0.654809	+	0.755795i	$0.272749\pi$
$457$			32.3141i			1.51159i	−0.654809	+	0.755795i	$0.727251\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−15.4884	−	15.4884i	−0.721367	−	0.721367i	0.247516	−	0.968884i	$-0.420386\pi$
$461$	−15.4884	−	15.4884i	−0.721367	−	0.721367i	−0.968884	+	0.247516i	$0.920386\pi$
$462$		0			0
$463$	−27.2150			−1.26479			−0.632394	−	0.774647i	$-0.717928\pi$
$463$	−27.2150			−1.26479			−0.632394	+	0.774647i	$0.717928\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.60373	+	7.60373i	0.351859	+	0.351859i	0.860801	−	0.508942i	$-0.169963\pi$
$467$	7.60373	+	7.60373i	0.351859	+	0.351859i	−0.508942	+	0.860801i	$0.669963\pi$
$468$		0			0
$469$	4.16804	−	4.16804i	0.192462	−	0.192462i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	41.4880i		−	1.90762i
$474$		0			0
$475$	−1.05941	+	1.05941i	−0.0486092	+	0.0486092i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	0.515092			0.0235352			0.0117676	−	0.999931i	$-0.496254\pi$
$479$	0.515092			0.0235352			0.0117676	+	0.999931i	$0.496254\pi$
$480$		0			0
$481$	−11.4325			−0.521279
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.85304	−	3.85304i	0.174957	−	0.174957i
$486$		0			0
$487$		−	18.2489i		−	0.826937i	−0.910518	−	0.413468i	$-0.864317\pi$
$487$		−	18.2489i		−	0.826937i	0.910518	−	0.413468i	$-0.135683\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	26.5557	−	26.5557i	1.19844	−	1.19844i	0.223807	−	0.974633i	$-0.428151\pi$
$491$	26.5557	−	26.5557i	1.19844	−	1.19844i	0.974633	−	0.223807i	$-0.0718486\pi$
$492$		0			0
$493$	−3.36237	−	3.36237i	−0.151433	−	0.151433i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.63001			−0.207684
$498$		0			0
$499$	11.2247	+	11.2247i	0.502486	+	0.502486i	0.912210	−	0.409723i	$-0.134375\pi$
$499$	11.2247	+	11.2247i	0.502486	+	0.502486i	−0.409723	+	0.912210i	$0.634375\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	27.3061i		−	1.21752i	−0.793355	−	0.608759i	$-0.791668\pi$
$503$		−	27.3061i		−	1.21752i	0.793355	−	0.608759i	$-0.208332\pi$
$504$		0			0
$505$			15.8249i			0.704199i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	17.9387	+	17.9387i	0.795120	+	0.795120i	0.982322	−	0.187201i	$-0.0599417\pi$
$509$	17.9387	+	17.9387i	0.795120	+	0.795120i	−0.187201	+	0.982322i	$0.559942\pi$
$510$		0			0
$511$	2.60411			0.115199
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12.9343	+	12.9343i	0.569953	+	0.569953i
$516$		0			0
$517$	−45.3200	+	45.3200i	−1.99317	+	1.99317i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	11.7686i		−	0.515593i	−0.966199	−	0.257796i	$-0.917004\pi$
$521$		−	11.7686i		−	0.515593i	0.966199	−	0.257796i	$-0.0829963\pi$
$522$		0			0
$523$	−2.02148	+	2.02148i	−0.0883929	+	0.0883929i	−0.749921	−	0.661528i	$-0.769908\pi$
$523$	−2.02148	+	2.02148i	−0.0883929	+	0.0883929i	0.661528	+	0.749921i	$0.269908\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.445487			−0.0194057
$528$		0			0
$529$	−37.4612			−1.62875
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.28649	+	7.28649i	−0.315613	+	0.315613i
$534$		0			0
$535$			9.89450i			0.427777i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	21.1137	−	21.1137i	0.909432	−	0.909432i
$540$		0			0
$541$	−28.9598	−	28.9598i	−1.24508	−	1.24508i	−0.957867	−	0.287213i	$-0.907271\pi$
$541$	−28.9598	−	28.9598i	−1.24508	−	1.24508i	−0.287213	−	0.957867i	$-0.592729\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−17.9222			−0.767705
$546$		0			0
$547$	11.3485	+	11.3485i	0.485227	+	0.485227i	0.906796	−	0.421569i	$-0.138520\pi$
$547$	11.3485	+	11.3485i	0.485227	+	0.485227i	−0.421569	+	0.906796i	$0.638520\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			6.23059i			0.265432i
$552$		0			0
$553$			16.1023i			0.684741i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18.8445	−	18.8445i	−0.798468	−	0.798468i	0.184386	−	0.982854i	$-0.440970\pi$
$557$	−18.8445	−	18.8445i	−0.798468	−	0.798468i	−0.982854	+	0.184386i	$0.940970\pi$
$558$		0			0
$559$	13.1153			0.554718
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−33.1082	−	33.1082i	−1.39535	−	1.39535i	−0.812793	−	0.582553i	$-0.802054\pi$
$563$	−33.1082	−	33.1082i	−1.39535	−	1.39535i	−0.582553	−	0.812793i	$-0.697946\pi$
$564$		0			0
$565$	12.5936	−	12.5936i	0.529818	−	0.529818i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			1.55688i			0.0652677i	0.999467	+	0.0326339i	$0.0103895\pi$
$569$			1.55688i			0.0652677i	−0.999467	+	0.0326339i	$0.989610\pi$
$570$		0			0
$571$	−33.1614	+	33.1614i	−1.38776	+	1.38776i	−0.557752	+	0.830008i	$0.688336\pi$
$571$	−33.1614	+	33.1614i	−1.38776	+	1.38776i	−0.830008	+	0.557752i	$0.811664\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−7.77568			−0.324268
$576$		0			0
$577$	−36.5706			−1.52245			−0.761227	−	0.648486i	$-0.775402\pi$
$577$	−36.5706			−1.52245			−0.761227	+	0.648486i	$0.775402\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−15.7194	+	15.7194i	−0.652151	+	0.652151i
$582$		0			0
$583$		−	7.29040i		−	0.301937i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.73235	+	6.73235i	−0.277874	+	0.277874i	−0.832260	−	0.554386i	$-0.812953\pi$
$587$	−6.73235	+	6.73235i	−0.277874	+	0.277874i	0.554386	+	0.832260i	$0.312953\pi$
$588$		0			0
$589$	0.412751	+	0.412751i	0.0170071	+	0.0170071i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−11.7154			−0.481092			−0.240546	−	0.970638i	$-0.577327\pi$
$593$	−11.7154			−0.481092			−0.240546	+	0.970638i	$0.577327\pi$
$594$		0			0
$595$	−1.14343	−	1.14343i	−0.0468762	−	0.0468762i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	5.86233i		−	0.239528i	−0.992802	−	0.119764i	$-0.961786\pi$
$599$		−	5.86233i		−	0.239528i	0.992802	−	0.119764i	$-0.0382138\pi$
$600$		0			0
$601$			5.00353i			0.204098i	0.994779	+	0.102049i	$0.0325399\pi$
$601$			5.00353i			0.204098i	−0.994779	+	0.102049i	$0.967460\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	17.4395	+	17.4395i	0.709015	+	0.709015i
$606$		0			0
$607$	−22.7109			−0.921806			−0.460903	−	0.887450i	$-0.652474\pi$
$607$	−22.7109			−0.921806			−0.460903	+	0.887450i	$0.652474\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−14.3267	−	14.3267i	−0.579595	−	0.579595i
$612$		0			0
$613$	−7.25531	+	7.25531i	−0.293039	+	0.293039i	−0.838280	−	0.545240i	$-0.816438\pi$
$613$	−7.25531	+	7.25531i	−0.293039	+	0.293039i	0.545240	+	0.838280i	$0.316438\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			46.0710i			1.85475i	0.374133	+	0.927375i	$0.377940\pi$
$617$			46.0710i			1.85475i	−0.374133	+	0.927375i	$0.622060\pi$
$618$		0			0
$619$	34.2931	−	34.2931i	1.37836	−	1.37836i	0.530962	−	0.847396i	$-0.321831\pi$
$619$	34.2931	−	34.2931i	1.37836	−	1.37836i	0.847396	−	0.530962i	$-0.178169\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	26.3798			1.05688
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.89636	+	4.89636i	−0.195231	+	0.195231i
$630$		0			0
$631$			25.9017i			1.03113i	0.856850	+	0.515566i	$0.172418\pi$
$631$			25.9017i			1.03113i	−0.856850	+	0.515566i	$0.827582\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.10234	+	7.10234i	−0.281848	+	0.281848i
$636$		0			0
$637$	6.67452	+	6.67452i	0.264454	+	0.264454i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	15.3341			0.605660			0.302830	−	0.953045i	$-0.402069\pi$
$641$	15.3341			0.605660			0.302830	+	0.953045i	$0.402069\pi$
$642$		0			0
$643$	16.7507	+	16.7507i	0.660582	+	0.660582i	0.955517	−	0.294935i	$-0.0952981\pi$
$643$	16.7507	+	16.7507i	0.660582	+	0.660582i	−0.294935	+	0.955517i	$0.595298\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			4.81900i			0.189455i	0.995503	+	0.0947273i	$0.0301979\pi$
$647$			4.81900i			0.189455i	−0.995503	+	0.0947273i	$0.969802\pi$
$648$		0			0
$649$		−	71.8678i		−	2.82106i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−16.7842	−	16.7842i	−0.656817	−	0.656817i	0.297809	−	0.954625i	$-0.403744\pi$
$653$	−16.7842	−	16.7842i	−0.656817	−	0.656817i	−0.954625	+	0.297809i	$0.903744\pi$
$654$		0			0
$655$	−10.7511			−0.420079
$656$		0			0
$657$		0			0
$658$		0			0
$659$	16.4376	+	16.4376i	0.640320	+	0.640320i	0.950634	−	0.310314i	$-0.100434\pi$
$659$	16.4376	+	16.4376i	0.640320	+	0.640320i	−0.310314	+	0.950634i	$0.600434\pi$
$660$		0			0
$661$	−17.5904	+	17.5904i	−0.684187	+	0.684187i	−0.960941	−	0.276754i	$-0.910741\pi$
$661$	−17.5904	+	17.5904i	−0.684187	+	0.684187i	0.276754	+	0.960941i	$0.410741\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			2.11882i			0.0821645i
$666$		0			0
$667$	−22.8651	+	22.8651i	−0.885339	+	0.885339i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	18.8119			0.726227
$672$		0			0
$673$	27.0762			1.04371			0.521856	−	0.853033i	$-0.325240\pi$
$673$	27.0762			1.04371			0.521856	+	0.853033i	$0.325240\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.9093	+	16.9093i	−0.649877	+	0.649877i	−0.952963	−	0.303086i	$-0.901983\pi$
$677$	−16.9093	+	16.9093i	−0.649877	+	0.649877i	0.303086	+	0.952963i	$0.401983\pi$
$678$		0			0
$679$		−	7.70607i		−	0.295732i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−14.0531	+	14.0531i	−0.537728	+	0.537728i	−0.922861	−	0.385133i	$-0.874155\pi$
$683$	−14.0531	+	14.0531i	−0.537728	+	0.537728i	0.385133	+	0.922861i	$0.374155\pi$
$684$		0			0
$685$	−10.4654	−	10.4654i	−0.399862	−	0.399862i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.30466			0.0878005
$690$		0			0
$691$	−25.9737	−	25.9737i	−0.988086	−	0.988086i	0.0118439	−	0.999930i	$-0.496230\pi$
$691$	−25.9737	−	25.9737i	−0.988086	−	0.988086i	−0.999930	+	0.0118439i	$0.996230\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			11.9304i			0.452546i
$696$		0			0
$697$			6.24136i			0.236409i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−20.8789	−	20.8789i	−0.788586	−	0.788586i	0.192676	−	0.981262i	$-0.438283\pi$
$701$	−20.8789	−	20.8789i	−0.788586	−	0.788586i	−0.981262	+	0.192676i	$0.938283\pi$
$702$		0			0
$703$	9.07314			0.342200
$704$		0			0
$705$		0			0
$706$		0			0
$707$	15.8249	+	15.8249i	0.595157	+	0.595157i
$708$		0			0
$709$	14.4580	−	14.4580i	0.542983	−	0.542983i	−0.381419	−	0.924402i	$-0.624564\pi$
$709$	14.4580	−	14.4580i	0.542983	−	0.542983i	0.924402	+	0.381419i	$0.124564\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			3.02944i			0.113453i
$714$		0			0
$715$	−7.97186	+	7.97186i	−0.298131	+	0.298131i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.53803			0.131946			0.0659731	−	0.997821i	$-0.478985\pi$
$719$	3.53803			0.131946			0.0659731	+	0.997821i	$0.478985\pi$
$720$		0			0
$721$	25.8686			0.963397
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.94059	+	2.94059i	−0.109211	+	0.109211i
$726$		0			0
$727$			23.6681i			0.877802i	0.898535	+	0.438901i	$0.144632\pi$
$727$			23.6681i			0.877802i	−0.898535	+	0.438901i	$0.855368\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.61706	−	5.61706i	0.207754	−	0.207754i
$732$		0			0
$733$	−20.3055	−	20.3055i	−0.750000	−	0.750000i	0.224479	−	0.974479i	$-0.427932\pi$
$733$	−20.3055	−	20.3055i	−0.750000	−	0.750000i	−0.974479	+	0.224479i	$0.927932\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−24.8910			−0.916871
$738$		0			0
$739$	−4.63944	−	4.63944i	−0.170664	−	0.170664i	0.616607	−	0.787271i	$-0.288507\pi$
$739$	−4.63944	−	4.63944i	−0.170664	−	0.170664i	−0.787271	+	0.616607i	$0.788507\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	27.3029i		−	1.00165i	−0.865549	−	0.500824i	$-0.833030\pi$
$743$		−	27.3029i		−	1.00165i	0.865549	−	0.500824i	$-0.166970\pi$
$744$		0			0
$745$		−	1.09254i		−	0.0400277i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	9.89450	+	9.89450i	0.361537	+	0.361537i
$750$		0			0
$751$	26.1582			0.954527			0.477264	−	0.878760i	$-0.341629\pi$
$751$	26.1582			0.954527			0.477264	+	0.878760i	$0.341629\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.05117	+	2.05117i	0.0746497	+	0.0746497i
$756$		0			0
$757$	−16.8559	+	16.8559i	−0.612638	+	0.612638i	−0.943633	−	0.330995i	$-0.892616\pi$
$757$	−16.8559	+	16.8559i	−0.612638	+	0.612638i	0.330995	+	0.943633i	$0.392616\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			34.6337i			1.25547i	0.778427	+	0.627735i	$0.216018\pi$
$761$			34.6337i			1.25547i	−0.778427	+	0.627735i	$0.783982\pi$
$762$		0			0
$763$	−17.9222	+	17.9222i	−0.648829	+	0.648829i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	22.7190			0.820337
$768$		0			0
$769$	−8.79179			−0.317040			−0.158520	−	0.987356i	$-0.550672\pi$
$769$	−8.79179			−0.317040			−0.158520	+	0.987356i	$0.550672\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7.17118	−	7.17118i	0.257929	−	0.257929i	−0.566282	−	0.824211i	$-0.691619\pi$
$773$	7.17118	−	7.17118i	0.257929	−	0.257929i	0.824211	+	0.566282i	$0.191619\pi$
$774$		0			0
$775$			0.389604i			0.0139950i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.78274	−	5.78274i	0.207188	−	0.207188i
$780$		0			0
$781$	13.8249	+	13.8249i	0.494694	+	0.494694i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−3.68943			−0.131681
$786$		0			0
$787$	−10.9687	−	10.9687i	−0.390993	−	0.390993i	0.484048	−	0.875041i	$-0.339166\pi$
$787$	−10.9687	−	10.9687i	−0.390993	−	0.390993i	−0.875041	+	0.484048i	$0.839166\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	25.1873i		−	0.895556i
$792$		0			0
$793$			5.94688i			0.211180i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.13138	−	1.13138i	−0.0400756	−	0.0400756i	0.686785	−	0.726861i	$-0.259021\pi$
$797$	−1.13138	−	1.13138i	−0.0400756	−	0.0400756i	−0.726861	+	0.686785i	$0.759021\pi$
$798$		0			0
$799$	−12.2717			−0.434143
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.77568	−	7.77568i	−0.274398	−	0.274398i
$804$		0			0
$805$	−7.77568	+	7.77568i	−0.274057	+	0.274057i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			20.1610i			0.708822i	0.935090	+	0.354411i	$0.115319\pi$
$809$			20.1610i			0.708822i	−0.935090	+	0.354411i	$0.884681\pi$
$810$		0			0
$811$	28.0206	−	28.0206i	0.983935	−	0.983935i	−0.0159383	−	0.999873i	$-0.505074\pi$
$811$	28.0206	−	28.0206i	0.983935	−	0.983935i	0.999873	+	0.0159383i	$0.00507355\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.01333			−0.0705238
$816$		0			0
$817$	−10.4086			−0.364151
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−31.0657	+	31.0657i	−1.08420	+	1.08420i	−0.0880869	+	0.996113i	$0.528075\pi$
$821$	−31.0657	+	31.0657i	−1.08420	+	1.08420i	−0.996113	+	0.0880869i	$0.971925\pi$
$822$		0			0
$823$		−	18.5484i		−	0.646555i	−0.946304	−	0.323278i	$-0.895215\pi$
$823$		−	18.5484i		−	0.646555i	0.946304	−	0.323278i	$-0.104785\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−16.6080	+	16.6080i	−0.577517	+	0.577517i	−0.934218	−	0.356701i	$-0.883901\pi$
$827$	−16.6080	+	16.6080i	−0.577517	+	0.577517i	0.356701	+	0.934218i	$0.383901\pi$
$828$		0			0
$829$	−19.0261	−	19.0261i	−0.660803	−	0.660803i	0.294767	−	0.955569i	$-0.404758\pi$
$829$	−19.0261	−	19.0261i	−0.660803	−	0.660803i	−0.955569	+	0.294767i	$0.904758\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	5.71717			0.198088
$834$		0			0
$835$	−2.98481	−	2.98481i	−0.103294	−	0.103294i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			15.5569i			0.537083i	0.963268	+	0.268542i	$0.0865417\pi$
$839$			15.5569i			0.537083i	−0.963268	+	0.268542i	$0.913458\pi$
$840$		0			0
$841$		−	11.7059i		−	0.403651i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.67230	+	6.67230i	0.229534	+	0.229534i
$846$		0			0
$847$	34.8789			1.19845
$848$		0			0
$849$		0			0
$850$		0			0
$851$	33.2967	+	33.2967i	1.14139	+	1.14139i
$852$		0			0
$853$	−15.5124	+	15.5124i	−0.531133	+	0.531133i	−0.920910	−	0.389776i	$-0.872552\pi$
$853$	−15.5124	+	15.5124i	−0.531133	+	0.531133i	0.389776	+	0.920910i	$0.372552\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			27.0005i			0.922320i	0.887317	+	0.461160i	$0.152567\pi$
$857$			27.0005i			0.922320i	−0.887317	+	0.461160i	$0.847433\pi$
$858$		0			0
$859$	−1.31020	+	1.31020i	−0.0447035	+	0.0447035i	−0.729105	−	0.684402i	$-0.760063\pi$
$859$	−1.31020	+	1.31020i	−0.0447035	+	0.0447035i	0.684402	+	0.729105i	$0.260063\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−45.1917			−1.53834			−0.769172	−	0.639042i	$-0.779331\pi$
$863$	−45.1917			−1.53834			−0.769172	+	0.639042i	$0.779331\pi$
$864$		0			0
$865$	−20.0228			−0.680794
$866$		0			0
$867$		0			0
$868$		0			0
$869$	48.0805	−	48.0805i	1.63102	−	1.63102i
$870$		0			0
$871$		−	7.86860i		−	0.266617i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.00000	+	1.00000i	−0.0338062	+	0.0338062i
$876$		0			0
$877$	−10.2259	−	10.2259i	−0.345303	−	0.345303i	0.513053	−	0.858357i	$-0.328514\pi$
$877$	−10.2259	−	10.2259i	−0.345303	−	0.345303i	−0.858357	+	0.513053i	$0.828514\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	7.61942			0.256705			0.128352	−	0.991729i	$-0.459031\pi$
$881$	7.61942			0.256705			0.128352	+	0.991729i	$0.459031\pi$
$882$		0			0
$883$	1.70424	+	1.70424i	0.0573521	+	0.0573521i	0.735201	−	0.677849i	$-0.237088\pi$
$883$	1.70424	+	1.70424i	0.0573521	+	0.0573521i	−0.677849	+	0.735201i	$0.737088\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			23.9647i			0.804655i	0.915496	+	0.402327i	$0.131799\pi$
$887$			23.9647i			0.804655i	−0.915496	+	0.402327i	$0.868201\pi$
$888$		0			0
$889$			14.2047i			0.476410i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	11.3700	+	11.3700i	0.380482	+	0.380482i
$894$		0			0
$895$	16.8003			0.561572
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.14567	+	1.14567i	0.0382101	+	0.0382101i
$900$		0			0
$901$	0.987046	−	0.987046i	0.0328833	−	0.0328833i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			9.96520i			0.331254i
$906$		0			0
$907$	−19.1197	+	19.1197i	−0.634860	+	0.634860i	−0.949283	−	0.314423i	$-0.898189\pi$
$907$	−19.1197	+	19.1197i	−0.634860	+	0.634860i	0.314423	+	0.949283i	$0.398189\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.15918			0.137800			0.0688999	−	0.997624i	$-0.478051\pi$
$911$	4.15918			0.137800			0.0688999	+	0.997624i	$0.478051\pi$
$912$		0			0
$913$	93.8741			3.10678
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.7511	+	10.7511i	−0.355032	+	0.355032i
$918$		0			0
$919$			43.6229i			1.43899i	0.694499	+	0.719494i	$0.255626\pi$
$919$			43.6229i			1.43899i	−0.694499	+	0.719494i	$0.744374\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.37036	+	4.37036i	−0.143852	+	0.143852i
$924$		0			0
$925$	4.28216	+	4.28216i	0.140796	+	0.140796i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4.24392			−0.139238			−0.0696192	−	0.997574i	$-0.522178\pi$
$929$	−4.24392			−0.139238			−0.0696192	+	0.997574i	$0.522178\pi$
$930$		0			0
$931$	−5.29706	−	5.29706i	−0.173604	−	0.173604i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			6.82843i			0.223313i
$936$		0			0
$937$		−	9.87491i		−	0.322599i	−0.986905	−	0.161300i	$-0.948431\pi$
$937$		−	9.87491i		−	0.322599i	0.986905	−	0.161300i	$-0.0515685\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−26.0675	−	26.0675i	−0.849777	−	0.849777i	0.140328	−	0.990105i	$-0.455184\pi$
$941$	−26.0675	−	26.0675i	−0.849777	−	0.849777i	−0.990105	+	0.140328i	$0.955184\pi$
$942$		0			0
$943$	42.4431			1.38214
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−10.7355	−	10.7355i	−0.348857	−	0.348857i	0.510827	−	0.859684i	$-0.329339\pi$
$947$	−10.7355	−	10.7355i	−0.348857	−	0.348857i	−0.859684	+	0.510827i	$0.829339\pi$
$948$		0			0
$949$	2.45807	−	2.45807i	0.0797922	−	0.0797922i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			34.0844i			1.10410i	0.833811	+	0.552051i	$0.186154\pi$
$953$			34.0844i			1.10410i	−0.833811	+	0.552051i	$0.813846\pi$
$954$		0			0
$955$	10.8793	−	10.8793i	0.352046	−	0.352046i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−20.9308			−0.675890
$960$		0			0
$961$	−30.8482			−0.995104
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5.31814	+	5.31814i	−0.171197	+	0.171197i
$966$		0			0
$967$		−	9.41342i		−	0.302715i	−0.988479	−	0.151358i	$-0.951635\pi$
$967$		−	9.41342i		−	0.302715i	0.988479	−	0.151358i	$-0.0483645\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6.84018	+	6.84018i	−0.219512	+	0.219512i	−0.808293	−	0.588781i	$-0.799608\pi$
$971$	−6.84018	+	6.84018i	−0.219512	+	0.219512i	0.588781	+	0.808293i	$0.299608\pi$
$972$		0			0
$973$	11.9304	+	11.9304i	0.382471	+	0.382471i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5.38555			−0.172299			−0.0861494	−	0.996282i	$-0.527456\pi$
$977$	−5.38555			−0.172299			−0.0861494	+	0.996282i	$0.527456\pi$
$978$		0			0
$979$	−78.7682	−	78.7682i	−2.51744	−	2.51744i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	9.31630i		−	0.297144i	−0.988902	−	0.148572i	$-0.952532\pi$
$983$		−	9.31630i		−	0.297144i	0.988902	−	0.148572i	$-0.0474677\pi$
$984$		0			0
$985$		−	13.6663i		−	0.435444i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−38.1976	−	38.1976i	−1.21461	−	1.21461i
$990$		0			0
$991$	1.25727			0.0399384			0.0199692	−	0.999801i	$-0.493643\pi$
$991$	1.25727			0.0399384			0.0199692	+	0.999801i	$0.493643\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.61548	−	6.61548i	−0.209725	−	0.209725i
$996$		0			0
$997$	25.1661	−	25.1661i	0.797017	−	0.797017i	−0.185607	−	0.982624i	$-0.559425\pi$
$997$	25.1661	−	25.1661i	0.797017	−	0.797017i	0.982624	+	0.185607i	$0.0594250\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.b.721.2		8
3.2	odd	2		960.2.s.b.721.3		8
4.3	odd	2		720.2.t.b.541.3		8
12.11	even	2		240.2.s.b.61.2	✓	8
16.5	even	4	inner	2880.2.t.b.2161.2		8
16.11	odd	4		720.2.t.b.181.3		8
24.5	odd	2		1920.2.s.d.1441.2		8
24.11	even	2		1920.2.s.c.1441.3		8
48.5	odd	4		960.2.s.b.241.3		8
48.11	even	4		240.2.s.b.181.2	yes	8
48.29	odd	4		1920.2.s.d.481.2		8
48.35	even	4		1920.2.s.c.481.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.s.b.61.2	✓	8	12.11	even	2
240.2.s.b.181.2	yes	8	48.11	even	4
720.2.t.b.181.3		8	16.11	odd	4
720.2.t.b.541.3		8	4.3	odd	2
960.2.s.b.241.3		8	48.5	odd	4
960.2.s.b.721.3		8	3.2	odd	2
1920.2.s.c.481.3		8	48.35	even	4
1920.2.s.c.1441.3		8	24.11	even	2
1920.2.s.d.481.2		8	48.29	odd	4
1920.2.s.d.1441.2		8	24.5	odd	2
2880.2.t.b.721.2		8	1.1	even	1	trivial
2880.2.t.b.2161.2		8	16.5	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{5} +1.41421i q^{7} +O(q^{10})\)	\(q+(-0.707107 + 0.707107i) q^{5} +1.41421i q^{7} +(4.22274 - 4.22274i) q^{11} +(1.33490 + 1.33490i) q^{13} +1.14343 q^{17} +(-1.05941 - 1.05941i) q^{19} -7.77568i q^{23} -1.00000i q^{25} +(-2.94059 - 2.94059i) q^{29} -0.389604 q^{31} +(-1.00000 - 1.00000i) q^{35} +(-4.28216 + 4.28216i) q^{37} +5.45844i q^{41} +(4.91245 - 4.91245i) q^{43} -10.7324 q^{47} +5.00000 q^{49} +(0.863230 - 0.863230i) q^{53} +5.97186i q^{55} +(8.50961 - 8.50961i) q^{59} +(2.22746 + 2.22746i) q^{61} -1.88784 q^{65} +(-2.94725 - 2.94725i) q^{67} +3.27391i q^{71} -1.84138i q^{73} +(5.97186 + 5.97186i) q^{77} +11.3861 q^{79} +(11.1153 + 11.1153i) q^{83} +(-0.808530 + 0.808530i) q^{85} -18.6533i q^{89} +(-1.88784 + 1.88784i) q^{91} +1.49824 q^{95} -5.44902 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{11} + 8 q^{13} - 8 q^{17} - 8 q^{19} - 24 q^{29} - 8 q^{31} - 8 q^{35} - 8 q^{37} + 40 q^{49} + 8 q^{59} - 16 q^{61} + 8 q^{65} + 8 q^{77} + 40 q^{79} + 32 q^{83} + 8 q^{85} + 8 q^{91} - 16 q^{95} - 48 q^{97}+O(q^{100}) \) 8 * q + 8 * q^11 + 8 * q^13 - 8 * q^17 - 8 * q^19 - 24 * q^29 - 8 * q^31 - 8 * q^35 - 8 * q^37 + 40 * q^49 + 8 * q^59 - 16 * q^61 + 8 * q^65 + 8 * q^77 + 40 * q^79 + 32 * q^83 + 8 * q^85 + 8 * q^91 - 16 * q^95 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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