LMFDB - Embedded newform 2880.2.t.b.2161.1

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			2161.1
Root			$0.500000 - 0.0297061i$ of defining polynomial
Character	$\chi$	$=$	2880.2161
Dual form			2880.2.t.b.721.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{5} -1.41421i q^{7} +O(q^{10})$	$q+(-0.707107 - 0.707107i) q^{5} -1.41421i q^{7} +(-0.808530 - 0.808530i) q^{11} +(-0.749118 + 0.749118i) q^{13} -5.97186 q^{17} +(1.88784 - 1.88784i) q^{19} +1.88118i q^{23} +1.00000i q^{25} +(-5.88784 + 5.88784i) q^{29} -1.61040 q^{31} +(-1.00000 + 1.00000i) q^{35} +(3.69637 + 3.69637i) q^{37} +8.77215i q^{41} +(0.744406 + 0.744406i) q^{43} +13.5608 q^{47} +5.00000 q^{49} +(-0.863230 - 0.863230i) q^{53} +1.14343i q^{55} +(-10.7523 - 10.7523i) q^{59} +(-9.05588 + 9.05588i) q^{61} +1.05941 q^{65} +(2.94725 - 2.94725i) q^{67} +6.78863i q^{71} -2.32666i q^{73} +(-1.14343 + 1.14343i) q^{77} +4.27078 q^{79} +(-3.11529 + 3.11529i) q^{83} +(4.22274 + 4.22274i) q^{85} +10.3172i q^{89} +(1.05941 + 1.05941i) q^{91} -2.66981 q^{95} -3.72256 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$6$		0			0
$7$		−	1.41421i		−	0.534522i	−0.963624	−	0.267261i	$-0.913881\pi$
$7$		−	1.41421i		−	0.534522i	0.963624	−	0.267261i	$-0.0861187\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.808530	−	0.808530i	−0.243781	−	0.243781i	0.574631	−	0.818412i	$-0.305145\pi$
$11$	−0.808530	−	0.808530i	−0.243781	−	0.243781i	−0.818412	+	0.574631i	$0.805145\pi$
$12$		0			0
$13$	−0.749118	+	0.749118i	−0.207768	+	0.207768i	−0.803318	−	0.595550i	$-0.796934\pi$
$13$	−0.749118	+	0.749118i	−0.207768	+	0.207768i	0.595550	+	0.803318i	$0.296934\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.97186			−1.44839			−0.724195	−	0.689596i	$-0.757788\pi$
$17$	−5.97186			−1.44839			−0.724195	+	0.689596i	$0.757788\pi$
$18$		0			0
$19$	1.88784	−	1.88784i	0.433100	−	0.433100i	−0.456582	−	0.889682i	$-0.650926\pi$
$19$	1.88784	−	1.88784i	0.433100	−	0.433100i	0.889682	+	0.456582i	$0.150926\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.88118i			0.392252i	0.980579	+	0.196126i	$0.0628362\pi$
$23$			1.88118i			0.392252i	−0.980579	+	0.196126i	$0.937164\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.88784	+	5.88784i	−1.09334	+	1.09334i	−0.0981752	+	0.995169i	$0.531301\pi$
$29$	−5.88784	+	5.88784i	−1.09334	+	1.09334i	−0.995169	+	0.0981752i	$0.968699\pi$
$30$		0			0
$31$	−1.61040			−0.289236			−0.144618	−	0.989488i	$-0.546195\pi$
$31$	−1.61040			−0.289236			−0.144618	+	0.989488i	$0.546195\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.00000	+	1.00000i	−0.169031	+	0.169031i
$36$		0			0
$37$	3.69637	+	3.69637i	0.607679	+	0.607679i	0.942339	−	0.334660i	$-0.108621\pi$
$37$	3.69637	+	3.69637i	0.607679	+	0.607679i	−0.334660	+	0.942339i	$0.608621\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			8.77215i			1.36998i	0.728553	+	0.684990i	$0.240193\pi$
$41$			8.77215i			1.36998i	−0.728553	+	0.684990i	$0.759807\pi$
$42$		0			0
$43$	0.744406	+	0.744406i	0.113521	+	0.113521i	0.761585	−	0.648065i	$-0.224421\pi$
$43$	0.744406	+	0.744406i	0.113521	+	0.113521i	−0.648065	+	0.761585i	$0.724421\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	13.5608			1.97804			0.989022	−	0.147771i	$-0.0472098\pi$
$47$	13.5608			1.97804			0.989022	+	0.147771i	$0.0472098\pi$
$48$		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	$-0.276720\pi$
$53$	−0.863230	−	0.863230i	−0.118574	−	0.118574i	−0.763904	+	0.645330i	$0.776720\pi$
$54$		0			0
$55$			1.14343i			0.154181i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.7523	−	10.7523i	−1.39982	−	1.39982i	−0.800526	−	0.599298i	$-0.795447\pi$
$59$	−10.7523	−	10.7523i	−1.39982	−	1.39982i	−0.599298	−	0.800526i	$-0.704553\pi$
$60$		0			0
$61$	−9.05588	+	9.05588i	−1.15949	+	1.15949i	−0.174901	+	0.984586i	$0.555960\pi$
$61$	−9.05588	+	9.05588i	−1.15949	+	1.15949i	−0.984586	+	0.174901i	$0.944040\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.05941			0.131404
$66$		0			0
$67$	2.94725	−	2.94725i	0.360064	−	0.360064i	−0.503772	−	0.863836i	$-0.668055\pi$
$67$	2.94725	−	2.94725i	0.360064	−	0.360064i	0.863836	+	0.503772i	$0.168055\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			6.78863i			0.805662i	0.915274	+	0.402831i	$0.131974\pi$
$71$			6.78863i			0.805662i	−0.915274	+	0.402831i	$0.868026\pi$
$72$		0			0
$73$		−	2.32666i		−	0.272315i	−0.990687	−	0.136158i	$-0.956525\pi$
$73$		−	2.32666i		−	0.272315i	0.990687	−	0.136158i	$-0.0434754\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.14343	+	1.14343i	−0.130306	+	0.130306i
$78$		0			0
$79$	4.27078			0.480500			0.240250	−	0.970711i	$-0.422771\pi$
$79$	4.27078			0.480500			0.240250	+	0.970711i	$0.422771\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.11529	+	3.11529i	−0.341948	+	0.341948i	−0.857099	−	0.515151i	$-0.827736\pi$
$83$	−3.11529	+	3.11529i	−0.341948	+	0.341948i	0.515151	+	0.857099i	$0.327736\pi$
$84$		0			0
$85$	4.22274	+	4.22274i	0.458021	+	0.458021i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			10.3172i			1.09363i	0.837255	+	0.546813i	$0.184159\pi$
$89$			10.3172i			1.09363i	−0.837255	+	0.546813i	$0.815841\pi$
$90$		0			0
$91$	1.05941	+	1.05941i	0.111057	+	0.111057i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.66981			−0.273917
$96$		0			0
$97$	−3.72256			−0.377968			−0.188984	−	0.981980i	$-0.560519\pi$
$97$	−3.72256			−0.377968			−0.188984	+	0.981980i	$0.560519\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.29539	+	5.29539i	0.526911	+	0.526911i	0.919650	−	0.392739i	$-0.128472\pi$
$101$	5.29539	+	5.29539i	0.526911	+	0.526911i	−0.392739	+	0.919650i	$0.628472\pi$
$102$		0			0
$103$			15.8503i			1.56177i	0.624672	+	0.780887i	$0.285233\pi$
$103$			15.8503i			1.56177i	−0.624672	+	0.780887i	$0.714767\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.33962	−	1.33962i	−0.129506	−	0.129506i	0.639383	−	0.768888i	$-0.279190\pi$
$107$	−1.33962	−	1.33962i	−0.129506	−	0.129506i	−0.768888	+	0.639383i	$0.779190\pi$
$108$		0			0
$109$	−8.67294	+	8.67294i	−0.830717	+	0.830717i	−0.987615	−	0.156898i	$-0.949851\pi$
$109$	−8.67294	+	8.67294i	−0.830717	+	0.830717i	0.156898	+	0.987615i	$0.449851\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	15.3248			1.44164			0.720818	−	0.693124i	$-0.243766\pi$
$113$	15.3248			1.44164			0.720818	+	0.693124i	$0.243766\pi$
$114$		0			0
$115$	1.33019	−	1.33019i	0.124041	−	0.124041i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			8.44549i			0.774196i
$120$		0			0
$121$		−	9.69256i		−	0.881142i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	−	0.707107i	0.0632456	−	0.0632456i
$126$		0			0
$127$	−4.18636			−0.371480			−0.185740	−	0.982599i	$-0.559468\pi$
$127$	−4.18636			−0.371480			−0.185740	+	0.982599i	$0.559468\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.29734	−	4.29734i	0.375460	−	0.375460i	−0.494001	−	0.869461i	$-0.664466\pi$
$131$	4.29734	−	4.29734i	0.375460	−	0.375460i	0.869461	+	0.494001i	$0.164466\pi$
$132$		0			0
$133$	−2.66981	−	2.66981i	−0.231502	−	0.231502i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	7.68499i		−	0.656573i	−0.944578	−	0.328287i	$-0.893529\pi$
$137$		−	7.68499i		−	0.656573i	0.944578	−	0.328287i	$-0.106471\pi$
$138$		0			0
$139$	10.8776	+	10.8776i	0.922630	+	0.922630i	0.997215	−	0.0745848i	$-0.0237631\pi$
$139$	10.8776	+	10.8776i	0.922630	+	0.922630i	−0.0745848	+	0.997215i	$0.523763\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.21137			0.101300
$144$		0			0
$145$	8.32666			0.691492
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.0559	−	12.0559i	−0.987656	−	0.987656i	0.0122684	−	0.999925i	$-0.496095\pi$
$149$	−12.0559	−	12.0559i	−0.987656	−	0.987656i	−0.999925	+	0.0122684i	$0.996095\pi$
$150$		0			0
$151$		−	4.21450i		−	0.342971i	−0.985187	−	0.171486i	$-0.945143\pi$
$151$		−	4.21450i		−	0.342971i	0.985187	−	0.171486i	$-0.0548567\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.13872	+	1.13872i	0.0914643	+	0.0914643i
$156$		0			0
$157$	−9.53775	+	9.53775i	−0.761195	+	0.761195i	−0.976538	−	0.215343i	$-0.930913\pi$
$157$	−9.53775	+	9.53775i	−0.761195	+	0.761195i	0.215343	+	0.976538i	$0.430913\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.66038			0.209668
$162$		0			0
$163$	−11.0805	+	11.0805i	−0.867891	+	0.867891i	−0.992239	−	0.124348i	$-0.960316\pi$
$163$	−11.0805	+	11.0805i	−0.867891	+	0.867891i	0.124348	+	0.992239i	$0.460316\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			11.7359i			0.908150i	0.890963	+	0.454075i	$0.150030\pi$
$167$			11.7359i			0.908150i	−0.890963	+	0.454075i	$0.849970\pi$
$168$		0			0
$169$			11.8776i			0.913665i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.18766	+	7.18766i	−0.546468	+	0.546468i	−0.925417	−	0.378950i	$-0.876285\pi$
$173$	−7.18766	+	7.18766i	−0.546468	+	0.546468i	0.378950	+	0.925417i	$0.376285\pi$
$174$		0			0
$175$	1.41421			0.106904
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.84832	+	6.84832i	−0.511868	+	0.511868i	−0.915098	−	0.403231i	$-0.867887\pi$
$179$	−6.84832	+	6.84832i	−0.511868	+	0.511868i	0.403231	+	0.915098i	$0.367887\pi$
$180$		0			0
$181$	8.26725	+	8.26725i	0.614500	+	0.614500i	0.944115	−	0.329615i	$-0.106919\pi$
$181$	8.26725	+	8.26725i	0.614500	+	0.614500i	−0.329615	+	0.944115i	$0.606919\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	5.22746i		−	0.384330i
$186$		0			0
$187$	4.82843	+	4.82843i	0.353090	+	0.353090i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	17.2435			1.24770			0.623849	−	0.781545i	$-0.285568\pi$
$191$	17.2435			1.24770			0.623849	+	0.781545i	$0.285568\pi$
$192$		0			0
$193$	−26.8347			−1.93160			−0.965802	−	0.259281i	$-0.916514\pi$
$193$	−26.8347			−1.93160			−0.965802	+	0.259281i	$0.916514\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−0.821763	−	0.821763i	−0.0585482	−	0.0585482i	0.677226	−	0.735775i	$-0.263182\pi$
$197$	−0.821763	−	0.821763i	−0.0585482	−	0.0585482i	−0.735775	+	0.677226i	$0.763182\pi$
$198$		0			0
$199$			20.3263i			1.44089i	0.693512	+	0.720445i	$0.256063\pi$
$199$			20.3263i			1.44089i	−0.693512	+	0.720445i	$0.743937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.32666	+	8.32666i	0.584417	+	0.584417i
$204$		0			0
$205$	6.20285	−	6.20285i	0.433226	−	0.433226i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.05275			−0.211163
$210$		0			0
$211$	−20.0625	+	20.0625i	−1.38116	+	1.38116i	−0.538603	+	0.842560i	$0.681048\pi$
$211$	−20.0625	+	20.0625i	−1.38116	+	1.38116i	−0.842560	+	0.538603i	$0.818952\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	1.05275i		−	0.0717969i
$216$		0			0
$217$			2.27744i			0.154603i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.47363	−	4.47363i	0.300929	−	0.300929i
$222$		0			0
$223$	−10.6787			−0.715099			−0.357549	−	0.933894i	$-0.616388\pi$
$223$	−10.6787			−0.715099			−0.357549	+	0.933894i	$0.616388\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	17.5514	−	17.5514i	1.16492	−	1.16492i	0.181541	−	0.983383i	$-0.441891\pi$
$227$	17.5514	−	17.5514i	1.16492	−	1.16492i	0.983383	−	0.181541i	$-0.0581086\pi$
$228$		0			0
$229$	−9.61077	−	9.61077i	−0.635098	−	0.635098i	0.314245	−	0.949342i	$-0.398249\pi$
$229$	−9.61077	−	9.61077i	−0.635098	−	0.635098i	−0.949342	+	0.314245i	$0.898249\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	16.4268i		−	1.07615i	−0.842896	−	0.538077i	$-0.819151\pi$
$233$		−	16.4268i		−	1.07615i	0.842896	−	0.538077i	$-0.180849\pi$
$234$		0			0
$235$	−9.58892	−	9.58892i	−0.625512	−	0.625512i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−14.6439			−0.947235			−0.473618	−	0.880731i	$-0.657052\pi$
$239$	−14.6439			−0.947235			−0.473618	+	0.880731i	$0.657052\pi$
$240$		0			0
$241$	23.3529			1.50430			0.752148	−	0.658995i	$-0.229018\pi$
$241$	23.3529			1.50430			0.752148	+	0.658995i	$0.229018\pi$
$242$		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
$251$	−12.8085	−	12.8085i	−0.808467	−	0.808467i	0.175935	−	0.984402i	$-0.443705\pi$
$251$	−12.8085	−	12.8085i	−0.808467	−	0.808467i	−0.984402	+	0.175935i	$0.943705\pi$
$252$		0			0
$253$	1.52099	−	1.52099i	0.0956236	−	0.0956236i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	22.3024			1.39119			0.695594	−	0.718436i	$-0.255141\pi$
$257$	22.3024			1.39119			0.695594	+	0.718436i	$0.255141\pi$
$258$		0			0
$259$	5.22746	−	5.22746i	0.324818	−	0.324818i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			2.49471i			0.153830i	0.997038	+	0.0769151i	$0.0245070\pi$
$263$			2.49471i			0.153830i	−0.997038	+	0.0769151i	$0.975493\pi$
$264$		0			0
$265$			1.22079i			0.0749926i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−19.0031	+	19.0031i	−1.15864	+	1.15864i	−0.173874	+	0.984768i	$0.555628\pi$
$269$	−19.0031	+	19.0031i	−1.15864	+	1.15864i	−0.984768	+	0.173874i	$0.944372\pi$
$270$		0			0
$271$	−21.7918			−1.32376			−0.661878	−	0.749612i	$-0.730240\pi$
$271$	−21.7918			−1.32376			−0.661878	+	0.749612i	$0.730240\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.808530	−	0.808530i	0.0487562	−	0.0487562i
$276$		0			0
$277$	−11.6401	−	11.6401i	−0.699385	−	0.699385i	0.264893	−	0.964278i	$-0.414663\pi$
$277$	−11.6401	−	11.6401i	−0.699385	−	0.699385i	−0.964278	+	0.264893i	$0.914663\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	1.55136i		−	0.0925462i	−0.998929	−	0.0462731i	$-0.985266\pi$
$281$		−	1.55136i		−	0.0925462i	0.998929	−	0.0462731i	$-0.0147344\pi$
$282$		0			0
$283$	−18.5737	−	18.5737i	−1.10409	−	1.10409i	−0.993911	−	0.110183i	$-0.964856\pi$
$283$	−18.5737	−	18.5737i	−1.10409	−	1.10409i	−0.110183	−	0.993911i	$-0.535144\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.4057			0.732285
$288$		0			0
$289$	18.6631			1.09783
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.5671	+	10.5671i	0.617335	+	0.617335i	0.944847	−	0.327512i	$-0.106210\pi$
$293$	10.5671	+	10.5671i	0.617335	+	0.617335i	−0.327512	+	0.944847i	$0.606210\pi$
$294$		0			0
$295$			15.2060i			0.885326i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.40922	−	1.40922i	−0.0814974	−	0.0814974i
$300$		0			0
$301$	1.05275	−	1.05275i	0.0606794	−	0.0606794i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	12.8070			0.733324
$306$		0			0
$307$	11.4584	−	11.4584i	0.653968	−	0.653968i	−0.299978	−	0.953946i	$-0.596979\pi$
$307$	11.4584	−	11.4584i	0.653968	−	0.653968i	0.953946	+	0.299978i	$0.0969794\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			6.80196i			0.385704i	0.981228	+	0.192852i	$0.0617737\pi$
$311$			6.80196i			0.385704i	−0.981228	+	0.192852i	$0.938226\pi$
$312$		0			0
$313$			15.8382i			0.895229i	0.894227	+	0.447615i	$0.147726\pi$
$313$			15.8382i			0.895229i	−0.894227	+	0.447615i	$0.852274\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.37665	−	9.37665i	0.526645	−	0.526645i	−0.392925	−	0.919570i	$-0.628537\pi$
$317$	9.37665	−	9.37665i	0.526645	−	0.526645i	0.919570	+	0.392925i	$0.128537\pi$
$318$		0			0
$319$	9.52099			0.533073
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−11.2739	+	11.2739i	−0.627297	+	0.627297i
$324$		0			0
$325$	−0.749118	−	0.749118i	−0.0415536	−	0.0415536i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	19.1778i		−	1.05731i
$330$		0			0
$331$	−9.26059	−	9.26059i	−0.509008	−	0.509008i	0.405214	−	0.914222i	$-0.367197\pi$
$331$	−9.26059	−	9.26059i	−0.509008	−	0.509008i	−0.914222	+	0.405214i	$0.867197\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.16804			−0.227725
$336$		0			0
$337$	2.99411			0.163099			0.0815497	−	0.996669i	$-0.474013\pi$
$337$	2.99411			0.163099			0.0815497	+	0.996669i	$0.474013\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.30205	+	1.30205i	0.0705101	+	0.0705101i
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−14.4455	−	14.4455i	−0.775474	−	0.775474i	0.203583	−	0.979058i	$-0.434741\pi$
$347$	−14.4455	−	14.4455i	−0.775474	−	0.775474i	−0.979058	+	0.203583i	$0.934741\pi$
$348$		0			0
$349$	−5.63668	+	5.63668i	−0.301724	+	0.301724i	−0.841688	−	0.539964i	$-0.818438\pi$
$349$	−5.63668	+	5.63668i	−0.301724	+	0.301724i	0.539964	+	0.841688i	$0.318438\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	26.9058			1.43205			0.716025	−	0.698074i	$-0.245960\pi$
$353$	26.9058			1.43205			0.716025	+	0.698074i	$0.245960\pi$
$354$		0			0
$355$	4.80029	−	4.80029i	0.254773	−	0.254773i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			14.7533i			0.778649i	0.921101	+	0.389325i	$0.127292\pi$
$359$			14.7533i			0.778649i	−0.921101	+	0.389325i	$0.872708\pi$
$360$		0			0
$361$			11.8721i			0.624849i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.64520	+	1.64520i	−0.0861136	+	0.0861136i
$366$		0			0
$367$	34.6779			1.81017			0.905086	−	0.425228i	$-0.139806\pi$
$367$	34.6779			1.81017			0.905086	+	0.425228i	$0.139806\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.22079	+	1.22079i	−0.0633803	+	0.0633803i
$372$		0			0
$373$	10.8679	+	10.8679i	0.562721	+	0.562721i	0.930079	−	0.367359i	$-0.119738\pi$
$373$	10.8679	+	10.8679i	0.562721	+	0.562721i	−0.367359	+	0.930079i	$0.619738\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	8.82137i		−	0.454324i
$378$		0			0
$379$	−12.9374	−	12.9374i	−0.664551	−	0.664551i	0.291898	−	0.956449i	$-0.405713\pi$
$379$	−12.9374	−	12.9374i	−0.664551	−	0.664551i	−0.956449	+	0.291898i	$0.905713\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12.1814			−0.622439			−0.311219	−	0.950338i	$-0.600737\pi$
$383$	−12.1814			−0.622439			−0.311219	+	0.950338i	$0.600737\pi$
$384$		0			0
$385$	1.61706			0.0824130
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.57246	+	2.57246i	0.130429	+	0.130429i	0.769308	−	0.638879i	$-0.220601\pi$
$389$	2.57246	+	2.57246i	0.130429	+	0.130429i	−0.638879	+	0.769308i	$0.720601\pi$
$390$		0			0
$391$		−	11.2341i		−	0.568134i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.01990	−	3.01990i	−0.151948	−	0.151948i
$396$		0			0
$397$	8.94753	−	8.94753i	0.449064	−	0.449064i	−0.445979	−	0.895043i	$-0.647145\pi$
$397$	8.94753	−	8.94753i	0.449064	−	0.449064i	0.895043	+	0.445979i	$0.147145\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−14.3306			−0.715634			−0.357817	−	0.933792i	$-0.616479\pi$
$401$	−14.3306			−0.715634			−0.357817	+	0.933792i	$0.616479\pi$
$402$		0			0
$403$	1.20638	−	1.20638i	0.0600939	−	0.0600939i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	5.97725i		−	0.296281i
$408$		0			0
$409$		−	11.2019i		−	0.553900i	−0.960884	−	0.276950i	$-0.910676\pi$
$409$		−	11.2019i		−	0.553900i	0.960884	−	0.276950i	$-0.0893237\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−15.2060	+	15.2060i	−0.748237	+	0.748237i
$414$		0			0
$415$	4.40569			0.216267
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.6405	+	12.6405i	−0.617528	+	0.617528i	−0.944897	−	0.327369i	$-0.893838\pi$
$419$	−12.6405	+	12.6405i	−0.617528	+	0.617528i	0.327369	+	0.944897i	$0.393838\pi$
$420$		0			0
$421$	−2.83862	−	2.83862i	−0.138346	−	0.138346i	0.634542	−	0.772888i	$-0.281189\pi$
$421$	−2.83862	−	2.83862i	−0.138346	−	0.138346i	−0.772888	+	0.634542i	$0.781189\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	5.97186i		−	0.289678i
$426$		0			0
$427$	12.8070	+	12.8070i	0.619772	+	0.619772i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−29.9874			−1.44444			−0.722222	−	0.691662i	$-0.756879\pi$
$431$	−29.9874			−1.44444			−0.722222	+	0.691662i	$0.756879\pi$
$432$		0			0
$433$	−23.2176			−1.11577			−0.557884	−	0.829919i	$-0.688387\pi$
$433$	−23.2176			−1.11577			−0.557884	+	0.829919i	$0.688387\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.55136	+	3.55136i	0.169884	+	0.169884i
$438$		0			0
$439$			11.2110i			0.535070i	0.963548	+	0.267535i	$0.0862092\pi$
$439$			11.2110i			0.535070i	−0.963548	+	0.267535i	$0.913791\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	21.0684	+	21.0684i	1.00099	+	1.00099i	1.00000		0.000992295i	$0.000315857\pi$
$443$	21.0684	+	21.0684i	1.00099	+	1.00099i	0.000992295		1.00000i	$0.499684\pi$
$444$		0			0
$445$	7.29539	−	7.29539i	0.345835	−	0.345835i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−20.6970			−0.976753			−0.488376	−	0.872633i	$-0.662411\pi$
$449$	−20.6970			−0.976753			−0.488376	+	0.872633i	$0.662411\pi$
$450$		0			0
$451$	7.09254	−	7.09254i	0.333975	−	0.333975i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	1.49824i		−	0.0702383i
$456$		0			0
$457$		−	18.7986i		−	0.879362i	−0.898154	−	0.439681i	$-0.855091\pi$
$457$		−	18.7986i		−	0.879362i	0.898154	−	0.439681i	$-0.144909\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.68943	−	1.68943i	0.0786844	−	0.0786844i	−0.666669	−	0.745354i	$-0.732281\pi$
$461$	1.68943	−	1.68943i	0.0786844	−	0.0786844i	0.745354	+	0.666669i	$0.232281\pi$
$462$		0			0
$463$	38.0434			1.76803			0.884014	−	0.467460i	$-0.154831\pi$
$463$	38.0434			1.76803			0.884014	+	0.467460i	$0.154831\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	15.2247	−	15.2247i	0.704515	−	0.704515i	−0.260861	−	0.965376i	$-0.584007\pi$
$467$	15.2247	−	15.2247i	0.704515	−	0.704515i	0.965376	+	0.260861i	$0.0840066\pi$
$468$		0			0
$469$	−4.16804	−	4.16804i	−0.192462	−	0.192462i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	1.20375i		−	0.0553484i
$474$		0			0
$475$	1.88784	+	1.88784i	0.0866200	+	0.0866200i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−13.0004			−0.594002			−0.297001	−	0.954877i	$-0.595987\pi$
$479$	−13.0004			−0.594002			−0.297001	+	0.954877i	$0.595987\pi$
$480$		0			0
$481$	−5.53803			−0.252512
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.63224	+	2.63224i	0.119524	+	0.119524i
$486$		0			0
$487$		−	16.1068i		−	0.729868i	−0.931033	−	0.364934i	$-0.881092\pi$
$487$		−	16.1068i		−	0.729868i	0.931033	−	0.364934i	$-0.118908\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−16.9993	−	16.9993i	−0.767169	−	0.767169i	0.210438	−	0.977607i	$-0.432511\pi$
$491$	−16.9993	−	16.9993i	−0.767169	−	0.767169i	−0.977607	+	0.210438i	$0.932511\pi$
$492$		0			0
$493$	35.1614	−	35.1614i	1.58359	−	1.58359i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	9.60058			0.430645
$498$		0			0
$499$	3.60373	−	3.60373i	0.161325	−	0.161325i	−0.621828	−	0.783154i	$-0.713610\pi$
$499$	3.60373	−	3.60373i	0.161325	−	0.161325i	0.783154	+	0.621828i	$0.213610\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	25.4482i		−	1.13468i	−0.823484	−	0.567340i	$-0.807972\pi$
$503$		−	25.4482i		−	1.13468i	0.823484	−	0.567340i	$-0.192028\pi$
$504$		0			0
$505$		−	7.48881i		−	0.333248i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8.08087	+	8.08087i	−0.358178	+	0.358178i	−0.863141	−	0.504963i	$-0.831506\pi$
$509$	−8.08087	+	8.08087i	−0.358178	+	0.358178i	0.504963	+	0.863141i	$0.331506\pi$
$510$		0			0
$511$	−3.29040			−0.145559
$512$		0			0
$513$		0			0
$514$		0			0
$515$	11.2078	−	11.2078i	0.493876	−	0.493876i
$516$		0			0
$517$	−10.9643	−	10.9643i	−0.482209	−	0.482209i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	10.7981i		−	0.473071i	−0.971623	−	0.236536i	$-0.923988\pi$
$521$		−	10.7981i		−	0.473071i	0.971623	−	0.236536i	$-0.0760120\pi$
$522$		0			0
$523$	−17.9785	−	17.9785i	−0.786146	−	0.786146i	0.194714	−	0.980860i	$-0.437622\pi$
$523$	−17.9785	−	17.9785i	−0.786146	−	0.786146i	−0.980860	+	0.194714i	$0.937622\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9.61706			0.418926
$528$		0			0
$529$	19.4612			0.846138
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.57137	−	6.57137i	−0.284638	−	0.284638i
$534$		0			0
$535$			1.89450i			0.0819065i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.04265	−	4.04265i	−0.174129	−	0.174129i
$540$		0			0
$541$	6.61666	−	6.61666i	0.284473	−	0.284473i	−0.550417	−	0.834890i	$-0.685531\pi$
$541$	6.61666	−	6.61666i	0.284473	−	0.284473i	0.834890	+	0.550417i	$0.185531\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	12.2654			0.525392
$546$		0			0
$547$	9.62205	−	9.62205i	0.411409	−	0.411409i	−0.470820	−	0.882229i	$-0.656042\pi$
$547$	9.62205	−	9.62205i	0.411409	−	0.411409i	0.882229	+	0.470820i	$0.156042\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			22.2306i			0.947055i
$552$		0			0
$553$		−	6.03979i		−	0.256838i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.50137	−	2.50137i	0.105986	−	0.105986i	−0.652125	−	0.758111i	$-0.726122\pi$
$557$	2.50137	−	2.50137i	0.105986	−	0.105986i	0.758111	+	0.652125i	$0.226122\pi$
$558$		0			0
$559$	−1.11529			−0.0471719
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.20547	+	2.20547i	−0.0929496	+	0.0929496i	−0.752053	−	0.659103i	$-0.770936\pi$
$563$	−2.20547	+	2.20547i	−0.0929496	+	0.0929496i	0.659103	+	0.752053i	$0.270936\pi$
$564$		0			0
$565$	−10.8363	−	10.8363i	−0.455885	−	0.455885i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			17.5569i			0.736023i	0.929821	+	0.368011i	$0.119961\pi$
$569$			17.5569i			0.736023i	−0.929821	+	0.368011i	$0.880039\pi$
$570$		0			0
$571$	5.36237	+	5.36237i	0.224408	+	0.224408i	0.810352	−	0.585944i	$-0.199276\pi$
$571$	5.36237	+	5.36237i	0.224408	+	0.224408i	−0.585944	+	0.810352i	$0.699276\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.88118			−0.0784504
$576$		0			0
$577$	13.7422			0.572093			0.286047	−	0.958216i	$-0.407659\pi$
$577$	13.7422			0.572093			0.286047	+	0.958216i	$0.407659\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.40569	+	4.40569i	0.182779	+	0.182779i
$582$		0			0
$583$			1.39589i			0.0578120i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	17.5608	+	17.5608i	0.724811	+	0.724811i	0.969581	−	0.244770i	$-0.0787125\pi$
$587$	17.5608	+	17.5608i	0.724811	+	0.724811i	−0.244770	+	0.969581i	$0.578712\pi$
$588$		0			0
$589$	−3.04017	+	3.04017i	−0.125268	+	0.125268i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−41.3973			−1.69998			−0.849992	−	0.526795i	$-0.823394\pi$
$593$	−41.3973			−1.69998			−0.849992	+	0.526795i	$0.823394\pi$
$594$		0			0
$595$	5.97186	−	5.97186i	0.244822	−	0.244822i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			36.7651i			1.50218i	0.660199	+	0.751090i	$0.270472\pi$
$599$			36.7651i			1.50218i	−0.660199	+	0.751090i	$0.729528\pi$
$600$		0			0
$601$		−	13.3396i		−	0.544134i	−0.962278	−	0.272067i	$-0.912293\pi$
$601$		−	13.3396i		−	0.544134i	0.962278	−	0.272067i	$-0.0877073\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.85367	+	6.85367i	−0.278641	+	0.278641i
$606$		0			0
$607$	17.5393			0.711898			0.355949	−	0.934505i	$-0.384158\pi$
$607$	17.5393			0.711898			0.355949	+	0.934505i	$0.384158\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−10.1586	+	10.1586i	−0.410974	+	0.410974i
$612$		0			0
$613$	14.9538	+	14.9538i	0.603978	+	0.603978i	0.941366	−	0.337388i	$-0.109543\pi$
$613$	14.9538	+	14.9538i	0.603978	+	0.603978i	−0.337388	+	0.941366i	$0.609543\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	7.04165i		−	0.283486i	−0.989903	−	0.141743i	$-0.954729\pi$
$617$		−	7.04165i		−	0.283486i	0.989903	−	0.141743i	$-0.0452707\pi$
$618$		0			0
$619$	−14.2931	−	14.2931i	−0.574490	−	0.574490i	0.358890	−	0.933380i	$-0.383155\pi$
$619$	−14.2931	−	14.2931i	−0.574490	−	0.574490i	−0.933380	+	0.358890i	$0.883155\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	14.5908			0.584567
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−22.0742	−	22.0742i	−0.880156	−	0.880156i
$630$		0			0
$631$		−	41.3531i		−	1.64624i	−0.567867	−	0.823121i	$-0.692231\pi$
$631$		−	41.3531i		−	1.64624i	0.567867	−	0.823121i	$-0.307769\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.96021	+	2.96021i	0.117472	+	0.117472i
$636$		0			0
$637$	−3.74559	+	3.74559i	−0.148406	+	0.148406i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	14.3228			0.565715			0.282857	−	0.959162i	$-0.408718\pi$
$641$	14.3228			0.565715			0.282857	+	0.959162i	$0.408718\pi$
$642$		0			0
$643$	−13.4370	+	13.4370i	−0.529902	+	0.529902i	−0.920543	−	0.390641i	$-0.872253\pi$
$643$	−13.4370	+	13.4370i	−0.529902	+	0.529902i	0.390641	+	0.920543i	$0.372253\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	17.3231i		−	0.681043i	−0.940237	−	0.340521i	$-0.889396\pi$
$647$		−	17.3231i		−	0.681043i	0.940237	−	0.340521i	$-0.110604\pi$
$648$		0			0
$649$			17.3870i			0.682501i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−31.0148	+	31.0148i	−1.21370	+	1.21370i	−0.243903	+	0.969800i	$0.578428\pi$
$653$	−31.0148	+	31.0148i	−1.21370	+	1.21370i	−0.969800	+	0.243903i	$0.921572\pi$
$654$		0			0
$655$	−6.07736			−0.237462
$656$		0			0
$657$		0			0
$658$		0			0
$659$	29.8050	−	29.8050i	1.16104	−	1.16104i	0.176789	−	0.984249i	$-0.443429\pi$
$659$	29.8050	−	29.8050i	1.16104	−	1.16104i	0.984249	−	0.176789i	$-0.0565711\pi$
$660$		0			0
$661$	−3.86546	−	3.86546i	−0.150349	−	0.150349i	0.627925	−	0.778274i	$-0.283904\pi$
$661$	−3.86546	−	3.86546i	−0.150349	−	0.150349i	−0.778274	+	0.627925i	$0.783904\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			3.77568i			0.146415i
$666$		0			0
$667$	−11.0761	−	11.0761i	−0.428867	−	0.428867i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.6439			0.565322
$672$		0			0
$673$	−24.2478			−0.934685			−0.467342	−	0.884076i	$-0.654788\pi$
$673$	−24.2478			−0.934685			−0.467342	+	0.884076i	$0.654788\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.11030	+	9.11030i	0.350137	+	0.350137i	0.860161	−	0.510023i	$-0.170363\pi$
$677$	9.11030	+	9.11030i	0.350137	+	0.350137i	−0.510023	+	0.860161i	$0.670363\pi$
$678$		0			0
$679$			5.26449i			0.202033i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.43216	−	6.43216i	−0.246120	−	0.246120i	0.573256	−	0.819376i	$-0.305680\pi$
$683$	−6.43216	−	6.43216i	−0.246120	−	0.246120i	−0.819376	+	0.573256i	$0.805680\pi$
$684$		0			0
$685$	−5.43411	+	5.43411i	−0.207627	+	0.207627i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.29332			0.0492716
$690$		0			0
$691$	−8.79586	+	8.79586i	−0.334610	+	0.334610i	−0.854334	−	0.519724i	$-0.826035\pi$
$691$	−8.79586	+	8.79586i	−0.334610	+	0.334610i	0.519724	+	0.854334i	$0.326035\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	15.3833i		−	0.583522i
$696$		0			0
$697$		−	52.3861i		−	1.98426i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	27.7073	−	27.7073i	1.04649	−	1.04649i	0.0476269	−	0.998865i	$-0.484834\pi$
$701$	27.7073	−	27.7073i	1.04649	−	1.04649i	0.998865	−	0.0476269i	$-0.0151659\pi$
$702$		0			0
$703$	13.9563			0.526372
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7.48881	−	7.48881i	0.281646	−	0.281646i
$708$		0			0
$709$	−25.2865	−	25.2865i	−0.949653	−	0.949653i	0.0491386	−	0.998792i	$-0.484352\pi$
$709$	−25.2865	−	25.2865i	−0.949653	−	0.949653i	−0.998792	+	0.0491386i	$0.984352\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	3.02944i		−	0.113453i
$714$		0			0
$715$	−0.856566	−	0.856566i	−0.0320338	−	0.0320338i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9.43253			0.351774			0.175887	−	0.984410i	$-0.443721\pi$
$719$	9.43253			0.351774			0.175887	+	0.984410i	$0.443721\pi$
$720$		0			0
$721$	22.4157			0.834803
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.88784	−	5.88784i	−0.218669	−	0.218669i
$726$		0			0
$727$		−	1.10144i		−	0.0408501i	−0.999791	−	0.0204250i	$-0.993498\pi$
$727$		−	1.10144i		−	0.0408501i	0.999791	−	0.0204250i	$-0.00650195\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4.44549	−	4.44549i	−0.164422	−	0.164422i
$732$		0			0
$733$	−18.2214	+	18.2214i	−0.673024	+	0.673024i	−0.958412	−	0.285388i	$-0.907878\pi$
$733$	−18.2214	+	18.2214i	−0.673024	+	0.673024i	0.285388	+	0.958412i	$0.407878\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.76588			−0.175553
$738$		0			0
$739$	22.0953	−	22.0953i	0.812788	−	0.812788i	−0.172263	−	0.985051i	$-0.555108\pi$
$739$	22.0953	−	22.0953i	0.812788	−	0.812788i	0.985051	+	0.172263i	$0.0551079\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	47.3029i		−	1.73538i	−0.497109	−	0.867688i	$-0.665605\pi$
$743$		−	47.3029i		−	1.73538i	0.497109	−	0.867688i	$-0.334395\pi$
$744$		0			0
$745$			17.0496i			0.624649i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.89450	+	1.89450i	−0.0692236	+	0.0692236i
$750$		0			0
$751$	4.81234			0.175605			0.0878024	−	0.996138i	$-0.472016\pi$
$751$	4.81234			0.175605			0.0878024	+	0.996138i	$0.472016\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.98010	+	2.98010i	−0.108457	+	0.108457i
$756$		0			0
$757$	19.5838	+	19.5838i	0.711786	+	0.711786i	0.966909	−	0.255123i	$-0.0821159\pi$
$757$	19.5838	+	19.5838i	0.711786	+	0.711786i	−0.255123	+	0.966909i	$0.582116\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	0.278003i		−	0.0100776i	−0.999987	−	0.00503881i	$-0.998396\pi$
$761$		−	0.278003i		−	0.0100776i	0.999987	−	0.00503881i	$-0.00160391\pi$
$762$		0			0
$763$	12.2654	+	12.2654i	0.444037	+	0.444037i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	16.1094			0.581677
$768$		0			0
$769$	−20.5808			−0.742162			−0.371081	−	0.928600i	$-0.621013\pi$
$769$	−20.5808			−0.742162			−0.371081	+	0.928600i	$0.621013\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−18.3427	−	18.3427i	−0.659743	−	0.659743i	0.295576	−	0.955319i	$-0.404488\pi$
$773$	−18.3427	−	18.3427i	−0.659743	−	0.659743i	−0.955319	+	0.295576i	$0.904488\pi$
$774$		0			0
$775$		−	1.61040i		−	0.0578471i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	16.5604	+	16.5604i	0.593338	+	0.593338i
$780$		0			0
$781$	5.48881	−	5.48881i	0.196405	−	0.196405i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	13.4884			0.481422
$786$		0			0
$787$	−21.0313	+	21.0313i	−0.749684	+	0.749684i	−0.974420	−	0.224736i	$-0.927848\pi$
$787$	−21.0313	+	21.0313i	−0.749684	+	0.749684i	0.224736	+	0.974420i	$0.427848\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	21.6725i		−	0.770587i
$792$		0			0
$793$		−	13.5678i		−	0.481808i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	34.4451	−	34.4451i	1.22011	−	1.22011i	0.252515	−	0.967593i	$-0.418742\pi$
$797$	34.4451	−	34.4451i	1.22011	−	1.22011i	0.967593	−	0.252515i	$-0.0812576\pi$
$798$		0			0
$799$	−80.9831			−2.86498
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.88118	+	1.88118i	−0.0663852	+	0.0663852i
$804$		0			0
$805$	−1.88118	−	1.88118i	−0.0663027	−	0.0663027i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			4.84727i			0.170421i	0.996363	+	0.0852106i	$0.0271563\pi$
$809$			4.84727i			0.170421i	−0.996363	+	0.0852106i	$0.972844\pi$
$810$		0			0
$811$	37.5774	+	37.5774i	1.31952	+	1.31952i	0.914153	+	0.405369i	$0.132857\pi$
$811$	37.5774	+	37.5774i	1.31952	+	1.31952i	0.405369	+	0.914153i	$0.367143\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	15.6702			0.548903
$816$		0			0
$817$	2.81064			0.0983317
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.23725	+	6.23725i	0.217682	+	0.217682i	0.807521	−	0.589839i	$-0.200809\pi$
$821$	6.23725	+	6.23725i	0.217682	+	0.217682i	−0.589839	+	0.807521i	$0.700809\pi$
$822$		0			0
$823$		−	20.6905i		−	0.721225i	−0.932716	−	0.360613i	$-0.882568\pi$
$823$		−	20.6905i		−	0.721225i	0.932716	−	0.360613i	$-0.117432\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−5.53413	−	5.53413i	−0.192440	−	0.192440i	0.604309	−	0.796750i	$-0.293449\pi$
$827$	−5.53413	−	5.53413i	−0.192440	−	0.192440i	−0.796750	+	0.604309i	$0.793449\pi$
$828$		0			0
$829$	−21.2582	+	21.2582i	−0.738328	+	0.738328i	−0.972254	−	0.233926i	$-0.924842\pi$
$829$	−21.2582	+	21.2582i	−0.738328	+	0.738328i	0.233926	+	0.972254i	$0.424842\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−29.8593			−1.03456
$834$		0			0
$835$	8.29852	−	8.29852i	0.287182	−	0.287182i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			3.55688i			0.122797i	0.998113	+	0.0613985i	$0.0195561\pi$
$839$			3.55688i			0.122797i	−0.998113	+	0.0613985i	$0.980444\pi$
$840$		0			0
$841$		−	40.3333i		−	1.39080i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.39876	−	8.39876i	0.288926	−	0.288926i
$846$		0			0
$847$	−13.7073			−0.470990
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.95352	+	6.95352i	−0.238364	+	0.238364i
$852$		0			0
$853$	7.41187	+	7.41187i	0.253777	+	0.253777i	0.822517	−	0.568740i	$-0.192569\pi$
$853$	7.41187	+	7.41187i	0.253777	+	0.253777i	−0.568740	+	0.822517i	$0.692569\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			31.1426i			1.06381i	0.846803	+	0.531906i	$0.178524\pi$
$857$			31.1426i			1.06381i	−0.846803	+	0.531906i	$0.821476\pi$
$858$		0			0
$859$	−32.0035	−	32.0035i	−1.09195	−	1.09195i	−0.995321	−	0.0966247i	$-0.969195\pi$
$859$	−32.0035	−	32.0035i	−1.09195	−	1.09195i	−0.0966247	−	0.995321i	$-0.530805\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−29.2347			−0.995160			−0.497580	−	0.867418i	$-0.665778\pi$
$863$	−29.2347			−0.995160			−0.497580	+	0.867418i	$0.665778\pi$
$864$		0			0
$865$	10.1649			0.345617
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.45305	−	3.45305i	−0.117137	−	0.117137i
$870$		0			0
$871$			4.41568i			0.149619i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.00000	−	1.00000i	−0.0338062	−	0.0338062i
$876$		0			0
$877$	11.9832	−	11.9832i	0.404645	−	0.404645i	−0.475221	−	0.879866i	$-0.657632\pi$
$877$	11.9832	−	11.9832i	0.404645	−	0.404645i	0.879866	+	0.475221i	$0.157632\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−31.6194			−1.06529			−0.532643	−	0.846340i	$-0.678801\pi$
$881$	−31.6194			−1.06529			−0.532643	+	0.846340i	$0.678801\pi$
$882$		0			0
$883$	9.32520	−	9.32520i	0.313818	−	0.313818i	−0.532569	−	0.846387i	$-0.678773\pi$
$883$	9.32520	−	9.32520i	0.313818	−	0.313818i	0.846387	+	0.532569i	$0.178773\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	44.8049i		−	1.50440i	−0.658934	−	0.752200i	$-0.728993\pi$
$887$		−	44.8049i		−	1.50440i	0.658934	−	0.752200i	$-0.271007\pi$
$888$		0			0
$889$			5.92041i			0.198564i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	25.6006	−	25.6006i	0.856691	−	0.856691i
$894$		0			0
$895$	9.68499			0.323734
$896$		0			0
$897$		0			0
$898$		0			0
$899$	9.48175	−	9.48175i	0.316234	−	0.316234i
$900$		0			0
$901$	5.15509	+	5.15509i	0.171741	+	0.171741i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	11.6917i		−	0.388644i
$906$		0			0
$907$	19.4040	+	19.4040i	0.644299	+	0.644299i	0.951609	−	0.307310i	$-0.0994289\pi$
$907$	19.4040	+	19.4040i	0.644299	+	0.644299i	−0.307310	+	0.951609i	$0.599429\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	44.4094			1.47135			0.735674	−	0.677336i	$-0.236866\pi$
$911$	44.4094			1.47135			0.735674	+	0.677336i	$0.236866\pi$
$912$		0			0
$913$	5.03762			0.166721
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−6.07736	−	6.07736i	−0.200692	−	0.200692i
$918$		0			0
$919$			26.3092i			0.867861i	0.900946	+	0.433931i	$0.142874\pi$
$919$			26.3092i			0.867861i	−0.900946	+	0.433931i	$0.857126\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.08548	−	5.08548i	−0.167391	−	0.167391i
$924$		0			0
$925$	−3.69637	+	3.69637i	−0.121536	+	0.121536i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	41.9008			1.37472			0.687360	−	0.726317i	$-0.258770\pi$
$929$	41.9008			1.37472			0.687360	+	0.726317i	$0.258770\pi$
$930$		0			0
$931$	9.43920	−	9.43920i	0.309357	−	0.309357i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	6.82843i		−	0.223313i
$936$		0			0
$937$			50.1251i			1.63752i	0.574139	+	0.818758i	$0.305337\pi$
$937$			50.1251i			1.63752i	−0.574139	+	0.818758i	$0.694663\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−17.7315	+	17.7315i	−0.578029	+	0.578029i	−0.934360	−	0.356331i	$-0.884028\pi$
$941$	−17.7315	+	17.7315i	−0.578029	+	0.578029i	0.356331	+	0.934360i	$0.384028\pi$
$942$		0			0
$943$	−16.5020			−0.537378
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−8.29393	+	8.29393i	−0.269516	+	0.269516i	−0.828905	−	0.559389i	$-0.811036\pi$
$947$	−8.29393	+	8.29393i	−0.269516	+	0.269516i	0.559389	+	0.828905i	$0.311036\pi$
$948$		0			0
$949$	1.74294	+	1.74294i	0.0565783	+	0.0565783i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			22.6285i			0.733010i	0.930416	+	0.366505i	$0.119446\pi$
$953$			22.6285i			0.733010i	−0.930416	+	0.366505i	$0.880554\pi$
$954$		0			0
$955$	−12.1930	−	12.1930i	−0.394557	−	0.394557i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.8682			−0.350953
$960$		0			0
$961$	−28.4066			−0.916343
$962$		0			0
$963$		0			0
$964$		0			0
$965$	18.9750	+	18.9750i	0.610827	+	0.610827i
$966$		0			0
$967$		−	41.6144i		−	1.33823i	−0.743159	−	0.669115i	$-0.766673\pi$
$967$		−	41.6144i		−	1.33823i	0.743159	−	0.669115i	$-0.233327\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	21.7691	+	21.7691i	0.698604	+	0.698604i	0.964109	−	0.265505i	$-0.0855388\pi$
$971$	21.7691	+	21.7691i	0.698604	+	0.698604i	−0.265505	+	0.964109i	$0.585539\pi$
$972$		0			0
$973$	15.3833	−	15.3833i	0.493166	−	0.493166i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−17.3840			−0.556164			−0.278082	−	0.960557i	$-0.589699\pi$
$977$	−17.3840			−0.556164			−0.278082	+	0.960557i	$0.589699\pi$
$978$		0			0
$979$	8.34179	−	8.34179i	0.266605	−	0.266605i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	4.91428i		−	0.156741i	−0.996924	−	0.0783707i	$-0.975028\pi$
$983$		−	4.91428i		−	0.156741i	0.996924	−	0.0783707i	$-0.0249718\pi$
$984$		0			0
$985$			1.16215i			0.0370291i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.40036	+	1.40036i	−0.0445288	+	0.0445288i
$990$		0			0
$991$	−23.5415			−0.747822			−0.373911	−	0.927465i	$-0.621983\pi$
$991$	−23.5415			−0.747822			−0.373911	+	0.927465i	$0.621983\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	14.3728	−	14.3728i	0.455650	−	0.455650i
$996$		0			0
$997$	−19.6097	−	19.6097i	−0.621046	−	0.621046i	0.324753	−	0.945799i	$-0.394719\pi$
$997$	−19.6097	−	19.6097i	−0.621046	−	0.621046i	−0.945799	+	0.324753i	$0.894719\pi$
$998$		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.2161.1		8
3.2	odd	2		960.2.s.b.241.4		8
4.3	odd	2		720.2.t.b.181.1		8
12.11	even	2		240.2.s.b.181.4	yes	8
16.3	odd	4		720.2.t.b.541.1		8
16.13	even	4	inner	2880.2.t.b.721.1		8
24.5	odd	2		1920.2.s.d.481.1		8
24.11	even	2		1920.2.s.c.481.4		8
48.5	odd	4		1920.2.s.d.1441.1		8
48.11	even	4		1920.2.s.c.1441.4		8
48.29	odd	4		960.2.s.b.721.4		8
48.35	even	4		240.2.s.b.61.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.s.b.61.4	✓	8	48.35	even	4
240.2.s.b.181.4	yes	8	12.11	even	2
720.2.t.b.181.1		8	4.3	odd	2
720.2.t.b.541.1		8	16.3	odd	4
960.2.s.b.241.4		8	3.2	odd	2
960.2.s.b.721.4		8	48.29	odd	4
1920.2.s.c.481.4		8	24.11	even	2
1920.2.s.c.1441.4		8	48.11	even	4
1920.2.s.d.481.1		8	24.5	odd	2
1920.2.s.d.1441.1		8	48.5	odd	4
2880.2.t.b.721.1		8	16.13	even	4	inner
2880.2.t.b.2161.1		8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{5} -1.41421i q^{7} +O(q^{10})\)	\(q+(-0.707107 - 0.707107i) q^{5} -1.41421i q^{7} +(-0.808530 - 0.808530i) q^{11} +(-0.749118 + 0.749118i) q^{13} -5.97186 q^{17} +(1.88784 - 1.88784i) q^{19} +1.88118i q^{23} +1.00000i q^{25} +(-5.88784 + 5.88784i) q^{29} -1.61040 q^{31} +(-1.00000 + 1.00000i) q^{35} +(3.69637 + 3.69637i) q^{37} +8.77215i q^{41} +(0.744406 + 0.744406i) q^{43} +13.5608 q^{47} +5.00000 q^{49} +(-0.863230 - 0.863230i) q^{53} +1.14343i q^{55} +(-10.7523 - 10.7523i) q^{59} +(-9.05588 + 9.05588i) q^{61} +1.05941 q^{65} +(2.94725 - 2.94725i) q^{67} +6.78863i q^{71} -2.32666i q^{73} +(-1.14343 + 1.14343i) q^{77} +4.27078 q^{79} +(-3.11529 + 3.11529i) q^{83} +(4.22274 + 4.22274i) q^{85} +10.3172i q^{89} +(1.05941 + 1.05941i) q^{91} -2.66981 q^{95} -3.72256 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 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

Related objects

Downloads

Learn more