LMFDB - Embedded newform 2880.2.t.a.2161.2

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:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2161.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2880.2161
Dual form			2880.2.t.a.721.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} -0.828427i q^{7} +O(q^{10})$	$q+(0.707107 + 0.707107i) q^{5} -0.828427i q^{7} +(-1.41421 - 1.41421i) q^{11} +(-3.41421 + 3.41421i) q^{13} -2.58579 q^{17} +(1.82843 - 1.82843i) q^{19} -2.58579i q^{23} +1.00000i q^{25} +(3.41421 - 3.41421i) q^{29} +7.65685 q^{31} +(0.585786 - 0.585786i) q^{35} +(7.41421 + 7.41421i) q^{37} +0.828427i q^{41} +(7.65685 + 7.65685i) q^{43} +7.07107 q^{47} +6.31371 q^{49} +(4.00000 + 4.00000i) q^{53} -2.00000i q^{55} +(-4.58579 - 4.58579i) q^{59} +(-7.48528 + 7.48528i) q^{61} -4.82843 q^{65} +(-3.65685 + 3.65685i) q^{67} -8.00000i q^{71} +8.82843i q^{73} +(-1.17157 + 1.17157i) q^{77} +2.34315 q^{79} +(11.0711 - 11.0711i) q^{83} +(-1.82843 - 1.82843i) q^{85} +7.65685i q^{89} +(2.82843 + 2.82843i) q^{91} +2.58579 q^{95} +9.31371 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 8 q^{13} - 16 q^{17} - 4 q^{19} + 8 q^{29} + 8 q^{31} + 8 q^{35} + 24 q^{37} + 8 q^{43} - 20 q^{49} + 16 q^{53} - 24 q^{59} + 4 q^{61} - 8 q^{65} + 8 q^{67} - 16 q^{77} + 32 q^{79} + 16 q^{83} + 4 q^{85} + 16 q^{95} - 8 q^{97}+O(q^{100}) $ 4 * q - 8 * q^13 - 16 * q^17 - 4 * q^19 + 8 * q^29 + 8 * q^31 + 8 * q^35 + 24 * q^37 + 8 * q^43 - 20 * q^49 + 16 * q^53 - 24 * q^59 + 4 * q^61 - 8 * q^65 + 8 * q^67 - 16 * q^77 + 32 * q^79 + 16 * q^83 + 4 * q^85 + 16 * q^95 - 8 * 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
$7$		−	0.828427i		−	0.313116i	−0.987669	−	0.156558i	$-0.949960\pi$
$7$		−	0.828427i		−	0.313116i	0.987669	−	0.156558i	$-0.0500398\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.41421	−	1.41421i	−0.426401	−	0.426401i	0.460999	−	0.887401i	$-0.347491\pi$
$11$	−1.41421	−	1.41421i	−0.426401	−	0.426401i	−0.887401	+	0.460999i	$0.847491\pi$
$12$		0			0
$13$	−3.41421	+	3.41421i	−0.946932	+	0.946932i	−0.998661	−	0.0517287i	$-0.983527\pi$
$13$	−3.41421	+	3.41421i	−0.946932	+	0.946932i	0.0517287	+	0.998661i	$0.483527\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.58579			−0.627145			−0.313573	−	0.949564i	$-0.601526\pi$
$17$	−2.58579			−0.627145			−0.313573	+	0.949564i	$0.601526\pi$
$18$		0			0
$19$	1.82843	−	1.82843i	0.419470	−	0.419470i	−0.465551	−	0.885021i	$-0.654144\pi$
$19$	1.82843	−	1.82843i	0.419470	−	0.419470i	0.885021	+	0.465551i	$0.154144\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.58579i		−	0.539174i	−0.962976	−	0.269587i	$-0.913113\pi$
$23$		−	2.58579i		−	0.539174i	0.962976	−	0.269587i	$-0.0868871\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.41421	−	3.41421i	0.634004	−	0.634004i	−0.315066	−	0.949070i	$-0.602027\pi$
$29$	3.41421	−	3.41421i	0.634004	−	0.634004i	0.949070	+	0.315066i	$0.102027\pi$
$30$		0			0
$31$	7.65685			1.37521			0.687606	−	0.726084i	$-0.258662\pi$
$31$	7.65685			1.37521			0.687606	+	0.726084i	$0.258662\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.585786	−	0.585786i	0.0990160	−	0.0990160i
$36$		0			0
$37$	7.41421	+	7.41421i	1.21889	+	1.21889i	0.968021	+	0.250868i	$0.0807160\pi$
$37$	7.41421	+	7.41421i	1.21889	+	1.21889i	0.250868	+	0.968021i	$0.419284\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			0.828427i			0.129379i	0.997905	+	0.0646893i	$0.0206056\pi$
$41$			0.828427i			0.129379i	−0.997905	+	0.0646893i	$0.979394\pi$
$42$		0			0
$43$	7.65685	+	7.65685i	1.16766	+	1.16766i	0.982757	+	0.184902i	$0.0591969\pi$
$43$	7.65685	+	7.65685i	1.16766	+	1.16766i	0.184902	+	0.982757i	$0.440803\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.07107			1.03142			0.515711	−	0.856763i	$-0.327528\pi$
$47$	7.07107			1.03142			0.515711	+	0.856763i	$0.327528\pi$
$48$		0			0
$49$	6.31371			0.901958
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.00000	+	4.00000i	0.549442	+	0.549442i	0.926279	−	0.376837i	$-0.122988\pi$
$53$	4.00000	+	4.00000i	0.549442	+	0.549442i	−0.376837	+	0.926279i	$0.622988\pi$
$54$		0			0
$55$		−	2.00000i		−	0.269680i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.58579	−	4.58579i	−0.597019	−	0.597019i	0.342499	−	0.939518i	$-0.388727\pi$
$59$	−4.58579	−	4.58579i	−0.597019	−	0.597019i	−0.939518	+	0.342499i	$0.888727\pi$
$60$		0			0
$61$	−7.48528	+	7.48528i	−0.958392	+	0.958392i	−0.999168	−	0.0407762i	$-0.987017\pi$
$61$	−7.48528	+	7.48528i	−0.958392	+	0.958392i	0.0407762	+	0.999168i	$0.487017\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.82843			−0.598893
$66$		0			0
$67$	−3.65685	+	3.65685i	−0.446756	+	0.446756i	−0.894275	−	0.447519i	$-0.852308\pi$
$67$	−3.65685	+	3.65685i	−0.446756	+	0.446756i	0.447519	+	0.894275i	$0.352308\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.00000i		−	0.949425i	−0.880141	−	0.474713i	$-0.842552\pi$
$71$		−	8.00000i		−	0.949425i	0.880141	−	0.474713i	$-0.157448\pi$
$72$		0			0
$73$			8.82843i			1.03329i	0.856200	+	0.516645i	$0.172819\pi$
$73$			8.82843i			1.03329i	−0.856200	+	0.516645i	$0.827181\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.17157	+	1.17157i	−0.133513	+	0.133513i
$78$		0			0
$79$	2.34315			0.263624			0.131812	−	0.991275i	$-0.457920\pi$
$79$	2.34315			0.263624			0.131812	+	0.991275i	$0.457920\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.0711	−	11.0711i	1.21521	−	1.21521i	0.245917	−	0.969291i	$-0.420911\pi$
$83$	11.0711	−	11.0711i	1.21521	−	1.21521i	0.969291	−	0.245917i	$-0.0790889\pi$
$84$		0			0
$85$	−1.82843	−	1.82843i	−0.198321	−	0.198321i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			7.65685i			0.811625i	0.913956	+	0.405812i	$0.133011\pi$
$89$			7.65685i			0.811625i	−0.913956	+	0.405812i	$0.866989\pi$
$90$		0			0
$91$	2.82843	+	2.82843i	0.296500	+	0.296500i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.58579			0.265296
$96$		0			0
$97$	9.31371			0.945664			0.472832	−	0.881153i	$-0.343232\pi$
$97$	9.31371			0.945664			0.472832	+	0.881153i	$0.343232\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.41421	+	3.41421i	0.339727	+	0.339727i	0.856265	−	0.516538i	$-0.172779\pi$
$101$	3.41421	+	3.41421i	0.339727	+	0.339727i	−0.516538	+	0.856265i	$0.672779\pi$
$102$		0			0
$103$		−	10.0000i		−	0.985329i	−0.870219	−	0.492665i	$-0.836023\pi$
$103$		−	10.0000i		−	0.985329i	0.870219	−	0.492665i	$-0.163977\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.75736	+	3.75736i	0.363238	+	0.363238i	0.865003	−	0.501766i	$-0.167316\pi$
$107$	3.75736	+	3.75736i	0.363238	+	0.363238i	−0.501766	+	0.865003i	$0.667316\pi$
$108$		0			0
$109$	11.4853	−	11.4853i	1.10009	−	1.10009i	0.105691	−	0.994399i	$-0.466294\pi$
$109$	11.4853	−	11.4853i	1.10009	−	1.10009i	0.994399	−	0.105691i	$-0.0337056\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14.5858			−1.37212			−0.686058	−	0.727547i	$-0.740660\pi$
$113$	−14.5858			−1.37212			−0.686058	+	0.727547i	$0.740660\pi$
$114$		0			0
$115$	1.82843	−	1.82843i	0.170502	−	0.170502i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			2.14214i			0.196369i
$120$		0			0
$121$		−	7.00000i		−	0.636364i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$	−8.34315			−0.740334			−0.370167	−	0.928965i	$-0.620700\pi$
$127$	−8.34315			−0.740334			−0.370167	+	0.928965i	$0.620700\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.5858	+	10.5858i	−0.924884	+	0.924884i	−0.997369	−	0.0724850i	$-0.976907\pi$
$131$	−10.5858	+	10.5858i	−0.924884	+	0.924884i	0.0724850	+	0.997369i	$0.476907\pi$
$132$		0			0
$133$	−1.51472	−	1.51472i	−0.131343	−	0.131343i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	4.24264i		−	0.362473i	−0.983440	−	0.181237i	$-0.941990\pi$
$137$		−	4.24264i		−	0.362473i	0.983440	−	0.181237i	$-0.0580100\pi$
$138$		0			0
$139$	4.65685	+	4.65685i	0.394989	+	0.394989i	0.876461	−	0.481472i	$-0.159898\pi$
$139$	4.65685	+	4.65685i	0.394989	+	0.394989i	−0.481472	+	0.876461i	$0.659898\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	9.65685			0.807547
$144$		0			0
$145$	4.82843			0.400979
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.07107	+	7.07107i	0.579284	+	0.579284i	0.934706	−	0.355422i	$-0.115663\pi$
$149$	7.07107	+	7.07107i	0.579284	+	0.579284i	−0.355422	+	0.934706i	$0.615663\pi$
$150$		0			0
$151$			7.65685i			0.623106i	0.950229	+	0.311553i	$0.100849\pi$
$151$			7.65685i			0.623106i	−0.950229	+	0.311553i	$0.899151\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	5.41421	+	5.41421i	0.434880	+	0.434880i
$156$		0			0
$157$	7.07107	−	7.07107i	0.564333	−	0.564333i	−0.366203	−	0.930535i	$-0.619342\pi$
$157$	7.07107	−	7.07107i	0.564333	−	0.564333i	0.930535	+	0.366203i	$0.119342\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.14214			−0.168824
$162$		0			0
$163$	−13.6569	+	13.6569i	−1.06969	+	1.06969i	−0.0723048	+	0.997383i	$0.523035\pi$
$163$	−13.6569	+	13.6569i	−1.06969	+	1.06969i	−0.997383	+	0.0723048i	$0.976965\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			11.0711i			0.856705i	0.903612	+	0.428352i	$0.140906\pi$
$167$			11.0711i			0.856705i	−0.903612	+	0.428352i	$0.859094\pi$
$168$		0			0
$169$		−	10.3137i		−	0.793362i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	11.3137	−	11.3137i	0.860165	−	0.860165i	−0.131192	−	0.991357i	$-0.541880\pi$
$173$	11.3137	−	11.3137i	0.860165	−	0.860165i	0.991357	+	0.131192i	$0.0418803\pi$
$174$		0			0
$175$	0.828427			0.0626232
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−13.4142	+	13.4142i	−1.00263	+	1.00263i	−0.00262885	+	0.999997i	$0.500837\pi$
$179$	−13.4142	+	13.4142i	−1.00263	+	1.00263i	−0.999997	+	0.00262885i	$0.999163\pi$
$180$		0			0
$181$	13.8284	+	13.8284i	1.02786	+	1.02786i	0.999601	+	0.0282582i	$0.00899607\pi$
$181$	13.8284	+	13.8284i	1.02786	+	1.02786i	0.0282582	+	0.999601i	$0.491004\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			10.4853i			0.770893i
$186$		0			0
$187$	3.65685	+	3.65685i	0.267416	+	0.267416i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.4853			0.903403			0.451702	−	0.892169i	$-0.350817\pi$
$191$	12.4853			0.903403			0.451702	+	0.892169i	$0.350817\pi$
$192$		0			0
$193$	−4.82843			−0.347558			−0.173779	−	0.984785i	$-0.555598\pi$
$193$	−4.82843			−0.347558			−0.173779	+	0.984785i	$0.555598\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	16.2426	+	16.2426i	1.15724	+	1.15724i	0.985067	+	0.172174i	$0.0550790\pi$
$197$	16.2426	+	16.2426i	1.15724	+	1.15724i	0.172174	+	0.985067i	$0.444921\pi$
$198$		0			0
$199$		−	17.6569i		−	1.25166i	−0.779959	−	0.625831i	$-0.784760\pi$
$199$		−	17.6569i		−	1.25166i	0.779959	−	0.625831i	$-0.215240\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.82843	−	2.82843i	−0.198517	−	0.198517i
$204$		0			0
$205$	−0.585786	+	0.585786i	−0.0409131	+	0.0409131i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.17157			−0.357725
$210$		0			0
$211$	8.65685	−	8.65685i	0.595962	−	0.595962i	−0.343273	−	0.939236i	$-0.611536\pi$
$211$	8.65685	−	8.65685i	0.595962	−	0.595962i	0.939236	+	0.343273i	$0.111536\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			10.8284i			0.738493i
$216$		0			0
$217$		−	6.34315i		−	0.430601i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.82843	−	8.82843i	0.593864	−	0.593864i
$222$		0			0
$223$	20.8284			1.39477			0.697387	−	0.716694i	$-0.254346\pi$
$223$	20.8284			1.39477			0.697387	+	0.716694i	$0.254346\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−17.4142	+	17.4142i	−1.15582	+	1.15582i	−0.170457	+	0.985365i	$0.554524\pi$
$227$	−17.4142	+	17.4142i	−1.15582	+	1.15582i	−0.985365	+	0.170457i	$0.945476\pi$
$228$		0			0
$229$	−13.0000	−	13.0000i	−0.859064	−	0.859064i	0.132164	−	0.991228i	$-0.457808\pi$
$229$	−13.0000	−	13.0000i	−0.859064	−	0.859064i	−0.991228	+	0.132164i	$0.957808\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	27.5563i		−	1.80528i	−0.430400	−	0.902638i	$-0.641628\pi$
$233$		−	27.5563i		−	1.80528i	0.430400	−	0.902638i	$-0.358372\pi$
$234$		0			0
$235$	5.00000	+	5.00000i	0.326164	+	0.326164i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.9706			−0.838996			−0.419498	−	0.907756i	$-0.637794\pi$
$239$	−12.9706			−0.838996			−0.419498	+	0.907756i	$0.637794\pi$
$240$		0			0
$241$	−21.6569			−1.39504			−0.697520	−	0.716565i	$-0.745713\pi$
$241$	−21.6569			−1.39504			−0.697520	+	0.716565i	$0.745713\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.46447	+	4.46447i	0.285224	+	0.285224i
$246$		0			0
$247$			12.4853i			0.794419i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.8995	+	15.8995i	1.00357	+	1.00357i	0.999994	+	0.00357361i	$0.00113752\pi$
$251$	15.8995	+	15.8995i	1.00357	+	1.00357i	0.00357361	+	0.999994i	$0.498862\pi$
$252$		0			0
$253$	−3.65685	+	3.65685i	−0.229904	+	0.229904i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	26.3848			1.64584			0.822919	−	0.568159i	$-0.192344\pi$
$257$	26.3848			1.64584			0.822919	+	0.568159i	$0.192344\pi$
$258$		0			0
$259$	6.14214	−	6.14214i	0.381654	−	0.381654i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	14.3848i		−	0.887003i	−0.896274	−	0.443502i	$-0.853736\pi$
$263$		−	14.3848i		−	0.887003i	0.896274	−	0.443502i	$-0.146264\pi$
$264$		0			0
$265$			5.65685i			0.347498i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.72792	+	6.72792i	−0.410209	+	0.410209i	−0.881811	−	0.471603i	$-0.843676\pi$
$269$	−6.72792	+	6.72792i	−0.410209	+	0.410209i	0.471603	+	0.881811i	$0.343676\pi$
$270$		0			0
$271$	−0.686292			−0.0416892			−0.0208446	−	0.999783i	$-0.506636\pi$
$271$	−0.686292			−0.0416892			−0.0208446	+	0.999783i	$0.506636\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.41421	−	1.41421i	0.0852803	−	0.0852803i
$276$		0			0
$277$	19.4142	+	19.4142i	1.16649	+	1.16649i	0.983027	+	0.183460i	$0.0587297\pi$
$277$	19.4142	+	19.4142i	1.16649	+	1.16649i	0.183460	+	0.983027i	$0.441270\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	12.1421i		−	0.724339i	−0.932112	−	0.362170i	$-0.882036\pi$
$281$		−	12.1421i		−	0.724339i	0.932112	−	0.362170i	$-0.117964\pi$
$282$		0			0
$283$	0.343146	+	0.343146i	0.0203979	+	0.0203979i	0.717232	−	0.696834i	$-0.245409\pi$
$283$	0.343146	+	0.343146i	0.0203979	+	0.0203979i	−0.696834	+	0.717232i	$0.745409\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.686292			0.0405105
$288$		0			0
$289$	−10.3137			−0.606689
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.6569	+	13.6569i	0.797842	+	0.797842i	0.982755	−	0.184913i	$-0.0592003\pi$
$293$	13.6569	+	13.6569i	0.797842	+	0.797842i	−0.184913	+	0.982755i	$0.559200\pi$
$294$		0			0
$295$		−	6.48528i		−	0.377588i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.82843	+	8.82843i	0.510561	+	0.510561i
$300$		0			0
$301$	6.34315	−	6.34315i	0.365613	−	0.365613i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−10.5858			−0.606140
$306$		0			0
$307$	−7.65685	+	7.65685i	−0.437000	+	0.437000i	−0.891001	−	0.454001i	$-0.849996\pi$
$307$	−7.65685	+	7.65685i	−0.437000	+	0.437000i	0.454001	+	0.891001i	$0.349996\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	32.9706i		−	1.86959i	−0.355189	−	0.934795i	$-0.615583\pi$
$311$		−	32.9706i		−	1.86959i	0.355189	−	0.934795i	$-0.384417\pi$
$312$		0			0
$313$		−	16.8284i		−	0.951199i	−0.879662	−	0.475599i	$-0.842231\pi$
$313$		−	16.8284i		−	0.951199i	0.879662	−	0.475599i	$-0.157769\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.41421	−	9.41421i	0.528755	−	0.528755i	−0.391446	−	0.920201i	$-0.628025\pi$
$317$	9.41421	−	9.41421i	0.528755	−	0.528755i	0.920201	+	0.391446i	$0.128025\pi$
$318$		0			0
$319$	−9.65685			−0.540680
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.72792	+	4.72792i	−0.263069	+	0.263069i
$324$		0			0
$325$	−3.41421	−	3.41421i	−0.189386	−	0.189386i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	5.85786i		−	0.322955i
$330$		0			0
$331$	−15.8284	−	15.8284i	−0.870009	−	0.870009i	0.122464	−	0.992473i	$-0.460920\pi$
$331$	−15.8284	−	15.8284i	−0.870009	−	0.870009i	−0.992473	+	0.122464i	$0.960920\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.17157			−0.282553
$336$		0			0
$337$	−14.4853			−0.789064			−0.394532	−	0.918882i	$-0.629093\pi$
$337$	−14.4853			−0.789064			−0.394532	+	0.918882i	$0.629093\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−10.8284	−	10.8284i	−0.586392	−	0.586392i
$342$		0			0
$343$		−	11.0294i		−	0.595534i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.8284	+	14.8284i	0.796032	+	0.796032i	0.982467	−	0.186436i	$-0.0596935\pi$
$347$	14.8284	+	14.8284i	0.796032	+	0.796032i	−0.186436	+	0.982467i	$0.559694\pi$
$348$		0			0
$349$	9.48528	−	9.48528i	0.507735	−	0.507735i	−0.406095	−	0.913831i	$-0.633110\pi$
$349$	9.48528	−	9.48528i	0.507735	−	0.507735i	0.913831	+	0.406095i	$0.133110\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−18.5858			−0.989222			−0.494611	−	0.869115i	$-0.664689\pi$
$353$	−18.5858			−0.989222			−0.494611	+	0.869115i	$0.664689\pi$
$354$		0			0
$355$	5.65685	−	5.65685i	0.300235	−	0.300235i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	13.1716i		−	0.695169i	−0.937649	−	0.347585i	$-0.887002\pi$
$359$		−	13.1716i		−	0.695169i	0.937649	−	0.347585i	$-0.112998\pi$
$360$		0			0
$361$			12.3137i			0.648090i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.24264	+	6.24264i	−0.326755	+	0.326755i
$366$		0			0
$367$	1.02944			0.0537362			0.0268681	−	0.999639i	$-0.491447\pi$
$367$	1.02944			0.0537362			0.0268681	+	0.999639i	$0.491447\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.31371	−	3.31371i	0.172039	−	0.172039i
$372$		0			0
$373$	3.07107	+	3.07107i	0.159014	+	0.159014i	0.782130	−	0.623116i	$-0.214133\pi$
$373$	3.07107	+	3.07107i	0.159014	+	0.159014i	−0.623116	+	0.782130i	$0.714133\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			23.3137i			1.20072i
$378$		0			0
$379$	−16.6569	−	16.6569i	−0.855605	−	0.855605i	0.135212	−	0.990817i	$-0.456829\pi$
$379$	−16.6569	−	16.6569i	−0.855605	−	0.855605i	−0.990817	+	0.135212i	$0.956829\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−8.92893			−0.456247			−0.228124	−	0.973632i	$-0.573259\pi$
$383$	−8.92893			−0.456247			−0.228124	+	0.973632i	$0.573259\pi$
$384$		0			0
$385$	−1.65685			−0.0844411
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−26.3848	−	26.3848i	−1.33776	−	1.33776i	−0.898225	−	0.439537i	$-0.855143\pi$
$389$	−26.3848	−	26.3848i	−1.33776	−	1.33776i	−0.439537	−	0.898225i	$-0.644857\pi$
$390$		0			0
$391$			6.68629i			0.338140i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.65685	+	1.65685i	0.0833654	+	0.0833654i
$396$		0			0
$397$	9.89949	−	9.89949i	0.496841	−	0.496841i	−0.413612	−	0.910453i	$-0.635733\pi$
$397$	9.89949	−	9.89949i	0.496841	−	0.496841i	0.910453	+	0.413612i	$0.135733\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.97056			0.348093			0.174047	−	0.984737i	$-0.444316\pi$
$401$	6.97056			0.348093			0.174047	+	0.984737i	$0.444316\pi$
$402$		0			0
$403$	−26.1421	+	26.1421i	−1.30223	+	1.30223i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	20.9706i		−	1.03947i
$408$		0			0
$409$			14.6274i			0.723279i	0.932318	+	0.361640i	$0.117783\pi$
$409$			14.6274i			0.723279i	−0.932318	+	0.361640i	$0.882217\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.79899	+	3.79899i	−0.186936	+	0.186936i
$414$		0			0
$415$	15.6569			0.768565
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.75736	−	1.75736i	0.0858526	−	0.0858526i	−0.662876	−	0.748729i	$-0.730664\pi$
$419$	1.75736	−	1.75736i	0.0858526	−	0.0858526i	0.748729	+	0.662876i	$0.230664\pi$
$420$		0			0
$421$	3.00000	+	3.00000i	0.146211	+	0.146211i	0.776423	−	0.630212i	$-0.217032\pi$
$421$	3.00000	+	3.00000i	0.146211	+	0.146211i	−0.630212	+	0.776423i	$0.717032\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	2.58579i		−	0.125429i
$426$		0			0
$427$	6.20101	+	6.20101i	0.300088	+	0.300088i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.48528			0.216048			0.108024	−	0.994148i	$-0.465548\pi$
$431$	4.48528			0.216048			0.108024	+	0.994148i	$0.465548\pi$
$432$		0			0
$433$	31.1716			1.49801			0.749005	−	0.662564i	$-0.230532\pi$
$433$	31.1716			1.49801			0.749005	+	0.662564i	$0.230532\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.72792	−	4.72792i	−0.226167	−	0.226167i
$438$		0			0
$439$			17.6569i			0.842716i	0.906894	+	0.421358i	$0.138446\pi$
$439$			17.6569i			0.842716i	−0.906894	+	0.421358i	$0.861554\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.4142	−	13.4142i	−0.637329	−	0.637329i	0.312567	−	0.949896i	$-0.398811\pi$
$443$	−13.4142	−	13.4142i	−0.637329	−	0.637329i	−0.949896	+	0.312567i	$0.898811\pi$
$444$		0			0
$445$	−5.41421	+	5.41421i	−0.256658	+	0.256658i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−3.17157			−0.149676			−0.0748379	−	0.997196i	$-0.523844\pi$
$449$	−3.17157			−0.149676			−0.0748379	+	0.997196i	$0.523844\pi$
$450$		0			0
$451$	1.17157	−	1.17157i	0.0551672	−	0.0551672i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			4.00000i			0.187523i
$456$		0			0
$457$			12.3431i			0.577388i	0.957421	+	0.288694i	$0.0932210\pi$
$457$			12.3431i			0.577388i	−0.957421	+	0.288694i	$0.906779\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−16.7279	+	16.7279i	−0.779097	+	0.779097i	−0.979677	−	0.200580i	$-0.935717\pi$
$461$	−16.7279	+	16.7279i	−0.779097	+	0.779097i	0.200580	+	0.979677i	$0.435717\pi$
$462$		0			0
$463$	−0.343146			−0.0159473			−0.00797367	−	0.999968i	$-0.502538\pi$
$463$	−0.343146			−0.0159473			−0.00797367	+	0.999968i	$0.502538\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−15.3137	+	15.3137i	−0.708634	+	0.708634i	−0.966248	−	0.257614i	$-0.917064\pi$
$467$	−15.3137	+	15.3137i	−0.708634	+	0.708634i	0.257614	+	0.966248i	$0.417064\pi$
$468$		0			0
$469$	3.02944	+	3.02944i	0.139886	+	0.139886i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	21.6569i		−	0.995783i
$474$		0			0
$475$	1.82843	+	1.82843i	0.0838940	+	0.0838940i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.00000			−0.182765			−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000			−0.182765			−0.0913823	+	0.995816i	$0.529129\pi$
$480$		0			0
$481$	−50.6274			−2.30841
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.58579	+	6.58579i	0.299045	+	0.299045i
$486$		0			0
$487$			26.0000i			1.17817i	0.808070	+	0.589086i	$0.200512\pi$
$487$			26.0000i			1.17817i	−0.808070	+	0.589086i	$0.799488\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.24264	−	6.24264i	−0.281726	−	0.281726i	0.552071	−	0.833797i	$-0.313838\pi$
$491$	−6.24264	−	6.24264i	−0.281726	−	0.281726i	−0.833797	+	0.552071i	$0.813838\pi$
$492$		0			0
$493$	−8.82843	+	8.82843i	−0.397612	+	0.397612i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.62742			−0.297280
$498$		0			0
$499$	21.4853	−	21.4853i	0.961813	−	0.961813i	−0.0374839	−	0.999297i	$-0.511934\pi$
$499$	21.4853	−	21.4853i	0.961813	−	0.961813i	0.999297	+	0.0374839i	$0.0119343\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			11.7574i			0.524235i	0.965036	+	0.262117i	$0.0844208\pi$
$503$			11.7574i			0.524235i	−0.965036	+	0.262117i	$0.915579\pi$
$504$		0			0
$505$			4.82843i			0.214862i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	11.4142	−	11.4142i	0.505926	−	0.505926i	−0.407347	−	0.913273i	$-0.633546\pi$
$509$	11.4142	−	11.4142i	0.505926	−	0.505926i	0.913273	+	0.407347i	$0.133546\pi$
$510$		0			0
$511$	7.31371			0.323539
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.07107	−	7.07107i	0.311588	−	0.311588i
$516$		0			0
$517$	−10.0000	−	10.0000i	−0.439799	−	0.439799i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	1.51472i		−	0.0663610i	−0.999449	−	0.0331805i	$-0.989436\pi$
$521$		−	1.51472i		−	0.0663610i	0.999449	−	0.0331805i	$-0.0105636\pi$
$522$		0			0
$523$	−28.6274	−	28.6274i	−1.25179	−	1.25179i	−0.954917	−	0.296872i	$-0.904056\pi$
$523$	−28.6274	−	28.6274i	−1.25179	−	1.25179i	−0.296872	−	0.954917i	$-0.595944\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−19.7990			−0.862458
$528$		0			0
$529$	16.3137			0.709292
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.82843	−	2.82843i	−0.122513	−	0.122513i
$534$		0			0
$535$			5.31371i			0.229732i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8.92893	−	8.92893i	−0.384596	−	0.384596i
$540$		0			0
$541$	−0.514719	+	0.514719i	−0.0221295	+	0.0221295i	−0.718085	−	0.695955i	$-0.754981\pi$
$541$	−0.514719	+	0.514719i	−0.0221295	+	0.0221295i	0.695955	+	0.718085i	$0.254981\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	16.2426			0.695758
$546$		0			0
$547$	−2.48528	+	2.48528i	−0.106263	+	0.106263i	−0.758239	−	0.651976i	$-0.773940\pi$
$547$	−2.48528	+	2.48528i	−0.106263	+	0.106263i	0.651976	+	0.758239i	$0.273940\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	12.4853i		−	0.531891i
$552$		0			0
$553$		−	1.94113i		−	0.0825451i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18.8284	+	18.8284i	−0.797786	+	0.797786i	−0.982746	−	0.184960i	$-0.940784\pi$
$557$	−18.8284	+	18.8284i	−0.797786	+	0.797786i	0.184960	+	0.982746i	$0.440784\pi$
$558$		0			0
$559$	−52.2843			−2.21139
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−8.48528	+	8.48528i	−0.357612	+	0.357612i	−0.862932	−	0.505320i	$-0.831374\pi$
$563$	−8.48528	+	8.48528i	−0.357612	+	0.357612i	0.505320	+	0.862932i	$0.331374\pi$
$564$		0			0
$565$	−10.3137	−	10.3137i	−0.433901	−	0.433901i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	33.3137i		−	1.39658i	−0.715813	−	0.698292i	$-0.753944\pi$
$569$		−	33.3137i		−	1.39658i	0.715813	−	0.698292i	$-0.246056\pi$
$570$		0			0
$571$	23.8284	+	23.8284i	0.997189	+	0.997189i	0.999996	−	0.00280742i	$-0.000893631\pi$
$571$	23.8284	+	23.8284i	0.997189	+	0.997189i	−0.00280742	+	0.999996i	$0.500894\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.58579			0.107835
$576$		0			0
$577$	−16.3431			−0.680374			−0.340187	−	0.940358i	$-0.610490\pi$
$577$	−16.3431			−0.680374			−0.340187	+	0.940358i	$0.610490\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−9.17157	−	9.17157i	−0.380501	−	0.380501i
$582$		0			0
$583$		−	11.3137i		−	0.468566i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	11.0711	+	11.0711i	0.456952	+	0.456952i	0.897654	−	0.440702i	$-0.145270\pi$
$587$	11.0711	+	11.0711i	0.456952	+	0.456952i	−0.440702	+	0.897654i	$0.645270\pi$
$588$		0			0
$589$	14.0000	−	14.0000i	0.576860	−	0.576860i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−14.1005			−0.579038			−0.289519	−	0.957172i	$-0.593495\pi$
$593$	−14.1005			−0.579038			−0.289519	+	0.957172i	$0.593495\pi$
$594$		0			0
$595$	−1.51472	+	1.51472i	−0.0620974	+	0.0620974i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	30.1421i		−	1.23157i	−0.787913	−	0.615787i	$-0.788838\pi$
$599$		−	30.1421i		−	1.23157i	0.787913	−	0.615787i	$-0.211162\pi$
$600$		0			0
$601$		−	28.9706i		−	1.18173i	−0.806769	−	0.590867i	$-0.798786\pi$
$601$		−	28.9706i		−	1.18173i	0.806769	−	0.590867i	$-0.201214\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.94975	−	4.94975i	0.201236	−	0.201236i
$606$		0			0
$607$	1.51472			0.0614805			0.0307403	−	0.999527i	$-0.490214\pi$
$607$	1.51472			0.0614805			0.0307403	+	0.999527i	$0.490214\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−24.1421	+	24.1421i	−0.976686	+	0.976686i
$612$		0			0
$613$	1.41421	+	1.41421i	0.0571195	+	0.0571195i	0.735090	−	0.677970i	$-0.237140\pi$
$613$	1.41421	+	1.41421i	0.0571195	+	0.0571195i	−0.677970	+	0.735090i	$0.737140\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			15.2721i			0.614831i	0.951575	+	0.307415i	$0.0994641\pi$
$617$			15.2721i			0.614831i	−0.951575	+	0.307415i	$0.900536\pi$
$618$		0			0
$619$	−6.31371	−	6.31371i	−0.253769	−	0.253769i	0.568745	−	0.822514i	$-0.307429\pi$
$619$	−6.31371	−	6.31371i	−0.253769	−	0.253769i	−0.822514	+	0.568745i	$0.807429\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.34315			0.254133
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−19.1716	−	19.1716i	−0.764421	−	0.764421i
$630$		0			0
$631$		−	11.3137i		−	0.450392i	−0.974314	−	0.225196i	$-0.927698\pi$
$631$		−	11.3137i		−	0.450392i	0.974314	−	0.225196i	$-0.0723022\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.89949	−	5.89949i	−0.234114	−	0.234114i
$636$		0			0
$637$	−21.5563	+	21.5563i	−0.854094	+	0.854094i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	13.3137			0.525860			0.262930	−	0.964815i	$-0.415311\pi$
$641$	13.3137			0.525860			0.262930	+	0.964815i	$0.415311\pi$
$642$		0			0
$643$	−32.9706	+	32.9706i	−1.30023	+	1.30023i	−0.371999	+	0.928233i	$0.621327\pi$
$643$	−32.9706	+	32.9706i	−1.30023	+	1.30023i	−0.928233	+	0.371999i	$0.878673\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			7.07107i			0.277992i	0.990293	+	0.138996i	$0.0443876\pi$
$647$			7.07107i			0.277992i	−0.990293	+	0.138996i	$0.955612\pi$
$648$		0			0
$649$			12.9706i			0.509139i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.65685	−	1.65685i	0.0648377	−	0.0648377i	−0.673944	−	0.738782i	$-0.735401\pi$
$653$	1.65685	−	1.65685i	0.0648377	−	0.0648377i	0.738782	+	0.673944i	$0.235401\pi$
$654$		0			0
$655$	−14.9706			−0.584948
$656$		0			0
$657$		0			0
$658$		0			0
$659$	34.7279	−	34.7279i	1.35281	−	1.35281i	0.470303	−	0.882505i	$-0.344145\pi$
$659$	34.7279	−	34.7279i	1.35281	−	1.35281i	0.882505	−	0.470303i	$-0.155855\pi$
$660$		0			0
$661$	6.17157	+	6.17157i	0.240046	+	0.240046i	0.816869	−	0.576823i	$-0.195708\pi$
$661$	6.17157	+	6.17157i	0.240046	+	0.240046i	−0.576823	+	0.816869i	$0.695708\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	2.14214i		−	0.0830685i
$666$		0			0
$667$	−8.82843	−	8.82843i	−0.341838	−	0.341838i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	21.1716			0.817320
$672$		0			0
$673$	−13.7990			−0.531912			−0.265956	−	0.963985i	$-0.585688\pi$
$673$	−13.7990			−0.531912			−0.265956	+	0.963985i	$0.585688\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.0000	+	20.0000i	0.768662	+	0.768662i	0.977871	−	0.209209i	$-0.0670888\pi$
$677$	20.0000	+	20.0000i	0.768662	+	0.768662i	−0.209209	+	0.977871i	$0.567089\pi$
$678$		0			0
$679$		−	7.71573i		−	0.296102i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−25.6569	−	25.6569i	−0.981732	−	0.981732i	0.0181039	−	0.999836i	$-0.494237\pi$
$683$	−25.6569	−	25.6569i	−0.981732	−	0.981732i	−0.999836	+	0.0181039i	$0.994237\pi$
$684$		0			0
$685$	3.00000	−	3.00000i	0.114624	−	0.114624i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−27.3137			−1.04057
$690$		0			0
$691$	9.82843	−	9.82843i	0.373891	−	0.373891i	−0.495001	−	0.868892i	$-0.664832\pi$
$691$	9.82843	−	9.82843i	0.373891	−	0.373891i	0.868892	+	0.495001i	$0.164832\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			6.58579i			0.249813i
$696$		0			0
$697$		−	2.14214i		−	0.0811392i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−14.9289	+	14.9289i	−0.563858	+	0.563858i	−0.930401	−	0.366543i	$-0.880541\pi$
$701$	−14.9289	+	14.9289i	−0.563858	+	0.563858i	0.366543	+	0.930401i	$0.380541\pi$
$702$		0			0
$703$	27.1127			1.02257
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.82843	−	2.82843i	0.106374	−	0.106374i
$708$		0			0
$709$	−27.2843	−	27.2843i	−1.02468	−	1.02468i	−0.999688	−	0.0249947i	$-0.992043\pi$
$709$	−27.2843	−	27.2843i	−1.02468	−	1.02468i	−0.0249947	−	0.999688i	$-0.507957\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	19.7990i		−	0.741478i
$714$		0			0
$715$	6.82843	+	6.82843i	0.255369	+	0.255369i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−24.9706			−0.931245			−0.465622	−	0.884983i	$-0.654170\pi$
$719$	−24.9706			−0.931245			−0.465622	+	0.884983i	$0.654170\pi$
$720$		0			0
$721$	−8.28427			−0.308522
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.41421	+	3.41421i	0.126801	+	0.126801i
$726$		0			0
$727$			5.31371i			0.197075i	0.995133	+	0.0985373i	$0.0314164\pi$
$727$			5.31371i			0.197075i	−0.995133	+	0.0985373i	$0.968584\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−19.7990	−	19.7990i	−0.732292	−	0.732292i
$732$		0			0
$733$	−4.58579	+	4.58579i	−0.169380	+	0.169380i	−0.786707	−	0.617327i	$-0.788216\pi$
$733$	−4.58579	+	4.58579i	−0.169380	+	0.169380i	0.617327	+	0.786707i	$0.288216\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	10.3431			0.380995
$738$		0			0
$739$	2.17157	−	2.17157i	0.0798826	−	0.0798826i	−0.666037	−	0.745919i	$-0.732011\pi$
$739$	2.17157	−	2.17157i	0.0798826	−	0.0798826i	0.745919	+	0.666037i	$0.232011\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			30.1838i			1.10733i	0.832738	+	0.553667i	$0.186772\pi$
$743$			30.1838i			1.10733i	−0.832738	+	0.553667i	$0.813228\pi$
$744$		0			0
$745$			10.0000i			0.366372i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.11270	−	3.11270i	0.113736	−	0.113736i
$750$		0			0
$751$	0.343146			0.0125216			0.00626078	−	0.999980i	$-0.498007\pi$
$751$	0.343146			0.0125216			0.00626078	+	0.999980i	$0.498007\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−5.41421	+	5.41421i	−0.197043	+	0.197043i
$756$		0			0
$757$	−7.41421	−	7.41421i	−0.269474	−	0.269474i	0.559414	−	0.828888i	$-0.311026\pi$
$757$	−7.41421	−	7.41421i	−0.269474	−	0.269474i	−0.828888	+	0.559414i	$0.811026\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			39.1716i			1.41997i	0.704218	+	0.709984i	$0.251298\pi$
$761$			39.1716i			1.41997i	−0.704218	+	0.709984i	$0.748702\pi$
$762$		0			0
$763$	−9.51472	−	9.51472i	−0.344456	−	0.344456i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	31.3137			1.13067
$768$		0			0
$769$	−7.02944			−0.253488			−0.126744	−	0.991935i	$-0.540453\pi$
$769$	−7.02944			−0.253488			−0.126744	+	0.991935i	$0.540453\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	28.7279	+	28.7279i	1.03327	+	1.03327i	0.999427	+	0.0338444i	$0.0107751\pi$
$773$	28.7279	+	28.7279i	1.03327	+	1.03327i	0.0338444	+	0.999427i	$0.489225\pi$
$774$		0			0
$775$			7.65685i			0.275042i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.51472	+	1.51472i	0.0542704	+	0.0542704i
$780$		0			0
$781$	−11.3137	+	11.3137i	−0.404836	+	0.404836i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	10.0000			0.356915
$786$		0			0
$787$	12.1421	−	12.1421i	0.432820	−	0.432820i	−0.456766	−	0.889587i	$-0.650992\pi$
$787$	12.1421	−	12.1421i	0.432820	−	0.432820i	0.889587	+	0.456766i	$0.150992\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			12.0833i			0.429631i
$792$		0			0
$793$		−	51.1127i		−	1.81507i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9.89949	+	9.89949i	−0.350658	+	0.350658i	−0.860354	−	0.509696i	$-0.829758\pi$
$797$	−9.89949	+	9.89949i	−0.350658	+	0.350658i	0.509696	+	0.860354i	$0.329758\pi$
$798$		0			0
$799$	−18.2843			−0.646851
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12.4853	−	12.4853i	0.440596	−	0.440596i
$804$		0			0
$805$	−1.51472	−	1.51472i	−0.0533868	−	0.0533868i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			10.9706i			0.385704i	0.981228	+	0.192852i	$0.0617738\pi$
$809$			10.9706i			0.385704i	−0.981228	+	0.192852i	$0.938226\pi$
$810$		0			0
$811$	−21.0000	−	21.0000i	−0.737410	−	0.737410i	0.234666	−	0.972076i	$-0.424600\pi$
$811$	−21.0000	−	21.0000i	−0.737410	−	0.737410i	−0.972076	+	0.234666i	$0.924600\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19.3137			−0.676530
$816$		0			0
$817$	28.0000			0.979596
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−17.0711	−	17.0711i	−0.595784	−	0.595784i	0.343404	−	0.939188i	$-0.388420\pi$
$821$	−17.0711	−	17.0711i	−0.595784	−	0.595784i	−0.939188	+	0.343404i	$0.888420\pi$
$822$		0			0
$823$		−	42.0833i		−	1.46693i	−0.679727	−	0.733465i	$-0.737902\pi$
$823$		−	42.0833i		−	1.46693i	0.679727	−	0.733465i	$-0.262098\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.14214	−	2.14214i	−0.0744894	−	0.0744894i	0.668881	−	0.743370i	$-0.266774\pi$
$827$	−2.14214	−	2.14214i	−0.0744894	−	0.0744894i	−0.743370	+	0.668881i	$0.766774\pi$
$828$		0			0
$829$	−29.4853	+	29.4853i	−1.02407	+	1.02407i	−0.0243630	+	0.999703i	$0.507756\pi$
$829$	−29.4853	+	29.4853i	−1.02407	+	1.02407i	−0.999703	+	0.0243630i	$0.992244\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−16.3259			−0.565659
$834$		0			0
$835$	−7.82843	+	7.82843i	−0.270914	+	0.270914i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	23.7990i		−	0.821632i	−0.911718	−	0.410816i	$-0.865244\pi$
$839$		−	23.7990i		−	0.821632i	0.911718	−	0.410816i	$-0.134756\pi$
$840$		0			0
$841$			5.68629i			0.196079i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.29289	−	7.29289i	0.250883	−	0.250883i
$846$		0			0
$847$	−5.79899			−0.199256
$848$		0			0
$849$		0			0
$850$		0			0
$851$	19.1716	−	19.1716i	0.657193	−	0.657193i
$852$		0			0
$853$	8.04163	+	8.04163i	0.275340	+	0.275340i	0.831246	−	0.555905i	$-0.187628\pi$
$853$	8.04163	+	8.04163i	0.275340	+	0.275340i	−0.555905	+	0.831246i	$0.687628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			20.4437i			0.698342i	0.937059	+	0.349171i	$0.113537\pi$
$857$			20.4437i			0.698342i	−0.937059	+	0.349171i	$0.886463\pi$
$858$		0			0
$859$	27.2843	+	27.2843i	0.930927	+	0.930927i	0.997764	−	0.0668366i	$-0.0212906\pi$
$859$	27.2843	+	27.2843i	0.930927	+	0.930927i	−0.0668366	+	0.997764i	$0.521291\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−26.5858			−0.904991			−0.452495	−	0.891767i	$-0.649466\pi$
$863$	−26.5858			−0.904991			−0.452495	+	0.891767i	$0.649466\pi$
$864$		0			0
$865$	16.0000			0.544016
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.31371	−	3.31371i	−0.112410	−	0.112410i
$870$		0			0
$871$		−	24.9706i		−	0.846095i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.585786	+	0.585786i	0.0198032	+	0.0198032i
$876$		0			0
$877$	4.92893	−	4.92893i	0.166438	−	0.166438i	−0.618974	−	0.785412i	$-0.712451\pi$
$877$	4.92893	−	4.92893i	0.166438	−	0.166438i	0.785412	+	0.618974i	$0.212451\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−27.9411			−0.941360			−0.470680	−	0.882304i	$-0.655991\pi$
$881$	−27.9411			−0.941360			−0.470680	+	0.882304i	$0.655991\pi$
$882$		0			0
$883$	36.4853	−	36.4853i	1.22783	−	1.22783i	0.263044	−	0.964784i	$-0.415274\pi$
$883$	36.4853	−	36.4853i	1.22783	−	1.22783i	0.964784	−	0.263044i	$-0.0847263\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			57.2132i			1.92103i	0.278226	+	0.960516i	$0.410254\pi$
$887$			57.2132i			1.92103i	−0.278226	+	0.960516i	$0.589746\pi$
$888$		0			0
$889$			6.91169i			0.231811i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	12.9289	−	12.9289i	0.432650	−	0.432650i
$894$		0			0
$895$	−18.9706			−0.634116
$896$		0			0
$897$		0			0
$898$		0			0
$899$	26.1421	−	26.1421i	0.871889	−	0.871889i
$900$		0			0
$901$	−10.3431	−	10.3431i	−0.344580	−	0.344580i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			19.5563i			0.650075i
$906$		0			0
$907$	−30.8284	−	30.8284i	−1.02364	−	1.02364i	−0.999714	−	0.0239274i	$-0.992383\pi$
$907$	−30.8284	−	30.8284i	−1.02364	−	1.02364i	−0.0239274	−	0.999714i	$-0.507617\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.17157			0.0388159			0.0194080	−	0.999812i	$-0.493822\pi$
$911$	1.17157			0.0388159			0.0194080	+	0.999812i	$0.493822\pi$
$912$		0			0
$913$	−31.3137			−1.03633
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.76955	+	8.76955i	0.289596	+	0.289596i
$918$		0			0
$919$			28.6274i			0.944331i	0.881510	+	0.472166i	$0.156528\pi$
$919$			28.6274i			0.944331i	−0.881510	+	0.472166i	$0.843472\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	27.3137	+	27.3137i	0.899042	+	0.899042i
$924$		0			0
$925$	−7.41421	+	7.41421i	−0.243778	+	0.243778i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	47.4558			1.55698			0.778488	−	0.627660i	$-0.215987\pi$
$929$	47.4558			1.55698			0.778488	+	0.627660i	$0.215987\pi$
$930$		0			0
$931$	11.5442	−	11.5442i	0.378344	−	0.378344i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			5.17157i			0.169129i
$936$		0			0
$937$		−	8.62742i		−	0.281845i	−0.990021	−	0.140923i	$-0.954993\pi$
$937$		−	8.62742i		−	0.281845i	0.990021	−	0.140923i	$-0.0450069\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−21.2132	+	21.2132i	−0.691531	+	0.691531i	−0.962569	−	0.271038i	$-0.912633\pi$
$941$	−21.2132	+	21.2132i	−0.691531	+	0.691531i	0.271038	+	0.962569i	$0.412633\pi$
$942$		0			0
$943$	2.14214			0.0697575
$944$		0			0
$945$		0			0
$946$		0			0
$947$	3.07107	−	3.07107i	0.0997963	−	0.0997963i	−0.655446	−	0.755242i	$-0.727519\pi$
$947$	3.07107	−	3.07107i	0.0997963	−	0.0997963i	0.755242	+	0.655446i	$0.227519\pi$
$948$		0			0
$949$	−30.1421	−	30.1421i	−0.978455	−	0.978455i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			57.9828i			1.87825i	0.343582	+	0.939123i	$0.388360\pi$
$953$			57.9828i			1.87825i	−0.343582	+	0.939123i	$0.611640\pi$
$954$		0			0
$955$	8.82843	+	8.82843i	0.285681	+	0.285681i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.51472			−0.113496
$960$		0			0
$961$	27.6274			0.891207
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.41421	−	3.41421i	−0.109907	−	0.109907i
$966$		0			0
$967$		−	31.4558i		−	1.01155i	−0.862665	−	0.505776i	$-0.831206\pi$
$967$		−	31.4558i		−	1.01155i	0.862665	−	0.505776i	$-0.168794\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	13.7574	+	13.7574i	0.441495	+	0.441495i	0.892514	−	0.451019i	$-0.148939\pi$
$971$	13.7574	+	13.7574i	0.441495	+	0.441495i	−0.451019	+	0.892514i	$0.648939\pi$
$972$		0			0
$973$	3.85786	−	3.85786i	0.123677	−	0.123677i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19.5563			−0.625663			−0.312832	−	0.949809i	$-0.601278\pi$
$977$	−19.5563			−0.625663			−0.312832	+	0.949809i	$0.601278\pi$
$978$		0			0
$979$	10.8284	−	10.8284i	0.346078	−	0.346078i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	4.72792i		−	0.150797i	−0.997153	−	0.0753986i	$-0.975977\pi$
$983$		−	4.72792i		−	0.150797i	0.997153	−	0.0753986i	$-0.0240229\pi$
$984$		0			0
$985$			22.9706i			0.731903i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	19.7990	−	19.7990i	0.629571	−	0.629571i
$990$		0			0
$991$	4.00000			0.127064			0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000			0.127064			0.0635321	+	0.997980i	$0.479763\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	12.4853	−	12.4853i	0.395810	−	0.395810i
$996$		0			0
$997$	18.2426	+	18.2426i	0.577750	+	0.577750i	0.934283	−	0.356533i	$-0.116041\pi$
$997$	18.2426	+	18.2426i	0.577750	+	0.577750i	−0.356533	+	0.934283i	$0.616041\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.a.2161.2		4
3.2	odd	2		960.2.s.a.241.1		4
4.3	odd	2		720.2.t.a.181.1		4
12.11	even	2		240.2.s.a.181.2	yes	4
16.3	odd	4		720.2.t.a.541.1		4
16.13	even	4	inner	2880.2.t.a.721.2		4
24.5	odd	2		1920.2.s.b.481.2		4
24.11	even	2		1920.2.s.a.481.1		4
48.5	odd	4		1920.2.s.b.1441.2		4
48.11	even	4		1920.2.s.a.1441.1		4
48.29	odd	4		960.2.s.a.721.1		4
48.35	even	4		240.2.s.a.61.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.s.a.61.2	✓	4	48.35	even	4
240.2.s.a.181.2	yes	4	12.11	even	2
720.2.t.a.181.1		4	4.3	odd	2
720.2.t.a.541.1		4	16.3	odd	4
960.2.s.a.241.1		4	3.2	odd	2
960.2.s.a.721.1		4	48.29	odd	4
1920.2.s.a.481.1		4	24.11	even	2
1920.2.s.a.1441.1		4	48.11	even	4
1920.2.s.b.481.2		4	24.5	odd	2
1920.2.s.b.1441.2		4	48.5	odd	4
2880.2.t.a.721.2		4	16.13	even	4	inner
2880.2.t.a.2161.2		4	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)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} -0.828427i q^{7} +O(q^{10})\)	\(q+(0.707107 + 0.707107i) q^{5} -0.828427i q^{7} +(-1.41421 - 1.41421i) q^{11} +(-3.41421 + 3.41421i) q^{13} -2.58579 q^{17} +(1.82843 - 1.82843i) q^{19} -2.58579i q^{23} +1.00000i q^{25} +(3.41421 - 3.41421i) q^{29} +7.65685 q^{31} +(0.585786 - 0.585786i) q^{35} +(7.41421 + 7.41421i) q^{37} +0.828427i q^{41} +(7.65685 + 7.65685i) q^{43} +7.07107 q^{47} +6.31371 q^{49} +(4.00000 + 4.00000i) q^{53} -2.00000i q^{55} +(-4.58579 - 4.58579i) q^{59} +(-7.48528 + 7.48528i) q^{61} -4.82843 q^{65} +(-3.65685 + 3.65685i) q^{67} -8.00000i q^{71} +8.82843i q^{73} +(-1.17157 + 1.17157i) q^{77} +2.34315 q^{79} +(11.0711 - 11.0711i) q^{83} +(-1.82843 - 1.82843i) q^{85} +7.65685i q^{89} +(2.82843 + 2.82843i) q^{91} +2.58579 q^{95} +9.31371 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 8 q^{13} - 16 q^{17} - 4 q^{19} + 8 q^{29} + 8 q^{31} + 8 q^{35} + 24 q^{37} + 8 q^{43} - 20 q^{49} + 16 q^{53} - 24 q^{59} + 4 q^{61} - 8 q^{65} + 8 q^{67} - 16 q^{77} + 32 q^{79} + 16 q^{83} + 4 q^{85} + 16 q^{95} - 8 q^{97}+O(q^{100}) \) 4 * q - 8 * q^13 - 16 * q^17 - 4 * q^19 + 8 * q^29 + 8 * q^31 + 8 * q^35 + 24 * q^37 + 8 * q^43 - 20 * q^49 + 16 * q^53 - 24 * q^59 + 4 * q^61 - 8 * q^65 + 8 * q^67 - 16 * q^77 + 32 * q^79 + 16 * q^83 + 4 * q^85 + 16 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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