LMFDB - Embedded newform 2880.2.t.d.2161.8

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:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + \cdots + 1024 $ x^20 - 2x^18 - 4x^17 + 7x^16 + 16x^15 + 6x^14 - 36x^13 - 42x^12 + 40x^11 + 136x^10 + 80x^9 - 168x^8 - 288x^7 + 96x^6 + 512x^5 + 448x^4 - 512x^3 - 512*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2161.8
Root			$-1.04932 + 0.948122i$ of defining polynomial
Character	$\chi$	$=$	2880.2161
Dual form			2880.2.t.d.721.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} +0.740019i q^{7} +O(q^{10})$	$q+(0.707107 + 0.707107i) q^{5} +0.740019i q^{7} +(-3.83476 - 3.83476i) q^{11} +(3.31314 - 3.31314i) q^{13} -2.93893 q^{17} +(-5.02789 + 5.02789i) q^{19} +5.45159i q^{23} +1.00000i q^{25} +(-2.64012 + 2.64012i) q^{29} +5.94837 q^{31} +(-0.523272 + 0.523272i) q^{35} +(-0.479352 - 0.479352i) q^{37} +10.1918i q^{41} +(4.93728 + 4.93728i) q^{43} -8.15706 q^{47} +6.45237 q^{49} +(5.05247 + 5.05247i) q^{53} -5.42317i q^{55} +(3.83709 + 3.83709i) q^{59} +(-4.87697 + 4.87697i) q^{61} +4.68548 q^{65} +(3.99222 - 3.99222i) q^{67} +3.55343i q^{71} +11.1655i q^{73} +(2.83780 - 2.83780i) q^{77} -10.7776 q^{79} +(4.61002 - 4.61002i) q^{83} +(-2.07814 - 2.07814i) q^{85} +2.62476i q^{89} +(2.45179 + 2.45179i) q^{91} -7.11052 q^{95} +1.67846 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q + 8 q^{11} + 24 q^{17} + 4 q^{19} - 16 q^{29} + 16 q^{37} + 8 q^{43} - 52 q^{49} + 16 q^{53} - 16 q^{59} - 4 q^{61} + 8 q^{67} + 40 q^{77} - 56 q^{79} - 48 q^{83} + 4 q^{85} + 8 q^{91} + 56 q^{97}+O(q^{100}) $ 20 * q + 8 * q^11 + 24 * q^17 + 4 * q^19 - 16 * q^29 + 16 * q^37 + 8 * q^43 - 52 * q^49 + 16 * q^53 - 16 * q^59 - 4 * q^61 + 8 * q^67 + 40 * q^77 - 56 * q^79 - 48 * q^83 + 4 * q^85 + 8 * q^91 + 56 * 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
$7$			0.740019i			0.279701i	0.990173	+	0.139850i	$0.0446622\pi$
$7$			0.740019i			0.279701i	−0.990173	+	0.139850i	$0.955338\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.83476	−	3.83476i	−1.15622	−	1.15622i	−0.985281	−	0.170943i	$-0.945318\pi$
$11$	−3.83476	−	3.83476i	−1.15622	−	1.15622i	−0.170943	−	0.985281i	$-0.554682\pi$
$12$		0			0
$13$	3.31314	−	3.31314i	0.918899	−	0.918899i	−0.0780503	−	0.996949i	$-0.524869\pi$
$13$	3.31314	−	3.31314i	0.918899	−	0.918899i	0.996949	+	0.0780503i	$0.0248695\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.93893			−0.712796			−0.356398	−	0.934334i	$-0.615995\pi$
$17$	−2.93893			−0.712796			−0.356398	+	0.934334i	$0.615995\pi$
$18$		0			0
$19$	−5.02789	+	5.02789i	−1.15348	+	1.15348i	−0.167628	+	0.985850i	$0.553611\pi$
$19$	−5.02789	+	5.02789i	−1.15348	+	1.15348i	−0.985850	+	0.167628i	$0.946389\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.45159i			1.13673i	0.822775	+	0.568367i	$0.192425\pi$
$23$			5.45159i			1.13673i	−0.822775	+	0.568367i	$0.807575\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.64012	+	2.64012i	−0.490258	+	0.490258i	−0.908388	−	0.418129i	$-0.862686\pi$
$29$	−2.64012	+	2.64012i	−0.490258	+	0.490258i	0.418129	+	0.908388i	$0.362686\pi$
$30$		0			0
$31$	5.94837			1.06836			0.534179	−	0.845371i	$-0.320621\pi$
$31$	5.94837			1.06836			0.534179	+	0.845371i	$0.320621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.523272	+	0.523272i	−0.0884492	+	0.0884492i
$36$		0			0
$37$	−0.479352	−	0.479352i	−0.0788049	−	0.0788049i	0.666606	−	0.745411i	$-0.267747\pi$
$37$	−0.479352	−	0.479352i	−0.0788049	−	0.0788049i	−0.745411	+	0.666606i	$0.767747\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			10.1918i			1.59169i	0.605497	+	0.795847i	$0.292974\pi$
$41$			10.1918i			1.59169i	−0.605497	+	0.795847i	$0.707026\pi$
$42$		0			0
$43$	4.93728	+	4.93728i	0.752929	+	0.752929i	0.975025	−	0.222096i	$-0.0712899\pi$
$43$	4.93728	+	4.93728i	0.752929	+	0.752929i	−0.222096	+	0.975025i	$0.571290\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.15706			−1.18983			−0.594915	−	0.803789i	$-0.702814\pi$
$47$	−8.15706			−1.18983			−0.594915	+	0.803789i	$0.702814\pi$
$48$		0			0
$49$	6.45237			0.921767
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.05247	+	5.05247i	0.694010	+	0.694010i	0.963112	−	0.269102i	$-0.0867267\pi$
$53$	5.05247	+	5.05247i	0.694010	+	0.694010i	−0.269102	+	0.963112i	$0.586727\pi$
$54$		0			0
$55$		−	5.42317i		−	0.731260i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.83709	+	3.83709i	0.499547	+	0.499547i	0.911297	−	0.411750i	$-0.135082\pi$
$59$	3.83709	+	3.83709i	0.499547	+	0.499547i	−0.411750	+	0.911297i	$0.635082\pi$
$60$		0			0
$61$	−4.87697	+	4.87697i	−0.624432	+	0.624432i	−0.946662	−	0.322229i	$-0.895568\pi$
$61$	−4.87697	+	4.87697i	−0.624432	+	0.624432i	0.322229	+	0.946662i	$0.395568\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.68548			0.581163
$66$		0			0
$67$	3.99222	−	3.99222i	0.487728	−	0.487728i	−0.419861	−	0.907588i	$-0.637921\pi$
$67$	3.99222	−	3.99222i	0.487728	−	0.487728i	0.907588	+	0.419861i	$0.137921\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.55343i			0.421715i	0.977517	+	0.210857i	$0.0676255\pi$
$71$			3.55343i			0.421715i	−0.977517	+	0.210857i	$0.932374\pi$
$72$		0			0
$73$			11.1655i			1.30683i	0.757002	+	0.653413i	$0.226663\pi$
$73$			11.1655i			1.30683i	−0.757002	+	0.653413i	$0.773337\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.83780	−	2.83780i	0.323397	−	0.323397i
$78$		0			0
$79$	−10.7776			−1.21258			−0.606288	−	0.795245i	$-0.707342\pi$
$79$	−10.7776			−1.21258			−0.606288	+	0.795245i	$0.707342\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.61002	−	4.61002i	0.506016	−	0.506016i	−0.407285	−	0.913301i	$-0.633525\pi$
$83$	4.61002	−	4.61002i	0.506016	−	0.506016i	0.913301	+	0.407285i	$0.133525\pi$
$84$		0			0
$85$	−2.07814	−	2.07814i	−0.225406	−	0.225406i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			2.62476i			0.278224i	0.990277	+	0.139112i	$0.0444247\pi$
$89$			2.62476i			0.278224i	−0.990277	+	0.139112i	$0.955575\pi$
$90$		0			0
$91$	2.45179	+	2.45179i	0.257017	+	0.257017i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.11052			−0.729524
$96$		0			0
$97$	1.67846			0.170422			0.0852108	−	0.996363i	$-0.472844\pi$
$97$	1.67846			0.170422			0.0852108	+	0.996363i	$0.472844\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.52161	−	6.52161i	−0.648925	−	0.648925i	0.303808	−	0.952733i	$-0.401742\pi$
$101$	−6.52161	−	6.52161i	−0.648925	−	0.648925i	−0.952733	+	0.303808i	$0.901742\pi$
$102$		0			0
$103$		−	0.302418i		−	0.0297981i	−0.999889	−	0.0148991i	$-0.995257\pi$
$103$		−	0.302418i		−	0.0297981i	0.999889	−	0.0148991i	$-0.00474269\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.20078	−	1.20078i	−0.116084	−	0.116084i	0.646679	−	0.762762i	$-0.276157\pi$
$107$	−1.20078	−	1.20078i	−0.116084	−	0.116084i	−0.762762	+	0.646679i	$0.776157\pi$
$108$		0			0
$109$	−6.99992	+	6.99992i	−0.670471	+	0.670471i	−0.957825	−	0.287353i	$-0.907225\pi$
$109$	−6.99992	+	6.99992i	−0.670471	+	0.670471i	0.287353	+	0.957825i	$0.407225\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−15.1350			−1.42378			−0.711892	−	0.702289i	$-0.752161\pi$
$113$	−15.1350			−1.42378			−0.711892	+	0.702289i	$0.752161\pi$
$114$		0			0
$115$	−3.85485	+	3.85485i	−0.359467	+	0.359467i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	2.17486i		−	0.199370i
$120$		0			0
$121$			18.4108i			1.67371i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$	−8.94547			−0.793782			−0.396891	−	0.917866i	$-0.629911\pi$
$127$	−8.94547			−0.793782			−0.396891	+	0.917866i	$0.629911\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.63786	−	9.63786i	0.842064	−	0.842064i	−0.147063	−	0.989127i	$-0.546982\pi$
$131$	9.63786	−	9.63786i	0.842064	−	0.842064i	0.989127	+	0.147063i	$0.0469821\pi$
$132$		0			0
$133$	−3.72074	−	3.72074i	−0.322629	−	0.322629i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			6.92180i			0.591370i	0.955286	+	0.295685i	$0.0955478\pi$
$137$			6.92180i			0.591370i	−0.955286	+	0.295685i	$0.904452\pi$
$138$		0			0
$139$	−6.50393	−	6.50393i	−0.551657	−	0.551657i	0.375262	−	0.926919i	$-0.377553\pi$
$139$	−6.50393	−	6.50393i	−0.551657	−	0.551657i	−0.926919	+	0.375262i	$0.877553\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−25.4102			−2.12491
$144$		0			0
$145$	−3.73370			−0.310067
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.75043	+	5.75043i	0.471094	+	0.471094i	0.902268	−	0.431175i	$-0.141901\pi$
$149$	5.75043	+	5.75043i	0.471094	+	0.471094i	−0.431175	+	0.902268i	$0.641901\pi$
$150$		0			0
$151$		−	0.185782i		−	0.0151187i	−0.999971	−	0.00755935i	$-0.997594\pi$
$151$		−	0.185782i		−	0.0151187i	0.999971	−	0.00755935i	$-0.00240624\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.20613	+	4.20613i	0.337845	+	0.337845i
$156$		0			0
$157$	−11.8717	+	11.8717i	−0.947462	+	0.947462i	−0.998687	−	0.0512256i	$-0.983687\pi$
$157$	−11.8717	+	11.8717i	−0.947462	+	0.947462i	0.0512256	+	0.998687i	$0.483687\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.03428			−0.317946
$162$		0			0
$163$	−11.3813	+	11.3813i	−0.891454	+	0.891454i	−0.994660	−	0.103206i	$-0.967090\pi$
$163$	−11.3813	+	11.3813i	−0.891454	+	0.891454i	0.103206	+	0.994660i	$0.467090\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			12.8924i			0.997644i	0.866705	+	0.498822i	$0.166234\pi$
$167$			12.8924i			0.997644i	−0.866705	+	0.498822i	$0.833766\pi$
$168$		0			0
$169$		−	8.95377i		−	0.688751i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	12.1023	−	12.1023i	0.920122	−	0.920122i	−0.0769156	−	0.997038i	$-0.524507\pi$
$173$	12.1023	−	12.1023i	0.920122	−	0.920122i	0.997038	+	0.0769156i	$0.0245072\pi$
$174$		0			0
$175$	−0.740019			−0.0559402
$176$		0			0
$177$		0			0
$178$		0			0
$179$	15.5558	−	15.5558i	1.16269	−	1.16269i	0.178809	−	0.983884i	$-0.442776\pi$
$179$	15.5558	−	15.5558i	1.16269	−	1.16269i	0.983884	−	0.178809i	$-0.0572244\pi$
$180$		0			0
$181$	10.5970	+	10.5970i	0.787670	+	0.787670i	0.981112	−	0.193442i	$-0.0619651\pi$
$181$	10.5970	+	10.5970i	0.787670	+	0.787670i	−0.193442	+	0.981112i	$0.561965\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	0.677905i		−	0.0498406i
$186$		0			0
$187$	11.2701	+	11.2701i	0.824151	+	0.824151i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.84439			0.639958			0.319979	−	0.947425i	$-0.396324\pi$
$191$	8.84439			0.639958			0.319979	+	0.947425i	$0.396324\pi$
$192$		0			0
$193$	14.0714			1.01289			0.506443	−	0.862274i	$-0.330960\pi$
$193$	14.0714			1.01289			0.506443	+	0.862274i	$0.330960\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.15230	−	8.15230i	−0.580827	−	0.580827i	0.354303	−	0.935131i	$-0.384718\pi$
$197$	−8.15230	−	8.15230i	−0.580827	−	0.580827i	−0.935131	+	0.354303i	$0.884718\pi$
$198$		0			0
$199$			11.3466i			0.804340i	0.915565	+	0.402170i	$0.131744\pi$
$199$			11.3466i			0.804340i	−0.915565	+	0.402170i	$0.868256\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.95374	−	1.95374i	−0.137126	−	0.137126i
$204$		0			0
$205$	−7.20670	+	7.20670i	−0.503338	+	0.503338i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	38.5616			2.66736
$210$		0			0
$211$	15.6416	−	15.6416i	1.07681	−	1.07681i	0.0800209	−	0.996793i	$-0.474501\pi$
$211$	15.6416	−	15.6416i	1.07681	−	1.07681i	0.996793	−	0.0800209i	$-0.0254987\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			6.98237i			0.476194i
$216$		0			0
$217$			4.40191i			0.298821i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−9.73708	+	9.73708i	−0.654987	+	0.654987i
$222$		0			0
$223$	10.2773			0.688219			0.344110	−	0.938929i	$-0.388181\pi$
$223$	10.2773			0.688219			0.344110	+	0.938929i	$0.388181\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.8323	+	13.8323i	−0.918082	+	0.918082i	−0.996890	−	0.0788077i	$-0.974889\pi$
$227$	−13.8323	+	13.8323i	−0.918082	+	0.918082i	0.0788077	+	0.996890i	$0.474889\pi$
$228$		0			0
$229$	−11.3025	−	11.3025i	−0.746890	−	0.746890i	0.227004	−	0.973894i	$-0.427107\pi$
$229$	−11.3025	−	11.3025i	−0.746890	−	0.746890i	−0.973894	+	0.227004i	$0.927107\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	0.638284i		−	0.0418154i	−0.999781	−	0.0209077i	$-0.993344\pi$
$233$		−	0.638284i		−	0.0418154i	0.999781	−	0.0209077i	$-0.00665561\pi$
$234$		0			0
$235$	−5.76791	−	5.76791i	−0.376257	−	0.376257i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	27.2255			1.76107			0.880534	−	0.473983i	$-0.157184\pi$
$239$	27.2255			1.76107			0.880534	+	0.473983i	$0.157184\pi$
$240$		0			0
$241$	−14.0821			−0.907106			−0.453553	−	0.891229i	$-0.649844\pi$
$241$	−14.0821			−0.907106			−0.453553	+	0.891229i	$0.649844\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.56252	+	4.56252i	0.291488	+	0.291488i
$246$		0			0
$247$			33.3162i			2.11986i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.61761	+	1.61761i	0.102103	+	0.102103i	0.756313	−	0.654210i	$-0.226999\pi$
$251$	1.61761	+	1.61761i	0.102103	+	0.102103i	−0.654210	+	0.756313i	$0.726999\pi$
$252$		0			0
$253$	20.9055	−	20.9055i	1.31432	−	1.31432i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.53176			−0.282683			−0.141342	−	0.989961i	$-0.545142\pi$
$257$	−4.53176			−0.282683			−0.141342	+	0.989961i	$0.545142\pi$
$258$		0			0
$259$	0.354729	−	0.354729i	0.0220418	−	0.0220418i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			8.25620i			0.509099i	0.967060	+	0.254550i	$0.0819272\pi$
$263$			8.25620i			0.509099i	−0.967060	+	0.254550i	$0.918073\pi$
$264$		0			0
$265$			7.14527i			0.438931i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	14.8352	−	14.8352i	0.904520	−	0.904520i	−0.0913029	−	0.995823i	$-0.529103\pi$
$269$	14.8352	−	14.8352i	0.904520	−	0.904520i	0.995823	+	0.0913029i	$0.0291031\pi$
$270$		0			0
$271$	−32.0786			−1.94864			−0.974319	−	0.225172i	$-0.927706\pi$
$271$	−32.0786			−1.94864			−0.974319	+	0.225172i	$0.927706\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.83476	−	3.83476i	0.231245	−	0.231245i
$276$		0			0
$277$	−14.6755	−	14.6755i	−0.881763	−	0.881763i	0.111951	−	0.993714i	$-0.464290\pi$
$277$	−14.6755	−	14.6755i	−0.881763	−	0.881763i	−0.993714	+	0.111951i	$0.964290\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			24.6456i			1.47023i	0.677942	+	0.735116i	$0.262872\pi$
$281$			24.6456i			1.47023i	−0.677942	+	0.735116i	$0.737128\pi$
$282$		0			0
$283$	0.116449	+	0.116449i	0.00692219	+	0.00692219i	0.710559	−	0.703637i	$-0.248442\pi$
$283$	0.116449	+	0.116449i	0.00692219	+	0.00692219i	−0.703637	+	0.710559i	$0.748442\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−7.54214			−0.445198
$288$		0			0
$289$	−8.36268			−0.491923
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−21.5697	−	21.5697i	−1.26012	−	1.26012i	−0.951036	−	0.309079i	$-0.899979\pi$
$293$	−21.5697	−	21.5697i	−1.26012	−	1.26012i	−0.309079	−	0.951036i	$-0.600021\pi$
$294$		0			0
$295$			5.42647i			0.315941i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	18.0619	+	18.0619i	1.04454	+	1.04454i
$300$		0			0
$301$	−3.65368	+	3.65368i	−0.210595	+	0.210595i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.89708			−0.394926
$306$		0			0
$307$	−11.7544	+	11.7544i	−0.670856	+	0.670856i	−0.957913	−	0.287057i	$-0.907323\pi$
$307$	−11.7544	+	11.7544i	−0.670856	+	0.670856i	0.287057	+	0.957913i	$0.407323\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	15.8798i		−	0.900462i	−0.892912	−	0.450231i	$-0.851342\pi$
$311$		−	15.8798i		−	0.900462i	0.892912	−	0.450231i	$-0.148658\pi$
$312$		0			0
$313$			32.5435i			1.83947i	0.392542	+	0.919734i	$0.371596\pi$
$313$			32.5435i			1.83947i	−0.392542	+	0.919734i	$0.628404\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−13.8078	+	13.8078i	−0.775523	+	0.775523i	−0.979066	−	0.203543i	$-0.934754\pi$
$317$	−13.8078	+	13.8078i	−0.775523	+	0.775523i	0.203543	+	0.979066i	$0.434754\pi$
$318$		0			0
$319$	20.2485			1.13370
$320$		0			0
$321$		0			0
$322$		0			0
$323$	14.7766	−	14.7766i	0.822194	−	0.822194i
$324$		0			0
$325$	3.31314	+	3.31314i	0.183780	+	0.183780i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	6.03638i		−	0.332796i
$330$		0			0
$331$	5.85148	+	5.85148i	0.321627	+	0.321627i	0.849391	−	0.527764i	$-0.176970\pi$
$331$	5.85148	+	5.85148i	0.321627	+	0.321627i	−0.527764	+	0.849391i	$0.676970\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.64585			0.308466
$336$		0			0
$337$	−19.3223			−1.05255			−0.526276	−	0.850314i	$-0.676412\pi$
$337$	−19.3223			−1.05255			−0.526276	+	0.850314i	$0.676412\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−22.8106	−	22.8106i	−1.23526	−	1.23526i
$342$		0			0
$343$			9.95501i			0.537520i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.92151	+	6.92151i	0.371566	+	0.371566i	0.868047	−	0.496481i	$-0.165375\pi$
$347$	6.92151	+	6.92151i	0.371566	+	0.371566i	−0.496481	+	0.868047i	$0.665375\pi$
$348$		0			0
$349$	13.2497	−	13.2497i	0.709241	−	0.709241i	−0.257135	−	0.966376i	$-0.582778\pi$
$349$	13.2497	−	13.2497i	0.709241	−	0.709241i	0.966376	+	0.257135i	$0.0827784\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	21.8789			1.16450			0.582249	−	0.813010i	$-0.302173\pi$
$353$	21.8789			1.16450			0.582249	+	0.813010i	$0.302173\pi$
$354$		0			0
$355$	−2.51265	+	2.51265i	−0.133358	+	0.133358i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			6.94782i			0.366692i	0.983048	+	0.183346i	$0.0586928\pi$
$359$			6.94782i			0.366692i	−0.983048	+	0.183346i	$0.941307\pi$
$360$		0			0
$361$		−	31.5595i		−	1.66102i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.89522	+	7.89522i	−0.413254	+	0.413254i
$366$		0			0
$367$	−17.0448			−0.889730			−0.444865	−	0.895598i	$-0.646748\pi$
$367$	−17.0448			−0.889730			−0.444865	+	0.895598i	$0.646748\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.73892	+	3.73892i	−0.194115	+	0.194115i
$372$		0			0
$373$	−14.3704	−	14.3704i	−0.744071	−	0.744071i	0.229287	−	0.973359i	$-0.426360\pi$
$373$	−14.3704	−	14.3704i	−0.744071	−	0.744071i	−0.973359	+	0.229287i	$0.926360\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			17.4942i			0.900996i
$378$		0			0
$379$	−2.97499	−	2.97499i	−0.152815	−	0.152815i	0.626559	−	0.779374i	$-0.284463\pi$
$379$	−2.97499	−	2.97499i	−0.152815	−	0.152815i	−0.779374	+	0.626559i	$0.784463\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−20.4810			−1.04653			−0.523266	−	0.852170i	$-0.675286\pi$
$383$	−20.4810			−1.04653			−0.523266	+	0.852170i	$0.675286\pi$
$384$		0			0
$385$	4.01325			0.204534
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−11.8703	−	11.8703i	−0.601848	−	0.601848i	0.338955	−	0.940803i	$-0.389927\pi$
$389$	−11.8703	−	11.8703i	−0.601848	−	0.601848i	−0.940803	+	0.338955i	$0.889927\pi$
$390$		0			0
$391$		−	16.0218i		−	0.810259i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.62092	−	7.62092i	−0.383450	−	0.383450i
$396$		0			0
$397$	19.1282	−	19.1282i	0.960019	−	0.960019i	−0.0392118	−	0.999231i	$-0.512485\pi$
$397$	19.1282	−	19.1282i	0.960019	−	0.960019i	0.999231	+	0.0392118i	$0.0124847\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	16.0874			0.803368			0.401684	−	0.915778i	$-0.368425\pi$
$401$	16.0874			0.803368			0.401684	+	0.915778i	$0.368425\pi$
$402$		0			0
$403$	19.7078	−	19.7078i	0.981714	−	0.981714i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.67640i			0.182232i
$408$		0			0
$409$		−	23.4524i		−	1.15964i	−0.814743	−	0.579822i	$-0.803122\pi$
$409$		−	23.4524i		−	1.15964i	0.814743	−	0.579822i	$-0.196878\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.83952	+	2.83952i	−0.139724	+	0.139724i
$414$		0			0
$415$	6.51956			0.320032
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−14.1654	+	14.1654i	−0.692027	+	0.692027i	−0.962678	−	0.270651i	$-0.912761\pi$
$419$	−14.1654	+	14.1654i	−0.692027	+	0.692027i	0.270651	+	0.962678i	$0.412761\pi$
$420$		0			0
$421$	21.2978	+	21.2978i	1.03799	+	1.03799i	0.999249	+	0.0387434i	$0.0123355\pi$
$421$	21.2978	+	21.2978i	1.03799	+	1.03799i	0.0387434	+	0.999249i	$0.487665\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	2.93893i		−	0.142559i
$426$		0			0
$427$	−3.60905	−	3.60905i	−0.174654	−	0.174654i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−21.0148			−1.01225			−0.506123	−	0.862462i	$-0.668922\pi$
$431$	−21.0148			−1.01225			−0.506123	+	0.862462i	$0.668922\pi$
$432$		0			0
$433$	16.2253			0.779738			0.389869	−	0.920870i	$-0.372520\pi$
$433$	16.2253			0.779738			0.389869	+	0.920870i	$0.372520\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−27.4100	−	27.4100i	−1.31120	−	1.31120i
$438$		0			0
$439$		−	3.33967i		−	0.159394i	−0.996819	−	0.0796970i	$-0.974605\pi$
$439$		−	3.33967i		−	0.159394i	0.996819	−	0.0796970i	$-0.0253953\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13.4671	+	13.4671i	0.639841	+	0.639841i	0.950516	−	0.310675i	$-0.100555\pi$
$443$	13.4671	+	13.4671i	0.639841	+	0.639841i	−0.310675	+	0.950516i	$0.600555\pi$
$444$		0			0
$445$	−1.85598	+	1.85598i	−0.0879820	+	0.0879820i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−19.2995			−0.910799			−0.455400	−	0.890287i	$-0.650504\pi$
$449$	−19.2995			−0.910799			−0.455400	+	0.890287i	$0.650504\pi$
$450$		0			0
$451$	39.0832	−	39.0832i	1.84036	−	1.84036i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			3.46735i			0.162552i
$456$		0			0
$457$		−	21.0222i		−	0.983377i	−0.870771	−	0.491688i	$-0.836380\pi$
$457$		−	21.0222i		−	0.983377i	0.870771	−	0.491688i	$-0.163620\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	23.3056	−	23.3056i	1.08545	−	1.08545i	0.0894616	−	0.995990i	$-0.471485\pi$
$461$	23.3056	−	23.3056i	1.08545	−	1.08545i	0.995990	−	0.0894616i	$-0.0285146\pi$
$462$		0			0
$463$	1.65187			0.0767687			0.0383844	−	0.999263i	$-0.487779\pi$
$463$	1.65187			0.0767687			0.0383844	+	0.999263i	$0.487779\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−14.4723	+	14.4723i	−0.669699	+	0.669699i	−0.957646	−	0.287948i	$-0.907027\pi$
$467$	−14.4723	+	14.4723i	−0.669699	+	0.669699i	0.287948	+	0.957646i	$0.407027\pi$
$468$		0			0
$469$	2.95432	+	2.95432i	0.136418	+	0.136418i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	37.8666i		−	1.74111i
$474$		0			0
$475$	−5.02789	−	5.02789i	−0.230696	−	0.230696i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.07727			0.277678			0.138839	−	0.990315i	$-0.455663\pi$
$479$	6.07727			0.277678			0.138839	+	0.990315i	$0.455663\pi$
$480$		0			0
$481$	−3.17632			−0.144828
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.18685	+	1.18685i	0.0538921	+	0.0538921i
$486$		0			0
$487$		−	10.1863i		−	0.461586i	−0.973003	−	0.230793i	$-0.925868\pi$
$487$		−	10.1863i		−	0.461586i	0.973003	−	0.230793i	$-0.0741320\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8.75035	+	8.75035i	0.394898	+	0.394898i	0.876429	−	0.481531i	$-0.159919\pi$
$491$	8.75035	+	8.75035i	0.394898	+	0.394898i	−0.481531	+	0.876429i	$0.659919\pi$
$492$		0			0
$493$	7.75914	−	7.75914i	0.349454	−	0.349454i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.62961			−0.117954
$498$		0			0
$499$	23.8260	−	23.8260i	1.06660	−	1.06660i	0.0689808	−	0.997618i	$-0.478025\pi$
$499$	23.8260	−	23.8260i	1.06660	−	1.06660i	0.997618	−	0.0689808i	$-0.0219747\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	9.42267i		−	0.420136i	−0.977687	−	0.210068i	$-0.932631\pi$
$503$		−	9.42267i		−	0.420136i	0.977687	−	0.210068i	$-0.0673685\pi$
$504$		0			0
$505$		−	9.22295i		−	0.410416i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	16.3129	−	16.3129i	0.723055	−	0.723055i	−0.246172	−	0.969226i	$-0.579173\pi$
$509$	16.3129	−	16.3129i	0.723055	−	0.723055i	0.969226	+	0.246172i	$0.0791727\pi$
$510$		0			0
$511$	−8.26270			−0.365520
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.213842	−	0.213842i	0.00942299	−	0.00942299i
$516$		0			0
$517$	31.2804	+	31.2804i	1.37571	+	1.37571i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			10.7287i			0.470033i	0.971991	+	0.235016i	$0.0755144\pi$
$521$			10.7287i			0.470033i	−0.971991	+	0.235016i	$0.924486\pi$
$522$		0			0
$523$	16.3868	+	16.3868i	0.716546	+	0.716546i	0.967896	−	0.251351i	$-0.0808747\pi$
$523$	16.3868	+	16.3868i	0.716546	+	0.716546i	−0.251351	+	0.967896i	$0.580875\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−17.4819			−0.761521
$528$		0			0
$529$	−6.71979			−0.292165
$530$		0			0
$531$		0			0
$532$		0			0
$533$	33.7669	+	33.7669i	1.46261	+	1.46261i
$534$		0			0
$535$		−	1.69816i		−	0.0734177i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−24.7433	−	24.7433i	−1.06577	−	1.06577i
$540$		0			0
$541$	11.4471	−	11.4471i	0.492148	−	0.492148i	−0.416834	−	0.908983i	$-0.636860\pi$
$541$	11.4471	−	11.4471i	0.492148	−	0.492148i	0.908983	+	0.416834i	$0.136860\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.89939			−0.424043
$546$		0			0
$547$	−9.67749	+	9.67749i	−0.413780	+	0.413780i	−0.883053	−	0.469273i	$-0.844516\pi$
$547$	−9.67749	+	9.67749i	−0.413780	+	0.413780i	0.469273	+	0.883053i	$0.344516\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	26.5485i		−	1.13100i
$552$		0			0
$553$		−	7.97564i		−	0.339159i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10.0484	+	10.0484i	−0.425762	+	0.425762i	−0.887182	−	0.461420i	$-0.847340\pi$
$557$	−10.0484	+	10.0484i	−0.425762	+	0.425762i	0.461420	+	0.887182i	$0.347340\pi$
$558$		0			0
$559$	32.7158			1.38373
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8.44120	−	8.44120i	0.355754	−	0.355754i	−0.506491	−	0.862245i	$-0.669058\pi$
$563$	8.44120	−	8.44120i	0.355754	−	0.355754i	0.862245	+	0.506491i	$0.169058\pi$
$564$		0			0
$565$	−10.7021	−	10.7021i	−0.450240	−	0.450240i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	7.27300i		−	0.304900i	−0.988311	−	0.152450i	$-0.951284\pi$
$569$		−	7.27300i		−	0.304900i	0.988311	−	0.152450i	$-0.0487163\pi$
$570$		0			0
$571$	5.28045	+	5.28045i	0.220980	+	0.220980i	0.808911	−	0.587931i	$-0.200057\pi$
$571$	5.28045	+	5.28045i	0.220980	+	0.220980i	−0.587931	+	0.808911i	$0.700057\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5.45159			−0.227347
$576$		0			0
$577$	27.0550			1.12631			0.563157	−	0.826350i	$-0.309587\pi$
$577$	27.0550			1.12631			0.563157	+	0.826350i	$0.309587\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.41150	+	3.41150i	0.141533	+	0.141533i
$582$		0			0
$583$		−	38.7500i		−	1.60486i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.62135	−	6.62135i	−0.273292	−	0.273292i	0.557132	−	0.830424i	$-0.311902\pi$
$587$	−6.62135	−	6.62135i	−0.273292	−	0.273292i	−0.830424	+	0.557132i	$0.811902\pi$
$588$		0			0
$589$	−29.9078	+	29.9078i	−1.23233	+	1.23233i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.02017			0.370414			0.185207	−	0.982700i	$-0.440704\pi$
$593$	9.02017			0.370414			0.185207	+	0.982700i	$0.440704\pi$
$594$		0			0
$595$	1.53786	−	1.53786i	0.0630462	−	0.0630462i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			7.68375i			0.313950i	0.987603	+	0.156975i	$0.0501742\pi$
$599$			7.68375i			0.313950i	−0.987603	+	0.156975i	$0.949826\pi$
$600$		0			0
$601$			31.7822i			1.29642i	0.761460	+	0.648212i	$0.224483\pi$
$601$			31.7822i			1.29642i	−0.761460	+	0.648212i	$0.775517\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−13.0184	+	13.0184i	−0.529273	+	0.529273i
$606$		0			0
$607$	−25.7518			−1.04524			−0.522618	−	0.852567i	$-0.675044\pi$
$607$	−25.7518			−1.04524			−0.522618	+	0.852567i	$0.675044\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−27.0255	+	27.0255i	−1.09333	+	1.09333i
$612$		0			0
$613$	−10.4967	−	10.4967i	−0.423956	−	0.423956i	0.462607	−	0.886563i	$-0.346914\pi$
$613$	−10.4967	−	10.4967i	−0.423956	−	0.423956i	−0.886563	+	0.462607i	$0.846914\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			29.2461i			1.17740i	0.808351	+	0.588701i	$0.200360\pi$
$617$			29.2461i			1.17740i	−0.808351	+	0.588701i	$0.799640\pi$
$618$		0			0
$619$	−21.9641	−	21.9641i	−0.882814	−	0.882814i	0.111006	−	0.993820i	$-0.464593\pi$
$619$	−21.9641	−	21.9641i	−0.882814	−	0.882814i	−0.993820	+	0.111006i	$0.964593\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.94237			−0.0778194
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.40878	+	1.40878i	0.0561718	+	0.0561718i
$630$		0			0
$631$			6.46257i			0.257271i	0.991692	+	0.128635i	$0.0410597\pi$
$631$			6.46257i			0.257271i	−0.991692	+	0.128635i	$0.958940\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.32540	−	6.32540i	−0.251016	−	0.251016i
$636$		0			0
$637$	21.3776	−	21.3776i	0.847011	−	0.847011i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	5.56318			0.219732			0.109866	−	0.993946i	$-0.464958\pi$
$641$	5.56318			0.219732			0.109866	+	0.993946i	$0.464958\pi$
$642$		0			0
$643$	−2.91681	+	2.91681i	−0.115028	+	0.115028i	−0.762278	−	0.647250i	$-0.775919\pi$
$643$	−2.91681	+	2.91681i	−0.115028	+	0.115028i	0.647250	+	0.762278i	$0.275919\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	22.9740i		−	0.903201i	−0.892220	−	0.451601i	$-0.850853\pi$
$647$		−	22.9740i		−	0.903201i	0.892220	−	0.451601i	$-0.149147\pi$
$648$		0			0
$649$		−	29.4287i		−	1.15518i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−25.9783	+	25.9783i	−1.01661	+	1.01661i	−0.0167495	+	0.999860i	$0.505332\pi$
$653$	−25.9783	+	25.9783i	−1.01661	+	1.01661i	−0.999860	+	0.0167495i	$0.994668\pi$
$654$		0			0
$655$	13.6300			0.532568
$656$		0			0
$657$		0			0
$658$		0			0
$659$	28.1599	−	28.1599i	1.09695	−	1.09695i	0.102189	−	0.994765i	$-0.467415\pi$
$659$	28.1599	−	28.1599i	1.09695	−	1.09695i	0.994765	−	0.102189i	$-0.0325848\pi$
$660$		0			0
$661$	−24.8805	−	24.8805i	−0.967741	−	0.967741i	0.0317548	−	0.999496i	$-0.489890\pi$
$661$	−24.8805	−	24.8805i	−0.967741	−	0.967741i	−0.999496	+	0.0317548i	$0.989890\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	5.26192i		−	0.204048i
$666$		0			0
$667$	−14.3929	−	14.3929i	−0.557294	−	0.557294i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	37.4041			1.44397
$672$		0			0
$673$	−4.37152			−0.168510			−0.0842548	−	0.996444i	$-0.526851\pi$
$673$	−4.37152			−0.168510			−0.0842548	+	0.996444i	$0.526851\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.19018	+	7.19018i	0.276341	+	0.276341i	0.831646	−	0.555305i	$-0.187399\pi$
$677$	7.19018	+	7.19018i	0.276341	+	0.276341i	−0.555305	+	0.831646i	$0.687399\pi$
$678$		0			0
$679$			1.24209i			0.0476671i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	27.3182	+	27.3182i	1.04530	+	1.04530i	0.998924	+	0.0463792i	$0.0147682\pi$
$683$	27.3182	+	27.3182i	1.04530	+	1.04530i	0.0463792	+	0.998924i	$0.485232\pi$
$684$		0			0
$685$	−4.89445	+	4.89445i	−0.187007	+	0.187007i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	33.4791			1.27545
$690$		0			0
$691$	−10.7859	+	10.7859i	−0.410316	+	0.410316i	−0.881849	−	0.471533i	$-0.843701\pi$
$691$	−10.7859	+	10.7859i	−0.410316	+	0.410316i	0.471533	+	0.881849i	$0.343701\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	9.19795i		−	0.348898i
$696$		0			0
$697$		−	29.9530i		−	1.13455i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	7.34496	−	7.34496i	0.277415	−	0.277415i	−0.554661	−	0.832076i	$-0.687152\pi$
$701$	7.34496	−	7.34496i	0.277415	−	0.277415i	0.832076	+	0.554661i	$0.187152\pi$
$702$		0			0
$703$	4.82026			0.181799
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4.82612	−	4.82612i	0.181505	−	0.181505i
$708$		0			0
$709$	36.8251	+	36.8251i	1.38299	+	1.38299i	0.839254	+	0.543740i	$0.182992\pi$
$709$	36.8251	+	36.8251i	1.38299	+	1.38299i	0.543740	+	0.839254i	$0.317008\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			32.4281i			1.21444i
$714$		0			0
$715$	−17.9677	−	17.9677i	−0.671955	−	0.671955i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	15.1515			0.565055			0.282527	−	0.959259i	$-0.408827\pi$
$719$	15.1515			0.565055			0.282527	+	0.959259i	$0.408827\pi$
$720$		0			0
$721$	0.223795			0.00833456
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.64012	−	2.64012i	−0.0980517	−	0.0980517i
$726$		0			0
$727$		−	18.4900i		−	0.685756i	−0.939380	−	0.342878i	$-0.888598\pi$
$727$		−	18.4900i		−	0.685756i	0.939380	−	0.342878i	$-0.111402\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−14.5103	−	14.5103i	−0.536684	−	0.536684i
$732$		0			0
$733$	−10.2992	+	10.2992i	−0.380409	+	0.380409i	−0.871250	−	0.490840i	$-0.836690\pi$
$733$	−10.2992	+	10.2992i	−0.380409	+	0.380409i	0.490840	+	0.871250i	$0.336690\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−30.6184			−1.12784
$738$		0			0
$739$	−25.4615	+	25.4615i	−0.936618	+	0.936618i	−0.998108	−	0.0614897i	$-0.980415\pi$
$739$	−25.4615	+	25.4615i	−0.936618	+	0.936618i	0.0614897	+	0.998108i	$0.480415\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	46.0798i		−	1.69050i	−0.534368	−	0.845252i	$-0.679450\pi$
$743$		−	46.0798i		−	1.69050i	0.534368	−	0.845252i	$-0.320550\pi$
$744$		0			0
$745$			8.13234i			0.297946i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.888598	−	0.888598i	0.0324687	−	0.0324687i
$750$		0			0
$751$	16.4699			0.600997			0.300498	−	0.953782i	$-0.402847\pi$
$751$	16.4699			0.600997			0.300498	+	0.953782i	$0.402847\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.131367	−	0.131367i	0.00478095	−	0.00478095i
$756$		0			0
$757$	−21.4819	−	21.4819i	−0.780772	−	0.780772i	0.199189	−	0.979961i	$-0.436169\pi$
$757$	−21.4819	−	21.4819i	−0.780772	−	0.780772i	−0.979961	+	0.199189i	$0.936169\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	10.1316i		−	0.367270i	−0.982994	−	0.183635i	$-0.941214\pi$
$761$		−	10.1316i		−	0.367270i	0.982994	−	0.183635i	$-0.0587865\pi$
$762$		0			0
$763$	−5.18008	−	5.18008i	−0.187531	−	0.187531i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	25.4256			0.918067
$768$		0			0
$769$	33.2758			1.19996			0.599979	−	0.800016i	$-0.295176\pi$
$769$	33.2758			1.19996			0.599979	+	0.800016i	$0.295176\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.50648	−	6.50648i	−0.234022	−	0.234022i	0.580347	−	0.814369i	$-0.302917\pi$
$773$	−6.50648	−	6.50648i	−0.234022	−	0.234022i	−0.814369	+	0.580347i	$0.802917\pi$
$774$		0			0
$775$			5.94837i			0.213672i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−51.2434	−	51.2434i	−1.83598	−	1.83598i
$780$		0			0
$781$	13.6266	−	13.6266i	0.487597	−	0.487597i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−16.7891			−0.599227
$786$		0			0
$787$	25.9368	−	25.9368i	0.924547	−	0.924547i	−0.0727995	−	0.997347i	$-0.523193\pi$
$787$	25.9368	−	25.9368i	0.924547	−	0.924547i	0.997347	+	0.0727995i	$0.0231933\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	11.2002i		−	0.398234i
$792$		0			0
$793$			32.3162i			1.14758i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.54315	−	1.54315i	0.0546611	−	0.0546611i	−0.679248	−	0.733909i	$-0.737694\pi$
$797$	1.54315	−	1.54315i	0.0546611	−	0.0546611i	0.733909	+	0.679248i	$0.237694\pi$
$798$		0			0
$799$	23.9730			0.848105
$800$		0			0
$801$		0			0
$802$		0			0
$803$	42.8171	−	42.8171i	1.51098	−	1.51098i
$804$		0			0
$805$	−2.85266	−	2.85266i	−0.100543	−	0.100543i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	25.5155i		−	0.897076i	−0.893764	−	0.448538i	$-0.851945\pi$
$809$		−	25.5155i		−	0.897076i	0.893764	−	0.448538i	$-0.148055\pi$
$810$		0			0
$811$	−10.6605	−	10.6605i	−0.374342	−	0.374342i	0.494714	−	0.869056i	$-0.335273\pi$
$811$	−10.6605	−	10.6605i	−0.374342	−	0.374342i	−0.869056	+	0.494714i	$0.835273\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−16.0956			−0.563805
$816$		0			0
$817$	−49.6483			−1.73697
$818$		0			0
$819$		0			0
$820$		0			0
$821$	34.9507	+	34.9507i	1.21979	+	1.21979i	0.967706	+	0.252081i	$0.0811148\pi$
$821$	34.9507	+	34.9507i	1.21979	+	1.21979i	0.252081	+	0.967706i	$0.418885\pi$
$822$		0			0
$823$		−	35.6125i		−	1.24137i	−0.784059	−	0.620686i	$-0.786854\pi$
$823$		−	35.6125i		−	1.24137i	0.784059	−	0.620686i	$-0.213146\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−15.2133	−	15.2133i	−0.529019	−	0.529019i	0.391261	−	0.920280i	$-0.372039\pi$
$827$	−15.2133	−	15.2133i	−0.529019	−	0.529019i	−0.920280	+	0.391261i	$0.872039\pi$
$828$		0			0
$829$	24.4188	−	24.4188i	0.848101	−	0.848101i	−0.141795	−	0.989896i	$-0.545287\pi$
$829$	24.4188	−	24.4188i	0.848101	−	0.848101i	0.989896	+	0.141795i	$0.0452874\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−18.9631			−0.657032
$834$		0			0
$835$	−9.11630	+	9.11630i	−0.315483	+	0.315483i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1.78206i			0.0615236i	0.999527	+	0.0307618i	$0.00979333\pi$
$839$			1.78206i			0.0615236i	−0.999527	+	0.0307618i	$0.990207\pi$
$840$		0			0
$841$			15.0595i			0.519293i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.33127	−	6.33127i	0.217802	−	0.217802i
$846$		0			0
$847$	−13.6243			−0.468138
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.61323	−	2.61323i	0.0895802	−	0.0895802i
$852$		0			0
$853$	−20.1759	−	20.1759i	−0.690809	−	0.690809i	0.271601	−	0.962410i	$-0.412447\pi$
$853$	−20.1759	−	20.1759i	−0.690809	−	0.690809i	−0.962410	+	0.271601i	$0.912447\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	40.0844i		−	1.36926i	−0.728893	−	0.684628i	$-0.759965\pi$
$857$		−	40.0844i		−	1.36926i	0.728893	−	0.684628i	$-0.240035\pi$
$858$		0			0
$859$	3.09121	+	3.09121i	0.105471	+	0.105471i	0.757873	−	0.652402i	$-0.226239\pi$
$859$	3.09121	+	3.09121i	0.105471	+	0.105471i	−0.652402	+	0.757873i	$0.726239\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	13.8844			0.472630			0.236315	−	0.971676i	$-0.424060\pi$
$863$	13.8844			0.472630			0.236315	+	0.971676i	$0.424060\pi$
$864$		0			0
$865$	17.1153			0.581936
$866$		0			0
$867$		0			0
$868$		0			0
$869$	41.3296	+	41.3296i	1.40201	+	1.40201i
$870$		0			0
$871$		−	26.4536i		−	0.896345i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.523272	−	0.523272i	−0.0176898	−	0.0176898i
$876$		0			0
$877$	−21.3550	+	21.3550i	−0.721107	+	0.721107i	−0.968831	−	0.247724i	$-0.920317\pi$
$877$	−21.3550	+	21.3550i	−0.721107	+	0.721107i	0.247724	+	0.968831i	$0.420317\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	25.1815			0.848387			0.424193	−	0.905572i	$-0.360558\pi$
$881$	25.1815			0.848387			0.424193	+	0.905572i	$0.360558\pi$
$882$		0			0
$883$	−4.36865	+	4.36865i	−0.147017	+	0.147017i	−0.776784	−	0.629767i	$-0.783150\pi$
$883$	−4.36865	+	4.36865i	−0.147017	+	0.147017i	0.629767	+	0.776784i	$0.283150\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			36.7638i			1.23441i	0.786803	+	0.617204i	$0.211735\pi$
$887$			36.7638i			1.23441i	−0.786803	+	0.617204i	$0.788265\pi$
$888$		0			0
$889$		−	6.61982i		−	0.222022i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	41.0129	−	41.0129i	1.37244	−	1.37244i
$894$		0			0
$895$	21.9992			0.735351
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−15.7044	+	15.7044i	−0.523772	+	0.523772i
$900$		0			0
$901$	−14.8489	−	14.8489i	−0.494687	−	0.494687i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			14.9864i			0.498166i
$906$		0			0
$907$	24.5327	+	24.5327i	0.814594	+	0.814594i	0.985319	−	0.170725i	$-0.0546110\pi$
$907$	24.5327	+	24.5327i	0.814594	+	0.814594i	−0.170725	+	0.985319i	$0.554611\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.221626			−0.00734279			−0.00367139	−	0.999993i	$-0.501169\pi$
$911$	−0.221626			−0.00734279			−0.00367139	+	0.999993i	$0.501169\pi$
$912$		0			0
$913$	−35.3567			−1.17014
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.13220	+	7.13220i	0.235526	+	0.235526i
$918$		0			0
$919$			22.8234i			0.752874i	0.926442	+	0.376437i	$0.122851\pi$
$919$			22.8234i			0.752874i	−0.926442	+	0.376437i	$0.877149\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	11.7730	+	11.7730i	0.387513	+	0.387513i
$924$		0			0
$925$	0.479352	−	0.479352i	0.0157610	−	0.0157610i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−12.9959			−0.426383			−0.213191	−	0.977010i	$-0.568386\pi$
$929$	−12.9959			−0.426383			−0.213191	+	0.977010i	$0.568386\pi$
$930$		0			0
$931$	−32.4418	+	32.4418i	−1.06324	+	1.06324i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			15.9383i			0.521239i
$936$		0			0
$937$			51.7244i			1.68976i	0.534954	+	0.844881i	$0.320329\pi$
$937$			51.7244i			1.68976i	−0.534954	+	0.844881i	$0.679671\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	11.9700	−	11.9700i	0.390210	−	0.390210i	−0.484552	−	0.874762i	$-0.661017\pi$
$941$	11.9700	−	11.9700i	0.390210	−	0.390210i	0.874762	+	0.484552i	$0.161017\pi$
$942$		0			0
$943$	−55.5616			−1.80933
$944$		0			0
$945$		0			0
$946$		0			0
$947$	3.97492	−	3.97492i	0.129168	−	0.129168i	−0.639567	−	0.768735i	$-0.720887\pi$
$947$	3.97492	−	3.97492i	0.129168	−	0.129168i	0.768735	+	0.639567i	$0.220887\pi$
$948$		0			0
$949$	36.9929	+	36.9929i	1.20084	+	1.20084i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	13.1913i		−	0.427308i	−0.976909	−	0.213654i	$-0.931463\pi$
$953$		−	13.1913i		−	0.427308i	0.976909	−	0.213654i	$-0.0685365\pi$
$954$		0			0
$955$	6.25393	+	6.25393i	0.202372	+	0.202372i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−5.12227			−0.165407
$960$		0			0
$961$	4.38311			0.141391
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9.95002	+	9.95002i	0.320302	+	0.320302i
$966$		0			0
$967$			5.07109i			0.163075i	0.996670	+	0.0815376i	$0.0259831\pi$
$967$			5.07109i			0.163075i	−0.996670	+	0.0815376i	$0.974017\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.07041	+	2.07041i	0.0664427	+	0.0664427i	0.739547	−	0.673105i	$-0.235040\pi$
$971$	2.07041	+	2.07041i	0.0664427	+	0.0664427i	−0.673105	+	0.739547i	$0.735040\pi$
$972$		0			0
$973$	4.81304	−	4.81304i	0.154299	−	0.154299i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	6.93371			0.221829			0.110914	−	0.993830i	$-0.464622\pi$
$977$	6.93371			0.221829			0.110914	+	0.993830i	$0.464622\pi$
$978$		0			0
$979$	10.0653	−	10.0653i	0.321689	−	0.321689i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			49.6290i			1.58292i	0.611221	+	0.791460i	$0.290679\pi$
$983$			49.6290i			1.58292i	−0.611221	+	0.791460i	$0.709321\pi$
$984$		0			0
$985$		−	11.5291i		−	0.367347i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−26.9160	+	26.9160i	−0.855880	+	0.855880i
$990$		0			0
$991$	−47.6664			−1.51417			−0.757086	−	0.653315i	$-0.773378\pi$
$991$	−47.6664			−1.51417			−0.757086	+	0.653315i	$0.773378\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.02327	+	8.02327i	−0.254355	+	0.254355i
$996$		0			0
$997$	4.46020	+	4.46020i	0.141256	+	0.141256i	0.774199	−	0.632943i	$-0.218153\pi$
$997$	4.46020	+	4.46020i	0.141256	+	0.141256i	−0.632943	+	0.774199i	$0.718153\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.d.2161.8		20
3.2	odd	2		960.2.s.c.241.8		20
4.3	odd	2		720.2.t.d.181.8		20
12.11	even	2		240.2.s.c.181.3	yes	20
16.3	odd	4		720.2.t.d.541.8		20
16.13	even	4	inner	2880.2.t.d.721.8		20
24.5	odd	2		1920.2.s.f.481.3		20
24.11	even	2		1920.2.s.e.481.8		20
48.5	odd	4		1920.2.s.f.1441.3		20
48.11	even	4		1920.2.s.e.1441.8		20
48.29	odd	4		960.2.s.c.721.8		20
48.35	even	4		240.2.s.c.61.3	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.s.c.61.3	✓	20	48.35	even	4
240.2.s.c.181.3	yes	20	12.11	even	2
720.2.t.d.181.8		20	4.3	odd	2
720.2.t.d.541.8		20	16.3	odd	4
960.2.s.c.241.8		20	3.2	odd	2
960.2.s.c.721.8		20	48.29	odd	4
1920.2.s.e.481.8		20	24.11	even	2
1920.2.s.e.1441.8		20	48.11	even	4
1920.2.s.f.481.3		20	24.5	odd	2
1920.2.s.f.1441.3		20	48.5	odd	4
2880.2.t.d.721.8		20	16.13	even	4	inner
2880.2.t.d.2161.8		20	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)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} +0.740019i q^{7} +O(q^{10})\)	\(q+(0.707107 + 0.707107i) q^{5} +0.740019i q^{7} +(-3.83476 - 3.83476i) q^{11} +(3.31314 - 3.31314i) q^{13} -2.93893 q^{17} +(-5.02789 + 5.02789i) q^{19} +5.45159i q^{23} +1.00000i q^{25} +(-2.64012 + 2.64012i) q^{29} +5.94837 q^{31} +(-0.523272 + 0.523272i) q^{35} +(-0.479352 - 0.479352i) q^{37} +10.1918i q^{41} +(4.93728 + 4.93728i) q^{43} -8.15706 q^{47} +6.45237 q^{49} +(5.05247 + 5.05247i) q^{53} -5.42317i q^{55} +(3.83709 + 3.83709i) q^{59} +(-4.87697 + 4.87697i) q^{61} +4.68548 q^{65} +(3.99222 - 3.99222i) q^{67} +3.55343i q^{71} +11.1655i q^{73} +(2.83780 - 2.83780i) q^{77} -10.7776 q^{79} +(4.61002 - 4.61002i) q^{83} +(-2.07814 - 2.07814i) q^{85} +2.62476i q^{89} +(2.45179 + 2.45179i) q^{91} -7.11052 q^{95} +1.67846 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q + 8 q^{11} + 24 q^{17} + 4 q^{19} - 16 q^{29} + 16 q^{37} + 8 q^{43} - 52 q^{49} + 16 q^{53} - 16 q^{59} - 4 q^{61} + 8 q^{67} + 40 q^{77} - 56 q^{79} - 48 q^{83} + 4 q^{85} + 8 q^{91} + 56 q^{97}+O(q^{100}) \) 20 * q + 8 * q^11 + 24 * q^17 + 4 * q^19 - 16 * q^29 + 16 * q^37 + 8 * q^43 - 52 * q^49 + 16 * q^53 - 16 * q^59 - 4 * q^61 + 8 * q^67 + 40 * q^77 - 56 * q^79 - 48 * q^83 + 4 * q^85 + 8 * q^91 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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