LMFDB - Embedded newform 2880.2.t.d.2161.10

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			2161.10
Root			$1.19834 - 0.750988i$ of defining polynomial
Character	$\chi$	$=$	2880.2161
Dual form			2880.2.t.d.721.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} +3.79862i q^{7} +O(q^{10})$	$q+(0.707107 + 0.707107i) q^{5} +3.79862i q^{7} +(3.08662 + 3.08662i) q^{11} +(1.54638 - 1.54638i) q^{13} -4.32428 q^{17} +(5.37165 - 5.37165i) q^{19} +3.91059i q^{23} +1.00000i q^{25} +(-1.84243 + 1.84243i) q^{29} +9.52790 q^{31} +(-2.68603 + 2.68603i) q^{35} +(4.55033 + 4.55033i) q^{37} +0.580195i q^{41} +(-0.994741 - 0.994741i) q^{43} +2.22461 q^{47} -7.42948 q^{49} +(-4.80257 - 4.80257i) q^{53} +4.36514i q^{55} +(7.26404 + 7.26404i) q^{59} +(0.301222 - 0.301222i) q^{61} +2.18691 q^{65} +(-6.97711 + 6.97711i) q^{67} -0.585051i q^{71} -11.9999i q^{73} +(-11.7249 + 11.7249i) q^{77} -12.6436 q^{79} +(-11.1632 + 11.1632i) q^{83} +(-3.05773 - 3.05773i) q^{85} -12.9706i q^{89} +(5.87409 + 5.87409i) q^{91} +7.59666 q^{95} -6.78553 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			3.79862i			1.43574i	0.696176	+	0.717871i	$0.254883\pi$
$7$			3.79862i			1.43574i	−0.696176	+	0.717871i	$0.745117\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.08662	+	3.08662i	0.930650	+	0.930650i	0.997746	−	0.0670964i	$-0.0213735\pi$
$11$	3.08662	+	3.08662i	0.930650	+	0.930650i	−0.0670964	+	0.997746i	$0.521374\pi$
$12$		0			0
$13$	1.54638	−	1.54638i	0.428888	−	0.428888i	−0.459361	−	0.888249i	$-0.651922\pi$
$13$	1.54638	−	1.54638i	0.428888	−	0.428888i	0.888249	+	0.459361i	$0.151922\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.32428			−1.04879			−0.524396	−	0.851474i	$-0.675709\pi$
$17$	−4.32428			−1.04879			−0.524396	+	0.851474i	$0.675709\pi$
$18$		0			0
$19$	5.37165	−	5.37165i	1.23234	−	1.23234i	0.269279	−	0.963062i	$-0.413215\pi$
$19$	5.37165	−	5.37165i	1.23234	−	1.23234i	0.963062	−	0.269279i	$-0.0867853\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.91059i			0.815414i	0.913113	+	0.407707i	$0.133671\pi$
$23$			3.91059i			0.815414i	−0.913113	+	0.407707i	$0.866329\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.84243	+	1.84243i	−0.342131	+	0.342131i	−0.857168	−	0.515037i	$-0.827778\pi$
$29$	−1.84243	+	1.84243i	−0.342131	+	0.342131i	0.515037	+	0.857168i	$0.327778\pi$
$30$		0			0
$31$	9.52790			1.71126			0.855630	−	0.517587i	$-0.173170\pi$
$31$	9.52790			1.71126			0.855630	+	0.517587i	$0.173170\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.68603	+	2.68603i	−0.454021	+	0.454021i
$36$		0			0
$37$	4.55033	+	4.55033i	0.748070	+	0.748070i	0.974116	−	0.226047i	$-0.0725802\pi$
$37$	4.55033	+	4.55033i	0.748070	+	0.748070i	−0.226047	+	0.974116i	$0.572580\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			0.580195i			0.0906112i	0.998973	+	0.0453056i	$0.0144262\pi$
$41$			0.580195i			0.0906112i	−0.998973	+	0.0453056i	$0.985574\pi$
$42$		0			0
$43$	−0.994741	−	0.994741i	−0.151697	−	0.151697i	0.627179	−	0.778875i	$-0.284210\pi$
$43$	−0.994741	−	0.994741i	−0.151697	−	0.151697i	−0.778875	+	0.627179i	$0.784210\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.22461			0.324492			0.162246	−	0.986750i	$-0.448126\pi$
$47$	2.22461			0.324492			0.162246	+	0.986750i	$0.448126\pi$
$48$		0			0
$49$	−7.42948			−1.06135
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.80257	−	4.80257i	−0.659684	−	0.659684i	0.295622	−	0.955305i	$-0.404473\pi$
$53$	−4.80257	−	4.80257i	−0.659684	−	0.659684i	−0.955305	+	0.295622i	$0.904473\pi$
$54$		0			0
$55$			4.36514i			0.588595i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.26404	+	7.26404i	0.945698	+	0.945698i	0.998600	−	0.0529020i	$-0.0168471\pi$
$59$	7.26404	+	7.26404i	0.945698	+	0.945698i	−0.0529020	+	0.998600i	$0.516847\pi$
$60$		0			0
$61$	0.301222	−	0.301222i	0.0385676	−	0.0385676i	−0.687560	−	0.726128i	$-0.741318\pi$
$61$	0.301222	−	0.301222i	0.0385676	−	0.0385676i	0.726128	+	0.687560i	$0.241318\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.18691			0.271253
$66$		0			0
$67$	−6.97711	+	6.97711i	−0.852389	+	0.852389i	−0.990427	−	0.138038i	$-0.955921\pi$
$67$	−6.97711	+	6.97711i	−0.852389	+	0.852389i	0.138038	+	0.990427i	$0.455921\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	0.585051i		−	0.0694327i	−0.999397	−	0.0347164i	$-0.988947\pi$
$71$		−	0.585051i		−	0.0694327i	0.999397	−	0.0347164i	$-0.0110528\pi$
$72$		0			0
$73$		−	11.9999i		−	1.40448i	−0.711939	−	0.702241i	$-0.752183\pi$
$73$		−	11.9999i		−	1.40448i	0.711939	−	0.702241i	$-0.247817\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−11.7249	+	11.7249i	−1.33617	+	1.33617i
$78$		0			0
$79$	−12.6436			−1.42252			−0.711260	−	0.702929i	$-0.751875\pi$
$79$	−12.6436			−1.42252			−0.711260	+	0.702929i	$0.751875\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−11.1632	+	11.1632i	−1.22532	+	1.22532i	−0.259610	+	0.965713i	$0.583594\pi$
$83$	−11.1632	+	11.1632i	−1.22532	+	1.22532i	−0.965713	+	0.259610i	$0.916406\pi$
$84$		0			0
$85$	−3.05773	−	3.05773i	−0.331657	−	0.331657i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	12.9706i		−	1.37488i	−0.726241	−	0.687440i	$-0.758734\pi$
$89$		−	12.9706i		−	1.37488i	0.726241	−	0.687440i	$-0.241266\pi$
$90$		0			0
$91$	5.87409	+	5.87409i	0.615772	+	0.615772i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.59666			0.779401
$96$		0			0
$97$	−6.78553			−0.688966			−0.344483	−	0.938793i	$-0.611946\pi$
$97$	−6.78553			−0.688966			−0.344483	+	0.938793i	$0.611946\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.19304	+	6.19304i	0.616231	+	0.616231i	0.944562	−	0.328332i	$-0.106486\pi$
$101$	6.19304	+	6.19304i	0.616231	+	0.616231i	−0.328332	+	0.944562i	$0.606486\pi$
$102$		0			0
$103$			11.3519i			1.11854i	0.828987	+	0.559269i	$0.188918\pi$
$103$			11.3519i			1.11854i	−0.828987	+	0.559269i	$0.811082\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.58488	−	8.58488i	−0.829932	−	0.829932i	0.157575	−	0.987507i	$-0.449632\pi$
$107$	−8.58488	−	8.58488i	−0.829932	−	0.829932i	−0.987507	+	0.157575i	$0.949632\pi$
$108$		0			0
$109$	−10.6624	+	10.6624i	−1.02127	+	1.02127i	−0.0215039	+	0.999769i	$0.506845\pi$
$109$	−10.6624	+	10.6624i	−1.02127	+	1.02127i	−0.999769	+	0.0215039i	$0.993155\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.11152			0.292707			0.146354	−	0.989232i	$-0.453246\pi$
$113$	3.11152			0.292707			0.146354	+	0.989232i	$0.453246\pi$
$114$		0			0
$115$	−2.76520	+	2.76520i	−0.257856	+	0.257856i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	16.4263i		−	1.50579i
$120$		0			0
$121$			8.05441i			0.732219i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$	7.16390			0.635693			0.317846	−	0.948142i	$-0.397040\pi$
$127$	7.16390			0.635693			0.317846	+	0.948142i	$0.397040\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.55090	+	6.55090i	−0.572355	+	0.572355i	−0.932786	−	0.360431i	$-0.882630\pi$
$131$	−6.55090	+	6.55090i	−0.572355	+	0.572355i	0.360431	+	0.932786i	$0.382630\pi$
$132$		0			0
$133$	20.4048	+	20.4048i	1.76932	+	1.76932i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			11.0502i			0.944082i	0.881577	+	0.472041i	$0.156482\pi$
$137$			11.0502i			0.944082i	−0.881577	+	0.472041i	$0.843518\pi$
$138$		0			0
$139$	5.17171	+	5.17171i	0.438659	+	0.438659i	0.891561	−	0.452902i	$-0.149611\pi$
$139$	5.17171	+	5.17171i	0.438659	+	0.438659i	−0.452902	+	0.891561i	$0.649611\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	9.54615			0.798289
$144$		0			0
$145$	−2.60559			−0.216383
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.80335	+	5.80335i	0.475429	+	0.475429i	0.903666	−	0.428237i	$-0.140865\pi$
$149$	5.80335	+	5.80335i	0.475429	+	0.475429i	−0.428237	+	0.903666i	$0.640865\pi$
$150$		0			0
$151$			13.2591i			1.07901i	0.841982	+	0.539505i	$0.181389\pi$
$151$			13.2591i			1.07901i	−0.841982	+	0.539505i	$0.818611\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.73724	+	6.73724i	0.541148	+	0.541148i
$156$		0			0
$157$	6.83329	−	6.83329i	0.545356	−	0.545356i	−0.379738	−	0.925094i	$-0.623986\pi$
$157$	6.83329	−	6.83329i	0.545356	−	0.545356i	0.925094	+	0.379738i	$0.123986\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−14.8548			−1.17072
$162$		0			0
$163$	7.11541	−	7.11541i	0.557322	−	0.557322i	−0.371222	−	0.928544i	$-0.621061\pi$
$163$	7.11541	−	7.11541i	0.557322	−	0.557322i	0.928544	+	0.371222i	$0.121061\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	6.14029i		−	0.475150i	−0.971369	−	0.237575i	$-0.923648\pi$
$167$		−	6.14029i		−	0.475150i	0.971369	−	0.237575i	$-0.0763525\pi$
$168$		0			0
$169$			8.21743i			0.632110i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.29269	−	7.29269i	0.554453	−	0.554453i	−0.373270	−	0.927723i	$-0.621763\pi$
$173$	7.29269	−	7.29269i	0.554453	−	0.554453i	0.927723	+	0.373270i	$0.121763\pi$
$174$		0			0
$175$	−3.79862			−0.287148
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.22897	−	6.22897i	0.465575	−	0.465575i	−0.434902	−	0.900478i	$-0.643217\pi$
$179$	6.22897	−	6.22897i	0.465575	−	0.465575i	0.900478	+	0.434902i	$0.143217\pi$
$180$		0			0
$181$	−1.27302	−	1.27302i	−0.0946227	−	0.0946227i	0.658211	−	0.752834i	$-0.271314\pi$
$181$	−1.27302	−	1.27302i	−0.0946227	−	0.0946227i	−0.752834	+	0.658211i	$0.771314\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			6.43514i			0.473121i
$186$		0			0
$187$	−13.3474	−	13.3474i	−0.976058	−	0.976058i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.63638			0.697264			0.348632	−	0.937260i	$-0.386646\pi$
$191$	9.63638			0.697264			0.348632	+	0.937260i	$0.386646\pi$
$192$		0			0
$193$	−3.64530			−0.262394			−0.131197	−	0.991356i	$-0.541882\pi$
$193$	−3.64530			−0.262394			−0.131197	+	0.991356i	$0.541882\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	7.55422	+	7.55422i	0.538216	+	0.538216i	0.923005	−	0.384789i	$-0.125726\pi$
$197$	7.55422	+	7.55422i	0.538216	+	0.538216i	−0.384789	+	0.923005i	$0.625726\pi$
$198$		0			0
$199$		−	23.7442i		−	1.68318i	−0.540116	−	0.841591i	$-0.681620\pi$
$199$		−	23.7442i		−	1.68318i	0.540116	−	0.841591i	$-0.318380\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.99870	−	6.99870i	−0.491212	−	0.491212i
$204$		0			0
$205$	−0.410260	+	0.410260i	−0.0286538	+	0.0286538i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	33.1605			2.29376
$210$		0			0
$211$	−14.5856	+	14.5856i	−1.00412	+	1.00412i	−0.00412399	+	0.999991i	$0.501313\pi$
$211$	−14.5856	+	14.5856i	−1.00412	+	1.00412i	−0.999991	+	0.00412399i	$0.998687\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	1.40678i		−	0.0959413i
$216$		0			0
$217$			36.1928i			2.45693i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.68697	+	6.68697i	−0.449814	+	0.449814i
$222$		0			0
$223$	13.6893			0.916702			0.458351	−	0.888771i	$-0.348440\pi$
$223$	13.6893			0.916702			0.458351	+	0.888771i	$0.348440\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−11.9490	+	11.9490i	−0.793085	+	0.793085i	−0.981994	−	0.188910i	$-0.939505\pi$
$227$	−11.9490	+	11.9490i	−0.793085	+	0.793085i	0.188910	+	0.981994i	$0.439505\pi$
$228$		0			0
$229$	9.40821	+	9.40821i	0.621712	+	0.621712i	0.945969	−	0.324257i	$-0.105114\pi$
$229$	9.40821	+	9.40821i	0.621712	+	0.621712i	−0.324257	+	0.945969i	$0.605114\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	11.8223i		−	0.774505i	−0.921974	−	0.387253i	$-0.873424\pi$
$233$		−	11.8223i		−	0.774505i	0.921974	−	0.387253i	$-0.126576\pi$
$234$		0			0
$235$	1.57304	+	1.57304i	0.102614	+	0.102614i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	11.2115			0.725213			0.362607	−	0.931942i	$-0.381887\pi$
$239$	11.2115			0.725213			0.362607	+	0.931942i	$0.381887\pi$
$240$		0			0
$241$	7.89997			0.508881			0.254441	−	0.967088i	$-0.418109\pi$
$241$	7.89997			0.508881			0.254441	+	0.967088i	$0.418109\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5.25343	−	5.25343i	−0.335630	−	0.335630i
$246$		0			0
$247$		−	16.6132i		−	1.05707i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.34286	−	5.34286i	−0.337238	−	0.337238i	0.518089	−	0.855327i	$-0.326644\pi$
$251$	−5.34286	−	5.34286i	−0.337238	−	0.337238i	−0.855327	+	0.518089i	$0.826644\pi$
$252$		0			0
$253$	−12.0705	+	12.0705i	−0.758865	+	0.758865i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−15.5250			−0.968420			−0.484210	−	0.874952i	$-0.660893\pi$
$257$	−15.5250			−0.968420			−0.484210	+	0.874952i	$0.660893\pi$
$258$		0			0
$259$	−17.2850	+	17.2850i	−1.07404	+	1.07404i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	14.3705i		−	0.886126i	−0.896491	−	0.443063i	$-0.853892\pi$
$263$		−	14.3705i		−	0.886126i	0.896491	−	0.443063i	$-0.146108\pi$
$264$		0			0
$265$		−	6.79186i		−	0.417221i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	19.0607	−	19.0607i	1.16215	−	1.16215i	0.178145	−	0.984004i	$-0.442990\pi$
$269$	19.0607	−	19.0607i	1.16215	−	1.16215i	0.984004	−	0.178145i	$-0.0570098\pi$
$270$		0			0
$271$	4.66889			0.283615			0.141808	−	0.989894i	$-0.454709\pi$
$271$	4.66889			0.283615			0.141808	+	0.989894i	$0.454709\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.08662	+	3.08662i	−0.186130	+	0.186130i
$276$		0			0
$277$	−6.24572	−	6.24572i	−0.375269	−	0.375269i	0.494123	−	0.869392i	$-0.335489\pi$
$277$	−6.24572	−	6.24572i	−0.375269	−	0.375269i	−0.869392	+	0.494123i	$0.835489\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	29.7389i		−	1.77407i	−0.461699	−	0.887037i	$-0.652760\pi$
$281$		−	29.7389i		−	1.77407i	0.461699	−	0.887037i	$-0.347240\pi$
$282$		0			0
$283$	9.49040	+	9.49040i	0.564146	+	0.564146i	0.930482	−	0.366337i	$-0.119388\pi$
$283$	9.49040	+	9.49040i	0.564146	+	0.564146i	−0.366337	+	0.930482i	$0.619388\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.20394			−0.130094
$288$		0			0
$289$	1.69940			0.0999648
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.72797	+	1.72797i	0.100949	+	0.100949i	0.755777	−	0.654829i	$-0.227259\pi$
$293$	1.72797	+	1.72797i	0.100949	+	0.100949i	−0.654829	+	0.755777i	$0.727259\pi$
$294$		0			0
$295$			10.2729i			0.598112i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.04724	+	6.04724i	0.349721	+	0.349721i
$300$		0			0
$301$	3.77864	−	3.77864i	0.217797	−	0.217797i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.425993			0.0243923
$306$		0			0
$307$	−17.5875	+	17.5875i	−1.00377	+	1.00377i	−0.00377977	+	0.999993i	$0.501203\pi$
$307$	−17.5875	+	17.5875i	−1.00377	+	1.00377i	−0.999993	+	0.00377977i	$0.998797\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.71809i			0.0974241i	0.998813	+	0.0487120i	$0.0155117\pi$
$311$			1.71809i			0.0974241i	−0.998813	+	0.0487120i	$0.984488\pi$
$312$		0			0
$313$		−	16.4421i		−	0.929363i	−0.885478	−	0.464682i	$-0.846169\pi$
$313$		−	16.4421i		−	0.929363i	0.885478	−	0.464682i	$-0.153831\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.96065	−	7.96065i	0.447114	−	0.447114i	−0.447280	−	0.894394i	$-0.647607\pi$
$317$	7.96065	−	7.96065i	0.447114	−	0.447114i	0.894394	+	0.447280i	$0.147607\pi$
$318$		0			0
$319$	−11.3738			−0.636809
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−23.2285	+	23.2285i	−1.29247	+	1.29247i
$324$		0			0
$325$	1.54638	+	1.54638i	0.0857776	+	0.0857776i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			8.45043i			0.465887i
$330$		0			0
$331$	−9.02535	−	9.02535i	−0.496078	−	0.496078i	0.414137	−	0.910215i	$-0.364084\pi$
$331$	−9.02535	−	9.02535i	−0.496078	−	0.496078i	−0.910215	+	0.414137i	$0.864084\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−9.86712			−0.539098
$336$		0			0
$337$	7.44173			0.405377			0.202688	−	0.979243i	$-0.435032\pi$
$337$	7.44173			0.405377			0.202688	+	0.979243i	$0.435032\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	29.4090	+	29.4090i	1.59258	+	1.59258i
$342$		0			0
$343$		−	1.63142i		−	0.0880882i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.659315	−	0.659315i	−0.0353939	−	0.0353939i	0.689188	−	0.724582i	$-0.257967\pi$
$347$	−0.659315	−	0.659315i	−0.0353939	−	0.0353939i	−0.724582	+	0.689188i	$0.757967\pi$
$348$		0			0
$349$	8.04705	−	8.04705i	0.430749	−	0.430749i	−0.458134	−	0.888883i	$-0.651482\pi$
$349$	8.04705	−	8.04705i	0.430749	−	0.430749i	0.888883	+	0.458134i	$0.151482\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−18.3339			−0.975815			−0.487907	−	0.872895i	$-0.662240\pi$
$353$	−18.3339			−0.975815			−0.487907	+	0.872895i	$0.662240\pi$
$354$		0			0
$355$	0.413693	−	0.413693i	0.0219566	−	0.0219566i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			11.8783i			0.626912i	0.949603	+	0.313456i	$0.101487\pi$
$359$			11.8783i			0.626912i	−0.949603	+	0.313456i	$0.898513\pi$
$360$		0			0
$361$		−	38.7093i		−	2.03733i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.48521	−	8.48521i	0.444136	−	0.444136i
$366$		0			0
$367$	24.0792			1.25692			0.628462	−	0.777841i	$-0.283685\pi$
$367$	24.0792			1.25692			0.628462	+	0.777841i	$0.283685\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	18.2431	−	18.2431i	0.947135	−	0.947135i
$372$		0			0
$373$	5.50986	+	5.50986i	0.285290	+	0.285290i	0.835214	−	0.549925i	$-0.185344\pi$
$373$	5.50986	+	5.50986i	0.285290	+	0.285290i	−0.549925	+	0.835214i	$0.685344\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			5.69820i			0.293472i
$378$		0			0
$379$	4.41212	+	4.41212i	0.226635	+	0.226635i	0.811285	−	0.584650i	$-0.198768\pi$
$379$	4.41212	+	4.41212i	0.226635	+	0.226635i	−0.584650	+	0.811285i	$0.698768\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.4394			0.635625			0.317812	−	0.948154i	$-0.397052\pi$
$383$	12.4394			0.635625			0.317812	+	0.948154i	$0.397052\pi$
$384$		0			0
$385$	−16.5815			−0.845070
$386$		0			0
$387$		0			0
$388$		0			0
$389$	22.7006	+	22.7006i	1.15097	+	1.15097i	0.986359	+	0.164609i	$0.0526363\pi$
$389$	22.7006	+	22.7006i	1.15097	+	1.15097i	0.164609	+	0.986359i	$0.447364\pi$
$390$		0			0
$391$		−	16.9105i		−	0.855199i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.94040	−	8.94040i	−0.449840	−	0.449840i
$396$		0			0
$397$	−23.2641	+	23.2641i	−1.16759	+	1.16759i	−0.184819	+	0.982773i	$0.559170\pi$
$397$	−23.2641	+	23.2641i	−1.16759	+	1.16759i	−0.982773	+	0.184819i	$0.940830\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−31.5965			−1.57786			−0.788928	−	0.614486i	$-0.789364\pi$
$401$	−31.5965			−1.57786			−0.788928	+	0.614486i	$0.789364\pi$
$402$		0			0
$403$	14.7337	−	14.7337i	0.733939	−	0.733939i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			28.0903i			1.39238i
$408$		0			0
$409$		−	14.4988i		−	0.716917i	−0.933546	−	0.358459i	$-0.883302\pi$
$409$		−	14.4988i		−	0.716917i	0.933546	−	0.358459i	$-0.116698\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−27.5933	+	27.5933i	−1.35778	+	1.35778i
$414$		0			0
$415$	−15.7872			−0.774963
$416$		0			0
$417$		0			0
$418$		0			0
$419$	24.0709	−	24.0709i	1.17594	−	1.17594i	0.195169	−	0.980770i	$-0.437475\pi$
$419$	24.0709	−	24.0709i	1.17594	−	1.17594i	0.980770	−	0.195169i	$-0.0625254\pi$
$420$		0			0
$421$	−20.1095	−	20.1095i	−0.980079	−	0.980079i	0.0197268	−	0.999805i	$-0.493720\pi$
$421$	−20.1095	−	20.1095i	−0.980079	−	0.980079i	−0.999805	+	0.0197268i	$0.993720\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	4.32428i		−	0.209758i
$426$		0			0
$427$	1.14423	+	1.14423i	0.0553730	+	0.0553730i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−29.9026			−1.44036			−0.720180	−	0.693787i	$-0.755941\pi$
$431$	−29.9026			−1.44036			−0.720180	+	0.693787i	$0.755941\pi$
$432$		0			0
$433$	17.7487			0.852950			0.426475	−	0.904499i	$-0.359755\pi$
$433$	17.7487			0.852950			0.426475	+	0.904499i	$0.359755\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	21.0063	+	21.0063i	1.00487	+	1.00487i
$438$		0			0
$439$			15.1376i			0.722478i	0.932473	+	0.361239i	$0.117646\pi$
$439$			15.1376i			0.722478i	−0.932473	+	0.361239i	$0.882354\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−16.4687	−	16.4687i	−0.782451	−	0.782451i	0.197793	−	0.980244i	$-0.436623\pi$
$443$	−16.4687	−	16.4687i	−0.782451	−	0.782451i	−0.980244	+	0.197793i	$0.936623\pi$
$444$		0			0
$445$	9.17160	−	9.17160i	0.434775	−	0.434775i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−35.6078			−1.68044			−0.840218	−	0.542249i	$-0.817573\pi$
$449$	−35.6078			−1.68044			−0.840218	+	0.542249i	$0.817573\pi$
$450$		0			0
$451$	−1.79084	+	1.79084i	−0.0843273	+	0.0843273i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			8.30722i			0.389449i
$456$		0			0
$457$		−	1.98064i		−	0.0926502i	−0.998926	−	0.0463251i	$-0.985249\pi$
$457$		−	1.98064i		−	0.0926502i	0.998926	−	0.0463251i	$-0.0147510\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.13532	−	4.13532i	0.192601	−	0.192601i	−0.604218	−	0.796819i	$-0.706514\pi$
$461$	4.13532	−	4.13532i	0.192601	−	0.192601i	0.796819	+	0.604218i	$0.206514\pi$
$462$		0			0
$463$	−23.4228			−1.08855			−0.544274	−	0.838907i	$-0.683195\pi$
$463$	−23.4228			−1.08855			−0.544274	+	0.838907i	$0.683195\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.20786	+	4.20786i	−0.194717	+	0.194717i	−0.797731	−	0.603014i	$-0.793966\pi$
$467$	−4.20786	+	4.20786i	−0.194717	+	0.194717i	0.603014	+	0.797731i	$0.293966\pi$
$468$		0			0
$469$	−26.5033	−	26.5033i	−1.22381	−	1.22381i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	6.14077i		−	0.282353i
$474$		0			0
$475$	5.37165	+	5.37165i	0.246468	+	0.246468i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.52216			0.252314			0.126157	−	0.992010i	$-0.459736\pi$
$479$	5.52216			0.252314			0.126157	+	0.992010i	$0.459736\pi$
$480$		0			0
$481$	14.0731			0.641676
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.79809	−	4.79809i	−0.217870	−	0.217870i
$486$		0			0
$487$		−	28.0612i		−	1.27158i	−0.771864	−	0.635788i	$-0.780675\pi$
$487$		−	28.0612i		−	1.27158i	0.771864	−	0.635788i	$-0.219325\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.27407	−	1.27407i	−0.0574979	−	0.0574979i	0.677773	−	0.735271i	$-0.262945\pi$
$491$	−1.27407	−	1.27407i	−0.0574979	−	0.0574979i	−0.735271	+	0.677773i	$0.762945\pi$
$492$		0			0
$493$	7.96720	−	7.96720i	0.358825	−	0.358825i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.22238			0.0996875
$498$		0			0
$499$	26.3351	−	26.3351i	1.17892	−	1.17892i	0.198901	−	0.980020i	$-0.436263\pi$
$499$	26.3351	−	26.3351i	1.17892	−	1.17892i	0.980020	−	0.198901i	$-0.0637373\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	33.2293i		−	1.48162i	−0.671715	−	0.740810i	$-0.734442\pi$
$503$		−	33.2293i		−	1.48162i	0.671715	−	0.740810i	$-0.265558\pi$
$504$		0			0
$505$			8.75828i			0.389739i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.03424	−	5.03424i	0.223139	−	0.223139i	−0.586680	−	0.809819i	$-0.699565\pi$
$509$	5.03424	−	5.03424i	0.223139	−	0.223139i	0.809819	+	0.586680i	$0.199565\pi$
$510$		0			0
$511$	45.5830			2.01647
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.02701	+	8.02701i	−0.353713	+	0.353713i
$516$		0			0
$517$	6.86651	+	6.86651i	0.301989	+	0.301989i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	29.6023i		−	1.29690i	−0.761257	−	0.648450i	$-0.775418\pi$
$521$		−	29.6023i		−	1.29690i	0.761257	−	0.648450i	$-0.224582\pi$
$522$		0			0
$523$	−6.90122	−	6.90122i	−0.301769	−	0.301769i	0.539937	−	0.841706i	$-0.318448\pi$
$523$	−6.90122	−	6.90122i	−0.301769	−	0.301769i	−0.841706	+	0.539937i	$0.818448\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−41.2013			−1.79476
$528$		0			0
$529$	7.70731			0.335100
$530$		0			0
$531$		0			0
$532$		0			0
$533$	0.897200	+	0.897200i	0.0388621	+	0.0388621i
$534$		0			0
$535$		−	12.1409i		−	0.524895i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−22.9319	−	22.9319i	−0.987749	−	0.987749i
$540$		0			0
$541$	−7.18248	+	7.18248i	−0.308799	+	0.308799i	−0.844443	−	0.535645i	$-0.820069\pi$
$541$	−7.18248	+	7.18248i	−0.308799	+	0.308799i	0.535645	+	0.844443i	$0.320069\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−15.0789			−0.645910
$546$		0			0
$547$	14.9525	−	14.9525i	0.639323	−	0.639323i	−0.311066	−	0.950388i	$-0.600686\pi$
$547$	14.9525	−	14.9525i	0.639323	−	0.639323i	0.950388	+	0.311066i	$0.100686\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			19.7938i			0.843245i
$552$		0			0
$553$		−	48.0283i		−	2.04237i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−24.8706	+	24.8706i	−1.05380	+	1.05380i	−0.0553344	+	0.998468i	$0.517622\pi$
$557$	−24.8706	+	24.8706i	−1.05380	+	1.05380i	−0.998468	+	0.0553344i	$0.982378\pi$
$558$		0			0
$559$	−3.07649			−0.130122
$560$		0			0
$561$		0			0
$562$		0			0
$563$	11.1111	−	11.1111i	0.468277	−	0.468277i	−0.433079	−	0.901356i	$-0.642573\pi$
$563$	11.1111	−	11.1111i	0.468277	−	0.468277i	0.901356	+	0.433079i	$0.142573\pi$
$564$		0			0
$565$	2.20018	+	2.20018i	0.0925621	+	0.0925621i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	12.9347i		−	0.542250i	−0.962544	−	0.271125i	$-0.912604\pi$
$569$		−	12.9347i		−	0.542250i	0.962544	−	0.271125i	$-0.0873956\pi$
$570$		0			0
$571$	14.5979	+	14.5979i	0.610903	+	0.610903i	0.943181	−	0.332278i	$-0.107817\pi$
$571$	14.5979	+	14.5979i	0.610903	+	0.610903i	−0.332278	+	0.943181i	$0.607817\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.91059			−0.163083
$576$		0			0
$577$	15.7906			0.657373			0.328687	−	0.944439i	$-0.393394\pi$
$577$	15.7906			0.657373			0.328687	+	0.944439i	$0.393394\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−42.4048	−	42.4048i	−1.75925	−	1.75925i
$582$		0			0
$583$		−	29.6474i		−	1.22787i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.9365	+	13.9365i	0.575221	+	0.575221i	0.933583	−	0.358362i	$-0.116665\pi$
$587$	13.9365	+	13.9365i	0.575221	+	0.575221i	−0.358362	+	0.933583i	$0.616665\pi$
$588$		0			0
$589$	51.1805	−	51.1805i	2.10886	−	2.10886i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	47.4176			1.94721			0.973604	−	0.228242i	$-0.0732979\pi$
$593$	47.4176			1.94721			0.973604	+	0.228242i	$0.0732979\pi$
$594$		0			0
$595$	11.6151	−	11.6151i	0.476174	−	0.476174i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	23.5791i		−	0.963415i	−0.876332	−	0.481707i	$-0.840017\pi$
$599$		−	23.5791i		−	0.963415i	0.876332	−	0.481707i	$-0.159983\pi$
$600$		0			0
$601$		−	3.86582i		−	0.157690i	−0.996887	−	0.0788450i	$-0.974877\pi$
$601$		−	3.86582i		−	0.157690i	0.996887	−	0.0788450i	$-0.0251232\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.69533	+	5.69533i	−0.231548	+	0.231548i
$606$		0			0
$607$	−32.4306			−1.31632			−0.658159	−	0.752879i	$-0.728665\pi$
$607$	−32.4306			−1.31632			−0.658159	+	0.752879i	$0.728665\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.44008	−	3.44008i	0.139171	−	0.139171i
$612$		0			0
$613$	21.7952	+	21.7952i	0.880298	+	0.880298i	0.993565	−	0.113267i	$-0.0361314\pi$
$613$	21.7952	+	21.7952i	0.880298	+	0.880298i	−0.113267	+	0.993565i	$0.536131\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.49837i		−	0.0603219i	−0.999545	−	0.0301610i	$-0.990398\pi$
$617$		−	1.49837i		−	0.0603219i	0.999545	−	0.0301610i	$-0.00960199\pi$
$618$		0			0
$619$	−16.7004	−	16.7004i	−0.671246	−	0.671246i	0.286757	−	0.958003i	$-0.407423\pi$
$619$	−16.7004	−	16.7004i	−0.671246	−	0.671246i	−0.958003	+	0.286757i	$0.907423\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	49.2703			1.97397
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−19.6769	−	19.6769i	−0.784570	−	0.784570i
$630$		0			0
$631$		−	28.6304i		−	1.13976i	−0.821729	−	0.569878i	$-0.806990\pi$
$631$		−	28.6304i		−	1.13976i	0.821729	−	0.569878i	$-0.193010\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5.06564	+	5.06564i	0.201024	+	0.201024i
$636$		0			0
$637$	−11.4888	+	11.4888i	−0.455202	+	0.455202i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	47.4097			1.87257			0.936286	−	0.351239i	$-0.114240\pi$
$641$	47.4097			1.87257			0.936286	+	0.351239i	$0.114240\pi$
$642$		0			0
$643$	−22.0744	+	22.0744i	−0.870528	+	0.870528i	−0.992530	−	0.122002i	$-0.961069\pi$
$643$	−22.0744	+	22.0744i	−0.870528	+	0.870528i	0.122002	+	0.992530i	$0.461069\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			28.8171i			1.13292i	0.824090	+	0.566459i	$0.191687\pi$
$647$			28.8171i			1.13292i	−0.824090	+	0.566459i	$0.808313\pi$
$648$		0			0
$649$			44.8426i			1.76023i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	25.1198	−	25.1198i	0.983015	−	0.983015i	−0.0168430	−	0.999858i	$-0.505362\pi$
$653$	25.1198	−	25.1198i	0.983015	−	0.983015i	0.999858	+	0.0168430i	$0.00536155\pi$
$654$		0			0
$655$	−9.26437			−0.361989
$656$		0			0
$657$		0			0
$658$		0			0
$659$	11.3103	−	11.3103i	0.440585	−	0.440585i	−0.451623	−	0.892209i	$-0.649155\pi$
$659$	11.3103	−	11.3103i	0.440585	−	0.440585i	0.892209	+	0.451623i	$0.149155\pi$
$660$		0			0
$661$	−19.4036	−	19.4036i	−0.754713	−	0.754713i	0.220642	−	0.975355i	$-0.429185\pi$
$661$	−19.4036	−	19.4036i	−0.754713	−	0.754713i	−0.975355	+	0.220642i	$0.929185\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			28.8568i			1.11902i
$666$		0			0
$667$	−7.20500	−	7.20500i	−0.278979	−	0.278979i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.85952			0.0717858
$672$		0			0
$673$	−6.76645			−0.260827			−0.130414	−	0.991460i	$-0.541631\pi$
$673$	−6.76645			−0.260827			−0.130414	+	0.991460i	$0.541631\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.12216	+	2.12216i	0.0815613	+	0.0815613i	0.746710	−	0.665149i	$-0.231632\pi$
$677$	2.12216	+	2.12216i	0.0815613	+	0.0815613i	−0.665149	+	0.746710i	$0.731632\pi$
$678$		0			0
$679$		−	25.7756i		−	0.989177i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	16.9525	+	16.9525i	0.648669	+	0.648669i	0.952671	−	0.304002i	$-0.0983230\pi$
$683$	16.9525	+	16.9525i	0.648669	+	0.648669i	−0.304002	+	0.952671i	$0.598323\pi$
$684$		0			0
$685$	−7.81367	+	7.81367i	−0.298545	+	0.298545i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−14.8532			−0.565861
$690$		0			0
$691$	−18.4439	+	18.4439i	−0.701640	+	0.701640i	−0.964763	−	0.263122i	$-0.915248\pi$
$691$	−18.4439	+	18.4439i	−0.701640	+	0.701640i	0.263122	+	0.964763i	$0.415248\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			7.31391i			0.277432i
$696$		0			0
$697$		−	2.50893i		−	0.0950323i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6.59102	+	6.59102i	−0.248940	+	0.248940i	−0.820535	−	0.571596i	$-0.806325\pi$
$701$	−6.59102	+	6.59102i	−0.248940	+	0.248940i	0.571596	+	0.820535i	$0.306325\pi$
$702$		0			0
$703$	48.8856			1.84375
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−23.5250	+	23.5250i	−0.884748	+	0.884748i
$708$		0			0
$709$	−12.7715	−	12.7715i	−0.479642	−	0.479642i	0.425375	−	0.905017i	$-0.360142\pi$
$709$	−12.7715	−	12.7715i	−0.479642	−	0.479642i	−0.905017	+	0.425375i	$0.860142\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			37.2597i			1.39539i
$714$		0			0
$715$	6.75015	+	6.75015i	0.252441	+	0.252441i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	17.1105			0.638115			0.319057	−	0.947735i	$-0.396634\pi$
$719$	17.1105			0.638115			0.319057	+	0.947735i	$0.396634\pi$
$720$		0			0
$721$	−43.1215			−1.60593
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.84243	−	1.84243i	−0.0684263	−	0.0684263i
$726$		0			0
$727$			43.9133i			1.62865i	0.580407	+	0.814327i	$0.302893\pi$
$727$			43.9133i			1.62865i	−0.580407	+	0.814327i	$0.697107\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.30154	+	4.30154i	0.159098	+	0.159098i
$732$		0			0
$733$	−16.5301	+	16.5301i	−0.610553	+	0.610553i	−0.943090	−	0.332537i	$-0.892095\pi$
$733$	−16.5301	+	16.5301i	−0.610553	+	0.610553i	0.332537	+	0.943090i	$0.392095\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−43.0713			−1.58655
$738$		0			0
$739$	17.2689	−	17.2689i	0.635246	−	0.635246i	−0.314133	−	0.949379i	$-0.601714\pi$
$739$	17.2689	−	17.2689i	0.635246	−	0.635246i	0.949379	+	0.314133i	$0.101714\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	20.9862i		−	0.769909i	−0.922936	−	0.384955i	$-0.874217\pi$
$743$		−	20.9862i		−	0.769909i	0.922936	−	0.384955i	$-0.125783\pi$
$744$		0			0
$745$			8.20718i			0.300688i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	32.6107	−	32.6107i	1.19157	−	1.19157i
$750$		0			0
$751$	1.64813			0.0601413			0.0300706	−	0.999548i	$-0.490427\pi$
$751$	1.64813			0.0601413			0.0300706	+	0.999548i	$0.490427\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−9.37560	+	9.37560i	−0.341213	+	0.341213i
$756$		0			0
$757$	2.50864	+	2.50864i	0.0911779	+	0.0911779i	0.751225	−	0.660047i	$-0.229464\pi$
$757$	2.50864	+	2.50864i	0.0911779	+	0.0911779i	−0.660047	+	0.751225i	$0.729464\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	19.9555i		−	0.723387i	−0.932297	−	0.361693i	$-0.882199\pi$
$761$		−	19.9555i		−	0.723387i	0.932297	−	0.361693i	$-0.117801\pi$
$762$		0			0
$763$	−40.5024	−	40.5024i	−1.46628	−	1.46628i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	22.4659			0.811197
$768$		0			0
$769$	28.8082			1.03885			0.519426	−	0.854516i	$-0.326146\pi$
$769$	28.8082			1.03885			0.519426	+	0.854516i	$0.326146\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−33.6461	−	33.6461i	−1.21017	−	1.21017i	−0.970972	−	0.239195i	$-0.923116\pi$
$773$	−33.6461	−	33.6461i	−1.21017	−	1.21017i	−0.239195	−	0.970972i	$-0.576884\pi$
$774$		0			0
$775$			9.52790i			0.342252i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.11660	+	3.11660i	0.111664	+	0.111664i
$780$		0			0
$781$	1.80583	−	1.80583i	0.0646176	−	0.0646176i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	9.66373			0.344913
$786$		0			0
$787$	11.0029	−	11.0029i	0.392211	−	0.392211i	−0.483264	−	0.875475i	$-0.660549\pi$
$787$	11.0029	−	11.0029i	0.392211	−	0.392211i	0.875475	+	0.483264i	$0.160549\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			11.8195i			0.420252i
$792$		0			0
$793$		−	0.931607i		−	0.0330823i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13.3729	+	13.3729i	−0.473692	+	0.473692i	−0.903107	−	0.429415i	$-0.858720\pi$
$797$	−13.3729	+	13.3729i	−0.473692	+	0.473692i	0.429415	+	0.903107i	$0.358720\pi$
$798$		0			0
$799$	−9.61983			−0.340325
$800$		0			0
$801$		0			0
$802$		0			0
$803$	37.0391	−	37.0391i	1.30708	−	1.30708i
$804$		0			0
$805$	−10.5039	−	10.5039i	−0.370215	−	0.370215i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			20.9970i			0.738214i	0.929387	+	0.369107i	$0.120336\pi$
$809$			20.9970i			0.738214i	−0.929387	+	0.369107i	$0.879664\pi$
$810$		0			0
$811$	−26.7257	−	26.7257i	−0.938468	−	0.938468i	0.0597459	−	0.998214i	$-0.480971\pi$
$811$	−26.7257	−	26.7257i	−0.938468	−	0.938468i	−0.998214	+	0.0597459i	$0.980971\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.0627			0.352481
$816$		0			0
$817$	−10.6868			−0.373884
$818$		0			0
$819$		0			0
$820$		0			0
$821$	9.81609	+	9.81609i	0.342584	+	0.342584i	0.857338	−	0.514754i	$-0.172117\pi$
$821$	9.81609	+	9.81609i	0.342584	+	0.342584i	−0.514754	+	0.857338i	$0.672117\pi$
$822$		0			0
$823$			23.7241i			0.826969i	0.910511	+	0.413484i	$0.135688\pi$
$823$			23.7241i			0.826969i	−0.910511	+	0.413484i	$0.864312\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	13.8903	+	13.8903i	0.483014	+	0.483014i	0.906093	−	0.423079i	$-0.139051\pi$
$827$	13.8903	+	13.8903i	0.483014	+	0.483014i	−0.423079	+	0.906093i	$0.639051\pi$
$828$		0			0
$829$	34.7927	−	34.7927i	1.20840	−	1.20840i	0.236856	−	0.971545i	$-0.423883\pi$
$829$	34.7927	−	34.7927i	1.20840	−	1.20840i	0.971545	−	0.236856i	$-0.0761169\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	32.1271			1.11314
$834$		0			0
$835$	4.34184	−	4.34184i	0.150256	−	0.150256i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	41.9136i		−	1.44702i	−0.690314	−	0.723510i	$-0.742528\pi$
$839$		−	41.9136i		−	1.44702i	0.690314	−	0.723510i	$-0.257472\pi$
$840$		0			0
$841$			22.2109i			0.765892i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5.81060	+	5.81060i	−0.199891	+	0.199891i
$846$		0			0
$847$	−30.5956			−1.05128
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−17.7945	+	17.7945i	−0.609987	+	0.609987i
$852$		0			0
$853$	−17.8552	−	17.8552i	−0.611349	−	0.611349i	0.331949	−	0.943297i	$-0.392294\pi$
$853$	−17.8552	−	17.8552i	−0.611349	−	0.611349i	−0.943297	+	0.331949i	$0.892294\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	37.8604i		−	1.29328i	−0.762793	−	0.646642i	$-0.776173\pi$
$857$		−	37.8604i		−	1.29328i	0.762793	−	0.646642i	$-0.223827\pi$
$858$		0			0
$859$	33.0486	+	33.0486i	1.12760	+	1.12760i	0.990566	+	0.137036i	$0.0437576\pi$
$859$	33.0486	+	33.0486i	1.12760	+	1.12760i	0.137036	+	0.990566i	$0.456242\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	7.90340			0.269035			0.134518	−	0.990911i	$-0.457052\pi$
$863$	7.90340			0.269035			0.134518	+	0.990911i	$0.457052\pi$
$864$		0			0
$865$	10.3134			0.350667
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−39.0260	−	39.0260i	−1.32387	−	1.32387i
$870$		0			0
$871$			21.5785i			0.731159i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.68603	−	2.68603i	−0.0908043	−	0.0908043i
$876$		0			0
$877$	−23.4020	+	23.4020i	−0.790229	+	0.790229i	−0.981531	−	0.191302i	$-0.938729\pi$
$877$	−23.4020	+	23.4020i	−0.790229	+	0.790229i	0.191302	+	0.981531i	$0.438729\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.99400			−0.168252			−0.0841261	−	0.996455i	$-0.526810\pi$
$881$	−4.99400			−0.168252			−0.0841261	+	0.996455i	$0.526810\pi$
$882$		0			0
$883$	22.8573	−	22.8573i	0.769209	−	0.769209i	−0.208758	−	0.977967i	$-0.566942\pi$
$883$	22.8573	−	22.8573i	0.769209	−	0.769209i	0.977967	+	0.208758i	$0.0669423\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	20.5245i		−	0.689146i	−0.938760	−	0.344573i	$-0.888024\pi$
$887$		−	20.5245i		−	0.689146i	0.938760	−	0.344573i	$-0.111976\pi$
$888$		0			0
$889$			27.2129i			0.912691i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	11.9498	−	11.9498i	0.399885	−	0.399885i
$894$		0			0
$895$	8.80909			0.294456
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−17.5545	+	17.5545i	−0.585476	+	0.585476i
$900$		0			0
$901$	20.7677	+	20.7677i	0.691871	+	0.691871i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1.80032i		−	0.0598446i
$906$		0			0
$907$	12.1526	+	12.1526i	0.403521	+	0.403521i	0.879472	−	0.475951i	$-0.157896\pi$
$907$	12.1526	+	12.1526i	0.403521	+	0.403521i	−0.475951	+	0.879472i	$0.657896\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	26.1222			0.865468			0.432734	−	0.901522i	$-0.357549\pi$
$911$	26.1222			0.865468			0.432734	+	0.901522i	$0.357549\pi$
$912$		0			0
$913$	−68.9132			−2.28070
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−24.8843	−	24.8843i	−0.821753	−	0.821753i
$918$		0			0
$919$			12.3754i			0.408228i	0.978947	+	0.204114i	$0.0654313\pi$
$919$			12.3754i			0.408228i	−0.978947	+	0.204114i	$0.934569\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−0.904709	−	0.904709i	−0.0297789	−	0.0297789i
$924$		0			0
$925$	−4.55033	+	4.55033i	−0.149614	+	0.149614i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.1300			0.463591			0.231795	−	0.972765i	$-0.425540\pi$
$929$	14.1300			0.463591			0.231795	+	0.972765i	$0.425540\pi$
$930$		0			0
$931$	−39.9085	+	39.9085i	−1.30795	+	1.30795i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	18.8761i		−	0.617314i
$936$		0			0
$937$		−	39.1538i		−	1.27910i	−0.768750	−	0.639549i	$-0.779121\pi$
$937$		−	39.1538i		−	1.27910i	0.768750	−	0.639549i	$-0.220879\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	17.1373	−	17.1373i	0.558661	−	0.558661i	−0.370265	−	0.928926i	$-0.620733\pi$
$941$	17.1373	−	17.1373i	0.558661	−	0.558661i	0.928926	+	0.370265i	$0.120733\pi$
$942$		0			0
$943$	−2.26890			−0.0738856
$944$		0			0
$945$		0			0
$946$		0			0
$947$	21.7252	−	21.7252i	0.705973	−	0.705973i	−0.259713	−	0.965686i	$-0.583628\pi$
$947$	21.7252	−	21.7252i	0.705973	−	0.705973i	0.965686	+	0.259713i	$0.0836279\pi$
$948$		0			0
$949$	−18.5564	−	18.5564i	−0.602365	−	0.602365i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	6.05956i		−	0.196288i	−0.995172	−	0.0981442i	$-0.968709\pi$
$953$		−	6.05956i		−	0.196288i	0.995172	−	0.0981442i	$-0.0312906\pi$
$954$		0			0
$955$	6.81395	+	6.81395i	0.220494	+	0.220494i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−41.9754			−1.35546
$960$		0			0
$961$	59.7808			1.92841
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.57761	−	2.57761i	−0.0829763	−	0.0829763i
$966$		0			0
$967$			32.6389i			1.04959i	0.851227	+	0.524797i	$0.175859\pi$
$967$			32.6389i			1.04959i	−0.851227	+	0.524797i	$0.824141\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−4.73662	−	4.73662i	−0.152005	−	0.152005i	0.627008	−	0.779013i	$-0.284279\pi$
$971$	−4.73662	−	4.73662i	−0.152005	−	0.152005i	−0.779013	+	0.627008i	$0.784279\pi$
$972$		0			0
$973$	−19.6453	+	19.6453i	−0.629801	+	0.629801i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	31.9605			1.02251			0.511254	−	0.859430i	$-0.329181\pi$
$977$	31.9605			1.02251			0.511254	+	0.859430i	$0.329181\pi$
$978$		0			0
$979$	40.0353	−	40.0353i	1.27953	−	1.27953i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			36.1044i			1.15155i	0.817607	+	0.575776i	$0.195300\pi$
$983$			36.1044i			1.15155i	−0.817607	+	0.575776i	$0.804700\pi$
$984$		0			0
$985$			10.6833i			0.340398i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.89002	−	3.89002i	0.123695	−	0.123695i
$990$		0			0
$991$	−19.3054			−0.613255			−0.306628	−	0.951830i	$-0.599201\pi$
$991$	−19.3054			−0.613255			−0.306628	+	0.951830i	$0.599201\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16.7897	−	16.7897i	0.532269	−	0.532269i
$996$		0			0
$997$	−6.99944	−	6.99944i	−0.221674	−	0.221674i	0.587529	−	0.809203i	$-0.300101\pi$
$997$	−6.99944	−	6.99944i	−0.221674	−	0.221674i	−0.809203	+	0.587529i	$0.800101\pi$
$998$		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.2161.10		20
3.2	odd	2		960.2.s.c.241.10		20
4.3	odd	2		720.2.t.d.181.2		20
12.11	even	2		240.2.s.c.181.9	yes	20
16.3	odd	4		720.2.t.d.541.2		20
16.13	even	4	inner	2880.2.t.d.721.6		20
24.5	odd	2		1920.2.s.f.481.5		20
24.11	even	2		1920.2.s.e.481.6		20
48.5	odd	4		1920.2.s.f.1441.1		20
48.11	even	4		1920.2.s.e.1441.10		20
48.29	odd	4		960.2.s.c.721.6		20
48.35	even	4		240.2.s.c.61.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.s.c.61.9	✓	20	48.35	even	4
240.2.s.c.181.9	yes	20	12.11	even	2
720.2.t.d.181.2		20	4.3	odd	2
720.2.t.d.541.2		20	16.3	odd	4
960.2.s.c.241.10		20	3.2	odd	2
960.2.s.c.721.6		20	48.29	odd	4
1920.2.s.e.481.6		20	24.11	even	2
1920.2.s.e.1441.10		20	48.11	even	4
1920.2.s.f.481.5		20	24.5	odd	2
1920.2.s.f.1441.1		20	48.5	odd	4
2880.2.t.d.721.6		20	16.13	even	4	inner
2880.2.t.d.2161.10		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)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} +3.79862i q^{7} +O(q^{10})\)	\(q+(0.707107 + 0.707107i) q^{5} +3.79862i q^{7} +(3.08662 + 3.08662i) q^{11} +(1.54638 - 1.54638i) q^{13} -4.32428 q^{17} +(5.37165 - 5.37165i) q^{19} +3.91059i q^{23} +1.00000i q^{25} +(-1.84243 + 1.84243i) q^{29} +9.52790 q^{31} +(-2.68603 + 2.68603i) q^{35} +(4.55033 + 4.55033i) q^{37} +0.580195i q^{41} +(-0.994741 - 0.994741i) q^{43} +2.22461 q^{47} -7.42948 q^{49} +(-4.80257 - 4.80257i) q^{53} +4.36514i q^{55} +(7.26404 + 7.26404i) q^{59} +(0.301222 - 0.301222i) q^{61} +2.18691 q^{65} +(-6.97711 + 6.97711i) q^{67} -0.585051i q^{71} -11.9999i q^{73} +(-11.7249 + 11.7249i) q^{77} -12.6436 q^{79} +(-11.1632 + 11.1632i) q^{83} +(-3.05773 - 3.05773i) q^{85} -12.9706i q^{89} +(5.87409 + 5.87409i) q^{91} +7.59666 q^{95} -6.78553 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 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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