LMFDB - Embedded newform 2880.2.bc.a.593.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(593,2880)]

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, 2, 3]))

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

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

chi := DirichletCharacter("2880.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.bc (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:	$96$
Relative dimension:	$48$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.10
Character	$\chi$	$=$	2880.593
Dual form			2880.2.bc.a.1457.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.78433 + 1.34765i) q^{5} +(3.52140 + 3.52140i) q^{7} +O(q^{10})$	$q+(-1.78433 + 1.34765i) q^{5} +(3.52140 + 3.52140i) q^{7} +(-0.794589 - 0.794589i) q^{11} -1.81495i q^{13} +(-0.579924 - 0.579924i) q^{17} +(4.07807 - 4.07807i) q^{19} +(4.47496 + 4.47496i) q^{23} +(1.36768 - 4.80931i) q^{25} +(2.53613 + 2.53613i) q^{29} +8.04252 q^{31} +(-11.0290 - 1.53773i) q^{35} -5.45883i q^{37} +3.39090 q^{41} +4.86388 q^{43} +(-5.83824 - 5.83824i) q^{47} +17.8006i q^{49} +7.71651i q^{53} +(2.48864 + 0.346982i) q^{55} +(3.16642 - 3.16642i) q^{59} +(2.48943 + 2.48943i) q^{61} +(2.44591 + 3.23847i) q^{65} -2.85214 q^{67} +7.31072 q^{71} +(-10.1613 + 10.1613i) q^{73} -5.59614i q^{77} +9.50285i q^{79} -10.8245 q^{83} +(1.81631 + 0.253242i) q^{85} +8.83938i q^{89} +(6.39116 - 6.39116i) q^{91} +(-1.78081 + 12.7724i) q^{95} +(2.78908 - 2.78908i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q+O(q^{10}) $ 96 * q	$ 96 q - 16 q^{19} - 64 q^{43} + 32 q^{61}+O(q^{100}) $ 96 * q - 16 * q^19 - 64 * q^43 + 32 * q^61

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.78433	+	1.34765i	−0.797977	+	0.602688i
$5$	−1.78433	+	1.34765i	−0.797977	+	0.602688i
$6$		0			0
$7$	3.52140	+	3.52140i	1.33097	+	1.33097i	0.904507	+	0.426458i	$0.140239\pi$
$7$	3.52140	+	3.52140i	1.33097	+	1.33097i	0.426458	+	0.904507i	$0.359761\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.794589	−	0.794589i	−0.239578	−	0.239578i	0.577098	−	0.816675i	$-0.304185\pi$
$11$	−0.794589	−	0.794589i	−0.239578	−	0.239578i	−0.816675	+	0.577098i	$0.804185\pi$
$12$		0			0
$13$		−	1.81495i		−	0.503376i	−0.967808	−	0.251688i	$-0.919014\pi$
$13$		−	1.81495i		−	0.503376i	0.967808	−	0.251688i	$-0.0809856\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.579924	−	0.579924i	−0.140652	−	0.140652i	0.633275	−	0.773927i	$-0.281710\pi$
$17$	−0.579924	−	0.579924i	−0.140652	−	0.140652i	−0.773927	+	0.633275i	$0.781710\pi$
$18$		0			0
$19$	4.07807	−	4.07807i	0.935573	−	0.935573i	−0.0624739	−	0.998047i	$-0.519899\pi$
$19$	4.07807	−	4.07807i	0.935573	−	0.935573i	0.998047	+	0.0624739i	$0.0198990\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.47496	+	4.47496i	0.933094	+	0.933094i	0.997898	−	0.0648038i	$-0.0206422\pi$
$23$	4.47496	+	4.47496i	0.933094	+	0.933094i	−0.0648038	+	0.997898i	$0.520642\pi$
$24$		0			0
$25$	1.36768	−	4.80931i	0.273535	−	0.961862i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.53613	+	2.53613i	0.470948	+	0.470948i	0.902221	−	0.431273i	$-0.141935\pi$
$29$	2.53613	+	2.53613i	0.470948	+	0.470948i	−0.431273	+	0.902221i	$0.641935\pi$
$30$		0			0
$31$	8.04252			1.44448			0.722239	−	0.691643i	$-0.243113\pi$
$31$	8.04252			1.44448			0.722239	+	0.691643i	$0.243113\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−11.0290	−	1.53773i	−1.86424	−	0.259924i
$36$		0			0
$37$		−	5.45883i		−	0.897426i	−0.893676	−	0.448713i	$-0.851883\pi$
$37$		−	5.45883i		−	0.897426i	0.893676	−	0.448713i	$-0.148117\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.39090			0.529569			0.264785	−	0.964308i	$-0.414699\pi$
$41$	3.39090			0.529569			0.264785	+	0.964308i	$0.414699\pi$
$42$		0			0
$43$	4.86388			0.741735			0.370868	−	0.928686i	$-0.379060\pi$
$43$	4.86388			0.741735			0.370868	+	0.928686i	$0.379060\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.83824	−	5.83824i	−0.851595	−	0.851595i	0.138735	−	0.990330i	$-0.455696\pi$
$47$	−5.83824	−	5.83824i	−0.851595	−	0.851595i	−0.990330	+	0.138735i	$0.955696\pi$
$48$		0			0
$49$			17.8006i			2.54294i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			7.71651i			1.05994i	0.848015	+	0.529972i	$0.177798\pi$
$53$			7.71651i			1.05994i	−0.848015	+	0.529972i	$0.822202\pi$
$54$		0			0
$55$	2.48864	+	0.346982i	0.335568	+	0.0467871i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.16642	−	3.16642i	0.412232	−	0.412232i	−0.470283	−	0.882516i	$-0.655848\pi$
$59$	3.16642	−	3.16642i	0.412232	−	0.412232i	0.882516	+	0.470283i	$0.155848\pi$
$60$		0			0
$61$	2.48943	+	2.48943i	0.318739	+	0.318739i	0.848283	−	0.529544i	$-0.177637\pi$
$61$	2.48943	+	2.48943i	0.318739	+	0.318739i	−0.529544	+	0.848283i	$0.677637\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.44591	+	3.23847i	0.303378	+	0.401682i
$66$		0			0
$67$	−2.85214			−0.348444			−0.174222	−	0.984706i	$-0.555741\pi$
$67$	−2.85214			−0.348444			−0.174222	+	0.984706i	$0.555741\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.31072			0.867623			0.433812	−	0.901004i	$-0.357168\pi$
$71$	7.31072			0.867623			0.433812	+	0.901004i	$0.357168\pi$
$72$		0			0
$73$	−10.1613	+	10.1613i	−1.18929	+	1.18929i	−0.212032	+	0.977263i	$0.568008\pi$
$73$	−10.1613	+	10.1613i	−1.18929	+	1.18929i	−0.977263	+	0.212032i	$0.931992\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	5.59614i		−	0.637739i
$78$		0			0
$79$			9.50285i			1.06915i	0.845120	+	0.534577i	$0.179529\pi$
$79$			9.50285i			1.06915i	−0.845120	+	0.534577i	$0.820471\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.8245			−1.18814			−0.594072	−	0.804412i	$-0.702481\pi$
$83$	−10.8245			−1.18814			−0.594072	+	0.804412i	$0.702481\pi$
$84$		0			0
$85$	1.81631	+	0.253242i	0.197007	+	0.0274679i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			8.83938i			0.936972i	0.883471	+	0.468486i	$0.155200\pi$
$89$			8.83938i			0.936972i	−0.883471	+	0.468486i	$0.844800\pi$
$90$		0			0
$91$	6.39116	−	6.39116i	0.669976	−	0.669976i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.78081	+	12.7724i	−0.182708	+	1.31042i
$96$		0			0
$97$	2.78908	−	2.78908i	0.283188	−	0.283188i	−0.551191	−	0.834379i	$-0.685826\pi$
$97$	2.78908	−	2.78908i	0.283188	−	0.283188i	0.834379	+	0.551191i	$0.185826\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.5540	−	10.5540i	−1.05016	−	1.05016i	−0.998674	−	0.0514876i	$-0.983604\pi$
$101$	−10.5540	−	10.5540i	−1.05016	−	1.05016i	−0.0514876	−	0.998674i	$-0.516396\pi$
$102$		0			0
$103$	2.12084	−	2.12084i	0.208973	−	0.208973i	−0.594858	−	0.803831i	$-0.702792\pi$
$103$	2.12084	−	2.12084i	0.208973	−	0.208973i	0.803831	+	0.594858i	$0.202792\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.88547			0.375622			0.187811	−	0.982205i	$-0.439861\pi$
$107$	3.88547			0.375622			0.187811	+	0.982205i	$0.439861\pi$
$108$		0			0
$109$	−12.4641	+	12.4641i	−1.19385	+	1.19385i	−0.217871	+	0.975978i	$0.569911\pi$
$109$	−12.4641	+	12.4641i	−1.19385	+	1.19385i	−0.975978	+	0.217871i	$0.930089\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−12.7492	+	12.7492i	−1.19934	+	1.19934i	−0.224978	+	0.974364i	$0.572231\pi$
$113$	−12.7492	+	12.7492i	−1.19934	+	1.19934i	−0.974364	+	0.224978i	$0.927769\pi$
$114$		0			0
$115$	−14.0155	−	1.95413i	−1.30695	−	0.182224i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	4.08429i		−	0.374406i
$120$		0			0
$121$		−	9.73726i		−	0.885205i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.04088	+	10.4246i	0.361427	+	0.932400i
$126$		0			0
$127$	−9.37383	+	9.37383i	−0.831793	+	0.831793i	−0.987762	−	0.155969i	$-0.950150\pi$
$127$	−9.37383	+	9.37383i	−0.831793	+	0.831793i	0.155969	+	0.987762i	$0.450150\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.885812	+	0.885812i	−0.0773937	+	0.0773937i	−0.744744	−	0.667350i	$-0.767428\pi$
$131$	−0.885812	+	0.885812i	−0.0773937	+	0.0773937i	0.667350	+	0.744744i	$0.267428\pi$
$132$		0			0
$133$	28.7210			2.49043
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.99530	−	1.99530i	0.170470	−	0.170470i	−0.616716	−	0.787186i	$-0.711537\pi$
$137$	1.99530	−	1.99530i	0.170470	−	0.170470i	0.787186	+	0.616716i	$0.211537\pi$
$138$		0			0
$139$	4.37150	+	4.37150i	0.370786	+	0.370786i	0.867763	−	0.496977i	$-0.165557\pi$
$139$	4.37150	+	4.37150i	0.370786	+	0.370786i	−0.496977	+	0.867763i	$0.665557\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.44214	+	1.44214i	−0.120598	+	0.120598i
$144$		0			0
$145$	−7.94312	−	1.10748i	−0.659641	−	0.0919713i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.12416	−	6.12416i	0.501711	−	0.501711i	−0.410259	−	0.911969i	$-0.634562\pi$
$149$	6.12416	−	6.12416i	0.501711	−	0.501711i	0.911969	+	0.410259i	$0.134562\pi$
$150$		0			0
$151$			5.97875i			0.486544i	0.969958	+	0.243272i	$0.0782207\pi$
$151$			5.97875i			0.486544i	−0.969958	+	0.243272i	$0.921779\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−14.3505	+	10.8385i	−1.15266	+	0.870569i
$156$		0			0
$157$	18.8879			1.50742			0.753708	−	0.657209i	$-0.228263\pi$
$157$	18.8879			1.50742			0.753708	+	0.657209i	$0.228263\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			31.5163i			2.48383i
$162$		0			0
$163$			14.5507i			1.13970i	0.821749	+	0.569849i	$0.192998\pi$
$163$			14.5507i			1.13970i	−0.821749	+	0.569849i	$0.807002\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	15.6578	−	15.6578i	1.21164	−	1.21164i	0.241151	−	0.970488i	$-0.422475\pi$
$167$	15.6578	−	15.6578i	1.21164	−	1.21164i	0.970488	−	0.241151i	$-0.0775248\pi$
$168$		0			0
$169$	9.70597			0.746613
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.10719			0.540350			0.270175	−	0.962811i	$-0.412919\pi$
$173$	7.10719			0.540350			0.270175	+	0.962811i	$0.412919\pi$
$174$		0			0
$175$	21.7517	−	12.1194i	1.64427	−	0.916139i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	15.6929	+	15.6929i	1.17294	+	1.17294i	0.981504	+	0.191440i	$0.0613157\pi$
$179$	15.6929	+	15.6929i	1.17294	+	1.17294i	0.191440	+	0.981504i	$0.438684\pi$
$180$		0			0
$181$	6.56063	−	6.56063i	0.487648	−	0.487648i	−0.419915	−	0.907563i	$-0.637940\pi$
$181$	6.56063	−	6.56063i	0.487648	−	0.487648i	0.907563	+	0.419915i	$0.137940\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.35659	+	9.74036i	0.540867	+	0.716125i
$186$		0			0
$187$			0.921603i			0.0673943i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	20.4598i		−	1.48042i	−0.672376	−	0.740210i	$-0.734726\pi$
$191$		−	20.4598i		−	1.48042i	0.672376	−	0.740210i	$-0.265274\pi$
$192$		0			0
$193$	−15.7677	−	15.7677i	−1.13498	−	1.13498i	−0.989337	−	0.145647i	$-0.953474\pi$
$193$	−15.7677	−	15.7677i	−1.13498	−	1.13498i	−0.145647	−	0.989337i	$-0.546526\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.69499			−0.334504			−0.167252	−	0.985914i	$-0.553489\pi$
$197$	−4.69499			−0.334504			−0.167252	+	0.985914i	$0.553489\pi$
$198$		0			0
$199$	4.71983			0.334580			0.167290	−	0.985908i	$-0.446498\pi$
$199$	4.71983			0.334580			0.167290	+	0.985908i	$0.446498\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			17.8615i			1.25363i
$204$		0			0
$205$	−6.05048	+	4.56974i	−0.422584	+	0.319165i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.48078			−0.448285
$210$		0			0
$211$	−8.98274	−	8.98274i	−0.618397	−	0.618397i	0.326723	−	0.945120i	$-0.394056\pi$
$211$	−8.98274	−	8.98274i	−0.618397	−	0.618397i	−0.945120	+	0.326723i	$0.894056\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.67878	+	6.55481i	−0.591888	+	0.447035i
$216$		0			0
$217$	28.3210	+	28.3210i	1.92255	+	1.92255i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.05253	+	1.05253i	−0.0708009	+	0.0708009i
$222$		0			0
$223$	3.38870	+	3.38870i	0.226924	+	0.226924i	0.811406	−	0.584482i	$-0.198702\pi$
$223$	3.38870	+	3.38870i	0.226924	+	0.226924i	−0.584482	+	0.811406i	$0.698702\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			12.7291i			0.844857i	0.906396	+	0.422429i	$0.138822\pi$
$227$			12.7291i			0.844857i	−0.906396	+	0.422429i	$0.861178\pi$
$228$		0			0
$229$	18.3978	+	18.3978i	1.21576	+	1.21576i	0.969102	+	0.246662i	$0.0793336\pi$
$229$	18.3978	+	18.3978i	1.21576	+	1.21576i	0.246662	+	0.969102i	$0.420666\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.78291	+	4.78291i	0.313339	+	0.313339i	0.846202	−	0.532863i	$-0.178884\pi$
$233$	4.78291	+	4.78291i	0.313339	+	0.313339i	−0.532863	+	0.846202i	$0.678884\pi$
$234$		0			0
$235$	18.2853	+	2.54945i	1.19280	+	0.166308i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.68868			−0.626709			−0.313354	−	0.949636i	$-0.601453\pi$
$239$	−9.68868			−0.626709			−0.313354	+	0.949636i	$0.601453\pi$
$240$		0			0
$241$	6.75923			0.435400			0.217700	−	0.976016i	$-0.430145\pi$
$241$	6.75923			0.435400			0.217700	+	0.976016i	$0.430145\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−23.9889	−	31.7621i	−1.53260	−	2.02921i
$246$		0			0
$247$	−7.40148	−	7.40148i	−0.470945	−	0.470945i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−11.0136	−	11.0136i	−0.695172	−	0.695172i	0.268193	−	0.963365i	$-0.413574\pi$
$251$	−11.0136	−	11.0136i	−0.695172	−	0.695172i	−0.963365	+	0.268193i	$0.913574\pi$
$252$		0			0
$253$		−	7.11152i		−	0.447097i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.29957	+	4.29957i	0.268200	+	0.268200i	0.828374	−	0.560175i	$-0.189266\pi$
$257$	4.29957	+	4.29957i	0.268200	+	0.268200i	−0.560175	+	0.828374i	$0.689266\pi$
$258$		0			0
$259$	19.2227	−	19.2227i	1.19444	−	1.19444i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	20.2348	+	20.2348i	1.24773	+	1.24773i	0.956720	+	0.291012i	$0.0939918\pi$
$263$	20.2348	+	20.2348i	1.24773	+	1.24773i	0.291012	+	0.956720i	$0.406008\pi$
$264$		0			0
$265$	−10.3992	−	13.7688i	−0.638815	−	0.845811i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−22.6091	−	22.6091i	−1.37850	−	1.37850i	−0.847152	−	0.531350i	$-0.821685\pi$
$269$	−22.6091	−	22.6091i	−1.37850	−	1.37850i	−0.531350	−	0.847152i	$-0.678315\pi$
$270$		0			0
$271$	−0.832633			−0.0505788			−0.0252894	−	0.999680i	$-0.508051\pi$
$271$	−0.832633			−0.0505788			−0.0252894	+	0.999680i	$0.508051\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.90817	+	2.73468i	−0.295974	+	0.164908i
$276$		0			0
$277$		−	24.8462i		−	1.49286i	−0.665463	−	0.746431i	$-0.731766\pi$
$277$		−	24.8462i		−	1.49286i	0.665463	−	0.746431i	$-0.268234\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	11.4754			0.684563			0.342281	−	0.939597i	$-0.388800\pi$
$281$	11.4754			0.684563			0.342281	+	0.939597i	$0.388800\pi$
$282$		0			0
$283$	5.19216			0.308642			0.154321	−	0.988021i	$-0.450681\pi$
$283$	5.19216			0.308642			0.154321	+	0.988021i	$0.450681\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	11.9407	+	11.9407i	0.704838	+	0.704838i
$288$		0			0
$289$		−	16.3274i		−	0.960434i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			17.0751i			0.997538i	0.866735	+	0.498769i	$0.166214\pi$
$293$			17.0751i			0.997538i	−0.866735	+	0.498769i	$0.833786\pi$
$294$		0			0
$295$	−1.38271	+	9.91716i	−0.0805048	+	0.577400i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.12182	−	8.12182i	0.469697	−	0.469697i
$300$		0			0
$301$	17.1277	+	17.1277i	0.987224	+	0.987224i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.79685	−	1.08709i	−0.446447	−	0.0622465i
$306$		0			0
$307$	−14.8391			−0.846909			−0.423455	−	0.905917i	$-0.639183\pi$
$307$	−14.8391			−0.846909			−0.423455	+	0.905917i	$0.639183\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	11.7454			0.666020			0.333010	−	0.942923i	$-0.391936\pi$
$311$	11.7454			0.666020			0.333010	+	0.942923i	$0.391936\pi$
$312$		0			0
$313$	−6.70099	+	6.70099i	−0.378762	+	0.378762i	−0.870655	−	0.491893i	$-0.836305\pi$
$313$	−6.70099	+	6.70099i	−0.378762	+	0.378762i	0.491893	+	0.870655i	$0.336305\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	18.6931i		−	1.04991i	−0.851130	−	0.524956i	$-0.824082\pi$
$317$		−	18.6931i		−	1.04991i	0.851130	−	0.524956i	$-0.175918\pi$
$318$		0			0
$319$		−	4.03037i		−	0.225657i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.72994			−0.263181
$324$		0			0
$325$	−8.72864	−	2.48226i	−0.484178	−	0.137691i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	41.1176i		−	2.26689i
$330$		0			0
$331$	−0.985138	+	0.985138i	−0.0541481	+	0.0541481i	−0.733662	−	0.679514i	$-0.762191\pi$
$331$	−0.985138	+	0.985138i	−0.0541481	+	0.0541481i	0.679514	+	0.733662i	$0.262191\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.08916	−	3.84368i	0.278051	−	0.210003i
$336$		0			0
$337$	−12.2732	+	12.2732i	−0.668565	+	0.668565i	−0.957384	−	0.288819i	$-0.906738\pi$
$337$	−12.2732	+	12.2732i	−0.668565	+	0.668565i	0.288819	+	0.957384i	$0.406738\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.39050	−	6.39050i	−0.346065	−	0.346065i
$342$		0			0
$343$	−38.0332	+	38.0332i	−2.05360	+	2.05360i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.51724			−0.510912			−0.255456	−	0.966821i	$-0.582226\pi$
$347$	−9.51724			−0.510912			−0.255456	+	0.966821i	$0.582226\pi$
$348$		0			0
$349$	6.54235	−	6.54235i	0.350204	−	0.350204i	−0.509981	−	0.860185i	$-0.670348\pi$
$349$	6.54235	−	6.54235i	0.350204	−	0.350204i	0.860185	+	0.509981i	$0.170348\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.78695	−	9.78695i	0.520907	−	0.520907i	−0.396938	−	0.917845i	$-0.629927\pi$
$353$	9.78695	−	9.78695i	0.520907	−	0.520907i	0.917845	+	0.396938i	$0.129927\pi$
$354$		0			0
$355$	−13.0448	+	9.85230i	−0.692344	+	0.522906i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	2.64535i		−	0.139616i	−0.997560	−	0.0698081i	$-0.977761\pi$
$359$		−	2.64535i		−	0.139616i	0.997560	−	0.0698081i	$-0.0222387\pi$
$360$		0			0
$361$		−	14.2613i		−	0.750593i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.43727	−	31.8251i	0.232257	−	1.66580i
$366$		0			0
$367$	−6.03995	+	6.03995i	−0.315283	+	0.315283i	−0.846952	−	0.531669i	$-0.821565\pi$
$367$	−6.03995	+	6.03995i	−0.315283	+	0.315283i	0.531669	+	0.846952i	$0.321565\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−27.1729	+	27.1729i	−1.41075	+	1.41075i
$372$		0			0
$373$	−14.7609			−0.764288			−0.382144	−	0.924103i	$-0.624814\pi$
$373$	−14.7609			−0.764288			−0.382144	+	0.924103i	$0.624814\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.60295	−	4.60295i	0.237064	−	0.237064i
$378$		0			0
$379$	−15.5335	−	15.5335i	−0.797902	−	0.797902i	0.184862	−	0.982764i	$-0.440816\pi$
$379$	−15.5335	−	15.5335i	−0.797902	−	0.797902i	−0.982764	+	0.184862i	$0.940816\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.61642	−	9.61642i	0.491376	−	0.491376i	−0.417363	−	0.908740i	$-0.637046\pi$
$383$	9.61642	−	9.61642i	0.491376	−	0.491376i	0.908740	+	0.417363i	$0.137046\pi$
$384$		0			0
$385$	7.54164	+	9.98537i	0.384358	+	0.508902i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.77405	+	2.77405i	−0.140650	+	0.140650i	−0.773926	−	0.633276i	$-0.781710\pi$
$389$	−2.77405	+	2.77405i	−0.140650	+	0.140650i	0.633276	+	0.773926i	$0.281710\pi$
$390$		0			0
$391$		−	5.19028i		−	0.262483i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−12.8065	−	16.9562i	−0.644366	−	0.853161i
$396$		0			0
$397$	−26.0603			−1.30793			−0.653963	−	0.756526i	$-0.726895\pi$
$397$	−26.0603			−1.30793			−0.653963	+	0.756526i	$0.726895\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	2.43479i		−	0.121587i	−0.998150	−	0.0607937i	$-0.980637\pi$
$401$		−	2.43479i		−	0.121587i	0.998150	−	0.0607937i	$-0.0193632\pi$
$402$		0			0
$403$		−	14.5967i		−	0.727116i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.33753	+	4.33753i	−0.215003	+	0.215003i
$408$		0			0
$409$	−11.9845			−0.592596			−0.296298	−	0.955096i	$-0.595752\pi$
$409$	−11.9845			−0.592596			−0.296298	+	0.955096i	$0.595752\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	22.3005			1.09733
$414$		0			0
$415$	19.3145	−	14.5877i	0.948112	−	0.716080i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−22.9083	−	22.9083i	−1.11914	−	1.11914i	−0.991867	−	0.127277i	$-0.959376\pi$
$419$	−22.9083	−	22.9083i	−1.11914	−	1.11914i	−0.127277	−	0.991867i	$-0.540624\pi$
$420$		0			0
$421$	−6.98144	+	6.98144i	−0.340255	+	0.340255i	−0.856463	−	0.516208i	$-0.827343\pi$
$421$	−6.98144	+	6.98144i	−0.340255	+	0.340255i	0.516208	+	0.856463i	$0.327343\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.58218	+	1.99588i	−0.173761	+	0.0968146i
$426$		0			0
$427$			17.5326i			0.848461i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	18.2408i		−	0.878628i	−0.898334	−	0.439314i	$-0.855222\pi$
$431$		−	18.2408i		−	0.878628i	0.898334	−	0.439314i	$-0.144778\pi$
$432$		0			0
$433$	−11.5917	−	11.5917i	−0.557061	−	0.557061i	0.371409	−	0.928469i	$-0.378875\pi$
$433$	−11.5917	−	11.5917i	−0.557061	−	0.557061i	−0.928469	+	0.371409i	$0.878875\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	36.4984			1.74595
$438$		0			0
$439$	1.15895			0.0553134			0.0276567	−	0.999617i	$-0.491195\pi$
$439$	1.15895			0.0553134			0.0276567	+	0.999617i	$0.491195\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	8.12743i		−	0.386146i	−0.981184	−	0.193073i	$-0.938155\pi$
$443$		−	8.12743i		−	0.386146i	0.981184	−	0.193073i	$-0.0618454\pi$
$444$		0			0
$445$	−11.9124	−	15.7724i	−0.564701	−	0.747682i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4.24882			−0.200514			−0.100257	−	0.994962i	$-0.531967\pi$
$449$	−4.24882			−0.200514			−0.100257	+	0.994962i	$0.531967\pi$
$450$		0			0
$451$	−2.69437	−	2.69437i	−0.126873	−	0.126873i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.79090	+	20.0170i	−0.130839	+	0.938411i
$456$		0			0
$457$	−8.58027	−	8.58027i	−0.401368	−	0.401368i	0.477347	−	0.878715i	$-0.341599\pi$
$457$	−8.58027	−	8.58027i	−0.401368	−	0.401368i	−0.878715	+	0.477347i	$0.841599\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−27.2381	+	27.2381i	−1.26861	+	1.26861i	−0.321797	+	0.946809i	$0.604287\pi$
$461$	−27.2381	+	27.2381i	−1.26861	+	1.26861i	−0.946809	+	0.321797i	$0.895713\pi$
$462$		0			0
$463$	20.5504	+	20.5504i	0.955059	+	0.955059i	0.999033	−	0.0439739i	$-0.0140019\pi$
$463$	20.5504	+	20.5504i	0.955059	+	0.955059i	−0.0439739	+	0.999033i	$0.514002\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			7.62383i			0.352789i	0.984320	+	0.176394i	$0.0564434\pi$
$467$			7.62383i			0.352789i	−0.984320	+	0.176394i	$0.943557\pi$
$468$		0			0
$469$	−10.0435	−	10.0435i	−0.463767	−	0.463767i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.86479	−	3.86479i	−0.177703	−	0.177703i
$474$		0			0
$475$	−14.0352	−	25.1902i	−0.643979	−	1.15580i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.9717			0.821147			0.410574	−	0.911827i	$-0.365328\pi$
$479$	17.9717			0.821147			0.410574	+	0.911827i	$0.365328\pi$
$480$		0			0
$481$	−9.90748			−0.451742
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.21794	+	8.73535i	−0.0553038	+	0.396652i
$486$		0			0
$487$	19.4087	+	19.4087i	0.879492	+	0.879492i	0.993482	−	0.113990i	$-0.0363633\pi$
$487$	19.4087	+	19.4087i	0.879492	+	0.879492i	−0.113990	+	0.993482i	$0.536363\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.388463	+	0.388463i	0.0175311	+	0.0175311i	0.715818	−	0.698287i	$-0.246054\pi$
$491$	0.388463	+	0.388463i	0.0175311	+	0.0175311i	−0.698287	+	0.715818i	$0.746054\pi$
$492$		0			0
$493$		−	2.94153i		−	0.132480i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	25.7440	+	25.7440i	1.15478	+	1.15478i
$498$		0			0
$499$	15.5827	−	15.5827i	0.697577	−	0.697577i	−0.266310	−	0.963887i	$-0.585805\pi$
$499$	15.5827	−	15.5827i	0.697577	−	0.697577i	0.963887	+	0.266310i	$0.0858045\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10.5169	+	10.5169i	0.468927	+	0.468927i	0.901567	−	0.432640i	$-0.142418\pi$
$503$	10.5169	+	10.5169i	0.468927	+	0.468927i	−0.432640	+	0.901567i	$0.642418\pi$
$504$		0			0
$505$	33.0549	+	4.60873i	1.47092	+	0.205086i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.82377	+	5.82377i	0.258134	+	0.258134i	0.824295	−	0.566161i	$-0.191572\pi$
$509$	5.82377	+	5.82377i	0.258134	+	0.258134i	−0.566161	+	0.824295i	$0.691572\pi$
$510$		0			0
$511$	−71.5644			−3.16582
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.926132	+	6.64244i	−0.0408103	+	0.292701i
$516$		0			0
$517$			9.27801i			0.408046i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8.44184			−0.369844			−0.184922	−	0.982753i	$-0.559203\pi$
$521$	−8.44184			−0.369844			−0.184922	+	0.982753i	$0.559203\pi$
$522$		0			0
$523$	−21.7627			−0.951616			−0.475808	−	0.879549i	$-0.657844\pi$
$523$	−21.7627			−0.951616			−0.475808	+	0.879549i	$0.657844\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.66405	−	4.66405i	−0.203169	−	0.203169i
$528$		0			0
$529$			17.0506i			0.741330i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	6.15430i		−	0.266572i
$534$		0			0
$535$	−6.93296	+	5.23625i	−0.299738	+	0.226383i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	14.1441	−	14.1441i	0.609231	−	0.609231i
$540$		0			0
$541$	−2.65146	−	2.65146i	−0.113995	−	0.113995i	0.647808	−	0.761803i	$-0.275686\pi$
$541$	−2.65146	−	2.65146i	−0.113995	−	0.113995i	−0.761803	+	0.647808i	$0.775686\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.44286	−	39.0375i	0.233146	−	1.67218i
$546$		0			0
$547$	−10.2297			−0.437388			−0.218694	−	0.975793i	$-0.570180\pi$
$547$	−10.2297			−0.437388			−0.218694	+	0.975793i	$0.570180\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	20.6850			0.881212
$552$		0			0
$553$	−33.4634	+	33.4634i	−1.42301	+	1.42301i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			40.1061i			1.69935i	0.527306	+	0.849676i	$0.323202\pi$
$557$			40.1061i			1.69935i	−0.527306	+	0.849676i	$0.676798\pi$
$558$		0			0
$559$		−	8.82769i		−	0.373372i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	40.1102			1.69044			0.845221	−	0.534417i	$-0.179469\pi$
$563$	40.1102			1.69044			0.845221	+	0.534417i	$0.179469\pi$
$564$		0			0
$565$	5.56733	−	39.9302i	0.234219	−	1.67988i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	4.26260i		−	0.178698i	−0.996000	−	0.0893488i	$-0.971521\pi$
$569$		−	4.26260i		−	0.178698i	0.996000	−	0.0893488i	$-0.0284786\pi$
$570$		0			0
$571$	1.00161	−	1.00161i	0.0419161	−	0.0419161i	−0.685838	−	0.727754i	$-0.740564\pi$
$571$	1.00161	−	1.00161i	0.0419161	−	0.0419161i	0.727754	+	0.685838i	$0.240564\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	27.6418	−	15.4012i	1.15274	−	0.642273i
$576$		0			0
$577$	−3.57536	+	3.57536i	−0.148844	+	0.148844i	−0.777602	−	0.628757i	$-0.783564\pi$
$577$	−3.57536	+	3.57536i	−0.148844	+	0.148844i	0.628757	+	0.777602i	$0.283564\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−38.1175	−	38.1175i	−1.58138	−	1.58138i
$582$		0			0
$583$	6.13146	−	6.13146i	0.253939	−	0.253939i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	31.4094			1.29640			0.648202	−	0.761468i	$-0.275521\pi$
$587$	31.4094			1.29640			0.648202	+	0.761468i	$0.275521\pi$
$588$		0			0
$589$	32.7979	−	32.7979i	1.35142	−	1.35142i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.711706	+	0.711706i	−0.0292263	+	0.0292263i	−0.721569	−	0.692343i	$-0.756579\pi$
$593$	−0.711706	+	0.711706i	−0.0292263	+	0.0292263i	0.692343	+	0.721569i	$0.256579\pi$
$594$		0			0
$595$	5.50420	+	7.28773i	0.225650	+	0.298768i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	22.7143i		−	0.928081i	−0.885814	−	0.464041i	$-0.846399\pi$
$599$		−	22.7143i		−	0.928081i	0.885814	−	0.464041i	$-0.153601\pi$
$600$		0			0
$601$		−	2.66757i		−	0.108812i	−0.998519	−	0.0544062i	$-0.982673\pi$
$601$		−	2.66757i		−	0.108812i	0.998519	−	0.0544062i	$-0.0173266\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.1224	+	17.3745i	0.533502	+	0.706373i
$606$		0			0
$607$	19.7506	−	19.7506i	0.801653	−	0.801653i	−0.181701	−	0.983354i	$-0.558160\pi$
$607$	19.7506	−	19.7506i	0.801653	−	0.801653i	0.983354	+	0.181701i	$0.0581604\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−10.5961	+	10.5961i	−0.428672	+	0.428672i
$612$		0			0
$613$	−37.6286			−1.51981			−0.759903	−	0.650037i	$-0.774753\pi$
$613$	−37.6286			−1.51981			−0.759903	+	0.650037i	$0.774753\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.68313	+	7.68313i	−0.309311	+	0.309311i	−0.844642	−	0.535331i	$-0.820187\pi$
$617$	−7.68313	+	7.68313i	−0.309311	+	0.309311i	0.535331	+	0.844642i	$0.320187\pi$
$618$		0			0
$619$	−0.363070	−	0.363070i	−0.0145930	−	0.0145930i	0.699773	−	0.714366i	$-0.253285\pi$
$619$	−0.363070	−	0.363070i	−0.0145930	−	0.0145930i	−0.714366	+	0.699773i	$0.753285\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−31.1270	+	31.1270i	−1.24708	+	1.24708i
$624$		0			0
$625$	−21.2589	−	13.1552i	−0.850357	−	0.526207i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.16570	+	3.16570i	−0.126225	+	0.126225i
$630$		0			0
$631$		−	38.2533i		−	1.52284i	−0.648260	−	0.761419i	$-0.724503\pi$
$631$		−	38.2533i		−	1.52284i	0.648260	−	0.761419i	$-0.275497\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.09337	−	29.3587i	0.162441	−	1.16506i
$636$		0			0
$637$	32.3071			1.28005
$638$		0			0
$639$		0			0
$640$		0			0
$641$			8.83990i			0.349155i	0.984643	+	0.174577i	$0.0558560\pi$
$641$			8.83990i			0.349155i	−0.984643	+	0.174577i	$0.944144\pi$
$642$		0			0
$643$		−	17.2140i		−	0.678853i	−0.940633	−	0.339427i	$-0.889767\pi$
$643$		−	17.2140i		−	0.678853i	0.940633	−	0.339427i	$-0.110233\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5.49908	−	5.49908i	0.216191	−	0.216191i	−0.590700	−	0.806891i	$-0.701148\pi$
$647$	5.49908	−	5.49908i	0.216191	−	0.216191i	0.806891	+	0.590700i	$0.201148\pi$
$648$		0			0
$649$	−5.03200			−0.197523
$650$		0			0
$651$		0			0
$652$		0			0
$653$	43.6753			1.70914			0.854572	−	0.519332i	$-0.173819\pi$
$653$	43.6753			1.70914			0.854572	+	0.519332i	$0.173819\pi$
$654$		0			0
$655$	0.386817	−	2.77435i	0.0151142	−	0.108403i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.1428	+	12.1428i	0.473018	+	0.473018i	0.902890	−	0.429872i	$-0.141441\pi$
$659$	12.1428	+	12.1428i	0.473018	+	0.473018i	−0.429872	+	0.902890i	$0.641441\pi$
$660$		0			0
$661$	−19.7181	+	19.7181i	−0.766947	+	0.766947i	−0.977568	−	0.210621i	$-0.932451\pi$
$661$	−19.7181	+	19.7181i	−0.766947	+	0.766947i	0.210621	+	0.977568i	$0.432451\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−51.2479	+	38.7059i	−1.98731	+	1.50095i
$666$		0			0
$667$			22.6982i			0.878878i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	3.95615i		−	0.152726i
$672$		0			0
$673$	−7.49777	−	7.49777i	−0.289018	−	0.289018i	0.547674	−	0.836692i	$-0.315513\pi$
$673$	−7.49777	−	7.49777i	−0.289018	−	0.289018i	−0.836692	+	0.547674i	$0.815513\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.66092			0.371299			0.185650	−	0.982616i	$-0.440561\pi$
$677$	9.66092			0.371299			0.185650	+	0.982616i	$0.440561\pi$
$678$		0			0
$679$	19.6430			0.753828
$680$		0			0
$681$		0			0
$682$		0			0
$683$			13.1017i			0.501323i	0.968075	+	0.250662i	$0.0806481\pi$
$683$			13.1017i			0.501323i	−0.968075	+	0.250662i	$0.919352\pi$
$684$		0			0
$685$	−0.871310	+	6.24925i	−0.0332911	+	0.238771i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	14.0051			0.533550
$690$		0			0
$691$	4.80979	+	4.80979i	0.182973	+	0.182973i	0.792650	−	0.609677i	$-0.208701\pi$
$691$	4.80979	+	4.80979i	0.182973	+	0.182973i	−0.609677	+	0.792650i	$0.708701\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−13.6915	−	1.90895i	−0.519347	−	0.0724107i
$696$		0			0
$697$	−1.96646	−	1.96646i	−0.0744851	−	0.0744851i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.13442	−	6.13442i	0.231694	−	0.231694i	−0.581706	−	0.813399i	$-0.697614\pi$
$701$	6.13442	−	6.13442i	0.231694	−	0.231694i	0.813399	+	0.581706i	$0.197614\pi$
$702$		0			0
$703$	−22.2615	−	22.2615i	−0.839607	−	0.839607i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	74.3297i		−	2.79546i
$708$		0			0
$709$	27.7679	+	27.7679i	1.04285	+	1.04285i	0.999040	+	0.0438050i	$0.0139480\pi$
$709$	27.7679	+	27.7679i	1.04285	+	1.04285i	0.0438050	+	0.999040i	$0.486052\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	35.9900	+	35.9900i	1.34783	+	1.34783i
$714$		0			0
$715$	0.629754	−	4.51675i	0.0235515	−	0.168917i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.11652			−0.0416391			−0.0208195	−	0.999783i	$-0.506628\pi$
$719$	−1.11652			−0.0416391			−0.0208195	+	0.999783i	$0.506628\pi$
$720$		0			0
$721$	14.9367			0.556271
$722$		0			0
$723$		0			0
$724$		0			0
$725$	15.6657	−	8.72844i	0.581808	−	0.324166i
$726$		0			0
$727$	22.7799	+	22.7799i	0.844859	+	0.844859i	0.989486	−	0.144627i	$-0.0461983\pi$
$727$	22.7799	+	22.7799i	0.844859	+	0.844859i	−0.144627	+	0.989486i	$0.546198\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.82068	−	2.82068i	−0.104327	−	0.104327i
$732$		0			0
$733$			14.3681i			0.530699i	0.964152	+	0.265350i	$0.0854873\pi$
$733$			14.3681i			0.530699i	−0.964152	+	0.265350i	$0.914513\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.26628	+	2.26628i	0.0834795	+	0.0834795i
$738$		0			0
$739$	−2.18646	+	2.18646i	−0.0804301	+	0.0804301i	−0.746177	−	0.665747i	$-0.768113\pi$
$739$	−2.18646	+	2.18646i	−0.0804301	+	0.0804301i	0.665747	+	0.746177i	$0.268113\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−28.0875	−	28.0875i	−1.03043	−	1.03043i	−0.999522	−	0.0309076i	$-0.990160\pi$
$743$	−28.0875	−	28.0875i	−1.03043	−	1.03043i	−0.0309076	−	0.999522i	$-0.509840\pi$
$744$		0			0
$745$	−2.67430	+	19.1808i	−0.0979790	+	0.702729i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	13.6823	+	13.6823i	0.499940	+	0.499940i
$750$		0			0
$751$	−15.9304			−0.581307			−0.290654	−	0.956828i	$-0.593873\pi$
$751$	−15.9304			−0.581307			−0.290654	+	0.956828i	$0.593873\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.05727	−	10.6681i	−0.293234	−	0.388251i
$756$		0			0
$757$			37.2691i			1.35457i	0.735721	+	0.677285i	$0.236844\pi$
$757$			37.2691i			1.35457i	−0.735721	+	0.677285i	$0.763156\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	17.3081			0.627419			0.313709	−	0.949519i	$-0.398428\pi$
$761$	17.3081			0.627419			0.313709	+	0.949519i	$0.398428\pi$
$762$		0			0
$763$	−87.7826			−3.17794
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.74688	−	5.74688i	−0.207508	−	0.207508i
$768$		0			0
$769$		−	24.2688i		−	0.875155i	−0.899181	−	0.437577i	$-0.855837\pi$
$769$		−	24.2688i		−	0.875155i	0.899181	−	0.437577i	$-0.144163\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	13.0667i		−	0.469977i	−0.971998	−	0.234989i	$-0.924495\pi$
$773$		−	13.0667i		−	0.469977i	0.971998	−	0.234989i	$-0.0755053\pi$
$774$		0			0
$775$	10.9996	−	38.6790i	0.395116	−	1.38939i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13.8283	−	13.8283i	0.495450	−	0.495450i
$780$		0			0
$781$	−5.80902	−	5.80902i	−0.207863	−	0.207863i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−33.7022	+	25.4543i	−1.20288	+	0.908501i
$786$		0			0
$787$	−19.0159			−0.677844			−0.338922	−	0.940814i	$-0.610062\pi$
$787$	−19.0159			−0.677844			−0.338922	+	0.940814i	$0.610062\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−89.7900			−3.19257
$792$		0			0
$793$	4.51819	−	4.51819i	0.160446	−	0.160446i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	19.8027i		−	0.701447i	−0.936479	−	0.350723i	$-0.885936\pi$
$797$		−	19.8027i		−	0.701447i	0.936479	−	0.350723i	$-0.114064\pi$
$798$		0			0
$799$			6.77147i			0.239557i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	16.1482			0.569857
$804$		0			0
$805$	−42.4730	−	56.2355i	−1.49697	−	1.98204i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	43.2999i		−	1.52234i	−0.648550	−	0.761172i	$-0.724624\pi$
$809$		−	43.2999i		−	1.52234i	0.648550	−	0.761172i	$-0.275376\pi$
$810$		0			0
$811$	36.4064	−	36.4064i	1.27840	−	1.27840i	0.336837	−	0.941563i	$-0.390643\pi$
$811$	36.4064	−	36.4064i	1.27840	−	1.27840i	0.941563	−	0.336837i	$-0.109357\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19.6092	−	25.9633i	−0.686882	−	0.909453i
$816$		0			0
$817$	19.8352	−	19.8352i	0.693947	−	0.693947i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.11395	+	2.11395i	0.0737773	+	0.0737773i	0.743033	−	0.669255i	$-0.233387\pi$
$821$	2.11395	+	2.11395i	0.0737773	+	0.0737773i	−0.669255	+	0.743033i	$0.733387\pi$
$822$		0			0
$823$	0.790237	−	0.790237i	0.0275459	−	0.0275459i	−0.693200	−	0.720746i	$-0.743800\pi$
$823$	0.790237	−	0.790237i	0.0275459	−	0.0275459i	0.720746	+	0.693200i	$0.243800\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−19.8813			−0.691342			−0.345671	−	0.938356i	$-0.612349\pi$
$827$	−19.8813			−0.691342			−0.345671	+	0.938356i	$0.612349\pi$
$828$		0			0
$829$	−29.8495	+	29.8495i	−1.03671	+	1.03671i	−0.0374151	+	0.999300i	$0.511912\pi$
$829$	−29.8495	+	29.8495i	−1.03671	+	1.03671i	−0.999300	+	0.0374151i	$0.988088\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	10.3230	−	10.3230i	0.357670	−	0.357670i
$834$		0			0
$835$	−6.83747	+	49.0400i	−0.236621	+	1.69710i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	38.6066i		−	1.33285i	−0.745573	−	0.666424i	$-0.767824\pi$
$839$		−	38.6066i		−	1.33285i	0.745573	−	0.666424i	$-0.232176\pi$
$840$		0			0
$841$		−	16.1361i		−	0.556416i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−17.3187	+	13.0802i	−0.595780	+	0.449974i
$846$		0			0
$847$	34.2888	−	34.2888i	1.17818	−	1.17818i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	24.4281	−	24.4281i	0.837383	−	0.837383i
$852$		0			0
$853$	6.72965			0.230419			0.115209	−	0.993341i	$-0.463246\pi$
$853$	6.72965			0.230419			0.115209	+	0.993341i	$0.463246\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14.8406	−	14.8406i	0.506945	−	0.506945i	−0.406642	−	0.913588i	$-0.633300\pi$
$857$	14.8406	−	14.8406i	0.506945	−	0.506945i	0.913588	+	0.406642i	$0.133300\pi$
$858$		0			0
$859$	−7.20839	−	7.20839i	−0.245947	−	0.245947i	0.573358	−	0.819305i	$-0.305640\pi$
$859$	−7.20839	−	7.20839i	−0.245947	−	0.245947i	−0.819305	+	0.573358i	$0.805640\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−28.9476	+	28.9476i	−0.985388	+	0.985388i	−0.999895	−	0.0145070i	$-0.995382\pi$
$863$	−28.9476	+	28.9476i	−0.985388	+	0.985388i	0.0145070	+	0.999895i	$0.495382\pi$
$864$		0			0
$865$	−12.6816	+	9.57801i	−0.431187	+	0.325662i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.55087	−	7.55087i	0.256146	−	0.256146i
$870$		0			0
$871$			5.17648i			0.175398i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−22.4795	+	50.9386i	−0.759946	+	1.72204i
$876$		0			0
$877$	−26.1006			−0.881355			−0.440677	−	0.897666i	$-0.645262\pi$
$877$	−26.1006			−0.881355			−0.440677	+	0.897666i	$0.645262\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	52.3645i		−	1.76421i	−0.471057	−	0.882103i	$-0.656127\pi$
$881$		−	52.3645i		−	1.76421i	0.471057	−	0.882103i	$-0.343873\pi$
$882$		0			0
$883$		−	47.1364i		−	1.58627i	−0.609047	−	0.793134i	$-0.708448\pi$
$883$		−	47.1364i		−	1.58627i	0.609047	−	0.793134i	$-0.291552\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	39.2668	−	39.2668i	1.31845	−	1.31845i	0.403444	−	0.915004i	$-0.367813\pi$
$887$	39.2668	−	39.2668i	1.31845	−	1.31845i	0.915004	−	0.403444i	$-0.132187\pi$
$888$		0			0
$889$	−66.0181			−2.21417
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−47.6175			−1.59346
$894$		0			0
$895$	−49.1500	−	6.85280i	−1.64290	−	0.229064i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	20.3969	+	20.3969i	0.680275	+	0.680275i
$900$		0			0
$901$	4.47499	−	4.47499i	0.149083	−	0.149083i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.86490	+	20.5478i	−0.0952326	+	0.683031i
$906$		0			0
$907$			0.853198i			0.0283300i	0.999900	+	0.0141650i	$0.00450900\pi$
$907$			0.853198i			0.0283300i	−0.999900	+	0.0141650i	$0.995491\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			0.383383i			0.0127020i	0.999980	+	0.00635102i	$0.00202161\pi$
$911$			0.383383i			0.0127020i	−0.999980	+	0.00635102i	$0.997978\pi$
$912$		0			0
$913$	8.60104	+	8.60104i	0.284653	+	0.284653i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−6.23860			−0.206017
$918$		0			0
$919$	−20.4070			−0.673165			−0.336582	−	0.941654i	$-0.609271\pi$
$919$	−20.4070			−0.673165			−0.336582	+	0.941654i	$0.609271\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	13.2686i		−	0.436741i
$924$		0			0
$925$	−26.2532	−	7.46592i	−0.863200	−	0.245478i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−5.66448			−0.185846			−0.0929228	−	0.995673i	$-0.529621\pi$
$929$	−5.66448			−0.185846			−0.0929228	+	0.995673i	$0.529621\pi$
$930$		0			0
$931$	72.5919	+	72.5919i	2.37910	+	2.37910i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.24200	−	1.64444i	−0.0406177	−	0.0537791i
$936$		0			0
$937$	4.52855	+	4.52855i	0.147941	+	0.147941i	0.777198	−	0.629256i	$-0.216640\pi$
$937$	4.52855	+	4.52855i	0.147941	+	0.147941i	−0.629256	+	0.777198i	$0.716640\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	33.1033	−	33.1033i	1.07914	−	1.07914i	0.0825482	−	0.996587i	$-0.473694\pi$
$941$	33.1033	−	33.1033i	1.07914	−	1.07914i	0.996587	−	0.0825482i	$-0.0263058\pi$
$942$		0			0
$943$	15.1741	+	15.1741i	0.494138	+	0.494138i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	42.2938i		−	1.37436i	−0.726486	−	0.687182i	$-0.758848\pi$
$947$		−	42.2938i		−	1.37436i	0.726486	−	0.687182i	$-0.241152\pi$
$948$		0			0
$949$	18.4423	+	18.4423i	0.598662	+	0.598662i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−7.70327	−	7.70327i	−0.249533	−	0.249533i	0.571246	−	0.820779i	$-0.306460\pi$
$953$	−7.70327	−	7.70327i	−0.249533	−	0.249533i	−0.820779	+	0.571246i	$0.806460\pi$
$954$		0			0
$955$	27.5727	+	36.5071i	0.892231	+	1.18134i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.0525			0.453780
$960$		0			0
$961$	33.6821			1.08652
$962$		0			0
$963$		0			0
$964$		0			0
$965$	49.3841	+	6.88545i	1.58973	+	0.221651i
$966$		0			0
$967$	−15.1520	−	15.1520i	−0.487255	−	0.487255i	0.420184	−	0.907439i	$-0.361966\pi$
$967$	−15.1520	−	15.1520i	−0.487255	−	0.487255i	−0.907439	+	0.420184i	$0.861966\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	18.9617	+	18.9617i	0.608508	+	0.608508i	0.942556	−	0.334048i	$-0.108415\pi$
$971$	18.9617	+	18.9617i	0.608508	+	0.608508i	−0.334048	+	0.942556i	$0.608415\pi$
$972$		0			0
$973$			30.7876i			0.987007i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−35.4983	−	35.4983i	−1.13569	−	1.13569i	−0.989214	−	0.146477i	$-0.953207\pi$
$977$	−35.4983	−	35.4983i	−1.13569	−	1.13569i	−0.146477	−	0.989214i	$-0.546793\pi$
$978$		0			0
$979$	7.02367	−	7.02367i	0.224478	−	0.224478i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−14.4359	−	14.4359i	−0.460434	−	0.460434i	0.438364	−	0.898798i	$-0.355558\pi$
$983$	−14.4359	−	14.4359i	−0.460434	−	0.460434i	−0.898798	+	0.438364i	$0.855558\pi$
$984$		0			0
$985$	8.37743	−	6.32721i	0.266927	−	0.201602i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	21.7657	+	21.7657i	0.692109	+	0.692109i
$990$		0			0
$991$	−4.78714			−0.152069			−0.0760343	−	0.997105i	$-0.524226\pi$
$991$	−4.78714			−0.152069			−0.0760343	+	0.997105i	$0.524226\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.42174	+	6.36068i	−0.266987	+	0.201647i
$996$		0			0
$997$		−	1.03678i		−	0.0328350i	−0.999865	−	0.0164175i	$-0.994774\pi$
$997$		−	1.03678i		−	0.0328350i	0.999865	−	0.0164175i	$-0.00522609\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bc.a.593.10		96
3.2	odd	2	inner	2880.2.bc.a.593.39		96
4.3	odd	2		720.2.bc.a.413.35	yes	96
5.2	odd	4		2880.2.bg.a.17.34		96
12.11	even	2		720.2.bc.a.413.14	yes	96
15.2	even	4		2880.2.bg.a.17.15		96
16.5	even	4		2880.2.bg.a.2033.15		96
16.11	odd	4		720.2.bg.a.53.38	yes	96
20.7	even	4		720.2.bg.a.557.11	yes	96
48.5	odd	4		2880.2.bg.a.2033.34		96
48.11	even	4		720.2.bg.a.53.11	yes	96
60.47	odd	4		720.2.bg.a.557.38	yes	96
80.27	even	4		720.2.bc.a.197.14	✓	96
80.37	odd	4	inner	2880.2.bc.a.1457.39		96
240.107	odd	4		720.2.bc.a.197.35	yes	96
240.197	even	4	inner	2880.2.bc.a.1457.10		96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bc.a.197.14	✓	96	80.27	even	4
720.2.bc.a.197.35	yes	96	240.107	odd	4
720.2.bc.a.413.14	yes	96	12.11	even	2
720.2.bc.a.413.35	yes	96	4.3	odd	2
720.2.bg.a.53.11	yes	96	48.11	even	4
720.2.bg.a.53.38	yes	96	16.11	odd	4
720.2.bg.a.557.11	yes	96	20.7	even	4
720.2.bg.a.557.38	yes	96	60.47	odd	4
2880.2.bc.a.593.10		96	1.1	even	1	trivial
2880.2.bc.a.593.39		96	3.2	odd	2	inner
2880.2.bc.a.1457.10		96	240.197	even	4	inner
2880.2.bc.a.1457.39		96	80.37	odd	4	inner
2880.2.bg.a.17.15		96	15.2	even	4
2880.2.bg.a.17.34		96	5.2	odd	4
2880.2.bg.a.2033.15		96	16.5	even	4
2880.2.bg.a.2033.34		96	48.5	odd	4

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.bc (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.78433 + 1.34765i) q^{5} +(3.52140 + 3.52140i) q^{7} +O(q^{10})\)	\(q+(-1.78433 + 1.34765i) q^{5} +(3.52140 + 3.52140i) q^{7} +(-0.794589 - 0.794589i) q^{11} -1.81495i q^{13} +(-0.579924 - 0.579924i) q^{17} +(4.07807 - 4.07807i) q^{19} +(4.47496 + 4.47496i) q^{23} +(1.36768 - 4.80931i) q^{25} +(2.53613 + 2.53613i) q^{29} +8.04252 q^{31} +(-11.0290 - 1.53773i) q^{35} -5.45883i q^{37} +3.39090 q^{41} +4.86388 q^{43} +(-5.83824 - 5.83824i) q^{47} +17.8006i q^{49} +7.71651i q^{53} +(2.48864 + 0.346982i) q^{55} +(3.16642 - 3.16642i) q^{59} +(2.48943 + 2.48943i) q^{61} +(2.44591 + 3.23847i) q^{65} -2.85214 q^{67} +7.31072 q^{71} +(-10.1613 + 10.1613i) q^{73} -5.59614i q^{77} +9.50285i q^{79} -10.8245 q^{83} +(1.81631 + 0.253242i) q^{85} +8.83938i q^{89} +(6.39116 - 6.39116i) q^{91} +(-1.78081 + 12.7724i) q^{95} +(2.78908 - 2.78908i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q+O(q^{10}) \) 96 * q	\( 96 q - 16 q^{19} - 64 q^{43} + 32 q^{61}+O(q^{100}) \) 96 * q - 16 * q^19 - 64 * q^43 + 32 * q^61

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more