LMFDB - Embedded newform 2880.2.bl.c.1871.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(431,2880)]

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([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("2880.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.bl (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:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1871.15
Character	$\chi$	$=$	2880.1871
Dual form			2880.2.bl.c.431.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{5} -2.45331 q^{7} +O(q^{10})$	$q+(0.707107 - 0.707107i) q^{5} -2.45331 q^{7} +(-0.168035 - 0.168035i) q^{11} +(2.45119 - 2.45119i) q^{13} +1.49677i q^{17} +(-1.78948 - 1.78948i) q^{19} -1.51548i q^{23} -1.00000i q^{25} +(-4.35565 - 4.35565i) q^{29} +3.96288i q^{31} +(-1.73475 + 1.73475i) q^{35} +(1.35596 + 1.35596i) q^{37} +2.13862 q^{41} +(-5.24501 + 5.24501i) q^{43} -7.46748 q^{47} -0.981264 q^{49} +(8.51466 - 8.51466i) q^{53} -0.237637 q^{55} +(0.862894 + 0.862894i) q^{59} +(-3.60904 + 3.60904i) q^{61} -3.46650i q^{65} +(-11.1375 - 11.1375i) q^{67} -10.5177i q^{71} +3.75145i q^{73} +(0.412242 + 0.412242i) q^{77} -8.09102i q^{79} +(-5.60586 + 5.60586i) q^{83} +(1.05838 + 1.05838i) q^{85} -3.16072 q^{89} +(-6.01353 + 6.01353i) q^{91} -2.53070 q^{95} -15.1791 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 8 q^{19} - 32 q^{49} + 16 q^{55} + 16 q^{61} - 16 q^{67} + 16 q^{85} + 16 q^{91}+O(q^{100}) $ 32 * q + 8 * q^19 - 32 * q^49 + 16 * q^55 + 16 * q^61 - 16 * q^67 + 16 * q^85 + 16 * q^91

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
$7$	−2.45331			−0.927265			−0.463632	−	0.886028i	$-0.653454\pi$
$7$	−2.45331			−0.927265			−0.463632	+	0.886028i	$0.653454\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.168035	−	0.168035i	−0.0506644	−	0.0506644i	0.681321	−	0.731985i	$-0.261406\pi$
$11$	−0.168035	−	0.168035i	−0.0506644	−	0.0506644i	−0.731985	+	0.681321i	$0.761406\pi$
$12$		0			0
$13$	2.45119	−	2.45119i	0.679837	−	0.679837i	−0.280126	−	0.959963i	$-0.590376\pi$
$13$	2.45119	−	2.45119i	0.679837	−	0.679837i	0.959963	+	0.280126i	$0.0903763\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.49677i			0.363020i	0.983389	+	0.181510i	$0.0580985\pi$
$17$			1.49677i			0.363020i	−0.983389	+	0.181510i	$0.941902\pi$
$18$		0			0
$19$	−1.78948	−	1.78948i	−0.410534	−	0.410534i	0.471391	−	0.881925i	$-0.343752\pi$
$19$	−1.78948	−	1.78948i	−0.410534	−	0.410534i	−0.881925	+	0.471391i	$0.843752\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.51548i		−	0.316000i	−0.987439	−	0.158000i	$-0.949495\pi$
$23$		−	1.51548i		−	0.316000i	0.987439	−	0.158000i	$-0.0505045\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.35565	−	4.35565i	−0.808823	−	0.808823i	0.175633	−	0.984456i	$-0.443803\pi$
$29$	−4.35565	−	4.35565i	−0.808823	−	0.808823i	−0.984456	+	0.175633i	$0.943803\pi$
$30$		0			0
$31$			3.96288i			0.711753i	0.934533	+	0.355877i	$0.115818\pi$
$31$			3.96288i			0.711753i	−0.934533	+	0.355877i	$0.884182\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.73475	+	1.73475i	−0.293227	+	0.293227i
$36$		0			0
$37$	1.35596	+	1.35596i	0.222918	+	0.222918i	0.809726	−	0.586808i	$-0.199616\pi$
$37$	1.35596	+	1.35596i	0.222918	+	0.222918i	−0.586808	+	0.809726i	$0.699616\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.13862			0.333997			0.166998	−	0.985957i	$-0.446593\pi$
$41$	2.13862			0.333997			0.166998	+	0.985957i	$0.446593\pi$
$42$		0			0
$43$	−5.24501	+	5.24501i	−0.799856	+	0.799856i	−0.983073	−	0.183217i	$-0.941349\pi$
$43$	−5.24501	+	5.24501i	−0.799856	+	0.799856i	0.183217	+	0.983073i	$0.441349\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.46748			−1.08924			−0.544622	−	0.838682i	$-0.683327\pi$
$47$	−7.46748			−1.08924			−0.544622	+	0.838682i	$0.683327\pi$
$48$		0			0
$49$	−0.981264			−0.140181
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8.51466	−	8.51466i	1.16958	−	1.16958i	0.187270	−	0.982309i	$-0.440036\pi$
$53$	8.51466	−	8.51466i	1.16958	−	1.16958i	0.982309	−	0.187270i	$-0.0599638\pi$
$54$		0			0
$55$	−0.237637			−0.0320430
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.862894	+	0.862894i	0.112339	+	0.112339i	0.761042	−	0.648703i	$-0.224688\pi$
$59$	0.862894	+	0.862894i	0.112339	+	0.112339i	−0.648703	+	0.761042i	$0.724688\pi$
$60$		0			0
$61$	−3.60904	+	3.60904i	−0.462091	+	0.462091i	−0.899340	−	0.437250i	$-0.855953\pi$
$61$	−3.60904	+	3.60904i	−0.462091	+	0.462091i	0.437250	+	0.899340i	$0.355953\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	3.46650i		−	0.429967i
$66$		0			0
$67$	−11.1375	−	11.1375i	−1.36067	−	1.36067i	−0.873068	−	0.487598i	$-0.837873\pi$
$67$	−11.1375	−	11.1375i	−1.36067	−	1.36067i	−0.487598	−	0.873068i	$-0.662127\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.5177i		−	1.24822i	−0.781338	−	0.624108i	$-0.785463\pi$
$71$		−	10.5177i		−	1.24822i	0.781338	−	0.624108i	$-0.214537\pi$
$72$		0			0
$73$			3.75145i			0.439074i	0.975604	+	0.219537i	$0.0704546\pi$
$73$			3.75145i			0.439074i	−0.975604	+	0.219537i	$0.929545\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.412242	+	0.412242i	0.0469793	+	0.0469793i
$78$		0			0
$79$		−	8.09102i		−	0.910311i	−0.890412	−	0.455155i	$-0.849584\pi$
$79$		−	8.09102i		−	0.910311i	0.890412	−	0.455155i	$-0.150416\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.60586	+	5.60586i	−0.615323	+	0.615323i	−0.944328	−	0.329005i	$-0.893287\pi$
$83$	−5.60586	+	5.60586i	−0.615323	+	0.615323i	0.329005	+	0.944328i	$0.393287\pi$
$84$		0			0
$85$	1.05838	+	1.05838i	0.114797	+	0.114797i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.16072			−0.335036			−0.167518	−	0.985869i	$-0.553575\pi$
$89$	−3.16072			−0.335036			−0.167518	+	0.985869i	$0.553575\pi$
$90$		0			0
$91$	−6.01353	+	6.01353i	−0.630389	+	0.630389i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.53070			−0.259644
$96$		0			0
$97$	−15.1791			−1.54120			−0.770602	−	0.637317i	$-0.780044\pi$
$97$	−15.1791			−1.54120			−0.770602	+	0.637317i	$0.780044\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.6318	−	11.6318i	1.15741	−	1.15741i	0.172381	−	0.985030i	$-0.444854\pi$
$101$	11.6318	−	11.6318i	1.15741	−	1.15741i	0.985030	−	0.172381i	$-0.0551461\pi$
$102$		0			0
$103$	−15.7995			−1.55678			−0.778388	−	0.627784i	$-0.783962\pi$
$103$	−15.7995			−1.55678			−0.778388	+	0.627784i	$0.783962\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.75829	+	7.75829i	0.750022	+	0.750022i	0.974483	−	0.224461i	$-0.0720621\pi$
$107$	7.75829	+	7.75829i	0.750022	+	0.750022i	−0.224461	+	0.974483i	$0.572062\pi$
$108$		0			0
$109$	−11.3270	+	11.3270i	−1.08493	+	1.08493i	−0.0888843	+	0.996042i	$0.528330\pi$
$109$	−11.3270	+	11.3270i	−1.08493	+	1.08493i	−0.996042	+	0.0888843i	$0.971670\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	9.93241i		−	0.934362i	−0.884162	−	0.467181i	$-0.845270\pi$
$113$		−	9.93241i		−	0.934362i	0.884162	−	0.467181i	$-0.154730\pi$
$114$		0			0
$115$	−1.07161	−	1.07161i	−0.0999279	−	0.0999279i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	3.67204i		−	0.336615i
$120$		0			0
$121$		−	10.9435i		−	0.994866i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	−	0.707107i	−0.0632456	−	0.0632456i
$126$		0			0
$127$		−	4.40291i		−	0.390695i	−0.980734	−	0.195347i	$-0.937417\pi$
$127$		−	4.40291i		−	0.390695i	0.980734	−	0.195347i	$-0.0625835\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.10481	−	5.10481i	0.446009	−	0.446009i	−0.448016	−	0.894025i	$-0.647869\pi$
$131$	5.10481	−	5.10481i	0.446009	−	0.446009i	0.894025	+	0.448016i	$0.147869\pi$
$132$		0			0
$133$	4.39014	+	4.39014i	0.380674	+	0.380674i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−20.9577			−1.79054			−0.895270	−	0.445524i	$-0.853017\pi$
$137$	−20.9577			−1.79054			−0.895270	+	0.445524i	$0.853017\pi$
$138$		0			0
$139$	−9.07493	+	9.07493i	−0.769725	+	0.769725i	−0.978058	−	0.208333i	$-0.933196\pi$
$139$	−9.07493	+	9.07493i	−0.769725	+	0.769725i	0.208333	+	0.978058i	$0.433196\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.823771			−0.0688872
$144$		0			0
$145$	−6.15981			−0.511545
$146$		0			0
$147$		0			0
$148$		0			0
$149$	11.9653	−	11.9653i	0.980237	−	0.980237i	−0.0195714	−	0.999808i	$-0.506230\pi$
$149$	11.9653	−	11.9653i	0.980237	−	0.980237i	0.999808	+	0.0195714i	$0.00623016\pi$
$150$		0			0
$151$	8.00035			0.651059			0.325530	−	0.945532i	$-0.394457\pi$
$151$	8.00035			0.651059			0.325530	+	0.945532i	$0.394457\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.80218	+	2.80218i	0.225076	+	0.225076i
$156$		0			0
$157$	−6.16568	+	6.16568i	−0.492075	+	0.492075i	−0.908959	−	0.416885i	$-0.863122\pi$
$157$	−6.16568	+	6.16568i	−0.492075	+	0.492075i	0.416885	+	0.908959i	$0.363122\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			3.71795i			0.293015i
$162$		0			0
$163$	1.70650	+	1.70650i	0.133664	+	0.133664i	0.770773	−	0.637110i	$-0.219870\pi$
$163$	1.70650	+	1.70650i	0.133664	+	0.133664i	−0.637110	+	0.770773i	$0.719870\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	8.51906i		−	0.659225i	−0.944116	−	0.329612i	$-0.893082\pi$
$167$		−	8.51906i		−	0.659225i	0.944116	−	0.329612i	$-0.106918\pi$
$168$		0			0
$169$			0.983355i			0.0756427i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.187551	+	0.187551i	0.0142592	+	0.0142592i	0.714200	−	0.699941i	$-0.246790\pi$
$173$	0.187551	+	0.187551i	0.0142592	+	0.0142592i	−0.699941	+	0.714200i	$0.746790\pi$
$174$		0			0
$175$			2.45331i			0.185453i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.1359	−	12.1359i	0.907083	−	0.907083i	−0.0889526	−	0.996036i	$-0.528352\pi$
$179$	12.1359	−	12.1359i	0.907083	−	0.907083i	0.996036	+	0.0889526i	$0.0283520\pi$
$180$		0			0
$181$	−11.1066	−	11.1066i	−0.825545	−	0.825545i	0.161352	−	0.986897i	$-0.448415\pi$
$181$	−11.1066	−	11.1066i	−0.825545	−	0.825545i	−0.986897	+	0.161352i	$0.948415\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.91761			0.140986
$186$		0			0
$187$	0.251510	−	0.251510i	0.0183922	−	0.0183922i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−15.6785			−1.13446			−0.567228	−	0.823560i	$-0.691984\pi$
$191$	−15.6785			−1.13446			−0.567228	+	0.823560i	$0.691984\pi$
$192$		0			0
$193$	−1.97072			−0.141855			−0.0709277	−	0.997481i	$-0.522596\pi$
$193$	−1.97072			−0.141855			−0.0709277	+	0.997481i	$0.522596\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.4333	+	15.4333i	−1.09958	+	1.09958i	−0.105117	+	0.994460i	$0.533522\pi$
$197$	−15.4333	+	15.4333i	−1.09958	+	1.09958i	−0.994460	+	0.105117i	$0.966478\pi$
$198$		0			0
$199$	20.6569			1.46433			0.732165	−	0.681127i	$-0.238510\pi$
$199$	20.6569			1.46433			0.732165	+	0.681127i	$0.238510\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.6858	+	10.6858i	0.749993	+	0.749993i
$204$		0			0
$205$	1.51223	−	1.51223i	0.105619	−	0.105619i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			0.601389i			0.0415990i
$210$		0			0
$211$	17.1103	+	17.1103i	1.17792	+	1.17792i	0.980274	+	0.197646i	$0.0633295\pi$
$211$	17.1103	+	17.1103i	1.17792	+	1.17792i	0.197646	+	0.980274i	$0.436670\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			7.41756i			0.505873i
$216$		0			0
$217$		−	9.72217i		−	0.659984i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.66886	+	3.66886i	0.246794	+	0.246794i
$222$		0			0
$223$		−	23.6260i		−	1.58211i	−0.611744	−	0.791055i	$-0.709532\pi$
$223$		−	23.6260i		−	1.58211i	0.611744	−	0.791055i	$-0.290468\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.193086	−	0.193086i	0.0128156	−	0.0128156i	−0.700670	−	0.713486i	$-0.747115\pi$
$227$	0.193086	−	0.193086i	0.0128156	−	0.0128156i	0.713486	+	0.700670i	$0.247115\pi$
$228$		0			0
$229$	13.5261	+	13.5261i	0.893832	+	0.893832i	0.994881	−	0.101050i	$-0.0322201\pi$
$229$	13.5261	+	13.5261i	0.893832	+	0.893832i	−0.101050	+	0.994881i	$0.532220\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−21.1553			−1.38593			−0.692967	−	0.720970i	$-0.743697\pi$
$233$	−21.1553			−1.38593			−0.692967	+	0.720970i	$0.743697\pi$
$234$		0			0
$235$	−5.28030	+	5.28030i	−0.344449	+	0.344449i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.1124			0.783484			0.391742	−	0.920075i	$-0.371873\pi$
$239$	12.1124			0.783484			0.391742	+	0.920075i	$0.371873\pi$
$240$		0			0
$241$	−28.2819			−1.82180			−0.910898	−	0.412631i	$-0.864610\pi$
$241$	−28.2819			−1.82180			−0.910898	+	0.412631i	$0.864610\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.693858	+	0.693858i	−0.0443290	+	0.0443290i
$246$		0			0
$247$	−8.77268			−0.558193
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.77156	−	2.77156i	−0.174940	−	0.174940i	0.614206	−	0.789146i	$-0.289476\pi$
$251$	−2.77156	−	2.77156i	−0.174940	−	0.174940i	−0.789146	+	0.614206i	$0.789476\pi$
$252$		0			0
$253$	−0.254654	+	0.254654i	−0.0160099	+	0.0160099i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			16.9820i			1.05931i	0.848215	+	0.529653i	$0.177678\pi$
$257$			16.9820i			1.05931i	−0.848215	+	0.529653i	$0.822322\pi$
$258$		0			0
$259$	−3.32658	−	3.32658i	−0.206704	−	0.206704i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	24.0141i		−	1.48077i	−0.672183	−	0.740385i	$-0.734643\pi$
$263$		−	24.0141i		−	1.48077i	0.672183	−	0.740385i	$-0.265357\pi$
$264$		0			0
$265$		−	12.0415i		−	0.739706i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.39927	−	2.39927i	−0.146286	−	0.146286i	0.630171	−	0.776457i	$-0.282985\pi$
$269$	−2.39927	−	2.39927i	−0.146286	−	0.146286i	−0.776457	+	0.630171i	$0.782985\pi$
$270$		0			0
$271$			11.4721i			0.696883i	0.937331	+	0.348441i	$0.113289\pi$
$271$			11.4721i			0.696883i	−0.937331	+	0.348441i	$0.886711\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.168035	+	0.168035i	−0.0101329	+	0.0101329i
$276$		0			0
$277$	5.80179	+	5.80179i	0.348596	+	0.348596i	0.859586	−	0.510990i	$-0.170721\pi$
$277$	5.80179	+	5.80179i	0.348596	+	0.348596i	−0.510990	+	0.859586i	$0.670721\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.56111			−0.331748			−0.165874	−	0.986147i	$-0.553044\pi$
$281$	−5.56111			−0.331748			−0.165874	+	0.986147i	$0.553044\pi$
$282$		0			0
$283$	−0.938908	+	0.938908i	−0.0558123	+	0.0558123i	−0.734462	−	0.678650i	$-0.762565\pi$
$283$	−0.938908	+	0.938908i	−0.0558123	+	0.0558123i	0.678650	+	0.734462i	$0.262565\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.24671			−0.309703
$288$		0			0
$289$	14.7597			0.868217
$290$		0			0
$291$		0			0
$292$		0			0
$293$	15.2476	−	15.2476i	0.890775	−	0.890775i	−0.103821	−	0.994596i	$-0.533107\pi$
$293$	15.2476	−	15.2476i	0.890775	−	0.890775i	0.994596	+	0.103821i	$0.0331069\pi$
$294$		0			0
$295$	1.22032			0.0710496
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.71473	−	3.71473i	−0.214828	−	0.214828i
$300$		0			0
$301$	12.8676	−	12.8676i	0.741678	−	0.741678i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			5.10396i			0.292252i
$306$		0			0
$307$	−7.90072	−	7.90072i	−0.450918	−	0.450918i	0.444741	−	0.895659i	$-0.353296\pi$
$307$	−7.90072	−	7.90072i	−0.450918	−	0.450918i	−0.895659	+	0.444741i	$0.853296\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	14.3378i		−	0.813022i	−0.913646	−	0.406511i	$-0.866745\pi$
$311$		−	14.3378i		−	0.813022i	0.913646	−	0.406511i	$-0.133255\pi$
$312$		0			0
$313$			11.3486i			0.641459i	0.947171	+	0.320730i	$0.103928\pi$
$313$			11.3486i			0.641459i	−0.947171	+	0.320730i	$0.896072\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.01582	+	6.01582i	0.337882	+	0.337882i	0.855570	−	0.517688i	$-0.173207\pi$
$317$	6.01582	+	6.01582i	0.337882	+	0.337882i	−0.517688	+	0.855570i	$0.673207\pi$
$318$		0			0
$319$			1.46380i			0.0819571i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.67843	−	2.67843i	0.149032	−	0.149032i
$324$		0			0
$325$	−2.45119	−	2.45119i	−0.135967	−	0.135967i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	18.3200			1.01002
$330$		0			0
$331$	−2.57633	+	2.57633i	−0.141608	+	0.141608i	−0.774357	−	0.632749i	$-0.781926\pi$
$331$	−2.57633	+	2.57633i	−0.141608	+	0.141608i	0.632749	+	0.774357i	$0.281926\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−15.7509			−0.860561
$336$		0			0
$337$	−18.1585			−0.989159			−0.494579	−	0.869132i	$-0.664678\pi$
$337$	−18.1585			−0.989159			−0.494579	+	0.869132i	$0.664678\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.665902	−	0.665902i	0.0360606	−	0.0360606i
$342$		0			0
$343$	19.5805			1.05725
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.30325	+	4.30325i	0.231011	+	0.231011i	0.813114	−	0.582104i	$-0.197770\pi$
$347$	4.30325	+	4.30325i	0.231011	+	0.231011i	−0.582104	+	0.813114i	$0.697770\pi$
$348$		0			0
$349$	−7.17028	+	7.17028i	−0.383816	+	0.383816i	−0.872475	−	0.488659i	$-0.837486\pi$
$349$	−7.17028	+	7.17028i	−0.383816	+	0.383816i	0.488659	+	0.872475i	$0.337486\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			11.4140i			0.607507i	0.952751	+	0.303753i	$0.0982399\pi$
$353$			11.4140i			0.607507i	−0.952751	+	0.303753i	$0.901760\pi$
$354$		0			0
$355$	−7.43710	−	7.43710i	−0.394720	−	0.394720i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	33.8939i		−	1.78885i	−0.447218	−	0.894425i	$-0.647585\pi$
$359$		−	33.8939i		−	1.78885i	0.447218	−	0.894425i	$-0.352415\pi$
$360$		0			0
$361$		−	12.5956i		−	0.662924i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.65267	+	2.65267i	0.138847	+	0.138847i
$366$		0			0
$367$		−	19.1553i		−	0.999898i	−0.866055	−	0.499949i	$-0.833352\pi$
$367$		−	19.1553i		−	0.999898i	0.866055	−	0.499949i	$-0.166648\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−20.8891	+	20.8891i	−1.08451	+	1.08451i
$372$		0			0
$373$	−0.0421542	−	0.0421542i	−0.00218266	−	0.00218266i	0.706015	−	0.708197i	$-0.250491\pi$
$373$	−0.0421542	−	0.0421542i	−0.00218266	−	0.00218266i	−0.708197	+	0.706015i	$0.750491\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−21.3530			−1.09974
$378$		0			0
$379$	−21.9336	+	21.9336i	−1.12665	+	1.12665i	−0.135934	+	0.990718i	$0.543403\pi$
$379$	−21.9336	+	21.9336i	−1.12665	+	1.12665i	−0.990718	+	0.135934i	$0.956597\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.21717			0.317683			0.158841	−	0.987304i	$-0.449224\pi$
$383$	6.21717			0.317683			0.158841	+	0.987304i	$0.449224\pi$
$384$		0			0
$385$	0.582998			0.0297123
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.56242	−	3.56242i	0.180622	−	0.180622i	−0.611005	−	0.791627i	$-0.709234\pi$
$389$	3.56242	−	3.56242i	0.180622	−	0.180622i	0.791627	+	0.611005i	$0.209234\pi$
$390$		0			0
$391$	2.26833			0.114714
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−5.72122	−	5.72122i	−0.287866	−	0.287866i
$396$		0			0
$397$	6.98826	−	6.98826i	0.350731	−	0.350731i	−0.509651	−	0.860381i	$-0.670225\pi$
$397$	6.98826	−	6.98826i	0.350731	−	0.350731i	0.860381	+	0.509651i	$0.170225\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			12.7576i			0.637082i	0.947909	+	0.318541i	$0.103193\pi$
$401$			12.7576i			0.637082i	−0.947909	+	0.318541i	$0.896807\pi$
$402$		0			0
$403$	9.71375	+	9.71375i	0.483876	+	0.483876i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	0.455696i		−	0.0225880i
$408$		0			0
$409$		−	32.9666i		−	1.63009i	−0.579396	−	0.815047i	$-0.696711\pi$
$409$		−	32.9666i		−	1.63009i	0.579396	−	0.815047i	$-0.303289\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.11695	−	2.11695i	−0.104168	−	0.104168i
$414$		0			0
$415$			7.92789i			0.389165i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	19.4628	−	19.4628i	0.950819	−	0.950819i	−0.0480275	−	0.998846i	$-0.515293\pi$
$419$	19.4628	−	19.4628i	0.950819	−	0.950819i	0.998846	+	0.0480275i	$0.0152935\pi$
$420$		0			0
$421$	−0.687352	−	0.687352i	−0.0334995	−	0.0334995i	0.690159	−	0.723658i	$-0.257541\pi$
$421$	−0.687352	−	0.687352i	−0.0334995	−	0.0334995i	−0.723658	+	0.690159i	$0.757541\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.49677			0.0726040
$426$		0			0
$427$	8.85411	−	8.85411i	0.428480	−	0.428480i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.56505			0.0753857			0.0376929	−	0.999289i	$-0.487999\pi$
$431$	1.56505			0.0753857			0.0376929	+	0.999289i	$0.487999\pi$
$432$		0			0
$433$	11.6062			0.557758			0.278879	−	0.960326i	$-0.410037\pi$
$433$	11.6062			0.557758			0.278879	+	0.960326i	$0.410037\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.71192	+	2.71192i	−0.129729	+	0.129729i
$438$		0			0
$439$	28.4745			1.35901			0.679507	−	0.733669i	$-0.262194\pi$
$439$	28.4745			1.35901			0.679507	+	0.733669i	$0.262194\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7.07278	+	7.07278i	0.336038	+	0.336038i	0.854874	−	0.518836i	$-0.173634\pi$
$443$	7.07278	+	7.07278i	0.336038	+	0.336038i	−0.518836	+	0.854874i	$0.673634\pi$
$444$		0			0
$445$	−2.23497	+	2.23497i	−0.105948	+	0.105948i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			15.5030i			0.731631i	0.930687	+	0.365816i	$0.119210\pi$
$449$			15.5030i			0.731631i	−0.930687	+	0.365816i	$0.880790\pi$
$450$		0			0
$451$	−0.359363	−	0.359363i	−0.0169218	−	0.0169218i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			8.50441i			0.398693i
$456$		0			0
$457$			19.8752i			0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$457$			19.8752i			0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	21.7593	+	21.7593i	1.01343	+	1.01343i	0.999909	+	0.0135210i	$0.00430398\pi$
$461$	21.7593	+	21.7593i	1.01343	+	1.01343i	0.0135210	+	0.999909i	$0.495696\pi$
$462$		0			0
$463$			34.3970i			1.59856i	0.600956	+	0.799282i	$0.294787\pi$
$463$			34.3970i			1.59856i	−0.600956	+	0.799282i	$0.705213\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−13.1620	+	13.1620i	−0.609066	+	0.609066i	−0.942702	−	0.333636i	$-0.891724\pi$
$467$	−13.1620	+	13.1620i	−0.609066	+	0.609066i	0.333636	+	0.942702i	$0.391724\pi$
$468$		0			0
$469$	27.3238	+	27.3238i	1.26170	+	1.26170i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.76269			0.0810485
$474$		0			0
$475$	−1.78948	+	1.78948i	−0.0821068	+	0.0821068i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.4850			0.616145			0.308073	−	0.951363i	$-0.400316\pi$
$479$	13.4850			0.616145			0.308073	+	0.951363i	$0.400316\pi$
$480$		0			0
$481$	6.64740			0.303096
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.7332	+	10.7332i	−0.487371	+	0.487371i
$486$		0			0
$487$	−8.27021			−0.374759			−0.187380	−	0.982288i	$-0.559999\pi$
$487$	−8.27021			−0.374759			−0.187380	+	0.982288i	$0.559999\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	22.7552	+	22.7552i	1.02693	+	1.02693i	0.999627	+	0.0272984i	$0.00869042\pi$
$491$	22.7552	+	22.7552i	1.02693	+	1.02693i	0.0272984	+	0.999627i	$0.491310\pi$
$492$		0			0
$493$	6.51940	−	6.51940i	0.293619	−	0.293619i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			25.8031i			1.15743i
$498$		0			0
$499$	7.23487	+	7.23487i	0.323877	+	0.323877i	0.850253	−	0.526375i	$-0.176449\pi$
$499$	7.23487	+	7.23487i	0.323877	+	0.323877i	−0.526375	+	0.850253i	$0.676449\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	6.69336i		−	0.298442i	−0.988804	−	0.149221i	$-0.952323\pi$
$503$		−	6.69336i		−	0.298442i	0.988804	−	0.149221i	$-0.0476766\pi$
$504$		0			0
$505$		−	16.4499i		−	0.732011i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.25890	−	7.25890i	−0.321745	−	0.321745i	0.527691	−	0.849436i	$-0.323058\pi$
$509$	−7.25890	−	7.25890i	−0.321745	−	0.321745i	−0.849436	+	0.527691i	$0.823058\pi$
$510$		0			0
$511$		−	9.20347i		−	0.407138i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−11.1720	+	11.1720i	−0.492296	+	0.492296i
$516$		0			0
$517$	1.25480	+	1.25480i	0.0551859	+	0.0551859i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−22.9907			−1.00724			−0.503621	−	0.863925i	$-0.667999\pi$
$521$	−22.9907			−1.00724			−0.503621	+	0.863925i	$0.667999\pi$
$522$		0			0
$523$	20.5585	−	20.5585i	0.898960	−	0.898960i	−0.0963841	−	0.995344i	$-0.530728\pi$
$523$	20.5585	−	20.5585i	0.898960	−	0.898960i	0.995344	+	0.0963841i	$0.0307277\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5.93151			−0.258381
$528$		0			0
$529$	20.7033			0.900144
$530$		0			0
$531$		0			0
$532$		0			0
$533$	5.24216	−	5.24216i	0.227063	−	0.227063i
$534$		0			0
$535$	10.9719			0.474356
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.164887	+	0.164887i	0.00710217	+	0.00710217i
$540$		0			0
$541$	13.8746	−	13.8746i	0.596515	−	0.596515i	−0.342869	−	0.939383i	$-0.611399\pi$
$541$	13.8746	−	13.8746i	0.596515	−	0.596515i	0.939383	+	0.342869i	$0.111399\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			16.0187i			0.686168i
$546$		0			0
$547$	−7.71736	−	7.71736i	−0.329971	−	0.329971i	0.522605	−	0.852575i	$-0.324960\pi$
$547$	−7.71736	−	7.71736i	−0.329971	−	0.329971i	−0.852575	+	0.522605i	$0.824960\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			15.5886i			0.664099i
$552$		0			0
$553$			19.8498i			0.844099i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−13.5541	−	13.5541i	−0.574305	−	0.574305i	0.359023	−	0.933329i	$-0.383110\pi$
$557$	−13.5541	−	13.5541i	−0.574305	−	0.574305i	−0.933329	+	0.359023i	$0.883110\pi$
$558$		0			0
$559$			25.7130i			1.08754i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.05619	−	7.05619i	0.297383	−	0.297383i	−0.542605	−	0.839988i	$-0.682562\pi$
$563$	7.05619	−	7.05619i	0.297383	−	0.297383i	0.839988	+	0.542605i	$0.182562\pi$
$564$		0			0
$565$	−7.02327	−	7.02327i	−0.295471	−	0.295471i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.4429			0.647402			0.323701	−	0.946160i	$-0.395073\pi$
$569$	15.4429			0.647402			0.323701	+	0.946160i	$0.395073\pi$
$570$		0			0
$571$	2.83841	−	2.83841i	0.118784	−	0.118784i	−0.645216	−	0.764000i	$-0.723233\pi$
$571$	2.83841	−	2.83841i	0.118784	−	0.118784i	0.764000	+	0.645216i	$0.223233\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.51548			−0.0631999
$576$		0			0
$577$	25.4978			1.06149			0.530745	−	0.847532i	$-0.321912\pi$
$577$	25.4978			1.06149			0.530745	+	0.847532i	$0.321912\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	13.7529	−	13.7529i	0.570568	−	0.570568i
$582$		0			0
$583$	−2.86152			−0.118512
$584$		0			0
$585$		0			0
$586$		0			0
$587$	21.0766	+	21.0766i	0.869926	+	0.869926i	0.992464	−	0.122538i	$-0.0391032\pi$
$587$	21.0766	+	21.0766i	0.869926	+	0.869926i	−0.122538	+	0.992464i	$0.539103\pi$
$588$		0			0
$589$	7.09147	−	7.09147i	0.292199	−	0.292199i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			0.273224i			0.0112200i	0.999984	+	0.00560998i	$0.00178572\pi$
$593$			0.273224i			0.0112200i	−0.999984	+	0.00560998i	$0.998214\pi$
$594$		0			0
$595$	−2.59653	−	2.59653i	−0.106447	−	0.106447i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			47.4914i			1.94045i	0.242213	+	0.970223i	$0.422127\pi$
$599$			47.4914i			1.94045i	−0.242213	+	0.970223i	$0.577873\pi$
$600$		0			0
$601$			28.3934i			1.15819i	0.815259	+	0.579096i	$0.196594\pi$
$601$			28.3934i			1.15819i	−0.815259	+	0.579096i	$0.803406\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.73824	−	7.73824i	−0.314604	−	0.314604i
$606$		0			0
$607$		−	44.2186i		−	1.79478i	−0.441242	−	0.897388i	$-0.645462\pi$
$607$		−	44.2186i		−	1.79478i	0.441242	−	0.897388i	$-0.354538\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−18.3042	+	18.3042i	−0.740508	+	0.740508i
$612$		0			0
$613$	−7.06668	−	7.06668i	−0.285421	−	0.285421i	0.549846	−	0.835266i	$-0.314687\pi$
$613$	−7.06668	−	7.06668i	−0.285421	−	0.285421i	−0.835266	+	0.549846i	$0.814687\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.63261			0.307277			0.153639	−	0.988127i	$-0.450901\pi$
$617$	7.63261			0.307277			0.153639	+	0.988127i	$0.450901\pi$
$618$		0			0
$619$	29.8806	−	29.8806i	1.20100	−	1.20100i	0.227143	−	0.973862i	$-0.427062\pi$
$619$	29.8806	−	29.8806i	1.20100	−	1.20100i	0.973862	−	0.227143i	$-0.0729383\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	7.75423			0.310667
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.02955	+	2.02955i	−0.0809236	+	0.0809236i
$630$		0			0
$631$	−3.65726			−0.145593			−0.0727966	−	0.997347i	$-0.523192\pi$
$631$	−3.65726			−0.145593			−0.0727966	+	0.997347i	$0.523192\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.11333	−	3.11333i	−0.123549	−	0.123549i
$636$		0			0
$637$	−2.40526	+	2.40526i	−0.0952999	+	0.0952999i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	36.6442i		−	1.44736i	−0.690135	−	0.723680i	$-0.742449\pi$
$641$		−	36.6442i		−	1.44736i	0.690135	−	0.723680i	$-0.257551\pi$
$642$		0			0
$643$	26.0326	+	26.0326i	1.02663	+	1.02663i	0.999636	+	0.0269896i	$0.00859211\pi$
$643$	26.0326	+	26.0326i	1.02663	+	1.02663i	0.0269896	+	0.999636i	$0.491408\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			47.1530i			1.85377i	0.375340	+	0.926887i	$0.377526\pi$
$647$			47.1530i			1.85377i	−0.375340	+	0.926887i	$0.622474\pi$
$648$		0			0
$649$		−	0.289993i		−	0.0113832i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	28.1688	+	28.1688i	1.10233	+	1.10233i	0.994129	+	0.108201i	$0.0345090\pi$
$653$	28.1688	+	28.1688i	1.10233	+	1.10233i	0.108201	+	0.994129i	$0.465491\pi$
$654$		0			0
$655$		−	7.21929i		−	0.282081i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	17.6993	−	17.6993i	0.689468	−	0.689468i	−0.272647	−	0.962114i	$-0.587899\pi$
$659$	17.6993	−	17.6993i	0.689468	−	0.689468i	0.962114	+	0.272647i	$0.0878990\pi$
$660$		0			0
$661$	−0.360305	−	0.360305i	−0.0140142	−	0.0140142i	0.700065	−	0.714079i	$-0.253154\pi$
$661$	−0.360305	−	0.360305i	−0.0140142	−	0.0140142i	−0.714079	+	0.700065i	$0.753154\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.20860			0.240759
$666$		0			0
$667$	−6.60090	+	6.60090i	−0.255588	+	0.255588i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.21289			0.0468231
$672$		0			0
$673$	−5.27511			−0.203340			−0.101670	−	0.994818i	$-0.532419\pi$
$673$	−5.27511			−0.203340			−0.101670	+	0.994818i	$0.532419\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12.2360	+	12.2360i	−0.470269	+	0.470269i	−0.902002	−	0.431733i	$-0.857902\pi$
$677$	−12.2360	+	12.2360i	−0.470269	+	0.470269i	0.431733	+	0.902002i	$0.357902\pi$
$678$		0			0
$679$	37.2390			1.42910
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15.8714	+	15.8714i	0.607302	+	0.607302i	0.942240	−	0.334938i	$-0.108715\pi$
$683$	15.8714	+	15.8714i	0.607302	+	0.607302i	−0.334938	+	0.942240i	$0.608715\pi$
$684$		0			0
$685$	−14.8194	+	14.8194i	−0.566219	+	0.566219i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	41.7421i		−	1.59025i
$690$		0			0
$691$	19.6017	+	19.6017i	0.745685	+	0.745685i	0.973666	−	0.227980i	$-0.0732122\pi$
$691$	19.6017	+	19.6017i	0.745685	+	0.745685i	−0.227980	+	0.973666i	$0.573212\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			12.8339i			0.486817i
$696$		0			0
$697$			3.20102i			0.121247i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−11.8100	−	11.8100i	−0.446058	−	0.446058i	0.447984	−	0.894042i	$-0.352142\pi$
$701$	−11.8100	−	11.8100i	−0.446058	−	0.446058i	−0.894042	+	0.447984i	$0.852142\pi$
$702$		0			0
$703$		−	4.85290i		−	0.183031i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−28.5365	+	28.5365i	−1.07323	+	1.07323i
$708$		0			0
$709$	29.2369	+	29.2369i	1.09802	+	1.09802i	0.994643	+	0.103373i	$0.0329637\pi$
$709$	29.2369	+	29.2369i	1.09802	+	1.09802i	0.103373	+	0.994643i	$0.467036\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.00566			0.224914
$714$		0			0
$715$	−0.582494	+	0.582494i	−0.0217840	+	0.0217840i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−38.2600			−1.42686			−0.713428	−	0.700728i	$-0.752859\pi$
$719$	−38.2600			−1.42686			−0.713428	+	0.700728i	$0.752859\pi$
$720$		0			0
$721$	38.7612			1.44354
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.35565	+	4.35565i	−0.161765	+	0.161765i
$726$		0			0
$727$	−0.854051			−0.0316750			−0.0158375	−	0.999875i	$-0.505041\pi$
$727$	−0.854051			−0.0316750			−0.0158375	+	0.999875i	$0.505041\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.85057	−	7.85057i	−0.290364	−	0.290364i
$732$		0			0
$733$	17.4638	−	17.4638i	0.645041	−	0.645041i	−0.306749	−	0.951790i	$-0.599241\pi$
$733$	17.4638	−	17.4638i	0.645041	−	0.645041i	0.951790	+	0.306749i	$0.0992413\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			3.74299i			0.137875i
$738$		0			0
$739$	−33.1490	−	33.1490i	−1.21941	−	1.21941i	−0.967840	−	0.251568i	$-0.919054\pi$
$739$	−33.1490	−	33.1490i	−1.21941	−	1.21941i	−0.251568	−	0.967840i	$-0.580946\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	17.9085i		−	0.657000i	−0.944504	−	0.328500i	$-0.893457\pi$
$743$		−	17.9085i		−	0.657000i	0.944504	−	0.328500i	$-0.106543\pi$
$744$		0			0
$745$		−	16.9215i		−	0.619956i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−19.0335	−	19.0335i	−0.695469	−	0.695469i
$750$		0			0
$751$		−	2.89154i		−	0.105514i	−0.998607	−	0.0527569i	$-0.983199\pi$
$751$		−	2.89154i		−	0.105514i	0.998607	−	0.0527569i	$-0.0168009\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.65710	−	5.65710i	0.205883	−	0.205883i
$756$		0			0
$757$	−19.2383	−	19.2383i	−0.699227	−	0.699227i	0.265017	−	0.964244i	$-0.414622\pi$
$757$	−19.2383	−	19.2383i	−0.699227	−	0.699227i	−0.964244	+	0.265017i	$0.914622\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	40.8693			1.48151			0.740756	−	0.671775i	$-0.234468\pi$
$761$	40.8693			1.48151			0.740756	+	0.671775i	$0.234468\pi$
$762$		0			0
$763$	27.7886	−	27.7886i	1.00601	−	1.00601i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.23023			0.152745
$768$		0			0
$769$	10.7254			0.386768			0.193384	−	0.981123i	$-0.438054\pi$
$769$	10.7254			0.386768			0.193384	+	0.981123i	$0.438054\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.76617	+	4.76617i	−0.171427	+	0.171427i	−0.787606	−	0.616179i	$-0.788680\pi$
$773$	−4.76617	+	4.76617i	−0.171427	+	0.171427i	0.616179	+	0.787606i	$0.288680\pi$
$774$		0			0
$775$	3.96288			0.142351
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.82701	−	3.82701i	−0.137117	−	0.137117i
$780$		0			0
$781$	−1.76733	+	1.76733i	−0.0632402	+	0.0632402i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			8.71958i			0.311215i
$786$		0			0
$787$	5.31791	+	5.31791i	0.189563	+	0.189563i	0.795507	−	0.605944i	$-0.207204\pi$
$787$	5.31791	+	5.31791i	0.189563	+	0.189563i	−0.605944	+	0.795507i	$0.707204\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			24.3673i			0.866401i
$792$		0			0
$793$			17.6929i			0.628293i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21.5582	+	21.5582i	0.763631	+	0.763631i	0.976977	−	0.213345i	$-0.0684360\pi$
$797$	21.5582	+	21.5582i	0.763631	+	0.763631i	−0.213345	+	0.976977i	$0.568436\pi$
$798$		0			0
$799$		−	11.1771i		−	0.395417i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.630374	−	0.630374i	0.0222454	−	0.0222454i
$804$		0			0
$805$	2.62899	+	2.62899i	0.0926596	+	0.0926596i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	40.3465			1.41851			0.709255	−	0.704952i	$-0.249032\pi$
$809$	40.3465			1.41851			0.709255	+	0.704952i	$0.249032\pi$
$810$		0			0
$811$	19.5855	−	19.5855i	0.687739	−	0.687739i	−0.273993	−	0.961732i	$-0.588344\pi$
$811$	19.5855	−	19.5855i	0.687739	−	0.687739i	0.961732	+	0.273993i	$0.0883445\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.41336			0.0845362
$816$		0			0
$817$	18.7716			0.656736
$818$		0			0
$819$		0			0
$820$		0			0
$821$	4.31714	−	4.31714i	0.150669	−	0.150669i	−0.627748	−	0.778417i	$-0.716023\pi$
$821$	4.31714	−	4.31714i	0.150669	−	0.150669i	0.778417	+	0.627748i	$0.216023\pi$
$822$		0			0
$823$	−28.2310			−0.984071			−0.492036	−	0.870575i	$-0.663747\pi$
$823$	−28.2310			−0.984071			−0.492036	+	0.870575i	$0.663747\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−30.6163	−	30.6163i	−1.06463	−	1.06463i	−0.997762	−	0.0668724i	$-0.978698\pi$
$827$	−30.6163	−	30.6163i	−1.06463	−	1.06463i	−0.0668724	−	0.997762i	$-0.521302\pi$
$828$		0			0
$829$	−29.0397	+	29.0397i	−1.00859	+	1.00859i	−0.00862935	+	0.999963i	$0.502747\pi$
$829$	−29.0397	+	29.0397i	−1.00859	+	1.00859i	−0.999963	+	0.00862935i	$0.997253\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	1.46873i		−	0.0508883i
$834$		0			0
$835$	−6.02389	−	6.02389i	−0.208465	−	0.208465i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			25.5700i			0.882775i	0.897317	+	0.441387i	$0.145513\pi$
$839$			25.5700i			0.882775i	−0.897317	+	0.441387i	$0.854487\pi$
$840$		0			0
$841$			8.94329i			0.308389i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.695337	+	0.695337i	0.0239203	+	0.0239203i
$846$		0			0
$847$			26.8479i			0.922504i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.05493	−	2.05493i	0.0704419	−	0.0704419i
$852$		0			0
$853$	3.76094	+	3.76094i	0.128772	+	0.128772i	0.768555	−	0.639783i	$-0.220976\pi$
$853$	3.76094	+	3.76094i	0.128772	+	0.128772i	−0.639783	+	0.768555i	$0.720976\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−51.1346			−1.74672			−0.873362	−	0.487071i	$-0.838065\pi$
$857$	−51.1346			−1.74672			−0.873362	+	0.487071i	$0.838065\pi$
$858$		0			0
$859$	−18.6138	+	18.6138i	−0.635094	+	0.635094i	−0.949341	−	0.314247i	$-0.898248\pi$
$859$	−18.6138	+	18.6138i	−0.635094	+	0.635094i	0.314247	+	0.949341i	$0.398248\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−7.40109			−0.251936			−0.125968	−	0.992034i	$-0.540204\pi$
$863$	−7.40109			−0.251936			−0.125968	+	0.992034i	$0.540204\pi$
$864$		0			0
$865$	0.265237			0.00901832
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.35957	+	1.35957i	−0.0461204	+	0.0461204i
$870$		0			0
$871$	−54.6004			−1.85006
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.73475	+	1.73475i	0.0586454	+	0.0586454i
$876$		0			0
$877$	9.57747	−	9.57747i	0.323408	−	0.323408i	−0.526665	−	0.850073i	$-0.676558\pi$
$877$	9.57747	−	9.57747i	0.323408	−	0.323408i	0.850073	+	0.526665i	$0.176558\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	8.18329i		−	0.275702i	−0.990453	−	0.137851i	$-0.955980\pi$
$881$		−	8.18329i		−	0.275702i	0.990453	−	0.137851i	$-0.0440195\pi$
$882$		0			0
$883$	11.8133	+	11.8133i	0.397548	+	0.397548i	0.877367	−	0.479819i	$-0.159298\pi$
$883$	11.8133	+	11.8133i	0.397548	+	0.397548i	−0.479819	+	0.877367i	$0.659298\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			30.5042i			1.02423i	0.858917	+	0.512115i	$0.171138\pi$
$887$			30.5042i			1.02423i	−0.858917	+	0.512115i	$0.828862\pi$
$888$		0			0
$889$			10.8017i			0.362278i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	13.3629	+	13.3629i	0.447171	+	0.447171i
$894$		0			0
$895$		−	17.1628i		−	0.573690i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	17.2609	−	17.2609i	0.575683	−	0.575683i
$900$		0			0
$901$	12.7445	+	12.7445i	0.424580	+	0.424580i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−15.7071			−0.522121
$906$		0			0
$907$	−6.07771	+	6.07771i	−0.201807	+	0.201807i	−0.800774	−	0.598967i	$-0.795578\pi$
$907$	−6.07771	+	6.07771i	−0.201807	+	0.201807i	0.598967	+	0.800774i	$0.295578\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−5.00898			−0.165955			−0.0829775	−	0.996551i	$-0.526443\pi$
$911$	−5.00898			−0.165955			−0.0829775	+	0.996551i	$0.526443\pi$
$912$		0			0
$913$	1.88396			0.0623500
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−12.5237	+	12.5237i	−0.413569	+	0.413569i
$918$		0			0
$919$	33.9710			1.12060			0.560299	−	0.828290i	$-0.310686\pi$
$919$	33.9710			1.12060			0.560299	+	0.828290i	$0.310686\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−25.7807	−	25.7807i	−0.848584	−	0.848584i
$924$		0			0
$925$	1.35596	−	1.35596i	0.0445835	−	0.0445835i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	41.8719i		−	1.37377i	−0.726764	−	0.686887i	$-0.758977\pi$
$929$		−	41.8719i		−	1.37377i	0.726764	−	0.686887i	$-0.241023\pi$
$930$		0			0
$931$	1.75595	+	1.75595i	0.0575489	+	0.0575489i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	0.355688i		−	0.0116323i
$936$		0			0
$937$		−	45.1739i		−	1.47577i	−0.674929	−	0.737883i	$-0.735826\pi$
$937$		−	45.1739i		−	1.47577i	0.674929	−	0.737883i	$-0.264174\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.45744	−	8.45744i	−0.275705	−	0.275705i	0.555687	−	0.831392i	$-0.312455\pi$
$941$	−8.45744	−	8.45744i	−0.275705	−	0.275705i	−0.831392	+	0.555687i	$0.812455\pi$
$942$		0			0
$943$		−	3.24104i		−	0.105543i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−22.8308	+	22.8308i	−0.741902	+	0.741902i	−0.972944	−	0.231042i	$-0.925787\pi$
$947$	−22.8308	+	22.8308i	−0.741902	+	0.741902i	0.231042	+	0.972944i	$0.425787\pi$
$948$		0			0
$949$	9.19550	+	9.19550i	0.298499	+	0.298499i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−29.2968			−0.949017			−0.474508	−	0.880251i	$-0.657374\pi$
$953$	−29.2968			−0.949017			−0.474508	+	0.880251i	$0.657374\pi$
$954$		0			0
$955$	−11.0864	+	11.0864i	−0.358747	+	0.358747i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	51.4158			1.66030
$960$		0			0
$961$	15.2956			0.493407
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.39351	+	1.39351i	−0.0448586	+	0.0448586i
$966$		0			0
$967$	−38.9662			−1.25307			−0.626535	−	0.779393i	$-0.715527\pi$
$967$	−38.9662			−1.25307			−0.626535	+	0.779393i	$0.715527\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−0.877003	−	0.877003i	−0.0281444	−	0.0281444i	0.692895	−	0.721039i	$-0.256335\pi$
$971$	−0.877003	−	0.877003i	−0.0281444	−	0.0281444i	−0.721039	+	0.692895i	$0.756335\pi$
$972$		0			0
$973$	22.2636	−	22.2636i	0.713739	−	0.713739i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	12.7599i		−	0.408225i	−0.978947	−	0.204113i	$-0.934569\pi$
$977$		−	12.7599i		−	0.408225i	0.978947	−	0.204113i	$-0.0654309\pi$
$978$		0			0
$979$	0.531112	+	0.531112i	0.0169744	+	0.0169744i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	45.8480i		−	1.46232i	−0.682203	−	0.731162i	$-0.738978\pi$
$983$		−	45.8480i		−	1.46232i	0.682203	−	0.731162i	$-0.261022\pi$
$984$		0			0
$985$			21.8260i			0.695434i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.94871	+	7.94871i	0.252754	+	0.252754i
$990$		0			0
$991$			0.736002i			0.0233799i	0.999932	+	0.0116899i	$0.00372111\pi$
$991$			0.736002i			0.0233799i	−0.999932	+	0.0116899i	$0.996279\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	14.6066	−	14.6066i	0.463062	−	0.463062i
$996$		0			0
$997$	−40.7975	−	40.7975i	−1.29207	−	1.29207i	−0.933504	−	0.358567i	$-0.883265\pi$
$997$	−40.7975	−	40.7975i	−1.29207	−	1.29207i	−0.358567	−	0.933504i	$-0.616735\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bl.c.1871.15		32
3.2	odd	2	inner	2880.2.bl.c.1871.4		32
4.3	odd	2		720.2.bl.c.251.3	✓	32
12.11	even	2		720.2.bl.c.251.14	yes	32
16.3	odd	4	inner	2880.2.bl.c.431.4		32
16.13	even	4		720.2.bl.c.611.14	yes	32
48.29	odd	4		720.2.bl.c.611.3	yes	32
48.35	even	4	inner	2880.2.bl.c.431.15		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bl.c.251.3	✓	32	4.3	odd	2
720.2.bl.c.251.14	yes	32	12.11	even	2
720.2.bl.c.611.3	yes	32	48.29	odd	4
720.2.bl.c.611.14	yes	32	16.13	even	4
2880.2.bl.c.431.4		32	16.3	odd	4	inner
2880.2.bl.c.431.15		32	48.35	even	4	inner
2880.2.bl.c.1871.4		32	3.2	odd	2	inner
2880.2.bl.c.1871.15		32	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.bl (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{5} -2.45331 q^{7} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{5} -2.45331 q^{7} +(-0.168035 - 0.168035i) q^{11} +(2.45119 - 2.45119i) q^{13} +1.49677i q^{17} +(-1.78948 - 1.78948i) q^{19} -1.51548i q^{23} -1.00000i q^{25} +(-4.35565 - 4.35565i) q^{29} +3.96288i q^{31} +(-1.73475 + 1.73475i) q^{35} +(1.35596 + 1.35596i) q^{37} +2.13862 q^{41} +(-5.24501 + 5.24501i) q^{43} -7.46748 q^{47} -0.981264 q^{49} +(8.51466 - 8.51466i) q^{53} -0.237637 q^{55} +(0.862894 + 0.862894i) q^{59} +(-3.60904 + 3.60904i) q^{61} -3.46650i q^{65} +(-11.1375 - 11.1375i) q^{67} -10.5177i q^{71} +3.75145i q^{73} +(0.412242 + 0.412242i) q^{77} -8.09102i q^{79} +(-5.60586 + 5.60586i) q^{83} +(1.05838 + 1.05838i) q^{85} -3.16072 q^{89} +(-6.01353 + 6.01353i) q^{91} -2.53070 q^{95} -15.1791 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 8 q^{19} - 32 q^{49} + 16 q^{55} + 16 q^{61} - 16 q^{67} + 16 q^{85} + 16 q^{91}+O(q^{100}) \) 32 * q + 8 * q^19 - 32 * q^49 + 16 * q^55 + 16 * q^61 - 16 * q^67 + 16 * q^85 + 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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