LMFDB - Embedded newform 384.4.j.b.97.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(97,384)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("384.97");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.j (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:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			97.5
Character	$\chi$	$=$	384.97
Dual form			384.4.j.b.289.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.12132 + 2.12132i) q^{3} +(7.29121 + 7.29121i) q^{5} -22.1610i q^{7} -9.00000i q^{9} +O(q^{10})$	\(q+(-2.12132 + 2.12132i) q^{3} +(7.29121 + 7.29121i) q^{5} -22.1610i q^{7} -9.00000i q^{9} +(-8.24116 - 8.24116i) q^{11} +(-51.9094 + 51.9094i) q^{13} -30.9340 q^{15} -58.7304 q^{17} +(54.5389 - 54.5389i) q^{19} +(47.0107 + 47.0107i) q^{21} +117.989i q^{23} -18.6766i q^{25} +(19.0919 + 19.0919i) q^{27} +(-175.283 + 175.283i) q^{29} -6.58699 q^{31} +34.9643 q^{33} +(161.581 - 161.581i) q^{35} +(-265.451 - 265.451i) q^{37} -220.233i q^{39} -98.7545i q^{41} +(-347.544 - 347.544i) q^{43} +(65.6209 - 65.6209i) q^{45} -141.880 q^{47} -148.112 q^{49} +(124.586 - 124.586i) q^{51} +(210.101 + 210.101i) q^{53} -120.176i q^{55} +231.389i q^{57} +(-427.318 - 427.318i) q^{59} +(-178.341 + 178.341i) q^{61} -199.449 q^{63} -756.965 q^{65} +(-480.715 + 480.715i) q^{67} +(-250.293 - 250.293i) q^{69} -884.186i q^{71} +794.543i q^{73} +(39.6190 + 39.6190i) q^{75} +(-182.633 + 182.633i) q^{77} -421.166 q^{79} -81.0000 q^{81} +(167.507 - 167.507i) q^{83} +(-428.215 - 428.215i) q^{85} -743.664i q^{87} -664.016i q^{89} +(1150.37 + 1150.37i) q^{91} +(13.9731 - 13.9731i) q^{93} +795.309 q^{95} -1083.81 q^{97} +(-74.1705 + 74.1705i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 40 q^{11} + 120 q^{15} - 24 q^{19} - 400 q^{29} - 744 q^{31} + 456 q^{35} - 16 q^{37} - 1240 q^{43} - 1176 q^{49} - 744 q^{51} - 752 q^{53} + 1376 q^{59} + 912 q^{61} - 504 q^{63} + 976 q^{65} + 2256 q^{67} + 528 q^{69} - 1104 q^{75} - 1904 q^{77} + 5992 q^{79} - 1944 q^{81} - 2680 q^{83} + 240 q^{85} + 3496 q^{91} - 7728 q^{95} + 360 q^{99}+O(q^{100}) $ 24 * q + 40 * q^11 + 120 * q^15 - 24 * q^19 - 400 * q^29 - 744 * q^31 + 456 * q^35 - 16 * q^37 - 1240 * q^43 - 1176 * q^49 - 744 * q^51 - 752 * q^53 + 1376 * q^59 + 912 * q^61 - 504 * q^63 + 976 * q^65 + 2256 * q^67 + 528 * q^69 - 1104 * q^75 - 1904 * q^77 + 5992 * q^79 - 1944 * q^81 - 2680 * q^83 + 240 * q^85 + 3496 * q^91 - 7728 * q^95 + 360 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$

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
$3$	−2.12132	+	2.12132i	−0.408248	+	0.408248i
$3$	−2.12132	+	2.12132i	−0.408248	+	0.408248i
$4$		0			0
$5$	7.29121	+	7.29121i	0.652145	+	0.652145i	0.953509	−	0.301364i	$-0.0974419\pi$
$5$	7.29121	+	7.29121i	0.652145	+	0.652145i	−0.301364	+	0.953509i	$0.597442\pi$
$6$		0			0
$7$		−	22.1610i		−	1.19658i	−0.801278	−	0.598292i	$-0.795846\pi$
$7$		−	22.1610i		−	1.19658i	0.801278	−	0.598292i	$-0.204154\pi$
$8$		0			0
$9$		−	9.00000i		−	0.333333i
$10$		0			0
$11$	−8.24116	−	8.24116i	−0.225891	−	0.225891i	0.585083	−	0.810974i	$-0.301062\pi$
$11$	−8.24116	−	8.24116i	−0.225891	−	0.225891i	−0.810974	+	0.585083i	$0.801062\pi$
$12$		0			0
$13$	−51.9094	+	51.9094i	−1.10747	+	1.10747i	−0.113986	+	0.993482i	$0.536362\pi$
$13$	−51.9094	+	51.9094i	−1.10747	+	1.10747i	−0.993482	+	0.113986i	$0.963638\pi$
$14$		0			0
$15$	−30.9340			−0.532475
$16$		0			0
$17$	−58.7304			−0.837894			−0.418947	−	0.908011i	$-0.637601\pi$
$17$	−58.7304			−0.837894			−0.418947	+	0.908011i	$0.637601\pi$
$18$		0			0
$19$	54.5389	−	54.5389i	0.658531	−	0.658531i	−0.296501	−	0.955032i	$-0.595820\pi$
$19$	54.5389	−	54.5389i	0.658531	−	0.658531i	0.955032	+	0.296501i	$0.0958199\pi$
$20$		0			0
$21$	47.0107	+	47.0107i	0.488503	+	0.488503i
$22$		0			0
$23$			117.989i			1.06967i	0.844955	+	0.534837i	$0.179627\pi$
$23$			117.989i			1.06967i	−0.844955	+	0.534837i	$0.820373\pi$
$24$		0			0
$25$		−	18.6766i		−	0.149413i
$26$		0			0
$27$	19.0919	+	19.0919i	0.136083	+	0.136083i
$28$		0			0
$29$	−175.283	+	175.283i	−1.12239	+	1.12239i	−0.131007	+	0.991381i	$0.541821\pi$
$29$	−175.283	+	175.283i	−1.12239	+	1.12239i	−0.991381	+	0.131007i	$0.958179\pi$
$30$		0			0
$31$	−6.58699			−0.0381631			−0.0190816	−	0.999818i	$-0.506074\pi$
$31$	−6.58699			−0.0381631			−0.0190816	+	0.999818i	$0.506074\pi$
$32$		0			0
$33$	34.9643			0.184439
$34$		0			0
$35$	161.581	−	161.581i	0.780347	−	0.780347i
$36$		0			0
$37$	−265.451	−	265.451i	−1.17946	−	1.17946i	−0.979883	−	0.199574i	$-0.936044\pi$
$37$	−265.451	−	265.451i	−1.17946	−	1.17946i	−0.199574	−	0.979883i	$-0.563956\pi$
$38$		0			0
$39$		−	220.233i		−	0.904244i
$40$		0			0
$41$		−	98.7545i		−	0.376167i	−0.982153	−	0.188084i	$-0.939772\pi$
$41$		−	98.7545i		−	0.376167i	0.982153	−	0.188084i	$-0.0602276\pi$
$42$		0			0
$43$	−347.544	−	347.544i	−1.23256	−	1.23256i	−0.962978	−	0.269580i	$-0.913115\pi$
$43$	−347.544	−	347.544i	−1.23256	−	1.23256i	−0.269580	−	0.962978i	$-0.586885\pi$
$44$		0			0
$45$	65.6209	−	65.6209i	0.217382	−	0.217382i
$46$		0			0
$47$	−141.880			−0.440327			−0.220164	−	0.975463i	$-0.570659\pi$
$47$	−141.880			−0.440327			−0.220164	+	0.975463i	$0.570659\pi$
$48$		0			0
$49$	−148.112			−0.431813
$50$		0			0
$51$	124.586	−	124.586i	0.342069	−	0.342069i
$52$		0			0
$53$	210.101	+	210.101i	0.544520	+	0.544520i	0.924850	−	0.380331i	$-0.124190\pi$
$53$	210.101	+	210.101i	0.544520	+	0.544520i	−0.380331	+	0.924850i	$0.624190\pi$
$54$		0			0
$55$		−	120.176i		−	0.294628i
$56$		0			0
$57$			231.389i			0.537688i
$58$		0			0
$59$	−427.318	−	427.318i	−0.942917	−	0.942917i	0.0555392	−	0.998457i	$-0.482312\pi$
$59$	−427.318	−	427.318i	−0.942917	−	0.942917i	−0.998457	+	0.0555392i	$0.982312\pi$
$60$		0			0
$61$	−178.341	+	178.341i	−0.374332	+	0.374332i	−0.869052	−	0.494720i	$-0.835270\pi$
$61$	−178.341	+	178.341i	−0.374332	+	0.374332i	0.494720	+	0.869052i	$0.335270\pi$
$62$		0			0
$63$	−199.449			−0.398861
$64$		0			0
$65$	−756.965			−1.44446
$66$		0			0
$67$	−480.715	+	480.715i	−0.876547	+	0.876547i	−0.993176	−	0.116628i	$-0.962791\pi$
$67$	−480.715	+	480.715i	−0.876547	+	0.876547i	0.116628	+	0.993176i	$0.462791\pi$
$68$		0			0
$69$	−250.293	−	250.293i	−0.436693	−	0.436693i
$70$		0			0
$71$		−	884.186i		−	1.47794i	−0.673740	−	0.738969i	$-0.735313\pi$
$71$		−	884.186i		−	1.47794i	0.673740	−	0.738969i	$-0.264687\pi$
$72$		0			0
$73$			794.543i			1.27389i	0.770908	+	0.636947i	$0.219803\pi$
$73$			794.543i			1.27389i	−0.770908	+	0.636947i	$0.780197\pi$
$74$		0			0
$75$	39.6190	+	39.6190i	0.0609975	+	0.0609975i
$76$		0			0
$77$	−182.633	+	182.633i	−0.270298	+	0.270298i
$78$		0			0
$79$	−421.166			−0.599809			−0.299904	−	0.953969i	$-0.596955\pi$
$79$	−421.166			−0.599809			−0.299904	+	0.953969i	$0.596955\pi$
$80$		0			0
$81$	−81.0000			−0.111111
$82$		0			0
$83$	167.507	−	167.507i	0.221522	−	0.221522i	−0.587617	−	0.809139i	$-0.699934\pi$
$83$	167.507	−	167.507i	0.221522	−	0.221522i	0.809139	+	0.587617i	$0.199934\pi$
$84$		0			0
$85$	−428.215	−	428.215i	−0.546429	−	0.546429i
$86$		0			0
$87$		−	743.664i		−	0.916427i
$88$		0			0
$89$		−	664.016i		−	0.790849i	−0.918498	−	0.395425i	$-0.870597\pi$
$89$		−	664.016i		−	0.790849i	0.918498	−	0.395425i	$-0.129403\pi$
$90$		0			0
$91$	1150.37	+	1150.37i	1.32518	+	1.32518i
$92$		0			0
$93$	13.9731	−	13.9731i	0.0155800	−	0.0155800i
$94$		0			0
$95$	795.309			0.858916
$96$		0			0
$97$	−1083.81			−1.13447			−0.567236	−	0.823555i	$-0.691987\pi$
$97$	−1083.81			−1.13447			−0.567236	+	0.823555i	$0.691987\pi$
$98$		0			0
$99$	−74.1705	+	74.1705i	−0.0752971	+	0.0752971i
$100$		0			0
$101$	600.838	+	600.838i	0.591937	+	0.591937i	0.938154	−	0.346217i	$-0.112534\pi$
$101$	600.838	+	600.838i	0.591937	+	0.591937i	−0.346217	+	0.938154i	$0.612534\pi$
$102$		0			0
$103$		−	30.6830i		−	0.0293523i	−0.999892	−	0.0146762i	$-0.995328\pi$
$103$		−	30.6830i		−	0.0293523i	0.999892	−	0.0146762i	$-0.00467173\pi$
$104$		0			0
$105$			685.529i			0.637150i
$106$		0			0
$107$	−812.356	−	812.356i	−0.733957	−	0.733957i	0.237444	−	0.971401i	$-0.423690\pi$
$107$	−812.356	−	812.356i	−0.733957	−	0.733957i	−0.971401	+	0.237444i	$0.923690\pi$
$108$		0			0
$109$	148.034	−	148.034i	0.130083	−	0.130083i	−0.639067	−	0.769151i	$-0.720679\pi$
$109$	148.034	−	148.034i	0.130083	−	0.130083i	0.769151	+	0.639067i	$0.220679\pi$
$110$		0			0
$111$	1126.21			0.963022
$112$		0			0
$113$	1801.39			1.49965			0.749826	−	0.661635i	$-0.230137\pi$
$113$	1801.39			1.49965			0.749826	+	0.661635i	$0.230137\pi$
$114$		0			0
$115$	−860.286	+	860.286i	−0.697583	+	0.697583i
$116$		0			0
$117$	467.185	+	467.185i	0.369156	+	0.369156i
$118$		0			0
$119$			1301.53i			1.00261i
$120$		0			0
$121$		−	1195.17i		−	0.897946i
$122$		0			0
$123$	209.490	+	209.490i	0.153570	+	0.153570i
$124$		0			0
$125$	1047.58	−	1047.58i	0.749584	−	0.749584i
$126$		0			0
$127$	−876.738			−0.612582			−0.306291	−	0.951938i	$-0.599088\pi$
$127$	−876.738			−0.612582			−0.306291	+	0.951938i	$0.599088\pi$
$128$		0			0
$129$	1474.51			1.00638
$130$		0			0
$131$	1355.99	−	1355.99i	0.904374	−	0.904374i	−0.0914368	−	0.995811i	$-0.529146\pi$
$131$	1355.99	−	1355.99i	0.904374	−	0.904374i	0.995811	+	0.0914368i	$0.0291459\pi$
$132$		0			0
$133$	−1208.64	−	1208.64i	−0.787988	−	0.787988i
$134$		0			0
$135$			278.406i			0.177492i
$136$		0			0
$137$			1833.19i			1.14321i	0.820529	+	0.571605i	$0.193679\pi$
$137$			1833.19i			1.14321i	−0.820529	+	0.571605i	$0.806321\pi$
$138$		0			0
$139$	705.601	+	705.601i	0.430563	+	0.430563i	0.888820	−	0.458257i	$-0.151526\pi$
$139$	705.601	+	705.601i	0.430563	+	0.430563i	−0.458257	+	0.888820i	$0.651526\pi$
$140$		0			0
$141$	300.974	−	300.974i	0.179763	−	0.179763i
$142$		0			0
$143$	855.588			0.500335
$144$		0			0
$145$	−2556.05			−1.46392
$146$		0			0
$147$	314.193	−	314.193i	0.176287	−	0.176287i
$148$		0			0
$149$	714.815	+	714.815i	0.393020	+	0.393020i	0.875762	−	0.482743i	$-0.160359\pi$
$149$	714.815	+	714.815i	0.393020	+	0.393020i	−0.482743	+	0.875762i	$0.660359\pi$
$150$		0			0
$151$			3190.79i			1.71962i	0.510614	+	0.859810i	$0.329418\pi$
$151$			3190.79i			1.71962i	−0.510614	+	0.859810i	$0.670582\pi$
$152$		0			0
$153$			528.573i			0.279298i
$154$		0			0
$155$	−48.0271	−	48.0271i	−0.0248879	−	0.0248879i
$156$		0			0
$157$	−471.880	+	471.880i	−0.239874	+	0.239874i	−0.816798	−	0.576924i	$-0.804253\pi$
$157$	−471.880	+	471.880i	−0.239874	+	0.239874i	0.576924	+	0.816798i	$0.304253\pi$
$158$		0			0
$159$	−891.381			−0.444598
$160$		0			0
$161$	2614.77			1.27995
$162$		0			0
$163$	−275.466	+	275.466i	−0.132369	+	0.132369i	−0.770187	−	0.637818i	$-0.779837\pi$
$163$	−275.466	+	275.466i	−0.132369	+	0.132369i	0.637818	+	0.770187i	$0.279837\pi$
$164$		0			0
$165$	254.932	+	254.932i	0.120281	+	0.120281i
$166$		0			0
$167$			1754.52i			0.812988i	0.913653	+	0.406494i	$0.133249\pi$
$167$			1754.52i			0.812988i	−0.913653	+	0.406494i	$0.866751\pi$
$168$		0			0
$169$		−	3192.18i		−	1.45297i
$170$		0			0
$171$	−490.850	−	490.850i	−0.219510	−	0.219510i
$172$		0			0
$173$	−972.532	+	972.532i	−0.427400	+	0.427400i	−0.887742	−	0.460342i	$-0.847727\pi$
$173$	−972.532	+	972.532i	−0.427400	+	0.427400i	0.460342	+	0.887742i	$0.347727\pi$
$174$		0			0
$175$	−413.893			−0.178785
$176$		0			0
$177$	1812.96			0.769889
$178$		0			0
$179$	−1944.54	+	1944.54i	−0.811964	+	0.811964i	−0.984928	−	0.172964i	$-0.944665\pi$
$179$	−1944.54	+	1944.54i	−0.811964	+	0.811964i	0.172964	+	0.984928i	$0.444665\pi$
$180$		0			0
$181$	1388.85	+	1388.85i	0.570345	+	0.570345i	0.932225	−	0.361880i	$-0.117865\pi$
$181$	1388.85	+	1388.85i	0.570345	+	0.570345i	−0.361880	+	0.932225i	$0.617865\pi$
$182$		0			0
$183$		−	756.637i		−	0.305641i
$184$		0			0
$185$		−	3870.92i		−	1.53835i
$186$		0			0
$187$	484.006	+	484.006i	0.189273	+	0.189273i
$188$		0			0
$189$	423.096	−	423.096i	0.162834	−	0.162834i
$190$		0			0
$191$	−1754.58			−0.664695			−0.332347	−	0.943157i	$-0.607841\pi$
$191$	−1754.58			−0.664695			−0.332347	+	0.943157i	$0.607841\pi$
$192$		0			0
$193$	4111.28			1.53335			0.766675	−	0.642036i	$-0.221910\pi$
$193$	4111.28			1.53335			0.766675	+	0.642036i	$0.221910\pi$
$194$		0			0
$195$	1605.76	−	1605.76i	0.589698	−	0.589698i
$196$		0			0
$197$	585.319	+	585.319i	0.211686	+	0.211686i	0.804984	−	0.593297i	$-0.202174\pi$
$197$	585.319	+	585.319i	0.211686	+	0.211686i	−0.593297	+	0.804984i	$0.702174\pi$
$198$		0			0
$199$			2620.65i			0.933533i	0.884381	+	0.466766i	$0.154581\pi$
$199$			2620.65i			0.933533i	−0.884381	+	0.466766i	$0.845419\pi$
$200$		0			0
$201$		−	2039.50i		−	0.715698i
$202$		0			0
$203$	3884.46	+	3884.46i	1.34303	+	1.34303i
$204$		0			0
$205$	720.039	−	720.039i	0.245316	−	0.245316i
$206$		0			0
$207$	1061.91			0.356558
$208$		0			0
$209$	−898.929			−0.297513
$210$		0			0
$211$	−969.705	+	969.705i	−0.316385	+	0.316385i	−0.847377	−	0.530992i	$-0.821819\pi$
$211$	−969.705	+	969.705i	−0.316385	+	0.316385i	0.530992	+	0.847377i	$0.321819\pi$
$212$		0			0
$213$	1875.64	+	1875.64i	0.603365	+	0.603365i
$214$		0			0
$215$		−	5068.04i		−	1.60761i
$216$		0			0
$217$			145.974i			0.0456654i
$218$		0			0
$219$	−1685.48	−	1685.48i	−0.520065	−	0.520065i
$220$		0			0
$221$	3048.66	−	3048.66i	0.927941	−	0.927941i
$222$		0			0
$223$	63.3747			0.0190309			0.00951544	−	0.999955i	$-0.496971\pi$
$223$	63.3747			0.0190309			0.00951544	+	0.999955i	$0.496971\pi$
$224$		0			0
$225$	−168.089			−0.0498042
$226$		0			0
$227$	1802.55	−	1802.55i	0.527047	−	0.527047i	−0.392644	−	0.919691i	$-0.628439\pi$
$227$	1802.55	−	1802.55i	0.527047	−	0.527047i	0.919691	+	0.392644i	$0.128439\pi$
$228$		0			0
$229$	−2895.47	−	2895.47i	−0.835538	−	0.835538i	0.152730	−	0.988268i	$-0.451193\pi$
$229$	−2895.47	−	2895.47i	−0.835538	−	0.835538i	−0.988268	+	0.152730i	$0.951193\pi$
$230$		0			0
$231$		−	774.845i		−	0.220697i
$232$		0			0
$233$			4262.01i			1.19834i	0.800621	+	0.599171i	$0.204503\pi$
$233$			4262.01i			1.19834i	−0.800621	+	0.599171i	$0.795497\pi$
$234$		0			0
$235$	−1034.48	−	1034.48i	−0.287157	−	0.287157i
$236$		0			0
$237$	893.428	−	893.428i	0.244871	−	0.244871i
$238$		0			0
$239$	3308.14			0.895337			0.447669	−	0.894200i	$-0.352254\pi$
$239$	3308.14			0.895337			0.447669	+	0.894200i	$0.352254\pi$
$240$		0			0
$241$	−3200.02			−0.855316			−0.427658	−	0.903941i	$-0.640661\pi$
$241$	−3200.02			−0.855316			−0.427658	+	0.903941i	$0.640661\pi$
$242$		0			0
$243$	171.827	−	171.827i	0.0453609	−	0.0453609i
$244$		0			0
$245$	−1079.91	−	1079.91i	−0.281605	−	0.281605i
$246$		0			0
$247$			5662.17i			1.45860i
$248$		0			0
$249$			710.673i			0.180872i
$250$		0			0
$251$	1945.31	+	1945.31i	0.489190	+	0.489190i	0.908051	−	0.418860i	$-0.137570\pi$
$251$	1945.31	+	1945.31i	0.489190	+	0.489190i	−0.418860	+	0.908051i	$0.637570\pi$
$252$		0			0
$253$	972.370	−	972.370i	0.241630	−	0.241630i
$254$		0			0
$255$	1816.76			0.446157
$256$		0			0
$257$	−82.9830			−0.0201414			−0.0100707	−	0.999949i	$-0.503206\pi$
$257$	−82.9830			−0.0201414			−0.0100707	+	0.999949i	$0.503206\pi$
$258$		0			0
$259$	−5882.67	+	5882.67i	−1.41132	+	1.41132i
$260$		0			0
$261$	1577.55	+	1577.55i	0.374130	+	0.374130i
$262$		0			0
$263$		−	2139.57i		−	0.501641i	−0.968034	−	0.250821i	$-0.919300\pi$
$263$		−	2139.57i		−	0.501641i	0.968034	−	0.250821i	$-0.0807004\pi$
$264$		0			0
$265$			3063.77i			0.710212i
$266$		0			0
$267$	1408.59	+	1408.59i	0.322863	+	0.322863i
$268$		0			0
$269$	−2564.40	+	2564.40i	−0.581243	+	0.581243i	−0.935245	−	0.354002i	$-0.884821\pi$
$269$	−2564.40	+	2564.40i	−0.581243	+	0.581243i	0.354002	+	0.935245i	$0.384821\pi$
$270$		0			0
$271$	1373.88			0.307960			0.153980	−	0.988074i	$-0.450791\pi$
$271$	1373.88			0.307960			0.153980	+	0.988074i	$0.450791\pi$
$272$		0			0
$273$	−4880.59			−1.08200
$274$		0			0
$275$	−153.917	+	153.917i	−0.0337510	+	0.0337510i
$276$		0			0
$277$	−1763.08	−	1763.08i	−0.382431	−	0.382431i	0.489546	−	0.871977i	$-0.337162\pi$
$277$	−1763.08	−	1763.08i	−0.382431	−	0.382431i	−0.871977	+	0.489546i	$0.837162\pi$
$278$		0			0
$279$			59.2829i			0.0127210i
$280$		0			0
$281$			8170.89i			1.73464i	0.497748	+	0.867321i	$0.334160\pi$
$281$			8170.89i			1.73464i	−0.497748	+	0.867321i	$0.665840\pi$
$282$		0			0
$283$	−1217.94	−	1217.94i	−0.255827	−	0.255827i	0.567528	−	0.823354i	$-0.307900\pi$
$283$	−1217.94	−	1217.94i	−0.255827	−	0.255827i	−0.823354	+	0.567528i	$0.807900\pi$
$284$		0			0
$285$	−1687.11	+	1687.11i	−0.350651	+	0.350651i
$286$		0			0
$287$	−2188.50			−0.450116
$288$		0			0
$289$	−1463.75			−0.297933
$290$		0			0
$291$	2299.10	−	2299.10i	0.463146	−	0.463146i
$292$		0			0
$293$	−2994.08	−	2994.08i	−0.596984	−	0.596984i	0.342525	−	0.939509i	$-0.388718\pi$
$293$	−2994.08	−	2994.08i	−0.596984	−	0.596984i	−0.939509	+	0.342525i	$0.888718\pi$
$294$		0			0
$295$		−	6231.33i		−	1.22984i
$296$		0			0
$297$		−	314.679i		−	0.0614798i
$298$		0			0
$299$	−6124.76	−	6124.76i	−1.18463	−	1.18463i
$300$		0			0
$301$	−7701.94	+	7701.94i	−1.47486	+	1.47486i
$302$		0			0
$303$	−2549.14			−0.483315
$304$		0			0
$305$	−2600.64			−0.488238
$306$		0			0
$307$	5382.02	−	5382.02i	1.00055	−	1.00055i	0.000548529	−	1.00000i	$-0.499825\pi$
$307$	5382.02	−	5382.02i	1.00055	−	1.00055i	1.00000		0.000548529i	$-0.000174602\pi$
$308$		0			0
$309$	65.0885	+	65.0885i	0.0119830	+	0.0119830i
$310$		0			0
$311$		−	7805.69i		−	1.42322i	−0.702577	−	0.711608i	$-0.747967\pi$
$311$		−	7805.69i		−	1.42322i	0.702577	−	0.711608i	$-0.252033\pi$
$312$		0			0
$313$		−	1549.25i		−	0.279772i	−0.990168	−	0.139886i	$-0.955326\pi$
$313$		−	1549.25i		−	0.279772i	0.990168	−	0.139886i	$-0.0446736\pi$
$314$		0			0
$315$	−1454.23	−	1454.23i	−0.260116	−	0.260116i
$316$		0			0
$317$	6112.35	−	6112.35i	1.08298	−	1.08298i	0.0867460	−	0.996230i	$-0.472353\pi$
$317$	6112.35	−	6112.35i	1.08298	−	1.08298i	0.996230	−	0.0867460i	$-0.0276468\pi$
$318$		0			0
$319$	2889.07			0.507076
$320$		0			0
$321$	3446.53			0.599273
$322$		0			0
$323$	−3203.09	+	3203.09i	−0.551779	+	0.551779i
$324$		0			0
$325$	969.491	+	969.491i	0.165470	+	0.165470i
$326$		0			0
$327$			628.056i			0.106213i
$328$		0			0
$329$			3144.22i			0.526888i
$330$		0			0
$331$	−3539.08	−	3539.08i	−0.587690	−	0.587690i	0.349316	−	0.937005i	$-0.386414\pi$
$331$	−3539.08	−	3539.08i	−0.587690	−	0.587690i	−0.937005	+	0.349316i	$0.886414\pi$
$332$		0			0
$333$	−2389.06	+	2389.06i	−0.393152	+	0.393152i
$334$		0			0
$335$	−7009.98			−1.14327
$336$		0			0
$337$	−7315.83			−1.18255			−0.591274	−	0.806471i	$-0.701375\pi$
$337$	−7315.83			−1.18255			−0.591274	+	0.806471i	$0.701375\pi$
$338$		0			0
$339$	−3821.33	+	3821.33i	−0.612231	+	0.612231i
$340$		0			0
$341$	54.2844	+	54.2844i	0.00862072	+	0.00862072i
$342$		0			0
$343$		−	4318.92i		−	0.679883i
$344$		0			0
$345$		−	3649.88i		−	0.569574i
$346$		0			0
$347$	1214.48	+	1214.48i	0.187886	+	0.187886i	0.794782	−	0.606895i	$-0.207585\pi$
$347$	1214.48	+	1214.48i	0.187886	+	0.187886i	−0.606895	+	0.794782i	$0.707585\pi$
$348$		0			0
$349$	3823.99	−	3823.99i	0.586514	−	0.586514i	−0.350171	−	0.936686i	$-0.613877\pi$
$349$	3823.99	−	3823.99i	0.586514	−	0.586514i	0.936686	+	0.350171i	$0.113877\pi$
$350$		0			0
$351$	−1982.10			−0.301415
$352$		0			0
$353$	4475.44			0.674797			0.337399	−	0.941362i	$-0.390453\pi$
$353$	4475.44			0.674797			0.337399	+	0.941362i	$0.390453\pi$
$354$		0			0
$355$	6446.78	−	6446.78i	0.963830	−	0.963830i
$356$		0			0
$357$	−2760.95	−	2760.95i	−0.409314	−	0.409314i
$358$		0			0
$359$			1340.95i			0.197138i	0.995130	+	0.0985688i	$0.0314264\pi$
$359$			1340.95i			0.197138i	−0.995130	+	0.0985688i	$0.968574\pi$
$360$		0			0
$361$			910.008i			0.132674i
$362$		0			0
$363$	2535.33	+	2535.33i	0.366585	+	0.366585i
$364$		0			0
$365$	−5793.18	+	5793.18i	−0.830764	+	0.830764i
$366$		0			0
$367$	−5859.22			−0.833376			−0.416688	−	0.909050i	$-0.636809\pi$
$367$	−5859.22			−0.833376			−0.416688	+	0.909050i	$0.636809\pi$
$368$		0			0
$369$	−888.790			−0.125389
$370$		0			0
$371$	4656.05	−	4656.05i	0.651563	−	0.651563i
$372$		0			0
$373$	4717.98	+	4717.98i	0.654927	+	0.654927i	0.954175	−	0.299248i	$-0.0967359\pi$
$373$	4717.98	+	4717.98i	0.654927	+	0.654927i	−0.299248	+	0.954175i	$0.596736\pi$
$374$		0			0
$375$			4444.49i			0.612033i
$376$		0			0
$377$		−	18197.7i		−	2.48602i
$378$		0			0
$379$	−252.484	−	252.484i	−0.0342197	−	0.0342197i	0.689790	−	0.724010i	$-0.257703\pi$
$379$	−252.484	−	252.484i	−0.0342197	−	0.0342197i	−0.724010	+	0.689790i	$0.757703\pi$
$380$		0			0
$381$	1859.84	−	1859.84i	0.250086	−	0.250086i
$382$		0			0
$383$	2526.32			0.337047			0.168524	−	0.985698i	$-0.446100\pi$
$383$	2526.32			0.337047			0.168524	+	0.985698i	$0.446100\pi$
$384$		0			0
$385$	−2663.23			−0.352547
$386$		0			0
$387$	−3127.90	+	3127.90i	−0.410853	+	0.410853i
$388$		0			0
$389$	−1772.59	−	1772.59i	−0.231038	−	0.231038i	0.582088	−	0.813126i	$-0.302236\pi$
$389$	−1772.59	−	1772.59i	−0.231038	−	0.231038i	−0.813126	+	0.582088i	$0.802236\pi$
$390$		0			0
$391$		−	6929.56i		−	0.896274i
$392$		0			0
$393$			5752.96i			0.738418i
$394$		0			0
$395$	−3070.81	−	3070.81i	−0.391163	−	0.391163i
$396$		0			0
$397$	−4182.28	+	4182.28i	−0.528722	+	0.528722i	−0.920191	−	0.391469i	$-0.871967\pi$
$397$	−4182.28	+	4182.28i	−0.528722	+	0.528722i	0.391469	+	0.920191i	$0.371967\pi$
$398$		0			0
$399$	5127.82			0.643389
$400$		0			0
$401$	−5929.46			−0.738411			−0.369206	−	0.929348i	$-0.620370\pi$
$401$	−5929.46			−0.738411			−0.369206	+	0.929348i	$0.620370\pi$
$402$		0			0
$403$	341.927	−	341.927i	0.0422645	−	0.0422645i
$404$		0			0
$405$	−590.588	−	590.588i	−0.0724606	−	0.0724606i
$406$		0			0
$407$			4375.25i			0.532858i
$408$		0			0
$409$		−	9651.64i		−	1.16685i	−0.812166	−	0.583427i	$-0.801712\pi$
$409$		−	9651.64i		−	1.16685i	0.812166	−	0.583427i	$-0.198288\pi$
$410$		0			0
$411$	−3888.78	−	3888.78i	−0.466713	−	0.466713i
$412$		0			0
$413$	−9469.82	+	9469.82i	−1.12828	+	1.12828i
$414$		0			0
$415$	2442.66			0.288929
$416$		0			0
$417$	−2993.61			−0.351553
$418$		0			0
$419$	−39.7232	+	39.7232i	−0.00463152	+	0.00463152i	−0.709419	−	0.704787i	$-0.751042\pi$
$419$	−39.7232	+	39.7232i	−0.00463152	+	0.00463152i	0.704787	+	0.709419i	$0.251042\pi$
$420$		0			0
$421$	−1741.60	−	1741.60i	−0.201616	−	0.201616i	0.599076	−	0.800692i	$-0.295535\pi$
$421$	−1741.60	−	1741.60i	−0.201616	−	0.201616i	−0.800692	+	0.599076i	$0.795535\pi$
$422$		0			0
$423$			1276.92i			0.146776i
$424$		0			0
$425$			1096.88i			0.125192i
$426$		0			0
$427$	3952.23	+	3952.23i	0.447919	+	0.447919i
$428$		0			0
$429$	−1814.98	+	1814.98i	−0.204261	+	0.204261i
$430$		0			0
$431$	115.905			0.0129535			0.00647673	−	0.999979i	$-0.497938\pi$
$431$	115.905			0.0129535			0.00647673	+	0.999979i	$0.497938\pi$
$432$		0			0
$433$	10706.1			1.18823			0.594115	−	0.804380i	$-0.297502\pi$
$433$	10706.1			1.18823			0.594115	+	0.804380i	$0.297502\pi$
$434$		0			0
$435$	5422.21	−	5422.21i	0.597643	−	0.597643i
$436$		0			0
$437$	6435.02	+	6435.02i	0.704414	+	0.704414i
$438$		0			0
$439$			1275.83i			0.138706i	0.997592	+	0.0693531i	$0.0220935\pi$
$439$			1275.83i			0.138706i	−0.997592	+	0.0693531i	$0.977906\pi$
$440$		0			0
$441$			1333.01i			0.143938i
$442$		0			0
$443$	−1291.45	−	1291.45i	−0.138507	−	0.138507i	0.634454	−	0.772961i	$-0.281225\pi$
$443$	−1291.45	−	1291.45i	−0.138507	−	0.138507i	−0.772961	+	0.634454i	$0.781225\pi$
$444$		0			0
$445$	4841.48	−	4841.48i	0.515749	−	0.515749i
$446$		0			0
$447$	−3032.70			−0.320899
$448$		0			0
$449$	2137.82			0.224699			0.112350	−	0.993669i	$-0.464162\pi$
$449$	2137.82			0.224699			0.112350	+	0.993669i	$0.464162\pi$
$450$		0			0
$451$	−813.852	+	813.852i	−0.0849729	+	0.0849729i
$452$		0			0
$453$	−6768.69	−	6768.69i	−0.702032	−	0.702032i
$454$		0			0
$455$			16775.1i			1.72842i
$456$		0			0
$457$			15629.9i			1.59986i	0.600092	+	0.799931i	$0.295131\pi$
$457$			15629.9i			1.59986i	−0.600092	+	0.799931i	$0.704869\pi$
$458$		0			0
$459$	−1121.27	−	1121.27i	−0.114023	−	0.114023i
$460$		0			0
$461$	2333.25	−	2333.25i	0.235727	−	0.235727i	−0.579351	−	0.815078i	$-0.696694\pi$
$461$	2333.25	−	2333.25i	0.235727	−	0.235727i	0.815078	+	0.579351i	$0.196694\pi$
$462$		0			0
$463$	−16917.7			−1.69812			−0.849061	−	0.528294i	$-0.822832\pi$
$463$	−16917.7			−1.69812			−0.849061	+	0.528294i	$0.822832\pi$
$464$		0			0
$465$	203.762			0.0203209
$466$		0			0
$467$	−3249.62	+	3249.62i	−0.322001	+	0.322001i	−0.849534	−	0.527533i	$-0.823117\pi$
$467$	−3249.62	+	3249.62i	−0.322001	+	0.322001i	0.527533	+	0.849534i	$0.323117\pi$
$468$		0			0
$469$	10653.1	+	10653.1i	1.04886	+	1.04886i
$470$		0			0
$471$		−	2002.02i		−	0.195856i
$472$		0			0
$473$			5728.34i			0.556848i
$474$		0			0
$475$	−1018.60	−	1018.60i	−0.0983929	−	0.0983929i
$476$		0			0
$477$	1890.91	−	1890.91i	0.181507	−	0.181507i
$478$		0			0
$479$	9780.09			0.932910			0.466455	−	0.884545i	$-0.345531\pi$
$479$	9780.09			0.932910			0.466455	+	0.884545i	$0.345531\pi$
$480$		0			0
$481$	27558.8			2.61242
$482$		0			0
$483$	−5546.76	+	5546.76i	−0.522539	+	0.522539i
$484$		0			0
$485$	−7902.25	−	7902.25i	−0.739840	−	0.739840i
$486$		0			0
$487$		−	2710.17i		−	0.252176i	−0.992019	−	0.126088i	$-0.959758\pi$
$487$		−	2710.17i		−	0.252176i	0.992019	−	0.126088i	$-0.0402421\pi$
$488$		0			0
$489$		−	1168.70i		−	0.108079i
$490$		0			0
$491$	1884.20	+	1884.20i	0.173183	+	0.173183i	0.788376	−	0.615193i	$-0.210922\pi$
$491$	1884.20	+	1884.20i	0.173183	+	0.173183i	−0.615193	+	0.788376i	$0.710922\pi$
$492$		0			0
$493$	10294.4	−	10294.4i	0.940443	−	0.940443i
$494$		0			0
$495$	−1081.58			−0.0982093
$496$		0			0
$497$	−19594.5			−1.76848
$498$		0			0
$499$	11085.6	−	11085.6i	0.994509	−	0.994509i	−0.00547579	−	0.999985i	$-0.501743\pi$
$499$	11085.6	−	11085.6i	0.994509	−	0.994509i	0.999985	+	0.00547579i	$0.00174301\pi$
$500$		0			0
$501$	−3721.90	−	3721.90i	−0.331901	−	0.331901i
$502$		0			0
$503$			9544.26i			0.846039i	0.906121	+	0.423020i	$0.139030\pi$
$503$			9544.26i			0.846039i	−0.906121	+	0.423020i	$0.860970\pi$
$504$		0			0
$505$			8761.67i			0.772058i
$506$		0			0
$507$	6771.63	+	6771.63i	0.593173	+	0.593173i
$508$		0			0
$509$	−1874.95	+	1874.95i	−0.163272	+	0.163272i	−0.784015	−	0.620742i	$-0.786831\pi$
$509$	−1874.95	+	1874.95i	−0.163272	+	0.163272i	0.620742	+	0.784015i	$0.286831\pi$
$510$		0			0
$511$	17607.9			1.52432
$512$		0			0
$513$	2082.50			0.179229
$514$		0			0
$515$	223.716	−	223.716i	0.0191420	−	0.0191420i
$516$		0			0
$517$	1169.26	+	1169.26i	0.0994660	+	0.0994660i
$518$		0			0
$519$		−	4126.10i		−	0.348971i
$520$		0			0
$521$		−	12549.3i		−	1.05527i	−0.849471	−	0.527635i	$-0.823079\pi$
$521$		−	12549.3i		−	1.05527i	0.849471	−	0.527635i	$-0.176921\pi$
$522$		0			0
$523$	5583.16	+	5583.16i	0.466797	+	0.466797i	0.900875	−	0.434078i	$-0.142926\pi$
$523$	5583.16	+	5583.16i	0.466797	+	0.466797i	−0.434078	+	0.900875i	$0.642926\pi$
$524$		0			0
$525$	877.999	−	877.999i	0.0729886	−	0.0729886i
$526$		0			0
$527$	386.856			0.0319767
$528$		0			0
$529$	−1754.51			−0.144202
$530$		0			0
$531$	−3845.87	+	3845.87i	−0.314306	+	0.314306i
$532$		0			0
$533$	5126.29	+	5126.29i	0.416593	+	0.416593i
$534$		0			0
$535$		−	11846.1i		−	0.957293i
$536$		0			0
$537$		−	8249.97i		−	0.662966i
$538$		0			0
$539$	1220.61	+	1220.61i	0.0975428	+	0.0975428i
$540$		0			0
$541$	7722.24	−	7722.24i	0.613688	−	0.613688i	−0.330217	−	0.943905i	$-0.607122\pi$
$541$	7722.24	−	7722.24i	0.613688	−	0.613688i	0.943905	+	0.330217i	$0.107122\pi$
$542$		0			0
$543$	−5892.40			−0.465685
$544$		0			0
$545$	2158.70			0.169667
$546$		0			0
$547$	−7464.17	+	7464.17i	−0.583446	+	0.583446i	−0.935848	−	0.352403i	$-0.885365\pi$
$547$	−7464.17	+	7464.17i	−0.583446	+	0.583446i	0.352403	+	0.935848i	$0.385365\pi$
$548$		0			0
$549$	1605.07	+	1605.07i	0.124777	+	0.124777i
$550$		0			0
$551$			19119.5i			1.47826i
$552$		0			0
$553$			9333.48i			0.717722i
$554$		0			0
$555$	8211.46	+	8211.46i	0.628031	+	0.628031i
$556$		0			0
$557$	10501.3	−	10501.3i	0.798841	−	0.798841i	−0.184072	−	0.982913i	$-0.558928\pi$
$557$	10501.3	−	10501.3i	0.798841	−	0.798841i	0.982913	+	0.184072i	$0.0589280\pi$
$558$		0			0
$559$	36081.6			2.73004
$560$		0			0
$561$	−2053.47			−0.154541
$562$		0			0
$563$	−16986.4	+	16986.4i	−1.27157	+	1.27157i	−0.326300	+	0.945266i	$0.605802\pi$
$563$	−16986.4	+	16986.4i	−1.27157	+	1.27157i	−0.945266	+	0.326300i	$0.894198\pi$
$564$		0			0
$565$	13134.3	+	13134.3i	0.977991	+	0.977991i
$566$		0			0
$567$			1795.04i			0.132954i
$568$		0			0
$569$		−	10274.2i		−	0.756971i	−0.925607	−	0.378486i	$-0.876445\pi$
$569$		−	10274.2i		−	0.756971i	0.925607	−	0.378486i	$-0.123555\pi$
$570$		0			0
$571$	5116.90	+	5116.90i	0.375019	+	0.375019i	0.869301	−	0.494283i	$-0.164569\pi$
$571$	5116.90	+	5116.90i	0.375019	+	0.375019i	−0.494283	+	0.869301i	$0.664569\pi$
$572$		0			0
$573$	3722.02	−	3722.02i	0.271361	−	0.271361i
$574$		0			0
$575$	2203.64			0.159823
$576$		0			0
$577$	4648.41			0.335383			0.167691	−	0.985840i	$-0.446369\pi$
$577$	4648.41			0.335383			0.167691	+	0.985840i	$0.446369\pi$
$578$		0			0
$579$	−8721.34	+	8721.34i	−0.625987	+	0.625987i
$580$		0			0
$581$	−3712.13	−	3712.13i	−0.265069	−	0.265069i
$582$		0			0
$583$		−	3462.95i		−	0.246004i
$584$		0			0
$585$			6812.68i			0.481487i
$586$		0			0
$587$	16839.3	+	16839.3i	1.18404	+	1.18404i	0.978688	+	0.205351i	$0.0658337\pi$
$587$	16839.3	+	16839.3i	1.18404	+	1.18404i	0.205351	+	0.978688i	$0.434166\pi$
$588$		0			0
$589$	−359.247	+	359.247i	−0.0251316	+	0.0251316i
$590$		0			0
$591$	−2483.30			−0.172841
$592$		0			0
$593$	15004.6			1.03906			0.519530	−	0.854452i	$-0.326107\pi$
$593$	15004.6			1.03906			0.519530	+	0.854452i	$0.326107\pi$
$594$		0			0
$595$	−9489.70	+	9489.70i	−0.653848	+	0.653848i
$596$		0			0
$597$	−5559.24	−	5559.24i	−0.381113	−	0.381113i
$598$		0			0
$599$		−	7740.90i		−	0.528021i	−0.964520	−	0.264011i	$-0.914955\pi$
$599$		−	7740.90i		−	0.528021i	0.964520	−	0.264011i	$-0.0850454\pi$
$600$		0			0
$601$		−	19210.1i		−	1.30382i	−0.758295	−	0.651912i	$-0.773967\pi$
$601$		−	19210.1i		−	1.30382i	0.758295	−	0.651912i	$-0.226033\pi$
$602$		0			0
$603$	4326.43	+	4326.43i	0.292182	+	0.292182i
$604$		0			0
$605$	8714.21	−	8714.21i	0.585592	−	0.585592i
$606$		0			0
$607$	−25349.2			−1.69505			−0.847524	−	0.530758i	$-0.821907\pi$
$607$	−25349.2			−1.69505			−0.847524	+	0.530758i	$0.821907\pi$
$608$		0			0
$609$	−16480.4			−1.09658
$610$		0			0
$611$	7364.93	−	7364.93i	0.487648	−	0.487648i
$612$		0			0
$613$	−12517.8	−	12517.8i	−0.824780	−	0.824780i	0.162010	−	0.986789i	$-0.448202\pi$
$613$	−12517.8	−	12517.8i	−0.824780	−	0.824780i	−0.986789	+	0.162010i	$0.948202\pi$
$614$		0			0
$615$			3054.87i			0.200300i
$616$		0			0
$617$			12650.0i			0.825399i	0.910867	+	0.412700i	$0.135414\pi$
$617$			12650.0i			0.825399i	−0.910867	+	0.412700i	$0.864586\pi$
$618$		0			0
$619$	−5166.55	−	5166.55i	−0.335479	−	0.335479i	0.519184	−	0.854663i	$-0.326236\pi$
$619$	−5166.55	−	5166.55i	−0.335479	−	0.335479i	−0.854663	+	0.519184i	$0.826236\pi$
$620$		0			0
$621$	−2252.64	+	2252.64i	−0.145564	+	0.145564i
$622$		0			0
$623$	−14715.3			−0.946318
$624$		0			0
$625$	12941.6			0.828263
$626$		0			0
$627$	1906.92	−	1906.92i	0.121459	−	0.121459i
$628$		0			0
$629$	15590.0	+	15590.0i	0.988260	+	0.988260i
$630$		0			0
$631$		−	12728.9i		−	0.803056i	−0.915847	−	0.401528i	$-0.868479\pi$
$631$		−	12728.9i		−	0.803056i	0.915847	−	0.401528i	$-0.131521\pi$
$632$		0			0
$633$		−	4114.11i		−	0.258327i
$634$		0			0
$635$	−6392.48	−	6392.48i	−0.399493	−	0.399493i
$636$		0			0
$637$	7688.40	−	7688.40i	0.478219	−	0.478219i
$638$		0			0
$639$	−7957.67			−0.492646
$640$		0			0
$641$	8725.04			0.537626			0.268813	−	0.963192i	$-0.413369\pi$
$641$	8725.04			0.537626			0.268813	+	0.963192i	$0.413369\pi$
$642$		0			0
$643$	11974.6	−	11974.6i	0.734418	−	0.734418i	−0.237073	−	0.971492i	$-0.576188\pi$
$643$	11974.6	−	11974.6i	0.734418	−	0.734418i	0.971492	+	0.237073i	$0.0761882\pi$
$644$		0			0
$645$	10750.9	+	10750.9i	0.656306	+	0.656306i
$646$		0			0
$647$		−	16791.7i		−	1.02033i	−0.860078	−	0.510163i	$-0.829585\pi$
$647$		−	16791.7i		−	1.02033i	0.860078	−	0.510163i	$-0.170415\pi$
$648$		0			0
$649$			7043.20i			0.425994i
$650$		0			0
$651$	−309.659	−	309.659i	−0.0186428	−	0.0186428i
$652$		0			0
$653$	−12379.7	+	12379.7i	−0.741894	+	0.741894i	−0.972942	−	0.231049i	$-0.925784\pi$
$653$	−12379.7	+	12379.7i	−0.741894	+	0.741894i	0.231049	+	0.972942i	$0.425784\pi$
$654$		0			0
$655$	19773.5			1.17957
$656$		0			0
$657$	7150.89			0.424631
$658$		0			0
$659$	2164.94	−	2164.94i	0.127973	−	0.127973i	−0.640219	−	0.768192i	$-0.721157\pi$
$659$	2164.94	−	2164.94i	0.127973	−	0.127973i	0.768192	+	0.640219i	$0.221157\pi$
$660$		0			0
$661$	−16951.5	−	16951.5i	−0.997481	−	0.997481i	0.00251552	−	0.999997i	$-0.499199\pi$
$661$	−16951.5	−	16951.5i	−0.997481	−	0.997481i	−0.999997	+	0.00251552i	$0.999199\pi$
$662$		0			0
$663$			12934.4i			0.757661i
$664$		0			0
$665$		−	17624.9i		−	1.02777i
$666$		0			0
$667$	−20681.6	−	20681.6i	−1.20059	−	1.20059i
$668$		0			0
$669$	−134.438	+	134.438i	−0.00776932	+	0.00776932i
$670$		0			0
$671$	2939.48			0.169117
$672$		0			0
$673$	−3167.92			−0.181448			−0.0907238	−	0.995876i	$-0.528918\pi$
$673$	−3167.92			−0.181448			−0.0907238	+	0.995876i	$0.528918\pi$
$674$		0			0
$675$	356.571	−	356.571i	0.0203325	−	0.0203325i
$676$		0			0
$677$	−8613.73	−	8613.73i	−0.488999	−	0.488999i	0.418991	−	0.907990i	$-0.362384\pi$
$677$	−8613.73	−	8613.73i	−0.488999	−	0.488999i	−0.907990	+	0.418991i	$0.862384\pi$
$678$		0			0
$679$			24018.3i			1.35749i
$680$		0			0
$681$			7647.58i			0.430332i
$682$		0			0
$683$	−23079.8	−	23079.8i	−1.29301	−	1.29301i	−0.932916	−	0.360093i	$-0.882745\pi$
$683$	−23079.8	−	23079.8i	−1.29301	−	1.29301i	−0.360093	−	0.932916i	$-0.617255\pi$
$684$		0			0
$685$	−13366.1	+	13366.1i	−0.745539	+	0.745539i
$686$		0			0
$687$	12284.4			0.682214
$688$		0			0
$689$	−21812.4			−1.20608
$690$		0			0
$691$	10996.2	−	10996.2i	0.605375	−	0.605375i	−0.336359	−	0.941734i	$-0.609196\pi$
$691$	10996.2	−	10996.2i	0.605375	−	0.605375i	0.941734	+	0.336359i	$0.109196\pi$
$692$		0			0
$693$	1643.69	+	1643.69i	0.0900993	+	0.0900993i
$694$		0			0
$695$			10289.4i			0.561580i
$696$		0			0
$697$			5799.89i			0.315188i
$698$		0			0
$699$	−9041.10	−	9041.10i	−0.489221	−	0.489221i
$700$		0			0
$701$	15045.8	−	15045.8i	0.810659	−	0.810659i	−0.174074	−	0.984733i	$-0.555693\pi$
$701$	15045.8	−	15045.8i	0.810659	−	0.810659i	0.984733	+	0.174074i	$0.0556931\pi$
$702$		0			0
$703$	−28954.8			−1.55342
$704$		0			0
$705$	4388.92			0.234463
$706$		0			0
$707$	13315.2	−	13315.2i	0.708302	−	0.708302i
$708$		0			0
$709$	13939.7	+	13939.7i	0.738388	+	0.738388i	0.972266	−	0.233878i	$-0.0751415\pi$
$709$	13939.7	+	13939.7i	0.738388	+	0.738388i	−0.233878	+	0.972266i	$0.575141\pi$
$710$		0			0
$711$			3790.50i			0.199936i
$712$		0			0
$713$		−	777.195i		−	0.0408221i
$714$		0			0
$715$	6238.27	+	6238.27i	0.326291	+	0.326291i
$716$		0			0
$717$	−7017.62	+	7017.62i	−0.365520	+	0.365520i
$718$		0			0
$719$	−32099.6			−1.66497			−0.832484	−	0.554050i	$-0.813082\pi$
$719$	−32099.6			−1.66497			−0.832484	+	0.554050i	$0.813082\pi$
$720$		0			0
$721$	−679.968			−0.0351225
$722$		0			0
$723$	6788.26	−	6788.26i	0.349181	−	0.349181i
$724$		0			0
$725$	3273.69	+	3273.69i	0.167699	+	0.167699i
$726$		0			0
$727$			1706.65i			0.0870646i	0.999052	+	0.0435323i	$0.0138611\pi$
$727$			1706.65i			0.0870646i	−0.999052	+	0.0435323i	$0.986139\pi$
$728$		0			0
$729$			729.000i			0.0370370i
$730$		0			0
$731$	20411.4	+	20411.4i	1.03275	+	1.03275i
$732$		0			0
$733$	−19995.6	+	19995.6i	−1.00758	+	1.00758i	−0.00760565	+	0.999971i	$0.502421\pi$
$733$	−19995.6	+	19995.6i	−1.00758	+	1.00758i	−0.999971	+	0.00760565i	$0.997579\pi$
$734$		0			0
$735$	4581.69			0.229930
$736$		0			0
$737$	7923.30			0.396009
$738$		0			0
$739$	3123.50	−	3123.50i	0.155480	−	0.155480i	−0.625080	−	0.780560i	$-0.714934\pi$
$739$	3123.50	−	3123.50i	0.155480	−	0.155480i	0.780560	+	0.625080i	$0.214934\pi$
$740$		0			0
$741$	−12011.3	−	12011.3i	−0.595473	−	0.595473i
$742$		0			0
$743$		−	7764.79i		−	0.383395i	−0.981454	−	0.191698i	$-0.938601\pi$
$743$		−	7764.79i		−	0.383395i	0.981454	−	0.191698i	$-0.0613993\pi$
$744$		0			0
$745$			10423.7i			0.512612i
$746$		0			0
$747$	−1507.56	−	1507.56i	−0.0738406	−	0.0738406i
$748$		0			0
$749$	−18002.7	+	18002.7i	−0.878241	+	0.878241i
$750$		0			0
$751$	7697.83			0.374032			0.187016	−	0.982357i	$-0.440118\pi$
$751$	7697.83			0.374032			0.187016	+	0.982357i	$0.440118\pi$
$752$		0			0
$753$	−8253.25			−0.399422
$754$		0			0
$755$	−23264.7	+	23264.7i	−1.12144	+	1.12144i
$756$		0			0
$757$	−3022.39	−	3022.39i	−0.145113	−	0.145113i	0.630818	−	0.775931i	$-0.282720\pi$
$757$	−3022.39	−	3022.39i	−0.145113	−	0.145113i	−0.775931	+	0.630818i	$0.782720\pi$
$758$		0			0
$759$			4125.42i			0.197290i
$760$		0			0
$761$			26737.7i			1.27364i	0.771012	+	0.636820i	$0.219751\pi$
$761$			26737.7i			1.27364i	−0.771012	+	0.636820i	$0.780249\pi$
$762$		0			0
$763$	−3280.59	−	3280.59i	−0.155656	−	0.155656i
$764$		0			0
$765$	−3853.94	+	3853.94i	−0.182143	+	0.182143i
$766$		0			0
$767$	44363.7			2.08850
$768$		0			0
$769$	−14158.4			−0.663934			−0.331967	−	0.943291i	$-0.607712\pi$
$769$	−14158.4			−0.663934			−0.331967	+	0.943291i	$0.607712\pi$
$770$		0			0
$771$	176.033	−	176.033i	0.00822269	−	0.00822269i
$772$		0			0
$773$	−15285.5	−	15285.5i	−0.711231	−	0.711231i	0.255562	−	0.966793i	$-0.417740\pi$
$773$	−15285.5	−	15285.5i	−0.711231	−	0.711231i	−0.966793	+	0.255562i	$0.917740\pi$
$774$		0			0
$775$			123.022i			0.00570206i
$776$		0			0
$777$		−	24958.1i		−	1.15234i
$778$		0			0
$779$	−5385.96	−	5385.96i	−0.247718	−	0.247718i
$780$		0			0
$781$	−7286.72	+	7286.72i	−0.333853	+	0.333853i
$782$		0			0
$783$	−6692.97			−0.305476
$784$		0			0
$785$	−6881.16			−0.312865
$786$		0			0
$787$	11317.7	−	11317.7i	0.512622	−	0.512622i	−0.402707	−	0.915329i	$-0.631931\pi$
$787$	11317.7	−	11317.7i	0.512622	−	0.512622i	0.915329	+	0.402707i	$0.131931\pi$
$788$		0			0
$789$	4538.72	+	4538.72i	0.204794	+	0.204794i
$790$		0			0
$791$		−	39920.7i		−	1.79446i
$792$		0			0
$793$		−	18515.2i		−	0.829121i
$794$		0			0
$795$	−6499.25	−	6499.25i	−0.289943	−	0.289943i
$796$		0			0
$797$	−24453.7	+	24453.7i	−1.08682	+	1.08682i	−0.0909639	+	0.995854i	$0.528995\pi$
$797$	−24453.7	+	24453.7i	−1.08682	+	1.08682i	−0.995854	+	0.0909639i	$0.971005\pi$
$798$		0			0
$799$	8332.68			0.368948
$800$		0			0
$801$	−5976.15			−0.263616
$802$		0			0
$803$	6547.96	−	6547.96i	0.287761	−	0.287761i
$804$		0			0
$805$	19064.8	+	19064.8i	0.834717	+	0.834717i
$806$		0			0
$807$		−	10879.8i		−	0.474583i
$808$		0			0
$809$			4516.01i			0.196260i	0.995174	+	0.0981301i	$0.0312861\pi$
$809$			4516.01i			0.196260i	−0.995174	+	0.0981301i	$0.968714\pi$
$810$		0			0
$811$	14460.0	+	14460.0i	0.626092	+	0.626092i	0.947082	−	0.320990i	$-0.104016\pi$
$811$	14460.0	+	14460.0i	0.626092	+	0.626092i	−0.320990	+	0.947082i	$0.604016\pi$
$812$		0			0
$813$	−2914.44	+	2914.44i	−0.125724	+	0.125724i
$814$		0			0
$815$	−4016.96			−0.172648
$816$		0			0
$817$	−37909.4			−1.62336
$818$		0			0
$819$	10353.3	−	10353.3i	0.441726	−	0.441726i
$820$		0			0
$821$	−20286.7	−	20286.7i	−0.862377	−	0.862377i	0.129237	−	0.991614i	$-0.458747\pi$
$821$	−20286.7	−	20286.7i	−0.862377	−	0.862377i	−0.991614	+	0.129237i	$0.958747\pi$
$822$		0			0
$823$			27490.2i			1.16434i	0.813068	+	0.582168i	$0.197796\pi$
$823$			27490.2i			1.16434i	−0.813068	+	0.582168i	$0.802204\pi$
$824$		0			0
$825$		−	653.013i		−	0.0275576i
$826$		0			0
$827$	4576.46	+	4576.46i	0.192429	+	0.192429i	0.796745	−	0.604316i	$-0.206553\pi$
$827$	4576.46	+	4576.46i	0.192429	+	0.192429i	−0.604316	+	0.796745i	$0.706553\pi$
$828$		0			0
$829$	14697.8	−	14697.8i	0.615772	−	0.615772i	−0.328672	−	0.944444i	$-0.606601\pi$
$829$	14697.8	−	14697.8i	0.615772	−	0.615772i	0.944444	+	0.328672i	$0.106601\pi$
$830$		0			0
$831$	7480.13			0.312254
$832$		0			0
$833$	8698.66			0.361814
$834$		0			0
$835$	−12792.6	+	12792.6i	−0.530186	+	0.530186i
$836$		0			0
$837$	−125.758	−	125.758i	−0.00519335	−	0.00519335i
$838$		0			0
$839$		−	9206.59i		−	0.378840i	−0.981896	−	0.189420i	$-0.939339\pi$
$839$		−	9206.59i		−	0.378840i	0.981896	−	0.189420i	$-0.0606608\pi$
$840$		0			0
$841$		−	37059.4i		−	1.51951i
$842$		0			0
$843$	−17333.1	−	17333.1i	−0.708165	−	0.708165i
$844$		0			0
$845$	23274.8	−	23274.8i	0.947548	−	0.947548i
$846$		0			0
$847$	−26486.1			−1.07447
$848$		0			0
$849$	5167.27			0.208881
$850$		0			0
$851$	31320.4	−	31320.4i	1.26163	−	1.26163i
$852$		0			0
$853$	17252.0	+	17252.0i	0.692494	+	0.692494i	0.962780	−	0.270286i	$-0.0871184\pi$
$853$	17252.0	+	17252.0i	0.692494	+	0.692494i	−0.270286	+	0.962780i	$0.587118\pi$
$854$		0			0
$855$		−	7157.79i		−	0.286305i
$856$		0			0
$857$			22957.2i			0.915057i	0.889195	+	0.457529i	$0.151265\pi$
$857$			22957.2i			0.915057i	−0.889195	+	0.457529i	$0.848735\pi$
$858$		0			0
$859$	−6376.26	−	6376.26i	−0.253266	−	0.253266i	0.569043	−	0.822308i	$-0.307314\pi$
$859$	−6376.26	−	6376.26i	−0.253266	−	0.253266i	−0.822308	+	0.569043i	$0.807314\pi$
$860$		0			0
$861$	4642.51	−	4642.51i	0.183759	−	0.183759i
$862$		0			0
$863$	−35795.9			−1.41194			−0.705971	−	0.708241i	$-0.749489\pi$
$863$	−35795.9			−1.41194			−0.705971	+	0.708241i	$0.749489\pi$
$864$		0			0
$865$	−14181.9			−0.557454
$866$		0			0
$867$	3105.07	−	3105.07i	0.121631	−	0.121631i
$868$		0			0
$869$	3470.90	+	3470.90i	0.135492	+	0.135492i
$870$		0			0
$871$		−	49907.3i		−	1.94150i
$872$		0			0
$873$			9754.25i			0.378157i
$874$		0			0
$875$	−23215.4	−	23215.4i	−0.896940	−	0.896940i
$876$		0			0
$877$	−9797.22	+	9797.22i	−0.377228	+	0.377228i	−0.870101	−	0.492873i	$-0.835947\pi$
$877$	−9797.22	+	9797.22i	−0.377228	+	0.377228i	0.492873	+	0.870101i	$0.335947\pi$
$878$		0			0
$879$	12702.8			0.487435
$880$		0			0
$881$	−3227.81			−0.123437			−0.0617183	−	0.998094i	$-0.519658\pi$
$881$	−3227.81			−0.123437			−0.0617183	+	0.998094i	$0.519658\pi$
$882$		0			0
$883$	−33226.7	+	33226.7i	−1.26633	+	1.26633i	−0.318354	+	0.947972i	$0.603130\pi$
$883$	−33226.7	+	33226.7i	−1.26633	+	1.26633i	−0.947972	+	0.318354i	$0.896870\pi$
$884$		0			0
$885$	13218.7	+	13218.7i	0.502079	+	0.502079i
$886$		0			0
$887$		−	3411.07i		−	0.129123i	−0.997914	−	0.0645617i	$-0.979435\pi$
$887$		−	3411.07i		−	0.129123i	0.997914	−	0.0645617i	$-0.0205649\pi$
$888$		0			0
$889$			19429.4i			0.733006i
$890$		0			0
$891$	667.534	+	667.534i	0.0250990	+	0.0250990i
$892$		0			0
$893$	−7738.00	+	7738.00i	−0.289969	+	0.289969i
$894$		0			0
$895$	−28356.0			−1.05904
$896$		0			0
$897$	25985.2			0.967246
$898$		0			0
$899$	1154.59	−	1154.59i	0.0428339	−	0.0428339i
$900$		0			0
$901$	−12339.3	−	12339.3i	−0.456250	−	0.456250i
$902$		0			0
$903$		−	32676.6i		−	1.20422i
$904$		0			0
$905$			20252.8i			0.743896i
$906$		0			0
$907$	−11317.0	−	11317.0i	−0.414305	−	0.414305i	0.468930	−	0.883235i	$-0.344640\pi$
$907$	−11317.0	−	11317.0i	−0.414305	−	0.414305i	−0.883235	+	0.468930i	$0.844640\pi$
$908$		0			0
$909$	5407.54	−	5407.54i	0.197312	−	0.197312i
$910$		0			0
$911$	5574.60			0.202738			0.101369	−	0.994849i	$-0.467678\pi$
$911$	5574.60			0.202738			0.101369	+	0.994849i	$0.467678\pi$
$912$		0			0
$913$	−2760.91			−0.100080
$914$		0			0
$915$	5516.80	−	5516.80i	0.199322	−	0.199322i
$916$		0			0
$917$	−30050.1	−	30050.1i	−1.08216	−	1.08216i
$918$		0			0
$919$			10717.9i			0.384713i	0.981325	+	0.192356i	$0.0616130\pi$
$919$			10717.9i			0.384713i	−0.981325	+	0.192356i	$0.938387\pi$
$920$		0			0
$921$			22834.0i			0.816944i
$922$		0			0
$923$	45897.6	+	45897.6i	1.63677	+	1.63677i
$924$		0			0
$925$	−4957.72	+	4957.72i	−0.176226	+	0.176226i
$926$		0			0
$927$	−276.147			−0.00978410
$928$		0			0
$929$	−5276.25			−0.186338			−0.0931692	−	0.995650i	$-0.529700\pi$
$929$	−5276.25			−0.186338			−0.0931692	+	0.995650i	$0.529700\pi$
$930$		0			0
$931$	−8077.87	+	8077.87i	−0.284362	+	0.284362i
$932$		0			0
$933$	16558.4	+	16558.4i	0.581025	+	0.581025i
$934$		0			0
$935$			7057.98i			0.246867i
$936$		0			0
$937$		−	34230.9i		−	1.19346i	−0.802441	−	0.596731i	$-0.796466\pi$
$937$		−	34230.9i		−	1.19346i	0.802441	−	0.596731i	$-0.203534\pi$
$938$		0			0
$939$	3286.45	+	3286.45i	0.114216	+	0.114216i
$940$		0			0
$941$	−2969.85	+	2969.85i	−0.102884	+	0.102884i	−0.756675	−	0.653791i	$-0.773178\pi$
$941$	−2969.85	+	2969.85i	−0.102884	+	0.102884i	0.653791	+	0.756675i	$0.273178\pi$
$942$		0			0
$943$	11652.0			0.402376
$944$		0			0
$945$	6169.76			0.212383
$946$		0			0
$947$	−37250.4	+	37250.4i	−1.27822	+	1.27822i	−0.336557	+	0.941663i	$0.609263\pi$
$947$	−37250.4	+	37250.4i	−1.27822	+	1.27822i	−0.941663	+	0.336557i	$0.890737\pi$
$948$		0			0
$949$	−41244.3	−	41244.3i	−1.41080	−	1.41080i
$950$		0			0
$951$			25932.5i			0.884247i
$952$		0			0
$953$			39434.0i			1.34039i	0.742185	+	0.670195i	$0.233790\pi$
$953$			39434.0i			1.34039i	−0.742185	+	0.670195i	$0.766210\pi$
$954$		0			0
$955$	−12793.0	−	12793.0i	−0.433478	−	0.433478i
$956$		0			0
$957$	−6128.65	+	6128.65i	−0.207013	+	0.207013i
$958$		0			0
$959$	40625.3			1.36795
$960$		0			0
$961$	−29747.6			−0.998544
$962$		0			0
$963$	−7311.20	+	7311.20i	−0.244652	+	0.244652i
$964$		0			0
$965$	29976.2	+	29976.2i	0.999967	+	0.999967i
$966$		0			0
$967$		−	2688.08i		−	0.0893927i	−0.999001	−	0.0446964i	$-0.985768\pi$
$967$		−	2688.08i		−	0.0893927i	0.999001	−	0.0446964i	$-0.0142320\pi$
$968$		0			0
$969$		−	13589.6i		−	0.450526i
$970$		0			0
$971$	−33370.7	−	33370.7i	−1.10290	−	1.10290i	−0.994059	−	0.108840i	$-0.965286\pi$
$971$	−33370.7	−	33370.7i	−1.10290	−	1.10290i	−0.108840	−	0.994059i	$-0.534714\pi$
$972$		0			0
$973$	15636.9	−	15636.9i	0.515205	−	0.515205i
$974$		0			0
$975$	−4113.20			−0.135105
$976$		0			0
$977$	8643.45			0.283039			0.141519	−	0.989936i	$-0.454801\pi$
$977$	8643.45			0.283039			0.141519	+	0.989936i	$0.454801\pi$
$978$		0			0
$979$	−5472.27	+	5472.27i	−0.178646	+	0.178646i
$980$		0			0
$981$	−1332.31	−	1332.31i	−0.0433612	−	0.0433612i
$982$		0			0
$983$		−	50643.1i		−	1.64320i	−0.570067	−	0.821598i	$-0.693083\pi$
$983$		−	50643.1i		−	1.64320i	0.570067	−	0.821598i	$-0.306917\pi$
$984$		0			0
$985$			8535.36i			0.276101i
$986$		0			0
$987$	−6669.89	−	6669.89i	−0.215101	−	0.215101i
$988$		0			0
$989$	41006.6	−	41006.6i	1.31844	−	1.31844i
$990$		0			0
$991$	−11424.2			−0.366198			−0.183099	−	0.983094i	$-0.558613\pi$
$991$	−11424.2			−0.366198			−0.183099	+	0.983094i	$0.558613\pi$
$992$		0			0
$993$	15015.0			0.479846
$994$		0			0
$995$	−19107.7	+	19107.7i	−0.608799	+	0.608799i
$996$		0			0
$997$	−35877.3	−	35877.3i	−1.13967	−	1.13967i	−0.988510	−	0.151155i	$-0.951701\pi$
$997$	−35877.3	−	35877.3i	−1.13967	−	1.13967i	−0.151155	−	0.988510i	$-0.548299\pi$
$998$		0			0
$999$		−	10135.9i		−	0.321007i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.j.b.97.5		24
4.3	odd	2		384.4.j.a.97.8		24
8.3	odd	2		192.4.j.a.49.2		24
8.5	even	2		48.4.j.a.37.3	yes	24
16.3	odd	4		384.4.j.a.289.8		24
16.5	even	4		48.4.j.a.13.3	✓	24
16.11	odd	4		192.4.j.a.145.2		24
16.13	even	4	inner	384.4.j.b.289.5		24
24.5	odd	2		144.4.k.b.37.10		24
24.11	even	2		576.4.k.b.433.9		24
48.5	odd	4		144.4.k.b.109.10		24
48.11	even	4		576.4.k.b.145.9		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.4.j.a.13.3	✓	24	16.5	even	4
48.4.j.a.37.3	yes	24	8.5	even	2
144.4.k.b.37.10		24	24.5	odd	2
144.4.k.b.109.10		24	48.5	odd	4
192.4.j.a.49.2		24	8.3	odd	2
192.4.j.a.145.2		24	16.11	odd	4
384.4.j.a.97.8		24	4.3	odd	2
384.4.j.a.289.8		24	16.3	odd	4
384.4.j.b.97.5		24	1.1	even	1	trivial
384.4.j.b.289.5		24	16.13	even	4	inner
576.4.k.b.145.9		24	48.11	even	4
576.4.k.b.433.9		24	24.11	even	2

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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.12132 + 2.12132i) q^{3} +(7.29121 + 7.29121i) q^{5} -22.1610i q^{7} -9.00000i q^{9} +O(q^{10})\)	\(q+(-2.12132 + 2.12132i) q^{3} +(7.29121 + 7.29121i) q^{5} -22.1610i q^{7} -9.00000i q^{9} +(-8.24116 - 8.24116i) q^{11} +(-51.9094 + 51.9094i) q^{13} -30.9340 q^{15} -58.7304 q^{17} +(54.5389 - 54.5389i) q^{19} +(47.0107 + 47.0107i) q^{21} +117.989i q^{23} -18.6766i q^{25} +(19.0919 + 19.0919i) q^{27} +(-175.283 + 175.283i) q^{29} -6.58699 q^{31} +34.9643 q^{33} +(161.581 - 161.581i) q^{35} +(-265.451 - 265.451i) q^{37} -220.233i q^{39} -98.7545i q^{41} +(-347.544 - 347.544i) q^{43} +(65.6209 - 65.6209i) q^{45} -141.880 q^{47} -148.112 q^{49} +(124.586 - 124.586i) q^{51} +(210.101 + 210.101i) q^{53} -120.176i q^{55} +231.389i q^{57} +(-427.318 - 427.318i) q^{59} +(-178.341 + 178.341i) q^{61} -199.449 q^{63} -756.965 q^{65} +(-480.715 + 480.715i) q^{67} +(-250.293 - 250.293i) q^{69} -884.186i q^{71} +794.543i q^{73} +(39.6190 + 39.6190i) q^{75} +(-182.633 + 182.633i) q^{77} -421.166 q^{79} -81.0000 q^{81} +(167.507 - 167.507i) q^{83} +(-428.215 - 428.215i) q^{85} -743.664i q^{87} -664.016i q^{89} +(1150.37 + 1150.37i) q^{91} +(13.9731 - 13.9731i) q^{93} +795.309 q^{95} -1083.81 q^{97} +(-74.1705 + 74.1705i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 40 q^{11} + 120 q^{15} - 24 q^{19} - 400 q^{29} - 744 q^{31} + 456 q^{35} - 16 q^{37} - 1240 q^{43} - 1176 q^{49} - 744 q^{51} - 752 q^{53} + 1376 q^{59} + 912 q^{61} - 504 q^{63} + 976 q^{65} + 2256 q^{67} + 528 q^{69} - 1104 q^{75} - 1904 q^{77} + 5992 q^{79} - 1944 q^{81} - 2680 q^{83} + 240 q^{85} + 3496 q^{91} - 7728 q^{95} + 360 q^{99}+O(q^{100}) \) 24 * q + 40 * q^11 + 120 * q^15 - 24 * q^19 - 400 * q^29 - 744 * q^31 + 456 * q^35 - 16 * q^37 - 1240 * q^43 - 1176 * q^49 - 744 * q^51 - 752 * q^53 + 1376 * q^59 + 912 * q^61 - 504 * q^63 + 976 * q^65 + 2256 * q^67 + 528 * q^69 - 1104 * q^75 - 1904 * q^77 + 5992 * q^79 - 1944 * q^81 - 2680 * q^83 + 240 * q^85 + 3496 * q^91 - 7728 * q^95 + 360 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more