LMFDB - Embedded newform 384.5.l.a.31.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,5,Mod(31,384)]

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([2, 1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("384.31");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	384.l (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:	$39.6940658242$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			31.6
Character	$\chi$	$=$	384.31
Dual form			384.5.l.a.223.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.67423 - 3.67423i) q^{3} +(2.10268 + 2.10268i) q^{5} -84.8276 q^{7} +27.0000i q^{9} +O(q^{10})$	\(q+(-3.67423 - 3.67423i) q^{3} +(2.10268 + 2.10268i) q^{5} -84.8276 q^{7} +27.0000i q^{9} +(-57.8744 + 57.8744i) q^{11} +(-192.435 + 192.435i) q^{13} -15.4515i q^{15} +507.642 q^{17} +(-126.683 - 126.683i) q^{19} +(311.677 + 311.677i) q^{21} +173.371 q^{23} -616.157i q^{25} +(99.2043 - 99.2043i) q^{27} +(64.7538 - 64.7538i) q^{29} +366.621i q^{31} +425.288 q^{33} +(-178.365 - 178.365i) q^{35} +(-801.887 - 801.887i) q^{37} +1414.11 q^{39} +461.370i q^{41} +(-1147.51 + 1147.51i) q^{43} +(-56.7723 + 56.7723i) q^{45} -4098.44i q^{47} +4794.73 q^{49} +(-1865.20 - 1865.20i) q^{51} +(2542.44 + 2542.44i) q^{53} -243.383 q^{55} +930.924i q^{57} +(959.072 - 959.072i) q^{59} +(162.570 - 162.570i) q^{61} -2290.35i q^{63} -809.260 q^{65} +(-771.580 - 771.580i) q^{67} +(-637.006 - 637.006i) q^{69} +2063.45 q^{71} -463.678i q^{73} +(-2263.91 + 2263.91i) q^{75} +(4909.35 - 4909.35i) q^{77} -8437.39i q^{79} -729.000 q^{81} +(-3103.77 - 3103.77i) q^{83} +(1067.41 + 1067.41i) q^{85} -475.841 q^{87} -7668.88i q^{89} +(16323.8 - 16323.8i) q^{91} +(1347.05 - 1347.05i) q^{93} -532.746i q^{95} +16387.4 q^{97} +(-1562.61 - 1562.61i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 192 q^{11} + 704 q^{19} + 2304 q^{23} + 1728 q^{29} - 5184 q^{35} - 3648 q^{37} + 1088 q^{43} + 10976 q^{49} - 4032 q^{51} - 960 q^{53} - 11776 q^{55} + 13056 q^{59} - 3776 q^{61} + 4032 q^{65} - 896 q^{67} + 9792 q^{69} + 39936 q^{71} - 1152 q^{75} - 9408 q^{77} - 23328 q^{81} + 24000 q^{83} + 11200 q^{85} + 30528 q^{91} - 5184 q^{99}+O(q^{100}) $ 32 * q - 192 * q^11 + 704 * q^19 + 2304 * q^23 + 1728 * q^29 - 5184 * q^35 - 3648 * q^37 + 1088 * q^43 + 10976 * q^49 - 4032 * q^51 - 960 * q^53 - 11776 * q^55 + 13056 * q^59 - 3776 * q^61 + 4032 * q^65 - 896 * q^67 + 9792 * q^69 + 39936 * q^71 - 1152 * q^75 - 9408 * q^77 - 23328 * q^81 + 24000 * q^83 + 11200 * q^85 + 30528 * q^91 - 5184 * 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
$3$	−3.67423	−	3.67423i	−0.408248	−	0.408248i
$3$	−3.67423	−	3.67423i	−0.408248	−	0.408248i
$4$		0			0
$5$	2.10268	+	2.10268i	0.0841071	+	0.0841071i	0.747909	−	0.663802i	$-0.231058\pi$
$5$	2.10268	+	2.10268i	0.0841071	+	0.0841071i	−0.663802	+	0.747909i	$0.731058\pi$
$6$		0			0
$7$	−84.8276			−1.73118			−0.865588	−	0.500757i	$-0.833055\pi$
$7$	−84.8276			−1.73118			−0.865588	+	0.500757i	$0.833055\pi$
$8$		0			0
$9$			27.0000i			0.333333i
$10$		0			0
$11$	−57.8744	+	57.8744i	−0.478301	+	0.478301i	−0.904588	−	0.426287i	$-0.859821\pi$
$11$	−57.8744	+	57.8744i	−0.478301	+	0.478301i	0.426287	+	0.904588i	$0.359821\pi$
$12$		0			0
$13$	−192.435	+	192.435i	−1.13867	+	1.13867i	−0.149983	+	0.988689i	$0.547922\pi$
$13$	−192.435	+	192.435i	−1.13867	+	1.13867i	−0.988689	+	0.149983i	$0.952078\pi$
$14$		0			0
$15$		−	15.4515i		−	0.0686732i
$16$		0			0
$17$	507.642			1.75655			0.878273	−	0.478159i	$-0.158696\pi$
$17$	507.642			1.75655			0.878273	+	0.478159i	$0.158696\pi$
$18$		0			0
$19$	−126.683	−	126.683i	−0.350922	−	0.350922i	0.509531	−	0.860452i	$-0.329819\pi$
$19$	−126.683	−	126.683i	−0.350922	−	0.350922i	−0.860452	+	0.509531i	$0.829819\pi$
$20$		0			0
$21$	311.677	+	311.677i	0.706750	+	0.706750i
$22$		0			0
$23$	173.371			0.327734			0.163867	−	0.986482i	$-0.447603\pi$
$23$	173.371			0.327734			0.163867	+	0.986482i	$0.447603\pi$
$24$		0			0
$25$		−	616.157i		−	0.985852i
$26$		0			0
$27$	99.2043	−	99.2043i	0.136083	−	0.136083i
$28$		0			0
$29$	64.7538	−	64.7538i	0.0769961	−	0.0769961i	−0.667560	−	0.744556i	$-0.732661\pi$
$29$	64.7538	−	64.7538i	0.0769961	−	0.0769961i	0.744556	+	0.667560i	$0.232661\pi$
$30$		0			0
$31$			366.621i			0.381500i	0.981639	+	0.190750i	$0.0610920\pi$
$31$			366.621i			0.381500i	−0.981639	+	0.190750i	$0.938908\pi$
$32$		0			0
$33$	425.288			0.390531
$34$		0			0
$35$	−178.365	−	178.365i	−0.145604	−	0.145604i
$36$		0			0
$37$	−801.887	−	801.887i	−0.585747	−	0.585747i	0.350730	−	0.936477i	$-0.385934\pi$
$37$	−801.887	−	801.887i	−0.585747	−	0.585747i	−0.936477	+	0.350730i	$0.885934\pi$
$38$		0			0
$39$	1414.11			0.929721
$40$		0			0
$41$			461.370i			0.274462i	0.990539	+	0.137231i	$0.0438202\pi$
$41$			461.370i			0.274462i	−0.990539	+	0.137231i	$0.956180\pi$
$42$		0			0
$43$	−1147.51	+	1147.51i	−0.620613	+	0.620613i	−0.945688	−	0.325075i	$-0.894610\pi$
$43$	−1147.51	+	1147.51i	−0.620613	+	0.620613i	0.325075	+	0.945688i	$0.394610\pi$
$44$		0			0
$45$	−56.7723	+	56.7723i	−0.0280357	+	0.0280357i
$46$		0			0
$47$		−	4098.44i		−	1.85534i	−0.373406	−	0.927668i	$-0.621810\pi$
$47$		−	4098.44i		−	1.85534i	0.373406	−	0.927668i	$-0.378190\pi$
$48$		0			0
$49$	4794.73			1.99697
$50$		0			0
$51$	−1865.20	−	1865.20i	−0.717107	−	0.717107i
$52$		0			0
$53$	2542.44	+	2542.44i	0.905106	+	0.905106i	0.995872	−	0.0907664i	$-0.0289317\pi$
$53$	2542.44	+	2542.44i	0.905106	+	0.905106i	−0.0907664	+	0.995872i	$0.528932\pi$
$54$		0			0
$55$	−243.383			−0.0804571
$56$		0			0
$57$			930.924i			0.286526i
$58$		0			0
$59$	959.072	−	959.072i	0.275516	−	0.275516i	−0.555800	−	0.831316i	$-0.687588\pi$
$59$	959.072	−	959.072i	0.275516	−	0.275516i	0.831316	+	0.555800i	$0.187588\pi$
$60$		0			0
$61$	162.570	−	162.570i	0.0436898	−	0.0436898i	−0.684924	−	0.728614i	$-0.740165\pi$
$61$	162.570	−	162.570i	0.0436898	−	0.0436898i	0.728614	+	0.684924i	$0.240165\pi$
$62$		0			0
$63$		−	2290.35i		−	0.577059i
$64$		0			0
$65$	−809.260			−0.191541
$66$		0			0
$67$	−771.580	−	771.580i	−0.171882	−	0.171882i	0.615924	−	0.787806i	$-0.288783\pi$
$67$	−771.580	−	771.580i	−0.171882	−	0.171882i	−0.787806	+	0.615924i	$0.788783\pi$
$68$		0			0
$69$	−637.006	−	637.006i	−0.133797	−	0.133797i
$70$		0			0
$71$	2063.45			0.409334			0.204667	−	0.978832i	$-0.434389\pi$
$71$	2063.45			0.409334			0.204667	+	0.978832i	$0.434389\pi$
$72$		0			0
$73$		−	463.678i		−	0.0870103i	−0.999053	−	0.0435052i	$-0.986148\pi$
$73$		−	463.678i		−	0.0870103i	0.999053	−	0.0435052i	$-0.0138525\pi$
$74$		0			0
$75$	−2263.91	+	2263.91i	−0.402472	+	0.402472i
$76$		0			0
$77$	4909.35	−	4909.35i	0.828023	−	0.828023i
$78$		0			0
$79$		−	8437.39i		−	1.35193i	−0.736934	−	0.675965i	$-0.763727\pi$
$79$		−	8437.39i		−	1.35193i	0.736934	−	0.675965i	$-0.236273\pi$
$80$		0			0
$81$	−729.000			−0.111111
$82$		0			0
$83$	−3103.77	−	3103.77i	−0.450540	−	0.450540i	0.444993	−	0.895534i	$-0.353206\pi$
$83$	−3103.77	−	3103.77i	−0.450540	−	0.450540i	−0.895534	+	0.444993i	$0.853206\pi$
$84$		0			0
$85$	1067.41	+	1067.41i	0.147738	+	0.147738i
$86$		0			0
$87$	−475.841			−0.0628671
$88$		0			0
$89$		−	7668.88i		−	0.968171i	−0.875021	−	0.484085i	$-0.839152\pi$
$89$		−	7668.88i		−	0.968171i	0.875021	−	0.484085i	$-0.160848\pi$
$90$		0			0
$91$	16323.8	−	16323.8i	1.97124	−	1.97124i
$92$		0			0
$93$	1347.05	−	1347.05i	0.155747	−	0.155747i
$94$		0			0
$95$		−	532.746i		−	0.0590300i
$96$		0			0
$97$	16387.4			1.74167			0.870835	−	0.491575i	$-0.163579\pi$
$97$	16387.4			1.74167			0.870835	+	0.491575i	$0.163579\pi$
$98$		0			0
$99$	−1562.61	−	1562.61i	−0.159434	−	0.159434i
$100$		0			0
$101$	7819.54	+	7819.54i	0.766546	+	0.766546i	0.977497	−	0.210951i	$-0.0676559\pi$
$101$	7819.54	+	7819.54i	0.766546	+	0.766546i	−0.210951	+	0.977497i	$0.567656\pi$
$102$		0			0
$103$	5373.34			0.506489			0.253244	−	0.967402i	$-0.418502\pi$
$103$	5373.34			0.506489			0.253244	+	0.967402i	$0.418502\pi$
$104$		0			0
$105$			1310.71i			0.118885i
$106$		0			0
$107$	9702.27	−	9702.27i	0.847434	−	0.847434i	−0.142378	−	0.989812i	$-0.545475\pi$
$107$	9702.27	−	9702.27i	0.847434	−	0.847434i	0.989812	+	0.142378i	$0.0454749\pi$
$108$		0			0
$109$	10200.4	−	10200.4i	0.858550	−	0.858550i	−0.132618	−	0.991167i	$-0.542338\pi$
$109$	10200.4	−	10200.4i	0.858550	−	0.858550i	0.991167	+	0.132618i	$0.0423382\pi$
$110$		0			0
$111$			5892.64i			0.478260i
$112$		0			0
$113$	−13057.9			−1.02262			−0.511311	−	0.859395i	$-0.670840\pi$
$113$	−13057.9			−1.02262			−0.511311	+	0.859395i	$0.670840\pi$
$114$		0			0
$115$	364.543	+	364.543i	0.0275647	+	0.0275647i
$116$		0			0
$117$	−5195.76	−	5195.76i	−0.379557	−	0.379557i
$118$		0			0
$119$	−43062.1			−3.04089
$120$		0			0
$121$			7942.10i			0.542456i
$122$		0			0
$123$	1695.18	−	1695.18i	0.112049	−	0.112049i
$124$		0			0
$125$	2609.75	−	2609.75i	0.167024	−	0.167024i
$126$		0			0
$127$			12757.2i			0.790948i	0.918477	+	0.395474i	$0.129420\pi$
$127$			12757.2i			0.790948i	−0.918477	+	0.395474i	$0.870580\pi$
$128$		0			0
$129$	8432.47			0.506728
$130$		0			0
$131$	6898.48	+	6898.48i	0.401986	+	0.401986i	0.878932	−	0.476946i	$-0.158256\pi$
$131$	6898.48	+	6898.48i	0.401986	+	0.401986i	−0.476946	+	0.878932i	$0.658256\pi$
$132$		0			0
$133$	10746.2	+	10746.2i	0.607507	+	0.607507i
$134$		0			0
$135$	417.190			0.0228911
$136$		0			0
$137$			19730.1i			1.05121i	0.850730	+	0.525603i	$0.176160\pi$
$137$			19730.1i			1.05121i	−0.850730	+	0.525603i	$0.823840\pi$
$138$		0			0
$139$	−22880.9	+	22880.9i	−1.18425	+	1.18425i	−0.205619	+	0.978632i	$0.565921\pi$
$139$	−22880.9	+	22880.9i	−1.18425	+	1.18425i	−0.978632	+	0.205619i	$0.934079\pi$
$140$		0			0
$141$	−15058.6	+	15058.6i	−0.757438	+	0.757438i
$142$		0			0
$143$		−	22274.2i		−	1.08926i
$144$		0			0
$145$	272.313			0.0129518
$146$		0			0
$147$	−17616.9	−	17616.9i	−0.815260	−	0.815260i
$148$		0			0
$149$	14167.4	+	14167.4i	0.638141	+	0.638141i	0.950097	−	0.311955i	$-0.100984\pi$
$149$	14167.4	+	14167.4i	0.638141	+	0.638141i	−0.311955	+	0.950097i	$0.600984\pi$
$150$		0			0
$151$	30206.8			1.32480			0.662402	−	0.749149i	$-0.269537\pi$
$151$	30206.8			1.32480			0.662402	+	0.749149i	$0.269537\pi$
$152$		0			0
$153$			13706.3i			0.585516i
$154$		0			0
$155$	−770.887	+	770.887i	−0.0320869	+	0.0320869i
$156$		0			0
$157$	17987.6	−	17987.6i	0.729749	−	0.729749i	−0.240820	−	0.970570i	$-0.577416\pi$
$157$	17987.6	−	17987.6i	0.729749	−	0.729749i	0.970570	+	0.240820i	$0.0774165\pi$
$158$		0			0
$159$		−	18683.1i		−	0.739016i
$160$		0			0
$161$	−14706.7			−0.567364
$162$		0			0
$163$	3229.11	+	3229.11i	0.121537	+	0.121537i	0.765259	−	0.643722i	$-0.222611\pi$
$163$	3229.11	+	3229.11i	0.121537	+	0.121537i	−0.643722	+	0.765259i	$0.722611\pi$
$164$		0			0
$165$	894.245	+	894.245i	0.0328465	+	0.0328465i
$166$		0			0
$167$	16592.4			0.594946			0.297473	−	0.954730i	$-0.403856\pi$
$167$	16592.4			0.594946			0.297473	+	0.954730i	$0.403856\pi$
$168$		0			0
$169$		−	45501.8i		−	1.59315i
$170$		0			0
$171$	3420.43	−	3420.43i	0.116974	−	0.116974i
$172$		0			0
$173$	−4707.05	+	4707.05i	−0.157274	+	0.157274i	−0.781358	−	0.624084i	$-0.785472\pi$
$173$	−4707.05	+	4707.05i	−0.157274	+	0.157274i	0.624084	+	0.781358i	$0.285472\pi$
$174$		0			0
$175$			52267.2i			1.70668i
$176$		0			0
$177$	−7047.71			−0.224958
$178$		0			0
$179$	26175.4	+	26175.4i	0.816934	+	0.816934i	0.985663	−	0.168729i	$-0.0539662\pi$
$179$	26175.4	+	26175.4i	0.816934	+	0.816934i	−0.168729	+	0.985663i	$0.553966\pi$
$180$		0			0
$181$	−40133.6	−	40133.6i	−1.22504	−	1.22504i	−0.965817	−	0.259223i	$-0.916533\pi$
$181$	−40133.6	−	40133.6i	−1.22504	−	1.22504i	−0.259223	−	0.965817i	$-0.583467\pi$
$182$		0			0
$183$	−1194.64			−0.0356726
$184$		0			0
$185$		−	3372.22i		−	0.0985310i
$186$		0			0
$187$	−29379.5	+	29379.5i	−0.840158	+	0.840158i
$188$		0			0
$189$	−8415.27	+	8415.27i	−0.235583	+	0.235583i
$190$		0			0
$191$		−	10298.6i		−	0.282299i	−0.989988	−	0.141150i	$-0.954920\pi$
$191$		−	10298.6i		−	0.282299i	0.989988	−	0.141150i	$-0.0450798\pi$
$192$		0			0
$193$	−21315.6			−0.572245			−0.286122	−	0.958193i	$-0.592366\pi$
$193$	−21315.6			−0.572245			−0.286122	+	0.958193i	$0.592366\pi$
$194$		0			0
$195$	2973.41	+	2973.41i	0.0781962	+	0.0781962i
$196$		0			0
$197$	−35224.5	−	35224.5i	−0.907638	−	0.907638i	0.0884429	−	0.996081i	$-0.471811\pi$
$197$	−35224.5	−	35224.5i	−0.907638	−	0.907638i	−0.996081	+	0.0884429i	$0.971811\pi$
$198$		0			0
$199$	31089.7			0.785074			0.392537	−	0.919736i	$-0.371598\pi$
$199$	31089.7			0.785074			0.392537	+	0.919736i	$0.371598\pi$
$200$		0			0
$201$			5669.93i			0.140341i
$202$		0			0
$203$	−5492.91	+	5492.91i	−0.133294	+	0.133294i
$204$		0			0
$205$	−970.113	+	970.113i	−0.0230842	+	0.0230842i
$206$		0			0
$207$			4681.02i			0.109245i
$208$		0			0
$209$	14663.4			0.335692
$210$		0			0
$211$	37821.8	+	37821.8i	0.849528	+	0.849528i	0.990074	−	0.140546i	$-0.0448858\pi$
$211$	37821.8	+	37821.8i	0.849528	+	0.849528i	−0.140546	+	0.990074i	$0.544886\pi$
$212$		0			0
$213$	−7581.61	−	7581.61i	−0.167110	−	0.167110i
$214$		0			0
$215$	−4825.70			−0.104396
$216$		0			0
$217$		−	31099.6i		−	0.660444i
$218$		0			0
$219$	−1703.66	+	1703.66i	−0.0355218	+	0.0355218i
$220$		0			0
$221$	−97688.3	+	97688.3i	−2.00013	+	2.00013i
$222$		0			0
$223$		−	2858.47i		−	0.0574809i	−0.999587	−	0.0287404i	$-0.990850\pi$
$223$		−	2858.47i		−	0.0574809i	0.999587	−	0.0287404i	$-0.00914963\pi$
$224$		0			0
$225$	16636.3			0.328617
$226$		0			0
$227$	−70688.1	−	70688.1i	−1.37181	−	1.37181i	−0.857758	−	0.514053i	$-0.828143\pi$
$227$	−70688.1	−	70688.1i	−1.37181	−	1.37181i	−0.514053	−	0.857758i	$-0.671857\pi$
$228$		0			0
$229$	394.626	+	394.626i	0.00752514	+	0.00752514i	0.710859	−	0.703334i	$-0.248306\pi$
$229$	394.626	+	394.626i	0.00752514	+	0.00752514i	−0.703334	+	0.710859i	$0.748306\pi$
$230$		0			0
$231$	−36076.2			−0.676078
$232$		0			0
$233$			21767.8i			0.400961i	0.979698	+	0.200481i	$0.0642504\pi$
$233$			21767.8i			0.400961i	−0.979698	+	0.200481i	$0.935750\pi$
$234$		0			0
$235$	8617.69	−	8617.69i	0.156047	−	0.156047i
$236$		0			0
$237$	−31001.0	+	31001.0i	−0.551923	+	0.551923i
$238$		0			0
$239$			81943.0i			1.43455i	0.696789	+	0.717276i	$0.254611\pi$
$239$			81943.0i			1.43455i	−0.696789	+	0.717276i	$0.745389\pi$
$240$		0			0
$241$	11957.8			0.205882			0.102941	−	0.994687i	$-0.467175\pi$
$241$	11957.8			0.205882			0.102941	+	0.994687i	$0.467175\pi$
$242$		0			0
$243$	2678.52	+	2678.52i	0.0453609	+	0.0453609i
$244$		0			0
$245$	10081.8	+	10081.8i	0.167959	+	0.167959i
$246$		0			0
$247$	48756.5			0.799169
$248$		0			0
$249$			22808.0i			0.367865i
$250$		0			0
$251$	−33809.8	+	33809.8i	−0.536655	+	0.536655i	−0.922545	−	0.385890i	$-0.873894\pi$
$251$	−33809.8	+	33809.8i	−0.536655	+	0.536655i	0.385890	+	0.922545i	$0.373894\pi$
$252$		0			0
$253$	−10033.7	+	10033.7i	−0.156755	+	0.156755i
$254$		0			0
$255$		−	7843.81i		−	0.120628i
$256$		0			0
$257$	−119487.			−1.80907			−0.904536	−	0.426396i	$-0.859783\pi$
$257$	−119487.			−1.80907			−0.904536	+	0.426396i	$0.859783\pi$
$258$		0			0
$259$	68022.2	+	68022.2i	1.01403	+	1.01403i
$260$		0			0
$261$	1748.35	+	1748.35i	0.0256654	+	0.0256654i
$262$		0			0
$263$	−34094.4			−0.492915			−0.246457	−	0.969154i	$-0.579267\pi$
$263$	−34094.4			−0.492915			−0.246457	+	0.969154i	$0.579267\pi$
$264$		0			0
$265$			10691.9i			0.152252i
$266$		0			0
$267$	−28177.3	+	28177.3i	−0.395254	+	0.395254i
$268$		0			0
$269$	46201.7	−	46201.7i	0.638489	−	0.638489i	−0.311694	−	0.950183i	$-0.600896\pi$
$269$	46201.7	−	46201.7i	0.638489	−	0.638489i	0.950183	+	0.311694i	$0.100896\pi$
$270$		0			0
$271$			90603.1i			1.23369i	0.787086	+	0.616843i	$0.211589\pi$
$271$			90603.1i			1.23369i	−0.787086	+	0.616843i	$0.788411\pi$
$272$		0			0
$273$	−119955.			−1.60951
$274$		0			0
$275$	35659.8	+	35659.8i	0.471534	+	0.471534i
$276$		0			0
$277$	−259.186	−	259.186i	−0.00337794	−	0.00337794i	0.705416	−	0.708794i	$-0.250760\pi$
$277$	−259.186	−	259.186i	−0.00337794	−	0.00337794i	−0.708794	+	0.705416i	$0.750760\pi$
$278$		0			0
$279$	−9898.78			−0.127167
$280$		0			0
$281$		−	114309.i		−	1.44766i	−0.689979	−	0.723829i	$-0.742380\pi$
$281$		−	114309.i		−	1.44766i	0.689979	−	0.723829i	$-0.257620\pi$
$282$		0			0
$283$	33739.3	−	33739.3i	0.421273	−	0.421273i	−0.464369	−	0.885642i	$-0.653719\pi$
$283$	33739.3	−	33739.3i	0.421273	−	0.421273i	0.885642	+	0.464369i	$0.153719\pi$
$284$		0			0
$285$	−1957.43	+	1957.43i	−0.0240989	+	0.0240989i
$286$		0			0
$287$		−	39136.9i		−	0.475142i
$288$		0			0
$289$	174179.			2.08546
$290$		0			0
$291$	−60211.1	−	60211.1i	−0.711034	−	0.711034i
$292$		0			0
$293$	−53831.9	−	53831.9i	−0.627054	−	0.627054i	0.320272	−	0.947326i	$-0.396226\pi$
$293$	−53831.9	−	53831.9i	−0.627054	−	0.627054i	−0.947326	+	0.320272i	$0.896226\pi$
$294$		0			0
$295$	4033.24			0.0463458
$296$		0			0
$297$			11482.8i			0.130177i
$298$		0			0
$299$	−33362.7	+	33362.7i	−0.373181	+	0.373181i
$300$		0			0
$301$	97340.8	−	97340.8i	1.07439	−	1.07439i
$302$		0			0
$303$		−	57461.6i		−	0.625882i
$304$		0			0
$305$	683.664			0.00734925
$306$		0			0
$307$	99665.1	+	99665.1i	1.05747	+	1.05747i	0.998245	+	0.0592206i	$0.0188615\pi$
$307$	99665.1	+	99665.1i	1.05747	+	1.05747i	0.0592206	+	0.998245i	$0.481138\pi$
$308$		0			0
$309$	−19742.9	−	19742.9i	−0.206773	−	0.206773i
$310$		0			0
$311$	81209.1			0.839622			0.419811	−	0.907612i	$-0.362096\pi$
$311$	81209.1			0.839622			0.419811	+	0.907612i	$0.362096\pi$
$312$		0			0
$313$		−	16053.4i		−	0.163862i	−0.996638	−	0.0819310i	$-0.973891\pi$
$313$		−	16053.4i		−	0.163862i	0.996638	−	0.0819310i	$-0.0261087\pi$
$314$		0			0
$315$	4815.86	−	4815.86i	0.0485347	−	0.0485347i
$316$		0			0
$317$	−5778.79	+	5778.79i	−0.0575067	+	0.0575067i	−0.735275	−	0.677769i	$-0.762947\pi$
$317$	−5778.79	+	5778.79i	−0.0575067	+	0.0575067i	0.677769	+	0.735275i	$0.262947\pi$
$318$		0			0
$319$			7495.17i			0.0736547i
$320$		0			0
$321$	−71296.9			−0.691927
$322$		0			0
$323$	−64309.4	−	64309.4i	−0.616410	−	0.616410i
$324$		0			0
$325$	118571.	+	118571.i	1.12256	+	1.12256i
$326$		0			0
$327$	−74957.5			−0.701003
$328$		0			0
$329$			347661.i			3.21191i
$330$		0			0
$331$	79971.6	−	79971.6i	0.729928	−	0.729928i	−0.240677	−	0.970605i	$-0.577370\pi$
$331$	79971.6	−	79971.6i	0.729928	−	0.729928i	0.970605	+	0.240677i	$0.0773695\pi$
$332$		0			0
$333$	21651.0	−	21651.0i	0.195249	−	0.195249i
$334$		0			0
$335$		−	3244.77i		−	0.0289131i
$336$		0			0
$337$	−129995.			−1.14464			−0.572319	−	0.820031i	$-0.693956\pi$
$337$	−129995.			−1.14464			−0.572319	+	0.820031i	$0.693956\pi$
$338$		0			0
$339$	47977.7	+	47977.7i	0.417484	+	0.417484i
$340$		0			0
$341$	−21218.0	−	21218.0i	−0.182472	−	0.182472i
$342$		0			0
$343$	−203054.			−1.72593
$344$		0			0
$345$		−	2678.84i		−	0.0225065i
$346$		0			0
$347$	22976.9	−	22976.9i	0.190824	−	0.190824i	−0.605228	−	0.796052i	$-0.706918\pi$
$347$	22976.9	−	22976.9i	0.190824	−	0.190824i	0.796052	+	0.605228i	$0.206918\pi$
$348$		0			0
$349$	87638.2	−	87638.2i	0.719520	−	0.719520i	−0.248987	−	0.968507i	$-0.580098\pi$
$349$	87638.2	−	87638.2i	0.719520	−	0.719520i	0.968507	+	0.248987i	$0.0800977\pi$
$350$		0			0
$351$			38180.9i			0.309907i
$352$		0			0
$353$	144112.			1.15651			0.578257	−	0.815854i	$-0.303733\pi$
$353$	144112.			1.15651			0.578257	+	0.815854i	$0.303733\pi$
$354$		0			0
$355$	4338.78	+	4338.78i	0.0344279	+	0.0344279i
$356$		0			0
$357$	158220.	+	158220.i	1.24144	+	1.24144i
$358$		0			0
$359$	−30554.2			−0.237073			−0.118536	−	0.992950i	$-0.537820\pi$
$359$	−30554.2			−0.237073			−0.118536	+	0.992950i	$0.537820\pi$
$360$		0			0
$361$		−	98224.0i		−	0.753708i
$362$		0			0
$363$	29181.1	−	29181.1i	0.221457	−	0.221457i
$364$		0			0
$365$	974.966	−	974.966i	0.00731819	−	0.00731819i
$366$		0			0
$367$		−	27735.4i		−	0.205922i	−0.994685	−	0.102961i	$-0.967168\pi$
$367$		−	27735.4i		−	0.205922i	0.994685	−	0.102961i	$-0.0328316\pi$
$368$		0			0
$369$	−12457.0			−0.0914873
$370$		0			0
$371$	−215669.	−	215669.i	−1.56690	−	1.56690i
$372$		0			0
$373$	−91259.4	−	91259.4i	−0.655934	−	0.655934i	0.298481	−	0.954415i	$-0.403520\pi$
$373$	−91259.4	−	91259.4i	−0.655934	−	0.655934i	−0.954415	+	0.298481i	$0.903520\pi$
$374$		0			0
$375$	−19177.7			−0.136375
$376$		0			0
$377$			24921.8i			0.175347i
$378$		0			0
$379$	−725.899	+	725.899i	−0.00505356	+	0.00505356i	−0.709629	−	0.704575i	$-0.751137\pi$
$379$	−725.899	+	725.899i	−0.00505356	+	0.00505356i	0.704575	+	0.709629i	$0.251137\pi$
$380$		0			0
$381$	46872.9	−	46872.9i	0.322903	−	0.322903i
$382$		0			0
$383$		−	81078.9i		−	0.552727i	−0.961053	−	0.276363i	$-0.910871\pi$
$383$		−	81078.9i		−	0.552727i	0.961053	−	0.276363i	$-0.0891293\pi$
$384$		0			0
$385$	20645.6			0.139285
$386$		0			0
$387$	−30982.9	−	30982.9i	−0.206871	−	0.206871i
$388$		0			0
$389$	77667.7	+	77667.7i	0.513265	+	0.513265i	0.915525	−	0.402260i	$-0.131775\pi$
$389$	77667.7	+	77667.7i	0.513265	+	0.513265i	−0.402260	+	0.915525i	$0.631775\pi$
$390$		0			0
$391$	88010.4			0.575679
$392$		0			0
$393$		−	50693.3i		−	0.328220i
$394$		0			0
$395$	17741.1	−	17741.1i	0.113707	−	0.113707i
$396$		0			0
$397$	−21877.8	+	21877.8i	−0.138811	+	0.138811i	−0.773098	−	0.634287i	$-0.781294\pi$
$397$	−21877.8	+	21877.8i	−0.138811	+	0.138811i	0.634287	+	0.773098i	$0.281294\pi$
$398$		0			0
$399$		−	78968.0i		−	0.496027i
$400$		0			0
$401$	175824.			1.09343			0.546714	−	0.837319i	$-0.315879\pi$
$401$	175824.			1.09343			0.546714	+	0.837319i	$0.315879\pi$
$402$		0			0
$403$	−70551.0	−	70551.0i	−0.434403	−	0.434403i
$404$		0			0
$405$	−1532.85	−	1532.85i	−0.00934524	−	0.00934524i
$406$		0			0
$407$	92817.6			0.560327
$408$		0			0
$409$		−	174646.i		−	1.04402i	−0.852938	−	0.522012i	$-0.825181\pi$
$409$		−	174646.i		−	1.04402i	0.852938	−	0.522012i	$-0.174819\pi$
$410$		0			0
$411$	72493.0	−	72493.0i	0.429153	−	0.429153i
$412$		0			0
$413$	−81355.8	+	81355.8i	−0.476967	+	0.476967i
$414$		0			0
$415$		−	13052.5i		−	0.0757873i
$416$		0			0
$417$	168140.			0.966937
$418$		0			0
$419$	167807.	+	167807.i	0.955831	+	0.955831i	0.999065	−	0.0432336i	$-0.0137660\pi$
$419$	167807.	+	167807.i	0.955831	+	0.955831i	−0.0432336	+	0.999065i	$0.513766\pi$
$420$		0			0
$421$	−18489.0	−	18489.0i	−0.104316	−	0.104316i	0.653023	−	0.757338i	$-0.273501\pi$
$421$	−18489.0	−	18489.0i	−0.104316	−	0.104316i	−0.757338	+	0.653023i	$0.773501\pi$
$422$		0			0
$423$	110658.			0.618445
$424$		0			0
$425$		−	312787.i		−	1.73169i
$426$		0			0
$427$	−13790.4	+	13790.4i	−0.0756348	+	0.0756348i
$428$		0			0
$429$	−81840.6	+	81840.6i	−0.444687	+	0.444687i
$430$		0			0
$431$		−	206513.i		−	1.11171i	−0.831278	−	0.555857i	$-0.812390\pi$
$431$		−	206513.i		−	1.11171i	0.831278	−	0.555857i	$-0.187610\pi$
$432$		0			0
$433$	29880.3			0.159371			0.0796855	−	0.996820i	$-0.474608\pi$
$433$	29880.3			0.159371			0.0796855	+	0.996820i	$0.474608\pi$
$434$		0			0
$435$	−1000.54	−	1000.54i	−0.00528757	−	0.00528757i
$436$		0			0
$437$	−21963.1	−	21963.1i	−0.115009	−	0.115009i
$438$		0			0
$439$	82988.1			0.430612			0.215306	−	0.976547i	$-0.430925\pi$
$439$	82988.1			0.430612			0.215306	+	0.976547i	$0.430925\pi$
$440$		0			0
$441$			129458.i			0.665657i
$442$		0			0
$443$	213195.	−	213195.i	1.08635	−	1.08635i	0.0904459	−	0.995901i	$-0.471171\pi$
$443$	213195.	−	213195.i	1.08635	−	1.08635i	0.995901	−	0.0904459i	$-0.0288292\pi$
$444$		0			0
$445$	16125.2	−	16125.2i	0.0814300	−	0.0814300i
$446$		0			0
$447$		−	104109.i		−	0.521040i
$448$		0			0
$449$	174583.			0.865984			0.432992	−	0.901398i	$-0.357458\pi$
$449$	174583.			0.865984			0.432992	+	0.901398i	$0.357458\pi$
$450$		0			0
$451$	−26701.5	−	26701.5i	−0.131275	−	0.131275i
$452$		0			0
$453$	−110987.	−	110987.i	−0.540849	−	0.540849i
$454$		0			0
$455$	68647.6			0.331591
$456$		0			0
$457$			129994.i			0.622429i	0.950340	+	0.311214i	$0.100736\pi$
$457$			129994.i			0.622429i	−0.950340	+	0.311214i	$0.899264\pi$
$458$		0			0
$459$	50360.3	−	50360.3i	0.239036	−	0.239036i
$460$		0			0
$461$	160419.	−	160419.i	0.754840	−	0.754840i	−0.220538	−	0.975378i	$-0.570781\pi$
$461$	160419.	−	160419.i	0.754840	−	0.754840i	0.975378	+	0.220538i	$0.0707814\pi$
$462$		0			0
$463$		−	237870.i		−	1.10963i	−0.831974	−	0.554815i	$-0.812789\pi$
$463$		−	237870.i		−	1.10963i	0.831974	−	0.554815i	$-0.187211\pi$
$464$		0			0
$465$	5664.84			0.0261988
$466$		0			0
$467$	−154822.	−	154822.i	−0.709902	−	0.709902i	0.256612	−	0.966514i	$-0.417394\pi$
$467$	−154822.	−	154822.i	−0.709902	−	0.709902i	−0.966514	+	0.256612i	$0.917394\pi$
$468$		0			0
$469$	65451.3	+	65451.3i	0.297559	+	0.297559i
$470$		0			0
$471$	−132181.			−0.595838
$472$		0			0
$473$		−	132823.i		−	0.593680i
$474$		0			0
$475$	−78056.5	+	78056.5i	−0.345957	+	0.345957i
$476$		0			0
$477$	−68645.9	+	68645.9i	−0.301702	+	0.301702i
$478$		0			0
$479$			75781.5i			0.330288i	0.986270	+	0.165144i	$0.0528088\pi$
$479$			75781.5i			0.330288i	−0.986270	+	0.165144i	$0.947191\pi$
$480$		0			0
$481$	308623.			1.33395
$482$		0			0
$483$	54035.7	+	54035.7i	0.231626	+	0.231626i
$484$		0			0
$485$	34457.4	+	34457.4i	0.146487	+	0.146487i
$486$		0			0
$487$	198624.			0.837481			0.418740	−	0.908106i	$-0.362472\pi$
$487$	198624.			0.837481			0.418740	+	0.908106i	$0.362472\pi$
$488$		0			0
$489$		−	23729.0i		−	0.0992345i
$490$		0			0
$491$	−160150.	+	160150.i	−0.664301	+	0.664301i	−0.956391	−	0.292090i	$-0.905649\pi$
$491$	−160150.	+	160150.i	−0.664301	+	0.664301i	0.292090	+	0.956391i	$0.405649\pi$
$492$		0			0
$493$	32871.7	−	32871.7i	0.135247	−	0.135247i
$494$		0			0
$495$		−	6571.33i		−	0.0268190i
$496$		0			0
$497$	−175038.			−0.708630
$498$		0			0
$499$	−169851.	−	169851.i	−0.682131	−	0.682131i	0.278349	−	0.960480i	$-0.410213\pi$
$499$	−169851.	−	169851.i	−0.682131	−	0.682131i	−0.960480	+	0.278349i	$0.910213\pi$
$500$		0			0
$501$	−60964.5	−	60964.5i	−0.242886	−	0.242886i
$502$		0			0
$503$	−115453.			−0.456320			−0.228160	−	0.973624i	$-0.573271\pi$
$503$	−115453.			−0.456320			−0.228160	+	0.973624i	$0.573271\pi$
$504$		0			0
$505$			32883.9i			0.128944i
$506$		0			0
$507$	−167184.	+	167184.i	−0.650399	+	0.650399i
$508$		0			0
$509$	−174996.	+	174996.i	−0.675449	+	0.675449i	−0.958967	−	0.283518i	$-0.908498\pi$
$509$	−174996.	+	174996.i	−0.675449	+	0.675449i	0.283518	+	0.958967i	$0.408498\pi$
$510$		0			0
$511$			39332.7i			0.150630i
$512$		0			0
$513$	−25134.9			−0.0955087
$514$		0			0
$515$	11298.4	+	11298.4i	0.0425993	+	0.0425993i
$516$		0			0
$517$	237195.	+	237195.i	0.887409	+	0.887409i
$518$		0			0
$519$	34589.6			0.128414
$520$		0			0
$521$			408830.i			1.50615i	0.657936	+	0.753074i	$0.271430\pi$
$521$			408830.i			1.50615i	−0.657936	+	0.753074i	$0.728570\pi$
$522$		0			0
$523$	−186677.	+	186677.i	−0.682477	+	0.682477i	−0.960558	−	0.278080i	$-0.910302\pi$
$523$	−186677.	+	186677.i	−0.682477	+	0.682477i	0.278080	+	0.960558i	$0.410302\pi$
$524$		0			0
$525$	192042.	−	192042.i	0.696751	−	0.696751i
$526$		0			0
$527$			186112.i			0.670122i
$528$		0			0
$529$	−249783.			−0.892591
$530$		0			0
$531$	25894.9	+	25894.9i	0.0918387	+	0.0918387i
$532$		0			0
$533$	−88784.0	−	88784.0i	−0.312522	−	0.312522i
$534$		0			0
$535$	40801.5			0.142550
$536$		0			0
$537$		−	192349.i		−	0.667024i
$538$		0			0
$539$	−277492.	+	277492.i	−0.955153	+	0.955153i
$540$		0			0
$541$	−133417.	+	133417.i	−0.455844	+	0.455844i	−0.897288	−	0.441445i	$-0.854466\pi$
$541$	−133417.	+	133417.i	−0.455844	+	0.455844i	0.441445	+	0.897288i	$0.354466\pi$
$542$		0			0
$543$			294920.i			1.00024i
$544$		0			0
$545$	42896.4			0.144420
$546$		0			0
$547$	−95416.9	−	95416.9i	−0.318897	−	0.318897i	0.529446	−	0.848343i	$-0.322400\pi$
$547$	−95416.9	−	95416.9i	−0.318897	−	0.318897i	−0.848343	+	0.529446i	$0.822400\pi$
$548$		0			0
$549$	4389.39	+	4389.39i	0.0145633	+	0.0145633i
$550$		0			0
$551$	−16406.4			−0.0540392
$552$		0			0
$553$			715724.i			2.34043i
$554$		0			0
$555$	−12390.3	+	12390.3i	−0.0402251	+	0.0402251i
$556$		0			0
$557$	175167.	−	175167.i	0.564600	−	0.564600i	−0.366011	−	0.930611i	$-0.619277\pi$
$557$	175167.	−	175167.i	0.564600	−	0.564600i	0.930611	+	0.366011i	$0.119277\pi$
$558$		0			0
$559$		−	441644.i		−	1.41335i
$560$		0			0
$561$	215894.			0.685986
$562$		0			0
$563$	163587.	+	163587.i	0.516097	+	0.516097i	0.916388	−	0.400291i	$-0.131091\pi$
$563$	163587.	+	163587.i	0.516097	+	0.516097i	−0.400291	+	0.916388i	$0.631091\pi$
$564$		0			0
$565$	−27456.5	−	27456.5i	−0.0860099	−	0.0860099i
$566$		0			0
$567$	61839.3			0.192353
$568$		0			0
$569$		−	361331.i		−	1.11604i	−0.829826	−	0.558022i	$-0.811561\pi$
$569$		−	361331.i		−	1.11604i	0.829826	−	0.558022i	$-0.188439\pi$
$570$		0			0
$571$	263234.	−	263234.i	0.807364	−	0.807364i	−0.176871	−	0.984234i	$-0.556597\pi$
$571$	263234.	−	263234.i	0.807364	−	0.807364i	0.984234	+	0.176871i	$0.0565974\pi$
$572$		0			0
$573$	−37839.3	+	37839.3i	−0.115248	+	0.115248i
$574$		0			0
$575$		−	106824.i		−	0.323097i
$576$		0			0
$577$	−328682.			−0.987243			−0.493622	−	0.869677i	$-0.664327\pi$
$577$	−328682.			−0.987243			−0.493622	+	0.869677i	$0.664327\pi$
$578$		0			0
$579$	78318.3	+	78318.3i	0.233618	+	0.233618i
$580$		0			0
$581$	263286.	+	263286.i	0.779965	+	0.779965i
$582$		0			0
$583$	−294285.			−0.865826
$584$		0			0
$585$		−	21850.0i		−	0.0638469i
$586$		0			0
$587$	349626.	−	349626.i	1.01468	−	1.01468i	0.0147866	−	0.999891i	$-0.495293\pi$
$587$	349626.	−	349626.i	1.01468	−	1.01468i	0.999891	−	0.0147866i	$-0.00470690\pi$
$588$		0			0
$589$	46444.6	−	46444.6i	0.133877	−	0.133877i
$590$		0			0
$591$			258846.i			0.741084i
$592$		0			0
$593$	−414406.			−1.17846			−0.589232	−	0.807964i	$-0.700570\pi$
$593$	−414406.			−1.17846			−0.589232	+	0.807964i	$0.700570\pi$
$594$		0			0
$595$	−90545.6	−	90545.6i	−0.255761	−	0.255761i
$596$		0			0
$597$	−114231.	−	114231.i	−0.320505	−	0.320505i
$598$		0			0
$599$	−369814.			−1.03069			−0.515347	−	0.856982i	$-0.672337\pi$
$599$	−369814.			−1.03069			−0.515347	+	0.856982i	$0.672337\pi$
$600$		0			0
$601$		−	318196.i		−	0.880940i	−0.897767	−	0.440470i	$-0.854812\pi$
$601$		−	318196.i		−	0.880940i	0.897767	−	0.440470i	$-0.145188\pi$
$602$		0			0
$603$	20832.7	−	20832.7i	0.0572941	−	0.0572941i
$604$		0			0
$605$	−16699.7	+	16699.7i	−0.0456244	+	0.0456244i
$606$		0			0
$607$			568888.i			1.54401i	0.635618	+	0.772003i	$0.280745\pi$
$607$			568888.i			1.54401i	−0.635618	+	0.772003i	$0.719255\pi$
$608$		0			0
$609$	40364.5			0.108834
$610$		0			0
$611$	788685.	+	788685.i	2.11262	+	2.11262i
$612$		0			0
$613$	−290102.	−	290102.i	−0.772023	−	0.772023i	0.206437	−	0.978460i	$-0.433813\pi$
$613$	−290102.	−	290102.i	−0.772023	−	0.772023i	−0.978460	+	0.206437i	$0.933813\pi$
$614$		0			0
$615$	7128.85			0.0188482
$616$		0			0
$617$		−	222944.i		−	0.585633i	−0.956169	−	0.292817i	$-0.905407\pi$
$617$		−	222944.i		−	0.585633i	0.956169	−	0.292817i	$-0.0945925\pi$
$618$		0			0
$619$	−52984.5	+	52984.5i	−0.138283	+	0.138283i	−0.772860	−	0.634577i	$-0.781174\pi$
$619$	−52984.5	+	52984.5i	−0.138283	+	0.138283i	0.634577	+	0.772860i	$0.281174\pi$
$620$		0			0
$621$	17199.2	−	17199.2i	0.0445989	−	0.0445989i
$622$		0			0
$623$			650533.i			1.67607i
$624$		0			0
$625$	−374123.			−0.957756
$626$		0			0
$627$	−53876.7	−	53876.7i	−0.137046	−	0.137046i
$628$		0			0
$629$	−407072.	−	407072.i	−1.02889	−	1.02889i
$630$		0			0
$631$	178806.			0.449079			0.224540	−	0.974465i	$-0.427912\pi$
$631$	178806.			0.449079			0.224540	+	0.974465i	$0.427912\pi$
$632$		0			0
$633$		−	277933.i		−	0.693637i
$634$		0			0
$635$	−26824.3	+	26824.3i	−0.0665243	+	0.0665243i
$636$		0			0
$637$	−922675.	+	922675.i	−2.27389	+	2.27389i
$638$		0			0
$639$			55713.3i			0.136445i
$640$		0			0
$641$	−20182.3			−0.0491196			−0.0245598	−	0.999698i	$-0.507818\pi$
$641$	−20182.3			−0.0491196			−0.0245598	+	0.999698i	$0.507818\pi$
$642$		0			0
$643$	−220638.	−	220638.i	−0.533652	−	0.533652i	0.388005	−	0.921657i	$-0.373164\pi$
$643$	−220638.	−	220638.i	−0.533652	−	0.533652i	−0.921657	+	0.388005i	$0.873164\pi$
$644$		0			0
$645$	17730.8	+	17730.8i	0.0426195	+	0.0426195i
$646$		0			0
$647$	29047.0			0.0693893			0.0346946	−	0.999398i	$-0.488954\pi$
$647$	29047.0			0.0693893			0.0346946	+	0.999398i	$0.488954\pi$
$648$		0			0
$649$			111011.i			0.263559i
$650$		0			0
$651$	−114267.	+	114267.i	−0.269625	+	0.269625i
$652$		0			0
$653$	438317.	−	438317.i	1.02793	−	1.02793i	0.0283280	−	0.999599i	$-0.490982\pi$
$653$	438317.	−	438317.i	1.02793	−	1.02793i	0.999599	−	0.0283280i	$-0.00901828\pi$
$654$		0			0
$655$			29010.6i			0.0676198i
$656$		0			0
$657$	12519.3			0.0290034
$658$		0			0
$659$	453511.	+	453511.i	1.04428	+	1.04428i	0.998973	+	0.0453069i	$0.0144266\pi$
$659$	453511.	+	453511.i	1.04428	+	1.04428i	0.0453069	+	0.998973i	$0.485573\pi$
$660$		0			0
$661$	−387740.	−	387740.i	−0.887436	−	0.887436i	0.106840	−	0.994276i	$-0.465927\pi$
$661$	−387740.	−	387740.i	−0.887436	−	0.887436i	−0.994276	+	0.106840i	$0.965927\pi$
$662$		0			0
$663$	717860.			1.63310
$664$		0			0
$665$			45191.6i			0.102191i
$666$		0			0
$667$	11226.4	−	11226.4i	0.0252342	−	0.0252342i
$668$		0			0
$669$	−10502.7	+	10502.7i	−0.0234665	+	0.0234665i
$670$		0			0
$671$			18817.3i			0.0417938i
$672$		0			0
$673$	82244.9			0.181585			0.0907923	−	0.995870i	$-0.471060\pi$
$673$	82244.9			0.181585			0.0907923	+	0.995870i	$0.471060\pi$
$674$		0			0
$675$	−61125.5	−	61125.5i	−0.134157	−	0.134157i
$676$		0			0
$677$	576584.	+	576584.i	1.25801	+	1.25801i	0.952041	+	0.305972i	$0.0989814\pi$
$677$	576584.	+	576584.i	1.25801	+	1.25801i	0.305972	+	0.952041i	$0.401019\pi$
$678$		0			0
$679$	−1.39010e6			−3.01514
$680$		0			0
$681$			519449.i			1.12008i
$682$		0			0
$683$	342109.	−	342109.i	0.733371	−	0.733371i	−0.237915	−	0.971286i	$-0.576464\pi$
$683$	342109.	−	342109.i	0.733371	−	0.733371i	0.971286	+	0.237915i	$0.0764641\pi$
$684$		0			0
$685$	−41486.0	+	41486.0i	−0.0884140	+	0.0884140i
$686$		0			0
$687$		−	2899.90i		−	0.00614425i
$688$		0			0
$689$	−978512.			−2.06124
$690$		0			0
$691$	213096.	+	213096.i	0.446293	+	0.446293i	0.894120	−	0.447827i	$-0.147802\pi$
$691$	213096.	+	213096.i	0.446293	+	0.446293i	−0.447827	+	0.894120i	$0.647802\pi$
$692$		0			0
$693$	132552.	+	132552.i	0.276008	+	0.276008i
$694$		0			0
$695$	−96222.4			−0.199208
$696$		0			0
$697$			234211.i			0.482105i
$698$		0			0
$699$	79980.0	−	79980.0i	0.163692	−	0.163692i
$700$		0			0
$701$	293532.	−	293532.i	0.597337	−	0.597337i	−0.342266	−	0.939603i	$-0.611194\pi$
$701$	293532.	−	293532.i	0.597337	−	0.597337i	0.939603	+	0.342266i	$0.111194\pi$
$702$		0			0
$703$			203170.i			0.411102i
$704$		0			0
$705$	−63326.8			−0.127412
$706$		0			0
$707$	−663313.	−	663313.i	−1.32703	−	1.32703i
$708$		0			0
$709$	13231.5	+	13231.5i	0.0263218	+	0.0263218i	0.720145	−	0.693823i	$-0.244075\pi$
$709$	13231.5	+	13231.5i	0.0263218	+	0.0263218i	−0.693823	+	0.720145i	$0.744075\pi$
$710$		0			0
$711$	227810.			0.450643
$712$		0			0
$713$			63561.5i			0.125030i
$714$		0			0
$715$	46835.4	−	46835.4i	0.0916142	−	0.0916142i
$716$		0			0
$717$	301078.	−	301078.i	0.585653	−	0.585653i
$718$		0			0
$719$		−	477459.i		−	0.923587i	−0.886987	−	0.461794i	$-0.847206\pi$
$719$		−	477459.i		−	0.923587i	0.886987	−	0.461794i	$-0.152794\pi$
$720$		0			0
$721$	−455808.			−0.876821
$722$		0			0
$723$	−43935.9	−	43935.9i	−0.0840511	−	0.0840511i
$724$		0			0
$725$	−39898.5	−	39898.5i	−0.0759068	−	0.0759068i
$726$		0			0
$727$	212060.			0.401227			0.200614	−	0.979670i	$-0.435706\pi$
$727$	212060.			0.401227			0.200614	+	0.979670i	$0.435706\pi$
$728$		0			0
$729$		−	19683.0i		−	0.0370370i
$730$		0			0
$731$	−582526.	+	582526.i	−1.09014	+	1.09014i
$732$		0			0
$733$	−84332.9	+	84332.9i	−0.156960	+	0.156960i	−0.781218	−	0.624258i	$-0.785401\pi$
$733$	−84332.9	+	84332.9i	−0.156960	+	0.156960i	0.624258	+	0.781218i	$0.285401\pi$
$734$		0			0
$735$		−	74085.5i		−	0.137138i
$736$		0			0
$737$	89309.5			0.164423
$738$		0			0
$739$	−190782.	−	190782.i	−0.349340	−	0.349340i	0.510524	−	0.859864i	$-0.329452\pi$
$739$	−190782.	−	190782.i	−0.349340	−	0.349340i	−0.859864	+	0.510524i	$0.829452\pi$
$740$		0			0
$741$	−179143.	−	179143.i	−0.326259	−	0.326259i
$742$		0			0
$743$	131122.			0.237518			0.118759	−	0.992923i	$-0.462108\pi$
$743$	131122.			0.237518			0.118759	+	0.992923i	$0.462108\pi$
$744$		0			0
$745$			59578.9i			0.107344i
$746$		0			0
$747$	83801.9	−	83801.9i	0.150180	−	0.150180i
$748$		0			0
$749$	−823021.	+	823021.i	−1.46706	+	1.46706i
$750$		0			0
$751$			704733.i			1.24952i	0.780815	+	0.624762i	$0.214804\pi$
$751$			704733.i			1.24952i	−0.780815	+	0.624762i	$0.785196\pi$
$752$		0			0
$753$	248450.			0.438177
$754$		0			0
$755$	63515.3	+	63515.3i	0.111425	+	0.111425i
$756$		0			0
$757$	−411751.	−	411751.i	−0.718526	−	0.718526i	0.249777	−	0.968303i	$-0.419643\pi$
$757$	−411751.	−	411751.i	−0.718526	−	0.718526i	−0.968303	+	0.249777i	$0.919643\pi$
$758$		0			0
$759$	73732.7			0.127990
$760$		0			0
$761$		−	82790.4i		−	0.142959i	−0.997442	−	0.0714793i	$-0.977228\pi$
$761$		−	82790.4i		−	0.142959i	0.997442	−	0.0714793i	$-0.0227720\pi$
$762$		0			0
$763$	−865278.	+	865278.i	−1.48630	+	1.48630i
$764$		0			0
$765$	−28820.0	+	28820.0i	−0.0492460	+	0.0492460i
$766$		0			0
$767$			369119.i			0.627445i
$768$		0			0
$769$	107378.			0.181578			0.0907888	−	0.995870i	$-0.471061\pi$
$769$	107378.			0.181578			0.0907888	+	0.995870i	$0.471061\pi$
$770$		0			0
$771$	439025.	+	439025.i	0.738551	+	0.738551i
$772$		0			0
$773$	−136137.	−	136137.i	−0.227833	−	0.227833i	0.583954	−	0.811787i	$-0.301505\pi$
$773$	−136137.	−	136137.i	−0.227833	−	0.227833i	−0.811787	+	0.583954i	$0.801505\pi$
$774$		0			0
$775$	225897.			0.376103
$776$		0			0
$777$		−	499859.i		−	0.827953i
$778$		0			0
$779$	58447.6	−	58447.6i	0.0963145	−	0.0963145i
$780$		0			0
$781$	−119421.	+	119421.i	−0.195785	+	0.195785i
$782$		0			0
$783$		−	12847.7i		−	0.0209557i
$784$		0			0
$785$	75644.2			0.122754
$786$		0			0
$787$	−39500.6	−	39500.6i	−0.0637756	−	0.0637756i	0.674500	−	0.738275i	$-0.264359\pi$
$787$	−39500.6	−	39500.6i	−0.0637756	−	0.0637756i	−0.738275	+	0.674500i	$0.764359\pi$
$788$		0			0
$789$	125271.	+	125271.i	0.201232	+	0.201232i
$790$		0			0
$791$	1.10767e6			1.77034
$792$		0			0
$793$			62568.4i			0.0994967i
$794$		0			0
$795$	39284.5	−	39284.5i	0.0621565	−	0.0621565i
$796$		0			0
$797$	504452.	−	504452.i	0.794151	−	0.794151i	−0.188015	−	0.982166i	$-0.560206\pi$
$797$	504452.	−	504452.i	0.794151	−	0.794151i	0.982166	+	0.188015i	$0.0602055\pi$
$798$		0			0
$799$		−	2.08054e6i		−	3.25898i
$800$		0			0
$801$	207060.			0.322724
$802$		0			0
$803$	26835.1	+	26835.1i	0.0416171	+	0.0416171i
$804$		0			0
$805$	−30923.4	−	30923.4i	−0.0477194	−	0.0477194i
$806$		0			0
$807$	−339512.			−0.521324
$808$		0			0
$809$		−	647248.i		−	0.988948i	−0.869192	−	0.494474i	$-0.835361\pi$
$809$		−	647248.i		−	0.988948i	0.869192	−	0.494474i	$-0.164639\pi$
$810$		0			0
$811$	149216.	−	149216.i	0.226868	−	0.226868i	−0.584515	−	0.811383i	$-0.698715\pi$
$811$	149216.	−	149216.i	0.226868	−	0.226868i	0.811383	+	0.584515i	$0.198715\pi$
$812$		0			0
$813$	332897.	−	332897.i	0.503650	−	0.503650i
$814$		0			0
$815$			13579.6i			0.0204442i
$816$		0			0
$817$	290740.			0.435573
$818$		0			0
$819$	440744.	+	440744.i	0.657080	+	0.657080i
$820$		0			0
$821$	−172848.	−	172848.i	−0.256435	−	0.256435i	0.567168	−	0.823602i	$-0.308039\pi$
$821$	−172848.	−	172848.i	−0.256435	−	0.256435i	−0.823602	+	0.567168i	$0.808039\pi$
$822$		0			0
$823$	−21410.9			−0.0316107			−0.0158054	−	0.999875i	$-0.505031\pi$
$823$	−21410.9			−0.0316107			−0.0158054	+	0.999875i	$0.505031\pi$
$824$		0			0
$825$		−	262045.i		−	0.385006i
$826$		0			0
$827$	−1742.56	+	1742.56i	−0.00254786	+	0.00254786i	−0.708380	−	0.705832i	$-0.750573\pi$
$827$	−1742.56	+	1742.56i	−0.00254786	+	0.00254786i	0.705832	+	0.708380i	$0.250573\pi$
$828$		0			0
$829$	−193828.	+	193828.i	−0.282037	+	0.282037i	−0.833921	−	0.551884i	$-0.813909\pi$
$829$	−193828.	+	193828.i	−0.282037	+	0.282037i	0.551884	+	0.833921i	$0.313909\pi$
$830$		0			0
$831$			1904.62i			0.00275808i
$832$		0			0
$833$	2.43400e6			3.50777
$834$		0			0
$835$	34888.6	+	34888.6i	0.0500392	+	0.0500392i
$836$		0			0
$837$	36370.4	+	36370.4i	0.0519156	+	0.0519156i
$838$		0			0
$839$	742266.			1.05447			0.527237	−	0.849718i	$-0.323228\pi$
$839$	742266.			1.05447			0.527237	+	0.849718i	$0.323228\pi$
$840$		0			0
$841$			698895.i			0.988143i
$842$		0			0
$843$	−419996.	+	419996.i	−0.591004	+	0.591004i
$844$		0			0
$845$	95675.7	−	95675.7i	0.133995	−	0.133995i
$846$		0			0
$847$		−	673709.i		−	0.939087i
$848$		0			0
$849$	−247932.			−0.343968
$850$		0			0
$851$	−139024.	−	139024.i	−0.191969	−	0.191969i
$852$		0			0
$853$	575553.	+	575553.i	0.791020	+	0.791020i	0.981660	−	0.190640i	$-0.0610565\pi$
$853$	575553.	+	575553.i	0.791020	+	0.791020i	−0.190640	+	0.981660i	$0.561056\pi$
$854$		0			0
$855$	14384.1			0.0196767
$856$		0			0
$857$			541248.i			0.736944i	0.929639	+	0.368472i	$0.120119\pi$
$857$			541248.i			0.736944i	−0.929639	+	0.368472i	$0.879881\pi$
$858$		0			0
$859$	971934.	−	971934.i	1.31720	−	1.31720i	0.401210	−	0.915986i	$-0.368590\pi$
$859$	971934.	−	971934.i	1.31720	−	1.31720i	0.915986	−	0.401210i	$-0.131410\pi$
$860$		0			0
$861$	−143798.	+	143798.i	−0.193976	+	0.193976i
$862$		0			0
$863$			373292.i			0.501219i	0.968088	+	0.250610i	$0.0806310\pi$
$863$			373292.i			0.501219i	−0.968088	+	0.250610i	$0.919369\pi$
$864$		0			0
$865$	−19794.8			−0.0264557
$866$		0			0
$867$	−639976.	−	639976.i	−0.851384	−	0.851384i
$868$		0			0
$869$	488309.	+	488309.i	0.646630	+	0.646630i
$870$		0			0
$871$	296959.			0.391435
$872$		0			0
$873$			442459.i			0.580557i
$874$		0			0
$875$	−221379.	+	221379.i	−0.289148	+	0.289148i
$876$		0			0
$877$	109748.	−	109748.i	0.142691	−	0.142691i	−0.632153	−	0.774844i	$-0.717828\pi$
$877$	109748.	−	109748.i	0.142691	−	0.142691i	0.774844	+	0.632153i	$0.217828\pi$
$878$		0			0
$879$			395582.i			0.511987i
$880$		0			0
$881$	−309685.			−0.398996			−0.199498	−	0.979898i	$-0.563931\pi$
$881$	−309685.			−0.398996			−0.199498	+	0.979898i	$0.563931\pi$
$882$		0			0
$883$	−931347.	−	931347.i	−1.19451	−	1.19451i	−0.975787	−	0.218724i	$-0.929811\pi$
$883$	−931347.	−	931347.i	−1.19451	−	1.19451i	−0.218724	−	0.975787i	$-0.570189\pi$
$884$		0			0
$885$	−14819.1	−	14819.1i	−0.0189206	−	0.0189206i
$886$		0			0
$887$	−1.22870e6			−1.56170			−0.780849	−	0.624720i	$-0.785213\pi$
$887$	−1.22870e6			−1.56170			−0.780849	+	0.624720i	$0.785213\pi$
$888$		0			0
$889$		−	1.08216e6i		−	1.36927i
$890$		0			0
$891$	42190.5	−	42190.5i	0.0531446	−	0.0531446i
$892$		0			0
$893$	−519201.	+	519201.i	−0.651077	+	0.651077i
$894$		0			0
$895$			110077.i			0.137420i
$896$		0			0
$897$	245165.			0.304701
$898$		0			0
$899$	23740.1	+	23740.1i	0.0293740	+	0.0293740i
$900$		0			0
$901$	1.29065e6	+	1.29065e6i	1.58986	+	1.58986i
$902$		0			0
$903$	−715306.			−0.877236
$904$		0			0
$905$		−	168776.i		−	0.206069i
$906$		0			0
$907$	−33259.1	+	33259.1i	−0.0404292	+	0.0404292i	−0.727032	−	0.686603i	$-0.759101\pi$
$907$	−33259.1	+	33259.1i	−0.0404292	+	0.0404292i	0.686603	+	0.727032i	$0.259101\pi$
$908$		0			0
$909$	−211127.	+	211127.i	−0.255515	+	0.255515i
$910$		0			0
$911$			224558.i			0.270577i	0.990806	+	0.135289i	$0.0431962\pi$
$911$			224558.i			0.270577i	−0.990806	+	0.135289i	$0.956804\pi$
$912$		0			0
$913$	359258.			0.430988
$914$		0			0
$915$	−2511.94	−	2511.94i	−0.00300032	−	0.00300032i
$916$		0			0
$917$	−585182.	−	585182.i	−0.695909	−	0.695909i
$918$		0			0
$919$	−596247.			−0.705984			−0.352992	−	0.935626i	$-0.614836\pi$
$919$	−596247.			−0.705984			−0.352992	+	0.935626i	$0.614836\pi$
$920$		0			0
$921$		−	732386.i		−	0.863417i
$922$		0			0
$923$	−397082.	+	397082.i	−0.466097	+	0.466097i
$924$		0			0
$925$	−494089.	+	494089.i	−0.577460	+	0.577460i
$926$		0			0
$927$			145080.i			0.168830i
$928$		0			0
$929$	−153462.			−0.177816			−0.0889079	−	0.996040i	$-0.528338\pi$
$929$	−153462.			−0.177816			−0.0889079	+	0.996040i	$0.528338\pi$
$930$		0			0
$931$	−607409.	−	607409.i	−0.700780	−	0.700780i
$932$		0			0
$933$	−298381.	−	298381.i	−0.342774	−	0.342774i
$934$		0			0
$935$	−123551.			−0.141327
$936$		0			0
$937$			1.23354e6i			1.40499i	0.711689	+	0.702494i	$0.247930\pi$
$937$			1.23354e6i			1.40499i	−0.711689	+	0.702494i	$0.752070\pi$
$938$		0			0
$939$	−58984.0	+	58984.0i	−0.0668964	+	0.0668964i
$940$		0			0
$941$	537413.	−	537413.i	0.606917	−	0.606917i	−0.335222	−	0.942139i	$-0.608811\pi$
$941$	537413.	−	537413.i	0.606917	−	0.606917i	0.942139	+	0.335222i	$0.108811\pi$
$942$		0			0
$943$			79988.2i			0.0899503i
$944$		0			0
$945$	−35389.2			−0.0396285
$946$		0			0
$947$	−621957.	−	621957.i	−0.693522	−	0.693522i	0.269483	−	0.963005i	$-0.413147\pi$
$947$	−621957.	−	621957.i	−0.693522	−	0.693522i	−0.963005	+	0.269483i	$0.913147\pi$
$948$		0			0
$949$	89228.1	+	89228.1i	0.0990762	+	0.0990762i
$950$		0			0
$951$	42465.3			0.0469540
$952$		0			0
$953$		−	609536.i		−	0.671140i	−0.942015	−	0.335570i	$-0.891071\pi$
$953$		−	609536.i		−	0.671140i	0.942015	−	0.335570i	$-0.108929\pi$
$954$		0			0
$955$	21654.5	−	21654.5i	0.0237434	−	0.0237434i
$956$		0			0
$957$	27539.0	−	27539.0i	0.0300694	−	0.0300694i
$958$		0			0
$959$		−	1.67366e6i		−	1.81982i
$960$		0			0
$961$	789110.			0.854458
$962$		0			0
$963$	261961.	+	261961.i	0.282478	+	0.282478i
$964$		0			0
$965$	−44819.7	−	44819.7i	−0.0481299	−	0.0481299i
$966$		0			0
$967$	1.17833e6			1.26013			0.630063	−	0.776544i	$-0.283029\pi$
$967$	1.17833e6			1.26013			0.630063	+	0.776544i	$0.283029\pi$
$968$		0			0
$969$			472576.i			0.503297i
$970$		0			0
$971$	1.17466e6	−	1.17466e6i	1.24588	−	1.24588i	0.288351	−	0.957525i	$-0.406893\pi$
$971$	1.17466e6	−	1.17466e6i	1.24588	−	1.24588i	0.957525	−	0.288351i	$-0.0931070\pi$
$972$		0			0
$973$	1.94093e6	−	1.94093e6i	2.05015	−	2.05015i
$974$		0			0
$975$		−	871312.i		−	0.916568i
$976$		0			0
$977$	1.03389e6			1.08314			0.541568	−	0.840657i	$-0.317831\pi$
$977$	1.03389e6			1.08314			0.541568	+	0.840657i	$0.317831\pi$
$978$		0			0
$979$	443832.	+	443832.i	0.463077	+	0.463077i
$980$		0			0
$981$	275412.	+	275412.i	0.286183	+	0.286183i
$982$		0			0
$983$	−782606.			−0.809909			−0.404954	−	0.914337i	$-0.632713\pi$
$983$	−782606.			−0.809909			−0.404954	+	0.914337i	$0.632713\pi$
$984$		0			0
$985$		−	148132.i		−	0.152678i
$986$		0			0
$987$	1.27739e6	−	1.27739e6i	1.31126	−	1.31126i
$988$		0			0
$989$	−198946.	+	198946.i	−0.203396	+	0.203396i
$990$		0			0
$991$			831804.i			0.846981i	0.905900	+	0.423491i	$0.139195\pi$
$991$			831804.i			0.846981i	−0.905900	+	0.423491i	$0.860805\pi$
$992$		0			0
$993$	−587669.			−0.595984
$994$		0			0
$995$	65371.7	+	65371.7i	0.0660303	+	0.0660303i
$996$		0			0
$997$	481944.	+	481944.i	0.484849	+	0.484849i	0.906676	−	0.421827i	$-0.138611\pi$
$997$	481944.	+	481944.i	0.484849	+	0.484849i	−0.421827	+	0.906676i	$0.638611\pi$
$998$		0			0
$999$	−159101.			−0.159420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.l.a.31.6		32
4.3	odd	2		384.5.l.b.31.11		32
8.3	odd	2		48.5.l.a.43.6	yes	32
8.5	even	2		192.5.l.a.79.11		32
16.3	odd	4	inner	384.5.l.a.223.6		32
16.5	even	4		48.5.l.a.19.6	✓	32
16.11	odd	4		192.5.l.a.175.11		32
16.13	even	4		384.5.l.b.223.11		32
24.5	odd	2		576.5.m.b.271.9		32
24.11	even	2		144.5.m.c.91.11		32
48.5	odd	4		144.5.m.c.19.11		32
48.11	even	4		576.5.m.b.559.9		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.5.l.a.19.6	✓	32	16.5	even	4
48.5.l.a.43.6	yes	32	8.3	odd	2
144.5.m.c.19.11		32	48.5	odd	4
144.5.m.c.91.11		32	24.11	even	2
192.5.l.a.79.11		32	8.5	even	2
192.5.l.a.175.11		32	16.11	odd	4
384.5.l.a.31.6		32	1.1	even	1	trivial
384.5.l.a.223.6		32	16.3	odd	4	inner
384.5.l.b.31.11		32	4.3	odd	2
384.5.l.b.223.11		32	16.13	even	4
576.5.m.b.271.9		32	24.5	odd	2
576.5.m.b.559.9		32	48.11	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	384.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.67423 - 3.67423i) q^{3} +(2.10268 + 2.10268i) q^{5} -84.8276 q^{7} +27.0000i q^{9} +O(q^{10})\)	\(q+(-3.67423 - 3.67423i) q^{3} +(2.10268 + 2.10268i) q^{5} -84.8276 q^{7} +27.0000i q^{9} +(-57.8744 + 57.8744i) q^{11} +(-192.435 + 192.435i) q^{13} -15.4515i q^{15} +507.642 q^{17} +(-126.683 - 126.683i) q^{19} +(311.677 + 311.677i) q^{21} +173.371 q^{23} -616.157i q^{25} +(99.2043 - 99.2043i) q^{27} +(64.7538 - 64.7538i) q^{29} +366.621i q^{31} +425.288 q^{33} +(-178.365 - 178.365i) q^{35} +(-801.887 - 801.887i) q^{37} +1414.11 q^{39} +461.370i q^{41} +(-1147.51 + 1147.51i) q^{43} +(-56.7723 + 56.7723i) q^{45} -4098.44i q^{47} +4794.73 q^{49} +(-1865.20 - 1865.20i) q^{51} +(2542.44 + 2542.44i) q^{53} -243.383 q^{55} +930.924i q^{57} +(959.072 - 959.072i) q^{59} +(162.570 - 162.570i) q^{61} -2290.35i q^{63} -809.260 q^{65} +(-771.580 - 771.580i) q^{67} +(-637.006 - 637.006i) q^{69} +2063.45 q^{71} -463.678i q^{73} +(-2263.91 + 2263.91i) q^{75} +(4909.35 - 4909.35i) q^{77} -8437.39i q^{79} -729.000 q^{81} +(-3103.77 - 3103.77i) q^{83} +(1067.41 + 1067.41i) q^{85} -475.841 q^{87} -7668.88i q^{89} +(16323.8 - 16323.8i) q^{91} +(1347.05 - 1347.05i) q^{93} -532.746i q^{95} +16387.4 q^{97} +(-1562.61 - 1562.61i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 192 q^{11} + 704 q^{19} + 2304 q^{23} + 1728 q^{29} - 5184 q^{35} - 3648 q^{37} + 1088 q^{43} + 10976 q^{49} - 4032 q^{51} - 960 q^{53} - 11776 q^{55} + 13056 q^{59} - 3776 q^{61} + 4032 q^{65} - 896 q^{67} + 9792 q^{69} + 39936 q^{71} - 1152 q^{75} - 9408 q^{77} - 23328 q^{81} + 24000 q^{83} + 11200 q^{85} + 30528 q^{91} - 5184 q^{99}+O(q^{100}) \) 32 * q - 192 * q^11 + 704 * q^19 + 2304 * q^23 + 1728 * q^29 - 5184 * q^35 - 3648 * q^37 + 1088 * q^43 + 10976 * q^49 - 4032 * q^51 - 960 * q^53 - 11776 * q^55 + 13056 * q^59 - 3776 * q^61 + 4032 * q^65 - 896 * q^67 + 9792 * q^69 + 39936 * q^71 - 1152 * q^75 - 9408 * q^77 - 23328 * q^81 + 24000 * q^83 + 11200 * q^85 + 30528 * q^91 - 5184 * q^99

Introduction

L-functions

Modular forms

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Database

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