LMFDB - Embedded newform 192.5.l.a.175.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,5,Mod(79,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, 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("192.79");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	192.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:	$19.8470329121$
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			175.11
Character	$\chi$	$=$	192.175
Dual form			192.5.l.a.79.11

$q$-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 $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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	$-0.768942\pi$
$5$	−2.10268	+	2.10268i	−0.0841071	+	0.0841071i	0.663802	+	0.747909i	$0.268942\pi$
$6$		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	$-0.140179\pi$
$11$	57.8744	+	57.8744i	0.478301	+	0.478301i	−0.426287	+	0.904588i	$0.640179\pi$
$12$		0			0
$13$	192.435	+	192.435i	1.13867	+	1.13867i	0.988689	+	0.149983i	$0.0479219\pi$
$13$	192.435	+	192.435i	1.13867	+	1.13867i	0.149983	+	0.988689i	$0.452078\pi$
$14$		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	$-0.670181\pi$
$19$	126.683	−	126.683i	0.350922	−	0.350922i	0.860452	+	0.509531i	$0.170181\pi$
$20$		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	$-0.267339\pi$
$29$	−64.7538	−	64.7538i	−0.0769961	−	0.0769961i	−0.744556	+	0.667560i	$0.767339\pi$
$30$		0			0
$31$		−	366.621i		−	0.381500i	−0.981639	−	0.190750i	$-0.938908\pi$
$31$		−	366.621i		−	0.381500i	0.981639	−	0.190750i	$-0.0610920\pi$
$32$		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	$-0.614066\pi$
$37$	801.887	−	801.887i	0.585747	−	0.585747i	0.936477	+	0.350730i	$0.114066\pi$
$38$		0			0
$39$	1414.11			0.929721
$40$		0			0
$41$		−	461.370i		−	0.274462i	−0.990539	−	0.137231i	$-0.956180\pi$
$41$		−	461.370i		−	0.274462i	0.990539	−	0.137231i	$-0.0438202\pi$
$42$		0			0
$43$	1147.51	+	1147.51i	0.620613	+	0.620613i	0.945688	−	0.325075i	$-0.105390\pi$
$43$	1147.51	+	1147.51i	0.620613	+	0.620613i	−0.325075	+	0.945688i	$0.605390\pi$
$44$		0			0
$45$	56.7723	+	56.7723i	0.0280357	+	0.0280357i
$46$		0			0
$47$			4098.44i			1.85534i	0.373406	+	0.927668i	$0.378190\pi$
$47$			4098.44i			1.85534i	−0.373406	+	0.927668i	$0.621810\pi$
$48$		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	$-0.971068\pi$
$53$	−2542.44	+	2542.44i	−0.905106	+	0.905106i	0.0907664	+	0.995872i	$0.471068\pi$
$54$		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	$-0.312412\pi$
$59$	−959.072	−	959.072i	−0.275516	−	0.275516i	−0.831316	+	0.555800i	$0.812412\pi$
$60$		0			0
$61$	−162.570	−	162.570i	−0.0436898	−	0.0436898i	0.684924	−	0.728614i	$-0.259835\pi$
$61$	−162.570	−	162.570i	−0.0436898	−	0.0436898i	−0.728614	+	0.684924i	$0.759835\pi$
$62$		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	$-0.711217\pi$
$67$	771.580	−	771.580i	0.171882	−	0.171882i	0.787806	+	0.615924i	$0.211217\pi$
$68$		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	$0.0138525\pi$
$73$			463.678i			0.0870103i	−0.999053	+	0.0435052i	$0.986148\pi$
$74$		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	$0.236273\pi$
$79$			8437.39i			1.35193i	−0.736934	+	0.675965i	$0.763727\pi$
$80$		0			0
$81$	−729.000			−0.111111
$82$		0			0
$83$	3103.77	−	3103.77i	0.450540	−	0.450540i	−0.444993	−	0.895534i	$-0.646794\pi$
$83$	3103.77	−	3103.77i	0.450540	−	0.450540i	0.895534	+	0.444993i	$0.146794\pi$
$84$		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	$0.160848\pi$
$89$			7668.88i			0.968171i	−0.875021	+	0.484085i	$0.839152\pi$
$90$		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	$-0.932344\pi$
$101$	−7819.54	+	7819.54i	−0.766546	+	0.766546i	0.210951	+	0.977497i	$0.432344\pi$
$102$		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	$-0.454525\pi$
$107$	−9702.27	−	9702.27i	−0.847434	−	0.847434i	−0.989812	+	0.142378i	$0.954525\pi$
$108$		0			0
$109$	−10200.4	−	10200.4i	−0.858550	−	0.858550i	0.132618	−	0.991167i	$-0.457662\pi$
$109$	−10200.4	−	10200.4i	−0.858550	−	0.858550i	−0.991167	+	0.132618i	$0.957662\pi$
$110$		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	$-0.870580\pi$
$127$		−	12757.2i		−	0.790948i	0.918477	−	0.395474i	$-0.129420\pi$
$128$		0			0
$129$	8432.47			0.506728
$130$		0			0
$131$	−6898.48	+	6898.48i	−0.401986	+	0.401986i	−0.878932	−	0.476946i	$-0.841744\pi$
$131$	−6898.48	+	6898.48i	−0.401986	+	0.401986i	0.476946	+	0.878932i	$0.341744\pi$
$132$		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	$-0.823840\pi$
$137$		−	19730.1i		−	1.05121i	0.850730	−	0.525603i	$-0.176160\pi$
$138$		0			0
$139$	22880.9	+	22880.9i	1.18425	+	1.18425i	0.978632	+	0.205619i	$0.0659208\pi$
$139$	22880.9	+	22880.9i	1.18425	+	1.18425i	0.205619	+	0.978632i	$0.434079\pi$
$140$		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	$-0.899016\pi$
$149$	−14167.4	+	14167.4i	−0.638141	+	0.638141i	0.311955	+	0.950097i	$0.399016\pi$
$150$		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	$-0.422584\pi$
$157$	−17987.6	−	17987.6i	−0.729749	−	0.729749i	−0.970570	+	0.240820i	$0.922584\pi$
$158$		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	$-0.777389\pi$
$163$	−3229.11	+	3229.11i	−0.121537	+	0.121537i	0.643722	+	0.765259i	$0.277389\pi$
$164$		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	$-0.214528\pi$
$173$	4707.05	+	4707.05i	0.157274	+	0.157274i	−0.624084	+	0.781358i	$0.714528\pi$
$174$		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	$-0.946034\pi$
$179$	−26175.4	+	26175.4i	−0.816934	+	0.816934i	0.168729	+	0.985663i	$0.446034\pi$
$180$		0			0
$181$	40133.6	−	40133.6i	1.22504	−	1.22504i	0.259223	−	0.965817i	$-0.416533\pi$
$181$	40133.6	−	40133.6i	1.22504	−	1.22504i	0.965817	−	0.259223i	$-0.0834665\pi$
$182$		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	$0.0450798\pi$
$191$			10298.6i			0.282299i	−0.989988	+	0.141150i	$0.954920\pi$
$192$		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	$-0.528189\pi$
$197$	35224.5	−	35224.5i	0.907638	−	0.907638i	0.996081	+	0.0884429i	$0.0281891\pi$
$198$		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	$-0.955114\pi$
$211$	−37821.8	+	37821.8i	−0.849528	+	0.849528i	0.140546	+	0.990074i	$0.455114\pi$
$212$		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	$0.00914963\pi$
$223$			2858.47i			0.0574809i	−0.999587	+	0.0287404i	$0.990850\pi$
$224$		0			0
$225$	16636.3			0.328617
$226$		0			0
$227$	70688.1	−	70688.1i	1.37181	−	1.37181i	0.514053	−	0.857758i	$-0.328143\pi$
$227$	70688.1	−	70688.1i	1.37181	−	1.37181i	0.857758	−	0.514053i	$-0.171857\pi$
$228$		0			0
$229$	−394.626	+	394.626i	−0.00752514	+	0.00752514i	−0.710859	−	0.703334i	$-0.751694\pi$
$229$	−394.626	+	394.626i	−0.00752514	+	0.00752514i	0.703334	+	0.710859i	$0.251694\pi$
$230$		0			0
$231$	−36076.2			−0.676078
$232$		0			0
$233$		−	21767.8i		−	0.400961i	−0.979698	−	0.200481i	$-0.935750\pi$
$233$		−	21767.8i		−	0.400961i	0.979698	−	0.200481i	$-0.0642504\pi$
$234$		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	$-0.745389\pi$
$239$		−	81943.0i		−	1.43455i	0.696789	−	0.717276i	$-0.254611\pi$
$240$		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	$-0.126106\pi$
$251$	33809.8	+	33809.8i	0.536655	+	0.536655i	−0.385890	+	0.922545i	$0.626106\pi$
$252$		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	$-0.399104\pi$
$269$	−46201.7	−	46201.7i	−0.638489	−	0.638489i	−0.950183	+	0.311694i	$0.899104\pi$
$270$		0			0
$271$		−	90603.1i		−	1.23369i	−0.787086	−	0.616843i	$-0.788411\pi$
$271$		−	90603.1i		−	1.23369i	0.787086	−	0.616843i	$-0.211589\pi$
$272$		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	$-0.749240\pi$
$277$	259.186	−	259.186i	0.00337794	−	0.00337794i	0.708794	+	0.705416i	$0.249240\pi$
$278$		0			0
$279$	−9898.78			−0.127167
$280$		0			0
$281$			114309.i			1.44766i	0.689979	+	0.723829i	$0.257620\pi$
$281$			114309.i			1.44766i	−0.689979	+	0.723829i	$0.742380\pi$
$282$		0			0
$283$	−33739.3	−	33739.3i	−0.421273	−	0.421273i	0.464369	−	0.885642i	$-0.346281\pi$
$283$	−33739.3	−	33739.3i	−0.421273	−	0.421273i	−0.885642	+	0.464369i	$0.846281\pi$
$284$		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	$-0.603774\pi$
$293$	53831.9	−	53831.9i	0.627054	−	0.627054i	0.947326	+	0.320272i	$0.103774\pi$
$294$		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	−0.0592206	+	0.998245i	$0.518862\pi$
$307$	−99665.1	+	99665.1i	−1.05747	+	1.05747i	−0.998245	+	0.0592206i	$0.981138\pi$
$308$		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	$0.0261087\pi$
$313$			16053.4i			0.163862i	−0.996638	+	0.0819310i	$0.973891\pi$
$314$		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	$-0.237053\pi$
$317$	5778.79	+	5778.79i	0.0575067	+	0.0575067i	−0.677769	+	0.735275i	$0.737053\pi$
$318$		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	$-0.422630\pi$
$331$	−79971.6	−	79971.6i	−0.729928	−	0.729928i	−0.970605	+	0.240677i	$0.922630\pi$
$332$		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	$-0.293082\pi$
$347$	−22976.9	−	22976.9i	−0.190824	−	0.190824i	−0.796052	+	0.605228i	$0.793082\pi$
$348$		0			0
$349$	−87638.2	−	87638.2i	−0.719520	−	0.719520i	0.248987	−	0.968507i	$-0.419902\pi$
$349$	−87638.2	−	87638.2i	−0.719520	−	0.719520i	−0.968507	+	0.248987i	$0.919902\pi$
$350$		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	$0.0328316\pi$
$367$			27735.4i			0.205922i	−0.994685	+	0.102961i	$0.967168\pi$
$368$		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	$-0.596480\pi$
$373$	91259.4	−	91259.4i	0.655934	−	0.655934i	0.954415	+	0.298481i	$0.0964801\pi$
$374$		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	$-0.248863\pi$
$379$	725.899	+	725.899i	0.00505356	+	0.00505356i	−0.704575	+	0.709629i	$0.748863\pi$
$380$		0			0
$381$	−46872.9	−	46872.9i	−0.322903	−	0.322903i
$382$		0			0
$383$			81078.9i			0.552727i	0.961053	+	0.276363i	$0.0891293\pi$
$383$			81078.9i			0.552727i	−0.961053	+	0.276363i	$0.910871\pi$
$384$		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	$-0.868225\pi$
$389$	−77667.7	+	77667.7i	−0.513265	+	0.513265i	0.402260	+	0.915525i	$0.368225\pi$
$390$		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	$-0.218706\pi$
$397$	21877.8	+	21877.8i	0.138811	+	0.138811i	−0.634287	+	0.773098i	$0.718706\pi$
$398$		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	$0.174819\pi$
$409$			174646.i			1.04402i	−0.852938	+	0.522012i	$0.825181\pi$
$410$		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	$-0.986234\pi$
$419$	−167807.	+	167807.i	−0.955831	+	0.955831i	0.0432336	+	0.999065i	$0.486234\pi$
$420$		0			0
$421$	18489.0	−	18489.0i	0.104316	−	0.104316i	−0.653023	−	0.757338i	$-0.726499\pi$
$421$	18489.0	−	18489.0i	0.104316	−	0.104316i	0.757338	+	0.653023i	$0.226499\pi$
$422$		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	$0.187610\pi$
$431$			206513.i			1.11171i	−0.831278	+	0.555857i	$0.812390\pi$
$432$		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	−0.995901	−	0.0904459i	$-0.971171\pi$
$443$	−213195.	−	213195.i	−1.08635	−	1.08635i	−0.0904459	−	0.995901i	$-0.528829\pi$
$444$		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	$-0.899264\pi$
$457$		−	129994.i		−	0.622429i	0.950340	−	0.311214i	$-0.100736\pi$
$458$		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	$-0.429219\pi$
$461$	−160419.	−	160419.i	−0.754840	−	0.754840i	−0.975378	+	0.220538i	$0.929219\pi$
$462$		0			0
$463$			237870.i			1.10963i	0.831974	+	0.554815i	$0.187211\pi$
$463$			237870.i			1.10963i	−0.831974	+	0.554815i	$0.812789\pi$
$464$		0			0
$465$	5664.84			0.0261988
$466$		0			0
$467$	154822.	−	154822.i	0.709902	−	0.709902i	−0.256612	−	0.966514i	$-0.582606\pi$
$467$	154822.	−	154822.i	0.709902	−	0.709902i	0.966514	+	0.256612i	$0.0826064\pi$
$468$		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	$-0.947191\pi$
$479$		−	75781.5i		−	0.330288i	0.986270	−	0.165144i	$-0.0528088\pi$
$480$		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	$-0.0943506\pi$
$491$	160150.	+	160150.i	0.664301	+	0.664301i	−0.292090	+	0.956391i	$0.594351\pi$
$492$		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	$-0.589787\pi$
$499$	169851.	−	169851.i	0.682131	−	0.682131i	0.960480	+	0.278349i	$0.0897872\pi$
$500$		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	$-0.0915016\pi$
$509$	174996.	+	174996.i	0.675449	+	0.675449i	−0.283518	+	0.958967i	$0.591502\pi$
$510$		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	$-0.728570\pi$
$521$		−	408830.i		−	1.50615i	0.657936	−	0.753074i	$-0.271430\pi$
$522$		0			0
$523$	186677.	+	186677.i	0.682477	+	0.682477i	0.960558	−	0.278080i	$-0.0896982\pi$
$523$	186677.	+	186677.i	0.682477	+	0.682477i	−0.278080	+	0.960558i	$0.589698\pi$
$524$		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	$-0.145534\pi$
$541$	133417.	+	133417.i	0.455844	+	0.455844i	−0.441445	+	0.897288i	$0.645534\pi$
$542$		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	$-0.677600\pi$
$547$	95416.9	−	95416.9i	0.318897	−	0.318897i	0.848343	+	0.529446i	$0.177600\pi$
$548$		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	$-0.380723\pi$
$557$	−175167.	−	175167.i	−0.564600	−	0.564600i	−0.930611	+	0.366011i	$0.880723\pi$
$558$		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	$-0.868909\pi$
$563$	−163587.	+	163587.i	−0.516097	+	0.516097i	0.400291	+	0.916388i	$0.368909\pi$
$564$		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	$0.188439\pi$
$569$			361331.i			1.11604i	−0.829826	+	0.558022i	$0.811561\pi$
$570$		0			0
$571$	−263234.	−	263234.i	−0.807364	−	0.807364i	0.176871	−	0.984234i	$-0.443403\pi$
$571$	−263234.	−	263234.i	−0.807364	−	0.807364i	−0.984234	+	0.176871i	$0.943403\pi$
$572$		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	−0.999891	−	0.0147866i	$-0.995293\pi$
$587$	−349626.	−	349626.i	−1.01468	−	1.01468i	−0.0147866	−	0.999891i	$-0.504707\pi$
$588$		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	$0.145188\pi$
$601$			318196.i			0.880940i	−0.897767	+	0.440470i	$0.854812\pi$
$602$		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	$-0.719255\pi$
$607$		−	568888.i		−	1.54401i	0.635618	−	0.772003i	$-0.280745\pi$
$608$		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	$-0.566187\pi$
$613$	290102.	−	290102.i	0.772023	−	0.772023i	0.978460	+	0.206437i	$0.0661868\pi$
$614$		0			0
$615$	7128.85			0.0188482
$616$		0			0
$617$			222944.i			0.585633i	0.956169	+	0.292817i	$0.0945925\pi$
$617$			222944.i			0.585633i	−0.956169	+	0.292817i	$0.905407\pi$
$618$		0			0
$619$	52984.5	+	52984.5i	0.138283	+	0.138283i	0.772860	−	0.634577i	$-0.218826\pi$
$619$	52984.5	+	52984.5i	0.138283	+	0.138283i	−0.634577	+	0.772860i	$0.718826\pi$
$620$		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	$-0.626836\pi$
$643$	220638.	−	220638.i	0.533652	−	0.533652i	0.921657	+	0.388005i	$0.126836\pi$
$644$		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	−0.999599	−	0.0283280i	$-0.990982\pi$
$653$	−438317.	−	438317.i	−1.02793	−	1.02793i	−0.0283280	−	0.999599i	$-0.509018\pi$
$654$		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	−0.0453069	+	0.998973i	$0.514427\pi$
$659$	−453511.	+	453511.i	−1.04428	+	1.04428i	−0.998973	+	0.0453069i	$0.985573\pi$
$660$		0			0
$661$	387740.	−	387740.i	0.887436	−	0.887436i	−0.106840	−	0.994276i	$-0.534073\pi$
$661$	387740.	−	387740.i	0.887436	−	0.887436i	0.994276	+	0.106840i	$0.0340732\pi$
$662$		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	−0.305972	+	0.952041i	$0.598981\pi$
$677$	−576584.	+	576584.i	−1.25801	+	1.25801i	−0.952041	+	0.305972i	$0.901019\pi$
$678$		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	$-0.423536\pi$
$683$	−342109.	−	342109.i	−0.733371	−	0.733371i	−0.971286	+	0.237915i	$0.923536\pi$
$684$		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	$-0.852198\pi$
$691$	−213096.	+	213096.i	−0.446293	+	0.446293i	0.447827	+	0.894120i	$0.352198\pi$
$692$		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	$-0.388806\pi$
$701$	−293532.	−	293532.i	−0.597337	−	0.597337i	−0.939603	+	0.342266i	$0.888806\pi$
$702$		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	$-0.755925\pi$
$709$	−13231.5	+	13231.5i	−0.0263218	+	0.0263218i	0.693823	+	0.720145i	$0.255925\pi$
$710$		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	$0.152794\pi$
$719$			477459.i			0.923587i	−0.886987	+	0.461794i	$0.847206\pi$
$720$		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	$-0.214599\pi$
$733$	84332.9	+	84332.9i	0.156960	+	0.156960i	−0.624258	+	0.781218i	$0.714599\pi$
$734$		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	$-0.670548\pi$
$739$	190782.	−	190782.i	0.349340	−	0.349340i	0.859864	+	0.510524i	$0.170548\pi$
$740$		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	$-0.785196\pi$
$751$		−	704733.i		−	1.24952i	0.780815	−	0.624762i	$-0.214804\pi$
$752$		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	$-0.580357\pi$
$757$	411751.	−	411751.i	0.718526	−	0.718526i	0.968303	+	0.249777i	$0.0803574\pi$
$758$		0			0
$759$	73732.7			0.127990
$760$		0			0
$761$			82790.4i			0.142959i	0.997442	+	0.0714793i	$0.0227720\pi$
$761$			82790.4i			0.142959i	−0.997442	+	0.0714793i	$0.977228\pi$
$762$		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	$-0.698495\pi$
$773$	136137.	−	136137.i	0.227833	−	0.227833i	0.811787	+	0.583954i	$0.198495\pi$
$774$		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	$-0.735641\pi$
$787$	39500.6	−	39500.6i	0.0637756	−	0.0637756i	0.738275	+	0.674500i	$0.235641\pi$
$788$		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	$-0.439794\pi$
$797$	−504452.	−	504452.i	−0.794151	−	0.794151i	−0.982166	+	0.188015i	$0.939794\pi$
$798$		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	$0.164639\pi$
$809$			647248.i			0.988948i	−0.869192	+	0.494474i	$0.835361\pi$
$810$		0			0
$811$	−149216.	−	149216.i	−0.226868	−	0.226868i	0.584515	−	0.811383i	$-0.301285\pi$
$811$	−149216.	−	149216.i	−0.226868	−	0.226868i	−0.811383	+	0.584515i	$0.801285\pi$
$812$		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	$-0.691961\pi$
$821$	172848.	−	172848.i	0.256435	−	0.256435i	0.823602	+	0.567168i	$0.191961\pi$
$822$		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	$-0.249427\pi$
$827$	1742.56	+	1742.56i	0.00254786	+	0.00254786i	−0.705832	+	0.708380i	$0.749427\pi$
$828$		0			0
$829$	193828.	+	193828.i	0.282037	+	0.282037i	0.833921	−	0.551884i	$-0.186091\pi$
$829$	193828.	+	193828.i	0.282037	+	0.282037i	−0.551884	+	0.833921i	$0.686091\pi$
$830$		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	$-0.938944\pi$
$853$	−575553.	+	575553.i	−0.791020	+	0.791020i	0.190640	+	0.981660i	$0.438944\pi$
$854$		0			0
$855$	14384.1			0.0196767
$856$		0			0
$857$		−	541248.i		−	0.736944i	−0.929639	−	0.368472i	$-0.879881\pi$
$857$		−	541248.i		−	0.736944i	0.929639	−	0.368472i	$-0.120119\pi$
$858$		0			0
$859$	−971934.	−	971934.i	−1.31720	−	1.31720i	−0.915986	−	0.401210i	$-0.868590\pi$
$859$	−971934.	−	971934.i	−1.31720	−	1.31720i	−0.401210	−	0.915986i	$-0.631410\pi$
$860$		0			0
$861$	143798.	+	143798.i	0.193976	+	0.193976i
$862$		0			0
$863$		−	373292.i		−	0.501219i	−0.968088	−	0.250610i	$-0.919369\pi$
$863$		−	373292.i		−	0.501219i	0.968088	−	0.250610i	$-0.0806310\pi$
$864$		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	$-0.282172\pi$
$877$	−109748.	−	109748.i	−0.142691	−	0.142691i	−0.774844	+	0.632153i	$0.782172\pi$
$878$		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	0.218724	−	0.975787i	$-0.429811\pi$
$883$	931347.	−	931347.i	1.19451	−	1.19451i	0.975787	−	0.218724i	$-0.0701895\pi$
$884$		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	$-0.240899\pi$
$907$	33259.1	+	33259.1i	0.0404292	+	0.0404292i	−0.686603	+	0.727032i	$0.740899\pi$
$908$		0			0
$909$	211127.	+	211127.i	0.255515	+	0.255515i
$910$		0			0
$911$		−	224558.i		−	0.270577i	−0.990806	−	0.135289i	$-0.956804\pi$
$911$		−	224558.i		−	0.270577i	0.990806	−	0.135289i	$-0.0431962\pi$
$912$		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	$-0.752070\pi$
$937$		−	1.23354e6i		−	1.40499i	0.711689	−	0.702494i	$-0.247930\pi$
$938$		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	$-0.391189\pi$
$941$	−537413.	−	537413.i	−0.606917	−	0.606917i	−0.942139	+	0.335222i	$0.891189\pi$
$942$		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	$-0.586853\pi$
$947$	621957.	−	621957.i	0.693522	−	0.693522i	0.963005	+	0.269483i	$0.0868529\pi$
$948$		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	$0.108929\pi$
$953$			609536.i			0.671140i	−0.942015	+	0.335570i	$0.891071\pi$
$954$		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	−0.957525	−	0.288351i	$-0.906893\pi$
$971$	−1.17466e6	−	1.17466e6i	−1.24588	−	1.24588i	−0.288351	−	0.957525i	$-0.593107\pi$
$972$		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	$-0.860805\pi$
$991$		−	831804.i		−	0.846981i	0.905900	−	0.423491i	$-0.139195\pi$
$992$		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	$-0.861389\pi$
$997$	−481944.	+	481944.i	−0.484849	+	0.484849i	0.421827	+	0.906676i	$0.361389\pi$
$998$		0			0
$999$	−159101.			−0.159420

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.5.l.a.19.6	✓	32	4.3	odd	2
48.5.l.a.43.6	yes	32	16.5	even	4
144.5.m.c.19.11		32	12.11	even	2
144.5.m.c.91.11		32	48.5	odd	4
192.5.l.a.79.11		32	16.11	odd	4	inner
192.5.l.a.175.11		32	1.1	even	1	trivial
384.5.l.a.31.6		32	16.3	odd	4
384.5.l.a.223.6		32	8.5	even	2
384.5.l.b.31.11		32	16.13	even	4
384.5.l.b.223.11		32	8.3	odd	2
576.5.m.b.271.9		32	48.11	even	4
576.5.m.b.559.9		32	3.2	odd	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	192.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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more