LMFDB - Embedded newform 648.5.m.g.593.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [648,5,Mod(377,648)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("648.377"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	648.m (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,-96] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$66.9837360783$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			593.6
Character	$\chi$	$=$	648.593
Dual form			648.5.m.g.377.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.13894 + 3.54432i) q^{5} +(26.4902 - 45.8823i) q^{7} +(20.6208 + 11.9054i) q^{11} +(-138.907 - 240.593i) q^{13} +91.5027i q^{17} -332.045 q^{19} +(347.849 - 200.831i) q^{23} +(-287.376 + 497.749i) q^{25} +(-316.307 - 182.620i) q^{29} +(175.532 + 304.031i) q^{31} +375.558i q^{35} -42.6630 q^{37} +(-37.4041 + 21.5953i) q^{41} +(-21.6099 + 37.4295i) q^{43} +(2431.98 + 1404.11i) q^{47} +(-202.957 - 351.532i) q^{49} -4128.48i q^{53} -168.786 q^{55} +(-5504.74 + 3178.16i) q^{59} +(-848.536 + 1469.71i) q^{61} +(1705.48 + 984.659i) q^{65} +(1269.40 + 2198.67i) q^{67} -1550.91i q^{71} -4521.97 q^{73} +(1092.50 - 630.752i) q^{77} +(-3500.87 + 6063.68i) q^{79} +(-4962.11 - 2864.87i) q^{83} +(-324.315 - 561.730i) q^{85} +6177.83i q^{89} -14718.6 q^{91} +(2038.41 - 1176.87i) q^{95} +(-5363.80 + 9290.37i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 96 q^{7} + 72 q^{13} + 672 q^{19} - 84 q^{25} + 1536 q^{31} - 984 q^{37} - 10128 q^{43} + 6828 q^{49} - 27936 q^{55} - 26268 q^{61} + 20784 q^{67} - 19968 q^{73} - 44592 q^{79} + 45348 q^{85} - 23616 q^{91}+ \cdots - 76608 q^{97}+O(q^{100}) $ 24 * q - 96 * q^7 + 72 * q^13 + 672 * q^19 - 84 * q^25 + 1536 * q^31 - 984 * q^37 - 10128 * q^43 + 6828 * q^49 - 27936 * q^55 - 26268 * q^61 + 20784 * q^67 - 19968 * q^73 - 44592 * q^79 + 45348 * q^85 - 23616 * q^91 - 76608 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−6.13894	+	3.54432i	−0.245558	+	0.141773i	−0.617728	−	0.786391i	$-0.711947\pi$
$5$	−6.13894	+	3.54432i	−0.245558	+	0.141773i	0.372171	+	0.928164i	$0.378614\pi$
$6$		0			0
$7$	26.4902	−	45.8823i	0.540615	−	0.936374i	−0.458253	−	0.888822i	$-0.651525\pi$
$7$	26.4902	−	45.8823i	0.540615	−	0.936374i	0.998869	−	0.0475519i	$-0.0151419\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	20.6208	+	11.9054i	0.170420	+	0.0983918i	0.582784	−	0.812627i	$-0.301963\pi$
$11$	20.6208	+	11.9054i	0.170420	+	0.0983918i	−0.412364	+	0.911019i	$0.635297\pi$
$12$		0			0
$13$	−138.907	−	240.593i	−0.821933	−	1.42363i	−0.904241	−	0.427022i	$-0.859563\pi$
$13$	−138.907	−	240.593i	−0.821933	−	1.42363i	0.0823084	−	0.996607i	$-0.473771\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			91.5027i			0.316618i	0.987390	+	0.158309i	$0.0506043\pi$
$17$			91.5027i			0.316618i	−0.987390	+	0.158309i	$0.949396\pi$
$18$		0			0
$19$	−332.045			−0.919793			−0.459896	−	0.887973i	$-0.652113\pi$
$19$	−332.045			−0.919793			−0.459896	+	0.887973i	$0.652113\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	347.849	−	200.831i	0.657560	−	0.379642i	−0.133787	−	0.991010i	$-0.542714\pi$
$23$	347.849	−	200.831i	0.657560	−	0.379642i	0.791347	+	0.611368i	$0.209380\pi$
$24$		0			0
$25$	−287.376	+	497.749i	−0.459801	+	0.796399i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−316.307	−	182.620i	−0.376108	−	0.217146i	0.300016	−	0.953934i	$-0.403008\pi$
$29$	−316.307	−	182.620i	−0.376108	−	0.217146i	−0.676124	+	0.736788i	$0.736341\pi$
$30$		0			0
$31$	175.532	+	304.031i	0.182656	+	0.316369i	0.942784	−	0.333404i	$-0.108197\pi$
$31$	175.532	+	304.031i	0.182656	+	0.316369i	−0.760128	+	0.649773i	$0.774864\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			375.558i			0.306578i
$36$		0			0
$37$	−42.6630			−0.0311636			−0.0155818	−	0.999879i	$-0.504960\pi$
$37$	−42.6630			−0.0311636			−0.0155818	+	0.999879i	$0.504960\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−37.4041	+	21.5953i	−0.0222511	+	0.0128467i	−0.511084	−	0.859531i	$-0.670756\pi$
$41$	−37.4041	+	21.5953i	−0.0222511	+	0.0128467i	0.488833	+	0.872377i	$0.337423\pi$
$42$		0			0
$43$	−21.6099	+	37.4295i	−0.0116874	+	0.0202431i	−0.871810	−	0.489844i	$-0.837054\pi$
$43$	−21.6099	+	37.4295i	−0.0116874	+	0.0202431i	0.860123	+	0.510087i	$0.170387\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2431.98	+	1404.11i	1.10094	+	0.635630i	0.936468	−	0.350752i	$-0.114074\pi$
$47$	2431.98	+	1404.11i	1.10094	+	0.635630i	0.164474	+	0.986381i	$0.447407\pi$
$48$		0			0
$49$	−202.957	−	351.532i	−0.0845302	−	0.146411i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	4128.48i		−	1.46973i	−0.678212	−	0.734866i	$-0.737245\pi$
$53$		−	4128.48i		−	1.46973i	0.678212	−	0.734866i	$-0.262755\pi$
$54$		0			0
$55$	−168.786			−0.0557971
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5504.74	+	3178.16i	−1.58137	+	0.913003i	−0.586707	+	0.809799i	$0.699576\pi$
$59$	−5504.74	+	3178.16i	−1.58137	+	0.913003i	−0.994660	+	0.103203i	$0.967091\pi$
$60$		0			0
$61$	−848.536	+	1469.71i	−0.228040	+	0.394976i	−0.957227	−	0.289338i	$-0.906565\pi$
$61$	−848.536	+	1469.71i	−0.228040	+	0.394976i	0.729187	+	0.684314i	$0.239898\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1705.48	+	984.659i	0.403664	+	0.233055i
$66$		0			0
$67$	1269.40	+	2198.67i	0.282781	+	0.489791i	0.972069	−	0.234697i	$-0.0754097\pi$
$67$	1269.40	+	2198.67i	0.282781	+	0.489791i	−0.689288	+	0.724488i	$0.742076\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1550.91i		−	0.307659i	−0.988097	−	0.153829i	$-0.950839\pi$
$71$		−	1550.91i		−	0.307659i	0.988097	−	0.153829i	$-0.0491606\pi$
$72$		0			0
$73$	−4521.97			−0.848558			−0.424279	−	0.905532i	$-0.639472\pi$
$73$	−4521.97			−0.848558			−0.424279	+	0.905532i	$0.639472\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1092.50	−	630.752i	0.184263	−	0.106384i
$78$		0			0
$79$	−3500.87	+	6063.68i	−0.560947	+	0.971588i	0.436467	+	0.899720i	$0.356229\pi$
$79$	−3500.87	+	6063.68i	−0.560947	+	0.971588i	−0.997414	+	0.0718683i	$0.977104\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4962.11	−	2864.87i	−0.720294	−	0.415862i	0.0945669	−	0.995519i	$-0.469853\pi$
$83$	−4962.11	−	2864.87i	−0.720294	−	0.415862i	−0.814861	+	0.579657i	$0.803187\pi$
$84$		0			0
$85$	−324.315	−	561.730i	−0.0448879	−	0.0777481i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			6177.83i			0.779930i	0.920830	+	0.389965i	$0.127513\pi$
$89$			6177.83i			0.779930i	−0.920830	+	0.389965i	$0.872487\pi$
$90$		0			0
$91$	−14718.6			−1.77740
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2038.41	−	1176.87i	0.225862	−	0.130402i
$96$		0			0
$97$	−5363.80	+	9290.37i	−0.570071	+	0.987392i	0.426487	+	0.904494i	$0.359751\pi$
$97$	−5363.80	+	9290.37i	−0.570071	+	0.987392i	−0.996558	+	0.0828985i	$0.973582\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	10729.1	+	6194.44i	1.05177	+	0.607238i	0.923144	−	0.384455i	$-0.125611\pi$
$101$	10729.1	+	6194.44i	1.05177	+	0.607238i	0.128624	+	0.991693i	$0.458944\pi$
$102$		0			0
$103$	−3461.49	−	5995.48i	−0.326279	−	0.565132i	0.655491	−	0.755203i	$-0.272462\pi$
$103$	−3461.49	−	5995.48i	−0.326279	−	0.565132i	−0.981770	+	0.190071i	$0.939128\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	18280.7i		−	1.59671i	−0.602190	−	0.798353i	$-0.705705\pi$
$107$		−	18280.7i		−	1.59671i	0.602190	−	0.798353i	$-0.294295\pi$
$108$		0			0
$109$	−16716.1			−1.40696			−0.703479	−	0.710716i	$-0.748371\pi$
$109$	−16716.1			−1.40696			−0.703479	+	0.710716i	$0.748371\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14026.0	+	8097.93i	−1.09844	+	0.634187i	−0.935812	−	0.352500i	$-0.885332\pi$
$113$	−14026.0	+	8097.93i	−1.09844	+	0.634187i	−0.162632	+	0.986687i	$0.551998\pi$
$114$		0			0
$115$	−1423.62	+	2465.78i	−0.107646	+	0.186448i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4198.36	+	2423.92i	0.296473	+	0.171169i
$120$		0			0
$121$	−7037.02	−	12188.5i	−0.480638	−	0.832490i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	8504.60i		−	0.544295i
$126$		0			0
$127$	−14891.2			−0.923254			−0.461627	−	0.887074i	$-0.652734\pi$
$127$	−14891.2			−0.923254			−0.461627	+	0.887074i	$0.652734\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−16833.2	+	9718.62i	−0.980896	+	0.566320i	−0.902540	−	0.430605i	$-0.858300\pi$
$131$	−16833.2	+	9718.62i	−0.980896	+	0.566320i	−0.0783553	+	0.996926i	$0.524967\pi$
$132$		0			0
$133$	−8795.93	+	15235.0i	−0.497254	+	0.861270i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	23083.4	+	13327.2i	1.22987	+	0.710065i	0.967002	−	0.254767i	$-0.0819989\pi$
$137$	23083.4	+	13327.2i	1.22987	+	0.710065i	0.262866	+	0.964832i	$0.415332\pi$
$138$		0			0
$139$	1934.20	+	3350.14i	0.100109	+	0.173394i	0.911729	−	0.410791i	$-0.134748\pi$
$139$	1934.20	+	3350.14i	0.100109	+	0.173394i	−0.811620	+	0.584185i	$0.801414\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	6614.96i		−	0.323486i
$144$		0			0
$145$	2589.05			0.123142
$146$		0			0
$147$		0			0
$148$		0			0
$149$	19682.8	−	11363.9i	0.886571	−	0.511862i	0.0137518	−	0.999905i	$-0.495623\pi$
$149$	19682.8	−	11363.9i	0.886571	−	0.511862i	0.872819	+	0.488043i	$0.162289\pi$
$150$		0			0
$151$	−13242.7	+	22937.1i	−0.580797	+	1.00597i	0.414588	+	0.910009i	$0.363926\pi$
$151$	−13242.7	+	22937.1i	−0.580797	+	1.00597i	−0.995385	+	0.0959606i	$0.969408\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2155.17	−	1244.29i	−0.0897051	−	0.0517913i
$156$		0			0
$157$	12596.7	+	21818.1i	0.511042	+	0.885151i	0.999918	+	0.0127975i	$0.00407369\pi$
$157$	12596.7	+	21818.1i	0.511042	+	0.885151i	−0.488876	+	0.872353i	$0.662593\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	21280.2i		−	0.820962i
$162$		0			0
$163$	−20189.7			−0.759897			−0.379948	−	0.925008i	$-0.624058\pi$
$163$	−20189.7			−0.759897			−0.379948	+	0.925008i	$0.624058\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−43387.6	+	25049.9i	−1.55573	+	0.898198i	−0.558067	+	0.829796i	$0.688457\pi$
$167$	−43387.6	+	25049.9i	−1.55573	+	0.898198i	−0.997658	+	0.0684028i	$0.978210\pi$
$168$		0			0
$169$	−24309.6	+	42105.5i	−0.851147	+	1.47423i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	37922.3	+	21894.4i	1.26707	+	0.731546i	0.974433	−	0.224677i	$-0.0721326\pi$
$173$	37922.3	+	21894.4i	1.26707	+	0.731546i	0.292641	+	0.956222i	$0.405466\pi$
$174$		0			0
$175$	15225.3	+	26370.9i	0.497151	+	0.861091i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			9037.18i			0.282050i	0.990006	+	0.141025i	$0.0450399\pi$
$179$			9037.18i			0.282050i	−0.990006	+	0.141025i	$0.954960\pi$
$180$		0			0
$181$	−13305.4			−0.406136			−0.203068	−	0.979165i	$-0.565091\pi$
$181$	−13305.4			−0.406136			−0.203068	+	0.979165i	$0.565091\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	261.906	−	151.211i	0.00765246	−	0.00441815i
$186$		0			0
$187$	−1089.38	+	1886.86i	−0.0311527	+	0.0539580i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−18267.5	−	10546.8i	−0.500741	−	0.289103i	0.228279	−	0.973596i	$-0.426690\pi$
$191$	−18267.5	−	10546.8i	−0.500741	−	0.289103i	−0.729019	+	0.684493i	$0.760024\pi$
$192$		0			0
$193$	−21813.9	−	37782.8i	−0.585624	−	1.01433i	−0.994797	−	0.101874i	$-0.967516\pi$
$193$	−21813.9	−	37782.8i	−0.585624	−	1.01433i	0.409173	−	0.912457i	$-0.365817\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	51851.1i		−	1.33606i	−0.744135	−	0.668029i	$-0.767138\pi$
$197$		−	51851.1i		−	1.33606i	0.744135	−	0.668029i	$-0.232862\pi$
$198$		0			0
$199$	−77783.6			−1.96418			−0.982092	−	0.188404i	$-0.939668\pi$
$199$	−77783.6			−1.96418			−0.982092	+	0.188404i	$0.939668\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−16758.0	+	9675.26i	−0.406660	+	0.234785i
$204$		0			0
$205$	153.081	−	265.144i	0.00364262	−	0.00630920i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6847.03	−	3953.13i	−0.156751	−	0.0905001i
$210$		0			0
$211$	−14540.7	−	25185.3i	−0.326604	−	0.565694i	0.655232	−	0.755428i	$-0.272571\pi$
$211$	−14540.7	−	25185.3i	−0.326604	−	0.565694i	−0.981836	+	0.189734i	$0.939238\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	306.370i		−	0.00662780i
$216$		0			0
$217$	18599.5			0.394986
$218$		0			0
$219$		0			0
$220$		0			0
$221$	22014.9	−	12710.3i	0.450747	−	0.260239i
$222$		0			0
$223$	33819.5	−	58577.1i	0.680076	−	1.17793i	−0.294881	−	0.955534i	$-0.595280\pi$
$223$	33819.5	−	58577.1i	0.680076	−	1.17793i	0.974957	−	0.222392i	$-0.0713865\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16720.9	+	9653.81i	0.324495	+	0.187347i	0.653394	−	0.757018i	$-0.273344\pi$
$227$	16720.9	+	9653.81i	0.324495	+	0.187347i	−0.328900	+	0.944365i	$0.606678\pi$
$228$		0			0
$229$	−40086.9	−	69432.5i	−0.764418	−	1.32401i	−0.940554	−	0.339645i	$-0.889693\pi$
$229$	−40086.9	−	69432.5i	−0.764418	−	1.32401i	0.176135	−	0.984366i	$-0.443640\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			64412.4i			1.18647i	0.805029	+	0.593236i	$0.202150\pi$
$233$			64412.4i			1.18647i	−0.805029	+	0.593236i	$0.797850\pi$
$234$		0			0
$235$	−19906.4			−0.360460
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−21955.5	+	12676.0i	−0.384368	+	0.221915i	−0.679717	−	0.733474i	$-0.737897\pi$
$239$	−21955.5	+	12676.0i	−0.384368	+	0.221915i	0.295349	+	0.955389i	$0.404564\pi$
$240$		0			0
$241$	−27366.4	+	47399.9i	−0.471176	+	0.816100i	−0.999456	−	0.0329695i	$-0.989504\pi$
$241$	−27366.4	+	47399.9i	−0.471176	+	0.816100i	0.528281	+	0.849070i	$0.322837\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2491.88	+	1438.69i	0.0415141	+	0.0239682i
$246$		0			0
$247$	46123.3	+	79887.9i	0.756008	+	1.30944i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	35904.7i		−	0.569908i	−0.958541	−	0.284954i	$-0.908022\pi$
$251$		−	35904.7i		−	0.569908i	0.958541	−	0.284954i	$-0.0919782\pi$
$252$		0			0
$253$	9563.89			0.149415
$254$		0			0
$255$		0			0
$256$		0			0
$257$	22706.3	−	13109.5i	0.343779	−	0.198481i	−0.318163	−	0.948036i	$-0.603066\pi$
$257$	22706.3	−	13109.5i	0.343779	−	0.198481i	0.661942	+	0.749555i	$0.269733\pi$
$258$		0			0
$259$	−1130.15	+	1957.48i	−0.0168475	+	0.0291808i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−112802.	−	65126.4i	−1.63082	−	0.941555i	−0.983840	−	0.179048i	$-0.942698\pi$
$263$	−112802.	−	65126.4i	−1.63082	−	0.941555i	−0.646980	−	0.762507i	$-0.723968\pi$
$264$		0			0
$265$	14632.6	+	25344.5i	0.208368	+	0.360904i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			56557.5i			0.781602i	0.920475	+	0.390801i	$0.127802\pi$
$269$			56557.5i			0.781602i	−0.920475	+	0.390801i	$0.872198\pi$
$270$		0			0
$271$	−8171.92			−0.111272			−0.0556359	−	0.998451i	$-0.517719\pi$
$271$	−8171.92			−0.111272			−0.0556359	+	0.998451i	$0.517719\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−11851.8	+	6842.65i	−0.156718	+	0.0904813i
$276$		0			0
$277$	−24220.2	+	41950.7i	−0.315660	+	0.546739i	−0.979577	−	0.201067i	$-0.935559\pi$
$277$	−24220.2	+	41950.7i	−0.315660	+	0.546739i	0.663918	+	0.747806i	$0.268892\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	118207.	+	68246.7i	1.49703	+	0.864309i	0.999994	−	0.00342359i	$-0.00108977\pi$
$281$	118207.	+	68246.7i	1.49703	+	0.864309i	0.497032	+	0.867732i	$0.334423\pi$
$282$		0			0
$283$	16828.8	+	29148.3i	0.210126	+	0.363949i	0.951754	−	0.306863i	$-0.0992792\pi$
$283$	16828.8	+	29148.3i	0.210126	+	0.363949i	−0.741628	+	0.670812i	$0.765946\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			2288.25i			0.0277805i
$288$		0			0
$289$	75148.3			0.899753
$290$		0			0
$291$		0			0
$292$		0			0
$293$	98630.0	−	56944.0i	1.14888	−	0.663305i	0.200264	−	0.979742i	$-0.435820\pi$
$293$	98630.0	−	56944.0i	1.14888	−	0.663305i	0.948613	+	0.316437i	$0.102487\pi$
$294$		0			0
$295$	22528.9	−	39021.1i	0.258878	−	0.448390i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−96637.1	−	55793.5i	−1.08094	−	0.624081i
$300$		0			0
$301$	1144.90	+	1983.03i	0.0126367	+	0.0218875i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	12029.9i		−	0.129319i
$306$		0			0
$307$	144481.			1.53297			0.766487	−	0.642260i	$-0.222003\pi$
$307$	144481.			1.53297			0.766487	+	0.642260i	$0.222003\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−88652.8	+	51183.7i	−0.916582	+	0.529189i	−0.882543	−	0.470231i	$-0.844170\pi$
$311$	−88652.8	+	51183.7i	−0.916582	+	0.529189i	−0.0340391	+	0.999421i	$0.510837\pi$
$312$		0			0
$313$	71202.8	−	123327.i	0.726789	−	1.25884i	−0.231444	−	0.972848i	$-0.574345\pi$
$313$	71202.8	−	123327.i	0.726789	−	1.25884i	0.958233	−	0.285987i	$-0.0923215\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	84337.2	+	48692.1i	0.839268	+	0.484552i	0.857015	−	0.515291i	$-0.172316\pi$
$317$	84337.2	+	48692.1i	0.839268	+	0.484552i	−0.0177473	+	0.999843i	$0.505649\pi$
$318$		0			0
$319$	−4348.33	−	7531.53i	−0.0427308	−	0.0740119i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	30383.0i		−	0.291223i
$324$		0			0
$325$	159673.			1.51170
$326$		0			0
$327$		0			0
$328$		0			0
$329$	128847.	−	74390.0i	1.19037	−	0.687262i
$330$		0			0
$331$	64347.8	−	111454.i	0.587324	−	1.01728i	−0.407257	−	0.913314i	$-0.633515\pi$
$331$	64347.8	−	111454.i	0.587324	−	1.01728i	0.994581	−	0.103962i	$-0.0331519\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−15585.6	−	8998.35i	−0.138878	−	0.0801813i
$336$		0			0
$337$	−7822.63	−	13549.2i	−0.0688799	−	0.119304i	0.829529	−	0.558464i	$-0.188609\pi$
$337$	−7822.63	−	13549.2i	−0.0688799	−	0.119304i	−0.898409	+	0.439161i	$0.855276\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			8359.14i			0.0718874i
$342$		0			0
$343$	105700.			0.898438
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−42986.0	+	24818.0i	−0.357000	+	0.206114i	−0.667764	−	0.744373i	$-0.732748\pi$
$347$	−42986.0	+	24818.0i	−0.357000	+	0.206114i	0.310764	+	0.950487i	$0.399415\pi$
$348$		0			0
$349$	42646.3	−	73865.6i	0.350131	−	0.606445i	−0.636141	−	0.771573i	$-0.719470\pi$
$349$	42646.3	−	73865.6i	0.350131	−	0.606445i	0.986272	+	0.165128i	$0.0528037\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−15023.7	−	8673.93i	−0.120567	−	0.0696091i	0.438504	−	0.898729i	$-0.355509\pi$
$353$	−15023.7	−	8673.93i	−0.120567	−	0.0696091i	−0.559070	+	0.829120i	$0.688842\pi$
$354$		0			0
$355$	5496.92	+	9520.94i	0.0436177	+	0.0755480i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			184846.i			1.43424i	0.696952	+	0.717118i	$0.254539\pi$
$359$			184846.i			1.43424i	−0.696952	+	0.717118i	$0.745461\pi$
$360$		0			0
$361$	−20067.0			−0.153981
$362$		0			0
$363$		0			0
$364$		0			0
$365$	27760.1	−	16027.3i	0.208370	−	0.120302i
$366$		0			0
$367$	25887.9	−	44839.1i	0.192205	−	0.332908i	−0.753776	−	0.657132i	$-0.771770\pi$
$367$	25887.9	−	44839.1i	0.192205	−	0.332908i	0.945981	+	0.324223i	$0.105103\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−189424.	−	109364.i	−1.37622	−	0.794560i
$372$		0			0
$373$	−27523.6	−	47672.2i	−0.197828	−	0.342647i	0.749996	−	0.661442i	$-0.230055\pi$
$373$	−27523.6	−	47672.2i	−0.197828	−	0.342647i	−0.947824	+	0.318795i	$0.896722\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			101468.i			0.713918i
$378$		0			0
$379$	−119817.			−0.834143			−0.417072	−	0.908874i	$-0.636944\pi$
$379$	−119817.			−0.834143			−0.417072	+	0.908874i	$0.636944\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	134871.	−	77868.0i	0.919437	−	0.530837i	0.0359819	−	0.999352i	$-0.488544\pi$
$383$	134871.	−	77868.0i	0.919437	−	0.530837i	0.883456	+	0.468515i	$0.155211\pi$
$384$		0			0
$385$	−4471.18	+	7744.31i	−0.0301648	+	0.0522470i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−134103.	−	77424.5i	−0.886217	−	0.511658i	−0.0135137	−	0.999909i	$-0.504302\pi$
$389$	−134103.	−	77424.5i	−0.886217	−	0.511658i	−0.872703	+	0.488251i	$0.837635\pi$
$390$		0			0
$391$	18376.6	+	31829.1i	0.120202	+	0.208196i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	49632.8i		−	0.318108i
$396$		0			0
$397$	148940.			0.944997			0.472498	−	0.881332i	$-0.343352\pi$
$397$	148940.			0.944997			0.472498	+	0.881332i	$0.343352\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	184816.	−	106704.i	1.14935	−	0.663576i	0.200619	−	0.979669i	$-0.435705\pi$
$401$	184816.	−	106704.i	1.14935	−	0.663576i	0.948728	+	0.316094i	$0.102371\pi$
$402$		0			0
$403$	48765.2	−	84463.8i	0.300262	−	0.520069i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−879.744	−	507.920i	−0.00531089	−	0.00306624i
$408$		0			0
$409$	−24821.4	−	42991.9i	−0.148381	−	0.257004i	0.782248	−	0.622967i	$-0.214073\pi$
$409$	−24821.4	−	42991.9i	−0.148381	−	0.257004i	−0.930629	+	0.365963i	$0.880740\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			336760.i			1.97433i
$414$		0			0
$415$	40616.1			0.235832
$416$		0			0
$417$		0			0
$418$		0			0
$419$	19474.6	−	11243.7i	0.110928	−	0.0640444i	−0.443509	−	0.896270i	$-0.646267\pi$
$419$	19474.6	−	11243.7i	0.110928	−	0.0640444i	0.554438	+	0.832225i	$0.312933\pi$
$420$		0			0
$421$	−51716.3	+	89575.3i	−0.291785	+	0.505387i	−0.974232	−	0.225549i	$-0.927583\pi$
$421$	−51716.3	+	89575.3i	−0.291785	+	0.505387i	0.682447	+	0.730935i	$0.260916\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−45545.4	−	26295.6i	−0.252154	−	0.145581i
$426$		0			0
$427$	44955.7	+	77865.6i	0.246564	+	0.427061i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			116100.i			0.624996i	0.949918	+	0.312498i	$0.101166\pi$
$431$			116100.i			0.624996i	−0.949918	+	0.312498i	$0.898834\pi$
$432$		0			0
$433$	296628.			1.58211			0.791054	−	0.611747i	$-0.209533\pi$
$433$	296628.			1.58211			0.791054	+	0.611747i	$0.209533\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−115502.	+	66684.9i	−0.604819	+	0.349192i
$438$		0			0
$439$	162410.	−	281303.i	0.842722	−	1.45964i	−0.0448636	−	0.998993i	$-0.514285\pi$
$439$	162410.	−	281303.i	0.842722	−	1.45964i	0.887585	−	0.460644i	$-0.152381\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−268280.	−	154892.i	−1.36704	−	0.789261i	−0.376491	−	0.926420i	$-0.622869\pi$
$443$	−268280.	−	154892.i	−1.36704	−	0.789261i	−0.990549	+	0.137160i	$0.956203\pi$
$444$		0			0
$445$	−21896.2	−	37925.3i	−0.110573	−	0.191518i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	367550.i		−	1.82316i	−0.411127	−	0.911578i	$-0.634865\pi$
$449$		−	367550.i		−	1.82316i	0.411127	−	0.911578i	$-0.365135\pi$
$450$		0			0
$451$	−1028.40			−0.00505603
$452$		0			0
$453$		0			0
$454$		0			0
$455$	90356.9	−	52167.6i	0.436454	−	0.251987i
$456$		0			0
$457$	87806.8	−	152086.i	0.420432	−	0.728209i	−0.575550	−	0.817767i	$-0.695212\pi$
$457$	87806.8	−	152086.i	0.420432	−	0.728209i	0.995982	+	0.0895575i	$0.0285453\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	90099.2	+	52018.8i	0.423955	+	0.244770i	0.696768	−	0.717297i	$-0.254621\pi$
$461$	90099.2	+	52018.8i	0.423955	+	0.244770i	−0.272813	+	0.962067i	$0.587954\pi$
$462$		0			0
$463$	22011.7	+	38125.3i	0.102681	+	0.177849i	0.912788	−	0.408433i	$-0.133925\pi$
$463$	22011.7	+	38125.3i	0.102681	+	0.177849i	−0.810107	+	0.586282i	$0.800591\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			85970.6i			0.394199i	0.980383	+	0.197100i	$0.0631523\pi$
$467$			85970.6i			0.394199i	−0.980383	+	0.197100i	$0.936848\pi$
$468$		0			0
$469$	134507.			0.611503
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−891.227	+	514.550i	−0.00398351	+	0.00229988i
$474$		0			0
$475$	95421.7	−	165275.i	0.422922	−	0.732522i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	329529.	+	190253.i	1.43622	+	0.829204i	0.997585	−	0.0694595i	$-0.0221274\pi$
$479$	329529.	+	190253.i	1.43622	+	0.829204i	0.438639	+	0.898663i	$0.355461\pi$
$480$		0			0
$481$	5926.17	+	10264.4i	0.0256144	+	0.0443654i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	76044.1i		−	0.323282i
$486$		0			0
$487$	−103685.			−0.437179			−0.218590	−	0.975817i	$-0.570146\pi$
$487$	−103685.			−0.437179			−0.218590	+	0.975817i	$0.570146\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−230938.	+	133332.i	−0.957927	+	0.553060i	−0.895534	−	0.444992i	$-0.853206\pi$
$491$	−230938.	+	133332.i	−0.957927	+	0.553060i	−0.0623928	+	0.998052i	$0.519873\pi$
$492$		0			0
$493$	16710.2	−	28942.9i	0.0687524	−	0.119083i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−71159.3	−	41083.8i	−0.288084	−	0.166325i
$498$		0			0
$499$	−9937.74	−	17212.7i	−0.0399104	−	0.0691269i	0.845380	−	0.534165i	$-0.179374\pi$
$499$	−9937.74	−	17212.7i	−0.0399104	−	0.0691269i	−0.885291	+	0.465038i	$0.846041\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	12736.5i		−	0.0503402i	−0.999683	−	0.0251701i	$-0.991987\pi$
$503$		−	12736.5i		−	0.0503402i	0.999683	−	0.0251701i	$-0.00801273\pi$
$504$		0			0
$505$	−87820.3			−0.344360
$506$		0			0
$507$		0			0
$508$		0			0
$509$	28315.3	−	16347.9i	0.109291	−	0.0630995i	−0.444358	−	0.895849i	$-0.646568\pi$
$509$	28315.3	−	16347.9i	0.109291	−	0.0630995i	0.553649	+	0.832750i	$0.313235\pi$
$510$		0			0
$511$	−119788.	+	207478.i	−0.458744	+	0.794567i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	42499.8	+	24537.3i	0.160241	+	0.0925149i
$516$		0			0
$517$	33432.9	+	57907.5i	0.125081	+	0.216648i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	1464.29i		−	0.00539451i	−0.999996	−	0.00269725i	$-0.999141\pi$
$521$		−	1464.29i		−	0.00539451i	0.999996	−	0.00269725i	$-0.000858564\pi$
$522$		0			0
$523$	443548.			1.62158			0.810788	−	0.585339i	$-0.199039\pi$
$523$	443548.			1.62158			0.810788	+	0.585339i	$0.199039\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−27819.7	+	16061.7i	−0.100168	+	0.0578322i
$528$		0			0
$529$	−59254.5	+	102632.i	−0.211743	+	0.366750i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10391.4	+	5999.45i	0.0365778	+	0.0211182i
$534$		0			0
$535$	64792.6	+	112224.i	0.226369	+	0.392083i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	9665.15i		−	0.0332683i
$540$		0			0
$541$	−74322.5			−0.253937			−0.126968	−	0.991907i	$-0.540525\pi$
$541$	−74322.5			−0.253937			−0.126968	+	0.991907i	$0.540525\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	102619.	−	59247.1i	0.345489	−	0.199468i
$546$		0			0
$547$	−296248.	+	513116.i	−0.990103	+	1.71491i	−0.373509	+	0.927627i	$0.621845\pi$
$547$	−296248.	+	513116.i	−0.990103	+	1.71491i	−0.616594	+	0.787282i	$0.711488\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	105028.	+	60638.0i	0.345941	+	0.199729i
$552$		0			0
$553$	185477.	+	321256.i	0.606513	+	1.05051i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	101727.i		−	0.327888i	−0.986470	−	0.163944i	$-0.947578\pi$
$557$		−	101727.i		−	0.327888i	0.986470	−	0.163944i	$-0.0524216\pi$
$558$		0			0
$559$	12007.1			0.0384249
$560$		0			0
$561$		0			0
$562$		0			0
$563$	375274.	−	216665.i	1.18395	−	0.683552i	0.227022	−	0.973890i	$-0.427101\pi$
$563$	375274.	−	216665.i	1.18395	−	0.683552i	0.956924	+	0.290338i	$0.0937676\pi$
$564$		0			0
$565$	57403.3	−	99425.5i	0.179821	−	0.311459i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−66241.5	−	38244.6i	−0.204600	−	0.118126i	0.394199	−	0.919025i	$-0.371022\pi$
$569$	−66241.5	−	38244.6i	−0.204600	−	0.118126i	−0.598799	+	0.800899i	$0.704355\pi$
$570$		0			0
$571$	−152369.	−	263911.i	−0.467330	−	0.809440i	0.531973	−	0.846761i	$-0.321451\pi$
$571$	−152369.	−	263911.i	−0.467330	−	0.809440i	−0.999303	+	0.0373215i	$0.988117\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			230856.i			0.698240i
$576$		0			0
$577$	−583342.			−1.75215			−0.876075	−	0.482174i	$-0.839847\pi$
$577$	−583342.			−1.75215			−0.876075	+	0.482174i	$0.839847\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−262894.	+	151782.i	−0.778804	+	0.449643i
$582$		0			0
$583$	49151.2	−	85132.4i	0.144610	−	0.250471i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−223895.	−	129266.i	−0.649782	−	0.375152i	0.138591	−	0.990350i	$-0.455743\pi$
$587$	−223895.	−	129266.i	−0.649782	−	0.375152i	−0.788373	+	0.615198i	$0.789076\pi$
$588$		0			0
$589$	−58284.7	−	100952.i	−0.168006	−	0.290994i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			348330.i			0.990563i	0.868733	+	0.495281i	$0.164935\pi$
$593$			348330.i			0.990563i	−0.868733	+	0.495281i	$0.835065\pi$
$594$		0			0
$595$	−34364.6			−0.0970683
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−193573.	+	111760.i	−0.539500	+	0.311481i	−0.744876	−	0.667202i	$-0.767492\pi$
$599$	−193573.	+	111760.i	−0.539500	+	0.311481i	0.205376	+	0.978683i	$0.434158\pi$
$600$		0			0
$601$	−17502.7	+	30315.6i	−0.0484570	+	0.0839299i	−0.889237	−	0.457448i	$-0.848764\pi$
$601$	−17502.7	+	30315.6i	−0.0484570	+	0.0839299i	0.840780	+	0.541378i	$0.182097\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	86399.8	+	49882.9i	0.236049	+	0.136283i
$606$		0			0
$607$	−206380.	−	357461.i	−0.560132	−	0.970177i	−0.997484	−	0.0708871i	$-0.977417\pi$
$607$	−206380.	−	357461.i	−0.560132	−	0.970177i	0.437352	−	0.899290i	$-0.355916\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	780158.i		−	2.08978i
$612$		0			0
$613$	−431167.			−1.14743			−0.573713	−	0.819056i	$-0.694497\pi$
$613$	−431167.			−1.14743			−0.573713	+	0.819056i	$0.694497\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−132463.	+	76477.6i	−0.347956	+	0.200893i	−0.663785	−	0.747924i	$-0.731051\pi$
$617$	−132463.	+	76477.6i	−0.347956	+	0.200893i	0.315828	+	0.948816i	$0.397718\pi$
$618$		0			0
$619$	−198033.	+	343003.i	−0.516840	+	0.895193i	0.482969	+	0.875638i	$0.339558\pi$
$619$	−198033.	+	343003.i	−0.516840	+	0.895193i	−0.999809	+	0.0195555i	$0.993775\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	283453.	+	163652.i	0.730306	+	0.421642i
$624$		0			0
$625$	−149467.	−	258884.i	−0.382635	−	0.662743i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	3903.78i		−	0.00986697i
$630$		0			0
$631$	−699103.			−1.75583			−0.877915	−	0.478817i	$-0.841066\pi$
$631$	−699103.			−1.75583			−0.877915	+	0.478817i	$0.841066\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	91416.0	−	52779.0i	0.226712	−	0.130892i
$636$		0			0
$637$	−56384.2	+	97660.2i	−0.138956	+	0.240679i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	91235.2	+	52674.7i	0.222048	+	0.128199i	0.606898	−	0.794780i	$-0.292414\pi$
$641$	91235.2	+	52674.7i	0.222048	+	0.128199i	−0.384850	+	0.922979i	$0.625747\pi$
$642$		0			0
$643$	−180074.	−	311898.i	−0.435542	−	0.754381i	0.561798	−	0.827275i	$-0.310110\pi$
$643$	−180074.	−	311898.i	−0.435542	−	0.754381i	−0.997340	+	0.0728939i	$0.976777\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	109548.i		−	0.261694i	−0.991403	−	0.130847i	$-0.958230\pi$
$647$		−	109548.i		−	0.261694i	0.991403	−	0.130847i	$-0.0417697\pi$
$648$		0			0
$649$	−151349.			−0.359328
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−262354.	+	151470.i	−0.615265	+	0.355223i	−0.775023	−	0.631933i	$-0.782262\pi$
$653$	−262354.	+	151470.i	−0.615265	+	0.355223i	0.159758	+	0.987156i	$0.448928\pi$
$654$		0			0
$655$	68891.8	−	119324.i	0.160578	−	0.278129i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	747985.	+	431850.i	1.72235	+	0.994401i	0.914012	+	0.405687i	$0.132968\pi$
$659$	747985.	+	431850.i	1.72235	+	0.994401i	0.808341	+	0.588715i	$0.200366\pi$
$660$		0			0
$661$	55209.0	+	95624.8i	0.126359	+	0.218861i	0.922263	−	0.386562i	$-0.126337\pi$
$661$	55209.0	+	95624.8i	0.126359	+	0.218861i	−0.795904	+	0.605423i	$0.793004\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	124702.i		−	0.281989i
$666$		0			0
$667$	−146703.			−0.329751
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−34994.9	+	20204.3i	−0.0777249	+	0.0448745i
$672$		0			0
$673$	−28177.8	+	48805.4i	−0.0622124	+	0.107755i	−0.895454	−	0.445154i	$-0.853149\pi$
$673$	−28177.8	+	48805.4i	−0.0622124	+	0.107755i	0.833242	+	0.552909i	$0.186482\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−300049.	−	173233.i	−0.654658	−	0.377967i	0.135580	−	0.990766i	$-0.456710\pi$
$677$	−300049.	−	173233.i	−0.654658	−	0.377967i	−0.790239	+	0.612799i	$0.790043\pi$
$678$		0			0
$679$	284176.	+	492207.i	0.616379	+	1.06760i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	421086.i		−	0.902670i	−0.892355	−	0.451335i	$-0.850948\pi$
$683$		−	421086.i		−	0.902670i	0.892355	−	0.451335i	$-0.149052\pi$
$684$		0			0
$685$	−188944.			−0.402672
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−993284.	+	573473.i	−2.09235	+	1.20802i
$690$		0			0
$691$	−389372.	+	674412.i	−0.815471	+	1.41244i	0.0935174	+	0.995618i	$0.470189\pi$
$691$	−389372.	+	674412.i	−0.815471	+	1.41244i	−0.908989	+	0.416820i	$0.863144\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−23747.9	−	13710.9i	−0.0491650	−	0.0283854i
$696$		0			0
$697$	−1976.03	−	3422.58i	−0.00406749	−	0.00704511i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	388023.i		−	0.789626i	−0.918761	−	0.394813i	$-0.870809\pi$
$701$		−	388023.i		−	0.789626i	0.918761	−	0.394813i	$-0.129191\pi$
$702$		0			0
$703$	14166.0			0.0286641
$704$		0			0
$705$		0			0
$706$		0			0
$707$	568430.	−	328183.i	1.13720	−	0.656565i
$708$		0			0
$709$	−162314.	+	281136.i	−0.322896	+	0.559273i	−0.981084	−	0.193581i	$-0.937990\pi$
$709$	−162314.	+	281136.i	−0.322896	+	0.559273i	0.658188	+	0.752854i	$0.271323\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	122118.	+	70504.6i	0.240214	+	0.138688i
$714$		0			0
$715$	23445.5	+	40608.9i	0.0458615	+	0.0794344i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			284337.i			0.550017i	0.961442	+	0.275008i	$0.0886807\pi$
$719$			284337.i			0.550017i	−0.961442	+	0.275008i	$0.911319\pi$
$720$		0			0
$721$	−366782.			−0.705566
$722$		0			0
$723$		0			0
$724$		0			0
$725$	181798.	−	104961.i	0.345870	−	0.199688i
$726$		0			0
$727$	−74675.8	+	129342.i	−0.141290	+	0.244721i	−0.927983	−	0.372624i	$-0.878458\pi$
$727$	−74675.8	+	129342.i	−0.141290	+	0.244721i	0.786693	+	0.617345i	$0.211792\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3424.90	−	1977.37i	−0.00640934	−	0.00370043i
$732$		0			0
$733$	289067.	+	500678.i	0.538009	+	0.931860i	0.999011	+	0.0444604i	$0.0141568\pi$
$733$	289067.	+	500678.i	0.538009	+	0.931860i	−0.461002	+	0.887399i	$0.652510\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			60451.1i			0.111293i
$738$		0			0
$739$	499172.			0.914031			0.457016	−	0.889459i	$-0.348918\pi$
$739$	499172.			0.914031			0.457016	+	0.889459i	$0.348918\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−470213.	+	271477.i	−0.851759	+	0.491763i	−0.861244	−	0.508192i	$-0.830314\pi$
$743$	−470213.	+	271477.i	−0.851759	+	0.491763i	0.00948514	+	0.999955i	$0.496981\pi$
$744$		0			0
$745$	−80554.3	+	139524.i	−0.145136	+	0.251383i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−838760.	−	484258.i	−1.49511	−	0.863204i
$750$		0			0
$751$	140872.	+	243998.i	0.249773	+	0.432620i	0.963463	−	0.267842i	$-0.0863106\pi$
$751$	140872.	+	243998.i	0.249773	+	0.432620i	−0.713690	+	0.700462i	$0.752977\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	187746.i		−	0.329365i
$756$		0			0
$757$	−1.00873e6			−1.76029			−0.880143	−	0.474708i	$-0.842554\pi$
$757$	−1.00873e6			−1.76029			−0.880143	+	0.474708i	$0.842554\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−566414.	+	327019.i	−0.978059	+	0.564682i	−0.901683	−	0.432397i	$-0.857668\pi$
$761$	−566414.	+	327019.i	−0.978059	+	0.564682i	−0.0763751	+	0.997079i	$0.524335\pi$
$762$		0			0
$763$	−442811.	+	766972.i	−0.760623	+	1.31744i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.52929e6	+	882936.i	2.59955	+	1.50085i
$768$		0			0
$769$	255331.	+	442247.i	0.431769	+	0.747846i	0.997026	−	0.0770689i	$-0.0245561\pi$
$769$	255331.	+	442247.i	0.431769	+	0.747846i	−0.565257	+	0.824915i	$0.691223\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	795361.i		−	1.33108i	−0.746360	−	0.665542i	$-0.768200\pi$
$773$		−	795361.i		−	1.33108i	0.746360	−	0.665542i	$-0.231800\pi$
$774$		0			0
$775$	−201775.			−0.335941
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12419.9	−	7170.60i	0.0204664	−	0.0118163i
$780$		0			0
$781$	18464.2	−	31980.9i	0.0302711	−	0.0524311i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−154661.	−	89293.3i	−0.250981	−	0.144904i
$786$		0			0
$787$	−518315.	−	897747.i	−0.836843	−	1.44945i	−0.892521	−	0.451006i	$-0.851065\pi$
$787$	−518315.	−	897747.i	−0.836843	−	1.44945i	0.0556776	−	0.998449i	$-0.482268\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			858062.i			1.37140i
$792$		0			0
$793$	471469.			0.749733
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−144113.	+	83203.8i	−0.226875	+	0.130986i	−0.609130	−	0.793071i	$-0.708481\pi$
$797$	−144113.	+	83203.8i	−0.226875	+	0.130986i	0.382254	+	0.924057i	$0.375148\pi$
$798$		0			0
$799$	−128479.	+	222533.i	−0.201252	+	0.348579i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−93246.4	−	53835.9i	−0.144611	−	0.0834912i
$804$		0			0
$805$	75423.7	+	130638.i	0.116390	+	0.201594i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	751726.i		−	1.14858i	−0.818651	−	0.574292i	$-0.805278\pi$
$809$		−	751726.i		−	1.14858i	0.818651	−	0.574292i	$-0.194722\pi$
$810$		0			0
$811$	−25761.8			−0.0391682			−0.0195841	−	0.999808i	$-0.506234\pi$
$811$	−25761.8			−0.0391682			−0.0195841	+	0.999808i	$0.506234\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	123943.	−	71558.7i	0.186598	−	0.107733i
$816$		0			0
$817$	7175.48	−	12428.3i	0.0107500	−	0.0186195i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.04276e6	+	602036.i	1.54702	+	0.893174i	0.998367	+	0.0571258i	$0.0181936\pi$
$821$	1.04276e6	+	602036.i	1.54702	+	0.893174i	0.548656	+	0.836048i	$0.315140\pi$
$822$		0			0
$823$	−200321.	−	346966.i	−0.295751	−	0.512257i	0.679408	−	0.733761i	$-0.262237\pi$
$823$	−200321.	−	346966.i	−0.295751	−	0.512257i	−0.975159	+	0.221504i	$0.928903\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	326642.i		−	0.477596i	−0.971069	−	0.238798i	$-0.923247\pi$
$827$		−	326642.i		−	0.477596i	0.971069	−	0.238798i	$-0.0767534\pi$
$828$		0			0
$829$	−296231.			−0.431043			−0.215522	−	0.976499i	$-0.569145\pi$
$829$	−296231.			−0.431043			−0.215522	+	0.976499i	$0.569145\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	32166.1	−	18571.1i	0.0463563	−	0.0267638i
$834$		0			0
$835$	177569.	−	307559.i	0.254680	−	0.441119i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	578914.	+	334236.i	0.822413	+	0.474821i	0.851248	−	0.524764i	$-0.175846\pi$
$839$	578914.	+	334236.i	0.822413	+	0.474821i	−0.0288346	+	0.999584i	$0.509180\pi$
$840$		0			0
$841$	−286940.	−	496995.i	−0.405695	−	0.702685i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	344644.i		−	0.482678i
$846$		0			0
$847$	−745647.			−1.03936
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−14840.3	+	8568.04i	−0.0204919	+	0.0118310i
$852$		0			0
$853$	453408.	−	785325.i	0.623147	−	1.07932i	−0.365748	−	0.930714i	$-0.619187\pi$
$853$	453408.	−	785325.i	0.623147	−	1.07932i	0.988896	−	0.148609i	$-0.0474797\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.22175e6	−	705377.i	−1.66349	−	0.960417i	−0.971029	−	0.238962i	$-0.923193\pi$
$857$	−1.22175e6	−	705377.i	−1.66349	−	0.960417i	−0.692462	−	0.721455i	$-0.743474\pi$
$858$		0			0
$859$	−182284.	−	315725.i	−0.247037	−	0.427881i	0.715665	−	0.698444i	$-0.246124\pi$
$859$	−182284.	−	315725.i	−0.247037	−	0.427881i	−0.962702	+	0.270562i	$0.912790\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			625950.i			0.840462i	0.907417	+	0.420231i	$0.138051\pi$
$863$			625950.i			0.840462i	−0.907417	+	0.420231i	$0.861949\pi$
$864$		0			0
$865$	−310404.			−0.414853
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−144381.	+	83358.6i	−0.191193	+	0.110385i
$870$		0			0
$871$	352657.	−	610820.i	0.464854	−	0.805151i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−390211.	−	225288.i	−0.509663	−	0.294254i
$876$		0			0
$877$	−124221.	−	215157.i	−0.161509	−	0.279741i	0.773901	−	0.633306i	$-0.218303\pi$
$877$	−124221.	−	215157.i	−0.161509	−	0.279741i	−0.935410	+	0.353565i	$0.884969\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			616016.i			0.793671i	0.917890	+	0.396835i	$0.129892\pi$
$881$			616016.i			0.793671i	−0.917890	+	0.396835i	$0.870108\pi$
$882$		0			0
$883$	498838.			0.639791			0.319895	−	0.947453i	$-0.396352\pi$
$883$	498838.			0.639791			0.319895	+	0.947453i	$0.396352\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.34742e6	−	777933.i	1.71260	−	0.988769i	0.781575	−	0.623811i	$-0.214417\pi$
$887$	1.34742e6	−	777933.i	1.71260	−	0.988769i	0.931024	−	0.364958i	$-0.118917\pi$
$888$		0			0
$889$	−394469.	+	683241.i	−0.499125	+	0.864510i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−807528.	−	466227.i	−1.01264	−	0.584647i
$894$		0			0
$895$	−32030.6	−	55478.7i	−0.0399871	−	0.0692596i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	128223.i		−	0.158652i
$900$		0			0
$901$	377767.			0.465344
$902$		0			0
$903$		0			0
$904$		0			0
$905$	81681.1	−	47158.6i	0.0997297	−	0.0575790i
$906$		0			0
$907$	−364695.	+	631670.i	−0.443318	+	0.767849i	−0.997933	−	0.0642580i	$-0.979532\pi$
$907$	−364695.	+	631670.i	−0.443318	+	0.767849i	0.554616	+	0.832107i	$0.312865\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	238701.	+	137814.i	0.287619	+	0.166057i	0.636868	−	0.770973i	$-0.280230\pi$
$911$	238701.	+	137814.i	0.287619	+	0.166057i	−0.349249	+	0.937030i	$0.613563\pi$
$912$		0			0
$913$	−68215.0	−	118152.i	−0.0818348	−	0.141742i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1.02979e6i			1.22465i
$918$		0			0
$919$	146750.			0.173759			0.0868794	−	0.996219i	$-0.472311\pi$
$919$	146750.			0.173759			0.0868794	+	0.996219i	$0.472311\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−373138.	+	215431.i	−0.437992	+	0.252875i
$924$		0			0
$925$	12260.3	−	21235.5i	0.0143291	−	0.0248187i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	344982.	+	199176.i	0.399729	+	0.230783i	0.686367	−	0.727255i	$-0.259204\pi$
$929$	344982.	+	199176.i	0.399729	+	0.230783i	−0.286638	+	0.958039i	$0.592538\pi$
$930$		0			0
$931$	67390.9	+	116725.i	0.0777503	+	0.134667i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	15444.4i		−	0.0176664i
$936$		0			0
$937$	100126.			0.114043			0.0570214	−	0.998373i	$-0.481840\pi$
$937$	100126.			0.114043			0.0570214	+	0.998373i	$0.481840\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	389767.	−	225032.i	0.440176	−	0.254136i	−0.263496	−	0.964660i	$-0.584876\pi$
$941$	389767.	−	225032.i	0.440176	−	0.254136i	0.703672	+	0.710525i	$0.251542\pi$
$942$		0			0
$943$	−8673.99	+	15023.8i	−0.00975429	+	0.0168949i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.20436e6	−	695338.i	−1.34294	−	0.775346i	−0.355702	−	0.934600i	$-0.615758\pi$
$947$	−1.20436e6	−	695338.i	−1.34294	−	0.775346i	−0.987238	+	0.159253i	$0.949091\pi$
$948$		0			0
$949$	628131.	+	1.08795e6i	0.697458	+	1.20803i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			965440.i			1.06301i	0.847054	+	0.531507i	$0.178374\pi$
$953$			965440.i			1.06301i	−0.847054	+	0.531507i	$0.821626\pi$
$954$		0			0
$955$	149524.			0.163948
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.22297e6	−	706080.i	1.32977	−	0.767744i
$960$		0			0
$961$	400137.	−	693058.i	0.433274	−	0.750452i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	267829.	+	154631.i	0.287609	+	0.166051i
$966$		0			0
$967$	442773.	+	766905.i	0.473509	+	0.820141i	0.999540	−	0.0303239i	$-0.00965389\pi$
$967$	442773.	+	766905.i	0.473509	+	0.820141i	−0.526031	+	0.850465i	$0.676321\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1.57335e6i		−	1.66873i	−0.551210	−	0.834366i	$-0.685834\pi$
$971$		−	1.57335e6i		−	1.66873i	0.551210	−	0.834366i	$-0.314166\pi$
$972$		0			0
$973$	204949.			0.216482
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.31749e6	+	760654.i	−1.38025	+	0.796889i	−0.992189	−	0.124743i	$-0.960189\pi$
$977$	−1.31749e6	+	760654.i	−1.38025	+	0.796889i	−0.388064	+	0.921633i	$0.626856\pi$
$978$		0			0
$979$	−73549.6	+	127392.i	−0.0767387	+	0.132915i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.36258e6	+	786688.i	1.41012	+	0.814133i	0.995399	−	0.0958157i	$-0.0305459\pi$
$983$	1.36258e6	+	786688.i	1.41012	+	0.814133i	0.414721	+	0.909949i	$0.363879\pi$
$984$		0			0
$985$	183777.	+	318311.i	0.189417	+	0.328079i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			17359.8i			0.0177481i
$990$		0			0
$991$	1.27581e6			1.29908			0.649542	−	0.760326i	$-0.274961\pi$
$991$	1.27581e6			1.29908			0.649542	+	0.760326i	$0.274961\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	477509.	−	275690.i	0.482320	−	0.278468i
$996$		0			0
$997$	344020.	−	595860.i	0.346093	−	0.599451i	−0.639459	−	0.768826i	$-0.720841\pi$
$997$	344020.	−	595860.i	0.346093	−	0.599451i	0.985552	+	0.169375i	$0.0541748\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.5.m.g.593.6		24
3.2	odd	2	inner	648.5.m.g.593.7		24
9.2	odd	6		648.5.e.b.161.7	yes	12
9.4	even	3	inner	648.5.m.g.377.7		24
9.5	odd	6	inner	648.5.m.g.377.6		24
9.7	even	3		648.5.e.b.161.6	✓	12
36.7	odd	6		1296.5.e.h.161.6		12
36.11	even	6		1296.5.e.h.161.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.5.e.b.161.6	✓	12	9.7	even	3
648.5.e.b.161.7	yes	12	9.2	odd	6
648.5.m.g.377.6		24	9.5	odd	6	inner
648.5.m.g.377.7		24	9.4	even	3	inner
648.5.m.g.593.6		24	1.1	even	1	trivial
648.5.m.g.593.7		24	3.2	odd	2	inner
1296.5.e.h.161.6		12	36.7	odd	6
1296.5.e.h.161.7		12	36.11	even	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	648.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-6.13894 + 3.54432i) q^{5} +(26.4902 - 45.8823i) q^{7} +(20.6208 + 11.9054i) q^{11} +(-138.907 - 240.593i) q^{13} +91.5027i q^{17} -332.045 q^{19} +(347.849 - 200.831i) q^{23} +(-287.376 + 497.749i) q^{25} +(-316.307 - 182.620i) q^{29} +(175.532 + 304.031i) q^{31} +375.558i q^{35} -42.6630 q^{37} +(-37.4041 + 21.5953i) q^{41} +(-21.6099 + 37.4295i) q^{43} +(2431.98 + 1404.11i) q^{47} +(-202.957 - 351.532i) q^{49} -4128.48i q^{53} -168.786 q^{55} +(-5504.74 + 3178.16i) q^{59} +(-848.536 + 1469.71i) q^{61} +(1705.48 + 984.659i) q^{65} +(1269.40 + 2198.67i) q^{67} -1550.91i q^{71} -4521.97 q^{73} +(1092.50 - 630.752i) q^{77} +(-3500.87 + 6063.68i) q^{79} +(-4962.11 - 2864.87i) q^{83} +(-324.315 - 561.730i) q^{85} +6177.83i q^{89} -14718.6 q^{91} +(2038.41 - 1176.87i) q^{95} +(-5363.80 + 9290.37i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 96 q^{7} + 72 q^{13} + 672 q^{19} - 84 q^{25} + 1536 q^{31} - 984 q^{37} - 10128 q^{43} + 6828 q^{49} - 27936 q^{55} - 26268 q^{61} + 20784 q^{67} - 19968 q^{73} - 44592 q^{79} + 45348 q^{85} - 23616 q^{91}+ \cdots - 76608 q^{97}+O(q^{100}) \) 24 * q - 96 * q^7 + 72 * q^13 + 672 * q^19 - 84 * q^25 + 1536 * q^31 - 984 * q^37 - 10128 * q^43 + 6828 * q^49 - 27936 * q^55 - 26268 * q^61 + 20784 * q^67 - 19968 * q^73 - 44592 * q^79 + 45348 * q^85 - 23616 * q^91 - 76608 * q^97

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