LMFDB - Embedded newform 432.8.i.b.145.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,8,Mod(145,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("432.145");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	432.i (of order $3$, 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:	$134.950331009$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1336x^{6} + 633664x^{4} + 125389995x^{2} + 8783438400 $ x^8 + 1336x^6 + 633664x^4 + 125389995*x^2 + 8783438400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{18} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			145.2
Root			$13.4379i$ of defining polynomial
Character	$\chi$	$=$	432.145
Dual form			432.8.i.b.289.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-103.252 + 178.837i) q^{5} +(808.604 + 1400.54i) q^{7} +(-2338.01 - 4049.55i) q^{11} +(-3336.65 + 5779.25i) q^{13} +25731.9 q^{17} -22384.5 q^{19} +(-11710.7 + 20283.5i) q^{23} +(17740.6 + 30727.7i) q^{25} +(81488.4 + 141142. i) q^{29} +(-116733. + 202187. i) q^{31} -333959. q^{35} +308867. q^{37} +(-157555. + 272893. i) q^{41} +(-61173.2 - 105955. i) q^{43} +(97801.6 + 169397. i) q^{47} +(-895910. + 1.55176e6i) q^{49} +369367. q^{53} +965615. q^{55} +(560669. - 971107. i) q^{59} +(-45155.7 - 78212.0i) q^{61} +(-689031. - 1.19344e6i) q^{65} +(1.76399e6 - 3.05533e6i) q^{67} -703989. q^{71} +4.06113e6 q^{73} +(3.78105e6 - 6.54897e6i) q^{77} +(2.33651e6 + 4.04696e6i) q^{79} +(-4.38570e6 - 7.59625e6i) q^{83} +(-2.65686e6 + 4.60182e6i) q^{85} -1.07368e7 q^{89} -1.07921e7 q^{91} +(2.31123e6 - 4.00317e6i) q^{95} +(-3.72029e6 - 6.44373e6i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 54 q^{5} + 44 q^{7} + 2172 q^{11} - 6398 q^{13} + 51972 q^{17} - 90712 q^{19} - 2028 q^{23} - 173446 q^{25} - 283098 q^{29} + 812 q^{31} - 1492992 q^{35} + 441688 q^{37} - 610704 q^{41} + 84380 q^{43}+ \cdots - 22003112 q^{97}+O(q^{100}) $ 8 * q - 54 * q^5 + 44 * q^7 + 2172 * q^11 - 6398 * q^13 + 51972 * q^17 - 90712 * q^19 - 2028 * q^23 - 173446 * q^25 - 283098 * q^29 + 812 * q^31 - 1492992 * q^35 + 441688 * q^37 - 610704 * q^41 + 84380 * q^43 + 855708 * q^47 - 1273038 * q^49 + 1842600 * q^53 - 7016760 * q^55 + 380796 * q^59 - 3151130 * q^61 + 326538 * q^65 + 9961580 * q^67 + 8970384 * q^71 + 29234932 * q^73 + 5954826 * q^77 + 17309396 * q^79 - 11520192 * q^83 - 14675148 * q^85 - 26066064 * q^89 - 26231936 * q^91 - 31016952 * q^95 - 22003112 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−103.252	+	178.837i	−0.369405	+	0.639828i	−0.989473	−	0.144720i	$-0.953772\pi$
$5$	−103.252	+	178.837i	−0.369405	+	0.639828i	0.620068	+	0.784548i	$0.287105\pi$
$6$		0			0
$7$	808.604	+	1400.54i	0.891031	+	1.54331i	0.838641	+	0.544684i	$0.183350\pi$
$7$	808.604	+	1400.54i	0.891031	+	1.54331i	0.0523898	+	0.998627i	$0.483316\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2338.01	−	4049.55i	−0.529629	−	0.917345i	−0.999403	−	0.0345577i	$-0.988998\pi$
$11$	−2338.01	−	4049.55i	−0.529629	−	0.917345i	0.469774	−	0.882787i	$-0.344336\pi$
$12$		0			0
$13$	−3336.65	+	5779.25i	−0.421220	+	0.729575i	−0.996059	−	0.0886915i	$-0.971731\pi$
$13$	−3336.65	+	5779.25i	−0.421220	+	0.729575i	0.574839	+	0.818267i	$0.305065\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	25731.9			1.27028			0.635141	−	0.772396i	$-0.280942\pi$
$17$	25731.9			1.27028			0.635141	+	0.772396i	$0.280942\pi$
$18$		0			0
$19$	−22384.5			−0.748702			−0.374351	−	0.927287i	$-0.622134\pi$
$19$	−22384.5			−0.748702			−0.374351	+	0.927287i	$0.622134\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−11710.7	+	20283.5i	−0.200695	+	0.347613i	−0.948752	−	0.316020i	$-0.897653\pi$
$23$	−11710.7	+	20283.5i	−0.200695	+	0.347613i	0.748058	+	0.663634i	$0.230987\pi$
$24$		0			0
$25$	17740.6	+	30727.7i	0.227080	+	0.393315i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	81488.4	+	141142.i	0.620444	+	1.07464i	0.989403	+	0.145195i	$0.0463811\pi$
$29$	81488.4	+	141142.i	0.620444	+	1.07464i	−0.368959	+	0.929446i	$0.620286\pi$
$30$		0			0
$31$	−116733.	+	202187.i	−0.703763	+	1.21895i	0.263373	+	0.964694i	$0.415165\pi$
$31$	−116733.	+	202187.i	−0.703763	+	1.21895i	−0.967136	+	0.254260i	$0.918168\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−333959.			−1.31660
$36$		0			0
$37$	308867.			1.00246			0.501229	−	0.865315i	$-0.332882\pi$
$37$	308867.			1.00246			0.501229	+	0.865315i	$0.332882\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−157555.	+	272893.i	−0.357017	+	0.618372i	−0.987461	−	0.157863i	$-0.949539\pi$
$41$	−157555.	+	272893.i	−0.357017	+	0.618372i	0.630444	+	0.776235i	$0.282873\pi$
$42$		0			0
$43$	−61173.2	−	105955.i	−0.117333	−	0.203228i	0.801377	−	0.598160i	$-0.204101\pi$
$43$	−61173.2	−	105955.i	−0.117333	−	0.203228i	−0.918710	+	0.394933i	$0.870768\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	97801.6	+	169397.i	0.137405	+	0.237993i	0.926514	−	0.376261i	$-0.122790\pi$
$47$	97801.6	+	169397.i	0.137405	+	0.237993i	−0.789108	+	0.614254i	$0.789457\pi$
$48$		0			0
$49$	−895910.	+	1.55176e6i	−1.08787	+	1.88425i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	369367.			0.340794			0.170397	−	0.985375i	$-0.445495\pi$
$53$	369367.			0.340794			0.170397	+	0.985375i	$0.445495\pi$
$54$		0			0
$55$	965615.			0.782590
$56$		0			0
$57$		0			0
$58$		0			0
$59$	560669.	−	971107.i	0.355406	−	0.615581i	−0.631781	−	0.775147i	$-0.717676\pi$
$59$	560669.	−	971107.i	0.355406	−	0.615581i	0.987187	+	0.159565i	$0.0510093\pi$
$60$		0			0
$61$	−45155.7	−	78212.0i	−0.0254717	−	0.0441183i	0.853009	−	0.521897i	$-0.174775\pi$
$61$	−45155.7	−	78212.0i	−0.0254717	−	0.0441183i	−0.878480	+	0.477779i	$0.841442\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−689031.	−	1.19344e6i	−0.311202	−	0.539017i
$66$		0			0
$67$	1.76399e6	−	3.05533e6i	0.716532	−	1.24107i	−0.245834	−	0.969312i	$-0.579062\pi$
$67$	1.76399e6	−	3.05533e6i	0.716532	−	1.24107i	0.962366	−	0.271758i	$-0.0876049\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−703989.			−0.233433			−0.116716	−	0.993165i	$-0.537237\pi$
$71$	−703989.			−0.233433			−0.116716	+	0.993165i	$0.537237\pi$
$72$		0			0
$73$	4.06113e6			1.22185			0.610923	−	0.791690i	$-0.290799\pi$
$73$	4.06113e6			1.22185			0.610923	+	0.791690i	$0.290799\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.78105e6	−	6.54897e6i	0.943832	−	1.63477i
$78$		0			0
$79$	2.33651e6	+	4.04696e6i	0.533180	+	0.923494i	0.999249	+	0.0387462i	$0.0123364\pi$
$79$	2.33651e6	+	4.04696e6i	0.533180	+	0.923494i	−0.466069	+	0.884748i	$0.654330\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.38570e6	−	7.59625e6i	−0.841910	−	1.45823i	−0.888278	−	0.459306i	$-0.848098\pi$
$83$	−4.38570e6	−	7.59625e6i	−0.841910	−	1.45823i	0.0463686	−	0.998924i	$-0.485235\pi$
$84$		0			0
$85$	−2.65686e6	+	4.60182e6i	−0.469248	+	0.812762i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.07368e7			−1.61439			−0.807194	−	0.590286i	$-0.799015\pi$
$89$	−1.07368e7			−1.61439			−0.807194	+	0.590286i	$0.799015\pi$
$90$		0			0
$91$	−1.07921e7			−1.50128
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.31123e6	−	4.00317e6i	0.276574	−	0.479040i
$96$		0			0
$97$	−3.72029e6	−	6.44373e6i	−0.413881	−	0.716863i	0.581429	−	0.813597i	$-0.302494\pi$
$97$	−3.72029e6	−	6.44373e6i	−0.413881	−	0.716863i	−0.995310	+	0.0967340i	$0.969160\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.40397e6	+	1.28241e7i	0.715056	+	1.23851i	0.962938	+	0.269723i	$0.0869321\pi$
$101$	7.40397e6	+	1.28241e7i	0.715056	+	1.23851i	−0.247882	+	0.968790i	$0.579735\pi$
$102$		0			0
$103$	−8.62804e6	+	1.49442e7i	−0.778004	+	1.34754i	0.155086	+	0.987901i	$0.450435\pi$
$103$	−8.62804e6	+	1.49442e7i	−0.778004	+	1.34754i	−0.933090	+	0.359642i	$0.882899\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.31928e7			−1.04110			−0.520552	−	0.853830i	$-0.674274\pi$
$107$	−1.31928e7			−1.04110			−0.520552	+	0.853830i	$0.674274\pi$
$108$		0			0
$109$	−2.46882e6			−0.182598			−0.0912990	−	0.995824i	$-0.529102\pi$
$109$	−2.46882e6			−0.182598			−0.0912990	+	0.995824i	$0.529102\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.72194e6	−	2.98249e6i	0.112265	−	0.194448i	−0.804418	−	0.594063i	$-0.797523\pi$
$113$	1.72194e6	−	2.98249e6i	0.112265	−	0.194448i	0.916683	+	0.399615i	$0.130856\pi$
$114$		0			0
$115$	−2.41830e6	−	4.18862e6i	−0.148275	−	0.256820i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.08069e7	+	3.60386e7i	1.13186	+	1.96044i
$120$		0			0
$121$	−1.18899e6	+	2.05940e6i	−0.0610141	+	0.105680i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.34601e7			−1.07435
$126$		0			0
$127$	119100.			0.00515940			0.00257970	−	0.999997i	$-0.499179\pi$
$127$	119100.			0.00515940			0.00257970	+	0.999997i	$0.499179\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.10622e7	+	1.91602e7i	−0.429923	+	0.744648i	−0.996866	−	0.0791089i	$-0.974793\pi$
$131$	−1.10622e7	+	1.91602e7i	−0.429923	+	0.744648i	0.566943	+	0.823757i	$0.308126\pi$
$132$		0			0
$133$	−1.81002e7	−	3.13504e7i	−0.667116	−	1.15548i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.94797e6	−	5.10604e6i	−0.0979493	−	0.169653i	0.812886	−	0.582422i	$-0.197895\pi$
$137$	−2.94797e6	−	5.10604e6i	−0.0979493	−	0.169653i	−0.910836	+	0.412769i	$0.864562\pi$
$138$		0			0
$139$	1.77894e7	−	3.08121e7i	0.561834	−	0.973126i	−0.435502	−	0.900188i	$-0.643429\pi$
$139$	1.77894e7	−	3.08121e7i	0.561834	−	0.973126i	0.997336	−	0.0729381i	$-0.0232376\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.12045e7			0.892363
$144$		0			0
$145$	−3.36553e7			−0.916780
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.67110e7	+	2.89443e7i	−0.413857	+	0.716821i	−0.995308	−	0.0967603i	$-0.969152\pi$
$149$	−1.67110e7	+	2.89443e7i	−0.413857	+	0.716821i	0.581451	+	0.813582i	$0.302485\pi$
$150$		0			0
$151$	−2.49541e7	−	4.32217e7i	−0.589823	−	1.02160i	−0.994255	−	0.107036i	$-0.965864\pi$
$151$	−2.49541e7	−	4.32217e7i	−0.589823	−	1.02160i	0.404432	−	0.914568i	$-0.367469\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.41057e7	−	4.17523e7i	−0.519947	−	0.900575i
$156$		0			0
$157$	2.13156e7	−	3.69197e7i	0.439590	−	0.761393i	−0.558067	−	0.829796i	$-0.688457\pi$
$157$	2.13156e7	−	3.69197e7i	0.439590	−	0.761393i	0.997658	+	0.0684026i	$0.0217902\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.78773e7			−0.715300
$162$		0			0
$163$	4.06125e7			0.734519			0.367259	−	0.930119i	$-0.380296\pi$
$163$	4.06125e7			0.734519			0.367259	+	0.930119i	$0.380296\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.62955e7	+	6.28656e7i	−0.603038	+	1.04449i	0.389320	+	0.921102i	$0.372710\pi$
$167$	−3.62955e7	+	6.28656e7i	−0.603038	+	1.04449i	−0.992358	+	0.123390i	$0.960623\pi$
$168$		0			0
$169$	9.10774e6	+	1.57751e7i	0.145147	+	0.251401i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.61254e7	+	2.79301e7i	0.236783	+	0.410120i	0.959789	−	0.280721i	$-0.0905736\pi$
$173$	1.61254e7	+	2.79301e7i	0.236783	+	0.410120i	−0.723006	+	0.690841i	$0.757240\pi$
$174$		0			0
$175$	−2.86903e7	+	4.96931e7i	−0.404671	+	0.700911i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.74576e7			−0.748793			−0.374396	−	0.927269i	$-0.622150\pi$
$179$	−5.74576e7			−0.748793			−0.374396	+	0.927269i	$0.622150\pi$
$180$		0			0
$181$	2.89626e7			0.363047			0.181524	−	0.983387i	$-0.441897\pi$
$181$	2.89626e7			0.363047			0.181524	+	0.983387i	$0.441897\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.18911e7	+	5.52370e7i	−0.370313	+	0.641400i
$186$		0			0
$187$	−6.01614e7	−	1.04203e8i	−0.672779	−	1.16529i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.94718e7	−	8.56877e7i	−0.513737	−	0.889819i	−0.999873	−	0.0159358i	$-0.994927\pi$
$191$	−4.94718e7	−	8.56877e7i	−0.513737	−	0.889819i	0.486136	−	0.873883i	$-0.338406\pi$
$192$		0			0
$193$	7.72389e6	−	1.33782e7i	0.0773367	−	0.133951i	−0.824763	−	0.565478i	$-0.808692\pi$
$193$	7.72389e6	−	1.33782e7i	0.0773367	−	0.133951i	0.902100	+	0.431527i	$0.142025\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.85117e7			0.918028			0.459014	−	0.888429i	$-0.348203\pi$
$197$	9.85117e7			0.918028			0.459014	+	0.888429i	$0.348203\pi$
$198$		0			0
$199$	−1.40825e8			−1.26676			−0.633380	−	0.773841i	$-0.718333\pi$
$199$	−1.40825e8			−1.26676			−0.633380	+	0.773841i	$0.718333\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.31784e8	+	2.28256e8i	−1.10567	+	1.91508i
$204$		0			0
$205$	−3.25357e7	−	5.63534e7i	−0.263768	−	0.456859i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.23351e7	+	9.06470e7i	0.396534	+	0.686818i
$210$		0			0
$211$	−3.02133e7	+	5.23309e7i	−0.221416	+	0.383504i	−0.955238	−	0.295838i	$-0.904401\pi$
$211$	−3.02133e7	+	5.23309e7i	−0.221416	+	0.383504i	0.733822	+	0.679342i	$0.237735\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.52650e7			0.173374
$216$		0			0
$217$	−3.77562e8			−2.50830
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.58584e7	+	1.48711e8i	−0.535069	+	0.926767i
$222$		0			0
$223$	−1.42446e8	−	2.46723e8i	−0.860165	−	1.48985i	−0.871769	−	0.489918i	$-0.837027\pi$
$223$	−1.42446e8	−	2.46723e8i	−0.860165	−	1.48985i	0.0116033	−	0.999933i	$-0.496306\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.13578e7	−	1.23595e8i	−0.404903	−	0.701312i	0.589407	−	0.807836i	$-0.299361\pi$
$227$	−7.13578e7	−	1.23595e8i	−0.404903	−	0.701312i	−0.994310	+	0.106524i	$0.966028\pi$
$228$		0			0
$229$	−6.39437e7	+	1.10754e8i	−0.351863	+	0.609445i	−0.986576	−	0.163303i	$-0.947785\pi$
$229$	−6.39437e7	+	1.10754e8i	−0.351863	+	0.609445i	0.634713	+	0.772748i	$0.281118\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.85503e8			−1.47865			−0.739324	−	0.673350i	$-0.764855\pi$
$233$	−2.85503e8			−1.47865			−0.739324	+	0.673350i	$0.764855\pi$
$234$		0			0
$235$	−4.03928e7			−0.203033
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5.90868e6	+	1.02341e7i	−0.0279961	+	0.0484907i	−0.879684	−	0.475559i	$-0.842246\pi$
$239$	−5.90868e6	+	1.02341e7i	−0.0279961	+	0.0484907i	0.851688	+	0.524049i	$0.175579\pi$
$240$		0			0
$241$	−7.41195e7	−	1.28379e8i	−0.341093	−	0.590791i	0.643543	−	0.765410i	$-0.277464\pi$
$241$	−7.41195e7	−	1.28379e8i	−0.341093	−	0.590791i	−0.984636	+	0.174619i	$0.944130\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.85009e8	−	3.20444e8i	−0.803731	−	1.39210i
$246$		0			0
$247$	7.46892e7	−	1.29365e8i	0.315368	−	0.546234i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.92645e7			0.276473			0.138237	−	0.990399i	$-0.455857\pi$
$251$	6.92645e7			0.276473			0.138237	+	0.990399i	$0.455857\pi$
$252$		0			0
$253$	1.09519e8			0.425175
$254$		0			0
$255$		0			0
$256$		0			0
$257$	8.27289e7	−	1.43291e8i	0.304013	−	0.526565i	−0.673028	−	0.739617i	$-0.735007\pi$
$257$	8.27289e7	−	1.43291e8i	0.304013	−	0.526565i	0.977041	+	0.213051i	$0.0683402\pi$
$258$		0			0
$259$	2.49751e8	+	4.32582e8i	0.893221	+	1.54710i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	8.54750e7	+	1.48047e8i	0.289730	+	0.501828i	0.973745	−	0.227640i	$-0.0731010\pi$
$263$	8.54750e7	+	1.48047e8i	0.289730	+	0.501828i	−0.684015	+	0.729468i	$0.739768\pi$
$264$		0			0
$265$	−3.81378e7	+	6.60566e7i	−0.125891	+	0.218050i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.40056e8			−0.751932			−0.375966	−	0.926633i	$-0.622689\pi$
$269$	−2.40056e8			−0.751932			−0.375966	+	0.926633i	$0.622689\pi$
$270$		0			0
$271$	−2.92612e8			−0.893098			−0.446549	−	0.894759i	$-0.647347\pi$
$271$	−2.92612e8			−0.893098			−0.446549	+	0.894759i	$0.647347\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.29556e7	−	1.43683e8i	0.240537	−	0.416622i
$276$		0			0
$277$	−4.35557e7	−	7.54407e7i	−0.123131	−	0.213269i	0.797870	−	0.602829i	$-0.205960\pi$
$277$	−4.35557e7	−	7.54407e7i	−0.123131	−	0.213269i	−0.921001	+	0.389561i	$0.872627\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.68148e7	−	1.33047e8i	−0.206525	−	0.357712i	0.744092	−	0.668077i	$-0.232882\pi$
$281$	−7.68148e7	−	1.33047e8i	−0.206525	−	0.357712i	−0.950618	+	0.310365i	$0.899549\pi$
$282$		0			0
$283$	−1.09494e8	+	1.89650e8i	−0.287170	+	0.497393i	−0.973133	−	0.230243i	$-0.926048\pi$
$283$	−1.09494e8	+	1.89650e8i	−0.287170	+	0.497393i	0.685963	+	0.727636i	$0.259381\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.09599e8			−1.27245
$288$		0			0
$289$	2.51791e8			0.613617
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.45004e8	+	2.51154e8i	−0.336777	+	0.583315i	−0.983825	−	0.179135i	$-0.942670\pi$
$293$	−1.45004e8	+	2.51154e8i	−0.336777	+	0.583315i	0.647048	+	0.762450i	$0.276004\pi$
$294$		0			0
$295$	1.15780e8	+	2.00537e8i	0.262577	+	0.454797i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.81492e7	−	1.35358e8i	−0.169073	−	0.292844i
$300$		0			0
$301$	9.89299e7	−	1.71352e8i	0.209095	−	0.362164i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.86496e7			0.0376375
$306$		0			0
$307$	−2.99914e7			−0.0591579			−0.0295789	−	0.999562i	$-0.509417\pi$
$307$	−2.99914e7			−0.0591579			−0.0295789	+	0.999562i	$0.509417\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	7.31419e6	−	1.26686e7i	0.0137881	−	0.0238817i	−0.859049	−	0.511893i	$-0.828944\pi$
$311$	7.31419e6	−	1.26686e7i	0.0137881	−	0.0238817i	0.872837	+	0.488012i	$0.162278\pi$
$312$		0			0
$313$	2.52793e7	+	4.37851e7i	0.0465973	+	0.0807089i	0.888383	−	0.459102i	$-0.151829\pi$
$313$	2.52793e7	+	4.37851e7i	0.0465973	+	0.0807089i	−0.841786	+	0.539811i	$0.818496\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.24798e7	−	5.62566e7i	−0.0572671	−	0.0991896i	0.835971	−	0.548774i	$-0.184905\pi$
$317$	−3.24798e7	−	5.62566e7i	−0.0572671	−	0.0991896i	−0.893238	+	0.449585i	$0.851572\pi$
$318$		0			0
$319$	3.81041e8	−	6.59983e8i	0.657211	−	1.13832i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5.75994e8			−0.951063
$324$		0			0
$325$	−2.36778e8			−0.382603
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.58166e8	+	2.73951e8i	−0.244865	+	0.424118i
$330$		0			0
$331$	4.42586e7	+	7.66582e7i	0.0670811	+	0.116188i	0.897615	−	0.440780i	$-0.145298\pi$
$331$	4.42586e7	+	7.66582e7i	0.0670811	+	0.116188i	−0.830534	+	0.556968i	$0.811965\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.64271e8	+	6.30936e8i	0.529381	+	0.916914i
$336$		0			0
$337$	2.27637e8	−	3.94279e8i	0.323995	−	0.561176i	−0.657313	−	0.753618i	$-0.728307\pi$
$337$	2.27637e8	−	3.94279e8i	0.323995	−	0.561176i	0.981309	+	0.192441i	$0.0616404\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.09169e9			1.49093
$342$		0			0
$343$	−1.56590e9			−2.09525
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.48016e8	+	2.56372e8i	−0.190176	+	0.329395i	−0.945309	−	0.326178i	$-0.894239\pi$
$347$	−1.48016e8	+	2.56372e8i	−0.190176	+	0.329395i	0.755132	+	0.655572i	$0.227573\pi$
$348$		0			0
$349$	−3.80328e8	−	6.58748e8i	−0.478927	−	0.829526i	0.520781	−	0.853691i	$-0.325641\pi$
$349$	−3.80328e8	−	6.58748e8i	−0.478927	−	0.829526i	−0.999708	+	0.0241640i	$0.992308\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.00254e8	+	1.21288e9i	0.847313	+	1.46759i	0.883597	+	0.468248i	$0.155115\pi$
$353$	7.00254e8	+	1.21288e9i	0.847313	+	1.46759i	−0.0362837	+	0.999342i	$0.511552\pi$
$354$		0			0
$355$	7.26882e7	−	1.25900e8i	0.0862312	−	0.149357i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.32132e9			1.50722			0.753612	−	0.657320i	$-0.228310\pi$
$359$	1.32132e9			1.50722			0.753612	+	0.657320i	$0.228310\pi$
$360$		0			0
$361$	−3.92808e8			−0.439446
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.19318e8	+	7.26281e8i	−0.451356	+	0.781771i
$366$		0			0
$367$	−3.70961e8	−	6.42523e8i	−0.391739	−	0.678511i	0.600940	−	0.799294i	$-0.294793\pi$
$367$	−3.70961e8	−	6.42523e8i	−0.391739	−	0.678511i	−0.992679	+	0.120783i	$0.961460\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.98672e8	+	5.17314e8i	0.303658	+	0.525952i
$372$		0			0
$373$	−4.39730e8	+	7.61635e8i	−0.438738	+	0.759916i	−0.997592	−	0.0693494i	$-0.977908\pi$
$373$	−4.39730e8	+	7.61635e8i	−0.438738	+	0.759916i	0.558855	+	0.829266i	$0.311241\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.08759e9			−1.04538
$378$		0			0
$379$	1.99736e9			1.88460			0.942301	−	0.334767i	$-0.108658\pi$
$379$	1.99736e9			1.88460			0.942301	+	0.334767i	$0.108658\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.54585e8	+	2.67750e8i	−0.140596	+	0.243519i	−0.927721	−	0.373274i	$-0.878235\pi$
$383$	−1.54585e8	+	2.67750e8i	−0.140596	+	0.243519i	0.787125	+	0.616793i	$0.211568\pi$
$384$		0			0
$385$	7.80800e8	+	1.35239e9i	0.697312	+	1.20778i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.64068e8	−	1.66982e9i	−0.830394	−	1.43828i	−0.897726	−	0.440554i	$-0.854782\pi$
$389$	−9.64068e8	−	1.66982e9i	−0.830394	−	1.43828i	0.0673322	−	0.997731i	$-0.478551\pi$
$390$		0			0
$391$	−3.01338e8	+	5.21934e8i	−0.254939	+	0.441567i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−9.64997e8			−0.787837
$396$		0			0
$397$	−1.08397e9			−0.869460			−0.434730	−	0.900561i	$-0.643156\pi$
$397$	−1.08397e9			−0.869460			−0.434730	+	0.900561i	$0.643156\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4.81857e8	−	8.34600e8i	0.373175	−	0.646358i	−0.616877	−	0.787060i	$-0.711602\pi$
$401$	4.81857e8	−	8.34600e8i	0.373175	−	0.646358i	0.990052	+	0.140701i	$0.0449357\pi$
$402$		0			0
$403$	−7.78993e8	−	1.34926e9i	−0.592879	−	1.02690i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−7.22135e8	−	1.25077e9i	−0.530931	−	0.919599i
$408$		0			0
$409$	6.07460e7	−	1.05215e8i	0.0439022	−	0.0760408i	−0.843239	−	0.537538i	$-0.819354\pi$
$409$	6.07460e7	−	1.05215e8i	0.0439022	−	0.0760408i	0.887141	+	0.461498i	$0.152688\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.81344e9			1.26671
$414$		0			0
$415$	1.81132e9			1.24402
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.52373e8	−	7.83532e8i	0.300433	−	0.520365i	−0.675801	−	0.737084i	$-0.736202\pi$
$419$	4.52373e8	−	7.83532e8i	0.300433	−	0.520365i	0.976234	+	0.216719i	$0.0695356\pi$
$420$		0			0
$421$	5.94491e8	+	1.02969e9i	0.388292	+	0.672541i	0.992220	−	0.124498i	$-0.0397320\pi$
$421$	5.94491e8	+	1.02969e9i	0.388292	+	0.672541i	−0.603928	+	0.797039i	$0.706399\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.56500e8	+	7.90681e8i	0.288456	+	0.499621i
$426$		0			0
$427$	7.30262e7	−	1.26485e8i	0.0453922	−	0.0786216i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.77750e9			1.06940			0.534700	−	0.845042i	$-0.320425\pi$
$431$	1.77750e9			1.06940			0.534700	+	0.845042i	$0.320425\pi$
$432$		0			0
$433$	1.98626e9			1.17579			0.587894	−	0.808938i	$-0.299957\pi$
$433$	1.98626e9			1.17579			0.587894	+	0.808938i	$0.299957\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.62138e8	−	4.54036e8i	0.150260	−	0.260259i
$438$		0			0
$439$	6.54719e8	+	1.13401e9i	0.369342	+	0.639719i	0.989463	−	0.144787i	$-0.0462497\pi$
$439$	6.54719e8	+	1.13401e9i	0.369342	+	0.639719i	−0.620121	+	0.784506i	$0.712916\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9.12222e8	−	1.58002e9i	−0.498526	−	0.863472i	0.501473	−	0.865173i	$-0.332792\pi$
$443$	−9.12222e8	−	1.58002e9i	−0.498526	−	0.863472i	−0.999999	+	0.00170136i	$0.999458\pi$
$444$		0			0
$445$	1.10859e9	−	1.92013e9i	0.596363	−	1.03293i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.58855e9			1.87093			0.935464	−	0.353423i	$-0.114982\pi$
$449$	3.58855e9			1.87093			0.935464	+	0.353423i	$0.114982\pi$
$450$		0			0
$451$	1.47346e9			0.756346
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.11431e9	−	1.93004e9i	0.554581	−	0.960562i
$456$		0			0
$457$	−1.59845e9	−	2.76860e9i	−0.783417	−	1.35692i	−0.929940	−	0.367711i	$-0.880141\pi$
$457$	−1.59845e9	−	2.76860e9i	−0.783417	−	1.35692i	0.146522	−	0.989207i	$-0.453192\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.87364e9	−	3.24524e9i	−0.890704	−	1.54275i	−0.839033	−	0.544081i	$-0.816878\pi$
$461$	−1.87364e9	−	3.24524e9i	−0.890704	−	1.54275i	−0.0516717	−	0.998664i	$-0.516455\pi$
$462$		0			0
$463$	−1.09312e9	+	1.89333e9i	−0.511838	+	0.886530i	0.488068	+	0.872806i	$0.337702\pi$
$463$	−1.09312e9	+	1.89333e9i	−0.511838	+	0.886530i	−0.999906	+	0.0137240i	$0.995631\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9.63843e8			0.437922			0.218961	−	0.975734i	$-0.429733\pi$
$467$	9.63843e8			0.437922			0.218961	+	0.975734i	$0.429733\pi$
$468$		0			0
$469$	5.70549e9			2.55381
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.86047e8	+	4.95448e8i	−0.124286	+	0.215270i
$474$		0			0
$475$	−3.97115e8	−	6.87823e8i	−0.170015	−	0.294475i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8.61804e8	+	1.49269e9i	0.358290	+	0.620576i	0.987675	−	0.156517i	$-0.0500267\pi$
$479$	8.61804e8	+	1.49269e9i	0.358290	+	0.620576i	−0.629386	+	0.777093i	$0.716693\pi$
$480$		0			0
$481$	−1.03058e9	+	1.78502e9i	−0.422256	+	0.731368i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.53651e9			0.611558
$486$		0			0
$487$	6.57115e7			0.0257804			0.0128902	−	0.999917i	$-0.495897\pi$
$487$	6.57115e7			0.0257804			0.0128902	+	0.999917i	$0.495897\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9.79411e8	+	1.69639e9i	−0.373405	+	0.646756i	−0.990087	−	0.140456i	$-0.955143\pi$
$491$	−9.79411e8	+	1.69639e9i	−0.373405	+	0.646756i	0.616682	+	0.787212i	$0.288476\pi$
$492$		0			0
$493$	2.09685e9	+	3.63185e9i	0.788140	+	1.36510i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5.69249e8	−	9.85968e8i	−0.207996	−	0.360259i
$498$		0			0
$499$	−1.54210e9	+	2.67100e9i	−0.555599	+	0.962325i	0.442258	+	0.896888i	$0.354178\pi$
$499$	−1.54210e9	+	2.67100e9i	−0.555599	+	0.962325i	−0.997857	+	0.0654372i	$0.979156\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.95457e9			1.38552			0.692758	−	0.721170i	$-0.256395\pi$
$503$	3.95457e9			1.38552			0.692758	+	0.721170i	$0.256395\pi$
$504$		0			0
$505$	−3.05789e9			−1.05658
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.92422e9	+	3.33285e9i	−0.646760	+	1.12022i	0.337132	+	0.941457i	$0.390543\pi$
$509$	−1.92422e9	+	3.33285e9i	−0.646760	+	1.12022i	−0.983892	+	0.178764i	$0.942790\pi$
$510$		0			0
$511$	3.28384e9	+	5.68778e9i	1.08870	+	1.88569i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.78172e9	−	3.08603e9i	−0.574797	−	0.995578i
$516$		0			0
$517$	4.57322e8	−	7.92105e8i	0.145548	−	0.252096i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.90239e8			0.0589344			0.0294672	−	0.999566i	$-0.490619\pi$
$521$	1.90239e8			0.0589344			0.0294672	+	0.999566i	$0.490619\pi$
$522$		0			0
$523$	9.89767e8			0.302536			0.151268	−	0.988493i	$-0.451664\pi$
$523$	9.89767e8			0.302536			0.151268	+	0.988493i	$0.451664\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.00375e9	+	5.20265e9i	−0.893978	+	1.54842i
$528$		0			0
$529$	1.42813e9	+	2.47360e9i	0.419443	+	0.726497i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.05141e9	−	1.82110e9i	−0.300766	−	0.520941i
$534$		0			0
$535$	1.36218e9	−	2.35937e9i	0.384589	−	0.666128i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	8.37858e9			2.30468
$540$		0			0
$541$	−3.06000e9			−0.830865			−0.415433	−	0.909624i	$-0.636370\pi$
$541$	−3.06000e9			−0.830865			−0.415433	+	0.909624i	$0.636370\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.54910e8	−	4.41517e8i	0.0674526	−	0.116831i
$546$		0			0
$547$	2.69716e9	+	4.67161e9i	0.704612	+	1.22042i	0.966831	+	0.255416i	$0.0822124\pi$
$547$	2.69716e9	+	4.67161e9i	0.704612	+	1.22042i	−0.262219	+	0.965008i	$0.584454\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.82407e9	−	3.15939e9i	−0.464528	−	0.804586i
$552$		0			0
$553$	−3.77863e9	+	6.54478e9i	−0.950159	+	1.64572i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	704537.			0.000172747			8.63736e−5	−	1.00000i	$-0.499973\pi$
$557$	704537.			0.000172747			8.63736e−5		1.00000i	$0.499973\pi$
$558$		0			0
$559$	8.16456e8			0.197693
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.58230e9	−	2.74063e9i	0.373688	−	0.647247i	−0.616441	−	0.787401i	$-0.711426\pi$
$563$	1.58230e9	−	2.74063e9i	0.373688	−	0.647247i	0.990130	+	0.140153i	$0.0447596\pi$
$564$		0			0
$565$	3.55587e8	+	6.15894e8i	0.0829423	+	0.143660i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.94504e8	+	1.20292e9i	0.158045	+	0.273742i	0.934164	−	0.356845i	$-0.116147\pi$
$569$	6.94504e8	+	1.20292e9i	0.158045	+	0.273742i	−0.776118	+	0.630587i	$0.782814\pi$
$570$		0			0
$571$	2.83085e9	−	4.90317e9i	0.636341	−	1.10217i	−0.349889	−	0.936791i	$-0.613781\pi$
$571$	2.83085e9	−	4.90317e9i	0.636341	−	1.10217i	0.986229	−	0.165383i	$-0.0528861\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.31022e8			−0.182295
$576$		0			0
$577$	3.72547e9			0.807358			0.403679	−	0.914901i	$-0.367731\pi$
$577$	3.72547e9			0.807358			0.403679	+	0.914901i	$0.367731\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.09259e9	−	1.22847e10i	1.50034	−	2.59866i
$582$		0			0
$583$	−8.63583e8	−	1.49577e9i	−0.180495	−	0.312626i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.23155e8	+	1.07934e9i	0.127164	+	0.220254i	0.922577	−	0.385814i	$-0.126079\pi$
$587$	6.23155e8	+	1.07934e9i	0.127164	+	0.220254i	−0.795413	+	0.606068i	$0.792746\pi$
$588$		0			0
$589$	2.61300e9	−	4.52584e9i	0.526909	−	0.912633i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2.28414e9			−0.449813			−0.224906	−	0.974380i	$-0.572208\pi$
$593$	−2.28414e9			−0.449813			−0.224906	+	0.974380i	$0.572208\pi$
$594$		0			0
$595$	−8.59340e9			−1.67246
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−9.17038e8	+	1.58836e9i	−0.174339	+	0.301963i	−0.939932	−	0.341361i	$-0.889112\pi$
$599$	−9.17038e8	+	1.58836e9i	−0.174339	+	0.301963i	0.765594	+	0.643325i	$0.222445\pi$
$600$		0			0
$601$	−1.04820e9	−	1.81554e9i	−0.196962	−	0.341149i	0.750580	−	0.660780i	$-0.229774\pi$
$601$	−1.04820e9	−	1.81554e9i	−0.196962	−	0.341149i	−0.947542	+	0.319631i	$0.896441\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.45531e8	−	4.25273e8i	−0.0450778	−	0.0780771i
$606$		0			0
$607$	1.96750e9	−	3.40781e9i	0.357070	−	0.618464i	−0.630400	−	0.776271i	$-0.717109\pi$
$607$	1.96750e9	−	3.40781e9i	0.357070	−	0.618464i	0.987470	+	0.157807i	$0.0504423\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.30532e9			−0.231512
$612$		0			0
$613$	8.35535e9			1.46505			0.732526	−	0.680739i	$-0.238341\pi$
$613$	8.35535e9			1.46505			0.732526	+	0.680739i	$0.238341\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5.33262e9	+	9.23638e9i	−0.913993	+	1.58308i	−0.105624	+	0.994406i	$0.533684\pi$
$617$	−5.33262e9	+	9.23638e9i	−0.913993	+	1.58308i	−0.808369	+	0.588676i	$0.799649\pi$
$618$		0			0
$619$	−3.59538e9	−	6.22739e9i	−0.609295	−	1.05533i	−0.991357	−	0.131194i	$-0.958119\pi$
$619$	−3.59538e9	−	6.22739e9i	−0.609295	−	1.05533i	0.382061	−	0.924137i	$-0.375214\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.68178e9	−	1.50373e10i	−1.43847	−	2.49150i
$624$		0			0
$625$	1.03631e9	−	1.79494e9i	0.169789	−	0.294083i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7.94774e9			1.27340
$630$		0			0
$631$	6.01287e9			0.952750			0.476375	−	0.879242i	$-0.341951\pi$
$631$	6.01287e9			0.952750			0.476375	+	0.879242i	$0.341951\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.22973e7	+	2.12995e7i	−0.00190591	+	0.00330113i
$636$		0			0
$637$	−5.97868e9	−	1.03554e10i	−0.916468	−	1.58737i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.26243e9	+	2.18659e9i	0.189323	+	0.327917i	0.945025	−	0.326999i	$-0.106037\pi$
$641$	1.26243e9	+	2.18659e9i	0.189323	+	0.327917i	−0.755702	+	0.654916i	$0.772704\pi$
$642$		0			0
$643$	−4.05815e9	+	7.02892e9i	−0.601990	+	1.04268i	0.390529	+	0.920590i	$0.372292\pi$
$643$	−4.05815e9	+	7.02892e9i	−0.601990	+	1.04268i	−0.992519	+	0.122087i	$0.961041\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.70712e9			−0.973578			−0.486789	−	0.873519i	$-0.661832\pi$
$647$	−6.70712e9			−0.973578			−0.486789	+	0.873519i	$0.661832\pi$
$648$		0			0
$649$	−5.24340e9			−0.752933
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.33252e9	+	5.77210e9i	−0.468357	+	0.811218i	−0.999346	−	0.0361606i	$-0.988487\pi$
$653$	−3.33252e9	+	5.77210e9i	−0.468357	+	0.811218i	0.530989	+	0.847379i	$0.321821\pi$
$654$		0			0
$655$	−2.28438e9	−	3.95665e9i	−0.317631	−	0.550153i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.56243e9	+	1.13665e10i	0.893235	+	1.54713i	0.835973	+	0.548770i	$0.184904\pi$
$659$	6.56243e9	+	1.13665e10i	0.893235	+	1.54713i	0.0572622	+	0.998359i	$0.481763\pi$
$660$		0			0
$661$	8.40716e8	−	1.45616e9i	0.113226	−	0.196112i	−0.803843	−	0.594841i	$-0.797215\pi$
$661$	8.40716e8	−	1.45616e9i	0.113226	−	0.196112i	0.917069	+	0.398728i	$0.130548\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7.47549e9			0.985744
$666$		0			0
$667$	−3.81715e9			−0.498079
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.11149e8	+	3.65721e8i	−0.0269811	+	0.0467327i
$672$		0			0
$673$	1.90889e9	+	3.30630e9i	0.241395	+	0.418109i	0.961112	−	0.276159i	$-0.0890616\pi$
$673$	1.90889e9	+	3.30630e9i	0.241395	+	0.418109i	−0.719717	+	0.694268i	$0.755728\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.08674e9	+	7.07845e9i	0.506194	+	0.876754i	0.999974	+	0.00716716i	$0.00228140\pi$
$677$	4.08674e9	+	7.07845e9i	0.506194	+	0.876754i	−0.493780	+	0.869587i	$0.664385\pi$
$678$		0			0
$679$	6.01648e9	−	1.04209e10i	0.737562	−	1.27749i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.11162e10			1.33501			0.667507	−	0.744604i	$-0.267362\pi$
$683$	1.11162e10			1.33501			0.667507	+	0.744604i	$0.267362\pi$
$684$		0			0
$685$	1.21753e9			0.144732
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.23245e9	+	2.13467e9i	−0.143550	+	0.248635i
$690$		0			0
$691$	−1.06038e9	−	1.83664e9i	−0.122262	−	0.211763i	0.798398	−	0.602131i	$-0.205681\pi$
$691$	−1.06038e9	−	1.83664e9i	−0.122262	−	0.211763i	−0.920659	+	0.390367i	$0.872348\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.67356e9	+	6.36280e9i	0.415089	+	0.718955i
$696$		0			0
$697$	−4.05419e9	+	7.02206e9i	−0.453512	+	0.785506i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.27398e10			−1.39684			−0.698422	−	0.715686i	$-0.746114\pi$
$701$	−1.27398e10			−1.39684			−0.698422	+	0.715686i	$0.746114\pi$
$702$		0			0
$703$	−6.91383e9			−0.750542
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.19738e10	+	2.07392e10i	−1.27427	+	2.20711i
$708$		0			0
$709$	3.84065e9	+	6.65221e9i	0.404710	+	0.700977i	0.994288	−	0.106734i	$-0.0340393\pi$
$709$	3.84065e9	+	6.65221e9i	0.404710	+	0.700977i	−0.589578	+	0.807711i	$0.700706\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.73405e9	−	4.73551e9i	−0.282483	−	0.489275i
$714$		0			0
$715$	−3.22192e9	+	5.58053e9i	−0.329643	+	0.570958i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.29899e10			−1.30333			−0.651666	−	0.758506i	$-0.725929\pi$
$719$	−1.29899e10			−1.30333			−0.651666	+	0.758506i	$0.725929\pi$
$720$		0			0
$721$	−2.79067e10			−2.77290
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.89131e9	+	5.00790e9i	−0.281781	+	0.488060i
$726$		0			0
$727$	−4.67747e8	−	8.10161e8i	−0.0451482	−	0.0781990i	0.842568	−	0.538590i	$-0.181043\pi$
$727$	−4.67747e8	−	8.10161e8i	−0.0451482	−	0.0781990i	−0.887716	+	0.460391i	$0.847709\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.57410e9	−	2.72643e9i	−0.149047	−	0.258156i
$732$		0			0
$733$	5.82375e9	−	1.00870e10i	0.546184	−	0.946019i	−0.452347	−	0.891842i	$-0.649413\pi$
$733$	5.82375e9	−	1.00870e10i	0.546184	−	0.946019i	0.998531	−	0.0541767i	$-0.0172534\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.64969e10			−1.51798
$738$		0			0
$739$	1.36168e10			1.24114			0.620570	−	0.784151i	$-0.286901\pi$
$739$	1.36168e10			1.24114			0.620570	+	0.784151i	$0.286901\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7.55834e8	+	1.30914e9i	−0.0676029	+	0.117092i	−0.897846	−	0.440310i	$-0.854868\pi$
$743$	−7.55834e8	+	1.30914e9i	−0.0676029	+	0.117092i	0.830243	+	0.557402i	$0.188202\pi$
$744$		0			0
$745$	−3.45088e9	−	5.97710e9i	−0.305762	−	0.529594i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.06678e10	−	1.84771e10i	−0.927657	−	1.60675i
$750$		0			0
$751$	−9.52029e9	+	1.64896e10i	−0.820183	+	1.42060i	0.0853635	+	0.996350i	$0.472795\pi$
$751$	−9.52029e9	+	1.64896e10i	−0.820183	+	1.42060i	−0.905546	+	0.424248i	$0.860538\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.03062e10			0.871534
$756$		0			0
$757$	−6.00984e9			−0.503532			−0.251766	−	0.967788i	$-0.581011\pi$
$757$	−6.00984e9			−0.503532			−0.251766	+	0.967788i	$0.581011\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.12107e10	+	1.94175e10i	−0.922116	+	1.59715i	−0.125981	+	0.992033i	$0.540208\pi$
$761$	−1.12107e10	+	1.94175e10i	−0.922116	+	1.59715i	−0.796135	+	0.605119i	$0.793126\pi$
$762$		0			0
$763$	−1.99630e9	−	3.45769e9i	−0.162701	−	0.281806i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.74152e9	+	6.48050e9i	0.299408	+	0.518591i
$768$		0			0
$769$	3.36877e9	−	5.83488e9i	0.267134	−	0.462690i	−0.700987	−	0.713175i	$-0.747257\pi$
$769$	3.36877e9	−	5.83488e9i	0.267134	−	0.462690i	0.968121	+	0.250485i	$0.0805900\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	9.36137e9			0.728973			0.364486	−	0.931209i	$-0.381245\pi$
$773$	9.36137e9			0.728973			0.364486	+	0.931209i	$0.381245\pi$
$774$		0			0
$775$	−8.28365e9			−0.639243
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.52678e9	−	6.10857e9i	0.267299	−	0.462976i
$780$		0			0
$781$	1.64593e9	+	2.85084e9i	0.123633	+	0.214138i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.40174e9	+	7.62404e9i	0.324774	+	0.562524i
$786$		0			0
$787$	3.25442e9	−	5.63682e9i	0.237992	−	0.412214i	−0.722146	−	0.691740i	$-0.756844\pi$
$787$	3.25442e9	−	5.63682e9i	0.237992	−	0.412214i	0.960138	+	0.279527i	$0.0901776\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5.56947e9			0.400125
$792$		0			0
$793$	6.02676e8			0.0429169
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8.12967e9	−	1.40810e10i	0.568812	−	0.985211i	−0.427872	−	0.903839i	$-0.640737\pi$
$797$	8.12967e9	−	1.40810e10i	0.568812	−	0.985211i	0.996684	−	0.0813716i	$-0.0259300\pi$
$798$		0			0
$799$	2.51662e9	+	4.35891e9i	0.174544	+	0.302318i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9.49495e9	−	1.64457e10i	−0.647125	−	1.12085i
$804$		0			0
$805$	3.91090e9	−	6.77387e9i	0.264235	−	0.457669i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.11275e10			1.40291			0.701454	−	0.712715i	$-0.252535\pi$
$809$	2.11275e10			1.40291			0.701454	+	0.712715i	$0.252535\pi$
$810$		0			0
$811$	−1.05784e10			−0.696383			−0.348191	−	0.937423i	$-0.613204\pi$
$811$	−1.05784e10			−0.696383			−0.348191	+	0.937423i	$0.613204\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.19331e9	+	7.26303e9i	−0.271335	+	0.469965i
$816$		0			0
$817$	1.36933e9	+	2.37175e9i	0.0878478	+	0.152157i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.46609e10	−	2.53934e10i	−0.924610	−	1.60147i	−0.792188	−	0.610277i	$-0.791058\pi$
$821$	−1.46609e10	−	2.53934e10i	−0.924610	−	1.60147i	−0.132422	−	0.991193i	$-0.542275\pi$
$822$		0			0
$823$	4.13709e9	−	7.16565e9i	0.258700	−	0.448081i	−0.707194	−	0.707019i	$-0.750039\pi$
$823$	4.13709e9	−	7.16565e9i	0.258700	−	0.448081i	0.965894	+	0.258939i	$0.0833727\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.68712e10			−1.03723			−0.518616	−	0.855007i	$-0.673552\pi$
$827$	−1.68712e10			−1.03723			−0.518616	+	0.855007i	$0.673552\pi$
$828$		0			0
$829$	1.62532e10			0.990830			0.495415	−	0.868657i	$-0.335016\pi$
$829$	1.62532e10			0.990830			0.495415	+	0.868657i	$0.335016\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.30534e10	+	3.99297e10i	−1.38191	+	2.39353i
$834$		0			0
$835$	−7.49514e9	−	1.29820e10i	−0.445530	−	0.771681i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.12134e10	−	1.94222e10i	−0.655498	−	1.13536i	−0.981769	−	0.190079i	$-0.939125\pi$
$839$	−1.12134e10	−	1.94222e10i	−0.655498	−	1.13536i	0.326271	−	0.945276i	$-0.394208\pi$
$840$		0			0
$841$	−4.65578e9	+	8.06405e9i	−0.269902	+	0.467485i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.76156e9			−0.214471
$846$		0			0
$847$	−3.84570e9			−0.217462
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.61706e9	+	6.26492e9i	−0.201188	+	0.348468i
$852$		0			0
$853$	−1.11195e10	−	1.92595e10i	−0.613428	−	1.06249i	−0.990658	−	0.136369i	$-0.956457\pi$
$853$	−1.11195e10	−	1.92595e10i	−0.613428	−	1.06249i	0.377230	−	0.926120i	$-0.376877\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.25516e9	−	1.42984e10i	−0.448015	−	0.775985i	0.550242	−	0.835005i	$-0.314536\pi$
$857$	−8.25516e9	−	1.42984e10i	−0.448015	−	0.775985i	−0.998257	+	0.0590205i	$0.981202\pi$
$858$		0			0
$859$	−1.01406e10	+	1.75640e10i	−0.545866	+	0.945468i	0.452686	+	0.891670i	$0.350466\pi$
$859$	−1.01406e10	+	1.75640e10i	−0.545866	+	0.945468i	−0.998552	+	0.0537978i	$0.982867\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.39028e10			1.26593			0.632967	−	0.774179i	$-0.281837\pi$
$863$	2.39028e10			1.26593			0.632967	+	0.774179i	$0.281837\pi$
$864$		0			0
$865$	−6.65992e9			−0.349875
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.09256e10	−	1.89237e10i	0.564775	−	0.978219i
$870$		0			0
$871$	1.17717e10	+	2.03891e10i	0.603636	+	1.04553i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.89699e10	−	3.28569e10i	−0.957277	−	1.65805i
$876$		0			0
$877$	1.28617e9	−	2.22771e9i	0.0643873	−	0.111522i	−0.832035	−	0.554724i	$-0.812824\pi$
$877$	1.28617e9	−	2.22771e9i	0.0643873	−	0.111522i	0.896422	+	0.443201i	$0.146157\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	3.40624e10			1.67826			0.839130	−	0.543931i	$-0.183065\pi$
$881$	3.40624e10			1.67826			0.839130	+	0.543931i	$0.183065\pi$
$882$		0			0
$883$	−5.47993e9			−0.267863			−0.133931	−	0.990991i	$-0.542760\pi$
$883$	−5.47993e9			−0.267863			−0.133931	+	0.990991i	$0.542760\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.08266e10	+	1.87522e10i	−0.520904	+	0.902233i	0.478800	+	0.877924i	$0.341072\pi$
$887$	−1.08266e10	+	1.87522e10i	−0.520904	+	0.902233i	−0.999704	+	0.0243090i	$0.992261\pi$
$888$		0			0
$889$	9.63048e7	+	1.66805e8i	0.00459718	+	0.00796255i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.18924e9	−	3.79187e9i	−0.102876	−	0.178186i
$894$		0			0
$895$	5.93260e9	−	1.02756e10i	0.276608	−	0.479098i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.80495e10			−1.74658
$900$		0			0
$901$	9.50451e9			0.432905
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.99044e9	+	5.17960e9i	−0.134111	+	0.232288i
$906$		0			0
$907$	−1.64024e10	−	2.84098e10i	−0.729931	−	1.26428i	−0.956912	−	0.290378i	$-0.906219\pi$
$907$	−1.64024e10	−	2.84098e10i	−0.729931	−	1.26428i	0.226982	−	0.973899i	$-0.427114\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	9.25174e9	+	1.60245e10i	0.405423	+	0.702214i	0.994371	−	0.105957i	$-0.0337907\pi$
$911$	9.25174e9	+	1.60245e10i	0.405423	+	0.702214i	−0.588947	+	0.808172i	$0.700457\pi$
$912$		0			0
$913$	−2.05076e10	+	3.55202e10i	−0.891800	+	1.54464i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.57796e10			−1.53230
$918$		0			0
$919$	−1.47226e10			−0.625720			−0.312860	−	0.949799i	$-0.601287\pi$
$919$	−1.47226e10			−0.625720			−0.312860	+	0.949799i	$0.601287\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.34897e9	−	4.06853e9i	0.0983267	−	0.170307i
$924$		0			0
$925$	5.47951e9	+	9.49078e9i	0.227638	+	0.394281i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.24590e9	−	3.89000e9i	−0.0919040	−	0.159182i	0.816408	−	0.577475i	$-0.195962\pi$
$929$	−2.24590e9	−	3.89000e9i	−0.0919040	−	0.159182i	−0.908312	+	0.418293i	$0.862629\pi$
$930$		0			0
$931$	2.00544e10	−	3.47353e10i	0.814492	−	1.41074i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.48471e10			0.994111
$936$		0			0
$937$	−3.45477e10			−1.37193			−0.685963	−	0.727637i	$-0.740619\pi$
$937$	−3.45477e10			−1.37193			−0.685963	+	0.727637i	$0.740619\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.06261e10	+	1.84050e10i	−0.415729	+	0.720065i	−0.995505	−	0.0947119i	$-0.969807\pi$
$941$	−1.06261e10	+	1.84050e10i	−0.415729	+	0.720065i	0.579775	+	0.814776i	$0.303140\pi$
$942$		0			0
$943$	−3.69016e9	−	6.39155e9i	−0.143303	−	0.248208i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.52119e10	+	2.63477e10i	0.582046	+	1.00813i	0.995237	+	0.0974893i	$0.0310812\pi$
$947$	1.52119e10	+	2.63477e10i	0.582046	+	1.00813i	−0.413190	+	0.910645i	$0.635585\pi$
$948$		0			0
$949$	−1.35506e10	+	2.34703e10i	−0.514666	+	0.891429i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	3.86478e10			1.44644			0.723218	−	0.690620i	$-0.242662\pi$
$953$	3.86478e10			1.44644			0.723218	+	0.690620i	$0.242662\pi$
$954$		0			0
$955$	2.04322e10			0.759108
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.76749e9	−	8.25753e9i	0.174552	−	0.302333i
$960$		0			0
$961$	−1.34967e10	−	2.33771e10i	−0.490566	−	0.849685i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.59501e9	+	2.76264e9i	0.0571371	+	0.0989643i
$966$		0			0
$967$	−1.70024e10	+	2.94491e10i	−0.604670	+	1.04732i	0.387434	+	0.921898i	$0.373362\pi$
$967$	−1.70024e10	+	2.94491e10i	−0.604670	+	1.04732i	−0.992104	+	0.125421i	$0.959972\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.08691e10			0.731538			0.365769	−	0.930706i	$-0.380806\pi$
$971$	2.08691e10			0.731538			0.365769	+	0.930706i	$0.380806\pi$
$972$		0			0
$973$	5.75382e10			2.00245
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.13334e10	+	3.69505e10i	−0.731862	+	1.26762i	0.224225	+	0.974537i	$0.428015\pi$
$977$	−2.13334e10	+	3.69505e10i	−0.731862	+	1.26762i	−0.956087	+	0.293085i	$0.905318\pi$
$978$		0			0
$979$	2.51026e10	+	4.34790e10i	0.855027	+	1.48095i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	9.99083e9	+	1.73046e10i	0.335478	+	0.581065i	0.983577	−	0.180491i	$-0.0577688\pi$
$983$	9.99083e9	+	1.73046e10i	0.335478	+	0.581065i	−0.648098	+	0.761557i	$0.724435\pi$
$984$		0			0
$985$	−1.01715e10	+	1.76176e10i	−0.339124	+	0.587380i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.86553e9			0.0941928
$990$		0			0
$991$	3.14380e10			1.02612			0.513059	−	0.858353i	$-0.328512\pi$
$991$	3.14380e10			1.02612			0.513059	+	0.858353i	$0.328512\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.45404e10	−	2.51848e10i	0.467947	−	0.810508i
$996$		0			0
$997$	−1.75392e10	−	3.03787e10i	−0.560500	−	0.970814i	−0.997453	−	0.0713299i	$-0.977276\pi$
$997$	−1.75392e10	−	3.03787e10i	−0.560500	−	0.970814i	0.436953	−	0.899484i	$-0.356058\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.8.i.b.145.2		8
3.2	odd	2		144.8.i.b.49.4		8
4.3	odd	2		54.8.c.b.37.2		8
9.2	odd	6		144.8.i.b.97.4		8
9.7	even	3	inner	432.8.i.b.289.2		8
12.11	even	2		18.8.c.b.13.1	yes	8
36.7	odd	6		54.8.c.b.19.2		8
36.11	even	6		18.8.c.b.7.1	✓	8
36.23	even	6		162.8.a.h.1.2		4
36.31	odd	6		162.8.a.i.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.8.c.b.7.1	✓	8	36.11	even	6
18.8.c.b.13.1	yes	8	12.11	even	2
54.8.c.b.19.2		8	36.7	odd	6
54.8.c.b.37.2		8	4.3	odd	2
144.8.i.b.49.4		8	3.2	odd	2
144.8.i.b.97.4		8	9.2	odd	6
162.8.a.h.1.2		4	36.23	even	6
162.8.a.i.1.3		4	36.31	odd	6
432.8.i.b.145.2		8	1.1	even	1	trivial
432.8.i.b.289.2		8	9.7	even	3	inner

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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	432.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-103.252 + 178.837i) q^{5} +(808.604 + 1400.54i) q^{7} +(-2338.01 - 4049.55i) q^{11} +(-3336.65 + 5779.25i) q^{13} +25731.9 q^{17} -22384.5 q^{19} +(-11710.7 + 20283.5i) q^{23} +(17740.6 + 30727.7i) q^{25} +(81488.4 + 141142. i) q^{29} +(-116733. + 202187. i) q^{31} -333959. q^{35} +308867. q^{37} +(-157555. + 272893. i) q^{41} +(-61173.2 - 105955. i) q^{43} +(97801.6 + 169397. i) q^{47} +(-895910. + 1.55176e6i) q^{49} +369367. q^{53} +965615. q^{55} +(560669. - 971107. i) q^{59} +(-45155.7 - 78212.0i) q^{61} +(-689031. - 1.19344e6i) q^{65} +(1.76399e6 - 3.05533e6i) q^{67} -703989. q^{71} +4.06113e6 q^{73} +(3.78105e6 - 6.54897e6i) q^{77} +(2.33651e6 + 4.04696e6i) q^{79} +(-4.38570e6 - 7.59625e6i) q^{83} +(-2.65686e6 + 4.60182e6i) q^{85} -1.07368e7 q^{89} -1.07921e7 q^{91} +(2.31123e6 - 4.00317e6i) q^{95} +(-3.72029e6 - 6.44373e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 54 q^{5} + 44 q^{7} + 2172 q^{11} - 6398 q^{13} + 51972 q^{17} - 90712 q^{19} - 2028 q^{23} - 173446 q^{25} - 283098 q^{29} + 812 q^{31} - 1492992 q^{35} + 441688 q^{37} - 610704 q^{41} + 84380 q^{43}+ \cdots - 22003112 q^{97}+O(q^{100}) \) 8 * q - 54 * q^5 + 44 * q^7 + 2172 * q^11 - 6398 * q^13 + 51972 * q^17 - 90712 * q^19 - 2028 * q^23 - 173446 * q^25 - 283098 * q^29 + 812 * q^31 - 1492992 * q^35 + 441688 * q^37 - 610704 * q^41 + 84380 * q^43 + 855708 * q^47 - 1273038 * q^49 + 1842600 * q^53 - 7016760 * q^55 + 380796 * q^59 - 3151130 * q^61 + 326538 * q^65 + 9961580 * q^67 + 8970384 * q^71 + 29234932 * q^73 + 5954826 * q^77 + 17309396 * q^79 - 11520192 * q^83 - 14675148 * q^85 - 26066064 * q^89 - 26231936 * q^91 - 31016952 * q^95 - 22003112 * q^97

Introduction

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