LMFDB - Embedded newform 252.9.bk.a.53.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(53,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("252.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.bk (of order $6$, degree $2$, 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:	$102.659409735$
Analytic rank:	$0$
Dimension:	$44$
Relative dimension:	$22$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.18
Character	$\chi$	$=$	252.53
Dual form			252.9.bk.a.233.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(674.478 - 389.410i) q^{5} +(-1634.98 - 1758.31i) q^{7} +O(q^{10})$	$q+(674.478 - 389.410i) q^{5} +(-1634.98 - 1758.31i) q^{7} +(-291.697 - 168.411i) q^{11} +46.7231 q^{13} +(-98146.7 - 56665.0i) q^{17} +(41181.7 + 71328.8i) q^{19} +(-326470. + 188487. i) q^{23} +(107968. - 187006. i) q^{25} -747068. i q^{29} +(-774022. + 1.34064e6i) q^{31} +(-1.78746e6 - 549266. i) q^{35} +(475345. + 823322. i) q^{37} +3.98956e6i q^{41} -178869. q^{43} +(4.60967e6 - 2.66139e6i) q^{47} +(-418507. + 5.74959e6i) q^{49} +(708441. + 409018. i) q^{53} -262325. q^{55} +(8.19042e6 + 4.72874e6i) q^{59} +(-1.08168e7 - 1.87353e7i) q^{61} +(31513.7 - 18194.5i) q^{65} +(7.78342e6 - 1.34813e7i) q^{67} +7.95425e6i q^{71} +(1.59210e7 - 2.75759e7i) q^{73} +(180798. + 788243. i) q^{77} +(2.44265e7 + 4.23079e7i) q^{79} +9.26637e6i q^{83} -8.82638e7 q^{85} +(-3.56105e7 + 2.05597e7i) q^{89} +(-76391.2 - 82153.7i) q^{91} +(5.55524e7 + 3.20732e7i) q^{95} -6.33379e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + 1230 q^{7}+O(q^{10}) $ 44 * q + 1230 * q^7	$ 44 q + 1230 q^{7} - 101380 q^{13} + 62770 q^{19} + 2247666 q^{25} + 1389254 q^{31} - 2136026 q^{37} + 11510140 q^{43} - 3824398 q^{49} - 42646528 q^{55} + 27346232 q^{61} + 14239194 q^{67} - 64344138 q^{73} + 7061786 q^{79} - 54198208 q^{85} - 45697066 q^{91} + 476543496 q^{97}+O(q^{100}) $ 44 * q + 1230 * q^7 - 101380 * q^13 + 62770 * q^19 + 2247666 * q^25 + 1389254 * q^31 - 2136026 * q^37 + 11510140 * q^43 - 3824398 * q^49 - 42646528 * q^55 + 27346232 * q^61 + 14239194 * q^67 - 64344138 * q^73 + 7061786 * q^79 - 54198208 * q^85 - 45697066 * q^91 + 476543496 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	674.478	−	389.410i	1.07917	−	0.623056i	0.148494	−	0.988913i	$-0.452557\pi$
$5$	674.478	−	389.410i	1.07917	−	0.623056i	0.930671	+	0.365857i	$0.119224\pi$
$6$		0			0
$7$	−1634.98	−	1758.31i	−0.680956	−	0.732324i
$7$	−1634.98	−	1758.31i	−0.680956	−	0.732324i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−291.697	−	168.411i	−0.0199233	−	0.0115027i	0.490005	−	0.871719i	$-0.336995\pi$
$11$	−291.697	−	168.411i	−0.0199233	−	0.0115027i	−0.509929	+	0.860217i	$0.670328\pi$
$12$		0			0
$13$	46.7231			0.00163591			0.000817953	−	1.00000i	$-0.499740\pi$
$13$	46.7231			0.00163591			0.000817953		1.00000i	$0.499740\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−98146.7	−	56665.0i	−1.17511	−	0.678453i	−0.220235	−	0.975447i	$-0.570682\pi$
$17$	−98146.7	−	56665.0i	−1.17511	−	0.678453i	−0.954879	+	0.296994i	$0.904016\pi$
$18$		0			0
$19$	41181.7	+	71328.8i	0.316002	+	0.547332i	0.979650	−	0.200713i	$-0.0643259\pi$
$19$	41181.7	+	71328.8i	0.316002	+	0.547332i	−0.663648	+	0.748045i	$0.730993\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−326470.	+	188487.i	−1.16663	+	0.673552i	−0.952883	−	0.303338i	$-0.901899\pi$
$23$	−326470.	+	188487.i	−1.16663	+	0.673552i	−0.213743	+	0.976890i	$0.568566\pi$
$24$		0			0
$25$	107968.	−	187006.i	0.276399	−	0.478736i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	747068.i		−	1.05625i	−0.849165	−	0.528127i	$-0.822894\pi$
$29$		−	747068.i		−	1.05625i	0.849165	−	0.528127i	$-0.177106\pi$
$30$		0			0
$31$	−774022.	+	1.34064e6i	−0.838120	+	1.45167i	0.0533442	+	0.998576i	$0.483012\pi$
$31$	−774022.	+	1.34064e6i	−0.838120	+	1.45167i	−0.891464	+	0.453091i	$0.850321\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.78746e6	−	549266.i	−1.19114	−	0.366024i
$36$		0			0
$37$	475345.	+	823322.i	0.253631	+	0.439302i	0.964523	−	0.264000i	$-0.0850418\pi$
$37$	475345.	+	823322.i	0.253631	+	0.439302i	−0.710892	+	0.703301i	$0.751708\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			3.98956e6i			1.41185i	0.708285	+	0.705927i	$0.249469\pi$
$41$			3.98956e6i			1.41185i	−0.708285	+	0.705927i	$0.750531\pi$
$42$		0			0
$43$	−178869.			−0.0523193			−0.0261596	−	0.999658i	$-0.508328\pi$
$43$	−178869.			−0.0523193			−0.0261596	+	0.999658i	$0.508328\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.60967e6	−	2.66139e6i	0.944666	−	0.545403i	0.0532463	−	0.998581i	$-0.483043\pi$
$47$	4.60967e6	−	2.66139e6i	0.944666	−	0.545403i	0.891420	+	0.453178i	$0.149710\pi$
$48$		0			0
$49$	−418507.	+	5.74959e6i	−0.0725969	+	0.997361i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	708441.	+	409018.i	0.0897842	+	0.0518369i	0.544220	−	0.838943i	$-0.316826\pi$
$53$	708441.	+	409018.i	0.0897842	+	0.0518369i	−0.454436	+	0.890780i	$0.650159\pi$
$54$		0			0
$55$	−262325.			−0.0286674
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.19042e6	+	4.72874e6i	0.675925	+	0.390245i	0.798318	−	0.602236i	$-0.205724\pi$
$59$	8.19042e6	+	4.72874e6i	0.675925	+	0.390245i	−0.122393	+	0.992482i	$0.539057\pi$
$60$		0			0
$61$	−1.08168e7	−	1.87353e7i	−0.781234	−	1.35314i	−0.931223	−	0.364450i	$-0.881257\pi$
$61$	−1.08168e7	−	1.87353e7i	−0.781234	−	1.35314i	0.149989	−	0.988688i	$-0.452076\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	31513.7	−	18194.5i	0.00176541	−	0.00101926i
$66$		0			0
$67$	7.78342e6	−	1.34813e7i	0.386252	−	0.669009i	−0.605690	−	0.795701i	$-0.707103\pi$
$67$	7.78342e6	−	1.34813e7i	0.386252	−	0.669009i	0.991942	+	0.126692i	$0.0404361\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.95425e6i			0.313015i	0.987677	+	0.156508i	$0.0500236\pi$
$71$			7.95425e6i			0.313015i	−0.987677	+	0.156508i	$0.949976\pi$
$72$		0			0
$73$	1.59210e7	−	2.75759e7i	0.560632	−	0.971043i	−0.436810	−	0.899554i	$-0.643892\pi$
$73$	1.59210e7	−	2.75759e7i	0.560632	−	0.971043i	0.997441	−	0.0714887i	$-0.0227750\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	180798.	+	788243.i	0.00514318	+	0.0224232i
$78$		0			0
$79$	2.44265e7	+	4.23079e7i	0.627122	+	1.08621i	0.988126	+	0.153643i	$0.0491007\pi$
$79$	2.44265e7	+	4.23079e7i	0.627122	+	1.08621i	−0.361004	+	0.932564i	$0.617566\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			9.26637e6i			0.195253i	0.995223	+	0.0976264i	$0.0311250\pi$
$83$			9.26637e6i			0.195253i	−0.995223	+	0.0976264i	$0.968875\pi$
$84$		0			0
$85$	−8.82638e7			−1.69086
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.56105e7	+	2.05597e7i	−0.567569	+	0.327686i	−0.756178	−	0.654366i	$-0.772935\pi$
$89$	−3.56105e7	+	2.05597e7i	−0.567569	+	0.327686i	0.188609	+	0.982052i	$0.439602\pi$
$90$		0			0
$91$	−76391.2	−	82153.7i	−0.00111398	−	0.00119801i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.55524e7	+	3.20732e7i	0.682037	+	0.393774i
$96$		0			0
$97$	−6.33379e7			−0.715446			−0.357723	−	0.933828i	$-0.616447\pi$
$97$	−6.33379e7			−0.715446			−0.357723	+	0.933828i	$0.616447\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.64407e7	+	3.83595e7i	0.638482	+	0.368628i	0.784030	−	0.620724i	$-0.213161\pi$
$101$	6.64407e7	+	3.83595e7i	0.638482	+	0.368628i	−0.145548	+	0.989351i	$0.546494\pi$
$102$		0			0
$103$	6.07200e7	+	1.05170e8i	0.539489	+	0.934423i	0.998932	+	0.0462153i	$0.0147160\pi$
$103$	6.07200e7	+	1.05170e8i	0.539489	+	0.934423i	−0.459442	+	0.888208i	$0.651951\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.07115e7	+	4.65988e7i	−0.615744	+	0.355500i	−0.775210	−	0.631703i	$-0.782356\pi$
$107$	−8.07115e7	+	4.65988e7i	−0.615744	+	0.355500i	0.159466	+	0.987203i	$0.449023\pi$
$108$		0			0
$109$	−1.00369e8	+	1.73845e8i	−0.711042	+	1.23156i	0.253425	+	0.967355i	$0.418443\pi$
$109$	−1.00369e8	+	1.73845e8i	−0.711042	+	1.23156i	−0.964466	+	0.264205i	$0.914890\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			6.47043e7i			0.396844i	0.980117	+	0.198422i	$0.0635816\pi$
$113$			6.47043e7i			0.396844i	−0.980117	+	0.198422i	$0.936418\pi$
$114$		0			0
$115$	−1.46798e8	+	2.54261e8i	−0.839322	+	1.45375i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.08329e7	+	2.65218e8i	0.303354	+	1.32256i
$120$		0			0
$121$	−1.07123e8	−	1.85542e8i	−0.499735	−	0.865567i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			1.36051e8i			0.557265i
$126$		0			0
$127$	9.72175e7			0.373706			0.186853	−	0.982388i	$-0.440171\pi$
$127$	9.72175e7			0.373706			0.186853	+	0.982388i	$0.440171\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.80952e7	−	4.50883e7i	0.265179	−	0.153101i	−0.361516	−	0.932366i	$-0.617741\pi$
$131$	7.80952e7	−	4.50883e7i	0.265179	−	0.153101i	0.626695	+	0.779265i	$0.284407\pi$
$132$		0			0
$133$	5.80871e7	−	1.89031e8i	0.185641	−	0.604125i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.80714e8	+	2.19806e8i	1.08073	+	0.623959i	0.931093	−	0.364782i	$-0.118856\pi$
$137$	3.80714e8	+	2.19806e8i	1.08073	+	0.623959i	0.149636	+	0.988741i	$0.452190\pi$
$138$		0			0
$139$	−5.94059e8			−1.59137			−0.795683	−	0.605713i	$-0.792888\pi$
$139$	−5.94059e8			−1.59137			−0.795683	+	0.605713i	$0.792888\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−13629.0	−	7868.71i	−3.25927e−5	−	1.88174e-5i
$144$		0			0
$145$	−2.90916e8	−	5.03881e8i	−0.658106	−	1.13987i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.93167e8	+	1.11525e8i	−0.391912	+	0.226270i	−0.682988	−	0.730429i	$-0.739320\pi$
$149$	−1.93167e8	+	1.11525e8i	−0.391912	+	0.226270i	0.291076	+	0.956700i	$0.405987\pi$
$150$		0			0
$151$	2.98322e8	−	5.16710e8i	0.573823	−	0.993891i	−0.422345	−	0.906435i	$-0.638793\pi$
$151$	2.98322e8	−	5.16710e8i	0.573823	−	0.993891i	0.996168	−	0.0874559i	$-0.0278737\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			1.20565e9i			2.08878i
$156$		0			0
$157$	−3.23899e8	+	5.61010e8i	−0.533103	+	0.923362i	0.466149	+	0.884706i	$0.345641\pi$
$157$	−3.23899e8	+	5.61010e8i	−0.533103	+	0.923362i	−0.999253	+	0.0386560i	$0.987692\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.65190e8	+	2.65863e8i	1.28768	+	0.395689i
$162$		0			0
$163$	4.84999e8	+	8.40043e8i	0.687054	+	1.19001i	0.972787	+	0.231702i	$0.0744296\pi$
$163$	4.84999e8	+	8.40043e8i	0.687054	+	1.19001i	−0.285733	+	0.958309i	$0.592237\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			5.10131e8i			0.655868i	0.944701	+	0.327934i	$0.106352\pi$
$167$			5.10131e8i			0.655868i	−0.944701	+	0.327934i	$0.893648\pi$
$168$		0			0
$169$	−8.15729e8			−0.999997
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.78976e7	−	1.03332e7i	0.0199807	−	0.0115358i	−0.489976	−	0.871736i	$-0.662995\pi$
$173$	1.78976e7	−	1.03332e7i	0.0199807	−	0.0115358i	0.509957	+	0.860200i	$0.329661\pi$
$174$		0			0
$175$	−5.05341e8	+	1.15909e8i	−0.538805	+	0.123585i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.33160e9	+	7.68801e8i	1.29707	+	0.748862i	0.979897	−	0.199507i	$-0.0639339\pi$
$179$	1.33160e9	+	7.68801e8i	1.29707	+	0.748862i	0.317171	+	0.948369i	$0.397267\pi$
$180$		0			0
$181$	−1.58836e9			−1.47991			−0.739954	−	0.672658i	$-0.765153\pi$
$181$	−1.58836e9			−1.47991			−0.739954	+	0.672658i	$0.765153\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.41220e8	+	3.70209e8i	0.547420	+	0.316053i
$186$		0			0
$187$	1.90861e7	+	3.30581e7i	0.0156081	+	0.0270340i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.16068e9	+	6.70118e8i	−0.872125	+	0.503521i	−0.868054	−	0.496470i	$-0.834629\pi$
$191$	−1.16068e9	+	6.70118e8i	−0.872125	+	0.503521i	−0.00407097	+	0.999992i	$0.501296\pi$
$192$		0			0
$193$	−6.28612e8	+	1.08879e9i	−0.453058	+	0.784719i	−0.998574	−	0.0533809i	$-0.983000\pi$
$193$	−6.28612e8	+	1.08879e9i	−0.453058	+	0.784719i	0.545516	+	0.838100i	$0.316334\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.61383e8i			0.306335i	0.988200	+	0.153167i	$0.0489474\pi$
$197$			4.61383e8i			0.306335i	−0.988200	+	0.153167i	$0.951053\pi$
$198$		0			0
$199$	−5.92807e8	+	1.02677e9i	−0.378008	+	0.654729i	−0.990772	−	0.135537i	$-0.956724\pi$
$199$	−5.92807e8	+	1.02677e9i	−0.378008	+	0.654729i	0.612764	+	0.790266i	$0.290057\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.31358e9	+	1.22144e9i	−0.773520	+	0.719263i
$204$		0			0
$205$	1.55358e9	+	2.69087e9i	0.879665	+	1.52362i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	2.77419e7i		−	0.0145396i
$210$		0			0
$211$	−2.99372e9			−1.51037			−0.755183	−	0.655514i	$-0.772452\pi$
$211$	−2.99372e9			−1.51037			−0.755183	+	0.655514i	$0.772452\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.20643e8	+	6.96535e7i	−0.0564611	+	0.0325979i
$216$		0			0
$217$	3.62278e9	−	8.30952e8i	1.63381	−	0.374746i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.58572e6	−	2.64757e6i	−0.00192238	−	0.00110988i
$222$		0			0
$223$	−2.79107e9			−1.12863			−0.564314	−	0.825560i	$-0.690859\pi$
$223$	−2.79107e9			−1.12863			−0.564314	+	0.825560i	$0.690859\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.92916e9	+	1.11380e9i	0.726549	+	0.419473i	0.817158	−	0.576413i	$-0.195548\pi$
$227$	1.92916e9	+	1.11380e9i	0.726549	+	0.419473i	−0.0906092	+	0.995887i	$0.528881\pi$
$228$		0			0
$229$	−1.29738e9	−	2.24713e9i	−0.471765	−	0.817122i	0.527713	−	0.849423i	$-0.323050\pi$
$229$	−1.29738e9	−	2.24713e9i	−0.471765	−	0.817122i	−0.999478	+	0.0323013i	$0.989716\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.86548e9	+	1.65438e9i	−0.972239	+	0.561322i	−0.899918	−	0.436059i	$-0.856374\pi$
$233$	−2.86548e9	+	1.65438e9i	−0.972239	+	0.561322i	−0.0723208	+	0.997381i	$0.523041\pi$
$234$		0			0
$235$	2.07275e9	−	3.59011e9i	0.679634	−	1.17716i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	5.57435e9i		−	1.70845i	−0.519902	−	0.854226i	$-0.674032\pi$
$239$		−	5.57435e9i		−	1.70845i	0.519902	−	0.854226i	$-0.325968\pi$
$240$		0			0
$241$	4.64155e7	−	8.03940e7i	0.0137592	−	0.0238317i	−0.859064	−	0.511869i	$-0.828953\pi$
$241$	4.64155e7	−	8.03940e7i	0.0137592	−	0.0238317i	0.872823	+	0.488037i	$0.162287\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.95668e9	+	4.04094e9i	0.543068	+	1.12155i
$246$		0			0
$247$	1.92414e6	+	3.33271e6i	0.000516950	+	0.000895384i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	2.50678e9i		−	0.631571i	−0.948831	−	0.315785i	$-0.897732\pi$
$251$		−	2.50678e9i		−	0.631571i	0.948831	−	0.315785i	$-0.102268\pi$
$252$		0			0
$253$	1.26974e8			0.0309907
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.74846e9	−	1.58683e9i	0.630024	−	0.363745i	−0.150737	−	0.988574i	$-0.548165\pi$
$257$	2.74846e9	−	1.58683e9i	0.630024	−	0.363745i	0.780762	+	0.624829i	$0.214831\pi$
$258$		0			0
$259$	6.70478e8	−	2.18192e9i	0.149000	−	0.484885i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.06026e9	+	3.49889e9i	1.26668	+	0.731320i	0.974359	−	0.224999i	$-0.0722378\pi$
$263$	6.06026e9	+	3.49889e9i	1.26668	+	0.731320i	0.292325	+	0.956319i	$0.405571\pi$
$264$		0			0
$265$	6.37104e8			0.129189
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.42291e9	−	3.70827e9i	−1.22666	−	0.708210i	−0.260327	−	0.965521i	$-0.583830\pi$
$269$	−6.42291e9	−	3.70827e9i	−1.22666	−	0.708210i	−0.966329	+	0.257311i	$0.917164\pi$
$270$		0			0
$271$	9.39114e8	+	1.62659e9i	0.174117	+	0.301580i	0.939855	−	0.341573i	$-0.110960\pi$
$271$	9.39114e8	+	1.62659e9i	0.174117	+	0.301580i	−0.765738	+	0.643152i	$0.777626\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.29880e7	+	3.63662e7i	−0.0110135	+	0.00635867i
$276$		0			0
$277$	2.49984e9	−	4.32985e9i	0.424613	−	0.735450i	−0.571772	−	0.820413i	$-0.693744\pi$
$277$	2.49984e9	−	4.32985e9i	0.424613	−	0.735450i	0.996384	+	0.0849624i	$0.0270770\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			7.63815e9i			1.22508i	0.790441	+	0.612538i	$0.209851\pi$
$281$			7.63815e9i			1.22508i	−0.790441	+	0.612538i	$0.790149\pi$
$282$		0			0
$283$	−1.13228e9	+	1.96116e9i	−0.176525	+	0.305751i	−0.940688	−	0.339273i	$-0.889819\pi$
$283$	−1.13228e9	+	1.96116e9i	−0.176525	+	0.305751i	0.764163	+	0.645023i	$0.223152\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	7.01489e9	−	6.52284e9i	1.03393	−	0.961411i
$288$		0			0
$289$	2.93398e9	+	5.08180e9i	0.420596	+	0.728494i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	1.08056e10i		−	1.46615i	−0.680148	−	0.733075i	$-0.738084\pi$
$293$		−	1.08056e10i		−	1.46615i	0.680148	−	0.733075i	$-0.261916\pi$
$294$		0			0
$295$	7.36568e9			0.972579
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.52537e7	+	8.80672e6i	−0.00190849	+	0.00110187i
$300$		0			0
$301$	2.92447e8	+	3.14507e8i	0.0356271	+	0.0383147i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.45915e10	−	8.42438e9i	−1.68616	−	0.973506i
$306$		0			0
$307$	−1.45717e9			−0.164043			−0.0820213	−	0.996631i	$-0.526138\pi$
$307$	−1.45717e9			−0.164043			−0.0820213	+	0.996631i	$0.526138\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.11591e10	−	6.44270e9i	−1.19285	−	0.688694i	−0.233901	−	0.972260i	$-0.575149\pi$
$311$	−1.11591e10	−	6.44270e9i	−1.19285	−	0.688694i	−0.958953	+	0.283566i	$0.908482\pi$
$312$		0			0
$313$	−1.96042e9	−	3.39554e9i	−0.204254	−	0.353779i	0.745641	−	0.666348i	$-0.232144\pi$
$313$	−1.96042e9	−	3.39554e9i	−0.204254	−	0.353779i	−0.949895	+	0.312570i	$0.898810\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.36309e9	+	2.51903e9i	−0.432073	+	0.249458i	−0.700229	−	0.713918i	$-0.746919\pi$
$317$	−4.36309e9	+	2.51903e9i	−0.432073	+	0.249458i	0.268156	+	0.963375i	$0.413586\pi$
$318$		0			0
$319$	−1.25815e8	+	2.17918e8i	−0.0121498	+	0.0210441i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	9.33426e9i		−	0.857570i
$324$		0			0
$325$	5.04461e6	−	8.73752e6i	0.000452162	−	0.000783167i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.22163e10	−	3.75391e9i	−1.04269	−	0.320406i
$330$		0			0
$331$	3.79211e9	+	6.56813e9i	0.315914	+	0.547180i	0.979631	−	0.200804i	$-0.0643554\pi$
$331$	3.79211e9	+	6.56813e9i	0.315914	+	0.547180i	−0.663717	+	0.747984i	$0.731022\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	1.21238e10i		−	0.962628i
$336$		0			0
$337$	2.10409e10			1.63134			0.815670	−	0.578517i	$-0.196368\pi$
$337$	2.10409e10			1.63134			0.815670	+	0.578517i	$0.196368\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.51560e8	−	2.60708e8i	0.0333963	−	0.0192813i
$342$		0			0
$343$	1.07938e10	−	8.66458e9i	0.779827	−	0.625995i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.13881e9	−	1.81219e9i	−0.216495	−	0.124993i	0.387832	−	0.921730i	$-0.373224\pi$
$347$	−3.13881e9	−	1.81219e9i	−0.216495	−	0.124993i	−0.604326	+	0.796737i	$0.706558\pi$
$348$		0			0
$349$	4.00609e9			0.270034			0.135017	−	0.990843i	$-0.456891\pi$
$349$	4.00609e9			0.270034			0.135017	+	0.990843i	$0.456891\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.38535e9	+	7.99830e8i	0.0892194	+	0.0515108i	0.543946	−	0.839120i	$-0.316930\pi$
$353$	1.38535e9	+	7.99830e8i	0.0892194	+	0.0515108i	−0.454727	+	0.890631i	$0.650263\pi$
$354$		0			0
$355$	3.09747e9	+	5.36497e9i	0.195026	+	0.337795i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4.59519e9	−	2.65303e9i	0.276647	−	0.159722i	−0.355258	−	0.934768i	$-0.615607\pi$
$359$	4.59519e9	−	2.65303e9i	0.276647	−	0.159722i	0.631904	+	0.775046i	$0.282274\pi$
$360$		0			0
$361$	5.09991e9	−	8.83331e9i	0.300285	−	0.520109i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	2.47991e10i		−	1.39722i
$366$		0			0
$367$	1.07704e9	−	1.86549e9i	0.0593700	−	0.102832i	−0.834813	−	0.550534i	$-0.814424\pi$
$367$	1.07704e9	−	1.86549e9i	0.0593700	−	0.102832i	0.894183	+	0.447702i	$0.147757\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.39102e8	−	1.91439e9i	−0.0231777	−	0.101050i
$372$		0			0
$373$	−1.54727e10	−	2.67995e10i	−0.799338	−	1.38449i	−0.920048	−	0.391806i	$-0.871850\pi$
$373$	−1.54727e10	−	2.67995e10i	−0.799338	−	1.38449i	0.120710	−	0.992688i	$-0.461483\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	3.49054e7i		−	0.00172793i
$378$		0			0
$379$	−2.48874e10			−1.20621			−0.603105	−	0.797661i	$-0.706070\pi$
$379$	−2.48874e10			−1.20621			−0.603105	+	0.797661i	$0.706070\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5.30297e8	+	3.06167e8i	−0.0246447	+	0.0142286i	−0.512272	−	0.858823i	$-0.671196\pi$
$383$	−5.30297e8	+	3.06167e8i	−0.0246447	+	0.0142286i	0.487627	+	0.873052i	$0.337863\pi$
$384$		0			0
$385$	4.28894e8	+	4.61248e8i	0.0195212	+	0.0209938i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.98599e9	+	1.14661e9i	0.0867320	+	0.0500747i	0.542739	−	0.839902i	$-0.317387\pi$
$389$	1.98599e9	+	1.14661e9i	0.0867320	+	0.0500747i	−0.456007	+	0.889976i	$0.650721\pi$
$390$		0			0
$391$	4.27226e10			1.82789
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.29502e10	+	1.90238e10i	1.35354	+	0.781465i
$396$		0			0
$397$	3.26321e9	+	5.65204e9i	0.131366	+	0.227532i	0.924203	−	0.381901i	$-0.124730\pi$
$397$	3.26321e9	+	5.65204e9i	0.131366	+	0.227532i	−0.792837	+	0.609433i	$0.791397\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.22563e10	−	1.86232e10i	1.24749	−	0.720238i	0.276881	−	0.960904i	$-0.410699\pi$
$401$	3.22563e10	−	1.86232e10i	1.24749	−	0.720238i	0.970608	+	0.240666i	$0.0773658\pi$
$402$		0			0
$403$	−3.61647e7	+	6.26391e7i	−0.00137109	+	0.00237479i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	3.20214e8i		−	0.0116698i
$408$		0			0
$409$	2.13828e10	−	3.70361e10i	0.764138	−	1.32353i	−0.176563	−	0.984289i	$-0.556498\pi$
$409$	2.13828e10	−	3.70361e10i	0.764138	−	1.32353i	0.940701	−	0.339237i	$-0.110169\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5.07655e9	−	2.21327e10i	−0.174489	−	0.760736i
$414$		0			0
$415$	3.60842e9	+	6.24997e9i	0.121653	+	0.210710i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			3.93333e10i			1.27616i	0.769971	+	0.638079i	$0.220271\pi$
$419$			3.93333e10i			1.27616i	−0.769971	+	0.638079i	$0.779729\pi$
$420$		0			0
$421$	−2.12925e10			−0.677795			−0.338898	−	0.940823i	$-0.610054\pi$
$421$	−2.12925e10			−0.677795			−0.338898	+	0.940823i	$0.610054\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.11934e10	+	1.22360e10i	−0.649600	+	0.375047i
$426$		0			0
$427$	−1.52572e10	+	4.96512e10i	−0.458949	+	1.49354i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3.33185e10	−	1.92364e10i	−0.965553	−	0.557462i	−0.0676756	−	0.997707i	$-0.521558\pi$
$431$	−3.33185e10	−	1.92364e10i	−0.965553	−	0.557462i	−0.897878	+	0.440245i	$0.854892\pi$
$432$		0			0
$433$	−3.32573e9			−0.0946097			−0.0473048	−	0.998880i	$-0.515063\pi$
$433$	−3.32573e9			−0.0946097			−0.0473048	+	0.998880i	$0.515063\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.68892e10	−	1.55245e10i	−0.737313	−	0.425688i
$438$		0			0
$439$	5.39336e8	+	9.34157e8i	0.0145212	+	0.0251514i	0.873195	−	0.487372i	$-0.162044\pi$
$439$	5.39336e8	+	9.34157e8i	0.0145212	+	0.0251514i	−0.858674	+	0.512523i	$0.828711\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.97276e10	+	1.71632e10i	−0.771870	+	0.445639i	−0.833541	−	0.552457i	$-0.813690\pi$
$443$	−2.97276e10	+	1.71632e10i	−0.771870	+	0.445639i	0.0616712	+	0.998097i	$0.480357\pi$
$444$		0			0
$445$	−1.60124e10	+	2.77342e10i	−0.408334	+	0.707254i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			3.67492e10i			0.904195i	0.891969	+	0.452098i	$0.149324\pi$
$449$			3.67492e10i			0.904195i	−0.891969	+	0.452098i	$0.850676\pi$
$450$		0			0
$451$	6.71888e8	−	1.16374e9i	0.0162402	−	0.0281288i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.35157e7	−	2.56634e7i	−0.00194860	−	0.000598782i
$456$		0			0
$457$	−2.21688e10	−	3.83974e10i	−0.508249	−	0.880314i	−0.999954	−	0.00955198i	$-0.996959\pi$
$457$	−2.21688e10	−	3.83974e10i	−0.508249	−	0.880314i	0.491705	−	0.870762i	$-0.336374\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	2.39093e10i		−	0.529374i	−0.964334	−	0.264687i	$-0.914731\pi$
$461$		−	2.39093e10i		−	0.529374i	0.964334	−	0.264687i	$-0.0852686\pi$
$462$		0			0
$463$	1.59365e10			0.346793			0.173396	−	0.984852i	$-0.444526\pi$
$463$	1.59365e10			0.346793			0.173396	+	0.984852i	$0.444526\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.63380e10	−	2.09797e10i	0.764000	−	0.441095i	−0.0667302	−	0.997771i	$-0.521257\pi$
$467$	3.63380e10	−	2.09797e10i	0.764000	−	0.441095i	0.830730	+	0.556676i	$0.187923\pi$
$468$		0			0
$469$	−3.64300e10	+	8.35590e9i	−0.752952	+	0.172704i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5.21756e7	+	3.01236e7i	0.00104237	+	0.000601814i
$474$		0			0
$475$	1.77853e10			0.349370
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.17379e10	+	2.98709e10i	0.982803	+	0.567422i	0.903115	−	0.429398i	$-0.141274\pi$
$479$	5.17379e10	+	2.98709e10i	0.982803	+	0.567422i	0.0796879	+	0.996820i	$0.474608\pi$
$480$		0			0
$481$	2.22096e7	+	3.84682e7i	0.000414917	+	0.000718656i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.27201e10	+	2.46644e10i	−0.772085	+	0.445763i
$486$		0			0
$487$	−5.18299e10	+	8.97720e10i	−0.921434	+	1.59597i	−0.124237	+	0.992253i	$0.539648\pi$
$487$	−5.18299e10	+	8.97720e10i	−0.921434	+	1.59597i	−0.797197	+	0.603719i	$0.793685\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			1.00236e11i			1.72464i	0.506365	+	0.862319i	$0.330989\pi$
$491$			1.00236e11i			1.72464i	−0.506365	+	0.862319i	$0.669011\pi$
$492$		0			0
$493$	−4.23327e10	+	7.33223e10i	−0.716618	+	1.24122i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.39860e10	−	1.30050e10i	0.229229	−	0.213150i
$498$		0			0
$499$	−2.28401e10	−	3.95601e10i	−0.368379	−	0.638051i	0.620933	−	0.783864i	$-0.286754\pi$
$499$	−2.28401e10	−	3.95601e10i	−0.368379	−	0.638051i	−0.989312	+	0.145812i	$0.953421\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			6.46334e10i			1.00968i	0.863212	+	0.504842i	$0.168449\pi$
$503$			6.46334e10i			1.00968i	−0.863212	+	0.504842i	$0.831551\pi$
$504$		0			0
$505$	5.97504e10			0.918703
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8.11934e10	+	4.68770e10i	−1.20962	+	0.698375i	−0.962677	−	0.270654i	$-0.912760\pi$
$509$	−8.11934e10	+	4.68770e10i	−1.20962	+	0.698375i	−0.246945	+	0.969029i	$0.579427\pi$
$510$		0			0
$511$	−7.45174e10	+	1.70920e10i	−1.09288	+	0.250674i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.19086e10	+	4.72900e10i	1.16440	+	0.672265i
$516$		0			0
$517$	−1.79284e9			−0.0250945
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8.63045e10	+	4.98279e10i	1.17134	+	0.676272i	0.953995	−	0.299823i	$-0.0969275\pi$
$521$	8.63045e10	+	4.98279e10i	1.17134	+	0.676272i	0.217343	+	0.976095i	$0.430261\pi$
$522$		0			0
$523$	3.25436e9	+	5.63672e9i	0.0434969	+	0.0753389i	0.886954	−	0.461857i	$-0.152817\pi$
$523$	3.25436e9	+	5.63672e9i	0.0434969	+	0.0753389i	−0.843457	+	0.537196i	$0.819483\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.51935e11	−	8.77199e10i	1.96977	−	1.13725i
$528$		0			0
$529$	3.18995e10	−	5.52516e10i	0.407344	−	0.705541i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			1.86405e8i			0.00230966i
$534$		0			0
$535$	−3.62921e10	+	6.28598e10i	−0.442993	+	0.767287i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.09037e9	−	1.60666e9i	0.0129187	−	0.0190357i
$540$		0			0
$541$	−8.36034e10	−	1.44805e11i	−0.975966	−	1.69042i	−0.676710	−	0.736250i	$-0.736595\pi$
$541$	−8.36034e10	−	1.44805e11i	−0.975966	−	1.69042i	−0.299256	−	0.954173i	$-0.596738\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			1.56339e11i			1.77208i
$546$		0			0
$547$	−8.53766e10			−0.953651			−0.476826	−	0.878998i	$-0.658213\pi$
$547$	−8.53766e10			−0.953651			−0.476826	+	0.878998i	$0.658213\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.32875e10	−	3.07656e10i	0.578122	−	0.333779i
$552$		0			0
$553$	3.44537e10	−	1.12122e11i	0.368413	−	1.19892i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.84279e10	−	1.64129e10i	−0.295341	−	0.170515i	0.345007	−	0.938600i	$-0.387877\pi$
$557$	−2.84279e10	−	1.64129e10i	−0.295341	−	0.170515i	−0.640348	+	0.768085i	$0.721210\pi$
$558$		0			0
$559$	−8.35732e6			−8.55894e−5
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.16276e11	+	6.71317e10i	1.15732	+	0.668181i	0.950661	−	0.310231i	$-0.100406\pi$
$563$	1.16276e11	+	6.71317e10i	1.15732	+	0.668181i	0.206663	+	0.978412i	$0.433740\pi$
$564$		0			0
$565$	2.51965e10	+	4.36416e10i	0.247256	+	0.428260i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.23300e11	+	7.11873e10i	−1.17629	+	0.679131i	−0.955153	−	0.296112i	$-0.904310\pi$
$569$	−1.23300e11	+	7.11873e10i	−1.17629	+	0.679131i	−0.221136	+	0.975243i	$0.570977\pi$
$570$		0			0
$571$	−3.17814e10	+	5.50470e10i	−0.298971	+	0.517833i	−0.975901	−	0.218215i	$-0.929977\pi$
$571$	−3.17814e10	+	5.50470e10i	−0.298971	+	0.517833i	0.676930	+	0.736047i	$0.263310\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			8.14026e10i			0.744675i
$576$		0			0
$577$	9.97886e8	−	1.72839e9i	0.00900281	−	0.0155933i	−0.861489	−	0.507776i	$-0.830468\pi$
$577$	9.97886e8	−	1.72839e9i	0.00900281	−	0.0155933i	0.870492	+	0.492183i	$0.163801\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.62931e10	−	1.51503e10i	0.142988	−	0.132959i
$582$		0			0
$583$	−1.37767e8	−	2.38619e8i	−0.00119253	−	0.00206553i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1.66874e11i		−	1.40551i	−0.711430	−	0.702757i	$-0.751952\pi$
$587$		−	1.66874e11i		−	1.40551i	0.711430	−	0.702757i	$-0.248048\pi$
$588$		0			0
$589$	−1.27502e11			−1.05939
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.57012e11	+	9.06509e10i	−1.26974	+	0.733084i	−0.974938	−	0.222475i	$-0.928586\pi$
$593$	−1.57012e11	+	9.06509e10i	−1.26974	+	0.733084i	−0.294800	+	0.955559i	$0.595253\pi$
$594$		0			0
$595$	1.44309e11	+	1.55195e11i	1.15140	+	1.23826i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	4.12369e9	+	2.38081e9i	0.0320316	+	0.0184935i	0.515930	−	0.856631i	$-0.327446\pi$
$599$	4.12369e9	+	2.38081e9i	0.0320316	+	0.0184935i	−0.483899	+	0.875124i	$0.660780\pi$
$600$		0			0
$601$	5.35010e10			0.410075			0.205038	−	0.978754i	$-0.434268\pi$
$601$	5.35010e10			0.410075			0.205038	+	0.978754i	$0.434268\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.44504e11	−	8.34294e10i	−1.07859	−	0.622727i
$606$		0			0
$607$	−3.98245e10	−	6.89781e10i	−0.293357	−	0.508109i	0.681245	−	0.732056i	$-0.261439\pi$
$607$	−3.98245e10	−	6.89781e10i	−0.293357	−	0.508109i	−0.974601	+	0.223947i	$0.928106\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.15378e8	−	1.24349e8i	0.00154539	−	0.000892229i
$612$		0			0
$613$	3.46589e9	−	6.00309e9i	0.0245455	−	0.0425141i	−0.853492	−	0.521106i	$-0.825519\pi$
$613$	3.46589e9	−	6.00309e9i	0.0245455	−	0.0425141i	0.878037	+	0.478592i	$0.158853\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1.18082e11i			0.814783i	0.913254	+	0.407392i	$0.133562\pi$
$617$			1.18082e11i			0.814783i	−0.913254	+	0.407392i	$0.866438\pi$
$618$		0			0
$619$	2.23245e10	−	3.86672e10i	0.152061	−	0.263378i	−0.779924	−	0.625875i	$-0.784742\pi$
$619$	2.23245e10	−	3.86672e10i	0.152061	−	0.263378i	0.931985	+	0.362496i	$0.118076\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9.43728e10	+	2.89996e10i	0.626462	+	0.192504i
$624$		0			0
$625$	9.51548e10	+	1.64813e11i	0.623606	+	1.08012i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.07742e11i		−	0.688307i
$630$		0			0
$631$	−8.91802e10			−0.562537			−0.281269	−	0.959629i	$-0.590755\pi$
$631$	−8.91802e10			−0.562537			−0.281269	+	0.959629i	$0.590755\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.55711e10	−	3.78575e10i	0.403290	−	0.232840i
$636$		0			0
$637$	−1.95539e7	+	2.68639e8i	−0.000118762	+	0.00163159i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.36113e11	−	1.36320e11i	−1.39858	−	0.807470i	−0.404336	−	0.914611i	$-0.632497\pi$
$641$	−2.36113e11	−	1.36320e11i	−1.39858	−	0.807470i	−0.994244	+	0.107140i	$0.965831\pi$
$642$		0			0
$643$	−2.69153e11			−1.57454			−0.787272	−	0.616606i	$-0.788507\pi$
$643$	−2.69153e11			−1.57454			−0.787272	+	0.616606i	$0.788507\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.43221e11	+	1.40424e11i	1.38798	+	0.801353i	0.993088	−	0.117372i	$-0.0374471\pi$
$647$	2.43221e11	+	1.40424e11i	1.38798	+	0.801353i	0.394896	+	0.918726i	$0.370780\pi$
$648$		0			0
$649$	−1.59275e9	−	2.75872e9i	−0.00897777	−	0.0155500i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.90008e11	+	1.09701e11i	−1.04501	+	0.603334i	−0.921247	−	0.388978i	$-0.872828\pi$
$653$	−1.90008e11	+	1.09701e11i	−1.04501	+	0.603334i	−0.123759	+	0.992312i	$0.539495\pi$
$654$		0			0
$655$	3.51157e10	−	6.08221e10i	0.190781	−	0.330443i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	2.75082e11i		−	1.45855i	−0.684223	−	0.729273i	$-0.739858\pi$
$659$		−	2.75082e11i		−	1.45855i	0.684223	−	0.729273i	$-0.260142\pi$
$660$		0			0
$661$	−5.90309e10	+	1.02245e11i	−0.309224	+	0.535592i	−0.978193	−	0.207698i	$-0.933403\pi$
$661$	−5.90309e10	+	1.02245e11i	−0.309224	+	0.535592i	0.668969	+	0.743291i	$0.266736\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.44322e10	−	1.50117e11i	−0.176067	−	0.767616i
$666$		0			0
$667$	1.40813e11	+	2.43895e11i	0.711442	+	1.23225i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			7.28672e9i			0.0359453i
$672$		0			0
$673$	1.00170e11			0.488292			0.244146	−	0.969739i	$-0.421492\pi$
$673$	1.00170e11			0.488292			0.244146	+	0.969739i	$0.421492\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.80353e11	−	1.04127e11i	0.858558	−	0.495689i	−0.00497123	−	0.999988i	$-0.501582\pi$
$677$	1.80353e11	−	1.04127e11i	0.858558	−	0.495689i	0.863529	+	0.504299i	$0.168249\pi$
$678$		0			0
$679$	1.03556e11	+	1.11368e11i	0.487188	+	0.523938i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	2.75907e11	+	1.59295e11i	1.26788	+	0.732013i	0.974587	−	0.224008i	$-0.0719141\pi$
$683$	2.75907e11	+	1.59295e11i	1.26788	+	0.732013i	0.293297	+	0.956021i	$0.405247\pi$
$684$		0			0
$685$	3.42378e11			1.55505
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.31006e7	+	1.91106e7i	0.000146879	+	8.48004e-5i
$690$		0			0
$691$	−3.42190e10	−	5.92691e10i	−0.150091	−	0.259966i	0.781170	−	0.624319i	$-0.214623\pi$
$691$	−3.42190e10	−	5.92691e10i	−0.150091	−	0.259966i	−0.931261	+	0.364353i	$0.881290\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.00680e11	+	2.31332e11i	−1.71735	+	0.991511i
$696$		0			0
$697$	2.26069e11	−	3.91563e11i	0.957876	−	1.65909i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	4.52825e10i		−	0.187525i	−0.995595	−	0.0937624i	$-0.970111\pi$
$701$		−	4.52825e10i		−	0.187525i	0.995595	−	0.0937624i	$-0.0298894\pi$
$702$		0			0
$703$	−3.91511e10	+	6.78117e10i	−0.160296	+	0.277641i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.11810e10	−	1.79540e11i	−0.164823	−	0.718595i
$708$		0			0
$709$	−1.79001e11	−	3.10038e11i	−0.708385	−	1.22696i	−0.965456	−	0.260566i	$-0.916091\pi$
$709$	−1.79001e11	−	3.10038e11i	−0.708385	−	1.22696i	0.257071	−	0.966393i	$-0.417243\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	5.83573e11i		−	2.25807i
$714$		0			0
$715$	−1.22566e7			−4.68972e−5
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8.08955e10	−	4.67051e10i	0.302698	−	0.174763i	−0.340956	−	0.940079i	$-0.610751\pi$
$719$	8.08955e10	−	4.67051e10i	0.302698	−	0.174763i	0.643654	+	0.765317i	$0.277418\pi$
$720$		0			0
$721$	8.56459e10	−	2.78715e11i	0.316932	−	1.03138i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.39707e11	−	8.06596e10i	−0.505667	−	0.291947i
$726$		0			0
$727$	2.36551e11			0.846811			0.423406	−	0.905940i	$-0.360835\pi$
$727$	2.36551e11			0.846811			0.423406	+	0.905940i	$0.360835\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.75554e10	+	1.01356e10i	0.0614811	+	0.0354962i
$732$		0			0
$733$	−2.17827e11	−	3.77288e11i	−0.754565	−	1.30694i	−0.945590	−	0.325360i	$-0.894515\pi$
$733$	−2.17827e11	−	3.77288e11i	−0.754565	−	1.30694i	0.191025	−	0.981585i	$-0.438819\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.54080e9	+	2.62163e9i	−0.0153908	+	0.00888591i
$738$		0			0
$739$	1.87063e11	−	3.24003e11i	0.627206	−	1.08635i	−0.360904	−	0.932603i	$-0.617532\pi$
$739$	1.87063e11	−	3.24003e11i	0.627206	−	1.08635i	0.988110	−	0.153750i	$-0.0491350\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	3.22304e11i		−	1.05757i	−0.848754	−	0.528787i	$-0.822647\pi$
$743$		−	3.22304e11i		−	1.05757i	0.848754	−	0.528787i	$-0.177353\pi$
$744$		0			0
$745$	−8.68581e10	+	1.50443e11i	−0.281958	+	0.488366i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.13897e11	+	6.57279e10i	0.679636	+	0.208844i
$750$		0			0
$751$	3.28833e10	+	5.69555e10i	0.103375	+	0.179051i	0.913073	−	0.407796i	$-0.133703\pi$
$751$	3.28833e10	+	5.69555e10i	0.103375	+	0.179051i	−0.809698	+	0.586847i	$0.800369\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	4.64679e11i		−	1.43010i
$756$		0			0
$757$	5.18611e11			1.57928			0.789638	−	0.613573i	$-0.210268\pi$
$757$	5.18611e11			1.57928			0.789638	+	0.613573i	$0.210268\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.03805e11	+	5.99319e10i	−0.309514	+	0.178698i	−0.646709	−	0.762737i	$-0.723855\pi$
$761$	−1.03805e11	+	5.99319e10i	−0.309514	+	0.178698i	0.337195	+	0.941435i	$0.390522\pi$
$762$		0			0
$763$	4.69775e11	−	1.07752e11i	1.38609	−	0.317926i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.82682e8	+	2.20942e8i	0.00110575	+	0.000638405i
$768$		0			0
$769$	−3.19093e11			−0.912455			−0.456227	−	0.889863i	$-0.650800\pi$
$769$	−3.19093e11			−0.912455			−0.456227	+	0.889863i	$0.650800\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5.64540e11	−	3.25938e11i	−1.58116	−	0.912886i	−0.994690	−	0.102918i	$-0.967182\pi$
$773$	−5.64540e11	−	3.25938e11i	−1.58116	−	0.912886i	−0.586475	−	0.809967i	$-0.699485\pi$
$774$		0			0
$775$	1.67139e11	+	2.89494e11i	0.463310	+	0.802477i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.84571e11	+	1.64297e11i	−0.772753	+	0.446149i
$780$		0			0
$781$	1.33959e9	−	2.32023e9i	0.00360053	−	0.00623630i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			5.04519e11i			1.32861i
$786$		0			0
$787$	−1.96638e11	+	3.40586e11i	−0.512587	+	0.887827i	0.487306	+	0.873231i	$0.337980\pi$
$787$	−1.96638e11	+	3.40586e11i	−0.512587	+	0.887827i	−0.999893	+	0.0145960i	$0.995354\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.13770e11	−	1.05790e11i	0.290618	−	0.270233i
$792$		0			0
$793$	−5.05397e8	−	8.75373e8i	−0.00127803	−	0.00221361i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	6.96733e11i		−	1.72677i	−0.504550	−	0.863383i	$-0.668341\pi$
$797$		−	6.96733e11i		−	1.72677i	0.504550	−	0.863383i	$-0.331659\pi$
$798$		0			0
$799$	−6.03232e11			−1.48012
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9.28820e9	+	5.36254e9i	−0.0223393	+	0.0128976i
$804$		0			0
$805$	6.87081e11	−	1.57595e11i	1.63616	−	0.375283i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−5.85430e11	−	3.37998e11i	−1.36672	−	0.789079i	−0.376216	−	0.926532i	$-0.622775\pi$
$809$	−5.85430e11	−	3.37998e11i	−1.36672	−	0.789079i	−0.990508	+	0.137453i	$0.956108\pi$
$810$		0			0
$811$	3.84102e11			0.887897			0.443949	−	0.896052i	$-0.353577\pi$
$811$	3.84102e11			0.887897			0.443949	+	0.896052i	$0.353577\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.54243e11	+	3.77727e11i	1.48289	+	0.856146i
$816$		0			0
$817$	−7.36614e9	−	1.27585e10i	−0.0165330	−	0.0286360i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.20317e10	−	6.94653e9i	0.0264823	−	0.0152896i	−0.486700	−	0.873569i	$-0.661800\pi$
$821$	1.20317e10	−	6.94653e9i	0.0264823	−	0.0152896i	0.513183	+	0.858279i	$0.328466\pi$
$822$		0			0
$823$	−7.39268e10	+	1.28045e11i	−0.161140	+	0.279102i	−0.935278	−	0.353915i	$-0.884850\pi$
$823$	−7.39268e10	+	1.28045e11i	−0.161140	+	0.279102i	0.774138	+	0.633017i	$0.218184\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			2.04733e11i			0.437689i	0.975760	+	0.218845i	$0.0702288\pi$
$827$			2.04733e11i			0.437689i	−0.975760	+	0.218845i	$0.929771\pi$
$828$		0			0
$829$	−1.99191e11	+	3.45009e11i	−0.421746	+	0.730486i	−0.996110	−	0.0881141i	$-0.971916\pi$
$829$	−1.99191e11	+	3.45009e11i	−0.421746	+	0.730486i	0.574364	+	0.818600i	$0.305249\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.66876e11	−	5.40589e11i	0.761972	−	1.12276i
$834$		0			0
$835$	1.98650e11	+	3.44073e11i	0.408643	+	0.707790i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			2.17098e11i			0.438134i	0.975710	+	0.219067i	$0.0703014\pi$
$839$			2.17098e11i			0.438134i	−0.975710	+	0.219067i	$0.929699\pi$
$840$		0			0
$841$	−5.78648e10			−0.115673
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5.50191e11	+	3.17653e11i	−1.07916	+	0.623055i
$846$		0			0
$847$	−1.51097e11	+	4.91712e11i	−0.293578	+	0.955382i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.10372e11	−	1.79193e11i	−0.591785	−	0.341667i
$852$		0			0
$853$	−6.80006e11			−1.28445			−0.642225	−	0.766517i	$-0.721988\pi$
$853$	−6.80006e11			−1.28445			−0.642225	+	0.766517i	$0.721988\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.16628e11	+	1.82805e11i	0.586984	+	0.338895i	0.763904	−	0.645330i	$-0.223280\pi$
$857$	3.16628e11	+	1.82805e11i	0.586984	+	0.338895i	−0.176920	+	0.984225i	$0.556613\pi$
$858$		0			0
$859$	1.66673e11	+	2.88687e11i	0.306121	+	0.530217i	0.977510	−	0.210888i	$-0.0676354\pi$
$859$	1.66673e11	+	2.88687e11i	0.306121	+	0.530217i	−0.671389	+	0.741105i	$0.734302\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−7.33254e10	+	4.23344e10i	−0.132194	+	0.0763221i	−0.564638	−	0.825338i	$-0.690984\pi$
$863$	−7.33254e10	+	4.23344e10i	−0.132194	+	0.0763221i	0.432445	+	0.901660i	$0.357651\pi$
$864$		0			0
$865$	8.04769e9	−	1.39390e10i	0.0143750	−	0.0248982i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1.64548e10i		−	0.0288545i
$870$		0			0
$871$	3.63665e8	−	6.29887e8i	0.000631872	−	0.00109444i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.39220e11	−	2.22440e11i	0.408099	−	0.379473i
$876$		0			0
$877$	−4.79184e11	−	8.29972e11i	−0.810036	−	1.40302i	−0.912838	−	0.408322i	$-0.866114\pi$
$877$	−4.79184e11	−	8.29972e11i	−0.810036	−	1.40302i	0.102802	−	0.994702i	$-0.467219\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1.87605e11i		−	0.311417i	−0.987803	−	0.155708i	$-0.950234\pi$
$881$		−	1.87605e11i		−	0.311417i	0.987803	−	0.155708i	$-0.0497660\pi$
$882$		0			0
$883$	−3.49167e11			−0.574369			−0.287184	−	0.957875i	$-0.592719\pi$
$883$	−3.49167e11			−0.574369			−0.287184	+	0.957875i	$0.592719\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4.62160e10	−	2.66828e10i	0.0746616	−	0.0431059i	−0.462204	−	0.886773i	$-0.652941\pi$
$887$	4.62160e10	−	2.66828e10i	0.0746616	−	0.0431059i	0.536866	+	0.843667i	$0.319608\pi$
$888$		0			0
$889$	−1.58948e11	−	1.70939e11i	−0.254477	−	0.273674i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.79668e11	+	2.19202e11i	0.597033	+	0.344697i
$894$		0			0
$895$	1.19752e12			1.86633
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.00155e12	+	5.78247e11i	1.53333	+	0.885268i
$900$		0			0
$901$	−4.63541e10	−	8.02877e10i	−0.0703378	−	0.121829i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.07131e12	+	6.18524e11i	−1.59707	+	0.922066i
$906$		0			0
$907$	3.30976e11	−	5.73267e11i	0.489066	−	0.847087i	−0.510855	−	0.859667i	$-0.670671\pi$
$907$	3.30976e11	−	5.73267e11i	0.489066	−	0.847087i	0.999921	+	0.0125802i	$0.00400451\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1.59034e11i			0.230896i	0.993314	+	0.115448i	$0.0368304\pi$
$911$			1.59034e11i			0.230896i	−0.993314	+	0.115448i	$0.963170\pi$
$912$		0			0
$913$	1.56056e9	−	2.70297e9i	0.00224594	−	0.00389008i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.06963e11	−	6.35973e10i	−0.292695	−	0.0899418i
$918$		0			0
$919$	−4.91908e10	−	8.52010e10i	−0.0689639	−	0.119449i	0.829482	−	0.558534i	$-0.188636\pi$
$919$	−4.91908e10	−	8.52010e10i	−0.0689639	−	0.119449i	−0.898446	+	0.439085i	$0.855303\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			3.71647e8i			0.000512064i
$924$		0			0
$925$	2.05289e11			0.280413
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−7.72137e10	+	4.45793e10i	−0.103665	+	0.0598509i	−0.550936	−	0.834547i	$-0.685729\pi$
$929$	−7.72137e10	+	4.45793e10i	−0.103665	+	0.0598509i	0.447271	+	0.894398i	$0.352396\pi$
$930$		0			0
$931$	−4.27346e11	+	2.06926e11i	−0.568829	+	0.275434i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.57463e10	+	1.48646e10i	0.0336875	+	0.0194495i
$936$		0			0
$937$	6.04170e10			0.0783792			0.0391896	−	0.999232i	$-0.487522\pi$
$937$	6.04170e10			0.0783792			0.0391896	+	0.999232i	$0.487522\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.15284e12	+	6.65592e11i	1.47031	+	0.848887i	0.999445	−	0.0333161i	$-0.0106068\pi$
$941$	1.15284e12	+	6.65592e11i	1.47031	+	0.848887i	0.470870	+	0.882203i	$0.343940\pi$
$942$		0			0
$943$	−7.51982e11	−	1.30247e12i	−0.950957	−	1.64711i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6.29771e11	+	3.63598e11i	−0.783037	+	0.452087i	−0.837506	−	0.546429i	$-0.815987\pi$
$947$	−6.29771e11	+	3.63598e11i	−0.783037	+	0.452087i	0.0544683	+	0.998516i	$0.482654\pi$
$948$		0			0
$949$	7.43876e8	−	1.28843e9i	0.000917141	−	0.00158853i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			8.43625e11i			1.02277i	0.859352	+	0.511385i	$0.170867\pi$
$953$			8.43625e11i			1.02277i	−0.859352	+	0.511385i	$0.829133\pi$
$954$		0			0
$955$	−5.21902e11	+	9.03960e11i	−0.627444	+	1.08677i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.35973e11	−	1.02879e12i	−0.278989	−	1.21633i
$960$		0			0
$961$	−7.71773e11	−	1.33675e12i	−0.904891	−	1.56732i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			9.79153e11i			1.12912i
$966$		0			0
$967$	−1.29950e12			−1.48617			−0.743087	−	0.669195i	$-0.766639\pi$
$967$	−1.29950e12			−1.48617			−0.743087	+	0.669195i	$0.766639\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.21619e12	−	7.02169e11i	1.36812	−	0.789887i	0.377436	−	0.926036i	$-0.376806\pi$
$971$	1.21619e12	−	7.02169e11i	1.36812	−	0.789887i	0.990688	+	0.136149i	$0.0434725\pi$
$972$		0			0
$973$	9.71272e11	+	1.04454e12i	1.08365	+	1.16540i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.18792e11	+	1.26320e11i	0.240134	+	0.138641i	0.615238	−	0.788341i	$-0.289060\pi$
$977$	2.18792e11	+	1.26320e11i	0.240134	+	0.138641i	−0.375104	+	0.926983i	$0.622393\pi$
$978$		0			0
$979$	1.38500e10			0.0150771
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.51236e12	+	8.73163e11i	1.61973	+	0.935150i	0.986990	+	0.160784i	$0.0514023\pi$
$983$	1.51236e12	+	8.73163e11i	1.61973	+	0.935150i	0.632738	+	0.774366i	$0.281931\pi$
$984$		0			0
$985$	1.79667e11	+	3.11193e11i	0.190864	+	0.330586i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.83954e10	−	3.37146e10i	0.0610370	−	0.0352397i
$990$		0			0
$991$	−4.26050e11	+	7.37941e11i	−0.441740	+	0.765116i	−0.997819	−	0.0660138i	$-0.978972\pi$
$991$	−4.26050e11	+	7.37941e11i	−0.441740	+	0.765116i	0.556079	+	0.831129i	$0.312305\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			9.23380e11i			0.942081i
$996$		0			0
$997$	−6.18782e11	+	1.07176e12i	−0.626264	+	1.08472i	0.362032	+	0.932166i	$0.382083\pi$
$997$	−6.18782e11	+	1.07176e12i	−0.626264	+	1.08472i	−0.988295	+	0.152554i	$0.951250\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.bk.a.53.18	yes	44
3.2	odd	2	inner	252.9.bk.a.53.5	✓	44
7.2	even	3	inner	252.9.bk.a.233.5	yes	44
21.2	odd	6	inner	252.9.bk.a.233.18	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bk.a.53.5	✓	44	3.2	odd	2	inner
252.9.bk.a.53.18	yes	44	1.1	even	1	trivial
252.9.bk.a.233.5	yes	44	7.2	even	3	inner
252.9.bk.a.233.18	yes	44	21.2	odd	6	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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.bk (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(674.478 - 389.410i) q^{5} +(-1634.98 - 1758.31i) q^{7} +O(q^{10})\)	\(q+(674.478 - 389.410i) q^{5} +(-1634.98 - 1758.31i) q^{7} +(-291.697 - 168.411i) q^{11} +46.7231 q^{13} +(-98146.7 - 56665.0i) q^{17} +(41181.7 + 71328.8i) q^{19} +(-326470. + 188487. i) q^{23} +(107968. - 187006. i) q^{25} -747068. i q^{29} +(-774022. + 1.34064e6i) q^{31} +(-1.78746e6 - 549266. i) q^{35} +(475345. + 823322. i) q^{37} +3.98956e6i q^{41} -178869. q^{43} +(4.60967e6 - 2.66139e6i) q^{47} +(-418507. + 5.74959e6i) q^{49} +(708441. + 409018. i) q^{53} -262325. q^{55} +(8.19042e6 + 4.72874e6i) q^{59} +(-1.08168e7 - 1.87353e7i) q^{61} +(31513.7 - 18194.5i) q^{65} +(7.78342e6 - 1.34813e7i) q^{67} +7.95425e6i q^{71} +(1.59210e7 - 2.75759e7i) q^{73} +(180798. + 788243. i) q^{77} +(2.44265e7 + 4.23079e7i) q^{79} +9.26637e6i q^{83} -8.82638e7 q^{85} +(-3.56105e7 + 2.05597e7i) q^{89} +(-76391.2 - 82153.7i) q^{91} +(5.55524e7 + 3.20732e7i) q^{95} -6.33379e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 1230 q^{7}+O(q^{10}) \) 44 * q + 1230 * q^7	\( 44 q + 1230 q^{7} - 101380 q^{13} + 62770 q^{19} + 2247666 q^{25} + 1389254 q^{31} - 2136026 q^{37} + 11510140 q^{43} - 3824398 q^{49} - 42646528 q^{55} + 27346232 q^{61} + 14239194 q^{67} - 64344138 q^{73} + 7061786 q^{79} - 54198208 q^{85} - 45697066 q^{91} + 476543496 q^{97}+O(q^{100}) \) 44 * q + 1230 * q^7 - 101380 * q^13 + 62770 * q^19 + 2247666 * q^25 + 1389254 * q^31 - 2136026 * q^37 + 11510140 * q^43 - 3824398 * q^49 - 42646528 * q^55 + 27346232 * q^61 + 14239194 * q^67 - 64344138 * q^73 + 7061786 * q^79 - 54198208 * q^85 - 45697066 * q^91 + 476543496 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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