LMFDB - Embedded newform 252.9.bk.a.53.13

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			53.13
Character	$\chi$	$=$	252.53
Dual form			252.9.bk.a.233.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(36.7886 - 21.2399i) q^{5} +(-115.865 - 2398.20i) q^{7} +O(q^{10})$	$q+(36.7886 - 21.2399i) q^{5} +(-115.865 - 2398.20i) q^{7} +(18785.5 + 10845.8i) q^{11} -33631.9 q^{13} +(-15399.5 - 8890.88i) q^{17} +(-57432.8 - 99476.6i) q^{19} +(72299.8 - 41742.3i) q^{23} +(-194410. + 336728. i) q^{25} -1.22285e6i q^{29} +(559905. - 969784. i) q^{31} +(-55200.1 - 85765.5i) q^{35} +(532365. + 922084. i) q^{37} +5.39326e6i q^{41} +504960. q^{43} +(-118413. + 68365.6i) q^{47} +(-5.73795e6 + 555735. i) q^{49} +(-4.58967e6 - 2.64985e6i) q^{53} +921457. q^{55} +(-3.36414e6 - 1.94229e6i) q^{59} +(7.93925e6 + 1.37512e7i) q^{61} +(-1.23727e6 + 714337. i) q^{65} +(1.00195e7 - 1.73543e7i) q^{67} -1.52625e7i q^{71} +(-1.41130e7 + 2.44444e7i) q^{73} +(2.38339e7 - 4.63081e7i) q^{77} +(-2.48788e7 - 4.30914e7i) q^{79} +2.84892e7i q^{83} -755366. q^{85} +(-6.02063e7 + 3.47601e7i) q^{89} +(3.89675e6 + 8.06561e7i) q^{91} +(-4.22574e6 - 2.43973e6i) q^{95} -1.48273e8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	36.7886	−	21.2399i	0.0588617	−	0.0339838i	−0.470280	−	0.882517i	$-0.655847\pi$
$5$	36.7886	−	21.2399i	0.0588617	−	0.0339838i	0.529142	+	0.848533i	$0.322514\pi$
$6$		0			0
$7$	−115.865	−	2398.20i	−0.0482569	−	0.998835i
$7$	−115.865	−	2398.20i	−0.0482569	−	0.998835i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	18785.5	+	10845.8i	1.28308	+	0.740784i	0.977410	−	0.211354i	$-0.0677873\pi$
$11$	18785.5	+	10845.8i	1.28308	+	0.740784i	0.305667	+	0.952139i	$0.401121\pi$
$12$		0			0
$13$	−33631.9			−1.17755			−0.588773	−	0.808299i	$-0.700389\pi$
$13$	−33631.9			−1.17755			−0.588773	+	0.808299i	$0.700389\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15399.5	−	8890.88i	−0.184378	−	0.106451i	0.404970	−	0.914330i	$-0.367282\pi$
$17$	−15399.5	−	8890.88i	−0.184378	−	0.106451i	−0.589348	+	0.807879i	$0.700615\pi$
$18$		0			0
$19$	−57432.8	−	99476.6i	−0.440703	−	0.763320i	0.557039	−	0.830486i	$-0.311937\pi$
$19$	−57432.8	−	99476.6i	−0.440703	−	0.763320i	−0.997742	+	0.0671667i	$0.978604\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	72299.8	−	41742.3i	0.258360	−	0.149164i	−0.365226	−	0.930919i	$-0.619008\pi$
$23$	72299.8	−	41742.3i	0.258360	−	0.149164i	0.623586	+	0.781754i	$0.285675\pi$
$24$		0			0
$25$	−194410.	+	336728.i	−0.497690	+	0.862025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	1.22285e6i		−	1.72894i	−0.502682	−	0.864471i	$-0.667653\pi$
$29$		−	1.22285e6i		−	1.72894i	0.502682	−	0.864471i	$-0.332347\pi$
$30$		0			0
$31$	559905.	−	969784.i	0.606272	−	1.05009i	−0.385577	−	0.922676i	$-0.625998\pi$
$31$	559905.	−	969784.i	0.606272	−	1.05009i	0.991849	−	0.127418i	$-0.0406691\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−55200.1	−	85765.5i	−0.0367847	−	0.0571532i
$36$		0			0
$37$	532365.	+	922084.i	0.284055	+	0.491998i	0.972380	−	0.233405i	$-0.0749868\pi$
$37$	532365.	+	922084.i	0.284055	+	0.491998i	−0.688324	+	0.725403i	$0.741653\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			5.39326e6i			1.90861i	0.298841	+	0.954303i	$0.403400\pi$
$41$			5.39326e6i			1.90861i	−0.298841	+	0.954303i	$0.596600\pi$
$42$		0			0
$43$	504960.			0.147701			0.0738504	−	0.997269i	$-0.476471\pi$
$43$	504960.			0.147701			0.0738504	+	0.997269i	$0.476471\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−118413.	+	68365.6i	−0.0242665	+	0.0140103i	−0.512084	−	0.858935i	$-0.671126\pi$
$47$	−118413.	+	68365.6i	−0.0242665	+	0.0140103i	0.487818	+	0.872946i	$0.337793\pi$
$48$		0			0
$49$	−5.73795e6	+	555735.i	−0.995343	+	0.0964014i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.58967e6	−	2.64985e6i	−0.581672	−	0.335828i	0.180126	−	0.983644i	$-0.442349\pi$
$53$	−4.58967e6	−	2.64985e6i	−0.581672	−	0.335828i	−0.761797	+	0.647815i	$0.775683\pi$
$54$		0			0
$55$	921457.			0.100699
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.36414e6	−	1.94229e6i	−0.277630	−	0.160290i	0.354720	−	0.934973i	$-0.384576\pi$
$59$	−3.36414e6	−	1.94229e6i	−0.277630	−	0.160290i	−0.632350	+	0.774683i	$0.717909\pi$
$60$		0			0
$61$	7.93925e6	+	1.37512e7i	0.573404	+	0.993164i	0.996213	+	0.0869455i	$0.0277106\pi$
$61$	7.93925e6	+	1.37512e7i	0.573404	+	0.993164i	−0.422809	+	0.906219i	$0.638956\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.23727e6	+	714337.i	−0.0693124	+	0.0400175i
$66$		0			0
$67$	1.00195e7	−	1.73543e7i	0.497218	−	0.861207i	−0.502777	−	0.864416i	$-0.667688\pi$
$67$	1.00195e7	−	1.73543e7i	0.497218	−	0.861207i	0.999995	+	0.00320907i	$0.00102148\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.52625e7i		−	0.600608i	−0.953844	−	0.300304i	$-0.902912\pi$
$71$		−	1.52625e7i		−	0.600608i	0.953844	−	0.300304i	$-0.0970881\pi$
$72$		0			0
$73$	−1.41130e7	+	2.44444e7i	−0.496967	+	0.860773i	−0.999994	−	0.00349825i	$-0.998886\pi$
$73$	−1.41130e7	+	2.44444e7i	−0.496967	+	0.860773i	0.503027	+	0.864271i	$0.332220\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.38339e7	−	4.63081e7i	0.678004	−	1.31733i
$78$		0			0
$79$	−2.48788e7	−	4.30914e7i	−0.638737	−	1.10632i	−0.985710	−	0.168449i	$-0.946124\pi$
$79$	−2.48788e7	−	4.30914e7i	−0.638737	−	1.10632i	0.346974	−	0.937875i	$-0.387209\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			2.84892e7i			0.600299i	0.953892	+	0.300150i	$0.0970366\pi$
$83$			2.84892e7i			0.600299i	−0.953892	+	0.300150i	$0.902963\pi$
$84$		0			0
$85$	−755366.			−0.0144704
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.02063e7	+	3.47601e7i	−0.959582	+	0.554015i	−0.896044	−	0.443964i	$-0.853572\pi$
$89$	−6.02063e7	+	3.47601e7i	−0.959582	+	0.554015i	−0.0635378	+	0.997979i	$0.520238\pi$
$90$		0			0
$91$	3.89675e6	+	8.06561e7i	0.0568247	+	1.17617i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.22574e6	−	2.43973e6i	−0.0518811	−	0.0299535i
$96$		0			0
$97$	−1.48273e8			−1.67485			−0.837426	−	0.546551i	$-0.815940\pi$
$97$	−1.48273e8			−1.67485			−0.837426	+	0.546551i	$0.815940\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.13350e8	−	6.54429e7i	−1.08928	−	0.628893i	−0.155892	−	0.987774i	$-0.549825\pi$
$101$	−1.13350e8	−	6.54429e7i	−1.08928	−	0.628893i	−0.933383	+	0.358881i	$0.883159\pi$
$102$		0			0
$103$	1.04450e8	+	1.80912e8i	0.928022	+	1.60738i	0.786628	+	0.617428i	$0.211825\pi$
$103$	1.04450e8	+	1.80912e8i	0.928022	+	1.60738i	0.141394	+	0.989953i	$0.454841\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.71997e8	+	9.93025e7i	−1.31216	+	0.757574i	−0.982453	−	0.186511i	$-0.940282\pi$
$107$	−1.71997e8	+	9.93025e7i	−1.31216	+	0.757574i	−0.329703	+	0.944085i	$0.606949\pi$
$108$		0			0
$109$	7.46870e7	−	1.29362e8i	0.529102	−	0.916431i	−0.470322	−	0.882495i	$-0.655862\pi$
$109$	7.46870e7	−	1.29362e8i	0.529102	−	0.916431i	0.999424	−	0.0339362i	$-0.0108043\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	2.38996e8i		−	1.46580i	−0.680334	−	0.732902i	$-0.738165\pi$
$113$		−	2.38996e8i		−	1.46580i	0.680334	−	0.732902i	$-0.261835\pi$
$114$		0			0
$115$	1.77321e6	−	3.07128e6i	0.0101384	−	0.0175602i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.95379e7	+	3.79612e7i	−0.0974293	+	0.189301i
$120$		0			0
$121$	1.28084e8	+	2.21849e8i	0.597523	+	1.03494i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			3.31107e7i			0.135621i
$126$		0			0
$127$	−1.59877e8			−0.614570			−0.307285	−	0.951617i	$-0.599421\pi$
$127$	−1.59877e8			−0.614570			−0.307285	+	0.951617i	$0.599421\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.80415e8	−	1.04163e8i	0.612616	−	0.353694i	−0.161372	−	0.986894i	$-0.551592\pi$
$131$	1.80415e8	−	1.04163e8i	0.612616	−	0.353694i	0.773989	+	0.633199i	$0.218259\pi$
$132$		0			0
$133$	−2.31911e8	+	1.49261e8i	−0.741163	+	0.477025i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.66138e8	−	3.26860e8i	−1.60709	−	0.927853i	−0.990017	−	0.140946i	$-0.954986\pi$
$137$	−5.66138e8	−	3.26860e8i	−1.60709	−	0.927853i	−0.617071	−	0.786907i	$-0.711681\pi$
$138$		0			0
$139$	−3.58488e8			−0.960318			−0.480159	−	0.877181i	$-0.659421\pi$
$139$	−3.58488e8			−0.960318			−0.480159	+	0.877181i	$0.659421\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.31792e8	−	3.64765e8i	−1.51088	−	0.872307i
$144$		0			0
$145$	−2.59732e7	−	4.49868e7i	−0.0587561	−	0.101769i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.80699e8	+	1.04327e8i	−0.366615	+	0.211665i	−0.671979	−	0.740570i	$-0.734555\pi$
$149$	−1.80699e8	+	1.04327e8i	−0.366615	+	0.211665i	0.305363	+	0.952236i	$0.401222\pi$
$150$		0			0
$151$	−4.05301e8	+	7.02001e8i	−0.779596	+	1.35030i	0.152579	+	0.988291i	$0.451242\pi$
$151$	−4.05301e8	+	7.02001e8i	−0.779596	+	1.35030i	−0.932175	+	0.362008i	$0.882091\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	4.75693e7i		−	0.0824138i
$156$		0			0
$157$	−3.51790e8	+	6.09318e8i	−0.579008	+	1.00287i	0.416585	+	0.909097i	$0.363227\pi$
$157$	−3.51790e8	+	6.09318e8i	−0.579008	+	1.00287i	−0.995593	+	0.0937753i	$0.970106\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.08484e8	−	1.68553e8i	−0.161458	−	0.250861i
$162$		0			0
$163$	−8.15949e7	−	1.41326e8i	−0.115588	−	0.200204i	0.802427	−	0.596751i	$-0.203542\pi$
$163$	−8.15949e7	−	1.41326e8i	−0.115588	−	0.200204i	−0.918015	+	0.396547i	$0.870209\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	8.62258e8i		−	1.10859i	−0.832320	−	0.554296i	$-0.812988\pi$
$167$		−	8.62258e8i		−	1.10859i	0.832320	−	0.554296i	$-0.187012\pi$
$168$		0			0
$169$	3.15372e8			0.386613
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.81642e8	+	5.66751e8i	−1.09589	+	0.632715i	−0.935139	−	0.354280i	$-0.884726\pi$
$173$	−9.81642e8	+	5.66751e8i	−1.09589	+	0.632715i	−0.160754	+	0.986994i	$0.551393\pi$
$174$		0			0
$175$	8.30068e8	+	4.27220e8i	0.885037	+	0.455512i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.42406e9	−	8.22183e8i	−1.38713	−	0.800860i	−0.394139	−	0.919051i	$-0.628957\pi$
$179$	−1.42406e9	−	8.22183e8i	−1.38713	−	0.800860i	−0.992991	+	0.118191i	$0.962291\pi$
$180$		0			0
$181$	7.31417e8			0.681476			0.340738	−	0.940158i	$-0.389323\pi$
$181$	7.31417e8			0.681476			0.340738	+	0.940158i	$0.389323\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.91699e7	+	2.26148e7i	0.0334400	+	0.0193066i
$186$		0			0
$187$	−1.92858e8	−	3.34040e8i	−0.157714	−	0.273169i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.70610e8	+	9.85016e7i	−0.128195	+	0.0740133i	−0.562726	−	0.826643i	$-0.690247\pi$
$191$	−1.70610e8	+	9.85016e7i	−0.128195	+	0.0740133i	0.434531	+	0.900657i	$0.356914\pi$
$192$		0			0
$193$	−1.24715e9	+	2.16013e9i	−0.898856	+	1.55686i	−0.0698981	+	0.997554i	$0.522267\pi$
$193$	−1.24715e9	+	2.16013e9i	−0.898856	+	1.55686i	−0.828958	+	0.559311i	$0.811066\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.04300e8i			0.534014i	0.963695	+	0.267007i	$0.0860348\pi$
$197$			8.04300e8i			0.534014i	−0.963695	+	0.267007i	$0.913965\pi$
$198$		0			0
$199$	−6.12218e8	+	1.06039e9i	−0.390386	+	0.676168i	−0.992500	−	0.122242i	$-0.960992\pi$
$199$	−6.12218e8	+	1.06039e9i	−0.390386	+	0.676168i	0.602115	+	0.798410i	$0.294325\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.93264e9	+	1.41685e8i	−1.72693	+	0.0834334i
$204$		0			0
$205$	1.14552e8	+	1.98410e8i	0.0648617	+	0.112344i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	2.49163e9i		−	1.30586i
$210$		0			0
$211$	−2.13755e9			−1.07842			−0.539209	−	0.842172i	$-0.681277\pi$
$211$	−2.13755e9			−1.07842			−0.539209	+	0.842172i	$0.681277\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.85768e7	−	1.07253e7i	0.00869393	−	0.00501944i
$216$		0			0
$217$	−2.39061e9	−	1.23040e9i	−1.07813	−	0.554892i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.17913e8	+	2.99017e8i	0.217114	+	0.125351i
$222$		0			0
$223$	1.10739e9			0.447796			0.223898	−	0.974613i	$-0.428122\pi$
$223$	1.10739e9			0.447796			0.223898	+	0.974613i	$0.428122\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.99760e8	−	2.88536e8i	−0.188216	−	0.108667i	0.402931	−	0.915230i	$-0.367992\pi$
$227$	−4.99760e8	−	2.88536e8i	−0.188216	−	0.108667i	−0.591147	+	0.806564i	$0.701325\pi$
$228$		0			0
$229$	2.10895e8	+	3.65281e8i	0.0766875	+	0.132827i	0.901819	−	0.432114i	$-0.142232\pi$
$229$	2.10895e8	+	3.65281e8i	0.0766875	+	0.132827i	−0.825131	+	0.564941i	$0.808899\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.05174e9	−	1.18457e9i	0.696143	−	0.401919i	−0.109766	−	0.993957i	$-0.535010\pi$
$233$	2.05174e9	−	1.18457e9i	0.696143	−	0.401919i	0.805909	+	0.592039i	$0.201677\pi$
$234$		0			0
$235$	−2.90416e6	+	5.03015e6i	−0.000952245	+	0.00164934i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	2.72360e9i		−	0.834741i	−0.908736	−	0.417371i	$-0.862952\pi$
$239$		−	2.72360e9i		−	0.834741i	0.908736	−	0.417371i	$-0.137048\pi$
$240$		0			0
$241$	1.47981e9	−	2.56311e9i	0.438671	−	0.759801i	−0.558916	−	0.829224i	$-0.688783\pi$
$241$	1.47981e9	−	2.56311e9i	0.438671	−	0.759801i	0.997587	+	0.0694233i	$0.0221159\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.99287e8	+	1.42318e8i	−0.0553115	+	0.0394999i
$246$		0			0
$247$	1.93157e9	+	3.34558e9i	0.518948	+	0.898844i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			1.86049e9i			0.468740i	0.972147	+	0.234370i	$0.0753027\pi$
$251$			1.86049e9i			0.468740i	−0.972147	+	0.234370i	$0.924697\pi$
$252$		0			0
$253$	1.81092e9			0.441995
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.11838e8	+	1.22305e8i	−0.0485593	+	0.0280357i	−0.524083	−	0.851667i	$-0.675592\pi$
$257$	−2.11838e8	+	1.22305e8i	−0.0485593	+	0.0280357i	0.475524	+	0.879703i	$0.342259\pi$
$258$		0			0
$259$	2.14966e9	−	1.38356e9i	0.477717	−	0.307467i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.55911e9	+	3.78691e9i	1.37095	+	0.791519i	0.991048	−	0.133507i	$-0.0426238\pi$
$263$	6.55911e9	+	3.78691e9i	1.37095	+	0.791519i	0.379904	+	0.925026i	$0.375957\pi$
$264$		0			0
$265$	−2.25130e8			−0.0456509
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.16447e9	−	1.82701e9i	−0.604354	−	0.348924i	0.166398	−	0.986059i	$-0.446786\pi$
$269$	−3.16447e9	−	1.82701e9i	−0.604354	−	0.348924i	−0.770753	+	0.637135i	$0.780120\pi$
$270$		0			0
$271$	−1.22002e9	−	2.11313e9i	−0.226198	−	0.391787i	0.730480	−	0.682934i	$-0.239296\pi$
$271$	−1.22002e9	−	2.11313e9i	−0.226198	−	0.391787i	−0.956678	+	0.291147i	$0.905963\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.30420e9	+	4.21708e9i	−1.27715	+	0.737362i
$276$		0			0
$277$	9.45825e8	−	1.63822e9i	0.160654	−	0.278261i	−0.774449	−	0.632636i	$-0.781973\pi$
$277$	9.45825e8	−	1.63822e9i	0.160654	−	0.278261i	0.935103	+	0.354375i	$0.115306\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			1.87749e9i			0.301129i	0.988600	+	0.150565i	$0.0481091\pi$
$281$			1.87749e9i			0.301129i	−0.988600	+	0.150565i	$0.951891\pi$
$282$		0			0
$283$	−2.80108e9	+	4.85161e9i	−0.436696	+	0.756380i	−0.997432	−	0.0716145i	$-0.977185\pi$
$283$	−2.80108e9	+	4.85161e9i	−0.436696	+	0.756380i	0.560736	+	0.827995i	$0.310518\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.29341e10	−	6.24890e8i	1.90638	−	0.0921034i
$288$		0			0
$289$	−3.32978e9	−	5.76735e9i	−0.477336	−	0.826771i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			7.86272e9i			1.06685i	0.845848	+	0.533423i	$0.179095\pi$
$293$			7.86272e9i			1.06685i	−0.845848	+	0.533423i	$0.820905\pi$
$294$		0			0
$295$	−1.65016e8			−0.0217890
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.43158e9	+	1.40387e9i	−0.304231	+	0.175648i
$300$		0			0
$301$	−5.85071e7	−	1.21100e9i	−0.00712759	−	0.147529i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5.84148e8	+	3.37258e8i	0.0675030	+	0.0389729i
$306$		0			0
$307$	5.09045e9			0.573063			0.286532	−	0.958071i	$-0.407498\pi$
$307$	5.09045e9			0.573063			0.286532	+	0.958071i	$0.407498\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.22409e10	+	7.06727e9i	1.30849	+	0.755458i	0.981844	−	0.189688i	$-0.0607477\pi$
$311$	1.22409e10	+	7.06727e9i	1.30849	+	0.755458i	0.326647	+	0.945146i	$0.394081\pi$
$312$		0			0
$313$	3.17243e9	+	5.49481e9i	0.330533	+	0.572500i	0.982616	−	0.185647i	$-0.0594381\pi$
$313$	3.17243e9	+	5.49481e9i	0.330533	+	0.572500i	−0.652083	+	0.758147i	$0.726105\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.47036e10	−	8.48911e9i	1.45608	−	0.840669i	0.457266	−	0.889330i	$-0.348829\pi$
$317$	1.47036e10	−	8.48911e9i	1.45608	−	0.840669i	0.998815	+	0.0486607i	$0.0154953\pi$
$318$		0			0
$319$	1.32628e10	−	2.29718e10i	1.28077	−	2.21836i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			2.04251e9i			0.187653i
$324$		0			0
$325$	6.53838e9	−	1.13248e10i	0.586053	−	1.01507i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.77674e8	+	2.76056e8i	0.0151650	+	0.0235621i
$330$		0			0
$331$	−5.71794e9	−	9.90376e9i	−0.476351	−	0.825065i	0.523281	−	0.852160i	$-0.324708\pi$
$331$	−5.71794e9	−	9.90376e9i	−0.476351	−	0.825065i	−0.999633	+	0.0270950i	$0.991374\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	8.51253e8i		−	0.0675895i
$336$		0			0
$337$	1.74137e8			0.0135011			0.00675057	−	0.999977i	$-0.497851\pi$
$337$	1.74137e8			0.0135011			0.00675057	+	0.999977i	$0.497851\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.10362e10	−	1.21453e10i	1.55579	−	0.898234i
$342$		0			0
$343$	1.99759e9	+	1.36964e10i	0.144321	+	0.989531i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.57377e9	+	3.21801e9i	0.384442	+	0.221958i	0.679749	−	0.733445i	$-0.262089\pi$
$347$	5.57377e9	+	3.21801e9i	0.384442	+	0.221958i	−0.295307	+	0.955402i	$0.595422\pi$
$348$		0			0
$349$	−1.50707e10			−1.01585			−0.507927	−	0.861400i	$-0.669588\pi$
$349$	−1.50707e10			−1.01585			−0.507927	+	0.861400i	$0.669588\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.16201e9	+	4.71234e9i	0.525652	+	0.303485i	0.739244	−	0.673438i	$-0.235183\pi$
$353$	8.16201e9	+	4.71234e9i	0.525652	+	0.303485i	−0.213592	+	0.976923i	$0.568516\pi$
$354$		0			0
$355$	−3.24173e8	−	5.61484e8i	−0.0204110	−	0.0353528i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.01053e10	+	5.83431e9i	−0.608377	+	0.351247i	−0.772330	−	0.635222i	$-0.780909\pi$
$359$	−1.01053e10	+	5.83431e9i	−0.608377	+	0.351247i	0.163953	+	0.986468i	$0.447575\pi$
$360$		0			0
$361$	1.89472e9	−	3.28175e9i	0.111562	−	0.193231i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			1.19903e9i			0.0675554i
$366$		0			0
$367$	5.81752e9	−	1.00762e10i	0.320682	−	0.555437i	−0.659947	−	0.751312i	$-0.729421\pi$
$367$	5.81752e9	−	1.00762e10i	0.320682	−	0.555437i	0.980629	+	0.195875i	$0.0627547\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.82309e9	+	1.13140e10i	−0.307367	+	0.597200i
$372$		0			0
$373$	−1.19418e9	−	2.06839e9i	−0.0616930	−	0.106855i	0.833529	−	0.552475i	$-0.186317\pi$
$373$	−1.19418e9	−	2.06839e9i	−0.0616930	−	0.106855i	−0.895222	+	0.445620i	$0.852983\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			4.11267e10i			2.03591i
$378$		0			0
$379$	2.64970e10			1.28422			0.642111	−	0.766612i	$-0.278059\pi$
$379$	2.64970e10			1.28422			0.642111	+	0.766612i	$0.278059\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.54637e9	+	8.92799e8i	−0.0718653	+	0.0414915i	−0.535502	−	0.844534i	$-0.679878\pi$
$383$	−1.54637e9	+	8.92799e8i	−0.0718653	+	0.0414915i	0.463637	+	0.886025i	$0.346544\pi$
$384$		0			0
$385$	−1.06764e8	−	2.20984e9i	−0.00485941	−	0.100581i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.68915e9	+	3.86198e9i	0.292127	+	0.168660i	0.638901	−	0.769289i	$-0.279389\pi$
$389$	6.68915e9	+	3.86198e9i	0.292127	+	0.168660i	−0.346773	+	0.937949i	$0.612723\pi$
$390$		0			0
$391$	−1.48450e9			−0.0635147
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.83051e9	−	1.05685e9i	−0.0751943	−	0.0434134i
$396$		0			0
$397$	−6.26609e9	−	1.08532e10i	−0.252252	−	0.436913i	0.711894	−	0.702287i	$-0.247838\pi$
$397$	−6.26609e9	−	1.08532e10i	−0.252252	−	0.436913i	−0.964145	+	0.265374i	$0.914504\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4.36803e10	−	2.52188e10i	1.68930	−	0.975321i	0.734256	−	0.678873i	$-0.237531\pi$
$401$	4.36803e10	−	2.52188e10i	1.68930	−	0.975321i	0.955049	−	0.296448i	$-0.0958020\pi$
$402$		0			0
$403$	−1.88307e10	+	3.26157e10i	−0.713913	+	1.23653i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			2.30958e10i			0.841695i
$408$		0			0
$409$	1.47481e10	−	2.55445e10i	0.527041	−	0.912861i	−0.472463	−	0.881351i	$-0.656635\pi$
$409$	1.47481e10	−	2.55445e10i	0.527041	−	0.912861i	0.999503	−	0.0315105i	$-0.0100318\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.26821e9	+	8.29293e9i	−0.146705	+	0.285041i
$414$		0			0
$415$	6.05107e8	+	1.04808e9i	0.0204005	+	0.0353346i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	4.10551e10i		−	1.33202i	−0.745943	−	0.666010i	$-0.768001\pi$
$419$		−	4.10551e10i		−	1.33202i	0.745943	−	0.666010i	$-0.231999\pi$
$420$		0			0
$421$	3.48209e9			0.110844			0.0554220	−	0.998463i	$-0.482350\pi$
$421$	3.48209e9			0.110844			0.0554220	+	0.998463i	$0.482350\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.98763e9	−	3.45696e9i	0.183527	−	0.105959i
$426$		0			0
$427$	3.20583e10	−	2.06332e10i	0.964336	−	0.620663i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3.06204e10	−	1.76787e10i	−0.887363	−	0.512319i	−0.0142842	−	0.999898i	$-0.504547\pi$
$431$	−3.06204e10	−	1.76787e10i	−0.887363	−	0.512319i	−0.873079	+	0.487578i	$0.837880\pi$
$432$		0			0
$433$	2.35769e10			0.670711			0.335356	−	0.942092i	$-0.391144\pi$
$433$	2.35769e10			0.670711			0.335356	+	0.942092i	$0.391144\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.30477e9	−	4.79476e9i	−0.227720	−	0.131474i
$438$		0			0
$439$	−5.29664e9	−	9.17406e9i	−0.142608	−	0.247004i	0.785870	−	0.618391i	$-0.212215\pi$
$439$	−5.29664e9	−	9.17406e9i	−0.142608	−	0.247004i	−0.928478	+	0.371388i	$0.878882\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.71355e10	+	2.14402e10i	−0.964217	+	0.556691i	−0.897468	−	0.441079i	$-0.854596\pi$
$443$	−3.71355e10	+	2.14402e10i	−0.964217	+	0.556691i	−0.0667485	+	0.997770i	$0.521263\pi$
$444$		0			0
$445$	−1.47660e9	+	2.55755e9i	−0.0376551	+	0.0652206i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	3.15632e10i		−	0.776598i	−0.921534	−	0.388299i	$-0.873063\pi$
$449$		−	3.15632e10i		−	0.776598i	0.921534	−	0.388299i	$-0.126937\pi$
$450$		0			0
$451$	−5.84944e10	+	1.01315e11i	−1.41387	+	2.44889i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.85648e9	+	2.88445e9i	0.0433157	+	0.0673005i
$456$		0			0
$457$	4.83048e8	+	8.36663e8i	0.0110745	+	0.0191816i	0.871510	−	0.490379i	$-0.163141\pi$
$457$	4.83048e8	+	8.36663e8i	0.0110745	+	0.0191816i	−0.860435	+	0.509560i	$0.829808\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	4.56175e10i		−	1.01002i	−0.863115	−	0.505008i	$-0.831490\pi$
$461$		−	4.56175e10i		−	1.01002i	0.863115	−	0.505008i	$-0.168510\pi$
$462$		0			0
$463$	−8.50463e10			−1.85068			−0.925341	−	0.379137i	$-0.876221\pi$
$463$	−8.50463e10			−1.85068			−0.925341	+	0.379137i	$0.876221\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.11963e10	+	6.46421e9i	−0.235401	+	0.135909i	−0.613061	−	0.790035i	$-0.710062\pi$
$467$	−1.11963e10	+	6.46421e9i	−0.235401	+	0.135909i	0.377660	+	0.925944i	$0.376729\pi$
$468$		0			0
$469$	−4.27800e10	−	2.20181e10i	−0.884198	−	0.455080i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.48593e9	+	5.47671e9i	0.189511	+	0.109415i
$474$		0			0
$475$	4.46621e10			0.877334
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.52914e10	−	2.03755e10i	−0.670389	−	0.387049i	0.125835	−	0.992051i	$-0.459839\pi$
$479$	−3.52914e10	−	2.03755e10i	−0.670389	−	0.387049i	−0.796224	+	0.605002i	$0.793172\pi$
$480$		0			0
$481$	−1.79044e10	−	3.10114e10i	−0.334488	−	0.579350i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.45477e9	+	3.14931e9i	−0.0985846	+	0.0569179i
$486$		0			0
$487$	2.71661e8	−	4.70530e8i	0.00482960	−	0.00836511i	−0.863601	−	0.504177i	$-0.831796\pi$
$487$	2.71661e8	−	4.70530e8i	0.00482960	−	0.00836511i	0.868430	+	0.495812i	$0.165129\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			2.77606e10i			0.477643i	0.971063	+	0.238822i	$0.0767611\pi$
$491$			2.77606e10i			0.477643i	−0.971063	+	0.238822i	$0.923239\pi$
$492$		0			0
$493$	−1.08722e10	+	1.88312e10i	−0.184047	+	0.318780i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.66025e10	+	1.76838e9i	−0.599908	+	0.0289835i
$498$		0			0
$499$	3.04720e10	+	5.27790e10i	0.491472	+	0.851254i	0.999952	−	0.00981965i	$-0.00312574\pi$
$499$	3.04720e10	+	5.27790e10i	0.491472	+	0.851254i	−0.508480	+	0.861074i	$0.669792\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			1.07172e10i			0.167421i	0.996490	+	0.0837103i	$0.0266770\pi$
$503$			1.07172e10i			0.167421i	−0.996490	+	0.0837103i	$0.973323\pi$
$504$		0			0
$505$	−5.56000e9			−0.0854888
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.18625e10	+	3.57163e10i	−0.921629	+	0.532103i	−0.884154	−	0.467195i	$-0.845265\pi$
$509$	−6.18625e10	+	3.57163e10i	−0.921629	+	0.532103i	−0.0374744	+	0.999298i	$0.511931\pi$
$510$		0			0
$511$	6.02579e10	+	3.10136e10i	0.883752	+	0.454850i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.68511e9	+	4.43700e9i	0.109250	+	0.0630755i
$516$		0			0
$517$	−2.96593e9			−0.0415143
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−7.55227e10	−	4.36031e10i	−1.02501	−	0.591788i	−0.109457	−	0.993992i	$-0.534911\pi$
$521$	−7.55227e10	−	4.36031e10i	−1.02501	−	0.591788i	−0.915550	+	0.402204i	$0.868244\pi$
$522$		0			0
$523$	3.58953e10	+	6.21724e10i	0.479767	+	0.830981i	0.999731	−	0.0232074i	$-0.00738780\pi$
$523$	3.58953e10	+	6.21724e10i	0.479767	+	0.830981i	−0.519964	+	0.854188i	$0.674054\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.72445e10	+	9.95610e9i	−0.223567	+	0.129076i
$528$		0			0
$529$	−3.56706e10	+	6.17834e10i	−0.455500	+	0.788949i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	1.81386e11i		−	2.24747i
$534$		0			0
$535$	−4.21835e9	+	7.30639e9i	−0.0514905	+	0.0891842i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.13818e11	−	5.17931e10i	−1.34851	−	0.613644i
$540$		0			0
$541$	1.84279e10	+	3.19180e10i	0.215123	+	0.372603i	0.953311	−	0.301992i	$-0.0976515\pi$
$541$	1.84279e10	+	3.19180e10i	0.215123	+	0.372603i	−0.738188	+	0.674595i	$0.764318\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	6.34538e9i		−	0.0719236i
$546$		0			0
$547$	1.06513e10			0.118974			0.0594870	−	0.998229i	$-0.481054\pi$
$547$	1.06513e10			0.118974			0.0594870	+	0.998229i	$0.481054\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.21645e11	+	7.02316e10i	−1.31974	+	0.761950i
$552$		0			0
$553$	−1.00459e11	+	6.46573e10i	−1.07421	+	0.691380i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	7.64764e10	+	4.41537e10i	0.794523	+	0.458718i	0.841553	−	0.540175i	$-0.181642\pi$
$557$	7.64764e10	+	4.41537e10i	0.794523	+	0.458718i	−0.0470291	+	0.998894i	$0.514975\pi$
$558$		0			0
$559$	−1.69827e10			−0.173924
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.39954e11	−	8.08027e10i	−1.39301	−	0.804253i	−0.399359	−	0.916794i	$-0.630767\pi$
$563$	−1.39954e11	−	8.08027e10i	−1.39301	−	0.804253i	−0.993647	+	0.112542i	$0.964101\pi$
$564$		0			0
$565$	−5.07624e9	−	8.79231e9i	−0.0498136	−	0.0862798i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.02934e11	−	5.94289e10i	0.981995	−	0.566955i	0.0791229	−	0.996865i	$-0.474788\pi$
$569$	1.02934e11	−	5.94289e10i	0.981995	−	0.566955i	0.902872	+	0.429910i	$0.141455\pi$
$570$		0			0
$571$	1.38083e9	−	2.39167e9i	0.0129896	−	0.0224987i	−0.859458	−	0.511207i	$-0.829198\pi$
$571$	1.38083e9	−	2.39167e9i	0.0129896	−	0.0224987i	0.872447	+	0.488708i	$0.162532\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			3.24605e10i			0.296951i
$576$		0			0
$577$	7.45450e9	−	1.29116e10i	0.0672536	−	0.116487i	−0.830438	−	0.557111i	$-0.811910\pi$
$577$	7.45450e9	−	1.29116e10i	0.0672536	−	0.116487i	0.897691	+	0.440625i	$0.145243\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	6.83228e10	−	3.30090e9i	0.599600	−	0.0289686i
$582$		0			0
$583$	−5.74795e10	−	9.95575e10i	−0.497553	−	0.861786i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1.54651e11i			1.30256i	0.758836	+	0.651282i	$0.225769\pi$
$587$			1.54651e11i			1.30256i	−0.758836	+	0.651282i	$0.774231\pi$
$588$		0			0
$589$	−1.28628e11			−1.06874
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.76058e11	−	1.01647e11i	1.42376	−	0.822007i	0.427140	−	0.904185i	$-0.359521\pi$
$593$	1.76058e11	−	1.01647e11i	1.42376	−	0.822007i	0.996618	+	0.0821785i	$0.0261878\pi$
$594$		0			0
$595$	8.75203e7	+	1.81152e9i	0.000698299	+	0.0144536i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.94736e11	−	1.12431e11i	−1.51265	−	0.873329i	−0.999891	−	0.0147963i	$-0.995290\pi$
$599$	−1.94736e11	−	1.12431e11i	−1.51265	−	0.873329i	−0.512759	−	0.858532i	$-0.671377\pi$
$600$		0			0
$601$	−1.77363e11			−1.35945			−0.679726	−	0.733466i	$-0.737901\pi$
$601$	−1.77363e11			−1.35945			−0.679726	+	0.733466i	$0.737901\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9.42409e9	+	5.44100e9i	0.0703425	+	0.0406123i
$606$		0			0
$607$	1.03220e11	+	1.78782e11i	0.760343	+	1.31695i	0.942674	+	0.333715i	$0.108302\pi$
$607$	1.03220e11	+	1.78782e11i	0.760343	+	1.31695i	−0.182331	+	0.983237i	$0.558364\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.98244e9	−	2.29926e9i	0.0285749	−	0.0164977i
$612$		0			0
$613$	8.92529e10	−	1.54591e11i	0.632092	−	1.09482i	−0.355031	−	0.934854i	$-0.615530\pi$
$613$	8.92529e10	−	1.54591e11i	0.632092	−	1.09482i	0.987123	−	0.159961i	$-0.0511369\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			7.92670e9i			0.0546956i	0.999626	+	0.0273478i	$0.00870615\pi$
$617$			7.92670e9i			0.0546956i	−0.999626	+	0.0273478i	$0.991294\pi$
$618$		0			0
$619$	−9.30080e10	+	1.61095e11i	−0.633516	+	1.09728i	0.353311	+	0.935506i	$0.385056\pi$
$619$	−9.30080e10	+	1.61095e11i	−0.633516	+	1.09728i	−0.986827	+	0.161776i	$0.948278\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9.03377e10	+	1.40360e11i	0.599676	+	0.931729i
$624$		0			0
$625$	−7.52382e10	−	1.30316e11i	−0.493081	−	0.854042i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.89328e10i		−	0.120952i
$630$		0			0
$631$	1.65626e11			1.04474			0.522372	−	0.852717i	$-0.325047\pi$
$631$	1.65626e11			1.04474			0.522372	+	0.852717i	$0.325047\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.88165e9	+	3.39577e9i	−0.0361747	+	0.0208855i
$636$		0			0
$637$	1.92978e11	−	1.86904e10i	1.17206	−	0.113517i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.35061e11	−	7.79773e10i	−0.800012	−	0.461887i	0.0434633	−	0.999055i	$-0.486161\pi$
$641$	−1.35061e11	−	7.79773e10i	−0.800012	−	0.461887i	−0.843475	+	0.537168i	$0.819494\pi$
$642$		0			0
$643$	2.44491e11			1.43027			0.715136	−	0.698986i	$-0.246365\pi$
$643$	2.44491e11			1.43027			0.715136	+	0.698986i	$0.246365\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7.45655e10	−	4.30504e10i	−0.425521	−	0.245675i	0.271916	−	0.962321i	$-0.412343\pi$
$647$	−7.45655e10	−	4.30504e10i	−0.425521	−	0.245675i	−0.697437	+	0.716647i	$0.745676\pi$
$648$		0			0
$649$	−4.21314e10	−	7.29738e10i	−0.237480	−	0.411328i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.96231e9	−	3.44234e9i	0.0327915	−	0.0189322i	−0.483515	−	0.875336i	$-0.660640\pi$
$653$	5.96231e9	−	3.44234e9i	0.0327915	−	0.0189322i	0.516306	+	0.856404i	$0.327307\pi$
$654$		0			0
$655$	4.42482e9	−	7.66401e9i	0.0240398	−	0.0416381i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	6.39086e10i		−	0.338858i	−0.985542	−	0.169429i	$-0.945808\pi$
$659$		−	6.39086e10i		−	0.338858i	0.985542	−	0.169429i	$-0.0541923\pi$
$660$		0			0
$661$	−6.36970e9	+	1.10326e10i	−0.0333667	+	0.0577928i	−0.882227	−	0.470825i	$-0.843956\pi$
$661$	−6.36970e9	+	1.10326e10i	−0.0333667	+	0.0577928i	0.848860	+	0.528618i	$0.177290\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5.36136e9	+	1.04169e10i	−0.0274150	+	0.0532661i
$666$		0			0
$667$	−5.10445e10	−	8.84117e10i	−0.257897	−	0.446690i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			3.44431e11i			1.69907i
$672$		0			0
$673$	3.95055e10			0.192574			0.0962870	−	0.995354i	$-0.469303\pi$
$673$	3.95055e10			0.192574			0.0962870	+	0.995354i	$0.469303\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.56598e11	−	1.48147e11i	1.22152	−	0.705243i	0.256275	−	0.966604i	$-0.417505\pi$
$677$	2.56598e11	−	1.48147e11i	1.22152	−	0.705243i	0.965241	+	0.261361i	$0.0841714\pi$
$678$		0			0
$679$	1.71797e10	+	3.55590e11i	0.0808232	+	1.67290i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	2.29563e11	+	1.32538e11i	1.05492	+	0.609058i	0.924023	−	0.382338i	$-0.124881\pi$
$683$	2.29563e11	+	1.32538e11i	1.05492	+	0.609058i	0.130897	+	0.991396i	$0.458214\pi$
$684$		0			0
$685$	−2.77699e10			−0.126128
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.54359e11	+	8.91193e10i	0.684945	+	0.395453i
$690$		0			0
$691$	−1.44204e11	−	2.49769e11i	−0.632507	−	1.09553i	−0.987038	−	0.160490i	$-0.948693\pi$
$691$	−1.44204e11	−	2.49769e11i	−0.632507	−	1.09553i	0.354531	−	0.935044i	$-0.384641\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.31883e10	+	7.61424e9i	−0.0565260	+	0.0326353i
$696$		0			0
$697$	4.79509e10	−	8.30534e10i	0.203173	−	0.351906i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			2.31307e11i			0.957894i	0.877844	+	0.478947i	$0.158981\pi$
$701$			2.31307e11i			0.957894i	−0.877844	+	0.478947i	$0.841019\pi$
$702$		0			0
$703$	6.11505e10	−	1.05916e11i	0.250368	−	0.433650i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.43812e11	+	2.79420e11i	−0.575596	+	1.11835i
$708$		0			0
$709$	−1.00350e11	−	1.73811e11i	−0.397129	−	0.687848i	0.596241	−	0.802805i	$-0.296660\pi$
$709$	−1.00350e11	−	1.73811e11i	−0.397129	−	0.687848i	−0.993370	+	0.114957i	$0.963327\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	9.34870e10i		−	0.361737i
$714$		0			0
$715$	−3.09903e10			−0.118577
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8.75820e10	−	5.05655e10i	0.327717	−	0.189208i	−0.327110	−	0.944986i	$-0.606075\pi$
$719$	8.75820e10	−	5.05655e10i	0.327717	−	0.189208i	0.654827	+	0.755779i	$0.272741\pi$
$720$		0			0
$721$	4.21762e11	−	2.71453e11i	1.56072	−	1.00451i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.11768e11	+	2.37734e11i	1.49039	+	0.860478i
$726$		0			0
$727$	−1.13682e11			−0.406963			−0.203481	−	0.979079i	$-0.565226\pi$
$727$	−1.13682e11			−0.406963			−0.203481	+	0.979079i	$0.565226\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.77611e9	−	4.48954e9i	−0.0272328	−	0.0157229i
$732$		0			0
$733$	−1.85533e11	−	3.21352e11i	−0.642695	−	1.11318i	−0.984829	−	0.173529i	$-0.944483\pi$
$733$	−1.85533e11	−	3.21352e11i	−0.642695	−	1.11318i	0.342133	−	0.939651i	$-0.388851\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.76443e11	−	2.17340e11i	1.27594	−	0.736663i
$738$		0			0
$739$	1.37581e10	−	2.38297e10i	0.0461297	−	0.0798990i	−0.842039	−	0.539417i	$-0.818645\pi$
$739$	1.37581e10	−	2.38297e10i	0.0461297	−	0.0798990i	0.888168	+	0.459518i	$0.151978\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	4.06856e11i		−	1.33501i	−0.744605	−	0.667506i	$-0.767362\pi$
$743$		−	4.06856e11i		−	1.33501i	0.744605	−	0.667506i	$-0.232638\pi$
$744$		0			0
$745$	−4.43177e9	+	7.67606e9i	−0.0143864	+	0.0249180i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.58076e11	+	4.00978e11i	0.820012	+	1.27407i
$750$		0			0
$751$	1.46757e11	+	2.54190e11i	0.461359	+	0.799097i	0.999029	−	0.0440583i	$-0.0140287\pi$
$751$	1.46757e11	+	2.54190e11i	0.461359	+	0.799097i	−0.537670	+	0.843155i	$0.680695\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			3.44342e10i			0.105975i
$756$		0			0
$757$	9.73267e9			0.0296380			0.0148190	−	0.999890i	$-0.495283\pi$
$757$	9.73267e9			0.0296380			0.0148190	+	0.999890i	$0.495283\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.39468e11	−	1.38257e11i	0.714018	−	0.412238i	−0.0985291	−	0.995134i	$-0.531414\pi$
$761$	2.39468e11	−	1.38257e11i	0.714018	−	0.412238i	0.812547	+	0.582896i	$0.198080\pi$
$762$		0			0
$763$	−3.18889e11	−	1.64126e11i	−0.940896	−	0.484261i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.13142e11	+	6.53228e10i	0.326922	+	0.188748i
$768$		0			0
$769$	3.57246e11			1.02156			0.510778	−	0.859713i	$-0.329358\pi$
$769$	3.57246e11			1.02156			0.510778	+	0.859713i	$0.329358\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−3.39187e11	−	1.95830e11i	−0.949995	−	0.548480i	−0.0569158	−	0.998379i	$-0.518127\pi$
$773$	−3.39187e11	−	1.95830e11i	−0.949995	−	0.548480i	−0.893079	+	0.449899i	$0.851460\pi$
$774$		0			0
$775$	2.17703e11	+	3.77072e11i	0.603471	+	1.04524i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.36503e11	−	3.09750e11i	1.45688	−	0.841128i
$780$		0			0
$781$	1.65534e11	−	2.86713e11i	0.444921	−	0.770626i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			2.98879e10i			0.0787077i
$786$		0			0
$787$	1.47643e11	−	2.55726e11i	0.384871	−	0.666616i	−0.606880	−	0.794793i	$-0.707579\pi$
$787$	1.47643e11	−	2.55726e11i	0.384871	−	0.666616i	0.991751	+	0.128177i	$0.0409127\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−5.73160e11	+	2.76912e10i	−1.46410	+	0.0707352i
$792$		0			0
$793$	−2.67012e11	−	4.62478e11i	−0.675209	−	1.16950i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			7.63028e10i			0.189107i	0.995520	+	0.0945534i	$0.0301423\pi$
$797$			7.63028e10i			0.189107i	−0.995520	+	0.0945534i	$0.969858\pi$
$798$		0			0
$799$	2.43132e9			0.00596562
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5.30240e11	+	3.06134e11i	−1.27529	+	0.736291i
$804$		0			0
$805$	−7.57101e9	−	3.89665e9i	−0.0180289	−	0.00927915i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.14896e11	+	3.55010e11i	1.43551	+	0.828795i	0.997534	−	0.0701899i	$-0.0223605\pi$
$809$	6.14896e11	+	3.55010e11i	1.43551	+	0.828795i	0.437981	+	0.898984i	$0.355694\pi$
$810$		0			0
$811$	7.28420e11			1.68383			0.841916	−	0.539609i	$-0.181428\pi$
$811$	7.28420e11			1.68383			0.841916	+	0.539609i	$0.181428\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.00352e9	−	3.46613e9i	−0.0136074	−	0.00785624i
$816$		0			0
$817$	−2.90013e10	−	5.02317e10i	−0.0650922	−	0.112743i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	4.30592e11	−	2.48602e11i	0.947748	−	0.547183i	0.0553673	−	0.998466i	$-0.482367\pi$
$821$	4.30592e11	−	2.48602e11i	0.947748	−	0.547183i	0.892381	+	0.451284i	$0.149034\pi$
$822$		0			0
$823$	−4.29315e10	+	7.43596e10i	−0.0935787	+	0.162083i	−0.909015	−	0.416764i	$-0.863164\pi$
$823$	−4.29315e10	+	7.43596e10i	−0.0935787	+	0.162083i	0.815436	+	0.578847i	$0.196497\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			9.26379e11i			1.98046i	0.139432	+	0.990232i	$0.455472\pi$
$827$			9.26379e11i			1.98046i	−0.139432	+	0.990232i	$0.544528\pi$
$828$		0			0
$829$	−4.36477e11	+	7.56001e11i	−0.924152	+	1.60068i	−0.131234	+	0.991351i	$0.541894\pi$
$829$	−4.36477e11	+	7.56001e11i	−0.924152	+	1.60068i	−0.792919	+	0.609328i	$0.791439\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.33024e10	+	4.24574e10i	0.193782	+	0.0881808i
$834$		0			0
$835$	−1.83143e10	−	3.17213e10i	−0.0376742	−	0.0652536i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			6.86462e11i			1.38538i	0.721236	+	0.692689i	$0.243574\pi$
$839$			6.86462e11i			1.38538i	−0.721236	+	0.692689i	$0.756426\pi$
$840$		0			0
$841$	−9.95111e11			−1.98924
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.16021e10	−	6.69847e9i	0.0227567	−	0.0131386i
$846$		0			0
$847$	5.17198e11	−	3.32877e11i	1.00490	−	0.646770i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7.69798e10	+	4.44443e10i	0.146777	+	0.0847419i
$852$		0			0
$853$	8.47904e11			1.60159			0.800793	−	0.598941i	$-0.204411\pi$
$853$	8.47904e11			1.60159			0.800793	+	0.598941i	$0.204411\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.03566e11	−	1.75264e11i	−0.562769	−	0.324915i	0.191487	−	0.981495i	$-0.438669\pi$
$857$	−3.03566e11	−	1.75264e11i	−0.562769	−	0.324915i	−0.754256	+	0.656580i	$0.772002\pi$
$858$		0			0
$859$	−3.15123e11	−	5.45809e11i	−0.578772	−	1.00246i	−0.995621	−	0.0934868i	$-0.970199\pi$
$859$	−3.15123e11	−	5.45809e11i	−0.578772	−	1.00246i	0.416848	−	0.908976i	$-0.363135\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−6.44008e10	+	3.71818e10i	−0.116104	+	0.0670328i	−0.556927	−	0.830561i	$-0.688020\pi$
$863$	−6.44008e10	+	3.71818e10i	−0.116104	+	0.0670328i	0.440823	+	0.897594i	$0.354687\pi$
$864$		0			0
$865$	−2.40755e10	+	4.16999e10i	−0.0430041	+	0.0744854i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1.07933e12i		−	1.89266i
$870$		0			0
$871$	−3.36975e11	+	5.83657e11i	−0.585497	+	1.01411i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7.94061e10	−	3.83636e9i	0.135463	−	0.00654467i
$876$		0			0
$877$	−3.03583e11	−	5.25822e11i	−0.513192	−	0.888874i	−0.999883	−	0.0152999i	$-0.995130\pi$
$877$	−3.03583e11	−	5.25822e11i	−0.513192	−	0.888874i	0.486691	−	0.873574i	$-0.338204\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1.79724e11i			0.298334i	0.988812	+	0.149167i	$0.0476593\pi$
$881$			1.79724e11i			0.298334i	−0.988812	+	0.149167i	$0.952341\pi$
$882$		0			0
$883$	3.97608e11			0.654051			0.327026	−	0.945015i	$-0.393954\pi$
$883$	3.97608e11			0.654051			0.327026	+	0.945015i	$0.393954\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.64234e10	+	3.25761e10i	−0.0911518	+	0.0526265i	−0.544883	−	0.838512i	$-0.683426\pi$
$887$	−5.64234e10	+	3.25761e10i	−0.0911518	+	0.0526265i	0.453731	+	0.891139i	$0.350093\pi$
$888$		0			0
$889$	1.85241e10	+	3.83418e11i	0.0296573	+	0.613854i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.36016e10	+	7.85286e9i	0.0213886	+	0.0123487i
$894$		0			0
$895$	−6.98523e10			−0.108865
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.18590e12	−	6.84679e11i	−1.81555	−	1.04821i
$900$		0			0
$901$	4.71190e10	+	8.16124e10i	0.0714984	+	0.123839i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.69078e10	−	1.55352e10i	0.0401129	−	0.0231592i
$906$		0			0
$907$	−2.26093e11	+	3.91605e11i	−0.334086	+	0.578653i	−0.983309	−	0.181945i	$-0.941761\pi$
$907$	−2.26093e11	+	3.91605e11i	−0.334086	+	0.578653i	0.649223	+	0.760598i	$0.275094\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1.36713e12i		−	1.98489i	−0.122678	−	0.992447i	$-0.539148\pi$
$911$		−	1.36713e12i		−	1.98489i	0.122678	−	0.992447i	$-0.460852\pi$
$912$		0			0
$913$	−3.08989e11	+	5.35184e11i	−0.444692	+	0.770230i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.70708e11	−	4.20604e11i	−0.382845	−	0.594834i
$918$		0			0
$919$	4.01410e10	+	6.95262e10i	0.0562763	+	0.0974735i	0.892791	−	0.450471i	$-0.148744\pi$
$919$	4.01410e10	+	6.95262e10i	0.0562763	+	0.0974735i	−0.836515	+	0.547944i	$0.815411\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			5.13305e11i			0.707243i
$924$		0			0
$925$	−4.13989e11			−0.565486
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.66173e11	+	9.59400e10i	−0.223099	+	0.128806i	−0.607384	−	0.794408i	$-0.707781\pi$
$929$	−1.66173e11	+	9.59400e10i	−0.223099	+	0.128806i	0.384285	+	0.923214i	$0.374448\pi$
$930$		0			0
$931$	3.84829e11	+	5.38874e11i	0.512235	+	0.717280i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.41899e10	−	8.19257e9i	−0.0185667	−	0.0107195i
$936$		0			0
$937$	6.82042e10			0.0884816			0.0442408	−	0.999021i	$-0.485913\pi$
$937$	6.82042e10			0.0884816			0.0442408	+	0.999021i	$0.485913\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−4.33449e11	−	2.50252e11i	−0.552814	−	0.319168i	0.197442	−	0.980315i	$-0.436737\pi$
$941$	−4.33449e11	−	2.50252e11i	−0.552814	−	0.319168i	−0.750256	+	0.661147i	$0.770070\pi$
$942$		0			0
$943$	2.25127e11	+	3.89932e11i	0.284696	+	0.493108i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4.85222e11	−	2.80143e11i	0.603310	−	0.348321i	−0.167033	−	0.985951i	$-0.553419\pi$
$947$	4.85222e11	−	2.80143e11i	0.603310	−	0.348321i	0.770343	+	0.637630i	$0.220085\pi$
$948$		0			0
$949$	4.74647e11	−	8.22112e11i	0.585202	−	1.01360i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	3.76236e11i		−	0.456131i	−0.973646	−	0.228065i	$-0.926760\pi$
$953$		−	3.76236e11i		−	0.456131i	0.973646	−	0.228065i	$-0.0732400\pi$
$954$		0			0
$955$	−4.18433e9	+	7.24747e9i	−0.00503051	+	0.00871310i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−7.18281e11	+	1.39558e12i	−0.849219	+	1.64999i
$960$		0			0
$961$	−2.00542e11	−	3.47349e11i	−0.235132	−	0.407260i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			1.05958e11i			0.122186i
$966$		0			0
$967$	−9.30742e11			−1.06445			−0.532223	−	0.846604i	$-0.678643\pi$
$967$	−9.30742e11			−1.06445			−0.532223	+	0.846604i	$0.678643\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.48063e12	−	8.54843e11i	1.66560	−	0.961633i	0.695629	−	0.718401i	$-0.255126\pi$
$971$	1.48063e12	−	8.54843e11i	1.66560	−	0.961633i	0.969968	−	0.243232i	$-0.0782075\pi$
$972$		0			0
$973$	4.15361e10	+	8.59726e11i	0.0463420	+	0.959199i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.06224e12	−	6.13285e11i	−1.16586	−	0.673107i	−0.213155	−	0.977018i	$-0.568374\pi$
$977$	−1.06224e12	−	6.13285e11i	−1.16586	−	0.673107i	−0.952700	+	0.303912i	$0.901707\pi$
$978$		0			0
$979$	−1.50801e12			−1.64162
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.11814e12	−	6.45558e11i	−1.19752	−	0.691386i	−0.237516	−	0.971384i	$-0.576333\pi$
$983$	−1.11814e12	−	6.45558e11i	−1.19752	−	0.691386i	−0.960001	+	0.279997i	$0.909666\pi$
$984$		0			0
$985$	1.70832e10	+	2.95890e10i	0.0181479	+	0.0314330i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.65085e10	−	2.10782e10i	0.0381601	−	0.0220317i
$990$		0			0
$991$	1.36853e11	−	2.37037e11i	0.141893	−	0.245766i	−0.786317	−	0.617824i	$-0.788014\pi$
$991$	1.36853e11	−	2.37037e11i	0.141893	−	0.245766i	0.928209	+	0.372058i	$0.121348\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			5.20138e10i			0.0530672i
$996$		0			0
$997$	−7.53655e11	+	1.30537e12i	−0.762767	+	1.32115i	0.178651	+	0.983912i	$0.442827\pi$
$997$	−7.53655e11	+	1.30537e12i	−0.762767	+	1.32115i	−0.941419	+	0.337240i	$0.890507\pi$
$998$		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.53.13	yes	44
3.2	odd	2	inner	252.9.bk.a.53.10	✓	44
7.2	even	3	inner	252.9.bk.a.233.10	yes	44
21.2	odd	6	inner	252.9.bk.a.233.13	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bk.a.53.10	✓	44	3.2	odd	2	inner
252.9.bk.a.53.13	yes	44	1.1	even	1	trivial
252.9.bk.a.233.10	yes	44	7.2	even	3	inner
252.9.bk.a.233.13	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)

\(f(q)\)	\(=\)	\(q+(36.7886 - 21.2399i) q^{5} +(-115.865 - 2398.20i) q^{7} +O(q^{10})\)	\(q+(36.7886 - 21.2399i) q^{5} +(-115.865 - 2398.20i) q^{7} +(18785.5 + 10845.8i) q^{11} -33631.9 q^{13} +(-15399.5 - 8890.88i) q^{17} +(-57432.8 - 99476.6i) q^{19} +(72299.8 - 41742.3i) q^{23} +(-194410. + 336728. i) q^{25} -1.22285e6i q^{29} +(559905. - 969784. i) q^{31} +(-55200.1 - 85765.5i) q^{35} +(532365. + 922084. i) q^{37} +5.39326e6i q^{41} +504960. q^{43} +(-118413. + 68365.6i) q^{47} +(-5.73795e6 + 555735. i) q^{49} +(-4.58967e6 - 2.64985e6i) q^{53} +921457. q^{55} +(-3.36414e6 - 1.94229e6i) q^{59} +(7.93925e6 + 1.37512e7i) q^{61} +(-1.23727e6 + 714337. i) q^{65} +(1.00195e7 - 1.73543e7i) q^{67} -1.52625e7i q^{71} +(-1.41130e7 + 2.44444e7i) q^{73} +(2.38339e7 - 4.63081e7i) q^{77} +(-2.48788e7 - 4.30914e7i) q^{79} +2.84892e7i q^{83} -755366. q^{85} +(-6.02063e7 + 3.47601e7i) q^{89} +(3.89675e6 + 8.06561e7i) q^{91} +(-4.22574e6 - 2.43973e6i) q^{95} -1.48273e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 1230 q^{7}+O(q^{10}) \) 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

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