LMFDB - Embedded newform 252.9.bk.a.233.8

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			233.8
Character	$\chi$	$=$	252.233
Dual form			252.9.bk.a.53.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-318.126 - 183.670i) q^{5} +(1593.54 - 1795.95i) q^{7} +O(q^{10})$	$q+(-318.126 - 183.670i) q^{5} +(1593.54 - 1795.95i) q^{7} +(-9689.38 + 5594.16i) q^{11} +39074.1 q^{13} +(-78224.9 + 45163.2i) q^{17} +(-47061.4 + 81512.8i) q^{19} +(-177185. - 102298. i) q^{23} +(-127843. - 221431. i) q^{25} +77113.6i q^{29} +(31704.9 + 54914.5i) q^{31} +(-836808. + 278655. i) q^{35} +(289978. - 502257. i) q^{37} +1.45775e6i q^{41} +76887.2 q^{43} +(-467791. - 270079. i) q^{47} +(-686092. - 5.72383e6i) q^{49} +(1.13764e7 - 6.56818e6i) q^{53} +4.10993e6 q^{55} +(-2.24503e6 + 1.29617e6i) q^{59} +(-6.68659e6 + 1.15815e7i) q^{61} +(-1.24305e7 - 7.17676e6i) q^{65} +(-191597. - 331856. i) q^{67} +3.97275e7i q^{71} +(2.16726e7 + 3.75380e7i) q^{73} +(-5.39351e6 + 2.63162e7i) q^{77} +(-7.49530e6 + 1.29822e7i) q^{79} +1.06260e7i q^{83} +3.31805e7 q^{85} +(-5.41475e7 - 3.12621e7i) q^{89} +(6.22660e7 - 7.01753e7i) q^{91} +(2.99429e7 - 1.72876e7i) q^{95} -7.35805e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

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

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
$5$	−318.126	−	183.670i	−0.509002	−	0.293872i	0.223421	−	0.974722i	$-0.428277\pi$
$5$	−318.126	−	183.670i	−0.509002	−	0.293872i	−0.732423	+	0.680850i	$0.761611\pi$
$6$		0			0
$7$	1593.54	−	1795.95i	0.663696	−	0.748002i
$7$	1593.54	−	1795.95i	0.663696	−	0.748002i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−9689.38	+	5594.16i	−0.661797	+	0.382089i	−0.792962	−	0.609272i	$-0.791462\pi$
$11$	−9689.38	+	5594.16i	−0.661797	+	0.382089i	0.131164	+	0.991361i	$0.458129\pi$
$12$		0			0
$13$	39074.1			1.36809			0.684047	−	0.729438i	$-0.260218\pi$
$13$	39074.1			1.36809			0.684047	+	0.729438i	$0.260218\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−78224.9	+	45163.2i	−0.936589	+	0.540740i	−0.888890	−	0.458122i	$-0.848522\pi$
$17$	−78224.9	+	45163.2i	−0.936589	+	0.540740i	−0.0476998	+	0.998862i	$0.515189\pi$
$18$		0			0
$19$	−47061.4	+	81512.8i	−0.361119	+	0.625477i	−0.988145	−	0.153521i	$-0.950939\pi$
$19$	−47061.4	+	81512.8i	−0.361119	+	0.625477i	0.627026	+	0.778998i	$0.284272\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−177185.	−	102298.i	−0.633164	−	0.365558i	0.148812	−	0.988865i	$-0.452455\pi$
$23$	−177185.	−	102298.i	−0.633164	−	0.365558i	−0.781977	+	0.623308i	$0.785788\pi$
$24$		0			0
$25$	−127843.	−	221431.i	−0.327278	−	0.566862i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			77113.6i			0.109028i	0.998513	+	0.0545142i	$0.0173610\pi$
$29$			77113.6i			0.109028i	−0.998513	+	0.0545142i	$0.982639\pi$
$30$		0			0
$31$	31704.9	+	54914.5i	0.0343304	+	0.0594621i	0.882680	−	0.469974i	$-0.155737\pi$
$31$	31704.9	+	54914.5i	0.0343304	+	0.0594621i	−0.848350	+	0.529436i	$0.822403\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−836808.	+	278655.i	−0.557640	+	0.185692i
$36$		0			0
$37$	289978.	−	502257.i	0.154724	−	0.267990i	−0.778234	−	0.627974i	$-0.783884\pi$
$37$	289978.	−	502257.i	0.154724	−	0.267990i	0.932959	+	0.359984i	$0.117218\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.45775e6i			0.515879i	0.966161	+	0.257940i	$0.0830435\pi$
$41$			1.45775e6i			0.515879i	−0.966161	+	0.257940i	$0.916956\pi$
$42$		0			0
$43$	76887.2			0.0224895			0.0112448	−	0.999937i	$-0.496421\pi$
$43$	76887.2			0.0224895			0.0112448	+	0.999937i	$0.496421\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−467791.	−	270079.i	−0.0958651	−	0.0553477i	0.451301	−	0.892372i	$-0.350960\pi$
$47$	−467791.	−	270079.i	−0.0958651	−	0.0553477i	−0.547166	+	0.837024i	$0.684293\pi$
$48$		0			0
$49$	−686092.	−	5.72383e6i	−0.119014	−	0.992893i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.13764e7	−	6.56818e6i	1.44179	−	0.832419i	0.443822	−	0.896115i	$-0.353622\pi$
$53$	1.13764e7	−	6.56818e6i	1.44179	−	0.832419i	0.997969	+	0.0636964i	$0.0202889\pi$
$54$		0			0
$55$	4.10993e6			0.449142
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.24503e6	+	1.29617e6i	−0.185274	+	0.106968i	−0.589768	−	0.807573i	$-0.700781\pi$
$59$	−2.24503e6	+	1.29617e6i	−0.185274	+	0.106968i	0.404494	+	0.914540i	$0.367448\pi$
$60$		0			0
$61$	−6.68659e6	+	1.15815e7i	−0.482932	+	0.836462i	−0.999808	−	0.0195980i	$-0.993761\pi$
$61$	−6.68659e6	+	1.15815e7i	−0.482932	+	0.836462i	0.516876	+	0.856060i	$0.327095\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.24305e7	−	7.17676e6i	−0.696363	−	0.402045i
$66$		0			0
$67$	−191597.	−	331856.i	−0.00950803	−	0.0164684i	0.861232	−	0.508212i	$-0.169693\pi$
$67$	−191597.	−	331856.i	−0.00950803	−	0.0164684i	−0.870740	+	0.491743i	$0.836360\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.97275e7i			1.56336i	0.623683	+	0.781678i	$0.285636\pi$
$71$			3.97275e7i			1.56336i	−0.623683	+	0.781678i	$0.714364\pi$
$72$		0			0
$73$	2.16726e7	+	3.75380e7i	0.763167	+	1.32184i	0.941210	+	0.337821i	$0.109690\pi$
$73$	2.16726e7	+	3.75380e7i	0.763167	+	1.32184i	−0.178044	+	0.984023i	$0.556977\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.39351e6	+	2.63162e7i	−0.153429	+	0.748617i
$78$		0			0
$79$	−7.49530e6	+	1.29822e7i	−0.192434	+	0.333305i	−0.946056	−	0.324003i	$-0.894971\pi$
$79$	−7.49530e6	+	1.29822e7i	−0.192434	+	0.333305i	0.753623	+	0.657307i	$0.228305\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1.06260e7i			0.223902i	0.993714	+	0.111951i	$0.0357100\pi$
$83$			1.06260e7i			0.223902i	−0.993714	+	0.111951i	$0.964290\pi$
$84$		0			0
$85$	3.31805e7			0.635634
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.41475e7	−	3.12621e7i	−0.863015	−	0.498262i	0.00200600	−	0.999998i	$-0.499361\pi$
$89$	−5.41475e7	−	3.12621e7i	−0.863015	−	0.498262i	−0.865021	+	0.501736i	$0.832695\pi$
$90$		0			0
$91$	6.22660e7	−	7.01753e7i	0.907999	−	1.02334i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.99429e7	−	1.72876e7i	0.367621	−	0.212246i
$96$		0			0
$97$	−7.35805e7			−0.831143			−0.415571	−	0.909561i	$-0.636418\pi$
$97$	−7.35805e7			−0.831143			−0.415571	+	0.909561i	$0.636418\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.28737e7	+	3.63002e7i	−0.604204	+	0.348838i	−0.770694	−	0.637206i	$-0.780090\pi$
$101$	−6.28737e7	+	3.63002e7i	−0.604204	+	0.348838i	0.166489	+	0.986043i	$0.446757\pi$
$102$		0			0
$103$	4.28747e7	−	7.42612e7i	0.380936	−	0.659801i	−0.610260	−	0.792201i	$-0.708935\pi$
$103$	4.28747e7	−	7.42612e7i	0.380936	−	0.659801i	0.991196	+	0.132400i	$0.0422684\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.44527e7	+	4.29853e7i	0.567996	+	0.327933i	0.756349	−	0.654169i	$-0.226981\pi$
$107$	7.44527e7	+	4.29853e7i	0.567996	+	0.327933i	−0.188352	+	0.982102i	$0.560315\pi$
$108$		0			0
$109$	1.41127e7	+	2.44439e7i	0.0999780	+	0.173167i	0.911675	−	0.410911i	$-0.134789\pi$
$109$	1.41127e7	+	2.44439e7i	0.0999780	+	0.173167i	−0.811697	+	0.584078i	$0.801456\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1.63762e8i			1.00438i	0.864757	+	0.502191i	$0.167473\pi$
$113$			1.63762e8i			1.00438i	−0.864757	+	0.502191i	$0.832527\pi$
$114$		0			0
$115$	3.75782e7	+	6.50874e7i	0.214855	+	0.372139i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.35432e7	+	2.12457e8i	−0.217136	+	1.05946i
$120$		0			0
$121$	−4.45901e7	+	7.72323e7i	−0.208016	+	0.360294i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			2.37416e8i			0.972457i
$126$		0			0
$127$	3.79953e8			1.46054			0.730272	−	0.683156i	$-0.239393\pi$
$127$	3.79953e8			1.46054			0.730272	+	0.683156i	$0.239393\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.65157e8	−	9.53532e7i	−0.560803	−	0.323780i	0.192665	−	0.981265i	$-0.438287\pi$
$131$	−1.65157e8	−	9.53532e7i	−0.560803	−	0.323780i	−0.753468	+	0.657485i	$0.771620\pi$
$132$		0			0
$133$	7.13991e7	+	2.14414e8i	0.228184	+	0.685245i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.22765e8	+	2.44084e8i	−1.20010	+	0.692877i	−0.960577	−	0.278014i	$-0.910324\pi$
$137$	−4.22765e8	+	2.44084e8i	−1.20010	+	0.692877i	−0.239521	+	0.970891i	$0.576990\pi$
$138$		0			0
$139$	3.92373e8			1.05109			0.525545	−	0.850766i	$-0.323862\pi$
$139$	3.92373e8			1.05109			0.525545	+	0.850766i	$0.323862\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.78604e8	+	2.18587e8i	−0.905401	+	0.522734i
$144$		0			0
$145$	1.41635e7	−	2.45319e7i	0.0320404	−	0.0554956i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.06188e8	+	2.92248e8i	1.02699	+	0.592934i	0.916122	−	0.400901i	$-0.131303\pi$
$149$	5.06188e8	+	2.92248e8i	1.02699	+	0.592934i	0.110871	+	0.993835i	$0.464636\pi$
$150$		0			0
$151$	3.35728e8	+	5.81497e8i	0.645772	+	1.11851i	0.984123	+	0.177490i	$0.0567978\pi$
$151$	3.35728e8	+	5.81497e8i	0.645772	+	1.11851i	−0.338350	+	0.941020i	$0.609869\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	2.32930e7i		−	0.0403551i
$156$		0			0
$157$	1.03816e8	+	1.79815e8i	0.170870	+	0.295956i	0.938724	−	0.344669i	$-0.112009\pi$
$157$	1.03816e8	+	1.79815e8i	0.170870	+	0.295956i	−0.767854	+	0.640625i	$0.778675\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.66073e8	+	1.55201e8i	−0.693667	+	0.230989i
$162$		0			0
$163$	−1.67092e8	+	2.89411e8i	−0.236704	+	0.409983i	−0.959766	−	0.280800i	$-0.909400\pi$
$163$	−1.67092e8	+	2.89411e8i	−0.236704	+	0.409983i	0.723063	+	0.690782i	$0.242734\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			1.25347e9i			1.61157i	0.592210	+	0.805784i	$0.298256\pi$
$167$			1.25347e9i			1.61157i	−0.592210	+	0.805784i	$0.701744\pi$
$168$		0			0
$169$	7.11058e8			0.871683
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.19781e9	+	6.91558e8i	1.33723	+	0.772048i	0.986395	−	0.164390i	$-0.0525657\pi$
$173$	1.19781e9	+	6.91558e8i	1.33723	+	0.772048i	0.350831	+	0.936439i	$0.385899\pi$
$174$		0			0
$175$	−6.01401e8	−	1.23257e8i	−0.641227	−	0.131420i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.43101e9	+	8.26191e8i	−1.39389	+	0.804764i	−0.993744	−	0.111686i	$-0.964375\pi$
$179$	−1.43101e9	+	8.26191e8i	−1.39389	+	0.804764i	−0.400149	+	0.916450i	$0.631042\pi$
$180$		0			0
$181$	1.81734e9			1.69325			0.846626	−	0.532188i	$-0.178630\pi$
$181$	1.81734e9			1.69325			0.846626	+	0.532188i	$0.178630\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.84499e8	+	1.06521e8i	−0.157510	+	0.0909384i
$186$		0			0
$187$	5.05300e8	−	8.75206e8i	0.413222	−	0.715721i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.51019e8	−	3.75866e8i	−0.489171	−	0.282423i	0.235060	−	0.971981i	$-0.424471\pi$
$191$	−6.51019e8	−	3.75866e8i	−0.489171	−	0.282423i	−0.724230	+	0.689558i	$0.757805\pi$
$192$		0			0
$193$	−2.75569e8	−	4.77300e8i	−0.198610	−	0.344003i	0.749468	−	0.662041i	$-0.230309\pi$
$193$	−2.75569e8	−	4.77300e8i	−0.198610	−	0.344003i	−0.948078	+	0.318038i	$0.896976\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			1.68647e9i			1.11973i	0.828584	+	0.559865i	$0.189147\pi$
$197$			1.68647e9i			1.11973i	−0.828584	+	0.559865i	$0.810853\pi$
$198$		0			0
$199$	6.98830e8	+	1.21041e9i	0.445614	+	0.771827i	0.998095	−	0.0616991i	$-0.0196519\pi$
$199$	6.98830e8	+	1.21041e9i	0.445614	+	0.771827i	−0.552480	+	0.833526i	$0.686319\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.38492e8	+	1.22883e8i	0.0815534	+	0.0723617i
$204$		0			0
$205$	2.67745e8	−	4.63749e8i	0.151603	−	0.262583i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	1.05308e9i		−	0.551919i
$210$		0			0
$211$	−1.95268e9			−0.985146			−0.492573	−	0.870271i	$-0.663944\pi$
$211$	−1.95268e9			−0.985146			−0.492573	+	0.870271i	$0.663944\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.44598e7	−	1.41219e7i	−0.0114472	−	0.00660905i
$216$		0			0
$217$	1.49147e8	+	3.05677e7i	0.0672627	+	0.0137855i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.05657e9	+	1.76471e9i	−1.28134	+	0.739784i
$222$		0			0
$223$	3.75094e9			1.51677			0.758387	−	0.651804i	$-0.225988\pi$
$223$	3.75094e9			1.51677			0.758387	+	0.651804i	$0.225988\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.88830e8	+	3.97696e8i	−0.259423	+	0.149778i	−0.624071	−	0.781367i	$-0.714523\pi$
$227$	−6.88830e8	+	3.97696e8i	−0.259423	+	0.149778i	0.364648	+	0.931145i	$0.381189\pi$
$228$		0			0
$229$	−4.13394e8	+	7.16019e8i	−0.150322	+	0.260365i	−0.931346	−	0.364136i	$-0.881364\pi$
$229$	−4.13394e8	+	7.16019e8i	−0.150322	+	0.260365i	0.781024	+	0.624501i	$0.214698\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.21585e9	+	7.01972e8i	0.412531	+	0.238175i	0.691877	−	0.722016i	$-0.256784\pi$
$233$	1.21585e9	+	7.01972e8i	0.412531	+	0.238175i	−0.279346	+	0.960191i	$0.590118\pi$
$234$		0			0
$235$	9.92110e7	+	1.71839e8i	0.0325303	+	0.0563442i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			1.96293e9i			0.601606i	0.953686	+	0.300803i	$0.0972547\pi$
$239$			1.96293e9i			0.601606i	−0.953686	+	0.300803i	$0.902745\pi$
$240$		0			0
$241$	1.20189e9	+	2.08174e9i	0.356284	+	0.617103i	0.987337	−	0.158637i	$-0.0507101\pi$
$241$	1.20189e9	+	2.08174e9i	0.356284	+	0.617103i	−0.631052	+	0.775740i	$0.717377\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−8.33033e8	+	1.94691e9i	−0.231205	+	0.540359i
$246$		0			0
$247$	−1.83888e9	+	3.18504e9i	−0.494045	+	0.855711i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	1.50763e9i		−	0.379839i	−0.981800	−	0.189920i	$-0.939177\pi$
$251$		−	1.50763e9i		−	0.379839i	0.981800	−	0.189920i	$-0.0608228\pi$
$252$		0			0
$253$	2.28909e9			0.558702
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.25869e9	−	1.30405e9i	−0.517754	−	0.298925i	0.218261	−	0.975890i	$-0.429961\pi$
$257$	−2.25869e9	−	1.30405e9i	−0.517754	−	0.298925i	−0.736015	+	0.676965i	$0.763295\pi$
$258$		0			0
$259$	−4.39939e8	−	1.32115e9i	−0.0977674	−	0.293598i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.87675e9	−	3.97029e9i	1.43734	−	0.829850i	0.439678	−	0.898155i	$-0.355093\pi$
$263$	6.87675e9	−	3.97029e9i	1.43734	−	0.829850i	0.997664	+	0.0683052i	$0.0217592\pi$
$264$		0			0
$265$	−4.82552e9			−0.978499
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.05431e9	−	3.49546e9i	1.15626	−	0.667567i	0.205856	−	0.978582i	$-0.434002\pi$
$269$	6.05431e9	−	3.49546e9i	1.15626	−	0.667567i	0.950405	+	0.311015i	$0.100669\pi$
$270$		0			0
$271$	−1.05192e9	+	1.82198e9i	−0.195032	+	0.337806i	−0.946911	−	0.321496i	$-0.895814\pi$
$271$	−1.05192e9	+	1.82198e9i	−0.195032	+	0.337806i	0.751879	+	0.659301i	$0.229148\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.47744e9	+	1.43035e9i	0.433184	+	0.250099i
$276$		0			0
$277$	−4.71012e9	−	8.15817e9i	−0.800043	−	1.38571i	−0.919587	−	0.392886i	$-0.871477\pi$
$277$	−4.71012e9	−	8.15817e9i	−0.800043	−	1.38571i	0.119544	−	0.992829i	$-0.461857\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	7.56622e9i		−	1.21354i	−0.794878	−	0.606769i	$-0.792465\pi$
$281$		−	7.56622e9i		−	1.21354i	0.794878	−	0.606769i	$-0.207535\pi$
$282$		0			0
$283$	−2.74735e9	−	4.75856e9i	−0.428320	−	0.741873i	0.568404	−	0.822750i	$-0.307561\pi$
$283$	−2.74735e9	−	4.75856e9i	−0.428320	−	0.741873i	−0.996724	+	0.0808771i	$0.974228\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.61805e9	+	2.32298e9i	0.385879	+	0.342387i
$288$		0			0
$289$	5.91542e8	−	1.02458e9i	0.0847997	−	0.146877i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	9.75152e9i		−	1.32313i	−0.749889	−	0.661564i	$-0.769893\pi$
$293$		−	9.75152e9i		−	1.32313i	0.749889	−	0.661564i	$-0.230107\pi$
$294$		0			0
$295$	9.52270e8			0.125740
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.92337e9	−	3.99721e9i	−0.866229	−	0.500117i
$300$		0			0
$301$	1.22522e8	−	1.38086e8i	0.0149262	−	0.0168222i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.25436e9	−	2.45626e9i	0.491626	−	0.283841i
$306$		0			0
$307$	−2.62857e9			−0.295914			−0.147957	−	0.988994i	$-0.547270\pi$
$307$	−2.62857e9			−0.295914			−0.147957	+	0.988994i	$0.547270\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−7.62593e9	+	4.40283e9i	−0.815175	+	0.470642i	−0.848750	−	0.528794i	$-0.822644\pi$
$311$	−7.62593e9	+	4.40283e9i	−0.815175	+	0.470642i	0.0335744	+	0.999436i	$0.489311\pi$
$312$		0			0
$313$	−2.72120e8	+	4.71326e8i	−0.0283520	+	0.0491071i	−0.879853	−	0.475245i	$-0.842359\pi$
$313$	−2.72120e8	+	4.71326e8i	−0.0283520	+	0.0491071i	0.851501	+	0.524353i	$0.175693\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.12025e8	+	2.37883e8i	0.0408025	+	0.0235573i	0.520262	−	0.854006i	$-0.325834\pi$
$317$	4.12025e8	+	2.37883e8i	0.0408025	+	0.0235573i	−0.479460	+	0.877564i	$0.659167\pi$
$318$		0			0
$319$	−4.31386e8	−	7.47183e8i	−0.0416585	−	0.0721547i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	8.50177e9i		−	0.781087i
$324$		0			0
$325$	−4.99536e9	−	8.65221e9i	−0.447747	−	0.775521i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.23049e9	+	4.09750e8i	−0.105025	+	0.0349732i
$330$		0			0
$331$	−7.64888e9	+	1.32482e10i	−0.637215	+	1.10369i	0.348826	+	0.937187i	$0.386580\pi$
$331$	−7.64888e9	+	1.32482e10i	−0.637215	+	1.10369i	−0.986041	+	0.166501i	$0.946753\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			1.40763e8i			0.0111766i
$336$		0			0
$337$	4.58082e8			0.0355159			0.0177580	−	0.999842i	$-0.494347\pi$
$337$	4.58082e8			0.0355159			0.0177580	+	0.999842i	$0.494347\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.14401e8	−	3.54725e8i	−0.0454396	−	0.0262346i
$342$		0			0
$343$	−1.13730e10	−	7.88893e9i	−0.821675	−	0.569956i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.37866e9	−	4.83742e9i	0.577905	−	0.333654i	−0.182395	−	0.983225i	$-0.558385\pi$
$347$	8.37866e9	−	4.83742e9i	0.577905	−	0.333654i	0.760300	+	0.649572i	$0.225052\pi$
$348$		0			0
$349$	−2.35028e10			−1.58423			−0.792114	−	0.610373i	$-0.791019\pi$
$349$	−2.35028e10			−1.58423			−0.792114	+	0.610373i	$0.791019\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.52218e10	−	1.45618e10i	1.62434	−	0.937813i	0.638599	−	0.769540i	$-0.279514\pi$
$353$	2.52218e10	−	1.45618e10i	1.62434	−	0.937813i	0.985740	−	0.168273i	$-0.0538190\pi$
$354$		0			0
$355$	7.29676e9	−	1.26384e10i	0.459427	−	0.795751i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.48265e10	−	8.56011e9i	−0.892611	−	0.515349i	−0.0178153	−	0.999841i	$-0.505671\pi$
$359$	−1.48265e10	−	8.56011e9i	−0.892611	−	0.515349i	−0.874796	+	0.484492i	$0.839004\pi$
$360$		0			0
$361$	4.06223e9	+	7.03599e9i	0.239186	+	0.414282i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	1.59224e10i		−	0.897094i
$366$		0			0
$367$	2.65407e9	+	4.59698e9i	0.146301	+	0.253401i	0.929858	−	0.367920i	$-0.119930\pi$
$367$	2.65407e9	+	4.59698e9i	0.146301	+	0.253401i	−0.783557	+	0.621321i	$0.786596\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6.33259e9	−	3.08982e10i	0.334261	−	1.63094i
$372$		0			0
$373$	4.59907e9	−	7.96582e9i	0.237593	−	0.411524i	−0.722430	−	0.691444i	$-0.756975\pi$
$373$	4.59907e9	−	7.96582e9i	0.237593	−	0.411524i	0.960023	+	0.279920i	$0.0903081\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			3.01315e9i			0.149161i
$378$		0			0
$379$	−2.62913e10			−1.27425			−0.637125	−	0.770760i	$-0.719877\pi$
$379$	−2.62913e10			−1.27425			−0.637125	+	0.770760i	$0.719877\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.33650e10	+	7.71627e9i	0.621116	+	0.358602i	0.777304	−	0.629126i	$-0.216587\pi$
$383$	1.33650e10	+	7.71627e9i	0.621116	+	0.358602i	−0.156187	+	0.987727i	$0.549920\pi$
$384$		0			0
$385$	6.54931e9	−	7.38123e9i	0.298094	−	0.335959i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.38797e9	−	8.01346e8i	0.0606153	−	0.0349963i	−0.469386	−	0.882993i	$-0.655525\pi$
$389$	1.38797e9	−	8.01346e8i	0.0606153	−	0.0349963i	0.530001	+	0.847997i	$0.322191\pi$
$390$		0			0
$391$	1.84804e10			0.790687
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.76890e9	−	2.75333e9i	0.195898	−	0.113102i
$396$		0			0
$397$	−1.74149e9	+	3.01636e9i	−0.0701067	+	0.121428i	−0.898948	−	0.438056i	$-0.855667\pi$
$397$	−1.74149e9	+	3.01636e9i	−0.0701067	+	0.121428i	0.828841	+	0.559484i	$0.189001\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.63717e10	+	9.45220e9i	0.633164	+	0.365557i	0.781976	−	0.623308i	$-0.214212\pi$
$401$	1.63717e10	+	9.45220e9i	0.633164	+	0.365557i	−0.148813	+	0.988865i	$0.547545\pi$
$402$		0			0
$403$	1.23884e9	+	2.14574e9i	0.0469673	+	0.0813497i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			6.48874e9i			0.236474i
$408$		0			0
$409$	−3.36242e9	−	5.82388e9i	−0.120160	−	0.208123i	0.799671	−	0.600439i	$-0.205007\pi$
$409$	−3.36242e9	−	5.82388e9i	−0.120160	−	0.208123i	−0.919831	+	0.392316i	$0.871674\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.24968e9	+	6.09746e9i	−0.0429534	+	0.209579i
$414$		0			0
$415$	1.95168e9	−	3.38042e9i	0.0657987	−	0.113967i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			1.76725e9i			0.0573380i	0.999589	+	0.0286690i	$0.00912687\pi$
$419$			1.76725e9i			0.0573380i	−0.999589	+	0.0286690i	$0.990873\pi$
$420$		0			0
$421$	−3.74949e10			−1.19356			−0.596780	−	0.802405i	$-0.703553\pi$
$421$	−3.74949e10			−1.19356			−0.596780	+	0.802405i	$0.703553\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.00010e10	+	1.15476e10i	0.613050	+	0.353945i
$426$		0			0
$427$	1.01445e10	+	3.04644e10i	0.305155	+	0.916391i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3.41757e10	+	1.97314e10i	−0.990395	+	0.571805i	−0.905392	−	0.424576i	$-0.860423\pi$
$431$	−3.41757e10	+	1.97314e10i	−0.990395	+	0.571805i	−0.0850029	+	0.996381i	$0.527090\pi$
$432$		0			0
$433$	6.51519e9			0.185343			0.0926713	−	0.995697i	$-0.470459\pi$
$433$	6.51519e9			0.185343			0.0926713	+	0.995697i	$0.470459\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.66772e10	−	9.62858e9i	0.457296	−	0.264020i
$438$		0			0
$439$	−1.57305e10	+	2.72461e10i	−0.423531	+	0.733578i	−0.996282	−	0.0861519i	$-0.972543\pi$
$439$	−1.57305e10	+	2.72461e10i	−0.423531	+	0.733578i	0.572751	+	0.819730i	$0.305876\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.44999e10	−	3.14655e10i	−1.41508	−	0.816996i	−0.419217	−	0.907886i	$-0.637695\pi$
$443$	−5.44999e10	−	3.14655e10i	−1.41508	−	0.816996i	−0.995861	+	0.0908900i	$0.971029\pi$
$444$		0			0
$445$	1.14838e10	+	1.98906e10i	0.292851	+	0.507232i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			2.92532e9i			0.0719760i	0.999352	+	0.0359880i	$0.0114578\pi$
$449$			2.92532e9i			0.0719760i	−0.999352	+	0.0359880i	$0.988542\pi$
$450$		0			0
$451$	−8.15490e9	−	1.41247e10i	−0.197112	−	0.341407i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.26976e10	+	1.08882e10i	−0.762904	+	0.254045i
$456$		0			0
$457$	3.80250e10	−	6.58612e10i	0.871774	−	1.50996i	0.0116142	−	0.999933i	$-0.496303\pi$
$457$	3.80250e10	−	6.58612e10i	0.871774	−	1.50996i	0.860160	−	0.510024i	$-0.170364\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			4.51927e10i			1.00061i	0.865850	+	0.500304i	$0.166779\pi$
$461$			4.51927e10i			1.00061i	−0.865850	+	0.500304i	$0.833221\pi$
$462$		0			0
$463$	5.20179e10			1.13195			0.565977	−	0.824421i	$-0.308499\pi$
$463$	5.20179e10			1.13195			0.565977	+	0.824421i	$0.308499\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.17329e10	−	1.25475e10i	−0.456930	−	0.263809i	0.253823	−	0.967251i	$-0.418312\pi$
$467$	−2.17329e10	−	1.25475e10i	−0.456930	−	0.263809i	−0.710753	+	0.703442i	$0.751645\pi$
$468$		0			0
$469$	−9.01316e8	−	1.84725e8i	−0.0186288	−	0.00381799i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.44989e8	+	4.30120e8i	−0.0148835	+	0.00859300i
$474$		0			0
$475$	2.40659e10			0.472746
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7.62124e10	+	4.40013e10i	−1.44772	+	0.835840i	−0.998345	−	0.0575044i	$-0.981686\pi$
$479$	−7.62124e10	+	4.40013e10i	−1.44772	+	0.835840i	−0.449372	+	0.893345i	$0.648352\pi$
$480$		0			0
$481$	1.13307e10	−	1.96253e10i	0.211677	−	0.366636i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.34079e10	+	1.35145e10i	0.423053	+	0.244250i
$486$		0			0
$487$	1.43037e10	+	2.47748e10i	0.254293	+	0.440448i	0.964703	−	0.263340i	$-0.0848240\pi$
$487$	1.43037e10	+	2.47748e10i	0.254293	+	0.440448i	−0.710410	+	0.703788i	$0.751491\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			4.22575e10i			0.727074i	0.931580	+	0.363537i	$0.118431\pi$
$491$			4.22575e10i			0.727074i	−0.931580	+	0.363537i	$0.881569\pi$
$492$		0			0
$493$	−3.48270e9	−	6.03221e9i	−0.0589560	−	0.102115i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7.13487e10	+	6.33071e10i	1.16939	+	1.03759i
$498$		0			0
$499$	−2.69799e10	+	4.67306e10i	−0.435150	+	0.753702i	−0.997308	−	0.0733282i	$-0.976638\pi$
$499$	−2.69799e10	+	4.67306e10i	−0.435150	+	0.753702i	0.562158	+	0.827030i	$0.309971\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			3.61297e10i			0.564407i	0.959355	+	0.282204i	$0.0910653\pi$
$503$			3.61297e10i			0.564407i	−0.959355	+	0.282204i	$0.908935\pi$
$504$		0			0
$505$	2.66690e10			0.410055
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.75605e9	+	1.59121e9i	0.0410597	+	0.0237058i	0.520389	−	0.853929i	$-0.325787\pi$
$509$	2.75605e9	+	1.59121e9i	0.0410597	+	0.0237058i	−0.479330	+	0.877635i	$0.659120\pi$
$510$		0			0
$511$	1.01953e11	+	2.08952e10i	1.49525	+	0.306453i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.72791e10	+	1.57496e10i	−0.387795	+	0.223893i
$516$		0			0
$517$	6.04347e9			0.0845910
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.29192e10	−	7.45891e9i	0.175342	−	0.101233i	−0.409761	−	0.912193i	$-0.634388\pi$
$521$	1.29192e10	−	7.45891e9i	0.175342	−	0.101233i	0.585102	+	0.810960i	$0.301054\pi$
$522$		0			0
$523$	−5.37970e10	+	9.31791e10i	−0.719037	+	1.24541i	0.242345	+	0.970190i	$0.422084\pi$
$523$	−5.37970e10	+	9.31791e10i	−0.719037	+	1.24541i	−0.961382	+	0.275219i	$0.911250\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.96022e9	−	2.86378e9i	−0.0643070	−	0.0371277i
$528$		0			0
$529$	−1.82257e10	−	3.15679e10i	−0.232735	−	0.403109i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			5.69604e10i			0.705771i
$534$		0			0
$535$	−1.57902e10	−	2.73495e10i	−0.192741	−	0.333837i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.86678e10	+	5.16222e10i	0.458137	+	0.611620i
$540$		0			0
$541$	9.75750e9	−	1.69005e10i	0.113907	−	0.197292i	−0.803435	−	0.595392i	$-0.796997\pi$
$541$	9.75750e9	−	1.69005e10i	0.113907	−	0.197292i	0.917342	+	0.398100i	$0.130330\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1.03683e10i		−	0.117523i
$546$		0			0
$547$	1.10400e11			1.23317			0.616583	−	0.787290i	$-0.288517\pi$
$547$	1.10400e11			1.23317			0.616583	+	0.787290i	$0.288517\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.28575e9	−	3.62908e9i	−0.0681947	−	0.0393722i
$552$		0			0
$553$	1.13715e10	+	3.41489e10i	0.121595	+	0.365154i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.32187e10	−	1.91788e10i	0.345114	−	0.199251i	−0.317417	−	0.948286i	$-0.602816\pi$
$557$	3.32187e10	−	1.91788e10i	0.345114	−	0.199251i	0.662531	+	0.749034i	$0.269482\pi$
$558$		0			0
$559$	3.00430e9			0.0307678
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.59731e10	+	3.23161e10i	−0.557116	+	0.321651i	−0.751987	−	0.659178i	$-0.770904\pi$
$563$	−5.59731e10	+	3.23161e10i	−0.557116	+	0.321651i	0.194871	+	0.980829i	$0.437571\pi$
$564$		0			0
$565$	3.00782e10	−	5.20969e10i	0.295160	−	0.511232i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.08908e11	−	6.28783e10i	−1.03899	−	0.599863i	−0.119445	−	0.992841i	$-0.538112\pi$
$569$	−1.08908e11	−	6.28783e10i	−1.03899	−	0.599863i	−0.919548	+	0.392978i	$0.871445\pi$
$570$		0			0
$571$	3.64229e10	+	6.30864e10i	0.342634	+	0.593460i	0.984921	−	0.173005i	$-0.0553476\pi$
$571$	3.64229e10	+	6.30864e10i	0.342634	+	0.593460i	−0.642287	+	0.766464i	$0.722014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			5.23123e10i			0.478556i
$576$		0			0
$577$	4.29136e10	+	7.43285e10i	0.387161	+	0.670583i	0.992066	−	0.125715i	$-0.0401224\pi$
$577$	4.29136e10	+	7.43285e10i	0.387161	+	0.670583i	−0.604905	+	0.796297i	$0.706789\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.90838e10	+	1.69329e10i	0.167479	+	0.148603i
$582$		0			0
$583$	−7.34870e10	+	1.27283e11i	−0.636116	+	1.10179i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			2.31204e11i			1.94735i	0.227950	+	0.973673i	$0.426798\pi$
$587$			2.31204e11i			1.94735i	−0.227950	+	0.973673i	$0.573202\pi$
$588$		0			0
$589$	−5.96831e9			−0.0495895
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.98024e10	−	2.87534e10i	−0.402746	−	0.232526i	0.284922	−	0.958551i	$-0.408032\pi$
$593$	−4.98024e10	−	2.87534e10i	−0.402746	−	0.232526i	−0.687668	+	0.726025i	$0.741366\pi$
$594$		0			0
$595$	5.28743e10	−	5.95906e10i	0.421868	−	0.475456i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.19135e11	+	6.87828e10i	−0.925408	+	0.534285i	−0.885356	−	0.464913i	$-0.846086\pi$
$599$	−1.19135e11	+	6.87828e10i	−0.925408	+	0.534285i	−0.0400517	+	0.999198i	$0.512752\pi$
$600$		0			0
$601$	9.87162e10			0.756642			0.378321	−	0.925674i	$-0.376502\pi$
$601$	9.87162e10			0.756642			0.378321	+	0.925674i	$0.376502\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.83706e10	−	1.63797e10i	0.211761	−	0.122260i
$606$		0			0
$607$	5.60446e10	−	9.70721e10i	0.412838	−	0.715056i	−0.582361	−	0.812930i	$-0.697871\pi$
$607$	5.60446e10	−	9.70721e10i	0.412838	−	0.715056i	0.995199	+	0.0978746i	$0.0312044\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.82785e10	−	1.05531e10i	−0.131152	−	0.0757209i
$612$		0			0
$613$	1.42634e10	+	2.47050e10i	0.101014	+	0.174962i	0.912103	−	0.409962i	$-0.134458\pi$
$613$	1.42634e10	+	2.47050e10i	0.101014	+	0.174962i	−0.811089	+	0.584923i	$0.801125\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.55361e10i		−	0.107202i	−0.998562	−	0.0536008i	$-0.982930\pi$
$617$		−	1.55361e10i		−	0.107202i	0.998562	−	0.0536008i	$-0.0170699\pi$
$618$		0			0
$619$	2.97047e10	+	5.14501e10i	0.202331	+	0.350448i	0.949279	−	0.314435i	$-0.101815\pi$
$619$	2.97047e10	+	5.14501e10i	0.202331	+	0.350448i	−0.746948	+	0.664883i	$0.768482\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.42431e11	+	4.74291e10i	−0.945481	+	0.314842i
$624$		0			0
$625$	−6.33237e9	+	1.09680e10i	−0.0414998	+	0.0718798i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			5.23853e10i			0.334663i
$630$		0			0
$631$	−2.14883e11			−1.35545			−0.677725	−	0.735315i	$-0.737034\pi$
$631$	−2.14883e11			−1.35545			−0.677725	+	0.735315i	$0.737034\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.20873e11	−	6.97860e10i	−0.743420	−	0.429214i
$636$		0			0
$637$	−2.68085e10	−	2.23654e11i	−0.162823	−	1.35837i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.96391e11	−	1.13386e11i	1.16329	−	0.671627i	0.211202	−	0.977442i	$-0.432262\pi$
$641$	1.96391e11	−	1.13386e11i	1.16329	−	0.671627i	0.952091	+	0.305815i	$0.0989289\pi$
$642$		0			0
$643$	−8.00949e10			−0.468556			−0.234278	−	0.972170i	$-0.575273\pi$
$643$	−8.00949e10			−0.468556			−0.234278	+	0.972170i	$0.575273\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.57558e11	−	1.48701e11i	1.46980	−	0.848588i	0.470372	−	0.882468i	$-0.344120\pi$
$647$	2.57558e11	−	1.48701e11i	1.46980	−	0.848588i	0.999426	+	0.0338796i	$0.0107863\pi$
$648$		0			0
$649$	1.45020e10	−	2.51181e10i	0.0817425	−	0.141582i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.07864e10	+	2.93215e10i	0.279315	+	0.161263i	0.633113	−	0.774059i	$-0.281777\pi$
$653$	5.07864e10	+	2.93215e10i	0.279315	+	0.161263i	−0.353798	+	0.935322i	$0.615110\pi$
$654$		0			0
$655$	3.50271e10	+	6.06687e10i	0.190300	+	0.329609i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	2.80006e11i		−	1.48465i	−0.670037	−	0.742327i	$-0.733722\pi$
$659$		−	2.80006e11i		−	1.48465i	0.670037	−	0.742327i	$-0.266278\pi$
$660$		0			0
$661$	−1.89573e11	−	3.28351e11i	−0.993051	−	1.72002i	−0.598440	−	0.801167i	$-0.704213\pi$
$661$	−1.89573e11	−	3.28351e11i	−0.993051	−	1.72002i	−0.394611	−	0.918848i	$-0.629121\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.66675e10	−	8.13244e10i	0.0852282	−	0.415848i
$666$		0			0
$667$	7.88857e9	−	1.36634e10i	0.0398561	−	0.0690328i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1.49624e11i		−	0.738091i
$672$		0			0
$673$	−1.75958e11			−0.857725			−0.428862	−	0.903370i	$-0.641086\pi$
$673$	−1.75958e11			−0.857725			−0.428862	+	0.903370i	$0.641086\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.38063e11	+	1.95181e11i	1.60932	+	0.929144i	0.989521	+	0.144386i	$0.0461207\pi$
$677$	3.38063e11	+	1.95181e11i	1.60932	+	0.929144i	0.619803	+	0.784758i	$0.287213\pi$
$678$		0			0
$679$	−1.17253e11	+	1.32147e11i	−0.551627	+	0.621696i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.03351e11	+	1.75140e11i	−1.39400	+	0.804825i	−0.993755	−	0.111585i	$-0.964407\pi$
$683$	−3.03351e11	+	1.75140e11i	−1.39400	+	0.804825i	−0.400242	+	0.916409i	$0.631074\pi$
$684$		0			0
$685$	1.79324e11			0.814470
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.44524e11	−	2.56646e11i	1.97251	−	1.13883i
$690$		0			0
$691$	−1.08602e11	+	1.88103e11i	−0.476347	+	0.825058i	−0.999633	−	0.0270997i	$-0.991373\pi$
$691$	−1.08602e11	+	1.88103e11i	−0.476347	+	0.825058i	0.523285	+	0.852157i	$0.324706\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.24824e11	−	7.20672e10i	−0.535007	−	0.308886i
$696$		0			0
$697$	−6.58366e10	−	1.14032e11i	−0.278956	−	0.483167i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1.65806e11i		−	0.686638i	−0.939219	−	0.343319i	$-0.888449\pi$
$701$		−	1.65806e11i		−	0.686638i	0.939219	−	0.343319i	$-0.111551\pi$
$702$		0			0
$703$	2.72936e10	+	4.72739e10i	0.111748	+	0.193553i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.49981e10	+	1.70764e11i	−0.140077	+	0.683468i
$708$		0			0
$709$	−3.04636e10	+	5.27645e10i	−0.120558	+	0.208813i	−0.919988	−	0.391947i	$-0.871802\pi$
$709$	−3.04636e10	+	5.27645e10i	−0.120558	+	0.208813i	0.799430	+	0.600760i	$0.205135\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1.29734e10i		−	0.0501990i
$714$		0			0
$715$	1.60592e11			0.614468
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2.85721e11	−	1.64961e11i	−1.06912	−	0.617257i	−0.141180	−	0.989984i	$-0.545090\pi$
$719$	−2.85721e11	−	1.64961e11i	−1.06912	−	0.617257i	−0.927941	+	0.372727i	$0.878423\pi$
$720$		0			0
$721$	−6.50472e10	−	1.95339e11i	−0.240706	−	0.722848i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.70753e10	−	9.85844e9i	0.0618040	−	0.0356826i
$726$		0			0
$727$	2.16348e9			0.00774489			0.00387244	−	0.999993i	$-0.498767\pi$
$727$	2.16348e9			0.00774489			0.00387244	+	0.999993i	$0.498767\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−6.01449e9	+	3.47247e9i	−0.0210634	+	0.0121610i
$732$		0			0
$733$	6.09762e10	−	1.05614e11i	0.211224	−	0.365851i	−0.740874	−	0.671645i	$-0.765588\pi$
$733$	6.09762e10	−	1.05614e11i	0.211224	−	0.365851i	0.952098	+	0.305793i	$0.0989216\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.71292e9	+	2.14366e9i	0.0125848	+	0.00726583i
$738$		0			0
$739$	1.51287e11	+	2.62037e11i	0.507253	+	0.878588i	0.999965	+	0.00839520i	$0.00267231\pi$
$739$	1.51287e11	+	2.62037e11i	0.507253	+	0.878588i	−0.492712	+	0.870192i	$0.663994\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	5.81127e11i		−	1.90685i	−0.301635	−	0.953423i	$-0.597532\pi$
$743$		−	5.81127e11i		−	1.90685i	0.301635	−	0.953423i	$-0.402468\pi$
$744$		0			0
$745$	−1.07355e11	−	1.85944e11i	−0.348494	−	0.603609i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.95843e11	−	6.52150e10i	0.622271	−	0.207215i
$750$		0			0
$751$	−2.64610e11	+	4.58318e11i	−0.831852	+	1.44081i	0.0647154	+	0.997904i	$0.479386\pi$
$751$	−2.64610e11	+	4.58318e11i	−0.831852	+	1.44081i	−0.896568	+	0.442907i	$0.853947\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	2.46653e11i		−	0.759099i
$756$		0			0
$757$	−5.48937e11			−1.67163			−0.835813	−	0.549015i	$-0.815003\pi$
$757$	−5.48937e11			−1.67163			−0.835813	+	0.549015i	$0.815003\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.55337e11	+	8.96836e10i	0.463164	+	0.267408i	0.713374	−	0.700784i	$-0.247166\pi$
$761$	1.55337e11	+	8.96836e10i	0.463164	+	0.267408i	−0.250210	+	0.968192i	$0.580500\pi$
$762$		0			0
$763$	6.63892e10	+	1.36065e10i	0.195884	+	0.0401465i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.77226e10	+	5.06467e10i	−0.253472	+	0.146342i
$768$		0			0
$769$	6.58883e9			0.0188409			0.00942047	−	0.999956i	$-0.497001\pi$
$769$	6.58883e9			0.0188409			0.00942047	+	0.999956i	$0.497001\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4.76170e11	−	2.74917e11i	1.33366	−	0.769987i	0.347798	−	0.937569i	$-0.386929\pi$
$773$	4.76170e11	−	2.74917e11i	1.33366	−	0.769987i	0.985858	+	0.167583i	$0.0535962\pi$
$774$		0			0
$775$	8.10649e9	−	1.40409e10i	0.0224712	−	0.0389212i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.18825e11	−	6.86038e10i	−0.322670	−	0.186294i
$780$		0			0
$781$	−2.22242e11	−	3.84935e11i	−0.597341	−	1.03462i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	7.62717e10i		−	0.200856i
$786$		0			0
$787$	−3.10681e11	−	5.38116e11i	−0.809871	−	1.40274i	−0.912953	−	0.408065i	$-0.866204\pi$
$787$	−3.10681e11	−	5.38116e11i	−0.809871	−	1.40274i	0.103082	−	0.994673i	$-0.467130\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2.94109e11	+	2.60960e11i	0.751280	+	0.666605i
$792$		0			0
$793$	−2.61273e11	+	4.52538e11i	−0.660696	+	1.14436i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1.48906e10i		−	0.0369044i	−0.999830	−	0.0184522i	$-0.994126\pi$
$797$		−	1.48906e10i		−	0.0369044i	0.999830	−	0.0184522i	$-0.00587385\pi$
$798$		0			0
$799$	4.87905e10			0.119715
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−4.19988e11	−	2.42480e11i	−1.01012	−	0.583195i
$804$		0			0
$805$	1.76776e11	+	3.62303e10i	0.420959	+	0.0862757i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.45466e11	−	1.41720e11i	0.573057	−	0.330855i	−0.185312	−	0.982680i	$-0.559330\pi$
$809$	2.45466e11	−	1.41720e11i	0.573057	−	0.330855i	0.758369	+	0.651825i	$0.225996\pi$
$810$		0			0
$811$	1.18370e10			0.0273626			0.0136813	−	0.999906i	$-0.495645\pi$
$811$	1.18370e10			0.0273626			0.0136813	+	0.999906i	$0.495645\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.06313e11	−	6.13796e10i	0.240965	−	0.139121i
$816$		0			0
$817$	−3.61842e9	+	6.26729e9i	−0.00812140	+	0.0140667i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.03235e11	−	2.90543e11i	−1.10764	−	0.639495i	−0.169422	−	0.985544i	$-0.554190\pi$
$821$	−5.03235e11	−	2.90543e11i	−1.10764	−	0.639495i	−0.938217	+	0.346048i	$0.887523\pi$
$822$		0			0
$823$	−2.88901e11	−	5.00391e11i	−0.629723	−	1.09071i	−0.987607	−	0.156946i	$-0.949835\pi$
$823$	−2.88901e11	−	5.00391e11i	−0.629723	−	1.09071i	0.357884	−	0.933766i	$-0.383498\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1.43219e11i			0.306182i	0.988212	+	0.153091i	$0.0489227\pi$
$827$			1.43219e11i			0.306182i	−0.988212	+	0.153091i	$0.951077\pi$
$828$		0			0
$829$	−9.91279e10	−	1.71695e11i	−0.209883	−	0.363529i	0.741794	−	0.670627i	$-0.233975\pi$
$829$	−9.91279e10	−	1.71695e11i	−0.209883	−	0.363529i	−0.951678	+	0.307099i	$0.900642\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.12176e11	+	4.16760e11i	0.648364	+	0.865577i
$834$		0			0
$835$	2.30225e11	−	3.98762e11i	0.473595	−	0.820291i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	5.66566e10i		−	0.114341i	−0.998364	−	0.0571706i	$-0.981792\pi$
$839$		−	5.66566e10i		−	0.114341i	0.998364	−	0.0571706i	$-0.0182079\pi$
$840$		0			0
$841$	4.94300e11			0.988113
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.26206e11	−	1.30600e11i	−0.443688	−	0.256163i
$846$		0			0
$847$	6.76497e10	+	2.03154e11i	0.131441	+	0.394723i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.02760e11	+	5.93284e10i	−0.195932	+	0.113121i
$852$		0			0
$853$	6.61843e11			1.25014			0.625070	−	0.780569i	$-0.285070\pi$
$853$	6.61843e11			1.25014			0.625070	+	0.780569i	$0.285070\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.50676e11	−	2.60198e11i	0.835489	−	0.482370i	−0.0202393	−	0.999795i	$-0.506443\pi$
$857$	4.50676e11	−	2.60198e11i	0.835489	−	0.482370i	0.855728	+	0.517425i	$0.173109\pi$
$858$		0			0
$859$	−1.65242e10	+	2.86207e10i	−0.0303492	+	0.0525663i	−0.880801	−	0.473487i	$-0.842995\pi$
$859$	−1.65242e10	+	2.86207e10i	−0.0303492	+	0.0525663i	0.850452	+	0.526053i	$0.176329\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.96909e11	−	1.71420e11i	−0.535279	−	0.309043i	0.207885	−	0.978153i	$-0.433342\pi$
$863$	−2.96909e11	−	1.71420e11i	−0.535279	−	0.309043i	−0.743163	+	0.669110i	$0.766675\pi$
$864$		0			0
$865$	−2.54037e11	−	4.40006e11i	−0.453767	−	0.785948i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1.67720e11i		−	0.294107i
$870$		0			0
$871$	−7.48651e9	−	1.29670e10i	−0.0130079	−	0.0225303i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.26388e11	+	3.78331e11i	0.727400	+	0.645416i
$876$		0			0
$877$	−3.20979e11	+	5.55951e11i	−0.542597	+	0.939806i	0.456156	+	0.889900i	$0.349226\pi$
$877$	−3.20979e11	+	5.55951e11i	−0.542597	+	0.939806i	−0.998754	+	0.0499068i	$0.984108\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			6.85313e11i			1.13759i	0.822480	+	0.568795i	$0.192590\pi$
$881$			6.85313e11i			1.13759i	−0.822480	+	0.568795i	$0.807410\pi$
$882$		0			0
$883$	−8.19942e11			−1.34878			−0.674389	−	0.738377i	$-0.735593\pi$
$883$	−8.19942e11			−1.34878			−0.674389	+	0.738377i	$0.735593\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.01011e11	−	5.83186e10i	−0.163182	−	0.0942133i	0.416185	−	0.909280i	$-0.363367\pi$
$887$	−1.01011e11	−	5.83186e10i	−0.163182	−	0.0942133i	−0.579367	+	0.815067i	$0.696700\pi$
$888$		0			0
$889$	6.05468e11	−	6.82378e11i	0.969358	−	1.09249i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.40298e10	−	2.54206e10i	0.0692374	−	0.0399742i
$894$		0			0
$895$	6.06987e11			0.945992
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.23465e9	+	2.44488e9i	−0.00648305	+	0.00374299i
$900$		0			0
$901$	−5.93280e11	+	1.02759e12i	−0.900244	+	1.55927i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5.78143e11	−	3.33791e11i	−0.861868	−	0.497600i
$906$		0			0
$907$	5.00470e11	+	8.66839e11i	0.739518	+	1.28088i	0.952713	+	0.303873i	$0.0982799\pi$
$907$	5.00470e11	+	8.66839e11i	0.739518	+	1.28088i	−0.213194	+	0.977010i	$0.568387\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1.33509e11i			0.193837i	0.995292	+	0.0969187i	$0.0308987\pi$
$911$			1.33509e11i			0.193837i	−0.995292	+	0.0969187i	$0.969101\pi$
$912$		0			0
$913$	−5.94437e10	−	1.02960e11i	−0.0855506	−	0.148178i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4.34433e11	+	1.44665e11i	−0.614391	+	0.204590i
$918$		0			0
$919$	−3.77396e11	+	6.53668e11i	−0.529096	+	0.916422i	0.470328	+	0.882492i	$0.344136\pi$
$919$	−3.77396e11	+	6.53668e11i	−0.529096	+	0.916422i	−0.999424	+	0.0339300i	$0.989198\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			1.55232e12i			2.13882i
$924$		0			0
$925$	−1.48287e11			−0.202551
$926$		0			0
$927$		0			0
$928$		0			0
$929$	8.90886e10	+	5.14353e10i	0.119608	+	0.0690556i	0.558610	−	0.829430i	$-0.311335\pi$
$929$	8.90886e10	+	5.14353e10i	0.119608	+	0.0690556i	−0.439003	+	0.898486i	$0.644668\pi$
$930$		0			0
$931$	4.98854e11	+	2.13446e11i	0.664010	+	0.284112i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.21498e11	+	1.85617e11i	−0.420661	+	0.242869i
$936$		0			0
$937$	−3.57056e11			−0.463210			−0.231605	−	0.972810i	$-0.574398\pi$
$937$	−3.57056e11			−0.463210			−0.231605	+	0.972810i	$0.574398\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4.74713e11	−	2.74076e11i	0.605443	−	0.349552i	−0.165737	−	0.986170i	$-0.553000\pi$
$941$	4.74713e11	−	2.74076e11i	0.605443	−	0.349552i	0.771180	+	0.636617i	$0.219667\pi$
$942$		0			0
$943$	1.49125e11	−	2.58292e11i	0.188584	−	0.326636i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−5.06769e11	−	2.92583e11i	−0.630101	−	0.363789i	0.150690	−	0.988581i	$-0.451850\pi$
$947$	−5.06769e11	−	2.92583e11i	−0.630101	−	0.363789i	−0.780791	+	0.624792i	$0.785184\pi$
$948$		0			0
$949$	8.46838e11	+	1.46677e12i	1.04408	+	1.80841i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			6.81353e11i			0.826039i	0.910722	+	0.413019i	$0.135526\pi$
$953$			6.81353e11i			0.826039i	−0.910722	+	0.413019i	$0.864474\pi$
$954$		0			0
$955$	1.38071e11	+	2.39146e11i	0.165993	+	0.287507i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.35328e11	+	1.14822e12i	−0.278227	+	1.35754i
$960$		0			0
$961$	4.24435e11	−	7.35143e11i	0.497643	−	0.861943i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			2.02456e11i			0.233464i
$966$		0			0
$967$	1.37573e12			1.57335			0.786676	−	0.617366i	$-0.211800\pi$
$967$	1.37573e12			1.57335			0.786676	+	0.617366i	$0.211800\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.61510e10	−	9.32477e9i	−0.0181686	−	0.0104897i	0.490888	−	0.871223i	$-0.336672\pi$
$971$	−1.61510e10	−	9.32477e9i	−0.0181686	−	0.0104897i	−0.509057	+	0.860733i	$0.670006\pi$
$972$		0			0
$973$	6.25260e11	−	7.04683e11i	0.697604	−	0.786217i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.00602e12	+	5.80824e11i	−1.10415	+	0.637480i	−0.937307	−	0.348504i	$-0.886690\pi$
$977$	−1.00602e12	+	5.80824e11i	−1.10415	+	0.637480i	−0.166840	+	0.985984i	$0.553356\pi$
$978$		0			0
$979$	6.99540e11			0.761521
$980$		0			0
$981$		0			0
$982$		0			0
$983$	8.37978e11	−	4.83807e11i	0.897467	−	0.518153i	0.0210897	−	0.999778i	$-0.493286\pi$
$983$	8.37978e11	−	4.83807e11i	0.897467	−	0.518153i	0.876378	+	0.481625i	$0.159953\pi$
$984$		0			0
$985$	3.09754e11	−	5.36510e11i	0.329058	−	0.569945i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.36233e10	−	7.86541e9i	−0.0142396	−	0.00822122i
$990$		0			0
$991$	3.36765e11	+	5.83294e11i	0.349166	+	0.604774i	0.986102	−	0.166143i	$-0.0531315\pi$
$991$	3.36765e11	+	5.83294e11i	0.349166	+	0.604774i	−0.636935	+	0.770917i	$0.719798\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	5.13417e11i		−	0.523815i
$996$		0			0
$997$	−4.86655e11	−	8.42911e11i	−0.492539	−	0.853102i	0.507424	−	0.861696i	$-0.330598\pi$
$997$	−4.86655e11	−	8.42911e11i	−0.492539	−	0.853102i	−0.999963	+	0.00859423i	$0.997264\pi$
$998$		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.233.8	yes	44
3.2	odd	2	inner	252.9.bk.a.233.15	yes	44
7.4	even	3	inner	252.9.bk.a.53.15	yes	44
21.11	odd	6	inner	252.9.bk.a.53.8	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bk.a.53.8	✓	44	21.11	odd	6	inner
252.9.bk.a.53.15	yes	44	7.4	even	3	inner
252.9.bk.a.233.8	yes	44	1.1	even	1	trivial
252.9.bk.a.233.15	yes	44	3.2	odd	2	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)

\(f(q)\)	\(=\)	\(q+(-318.126 - 183.670i) q^{5} +(1593.54 - 1795.95i) q^{7} +O(q^{10})\)	\(q+(-318.126 - 183.670i) q^{5} +(1593.54 - 1795.95i) q^{7} +(-9689.38 + 5594.16i) q^{11} +39074.1 q^{13} +(-78224.9 + 45163.2i) q^{17} +(-47061.4 + 81512.8i) q^{19} +(-177185. - 102298. i) q^{23} +(-127843. - 221431. i) q^{25} +77113.6i q^{29} +(31704.9 + 54914.5i) q^{31} +(-836808. + 278655. i) q^{35} +(289978. - 502257. i) q^{37} +1.45775e6i q^{41} +76887.2 q^{43} +(-467791. - 270079. i) q^{47} +(-686092. - 5.72383e6i) q^{49} +(1.13764e7 - 6.56818e6i) q^{53} +4.10993e6 q^{55} +(-2.24503e6 + 1.29617e6i) q^{59} +(-6.68659e6 + 1.15815e7i) q^{61} +(-1.24305e7 - 7.17676e6i) q^{65} +(-191597. - 331856. i) q^{67} +3.97275e7i q^{71} +(2.16726e7 + 3.75380e7i) q^{73} +(-5.39351e6 + 2.63162e7i) q^{77} +(-7.49530e6 + 1.29822e7i) q^{79} +1.06260e7i q^{83} +3.31805e7 q^{85} +(-5.41475e7 - 3.12621e7i) q^{89} +(6.22660e7 - 7.01753e7i) q^{91} +(2.99429e7 - 1.72876e7i) q^{95} -7.35805e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 1230 q^{7}+O(q^{10}) \) 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

Related objects

Downloads

Learn more