LMFDB - Embedded newform 252.12.k.e.37.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,12,Mod(37,252)]

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, 0, 2]))

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("252.37");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, 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:	$193.622481501$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(\zeta_{3})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.12
Character	$\chi$	$=$	252.37
Dual form			252.12.k.e.109.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4415.92 - 7648.60i) q^{5} +(-36565.0 - 25304.7i) q^{7} +O(q^{10})$	$q+(4415.92 - 7648.60i) q^{5} +(-36565.0 - 25304.7i) q^{7} +(-163744. - 283613. i) q^{11} +1.62015e6 q^{13} +(1.31028e6 + 2.26947e6i) q^{17} +(3.70191e6 - 6.41189e6i) q^{19} +(-5.76513e6 + 9.98551e6i) q^{23} +(-1.45866e7 - 2.52648e7i) q^{25} +1.97985e8 q^{29} +(1.08755e8 + 1.88370e8i) q^{31} +(-3.55014e8 + 1.67927e8i) q^{35} +(-3.38116e8 + 5.85635e8i) q^{37} +1.07092e9 q^{41} +3.16356e8 q^{43} +(1.43982e9 - 2.49384e9i) q^{47} +(6.96670e8 + 1.85053e9i) q^{49} +(6.06271e8 + 1.05009e9i) q^{53} -2.89232e9 q^{55} +(5.29381e9 + 9.16914e9i) q^{59} +(5.87221e9 - 1.01710e10i) q^{61} +(7.15446e9 - 1.23919e10i) q^{65} +(-3.68558e8 - 6.38361e8i) q^{67} +1.60970e10 q^{71} +(-5.44830e9 - 9.43674e9i) q^{73} +(-1.18945e9 + 1.45138e10i) q^{77} +(-1.18304e10 + 2.04908e10i) q^{79} -5.25042e10 q^{83} +2.31444e10 q^{85} +(8.77325e8 - 1.51957e9i) q^{89} +(-5.92408e10 - 4.09975e10i) q^{91} +(-3.26946e10 - 5.66288e10i) q^{95} +7.32713e10 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 85666 q^{7}+O(q^{10}) $ 28 * q + 85666 * q^7	$ 28 q + 85666 q^{7} + 2794700 q^{13} + 37414826 q^{19} - 135469250 q^{25} + 71052098 q^{31} - 395898874 q^{37} + 980170268 q^{43} - 1253333270 q^{49} - 8844657360 q^{55} + 16692685100 q^{61} + 10934009870 q^{67} + 25578690650 q^{73} - 113974174090 q^{79} + 111789733440 q^{85} - 267181531078 q^{91} + 279658007384 q^{97}+O(q^{100}) $ 28 * q + 85666 * q^7 + 2794700 * q^13 + 37414826 * q^19 - 135469250 * q^25 + 71052098 * q^31 - 395898874 * q^37 + 980170268 * q^43 - 1253333270 * q^49 - 8844657360 * q^55 + 16692685100 * q^61 + 10934009870 * q^67 + 25578690650 * q^73 - 113974174090 * q^79 + 111789733440 * q^85 - 267181531078 * q^91 + 279658007384 * 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
$5$	4415.92	−	7648.60i	0.631955	−	1.09458i	−0.355197	−	0.934792i	$-0.615586\pi$
$5$	4415.92	−	7648.60i	0.631955	−	1.09458i	0.987152	−	0.159786i	$-0.0510805\pi$
$6$		0			0
$7$	−36565.0	−	25304.7i	−0.822292	−	0.569065i
$7$	−36565.0	−	25304.7i	−0.822292	−	0.569065i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−163744.	−	283613.i	−0.306553	−	0.530966i	0.671053	−	0.741410i	$-0.265842\pi$
$11$	−163744.	−	283613.i	−0.306553	−	0.530966i	−0.977606	+	0.210444i	$0.932509\pi$
$12$		0			0
$13$	1.62015e6			1.21023			0.605114	−	0.796139i	$-0.293128\pi$
$13$	1.62015e6			1.21023			0.605114	+	0.796139i	$0.293128\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.31028e6	+	2.26947e6i	0.223818	+	0.387664i	0.955964	−	0.293483i	$-0.0948145\pi$
$17$	1.31028e6	+	2.26947e6i	0.223818	+	0.387664i	−0.732146	+	0.681148i	$0.761481\pi$
$18$		0			0
$19$	3.70191e6	−	6.41189e6i	0.342989	−	0.594075i	−0.641997	−	0.766707i	$-0.721894\pi$
$19$	3.70191e6	−	6.41189e6i	0.342989	−	0.594075i	0.984986	+	0.172632i	$0.0552272\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.76513e6	+	9.98551e6i	−0.186770	+	0.323495i	−0.944171	−	0.329455i	$-0.893135\pi$
$23$	−5.76513e6	+	9.98551e6i	−0.186770	+	0.323495i	0.757402	+	0.652949i	$0.226468\pi$
$24$		0			0
$25$	−1.45866e7	−	2.52648e7i	−0.298734	−	0.517422i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.97985e8			1.79244			0.896219	−	0.443612i	$-0.146303\pi$
$29$	1.97985e8			1.79244			0.896219	+	0.443612i	$0.146303\pi$
$30$		0			0
$31$	1.08755e8	+	1.88370e8i	0.682278	+	1.18174i	0.974284	+	0.225323i	$0.0723438\pi$
$31$	1.08755e8	+	1.88370e8i	0.682278	+	1.18174i	−0.292007	+	0.956416i	$0.594323\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.55014e8	+	1.67927e8i	−1.14254	+	0.540439i
$36$		0			0
$37$	−3.38116e8	+	5.85635e8i	−0.801598	+	1.38841i	0.116965	+	0.993136i	$0.462683\pi$
$37$	−3.38116e8	+	5.85635e8i	−0.801598	+	1.38841i	−0.918564	+	0.395273i	$0.870650\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.07092e9			1.44360			0.721799	−	0.692102i	$-0.243315\pi$
$41$	1.07092e9			1.44360			0.721799	+	0.692102i	$0.243315\pi$
$42$		0			0
$43$	3.16356e8			0.328171			0.164085	−	0.986446i	$-0.447533\pi$
$43$	3.16356e8			0.328171			0.164085	+	0.986446i	$0.447533\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.43982e9	−	2.49384e9i	0.915735	−	1.58610i	0.109915	−	0.993941i	$-0.464942\pi$
$47$	1.43982e9	−	2.49384e9i	0.915735	−	1.58610i	0.805821	−	0.592159i	$-0.201724\pi$
$48$		0			0
$49$	6.96670e8	+	1.85053e9i	0.352329	+	0.935876i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.06271e8	+	1.05009e9i	0.199136	+	0.344913i	0.948248	−	0.317529i	$-0.102853\pi$
$53$	6.06271e8	+	1.05009e9i	0.199136	+	0.344913i	−0.749113	+	0.662443i	$0.769520\pi$
$54$		0			0
$55$	−2.89232e9			−0.774912
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.29381e9	+	9.16914e9i	0.964011	+	1.66972i	0.712249	+	0.701926i	$0.247676\pi$
$59$	5.29381e9	+	9.16914e9i	0.964011	+	1.66972i	0.251761	+	0.967789i	$0.418990\pi$
$60$		0			0
$61$	5.87221e9	−	1.01710e10i	0.890199	−	1.54187i	0.0505632	−	0.998721i	$-0.483898\pi$
$61$	5.87221e9	−	1.01710e10i	0.890199	−	1.54187i	0.839636	−	0.543149i	$-0.182768\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.15446e9	−	1.23919e10i	0.764809	−	1.32469i
$66$		0			0
$67$	−3.68558e8	−	6.38361e8i	−0.0333499	−	0.0577637i	0.848869	−	0.528604i	$-0.177284\pi$
$67$	−3.68558e8	−	6.38361e8i	−0.0333499	−	0.0577637i	−0.882219	+	0.470840i	$0.843951\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.60970e10			1.05882			0.529411	−	0.848365i	$-0.322413\pi$
$71$	1.60970e10			1.05882			0.529411	+	0.848365i	$0.322413\pi$
$72$		0			0
$73$	−5.44830e9	−	9.43674e9i	−0.307599	−	0.532778i	0.670237	−	0.742147i	$-0.266192\pi$
$73$	−5.44830e9	−	9.43674e9i	−0.307599	−	0.532778i	−0.977837	+	0.209369i	$0.932859\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.18945e9	+	1.45138e10i	−0.0500780	+	0.611058i
$78$		0			0
$79$	−1.18304e10	+	2.04908e10i	−0.432564	+	0.749222i	−0.997093	−	0.0761906i	$-0.975724\pi$
$79$	−1.18304e10	+	2.04908e10i	−0.432564	+	0.749222i	0.564530	+	0.825413i	$0.309058\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.25042e10			−1.46307			−0.731534	−	0.681805i	$-0.761195\pi$
$83$	−5.25042e10			−1.46307			−0.731534	+	0.681805i	$0.761195\pi$
$84$		0			0
$85$	2.31444e10			0.565772
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.77325e8	−	1.51957e9i	0.0166539	−	0.0288454i	−0.857578	−	0.514353i	$-0.828032\pi$
$89$	8.77325e8	−	1.51957e9i	0.0166539	−	0.0288454i	0.874232	+	0.485508i	$0.161365\pi$
$90$		0			0
$91$	−5.92408e10	−	4.09975e10i	−0.995161	−	0.688699i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.26946e10	−	5.66288e10i	−0.433508	−	0.750857i
$96$		0			0
$97$	7.32713e10			0.866342			0.433171	−	0.901312i	$-0.357395\pi$
$97$	7.32713e10			0.866342			0.433171	+	0.901312i	$0.357395\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.59199e10	−	6.22151e10i	−0.340069	−	0.589017i	0.644376	−	0.764709i	$-0.277117\pi$
$101$	−3.59199e10	−	6.22151e10i	−0.340069	−	0.589017i	−0.984445	+	0.175692i	$0.943784\pi$
$102$		0			0
$103$	1.59342e9	−	2.75988e9i	0.0135433	−	0.0234577i	−0.859174	−	0.511683i	$-0.829022\pi$
$103$	1.59342e9	−	2.75988e9i	0.0135433	−	0.0234577i	0.872718	+	0.488225i	$0.162356\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.27836e11	+	2.21418e11i	−0.881132	+	1.52616i	−0.0310477	+	0.999518i	$0.509884\pi$
$107$	−1.27836e11	+	2.21418e11i	−0.881132	+	1.52616i	−0.850084	+	0.526647i	$0.823449\pi$
$108$		0			0
$109$	8.33578e10	+	1.44380e11i	0.518920	+	0.898796i	0.999758	+	0.0219865i	$0.00699909\pi$
$109$	8.33578e10	+	1.44380e11i	0.518920	+	0.898796i	−0.480838	+	0.876809i	$0.659668\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.44148e11			−0.735997			−0.367999	−	0.929826i	$-0.619957\pi$
$113$	−1.44148e11			−0.735997			−0.367999	+	0.929826i	$0.619957\pi$
$114$		0			0
$115$	5.09167e10	+	8.81904e10i	0.236060	+	0.408868i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9.51796e9	−	1.16139e11i	0.0365625	−	0.446140i
$120$		0			0
$121$	8.90315e10	−	1.54207e11i	0.312050	−	0.540486i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.73589e11			0.508764
$126$		0			0
$127$	−5.74607e11			−1.54330			−0.771651	−	0.636047i	$-0.780569\pi$
$127$	−5.74607e11			−1.54330			−0.771651	+	0.636047i	$0.780569\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.42166e11	−	2.46239e11i	0.321962	−	0.557654i	−0.658931	−	0.752203i	$-0.728991\pi$
$131$	1.42166e11	−	2.46239e11i	0.321962	−	0.557654i	0.980893	+	0.194550i	$0.0623245\pi$
$132$		0			0
$133$	−2.97611e11	+	1.40775e11i	−0.620105	+	0.293320i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.41807e11	+	7.65232e11i	0.782112	+	1.35466i	0.930709	+	0.365760i	$0.119191\pi$
$137$	4.41807e11	+	7.65232e11i	0.782112	+	1.35466i	−0.148597	+	0.988898i	$0.547476\pi$
$138$		0			0
$139$	7.35885e11			1.20290			0.601449	−	0.798912i	$-0.294590\pi$
$139$	7.35885e11			1.20290			0.601449	+	0.798912i	$0.294590\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.65290e11	−	4.59496e11i	−0.370999	−	0.642590i
$144$		0			0
$145$	8.74287e11	−	1.51431e12i	1.13274	−	1.96196i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.38720e11	−	2.40271e11i	0.154745	−	0.268026i	−0.778221	−	0.627990i	$-0.783878\pi$
$149$	1.38720e11	−	2.40271e11i	0.154745	−	0.268026i	0.932966	+	0.359964i	$0.117211\pi$
$150$		0			0
$151$	−3.54520e10	−	6.14047e10i	−0.0367509	−	0.0636544i	0.847065	−	0.531489i	$-0.178367\pi$
$151$	−3.54520e10	−	6.14047e10i	−0.0367509	−	0.0636544i	−0.883816	+	0.467835i	$0.845034\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.92102e12			1.72467
$156$		0			0
$157$	−7.60381e11	−	1.31702e12i	−0.636184	−	1.10190i	−0.986263	−	0.165183i	$-0.947178\pi$
$157$	−7.60381e11	−	1.31702e12i	−0.636184	−	1.10190i	0.350079	−	0.936720i	$-0.386155\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.63482e11	−	2.19235e11i	0.337669	−	0.159723i
$162$		0			0
$163$	4.26239e11	−	7.38268e11i	0.290149	−	0.502553i	−0.683696	−	0.729767i	$-0.739628\pi$
$163$	4.26239e11	−	7.38268e11i	0.290149	−	0.502553i	0.973845	+	0.227214i	$0.0729617\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.44464e12			−0.860636			−0.430318	−	0.902677i	$-0.641599\pi$
$167$	−1.44464e12			−0.860636			−0.430318	+	0.902677i	$0.641599\pi$
$168$		0			0
$169$	8.32729e11			0.464651
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.42009e11	−	2.45967e11i	0.0696726	−	0.120676i	−0.829085	−	0.559123i	$-0.811138\pi$
$173$	1.42009e11	−	2.45967e11i	0.0696726	−	0.120676i	0.898757	+	0.438447i	$0.144471\pi$
$174$		0			0
$175$	−1.05958e11	+	1.29292e12i	−0.0488006	+	0.595471i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.05724e12	−	1.83119e12i	−0.430012	−	0.744803i	0.566862	−	0.823813i	$-0.308157\pi$
$179$	−1.05724e12	−	1.83119e12i	−0.430012	−	0.744803i	−0.996874	+	0.0790100i	$0.974824\pi$
$180$		0			0
$181$	−1.96228e11			−0.0750808			−0.0375404	−	0.999295i	$-0.511952\pi$
$181$	−1.96228e11			−0.0750808			−0.0375404	+	0.999295i	$0.511952\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.98619e12	+	5.17223e12i	1.01315	+	1.75482i
$186$		0			0
$187$	4.29102e11	−	7.43226e11i	0.137224	−	0.237680i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.17934e11	+	1.07029e12i	−0.175897	+	0.304663i	−0.940471	−	0.339873i	$-0.889616\pi$
$191$	−6.17934e11	+	1.07029e12i	−0.175897	+	0.304663i	0.764574	+	0.644536i	$0.222949\pi$
$192$		0			0
$193$	2.76975e12	+	4.79734e12i	0.744517	+	1.28954i	0.950420	+	0.310969i	$0.100654\pi$
$193$	2.76975e12	+	4.79734e12i	0.744517	+	1.28954i	−0.205903	+	0.978572i	$0.566013\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.68356e12			−0.884511			−0.442255	−	0.896889i	$-0.645821\pi$
$197$	−3.68356e12			−0.884511			−0.442255	+	0.896889i	$0.645821\pi$
$198$		0			0
$199$	−2.12062e12	−	3.67301e12i	−0.481693	−	0.834316i	0.518086	−	0.855328i	$-0.326645\pi$
$199$	−2.12062e12	−	3.67301e12i	−0.481693	−	0.834316i	−0.999779	+	0.0210119i	$0.993311\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−7.23933e12	−	5.00996e12i	−1.47391	−	1.02001i
$204$		0			0
$205$	4.72910e12	−	8.19105e12i	0.912289	−	1.58013i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.42466e12			−0.420578
$210$		0			0
$211$	3.11511e12			0.512766			0.256383	−	0.966575i	$-0.417469\pi$
$211$	3.11511e12			0.512766			0.256383	+	0.966575i	$0.417469\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.39700e12	−	2.41968e12i	0.207389	−	0.359208i
$216$		0			0
$217$	7.90006e11	−	9.63976e12i	0.111456	−	1.36000i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.12285e12	+	3.67689e12i	0.270871	+	0.469162i
$222$		0			0
$223$	4.50845e12			0.547457			0.273729	−	0.961807i	$-0.411743\pi$
$223$	4.50845e12			0.547457			0.273729	+	0.961807i	$0.411743\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.29604e12	−	9.17302e12i	−0.583189	−	1.01011i	−0.995099	−	0.0988883i	$-0.968471\pi$
$227$	−5.29604e12	−	9.17302e12i	−0.583189	−	1.01011i	0.411909	−	0.911225i	$-0.364862\pi$
$228$		0			0
$229$	8.32440e12	−	1.44183e13i	0.873489	−	1.51293i	0.0151263	−	0.999886i	$-0.495185\pi$
$229$	8.32440e12	−	1.44183e13i	0.873489	−	1.51293i	0.858363	−	0.513043i	$-0.171482\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.03434e12	+	1.04518e13i	−0.575668	+	0.997087i	0.420300	+	0.907385i	$0.361925\pi$
$233$	−6.03434e12	+	1.04518e13i	−0.575668	+	0.997087i	−0.995969	+	0.0897017i	$0.971409\pi$
$234$		0			0
$235$	−1.27163e13	−	2.20252e13i	−1.15741	−	2.00469i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.97960e12			−0.164206			−0.0821030	−	0.996624i	$-0.526164\pi$
$239$	−1.97960e12			−0.164206			−0.0821030	+	0.996624i	$0.526164\pi$
$240$		0			0
$241$	3.01347e12	+	5.21948e12i	0.238766	+	0.413555i	0.960361	−	0.278761i	$-0.0899237\pi$
$241$	3.01347e12	+	5.21948e12i	0.238766	+	0.413555i	−0.721594	+	0.692316i	$0.756590\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.72304e13	+	2.84326e12i	1.24705	+	0.205780i
$246$		0			0
$247$	5.99765e12	−	1.03882e13i	0.415095	−	0.718966i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.34626e13			1.48652			0.743258	−	0.669004i	$-0.233279\pi$
$251$	2.34626e13			1.48652			0.743258	+	0.669004i	$0.233279\pi$
$252$		0			0
$253$	3.77603e12			0.229020
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.73214e12	−	8.19631e12i	0.263285	−	0.456023i	−0.703828	−	0.710370i	$-0.748527\pi$
$257$	4.73214e12	−	8.19631e12i	0.263285	−	0.456023i	0.967113	+	0.254348i	$0.0818608\pi$
$258$		0			0
$259$	2.71825e13	−	1.28578e13i	1.44924	−	0.685516i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.67219e12	−	4.62837e12i	−0.130952	−	0.226815i	0.793092	−	0.609102i	$-0.208470\pi$
$263$	−2.67219e12	−	4.62837e12i	−0.130952	−	0.226815i	−0.924044	+	0.382287i	$0.875137\pi$
$264$		0			0
$265$	1.07090e13			0.503379
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.18839e13	+	2.05835e13i	0.514425	+	0.891009i	0.999860	+	0.0167369i	$0.00532776\pi$
$269$	1.18839e13	+	2.05835e13i	0.514425	+	0.891009i	−0.485435	+	0.874273i	$0.661339\pi$
$270$		0			0
$271$	5.12183e12	−	8.87127e12i	0.212860	−	0.368684i	−0.739748	−	0.672884i	$-0.765056\pi$
$271$	5.12183e12	−	8.87127e12i	0.212860	−	0.368684i	0.952608	+	0.304199i	$0.0983888\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.77695e12	+	8.27392e12i	−0.183156	+	0.317235i
$276$		0			0
$277$	−1.76880e13	−	3.06365e13i	−0.651687	−	1.12876i	−0.982713	−	0.185134i	$-0.940728\pi$
$277$	−1.76880e13	−	3.06365e13i	−0.651687	−	1.12876i	0.331026	−	0.943622i	$-0.392605\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.68535e13			0.914358			0.457179	−	0.889375i	$-0.348860\pi$
$281$	2.68535e13			0.914358			0.457179	+	0.889375i	$0.348860\pi$
$282$		0			0
$283$	−1.44474e13	−	2.50236e13i	−0.473111	−	0.819452i	0.526415	−	0.850228i	$-0.323536\pi$
$283$	−1.44474e13	−	2.50236e13i	−0.473111	−	0.819452i	−0.999526	+	0.0307754i	$0.990202\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.91582e13	−	2.70994e13i	−1.18706	−	0.821502i
$288$		0			0
$289$	1.37023e13	−	2.37330e13i	0.399811	−	0.692493i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.43381e13			1.19951			0.599757	−	0.800183i	$-0.295264\pi$
$293$	4.43381e13			1.19951			0.599757	+	0.800183i	$0.295264\pi$
$294$		0			0
$295$	9.35081e13			2.43685
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9.34039e12	+	1.61780e13i	−0.226034	+	0.391502i
$300$		0			0
$301$	−1.15676e13	−	8.00530e12i	−0.269852	−	0.186751i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.18624e13	−	8.98283e13i	−1.12513	−	1.94878i
$306$		0			0
$307$	4.12666e13			0.863650			0.431825	−	0.901957i	$-0.357870\pi$
$307$	4.12666e13			0.863650			0.431825	+	0.901957i	$0.357870\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.05188e13	+	1.82192e13i	0.205015	+	0.355096i	0.950138	−	0.311831i	$-0.100942\pi$
$311$	1.05188e13	+	1.82192e13i	0.205015	+	0.355096i	−0.745123	+	0.666928i	$0.767609\pi$
$312$		0			0
$313$	1.57700e13	−	2.73145e13i	0.296715	−	0.513925i	−0.678668	−	0.734446i	$-0.737442\pi$
$313$	1.57700e13	−	2.73145e13i	0.296715	−	0.513925i	0.975382	+	0.220521i	$0.0707756\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.15937e13	+	7.20425e13i	−0.729797	+	1.26405i	0.227172	+	0.973855i	$0.427052\pi$
$317$	−4.15937e13	+	7.20425e13i	−0.729797	+	1.26405i	−0.956969	+	0.290191i	$0.906281\pi$
$318$		0			0
$319$	−3.24190e13	−	5.61513e13i	−0.549478	−	0.951724i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.94021e13			0.307069
$324$		0			0
$325$	−2.36325e13	−	4.09327e13i	−0.361536	−	0.626199i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.15753e14	+	5.47530e13i	−1.65560	+	0.783125i
$330$		0			0
$331$	2.69356e13	−	4.66537e13i	0.372625	−	0.645405i	−0.617344	−	0.786694i	$-0.711791\pi$
$331$	2.69356e13	−	4.66537e13i	0.372625	−	0.645405i	0.989968	+	0.141288i	$0.0451245\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.51008e12			−0.0843025
$336$		0			0
$337$	−1.25689e12			−0.0157519			−0.00787597	−	0.999969i	$-0.502507\pi$
$337$	−1.25689e12			−0.0157519			−0.00787597	+	0.999969i	$0.502507\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.56161e13	−	6.16889e13i	0.418309	−	0.724533i
$342$		0			0
$343$	2.13535e13	−	8.52937e13i	0.242857	−	0.970062i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.32290e12	−	2.29133e12i	−0.0141161	−	0.0244498i	0.858881	−	0.512175i	$-0.171160\pi$
$347$	−1.32290e12	−	2.29133e12i	−0.0141161	−	0.0244498i	−0.872997	+	0.487725i	$0.837827\pi$
$348$		0			0
$349$	−1.32862e13			−0.137360			−0.0686802	−	0.997639i	$-0.521879\pi$
$349$	−1.32862e13			−0.137360			−0.0686802	+	0.997639i	$0.521879\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.88045e12	−	6.72114e12i	−0.0376809	−	0.0652653i	0.846570	−	0.532278i	$-0.178664\pi$
$353$	−3.88045e12	−	6.72114e12i	−0.0376809	−	0.0652653i	−0.884251	+	0.467012i	$0.845330\pi$
$354$		0			0
$355$	7.10829e13	−	1.23119e14i	0.669128	−	1.15896i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.38673e13	+	4.13393e13i	−0.211243	+	0.365884i	−0.952104	−	0.305775i	$-0.901085\pi$
$359$	−2.38673e13	+	4.13393e13i	−0.211243	+	0.365884i	0.740861	+	0.671659i	$0.234418\pi$
$360$		0			0
$361$	3.08369e13	+	5.34111e13i	0.264717	+	0.458503i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−9.62371e13			−0.777556
$366$		0			0
$367$	1.28363e12	+	2.22331e12i	0.0100641	+	0.0174316i	0.871014	−	0.491259i	$-0.163463\pi$
$367$	1.28363e12	+	2.22331e12i	0.0100641	+	0.0174316i	−0.860949	+	0.508690i	$0.830130\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.40399e12	−	5.37381e13i	0.0325304	−	0.396941i
$372$		0			0
$373$	1.26873e14	−	2.19751e14i	0.909853	−	1.57591i	0.0955862	−	0.995421i	$-0.469527\pi$
$373$	1.26873e14	−	2.19751e14i	0.909853	−	1.57591i	0.814267	−	0.580491i	$-0.197139\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.20766e14			2.16926
$378$		0			0
$379$	−2.00455e14			−1.31674			−0.658372	−	0.752693i	$-0.728755\pi$
$379$	−2.00455e14			−1.31674			−0.658372	+	0.752693i	$0.728755\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.48476e12	−	7.76784e12i	0.0278065	−	0.0481623i	−0.851787	−	0.523888i	$-0.824481\pi$
$383$	4.48476e12	−	7.76784e12i	0.0278065	−	0.0481623i	0.879594	+	0.475726i	$0.157814\pi$
$384$		0			0
$385$	1.05758e14	+	7.31895e13i	0.637204	+	0.440976i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.05989e14	−	1.83578e14i	−0.603307	−	1.04496i	−0.992317	−	0.123725i	$-0.960516\pi$
$389$	−1.05989e14	−	1.83578e14i	−0.603307	−	1.04496i	0.389010	−	0.921234i	$-0.372817\pi$
$390$		0			0
$391$	−3.02158e13			−0.167210
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.04484e14	+	1.80972e14i	0.546721	+	0.946949i
$396$		0			0
$397$	4.03478e12	−	6.98845e12i	0.0205339	−	0.0355658i	−0.855576	−	0.517677i	$-0.826797\pi$
$397$	4.03478e12	−	6.98845e12i	0.0205339	−	0.0355658i	0.876110	+	0.482112i	$0.160130\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.79341e14	+	3.10627e14i	−0.863743	+	1.49605i	0.00454658	+	0.999990i	$0.498553\pi$
$401$	−1.79341e14	+	3.10627e14i	−0.863743	+	1.49605i	−0.868290	+	0.496057i	$0.834781\pi$
$402$		0			0
$403$	1.76200e14	+	3.05187e14i	0.825711	+	1.43017i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.21458e14			0.982931
$408$		0			0
$409$	4.76470e13	+	8.25271e13i	0.205853	+	0.356548i	0.950404	−	0.311017i	$-0.100670\pi$
$409$	4.76470e13	+	8.25271e13i	0.205853	+	0.356548i	−0.744551	+	0.667566i	$0.767336\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.84546e13	−	4.69228e14i	0.157479	−	1.92158i
$414$		0			0
$415$	−2.31854e14	+	4.01583e14i	−0.924592	+	1.60144i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.91297e14			1.85852			0.929259	−	0.369429i	$-0.120447\pi$
$419$	4.91297e14			1.85852			0.929259	+	0.369429i	$0.120447\pi$
$420$		0			0
$421$	−8.91249e13			−0.328434			−0.164217	−	0.986424i	$-0.552510\pi$
$421$	−8.91249e13			−0.328434			−0.164217	+	0.986424i	$0.552510\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.82251e13	−	6.62078e13i	0.133724	−	0.231617i
$426$		0			0
$427$	−4.72090e14	+	2.23306e14i	−1.60943	+	0.761286i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5.88781e13	−	1.01980e14i	−0.190690	−	0.330285i	0.754789	−	0.655968i	$-0.227739\pi$
$431$	−5.88781e13	−	1.01980e14i	−0.190690	−	0.330285i	−0.945479	+	0.325683i	$0.894406\pi$
$432$		0			0
$433$	1.72536e14			0.544747			0.272374	−	0.962192i	$-0.412191\pi$
$433$	1.72536e14			0.544747			0.272374	+	0.962192i	$0.412191\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.26840e13	+	7.39308e13i	0.128120	+	0.221910i
$438$		0			0
$439$	7.65037e12	−	1.32508e13i	0.0223938	−	0.0387872i	−0.854611	−	0.519268i	$-0.826205\pi$
$439$	7.65037e12	−	1.32508e13i	0.0223938	−	0.0387872i	0.877005	+	0.480481i	$0.159538\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.45030e14	+	5.97610e14i	−0.960808	+	1.66417i	−0.240328	+	0.970692i	$0.577255\pi$
$443$	−3.45030e14	+	5.97610e14i	−0.960808	+	1.66417i	−0.720480	+	0.693476i	$0.756078\pi$
$444$		0			0
$445$	−7.74840e12	−	1.34206e13i	−0.0210490	−	0.0364580i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.48970e13			0.193691			0.0968455	−	0.995299i	$-0.469125\pi$
$449$	7.48970e13			0.193691			0.0968455	+	0.995299i	$0.469125\pi$
$450$		0			0
$451$	−1.75357e14	−	3.03728e14i	−0.442540	−	0.766502i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5.75176e14	+	2.72067e14i	−1.38273	+	0.654055i
$456$		0			0
$457$	4.73909e13	−	8.20835e13i	0.111213	−	0.192627i	−0.805047	−	0.593212i	$-0.797860\pi$
$457$	4.73909e13	−	8.20835e13i	0.111213	−	0.192627i	0.916260	+	0.400585i	$0.131193\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−5.64437e14			−1.26259			−0.631293	−	0.775545i	$-0.717475\pi$
$461$	−5.64437e14			−1.26259			−0.631293	+	0.775545i	$0.717475\pi$
$462$		0			0
$463$	3.35145e14			0.732043			0.366022	−	0.930606i	$-0.380720\pi$
$463$	3.35145e14			0.732043			0.366022	+	0.930606i	$0.380720\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.94285e14	−	6.82921e14i	0.821423	−	1.42275i	−0.0831992	−	0.996533i	$-0.526514\pi$
$467$	3.94285e14	−	6.82921e14i	0.821423	−	1.42275i	0.904622	−	0.426214i	$-0.140153\pi$
$468$		0			0
$469$	−2.67723e12	+	3.26679e13i	−0.00544797	+	0.0664769i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−5.18015e13	−	8.97228e13i	−0.100602	−	0.174247i
$474$		0			0
$475$	−2.15993e14			−0.409850
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.59097e14	+	4.48770e14i	0.469481	+	0.813165i	0.999391	−	0.0348891i	$-0.0111078\pi$
$479$	2.59097e14	+	4.48770e14i	0.469481	+	0.813165i	−0.529910	+	0.848054i	$0.677774\pi$
$480$		0			0
$481$	−5.47800e14	+	9.48817e14i	−0.970116	+	1.68029i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.23560e14	−	5.60423e14i	0.547489	−	0.948279i
$486$		0			0
$487$	−1.41539e14	−	2.45152e14i	−0.234135	−	0.405534i	0.724886	−	0.688869i	$-0.241892\pi$
$487$	−1.41539e14	−	2.45152e14i	−0.234135	−	0.405534i	−0.959021	+	0.283335i	$0.908559\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.25190e14			−0.988698			−0.494349	−	0.869264i	$-0.664593\pi$
$491$	−6.25190e14			−0.988698			−0.494349	+	0.869264i	$0.664593\pi$
$492$		0			0
$493$	2.59416e14	+	4.49322e14i	0.401180	+	0.694864i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5.88585e14	−	4.07329e14i	−0.870661	−	0.602539i
$498$		0			0
$499$	−5.74287e14	+	9.94695e14i	−0.830953	+	1.43925i	0.0663310	+	0.997798i	$0.478871\pi$
$499$	−5.74287e14	+	9.94695e14i	−0.830953	+	1.43925i	−0.897284	+	0.441454i	$0.854463\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.80721e14			−0.527209			−0.263604	−	0.964631i	$-0.584911\pi$
$503$	−3.80721e14			−0.527209			−0.263604	+	0.964631i	$0.584911\pi$
$504$		0			0
$505$	−6.34477e14			−0.859634
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5.78564e14	+	1.00210e15i	−0.750592	+	1.30006i	0.196945	+	0.980415i	$0.436898\pi$
$509$	−5.78564e14	+	1.00210e15i	−0.750592	+	1.30006i	−0.947536	+	0.319648i	$0.896435\pi$
$510$		0			0
$511$	−3.95768e13	+	4.82922e14i	−0.0502488	+	0.613143i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.40728e13	−	2.43748e13i	−0.0171175	−	0.0296484i
$516$		0			0
$517$	−9.43049e14			−1.12289
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.63015e14	−	2.82351e14i	−0.186046	−	0.322242i	0.757882	−	0.652391i	$-0.226234\pi$
$521$	−1.63015e14	−	2.82351e14i	−0.186046	−	0.322242i	−0.943929	+	0.330150i	$0.892901\pi$
$522$		0			0
$523$	−2.85077e14	+	4.93768e14i	−0.318569	+	0.551777i	−0.980190	−	0.198061i	$-0.936535\pi$
$523$	−2.85077e14	+	4.93768e14i	−0.318569	+	0.551777i	0.661621	+	0.749838i	$0.269869\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.85000e14	+	4.93634e14i	−0.305412	+	0.528989i
$528$		0			0
$529$	4.09931e14	+	7.10022e14i	0.430234	+	0.745187i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.73506e15			1.74708
$534$		0			0
$535$	1.12902e15	+	1.95552e15i	1.11367	+	1.92893i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.10760e14	−	5.00599e14i	0.388911	−	0.473971i
$540$		0			0
$541$	−8.15976e14	+	1.41331e15i	−0.756994	+	1.31115i	0.187383	+	0.982287i	$0.439999\pi$
$541$	−8.15976e14	+	1.41331e15i	−0.756994	+	1.31115i	−0.944377	+	0.328865i	$0.893334\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.47240e15			1.31174
$546$		0			0
$547$	−1.11864e15			−0.976700			−0.488350	−	0.872648i	$-0.662401\pi$
$547$	−1.11864e15			−0.976700			−0.488350	+	0.872648i	$0.662401\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7.32923e14	−	1.26946e15i	0.614787	−	1.06484i
$552$		0			0
$553$	9.51093e14	−	4.49882e14i	0.782050	−	0.369923i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.91063e14	+	3.30930e14i	0.150998	+	0.261537i	0.931595	−	0.363498i	$-0.118418\pi$
$557$	1.91063e14	+	3.30930e14i	0.150998	+	0.261537i	−0.780596	+	0.625036i	$0.785084\pi$
$558$		0			0
$559$	5.12545e14			0.397161
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.55105e12	−	9.61470e12i	−0.00413598	−	0.00716373i	0.863950	−	0.503578i	$-0.167983\pi$
$563$	−5.55105e12	−	9.61470e12i	−0.00413598	−	0.00716373i	−0.868086	+	0.496414i	$0.834650\pi$
$564$		0			0
$565$	−6.36544e14	+	1.10253e15i	−0.465117	+	0.805606i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.19121e15	+	2.06323e15i	−0.837277	+	1.45021i	0.0548861	+	0.998493i	$0.482520\pi$
$569$	−1.19121e15	+	2.06323e15i	−0.837277	+	1.45021i	−0.892163	+	0.451714i	$0.850813\pi$
$570$		0			0
$571$	−7.21148e14	−	1.24907e15i	−0.497194	−	0.861166i	0.502801	−	0.864402i	$-0.332303\pi$
$571$	−7.21148e14	−	1.24907e15i	−0.497194	−	0.861166i	−0.999995	+	0.00323683i	$0.998970\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.36375e14			0.223178
$576$		0			0
$577$	−7.94922e13	−	1.37685e14i	−0.0517437	−	0.0896227i	0.838993	−	0.544142i	$-0.183145\pi$
$577$	−7.94922e13	−	1.37685e14i	−0.0517437	−	0.0896227i	−0.890737	+	0.454519i	$0.849811\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.91981e15	+	1.32860e15i	1.20307	+	0.832581i
$582$		0			0
$583$	1.98547e14	−	3.43893e14i	0.122092	−	0.211469i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.01493e13			0.0119330			0.00596651	−	0.999982i	$-0.498101\pi$
$587$	2.01493e13			0.0119330			0.00596651	+	0.999982i	$0.498101\pi$
$588$		0			0
$589$	1.61041e15			0.936056
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.54276e14	−	2.67213e14i	0.0863966	−	0.149643i	−0.819589	−	0.572952i	$-0.805798\pi$
$593$	1.54276e14	−	2.67213e14i	0.0863966	−	0.149643i	0.905985	+	0.423309i	$0.139131\pi$
$594$		0			0
$595$	−8.46273e14	−	5.85662e14i	−0.465230	−	0.321961i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.66801e14	−	1.15493e15i	−0.353304	−	0.611941i	0.633522	−	0.773725i	$-0.281609\pi$
$599$	−6.66801e14	−	1.15493e15i	−0.353304	−	0.611941i	−0.986826	+	0.161784i	$0.948275\pi$
$600$		0			0
$601$	−3.62651e15			−1.88660			−0.943298	−	0.331946i	$-0.892295\pi$
$601$	−3.62651e15			−1.88660			−0.943298	+	0.331946i	$0.892295\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.86312e14	−	1.36193e15i	−0.394403	−	0.683126i
$606$		0			0
$607$	−1.11219e15	+	1.92636e15i	−0.547823	+	0.948857i	0.450601	+	0.892726i	$0.351210\pi$
$607$	−1.11219e15	+	1.92636e15i	−0.547823	+	0.948857i	−0.998423	+	0.0561313i	$0.982123\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.33273e15	−	4.04040e15i	1.10825	−	1.91954i
$612$		0			0
$613$	1.10277e15	+	1.91005e15i	0.514579	+	0.891278i	0.999857	+	0.0169174i	$0.00538523\pi$
$613$	1.10277e15	+	1.91005e15i	0.514579	+	0.891278i	−0.485278	+	0.874360i	$0.661281\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.81703e15			−1.26830			−0.634151	−	0.773210i	$-0.718650\pi$
$617$	−2.81703e15			−1.26830			−0.634151	+	0.773210i	$0.718650\pi$
$618$		0			0
$619$	−1.08191e15	−	1.87393e15i	−0.478514	−	0.828810i	0.521183	−	0.853445i	$-0.325491\pi$
$619$	−1.08191e15	−	1.87393e15i	−0.478514	−	0.828810i	−0.999697	+	0.0246351i	$0.992158\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.05317e13	+	3.33627e13i	−0.0301093	+	0.0142422i
$624$		0			0
$625$	1.47879e15	−	2.56134e15i	0.620250	−	1.07430i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.77211e15			−0.717649
$630$		0			0
$631$	1.72755e15			0.687494			0.343747	−	0.939062i	$-0.388304\pi$
$631$	1.72755e15			0.687494			0.343747	+	0.939062i	$0.388304\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.53742e15	+	4.39494e15i	−0.975297	+	1.68926i
$636$		0			0
$637$	1.12871e15	+	2.99814e15i	0.426398	+	1.13262i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.24235e15	+	3.88386e15i	0.818434	+	1.41757i	0.906835	+	0.421485i	$0.138491\pi$
$641$	2.24235e15	+	3.88386e15i	0.818434	+	1.41757i	−0.0884009	+	0.996085i	$0.528176\pi$
$642$		0			0
$643$	1.10216e15			0.395444			0.197722	−	0.980258i	$-0.436646\pi$
$643$	1.10216e15			0.395444			0.197722	+	0.980258i	$0.436646\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.53359e14	−	1.13165e15i	−0.226557	−	0.392409i	0.730228	−	0.683203i	$-0.239414\pi$
$647$	−6.53359e14	−	1.13165e15i	−0.226557	−	0.392409i	−0.956786	+	0.290794i	$0.906080\pi$
$648$		0			0
$649$	1.73366e15	−	3.00279e15i	0.591042	−	1.02371i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.49631e15	−	2.59168e15i	0.493172	−	0.854199i	−0.506797	−	0.862065i	$-0.669171\pi$
$653$	1.49631e15	−	2.59168e15i	0.493172	−	0.854199i	0.999969	+	0.00786625i	$0.00250393\pi$
$654$		0			0
$655$	−1.25559e15	−	2.17474e15i	−0.406930	−	0.704824i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4.10226e15			−1.28574			−0.642871	−	0.765974i	$-0.722257\pi$
$659$	−4.10226e15			−1.28574			−0.642871	+	0.765974i	$0.722257\pi$
$660$		0			0
$661$	−1.46524e15	−	2.53787e15i	−0.451649	−	0.782278i	0.546840	−	0.837237i	$-0.315831\pi$
$661$	−1.46524e15	−	2.53787e15i	−0.451649	−	0.782278i	−0.998489	+	0.0549588i	$0.982497\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.37496e14	+	2.89796e15i	−0.0708170	+	0.864118i
$666$		0			0
$667$	−1.14141e15	+	1.97698e15i	−0.334773	+	0.579844i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.84616e15			−1.09157
$672$		0			0
$673$	−9.27300e14			−0.258903			−0.129452	−	0.991586i	$-0.541322\pi$
$673$	−9.27300e14			−0.258903			−0.129452	+	0.991586i	$0.541322\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8.76335e14	−	1.51786e15i	0.236828	−	0.410198i	−0.722975	−	0.690875i	$-0.757226\pi$
$677$	8.76335e14	−	1.51786e15i	0.236828	−	0.410198i	0.959802	+	0.280677i	$0.0905590\pi$
$678$		0			0
$679$	−2.67917e15	−	1.85411e15i	−0.712387	−	0.493005i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.48295e14	−	4.30059e14i	−0.0639224	−	0.110717i	0.832293	−	0.554336i	$-0.187028\pi$
$683$	−2.48295e14	−	4.30059e14i	−0.0639224	−	0.110717i	−0.896215	+	0.443619i	$0.853694\pi$
$684$		0			0
$685$	7.80393e15			1.97704
$686$		0			0
$687$		0			0
$688$		0			0
$689$	9.82251e14	+	1.70131e15i	0.241000	+	0.417424i
$690$		0			0
$691$	2.52516e15	−	4.37371e15i	0.609762	−	1.05614i	−0.381518	−	0.924361i	$-0.624599\pi$
$691$	2.52516e15	−	4.37371e15i	0.609762	−	1.05614i	0.991279	−	0.131777i	$-0.0420682\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.24961e15	−	5.62849e15i	0.760177	−	1.31666i
$696$		0			0
$697$	1.40321e15	+	2.43043e15i	0.323103	+	0.559632i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.91610e15			0.427533			0.213767	−	0.976885i	$-0.431427\pi$
$701$	1.91610e15			0.427533			0.213767	+	0.976885i	$0.431427\pi$
$702$		0			0
$703$	2.50335e15	+	4.33593e15i	0.549879	+	0.952419i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.60924e14	+	3.18384e15i	−0.0555531	+	0.677866i
$708$		0			0
$709$	3.20717e15	−	5.55499e15i	0.672308	−	1.16447i	−0.304940	−	0.952372i	$-0.598636\pi$
$709$	3.20717e15	−	5.55499e15i	0.672308	−	1.16447i	0.977248	−	0.212100i	$-0.0680303\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.50796e15			−0.509715
$714$		0			0
$715$	−4.68600e15			−0.937820
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.63350e15	+	2.82931e15i	−0.317038	+	0.549126i	−0.979869	−	0.199643i	$-0.936022\pi$
$719$	−1.63350e15	+	2.82931e15i	−0.317038	+	0.549126i	0.662831	+	0.748769i	$0.269355\pi$
$720$		0			0
$721$	−1.28101e14	+	6.05940e13i	−0.0244855	+	0.0115821i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.88794e15	−	5.00205e15i	−0.535462	−	0.927447i
$726$		0			0
$727$	−4.83415e14			−0.0882838			−0.0441419	−	0.999025i	$-0.514055\pi$
$727$	−4.83415e14			−0.0882838			−0.0441419	+	0.999025i	$0.514055\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.14515e14	+	7.17961e14i	0.0734505	+	0.127220i
$732$		0			0
$733$	4.48117e15	−	7.76162e15i	0.782204	−	1.35482i	−0.148452	−	0.988920i	$-0.547429\pi$
$733$	4.48117e15	−	7.76162e15i	0.782204	−	1.35482i	0.930655	−	0.365897i	$-0.119238\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.20698e14	+	2.09056e14i	−0.0204470	+	0.0354153i
$738$		0			0
$739$	−1.21654e15	−	2.10711e15i	−0.203041	−	0.351677i	0.746466	−	0.665424i	$-0.231749\pi$
$739$	−1.21654e15	−	2.10711e15i	−0.203041	−	0.351677i	−0.949507	+	0.313747i	$0.898416\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.06507e16			1.72560			0.862798	−	0.505549i	$-0.168710\pi$
$743$	1.06507e16			1.72560			0.862798	+	0.505549i	$0.168710\pi$
$744$		0			0
$745$	−1.22516e15	−	2.12203e15i	−0.195583	−	0.338760i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.02772e16	−	4.86129e15i	1.59304	−	0.753532i
$750$		0			0
$751$	1.80280e15	−	3.12254e15i	0.275377	−	0.476967i	−0.694853	−	0.719152i	$-0.744531\pi$
$751$	1.80280e15	−	3.12254e15i	0.275377	−	0.476967i	0.970230	+	0.242185i	$0.0778639\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.26213e14			−0.0928996
$756$		0			0
$757$	2.35449e15			0.344247			0.172123	−	0.985075i	$-0.444937\pi$
$757$	2.35449e15			0.344247			0.172123	+	0.985075i	$0.444937\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.27438e14	+	1.08675e15i	−0.0891159	+	0.154353i	−0.907138	−	0.420834i	$-0.861737\pi$
$761$	−6.27438e14	+	1.08675e15i	−0.0891159	+	0.154353i	0.818022	+	0.575187i	$0.195071\pi$
$762$		0			0
$763$	6.05516e14	−	7.38859e15i	0.0847698	−	1.03437i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.57676e15	+	1.48554e16i	1.16667	+	2.02074i
$768$		0			0
$769$	5.40294e15			0.724495			0.362247	−	0.932082i	$-0.382010\pi$
$769$	5.40294e15			0.724495			0.362247	+	0.932082i	$0.382010\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.91299e15	−	8.50955e15i	−0.640263	−	1.10897i	−0.985374	−	0.170406i	$-0.945492\pi$
$773$	−4.91299e15	−	8.50955e15i	−0.640263	−	1.10897i	0.345111	−	0.938562i	$-0.387841\pi$
$774$		0			0
$775$	3.17274e15	−	5.49535e15i	0.407639	−	0.706051i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.96445e15	−	6.86663e15i	0.495139	−	0.857606i
$780$		0			0
$781$	−2.63578e15	−	4.56531e15i	−0.324586	−	0.562199i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.34311e16			−1.60816
$786$		0			0
$787$	−5.14201e15	−	8.90622e15i	−0.607116	−	1.05156i	−0.991713	−	0.128471i	$-0.958993\pi$
$787$	−5.14201e15	−	8.90622e15i	−0.607116	−	1.05156i	0.384597	−	0.923084i	$-0.374340\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5.27076e15	+	3.64761e15i	0.605205	+	0.418830i
$792$		0			0
$793$	9.51386e15	−	1.64785e16i	1.07734	−	1.86601i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.64604e15			0.181309			0.0906545	−	0.995882i	$-0.471104\pi$
$797$	1.64604e15			0.181309			0.0906545	+	0.995882i	$0.471104\pi$
$798$		0			0
$799$	7.54627e15			0.819833
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.78426e15	+	3.09042e15i	−0.188591	+	0.326650i
$804$		0			0
$805$	3.69863e14	−	4.51311e15i	0.0385623	−	0.470542i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.19082e15	+	1.07228e16i	0.628103	+	1.08791i	0.987932	+	0.154889i	$0.0495019\pi$
$809$	6.19082e15	+	1.07228e16i	0.628103	+	1.08791i	−0.359828	+	0.933018i	$0.617165\pi$
$810$		0			0
$811$	−6.77561e15			−0.678162			−0.339081	−	0.940757i	$-0.610116\pi$
$811$	−6.77561e15			−0.678162			−0.339081	+	0.940757i	$0.610116\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.76447e15	−	6.52026e15i	−0.366722	−	0.635182i
$816$		0			0
$817$	1.17112e15	−	2.02844e15i	0.112559	−	0.194958i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.79379e15	−	3.10694e15i	0.167836	−	0.290700i	−0.769823	−	0.638258i	$-0.779655\pi$
$821$	1.79379e15	−	3.10694e15i	0.167836	−	0.290700i	0.937659	+	0.347557i	$0.112989\pi$
$822$		0			0
$823$	−5.40105e14	−	9.35490e14i	−0.0498631	−	0.0863655i	0.840017	−	0.542561i	$-0.182545\pi$
$823$	−5.40105e14	−	9.35490e14i	−0.0498631	−	0.0863655i	−0.889880	+	0.456195i	$0.849212\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.77767e15			−0.429473			−0.214736	−	0.976672i	$-0.568889\pi$
$827$	−4.77767e15			−0.429473			−0.214736	+	0.976672i	$0.568889\pi$
$828$		0			0
$829$	3.09609e15	+	5.36259e15i	0.274640	+	0.475691i	0.970044	−	0.242928i	$-0.0781079\pi$
$829$	3.09609e15	+	5.36259e15i	0.274640	+	0.475691i	−0.695404	+	0.718619i	$0.744775\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.28690e15	+	4.00579e15i	−0.283948	+	0.346051i
$834$		0			0
$835$	−6.37942e15	+	1.10495e16i	−0.543883	+	0.942033i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.68282e16			−1.39749			−0.698744	−	0.715372i	$-0.746257\pi$
$839$	−1.68282e16			−1.39749			−0.698744	+	0.715372i	$0.746257\pi$
$840$		0			0
$841$	2.69977e16			2.21283
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.67726e15	−	6.36921e15i	0.293638	−	0.508597i
$846$		0			0
$847$	−7.15760e15	+	3.38566e15i	−0.564168	+	0.266861i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.89857e15	−	6.75253e15i	−0.299429	−	0.518625i
$852$		0			0
$853$	4.37357e15			0.331602			0.165801	−	0.986159i	$-0.446979\pi$
$853$	4.37357e15			0.331602			0.165801	+	0.986159i	$0.446979\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4.44612e15	−	7.70091e15i	−0.328539	−	0.569046i	0.653683	−	0.756768i	$-0.273223\pi$
$857$	−4.44612e15	−	7.70091e15i	−0.328539	−	0.569046i	−0.982222	+	0.187722i	$0.939890\pi$
$858$		0			0
$859$	−6.35078e15	+	1.09999e16i	−0.463303	+	0.802464i	−0.999123	−	0.0418683i	$-0.986669\pi$
$859$	−6.35078e15	+	1.09999e16i	−0.463303	+	0.802464i	0.535821	+	0.844332i	$0.320002\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.03389e16	+	1.79075e16i	−0.735217	+	1.27343i	0.219411	+	0.975632i	$0.429586\pi$
$863$	−1.03389e16	+	1.79075e16i	−0.735217	+	1.27343i	−0.954628	+	0.297800i	$0.903747\pi$
$864$		0			0
$865$	−1.25420e15	−	2.17234e15i	−0.0880599	−	0.152524i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.74863e15			0.530416
$870$		0			0
$871$	−5.97119e14	−	1.03424e15i	−0.0403609	−	0.0699072i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6.34727e15	−	4.39262e15i	−0.418353	−	0.289520i
$876$		0			0
$877$	−4.62530e15	+	8.01126e15i	−0.301053	+	0.521438i	−0.976375	−	0.216085i	$-0.930671\pi$
$877$	−4.62530e15	+	8.01126e15i	−0.301053	+	0.521438i	0.675322	+	0.737523i	$0.264005\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.39184e16			−1.51833			−0.759163	−	0.650901i	$-0.774391\pi$
$881$	−2.39184e16			−1.51833			−0.759163	+	0.650901i	$0.774391\pi$
$882$		0			0
$883$	−2.78563e16			−1.74638			−0.873192	−	0.487376i	$-0.837954\pi$
$883$	−2.78563e16			−1.74638			−0.873192	+	0.487376i	$0.837954\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8.95498e15	−	1.55105e16i	0.547627	−	0.948517i	−0.450810	−	0.892620i	$-0.648865\pi$
$887$	8.95498e15	−	1.55105e16i	0.547627	−	0.948517i	0.998437	−	0.0558972i	$-0.0178019\pi$
$888$		0			0
$889$	2.10105e16	+	1.45403e16i	1.26904	+	0.878239i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.06602e16	−	1.84639e16i	−0.628175	−	1.08803i
$894$		0			0
$895$	−1.86747e16			−1.08699
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.15320e16	+	3.72944e16i	1.22294	+	2.11819i
$900$		0			0
$901$	−1.58877e15	+	2.75183e15i	−0.0891404	+	0.154396i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−8.66527e14	+	1.50087e15i	−0.0474477	+	0.0821818i
$906$		0			0
$907$	−7.35821e15	−	1.27448e16i	−0.398045	−	0.689434i	0.595440	−	0.803400i	$-0.296978\pi$
$907$	−7.35821e15	−	1.27448e16i	−0.398045	−	0.689434i	−0.993485	+	0.113966i	$0.963645\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	9.13912e15			0.482562			0.241281	−	0.970455i	$-0.422432\pi$
$911$	9.13912e15			0.482562			0.241281	+	0.970455i	$0.422432\pi$
$912$		0			0
$913$	8.59725e15	+	1.48909e16i	0.448508	+	0.776839i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.14293e16	+	5.40625e15i	−0.582088	+	0.275337i
$918$		0			0
$919$	1.56481e16	−	2.71033e16i	0.787456	−	1.36391i	−0.140064	−	0.990142i	$-0.544731\pi$
$919$	1.56481e16	−	2.71033e16i	0.787456	−	1.36391i	0.927521	−	0.373772i	$-0.121936\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.60795e16			1.28142
$924$		0			0
$925$	1.97279e16			0.957858
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6.93247e15	+	1.20074e16i	−0.328701	+	0.569328i	−0.982254	−	0.187553i	$-0.939944\pi$
$929$	−6.93247e15	+	1.20074e16i	−0.328701	+	0.569328i	0.653553	+	0.756881i	$0.273278\pi$
$930$		0			0
$931$	1.44444e16	+	2.38353e15i	0.676826	+	0.111686i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.78976e15	−	6.56405e15i	−0.173439	−	0.300406i
$936$		0			0
$937$	−1.82662e16			−0.826191			−0.413095	−	0.910688i	$-0.635552\pi$
$937$	−1.82662e16			−0.826191			−0.413095	+	0.910688i	$0.635552\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−2.20697e15	−	3.82258e15i	−0.0975108	−	0.168894i	0.813143	−	0.582064i	$-0.197755\pi$
$941$	−2.20697e15	−	3.82258e15i	−0.0975108	−	0.168894i	−0.910654	+	0.413170i	$0.864421\pi$
$942$		0			0
$943$	−6.17401e15	+	1.06937e16i	−0.269620	+	0.466996i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	7.48424e15	−	1.29631e16i	0.319318	−	0.553075i	−0.661028	−	0.750361i	$-0.729880\pi$
$947$	7.48424e15	−	1.29631e16i	0.319318	−	0.553075i	0.980346	+	0.197287i	$0.0632129\pi$
$948$		0			0
$949$	−8.82707e15	−	1.52889e16i	−0.372265	−	0.644782i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2.05497e16			0.846827			0.423413	−	0.905937i	$-0.360832\pi$
$953$	2.05497e16			0.846827			0.423413	+	0.905937i	$0.360832\pi$
$954$		0			0
$955$	5.45749e15	+	9.45266e15i	0.222318	+	0.385066i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.20931e15	−	3.91605e16i	0.127764	−	1.55900i
$960$		0			0
$961$	−1.09512e16	+	1.89680e16i	−0.431005	+	0.746523i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.89239e16			1.88200
$966$		0			0
$967$	−3.58563e16			−1.36370			−0.681851	−	0.731491i	$-0.738825\pi$
$967$	−3.58563e16			−1.36370			−0.681851	+	0.731491i	$0.738825\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.03957e16	+	1.80059e16i	−0.386499	+	0.669435i	−0.991976	−	0.126427i	$-0.959649\pi$
$971$	−1.03957e16	+	1.80059e16i	−0.386499	+	0.669435i	0.605477	+	0.795863i	$0.292982\pi$
$972$		0			0
$973$	−2.69076e16	−	1.86214e16i	−0.989133	−	0.684527i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.91632e16	+	3.31916e16i	0.688728	+	1.19291i	0.972250	+	0.233946i	$0.0751638\pi$
$977$	1.91632e16	+	3.31916e16i	0.688728	+	1.19291i	−0.283522	+	0.958966i	$0.591503\pi$
$978$		0			0
$979$	−5.74628e14			−0.0204212
$980$		0			0
$981$		0			0
$982$		0			0
$983$	9.05217e15	+	1.56788e16i	0.314564	+	0.544840i	0.979345	−	0.202198i	$-0.0648085\pi$
$983$	9.05217e15	+	1.56788e16i	0.314564	+	0.544840i	−0.664781	+	0.747038i	$0.731475\pi$
$984$		0			0
$985$	−1.62663e16	+	2.81740e16i	−0.558971	+	0.968166i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.82383e15	+	3.15897e15i	−0.0612923	+	0.106161i
$990$		0			0
$991$	1.49937e16	+	2.59698e16i	0.498314	+	0.863105i	0.999998	−	0.00194607i	$-0.000619453\pi$
$991$	1.49937e16	+	2.59698e16i	0.498314	+	0.863105i	−0.501684	+	0.865051i	$0.667286\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.74579e16			−1.21763
$996$		0			0
$997$	2.52903e16	+	4.38040e16i	0.813074	+	1.40829i	0.910702	+	0.413063i	$0.135541\pi$
$997$	2.52903e16	+	4.38040e16i	0.813074	+	1.40829i	−0.0976282	+	0.995223i	$0.531126\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.12.k.e.37.12	yes	28
3.2	odd	2	inner	252.12.k.e.37.3	✓	28
7.4	even	3	inner	252.12.k.e.109.12	yes	28
21.11	odd	6	inner	252.12.k.e.109.3	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.12.k.e.37.3	✓	28	3.2	odd	2	inner
252.12.k.e.37.12	yes	28	1.1	even	1	trivial
252.12.k.e.109.3	yes	28	21.11	odd	6	inner
252.12.k.e.109.12	yes	28	7.4	even	3	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4415.92 - 7648.60i) q^{5} +(-36565.0 - 25304.7i) q^{7} +O(q^{10})\)	\(q+(4415.92 - 7648.60i) q^{5} +(-36565.0 - 25304.7i) q^{7} +(-163744. - 283613. i) q^{11} +1.62015e6 q^{13} +(1.31028e6 + 2.26947e6i) q^{17} +(3.70191e6 - 6.41189e6i) q^{19} +(-5.76513e6 + 9.98551e6i) q^{23} +(-1.45866e7 - 2.52648e7i) q^{25} +1.97985e8 q^{29} +(1.08755e8 + 1.88370e8i) q^{31} +(-3.55014e8 + 1.67927e8i) q^{35} +(-3.38116e8 + 5.85635e8i) q^{37} +1.07092e9 q^{41} +3.16356e8 q^{43} +(1.43982e9 - 2.49384e9i) q^{47} +(6.96670e8 + 1.85053e9i) q^{49} +(6.06271e8 + 1.05009e9i) q^{53} -2.89232e9 q^{55} +(5.29381e9 + 9.16914e9i) q^{59} +(5.87221e9 - 1.01710e10i) q^{61} +(7.15446e9 - 1.23919e10i) q^{65} +(-3.68558e8 - 6.38361e8i) q^{67} +1.60970e10 q^{71} +(-5.44830e9 - 9.43674e9i) q^{73} +(-1.18945e9 + 1.45138e10i) q^{77} +(-1.18304e10 + 2.04908e10i) q^{79} -5.25042e10 q^{83} +2.31444e10 q^{85} +(8.77325e8 - 1.51957e9i) q^{89} +(-5.92408e10 - 4.09975e10i) q^{91} +(-3.26946e10 - 5.66288e10i) q^{95} +7.32713e10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 85666 q^{7}+O(q^{10}) \) 28 * q + 85666 * q^7	\( 28 q + 85666 q^{7} + 2794700 q^{13} + 37414826 q^{19} - 135469250 q^{25} + 71052098 q^{31} - 395898874 q^{37} + 980170268 q^{43} - 1253333270 q^{49} - 8844657360 q^{55} + 16692685100 q^{61} + 10934009870 q^{67} + 25578690650 q^{73} - 113974174090 q^{79} + 111789733440 q^{85} - 267181531078 q^{91} + 279658007384 q^{97}+O(q^{100}) \) 28 * q + 85666 * q^7 + 2794700 * q^13 + 37414826 * q^19 - 135469250 * q^25 + 71052098 * q^31 - 395898874 * q^37 + 980170268 * q^43 - 1253333270 * q^49 - 8844657360 * q^55 + 16692685100 * q^61 + 10934009870 * q^67 + 25578690650 * q^73 - 113974174090 * q^79 + 111789733440 * q^85 - 267181531078 * q^91 + 279658007384 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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