LMFDB - Embedded newform 252.12.k.e.109.1

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			109.1
Character	$\chi$	$=$	252.109
Dual form			252.12.k.e.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5969.07 - 10338.7i) q^{5} +(-19562.4 - 39932.9i) q^{7} +O(q^{10})$	$q+(-5969.07 - 10338.7i) q^{5} +(-19562.4 - 39932.9i) q^{7} +(225827. - 391144. i) q^{11} -2.01567e6 q^{13} +(-1.36489e6 + 2.36406e6i) q^{17} +(3.24572e6 + 5.62175e6i) q^{19} +(1.08704e7 + 1.88281e7i) q^{23} +(-4.68455e7 + 8.11387e7i) q^{25} +1.78124e8 q^{29} +(3.08267e7 - 5.33933e7i) q^{31} +(-2.96086e8 + 4.40613e8i) q^{35} +(1.65389e8 + 2.86462e8i) q^{37} -1.76606e8 q^{41} +1.54307e9 q^{43} +(1.20891e9 + 2.09389e9i) q^{47} +(-1.21195e9 + 1.56237e9i) q^{49} +(-2.67377e9 + 4.63110e9i) q^{53} -5.39190e9 q^{55} +(4.67094e7 - 8.09031e7i) q^{59} +(2.30307e9 + 3.98904e9i) q^{61} +(1.20317e10 + 2.08394e10i) q^{65} +(6.33774e9 - 1.09773e10i) q^{67} +1.03314e10 q^{71} +(2.83940e9 - 4.91798e9i) q^{73} +(-2.00372e10 - 1.36621e9i) q^{77} +(-2.64174e10 - 4.57562e10i) q^{79} -8.41837e9 q^{83} +3.25885e10 q^{85} +(-3.71018e10 - 6.42622e10i) q^{89} +(3.94313e10 + 8.04915e10i) q^{91} +(3.87478e10 - 6.71132e10i) q^{95} -6.08664e10 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−5969.07	−	10338.7i	−0.854223	−	1.47956i	−0.877364	−	0.479826i	$-0.840700\pi$
$5$	−5969.07	−	10338.7i	−0.854223	−	1.47956i	0.0231403	−	0.999732i	$-0.492634\pi$
$6$		0			0
$7$	−19562.4	−	39932.9i	−0.439930	−	0.898032i
$7$	−19562.4	−	39932.9i	−0.439930	−	0.898032i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	225827.	−	391144.i	0.422781	−	0.732279i	−0.573429	−	0.819255i	$-0.694387\pi$
$11$	225827.	−	391144.i	0.422781	−	0.732279i	0.996210	+	0.0869763i	$0.0277204\pi$
$12$		0			0
$13$	−2.01567e6			−1.50567			−0.752836	−	0.658208i	$-0.771315\pi$
$13$	−2.01567e6			−1.50567			−0.752836	+	0.658208i	$0.771315\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.36489e6	+	2.36406e6i	−0.233147	+	0.403822i	−0.958732	−	0.284310i	$-0.908236\pi$
$17$	−1.36489e6	+	2.36406e6i	−0.233147	+	0.403822i	0.725586	+	0.688132i	$0.241569\pi$
$18$		0			0
$19$	3.24572e6	+	5.62175e6i	0.300723	+	0.520867i	0.976300	−	0.216422i	$-0.0694388\pi$
$19$	3.24572e6	+	5.62175e6i	0.300723	+	0.520867i	−0.675577	+	0.737289i	$0.736105\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.08704e7	+	1.88281e7i	0.352162	+	0.609962i	0.986628	−	0.162989i	$-0.0521134\pi$
$23$	1.08704e7	+	1.88281e7i	0.352162	+	0.609962i	−0.634466	+	0.772950i	$0.718780\pi$
$24$		0			0
$25$	−4.68455e7	+	8.11387e7i	−0.959395	+	1.66172i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.78124e8			1.61262			0.806310	−	0.591493i	$-0.201461\pi$
$29$	1.78124e8			1.61262			0.806310	+	0.591493i	$0.201461\pi$
$30$		0			0
$31$	3.08267e7	−	5.33933e7i	0.193391	−	0.334964i	−0.752981	−	0.658043i	$-0.771385\pi$
$31$	3.08267e7	−	5.33933e7i	0.193391	−	0.334964i	0.946372	+	0.323079i	$0.104718\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.96086e8	+	4.40613e8i	−0.952892	+	1.41802i
$36$		0			0
$37$	1.65389e8	+	2.86462e8i	0.392101	+	0.679138i	0.992726	−	0.120392i	$-0.0384152\pi$
$37$	1.65389e8	+	2.86462e8i	0.392101	+	0.679138i	−0.600626	+	0.799530i	$0.705082\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.76606e8			−0.238064			−0.119032	−	0.992890i	$-0.537979\pi$
$41$	−1.76606e8			−0.238064			−0.119032	+	0.992890i	$0.537979\pi$
$42$		0			0
$43$	1.54307e9			1.60070			0.800348	−	0.599536i	$-0.204648\pi$
$43$	1.54307e9			1.60070			0.800348	+	0.599536i	$0.204648\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.20891e9	+	2.09389e9i	0.768875	+	1.33173i	0.938173	+	0.346166i	$0.112517\pi$
$47$	1.20891e9	+	2.09389e9i	0.768875	+	1.33173i	−0.169298	+	0.985565i	$0.554150\pi$
$48$		0			0
$49$	−1.21195e9	+	1.56237e9i	−0.612923	+	0.790142i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.67377e9	+	4.63110e9i	−0.878225	+	1.52113i	−0.0249389	+	0.999689i	$0.507939\pi$
$53$	−2.67377e9	+	4.63110e9i	−0.878225	+	1.52113i	−0.853287	+	0.521442i	$0.825394\pi$
$54$		0			0
$55$	−5.39190e9			−1.44460
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.67094e7	−	8.09031e7i	0.00850586	−	0.0147326i	−0.861741	−	0.507348i	$-0.830626\pi$
$59$	4.67094e7	−	8.09031e7i	0.00850586	−	0.0147326i	0.870247	+	0.492616i	$0.163959\pi$
$60$		0			0
$61$	2.30307e9	+	3.98904e9i	0.349135	+	0.604720i	0.986096	−	0.166176i	$-0.0531421\pi$
$61$	2.30307e9	+	3.98904e9i	0.349135	+	0.604720i	−0.636961	+	0.770896i	$0.719809\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.20317e10	+	2.08394e10i	1.28618	+	2.22773i
$66$		0			0
$67$	6.33774e9	−	1.09773e10i	0.573486	−	0.993307i	−0.422718	−	0.906261i	$-0.638924\pi$
$67$	6.33774e9	−	1.09773e10i	0.573486	−	0.993307i	0.996204	−	0.0870460i	$-0.0277427\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.03314e10			0.679578			0.339789	−	0.940502i	$-0.389644\pi$
$71$	1.03314e10			0.679578			0.339789	+	0.940502i	$0.389644\pi$
$72$		0			0
$73$	2.83940e9	−	4.91798e9i	0.160306	−	0.277658i	−0.774672	−	0.632363i	$-0.782085\pi$
$73$	2.83940e9	−	4.91798e9i	0.160306	−	0.277658i	0.934978	+	0.354704i	$0.115419\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.00372e10	−	1.36621e9i	−0.843604	−	0.0575199i
$78$		0			0
$79$	−2.64174e10	−	4.57562e10i	−0.965919	−	1.67302i	−0.707126	−	0.707087i	$-0.750009\pi$
$79$	−2.64174e10	−	4.57562e10i	−0.965919	−	1.67302i	−0.258792	−	0.965933i	$-0.583325\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−8.41837e9			−0.234584			−0.117292	−	0.993097i	$-0.537421\pi$
$83$	−8.41837e9			−0.234584			−0.117292	+	0.993097i	$0.537421\pi$
$84$		0			0
$85$	3.25885e10			0.796638
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.71018e10	−	6.42622e10i	−0.704287	−	1.21986i	−0.966948	−	0.254973i	$-0.917933\pi$
$89$	−3.71018e10	−	6.42622e10i	−0.704287	−	1.21986i	0.262661	−	0.964888i	$-0.415400\pi$
$90$		0			0
$91$	3.94313e10	+	8.04915e10i	0.662390	+	1.35214i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.87478e10	−	6.71132e10i	0.513769	−	0.889874i
$96$		0			0
$97$	−6.08664e10			−0.719670			−0.359835	−	0.933016i	$-0.617167\pi$
$97$	−6.08664e10			−0.719670			−0.359835	+	0.933016i	$0.617167\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.68604e10	−	6.38442e10i	0.348974	−	0.604441i	−0.637094	−	0.770787i	$-0.719864\pi$
$101$	3.68604e10	−	6.38442e10i	0.348974	−	0.604441i	0.986067	+	0.166346i	$0.0531968\pi$
$102$		0			0
$103$	6.53654e9	+	1.13216e10i	0.0555576	+	0.0962286i	0.892467	−	0.451113i	$-0.148973\pi$
$103$	6.53654e9	+	1.13216e10i	0.0555576	+	0.0962286i	−0.836909	+	0.547342i	$0.815640\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.70692e10	−	1.50808e11i	−0.600141	−	1.03948i	−0.992799	−	0.119791i	$-0.961778\pi$
$107$	−8.70692e10	−	1.50808e11i	−0.600141	−	1.03948i	0.392658	−	0.919685i	$-0.371556\pi$
$108$		0			0
$109$	−1.23148e11	+	2.13298e11i	−0.766620	+	1.32782i	0.172766	+	0.984963i	$0.444730\pi$
$109$	−1.23148e11	+	2.13298e11i	−0.766620	+	1.32782i	−0.939386	+	0.342862i	$0.888604\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.80489e11			−0.921549			−0.460775	−	0.887517i	$-0.652428\pi$
$113$	−1.80489e11			−0.921549			−0.460775	+	0.887517i	$0.652428\pi$
$114$		0			0
$115$	1.29772e11	−	2.24772e11i	0.601649	−	1.04209i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.21105e11	+	8.25733e9i	0.465214	+	0.0317199i
$120$		0			0
$121$	4.06602e10	+	7.04256e10i	0.142512	+	0.246837i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	5.35578e11			1.56970
$126$		0			0
$127$	3.28336e11			0.881856			0.440928	−	0.897543i	$-0.354649\pi$
$127$	3.28336e11			0.881856			0.440928	+	0.897543i	$0.354649\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.26305e11	−	2.18766e11i	−0.286040	−	0.495436i	0.686821	−	0.726827i	$-0.259006\pi$
$131$	−1.26305e11	−	2.18766e11i	−0.286040	−	0.495436i	−0.972861	+	0.231391i	$0.925672\pi$
$132$		0			0
$133$	1.60999e11	−	2.39586e11i	0.335458	−	0.499204i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.99988e11	+	5.19595e11i	−0.531057	+	0.919818i	0.468286	+	0.883577i	$0.344872\pi$
$137$	−2.99988e11	+	5.19595e11i	−0.531057	+	0.919818i	−0.999343	+	0.0362408i	$0.988462\pi$
$138$		0			0
$139$	6.62834e11			1.08349			0.541743	−	0.840544i	$-0.317765\pi$
$139$	6.62834e11			1.08349			0.541743	+	0.840544i	$0.317765\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.55192e11	+	7.88415e11i	−0.636570	+	1.10257i
$144$		0			0
$145$	−1.06323e12	−	1.84157e12i	−1.37754	−	2.38597i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.54738e11	+	1.48045e12i	0.953473	+	1.65146i	0.737824	+	0.674993i	$0.235853\pi$
$149$	8.54738e11	+	1.48045e12i	0.953473	+	1.65146i	0.215649	+	0.976471i	$0.430813\pi$
$150$		0			0
$151$	−7.51328e11	+	1.30134e12i	−0.778855	+	1.34902i	0.153747	+	0.988110i	$0.450866\pi$
$151$	−7.51328e11	+	1.30134e12i	−0.778855	+	1.34902i	−0.932602	+	0.360906i	$0.882468\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.36025e11			−0.660798
$156$		0			0
$157$	1.96019e11	−	3.39515e11i	0.164002	−	0.284060i	−0.772298	−	0.635260i	$-0.780893\pi$
$157$	1.96019e11	−	3.39515e11i	0.164002	−	0.284060i	0.936301	+	0.351200i	$0.114226\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.39208e11	−	8.02409e11i	0.392839	−	0.584593i
$162$		0			0
$163$	−3.82606e11	−	6.62693e11i	−0.260447	−	0.451108i	0.705913	−	0.708298i	$-0.250537\pi$
$163$	−3.82606e11	−	6.62693e11i	−0.260447	−	0.451108i	−0.966361	+	0.257190i	$0.917203\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.95935e11			−0.355024			−0.177512	−	0.984119i	$-0.556805\pi$
$167$	−5.95935e11			−0.355024			−0.177512	+	0.984119i	$0.556805\pi$
$168$		0			0
$169$	2.27075e12			1.26705
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.27826e11	+	1.43384e12i	0.406149	+	0.703471i	0.994454	−	0.105168i	$-0.0335380\pi$
$173$	8.27826e11	+	1.43384e12i	0.406149	+	0.703471i	−0.588305	+	0.808639i	$0.700205\pi$
$174$		0			0
$175$	4.15652e12	+	2.83406e11i	1.91435	+	0.130527i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−7.46999e11	+	1.29384e12i	−0.303828	+	0.526246i	−0.977000	−	0.213240i	$-0.931598\pi$
$179$	−7.46999e11	+	1.29384e12i	−0.303828	+	0.526246i	0.673171	+	0.739486i	$0.264932\pi$
$180$		0			0
$181$	−1.48892e12			−0.569689			−0.284844	−	0.958574i	$-0.591942\pi$
$181$	−1.48892e12			−0.569689			−0.284844	+	0.958574i	$0.591942\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.97444e12	−	3.41983e12i	0.669883	−	1.16027i
$186$		0			0
$187$	6.16459e11	+	1.06774e12i	0.197140	+	0.341457i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.68246e11	−	4.64616e11i	−0.0763572	−	0.132255i	0.825318	−	0.564668i	$-0.190996\pi$
$191$	−2.68246e11	−	4.64616e11i	−0.0763572	−	0.132255i	−0.901676	+	0.432413i	$0.857662\pi$
$192$		0			0
$193$	1.89982e12	−	3.29058e12i	0.510677	−	0.884519i	−0.489246	−	0.872146i	$-0.662728\pi$
$193$	1.89982e12	−	3.29058e12i	0.510677	−	0.884519i	0.999923	−	0.0123734i	$-0.00393867\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.67796e12			0.402918			0.201459	−	0.979497i	$-0.435432\pi$
$197$	1.67796e12			0.402918			0.201459	+	0.979497i	$0.435432\pi$
$198$		0			0
$199$	2.59078e12	−	4.48735e12i	0.588488	−	1.01929i	−0.405942	−	0.913899i	$-0.633057\pi$
$199$	2.59078e12	−	4.48735e12i	0.588488	−	1.01929i	0.994431	−	0.105393i	$-0.0336101\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.48453e12	−	7.11299e12i	−0.709440	−	1.44819i
$204$		0			0
$205$	1.05417e12	+	1.82588e12i	0.203360	+	0.352229i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.93188e12			0.508560
$210$		0			0
$211$	6.05179e12			0.996162			0.498081	−	0.867130i	$-0.334038\pi$
$211$	6.05179e12			0.996162			0.498081	+	0.867130i	$0.334038\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−9.21068e12	−	1.59534e13i	−1.36735	−	2.36832i
$216$		0			0
$217$	−2.73520e12	−	1.86495e11i	−0.385887	−	0.0263111i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.75117e12	−	4.76516e12i	0.351043	−	0.608024i
$222$		0			0
$223$	9.45610e11			0.114825			0.0574123	−	0.998351i	$-0.481715\pi$
$223$	9.45610e11			0.114825			0.0574123	+	0.998351i	$0.481715\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.28901e12	+	1.43570e13i	−0.912768	+	1.58096i	−0.102632	+	0.994719i	$0.532726\pi$
$227$	−8.28901e12	+	1.43570e13i	−0.912768	+	1.58096i	−0.810136	+	0.586241i	$0.800607\pi$
$228$		0			0
$229$	1.26416e12	+	2.18958e12i	0.132649	+	0.229755i	0.924697	−	0.380704i	$-0.124318\pi$
$229$	1.26416e12	+	2.18958e12i	0.132649	+	0.229755i	−0.792048	+	0.610459i	$0.790985\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.48108e12	+	7.76147e12i	0.427490	+	0.740434i	0.996649	−	0.0817932i	$-0.0260647\pi$
$233$	4.48108e12	+	7.76147e12i	0.427490	+	0.740434i	−0.569160	+	0.822227i	$0.692731\pi$
$234$		0			0
$235$	1.44321e13	−	2.49972e13i	1.31358	−	2.27519i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.36072e12			−0.278768			−0.139384	−	0.990238i	$-0.544512\pi$
$239$	−3.36072e12			−0.278768			−0.139384	+	0.990238i	$0.544512\pi$
$240$		0			0
$241$	1.11170e13	−	1.92552e13i	0.880835	−	1.52565i	0.0304214	−	0.999537i	$-0.490315\pi$
$241$	1.11170e13	−	1.92552e13i	0.880835	−	1.52565i	0.850414	−	0.526114i	$-0.176352\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.33871e13	+	3.20413e12i	1.69264	+	0.231898i
$246$		0			0
$247$	−6.54229e12	−	1.13316e13i	−0.452790	−	0.784255i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.36923e13			−0.867505			−0.433752	−	0.901032i	$-0.642811\pi$
$251$	−1.36923e13			−0.867505			−0.433752	+	0.901032i	$0.642811\pi$
$252$		0			0
$253$	9.81930e12			0.595550
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.50476e13	+	2.60632e13i	0.837211	+	1.45009i	0.892217	+	0.451607i	$0.149149\pi$
$257$	1.50476e13	+	2.60632e13i	0.837211	+	1.45009i	−0.0550056	+	0.998486i	$0.517518\pi$
$258$		0			0
$259$	8.20387e12	−	1.22084e13i	0.437391	−	0.650892i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.52913e12	−	9.57674e12i	0.270957	−	0.469311i	−0.698150	−	0.715951i	$-0.745993\pi$
$263$	5.52913e12	−	9.57674e12i	0.270957	−	0.469311i	0.969107	+	0.246640i	$0.0793265\pi$
$264$		0			0
$265$	6.38395e13			3.00080
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.17940e13	+	2.04279e13i	−0.510535	+	0.884272i	0.489391	+	0.872065i	$0.337219\pi$
$269$	−1.17940e13	+	2.04279e13i	−0.510535	+	0.884272i	−0.999925	+	0.0122074i	$0.996114\pi$
$270$		0			0
$271$	−1.10496e12	−	1.91385e12i	−0.0459215	−	0.0795384i	0.842151	−	0.539242i	$-0.181289\pi$
$271$	−1.10496e12	−	1.91385e12i	−0.0459215	−	0.0795384i	−0.888073	+	0.459703i	$0.847956\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.11579e13	+	3.66466e13i	0.811229	+	1.40509i
$276$		0			0
$277$	1.18708e13	−	2.05609e13i	0.437363	−	0.757536i	−0.560122	−	0.828410i	$-0.689246\pi$
$277$	1.18708e13	−	2.05609e13i	0.437363	−	0.757536i	0.997485	+	0.0708746i	$0.0225790\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.96374e12			−0.305214			−0.152607	−	0.988287i	$-0.548767\pi$
$281$	−8.96374e12			−0.305214			−0.152607	+	0.988287i	$0.548767\pi$
$282$		0			0
$283$	−1.25161e13	+	2.16785e13i	−0.409868	+	0.709912i	−0.994875	−	0.101116i	$-0.967759\pi$
$283$	−1.25161e13	+	2.16785e13i	−0.409868	+	0.709912i	0.585007	+	0.811029i	$0.301092\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.45484e12	+	7.05238e12i	0.104731	+	0.213789i
$288$		0			0
$289$	1.34101e13	+	2.32269e13i	0.391285	+	0.677726i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5.69210e13			−1.53993			−0.769964	−	0.638087i	$-0.779726\pi$
$293$	−5.69210e13			−1.53993			−0.769964	+	0.638087i	$0.779726\pi$
$294$		0			0
$295$	−1.11525e12			−0.0290636
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.19111e13	−	3.79511e13i	−0.530240	−	0.918402i
$300$		0			0
$301$	−3.01862e13	−	6.16193e13i	−0.704194	−	1.43748i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.74944e13	−	4.76217e13i	0.596479	−	1.03313i
$306$		0			0
$307$	9.07497e13			1.89926			0.949630	−	0.313375i	$-0.101460\pi$
$307$	9.07497e13			1.89926			0.949630	+	0.313375i	$0.101460\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2.88354e13	−	4.99443e13i	0.562009	−	0.973428i	−0.435312	−	0.900280i	$-0.643362\pi$
$311$	2.88354e13	−	4.99443e13i	0.562009	−	0.973428i	0.997321	−	0.0731482i	$-0.0233046\pi$
$312$		0			0
$313$	2.94246e13	+	5.09649e13i	0.553626	+	0.958909i	0.998009	+	0.0630720i	$0.0200898\pi$
$313$	2.94246e13	+	5.09649e13i	0.553626	+	0.958909i	−0.444383	+	0.895837i	$0.646577\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.48095e13	+	4.29714e13i	0.435304	+	0.753969i	0.997320	−	0.0731575i	$-0.0233076\pi$
$317$	2.48095e13	+	4.29714e13i	0.435304	+	0.753969i	−0.562016	+	0.827126i	$0.689974\pi$
$318$		0			0
$319$	4.02251e13	−	6.96719e13i	0.681786	−	1.18089i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.77202e13			−0.280450
$324$		0			0
$325$	9.44248e13	−	1.63549e14i	1.44453	−	2.50201i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	5.99661e13	−	8.92370e13i	0.857686	−	1.27634i
$330$		0			0
$331$	−2.31334e13	−	4.00682e13i	−0.320026	−	0.554302i	0.660467	−	0.750855i	$-0.270358\pi$
$331$	−2.31334e13	−	4.00682e13i	−0.320026	−	0.554302i	−0.980493	+	0.196554i	$0.937025\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.51322e14			−1.95954
$336$		0			0
$337$	8.64502e13			1.08343			0.541716	−	0.840562i	$-0.317775\pi$
$337$	8.64502e13			1.08343			0.541716	+	0.840562i	$0.317775\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.39230e13	−	2.41153e13i	−0.163525	−	0.283233i
$342$		0			0
$343$	8.60987e13	+	1.78329e13i	0.979217	+	0.202817i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.90451e13	−	1.54231e14i	0.950162	−	1.64573i	0.205093	−	0.978742i	$-0.434250\pi$
$347$	8.90451e13	−	1.54231e14i	0.950162	−	1.64573i	0.745069	−	0.666987i	$-0.232416\pi$
$348$		0			0
$349$	−1.41782e14			−1.46582			−0.732911	−	0.680325i	$-0.761839\pi$
$349$	−1.41782e14			−1.46582			−0.732911	+	0.680325i	$0.761839\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.33437e12	+	5.77530e12i	−0.0323782	+	0.0560807i	−0.881760	−	0.471697i	$-0.843641\pi$
$353$	−3.33437e12	+	5.77530e12i	−0.0323782	+	0.0560807i	0.849382	+	0.527778i	$0.176975\pi$
$354$		0			0
$355$	−6.16690e13	−	1.06814e14i	−0.580511	−	1.00548i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.00357e14	+	1.73823e14i	0.888235	+	1.53847i	0.841960	+	0.539539i	$0.181402\pi$
$359$	1.00357e14	+	1.73823e14i	0.888235	+	1.53847i	0.0462744	+	0.998929i	$0.485265\pi$
$360$		0			0
$361$	3.71757e13	−	6.43903e13i	0.319132	−	0.552752i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.77942e13			−0.547749
$366$		0			0
$367$	−4.52622e13	+	7.83965e13i	−0.354873	+	0.614657i	−0.987096	−	0.160129i	$-0.948809\pi$
$367$	−4.52622e13	+	7.83965e13i	−0.354873	+	0.614657i	0.632224	+	0.774786i	$0.282142\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.37239e14	+	1.61758e13i	1.75238	+	0.119484i
$372$		0			0
$373$	−4.41099e12	−	7.64007e12i	−0.0316328	−	0.0547897i	0.849776	−	0.527145i	$-0.176737\pi$
$373$	−4.41099e12	−	7.64007e12i	−0.0316328	−	0.0547897i	−0.881408	+	0.472355i	$0.843404\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.59038e14			−2.42808
$378$		0			0
$379$	1.50337e14			0.987530			0.493765	−	0.869595i	$-0.335620\pi$
$379$	1.50337e14			0.987530			0.493765	+	0.869595i	$0.335620\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.71937e13	−	2.97804e13i	−0.106605	−	0.184645i	0.807788	−	0.589473i	$-0.200665\pi$
$383$	−1.71937e13	−	2.97804e13i	−0.106605	−	0.184645i	−0.914393	+	0.404828i	$0.867331\pi$
$384$		0			0
$385$	1.05479e14	+	2.15314e14i	0.635522	+	1.29730i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.18167e14	−	2.04671e14i	0.672626	−	1.16502i	−0.304531	−	0.952502i	$-0.598500\pi$
$389$	1.18167e14	−	2.04671e14i	0.672626	−	1.16502i	0.977157	−	0.212520i	$-0.0681669\pi$
$390$		0			0
$391$	−5.93477e13			−0.328421
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.15374e14	+	5.46244e14i	−1.65022	+	2.85827i
$396$		0			0
$397$	−3.79776e13	−	6.57792e13i	−0.193277	−	0.334765i	0.753057	−	0.657955i	$-0.228578\pi$
$397$	−3.79776e13	−	6.57792e13i	−0.193277	−	0.334765i	−0.946334	+	0.323189i	$0.895245\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.73436e14	+	3.00401e14i	0.835307	+	1.44679i	0.893780	+	0.448505i	$0.148044\pi$
$401$	1.73436e14	+	3.00401e14i	0.835307	+	1.44679i	−0.0584730	+	0.998289i	$0.518623\pi$
$402$		0			0
$403$	−6.21363e13	+	1.07623e14i	−0.291184	+	0.504345i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.49397e14			0.663092
$408$		0			0
$409$	4.50933e11	−	7.81040e11i	0.00194820	−	0.00337439i	−0.865050	−	0.501686i	$-0.832713\pi$
$409$	4.50933e11	−	7.81040e11i	0.00194820	−	0.00337439i	0.866998	+	0.498312i	$0.166047\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.14445e12	−	2.82583e11i	−0.0169723	−	0.00115723i
$414$		0			0
$415$	5.02498e13	+	8.70352e13i	0.200387	+	0.347081i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.32742e14			1.25872			0.629362	−	0.777112i	$-0.283316\pi$
$419$	3.32742e14			1.25872			0.629362	+	0.777112i	$0.283316\pi$
$420$		0			0
$421$	−1.28827e14			−0.474738			−0.237369	−	0.971420i	$-0.576285\pi$
$421$	−1.28827e14			−0.474738			−0.237369	+	0.971420i	$0.576285\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.27878e14	−	2.21491e14i	−0.447360	−	0.774850i
$426$		0			0
$427$	1.14240e14	−	1.70004e14i	0.389463	−	0.579569i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.94614e13	+	1.54952e14i	−0.289742	+	0.501847i	−0.973748	−	0.227629i	$-0.926903\pi$
$431$	−8.94614e13	+	1.54952e14i	−0.289742	+	0.501847i	0.684006	+	0.729476i	$0.260236\pi$
$432$		0			0
$433$	−6.20495e12			−0.0195909			−0.00979545	−	0.999952i	$-0.503118\pi$
$433$	−6.20495e12			−0.0195909			−0.00979545	+	0.999952i	$0.503118\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7.05645e13	+	1.22221e14i	−0.211806	+	0.366859i
$438$		0			0
$439$	2.82166e13	+	4.88725e13i	0.0825942	+	0.143057i	0.904364	−	0.426763i	$-0.140346\pi$
$439$	2.82166e13	+	4.88725e13i	0.0825942	+	0.143057i	−0.821769	+	0.569820i	$0.807013\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.16038e14	+	3.74189e14i	0.601603	+	1.04201i	0.992579	+	0.121605i	$0.0388042\pi$
$443$	2.16038e14	+	3.74189e14i	0.601603	+	1.04201i	−0.390976	+	0.920401i	$0.627862\pi$
$444$		0			0
$445$	−4.42926e14	+	7.67170e14i	−1.20324	+	2.08407i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−3.50387e14			−0.906136			−0.453068	−	0.891476i	$-0.649671\pi$
$449$	−3.50387e14			−0.906136			−0.453068	+	0.891476i	$0.649671\pi$
$450$		0			0
$451$	−3.98823e13	+	6.90782e13i	−0.100649	+	0.174329i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.96811e14	−	8.88129e14i	1.43474	−	2.13508i
$456$		0			0
$457$	−1.45720e14	−	2.52395e14i	−0.341964	−	0.592299i	0.642833	−	0.766006i	$-0.277759\pi$
$457$	−1.45720e14	−	2.52395e14i	−0.341964	−	0.592299i	−0.984797	+	0.173707i	$0.944426\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8.20135e14			−1.83455			−0.917277	−	0.398250i	$-0.869618\pi$
$461$	−8.20135e14			−1.83455			−0.917277	+	0.398250i	$0.869618\pi$
$462$		0			0
$463$	−1.84091e14			−0.402103			−0.201052	−	0.979581i	$-0.564436\pi$
$463$	−1.84091e14			−0.402103			−0.201052	+	0.979581i	$0.564436\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.27157e14	−	2.20243e14i	−0.264910	−	0.458838i	0.702630	−	0.711555i	$-0.252009\pi$
$467$	−1.27157e14	−	2.20243e14i	−0.264910	−	0.458838i	−0.967540	+	0.252718i	$0.918676\pi$
$468$		0			0
$469$	−5.62337e14	−	3.83421e13i	−1.14432	−	0.0780234i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.48467e14	−	6.03562e14i	0.676745	−	1.17216i
$474$		0			0
$475$	−6.08189e14			−1.15405
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.19304e14	−	7.26255e14i	0.759772	−	1.31596i	−0.183195	−	0.983077i	$-0.558644\pi$
$479$	4.19304e14	−	7.26255e14i	0.759772	−	1.31596i	0.942967	−	0.332887i	$-0.108023\pi$
$480$		0			0
$481$	−3.33369e14	−	5.77413e14i	−0.590375	−	1.02256i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.63316e14	+	6.29281e14i	0.614759	+	1.06479i
$486$		0			0
$487$	−2.94255e14	+	5.09665e14i	−0.486760	+	0.843093i	−0.999884	−	0.0152212i	$-0.995155\pi$
$487$	−2.94255e14	+	5.09665e14i	−0.486760	+	0.843093i	0.513124	+	0.858314i	$0.328488\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.65065e14			−0.419184			−0.209592	−	0.977789i	$-0.567214\pi$
$491$	−2.65065e14			−0.419184			−0.209592	+	0.977789i	$0.567214\pi$
$492$		0			0
$493$	−2.43120e14	+	4.21095e14i	−0.375977	+	0.651212i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.02108e14	−	4.12564e14i	−0.298967	−	0.610283i
$498$		0			0
$499$	4.35155e14	+	7.53710e14i	0.629637	+	1.09056i	0.987624	+	0.156837i	$0.0501299\pi$
$499$	4.35155e14	+	7.53710e14i	0.629637	+	1.09056i	−0.357987	+	0.933727i	$0.616537\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.07465e15			−1.48814			−0.744068	−	0.668104i	$-0.767106\pi$
$503$	−1.07465e15			−1.48814			−0.744068	+	0.668104i	$0.767106\pi$
$504$		0			0
$505$	−8.80090e14			−1.19241
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4.34094e14	−	7.51873e14i	−0.563165	−	0.975431i	−0.997218	−	0.0745436i	$-0.976250\pi$
$509$	−4.34094e14	−	7.51873e14i	−0.563165	−	0.975431i	0.434052	−	0.900888i	$-0.357083\pi$
$510$		0			0
$511$	−2.51935e14	−	1.71778e13i	−0.319870	−	0.0218098i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.80342e13	−	1.35159e14i	0.0949172	−	0.164401i
$516$		0			0
$517$	1.09202e15			1.30027
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2.86866e14	+	4.96866e14i	−0.327395	+	0.567064i	−0.981994	−	0.188912i	$-0.939504\pi$
$521$	−2.86866e14	+	4.96866e14i	−0.327395	+	0.567064i	0.654599	+	0.755976i	$0.272837\pi$
$522$		0			0
$523$	7.94860e14	+	1.37674e15i	0.888242	+	1.53848i	0.841952	+	0.539553i	$0.181407\pi$
$523$	7.94860e14	+	1.37674e15i	0.888242	+	1.53848i	0.0462909	+	0.998928i	$0.485260\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.41502e13	+	1.45752e14i	0.0901772	+	0.156191i
$528$		0			0
$529$	2.40074e14	−	4.15821e14i	0.251964	−	0.436415i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.55978e14			0.358446
$534$		0			0
$535$	−1.03944e15	+	1.80037e15i	−1.02531	+	1.77589i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.37420e14	+	8.26872e14i	0.319472	+	0.782888i
$540$		0			0
$541$	−6.86963e14	−	1.18985e15i	−0.637306	−	1.10385i	−0.986021	−	0.166618i	$-0.946715\pi$
$541$	−6.86963e14	−	1.18985e15i	−0.637306	−	1.10385i	0.348715	−	0.937229i	$-0.386618\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.94030e15			2.61946
$546$		0			0
$547$	−1.72422e15			−1.50544			−0.752719	−	0.658342i	$-0.771258\pi$
$547$	−1.72422e15			−1.50544			−0.752719	+	0.658342i	$0.771258\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.78139e14	+	1.00137e15i	0.484952	+	0.839961i
$552$		0			0
$553$	−1.31039e15	+	1.95003e15i	−1.07749	+	1.60344i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9.11166e14	+	1.57819e15i	−0.720102	+	1.24725i	0.240856	+	0.970561i	$0.422572\pi$
$557$	−9.11166e14	+	1.57819e15i	−0.720102	+	1.24725i	−0.960958	+	0.276693i	$0.910762\pi$
$558$		0			0
$559$	−3.11031e15			−2.41012
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.73498e14	+	9.93328e14i	−0.427303	+	0.740110i	−0.996632	−	0.0819991i	$-0.973870\pi$
$563$	−5.73498e14	+	9.93328e14i	−0.427303	+	0.740110i	0.569329	+	0.822109i	$0.307203\pi$
$564$		0			0
$565$	1.07735e15	+	1.86602e15i	0.787209	+	1.36349i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.79577e14	+	6.57447e14i	0.266798	+	0.462107i	0.968033	−	0.250823i	$-0.0807011\pi$
$569$	3.79577e14	+	6.57447e14i	0.266798	+	0.462107i	−0.701235	+	0.712930i	$0.747368\pi$
$570$		0			0
$571$	1.00511e15	−	1.74090e15i	0.692968	−	1.20026i	−0.277892	−	0.960612i	$-0.589636\pi$
$571$	1.00511e15	−	1.74090e15i	0.692968	−	1.20026i	0.970861	−	0.239644i	$-0.0770308\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.03691e15			−1.35145
$576$		0			0
$577$	−2.80464e14	+	4.85777e14i	−0.182562	+	0.316206i	−0.942752	−	0.333494i	$-0.891772\pi$
$577$	−2.80464e14	+	4.85777e14i	−0.182562	+	0.316206i	0.760191	+	0.649700i	$0.225106\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.64684e14	+	3.36170e14i	0.103201	+	0.210664i
$582$		0			0
$583$	1.20762e15	+	2.09165e15i	0.742595	+	1.28621i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.38574e14			−0.318959			−0.159480	−	0.987201i	$-0.550982\pi$
$587$	−5.38574e14			−0.318959			−0.159480	+	0.987201i	$0.550982\pi$
$588$		0			0
$589$	4.00219e14			0.232629
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.86885e14	+	1.53613e15i	0.496668	+	0.860255i	0.999993	−	0.00384280i	$-0.00122320\pi$
$593$	8.86885e14	+	1.53613e15i	0.496668	+	0.860255i	−0.503324	+	0.864098i	$0.667890\pi$
$594$		0			0
$595$	−6.37511e14	−	1.30136e15i	−0.350465	−	0.715406i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.30508e14	−	3.99252e14i	0.122135	−	0.211543i	−0.798475	−	0.602028i	$-0.794359\pi$
$599$	2.30508e14	−	3.99252e14i	0.122135	−	0.211543i	0.920609	+	0.390485i	$0.127693\pi$
$600$		0			0
$601$	6.44628e13			0.0335351			0.0167675	−	0.999859i	$-0.494662\pi$
$601$	6.44628e13			0.0335351			0.0167675	+	0.999859i	$0.494662\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.85407e14	−	8.40750e14i	0.243474	−	0.421709i
$606$		0			0
$607$	−1.19914e15	−	2.07697e15i	−0.590653	−	1.02304i	−0.994145	−	0.108058i	$-0.965537\pi$
$607$	−1.19914e15	−	2.07697e15i	−0.590653	−	1.02304i	0.403492	−	0.914983i	$-0.367796\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.43676e15	−	4.22059e15i	−1.15767	−	2.00515i
$612$		0			0
$613$	−5.69159e14	+	9.85812e14i	−0.265583	+	0.460004i	−0.967716	−	0.252042i	$-0.918898\pi$
$613$	−5.69159e14	+	9.85812e14i	−0.265583	+	0.460004i	0.702133	+	0.712046i	$0.252231\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.22202e15			1.45064			0.725320	−	0.688412i	$-0.241692\pi$
$617$	3.22202e15			1.45064			0.725320	+	0.688412i	$0.241692\pi$
$618$		0			0
$619$	1.52931e15	−	2.64884e15i	0.676390	−	1.17154i	−0.299671	−	0.954043i	$-0.596877\pi$
$619$	1.52931e15	−	2.64884e15i	0.676390	−	1.17154i	0.976061	−	0.217499i	$-0.0697899\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.84038e15	+	2.73871e15i	−0.785637	+	1.16913i
$624$		0			0
$625$	−9.09526e14	−	1.57534e15i	−0.381483	−	0.660747i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9.02954e14			−0.365668
$630$		0			0
$631$	7.07128e14			0.281408			0.140704	−	0.990052i	$-0.455063\pi$
$631$	7.07128e14			0.281408			0.140704	+	0.990052i	$0.455063\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.95986e15	−	3.39457e15i	−0.753302	−	1.30476i
$636$		0			0
$637$	2.44289e15	−	3.14922e15i	0.922861	−	1.18970i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.36624e15	−	2.36639e15i	0.498662	−	0.863708i	−0.501337	−	0.865252i	$-0.667158\pi$
$641$	1.36624e15	−	2.36639e15i	0.498662	−	0.863708i	0.999999	+	0.00154396i	$0.000491458\pi$
$642$		0			0
$643$	−3.26191e14			−0.117034			−0.0585170	−	0.998286i	$-0.518637\pi$
$643$	−3.26191e14			−0.117034			−0.0585170	+	0.998286i	$0.518637\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.59121e14	−	6.22016e14i	0.124528	−	0.215689i	−0.797020	−	0.603953i	$-0.793592\pi$
$647$	3.59121e14	−	6.22016e14i	0.124528	−	0.215689i	0.921548	+	0.388263i	$0.126925\pi$
$648$		0			0
$649$	−2.10965e13	−	3.65402e13i	−0.00719224	−	0.0124573i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.51647e14	−	1.64830e15i	−0.313656	−	0.543268i	0.665495	−	0.746402i	$-0.268220\pi$
$653$	−9.51647e14	−	1.64830e15i	−0.313656	−	0.543268i	−0.979151	+	0.203134i	$0.934887\pi$
$654$		0			0
$655$	−1.50784e15	+	2.61166e15i	−0.488684	+	0.846426i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.51100e15			1.10043			0.550213	−	0.835024i	$-0.314546\pi$
$659$	3.51100e15			1.10043			0.550213	+	0.835024i	$0.314546\pi$
$660$		0			0
$661$	−2.78986e14	+	4.83218e14i	−0.0859952	+	0.148948i	−0.905815	−	0.423674i	$-0.860740\pi$
$661$	−2.78986e14	+	4.83218e14i	−0.0859952	+	0.148948i	0.819820	+	0.572622i	$0.194074\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.43803e15	−	2.34417e14i	−1.02516	−	0.0698988i
$666$		0			0
$667$	1.93627e15	+	3.35372e15i	0.567903	+	0.983637i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.08038e15			0.590432
$672$		0			0
$673$	−8.28706e14			−0.231376			−0.115688	−	0.993286i	$-0.536907\pi$
$673$	−8.28706e14			−0.231376			−0.115688	+	0.993286i	$0.536907\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.60125e15	+	4.50550e15i	0.702982	+	1.21760i	0.967415	+	0.253197i	$0.0814821\pi$
$677$	2.60125e15	+	4.50550e15i	0.702982	+	1.21760i	−0.264432	+	0.964404i	$0.585185\pi$
$678$		0			0
$679$	1.19069e15	+	2.43057e15i	0.316604	+	0.646286i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.03530e15	+	1.79320e15i	−0.266535	+	0.461651i	−0.967965	−	0.251087i	$-0.919212\pi$
$683$	−1.03530e15	+	1.79320e15i	−0.266535	+	0.461651i	0.701430	+	0.712738i	$0.252545\pi$
$684$		0			0
$685$	7.16260e15			1.81457
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5.38942e15	−	9.33475e15i	1.32232	−	2.29032i
$690$		0			0
$691$	−1.20456e15	−	2.08636e15i	−0.290870	−	0.503801i	0.683146	−	0.730282i	$-0.260611\pi$
$691$	−1.20456e15	−	2.08636e15i	−0.290870	−	0.503801i	−0.974016	+	0.226481i	$0.927278\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.95650e15	−	6.85286e15i	−0.925539	−	1.60308i
$696$		0			0
$697$	2.41048e14	−	4.17507e14i	0.0555038	−	0.0961355i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.65410e15			−0.369074			−0.184537	−	0.982826i	$-0.559079\pi$
$701$	−1.65410e15			−0.369074			−0.184537	+	0.982826i	$0.559079\pi$
$702$		0			0
$703$	−1.07361e15	+	1.85955e15i	−0.235827	+	0.408465i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.27056e15	−	2.22998e14i	−0.696331	−	0.0474783i
$708$		0			0
$709$	−3.17562e15	−	5.50034e15i	−0.665694	−	1.15302i	−0.979097	−	0.203395i	$-0.934802\pi$
$709$	−3.17562e15	−	5.50034e15i	−0.665694	−	1.15302i	0.313403	−	0.949620i	$-0.398531\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.34039e15			0.272420
$714$		0			0
$715$	1.08683e16			2.17509
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.08272e15	+	7.07148e15i	0.792393	+	1.37246i	0.924481	+	0.381227i	$0.124498\pi$
$719$	4.08272e15	+	7.07148e15i	0.792393	+	1.37246i	−0.132088	+	0.991238i	$0.542168\pi$
$720$		0			0
$721$	3.24235e14	−	4.82502e14i	0.0619749	−	0.0922264i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8.34428e15	+	1.44527e16i	−1.54714	+	2.67973i
$726$		0			0
$727$	4.74774e15			0.867058			0.433529	−	0.901140i	$-0.357268\pi$
$727$	4.74774e15			0.867058			0.433529	+	0.901140i	$0.357268\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.10612e15	+	3.64791e15i	−0.373197	+	0.646396i
$732$		0			0
$733$	−4.64434e15	−	8.04422e15i	−0.810684	−	1.40415i	−0.912386	−	0.409331i	$-0.865762\pi$
$733$	−4.64434e15	−	8.04422e15i	−0.810684	−	1.40415i	0.101702	−	0.994815i	$-0.467571\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.86246e15	−	4.95793e15i	−0.484919	−	0.839904i
$738$		0			0
$739$	9.58776e14	−	1.66065e15i	0.160019	−	0.277162i	−0.774856	−	0.632138i	$-0.782178\pi$
$739$	9.58776e14	−	1.66065e15i	0.160019	−	0.277162i	0.934875	+	0.354976i	$0.115511\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.82093e15			0.295022			0.147511	−	0.989060i	$-0.452874\pi$
$743$	1.82093e15			0.295022			0.147511	+	0.989060i	$0.452874\pi$
$744$		0			0
$745$	1.02040e16	−	1.76738e16i	1.62896	−	2.82144i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.31893e15	+	6.42710e15i	−0.669462	+	0.996243i
$750$		0			0
$751$	1.45306e15	+	2.51677e15i	0.221954	+	0.384435i	0.955401	−	0.295311i	$-0.0954233\pi$
$751$	1.45306e15	+	2.51677e15i	0.221954	+	0.384435i	−0.733447	+	0.679746i	$0.762090\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.79389e16			2.66126
$756$		0			0
$757$	8.56752e15			1.25264			0.626322	−	0.779564i	$-0.284559\pi$
$757$	8.56752e15			1.25264			0.626322	+	0.779564i	$0.284559\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.71057e15	+	2.96279e15i	0.242954	+	0.420809i	0.961554	−	0.274614i	$-0.0885501\pi$
$761$	1.71057e15	+	2.96279e15i	0.242954	+	0.420809i	−0.718600	+	0.695424i	$0.755217\pi$
$762$		0			0
$763$	1.09267e16	+	7.45018e14i	1.52969	+	0.104299i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.41506e13	+	1.63074e14i	−0.0128070	+	0.0221824i
$768$		0			0
$769$	−1.09800e16			−1.47234			−0.736168	−	0.676799i	$-0.763367\pi$
$769$	−1.09800e16			−1.47234			−0.736168	+	0.676799i	$0.763367\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.28165e14	+	7.41604e14i	−0.0557987	+	0.0966462i	−0.892576	−	0.450898i	$-0.851104\pi$
$773$	−4.28165e14	+	7.41604e14i	−0.0557987	+	0.0966462i	0.836777	+	0.547544i	$0.184437\pi$
$774$		0			0
$775$	2.88818e15	+	5.00247e15i	0.371077	+	0.642725i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.73213e14	−	9.92834e14i	−0.0715912	−	0.124000i
$780$		0			0
$781$	2.33311e15	−	4.04107e15i	0.287313	−	0.497641i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.68020e15			−0.560378
$786$		0			0
$787$	3.15376e15	−	5.46248e15i	0.372364	−	0.644954i	−0.617565	−	0.786520i	$-0.711881\pi$
$787$	3.15376e15	−	5.46248e15i	0.372364	−	0.644954i	0.989929	+	0.141567i	$0.0452139\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3.53080e15	+	7.20744e15i	0.405417	+	0.827581i
$792$		0			0
$793$	−4.64223e15	−	8.04057e15i	−0.525683	−	0.910510i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.47160e15			−0.162095			−0.0810476	−	0.996710i	$-0.525827\pi$
$797$	−1.47160e15			−0.162095			−0.0810476	+	0.996710i	$0.525827\pi$
$798$		0			0
$799$	−6.60013e15			−0.717043
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.28242e15	−	2.22122e15i	−0.135549	−	0.234778i
$804$		0			0
$805$	−1.15145e16	−	7.85096e14i	−1.20051	−	0.0818550i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.74828e15	−	6.49221e15i	0.380290	−	0.658681i	−0.610814	−	0.791774i	$-0.709158\pi$
$809$	3.74828e15	−	6.49221e15i	0.380290	−	0.658681i	0.991104	+	0.133093i	$0.0424909\pi$
$810$		0			0
$811$	3.15860e15			0.316140			0.158070	−	0.987428i	$-0.449473\pi$
$811$	3.15860e15			0.316140			0.158070	+	0.987428i	$0.449473\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.56760e15	+	7.91132e15i	−0.444961	+	0.770694i
$816$		0			0
$817$	5.00837e15	+	8.67475e15i	0.481366	+	0.833750i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−2.52754e15	−	4.37783e15i	−0.236489	−	0.409611i	0.723216	−	0.690622i	$-0.242663\pi$
$821$	−2.52754e15	−	4.37783e15i	−0.236489	−	0.409611i	−0.959704	+	0.281012i	$0.909330\pi$
$822$		0			0
$823$	6.74253e15	−	1.16784e16i	0.622478	−	1.07816i	−0.366545	−	0.930400i	$-0.619460\pi$
$823$	6.74253e15	−	1.16784e16i	0.622478	−	1.07816i	0.989023	−	0.147763i	$-0.0472072\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	9.22293e15			0.829065			0.414533	−	0.910034i	$-0.363945\pi$
$827$	9.22293e15			0.829065			0.414533	+	0.910034i	$0.363945\pi$
$828$		0			0
$829$	2.61350e15	−	4.52671e15i	0.231831	−	0.401543i	−0.726516	−	0.687150i	$-0.758862\pi$
$829$	2.61350e15	−	4.52671e15i	0.231831	−	0.401543i	0.958347	+	0.285606i	$0.0921950\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.03936e15	−	4.99759e15i	−0.176176	−	0.431731i
$834$		0			0
$835$	3.55718e15	+	6.16121e15i	0.303270	+	0.525279i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	6.20331e15			0.515149			0.257574	−	0.966259i	$-0.417077\pi$
$839$	6.20331e15			0.515149			0.257574	+	0.966259i	$0.417077\pi$
$840$		0			0
$841$	1.95275e16			1.60055
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.35543e16	−	2.34767e16i	−1.08234	−	1.87467i
$846$		0			0
$847$	2.01689e15	−	3.00138e15i	0.158973	−	0.236571i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.59569e15	+	6.22792e15i	−0.276166	+	0.478333i
$852$		0			0
$853$	5.91729e15			0.448645			0.224323	−	0.974515i	$-0.427983\pi$
$853$	5.91729e15			0.448645			0.224323	+	0.974515i	$0.427983\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.29414e16	−	2.24152e16i	0.956284	−	1.65633i	0.224882	−	0.974386i	$-0.427800\pi$
$857$	1.29414e16	−	2.24152e16i	0.956284	−	1.65633i	0.731402	−	0.681947i	$-0.238866\pi$
$858$		0			0
$859$	6.50982e15	+	1.12753e16i	0.474905	+	0.822559i	0.999587	−	0.0287388i	$-0.00914910\pi$
$859$	6.50982e15	+	1.12753e16i	0.474905	+	0.822559i	−0.524682	+	0.851298i	$0.675816\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.90083e15	−	8.48848e15i	−0.348506	−	0.603630i	0.637478	−	0.770468i	$-0.279978\pi$
$863$	−4.90083e15	−	8.48848e15i	−0.348506	−	0.603630i	−0.985984	+	0.166838i	$0.946644\pi$
$864$		0			0
$865$	9.88270e15	−	1.71173e16i	0.693884	−	1.20184i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.38630e16			−1.63349
$870$		0			0
$871$	−1.27748e16	+	2.21266e16i	−0.863482	+	1.49559i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.04772e16	−	2.13872e16i	−0.690560	−	1.40964i
$876$		0			0
$877$	7.18835e15	+	1.24506e16i	0.467877	+	0.810387i	0.999326	−	0.0367035i	$-0.0116857\pi$
$877$	7.18835e15	+	1.24506e16i	0.467877	+	0.810387i	−0.531449	+	0.847090i	$0.678352\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.76989e16			1.12351			0.561757	−	0.827302i	$-0.310126\pi$
$881$	1.76989e16			1.12351			0.561757	+	0.827302i	$0.310126\pi$
$882$		0			0
$883$	−2.61552e16			−1.63974			−0.819869	−	0.572551i	$-0.805954\pi$
$883$	−2.61552e16			−1.63974			−0.819869	+	0.572551i	$0.805954\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4.87889e15	+	8.45048e15i	0.298360	+	0.516775i	0.975761	−	0.218839i	$-0.0702271\pi$
$887$	4.87889e15	+	8.45048e15i	0.298360	+	0.516775i	−0.677401	+	0.735614i	$0.736894\pi$
$888$		0			0
$889$	−6.42304e15	−	1.31114e16i	−0.387955	−	0.791935i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.84757e15	+	1.35924e16i	−0.462437	+	0.800964i
$894$		0			0
$895$	1.78355e16			1.03815
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.49095e15	−	9.51061e15i	0.311867	−	0.540169i
$900$		0			0
$901$	−7.29881e15	−	1.26419e16i	−0.409511	−	0.709294i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8.88743e15	+	1.53935e16i	0.486642	+	0.842888i
$906$		0			0
$907$	−1.21520e16	+	2.10479e16i	−0.657368	+	1.13859i	0.323926	+	0.946082i	$0.394997\pi$
$907$	−1.21520e16	+	2.10479e16i	−0.657368	+	1.13859i	−0.981294	+	0.192513i	$0.938336\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.23378e16			0.651460			0.325730	−	0.945463i	$-0.394390\pi$
$911$	1.23378e16			0.651460			0.325730	+	0.945463i	$0.394390\pi$
$912$		0			0
$913$	−1.90109e15	+	3.29279e15i	−0.0991778	+	0.171781i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−6.26514e15	+	9.32331e15i	−0.319080	+	0.474831i
$918$		0			0
$919$	2.86000e15	+	4.95366e15i	0.143923	+	0.249282i	0.928971	−	0.370153i	$-0.120695\pi$
$919$	2.86000e15	+	4.95366e15i	0.143923	+	0.249282i	−0.785048	+	0.619435i	$0.787362\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2.08247e16			−1.02322
$924$		0			0
$925$	−3.09909e16			−1.50472
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.78204e15	−	3.08658e15i	−0.0844948	−	0.146349i	0.820681	−	0.571387i	$-0.193594\pi$
$929$	−1.78204e15	−	3.08658e15i	−0.0844948	−	0.146349i	−0.905176	+	0.425037i	$0.860261\pi$
$930$		0			0
$931$	−1.27169e16	−	1.74227e15i	−0.595879	−	0.0816378i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	7.35937e15	−	1.27468e16i	0.336804	−	0.583361i
$936$		0			0
$937$	−1.07666e16			−0.486978			−0.243489	−	0.969904i	$-0.578292\pi$
$937$	−1.07666e16			−0.486978			−0.243489	+	0.969904i	$0.578292\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.99923e16	+	3.46278e16i	−0.883325	+	1.52996i	−0.0357044	+	0.999362i	$0.511367\pi$
$941$	−1.99923e16	+	3.46278e16i	−0.883325	+	1.52996i	−0.847621	+	0.530602i	$0.821966\pi$
$942$		0			0
$943$	−1.91977e15	−	3.32514e15i	−0.0838370	−	0.145210i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.26575e16	+	3.92439e16i	0.966690	+	1.67436i	0.705006	+	0.709202i	$0.250945\pi$
$947$	2.26575e16	+	3.92439e16i	0.966690	+	1.67436i	0.261684	+	0.965154i	$0.415722\pi$
$948$		0			0
$949$	−5.72328e15	+	9.91301e15i	−0.241369	+	0.418063i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	4.16824e16			1.71768			0.858839	−	0.512245i	$-0.171186\pi$
$953$	4.16824e16			1.71768			0.858839	+	0.512245i	$0.171186\pi$
$954$		0			0
$955$	−3.20236e15	+	5.54665e15i	−0.130452	+	0.225950i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.66174e16	+	1.81487e15i	1.05965	+	0.0722509i
$960$		0			0
$961$	1.08037e16	+	1.87125e16i	0.425200	+	0.736467i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.53605e16			−1.74493
$966$		0			0
$967$	1.85340e16			0.704893			0.352446	−	0.935832i	$-0.385350\pi$
$967$	1.85340e16			0.704893			0.352446	+	0.935832i	$0.385350\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	8.64967e15	+	1.49817e16i	0.321584	+	0.556999i	0.980815	−	0.194941i	$-0.0624515\pi$
$971$	8.64967e15	+	1.49817e16i	0.321584	+	0.556999i	−0.659231	+	0.751940i	$0.729118\pi$
$972$		0			0
$973$	−1.29666e16	−	2.64689e16i	−0.476658	−	0.973005i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.06376e16	−	3.57453e16i	0.741717	−	1.28469i	−0.209996	−	0.977702i	$-0.567345\pi$
$977$	2.06376e16	−	3.57453e16i	0.741717	−	1.28469i	0.951713	−	0.306989i	$-0.0993215\pi$
$978$		0			0
$979$	−3.35143e16			−1.19104
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.87035e16	+	3.23953e16i	−0.649947	+	1.12574i	0.333189	+	0.942860i	$0.391875\pi$
$983$	−1.87035e16	+	3.23953e16i	−0.649947	+	1.12574i	−0.983135	+	0.182880i	$0.941458\pi$
$984$		0			0
$985$	−1.00158e16	−	1.73479e16i	−0.344182	−	0.596140i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.67738e16	+	2.90530e16i	0.563704	+	0.976363i
$990$		0			0
$991$	−1.48476e16	+	2.57168e16i	−0.493458	+	0.854695i	−0.999972	−	0.00753743i	$-0.997601\pi$
$991$	−1.48476e16	+	2.57168e16i	−0.493458	+	0.854695i	0.506513	+	0.862232i	$0.330934\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.18581e16			−2.01080
$996$		0			0
$997$	1.83287e16	−	3.17463e16i	0.589263	−	1.02063i	−0.405066	−	0.914287i	$-0.632751\pi$
$997$	1.83287e16	−	3.17463e16i	0.589263	−	1.02063i	0.994329	−	0.106346i	$-0.0339152\pi$
$998$		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.109.1	yes	28
3.2	odd	2	inner	252.12.k.e.109.14	yes	28
7.2	even	3	inner	252.12.k.e.37.1	✓	28
21.2	odd	6	inner	252.12.k.e.37.14	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.12.k.e.37.1	✓	28	7.2	even	3	inner
252.12.k.e.37.14	yes	28	21.2	odd	6	inner
252.12.k.e.109.1	yes	28	1.1	even	1	trivial
252.12.k.e.109.14	yes	28	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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-5969.07 - 10338.7i) q^{5} +(-19562.4 - 39932.9i) q^{7} +O(q^{10})\)	\(q+(-5969.07 - 10338.7i) q^{5} +(-19562.4 - 39932.9i) q^{7} +(225827. - 391144. i) q^{11} -2.01567e6 q^{13} +(-1.36489e6 + 2.36406e6i) q^{17} +(3.24572e6 + 5.62175e6i) q^{19} +(1.08704e7 + 1.88281e7i) q^{23} +(-4.68455e7 + 8.11387e7i) q^{25} +1.78124e8 q^{29} +(3.08267e7 - 5.33933e7i) q^{31} +(-2.96086e8 + 4.40613e8i) q^{35} +(1.65389e8 + 2.86462e8i) q^{37} -1.76606e8 q^{41} +1.54307e9 q^{43} +(1.20891e9 + 2.09389e9i) q^{47} +(-1.21195e9 + 1.56237e9i) q^{49} +(-2.67377e9 + 4.63110e9i) q^{53} -5.39190e9 q^{55} +(4.67094e7 - 8.09031e7i) q^{59} +(2.30307e9 + 3.98904e9i) q^{61} +(1.20317e10 + 2.08394e10i) q^{65} +(6.33774e9 - 1.09773e10i) q^{67} +1.03314e10 q^{71} +(2.83940e9 - 4.91798e9i) q^{73} +(-2.00372e10 - 1.36621e9i) q^{77} +(-2.64174e10 - 4.57562e10i) q^{79} -8.41837e9 q^{83} +3.25885e10 q^{85} +(-3.71018e10 - 6.42622e10i) q^{89} +(3.94313e10 + 8.04915e10i) q^{91} +(3.87478e10 - 6.71132e10i) q^{95} -6.08664e10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 85666 q^{7}+O(q^{10}) \) 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

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Groups

Database

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