LMFDB - Embedded newform 20.18.c.a.9.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,18,Mod(9,20)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 18, names="a")

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

chi := DirichletCharacter("20.9");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	20.c (of order $2$, degree $1$, 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:	$36.6444174689$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 10513788 x^{6} + 47438777752 x^{5} - 249513269598475 x^{4} + \cdots + 12\!\cdots\!68 $ x^8 - x^7 - 10513788x^6 + 47438777752x^5 - 249513269598475x^4 + 2482001120988967319x^3 + 4452658897796563124186x^2 - 53508982948055868249424962x + 129749232836567172546933042968
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{35}\cdot 3^{4}\cdot 5^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			9.2
Root			$-5032.46 + 787.915i$ of defining polynomial
Character	$\chi$	$=$	20.9
Dual form			20.18.c.a.9.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-14851.7i q^{3} +(-846917. + 213710. i) q^{5} -2.81651e6i q^{7} -9.14330e7 q^{9} -4.42439e8 q^{11} +2.52490e9i q^{13} +(3.17395e9 + 1.25782e10i) q^{15} +2.85914e10i q^{17} -3.44100e10 q^{19} -4.18300e10 q^{21} -2.27367e11i q^{23} +(6.71596e11 - 3.61989e11i) q^{25} -5.60016e11i q^{27} +2.03177e12 q^{29} -3.80783e11 q^{31} +6.57098e12i q^{33} +(6.01916e11 + 2.38535e12i) q^{35} +2.15801e13i q^{37} +3.74991e13 q^{39} +9.27031e13 q^{41} -2.13669e13i q^{43} +(7.74361e13 - 1.95401e13i) q^{45} -9.84587e13i q^{47} +2.24698e14 q^{49} +4.24631e14 q^{51} +8.96966e14i q^{53} +(3.74709e14 - 9.45536e13i) q^{55} +5.11047e14i q^{57} +1.02355e15 q^{59} -8.90426e14 q^{61} +2.57522e14i q^{63} +(-5.39596e14 - 2.13838e15i) q^{65} -5.11536e15i q^{67} -3.37678e15 q^{69} -1.86518e13 q^{71} +1.21487e16i q^{73} +(-5.37615e15 - 9.97434e15i) q^{75} +1.24614e15i q^{77} -2.18708e16 q^{79} -2.01249e16 q^{81} +2.97110e16i q^{83} +(-6.11026e15 - 2.42145e16i) q^{85} -3.01752e16i q^{87} +3.50927e16 q^{89} +7.11142e15 q^{91} +5.65527e15i q^{93} +(2.91424e16 - 7.35375e15i) q^{95} -3.98932e16i q^{97} +4.04536e16 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 1276800 q^{5} - 376848248 q^{9} - 396200640 q^{11} + 5677983200 q^{15} + 11821646592 q^{19} - 420670059472 q^{21} + 1642212165000 q^{25} + 1543712861232 q^{29} - 13722543013312 q^{31} - 13325691076800 q^{35}+ \cdots - 23\!\cdots\!60 q^{99}+O(q^{100}) $ 8 * q + 1276800 * q^5 - 376848248 * q^9 - 396200640 * q^11 + 5677983200 * q^15 + 11821646592 * q^19 - 420670059472 * q^21 + 1642212165000 * q^25 + 1543712861232 * q^29 - 13722543013312 * q^31 - 13325691076800 * q^35 - 79298226023808 * q^39 + 29824965017568 * q^41 - 238216983006800 * q^45 - 535615080275736 * q^49 - 764079320419264 * q^51 + 339212427988000 * q^55 + 94539545151744 * q^59 - 3380827226969024 * q^61 + 130929936568800 * q^65 + 5559752175554416 * q^69 + 20155606060098624 * q^71 + 938454087760000 * q^75 - 3634057920126144 * q^79 + 15949137882627512 * q^81 + 3575192471022400 * q^85 + 103487949043655568 * q^89 - 69783870139121088 * q^91 + 67219016711959200 * q^95 - 239420571619653760 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

$n$	$11$	$17$
$\chi(n)$	$1$	$-1$

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
$3$		−	14851.7i		−	1.30691i	−0.756965	−	0.653455i	$-0.773319\pi$
$3$		−	14851.7i		−	1.30691i	0.756965	−	0.653455i	$-0.226681\pi$
$4$		0			0
$5$	−846917.	+	213710.i	−0.969607	+	0.244669i
$5$	−846917.	+	213710.i	−0.969607	+	0.244669i
$6$		0			0
$7$		−	2.81651e6i		−	0.184662i	−0.995728	−	0.0923312i	$-0.970568\pi$
$7$		−	2.81651e6i		−	0.184662i	0.995728	−	0.0923312i	$-0.0294318\pi$
$8$		0			0
$9$	−9.14330e7			−0.708014
$10$		0			0
$11$	−4.42439e8			−0.622323			−0.311162	−	0.950357i	$-0.600718\pi$
$11$	−4.42439e8			−0.622323			−0.311162	+	0.950357i	$0.600718\pi$
$12$		0			0
$13$			2.52490e9i			0.858472i	0.903192	+	0.429236i	$0.141217\pi$
$13$			2.52490e9i			0.858472i	−0.903192	+	0.429236i	$0.858783\pi$
$14$		0			0
$15$	3.17395e9	+	1.25782e10i	0.319761	+	1.26719i
$16$		0			0
$17$			2.85914e10i			0.994076i	0.867729	+	0.497038i	$0.165579\pi$
$17$			2.85914e10i			0.994076i	−0.867729	+	0.497038i	$0.834421\pi$
$18$		0			0
$19$	−3.44100e10			−0.464814			−0.232407	−	0.972619i	$-0.574660\pi$
$19$	−3.44100e10			−0.464814			−0.232407	+	0.972619i	$0.574660\pi$
$20$		0			0
$21$	−4.18300e10			−0.241337
$22$		0			0
$23$		−	2.27367e11i		−	0.605397i	−0.953086	−	0.302698i	$-0.902113\pi$
$23$		−	2.27367e11i		−	0.605397i	0.953086	−	0.302698i	$-0.0978875\pi$
$24$		0			0
$25$	6.71596e11	−	3.61989e11i	0.880274	−	0.474466i
$26$		0			0
$27$		−	5.60016e11i		−	0.381600i
$28$		0			0
$29$	2.03177e12			0.754208			0.377104	−	0.926171i	$-0.376920\pi$
$29$	2.03177e12			0.754208			0.377104	+	0.926171i	$0.376920\pi$
$30$		0			0
$31$	−3.80783e11			−0.0801868			−0.0400934	−	0.999196i	$-0.512766\pi$
$31$	−3.80783e11			−0.0801868			−0.0400934	+	0.999196i	$0.512766\pi$
$32$		0			0
$33$			6.57098e12i			0.813320i
$34$		0			0
$35$	6.01916e11	+	2.38535e12i	0.0451812	+	0.179050i
$36$		0			0
$37$			2.15801e13i			1.01004i	0.863108	+	0.505019i	$0.168515\pi$
$37$			2.15801e13i			1.01004i	−0.863108	+	0.505019i	$0.831485\pi$
$38$		0			0
$39$	3.74991e13			1.12195
$40$		0			0
$41$	9.27031e13			1.81314			0.906570	−	0.422056i	$-0.138691\pi$
$41$	9.27031e13			1.81314			0.906570	+	0.422056i	$0.138691\pi$
$42$		0			0
$43$		−	2.13669e13i		−	0.278778i	−0.990238	−	0.139389i	$-0.955486\pi$
$43$		−	2.13669e13i		−	0.278778i	0.990238	−	0.139389i	$-0.0445139\pi$
$44$		0			0
$45$	7.74361e13	−	1.95401e13i	0.686495	−	0.173229i
$46$		0			0
$47$		−	9.84587e13i		−	0.603146i	−0.953443	−	0.301573i	$-0.902488\pi$
$47$		−	9.84587e13i		−	0.603146i	0.953443	−	0.301573i	$-0.0975117\pi$
$48$		0			0
$49$	2.24698e14			0.965900
$50$		0			0
$51$	4.24631e14			1.29917
$52$		0			0
$53$			8.96966e14i			1.97893i	0.144762	+	0.989467i	$0.453758\pi$
$53$			8.96966e14i			1.97893i	−0.144762	+	0.989467i	$0.546242\pi$
$54$		0			0
$55$	3.74709e14	−	9.45536e13i	0.603409	−	0.152263i
$56$		0			0
$57$			5.11047e14i			0.607470i
$58$		0			0
$59$	1.02355e15			0.907546			0.453773	−	0.891117i	$-0.350078\pi$
$59$	1.02355e15			0.907546			0.453773	+	0.891117i	$0.350078\pi$
$60$		0			0
$61$	−8.90426e14			−0.594695			−0.297348	−	0.954769i	$-0.596102\pi$
$61$	−8.90426e14			−0.594695			−0.297348	+	0.954769i	$0.596102\pi$
$62$		0			0
$63$			2.57522e14i			0.130743i
$64$		0			0
$65$	−5.39596e14	−	2.13838e15i	−0.210042	−	0.832380i
$66$		0			0
$67$		−	5.11536e15i		−	1.53901i	−0.638643	−	0.769503i	$-0.720504\pi$
$67$		−	5.11536e15i		−	1.53901i	0.638643	−	0.769503i	$-0.279496\pi$
$68$		0			0
$69$	−3.37678e15			−0.791199
$70$		0			0
$71$	−1.86518e13			−0.00342786			−0.00171393	−	0.999999i	$-0.500546\pi$
$71$	−1.86518e13			−0.00342786			−0.00171393	+	0.999999i	$0.500546\pi$
$72$		0			0
$73$			1.21487e16i			1.76314i	0.472057	+	0.881568i	$0.343512\pi$
$73$			1.21487e16i			1.76314i	−0.472057	+	0.881568i	$0.656488\pi$
$74$		0			0
$75$	−5.37615e15	−	9.97434e15i	−0.620084	−	1.15044i
$76$		0			0
$77$			1.24614e15i			0.114920i
$78$		0			0
$79$	−2.18708e16			−1.62194			−0.810968	−	0.585091i	$-0.801059\pi$
$79$	−2.18708e16			−1.62194			−0.810968	+	0.585091i	$0.801059\pi$
$80$		0			0
$81$	−2.01249e16			−1.20673
$82$		0			0
$83$			2.97110e16i			1.44795i	0.689827	+	0.723974i	$0.257687\pi$
$83$			2.97110e16i			1.44795i	−0.689827	+	0.723974i	$0.742313\pi$
$84$		0			0
$85$	−6.11026e15	−	2.42145e16i	−0.243220	−	0.963863i
$86$		0			0
$87$		−	3.01752e16i		−	0.985682i
$88$		0			0
$89$	3.50927e16			0.944934			0.472467	−	0.881348i	$-0.343364\pi$
$89$	3.50927e16			0.944934			0.472467	+	0.881348i	$0.343364\pi$
$90$		0			0
$91$	7.11142e15			0.158527
$92$		0			0
$93$			5.65527e15i			0.104797i
$94$		0			0
$95$	2.91424e16	−	7.35375e15i	0.450687	−	0.113726i
$96$		0			0
$97$		−	3.98932e16i		−	0.516820i	−0.966035	−	0.258410i	$-0.916801\pi$
$97$		−	3.98932e16i		−	0.516820i	0.966035	−	0.258410i	$-0.0831986\pi$
$98$		0			0
$99$	4.04536e16			0.440613
$100$		0			0
$101$	5.49755e16			0.505170			0.252585	−	0.967575i	$-0.418719\pi$
$101$	5.49755e16			0.505170			0.252585	+	0.967575i	$0.418719\pi$
$102$		0			0
$103$			1.68517e17i			1.31077i	0.755295	+	0.655385i	$0.227494\pi$
$103$			1.68517e17i			1.31077i	−0.755295	+	0.655385i	$0.772506\pi$
$104$		0			0
$105$	3.54265e16	−	8.93948e15i	0.234002	−	0.0590477i
$106$		0			0
$107$			2.41403e17i			1.35825i	0.734022	+	0.679126i	$0.237641\pi$
$107$			2.41403e17i			1.35825i	−0.734022	+	0.679126i	$0.762359\pi$
$108$		0			0
$109$	1.12604e17			0.541288			0.270644	−	0.962679i	$-0.412763\pi$
$109$	1.12604e17			0.541288			0.270644	+	0.962679i	$0.412763\pi$
$110$		0			0
$111$	3.20501e17			1.32003
$112$		0			0
$113$		−	2.03615e17i		−	0.720513i	−0.932853	−	0.360257i	$-0.882689\pi$
$113$		−	2.03615e17i		−	0.720513i	0.932853	−	0.360257i	$-0.117311\pi$
$114$		0			0
$115$	4.85905e16	+	1.92561e17i	0.148122	+	0.586997i
$116$		0			0
$117$		−	2.30859e17i		−	0.607810i
$118$		0			0
$119$	8.05281e16			0.183568
$120$		0			0
$121$	−3.09694e17			−0.612714
$122$		0			0
$123$		−	1.37680e18i		−	2.36961i
$124$		0			0
$125$	−4.91425e17	+	4.50101e17i	−0.737432	+	0.675421i
$126$		0			0
$127$			4.39596e17i			0.576398i	0.957571	+	0.288199i	$0.0930564\pi$
$127$			4.39596e17i			0.576398i	−0.957571	+	0.288199i	$0.906944\pi$
$128$		0			0
$129$	−3.17335e17			−0.364338
$130$		0			0
$131$	−1.27789e18			−1.28733			−0.643663	−	0.765309i	$-0.722586\pi$
$131$	−1.27789e18			−1.28733			−0.643663	+	0.765309i	$0.722586\pi$
$132$		0			0
$133$			9.69162e16i			0.0858336i
$134$		0			0
$135$	1.19681e17	+	4.74287e17i	0.0933657	+	0.370002i
$136$		0			0
$137$			1.27147e18i			0.875346i	0.899134	+	0.437673i	$0.144197\pi$
$137$			1.27147e18i			0.875346i	−0.899134	+	0.437673i	$0.855803\pi$
$138$		0			0
$139$	−4.65518e17			−0.283342			−0.141671	−	0.989914i	$-0.545247\pi$
$139$	−4.65518e17			−0.283342			−0.141671	+	0.989914i	$0.545247\pi$
$140$		0			0
$141$	−1.46228e18			−0.788257
$142$		0			0
$143$		−	1.11712e18i		−	0.534247i
$144$		0			0
$145$	−1.72074e18	+	4.34208e17i	−0.731285	+	0.184531i
$146$		0			0
$147$		−	3.33715e18i		−	1.26234i
$148$		0			0
$149$	−1.66540e18			−0.561612			−0.280806	−	0.959765i	$-0.590602\pi$
$149$	−1.66540e18			−0.561612			−0.280806	+	0.959765i	$0.590602\pi$
$150$		0			0
$151$	3.74893e18			1.12876			0.564382	−	0.825513i	$-0.309114\pi$
$151$	3.74893e18			1.12876			0.564382	+	0.825513i	$0.309114\pi$
$152$		0			0
$153$		−	2.61420e18i		−	0.703820i
$154$		0			0
$155$	3.22491e17	−	8.13770e16i	0.0777497	−	0.0196192i
$156$		0			0
$157$			1.10356e18i			0.238588i	0.992859	+	0.119294i	$0.0380632\pi$
$157$			1.10356e18i			0.238588i	−0.992859	+	0.119294i	$0.961937\pi$
$158$		0			0
$159$	1.33215e19			2.58629
$160$		0			0
$161$	−6.40381e17			−0.111794
$162$		0			0
$163$			1.16561e19i			1.83214i	0.401023	+	0.916068i	$0.368655\pi$
$163$			1.16561e19i			1.83214i	−0.401023	+	0.916068i	$0.631345\pi$
$164$		0			0
$165$	−1.40428e18	−	5.56507e18i	−0.198994	−	0.788601i
$166$		0			0
$167$			7.25594e18i			0.928119i	0.885804	+	0.464060i	$0.153608\pi$
$167$			7.25594e18i			0.928119i	−0.885804	+	0.464060i	$0.846392\pi$
$168$		0			0
$169$	2.27529e18			0.263027
$170$		0			0
$171$	3.14621e18			0.329095
$172$		0			0
$173$			1.18547e19i			1.12330i	0.827374	+	0.561652i	$0.189834\pi$
$173$			1.18547e19i			1.12330i	−0.827374	+	0.561652i	$0.810166\pi$
$174$		0			0
$175$	−1.01955e18	−	1.89156e18i	−0.0876159	−	0.162553i
$176$		0			0
$177$		−	1.52015e19i		−	1.18608i
$178$		0			0
$179$	2.06752e19			1.46622			0.733111	−	0.680109i	$-0.238067\pi$
$179$	2.06752e19			1.46622			0.733111	+	0.680109i	$0.238067\pi$
$180$		0			0
$181$	−1.19212e19			−0.769226			−0.384613	−	0.923078i	$-0.625665\pi$
$181$	−1.19212e19			−0.769226			−0.384613	+	0.923078i	$0.625665\pi$
$182$		0			0
$183$			1.32243e19i			0.777213i
$184$		0			0
$185$	−4.61187e18	−	1.82765e19i	−0.247125	−	0.979340i
$186$		0			0
$187$		−	1.26500e19i		−	0.618637i
$188$		0			0
$189$	−1.57729e18			−0.0704671
$190$		0			0
$191$	1.84418e18			0.0753391			0.0376696	−	0.999290i	$-0.488007\pi$
$191$	1.84418e18			0.0753391			0.0376696	+	0.999290i	$0.488007\pi$
$192$		0			0
$193$		−	2.14178e19i		−	0.800825i	−0.916335	−	0.400412i	$-0.868867\pi$
$193$		−	2.14178e19i		−	0.800825i	0.916335	−	0.400412i	$-0.131133\pi$
$194$		0			0
$195$	−3.17586e19	+	8.01392e18i	−1.08785	+	0.274505i
$196$		0			0
$197$			1.01798e19i			0.319724i	0.987139	+	0.159862i	$0.0511050\pi$
$197$			1.01798e19i			0.319724i	−0.987139	+	0.159862i	$0.948895\pi$
$198$		0			0
$199$	1.53363e19			0.442048			0.221024	−	0.975268i	$-0.429060\pi$
$199$	1.53363e19			0.442048			0.221024	+	0.975268i	$0.429060\pi$
$200$		0			0
$201$	−7.59718e19			−2.01134
$202$		0			0
$203$		−	5.72250e18i		−	0.139274i
$204$		0			0
$205$	−7.85118e19	+	1.98115e19i	−1.75803	+	0.443619i
$206$		0			0
$207$			2.07888e19i			0.428629i
$208$		0			0
$209$	1.52243e19			0.289265
$210$		0			0
$211$	1.63263e19			0.286079			0.143039	−	0.989717i	$-0.454312\pi$
$211$	1.63263e19			0.286079			0.143039	+	0.989717i	$0.454312\pi$
$212$		0			0
$213$			2.77010e17i			0.00447991i
$214$		0			0
$215$	4.56631e18	+	1.80960e19i	0.0682085	+	0.270305i
$216$		0			0
$217$			1.07248e18i			0.0148075i
$218$		0			0
$219$	1.80429e20			2.30426
$220$		0			0
$221$	−7.21905e19			−0.853386
$222$		0			0
$223$		−	1.27843e20i		−	1.39987i	−0.714207	−	0.699934i	$-0.753213\pi$
$223$		−	1.27843e20i		−	1.39987i	0.714207	−	0.699934i	$-0.246787\pi$
$224$		0			0
$225$	−6.14060e19	+	3.30977e19i	−0.623246	+	0.335928i
$226$		0			0
$227$		−	5.56276e19i		−	0.523686i	−0.965110	−	0.261843i	$-0.915670\pi$
$227$		−	5.56276e19i		−	0.523686i	0.965110	−	0.261843i	$-0.0843303\pi$
$228$		0			0
$229$	1.26901e20			1.10883			0.554414	−	0.832241i	$-0.312943\pi$
$229$	1.26901e20			1.10883			0.554414	+	0.832241i	$0.312943\pi$
$230$		0			0
$231$	1.85072e19			0.150190
$232$		0			0
$233$		−	2.01748e20i		−	1.52154i	−0.649020	−	0.760771i	$-0.724821\pi$
$233$		−	2.01748e20i		−	1.52154i	0.649020	−	0.760771i	$-0.275179\pi$
$234$		0			0
$235$	2.10416e19	+	8.33863e19i	0.147571	+	0.584814i
$236$		0			0
$237$			3.24818e20i			2.11972i
$238$		0			0
$239$	2.59105e20			1.57432			0.787160	−	0.616749i	$-0.211551\pi$
$239$	2.59105e20			1.57432			0.787160	+	0.616749i	$0.211551\pi$
$240$		0			0
$241$	6.65745e19			0.376845			0.188423	−	0.982088i	$-0.439663\pi$
$241$	6.65745e19			0.376845			0.188423	+	0.982088i	$0.439663\pi$
$242$		0			0
$243$			2.26568e20i			1.19549i
$244$		0			0
$245$	−1.90300e20	+	4.80201e19i	−0.936543	+	0.236326i
$246$		0			0
$247$		−	8.68819e19i		−	0.399030i
$248$		0			0
$249$	4.41259e20			1.89234
$250$		0			0
$251$	−5.06363e19			−0.202878			−0.101439	−	0.994842i	$-0.532345\pi$
$251$	−5.06363e19			−0.202878			−0.101439	+	0.994842i	$0.532345\pi$
$252$		0			0
$253$			1.00596e20i			0.376752i
$254$		0			0
$255$	−3.59627e20	+	9.07478e19i	−1.25968	+	0.317866i
$256$		0			0
$257$		−	2.25858e20i		−	0.740293i	−0.928973	−	0.370147i	$-0.879307\pi$
$257$		−	2.25858e20i		−	0.740293i	0.928973	−	0.370147i	$-0.120693\pi$
$258$		0			0
$259$	6.07805e19			0.186516
$260$		0			0
$261$	−1.85771e20			−0.533989
$262$		0			0
$263$			1.80364e20i			0.485877i	0.970042	+	0.242939i	$0.0781114\pi$
$263$			1.80364e20i			0.485877i	−0.970042	+	0.242939i	$0.921889\pi$
$264$		0			0
$265$	−1.91690e20	−	7.59655e20i	−0.484184	−	1.91879i
$266$		0			0
$267$		−	5.21186e20i		−	1.23494i
$268$		0			0
$269$	2.52208e20			0.560874			0.280437	−	0.959872i	$-0.409521\pi$
$269$	2.52208e20			0.560874			0.280437	+	0.959872i	$0.409521\pi$
$270$		0			0
$271$	−8.16795e20			−1.70559			−0.852794	−	0.522247i	$-0.825094\pi$
$271$	−8.16795e20			−1.70559			−0.852794	+	0.522247i	$0.825094\pi$
$272$		0			0
$273$		−	1.05617e20i		−	0.207181i
$274$		0			0
$275$	−2.97140e20	+	1.60158e20i	−0.547815	+	0.295271i
$276$		0			0
$277$			4.58257e20i			0.794385i	0.917735	+	0.397193i	$0.130016\pi$
$277$			4.58257e20i			0.794385i	−0.917735	+	0.397193i	$0.869984\pi$
$278$		0			0
$279$	3.48161e19			0.0567734
$280$		0			0
$281$	−7.34073e19			−0.112651			−0.0563256	−	0.998412i	$-0.517938\pi$
$281$	−7.34073e19			−0.112651			−0.0563256	+	0.998412i	$0.517938\pi$
$282$		0			0
$283$			7.85417e20i			1.13479i	0.823446	+	0.567395i	$0.192049\pi$
$283$			7.85417e20i			1.13479i	−0.823446	+	0.567395i	$0.807951\pi$
$284$		0			0
$285$	−1.09216e20	−	4.32814e20i	−0.148629	−	0.589007i
$286$		0			0
$287$		−	2.61099e20i		−	0.334819i
$288$		0			0
$289$	9.77131e18			0.0118119
$290$		0			0
$291$	−5.92483e20			−0.675438
$292$		0			0
$293$			9.03327e20i			0.971562i	0.874081	+	0.485781i	$0.161465\pi$
$293$			9.03327e20i			0.971562i	−0.874081	+	0.485781i	$0.838535\pi$
$294$		0			0
$295$	−8.66865e20	+	2.18743e20i	−0.879962	+	0.222048i
$296$		0			0
$297$			2.47773e20i			0.237478i
$298$		0			0
$299$	5.74078e20			0.519716
$300$		0			0
$301$	−6.01801e19			−0.0514799
$302$		0			0
$303$		−	8.16480e20i		−	0.660211i
$304$		0			0
$305$	7.54117e20	−	1.90293e20i	0.576620	−	0.145504i
$306$		0			0
$307$		−	1.63564e21i		−	1.18308i	−0.806277	−	0.591538i	$-0.798521\pi$
$307$		−	1.63564e21i		−	1.18308i	0.806277	−	0.591538i	$-0.201479\pi$
$308$		0			0
$309$	2.50276e21			1.71306
$310$		0			0
$311$	1.89136e21			1.22549			0.612746	−	0.790280i	$-0.290065\pi$
$311$	1.89136e21			1.22549			0.612746	+	0.790280i	$0.290065\pi$
$312$		0			0
$313$		−	6.94294e20i		−	0.426007i	−0.977051	−	0.213003i	$-0.931675\pi$
$313$		−	6.94294e20i		−	0.426007i	0.977051	−	0.213003i	$-0.0683246\pi$
$314$		0			0
$315$	−5.50350e19	−	2.18100e20i	−0.0319889	−	0.126770i
$316$		0			0
$317$			3.30557e21i			1.82072i	0.413817	+	0.910360i	$0.364195\pi$
$317$			3.30557e21i			1.82072i	−0.413817	+	0.910360i	$0.635805\pi$
$318$		0			0
$319$	−8.98934e20			−0.469361
$320$		0			0
$321$	3.58525e21			1.77511
$322$		0			0
$323$		−	9.83831e20i		−	0.462061i
$324$		0			0
$325$	9.13986e20	+	1.69571e21i	0.407315	+	0.755690i
$326$		0			0
$327$		−	1.67236e21i		−	0.707415i
$328$		0			0
$329$	−2.77310e20			−0.111378
$330$		0			0
$331$	−3.89557e21			−1.48605			−0.743024	−	0.669265i	$-0.766609\pi$
$331$	−3.89557e21			−1.48605			−0.743024	+	0.669265i	$0.766609\pi$
$332$		0			0
$333$		−	1.97313e21i		−	0.715121i
$334$		0			0
$335$	1.09320e21	+	4.33228e21i	0.376547	+	1.49223i
$336$		0			0
$337$			3.92338e21i			1.28471i	0.766406	+	0.642357i	$0.222043\pi$
$337$			3.92338e21i			1.28471i	−0.766406	+	0.642357i	$0.777957\pi$
$338$		0			0
$339$	−3.02402e21			−0.941646
$340$		0			0
$341$	1.68473e20			0.0499021
$342$		0			0
$343$		−	1.28807e21i		−	0.363028i
$344$		0			0
$345$	2.85985e21	−	7.21651e20i	0.767152	−	0.193582i
$346$		0			0
$347$		−	1.38286e21i		−	0.353165i	−0.984286	−	0.176583i	$-0.943496\pi$
$347$		−	1.38286e21i		−	0.353165i	0.984286	−	0.176583i	$-0.0565043\pi$
$348$		0			0
$349$	−4.46619e21			−1.08623			−0.543114	−	0.839659i	$-0.682755\pi$
$349$	−4.46619e21			−1.08623			−0.543114	+	0.839659i	$0.682755\pi$
$350$		0			0
$351$	1.41398e21			0.327593
$352$		0			0
$353$		−	4.75052e21i		−	1.04871i	−0.851499	−	0.524356i	$-0.824306\pi$
$353$		−	4.75052e21i		−	1.04871i	0.851499	−	0.524356i	$-0.175694\pi$
$354$		0			0
$355$	1.57965e19	−	3.98606e18i	0.00332368	−	0.000838693i
$356$		0			0
$357$		−	1.19598e21i		−	0.239907i
$358$		0			0
$359$	−4.17436e21			−0.798523			−0.399261	−	0.916837i	$-0.630733\pi$
$359$	−4.17436e21			−0.798523			−0.399261	+	0.916837i	$0.630733\pi$
$360$		0			0
$361$	−4.29634e21			−0.783948
$362$		0			0
$363$			4.59949e21i			0.800762i
$364$		0			0
$365$	−2.59630e21	−	1.02889e22i	−0.431385	−	1.70955i
$366$		0			0
$367$		−	5.37278e21i		−	0.852192i	−0.904678	−	0.426096i	$-0.859888\pi$
$367$		−	5.37278e21i		−	0.852192i	0.904678	−	0.426096i	$-0.140112\pi$
$368$		0			0
$369$	−8.47612e21			−1.28373
$370$		0			0
$371$	2.52632e21			0.365434
$372$		0			0
$373$		−	3.85250e21i		−	0.532375i	−0.963921	−	0.266187i	$-0.914236\pi$
$373$		−	3.85250e21i		−	0.532375i	0.963921	−	0.266187i	$-0.0857640\pi$
$374$		0			0
$375$	6.68476e21	+	7.29850e21i	0.882714	+	0.963758i
$376$		0			0
$377$			5.13001e21i			0.647466i
$378$		0			0
$379$	−1.47145e22			−1.77547			−0.887735	−	0.460354i	$-0.847723\pi$
$379$	−1.47145e22			−1.77547			−0.887735	+	0.460354i	$0.847723\pi$
$380$		0			0
$381$	6.52875e21			0.753300
$382$		0			0
$383$			1.14710e22i			1.26594i	0.774178	+	0.632968i	$0.218164\pi$
$383$			1.14710e22i			1.26594i	−0.774178	+	0.632968i	$0.781836\pi$
$384$		0			0
$385$	−2.66311e20	−	1.05537e21i	−0.0281173	−	0.111427i
$386$		0			0
$387$			1.95364e21i			0.197379i
$388$		0			0
$389$	9.20323e20			0.0889957			0.0444978	−	0.999009i	$-0.485831\pi$
$389$	9.20323e20			0.0889957			0.0444978	+	0.999009i	$0.485831\pi$
$390$		0			0
$391$	6.50073e21			0.601811
$392$		0			0
$393$			1.89789e22i			1.68242i
$394$		0			0
$395$	1.85227e22	−	4.67399e21i	1.57264	−	0.396838i
$396$		0			0
$397$		−	8.71764e21i		−	0.709055i	−0.935046	−	0.354527i	$-0.884642\pi$
$397$		−	8.71764e21i		−	0.709055i	0.935046	−	0.354527i	$-0.115358\pi$
$398$		0			0
$399$	1.43937e21			0.112177
$400$		0			0
$401$	7.08659e21			0.529310			0.264655	−	0.964343i	$-0.414742\pi$
$401$	7.08659e21			0.529310			0.264655	+	0.964343i	$0.414742\pi$
$402$		0			0
$403$		−	9.61439e20i		−	0.0688381i
$404$		0			0
$405$	1.70441e22	−	4.30088e21i	1.17005	−	0.295250i
$406$		0			0
$407$		−	9.54787e21i		−	0.628570i
$408$		0			0
$409$	−2.23529e22			−1.41151			−0.705757	−	0.708454i	$-0.749393\pi$
$409$	−2.23529e22			−1.41151			−0.705757	+	0.708454i	$0.749393\pi$
$410$		0			0
$411$	1.88834e22			1.14400
$412$		0			0
$413$		−	2.88285e21i		−	0.167590i
$414$		0			0
$415$	−6.34952e21	−	2.51627e22i	−0.354268	−	1.40394i
$416$		0			0
$417$			6.91374e21i			0.370302i
$418$		0			0
$419$	3.04783e22			1.56737			0.783686	−	0.621157i	$-0.213337\pi$
$419$	3.04783e22			1.56737			0.783686	+	0.621157i	$0.213337\pi$
$420$		0			0
$421$	−1.97241e22			−0.974090			−0.487045	−	0.873377i	$-0.661925\pi$
$421$	−1.97241e22			−0.974090			−0.487045	+	0.873377i	$0.661925\pi$
$422$		0			0
$423$			9.00238e21i			0.427036i
$424$		0			0
$425$	1.03498e22	+	1.92019e22i	0.471655	+	0.875060i
$426$		0			0
$427$			2.50790e21i			0.109818i
$428$		0			0
$429$	−1.65911e22			−0.698212
$430$		0			0
$431$	3.38616e22			1.36978			0.684889	−	0.728648i	$-0.259851\pi$
$431$	3.38616e22			1.36978			0.684889	+	0.728648i	$0.259851\pi$
$432$		0			0
$433$			1.91463e21i			0.0744623i	0.999307	+	0.0372312i	$0.0118538\pi$
$433$			1.91463e21i			0.0744623i	−0.999307	+	0.0372312i	$0.988146\pi$
$434$		0			0
$435$	6.44874e21	+	2.55559e22i	0.241166	+	0.955723i
$436$		0			0
$437$			7.82369e21i			0.281397i
$438$		0			0
$439$	−3.41225e22			−1.18057			−0.590286	−	0.807194i	$-0.700985\pi$
$439$	−3.41225e22			−1.18057			−0.590286	+	0.807194i	$0.700985\pi$
$440$		0			0
$441$	−2.05448e22			−0.683870
$442$		0			0
$443$		−	2.42177e22i		−	0.775713i	−0.921720	−	0.387856i	$-0.873216\pi$
$443$		−	2.42177e22i		−	0.775713i	0.921720	−	0.387856i	$-0.126784\pi$
$444$		0			0
$445$	−2.97206e22	+	7.49965e21i	−0.916214	+	0.231196i
$446$		0			0
$447$			2.47341e22i			0.733976i
$448$		0			0
$449$	−3.63234e22			−1.03775			−0.518875	−	0.854850i	$-0.673649\pi$
$449$	−3.63234e22			−1.03775			−0.518875	+	0.854850i	$0.673649\pi$
$450$		0			0
$451$	−4.10155e22			−1.12836
$452$		0			0
$453$		−	5.56780e22i		−	1.47519i
$454$		0			0
$455$	−6.02278e21	+	1.51978e21i	−0.153709	+	0.0387868i
$456$		0			0
$457$		−	2.30628e22i		−	0.567054i	−0.958964	−	0.283527i	$-0.908495\pi$
$457$		−	2.30628e22i		−	0.567054i	0.958964	−	0.283527i	$-0.0915046\pi$
$458$		0			0
$459$	1.60116e22			0.379339
$460$		0			0
$461$	2.76356e21			0.0630974			0.0315487	−	0.999502i	$-0.489956\pi$
$461$	2.76356e21			0.0630974			0.0315487	+	0.999502i	$0.489956\pi$
$462$		0			0
$463$		−	5.84880e22i		−	1.28715i	−0.765384	−	0.643574i	$-0.777451\pi$
$463$		−	5.84880e22i		−	1.28715i	0.765384	−	0.643574i	$-0.222549\pi$
$464$		0			0
$465$	−1.20859e21	−	4.78955e21i	−0.0256406	−	0.101612i
$466$		0			0
$467$			3.20324e22i			0.655233i	0.944811	+	0.327617i	$0.106245\pi$
$467$			3.20324e22i			0.655233i	−0.944811	+	0.327617i	$0.893755\pi$
$468$		0			0
$469$	−1.44075e22			−0.284197
$470$		0			0
$471$	1.63898e22			0.311814
$472$		0			0
$473$			9.45355e21i			0.173490i
$474$		0			0
$475$	−2.31096e22	+	1.24560e22i	−0.409164	+	0.220538i
$476$		0			0
$477$		−	8.20123e22i		−	1.40111i
$478$		0			0
$479$	5.36584e22			0.884679			0.442340	−	0.896848i	$-0.354149\pi$
$479$	5.36584e22			0.884679			0.442340	+	0.896848i	$0.354149\pi$
$480$		0			0
$481$	−5.44875e22			−0.867089
$482$		0			0
$483$			9.51075e21i			0.146105i
$484$		0			0
$485$	8.52557e21	+	3.37862e22i	0.126450	+	0.501112i
$486$		0			0
$487$			9.26954e22i			1.32758i	0.747917	+	0.663792i	$0.231054\pi$
$487$			9.26954e22i			1.32758i	−0.747917	+	0.663792i	$0.768946\pi$
$488$		0			0
$489$	1.73113e23			2.39444
$490$		0			0
$491$	−7.67444e22			−1.02531			−0.512653	−	0.858596i	$-0.671337\pi$
$491$	−7.67444e22			−1.02531			−0.512653	+	0.858596i	$0.671337\pi$
$492$		0			0
$493$			5.80911e22i			0.749740i
$494$		0			0
$495$	−3.42608e22	+	8.64532e21i	−0.427222	+	0.107804i
$496$		0			0
$497$			5.25329e19i			0.000632997i
$498$		0			0
$499$	−3.02542e22			−0.352315			−0.176158	−	0.984362i	$-0.556367\pi$
$499$	−3.02542e22			−0.352315			−0.176158	+	0.984362i	$0.556367\pi$
$500$		0			0
$501$	1.07763e23			1.21297
$502$		0			0
$503$			6.06036e22i			0.659432i	0.944080	+	0.329716i	$0.106953\pi$
$503$			6.06036e22i			0.659432i	−0.944080	+	0.329716i	$0.893047\pi$
$504$		0			0
$505$	−4.65596e22	+	1.17488e22i	−0.489816	+	0.123599i
$506$		0			0
$507$		−	3.37919e22i		−	0.343752i
$508$		0			0
$509$	1.47131e23			1.44745			0.723723	−	0.690091i	$-0.242429\pi$
$509$	1.47131e23			1.44745			0.723723	+	0.690091i	$0.242429\pi$
$510$		0			0
$511$	3.42170e22			0.325585
$512$		0			0
$513$			1.92701e22i			0.177373i
$514$		0			0
$515$	−3.60137e22	−	1.42720e23i	−0.320705	−	1.27093i
$516$		0			0
$517$			4.35620e22i			0.375352i
$518$		0			0
$519$	1.76062e23			1.46806
$520$		0			0
$521$	1.22760e23			0.990689			0.495344	−	0.868697i	$-0.335042\pi$
$521$	1.22760e23			0.990689			0.495344	+	0.868697i	$0.335042\pi$
$522$		0			0
$523$		−	6.60040e22i		−	0.515591i	−0.966199	−	0.257796i	$-0.917004\pi$
$523$		−	6.60040e22i		−	0.515591i	0.966199	−	0.257796i	$-0.0829961\pi$
$524$		0			0
$525$	−2.80929e22	+	1.51420e22i	−0.212443	+	0.114506i
$526$		0			0
$527$		−	1.08871e22i		−	0.0797118i
$528$		0			0
$529$	8.93545e22			0.633495
$530$		0			0
$531$	−9.35866e22			−0.642555
$532$		0			0
$533$			2.34066e23i			1.55653i
$534$		0			0
$535$	−5.15902e22	−	2.04448e23i	−0.332322	−	1.31697i
$536$		0			0
$537$		−	3.07063e23i		−	1.91622i
$538$		0			0
$539$	−9.94151e22			−0.601102
$540$		0			0
$541$	−3.00499e23			−1.76062			−0.880311	−	0.474397i	$-0.842666\pi$
$541$	−3.00499e23			−1.76062			−0.880311	+	0.474397i	$0.842666\pi$
$542$		0			0
$543$			1.77051e23i			1.00531i
$544$		0			0
$545$	−9.53662e22	+	2.40646e22i	−0.524837	+	0.132437i
$546$		0			0
$547$			4.51927e22i			0.241088i	0.992708	+	0.120544i	$0.0384639\pi$
$547$			4.51927e22i			0.241088i	−0.992708	+	0.120544i	$0.961536\pi$
$548$		0			0
$549$	8.14143e22			0.421052
$550$		0			0
$551$	−6.99131e22			−0.350566
$552$		0			0
$553$			6.15993e22i			0.299510i
$554$		0			0
$555$	−2.71437e23	+	6.84941e22i	−1.27991	+	0.322970i
$556$		0			0
$557$			2.27372e23i			1.03984i	0.854214	+	0.519921i	$0.174039\pi$
$557$			2.27372e23i			1.03984i	−0.854214	+	0.519921i	$0.825961\pi$
$558$		0			0
$559$	5.39492e22			0.239323
$560$		0			0
$561$	−1.87874e23			−0.808503
$562$		0			0
$563$		−	3.02864e23i		−	1.26452i	−0.774755	−	0.632261i	$-0.782127\pi$
$563$		−	3.02864e23i		−	1.26452i	0.774755	−	0.632261i	$-0.217873\pi$
$564$		0			0
$565$	4.35144e22	+	1.72445e23i	0.176287	+	0.698615i
$566$		0			0
$567$			5.66819e22i			0.222838i
$568$		0			0
$569$	2.67640e23			1.02117			0.510583	−	0.859828i	$-0.329430\pi$
$569$	2.67640e23			1.02117			0.510583	+	0.859828i	$0.329430\pi$
$570$		0			0
$571$	−1.94644e23			−0.720832			−0.360416	−	0.932792i	$-0.617365\pi$
$571$	−1.94644e23			−0.720832			−0.360416	+	0.932792i	$0.617365\pi$
$572$		0			0
$573$		−	2.73893e22i		−	0.0984615i
$574$		0			0
$575$	−8.23041e22	−	1.52698e23i	−0.287240	−	0.532915i
$576$		0			0
$577$		−	2.28616e23i		−	0.774660i	−0.921941	−	0.387330i	$-0.873397\pi$
$577$		−	2.28616e23i		−	0.774660i	0.921941	−	0.387330i	$-0.126603\pi$
$578$		0			0
$579$	−3.18091e23			−1.04661
$580$		0			0
$581$	8.36813e22			0.267381
$582$		0			0
$583$		−	3.96853e23i		−	1.23154i
$584$		0			0
$585$	4.93369e22	+	1.95519e23i	0.148712	+	0.589336i
$586$		0			0
$587$			1.19604e23i			0.350204i	0.984550	+	0.175102i	$0.0560256\pi$
$587$			1.19604e23i			0.350204i	−0.984550	+	0.175102i	$0.943974\pi$
$588$		0			0
$589$	1.31027e22			0.0372720
$590$		0			0
$591$	1.51187e23			0.417851
$592$		0			0
$593$		−	5.15740e23i		−	1.38505i	−0.721393	−	0.692526i	$-0.756498\pi$
$593$		−	5.15740e23i		−	1.38505i	0.721393	−	0.692526i	$-0.243502\pi$
$594$		0			0
$595$	−6.82005e22	+	1.72096e22i	−0.177989	+	0.0449135i
$596$		0			0
$597$		−	2.27770e23i		−	0.577717i
$598$		0			0
$599$	1.13746e23			0.280420			0.140210	−	0.990122i	$-0.455222\pi$
$599$	1.13746e23			0.280420			0.140210	+	0.990122i	$0.455222\pi$
$600$		0			0
$601$	5.02442e23			1.20407			0.602037	−	0.798468i	$-0.294356\pi$
$601$	5.02442e23			1.20407			0.602037	+	0.798468i	$0.294356\pi$
$602$		0			0
$603$			4.67713e23i			1.08964i
$604$		0			0
$605$	2.62285e23	−	6.61847e22i	0.594091	−	0.149912i
$606$		0			0
$607$			4.26808e23i			0.940001i	0.882666	+	0.470001i	$0.155746\pi$
$607$			4.26808e23i			0.940001i	−0.882666	+	0.470001i	$0.844254\pi$
$608$		0			0
$609$	−8.49888e22			−0.182018
$610$		0			0
$611$	2.48599e23			0.517784
$612$		0			0
$613$		−	2.42466e22i		−	0.0491175i	−0.999698	−	0.0245587i	$-0.992182\pi$
$613$		−	2.42466e22i		−	0.0491175i	0.999698	−	0.0245587i	$-0.00781808\pi$
$614$		0			0
$615$	2.94235e23	+	1.16603e24i	0.579771	+	2.29759i
$616$		0			0
$617$			6.47875e23i			1.24184i	0.783872	+	0.620922i	$0.213242\pi$
$617$			6.47875e23i			1.24184i	−0.783872	+	0.620922i	$0.786758\pi$
$618$		0			0
$619$	4.74426e23			0.884705			0.442352	−	0.896841i	$-0.354144\pi$
$619$	4.74426e23			0.884705			0.442352	+	0.896841i	$0.354144\pi$
$620$		0			0
$621$	−1.27329e23			−0.231019
$622$		0			0
$623$		−	9.88390e22i		−	0.174494i
$624$		0			0
$625$	3.20005e23	−	4.86220e23i	0.549765	−	0.835320i
$626$		0			0
$627$		−	2.26107e23i		−	0.378043i
$628$		0			0
$629$	−6.17004e23			−1.00406
$630$		0			0
$631$	1.06676e24			1.68973			0.844863	−	0.534982i	$-0.179682\pi$
$631$	1.06676e24			1.68973			0.844863	+	0.534982i	$0.179682\pi$
$632$		0			0
$633$		−	2.42473e23i		−	0.373879i
$634$		0			0
$635$	−9.39459e22	−	3.72301e23i	−0.141027	−	0.558879i
$636$		0			0
$637$			5.67340e23i			0.829198i
$638$		0			0
$639$	1.70539e21			0.00242697
$640$		0			0
$641$	−1.11178e24			−1.54072			−0.770361	−	0.637608i	$-0.779924\pi$
$641$	−1.11178e24			−1.54072			−0.770361	+	0.637608i	$0.779924\pi$
$642$		0			0
$643$		−	4.81844e21i		−	0.00650300i	−0.999995	−	0.00325150i	$-0.998965\pi$
$643$		−	4.81844e21i		−	0.00650300i	0.999995	−	0.00325150i	$-0.00103499\pi$
$644$		0			0
$645$	2.68756e23	−	6.78175e22i	0.353265	−	0.0891423i
$646$		0			0
$647$		−	7.81396e23i		−	1.00043i	−0.865902	−	0.500213i	$-0.833255\pi$
$647$		−	7.81396e23i		−	1.00043i	0.865902	−	0.500213i	$-0.166745\pi$
$648$		0			0
$649$	−4.52861e23			−0.564787
$650$		0			0
$651$	1.59281e22			0.0193520
$652$		0			0
$653$		−	5.73789e23i		−	0.679188i	−0.940572	−	0.339594i	$-0.889710\pi$
$653$		−	5.73789e23i		−	0.679188i	0.940572	−	0.339594i	$-0.110290\pi$
$654$		0			0
$655$	1.08227e24	−	2.73098e23i	1.24820	−	0.314969i
$656$		0			0
$657$		−	1.11079e24i		−	1.24832i
$658$		0			0
$659$	−1.34401e24			−1.47189			−0.735945	−	0.677041i	$-0.763262\pi$
$659$	−1.34401e24			−1.47189			−0.735945	+	0.677041i	$0.763262\pi$
$660$		0			0
$661$	2.97094e23			0.317089			0.158545	−	0.987352i	$-0.449320\pi$
$661$	2.97094e23			0.317089			0.158545	+	0.987352i	$0.449320\pi$
$662$		0			0
$663$			1.07215e24i			1.11530i
$664$		0			0
$665$	−2.07119e22	−	8.20799e22i	−0.0210008	−	0.0832249i
$666$		0			0
$667$		−	4.61956e23i		−	0.456595i
$668$		0			0
$669$	−1.89869e24			−1.82950
$670$		0			0
$671$	3.93960e23			0.370093
$672$		0			0
$673$			1.99614e24i			1.82836i	0.405308	+	0.914180i	$0.367164\pi$
$673$			1.99614e24i			1.82836i	−0.405308	+	0.914180i	$0.632836\pi$
$674$		0			0
$675$	−2.02719e23	−	3.76104e23i	−0.181056	−	0.335912i
$676$		0			0
$677$			9.84976e23i			0.857871i	0.903335	+	0.428935i	$0.141111\pi$
$677$			9.84976e23i			0.857871i	−0.903335	+	0.428935i	$0.858889\pi$
$678$		0			0
$679$	−1.12360e23			−0.0954372
$680$		0			0
$681$	−8.26165e23			−0.684410
$682$		0			0
$683$			8.79534e23i			0.710684i	0.934736	+	0.355342i	$0.115636\pi$
$683$			8.79534e23i			0.710684i	−0.934736	+	0.355342i	$0.884364\pi$
$684$		0			0
$685$	−2.71724e23	−	1.07682e24i	−0.214170	−	0.848741i
$686$		0			0
$687$		−	1.88470e24i		−	1.44914i
$688$		0			0
$689$	−2.26475e24			−1.69886
$690$		0			0
$691$	1.10966e24			0.812132			0.406066	−	0.913844i	$-0.366900\pi$
$691$	1.10966e24			0.812132			0.406066	+	0.913844i	$0.366900\pi$
$692$		0			0
$693$		−	1.13938e23i		−	0.0813647i
$694$		0			0
$695$	3.94255e23	−	9.94857e22i	0.274730	−	0.0693250i
$696$		0			0
$697$			2.65051e24i			1.80240i
$698$		0			0
$699$	−2.99630e24			−1.98852
$700$		0			0
$701$	−1.06140e24			−0.687502			−0.343751	−	0.939061i	$-0.611698\pi$
$701$	−1.06140e24			−0.687502			−0.343751	+	0.939061i	$0.611698\pi$
$702$		0			0
$703$		−	7.42570e23i		−	0.469480i
$704$		0			0
$705$	1.23843e24	−	3.12503e23i	0.764300	−	0.192862i
$706$		0			0
$707$		−	1.54839e23i		−	0.0932858i
$708$		0			0
$709$	2.92308e24			1.71929			0.859643	−	0.510895i	$-0.170686\pi$
$709$	2.92308e24			1.71929			0.859643	+	0.510895i	$0.170686\pi$
$710$		0			0
$711$	1.99971e24			1.14835
$712$		0			0
$713$			8.65773e22i			0.0485448i
$714$		0			0
$715$	2.38739e23	+	9.46104e23i	0.130714	+	0.518009i
$716$		0			0
$717$		−	3.84814e24i		−	2.05749i
$718$		0			0
$719$	2.49947e24			1.30513			0.652564	−	0.757734i	$-0.273693\pi$
$719$	2.49947e24			1.30513			0.652564	+	0.757734i	$0.273693\pi$
$720$		0			0
$721$	4.74629e23			0.242050
$722$		0			0
$723$		−	9.88745e23i		−	0.492503i
$724$		0			0
$725$	1.36453e24	−	7.35477e23i	0.663909	−	0.357846i
$726$		0			0
$727$			2.86353e24i			1.36100i	0.732747	+	0.680501i	$0.238238\pi$
$727$			2.86353e24i			1.36100i	−0.732747	+	0.680501i	$0.761762\pi$
$728$		0			0
$729$	7.65993e23			0.355665
$730$		0			0
$731$	6.10909e23			0.277127
$732$		0			0
$733$		−	2.89953e24i		−	1.28512i	−0.766236	−	0.642560i	$-0.777872\pi$
$733$		−	2.89953e24i		−	1.28512i	0.766236	−	0.642560i	$-0.222128\pi$
$734$		0			0
$735$	7.13180e23	+	2.82628e24i	0.308857	+	1.22398i
$736$		0			0
$737$			2.26324e24i			0.957759i
$738$		0			0
$739$	−2.59418e24			−1.07281			−0.536404	−	0.843961i	$-0.680218\pi$
$739$	−2.59418e24			−1.07281			−0.536404	+	0.843961i	$0.680218\pi$
$740$		0			0
$741$	−1.29034e24			−0.521496
$742$		0			0
$743$			1.98231e24i			0.783010i	0.920176	+	0.391505i	$0.128045\pi$
$743$			1.98231e24i			0.783010i	−0.920176	+	0.391505i	$0.871955\pi$
$744$		0			0
$745$	1.41046e24	−	3.55913e23i	0.544543	−	0.137409i
$746$		0			0
$747$		−	2.71656e24i		−	1.02517i
$748$		0			0
$749$	6.79914e23			0.250818
$750$		0			0
$751$	4.22901e24			1.52510			0.762551	−	0.646928i	$-0.223946\pi$
$751$	4.22901e24			1.52510			0.762551	+	0.646928i	$0.223946\pi$
$752$		0			0
$753$			7.52035e23i			0.265143i
$754$		0			0
$755$	−3.17503e24	+	8.01183e23i	−1.09446	+	0.276174i
$756$		0			0
$757$			4.07711e24i			1.37416i	0.726582	+	0.687080i	$0.241108\pi$
$757$			4.07711e24i			1.37416i	−0.726582	+	0.687080i	$0.758892\pi$
$758$		0			0
$759$	1.49402e24			0.492381
$760$		0			0
$761$	−8.57620e23			−0.276392			−0.138196	−	0.990405i	$-0.544130\pi$
$761$	−8.57620e23			−0.276392			−0.138196	+	0.990405i	$0.544130\pi$
$762$		0			0
$763$		−	3.17150e23i		−	0.0999555i
$764$		0			0
$765$	5.58680e23	+	2.21401e24i	0.172203	+	0.682428i
$766$		0			0
$767$			2.58437e24i			0.779102i
$768$		0			0
$769$	2.33285e24			0.687881			0.343941	−	0.938991i	$-0.388238\pi$
$769$	2.33285e24			0.687881			0.343941	+	0.938991i	$0.388238\pi$
$770$		0			0
$771$	−3.35438e24			−0.967497
$772$		0			0
$773$		−	6.45600e24i		−	1.82154i	−0.412919	−	0.910768i	$-0.635491\pi$
$773$		−	6.45600e24i		−	1.82154i	0.412919	−	0.910768i	$-0.364509\pi$
$774$		0			0
$775$	−2.55732e23	+	1.37839e23i	−0.0705864	+	0.0380459i
$776$		0			0
$777$		−	9.02694e23i		−	0.243760i
$778$		0			0
$779$	−3.18991e24			−0.842773
$780$		0			0
$781$	8.25227e21			0.00213324
$782$		0			0
$783$		−	1.13782e24i		−	0.287806i
$784$		0			0
$785$	−2.35842e23	−	9.34624e23i	−0.0583752	−	0.231337i
$786$		0			0
$787$		−	5.78611e24i		−	1.40153i	−0.713393	−	0.700764i	$-0.752843\pi$
$787$		−	5.78611e24i		−	1.40153i	0.713393	−	0.700764i	$-0.247157\pi$
$788$		0			0
$789$	2.67872e24			0.634998
$790$		0			0
$791$	−5.73483e23			−0.133052
$792$		0			0
$793$		−	2.24824e24i		−	0.510529i
$794$		0			0
$795$	−1.12822e25	+	2.84693e24i	−2.50768	+	0.632785i
$796$		0			0
$797$			8.42631e23i			0.183333i	0.995790	+	0.0916667i	$0.0292194\pi$
$797$			8.42631e23i			0.183333i	−0.995790	+	0.0916667i	$0.970781\pi$
$798$		0			0
$799$	2.81507e24			0.599573
$800$		0			0
$801$	−3.20863e24			−0.669026
$802$		0			0
$803$		−	5.37507e24i		−	1.09724i
$804$		0			0
$805$	5.42349e23	−	1.36856e23i	0.108396	−	0.0273525i
$806$		0			0
$807$		−	3.74573e24i		−	0.733012i
$808$		0			0
$809$	1.58012e24			0.302780			0.151390	−	0.988474i	$-0.451625\pi$
$809$	1.58012e24			0.302780			0.151390	+	0.988474i	$0.451625\pi$
$810$		0			0
$811$	−9.76054e24			−1.83146			−0.915728	−	0.401799i	$-0.868385\pi$
$811$	−9.76054e24			−1.83146			−0.915728	+	0.401799i	$0.868385\pi$
$812$		0			0
$813$			1.21308e25i			2.22905i
$814$		0			0
$815$	−2.49102e24	−	9.87172e24i	−0.448267	−	1.77645i
$816$		0			0
$817$			7.35234e23i			0.129580i
$818$		0			0
$819$	−6.50218e23			−0.112240
$820$		0			0
$821$	−4.02137e24			−0.679918			−0.339959	−	0.940440i	$-0.610413\pi$
$821$	−4.02137e24			−0.679918			−0.339959	+	0.940440i	$0.610413\pi$
$822$		0			0
$823$			6.02144e24i			0.997246i	0.866819	+	0.498623i	$0.166161\pi$
$823$			6.02144e24i			0.997246i	−0.866819	+	0.498623i	$0.833839\pi$
$824$		0			0
$825$	2.37862e24	+	4.41304e24i	0.385893	+	0.715945i
$826$		0			0
$827$			6.58679e24i			1.04683i	0.852078	+	0.523416i	$0.175342\pi$
$827$			6.58679e24i			1.04683i	−0.852078	+	0.523416i	$0.824658\pi$
$828$		0			0
$829$	7.40221e24			1.15252			0.576259	−	0.817267i	$-0.304512\pi$
$829$	7.40221e24			1.15252			0.576259	+	0.817267i	$0.304512\pi$
$830$		0			0
$831$	6.80590e24			1.03819
$832$		0			0
$833$			6.42443e24i			0.960178i
$834$		0			0
$835$	−1.55067e24	−	6.14518e24i	−0.227082	−	0.899911i
$836$		0			0
$837$			2.13244e23i			0.0305993i
$838$		0			0
$839$	4.72687e24			0.664656			0.332328	−	0.943164i	$-0.392166\pi$
$839$	4.72687e24			0.664656			0.332328	+	0.943164i	$0.392166\pi$
$840$		0			0
$841$	−3.12907e24			−0.431171
$842$		0			0
$843$			1.09022e24i			0.147225i
$844$		0			0
$845$	−1.92698e24	+	4.86251e23i	−0.255032	+	0.0643545i
$846$		0			0
$847$			8.72258e23i			0.113145i
$848$		0			0
$849$	1.16648e25			1.48307
$850$		0			0
$851$	4.90658e24			0.611474
$852$		0			0
$853$			1.37911e24i			0.168474i	0.996446	+	0.0842369i	$0.0268452\pi$
$853$			1.37911e24i			0.168474i	−0.996446	+	0.0842369i	$0.973155\pi$
$854$		0			0
$855$	−2.66458e24	+	6.72376e23i	−0.319092	+	0.0805193i
$856$		0			0
$857$		−	1.45049e25i		−	1.70285i	−0.524474	−	0.851427i	$-0.675738\pi$
$857$		−	1.45049e25i		−	1.70285i	0.524474	−	0.851427i	$-0.324262\pi$
$858$		0			0
$859$	−1.67188e24			−0.192426			−0.0962128	−	0.995361i	$-0.530673\pi$
$859$	−1.67188e24			−0.192426			−0.0962128	+	0.995361i	$0.530673\pi$
$860$		0			0
$861$	−3.87777e24			−0.437578
$862$		0			0
$863$		−	1.17508e24i		−	0.130010i	−0.997885	−	0.0650048i	$-0.979294\pi$
$863$		−	1.17508e24i		−	0.130010i	0.997885	−	0.0650048i	$-0.0207063\pi$
$864$		0			0
$865$	−2.53346e24	−	1.00399e25i	−0.274838	−	1.08916i
$866$		0			0
$867$		−	1.45121e23i		−	0.0154371i
$868$		0			0
$869$	9.67649e24			1.00937
$870$		0			0
$871$	1.29158e25			1.32119
$872$		0			0
$873$			3.64756e24i			0.365916i
$874$		0			0
$875$	1.26771e24	+	1.38410e24i	0.124725	+	0.136176i
$876$		0			0
$877$		−	6.28619e24i		−	0.606584i	−0.952898	−	0.303292i	$-0.901914\pi$
$877$		−	6.28619e24i		−	0.606584i	0.952898	−	0.303292i	$-0.0980857\pi$
$878$		0			0
$879$	1.34160e25			1.26974
$880$		0			0
$881$	2.08197e24			0.193276			0.0966382	−	0.995320i	$-0.469191\pi$
$881$	2.08197e24			0.193276			0.0966382	+	0.995320i	$0.469191\pi$
$882$		0			0
$883$		−	2.99854e24i		−	0.273051i	−0.990637	−	0.136525i	$-0.956406\pi$
$883$		−	2.99854e24i		−	0.273051i	0.990637	−	0.136525i	$-0.0435935\pi$
$884$		0			0
$885$	3.24871e24	+	1.28744e25i	0.290197	+	1.15003i
$886$		0			0
$887$			1.75523e25i			1.53809i	0.639193	+	0.769046i	$0.279269\pi$
$887$			1.75523e25i			1.53809i	−0.639193	+	0.769046i	$0.720731\pi$
$888$		0			0
$889$	1.23813e24			0.106439
$890$		0			0
$891$	8.90403e24			0.750976
$892$		0			0
$893$			3.38797e24i			0.280351i
$894$		0			0
$895$	−1.75102e25	+	4.41850e24i	−1.42166	+	0.358739i
$896$		0			0
$897$		−	8.52604e24i		−	0.679222i
$898$		0			0
$899$	−7.73662e23			−0.0604775
$900$		0			0
$901$	−2.56455e25			−1.96721
$902$		0			0
$903$			8.93777e23i			0.0672795i
$904$		0			0
$905$	1.00963e25	−	2.54769e24i	0.745846	−	0.188206i
$906$		0			0
$907$			2.30085e25i			1.66812i	0.551676	+	0.834059i	$0.313989\pi$
$907$			2.30085e25i			1.66812i	−0.551676	+	0.834059i	$0.686011\pi$
$908$		0			0
$909$	−5.02657e24			−0.357667
$910$		0			0
$911$	9.13278e24			0.637818			0.318909	−	0.947785i	$-0.396684\pi$
$911$	9.13278e24			0.637818			0.318909	+	0.947785i	$0.396684\pi$
$912$		0			0
$913$		−	1.31453e25i		−	0.901092i
$914$		0			0
$915$	−2.82617e24	−	1.11999e25i	−0.190160	−	0.753591i
$916$		0			0
$917$			3.59920e24i			0.237721i
$918$		0			0
$919$	−2.78711e25			−1.80706			−0.903528	−	0.428529i	$-0.859032\pi$
$919$	−2.78711e25			−1.80706			−0.903528	+	0.428529i	$0.859032\pi$
$920$		0			0
$921$	−2.42921e25			−1.54617
$922$		0			0
$923$		−	4.70938e22i		−	0.00294272i
$924$		0			0
$925$	7.81173e24	+	1.44931e25i	0.479229	+	0.889111i
$926$		0			0
$927$		−	1.54080e25i		−	0.928043i
$928$		0			0
$929$	2.77762e24			0.164263			0.0821313	−	0.996622i	$-0.473827\pi$
$929$	2.77762e24			0.164263			0.0821313	+	0.996622i	$0.473827\pi$
$930$		0			0
$931$	−7.73185e24			−0.448964
$932$		0			0
$933$		−	2.80899e25i		−	1.60161i
$934$		0			0
$935$	2.70342e24	+	1.07135e25i	0.151361	+	0.599834i
$936$		0			0
$937$		−	1.29460e25i		−	0.711787i	−0.934526	−	0.355894i	$-0.884177\pi$
$937$		−	1.29460e25i		−	0.711787i	0.934526	−	0.355894i	$-0.115823\pi$
$938$		0			0
$939$	−1.03115e25			−0.556753
$940$		0			0
$941$	−6.21590e24			−0.329604			−0.164802	−	0.986327i	$-0.552699\pi$
$941$	−6.21590e24			−0.329604			−0.164802	+	0.986327i	$0.552699\pi$
$942$		0			0
$943$		−	2.10776e25i		−	1.09767i
$944$		0			0
$945$	1.33583e24	−	3.37082e23i	0.0683254	−	0.0172411i
$946$		0			0
$947$		−	3.07356e24i		−	0.154407i	−0.997015	−	0.0772035i	$-0.975401\pi$
$947$		−	3.07356e24i		−	0.154407i	0.997015	−	0.0772035i	$-0.0245991\pi$
$948$		0			0
$949$	−3.06743e25			−1.51360
$950$		0			0
$951$	4.90934e25			2.37952
$952$		0			0
$953$			3.87775e24i			0.184625i	0.995730	+	0.0923124i	$0.0294258\pi$
$953$			3.87775e24i			0.184625i	−0.995730	+	0.0923124i	$0.970574\pi$
$954$		0			0
$955$	−1.56187e24	+	3.94120e23i	−0.0730493	+	0.0184332i
$956$		0			0
$957$			1.33507e25i			0.613412i
$958$		0			0
$959$	3.58110e24			0.161643
$960$		0			0
$961$	−2.24051e25			−0.993570
$962$		0			0
$963$		−	2.20722e25i		−	0.961661i
$964$		0			0
$965$	4.57719e24	+	1.81391e25i	0.195937	+	0.776485i
$966$		0			0
$967$			2.05298e25i			0.863496i	0.901994	+	0.431748i	$0.142103\pi$
$967$			2.05298e25i			0.863496i	−0.901994	+	0.431748i	$0.857897\pi$
$968$		0			0
$969$	−1.46116e25			−0.603872
$970$		0			0
$971$	1.50060e25			0.609398			0.304699	−	0.952449i	$-0.401444\pi$
$971$	1.50060e25			0.609398			0.304699	+	0.952449i	$0.401444\pi$
$972$		0			0
$973$			1.31114e24i			0.0523226i
$974$		0			0
$975$	2.51842e25	−	1.35742e25i	0.987619	−	0.532324i
$976$		0			0
$977$			4.52504e24i			0.174389i	0.996191	+	0.0871945i	$0.0277901\pi$
$977$			4.52504e24i			0.174389i	−0.996191	+	0.0871945i	$0.972210\pi$
$978$		0			0
$979$	−1.55264e25			−0.588054
$980$		0			0
$981$	−1.02957e25			−0.383239
$982$		0			0
$983$			2.62506e25i			0.960360i	0.877170	+	0.480180i	$0.159429\pi$
$983$			2.62506e25i			0.960360i	−0.877170	+	0.480180i	$0.840571\pi$
$984$		0			0
$985$	−2.17552e24	−	8.62142e24i	−0.0782266	−	0.310007i
$986$		0			0
$987$			4.11853e24i			0.145561i
$988$		0			0
$989$	−4.85811e24			−0.168772
$990$		0			0
$991$	4.16613e25			1.42268			0.711339	−	0.702849i	$-0.248089\pi$
$991$	4.16613e25			1.42268			0.711339	+	0.702849i	$0.248089\pi$
$992$		0			0
$993$			5.78558e25i			1.94213i
$994$		0			0
$995$	−1.29886e25	+	3.27752e24i	−0.428612	+	0.108155i
$996$		0			0
$997$		−	1.14127e25i		−	0.370236i	−0.982716	−	0.185118i	$-0.940733\pi$
$997$		−	1.14127e25i		−	0.370236i	0.982716	−	0.185118i	$-0.0592668\pi$
$998$		0			0
$999$	1.20852e25			0.385431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.18.c.a.9.2	✓	8
4.3	odd	2		80.18.c.c.49.7		8
5.2	odd	4		100.18.a.f.1.2		8
5.3	odd	4		100.18.a.f.1.7		8
5.4	even	2	inner	20.18.c.a.9.7	yes	8
20.19	odd	2		80.18.c.c.49.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.18.c.a.9.2	✓	8	1.1	even	1	trivial
20.18.c.a.9.7	yes	8	5.4	even	2	inner
80.18.c.c.49.2		8	20.19	odd	2
80.18.c.c.49.7		8	4.3	odd	2
100.18.a.f.1.2		8	5.2	odd	4
100.18.a.f.1.7		8	5.3	odd	4

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:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	20.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-14851.7i q^{3} +(-846917. + 213710. i) q^{5} -2.81651e6i q^{7} -9.14330e7 q^{9} -4.42439e8 q^{11} +2.52490e9i q^{13} +(3.17395e9 + 1.25782e10i) q^{15} +2.85914e10i q^{17} -3.44100e10 q^{19} -4.18300e10 q^{21} -2.27367e11i q^{23} +(6.71596e11 - 3.61989e11i) q^{25} -5.60016e11i q^{27} +2.03177e12 q^{29} -3.80783e11 q^{31} +6.57098e12i q^{33} +(6.01916e11 + 2.38535e12i) q^{35} +2.15801e13i q^{37} +3.74991e13 q^{39} +9.27031e13 q^{41} -2.13669e13i q^{43} +(7.74361e13 - 1.95401e13i) q^{45} -9.84587e13i q^{47} +2.24698e14 q^{49} +4.24631e14 q^{51} +8.96966e14i q^{53} +(3.74709e14 - 9.45536e13i) q^{55} +5.11047e14i q^{57} +1.02355e15 q^{59} -8.90426e14 q^{61} +2.57522e14i q^{63} +(-5.39596e14 - 2.13838e15i) q^{65} -5.11536e15i q^{67} -3.37678e15 q^{69} -1.86518e13 q^{71} +1.21487e16i q^{73} +(-5.37615e15 - 9.97434e15i) q^{75} +1.24614e15i q^{77} -2.18708e16 q^{79} -2.01249e16 q^{81} +2.97110e16i q^{83} +(-6.11026e15 - 2.42145e16i) q^{85} -3.01752e16i q^{87} +3.50927e16 q^{89} +7.11142e15 q^{91} +5.65527e15i q^{93} +(2.91424e16 - 7.35375e15i) q^{95} -3.98932e16i q^{97} +4.04536e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 1276800 q^{5} - 376848248 q^{9} - 396200640 q^{11} + 5677983200 q^{15} + 11821646592 q^{19} - 420670059472 q^{21} + 1642212165000 q^{25} + 1543712861232 q^{29} - 13722543013312 q^{31} - 13325691076800 q^{35}+ \cdots - 23\!\cdots\!60 q^{99}+O(q^{100}) \) 8 * q + 1276800 * q^5 - 376848248 * q^9 - 396200640 * q^11 + 5677983200 * q^15 + 11821646592 * q^19 - 420670059472 * q^21 + 1642212165000 * q^25 + 1543712861232 * q^29 - 13722543013312 * q^31 - 13325691076800 * q^35 - 79298226023808 * q^39 + 29824965017568 * q^41 - 238216983006800 * q^45 - 535615080275736 * q^49 - 764079320419264 * q^51 + 339212427988000 * q^55 + 94539545151744 * q^59 - 3380827226969024 * q^61 + 130929936568800 * q^65 + 5559752175554416 * q^69 + 20155606060098624 * q^71 + 938454087760000 * q^75 - 3634057920126144 * q^79 + 15949137882627512 * q^81 + 3575192471022400 * q^85 + 103487949043655568 * q^89 - 69783870139121088 * q^91 + 67219016711959200 * q^95 - 239420571619653760 * q^99

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