LMFDB - Embedded newform 144.11.e.b.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,11,Mod(17,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("144.17");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	144.e (of order $2$, degree $1$, not 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:	$91.4914443850$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{865})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 427x^{2} + 428x + 47526 $ x^4 - 2x^3 - 427x^2 + 428*x + 47526
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{15} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.2
Root			$-14.2054 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	144.17
Dual form			144.11.e.b.17.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3155.64i q^{5} +13662.3 q^{7} +O(q^{10})$	$q-3155.64i q^{5} +13662.3 q^{7} -62736.3i q^{11} +200379. q^{13} +556674. i q^{17} +3.25649e6 q^{19} -6.15671e6i q^{23} -192446. q^{25} +4.03283e7i q^{29} +4.77843e7 q^{31} -4.31132e7i q^{35} -7.11048e7 q^{37} +1.39960e8i q^{41} -2.17639e7 q^{43} -1.67168e8i q^{47} -9.58181e7 q^{49} -7.91424e8i q^{53} -1.97973e8 q^{55} +2.86465e8i q^{59} +1.01308e9 q^{61} -6.32324e8i q^{65} -9.27728e8 q^{67} +2.49137e9i q^{71} +9.45207e8 q^{73} -8.57119e8i q^{77} +3.35970e9 q^{79} -5.13533e9i q^{83} +1.75666e9 q^{85} +3.50859e9i q^{89} +2.73763e9 q^{91} -1.02763e10i q^{95} -6.06949e9 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 21584 q^{7}+O(q^{10}) $ 4 * q - 21584 * q^7	$ 4 q - 21584 q^{7} - 646912 q^{13} + 1438528 q^{19} - 48796580 q^{25} + 99886192 q^{31} - 243863176 q^{37} + 460449760 q^{43} + 439434108 q^{49} - 1130824800 q^{55} + 2750573000 q^{61} - 2190062816 q^{67} + 5049649472 q^{73} + 795481360 q^{79} - 8212243320 q^{85} + 31095226112 q^{91} + 632084096 q^{97}+O(q^{100}) $ 4 * q - 21584 * q^7 - 646912 * q^13 + 1438528 * q^19 - 48796580 * q^25 + 99886192 * q^31 - 243863176 * q^37 + 460449760 * q^43 + 439434108 * q^49 - 1130824800 * q^55 + 2750573000 * q^61 - 2190062816 * q^67 + 5049649472 * q^73 + 795481360 * q^79 - 8212243320 * q^85 + 31095226112 * q^91 + 632084096 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 3155.64i	− 1.00981i	−0.863176	−	0.504903i	$-0.831528\pi$
$5$	− 3155.64i	− 1.00981i	0.863176	−	0.504903i	$-0.168472\pi$
$6$	0	0
$7$	13662.3	0.812891	0.406445	−	0.913675i	$-0.366768\pi$
$7$	13662.3	0.812891	0.406445	+	0.913675i	$0.366768\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 62736.3i	− 0.389543i	−0.980849	−	0.194771i	$-0.937603\pi$
$11$	− 62736.3i	− 0.389543i	0.980849	−	0.194771i	$-0.0623965\pi$
$12$	0	0
$13$	200379.	0.539678	0.269839	−	0.962905i	$-0.413029\pi$
$13$	200379.	0.539678	0.269839	+	0.962905i	$0.413029\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	556674.i	0.392064i	0.980598	+	0.196032i	$0.0628056\pi$
$17$	556674.i	0.392064i	−0.980598	+	0.196032i	$0.937194\pi$
$18$	0	0
$19$	3.25649e6	1.31517	0.657584	−	0.753381i	$-0.271578\pi$
$19$	3.25649e6	1.31517	0.657584	+	0.753381i	$0.271578\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 6.15671e6i	− 0.956553i	−0.878209	−	0.478277i	$-0.841262\pi$
$23$	− 6.15671e6i	− 0.956553i	0.878209	−	0.478277i	$-0.158738\pi$
$24$	0	0
$25$	−192446.	−0.0197065
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.03283e7i	1.96617i	0.183158	+	0.983083i	$0.441368\pi$
$29$	4.03283e7i	1.96617i	−0.183158	+	0.983083i	$0.558632\pi$
$30$	0	0
$31$	4.77843e7	1.66908	0.834539	−	0.550949i	$-0.185734\pi$
$31$	4.77843e7	1.66908	0.834539	+	0.550949i	$0.185734\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 4.31132e7i	− 0.820861i
$36$	0	0
$37$	−7.11048e7	−1.02539	−0.512696	−	0.858570i	$-0.671353\pi$
$37$	−7.11048e7	−1.02539	−0.512696	+	0.858570i	$0.671353\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.39960e8i	1.20805i	0.796965	+	0.604026i	$0.206438\pi$
$41$	1.39960e8i	1.20805i	−0.796965	+	0.604026i	$0.793562\pi$
$42$	0	0
$43$	−2.17639e7	−0.148045	−0.0740227	−	0.997257i	$-0.523584\pi$
$43$	−2.17639e7	−0.148045	−0.0740227	+	0.997257i	$0.523584\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.67168e8i	− 0.728891i	−0.931225	−	0.364446i	$-0.881258\pi$
$47$	− 1.67168e8i	− 0.728891i	0.931225	−	0.364446i	$-0.118742\pi$
$48$	0	0
$49$	−9.58181e7	−0.339209
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.91424e8i	− 1.89247i	−0.323474	−	0.946237i	$-0.604851\pi$
$53$	− 7.91424e8i	− 1.89247i	0.323474	−	0.946237i	$-0.395149\pi$
$54$	0	0
$55$	−1.97973e8	−0.393363
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.86465e8i	0.400692i	0.979725	+	0.200346i	$0.0642067\pi$
$59$	2.86465e8i	0.400692i	−0.979725	+	0.200346i	$0.935793\pi$
$60$	0	0
$61$	1.01308e9	1.19949	0.599743	−	0.800192i	$-0.295269\pi$
$61$	1.01308e9	1.19949	0.599743	+	0.800192i	$0.295269\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 6.32324e8i	− 0.544970i
$66$	0	0
$67$	−9.27728e8	−0.687142	−0.343571	−	0.939127i	$-0.611637\pi$
$67$	−9.27728e8	−0.687142	−0.343571	+	0.939127i	$0.611637\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.49137e9i	1.38085i	0.723405	+	0.690424i	$0.242576\pi$
$71$	2.49137e9i	1.38085i	−0.723405	+	0.690424i	$0.757424\pi$
$72$	0	0
$73$	9.45207e8	0.455945	0.227973	−	0.973668i	$-0.426790\pi$
$73$	9.45207e8	0.455945	0.227973	+	0.973668i	$0.426790\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 8.57119e8i	− 0.316656i
$78$	0	0
$79$	3.35970e9	1.09186	0.545928	−	0.837832i	$-0.316177\pi$
$79$	3.35970e9	1.09186	0.545928	+	0.837832i	$0.316177\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 5.13533e9i	− 1.30370i	−0.758348	−	0.651850i	$-0.773993\pi$
$83$	− 5.13533e9i	− 1.30370i	0.758348	−	0.651850i	$-0.226007\pi$
$84$	0	0
$85$	1.75666e9	0.395908
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.50859e9i	0.628322i	0.949370	+	0.314161i	$0.101723\pi$
$89$	3.50859e9i	0.628322i	−0.949370	+	0.314161i	$0.898277\pi$
$90$	0	0
$91$	2.73763e9	0.438699
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 1.02763e10i	− 1.32806i
$96$	0	0
$97$	−6.06949e9	−0.706795	−0.353398	−	0.935473i	$-0.614974\pi$
$97$	−6.06949e9	−0.706795	−0.353398	+	0.935473i	$0.614974\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 6.80727e9i	− 0.647689i	−0.946110	−	0.323844i	$-0.895025\pi$
$101$	− 6.80727e9i	− 0.647689i	0.946110	−	0.323844i	$-0.104975\pi$
$102$	0	0
$103$	1.74498e10	1.50524	0.752619	−	0.658456i	$-0.228790\pi$
$103$	1.74498e10	1.50524	0.752619	+	0.658456i	$0.228790\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.69087e10i	− 1.91855i	−0.282473	−	0.959275i	$-0.591155\pi$
$107$	− 2.69087e10i	− 1.91855i	0.282473	−	0.959275i	$-0.408845\pi$
$108$	0	0
$109$	4.93582e9	0.320794	0.160397	−	0.987053i	$-0.448723\pi$
$109$	4.93582e9	0.320794	0.160397	+	0.987053i	$0.448723\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.82710e10i	− 0.991675i	−0.868415	−	0.495837i	$-0.834861\pi$
$113$	− 1.82710e10i	− 0.991675i	0.868415	−	0.495837i	$-0.165139\pi$
$114$	0	0
$115$	−1.94284e10	−0.965933
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.60543e9i	0.318705i
$120$	0	0
$121$	2.20016e10	0.848256
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 3.02095e10i	− 0.989905i
$126$	0	0
$127$	2.34860e10	0.710871	0.355436	−	0.934701i	$-0.384333\pi$
$127$	2.34860e10	0.710871	0.355436	+	0.934701i	$0.384333\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.97294e10i	0.511396i	0.966757	+	0.255698i	$0.0823053\pi$
$131$	1.97294e10i	0.511396i	−0.966757	+	0.255698i	$0.917695\pi$
$132$	0	0
$133$	4.44909e10	1.06909
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 7.18908e10i	− 1.48960i	−0.667286	−	0.744802i	$-0.732544\pi$
$137$	− 7.18908e10i	− 1.48960i	0.667286	−	0.744802i	$-0.267456\pi$
$138$	0	0
$139$	2.40349e10	0.463199	0.231600	−	0.972811i	$-0.425604\pi$
$139$	2.40349e10	0.463199	0.231600	+	0.972811i	$0.425604\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 1.25710e10i	− 0.210228i
$144$	0	0
$145$	1.27262e11	1.98545
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 1.97308e10i	− 0.268667i	−0.990936	−	0.134333i	$-0.957111\pi$
$149$	− 1.97308e10i	− 0.268667i	0.990936	−	0.134333i	$-0.0428893\pi$
$150$	0	0
$151$	4.12033e10	0.524864	0.262432	−	0.964950i	$-0.415475\pi$
$151$	4.12033e10	0.524864	0.262432	+	0.964950i	$0.415475\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.50790e11i	− 1.68544i
$156$	0	0
$157$	−8.30056e10	−0.870180	−0.435090	−	0.900387i	$-0.643283\pi$
$157$	−8.30056e10	−0.870180	−0.435090	+	0.900387i	$0.643283\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 8.41145e10i	− 0.777573i
$162$	0	0
$163$	1.05593e11	0.917696	0.458848	−	0.888515i	$-0.348262\pi$
$163$	1.05593e11	0.917696	0.458848	+	0.888515i	$0.348262\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1.50592e11i	− 1.15936i	−0.814844	−	0.579681i	$-0.803177\pi$
$167$	− 1.50592e11i	− 1.15936i	0.814844	−	0.579681i	$-0.196823\pi$
$168$	0	0
$169$	−9.77068e10	−0.708747
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.21165e10i	0.529908i	0.964261	+	0.264954i	$0.0853567\pi$
$173$	8.21165e10i	0.529908i	−0.964261	+	0.264954i	$0.914643\pi$
$174$	0	0
$175$	−2.62925e9	−0.0160192
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1.13910e11i	− 0.619867i	−0.950758	−	0.309933i	$-0.899693\pi$
$179$	− 1.13910e11i	− 0.619867i	0.950758	−	0.309933i	$-0.100307\pi$
$180$	0	0
$181$	5.06895e10	0.260930	0.130465	−	0.991453i	$-0.458353\pi$
$181$	5.06895e10	0.260930	0.130465	+	0.991453i	$0.458353\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.24381e11i	1.03545i
$186$	0	0
$187$	3.49237e10	0.152726
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 4.96647e10i	− 0.195381i	−0.995217	−	0.0976903i	$-0.968855\pi$
$191$	− 4.96647e10i	− 0.195381i	0.995217	−	0.0976903i	$-0.0311455\pi$
$192$	0	0
$193$	−2.72346e11	−1.01703	−0.508515	−	0.861053i	$-0.669806\pi$
$193$	−2.72346e11	−1.01703	−0.508515	+	0.861053i	$0.669806\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.60900e11i	− 0.879313i	−0.898166	−	0.439656i	$-0.855100\pi$
$197$	− 2.60900e11i	− 0.879313i	0.898166	−	0.439656i	$-0.144900\pi$
$198$	0	0
$199$	−2.08123e11	−0.666891	−0.333445	−	0.942769i	$-0.608211\pi$
$199$	−2.08123e11	−0.666891	−0.333445	+	0.942769i	$0.608211\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.50976e11i	1.59828i
$204$	0	0
$205$	4.41664e11	1.21990
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 2.04300e11i	− 0.512314i
$210$	0	0
$211$	4.08716e11	0.977259	0.488629	−	0.872491i	$-0.337497\pi$
$211$	4.08716e11	0.977259	0.488629	+	0.872491i	$0.337497\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.86791e10i	0.149497i
$216$	0	0
$217$	6.52841e11	1.35678
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.11546e11i	0.211588i
$222$	0	0
$223$	−8.93428e11	−1.62008	−0.810038	−	0.586378i	$-0.800554\pi$
$223$	−8.93428e11	−1.62008	−0.810038	+	0.586378i	$0.800554\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.25294e11i	1.20333i	0.798748	+	0.601665i	$0.205496\pi$
$227$	7.25294e11i	1.20333i	−0.798748	+	0.601665i	$0.794504\pi$
$228$	0	0
$229$	−6.73465e11	−1.06939	−0.534697	−	0.845044i	$-0.679574\pi$
$229$	−6.73465e11	−1.06939	−0.534697	+	0.845044i	$0.679574\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.23198e11i	0.470640i	0.971918	+	0.235320i	$0.0756138\pi$
$233$	3.23198e11i	0.470640i	−0.971918	+	0.235320i	$0.924386\pi$
$234$	0	0
$235$	−5.27521e11	−0.736038
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.06713e12i	1.36845i	0.729273	+	0.684223i	$0.239859\pi$
$239$	1.06713e12i	1.36845i	−0.729273	+	0.684223i	$0.760141\pi$
$240$	0	0
$241$	−6.61735e11	−0.813952	−0.406976	−	0.913439i	$-0.633417\pi$
$241$	−6.61735e11	−0.813952	−0.406976	+	0.913439i	$0.633417\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.02368e11i	0.342535i
$246$	0	0
$247$	6.52531e11	0.709768
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 1.09896e12i	− 1.10309i	−0.834145	−	0.551546i	$-0.814038\pi$
$251$	− 1.09896e12i	− 1.10309i	0.834145	−	0.551546i	$-0.185962\pi$
$252$	0	0
$253$	−3.86249e11	−0.372619
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.88259e12i	1.67915i	0.543241	+	0.839577i	$0.317197\pi$
$257$	1.88259e12i	1.67915i	−0.543241	+	0.839577i	$0.682803\pi$
$258$	0	0
$259$	−9.71451e11	−0.833532
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2.63653e11i	− 0.209534i	−0.994497	−	0.104767i	$-0.966590\pi$
$263$	− 2.63653e11i	− 0.209534i	0.994497	−	0.104767i	$-0.0334097\pi$
$264$	0	0
$265$	−2.49745e12	−1.91103
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.21572e12i	0.863123i	0.902084	+	0.431561i	$0.142037\pi$
$269$	1.21572e12i	0.863123i	−0.902084	+	0.431561i	$0.857963\pi$
$270$	0	0
$271$	−1.65179e12	−1.13008	−0.565039	−	0.825064i	$-0.691139\pi$
$271$	−1.65179e12	−1.13008	−0.565039	+	0.825064i	$0.691139\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.20734e10i	0.00767653i
$276$	0	0
$277$	−8.23398e11	−0.504907	−0.252453	−	0.967609i	$-0.581237\pi$
$277$	−8.23398e11	−0.504907	−0.252453	+	0.967609i	$0.581237\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.80283e11i	0.274136i	0.990562	+	0.137068i	$0.0437679\pi$
$281$	4.80283e11i	0.274136i	−0.990562	+	0.137068i	$0.956232\pi$
$282$	0	0
$283$	2.89187e12	1.59311	0.796557	−	0.604563i	$-0.206652\pi$
$283$	2.89187e12	1.59311	0.796557	+	0.604563i	$0.206652\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.91217e12i	0.982014i
$288$	0	0
$289$	1.70611e12	0.846286
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.43864e12i	0.666213i	0.942889	+	0.333106i	$0.108097\pi$
$293$	1.43864e12i	0.666213i	−0.942889	+	0.333106i	$0.891903\pi$
$294$	0	0
$295$	9.03980e11	0.404621
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 1.23367e12i	− 0.516231i
$300$	0	0
$301$	−2.97344e11	−0.120345
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 3.19692e12i	− 1.21125i
$306$	0	0
$307$	−4.68857e12	−1.71929	−0.859644	−	0.510893i	$-0.829315\pi$
$307$	−4.68857e12	−1.71929	−0.859644	+	0.510893i	$0.829315\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.01870e12i	1.38129i	0.723196	+	0.690643i	$0.242672\pi$
$311$	4.01870e12i	1.38129i	−0.723196	+	0.690643i	$0.757328\pi$
$312$	0	0
$313$	5.61453e12	1.86892	0.934461	−	0.356064i	$-0.115882\pi$
$313$	5.61453e12	1.86892	0.934461	+	0.356064i	$0.115882\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.03202e11i	0.0322399i	0.999870	+	0.0161200i	$0.00513136\pi$
$317$	1.03202e11i	0.0322399i	−0.999870	+	0.0161200i	$0.994869\pi$
$318$	0	0
$319$	2.53005e12	0.765906
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.81280e12i	0.515630i
$324$	0	0
$325$	−3.85622e10	−0.0106352
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 2.28389e12i	− 0.592509i
$330$	0	0
$331$	−1.96076e12	−0.493497	−0.246748	−	0.969080i	$-0.579362\pi$
$331$	−1.96076e12	−0.493497	−0.246748	+	0.969080i	$0.579362\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.92758e12i	0.693880i
$336$	0	0
$337$	−7.53374e12	−1.73325	−0.866624	−	0.498961i	$-0.833715\pi$
$337$	−7.53374e12	−1.73325	−0.866624	+	0.498961i	$0.833715\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 2.99781e12i	− 0.650177i
$342$	0	0
$343$	−5.16834e12	−1.08863
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8.15399e12i	− 1.62078i	−0.585894	−	0.810388i	$-0.699256\pi$
$347$	− 8.15399e12i	− 1.62078i	0.585894	−	0.810388i	$-0.300744\pi$
$348$	0	0
$349$	6.13886e12	1.18566	0.592831	−	0.805327i	$-0.298010\pi$
$349$	6.13886e12	1.18566	0.592831	+	0.805327i	$0.298010\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 2.00965e12i	− 0.366647i	−0.983053	−	0.183323i	$-0.941314\pi$
$353$	− 2.00965e12i	− 0.366647i	0.983053	−	0.183323i	$-0.0586855\pi$
$354$	0	0
$355$	7.86186e12	1.39439
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 6.41488e12i	− 1.07576i	−0.843021	−	0.537881i	$-0.819225\pi$
$359$	− 6.41488e12i	− 1.07576i	0.843021	−	0.537881i	$-0.180775\pi$
$360$	0	0
$361$	4.47364e12	0.729667
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2.98273e12i	− 0.460416i
$366$	0	0
$367$	2.16002e12	0.324435	0.162217	−	0.986755i	$-0.448135\pi$
$367$	2.16002e12	0.324435	0.162217	+	0.986755i	$0.448135\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.08126e13i	− 1.53837i
$372$	0	0
$373$	−6.73800e12	−0.933226	−0.466613	−	0.884461i	$-0.654526\pi$
$373$	−6.73800e12	−0.933226	−0.466613	+	0.884461i	$0.654526\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.08094e12i	1.06110i
$378$	0	0
$379$	1.21474e13	1.55342	0.776709	−	0.629859i	$-0.216887\pi$
$379$	1.21474e13	1.55342	0.776709	+	0.629859i	$0.216887\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 2.30523e12i	− 0.279718i	−0.990171	−	0.139859i	$-0.955335\pi$
$383$	− 2.30523e12i	− 0.279718i	0.990171	−	0.139859i	$-0.0446650\pi$
$384$	0	0
$385$	−2.70476e12	−0.319761
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 8.14017e12i	− 0.913872i	−0.889500	−	0.456936i	$-0.848947\pi$
$389$	− 8.14017e12i	− 0.913872i	0.889500	−	0.456936i	$-0.151053\pi$
$390$	0	0
$391$	3.42728e12	0.375030
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1.06020e13i	− 1.10256i
$396$	0	0
$397$	1.04376e13	1.05840	0.529198	−	0.848498i	$-0.322493\pi$
$397$	1.04376e13	1.05840	0.529198	+	0.848498i	$0.322493\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 2.81563e12i	− 0.271552i	−0.990740	−	0.135776i	$-0.956647\pi$
$401$	− 2.81563e12i	− 0.271552i	0.990740	−	0.135776i	$-0.0433528\pi$
$402$	0	0
$403$	9.57495e12	0.900765
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.46085e12i	0.399435i
$408$	0	0
$409$	9.69304e12	0.846922	0.423461	−	0.905914i	$-0.360815\pi$
$409$	9.69304e12	0.846922	0.423461	+	0.905914i	$0.360815\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.91375e12i	0.325719i
$414$	0	0
$415$	−1.62052e13	−1.31648
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 9.71440e11i	− 0.0752222i	−0.999292	−	0.0376111i	$-0.988025\pi$
$419$	− 9.71440e11i	− 0.0752222i	0.999292	−	0.0376111i	$-0.0119748\pi$
$420$	0	0
$421$	1.11983e12	0.0846725	0.0423362	−	0.999103i	$-0.486520\pi$
$421$	1.11983e12	0.0846725	0.0423362	+	0.999103i	$0.486520\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1.07130e11i	− 0.00772621i
$426$	0	0
$427$	1.38410e13	0.975051
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.25045e13i	1.51316i	0.653904	+	0.756578i	$0.273130\pi$
$431$	2.25045e13i	1.51316i	−0.653904	+	0.756578i	$0.726870\pi$
$432$	0	0
$433$	1.09000e12	0.0716125	0.0358062	−	0.999359i	$-0.488600\pi$
$433$	1.09000e12	0.0716125	0.0358062	+	0.999359i	$0.488600\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.00492e13i	− 1.25803i
$438$	0	0
$439$	−1.88255e13	−1.15458	−0.577291	−	0.816539i	$-0.695890\pi$
$439$	−1.88255e13	−1.15458	−0.577291	+	0.816539i	$0.695890\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.79769e12i	0.457033i	0.973540	+	0.228516i	$0.0733874\pi$
$443$	7.79769e12i	0.457033i	−0.973540	+	0.228516i	$0.926613\pi$
$444$	0	0
$445$	1.10718e13	0.634483
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 2.97774e13i	− 1.63176i	−0.578224	−	0.815878i	$-0.696254\pi$
$449$	− 2.97774e13i	− 1.63176i	0.578224	−	0.815878i	$-0.303746\pi$
$450$	0	0
$451$	8.78059e12	0.470588
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 8.63896e12i	− 0.443001i
$456$	0	0
$457$	1.67367e12	0.0839631	0.0419815	−	0.999118i	$-0.486633\pi$
$457$	1.67367e12	0.0839631	0.0419815	+	0.999118i	$0.486633\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.40086e13i	0.672805i	0.941718	+	0.336403i	$0.109210\pi$
$461$	1.40086e13i	0.672805i	−0.941718	+	0.336403i	$0.890790\pi$
$462$	0	0
$463$	−3.45771e13	−1.62511	−0.812557	−	0.582881i	$-0.801925\pi$
$463$	−3.45771e13	−1.62511	−0.812557	+	0.582881i	$0.801925\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.98869e13i	− 0.895330i	−0.894201	−	0.447665i	$-0.852256\pi$
$467$	− 1.98869e13i	− 0.895330i	0.894201	−	0.447665i	$-0.147744\pi$
$468$	0	0
$469$	−1.26749e13	−0.558571
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.36539e12i	0.0576700i
$474$	0	0
$475$	−6.26699e11	−0.0259174
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.08277e13i	0.825969i	0.910738	+	0.412984i	$0.135514\pi$
$479$	2.08277e13i	0.825969i	−0.910738	+	0.412984i	$0.864486\pi$
$480$	0	0
$481$	−1.42479e13	−0.553382
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.91531e13i	0.713726i
$486$	0	0
$487$	1.83888e13	0.671286	0.335643	−	0.941989i	$-0.391046\pi$
$487$	1.83888e13	0.671286	0.335643	+	0.941989i	$0.391046\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 2.40429e13i	− 0.842519i	−0.906940	−	0.421259i	$-0.861588\pi$
$491$	− 2.40429e13i	− 0.842519i	0.906940	−	0.421259i	$-0.138412\pi$
$492$	0	0
$493$	−2.24498e13	−0.770863
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.40377e13i	1.12248i
$498$	0	0
$499$	−1.20029e12	−0.0387957	−0.0193979	−	0.999812i	$-0.506175\pi$
$499$	−1.20029e12	−0.0387957	−0.0193979	+	0.999812i	$0.506175\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.28291e13i	1.95128i	0.219367	+	0.975642i	$0.429601\pi$
$503$	6.28291e13i	1.95128i	−0.219367	+	0.975642i	$0.570399\pi$
$504$	0	0
$505$	−2.14813e13	−0.654039
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.41227e13i	0.998745i	0.866387	+	0.499373i	$0.166436\pi$
$509$	3.41227e13i	0.998745i	−0.866387	+	0.499373i	$0.833564\pi$
$510$	0	0
$511$	1.29137e13	0.370633
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 5.50654e13i	− 1.52000i
$516$	0	0
$517$	−1.04875e13	−0.283935
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6.35549e13i	1.65562i	0.561009	+	0.827810i	$0.310413\pi$
$521$	6.35549e13i	1.65562i	−0.561009	+	0.827810i	$0.689587\pi$
$522$	0	0
$523$	2.68002e13	0.684903	0.342451	−	0.939536i	$-0.388743\pi$
$523$	2.68002e13	0.684903	0.342451	+	0.939536i	$0.388743\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.66003e13i	0.654385i
$528$	0	0
$529$	3.52149e12	0.0850057
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.80451e13i	0.651959i
$534$	0	0
$535$	−8.49141e13	−1.93736
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.01127e12i	0.132136i
$540$	0	0
$541$	−2.13159e13	−0.459958	−0.229979	−	0.973196i	$-0.573866\pi$
$541$	−2.13159e13	−0.459958	−0.229979	+	0.973196i	$0.573866\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1.55757e13i	− 0.323940i
$546$	0	0
$547$	4.47766e13	0.914355	0.457177	−	0.889376i	$-0.348860\pi$
$547$	4.47766e13	0.914355	0.457177	+	0.889376i	$0.348860\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.31329e14i	2.58584i
$552$	0	0
$553$	4.59011e13	0.887559
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.37300e13i	0.442611i	0.975205	+	0.221305i	$0.0710317\pi$
$557$	2.37300e13i	0.442611i	−0.975205	+	0.221305i	$0.928968\pi$
$558$	0	0
$559$	−4.36103e12	−0.0798969
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 7.05157e12i	− 0.124665i	−0.998055	−	0.0623324i	$-0.980146\pi$
$563$	− 7.05157e12i	− 0.124665i	0.998055	−	0.0623324i	$-0.0198539\pi$
$564$	0	0
$565$	−5.76566e13	−1.00140
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 7.59133e13i	− 1.27279i	−0.771364	−	0.636395i	$-0.780425\pi$
$569$	− 7.59133e13i	− 1.27279i	0.771364	−	0.636395i	$-0.219575\pi$
$570$	0	0
$571$	−1.05526e14	−1.73852	−0.869260	−	0.494356i	$-0.835404\pi$
$571$	−1.05526e14	−1.73852	−0.869260	+	0.494356i	$0.835404\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.18484e12i	0.0188503i
$576$	0	0
$577$	−5.22466e13	−0.816919	−0.408459	−	0.912777i	$-0.633934\pi$
$577$	−5.22466e13	−0.816919	−0.408459	+	0.912777i	$0.633934\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 7.01601e13i	− 1.05977i
$582$	0	0
$583$	−4.96510e13	−0.737200
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.42445e13i	− 1.35228i	−0.736775	−	0.676138i	$-0.763652\pi$
$587$	− 9.42445e13i	− 1.35228i	0.736775	−	0.676138i	$-0.236348\pi$
$588$	0	0
$589$	1.55609e14	2.19512
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 2.00339e13i	− 0.273208i	−0.990626	−	0.136604i	$-0.956381\pi$
$593$	− 2.00339e13i	− 0.273208i	0.990626	−	0.136604i	$-0.0436188\pi$
$594$	0	0
$595$	2.40000e13	0.321830
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.27695e14i	1.65592i	0.560789	+	0.827959i	$0.310498\pi$
$599$	1.27695e14i	1.65592i	−0.560789	+	0.827959i	$0.689502\pi$
$600$	0	0
$601$	8.41606e12	0.107334	0.0536669	−	0.998559i	$-0.482909\pi$
$601$	8.41606e12	0.107334	0.0536669	+	0.998559i	$0.482909\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 6.94291e13i	− 0.856574i
$606$	0	0
$607$	−3.29765e13	−0.400185	−0.200092	−	0.979777i	$-0.564124\pi$
$607$	−3.29765e13	−0.400185	−0.200092	+	0.979777i	$0.564124\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 3.34968e13i	− 0.393367i
$612$	0	0
$613$	−4.25352e13	−0.491412	−0.245706	−	0.969344i	$-0.579020\pi$
$613$	−4.25352e13	−0.491412	−0.245706	+	0.969344i	$0.579020\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.59920e14i	1.78845i	0.447615	+	0.894226i	$0.352274\pi$
$617$	1.59920e14i	1.78845i	−0.447615	+	0.894226i	$0.647726\pi$
$618$	0	0
$619$	7.40577e13	0.814924	0.407462	−	0.913222i	$-0.366414\pi$
$619$	7.40577e13	0.814924	0.407462	+	0.913222i	$0.366414\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.79352e13i	0.510757i
$624$	0	0
$625$	−9.72098e13	−1.01932
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 3.95822e13i	− 0.402019i
$630$	0	0
$631$	9.16626e12	0.0916316	0.0458158	−	0.998950i	$-0.485411\pi$
$631$	9.16626e12	0.0916316	0.0458158	+	0.998950i	$0.485411\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 7.41135e13i	− 0.717842i
$636$	0	0
$637$	−1.91999e13	−0.183064
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.38052e13i	0.682019i	0.940060	+	0.341010i	$0.110769\pi$
$641$	7.38052e13i	0.682019i	−0.940060	+	0.341010i	$0.889231\pi$
$642$	0	0
$643$	6.75150e13	0.614250	0.307125	−	0.951669i	$-0.400633\pi$
$643$	6.75150e13	0.614250	0.307125	+	0.951669i	$0.400633\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.48268e14i	− 1.30775i	−0.756601	−	0.653877i	$-0.773141\pi$
$647$	− 1.48268e14i	− 1.30775i	0.756601	−	0.653877i	$-0.226859\pi$
$648$	0	0
$649$	1.79717e13	0.156087
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.64272e13i	0.306803i	0.988164	+	0.153402i	$0.0490228\pi$
$653$	3.64272e13i	0.306803i	−0.988164	+	0.153402i	$0.950977\pi$
$654$	0	0
$655$	6.22589e13	0.516410
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1.43074e14i	− 1.15115i	−0.817748	−	0.575577i	$-0.804778\pi$
$659$	− 1.43074e14i	− 1.15115i	0.817748	−	0.575577i	$-0.195222\pi$
$660$	0	0
$661$	1.53905e14	1.21968	0.609838	−	0.792526i	$-0.291234\pi$
$661$	1.53905e14	1.21968	0.609838	+	0.792526i	$0.291234\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.40397e14i	− 1.07957i
$666$	0	0
$667$	2.48290e14	1.88074
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 6.35570e13i	− 0.467252i
$672$	0	0
$673$	1.95446e14	1.41563	0.707817	−	0.706395i	$-0.249680\pi$
$673$	1.95446e14	1.41563	0.707817	+	0.706395i	$0.249680\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.79249e14i	1.26042i	0.776426	+	0.630208i	$0.217031\pi$
$677$	1.79249e14i	1.26042i	−0.776426	+	0.630208i	$0.782969\pi$
$678$	0	0
$679$	−8.29229e13	−0.574547
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 2.50626e13i	− 0.168625i	−0.996439	−	0.0843127i	$-0.973131\pi$
$683$	− 2.50626e13i	− 0.168625i	0.996439	−	0.0843127i	$-0.0268695\pi$
$684$	0	0
$685$	−2.26862e14	−1.50421
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1.58585e14i	− 1.02133i
$690$	0	0
$691$	2.97222e14	1.88665	0.943324	−	0.331873i	$-0.107680\pi$
$691$	2.97222e14	1.88665	0.943324	+	0.331873i	$0.107680\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 7.58454e13i	− 0.467741i
$696$	0	0
$697$	−7.79123e13	−0.473633
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 2.01709e14i	− 1.19161i	−0.803129	−	0.595805i	$-0.796833\pi$
$701$	− 2.01709e14i	− 1.19161i	0.803129	−	0.595805i	$-0.203167\pi$
$702$	0	0
$703$	−2.31552e14	−1.34856
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 9.30027e13i	− 0.526500i
$708$	0	0
$709$	2.54114e14	1.41839	0.709197	−	0.705010i	$-0.249058\pi$
$709$	2.54114e14	1.41839	0.709197	+	0.705010i	$0.249058\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2.94194e14i	− 1.59656i
$714$	0	0
$715$	−3.96696e13	−0.212289
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.42854e14i	0.743445i	0.928344	+	0.371723i	$0.121233\pi$
$719$	1.42854e14i	0.743445i	−0.928344	+	0.371723i	$0.878767\pi$
$720$	0	0
$721$	2.38404e14	1.22359
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 7.76104e12i	− 0.0387463i
$726$	0	0
$727$	−1.69339e13	−0.0833845	−0.0416923	−	0.999130i	$-0.513275\pi$
$727$	−1.69339e13	−0.0833845	−0.0416923	+	0.999130i	$0.513275\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1.21154e13i	− 0.0580432i
$732$	0	0
$733$	−2.86833e13	−0.135553	−0.0677766	−	0.997701i	$-0.521590\pi$
$733$	−2.86833e13	−0.135553	−0.0677766	+	0.997701i	$0.521590\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.82022e13i	0.267671i
$738$	0	0
$739$	−3.36378e14	−1.52618	−0.763089	−	0.646294i	$-0.776318\pi$
$739$	−3.36378e14	−1.52618	−0.763089	+	0.646294i	$0.776318\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.33755e14i	0.590701i	0.955389	+	0.295350i	$0.0954364\pi$
$743$	1.33755e14i	0.590701i	−0.955389	+	0.295350i	$0.904564\pi$
$744$	0	0
$745$	−6.22634e13	−0.271301
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 3.67633e14i	− 1.55957i
$750$	0	0
$751$	−7.52961e12	−0.0315190	−0.0157595	−	0.999876i	$-0.505017\pi$
$751$	−7.52961e12	−0.0315190	−0.0157595	+	0.999876i	$0.505017\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1.30023e14i	− 0.530011i
$756$	0	0
$757$	−1.62040e14	−0.651844	−0.325922	−	0.945397i	$-0.605675\pi$
$757$	−1.62040e14	−0.651844	−0.325922	+	0.945397i	$0.605675\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 2.46006e13i	− 0.0963878i	−0.998838	−	0.0481939i	$-0.984653\pi$
$761$	− 2.46006e13i	− 0.0963878i	0.998838	−	0.0481939i	$-0.0153465\pi$
$762$	0	0
$763$	6.74344e13	0.260771
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.74015e13i	0.216245i
$768$	0	0
$769$	5.88347e12	0.0218777	0.0109388	−	0.999940i	$-0.496518\pi$
$769$	5.88347e12	0.0218777	0.0109388	+	0.999940i	$0.496518\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.52416e13i	0.0552247i	0.999619	+	0.0276124i	$0.00879041\pi$
$773$	1.52416e13i	0.0552247i	−0.999619	+	0.0276124i	$0.991210\pi$
$774$	0	0
$775$	−9.19591e12	−0.0328917
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.55779e14i	1.58879i
$780$	0	0
$781$	1.56299e14	0.537900
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.61936e14i	0.878712i
$786$	0	0
$787$	3.05775e14	1.01281	0.506406	−	0.862295i	$-0.330974\pi$
$787$	3.05775e14	1.01281	0.506406	+	0.862295i	$0.330974\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 2.49623e14i	− 0.806123i
$792$	0	0
$793$	2.03000e14	0.647337
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.28192e14i	1.64248i	0.570582	+	0.821240i	$0.306717\pi$
$797$	5.28192e14i	1.64248i	−0.570582	+	0.821240i	$0.693283\pi$
$798$	0	0
$799$	9.30579e13	0.285772
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 5.92988e13i	− 0.177610i
$804$	0	0
$805$	−2.65435e14	−0.785197
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.65656e14i	0.478042i	0.971014	+	0.239021i	$0.0768264\pi$
$809$	1.65656e14i	0.478042i	−0.971014	+	0.239021i	$0.923174\pi$
$810$	0	0
$811$	−1.08318e14	−0.308742	−0.154371	−	0.988013i	$-0.549335\pi$
$811$	−1.08318e14	−0.308742	−0.154371	+	0.988013i	$0.549335\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 3.33215e14i	− 0.926694i
$816$	0	0
$817$	−7.08739e13	−0.194705
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.35605e14i	0.631638i	0.948819	+	0.315819i	$0.102279\pi$
$821$	2.35605e14i	0.631638i	−0.948819	+	0.315819i	$0.897721\pi$
$822$	0	0
$823$	5.26284e14	1.39387	0.696933	−	0.717136i	$-0.254547\pi$
$823$	5.26284e14	1.39387	0.696933	+	0.717136i	$0.254547\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.77432e14i	− 0.975689i	−0.872931	−	0.487845i	$-0.837783\pi$
$827$	− 3.77432e14i	− 0.975689i	0.872931	−	0.487845i	$-0.162217\pi$
$828$	0	0
$829$	4.96486e13	0.126804	0.0634022	−	0.997988i	$-0.479805\pi$
$829$	4.96486e13	0.126804	0.0634022	+	0.997988i	$0.479805\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 5.33395e13i	− 0.132992i
$834$	0	0
$835$	−4.75213e14	−1.17073
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.26745e14i	1.26704i	0.773726	+	0.633520i	$0.218391\pi$
$839$	5.26745e14i	1.26704i	−0.773726	+	0.633520i	$0.781609\pi$
$840$	0	0
$841$	−1.20567e15	−2.86581
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.08328e14i	0.715697i
$846$	0	0
$847$	3.00591e14	0.689540
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.37771e14i	0.980843i
$852$	0	0
$853$	3.71861e14	0.823447	0.411724	−	0.911309i	$-0.364927\pi$
$853$	3.71861e14	0.823447	0.411724	+	0.911309i	$0.364927\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.50380e14i	1.19058i	0.803511	+	0.595290i	$0.202963\pi$
$857$	5.50380e14i	1.19058i	−0.803511	+	0.595290i	$0.797037\pi$
$858$	0	0
$859$	−4.90341e14	−1.04841	−0.524207	−	0.851591i	$-0.675638\pi$
$859$	−4.90341e14	−1.04841	−0.524207	+	0.851591i	$0.675638\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5.36959e14i	1.12173i	0.827908	+	0.560863i	$0.189531\pi$
$863$	5.36959e14i	1.12173i	−0.827908	+	0.560863i	$0.810469\pi$
$864$	0	0
$865$	2.59130e14	0.535103
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 2.10775e14i	− 0.425325i
$870$	0	0
$871$	−1.85897e14	−0.370836
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 4.12730e14i	− 0.804685i
$876$	0	0
$877$	−9.30905e14	−1.79435	−0.897175	−	0.441675i	$-0.854384\pi$
$877$	−9.30905e14	−1.79435	−0.897175	+	0.441675i	$0.854384\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.41915e14i	0.267392i	0.991022	+	0.133696i	$0.0426846\pi$
$881$	1.41915e14i	0.267392i	−0.991022	+	0.133696i	$0.957315\pi$
$882$	0	0
$883$	−9.56307e14	−1.78153	−0.890766	−	0.454462i	$-0.849832\pi$
$883$	−9.56307e14	−1.78153	−0.890766	+	0.454462i	$0.849832\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 8.67599e14i	− 1.58016i	−0.613004	−	0.790080i	$-0.710039\pi$
$887$	− 8.67599e14i	− 1.58016i	0.613004	−	0.790080i	$-0.289961\pi$
$888$	0	0
$889$	3.20872e14	0.577861
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 5.44379e14i	− 0.958615i
$894$	0	0
$895$	−3.59460e14	−0.625945
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.92706e15i	3.28169i
$900$	0	0
$901$	4.40566e14	0.741970
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1.59958e14i	− 0.263489i
$906$	0	0
$907$	1.95057e14	0.317778	0.158889	−	0.987296i	$-0.449209\pi$
$907$	1.95057e14	0.317778	0.158889	+	0.987296i	$0.449209\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	7.67274e14i	1.22281i	0.791318	+	0.611404i	$0.209395\pi$
$911$	7.67274e14i	1.22281i	−0.791318	+	0.611404i	$0.790605\pi$
$912$	0	0
$913$	−3.22171e14	−0.507847
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.69548e14i	0.415709i
$918$	0	0
$919$	−5.50077e14	−0.839162	−0.419581	−	0.907718i	$-0.637823\pi$
$919$	−5.50077e14	−0.839162	−0.419581	+	0.907718i	$0.637823\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.99217e14i	0.745214i
$924$	0	0
$925$	1.36839e13	0.0202069
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9.17885e13i	0.132651i	0.997798	+	0.0663254i	$0.0211275\pi$
$929$	9.17885e13i	0.132651i	−0.997798	+	0.0663254i	$0.978872\pi$
$930$	0	0
$931$	−3.12030e14	−0.446117
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1.10207e14i	− 0.154223i
$936$	0	0
$937$	−1.24541e14	−0.172431	−0.0862154	−	0.996277i	$-0.527477\pi$
$937$	−1.24541e14	−0.172431	−0.0862154	+	0.996277i	$0.527477\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 7.11466e14i	− 0.964286i	−0.876093	−	0.482143i	$-0.839859\pi$
$941$	− 7.11466e14i	− 0.964286i	0.876093	−	0.482143i	$-0.160141\pi$
$942$	0	0
$943$	8.61694e14	1.15557
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.18615e14i	− 0.549624i	−0.961498	−	0.274812i	$-0.911384\pi$
$947$	− 4.18615e14i	− 0.549624i	0.961498	−	0.274812i	$-0.0886155\pi$
$948$	0	0
$949$	1.89399e14	0.246064
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1.44086e14i	− 0.183298i	−0.995791	−	0.0916488i	$-0.970786\pi$
$953$	− 1.44086e14i	− 0.183298i	0.995791	−	0.0916488i	$-0.0292137\pi$
$954$	0	0
$955$	−1.56724e14	−0.197296
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 9.82191e14i	− 1.21088i
$960$	0	0
$961$	1.46371e15	1.78582
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8.59426e14i	1.02700i
$966$	0	0
$967$	1.09698e15	1.29737	0.648686	−	0.761056i	$-0.275319\pi$
$967$	1.09698e15	1.29737	0.648686	+	0.761056i	$0.275319\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.77841e13i	0.0901145i	0.998984	+	0.0450572i	$0.0143470\pi$
$971$	7.77841e13i	0.0901145i	−0.998984	+	0.0450572i	$0.985653\pi$
$972$	0	0
$973$	3.28370e14	0.376530
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.68951e14i	0.189796i	0.995487	+	0.0948982i	$0.0302526\pi$
$977$	1.68951e14i	0.189796i	−0.995487	+	0.0948982i	$0.969747\pi$
$978$	0	0
$979$	2.20116e14	0.244758
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 5.30854e14i	− 0.578373i	−0.957273	−	0.289186i	$-0.906615\pi$
$983$	− 5.30854e14i	− 0.578373i	0.957273	−	0.289186i	$-0.0933847\pi$
$984$	0	0
$985$	−8.23308e14	−0.887935
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.33994e14i	0.141613i
$990$	0	0
$991$	9.48887e13	0.0992765	0.0496382	−	0.998767i	$-0.484193\pi$
$991$	9.48887e13	0.0992765	0.0496382	+	0.998767i	$0.484193\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.56761e14i	0.673430i
$996$	0	0
$997$	−7.37360e14	−0.748520	−0.374260	−	0.927324i	$-0.622103\pi$
$997$	−7.37360e14	−0.748520	−0.374260	+	0.927324i	$0.622103\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.11.e.b.17.2		4
3.2	odd	2	inner	144.11.e.b.17.3		4
4.3	odd	2		36.11.c.a.17.2	✓	4
12.11	even	2		36.11.c.a.17.3	yes	4
36.7	odd	6		324.11.g.e.53.3		8
36.11	even	6		324.11.g.e.53.2		8
36.23	even	6		324.11.g.e.269.3		8
36.31	odd	6		324.11.g.e.269.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.11.c.a.17.2	✓	4	4.3	odd	2
36.11.c.a.17.3	yes	4	12.11	even	2
144.11.e.b.17.2		4	1.1	even	1	trivial
144.11.e.b.17.3		4	3.2	odd	2	inner
324.11.g.e.53.2		8	36.11	even	6
324.11.g.e.53.3		8	36.7	odd	6
324.11.g.e.269.2		8	36.31	odd	6
324.11.g.e.269.3		8	36.23	even	6

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:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	144.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3155.64i q^{5} +13662.3 q^{7} +O(q^{10})\)	\(q-3155.64i q^{5} +13662.3 q^{7} -62736.3i q^{11} +200379. q^{13} +556674. i q^{17} +3.25649e6 q^{19} -6.15671e6i q^{23} -192446. q^{25} +4.03283e7i q^{29} +4.77843e7 q^{31} -4.31132e7i q^{35} -7.11048e7 q^{37} +1.39960e8i q^{41} -2.17639e7 q^{43} -1.67168e8i q^{47} -9.58181e7 q^{49} -7.91424e8i q^{53} -1.97973e8 q^{55} +2.86465e8i q^{59} +1.01308e9 q^{61} -6.32324e8i q^{65} -9.27728e8 q^{67} +2.49137e9i q^{71} +9.45207e8 q^{73} -8.57119e8i q^{77} +3.35970e9 q^{79} -5.13533e9i q^{83} +1.75666e9 q^{85} +3.50859e9i q^{89} +2.73763e9 q^{91} -1.02763e10i q^{95} -6.06949e9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 21584 q^{7}+O(q^{10}) \) 4 * q - 21584 * q^7	\( 4 q - 21584 q^{7} - 646912 q^{13} + 1438528 q^{19} - 48796580 q^{25} + 99886192 q^{31} - 243863176 q^{37} + 460449760 q^{43} + 439434108 q^{49} - 1130824800 q^{55} + 2750573000 q^{61} - 2190062816 q^{67} + 5049649472 q^{73} + 795481360 q^{79} - 8212243320 q^{85} + 31095226112 q^{91} + 632084096 q^{97}+O(q^{100}) \) 4 * q - 21584 * q^7 - 646912 * q^13 + 1438528 * q^19 - 48796580 * q^25 + 99886192 * q^31 - 243863176 * q^37 + 460449760 * q^43 + 439434108 * q^49 - 1130824800 * q^55 + 2750573000 * q^61 - 2190062816 * q^67 + 5049649472 * q^73 + 795481360 * q^79 - 8212243320 * q^85 + 31095226112 * q^91 + 632084096 * q^97

Introduction

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