LMFDB - Embedded newform 144.13.g.g.127.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,13,Mod(127,144)]

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([1, 0, 0]))

N = Newforms(chi, 13, names="a")

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

chi := DirichletCharacter("144.127");

S:= CuspForms(chi, 13);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 13 $
Character orbit:	$[\chi]$	$=$	144.g (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:	$131.615109688$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{2521})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 631x^{2} + 630x + 396900 $ x^4 - x^3 + 631x^2 + 630x + 396900
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.1
Root			$-12.3024 + 21.3084i$ of defining polynomial
Character	$\chi$	$=$	144.127
Dual form			144.13.g.g.127.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15554.5 q^{5} -81391.8i q^{7} +O(q^{10})$	$q-15554.5 q^{5} -81391.8i q^{7} -1.83066e6i q^{11} -1.54702e6 q^{13} -4.03088e7 q^{17} +8.21439e7i q^{19} -1.41567e8i q^{23} -2.19905e6 q^{25} -4.11796e8 q^{29} -1.49024e9i q^{31} +1.26601e9i q^{35} -2.50208e9 q^{37} +1.24249e9 q^{41} -6.93112e9i q^{43} -8.07836e9i q^{47} +7.21666e9 q^{49} -1.93041e10 q^{53} +2.84749e10i q^{55} +8.78748e9i q^{59} +8.42744e10 q^{61} +2.40631e10 q^{65} -2.92509e10i q^{67} +2.13890e11i q^{71} -2.24360e11 q^{73} -1.49001e11 q^{77} +2.95086e10i q^{79} -5.72394e10i q^{83} +6.26982e11 q^{85} +4.16334e11 q^{89} +1.25915e11i q^{91} -1.27770e12i q^{95} -7.23049e11 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 14904 q^{5}+O(q^{10}) $ 4 * q + 14904 * q^5	$ 4 q + 14904 q^{5} + 12552520 q^{13} - 57429000 q^{17} + 565916108 q^{25} + 597450168 q^{29} - 3792664120 q^{37} + 11480589432 q^{41} + 630768772 q^{49} - 94557783240 q^{53} + 87441423944 q^{61} + 408098641392 q^{65} - 759419897720 q^{73} - 498141135360 q^{77} + 1787449271184 q^{85} + 681837812856 q^{89} - 407801291000 q^{97}+O(q^{100}) $ 4 * q + 14904 * q^5 + 12552520 * q^13 - 57429000 * q^17 + 565916108 * q^25 + 597450168 * q^29 - 3792664120 * q^37 + 11480589432 * q^41 + 630768772 * q^49 - 94557783240 * q^53 + 87441423944 * q^61 + 408098641392 * q^65 - 759419897720 * q^73 - 498141135360 * q^77 + 1787449271184 * q^85 + 681837812856 * q^89 - 407801291000 * 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
$5$	−15554.5	−0.995486	−0.497743	−	0.867325i	$-0.665838\pi$
$5$	−15554.5	−0.995486	−0.497743	+	0.867325i	$0.665838\pi$
$6$	0	0
$7$	− 81391.8i	− 0.691819i	−0.938268	−	0.345910i	$-0.887570\pi$
$7$	− 81391.8i	− 0.691819i	0.938268	−	0.345910i	$-0.112430\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.83066e6i	− 1.03336i	−0.856179	−	0.516679i	$-0.827168\pi$
$11$	− 1.83066e6i	− 1.03336i	0.856179	−	0.516679i	$-0.172832\pi$
$12$	0	0
$13$	−1.54702e6	−0.320507	−0.160253	−	0.987076i	$-0.551231\pi$
$13$	−1.54702e6	−0.320507	−0.160253	+	0.987076i	$0.551231\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.03088e7	−1.66996	−0.834980	−	0.550281i	$-0.814521\pi$
$17$	−4.03088e7	−1.66996	−0.834980	+	0.550281i	$0.814521\pi$
$18$	0	0
$19$	8.21439e7i	1.74604i	0.487687	+	0.873019i	$0.337841\pi$
$19$	8.21439e7i	1.74604i	−0.487687	+	0.873019i	$0.662159\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1.41567e8i	− 0.956305i	−0.878277	−	0.478153i	$-0.841307\pi$
$23$	− 1.41567e8i	− 0.956305i	0.878277	−	0.478153i	$-0.158693\pi$
$24$	0	0
$25$	−2.19905e6	−0.00900729
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.11796e8	−0.692299	−0.346149	−	0.938179i	$-0.612511\pi$
$29$	−4.11796e8	−0.692299	−0.346149	+	0.938179i	$0.612511\pi$
$30$	0	0
$31$	− 1.49024e9i	− 1.67913i	−0.543257	−	0.839567i	$-0.682809\pi$
$31$	− 1.49024e9i	− 1.67913i	0.543257	−	0.839567i	$-0.317191\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.26601e9i	0.688696i
$36$	0	0
$37$	−2.50208e9	−0.975192	−0.487596	−	0.873069i	$-0.662126\pi$
$37$	−2.50208e9	−0.975192	−0.487596	+	0.873069i	$0.662126\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.24249e9	0.261571	0.130786	−	0.991411i	$-0.458250\pi$
$41$	1.24249e9	0.261571	0.130786	+	0.991411i	$0.458250\pi$
$42$	0	0
$43$	− 6.93112e9i	− 1.09646i	−0.836328	−	0.548230i	$-0.815302\pi$
$43$	− 6.93112e9i	− 1.09646i	0.836328	−	0.548230i	$-0.184698\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.07836e9i	− 0.749439i	−0.927138	−	0.374719i	$-0.877739\pi$
$47$	− 8.07836e9i	− 0.749439i	0.927138	−	0.374719i	$-0.122261\pi$
$48$	0	0
$49$	7.21666e9	0.521386
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.93041e10	−0.870950	−0.435475	−	0.900201i	$-0.643420\pi$
$53$	−1.93041e10	−0.870950	−0.435475	+	0.900201i	$0.643420\pi$
$54$	0	0
$55$	2.84749e10i	1.02869i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.78748e9i	0.208330i	0.994560	+	0.104165i	$0.0332170\pi$
$59$	8.78748e9i	0.208330i	−0.994560	+	0.104165i	$0.966783\pi$
$60$	0	0
$61$	8.42744e10	1.63575	0.817875	−	0.575396i	$-0.195152\pi$
$61$	8.42744e10	1.63575	0.817875	+	0.575396i	$0.195152\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.40631e10	0.319060
$66$	0	0
$67$	− 2.92509e10i	− 0.323363i	−0.986843	−	0.161682i	$-0.948308\pi$
$67$	− 2.92509e10i	− 0.323363i	0.986843	−	0.161682i	$-0.0516918\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.13890e11i	1.66970i	0.550475	+	0.834852i	$0.314447\pi$
$71$	2.13890e11i	1.66970i	−0.550475	+	0.834852i	$0.685553\pi$
$72$	0	0
$73$	−2.24360e11	−1.48254	−0.741272	−	0.671205i	$-0.765777\pi$
$73$	−2.24360e11	−1.48254	−0.741272	+	0.671205i	$0.765777\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.49001e11	−0.714897
$78$	0	0
$79$	2.95086e10i	0.121391i	0.998156	+	0.0606954i	$0.0193318\pi$
$79$	2.95086e10i	0.121391i	−0.998156	+	0.0606954i	$0.980668\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 5.72394e10i	− 0.175076i	−0.996161	−	0.0875380i	$-0.972100\pi$
$83$	− 5.72394e10i	− 0.175076i	0.996161	−	0.0875380i	$-0.0278999\pi$
$84$	0	0
$85$	6.26982e11	1.66242
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.16334e11	0.837725	0.418863	−	0.908050i	$-0.362429\pi$
$89$	4.16334e11	0.837725	0.418863	+	0.908050i	$0.362429\pi$
$90$	0	0
$91$	1.25915e11i	0.221733i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 1.27770e12i	− 1.73816i
$96$	0	0
$97$	−7.23049e11	−0.868035	−0.434017	−	0.900904i	$-0.642904\pi$
$97$	−7.23049e11	−0.868035	−0.434017	+	0.900904i	$0.642904\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.87665e11	−0.176789	−0.0883945	−	0.996086i	$-0.528174\pi$
$101$	−1.87665e11	−0.176789	−0.0883945	+	0.996086i	$0.528174\pi$
$102$	0	0
$103$	− 9.01520e11i	− 0.755009i	−0.926008	−	0.377504i	$-0.876782\pi$
$103$	− 9.01520e11i	− 0.755009i	0.926008	−	0.377504i	$-0.123218\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.48141e12i	1.65347i	0.562591	+	0.826735i	$0.309804\pi$
$107$	2.48141e12i	1.65347i	−0.562591	+	0.826735i	$0.690196\pi$
$108$	0	0
$109$	−9.93786e11	−0.592562	−0.296281	−	0.955101i	$-0.595747\pi$
$109$	−9.93786e11	−0.592562	−0.296281	+	0.955101i	$0.595747\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.47484e12	0.708394	0.354197	−	0.935171i	$-0.384754\pi$
$113$	1.47484e12	0.708394	0.354197	+	0.935171i	$0.384754\pi$
$114$	0	0
$115$	2.20201e12i	0.951988i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.28080e12i	1.15531i
$120$	0	0
$121$	−2.12880e11	−0.0678301
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.83168e12	1.00445
$126$	0	0
$127$	2.84255e12i	0.677464i	0.940883	+	0.338732i	$0.109998\pi$
$127$	2.84255e12i	0.677464i	−0.940883	+	0.338732i	$0.890002\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.26466e11i	− 0.0250234i	−0.999922	−	0.0125117i	$-0.996017\pi$
$131$	− 1.26466e11i	− 0.0250234i	0.999922	−	0.0125117i	$-0.00398270\pi$
$132$	0	0
$133$	6.68584e12	1.20794
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.68838e12	−0.860330	−0.430165	−	0.902750i	$-0.641545\pi$
$137$	−5.68838e12	−0.860330	−0.430165	+	0.902750i	$0.641545\pi$
$138$	0	0
$139$	7.81902e12i	1.08408i	0.840351	+	0.542042i	$0.182349\pi$
$139$	7.81902e12i	1.08408i	−0.840351	+	0.542042i	$0.817651\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.83207e12i	0.331198i
$144$	0	0
$145$	6.40526e12	0.689174
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.17409e13	1.98683	0.993415	−	0.114573i	$-0.0365500\pi$
$149$	2.17409e13	1.98683	0.993415	+	0.114573i	$0.0365500\pi$
$150$	0	0
$151$	2.05592e12i	0.173439i	0.996233	+	0.0867193i	$0.0276383\pi$
$151$	2.05592e12i	0.173439i	−0.996233	+	0.0867193i	$0.972362\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.31798e13i	1.67155i
$156$	0	0
$157$	−6.41353e11	−0.0428252	−0.0214126	−	0.999771i	$-0.506816\pi$
$157$	−6.41353e11	−0.0428252	−0.0214126	+	0.999771i	$0.506816\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.15224e13	−0.661590
$162$	0	0
$163$	− 1.98418e13i	− 1.05792i	−0.848645	−	0.528962i	$-0.822581\pi$
$163$	− 1.98418e13i	− 1.05792i	0.848645	−	0.528962i	$-0.177419\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.93970e13i	1.81620i	0.418749	+	0.908102i	$0.362469\pi$
$167$	3.93970e13i	1.81620i	−0.418749	+	0.908102i	$0.637531\pi$
$168$	0	0
$169$	−2.09048e13	−0.897275
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.38842e12	0.312898	0.156449	−	0.987686i	$-0.449995\pi$
$173$	8.38842e12	0.312898	0.156449	+	0.987686i	$0.449995\pi$
$174$	0	0
$175$	1.78984e11i	0.00623142i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 7.52667e12i	− 0.228815i	−0.993434	−	0.114408i	$-0.963503\pi$
$179$	− 7.52667e12i	− 0.228815i	0.993434	−	0.114408i	$-0.0364970\pi$
$180$	0	0
$181$	4.37174e13	1.24332	0.621659	−	0.783288i	$-0.286459\pi$
$181$	4.37174e13	1.24332	0.621659	+	0.783288i	$0.286459\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.89185e13	0.970790
$186$	0	0
$187$	7.37916e13i	1.72567i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.27054e13i	1.49750i	0.662853	+	0.748749i	$0.269345\pi$
$191$	7.27054e13i	1.49750i	−0.662853	+	0.748749i	$0.730655\pi$
$192$	0	0
$193$	6.08414e13	1.17721	0.588607	−	0.808419i	$-0.299677\pi$
$193$	6.08414e13	1.17721	0.588607	+	0.808419i	$0.299677\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.45880e12	−0.110498	−0.0552490	−	0.998473i	$-0.517595\pi$
$197$	−6.45880e12	−0.110498	−0.0552490	+	0.998473i	$0.517595\pi$
$198$	0	0
$199$	1.04647e14i	1.68503i	0.538674	+	0.842515i	$0.318926\pi$
$199$	1.04647e14i	1.68503i	−0.538674	+	0.842515i	$0.681074\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.35168e13i	0.478946i
$204$	0	0
$205$	−1.93263e13	−0.260390
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.50377e14	1.80428
$210$	0	0
$211$	9.50396e11i	0.0107699i	0.999986	+	0.00538493i	$0.00171408\pi$
$211$	9.50396e11i	0.0107699i	−0.999986	+	0.00538493i	$0.998286\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.07810e14i	1.09151i
$216$	0	0
$217$	−1.21293e14	−1.16166
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.23586e13	0.535233
$222$	0	0
$223$	− 8.18837e13i	− 0.665837i	−0.942956	−	0.332919i	$-0.891967\pi$
$223$	− 8.18837e13i	− 0.665837i	0.942956	−	0.332919i	$-0.108033\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.39096e14i	1.01662i	0.861173	+	0.508312i	$0.169730\pi$
$227$	1.39096e14i	1.01662i	−0.861173	+	0.508312i	$0.830270\pi$
$228$	0	0
$229$	1.73493e14	1.20301	0.601506	−	0.798868i	$-0.294567\pi$
$229$	1.73493e14	1.20301	0.601506	+	0.798868i	$0.294567\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.06486e14	0.665514	0.332757	−	0.943013i	$-0.392021\pi$
$233$	1.06486e14	0.665514	0.332757	+	0.943013i	$0.392021\pi$
$234$	0	0
$235$	1.25655e14i	0.746056i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 4.00355e13i	− 0.214812i	−0.994215	−	0.107406i	$-0.965746\pi$
$239$	− 4.00355e13i	− 0.214812i	0.994215	−	0.107406i	$-0.0342544\pi$
$240$	0	0
$241$	1.29194e13	0.0659388	0.0329694	−	0.999456i	$-0.489504\pi$
$241$	1.29194e13	0.0659388	0.0329694	+	0.999456i	$0.489504\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.12251e14	−0.519033
$246$	0	0
$247$	− 1.27079e14i	− 0.559617i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 1.81611e14i	− 0.726272i	−0.931736	−	0.363136i	$-0.881706\pi$
$251$	− 1.81611e14i	− 0.726272i	0.931736	−	0.363136i	$-0.118294\pi$
$252$	0	0
$253$	−2.59162e14	−0.988206
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.46206e14	−1.89565	−0.947824	−	0.318794i	$-0.896722\pi$
$257$	−5.46206e14	−1.89565	−0.947824	+	0.318794i	$0.896722\pi$
$258$	0	0
$259$	2.03648e14i	0.674656i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.84090e13i	0.0556282i	0.999613	+	0.0278141i	$0.00885464\pi$
$263$	1.84090e13i	0.0556282i	−0.999613	+	0.0278141i	$0.991145\pi$
$264$	0	0
$265$	3.00264e14	0.867019
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.50817e14	0.398050	0.199025	−	0.979994i	$-0.436223\pi$
$269$	1.50817e14	0.398050	0.199025	+	0.979994i	$0.436223\pi$
$270$	0	0
$271$	− 2.49717e14i	− 0.630423i	−0.949021	−	0.315212i	$-0.897924\pi$
$271$	− 2.49717e14i	− 0.630423i	0.949021	−	0.315212i	$-0.102076\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.02570e12i	0.00930776i
$276$	0	0
$277$	−4.06823e14	−0.900589	−0.450295	−	0.892880i	$-0.648681\pi$
$277$	−4.06823e14	−0.900589	−0.450295	+	0.892880i	$0.648681\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.48668e14	1.31760	0.658802	−	0.752317i	$-0.271064\pi$
$281$	6.48668e14	1.31760	0.658802	+	0.752317i	$0.271064\pi$
$282$	0	0
$283$	− 2.83689e14i	− 0.552235i	−0.961124	−	0.276117i	$-0.910952\pi$
$283$	− 2.83689e14i	− 0.552235i	0.961124	−	0.276117i	$-0.0890478\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.01129e14i	− 0.180960i
$288$	0	0
$289$	1.04217e15	1.78876
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.25004e14	−0.513668	−0.256834	−	0.966456i	$-0.582679\pi$
$293$	−3.25004e14	−0.513668	−0.256834	+	0.966456i	$0.582679\pi$
$294$	0	0
$295$	− 1.36685e14i	− 0.207390i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.19008e14i	0.306502i
$300$	0	0
$301$	−5.64136e14	−0.758551
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.31084e15	−1.62837
$306$	0	0
$307$	− 4.13833e14i	− 0.494305i	−0.968977	−	0.247153i	$-0.920505\pi$
$307$	− 4.13833e14i	− 0.494305i	0.968977	−	0.247153i	$-0.0794949\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 9.11794e14i	− 1.00771i	−0.863789	−	0.503853i	$-0.831915\pi$
$311$	− 9.11794e14i	− 1.00771i	0.863789	−	0.503853i	$-0.168085\pi$
$312$	0	0
$313$	−5.82226e14	−0.619192	−0.309596	−	0.950868i	$-0.600194\pi$
$313$	−5.82226e14	−0.619192	−0.309596	+	0.950868i	$0.600194\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.83821e14	0.476792	0.238396	−	0.971168i	$-0.423378\pi$
$317$	4.83821e14	0.476792	0.238396	+	0.971168i	$0.423378\pi$
$318$	0	0
$319$	7.53857e14i	0.715393i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 3.31112e15i	− 2.91581i
$324$	0	0
$325$	3.40198e12	0.00288690
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.57512e14	−0.518476
$330$	0	0
$331$	4.25667e14i	0.323670i	0.986818	+	0.161835i	$0.0517412\pi$
$331$	4.25667e14i	0.323670i	−0.986818	+	0.161835i	$0.948259\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.54983e14i	0.321904i
$336$	0	0
$337$	−1.74188e15	−1.18916	−0.594579	−	0.804037i	$-0.702681\pi$
$337$	−1.74188e15	−1.18916	−0.594579	+	0.804037i	$0.702681\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.72811e15	−1.73515
$342$	0	0
$343$	− 1.71394e15i	− 1.05252i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.51093e14i	− 0.143833i	−0.997411	−	0.0719165i	$-0.977088\pi$
$347$	− 2.51093e14i	− 0.143833i	0.997411	−	0.0719165i	$-0.0229115\pi$
$348$	0	0
$349$	7.29263e14	0.403582	0.201791	−	0.979429i	$-0.435324\pi$
$349$	7.29263e14	0.403582	0.201791	+	0.979429i	$0.435324\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.24515e14	0.322771	0.161386	−	0.986891i	$-0.448404\pi$
$353$	6.24515e14	0.322771	0.161386	+	0.986891i	$0.448404\pi$
$354$	0	0
$355$	− 3.32694e15i	− 1.66217i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 3.43108e15i	− 1.60275i	−0.598165	−	0.801373i	$-0.704103\pi$
$359$	− 3.43108e15i	− 1.60275i	0.598165	−	0.801373i	$-0.295897\pi$
$360$	0	0
$361$	−4.53430e15	−2.04865
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.48979e15	1.47585
$366$	0	0
$367$	− 3.12590e15i	− 1.27932i	−0.768658	−	0.639660i	$-0.779075\pi$
$367$	− 3.12590e15i	− 1.27932i	0.768658	−	0.639660i	$-0.220925\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.57119e15i	0.602540i
$372$	0	0
$373$	3.60528e15	1.33871	0.669354	−	0.742944i	$-0.266571\pi$
$373$	3.60528e15	1.33871	0.669354	+	0.742944i	$0.266571\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.37058e14	0.221886
$378$	0	0
$379$	2.94829e15i	0.994797i	0.867522	+	0.497399i	$0.165711\pi$
$379$	2.94829e15i	0.994797i	−0.867522	+	0.497399i	$0.834289\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 1.60473e15i	− 0.508404i	−0.967151	−	0.254202i	$-0.918187\pi$
$383$	− 1.60473e15i	− 0.508404i	0.967151	−	0.254202i	$-0.0818128\pi$
$384$	0	0
$385$	2.31763e15	0.711670
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.95141e15	−1.14039	−0.570197	−	0.821508i	$-0.693133\pi$
$389$	−3.95141e15	−1.14039	−0.570197	+	0.821508i	$0.693133\pi$
$390$	0	0
$391$	5.70641e15i	1.59699i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 4.58990e14i	− 0.120843i
$396$	0	0
$397$	1.10374e15	0.281919	0.140959	−	0.990015i	$-0.454981\pi$
$397$	1.10374e15	0.281919	0.140959	+	0.990015i	$0.454981\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.98081e15	−1.19794	−0.598968	−	0.800773i	$-0.704423\pi$
$401$	−4.98081e15	−1.19794	−0.598968	+	0.800773i	$0.704423\pi$
$402$	0	0
$403$	2.30543e15i	0.538173i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.58044e15i	1.00772i
$408$	0	0
$409$	−2.94769e15	−0.629712	−0.314856	−	0.949139i	$-0.601956\pi$
$409$	−2.94769e15	−0.629712	−0.314856	+	0.949139i	$0.601956\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.15229e14	0.144127
$414$	0	0
$415$	8.90328e14i	0.174286i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 2.91711e15i	− 0.539098i	−0.962987	−	0.269549i	$-0.913125\pi$
$419$	− 2.91711e15i	− 0.539098i	0.962987	−	0.269549i	$-0.0868747\pi$
$420$	0	0
$421$	1.61187e15	0.289492	0.144746	−	0.989469i	$-0.453764\pi$
$421$	1.61187e15	0.289492	0.144746	+	0.989469i	$0.453764\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.86408e13	0.0150418
$426$	0	0
$427$	− 6.85925e15i	− 1.13164i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.36128e15i	1.14839i	0.818718	+	0.574196i	$0.194685\pi$
$431$	7.36128e15i	1.14839i	−0.818718	+	0.574196i	$0.805315\pi$
$432$	0	0
$433$	9.33898e15	1.41701	0.708504	−	0.705707i	$-0.249371\pi$
$433$	9.33898e15	1.41701	0.708504	+	0.705707i	$0.249371\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.16289e16	1.66974
$438$	0	0
$439$	− 3.11346e15i	− 0.434967i	−0.976064	−	0.217483i	$-0.930215\pi$
$439$	− 3.11346e15i	− 0.434967i	0.976064	−	0.217483i	$-0.0697848\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.08063e16i	1.42974i	0.699259	+	0.714868i	$0.253513\pi$
$443$	1.08063e16i	1.42974i	−0.699259	+	0.714868i	$0.746487\pi$
$444$	0	0
$445$	−6.47585e15	−0.833944
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.96447e14	0.0727937	0.0363968	−	0.999337i	$-0.488412\pi$
$449$	5.96447e14	0.0727937	0.0363968	+	0.999337i	$0.488412\pi$
$450$	0	0
$451$	− 2.27457e15i	− 0.270297i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 1.95854e15i	− 0.220732i
$456$	0	0
$457$	−4.87925e15	−0.535619	−0.267810	−	0.963472i	$-0.586300\pi$
$457$	−4.87925e15	−0.535619	−0.267810	+	0.963472i	$0.586300\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.12473e15	0.950636	0.475318	−	0.879814i	$-0.342333\pi$
$461$	9.12473e15	0.950636	0.475318	+	0.879814i	$0.342333\pi$
$462$	0	0
$463$	1.31074e16i	1.33054i	0.746601	+	0.665272i	$0.231685\pi$
$463$	1.31074e16i	1.33054i	−0.746601	+	0.665272i	$0.768315\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.13072e16i	1.09006i	0.838415	+	0.545032i	$0.183483\pi$
$467$	1.13072e16i	1.09006i	−0.838415	+	0.545032i	$0.816517\pi$
$468$	0	0
$469$	−2.38079e15	−0.223709
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.26885e16	−1.13304
$474$	0	0
$475$	− 1.80638e14i	− 0.0157271i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 1.57501e16i	− 1.30398i	−0.758227	−	0.651990i	$-0.773934\pi$
$479$	− 1.57501e16i	− 1.30398i	0.758227	−	0.651990i	$-0.226066\pi$
$480$	0	0
$481$	3.87077e15	0.312556
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.12466e16	0.864117
$486$	0	0
$487$	2.25680e14i	0.0169169i	0.999964	+	0.00845843i	$0.00269243\pi$
$487$	2.25680e14i	0.0169169i	−0.999964	+	0.00845843i	$0.997308\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.16760e16i	0.833307i	0.909065	+	0.416654i	$0.136797\pi$
$491$	1.16760e16i	0.833307i	−0.909065	+	0.416654i	$0.863203\pi$
$492$	0	0
$493$	1.65990e16	1.15611
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.74089e16	1.15513
$498$	0	0
$499$	2.70706e16i	1.75346i	0.480986	+	0.876728i	$0.340279\pi$
$499$	2.70706e16i	1.75346i	−0.480986	+	0.876728i	$0.659721\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.59616e15i	0.407271i	0.979047	+	0.203636i	$0.0652758\pi$
$503$	6.59616e15i	0.407271i	−0.979047	+	0.203636i	$0.934724\pi$
$504$	0	0
$505$	2.91903e15	0.175991
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.89831e15	0.109159	0.0545797	−	0.998509i	$-0.482618\pi$
$509$	1.89831e15	0.109159	0.0545797	+	0.998509i	$0.482618\pi$
$510$	0	0
$511$	1.82610e16i	1.02565i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.40227e16i	0.751601i
$516$	0	0
$517$	−1.47887e16	−0.774439
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.88876e15	0.144439	0.0722195	−	0.997389i	$-0.476992\pi$
$521$	2.88876e15	0.144439	0.0722195	+	0.997389i	$0.476992\pi$
$522$	0	0
$523$	8.02571e15i	0.392169i	0.980587	+	0.196085i	$0.0628227\pi$
$523$	8.02571e15i	0.392169i	−0.980587	+	0.196085i	$0.937177\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.00696e16i	2.80408i
$528$	0	0
$529$	1.87328e15	0.0854807
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.92216e15	−0.0838353
$534$	0	0
$535$	− 3.85971e16i	− 1.64601i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1.32112e16i	− 0.538779i
$540$	0	0
$541$	1.62612e16	0.648587	0.324294	−	0.945956i	$-0.394873\pi$
$541$	1.62612e16	0.648587	0.324294	+	0.945956i	$0.394873\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.54578e16	0.589887
$546$	0	0
$547$	− 2.07666e16i	− 0.775249i	−0.921817	−	0.387624i	$-0.873296\pi$
$547$	− 2.07666e16i	− 0.775249i	0.921817	−	0.387624i	$-0.126704\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 3.38265e16i	− 1.20878i
$552$	0	0
$553$	2.40176e15	0.0839805
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.37138e13	−0.00179868	−0.000899341	−	1.00000i	$-0.500286\pi$
$557$	−5.37138e13	−0.00179868	−0.000899341		1.00000i	$0.500286\pi$
$558$	0	0
$559$	1.07226e16i	0.351422i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 1.85111e16i	− 0.581276i	−0.956833	−	0.290638i	$-0.906132\pi$
$563$	− 1.85111e16i	− 0.581276i	0.956833	−	0.290638i	$-0.0938675\pi$
$564$	0	0
$565$	−2.29404e16	−0.705197
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.88964e16	−1.44080	−0.720400	−	0.693559i	$-0.756042\pi$
$569$	−4.88964e16	−1.44080	−0.720400	+	0.693559i	$0.756042\pi$
$570$	0	0
$571$	8.66666e15i	0.250055i	0.992153	+	0.125027i	$0.0399019\pi$
$571$	8.66666e15i	0.250055i	−0.992153	+	0.125027i	$0.960098\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.11313e14i	0.00861372i
$576$	0	0
$577$	−3.97492e16	−1.07714	−0.538572	−	0.842579i	$-0.681036\pi$
$577$	−3.97492e16	−1.07714	−0.538572	+	0.842579i	$0.681036\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.65882e15	−0.121121
$582$	0	0
$583$	3.53391e16i	0.900004i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3.81976e16i	− 0.933701i	−0.884336	−	0.466850i	$-0.845389\pi$
$587$	− 3.81976e16i	− 0.933701i	0.884336	−	0.466850i	$-0.154611\pi$
$588$	0	0
$589$	1.22414e17	2.93183
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.02258e16	−0.465133	−0.232566	−	0.972581i	$-0.574712\pi$
$593$	−2.02258e16	−0.465133	−0.232566	+	0.972581i	$0.574712\pi$
$594$	0	0
$595$	− 5.10312e16i	− 1.15010i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 4.30315e16i	− 0.931591i	−0.884892	−	0.465795i	$-0.845768\pi$
$599$	− 4.30315e16i	− 0.931591i	0.884892	−	0.465795i	$-0.154232\pi$
$600$	0	0
$601$	−2.70930e16	−0.574924	−0.287462	−	0.957792i	$-0.592812\pi$
$601$	−2.70930e16	−0.574924	−0.287462	+	0.957792i	$0.592812\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.31123e15	0.0675239
$606$	0	0
$607$	− 8.64234e16i	− 1.72782i	−0.503644	−	0.863911i	$-0.668008\pi$
$607$	− 8.64234e16i	− 1.72782i	0.503644	−	0.863911i	$-0.331992\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.24974e16i	0.240200i
$612$	0	0
$613$	−6.29725e16	−1.18683	−0.593414	−	0.804897i	$-0.702220\pi$
$613$	−6.29725e16	−1.18683	−0.593414	+	0.804897i	$0.702220\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7.02772e16	−1.27381	−0.636903	−	0.770944i	$-0.719785\pi$
$617$	−7.02772e16	−1.27381	−0.636903	+	0.770944i	$0.719785\pi$
$618$	0	0
$619$	− 1.87727e16i	− 0.333721i	−0.985980	−	0.166861i	$-0.946637\pi$
$619$	− 1.87727e16i	− 0.333721i	0.985980	−	0.166861i	$-0.0533630\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 3.38862e16i	− 0.579554i
$624$	0	0
$625$	−5.90629e16	−0.990912
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.00856e17	1.62853
$630$	0	0
$631$	8.24626e15i	0.130641i	0.997864	+	0.0653207i	$0.0208070\pi$
$631$	8.24626e15i	0.130641i	−0.997864	+	0.0653207i	$0.979193\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 4.42144e16i	− 0.674406i
$636$	0	0
$637$	−1.11643e16	−0.167108
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.21355e17	−1.74949	−0.874744	−	0.484585i	$-0.838971\pi$
$641$	−1.21355e17	−1.74949	−0.874744	+	0.484585i	$0.838971\pi$
$642$	0	0
$643$	4.19223e16i	0.593170i	0.955006	+	0.296585i	$0.0958478\pi$
$643$	4.19223e16i	0.593170i	−0.955006	+	0.296585i	$0.904152\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.27957e17i	1.74437i	0.489174	+	0.872186i	$0.337298\pi$
$647$	1.27957e17i	1.74437i	−0.489174	+	0.872186i	$0.662702\pi$
$648$	0	0
$649$	1.60869e16	0.215280
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.21056e16	0.801035	0.400517	−	0.916289i	$-0.368830\pi$
$653$	6.21056e16	0.801035	0.400517	+	0.916289i	$0.368830\pi$
$654$	0	0
$655$	1.96711e15i	0.0249105i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1.05753e17i	− 1.29116i	−0.763694	−	0.645579i	$-0.776616\pi$
$659$	− 1.05753e17i	− 1.29116i	0.763694	−	0.645579i	$-0.223384\pi$
$660$	0	0
$661$	−2.84542e16	−0.341144	−0.170572	−	0.985345i	$-0.554562\pi$
$661$	−2.84542e16	−0.341144	−0.170572	+	0.985345i	$0.554562\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.03995e17	−1.20249
$666$	0	0
$667$	5.82969e16i	0.662049i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1.54278e17i	− 1.69032i
$672$	0	0
$673$	−8.30595e14	−0.00893920	−0.00446960	−	0.999990i	$-0.501423\pi$
$673$	−8.30595e14	−0.00893920	−0.00446960	+	0.999990i	$0.501423\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.91998e16	0.718742	0.359371	−	0.933195i	$-0.382991\pi$
$677$	6.91998e16	0.718742	0.359371	+	0.933195i	$0.382991\pi$
$678$	0	0
$679$	5.88503e16i	0.600523i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1.01418e17i	− 0.999058i	−0.866297	−	0.499529i	$-0.833506\pi$
$683$	− 1.01418e17i	− 0.999058i	0.866297	−	0.499529i	$-0.166494\pi$
$684$	0	0
$685$	8.84798e16	0.856447
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.98639e16	0.279145
$690$	0	0
$691$	− 7.41152e16i	− 0.680830i	−0.940275	−	0.340415i	$-0.889432\pi$
$691$	− 7.41152e16i	− 0.680830i	0.940275	−	0.340415i	$-0.110568\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.21621e17i	− 1.07919i
$696$	0	0
$697$	−5.00832e16	−0.436813
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5.21853e16	0.439785	0.219892	−	0.975524i	$-0.429429\pi$
$701$	5.21853e16	0.439785	0.219892	+	0.975524i	$0.429429\pi$
$702$	0	0
$703$	− 2.05530e17i	− 1.70272i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.52744e16i	0.122306i
$708$	0	0
$709$	3.85999e16	0.303885	0.151942	−	0.988389i	$-0.451447\pi$
$709$	3.85999e16	0.303885	0.151942	+	0.988389i	$0.451447\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.10969e17	−1.60576
$714$	0	0
$715$	− 4.40514e16i	− 0.329703i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.30866e15i	0.0673774i	0.999432	+	0.0336887i	$0.0107255\pi$
$719$	9.30866e15i	0.0673774i	−0.999432	+	0.0336887i	$0.989275\pi$
$720$	0	0
$721$	−7.33763e16	−0.522329
$722$	0	0
$723$	0	0
$724$	0	0
$725$	9.05557e14	0.00623574
$726$	0	0
$727$	− 6.14612e15i	− 0.0416289i	−0.999783	−	0.0208144i	$-0.993374\pi$
$727$	− 6.14612e15i	− 0.0416289i	0.999783	−	0.0208144i	$-0.00662592\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.79385e17i	1.83104i
$732$	0	0
$733$	−5.13778e16	−0.331247	−0.165623	−	0.986189i	$-0.552964\pi$
$733$	−5.13778e16	−0.331247	−0.165623	+	0.986189i	$0.552964\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.35484e16	−0.334150
$738$	0	0
$739$	5.29636e16i	0.325170i	0.986695	+	0.162585i	$0.0519832\pi$
$739$	5.29636e16i	0.325170i	−0.986695	+	0.162585i	$0.948017\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.82098e17i	1.08236i	0.840906	+	0.541181i	$0.182023\pi$
$743$	1.82098e17i	1.08236i	−0.840906	+	0.541181i	$0.817977\pi$
$744$	0	0
$745$	−3.38169e17	−1.97786
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.01967e17	1.14390
$750$	0	0
$751$	− 4.12833e16i	− 0.230110i	−0.993359	−	0.115055i	$-0.963296\pi$
$751$	− 4.12833e16i	− 0.230110i	0.993359	−	0.115055i	$-0.0367044\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 3.19788e16i	− 0.172656i
$756$	0	0
$757$	2.63772e17	1.40170	0.700848	−	0.713311i	$-0.252805\pi$
$757$	2.63772e17	1.40170	0.700848	+	0.713311i	$0.252805\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.64979e16	0.187914	0.0939571	−	0.995576i	$-0.470048\pi$
$761$	3.64979e16	0.187914	0.0939571	+	0.995576i	$0.470048\pi$
$762$	0	0
$763$	8.08861e16i	0.409946i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.35945e16i	− 0.0667712i
$768$	0	0
$769$	1.68115e17	0.812923	0.406462	−	0.913668i	$-0.366762\pi$
$769$	1.68115e17	0.812923	0.406462	+	0.913668i	$0.366762\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.86198e17	−0.872768	−0.436384	−	0.899760i	$-0.643741\pi$
$773$	−1.86198e17	−0.872768	−0.436384	+	0.899760i	$0.643741\pi$
$774$	0	0
$775$	3.27710e15i	0.0151244i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.02063e17i	0.456713i
$780$	0	0
$781$	3.91559e17	1.72540
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.97591e15	0.0426319
$786$	0	0
$787$	2.00856e17i	0.845351i	0.906281	+	0.422676i	$0.138909\pi$
$787$	2.00856e17i	0.845351i	−0.906281	+	0.422676i	$0.861091\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1.20040e17i	− 0.490081i
$792$	0	0
$793$	−1.30375e17	−0.524269
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.33457e17	0.520706	0.260353	−	0.965514i	$-0.416161\pi$
$797$	1.33457e17	0.520706	0.260353	+	0.965514i	$0.416161\pi$
$798$	0	0
$799$	3.25629e17i	1.25153i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.10726e17i	1.53200i
$804$	0	0
$805$	1.79225e17	0.658604
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.28849e17	0.816316	0.408158	−	0.912911i	$-0.366171\pi$
$809$	2.28849e17	0.816316	0.408158	+	0.912911i	$0.366171\pi$
$810$	0	0
$811$	1.29269e17i	0.454327i	0.973857	+	0.227164i	$0.0729453\pi$
$811$	1.29269e17i	0.454327i	−0.973857	+	0.227164i	$0.927055\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.08628e17i	1.05315i
$816$	0	0
$817$	5.69349e17	1.91446
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.45237e17	1.78044	0.890218	−	0.455535i	$-0.150552\pi$
$821$	5.45237e17	1.78044	0.890218	+	0.455535i	$0.150552\pi$
$822$	0	0
$823$	5.21467e17i	1.67814i	0.544024	+	0.839070i	$0.316900\pi$
$823$	5.21467e17i	1.67814i	−0.544024	+	0.839070i	$0.683100\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 4.18104e17i	− 1.30693i	−0.756958	−	0.653463i	$-0.773315\pi$
$827$	− 4.18104e17i	− 1.30693i	0.756958	−	0.653463i	$-0.226685\pi$
$828$	0	0
$829$	−6.37037e17	−1.96262	−0.981312	−	0.192422i	$-0.938366\pi$
$829$	−6.37037e17	−1.96262	−0.981312	+	0.192422i	$0.938366\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.90895e17	−0.870694
$834$	0	0
$835$	− 6.12800e17i	− 1.80801i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.74661e17i	1.64755i	0.566914	+	0.823777i	$0.308137\pi$
$839$	5.74661e17i	1.64755i	−0.566914	+	0.823777i	$0.691863\pi$
$840$	0	0
$841$	−1.84239e17	−0.520722
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.25163e17	0.893225
$846$	0	0
$847$	1.73267e16i	0.0469262i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.54212e17i	0.932581i
$852$	0	0
$853$	−2.56302e17	−0.665362	−0.332681	−	0.943039i	$-0.607953\pi$
$853$	−2.56302e17	−0.665362	−0.332681	+	0.943039i	$0.607953\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.74532e17	0.440545	0.220272	−	0.975438i	$-0.429305\pi$
$857$	1.74532e17	0.440545	0.220272	+	0.975438i	$0.429305\pi$
$858$	0	0
$859$	− 3.82606e17i	− 0.952341i	−0.879353	−	0.476171i	$-0.842024\pi$
$859$	− 3.82606e17i	− 0.952341i	0.879353	−	0.476171i	$-0.157976\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.74592e17i	1.14883i	0.818564	+	0.574415i	$0.194770\pi$
$863$	4.74592e17i	1.14883i	−0.818564	+	0.574415i	$0.805230\pi$
$864$	0	0
$865$	−1.30477e17	−0.311486
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.40201e16	0.125440
$870$	0	0
$871$	4.52519e16i	0.103640i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 3.11868e17i	− 0.694900i
$876$	0	0
$877$	−2.46703e17	−0.542221	−0.271110	−	0.962548i	$-0.587391\pi$
$877$	−2.46703e17	−0.542221	−0.271110	+	0.962548i	$0.587391\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−5.98230e17	−1.27942	−0.639709	−	0.768617i	$-0.720945\pi$
$881$	−5.98230e17	−1.27942	−0.639709	+	0.768617i	$0.720945\pi$
$882$	0	0
$883$	2.18140e17i	0.460227i	0.973164	+	0.230113i	$0.0739097\pi$
$883$	2.18140e17i	0.460227i	−0.973164	+	0.230113i	$0.926090\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 6.98691e17i	− 1.43464i	−0.696743	−	0.717321i	$-0.745368\pi$
$887$	− 6.98691e17i	− 1.43464i	0.696743	−	0.717321i	$-0.254632\pi$
$888$	0	0
$889$	2.31361e17	0.468683
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.63588e17	1.30855
$894$	0	0
$895$	1.17073e17i	0.227782i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.13673e17i	1.16246i
$900$	0	0
$901$	7.78123e17	1.45445
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.80000e17	−1.23771
$906$	0	0
$907$	9.99055e17i	1.79451i	0.441513	+	0.897255i	$0.354442\pi$
$907$	9.99055e17i	1.79451i	−0.441513	+	0.897255i	$0.645558\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 2.47310e17i	− 0.432644i	−0.976322	−	0.216322i	$-0.930594\pi$
$911$	− 2.47310e17i	− 0.432644i	0.976322	−	0.216322i	$-0.0694061\pi$
$912$	0	0
$913$	−1.04786e17	−0.180916
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.02933e16	−0.0173117
$918$	0	0
$919$	− 1.00235e18i	− 1.66389i	−0.554855	−	0.831947i	$-0.687226\pi$
$919$	− 1.00235e18i	− 1.66389i	0.554855	−	0.831947i	$-0.312774\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 3.30892e17i	− 0.535151i
$924$	0	0
$925$	5.50218e15	0.00878384
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.61280e17	−0.406454	−0.203227	−	0.979132i	$-0.565143\pi$
$929$	−2.61280e17	−0.406454	−0.203227	+	0.979132i	$0.565143\pi$
$930$	0	0
$931$	5.92804e17i	0.910360i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1.14779e18i	− 1.71788i
$936$	0	0
$937$	1.33555e17	0.197343	0.0986714	−	0.995120i	$-0.468541\pi$
$937$	1.33555e17	0.197343	0.0986714	+	0.995120i	$0.468541\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	9.33135e17	1.34402	0.672012	−	0.740541i	$-0.265430\pi$
$941$	9.33135e17	1.34402	0.672012	+	0.740541i	$0.265430\pi$
$942$	0	0
$943$	− 1.75896e17i	− 0.250142i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.16393e17i	0.715946i	0.933732	+	0.357973i	$0.116532\pi$
$947$	5.16393e17i	0.715946i	−0.933732	+	0.357973i	$0.883468\pi$
$948$	0	0
$949$	3.47090e17	0.475165
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.26980e17	0.569967	0.284984	−	0.958532i	$-0.408012\pi$
$953$	4.26980e17	0.569967	0.284984	+	0.958532i	$0.408012\pi$
$954$	0	0
$955$	− 1.13089e18i	− 1.49074i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.62988e17i	0.595193i
$960$	0	0
$961$	−1.43314e18	−1.81949
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−9.46356e17	−1.17190
$966$	0	0
$967$	7.27292e17i	0.889508i	0.895653	+	0.444754i	$0.146709\pi$
$967$	7.27292e17i	0.889508i	−0.895653	+	0.444754i	$0.853291\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.49312e17i	1.01333i	0.862143	+	0.506666i	$0.169122\pi$
$971$	8.49312e17i	1.01333i	−0.862143	+	0.506666i	$0.830878\pi$
$972$	0	0
$973$	6.36404e17	0.749991
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.76879e17	−1.12324	−0.561621	−	0.827395i	$-0.689822\pi$
$977$	−9.76879e17	−1.12324	−0.561621	+	0.827395i	$0.689822\pi$
$978$	0	0
$979$	− 7.62165e17i	− 0.865671i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	8.30623e17i	0.920625i	0.887757	+	0.460313i	$0.152263\pi$
$983$	8.30623e17i	0.920625i	−0.887757	+	0.460313i	$0.847737\pi$
$984$	0	0
$985$	1.00463e17	0.109999
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.81221e17	−1.04855
$990$	0	0
$991$	1.15773e17i	0.122226i	0.998131	+	0.0611131i	$0.0194650\pi$
$991$	1.15773e17i	0.122226i	−0.998131	+	0.0611131i	$0.980535\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.62773e18i	− 1.67742i
$996$	0	0
$997$	7.47230e17	0.760823	0.380411	−	0.924817i	$-0.375782\pi$
$997$	7.47230e17	0.760823	0.380411	+	0.924817i	$0.375782\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.13.g.g.127.1		4
3.2	odd	2		16.13.c.b.15.3	yes	4
4.3	odd	2	inner	144.13.g.g.127.2		4
12.11	even	2		16.13.c.b.15.2	✓	4
24.5	odd	2		64.13.c.e.63.2		4
24.11	even	2		64.13.c.e.63.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.13.c.b.15.2	✓	4	12.11	even	2
16.13.c.b.15.3	yes	4	3.2	odd	2
64.13.c.e.63.2		4	24.5	odd	2
64.13.c.e.63.3		4	24.11	even	2
144.13.g.g.127.1		4	1.1	even	1	trivial
144.13.g.g.127.2		4	4.3	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:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	144.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-15554.5 q^{5} -81391.8i q^{7} +O(q^{10})\)	\(q-15554.5 q^{5} -81391.8i q^{7} -1.83066e6i q^{11} -1.54702e6 q^{13} -4.03088e7 q^{17} +8.21439e7i q^{19} -1.41567e8i q^{23} -2.19905e6 q^{25} -4.11796e8 q^{29} -1.49024e9i q^{31} +1.26601e9i q^{35} -2.50208e9 q^{37} +1.24249e9 q^{41} -6.93112e9i q^{43} -8.07836e9i q^{47} +7.21666e9 q^{49} -1.93041e10 q^{53} +2.84749e10i q^{55} +8.78748e9i q^{59} +8.42744e10 q^{61} +2.40631e10 q^{65} -2.92509e10i q^{67} +2.13890e11i q^{71} -2.24360e11 q^{73} -1.49001e11 q^{77} +2.95086e10i q^{79} -5.72394e10i q^{83} +6.26982e11 q^{85} +4.16334e11 q^{89} +1.25915e11i q^{91} -1.27770e12i q^{95} -7.23049e11 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 14904 q^{5}+O(q^{10}) \) 4 * q + 14904 * q^5	\( 4 q + 14904 q^{5} + 12552520 q^{13} - 57429000 q^{17} + 565916108 q^{25} + 597450168 q^{29} - 3792664120 q^{37} + 11480589432 q^{41} + 630768772 q^{49} - 94557783240 q^{53} + 87441423944 q^{61} + 408098641392 q^{65} - 759419897720 q^{73} - 498141135360 q^{77} + 1787449271184 q^{85} + 681837812856 q^{89} - 407801291000 q^{97}+O(q^{100}) \) 4 * q + 14904 * q^5 + 12552520 * q^13 - 57429000 * q^17 + 565916108 * q^25 + 597450168 * q^29 - 3792664120 * q^37 + 11480589432 * q^41 + 630768772 * q^49 - 94557783240 * q^53 + 87441423944 * q^61 + 408098641392 * q^65 - 759419897720 * q^73 - 498141135360 * q^77 + 1787449271184 * q^85 + 681837812856 * q^89 - 407801291000 * q^97

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