LMFDB - Embedded newform 144.9.e.b.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,9,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, 9, names="a")

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

chi := DirichletCharacter("144.17");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 9 $
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:	$58.6625198488$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	144.17
Dual form			144.9.e.b.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+912.168i q^{5} -1652.00 q^{7} +O(q^{10})$	$q+912.168i q^{5} -1652.00 q^{7} -13763.1i q^{11} +46304.0 q^{13} +110262. i q^{17} +243664. q^{19} +143282. i q^{23} -441425. q^{25} +305169. i q^{29} -384164. q^{31} -1.50690e6i q^{35} +496982. q^{37} -1.00800e6i q^{41} -5.33444e6 q^{43} +6.45314e6i q^{47} -3.03570e6 q^{49} -2.70826e6i q^{53} +1.25543e7 q^{55} +1.24442e7i q^{59} +2.33537e6 q^{61} +4.22370e7i q^{65} -3.06745e7 q^{67} +1.21807e7i q^{71} -1.15197e7 q^{73} +2.27367e7i q^{77} +2.65824e6 q^{79} -5.18454e7i q^{83} -1.00577e8 q^{85} -3.86612e7i q^{89} -7.64942e7 q^{91} +2.22262e8i q^{95} -5.15952e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3304 q^{7}+O(q^{10}) $ 2 * q - 3304 * q^7	$ 2 q - 3304 q^{7} + 92608 q^{13} + 487328 q^{19} - 882850 q^{25} - 768328 q^{31} + 993964 q^{37} - 10668880 q^{43} - 6071394 q^{49} + 25108560 q^{55} + 4670740 q^{61} - 61348912 q^{67} - 23039456 q^{73} + 5316488 q^{79} - 201154860 q^{85} - 152988416 q^{91} - 103190336 q^{97}+O(q^{100}) $ 2 * q - 3304 * q^7 + 92608 * q^13 + 487328 * q^19 - 882850 * q^25 - 768328 * q^31 + 993964 * q^37 - 10668880 * q^43 - 6071394 * q^49 + 25108560 * q^55 + 4670740 * q^61 - 61348912 * q^67 - 23039456 * q^73 + 5316488 * q^79 - 201154860 * q^85 - 152988416 * q^91 - 103190336 * 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
$5$	912.168i	1.45947i	0.683731	+	0.729734i	$0.260356\pi$
$5$	912.168i	1.45947i	−0.683731	+	0.729734i	$0.739644\pi$
$6$	0	0
$7$	−1652.00	−0.688047	−0.344023	−	0.938961i	$-0.611790\pi$
$7$	−1652.00	−0.688047	−0.344023	+	0.938961i	$0.611790\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 13763.1i	− 0.940040i	−0.882656	−	0.470020i	$-0.844247\pi$
$11$	− 13763.1i	− 0.940040i	0.882656	−	0.470020i	$-0.155753\pi$
$12$	0	0
$13$	46304.0	1.62123	0.810616	−	0.585578i	$-0.199133\pi$
$13$	46304.0	1.62123	0.810616	+	0.585578i	$0.199133\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	110262.i	1.32017i	0.751191	+	0.660085i	$0.229480\pi$
$17$	110262.i	1.32017i	−0.751191	+	0.660085i	$0.770520\pi$
$18$	0	0
$19$	243664.	1.86972	0.934861	−	0.355014i	$-0.115524\pi$
$19$	243664.	1.86972	0.934861	+	0.355014i	$0.115524\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	143282.i	0.512014i	0.966675	+	0.256007i	$0.0824070\pi$
$23$	143282.i	0.512014i	−0.966675	+	0.256007i	$0.917593\pi$
$24$	0	0
$25$	−441425.	−1.13005
$26$	0	0
$27$	0	0
$28$	0	0
$29$	305169.i	0.431468i	0.976452	+	0.215734i	$0.0692144\pi$
$29$	305169.i	0.431468i	−0.976452	+	0.215734i	$0.930786\pi$
$30$	0	0
$31$	−384164.	−0.415978	−0.207989	−	0.978131i	$-0.566692\pi$
$31$	−384164.	−0.415978	−0.207989	+	0.978131i	$0.566692\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.50690e6i	− 1.00418i
$36$	0	0
$37$	496982.	0.265176	0.132588	−	0.991171i	$-0.457671\pi$
$37$	496982.	0.265176	0.132588	+	0.991171i	$0.457671\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1.00800e6i	− 0.356720i	−0.983965	−	0.178360i	$-0.942921\pi$
$41$	− 1.00800e6i	− 0.356720i	0.983965	−	0.178360i	$-0.0570791\pi$
$42$	0	0
$43$	−5.33444e6	−1.56032	−0.780162	−	0.625577i	$-0.784864\pi$
$43$	−5.33444e6	−1.56032	−0.780162	+	0.625577i	$0.784864\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.45314e6i	1.32245i	0.750187	+	0.661226i	$0.229963\pi$
$47$	6.45314e6i	1.32245i	−0.750187	+	0.661226i	$0.770037\pi$
$48$	0	0
$49$	−3.03570e6	−0.526592
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.70826e6i	− 0.343231i	−0.985164	−	0.171616i	$-0.945101\pi$
$53$	− 2.70826e6i	− 0.343231i	0.985164	−	0.171616i	$-0.0548987\pi$
$54$	0	0
$55$	1.25543e7	1.37196
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.24442e7i	1.02697i	0.858098	+	0.513487i	$0.171646\pi$
$59$	1.24442e7i	1.02697i	−0.858098	+	0.513487i	$0.828354\pi$
$60$	0	0
$61$	2.33537e6	0.168669	0.0843347	−	0.996437i	$-0.473124\pi$
$61$	2.33537e6	0.168669	0.0843347	+	0.996437i	$0.473124\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.22370e7i	2.36614i
$66$	0	0
$67$	−3.06745e7	−1.52222	−0.761110	−	0.648622i	$-0.775345\pi$
$67$	−3.06745e7	−1.52222	−0.761110	+	0.648622i	$0.775345\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.21807e7i	0.479334i	0.970855	+	0.239667i	$0.0770382\pi$
$71$	1.21807e7i	0.479334i	−0.970855	+	0.239667i	$0.922962\pi$
$72$	0	0
$73$	−1.15197e7	−0.405649	−0.202825	−	0.979215i	$-0.565012\pi$
$73$	−1.15197e7	−0.405649	−0.202825	+	0.979215i	$0.565012\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.27367e7i	0.646791i
$78$	0	0
$79$	2.65824e6	0.0682475	0.0341237	−	0.999418i	$-0.489136\pi$
$79$	2.65824e6	0.0682475	0.0341237	+	0.999418i	$0.489136\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 5.18454e7i	− 1.09244i	−0.837641	−	0.546221i	$-0.816066\pi$
$83$	− 5.18454e7i	− 1.09244i	0.837641	−	0.546221i	$-0.183934\pi$
$84$	0	0
$85$	−1.00577e8	−1.92675
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 3.86612e7i	− 0.616191i	−0.951355	−	0.308095i	$-0.900308\pi$
$89$	− 3.86612e7i	− 0.616191i	0.951355	−	0.308095i	$-0.0996916\pi$
$90$	0	0
$91$	−7.64942e7	−1.11548
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.22262e8i	2.72880i
$96$	0	0
$97$	−5.15952e7	−0.582803	−0.291402	−	0.956601i	$-0.594122\pi$
$97$	−5.15952e7	−0.582803	−0.291402	+	0.956601i	$0.594122\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.68690e8i	1.62108i	0.585686	+	0.810538i	$0.300825\pi$
$101$	1.68690e8i	1.62108i	−0.585686	+	0.810538i	$0.699175\pi$
$102$	0	0
$103$	−2.89769e7	−0.257456	−0.128728	−	0.991680i	$-0.541089\pi$
$103$	−2.89769e7	−0.257456	−0.128728	+	0.991680i	$0.541089\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.47848e8i	1.12792i	0.825801	+	0.563961i	$0.190723\pi$
$107$	1.47848e8i	1.12792i	−0.825801	+	0.563961i	$0.809277\pi$
$108$	0	0
$109$	1.67988e8	1.19007	0.595035	−	0.803699i	$-0.297138\pi$
$109$	1.67988e8	1.19007	0.595035	+	0.803699i	$0.297138\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.27408e8i	0.781416i	0.920515	+	0.390708i	$0.127770\pi$
$113$	1.27408e8i	0.781416i	−0.920515	+	0.390708i	$0.872230\pi$
$114$	0	0
$115$	−1.30698e8	−0.747268
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 1.82153e8i	− 0.908339i
$120$	0	0
$121$	2.49352e7	0.116325
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 4.63381e7i	− 0.189801i
$126$	0	0
$127$	−1.23676e7	−0.0475413	−0.0237706	−	0.999717i	$-0.507567\pi$
$127$	−1.23676e7	−0.0475413	−0.0237706	+	0.999717i	$0.507567\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.28395e8i	1.45465i	0.686293	+	0.727325i	$0.259237\pi$
$131$	4.28395e8i	1.45465i	−0.686293	+	0.727325i	$0.740763\pi$
$132$	0	0
$133$	−4.02533e8	−1.28646
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.12221e7i	− 0.0602429i	−0.999546	−	0.0301214i	$-0.990411\pi$
$137$	− 2.12221e7i	− 0.0602429i	0.999546	−	0.0301214i	$-0.00958940\pi$
$138$	0	0
$139$	5.68460e8	1.52279	0.761396	−	0.648287i	$-0.224514\pi$
$139$	5.68460e8	1.52279	0.761396	+	0.648287i	$0.224514\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 6.37288e8i	− 1.52402i
$144$	0	0
$145$	−2.78365e8	−0.629713
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.01222e7i	0.0814030i	0.999171	+	0.0407015i	$0.0129593\pi$
$149$	4.01222e7i	0.0814030i	−0.999171	+	0.0407015i	$0.987041\pi$
$150$	0	0
$151$	−6.58716e8	−1.26704	−0.633520	−	0.773726i	$-0.718391\pi$
$151$	−6.58716e8	−1.26704	−0.633520	+	0.773726i	$0.718391\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 3.50422e8i	− 0.607106i
$156$	0	0
$157$	6.20689e7	0.102159	0.0510793	−	0.998695i	$-0.483734\pi$
$157$	6.20689e7	0.102159	0.0510793	+	0.998695i	$0.483734\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2.36703e8i	− 0.352289i
$162$	0	0
$163$	−1.02683e8	−0.145461	−0.0727305	−	0.997352i	$-0.523171\pi$
$163$	−1.02683e8	−0.145461	−0.0727305	+	0.997352i	$0.523171\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 2.77371e8i	− 0.356612i	−0.983975	−	0.178306i	$-0.942938\pi$
$167$	− 2.77371e8i	− 0.356612i	0.983975	−	0.178306i	$-0.0570616\pi$
$168$	0	0
$169$	1.32833e9	1.62839
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 9.63632e8i	− 1.07579i	−0.843012	−	0.537894i	$-0.819220\pi$
$173$	− 9.63632e8i	− 1.07579i	0.843012	−	0.537894i	$-0.180780\pi$
$174$	0	0
$175$	7.29234e8	0.777526
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.01680e8i	0.293856i	0.989147	+	0.146928i	$0.0469386\pi$
$179$	3.01680e8i	0.293856i	−0.989147	+	0.146928i	$0.953061\pi$
$180$	0	0
$181$	1.63834e9	1.52647	0.763236	−	0.646120i	$-0.223609\pi$
$181$	1.63834e9	1.52647	0.763236	+	0.646120i	$0.223609\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.53331e8i	0.387016i
$186$	0	0
$187$	1.51755e9	1.24101
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.67433e7i	0.0576643i	0.999584	+	0.0288321i	$0.00917883\pi$
$191$	7.67433e7i	0.0576643i	−0.999584	+	0.0288321i	$0.990821\pi$
$192$	0	0
$193$	−1.26902e9	−0.914614	−0.457307	−	0.889309i	$-0.651186\pi$
$193$	−1.26902e9	−0.914614	−0.457307	+	0.889309i	$0.651186\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.47220e9i	− 1.64142i	−0.571347	−	0.820709i	$-0.693579\pi$
$197$	− 2.47220e9i	− 1.64142i	0.571347	−	0.820709i	$-0.306421\pi$
$198$	0	0
$199$	1.55476e9	0.991408	0.495704	−	0.868492i	$-0.334910\pi$
$199$	1.55476e9	0.991408	0.495704	+	0.868492i	$0.334910\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 5.04139e8i	− 0.296870i
$204$	0	0
$205$	9.19469e8	0.520621
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 3.35358e9i	− 1.75761i
$210$	0	0
$211$	−1.99692e9	−1.00747	−0.503734	−	0.863859i	$-0.668041\pi$
$211$	−1.99692e9	−1.00747	−0.503734	+	0.863859i	$0.668041\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 4.86590e9i	− 2.27724i
$216$	0	0
$217$	6.34639e8	0.286212
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.10557e9i	2.14030i
$222$	0	0
$223$	2.30092e8	0.0930428	0.0465214	−	0.998917i	$-0.485186\pi$
$223$	2.30092e8	0.0930428	0.0465214	+	0.998917i	$0.485186\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.57317e8i	0.0969091i	0.998825	+	0.0484546i	$0.0154296\pi$
$227$	2.57317e8i	0.0969091i	−0.998825	+	0.0484546i	$0.984570\pi$
$228$	0	0
$229$	−3.57540e9	−1.30012	−0.650058	−	0.759884i	$-0.725255\pi$
$229$	−3.57540e9	−1.30012	−0.650058	+	0.759884i	$0.725255\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.85180e9i	0.628303i	0.949373	+	0.314152i	$0.101720\pi$
$233$	1.85180e9i	0.628303i	−0.949373	+	0.314152i	$0.898280\pi$
$234$	0	0
$235$	−5.88635e9	−1.93008
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.40618e6i	0.00196340i	1.00000		0.000981698i	$0.000312484\pi$
$239$	6.40618e6i	0.00196340i	−1.00000		0.000981698i	$0.999688\pi$
$240$	0	0
$241$	1.12087e9	0.332266	0.166133	−	0.986103i	$-0.446872\pi$
$241$	1.12087e9	0.332266	0.166133	+	0.986103i	$0.446872\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2.76906e9i	− 0.768544i
$246$	0	0
$247$	1.12826e10	3.03125
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.73672e9i	0.437556i	0.975775	+	0.218778i	$0.0702071\pi$
$251$	1.73672e9i	0.437556i	−0.975775	+	0.218778i	$0.929793\pi$
$252$	0	0
$253$	1.97201e9	0.481314
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3.28322e9i	− 0.752605i	−0.926497	−	0.376303i	$-0.877195\pi$
$257$	− 3.28322e9i	− 0.752605i	0.926497	−	0.376303i	$-0.122805\pi$
$258$	0	0
$259$	−8.21014e8	−0.182453
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.51385e9i	1.15248i	0.817282	+	0.576238i	$0.195480\pi$
$263$	5.51385e9i	1.15248i	−0.817282	+	0.576238i	$0.804520\pi$
$264$	0	0
$265$	2.47039e9	0.500935
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.60854e9i	0.498182i	0.968480	+	0.249091i	$0.0801318\pi$
$269$	2.60854e9i	0.498182i	−0.968480	+	0.249091i	$0.919868\pi$
$270$	0	0
$271$	−2.63873e9	−0.489235	−0.244618	−	0.969620i	$-0.578662\pi$
$271$	−2.63873e9	−0.489235	−0.244618	+	0.969620i	$0.578662\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.07539e9i	1.06229i
$276$	0	0
$277$	−5.78673e9	−0.982911	−0.491456	−	0.870903i	$-0.663535\pi$
$277$	−5.78673e9	−0.982911	−0.491456	+	0.870903i	$0.663535\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.20895e10i	1.93903i	0.245030	+	0.969516i	$0.421202\pi$
$281$	1.20895e10i	1.93903i	−0.245030	+	0.969516i	$0.578798\pi$
$282$	0	0
$283$	−7.55769e9	−1.17827	−0.589133	−	0.808036i	$-0.700531\pi$
$283$	−7.55769e9	−1.17827	−0.589133	+	0.808036i	$0.700531\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.66522e9i	0.245440i
$288$	0	0
$289$	−5.18195e9	−0.742851
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 3.17354e9i	− 0.430599i	−0.976548	−	0.215300i	$-0.930927\pi$
$293$	− 3.17354e9i	− 0.430599i	0.976548	−	0.215300i	$-0.0690728\pi$
$294$	0	0
$295$	−1.13512e10	−1.49884
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.63455e9i	0.830093i
$300$	0	0
$301$	8.81249e9	1.07358
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.13025e9i	0.246168i
$306$	0	0
$307$	−1.93718e9	−0.218081	−0.109040	−	0.994037i	$-0.534778\pi$
$307$	−1.93718e9	−0.218081	−0.109040	+	0.994037i	$0.534778\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 1.80970e10i	− 1.93448i	−0.253856	−	0.967242i	$-0.581699\pi$
$311$	− 1.80970e10i	− 1.93448i	0.253856	−	0.967242i	$-0.418301\pi$
$312$	0	0
$313$	−5.98915e9	−0.624005	−0.312002	−	0.950081i	$-0.601000\pi$
$313$	−5.98915e9	−0.624005	−0.312002	+	0.950081i	$0.601000\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.20671e10i	1.19499i	0.801872	+	0.597496i	$0.203838\pi$
$317$	1.20671e10i	1.19499i	−0.801872	+	0.597496i	$0.796162\pi$
$318$	0	0
$319$	4.20008e9	0.405597
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.68669e10i	2.46835i
$324$	0	0
$325$	−2.04397e10	−1.83207
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 1.06606e10i	− 0.909908i
$330$	0	0
$331$	1.08687e9	0.0905452	0.0452726	−	0.998975i	$-0.485584\pi$
$331$	1.08687e9	0.0905452	0.0452726	+	0.998975i	$0.485584\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 2.79802e10i	− 2.22163i
$336$	0	0
$337$	−2.18298e10	−1.69251	−0.846255	−	0.532778i	$-0.821148\pi$
$337$	−2.18298e10	−1.69251	−0.846255	+	0.532778i	$0.821148\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.28730e9i	0.391036i
$342$	0	0
$343$	1.45384e10	1.05037
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.29991e10i	1.58633i	0.609007	+	0.793165i	$0.291568\pi$
$347$	2.29991e10i	1.58633i	−0.609007	+	0.793165i	$0.708432\pi$
$348$	0	0
$349$	1.68033e10	1.13265	0.566323	−	0.824184i	$-0.308366\pi$
$349$	1.68033e10	1.13265	0.566323	+	0.824184i	$0.308366\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.44522e10i	0.930751i	0.885113	+	0.465376i	$0.154081\pi$
$353$	1.44522e10i	0.930751i	−0.885113	+	0.465376i	$0.845919\pi$
$354$	0	0
$355$	−1.11108e10	−0.699572
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.34135e8i	0.0140958i	0.999975	+	0.00704789i	$0.00224343\pi$
$359$	2.34135e8i	0.0140958i	−0.999975	+	0.00704789i	$0.997757\pi$
$360$	0	0
$361$	4.23886e10	2.49586
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 1.05079e10i	− 0.592032i
$366$	0	0
$367$	1.03738e10	0.571841	0.285921	−	0.958253i	$-0.407701\pi$
$367$	1.03738e10	0.571841	0.285921	+	0.958253i	$0.407701\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.47405e9i	0.236159i
$372$	0	0
$373$	1.97882e10	1.02228	0.511141	−	0.859497i	$-0.329223\pi$
$373$	1.97882e10	1.02228	0.511141	+	0.859497i	$0.329223\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.41305e10i	0.699509i
$378$	0	0
$379$	1.53148e9	0.0742257	0.0371128	−	0.999311i	$-0.488184\pi$
$379$	1.53148e9	0.0742257	0.0371128	+	0.999311i	$0.488184\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 2.22097e10i	− 1.03216i	−0.856540	−	0.516081i	$-0.827390\pi$
$383$	− 2.22097e10i	− 1.03216i	0.856540	−	0.516081i	$-0.172610\pi$
$384$	0	0
$385$	−2.07397e10	−0.943972
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 2.12108e10i	− 0.926315i	−0.886276	−	0.463158i	$-0.846716\pi$
$389$	− 2.12108e10i	− 0.926315i	0.886276	−	0.463158i	$-0.153284\pi$
$390$	0	0
$391$	−1.57986e10	−0.675946
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.42476e9i	0.0996050i
$396$	0	0
$397$	−1.65632e10	−0.666778	−0.333389	−	0.942789i	$-0.608192\pi$
$397$	−1.65632e10	−0.666778	−0.333389	+	0.942789i	$0.608192\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 4.19690e10i	− 1.62312i	−0.584269	−	0.811560i	$-0.698619\pi$
$401$	− 4.19690e10i	− 1.62312i	0.584269	−	0.811560i	$-0.301381\pi$
$402$	0	0
$403$	−1.77883e10	−0.674396
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.84003e9i	− 0.249276i
$408$	0	0
$409$	2.24201e10	0.801206	0.400603	−	0.916252i	$-0.368801\pi$
$409$	2.24201e10	0.801206	0.400603	+	0.916252i	$0.368801\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 2.05578e10i	− 0.706606i
$414$	0	0
$415$	4.72917e10	1.59438
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.43586e10i	− 0.465859i	−0.972494	−	0.232930i	$-0.925169\pi$
$419$	− 1.43586e10i	− 0.465859i	0.972494	−	0.232930i	$-0.0748311\pi$
$420$	0	0
$421$	−3.59143e10	−1.14324	−0.571622	−	0.820517i	$-0.693686\pi$
$421$	−3.59143e10	−1.14324	−0.571622	+	0.820517i	$0.693686\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 4.86724e10i	− 1.49186i
$426$	0	0
$427$	−3.85803e9	−0.116052
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 4.79083e10i	− 1.38836i	−0.719802	−	0.694179i	$-0.755767\pi$
$431$	− 4.79083e10i	− 1.38836i	0.719802	−	0.694179i	$-0.244233\pi$
$432$	0	0
$433$	2.27883e10	0.648278	0.324139	−	0.946010i	$-0.394925\pi$
$433$	2.27883e10	0.648278	0.324139	+	0.946010i	$0.394925\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.49128e10i	0.957323i
$438$	0	0
$439$	5.10762e10	1.37518	0.687591	−	0.726098i	$-0.258668\pi$
$439$	5.10762e10	1.37518	0.687591	+	0.726098i	$0.258668\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.11525e10i	0.289572i	0.989463	+	0.144786i	$0.0462493\pi$
$443$	1.11525e10i	0.289572i	−0.989463	+	0.144786i	$0.953751\pi$
$444$	0	0
$445$	3.52655e10	0.899311
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.12386e9i	0.0768611i	0.999261	+	0.0384305i	$0.0122358\pi$
$449$	3.12386e9i	0.0768611i	−0.999261	+	0.0384305i	$0.987764\pi$
$450$	0	0
$451$	−1.38733e10	−0.335331
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 6.97755e10i	− 1.62801i
$456$	0	0
$457$	5.74808e7	0.00131783	0.000658913	−	1.00000i	$-0.499790\pi$
$457$	5.74808e7	0.00131783	0.000658913		1.00000i	$0.499790\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.00093e10i	0.443026i	0.975157	+	0.221513i	$0.0710995\pi$
$461$	2.00093e10i	0.443026i	−0.975157	+	0.221513i	$0.928900\pi$
$462$	0	0
$463$	5.85446e10	1.27398	0.636991	−	0.770871i	$-0.280179\pi$
$463$	5.85446e10	1.27398	0.636991	+	0.770871i	$0.280179\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.57896e10i	− 0.752471i	−0.926524	−	0.376236i	$-0.877218\pi$
$467$	− 3.57896e10i	− 0.752471i	0.926524	−	0.376236i	$-0.122782\pi$
$468$	0	0
$469$	5.06742e10	1.04736
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.34186e10i	1.46677i
$474$	0	0
$475$	−1.07559e11	−2.11288
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.97409e10i	1.51474i	0.652983	+	0.757372i	$0.273517\pi$
$479$	7.97409e10i	1.51474i	−0.652983	+	0.757372i	$0.726483\pi$
$480$	0	0
$481$	2.30123e10	0.429911
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 4.70634e10i	− 0.850583i
$486$	0	0
$487$	5.71961e10	1.01683	0.508417	−	0.861111i	$-0.330231\pi$
$487$	5.71961e10	1.01683	0.508417	+	0.861111i	$0.330231\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.12512e11i	1.93585i	0.251237	+	0.967926i	$0.419163\pi$
$491$	1.12512e11i	1.93585i	−0.251237	+	0.967926i	$0.580837\pi$
$492$	0	0
$493$	−3.36485e10	−0.569611
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.01225e10i	− 0.329804i
$498$	0	0
$499$	9.93290e9	0.160204	0.0801021	−	0.996787i	$-0.474475\pi$
$499$	9.93290e9	0.160204	0.0801021	+	0.996787i	$0.474475\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 1.96515e10i	− 0.306990i	−0.988149	−	0.153495i	$-0.950947\pi$
$503$	− 1.96515e10i	− 0.306990i	0.988149	−	0.153495i	$-0.0490529\pi$
$504$	0	0
$505$	−1.53873e11	−2.36591
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 2.11996e10i	− 0.315832i	−0.987453	−	0.157916i	$-0.949523\pi$
$509$	− 2.11996e10i	− 0.315832i	0.987453	−	0.157916i	$-0.0504775\pi$
$510$	0	0
$511$	1.90306e10	0.279106
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 2.64318e10i	− 0.375749i
$516$	0	0
$517$	8.88154e10	1.24316
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 7.32537e10i	− 0.994212i	−0.867690	−	0.497106i	$-0.834396\pi$
$521$	− 7.32537e10i	− 0.994212i	0.867690	−	0.497106i	$-0.165604\pi$
$522$	0	0
$523$	−4.82939e9	−0.0645485	−0.0322742	−	0.999479i	$-0.510275\pi$
$523$	−4.82939e9	−0.0645485	−0.0322742	+	0.999479i	$0.510275\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.23587e10i	− 0.549161i
$528$	0	0
$529$	5.77811e10	0.737842
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4.66747e10i	− 0.578325i
$534$	0	0
$535$	−1.34862e11	−1.64617
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.17807e10i	0.495017i
$540$	0	0
$541$	−1.11911e11	−1.30643	−0.653213	−	0.757174i	$-0.726579\pi$
$541$	−1.11911e11	−1.30643	−0.653213	+	0.757174i	$0.726579\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.53233e11i	1.73687i
$546$	0	0
$547$	−4.67934e10	−0.522679	−0.261340	−	0.965247i	$-0.584164\pi$
$547$	−4.67934e10	−0.522679	−0.261340	+	0.965247i	$0.584164\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.43587e10i	0.806725i
$552$	0	0
$553$	−4.39142e9	−0.0469574
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.40745e11i	1.46222i	0.682260	+	0.731109i	$0.260997\pi$
$557$	1.40745e11i	1.46222i	−0.682260	+	0.731109i	$0.739003\pi$
$558$	0	0
$559$	−2.47006e11	−2.52965
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.58622e11i	1.57881i	0.613873	+	0.789405i	$0.289611\pi$
$563$	1.58622e11i	1.57881i	−0.613873	+	0.789405i	$0.710389\pi$
$564$	0	0
$565$	−1.16217e11	−1.14045
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.72545e10i	0.355410i	0.984084	+	0.177705i	$0.0568673\pi$
$569$	3.72545e10i	0.355410i	−0.984084	+	0.177705i	$0.943133\pi$
$570$	0	0
$571$	1.93909e11	1.82412	0.912060	−	0.410058i	$-0.134491\pi$
$571$	1.93909e11	1.82412	0.912060	+	0.410058i	$0.134491\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 6.32485e10i	− 0.578600i
$576$	0	0
$577$	1.24181e11	1.12034	0.560171	−	0.828377i	$-0.310735\pi$
$577$	1.24181e11	1.12034	0.560171	+	0.828377i	$0.310735\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.56486e10i	0.751651i
$582$	0	0
$583$	−3.72741e10	−0.322651
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.31313e11i	− 1.94826i	−0.225989	−	0.974130i	$-0.572561\pi$
$587$	− 2.31313e11i	− 1.94826i	0.225989	−	0.974130i	$-0.427439\pi$
$588$	0	0
$589$	−9.36069e10	−0.777762
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.09925e10i	0.574108i	0.957914	+	0.287054i	$0.0926760\pi$
$593$	7.09925e10i	0.574108i	−0.957914	+	0.287054i	$0.907324\pi$
$594$	0	0
$595$	1.66154e11	1.32569
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.60656e11i	1.24793i	0.781453	+	0.623964i	$0.214479\pi$
$599$	1.60656e11i	1.24793i	−0.781453	+	0.623964i	$0.785521\pi$
$600$	0	0
$601$	−1.51515e11	−1.16134	−0.580669	−	0.814140i	$-0.697209\pi$
$601$	−1.51515e11	−1.16134	−0.580669	+	0.814140i	$0.697209\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.27451e10i	0.169772i
$606$	0	0
$607$	1.69914e11	1.25162	0.625811	−	0.779975i	$-0.284768\pi$
$607$	1.69914e11	1.25162	0.625811	+	0.779975i	$0.284768\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.98806e11i	2.14400i
$612$	0	0
$613$	2.40508e11	1.70329	0.851643	−	0.524123i	$-0.175607\pi$
$613$	2.40508e11	1.70329	0.851643	+	0.524123i	$0.175607\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.89685e10i	− 0.475894i	−0.971278	−	0.237947i	$-0.923526\pi$
$617$	− 6.89685e10i	− 0.475894i	0.971278	−	0.237947i	$-0.0764744\pi$
$618$	0	0
$619$	−1.09219e11	−0.743935	−0.371967	−	0.928246i	$-0.621317\pi$
$619$	−1.09219e11	−0.743935	−0.371967	+	0.928246i	$0.621317\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.38683e10i	0.423968i
$624$	0	0
$625$	−1.30164e11	−0.853040
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.47982e10i	0.350077i
$630$	0	0
$631$	5.73490e10	0.361750	0.180875	−	0.983506i	$-0.442107\pi$
$631$	5.73490e10	0.361750	0.180875	+	0.983506i	$0.442107\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.12813e10i	− 0.0693850i
$636$	0	0
$637$	−1.40565e11	−0.853727
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.16494e10i	0.483639i	0.970321	+	0.241819i	$0.0777441\pi$
$641$	8.16494e10i	0.483639i	−0.970321	+	0.241819i	$0.922256\pi$
$642$	0	0
$643$	−1.24348e11	−0.727438	−0.363719	−	0.931509i	$-0.618493\pi$
$643$	−1.24348e11	−0.727438	−0.363719	+	0.931509i	$0.618493\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.93727e11i	− 1.10554i	−0.833335	−	0.552768i	$-0.813572\pi$
$647$	− 1.93727e11i	− 1.10554i	0.833335	−	0.552768i	$-0.186428\pi$
$648$	0	0
$649$	1.71271e11	0.965396
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.96545e11i	− 1.63094i	−0.578800	−	0.815469i	$-0.696479\pi$
$653$	− 2.96545e11i	− 1.63094i	0.578800	−	0.815469i	$-0.303521\pi$
$654$	0	0
$655$	−3.90768e11	−2.12302
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 3.76214e11i	− 1.99477i	−0.0722538	−	0.997386i	$-0.523019\pi$
$659$	− 3.76214e11i	− 1.99477i	0.0722538	−	0.997386i	$-0.476981\pi$
$660$	0	0
$661$	1.32954e11	0.696459	0.348229	−	0.937409i	$-0.386783\pi$
$661$	1.32954e11	0.696459	0.348229	+	0.937409i	$0.386783\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 3.67178e11i	− 1.87754i
$666$	0	0
$667$	−4.37254e10	−0.220917
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 3.21420e10i	− 0.158556i
$672$	0	0
$673$	−9.72751e9	−0.0474178	−0.0237089	−	0.999719i	$-0.507547\pi$
$673$	−9.72751e9	−0.0474178	−0.0237089	+	0.999719i	$0.507547\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.98501e11i	1.42099i	0.703702	+	0.710495i	$0.251529\pi$
$677$	2.98501e11i	1.42099i	−0.703702	+	0.710495i	$0.748471\pi$
$678$	0	0
$679$	8.52352e10	0.400996
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1.13498e11i	− 0.521559i	−0.965398	−	0.260780i	$-0.916020\pi$
$683$	− 1.13498e11i	− 0.521559i	0.965398	−	0.260780i	$-0.0839796\pi$
$684$	0	0
$685$	1.93581e10	0.0879225
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1.25403e11i	− 0.556457i
$690$	0	0
$691$	1.04515e11	0.458423	0.229212	−	0.973377i	$-0.426385\pi$
$691$	1.04515e11	0.458423	0.229212	+	0.973377i	$0.426385\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.18531e11i	2.22247i
$696$	0	0
$697$	1.11145e11	0.470931
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1.73022e11i	− 0.716522i	−0.933621	−	0.358261i	$-0.883370\pi$
$701$	− 1.73022e11i	− 0.716522i	0.933621	−	0.358261i	$-0.116630\pi$
$702$	0	0
$703$	1.21097e11	0.495805
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2.78676e11i	− 1.11538i
$708$	0	0
$709$	−4.03858e10	−0.159825	−0.0799123	−	0.996802i	$-0.525464\pi$
$709$	−4.03858e10	−0.159825	−0.0799123	+	0.996802i	$0.525464\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 5.50440e10i	− 0.212986i
$714$	0	0
$715$	5.81313e11	2.22426
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.84303e11i	1.81218i	0.423085	+	0.906090i	$0.360947\pi$
$719$	4.84303e11i	1.81218i	−0.423085	+	0.906090i	$0.639053\pi$
$720$	0	0
$721$	4.78698e10	0.177142
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1.34709e11i	− 0.487579i
$726$	0	0
$727$	−7.60391e10	−0.272207	−0.136104	−	0.990695i	$-0.543458\pi$
$727$	−7.60391e10	−0.272207	−0.136104	+	0.990695i	$0.543458\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 5.88186e11i	− 2.05990i
$732$	0	0
$733$	2.69342e10	0.0933015	0.0466507	−	0.998911i	$-0.485145\pi$
$733$	2.69342e10	0.0933015	0.0466507	+	0.998911i	$0.485145\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.22176e11i	1.43095i
$738$	0	0
$739$	1.91084e11	0.640690	0.320345	−	0.947301i	$-0.396201\pi$
$739$	1.91084e11	0.640690	0.320345	+	0.947301i	$0.396201\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 4.04708e11i	− 1.32796i	−0.747749	−	0.663982i	$-0.768865\pi$
$743$	− 4.04708e11i	− 1.32796i	0.747749	−	0.663982i	$-0.231135\pi$
$744$	0	0
$745$	−3.65982e10	−0.118805
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 2.44244e11i	− 0.776063i
$750$	0	0
$751$	3.81041e11	1.19788	0.598939	−	0.800795i	$-0.295589\pi$
$751$	3.81041e11	1.19788	0.598939	+	0.800795i	$0.295589\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 6.00859e11i	− 1.84921i
$756$	0	0
$757$	1.75221e11	0.533584	0.266792	−	0.963754i	$-0.414036\pi$
$757$	1.75221e11	0.533584	0.266792	+	0.963754i	$0.414036\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 6.32798e10i	− 0.188680i	−0.995540	−	0.0943401i	$-0.969926\pi$
$761$	− 6.32798e10i	− 0.188680i	0.995540	−	0.0943401i	$-0.0300741\pi$
$762$	0	0
$763$	−2.77517e11	−0.818824
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.76217e11i	1.66496i
$768$	0	0
$769$	3.62797e11	1.03743	0.518715	−	0.854947i	$-0.326411\pi$
$769$	3.62797e11	1.03743	0.518715	+	0.854947i	$0.326411\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 2.37965e11i	− 0.666492i	−0.942840	−	0.333246i	$-0.891856\pi$
$773$	− 2.37965e11i	− 0.666492i	0.942840	−	0.333246i	$-0.108144\pi$
$774$	0	0
$775$	1.69580e11	0.470075
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2.45614e11i	− 0.666967i
$780$	0	0
$781$	1.67644e11	0.450593
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5.66172e10i	0.149097i
$786$	0	0
$787$	−6.18602e10	−0.161255	−0.0806274	−	0.996744i	$-0.525692\pi$
$787$	−6.18602e10	−0.161255	−0.0806274	+	0.996744i	$0.525692\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 2.10478e11i	− 0.537650i
$792$	0	0
$793$	1.08137e11	0.273452
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.13523e11i	0.529191i	0.964360	+	0.264595i	$0.0852384\pi$
$797$	2.13523e11i	0.529191i	−0.964360	+	0.264595i	$0.914762\pi$
$798$	0	0
$799$	−7.11536e11	−1.74586
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.58547e11i	0.381327i
$804$	0	0
$805$	2.15913e11	0.514155
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.08277e11i	0.252779i	0.991981	+	0.126389i	$0.0403389\pi$
$809$	1.08277e11i	0.252779i	−0.991981	+	0.126389i	$0.959661\pi$
$810$	0	0
$811$	−1.10571e11	−0.255597	−0.127799	−	0.991800i	$-0.540791\pi$
$811$	−1.10571e11	−0.255597	−0.127799	+	0.991800i	$0.540791\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 9.36637e10i	− 0.212296i
$816$	0	0
$817$	−1.29981e12	−2.91737
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 7.32720e11i	− 1.61274i	−0.591409	−	0.806372i	$-0.701428\pi$
$821$	− 7.32720e11i	− 1.61274i	0.591409	−	0.806372i	$-0.298572\pi$
$822$	0	0
$823$	−2.30116e11	−0.501588	−0.250794	−	0.968040i	$-0.580692\pi$
$823$	−2.30116e11	−0.501588	−0.250794	+	0.968040i	$0.580692\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.39758e11i	1.15392i	0.816771	+	0.576961i	$0.195762\pi$
$827$	5.39758e11i	1.15392i	−0.816771	+	0.576961i	$0.804238\pi$
$828$	0	0
$829$	−4.52429e11	−0.957928	−0.478964	−	0.877835i	$-0.658988\pi$
$829$	−4.52429e11	−0.957928	−0.478964	+	0.877835i	$0.658988\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3.34722e11i	− 0.695191i
$834$	0	0
$835$	2.53009e11	0.520463
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.80073e11i	1.37249i	0.727373	+	0.686243i	$0.240741\pi$
$839$	6.80073e11i	1.37249i	−0.727373	+	0.686243i	$0.759259\pi$
$840$	0	0
$841$	4.07118e11	0.813836
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.21166e12i	2.37659i
$846$	0	0
$847$	−4.11930e10	−0.0800368
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.12088e10i	0.135774i
$852$	0	0
$853$	−7.22567e11	−1.36484	−0.682420	−	0.730960i	$-0.739072\pi$
$853$	−7.22567e11	−1.36484	−0.682420	+	0.730960i	$0.739072\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 8.52977e11i	− 1.58130i	−0.612269	−	0.790649i	$-0.709743\pi$
$857$	− 8.52977e11i	− 1.58130i	0.612269	−	0.790649i	$-0.290257\pi$
$858$	0	0
$859$	8.02293e11	1.47353	0.736767	−	0.676147i	$-0.236352\pi$
$859$	8.02293e11	1.47353	0.736767	+	0.676147i	$0.236352\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.50140e11i	0.270678i	0.990799	+	0.135339i	$0.0432124\pi$
$863$	1.50140e11i	0.270678i	−0.990799	+	0.135339i	$0.956788\pi$
$864$	0	0
$865$	8.78994e11	1.57008
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 3.65857e10i	− 0.0641553i
$870$	0	0
$871$	−1.42035e12	−2.46787
$872$	0	0
$873$	0	0
$874$	0	0
$875$	7.65506e10i	0.130592i
$876$	0	0
$877$	5.12912e11	0.867050	0.433525	−	0.901141i	$-0.357270\pi$
$877$	5.12912e11	0.867050	0.433525	+	0.901141i	$0.357270\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.61215e11i	0.267610i	0.991008	+	0.133805i	$0.0427196\pi$
$881$	1.61215e11i	0.267610i	−0.991008	+	0.133805i	$0.957280\pi$
$882$	0	0
$883$	5.22475e11	0.859455	0.429727	−	0.902959i	$-0.358610\pi$
$883$	5.22475e11	0.859455	0.429727	+	0.902959i	$0.358610\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.45677e10i	− 0.0235340i	−0.999931	−	0.0117670i	$-0.996254\pi$
$887$	− 1.45677e10i	− 0.0235340i	0.999931	−	0.0117670i	$-0.00374564\pi$
$888$	0	0
$889$	2.04313e10	0.0327106
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.57240e12i	2.47262i
$894$	0	0
$895$	−2.75183e11	−0.428873
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1.17235e11i	− 0.179481i
$900$	0	0
$901$	2.98618e11	0.453124
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.49444e12i	2.22784i
$906$	0	0
$907$	5.71674e11	0.844732	0.422366	−	0.906425i	$-0.361200\pi$
$907$	5.71674e11	0.844732	0.422366	+	0.906425i	$0.361200\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 3.97204e11i	− 0.576687i	−0.957527	−	0.288343i	$-0.906896\pi$
$911$	− 3.97204e11i	− 0.576687i	0.957527	−	0.288343i	$-0.0931045\pi$
$912$	0	0
$913$	−7.13555e11	−1.02694
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 7.07708e11i	− 1.00087i
$918$	0	0
$919$	−1.81757e11	−0.254818	−0.127409	−	0.991850i	$-0.540666\pi$
$919$	−1.81757e11	−0.254818	−0.127409	+	0.991850i	$0.540666\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.64014e11i	0.777111i
$924$	0	0
$925$	−2.19380e11	−0.299661
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.94884e11i	1.20144i	0.799458	+	0.600722i	$0.205120\pi$
$929$	8.94884e11i	1.20144i	−0.799458	+	0.600722i	$0.794880\pi$
$930$	0	0
$931$	−7.39690e11	−0.984580
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.38426e12i	1.81122i
$936$	0	0
$937$	−8.28536e11	−1.07486	−0.537431	−	0.843308i	$-0.680605\pi$
$937$	−8.28536e11	−1.07486	−0.537431	+	0.843308i	$0.680605\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.12228e11i	0.525749i	0.964830	+	0.262875i	$0.0846706\pi$
$941$	4.12228e11i	0.525749i	−0.964830	+	0.262875i	$0.915329\pi$
$942$	0	0
$943$	1.44429e11	0.182645
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.04875e11i	− 0.503410i	−0.967804	−	0.251705i	$-0.919009\pi$
$947$	− 4.04875e11i	− 0.503410i	0.967804	−	0.251705i	$-0.0809912\pi$
$948$	0	0
$949$	−5.33409e11	−0.657652
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 2.71037e11i	− 0.328593i	−0.986411	−	0.164296i	$-0.947465\pi$
$953$	− 2.71037e11i	− 0.328593i	0.986411	−	0.164296i	$-0.0525353\pi$
$954$	0	0
$955$	−7.00028e10	−0.0841592
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.50589e10i	0.0414499i
$960$	0	0
$961$	−7.05309e11	−0.826963
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1.15756e12i	− 1.33485i
$966$	0	0
$967$	6.46243e9	0.00739078	0.00369539	−	0.999993i	$-0.498824\pi$
$967$	6.46243e9	0.00739078	0.00369539	+	0.999993i	$0.498824\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 8.40945e11i	− 0.945999i	−0.881063	−	0.472999i	$-0.843171\pi$
$971$	− 8.40945e11i	− 0.945999i	0.881063	−	0.472999i	$-0.156829\pi$
$972$	0	0
$973$	−9.39096e11	−1.04775
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.47264e12i	− 1.61629i	−0.588984	−	0.808144i	$-0.700472\pi$
$977$	− 1.47264e12i	− 1.61629i	0.588984	−	0.808144i	$-0.299528\pi$
$978$	0	0
$979$	−5.32099e11	−0.579244
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 2.99994e11i	− 0.321291i	−0.987012	−	0.160645i	$-0.948642\pi$
$983$	− 2.99994e11i	− 0.321291i	0.987012	−	0.160645i	$-0.0513576\pi$
$984$	0	0
$985$	2.25506e12	2.39560
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 7.64332e11i	− 0.798908i
$990$	0	0
$991$	3.61468e11	0.374779	0.187390	−	0.982286i	$-0.439997\pi$
$991$	3.61468e11	0.374779	0.187390	+	0.982286i	$0.439997\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.41821e12i	1.44693i
$996$	0	0
$997$	−7.97655e11	−0.807299	−0.403649	−	0.914914i	$-0.632258\pi$
$997$	−7.97655e11	−0.807299	−0.403649	+	0.914914i	$0.632258\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.9.e.b.17.2		2
3.2	odd	2	inner	144.9.e.b.17.1		2
4.3	odd	2		18.9.b.b.17.1	✓	2
12.11	even	2		18.9.b.b.17.2	yes	2
20.3	even	4		450.9.b.b.449.1		4
20.7	even	4		450.9.b.b.449.4		4
20.19	odd	2		450.9.d.a.251.2		2
36.7	odd	6		162.9.d.b.53.1		4
36.11	even	6		162.9.d.b.53.2		4
36.23	even	6		162.9.d.b.107.1		4
36.31	odd	6		162.9.d.b.107.2		4
60.23	odd	4		450.9.b.b.449.3		4
60.47	odd	4		450.9.b.b.449.2		4
60.59	even	2		450.9.d.a.251.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.9.b.b.17.1	✓	2	4.3	odd	2
18.9.b.b.17.2	yes	2	12.11	even	2
144.9.e.b.17.1		2	3.2	odd	2	inner
144.9.e.b.17.2		2	1.1	even	1	trivial
162.9.d.b.53.1		4	36.7	odd	6
162.9.d.b.53.2		4	36.11	even	6
162.9.d.b.107.1		4	36.23	even	6
162.9.d.b.107.2		4	36.31	odd	6
450.9.b.b.449.1		4	20.3	even	4
450.9.b.b.449.2		4	60.47	odd	4
450.9.b.b.449.3		4	60.23	odd	4
450.9.b.b.449.4		4	20.7	even	4
450.9.d.a.251.1		2	60.59	even	2
450.9.d.a.251.2		2	20.19	odd	2

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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	144.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+912.168i q^{5} -1652.00 q^{7} +O(q^{10})\)	\(q+912.168i q^{5} -1652.00 q^{7} -13763.1i q^{11} +46304.0 q^{13} +110262. i q^{17} +243664. q^{19} +143282. i q^{23} -441425. q^{25} +305169. i q^{29} -384164. q^{31} -1.50690e6i q^{35} +496982. q^{37} -1.00800e6i q^{41} -5.33444e6 q^{43} +6.45314e6i q^{47} -3.03570e6 q^{49} -2.70826e6i q^{53} +1.25543e7 q^{55} +1.24442e7i q^{59} +2.33537e6 q^{61} +4.22370e7i q^{65} -3.06745e7 q^{67} +1.21807e7i q^{71} -1.15197e7 q^{73} +2.27367e7i q^{77} +2.65824e6 q^{79} -5.18454e7i q^{83} -1.00577e8 q^{85} -3.86612e7i q^{89} -7.64942e7 q^{91} +2.22262e8i q^{95} -5.15952e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3304 q^{7}+O(q^{10}) \) 2 * q - 3304 * q^7	\( 2 q - 3304 q^{7} + 92608 q^{13} + 487328 q^{19} - 882850 q^{25} - 768328 q^{31} + 993964 q^{37} - 10668880 q^{43} - 6071394 q^{49} + 25108560 q^{55} + 4670740 q^{61} - 61348912 q^{67} - 23039456 q^{73} + 5316488 q^{79} - 201154860 q^{85} - 152988416 q^{91} - 103190336 q^{97}+O(q^{100}) \) 2 * q - 3304 * q^7 + 92608 * q^13 + 487328 * q^19 - 882850 * q^25 - 768328 * q^31 + 993964 * q^37 - 10668880 * q^43 - 6071394 * q^49 + 25108560 * q^55 + 4670740 * q^61 - 61348912 * q^67 - 23039456 * q^73 + 5316488 * q^79 - 201154860 * q^85 - 152988416 * q^91 - 103190336 * q^97

Introduction

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