LMFDB - Embedded newform 1656.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1656,2,Mod(1,1656)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1656, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1656.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1656 = 2^{3} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1656.a (trivial)

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:	yes
Analytic conductor:	$13.2232265747$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1656.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{5} +O(q^{10})$	$q-2.00000 q^{5} -5.12311 q^{11} +4.56155 q^{13} +3.12311 q^{17} +5.12311 q^{19} +1.00000 q^{23} -1.00000 q^{25} +0.561553 q^{29} -6.56155 q^{31} -8.24621 q^{37} -10.8078 q^{41} -8.00000 q^{43} -11.6847 q^{47} -7.00000 q^{49} -2.00000 q^{53} +10.2462 q^{55} +6.24621 q^{59} +12.2462 q^{61} -9.12311 q^{65} -5.12311 q^{67} -9.43845 q^{71} -2.31534 q^{73} -5.12311 q^{79} +2.24621 q^{83} -6.24621 q^{85} +13.3693 q^{89} -10.2462 q^{95} -13.3693 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5}+O(q^{10}) $ 2 * q - 4 * q^5	$ 2 q - 4 q^{5} - 2 q^{11} + 5 q^{13} - 2 q^{17} + 2 q^{19} + 2 q^{23} - 2 q^{25} - 3 q^{29} - 9 q^{31} - q^{41} - 16 q^{43} - 11 q^{47} - 14 q^{49} - 4 q^{53} + 4 q^{55} - 4 q^{59} + 8 q^{61} - 10 q^{65} - 2 q^{67} - 23 q^{71} - 17 q^{73} - 2 q^{79} - 12 q^{83} + 4 q^{85} + 2 q^{89} - 4 q^{95} - 2 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - 2 * q^11 + 5 * q^13 - 2 * q^17 + 2 * q^19 + 2 * q^23 - 2 * q^25 - 3 * q^29 - 9 * q^31 - q^41 - 16 * q^43 - 11 * q^47 - 14 * q^49 - 4 * q^53 + 4 * q^55 - 4 * q^59 + 8 * q^61 - 10 * q^65 - 2 * q^67 - 23 * q^71 - 17 * q^73 - 2 * q^79 - 12 * q^83 + 4 * q^85 + 2 * q^89 - 4 * q^95 - 2 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.12311	−1.54467	−0.772337	−	0.635213i	$-0.780912\pi$
$11$	−5.12311	−1.54467	−0.772337	+	0.635213i	$0.780912\pi$
$12$	0	0
$13$	4.56155	1.26515	0.632574	−	0.774500i	$-0.281999\pi$
$13$	4.56155	1.26515	0.632574	+	0.774500i	$0.281999\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.12311	0.757464	0.378732	−	0.925506i	$-0.376360\pi$
$17$	3.12311	0.757464	0.378732	+	0.925506i	$0.376360\pi$
$18$	0	0
$19$	5.12311	1.17532	0.587661	−	0.809108i	$-0.300049\pi$
$19$	5.12311	1.17532	0.587661	+	0.809108i	$0.300049\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.561553	0.104278	0.0521389	−	0.998640i	$-0.483396\pi$
$29$	0.561553	0.104278	0.0521389	+	0.998640i	$0.483396\pi$
$30$	0	0
$31$	−6.56155	−1.17849	−0.589245	−	0.807955i	$-0.700575\pi$
$31$	−6.56155	−1.17849	−0.589245	+	0.807955i	$0.700575\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.24621	−1.35567	−0.677834	−	0.735215i	$-0.737081\pi$
$37$	−8.24621	−1.35567	−0.677834	+	0.735215i	$0.737081\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.8078	−1.68789	−0.843945	−	0.536430i	$-0.819772\pi$
$41$	−10.8078	−1.68789	−0.843945	+	0.536430i	$0.819772\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.6847	−1.70438	−0.852191	−	0.523230i	$-0.824727\pi$
$47$	−11.6847	−1.70438	−0.852191	+	0.523230i	$0.824727\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	10.2462	1.38160
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.24621	0.813187	0.406594	−	0.913609i	$-0.366716\pi$
$59$	6.24621	0.813187	0.406594	+	0.913609i	$0.366716\pi$
$60$	0	0
$61$	12.2462	1.56797	0.783983	−	0.620782i	$-0.213185\pi$
$61$	12.2462	1.56797	0.783983	+	0.620782i	$0.213185\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−9.12311	−1.13158
$66$	0	0
$67$	−5.12311	−0.625887	−0.312943	−	0.949772i	$-0.601315\pi$
$67$	−5.12311	−0.625887	−0.312943	+	0.949772i	$0.601315\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.43845	−1.12014	−0.560069	−	0.828446i	$-0.689225\pi$
$71$	−9.43845	−1.12014	−0.560069	+	0.828446i	$0.689225\pi$
$72$	0	0
$73$	−2.31534	−0.270990	−0.135495	−	0.990778i	$-0.543263\pi$
$73$	−2.31534	−0.270990	−0.135495	+	0.990778i	$0.543263\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.12311	−0.576394	−0.288197	−	0.957571i	$-0.593056\pi$
$79$	−5.12311	−0.576394	−0.288197	+	0.957571i	$0.593056\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.24621	0.246554	0.123277	−	0.992372i	$-0.460660\pi$
$83$	2.24621	0.246554	0.123277	+	0.992372i	$0.460660\pi$
$84$	0	0
$85$	−6.24621	−0.677497
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.3693	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$89$	13.3693	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−10.2462	−1.05124
$96$	0	0
$97$	−13.3693	−1.35745	−0.678724	−	0.734393i	$-0.737467\pi$
$97$	−13.3693	−1.35745	−0.678724	+	0.734393i	$0.737467\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.24621	0.422514	0.211257	−	0.977431i	$-0.432244\pi$
$101$	4.24621	0.422514	0.211257	+	0.977431i	$0.432244\pi$
$102$	0	0
$103$	2.24621	0.221326	0.110663	−	0.993858i	$-0.464703\pi$
$103$	2.24621	0.221326	0.110663	+	0.993858i	$0.464703\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.87689	−0.278120	−0.139060	−	0.990284i	$-0.544408\pi$
$107$	−2.87689	−0.278120	−0.139060	+	0.990284i	$0.544408\pi$
$108$	0	0
$109$	4.87689	0.467122	0.233561	−	0.972342i	$-0.424962\pi$
$109$	4.87689	0.467122	0.233561	+	0.972342i	$0.424962\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.1231	1.04637	0.523187	−	0.852218i	$-0.324743\pi$
$113$	11.1231	1.04637	0.523187	+	0.852218i	$0.324743\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	15.2462	1.38602
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−11.6847	−1.03685	−0.518423	−	0.855124i	$-0.673481\pi$
$127$	−11.6847	−1.03685	−0.518423	+	0.855124i	$0.673481\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.6847	−1.37037	−0.685187	−	0.728367i	$-0.740280\pi$
$131$	−15.6847	−1.37037	−0.685187	+	0.728367i	$0.740280\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.1231	−1.29205	−0.646027	−	0.763315i	$-0.723571\pi$
$137$	−15.1231	−1.29205	−0.646027	+	0.763315i	$0.723571\pi$
$138$	0	0
$139$	−15.6847	−1.33036	−0.665178	−	0.746685i	$-0.731644\pi$
$139$	−15.6847	−1.33036	−0.665178	+	0.746685i	$0.731644\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−23.3693	−1.95424
$144$	0	0
$145$	−1.12311	−0.0932688
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.75379	0.307522	0.153761	−	0.988108i	$-0.450861\pi$
$149$	3.75379	0.307522	0.153761	+	0.988108i	$0.450861\pi$
$150$	0	0
$151$	−14.5616	−1.18500	−0.592501	−	0.805570i	$-0.701859\pi$
$151$	−14.5616	−1.18500	−0.592501	+	0.805570i	$0.701859\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	13.1231	1.05407
$156$	0	0
$157$	12.8769	1.02769	0.513844	−	0.857884i	$-0.328221\pi$
$157$	12.8769	1.02769	0.513844	+	0.857884i	$0.328221\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−9.93087	−0.777846	−0.388923	−	0.921270i	$-0.627153\pi$
$163$	−9.93087	−0.777846	−0.388923	+	0.921270i	$0.627153\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	7.80776	0.600597
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.6847	1.17233	0.586163	−	0.810193i	$-0.300638\pi$
$179$	15.6847	1.17233	0.586163	+	0.810193i	$0.300638\pi$
$180$	0	0
$181$	20.2462	1.50489	0.752445	−	0.658656i	$-0.228875\pi$
$181$	20.2462	1.50489	0.752445	+	0.658656i	$0.228875\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	16.4924	1.21255
$186$	0	0
$187$	−16.0000	−1.17004
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.630683	−0.0456346	−0.0228173	−	0.999740i	$-0.507264\pi$
$191$	−0.630683	−0.0456346	−0.0228173	+	0.999740i	$0.507264\pi$
$192$	0	0
$193$	−4.56155	−0.328348	−0.164174	−	0.986431i	$-0.552496\pi$
$193$	−4.56155	−0.328348	−0.164174	+	0.986431i	$0.552496\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.9309	−0.850039	−0.425020	−	0.905184i	$-0.639733\pi$
$197$	−11.9309	−0.850039	−0.425020	+	0.905184i	$0.639733\pi$
$198$	0	0
$199$	2.87689	0.203938	0.101969	−	0.994788i	$-0.467486\pi$
$199$	2.87689	0.203938	0.101969	+	0.994788i	$0.467486\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	21.6155	1.50969
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−26.2462	−1.81549
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.0000	1.09119
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	14.2462	0.958304
$222$	0	0
$223$	4.49242	0.300835	0.150417	−	0.988623i	$-0.451938\pi$
$223$	4.49242	0.300835	0.150417	+	0.988623i	$0.451938\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.2462	−0.680065	−0.340032	−	0.940414i	$-0.610438\pi$
$227$	−10.2462	−0.680065	−0.340032	+	0.940414i	$0.610438\pi$
$228$	0	0
$229$	23.1231	1.52802	0.764009	−	0.645206i	$-0.223228\pi$
$229$	23.1231	1.52802	0.764009	+	0.645206i	$0.223228\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.0540	−1.37929	−0.689646	−	0.724147i	$-0.742234\pi$
$233$	−21.0540	−1.37929	−0.689646	+	0.724147i	$0.742234\pi$
$234$	0	0
$235$	23.3693	1.52445
$236$	0	0
$237$	0	0
$238$	0	0
$239$	11.0540	0.715022	0.357511	−	0.933909i	$-0.383625\pi$
$239$	11.0540	0.715022	0.357511	+	0.933909i	$0.383625\pi$
$240$	0	0
$241$	−26.4924	−1.70653	−0.853263	−	0.521480i	$-0.825380\pi$
$241$	−26.4924	−1.70653	−0.853263	+	0.521480i	$0.825380\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	14.0000	0.894427
$246$	0	0
$247$	23.3693	1.48695
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−10.2462	−0.646735	−0.323368	−	0.946273i	$-0.604815\pi$
$251$	−10.2462	−0.646735	−0.323368	+	0.946273i	$0.604815\pi$
$252$	0	0
$253$	−5.12311	−0.322087
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.6847	1.10314	0.551569	−	0.834129i	$-0.314029\pi$
$257$	17.6847	1.10314	0.551569	+	0.834129i	$0.314029\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	10.8769	0.670698	0.335349	−	0.942094i	$-0.391146\pi$
$263$	10.8769	0.670698	0.335349	+	0.942094i	$0.391146\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−25.6847	−1.56602	−0.783011	−	0.622008i	$-0.786317\pi$
$269$	−25.6847	−1.56602	−0.783011	+	0.622008i	$0.786317\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.12311	0.308935
$276$	0	0
$277$	−2.80776	−0.168702	−0.0843511	−	0.996436i	$-0.526882\pi$
$277$	−2.80776	−0.168702	−0.0843511	+	0.996436i	$0.526882\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.12311	0.186309	0.0931544	−	0.995652i	$-0.470305\pi$
$281$	3.12311	0.186309	0.0931544	+	0.995652i	$0.470305\pi$
$282$	0	0
$283$	−5.12311	−0.304537	−0.152269	−	0.988339i	$-0.548658\pi$
$283$	−5.12311	−0.304537	−0.152269	+	0.988339i	$0.548658\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−7.24621	−0.426248
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.36932	−0.547361	−0.273681	−	0.961821i	$-0.588241\pi$
$293$	−9.36932	−0.547361	−0.273681	+	0.961821i	$0.588241\pi$
$294$	0	0
$295$	−12.4924	−0.727337
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.56155	0.263801
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−24.4924	−1.40243
$306$	0	0
$307$	−1.75379	−0.100094	−0.0500470	−	0.998747i	$-0.515937\pi$
$307$	−1.75379	−0.100094	−0.0500470	+	0.998747i	$0.515937\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.43845	−0.0815669	−0.0407834	−	0.999168i	$-0.512985\pi$
$311$	−1.43845	−0.0815669	−0.0407834	+	0.999168i	$0.512985\pi$
$312$	0	0
$313$	9.36932	0.529585	0.264793	−	0.964305i	$-0.414697\pi$
$313$	9.36932	0.529585	0.264793	+	0.964305i	$0.414697\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.2462	1.13714	0.568570	−	0.822635i	$-0.307497\pi$
$317$	20.2462	1.13714	0.568570	+	0.822635i	$0.307497\pi$
$318$	0	0
$319$	−2.87689	−0.161075
$320$	0	0
$321$	0	0
$322$	0	0
$323$	16.0000	0.890264
$324$	0	0
$325$	−4.56155	−0.253029
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−30.4233	−1.67222	−0.836108	−	0.548565i	$-0.815174\pi$
$331$	−30.4233	−1.67222	−0.836108	+	0.548565i	$0.815174\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.2462	0.559810
$336$	0	0
$337$	19.6155	1.06853	0.534263	−	0.845318i	$-0.320589\pi$
$337$	19.6155	1.06853	0.534263	+	0.845318i	$0.320589\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	33.6155	1.82038
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.4924	−0.885360	−0.442680	−	0.896680i	$-0.645972\pi$
$347$	−16.4924	−0.885360	−0.442680	+	0.896680i	$0.645972\pi$
$348$	0	0
$349$	−22.3153	−1.19451	−0.597256	−	0.802050i	$-0.703743\pi$
$349$	−22.3153	−1.19451	−0.597256	+	0.802050i	$0.703743\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−11.4384	−0.608807	−0.304404	−	0.952543i	$-0.598457\pi$
$353$	−11.4384	−0.608807	−0.304404	+	0.952543i	$0.598457\pi$
$354$	0	0
$355$	18.8769	1.00188
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.6155	−0.929712	−0.464856	−	0.885386i	$-0.653894\pi$
$359$	−17.6155	−0.929712	−0.464856	+	0.885386i	$0.653894\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.63068	0.242381
$366$	0	0
$367$	2.24621	0.117251	0.0586256	−	0.998280i	$-0.481328\pi$
$367$	2.24621	0.117251	0.0586256	+	0.998280i	$0.481328\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	22.4924	1.16461	0.582307	−	0.812969i	$-0.302150\pi$
$373$	22.4924	1.16461	0.582307	+	0.812969i	$0.302150\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.56155	0.131927
$378$	0	0
$379$	20.4924	1.05263	0.526313	−	0.850291i	$-0.323574\pi$
$379$	20.4924	1.05263	0.526313	+	0.850291i	$0.323574\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−26.2462	−1.34112	−0.670559	−	0.741856i	$-0.733946\pi$
$383$	−26.2462	−1.34112	−0.670559	+	0.741856i	$0.733946\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.36932	−0.0694271	−0.0347136	−	0.999397i	$-0.511052\pi$
$389$	−1.36932	−0.0694271	−0.0347136	+	0.999397i	$0.511052\pi$
$390$	0	0
$391$	3.12311	0.157942
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10.2462	0.515543
$396$	0	0
$397$	20.5616	1.03195	0.515977	−	0.856602i	$-0.327429\pi$
$397$	20.5616	1.03195	0.515977	+	0.856602i	$0.327429\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.7538	0.586956	0.293478	−	0.955966i	$-0.405187\pi$
$401$	11.7538	0.586956	0.293478	+	0.955966i	$0.405187\pi$
$402$	0	0
$403$	−29.9309	−1.49096
$404$	0	0
$405$	0	0
$406$	0	0
$407$	42.2462	2.09407
$408$	0	0
$409$	−27.3002	−1.34991	−0.674954	−	0.737860i	$-0.735836\pi$
$409$	−27.3002	−1.34991	−0.674954	+	0.737860i	$0.735836\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.49242	−0.220524
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.4924	0.610295	0.305147	−	0.952305i	$-0.401294\pi$
$419$	12.4924	0.610295	0.305147	+	0.952305i	$0.401294\pi$
$420$	0	0
$421$	−27.1231	−1.32190	−0.660950	−	0.750430i	$-0.729846\pi$
$421$	−27.1231	−1.32190	−0.660950	+	0.750430i	$0.729846\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.12311	−0.151493
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	28.4924	1.37243	0.686216	−	0.727398i	$-0.259271\pi$
$431$	28.4924	1.37243	0.686216	+	0.727398i	$0.259271\pi$
$432$	0	0
$433$	24.7386	1.18886	0.594431	−	0.804146i	$-0.297377\pi$
$433$	24.7386	1.18886	0.594431	+	0.804146i	$0.297377\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.12311	0.245071
$438$	0	0
$439$	11.0540	0.527577	0.263789	−	0.964580i	$-0.415028\pi$
$439$	11.0540	0.527577	0.263789	+	0.964580i	$0.415028\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.06913	0.288353	0.144177	−	0.989552i	$-0.453947\pi$
$443$	6.06913	0.288353	0.144177	+	0.989552i	$0.453947\pi$
$444$	0	0
$445$	−26.7386	−1.26753
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.24621	0.389163	0.194581	−	0.980886i	$-0.437665\pi$
$449$	8.24621	0.389163	0.194581	+	0.980886i	$0.437665\pi$
$450$	0	0
$451$	55.3693	2.60724
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.36932	−0.251166	−0.125583	−	0.992083i	$-0.540080\pi$
$457$	−5.36932	−0.251166	−0.125583	+	0.992083i	$0.540080\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.8078	0.503368	0.251684	−	0.967809i	$-0.419016\pi$
$461$	10.8078	0.503368	0.251684	+	0.967809i	$0.419016\pi$
$462$	0	0
$463$	12.4924	0.580572	0.290286	−	0.956940i	$-0.406250\pi$
$463$	12.4924	0.580572	0.290286	+	0.956940i	$0.406250\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.3693	0.711207	0.355604	−	0.934637i	$-0.384275\pi$
$467$	15.3693	0.711207	0.355604	+	0.934637i	$0.384275\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	40.9848	1.88449
$474$	0	0
$475$	−5.12311	−0.235064
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.2462	−0.468161	−0.234081	−	0.972217i	$-0.575208\pi$
$479$	−10.2462	−0.468161	−0.234081	+	0.972217i	$0.575208\pi$
$480$	0	0
$481$	−37.6155	−1.71512
$482$	0	0
$483$	0	0
$484$	0	0
$485$	26.7386	1.21414
$486$	0	0
$487$	12.3153	0.558061	0.279031	−	0.960282i	$-0.409987\pi$
$487$	12.3153	0.558061	0.279031	+	0.960282i	$0.409987\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−12.8078	−0.578006	−0.289003	−	0.957328i	$-0.593324\pi$
$491$	−12.8078	−0.578006	−0.289003	+	0.957328i	$0.593324\pi$
$492$	0	0
$493$	1.75379	0.0789867
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−23.0540	−1.03204	−0.516019	−	0.856577i	$-0.672587\pi$
$499$	−23.0540	−1.03204	−0.516019	+	0.856577i	$0.672587\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−7.36932	−0.328582	−0.164291	−	0.986412i	$-0.552534\pi$
$503$	−7.36932	−0.328582	−0.164291	+	0.986412i	$0.552534\pi$
$504$	0	0
$505$	−8.49242	−0.377908
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.3002	0.678169	0.339084	−	0.940756i	$-0.389883\pi$
$509$	15.3002	0.678169	0.339084	+	0.940756i	$0.389883\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.49242	−0.197960
$516$	0	0
$517$	59.8617	2.63272
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−19.6155	−0.859372	−0.429686	−	0.902978i	$-0.641376\pi$
$521$	−19.6155	−0.859372	−0.429686	+	0.902978i	$0.641376\pi$
$522$	0	0
$523$	5.75379	0.251596	0.125798	−	0.992056i	$-0.459851\pi$
$523$	5.75379	0.251596	0.125798	+	0.992056i	$0.459851\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20.4924	−0.892664
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−49.3002	−2.13543
$534$	0	0
$535$	5.75379	0.248758
$536$	0	0
$537$	0	0
$538$	0	0
$539$	35.8617	1.54467
$540$	0	0
$541$	35.9309	1.54479	0.772394	−	0.635143i	$-0.219059\pi$
$541$	35.9309	1.54479	0.772394	+	0.635143i	$0.219059\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.75379	−0.417806
$546$	0	0
$547$	43.5464	1.86191	0.930955	−	0.365135i	$-0.118977\pi$
$547$	43.5464	1.86191	0.930955	+	0.365135i	$0.118977\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.87689	0.122560
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.876894	0.0371552	0.0185776	−	0.999827i	$-0.494086\pi$
$557$	0.876894	0.0371552	0.0185776	+	0.999827i	$0.494086\pi$
$558$	0	0
$559$	−36.4924	−1.54347
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.75379	−0.242493	−0.121247	−	0.992622i	$-0.538689\pi$
$563$	−5.75379	−0.242493	−0.121247	+	0.992622i	$0.538689\pi$
$564$	0	0
$565$	−22.2462	−0.935905
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.24621	0.345699	0.172850	−	0.984948i	$-0.444703\pi$
$569$	8.24621	0.345699	0.172850	+	0.984948i	$0.444703\pi$
$570$	0	0
$571$	−11.5076	−0.481577	−0.240789	−	0.970578i	$-0.577406\pi$
$571$	−11.5076	−0.481577	−0.240789	+	0.970578i	$0.577406\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	15.9309	0.663211	0.331605	−	0.943418i	$-0.392410\pi$
$577$	15.9309	0.663211	0.331605	+	0.943418i	$0.392410\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.2462	0.424355
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.0540	0.951539	0.475770	−	0.879570i	$-0.342170\pi$
$587$	23.0540	0.951539	0.475770	+	0.879570i	$0.342170\pi$
$588$	0	0
$589$	−33.6155	−1.38510
$590$	0	0
$591$	0	0
$592$	0	0
$593$	44.7386	1.83720	0.918598	−	0.395194i	$-0.129323\pi$
$593$	44.7386	1.83720	0.918598	+	0.395194i	$0.129323\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	−38.8078	−1.58300	−0.791501	−	0.611168i	$-0.790700\pi$
$601$	−38.8078	−1.58300	−0.791501	+	0.611168i	$0.790700\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−30.4924	−1.23969
$606$	0	0
$607$	−14.7386	−0.598223	−0.299111	−	0.954218i	$-0.596690\pi$
$607$	−14.7386	−0.598223	−0.299111	+	0.954218i	$0.596690\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−53.3002	−2.15629
$612$	0	0
$613$	−28.1080	−1.13527	−0.567635	−	0.823281i	$-0.692141\pi$
$613$	−28.1080	−1.13527	−0.567635	+	0.823281i	$0.692141\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−34.9848	−1.40844	−0.704218	−	0.709983i	$-0.748702\pi$
$617$	−34.9848	−1.40844	−0.704218	+	0.709983i	$0.748702\pi$
$618$	0	0
$619$	28.4924	1.14521	0.572604	−	0.819832i	$-0.305933\pi$
$619$	28.4924	1.14521	0.572604	+	0.819832i	$0.305933\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−25.7538	−1.02687
$630$	0	0
$631$	−6.73863	−0.268261	−0.134130	−	0.990964i	$-0.542824\pi$
$631$	−6.73863	−0.268261	−0.134130	+	0.990964i	$0.542824\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	23.3693	0.927383
$636$	0	0
$637$	−31.9309	−1.26515
$638$	0	0
$639$	0	0
$640$	0	0
$641$	13.3693	0.528056	0.264028	−	0.964515i	$-0.414949\pi$
$641$	13.3693	0.528056	0.264028	+	0.964515i	$0.414949\pi$
$642$	0	0
$643$	−23.3693	−0.921596	−0.460798	−	0.887505i	$-0.652437\pi$
$643$	−23.3693	−0.921596	−0.460798	+	0.887505i	$0.652437\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	29.3002	1.15191	0.575955	−	0.817482i	$-0.304630\pi$
$647$	29.3002	1.15191	0.575955	+	0.817482i	$0.304630\pi$
$648$	0	0
$649$	−32.0000	−1.25611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25.0540	−0.980438	−0.490219	−	0.871599i	$-0.663083\pi$
$653$	−25.0540	−0.980438	−0.490219	+	0.871599i	$0.663083\pi$
$654$	0	0
$655$	31.3693	1.22570
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−2.24621	−0.0875000	−0.0437500	−	0.999043i	$-0.513930\pi$
$659$	−2.24621	−0.0875000	−0.0437500	+	0.999043i	$0.513930\pi$
$660$	0	0
$661$	−0.246211	−0.00957651	−0.00478825	−	0.999989i	$-0.501524\pi$
$661$	−0.246211	−0.00957651	−0.00478825	+	0.999989i	$0.501524\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.561553	0.0217434
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−62.7386	−2.42200
$672$	0	0
$673$	−2.94602	−0.113561	−0.0567805	−	0.998387i	$-0.518084\pi$
$673$	−2.94602	−0.113561	−0.0567805	+	0.998387i	$0.518084\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.9848	1.49831	0.749155	−	0.662395i	$-0.230460\pi$
$677$	38.9848	1.49831	0.749155	+	0.662395i	$0.230460\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−16.3153	−0.624289	−0.312145	−	0.950035i	$-0.601047\pi$
$683$	−16.3153	−0.624289	−0.312145	+	0.950035i	$0.601047\pi$
$684$	0	0
$685$	30.2462	1.15565
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−9.12311	−0.347563
$690$	0	0
$691$	20.9848	0.798301	0.399151	−	0.916885i	$-0.369305\pi$
$691$	20.9848	0.798301	0.399151	+	0.916885i	$0.369305\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	31.3693	1.18991
$696$	0	0
$697$	−33.7538	−1.27852
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−29.8617	−1.12786	−0.563931	−	0.825822i	$-0.690712\pi$
$701$	−29.8617	−1.12786	−0.563931	+	0.825822i	$0.690712\pi$
$702$	0	0
$703$	−42.2462	−1.59335
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−37.3693	−1.40343	−0.701717	−	0.712456i	$-0.747583\pi$
$709$	−37.3693	−1.40343	−0.701717	+	0.712456i	$0.747583\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.56155	−0.245732
$714$	0	0
$715$	46.7386	1.74793
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.49242	−0.167539	−0.0837695	−	0.996485i	$-0.526696\pi$
$719$	−4.49242	−0.167539	−0.0837695	+	0.996485i	$0.526696\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.561553	−0.0208555
$726$	0	0
$727$	−39.3693	−1.46013	−0.730064	−	0.683379i	$-0.760510\pi$
$727$	−39.3693	−1.46013	−0.730064	+	0.683379i	$0.760510\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−24.9848	−0.924098
$732$	0	0
$733$	49.3693	1.82350	0.911749	−	0.410749i	$-0.134733\pi$
$733$	49.3693	1.82350	0.911749	+	0.410749i	$0.134733\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	26.2462	0.966792
$738$	0	0
$739$	0.315342	0.0116000	0.00580001	−	0.999983i	$-0.498154\pi$
$739$	0.315342	0.0116000	0.00580001	+	0.999983i	$0.498154\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−34.2462	−1.25637	−0.628186	−	0.778063i	$-0.716202\pi$
$743$	−34.2462	−1.25637	−0.628186	+	0.778063i	$0.716202\pi$
$744$	0	0
$745$	−7.50758	−0.275056
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	40.9848	1.49556	0.747779	−	0.663948i	$-0.231120\pi$
$751$	40.9848	1.49556	0.747779	+	0.663948i	$0.231120\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	29.1231	1.05990
$756$	0	0
$757$	−1.50758	−0.0547938	−0.0273969	−	0.999625i	$-0.508722\pi$
$757$	−1.50758	−0.0547938	−0.0273969	+	0.999625i	$0.508722\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.3153	0.663931	0.331965	−	0.943292i	$-0.392288\pi$
$761$	18.3153	0.663931	0.331965	+	0.943292i	$0.392288\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	28.4924	1.02880
$768$	0	0
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.63068	−0.0946191	−0.0473095	−	0.998880i	$-0.515065\pi$
$773$	−2.63068	−0.0946191	−0.0473095	+	0.998880i	$0.515065\pi$
$774$	0	0
$775$	6.56155	0.235698
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−55.3693	−1.98381
$780$	0	0
$781$	48.3542	1.73025
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−25.7538	−0.919192
$786$	0	0
$787$	−26.2462	−0.935576	−0.467788	−	0.883841i	$-0.654949\pi$
$787$	−26.2462	−0.935576	−0.467788	+	0.883841i	$0.654949\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	55.8617	1.98371
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12.2462	−0.433783	−0.216892	−	0.976196i	$-0.569592\pi$
$797$	−12.2462	−0.433783	−0.216892	+	0.976196i	$0.569592\pi$
$798$	0	0
$799$	−36.4924	−1.29101
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.8617	0.418592
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	33.9309	1.19147	0.595737	−	0.803180i	$-0.296860\pi$
$811$	33.9309	1.19147	0.595737	+	0.803180i	$0.296860\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	19.8617	0.695726
$816$	0	0
$817$	−40.9848	−1.43388
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24.7386	0.863384	0.431692	−	0.902021i	$-0.357917\pi$
$821$	24.7386	0.863384	0.431692	+	0.902021i	$0.357917\pi$
$822$	0	0
$823$	49.4384	1.72332	0.861658	−	0.507489i	$-0.169426\pi$
$823$	49.4384	1.72332	0.861658	+	0.507489i	$0.169426\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.49242	−0.156217	−0.0781084	−	0.996945i	$-0.524888\pi$
$827$	−4.49242	−0.156217	−0.0781084	+	0.996945i	$0.524888\pi$
$828$	0	0
$829$	−20.2462	−0.703180	−0.351590	−	0.936154i	$-0.614359\pi$
$829$	−20.2462	−0.703180	−0.351590	+	0.936154i	$0.614359\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−21.8617	−0.757464
$834$	0	0
$835$	−32.0000	−1.10741
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−50.2462	−1.73469	−0.867346	−	0.497706i	$-0.834176\pi$
$839$	−50.2462	−1.73469	−0.867346	+	0.497706i	$0.834176\pi$
$840$	0	0
$841$	−28.6847	−0.989126
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−15.6155	−0.537190
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.24621	−0.282676
$852$	0	0
$853$	−34.9848	−1.19786	−0.598929	−	0.800802i	$-0.704407\pi$
$853$	−34.9848	−1.19786	−0.598929	+	0.800802i	$0.704407\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16.5616	−0.565732	−0.282866	−	0.959159i	$-0.591285\pi$
$857$	−16.5616	−0.565732	−0.282866	+	0.959159i	$0.591285\pi$
$858$	0	0
$859$	16.9460	0.578191	0.289095	−	0.957300i	$-0.406646\pi$
$859$	16.9460	0.578191	0.289095	+	0.957300i	$0.406646\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8.17708	0.278351	0.139176	−	0.990268i	$-0.455555\pi$
$863$	8.17708	0.278351	0.139176	+	0.990268i	$0.455555\pi$
$864$	0	0
$865$	−20.0000	−0.680020
$866$	0	0
$867$	0	0
$868$	0	0
$869$	26.2462	0.890342
$870$	0	0
$871$	−23.3693	−0.791839
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.9848	1.04628	0.523142	−	0.852246i	$-0.324760\pi$
$877$	30.9848	1.04628	0.523142	+	0.852246i	$0.324760\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.87689	−0.164307	−0.0821534	−	0.996620i	$-0.526180\pi$
$881$	−4.87689	−0.164307	−0.0821534	+	0.996620i	$0.526180\pi$
$882$	0	0
$883$	36.9848	1.24464	0.622320	−	0.782763i	$-0.286190\pi$
$883$	36.9848	1.24464	0.622320	+	0.782763i	$0.286190\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.6847	−0.660946	−0.330473	−	0.943815i	$-0.607208\pi$
$887$	−19.6847	−0.660946	−0.330473	+	0.943815i	$0.607208\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−59.8617	−2.00320
$894$	0	0
$895$	−31.3693	−1.04856
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.68466	−0.122890
$900$	0	0
$901$	−6.24621	−0.208091
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−40.4924	−1.34601
$906$	0	0
$907$	42.8769	1.42370	0.711852	−	0.702330i	$-0.247857\pi$
$907$	42.8769	1.42370	0.711852	+	0.702330i	$0.247857\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	37.1231	1.22994	0.614972	−	0.788549i	$-0.289167\pi$
$911$	37.1231	1.22994	0.614972	+	0.788549i	$0.289167\pi$
$912$	0	0
$913$	−11.5076	−0.380845
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	37.7538	1.24538	0.622691	−	0.782468i	$-0.286039\pi$
$919$	37.7538	1.24538	0.622691	+	0.782468i	$0.286039\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−43.0540	−1.41714
$924$	0	0
$925$	8.24621	0.271134
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−34.8078	−1.14201	−0.571003	−	0.820948i	$-0.693445\pi$
$929$	−34.8078	−1.14201	−0.571003	+	0.820948i	$0.693445\pi$
$930$	0	0
$931$	−35.8617	−1.17532
$932$	0	0
$933$	0	0
$934$	0	0
$935$	32.0000	1.04651
$936$	0	0
$937$	15.7538	0.514654	0.257327	−	0.966324i	$-0.417158\pi$
$937$	15.7538	0.514654	0.257327	+	0.966324i	$0.417158\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−39.1231	−1.27538	−0.637688	−	0.770294i	$-0.720109\pi$
$941$	−39.1231	−1.27538	−0.637688	+	0.770294i	$0.720109\pi$
$942$	0	0
$943$	−10.8078	−0.351949
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.56155	0.0832393	0.0416196	−	0.999134i	$-0.486748\pi$
$947$	2.56155	0.0832393	0.0416196	+	0.999134i	$0.486748\pi$
$948$	0	0
$949$	−10.5616	−0.342843
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−44.2462	−1.43328	−0.716638	−	0.697446i	$-0.754320\pi$
$953$	−44.2462	−1.43328	−0.716638	+	0.697446i	$0.754320\pi$
$954$	0	0
$955$	1.26137	0.0408169
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	12.0540	0.388838
$962$	0	0
$963$	0	0
$964$	0	0
$965$	9.12311	0.293683
$966$	0	0
$967$	35.0540	1.12726	0.563630	−	0.826027i	$-0.309404\pi$
$967$	35.0540	1.12726	0.563630	+	0.826027i	$0.309404\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−31.3693	−1.00669	−0.503345	−	0.864086i	$-0.667897\pi$
$971$	−31.3693	−1.00669	−0.503345	+	0.864086i	$0.667897\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	57.2311	1.83098	0.915492	−	0.402337i	$-0.131802\pi$
$977$	57.2311	1.83098	0.915492	+	0.402337i	$0.131802\pi$
$978$	0	0
$979$	−68.4924	−2.18903
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−22.1080	−0.705134	−0.352567	−	0.935787i	$-0.614691\pi$
$983$	−22.1080	−0.705134	−0.352567	+	0.935787i	$0.614691\pi$
$984$	0	0
$985$	23.8617	0.760298
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	8.98485	0.285413	0.142707	−	0.989765i	$-0.454419\pi$
$991$	8.98485	0.285413	0.142707	+	0.989765i	$0.454419\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5.75379	−0.182407
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1656.2.a.j.1.1		2
3.2	odd	2		184.2.a.e.1.1	✓	2
4.3	odd	2		3312.2.a.t.1.2		2
12.11	even	2		368.2.a.i.1.2		2
15.2	even	4		4600.2.e.o.4049.4		4
15.8	even	4		4600.2.e.o.4049.1		4
15.14	odd	2		4600.2.a.s.1.2		2
21.20	even	2		9016.2.a.w.1.2		2
24.5	odd	2		1472.2.a.u.1.2		2
24.11	even	2		1472.2.a.p.1.1		2
60.59	even	2		9200.2.a.br.1.1		2
69.68	even	2		4232.2.a.o.1.1		2
276.275	odd	2		8464.2.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.a.e.1.1	✓	2	3.2	odd	2
368.2.a.i.1.2		2	12.11	even	2
1472.2.a.p.1.1		2	24.11	even	2
1472.2.a.u.1.2		2	24.5	odd	2
1656.2.a.j.1.1		2	1.1	even	1	trivial
3312.2.a.t.1.2		2	4.3	odd	2
4232.2.a.o.1.1		2	69.68	even	2
4600.2.a.s.1.2		2	15.14	odd	2
4600.2.e.o.4049.1		4	15.8	even	4
4600.2.e.o.4049.4		4	15.2	even	4
8464.2.a.bd.1.2		2	276.275	odd	2
9016.2.a.w.1.2		2	21.20	even	2
9200.2.a.br.1.1		2	60.59	even	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:	\( N \)	\(=\)	\( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1656.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{5} +O(q^{10})\)	\(q-2.00000 q^{5} -5.12311 q^{11} +4.56155 q^{13} +3.12311 q^{17} +5.12311 q^{19} +1.00000 q^{23} -1.00000 q^{25} +0.561553 q^{29} -6.56155 q^{31} -8.24621 q^{37} -10.8078 q^{41} -8.00000 q^{43} -11.6847 q^{47} -7.00000 q^{49} -2.00000 q^{53} +10.2462 q^{55} +6.24621 q^{59} +12.2462 q^{61} -9.12311 q^{65} -5.12311 q^{67} -9.43845 q^{71} -2.31534 q^{73} -5.12311 q^{79} +2.24621 q^{83} -6.24621 q^{85} +13.3693 q^{89} -10.2462 q^{95} -13.3693 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5}+O(q^{10}) \) 2 * q - 4 * q^5	\( 2 q - 4 q^{5} - 2 q^{11} + 5 q^{13} - 2 q^{17} + 2 q^{19} + 2 q^{23} - 2 q^{25} - 3 q^{29} - 9 q^{31} - q^{41} - 16 q^{43} - 11 q^{47} - 14 q^{49} - 4 q^{53} + 4 q^{55} - 4 q^{59} + 8 q^{61} - 10 q^{65} - 2 q^{67} - 23 q^{71} - 17 q^{73} - 2 q^{79} - 12 q^{83} + 4 q^{85} + 2 q^{89} - 4 q^{95} - 2 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - 2 * q^11 + 5 * q^13 - 2 * q^17 + 2 * q^19 + 2 * q^23 - 2 * q^25 - 3 * q^29 - 9 * q^31 - q^41 - 16 * q^43 - 11 * q^47 - 14 * q^49 - 4 * q^53 + 4 * q^55 - 4 * q^59 + 8 * q^61 - 10 * q^65 - 2 * q^67 - 23 * q^71 - 17 * q^73 - 2 * q^79 - 12 * q^83 + 4 * q^85 + 2 * q^89 - 4 * q^95 - 2 * q^97

Introduction

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