LMFDB - Embedded newform 1656.2.b.b.413.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1656,2,Mod(413,1656)]

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, 1, 1, 1]))

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

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

chi := DirichletCharacter("1656.413");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1656 = 2^{3} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1656.b (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:	$13.2232265747$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1173738225664.17
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{4} + 256 $ x^8 - 14*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			413.8
Root			$0.551740 + 1.92239i$ of defining polynomial
Character	$\chi$	$=$	1656.413
Dual form			1656.2.b.b.413.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} +3.84478i q^{5} -2.82843i q^{8} +O(q^{10})$	$q+1.41421i q^{2} -2.00000 q^{4} +3.84478i q^{5} -2.82843i q^{8} -5.43734 q^{10} -3.84478i q^{11} +4.00000 q^{16} -7.56424 q^{19} -7.68956i q^{20} +5.43734 q^{22} -4.79583i q^{23} -9.78233 q^{25} -10.7823 q^{31} +5.65685i q^{32} +2.12690 q^{37} -10.6974i q^{38} +10.8747 q^{40} -9.59166i q^{41} -8.74778 q^{43} +7.68956i q^{44} +6.78233 q^{46} -11.0059i q^{47} +7.00000 q^{49} -13.8343i q^{50} +14.5422i q^{53} +14.7823 q^{55} +14.1851 q^{61} -15.2485i q^{62} -8.00000 q^{64} -2.12690 q^{67} -2.52060i q^{71} -16.7823 q^{73} +3.00789i q^{74} +15.1285 q^{76} +15.3791i q^{80} +13.5647 q^{82} +8.52645i q^{83} -12.3712i q^{86} -10.8747 q^{88} +9.59166i q^{92} +15.5647 q^{94} -29.0828i q^{95} +9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4}+O(q^{10}) $ 8 * q - 16 * q^4	$ 8 q - 16 q^{4} + 32 q^{16} - 24 q^{25} - 32 q^{31} + 56 q^{49} + 64 q^{55} - 64 q^{64} - 80 q^{73} + 16 q^{94}+O(q^{100}) $ 8 * q - 16 * q^4 + 32 * q^16 - 24 * q^25 - 32 * q^31 + 56 * q^49 + 64 * q^55 - 64 * q^64 - 80 * q^73 + 16 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1656\mathbb{Z}\right)^\times$.

$n$	$415$	$649$	$829$	$1289$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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 $

$2$	1.41421i	1.00000i
$2$	1.41421i	1.00000i
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	3.84478i	1.71944i	0.510768	+	0.859719i	$0.329361\pi$
$5$	3.84478i	1.71944i	−0.510768	+	0.859719i	$0.670639\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 2.82843i	− 1.00000i
$9$	0	0
$10$	−5.43734	−1.71944
$11$	− 3.84478i	− 1.15924i	−0.814885	−	0.579622i	$-0.803200\pi$
$11$	− 3.84478i	− 1.15924i	0.814885	−	0.579622i	$-0.196800\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−7.56424	−1.73535	−0.867677	−	0.497128i	$-0.834388\pi$
$19$	−7.56424	−1.73535	−0.867677	+	0.497128i	$0.834388\pi$
$20$	− 7.68956i	− 1.71944i
$21$	0	0
$22$	5.43734	1.15924
$23$	− 4.79583i	− 1.00000i
$23$	− 4.79583i	− 1.00000i
$24$	0	0
$25$	−9.78233	−1.95647
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−10.7823	−1.93656	−0.968282	−	0.249861i	$-0.919615\pi$
$31$	−10.7823	−1.93656	−0.968282	+	0.249861i	$0.919615\pi$
$32$	5.65685i	1.00000i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.12690	0.349660	0.174830	−	0.984599i	$-0.444062\pi$
$37$	2.12690	0.349660	0.174830	+	0.984599i	$0.444062\pi$
$38$	− 10.6974i	− 1.73535i
$39$	0	0
$40$	10.8747	1.71944
$41$	− 9.59166i	− 1.49797i	−0.662589	−	0.748983i	$-0.730542\pi$
$41$	− 9.59166i	− 1.49797i	0.662589	−	0.748983i	$-0.269458\pi$
$42$	0	0
$43$	−8.74778	−1.33402	−0.667012	−	0.745047i	$-0.732427\pi$
$43$	−8.74778	−1.33402	−0.667012	+	0.745047i	$0.732427\pi$
$44$	7.68956i	1.15924i
$45$	0	0
$46$	6.78233	1.00000
$47$	− 11.0059i	− 1.60537i	−0.596402	−	0.802686i	$-0.703403\pi$
$47$	− 11.0059i	− 1.60537i	0.596402	−	0.802686i	$-0.296597\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	− 13.8343i	− 1.95647i
$51$	0	0
$52$	0	0
$53$	14.5422i	1.99753i	0.0497015	+	0.998764i	$0.484173\pi$
$53$	14.5422i	1.99753i	−0.0497015	+	0.998764i	$0.515827\pi$
$54$	0	0
$55$	14.7823	1.99325
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	14.1851	1.81622	0.908109	−	0.418733i	$-0.137526\pi$
$61$	14.1851	1.81622	0.908109	+	0.418733i	$0.137526\pi$
$62$	− 15.2485i	− 1.93656i
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−2.12690	−0.259842	−0.129921	−	0.991524i	$-0.541472\pi$
$67$	−2.12690	−0.259842	−0.129921	+	0.991524i	$0.541472\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 2.52060i	− 0.299140i	−0.988751	−	0.149570i	$-0.952211\pi$
$71$	− 2.52060i	− 0.299140i	0.988751	−	0.149570i	$-0.0477889\pi$
$72$	0	0
$73$	−16.7823	−1.96422	−0.982112	−	0.188300i	$-0.939702\pi$
$73$	−16.7823	−1.96422	−0.982112	+	0.188300i	$0.939702\pi$
$74$	3.00789i	0.349660i
$75$	0	0
$76$	15.1285	1.73535
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	15.3791i	1.71944i
$81$	0	0
$82$	13.5647	1.49797
$83$	8.52645i	0.935900i	0.883755	+	0.467950i	$0.155007\pi$
$83$	8.52645i	0.935900i	−0.883755	+	0.467950i	$0.844993\pi$
$84$	0	0
$85$	0	0
$86$	− 12.3712i	− 1.33402i
$87$	0	0
$88$	−10.8747	−1.15924
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	9.59166i	1.00000i
$93$	0	0
$94$	15.5647	1.60537
$95$	− 29.0828i	− 2.98383i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	9.89949i	1.00000i
$99$	0	0
$100$	19.5647	1.95647
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	−20.5658	−1.99753
$107$	− 0.836893i	− 0.0809055i	−0.999181	−	0.0404528i	$-0.987120\pi$
$107$	− 0.836893i	− 0.0809055i	0.999181	−	0.0404528i	$-0.0128800\pi$
$108$	0	0
$109$	7.56424	0.724522	0.362261	−	0.932077i	$-0.382005\pi$
$109$	7.56424	0.724522	0.362261	+	0.932077i	$0.382005\pi$
$110$	20.9054i	1.99325i
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	18.4389	1.71944
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3.78233	−0.343848
$122$	20.0608i	1.81622i
$123$	0	0
$124$	21.5647	1.93656
$125$	− 18.3870i	− 1.64458i
$126$	0	0
$127$	13.5647	1.20367	0.601834	−	0.798621i	$-0.294437\pi$
$127$	13.5647	1.20367	0.601834	+	0.798621i	$0.294437\pi$
$128$	− 11.3137i	− 1.00000i
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	− 3.00789i	− 0.259842i
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	3.56466	0.299140
$143$	0	0
$144$	0	0
$145$	0	0
$146$	− 23.7338i	− 1.96422i
$147$	0	0
$148$	−4.25379	−0.349660
$149$	− 22.2318i	− 1.82130i	−0.413180	−	0.910649i	$-0.635582\pi$
$149$	− 22.2318i	− 1.82130i	0.413180	−	0.910649i	$-0.364418\pi$
$150$	0	0
$151$	−22.7823	−1.85400	−0.927000	−	0.375062i	$-0.877622\pi$
$151$	−22.7823	−1.85400	−0.927000	+	0.375062i	$0.877622\pi$
$152$	21.3949i	1.73535i
$153$	0	0
$154$	0	0
$155$	− 41.4557i	− 3.32980i
$156$	0	0
$157$	−25.0598	−1.99999	−0.999995	−	0.00319494i	$-0.998983\pi$
$157$	−25.0598	−1.99999	−0.999995	+	0.00319494i	$0.998983\pi$
$158$	0	0
$159$	0	0
$160$	−21.7494	−1.71944
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	19.1833i	1.49797i
$165$	0	0
$166$	−12.0582	−0.935900
$167$	5.96469i	0.461561i	0.973006	+	0.230781i	$0.0741280\pi$
$167$	5.96469i	0.461561i	−0.973006	+	0.230781i	$0.925872\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	17.4956	1.33402
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	− 15.3791i	− 1.15924i
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−17.2554	−1.28258	−0.641291	−	0.767298i	$-0.721601\pi$
$181$	−17.2554	−1.28258	−0.641291	+	0.767298i	$0.721601\pi$
$182$	0	0
$183$	0	0
$184$	−13.5647	−1.00000
$185$	8.17745i	0.601218i
$186$	0	0
$187$	0	0
$188$	22.0118i	1.60537i
$189$	0	0
$190$	41.1293	2.98383
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−27.1293	−1.95281	−0.976406	−	0.215945i	$-0.930717\pi$
$193$	−27.1293	−1.95281	−0.976406	+	0.215945i	$0.930717\pi$
$194$	0	0
$195$	0	0
$196$	−14.0000	−1.00000
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	27.6686i	1.95647i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	36.8778	2.57566
$206$	0	0
$207$	0	0
$208$	0	0
$209$	29.0828i	2.01170i
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	− 29.0845i	− 1.99753i
$213$	0	0
$214$	1.18355	0.0809055
$215$	− 33.6333i	− 2.29377i
$216$	0	0
$217$	0	0
$218$	10.6974i	0.724522i
$219$	0	0
$220$	−29.5647	−1.99325
$221$	0	0
$222$	0	0
$223$	13.2177	0.885121	0.442560	−	0.896739i	$-0.354070\pi$
$223$	13.2177	0.885121	0.442560	+	0.896739i	$0.354070\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.86055i	0.654468i	0.944943	+	0.327234i	$0.106117\pi$
$227$	9.86055i	0.654468i	−0.944943	+	0.327234i	$0.893883\pi$
$228$	0	0
$229$	−23.8763	−1.57779	−0.788894	−	0.614530i	$-0.789346\pi$
$229$	−23.8763	−1.57779	−0.788894	+	0.614530i	$0.789346\pi$
$230$	26.0766i	1.71944i
$231$	0	0
$232$	0	0
$233$	9.28383i	0.608204i	0.952639	+	0.304102i	$0.0983564\pi$
$233$	9.28383i	0.608204i	−0.952639	+	0.304102i	$0.901644\pi$
$234$	0	0
$235$	42.3152	2.76034
$236$	0	0
$237$	0	0
$238$	0	0
$239$	11.3137i	0.731823i	0.930650	+	0.365911i	$0.119243\pi$
$239$	11.3137i	0.731823i	−0.930650	+	0.365911i	$0.880757\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	− 5.34902i	− 0.343848i
$243$	0	0
$244$	−28.3702	−1.81622
$245$	26.9135i	1.71944i
$246$	0	0
$247$	0	0
$248$	30.4970i	1.93656i
$249$	0	0
$250$	26.0032	1.64458
$251$	31.5951i	1.99427i	0.0756629	+	0.997133i	$0.475893\pi$
$251$	31.5951i	1.99427i	−0.0756629	+	0.997133i	$0.524107\pi$
$252$	0	0
$253$	−18.4389	−1.15924
$254$	19.1833i	1.20367i
$255$	0	0
$256$	16.0000	1.00000
$257$	− 14.1421i	− 0.882162i	−0.897467	−	0.441081i	$-0.854595\pi$
$257$	− 14.1421i	− 0.882162i	0.897467	−	0.441081i	$-0.145405\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−55.9116	−3.43463
$266$	0	0
$267$	0	0
$268$	4.25379	0.259842
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	13.5647	0.823994	0.411997	−	0.911185i	$-0.364831\pi$
$271$	13.5647	0.823994	0.411997	+	0.911185i	$0.364831\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	37.6109i	2.26802i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−33.5674	−1.99537	−0.997687	−	0.0679748i	$-0.978346\pi$
$283$	−33.5674	−1.99537	−0.997687	+	0.0679748i	$0.978346\pi$
$284$	5.04119i	0.299140i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	33.5647	1.96422
$293$	− 8.52645i	− 0.498121i	−0.968488	−	0.249060i	$-0.919878\pi$
$293$	− 8.52645i	− 0.498121i	0.968488	−	0.249060i	$-0.0801217\pi$
$294$	0	0
$295$	0	0
$296$	− 6.01577i	− 0.349660i
$297$	0	0
$298$	31.4405	1.82130
$299$	0	0
$300$	0	0
$301$	0	0
$302$	− 32.2191i	− 1.85400i
$303$	0	0
$304$	−30.2569	−1.73535
$305$	54.5387i	3.12287i
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	58.6272	3.32980
$311$	− 9.59166i	− 0.543893i	−0.962312	−	0.271947i	$-0.912333\pi$
$311$	− 9.59166i	− 0.543893i	0.962312	−	0.271947i	$-0.0876674\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	− 35.4399i	− 1.99999i
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	− 30.7582i	− 1.71944i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	−27.1293	−1.49797
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	− 17.0529i	− 0.935900i
$333$	0	0
$334$	−8.43534	−0.461561
$335$	− 8.17745i	− 0.446782i
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	18.3848i	1.00000i
$339$	0	0
$340$	0	0
$341$	41.4557i	2.24495i
$342$	0	0
$343$	0	0
$344$	24.7425i	1.33402i
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	21.7494	1.15924
$353$	0.798550i	0.0425025i	0.999774	+	0.0212513i	$0.00676499\pi$
$353$	0.798550i	0.0425025i	−0.999774	+	0.0212513i	$0.993235\pi$
$354$	0	0
$355$	9.69113	0.514352
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	38.2177	2.01146
$362$	− 24.4028i	− 1.28258i
$363$	0	0
$364$	0	0
$365$	− 64.5244i	− 3.37736i
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	− 19.1833i	− 1.00000i
$369$	0	0
$370$	−11.5647	−0.601218
$371$	0	0
$372$	0	0
$373$	20.8060	1.07729	0.538647	−	0.842531i	$-0.318936\pi$
$373$	20.8060	1.07729	0.538647	+	0.842531i	$0.318936\pi$
$374$	0	0
$375$	0	0
$376$	−31.1293	−1.60537
$377$	0	0
$378$	0	0
$379$	−15.3687	−0.789435	−0.394718	−	0.918802i	$-0.629158\pi$
$379$	−15.3687	−0.789435	−0.394718	+	0.918802i	$0.629158\pi$
$380$	58.1656i	2.98383i
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	− 38.3667i	− 1.95281i
$387$	0	0
$388$	0	0
$389$	− 5.51857i	− 0.279802i	−0.990165	−	0.139901i	$-0.955322\pi$
$389$	− 5.51857i	− 0.279802i	0.990165	−	0.139901i	$-0.0446785\pi$
$390$	0	0
$391$	0	0
$392$	− 19.7990i	− 1.00000i
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	−39.1293	−1.95647
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8.17745i	− 0.405341i
$408$	0	0
$409$	−27.1293	−1.34146	−0.670729	−	0.741702i	$-0.734019\pi$
$409$	−27.1293	−1.34146	−0.670729	+	0.741702i	$0.734019\pi$
$410$	52.1531i	2.57566i
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−32.7823	−1.60922
$416$	0	0
$417$	0	0
$418$	−41.1293	−2.01170
$419$	22.2318i	1.08609i	0.839702	+	0.543047i	$0.182729\pi$
$419$	22.2318i	1.08609i	−0.839702	+	0.543047i	$0.817271\pi$
$420$	0	0
$421$	−31.6807	−1.54402	−0.772011	−	0.635609i	$-0.780749\pi$
$421$	−31.6807	−1.54402	−0.772011	+	0.635609i	$0.780749\pi$
$422$	0	0
$423$	0	0
$424$	41.1316	1.99753
$425$	0	0
$426$	0	0
$427$	0	0
$428$	1.67379i	0.0809055i
$429$	0	0
$430$	47.5647	2.29377
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	−15.1285	−0.724522
$437$	36.2768i	1.73535i
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	− 41.8107i	− 1.99325i
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	18.6926i	0.885121i
$447$	0	0
$448$	0	0
$449$	− 31.1127i	− 1.46830i	−0.678988	−	0.734150i	$-0.737581\pi$
$449$	− 31.1127i	− 1.46830i	0.678988	−	0.734150i	$-0.262419\pi$
$450$	0	0
$451$	−36.8778	−1.73651
$452$	0	0
$453$	0	0
$454$	−13.9449	−0.654468
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	− 33.7661i	− 1.57779i
$459$	0	0
$460$	−36.8778	−1.71944
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	0	0
$465$	0	0
$466$	−13.1293	−0.608204
$467$	− 40.6188i	− 1.87961i	−0.341707	−	0.939807i	$-0.611005\pi$
$467$	− 40.6188i	− 1.87961i	0.341707	−	0.939807i	$-0.388995\pi$
$468$	0	0
$469$	0	0
$470$	59.8427i	2.76034i
$471$	0	0
$472$	0	0
$473$	33.6333i	1.54646i
$474$	0	0
$475$	73.9959	3.39516
$476$	0	0
$477$	0	0
$478$	−16.0000	−0.731823
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	7.56466	0.343848
$485$	0	0
$486$	0	0
$487$	13.5647	0.614673	0.307337	−	0.951601i	$-0.400562\pi$
$487$	13.5647	0.614673	0.307337	+	0.951601i	$0.400562\pi$
$488$	− 40.1216i	− 1.81622i
$489$	0	0
$490$	−38.0614	−1.71944
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−43.1293	−1.93656
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	36.7740i	1.64458i
$501$	0	0
$502$	−44.6823	−1.99427
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	− 26.0766i	− 1.15924i
$507$	0	0
$508$	−27.1293	−1.20367
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	22.6274i	1.00000i
$513$	0	0
$514$	20.0000	0.882162
$515$	0	0
$516$	0	0
$517$	−42.3152	−1.86102
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−26.9465	−1.17829	−0.589144	−	0.808028i	$-0.700535\pi$
$523$	−26.9465	−1.17829	−0.589144	+	0.808028i	$0.700535\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	− 79.0710i	− 3.43463i
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	3.21767	0.139112
$536$	6.01577i	0.259842i
$537$	0	0
$538$	0	0
$539$	− 26.9135i	− 1.15924i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	19.1833i	0.823994i
$543$	0	0
$544$	0	0
$545$	29.0828i	1.24577i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	−53.1898	−2.26802
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 12.8684i	− 0.545253i	−0.962120	−	0.272627i	$-0.912108\pi$
$557$	− 12.8684i	− 0.545253i	0.962120	−	0.272627i	$-0.0878924\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 46.9742i	− 1.97973i	−0.142017	−	0.989864i	$-0.545359\pi$
$563$	− 46.9742i	− 1.97973i	0.142017	−	0.989864i	$-0.454641\pi$
$564$	0	0
$565$	0	0
$566$	− 47.4715i	− 1.99537i
$567$	0	0
$568$	−7.12932	−0.299140
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	44.4421	1.85984	0.929921	−	0.367759i	$-0.119875\pi$
$571$	44.4421	1.85984	0.929921	+	0.367759i	$0.119875\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	46.9144i	1.95647i
$576$	0	0
$577$	35.9116	1.49502	0.747511	−	0.664250i	$-0.231249\pi$
$577$	35.9116	1.49502	0.747511	+	0.664250i	$0.231249\pi$
$578$	− 24.0416i	− 1.00000i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	55.9116	2.31562
$584$	47.4676i	1.96422i
$585$	0	0
$586$	12.0582	0.498121
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	81.5601	3.36063
$590$	0	0
$591$	0	0
$592$	8.50759	0.349660
$593$	47.9583i	1.96941i	0.174224	+	0.984706i	$0.444258\pi$
$593$	47.9583i	1.96941i	−0.174224	+	0.984706i	$0.555742\pi$
$594$	0	0
$595$	0	0
$596$	44.4636i	1.82130i
$597$	0	0
$598$	0	0
$599$	− 39.5980i	− 1.61793i	−0.587857	−	0.808965i	$-0.700028\pi$
$599$	− 39.5980i	− 1.61793i	0.587857	−	0.808965i	$-0.299972\pi$
$600$	0	0
$601$	−40.7823	−1.66355	−0.831773	−	0.555116i	$-0.812674\pi$
$601$	−40.7823	−1.66355	−0.831773	+	0.555116i	$0.812674\pi$
$602$	0	0
$603$	0	0
$604$	45.5647	1.85400
$605$	− 14.5422i	− 0.591226i
$606$	0	0
$607$	41.9116	1.70114	0.850571	−	0.525861i	$-0.176257\pi$
$607$	41.9116	1.70114	0.850571	+	0.525861i	$0.176257\pi$
$608$	− 42.7898i	− 1.73535i
$609$	0	0
$610$	−77.1293	−3.12287
$611$	0	0
$612$	0	0
$613$	21.5092	0.868747	0.434373	−	0.900733i	$-0.356970\pi$
$613$	21.5092	0.868747	0.434373	+	0.900733i	$0.356970\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−34.7509	−1.39676	−0.698379	−	0.715728i	$-0.746095\pi$
$619$	−34.7509	−1.39676	−0.698379	+	0.715728i	$0.746095\pi$
$620$	82.9114i	3.32980i
$621$	0	0
$622$	13.5647	0.543893
$623$	0	0
$624$	0	0
$625$	21.7823	0.871293
$626$	0	0
$627$	0	0
$628$	50.1196	1.99999
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	52.1531i	2.06963i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	43.4987	1.71944
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	−47.9927	−1.89265	−0.946324	−	0.323220i	$-0.895235\pi$
$643$	−47.9927	−1.89265	−0.946324	+	0.323220i	$0.895235\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	47.9583i	1.88544i	0.333591	+	0.942718i	$0.391740\pi$
$647$	47.9583i	1.88544i	−0.333591	+	0.942718i	$0.608260\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	− 38.3667i	− 1.49797i
$657$	0	0
$658$	0	0
$659$	− 31.2554i	− 1.21754i	−0.793347	−	0.608770i	$-0.791663\pi$
$659$	− 31.2554i	− 1.21754i	0.793347	−	0.608770i	$-0.208337\pi$
$660$	0	0
$661$	−4.49399	−0.174796	−0.0873979	−	0.996173i	$-0.527855\pi$
$661$	−4.49399	−0.174796	−0.0873979	+	0.996173i	$0.527855\pi$
$662$	0	0
$663$	0	0
$664$	24.1164	0.935900
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 11.9294i	− 0.461561i
$669$	0	0
$670$	11.5647	0.446782
$671$	− 54.5387i	− 2.10544i
$672$	0	0
$673$	47.9116	1.84686	0.923429	−	0.383769i	$-0.125374\pi$
$673$	47.9116	1.84686	0.923429	+	0.383769i	$0.125374\pi$
$674$	0	0
$675$	0	0
$676$	−26.0000	−1.00000
$677$	51.3162i	1.97224i	0.166025	+	0.986122i	$0.446907\pi$
$677$	51.3162i	1.97224i	−0.166025	+	0.986122i	$0.553093\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	−58.6272	−2.24495
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−34.9911	−1.33402
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 51.6559i	− 1.95102i	−0.219961	−	0.975509i	$-0.570593\pi$
$701$	− 51.6559i	− 1.95102i	0.219961	−	0.975509i	$-0.429407\pi$
$702$	0	0
$703$	−16.0884	−0.606784
$704$	30.7582i	1.15924i
$705$	0	0
$706$	−1.12932	−0.0425025
$707$	0	0
$708$	0	0
$709$	−11.8180	−0.443835	−0.221918	−	0.975065i	$-0.571232\pi$
$709$	−11.8180	−0.443835	−0.221918	+	0.975065i	$0.571232\pi$
$710$	13.7053i	0.514352i
$711$	0	0
$712$	0	0
$713$	51.7102i	1.93656i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	45.2548i	1.68772i	0.536563	+	0.843860i	$0.319722\pi$
$719$	45.2548i	1.68772i	−0.536563	+	0.843860i	$0.680278\pi$
$720$	0	0
$721$	0	0
$722$	54.0479i	2.01146i
$723$	0	0
$724$	34.5107	1.28258
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	91.2512	3.37736
$731$	0	0
$732$	0	0
$733$	54.1332	1.99946	0.999728	−	0.0233363i	$-0.00742884\pi$
$733$	54.1332	1.99946	0.999728	+	0.0233363i	$0.00742884\pi$
$734$	0	0
$735$	0	0
$736$	27.1293	1.00000
$737$	8.17745i	0.301220i
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	− 16.3549i	− 0.601218i
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	85.4763	3.13161
$746$	29.4241i	1.07729i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	− 44.0235i	− 1.60537i
$753$	0	0
$754$	0	0
$755$	− 87.5930i	− 3.18784i
$756$	0	0
$757$	53.4300	1.94195	0.970974	−	0.239185i	$-0.0768803\pi$
$757$	53.4300	1.94195	0.970974	+	0.239185i	$0.0768803\pi$
$758$	− 21.7346i	− 0.789435i
$759$	0	0
$760$	−82.2586	−2.98383
$761$	− 31.2956i	− 1.13446i	−0.823558	−	0.567232i	$-0.808014\pi$
$761$	− 31.2956i	− 1.13446i	0.823558	−	0.567232i	$-0.191986\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	54.2586	1.95281
$773$	17.5501i	0.631234i	0.948887	+	0.315617i	$0.102211\pi$
$773$	17.5501i	0.631234i	−0.948887	+	0.315617i	$0.897789\pi$
$774$	0	0
$775$	105.476	3.78882
$776$	0	0
$777$	0	0
$778$	7.80443	0.279802
$779$	72.5536i	2.59950i
$780$	0	0
$781$	−9.69113	−0.346776
$782$	0	0
$783$	0	0
$784$	28.0000	1.00000
$785$	− 96.3494i	− 3.43886i
$786$	0	0
$787$	25.0598	0.893285	0.446643	−	0.894712i	$-0.352620\pi$
$787$	25.0598	0.893285	0.446643	+	0.894712i	$0.352620\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.17888i	0.183445i	0.995785	+	0.0917227i	$0.0292373\pi$
$797$	5.17888i	0.183445i	−0.995785	+	0.0917227i	$0.970763\pi$
$798$	0	0
$799$	0	0
$800$	− 55.3372i	− 1.95647i
$801$	0	0
$802$	0	0
$803$	64.5244i	2.27702i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 16.1720i	− 0.568578i	−0.958739	−	0.284289i	$-0.908243\pi$
$809$	− 16.1720i	− 0.568578i	0.958739	−	0.284289i	$-0.0917575\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	11.5647	0.405341
$815$	0	0
$816$	0	0
$817$	66.1703	2.31501
$818$	− 38.3667i	− 1.34146i
$819$	0	0
$820$	−73.7557	−2.57566
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	−46.7823	−1.63073	−0.815365	−	0.578948i	$-0.803464\pi$
$823$	−46.7823	−1.63073	−0.815365	+	0.578948i	$0.803464\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 37.6109i	− 1.30786i	−0.756555	−	0.653930i	$-0.773119\pi$
$827$	− 37.6109i	− 1.30786i	0.756555	−	0.653930i	$-0.226881\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	− 46.3612i	− 1.60922i
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−22.9329	−0.793626
$836$	− 58.1656i	− 2.01170i
$837$	0	0
$838$	−31.4405	−1.08609
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 44.8033i	− 1.54402i
$843$	0	0
$844$	0	0
$845$	49.9821i	1.71944i
$846$	0	0
$847$	0	0
$848$	58.1689i	1.99753i
$849$	0	0
$850$	0	0
$851$	− 10.2002i	− 0.349660i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−2.36709	−0.0809055
$857$	− 22.8103i	− 0.779185i	−0.920987	−	0.389593i	$-0.872616\pi$
$857$	− 22.8103i	− 0.779185i	0.920987	−	0.389593i	$-0.127384\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	67.2666i	2.29377i
$861$	0	0
$862$	0	0
$863$	47.9583i	1.63252i	0.577685	+	0.816260i	$0.303956\pi$
$863$	47.9583i	1.63252i	−0.577685	+	0.816260i	$0.696044\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 21.3949i	− 0.724522i
$873$	0	0
$874$	−51.3031	−1.73535
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	− 14.1421i	− 0.477274i
$879$	0	0
$880$	59.1293	1.99325
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.4205i	1.05500i	0.849556	+	0.527499i	$0.176870\pi$
$887$	31.4205i	1.05500i	−0.849556	+	0.527499i	$0.823130\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−26.4353	−0.885121
$893$	83.2511i	2.78589i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	44.0000	1.46830
$899$	0	0
$900$	0	0
$901$	0	0
$902$	− 52.1531i	− 1.73651i
$903$	0	0
$904$	0	0
$905$	− 66.3431i	− 2.20532i
$906$	0	0
$907$	−59.5705	−1.97801	−0.989004	−	0.147891i	$-0.952751\pi$
$907$	−59.5705	−1.97801	−0.989004	+	0.147891i	$0.952751\pi$
$908$	− 19.7211i	− 0.654468i
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	32.7823	1.08494
$914$	0	0
$915$	0	0
$916$	47.7525	1.57779
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	− 52.1531i	− 1.71944i
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−20.8060	−0.684097
$926$	19.7990i	0.650635i
$927$	0	0
$928$	0	0
$929$	53.7401i	1.76316i	0.472038	+	0.881578i	$0.343518\pi$
$929$	53.7401i	1.76316i	−0.472038	+	0.881578i	$0.656482\pi$
$930$	0	0
$931$	−52.9497	−1.73535
$932$	− 18.5677i	− 0.608204i
$933$	0	0
$934$	57.4436	1.87961
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	−84.6303	−2.76034
$941$	− 49.6424i	− 1.61830i	−0.587604	−	0.809149i	$-0.699929\pi$
$941$	− 49.6424i	− 1.61830i	0.587604	−	0.809149i	$-0.300071\pi$
$942$	0	0
$943$	−46.0000	−1.49797
$944$	0	0
$945$	0	0
$946$	−47.5647	−1.54646
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	104.646i	3.39516i
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	− 22.6274i	− 0.731823i
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	85.2586	2.75028
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 104.306i	− 3.35774i
$966$	0	0
$967$	5.91165	0.190106	0.0950529	−	0.995472i	$-0.469698\pi$
$967$	5.91165	0.190106	0.0950529	+	0.995472i	$0.469698\pi$
$968$	10.6980i	0.343848i
$969$	0	0
$970$	0	0
$971$	− 13.2081i	− 0.423869i	−0.977284	−	0.211934i	$-0.932024\pi$
$971$	− 13.2081i	− 0.423869i	0.977284	−	0.211934i	$-0.0679763\pi$
$972$	0	0
$973$	0	0
$974$	19.1833i	0.614673i
$975$	0	0
$976$	56.7405	1.81622
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	− 53.8269i	− 1.71944i
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	41.9529i	1.33402i
$990$	0	0
$991$	37.2177	1.18226	0.591129	−	0.806577i	$-0.298682\pi$
$991$	37.2177	1.18226	0.591129	+	0.806577i	$0.298682\pi$
$992$	− 60.9941i	− 1.93656i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1656.2.b.b.413.8	yes	8
3.2	odd	2	inner	1656.2.b.b.413.1	✓	8
4.3	odd	2		6624.2.b.b.2897.8		8
8.3	odd	2		6624.2.b.b.2897.1		8
8.5	even	2	inner	1656.2.b.b.413.5	yes	8
12.11	even	2		6624.2.b.b.2897.2		8
23.22	odd	2	inner	1656.2.b.b.413.5	yes	8
24.5	odd	2	inner	1656.2.b.b.413.4	yes	8
24.11	even	2		6624.2.b.b.2897.7		8
69.68	even	2	inner	1656.2.b.b.413.4	yes	8
92.91	even	2		6624.2.b.b.2897.1		8
184.45	odd	2	CM	1656.2.b.b.413.8	yes	8
184.91	even	2		6624.2.b.b.2897.8		8
276.275	odd	2		6624.2.b.b.2897.7		8
552.275	odd	2		6624.2.b.b.2897.2		8
552.413	even	2	inner	1656.2.b.b.413.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1656.2.b.b.413.1	✓	8	3.2	odd	2	inner
1656.2.b.b.413.1	✓	8	552.413	even	2	inner
1656.2.b.b.413.4	yes	8	24.5	odd	2	inner
1656.2.b.b.413.4	yes	8	69.68	even	2	inner
1656.2.b.b.413.5	yes	8	8.5	even	2	inner
1656.2.b.b.413.5	yes	8	23.22	odd	2	inner
1656.2.b.b.413.8	yes	8	1.1	even	1	trivial
1656.2.b.b.413.8	yes	8	184.45	odd	2	CM
6624.2.b.b.2897.1		8	8.3	odd	2
6624.2.b.b.2897.1		8	92.91	even	2
6624.2.b.b.2897.2		8	12.11	even	2
6624.2.b.b.2897.2		8	552.275	odd	2
6624.2.b.b.2897.7		8	24.11	even	2
6624.2.b.b.2897.7		8	276.275	odd	2
6624.2.b.b.2897.8		8	4.3	odd	2
6624.2.b.b.2897.8		8	184.91	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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +3.84478i q^{5} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} +3.84478i q^{5} -2.82843i q^{8} -5.43734 q^{10} -3.84478i q^{11} +4.00000 q^{16} -7.56424 q^{19} -7.68956i q^{20} +5.43734 q^{22} -4.79583i q^{23} -9.78233 q^{25} -10.7823 q^{31} +5.65685i q^{32} +2.12690 q^{37} -10.6974i q^{38} +10.8747 q^{40} -9.59166i q^{41} -8.74778 q^{43} +7.68956i q^{44} +6.78233 q^{46} -11.0059i q^{47} +7.00000 q^{49} -13.8343i q^{50} +14.5422i q^{53} +14.7823 q^{55} +14.1851 q^{61} -15.2485i q^{62} -8.00000 q^{64} -2.12690 q^{67} -2.52060i q^{71} -16.7823 q^{73} +3.00789i q^{74} +15.1285 q^{76} +15.3791i q^{80} +13.5647 q^{82} +8.52645i q^{83} -12.3712i q^{86} -10.8747 q^{88} +9.59166i q^{92} +15.5647 q^{94} -29.0828i q^{95} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4}+O(q^{10}) \) 8 * q - 16 * q^4	\( 8 q - 16 q^{4} + 32 q^{16} - 24 q^{25} - 32 q^{31} + 56 q^{49} + 64 q^{55} - 64 q^{64} - 80 q^{73} + 16 q^{94}+O(q^{100}) \) 8 * q - 16 * q^4 + 32 * q^16 - 24 * q^25 - 32 * q^31 + 56 * q^49 + 64 * q^55 - 64 * q^64 - 80 * q^73 + 16 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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