LMFDB - Embedded newform 8.9.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,9,Mod(3,8)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	8.d (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:	yes
Analytic conductor:	$3.25902888049$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			3.1
Character	$\chi$	$=$	8.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+16.0000 q^{2} +34.0000 q^{3} +256.000 q^{4} +544.000 q^{6} +4096.00 q^{8} -5405.00 q^{9} +O(q^{10})$

$q+16.0000 q^{2} +34.0000 q^{3} +256.000 q^{4} +544.000 q^{6} +4096.00 q^{8} -5405.00 q^{9} -27166.0 q^{11} +8704.00 q^{12} +65536.0 q^{16} +162434. q^{17} -86480.0 q^{18} -72286.0 q^{19} -434656. q^{22} +139264. q^{24} +390625. q^{25} -406844. q^{27} +1.04858e6 q^{32} -923644. q^{33} +2.59894e6 q^{34} -1.38368e6 q^{36} -1.15658e6 q^{38} -4.09901e6 q^{41} +5.42640e6 q^{43} -6.95450e6 q^{44} +2.22822e6 q^{48} +5.76480e6 q^{49} +6.25000e6 q^{50} +5.52276e6 q^{51} -6.50950e6 q^{54} -2.45772e6 q^{57} -2.41781e7 q^{59} +1.67772e7 q^{64} -1.47783e7 q^{66} -1.39443e7 q^{67} +4.15831e7 q^{68} -2.21389e7 q^{72} +3.35676e7 q^{73} +1.32812e7 q^{75} -1.85052e7 q^{76} +2.16295e7 q^{81} -6.55841e7 q^{82} +3.02100e7 q^{83} +8.68224e7 q^{86} -1.11272e8 q^{88} -9.55198e7 q^{89} +3.56516e7 q^{96} -7.74182e7 q^{97} +9.22368e7 q^{98} +1.46832e8 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8\mathbb{Z}\right)^\times$.

$n$	$5$	$7$
$\chi(n)$	$-1$	$-1$

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 $

$2$	16.0000	1.00000
$2$	16.0000	1.00000
$3$	34.0000	0.419753	0.209877	−	0.977728i	$-0.432694\pi$
$3$	34.0000	0.419753	0.209877	+	0.977728i	$0.432694\pi$
$4$	256.000	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	544.000	0.419753
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	4096.00	1.00000
$9$	−5405.00	−0.823807
$10$	0	0
$11$	−27166.0	−1.85547	−0.927737	−	0.373234i	$-0.878249\pi$
$11$	−27166.0	−1.85547	−0.927737	+	0.373234i	$0.878249\pi$
$12$	8704.00	0.419753
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	65536.0	1.00000
$17$	162434.	1.94483	0.972414	−	0.233261i	$-0.0749397\pi$
$17$	162434.	1.94483	0.972414	+	0.233261i	$0.0749397\pi$
$18$	−86480.0	−0.823807
$19$	−72286.0	−0.554677	−0.277338	−	0.960772i	$-0.589452\pi$
$19$	−72286.0	−0.554677	−0.277338	+	0.960772i	$0.589452\pi$
$20$	0	0
$21$	0	0
$22$	−434656.	−1.85547
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	139264.	0.419753
$25$	390625.	1.00000
$26$	0	0
$27$	−406844.	−0.765549
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	1.04858e6	1.00000
$33$	−923644.	−0.778841
$34$	2.59894e6	1.94483
$35$	0	0
$36$	−1.38368e6	−0.823807
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−1.15658e6	−0.554677
$39$	0	0
$40$	0	0
$41$	−4.09901e6	−1.45058	−0.725292	−	0.688441i	$-0.758295\pi$
$41$	−4.09901e6	−1.45058	−0.725292	+	0.688441i	$0.758295\pi$
$42$	0	0
$43$	5.42640e6	1.58722	0.793612	−	0.608424i	$-0.208198\pi$
$43$	5.42640e6	1.58722	0.793612	+	0.608424i	$0.208198\pi$
$44$	−6.95450e6	−1.85547
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	2.22822e6	0.419753
$49$	5.76480e6	1.00000
$50$	6.25000e6	1.00000
$51$	5.52276e6	0.816348
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−6.50950e6	−0.765549
$55$	0	0
$56$	0	0
$57$	−2.45772e6	−0.232827
$58$	0	0
$59$	−2.41781e7	−1.99533	−0.997663	−	0.0683312i	$-0.978233\pi$
$59$	−2.41781e7	−1.99533	−0.997663	+	0.0683312i	$0.978233\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	1.67772e7	1.00000
$65$	0	0
$66$	−1.47783e7	−0.778841
$67$	−1.39443e7	−0.691986	−0.345993	−	0.938237i	$-0.612458\pi$
$67$	−1.39443e7	−0.691986	−0.345993	+	0.938237i	$0.612458\pi$
$68$	4.15831e7	1.94483
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−2.21389e7	−0.823807
$73$	3.35676e7	1.18203	0.591015	−	0.806661i	$-0.298728\pi$
$73$	3.35676e7	1.18203	0.591015	+	0.806661i	$0.298728\pi$
$74$	0	0
$75$	1.32812e7	0.419753
$76$	−1.85052e7	−0.554677
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	2.16295e7	0.502466
$82$	−6.55841e7	−1.45058
$83$	3.02100e7	0.636558	0.318279	−	0.947997i	$-0.396895\pi$
$83$	3.02100e7	0.636558	0.318279	+	0.947997i	$0.396895\pi$
$84$	0	0
$85$	0	0
$86$	8.68224e7	1.58722
$87$	0	0
$88$	−1.11272e8	−1.85547
$89$	−9.55198e7	−1.52242	−0.761208	−	0.648508i	$-0.775393\pi$
$89$	−9.55198e7	−1.52242	−0.761208	+	0.648508i	$0.775393\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	3.56516e7	0.419753
$97$	−7.74182e7	−0.874493	−0.437247	−	0.899342i	$-0.644046\pi$
$97$	−7.74182e7	−0.874493	−0.437247	+	0.899342i	$0.644046\pi$
$98$	9.22368e7	1.00000
$99$	1.46832e8	1.52855
$100$	1.00000e8	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	8.83641e7	0.816348
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.84965e8	−1.41109	−0.705546	−	0.708664i	$-0.749298\pi$
$107$	−1.84965e8	−1.41109	−0.705546	+	0.708664i	$0.749298\pi$
$108$	−1.04152e8	−0.765549
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.22024e7	−0.442831	−0.221415	−	0.975180i	$-0.571068\pi$
$113$	−7.22024e7	−0.442831	−0.221415	+	0.975180i	$0.571068\pi$
$114$	−3.93236e7	−0.232827
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−3.86849e8	−1.99533
$119$	0	0
$120$	0	0
$121$	5.23633e8	2.44279
$122$	0	0
$123$	−1.39366e8	−0.608887
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	2.68435e8	1.00000
$129$	1.84498e8	0.666242
$130$	0	0
$131$	3.39909e8	1.15419	0.577095	−	0.816677i	$-0.304186\pi$
$131$	3.39909e8	1.15419	0.577095	+	0.816677i	$0.304186\pi$
$132$	−2.36453e8	−0.778841
$133$	0	0
$134$	−2.23109e8	−0.691986
$135$	0	0
$136$	6.65330e8	1.94483
$137$	−3.39511e8	−0.963767	−0.481884	−	0.876235i	$-0.660047\pi$
$137$	−3.39511e8	−0.963767	−0.481884	+	0.876235i	$0.660047\pi$
$138$	0	0
$139$	−7.32208e8	−1.96144	−0.980720	−	0.195418i	$-0.937394\pi$
$139$	−7.32208e8	−1.96144	−0.980720	+	0.195418i	$0.937394\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−3.54222e8	−0.823807
$145$	0	0
$146$	5.37081e8	1.18203
$147$	1.96003e8	0.419753
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	2.12500e8	0.419753
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−2.96083e8	−0.554677
$153$	−8.77956e8	−1.60216
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	3.46072e8	0.502466
$163$	1.14307e9	1.61929	0.809644	−	0.586921i	$-0.199660\pi$
$163$	1.14307e9	1.61929	0.809644	+	0.586921i	$0.199660\pi$
$164$	−1.04935e9	−1.45058
$165$	0	0
$166$	4.83359e8	0.636558
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	8.15731e8	1.00000
$170$	0	0
$171$	3.90706e8	0.456947
$172$	1.38916e9	1.58722
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	−1.78035e9	−1.85547
$177$	−8.22055e8	−0.837544
$178$	−1.52832e9	−1.52242
$179$	1.90643e9	1.85699	0.928493	−	0.371349i	$-0.121105\pi$
$179$	1.90643e9	1.85699	0.928493	+	0.371349i	$0.121105\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.41268e9	−3.60858
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	5.70425e8	0.419753
$193$	1.43626e9	1.03515	0.517574	−	0.855638i	$-0.326835\pi$
$193$	1.43626e9	1.03515	0.517574	+	0.855638i	$0.326835\pi$
$194$	−1.23869e9	−0.874493
$195$	0	0
$196$	1.47579e9	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	2.34932e9	1.52855
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	1.60000e9	1.00000
$201$	−4.74106e8	−0.290463
$202$	0	0
$203$	0	0
$204$	1.41383e9	0.816348
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.96372e9	1.02919
$210$	0	0
$211$	−2.52282e9	−1.27279	−0.636395	−	0.771363i	$-0.719575\pi$
$211$	−2.52282e9	−1.27279	−0.636395	+	0.771363i	$0.719575\pi$
$212$	0	0
$213$	0	0
$214$	−2.95945e9	−1.41109
$215$	0	0
$216$	−1.66643e9	−0.765549
$217$	0	0
$218$	0	0
$219$	1.14130e9	0.496160
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−2.11133e9	−0.823807
$226$	−1.15524e9	−0.442831
$227$	3.87828e9	1.46062	0.730308	−	0.683118i	$-0.239377\pi$
$227$	3.87828e9	1.46062	0.730308	+	0.683118i	$0.239377\pi$
$228$	−6.29177e8	−0.232827
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.69938e8	0.159447	0.0797236	−	0.996817i	$-0.474596\pi$
$233$	4.69938e8	0.159447	0.0797236	+	0.996817i	$0.474596\pi$
$234$	0	0
$235$	0	0
$236$	−6.18959e9	−1.99533
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−5.80472e8	−0.172073	−0.0860366	−	0.996292i	$-0.527420\pi$
$241$	−5.80472e8	−0.172073	−0.0860366	+	0.996292i	$0.527420\pi$
$242$	8.37812e9	2.44279
$243$	3.40471e9	0.976460
$244$	0	0
$245$	0	0
$246$	−2.22986e9	−0.608887
$247$	0	0
$248$	0	0
$249$	1.02714e9	0.267197
$250$	0	0
$251$	3.70800e8	0.0934210	0.0467105	−	0.998908i	$-0.485126\pi$
$251$	3.70800e8	0.0934210	0.0467105	+	0.998908i	$0.485126\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	4.29497e9	1.00000
$257$	−8.43940e9	−1.93455	−0.967273	−	0.253738i	$-0.918340\pi$
$257$	−8.43940e9	−1.93455	−0.967273	+	0.253738i	$0.918340\pi$
$258$	2.95196e9	0.666242
$259$	0	0
$260$	0	0
$261$	0	0
$262$	5.43854e9	1.15419
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−3.78325e9	−0.778841
$265$	0	0
$266$	0	0
$267$	−3.24767e9	−0.639039
$268$	−3.56974e9	−0.691986
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	1.06453e10	1.94483
$273$	0	0
$274$	−5.43218e9	−0.963767
$275$	−1.06117e10	−1.85547
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−1.17153e10	−1.96144
$279$	0	0
$280$	0	0
$281$	−2.21245e9	−0.354852	−0.177426	−	0.984134i	$-0.556777\pi$
$281$	−2.21245e9	−0.354852	−0.177426	+	0.984134i	$0.556777\pi$
$282$	0	0
$283$	1.07196e10	1.67122	0.835611	−	0.549321i	$-0.185114\pi$
$283$	1.07196e10	1.67122	0.835611	+	0.549321i	$0.185114\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−5.66755e9	−0.823807
$289$	1.94090e10	2.78236
$290$	0	0
$291$	−2.63222e9	−0.367071
$292$	8.59329e9	1.18203
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	3.13605e9	0.419753
$295$	0	0
$296$	0	0
$297$	1.10523e10	1.42046
$298$	0	0
$299$	0	0
$300$	3.40000e9	0.419753
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−4.73734e9	−0.554677
$305$	0	0
$306$	−1.40473e10	−1.60216
$307$	−6.68482e9	−0.752551	−0.376276	−	0.926508i	$-0.622795\pi$
$307$	−6.68482e9	−0.752551	−0.376276	+	0.926508i	$0.622795\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−1.26772e10	−1.32083	−0.660415	−	0.750901i	$-0.729620\pi$
$313$	−1.26772e10	−1.32083	−0.660415	+	0.750901i	$0.729620\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−6.28882e9	−0.592310
$322$	0	0
$323$	−1.17417e10	−1.07875
$324$	5.53715e9	0.502466
$325$	0	0
$326$	1.82892e10	1.61929
$327$	0	0
$328$	−1.67895e10	−1.45058
$329$	0	0
$330$	0	0
$331$	2.39214e10	1.99285	0.996424	−	0.0844976i	$-0.0269285\pi$
$331$	2.39214e10	1.99285	0.996424	+	0.0844976i	$0.0269285\pi$
$332$	7.73375e9	0.636558
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.57940e10	−1.99986	−0.999930	−	0.0118516i	$-0.996227\pi$
$337$	−2.57940e10	−1.99986	−0.999930	+	0.0118516i	$0.996227\pi$
$338$	1.30517e10	1.00000
$339$	−2.45488e9	−0.185880
$340$	0	0
$341$	0	0
$342$	6.25129e9	0.456947
$343$	0	0
$344$	2.22265e10	1.58722
$345$	0	0
$346$	0	0
$347$	7.92343e9	0.546507	0.273253	−	0.961942i	$-0.411900\pi$
$347$	7.92343e9	0.546507	0.273253	+	0.961942i	$0.411900\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−2.84856e10	−1.85547
$353$	1.37121e10	0.883093	0.441546	−	0.897238i	$-0.354430\pi$
$353$	1.37121e10	0.883093	0.441546	+	0.897238i	$0.354430\pi$
$354$	−1.31529e10	−0.837544
$355$	0	0
$356$	−2.44531e10	−1.52242
$357$	0	0
$358$	3.05029e10	1.85699
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−1.17583e10	−0.692334
$362$	0	0
$363$	1.78035e10	1.02537
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	2.21551e10	1.19500
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−7.06029e10	−3.60858
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−7.66145e9	−0.371325	−0.185663	−	0.982614i	$-0.559443\pi$
$379$	−7.66145e9	−0.371325	−0.185663	+	0.982614i	$0.559443\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	9.12681e9	0.419753
$385$	0	0
$386$	2.29801e10	1.03515
$387$	−2.93297e10	−1.30757
$388$	−1.98191e10	−0.874493
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	2.36126e10	1.00000
$393$	1.15569e10	0.484474
$394$	0	0
$395$	0	0
$396$	3.75891e10	1.52855
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	2.56000e10	1.00000
$401$	1.85927e10	0.719061	0.359530	−	0.933133i	$-0.382937\pi$
$401$	1.85927e10	0.719061	0.359530	+	0.933133i	$0.382937\pi$
$402$	−7.58569e9	−0.290463
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	2.26212e10	0.816348
$409$	−6.23419e9	−0.222785	−0.111393	−	0.993776i	$-0.535531\pi$
$409$	−6.23419e9	−0.222785	−0.111393	+	0.993776i	$0.535531\pi$
$410$	0	0
$411$	−1.15434e10	−0.404544
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.48951e10	−0.823321
$418$	3.14195e10	1.02919
$419$	−5.40872e10	−1.75484	−0.877422	−	0.479720i	$-0.840738\pi$
$419$	−5.40872e10	−1.75484	−0.877422	+	0.479720i	$0.840738\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−4.03651e10	−1.27279
$423$	0	0
$424$	0	0
$425$	6.34508e10	1.94483
$426$	0	0
$427$	0	0
$428$	−4.73511e10	−1.41109
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−2.66629e10	−0.765549
$433$	−6.86319e10	−1.95243	−0.976213	−	0.216813i	$-0.930434\pi$
$433$	−6.86319e10	−1.95243	−0.976213	+	0.216813i	$0.930434\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	1.82607e10	0.496160
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−3.11587e10	−0.823807
$442$	0	0
$443$	6.61486e10	1.71754	0.858768	−	0.512364i	$-0.171230\pi$
$443$	6.61486e10	1.71754	0.858768	+	0.512364i	$0.171230\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.89524e10	0.958405	0.479202	−	0.877704i	$-0.340926\pi$
$449$	3.89524e10	0.958405	0.479202	+	0.877704i	$0.340926\pi$
$450$	−3.37812e10	−0.823807
$451$	1.11354e11	2.69152
$452$	−1.84838e10	−0.442831
$453$	0	0
$454$	6.20525e10	1.46062
$455$	0	0
$456$	−1.00668e10	−0.232827
$457$	4.31242e10	0.988681	0.494340	−	0.869268i	$-0.335410\pi$
$457$	4.31242e10	0.988681	0.494340	+	0.869268i	$0.335410\pi$
$458$	0	0
$459$	−6.60853e10	−1.48886
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	7.51901e9	0.159447
$467$	9.41185e10	1.97883	0.989413	−	0.145128i	$-0.0463594\pi$
$467$	9.41185e10	1.97883	0.989413	+	0.145128i	$0.0463594\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−9.90334e10	−1.99533
$473$	−1.47414e11	−2.94505
$474$	0	0
$475$	−2.82367e10	−0.554677
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	−9.28756e9	−0.172073
$483$	0	0
$484$	1.34050e11	2.44279
$485$	0	0
$486$	5.44753e10	0.976460
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	3.88645e10	0.679701
$490$	0	0
$491$	−9.95056e10	−1.71207	−0.856035	−	0.516918i	$-0.827079\pi$
$491$	−9.95056e10	−1.71207	−0.856035	+	0.516918i	$0.827079\pi$
$492$	−3.56777e10	−0.608887
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	1.64342e10	0.267197
$499$	−1.02919e11	−1.65995	−0.829975	−	0.557801i	$-0.811645\pi$
$499$	−1.02919e11	−1.65995	−0.829975	+	0.557801i	$0.811645\pi$
$500$	0	0
$501$	0	0
$502$	5.93280e9	0.0934210
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.77348e10	0.419753
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	6.87195e10	1.00000
$513$	2.94091e10	0.424632
$514$	−1.35030e11	−1.93455
$515$	0	0
$516$	4.72314e10	0.666242
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.27450e10	0.987308	0.493654	−	0.869658i	$-0.335661\pi$
$521$	7.27450e10	0.987308	0.493654	+	0.869658i	$0.335661\pi$
$522$	0	0
$523$	1.41570e9	0.0189219	0.00946097	−	0.999955i	$-0.496988\pi$
$523$	1.41570e9	0.0189219	0.00946097	+	0.999955i	$0.496988\pi$
$524$	8.70166e10	1.15419
$525$	0	0
$526$	0	0
$527$	0	0
$528$	−6.05319e10	−0.778841
$529$	7.83110e10	1.00000
$530$	0	0
$531$	1.30683e11	1.64376
$532$	0	0
$533$	0	0
$534$	−5.19628e10	−0.639039
$535$	0	0
$536$	−5.71158e10	−0.691986
$537$	6.48186e10	0.779476
$538$	0	0
$539$	−1.56607e11	−1.85547
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	1.70324e11	1.94483
$545$	0	0
$546$	0	0
$547$	1.99228e10	0.222537	0.111268	−	0.993790i	$-0.464509\pi$
$547$	1.99228e10	0.222537	0.111268	+	0.993790i	$0.464509\pi$
$548$	−8.69149e10	−0.963767
$549$	0	0
$550$	−1.69788e11	−1.85547
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−1.87445e11	−1.96144
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−1.50031e11	−1.51471
$562$	−3.53991e10	−0.354852
$563$	1.38789e11	1.38141	0.690705	−	0.723136i	$-0.257300\pi$
$563$	1.38789e11	1.38141	0.690705	+	0.723136i	$0.257300\pi$
$564$	0	0
$565$	0	0
$566$	1.71514e11	1.67122
$567$	0	0
$568$	0	0
$569$	−1.44287e11	−1.37651	−0.688255	−	0.725468i	$-0.741623\pi$
$569$	−1.44287e11	−1.37651	−0.688255	+	0.725468i	$0.741623\pi$
$570$	0	0
$571$	−1.64593e11	−1.54834	−0.774170	−	0.632978i	$-0.781832\pi$
$571$	−1.64593e11	−1.54834	−0.774170	+	0.632978i	$0.781832\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−9.06809e10	−0.823807
$577$	2.21678e11	1.99995	0.999976	−	0.00693236i	$-0.00220666\pi$
$577$	2.21678e11	1.99995	0.999976	+	0.00693236i	$0.00220666\pi$
$578$	3.10545e11	2.78236
$579$	4.88327e10	0.434507
$580$	0	0
$581$	0	0
$582$	−4.21155e10	−0.367071
$583$	0	0
$584$	1.37493e11	1.18203
$585$	0	0
$586$	0	0
$587$	1.29666e11	1.09213	0.546066	−	0.837742i	$-0.316125\pi$
$587$	1.29666e11	1.09213	0.546066	+	0.837742i	$0.316125\pi$
$588$	5.01768e10	0.419753
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.45734e11	−1.98722	−0.993612	−	0.112847i	$-0.964003\pi$
$593$	−2.45734e11	−1.98722	−0.993612	+	0.112847i	$0.964003\pi$
$594$	1.76837e11	1.42046
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	5.44000e10	0.419753
$601$	−2.48165e11	−1.90214	−0.951069	−	0.308978i	$-0.900013\pi$
$601$	−2.48165e11	−1.90214	−0.951069	+	0.308978i	$0.900013\pi$
$602$	0	0
$603$	7.53689e10	0.570063
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−7.57974e10	−0.554677
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−2.24757e11	−1.60216
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−1.06957e11	−0.752551
$615$	0	0
$616$	0	0
$617$	1.32420e11	0.913722	0.456861	−	0.889538i	$-0.348974\pi$
$617$	1.32420e11	0.913722	0.456861	+	0.889538i	$0.348974\pi$
$618$	0	0
$619$	−9.06943e10	−0.617756	−0.308878	−	0.951102i	$-0.599954\pi$
$619$	−9.06943e10	−0.617756	−0.308878	+	0.951102i	$0.599954\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.52588e11	1.00000
$626$	−2.02836e11	−1.32083
$627$	6.67665e10	0.432005
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−8.57759e10	−0.534258
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	9.79076e9	0.0579942	0.0289971	−	0.999579i	$-0.490769\pi$
$641$	9.79076e9	0.0579942	0.0289971	+	0.999579i	$0.490769\pi$
$642$	−1.00621e11	−0.592310
$643$	7.65969e10	0.448092	0.224046	−	0.974579i	$-0.428073\pi$
$643$	7.65969e10	0.448092	0.224046	+	0.974579i	$0.428073\pi$
$644$	0	0
$645$	0	0
$646$	−1.87867e11	−1.07875
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	8.85945e10	0.502466
$649$	6.56822e11	3.70228
$650$	0	0
$651$	0	0
$652$	2.92627e11	1.61929
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−2.68632e11	−1.45058
$657$	−1.81433e11	−0.973764
$658$	0	0
$659$	−3.62926e11	−1.92432	−0.962158	−	0.272493i	$-0.912152\pi$
$659$	−3.62926e11	−1.92432	−0.962158	+	0.272493i	$0.912152\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	3.82742e11	1.99285
$663$	0	0
$664$	1.23740e11	0.636558
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	7.87721e9	0.0383983	0.0191992	−	0.999816i	$-0.493888\pi$
$673$	7.87721e9	0.0383983	0.0191992	+	0.999816i	$0.493888\pi$
$674$	−4.12704e11	−1.99986
$675$	−1.58923e11	−0.765549
$676$	2.08827e11	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	−3.92781e10	−0.185880
$679$	0	0
$680$	0	0
$681$	1.31862e11	0.613098
$682$	0	0
$683$	1.64741e11	0.757040	0.378520	−	0.925593i	$-0.376433\pi$
$683$	1.64741e11	0.757040	0.378520	+	0.925593i	$0.376433\pi$
$684$	1.00021e11	0.456947
$685$	0	0
$686$	0	0
$687$	0	0
$688$	3.55625e11	1.58722
$689$	0	0
$690$	0	0
$691$	−2.82749e11	−1.24019	−0.620095	−	0.784526i	$-0.712906\pi$
$691$	−2.82749e11	−1.24019	−0.620095	+	0.784526i	$0.712906\pi$
$692$	0	0
$693$	0	0
$694$	1.26775e11	0.546507
$695$	0	0
$696$	0	0
$697$	−6.65818e11	−2.82114
$698$	0	0
$699$	1.59779e10	0.0669285
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−4.55770e11	−1.85547
$705$	0	0
$706$	2.19394e11	0.883093
$707$	0	0
$708$	−2.10446e11	−0.837544
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−3.91249e11	−1.52242
$713$	0	0
$714$	0	0
$715$	0	0
$716$	4.88046e11	1.85699
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	−1.88133e11	−0.692334
$723$	−1.97361e10	−0.0722283
$724$	0	0
$725$	0	0
$726$	2.84856e11	1.02537
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−2.61512e10	−0.0925936
$730$	0	0
$731$	8.81432e11	3.08688
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.78810e11	1.28396
$738$	3.54482e11	1.19500
$739$	3.80751e11	1.27662	0.638312	−	0.769778i	$-0.279633\pi$
$739$	3.80751e11	1.27662	0.638312	+	0.769778i	$0.279633\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.63285e11	−0.524401
$748$	−1.12965e12	−3.60858
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	1.26072e10	0.0392138
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−1.22583e11	−0.371325
$759$	0	0
$760$	0	0
$761$	−5.45467e10	−0.162641	−0.0813204	−	0.996688i	$-0.525914\pi$
$761$	−5.45467e10	−0.162641	−0.0813204	+	0.996688i	$0.525914\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	1.46029e11	0.419753
$769$	−6.94260e11	−1.98526	−0.992628	−	0.121201i	$-0.961325\pi$
$769$	−6.94260e11	−1.98526	−0.992628	+	0.121201i	$0.961325\pi$
$770$	0	0
$771$	−2.86940e11	−0.812032
$772$	3.67681e11	1.03515
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−4.69275e11	−1.30757
$775$	0	0
$776$	−3.17105e11	−0.874493
$777$	0	0
$778$	0	0
$779$	2.96301e11	0.804605
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	3.77802e11	1.00000
$785$	0	0
$786$	1.84910e11	0.484474
$787$	−6.21875e11	−1.62108	−0.810540	−	0.585684i	$-0.800826\pi$
$787$	−6.21875e11	−1.62108	−0.810540	+	0.585684i	$0.800826\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	6.01425e11	1.52855
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	4.09600e11	1.00000
$801$	5.16285e11	1.25418
$802$	2.97484e11	0.719061
$803$	−9.11896e11	−2.19323
$804$	−1.21371e11	−0.290463
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5.68377e11	1.32691	0.663456	−	0.748215i	$-0.269089\pi$
$809$	5.68377e11	1.32691	0.663456	+	0.748215i	$0.269089\pi$
$810$	0	0
$811$	7.82841e11	1.80963	0.904815	−	0.425804i	$-0.140009\pi$
$811$	7.82841e11	1.80963	0.904815	+	0.425804i	$0.140009\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	3.61939e11	0.816348
$817$	−3.92253e11	−0.880396
$818$	−9.97470e10	−0.222785
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−1.84694e11	−0.404544
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	−3.60798e11	−0.778841
$826$	0	0
$827$	−8.84989e11	−1.89198	−0.945988	−	0.324200i	$-0.894905\pi$
$827$	−8.84989e11	−1.89198	−0.945988	+	0.324200i	$0.894905\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	9.36400e11	1.94483
$834$	−3.98321e11	−0.823321
$835$	0	0
$836$	5.02713e11	1.02919
$837$	0	0
$838$	−8.65395e11	−1.75484
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	5.00246e11	1.00000
$842$	0	0
$843$	−7.52232e10	−0.148950
$844$	−6.45842e11	−1.27279
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	3.64468e11	0.701501
$850$	1.01521e12	1.94483
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−7.57618e11	−1.41109
$857$	−1.07825e12	−1.99892	−0.999462	−	0.0328011i	$-0.989557\pi$
$857$	−1.07825e12	−1.99892	−0.999462	+	0.0328011i	$0.989557\pi$
$858$	0	0
$859$	4.32719e11	0.794755	0.397377	−	0.917655i	$-0.369920\pi$
$859$	4.32719e11	0.794755	0.397377	+	0.917655i	$0.369920\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−4.26607e11	−0.765549
$865$	0	0
$866$	−1.09811e12	−1.95243
$867$	6.59908e11	1.16790
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	4.18446e11	0.720414
$874$	0	0
$875$	0	0
$876$	2.92172e11	0.496160
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9.39089e11	−1.55885	−0.779423	−	0.626498i	$-0.784488\pi$
$881$	−9.39089e11	−1.55885	−0.779423	+	0.626498i	$0.784488\pi$
$882$	−4.98540e11	−0.823807
$883$	1.17202e12	1.92793	0.963965	−	0.266030i	$-0.0857120\pi$
$883$	1.17202e12	1.92793	0.963965	+	0.266030i	$0.0857120\pi$
$884$	0	0
$885$	0	0
$886$	1.05838e12	1.71754
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−5.87587e11	−0.932313
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	6.23239e11	0.958405
$899$	0	0
$900$	−5.40500e11	−0.823807
$901$	0	0
$902$	1.78166e12	2.69152
$903$	0	0
$904$	−2.95741e11	−0.442831
$905$	0	0
$906$	0	0
$907$	3.17048e11	0.468486	0.234243	−	0.972178i	$-0.424739\pi$
$907$	3.17048e11	0.468486	0.234243	+	0.972178i	$0.424739\pi$
$908$	9.92840e11	1.46062
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−1.61069e11	−0.232827
$913$	−8.20684e11	−1.18112
$914$	6.89987e11	0.988681
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−1.05736e12	−1.48886
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−2.27284e11	−0.315886
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.12157e12	−1.50579	−0.752895	−	0.658140i	$-0.771343\pi$
$929$	−1.12157e12	−1.50579	−0.752895	+	0.658140i	$0.771343\pi$
$930$	0	0
$931$	−4.16714e11	−0.554677
$932$	1.20304e11	0.159447
$933$	0	0
$934$	1.50590e12	1.97883
$935$	0	0
$936$	0	0
$937$	−3.03224e9	−0.00393373	−0.00196687	−	0.999998i	$-0.500626\pi$
$937$	−3.03224e9	−0.00393373	−0.00196687	+	0.999998i	$0.500626\pi$
$938$	0	0
$939$	−4.31026e11	−0.554422
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	−1.58453e12	−1.99533
$945$	0	0
$946$	−2.35862e12	−2.94505
$947$	−9.59570e11	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$947$	−9.59570e11	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$948$	0	0
$949$	0	0
$950$	−4.51787e11	−0.554677
$951$	0	0
$952$	0	0
$953$	1.57684e12	1.91169	0.955843	−	0.293879i	$-0.0949463\pi$
$953$	1.57684e12	1.91169	0.955843	+	0.293879i	$0.0949463\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	8.52891e11	1.00000
$962$	0	0
$963$	9.99738e11	1.16247
$964$	−1.48601e11	−0.172073
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	2.14480e12	2.44279
$969$	−3.99218e11	−0.452809
$970$	0	0
$971$	−8.99918e11	−1.01234	−0.506170	−	0.862434i	$-0.668939\pi$
$971$	−8.99918e11	−1.01234	−0.506170	+	0.862434i	$0.668939\pi$
$972$	8.71605e11	0.976460
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.30947e12	1.43720	0.718598	−	0.695425i	$-0.244784\pi$
$977$	1.30947e12	1.43720	0.718598	+	0.695425i	$0.244784\pi$
$978$	6.21833e11	0.679701
$979$	2.59489e12	2.82480
$980$	0	0
$981$	0	0
$982$	−1.59209e12	−1.71207
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−5.70844e11	−0.608887
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	8.13327e11	0.836504
$994$	0	0
$995$	0	0
$996$	2.62947e11	0.267197
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−1.64671e12	−1.65995
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.9.d.a.3.1	✓	1
3.2	odd	2		72.9.b.a.19.1		1
4.3	odd	2		32.9.d.a.15.1		1
8.3	odd	2	CM	8.9.d.a.3.1	✓	1
8.5	even	2		32.9.d.a.15.1		1
12.11	even	2		288.9.b.a.271.1		1
16.3	odd	4		256.9.c.f.255.1		2
16.5	even	4		256.9.c.f.255.1		2
16.11	odd	4		256.9.c.f.255.2		2
16.13	even	4		256.9.c.f.255.2		2
24.5	odd	2		288.9.b.a.271.1		1
24.11	even	2		72.9.b.a.19.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.9.d.a.3.1	✓	1	1.1	even	1	trivial
8.9.d.a.3.1	✓	1	8.3	odd	2	CM
32.9.d.a.15.1		1	4.3	odd	2
32.9.d.a.15.1		1	8.5	even	2
72.9.b.a.19.1		1	3.2	odd	2
72.9.b.a.19.1		1	24.11	even	2
256.9.c.f.255.1		2	16.3	odd	4
256.9.c.f.255.1		2	16.5	even	4
256.9.c.f.255.2		2	16.11	odd	4
256.9.c.f.255.2		2	16.13	even	4
288.9.b.a.271.1		1	12.11	even	2
288.9.b.a.271.1		1	24.5	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:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	8.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(5\)	\(7\)
\(\chi(n)\)	\(-1\)	\(-1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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