LMFDB - Embedded newform 8.19.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,19,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, 19, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 19);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 19 $
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:	$16.4308910168$
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)

$f(q)$

$=$

$q-512.000 q^{2} -3266.00 q^{3} +262144. q^{4} +1.67219e6 q^{6} -1.34218e8 q^{8} -3.76754e8 q^{9} +O(q^{10})$

$q-512.000 q^{2} -3266.00 q^{3} +262144. q^{4} +1.67219e6 q^{6} -1.34218e8 q^{8} -3.76754e8 q^{9} -3.54350e8 q^{11} -8.56162e8 q^{12} +6.87195e10 q^{16} +1.19842e11 q^{17} +1.92898e11 q^{18} +3.35014e11 q^{19} +1.81427e11 q^{22} +4.38355e11 q^{24} +3.81470e12 q^{25} +2.49579e12 q^{27} -3.51844e13 q^{32} +1.15731e12 q^{33} -6.13593e13 q^{34} -9.87637e13 q^{36} -1.71527e14 q^{38} +5.22163e14 q^{41} +1.00025e15 q^{43} -9.28906e13 q^{44} -2.24438e14 q^{48} +1.62841e15 q^{49} -1.95312e15 q^{50} -3.91405e14 q^{51} -1.27785e15 q^{54} -1.09415e15 q^{57} -1.02290e16 q^{59} +1.80144e16 q^{64} -5.92541e14 q^{66} -5.04671e16 q^{67} +3.14160e16 q^{68} +5.05670e16 q^{72} +6.06152e16 q^{73} -1.24588e16 q^{75} +8.78218e16 q^{76} +1.37811e17 q^{81} -2.67347e17 q^{82} -3.52960e17 q^{83} -5.12128e17 q^{86} +4.75600e16 q^{88} +4.87058e17 q^{89} +1.14912e17 q^{96} +1.50107e18 q^{97} -8.33748e17 q^{98} +1.33503e17 q^{99} +O(q^{100})$

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 $

$2$	−512.000	−1.00000
$2$	−512.000	−1.00000
$3$	−3266.00	−0.165930	−0.0829650	−	0.996552i	$-0.526439\pi$
$3$	−3266.00	−0.165930	−0.0829650	+	0.996552i	$0.526439\pi$
$4$	262144.	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	1.67219e6	0.165930
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−1.34218e8	−1.00000
$9$	−3.76754e8	−0.972467
$10$	0	0
$11$	−3.54350e8	−0.150279	−0.0751394	−	0.997173i	$-0.523940\pi$
$11$	−3.54350e8	−0.150279	−0.0751394	+	0.997173i	$0.523940\pi$
$12$	−8.56162e8	−0.165930
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	6.87195e10	1.00000
$17$	1.19842e11	1.01058	0.505290	−	0.862950i	$-0.331386\pi$
$17$	1.19842e11	1.01058	0.505290	+	0.862950i	$0.331386\pi$
$18$	1.92898e11	0.972467
$19$	3.35014e11	1.03820	0.519099	−	0.854714i	$-0.326268\pi$
$19$	3.35014e11	1.03820	0.519099	+	0.854714i	$0.326268\pi$
$20$	0	0
$21$	0	0
$22$	1.81427e11	0.150279
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	4.38355e11	0.165930
$25$	3.81470e12	1.00000
$26$	0	0
$27$	2.49579e12	0.327291
$28$	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			$\pi$
$32$	−3.51844e13	−1.00000
$33$	1.15731e12	0.0249358
$34$	−6.13593e13	−1.01058
$35$	0	0
$36$	−9.87637e13	−0.972467
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−1.71527e14	−1.03820
$39$	0	0
$40$	0	0
$41$	5.22163e14	1.59496	0.797482	−	0.603342i	$-0.206165\pi$
$41$	5.22163e14	1.59496	0.797482	+	0.603342i	$0.206165\pi$
$42$	0	0
$43$	1.00025e15	1.99018	0.995091	−	0.0989682i	$-0.0315542\pi$
$43$	1.00025e15	1.99018	0.995091	+	0.0989682i	$0.0315542\pi$
$44$	−9.28906e13	−0.150279
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−2.24438e14	−0.165930
$49$	1.62841e15	1.00000
$50$	−1.95312e15	−1.00000
$51$	−3.91405e14	−0.167685
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−1.27785e15	−0.327291
$55$	0	0
$56$	0	0
$57$	−1.09415e15	−0.172268
$58$	0	0
$59$	−1.02290e16	−1.18077	−0.590383	−	0.807123i	$-0.701023\pi$
$59$	−1.02290e16	−1.18077	−0.590383	+	0.807123i	$0.701023\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	1.80144e16	1.00000
$65$	0	0
$66$	−5.92541e14	−0.0249358
$67$	−5.04671e16	−1.85496	−0.927481	−	0.373871i	$-0.878030\pi$
$67$	−5.04671e16	−1.85496	−0.927481	+	0.373871i	$0.878030\pi$
$68$	3.14160e16	1.01058
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	5.05670e16	0.972467
$73$	6.06152e16	1.02962	0.514808	−	0.857305i	$-0.327863\pi$
$73$	6.06152e16	1.02962	0.514808	+	0.857305i	$0.327863\pi$
$74$	0	0
$75$	−1.24588e16	−0.165930
$76$	8.78218e16	1.03820
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	1.37811e17	0.918160
$82$	−2.67347e17	−1.59496
$83$	−3.52960e17	−1.88809	−0.944046	−	0.329813i	$-0.893014\pi$
$83$	−3.52960e17	−1.88809	−0.944046	+	0.329813i	$0.893014\pi$
$84$	0	0
$85$	0	0
$86$	−5.12128e17	−1.99018
$87$	0	0
$88$	4.75600e16	0.150279
$89$	4.87058e17	1.39018	0.695089	−	0.718923i	$-0.255365\pi$
$89$	4.87058e17	1.39018	0.695089	+	0.718923i	$0.255365\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	1.14912e17	0.165930
$97$	1.50107e18	1.97449	0.987244	−	0.159214i	$-0.0508959\pi$
$97$	1.50107e18	1.97449	0.987244	+	0.159214i	$0.0508959\pi$
$98$	−8.33748e17	−1.00000
$99$	1.33503e17	0.146141
$100$	1.00000e18	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	2.00400e17	0.167685
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.02760e18	−1.10288	−0.551439	−	0.834215i	$-0.685921\pi$
$107$	−2.02760e18	−1.10288	−0.551439	+	0.834215i	$0.685921\pi$
$108$	6.54257e17	0.327291
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.68774e18	−1.56048	−0.780238	−	0.625482i	$-0.784902\pi$
$113$	−4.68774e18	−1.56048	−0.780238	+	0.625482i	$0.784902\pi$
$114$	5.60207e17	0.172268
$115$	0	0
$116$	0	0
$117$	0	0
$118$	5.23723e18	1.18077
$119$	0	0
$120$	0	0
$121$	−5.43435e18	−0.977416
$122$	0	0
$123$	−1.70538e18	−0.264652
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−9.22337e18	−1.00000
$129$	−3.26682e18	−0.330231
$130$	0	0
$131$	1.90150e19	1.67361	0.836807	−	0.547497i	$-0.184419\pi$
$131$	1.90150e19	1.67361	0.836807	+	0.547497i	$0.184419\pi$
$132$	3.03381e17	0.0249358
$133$	0	0
$134$	2.58391e19	1.85496
$135$	0	0
$136$	−1.60850e19	−1.01058
$137$	1.59042e18	0.0935461	0.0467730	−	0.998906i	$-0.485106\pi$
$137$	1.59042e18	0.0935461	0.0467730	+	0.998906i	$0.485106\pi$
$138$	0	0
$139$	3.64855e19	1.88359	0.941797	−	0.336182i	$-0.109136\pi$
$139$	3.64855e19	1.88359	0.941797	+	0.336182i	$0.109136\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−2.58903e19	−0.972467
$145$	0	0
$146$	−3.10350e19	−1.02962
$147$	−5.31840e18	−0.165930
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	6.37891e18	0.165930
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−4.49648e19	−1.03820
$153$	−4.51511e19	−0.982755
$154$	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
$162$	−7.05592e19	−0.918160
$163$	−2.57981e19	−0.317613	−0.158807	−	0.987310i	$-0.550765\pi$
$163$	−2.57981e19	−0.317613	−0.158807	+	0.987310i	$0.550765\pi$
$164$	1.36882e20	1.59496
$165$	0	0
$166$	1.80716e20	1.88809
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.12455e20	1.00000
$170$	0	0
$171$	−1.26218e20	−1.00961
$172$	2.62210e20	1.99018
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	−2.43507e19	−0.150279
$177$	3.34078e19	0.195924
$178$	−2.49374e20	−1.39018
$179$	−2.84970e20	−1.51050	−0.755252	−	0.655434i	$-0.772486\pi$
$179$	−2.84970e20	−1.51050	−0.755252	+	0.655434i	$0.772486\pi$
$180$	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
$187$	−4.24661e19	−0.151869
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−5.88350e19	−0.165930
$193$	5.49022e20	1.47766	0.738829	−	0.673893i	$-0.235379\pi$
$193$	5.49022e20	1.47766	0.738829	+	0.673893i	$0.235379\pi$
$194$	−7.68546e20	−1.97449
$195$	0	0
$196$	4.26879e20	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−6.83533e19	−0.146141
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−5.12000e20	−1.00000
$201$	1.64825e20	0.307794
$202$	0	0
$203$	0	0
$204$	−1.02605e20	−0.167685
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.18712e20	−0.156019
$210$	0	0
$211$	1.54335e21	1.86176	0.930878	−	0.365331i	$-0.119044\pi$
$211$	1.54335e21	1.86176	0.930878	+	0.365331i	$0.119044\pi$
$212$	0	0
$213$	0	0
$214$	1.03813e21	1.10288
$215$	0	0
$216$	−3.34980e20	−0.327291
$217$	0	0
$218$	0	0
$219$	−1.97969e20	−0.170844
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−1.43720e21	−0.972467
$226$	2.40012e21	1.56048
$227$	−3.87177e20	−0.241923	−0.120961	−	0.992657i	$-0.538598\pi$
$227$	−3.87177e20	−0.241923	−0.120961	+	0.992657i	$0.538598\pi$
$228$	−2.86826e20	−0.172268
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.95634e21	−1.95475	−0.977374	−	0.211520i	$-0.932159\pi$
$233$	−3.95634e21	−1.95475	−0.977374	+	0.211520i	$0.932159\pi$
$234$	0	0
$235$	0	0
$236$	−2.68146e21	−1.18077
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−3.03581e21	−1.10693	−0.553467	−	0.832871i	$-0.686695\pi$
$241$	−3.03581e21	−1.10693	−0.553467	+	0.832871i	$0.686695\pi$
$242$	2.78239e21	0.977416
$243$	−1.41701e21	−0.479642
$244$	0	0
$245$	0	0
$246$	8.73156e20	0.264652
$247$	0	0
$248$	0	0
$249$	1.15277e21	0.313291
$250$	0	0
$251$	7.58375e21	1.91788	0.958938	−	0.283615i	$-0.0915339\pi$
$251$	7.58375e21	1.91788	0.958938	+	0.283615i	$0.0915339\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	4.72237e21	1.00000
$257$	9.57081e21	1.95682	0.978410	−	0.206675i	$-0.0662644\pi$
$257$	9.57081e21	1.95682	0.978410	+	0.206675i	$0.0662644\pi$
$258$	1.67261e21	0.330231
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−9.73570e21	−1.67361
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−1.55331e20	−0.0249358
$265$	0	0
$266$	0	0
$267$	−1.59073e21	−0.230672
$268$	−1.32296e22	−1.85496
$269$	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			$\pi$
$272$	8.23551e21	1.01058
$273$	0	0
$274$	−8.14293e20	−0.0935461
$275$	−1.35174e21	−0.150279
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−1.86806e22	−1.88359
$279$	0	0
$280$	0	0
$281$	1.55808e22	1.42637	0.713186	−	0.700975i	$-0.247252\pi$
$281$	1.55808e22	1.42637	0.713186	+	0.700975i	$0.247252\pi$
$282$	0	0
$283$	−2.24902e22	−1.93159	−0.965795	−	0.259307i	$-0.916506\pi$
$283$	−2.24902e22	−1.93159	−0.965795	+	0.259307i	$0.916506\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.32558e22	0.972467
$289$	2.99128e20	0.0212704
$290$	0	0
$291$	−4.90249e21	−0.327627
$292$	1.58899e22	1.02962
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	2.72302e21	0.165930
$295$	0	0
$296$	0	0
$297$	−8.84383e20	−0.0491850
$298$	0	0
$299$	0	0
$300$	−3.26600e21	−0.165930
$301$	0	0
$302$	0	0
$303$	0	0
$304$	2.30220e22	1.03820
$305$	0	0
$306$	2.31174e22	0.982755
$307$	−1.47844e22	−0.610322	−0.305161	−	0.952301i	$-0.598710\pi$
$307$	−1.47844e22	−0.610322	−0.305161	+	0.952301i	$0.598710\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−2.48227e22	−0.860894	−0.430447	−	0.902616i	$-0.641644\pi$
$313$	−2.48227e22	−0.860894	−0.430447	+	0.902616i	$0.641644\pi$
$314$	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
$321$	6.62213e21	0.183000
$322$	0	0
$323$	4.01489e22	1.04918
$324$	3.61263e22	0.918160
$325$	0	0
$326$	1.32086e22	0.317613
$327$	0	0
$328$	−7.00835e22	−1.59496
$329$	0	0
$330$	0	0
$331$	1.80468e22	0.378398	0.189199	−	0.981939i	$-0.439411\pi$
$331$	1.80468e22	0.378398	0.189199	+	0.981939i	$0.439411\pi$
$332$	−9.25265e22	−1.88809
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−8.13695e22	−1.45142	−0.725709	−	0.688001i	$-0.758488\pi$
$337$	−8.13695e22	−1.45142	−0.725709	+	0.688001i	$0.758488\pi$
$338$	−5.75772e22	−1.00000
$339$	1.53101e22	0.258930
$340$	0	0
$341$	0	0
$342$	6.46234e22	1.00961
$343$	0	0
$344$	−1.34251e23	−1.99018
$345$	0	0
$346$	0	0
$347$	1.42037e23	1.94731	0.973657	−	0.228018i	$-0.0732243\pi$
$347$	1.42037e23	1.94731	0.973657	+	0.228018i	$0.0732243\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	1.24676e22	0.150279
$353$	1.01143e23	1.18841	0.594203	−	0.804315i	$-0.297468\pi$
$353$	1.01143e23	1.18841	0.594203	+	0.804315i	$0.297468\pi$
$354$	−1.71048e22	−0.195924
$355$	0	0
$356$	1.27679e23	1.39018
$357$	0	0
$358$	1.45905e23	1.51050
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	8.10683e21	0.0778550
$362$	0	0
$363$	1.77486e22	0.162183
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−1.96727e23	−1.55105
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	2.17427e22	0.151869
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−2.21819e23	−1.37482	−0.687411	−	0.726268i	$-0.741253\pi$
$379$	−2.21819e23	−1.37482	−0.687411	+	0.726268i	$0.741253\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	3.01235e22	0.165930
$385$	0	0
$386$	−2.81099e23	−1.47766
$387$	−3.76848e23	−1.93539
$388$	3.93496e23	1.97449
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−2.18562e23	−1.00000
$393$	−6.21031e22	−0.277703
$394$	0	0
$395$	0	0
$396$	3.49969e22	0.146141
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	2.62144e23	1.00000
$401$	4.86318e23	1.81393	0.906967	−	0.421203i	$-0.138392\pi$
$401$	4.86318e23	1.81393	0.906967	+	0.421203i	$0.138392\pi$
$402$	−8.43906e22	−0.307794
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	5.25335e22	0.167685
$409$	3.84499e23	1.20056	0.600282	−	0.799788i	$-0.295055\pi$
$409$	3.84499e23	1.20056	0.600282	+	0.799788i	$0.295055\pi$
$410$	0	0
$411$	−5.19430e21	−0.0155221
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.19162e23	−0.312545
$418$	6.07805e22	0.156019
$419$	2.65749e23	0.667646	0.333823	−	0.942636i	$-0.391661\pi$
$419$	2.65749e23	0.667646	0.333823	+	0.942636i	$0.391661\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−7.90196e23	−1.86176
$423$	0	0
$424$	0	0
$425$	4.57163e23	1.01058
$426$	0	0
$427$	0	0
$428$	−5.31522e23	−1.10288
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.71510e23	0.327291
$433$	3.09750e23	0.578921	0.289461	−	0.957190i	$-0.406524\pi$
$433$	3.09750e23	0.578921	0.289461	+	0.957190i	$0.406524\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	1.01360e23	0.170844
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−6.13511e23	−0.972467
$442$	0	0
$443$	4.63503e23	0.705373	0.352687	−	0.935741i	$-0.385268\pi$
$443$	4.63503e23	0.705373	0.352687	+	0.935741i	$0.385268\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.10365e24	−1.48804	−0.744020	−	0.668157i	$-0.767083\pi$
$449$	−1.10365e24	−1.48804	−0.744020	+	0.668157i	$0.767083\pi$
$450$	7.35847e23	0.972467
$451$	−1.85028e23	−0.239689
$452$	−1.22886e24	−1.56048
$453$	0	0
$454$	1.98235e23	0.241923
$455$	0	0
$456$	1.46855e23	0.172268
$457$	−1.21141e24	−1.39331	−0.696653	−	0.717408i	$-0.745328\pi$
$457$	−1.21141e24	−1.39331	−0.696653	+	0.717408i	$0.745328\pi$
$458$	0	0
$459$	2.99102e23	0.330754
$460$	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
$466$	2.02565e24	1.95475
$467$	−6.80067e23	−0.643724	−0.321862	−	0.946787i	$-0.604309\pi$
$467$	−6.80067e23	−0.643724	−0.321862	+	0.946787i	$0.604309\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	1.37291e24	1.18077
$473$	−3.54438e23	−0.299082
$474$	0	0
$475$	1.27798e24	1.03820
$476$	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
$482$	1.55434e24	1.10693
$483$	0	0
$484$	−1.42458e24	−0.977416
$485$	0	0
$486$	7.25510e23	0.479642
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	8.42565e22	0.0527016
$490$	0	0
$491$	3.00554e24	1.81213	0.906065	−	0.423139i	$-0.139072\pi$
$491$	3.00554e24	1.81213	0.906065	+	0.423139i	$0.139072\pi$
$492$	−4.47056e23	−0.264652
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−5.90218e23	−0.313291
$499$	−3.27870e24	−1.70921	−0.854607	−	0.519275i	$-0.826202\pi$
$499$	−3.27870e24	−1.70921	−0.854607	+	0.519275i	$0.826202\pi$
$500$	0	0
$501$	0	0
$502$	−3.88288e24	−1.91788
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.67279e23	−0.165930
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−2.41785e24	−1.00000
$513$	8.36125e23	0.339793
$514$	−4.90026e24	−1.95682
$515$	0	0
$516$	−8.56377e23	−0.330231
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.06522e24	1.43729	0.718645	−	0.695377i	$-0.244763\pi$
$521$	4.06522e24	1.43729	0.718645	+	0.695377i	$0.244763\pi$
$522$	0	0
$523$	−2.12505e24	−0.725862	−0.362931	−	0.931816i	$-0.618224\pi$
$523$	−2.12505e24	−0.725862	−0.362931	+	0.931816i	$0.618224\pi$
$524$	4.98468e24	1.67361
$525$	0	0
$526$	0	0
$527$	0	0
$528$	7.95295e22	0.0249358
$529$	3.24415e24	1.00000
$530$	0	0
$531$	3.85380e24	1.14826
$532$	0	0
$533$	0	0
$534$	8.14455e23	0.230672
$535$	0	0
$536$	6.77357e24	1.85496
$537$	9.30712e23	0.250638
$538$	0	0
$539$	−5.77028e23	−0.150279
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−4.21658e24	−1.01058
$545$	0	0
$546$	0	0
$547$	−8.68027e24	−1.97992	−0.989960	−	0.141351i	$-0.954855\pi$
$547$	−8.68027e24	−1.97992	−0.989960	+	0.141351i	$0.954855\pi$
$548$	4.16918e23	0.0935461
$549$	0	0
$550$	6.92089e23	0.150279
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	9.56446e24	1.88359
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.38694e23	0.0251996
$562$	−7.97736e24	−1.42637
$563$	−1.10183e25	−1.93883	−0.969417	−	0.245419i	$-0.921075\pi$
$563$	−1.10183e25	−1.93883	−0.969417	+	0.245419i	$0.921075\pi$
$564$	0	0
$565$	0	0
$566$	1.15150e25	1.93159
$567$	0	0
$568$	0	0
$569$	−1.07894e25	−1.72578	−0.862890	−	0.505392i	$-0.831348\pi$
$569$	−1.07894e25	−1.72578	−0.862890	+	0.505392i	$0.831348\pi$
$570$	0	0
$571$	8.85990e24	1.37311	0.686553	−	0.727080i	$-0.259123\pi$
$571$	8.85990e24	1.37311	0.686553	+	0.727080i	$0.259123\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−6.78699e24	−0.972467
$577$	2.21137e23	0.0311946	0.0155973	−	0.999878i	$-0.495035\pi$
$577$	2.21137e23	0.0311946	0.0155973	+	0.999878i	$0.495035\pi$
$578$	−1.53153e23	−0.0212704
$579$	−1.79310e24	−0.245188
$580$	0	0
$581$	0	0
$582$	2.51007e24	0.327627
$583$	0	0
$584$	−8.13563e24	−1.02962
$585$	0	0
$586$	0	0
$587$	1.01955e25	1.23216	0.616081	−	0.787683i	$-0.288719\pi$
$587$	1.01955e25	1.23216	0.616081	+	0.787683i	$0.288719\pi$
$588$	−1.39419e24	−0.165930
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.56385e25	−1.72465	−0.862326	−	0.506353i	$-0.830993\pi$
$593$	−1.56385e25	−1.72465	−0.862326	+	0.506353i	$0.830993\pi$
$594$	4.52804e23	0.0491850
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	1.67219e24	0.165930
$601$	2.03965e25	1.99382	0.996910	−	0.0785507i	$-0.0250293\pi$
$601$	2.03965e25	1.99382	0.996910	+	0.0785507i	$0.0250293\pi$
$602$	0	0
$603$	1.90137e25	1.80389
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−1.17872e25	−1.03820
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−1.18361e25	−0.982755
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	7.56963e24	0.610322
$615$	0	0
$616$	0	0
$617$	−1.61927e25	−1.24955	−0.624775	−	0.780805i	$-0.714809\pi$
$617$	−1.61927e25	−1.24955	−0.624775	+	0.780805i	$0.714809\pi$
$618$	0	0
$619$	2.37714e25	1.78171	0.890857	−	0.454283i	$-0.150105\pi$
$619$	2.37714e25	1.78171	0.890857	+	0.454283i	$0.150105\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.45519e25	1.00000
$626$	1.27092e25	0.860894
$627$	3.87713e23	0.0258883
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−5.04059e24	−0.308921
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.17518e25	−0.643252	−0.321626	−	0.946867i	$-0.604229\pi$
$641$	−1.17518e25	−0.643252	−0.321626	+	0.946867i	$0.604229\pi$
$642$	−3.39053e24	−0.183000
$643$	−3.73263e25	−1.98662	−0.993312	−	0.115462i	$-0.963165\pi$
$643$	−3.73263e25	−1.98662	−0.993312	+	0.115462i	$0.963165\pi$
$644$	0	0
$645$	0	0
$646$	−2.05562e25	−1.04918
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.84967e25	−0.918160
$649$	3.62463e24	0.177444
$650$	0	0
$651$	0	0
$652$	−6.76281e24	−0.317613
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	3.58827e25	1.59496
$657$	−2.28370e25	−1.00127
$658$	0	0
$659$	−4.62488e25	−1.97302	−0.986511	−	0.163697i	$-0.947658\pi$
$659$	−4.62488e25	−1.97302	−0.986511	+	0.163697i	$0.947658\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−9.23998e24	−0.378398
$663$	0	0
$664$	4.73736e25	1.88809
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	5.32210e25	1.87909	0.939545	−	0.342426i	$-0.111249\pi$
$673$	5.32210e25	1.87909	0.939545	+	0.342426i	$0.111249\pi$
$674$	4.16612e25	1.45142
$675$	9.52069e24	0.327291
$676$	2.94795e25	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	−7.83880e24	−0.258930
$679$	0	0
$680$	0	0
$681$	1.26452e24	0.0401422
$682$	0	0
$683$	2.99137e25	0.924877	0.462438	−	0.886651i	$-0.346975\pi$
$683$	2.99137e25	0.924877	0.462438	+	0.886651i	$0.346975\pi$
$684$	−3.30872e25	−1.00961
$685$	0	0
$686$	0	0
$687$	0	0
$688$	6.87367e25	1.99018
$689$	0	0
$690$	0	0
$691$	−6.80314e25	−1.89412	−0.947059	−	0.321059i	$-0.895961\pi$
$691$	−6.80314e25	−1.89412	−0.947059	+	0.321059i	$0.895961\pi$
$692$	0	0
$693$	0	0
$694$	−7.27229e25	−1.94731
$695$	0	0
$696$	0	0
$697$	6.25772e25	1.61184
$698$	0	0
$699$	1.29214e25	0.324351
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−6.38340e24	−0.150279
$705$	0	0
$706$	−5.17854e25	−1.18841
$707$	0	0
$708$	8.75766e24	0.195924
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−6.53718e25	−1.39018
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−7.47032e25	−1.51050
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	−4.15070e24	−0.0778550
$723$	9.91497e24	0.183674
$724$	0	0
$725$	0	0
$726$	−9.08728e24	−0.162183
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−4.87628e25	−0.838573
$730$	0	0
$731$	1.19872e26	2.01124
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.78830e25	0.278761
$738$	1.00724e26	1.55105
$739$	−1.20793e26	−1.83756	−0.918782	−	0.394766i	$-0.870826\pi$
$739$	−1.20793e26	−1.83756	−0.918782	+	0.394766i	$0.870826\pi$
$740$	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
$747$	1.32979e26	1.83611
$748$	−1.11322e25	−0.151869
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	−2.47685e25	−0.318233
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	1.13572e26	1.37482
$759$	0	0
$760$	0	0
$761$	−1.43585e26	−1.67744	−0.838718	−	0.544567i	$-0.816694\pi$
$761$	−1.43585e26	−1.67744	−0.838718	+	0.544567i	$0.816694\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−1.54232e25	−0.165930
$769$	−9.21536e25	−0.979885	−0.489943	−	0.871755i	$-0.662982\pi$
$769$	−9.21536e25	−0.979885	−0.489943	+	0.871755i	$0.662982\pi$
$770$	0	0
$771$	−3.12583e25	−0.324695
$772$	1.43923e26	1.47766
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	1.92946e26	1.93539
$775$	0	0
$776$	−2.01470e26	−1.97449
$777$	0	0
$778$	0	0
$779$	1.74932e26	1.65589
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.11904e26	1.00000
$785$	0	0
$786$	3.17968e25	0.277703
$787$	−1.35055e26	−1.16610	−0.583051	−	0.812436i	$-0.698141\pi$
$787$	−1.35055e26	−1.16610	−0.583051	+	0.812436i	$0.698141\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−1.79184e25	−0.146141
$793$	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
$800$	−1.34218e26	−1.00000
$801$	−1.83501e26	−1.35190
$802$	−2.48995e26	−1.81393
$803$	−2.14790e25	−0.154730
$804$	4.32080e25	0.307794
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	9.65583e25	0.650508	0.325254	−	0.945627i	$-0.394550\pi$
$809$	9.65583e25	0.650508	0.325254	+	0.945627i	$0.394550\pi$
$810$	0	0
$811$	−2.53712e26	−1.67168	−0.835841	−	0.548972i	$-0.815019\pi$
$811$	−2.53712e26	−1.67168	−0.835841	+	0.548972i	$0.815019\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−2.68972e25	−0.167685
$817$	3.35098e26	2.06620
$818$	−1.96864e26	−1.20056
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	2.65948e24	0.0155221
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	4.41477e24	0.0249358
$826$	0	0
$827$	3.61565e26	1.99819	0.999096	−	0.0425114i	$-0.0135359\pi$
$827$	3.61565e26	1.99819	0.999096	+	0.0425114i	$0.0135359\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.95153e26	1.01058
$834$	6.10108e25	0.312545
$835$	0	0
$836$	−3.11196e25	−0.156019
$837$	0	0
$838$	−1.36064e26	−0.667646
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2.10457e26	1.00000
$842$	0	0
$843$	−5.08869e25	−0.236678
$844$	4.04580e26	1.86176
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	7.34529e25	0.320509
$850$	−2.34067e26	−1.01058
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	2.72139e26	1.10288
$857$	3.25681e26	1.30607	0.653033	−	0.757329i	$-0.273496\pi$
$857$	3.25681e26	1.30607	0.653033	+	0.757329i	$0.273496\pi$
$858$	0	0
$859$	−4.40249e26	−1.72886	−0.864431	−	0.502752i	$-0.832321\pi$
$859$	−4.40249e26	−1.72886	−0.864431	+	0.502752i	$0.832321\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−8.78129e25	−0.327291
$865$	0	0
$866$	−1.58592e26	−0.578921
$867$	−9.76951e23	−0.00352940
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−5.65533e26	−1.92013
$874$	0	0
$875$	0	0
$876$	−5.18964e25	−0.170844
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.73095e26	−1.47967	−0.739836	−	0.672787i	$-0.765097\pi$
$881$	−4.73095e26	−1.47967	−0.739836	+	0.672787i	$0.765097\pi$
$882$	3.14118e26	0.972467
$883$	−5.36481e26	−1.64402	−0.822011	−	0.569471i	$-0.807148\pi$
$883$	−5.36481e26	−1.64402	−0.822011	+	0.569471i	$0.807148\pi$
$884$	0	0
$885$	0	0
$886$	−2.37314e26	−0.705373
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.88332e25	−0.137980
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	5.65069e26	1.48804
$899$	0	0
$900$	−3.76754e26	−0.972467
$901$	0	0
$902$	9.47344e25	0.239689
$903$	0	0
$904$	6.29177e26	1.56048
$905$	0	0
$906$	0	0
$907$	8.22752e26	1.98063	0.990315	−	0.138840i	$-0.0443374\pi$
$907$	8.22752e26	1.98063	0.990315	+	0.138840i	$0.0443374\pi$
$908$	−1.01496e26	−0.241923
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−7.51897e25	−0.172268
$913$	1.25071e26	0.283740
$914$	6.20244e26	1.39331
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−1.53140e26	−0.330754
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	4.82860e25	0.101271
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−7.61266e26	−1.47705	−0.738525	−	0.674226i	$-0.764477\pi$
$929$	−7.61266e26	−1.47705	−0.738525	+	0.674226i	$0.764477\pi$
$930$	0	0
$931$	5.45541e26	1.03820
$932$	−1.03713e27	−1.95475
$933$	0	0
$934$	3.48194e26	0.643724
$935$	0	0
$936$	0	0
$937$	4.30663e26	0.773537	0.386768	−	0.922177i	$-0.373591\pi$
$937$	4.30663e26	0.773537	0.386768	+	0.922177i	$0.373591\pi$
$938$	0	0
$939$	8.10709e25	0.142848
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	−7.02929e26	−1.18077
$945$	0	0
$946$	1.81472e26	0.299082
$947$	1.18392e27	1.93273	0.966366	−	0.257170i	$-0.0827902\pi$
$947$	1.18392e27	1.93273	0.966366	+	0.257170i	$0.0827902\pi$
$948$	0	0
$949$	0	0
$950$	−6.54324e26	−1.03820
$951$	0	0
$952$	0	0
$953$	−8.06436e26	−1.24375	−0.621876	−	0.783116i	$-0.713629\pi$
$953$	−8.06436e26	−1.24375	−0.621876	+	0.783116i	$0.713629\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	6.99054e26	1.00000
$962$	0	0
$963$	7.63904e26	1.07251
$964$	−7.95820e26	−1.10693
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	7.29387e26	0.977416
$969$	−1.31126e26	−0.174091
$970$	0	0
$971$	−1.53443e27	−1.99974	−0.999871	−	0.0160616i	$-0.994887\pi$
$971$	−1.53443e27	−1.99974	−0.999871	+	0.0160616i	$0.994887\pi$
$972$	−3.71461e26	−0.479642
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.57600e26	0.317610	0.158805	−	0.987310i	$-0.449236\pi$
$977$	2.57600e26	0.317610	0.158805	+	0.987310i	$0.449236\pi$
$978$	−4.31393e25	−0.0527016
$979$	−1.72589e26	−0.208914
$980$	0	0
$981$	0	0
$982$	−1.53884e27	−1.81213
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	2.28893e26	0.264652
$985$	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
$993$	−5.89410e25	−0.0627876
$994$	0	0
$995$	0	0
$996$	3.02191e26	0.313291
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	1.67869e27	1.70921
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.19.d.a.3.1	✓	1
3.2	odd	2		72.19.b.a.19.1		1
4.3	odd	2		32.19.d.a.15.1		1
8.3	odd	2	CM	8.19.d.a.3.1	✓	1
8.5	even	2		32.19.d.a.15.1		1
24.11	even	2		72.19.b.a.19.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.19.d.a.3.1	✓	1	1.1	even	1	trivial
8.19.d.a.3.1	✓	1	8.3	odd	2	CM
32.19.d.a.15.1		1	4.3	odd	2
32.19.d.a.15.1		1	8.5	even	2
72.19.b.a.19.1		1	3.2	odd	2
72.19.b.a.19.1		1	24.11	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:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 19 \)
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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