LMFDB - Embedded newform 8.7.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,7,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, 7, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 7 $
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:	$1.84043266896$
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)

$f(q)$

$=$

$q-8.00000 q^{2} +46.0000 q^{3} +64.0000 q^{4} -368.000 q^{6} -512.000 q^{8} +1387.00 q^{9} +O(q^{10})$

$q-8.00000 q^{2} +46.0000 q^{3} +64.0000 q^{4} -368.000 q^{6} -512.000 q^{8} +1387.00 q^{9} -2338.00 q^{11} +2944.00 q^{12} +4096.00 q^{16} -1726.00 q^{17} -11096.0 q^{18} -2482.00 q^{19} +18704.0 q^{22} -23552.0 q^{24} +15625.0 q^{25} +30268.0 q^{27} -32768.0 q^{32} -107548. q^{33} +13808.0 q^{34} +88768.0 q^{36} +19856.0 q^{38} +134642. q^{41} -74914.0 q^{43} -149632. q^{44} +188416. q^{48} +117649. q^{49} -125000. q^{50} -79396.0 q^{51} -242144. q^{54} -114172. q^{57} +304958. q^{59} +262144. q^{64} +860384. q^{66} -596626. q^{67} -110464. q^{68} -710144. q^{72} -593134. q^{73} +718750. q^{75} -158848. q^{76} +381205. q^{81} -1.07714e6 q^{82} +678926. q^{83} +599312. q^{86} +1.19706e6 q^{88} -357262. q^{89} -1.50733e6 q^{96} +1.82275e6 q^{97} -941192. q^{98} -3.24281e6 q^{99} +O(q^{100})$

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 $

$2$	−8.00000	−1.00000
$2$	−8.00000	−1.00000
$3$	46.0000	1.70370	0.851852	−	0.523783i	$-0.175480\pi$
$3$	46.0000	1.70370	0.851852	+	0.523783i	$0.175480\pi$
$4$	64.0000	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−368.000	−1.70370
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−512.000	−1.00000
$9$	1387.00	1.90261
$10$	0	0
$11$	−2338.00	−1.75657	−0.878287	−	0.478134i	$-0.841313\pi$
$11$	−2338.00	−1.75657	−0.878287	+	0.478134i	$0.841313\pi$
$12$	2944.00	1.70370
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	4096.00	1.00000
$17$	−1726.00	−0.351313	−0.175656	−	0.984452i	$-0.556205\pi$
$17$	−1726.00	−0.351313	−0.175656	+	0.984452i	$0.556205\pi$
$18$	−11096.0	−1.90261
$19$	−2482.00	−0.361860	−0.180930	−	0.983496i	$-0.557911\pi$
$19$	−2482.00	−0.361860	−0.180930	+	0.983496i	$0.557911\pi$
$20$	0	0
$21$	0	0
$22$	18704.0	1.75657
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−23552.0	−1.70370
$25$	15625.0	1.00000
$26$	0	0
$27$	30268.0	1.53777
$28$	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			$\pi$
$32$	−32768.0	−1.00000
$33$	−107548.	−2.99268
$34$	13808.0	0.351313
$35$	0	0
$36$	88768.0	1.90261
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	19856.0	0.361860
$39$	0	0
$40$	0	0
$41$	134642.	1.95357	0.976785	−	0.214222i	$-0.0687216\pi$
$41$	134642.	1.95357	0.976785	+	0.214222i	$0.0687216\pi$
$42$	0	0
$43$	−74914.0	−0.942232	−0.471116	−	0.882071i	$-0.656149\pi$
$43$	−74914.0	−0.942232	−0.471116	+	0.882071i	$0.656149\pi$
$44$	−149632.	−1.75657
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	188416.	1.70370
$49$	117649.	1.00000
$50$	−125000.	−1.00000
$51$	−79396.0	−0.598533
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−242144.	−1.53777
$55$	0	0
$56$	0	0
$57$	−114172.	−0.616503
$58$	0	0
$59$	304958.	1.48485	0.742427	−	0.669927i	$-0.233674\pi$
$59$	304958.	1.48485	0.742427	+	0.669927i	$0.233674\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	262144.	1.00000
$65$	0	0
$66$	860384.	2.99268
$67$	−596626.	−1.98371	−0.991854	−	0.127380i	$-0.959343\pi$
$67$	−596626.	−1.98371	−0.991854	+	0.127380i	$0.959343\pi$
$68$	−110464.	−0.351313
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−710144.	−1.90261
$73$	−593134.	−1.52470	−0.762350	−	0.647165i	$-0.775954\pi$
$73$	−593134.	−1.52470	−0.762350	+	0.647165i	$0.775954\pi$
$74$	0	0
$75$	718750.	1.70370
$76$	−158848.	−0.361860
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	381205.	0.717304
$82$	−1.07714e6	−1.95357
$83$	678926.	1.18738	0.593688	−	0.804695i	$-0.297671\pi$
$83$	678926.	1.18738	0.593688	+	0.804695i	$0.297671\pi$
$84$	0	0
$85$	0	0
$86$	599312.	0.942232
$87$	0	0
$88$	1.19706e6	1.75657
$89$	−357262.	−0.506777	−0.253388	−	0.967365i	$-0.581545\pi$
$89$	−357262.	−0.506777	−0.253388	+	0.967365i	$0.581545\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−1.50733e6	−1.70370
$97$	1.82275e6	1.99716	0.998580	−	0.0532728i	$-0.0169653\pi$
$97$	1.82275e6	1.99716	0.998580	+	0.0532728i	$0.0169653\pi$
$98$	−941192.	−1.00000
$99$	−3.24281e6	−3.34207
$100$	1.00000e6	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	635168.	0.598533
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	474014.	0.386937	0.193468	−	0.981107i	$-0.438026\pi$
$107$	474014.	0.386937	0.193468	+	0.981107i	$0.438026\pi$
$108$	1.93715e6	1.53777
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.81289e6	−1.94948	−0.974738	−	0.223350i	$-0.928301\pi$
$113$	−2.81289e6	−1.94948	−0.974738	+	0.223350i	$0.928301\pi$
$114$	913376.	0.616503
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−2.43966e6	−1.48485
$119$	0	0
$120$	0	0
$121$	3.69468e6	2.08555
$122$	0	0
$123$	6.19353e6	3.32830
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−2.09715e6	−1.00000
$129$	−3.44604e6	−1.60528
$130$	0	0
$131$	−2.95362e6	−1.31383	−0.656917	−	0.753963i	$-0.728140\pi$
$131$	−2.95362e6	−1.31383	−0.656917	+	0.753963i	$0.728140\pi$
$132$	−6.88307e6	−2.99268
$133$	0	0
$134$	4.77301e6	1.98371
$135$	0	0
$136$	883712.	0.351313
$137$	−80206.0	−0.0311921	−0.0155961	−	0.999878i	$-0.504965\pi$
$137$	−80206.0	−0.0311921	−0.0155961	+	0.999878i	$0.504965\pi$
$138$	0	0
$139$	−3.19856e6	−1.19100	−0.595498	−	0.803357i	$-0.703045\pi$
$139$	−3.19856e6	−1.19100	−0.595498	+	0.803357i	$0.703045\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	5.68115e6	1.90261
$145$	0	0
$146$	4.74507e6	1.52470
$147$	5.41185e6	1.70370
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−5.75000e6	−1.70370
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	1.27078e6	0.361860
$153$	−2.39396e6	−0.668410
$154$	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
$162$	−3.04964e6	−0.717304
$163$	−7.72059e6	−1.78274	−0.891370	−	0.453277i	$-0.850255\pi$
$163$	−7.72059e6	−1.78274	−0.891370	+	0.453277i	$0.850255\pi$
$164$	8.61709e6	1.95357
$165$	0	0
$166$	−5.43141e6	−1.18738
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	4.82681e6	1.00000
$170$	0	0
$171$	−3.44253e6	−0.688478
$172$	−4.79450e6	−0.942232
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	−9.57645e6	−1.75657
$177$	1.40281e7	2.52975
$178$	2.85810e6	0.506777
$179$	3.22888e6	0.562979	0.281490	−	0.959564i	$-0.409171\pi$
$179$	3.22888e6	0.562979	0.281490	+	0.959564i	$0.409171\pi$
$180$	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
$187$	4.03539e6	0.617107
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	1.20586e7	1.70370
$193$	−1.00100e7	−1.39240	−0.696198	−	0.717850i	$-0.745126\pi$
$193$	−1.00100e7	−1.39240	−0.696198	+	0.717850i	$0.745126\pi$
$194$	−1.45820e7	−1.99716
$195$	0	0
$196$	7.52954e6	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	2.59424e7	3.34207
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−8.00000e6	−1.00000
$201$	−2.74448e7	−3.37965
$202$	0	0
$203$	0	0
$204$	−5.08134e6	−0.598533
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.80292e6	0.635634
$210$	0	0
$211$	1.86421e7	1.98448	0.992240	−	0.124340i	$-0.0396814\pi$
$211$	1.86421e7	1.98448	0.992240	+	0.124340i	$0.0396814\pi$
$212$	0	0
$213$	0	0
$214$	−3.79211e6	−0.386937
$215$	0	0
$216$	−1.54972e7	−1.53777
$217$	0	0
$218$	0	0
$219$	−2.72842e7	−2.59764
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	2.16719e7	1.90261
$226$	2.25032e7	1.94948
$227$	1.97707e7	1.69023	0.845114	−	0.534586i	$-0.179533\pi$
$227$	1.97707e7	1.69023	0.845114	+	0.534586i	$0.179533\pi$
$228$	−7.30701e6	−0.616503
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.10622e7	0.874530	0.437265	−	0.899333i	$-0.355947\pi$
$233$	1.10622e7	0.874530	0.437265	+	0.899333i	$0.355947\pi$
$234$	0	0
$235$	0	0
$236$	1.95173e7	1.48485
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−2.65018e7	−1.89332	−0.946659	−	0.322237i	$-0.895565\pi$
$241$	−2.65018e7	−1.89332	−0.946659	+	0.322237i	$0.895565\pi$
$242$	−2.95575e7	−2.08555
$243$	−4.52994e6	−0.315699
$244$	0	0
$245$	0	0
$246$	−4.95483e7	−3.32830
$247$	0	0
$248$	0	0
$249$	3.12306e7	2.02294
$250$	0	0
$251$	−1.31193e7	−0.829640	−0.414820	−	0.909904i	$-0.636155\pi$
$251$	−1.31193e7	−0.829640	−0.414820	+	0.909904i	$0.636155\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.67772e7	1.00000
$257$	−1.89723e7	−1.11769	−0.558844	−	0.829273i	$-0.688755\pi$
$257$	−1.89723e7	−1.11769	−0.558844	+	0.829273i	$0.688755\pi$
$258$	2.75684e7	1.60528
$259$	0	0
$260$	0	0
$261$	0	0
$262$	2.36289e7	1.31383
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	5.50646e7	2.99268
$265$	0	0
$266$	0	0
$267$	−1.64341e7	−0.863398
$268$	−3.81841e7	−1.98371
$269$	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			$\pi$
$272$	−7.06970e6	−0.351313
$273$	0	0
$274$	641648.	0.0311921
$275$	−3.65312e7	−1.75657
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	2.55885e7	1.19100
$279$	0	0
$280$	0	0
$281$	4.28969e7	1.93333	0.966667	−	0.256038i	$-0.0824172\pi$
$281$	4.28969e7	1.93333	0.966667	+	0.256038i	$0.0824172\pi$
$282$	0	0
$283$	1.91505e7	0.844931	0.422466	−	0.906379i	$-0.361165\pi$
$283$	1.91505e7	0.844931	0.422466	+	0.906379i	$0.361165\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−4.54492e7	−1.90261
$289$	−2.11585e7	−0.876579
$290$	0	0
$291$	8.38467e7	3.40257
$292$	−3.79606e7	−1.52470
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−4.32948e7	−1.70370
$295$	0	0
$296$	0	0
$297$	−7.07666e7	−2.70121
$298$	0	0
$299$	0	0
$300$	4.60000e7	1.70370
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−1.01663e7	−0.361860
$305$	0	0
$306$	1.91517e7	0.668410
$307$	5.97121e6	0.206370	0.103185	−	0.994662i	$-0.467097\pi$
$307$	5.97121e6	0.206370	0.103185	+	0.994662i	$0.467097\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	9.06351e6	0.295572	0.147786	−	0.989019i	$-0.452785\pi$
$313$	9.06351e6	0.295572	0.147786	+	0.989019i	$0.452785\pi$
$314$	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
$321$	2.18046e7	0.659225
$322$	0	0
$323$	4.28393e6	0.127126
$324$	2.43971e7	0.717304
$325$	0	0
$326$	6.17648e7	1.78274
$327$	0	0
$328$	−6.89367e7	−1.95357
$329$	0	0
$330$	0	0
$331$	−4.59882e6	−0.126813	−0.0634063	−	0.997988i	$-0.520196\pi$
$331$	−4.59882e6	−0.126813	−0.0634063	+	0.997988i	$0.520196\pi$
$332$	4.34513e7	1.18738
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5.36426e7	1.40159	0.700794	−	0.713364i	$-0.252829\pi$
$337$	5.36426e7	1.40159	0.700794	+	0.713364i	$0.252829\pi$
$338$	−3.86145e7	−1.00000
$339$	−1.29393e8	−3.32133
$340$	0	0
$341$	0	0
$342$	2.75403e7	0.688478
$343$	0	0
$344$	3.83560e7	0.942232
$345$	0	0
$346$	0	0
$347$	−4.72029e7	−1.12975	−0.564873	−	0.825178i	$-0.691075\pi$
$347$	−4.72029e7	−1.12975	−0.564873	+	0.825178i	$0.691075\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	7.66116e7	1.75657
$353$	−6.52211e7	−1.48274	−0.741368	−	0.671099i	$-0.765823\pi$
$353$	−6.52211e7	−1.48274	−0.741368	+	0.671099i	$0.765823\pi$
$354$	−1.12225e8	−2.52975
$355$	0	0
$356$	−2.28648e7	−0.506777
$357$	0	0
$358$	−2.58310e7	−0.562979
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−4.08856e7	−0.869057
$362$	0	0
$363$	1.69955e8	3.55316
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	1.86748e8	3.71687
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−3.22831e7	−0.617107
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	2.72157e7	0.499921	0.249961	−	0.968256i	$-0.419582\pi$
$379$	2.72157e7	0.499921	0.249961	+	0.968256i	$0.419582\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−9.64690e7	−1.70370
$385$	0	0
$386$	8.00801e7	1.39240
$387$	−1.03906e8	−1.79270
$388$	1.16656e8	1.99716
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−6.02363e7	−1.00000
$393$	−1.35866e8	−2.23838
$394$	0	0
$395$	0	0
$396$	−2.07540e8	−3.34207
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	6.40000e7	1.00000
$401$	−7.99344e7	−1.23965	−0.619827	−	0.784738i	$-0.712797\pi$
$401$	−7.99344e7	−1.23965	−0.619827	+	0.784738i	$0.712797\pi$
$402$	2.19558e8	3.37965
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	4.06508e7	0.598533
$409$	1.30356e8	1.90529	0.952644	−	0.304088i	$-0.0983516\pi$
$409$	1.30356e8	1.90529	0.952644	+	0.304088i	$0.0983516\pi$
$410$	0	0
$411$	−3.68948e6	−0.0531422
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.47134e8	−2.02910
$418$	−4.64233e7	−0.635634
$419$	1.34918e8	1.83412	0.917062	−	0.398744i	$-0.130554\pi$
$419$	1.34918e8	1.83412	0.917062	+	0.398744i	$0.130554\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−1.49136e8	−1.98448
$423$	0	0
$424$	0	0
$425$	−2.69688e7	−0.351313
$426$	0	0
$427$	0	0
$428$	3.03369e7	0.386937
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.23978e8	1.53777
$433$	−1.32005e8	−1.62603	−0.813014	−	0.582244i	$-0.802175\pi$
$433$	−1.32005e8	−1.62603	−0.813014	+	0.582244i	$0.802175\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	2.18273e8	2.59764
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	1.63179e8	1.90261
$442$	0	0
$443$	1.59916e8	1.83942	0.919711	−	0.392595i	$-0.128423\pi$
$443$	1.59916e8	1.83942	0.919711	+	0.392595i	$0.128423\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.25702e8	1.38869	0.694344	−	0.719643i	$-0.255694\pi$
$449$	1.25702e8	1.38869	0.694344	+	0.719643i	$0.255694\pi$
$450$	−1.73375e8	−1.90261
$451$	−3.14793e8	−3.43159
$452$	−1.80025e8	−1.94948
$453$	0	0
$454$	−1.58166e8	−1.69023
$455$	0	0
$456$	5.84561e7	0.616503
$457$	1.35637e8	1.42112	0.710558	−	0.703639i	$-0.248443\pi$
$457$	1.35637e8	1.42112	0.710558	+	0.703639i	$0.248443\pi$
$458$	0	0
$459$	−5.22426e7	−0.540240
$460$	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
$466$	−8.84978e7	−0.874530
$467$	2.22058e7	0.218030	0.109015	−	0.994040i	$-0.465230\pi$
$467$	2.22058e7	0.218030	0.109015	+	0.994040i	$0.465230\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−1.56138e8	−1.48485
$473$	1.75149e8	1.65510
$474$	0	0
$475$	−3.87812e7	−0.361860
$476$	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
$482$	2.12014e8	1.89332
$483$	0	0
$484$	2.36460e8	2.08555
$485$	0	0
$486$	3.62395e7	0.315699
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	−3.55147e8	−3.03726
$490$	0	0
$491$	−8.73643e7	−0.738056	−0.369028	−	0.929418i	$-0.620309\pi$
$491$	−8.73643e7	−0.738056	−0.369028	+	0.929418i	$0.620309\pi$
$492$	3.96386e8	3.32830
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−2.49845e8	−2.02294
$499$	8.32468e7	0.669986	0.334993	−	0.942221i	$-0.391266\pi$
$499$	8.32468e7	0.669986	0.334993	+	0.942221i	$0.391266\pi$
$500$	0	0
$501$	0	0
$502$	1.04954e8	0.829640
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.22033e8	1.70370
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−1.34218e8	−1.00000
$513$	−7.51252e7	−0.556459
$514$	1.51778e8	1.11769
$515$	0	0
$516$	−2.20547e8	−1.60528
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.98899e8	−1.40644	−0.703218	−	0.710974i	$-0.748254\pi$
$521$	−1.98899e8	−1.40644	−0.703218	+	0.710974i	$0.748254\pi$
$522$	0	0
$523$	−2.63549e8	−1.84228	−0.921141	−	0.389229i	$-0.872741\pi$
$523$	−2.63549e8	−1.84228	−0.921141	+	0.389229i	$0.872741\pi$
$524$	−1.89032e8	−1.31383
$525$	0	0
$526$	0	0
$527$	0	0
$528$	−4.40517e8	−2.99268
$529$	1.48036e8	1.00000
$530$	0	0
$531$	4.22977e8	2.82509
$532$	0	0
$533$	0	0
$534$	1.31472e8	0.863398
$535$	0	0
$536$	3.05473e8	1.98371
$537$	1.48528e8	0.959150
$538$	0	0
$539$	−2.75063e8	−1.75657
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	5.65576e7	0.351313
$545$	0	0
$546$	0	0
$547$	1.50088e8	0.917030	0.458515	−	0.888687i	$-0.348382\pi$
$547$	1.50088e8	0.917030	0.458515	+	0.888687i	$0.348382\pi$
$548$	−5.13318e6	−0.0311921
$549$	0	0
$550$	2.92250e8	1.75657
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−2.04708e8	−1.19100
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.85628e8	1.05137
$562$	−3.43175e8	−1.93333
$563$	2.03362e8	1.13958	0.569789	−	0.821791i	$-0.307025\pi$
$563$	2.03362e8	1.13958	0.569789	+	0.821791i	$0.307025\pi$
$564$	0	0
$565$	0	0
$566$	−1.53204e8	−0.844931
$567$	0	0
$568$	0	0
$569$	−3.62709e8	−1.96889	−0.984445	−	0.175695i	$-0.943783\pi$
$569$	−3.62709e8	−1.96889	−0.984445	+	0.175695i	$0.943783\pi$
$570$	0	0
$571$	3.58715e8	1.92682	0.963409	−	0.268035i	$-0.0863743\pi$
$571$	3.58715e8	1.92682	0.963409	+	0.268035i	$0.0863743\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	3.63594e8	1.90261
$577$	−1.99757e6	−0.0103986	−0.00519929	−	0.999986i	$-0.501655\pi$
$577$	−1.99757e6	−0.0103986	−0.00519929	+	0.999986i	$0.501655\pi$
$578$	1.69268e8	0.876579
$579$	−4.60461e8	−2.37223
$580$	0	0
$581$	0	0
$582$	−6.70773e8	−3.40257
$583$	0	0
$584$	3.03685e8	1.52470
$585$	0	0
$586$	0	0
$587$	−2.97402e8	−1.47038	−0.735189	−	0.677862i	$-0.762906\pi$
$587$	−2.97402e8	−1.47038	−0.735189	+	0.677862i	$0.762906\pi$
$588$	3.46359e8	1.70370
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.68860e8	1.28933	0.644663	−	0.764467i	$-0.276998\pi$
$593$	2.68860e8	1.28933	0.644663	+	0.764467i	$0.276998\pi$
$594$	5.66133e8	2.70121
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−3.68000e8	−1.70370
$601$	−2.26861e8	−1.04505	−0.522525	−	0.852624i	$-0.675010\pi$
$601$	−2.26861e8	−1.04505	−0.522525	+	0.852624i	$0.675010\pi$
$602$	0	0
$603$	−8.27520e8	−3.77422
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	8.13302e7	0.361860
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−1.53214e8	−0.668410
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−4.77697e7	−0.206370
$615$	0	0
$616$	0	0
$617$	3.44191e8	1.46536	0.732679	−	0.680575i	$-0.238270\pi$
$617$	3.44191e8	1.46536	0.732679	+	0.680575i	$0.238270\pi$
$618$	0	0
$619$	4.68505e8	1.97534	0.987671	−	0.156543i	$-0.0500348\pi$
$619$	4.68505e8	1.97534	0.987671	+	0.156543i	$0.0500348\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.44141e8	1.00000
$626$	−7.25080e7	−0.295572
$627$	2.66934e8	1.08293
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	8.57535e8	3.38096
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−4.82154e8	−1.83068	−0.915338	−	0.402687i	$-0.868076\pi$
$641$	−4.82154e8	−1.83068	−0.915338	+	0.402687i	$0.868076\pi$
$642$	−1.74437e8	−0.659225
$643$	2.83407e8	1.06605	0.533025	−	0.846099i	$-0.321055\pi$
$643$	2.83407e8	1.06605	0.533025	+	0.846099i	$0.321055\pi$
$644$	0	0
$645$	0	0
$646$	−3.42715e7	−0.127126
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.95177e8	−0.717304
$649$	−7.12992e8	−2.60826
$650$	0	0
$651$	0	0
$652$	−4.94118e8	−1.78274
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	5.51494e8	1.95357
$657$	−8.22677e8	−2.90090
$658$	0	0
$659$	3.12918e8	1.09339	0.546694	−	0.837332i	$-0.315886\pi$
$659$	3.12918e8	1.09339	0.546694	+	0.837332i	$0.315886\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	3.67905e7	0.126813
$663$	0	0
$664$	−3.47610e8	−1.18738
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−2.41386e8	−0.791895	−0.395947	−	0.918273i	$-0.629584\pi$
$673$	−2.41386e8	−0.791895	−0.395947	+	0.918273i	$0.629584\pi$
$674$	−4.29141e8	−1.40159
$675$	4.72938e8	1.53777
$676$	3.08916e8	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	1.03514e9	3.32133
$679$	0	0
$680$	0	0
$681$	9.09454e8	2.87965
$682$	0	0
$683$	−4.93943e8	−1.55030	−0.775148	−	0.631779i	$-0.782325\pi$
$683$	−4.93943e8	−1.55030	−0.775148	+	0.631779i	$0.782325\pi$
$684$	−2.20322e8	−0.688478
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−3.06848e8	−0.942232
$689$	0	0
$690$	0	0
$691$	−6.55966e8	−1.98814	−0.994071	−	0.108734i	$-0.965320\pi$
$691$	−6.55966e8	−1.98814	−0.994071	+	0.108734i	$0.965320\pi$
$692$	0	0
$693$	0	0
$694$	3.77624e8	1.12975
$695$	0	0
$696$	0	0
$697$	−2.32392e8	−0.686314
$698$	0	0
$699$	5.08862e8	1.48994
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−6.12893e8	−1.75657
$705$	0	0
$706$	5.21768e8	1.48274
$707$	0	0
$708$	8.97796e8	2.52975
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	1.82918e8	0.506777
$713$	0	0
$714$	0	0
$715$	0	0
$716$	2.06648e8	0.562979
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	3.27084e8	0.869057
$723$	−1.21908e9	−3.22565
$724$	0	0
$725$	0	0
$726$	−1.35964e9	−3.55316
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−4.86276e8	−1.25516
$730$	0	0
$731$	1.29302e8	0.331018
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.39491e9	3.48453
$738$	−1.49399e9	−3.71687
$739$	4.94167e8	1.22445	0.612224	−	0.790685i	$-0.290275\pi$
$739$	4.94167e8	1.22445	0.612224	+	0.790685i	$0.290275\pi$
$740$	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
$747$	9.41670e8	2.25911
$748$	2.58265e8	0.617107
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	−6.03488e8	−1.41346
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−2.17725e8	−0.499921
$759$	0	0
$760$	0	0
$761$	2.86987e8	0.651191	0.325595	−	0.945509i	$-0.394435\pi$
$761$	2.86987e8	0.651191	0.325595	+	0.945509i	$0.394435\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	7.71752e8	1.70370
$769$	6.98978e8	1.53704	0.768519	−	0.639827i	$-0.220994\pi$
$769$	6.98978e8	1.53704	0.768519	+	0.639827i	$0.220994\pi$
$770$	0	0
$771$	−8.72725e8	−1.90421
$772$	−6.40641e8	−1.39240
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	8.31246e8	1.79270
$775$	0	0
$776$	−9.33250e8	−1.99716
$777$	0	0
$778$	0	0
$779$	−3.34181e8	−0.706919
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	4.81890e8	1.00000
$785$	0	0
$786$	1.08693e9	2.23838
$787$	−9.26584e8	−1.90091	−0.950453	−	0.310868i	$-0.899380\pi$
$787$	−9.26584e8	−1.90091	−0.950453	+	0.310868i	$0.899380\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	1.66032e9	3.34207
$793$	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
$800$	−5.12000e8	−1.00000
$801$	−4.95522e8	−0.964197
$802$	6.39475e8	1.23965
$803$	1.38675e9	2.67825
$804$	−1.75647e9	−3.37965
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.53143e8	−1.61130	−0.805649	−	0.592393i	$-0.798183\pi$
$809$	−8.53143e8	−1.61130	−0.805649	+	0.592393i	$0.798183\pi$
$810$	0	0
$811$	3.45583e8	0.647873	0.323937	−	0.946079i	$-0.394994\pi$
$811$	3.45583e8	0.647873	0.323937	+	0.946079i	$0.394994\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−3.25206e8	−0.598533
$817$	1.85937e8	0.340956
$818$	−1.04285e9	−1.90529
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	2.95158e7	0.0531422
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	−1.68044e9	−2.99268
$826$	0	0
$827$	−5.79438e8	−1.02445	−0.512225	−	0.858851i	$-0.671179\pi$
$827$	−5.79438e8	−1.02445	−0.512225	+	0.858851i	$0.671179\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.03062e8	−0.351313
$834$	1.17707e9	2.02910
$835$	0	0
$836$	3.71387e8	0.635634
$837$	0	0
$838$	−1.07935e9	−1.83412
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	5.94823e8	1.00000
$842$	0	0
$843$	1.97326e9	3.29383
$844$	1.19309e9	1.98448
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	8.80924e8	1.43951
$850$	2.15750e8	0.351313
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−2.42695e8	−0.386937
$857$	−9.11769e8	−1.44858	−0.724290	−	0.689496i	$-0.757832\pi$
$857$	−9.11769e8	−1.44858	−0.724290	+	0.689496i	$0.757832\pi$
$858$	0	0
$859$	8.15878e8	1.28720	0.643599	−	0.765363i	$-0.277440\pi$
$859$	8.15878e8	1.28720	0.643599	+	0.765363i	$0.277440\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−9.91822e8	−1.53777
$865$	0	0
$866$	1.05604e9	1.62603
$867$	−9.73291e8	−1.49343
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.52816e9	3.79981
$874$	0	0
$875$	0	0
$876$	−1.74619e9	−2.59764
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.74799e8	0.548114	0.274057	−	0.961714i	$-0.411634\pi$
$881$	3.74799e8	0.548114	0.274057	+	0.961714i	$0.411634\pi$
$882$	−1.30543e9	−1.90261
$883$	−1.34895e9	−1.95935	−0.979676	−	0.200584i	$-0.935716\pi$
$883$	−1.34895e9	−1.95935	−0.979676	+	0.200584i	$0.935716\pi$
$884$	0	0
$885$	0	0
$886$	−1.27933e9	−1.83942
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−8.91257e8	−1.26000
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−1.00562e9	−1.38869
$899$	0	0
$900$	1.38700e9	1.90261
$901$	0	0
$902$	2.51834e9	3.43159
$903$	0	0
$904$	1.44020e9	1.94948
$905$	0	0
$906$	0	0
$907$	−8.05321e8	−1.07931	−0.539656	−	0.841885i	$-0.681446\pi$
$907$	−8.05321e8	−1.07931	−0.539656	+	0.841885i	$0.681446\pi$
$908$	1.26533e9	1.69023
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−4.67649e8	−0.616503
$913$	−1.58733e9	−2.08571
$914$	−1.08510e9	−1.42112
$915$	0	0
$916$	0	0
$917$	0	0
$918$	4.17941e8	0.540240
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	2.74676e8	0.351594
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.55501e9	−1.93948	−0.969739	−	0.244146i	$-0.921492\pi$
$929$	−1.55501e9	−1.93948	−0.969739	+	0.244146i	$0.921492\pi$
$930$	0	0
$931$	−2.92005e8	−0.361860
$932$	7.07982e8	0.874530
$933$	0	0
$934$	−1.77646e8	−0.218030
$935$	0	0
$936$	0	0
$937$	1.52100e9	1.84889	0.924443	−	0.381320i	$-0.124530\pi$
$937$	1.52100e9	1.84889	0.924443	+	0.381320i	$0.124530\pi$
$938$	0	0
$939$	4.16921e8	0.503567
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	1.24911e9	1.48485
$945$	0	0
$946$	−1.40119e9	−1.65510
$947$	1.69218e9	1.99249	0.996244	−	0.0865891i	$-0.0275967\pi$
$947$	1.69218e9	1.99249	0.996244	+	0.0865891i	$0.0275967\pi$
$948$	0	0
$949$	0	0
$950$	3.10250e8	0.361860
$951$	0	0
$952$	0	0
$953$	3.84034e8	0.443701	0.221851	−	0.975081i	$-0.428790\pi$
$953$	3.84034e8	0.443701	0.221851	+	0.975081i	$0.428790\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	8.87504e8	1.00000
$962$	0	0
$963$	6.57457e8	0.736188
$964$	−1.69611e9	−1.89332
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−1.89168e9	−2.08555
$969$	1.97061e8	0.216585
$970$	0	0
$971$	−1.83097e9	−1.99997	−0.999986	−	0.00535408i	$-0.998296\pi$
$971$	−1.83097e9	−1.99997	−0.999986	+	0.00535408i	$0.998296\pi$
$972$	−2.89916e8	−0.315699
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.56343e9	−1.67647	−0.838234	−	0.545310i	$-0.816412\pi$
$977$	−1.56343e9	−1.67647	−0.838234	+	0.545310i	$0.816412\pi$
$978$	2.84118e9	3.03726
$979$	8.35279e8	0.890191
$980$	0	0
$981$	0	0
$982$	6.98914e8	0.738056
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−3.17109e9	−3.32830
$985$	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
$993$	−2.11546e8	−0.216051
$994$	0	0
$995$	0	0
$996$	1.99876e9	2.02294
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−6.65974e8	−0.669986
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.7.d.a.3.1	✓	1
3.2	odd	2		72.7.b.a.19.1		1
4.3	odd	2		32.7.d.a.15.1		1
8.3	odd	2	CM	8.7.d.a.3.1	✓	1
8.5	even	2		32.7.d.a.15.1		1
12.11	even	2		288.7.b.a.271.1		1
16.3	odd	4		256.7.c.d.255.1		2
16.5	even	4		256.7.c.d.255.1		2
16.11	odd	4		256.7.c.d.255.2		2
16.13	even	4		256.7.c.d.255.2		2
24.5	odd	2		288.7.b.a.271.1		1
24.11	even	2		72.7.b.a.19.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.7.d.a.3.1	✓	1	1.1	even	1	trivial
8.7.d.a.3.1	✓	1	8.3	odd	2	CM
32.7.d.a.15.1		1	4.3	odd	2
32.7.d.a.15.1		1	8.5	even	2
72.7.b.a.19.1		1	3.2	odd	2
72.7.b.a.19.1		1	24.11	even	2
256.7.c.d.255.1		2	16.3	odd	4
256.7.c.d.255.1		2	16.5	even	4
256.7.c.d.255.2		2	16.11	odd	4
256.7.c.d.255.2		2	16.13	even	4
288.7.b.a.271.1		1	12.11	even	2
288.7.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:	\( k \)	\(=\)	\( 7 \)
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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