LMFDB - Embedded newform 448.8.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,8,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("448.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	448.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$139.948491417$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-18.0000 q^{3} -160.000 q^{5} +343.000 q^{7} -1863.00 q^{9} +5704.00 q^{11} -1388.00 q^{13} +2880.00 q^{15} -31434.0 q^{17} -19966.0 q^{19} -6174.00 q^{21} +77136.0 q^{23} -52525.0 q^{25} +72900.0 q^{27} +193374. q^{29} +26356.0 q^{31} -102672. q^{33} -54880.0 q^{35} -204346. q^{37} +24984.0 q^{39} -663050. q^{41} -335920. q^{43} +298080. q^{45} -1.11981e6 q^{47} +117649. q^{49} +565812. q^{51} -112782. q^{53} -912640. q^{55} +359388. q^{57} +536154. q^{59} +1.17026e6 q^{61} -639009. q^{63} +222080. q^{65} +3.89066e6 q^{67} -1.38845e6 q^{69} -2.50534e6 q^{71} -1.43507e6 q^{73} +945450. q^{75} +1.95647e6 q^{77} -176536. q^{79} +2.76218e6 q^{81} -6.21162e6 q^{83} +5.02944e6 q^{85} -3.48073e6 q^{87} -4.72906e6 q^{89} -476084. q^{91} -474408. q^{93} +3.19456e6 q^{95} -2.12956e6 q^{97} -1.06266e7 q^{99} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−18.0000	−0.384900	−0.192450	−	0.981307i	$-0.561643\pi$
$3$	−18.0000	−0.384900	−0.192450	+	0.981307i	$0.561643\pi$
$4$	0	0
$5$	−160.000	−0.572433	−0.286217	−	0.958165i	$-0.592398\pi$
$5$	−160.000	−0.572433	−0.286217	+	0.958165i	$0.592398\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	−1863.00	−0.851852
$10$	0	0
$11$	5704.00	1.29213	0.646063	−	0.763284i	$-0.276414\pi$
$11$	5704.00	1.29213	0.646063	+	0.763284i	$0.276414\pi$
$12$	0	0
$13$	−1388.00	−0.175222	−0.0876108	−	0.996155i	$-0.527923\pi$
$13$	−1388.00	−0.175222	−0.0876108	+	0.996155i	$0.527923\pi$
$14$	0	0
$15$	2880.00	0.220330
$16$	0	0
$17$	−31434.0	−1.55177	−0.775887	−	0.630872i	$-0.782697\pi$
$17$	−31434.0	−1.55177	−0.775887	+	0.630872i	$0.782697\pi$
$18$	0	0
$19$	−19966.0	−0.667811	−0.333905	−	0.942607i	$-0.608367\pi$
$19$	−19966.0	−0.667811	−0.333905	+	0.942607i	$0.608367\pi$
$20$	0	0
$21$	−6174.00	−0.145479
$22$	0	0
$23$	77136.0	1.32193	0.660967	−	0.750415i	$-0.270146\pi$
$23$	77136.0	1.32193	0.660967	+	0.750415i	$0.270146\pi$
$24$	0	0
$25$	−52525.0	−0.672320
$26$	0	0
$27$	72900.0	0.712778
$28$	0	0
$29$	193374.	1.47233	0.736165	−	0.676802i	$-0.236635\pi$
$29$	193374.	1.47233	0.736165	+	0.676802i	$0.236635\pi$
$30$	0	0
$31$	26356.0	0.158896	0.0794481	−	0.996839i	$-0.474684\pi$
$31$	26356.0	0.158896	0.0794481	+	0.996839i	$0.474684\pi$
$32$	0	0
$33$	−102672.	−0.497340
$34$	0	0
$35$	−54880.0	−0.216359
$36$	0	0
$37$	−204346.	−0.663224	−0.331612	−	0.943416i	$-0.607592\pi$
$37$	−204346.	−0.663224	−0.331612	+	0.943416i	$0.607592\pi$
$38$	0	0
$39$	24984.0	0.0674428
$40$	0	0
$41$	−663050.	−1.50246	−0.751230	−	0.660041i	$-0.770539\pi$
$41$	−663050.	−1.50246	−0.751230	+	0.660041i	$0.770539\pi$
$42$	0	0
$43$	−335920.	−0.644312	−0.322156	−	0.946687i	$-0.604408\pi$
$43$	−335920.	−0.644312	−0.322156	+	0.946687i	$0.604408\pi$
$44$	0	0
$45$	298080.	0.487628
$46$	0	0
$47$	−1.11981e6	−1.57327	−0.786634	−	0.617420i	$-0.788178\pi$
$47$	−1.11981e6	−1.57327	−0.786634	+	0.617420i	$0.788178\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	565812.	0.597278
$52$	0	0
$53$	−112782.	−0.104058	−0.0520289	−	0.998646i	$-0.516569\pi$
$53$	−112782.	−0.104058	−0.0520289	+	0.998646i	$0.516569\pi$
$54$	0	0
$55$	−912640.	−0.739657
$56$	0	0
$57$	359388.	0.257041
$58$	0	0
$59$	536154.	0.339866	0.169933	−	0.985456i	$-0.445645\pi$
$59$	536154.	0.339866	0.169933	+	0.985456i	$0.445645\pi$
$60$	0	0
$61$	1.17026e6	0.660130	0.330065	−	0.943958i	$-0.392929\pi$
$61$	1.17026e6	0.660130	0.330065	+	0.943958i	$0.392929\pi$
$62$	0	0
$63$	−639009.	−0.321970
$64$	0	0
$65$	222080.	0.100303
$66$	0	0
$67$	3.89066e6	1.58038	0.790190	−	0.612862i	$-0.209982\pi$
$67$	3.89066e6	1.58038	0.790190	+	0.612862i	$0.209982\pi$
$68$	0	0
$69$	−1.38845e6	−0.508813
$70$	0	0
$71$	−2.50534e6	−0.830736	−0.415368	−	0.909653i	$-0.636347\pi$
$71$	−2.50534e6	−0.830736	−0.415368	+	0.909653i	$0.636347\pi$
$72$	0	0
$73$	−1.43507e6	−0.431761	−0.215880	−	0.976420i	$-0.569262\pi$
$73$	−1.43507e6	−0.431761	−0.215880	+	0.976420i	$0.569262\pi$
$74$	0	0
$75$	945450.	0.258776
$76$	0	0
$77$	1.95647e6	0.488378
$78$	0	0
$79$	−176536.	−0.0402845	−0.0201423	−	0.999797i	$-0.506412\pi$
$79$	−176536.	−0.0402845	−0.0201423	+	0.999797i	$0.506412\pi$
$80$	0	0
$81$	2.76218e6	0.577503
$82$	0	0
$83$	−6.21162e6	−1.19243	−0.596213	−	0.802826i	$-0.703329\pi$
$83$	−6.21162e6	−1.19243	−0.596213	+	0.802826i	$0.703329\pi$
$84$	0	0
$85$	5.02944e6	0.888287
$86$	0	0
$87$	−3.48073e6	−0.566700
$88$	0	0
$89$	−4.72906e6	−0.711066	−0.355533	−	0.934664i	$-0.615701\pi$
$89$	−4.72906e6	−0.711066	−0.355533	+	0.934664i	$0.615701\pi$
$90$	0	0
$91$	−476084.	−0.0662276
$92$	0	0
$93$	−474408.	−0.0611592
$94$	0	0
$95$	3.19456e6	0.382277
$96$	0	0
$97$	−2.12956e6	−0.236913	−0.118457	−	0.992959i	$-0.537795\pi$
$97$	−2.12956e6	−0.236913	−0.118457	+	0.992959i	$0.537795\pi$
$98$	0	0
$99$	−1.06266e7	−1.10070
$100$	0	0
$101$	−5.66032e6	−0.546659	−0.273329	−	0.961921i	$-0.588125\pi$
$101$	−5.66032e6	−0.546659	−0.273329	+	0.961921i	$0.588125\pi$
$102$	0	0
$103$	1.40257e7	1.26472	0.632362	−	0.774673i	$-0.282086\pi$
$103$	1.40257e7	1.26472	0.632362	+	0.774673i	$0.282086\pi$
$104$	0	0
$105$	987840.	0.0832768
$106$	0	0
$107$	−2.07289e6	−0.163581	−0.0817906	−	0.996650i	$-0.526064\pi$
$107$	−2.07289e6	−0.163581	−0.0817906	+	0.996650i	$0.526064\pi$
$108$	0	0
$109$	6.87247e6	0.508300	0.254150	−	0.967165i	$-0.418204\pi$
$109$	6.87247e6	0.508300	0.254150	+	0.967165i	$0.418204\pi$
$110$	0	0
$111$	3.67823e6	0.255275
$112$	0	0
$113$	2.27437e7	1.48281	0.741407	−	0.671055i	$-0.234159\pi$
$113$	2.27437e7	1.48281	0.741407	+	0.671055i	$0.234159\pi$
$114$	0	0
$115$	−1.23418e7	−0.756719
$116$	0	0
$117$	2.58584e6	0.149263
$118$	0	0
$119$	−1.07819e7	−0.586515
$120$	0	0
$121$	1.30484e7	0.669592
$122$	0	0
$123$	1.19349e7	0.578297
$124$	0	0
$125$	2.09040e7	0.957292
$126$	0	0
$127$	2.51834e7	1.09094	0.545470	−	0.838130i	$-0.316351\pi$
$127$	2.51834e7	1.09094	0.545470	+	0.838130i	$0.316351\pi$
$128$	0	0
$129$	6.04656e6	0.247996
$130$	0	0
$131$	−1.74187e7	−0.676966	−0.338483	−	0.940972i	$-0.609914\pi$
$131$	−1.74187e7	−0.676966	−0.338483	+	0.940972i	$0.609914\pi$
$132$	0	0
$133$	−6.84834e6	−0.252409
$134$	0	0
$135$	−1.16640e7	−0.408018
$136$	0	0
$137$	4.49972e7	1.49508	0.747538	−	0.664219i	$-0.231236\pi$
$137$	4.49972e7	1.49508	0.747538	+	0.664219i	$0.231236\pi$
$138$	0	0
$139$	4.26865e7	1.34815	0.674076	−	0.738662i	$-0.264542\pi$
$139$	4.26865e7	1.34815	0.674076	+	0.738662i	$0.264542\pi$
$140$	0	0
$141$	2.01566e7	0.605551
$142$	0	0
$143$	−7.91715e6	−0.226409
$144$	0	0
$145$	−3.09398e7	−0.842811
$146$	0	0
$147$	−2.11768e6	−0.0549857
$148$	0	0
$149$	−4.19364e7	−1.03858	−0.519289	−	0.854599i	$-0.673803\pi$
$149$	−4.19364e7	−1.03858	−0.519289	+	0.854599i	$0.673803\pi$
$150$	0	0
$151$	7.60087e6	0.179657	0.0898285	−	0.995957i	$-0.471368\pi$
$151$	7.60087e6	0.179657	0.0898285	+	0.995957i	$0.471368\pi$
$152$	0	0
$153$	5.85615e7	1.32188
$154$	0	0
$155$	−4.21696e6	−0.0909575
$156$	0	0
$157$	−8.46198e7	−1.74511	−0.872556	−	0.488515i	$-0.837539\pi$
$157$	−8.46198e7	−1.74511	−0.872556	+	0.488515i	$0.837539\pi$
$158$	0	0
$159$	2.03008e6	0.0400518
$160$	0	0
$161$	2.64576e7	0.499644
$162$	0	0
$163$	1.54856e7	0.280074	0.140037	−	0.990146i	$-0.455278\pi$
$163$	1.54856e7	0.280074	0.140037	+	0.990146i	$0.455278\pi$
$164$	0	0
$165$	1.64275e7	0.284694
$166$	0	0
$167$	4.48893e7	0.745823	0.372911	−	0.927867i	$-0.378360\pi$
$167$	4.48893e7	0.745823	0.372911	+	0.927867i	$0.378360\pi$
$168$	0	0
$169$	−6.08220e7	−0.969297
$170$	0	0
$171$	3.71967e7	0.568876
$172$	0	0
$173$	1.02900e8	1.51096	0.755479	−	0.655173i	$-0.227404\pi$
$173$	1.02900e8	1.51096	0.755479	+	0.655173i	$0.227404\pi$
$174$	0	0
$175$	−1.80161e7	−0.254113
$176$	0	0
$177$	−9.65077e6	−0.130814
$178$	0	0
$179$	5.01780e7	0.653924	0.326962	−	0.945037i	$-0.393975\pi$
$179$	5.01780e7	0.653924	0.326962	+	0.945037i	$0.393975\pi$
$180$	0	0
$181$	1.32769e8	1.66426	0.832128	−	0.554583i	$-0.187122\pi$
$181$	1.32769e8	1.66426	0.832128	+	0.554583i	$0.187122\pi$
$182$	0	0
$183$	−2.10648e7	−0.254084
$184$	0	0
$185$	3.26954e7	0.379651
$186$	0	0
$187$	−1.79300e8	−2.00509
$188$	0	0
$189$	2.50047e7	0.269405
$190$	0	0
$191$	−1.43977e8	−1.49512	−0.747559	−	0.664195i	$-0.768775\pi$
$191$	−1.43977e8	−1.49512	−0.747559	+	0.664195i	$0.768775\pi$
$192$	0	0
$193$	9.17033e7	0.918194	0.459097	−	0.888386i	$-0.348173\pi$
$193$	9.17033e7	0.918194	0.459097	+	0.888386i	$0.348173\pi$
$194$	0	0
$195$	−3.99744e6	−0.0386065
$196$	0	0
$197$	1.10905e8	1.03352	0.516761	−	0.856130i	$-0.327138\pi$
$197$	1.10905e8	1.03352	0.516761	+	0.856130i	$0.327138\pi$
$198$	0	0
$199$	1.61688e8	1.45443	0.727214	−	0.686411i	$-0.240815\pi$
$199$	1.61688e8	1.45443	0.727214	+	0.686411i	$0.240815\pi$
$200$	0	0
$201$	−7.00319e7	−0.608288
$202$	0	0
$203$	6.63273e7	0.556488
$204$	0	0
$205$	1.06088e8	0.860058
$206$	0	0
$207$	−1.43704e8	−1.12609
$208$	0	0
$209$	−1.13886e8	−0.862896
$210$	0	0
$211$	1.74772e6	0.0128080	0.00640401	−	0.999979i	$-0.497962\pi$
$211$	1.74772e6	0.0128080	0.00640401	+	0.999979i	$0.497962\pi$
$212$	0	0
$213$	4.50962e7	0.319751
$214$	0	0
$215$	5.37472e7	0.368826
$216$	0	0
$217$	9.04011e6	0.0600571
$218$	0	0
$219$	2.58313e7	0.166185
$220$	0	0
$221$	4.36304e7	0.271904
$222$	0	0
$223$	4.83064e7	0.291701	0.145850	−	0.989307i	$-0.453408\pi$
$223$	4.83064e7	0.291701	0.145850	+	0.989307i	$0.453408\pi$
$224$	0	0
$225$	9.78541e7	0.572717
$226$	0	0
$227$	2.05296e8	1.16491	0.582453	−	0.812864i	$-0.302093\pi$
$227$	2.05296e8	1.16491	0.582453	+	0.812864i	$0.302093\pi$
$228$	0	0
$229$	2.81108e8	1.54685	0.773426	−	0.633886i	$-0.218541\pi$
$229$	2.81108e8	1.54685	0.773426	+	0.633886i	$0.218541\pi$
$230$	0	0
$231$	−3.52165e7	−0.187977
$232$	0	0
$233$	3.08493e8	1.59772	0.798858	−	0.601519i	$-0.205438\pi$
$233$	3.08493e8	1.59772	0.798858	+	0.601519i	$0.205438\pi$
$234$	0	0
$235$	1.79170e8	0.900591
$236$	0	0
$237$	3.17765e6	0.0155055
$238$	0	0
$239$	1.31783e8	0.624408	0.312204	−	0.950015i	$-0.398933\pi$
$239$	1.31783e8	0.624408	0.312204	+	0.950015i	$0.398933\pi$
$240$	0	0
$241$	2.79725e8	1.28727	0.643637	−	0.765331i	$-0.277425\pi$
$241$	2.79725e8	1.28727	0.643637	+	0.765331i	$0.277425\pi$
$242$	0	0
$243$	−2.09152e8	−0.935059
$244$	0	0
$245$	−1.88238e7	−0.0817762
$246$	0	0
$247$	2.77128e7	0.117015
$248$	0	0
$249$	1.11809e8	0.458965
$250$	0	0
$251$	1.24213e8	0.495805	0.247902	−	0.968785i	$-0.420259\pi$
$251$	1.24213e8	0.495805	0.247902	+	0.968785i	$0.420259\pi$
$252$	0	0
$253$	4.39984e8	1.70811
$254$	0	0
$255$	−9.05299e7	−0.341902
$256$	0	0
$257$	3.83682e8	1.40996	0.704978	−	0.709229i	$-0.250957\pi$
$257$	3.83682e8	1.40996	0.704978	+	0.709229i	$0.250957\pi$
$258$	0	0
$259$	−7.00907e7	−0.250675
$260$	0	0
$261$	−3.60256e8	−1.25421
$262$	0	0
$263$	−4.54839e8	−1.54174	−0.770872	−	0.636990i	$-0.780179\pi$
$263$	−4.54839e8	−1.54174	−0.770872	+	0.636990i	$0.780179\pi$
$264$	0	0
$265$	1.80451e7	0.0595661
$266$	0	0
$267$	8.51231e7	0.273689
$268$	0	0
$269$	−8.19052e7	−0.256554	−0.128277	−	0.991738i	$-0.540945\pi$
$269$	−8.19052e7	−0.256554	−0.128277	+	0.991738i	$0.540945\pi$
$270$	0	0
$271$	2.41677e8	0.737638	0.368819	−	0.929501i	$-0.379762\pi$
$271$	2.41677e8	0.737638	0.368819	+	0.929501i	$0.379762\pi$
$272$	0	0
$273$	8.56951e6	0.0254910
$274$	0	0
$275$	−2.99603e8	−0.868723
$276$	0	0
$277$	2.02562e7	0.0572636	0.0286318	−	0.999590i	$-0.490885\pi$
$277$	2.02562e7	0.0572636	0.0286318	+	0.999590i	$0.490885\pi$
$278$	0	0
$279$	−4.91012e7	−0.135356
$280$	0	0
$281$	−2.24402e8	−0.603330	−0.301665	−	0.953414i	$-0.597542\pi$
$281$	−2.24402e8	−0.603330	−0.301665	+	0.953414i	$0.597542\pi$
$282$	0	0
$283$	−5.79761e8	−1.52054	−0.760268	−	0.649610i	$-0.774932\pi$
$283$	−5.79761e8	−1.52054	−0.760268	+	0.649610i	$0.774932\pi$
$284$	0	0
$285$	−5.75021e7	−0.147139
$286$	0	0
$287$	−2.27426e8	−0.567876
$288$	0	0
$289$	5.77758e8	1.40800
$290$	0	0
$291$	3.83321e7	0.0911879
$292$	0	0
$293$	−9.69369e7	−0.225140	−0.112570	−	0.993644i	$-0.535908\pi$
$293$	−9.69369e7	−0.225140	−0.112570	+	0.993644i	$0.535908\pi$
$294$	0	0
$295$	−8.57846e7	−0.194551
$296$	0	0
$297$	4.15822e8	0.921000
$298$	0	0
$299$	−1.07065e8	−0.231631
$300$	0	0
$301$	−1.15221e8	−0.243527
$302$	0	0
$303$	1.01886e8	0.210409
$304$	0	0
$305$	−1.87242e8	−0.377880
$306$	0	0
$307$	−6.23962e8	−1.23076	−0.615381	−	0.788230i	$-0.710998\pi$
$307$	−6.23962e8	−1.23076	−0.615381	+	0.788230i	$0.710998\pi$
$308$	0	0
$309$	−2.52463e8	−0.486792
$310$	0	0
$311$	4.83166e8	0.910824	0.455412	−	0.890281i	$-0.349492\pi$
$311$	4.83166e8	0.910824	0.455412	+	0.890281i	$0.349492\pi$
$312$	0	0
$313$	−4.88641e8	−0.900709	−0.450354	−	0.892850i	$-0.648702\pi$
$313$	−4.88641e8	−0.900709	−0.450354	+	0.892850i	$0.648702\pi$
$314$	0	0
$315$	1.02241e8	0.184306
$316$	0	0
$317$	5.89516e8	1.03941	0.519707	−	0.854345i	$-0.326041\pi$
$317$	5.89516e8	1.03941	0.519707	+	0.854345i	$0.326041\pi$
$318$	0	0
$319$	1.10301e9	1.90244
$320$	0	0
$321$	3.73121e7	0.0629624
$322$	0	0
$323$	6.27611e8	1.03629
$324$	0	0
$325$	7.29047e7	0.117805
$326$	0	0
$327$	−1.23704e8	−0.195645
$328$	0	0
$329$	−3.84096e8	−0.594639
$330$	0	0
$331$	−4.96907e8	−0.753143	−0.376571	−	0.926388i	$-0.622897\pi$
$331$	−4.96907e8	−0.753143	−0.376571	+	0.926388i	$0.622897\pi$
$332$	0	0
$333$	3.80697e8	0.564968
$334$	0	0
$335$	−6.22506e8	−0.904662
$336$	0	0
$337$	−8.00789e8	−1.13976	−0.569880	−	0.821728i	$-0.693010\pi$
$337$	−8.00789e8	−1.13976	−0.569880	+	0.821728i	$0.693010\pi$
$338$	0	0
$339$	−4.09387e8	−0.570736
$340$	0	0
$341$	1.50335e8	0.205314
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	0	0
$345$	2.22152e8	0.291261
$346$	0	0
$347$	9.36814e8	1.20365	0.601825	−	0.798628i	$-0.294440\pi$
$347$	9.36814e8	1.20365	0.601825	+	0.798628i	$0.294440\pi$
$348$	0	0
$349$	3.18551e8	0.401135	0.200567	−	0.979680i	$-0.435721\pi$
$349$	3.18551e8	0.401135	0.200567	+	0.979680i	$0.435721\pi$
$350$	0	0
$351$	−1.01185e8	−0.124894
$352$	0	0
$353$	1.47778e9	1.78813	0.894065	−	0.447938i	$-0.147841\pi$
$353$	1.47778e9	1.78813	0.894065	+	0.447938i	$0.147841\pi$
$354$	0	0
$355$	4.00855e8	0.475541
$356$	0	0
$357$	1.94074e8	0.225750
$358$	0	0
$359$	7.63543e8	0.870969	0.435485	−	0.900196i	$-0.356577\pi$
$359$	7.63543e8	0.870969	0.435485	+	0.900196i	$0.356577\pi$
$360$	0	0
$361$	−4.95231e8	−0.554029
$362$	0	0
$363$	−2.34872e8	−0.257726
$364$	0	0
$365$	2.29611e8	0.247154
$366$	0	0
$367$	3.62182e8	0.382469	0.191234	−	0.981544i	$-0.438751\pi$
$367$	3.62182e8	0.382469	0.191234	+	0.981544i	$0.438751\pi$
$368$	0	0
$369$	1.23526e9	1.27987
$370$	0	0
$371$	−3.86842e7	−0.0393301
$372$	0	0
$373$	−4.93379e8	−0.492266	−0.246133	−	0.969236i	$-0.579160\pi$
$373$	−4.93379e8	−0.492266	−0.246133	+	0.969236i	$0.579160\pi$
$374$	0	0
$375$	−3.76272e8	−0.368462
$376$	0	0
$377$	−2.68403e8	−0.257984
$378$	0	0
$379$	1.76712e9	1.66735	0.833677	−	0.552252i	$-0.186231\pi$
$379$	1.76712e9	1.66735	0.833677	+	0.552252i	$0.186231\pi$
$380$	0	0
$381$	−4.53301e8	−0.419903
$382$	0	0
$383$	−1.83455e6	−0.00166853	−0.000834264	−	1.00000i	$-0.500266\pi$
$383$	−1.83455e6	−0.00166853	−0.000834264		1.00000i	$0.500266\pi$
$384$	0	0
$385$	−3.13036e8	−0.279564
$386$	0	0
$387$	6.25819e8	0.548858
$388$	0	0
$389$	1.74462e9	1.50272	0.751359	−	0.659894i	$-0.229399\pi$
$389$	1.74462e9	1.50272	0.751359	+	0.659894i	$0.229399\pi$
$390$	0	0
$391$	−2.42469e9	−2.05134
$392$	0	0
$393$	3.13537e8	0.260564
$394$	0	0
$395$	2.82458e7	0.0230602
$396$	0	0
$397$	−2.11474e9	−1.69625	−0.848124	−	0.529798i	$-0.822268\pi$
$397$	−2.11474e9	−1.69625	−0.848124	+	0.529798i	$0.822268\pi$
$398$	0	0
$399$	1.23270e8	0.0971522
$400$	0	0
$401$	−1.91213e9	−1.48085	−0.740426	−	0.672138i	$-0.765376\pi$
$401$	−1.91213e9	−1.48085	−0.740426	+	0.672138i	$0.765376\pi$
$402$	0	0
$403$	−3.65821e7	−0.0278421
$404$	0	0
$405$	−4.41949e8	−0.330582
$406$	0	0
$407$	−1.16559e9	−0.856969
$408$	0	0
$409$	−1.25751e9	−0.908824	−0.454412	−	0.890792i	$-0.650151\pi$
$409$	−1.25751e9	−0.908824	−0.454412	+	0.890792i	$0.650151\pi$
$410$	0	0
$411$	−8.09949e8	−0.575455
$412$	0	0
$413$	1.83901e8	0.128457
$414$	0	0
$415$	9.93860e8	0.682585
$416$	0	0
$417$	−7.68357e8	−0.518904
$418$	0	0
$419$	1.16244e9	0.772010	0.386005	−	0.922497i	$-0.373855\pi$
$419$	1.16244e9	0.772010	0.386005	+	0.922497i	$0.373855\pi$
$420$	0	0
$421$	−1.16212e9	−0.759039	−0.379519	−	0.925184i	$-0.623911\pi$
$421$	−1.16212e9	−0.759039	−0.379519	+	0.925184i	$0.623911\pi$
$422$	0	0
$423$	2.08621e9	1.34019
$424$	0	0
$425$	1.65107e9	1.04329
$426$	0	0
$427$	4.01401e8	0.249506
$428$	0	0
$429$	1.42509e8	0.0871447
$430$	0	0
$431$	−1.81487e9	−1.09188	−0.545942	−	0.837823i	$-0.683828\pi$
$431$	−1.81487e9	−1.09188	−0.545942	+	0.837823i	$0.683828\pi$
$432$	0	0
$433$	−1.99774e9	−1.18258	−0.591292	−	0.806458i	$-0.701382\pi$
$433$	−1.99774e9	−1.18258	−0.591292	+	0.806458i	$0.701382\pi$
$434$	0	0
$435$	5.56917e8	0.324398
$436$	0	0
$437$	−1.54010e9	−0.882802
$438$	0	0
$439$	−3.34073e9	−1.88458	−0.942291	−	0.334794i	$-0.891333\pi$
$439$	−3.34073e9	−1.88458	−0.942291	+	0.334794i	$0.891333\pi$
$440$	0	0
$441$	−2.19180e8	−0.121693
$442$	0	0
$443$	3.83408e8	0.209531	0.104766	−	0.994497i	$-0.466591\pi$
$443$	3.83408e8	0.209531	0.104766	+	0.994497i	$0.466591\pi$
$444$	0	0
$445$	7.56650e8	0.407038
$446$	0	0
$447$	7.54855e8	0.399749
$448$	0	0
$449$	−2.18623e9	−1.13982	−0.569908	−	0.821709i	$-0.693021\pi$
$449$	−2.18623e9	−1.13982	−0.569908	+	0.821709i	$0.693021\pi$
$450$	0	0
$451$	−3.78204e9	−1.94137
$452$	0	0
$453$	−1.36816e8	−0.0691500
$454$	0	0
$455$	7.61734e7	0.0379109
$456$	0	0
$457$	1.48419e9	0.727416	0.363708	−	0.931513i	$-0.381511\pi$
$457$	1.48419e9	0.727416	0.363708	+	0.931513i	$0.381511\pi$
$458$	0	0
$459$	−2.29154e9	−1.10607
$460$	0	0
$461$	1.31392e9	0.624621	0.312311	−	0.949980i	$-0.398897\pi$
$461$	1.31392e9	0.624621	0.312311	+	0.949980i	$0.398897\pi$
$462$	0	0
$463$	1.45649e9	0.681984	0.340992	−	0.940066i	$-0.389237\pi$
$463$	1.45649e9	0.681984	0.340992	+	0.940066i	$0.389237\pi$
$464$	0	0
$465$	7.59053e7	0.0350096
$466$	0	0
$467$	3.40374e9	1.54649	0.773244	−	0.634108i	$-0.218633\pi$
$467$	3.40374e9	1.54649	0.773244	+	0.634108i	$0.218633\pi$
$468$	0	0
$469$	1.33450e9	0.597327
$470$	0	0
$471$	1.52316e9	0.671694
$472$	0	0
$473$	−1.91609e9	−0.832533
$474$	0	0
$475$	1.04871e9	0.448983
$476$	0	0
$477$	2.10113e8	0.0886417
$478$	0	0
$479$	3.44352e9	1.43162	0.715811	−	0.698294i	$-0.246057\pi$
$479$	3.44352e9	1.43162	0.715811	+	0.698294i	$0.246057\pi$
$480$	0	0
$481$	2.83632e8	0.116211
$482$	0	0
$483$	−4.76238e8	−0.192313
$484$	0	0
$485$	3.40730e8	0.135617
$486$	0	0
$487$	1.21791e9	0.477818	0.238909	−	0.971042i	$-0.423210\pi$
$487$	1.21791e9	0.477818	0.238909	+	0.971042i	$0.423210\pi$
$488$	0	0
$489$	−2.78742e8	−0.107800
$490$	0	0
$491$	7.94149e7	0.0302773	0.0151386	−	0.999885i	$-0.495181\pi$
$491$	7.94149e7	0.0302773	0.0151386	+	0.999885i	$0.495181\pi$
$492$	0	0
$493$	−6.07852e9	−2.28472
$494$	0	0
$495$	1.70025e9	0.630078
$496$	0	0
$497$	−8.59333e8	−0.313989
$498$	0	0
$499$	−5.13507e9	−1.85010	−0.925048	−	0.379850i	$-0.875976\pi$
$499$	−5.13507e9	−1.85010	−0.925048	+	0.379850i	$0.875976\pi$
$500$	0	0
$501$	−8.08008e8	−0.287067
$502$	0	0
$503$	6.91853e8	0.242396	0.121198	−	0.992628i	$-0.461326\pi$
$503$	6.91853e8	0.242396	0.121198	+	0.992628i	$0.461326\pi$
$504$	0	0
$505$	9.05651e8	0.312926
$506$	0	0
$507$	1.09480e9	0.373083
$508$	0	0
$509$	−1.84233e9	−0.619234	−0.309617	−	0.950861i	$-0.600201\pi$
$509$	−1.84233e9	−0.619234	−0.309617	+	0.950861i	$0.600201\pi$
$510$	0	0
$511$	−4.92229e8	−0.163190
$512$	0	0
$513$	−1.45552e9	−0.476001
$514$	0	0
$515$	−2.24412e9	−0.723970
$516$	0	0
$517$	−6.38741e9	−2.03286
$518$	0	0
$519$	−1.85219e9	−0.581568
$520$	0	0
$521$	−4.74854e9	−1.47105	−0.735527	−	0.677496i	$-0.763065\pi$
$521$	−4.74854e9	−1.47105	−0.735527	+	0.677496i	$0.763065\pi$
$522$	0	0
$523$	−2.41943e9	−0.739533	−0.369766	−	0.929125i	$-0.620562\pi$
$523$	−2.41943e9	−0.739533	−0.369766	+	0.929125i	$0.620562\pi$
$524$	0	0
$525$	3.24289e8	0.0978082
$526$	0	0
$527$	−8.28475e8	−0.246571
$528$	0	0
$529$	2.54514e9	0.747509
$530$	0	0
$531$	−9.98855e8	−0.289515
$532$	0	0
$533$	9.20313e8	0.263263
$534$	0	0
$535$	3.31663e8	0.0936394
$536$	0	0
$537$	−9.03204e8	−0.251696
$538$	0	0
$539$	6.71070e8	0.184590
$540$	0	0
$541$	−4.41722e9	−1.19938	−0.599692	−	0.800231i	$-0.704710\pi$
$541$	−4.41722e9	−1.19938	−0.599692	+	0.800231i	$0.704710\pi$
$542$	0	0
$543$	−2.38983e9	−0.640573
$544$	0	0
$545$	−1.09960e9	−0.290968
$546$	0	0
$547$	3.48875e9	0.911412	0.455706	−	0.890130i	$-0.349387\pi$
$547$	3.48875e9	0.911412	0.455706	+	0.890130i	$0.349387\pi$
$548$	0	0
$549$	−2.18020e9	−0.562333
$550$	0	0
$551$	−3.86091e9	−0.983238
$552$	0	0
$553$	−6.05518e7	−0.0152261
$554$	0	0
$555$	−5.88516e8	−0.146128
$556$	0	0
$557$	2.78374e9	0.682551	0.341276	−	0.939963i	$-0.389141\pi$
$557$	2.78374e9	0.682551	0.341276	+	0.939963i	$0.389141\pi$
$558$	0	0
$559$	4.66257e8	0.112897
$560$	0	0
$561$	3.22739e9	0.771759
$562$	0	0
$563$	−2.46624e9	−0.582446	−0.291223	−	0.956655i	$-0.594062\pi$
$563$	−2.46624e9	−0.582446	−0.291223	+	0.956655i	$0.594062\pi$
$564$	0	0
$565$	−3.63899e9	−0.848813
$566$	0	0
$567$	9.47428e8	0.218276
$568$	0	0
$569$	−3.52045e8	−0.0801133	−0.0400567	−	0.999197i	$-0.512754\pi$
$569$	−3.52045e8	−0.0801133	−0.0400567	+	0.999197i	$0.512754\pi$
$570$	0	0
$571$	7.44085e9	1.67262	0.836308	−	0.548260i	$-0.184709\pi$
$571$	7.44085e9	1.67262	0.836308	+	0.548260i	$0.184709\pi$
$572$	0	0
$573$	2.59158e9	0.575472
$574$	0	0
$575$	−4.05157e9	−0.888762
$576$	0	0
$577$	3.05727e9	0.662550	0.331275	−	0.943534i	$-0.392521\pi$
$577$	3.05727e9	0.662550	0.331275	+	0.943534i	$0.392521\pi$
$578$	0	0
$579$	−1.65066e9	−0.353413
$580$	0	0
$581$	−2.13059e9	−0.450695
$582$	0	0
$583$	−6.43309e8	−0.134456
$584$	0	0
$585$	−4.13735e8	−0.0854431
$586$	0	0
$587$	2.88379e9	0.588478	0.294239	−	0.955732i	$-0.404934\pi$
$587$	2.88379e9	0.588478	0.294239	+	0.955732i	$0.404934\pi$
$588$	0	0
$589$	−5.26224e8	−0.106113
$590$	0	0
$591$	−1.99629e9	−0.397803
$592$	0	0
$593$	−4.70076e9	−0.925714	−0.462857	−	0.886433i	$-0.653176\pi$
$593$	−4.70076e9	−0.925714	−0.462857	+	0.886433i	$0.653176\pi$
$594$	0	0
$595$	1.72510e9	0.335741
$596$	0	0
$597$	−2.91039e9	−0.559810
$598$	0	0
$599$	4.06323e9	0.772464	0.386232	−	0.922402i	$-0.373776\pi$
$599$	4.06323e9	0.772464	0.386232	+	0.922402i	$0.373776\pi$
$600$	0	0
$601$	−2.06956e9	−0.388881	−0.194441	−	0.980914i	$-0.562289\pi$
$601$	−2.06956e9	−0.388881	−0.194441	+	0.980914i	$0.562289\pi$
$602$	0	0
$603$	−7.24830e9	−1.34625
$604$	0	0
$605$	−2.08775e9	−0.383297
$606$	0	0
$607$	5.56828e9	1.01056	0.505278	−	0.862956i	$-0.331390\pi$
$607$	5.56828e9	1.01056	0.505278	+	0.862956i	$0.331390\pi$
$608$	0	0
$609$	−1.19389e9	−0.214192
$610$	0	0
$611$	1.55430e9	0.275671
$612$	0	0
$613$	3.79664e9	0.665714	0.332857	−	0.942977i	$-0.391987\pi$
$613$	3.79664e9	0.665714	0.332857	+	0.942977i	$0.391987\pi$
$614$	0	0
$615$	−1.90958e9	−0.331036
$616$	0	0
$617$	−3.96235e9	−0.679133	−0.339566	−	0.940582i	$-0.610280\pi$
$617$	−3.96235e9	−0.679133	−0.339566	+	0.940582i	$0.610280\pi$
$618$	0	0
$619$	5.55082e9	0.940676	0.470338	−	0.882486i	$-0.344132\pi$
$619$	5.55082e9	0.940676	0.470338	+	0.882486i	$0.344132\pi$
$620$	0	0
$621$	5.62321e9	0.942245
$622$	0	0
$623$	−1.62207e9	−0.268758
$624$	0	0
$625$	7.58876e8	0.124334
$626$	0	0
$627$	2.04995e9	0.332129
$628$	0	0
$629$	6.42341e9	1.02917
$630$	0	0
$631$	2.43356e8	0.0385602	0.0192801	−	0.999814i	$-0.493863\pi$
$631$	2.43356e8	0.0385602	0.0192801	+	0.999814i	$0.493863\pi$
$632$	0	0
$633$	−3.14589e7	−0.00492981
$634$	0	0
$635$	−4.02934e9	−0.624491
$636$	0	0
$637$	−1.63297e8	−0.0250317
$638$	0	0
$639$	4.66746e9	0.707664
$640$	0	0
$641$	−3.89227e9	−0.583714	−0.291857	−	0.956462i	$-0.594273\pi$
$641$	−3.89227e9	−0.583714	−0.291857	+	0.956462i	$0.594273\pi$
$642$	0	0
$643$	2.22698e9	0.330353	0.165177	−	0.986264i	$-0.447181\pi$
$643$	2.22698e9	0.330353	0.165177	+	0.986264i	$0.447181\pi$
$644$	0	0
$645$	−9.67450e8	−0.141961
$646$	0	0
$647$	−7.46182e9	−1.08313	−0.541564	−	0.840659i	$-0.682168\pi$
$647$	−7.46182e9	−1.08313	−0.541564	+	0.840659i	$0.682168\pi$
$648$	0	0
$649$	3.05822e9	0.439150
$650$	0	0
$651$	−1.62722e8	−0.0231160
$652$	0	0
$653$	−1.27961e8	−0.0179839	−0.00899193	−	0.999960i	$-0.502862\pi$
$653$	−1.27961e8	−0.0179839	−0.00899193	+	0.999960i	$0.502862\pi$
$654$	0	0
$655$	2.78700e9	0.387518
$656$	0	0
$657$	2.67354e9	0.367796
$658$	0	0
$659$	−1.10052e10	−1.49796	−0.748978	−	0.662595i	$-0.769455\pi$
$659$	−1.10052e10	−1.49796	−0.748978	+	0.662595i	$0.769455\pi$
$660$	0	0
$661$	1.22326e10	1.64745	0.823727	−	0.566987i	$-0.191891\pi$
$661$	1.22326e10	1.64745	0.823727	+	0.566987i	$0.191891\pi$
$662$	0	0
$663$	−7.85347e8	−0.104656
$664$	0	0
$665$	1.09573e9	0.144487
$666$	0	0
$667$	1.49161e10	1.94632
$668$	0	0
$669$	−8.69514e8	−0.112276
$670$	0	0
$671$	6.67519e9	0.852972
$672$	0	0
$673$	−1.20862e10	−1.52840	−0.764201	−	0.644978i	$-0.776867\pi$
$673$	−1.20862e10	−1.52840	−0.764201	+	0.644978i	$0.776867\pi$
$674$	0	0
$675$	−3.82907e9	−0.479215
$676$	0	0
$677$	1.16275e10	1.44021	0.720104	−	0.693867i	$-0.244094\pi$
$677$	1.16275e10	1.44021	0.720104	+	0.693867i	$0.244094\pi$
$678$	0	0
$679$	−7.30440e8	−0.0895448
$680$	0	0
$681$	−3.69534e9	−0.448373
$682$	0	0
$683$	1.42381e10	1.70994	0.854970	−	0.518678i	$-0.173576\pi$
$683$	1.42381e10	1.70994	0.854970	+	0.518678i	$0.173576\pi$
$684$	0	0
$685$	−7.19955e9	−0.855832
$686$	0	0
$687$	−5.05994e9	−0.595384
$688$	0	0
$689$	1.56541e8	0.0182332
$690$	0	0
$691$	−9.50918e9	−1.09640	−0.548201	−	0.836347i	$-0.684687\pi$
$691$	−9.50918e9	−1.09640	−0.548201	+	0.836347i	$0.684687\pi$
$692$	0	0
$693$	−3.64491e9	−0.416026
$694$	0	0
$695$	−6.82984e9	−0.771727
$696$	0	0
$697$	2.08423e10	2.33148
$698$	0	0
$699$	−5.55288e9	−0.614961
$700$	0	0
$701$	9.29838e9	1.01952	0.509758	−	0.860318i	$-0.329735\pi$
$701$	9.29838e9	1.01952	0.509758	+	0.860318i	$0.329735\pi$
$702$	0	0
$703$	4.07997e9	0.442908
$704$	0	0
$705$	−3.22506e9	−0.346638
$706$	0	0
$707$	−1.94149e9	−0.206618
$708$	0	0
$709$	1.42059e10	1.49695	0.748476	−	0.663162i	$-0.230786\pi$
$709$	1.42059e10	1.49695	0.748476	+	0.663162i	$0.230786\pi$
$710$	0	0
$711$	3.28887e8	0.0343165
$712$	0	0
$713$	2.03300e9	0.210050
$714$	0	0
$715$	1.26674e9	0.129604
$716$	0	0
$717$	−2.37210e9	−0.240335
$718$	0	0
$719$	1.13805e10	1.14186	0.570928	−	0.821000i	$-0.306584\pi$
$719$	1.13805e10	1.14186	0.570928	+	0.821000i	$0.306584\pi$
$720$	0	0
$721$	4.81083e9	0.478021
$722$	0	0
$723$	−5.03504e9	−0.495472
$724$	0	0
$725$	−1.01570e10	−0.989877
$726$	0	0
$727$	2.05150e9	0.198016	0.0990079	−	0.995087i	$-0.468433\pi$
$727$	2.05150e9	0.198016	0.0990079	+	0.995087i	$0.468433\pi$
$728$	0	0
$729$	−2.27616e9	−0.217599
$730$	0	0
$731$	1.05593e10	0.999827
$732$	0	0
$733$	2.80160e9	0.262750	0.131375	−	0.991333i	$-0.458061\pi$
$733$	2.80160e9	0.262750	0.131375	+	0.991333i	$0.458061\pi$
$734$	0	0
$735$	3.38829e8	0.0314757
$736$	0	0
$737$	2.21923e10	2.04205
$738$	0	0
$739$	−1.17528e10	−1.07124	−0.535620	−	0.844459i	$-0.679922\pi$
$739$	−1.17528e10	−1.07124	−0.535620	+	0.844459i	$0.679922\pi$
$740$	0	0
$741$	−4.98831e8	−0.0450391
$742$	0	0
$743$	−1.27682e10	−1.14201	−0.571005	−	0.820947i	$-0.693446\pi$
$743$	−1.27682e10	−1.14201	−0.571005	+	0.820947i	$0.693446\pi$
$744$	0	0
$745$	6.70982e9	0.594517
$746$	0	0
$747$	1.15723e10	1.01577
$748$	0	0
$749$	−7.11002e8	−0.0618279
$750$	0	0
$751$	−8.94826e9	−0.770901	−0.385451	−	0.922728i	$-0.625954\pi$
$751$	−8.94826e9	−0.770901	−0.385451	+	0.922728i	$0.625954\pi$
$752$	0	0
$753$	−2.23584e9	−0.190835
$754$	0	0
$755$	−1.21614e9	−0.102842
$756$	0	0
$757$	−1.12241e10	−0.940407	−0.470203	−	0.882558i	$-0.655819\pi$
$757$	−1.12241e10	−0.940407	−0.470203	+	0.882558i	$0.655819\pi$
$758$	0	0
$759$	−7.91971e9	−0.657450
$760$	0	0
$761$	−3.06542e9	−0.252141	−0.126070	−	0.992021i	$-0.540237\pi$
$761$	−3.06542e9	−0.252141	−0.126070	+	0.992021i	$0.540237\pi$
$762$	0	0
$763$	2.35726e9	0.192119
$764$	0	0
$765$	−9.36985e9	−0.756689
$766$	0	0
$767$	−7.44182e8	−0.0595518
$768$	0	0
$769$	−1.33911e10	−1.06188	−0.530939	−	0.847410i	$-0.678161\pi$
$769$	−1.33911e10	−1.06188	−0.530939	+	0.847410i	$0.678161\pi$
$770$	0	0
$771$	−6.90628e9	−0.542692
$772$	0	0
$773$	2.07529e9	0.161604	0.0808018	−	0.996730i	$-0.474252\pi$
$773$	2.07529e9	0.161604	0.0808018	+	0.996730i	$0.474252\pi$
$774$	0	0
$775$	−1.38435e9	−0.106829
$776$	0	0
$777$	1.26163e9	0.0964849
$778$	0	0
$779$	1.32385e10	1.00336
$780$	0	0
$781$	−1.42905e10	−1.07342
$782$	0	0
$783$	1.40970e10	1.04944
$784$	0	0
$785$	1.35392e10	0.998960
$786$	0	0
$787$	1.02989e10	0.753145	0.376572	−	0.926387i	$-0.377103\pi$
$787$	1.02989e10	0.753145	0.376572	+	0.926387i	$0.377103\pi$
$788$	0	0
$789$	8.18710e9	0.593418
$790$	0	0
$791$	7.80110e9	0.560451
$792$	0	0
$793$	−1.62433e9	−0.115669
$794$	0	0
$795$	−3.24812e8	−0.0229270
$796$	0	0
$797$	−6.69270e9	−0.468271	−0.234135	−	0.972204i	$-0.575226\pi$
$797$	−6.69270e9	−0.468271	−0.234135	+	0.972204i	$0.575226\pi$
$798$	0	0
$799$	3.52002e10	2.44136
$800$	0	0
$801$	8.81024e9	0.605723
$802$	0	0
$803$	−8.18564e9	−0.557890
$804$	0	0
$805$	−4.23322e9	−0.286013
$806$	0	0
$807$	1.47429e9	0.0987476
$808$	0	0
$809$	1.56291e10	1.03780	0.518901	−	0.854834i	$-0.326341\pi$
$809$	1.56291e10	1.03780	0.518901	+	0.854834i	$0.326341\pi$
$810$	0	0
$811$	−3.31300e9	−0.218096	−0.109048	−	0.994036i	$-0.534780\pi$
$811$	−3.31300e9	−0.218096	−0.109048	+	0.994036i	$0.534780\pi$
$812$	0	0
$813$	−4.35019e9	−0.283917
$814$	0	0
$815$	−2.47770e9	−0.160324
$816$	0	0
$817$	6.70698e9	0.430279
$818$	0	0
$819$	8.86944e8	0.0564161
$820$	0	0
$821$	1.79884e10	1.13446	0.567232	−	0.823558i	$-0.308014\pi$
$821$	1.79884e10	1.13446	0.567232	+	0.823558i	$0.308014\pi$
$822$	0	0
$823$	−9.56219e9	−0.597940	−0.298970	−	0.954262i	$-0.596643\pi$
$823$	−9.56219e9	−0.597940	−0.298970	+	0.954262i	$0.596643\pi$
$824$	0	0
$825$	5.39285e9	0.334372
$826$	0	0
$827$	−1.65562e10	−1.01787	−0.508934	−	0.860805i	$-0.669960\pi$
$827$	−1.65562e10	−1.01787	−0.508934	+	0.860805i	$0.669960\pi$
$828$	0	0
$829$	3.10473e10	1.89271	0.946353	−	0.323134i	$-0.104736\pi$
$829$	3.10473e10	1.89271	0.946353	+	0.323134i	$0.104736\pi$
$830$	0	0
$831$	−3.64611e8	−0.0220408
$832$	0	0
$833$	−3.69818e9	−0.221682
$834$	0	0
$835$	−7.18229e9	−0.426934
$836$	0	0
$837$	1.92135e9	0.113258
$838$	0	0
$839$	−1.11255e10	−0.650358	−0.325179	−	0.945652i	$-0.605425\pi$
$839$	−1.11255e10	−0.650358	−0.325179	+	0.945652i	$0.605425\pi$
$840$	0	0
$841$	2.01436e10	1.16775
$842$	0	0
$843$	4.03924e9	0.232222
$844$	0	0
$845$	9.73152e9	0.554858
$846$	0	0
$847$	4.47562e9	0.253082
$848$	0	0
$849$	1.04357e10	0.585254
$850$	0	0
$851$	−1.57624e10	−0.876738
$852$	0	0
$853$	1.12283e10	0.619430	0.309715	−	0.950829i	$-0.399766\pi$
$853$	1.12283e10	0.619430	0.309715	+	0.950829i	$0.399766\pi$
$854$	0	0
$855$	−5.95147e9	−0.325644
$856$	0	0
$857$	−1.34110e10	−0.727828	−0.363914	−	0.931433i	$-0.618560\pi$
$857$	−1.34110e10	−0.727828	−0.363914	+	0.931433i	$0.618560\pi$
$858$	0	0
$859$	−3.30638e10	−1.77982	−0.889912	−	0.456133i	$-0.849234\pi$
$859$	−3.30638e10	−1.77982	−0.889912	+	0.456133i	$0.849234\pi$
$860$	0	0
$861$	4.09367e9	0.218576
$862$	0	0
$863$	1.86096e10	0.985598	0.492799	−	0.870143i	$-0.335974\pi$
$863$	1.86096e10	0.985598	0.492799	+	0.870143i	$0.335974\pi$
$864$	0	0
$865$	−1.64639e10	−0.864923
$866$	0	0
$867$	−1.03996e10	−0.541940
$868$	0	0
$869$	−1.00696e9	−0.0520527
$870$	0	0
$871$	−5.40024e9	−0.276917
$872$	0	0
$873$	3.96737e9	0.201815
$874$	0	0
$875$	7.17007e9	0.361822
$876$	0	0
$877$	2.01231e10	1.00739	0.503694	−	0.863882i	$-0.331974\pi$
$877$	2.01231e10	1.00739	0.503694	+	0.863882i	$0.331974\pi$
$878$	0	0
$879$	1.74486e9	0.0866564
$880$	0	0
$881$	3.55752e10	1.75280	0.876398	−	0.481587i	$-0.159939\pi$
$881$	3.55752e10	1.75280	0.876398	+	0.481587i	$0.159939\pi$
$882$	0	0
$883$	4.88663e8	0.0238862	0.0119431	−	0.999929i	$-0.496198\pi$
$883$	4.88663e8	0.0238862	0.0119431	+	0.999929i	$0.496198\pi$
$884$	0	0
$885$	1.54412e9	0.0748825
$886$	0	0
$887$	−3.66732e10	−1.76448	−0.882238	−	0.470804i	$-0.843964\pi$
$887$	−3.66732e10	−1.76448	−0.882238	+	0.470804i	$0.843964\pi$
$888$	0	0
$889$	8.63790e9	0.412337
$890$	0	0
$891$	1.57555e10	0.746208
$892$	0	0
$893$	2.23582e10	1.05065
$894$	0	0
$895$	−8.02848e9	−0.374328
$896$	0	0
$897$	1.92717e9	0.0891550
$898$	0	0
$899$	5.09657e9	0.233948
$900$	0	0
$901$	3.54519e9	0.161474
$902$	0	0
$903$	2.07397e9	0.0937336
$904$	0	0
$905$	−2.12430e10	−0.952676
$906$	0	0
$907$	−9.61954e9	−0.428084	−0.214042	−	0.976824i	$-0.568663\pi$
$907$	−9.61954e9	−0.428084	−0.214042	+	0.976824i	$0.568663\pi$
$908$	0	0
$909$	1.05452e10	0.465672
$910$	0	0
$911$	1.36974e10	0.600239	0.300119	−	0.953902i	$-0.402973\pi$
$911$	1.36974e10	0.600239	0.300119	+	0.953902i	$0.402973\pi$
$912$	0	0
$913$	−3.54311e10	−1.54077
$914$	0	0
$915$	3.37036e9	0.145446
$916$	0	0
$917$	−5.97463e9	−0.255869
$918$	0	0
$919$	9.66864e9	0.410923	0.205462	−	0.978665i	$-0.434130\pi$
$919$	9.66864e9	0.410923	0.205462	+	0.978665i	$0.434130\pi$
$920$	0	0
$921$	1.12313e10	0.473720
$922$	0	0
$923$	3.47742e9	0.145563
$924$	0	0
$925$	1.07333e10	0.445899
$926$	0	0
$927$	−2.61300e10	−1.07736
$928$	0	0
$929$	5.56916e9	0.227895	0.113948	−	0.993487i	$-0.463650\pi$
$929$	5.56916e9	0.227895	0.113948	+	0.993487i	$0.463650\pi$
$930$	0	0
$931$	−2.34898e9	−0.0954015
$932$	0	0
$933$	−8.69698e9	−0.350576
$934$	0	0
$935$	2.86879e10	1.14778
$936$	0	0
$937$	3.10166e9	0.123170	0.0615850	−	0.998102i	$-0.480384\pi$
$937$	3.10166e9	0.123170	0.0615850	+	0.998102i	$0.480384\pi$
$938$	0	0
$939$	8.79553e9	0.346683
$940$	0	0
$941$	1.16729e10	0.456683	0.228341	−	0.973581i	$-0.426670\pi$
$941$	1.16729e10	0.456683	0.228341	+	0.973581i	$0.426670\pi$
$942$	0	0
$943$	−5.11450e10	−1.98615
$944$	0	0
$945$	−4.00075e9	−0.154216
$946$	0	0
$947$	4.15512e10	1.58986	0.794929	−	0.606703i	$-0.207508\pi$
$947$	4.15512e10	1.58986	0.794929	+	0.606703i	$0.207508\pi$
$948$	0	0
$949$	1.99188e9	0.0756538
$950$	0	0
$951$	−1.06113e10	−0.400070
$952$	0	0
$953$	1.50143e10	0.561926	0.280963	−	0.959719i	$-0.409346\pi$
$953$	1.50143e10	0.561926	0.280963	+	0.959719i	$0.409346\pi$
$954$	0	0
$955$	2.30363e10	0.855856
$956$	0	0
$957$	−1.98541e10	−0.732248
$958$	0	0
$959$	1.54340e10	0.565086
$960$	0	0
$961$	−2.68180e10	−0.974752
$962$	0	0
$963$	3.86180e9	0.139347
$964$	0	0
$965$	−1.46725e10	−0.525605
$966$	0	0
$967$	−3.06060e10	−1.08846	−0.544232	−	0.838935i	$-0.683179\pi$
$967$	−3.06060e10	−1.08846	−0.544232	+	0.838935i	$0.683179\pi$
$968$	0	0
$969$	−1.12970e10	−0.398869
$970$	0	0
$971$	4.67597e10	1.63909	0.819547	−	0.573011i	$-0.194225\pi$
$971$	4.67597e10	1.63909	0.819547	+	0.573011i	$0.194225\pi$
$972$	0	0
$973$	1.46415e10	0.509553
$974$	0	0
$975$	−1.31228e9	−0.0453432
$976$	0	0
$977$	1.03915e10	0.356491	0.178246	−	0.983986i	$-0.442958\pi$
$977$	1.03915e10	0.356491	0.178246	+	0.983986i	$0.442958\pi$
$978$	0	0
$979$	−2.69746e10	−0.918788
$980$	0	0
$981$	−1.28034e10	−0.432996
$982$	0	0
$983$	−3.05164e8	−0.0102470	−0.00512350	−	0.999987i	$-0.501631\pi$
$983$	−3.05164e8	−0.0102470	−0.00512350	+	0.999987i	$0.501631\pi$
$984$	0	0
$985$	−1.77448e10	−0.591623
$986$	0	0
$987$	6.91372e9	0.228877
$988$	0	0
$989$	−2.59115e10	−0.851738
$990$	0	0
$991$	5.48412e10	1.78999	0.894993	−	0.446081i	$-0.147181\pi$
$991$	5.48412e10	1.78999	0.894993	+	0.446081i	$0.147181\pi$
$992$	0	0
$993$	8.94433e9	0.289885
$994$	0	0
$995$	−2.58701e10	−0.832563
$996$	0	0
$997$	7.31634e9	0.233809	0.116904	−	0.993143i	$-0.462703\pi$
$997$	7.31634e9	0.233809	0.116904	+	0.993143i	$0.462703\pi$
$998$	0	0
$999$	−1.48968e10	−0.472731

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.8.a.e.1.1		1
4.3	odd	2		448.8.a.f.1.1		1
8.3	odd	2		56.8.a.a.1.1	✓	1
8.5	even	2		112.8.a.b.1.1		1
24.11	even	2		504.8.a.a.1.1		1
56.27	even	2		392.8.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.8.a.a.1.1	✓	1	8.3	odd	2
112.8.a.b.1.1		1	8.5	even	2
392.8.a.c.1.1		1	56.27	even	2
448.8.a.e.1.1		1	1.1	even	1	trivial
448.8.a.f.1.1		1	4.3	odd	2
504.8.a.a.1.1		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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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