LMFDB - Embedded newform 588.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,8,Mod(1,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, 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("588.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	588.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:	$183.682394985$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	588.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-27.0000 q^{3} +240.000 q^{5} +729.000 q^{9} +O(q^{10})$

$q-27.0000 q^{3} +240.000 q^{5} +729.000 q^{9} +702.000 q^{11} +3958.00 q^{13} -6480.00 q^{15} +3408.00 q^{17} +49036.0 q^{19} -11514.0 q^{23} -20525.0 q^{25} -19683.0 q^{27} +49662.0 q^{29} +113320. q^{31} -18954.0 q^{33} -66886.0 q^{37} -106866. q^{39} +360900. q^{41} -765292. q^{43} +174960. q^{45} +1.34488e6 q^{47} -92016.0 q^{51} +358962. q^{53} +168480. q^{55} -1.32397e6 q^{57} -930528. q^{59} +1.31883e6 q^{61} +949920. q^{65} +1.89346e6 q^{67} +310878. q^{69} +227994. q^{71} -784934. q^{73} +554175. q^{75} -2.10089e6 q^{79} +531441. q^{81} -8.62931e6 q^{83} +817920. q^{85} -1.34087e6 q^{87} -5.90310e6 q^{89} -3.05964e6 q^{93} +1.17686e7 q^{95} -773846. q^{97} +511758. q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	240.000	0.858650	0.429325	−	0.903150i	$-0.358751\pi$
$5$	240.000	0.858650	0.429325	+	0.903150i	$0.358751\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	702.000	0.159024	0.0795120	−	0.996834i	$-0.474664\pi$
$11$	702.000	0.159024	0.0795120	+	0.996834i	$0.474664\pi$
$12$	0	0
$13$	3958.00	0.499659	0.249830	−	0.968290i	$-0.419625\pi$
$13$	3958.00	0.499659	0.249830	+	0.968290i	$0.419625\pi$
$14$	0	0
$15$	−6480.00	−0.495742
$16$	0	0
$17$	3408.00	0.168240	0.0841198	−	0.996456i	$-0.473192\pi$
$17$	3408.00	0.168240	0.0841198	+	0.996456i	$0.473192\pi$
$18$	0	0
$19$	49036.0	1.64013	0.820063	−	0.572273i	$-0.193938\pi$
$19$	49036.0	1.64013	0.820063	+	0.572273i	$0.193938\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−11514.0	−0.197323	−0.0986617	−	0.995121i	$-0.531456\pi$
$23$	−11514.0	−0.197323	−0.0986617	+	0.995121i	$0.531456\pi$
$24$	0	0
$25$	−20525.0	−0.262720
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	49662.0	0.378121	0.189061	−	0.981965i	$-0.439456\pi$
$29$	49662.0	0.378121	0.189061	+	0.981965i	$0.439456\pi$
$30$	0	0
$31$	113320.	0.683189	0.341594	−	0.939847i	$-0.389033\pi$
$31$	113320.	0.683189	0.341594	+	0.939847i	$0.389033\pi$
$32$	0	0
$33$	−18954.0	−0.0918126
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−66886.0	−0.217085	−0.108542	−	0.994092i	$-0.534618\pi$
$37$	−66886.0	−0.217085	−0.108542	+	0.994092i	$0.534618\pi$
$38$	0	0
$39$	−106866.	−0.288478
$40$	0	0
$41$	360900.	0.817793	0.408896	−	0.912581i	$-0.365914\pi$
$41$	360900.	0.817793	0.408896	+	0.912581i	$0.365914\pi$
$42$	0	0
$43$	−765292.	−1.46787	−0.733935	−	0.679220i	$-0.762318\pi$
$43$	−765292.	−1.46787	−0.733935	+	0.679220i	$0.762318\pi$
$44$	0	0
$45$	174960.	0.286217
$46$	0	0
$47$	1.34488e6	1.88947	0.944734	−	0.327837i	$-0.106320\pi$
$47$	1.34488e6	1.88947	0.944734	+	0.327837i	$0.106320\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−92016.0	−0.0971332
$52$	0	0
$53$	358962.	0.331194	0.165597	−	0.986193i	$-0.447045\pi$
$53$	358962.	0.331194	0.165597	+	0.986193i	$0.447045\pi$
$54$	0	0
$55$	168480.	0.136546
$56$	0	0
$57$	−1.32397e6	−0.946928
$58$	0	0
$59$	−930528.	−0.589858	−0.294929	−	0.955519i	$-0.595296\pi$
$59$	−930528.	−0.589858	−0.294929	+	0.955519i	$0.595296\pi$
$60$	0	0
$61$	1.31883e6	0.743936	0.371968	−	0.928246i	$-0.378683\pi$
$61$	1.31883e6	0.743936	0.371968	+	0.928246i	$0.378683\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	949920.	0.429033
$66$	0	0
$67$	1.89346e6	0.769122	0.384561	−	0.923100i	$-0.374353\pi$
$67$	1.89346e6	0.769122	0.384561	+	0.923100i	$0.374353\pi$
$68$	0	0
$69$	310878.	0.113925
$70$	0	0
$71$	227994.	0.0755995	0.0377998	−	0.999285i	$-0.487965\pi$
$71$	227994.	0.0755995	0.0377998	+	0.999285i	$0.487965\pi$
$72$	0	0
$73$	−784934.	−0.236158	−0.118079	−	0.993004i	$-0.537674\pi$
$73$	−784934.	−0.236158	−0.118079	+	0.993004i	$0.537674\pi$
$74$	0	0
$75$	554175.	0.151681
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.10089e6	−0.479412	−0.239706	−	0.970846i	$-0.577051\pi$
$79$	−2.10089e6	−0.479412	−0.239706	+	0.970846i	$0.577051\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−8.62931e6	−1.65654	−0.828271	−	0.560327i	$-0.810675\pi$
$83$	−8.62931e6	−1.65654	−0.828271	+	0.560327i	$0.810675\pi$
$84$	0	0
$85$	817920.	0.144459
$86$	0	0
$87$	−1.34087e6	−0.218308
$88$	0	0
$89$	−5.90310e6	−0.887596	−0.443798	−	0.896127i	$-0.646369\pi$
$89$	−5.90310e6	−0.887596	−0.443798	+	0.896127i	$0.646369\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.05964e6	−0.394439
$94$	0	0
$95$	1.17686e7	1.40830
$96$	0	0
$97$	−773846.	−0.0860902	−0.0430451	−	0.999073i	$-0.513706\pi$
$97$	−773846.	−0.0860902	−0.0430451	+	0.999073i	$0.513706\pi$
$98$	0	0
$99$	511758.	0.0530080
$100$	0	0
$101$	−3.33340e6	−0.321930	−0.160965	−	0.986960i	$-0.551461\pi$
$101$	−3.33340e6	−0.321930	−0.160965	+	0.986960i	$0.551461\pi$
$102$	0	0
$103$	5.65905e6	0.510285	0.255143	−	0.966903i	$-0.417878\pi$
$103$	5.65905e6	0.510285	0.255143	+	0.966903i	$0.417878\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.53822e7	1.21388	0.606940	−	0.794748i	$-0.292397\pi$
$107$	1.53822e7	1.21388	0.606940	+	0.794748i	$0.292397\pi$
$108$	0	0
$109$	4.47690e6	0.331120	0.165560	−	0.986200i	$-0.447057\pi$
$109$	4.47690e6	0.331120	0.165560	+	0.986200i	$0.447057\pi$
$110$	0	0
$111$	1.80592e6	0.125334
$112$	0	0
$113$	−9.72279e6	−0.633894	−0.316947	−	0.948443i	$-0.602658\pi$
$113$	−9.72279e6	−0.633894	−0.316947	+	0.948443i	$0.602658\pi$
$114$	0	0
$115$	−2.76336e6	−0.169432
$116$	0	0
$117$	2.88538e6	0.166553
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.89944e7	−0.974711
$122$	0	0
$123$	−9.74430e6	−0.472153
$124$	0	0
$125$	−2.36760e7	−1.08423
$126$	0	0
$127$	1.52949e7	0.662572	0.331286	−	0.943530i	$-0.392517\pi$
$127$	1.52949e7	0.662572	0.331286	+	0.943530i	$0.392517\pi$
$128$	0	0
$129$	2.06629e7	0.847475
$130$	0	0
$131$	−4.36966e7	−1.69824	−0.849119	−	0.528202i	$-0.822867\pi$
$131$	−4.36966e7	−1.69824	−0.849119	+	0.528202i	$0.822867\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.72392e6	−0.165247
$136$	0	0
$137$	−3.00071e6	−0.0997017	−0.0498509	−	0.998757i	$-0.515875\pi$
$137$	−3.00071e6	−0.0997017	−0.0498509	+	0.998757i	$0.515875\pi$
$138$	0	0
$139$	1.73192e7	0.546987	0.273494	−	0.961874i	$-0.411821\pi$
$139$	1.73192e7	0.546987	0.273494	+	0.961874i	$0.411821\pi$
$140$	0	0
$141$	−3.63117e7	−1.09089
$142$	0	0
$143$	2.77852e6	0.0794578
$144$	0	0
$145$	1.19189e7	0.324674
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.90458e7	−0.966990	−0.483495	−	0.875347i	$-0.660633\pi$
$149$	−3.90458e7	−0.966990	−0.483495	+	0.875347i	$0.660633\pi$
$150$	0	0
$151$	−1.06602e7	−0.251968	−0.125984	−	0.992032i	$-0.540209\pi$
$151$	−1.06602e7	−0.251968	−0.125984	+	0.992032i	$0.540209\pi$
$152$	0	0
$153$	2.48443e6	0.0560799
$154$	0	0
$155$	2.71968e7	0.586620
$156$	0	0
$157$	5.52292e7	1.13899	0.569496	−	0.821994i	$-0.307139\pi$
$157$	5.52292e7	1.13899	0.569496	+	0.821994i	$0.307139\pi$
$158$	0	0
$159$	−9.69197e6	−0.191215
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1.65741e7	−0.299760	−0.149880	−	0.988704i	$-0.547889\pi$
$163$	−1.65741e7	−0.299760	−0.149880	+	0.988704i	$0.547889\pi$
$164$	0	0
$165$	−4.54896e6	−0.0788349
$166$	0	0
$167$	5.70004e7	0.947044	0.473522	−	0.880782i	$-0.342982\pi$
$167$	5.70004e7	0.947044	0.473522	+	0.880782i	$0.342982\pi$
$168$	0	0
$169$	−4.70828e7	−0.750340
$170$	0	0
$171$	3.57472e7	0.546709
$172$	0	0
$173$	2.84256e7	0.417396	0.208698	−	0.977980i	$-0.433077\pi$
$173$	2.84256e7	0.417396	0.208698	+	0.977980i	$0.433077\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.51243e7	0.340555
$178$	0	0
$179$	1.44314e8	1.88071	0.940357	−	0.340190i	$-0.110491\pi$
$179$	1.44314e8	1.88071	0.940357	+	0.340190i	$0.110491\pi$
$180$	0	0
$181$	1.17741e8	1.47588	0.737942	−	0.674865i	$-0.235798\pi$
$181$	1.17741e8	1.47588	0.737942	+	0.674865i	$0.235798\pi$
$182$	0	0
$183$	−3.56085e7	−0.429512
$184$	0	0
$185$	−1.60526e7	−0.186400
$186$	0	0
$187$	2.39242e6	0.0267541
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.47184e8	1.52842	0.764211	−	0.644966i	$-0.223128\pi$
$191$	1.47184e8	1.52842	0.764211	+	0.644966i	$0.223128\pi$
$192$	0	0
$193$	1.20940e8	1.21093	0.605465	−	0.795872i	$-0.292987\pi$
$193$	1.20940e8	1.21093	0.605465	+	0.795872i	$0.292987\pi$
$194$	0	0
$195$	−2.56478e7	−0.247702
$196$	0	0
$197$	−5.80341e7	−0.540818	−0.270409	−	0.962745i	$-0.587159\pi$
$197$	−5.80341e7	−0.540818	−0.270409	+	0.962745i	$0.587159\pi$
$198$	0	0
$199$	−7.80160e7	−0.701775	−0.350888	−	0.936418i	$-0.614120\pi$
$199$	−7.80160e7	−0.701775	−0.350888	+	0.936418i	$0.614120\pi$
$200$	0	0
$201$	−5.11235e7	−0.444053
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.66160e7	0.702198
$206$	0	0
$207$	−8.39371e6	−0.0657745
$208$	0	0
$209$	3.44233e7	0.260820
$210$	0	0
$211$	2.11854e8	1.55256	0.776279	−	0.630390i	$-0.217105\pi$
$211$	2.11854e8	1.55256	0.776279	+	0.630390i	$0.217105\pi$
$212$	0	0
$213$	−6.15584e6	−0.0436474
$214$	0	0
$215$	−1.83670e8	−1.26039
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.11932e7	0.136346
$220$	0	0
$221$	1.34889e7	0.0840625
$222$	0	0
$223$	1.71323e8	1.03454	0.517271	−	0.855822i	$-0.326948\pi$
$223$	1.71323e8	1.03454	0.517271	+	0.855822i	$0.326948\pi$
$224$	0	0
$225$	−1.49627e7	−0.0875733
$226$	0	0
$227$	2.64284e8	1.49962	0.749808	−	0.661656i	$-0.230146\pi$
$227$	2.64284e8	1.49962	0.749808	+	0.661656i	$0.230146\pi$
$228$	0	0
$229$	−5.30633e6	−0.0291991	−0.0145996	−	0.999893i	$-0.504647\pi$
$229$	−5.30633e6	−0.0291991	−0.0145996	+	0.999893i	$0.504647\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.38215e8	0.715829	0.357914	−	0.933754i	$-0.383488\pi$
$233$	1.38215e8	0.715829	0.357914	+	0.933754i	$0.383488\pi$
$234$	0	0
$235$	3.22770e8	1.62239
$236$	0	0
$237$	5.67241e7	0.276789
$238$	0	0
$239$	−2.71430e8	−1.28607	−0.643037	−	0.765836i	$-0.722326\pi$
$239$	−2.71430e8	−1.28607	−0.643037	+	0.765836i	$0.722326\pi$
$240$	0	0
$241$	−2.46061e7	−0.113236	−0.0566179	−	0.998396i	$-0.518032\pi$
$241$	−2.46061e7	−0.113236	−0.0566179	+	0.998396i	$0.518032\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.94084e8	0.819505
$248$	0	0
$249$	2.32991e8	0.956405
$250$	0	0
$251$	2.08916e8	0.833900	0.416950	−	0.908929i	$-0.363099\pi$
$251$	2.08916e8	0.833900	0.416950	+	0.908929i	$0.363099\pi$
$252$	0	0
$253$	−8.08283e6	−0.0313792
$254$	0	0
$255$	−2.20838e7	−0.0834034
$256$	0	0
$257$	1.12952e8	0.415075	0.207537	−	0.978227i	$-0.433455\pi$
$257$	1.12952e8	0.415075	0.207537	+	0.978227i	$0.433455\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	3.62036e7	0.126040
$262$	0	0
$263$	1.61527e8	0.547520	0.273760	−	0.961798i	$-0.411733\pi$
$263$	1.61527e8	0.547520	0.273760	+	0.961798i	$0.411733\pi$
$264$	0	0
$265$	8.61509e7	0.284380
$266$	0	0
$267$	1.59384e8	0.512454
$268$	0	0
$269$	2.84015e8	0.889628	0.444814	−	0.895623i	$-0.353270\pi$
$269$	2.84015e8	0.889628	0.444814	+	0.895623i	$0.353270\pi$
$270$	0	0
$271$	−2.88374e8	−0.880165	−0.440083	−	0.897957i	$-0.645051\pi$
$271$	−2.88374e8	−0.880165	−0.440083	+	0.897957i	$0.645051\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.44086e7	−0.0417788
$276$	0	0
$277$	3.07325e8	0.868797	0.434398	−	0.900721i	$-0.356961\pi$
$277$	3.07325e8	0.868797	0.434398	+	0.900721i	$0.356961\pi$
$278$	0	0
$279$	8.26103e7	0.227730
$280$	0	0
$281$	−3.81803e8	−1.02652	−0.513260	−	0.858233i	$-0.671562\pi$
$281$	−3.81803e8	−1.02652	−0.513260	+	0.858233i	$0.671562\pi$
$282$	0	0
$283$	6.05387e8	1.58775	0.793873	−	0.608084i	$-0.208062\pi$
$283$	6.05387e8	1.58775	0.793873	+	0.608084i	$0.208062\pi$
$284$	0	0
$285$	−3.17753e8	−0.813080
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3.98724e8	−0.971695
$290$	0	0
$291$	2.08938e7	0.0497042
$292$	0	0
$293$	6.33619e8	1.47161	0.735804	−	0.677195i	$-0.236805\pi$
$293$	6.33619e8	1.47161	0.735804	+	0.677195i	$0.236805\pi$
$294$	0	0
$295$	−2.23327e8	−0.506482
$296$	0	0
$297$	−1.38175e7	−0.0306042
$298$	0	0
$299$	−4.55724e7	−0.0985945
$300$	0	0
$301$	0	0
$302$	0	0
$303$	9.00017e7	0.185867
$304$	0	0
$305$	3.16520e8	0.638781
$306$	0	0
$307$	3.79251e8	0.748070	0.374035	−	0.927415i	$-0.377974\pi$
$307$	3.79251e8	0.748070	0.374035	+	0.927415i	$0.377974\pi$
$308$	0	0
$309$	−1.52794e8	−0.294613
$310$	0	0
$311$	4.80859e8	0.906476	0.453238	−	0.891390i	$-0.350269\pi$
$311$	4.80859e8	0.906476	0.453238	+	0.891390i	$0.350269\pi$
$312$	0	0
$313$	−8.04070e8	−1.48214	−0.741069	−	0.671429i	$-0.765681\pi$
$313$	−8.04070e8	−1.48214	−0.741069	+	0.671429i	$0.765681\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.14707e8	−1.26015	−0.630073	−	0.776536i	$-0.716975\pi$
$317$	−7.14707e8	−1.26015	−0.630073	+	0.776536i	$0.716975\pi$
$318$	0	0
$319$	3.48627e7	0.0601304
$320$	0	0
$321$	−4.15320e8	−0.700833
$322$	0	0
$323$	1.67115e8	0.275934
$324$	0	0
$325$	−8.12380e7	−0.131271
$326$	0	0
$327$	−1.20876e8	−0.191172
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.00218e8	0.303462	0.151731	−	0.988422i	$-0.451515\pi$
$331$	2.00218e8	0.303462	0.151731	+	0.988422i	$0.451515\pi$
$332$	0	0
$333$	−4.87599e7	−0.0723616
$334$	0	0
$335$	4.54431e8	0.660407
$336$	0	0
$337$	−6.36870e8	−0.906455	−0.453228	−	0.891395i	$-0.649728\pi$
$337$	−6.36870e8	−0.906455	−0.453228	+	0.891395i	$0.649728\pi$
$338$	0	0
$339$	2.62515e8	0.365979
$340$	0	0
$341$	7.95506e7	0.108643
$342$	0	0
$343$	0	0
$344$	0	0
$345$	7.46107e7	0.0978215
$346$	0	0
$347$	1.00147e9	1.28672	0.643361	−	0.765563i	$-0.277540\pi$
$347$	1.00147e9	1.28672	0.643361	+	0.765563i	$0.277540\pi$
$348$	0	0
$349$	−9.60061e8	−1.20895	−0.604477	−	0.796623i	$-0.706618\pi$
$349$	−9.60061e8	−1.20895	−0.604477	+	0.796623i	$0.706618\pi$
$350$	0	0
$351$	−7.79053e7	−0.0961595
$352$	0	0
$353$	−3.88199e8	−0.469724	−0.234862	−	0.972029i	$-0.575464\pi$
$353$	−3.88199e8	−0.469724	−0.234862	+	0.972029i	$0.575464\pi$
$354$	0	0
$355$	5.47186e7	0.0649136
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.18153e7	0.0933263	0.0466632	−	0.998911i	$-0.485141\pi$
$359$	8.18153e7	0.0933263	0.0466632	+	0.998911i	$0.485141\pi$
$360$	0	0
$361$	1.51066e9	1.69002
$362$	0	0
$363$	5.12848e8	0.562750
$364$	0	0
$365$	−1.88384e8	−0.202777
$366$	0	0
$367$	1.09898e9	1.16053	0.580266	−	0.814427i	$-0.302949\pi$
$367$	1.09898e9	1.16053	0.580266	+	0.814427i	$0.302949\pi$
$368$	0	0
$369$	2.63096e8	0.272598
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.90918e9	1.90488	0.952438	−	0.304733i	$-0.0985671\pi$
$373$	1.90918e9	1.90488	0.952438	+	0.304733i	$0.0985671\pi$
$374$	0	0
$375$	6.39252e8	0.625983
$376$	0	0
$377$	1.96562e8	0.188932
$378$	0	0
$379$	−1.07421e9	−1.01356	−0.506781	−	0.862075i	$-0.669165\pi$
$379$	−1.07421e9	−1.01356	−0.506781	+	0.862075i	$0.669165\pi$
$380$	0	0
$381$	−4.12962e8	−0.382536
$382$	0	0
$383$	2.16642e8	0.197037	0.0985184	−	0.995135i	$-0.468590\pi$
$383$	2.16642e8	0.197037	0.0985184	+	0.995135i	$0.468590\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.57898e8	−0.489290
$388$	0	0
$389$	2.04803e9	1.76406	0.882028	−	0.471197i	$-0.156178\pi$
$389$	2.04803e9	1.76406	0.882028	+	0.471197i	$0.156178\pi$
$390$	0	0
$391$	−3.92397e7	−0.0331976
$392$	0	0
$393$	1.17981e9	0.980478
$394$	0	0
$395$	−5.04214e8	−0.411647
$396$	0	0
$397$	−1.73429e9	−1.39109	−0.695544	−	0.718483i	$-0.744837\pi$
$397$	−1.73429e9	−1.39109	−0.695544	+	0.718483i	$0.744837\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.64740e9	1.27583	0.637917	−	0.770105i	$-0.279796\pi$
$401$	1.64740e9	1.27583	0.637917	+	0.770105i	$0.279796\pi$
$402$	0	0
$403$	4.48521e8	0.341362
$404$	0	0
$405$	1.27546e8	0.0954056
$406$	0	0
$407$	−4.69540e7	−0.0345217
$408$	0	0
$409$	1.95391e8	0.141213	0.0706063	−	0.997504i	$-0.477507\pi$
$409$	1.95391e8	0.141213	0.0706063	+	0.997504i	$0.477507\pi$
$410$	0	0
$411$	8.10193e7	0.0575628
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−2.07103e9	−1.42239
$416$	0	0
$417$	−4.67620e8	−0.315803
$418$	0	0
$419$	−1.63659e9	−1.08690	−0.543451	−	0.839441i	$-0.682883\pi$
$419$	−1.63659e9	−1.08690	−0.543451	+	0.839441i	$0.682883\pi$
$420$	0	0
$421$	−1.40514e8	−0.0917765	−0.0458883	−	0.998947i	$-0.514612\pi$
$421$	−1.40514e8	−0.0917765	−0.0458883	+	0.998947i	$0.514612\pi$
$422$	0	0
$423$	9.80415e8	0.629823
$424$	0	0
$425$	−6.99492e7	−0.0441999
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−7.50199e7	−0.0458750
$430$	0	0
$431$	1.14319e9	0.687780	0.343890	−	0.939010i	$-0.388255\pi$
$431$	1.14319e9	0.687780	0.343890	+	0.939010i	$0.388255\pi$
$432$	0	0
$433$	2.24052e9	1.32630	0.663148	−	0.748488i	$-0.269220\pi$
$433$	2.24052e9	1.32630	0.663148	+	0.748488i	$0.269220\pi$
$434$	0	0
$435$	−3.21810e8	−0.187451
$436$	0	0
$437$	−5.64601e8	−0.323636
$438$	0	0
$439$	1.69630e9	0.956921	0.478461	−	0.878109i	$-0.341195\pi$
$439$	1.69630e9	0.956921	0.478461	+	0.878109i	$0.341195\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.04940e9	1.11999	0.559994	−	0.828497i	$-0.310803\pi$
$443$	2.04940e9	1.11999	0.559994	+	0.828497i	$0.310803\pi$
$444$	0	0
$445$	−1.41674e9	−0.762134
$446$	0	0
$447$	1.05424e9	0.558292
$448$	0	0
$449$	−1.62831e9	−0.848937	−0.424469	−	0.905443i	$-0.639539\pi$
$449$	−1.62831e9	−0.848937	−0.424469	+	0.905443i	$0.639539\pi$
$450$	0	0
$451$	2.53352e8	0.130049
$452$	0	0
$453$	2.87825e8	0.145474
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.17242e8	−0.0574612	−0.0287306	−	0.999587i	$-0.509146\pi$
$457$	−1.17242e8	−0.0574612	−0.0287306	+	0.999587i	$0.509146\pi$
$458$	0	0
$459$	−6.70797e7	−0.0323777
$460$	0	0
$461$	2.50354e9	1.19015	0.595075	−	0.803670i	$-0.297122\pi$
$461$	2.50354e9	1.19015	0.595075	+	0.803670i	$0.297122\pi$
$462$	0	0
$463$	−1.01316e9	−0.474398	−0.237199	−	0.971461i	$-0.576229\pi$
$463$	−1.01316e9	−0.474398	−0.237199	+	0.971461i	$0.576229\pi$
$464$	0	0
$465$	−7.34314e8	−0.338685
$466$	0	0
$467$	7.82198e8	0.355392	0.177696	−	0.984085i	$-0.443136\pi$
$467$	7.82198e8	0.355392	0.177696	+	0.984085i	$0.443136\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.49119e9	−0.657597
$472$	0	0
$473$	−5.37235e8	−0.233427
$474$	0	0
$475$	−1.00646e9	−0.430894
$476$	0	0
$477$	2.61683e8	0.110398
$478$	0	0
$479$	−1.85704e8	−0.0772055	−0.0386027	−	0.999255i	$-0.512291\pi$
$479$	−1.85704e8	−0.0772055	−0.0386027	+	0.999255i	$0.512291\pi$
$480$	0	0
$481$	−2.64735e8	−0.108468
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.85723e8	−0.0739213
$486$	0	0
$487$	−1.68521e9	−0.661155	−0.330577	−	0.943779i	$-0.607243\pi$
$487$	−1.68521e9	−0.661155	−0.330577	+	0.943779i	$0.607243\pi$
$488$	0	0
$489$	4.47501e8	0.173067
$490$	0	0
$491$	3.19408e9	1.21776	0.608879	−	0.793263i	$-0.291620\pi$
$491$	3.19408e9	1.21776	0.608879	+	0.793263i	$0.291620\pi$
$492$	0	0
$493$	1.69248e8	0.0636150
$494$	0	0
$495$	1.22822e8	0.0455153
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.03224e8	0.145276	0.0726380	−	0.997358i	$-0.476858\pi$
$499$	4.03224e8	0.145276	0.0726380	+	0.997358i	$0.476858\pi$
$500$	0	0
$501$	−1.53901e9	−0.546776
$502$	0	0
$503$	−3.05407e8	−0.107002	−0.0535009	−	0.998568i	$-0.517038\pi$
$503$	−3.05407e8	−0.107002	−0.0535009	+	0.998568i	$0.517038\pi$
$504$	0	0
$505$	−8.00015e8	−0.276426
$506$	0	0
$507$	1.27123e9	0.433209
$508$	0	0
$509$	1.73298e9	0.582479	0.291240	−	0.956650i	$-0.405932\pi$
$509$	1.73298e9	0.582479	0.291240	+	0.956650i	$0.405932\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−9.65176e8	−0.315643
$514$	0	0
$515$	1.35817e9	0.438157
$516$	0	0
$517$	9.44103e8	0.300471
$518$	0	0
$519$	−7.67492e8	−0.240984
$520$	0	0
$521$	5.29373e8	0.163995	0.0819974	−	0.996633i	$-0.473870\pi$
$521$	5.29373e8	0.163995	0.0819974	+	0.996633i	$0.473870\pi$
$522$	0	0
$523$	−3.60373e9	−1.10153	−0.550765	−	0.834661i	$-0.685664\pi$
$523$	−3.60373e9	−1.10153	−0.550765	+	0.834661i	$0.685664\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.86195e8	0.114939
$528$	0	0
$529$	−3.27225e9	−0.961063
$530$	0	0
$531$	−6.78355e8	−0.196619
$532$	0	0
$533$	1.42844e9	0.408618
$534$	0	0
$535$	3.69173e9	1.04230
$536$	0	0
$537$	−3.89648e9	−1.08583
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−2.75672e9	−0.748517	−0.374259	−	0.927324i	$-0.622103\pi$
$541$	−2.75672e9	−0.748517	−0.374259	+	0.927324i	$0.622103\pi$
$542$	0	0
$543$	−3.17900e9	−0.852102
$544$	0	0
$545$	1.07446e9	0.284316
$546$	0	0
$547$	6.28471e9	1.64183	0.820917	−	0.571047i	$-0.193463\pi$
$547$	6.28471e9	1.64183	0.820917	+	0.571047i	$0.193463\pi$
$548$	0	0
$549$	9.61430e8	0.247979
$550$	0	0
$551$	2.43523e9	0.620167
$552$	0	0
$553$	0	0
$554$	0	0
$555$	4.33421e8	0.107618
$556$	0	0
$557$	−6.01153e9	−1.47398	−0.736991	−	0.675903i	$-0.763754\pi$
$557$	−6.01153e9	−1.47398	−0.736991	+	0.675903i	$0.763754\pi$
$558$	0	0
$559$	−3.02903e9	−0.733435
$560$	0	0
$561$	−6.45952e7	−0.0154465
$562$	0	0
$563$	−5.21783e9	−1.23228	−0.616142	−	0.787635i	$-0.711305\pi$
$563$	−5.21783e9	−1.23228	−0.616142	+	0.787635i	$0.711305\pi$
$564$	0	0
$565$	−2.33347e9	−0.544293
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.35896e8	0.121952	0.0609758	−	0.998139i	$-0.480579\pi$
$569$	5.35896e8	0.121952	0.0609758	+	0.998139i	$0.480579\pi$
$570$	0	0
$571$	−5.46646e9	−1.22880	−0.614398	−	0.788997i	$-0.710601\pi$
$571$	−5.46646e9	−1.22880	−0.614398	+	0.788997i	$0.710601\pi$
$572$	0	0
$573$	−3.97396e9	−0.882435
$574$	0	0
$575$	2.36325e8	0.0518408
$576$	0	0
$577$	−3.69396e9	−0.800528	−0.400264	−	0.916400i	$-0.631082\pi$
$577$	−3.69396e9	−0.800528	−0.400264	+	0.916400i	$0.631082\pi$
$578$	0	0
$579$	−3.26538e9	−0.699130
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.51991e8	0.0526679
$584$	0	0
$585$	6.92492e8	0.143011
$586$	0	0
$587$	−5.08755e9	−1.03819	−0.519093	−	0.854718i	$-0.673730\pi$
$587$	−5.08755e9	−1.03819	−0.519093	+	0.854718i	$0.673730\pi$
$588$	0	0
$589$	5.55676e9	1.12052
$590$	0	0
$591$	1.56692e9	0.312242
$592$	0	0
$593$	5.88519e9	1.15896	0.579481	−	0.814986i	$-0.303255\pi$
$593$	5.88519e9	1.15896	0.579481	+	0.814986i	$0.303255\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.10643e9	0.405170
$598$	0	0
$599$	4.10449e9	0.780306	0.390153	−	0.920750i	$-0.372422\pi$
$599$	4.10449e9	0.780306	0.390153	+	0.920750i	$0.372422\pi$
$600$	0	0
$601$	−6.08667e8	−0.114372	−0.0571859	−	0.998364i	$-0.518213\pi$
$601$	−6.08667e8	−0.114372	−0.0571859	+	0.998364i	$0.518213\pi$
$602$	0	0
$603$	1.38034e9	0.256374
$604$	0	0
$605$	−4.55865e9	−0.836936
$606$	0	0
$607$	6.46963e9	1.17414	0.587069	−	0.809537i	$-0.300282\pi$
$607$	6.46963e9	1.17414	0.587069	+	0.809537i	$0.300282\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.32302e9	0.944091
$612$	0	0
$613$	4.74144e9	0.831378	0.415689	−	0.909507i	$-0.363541\pi$
$613$	4.74144e9	0.831378	0.415689	+	0.909507i	$0.363541\pi$
$614$	0	0
$615$	−2.33863e9	−0.405414
$616$	0	0
$617$	−8.22535e9	−1.40980	−0.704898	−	0.709308i	$-0.749007\pi$
$617$	−8.22535e9	−1.40980	−0.704898	+	0.709308i	$0.749007\pi$
$618$	0	0
$619$	−1.09006e10	−1.84728	−0.923640	−	0.383262i	$-0.874801\pi$
$619$	−1.09006e10	−1.84728	−0.923640	+	0.383262i	$0.874801\pi$
$620$	0	0
$621$	2.26630e8	0.0379749
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4.07872e9	−0.668258
$626$	0	0
$627$	−9.29428e8	−0.150584
$628$	0	0
$629$	−2.27947e8	−0.0365223
$630$	0	0
$631$	5.18857e9	0.822138	0.411069	−	0.911604i	$-0.365156\pi$
$631$	5.18857e9	0.822138	0.411069	+	0.911604i	$0.365156\pi$
$632$	0	0
$633$	−5.72005e9	−0.896370
$634$	0	0
$635$	3.67077e9	0.568918
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.66208e8	0.0251998
$640$	0	0
$641$	−1.12275e10	−1.68375	−0.841877	−	0.539669i	$-0.818549\pi$
$641$	−1.12275e10	−1.68375	−0.841877	+	0.539669i	$0.818549\pi$
$642$	0	0
$643$	−8.49079e9	−1.25953	−0.629767	−	0.776784i	$-0.716849\pi$
$643$	−8.49079e9	−1.25953	−0.629767	+	0.776784i	$0.716849\pi$
$644$	0	0
$645$	4.95909e9	0.727685
$646$	0	0
$647$	7.96743e8	0.115652	0.0578260	−	0.998327i	$-0.481583\pi$
$647$	7.96743e8	0.115652	0.0578260	+	0.998327i	$0.481583\pi$
$648$	0	0
$649$	−6.53231e8	−0.0938016
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.80739e9	−1.37834	−0.689171	−	0.724598i	$-0.742025\pi$
$653$	−9.80739e9	−1.37834	−0.689171	+	0.724598i	$0.742025\pi$
$654$	0	0
$655$	−1.04872e10	−1.45819
$656$	0	0
$657$	−5.72217e8	−0.0787194
$658$	0	0
$659$	−1.00172e10	−1.36348	−0.681741	−	0.731594i	$-0.738777\pi$
$659$	−1.00172e10	−1.36348	−0.681741	+	0.731594i	$0.738777\pi$
$660$	0	0
$661$	1.49593e9	0.201469	0.100734	−	0.994913i	$-0.467881\pi$
$661$	1.49593e9	0.201469	0.100734	+	0.994913i	$0.467881\pi$
$662$	0	0
$663$	−3.64199e8	−0.0485335
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.71808e8	−0.0746122
$668$	0	0
$669$	−4.62571e9	−0.597293
$670$	0	0
$671$	9.25821e8	0.118304
$672$	0	0
$673$	7.25457e9	0.917401	0.458700	−	0.888591i	$-0.348315\pi$
$673$	7.25457e9	0.917401	0.458700	+	0.888591i	$0.348315\pi$
$674$	0	0
$675$	4.03994e8	0.0505605
$676$	0	0
$677$	1.47562e10	1.82774	0.913870	−	0.406006i	$-0.133079\pi$
$677$	1.47562e10	1.82774	0.913870	+	0.406006i	$0.133079\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−7.13566e9	−0.865804
$682$	0	0
$683$	5.53531e9	0.664767	0.332383	−	0.943144i	$-0.392147\pi$
$683$	5.53531e9	0.664767	0.332383	+	0.943144i	$0.392147\pi$
$684$	0	0
$685$	−7.20171e8	−0.0856089
$686$	0	0
$687$	1.43271e8	0.0168581
$688$	0	0
$689$	1.42077e9	0.165484
$690$	0	0
$691$	5.55281e9	0.640236	0.320118	−	0.947378i	$-0.396277\pi$
$691$	5.55281e9	0.640236	0.320118	+	0.947378i	$0.396277\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.15662e9	0.469671
$696$	0	0
$697$	1.22995e9	0.137585
$698$	0	0
$699$	−3.73180e9	−0.413284
$700$	0	0
$701$	−1.06044e10	−1.16271	−0.581357	−	0.813648i	$-0.697478\pi$
$701$	−1.06044e10	−1.16271	−0.581357	+	0.813648i	$0.697478\pi$
$702$	0	0
$703$	−3.27982e9	−0.356046
$704$	0	0
$705$	−8.71480e9	−0.936689
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−5.44638e9	−0.573913	−0.286957	−	0.957944i	$-0.592644\pi$
$709$	−5.44638e9	−0.573913	−0.286957	+	0.957944i	$0.592644\pi$
$710$	0	0
$711$	−1.53155e9	−0.159804
$712$	0	0
$713$	−1.30477e9	−0.134809
$714$	0	0
$715$	6.66844e8	0.0682265
$716$	0	0
$717$	7.32862e9	0.742515
$718$	0	0
$719$	1.90216e10	1.90851	0.954257	−	0.298986i	$-0.0966484\pi$
$719$	1.90216e10	1.90851	0.954257	+	0.298986i	$0.0966484\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	6.64366e8	0.0653767
$724$	0	0
$725$	−1.01931e9	−0.0993400
$726$	0	0
$727$	−7.80283e9	−0.753150	−0.376575	−	0.926386i	$-0.622898\pi$
$727$	−7.80283e9	−0.753150	−0.376575	+	0.926386i	$0.622898\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−2.60812e9	−0.246954
$732$	0	0
$733$	1.20743e9	0.113239	0.0566197	−	0.998396i	$-0.481968\pi$
$733$	1.20743e9	0.113239	0.0566197	+	0.998396i	$0.481968\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.32921e9	0.122309
$738$	0	0
$739$	−1.64892e10	−1.50294	−0.751472	−	0.659765i	$-0.770656\pi$
$739$	−1.64892e10	−1.50294	−0.751472	+	0.659765i	$0.770656\pi$
$740$	0	0
$741$	−5.24028e9	−0.473141
$742$	0	0
$743$	−2.43983e9	−0.218222	−0.109111	−	0.994030i	$-0.534800\pi$
$743$	−2.43983e9	−0.218222	−0.109111	+	0.994030i	$0.534800\pi$
$744$	0	0
$745$	−9.37099e9	−0.830306
$746$	0	0
$747$	−6.29077e9	−0.552181
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.51959e10	−1.30914	−0.654571	−	0.756000i	$-0.727151\pi$
$751$	−1.51959e10	−1.30914	−0.654571	+	0.756000i	$0.727151\pi$
$752$	0	0
$753$	−5.64073e9	−0.481452
$754$	0	0
$755$	−2.55844e9	−0.216352
$756$	0	0
$757$	−1.52456e9	−0.127735	−0.0638673	−	0.997958i	$-0.520343\pi$
$757$	−1.52456e9	−0.127735	−0.0638673	+	0.997958i	$0.520343\pi$
$758$	0	0
$759$	2.18236e8	0.0181168
$760$	0	0
$761$	2.33134e10	1.91761	0.958803	−	0.284072i	$-0.0916854\pi$
$761$	2.33134e10	1.91761	0.958803	+	0.284072i	$0.0916854\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	5.96264e8	0.0481530
$766$	0	0
$767$	−3.68303e9	−0.294728
$768$	0	0
$769$	5.50489e9	0.436523	0.218261	−	0.975890i	$-0.429961\pi$
$769$	5.50489e9	0.436523	0.218261	+	0.975890i	$0.429961\pi$
$770$	0	0
$771$	−3.04969e9	−0.239643
$772$	0	0
$773$	−1.72465e10	−1.34299	−0.671494	−	0.741010i	$-0.734347\pi$
$773$	−1.72465e10	−1.34299	−0.671494	+	0.741010i	$0.734347\pi$
$774$	0	0
$775$	−2.32589e9	−0.179487
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.76971e10	1.34128
$780$	0	0
$781$	1.60052e8	0.0120221
$782$	0	0
$783$	−9.77497e8	−0.0727695
$784$	0	0
$785$	1.32550e10	0.977995
$786$	0	0
$787$	8.41133e9	0.615111	0.307555	−	0.951530i	$-0.400489\pi$
$787$	8.41133e9	0.615111	0.307555	+	0.951530i	$0.400489\pi$
$788$	0	0
$789$	−4.36123e9	−0.316111
$790$	0	0
$791$	0	0
$792$	0	0
$793$	5.21994e9	0.371715
$794$	0	0
$795$	−2.32607e9	−0.164187
$796$	0	0
$797$	−7.89074e9	−0.552095	−0.276047	−	0.961144i	$-0.589025\pi$
$797$	−7.89074e9	−0.552095	−0.276047	+	0.961144i	$0.589025\pi$
$798$	0	0
$799$	4.58334e9	0.317884
$800$	0	0
$801$	−4.30336e9	−0.295865
$802$	0	0
$803$	−5.51024e8	−0.0375548
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.66841e9	−0.513627
$808$	0	0
$809$	1.56202e10	1.03721	0.518606	−	0.855013i	$-0.326451\pi$
$809$	1.56202e10	1.03721	0.518606	+	0.855013i	$0.326451\pi$
$810$	0	0
$811$	−8.33712e9	−0.548836	−0.274418	−	0.961610i	$-0.588485\pi$
$811$	−8.33712e9	−0.548836	−0.274418	+	0.961610i	$0.588485\pi$
$812$	0	0
$813$	7.78611e9	0.508164
$814$	0	0
$815$	−3.97779e9	−0.257389
$816$	0	0
$817$	−3.75269e10	−2.40749
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.17639e10	−0.741907	−0.370954	−	0.928651i	$-0.620969\pi$
$821$	−1.17639e10	−0.741907	−0.370954	+	0.928651i	$0.620969\pi$
$822$	0	0
$823$	−2.54212e10	−1.58963	−0.794816	−	0.606851i	$-0.792433\pi$
$823$	−2.54212e10	−1.58963	−0.794816	+	0.606851i	$0.792433\pi$
$824$	0	0
$825$	3.89031e8	0.0241210
$826$	0	0
$827$	1.92118e10	1.18113	0.590566	−	0.806990i	$-0.298905\pi$
$827$	1.92118e10	1.18113	0.590566	+	0.806990i	$0.298905\pi$
$828$	0	0
$829$	2.08097e10	1.26860	0.634299	−	0.773088i	$-0.281289\pi$
$829$	2.08097e10	1.26860	0.634299	+	0.773088i	$0.281289\pi$
$830$	0	0
$831$	−8.29777e9	−0.501600
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.36801e10	0.813179
$836$	0	0
$837$	−2.23048e9	−0.131480
$838$	0	0
$839$	2.17898e10	1.27376	0.636878	−	0.770964i	$-0.280225\pi$
$839$	2.17898e10	1.27376	0.636878	+	0.770964i	$0.280225\pi$
$840$	0	0
$841$	−1.47836e10	−0.857024
$842$	0	0
$843$	1.03087e10	0.592661
$844$	0	0
$845$	−1.12999e10	−0.644280
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.63455e10	−0.916685
$850$	0	0
$851$	7.70125e8	0.0428359
$852$	0	0
$853$	−2.42027e10	−1.33519	−0.667594	−	0.744526i	$-0.732676\pi$
$853$	−2.42027e10	−1.33519	−0.667594	+	0.744526i	$0.732676\pi$
$854$	0	0
$855$	8.57934e9	0.469432
$856$	0	0
$857$	−2.21583e10	−1.20255	−0.601276	−	0.799042i	$-0.705341\pi$
$857$	−2.21583e10	−1.20255	−0.601276	+	0.799042i	$0.705341\pi$
$858$	0	0
$859$	−2.12882e10	−1.14594	−0.572970	−	0.819576i	$-0.694209\pi$
$859$	−2.12882e10	−1.14594	−0.572970	+	0.819576i	$0.694209\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.55835e10	1.88456	0.942281	−	0.334823i	$-0.108677\pi$
$863$	3.55835e10	1.88456	0.942281	+	0.334823i	$0.108677\pi$
$864$	0	0
$865$	6.82215e9	0.358397
$866$	0	0
$867$	1.07656e10	0.561009
$868$	0	0
$869$	−1.47483e9	−0.0762380
$870$	0	0
$871$	7.49433e9	0.384299
$872$	0	0
$873$	−5.64134e8	−0.0286967
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.86385e10	−1.93429	−0.967146	−	0.254223i	$-0.918180\pi$
$877$	−3.86385e10	−1.93429	−0.967146	+	0.254223i	$0.918180\pi$
$878$	0	0
$879$	−1.71077e10	−0.849633
$880$	0	0
$881$	1.74469e9	0.0859615	0.0429807	−	0.999076i	$-0.486315\pi$
$881$	1.74469e9	0.0859615	0.0429807	+	0.999076i	$0.486315\pi$
$882$	0	0
$883$	2.42096e10	1.18338	0.591691	−	0.806165i	$-0.298461\pi$
$883$	2.42096e10	1.18338	0.591691	+	0.806165i	$0.298461\pi$
$884$	0	0
$885$	6.02982e9	0.292417
$886$	0	0
$887$	2.46025e10	1.18371	0.591857	−	0.806043i	$-0.298395\pi$
$887$	2.46025e10	1.18371	0.591857	+	0.806043i	$0.298395\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.73072e8	0.0176693
$892$	0	0
$893$	6.59473e10	3.09897
$894$	0	0
$895$	3.46353e10	1.61487
$896$	0	0
$897$	1.23046e9	0.0569236
$898$	0	0
$899$	5.62770e9	0.258328
$900$	0	0
$901$	1.22334e9	0.0557200
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.82578e10	1.26727
$906$	0	0
$907$	8.41700e9	0.374569	0.187284	−	0.982306i	$-0.440031\pi$
$907$	8.41700e9	0.374569	0.187284	+	0.982306i	$0.440031\pi$
$908$	0	0
$909$	−2.43005e9	−0.107310
$910$	0	0
$911$	−3.62674e10	−1.58929	−0.794644	−	0.607076i	$-0.792342\pi$
$911$	−3.62674e10	−1.58929	−0.794644	+	0.607076i	$0.792342\pi$
$912$	0	0
$913$	−6.05777e9	−0.263430
$914$	0	0
$915$	−8.54604e9	−0.368800
$916$	0	0
$917$	0	0
$918$	0	0
$919$	3.50569e10	1.48994	0.744971	−	0.667097i	$-0.232463\pi$
$919$	3.50569e10	1.48994	0.744971	+	0.667097i	$0.232463\pi$
$920$	0	0
$921$	−1.02398e10	−0.431898
$922$	0	0
$923$	9.02400e8	0.0377740
$924$	0	0
$925$	1.37284e9	0.0570325
$926$	0	0
$927$	4.12545e9	0.170095
$928$	0	0
$929$	−5.98739e9	−0.245009	−0.122505	−	0.992468i	$-0.539093\pi$
$929$	−5.98739e9	−0.245009	−0.122505	+	0.992468i	$0.539093\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.29832e10	−0.523354
$934$	0	0
$935$	5.74180e8	0.0229724
$936$	0	0
$937$	1.27744e10	0.507283	0.253642	−	0.967298i	$-0.418372\pi$
$937$	1.27744e10	0.507283	0.253642	+	0.967298i	$0.418372\pi$
$938$	0	0
$939$	2.17099e10	0.855713
$940$	0	0
$941$	−2.08847e9	−0.0817080	−0.0408540	−	0.999165i	$-0.513008\pi$
$941$	−2.08847e9	−0.0817080	−0.0408540	+	0.999165i	$0.513008\pi$
$942$	0	0
$943$	−4.15540e9	−0.161370
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.26786e9	0.201562	0.100781	−	0.994909i	$-0.467866\pi$
$947$	5.26786e9	0.201562	0.100781	+	0.994909i	$0.467866\pi$
$948$	0	0
$949$	−3.10677e9	−0.117999
$950$	0	0
$951$	1.92971e10	0.727545
$952$	0	0
$953$	−1.40878e10	−0.527252	−0.263626	−	0.964625i	$-0.584918\pi$
$953$	−1.40878e10	−0.527252	−0.263626	+	0.964625i	$0.584918\pi$
$954$	0	0
$955$	3.53241e10	1.31238
$956$	0	0
$957$	−9.41294e8	−0.0347163
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.46712e10	−0.533253
$962$	0	0
$963$	1.12136e10	0.404626
$964$	0	0
$965$	2.90256e10	1.03976
$966$	0	0
$967$	−4.95778e10	−1.76317	−0.881587	−	0.472022i	$-0.843524\pi$
$967$	−4.95778e10	−1.76317	−0.881587	+	0.472022i	$0.843524\pi$
$968$	0	0
$969$	−4.51210e9	−0.159311
$970$	0	0
$971$	4.72967e10	1.65792	0.828960	−	0.559308i	$-0.188933\pi$
$971$	4.72967e10	1.65792	0.828960	+	0.559308i	$0.188933\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2.19342e9	0.0757891
$976$	0	0
$977$	1.99103e10	0.683041	0.341520	−	0.939874i	$-0.389058\pi$
$977$	1.99103e10	0.683041	0.341520	+	0.939874i	$0.389058\pi$
$978$	0	0
$979$	−4.14398e9	−0.141149
$980$	0	0
$981$	3.26366e9	0.110373
$982$	0	0
$983$	−1.84811e9	−0.0620568	−0.0310284	−	0.999519i	$-0.509878\pi$
$983$	−1.84811e9	−0.0620568	−0.0310284	+	0.999519i	$0.509878\pi$
$984$	0	0
$985$	−1.39282e10	−0.464374
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.81157e9	0.289645
$990$	0	0
$991$	−1.39246e10	−0.454489	−0.227245	−	0.973838i	$-0.572972\pi$
$991$	−1.39246e10	−0.454489	−0.227245	+	0.973838i	$0.572972\pi$
$992$	0	0
$993$	−5.40588e9	−0.175204
$994$	0	0
$995$	−1.87238e10	−0.602579
$996$	0	0
$997$	3.58707e10	1.14632	0.573161	−	0.819443i	$-0.305717\pi$
$997$	3.58707e10	1.14632	0.573161	+	0.819443i	$0.305717\pi$
$998$	0	0
$999$	1.31652e9	0.0417780

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.8.a.b.1.1		1
7.2	even	3		588.8.i.e.361.1		2
7.3	odd	6		588.8.i.d.373.1		2
7.4	even	3		588.8.i.e.373.1		2
7.5	odd	6		588.8.i.d.361.1		2
7.6	odd	2		84.8.a.b.1.1	✓	1
21.20	even	2		252.8.a.b.1.1		1
28.27	even	2		336.8.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.8.a.b.1.1	✓	1	7.6	odd	2
252.8.a.b.1.1		1	21.20	even	2
336.8.a.c.1.1		1	28.27	even	2
588.8.a.b.1.1		1	1.1	even	1	trivial
588.8.i.d.361.1		2	7.5	odd	6
588.8.i.d.373.1		2	7.3	odd	6
588.8.i.e.361.1		2	7.2	even	3
588.8.i.e.373.1		2	7.4	even	3

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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	588.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more