LMFDB - Embedded newform 17.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [17,8,Mod(1,17)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(17, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("17.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 17 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	17.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:	$5.31054543323$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +18.0000 q^{3} -124.000 q^{4} -10.0000 q^{5} -36.0000 q^{6} -902.000 q^{7} +504.000 q^{8} -1863.00 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +18.0000 q^{3} -124.000 q^{4} -10.0000 q^{5} -36.0000 q^{6} -902.000 q^{7} +504.000 q^{8} -1863.00 q^{9} +20.0000 q^{10} -8634.00 q^{11} -2232.00 q^{12} +10858.0 q^{13} +1804.00 q^{14} -180.000 q^{15} +14864.0 q^{16} +4913.00 q^{17} +3726.00 q^{18} -784.000 q^{19} +1240.00 q^{20} -16236.0 q^{21} +17268.0 q^{22} +77330.0 q^{23} +9072.00 q^{24} -78025.0 q^{25} -21716.0 q^{26} -72900.0 q^{27} +111848. q^{28} -18210.0 q^{29} +360.000 q^{30} -237002. q^{31} -94240.0 q^{32} -155412. q^{33} -9826.00 q^{34} +9020.00 q^{35} +231012. q^{36} +230878. q^{37} +1568.00 q^{38} +195444. q^{39} -5040.00 q^{40} -304182. q^{41} +32472.0 q^{42} -525032. q^{43} +1.07062e6 q^{44} +18630.0 q^{45} -154660. q^{46} +802752. q^{47} +267552. q^{48} -9939.00 q^{49} +156050. q^{50} +88434.0 q^{51} -1.34639e6 q^{52} +152862. q^{53} +145800. q^{54} +86340.0 q^{55} -454608. q^{56} -14112.0 q^{57} +36420.0 q^{58} -1.60241e6 q^{59} +22320.0 q^{60} -2.60161e6 q^{61} +474004. q^{62} +1.68043e6 q^{63} -1.71411e6 q^{64} -108580. q^{65} +310824. q^{66} +1.07460e6 q^{67} -609212. q^{68} +1.39194e6 q^{69} -18040.0 q^{70} -502298. q^{71} -938952. q^{72} +3.64826e6 q^{73} -461756. q^{74} -1.40445e6 q^{75} +97216.0 q^{76} +7.78787e6 q^{77} -390888. q^{78} -2.89217e6 q^{79} -148640. q^{80} +2.76218e6 q^{81} +608364. q^{82} +728104. q^{83} +2.01326e6 q^{84} -49130.0 q^{85} +1.05006e6 q^{86} -327780. q^{87} -4.35154e6 q^{88} +7.93185e6 q^{89} -37260.0 q^{90} -9.79392e6 q^{91} -9.58892e6 q^{92} -4.26604e6 q^{93} -1.60550e6 q^{94} +7840.00 q^{95} -1.69632e6 q^{96} -6.55104e6 q^{97} +19878.0 q^{98} +1.60851e7 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−0.176777	−0.0883883	−	0.996086i	$-0.528172\pi$
$2$	−2.00000	−0.176777	−0.0883883	+	0.996086i	$0.528172\pi$
$3$	18.0000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	18.0000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	−124.000	−0.968750
$5$	−10.0000	−0.0357771	−0.0178885	−	0.999840i	$-0.505694\pi$
$5$	−10.0000	−0.0357771	−0.0178885	+	0.999840i	$0.505694\pi$
$6$	−36.0000	−0.0680414
$7$	−902.000	−0.993947	−0.496974	−	0.867766i	$-0.665555\pi$
$7$	−902.000	−0.993947	−0.496974	+	0.867766i	$0.665555\pi$
$8$	504.000	0.348029
$9$	−1863.00	−0.851852
$10$	20.0000	0.00632456
$11$	−8634.00	−1.95586	−0.977930	−	0.208934i	$-0.933001\pi$
$11$	−8634.00	−1.95586	−0.977930	+	0.208934i	$0.933001\pi$
$12$	−2232.00	−0.372872
$13$	10858.0	1.37072	0.685359	−	0.728205i	$-0.259645\pi$
$13$	10858.0	1.37072	0.685359	+	0.728205i	$0.259645\pi$
$14$	1804.00	0.175707
$15$	−180.000	−0.0137706
$16$	14864.0	0.907227
$17$	4913.00	0.242536
$17$	4913.00	0.242536
$18$	3726.00	0.150588
$19$	−784.000	−0.0262228	−0.0131114	−	0.999914i	$-0.504174\pi$
$19$	−784.000	−0.0262228	−0.0131114	+	0.999914i	$0.504174\pi$
$20$	1240.00	0.0346591
$21$	−16236.0	−0.382571
$22$	17268.0	0.345750
$23$	77330.0	1.32526	0.662629	−	0.748948i	$-0.269441\pi$
$23$	77330.0	1.32526	0.662629	+	0.748948i	$0.269441\pi$
$24$	9072.00	0.133956
$25$	−78025.0	−0.998720
$26$	−21716.0	−0.242311
$27$	−72900.0	−0.712778
$28$	111848.	0.962887
$29$	−18210.0	−0.138649	−0.0693245	−	0.997594i	$-0.522084\pi$
$29$	−18210.0	−0.138649	−0.0693245	+	0.997594i	$0.522084\pi$
$30$	360.000	0.00243432
$31$	−237002.	−1.42885	−0.714424	−	0.699713i	$-0.753311\pi$
$31$	−237002.	−1.42885	−0.714424	+	0.699713i	$0.753311\pi$
$32$	−94240.0	−0.508406
$33$	−155412.	−0.752811
$34$	−9826.00	−0.0428746
$35$	9020.00	0.0355605
$36$	231012.	0.825231
$37$	230878.	0.749336	0.374668	−	0.927159i	$-0.377757\pi$
$37$	230878.	0.749336	0.374668	+	0.927159i	$0.377757\pi$
$38$	1568.00	0.00463557
$39$	195444.	0.527590
$40$	−5040.00	−0.0124515
$41$	−304182.	−0.689271	−0.344636	−	0.938737i	$-0.611997\pi$
$41$	−304182.	−0.689271	−0.344636	+	0.938737i	$0.611997\pi$
$42$	32472.0	0.0676296
$43$	−525032.	−1.00704	−0.503519	−	0.863984i	$-0.667962\pi$
$43$	−525032.	−1.00704	−0.503519	+	0.863984i	$0.667962\pi$
$44$	1.07062e6	1.89474
$45$	18630.0	0.0304768
$46$	−154660.	−0.234275
$47$	802752.	1.12782	0.563909	−	0.825837i	$-0.309297\pi$
$47$	802752.	1.12782	0.563909	+	0.825837i	$0.309297\pi$
$48$	267552.	0.349192
$49$	−9939.00	−0.0120686
$50$	156050.	0.176550
$51$	88434.0	0.0933520
$52$	−1.34639e6	−1.32788
$53$	152862.	0.141037	0.0705187	−	0.997510i	$-0.477535\pi$
$53$	152862.	0.141037	0.0705187	+	0.997510i	$0.477535\pi$
$54$	145800.	0.126003
$55$	86340.0	0.0699750
$56$	−454608.	−0.345923
$57$	−14112.0	−0.0100931
$58$	36420.0	0.0245099
$59$	−1.60241e6	−1.01576	−0.507880	−	0.861428i	$-0.669571\pi$
$59$	−1.60241e6	−1.01576	−0.507880	+	0.861428i	$0.669571\pi$
$60$	22320.0	0.0133403
$61$	−2.60161e6	−1.46753	−0.733766	−	0.679402i	$-0.762239\pi$
$61$	−2.60161e6	−1.46753	−0.733766	+	0.679402i	$0.762239\pi$
$62$	474004.	0.252587
$63$	1.68043e6	0.846696
$64$	−1.71411e6	−0.817352
$65$	−108580.	−0.0490403
$66$	310824.	0.133079
$67$	1.07460e6	0.436502	0.218251	−	0.975893i	$-0.429965\pi$
$67$	1.07460e6	0.436502	0.218251	+	0.975893i	$0.429965\pi$
$68$	−609212.	−0.234956
$69$	1.39194e6	0.510092
$70$	−18040.0	−0.00628628
$71$	−502298.	−0.166555	−0.0832774	−	0.996526i	$-0.526539\pi$
$71$	−502298.	−0.166555	−0.0832774	+	0.996526i	$0.526539\pi$
$72$	−938952.	−0.296469
$73$	3.64826e6	1.09763	0.548814	−	0.835944i	$-0.315079\pi$
$73$	3.64826e6	1.09763	0.548814	+	0.835944i	$0.315079\pi$
$74$	−461756.	−0.132465
$75$	−1.40445e6	−0.384408
$76$	97216.0	0.0254033
$77$	7.78787e6	1.94402
$78$	−390888.	−0.0932655
$79$	−2.89217e6	−0.659978	−0.329989	−	0.943985i	$-0.607045\pi$
$79$	−2.89217e6	−0.659978	−0.329989	+	0.943985i	$0.607045\pi$
$80$	−148640.	−0.0324579
$81$	2.76218e6	0.577503
$82$	608364.	0.121847
$83$	728104.	0.139772	0.0698860	−	0.997555i	$-0.477736\pi$
$83$	728104.	0.139772	0.0698860	+	0.997555i	$0.477736\pi$
$84$	2.01326e6	0.370615
$85$	−49130.0	−0.00867722
$86$	1.05006e6	0.178021
$87$	−327780.	−0.0533661
$88$	−4.35154e6	−0.680696
$89$	7.93185e6	1.19264	0.596320	−	0.802747i	$-0.296629\pi$
$89$	7.93185e6	1.19264	0.596320	+	0.802747i	$0.296629\pi$
$90$	−37260.0	−0.00538758
$91$	−9.79392e6	−1.36242
$92$	−9.58892e6	−1.28384
$93$	−4.26604e6	−0.549964
$94$	−1.60550e6	−0.199372
$95$	7840.00	0.000938174 0
$96$	−1.69632e6	−0.195685
$97$	−6.55104e6	−0.728801	−0.364401	−	0.931242i	$-0.618726\pi$
$97$	−6.55104e6	−0.728801	−0.364401	+	0.931242i	$0.618726\pi$
$98$	19878.0	0.00213344
$99$	1.60851e7	1.66610
$100$	9.67510e6	0.967510
$101$	−4.98083e6	−0.481035	−0.240518	−	0.970645i	$-0.577317\pi$
$101$	−4.98083e6	−0.481035	−0.240518	+	0.970645i	$0.577317\pi$
$102$	−176868.	−0.0165025
$103$	−1.89286e7	−1.70683	−0.853413	−	0.521235i	$-0.825471\pi$
$103$	−1.89286e7	−1.70683	−0.853413	+	0.521235i	$0.825471\pi$
$104$	5.47243e6	0.477050
$105$	162360.	0.0136873
$106$	−305724.	−0.0249321
$107$	−356942.	−0.0281679	−0.0140839	−	0.999901i	$-0.504483\pi$
$107$	−356942.	−0.0281679	−0.0140839	+	0.999901i	$0.504483\pi$
$108$	9.03960e6	0.690504
$109$	−9.43951e6	−0.698163	−0.349081	−	0.937092i	$-0.613506\pi$
$109$	−9.43951e6	−0.698163	−0.349081	+	0.937092i	$0.613506\pi$
$110$	−172680.	−0.0123699
$111$	4.15580e6	0.288420
$112$	−1.34073e7	−0.901735
$113$	1.96913e7	1.28381	0.641904	−	0.766785i	$-0.278145\pi$
$113$	1.96913e7	1.28381	0.641904	+	0.766785i	$0.278145\pi$
$114$	28224.0	0.00178423
$115$	−773300.	−0.0474139
$116$	2.25804e6	0.134316
$117$	−2.02285e7	−1.16765
$118$	3.20482e6	0.179563
$119$	−4.43153e6	−0.241068
$120$	−90720.0	−0.00479257
$121$	5.50588e7	2.82539
$122$	5.20322e6	0.259426
$123$	−5.47528e6	−0.265301
$124$	2.93882e7	1.38420
$125$	1.56150e6	0.0715084
$126$	−3.36085e6	−0.149676
$127$	−2.05768e7	−0.891383	−0.445692	−	0.895187i	$-0.647042\pi$
$127$	−2.05768e7	−0.891383	−0.445692	+	0.895187i	$0.647042\pi$
$128$	1.54909e7	0.652894
$129$	−9.45058e6	−0.387609
$130$	217160.	0.00866918
$131$	−3.18813e7	−1.23904	−0.619522	−	0.784979i	$-0.712673\pi$
$131$	−3.18813e7	−1.23904	−0.619522	+	0.784979i	$0.712673\pi$
$132$	1.92711e7	0.729285
$133$	707168.	0.0260640
$134$	−2.14921e6	−0.0771635
$135$	729000.	0.0255011
$136$	2.47615e6	0.0844095
$137$	−7.36814e6	−0.244814	−0.122407	−	0.992480i	$-0.539061\pi$
$137$	−7.36814e6	−0.244814	−0.122407	+	0.992480i	$0.539061\pi$
$138$	−2.78388e6	−0.0901724
$139$	−1.90889e7	−0.602879	−0.301439	−	0.953485i	$-0.597467\pi$
$139$	−1.90889e7	−0.602879	−0.301439	+	0.953485i	$0.597467\pi$
$140$	−1.11848e6	−0.0344493
$141$	1.44495e7	0.434097
$142$	1.00460e6	0.0294430
$143$	−9.37480e7	−2.68093
$144$	−2.76916e7	−0.772823
$145$	182100.	0.00496046
$146$	−7.29652e6	−0.194035
$147$	−178902.	−0.00464520
$148$	−2.86289e7	−0.725919
$149$	7.08313e7	1.75418	0.877088	−	0.480329i	$-0.159483\pi$
$149$	7.08313e7	1.75418	0.877088	+	0.480329i	$0.159483\pi$
$150$	2.80890e6	0.0679543
$151$	6.20655e7	1.46700	0.733502	−	0.679688i	$-0.237885\pi$
$151$	6.20655e7	1.46700	0.733502	+	0.679688i	$0.237885\pi$
$152$	−395136.	−0.00912629
$153$	−9.15292e6	−0.206604
$154$	−1.55757e7	−0.343658
$155$	2.37002e6	0.0511200
$156$	−2.42351e7	−0.511102
$157$	3.85566e7	0.795152	0.397576	−	0.917569i	$-0.369852\pi$
$157$	3.85566e7	0.795152	0.397576	+	0.917569i	$0.369852\pi$
$158$	5.78435e6	0.116669
$159$	2.75152e6	0.0542853
$160$	942400.	0.0181893
$161$	−6.97517e7	−1.31724
$162$	−5.52436e6	−0.102089
$163$	−3.95993e7	−0.716195	−0.358097	−	0.933684i	$-0.616574\pi$
$163$	−3.95993e7	−0.716195	−0.358097	+	0.933684i	$0.616574\pi$
$164$	3.77186e7	0.667731
$165$	1.55412e6	0.0269334
$166$	−1.45621e6	−0.0247084
$167$	7.45237e7	1.23819	0.619094	−	0.785317i	$-0.287500\pi$
$167$	7.45237e7	1.23819	0.619094	+	0.785317i	$0.287500\pi$
$168$	−8.18294e6	−0.133146
$169$	5.51476e7	0.878868
$170$	98260.0	0.00153393
$171$	1.46059e6	0.0223379
$172$	6.51040e7	0.975569
$173$	3.97054e7	0.583027	0.291513	−	0.956567i	$-0.405841\pi$
$173$	3.97054e7	0.583027	0.291513	+	0.956567i	$0.405841\pi$
$174$	655560.	0.00943387
$175$	7.03786e7	0.992675
$176$	−1.28336e8	−1.77441
$177$	−2.88433e7	−0.390966
$178$	−1.58637e7	−0.210831
$179$	1.17211e8	1.52751	0.763753	−	0.645509i	$-0.223355\pi$
$179$	1.17211e8	1.52751	0.763753	+	0.645509i	$0.223355\pi$
$180$	−2.31012e6	−0.0295244
$181$	1.35726e7	0.170133	0.0850664	−	0.996375i	$-0.472890\pi$
$181$	1.35726e7	0.170133	0.0850664	+	0.996375i	$0.472890\pi$
$182$	1.95878e7	0.240844
$183$	−4.68290e7	−0.564854
$184$	3.89743e7	0.461229
$185$	−2.30878e6	−0.0268091
$186$	8.53207e6	0.0972208
$187$	−4.24188e7	−0.474366
$188$	−9.95412e7	−1.09257
$189$	6.57558e7	0.708464
$190$	−15680.0	−0.000165847 0
$191$	−1.39487e8	−1.44850	−0.724249	−	0.689538i	$-0.757814\pi$
$191$	−1.39487e8	−1.44850	−0.724249	+	0.689538i	$0.757814\pi$
$192$	−3.08540e7	−0.314599
$193$	−5.24931e7	−0.525595	−0.262798	−	0.964851i	$-0.584645\pi$
$193$	−5.24931e7	−0.525595	−0.262798	+	0.964851i	$0.584645\pi$
$194$	1.31021e7	0.128835
$195$	−1.95444e6	−0.0188756
$196$	1.23244e6	0.0116914
$197$	−7.25482e7	−0.676075	−0.338037	−	0.941133i	$-0.609763\pi$
$197$	−7.25482e7	−0.676075	−0.338037	+	0.941133i	$0.609763\pi$
$198$	−3.21703e7	−0.294528
$199$	3.85453e7	0.346725	0.173363	−	0.984858i	$-0.444537\pi$
$199$	3.85453e7	0.346725	0.173363	+	0.984858i	$0.444537\pi$
$200$	−3.93246e7	−0.347584
$201$	1.93429e7	0.168010
$202$	9.96166e6	0.0850358
$203$	1.64254e7	0.137810
$204$	−1.09658e7	−0.0904348
$205$	3.04182e6	0.0246601
$206$	3.78573e7	0.301727
$207$	−1.44066e8	−1.12892
$208$	1.61393e8	1.24355
$209$	6.76906e6	0.0512880
$210$	−324720.	−0.00241959
$211$	−1.28790e8	−0.943831	−0.471916	−	0.881644i	$-0.656437\pi$
$211$	−1.28790e8	−0.943831	−0.471916	+	0.881644i	$0.656437\pi$
$212$	−1.89549e7	−0.136630
$213$	−9.04136e6	−0.0641070
$214$	713884.	0.00497943
$215$	5.25032e6	0.0360289
$216$	−3.67416e7	−0.248068
$217$	2.13776e8	1.42020
$218$	1.88790e7	0.123419
$219$	6.56686e7	0.422478
$220$	−1.07062e7	−0.0677882
$221$	5.33454e7	0.332448
$222$	−8.31161e6	−0.0509858
$223$	−1.81673e8	−1.09704	−0.548520	−	0.836137i	$-0.684809\pi$
$223$	−1.81673e8	−1.09704	−0.548520	+	0.836137i	$0.684809\pi$
$224$	8.50045e7	0.505328
$225$	1.45361e8	0.850761
$226$	−3.93826e7	−0.226947
$227$	5.81830e7	0.330146	0.165073	−	0.986281i	$-0.447214\pi$
$227$	5.81830e7	0.330146	0.165073	+	0.986281i	$0.447214\pi$
$228$	1.74989e6	0.00977774
$229$	−1.70767e7	−0.0939681	−0.0469840	−	0.998896i	$-0.514961\pi$
$229$	−1.70767e7	−0.0939681	−0.0469840	+	0.998896i	$0.514961\pi$
$230$	1.54660e6	0.00838167
$231$	1.40182e8	0.748254
$232$	−9.17784e6	−0.0482539
$233$	−1.55038e8	−0.802955	−0.401478	−	0.915869i	$-0.631503\pi$
$233$	−1.55038e8	−0.802955	−0.401478	+	0.915869i	$0.631503\pi$
$234$	4.04569e7	0.206413
$235$	−8.02752e6	−0.0403500
$236$	1.98699e8	0.984017
$237$	−5.20591e7	−0.254026
$238$	8.86305e6	0.0426151
$239$	8.87298e7	0.420414	0.210207	−	0.977657i	$-0.432586\pi$
$239$	8.87298e7	0.420414	0.210207	+	0.977657i	$0.432586\pi$
$240$	−2.67552e6	−0.0124931
$241$	−5.45231e7	−0.250912	−0.125456	−	0.992099i	$-0.540039\pi$
$241$	−5.45231e7	−0.250912	−0.125456	+	0.992099i	$0.540039\pi$
$242$	−1.10118e8	−0.499462
$243$	2.09152e8	0.935059
$244$	3.22600e8	1.42167
$245$	99390.0	0.000431779 0
$246$	1.09506e7	0.0468990
$247$	−8.51267e6	−0.0359440
$248$	−1.19449e8	−0.497281
$249$	1.31059e7	0.0537983
$250$	−3.12300e6	−0.0126410
$251$	−3.13299e8	−1.25055	−0.625275	−	0.780404i	$-0.715013\pi$
$251$	−3.13299e8	−1.25055	−0.625275	+	0.780404i	$0.715013\pi$
$252$	−2.08373e8	−0.820237
$253$	−6.67667e8	−2.59202
$254$	4.11536e7	0.157576
$255$	−884340.	−0.00333986
$256$	1.88424e8	0.701936
$257$	1.99287e8	0.732342	0.366171	−	0.930548i	$-0.380668\pi$
$257$	1.99287e8	0.732342	0.366171	+	0.930548i	$0.380668\pi$
$258$	1.89012e7	0.0685203
$259$	−2.08252e8	−0.744800
$260$	1.34639e7	0.0475078
$261$	3.39252e7	0.118108
$262$	6.37626e7	0.219034
$263$	2.21715e8	0.751535	0.375768	−	0.926714i	$-0.377379\pi$
$263$	2.21715e8	0.751535	0.375768	+	0.926714i	$0.377379\pi$
$264$	−7.83276e7	−0.262000
$265$	−1.52862e6	−0.00504590
$266$	−1.41434e6	−0.00460752
$267$	1.42773e8	0.459047
$268$	−1.33251e8	−0.422862
$269$	−2.07584e8	−0.650220	−0.325110	−	0.945676i	$-0.605401\pi$
$269$	−2.07584e8	−0.650220	−0.325110	+	0.945676i	$0.605401\pi$
$270$	−1.45800e6	−0.00450800
$271$	−4.53261e8	−1.38343	−0.691713	−	0.722172i	$-0.743144\pi$
$271$	−4.53261e8	−1.38343	−0.691713	+	0.722172i	$0.743144\pi$
$272$	7.30268e7	0.220035
$273$	−1.76290e8	−0.524396
$274$	1.47363e7	0.0432774
$275$	6.73668e8	1.95336
$276$	−1.72601e8	−0.494152
$277$	−5.50284e8	−1.55563	−0.777817	−	0.628490i	$-0.783673\pi$
$277$	−5.50284e8	−1.55563	−0.777817	+	0.628490i	$0.783673\pi$
$278$	3.81779e7	0.106575
$279$	4.41535e8	1.21717
$280$	4.54608e6	0.0123761
$281$	−3.35160e8	−0.901113	−0.450557	−	0.892748i	$-0.648774\pi$
$281$	−3.35160e8	−0.901113	−0.450557	+	0.892748i	$0.648774\pi$
$282$	−2.88991e7	−0.0767383
$283$	−1.32532e8	−0.347590	−0.173795	−	0.984782i	$-0.555603\pi$
$283$	−1.32532e8	−0.347590	−0.173795	+	0.984782i	$0.555603\pi$
$284$	6.22850e7	0.161350
$285$	141120.	0.000361103 0
$286$	1.87496e8	0.473926
$287$	2.74372e8	0.685099
$288$	1.75569e8	0.433086
$289$	2.41376e7	0.0588235
$290$	−364200.	−0.000876894 0
$291$	−1.17919e8	−0.280516
$292$	−4.52384e8	−1.06333
$293$	5.09454e8	1.18323	0.591614	−	0.806222i	$-0.298491\pi$
$293$	5.09454e8	1.18323	0.591614	+	0.806222i	$0.298491\pi$
$294$	357804.	0.000821163 0
$295$	1.60241e7	0.0363409
$296$	1.16363e8	0.260791
$297$	6.29419e8	1.39409
$298$	−1.41663e8	−0.310097
$299$	8.39649e8	1.81656
$300$	1.74152e8	0.372395
$301$	4.73579e8	1.00094
$302$	−1.24131e8	−0.259332
$303$	−8.96549e7	−0.185151
$304$	−1.16534e7	−0.0237900
$305$	2.60161e7	0.0525040
$306$	1.83058e7	0.0365228
$307$	1.42069e8	0.280230	0.140115	−	0.990135i	$-0.455253\pi$
$307$	1.42069e8	0.280230	0.140115	+	0.990135i	$0.455253\pi$
$308$	−9.65696e8	−1.88327
$309$	−3.40716e8	−0.656958
$310$	−4.74004e6	−0.00903683
$311$	−3.78883e8	−0.714239	−0.357119	−	0.934059i	$-0.616241\pi$
$311$	−3.78883e8	−0.714239	−0.357119	+	0.934059i	$0.616241\pi$
$312$	9.85038e7	0.183617
$313$	6.89702e8	1.27132	0.635662	−	0.771968i	$-0.280727\pi$
$313$	6.89702e8	1.27132	0.635662	+	0.771968i	$0.280727\pi$
$314$	−7.71132e7	−0.140564
$315$	−1.68043e7	−0.0302923
$316$	3.58630e8	0.639354
$317$	−4.58951e8	−0.809205	−0.404602	−	0.914493i	$-0.632590\pi$
$317$	−4.58951e8	−0.809205	−0.404602	+	0.914493i	$0.632590\pi$
$318$	−5.50303e6	−0.00959637
$319$	1.57225e8	0.271178
$320$	1.71411e7	0.0292425
$321$	−6.42496e6	−0.0108418
$322$	1.39503e8	0.232857
$323$	−3.85179e6	−0.00635995
$324$	−3.42510e8	−0.559456
$325$	−8.47195e8	−1.36896
$326$	7.91986e7	0.126607
$327$	−1.69911e8	−0.268723
$328$	−1.53308e8	−0.239886
$329$	−7.24082e8	−1.12099
$330$	−3.10824e6	−0.00476119
$331$	4.18177e8	0.633815	0.316907	−	0.948457i	$-0.397356\pi$
$331$	4.18177e8	0.633815	0.316907	+	0.948457i	$0.397356\pi$
$332$	−9.02849e7	−0.135404
$333$	−4.30126e8	−0.638323
$334$	−1.49047e8	−0.218883
$335$	−1.07460e7	−0.0156168
$336$	−2.41332e8	−0.347078
$337$	−1.29839e9	−1.84799	−0.923997	−	0.382400i	$-0.875098\pi$
$337$	−1.29839e9	−1.84799	−0.923997	+	0.382400i	$0.875098\pi$
$338$	−1.10295e8	−0.155363
$339$	3.54444e8	0.494138
$340$	6.09212e6	0.00840606
$341$	2.04628e9	2.79463
$342$	−2.92118e6	−0.00394882
$343$	7.51801e8	1.00594
$344$	−2.64616e8	−0.350479
$345$	−1.39194e7	−0.0182496
$346$	−7.94108e7	−0.103066
$347$	6.11236e8	0.785336	0.392668	−	0.919680i	$-0.371552\pi$
$347$	6.11236e8	0.785336	0.392668	+	0.919680i	$0.371552\pi$
$348$	4.06447e7	0.0516984
$349$	1.94590e8	0.245037	0.122519	−	0.992466i	$-0.460903\pi$
$349$	1.94590e8	0.245037	0.122519	+	0.992466i	$0.460903\pi$
$350$	−1.40757e8	−0.175482
$351$	−7.91548e8	−0.977018
$352$	8.13668e8	0.994370
$353$	−1.24068e9	−1.50124	−0.750619	−	0.660735i	$-0.770245\pi$
$353$	−1.24068e9	−1.50124	−0.750619	+	0.660735i	$0.770245\pi$
$354$	5.76867e7	0.0691137
$355$	5.02298e6	0.00595885
$356$	−9.83549e8	−1.15537
$357$	−7.97675e7	−0.0927870
$358$	−2.34422e8	−0.270027
$359$	−3.83631e8	−0.437606	−0.218803	−	0.975769i	$-0.570215\pi$
$359$	−3.83631e8	−0.437606	−0.218803	+	0.975769i	$0.570215\pi$
$360$	9.38952e6	0.0106068
$361$	−8.93257e8	−0.999312
$362$	−2.71452e7	−0.0300755
$363$	9.91058e8	1.08749
$364$	1.21445e9	1.31985
$365$	−3.64826e7	−0.0392700
$366$	9.36580e7	0.0998529
$367$	−2.60781e8	−0.275388	−0.137694	−	0.990475i	$-0.543969\pi$
$367$	−2.60781e8	−0.275388	−0.137694	+	0.990475i	$0.543969\pi$
$368$	1.14943e9	1.20231
$369$	5.66691e8	0.587157
$370$	4.61756e6	0.00473922
$371$	−1.37882e8	−0.140184
$372$	5.28988e8	0.532778
$373$	1.38439e9	1.38127	0.690636	−	0.723203i	$-0.257331\pi$
$373$	1.38439e9	1.38127	0.690636	+	0.723203i	$0.257331\pi$
$374$	8.48377e7	0.0838568
$375$	2.81070e7	0.0275236
$376$	4.04587e8	0.392513
$377$	−1.97724e8	−0.190049
$378$	−1.31512e8	−0.125240
$379$	3.20542e7	0.0302446	0.0151223	−	0.999886i	$-0.495186\pi$
$379$	3.20542e7	0.0302446	0.0151223	+	0.999886i	$0.495186\pi$
$380$	−972160.	−0.000908856 0
$381$	−3.70382e8	−0.343094
$382$	2.78975e8	0.256061
$383$	−8.57455e8	−0.779858	−0.389929	−	0.920845i	$-0.627500\pi$
$383$	−8.57455e8	−0.779858	−0.389929	+	0.920845i	$0.627500\pi$
$384$	2.78837e8	0.251299
$385$	−7.78787e7	−0.0695514
$386$	1.04986e8	0.0929130
$387$	9.78135e8	0.857848
$388$	8.12329e8	0.706026
$389$	−3.96303e8	−0.341353	−0.170676	−	0.985327i	$-0.554595\pi$
$389$	−3.96303e8	−0.341353	−0.170676	+	0.985327i	$0.554595\pi$
$390$	3.90888e6	0.00333677
$391$	3.79922e8	0.321422
$392$	−5.00926e6	−0.00420022
$393$	−5.73864e8	−0.476908
$394$	1.45096e8	0.119514
$395$	2.89217e7	0.0236121
$396$	−1.99456e9	−1.61404
$397$	6.18546e7	0.0496141	0.0248070	−	0.999692i	$-0.492103\pi$
$397$	6.18546e7	0.0496141	0.0248070	+	0.999692i	$0.492103\pi$
$398$	−7.70905e7	−0.0612929
$399$	1.27290e7	0.0100321
$400$	−1.15976e9	−0.906065
$401$	1.69703e9	1.31427	0.657134	−	0.753774i	$-0.271769\pi$
$401$	1.69703e9	1.31427	0.657134	+	0.753774i	$0.271769\pi$
$402$	−3.86857e7	−0.0297002
$403$	−2.57337e9	−1.95855
$404$	6.17623e8	0.466003
$405$	−2.76218e7	−0.0206614
$406$	−3.28508e7	−0.0243616
$407$	−1.99340e9	−1.46560
$408$	4.45707e7	0.0324892
$409$	−1.65566e9	−1.19658	−0.598288	−	0.801281i	$-0.704152\pi$
$409$	−1.65566e9	−1.19658	−0.598288	+	0.801281i	$0.704152\pi$
$410$	−6.08364e6	−0.00435933
$411$	−1.32626e8	−0.0942288
$412$	2.34715e9	1.65349
$413$	1.44537e9	1.00961
$414$	2.88132e8	0.199567
$415$	−7.28104e6	−0.00500063
$416$	−1.02326e9	−0.696881
$417$	−3.43601e8	−0.232048
$418$	−1.35381e7	−0.00906653
$419$	−5.53605e8	−0.367664	−0.183832	−	0.982958i	$-0.558850\pi$
$419$	−5.53605e8	−0.367664	−0.183832	+	0.982958i	$0.558850\pi$
$420$	−2.01326e7	−0.0132595
$421$	1.99245e8	0.130137	0.0650684	−	0.997881i	$-0.479273\pi$
$421$	1.99245e8	0.130137	0.0650684	+	0.997881i	$0.479273\pi$
$422$	2.57581e8	0.166847
$423$	−1.49553e9	−0.960734
$424$	7.70424e7	0.0490851
$425$	−3.83337e8	−0.242225
$426$	1.80827e7	0.0113326
$427$	2.34665e9	1.45865
$428$	4.42608e7	0.0272877
$429$	−1.68746e9	−1.03189
$430$	−1.05006e7	−0.00636907
$431$	−7.32044e8	−0.440420	−0.220210	−	0.975453i	$-0.570674\pi$
$431$	−7.32044e8	−0.440420	−0.220210	+	0.975453i	$0.570674\pi$
$432$	−1.08359e9	−0.646651
$433$	2.94958e9	1.74604	0.873018	−	0.487689i	$-0.162160\pi$
$433$	2.94958e9	1.74604	0.873018	+	0.487689i	$0.162160\pi$
$434$	−4.27552e8	−0.251058
$435$	3.27780e6	0.00190928
$436$	1.17050e9	0.676345
$437$	−6.06267e7	−0.0347519
$438$	−1.31337e8	−0.0746842
$439$	−2.08951e9	−1.17874	−0.589371	−	0.807862i	$-0.700624\pi$
$439$	−2.08951e9	−1.17874	−0.589371	+	0.807862i	$0.700624\pi$
$440$	4.35154e7	0.0243533
$441$	1.85164e7	0.0102806
$442$	−1.06691e8	−0.0587690
$443$	7.44873e8	0.407070	0.203535	−	0.979068i	$-0.434757\pi$
$443$	7.44873e8	0.407070	0.203535	+	0.979068i	$0.434757\pi$
$444$	−5.15320e8	−0.279406
$445$	−7.93185e7	−0.0426692
$446$	3.63345e8	0.193931
$447$	1.27496e9	0.675183
$448$	1.54613e9	0.812405
$449$	−1.61417e9	−0.841562	−0.420781	−	0.907162i	$-0.638244\pi$
$449$	−1.61417e9	−0.841562	−0.420781	+	0.907162i	$0.638244\pi$
$450$	−2.90721e8	−0.150395
$451$	2.62631e9	1.34812
$452$	−2.44172e9	−1.24369
$453$	1.11718e9	0.564650
$454$	−1.16366e8	−0.0583621
$455$	9.79392e7	0.0487435
$456$	−7.11245e6	−0.00351271
$457$	1.05126e9	0.515233	0.257616	−	0.966247i	$-0.417063\pi$
$457$	1.05126e9	0.515233	0.257616	+	0.966247i	$0.417063\pi$
$458$	3.41535e7	0.0166114
$459$	−3.58158e8	−0.172874
$460$	9.58892e7	0.0459322
$461$	2.78477e9	1.32384	0.661921	−	0.749574i	$-0.269742\pi$
$461$	2.78477e9	1.32384	0.661921	+	0.749574i	$0.269742\pi$
$462$	−2.80363e8	−0.132274
$463$	1.33904e8	0.0626988	0.0313494	−	0.999508i	$-0.490020\pi$
$463$	1.33904e8	0.0626988	0.0313494	+	0.999508i	$0.490020\pi$
$464$	−2.70673e8	−0.125786
$465$	4.26604e7	0.0196761
$466$	3.10075e8	0.141944
$467$	−1.30461e9	−0.592749	−0.296375	−	0.955072i	$-0.595778\pi$
$467$	−1.30461e9	−0.592749	−0.296375	+	0.955072i	$0.595778\pi$
$468$	2.50833e9	1.13116
$469$	−9.69293e8	−0.433860
$470$	1.60550e7	0.00713295
$471$	6.94019e8	0.306054
$472$	−8.07614e8	−0.353514
$473$	4.53313e9	1.96963
$474$	1.04118e8	0.0449058
$475$	6.11716e7	0.0261892
$476$	5.49509e8	0.233534
$477$	−2.84782e8	−0.120143
$478$	−1.77460e8	−0.0743193
$479$	6.53759e8	0.271796	0.135898	−	0.990723i	$-0.456608\pi$
$479$	6.53759e8	0.271796	0.135898	+	0.990723i	$0.456608\pi$
$480$	1.69632e7	0.00700105
$481$	2.50687e9	1.02713
$482$	1.09046e8	0.0443553
$483$	−1.25553e9	−0.507005
$484$	−6.82729e9	−2.73709
$485$	6.55104e7	0.0260744
$486$	−4.18303e8	−0.165297
$487$	−1.33317e9	−0.523039	−0.261520	−	0.965198i	$-0.584224\pi$
$487$	−1.33317e9	−0.523039	−0.261520	+	0.965198i	$0.584224\pi$
$488$	−1.31121e9	−0.510744
$489$	−7.12788e8	−0.275663
$490$	−198780.	−7.63284e−5 0
$491$	3.37872e8	0.128815	0.0644075	−	0.997924i	$-0.479484\pi$
$491$	3.37872e8	0.128815	0.0644075	+	0.997924i	$0.479484\pi$
$492$	6.78934e8	0.257010
$493$	−8.94657e7	−0.0336273
$494$	1.70253e7	0.00635406
$495$	−1.60851e8	−0.0596083
$496$	−3.52280e9	−1.29629
$497$	4.53073e8	0.165547
$498$	−2.62117e7	−0.00951028
$499$	−4.33968e9	−1.56353	−0.781764	−	0.623574i	$-0.785680\pi$
$499$	−4.33968e9	−1.56353	−0.781764	+	0.623574i	$0.785680\pi$
$500$	−1.93626e8	−0.0692737
$501$	1.34143e9	0.476579
$502$	6.26598e8	0.221068
$503$	−2.78322e9	−0.975122	−0.487561	−	0.873089i	$-0.662113\pi$
$503$	−2.78322e9	−0.975122	−0.487561	+	0.873089i	$0.662113\pi$
$504$	8.46935e8	0.294675
$505$	4.98083e7	0.0172100
$506$	1.33533e9	0.458209
$507$	9.92658e8	0.338276
$508$	2.55152e9	0.863528
$509$	4.16198e9	1.39890	0.699452	−	0.714680i	$-0.253428\pi$
$509$	4.16198e9	1.39890	0.699452	+	0.714680i	$0.253428\pi$
$510$	1.76868e6	0.000590410 0
$511$	−3.29073e9	−1.09099
$512$	−2.35969e9	−0.776980
$513$	5.71536e7	0.0186910
$514$	−3.98575e8	−0.129461
$515$	1.89286e8	0.0610653
$516$	1.17187e9	0.375497
$517$	−6.93096e9	−2.20585
$518$	4.16504e8	0.131663
$519$	7.14698e8	0.224407
$520$	−5.47243e7	−0.0170675
$521$	−4.28862e9	−1.32857	−0.664286	−	0.747478i	$-0.731264\pi$
$521$	−4.28862e9	−1.32857	−0.664286	+	0.747478i	$0.731264\pi$
$522$	−6.78505e7	−0.0208788
$523$	3.92321e9	1.19918	0.599592	−	0.800306i	$-0.295330\pi$
$523$	3.92321e9	1.19918	0.599592	+	0.800306i	$0.295330\pi$
$524$	3.95328e9	1.20032
$525$	1.26681e9	0.382081
$526$	−4.43429e8	−0.132854
$527$	−1.16439e9	−0.346547
$528$	−2.31004e9	−0.682970
$529$	2.57510e9	0.756310
$530$	3.05724e6	0.000891998 0
$531$	2.98529e9	0.865277
$532$	−8.76888e7	−0.0252495
$533$	−3.30281e9	−0.944796
$534$	−2.85546e8	−0.0811489
$535$	3.56942e6	0.00100777
$536$	5.41600e8	0.151916
$537$	2.10980e9	0.587937
$538$	4.15167e8	0.114944
$539$	8.58133e7	0.0236045
$540$	−9.03960e7	−0.0247042
$541$	1.34264e8	0.0364561	0.0182280	−	0.999834i	$-0.494198\pi$
$541$	1.34264e8	0.0364561	0.0182280	+	0.999834i	$0.494198\pi$
$542$	9.06522e8	0.244558
$543$	2.44307e8	0.0654841
$544$	−4.63001e8	−0.123306
$545$	9.43951e7	0.0249782
$546$	3.52581e8	0.0927010
$547$	3.46681e9	0.905678	0.452839	−	0.891592i	$-0.350411\pi$
$547$	3.46681e9	0.905678	0.452839	+	0.891592i	$0.350411\pi$
$548$	9.13649e8	0.237163
$549$	4.84680e9	1.25012
$550$	−1.34734e9	−0.345308
$551$	1.42766e7	0.00363576
$552$	7.01538e8	0.177527
$553$	2.60874e9	0.655984
$554$	1.10057e9	0.275000
$555$	−4.15580e7	−0.0103188
$556$	2.36703e9	0.584039
$557$	7.44194e8	0.182471	0.0912353	−	0.995829i	$-0.470918\pi$
$557$	7.44194e8	0.182471	0.0912353	+	0.995829i	$0.470918\pi$
$558$	−8.83069e8	−0.215167
$559$	−5.70080e9	−1.38037
$560$	1.34073e8	0.0322615
$561$	−7.63539e8	−0.182583
$562$	6.70319e8	0.159296
$563$	8.26316e9	1.95149	0.975746	−	0.218908i	$-0.0702494\pi$
$563$	8.26316e9	1.95149	0.975746	+	0.218908i	$0.0702494\pi$
$564$	−1.79174e9	−0.420532
$565$	−1.96913e8	−0.0459309
$566$	2.65063e8	0.0614458
$567$	−2.49149e9	−0.574008
$568$	−2.53158e8	−0.0579659
$569$	−6.15288e9	−1.40018	−0.700092	−	0.714052i	$-0.746858\pi$
$569$	−6.15288e9	−1.40018	−0.700092	+	0.714052i	$0.746858\pi$
$570$	−282240.	−6.38347e−5 0
$571$	−3.21445e9	−0.722571	−0.361286	−	0.932455i	$-0.617662\pi$
$571$	−3.21445e9	−0.722571	−0.361286	+	0.932455i	$0.617662\pi$
$572$	1.16247e10	2.59715
$573$	−2.51077e9	−0.557527
$574$	−5.48744e8	−0.121110
$575$	−6.03367e9	−1.32356
$576$	3.19339e9	0.696263
$577$	−5.41775e9	−1.17410	−0.587048	−	0.809552i	$-0.699710\pi$
$577$	−5.41775e9	−1.17410	−0.587048	+	0.809552i	$0.699710\pi$
$578$	−4.82751e7	−0.0103986
$579$	−9.44875e8	−0.202302
$580$	−2.25804e7	−0.00480545
$581$	−6.56750e8	−0.138926
$582$	2.35837e8	0.0495886
$583$	−1.31981e9	−0.275849
$584$	1.83872e9	0.382007
$585$	2.02285e8	0.0417751
$586$	−1.01891e9	−0.209167
$587$	4.72616e9	0.964439	0.482219	−	0.876050i	$-0.339831\pi$
$587$	4.72616e9	0.964439	0.482219	+	0.876050i	$0.339831\pi$
$588$	2.21838e7	0.00450004
$589$	1.85810e8	0.0374683
$590$	−3.20482e7	−0.00642423
$591$	−1.30587e9	−0.260221
$592$	3.43177e9	0.679817
$593$	−1.99813e9	−0.393489	−0.196744	−	0.980455i	$-0.563037\pi$
$593$	−1.99813e9	−0.393489	−0.196744	+	0.980455i	$0.563037\pi$
$594$	−1.25884e9	−0.246443
$595$	4.43153e7	0.00862470
$596$	−8.78308e9	−1.69936
$597$	6.93815e8	0.133455
$598$	−1.67930e9	−0.321125
$599$	−6.30383e9	−1.19842	−0.599212	−	0.800590i	$-0.704519\pi$
$599$	−6.30383e9	−1.19842	−0.599212	+	0.800590i	$0.704519\pi$
$600$	−7.07843e8	−0.133785
$601$	6.82907e9	1.28322	0.641609	−	0.767031i	$-0.278267\pi$
$601$	6.82907e9	1.28322	0.641609	+	0.767031i	$0.278267\pi$
$602$	−9.47158e8	−0.176943
$603$	−2.00199e9	−0.371835
$604$	−7.69613e9	−1.42116
$605$	−5.50588e8	−0.101084
$606$	1.79310e8	0.0327303
$607$	−2.39101e9	−0.433931	−0.216965	−	0.976179i	$-0.569616\pi$
$607$	−2.39101e9	−0.433931	−0.216965	+	0.976179i	$0.569616\pi$
$608$	7.38842e7	0.0133318
$609$	2.95658e8	0.0530430
$610$	−5.20322e7	−0.00928149
$611$	8.71628e9	1.54592
$612$	1.13496e9	0.200148
$613$	7.47316e9	1.31037	0.655183	−	0.755470i	$-0.272592\pi$
$613$	7.47316e9	1.31037	0.655183	+	0.755470i	$0.272592\pi$
$614$	−2.84138e8	−0.0495381
$615$	5.47528e7	0.00949168
$616$	3.92509e9	0.676576
$617$	1.00330e9	0.171961	0.0859806	−	0.996297i	$-0.472598\pi$
$617$	1.00330e9	0.171961	0.0859806	+	0.996297i	$0.472598\pi$
$618$	6.81431e8	0.116135
$619$	−4.24625e9	−0.719595	−0.359798	−	0.933030i	$-0.617154\pi$
$619$	−4.24625e9	−0.719595	−0.359798	+	0.933030i	$0.617154\pi$
$620$	−2.93882e8	−0.0495225
$621$	−5.63736e9	−0.944615
$622$	7.57765e8	0.126261
$623$	−7.15453e9	−1.18542
$624$	2.90508e9	0.478643
$625$	6.08009e9	0.996162
$626$	−1.37940e9	−0.224740
$627$	1.21843e8	0.0197408
$628$	−4.78102e9	−0.770303
$629$	1.13430e9	0.181741
$630$	3.36085e7	0.00535498
$631$	−3.97776e9	−0.630284	−0.315142	−	0.949045i	$-0.602052\pi$
$631$	−3.97776e9	−0.630284	−0.315142	+	0.949045i	$0.602052\pi$
$632$	−1.45766e9	−0.229692
$633$	−2.31822e9	−0.363281
$634$	9.17901e8	0.143049
$635$	2.05768e8	0.0318911
$636$	−3.41188e8	−0.0525889
$637$	−1.07918e8	−0.0165426
$638$	−3.14450e8	−0.0479380
$639$	9.35781e8	0.141880
$640$	−1.54909e8	−0.0233587
$641$	−1.14888e10	−1.72295	−0.861473	−	0.507803i	$-0.830458\pi$
$641$	−1.14888e10	−1.72295	−0.861473	+	0.507803i	$0.830458\pi$
$642$	1.28499e7	0.00191658
$643$	6.31597e9	0.936918	0.468459	−	0.883485i	$-0.344809\pi$
$643$	6.31597e9	0.936918	0.468459	+	0.883485i	$0.344809\pi$
$644$	8.64921e9	1.27607
$645$	9.45058e7	0.0138675
$646$	7.70358e6	0.00112429
$647$	7.62179e9	1.10635	0.553174	−	0.833066i	$-0.313416\pi$
$647$	7.62179e9	1.10635	0.553174	+	0.833066i	$0.313416\pi$
$648$	1.39214e9	0.200988
$649$	1.38352e10	1.98668
$650$	1.69439e9	0.242001
$651$	3.84796e9	0.546635
$652$	4.91032e9	0.693814
$653$	5.93777e9	0.834502	0.417251	−	0.908791i	$-0.362994\pi$
$653$	5.93777e9	0.834502	0.417251	+	0.908791i	$0.362994\pi$
$654$	3.39822e8	0.0475039
$655$	3.18813e8	0.0443294
$656$	−4.52136e9	−0.625325
$657$	−6.79670e9	−0.935017
$658$	1.44816e9	0.198165
$659$	6.71216e9	0.913615	0.456808	−	0.889565i	$-0.348993\pi$
$659$	6.71216e9	0.913615	0.456808	+	0.889565i	$0.348993\pi$
$660$	−1.92711e8	−0.0260917
$661$	−1.09016e10	−1.46820	−0.734101	−	0.679040i	$-0.762396\pi$
$661$	−1.09016e10	−1.46820	−0.734101	+	0.679040i	$0.762396\pi$
$662$	−8.36354e8	−0.112044
$663$	9.60216e8	0.127959
$664$	3.66964e8	0.0486447
$665$	−7.07168e6	−0.000932496 0
$666$	8.60251e8	0.112841
$667$	−1.40818e9	−0.183746
$668$	−9.24094e9	−1.19949
$669$	−3.27011e9	−0.422251
$670$	2.14921e7	0.00276068
$671$	2.24623e10	2.87029
$672$	1.53008e9	0.194501
$673$	6.39542e9	0.808754	0.404377	−	0.914592i	$-0.367488\pi$
$673$	6.39542e9	0.808754	0.404377	+	0.914592i	$0.367488\pi$
$674$	2.59678e9	0.326682
$675$	5.68802e9	0.711866
$676$	−6.83831e9	−0.851403
$677$	−1.16017e10	−1.43702	−0.718508	−	0.695519i	$-0.755175\pi$
$677$	−1.16017e10	−1.43702	−0.718508	+	0.695519i	$0.755175\pi$
$678$	−7.08887e8	−0.0873520
$679$	5.90904e9	0.724390
$680$	−2.47615e7	−0.00301992
$681$	1.04729e9	0.127073
$682$	−4.09255e9	−0.494025
$683$	−1.24796e10	−1.49875	−0.749373	−	0.662149i	$-0.769645\pi$
$683$	−1.24796e10	−1.49875	−0.749373	+	0.662149i	$0.769645\pi$
$684$	−1.81113e8	−0.0216399
$685$	7.36814e7	0.00875872
$686$	−1.50360e9	−0.177827
$687$	−3.07381e8	−0.0361683
$688$	−7.80408e9	−0.913612
$689$	1.65978e9	0.193322
$690$	2.78388e7	0.00322611
$691$	3.42198e9	0.394552	0.197276	−	0.980348i	$-0.436790\pi$
$691$	3.42198e9	0.394552	0.197276	+	0.980348i	$0.436790\pi$
$692$	−4.92347e9	−0.564807
$693$	−1.45088e10	−1.65602
$694$	−1.22247e9	−0.138829
$695$	1.90889e8	0.0215692
$696$	−1.65201e8	−0.0185729
$697$	−1.49445e9	−0.167173
$698$	−3.89181e8	−0.0433169
$699$	−2.79068e9	−0.309058
$700$	−8.72694e9	−0.961654
$701$	−3.02723e9	−0.331919	−0.165960	−	0.986133i	$-0.553072\pi$
$701$	−3.02723e9	−0.331919	−0.165960	+	0.986133i	$0.553072\pi$
$702$	1.58310e9	0.172714
$703$	−1.81008e8	−0.0196497
$704$	1.47996e10	1.59863
$705$	−1.44495e8	−0.0155307
$706$	2.48137e9	0.265384
$707$	4.49271e9	0.478124
$708$	3.57657e9	0.378748
$709$	1.27703e10	1.34568	0.672838	−	0.739790i	$-0.265075\pi$
$709$	1.27703e10	1.34568	0.672838	+	0.739790i	$0.265075\pi$
$710$	−1.00460e7	−0.00105339
$711$	5.38812e9	0.562204
$712$	3.99765e9	0.415073
$713$	−1.83274e10	−1.89359
$714$	1.59535e8	0.0164026
$715$	9.37480e8	0.0959159
$716$	−1.45342e10	−1.47977
$717$	1.59714e9	0.161817
$718$	7.67263e8	0.0773586
$719$	−1.47159e10	−1.47650	−0.738252	−	0.674526i	$-0.764348\pi$
$719$	−1.47159e10	−1.47650	−0.738252	+	0.674526i	$0.764348\pi$
$720$	2.76916e8	0.0276493
$721$	1.70736e10	1.69650
$722$	1.78651e9	0.176655
$723$	−9.81415e8	−0.0965759
$724$	−1.68300e9	−0.164816
$725$	1.42084e9	0.138472
$726$	−1.98212e9	−0.192243
$727$	−1.65109e10	−1.59367	−0.796836	−	0.604195i	$-0.793495\pi$
$727$	−1.65109e10	−1.59367	−0.796836	+	0.604195i	$0.793495\pi$
$728$	−4.93613e9	−0.474162
$729$	−2.27616e9	−0.217599
$730$	7.29652e7	0.00694201
$731$	−2.57948e9	−0.244243
$732$	5.80679e9	0.547202
$733$	2.72589e9	0.255649	0.127825	−	0.991797i	$-0.459201\pi$
$733$	2.72589e9	0.255649	0.127825	+	0.991797i	$0.459201\pi$
$734$	5.21562e8	0.0486822
$735$	1.78902e6	0.000166192 0
$736$	−7.28758e9	−0.673769
$737$	−9.27813e9	−0.853737
$738$	−1.13338e9	−0.103796
$739$	2.00377e10	1.82639	0.913193	−	0.407527i	$-0.133609\pi$
$739$	2.00377e10	1.82639	0.913193	+	0.407527i	$0.133609\pi$
$740$	2.86289e8	0.0259713
$741$	−1.53228e8	−0.0138349
$742$	2.75763e8	0.0247812
$743$	1.60033e10	1.43136	0.715681	−	0.698428i	$-0.246117\pi$
$743$	1.60033e10	1.43136	0.715681	+	0.698428i	$0.246117\pi$
$744$	−2.15008e9	−0.191403
$745$	−7.08313e8	−0.0627593
$746$	−2.76879e9	−0.244177
$747$	−1.35646e9	−0.119065
$748$	5.25994e9	0.459542
$749$	3.21962e8	0.0279974
$750$	−5.62140e7	−0.00486553
$751$	1.30391e10	1.12334	0.561668	−	0.827363i	$-0.310160\pi$
$751$	1.30391e10	1.12334	0.561668	+	0.827363i	$0.310160\pi$
$752$	1.19321e10	1.02319
$753$	−5.63938e9	−0.481337
$754$	3.95448e8	0.0335962
$755$	−6.20655e8	−0.0524851
$756$	−8.15372e9	−0.686324
$757$	−8.72547e9	−0.731060	−0.365530	−	0.930800i	$-0.619112\pi$
$757$	−8.72547e9	−0.731060	−0.365530	+	0.930800i	$0.619112\pi$
$758$	−6.41083e7	−0.00534653
$759$	−1.20180e10	−0.997669
$760$	3.95136e6	0.000326512 0
$761$	1.71731e10	1.41254	0.706271	−	0.707942i	$-0.250376\pi$
$761$	1.71731e10	1.41254	0.706271	+	0.707942i	$0.250376\pi$
$762$	7.40764e8	0.0606510
$763$	8.51443e9	0.693937
$764$	1.72964e10	1.40323
$765$	9.15292e7	0.00739170
$766$	1.71491e9	0.137861
$767$	−1.73989e10	−1.39232
$768$	3.39164e9	0.270175
$769$	−3.85393e9	−0.305606	−0.152803	−	0.988257i	$-0.548830\pi$
$769$	−3.85393e9	−0.305606	−0.152803	+	0.988257i	$0.548830\pi$
$770$	1.55757e8	0.0122951
$771$	3.58717e9	0.281879
$772$	6.50914e9	0.509170
$773$	1.53031e10	1.19166	0.595830	−	0.803111i	$-0.296823\pi$
$773$	1.53031e10	1.19166	0.595830	+	0.803111i	$0.296823\pi$
$774$	−1.95627e9	−0.151648
$775$	1.84921e10	1.42702
$776$	−3.30172e9	−0.253644
$777$	−3.74854e9	−0.286674
$778$	7.92605e8	0.0603432
$779$	2.38479e8	0.0180746
$780$	2.42351e8	0.0182858
$781$	4.33684e9	0.325758
$782$	−7.59845e8	−0.0568200
$783$	1.32751e9	0.0988260
$784$	−1.47733e8	−0.0109489
$785$	−3.85566e8	−0.0284482
$786$	1.14773e9	0.0843062
$787$	1.31477e10	0.961476	0.480738	−	0.876864i	$-0.340369\pi$
$787$	1.31477e10	0.961476	0.480738	+	0.876864i	$0.340369\pi$
$788$	8.99597e9	0.654947
$789$	3.99086e9	0.289266
$790$	−5.78435e7	−0.00417407
$791$	−1.77616e10	−1.27604
$792$	8.10691e9	0.579852
$793$	−2.82483e10	−2.01157
$794$	−1.23709e8	−0.00877062
$795$	−2.75152e7	−0.00194217
$796$	−4.77961e9	−0.335890
$797$	−1.47519e10	−1.03216	−0.516078	−	0.856542i	$-0.672608\pi$
$797$	−1.47519e10	−1.03216	−0.516078	+	0.856542i	$0.672608\pi$
$798$	−2.54580e7	−0.00177343
$799$	3.94392e9	0.273536
$800$	7.35308e9	0.507755
$801$	−1.47770e10	−1.01595
$802$	−3.39406e9	−0.232332
$803$	−3.14991e10	−2.14681
$804$	−2.39852e9	−0.162760
$805$	6.97517e8	0.0471269
$806$	5.14674e9	0.346226
$807$	−3.73651e9	−0.250270
$808$	−2.51034e9	−0.167414
$809$	−6.85091e9	−0.454913	−0.227457	−	0.973788i	$-0.573041\pi$
$809$	−6.85091e9	−0.454913	−0.227457	+	0.973788i	$0.573041\pi$
$810$	5.52436e7	0.00365245
$811$	−7.43215e9	−0.489262	−0.244631	−	0.969616i	$-0.578667\pi$
$811$	−7.43215e9	−0.489262	−0.244631	+	0.969616i	$0.578667\pi$
$812$	−2.03675e9	−0.133503
$813$	−8.15870e9	−0.532481
$814$	3.98680e9	0.259083
$815$	3.95993e8	0.0256234
$816$	1.31448e9	0.0846914
$817$	4.11625e8	0.0264073
$818$	3.31133e9	0.211527
$819$	1.82461e10	1.16058
$820$	−3.77186e8	−0.0238895
$821$	−6.17127e9	−0.389201	−0.194600	−	0.980883i	$-0.562341\pi$
$821$	−6.17127e9	−0.389201	−0.194600	+	0.980883i	$0.562341\pi$
$822$	2.65253e8	0.0166575
$823$	−9.68073e9	−0.605353	−0.302677	−	0.953093i	$-0.597880\pi$
$823$	−9.68073e9	−0.605353	−0.302677	+	0.953093i	$0.597880\pi$
$824$	−9.54004e9	−0.594025
$825$	1.21260e10	0.751847
$826$	−2.89074e9	−0.178476
$827$	2.56143e10	1.57475	0.787377	−	0.616472i	$-0.211439\pi$
$827$	2.56143e10	1.57475	0.787377	+	0.616472i	$0.211439\pi$
$828$	1.78642e10	1.09364
$829$	−1.57993e10	−0.963157	−0.481579	−	0.876403i	$-0.659936\pi$
$829$	−1.57993e10	−0.963157	−0.481579	+	0.876403i	$0.659936\pi$
$830$	1.45621e7	0.000883996 0
$831$	−9.90511e9	−0.598764
$832$	−1.86118e10	−1.12036
$833$	−4.88303e7	−0.00292706
$834$	6.87202e8	0.0410207
$835$	−7.45237e8	−0.0442988
$836$	−8.39363e8	−0.0496853
$837$	1.72774e10	1.01845
$838$	1.10721e9	0.0649943
$839$	2.08451e10	1.21853	0.609265	−	0.792967i	$-0.291465\pi$
$839$	2.08451e10	1.21853	0.609265	+	0.792967i	$0.291465\pi$
$840$	8.18294e7	0.00476356
$841$	−1.69183e10	−0.980776
$842$	−3.98490e8	−0.0230052
$843$	−6.03287e9	−0.346839
$844$	1.59700e10	0.914337
$845$	−5.51476e8	−0.0314433
$846$	2.99105e9	0.169835
$847$	−4.96630e10	−2.80829
$848$	2.27214e9	0.127953
$849$	−2.38557e9	−0.133787
$850$	7.66674e8	0.0428198
$851$	1.78538e10	0.993064
$852$	1.12113e9	0.0621036
$853$	−8.47971e9	−0.467799	−0.233899	−	0.972261i	$-0.575149\pi$
$853$	−8.47971e9	−0.467799	−0.233899	+	0.972261i	$0.575149\pi$
$854$	−4.69330e9	−0.257855
$855$	−1.46059e7	−0.000799185 0
$856$	−1.79899e8	−0.00980325
$857$	2.46060e10	1.33539	0.667694	−	0.744436i	$-0.267281\pi$
$857$	2.46060e10	1.33539	0.667694	+	0.744436i	$0.267281\pi$
$858$	3.37493e9	0.182414
$859$	−8.92710e9	−0.480546	−0.240273	−	0.970705i	$-0.577237\pi$
$859$	−8.92710e9	−0.480546	−0.240273	+	0.970705i	$0.577237\pi$
$860$	−6.51040e8	−0.0349030
$861$	4.93870e9	0.263695
$862$	1.46409e9	0.0778559
$863$	4.48105e9	0.237324	0.118662	−	0.992935i	$-0.462139\pi$
$863$	4.48105e9	0.237324	0.118662	+	0.992935i	$0.462139\pi$
$864$	6.87010e9	0.362380
$865$	−3.97054e8	−0.0208590
$866$	−5.89917e9	−0.308658
$867$	4.34476e8	0.0226412
$868$	−2.65082e10	−1.37582
$869$	2.49710e10	1.29082
$870$	−6.55560e6	−0.000337517 0
$871$	1.16681e10	0.598322
$872$	−4.75751e9	−0.242981
$873$	1.22046e10	0.620831
$874$	1.21253e8	0.00614333
$875$	−1.40847e9	−0.0710756
$876$	−8.14291e9	−0.409275
$877$	1.56634e10	0.784129	0.392064	−	0.919938i	$-0.371761\pi$
$877$	1.56634e10	0.784129	0.392064	+	0.919938i	$0.371761\pi$
$878$	4.17902e9	0.208374
$879$	9.17017e9	0.455424
$880$	1.28336e9	0.0634831
$881$	2.27836e10	1.12255	0.561276	−	0.827629i	$-0.310311\pi$
$881$	2.27836e10	1.12255	0.561276	+	0.827629i	$0.310311\pi$
$882$	−3.70327e7	−0.00181738
$883$	5.75925e9	0.281516	0.140758	−	0.990044i	$-0.455046\pi$
$883$	5.75925e9	0.281516	0.140758	+	0.990044i	$0.455046\pi$
$884$	−6.61482e9	−0.322059
$885$	2.88433e8	0.0139876
$886$	−1.48975e9	−0.0719605
$887$	−9.51146e9	−0.457630	−0.228815	−	0.973470i	$-0.573485\pi$
$887$	−9.51146e9	−0.457630	−0.228815	+	0.973470i	$0.573485\pi$
$888$	2.09453e9	0.100378
$889$	1.85603e10	0.885988
$890$	1.58637e8	0.00754292
$891$	−2.38487e10	−1.12952
$892$	2.25274e10	1.06276
$893$	−6.29358e8	−0.0295745
$894$	−2.54993e9	−0.119357
$895$	−1.17211e9	−0.0546497
$896$	−1.39728e10	−0.648943
$897$	1.51137e10	0.699193
$898$	3.22833e9	0.148769
$899$	4.31581e9	0.198108
$900$	−1.80247e10	−0.824175
$901$	7.51011e8	0.0342066
$902$	−5.25261e9	−0.238316
$903$	8.52442e9	0.385263
$904$	9.92442e9	0.446802
$905$	−1.35726e8	−0.00608686
$906$	−2.23436e9	−0.0998169
$907$	−1.30542e10	−0.580929	−0.290465	−	0.956886i	$-0.593810\pi$
$907$	−1.30542e10	−0.580929	−0.290465	+	0.956886i	$0.593810\pi$
$908$	−7.21469e9	−0.319829
$909$	9.27929e9	0.409771
$910$	−1.95878e8	−0.00861671
$911$	5.80091e9	0.254204	0.127102	−	0.991890i	$-0.459432\pi$
$911$	5.80091e9	0.254204	0.127102	+	0.991890i	$0.459432\pi$
$912$	−2.09761e8	−0.00915677
$913$	−6.28645e9	−0.273374
$914$	−2.10252e9	−0.0910811
$915$	4.68290e8	0.0202088
$916$	2.11751e9	0.0910316
$917$	2.87569e10	1.23154
$918$	7.16315e8	0.0305601
$919$	−1.96504e10	−0.835157	−0.417578	−	0.908641i	$-0.637121\pi$
$919$	−1.96504e10	−0.835157	−0.417578	+	0.908641i	$0.637121\pi$
$920$	−3.89743e8	−0.0165014
$921$	2.55724e9	0.107861
$922$	−5.56954e9	−0.234024
$923$	−5.45395e9	−0.228300
$924$	−1.73825e10	−0.724871
$925$	−1.80143e10	−0.748377
$926$	−2.67808e8	−0.0110837
$927$	3.52641e10	1.45396
$928$	1.71611e9	0.0704900
$929$	−3.30645e10	−1.35303	−0.676514	−	0.736429i	$-0.736510\pi$
$929$	−3.30645e10	−1.35303	−0.676514	+	0.736429i	$0.736510\pi$
$930$	−8.53207e7	−0.00347828
$931$	7.79218e6	0.000316472 0
$932$	1.92247e10	0.777863
$933$	−6.81989e9	−0.274911
$934$	2.60922e9	0.104784
$935$	4.24188e8	0.0169714
$936$	−1.01951e10	−0.406376
$937$	−1.26924e10	−0.504029	−0.252014	−	0.967723i	$-0.581093\pi$
$937$	−1.26924e10	−0.504029	−0.252014	+	0.967723i	$0.581093\pi$
$938$	1.93859e9	0.0766964
$939$	1.24146e10	0.489333
$940$	9.95412e8	0.0390891
$941$	−3.46573e10	−1.35591	−0.677954	−	0.735104i	$-0.737133\pi$
$941$	−3.46573e10	−1.35591	−0.677954	+	0.735104i	$0.737133\pi$
$942$	−1.38804e9	−0.0541032
$943$	−2.35224e10	−0.913462
$944$	−2.38182e10	−0.921524
$945$	−6.57558e8	−0.0253468
$946$	−9.06625e9	−0.348184
$947$	−4.21831e10	−1.61404	−0.807019	−	0.590526i	$-0.798920\pi$
$947$	−4.21831e10	−1.61404	−0.807019	+	0.590526i	$0.798920\pi$
$948$	6.45533e9	0.246087
$949$	3.96128e10	1.50454
$950$	−1.22343e8	−0.00462964
$951$	−8.26111e9	−0.311463
$952$	−2.23349e9	−0.0838986
$953$	1.28421e10	0.480629	0.240315	−	0.970695i	$-0.422749\pi$
$953$	1.28421e10	0.480629	0.240315	+	0.970695i	$0.422749\pi$
$954$	5.69564e8	0.0212385
$955$	1.39487e9	0.0518231
$956$	−1.10025e10	−0.407276
$957$	2.83005e9	0.104376
$958$	−1.30752e9	−0.0480472
$959$	6.64606e9	0.243332
$960$	3.08540e8	0.0112554
$961$	2.86573e10	1.04161
$962$	−5.01375e9	−0.181572
$963$	6.64983e8	0.0239949
$964$	6.76086e9	0.243071
$965$	5.24931e8	0.0188043
$966$	2.51106e9	0.0896266
$967$	−1.39117e9	−0.0494753	−0.0247377	−	0.999694i	$-0.507875\pi$
$967$	−1.39117e9	−0.0494753	−0.0247377	+	0.999694i	$0.507875\pi$
$968$	2.77496e10	0.983317
$969$	−6.93323e7	−0.00244795
$970$	−1.31021e8	−0.00460934
$971$	3.86650e10	1.35535	0.677674	−	0.735362i	$-0.262988\pi$
$971$	3.86650e10	1.35535	0.677674	+	0.735362i	$0.262988\pi$
$972$	−2.59348e10	−0.905839
$973$	1.72182e10	0.599230
$974$	2.66634e9	0.0924612
$975$	−1.52495e10	−0.526914
$976$	−3.86703e10	−1.33138
$977$	−1.98228e10	−0.680038	−0.340019	−	0.940419i	$-0.610434\pi$
$977$	−1.98228e10	−0.680038	−0.340019	+	0.940419i	$0.610434\pi$
$978$	1.42558e9	0.0487309
$979$	−6.84836e10	−2.33264
$980$	−1.23244e7	−0.000418286 0
$981$	1.75858e10	0.594731
$982$	−6.75743e8	−0.0227715
$983$	3.28878e10	1.10433	0.552163	−	0.833736i	$-0.313802\pi$
$983$	3.28878e10	1.10433	0.552163	+	0.833736i	$0.313802\pi$
$984$	−2.75954e9	−0.0923323
$985$	7.25482e8	0.0241880
$986$	1.78931e8	0.00594453
$987$	−1.30335e10	−0.431470
$988$	1.05557e9	0.0348208
$989$	−4.06007e10	−1.33459
$990$	3.21703e8	0.0105374
$991$	4.22581e9	0.137928	0.0689640	−	0.997619i	$-0.478031\pi$
$991$	4.22581e9	0.137928	0.0689640	+	0.997619i	$0.478031\pi$
$992$	2.23351e10	0.726434
$993$	7.52719e9	0.243955
$994$	−9.06146e8	−0.0292648
$995$	−3.85453e8	−0.0124048
$996$	−1.62513e9	−0.0521171
$997$	4.92754e10	1.57470	0.787349	−	0.616507i	$-0.211453\pi$
$997$	4.92754e10	1.57470	0.787349	+	0.616507i	$0.211453\pi$
$998$	8.67936e9	0.276395
$999$	−1.68310e10	−0.534110

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.8.a.a.1.1	✓	1
3.2	odd	2		153.8.a.a.1.1		1
4.3	odd	2		272.8.a.b.1.1		1
5.4	even	2		425.8.a.a.1.1		1
17.16	even	2		289.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.8.a.a.1.1	✓	1	1.1	even	1	trivial
153.8.a.a.1.1		1	3.2	odd	2
272.8.a.b.1.1		1	4.3	odd	2
289.8.a.a.1.1		1	17.16	even	2
425.8.a.a.1.1		1	5.4	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:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	17.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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