LMFDB - Embedded newform 126.8.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [126,8,Mod(1,126)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("126.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	126.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,16,0,128,-168] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.3605132110$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3691}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3691 $ x^2 - 3691
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$60.7536$ of defining polynomial
Character	$\chi$	$=$	126.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.00000 q^{2} +64.0000 q^{4} +280.522 q^{5} +343.000 q^{7} +512.000 q^{8} +2244.17 q^{10} +196.349 q^{11} +5990.52 q^{13} +2744.00 q^{14} +4096.00 q^{16} +3301.39 q^{17} +12080.5 q^{19} +17953.4 q^{20} +1570.79 q^{22} -5184.16 q^{23} +567.370 q^{25} +47924.1 q^{26} +21952.0 q^{28} -48572.5 q^{29} +49276.4 q^{31} +32768.0 q^{32} +26411.1 q^{34} +96218.9 q^{35} +138418. q^{37} +96644.1 q^{38} +143627. q^{40} +156453. q^{41} +413137. q^{43} +12566.3 q^{44} -41473.3 q^{46} +760991. q^{47} +117649. q^{49} +4538.96 q^{50} +383393. q^{52} +1.43961e6 q^{53} +55080.1 q^{55} +175616. q^{56} -388580. q^{58} +2.24055e6 q^{59} -1.90900e6 q^{61} +394211. q^{62} +262144. q^{64} +1.68047e6 q^{65} -4.83392e6 q^{67} +211289. q^{68} +769751. q^{70} +3.41753e6 q^{71} -1.07075e6 q^{73} +1.10735e6 q^{74} +773153. q^{76} +67347.6 q^{77} -1.90535e6 q^{79} +1.14902e6 q^{80} +1.25162e6 q^{82} +1.09310e6 q^{83} +926112. q^{85} +3.30510e6 q^{86} +100531. q^{88} -6.70115e6 q^{89} +2.05475e6 q^{91} -331786. q^{92} +6.08793e6 q^{94} +3.38885e6 q^{95} +8.04200e6 q^{97} +941192. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{2} + 128 q^{4} - 168 q^{5} + 686 q^{7} + 1024 q^{8} - 1344 q^{10} + 5496 q^{11} - 5516 q^{13} + 5488 q^{14} + 8192 q^{16} + 10248 q^{17} + 6664 q^{19} - 10752 q^{20} + 43968 q^{22} + 45768 q^{23}+ \cdots + 1882384 q^{98}+O(q^{100}) $ 2 * q + 16 * q^2 + 128 * q^4 - 168 * q^5 + 686 * q^7 + 1024 * q^8 - 1344 * q^10 + 5496 * q^11 - 5516 * q^13 + 5488 * q^14 + 8192 * q^16 + 10248 * q^17 + 6664 * q^19 - 10752 * q^20 + 43968 * q^22 + 45768 * q^23 + 123614 * q^25 - 44128 * q^26 + 43904 * q^28 + 188640 * q^29 + 221032 * q^31 + 65536 * q^32 + 81984 * q^34 - 57624 * q^35 + 399316 * q^37 + 53312 * q^38 - 86016 * q^40 + 562968 * q^41 - 153560 * q^43 + 351744 * q^44 + 366144 * q^46 + 651504 * q^47 + 235298 * q^49 + 988912 * q^50 - 353024 * q^52 + 2032080 * q^53 - 2321928 * q^55 + 351232 * q^56 + 1509120 * q^58 + 3864336 * q^59 - 721028 * q^61 + 1768256 * q^62 + 524288 * q^64 + 6841392 * q^65 - 604376 * q^67 + 655872 * q^68 - 460992 * q^70 + 389592 * q^71 - 6935684 * q^73 + 3194528 * q^74 + 426496 * q^76 + 1885128 * q^77 - 7974992 * q^79 - 688128 * q^80 + 4503744 * q^82 + 2209536 * q^83 - 2189592 * q^85 - 1228480 * q^86 + 2813952 * q^88 + 1659000 * q^89 - 1891988 * q^91 + 2929152 * q^92 + 5212032 * q^94 + 5818272 * q^95 - 4807460 * q^97 + 1882384 * q^98

Coefficient data

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

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

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

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	280.522	1.00362	0.501812	−	0.864977i	$-0.332667\pi$
$5$	280.522	1.00362	0.501812	+	0.864977i	$0.332667\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	512.000	0.353553
$9$	0	0
$10$	2244.17	0.709670
$11$	196.349	0.0444789	0.0222394	−	0.999753i	$-0.492920\pi$
$11$	196.349	0.0444789	0.0222394	+	0.999753i	$0.492920\pi$
$12$	0	0
$13$	5990.52	0.756245	0.378123	−	0.925756i	$-0.376570\pi$
$13$	5990.52	0.756245	0.378123	+	0.925756i	$0.376570\pi$
$14$	2744.00	0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	3301.39	0.162977	0.0814884	−	0.996674i	$-0.474033\pi$
$17$	3301.39	0.162977	0.0814884	+	0.996674i	$0.474033\pi$
$18$	0	0
$19$	12080.5	0.404062	0.202031	−	0.979379i	$-0.435246\pi$
$19$	12080.5	0.404062	0.202031	+	0.979379i	$0.435246\pi$
$20$	17953.4	0.501812
$21$	0	0
$22$	1570.79	0.0314513
$23$	−5184.16	−0.0888446	−0.0444223	−	0.999013i	$-0.514145\pi$
$23$	−5184.16	−0.0888446	−0.0444223	+	0.999013i	$0.514145\pi$
$24$	0	0
$25$	567.370	0.00726234
$26$	47924.1	0.534746
$27$	0	0
$28$	21952.0	0.188982
$29$	−48572.5	−0.369826	−0.184913	−	0.982755i	$-0.559200\pi$
$29$	−48572.5	−0.369826	−0.184913	+	0.982755i	$0.559200\pi$
$30$	0	0
$31$	49276.4	0.297080	0.148540	−	0.988906i	$-0.452543\pi$
$31$	49276.4	0.297080	0.148540	+	0.988906i	$0.452543\pi$
$32$	32768.0	0.176777
$33$	0	0
$34$	26411.1	0.115242
$35$	96218.9	0.379334
$36$	0	0
$37$	138418.	0.449250	0.224625	−	0.974445i	$-0.427884\pi$
$37$	138418.	0.449250	0.224625	+	0.974445i	$0.427884\pi$
$38$	96644.1	0.285715
$39$	0	0
$40$	143627.	0.354835
$41$	156453.	0.354520	0.177260	−	0.984164i	$-0.443277\pi$
$41$	156453.	0.354520	0.177260	+	0.984164i	$0.443277\pi$
$42$	0	0
$43$	413137.	0.792418	0.396209	−	0.918160i	$-0.370326\pi$
$43$	413137.	0.792418	0.396209	+	0.918160i	$0.370326\pi$
$44$	12566.3	0.0222394
$45$	0	0
$46$	−41473.3	−0.0628227
$47$	760991.	1.06915	0.534573	−	0.845122i	$-0.320473\pi$
$47$	760991.	1.06915	0.534573	+	0.845122i	$0.320473\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	4538.96	0.00513525
$51$	0	0
$52$	383393.	0.378123
$53$	1.43961e6	1.32825	0.664126	−	0.747621i	$-0.268804\pi$
$53$	1.43961e6	1.32825	0.664126	+	0.747621i	$0.268804\pi$
$54$	0	0
$55$	55080.1	0.0446401
$56$	175616.	0.133631
$57$	0	0
$58$	−388580.	−0.261506
$59$	2.24055e6	1.42028	0.710139	−	0.704062i	$-0.248632\pi$
$59$	2.24055e6	1.42028	0.710139	+	0.704062i	$0.248632\pi$
$60$	0	0
$61$	−1.90900e6	−1.07684	−0.538421	−	0.842676i	$-0.680979\pi$
$61$	−1.90900e6	−1.07684	−0.538421	+	0.842676i	$0.680979\pi$
$62$	394211.	0.210067
$63$	0	0
$64$	262144.	0.125000
$65$	1.68047e6	0.758986
$66$	0	0
$67$	−4.83392e6	−1.96353	−0.981765	−	0.190097i	$-0.939120\pi$
$67$	−4.83392e6	−1.96353	−0.981765	+	0.190097i	$0.939120\pi$
$68$	211289.	0.0814884
$69$	0	0
$70$	769751.	0.268230
$71$	3.41753e6	1.13320	0.566602	−	0.823991i	$-0.308258\pi$
$71$	3.41753e6	1.13320	0.566602	+	0.823991i	$0.308258\pi$
$72$	0	0
$73$	−1.07075e6	−0.322149	−0.161075	−	0.986942i	$-0.551496\pi$
$73$	−1.07075e6	−0.322149	−0.161075	+	0.986942i	$0.551496\pi$
$74$	1.10735e6	0.317667
$75$	0	0
$76$	773153.	0.202031
$77$	67347.6	0.0168114
$78$	0	0
$79$	−1.90535e6	−0.434790	−0.217395	−	0.976084i	$-0.569756\pi$
$79$	−1.90535e6	−0.434790	−0.217395	+	0.976084i	$0.569756\pi$
$80$	1.14902e6	0.250906
$81$	0	0
$82$	1.25162e6	0.250683
$83$	1.09310e6	0.209840	0.104920	−	0.994481i	$-0.466541\pi$
$83$	1.09310e6	0.209840	0.104920	+	0.994481i	$0.466541\pi$
$84$	0	0
$85$	926112.	0.163568
$86$	3.30510e6	0.560324
$87$	0	0
$88$	100531.	0.0157257
$89$	−6.70115e6	−1.00759	−0.503796	−	0.863823i	$-0.668064\pi$
$89$	−6.70115e6	−1.00759	−0.503796	+	0.863823i	$0.668064\pi$
$90$	0	0
$91$	2.05475e6	0.285834
$92$	−331786.	−0.0444223
$93$	0	0
$94$	6.08793e6	0.756000
$95$	3.38885e6	0.405527
$96$	0	0
$97$	8.04200e6	0.894670	0.447335	−	0.894366i	$-0.352373\pi$
$97$	8.04200e6	0.894670	0.447335	+	0.894366i	$0.352373\pi$
$98$	941192.	0.101015
$99$	0	0
$100$	36311.7	0.00363117
$101$	−397856.	−0.0384239	−0.0192119	−	0.999815i	$-0.506116\pi$
$101$	−397856.	−0.0384239	−0.0192119	+	0.999815i	$0.506116\pi$
$102$	0	0
$103$	−1.07693e7	−0.971086	−0.485543	−	0.874213i	$-0.661378\pi$
$103$	−1.07693e7	−0.971086	−0.485543	+	0.874213i	$0.661378\pi$
$104$	3.06715e6	0.267373
$105$	0	0
$106$	1.15169e7	0.939216
$107$	8.73645e6	0.689432	0.344716	−	0.938707i	$-0.387975\pi$
$107$	8.73645e6	0.689432	0.344716	+	0.938707i	$0.387975\pi$
$108$	0	0
$109$	−1.16892e7	−0.864555	−0.432277	−	0.901741i	$-0.642290\pi$
$109$	−1.16892e7	−0.864555	−0.432277	+	0.901741i	$0.642290\pi$
$110$	440641.	0.0315653
$111$	0	0
$112$	1.40493e6	0.0944911
$113$	−6.69795e6	−0.436684	−0.218342	−	0.975872i	$-0.570065\pi$
$113$	−6.69795e6	−0.436684	−0.218342	+	0.975872i	$0.570065\pi$
$114$	0	0
$115$	−1.45427e6	−0.0891667
$116$	−3.10864e6	−0.184913
$117$	0	0
$118$	1.79244e7	1.00429
$119$	1.13238e6	0.0615995
$120$	0	0
$121$	−1.94486e7	−0.998022
$122$	−1.52720e7	−0.761442
$123$	0	0
$124$	3.15369e6	0.148540
$125$	−2.17566e7	−0.996336
$126$	0	0
$127$	3.28044e6	0.142108	0.0710541	−	0.997472i	$-0.477364\pi$
$127$	3.28044e6	0.142108	0.0710541	+	0.997472i	$0.477364\pi$
$128$	2.09715e6	0.0883883
$129$	0	0
$130$	1.34438e7	0.536684
$131$	−2.94925e7	−1.14621	−0.573103	−	0.819484i	$-0.694260\pi$
$131$	−2.94925e7	−1.14621	−0.573103	+	0.819484i	$0.694260\pi$
$132$	0	0
$133$	4.14362e6	0.152721
$134$	−3.86714e7	−1.38843
$135$	0	0
$136$	1.69031e6	0.0576210
$137$	−9.07574e6	−0.301551	−0.150775	−	0.988568i	$-0.548177\pi$
$137$	−9.07574e6	−0.301551	−0.150775	+	0.988568i	$0.548177\pi$
$138$	0	0
$139$	−3.99435e6	−0.126152	−0.0630760	−	0.998009i	$-0.520091\pi$
$139$	−3.99435e6	−0.126152	−0.0630760	+	0.998009i	$0.520091\pi$
$140$	6.15801e6	0.189667
$141$	0	0
$142$	2.73403e7	0.801297
$143$	1.17623e6	0.0336369
$144$	0	0
$145$	−1.36256e7	−0.371166
$146$	−8.56598e6	−0.227794
$147$	0	0
$148$	8.85878e6	0.224625
$149$	−4.40528e7	−1.09099	−0.545496	−	0.838114i	$-0.683659\pi$
$149$	−4.40528e7	−1.09099	−0.545496	+	0.838114i	$0.683659\pi$
$150$	0	0
$151$	1.98489e7	0.469156	0.234578	−	0.972097i	$-0.424629\pi$
$151$	1.98489e7	0.469156	0.234578	+	0.972097i	$0.424629\pi$
$152$	6.18523e6	0.142857
$153$	0	0
$154$	538781.	0.0118875
$155$	1.38231e7	0.298156
$156$	0	0
$157$	−8.20955e7	−1.69305	−0.846527	−	0.532346i	$-0.821310\pi$
$157$	−8.20955e7	−1.69305	−0.846527	+	0.532346i	$0.821310\pi$
$158$	−1.52428e7	−0.307443
$159$	0	0
$160$	9.19213e6	0.177417
$161$	−1.77817e6	−0.0335801
$162$	0	0
$163$	−6.14662e7	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$163$	−6.14662e7	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$164$	1.00130e7	0.177260
$165$	0	0
$166$	8.74483e6	0.148379
$167$	−8.68806e7	−1.44350	−0.721748	−	0.692156i	$-0.756661\pi$
$167$	−8.68806e7	−1.44350	−0.721748	+	0.692156i	$0.756661\pi$
$168$	0	0
$169$	−2.68622e7	−0.428093
$170$	7.40889e6	0.115660
$171$	0	0
$172$	2.64408e7	0.396209
$173$	−2.88102e7	−0.423043	−0.211521	−	0.977373i	$-0.567842\pi$
$173$	−2.88102e7	−0.423043	−0.211521	+	0.977373i	$0.567842\pi$
$174$	0	0
$175$	194608.	0.00274491
$176$	804245.	0.0111197
$177$	0	0
$178$	−5.36092e7	−0.712475
$179$	1.01430e7	0.132185	0.0660924	−	0.997814i	$-0.478947\pi$
$179$	1.01430e7	0.132185	0.0660924	+	0.997814i	$0.478947\pi$
$180$	0	0
$181$	−7.72824e7	−0.968737	−0.484368	−	0.874864i	$-0.660951\pi$
$181$	−7.72824e7	−0.968737	−0.484368	+	0.874864i	$0.660951\pi$
$182$	1.64380e7	0.202115
$183$	0	0
$184$	−2.65429e6	−0.0314113
$185$	3.88293e7	0.450878
$186$	0	0
$187$	648224.	0.00724903
$188$	4.87034e7	0.534573
$189$	0	0
$190$	2.71108e7	0.286751
$191$	1.22532e7	0.127242	0.0636212	−	0.997974i	$-0.479735\pi$
$191$	1.22532e7	0.127242	0.0636212	+	0.997974i	$0.479735\pi$
$192$	0	0
$193$	1.43432e8	1.43614	0.718069	−	0.695972i	$-0.245026\pi$
$193$	1.43432e8	1.43614	0.718069	+	0.695972i	$0.245026\pi$
$194$	6.43360e7	0.632627
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	1.83570e8	1.71069	0.855344	−	0.518061i	$-0.173346\pi$
$197$	1.83570e8	1.71069	0.855344	+	0.518061i	$0.173346\pi$
$198$	0	0
$199$	−1.36806e8	−1.23060	−0.615301	−	0.788292i	$-0.710966\pi$
$199$	−1.36806e8	−1.23060	−0.615301	+	0.788292i	$0.710966\pi$
$200$	290494.	0.00256763
$201$	0	0
$202$	−3.18285e6	−0.0271698
$203$	−1.66604e7	−0.139781
$204$	0	0
$205$	4.38885e7	0.355805
$206$	−8.61545e7	−0.686662
$207$	0	0
$208$	2.45372e7	0.189061
$209$	2.37199e6	0.0179722
$210$	0	0
$211$	1.94368e8	1.42441	0.712206	−	0.701970i	$-0.247696\pi$
$211$	1.94368e8	1.42441	0.712206	+	0.701970i	$0.247696\pi$
$212$	9.21353e7	0.664126
$213$	0	0
$214$	6.98916e7	0.487502
$215$	1.15894e8	0.795291
$216$	0	0
$217$	1.69018e7	0.112286
$218$	−9.35137e7	−0.611333
$219$	0	0
$220$	3.52512e6	0.0223200
$221$	1.97770e7	0.123250
$222$	0	0
$223$	9.02307e7	0.544863	0.272432	−	0.962175i	$-0.412172\pi$
$223$	9.02307e7	0.544863	0.272432	+	0.962175i	$0.412172\pi$
$224$	1.12394e7	0.0668153
$225$	0	0
$226$	−5.35836e7	−0.308782
$227$	−2.23570e7	−0.126860	−0.0634298	−	0.997986i	$-0.520204\pi$
$227$	−2.23570e7	−0.126860	−0.0634298	+	0.997986i	$0.520204\pi$
$228$	0	0
$229$	1.32826e8	0.730904	0.365452	−	0.930830i	$-0.380914\pi$
$229$	1.32826e8	0.730904	0.365452	+	0.930830i	$0.380914\pi$
$230$	−1.16342e7	−0.0630504
$231$	0	0
$232$	−2.48691e7	−0.130753
$233$	2.78915e8	1.44453	0.722264	−	0.691617i	$-0.243102\pi$
$233$	2.78915e8	1.44453	0.722264	+	0.691617i	$0.243102\pi$
$234$	0	0
$235$	2.13474e8	1.07302
$236$	1.43395e8	0.710139
$237$	0	0
$238$	9.05902e6	0.0435574
$239$	2.23103e8	1.05709	0.528547	−	0.848904i	$-0.322737\pi$
$239$	2.23103e8	1.05709	0.528547	+	0.848904i	$0.322737\pi$
$240$	0	0
$241$	−2.80301e8	−1.28993	−0.644964	−	0.764213i	$-0.723128\pi$
$241$	−2.80301e8	−1.28993	−0.644964	+	0.764213i	$0.723128\pi$
$242$	−1.55589e8	−0.705708
$243$	0	0
$244$	−1.22176e8	−0.538421
$245$	3.30031e7	0.143375
$246$	0	0
$247$	7.23686e7	0.305570
$248$	2.52295e7	0.105033
$249$	0	0
$250$	−1.74053e8	−0.704516
$251$	−3.21735e7	−0.128422	−0.0642111	−	0.997936i	$-0.520453\pi$
$251$	−3.21735e7	−0.128422	−0.0642111	+	0.997936i	$0.520453\pi$
$252$	0	0
$253$	−1.01790e6	−0.00395171
$254$	2.62435e7	0.100486
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	3.09856e8	1.13866	0.569330	−	0.822109i	$-0.307203\pi$
$257$	3.09856e8	1.13866	0.569330	+	0.822109i	$0.307203\pi$
$258$	0	0
$259$	4.74775e7	0.169800
$260$	1.07550e8	0.379493
$261$	0	0
$262$	−2.35940e8	−0.810490
$263$	−2.73930e8	−0.928528	−0.464264	−	0.885697i	$-0.653681\pi$
$263$	−2.73930e8	−0.928528	−0.464264	+	0.885697i	$0.653681\pi$
$264$	0	0
$265$	4.03843e8	1.33307
$266$	3.31489e7	0.107990
$267$	0	0
$268$	−3.09371e8	−0.981765
$269$	−3.00741e7	−0.0942019	−0.0471009	−	0.998890i	$-0.514998\pi$
$269$	−3.00741e7	−0.0942019	−0.0471009	+	0.998890i	$0.514998\pi$
$270$	0	0
$271$	−1.97765e8	−0.603611	−0.301806	−	0.953370i	$-0.597589\pi$
$271$	−1.97765e8	−0.603611	−0.301806	+	0.953370i	$0.597589\pi$
$272$	1.35225e7	0.0407442
$273$	0	0
$274$	−7.26059e7	−0.213228
$275$	111402.	0.000323021 0
$276$	0	0
$277$	−1.60160e8	−0.452768	−0.226384	−	0.974038i	$-0.572690\pi$
$277$	−1.60160e8	−0.452768	−0.226384	+	0.974038i	$0.572690\pi$
$278$	−3.19548e7	−0.0892029
$279$	0	0
$280$	4.92641e7	0.134115
$281$	1.66702e8	0.448196	0.224098	−	0.974567i	$-0.428056\pi$
$281$	1.66702e8	0.448196	0.224098	+	0.974567i	$0.428056\pi$
$282$	0	0
$283$	1.51666e8	0.397774	0.198887	−	0.980022i	$-0.436267\pi$
$283$	1.51666e8	0.397774	0.198887	+	0.980022i	$0.436267\pi$
$284$	2.18722e8	0.566602
$285$	0	0
$286$	9.40985e6	0.0237849
$287$	5.36634e7	0.133996
$288$	0	0
$289$	−3.99439e8	−0.973439
$290$	−1.09005e8	−0.262454
$291$	0	0
$292$	−6.85279e7	−0.161075
$293$	−8.31560e8	−1.93133	−0.965665	−	0.259789i	$-0.916347\pi$
$293$	−8.31560e8	−1.93133	−0.965665	+	0.259789i	$0.916347\pi$
$294$	0	0
$295$	6.28524e8	1.42543
$296$	7.08702e7	0.158834
$297$	0	0
$298$	−3.52422e8	−0.771448
$299$	−3.10558e7	−0.0671883
$300$	0	0
$301$	1.41706e8	0.299506
$302$	1.58791e8	0.331744
$303$	0	0
$304$	4.94818e7	0.101015
$305$	−5.35516e8	−1.08074
$306$	0	0
$307$	2.70903e8	0.534354	0.267177	−	0.963647i	$-0.413909\pi$
$307$	2.70903e8	0.534354	0.267177	+	0.963647i	$0.413909\pi$
$308$	4.31025e6	0.00840572
$309$	0	0
$310$	1.10585e8	0.210828
$311$	−3.70799e8	−0.699000	−0.349500	−	0.936936i	$-0.613649\pi$
$311$	−3.70799e8	−0.699000	−0.349500	+	0.936936i	$0.613649\pi$
$312$	0	0
$313$	−2.28334e8	−0.420887	−0.210443	−	0.977606i	$-0.567491\pi$
$313$	−2.28334e8	−0.420887	−0.210443	+	0.977606i	$0.567491\pi$
$314$	−6.56764e8	−1.19717
$315$	0	0
$316$	−1.21942e8	−0.217395
$317$	−3.72789e8	−0.657288	−0.328644	−	0.944454i	$-0.606592\pi$
$317$	−3.72789e8	−0.657288	−0.328644	+	0.944454i	$0.606592\pi$
$318$	0	0
$319$	−9.53714e6	−0.0164494
$320$	7.35371e7	0.125453
$321$	0	0
$322$	−1.42253e7	−0.0237447
$323$	3.98825e7	0.0658527
$324$	0	0
$325$	3.39884e6	0.00549211
$326$	−4.91730e8	−0.786076
$327$	0	0
$328$	8.01040e7	0.125342
$329$	2.61020e8	0.404099
$330$	0	0
$331$	−7.75556e7	−0.117548	−0.0587740	−	0.998271i	$-0.518719\pi$
$331$	−7.75556e7	−0.117548	−0.0587740	+	0.998271i	$0.518719\pi$
$332$	6.99586e7	0.104920
$333$	0	0
$334$	−6.95045e8	−1.02071
$335$	−1.35602e9	−1.97065
$336$	0	0
$337$	−1.49893e6	−0.00213343	−0.00106671	−	0.999999i	$-0.500340\pi$
$337$	−1.49893e6	−0.00213343	−0.00106671	+	0.999999i	$0.500340\pi$
$338$	−2.14898e8	−0.302708
$339$	0	0
$340$	5.92712e7	0.0817838
$341$	9.67535e6	0.0132138
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	2.11526e8	0.280162
$345$	0	0
$346$	−2.30481e8	−0.299136
$347$	1.29273e8	0.166094	0.0830471	−	0.996546i	$-0.473535\pi$
$347$	1.29273e8	0.166094	0.0830471	+	0.996546i	$0.473535\pi$
$348$	0	0
$349$	1.34850e8	0.169810	0.0849049	−	0.996389i	$-0.472941\pi$
$349$	1.34850e8	0.169810	0.0849049	+	0.996389i	$0.472941\pi$
$350$	1.55686e6	0.00194094
$351$	0	0
$352$	6.43396e6	0.00786283
$353$	−6.10080e8	−0.738202	−0.369101	−	0.929389i	$-0.620334\pi$
$353$	−6.10080e8	−0.738202	−0.369101	+	0.929389i	$0.620334\pi$
$354$	0	0
$355$	9.58691e8	1.13731
$356$	−4.28874e8	−0.503796
$357$	0	0
$358$	8.11442e7	0.0934688
$359$	1.72057e9	1.96264	0.981320	−	0.192380i	$-0.0616207\pi$
$359$	1.72057e9	1.96264	0.981320	+	0.192380i	$0.0616207\pi$
$360$	0	0
$361$	−7.47933e8	−0.836734
$362$	−6.18260e8	−0.685000
$363$	0	0
$364$	1.31504e8	0.142917
$365$	−3.00368e8	−0.323317
$366$	0	0
$367$	9.02388e8	0.952932	0.476466	−	0.879193i	$-0.341917\pi$
$367$	9.02388e8	0.952932	0.476466	+	0.879193i	$0.341917\pi$
$368$	−2.12343e7	−0.0222112
$369$	0	0
$370$	3.10635e8	0.318819
$371$	4.93788e8	0.502032
$372$	0	0
$373$	−4.87618e7	−0.0486518	−0.0243259	−	0.999704i	$-0.507744\pi$
$373$	−4.87618e7	−0.0486518	−0.0243259	+	0.999704i	$0.507744\pi$
$374$	5.18579e6	0.00512584
$375$	0	0
$376$	3.89627e8	0.378000
$377$	−2.90974e8	−0.279679
$378$	0	0
$379$	5.62104e8	0.530370	0.265185	−	0.964198i	$-0.414567\pi$
$379$	5.62104e8	0.530370	0.265185	+	0.964198i	$0.414567\pi$
$380$	2.16886e8	0.202763
$381$	0	0
$382$	9.80254e7	0.0899740
$383$	1.11329e9	1.01254	0.506270	−	0.862375i	$-0.331024\pi$
$383$	1.11329e9	1.01254	0.506270	+	0.862375i	$0.331024\pi$
$384$	0	0
$385$	1.88925e7	0.0168724
$386$	1.14746e9	1.01550
$387$	0	0
$388$	5.14688e8	0.447335
$389$	−2.00360e9	−1.72579	−0.862894	−	0.505385i	$-0.831350\pi$
$389$	−2.00360e9	−1.72579	−0.862894	+	0.505385i	$0.831350\pi$
$390$	0	0
$391$	−1.71150e7	−0.0144796
$392$	6.02363e7	0.0505076
$393$	0	0
$394$	1.46856e9	1.20964
$395$	−5.34491e8	−0.436366
$396$	0	0
$397$	−2.24640e8	−0.180186	−0.0900930	−	0.995933i	$-0.528716\pi$
$397$	−2.24640e8	−0.180186	−0.0900930	+	0.995933i	$0.528716\pi$
$398$	−1.09444e9	−0.870168
$399$	0	0
$400$	2.32395e6	0.00181559
$401$	2.47523e9	1.91695	0.958475	−	0.285176i	$-0.0920519\pi$
$401$	2.47523e9	1.91695	0.958475	+	0.285176i	$0.0920519\pi$
$402$	0	0
$403$	2.95191e8	0.224665
$404$	−2.54628e7	−0.0192119
$405$	0	0
$406$	−1.33283e8	−0.0988401
$407$	2.71783e7	0.0199821
$408$	0	0
$409$	−1.11095e9	−0.802905	−0.401452	−	0.915880i	$-0.631494\pi$
$409$	−1.11095e9	−0.802905	−0.401452	+	0.915880i	$0.631494\pi$
$410$	3.51108e8	0.251592
$411$	0	0
$412$	−6.89236e8	−0.485543
$413$	7.68510e8	0.536814
$414$	0	0
$415$	3.06639e8	0.210600
$416$	1.96297e8	0.133687
$417$	0	0
$418$	1.89760e7	0.0127083
$419$	−1.82836e9	−1.21426	−0.607132	−	0.794601i	$-0.707680\pi$
$419$	−1.82836e9	−1.21426	−0.607132	+	0.794601i	$0.707680\pi$
$420$	0	0
$421$	−2.26838e9	−1.48159	−0.740796	−	0.671730i	$-0.765551\pi$
$421$	−2.26838e9	−1.48159	−0.740796	+	0.671730i	$0.765551\pi$
$422$	1.55494e9	1.00721
$423$	0	0
$424$	7.37082e8	0.469608
$425$	1.87311e6	0.00118359
$426$	0	0
$427$	−6.54788e8	−0.407008
$428$	5.59133e8	0.344716
$429$	0	0
$430$	9.27151e8	0.562355
$431$	5.62754e8	0.338570	0.169285	−	0.985567i	$-0.445854\pi$
$431$	5.62754e8	0.338570	0.169285	+	0.985567i	$0.445854\pi$
$432$	0	0
$433$	−3.19554e9	−1.89163	−0.945817	−	0.324700i	$-0.894737\pi$
$433$	−3.19554e9	−1.89163	−0.945817	+	0.324700i	$0.894737\pi$
$434$	1.35214e8	0.0793979
$435$	0	0
$436$	−7.48110e8	−0.432277
$437$	−6.26274e7	−0.0358987
$438$	0	0
$439$	9.19706e8	0.518828	0.259414	−	0.965766i	$-0.416471\pi$
$439$	9.19706e8	0.518828	0.259414	+	0.965766i	$0.416471\pi$
$440$	2.82010e7	0.0157827
$441$	0	0
$442$	1.58216e8	0.0871512
$443$	1.67060e9	0.912975	0.456488	−	0.889730i	$-0.349107\pi$
$443$	1.67060e9	0.912975	0.456488	+	0.889730i	$0.349107\pi$
$444$	0	0
$445$	−1.87982e9	−1.01124
$446$	7.21846e8	0.385276
$447$	0	0
$448$	8.99154e7	0.0472456
$449$	3.15186e9	1.64325	0.821626	−	0.570027i	$-0.193067\pi$
$449$	3.15186e9	1.64325	0.821626	+	0.570027i	$0.193067\pi$
$450$	0	0
$451$	3.07194e7	0.0157686
$452$	−4.28669e8	−0.218342
$453$	0	0
$454$	−1.78856e8	−0.0897034
$455$	5.76401e8	0.286870
$456$	0	0
$457$	−8.74505e8	−0.428603	−0.214302	−	0.976768i	$-0.568748\pi$
$457$	−8.74505e8	−0.428603	−0.214302	+	0.976768i	$0.568748\pi$
$458$	1.06261e9	0.516827
$459$	0	0
$460$	−9.30733e7	−0.0445833
$461$	−2.25666e9	−1.07279	−0.536393	−	0.843969i	$-0.680213\pi$
$461$	−2.25666e9	−1.07279	−0.536393	+	0.843969i	$0.680213\pi$
$462$	0	0
$463$	−1.73804e9	−0.813815	−0.406907	−	0.913469i	$-0.633393\pi$
$463$	−1.73804e9	−0.813815	−0.406907	+	0.913469i	$0.633393\pi$
$464$	−1.98953e8	−0.0924564
$465$	0	0
$466$	2.23132e9	1.02144
$467$	3.03539e9	1.37913	0.689566	−	0.724222i	$-0.257801\pi$
$467$	3.03539e9	1.37913	0.689566	+	0.724222i	$0.257801\pi$
$468$	0	0
$469$	−1.65803e9	−0.742145
$470$	1.70779e9	0.758740
$471$	0	0
$472$	1.14716e9	0.502144
$473$	8.11189e7	0.0352459
$474$	0	0
$475$	6.85413e6	0.00293444
$476$	7.24722e7	0.0307997
$477$	0	0
$478$	1.78483e9	0.747478
$479$	−8.71018e8	−0.362120	−0.181060	−	0.983472i	$-0.557953\pi$
$479$	−8.71018e8	−0.362120	−0.181060	+	0.983472i	$0.557953\pi$
$480$	0	0
$481$	8.29198e8	0.339743
$482$	−2.24241e9	−0.912117
$483$	0	0
$484$	−1.24471e9	−0.499011
$485$	2.25596e9	0.897913
$486$	0	0
$487$	1.54718e9	0.607001	0.303500	−	0.952831i	$-0.401845\pi$
$487$	1.54718e9	0.607001	0.303500	+	0.952831i	$0.401845\pi$
$488$	−9.77409e8	−0.380721
$489$	0	0
$490$	2.64025e8	0.101381
$491$	2.31714e9	0.883421	0.441711	−	0.897158i	$-0.354372\pi$
$491$	2.31714e9	0.883421	0.441711	+	0.897158i	$0.354372\pi$
$492$	0	0
$493$	−1.60357e8	−0.0602730
$494$	5.78949e8	0.216071
$495$	0	0
$496$	2.01836e8	0.0742699
$497$	1.17221e9	0.428311
$498$	0	0
$499$	2.72420e9	0.981493	0.490746	−	0.871302i	$-0.336724\pi$
$499$	2.72420e9	0.981493	0.490746	+	0.871302i	$0.336724\pi$
$500$	−1.39242e9	−0.498168
$501$	0	0
$502$	−2.57388e8	−0.0908082
$503$	−4.39360e9	−1.53933	−0.769666	−	0.638447i	$-0.779577\pi$
$503$	−4.39360e9	−1.53933	−0.769666	+	0.638447i	$0.779577\pi$
$504$	0	0
$505$	−1.11607e8	−0.0385631
$506$	−8.14323e6	−0.00279428
$507$	0	0
$508$	2.09948e8	0.0710541
$509$	1.88221e9	0.632639	0.316320	−	0.948653i	$-0.397553\pi$
$509$	1.88221e9	0.632639	0.316320	+	0.948653i	$0.397553\pi$
$510$	0	0
$511$	−3.67267e8	−0.121761
$512$	1.34218e8	0.0441942
$513$	0	0
$514$	2.47885e9	0.805155
$515$	−3.02103e9	−0.974606
$516$	0	0
$517$	1.49420e8	0.0475544
$518$	3.79820e8	0.120067
$519$	0	0
$520$	8.60401e8	0.268342
$521$	2.44662e9	0.757939	0.378970	−	0.925409i	$-0.376278\pi$
$521$	2.44662e9	0.757939	0.378970	+	0.925409i	$0.376278\pi$
$522$	0	0
$523$	3.39937e9	1.03907	0.519533	−	0.854451i	$-0.326106\pi$
$523$	3.39937e9	1.03907	0.519533	+	0.854451i	$0.326106\pi$
$524$	−1.88752e9	−0.573103
$525$	0	0
$526$	−2.19144e9	−0.656569
$527$	1.62681e8	0.0484171
$528$	0	0
$529$	−3.37795e9	−0.992107
$530$	3.23074e9	0.942620
$531$	0	0
$532$	2.65192e8	0.0763605
$533$	9.37235e8	0.268104
$534$	0	0
$535$	2.45076e9	0.691931
$536$	−2.47497e9	−0.694213
$537$	0	0
$538$	−2.40593e8	−0.0666108
$539$	2.31002e7	0.00635412
$540$	0	0
$541$	−2.21284e9	−0.600841	−0.300421	−	0.953807i	$-0.597127\pi$
$541$	−2.21284e9	−0.600841	−0.300421	+	0.953807i	$0.597127\pi$
$542$	−1.58212e9	−0.426817
$543$	0	0
$544$	1.08180e8	0.0288105
$545$	−3.27908e9	−0.867689
$546$	0	0
$547$	6.02381e8	0.157368	0.0786838	−	0.996900i	$-0.474928\pi$
$547$	6.02381e8	0.157368	0.0786838	+	0.996900i	$0.474928\pi$
$548$	−5.80847e8	−0.150775
$549$	0	0
$550$	891220.	0.000228410 0
$551$	−5.86781e8	−0.149433
$552$	0	0
$553$	−6.53535e8	−0.164335
$554$	−1.28128e9	−0.320155
$555$	0	0
$556$	−2.55638e8	−0.0630760
$557$	−6.49640e9	−1.59287	−0.796434	−	0.604725i	$-0.793283\pi$
$557$	−6.49640e9	−1.59287	−0.796434	+	0.604725i	$0.793283\pi$
$558$	0	0
$559$	2.47491e9	0.599263
$560$	3.94113e8	0.0948336
$561$	0	0
$562$	1.33361e9	0.316922
$563$	−6.47285e9	−1.52868	−0.764339	−	0.644814i	$-0.776935\pi$
$563$	−6.47285e9	−1.52868	−0.764339	+	0.644814i	$0.776935\pi$
$564$	0	0
$565$	−1.87892e9	−0.438267
$566$	1.21333e9	0.281268
$567$	0	0
$568$	1.74978e9	0.400648
$569$	−3.11762e9	−0.709464	−0.354732	−	0.934968i	$-0.615428\pi$
$569$	−3.11762e9	−0.709464	−0.354732	+	0.934968i	$0.615428\pi$
$570$	0	0
$571$	−3.37414e9	−0.758467	−0.379234	−	0.925301i	$-0.623812\pi$
$571$	−3.37414e9	−0.758467	−0.379234	+	0.925301i	$0.623812\pi$
$572$	7.52788e7	0.0168185
$573$	0	0
$574$	4.29307e8	0.0947494
$575$	−2.94134e6	−0.000645220 0
$576$	0	0
$577$	3.99174e9	0.865062	0.432531	−	0.901619i	$-0.357621\pi$
$577$	3.99174e9	0.865062	0.432531	+	0.901619i	$0.357621\pi$
$578$	−3.19552e9	−0.688325
$579$	0	0
$580$	−8.72040e8	−0.185583
$581$	3.74934e8	0.0793120
$582$	0	0
$583$	2.82666e8	0.0590792
$584$	−5.48223e8	−0.113897
$585$	0	0
$586$	−6.65248e9	−1.36566
$587$	−9.66439e9	−1.97215	−0.986077	−	0.166287i	$-0.946822\pi$
$587$	−9.66439e9	−1.97215	−0.986077	+	0.166287i	$0.946822\pi$
$588$	0	0
$589$	5.95284e8	0.120039
$590$	5.02819e9	1.00793
$591$	0	0
$592$	5.66962e8	0.112312
$593$	1.36501e9	0.268810	0.134405	−	0.990927i	$-0.457088\pi$
$593$	1.36501e9	0.268810	0.134405	+	0.990927i	$0.457088\pi$
$594$	0	0
$595$	3.17656e8	0.0618227
$596$	−2.81938e9	−0.545496
$597$	0	0
$598$	−2.48447e8	−0.0475093
$599$	9.30699e8	0.176936	0.0884679	−	0.996079i	$-0.471803\pi$
$599$	9.30699e8	0.176936	0.0884679	+	0.996079i	$0.471803\pi$
$600$	0	0
$601$	−1.68577e9	−0.316765	−0.158383	−	0.987378i	$-0.550628\pi$
$601$	−1.68577e9	−0.316765	−0.158383	+	0.987378i	$0.550628\pi$
$602$	1.13365e9	0.211783
$603$	0	0
$604$	1.27033e9	0.234578
$605$	−5.45576e9	−1.00164
$606$	0	0
$607$	8.90039e9	1.61528	0.807641	−	0.589674i	$-0.200744\pi$
$607$	8.90039e9	1.61528	0.807641	+	0.589674i	$0.200744\pi$
$608$	3.95854e8	0.0714287
$609$	0	0
$610$	−4.28413e9	−0.764202
$611$	4.55873e9	0.808536
$612$	0	0
$613$	−3.31058e9	−0.580487	−0.290244	−	0.956953i	$-0.593736\pi$
$613$	−3.31058e9	−0.580487	−0.290244	+	0.956953i	$0.593736\pi$
$614$	2.16722e9	0.377845
$615$	0	0
$616$	3.44820e7	0.00594374
$617$	7.49974e9	1.28543	0.642715	−	0.766106i	$-0.277808\pi$
$617$	7.49974e9	1.28543	0.642715	+	0.766106i	$0.277808\pi$
$618$	0	0
$619$	2.53875e9	0.430232	0.215116	−	0.976589i	$-0.430987\pi$
$619$	2.53875e9	0.430232	0.215116	+	0.976589i	$0.430987\pi$
$620$	8.84678e8	0.149078
$621$	0	0
$622$	−2.96639e9	−0.494268
$623$	−2.29850e9	−0.380834
$624$	0	0
$625$	−6.14752e9	−1.00721
$626$	−1.82667e9	−0.297612
$627$	0	0
$628$	−5.25411e9	−0.846527
$629$	4.56973e8	0.0732173
$630$	0	0
$631$	9.36480e9	1.48387	0.741935	−	0.670472i	$-0.233908\pi$
$631$	9.36480e9	1.48387	0.741935	+	0.670472i	$0.233908\pi$
$632$	−9.75538e8	−0.153722
$633$	0	0
$634$	−2.98231e9	−0.464773
$635$	9.20234e8	0.142623
$636$	0	0
$637$	7.04779e8	0.108035
$638$	−7.62972e7	−0.0116315
$639$	0	0
$640$	5.88296e8	0.0887087
$641$	−8.69833e9	−1.30447	−0.652233	−	0.758018i	$-0.726168\pi$
$641$	−8.69833e9	−1.30447	−0.652233	+	0.758018i	$0.726168\pi$
$642$	0	0
$643$	8.35753e9	1.23977	0.619883	−	0.784694i	$-0.287180\pi$
$643$	8.35753e9	1.23977	0.619883	+	0.784694i	$0.287180\pi$
$644$	−1.13803e8	−0.0167901
$645$	0	0
$646$	3.19060e8	0.0465649
$647$	4.40456e9	0.639349	0.319674	−	0.947527i	$-0.396426\pi$
$647$	4.40456e9	0.639349	0.319674	+	0.947527i	$0.396426\pi$
$648$	0	0
$649$	4.39930e8	0.0631723
$650$	2.71907e7	0.00388351
$651$	0	0
$652$	−3.93384e9	−0.555840
$653$	2.34457e9	0.329509	0.164755	−	0.986335i	$-0.447317\pi$
$653$	2.34457e9	0.329509	0.164755	+	0.986335i	$0.447317\pi$
$654$	0	0
$655$	−8.27329e9	−1.15036
$656$	6.40832e8	0.0886300
$657$	0	0
$658$	2.08816e9	0.285741
$659$	3.06032e9	0.416551	0.208275	−	0.978070i	$-0.433215\pi$
$659$	3.06032e9	0.416551	0.208275	+	0.978070i	$0.433215\pi$
$660$	0	0
$661$	−7.61767e9	−1.02593	−0.512964	−	0.858410i	$-0.671453\pi$
$661$	−7.61767e9	−1.02593	−0.512964	+	0.858410i	$0.671453\pi$
$662$	−6.20445e8	−0.0831190
$663$	0	0
$664$	5.59669e8	0.0741896
$665$	1.16237e9	0.153275
$666$	0	0
$667$	2.51808e8	0.0328570
$668$	−5.56036e9	−0.721748
$669$	0	0
$670$	−1.08482e10	−1.39346
$671$	−3.74830e8	−0.0478967
$672$	0	0
$673$	1.88715e9	0.238646	0.119323	−	0.992855i	$-0.461928\pi$
$673$	1.88715e9	0.238646	0.119323	+	0.992855i	$0.461928\pi$
$674$	−1.19915e7	−0.00150856
$675$	0	0
$676$	−1.71918e9	−0.214047
$677$	−8.23130e9	−1.01955	−0.509775	−	0.860308i	$-0.670271\pi$
$677$	−8.23130e9	−1.01955	−0.509775	+	0.860308i	$0.670271\pi$
$678$	0	0
$679$	2.75841e9	0.338154
$680$	4.74169e8	0.0578299
$681$	0	0
$682$	7.74028e7	0.00934354
$683$	1.53495e10	1.84341	0.921705	−	0.387892i	$-0.126797\pi$
$683$	1.53495e10	1.84341	0.921705	+	0.387892i	$0.126797\pi$
$684$	0	0
$685$	−2.54594e9	−0.302644
$686$	3.22829e8	0.0381802
$687$	0	0
$688$	1.69221e9	0.198105
$689$	8.62403e9	1.00448
$690$	0	0
$691$	5.36543e9	0.618630	0.309315	−	0.950960i	$-0.399900\pi$
$691$	5.36543e9	0.618630	0.309315	+	0.950960i	$0.399900\pi$
$692$	−1.84385e9	−0.211521
$693$	0	0
$694$	1.03418e9	0.117446
$695$	−1.12050e9	−0.126609
$696$	0	0
$697$	5.16513e8	0.0577785
$698$	1.07880e9	0.120074
$699$	0	0
$700$	1.24549e7	0.00137245
$701$	8.98780e9	0.985463	0.492732	−	0.870181i	$-0.335998\pi$
$701$	8.98780e9	0.985463	0.492732	+	0.870181i	$0.335998\pi$
$702$	0	0
$703$	1.67217e9	0.181525
$704$	5.14717e7	0.00555986
$705$	0	0
$706$	−4.88064e9	−0.521988
$707$	−1.36465e8	−0.0145229
$708$	0	0
$709$	−2.23219e9	−0.235218	−0.117609	−	0.993060i	$-0.537523\pi$
$709$	−2.23219e9	−0.235218	−0.117609	+	0.993060i	$0.537523\pi$
$710$	7.66953e9	0.804201
$711$	0	0
$712$	−3.43099e9	−0.356237
$713$	−2.55457e8	−0.0263939
$714$	0	0
$715$	3.29958e8	0.0337589
$716$	6.49153e8	0.0660924
$717$	0	0
$718$	1.37645e10	1.38780
$719$	1.00409e10	1.00745	0.503723	−	0.863865i	$-0.331963\pi$
$719$	1.00409e10	1.00745	0.503723	+	0.863865i	$0.331963\pi$
$720$	0	0
$721$	−3.69388e9	−0.367036
$722$	−5.98346e9	−0.591660
$723$	0	0
$724$	−4.94608e9	−0.484368
$725$	−2.75586e7	−0.00268580
$726$	0	0
$727$	1.40935e10	1.36034	0.680172	−	0.733053i	$-0.261905\pi$
$727$	1.40935e10	1.36034	0.680172	+	0.733053i	$0.261905\pi$
$728$	1.05203e9	0.101058
$729$	0	0
$730$	−2.40294e9	−0.228620
$731$	1.36393e9	0.129146
$732$	0	0
$733$	1.38425e10	1.29823	0.649115	−	0.760690i	$-0.275139\pi$
$733$	1.38425e10	1.29823	0.649115	+	0.760690i	$0.275139\pi$
$734$	7.21910e9	0.673825
$735$	0	0
$736$	−1.69875e8	−0.0157057
$737$	−9.49134e8	−0.0873356
$738$	0	0
$739$	1.62204e10	1.47844	0.739222	−	0.673462i	$-0.235194\pi$
$739$	1.62204e10	1.47844	0.739222	+	0.673462i	$0.235194\pi$
$740$	2.48508e9	0.225439
$741$	0	0
$742$	3.95030e9	0.354990
$743$	1.36993e9	0.122529	0.0612643	−	0.998122i	$-0.480487\pi$
$743$	1.36993e9	0.122529	0.0612643	+	0.998122i	$0.480487\pi$
$744$	0	0
$745$	−1.23578e10	−1.09495
$746$	−3.90095e8	−0.0344020
$747$	0	0
$748$	4.14864e7	0.00362451
$749$	2.99660e9	0.260581
$750$	0	0
$751$	−1.56987e9	−0.135246	−0.0676228	−	0.997711i	$-0.521541\pi$
$751$	−1.56987e9	−0.135246	−0.0676228	+	0.997711i	$0.521541\pi$
$752$	3.11702e9	0.267286
$753$	0	0
$754$	−2.32779e9	−0.197763
$755$	5.56805e9	0.470857
$756$	0	0
$757$	2.22871e10	1.86732	0.933658	−	0.358166i	$-0.116598\pi$
$757$	2.22871e10	1.86732	0.933658	+	0.358166i	$0.116598\pi$
$758$	4.49683e9	0.375028
$759$	0	0
$760$	1.73509e9	0.143375
$761$	−9.35667e9	−0.769617	−0.384809	−	0.922996i	$-0.625733\pi$
$761$	−9.35667e9	−0.769617	−0.384809	+	0.922996i	$0.625733\pi$
$762$	0	0
$763$	−4.00940e9	−0.326771
$764$	7.84203e8	0.0636212
$765$	0	0
$766$	8.90630e9	0.715973
$767$	1.34221e10	1.07408
$768$	0	0
$769$	6.34341e9	0.503015	0.251507	−	0.967855i	$-0.419074\pi$
$769$	6.34341e9	0.503015	0.251507	+	0.967855i	$0.419074\pi$
$770$	1.51140e8	0.0119306
$771$	0	0
$772$	9.17966e9	0.718069
$773$	1.80394e10	1.40474	0.702368	−	0.711814i	$-0.252126\pi$
$773$	1.80394e10	1.40474	0.702368	+	0.711814i	$0.252126\pi$
$774$	0	0
$775$	2.79580e7	0.00215749
$776$	4.11750e9	0.316314
$777$	0	0
$778$	−1.60288e10	−1.22032
$779$	1.89003e9	0.143248
$780$	0	0
$781$	6.71028e8	0.0504037
$782$	−1.36920e8	−0.0102386
$783$	0	0
$784$	4.81890e8	0.0357143
$785$	−2.30296e10	−1.69919
$786$	0	0
$787$	−2.08167e10	−1.52230	−0.761151	−	0.648575i	$-0.775365\pi$
$787$	−2.08167e10	−1.52230	−0.761151	+	0.648575i	$0.775365\pi$
$788$	1.17485e10	0.855344
$789$	0	0
$790$	−4.27593e9	−0.308557
$791$	−2.29740e9	−0.165051
$792$	0	0
$793$	−1.14359e10	−0.814357
$794$	−1.79712e9	−0.127411
$795$	0	0
$796$	−8.75555e9	−0.615301
$797$	1.98274e10	1.38728	0.693638	−	0.720324i	$-0.256007\pi$
$797$	1.98274e10	1.38728	0.693638	+	0.720324i	$0.256007\pi$
$798$	0	0
$799$	2.51233e9	0.174246
$800$	1.85916e7	0.00128381
$801$	0	0
$802$	1.98019e10	1.35549
$803$	−2.10240e8	−0.0143288
$804$	0	0
$805$	−4.98815e8	−0.0337018
$806$	2.36153e9	0.158862
$807$	0	0
$808$	−2.03702e8	−0.0135849
$809$	−2.72229e9	−0.180765	−0.0903824	−	0.995907i	$-0.528809\pi$
$809$	−2.72229e9	−0.180765	−0.0903824	+	0.995907i	$0.528809\pi$
$810$	0	0
$811$	−1.89351e10	−1.24650	−0.623252	−	0.782021i	$-0.714189\pi$
$811$	−1.89351e10	−1.24650	−0.623252	+	0.782021i	$0.714189\pi$
$812$	−1.06626e9	−0.0698905
$813$	0	0
$814$	2.17426e8	0.0141295
$815$	−1.72426e10	−1.11571
$816$	0	0
$817$	4.99091e9	0.320186
$818$	−8.88762e9	−0.567739
$819$	0	0
$820$	2.80886e9	0.177902
$821$	−1.35074e10	−0.851865	−0.425933	−	0.904755i	$-0.640054\pi$
$821$	−1.35074e10	−0.851865	−0.425933	+	0.904755i	$0.640054\pi$
$822$	0	0
$823$	−2.88388e9	−0.180334	−0.0901671	−	0.995927i	$-0.528740\pi$
$823$	−2.88388e9	−0.180334	−0.0901671	+	0.995927i	$0.528740\pi$
$824$	−5.51389e9	−0.343331
$825$	0	0
$826$	6.14808e9	0.379585
$827$	−2.42144e10	−1.48869	−0.744346	−	0.667795i	$-0.767238\pi$
$827$	−2.42144e10	−1.48869	−0.744346	+	0.667795i	$0.767238\pi$
$828$	0	0
$829$	−1.94299e10	−1.18449	−0.592243	−	0.805759i	$-0.701758\pi$
$829$	−1.94299e10	−1.18449	−0.592243	+	0.805759i	$0.701758\pi$
$830$	2.45311e9	0.148917
$831$	0	0
$832$	1.57038e9	0.0945307
$833$	3.88405e8	0.0232824
$834$	0	0
$835$	−2.43719e10	−1.44873
$836$	1.51808e8	0.00898611
$837$	0	0
$838$	−1.46269e10	−0.858615
$839$	2.30928e9	0.134992	0.0674961	−	0.997720i	$-0.478499\pi$
$839$	2.30928e9	0.134992	0.0674961	+	0.997720i	$0.478499\pi$
$840$	0	0
$841$	−1.48906e10	−0.863229
$842$	−1.81470e10	−1.04764
$843$	0	0
$844$	1.24395e10	0.712206
$845$	−7.53543e9	−0.429645
$846$	0	0
$847$	−6.67088e9	−0.377217
$848$	5.89666e9	0.332063
$849$	0	0
$850$	1.49849e7	0.000836927 0
$851$	−7.17583e8	−0.0399134
$852$	0	0
$853$	5.09877e9	0.281283	0.140642	−	0.990061i	$-0.455083\pi$
$853$	5.09877e9	0.281283	0.140642	+	0.990061i	$0.455083\pi$
$854$	−5.23830e9	−0.287798
$855$	0	0
$856$	4.47306e9	0.243751
$857$	−9.73503e9	−0.528329	−0.264164	−	0.964478i	$-0.585096\pi$
$857$	−9.73503e9	−0.528329	−0.264164	+	0.964478i	$0.585096\pi$
$858$	0	0
$859$	1.98547e10	1.06878	0.534388	−	0.845239i	$-0.320542\pi$
$859$	1.98547e10	1.06878	0.534388	+	0.845239i	$0.320542\pi$
$860$	7.41721e9	0.397645
$861$	0	0
$862$	4.50203e9	0.239405
$863$	3.31264e9	0.175443	0.0877217	−	0.996145i	$-0.472041\pi$
$863$	3.31264e9	0.175443	0.0877217	+	0.996145i	$0.472041\pi$
$864$	0	0
$865$	−8.08187e9	−0.424576
$866$	−2.55643e10	−1.33759
$867$	0	0
$868$	1.08171e9	0.0561428
$869$	−3.74113e8	−0.0193390
$870$	0	0
$871$	−2.89577e10	−1.48491
$872$	−5.98488e9	−0.305666
$873$	0	0
$874$	−5.01019e8	−0.0253842
$875$	−7.46251e9	−0.376580
$876$	0	0
$877$	9.94078e9	0.497648	0.248824	−	0.968549i	$-0.419956\pi$
$877$	9.94078e9	0.497648	0.248824	+	0.968549i	$0.419956\pi$
$878$	7.35764e9	0.366866
$879$	0	0
$880$	2.25608e8	0.0111600
$881$	2.51302e10	1.23817	0.619086	−	0.785324i	$-0.287503\pi$
$881$	2.51302e10	1.23817	0.619086	+	0.785324i	$0.287503\pi$
$882$	0	0
$883$	1.44118e10	0.704458	0.352229	−	0.935914i	$-0.385424\pi$
$883$	1.44118e10	0.704458	0.352229	+	0.935914i	$0.385424\pi$
$884$	1.26573e9	0.0616252
$885$	0	0
$886$	1.33648e10	0.645571
$887$	−2.40854e10	−1.15883	−0.579416	−	0.815032i	$-0.696719\pi$
$887$	−2.40854e10	−1.15883	−0.579416	+	0.815032i	$0.696719\pi$
$888$	0	0
$889$	1.12519e9	0.0537118
$890$	−1.50385e10	−0.715057
$891$	0	0
$892$	5.77477e9	0.272432
$893$	9.19316e9	0.432001
$894$	0	0
$895$	2.84534e9	0.132664
$896$	7.19323e8	0.0334077
$897$	0	0
$898$	2.52148e10	1.16195
$899$	−2.39347e9	−0.109868
$900$	0	0
$901$	4.75273e9	0.216474
$902$	2.45755e8	0.0111501
$903$	0	0
$904$	−3.42935e9	−0.154391
$905$	−2.16794e10	−0.972248
$906$	0	0
$907$	−1.02624e10	−0.456690	−0.228345	−	0.973580i	$-0.573331\pi$
$907$	−1.02624e10	−0.456690	−0.228345	+	0.973580i	$0.573331\pi$
$908$	−1.43085e9	−0.0634298
$909$	0	0
$910$	4.61121e9	0.202848
$911$	−1.93711e10	−0.848866	−0.424433	−	0.905459i	$-0.639527\pi$
$911$	−1.93711e10	−0.848866	−0.424433	+	0.905459i	$0.639527\pi$
$912$	0	0
$913$	2.14629e8	0.00933344
$914$	−6.99604e9	−0.303068
$915$	0	0
$916$	8.50089e9	0.365452
$917$	−1.01159e10	−0.433225
$918$	0	0
$919$	−2.63726e10	−1.12085	−0.560426	−	0.828205i	$-0.689363\pi$
$919$	−2.63726e10	−1.12085	−0.560426	+	0.828205i	$0.689363\pi$
$920$	−7.44586e8	−0.0315252
$921$	0	0
$922$	−1.80533e10	−0.758574
$923$	2.04728e10	0.856981
$924$	0	0
$925$	7.85345e7	0.00326260
$926$	−1.39043e10	−0.575454
$927$	0	0
$928$	−1.59162e9	−0.0653766
$929$	−3.77372e9	−0.154424	−0.0772121	−	0.997015i	$-0.524602\pi$
$929$	−3.77372e9	−0.154424	−0.0772121	+	0.997015i	$0.524602\pi$
$930$	0	0
$931$	1.42126e9	0.0577231
$932$	1.78506e10	0.722264
$933$	0	0
$934$	2.42832e10	0.975194
$935$	1.81841e8	0.00727530
$936$	0	0
$937$	−4.40888e9	−0.175081	−0.0875406	−	0.996161i	$-0.527901\pi$
$937$	−4.40888e9	−0.175081	−0.0875406	+	0.996161i	$0.527901\pi$
$938$	−1.32643e10	−0.524776
$939$	0	0
$940$	1.36624e10	0.536510
$941$	8.51871e9	0.333281	0.166640	−	0.986018i	$-0.446708\pi$
$941$	8.51871e9	0.333281	0.166640	+	0.986018i	$0.446708\pi$
$942$	0	0
$943$	−8.11078e8	−0.0314972
$944$	9.17731e9	0.355069
$945$	0	0
$946$	6.48952e8	0.0249226
$947$	−2.94470e10	−1.12672	−0.563360	−	0.826212i	$-0.690491\pi$
$947$	−2.94470e10	−1.12672	−0.563360	+	0.826212i	$0.690491\pi$
$948$	0	0
$949$	−6.41434e9	−0.243624
$950$	5.48330e7	0.00207496
$951$	0	0
$952$	5.79777e8	0.0217787
$953$	−1.86093e10	−0.696473	−0.348236	−	0.937407i	$-0.613219\pi$
$953$	−1.86093e10	−0.696473	−0.348236	+	0.937407i	$0.613219\pi$
$954$	0	0
$955$	3.43728e9	0.127704
$956$	1.42786e10	0.528547
$957$	0	0
$958$	−6.96814e9	−0.256058
$959$	−3.11298e9	−0.113975
$960$	0	0
$961$	−2.50845e10	−0.911744
$962$	6.63358e9	0.240235
$963$	0	0
$964$	−1.79393e10	−0.644964
$965$	4.02358e10	1.44134
$966$	0	0
$967$	2.05267e10	0.730006	0.365003	−	0.931006i	$-0.381068\pi$
$967$	2.05267e10	0.730006	0.365003	+	0.931006i	$0.381068\pi$
$968$	−9.95769e9	−0.352854
$969$	0	0
$970$	1.80476e10	0.634921
$971$	−2.67970e8	−0.00939333	−0.00469667	−	0.999989i	$-0.501495\pi$
$971$	−2.67970e8	−0.00939333	−0.00469667	+	0.999989i	$0.501495\pi$
$972$	0	0
$973$	−1.37006e9	−0.0476810
$974$	1.23774e10	0.429214
$975$	0	0
$976$	−7.81927e9	−0.269210
$977$	−1.53498e10	−0.526590	−0.263295	−	0.964715i	$-0.584809\pi$
$977$	−1.53498e10	−0.526590	−0.263295	+	0.964715i	$0.584809\pi$
$978$	0	0
$979$	−1.31576e9	−0.0448165
$980$	2.11220e9	0.0716875
$981$	0	0
$982$	1.85372e10	0.624673
$983$	1.23280e10	0.413958	0.206979	−	0.978345i	$-0.433637\pi$
$983$	1.23280e10	0.413958	0.206979	+	0.978345i	$0.433637\pi$
$984$	0	0
$985$	5.14954e10	1.71689
$986$	−1.28285e9	−0.0426195
$987$	0	0
$988$	4.63159e9	0.152785
$989$	−2.14177e9	−0.0704021
$990$	0	0
$991$	−5.90638e10	−1.92781	−0.963904	−	0.266249i	$-0.914216\pi$
$991$	−5.90638e10	−1.92781	−0.963904	+	0.266249i	$0.914216\pi$
$992$	1.61469e9	0.0525167
$993$	0	0
$994$	9.37771e9	0.302862
$995$	−3.83769e10	−1.23506
$996$	0	0
$997$	2.68495e10	0.858031	0.429015	−	0.903297i	$-0.358861\pi$
$997$	2.68495e10	0.858031	0.429015	+	0.903297i	$0.358861\pi$
$998$	2.17936e10	0.694020
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.8.a.l.1.2	yes	2
3.2	odd	2		126.8.a.k.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.8.a.k.1.1	✓	2	3.2	odd	2
126.8.a.l.1.2	yes	2	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} +64.0000 q^{4} +280.522 q^{5} +343.000 q^{7} +512.000 q^{8} +2244.17 q^{10} +196.349 q^{11} +5990.52 q^{13} +2744.00 q^{14} +4096.00 q^{16} +3301.39 q^{17} +12080.5 q^{19} +17953.4 q^{20} +1570.79 q^{22} -5184.16 q^{23} +567.370 q^{25} +47924.1 q^{26} +21952.0 q^{28} -48572.5 q^{29} +49276.4 q^{31} +32768.0 q^{32} +26411.1 q^{34} +96218.9 q^{35} +138418. q^{37} +96644.1 q^{38} +143627. q^{40} +156453. q^{41} +413137. q^{43} +12566.3 q^{44} -41473.3 q^{46} +760991. q^{47} +117649. q^{49} +4538.96 q^{50} +383393. q^{52} +1.43961e6 q^{53} +55080.1 q^{55} +175616. q^{56} -388580. q^{58} +2.24055e6 q^{59} -1.90900e6 q^{61} +394211. q^{62} +262144. q^{64} +1.68047e6 q^{65} -4.83392e6 q^{67} +211289. q^{68} +769751. q^{70} +3.41753e6 q^{71} -1.07075e6 q^{73} +1.10735e6 q^{74} +773153. q^{76} +67347.6 q^{77} -1.90535e6 q^{79} +1.14902e6 q^{80} +1.25162e6 q^{82} +1.09310e6 q^{83} +926112. q^{85} +3.30510e6 q^{86} +100531. q^{88} -6.70115e6 q^{89} +2.05475e6 q^{91} -331786. q^{92} +6.08793e6 q^{94} +3.38885e6 q^{95} +8.04200e6 q^{97} +941192. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{2} + 128 q^{4} - 168 q^{5} + 686 q^{7} + 1024 q^{8} - 1344 q^{10} + 5496 q^{11} - 5516 q^{13} + 5488 q^{14} + 8192 q^{16} + 10248 q^{17} + 6664 q^{19} - 10752 q^{20} + 43968 q^{22} + 45768 q^{23}+ \cdots + 1882384 q^{98}+O(q^{100}) \) 2 * q + 16 * q^2 + 128 * q^4 - 168 * q^5 + 686 * q^7 + 1024 * q^8 - 1344 * q^10 + 5496 * q^11 - 5516 * q^13 + 5488 * q^14 + 8192 * q^16 + 10248 * q^17 + 6664 * q^19 - 10752 * q^20 + 43968 * q^22 + 45768 * q^23 + 123614 * q^25 - 44128 * q^26 + 43904 * q^28 + 188640 * q^29 + 221032 * q^31 + 65536 * q^32 + 81984 * q^34 - 57624 * q^35 + 399316 * q^37 + 53312 * q^38 - 86016 * q^40 + 562968 * q^41 - 153560 * q^43 + 351744 * q^44 + 366144 * q^46 + 651504 * q^47 + 235298 * q^49 + 988912 * q^50 - 353024 * q^52 + 2032080 * q^53 - 2321928 * q^55 + 351232 * q^56 + 1509120 * q^58 + 3864336 * q^59 - 721028 * q^61 + 1768256 * q^62 + 524288 * q^64 + 6841392 * q^65 - 604376 * q^67 + 655872 * q^68 - 460992 * q^70 + 389592 * q^71 - 6935684 * q^73 + 3194528 * q^74 + 426496 * q^76 + 1885128 * q^77 - 7974992 * q^79 - 688128 * q^80 + 4503744 * q^82 + 2209536 * q^83 - 2189592 * q^85 - 1228480 * q^86 + 2813952 * q^88 + 1659000 * q^89 - 1891988 * q^91 + 2929152 * q^92 + 5212032 * q^94 + 5818272 * q^95 - 4807460 * q^97 + 1882384 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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