LMFDB - Embedded newform 126.16.a.n.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [126,16,Mod(1,126)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 16, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("126.1"); S:= CuspForms(chi, 16); N := Newforms(S);

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	126.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-512,0,65536,-291312] 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:	$179.793816426$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 201842416x^{2} + 1462712685511x - 2792107035723495 $ x^4 - x^3 - 201842416x^2 + 1462712685511x - 2792107035723495
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{8}\cdot 5^{2}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-17211.5$ 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)

$f(q)$	$=$	\(q-128.000 q^{2} +16384.0 q^{4} +182011. q^{5} +823543. q^{7} -2.09715e6 q^{8} -2.32974e7 q^{10} -1.07710e8 q^{11} -9.14826e7 q^{13} -1.05414e8 q^{14} +2.68435e8 q^{16} +1.71768e9 q^{17} +5.58597e9 q^{19} +2.98207e9 q^{20} +1.37869e10 q^{22} +7.44596e9 q^{23} +2.61047e9 q^{25} +1.17098e10 q^{26} +1.34929e10 q^{28} -1.05345e11 q^{29} -4.89081e10 q^{31} -3.43597e10 q^{32} -2.19863e11 q^{34} +1.49894e11 q^{35} -6.73470e11 q^{37} -7.15005e11 q^{38} -3.81705e11 q^{40} +4.82615e11 q^{41} -1.35876e11 q^{43} -1.76473e12 q^{44} -9.53083e11 q^{46} -9.81674e11 q^{47} +6.78223e11 q^{49} -3.34140e11 q^{50} -1.49885e12 q^{52} +5.44704e12 q^{53} -1.96045e13 q^{55} -1.72709e12 q^{56} +1.34841e13 q^{58} -7.80779e12 q^{59} +2.96698e11 q^{61} +6.26024e12 q^{62} +4.39805e12 q^{64} -1.66509e13 q^{65} -1.32958e13 q^{67} +2.81425e13 q^{68} -1.91864e13 q^{70} -1.39178e14 q^{71} +3.62188e13 q^{73} +8.62041e13 q^{74} +9.15206e13 q^{76} -8.87041e13 q^{77} +2.75120e14 q^{79} +4.88582e13 q^{80} -6.17747e13 q^{82} -7.63360e13 q^{83} +3.12637e14 q^{85} +1.73921e13 q^{86} +2.25885e14 q^{88} -4.96006e13 q^{89} -7.53399e13 q^{91} +1.21995e14 q^{92} +1.25654e14 q^{94} +1.01671e15 q^{95} +4.07266e14 q^{97} -8.68126e13 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 512 q^{2} + 65536 q^{4} - 291312 q^{5} + 3294172 q^{7} - 8388608 q^{8} + 37287936 q^{10} - 121144752 q^{11} + 170165240 q^{13} - 421654016 q^{14} + 1073741824 q^{16} - 345685200 q^{17} + 4738793696 q^{19}+ \cdots - 347250213298688 q^{98}+O(q^{100}) $ 4 * q - 512 * q^2 + 65536 * q^4 - 291312 * q^5 + 3294172 * q^7 - 8388608 * q^8 + 37287936 * q^10 - 121144752 * q^11 + 170165240 * q^13 - 421654016 * q^14 + 1073741824 * q^16 - 345685200 * q^17 + 4738793696 * q^19 - 4772855808 * q^20 + 15506528256 * q^22 - 17023827024 * q^23 + 38327672524 * q^25 - 21781150720 * q^26 + 53971714048 * q^28 - 96242392128 * q^29 + 65665329632 * q^31 - 137438953472 * q^32 + 44247705600 * q^34 - 239907958416 * q^35 + 279650496248 * q^37 - 606565593088 * q^38 + 610925543424 * q^40 - 274507805040 * q^41 + 1551932654480 * q^43 - 1984835616768 * q^44 + 2179049859072 * q^46 - 686050223712 * q^47 + 2712892291396 * q^49 - 4905942083072 * q^50 + 2787987292160 * q^52 - 4929897017760 * q^53 + 3835130598432 * q^55 - 6908379398144 * q^56 + 12319026192384 * q^58 - 15661604798112 * q^59 + 7270740102248 * q^61 - 8405162192896 * q^62 + 17592186044416 * q^64 - 39986092580832 * q^65 + 2642113680560 * q^67 - 5663706316800 * q^68 + 30708218677248 * q^70 - 62559334721520 * q^71 - 53486860270312 * q^73 - 35795263519744 * q^74 + 77640395915264 * q^76 - 99767912496336 * q^77 + 77560377412544 * q^79 - 78198469558272 * q^80 + 35136999045120 * q^82 + 58088393760768 * q^83 + 48889916091648 * q^85 - 198647379773440 * q^86 + 254058958946304 * q^88 + 296914762526928 * q^89 + 140138392245320 * q^91 - 278918381961216 * q^92 + 87814428635136 * q^94 + 1325087451258048 * q^95 + 1391306483388248 * q^97 - 347250213298688 * 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 $

$2$	−128.000	−0.707107
$2$	−128.000	−0.707107
$3$	0	0
$3$	0	0
$4$	16384.0	0.500000
$5$	182011.	1.04189	0.520946	−	0.853589i	$-0.325579\pi$
$5$	182011.	1.04189	0.520946	+	0.853589i	$0.325579\pi$
$6$	0	0
$7$	823543.	0.377964
$7$	823543.	0.377964
$8$	−2.09715e6	−0.353553
$9$	0	0
$10$	−2.32974e7	−0.736729
$11$	−1.07710e8	−1.66653	−0.833263	−	0.552877i	$-0.813530\pi$
$11$	−1.07710e8	−1.66653	−0.833263	+	0.552877i	$0.813530\pi$
$12$	0	0
$13$	−9.14826e7	−0.404356	−0.202178	−	0.979349i	$-0.564802\pi$
$13$	−9.14826e7	−0.404356	−0.202178	+	0.979349i	$0.564802\pi$
$14$	−1.05414e8	−0.267261
$15$	0	0
$16$	2.68435e8	0.250000
$17$	1.71768e9	1.01526	0.507628	−	0.861576i	$-0.330522\pi$
$17$	1.71768e9	1.01526	0.507628	+	0.861576i	$0.330522\pi$
$18$	0	0
$19$	5.58597e9	1.43366	0.716831	−	0.697247i	$-0.245592\pi$
$19$	5.58597e9	1.43366	0.716831	+	0.697247i	$0.245592\pi$
$20$	2.98207e9	0.520946
$21$	0	0
$22$	1.37869e10	1.17841
$23$	7.44596e9	0.455997	0.227998	−	0.973661i	$-0.426782\pi$
$23$	7.44596e9	0.455997	0.227998	+	0.973661i	$0.426782\pi$
$24$	0	0
$25$	2.61047e9	0.0855400
$26$	1.17098e10	0.285923
$27$	0	0
$28$	1.34929e10	0.188982
$29$	−1.05345e11	−1.13404	−0.567020	−	0.823704i	$-0.691904\pi$
$29$	−1.05345e11	−1.13404	−0.567020	+	0.823704i	$0.691904\pi$
$30$	0	0
$31$	−4.89081e10	−0.319277	−0.159639	−	0.987176i	$-0.551033\pi$
$31$	−4.89081e10	−0.319277	−0.159639	+	0.987176i	$0.551033\pi$
$32$	−3.43597e10	−0.176777
$33$	0	0
$34$	−2.19863e11	−0.717895
$35$	1.49894e11	0.393798
$36$	0	0
$37$	−6.73470e11	−1.16629	−0.583143	−	0.812369i	$-0.698177\pi$
$37$	−6.73470e11	−1.16629	−0.583143	+	0.812369i	$0.698177\pi$
$38$	−7.15005e11	−1.01375
$39$	0	0
$40$	−3.81705e11	−0.368365
$41$	4.82615e11	0.387010	0.193505	−	0.981099i	$-0.438014\pi$
$41$	4.82615e11	0.387010	0.193505	+	0.981099i	$0.438014\pi$
$42$	0	0
$43$	−1.35876e11	−0.0762306	−0.0381153	−	0.999273i	$-0.512135\pi$
$43$	−1.35876e11	−0.0762306	−0.0381153	+	0.999273i	$0.512135\pi$
$44$	−1.76473e12	−0.833263
$45$	0	0
$46$	−9.53083e11	−0.322439
$47$	−9.81674e11	−0.282640	−0.141320	−	0.989964i	$-0.545135\pi$
$47$	−9.81674e11	−0.282640	−0.141320	+	0.989964i	$0.545135\pi$
$48$	0	0
$49$	6.78223e11	0.142857
$50$	−3.34140e11	−0.0604859
$51$	0	0
$52$	−1.49885e12	−0.202178
$53$	5.44704e12	0.636930	0.318465	−	0.947935i	$-0.396833\pi$
$53$	5.44704e12	0.636930	0.318465	+	0.947935i	$0.396833\pi$
$54$	0	0
$55$	−1.96045e13	−1.73634
$56$	−1.72709e12	−0.133631
$57$	0	0
$58$	1.34841e13	0.801888
$59$	−7.80779e12	−0.408449	−0.204225	−	0.978924i	$-0.565467\pi$
$59$	−7.80779e12	−0.408449	−0.204225	+	0.978924i	$0.565467\pi$
$60$	0	0
$61$	2.96698e11	0.0120876	0.00604380	−	0.999982i	$-0.498076\pi$
$61$	2.96698e11	0.0120876	0.00604380	+	0.999982i	$0.498076\pi$
$62$	6.26024e12	0.225763
$63$	0	0
$64$	4.39805e12	0.125000
$65$	−1.66509e13	−0.421295
$66$	0	0
$67$	−1.32958e13	−0.268011	−0.134005	−	0.990981i	$-0.542784\pi$
$67$	−1.32958e13	−0.268011	−0.134005	+	0.990981i	$0.542784\pi$
$68$	2.81425e13	0.507628
$69$	0	0
$70$	−1.91864e13	−0.278457
$71$	−1.39178e14	−1.81607	−0.908034	−	0.418896i	$-0.862417\pi$
$71$	−1.39178e14	−1.81607	−0.908034	+	0.418896i	$0.862417\pi$
$72$	0	0
$73$	3.62188e13	0.383719	0.191859	−	0.981422i	$-0.438548\pi$
$73$	3.62188e13	0.383719	0.191859	+	0.981422i	$0.438548\pi$
$74$	8.62041e13	0.824689
$75$	0	0
$76$	9.15206e13	0.716831
$77$	−8.87041e13	−0.629888
$78$	0	0
$79$	2.75120e14	1.61183	0.805914	−	0.592032i	$-0.201674\pi$
$79$	2.75120e14	1.61183	0.805914	+	0.592032i	$0.201674\pi$
$80$	4.88582e13	0.260473
$81$	0	0
$82$	−6.17747e13	−0.273657
$83$	−7.63360e13	−0.308776	−0.154388	−	0.988010i	$-0.549341\pi$
$83$	−7.63360e13	−0.308776	−0.154388	+	0.988010i	$0.549341\pi$
$84$	0	0
$85$	3.12637e14	1.05779
$86$	1.73921e13	0.0539031
$87$	0	0
$88$	2.25885e14	0.589206
$89$	−4.96006e13	−0.118867	−0.0594336	−	0.998232i	$-0.518929\pi$
$89$	−4.96006e13	−0.118867	−0.0594336	+	0.998232i	$0.518929\pi$
$90$	0	0
$91$	−7.53399e13	−0.152832
$92$	1.21995e14	0.227998
$93$	0	0
$94$	1.25654e14	0.199857
$95$	1.01671e15	1.49372
$96$	0	0
$97$	4.07266e14	0.511789	0.255894	−	0.966705i	$-0.417630\pi$
$97$	4.07266e14	0.511789	0.255894	+	0.966705i	$0.417630\pi$
$98$	−8.68126e13	−0.101015
$99$	0	0
$100$	4.27700e13	0.0427700
$101$	−5.65816e14	−0.525127	−0.262564	−	0.964915i	$-0.584568\pi$
$101$	−5.65816e14	−0.525127	−0.262564	+	0.964915i	$0.584568\pi$
$102$	0	0
$103$	−1.78833e15	−1.43274	−0.716370	−	0.697721i	$-0.754198\pi$
$103$	−1.78833e15	−1.43274	−0.716370	+	0.697721i	$0.754198\pi$
$104$	1.91853e14	0.142961
$105$	0	0
$106$	−6.97222e14	−0.450378
$107$	3.74822e14	0.225656	0.112828	−	0.993615i	$-0.464009\pi$
$107$	3.74822e14	0.225656	0.112828	+	0.993615i	$0.464009\pi$
$108$	0	0
$109$	1.18462e15	0.620695	0.310348	−	0.950623i	$-0.399555\pi$
$109$	1.18462e15	0.620695	0.310348	+	0.950623i	$0.399555\pi$
$110$	2.50937e15	1.22778
$111$	0	0
$112$	2.21068e14	0.0944911
$113$	−2.91865e15	−1.16706	−0.583530	−	0.812092i	$-0.698329\pi$
$113$	−2.91865e15	−1.16706	−0.583530	+	0.812092i	$0.698329\pi$
$114$	0	0
$115$	1.35525e15	0.475100
$116$	−1.72597e15	−0.567020
$117$	0	0
$118$	9.99397e14	0.288817
$119$	1.41458e15	0.383731
$120$	0	0
$121$	7.42426e15	1.77731
$122$	−3.79773e13	−0.00854723
$123$	0	0
$124$	−8.01310e14	−0.159639
$125$	−5.07940e15	−0.952769
$126$	0	0
$127$	1.94991e15	0.324703	0.162352	−	0.986733i	$-0.448092\pi$
$127$	1.94991e15	0.324703	0.162352	+	0.986733i	$0.448092\pi$
$128$	−5.62950e14	−0.0883883
$129$	0	0
$130$	2.13131e15	0.297901
$131$	−1.15882e13	−0.00152926	−0.000764629	−	1.00000i	$-0.500243\pi$
$131$	−1.15882e13	−0.00152926	−0.000764629		1.00000i	$0.500243\pi$
$132$	0	0
$133$	4.60029e15	0.541873
$134$	1.70186e15	0.189512
$135$	0	0
$136$	−3.60224e15	−0.358947
$137$	−7.56436e15	−0.713458	−0.356729	−	0.934208i	$-0.616108\pi$
$137$	−7.56436e15	−0.713458	−0.356729	+	0.934208i	$0.616108\pi$
$138$	0	0
$139$	−6.89957e15	−0.583729	−0.291864	−	0.956460i	$-0.594276\pi$
$139$	−6.89957e15	−0.583729	−0.291864	+	0.956460i	$0.594276\pi$
$140$	2.45586e15	0.196899
$141$	0	0
$142$	1.78148e16	1.28415
$143$	9.85362e15	0.673869
$144$	0	0
$145$	−1.91739e16	−1.18155
$146$	−4.63601e15	−0.271330
$147$	0	0
$148$	−1.10341e16	−0.583143
$149$	−2.34549e16	−1.17852	−0.589261	−	0.807943i	$-0.700581\pi$
$149$	−2.34549e16	−1.17852	−0.589261	+	0.807943i	$0.700581\pi$
$150$	0	0
$151$	4.34974e16	1.97759	0.988794	−	0.149287i	$-0.0476977\pi$
$151$	4.34974e16	1.97759	0.988794	+	0.149287i	$0.0476977\pi$
$152$	−1.17146e16	−0.506876
$153$	0	0
$154$	1.13541e16	0.445398
$155$	−8.90182e15	−0.332653
$156$	0	0
$157$	4.92741e16	1.67252	0.836260	−	0.548334i	$-0.184738\pi$
$157$	4.92741e16	1.67252	0.836260	+	0.548334i	$0.184738\pi$
$158$	−3.52153e16	−1.13973
$159$	0	0
$160$	−6.25385e15	−0.184182
$161$	6.13207e15	0.172351
$162$	0	0
$163$	−1.68432e16	−0.431536	−0.215768	−	0.976445i	$-0.569225\pi$
$163$	−1.68432e16	−0.431536	−0.215768	+	0.976445i	$0.569225\pi$
$164$	7.90717e15	0.193505
$165$	0	0
$166$	9.77101e15	0.218338
$167$	2.93793e16	0.627578	0.313789	−	0.949493i	$-0.398401\pi$
$167$	2.93793e16	0.627578	0.313789	+	0.949493i	$0.398401\pi$
$168$	0	0
$169$	−4.28168e16	−0.836497
$170$	−4.00175e16	−0.747969
$171$	0	0
$172$	−2.22619e15	−0.0381153
$173$	1.72022e16	0.281993	0.140997	−	0.990010i	$-0.454969\pi$
$173$	1.72022e16	0.281993	0.140997	+	0.990010i	$0.454969\pi$
$174$	0	0
$175$	2.14984e15	0.0323311
$176$	−2.89133e16	−0.416632
$177$	0	0
$178$	6.34888e15	0.0840518
$179$	1.00380e17	1.27424	0.637119	−	0.770765i	$-0.280126\pi$
$179$	1.00380e17	1.27424	0.637119	+	0.770765i	$0.280126\pi$
$180$	0	0
$181$	−1.49173e17	−1.74221	−0.871103	−	0.491100i	$-0.836595\pi$
$181$	−1.49173e17	−1.74221	−0.871103	+	0.491100i	$0.836595\pi$
$182$	9.64350e15	0.108069
$183$	0	0
$184$	−1.56153e16	−0.161219
$185$	−1.22579e17	−1.21514
$186$	0	0
$187$	−1.85012e17	−1.69195
$188$	−1.60837e16	−0.141320
$189$	0	0
$190$	−1.30139e17	−1.05622
$191$	−1.96712e17	−1.53490	−0.767450	−	0.641108i	$-0.778475\pi$
$191$	−1.96712e17	−1.53490	−0.767450	+	0.641108i	$0.778475\pi$
$192$	0	0
$193$	1.62414e16	0.117205	0.0586023	−	0.998281i	$-0.481336\pi$
$193$	1.62414e16	0.117205	0.0586023	+	0.998281i	$0.481336\pi$
$194$	−5.21301e16	−0.361889
$195$	0	0
$196$	1.11120e16	0.0714286
$197$	−1.32502e17	−0.819831	−0.409915	−	0.912124i	$-0.634442\pi$
$197$	−1.32502e17	−0.819831	−0.409915	+	0.912124i	$0.634442\pi$
$198$	0	0
$199$	−1.01048e17	−0.579600	−0.289800	−	0.957087i	$-0.593589\pi$
$199$	−1.01048e17	−0.579600	−0.289800	+	0.957087i	$0.593589\pi$
$200$	−5.47456e15	−0.0302429
$201$	0	0
$202$	7.24244e16	0.371321
$203$	−8.67560e16	−0.428627
$204$	0	0
$205$	8.78413e16	0.403223
$206$	2.28906e17	1.01310
$207$	0	0
$208$	−2.45572e16	−0.101089
$209$	−6.01667e17	−2.38924
$210$	0	0
$211$	3.94434e17	1.45833	0.729165	−	0.684338i	$-0.239909\pi$
$211$	3.94434e17	1.45833	0.729165	+	0.684338i	$0.239909\pi$
$212$	8.92444e16	0.318465
$213$	0	0
$214$	−4.79772e16	−0.159563
$215$	−2.47309e16	−0.0794240
$216$	0	0
$217$	−4.02779e16	−0.120675
$218$	−1.51631e17	−0.438898
$219$	0	0
$220$	−3.21200e17	−0.868171
$221$	−1.57138e17	−0.410525
$222$	0	0
$223$	−4.39611e17	−1.07345	−0.536725	−	0.843757i	$-0.680339\pi$
$223$	−4.39611e17	−1.07345	−0.536725	+	0.843757i	$0.680339\pi$
$224$	−2.82967e16	−0.0668153
$225$	0	0
$226$	3.73587e17	0.825236
$227$	−8.61734e17	−1.84153	−0.920766	−	0.390114i	$-0.872435\pi$
$227$	−8.61734e17	−1.84153	−0.920766	+	0.390114i	$0.872435\pi$
$228$	0	0
$229$	8.53953e17	1.70871	0.854354	−	0.519691i	$-0.173953\pi$
$229$	8.53953e17	1.70871	0.854354	+	0.519691i	$0.173953\pi$
$230$	−1.73472e17	−0.335946
$231$	0	0
$232$	2.20924e17	0.400944
$233$	−4.47950e17	−0.787155	−0.393578	−	0.919291i	$-0.628763\pi$
$233$	−4.47950e17	−0.787155	−0.393578	+	0.919291i	$0.628763\pi$
$234$	0	0
$235$	−1.78676e17	−0.294480
$236$	−1.27923e17	−0.204225
$237$	0	0
$238$	−1.81067e17	−0.271339
$239$	1.86410e17	0.270698	0.135349	−	0.990798i	$-0.456784\pi$
$239$	1.86410e17	0.270698	0.135349	+	0.990798i	$0.456784\pi$
$240$	0	0
$241$	−7.85126e17	−1.07105	−0.535527	−	0.844518i	$-0.679887\pi$
$241$	−7.85126e17	−1.07105	−0.535527	+	0.844518i	$0.679887\pi$
$242$	−9.50306e17	−1.25675
$243$	0	0
$244$	4.86109e15	0.00604380
$245$	1.23444e17	0.148842
$246$	0	0
$247$	−5.11020e17	−0.579709
$248$	1.02568e17	0.112882
$249$	0	0
$250$	6.50164e17	0.673709
$251$	−3.54416e17	−0.356418	−0.178209	−	0.983993i	$-0.557030\pi$
$251$	−3.54416e17	−0.356418	−0.178209	+	0.983993i	$0.557030\pi$
$252$	0	0
$253$	−8.02007e17	−0.759931
$254$	−2.49588e17	−0.229600
$255$	0	0
$256$	7.20576e16	0.0625000
$257$	−1.97949e18	−1.66746	−0.833730	−	0.552172i	$-0.813799\pi$
$257$	−1.97949e18	−1.66746	−0.833730	+	0.552172i	$0.813799\pi$
$258$	0	0
$259$	−5.54631e17	−0.440815
$260$	−2.72808e17	−0.210648
$261$	0	0
$262$	1.48329e15	0.00108135
$263$	−1.27171e18	−0.900991	−0.450496	−	0.892779i	$-0.648753\pi$
$263$	−1.27171e18	−0.900991	−0.450496	+	0.892779i	$0.648753\pi$
$264$	0	0
$265$	9.91423e17	0.663613
$266$	−5.88837e17	−0.383162
$267$	0	0
$268$	−2.17838e17	−0.134005
$269$	−1.86054e18	−1.11300	−0.556501	−	0.830847i	$-0.687856\pi$
$269$	−1.86054e18	−1.11300	−0.556501	+	0.830847i	$0.687856\pi$
$270$	0	0
$271$	2.95868e18	1.67428	0.837141	−	0.546986i	$-0.184225\pi$
$271$	2.95868e18	1.67428	0.837141	+	0.546986i	$0.184225\pi$
$272$	4.61087e17	0.253814
$273$	0	0
$274$	9.68238e17	0.504491
$275$	−2.81175e17	−0.142555
$276$	0	0
$277$	1.22524e18	0.588331	0.294165	−	0.955755i	$-0.404958\pi$
$277$	1.22524e18	0.588331	0.294165	+	0.955755i	$0.404958\pi$
$278$	8.83145e17	0.412759
$279$	0	0
$280$	−3.14350e17	−0.139229
$281$	−6.54984e17	−0.282445	−0.141222	−	0.989978i	$-0.545103\pi$
$281$	−6.54984e17	−0.282445	−0.141222	+	0.989978i	$0.545103\pi$
$282$	0	0
$283$	−1.23781e18	−0.506123	−0.253062	−	0.967450i	$-0.581438\pi$
$283$	−1.23781e18	−0.506123	−0.253062	+	0.967450i	$0.581438\pi$
$284$	−2.28029e18	−0.908034
$285$	0	0
$286$	−1.26126e18	−0.476497
$287$	3.97454e17	0.146276
$288$	0	0
$289$	8.80055e16	0.0307451
$290$	2.45426e18	0.835481
$291$	0	0
$292$	5.93410e17	0.191859
$293$	5.36204e18	1.68975	0.844876	−	0.534962i	$-0.179674\pi$
$293$	5.36204e18	1.68975	0.844876	+	0.534962i	$0.179674\pi$
$294$	0	0
$295$	−1.42110e18	−0.425560
$296$	1.41237e18	0.412344
$297$	0	0
$298$	3.00223e18	0.833340
$299$	−6.81176e17	−0.184385
$300$	0	0
$301$	−1.11900e17	−0.0288124
$302$	−5.56766e18	−1.39837
$303$	0	0
$304$	1.49947e18	0.358416
$305$	5.40023e16	0.0125940
$306$	0	0
$307$	−3.92173e18	−0.870844	−0.435422	−	0.900226i	$-0.643401\pi$
$307$	−3.92173e18	−0.870844	−0.435422	+	0.900226i	$0.643401\pi$
$308$	−1.45333e18	−0.314944
$309$	0	0
$310$	1.13943e18	0.235221
$311$	−6.88321e18	−1.38704	−0.693518	−	0.720439i	$-0.743940\pi$
$311$	−6.88321e18	−1.38704	−0.693518	+	0.720439i	$0.743940\pi$
$312$	0	0
$313$	4.26355e18	0.818822	0.409411	−	0.912350i	$-0.365734\pi$
$313$	4.26355e18	0.818822	0.409411	+	0.912350i	$0.365734\pi$
$314$	−6.30708e18	−1.18265
$315$	0	0
$316$	4.50756e18	0.805914
$317$	−5.61923e18	−0.981143	−0.490572	−	0.871401i	$-0.663212\pi$
$317$	−5.61923e18	−0.981143	−0.490572	+	0.871401i	$0.663212\pi$
$318$	0	0
$319$	1.13467e19	1.88991
$320$	8.00493e17	0.130237
$321$	0	0
$322$	−7.84905e17	−0.121870
$323$	9.59492e18	1.45553
$324$	0	0
$325$	−2.38813e17	−0.0345886
$326$	2.15593e18	0.305142
$327$	0	0
$328$	−1.01212e18	−0.136829
$329$	−8.08451e17	−0.106828
$330$	0	0
$331$	−5.31918e17	−0.0671638	−0.0335819	−	0.999436i	$-0.510691\pi$
$331$	−5.31918e17	−0.0671638	−0.0335819	+	0.999436i	$0.510691\pi$
$332$	−1.25069e18	−0.154388
$333$	0	0
$334$	−3.76055e18	−0.443765
$335$	−2.41998e18	−0.279238
$336$	0	0
$337$	−1.42988e19	−1.57789	−0.788943	−	0.614466i	$-0.789371\pi$
$337$	−1.42988e19	−1.57789	−0.788943	+	0.614466i	$0.789371\pi$
$338$	5.48055e18	0.591492
$339$	0	0
$340$	5.12225e18	0.528894
$341$	5.26791e18	0.532084
$342$	0	0
$343$	5.58546e17	0.0539949
$344$	2.84952e17	0.0269516
$345$	0	0
$346$	−2.20188e18	−0.199399
$347$	5.48427e18	0.486013	0.243007	−	0.970025i	$-0.421866\pi$
$347$	5.48427e18	0.486013	0.243007	+	0.970025i	$0.421866\pi$
$348$	0	0
$349$	−2.13307e19	−1.81057	−0.905284	−	0.424808i	$-0.860342\pi$
$349$	−2.13307e19	−1.81057	−0.905284	+	0.424808i	$0.860342\pi$
$350$	−2.75179e17	−0.0228615
$351$	0	0
$352$	3.70090e18	0.294603
$353$	−1.04779e19	−0.816514	−0.408257	−	0.912867i	$-0.633863\pi$
$353$	−1.04779e19	−0.816514	−0.408257	+	0.912867i	$0.633863\pi$
$354$	0	0
$355$	−2.53319e19	−1.89215
$356$	−8.12657e17	−0.0594336
$357$	0	0
$358$	−1.28487e19	−0.901023
$359$	−7.77860e17	−0.0534187	−0.0267093	−	0.999643i	$-0.508503\pi$
$359$	−7.77860e17	−0.0534187	−0.0267093	+	0.999643i	$0.508503\pi$
$360$	0	0
$361$	1.60220e19	1.05539
$362$	1.90941e19	1.23193
$363$	0	0
$364$	−1.23437e18	−0.0764160
$365$	6.59223e18	0.399794
$366$	0	0
$367$	−3.42327e19	−1.99272	−0.996358	−	0.0852668i	$-0.972826\pi$
$367$	−3.42327e19	−1.99272	−0.996358	+	0.0852668i	$0.972826\pi$
$368$	1.99876e18	0.113999
$369$	0	0
$370$	1.56901e19	0.859237
$371$	4.48588e18	0.240737
$372$	0	0
$373$	1.52798e19	0.787594	0.393797	−	0.919197i	$-0.371161\pi$
$373$	1.52798e19	0.787594	0.393797	+	0.919197i	$0.371161\pi$
$374$	2.36815e19	1.19639
$375$	0	0
$376$	2.05872e18	0.0999283
$377$	9.63723e18	0.458556
$378$	0	0
$379$	−2.04517e19	−0.935265	−0.467633	−	0.883923i	$-0.654893\pi$
$379$	−2.04517e19	−0.935265	−0.467633	+	0.883923i	$0.654893\pi$
$380$	1.66578e19	0.746861
$381$	0	0
$382$	2.51791e19	1.08534
$383$	3.67683e19	1.55411	0.777057	−	0.629430i	$-0.216712\pi$
$383$	3.67683e19	1.55411	0.777057	+	0.629430i	$0.216712\pi$
$384$	0	0
$385$	−1.61451e19	−0.656275
$386$	−2.07890e18	−0.0828761
$387$	0	0
$388$	6.67265e18	0.255894
$389$	−9.40953e18	−0.353953	−0.176977	−	0.984215i	$-0.556632\pi$
$389$	−9.40953e18	−0.353953	−0.176977	+	0.984215i	$0.556632\pi$
$390$	0	0
$391$	1.27898e19	0.462954
$392$	−1.42234e18	−0.0505076
$393$	0	0
$394$	1.69602e19	0.579708
$395$	5.00749e19	1.67935
$396$	0	0
$397$	−2.55384e19	−0.824640	−0.412320	−	0.911039i	$-0.635281\pi$
$397$	−2.55384e19	−0.824640	−0.412320	+	0.911039i	$0.635281\pi$
$398$	1.29341e19	0.409839
$399$	0	0
$400$	7.00743e17	0.0213850
$401$	−4.47453e19	−1.34019	−0.670093	−	0.742277i	$-0.733746\pi$
$401$	−4.47453e19	−1.34019	−0.670093	+	0.742277i	$0.733746\pi$
$402$	0	0
$403$	4.47424e18	0.129102
$404$	−9.27032e18	−0.262564
$405$	0	0
$406$	1.11048e19	0.303085
$407$	7.25396e19	1.94365
$408$	0	0
$409$	1.48506e19	0.383547	0.191774	−	0.981439i	$-0.438576\pi$
$409$	1.48506e19	0.383547	0.191774	+	0.981439i	$0.438576\pi$
$410$	−1.12437e19	−0.285121
$411$	0	0
$412$	−2.92999e19	−0.716370
$413$	−6.43005e18	−0.154379
$414$	0	0
$415$	−1.38940e19	−0.321711
$416$	3.14332e18	0.0714806
$417$	0	0
$418$	7.70134e19	1.68944
$419$	5.02854e19	1.08352	0.541760	−	0.840533i	$-0.317758\pi$
$419$	5.02854e19	1.08352	0.541760	+	0.840533i	$0.317758\pi$
$420$	0	0
$421$	2.06177e19	0.428672	0.214336	−	0.976760i	$-0.431241\pi$
$421$	2.06177e19	0.428672	0.214336	+	0.976760i	$0.431241\pi$
$422$	−5.04876e19	−1.03120
$423$	0	0
$424$	−1.14233e19	−0.225189
$425$	4.48396e18	0.0868450
$426$	0	0
$427$	2.44343e17	0.00456868
$428$	6.14108e18	0.112828
$429$	0	0
$430$	3.16556e18	0.0561613
$431$	4.66277e19	0.812952	0.406476	−	0.913662i	$-0.366758\pi$
$431$	4.66277e19	0.812952	0.406476	+	0.913662i	$0.366758\pi$
$432$	0	0
$433$	−7.04988e19	−1.18720	−0.593598	−	0.804761i	$-0.702293\pi$
$433$	−7.04988e19	−1.18720	−0.593598	+	0.804761i	$0.702293\pi$
$434$	5.15557e18	0.0853305
$435$	0	0
$436$	1.94087e19	0.310348
$437$	4.15929e19	0.653746
$438$	0	0
$439$	7.86521e19	1.19461	0.597305	−	0.802014i	$-0.296238\pi$
$439$	7.86521e19	1.19461	0.597305	+	0.802014i	$0.296238\pi$
$440$	4.11136e19	0.613889
$441$	0	0
$442$	2.01137e19	0.290285
$443$	7.84965e19	1.11384	0.556920	−	0.830566i	$-0.311983\pi$
$443$	7.84965e19	1.11384	0.556920	+	0.830566i	$0.311983\pi$
$444$	0	0
$445$	−9.02786e18	−0.123847
$446$	5.62702e19	0.759044
$447$	0	0
$448$	3.62198e18	0.0472456
$449$	−3.72442e19	−0.477762	−0.238881	−	0.971049i	$-0.576781\pi$
$449$	−3.72442e19	−0.477762	−0.238881	+	0.971049i	$0.576781\pi$
$450$	0	0
$451$	−5.19826e19	−0.644962
$452$	−4.78191e19	−0.583530
$453$	0	0
$454$	1.10302e20	1.30216
$455$	−1.37127e19	−0.159235
$456$	0	0
$457$	−1.52565e20	−1.71429	−0.857144	−	0.515076i	$-0.827764\pi$
$457$	−1.52565e20	−1.71429	−0.857144	+	0.515076i	$0.827764\pi$
$458$	−1.09306e20	−1.20824
$459$	0	0
$460$	2.22044e19	0.237550
$461$	−7.19267e19	−0.757065	−0.378533	−	0.925588i	$-0.623571\pi$
$461$	−7.19267e19	−0.757065	−0.378533	+	0.925588i	$0.623571\pi$
$462$	0	0
$463$	1.12558e20	1.14688	0.573439	−	0.819248i	$-0.305609\pi$
$463$	1.12558e20	1.14688	0.573439	+	0.819248i	$0.305609\pi$
$464$	−2.82783e19	−0.283510
$465$	0	0
$466$	5.73376e19	0.556603
$467$	9.05226e19	0.864730	0.432365	−	0.901699i	$-0.357679\pi$
$467$	9.05226e19	0.864730	0.432365	+	0.901699i	$0.357679\pi$
$468$	0	0
$469$	−1.09496e19	−0.101298
$470$	2.28705e19	0.208229
$471$	0	0
$472$	1.63741e19	0.144409
$473$	1.46352e19	0.127040
$474$	0	0
$475$	1.45820e19	0.122635
$476$	2.31765e19	0.191865
$477$	0	0
$478$	−2.38605e19	−0.191412
$479$	−1.25973e20	−0.994862	−0.497431	−	0.867504i	$-0.665723\pi$
$479$	−1.25973e20	−0.994862	−0.497431	+	0.867504i	$0.665723\pi$
$480$	0	0
$481$	6.16108e19	0.471594
$482$	1.00496e20	0.757350
$483$	0	0
$484$	1.21639e20	0.888655
$485$	7.41270e19	0.533229
$486$	0	0
$487$	−2.16573e20	−1.51055	−0.755277	−	0.655406i	$-0.772498\pi$
$487$	−2.16573e20	−1.51055	−0.755277	+	0.655406i	$0.772498\pi$
$488$	−6.22220e17	−0.00427361
$489$	0	0
$490$	−1.58009e19	−0.105247
$491$	1.03189e19	0.0676897	0.0338448	−	0.999427i	$-0.489225\pi$
$491$	1.03189e19	0.0676897	0.0338448	+	0.999427i	$0.489225\pi$
$492$	0	0
$493$	−1.80949e20	−1.15134
$494$	6.54105e19	0.409916
$495$	0	0
$496$	−1.31287e19	−0.0798193
$497$	−1.14619e20	−0.686409
$498$	0	0
$499$	−1.32540e19	−0.0770184	−0.0385092	−	0.999258i	$-0.512261\pi$
$499$	−1.32540e19	−0.0770184	−0.0385092	+	0.999258i	$0.512261\pi$
$500$	−8.32210e19	−0.476385
$501$	0	0
$502$	4.53652e19	0.252026
$503$	−2.57109e20	−1.40720	−0.703601	−	0.710595i	$-0.748426\pi$
$503$	−2.57109e20	−1.40720	−0.703601	+	0.710595i	$0.748426\pi$
$504$	0	0
$505$	−1.02985e20	−0.547126
$506$	1.02657e20	0.537352
$507$	0	0
$508$	3.19473e19	0.162352
$509$	1.28098e20	0.641445	0.320722	−	0.947173i	$-0.396074\pi$
$509$	1.28098e20	0.641445	0.320722	+	0.947173i	$0.396074\pi$
$510$	0	0
$511$	2.98278e19	0.145032
$512$	−9.22337e18	−0.0441942
$513$	0	0
$514$	2.53375e20	1.17907
$515$	−3.25495e20	−1.49276
$516$	0	0
$517$	1.05736e20	0.471027
$518$	7.09928e19	0.311703
$519$	0	0
$520$	3.49194e19	0.148950
$521$	1.58752e20	0.667479	0.333739	−	0.942665i	$-0.391689\pi$
$521$	1.58752e20	0.667479	0.333739	+	0.942665i	$0.391689\pi$
$522$	0	0
$523$	−3.26199e20	−1.33266	−0.666332	−	0.745655i	$-0.732137\pi$
$523$	−3.26199e20	−1.33266	−0.666332	+	0.745655i	$0.732137\pi$
$524$	−1.89861e17	−0.000764629 0
$525$	0	0
$526$	1.62779e20	0.637097
$527$	−8.40085e19	−0.324148
$528$	0	0
$529$	−2.11193e20	−0.792067
$530$	−1.26902e20	−0.469245
$531$	0	0
$532$	7.53711e19	0.270937
$533$	−4.41509e19	−0.156490
$534$	0	0
$535$	6.82217e19	0.235109
$536$	2.78832e19	0.0947561
$537$	0	0
$538$	2.38149e20	0.787011
$539$	−7.30516e19	−0.238075
$540$	0	0
$541$	1.98570e20	0.629410	0.314705	−	0.949189i	$-0.398094\pi$
$541$	1.98570e20	0.629410	0.314705	+	0.949189i	$0.398094\pi$
$542$	−3.78711e20	−1.18390
$543$	0	0
$544$	−5.90191e19	−0.179474
$545$	2.15613e20	0.646698
$546$	0	0
$547$	−2.10634e20	−0.614644	−0.307322	−	0.951606i	$-0.599433\pi$
$547$	−2.10634e20	−0.614644	−0.307322	+	0.951606i	$0.599433\pi$
$548$	−1.23935e20	−0.356729
$549$	0	0
$550$	3.59904e19	0.100801
$551$	−5.88454e20	−1.62583
$552$	0	0
$553$	2.26573e20	0.609214
$554$	−1.56830e20	−0.416013
$555$	0	0
$556$	−1.13043e20	−0.291864
$557$	−2.22267e20	−0.566190	−0.283095	−	0.959092i	$-0.591361\pi$
$557$	−2.22267e20	−0.566190	−0.283095	+	0.959092i	$0.591361\pi$
$558$	0	0
$559$	1.24303e19	0.0308243
$560$	4.02369e19	0.0984496
$561$	0	0
$562$	8.38380e19	0.199719
$563$	−1.15730e20	−0.272041	−0.136021	−	0.990706i	$-0.543431\pi$
$563$	−1.15730e20	−0.272041	−0.136021	+	0.990706i	$0.543431\pi$
$564$	0	0
$565$	−5.31226e20	−1.21595
$566$	1.58440e20	0.357883
$567$	0	0
$568$	2.91877e20	0.642077
$569$	−7.79708e20	−1.69274	−0.846369	−	0.532596i	$-0.821216\pi$
$569$	−7.79708e20	−1.69274	−0.846369	+	0.532596i	$0.821216\pi$
$570$	0	0
$571$	−7.31712e20	−1.54728	−0.773641	−	0.633625i	$-0.781566\pi$
$571$	−7.31712e20	−1.54728	−0.773641	+	0.633625i	$0.781566\pi$
$572$	1.61442e20	0.336935
$573$	0	0
$574$	−5.08741e19	−0.103433
$575$	1.94375e19	0.0390060
$576$	0	0
$577$	−5.47917e20	−1.07126	−0.535631	−	0.844452i	$-0.679926\pi$
$577$	−5.47917e20	−1.07126	−0.535631	+	0.844452i	$0.679926\pi$
$578$	−1.12647e19	−0.0217401
$579$	0	0
$580$	−3.14146e20	−0.590774
$581$	−6.28660e19	−0.116706
$582$	0	0
$583$	−5.86703e20	−1.06146
$584$	−7.59564e19	−0.135665
$585$	0	0
$586$	−6.86341e20	−1.19484
$587$	−2.66561e20	−0.458153	−0.229077	−	0.973408i	$-0.573571\pi$
$587$	−2.66561e20	−0.458153	−0.229077	+	0.973408i	$0.573571\pi$
$588$	0	0
$589$	−2.73199e20	−0.457736
$590$	1.81901e20	0.300916
$591$	0	0
$592$	−1.80783e20	−0.291572
$593$	−9.42224e20	−1.50053	−0.750264	−	0.661138i	$-0.770074\pi$
$593$	−9.42224e20	−1.50053	−0.750264	+	0.661138i	$0.770074\pi$
$594$	0	0
$595$	2.57470e20	0.399806
$596$	−3.84286e20	−0.589261
$597$	0	0
$598$	8.71905e19	0.130380
$599$	−7.18086e20	−1.06041	−0.530207	−	0.847868i	$-0.677886\pi$
$599$	−7.18086e20	−1.06041	−0.530207	+	0.847868i	$0.677886\pi$
$600$	0	0
$601$	1.57131e20	0.226310	0.113155	−	0.993577i	$-0.463904\pi$
$601$	1.57131e20	0.226310	0.113155	+	0.993577i	$0.463904\pi$
$602$	1.43232e19	0.0203735
$603$	0	0
$604$	7.12661e20	0.988794
$605$	1.35130e21	1.85177
$606$	0	0
$607$	−1.33189e21	−1.78055	−0.890273	−	0.455428i	$-0.849486\pi$
$607$	−1.33189e21	−1.78055	−0.890273	+	0.455428i	$0.849486\pi$
$608$	−1.91933e20	−0.253438
$609$	0	0
$610$	−6.91229e18	−0.00890529
$611$	8.98061e19	0.114287
$612$	0	0
$613$	1.44420e21	1.79338	0.896692	−	0.442654i	$-0.145963\pi$
$613$	1.44420e21	1.79338	0.896692	+	0.442654i	$0.145963\pi$
$614$	5.01982e20	0.615780
$615$	0	0
$616$	1.86026e20	0.222699
$617$	1.10371e20	0.130531	0.0652657	−	0.997868i	$-0.479211\pi$
$617$	1.10371e20	0.130531	0.0652657	+	0.997868i	$0.479211\pi$
$618$	0	0
$619$	1.26529e21	1.46053	0.730265	−	0.683164i	$-0.239396\pi$
$619$	1.26529e21	1.46053	0.730265	+	0.683164i	$0.239396\pi$
$620$	−1.45847e20	−0.166326
$621$	0	0
$622$	8.81050e20	0.980783
$623$	−4.08482e19	−0.0449276
$624$	0	0
$625$	−1.00417e21	−1.07822
$626$	−5.45735e20	−0.578994
$627$	0	0
$628$	8.07306e20	0.836260
$629$	−1.15681e21	−1.18408
$630$	0	0
$631$	9.83019e20	0.982521	0.491261	−	0.871013i	$-0.336536\pi$
$631$	9.83019e20	0.982521	0.491261	+	0.871013i	$0.336536\pi$
$632$	−5.76968e20	−0.569867
$633$	0	0
$634$	7.19261e20	0.693773
$635$	3.54905e20	0.338306
$636$	0	0
$637$	−6.20456e19	−0.0577651
$638$	−1.45238e21	−1.33637
$639$	0	0
$640$	−1.02463e20	−0.0920912
$641$	2.00102e21	1.77753	0.888763	−	0.458367i	$-0.151566\pi$
$641$	2.00102e21	1.77753	0.888763	+	0.458367i	$0.151566\pi$
$642$	0	0
$643$	1.47017e21	1.27580	0.637902	−	0.770118i	$-0.279803\pi$
$643$	1.47017e21	1.27580	0.637902	+	0.770118i	$0.279803\pi$
$644$	1.00468e20	0.0861753
$645$	0	0
$646$	−1.22815e21	−1.02922
$647$	−5.83262e20	−0.483149	−0.241575	−	0.970382i	$-0.577664\pi$
$647$	−5.83262e20	−0.483149	−0.241575	+	0.970382i	$0.577664\pi$
$648$	0	0
$649$	8.40979e20	0.680691
$650$	3.05680e19	0.0244578
$651$	0	0
$652$	−2.75959e20	−0.215768
$653$	3.61890e20	0.279723	0.139861	−	0.990171i	$-0.455334\pi$
$653$	3.61890e20	0.279723	0.139861	+	0.990171i	$0.455334\pi$
$654$	0	0
$655$	−2.10918e18	−0.00159332
$656$	1.29551e20	0.0967524
$657$	0	0
$658$	1.03482e20	0.0755387
$659$	2.22311e21	1.60443	0.802214	−	0.597036i	$-0.203655\pi$
$659$	2.22311e21	1.60443	0.802214	+	0.597036i	$0.203655\pi$
$660$	0	0
$661$	−8.08136e20	−0.570129	−0.285065	−	0.958508i	$-0.592015\pi$
$661$	−8.08136e20	−0.570129	−0.285065	+	0.958508i	$0.592015\pi$
$662$	6.80855e19	0.0474920
$663$	0	0
$664$	1.60088e20	0.109169
$665$	8.37304e20	0.564574
$666$	0	0
$667$	−7.84394e20	−0.517119
$668$	4.81351e20	0.313789
$669$	0	0
$670$	3.09757e20	0.197451
$671$	−3.19574e19	−0.0201443
$672$	0	0
$673$	2.42668e21	1.49589	0.747945	−	0.663760i	$-0.231040\pi$
$673$	2.42668e21	1.49589	0.747945	+	0.663760i	$0.231040\pi$
$674$	1.83025e21	1.11573
$675$	0	0
$676$	−7.01511e20	−0.418248
$677$	−2.15809e20	−0.127249	−0.0636244	−	0.997974i	$-0.520266\pi$
$677$	−2.15809e20	−0.127249	−0.0636244	+	0.997974i	$0.520266\pi$
$678$	0	0
$679$	3.35401e20	0.193438
$680$	−6.55647e20	−0.373984
$681$	0	0
$682$	−6.74292e20	−0.376240
$683$	3.50561e21	1.93467	0.967337	−	0.253493i	$-0.0815795\pi$
$683$	3.50561e21	1.93467	0.967337	+	0.253493i	$0.0815795\pi$
$684$	0	0
$685$	−1.37680e21	−0.743346
$686$	−7.14939e19	−0.0381802
$687$	0	0
$688$	−3.64739e19	−0.0190576
$689$	−4.98310e20	−0.257546
$690$	0	0
$691$	2.68683e21	1.35880	0.679399	−	0.733769i	$-0.262240\pi$
$691$	2.68683e21	1.35880	0.679399	+	0.733769i	$0.262240\pi$
$692$	2.81841e20	0.140997
$693$	0	0
$694$	−7.01987e20	−0.343663
$695$	−1.25580e21	−0.608183
$696$	0	0
$697$	8.28979e20	0.392914
$698$	2.73033e21	1.28026
$699$	0	0
$700$	3.52229e19	0.0161655
$701$	1.21386e21	0.551165	0.275583	−	0.961277i	$-0.411129\pi$
$701$	1.21386e21	0.551165	0.275583	+	0.961277i	$0.411129\pi$
$702$	0	0
$703$	−3.76198e21	−1.67206
$704$	−4.73715e20	−0.208316
$705$	0	0
$706$	1.34117e21	0.577362
$707$	−4.65973e20	−0.198479
$708$	0	0
$709$	9.43378e20	0.393404	0.196702	−	0.980463i	$-0.436977\pi$
$709$	9.43378e20	0.393404	0.196702	+	0.980463i	$0.436977\pi$
$710$	3.24248e21	1.33795
$711$	0	0
$712$	1.04020e20	0.0420259
$713$	−3.64168e20	−0.145590
$714$	0	0
$715$	1.79347e21	0.702099
$716$	1.64463e21	0.637119
$717$	0	0
$718$	9.95661e19	0.0377727
$719$	−4.15024e21	−1.55814	−0.779069	−	0.626938i	$-0.784308\pi$
$719$	−4.15024e21	−1.55814	−0.779069	+	0.626938i	$0.784308\pi$
$720$	0	0
$721$	−1.47276e21	−0.541525
$722$	−2.05081e21	−0.746272
$723$	0	0
$724$	−2.44404e21	−0.871103
$725$	−2.75000e20	−0.0970058
$726$	0	0
$727$	1.48472e21	0.513023	0.256512	−	0.966541i	$-0.417427\pi$
$727$	1.48472e21	0.513023	0.256512	+	0.966541i	$0.417427\pi$
$728$	1.57999e20	0.0540343
$729$	0	0
$730$	−8.43806e20	−0.282697
$731$	−2.33392e20	−0.0773935
$732$	0	0
$733$	7.93240e20	0.257706	0.128853	−	0.991664i	$-0.458870\pi$
$733$	7.93240e20	0.257706	0.128853	+	0.991664i	$0.458870\pi$
$734$	4.38178e21	1.40906
$735$	0	0
$736$	−2.55841e20	−0.0806096
$737$	1.43209e21	0.446647
$738$	0	0
$739$	−7.75584e20	−0.237026	−0.118513	−	0.992953i	$-0.537813\pi$
$739$	−7.75584e20	−0.237026	−0.118513	+	0.992953i	$0.537813\pi$
$740$	−2.00833e21	−0.607572
$741$	0	0
$742$	−5.74192e20	−0.170227
$743$	−9.10804e20	−0.267306	−0.133653	−	0.991028i	$-0.542671\pi$
$743$	−9.10804e20	−0.267306	−0.133653	+	0.991028i	$0.542671\pi$
$744$	0	0
$745$	−4.26906e21	−1.22789
$746$	−1.95582e21	−0.556913
$747$	0	0
$748$	−3.03124e21	−0.845976
$749$	3.08682e20	0.0852899
$750$	0	0
$751$	−9.40597e19	−0.0254744	−0.0127372	−	0.999919i	$-0.504054\pi$
$751$	−9.40597e19	−0.0254744	−0.0127372	+	0.999919i	$0.504054\pi$
$752$	−2.63516e20	−0.0706600
$753$	0	0
$754$	−1.23356e21	−0.324248
$755$	7.91700e21	2.06043
$756$	0	0
$757$	6.21339e20	0.158529	0.0792646	−	0.996854i	$-0.474743\pi$
$757$	6.21339e20	0.158529	0.0792646	+	0.996854i	$0.474743\pi$
$758$	2.61781e21	0.661332
$759$	0	0
$760$	−2.13219e21	−0.528110
$761$	2.60068e21	0.637827	0.318913	−	0.947784i	$-0.396682\pi$
$761$	2.60068e21	0.637827	0.318913	+	0.947784i	$0.396682\pi$
$762$	0	0
$763$	9.75581e20	0.234601
$764$	−3.22293e21	−0.767450
$765$	0	0
$766$	−4.70634e21	−1.09892
$767$	7.14277e20	0.165159
$768$	0	0
$769$	1.73183e21	0.392697	0.196348	−	0.980534i	$-0.437092\pi$
$769$	1.73183e21	0.392697	0.196348	+	0.980534i	$0.437092\pi$
$770$	2.06658e21	0.464057
$771$	0	0
$772$	2.66099e20	0.0586023
$773$	4.38561e21	0.956497	0.478249	−	0.878225i	$-0.341272\pi$
$773$	4.38561e21	0.956497	0.478249	+	0.878225i	$0.341272\pi$
$774$	0	0
$775$	−1.27673e20	−0.0273110
$776$	−8.54100e20	−0.180945
$777$	0	0
$778$	1.20442e21	0.250283
$779$	2.69588e21	0.554841
$780$	0	0
$781$	1.49909e22	3.02653
$782$	−1.63709e21	−0.327358
$783$	0	0
$784$	1.82059e20	0.0357143
$785$	8.96843e21	1.74259
$786$	0	0
$787$	−3.14503e21	−0.599535	−0.299767	−	0.954012i	$-0.596909\pi$
$787$	−3.14503e21	−0.599535	−0.299767	+	0.954012i	$0.596909\pi$
$788$	−2.17091e21	−0.409915
$789$	0	0
$790$	−6.40958e21	−1.18748
$791$	−2.40363e21	−0.441107
$792$	0	0
$793$	−2.71427e19	−0.00488769
$794$	3.26891e21	0.583108
$795$	0	0
$796$	−1.65557e21	−0.289800
$797$	−4.17857e21	−0.724587	−0.362294	−	0.932064i	$-0.618006\pi$
$797$	−4.17857e21	−0.724587	−0.362294	+	0.932064i	$0.618006\pi$
$798$	0	0
$799$	−1.68620e21	−0.286952
$800$	−8.96951e19	−0.0151215
$801$	0	0
$802$	5.72740e21	0.947655
$803$	−3.90114e21	−0.639478
$804$	0	0
$805$	1.11610e21	0.179571
$806$	−5.72703e20	−0.0912886
$807$	0	0
$808$	1.18660e21	0.185661
$809$	8.36645e21	1.29696	0.648481	−	0.761231i	$-0.275405\pi$
$809$	8.36645e21	1.29696	0.648481	+	0.761231i	$0.275405\pi$
$810$	0	0
$811$	3.04325e21	0.463106	0.231553	−	0.972822i	$-0.425619\pi$
$811$	3.04325e21	0.463106	0.231553	+	0.972822i	$0.425619\pi$
$812$	−1.42141e21	−0.214313
$813$	0	0
$814$	−9.28507e21	−1.37437
$815$	−3.06565e21	−0.449614
$816$	0	0
$817$	−7.58999e20	−0.109289
$818$	−1.90087e21	−0.271209
$819$	0	0
$820$	1.43919e21	0.201611
$821$	−1.85475e21	−0.257462	−0.128731	−	0.991680i	$-0.541090\pi$
$821$	−1.85475e21	−0.257462	−0.128731	+	0.991680i	$0.541090\pi$
$822$	0	0
$823$	7.38862e21	1.00708	0.503541	−	0.863972i	$-0.332030\pi$
$823$	7.38862e21	1.00708	0.503541	+	0.863972i	$0.332030\pi$
$824$	3.75039e21	0.506550
$825$	0	0
$826$	8.23046e20	0.109163
$827$	6.93728e21	0.911797	0.455899	−	0.890032i	$-0.349318\pi$
$827$	6.93728e21	0.911797	0.455899	+	0.890032i	$0.349318\pi$
$828$	0	0
$829$	−1.14538e22	−1.47839	−0.739197	−	0.673490i	$-0.764795\pi$
$829$	−1.14538e22	−1.47839	−0.739197	+	0.673490i	$0.764795\pi$
$830$	1.77843e21	0.227484
$831$	0	0
$832$	−4.02345e20	−0.0505445
$833$	1.16497e21	0.145037
$834$	0	0
$835$	5.34736e21	0.653869
$836$	−9.85771e21	−1.19462
$837$	0	0
$838$	−6.43653e21	−0.766164
$839$	−5.76444e21	−0.680052	−0.340026	−	0.940416i	$-0.610436\pi$
$839$	−5.76444e21	−0.680052	−0.340026	+	0.940416i	$0.610436\pi$
$840$	0	0
$841$	2.46835e21	0.286047
$842$	−2.63907e21	−0.303117
$843$	0	0
$844$	6.46241e21	0.729165
$845$	−7.79314e21	−0.871539
$846$	0	0
$847$	6.11420e21	0.671760
$848$	1.46218e21	0.159233
$849$	0	0
$850$	−5.73947e20	−0.0614087
$851$	−5.01463e21	−0.531823
$852$	0	0
$853$	−1.50972e22	−1.57318	−0.786589	−	0.617477i	$-0.788155\pi$
$853$	−1.50972e22	−1.57318	−0.786589	+	0.617477i	$0.788155\pi$
$854$	−3.12759e19	−0.00323055
$855$	0	0
$856$	−7.86058e20	−0.0797814
$857$	2.14000e21	0.215307	0.107654	−	0.994188i	$-0.465666\pi$
$857$	2.14000e21	0.215307	0.107654	+	0.994188i	$0.465666\pi$
$858$	0	0
$859$	−6.48884e21	−0.641532	−0.320766	−	0.947158i	$-0.603940\pi$
$859$	−6.48884e21	−0.641532	−0.320766	+	0.947158i	$0.603940\pi$
$860$	−4.05192e20	−0.0397120
$861$	0	0
$862$	−5.96835e21	−0.574844
$863$	3.80444e21	0.363253	0.181627	−	0.983368i	$-0.441864\pi$
$863$	3.80444e21	0.363253	0.181627	+	0.983368i	$0.441864\pi$
$864$	0	0
$865$	3.13100e21	0.293807
$866$	9.02385e21	0.839475
$867$	0	0
$868$	−6.59914e20	−0.0603377
$869$	−2.96332e22	−2.68615
$870$	0	0
$871$	1.21633e21	0.108372
$872$	−2.48432e21	−0.219449
$873$	0	0
$874$	−5.32390e21	−0.462268
$875$	−4.18311e21	−0.360113
$876$	0	0
$877$	9.12529e21	0.772235	0.386118	−	0.922450i	$-0.373816\pi$
$877$	9.12529e21	0.772235	0.386118	+	0.922450i	$0.373816\pi$
$878$	−1.00675e22	−0.844717
$879$	0	0
$880$	−5.26254e21	−0.434085
$881$	−9.49048e21	−0.776192	−0.388096	−	0.921619i	$-0.626867\pi$
$881$	−9.49048e21	−0.776192	−0.388096	+	0.921619i	$0.626867\pi$
$882$	0	0
$883$	1.29030e22	1.03749	0.518747	−	0.854928i	$-0.326398\pi$
$883$	1.29030e22	1.03749	0.518747	+	0.854928i	$0.326398\pi$
$884$	−2.57455e21	−0.205262
$885$	0	0
$886$	−1.00476e22	−0.787603
$887$	1.12969e22	0.878079	0.439040	−	0.898468i	$-0.355319\pi$
$887$	1.12969e22	0.878079	0.439040	+	0.898468i	$0.355319\pi$
$888$	0	0
$889$	1.60583e21	0.122726
$890$	1.15557e21	0.0875729
$891$	0	0
$892$	−7.20259e21	−0.536725
$893$	−5.48361e21	−0.405210
$894$	0	0
$895$	1.82703e22	1.32762
$896$	−4.63613e20	−0.0334077
$897$	0	0
$898$	4.76726e21	0.337829
$899$	5.15222e21	0.362073
$900$	0	0
$901$	9.35629e21	0.646647
$902$	6.65378e21	0.456057
$903$	0	0
$904$	6.12085e21	0.412618
$905$	−2.71511e22	−1.81519
$906$	0	0
$907$	1.91938e22	1.26214	0.631068	−	0.775728i	$-0.282617\pi$
$907$	1.91938e22	1.26214	0.631068	+	0.775728i	$0.282617\pi$
$908$	−1.41187e22	−0.920766
$909$	0	0
$910$	1.75523e21	0.112596
$911$	2.66478e21	0.169540	0.0847702	−	0.996401i	$-0.472984\pi$
$911$	2.66478e21	0.169540	0.0847702	+	0.996401i	$0.472984\pi$
$912$	0	0
$913$	8.22217e21	0.514583
$914$	1.95283e22	1.21219
$915$	0	0
$916$	1.39912e22	0.854354
$917$	−9.54338e18	−0.000578006 0
$918$	0	0
$919$	−2.10988e22	−1.25716	−0.628580	−	0.777745i	$-0.716364\pi$
$919$	−2.10988e22	−1.25716	−0.628580	+	0.777745i	$0.716364\pi$
$920$	−2.84216e21	−0.167973
$921$	0	0
$922$	9.20662e21	0.535326
$923$	1.27323e22	0.734338
$924$	0	0
$925$	−1.75807e21	−0.0997641
$926$	−1.44074e22	−0.810965
$927$	0	0
$928$	3.61962e21	0.200472
$929$	−2.26606e22	−1.24495	−0.622477	−	0.782638i	$-0.713873\pi$
$929$	−2.26606e22	−1.24495	−0.622477	+	0.782638i	$0.713873\pi$
$930$	0	0
$931$	3.78854e21	0.204809
$932$	−7.33922e21	−0.393578
$933$	0	0
$934$	−1.15869e22	−0.611456
$935$	−3.36742e22	−1.76283
$936$	0	0
$937$	2.99548e22	1.54319	0.771595	−	0.636114i	$-0.219459\pi$
$937$	2.99548e22	1.54319	0.771595	+	0.636114i	$0.219459\pi$
$938$	1.40155e21	0.0716289
$939$	0	0
$940$	−2.92742e21	−0.147240
$941$	7.54408e21	0.376430	0.188215	−	0.982128i	$-0.439730\pi$
$941$	7.54408e21	0.376430	0.188215	+	0.982128i	$0.439730\pi$
$942$	0	0
$943$	3.59353e21	0.176475
$944$	−2.09589e21	−0.102112
$945$	0	0
$946$	−1.87331e21	−0.0898310
$947$	−2.10087e22	−0.999481	−0.499741	−	0.866175i	$-0.666571\pi$
$947$	−2.10087e22	−0.999481	−0.499741	+	0.866175i	$0.666571\pi$
$948$	0	0
$949$	−3.31339e21	−0.155159
$950$	−1.86650e21	−0.0867163
$951$	0	0
$952$	−2.96660e21	−0.135669
$953$	−1.33670e22	−0.606508	−0.303254	−	0.952910i	$-0.598073\pi$
$953$	−1.33670e22	−0.606508	−0.303254	+	0.952910i	$0.598073\pi$
$954$	0	0
$955$	−3.58037e22	−1.59920
$956$	3.05414e21	0.135349
$957$	0	0
$958$	1.61246e22	0.703474
$959$	−6.22958e21	−0.269662
$960$	0	0
$961$	−2.10733e22	−0.898062
$962$	−7.88618e21	−0.333468
$963$	0	0
$964$	−1.28635e22	−0.535527
$965$	2.95612e21	0.122114
$966$	0	0
$967$	4.36244e22	1.77432	0.887159	−	0.461464i	$-0.152676\pi$
$967$	4.36244e22	1.77432	0.887159	+	0.461464i	$0.152676\pi$
$968$	−1.55698e22	−0.628374
$969$	0	0
$970$	−9.48826e21	−0.377050
$971$	4.14924e22	1.63616	0.818078	−	0.575108i	$-0.195040\pi$
$971$	4.14924e22	1.63616	0.818078	+	0.575108i	$0.195040\pi$
$972$	0	0
$973$	−5.68210e21	−0.220629
$974$	2.77213e22	1.06812
$975$	0	0
$976$	7.96441e19	0.00302190
$977$	−9.95035e21	−0.374653	−0.187326	−	0.982298i	$-0.559982\pi$
$977$	−9.95035e21	−0.374653	−0.187326	+	0.982298i	$0.559982\pi$
$978$	0	0
$979$	5.34250e21	0.198095
$980$	2.02251e21	0.0744209
$981$	0	0
$982$	−1.32082e21	−0.0478638
$983$	3.10738e21	0.111749	0.0558744	−	0.998438i	$-0.482205\pi$
$983$	3.10738e21	0.111749	0.0558744	+	0.998438i	$0.482205\pi$
$984$	0	0
$985$	−2.41168e22	−0.854176
$986$	2.31615e22	0.814121
$987$	0	0
$988$	−8.37254e21	−0.289855
$989$	−1.01173e21	−0.0347609
$990$	0	0
$991$	−4.17077e22	−1.41144	−0.705721	−	0.708489i	$-0.749377\pi$
$991$	−4.17077e22	−1.41144	−0.705721	+	0.708489i	$0.749377\pi$
$992$	1.68047e21	0.0564408
$993$	0	0
$994$	1.46712e22	0.485365
$995$	−1.83918e22	−0.603881
$996$	0	0
$997$	−1.12037e22	−0.362367	−0.181184	−	0.983449i	$-0.557993\pi$
$997$	−1.12037e22	−0.362367	−0.181184	+	0.983449i	$0.557993\pi$
$998$	1.69652e21	0.0544602
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.16.a.n.1.4	✓	4
3.2	odd	2		126.16.a.q.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.16.a.n.1.4	✓	4	1.1	even	1	trivial
126.16.a.q.1.1	yes	4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)

\(f(q)\)	\(=\)	\(q-128.000 q^{2} +16384.0 q^{4} +182011. q^{5} +823543. q^{7} -2.09715e6 q^{8} -2.32974e7 q^{10} -1.07710e8 q^{11} -9.14826e7 q^{13} -1.05414e8 q^{14} +2.68435e8 q^{16} +1.71768e9 q^{17} +5.58597e9 q^{19} +2.98207e9 q^{20} +1.37869e10 q^{22} +7.44596e9 q^{23} +2.61047e9 q^{25} +1.17098e10 q^{26} +1.34929e10 q^{28} -1.05345e11 q^{29} -4.89081e10 q^{31} -3.43597e10 q^{32} -2.19863e11 q^{34} +1.49894e11 q^{35} -6.73470e11 q^{37} -7.15005e11 q^{38} -3.81705e11 q^{40} +4.82615e11 q^{41} -1.35876e11 q^{43} -1.76473e12 q^{44} -9.53083e11 q^{46} -9.81674e11 q^{47} +6.78223e11 q^{49} -3.34140e11 q^{50} -1.49885e12 q^{52} +5.44704e12 q^{53} -1.96045e13 q^{55} -1.72709e12 q^{56} +1.34841e13 q^{58} -7.80779e12 q^{59} +2.96698e11 q^{61} +6.26024e12 q^{62} +4.39805e12 q^{64} -1.66509e13 q^{65} -1.32958e13 q^{67} +2.81425e13 q^{68} -1.91864e13 q^{70} -1.39178e14 q^{71} +3.62188e13 q^{73} +8.62041e13 q^{74} +9.15206e13 q^{76} -8.87041e13 q^{77} +2.75120e14 q^{79} +4.88582e13 q^{80} -6.17747e13 q^{82} -7.63360e13 q^{83} +3.12637e14 q^{85} +1.73921e13 q^{86} +2.25885e14 q^{88} -4.96006e13 q^{89} -7.53399e13 q^{91} +1.21995e14 q^{92} +1.25654e14 q^{94} +1.01671e15 q^{95} +4.07266e14 q^{97} -8.68126e13 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 512 q^{2} + 65536 q^{4} - 291312 q^{5} + 3294172 q^{7} - 8388608 q^{8} + 37287936 q^{10} - 121144752 q^{11} + 170165240 q^{13} - 421654016 q^{14} + 1073741824 q^{16} - 345685200 q^{17} + 4738793696 q^{19}+ \cdots - 347250213298688 q^{98}+O(q^{100}) \) 4 * q - 512 * q^2 + 65536 * q^4 - 291312 * q^5 + 3294172 * q^7 - 8388608 * q^8 + 37287936 * q^10 - 121144752 * q^11 + 170165240 * q^13 - 421654016 * q^14 + 1073741824 * q^16 - 345685200 * q^17 + 4738793696 * q^19 - 4772855808 * q^20 + 15506528256 * q^22 - 17023827024 * q^23 + 38327672524 * q^25 - 21781150720 * q^26 + 53971714048 * q^28 - 96242392128 * q^29 + 65665329632 * q^31 - 137438953472 * q^32 + 44247705600 * q^34 - 239907958416 * q^35 + 279650496248 * q^37 - 606565593088 * q^38 + 610925543424 * q^40 - 274507805040 * q^41 + 1551932654480 * q^43 - 1984835616768 * q^44 + 2179049859072 * q^46 - 686050223712 * q^47 + 2712892291396 * q^49 - 4905942083072 * q^50 + 2787987292160 * q^52 - 4929897017760 * q^53 + 3835130598432 * q^55 - 6908379398144 * q^56 + 12319026192384 * q^58 - 15661604798112 * q^59 + 7270740102248 * q^61 - 8405162192896 * q^62 + 17592186044416 * q^64 - 39986092580832 * q^65 + 2642113680560 * q^67 - 5663706316800 * q^68 + 30708218677248 * q^70 - 62559334721520 * q^71 - 53486860270312 * q^73 - 35795263519744 * q^74 + 77640395915264 * q^76 - 99767912496336 * q^77 + 77560377412544 * q^79 - 78198469558272 * q^80 + 35136999045120 * q^82 + 58088393760768 * q^83 + 48889916091648 * q^85 - 198647379773440 * q^86 + 254058958946304 * q^88 + 296914762526928 * q^89 + 140138392245320 * q^91 - 278918381961216 * q^92 + 87814428635136 * q^94 + 1325087451258048 * q^95 + 1391306483388248 * q^97 - 347250213298688 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more