LMFDB - Embedded newform 81.8.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,8,Mod(1,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, 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("81.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	81.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:	$25.3031870642$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 156x^{4} + 388x^{3} + 5992x^{2} - 18174x + 6597 $ x^6 - 2x^5 - 156x^4 + 388x^3 + 5992x^2 - 18174*x + 6597
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{9} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.423455$ of defining polynomial
Character	$\chi$	$=$	81.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-19.7742 q^{2} +263.020 q^{4} +473.605 q^{5} -1026.85 q^{7} -2669.91 q^{8} -9365.17 q^{10} -2226.85 q^{11} -3322.33 q^{13} +20305.1 q^{14} +19128.8 q^{16} +36094.5 q^{17} -37646.7 q^{19} +124567. q^{20} +44034.3 q^{22} -70860.6 q^{23} +146177. q^{25} +65696.4 q^{26} -270081. q^{28} -28859.0 q^{29} +15630.3 q^{31} -36508.7 q^{32} -713739. q^{34} -486321. q^{35} +41084.0 q^{37} +744433. q^{38} -1.26448e6 q^{40} -63463.6 q^{41} -151661. q^{43} -585706. q^{44} +1.40121e6 q^{46} -714732. q^{47} +230874. q^{49} -2.89054e6 q^{50} -873837. q^{52} +490289. q^{53} -1.05465e6 q^{55} +2.74159e6 q^{56} +570665. q^{58} -1.33623e6 q^{59} -1.79614e6 q^{61} -309077. q^{62} -1.72655e6 q^{64} -1.57347e6 q^{65} +2.54568e6 q^{67} +9.49355e6 q^{68} +9.61661e6 q^{70} -700407. q^{71} -4.69075e6 q^{73} -812403. q^{74} -9.90181e6 q^{76} +2.28664e6 q^{77} -4.98011e6 q^{79} +9.05950e6 q^{80} +1.25494e6 q^{82} +3.39027e6 q^{83} +1.70945e7 q^{85} +2.99898e6 q^{86} +5.94549e6 q^{88} +1.27747e6 q^{89} +3.41152e6 q^{91} -1.86377e7 q^{92} +1.41333e7 q^{94} -1.78297e7 q^{95} +7.65707e6 q^{97} -4.56536e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 438 q^{4} - 1932 q^{7} - 14922 q^{10} - 11886 q^{13} + 7314 q^{16} - 67164 q^{19} - 23364 q^{22} + 42324 q^{25} - 693348 q^{28} - 470832 q^{31} - 1834866 q^{34} - 1026258 q^{37} - 3091374 q^{40} - 1502268 q^{43}+ \cdots + 26307948 q^{97}+O(q^{100}) $ 6 * q + 438 * q^4 - 1932 * q^7 - 14922 * q^10 - 11886 * q^13 + 7314 * q^16 - 67164 * q^19 - 23364 * q^22 + 42324 * q^25 - 693348 * q^28 - 470832 * q^31 - 1834866 * q^34 - 1026258 * q^37 - 3091374 * q^40 - 1502268 * q^43 + 104580 * q^46 - 1769490 * q^49 - 2993334 * q^52 - 3667428 * q^55 + 659898 * q^58 - 4564770 * q^61 - 1047594 * q^64 + 2850060 * q^67 + 19286604 * q^70 - 1846842 * q^73 - 5520516 * q^76 - 2878188 * q^79 + 15068952 * q^82 + 32410098 * q^85 + 26598276 * q^88 + 16242108 * q^91 + 51428376 * q^94 + 26307948 * q^97

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 $

$2$	−19.7742	−1.74781	−0.873905	−	0.486097i	$-0.838420\pi$
$2$	−19.7742	−1.74781	−0.873905	+	0.486097i	$0.838420\pi$
$3$	0	0
$3$	0	0
$4$	263.020	2.05484
$5$	473.605	1.69442	0.847211	−	0.531256i	$-0.178280\pi$
$5$	473.605	1.69442	0.847211	+	0.531256i	$0.178280\pi$
$6$	0	0
$7$	−1026.85	−1.13152	−0.565761	−	0.824569i	$-0.691418\pi$
$7$	−1026.85	−1.13152	−0.565761	+	0.824569i	$0.691418\pi$
$8$	−2669.91	−1.84366
$9$	0	0
$10$	−9365.17	−2.96153
$11$	−2226.85	−0.504449	−0.252224	−	0.967669i	$-0.581162\pi$
$11$	−2226.85	−0.504449	−0.252224	+	0.967669i	$0.581162\pi$
$12$	0	0
$13$	−3322.33	−0.419412	−0.209706	−	0.977765i	$-0.567251\pi$
$13$	−3322.33	−0.419412	−0.209706	+	0.977765i	$0.567251\pi$
$14$	20305.1	1.97769
$15$	0	0
$16$	19128.8	1.16753
$17$	36094.5	1.78184	0.890921	−	0.454158i	$-0.150060\pi$
$17$	36094.5	1.78184	0.890921	+	0.454158i	$0.150060\pi$
$18$	0	0
$19$	−37646.7	−1.25918	−0.629592	−	0.776926i	$-0.716778\pi$
$19$	−37646.7	−1.25918	−0.629592	+	0.776926i	$0.716778\pi$
$20$	124567.	3.48177
$21$	0	0
$22$	44034.3	0.881681
$23$	−70860.6	−1.21439	−0.607194	−	0.794554i	$-0.707705\pi$
$23$	−70860.6	−1.21439	−0.607194	+	0.794554i	$0.707705\pi$
$24$	0	0
$25$	146177.	1.87107
$26$	65696.4	0.733052
$27$	0	0
$28$	−270081.	−2.32510
$29$	−28859.0	−0.219730	−0.109865	−	0.993947i	$-0.535042\pi$
$29$	−28859.0	−0.219730	−0.109865	+	0.993947i	$0.535042\pi$
$30$	0	0
$31$	15630.3	0.0942326	0.0471163	−	0.998889i	$-0.484997\pi$
$31$	15630.3	0.0942326	0.0471163	+	0.998889i	$0.484997\pi$
$32$	−36508.7	−0.196957
$33$	0	0
$34$	−713739.	−3.11432
$35$	−486321.	−1.91728
$36$	0	0
$37$	41084.0	0.133342	0.0666709	−	0.997775i	$-0.478762\pi$
$37$	41084.0	0.133342	0.0666709	+	0.997775i	$0.478762\pi$
$38$	744433.	2.20081
$39$	0	0
$40$	−1.26448e6	−3.12394
$41$	−63463.6	−0.143807	−0.0719037	−	0.997412i	$-0.522907\pi$
$41$	−63463.6	−0.143807	−0.0719037	+	0.997412i	$0.522907\pi$
$42$	0	0
$43$	−151661.	−0.290894	−0.145447	−	0.989366i	$-0.546462\pi$
$43$	−151661.	−0.290894	−0.145447	+	0.989366i	$0.546462\pi$
$44$	−585706.	−1.03656
$45$	0	0
$46$	1.40121e6	2.12252
$47$	−714732.	−1.00415	−0.502077	−	0.864823i	$-0.667431\pi$
$47$	−714732.	−1.00415	−0.502077	+	0.864823i	$0.667431\pi$
$48$	0	0
$49$	230874.	0.280343
$50$	−2.89054e6	−3.27027
$51$	0	0
$52$	−873837.	−0.861824
$53$	490289.	0.452363	0.226181	−	0.974085i	$-0.427376\pi$
$53$	490289.	0.452363	0.226181	+	0.974085i	$0.427376\pi$
$54$	0	0
$55$	−1.05465e6	−0.854749
$56$	2.74159e6	2.08614
$57$	0	0
$58$	570665.	0.384046
$59$	−1.33623e6	−0.847030	−0.423515	−	0.905889i	$-0.639204\pi$
$59$	−1.33623e6	−0.847030	−0.423515	+	0.905889i	$0.639204\pi$
$60$	0	0
$61$	−1.79614e6	−1.01318	−0.506588	−	0.862188i	$-0.669094\pi$
$61$	−1.79614e6	−1.01318	−0.506588	+	0.862188i	$0.669094\pi$
$62$	−309077.	−0.164701
$63$	0	0
$64$	−1.72655e6	−0.823284
$65$	−1.57347e6	−0.710660
$66$	0	0
$67$	2.54568e6	1.03405	0.517026	−	0.855970i	$-0.327039\pi$
$67$	2.54568e6	1.03405	0.517026	+	0.855970i	$0.327039\pi$
$68$	9.49355e6	3.66140
$69$	0	0
$70$	9.61661e6	3.35104
$71$	−700407.	−0.232245	−0.116122	−	0.993235i	$-0.537047\pi$
$71$	−700407.	−0.232245	−0.116122	+	0.993235i	$0.537047\pi$
$72$	0	0
$73$	−4.69075e6	−1.41128	−0.705638	−	0.708572i	$-0.749340\pi$
$73$	−4.69075e6	−1.41128	−0.705638	+	0.708572i	$0.749340\pi$
$74$	−812403.	−0.233056
$75$	0	0
$76$	−9.90181e6	−2.58742
$77$	2.28664e6	0.570795
$78$	0	0
$79$	−4.98011e6	−1.13643	−0.568217	−	0.822879i	$-0.692367\pi$
$79$	−4.98011e6	−1.13643	−0.568217	+	0.822879i	$0.692367\pi$
$80$	9.05950e6	1.97829
$81$	0	0
$82$	1.25494e6	0.251348
$83$	3.39027e6	0.650820	0.325410	−	0.945573i	$-0.394498\pi$
$83$	3.39027e6	0.650820	0.325410	+	0.945573i	$0.394498\pi$
$84$	0	0
$85$	1.70945e7	3.01919
$86$	2.99898e6	0.508427
$87$	0	0
$88$	5.94549e6	0.930032
$89$	1.27747e6	0.192081	0.0960405	−	0.995377i	$-0.469382\pi$
$89$	1.27747e6	0.192081	0.0960405	+	0.995377i	$0.469382\pi$
$90$	0	0
$91$	3.41152e6	0.474574
$92$	−1.86377e7	−2.49537
$93$	0	0
$94$	1.41333e7	1.75507
$95$	−1.78297e7	−2.13359
$96$	0	0
$97$	7.65707e6	0.851847	0.425924	−	0.904759i	$-0.359949\pi$
$97$	7.65707e6	0.851847	0.425924	+	0.904759i	$0.359949\pi$
$98$	−4.56536e6	−0.489986
$99$	0	0
$100$	3.84474e7	3.84474
$101$	−1.70415e7	−1.64582	−0.822911	−	0.568171i	$-0.807651\pi$
$101$	−1.70415e7	−1.64582	−0.822911	+	0.568171i	$0.807651\pi$
$102$	0	0
$103$	−1.75683e7	−1.58416	−0.792082	−	0.610414i	$-0.791003\pi$
$103$	−1.75683e7	−1.58416	−0.792082	+	0.610414i	$0.791003\pi$
$104$	8.87030e6	0.773253
$105$	0	0
$106$	−9.69508e6	−0.790644
$107$	−1.16418e7	−0.918710	−0.459355	−	0.888253i	$-0.651919\pi$
$107$	−1.16418e7	−0.918710	−0.459355	+	0.888253i	$0.651919\pi$
$108$	0	0
$109$	−2.29184e7	−1.69509	−0.847544	−	0.530725i	$-0.821920\pi$
$109$	−2.29184e7	−1.69509	−0.847544	+	0.530725i	$0.821920\pi$
$110$	2.08549e7	1.49394
$111$	0	0
$112$	−1.96424e7	−1.32108
$113$	342408.	0.0223239	0.0111619	−	0.999938i	$-0.496447\pi$
$113$	342408.	0.0223239	0.0111619	+	0.999938i	$0.496447\pi$
$114$	0	0
$115$	−3.35600e7	−2.05769
$116$	−7.59049e6	−0.451510
$117$	0	0
$118$	2.64229e7	1.48045
$119$	−3.70635e7	−2.01619
$120$	0	0
$121$	−1.45283e7	−0.745531
$122$	3.55172e7	1.77084
$123$	0	0
$124$	4.11107e6	0.193633
$125$	3.22298e7	1.47595
$126$	0	0
$127$	−1.04923e7	−0.454524	−0.227262	−	0.973834i	$-0.572977\pi$
$127$	−1.04923e7	−0.454524	−0.227262	+	0.973834i	$0.572977\pi$
$128$	3.88143e7	1.63590
$129$	0	0
$130$	3.11142e7	1.24210
$131$	−6.65490e6	−0.258638	−0.129319	−	0.991603i	$-0.541279\pi$
$131$	−6.65490e6	−0.258638	−0.129319	+	0.991603i	$0.541279\pi$
$132$	0	0
$133$	3.86574e7	1.42479
$134$	−5.03388e7	−1.80733
$135$	0	0
$136$	−9.63688e7	−3.28511
$137$	4.00135e7	1.32949	0.664744	−	0.747071i	$-0.268540\pi$
$137$	4.00135e7	1.32949	0.664744	+	0.747071i	$0.268540\pi$
$138$	0	0
$139$	1.88262e7	0.594581	0.297291	−	0.954787i	$-0.403917\pi$
$139$	1.88262e7	0.594581	0.297291	+	0.954787i	$0.403917\pi$
$140$	−1.27912e8	−3.93970
$141$	0	0
$142$	1.38500e7	0.405920
$143$	7.39833e6	0.211572
$144$	0	0
$145$	−1.36678e7	−0.372315
$146$	9.27559e7	2.46664
$147$	0	0
$148$	1.08059e7	0.273996
$149$	2.58112e7	0.639229	0.319614	−	0.947548i	$-0.396447\pi$
$149$	2.58112e7	0.639229	0.319614	+	0.947548i	$0.396447\pi$
$150$	0	0
$151$	−9.13502e6	−0.215919	−0.107959	−	0.994155i	$-0.534432\pi$
$151$	−9.13502e6	−0.215919	−0.107959	+	0.994155i	$0.534432\pi$
$152$	1.00513e8	2.32151
$153$	0	0
$154$	−4.52165e7	−0.997641
$155$	7.40259e6	0.159670
$156$	0	0
$157$	−3.30247e7	−0.681067	−0.340533	−	0.940232i	$-0.610608\pi$
$157$	−3.30247e7	−0.681067	−0.340533	+	0.940232i	$0.610608\pi$
$158$	9.84778e7	1.98627
$159$	0	0
$160$	−1.72907e7	−0.333729
$161$	7.27631e7	1.37411
$162$	0	0
$163$	9.98787e7	1.80641	0.903205	−	0.429210i	$-0.141208\pi$
$163$	9.98787e7	1.80641	0.903205	+	0.429210i	$0.141208\pi$
$164$	−1.66922e7	−0.295501
$165$	0	0
$166$	−6.70399e7	−1.13751
$167$	−1.22615e7	−0.203721	−0.101860	−	0.994799i	$-0.532480\pi$
$167$	−1.22615e7	−0.203721	−0.101860	+	0.994799i	$0.532480\pi$
$168$	0	0
$169$	−5.17107e7	−0.824094
$170$	−3.38031e8	−5.27698
$171$	0	0
$172$	−3.98898e7	−0.597740
$173$	−2.40046e7	−0.352479	−0.176240	−	0.984347i	$-0.556393\pi$
$173$	−2.40046e7	−0.352479	−0.176240	+	0.984347i	$0.556393\pi$
$174$	0	0
$175$	−1.50102e8	−2.11715
$176$	−4.25970e7	−0.588958
$177$	0	0
$178$	−2.52609e7	−0.335721
$179$	1.35827e8	1.77011	0.885054	−	0.465488i	$-0.154121\pi$
$179$	1.35827e8	1.77011	0.885054	+	0.465488i	$0.154121\pi$
$180$	0	0
$181$	−7.74602e7	−0.970965	−0.485482	−	0.874246i	$-0.661356\pi$
$181$	−7.74602e7	−0.970965	−0.485482	+	0.874246i	$0.661356\pi$
$182$	−6.74602e7	−0.829465
$183$	0	0
$184$	1.89191e8	2.23892
$185$	1.94576e7	0.225937
$186$	0	0
$187$	−8.03770e7	−0.898848
$188$	−1.87988e8	−2.06338
$189$	0	0
$190$	3.52568e8	3.72911
$191$	−3.78490e7	−0.393041	−0.196520	−	0.980500i	$-0.562964\pi$
$191$	−3.78490e7	−0.393041	−0.196520	+	0.980500i	$0.562964\pi$
$192$	0	0
$193$	−4.31117e7	−0.431663	−0.215831	−	0.976431i	$-0.569246\pi$
$193$	−4.31117e7	−0.431663	−0.215831	+	0.976431i	$0.569246\pi$
$194$	−1.51413e8	−1.48887
$195$	0	0
$196$	6.07245e7	0.576060
$197$	1.35255e8	1.26044	0.630221	−	0.776416i	$-0.282964\pi$
$197$	1.35255e8	1.26044	0.630221	+	0.776416i	$0.282964\pi$
$198$	0	0
$199$	9.83246e7	0.884456	0.442228	−	0.896903i	$-0.354188\pi$
$199$	9.83246e7	0.884456	0.442228	+	0.896903i	$0.354188\pi$
$200$	−3.90279e8	−3.44961
$201$	0	0
$202$	3.36982e8	2.87658
$203$	2.96339e7	0.248629
$204$	0	0
$205$	−3.00567e7	−0.243670
$206$	3.47400e8	2.76882
$207$	0	0
$208$	−6.35520e7	−0.489675
$209$	8.38336e7	0.635193
$210$	0	0
$211$	−6.82781e7	−0.500372	−0.250186	−	0.968198i	$-0.580492\pi$
$211$	−6.82781e7	−0.500372	−0.250186	+	0.968198i	$0.580492\pi$
$212$	1.28956e8	0.929533
$213$	0	0
$214$	2.30208e8	1.60573
$215$	−7.18275e7	−0.492897
$216$	0	0
$217$	−1.60499e7	−0.106626
$218$	4.53194e8	2.96269
$219$	0	0
$220$	−2.77393e8	−1.75637
$221$	−1.19918e8	−0.747325
$222$	0	0
$223$	2.91860e8	1.76241	0.881207	−	0.472731i	$-0.156732\pi$
$223$	2.91860e8	1.76241	0.881207	+	0.472731i	$0.156732\pi$
$224$	3.74889e7	0.222862
$225$	0	0
$226$	−6.77085e6	−0.0390179
$227$	−2.38586e8	−1.35380	−0.676901	−	0.736074i	$-0.736677\pi$
$227$	−2.38586e8	−1.35380	−0.676901	+	0.736074i	$0.736677\pi$
$228$	0	0
$229$	−1.53030e8	−0.842080	−0.421040	−	0.907042i	$-0.638335\pi$
$229$	−1.53030e8	−0.842080	−0.421040	+	0.907042i	$0.638335\pi$
$230$	6.63622e8	3.59644
$231$	0	0
$232$	7.70509e7	0.405107
$233$	−2.54013e8	−1.31556	−0.657779	−	0.753211i	$-0.728504\pi$
$233$	−2.54013e8	−1.31556	−0.657779	+	0.753211i	$0.728504\pi$
$234$	0	0
$235$	−3.38501e8	−1.70146
$236$	−3.51454e8	−1.74051
$237$	0	0
$238$	7.32902e8	3.52392
$239$	3.16138e8	1.49791	0.748953	−	0.662624i	$-0.230557\pi$
$239$	3.16138e8	1.49791	0.748953	+	0.662624i	$0.230557\pi$
$240$	0	0
$241$	2.15535e7	0.0991878	0.0495939	−	0.998769i	$-0.484207\pi$
$241$	2.15535e7	0.0991878	0.0495939	+	0.998769i	$0.484207\pi$
$242$	2.87286e8	1.30305
$243$	0	0
$244$	−4.72419e8	−2.08192
$245$	1.09343e8	0.475019
$246$	0	0
$247$	1.25074e8	0.528116
$248$	−4.17314e7	−0.173733
$249$	0	0
$250$	−6.37320e8	−2.57969
$251$	1.38647e8	0.553419	0.276709	−	0.960954i	$-0.410756\pi$
$251$	1.38647e8	0.553419	0.276709	+	0.960954i	$0.410756\pi$
$252$	0	0
$253$	1.57796e8	0.612596
$254$	2.07476e8	0.794421
$255$	0	0
$256$	−5.46524e8	−2.03596
$257$	−4.54276e8	−1.66938	−0.834688	−	0.550723i	$-0.814352\pi$
$257$	−4.54276e8	−1.66938	−0.834688	+	0.550723i	$0.814352\pi$
$258$	0	0
$259$	−4.21870e7	−0.150879
$260$	−4.13854e8	−1.46029
$261$	0	0
$262$	1.31595e8	0.452050
$263$	1.48149e8	0.502174	0.251087	−	0.967965i	$-0.419212\pi$
$263$	1.48149e8	0.502174	0.251087	+	0.967965i	$0.419212\pi$
$264$	0	0
$265$	2.32204e8	0.766493
$266$	−7.64420e8	−2.49027
$267$	0	0
$268$	6.69564e8	2.12481
$269$	2.17316e8	0.680703	0.340352	−	0.940298i	$-0.389454\pi$
$269$	2.17316e8	0.680703	0.340352	+	0.940298i	$0.389454\pi$
$270$	0	0
$271$	−3.83874e8	−1.17165	−0.585823	−	0.810439i	$-0.699229\pi$
$271$	−3.83874e8	−1.17165	−0.585823	+	0.810439i	$0.699229\pi$
$272$	6.90443e8	2.08035
$273$	0	0
$274$	−7.91236e8	−2.32369
$275$	−3.25515e8	−0.943857
$276$	0	0
$277$	−4.16650e8	−1.17786	−0.588928	−	0.808185i	$-0.700450\pi$
$277$	−4.16650e8	−1.17786	−0.588928	+	0.808185i	$0.700450\pi$
$278$	−3.72274e8	−1.03921
$279$	0	0
$280$	1.29843e9	3.53481
$281$	−6.34305e8	−1.70540	−0.852700	−	0.522401i	$-0.825036\pi$
$281$	−6.34305e8	−1.70540	−0.852700	+	0.522401i	$0.825036\pi$
$282$	0	0
$283$	7.08806e8	1.85898	0.929490	−	0.368847i	$-0.120247\pi$
$283$	7.08806e8	1.85898	0.929490	+	0.368847i	$0.120247\pi$
$284$	−1.84221e8	−0.477226
$285$	0	0
$286$	−1.46296e8	−0.369787
$287$	6.51675e7	0.162721
$288$	0	0
$289$	8.92471e8	2.17496
$290$	2.70270e8	0.650736
$291$	0	0
$292$	−1.23376e9	−2.89995
$293$	3.64597e7	0.0846791	0.0423395	−	0.999103i	$-0.486519\pi$
$293$	3.64597e7	0.0846791	0.0423395	+	0.999103i	$0.486519\pi$
$294$	0	0
$295$	−6.32845e8	−1.43523
$296$	−1.09690e8	−0.245837
$297$	0	0
$298$	−5.10396e8	−1.11725
$299$	2.35422e8	0.509328
$300$	0	0
$301$	1.55733e8	0.329153
$302$	1.80638e8	0.377385
$303$	0	0
$304$	−7.20135e8	−1.47013
$305$	−8.50661e8	−1.71675
$306$	0	0
$307$	−7.16485e8	−1.41326	−0.706631	−	0.707583i	$-0.749786\pi$
$307$	−7.16485e8	−1.41326	−0.706631	+	0.707583i	$0.749786\pi$
$308$	6.01431e8	1.17289
$309$	0	0
$310$	−1.46380e8	−0.279073
$311$	2.62929e8	0.495652	0.247826	−	0.968805i	$-0.420284\pi$
$311$	2.62929e8	0.495652	0.247826	+	0.968805i	$0.420284\pi$
$312$	0	0
$313$	2.99480e8	0.552029	0.276015	−	0.961153i	$-0.410986\pi$
$313$	2.99480e8	0.552029	0.276015	+	0.961153i	$0.410986\pi$
$314$	6.53037e8	1.19038
$315$	0	0
$316$	−1.30987e9	−2.33519
$317$	−1.77518e8	−0.312993	−0.156497	−	0.987679i	$-0.550020\pi$
$317$	−1.77518e8	−0.312993	−0.156497	+	0.987679i	$0.550020\pi$
$318$	0	0
$319$	6.42648e7	0.110842
$320$	−8.17705e8	−1.39499
$321$	0	0
$322$	−1.43883e9	−2.40168
$323$	−1.35884e9	−2.24367
$324$	0	0
$325$	−4.85648e8	−0.784747
$326$	−1.97502e9	−3.15726
$327$	0	0
$328$	1.69442e8	0.265132
$329$	7.33921e8	1.13622
$330$	0	0
$331$	8.18919e8	1.24120	0.620602	−	0.784126i	$-0.286888\pi$
$331$	8.18919e8	1.24120	0.620602	+	0.784126i	$0.286888\pi$
$332$	8.91707e8	1.33733
$333$	0	0
$334$	2.42461e8	0.356065
$335$	1.20565e9	1.75212
$336$	0	0
$337$	1.13506e9	1.61553	0.807763	−	0.589508i	$-0.200678\pi$
$337$	1.13506e9	1.61553	0.807763	+	0.589508i	$0.200678\pi$
$338$	1.02254e9	1.44036
$339$	0	0
$340$	4.49620e9	6.20396
$341$	−3.48064e7	−0.0475355
$342$	0	0
$343$	6.08581e8	0.814308
$344$	4.04920e8	0.536309
$345$	0	0
$346$	4.74673e8	0.616067
$347$	5.23134e8	0.672140	0.336070	−	0.941837i	$-0.390902\pi$
$347$	5.23134e8	0.672140	0.336070	+	0.941837i	$0.390902\pi$
$348$	0	0
$349$	2.23584e8	0.281548	0.140774	−	0.990042i	$-0.455041\pi$
$349$	2.23584e8	0.281548	0.140774	+	0.990042i	$0.455041\pi$
$350$	2.96814e9	3.70038
$351$	0	0
$352$	8.12996e7	0.0993548
$353$	3.85474e8	0.466427	0.233214	−	0.972426i	$-0.425076\pi$
$353$	3.85474e8	0.466427	0.233214	+	0.972426i	$0.425076\pi$
$354$	0	0
$355$	−3.31717e8	−0.393521
$356$	3.35999e8	0.394696
$357$	0	0
$358$	−2.68587e9	−3.09381
$359$	1.05112e9	1.19901	0.599504	−	0.800372i	$-0.295365\pi$
$359$	1.05112e9	1.19901	0.599504	+	0.800372i	$0.295365\pi$
$360$	0	0
$361$	5.23400e8	0.585542
$362$	1.53171e9	1.69706
$363$	0	0
$364$	8.97298e8	0.975173
$365$	−2.22156e9	−2.39130
$366$	0	0
$367$	1.93592e8	0.204435	0.102218	−	0.994762i	$-0.467406\pi$
$367$	1.93592e8	0.204435	0.102218	+	0.994762i	$0.467406\pi$
$368$	−1.35548e9	−1.41783
$369$	0	0
$370$	−3.84758e8	−0.394895
$371$	−5.03453e8	−0.511859
$372$	0	0
$373$	−3.53893e8	−0.353094	−0.176547	−	0.984292i	$-0.556493\pi$
$373$	−3.53893e8	−0.353094	−0.176547	+	0.984292i	$0.556493\pi$
$374$	1.58939e9	1.57102
$375$	0	0
$376$	1.90827e9	1.85132
$377$	9.58792e7	0.0921572
$378$	0	0
$379$	−4.69956e8	−0.443425	−0.221712	−	0.975112i	$-0.571165\pi$
$379$	−4.69956e8	−0.443425	−0.221712	+	0.975112i	$0.571165\pi$
$380$	−4.68955e9	−4.38418
$381$	0	0
$382$	7.48434e8	0.686961
$383$	1.53089e9	1.39235	0.696177	−	0.717871i	$-0.254883\pi$
$383$	1.53089e9	1.39235	0.696177	+	0.717871i	$0.254883\pi$
$384$	0	0
$385$	1.08296e9	0.967168
$386$	8.52500e8	0.754465
$387$	0	0
$388$	2.01396e9	1.75041
$389$	−8.33450e8	−0.717887	−0.358943	−	0.933359i	$-0.616863\pi$
$389$	−8.33450e8	−0.717887	−0.358943	+	0.933359i	$0.616863\pi$
$390$	0	0
$391$	−2.55767e9	−2.16385
$392$	−6.16413e8	−0.516857
$393$	0	0
$394$	−2.67457e9	−2.20301
$395$	−2.35861e9	−1.92560
$396$	0	0
$397$	8.58150e8	0.688330	0.344165	−	0.938909i	$-0.388162\pi$
$397$	8.58150e8	0.688330	0.344165	+	0.938909i	$0.388162\pi$
$398$	−1.94429e9	−1.54586
$399$	0	0
$400$	2.79619e9	2.18452
$401$	2.07167e8	0.160441	0.0802205	−	0.996777i	$-0.474438\pi$
$401$	2.07167e8	0.160441	0.0802205	+	0.996777i	$0.474438\pi$
$402$	0	0
$403$	−5.19289e7	−0.0395223
$404$	−4.48225e9	−3.38190
$405$	0	0
$406$	−5.85986e8	−0.434557
$407$	−9.14879e7	−0.0672641
$408$	0	0
$409$	−4.47164e8	−0.323173	−0.161586	−	0.986859i	$-0.551661\pi$
$409$	−4.47164e8	−0.323173	−0.161586	+	0.986859i	$0.551661\pi$
$410$	5.94348e8	0.425890
$411$	0	0
$412$	−4.62082e9	−3.25521
$413$	1.37210e9	0.958433
$414$	0	0
$415$	1.60565e9	1.10276
$416$	1.21294e8	0.0826062
$417$	0	0
$418$	−1.65774e9	−1.11020
$419$	−8.04489e8	−0.534283	−0.267141	−	0.963657i	$-0.586079\pi$
$419$	−8.04489e8	−0.534283	−0.267141	+	0.963657i	$0.586079\pi$
$420$	0	0
$421$	1.19625e9	0.781333	0.390666	−	0.920532i	$-0.372245\pi$
$421$	1.19625e9	0.781333	0.390666	+	0.920532i	$0.372245\pi$
$422$	1.35015e9	0.874555
$423$	0	0
$424$	−1.30903e9	−0.834003
$425$	5.27618e9	3.33395
$426$	0	0
$427$	1.84436e9	1.14643
$428$	−3.06203e9	−1.88780
$429$	0	0
$430$	1.42033e9	0.861490
$431$	−8.02879e8	−0.483036	−0.241518	−	0.970396i	$-0.577645\pi$
$431$	−8.02879e8	−0.483036	−0.241518	+	0.970396i	$0.577645\pi$
$432$	0	0
$433$	2.54405e9	1.50597	0.752987	−	0.658035i	$-0.228612\pi$
$433$	2.54405e9	1.50597	0.752987	+	0.658035i	$0.228612\pi$
$434$	3.17375e8	0.186363
$435$	0	0
$436$	−6.02800e9	−3.48314
$437$	2.66767e9	1.52914
$438$	0	0
$439$	−8.00679e8	−0.451682	−0.225841	−	0.974164i	$-0.572513\pi$
$439$	−8.00679e8	−0.451682	−0.225841	+	0.974164i	$0.572513\pi$
$440$	2.81581e9	1.57587
$441$	0	0
$442$	2.37127e9	1.30618
$443$	−1.07872e9	−0.589517	−0.294758	−	0.955572i	$-0.595239\pi$
$443$	−1.07872e9	−0.589517	−0.294758	+	0.955572i	$0.595239\pi$
$444$	0	0
$445$	6.05015e8	0.325466
$446$	−5.77131e9	−3.08036
$447$	0	0
$448$	1.77291e9	0.931565
$449$	1.67071e9	0.871042	0.435521	−	0.900179i	$-0.356564\pi$
$449$	1.67071e9	0.871042	0.435521	+	0.900179i	$0.356564\pi$
$450$	0	0
$451$	1.41324e8	0.0725434
$452$	9.00600e7	0.0458720
$453$	0	0
$454$	4.71786e9	2.36619
$455$	1.61572e9	0.804128
$456$	0	0
$457$	1.81152e9	0.887844	0.443922	−	0.896065i	$-0.353587\pi$
$457$	1.81152e9	0.887844	0.443922	+	0.896065i	$0.353587\pi$
$458$	3.02605e9	1.47180
$459$	0	0
$460$	−8.82693e9	−4.22822
$461$	9.08114e8	0.431705	0.215852	−	0.976426i	$-0.430747\pi$
$461$	9.08114e8	0.431705	0.215852	+	0.976426i	$0.430747\pi$
$462$	0	0
$463$	−2.96452e9	−1.38810	−0.694051	−	0.719926i	$-0.744176\pi$
$463$	−2.96452e9	−1.38810	−0.694051	+	0.719926i	$0.744176\pi$
$464$	−5.52038e8	−0.256541
$465$	0	0
$466$	5.02291e9	2.29935
$467$	−1.15258e9	−0.523675	−0.261837	−	0.965112i	$-0.584328\pi$
$467$	−1.15258e9	−0.523675	−0.261837	+	0.965112i	$0.584328\pi$
$468$	0	0
$469$	−2.61403e9	−1.17005
$470$	6.69359e9	2.97383
$471$	0	0
$472$	3.56760e9	1.56164
$473$	3.37727e8	0.146741
$474$	0	0
$475$	−5.50308e9	−2.35602
$476$	−9.74843e9	−4.14296
$477$	0	0
$478$	−6.25139e9	−2.61805
$479$	−5.66663e8	−0.235587	−0.117793	−	0.993038i	$-0.537582\pi$
$479$	−5.66663e8	−0.235587	−0.117793	+	0.993038i	$0.537582\pi$
$480$	0	0
$481$	−1.36494e8	−0.0559251
$482$	−4.26204e8	−0.173361
$483$	0	0
$484$	−3.82123e9	−1.53195
$485$	3.62643e9	1.44339
$486$	0	0
$487$	3.19934e9	1.25519	0.627595	−	0.778540i	$-0.284040\pi$
$487$	3.19934e9	1.25519	0.627595	+	0.778540i	$0.284040\pi$
$488$	4.79552e9	1.86795
$489$	0	0
$490$	−2.16218e9	−0.830243
$491$	2.13235e9	0.812966	0.406483	−	0.913658i	$-0.366755\pi$
$491$	2.13235e9	0.812966	0.406483	+	0.913658i	$0.366755\pi$
$492$	0	0
$493$	−1.04165e9	−0.391524
$494$	−2.47325e9	−0.923047
$495$	0	0
$496$	2.98989e8	0.110019
$497$	7.19212e8	0.262790
$498$	0	0
$499$	1.04291e9	0.375747	0.187874	−	0.982193i	$-0.439840\pi$
$499$	1.04291e9	0.375747	0.187874	+	0.982193i	$0.439840\pi$
$500$	8.47708e9	3.03285
$501$	0	0
$502$	−2.74164e9	−0.967271
$503$	2.59463e9	0.909051	0.454526	−	0.890734i	$-0.349809\pi$
$503$	2.59463e9	0.909051	0.454526	+	0.890734i	$0.349809\pi$
$504$	0	0
$505$	−8.07094e9	−2.78872
$506$	−3.12029e9	−1.07070
$507$	0	0
$508$	−2.75967e9	−0.933973
$509$	−5.04285e9	−1.69498	−0.847488	−	0.530815i	$-0.821886\pi$
$509$	−5.04285e9	−1.69498	−0.847488	+	0.530815i	$0.821886\pi$
$510$	0	0
$511$	4.81669e9	1.59689
$512$	5.83885e9	1.92257
$513$	0	0
$514$	8.98296e9	2.91775
$515$	−8.32046e9	−2.68424
$516$	0	0
$517$	1.59160e9	0.506545
$518$	8.34215e8	0.263708
$519$	0	0
$520$	4.20102e9	1.31022
$521$	−3.81880e9	−1.18303	−0.591514	−	0.806295i	$-0.701469\pi$
$521$	−3.81880e9	−1.18303	−0.591514	+	0.806295i	$0.701469\pi$
$522$	0	0
$523$	−1.81806e9	−0.555715	−0.277857	−	0.960622i	$-0.589624\pi$
$523$	−1.81806e9	−0.555715	−0.277857	+	0.960622i	$0.589624\pi$
$524$	−1.75037e9	−0.531460
$525$	0	0
$526$	−2.92953e9	−0.877704
$527$	5.64167e8	0.167908
$528$	0	0
$529$	1.61640e9	0.474738
$530$	−4.59164e9	−1.33968
$531$	0	0
$532$	1.01677e10	2.92772
$533$	2.10847e8	0.0603145
$534$	0	0
$535$	−5.51364e9	−1.55668
$536$	−6.79673e9	−1.90644
$537$	0	0
$538$	−4.29724e9	−1.18974
$539$	−5.14123e8	−0.141419
$540$	0	0
$541$	3.76234e9	1.02157	0.510784	−	0.859709i	$-0.329355\pi$
$541$	3.76234e9	1.02157	0.510784	+	0.859709i	$0.329355\pi$
$542$	7.59081e9	2.04781
$543$	0	0
$544$	−1.31776e9	−0.350947
$545$	−1.08543e10	−2.87220
$546$	0	0
$547$	−5.99456e9	−1.56604	−0.783018	−	0.621999i	$-0.786321\pi$
$547$	−5.99456e9	−1.56604	−0.783018	+	0.621999i	$0.786321\pi$
$548$	1.05243e10	2.73189
$549$	0	0
$550$	6.43680e9	1.64968
$551$	1.08645e9	0.276680
$552$	0	0
$553$	5.11382e9	1.28590
$554$	8.23893e9	2.05867
$555$	0	0
$556$	4.95166e9	1.22177
$557$	4.74153e9	1.16259	0.581294	−	0.813694i	$-0.302547\pi$
$557$	4.74153e9	1.16259	0.581294	+	0.813694i	$0.302547\pi$
$558$	0	0
$559$	5.03867e8	0.122004
$560$	−9.30273e9	−2.23847
$561$	0	0
$562$	1.25429e10	2.98072
$563$	−3.33143e7	−0.00786776	−0.00393388	−	0.999992i	$-0.501252\pi$
$563$	−3.33143e7	−0.00786776	−0.00393388	+	0.999992i	$0.501252\pi$
$564$	0	0
$565$	1.62166e8	0.0378260
$566$	−1.40161e10	−3.24914
$567$	0	0
$568$	1.87002e9	0.428181
$569$	−3.70701e9	−0.843589	−0.421794	−	0.906692i	$-0.638600\pi$
$569$	−3.70701e9	−0.843589	−0.421794	+	0.906692i	$0.638600\pi$
$570$	0	0
$571$	6.83721e9	1.53692	0.768462	−	0.639896i	$-0.221022\pi$
$571$	6.83721e9	1.53692	0.768462	+	0.639896i	$0.221022\pi$
$572$	1.94591e9	0.434746
$573$	0	0
$574$	−1.28864e9	−0.284406
$575$	−1.03582e10	−2.27220
$576$	0	0
$577$	4.39006e9	0.951382	0.475691	−	0.879612i	$-0.342198\pi$
$577$	4.39006e9	0.951382	0.475691	+	0.879612i	$0.342198\pi$
$578$	−1.76479e10	−3.80142
$579$	0	0
$580$	−3.59490e9	−0.765048
$581$	−3.48129e9	−0.736417
$582$	0	0
$583$	−1.09180e9	−0.228194
$584$	1.25239e10	2.60192
$585$	0	0
$586$	−7.20961e8	−0.148003
$587$	8.19912e8	0.167315	0.0836573	−	0.996495i	$-0.473340\pi$
$587$	8.19912e8	0.167315	0.0836573	+	0.996495i	$0.473340\pi$
$588$	0	0
$589$	−5.88428e8	−0.118656
$590$	1.25140e10	2.50850
$591$	0	0
$592$	7.85886e8	0.155680
$593$	4.33599e9	0.853880	0.426940	−	0.904280i	$-0.359592\pi$
$593$	4.33599e9	0.853880	0.426940	+	0.904280i	$0.359592\pi$
$594$	0	0
$595$	−1.75535e10	−3.41628
$596$	6.78885e9	1.31351
$597$	0	0
$598$	−4.65528e9	−0.890209
$599$	9.92412e9	1.88668	0.943339	−	0.331829i	$-0.107666\pi$
$599$	9.92412e9	1.88668	0.943339	+	0.331829i	$0.107666\pi$
$600$	0	0
$601$	1.94494e9	0.365465	0.182733	−	0.983163i	$-0.441506\pi$
$601$	1.94494e9	0.365465	0.182733	+	0.983163i	$0.441506\pi$
$602$	−3.07949e9	−0.575296
$603$	0	0
$604$	−2.40269e9	−0.443678
$605$	−6.88068e9	−1.26325
$606$	0	0
$607$	−7.99592e9	−1.45114	−0.725568	−	0.688151i	$-0.758423\pi$
$607$	−7.99592e9	−1.45114	−0.725568	+	0.688151i	$0.758423\pi$
$608$	1.37443e9	0.248005
$609$	0	0
$610$	1.68211e10	3.00055
$611$	2.37457e9	0.421154
$612$	0	0
$613$	5.19028e8	0.0910079	0.0455040	−	0.998964i	$-0.485511\pi$
$613$	5.19028e8	0.0910079	0.0455040	+	0.998964i	$0.485511\pi$
$614$	1.41679e10	2.47011
$615$	0	0
$616$	−6.10511e9	−1.05235
$617$	2.33763e9	0.400662	0.200331	−	0.979728i	$-0.435798\pi$
$617$	2.33763e9	0.400662	0.200331	+	0.979728i	$0.435798\pi$
$618$	0	0
$619$	−6.75307e9	−1.14442	−0.572208	−	0.820109i	$-0.693913\pi$
$619$	−6.75307e9	−1.14442	−0.572208	+	0.820109i	$0.693913\pi$
$620$	1.94703e9	0.328096
$621$	0	0
$622$	−5.19921e9	−0.866306
$623$	−1.31176e9	−0.217344
$624$	0	0
$625$	3.84414e9	0.629823
$626$	−5.92198e9	−0.964842
$627$	0	0
$628$	−8.68613e9	−1.39948
$629$	1.48290e9	0.237594
$630$	0	0
$631$	−9.84840e9	−1.56050	−0.780249	−	0.625469i	$-0.784908\pi$
$631$	−9.84840e9	−1.56050	−0.780249	+	0.625469i	$0.784908\pi$
$632$	1.32964e10	2.09520
$633$	0	0
$634$	3.51028e9	0.547052
$635$	−4.96920e9	−0.770155
$636$	0	0
$637$	−7.67040e8	−0.117579
$638$	−1.27079e9	−0.193732
$639$	0	0
$640$	1.83827e10	2.77191
$641$	−1.92918e9	−0.289314	−0.144657	−	0.989482i	$-0.546208\pi$
$641$	−1.92918e9	−0.289314	−0.144657	+	0.989482i	$0.546208\pi$
$642$	0	0
$643$	3.69430e9	0.548017	0.274009	−	0.961727i	$-0.411650\pi$
$643$	3.69430e9	0.548017	0.274009	+	0.961727i	$0.411650\pi$
$644$	1.91381e10	2.82357
$645$	0	0
$646$	2.68699e10	3.92150
$647$	1.04870e10	1.52225	0.761127	−	0.648602i	$-0.224646\pi$
$647$	1.04870e10	1.52225	0.761127	+	0.648602i	$0.224646\pi$
$648$	0	0
$649$	2.97558e9	0.427283
$650$	9.60330e9	1.37159
$651$	0	0
$652$	2.62700e10	3.71188
$653$	−6.76210e9	−0.950354	−0.475177	−	0.879890i	$-0.657616\pi$
$653$	−6.76210e9	−0.950354	−0.475177	+	0.879890i	$0.657616\pi$
$654$	0	0
$655$	−3.15180e9	−0.438242
$656$	−1.21398e9	−0.167899
$657$	0	0
$658$	−1.45127e10	−1.98590
$659$	−6.22939e8	−0.0847904	−0.0423952	−	0.999101i	$-0.513499\pi$
$659$	−6.22939e8	−0.0847904	−0.0423952	+	0.999101i	$0.513499\pi$
$660$	0	0
$661$	1.52850e9	0.205854	0.102927	−	0.994689i	$-0.467179\pi$
$661$	1.52850e9	0.205854	0.102927	+	0.994689i	$0.467179\pi$
$662$	−1.61935e10	−2.16939
$663$	0	0
$664$	−9.05169e9	−1.19989
$665$	1.83084e10	2.41420
$666$	0	0
$667$	2.04497e9	0.266837
$668$	−3.22501e9	−0.418614
$669$	0	0
$670$	−2.38407e10	−3.06237
$671$	3.99973e9	0.511096
$672$	0	0
$673$	−1.12950e10	−1.42835	−0.714176	−	0.699966i	$-0.753198\pi$
$673$	−1.12950e10	−1.42835	−0.714176	+	0.699966i	$0.753198\pi$
$674$	−2.24449e10	−2.82363
$675$	0	0
$676$	−1.36009e10	−1.69338
$677$	8.05475e9	0.997681	0.498841	−	0.866694i	$-0.333759\pi$
$677$	8.05475e9	0.997681	0.498841	+	0.866694i	$0.333759\pi$
$678$	0	0
$679$	−7.86265e9	−0.963884
$680$	−4.56408e10	−5.56637
$681$	0	0
$682$	6.88268e8	0.0830831
$683$	−1.53230e10	−1.84023	−0.920113	−	0.391652i	$-0.871904\pi$
$683$	−1.53230e10	−1.84023	−0.920113	+	0.391652i	$0.871904\pi$
$684$	0	0
$685$	1.89506e10	2.25272
$686$	−1.20342e10	−1.42326
$687$	0	0
$688$	−2.90109e9	−0.339627
$689$	−1.62890e9	−0.189726
$690$	0	0
$691$	3.65174e9	0.421043	0.210521	−	0.977589i	$-0.432484\pi$
$691$	3.65174e9	0.421043	0.210521	+	0.977589i	$0.432484\pi$
$692$	−6.31369e9	−0.724289
$693$	0	0
$694$	−1.03446e10	−1.17477
$695$	8.91620e9	1.00747
$696$	0	0
$697$	−2.29068e9	−0.256242
$698$	−4.42121e9	−0.492093
$699$	0	0
$700$	−3.94797e10	−4.35041
$701$	−1.42036e10	−1.55735	−0.778675	−	0.627427i	$-0.784108\pi$
$701$	−1.42036e10	−1.55735	−0.778675	+	0.627427i	$0.784108\pi$
$702$	0	0
$703$	−1.54667e9	−0.167902
$704$	3.84478e9	0.415305
$705$	0	0
$706$	−7.62245e9	−0.815226
$707$	1.74990e10	1.86228
$708$	0	0
$709$	1.16843e10	1.23123	0.615617	−	0.788045i	$-0.288907\pi$
$709$	1.16843e10	1.23123	0.615617	+	0.788045i	$0.288907\pi$
$710$	6.55943e9	0.687800
$711$	0	0
$712$	−3.41071e9	−0.354132
$713$	−1.10757e9	−0.114435
$714$	0	0
$715$	3.50389e9	0.358492
$716$	3.57251e10	3.63729
$717$	0	0
$718$	−2.07851e10	−2.09564
$719$	8.68481e9	0.871383	0.435691	−	0.900096i	$-0.356504\pi$
$719$	8.68481e9	0.871383	0.435691	+	0.900096i	$0.356504\pi$
$720$	0	0
$721$	1.80400e10	1.79252
$722$	−1.03498e10	−1.02342
$723$	0	0
$724$	−2.03735e10	−1.99518
$725$	−4.21853e9	−0.411129
$726$	0	0
$727$	3.49630e9	0.337472	0.168736	−	0.985661i	$-0.446031\pi$
$727$	3.49630e9	0.337472	0.168736	+	0.985661i	$0.446031\pi$
$728$	−9.10845e9	−0.874953
$729$	0	0
$730$	4.39297e10	4.17954
$731$	−5.47412e9	−0.518327
$732$	0	0
$733$	−9.31103e9	−0.873240	−0.436620	−	0.899646i	$-0.643825\pi$
$733$	−9.31103e9	−0.873240	−0.436620	+	0.899646i	$0.643825\pi$
$734$	−3.82812e9	−0.357314
$735$	0	0
$736$	2.58703e9	0.239182
$737$	−5.66885e9	−0.521626
$738$	0	0
$739$	−7.46338e9	−0.680268	−0.340134	−	0.940377i	$-0.610472\pi$
$739$	−7.46338e9	−0.680268	−0.340134	+	0.940377i	$0.610472\pi$
$740$	5.11773e9	0.464265
$741$	0	0
$742$	9.95538e9	0.894631
$743$	7.68972e9	0.687781	0.343890	−	0.939010i	$-0.388255\pi$
$743$	7.68972e9	0.687781	0.343890	+	0.939010i	$0.388255\pi$
$744$	0	0
$745$	1.22243e10	1.08312
$746$	6.99795e9	0.617141
$747$	0	0
$748$	−2.11407e10	−1.84699
$749$	1.19544e10	1.03954
$750$	0	0
$751$	−1.94830e10	−1.67848	−0.839238	−	0.543764i	$-0.816998\pi$
$751$	−1.94830e10	−1.67848	−0.839238	+	0.543764i	$0.816998\pi$
$752$	−1.36720e10	−1.17238
$753$	0	0
$754$	−1.89594e9	−0.161073
$755$	−4.32639e9	−0.365857
$756$	0	0
$757$	−4.45370e9	−0.373152	−0.186576	−	0.982441i	$-0.559739\pi$
$757$	−4.45370e9	−0.373152	−0.186576	+	0.982441i	$0.559739\pi$
$758$	9.29301e9	0.775022
$759$	0	0
$760$	4.76035e10	3.93361
$761$	−1.84866e10	−1.52058	−0.760292	−	0.649581i	$-0.774944\pi$
$761$	−1.84866e10	−1.52058	−0.760292	+	0.649581i	$0.774944\pi$
$762$	0	0
$763$	2.35338e10	1.91803
$764$	−9.95503e9	−0.807636
$765$	0	0
$766$	−3.02722e10	−2.43357
$767$	4.43939e9	0.355254
$768$	0	0
$769$	−1.10733e9	−0.0878079	−0.0439039	−	0.999036i	$-0.513980\pi$
$769$	−1.10733e9	−0.0878079	−0.0439039	+	0.999036i	$0.513980\pi$
$770$	−2.14148e10	−1.69043
$771$	0	0
$772$	−1.13392e10	−0.886998
$773$	1.36030e10	1.05927	0.529633	−	0.848227i	$-0.322330\pi$
$773$	1.36030e10	1.05927	0.529633	+	0.848227i	$0.322330\pi$
$774$	0	0
$775$	2.28479e9	0.176315
$776$	−2.04437e10	−1.57052
$777$	0	0
$778$	1.64808e10	1.25473
$779$	2.38919e9	0.181080
$780$	0	0
$781$	1.55970e9	0.117156
$782$	5.05760e10	3.78199
$783$	0	0
$784$	4.41635e9	0.327308
$785$	−1.56407e10	−1.15401
$786$	0	0
$787$	−1.99593e10	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$787$	−1.99593e10	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$788$	3.55748e10	2.59001
$789$	0	0
$790$	4.66396e10	3.36558
$791$	−3.51601e8	−0.0252599
$792$	0	0
$793$	5.96736e9	0.424938
$794$	−1.69692e10	−1.20307
$795$	0	0
$796$	2.58613e10	1.81742
$797$	−1.23171e10	−0.861799	−0.430899	−	0.902400i	$-0.641804\pi$
$797$	−1.23171e10	−0.861799	−0.430899	+	0.902400i	$0.641804\pi$
$798$	0	0
$799$	−2.57979e10	−1.78925
$800$	−5.33674e9	−0.368520
$801$	0	0
$802$	−4.09657e9	−0.280421
$803$	1.04456e10	0.711917
$804$	0	0
$805$	3.44610e10	2.32832
$806$	1.02685e9	0.0690774
$807$	0	0
$808$	4.54992e10	3.03434
$809$	1.27792e10	0.848559	0.424280	−	0.905531i	$-0.360527\pi$
$809$	1.27792e10	0.848559	0.424280	+	0.905531i	$0.360527\pi$
$810$	0	0
$811$	−2.84223e9	−0.187105	−0.0935526	−	0.995614i	$-0.529822\pi$
$811$	−2.84223e9	−0.187105	−0.0935526	+	0.995614i	$0.529822\pi$
$812$	7.79429e9	0.510893
$813$	0	0
$814$	1.80910e9	0.117565
$815$	4.73031e10	3.06082
$816$	0	0
$817$	5.70953e9	0.366288
$818$	8.84231e9	0.564845
$819$	0	0
$820$	−7.90550e9	−0.500704
$821$	1.77841e10	1.12158	0.560791	−	0.827957i	$-0.310497\pi$
$821$	1.77841e10	1.12158	0.560791	+	0.827957i	$0.310497\pi$
$822$	0	0
$823$	4.86118e9	0.303978	0.151989	−	0.988382i	$-0.451432\pi$
$823$	4.86118e9	0.303978	0.151989	+	0.988382i	$0.451432\pi$
$824$	4.69058e10	2.92066
$825$	0	0
$826$	−2.71323e10	−1.67516
$827$	−2.39461e10	−1.47220	−0.736099	−	0.676874i	$-0.763334\pi$
$827$	−2.39461e10	−1.47220	−0.736099	+	0.676874i	$0.763334\pi$
$828$	0	0
$829$	−1.19577e10	−0.728968	−0.364484	−	0.931210i	$-0.618755\pi$
$829$	−1.19577e10	−0.728968	−0.364484	+	0.931210i	$0.618755\pi$
$830$	−3.17504e10	−1.92742
$831$	0	0
$832$	5.73617e9	0.345295
$833$	8.33329e9	0.499527
$834$	0	0
$835$	−5.80711e9	−0.345189
$836$	2.20499e10	1.30522
$837$	0	0
$838$	1.59081e10	0.933825
$839$	−9.87176e9	−0.577069	−0.288535	−	0.957469i	$-0.593168\pi$
$839$	−9.87176e9	−0.577069	−0.288535	+	0.957469i	$0.593168\pi$
$840$	0	0
$841$	−1.64170e10	−0.951719
$842$	−2.36550e10	−1.36562
$843$	0	0
$844$	−1.79585e10	−1.02818
$845$	−2.44905e10	−1.39636
$846$	0	0
$847$	1.49184e10	0.843586
$848$	9.37864e9	0.528146
$849$	0	0
$850$	−1.04332e11	−5.82710
$851$	−2.91123e9	−0.161929
$852$	0	0
$853$	1.23979e10	0.683952	0.341976	−	0.939709i	$-0.388904\pi$
$853$	1.23979e10	0.683952	0.341976	+	0.939709i	$0.388904\pi$
$854$	−3.64708e10	−2.00375
$855$	0	0
$856$	3.10826e10	1.69379
$857$	3.52007e10	1.91037	0.955186	−	0.296007i	$-0.0956551\pi$
$857$	3.52007e10	1.91037	0.955186	+	0.296007i	$0.0956551\pi$
$858$	0	0
$859$	1.30137e10	0.700528	0.350264	−	0.936651i	$-0.386092\pi$
$859$	1.30137e10	0.700528	0.350264	+	0.936651i	$0.386092\pi$
$860$	−1.88920e10	−1.01282
$861$	0	0
$862$	1.58763e10	0.844256
$863$	−3.11852e10	−1.65162	−0.825811	−	0.563947i	$-0.809282\pi$
$863$	−3.11852e10	−1.65162	−0.825811	+	0.563947i	$0.809282\pi$
$864$	0	0
$865$	−1.13687e10	−0.597249
$866$	−5.03065e10	−2.63216
$867$	0	0
$868$	−4.22145e9	−0.219100
$869$	1.10900e10	0.573273
$870$	0	0
$871$	−8.45758e9	−0.433693
$872$	6.11901e10	3.12517
$873$	0	0
$874$	−5.27510e10	−2.67264
$875$	−3.30951e10	−1.67008
$876$	0	0
$877$	2.03806e10	1.02028	0.510140	−	0.860092i	$-0.329594\pi$
$877$	2.03806e10	1.02028	0.510140	+	0.860092i	$0.329594\pi$
$878$	1.58328e10	0.789454
$879$	0	0
$880$	−2.01742e10	−0.997944
$881$	1.59058e10	0.783681	0.391841	−	0.920033i	$-0.371838\pi$
$881$	1.59058e10	0.783681	0.391841	+	0.920033i	$0.371838\pi$
$882$	0	0
$883$	3.50773e10	1.71460	0.857302	−	0.514814i	$-0.172139\pi$
$883$	3.50773e10	1.71460	0.857302	+	0.514814i	$0.172139\pi$
$884$	−3.15407e10	−1.53563
$885$	0	0
$886$	2.13309e10	1.03036
$887$	−2.89660e10	−1.39366	−0.696828	−	0.717238i	$-0.745406\pi$
$887$	−2.89660e10	−1.39366	−0.696828	+	0.717238i	$0.745406\pi$
$888$	0	0
$889$	1.07740e10	0.514304
$890$	−1.19637e10	−0.568853
$891$	0	0
$892$	7.67649e10	3.62148
$893$	2.69073e10	1.26441
$894$	0	0
$895$	6.43283e10	2.99931
$896$	−3.98564e10	−1.85106
$897$	0	0
$898$	−3.30370e10	−1.52242
$899$	−4.51075e8	−0.0207057
$900$	0	0
$901$	1.76967e10	0.806039
$902$	−2.79457e9	−0.126792
$903$	0	0
$904$	−9.14197e8	−0.0411576
$905$	−3.66856e10	−1.64522
$906$	0	0
$907$	−2.43657e10	−1.08431	−0.542155	−	0.840278i	$-0.682392\pi$
$907$	−2.43657e10	−1.08431	−0.542155	+	0.840278i	$0.682392\pi$
$908$	−6.27529e10	−2.78185
$909$	0	0
$910$	−3.19495e10	−1.40546
$911$	−2.76976e9	−0.121374	−0.0606872	−	0.998157i	$-0.519329\pi$
$911$	−2.76976e9	−0.121374	−0.0606872	+	0.998157i	$0.519329\pi$
$912$	0	0
$913$	−7.54962e9	−0.328305
$914$	−3.58214e10	−1.55178
$915$	0	0
$916$	−4.02500e10	−1.73034
$917$	6.83358e9	0.292655
$918$	0	0
$919$	6.03812e9	0.256624	0.128312	−	0.991734i	$-0.459044\pi$
$919$	6.03812e9	0.256624	0.128312	+	0.991734i	$0.459044\pi$
$920$	8.96019e10	3.79367
$921$	0	0
$922$	−1.79572e10	−0.754538
$923$	2.32698e9	0.0974062
$924$	0	0
$925$	6.00553e9	0.249491
$926$	5.86211e10	2.42614
$927$	0	0
$928$	1.05361e9	0.0432774
$929$	−3.05571e9	−0.125042	−0.0625212	−	0.998044i	$-0.519914\pi$
$929$	−3.05571e9	−0.125042	−0.0625212	+	0.998044i	$0.519914\pi$
$930$	0	0
$931$	−8.69165e9	−0.353003
$932$	−6.68104e10	−2.70326
$933$	0	0
$934$	2.27914e10	0.915284
$935$	−3.80670e10	−1.52303
$936$	0	0
$937$	−4.12174e10	−1.63679	−0.818393	−	0.574658i	$-0.805135\pi$
$937$	−4.12174e10	−1.63679	−0.818393	+	0.574658i	$0.805135\pi$
$938$	5.16903e10	2.04503
$939$	0	0
$940$	−8.90324e10	−3.49623
$941$	2.44849e10	0.957933	0.478967	−	0.877833i	$-0.341011\pi$
$941$	2.44849e10	0.957933	0.478967	+	0.877833i	$0.341011\pi$
$942$	0	0
$943$	4.49707e9	0.174638
$944$	−2.55604e10	−0.988931
$945$	0	0
$946$	−6.67828e9	−0.256475
$947$	−3.75033e10	−1.43498	−0.717488	−	0.696571i	$-0.754708\pi$
$947$	−3.75033e10	−1.43498	−0.717488	+	0.696571i	$0.754708\pi$
$948$	0	0
$949$	1.55842e10	0.591906
$950$	1.08819e11	4.11787
$951$	0	0
$952$	9.89561e10	3.71718
$953$	−1.27579e10	−0.477480	−0.238740	−	0.971084i	$-0.576734\pi$
$953$	−1.27579e10	−0.477480	−0.238740	+	0.971084i	$0.576734\pi$
$954$	0	0
$955$	−1.79255e10	−0.665977
$956$	8.31505e10	3.07796
$957$	0	0
$958$	1.12053e10	0.411761
$959$	−4.10878e10	−1.50435
$960$	0	0
$961$	−2.72683e10	−0.991120
$962$	2.69907e9	0.0977464
$963$	0	0
$964$	5.66899e9	0.203815
$965$	−2.04179e10	−0.731419
$966$	0	0
$967$	−2.16883e10	−0.771316	−0.385658	−	0.922642i	$-0.626026\pi$
$967$	−2.16883e10	−0.771316	−0.385658	+	0.922642i	$0.626026\pi$
$968$	3.87892e10	1.37451
$969$	0	0
$970$	−7.17098e10	−2.52277
$971$	4.12029e10	1.44431	0.722155	−	0.691732i	$-0.243152\pi$
$971$	4.12029e10	1.44431	0.722155	+	0.691732i	$0.243152\pi$
$972$	0	0
$973$	−1.93317e10	−0.672782
$974$	−6.32645e10	−2.19383
$975$	0	0
$976$	−3.43579e10	−1.18291
$977$	−4.70697e8	−0.0161477	−0.00807384	−	0.999967i	$-0.502570\pi$
$977$	−4.70697e8	−0.0161477	−0.00807384	+	0.999967i	$0.502570\pi$
$978$	0	0
$979$	−2.84473e9	−0.0968950
$980$	2.87594e10	0.976089
$981$	0	0
$982$	−4.21655e10	−1.42091
$983$	6.80573e9	0.228527	0.114264	−	0.993450i	$-0.463549\pi$
$983$	6.80573e9	0.228527	0.114264	+	0.993450i	$0.463549\pi$
$984$	0	0
$985$	6.40577e10	2.13572
$986$	2.05978e10	0.684309
$987$	0	0
$988$	3.28970e10	1.08519
$989$	1.07468e10	0.353258
$990$	0	0
$991$	−5.55681e10	−1.81371	−0.906855	−	0.421442i	$-0.861524\pi$
$991$	−5.55681e10	−1.81371	−0.906855	+	0.421442i	$0.861524\pi$
$992$	−5.70642e8	−0.0185598
$993$	0	0
$994$	−1.42218e10	−0.459308
$995$	4.65671e10	1.49864
$996$	0	0
$997$	3.37314e10	1.07796	0.538979	−	0.842319i	$-0.318810\pi$
$997$	3.37314e10	1.07796	0.538979	+	0.842319i	$0.318810\pi$
$998$	−2.06228e10	−0.656735
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.8.a.d.1.1	✓	6
3.2	odd	2	inner	81.8.a.d.1.6	yes	6
9.2	odd	6		81.8.c.k.28.1		12
9.4	even	3		81.8.c.k.55.6		12
9.5	odd	6		81.8.c.k.55.1		12
9.7	even	3		81.8.c.k.28.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.8.a.d.1.1	✓	6	1.1	even	1	trivial
81.8.a.d.1.6	yes	6	3.2	odd	2	inner
81.8.c.k.28.1		12	9.2	odd	6
81.8.c.k.28.6		12	9.7	even	3
81.8.c.k.55.1		12	9.5	odd	6
81.8.c.k.55.6		12	9.4	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q-19.7742 q^{2} +263.020 q^{4} +473.605 q^{5} -1026.85 q^{7} -2669.91 q^{8} -9365.17 q^{10} -2226.85 q^{11} -3322.33 q^{13} +20305.1 q^{14} +19128.8 q^{16} +36094.5 q^{17} -37646.7 q^{19} +124567. q^{20} +44034.3 q^{22} -70860.6 q^{23} +146177. q^{25} +65696.4 q^{26} -270081. q^{28} -28859.0 q^{29} +15630.3 q^{31} -36508.7 q^{32} -713739. q^{34} -486321. q^{35} +41084.0 q^{37} +744433. q^{38} -1.26448e6 q^{40} -63463.6 q^{41} -151661. q^{43} -585706. q^{44} +1.40121e6 q^{46} -714732. q^{47} +230874. q^{49} -2.89054e6 q^{50} -873837. q^{52} +490289. q^{53} -1.05465e6 q^{55} +2.74159e6 q^{56} +570665. q^{58} -1.33623e6 q^{59} -1.79614e6 q^{61} -309077. q^{62} -1.72655e6 q^{64} -1.57347e6 q^{65} +2.54568e6 q^{67} +9.49355e6 q^{68} +9.61661e6 q^{70} -700407. q^{71} -4.69075e6 q^{73} -812403. q^{74} -9.90181e6 q^{76} +2.28664e6 q^{77} -4.98011e6 q^{79} +9.05950e6 q^{80} +1.25494e6 q^{82} +3.39027e6 q^{83} +1.70945e7 q^{85} +2.99898e6 q^{86} +5.94549e6 q^{88} +1.27747e6 q^{89} +3.41152e6 q^{91} -1.86377e7 q^{92} +1.41333e7 q^{94} -1.78297e7 q^{95} +7.65707e6 q^{97} -4.56536e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 438 q^{4} - 1932 q^{7} - 14922 q^{10} - 11886 q^{13} + 7314 q^{16} - 67164 q^{19} - 23364 q^{22} + 42324 q^{25} - 693348 q^{28} - 470832 q^{31} - 1834866 q^{34} - 1026258 q^{37} - 3091374 q^{40} - 1502268 q^{43}+ \cdots + 26307948 q^{97}+O(q^{100}) \) 6 * q + 438 * q^4 - 1932 * q^7 - 14922 * q^10 - 11886 * q^13 + 7314 * q^16 - 67164 * q^19 - 23364 * q^22 + 42324 * q^25 - 693348 * q^28 - 470832 * q^31 - 1834866 * q^34 - 1026258 * q^37 - 3091374 * q^40 - 1502268 * q^43 + 104580 * q^46 - 1769490 * q^49 - 2993334 * q^52 - 3667428 * q^55 + 659898 * q^58 - 4564770 * q^61 - 1047594 * q^64 + 2850060 * q^67 + 19286604 * q^70 - 1846842 * q^73 - 5520516 * q^76 - 2878188 * q^79 + 15068952 * q^82 + 32410098 * q^85 + 26598276 * q^88 + 16242108 * q^91 + 51428376 * q^94 + 26307948 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more