LMFDB - Embedded newform 8046.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 17x^{6} - 2x^{5} + 71x^{4} - 18x^{3} - 81x^{2} + 36x + 9 $ x^8 - 17x^6 - 2x^5 + 71x^4 - 18x^3 - 81x^2 + 36x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.17278$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.17278 q^{5} +1.96150 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.17278 q^{5} +1.96150 q^{7} +1.00000 q^{8} -1.17278 q^{10} -0.465557 q^{11} -2.12563 q^{13} +1.96150 q^{14} +1.00000 q^{16} +0.590988 q^{17} -7.81877 q^{19} -1.17278 q^{20} -0.465557 q^{22} +4.36934 q^{23} -3.62458 q^{25} -2.12563 q^{26} +1.96150 q^{28} -1.44210 q^{29} +9.39723 q^{31} +1.00000 q^{32} +0.590988 q^{34} -2.30041 q^{35} -3.39629 q^{37} -7.81877 q^{38} -1.17278 q^{40} -12.4113 q^{41} +4.71912 q^{43} -0.465557 q^{44} +4.36934 q^{46} +4.33354 q^{47} -3.15253 q^{49} -3.62458 q^{50} -2.12563 q^{52} -10.5164 q^{53} +0.545998 q^{55} +1.96150 q^{56} -1.44210 q^{58} -3.95347 q^{59} -10.4572 q^{61} +9.39723 q^{62} +1.00000 q^{64} +2.49291 q^{65} +0.763258 q^{67} +0.590988 q^{68} -2.30041 q^{70} +5.81196 q^{71} -7.40357 q^{73} -3.39629 q^{74} -7.81877 q^{76} -0.913187 q^{77} +0.711456 q^{79} -1.17278 q^{80} -12.4113 q^{82} +9.65399 q^{83} -0.693101 q^{85} +4.71912 q^{86} -0.465557 q^{88} +2.93347 q^{89} -4.16942 q^{91} +4.36934 q^{92} +4.33354 q^{94} +9.16973 q^{95} -6.58408 q^{97} -3.15253 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8	$ 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8} - 6 q^{11} - 8 q^{13} - 5 q^{14} + 8 q^{16} + 5 q^{17} - 14 q^{19} - 6 q^{22} - 21 q^{23} - 6 q^{25} - 8 q^{26} - 5 q^{28} - 3 q^{29} - 4 q^{31} + 8 q^{32} + 5 q^{34} + 2 q^{35} - 3 q^{37} - 14 q^{38} - 7 q^{41} - 12 q^{43} - 6 q^{44} - 21 q^{46} - 25 q^{47} - 7 q^{49} - 6 q^{50} - 8 q^{52} - 3 q^{53} - 9 q^{55} - 5 q^{56} - 3 q^{58} - 2 q^{59} - 17 q^{61} - 4 q^{62} + 8 q^{64} - 32 q^{65} - 14 q^{67} + 5 q^{68} + 2 q^{70} - 7 q^{71} - 10 q^{73} - 3 q^{74} - 14 q^{76} - 12 q^{77} - 33 q^{79} - 7 q^{82} - 13 q^{83} - 33 q^{85} - 12 q^{86} - 6 q^{88} + 22 q^{89} - 22 q^{91} - 21 q^{92} - 25 q^{94} + 14 q^{95} - 11 q^{97} - 7 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8 - 6 * q^11 - 8 * q^13 - 5 * q^14 + 8 * q^16 + 5 * q^17 - 14 * q^19 - 6 * q^22 - 21 * q^23 - 6 * q^25 - 8 * q^26 - 5 * q^28 - 3 * q^29 - 4 * q^31 + 8 * q^32 + 5 * q^34 + 2 * q^35 - 3 * q^37 - 14 * q^38 - 7 * q^41 - 12 * q^43 - 6 * q^44 - 21 * q^46 - 25 * q^47 - 7 * q^49 - 6 * q^50 - 8 * q^52 - 3 * q^53 - 9 * q^55 - 5 * q^56 - 3 * q^58 - 2 * q^59 - 17 * q^61 - 4 * q^62 + 8 * q^64 - 32 * q^65 - 14 * q^67 + 5 * q^68 + 2 * q^70 - 7 * q^71 - 10 * q^73 - 3 * q^74 - 14 * q^76 - 12 * q^77 - 33 * q^79 - 7 * q^82 - 13 * q^83 - 33 * q^85 - 12 * q^86 - 6 * q^88 + 22 * q^89 - 22 * q^91 - 21 * q^92 - 25 * q^94 + 14 * q^95 - 11 * q^97 - 7 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.17278	−0.524485	−0.262243	−	0.965002i	$-0.584462\pi$
$5$	−1.17278	−0.524485	−0.262243	+	0.965002i	$0.584462\pi$
$6$	0	0
$7$	1.96150	0.741376	0.370688	−	0.928757i	$-0.379122\pi$
$7$	1.96150	0.741376	0.370688	+	0.928757i	$0.379122\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.17278	−0.370867
$11$	−0.465557	−0.140371	−0.0701853	−	0.997534i	$-0.522359\pi$
$11$	−0.465557	−0.140371	−0.0701853	+	0.997534i	$0.522359\pi$
$12$	0	0
$13$	−2.12563	−0.589544	−0.294772	−	0.955568i	$-0.595244\pi$
$13$	−2.12563	−0.589544	−0.294772	+	0.955568i	$0.595244\pi$
$14$	1.96150	0.524232
$15$	0	0
$16$	1.00000	0.250000
$17$	0.590988	0.143336	0.0716678	−	0.997429i	$-0.477168\pi$
$17$	0.590988	0.143336	0.0716678	+	0.997429i	$0.477168\pi$
$18$	0	0
$19$	−7.81877	−1.79375	−0.896874	−	0.442286i	$-0.854168\pi$
$19$	−7.81877	−1.79375	−0.896874	+	0.442286i	$0.854168\pi$
$20$	−1.17278	−0.262243
$21$	0	0
$22$	−0.465557	−0.0992570
$23$	4.36934	0.911071	0.455535	−	0.890218i	$-0.349448\pi$
$23$	4.36934	0.911071	0.455535	+	0.890218i	$0.349448\pi$
$24$	0	0
$25$	−3.62458	−0.724915
$26$	−2.12563	−0.416871
$27$	0	0
$28$	1.96150	0.370688
$29$	−1.44210	−0.267791	−0.133896	−	0.990995i	$-0.542749\pi$
$29$	−1.44210	−0.267791	−0.133896	+	0.990995i	$0.542749\pi$
$30$	0	0
$31$	9.39723	1.68779	0.843897	−	0.536506i	$-0.180256\pi$
$31$	9.39723	1.68779	0.843897	+	0.536506i	$0.180256\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.590988	0.101354
$35$	−2.30041	−0.388841
$36$	0	0
$37$	−3.39629	−0.558347	−0.279173	−	0.960241i	$-0.590060\pi$
$37$	−3.39629	−0.558347	−0.279173	+	0.960241i	$0.590060\pi$
$38$	−7.81877	−1.26837
$39$	0	0
$40$	−1.17278	−0.185434
$41$	−12.4113	−1.93832	−0.969158	−	0.246440i	$-0.920739\pi$
$41$	−12.4113	−1.93832	−0.969158	+	0.246440i	$0.920739\pi$
$42$	0	0
$43$	4.71912	0.719659	0.359829	−	0.933018i	$-0.382835\pi$
$43$	4.71912	0.719659	0.359829	+	0.933018i	$0.382835\pi$
$44$	−0.465557	−0.0701853
$45$	0	0
$46$	4.36934	0.644224
$47$	4.33354	0.632112	0.316056	−	0.948741i	$-0.397641\pi$
$47$	4.33354	0.632112	0.316056	+	0.948741i	$0.397641\pi$
$48$	0	0
$49$	−3.15253	−0.450362
$50$	−3.62458	−0.512592
$51$	0	0
$52$	−2.12563	−0.294772
$53$	−10.5164	−1.44454	−0.722268	−	0.691613i	$-0.756900\pi$
$53$	−10.5164	−1.44454	−0.722268	+	0.691613i	$0.756900\pi$
$54$	0	0
$55$	0.545998	0.0736223
$56$	1.96150	0.262116
$57$	0	0
$58$	−1.44210	−0.189357
$59$	−3.95347	−0.514699	−0.257349	−	0.966318i	$-0.582849\pi$
$59$	−3.95347	−0.514699	−0.257349	+	0.966318i	$0.582849\pi$
$60$	0	0
$61$	−10.4572	−1.33890	−0.669452	−	0.742855i	$-0.733471\pi$
$61$	−10.4572	−1.33890	−0.669452	+	0.742855i	$0.733471\pi$
$62$	9.39723	1.19345
$63$	0	0
$64$	1.00000	0.125000
$65$	2.49291	0.309207
$66$	0	0
$67$	0.763258	0.0932468	0.0466234	−	0.998913i	$-0.485154\pi$
$67$	0.763258	0.0932468	0.0466234	+	0.998913i	$0.485154\pi$
$68$	0.590988	0.0716678
$69$	0	0
$70$	−2.30041	−0.274952
$71$	5.81196	0.689752	0.344876	−	0.938648i	$-0.387921\pi$
$71$	5.81196	0.689752	0.344876	+	0.938648i	$0.387921\pi$
$72$	0	0
$73$	−7.40357	−0.866522	−0.433261	−	0.901268i	$-0.642637\pi$
$73$	−7.40357	−0.866522	−0.433261	+	0.901268i	$0.642637\pi$
$74$	−3.39629	−0.394811
$75$	0	0
$76$	−7.81877	−0.896874
$77$	−0.913187	−0.104067
$78$	0	0
$79$	0.711456	0.0800450	0.0400225	−	0.999199i	$-0.487257\pi$
$79$	0.711456	0.0800450	0.0400225	+	0.999199i	$0.487257\pi$
$80$	−1.17278	−0.131121
$81$	0	0
$82$	−12.4113	−1.37060
$83$	9.65399	1.05966	0.529832	−	0.848103i	$-0.322255\pi$
$83$	9.65399	1.05966	0.529832	+	0.848103i	$0.322255\pi$
$84$	0	0
$85$	−0.693101	−0.0751774
$86$	4.71912	0.508876
$87$	0	0
$88$	−0.465557	−0.0496285
$89$	2.93347	0.310947	0.155474	−	0.987840i	$-0.450310\pi$
$89$	2.93347	0.310947	0.155474	+	0.987840i	$0.450310\pi$
$90$	0	0
$91$	−4.16942	−0.437074
$92$	4.36934	0.455535
$93$	0	0
$94$	4.33354	0.446971
$95$	9.16973	0.940795
$96$	0	0
$97$	−6.58408	−0.668512	−0.334256	−	0.942482i	$-0.608485\pi$
$97$	−6.58408	−0.668512	−0.334256	+	0.942482i	$0.608485\pi$
$98$	−3.15253	−0.318454
$99$	0	0
$100$	−3.62458	−0.362458
$101$	7.44638	0.740943	0.370471	−	0.928844i	$-0.379196\pi$
$101$	7.44638	0.740943	0.370471	+	0.928844i	$0.379196\pi$
$102$	0	0
$103$	−2.39959	−0.236439	−0.118219	−	0.992988i	$-0.537719\pi$
$103$	−2.39959	−0.236439	−0.118219	+	0.992988i	$0.537719\pi$
$104$	−2.12563	−0.208435
$105$	0	0
$106$	−10.5164	−1.02144
$107$	−10.9679	−1.06031	−0.530153	−	0.847902i	$-0.677865\pi$
$107$	−10.9679	−1.06031	−0.530153	+	0.847902i	$0.677865\pi$
$108$	0	0
$109$	−5.64880	−0.541057	−0.270529	−	0.962712i	$-0.587198\pi$
$109$	−5.64880	−0.541057	−0.270529	+	0.962712i	$0.587198\pi$
$110$	0.545998	0.0520588
$111$	0	0
$112$	1.96150	0.185344
$113$	0.782466	0.0736082	0.0368041	−	0.999323i	$-0.488282\pi$
$113$	0.782466	0.0736082	0.0368041	+	0.999323i	$0.488282\pi$
$114$	0	0
$115$	−5.12430	−0.477843
$116$	−1.44210	−0.133896
$117$	0	0
$118$	−3.95347	−0.363947
$119$	1.15922	0.106265
$120$	0	0
$121$	−10.7833	−0.980296
$122$	−10.4572	−0.946748
$123$	0	0
$124$	9.39723	0.843897
$125$	10.1148	0.904693
$126$	0	0
$127$	−13.9281	−1.23592	−0.617959	−	0.786210i	$-0.712040\pi$
$127$	−13.9281	−1.23592	−0.617959	+	0.786210i	$0.712040\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	2.49291	0.218643
$131$	19.8954	1.73827	0.869134	−	0.494576i	$-0.164677\pi$
$131$	19.8954	1.73827	0.869134	+	0.494576i	$0.164677\pi$
$132$	0	0
$133$	−15.3365	−1.32984
$134$	0.763258	0.0659354
$135$	0	0
$136$	0.590988	0.0506768
$137$	−13.8729	−1.18525	−0.592623	−	0.805480i	$-0.701907\pi$
$137$	−13.8729	−1.18525	−0.592623	+	0.805480i	$0.701907\pi$
$138$	0	0
$139$	−7.01887	−0.595333	−0.297666	−	0.954670i	$-0.596208\pi$
$139$	−7.01887	−0.595333	−0.297666	+	0.954670i	$0.596208\pi$
$140$	−2.30041	−0.194420
$141$	0	0
$142$	5.81196	0.487728
$143$	0.989602	0.0827547
$144$	0	0
$145$	1.69127	0.140453
$146$	−7.40357	−0.612724
$147$	0	0
$148$	−3.39629	−0.279173
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−12.5110	−1.01813	−0.509064	−	0.860728i	$-0.670008\pi$
$151$	−12.5110	−1.01813	−0.509064	+	0.860728i	$0.670008\pi$
$152$	−7.81877	−0.634186
$153$	0	0
$154$	−0.913187	−0.0735867
$155$	−11.0209	−0.885223
$156$	0	0
$157$	10.4396	0.833175	0.416587	−	0.909096i	$-0.363226\pi$
$157$	10.4396	0.833175	0.416587	+	0.909096i	$0.363226\pi$
$158$	0.711456	0.0566004
$159$	0	0
$160$	−1.17278	−0.0927168
$161$	8.57045	0.675446
$162$	0	0
$163$	4.26141	0.333779	0.166890	−	0.985976i	$-0.446628\pi$
$163$	4.26141	0.333779	0.166890	+	0.985976i	$0.446628\pi$
$164$	−12.4113	−0.969158
$165$	0	0
$166$	9.65399	0.749295
$167$	−5.65582	−0.437660	−0.218830	−	0.975763i	$-0.570224\pi$
$167$	−5.65582	−0.437660	−0.218830	+	0.975763i	$0.570224\pi$
$168$	0	0
$169$	−8.48169	−0.652438
$170$	−0.693101	−0.0531584
$171$	0	0
$172$	4.71912	0.359829
$173$	−13.0312	−0.990747	−0.495373	−	0.868680i	$-0.664969\pi$
$173$	−13.0312	−0.990747	−0.495373	+	0.868680i	$0.664969\pi$
$174$	0	0
$175$	−7.10959	−0.537435
$176$	−0.465557	−0.0350927
$177$	0	0
$178$	2.93347	0.219873
$179$	−12.3346	−0.921930	−0.460965	−	0.887418i	$-0.652497\pi$
$179$	−12.3346	−0.921930	−0.460965	+	0.887418i	$0.652497\pi$
$180$	0	0
$181$	8.05821	0.598962	0.299481	−	0.954102i	$-0.403187\pi$
$181$	8.05821	0.598962	0.299481	+	0.954102i	$0.403187\pi$
$182$	−4.16942	−0.309058
$183$	0	0
$184$	4.36934	0.322112
$185$	3.98312	0.292845
$186$	0	0
$187$	−0.275138	−0.0201201
$188$	4.33354	0.316056
$189$	0	0
$190$	9.16973	0.665242
$191$	−1.92514	−0.139298	−0.0696491	−	0.997572i	$-0.522188\pi$
$191$	−1.92514	−0.139298	−0.0696491	+	0.997572i	$0.522188\pi$
$192$	0	0
$193$	−9.91984	−0.714046	−0.357023	−	0.934096i	$-0.616208\pi$
$193$	−9.91984	−0.714046	−0.357023	+	0.934096i	$0.616208\pi$
$194$	−6.58408	−0.472709
$195$	0	0
$196$	−3.15253	−0.225181
$197$	12.1371	0.864733	0.432367	−	0.901698i	$-0.357679\pi$
$197$	12.1371	0.864733	0.432367	+	0.901698i	$0.357679\pi$
$198$	0	0
$199$	20.4028	1.44632	0.723158	−	0.690683i	$-0.242690\pi$
$199$	20.4028	1.44632	0.723158	+	0.690683i	$0.242690\pi$
$200$	−3.62458	−0.256296
$201$	0	0
$202$	7.44638	0.523926
$203$	−2.82867	−0.198534
$204$	0	0
$205$	14.5558	1.01662
$206$	−2.39959	−0.167187
$207$	0	0
$208$	−2.12563	−0.147386
$209$	3.64008	0.251790
$210$	0	0
$211$	−28.8908	−1.98893	−0.994463	−	0.105090i	$-0.966487\pi$
$211$	−28.8908	−1.98893	−0.994463	+	0.105090i	$0.966487\pi$
$212$	−10.5164	−0.722268
$213$	0	0
$214$	−10.9679	−0.749749
$215$	−5.53451	−0.377451
$216$	0	0
$217$	18.4326	1.25129
$218$	−5.64880	−0.382585
$219$	0	0
$220$	0.545998	0.0368112
$221$	−1.25622	−0.0845026
$222$	0	0
$223$	3.20529	0.214642	0.107321	−	0.994224i	$-0.465773\pi$
$223$	3.20529	0.214642	0.107321	+	0.994224i	$0.465773\pi$
$224$	1.96150	0.131058
$225$	0	0
$226$	0.782466	0.0520488
$227$	−16.9812	−1.12708	−0.563540	−	0.826089i	$-0.690561\pi$
$227$	−16.9812	−1.12708	−0.563540	+	0.826089i	$0.690561\pi$
$228$	0	0
$229$	−17.2500	−1.13991	−0.569957	−	0.821675i	$-0.693040\pi$
$229$	−17.2500	−1.13991	−0.569957	+	0.821675i	$0.693040\pi$
$230$	−5.12430	−0.337886
$231$	0	0
$232$	−1.44210	−0.0946785
$233$	23.4376	1.53545	0.767723	−	0.640782i	$-0.221390\pi$
$233$	23.4376	1.53545	0.767723	+	0.640782i	$0.221390\pi$
$234$	0	0
$235$	−5.08231	−0.331533
$236$	−3.95347	−0.257349
$237$	0	0
$238$	1.15922	0.0751411
$239$	18.7356	1.21190	0.605952	−	0.795502i	$-0.292793\pi$
$239$	18.7356	1.21190	0.605952	+	0.795502i	$0.292793\pi$
$240$	0	0
$241$	−5.08766	−0.327725	−0.163863	−	0.986483i	$-0.552395\pi$
$241$	−5.08766	−0.327725	−0.163863	+	0.986483i	$0.552395\pi$
$242$	−10.7833	−0.693174
$243$	0	0
$244$	−10.4572	−0.669452
$245$	3.69724	0.236208
$246$	0	0
$247$	16.6198	1.05749
$248$	9.39723	0.596725
$249$	0	0
$250$	10.1148	0.639714
$251$	17.2167	1.08671	0.543354	−	0.839504i	$-0.317154\pi$
$251$	17.2167	1.08671	0.543354	+	0.839504i	$0.317154\pi$
$252$	0	0
$253$	−2.03418	−0.127888
$254$	−13.9281	−0.873926
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.29216	0.142981	0.0714906	−	0.997441i	$-0.477224\pi$
$257$	2.29216	0.142981	0.0714906	+	0.997441i	$0.477224\pi$
$258$	0	0
$259$	−6.66181	−0.413945
$260$	2.49291	0.154604
$261$	0	0
$262$	19.8954	1.22914
$263$	−7.67339	−0.473161	−0.236581	−	0.971612i	$-0.576027\pi$
$263$	−7.67339	−0.473161	−0.236581	+	0.971612i	$0.576027\pi$
$264$	0	0
$265$	12.3335	0.757638
$266$	−15.3365	−0.940340
$267$	0	0
$268$	0.763258	0.0466234
$269$	2.18336	0.133122	0.0665608	−	0.997782i	$-0.478797\pi$
$269$	2.18336	0.133122	0.0665608	+	0.997782i	$0.478797\pi$
$270$	0	0
$271$	6.34926	0.385690	0.192845	−	0.981229i	$-0.438229\pi$
$271$	6.34926	0.385690	0.192845	+	0.981229i	$0.438229\pi$
$272$	0.590988	0.0358339
$273$	0	0
$274$	−13.8729	−0.838095
$275$	1.68745	0.101757
$276$	0	0
$277$	−0.0281068	−0.00168878	−0.000844388	−	1.00000i	$-0.500269\pi$
$277$	−0.0281068	−0.00168878	−0.000844388		1.00000i	$0.500269\pi$
$278$	−7.01887	−0.420964
$279$	0	0
$280$	−2.30041	−0.137476
$281$	−1.14739	−0.0684477	−0.0342238	−	0.999414i	$-0.510896\pi$
$281$	−1.14739	−0.0684477	−0.0342238	+	0.999414i	$0.510896\pi$
$282$	0	0
$283$	−12.3471	−0.733962	−0.366981	−	0.930229i	$-0.619609\pi$
$283$	−12.3471	−0.733962	−0.366981	+	0.930229i	$0.619609\pi$
$284$	5.81196	0.344876
$285$	0	0
$286$	0.989602	0.0585164
$287$	−24.3447	−1.43702
$288$	0	0
$289$	−16.6507	−0.979455
$290$	1.69127	0.0993149
$291$	0	0
$292$	−7.40357	−0.433261
$293$	11.9440	0.697778	0.348889	−	0.937164i	$-0.386559\pi$
$293$	11.9440	0.697778	0.348889	+	0.937164i	$0.386559\pi$
$294$	0	0
$295$	4.63658	0.269952
$296$	−3.39629	−0.197405
$297$	0	0
$298$	1.00000	0.0579284
$299$	−9.28761	−0.537116
$300$	0	0
$301$	9.25653	0.533538
$302$	−12.5110	−0.719926
$303$	0	0
$304$	−7.81877	−0.448437
$305$	12.2640	0.702236
$306$	0	0
$307$	9.99534	0.570464	0.285232	−	0.958458i	$-0.407929\pi$
$307$	9.99534	0.570464	0.285232	+	0.958458i	$0.407929\pi$
$308$	−0.913187	−0.0520337
$309$	0	0
$310$	−11.0209	−0.625947
$311$	29.3429	1.66388	0.831942	−	0.554862i	$-0.187229\pi$
$311$	29.3429	1.66388	0.831942	+	0.554862i	$0.187229\pi$
$312$	0	0
$313$	−0.470972	−0.0266209	−0.0133105	−	0.999911i	$-0.504237\pi$
$313$	−0.470972	−0.0266209	−0.0133105	+	0.999911i	$0.504237\pi$
$314$	10.4396	0.589143
$315$	0	0
$316$	0.711456	0.0400225
$317$	−12.0356	−0.675987	−0.337993	−	0.941148i	$-0.609748\pi$
$317$	−12.0356	−0.675987	−0.337993	+	0.941148i	$0.609748\pi$
$318$	0	0
$319$	0.671379	0.0375900
$320$	−1.17278	−0.0655607
$321$	0	0
$322$	8.57045	0.477612
$323$	−4.62079	−0.257108
$324$	0	0
$325$	7.70451	0.427369
$326$	4.26141	0.236018
$327$	0	0
$328$	−12.4113	−0.685298
$329$	8.50022	0.468633
$330$	0	0
$331$	1.30922	0.0719615	0.0359807	−	0.999352i	$-0.488545\pi$
$331$	1.30922	0.0719615	0.0359807	+	0.999352i	$0.488545\pi$
$332$	9.65399	0.529832
$333$	0	0
$334$	−5.65582	−0.309473
$335$	−0.895137	−0.0489066
$336$	0	0
$337$	−0.0393881	−0.00214561	−0.00107280	−	0.999999i	$-0.500341\pi$
$337$	−0.0393881	−0.00214561	−0.00107280	+	0.999999i	$0.500341\pi$
$338$	−8.48169	−0.461343
$339$	0	0
$340$	−0.693101	−0.0375887
$341$	−4.37494	−0.236917
$342$	0	0
$343$	−19.9142	−1.07526
$344$	4.71912	0.254438
$345$	0	0
$346$	−13.0312	−0.700564
$347$	−11.3284	−0.608139	−0.304069	−	0.952650i	$-0.598345\pi$
$347$	−11.3284	−0.608139	−0.304069	+	0.952650i	$0.598345\pi$
$348$	0	0
$349$	−13.9483	−0.746636	−0.373318	−	0.927704i	$-0.621780\pi$
$349$	−13.9483	−0.746636	−0.373318	+	0.927704i	$0.621780\pi$
$350$	−7.10959	−0.380024
$351$	0	0
$352$	−0.465557	−0.0248143
$353$	−30.0416	−1.59895	−0.799476	−	0.600697i	$-0.794890\pi$
$353$	−30.0416	−1.59895	−0.799476	+	0.600697i	$0.794890\pi$
$354$	0	0
$355$	−6.81617	−0.361765
$356$	2.93347	0.155474
$357$	0	0
$358$	−12.3346	−0.651903
$359$	−4.62666	−0.244186	−0.122093	−	0.992519i	$-0.538961\pi$
$359$	−4.62666	−0.244186	−0.122093	+	0.992519i	$0.538961\pi$
$360$	0	0
$361$	42.1331	2.21753
$362$	8.05821	0.423530
$363$	0	0
$364$	−4.16942	−0.218537
$365$	8.68279	0.454478
$366$	0	0
$367$	10.0899	0.526689	0.263344	−	0.964702i	$-0.415174\pi$
$367$	10.0899	0.526689	0.263344	+	0.964702i	$0.415174\pi$
$368$	4.36934	0.227768
$369$	0	0
$370$	3.98312	0.207072
$371$	−20.6278	−1.07094
$372$	0	0
$373$	−26.1388	−1.35342	−0.676708	−	0.736251i	$-0.736594\pi$
$373$	−26.1388	−1.35342	−0.676708	+	0.736251i	$0.736594\pi$
$374$	−0.275138	−0.0142271
$375$	0	0
$376$	4.33354	0.223485
$377$	3.06537	0.157875
$378$	0	0
$379$	−11.9330	−0.612958	−0.306479	−	0.951877i	$-0.599151\pi$
$379$	−11.9330	−0.612958	−0.306479	+	0.951877i	$0.599151\pi$
$380$	9.16973	0.470397
$381$	0	0
$382$	−1.92514	−0.0984988
$383$	35.9911	1.83906	0.919530	−	0.393021i	$-0.128570\pi$
$383$	35.9911	1.83906	0.919530	+	0.393021i	$0.128570\pi$
$384$	0	0
$385$	1.07097	0.0545818
$386$	−9.91984	−0.504907
$387$	0	0
$388$	−6.58408	−0.334256
$389$	−15.1761	−0.769459	−0.384729	−	0.923029i	$-0.625705\pi$
$389$	−15.1761	−0.769459	−0.384729	+	0.923029i	$0.625705\pi$
$390$	0	0
$391$	2.58223	0.130589
$392$	−3.15253	−0.159227
$393$	0	0
$394$	12.1371	0.611459
$395$	−0.834384	−0.0419824
$396$	0	0
$397$	−13.6990	−0.687534	−0.343767	−	0.939055i	$-0.611703\pi$
$397$	−13.6990	−0.687534	−0.343767	+	0.939055i	$0.611703\pi$
$398$	20.4028	1.02270
$399$	0	0
$400$	−3.62458	−0.181229
$401$	17.5024	0.874026	0.437013	−	0.899455i	$-0.356036\pi$
$401$	17.5024	0.874026	0.437013	+	0.899455i	$0.356036\pi$
$402$	0	0
$403$	−19.9751	−0.995028
$404$	7.44638	0.370471
$405$	0	0
$406$	−2.82867	−0.140385
$407$	1.58117	0.0783755
$408$	0	0
$409$	14.4353	0.713781	0.356890	−	0.934146i	$-0.383837\pi$
$409$	14.4353	0.713781	0.356890	+	0.934146i	$0.383837\pi$
$410$	14.5558	0.718858
$411$	0	0
$412$	−2.39959	−0.118219
$413$	−7.75473	−0.381585
$414$	0	0
$415$	−11.3221	−0.555778
$416$	−2.12563	−0.104218
$417$	0	0
$418$	3.64008	0.178042
$419$	−29.4434	−1.43840	−0.719201	−	0.694802i	$-0.755492\pi$
$419$	−29.4434	−1.43840	−0.719201	+	0.694802i	$0.755492\pi$
$420$	0	0
$421$	13.4926	0.657588	0.328794	−	0.944402i	$-0.393358\pi$
$421$	13.4926	0.657588	0.328794	+	0.944402i	$0.393358\pi$
$422$	−28.8908	−1.40638
$423$	0	0
$424$	−10.5164	−0.510721
$425$	−2.14208	−0.103906
$426$	0	0
$427$	−20.5117	−0.992631
$428$	−10.9679	−0.530153
$429$	0	0
$430$	−5.53451	−0.266898
$431$	−21.4167	−1.03161	−0.515803	−	0.856707i	$-0.672507\pi$
$431$	−21.4167	−1.03161	−0.515803	+	0.856707i	$0.672507\pi$
$432$	0	0
$433$	−34.2202	−1.64452	−0.822258	−	0.569114i	$-0.807286\pi$
$433$	−34.2202	−1.64452	−0.822258	+	0.569114i	$0.807286\pi$
$434$	18.4326	0.884795
$435$	0	0
$436$	−5.64880	−0.270529
$437$	−34.1629	−1.63423
$438$	0	0
$439$	26.8765	1.28274	0.641372	−	0.767230i	$-0.278365\pi$
$439$	26.8765	1.28274	0.641372	+	0.767230i	$0.278365\pi$
$440$	0.545998	0.0260294
$441$	0	0
$442$	−1.25622	−0.0597524
$443$	−7.02870	−0.333944	−0.166972	−	0.985962i	$-0.553399\pi$
$443$	−7.02870	−0.333944	−0.166972	+	0.985962i	$0.553399\pi$
$444$	0	0
$445$	−3.44033	−0.163087
$446$	3.20529	0.151775
$447$	0	0
$448$	1.96150	0.0926720
$449$	15.9065	0.750672	0.375336	−	0.926889i	$-0.377527\pi$
$449$	15.9065	0.750672	0.375336	+	0.926889i	$0.377527\pi$
$450$	0	0
$451$	5.77815	0.272083
$452$	0.782466	0.0368041
$453$	0	0
$454$	−16.9812	−0.796966
$455$	4.88983	0.229239
$456$	0	0
$457$	−13.2094	−0.617911	−0.308955	−	0.951077i	$-0.599979\pi$
$457$	−13.2094	−0.617911	−0.308955	+	0.951077i	$0.599979\pi$
$458$	−17.2500	−0.806040
$459$	0	0
$460$	−5.12430	−0.238922
$461$	−5.10499	−0.237763	−0.118881	−	0.992908i	$-0.537931\pi$
$461$	−5.10499	−0.237763	−0.118881	+	0.992908i	$0.537931\pi$
$462$	0	0
$463$	−38.4668	−1.78771	−0.893853	−	0.448361i	$-0.852008\pi$
$463$	−38.4668	−1.78771	−0.893853	+	0.448361i	$0.852008\pi$
$464$	−1.44210	−0.0669478
$465$	0	0
$466$	23.4376	1.08572
$467$	−31.8805	−1.47525	−0.737627	−	0.675209i	$-0.764054\pi$
$467$	−31.8805	−1.47525	−0.737627	+	0.675209i	$0.764054\pi$
$468$	0	0
$469$	1.49713	0.0691309
$470$	−5.08231	−0.234430
$471$	0	0
$472$	−3.95347	−0.181973
$473$	−2.19702	−0.101019
$474$	0	0
$475$	28.3397	1.30032
$476$	1.15922	0.0531327
$477$	0	0
$478$	18.7356	0.856945
$479$	21.9064	1.00093	0.500466	−	0.865756i	$-0.333162\pi$
$479$	21.9064	1.00093	0.500466	+	0.865756i	$0.333162\pi$
$480$	0	0
$481$	7.21926	0.329170
$482$	−5.08766	−0.231737
$483$	0	0
$484$	−10.7833	−0.490148
$485$	7.72170	0.350625
$486$	0	0
$487$	−9.48325	−0.429727	−0.214863	−	0.976644i	$-0.568931\pi$
$487$	−9.48325	−0.429727	−0.214863	+	0.976644i	$0.568931\pi$
$488$	−10.4572	−0.473374
$489$	0	0
$490$	3.69724	0.167024
$491$	−8.06415	−0.363930	−0.181965	−	0.983305i	$-0.558246\pi$
$491$	−8.06415	−0.363930	−0.181965	+	0.983305i	$0.558246\pi$
$492$	0	0
$493$	−0.852263	−0.0383840
$494$	16.6198	0.747761
$495$	0	0
$496$	9.39723	0.421948
$497$	11.4001	0.511366
$498$	0	0
$499$	8.27684	0.370522	0.185261	−	0.982689i	$-0.440687\pi$
$499$	8.27684	0.370522	0.185261	+	0.982689i	$0.440687\pi$
$500$	10.1148	0.452346
$501$	0	0
$502$	17.2167	0.768419
$503$	30.8305	1.37466	0.687331	−	0.726344i	$-0.258782\pi$
$503$	30.8305	1.37466	0.687331	+	0.726344i	$0.258782\pi$
$504$	0	0
$505$	−8.73301	−0.388614
$506$	−2.03418	−0.0904301
$507$	0	0
$508$	−13.9281	−0.617959
$509$	−12.9950	−0.575993	−0.287996	−	0.957631i	$-0.592989\pi$
$509$	−12.9950	−0.575993	−0.287996	+	0.957631i	$0.592989\pi$
$510$	0	0
$511$	−14.5221	−0.642419
$512$	1.00000	0.0441942
$513$	0	0
$514$	2.29216	0.101103
$515$	2.81420	0.124009
$516$	0	0
$517$	−2.01751	−0.0887299
$518$	−6.66181	−0.292703
$519$	0	0
$520$	2.49291	0.109321
$521$	19.1553	0.839209	0.419604	−	0.907707i	$-0.362169\pi$
$521$	19.1553	0.839209	0.419604	+	0.907707i	$0.362169\pi$
$522$	0	0
$523$	9.99372	0.436995	0.218497	−	0.975838i	$-0.429884\pi$
$523$	9.99372	0.436995	0.218497	+	0.975838i	$0.429884\pi$
$524$	19.8954	0.869134
$525$	0	0
$526$	−7.67339	−0.334576
$527$	5.55365	0.241921
$528$	0	0
$529$	−3.90886	−0.169950
$530$	12.3335	0.535731
$531$	0	0
$532$	−15.3365	−0.664921
$533$	26.3818	1.14272
$534$	0	0
$535$	12.8630	0.556115
$536$	0.763258	0.0329677
$537$	0	0
$538$	2.18336	0.0941312
$539$	1.46768	0.0632176
$540$	0	0
$541$	41.7951	1.79691	0.898456	−	0.439064i	$-0.144690\pi$
$541$	41.7951	1.79691	0.898456	+	0.439064i	$0.144690\pi$
$542$	6.34926	0.272724
$543$	0	0
$544$	0.590988	0.0253384
$545$	6.62483	0.283776
$546$	0	0
$547$	−23.8527	−1.01987	−0.509933	−	0.860214i	$-0.670330\pi$
$547$	−23.8527	−1.01987	−0.509933	+	0.860214i	$0.670330\pi$
$548$	−13.8729	−0.592623
$549$	0	0
$550$	1.68745	0.0719529
$551$	11.2754	0.480350
$552$	0	0
$553$	1.39552	0.0593434
$554$	−0.0281068	−0.00119415
$555$	0	0
$556$	−7.01887	−0.297666
$557$	9.13595	0.387103	0.193551	−	0.981090i	$-0.437999\pi$
$557$	9.13595	0.387103	0.193551	+	0.981090i	$0.437999\pi$
$558$	0	0
$559$	−10.0311	−0.424271
$560$	−2.30041	−0.0972102
$561$	0	0
$562$	−1.14739	−0.0483998
$563$	10.2441	0.431736	0.215868	−	0.976423i	$-0.430742\pi$
$563$	10.2441	0.431736	0.215868	+	0.976423i	$0.430742\pi$
$564$	0	0
$565$	−0.917664	−0.0386064
$566$	−12.3471	−0.518989
$567$	0	0
$568$	5.81196	0.243864
$569$	−41.0641	−1.72149	−0.860747	−	0.509032i	$-0.830003\pi$
$569$	−41.0641	−1.72149	−0.860747	+	0.509032i	$0.830003\pi$
$570$	0	0
$571$	1.24224	0.0519863	0.0259931	−	0.999662i	$-0.491725\pi$
$571$	1.24224	0.0519863	0.0259931	+	0.999662i	$0.491725\pi$
$572$	0.989602	0.0413773
$573$	0	0
$574$	−24.3447	−1.01613
$575$	−15.8370	−0.660449
$576$	0	0
$577$	17.3479	0.722204	0.361102	−	0.932526i	$-0.382401\pi$
$577$	17.3479	0.722204	0.361102	+	0.932526i	$0.382401\pi$
$578$	−16.6507	−0.692579
$579$	0	0
$580$	1.69127	0.0702263
$581$	18.9363	0.785609
$582$	0	0
$583$	4.89597	0.202770
$584$	−7.40357	−0.306362
$585$	0	0
$586$	11.9440	0.493404
$587$	−8.23024	−0.339698	−0.169849	−	0.985470i	$-0.554328\pi$
$587$	−8.23024	−0.339698	−0.169849	+	0.985470i	$0.554328\pi$
$588$	0	0
$589$	−73.4748	−3.02748
$590$	4.63658	0.190885
$591$	0	0
$592$	−3.39629	−0.139587
$593$	−13.8569	−0.569036	−0.284518	−	0.958671i	$-0.591834\pi$
$593$	−13.8569	−0.569036	−0.284518	+	0.958671i	$0.591834\pi$
$594$	0	0
$595$	−1.35952	−0.0557347
$596$	1.00000	0.0409616
$597$	0	0
$598$	−9.28761	−0.379799
$599$	36.6430	1.49719	0.748597	−	0.663025i	$-0.230728\pi$
$599$	36.6430	1.49719	0.748597	+	0.663025i	$0.230728\pi$
$600$	0	0
$601$	−10.5595	−0.430732	−0.215366	−	0.976533i	$-0.569094\pi$
$601$	−10.5595	−0.430732	−0.215366	+	0.976533i	$0.569094\pi$
$602$	9.25653	0.377268
$603$	0	0
$604$	−12.5110	−0.509064
$605$	12.6464	0.514151
$606$	0	0
$607$	45.0078	1.82681	0.913405	−	0.407051i	$-0.133443\pi$
$607$	45.0078	1.82681	0.913405	+	0.407051i	$0.133443\pi$
$608$	−7.81877	−0.317093
$609$	0	0
$610$	12.2640	0.496556
$611$	−9.21151	−0.372658
$612$	0	0
$613$	−3.42147	−0.138192	−0.0690960	−	0.997610i	$-0.522011\pi$
$613$	−3.42147	−0.138192	−0.0690960	+	0.997610i	$0.522011\pi$
$614$	9.99534	0.403379
$615$	0	0
$616$	−0.913187	−0.0367934
$617$	13.3906	0.539085	0.269542	−	0.962989i	$-0.413128\pi$
$617$	13.3906	0.539085	0.269542	+	0.962989i	$0.413128\pi$
$618$	0	0
$619$	47.6598	1.91561	0.957805	−	0.287418i	$-0.0927968\pi$
$619$	47.6598	1.91561	0.957805	+	0.287418i	$0.0927968\pi$
$620$	−11.0209	−0.442611
$621$	0	0
$622$	29.3429	1.17654
$623$	5.75399	0.230529
$624$	0	0
$625$	6.26043	0.250417
$626$	−0.470972	−0.0188238
$627$	0	0
$628$	10.4396	0.416587
$629$	−2.00717	−0.0800309
$630$	0	0
$631$	19.0743	0.759335	0.379668	−	0.925123i	$-0.376038\pi$
$631$	19.0743	0.759335	0.379668	+	0.925123i	$0.376038\pi$
$632$	0.711456	0.0283002
$633$	0	0
$634$	−12.0356	−0.477995
$635$	16.3347	0.648221
$636$	0	0
$637$	6.70112	0.265508
$638$	0.671379	0.0265802
$639$	0	0
$640$	−1.17278	−0.0463584
$641$	−12.1369	−0.479379	−0.239690	−	0.970850i	$-0.577046\pi$
$641$	−12.1369	−0.479379	−0.239690	+	0.970850i	$0.577046\pi$
$642$	0	0
$643$	−20.4358	−0.805910	−0.402955	−	0.915220i	$-0.632017\pi$
$643$	−20.4358	−0.805910	−0.402955	+	0.915220i	$0.632017\pi$
$644$	8.57045	0.337723
$645$	0	0
$646$	−4.62079	−0.181803
$647$	37.7865	1.48554	0.742770	−	0.669547i	$-0.233512\pi$
$647$	37.7865	1.48554	0.742770	+	0.669547i	$0.233512\pi$
$648$	0	0
$649$	1.84057	0.0722486
$650$	7.70451	0.302196
$651$	0	0
$652$	4.26141	0.166890
$653$	7.97309	0.312011	0.156006	−	0.987756i	$-0.450138\pi$
$653$	7.97309	0.312011	0.156006	+	0.987756i	$0.450138\pi$
$654$	0	0
$655$	−23.3330	−0.911697
$656$	−12.4113	−0.484579
$657$	0	0
$658$	8.50022	0.331373
$659$	−26.8034	−1.04411	−0.522056	−	0.852911i	$-0.674835\pi$
$659$	−26.8034	−1.04411	−0.522056	+	0.852911i	$0.674835\pi$
$660$	0	0
$661$	−37.1346	−1.44437	−0.722184	−	0.691701i	$-0.756862\pi$
$661$	−37.1346	−1.44437	−0.722184	+	0.691701i	$0.756862\pi$
$662$	1.30922	0.0508844
$663$	0	0
$664$	9.65399	0.374648
$665$	17.9864	0.697482
$666$	0	0
$667$	−6.30103	−0.243977
$668$	−5.65582	−0.218830
$669$	0	0
$670$	−0.895137	−0.0345822
$671$	4.86841	0.187943
$672$	0	0
$673$	−20.2765	−0.781603	−0.390802	−	0.920475i	$-0.627802\pi$
$673$	−20.2765	−0.781603	−0.390802	+	0.920475i	$0.627802\pi$
$674$	−0.0393881	−0.00151717
$675$	0	0
$676$	−8.48169	−0.326219
$677$	0.221466	0.00851163	0.00425582	−	0.999991i	$-0.498645\pi$
$677$	0.221466	0.00851163	0.00425582	+	0.999991i	$0.498645\pi$
$678$	0	0
$679$	−12.9146	−0.495618
$680$	−0.693101	−0.0265792
$681$	0	0
$682$	−4.37494	−0.167525
$683$	−47.7257	−1.82617	−0.913086	−	0.407767i	$-0.866307\pi$
$683$	−47.7257	−1.82617	−0.913086	+	0.407767i	$0.866307\pi$
$684$	0	0
$685$	16.2700	0.621644
$686$	−19.9142	−0.760326
$687$	0	0
$688$	4.71912	0.179915
$689$	22.3539	0.851618
$690$	0	0
$691$	−15.9678	−0.607445	−0.303723	−	0.952760i	$-0.598230\pi$
$691$	−15.9678	−0.607445	−0.303723	+	0.952760i	$0.598230\pi$
$692$	−13.0312	−0.495373
$693$	0	0
$694$	−11.3284	−0.430019
$695$	8.23162	0.312243
$696$	0	0
$697$	−7.33491	−0.277830
$698$	−13.9483	−0.527951
$699$	0	0
$700$	−7.10959	−0.268717
$701$	−1.42727	−0.0539073	−0.0269537	−	0.999637i	$-0.508581\pi$
$701$	−1.42727	−0.0539073	−0.0269537	+	0.999637i	$0.508581\pi$
$702$	0	0
$703$	26.5548	1.00153
$704$	−0.465557	−0.0175463
$705$	0	0
$706$	−30.0416	−1.13063
$707$	14.6061	0.549317
$708$	0	0
$709$	−4.82990	−0.181391	−0.0906954	−	0.995879i	$-0.528909\pi$
$709$	−4.82990	−0.181391	−0.0906954	+	0.995879i	$0.528909\pi$
$710$	−6.81617	−0.255806
$711$	0	0
$712$	2.93347	0.109936
$713$	41.0597	1.53770
$714$	0	0
$715$	−1.16059	−0.0434036
$716$	−12.3346	−0.460965
$717$	0	0
$718$	−4.62666	−0.172665
$719$	29.6646	1.10630	0.553151	−	0.833081i	$-0.313425\pi$
$719$	29.6646	1.10630	0.553151	+	0.833081i	$0.313425\pi$
$720$	0	0
$721$	−4.70679	−0.175290
$722$	42.1331	1.56803
$723$	0	0
$724$	8.05821	0.299481
$725$	5.22700	0.194126
$726$	0	0
$727$	5.34474	0.198225	0.0991127	−	0.995076i	$-0.468400\pi$
$727$	5.34474	0.198225	0.0991127	+	0.995076i	$0.468400\pi$
$728$	−4.16942	−0.154529
$729$	0	0
$730$	8.68279	0.321365
$731$	2.78894	0.103153
$732$	0	0
$733$	39.2738	1.45061	0.725306	−	0.688427i	$-0.241698\pi$
$733$	39.2738	1.45061	0.725306	+	0.688427i	$0.241698\pi$
$734$	10.0899	0.372425
$735$	0	0
$736$	4.36934	0.161056
$737$	−0.355340	−0.0130891
$738$	0	0
$739$	−17.3998	−0.640062	−0.320031	−	0.947407i	$-0.603693\pi$
$739$	−17.3998	−0.640062	−0.320031	+	0.947407i	$0.603693\pi$
$740$	3.98312	0.146422
$741$	0	0
$742$	−20.6278	−0.757272
$743$	20.6448	0.757384	0.378692	−	0.925523i	$-0.376374\pi$
$743$	20.6448	0.757384	0.378692	+	0.925523i	$0.376374\pi$
$744$	0	0
$745$	−1.17278	−0.0429675
$746$	−26.1388	−0.957010
$747$	0	0
$748$	−0.275138	−0.0100600
$749$	−21.5135	−0.786085
$750$	0	0
$751$	−21.7970	−0.795385	−0.397692	−	0.917519i	$-0.630189\pi$
$751$	−21.7970	−0.795385	−0.397692	+	0.917519i	$0.630189\pi$
$752$	4.33354	0.158028
$753$	0	0
$754$	3.06537	0.111634
$755$	14.6727	0.533994
$756$	0	0
$757$	−25.2702	−0.918462	−0.459231	−	0.888317i	$-0.651875\pi$
$757$	−25.2702	−0.918462	−0.459231	+	0.888317i	$0.651875\pi$
$758$	−11.9330	−0.433427
$759$	0	0
$760$	9.16973	0.332621
$761$	−19.5264	−0.707832	−0.353916	−	0.935277i	$-0.615150\pi$
$761$	−19.5264	−0.707832	−0.353916	+	0.935277i	$0.615150\pi$
$762$	0	0
$763$	−11.0801	−0.401127
$764$	−1.92514	−0.0696491
$765$	0	0
$766$	35.9911	1.30041
$767$	8.40363	0.303438
$768$	0	0
$769$	−23.6922	−0.854364	−0.427182	−	0.904166i	$-0.640494\pi$
$769$	−23.6922	−0.854364	−0.427182	+	0.904166i	$0.640494\pi$
$770$	1.07097	0.0385952
$771$	0	0
$772$	−9.91984	−0.357023
$773$	33.0112	1.18733	0.593665	−	0.804713i	$-0.297681\pi$
$773$	33.0112	1.18733	0.593665	+	0.804713i	$0.297681\pi$
$774$	0	0
$775$	−34.0610	−1.22351
$776$	−6.58408	−0.236355
$777$	0	0
$778$	−15.1761	−0.544090
$779$	97.0409	3.47685
$780$	0	0
$781$	−2.70579	−0.0968209
$782$	2.58223	0.0923402
$783$	0	0
$784$	−3.15253	−0.112590
$785$	−12.2435	−0.436988
$786$	0	0
$787$	14.8063	0.527789	0.263894	−	0.964552i	$-0.414993\pi$
$787$	14.8063	0.527789	0.263894	+	0.964552i	$0.414993\pi$
$788$	12.1371	0.432367
$789$	0	0
$790$	−0.834384	−0.0296861
$791$	1.53480	0.0545713
$792$	0	0
$793$	22.2281	0.789343
$794$	−13.6990	−0.486160
$795$	0	0
$796$	20.4028	0.723158
$797$	−31.3982	−1.11218	−0.556090	−	0.831122i	$-0.687699\pi$
$797$	−31.3982	−1.11218	−0.556090	+	0.831122i	$0.687699\pi$
$798$	0	0
$799$	2.56107	0.0906041
$800$	−3.62458	−0.128148
$801$	0	0
$802$	17.5024	0.618030
$803$	3.44678	0.121634
$804$	0	0
$805$	−10.0513	−0.354261
$806$	−19.9751	−0.703591
$807$	0	0
$808$	7.44638	0.261963
$809$	−6.70143	−0.235610	−0.117805	−	0.993037i	$-0.537586\pi$
$809$	−6.70143	−0.235610	−0.117805	+	0.993037i	$0.537586\pi$
$810$	0	0
$811$	17.2813	0.606828	0.303414	−	0.952859i	$-0.401874\pi$
$811$	17.2813	0.606828	0.303414	+	0.952859i	$0.401874\pi$
$812$	−2.82867	−0.0992670
$813$	0	0
$814$	1.58117	0.0554198
$815$	−4.99771	−0.175062
$816$	0	0
$817$	−36.8977	−1.29089
$818$	14.4353	0.504719
$819$	0	0
$820$	14.5558	0.508309
$821$	46.6670	1.62869	0.814346	−	0.580380i	$-0.197096\pi$
$821$	46.6670	1.62869	0.814346	+	0.580380i	$0.197096\pi$
$822$	0	0
$823$	41.4086	1.44341	0.721706	−	0.692200i	$-0.243358\pi$
$823$	41.4086	1.44341	0.721706	+	0.692200i	$0.243358\pi$
$824$	−2.39959	−0.0835937
$825$	0	0
$826$	−7.75473	−0.269821
$827$	−5.78285	−0.201089	−0.100545	−	0.994933i	$-0.532059\pi$
$827$	−5.78285	−0.201089	−0.100545	+	0.994933i	$0.532059\pi$
$828$	0	0
$829$	−4.00949	−0.139255	−0.0696277	−	0.997573i	$-0.522181\pi$
$829$	−4.00949	−0.139255	−0.0696277	+	0.997573i	$0.522181\pi$
$830$	−11.3221	−0.392994
$831$	0	0
$832$	−2.12563	−0.0736930
$833$	−1.86311	−0.0645529
$834$	0	0
$835$	6.63306	0.229546
$836$	3.64008	0.125895
$837$	0	0
$838$	−29.4434	−1.01710
$839$	25.0132	0.863552	0.431776	−	0.901981i	$-0.357887\pi$
$839$	25.0132	0.863552	0.431776	+	0.901981i	$0.357887\pi$
$840$	0	0
$841$	−26.9203	−0.928288
$842$	13.4926	0.464985
$843$	0	0
$844$	−28.8908	−0.994463
$845$	9.94720	0.342194
$846$	0	0
$847$	−21.1513	−0.726768
$848$	−10.5164	−0.361134
$849$	0	0
$850$	−2.14208	−0.0734727
$851$	−14.8396	−0.508693
$852$	0	0
$853$	−55.5157	−1.90082	−0.950410	−	0.310998i	$-0.899337\pi$
$853$	−55.5157	−1.90082	−0.950410	+	0.310998i	$0.899337\pi$
$854$	−20.5117	−0.701896
$855$	0	0
$856$	−10.9679	−0.374875
$857$	21.1229	0.721544	0.360772	−	0.932654i	$-0.382513\pi$
$857$	21.1229	0.721544	0.360772	+	0.932654i	$0.382513\pi$
$858$	0	0
$859$	−37.0273	−1.26335	−0.631677	−	0.775232i	$-0.717633\pi$
$859$	−37.0273	−1.26335	−0.631677	+	0.775232i	$0.717633\pi$
$860$	−5.53451	−0.188725
$861$	0	0
$862$	−21.4167	−0.729456
$863$	−12.8278	−0.436664	−0.218332	−	0.975874i	$-0.570062\pi$
$863$	−12.8278	−0.436664	−0.218332	+	0.975874i	$0.570062\pi$
$864$	0	0
$865$	15.2828	0.519632
$866$	−34.2202	−1.16285
$867$	0	0
$868$	18.4326	0.625644
$869$	−0.331223	−0.0112360
$870$	0	0
$871$	−1.62240	−0.0549731
$872$	−5.64880	−0.191293
$873$	0	0
$874$	−34.1629	−1.15558
$875$	19.8401	0.670717
$876$	0	0
$877$	22.2849	0.752506	0.376253	−	0.926517i	$-0.377212\pi$
$877$	22.2849	0.752506	0.376253	+	0.926517i	$0.377212\pi$
$878$	26.8765	0.907038
$879$	0	0
$880$	0.545998	0.0184056
$881$	46.3434	1.56135	0.780675	−	0.624937i	$-0.214875\pi$
$881$	46.3434	1.56135	0.780675	+	0.624937i	$0.214875\pi$
$882$	0	0
$883$	47.6811	1.60460	0.802298	−	0.596923i	$-0.203610\pi$
$883$	47.6811	1.60460	0.802298	+	0.596923i	$0.203610\pi$
$884$	−1.25622	−0.0422513
$885$	0	0
$886$	−7.02870	−0.236134
$887$	−49.4448	−1.66019	−0.830097	−	0.557620i	$-0.811715\pi$
$887$	−49.4448	−1.66019	−0.830097	+	0.557620i	$0.811715\pi$
$888$	0	0
$889$	−27.3199	−0.916280
$890$	−3.44033	−0.115320
$891$	0	0
$892$	3.20529	0.107321
$893$	−33.8830	−1.13385
$894$	0	0
$895$	14.4658	0.483539
$896$	1.96150	0.0655290
$897$	0	0
$898$	15.9065	0.530806
$899$	−13.5517	−0.451976
$900$	0	0
$901$	−6.21505	−0.207053
$902$	5.77815	0.192391
$903$	0	0
$904$	0.782466	0.0260244
$905$	−9.45054	−0.314147
$906$	0	0
$907$	−13.9456	−0.463058	−0.231529	−	0.972828i	$-0.574373\pi$
$907$	−13.9456	−0.463058	−0.231529	+	0.972828i	$0.574373\pi$
$908$	−16.9812	−0.563540
$909$	0	0
$910$	4.88983	0.162096
$911$	−22.4420	−0.743536	−0.371768	−	0.928326i	$-0.621248\pi$
$911$	−22.4420	−0.743536	−0.371768	+	0.928326i	$0.621248\pi$
$912$	0	0
$913$	−4.49448	−0.148746
$914$	−13.2094	−0.436929
$915$	0	0
$916$	−17.2500	−0.569957
$917$	39.0247	1.28871
$918$	0	0
$919$	52.6902	1.73809	0.869044	−	0.494735i	$-0.164735\pi$
$919$	52.6902	1.73809	0.869044	+	0.494735i	$0.164735\pi$
$920$	−5.12430	−0.168943
$921$	0	0
$922$	−5.10499	−0.168124
$923$	−12.3541	−0.406639
$924$	0	0
$925$	12.3101	0.404754
$926$	−38.4668	−1.26410
$927$	0	0
$928$	−1.44210	−0.0473392
$929$	7.70208	0.252697	0.126348	−	0.991986i	$-0.459674\pi$
$929$	7.70208	0.252697	0.126348	+	0.991986i	$0.459674\pi$
$930$	0	0
$931$	24.6489	0.807836
$932$	23.4376	0.767723
$933$	0	0
$934$	−31.8805	−1.04316
$935$	0.322678	0.0105527
$936$	0	0
$937$	55.1850	1.80282	0.901408	−	0.432970i	$-0.142534\pi$
$937$	55.1850	1.80282	0.901408	+	0.432970i	$0.142534\pi$
$938$	1.49713	0.0488829
$939$	0	0
$940$	−5.08231	−0.165767
$941$	16.6499	0.542771	0.271385	−	0.962471i	$-0.412518\pi$
$941$	16.6499	0.542771	0.271385	+	0.962471i	$0.412518\pi$
$942$	0	0
$943$	−54.2291	−1.76594
$944$	−3.95347	−0.128675
$945$	0	0
$946$	−2.19702	−0.0714312
$947$	3.22190	0.104698	0.0523489	−	0.998629i	$-0.483329\pi$
$947$	3.22190	0.104698	0.0523489	+	0.998629i	$0.483329\pi$
$948$	0	0
$949$	15.7373	0.510853
$950$	28.3397	0.919462
$951$	0	0
$952$	1.15922	0.0375705
$953$	−4.46386	−0.144599	−0.0722993	−	0.997383i	$-0.523034\pi$
$953$	−4.46386	−0.144599	−0.0722993	+	0.997383i	$0.523034\pi$
$954$	0	0
$955$	2.25778	0.0730599
$956$	18.7356	0.605952
$957$	0	0
$958$	21.9064	0.707765
$959$	−27.2117	−0.878712
$960$	0	0
$961$	57.3080	1.84865
$962$	7.21926	0.232758
$963$	0	0
$964$	−5.08766	−0.163863
$965$	11.6338	0.374506
$966$	0	0
$967$	−25.1333	−0.808233	−0.404117	−	0.914707i	$-0.632421\pi$
$967$	−25.1333	−0.808233	−0.404117	+	0.914707i	$0.632421\pi$
$968$	−10.7833	−0.346587
$969$	0	0
$970$	7.72170	0.247929
$971$	34.6186	1.11096	0.555482	−	0.831529i	$-0.312534\pi$
$971$	34.6186	1.11096	0.555482	+	0.831529i	$0.312534\pi$
$972$	0	0
$973$	−13.7675	−0.441365
$974$	−9.48325	−0.303863
$975$	0	0
$976$	−10.4572	−0.334726
$977$	7.41543	0.237241	0.118620	−	0.992940i	$-0.462153\pi$
$977$	7.41543	0.237241	0.118620	+	0.992940i	$0.462153\pi$
$978$	0	0
$979$	−1.36570	−0.0436478
$980$	3.69724	0.118104
$981$	0	0
$982$	−8.06415	−0.257337
$983$	−1.33151	−0.0424686	−0.0212343	−	0.999775i	$-0.506760\pi$
$983$	−1.33151	−0.0424686	−0.0212343	+	0.999775i	$0.506760\pi$
$984$	0	0
$985$	−14.2342	−0.453540
$986$	−0.852263	−0.0271416
$987$	0	0
$988$	16.6198	0.528747
$989$	20.6194	0.655660
$990$	0	0
$991$	−20.2348	−0.642781	−0.321390	−	0.946947i	$-0.604150\pi$
$991$	−20.2348	−0.642781	−0.321390	+	0.946947i	$0.604150\pi$
$992$	9.39723	0.298362
$993$	0	0
$994$	11.4001	0.361590
$995$	−23.9281	−0.758571
$996$	0	0
$997$	0.100798	0.00319231	0.00159615	−	0.999999i	$-0.499492\pi$
$997$	0.100798	0.00319231	0.00159615	+	0.999999i	$0.499492\pi$
$998$	8.27684	0.261999
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.f.1.3	yes	8
3.2	odd	2		8046.2.a.e.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.e.1.6	✓	8	3.2	odd	2
8046.2.a.f.1.3	yes	8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.17278 q^{5} +1.96150 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.17278 q^{5} +1.96150 q^{7} +1.00000 q^{8} -1.17278 q^{10} -0.465557 q^{11} -2.12563 q^{13} +1.96150 q^{14} +1.00000 q^{16} +0.590988 q^{17} -7.81877 q^{19} -1.17278 q^{20} -0.465557 q^{22} +4.36934 q^{23} -3.62458 q^{25} -2.12563 q^{26} +1.96150 q^{28} -1.44210 q^{29} +9.39723 q^{31} +1.00000 q^{32} +0.590988 q^{34} -2.30041 q^{35} -3.39629 q^{37} -7.81877 q^{38} -1.17278 q^{40} -12.4113 q^{41} +4.71912 q^{43} -0.465557 q^{44} +4.36934 q^{46} +4.33354 q^{47} -3.15253 q^{49} -3.62458 q^{50} -2.12563 q^{52} -10.5164 q^{53} +0.545998 q^{55} +1.96150 q^{56} -1.44210 q^{58} -3.95347 q^{59} -10.4572 q^{61} +9.39723 q^{62} +1.00000 q^{64} +2.49291 q^{65} +0.763258 q^{67} +0.590988 q^{68} -2.30041 q^{70} +5.81196 q^{71} -7.40357 q^{73} -3.39629 q^{74} -7.81877 q^{76} -0.913187 q^{77} +0.711456 q^{79} -1.17278 q^{80} -12.4113 q^{82} +9.65399 q^{83} -0.693101 q^{85} +4.71912 q^{86} -0.465557 q^{88} +2.93347 q^{89} -4.16942 q^{91} +4.36934 q^{92} +4.33354 q^{94} +9.16973 q^{95} -6.58408 q^{97} -3.15253 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8	\( 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8} - 6 q^{11} - 8 q^{13} - 5 q^{14} + 8 q^{16} + 5 q^{17} - 14 q^{19} - 6 q^{22} - 21 q^{23} - 6 q^{25} - 8 q^{26} - 5 q^{28} - 3 q^{29} - 4 q^{31} + 8 q^{32} + 5 q^{34} + 2 q^{35} - 3 q^{37} - 14 q^{38} - 7 q^{41} - 12 q^{43} - 6 q^{44} - 21 q^{46} - 25 q^{47} - 7 q^{49} - 6 q^{50} - 8 q^{52} - 3 q^{53} - 9 q^{55} - 5 q^{56} - 3 q^{58} - 2 q^{59} - 17 q^{61} - 4 q^{62} + 8 q^{64} - 32 q^{65} - 14 q^{67} + 5 q^{68} + 2 q^{70} - 7 q^{71} - 10 q^{73} - 3 q^{74} - 14 q^{76} - 12 q^{77} - 33 q^{79} - 7 q^{82} - 13 q^{83} - 33 q^{85} - 12 q^{86} - 6 q^{88} + 22 q^{89} - 22 q^{91} - 21 q^{92} - 25 q^{94} + 14 q^{95} - 11 q^{97} - 7 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8 - 6 * q^11 - 8 * q^13 - 5 * q^14 + 8 * q^16 + 5 * q^17 - 14 * q^19 - 6 * q^22 - 21 * q^23 - 6 * q^25 - 8 * q^26 - 5 * q^28 - 3 * q^29 - 4 * q^31 + 8 * q^32 + 5 * q^34 + 2 * q^35 - 3 * q^37 - 14 * q^38 - 7 * q^41 - 12 * q^43 - 6 * q^44 - 21 * q^46 - 25 * q^47 - 7 * q^49 - 6 * q^50 - 8 * q^52 - 3 * q^53 - 9 * q^55 - 5 * q^56 - 3 * q^58 - 2 * q^59 - 17 * q^61 - 4 * q^62 + 8 * q^64 - 32 * q^65 - 14 * q^67 + 5 * q^68 + 2 * q^70 - 7 * q^71 - 10 * q^73 - 3 * q^74 - 14 * q^76 - 12 * q^77 - 33 * q^79 - 7 * q^82 - 13 * q^83 - 33 * q^85 - 12 * q^86 - 6 * q^88 + 22 * q^89 - 22 * q^91 - 21 * q^92 - 25 * q^94 + 14 * q^95 - 11 * q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more