LMFDB - Embedded newform 8046.2.a.r.1.6

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:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 42 x^{12} + 70 x^{11} + 680 x^{10} - 882 x^{9} - 5559 x^{8} + 5066 x^{7} + \cdots - 7083 $ x^14 - 2x^13 - 42x^12 + 70x^11 + 680x^10 - 882x^9 - 5559x^8 + 5066x^7 + 24455x^6 - 12990x^5 - 55580x^4 + 9808x^3 + 53551x^2 + 6282*x - 7083
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.46245$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.46245 q^{5} -0.313566 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.46245 q^{5} -0.313566 q^{7} +1.00000 q^{8} -1.46245 q^{10} -2.75836 q^{11} +1.05198 q^{13} -0.313566 q^{14} +1.00000 q^{16} +2.55151 q^{17} +6.36814 q^{19} -1.46245 q^{20} -2.75836 q^{22} -3.56676 q^{23} -2.86125 q^{25} +1.05198 q^{26} -0.313566 q^{28} -5.75361 q^{29} +8.68386 q^{31} +1.00000 q^{32} +2.55151 q^{34} +0.458574 q^{35} +3.72853 q^{37} +6.36814 q^{38} -1.46245 q^{40} +1.25379 q^{41} +1.78536 q^{43} -2.75836 q^{44} -3.56676 q^{46} +0.965913 q^{47} -6.90168 q^{49} -2.86125 q^{50} +1.05198 q^{52} +3.30919 q^{53} +4.03396 q^{55} -0.313566 q^{56} -5.75361 q^{58} -4.07118 q^{59} +6.28296 q^{61} +8.68386 q^{62} +1.00000 q^{64} -1.53846 q^{65} -10.3399 q^{67} +2.55151 q^{68} +0.458574 q^{70} -6.70274 q^{71} -3.93510 q^{73} +3.72853 q^{74} +6.36814 q^{76} +0.864930 q^{77} -0.146204 q^{79} -1.46245 q^{80} +1.25379 q^{82} +1.89312 q^{83} -3.73144 q^{85} +1.78536 q^{86} -2.75836 q^{88} +16.1891 q^{89} -0.329865 q^{91} -3.56676 q^{92} +0.965913 q^{94} -9.31308 q^{95} +16.7176 q^{97} -6.90168 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 14 q^{2} + 14 q^{4} + 2 q^{5} + 4 q^{7} + 14 q^{8}+O(q^{10}) $ 14 * q + 14 * q^2 + 14 * q^4 + 2 * q^5 + 4 * q^7 + 14 * q^8	$ 14 q + 14 q^{2} + 14 q^{4} + 2 q^{5} + 4 q^{7} + 14 q^{8} + 2 q^{10} + 2 q^{11} + 4 q^{13} + 4 q^{14} + 14 q^{16} + 9 q^{17} + 14 q^{19} + 2 q^{20} + 2 q^{22} + 30 q^{23} + 18 q^{25} + 4 q^{26} + 4 q^{28} + 6 q^{29} + 11 q^{31} + 14 q^{32} + 9 q^{34} - 18 q^{35} + 13 q^{37} + 14 q^{38} + 2 q^{40} - 2 q^{41} + 12 q^{43} + 2 q^{44} + 30 q^{46} + 21 q^{47} + 32 q^{49} + 18 q^{50} + 4 q^{52} + 22 q^{53} - 7 q^{55} + 4 q^{56} + 6 q^{58} + 14 q^{59} + 31 q^{61} + 11 q^{62} + 14 q^{64} + 24 q^{67} + 9 q^{68} - 18 q^{70} + 28 q^{71} + 24 q^{73} + 13 q^{74} + 14 q^{76} + 16 q^{77} + 65 q^{79} + 2 q^{80} - 2 q^{82} - 15 q^{83} - 19 q^{85} + 12 q^{86} + 2 q^{88} - 11 q^{89} + 68 q^{91} + 30 q^{92} + 21 q^{94} + 8 q^{95} + 23 q^{97} + 32 q^{98}+O(q^{100}) $ 14 * q + 14 * q^2 + 14 * q^4 + 2 * q^5 + 4 * q^7 + 14 * q^8 + 2 * q^10 + 2 * q^11 + 4 * q^13 + 4 * q^14 + 14 * q^16 + 9 * q^17 + 14 * q^19 + 2 * q^20 + 2 * q^22 + 30 * q^23 + 18 * q^25 + 4 * q^26 + 4 * q^28 + 6 * q^29 + 11 * q^31 + 14 * q^32 + 9 * q^34 - 18 * q^35 + 13 * q^37 + 14 * q^38 + 2 * q^40 - 2 * q^41 + 12 * q^43 + 2 * q^44 + 30 * q^46 + 21 * q^47 + 32 * q^49 + 18 * q^50 + 4 * q^52 + 22 * q^53 - 7 * q^55 + 4 * q^56 + 6 * q^58 + 14 * q^59 + 31 * q^61 + 11 * q^62 + 14 * q^64 + 24 * q^67 + 9 * q^68 - 18 * q^70 + 28 * q^71 + 24 * q^73 + 13 * q^74 + 14 * q^76 + 16 * q^77 + 65 * q^79 + 2 * q^80 - 2 * q^82 - 15 * q^83 - 19 * q^85 + 12 * q^86 + 2 * q^88 - 11 * q^89 + 68 * q^91 + 30 * q^92 + 21 * q^94 + 8 * q^95 + 23 * q^97 + 32 * 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
$5$	−1.46245	−0.654026	−0.327013	−	0.945020i	$-0.606042\pi$
$5$	−1.46245	−0.654026	−0.327013	+	0.945020i	$0.606042\pi$
$6$	0	0
$7$	−0.313566	−0.118517	−0.0592585	−	0.998243i	$-0.518874\pi$
$7$	−0.313566	−0.118517	−0.0592585	+	0.998243i	$0.518874\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.46245	−0.462466
$11$	−2.75836	−0.831677	−0.415839	−	0.909438i	$-0.636512\pi$
$11$	−2.75836	−0.831677	−0.415839	+	0.909438i	$0.636512\pi$
$12$	0	0
$13$	1.05198	0.291766	0.145883	−	0.989302i	$-0.453398\pi$
$13$	1.05198	0.291766	0.145883	+	0.989302i	$0.453398\pi$
$14$	−0.313566	−0.0838042
$15$	0	0
$16$	1.00000	0.250000
$17$	2.55151	0.618831	0.309416	−	0.950927i	$-0.399867\pi$
$17$	2.55151	0.618831	0.309416	+	0.950927i	$0.399867\pi$
$18$	0	0
$19$	6.36814	1.46095	0.730476	−	0.682938i	$-0.239298\pi$
$19$	6.36814	1.46095	0.730476	+	0.682938i	$0.239298\pi$
$20$	−1.46245	−0.327013
$21$	0	0
$22$	−2.75836	−0.588085
$23$	−3.56676	−0.743721	−0.371861	−	0.928289i	$-0.621280\pi$
$23$	−3.56676	−0.743721	−0.371861	+	0.928289i	$0.621280\pi$
$24$	0	0
$25$	−2.86125	−0.572250
$26$	1.05198	0.206310
$27$	0	0
$28$	−0.313566	−0.0592585
$29$	−5.75361	−1.06842	−0.534210	−	0.845352i	$-0.679391\pi$
$29$	−5.75361	−1.06842	−0.534210	+	0.845352i	$0.679391\pi$
$30$	0	0
$31$	8.68386	1.55967	0.779833	−	0.625987i	$-0.215304\pi$
$31$	8.68386	1.55967	0.779833	+	0.625987i	$0.215304\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.55151	0.437580
$35$	0.458574	0.0775132
$36$	0	0
$37$	3.72853	0.612967	0.306484	−	0.951876i	$-0.400848\pi$
$37$	3.72853	0.612967	0.306484	+	0.951876i	$0.400848\pi$
$38$	6.36814	1.03305
$39$	0	0
$40$	−1.46245	−0.231233
$41$	1.25379	0.195809	0.0979047	−	0.995196i	$-0.468786\pi$
$41$	1.25379	0.195809	0.0979047	+	0.995196i	$0.468786\pi$
$42$	0	0
$43$	1.78536	0.272265	0.136132	−	0.990691i	$-0.456533\pi$
$43$	1.78536	0.272265	0.136132	+	0.990691i	$0.456533\pi$
$44$	−2.75836	−0.415839
$45$	0	0
$46$	−3.56676	−0.525891
$47$	0.965913	0.140893	0.0704465	−	0.997516i	$-0.477558\pi$
$47$	0.965913	0.140893	0.0704465	+	0.997516i	$0.477558\pi$
$48$	0	0
$49$	−6.90168	−0.985954
$50$	−2.86125	−0.404642
$51$	0	0
$52$	1.05198	0.145883
$53$	3.30919	0.454552	0.227276	−	0.973830i	$-0.427018\pi$
$53$	3.30919	0.454552	0.227276	+	0.973830i	$0.427018\pi$
$54$	0	0
$55$	4.03396	0.543939
$56$	−0.313566	−0.0419021
$57$	0	0
$58$	−5.75361	−0.755486
$59$	−4.07118	−0.530022	−0.265011	−	0.964245i	$-0.585376\pi$
$59$	−4.07118	−0.530022	−0.265011	+	0.964245i	$0.585376\pi$
$60$	0	0
$61$	6.28296	0.804450	0.402225	−	0.915541i	$-0.368237\pi$
$61$	6.28296	0.804450	0.402225	+	0.915541i	$0.368237\pi$
$62$	8.68386	1.10285
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.53846	−0.190823
$66$	0	0
$67$	−10.3399	−1.26323	−0.631613	−	0.775284i	$-0.717607\pi$
$67$	−10.3399	−1.26323	−0.631613	+	0.775284i	$0.717607\pi$
$68$	2.55151	0.309416
$69$	0	0
$70$	0.458574	0.0548101
$71$	−6.70274	−0.795469	−0.397735	−	0.917500i	$-0.630204\pi$
$71$	−6.70274	−0.795469	−0.397735	+	0.917500i	$0.630204\pi$
$72$	0	0
$73$	−3.93510	−0.460569	−0.230285	−	0.973123i	$-0.573966\pi$
$73$	−3.93510	−0.460569	−0.230285	+	0.973123i	$0.573966\pi$
$74$	3.72853	0.433433
$75$	0	0
$76$	6.36814	0.730476
$77$	0.864930	0.0985679
$78$	0	0
$79$	−0.146204	−0.0164492	−0.00822461	−	0.999966i	$-0.502618\pi$
$79$	−0.146204	−0.0164492	−0.00822461	+	0.999966i	$0.502618\pi$
$80$	−1.46245	−0.163507
$81$	0	0
$82$	1.25379	0.138458
$83$	1.89312	0.207797	0.103898	−	0.994588i	$-0.466868\pi$
$83$	1.89312	0.207797	0.103898	+	0.994588i	$0.466868\pi$
$84$	0	0
$85$	−3.73144	−0.404732
$86$	1.78536	0.192520
$87$	0	0
$88$	−2.75836	−0.294042
$89$	16.1891	1.71604	0.858019	−	0.513618i	$-0.171695\pi$
$89$	16.1891	1.71604	0.858019	+	0.513618i	$0.171695\pi$
$90$	0	0
$91$	−0.329865	−0.0345793
$92$	−3.56676	−0.371861
$93$	0	0
$94$	0.965913	0.0996264
$95$	−9.31308	−0.955501
$96$	0	0
$97$	16.7176	1.69741	0.848706	−	0.528866i	$-0.177382\pi$
$97$	16.7176	1.69741	0.848706	+	0.528866i	$0.177382\pi$
$98$	−6.90168	−0.697175
$99$	0	0
$100$	−2.86125	−0.286125
$101$	−1.72005	−0.171151	−0.0855757	−	0.996332i	$-0.527273\pi$
$101$	−1.72005	−0.171151	−0.0855757	+	0.996332i	$0.527273\pi$
$102$	0	0
$103$	10.6160	1.04602	0.523011	−	0.852326i	$-0.324809\pi$
$103$	10.6160	1.04602	0.523011	+	0.852326i	$0.324809\pi$
$104$	1.05198	0.103155
$105$	0	0
$106$	3.30919	0.321417
$107$	13.8852	1.34233	0.671166	−	0.741307i	$-0.265794\pi$
$107$	13.8852	1.34233	0.671166	+	0.741307i	$0.265794\pi$
$108$	0	0
$109$	9.01428	0.863411	0.431706	−	0.902015i	$-0.357912\pi$
$109$	9.01428	0.863411	0.431706	+	0.902015i	$0.357912\pi$
$110$	4.03396	0.384623
$111$	0	0
$112$	−0.313566	−0.0296292
$113$	5.11414	0.481098	0.240549	−	0.970637i	$-0.422672\pi$
$113$	5.11414	0.481098	0.240549	+	0.970637i	$0.422672\pi$
$114$	0	0
$115$	5.21620	0.486413
$116$	−5.75361	−0.534210
$117$	0	0
$118$	−4.07118	−0.374782
$119$	−0.800067	−0.0733420
$120$	0	0
$121$	−3.39144	−0.308313
$122$	6.28296	0.568832
$123$	0	0
$124$	8.68386	0.779833
$125$	11.4967	1.02829
$126$	0	0
$127$	−3.29554	−0.292431	−0.146216	−	0.989253i	$-0.546709\pi$
$127$	−3.29554	−0.292431	−0.146216	+	0.989253i	$0.546709\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.53846	−0.134932
$131$	−4.06559	−0.355213	−0.177606	−	0.984102i	$-0.556835\pi$
$131$	−4.06559	−0.355213	−0.177606	+	0.984102i	$0.556835\pi$
$132$	0	0
$133$	−1.99684	−0.173148
$134$	−10.3399	−0.893235
$135$	0	0
$136$	2.55151	0.218790
$137$	4.66733	0.398757	0.199378	−	0.979923i	$-0.436108\pi$
$137$	4.66733	0.398757	0.199378	+	0.979923i	$0.436108\pi$
$138$	0	0
$139$	18.3735	1.55842	0.779210	−	0.626762i	$-0.215620\pi$
$139$	18.3735	1.55842	0.779210	+	0.626762i	$0.215620\pi$
$140$	0.458574	0.0387566
$141$	0	0
$142$	−6.70274	−0.562482
$143$	−2.90174	−0.242656
$144$	0	0
$145$	8.41436	0.698774
$146$	−3.93510	−0.325672
$147$	0	0
$148$	3.72853	0.306484
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	2.98833	0.243187	0.121593	−	0.992580i	$-0.461200\pi$
$151$	2.98833	0.243187	0.121593	+	0.992580i	$0.461200\pi$
$152$	6.36814	0.516525
$153$	0	0
$154$	0.864930	0.0696980
$155$	−12.6997	−1.02006
$156$	0	0
$157$	8.45278	0.674605	0.337303	−	0.941396i	$-0.390485\pi$
$157$	8.45278	0.674605	0.337303	+	0.941396i	$0.390485\pi$
$158$	−0.146204	−0.0116314
$159$	0	0
$160$	−1.46245	−0.115617
$161$	1.11842	0.0881436
$162$	0	0
$163$	1.95884	0.153428	0.0767140	−	0.997053i	$-0.475557\pi$
$163$	1.95884	0.153428	0.0767140	+	0.997053i	$0.475557\pi$
$164$	1.25379	0.0979047
$165$	0	0
$166$	1.89312	0.146934
$167$	6.61409	0.511814	0.255907	−	0.966701i	$-0.417626\pi$
$167$	6.61409	0.511814	0.255907	+	0.966701i	$0.417626\pi$
$168$	0	0
$169$	−11.8933	−0.914872
$170$	−3.73144	−0.286189
$171$	0	0
$172$	1.78536	0.136132
$173$	−4.98204	−0.378778	−0.189389	−	0.981902i	$-0.560651\pi$
$173$	−4.98204	−0.378778	−0.189389	+	0.981902i	$0.560651\pi$
$174$	0	0
$175$	0.897191	0.0678213
$176$	−2.75836	−0.207919
$177$	0	0
$178$	16.1891	1.21342
$179$	19.5311	1.45982	0.729911	−	0.683543i	$-0.239562\pi$
$179$	19.5311	1.45982	0.729911	+	0.683543i	$0.239562\pi$
$180$	0	0
$181$	16.0544	1.19332	0.596658	−	0.802496i	$-0.296495\pi$
$181$	16.0544	1.19332	0.596658	+	0.802496i	$0.296495\pi$
$182$	−0.329865	−0.0244512
$183$	0	0
$184$	−3.56676	−0.262945
$185$	−5.45278	−0.400897
$186$	0	0
$187$	−7.03798	−0.514668
$188$	0.965913	0.0704465
$189$	0	0
$190$	−9.31308	−0.675642
$191$	9.14557	0.661750	0.330875	−	0.943675i	$-0.392656\pi$
$191$	9.14557	0.661750	0.330875	+	0.943675i	$0.392656\pi$
$192$	0	0
$193$	−1.13247	−0.0815173	−0.0407586	−	0.999169i	$-0.512977\pi$
$193$	−1.13247	−0.0815173	−0.0407586	+	0.999169i	$0.512977\pi$
$194$	16.7176	1.20025
$195$	0	0
$196$	−6.90168	−0.492977
$197$	−19.7062	−1.40401	−0.702004	−	0.712173i	$-0.747711\pi$
$197$	−19.7062	−1.40401	−0.702004	+	0.712173i	$0.747711\pi$
$198$	0	0
$199$	18.2259	1.29200	0.645999	−	0.763338i	$-0.276441\pi$
$199$	18.2259	1.29200	0.645999	+	0.763338i	$0.276441\pi$
$200$	−2.86125	−0.202321
$201$	0	0
$202$	−1.72005	−0.121022
$203$	1.80414	0.126626
$204$	0	0
$205$	−1.83361	−0.128065
$206$	10.6160	0.739650
$207$	0	0
$208$	1.05198	0.0729416
$209$	−17.5656	−1.21504
$210$	0	0
$211$	−18.6031	−1.28069	−0.640347	−	0.768086i	$-0.721209\pi$
$211$	−18.6031	−1.28069	−0.640347	+	0.768086i	$0.721209\pi$
$212$	3.30919	0.227276
$213$	0	0
$214$	13.8852	0.949171
$215$	−2.61099	−0.178068
$216$	0	0
$217$	−2.72297	−0.184847
$218$	9.01428	0.610524
$219$	0	0
$220$	4.03396	0.271969
$221$	2.68413	0.180554
$222$	0	0
$223$	−4.12033	−0.275918	−0.137959	−	0.990438i	$-0.544054\pi$
$223$	−4.12033	−0.275918	−0.137959	+	0.990438i	$0.544054\pi$
$224$	−0.313566	−0.0209510
$225$	0	0
$226$	5.11414	0.340188
$227$	1.58651	0.105300	0.0526501	−	0.998613i	$-0.483233\pi$
$227$	1.58651	0.105300	0.0526501	+	0.998613i	$0.483233\pi$
$228$	0	0
$229$	17.5566	1.16017	0.580086	−	0.814556i	$-0.303019\pi$
$229$	17.5566	1.16017	0.580086	+	0.814556i	$0.303019\pi$
$230$	5.21620	0.343946
$231$	0	0
$232$	−5.75361	−0.377743
$233$	24.2618	1.58945	0.794723	−	0.606973i	$-0.207616\pi$
$233$	24.2618	1.58945	0.794723	+	0.606973i	$0.207616\pi$
$234$	0	0
$235$	−1.41260	−0.0921477
$236$	−4.07118	−0.265011
$237$	0	0
$238$	−0.800067	−0.0518606
$239$	1.13063	0.0731347	0.0365673	−	0.999331i	$-0.488358\pi$
$239$	1.13063	0.0731347	0.0365673	+	0.999331i	$0.488358\pi$
$240$	0	0
$241$	28.6226	1.84374	0.921872	−	0.387494i	$-0.126659\pi$
$241$	28.6226	1.84374	0.921872	+	0.387494i	$0.126659\pi$
$242$	−3.39144	−0.218010
$243$	0	0
$244$	6.28296	0.402225
$245$	10.0933	0.644840
$246$	0	0
$247$	6.69915	0.426257
$248$	8.68386	0.551425
$249$	0	0
$250$	11.4967	0.727113
$251$	−26.2669	−1.65795	−0.828977	−	0.559282i	$-0.811077\pi$
$251$	−26.2669	−1.65795	−0.828977	+	0.559282i	$0.811077\pi$
$252$	0	0
$253$	9.83842	0.618536
$254$	−3.29554	−0.206780
$255$	0	0
$256$	1.00000	0.0625000
$257$	17.2472	1.07585	0.537927	−	0.842992i	$-0.319208\pi$
$257$	17.2472	1.07585	0.537927	+	0.842992i	$0.319208\pi$
$258$	0	0
$259$	−1.16914	−0.0726470
$260$	−1.53846	−0.0954115
$261$	0	0
$262$	−4.06559	−0.251173
$263$	−19.5088	−1.20297	−0.601483	−	0.798886i	$-0.705423\pi$
$263$	−19.5088	−1.20297	−0.601483	+	0.798886i	$0.705423\pi$
$264$	0	0
$265$	−4.83951	−0.297289
$266$	−1.99684	−0.122434
$267$	0	0
$268$	−10.3399	−0.631613
$269$	−10.5180	−0.641294	−0.320647	−	0.947199i	$-0.603900\pi$
$269$	−10.5180	−0.641294	−0.320647	+	0.947199i	$0.603900\pi$
$270$	0	0
$271$	−7.80465	−0.474099	−0.237049	−	0.971498i	$-0.576180\pi$
$271$	−7.80465	−0.474099	−0.237049	+	0.971498i	$0.576180\pi$
$272$	2.55151	0.154708
$273$	0	0
$274$	4.66733	0.281963
$275$	7.89236	0.475927
$276$	0	0
$277$	−1.58340	−0.0951372	−0.0475686	−	0.998868i	$-0.515147\pi$
$277$	−1.58340	−0.0951372	−0.0475686	+	0.998868i	$0.515147\pi$
$278$	18.3735	1.10197
$279$	0	0
$280$	0.458574	0.0274051
$281$	5.62534	0.335580	0.167790	−	0.985823i	$-0.446337\pi$
$281$	5.62534	0.335580	0.167790	+	0.985823i	$0.446337\pi$
$282$	0	0
$283$	21.4209	1.27334	0.636669	−	0.771137i	$-0.280312\pi$
$283$	21.4209	1.27334	0.636669	+	0.771137i	$0.280312\pi$
$284$	−6.70274	−0.397735
$285$	0	0
$286$	−2.90174	−0.171583
$287$	−0.393147	−0.0232067
$288$	0	0
$289$	−10.4898	−0.617048
$290$	8.41436	0.494108
$291$	0	0
$292$	−3.93510	−0.230285
$293$	32.7365	1.91249	0.956243	−	0.292572i	$-0.0945112\pi$
$293$	32.7365	1.91249	0.956243	+	0.292572i	$0.0945112\pi$
$294$	0	0
$295$	5.95388	0.346648
$296$	3.72853	0.216717
$297$	0	0
$298$	1.00000	0.0579284
$299$	−3.75216	−0.216993
$300$	0	0
$301$	−0.559829	−0.0322680
$302$	2.98833	0.171959
$303$	0	0
$304$	6.36814	0.365238
$305$	−9.18849	−0.526131
$306$	0	0
$307$	12.2795	0.700826	0.350413	−	0.936595i	$-0.386041\pi$
$307$	12.2795	0.700826	0.350413	+	0.936595i	$0.386041\pi$
$308$	0.864930	0.0492839
$309$	0	0
$310$	−12.6997	−0.721293
$311$	2.31798	0.131441	0.0657203	−	0.997838i	$-0.479065\pi$
$311$	2.31798	0.131441	0.0657203	+	0.997838i	$0.479065\pi$
$312$	0	0
$313$	−6.82142	−0.385570	−0.192785	−	0.981241i	$-0.561752\pi$
$313$	−6.82142	−0.385570	−0.192785	+	0.981241i	$0.561752\pi$
$314$	8.45278	0.477018
$315$	0	0
$316$	−0.146204	−0.00822461
$317$	−18.9467	−1.06415	−0.532076	−	0.846697i	$-0.678588\pi$
$317$	−18.9467	−1.06415	−0.532076	+	0.846697i	$0.678588\pi$
$318$	0	0
$319$	15.8705	0.888580
$320$	−1.46245	−0.0817533
$321$	0	0
$322$	1.11842	0.0623270
$323$	16.2484	0.904083
$324$	0	0
$325$	−3.00997	−0.166963
$326$	1.95884	0.108490
$327$	0	0
$328$	1.25379	0.0692291
$329$	−0.302878	−0.0166982
$330$	0	0
$331$	26.0452	1.43158	0.715788	−	0.698318i	$-0.246068\pi$
$331$	26.0452	1.43158	0.715788	+	0.698318i	$0.246068\pi$
$332$	1.89312	0.103898
$333$	0	0
$334$	6.61409	0.361907
$335$	15.1216	0.826183
$336$	0	0
$337$	−17.9037	−0.975279	−0.487639	−	0.873045i	$-0.662142\pi$
$337$	−17.9037	−0.975279	−0.487639	+	0.873045i	$0.662142\pi$
$338$	−11.8933	−0.646912
$339$	0	0
$340$	−3.73144	−0.202366
$341$	−23.9532	−1.29714
$342$	0	0
$343$	4.35910	0.235369
$344$	1.78536	0.0962601
$345$	0	0
$346$	−4.98204	−0.267836
$347$	29.4410	1.58048	0.790239	−	0.612799i	$-0.209956\pi$
$347$	29.4410	1.58048	0.790239	+	0.612799i	$0.209956\pi$
$348$	0	0
$349$	−10.7661	−0.576294	−0.288147	−	0.957586i	$-0.593039\pi$
$349$	−10.7661	−0.576294	−0.288147	+	0.957586i	$0.593039\pi$
$350$	0.897191	0.0479569
$351$	0	0
$352$	−2.75836	−0.147021
$353$	−14.5625	−0.775083	−0.387541	−	0.921852i	$-0.626676\pi$
$353$	−14.5625	−0.775083	−0.387541	+	0.921852i	$0.626676\pi$
$354$	0	0
$355$	9.80241	0.520258
$356$	16.1891	0.858019
$357$	0	0
$358$	19.5311	1.03225
$359$	12.6861	0.669547	0.334773	−	0.942299i	$-0.391340\pi$
$359$	12.6861	0.669547	0.334773	+	0.942299i	$0.391340\pi$
$360$	0	0
$361$	21.5533	1.13438
$362$	16.0544	0.843802
$363$	0	0
$364$	−0.329865	−0.0172896
$365$	5.75488	0.301224
$366$	0	0
$367$	9.51201	0.496523	0.248262	−	0.968693i	$-0.420141\pi$
$367$	9.51201	0.496523	0.248262	+	0.968693i	$0.420141\pi$
$368$	−3.56676	−0.185930
$369$	0	0
$370$	−5.45278	−0.283477
$371$	−1.03765	−0.0538721
$372$	0	0
$373$	13.6335	0.705913	0.352957	−	0.935640i	$-0.385176\pi$
$373$	13.6335	0.705913	0.352957	+	0.935640i	$0.385176\pi$
$374$	−7.03798	−0.363925
$375$	0	0
$376$	0.965913	0.0498132
$377$	−6.05268	−0.311729
$378$	0	0
$379$	8.91740	0.458056	0.229028	−	0.973420i	$-0.426445\pi$
$379$	8.91740	0.458056	0.229028	+	0.973420i	$0.426445\pi$
$380$	−9.31308	−0.477751
$381$	0	0
$382$	9.14557	0.467928
$383$	14.8064	0.756574	0.378287	−	0.925688i	$-0.376513\pi$
$383$	14.8064	0.756574	0.378287	+	0.925688i	$0.376513\pi$
$384$	0	0
$385$	−1.26491	−0.0644660
$386$	−1.13247	−0.0576414
$387$	0	0
$388$	16.7176	0.848706
$389$	−29.6168	−1.50163	−0.750816	−	0.660512i	$-0.770339\pi$
$389$	−29.6168	−1.50163	−0.750816	+	0.660512i	$0.770339\pi$
$390$	0	0
$391$	−9.10062	−0.460238
$392$	−6.90168	−0.348587
$393$	0	0
$394$	−19.7062	−0.992783
$395$	0.213815	0.0107582
$396$	0	0
$397$	−7.89296	−0.396137	−0.198068	−	0.980188i	$-0.563467\pi$
$397$	−7.89296	−0.396137	−0.198068	+	0.980188i	$0.563467\pi$
$398$	18.2259	0.913580
$399$	0	0
$400$	−2.86125	−0.143062
$401$	−23.9408	−1.19554	−0.597772	−	0.801666i	$-0.703947\pi$
$401$	−23.9408	−1.19554	−0.597772	+	0.801666i	$0.703947\pi$
$402$	0	0
$403$	9.13523	0.455058
$404$	−1.72005	−0.0855757
$405$	0	0
$406$	1.80414	0.0895380
$407$	−10.2846	−0.509791
$408$	0	0
$409$	−17.7283	−0.876606	−0.438303	−	0.898827i	$-0.644420\pi$
$409$	−17.7283	−0.876606	−0.438303	+	0.898827i	$0.644420\pi$
$410$	−1.83361	−0.0905553
$411$	0	0
$412$	10.6160	0.523011
$413$	1.27658	0.0628166
$414$	0	0
$415$	−2.76858	−0.135904
$416$	1.05198	0.0515775
$417$	0	0
$418$	−17.5656	−0.859164
$419$	4.45345	0.217565	0.108783	−	0.994066i	$-0.465305\pi$
$419$	4.45345	0.217565	0.108783	+	0.994066i	$0.465305\pi$
$420$	0	0
$421$	26.5721	1.29505	0.647523	−	0.762046i	$-0.275805\pi$
$421$	26.5721	1.29505	0.647523	+	0.762046i	$0.275805\pi$
$422$	−18.6031	−0.905587
$423$	0	0
$424$	3.30919	0.160708
$425$	−7.30049	−0.354126
$426$	0	0
$427$	−1.97012	−0.0953410
$428$	13.8852	0.671166
$429$	0	0
$430$	−2.61099	−0.125913
$431$	17.0238	0.820010	0.410005	−	0.912083i	$-0.365527\pi$
$431$	17.0238	0.820010	0.410005	+	0.912083i	$0.365527\pi$
$432$	0	0
$433$	−26.5129	−1.27413	−0.637064	−	0.770811i	$-0.719851\pi$
$433$	−26.5129	−1.27413	−0.637064	+	0.770811i	$0.719851\pi$
$434$	−2.72297	−0.130707
$435$	0	0
$436$	9.01428	0.431706
$437$	−22.7137	−1.08654
$438$	0	0
$439$	2.95153	0.140869	0.0704344	−	0.997516i	$-0.477561\pi$
$439$	2.95153	0.140869	0.0704344	+	0.997516i	$0.477561\pi$
$440$	4.03396	0.192311
$441$	0	0
$442$	2.68413	0.127671
$443$	−26.8435	−1.27538	−0.637688	−	0.770295i	$-0.720109\pi$
$443$	−26.8435	−1.27538	−0.637688	+	0.770295i	$0.720109\pi$
$444$	0	0
$445$	−23.6757	−1.12233
$446$	−4.12033	−0.195103
$447$	0	0
$448$	−0.313566	−0.0148146
$449$	−15.9574	−0.753077	−0.376539	−	0.926401i	$-0.622886\pi$
$449$	−15.9574	−0.753077	−0.376539	+	0.926401i	$0.622886\pi$
$450$	0	0
$451$	−3.45841	−0.162850
$452$	5.11414	0.240549
$453$	0	0
$454$	1.58651	0.0744585
$455$	0.482411	0.0226158
$456$	0	0
$457$	15.2612	0.713887	0.356943	−	0.934126i	$-0.383819\pi$
$457$	15.2612	0.713887	0.356943	+	0.934126i	$0.383819\pi$
$458$	17.5566	0.820365
$459$	0	0
$460$	5.21620	0.243207
$461$	−8.85470	−0.412404	−0.206202	−	0.978509i	$-0.566110\pi$
$461$	−8.85470	−0.412404	−0.206202	+	0.978509i	$0.566110\pi$
$462$	0	0
$463$	−42.5568	−1.97778	−0.988891	−	0.148646i	$-0.952509\pi$
$463$	−42.5568	−1.97778	−0.988891	+	0.148646i	$0.952509\pi$
$464$	−5.75361	−0.267105
$465$	0	0
$466$	24.2618	1.12391
$467$	−10.4138	−0.481893	−0.240947	−	0.970538i	$-0.577458\pi$
$467$	−10.4138	−0.481893	−0.240947	+	0.970538i	$0.577458\pi$
$468$	0	0
$469$	3.24226	0.149714
$470$	−1.41260	−0.0651583
$471$	0	0
$472$	−4.07118	−0.187391
$473$	−4.92467	−0.226436
$474$	0	0
$475$	−18.2208	−0.836029
$476$	−0.800067	−0.0366710
$477$	0	0
$478$	1.13063	0.0517140
$479$	28.8406	1.31776	0.658880	−	0.752248i	$-0.271030\pi$
$479$	28.8406	1.31776	0.658880	+	0.752248i	$0.271030\pi$
$480$	0	0
$481$	3.92234	0.178843
$482$	28.6226	1.30372
$483$	0	0
$484$	−3.39144	−0.154156
$485$	−24.4486	−1.11015
$486$	0	0
$487$	−5.12433	−0.232205	−0.116103	−	0.993237i	$-0.537040\pi$
$487$	−5.12433	−0.232205	−0.116103	+	0.993237i	$0.537040\pi$
$488$	6.28296	0.284416
$489$	0	0
$490$	10.0933	0.455971
$491$	6.54841	0.295526	0.147763	−	0.989023i	$-0.452793\pi$
$491$	6.54841	0.295526	0.147763	+	0.989023i	$0.452793\pi$
$492$	0	0
$493$	−14.6804	−0.661171
$494$	6.69915	0.301409
$495$	0	0
$496$	8.68386	0.389917
$497$	2.10176	0.0942766
$498$	0	0
$499$	−33.6700	−1.50728	−0.753638	−	0.657290i	$-0.771703\pi$
$499$	−33.6700	−1.50728	−0.753638	+	0.657290i	$0.771703\pi$
$500$	11.4967	0.514146
$501$	0	0
$502$	−26.2669	−1.17235
$503$	−22.1619	−0.988152	−0.494076	−	0.869419i	$-0.664493\pi$
$503$	−22.1619	−0.988152	−0.494076	+	0.869419i	$0.664493\pi$
$504$	0	0
$505$	2.51548	0.111938
$506$	9.83842	0.437371
$507$	0	0
$508$	−3.29554	−0.146216
$509$	−0.658935	−0.0292068	−0.0146034	−	0.999893i	$-0.504649\pi$
$509$	−0.658935	−0.0292068	−0.0146034	+	0.999893i	$0.504649\pi$
$510$	0	0
$511$	1.23392	0.0545853
$512$	1.00000	0.0441942
$513$	0	0
$514$	17.2472	0.760743
$515$	−15.5253	−0.684126
$516$	0	0
$517$	−2.66434	−0.117177
$518$	−1.16914	−0.0513692
$519$	0	0
$520$	−1.53846	−0.0674661
$521$	5.43286	0.238018	0.119009	−	0.992893i	$-0.462028\pi$
$521$	5.43286	0.238018	0.119009	+	0.992893i	$0.462028\pi$
$522$	0	0
$523$	−9.73779	−0.425804	−0.212902	−	0.977074i	$-0.568291\pi$
$523$	−9.73779	−0.425804	−0.212902	+	0.977074i	$0.568291\pi$
$524$	−4.06559	−0.177606
$525$	0	0
$526$	−19.5088	−0.850625
$527$	22.1569	0.965170
$528$	0	0
$529$	−10.2782	−0.446878
$530$	−4.83951	−0.210215
$531$	0	0
$532$	−1.99684	−0.0865738
$533$	1.31896	0.0571306
$534$	0	0
$535$	−20.3063	−0.877920
$536$	−10.3399	−0.446618
$537$	0	0
$538$	−10.5180	−0.453463
$539$	19.0373	0.819995
$540$	0	0
$541$	14.5606	0.626008	0.313004	−	0.949752i	$-0.398665\pi$
$541$	14.5606	0.626008	0.313004	+	0.949752i	$0.398665\pi$
$542$	−7.80465	−0.335238
$543$	0	0
$544$	2.55151	0.109395
$545$	−13.1829	−0.564694
$546$	0	0
$547$	−7.03971	−0.300996	−0.150498	−	0.988610i	$-0.548088\pi$
$547$	−7.03971	−0.300996	−0.150498	+	0.988610i	$0.548088\pi$
$548$	4.66733	0.199378
$549$	0	0
$550$	7.89236	0.336531
$551$	−36.6398	−1.56091
$552$	0	0
$553$	0.0458446	0.00194951
$554$	−1.58340	−0.0672721
$555$	0	0
$556$	18.3735	0.779210
$557$	−40.1965	−1.70318	−0.851590	−	0.524208i	$-0.824361\pi$
$557$	−40.1965	−1.70318	−0.851590	+	0.524208i	$0.824361\pi$
$558$	0	0
$559$	1.87816	0.0794377
$560$	0.458574	0.0193783
$561$	0	0
$562$	5.62534	0.237291
$563$	−11.2429	−0.473830	−0.236915	−	0.971530i	$-0.576136\pi$
$563$	−11.2429	−0.473830	−0.236915	+	0.971530i	$0.576136\pi$
$564$	0	0
$565$	−7.47917	−0.314651
$566$	21.4209	0.900386
$567$	0	0
$568$	−6.70274	−0.281241
$569$	−10.7838	−0.452082	−0.226041	−	0.974118i	$-0.572578\pi$
$569$	−10.7838	−0.452082	−0.226041	+	0.974118i	$0.572578\pi$
$570$	0	0
$571$	−15.8319	−0.662544	−0.331272	−	0.943535i	$-0.607478\pi$
$571$	−15.8319	−0.662544	−0.331272	+	0.943535i	$0.607478\pi$
$572$	−2.90174	−0.121328
$573$	0	0
$574$	−0.393147	−0.0164096
$575$	10.2054	0.425594
$576$	0	0
$577$	38.3006	1.59448	0.797238	−	0.603665i	$-0.206294\pi$
$577$	38.3006	1.59448	0.797238	+	0.603665i	$0.206294\pi$
$578$	−10.4898	−0.436319
$579$	0	0
$580$	8.41436	0.349387
$581$	−0.593618	−0.0246274
$582$	0	0
$583$	−9.12793	−0.378040
$584$	−3.93510	−0.162836
$585$	0	0
$586$	32.7365	1.35233
$587$	18.1965	0.751050	0.375525	−	0.926812i	$-0.377462\pi$
$587$	18.1965	0.751050	0.375525	+	0.926812i	$0.377462\pi$
$588$	0	0
$589$	55.3000	2.27860
$590$	5.95388	0.245117
$591$	0	0
$592$	3.72853	0.153242
$593$	21.8859	0.898745	0.449373	−	0.893344i	$-0.351648\pi$
$593$	21.8859	0.898745	0.449373	+	0.893344i	$0.351648\pi$
$594$	0	0
$595$	1.17006	0.0479676
$596$	1.00000	0.0409616
$597$	0	0
$598$	−3.75216	−0.153437
$599$	8.96636	0.366356	0.183178	−	0.983080i	$-0.441362\pi$
$599$	8.96636	0.366356	0.183178	+	0.983080i	$0.441362\pi$
$600$	0	0
$601$	−20.4281	−0.833278	−0.416639	−	0.909072i	$-0.636792\pi$
$601$	−20.4281	−0.833278	−0.416639	+	0.909072i	$0.636792\pi$
$602$	−0.559829	−0.0228169
$603$	0	0
$604$	2.98833	0.121593
$605$	4.95980	0.201645
$606$	0	0
$607$	−16.5379	−0.671253	−0.335627	−	0.941995i	$-0.608948\pi$
$607$	−16.5379	−0.671253	−0.335627	+	0.941995i	$0.608948\pi$
$608$	6.36814	0.258262
$609$	0	0
$610$	−9.18849	−0.372031
$611$	1.01612	0.0411078
$612$	0	0
$613$	45.4542	1.83588	0.917939	−	0.396722i	$-0.129852\pi$
$613$	45.4542	1.83588	0.917939	+	0.396722i	$0.129852\pi$
$614$	12.2795	0.495559
$615$	0	0
$616$	0.864930	0.0348490
$617$	−3.78328	−0.152309	−0.0761546	−	0.997096i	$-0.524264\pi$
$617$	−3.78328	−0.152309	−0.0761546	+	0.997096i	$0.524264\pi$
$618$	0	0
$619$	−15.1800	−0.610135	−0.305067	−	0.952331i	$-0.598679\pi$
$619$	−15.1800	−0.610135	−0.305067	+	0.952331i	$0.598679\pi$
$620$	−12.6997	−0.510032
$621$	0	0
$622$	2.31798	0.0929426
$623$	−5.07635	−0.203380
$624$	0	0
$625$	−2.50702	−0.100281
$626$	−6.82142	−0.272639
$627$	0	0
$628$	8.45278	0.337303
$629$	9.51338	0.379323
$630$	0	0
$631$	35.4512	1.41129	0.705646	−	0.708565i	$-0.250657\pi$
$631$	35.4512	1.41129	0.705646	+	0.708565i	$0.250657\pi$
$632$	−0.146204	−0.00581568
$633$	0	0
$634$	−18.9467	−0.752469
$635$	4.81955	0.191258
$636$	0	0
$637$	−7.26042	−0.287668
$638$	15.8705	0.628321
$639$	0	0
$640$	−1.46245	−0.0578083
$641$	9.82073	0.387896	0.193948	−	0.981012i	$-0.437871\pi$
$641$	9.82073	0.387896	0.193948	+	0.981012i	$0.437871\pi$
$642$	0	0
$643$	25.5640	1.00815	0.504073	−	0.863661i	$-0.331834\pi$
$643$	25.5640	1.00815	0.504073	+	0.863661i	$0.331834\pi$
$644$	1.11842	0.0440718
$645$	0	0
$646$	16.2484	0.639283
$647$	19.0924	0.750599	0.375299	−	0.926904i	$-0.377540\pi$
$647$	19.0924	0.750599	0.375299	+	0.926904i	$0.377540\pi$
$648$	0	0
$649$	11.2298	0.440807
$650$	−3.00997	−0.118061
$651$	0	0
$652$	1.95884	0.0767140
$653$	2.57138	0.100626	0.0503129	−	0.998734i	$-0.483978\pi$
$653$	2.57138	0.100626	0.0503129	+	0.998734i	$0.483978\pi$
$654$	0	0
$655$	5.94572	0.232318
$656$	1.25379	0.0489524
$657$	0	0
$658$	−0.302878	−0.0118074
$659$	5.59304	0.217874	0.108937	−	0.994049i	$-0.465255\pi$
$659$	5.59304	0.217874	0.108937	+	0.994049i	$0.465255\pi$
$660$	0	0
$661$	−36.2021	−1.40810	−0.704049	−	0.710151i	$-0.748626\pi$
$661$	−36.2021	−1.40810	−0.704049	+	0.710151i	$0.748626\pi$
$662$	26.0452	1.01228
$663$	0	0
$664$	1.89312	0.0734672
$665$	2.92027	0.113243
$666$	0	0
$667$	20.5218	0.794606
$668$	6.61409	0.255907
$669$	0	0
$670$	15.1216	0.584199
$671$	−17.3307	−0.669043
$672$	0	0
$673$	−32.1290	−1.23848	−0.619241	−	0.785201i	$-0.712560\pi$
$673$	−32.1290	−1.23848	−0.619241	+	0.785201i	$0.712560\pi$
$674$	−17.9037	−0.689626
$675$	0	0
$676$	−11.8933	−0.457436
$677$	−10.0850	−0.387598	−0.193799	−	0.981041i	$-0.562081\pi$
$677$	−10.0850	−0.387598	−0.193799	+	0.981041i	$0.562081\pi$
$678$	0	0
$679$	−5.24207	−0.201172
$680$	−3.73144	−0.143094
$681$	0	0
$682$	−23.9532	−0.917216
$683$	−29.8814	−1.14338	−0.571690	−	0.820470i	$-0.693712\pi$
$683$	−29.8814	−1.14338	−0.571690	+	0.820470i	$0.693712\pi$
$684$	0	0
$685$	−6.82572	−0.260797
$686$	4.35910	0.166431
$687$	0	0
$688$	1.78536	0.0680662
$689$	3.48119	0.132623
$690$	0	0
$691$	47.2350	1.79690	0.898452	−	0.439072i	$-0.144693\pi$
$691$	47.2350	1.79690	0.898452	+	0.439072i	$0.144693\pi$
$692$	−4.98204	−0.189389
$693$	0	0
$694$	29.4410	1.11757
$695$	−26.8703	−1.01925
$696$	0	0
$697$	3.19906	0.121173
$698$	−10.7661	−0.407501
$699$	0	0
$700$	0.897191	0.0339106
$701$	−2.85982	−0.108014	−0.0540070	−	0.998541i	$-0.517199\pi$
$701$	−2.85982	−0.108014	−0.0540070	+	0.998541i	$0.517199\pi$
$702$	0	0
$703$	23.7438	0.895516
$704$	−2.75836	−0.103960
$705$	0	0
$706$	−14.5625	−0.548066
$707$	0.539350	0.0202844
$708$	0	0
$709$	−11.5104	−0.432282	−0.216141	−	0.976362i	$-0.569347\pi$
$709$	−11.5104	−0.432282	−0.216141	+	0.976362i	$0.569347\pi$
$710$	9.80241	0.367878
$711$	0	0
$712$	16.1891	0.606711
$713$	−30.9733	−1.15996
$714$	0	0
$715$	4.24364	0.158703
$716$	19.5311	0.729911
$717$	0	0
$718$	12.6861	0.473441
$719$	17.5630	0.654989	0.327494	−	0.944853i	$-0.393796\pi$
$719$	17.5630	0.654989	0.327494	+	0.944853i	$0.393796\pi$
$720$	0	0
$721$	−3.32881	−0.123971
$722$	21.5533	0.802130
$723$	0	0
$724$	16.0544	0.596658
$725$	16.4625	0.611402
$726$	0	0
$727$	−7.40478	−0.274628	−0.137314	−	0.990528i	$-0.543847\pi$
$727$	−7.40478	−0.274628	−0.137314	+	0.990528i	$0.543847\pi$
$728$	−0.329865	−0.0122256
$729$	0	0
$730$	5.75488	0.212998
$731$	4.55536	0.168486
$732$	0	0
$733$	28.9446	1.06909	0.534546	−	0.845140i	$-0.320483\pi$
$733$	28.9446	1.06909	0.534546	+	0.845140i	$0.320483\pi$
$734$	9.51201	0.351095
$735$	0	0
$736$	−3.56676	−0.131473
$737$	28.5213	1.05060
$738$	0	0
$739$	42.6668	1.56952	0.784761	−	0.619798i	$-0.212785\pi$
$739$	42.6668	1.56952	0.784761	+	0.619798i	$0.212785\pi$
$740$	−5.45278	−0.200448
$741$	0	0
$742$	−1.03765	−0.0380933
$743$	7.70836	0.282792	0.141396	−	0.989953i	$-0.454841\pi$
$743$	7.70836	0.282792	0.141396	+	0.989953i	$0.454841\pi$
$744$	0	0
$745$	−1.46245	−0.0535799
$746$	13.6335	0.499156
$747$	0	0
$748$	−7.03798	−0.257334
$749$	−4.35393	−0.159089
$750$	0	0
$751$	11.5623	0.421915	0.210958	−	0.977495i	$-0.432342\pi$
$751$	11.5623	0.421915	0.210958	+	0.977495i	$0.432342\pi$
$752$	0.965913	0.0352232
$753$	0	0
$754$	−6.05268	−0.220426
$755$	−4.37028	−0.159051
$756$	0	0
$757$	37.6358	1.36790	0.683948	−	0.729530i	$-0.260261\pi$
$757$	37.6358	1.36790	0.683948	+	0.729530i	$0.260261\pi$
$758$	8.91740	0.323894
$759$	0	0
$760$	−9.31308	−0.337821
$761$	−42.9837	−1.55816	−0.779079	−	0.626926i	$-0.784313\pi$
$761$	−42.9837	−1.55816	−0.779079	+	0.626926i	$0.784313\pi$
$762$	0	0
$763$	−2.82658	−0.102329
$764$	9.14557	0.330875
$765$	0	0
$766$	14.8064	0.534979
$767$	−4.28279	−0.154643
$768$	0	0
$769$	11.1193	0.400972	0.200486	−	0.979697i	$-0.435748\pi$
$769$	11.1193	0.400972	0.200486	+	0.979697i	$0.435748\pi$
$770$	−1.26491	−0.0455843
$771$	0	0
$772$	−1.13247	−0.0407586
$773$	−19.7707	−0.711102	−0.355551	−	0.934657i	$-0.615707\pi$
$773$	−19.7707	−0.711102	−0.355551	+	0.934657i	$0.615707\pi$
$774$	0	0
$775$	−24.8467	−0.892518
$776$	16.7176	0.600125
$777$	0	0
$778$	−29.6168	−1.06181
$779$	7.98433	0.286068
$780$	0	0
$781$	18.4886	0.661574
$782$	−9.10062	−0.325437
$783$	0	0
$784$	−6.90168	−0.246488
$785$	−12.3617	−0.441210
$786$	0	0
$787$	32.3817	1.15428	0.577141	−	0.816644i	$-0.304168\pi$
$787$	32.3817	1.15428	0.577141	+	0.816644i	$0.304168\pi$
$788$	−19.7062	−0.702004
$789$	0	0
$790$	0.213815	0.00760721
$791$	−1.60362	−0.0570183
$792$	0	0
$793$	6.60954	0.234712
$794$	−7.89296	−0.280111
$795$	0	0
$796$	18.2259	0.645999
$797$	−37.1831	−1.31709	−0.658546	−	0.752541i	$-0.728828\pi$
$797$	−37.1831	−1.31709	−0.658546	+	0.752541i	$0.728828\pi$
$798$	0	0
$799$	2.46453	0.0871890
$800$	−2.86125	−0.101160
$801$	0	0
$802$	−23.9408	−0.845377
$803$	10.8544	0.383045
$804$	0	0
$805$	−1.63563	−0.0576483
$806$	9.13523	0.321775
$807$	0	0
$808$	−1.72005	−0.0605112
$809$	−18.7167	−0.658045	−0.329023	−	0.944322i	$-0.606719\pi$
$809$	−18.7167	−0.658045	−0.329023	+	0.944322i	$0.606719\pi$
$810$	0	0
$811$	11.8307	0.415433	0.207717	−	0.978189i	$-0.433397\pi$
$811$	11.8307	0.415433	0.207717	+	0.978189i	$0.433397\pi$
$812$	1.80414	0.0633129
$813$	0	0
$814$	−10.2846	−0.360477
$815$	−2.86470	−0.100346
$816$	0	0
$817$	11.3694	0.397766
$818$	−17.7283	−0.619854
$819$	0	0
$820$	−1.83361	−0.0640323
$821$	−40.7535	−1.42231	−0.711153	−	0.703037i	$-0.751827\pi$
$821$	−40.7535	−1.42231	−0.711153	+	0.703037i	$0.751827\pi$
$822$	0	0
$823$	−29.0663	−1.01319	−0.506594	−	0.862185i	$-0.669096\pi$
$823$	−29.0663	−1.01319	−0.506594	+	0.862185i	$0.669096\pi$
$824$	10.6160	0.369825
$825$	0	0
$826$	1.27658	0.0444181
$827$	−6.91429	−0.240433	−0.120217	−	0.992748i	$-0.538359\pi$
$827$	−6.91429	−0.240433	−0.120217	+	0.992748i	$0.538359\pi$
$828$	0	0
$829$	13.1050	0.455155	0.227577	−	0.973760i	$-0.426920\pi$
$829$	13.1050	0.455155	0.227577	+	0.973760i	$0.426920\pi$
$830$	−2.76858	−0.0960990
$831$	0	0
$832$	1.05198	0.0364708
$833$	−17.6097	−0.610139
$834$	0	0
$835$	−9.67276	−0.334740
$836$	−17.5656	−0.607521
$837$	0	0
$838$	4.45345	0.153842
$839$	−13.6882	−0.472568	−0.236284	−	0.971684i	$-0.575930\pi$
$839$	−13.6882	−0.472568	−0.236284	+	0.971684i	$0.575930\pi$
$840$	0	0
$841$	4.10406	0.141519
$842$	26.5721	0.915736
$843$	0	0
$844$	−18.6031	−0.640347
$845$	17.3934	0.598351
$846$	0	0
$847$	1.06344	0.0365403
$848$	3.30919	0.113638
$849$	0	0
$850$	−7.30049	−0.250405
$851$	−13.2988	−0.455877
$852$	0	0
$853$	9.98601	0.341915	0.170957	−	0.985278i	$-0.445314\pi$
$853$	9.98601	0.341915	0.170957	+	0.985278i	$0.445314\pi$
$854$	−1.97012	−0.0674163
$855$	0	0
$856$	13.8852	0.474586
$857$	−46.7246	−1.59608	−0.798042	−	0.602602i	$-0.794130\pi$
$857$	−46.7246	−1.59608	−0.798042	+	0.602602i	$0.794130\pi$
$858$	0	0
$859$	−33.9698	−1.15903	−0.579517	−	0.814960i	$-0.696759\pi$
$859$	−33.9698	−1.15903	−0.579517	+	0.814960i	$0.696759\pi$
$860$	−2.61099	−0.0890341
$861$	0	0
$862$	17.0238	0.579834
$863$	−43.1637	−1.46931	−0.734655	−	0.678441i	$-0.762656\pi$
$863$	−43.1637	−1.46931	−0.734655	+	0.678441i	$0.762656\pi$
$864$	0	0
$865$	7.28597	0.247731
$866$	−26.5129	−0.900945
$867$	0	0
$868$	−2.72297	−0.0924235
$869$	0.403283	0.0136804
$870$	0	0
$871$	−10.8774	−0.368567
$872$	9.01428	0.305262
$873$	0	0
$874$	−22.7137	−0.768301
$875$	−3.60497	−0.121870
$876$	0	0
$877$	−0.581697	−0.0196425	−0.00982125	−	0.999952i	$-0.503126\pi$
$877$	−0.581697	−0.0196425	−0.00982125	+	0.999952i	$0.503126\pi$
$878$	2.95153	0.0996093
$879$	0	0
$880$	4.03396	0.135985
$881$	−41.6685	−1.40385	−0.701924	−	0.712252i	$-0.747675\pi$
$881$	−41.6685	−1.40385	−0.701924	+	0.712252i	$0.747675\pi$
$882$	0	0
$883$	−5.78325	−0.194622	−0.0973109	−	0.995254i	$-0.531024\pi$
$883$	−5.78325	−0.194622	−0.0973109	+	0.995254i	$0.531024\pi$
$884$	2.68413	0.0902771
$885$	0	0
$886$	−26.8435	−0.901826
$887$	42.0468	1.41179	0.705896	−	0.708316i	$-0.250545\pi$
$887$	42.0468	1.41179	0.705896	+	0.708316i	$0.250545\pi$
$888$	0	0
$889$	1.03337	0.0346581
$890$	−23.6757	−0.793610
$891$	0	0
$892$	−4.12033	−0.137959
$893$	6.15108	0.205838
$894$	0	0
$895$	−28.5632	−0.954762
$896$	−0.313566	−0.0104755
$897$	0	0
$898$	−15.9574	−0.532506
$899$	−49.9635	−1.66638
$900$	0	0
$901$	8.44341	0.281291
$902$	−3.45841	−0.115153
$903$	0	0
$904$	5.11414	0.170094
$905$	−23.4788	−0.780460
$906$	0	0
$907$	−9.55205	−0.317171	−0.158585	−	0.987345i	$-0.550693\pi$
$907$	−9.55205	−0.317171	−0.158585	+	0.987345i	$0.550693\pi$
$908$	1.58651	0.0526501
$909$	0	0
$910$	0.482411	0.0159918
$911$	50.1964	1.66308	0.831540	−	0.555465i	$-0.187460\pi$
$911$	50.1964	1.66308	0.831540	+	0.555465i	$0.187460\pi$
$912$	0	0
$913$	−5.22190	−0.172820
$914$	15.2612	0.504794
$915$	0	0
$916$	17.5566	0.580086
$917$	1.27483	0.0420987
$918$	0	0
$919$	−30.7455	−1.01420	−0.507100	−	0.861887i	$-0.669282\pi$
$919$	−30.7455	−1.01420	−0.507100	+	0.861887i	$0.669282\pi$
$920$	5.21620	0.171973
$921$	0	0
$922$	−8.85470	−0.291614
$923$	−7.05114	−0.232091
$924$	0	0
$925$	−10.6683	−0.350770
$926$	−42.5568	−1.39850
$927$	0	0
$928$	−5.75361	−0.188872
$929$	54.4800	1.78743	0.893715	−	0.448635i	$-0.148090\pi$
$929$	54.4800	1.78743	0.893715	+	0.448635i	$0.148090\pi$
$930$	0	0
$931$	−43.9509	−1.44043
$932$	24.2618	0.794723
$933$	0	0
$934$	−10.4138	−0.340750
$935$	10.2927	0.336606
$936$	0	0
$937$	−6.75685	−0.220737	−0.110368	−	0.993891i	$-0.535203\pi$
$937$	−6.75685	−0.220737	−0.110368	+	0.993891i	$0.535203\pi$
$938$	3.24226	0.105864
$939$	0	0
$940$	−1.41260	−0.0460739
$941$	−15.1478	−0.493805	−0.246902	−	0.969040i	$-0.579413\pi$
$941$	−15.1478	−0.493805	−0.246902	+	0.969040i	$0.579413\pi$
$942$	0	0
$943$	−4.47198	−0.145628
$944$	−4.07118	−0.132506
$945$	0	0
$946$	−4.92467	−0.160115
$947$	−18.7274	−0.608560	−0.304280	−	0.952583i	$-0.598416\pi$
$947$	−18.7274	−0.608560	−0.304280	+	0.952583i	$0.598416\pi$
$948$	0	0
$949$	−4.13965	−0.134379
$950$	−18.2208	−0.591162
$951$	0	0
$952$	−0.800067	−0.0259303
$953$	−1.63703	−0.0530287	−0.0265143	−	0.999648i	$-0.508441\pi$
$953$	−1.63703	−0.0530287	−0.0265143	+	0.999648i	$0.508441\pi$
$954$	0	0
$955$	−13.3749	−0.432802
$956$	1.13063	0.0365673
$957$	0	0
$958$	28.8406	0.931798
$959$	−1.46352	−0.0472594
$960$	0	0
$961$	44.4093	1.43256
$962$	3.92234	0.126461
$963$	0	0
$964$	28.6226	0.921872
$965$	1.65618	0.0533144
$966$	0	0
$967$	−1.53179	−0.0492591	−0.0246295	−	0.999697i	$-0.507841\pi$
$967$	−1.53179	−0.0492591	−0.0246295	+	0.999697i	$0.507841\pi$
$968$	−3.39144	−0.109005
$969$	0	0
$970$	−24.4486	−0.784996
$971$	−8.75164	−0.280854	−0.140427	−	0.990091i	$-0.544847\pi$
$971$	−8.75164	−0.280854	−0.140427	+	0.990091i	$0.544847\pi$
$972$	0	0
$973$	−5.76132	−0.184699
$974$	−5.12433	−0.164194
$975$	0	0
$976$	6.28296	0.201112
$977$	43.2387	1.38333	0.691664	−	0.722220i	$-0.256878\pi$
$977$	43.2387	1.38333	0.691664	+	0.722220i	$0.256878\pi$
$978$	0	0
$979$	−44.6553	−1.42719
$980$	10.0933	0.322420
$981$	0	0
$982$	6.54841	0.208968
$983$	−25.3531	−0.808637	−0.404318	−	0.914618i	$-0.632491\pi$
$983$	−25.3531	−0.808637	−0.404318	+	0.914618i	$0.632491\pi$
$984$	0	0
$985$	28.8193	0.918258
$986$	−14.6804	−0.467519
$987$	0	0
$988$	6.69915	0.213128
$989$	−6.36795	−0.202489
$990$	0	0
$991$	−30.4271	−0.966549	−0.483275	−	0.875469i	$-0.660553\pi$
$991$	−30.4271	−0.966549	−0.483275	+	0.875469i	$0.660553\pi$
$992$	8.68386	0.275713
$993$	0	0
$994$	2.10176	0.0666636
$995$	−26.6544	−0.845000
$996$	0	0
$997$	−35.2729	−1.11710	−0.558551	−	0.829470i	$-0.688643\pi$
$997$	−35.2729	−1.11710	−0.558551	+	0.829470i	$0.688643\pi$
$998$	−33.6700	−1.06581
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.r.1.6	yes	14
3.2	odd	2		8046.2.a.q.1.9	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.q.1.9	✓	14	3.2	odd	2
8046.2.a.r.1.6	yes	14	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.46245 q^{5} -0.313566 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.46245 q^{5} -0.313566 q^{7} +1.00000 q^{8} -1.46245 q^{10} -2.75836 q^{11} +1.05198 q^{13} -0.313566 q^{14} +1.00000 q^{16} +2.55151 q^{17} +6.36814 q^{19} -1.46245 q^{20} -2.75836 q^{22} -3.56676 q^{23} -2.86125 q^{25} +1.05198 q^{26} -0.313566 q^{28} -5.75361 q^{29} +8.68386 q^{31} +1.00000 q^{32} +2.55151 q^{34} +0.458574 q^{35} +3.72853 q^{37} +6.36814 q^{38} -1.46245 q^{40} +1.25379 q^{41} +1.78536 q^{43} -2.75836 q^{44} -3.56676 q^{46} +0.965913 q^{47} -6.90168 q^{49} -2.86125 q^{50} +1.05198 q^{52} +3.30919 q^{53} +4.03396 q^{55} -0.313566 q^{56} -5.75361 q^{58} -4.07118 q^{59} +6.28296 q^{61} +8.68386 q^{62} +1.00000 q^{64} -1.53846 q^{65} -10.3399 q^{67} +2.55151 q^{68} +0.458574 q^{70} -6.70274 q^{71} -3.93510 q^{73} +3.72853 q^{74} +6.36814 q^{76} +0.864930 q^{77} -0.146204 q^{79} -1.46245 q^{80} +1.25379 q^{82} +1.89312 q^{83} -3.73144 q^{85} +1.78536 q^{86} -2.75836 q^{88} +16.1891 q^{89} -0.329865 q^{91} -3.56676 q^{92} +0.965913 q^{94} -9.31308 q^{95} +16.7176 q^{97} -6.90168 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{2} + 14 q^{4} + 2 q^{5} + 4 q^{7} + 14 q^{8}+O(q^{10}) \) 14 * q + 14 * q^2 + 14 * q^4 + 2 * q^5 + 4 * q^7 + 14 * q^8	\( 14 q + 14 q^{2} + 14 q^{4} + 2 q^{5} + 4 q^{7} + 14 q^{8} + 2 q^{10} + 2 q^{11} + 4 q^{13} + 4 q^{14} + 14 q^{16} + 9 q^{17} + 14 q^{19} + 2 q^{20} + 2 q^{22} + 30 q^{23} + 18 q^{25} + 4 q^{26} + 4 q^{28} + 6 q^{29} + 11 q^{31} + 14 q^{32} + 9 q^{34} - 18 q^{35} + 13 q^{37} + 14 q^{38} + 2 q^{40} - 2 q^{41} + 12 q^{43} + 2 q^{44} + 30 q^{46} + 21 q^{47} + 32 q^{49} + 18 q^{50} + 4 q^{52} + 22 q^{53} - 7 q^{55} + 4 q^{56} + 6 q^{58} + 14 q^{59} + 31 q^{61} + 11 q^{62} + 14 q^{64} + 24 q^{67} + 9 q^{68} - 18 q^{70} + 28 q^{71} + 24 q^{73} + 13 q^{74} + 14 q^{76} + 16 q^{77} + 65 q^{79} + 2 q^{80} - 2 q^{82} - 15 q^{83} - 19 q^{85} + 12 q^{86} + 2 q^{88} - 11 q^{89} + 68 q^{91} + 30 q^{92} + 21 q^{94} + 8 q^{95} + 23 q^{97} + 32 q^{98}+O(q^{100}) \) 14 * q + 14 * q^2 + 14 * q^4 + 2 * q^5 + 4 * q^7 + 14 * q^8 + 2 * q^10 + 2 * q^11 + 4 * q^13 + 4 * q^14 + 14 * q^16 + 9 * q^17 + 14 * q^19 + 2 * q^20 + 2 * q^22 + 30 * q^23 + 18 * q^25 + 4 * q^26 + 4 * q^28 + 6 * q^29 + 11 * q^31 + 14 * q^32 + 9 * q^34 - 18 * q^35 + 13 * q^37 + 14 * q^38 + 2 * q^40 - 2 * q^41 + 12 * q^43 + 2 * q^44 + 30 * q^46 + 21 * q^47 + 32 * q^49 + 18 * q^50 + 4 * q^52 + 22 * q^53 - 7 * q^55 + 4 * q^56 + 6 * q^58 + 14 * q^59 + 31 * q^61 + 11 * q^62 + 14 * q^64 + 24 * q^67 + 9 * q^68 - 18 * q^70 + 28 * q^71 + 24 * q^73 + 13 * q^74 + 14 * q^76 + 16 * q^77 + 65 * q^79 + 2 * q^80 - 2 * q^82 - 15 * q^83 - 19 * q^85 + 12 * q^86 + 2 * q^88 - 11 * q^89 + 68 * q^91 + 30 * q^92 + 21 * q^94 + 8 * q^95 + 23 * q^97 + 32 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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