LMFDB - Embedded newform 6039.2.a.j.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
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} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} + \cdots - 1 $ x^14 - x^13 - 21x^12 + 20x^11 + 167x^10 - 148x^9 - 627x^8 + 497x^7 + 1123x^6 - 745x^5 - 802x^4 + 386x^3 + 74x^2 - 15x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.179763$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.179763 q^{2} -1.96769 q^{4} -2.20477 q^{5} +3.17177 q^{7} -0.713244 q^{8} +O(q^{10})$	\(q+0.179763 q^{2} -1.96769 q^{4} -2.20477 q^{5} +3.17177 q^{7} -0.713244 q^{8} -0.396336 q^{10} +1.00000 q^{11} +5.02358 q^{13} +0.570167 q^{14} +3.80716 q^{16} -1.08785 q^{17} +3.53893 q^{19} +4.33828 q^{20} +0.179763 q^{22} -1.91419 q^{23} -0.139010 q^{25} +0.903056 q^{26} -6.24104 q^{28} +5.13495 q^{29} -0.976010 q^{31} +2.11088 q^{32} -0.195555 q^{34} -6.99300 q^{35} +7.67629 q^{37} +0.636169 q^{38} +1.57254 q^{40} +2.80345 q^{41} -7.46749 q^{43} -1.96769 q^{44} -0.344102 q^{46} -5.09413 q^{47} +3.06010 q^{49} -0.0249889 q^{50} -9.88482 q^{52} +10.9620 q^{53} -2.20477 q^{55} -2.26224 q^{56} +0.923076 q^{58} -8.89797 q^{59} +1.00000 q^{61} -0.175451 q^{62} -7.23485 q^{64} -11.0758 q^{65} -6.12066 q^{67} +2.14054 q^{68} -1.25709 q^{70} -10.4162 q^{71} +12.2613 q^{73} +1.37992 q^{74} -6.96349 q^{76} +3.17177 q^{77} +15.6773 q^{79} -8.39388 q^{80} +0.503958 q^{82} -10.7598 q^{83} +2.39845 q^{85} -1.34238 q^{86} -0.713244 q^{88} +14.2391 q^{89} +15.9336 q^{91} +3.76653 q^{92} -0.915737 q^{94} -7.80250 q^{95} -13.8759 q^{97} +0.550094 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) $ 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7	$ 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7} + 6 q^{10} + 14 q^{11} + q^{13} + 7 q^{14} + 17 q^{16} + 9 q^{17} + 22 q^{19} - 23 q^{20} + q^{22} - q^{23} + 25 q^{25} - 4 q^{26} + 37 q^{28} + 6 q^{29} + 9 q^{31} - 4 q^{32} + 8 q^{34} - 18 q^{35} + 18 q^{37} - 8 q^{38} + 16 q^{40} + 25 q^{41} + 25 q^{43} + 15 q^{44} + 20 q^{46} - 36 q^{47} + 25 q^{49} - 2 q^{50} - 13 q^{52} - q^{55} + 40 q^{56} + 33 q^{58} - 17 q^{59} + 14 q^{61} + 13 q^{62} - 6 q^{64} + 61 q^{65} + 22 q^{67} - 66 q^{68} + 44 q^{70} + 13 q^{71} + 20 q^{73} + 12 q^{74} + 49 q^{76} + 9 q^{77} + 31 q^{79} - 88 q^{80} + 2 q^{82} - 32 q^{83} + 2 q^{85} + 14 q^{86} + 21 q^{89} + 45 q^{91} + 14 q^{92} - 31 q^{94} - 23 q^{95} + 37 q^{97} + 38 q^{98}+O(q^{100}) $ 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7 + 6 * q^10 + 14 * q^11 + q^13 + 7 * q^14 + 17 * q^16 + 9 * q^17 + 22 * q^19 - 23 * q^20 + q^22 - q^23 + 25 * q^25 - 4 * q^26 + 37 * q^28 + 6 * q^29 + 9 * q^31 - 4 * q^32 + 8 * q^34 - 18 * q^35 + 18 * q^37 - 8 * q^38 + 16 * q^40 + 25 * q^41 + 25 * q^43 + 15 * q^44 + 20 * q^46 - 36 * q^47 + 25 * q^49 - 2 * q^50 - 13 * q^52 - q^55 + 40 * q^56 + 33 * q^58 - 17 * q^59 + 14 * q^61 + 13 * q^62 - 6 * q^64 + 61 * q^65 + 22 * q^67 - 66 * q^68 + 44 * q^70 + 13 * q^71 + 20 * q^73 + 12 * q^74 + 49 * q^76 + 9 * q^77 + 31 * q^79 - 88 * q^80 + 2 * q^82 - 32 * q^83 + 2 * q^85 + 14 * q^86 + 21 * q^89 + 45 * q^91 + 14 * q^92 - 31 * q^94 - 23 * q^95 + 37 * q^97 + 38 * 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 $

$2$	0.179763	0.127112	0.0635560	−	0.997978i	$-0.479756\pi$
$2$	0.179763	0.127112	0.0635560	+	0.997978i	$0.479756\pi$
$3$	0	0
$3$	0	0
$4$	−1.96769	−0.983843
$5$	−2.20477	−0.986001	−0.493001	−	0.870029i	$-0.664100\pi$
$5$	−2.20477	−0.986001	−0.493001	+	0.870029i	$0.664100\pi$
$6$	0	0
$7$	3.17177	1.19881	0.599407	−	0.800444i	$-0.295403\pi$
$7$	3.17177	1.19881	0.599407	+	0.800444i	$0.295403\pi$
$8$	−0.713244	−0.252170
$9$	0	0
$10$	−0.396336	−0.125332
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.02358	1.39329	0.696645	−	0.717416i	$-0.254675\pi$
$13$	5.02358	1.39329	0.696645	+	0.717416i	$0.254675\pi$
$14$	0.570167	0.152384
$15$	0	0
$16$	3.80716	0.951789
$17$	−1.08785	−0.263842	−0.131921	−	0.991260i	$-0.542114\pi$
$17$	−1.08785	−0.263842	−0.131921	+	0.991260i	$0.542114\pi$
$18$	0	0
$19$	3.53893	0.811885	0.405943	−	0.913899i	$-0.366943\pi$
$19$	3.53893	0.811885	0.405943	+	0.913899i	$0.366943\pi$
$20$	4.33828	0.970070
$21$	0	0
$22$	0.179763	0.0383257
$23$	−1.91419	−0.399137	−0.199568	−	0.979884i	$-0.563954\pi$
$23$	−1.91419	−0.399137	−0.199568	+	0.979884i	$0.563954\pi$
$24$	0	0
$25$	−0.139010	−0.0278020
$26$	0.903056	0.177104
$27$	0	0
$28$	−6.24104	−1.17944
$29$	5.13495	0.953537	0.476768	−	0.879029i	$-0.341808\pi$
$29$	5.13495	0.953537	0.476768	+	0.879029i	$0.341808\pi$
$30$	0	0
$31$	−0.976010	−0.175297	−0.0876483	−	0.996151i	$-0.527935\pi$
$31$	−0.976010	−0.175297	−0.0876483	+	0.996151i	$0.527935\pi$
$32$	2.11088	0.373154
$33$	0	0
$34$	−0.195555	−0.0335374
$35$	−6.99300	−1.18203
$36$	0	0
$37$	7.67629	1.26197	0.630987	−	0.775793i	$-0.282650\pi$
$37$	7.67629	1.26197	0.630987	+	0.775793i	$0.282650\pi$
$38$	0.636169	0.103200
$39$	0	0
$40$	1.57254	0.248640
$41$	2.80345	0.437826	0.218913	−	0.975744i	$-0.429749\pi$
$41$	2.80345	0.437826	0.218913	+	0.975744i	$0.429749\pi$
$42$	0	0
$43$	−7.46749	−1.13878	−0.569391	−	0.822067i	$-0.692821\pi$
$43$	−7.46749	−1.13878	−0.569391	+	0.822067i	$0.692821\pi$
$44$	−1.96769	−0.296640
$45$	0	0
$46$	−0.344102	−0.0507350
$47$	−5.09413	−0.743055	−0.371527	−	0.928422i	$-0.621166\pi$
$47$	−5.09413	−0.743055	−0.371527	+	0.928422i	$0.621166\pi$
$48$	0	0
$49$	3.06010	0.437157
$50$	−0.0249889	−0.00353396
$51$	0	0
$52$	−9.88482	−1.37078
$53$	10.9620	1.50574	0.752870	−	0.658169i	$-0.228669\pi$
$53$	10.9620	1.50574	0.752870	+	0.658169i	$0.228669\pi$
$54$	0	0
$55$	−2.20477	−0.297290
$56$	−2.26224	−0.302305
$57$	0	0
$58$	0.923076	0.121206
$59$	−8.89797	−1.15842	−0.579209	−	0.815179i	$-0.696638\pi$
$59$	−8.89797	−1.15842	−0.579209	+	0.815179i	$0.696638\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−0.175451	−0.0222823
$63$	0	0
$64$	−7.23485	−0.904356
$65$	−11.0758	−1.37379
$66$	0	0
$67$	−6.12066	−0.747757	−0.373879	−	0.927478i	$-0.621972\pi$
$67$	−6.12066	−0.747757	−0.373879	+	0.927478i	$0.621972\pi$
$68$	2.14054	0.259579
$69$	0	0
$70$	−1.25709	−0.150250
$71$	−10.4162	−1.23618	−0.618091	−	0.786107i	$-0.712093\pi$
$71$	−10.4162	−1.23618	−0.618091	+	0.786107i	$0.712093\pi$
$72$	0	0
$73$	12.2613	1.43508	0.717538	−	0.696519i	$-0.245269\pi$
$73$	12.2613	1.43508	0.717538	+	0.696519i	$0.245269\pi$
$74$	1.37992	0.160412
$75$	0	0
$76$	−6.96349	−0.798767
$77$	3.17177	0.361456
$78$	0	0
$79$	15.6773	1.76384	0.881918	−	0.471402i	$-0.156252\pi$
$79$	15.6773	1.76384	0.881918	+	0.471402i	$0.156252\pi$
$80$	−8.39388	−0.938465
$81$	0	0
$82$	0.503958	0.0556529
$83$	−10.7598	−1.18105	−0.590523	−	0.807021i	$-0.701078\pi$
$83$	−10.7598	−1.18105	−0.590523	+	0.807021i	$0.701078\pi$
$84$	0	0
$85$	2.39845	0.260148
$86$	−1.34238	−0.144753
$87$	0	0
$88$	−0.713244	−0.0760321
$89$	14.2391	1.50934	0.754669	−	0.656106i	$-0.227797\pi$
$89$	14.2391	1.50934	0.754669	+	0.656106i	$0.227797\pi$
$90$	0	0
$91$	15.9336	1.67030
$92$	3.76653	0.392688
$93$	0	0
$94$	−0.915737	−0.0944511
$95$	−7.80250	−0.800520
$96$	0	0
$97$	−13.8759	−1.40888	−0.704440	−	0.709763i	$-0.748802\pi$
$97$	−13.8759	−1.40888	−0.704440	+	0.709763i	$0.748802\pi$
$98$	0.550094	0.0555678
$99$	0	0
$100$	0.273528	0.0273528
$101$	3.48657	0.346927	0.173463	−	0.984840i	$-0.444504\pi$
$101$	3.48657	0.346927	0.173463	+	0.984840i	$0.444504\pi$
$102$	0	0
$103$	−19.2415	−1.89592	−0.947960	−	0.318389i	$-0.896858\pi$
$103$	−19.2415	−1.89592	−0.947960	+	0.318389i	$0.896858\pi$
$104$	−3.58304	−0.351346
$105$	0	0
$106$	1.97056	0.191398
$107$	−2.82984	−0.273571	−0.136786	−	0.990601i	$-0.543677\pi$
$107$	−2.82984	−0.273571	−0.136786	+	0.990601i	$0.543677\pi$
$108$	0	0
$109$	7.46781	0.715287	0.357643	−	0.933858i	$-0.383580\pi$
$109$	7.46781	0.715287	0.357643	+	0.933858i	$0.383580\pi$
$110$	−0.396336	−0.0377892
$111$	0	0
$112$	12.0754	1.14102
$113$	9.07728	0.853919	0.426959	−	0.904271i	$-0.359585\pi$
$113$	9.07728	0.853919	0.426959	+	0.904271i	$0.359585\pi$
$114$	0	0
$115$	4.22035	0.393549
$116$	−10.1040	−0.938130
$117$	0	0
$118$	−1.59953	−0.147249
$119$	−3.45039	−0.316297
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.179763	0.0162750
$123$	0	0
$124$	1.92048	0.172464
$125$	11.3303	1.01341
$126$	0	0
$127$	8.94879	0.794077	0.397039	−	0.917802i	$-0.370038\pi$
$127$	8.94879	0.794077	0.397039	+	0.917802i	$0.370038\pi$
$128$	−5.52231	−0.488108
$129$	0	0
$130$	−1.99103	−0.174625
$131$	−13.9885	−1.22218	−0.611090	−	0.791561i	$-0.709268\pi$
$131$	−13.9885	−1.22218	−0.611090	+	0.791561i	$0.709268\pi$
$132$	0	0
$133$	11.2246	0.973300
$134$	−1.10027	−0.0950488
$135$	0	0
$136$	0.775901	0.0665329
$137$	−19.7992	−1.69156	−0.845781	−	0.533530i	$-0.820865\pi$
$137$	−19.7992	−1.69156	−0.845781	+	0.533530i	$0.820865\pi$
$138$	0	0
$139$	−4.24898	−0.360394	−0.180197	−	0.983631i	$-0.557674\pi$
$139$	−4.24898	−0.360394	−0.180197	+	0.983631i	$0.557674\pi$
$140$	13.7600	1.16293
$141$	0	0
$142$	−1.87246	−0.157133
$143$	5.02358	0.420093
$144$	0	0
$145$	−11.3214	−0.940188
$146$	2.20413	0.182415
$147$	0	0
$148$	−15.1045	−1.24158
$149$	−4.17125	−0.341722	−0.170861	−	0.985295i	$-0.554655\pi$
$149$	−4.17125	−0.341722	−0.170861	+	0.985295i	$0.554655\pi$
$150$	0	0
$151$	18.0934	1.47242	0.736212	−	0.676751i	$-0.236613\pi$
$151$	18.0934	1.47242	0.736212	+	0.676751i	$0.236613\pi$
$152$	−2.52412	−0.204733
$153$	0	0
$154$	0.570167	0.0459454
$155$	2.15187	0.172843
$156$	0	0
$157$	19.5799	1.56265	0.781323	−	0.624126i	$-0.214545\pi$
$157$	19.5799	1.56265	0.781323	+	0.624126i	$0.214545\pi$
$158$	2.81821	0.224205
$159$	0	0
$160$	−4.65399	−0.367930
$161$	−6.07137	−0.478491
$162$	0	0
$163$	14.3214	1.12174	0.560870	−	0.827904i	$-0.310467\pi$
$163$	14.3214	1.12174	0.560870	+	0.827904i	$0.310467\pi$
$164$	−5.51632	−0.430752
$165$	0	0
$166$	−1.93422	−0.150125
$167$	−0.178130	−0.0137841	−0.00689205	−	0.999976i	$-0.502194\pi$
$167$	−0.178130	−0.0137841	−0.00689205	+	0.999976i	$0.502194\pi$
$168$	0	0
$169$	12.2363	0.941258
$170$	0.431153	0.0330679
$171$	0	0
$172$	14.6937	1.12038
$173$	−0.504790	−0.0383785	−0.0191892	−	0.999816i	$-0.506108\pi$
$173$	−0.504790	−0.0383785	−0.0191892	+	0.999816i	$0.506108\pi$
$174$	0	0
$175$	−0.440907	−0.0333294
$176$	3.80716	0.286975
$177$	0	0
$178$	2.55966	0.191855
$179$	−12.0518	−0.900795	−0.450397	−	0.892828i	$-0.648718\pi$
$179$	−12.0518	−0.900795	−0.450397	+	0.892828i	$0.648718\pi$
$180$	0	0
$181$	23.8425	1.77220	0.886098	−	0.463497i	$-0.153406\pi$
$181$	23.8425	1.77220	0.886098	+	0.463497i	$0.153406\pi$
$182$	2.86428	0.212315
$183$	0	0
$184$	1.36529	0.100650
$185$	−16.9244	−1.24431
$186$	0	0
$187$	−1.08785	−0.0795512
$188$	10.0236	0.731049
$189$	0	0
$190$	−1.40260	−0.101756
$191$	26.0154	1.88241	0.941205	−	0.337836i	$-0.109695\pi$
$191$	26.0154	1.88241	0.941205	+	0.337836i	$0.109695\pi$
$192$	0	0
$193$	−3.89975	−0.280710	−0.140355	−	0.990101i	$-0.544824\pi$
$193$	−3.89975	−0.280710	−0.140355	+	0.990101i	$0.544824\pi$
$194$	−2.49437	−0.179086
$195$	0	0
$196$	−6.02131	−0.430093
$197$	21.9665	1.56505	0.782526	−	0.622618i	$-0.213931\pi$
$197$	21.9665	1.56505	0.782526	+	0.622618i	$0.213931\pi$
$198$	0	0
$199$	1.14129	0.0809040	0.0404520	−	0.999181i	$-0.487120\pi$
$199$	1.14129	0.0809040	0.0404520	+	0.999181i	$0.487120\pi$
$200$	0.0991481	0.00701083
$201$	0	0
$202$	0.626758	0.0440985
$203$	16.2869	1.14311
$204$	0	0
$205$	−6.18096	−0.431697
$206$	−3.45892	−0.240994
$207$	0	0
$208$	19.1255	1.32612
$209$	3.53893	0.244793
$210$	0	0
$211$	9.40163	0.647235	0.323617	−	0.946188i	$-0.395101\pi$
$211$	9.40163	0.647235	0.323617	+	0.946188i	$0.395101\pi$
$212$	−21.5697	−1.48141
$213$	0	0
$214$	−0.508702	−0.0347742
$215$	16.4641	1.12284
$216$	0	0
$217$	−3.09568	−0.210148
$218$	1.34244	0.0909215
$219$	0	0
$220$	4.33828	0.292487
$221$	−5.46488	−0.367608
$222$	0	0
$223$	2.42729	0.162543	0.0812715	−	0.996692i	$-0.474102\pi$
$223$	2.42729	0.162543	0.0812715	+	0.996692i	$0.474102\pi$
$224$	6.69520	0.447342
$225$	0	0
$226$	1.63176	0.108543
$227$	19.4929	1.29379	0.646893	−	0.762581i	$-0.276068\pi$
$227$	19.4929	1.29379	0.646893	+	0.762581i	$0.276068\pi$
$228$	0	0
$229$	14.8284	0.979889	0.489944	−	0.871754i	$-0.337017\pi$
$229$	14.8284	0.979889	0.489944	+	0.871754i	$0.337017\pi$
$230$	0.758664	0.0500248
$231$	0	0
$232$	−3.66248	−0.240453
$233$	9.17826	0.601288	0.300644	−	0.953736i	$-0.402798\pi$
$233$	9.17826	0.601288	0.300644	+	0.953736i	$0.402798\pi$
$234$	0	0
$235$	11.2314	0.732653
$236$	17.5084	1.13970
$237$	0	0
$238$	−0.620255	−0.0402051
$239$	−17.9788	−1.16296	−0.581478	−	0.813562i	$-0.697525\pi$
$239$	−17.9788	−1.16296	−0.581478	+	0.813562i	$0.697525\pi$
$240$	0	0
$241$	−11.8019	−0.760226	−0.380113	−	0.924940i	$-0.624115\pi$
$241$	−11.8019	−0.760226	−0.380113	+	0.924940i	$0.624115\pi$
$242$	0.179763	0.0115556
$243$	0	0
$244$	−1.96769	−0.125968
$245$	−6.74680	−0.431037
$246$	0	0
$247$	17.7781	1.13119
$248$	0.696134	0.0442045
$249$	0	0
$250$	2.03677	0.128817
$251$	17.0765	1.07786	0.538929	−	0.842351i	$-0.318829\pi$
$251$	17.0765	1.07786	0.538929	+	0.842351i	$0.318829\pi$
$252$	0	0
$253$	−1.91419	−0.120344
$254$	1.60867	0.100937
$255$	0	0
$256$	13.4770	0.842312
$257$	−18.9734	−1.18353	−0.591763	−	0.806112i	$-0.701568\pi$
$257$	−18.9734	−1.18353	−0.591763	+	0.806112i	$0.701568\pi$
$258$	0	0
$259$	24.3474	1.51287
$260$	21.7937	1.35159
$261$	0	0
$262$	−2.51462	−0.155354
$263$	−14.6726	−0.904753	−0.452376	−	0.891827i	$-0.649424\pi$
$263$	−14.6726	−0.904753	−0.452376	+	0.891827i	$0.649424\pi$
$264$	0	0
$265$	−24.1685	−1.48466
$266$	2.01778	0.123718
$267$	0	0
$268$	12.0435	0.735675
$269$	16.9485	1.03337	0.516685	−	0.856176i	$-0.327166\pi$
$269$	16.9485	1.03337	0.516685	+	0.856176i	$0.327166\pi$
$270$	0	0
$271$	6.59936	0.400883	0.200441	−	0.979706i	$-0.435762\pi$
$271$	6.59936	0.400883	0.200441	+	0.979706i	$0.435762\pi$
$272$	−4.14160	−0.251121
$273$	0	0
$274$	−3.55918	−0.215018
$275$	−0.139010	−0.00838262
$276$	0	0
$277$	−19.7287	−1.18538	−0.592692	−	0.805429i	$-0.701935\pi$
$277$	−19.7287	−1.18538	−0.592692	+	0.805429i	$0.701935\pi$
$278$	−0.763811	−0.0458104
$279$	0	0
$280$	4.98772	0.298073
$281$	23.8313	1.42165	0.710827	−	0.703367i	$-0.248321\pi$
$281$	23.8313	1.42165	0.710827	+	0.703367i	$0.248321\pi$
$282$	0	0
$283$	−14.5254	−0.863447	−0.431724	−	0.902006i	$-0.642094\pi$
$283$	−14.5254	−0.863447	−0.431724	+	0.902006i	$0.642094\pi$
$284$	20.4959	1.21621
$285$	0	0
$286$	0.903056	0.0533988
$287$	8.89190	0.524872
$288$	0	0
$289$	−15.8166	−0.930388
$290$	−2.03517	−0.119509
$291$	0	0
$292$	−24.1264	−1.41189
$293$	3.05620	0.178545	0.0892724	−	0.996007i	$-0.471546\pi$
$293$	3.05620	0.178545	0.0892724	+	0.996007i	$0.471546\pi$
$294$	0	0
$295$	19.6179	1.14220
$296$	−5.47507	−0.318232
$297$	0	0
$298$	−0.749838	−0.0434369
$299$	−9.61610	−0.556113
$300$	0	0
$301$	−23.6851	−1.36519
$302$	3.25254	0.187163
$303$	0	0
$304$	13.4732	0.772743
$305$	−2.20477	−0.126244
$306$	0	0
$307$	−18.3076	−1.04487	−0.522436	−	0.852679i	$-0.674976\pi$
$307$	−18.3076	−1.04487	−0.522436	+	0.852679i	$0.674976\pi$
$308$	−6.24104	−0.355616
$309$	0	0
$310$	0.386828	0.0219704
$311$	20.8499	1.18229	0.591145	−	0.806565i	$-0.298676\pi$
$311$	20.8499	1.18229	0.591145	+	0.806565i	$0.298676\pi$
$312$	0	0
$313$	−3.49928	−0.197791	−0.0988956	−	0.995098i	$-0.531531\pi$
$313$	−3.49928	−0.197791	−0.0988956	+	0.995098i	$0.531531\pi$
$314$	3.51975	0.198631
$315$	0	0
$316$	−30.8480	−1.73534
$317$	18.4416	1.03578	0.517891	−	0.855447i	$-0.326717\pi$
$317$	18.4416	1.03578	0.517891	+	0.855447i	$0.326717\pi$
$318$	0	0
$319$	5.13495	0.287502
$320$	15.9512	0.891696
$321$	0	0
$322$	−1.09141	−0.0608219
$323$	−3.84981	−0.214209
$324$	0	0
$325$	−0.698328	−0.0387362
$326$	2.57447	0.142587
$327$	0	0
$328$	−1.99955	−0.110407
$329$	−16.1574	−0.890785
$330$	0	0
$331$	6.78274	0.372813	0.186407	−	0.982473i	$-0.440316\pi$
$331$	6.78274	0.372813	0.186407	+	0.982473i	$0.440316\pi$
$332$	21.1720	1.16196
$333$	0	0
$334$	−0.0320212	−0.00175212
$335$	13.4946	0.737289
$336$	0	0
$337$	35.9736	1.95961	0.979803	−	0.199965i	$-0.0640827\pi$
$337$	35.9736	1.95961	0.979803	+	0.199965i	$0.0640827\pi$
$338$	2.19965	0.119645
$339$	0	0
$340$	−4.71939	−0.255945
$341$	−0.976010	−0.0528539
$342$	0	0
$343$	−12.4964	−0.674745
$344$	5.32615	0.287167
$345$	0	0
$346$	−0.0907427	−0.00487836
$347$	8.50844	0.456757	0.228379	−	0.973572i	$-0.426658\pi$
$347$	8.50844	0.456757	0.228379	+	0.973572i	$0.426658\pi$
$348$	0	0
$349$	13.9195	0.745096	0.372548	−	0.928013i	$-0.378484\pi$
$349$	13.9195	0.745096	0.372548	+	0.928013i	$0.378484\pi$
$350$	−0.0792589	−0.00423657
$351$	0	0
$352$	2.11088	0.112510
$353$	21.3602	1.13689	0.568444	−	0.822722i	$-0.307546\pi$
$353$	21.3602	1.13689	0.568444	+	0.822722i	$0.307546\pi$
$354$	0	0
$355$	22.9654	1.21888
$356$	−28.0180	−1.48495
$357$	0	0
$358$	−2.16647	−0.114502
$359$	10.1841	0.537495	0.268748	−	0.963211i	$-0.413390\pi$
$359$	10.1841	0.537495	0.268748	+	0.963211i	$0.413390\pi$
$360$	0	0
$361$	−6.47601	−0.340842
$362$	4.28600	0.225267
$363$	0	0
$364$	−31.3523	−1.64331
$365$	−27.0333	−1.41499
$366$	0	0
$367$	−7.71600	−0.402772	−0.201386	−	0.979512i	$-0.564545\pi$
$367$	−7.71600	−0.402772	−0.201386	+	0.979512i	$0.564545\pi$
$368$	−7.28763	−0.379894
$369$	0	0
$370$	−3.04239	−0.158166
$371$	34.7688	1.80510
$372$	0	0
$373$	−17.3992	−0.900897	−0.450448	−	0.892802i	$-0.648736\pi$
$373$	−17.3992	−0.900897	−0.450448	+	0.892802i	$0.648736\pi$
$374$	−0.195555	−0.0101119
$375$	0	0
$376$	3.63336	0.187376
$377$	25.7958	1.32855
$378$	0	0
$379$	−2.31368	−0.118846	−0.0594230	−	0.998233i	$-0.518926\pi$
$379$	−2.31368	−0.118846	−0.0594230	+	0.998233i	$0.518926\pi$
$380$	15.3529	0.787585
$381$	0	0
$382$	4.67662	0.239277
$383$	8.99228	0.459484	0.229742	−	0.973252i	$-0.426212\pi$
$383$	8.99228	0.459484	0.229742	+	0.973252i	$0.426212\pi$
$384$	0	0
$385$	−6.99300	−0.356396
$386$	−0.701032	−0.0356816
$387$	0	0
$388$	27.3033	1.38612
$389$	7.65447	0.388097	0.194048	−	0.980992i	$-0.437838\pi$
$389$	7.65447	0.388097	0.194048	+	0.980992i	$0.437838\pi$
$390$	0	0
$391$	2.08235	0.105309
$392$	−2.18260	−0.110238
$393$	0	0
$394$	3.94878	0.198937
$395$	−34.5648	−1.73914
$396$	0	0
$397$	26.1237	1.31111	0.655555	−	0.755147i	$-0.272435\pi$
$397$	26.1237	1.31111	0.655555	+	0.755147i	$0.272435\pi$
$398$	0.205162	0.0102839
$399$	0	0
$400$	−0.529232	−0.0264616
$401$	27.3416	1.36537	0.682686	−	0.730712i	$-0.260812\pi$
$401$	27.3416	1.36537	0.682686	+	0.730712i	$0.260812\pi$
$402$	0	0
$403$	−4.90306	−0.244239
$404$	−6.86047	−0.341321
$405$	0	0
$406$	2.92778	0.145303
$407$	7.67629	0.380500
$408$	0	0
$409$	−24.2671	−1.19993	−0.599965	−	0.800027i	$-0.704819\pi$
$409$	−24.2671	−1.19993	−0.599965	+	0.800027i	$0.704819\pi$
$410$	−1.11111	−0.0548738
$411$	0	0
$412$	37.8612	1.86529
$413$	−28.2223	−1.38873
$414$	0	0
$415$	23.7229	1.16451
$416$	10.6042	0.519911
$417$	0	0
$418$	0.636169	0.0311161
$419$	−16.5732	−0.809655	−0.404828	−	0.914393i	$-0.632668\pi$
$419$	−16.5732	−0.809655	−0.404828	+	0.914393i	$0.632668\pi$
$420$	0	0
$421$	0.423105	0.0206209	0.0103104	−	0.999947i	$-0.496718\pi$
$421$	0.423105	0.0206209	0.0103104	+	0.999947i	$0.496718\pi$
$422$	1.69007	0.0822713
$423$	0	0
$424$	−7.81856	−0.379703
$425$	0.151222	0.00733532
$426$	0	0
$427$	3.17177	0.153493
$428$	5.56824	0.269151
$429$	0	0
$430$	2.95964	0.142726
$431$	−24.3394	−1.17239	−0.586195	−	0.810170i	$-0.699375\pi$
$431$	−24.3394	−1.17239	−0.586195	+	0.810170i	$0.699375\pi$
$432$	0	0
$433$	5.45603	0.262200	0.131100	−	0.991369i	$-0.458149\pi$
$433$	5.45603	0.262200	0.131100	+	0.991369i	$0.458149\pi$
$434$	−0.556489	−0.0267123
$435$	0	0
$436$	−14.6943	−0.703730
$437$	−6.77419	−0.324053
$438$	0	0
$439$	5.71582	0.272801	0.136401	−	0.990654i	$-0.456447\pi$
$439$	5.71582	0.272801	0.136401	+	0.990654i	$0.456447\pi$
$440$	1.57254	0.0749677
$441$	0	0
$442$	−0.982386	−0.0467273
$443$	−15.1986	−0.722106	−0.361053	−	0.932545i	$-0.617583\pi$
$443$	−15.1986	−0.722106	−0.361053	+	0.932545i	$0.617583\pi$
$444$	0	0
$445$	−31.3938	−1.48821
$446$	0.436337	0.0206612
$447$	0	0
$448$	−22.9473	−1.08416
$449$	1.16467	0.0549641	0.0274820	−	0.999622i	$-0.491251\pi$
$449$	1.16467	0.0549641	0.0274820	+	0.999622i	$0.491251\pi$
$450$	0	0
$451$	2.80345	0.132010
$452$	−17.8612	−0.840122
$453$	0	0
$454$	3.50410	0.164456
$455$	−35.1299	−1.64691
$456$	0	0
$457$	15.8354	0.740748	0.370374	−	0.928883i	$-0.379230\pi$
$457$	15.8354	0.740748	0.370374	+	0.928883i	$0.379230\pi$
$458$	2.66560	0.124556
$459$	0	0
$460$	−8.30431	−0.387191
$461$	−14.3447	−0.668097	−0.334048	−	0.942556i	$-0.608415\pi$
$461$	−14.3447	−0.668097	−0.334048	+	0.942556i	$0.608415\pi$
$462$	0	0
$463$	24.6422	1.14522	0.572611	−	0.819827i	$-0.305931\pi$
$463$	24.6422	1.14522	0.572611	+	0.819827i	$0.305931\pi$
$464$	19.5496	0.907566
$465$	0	0
$466$	1.64992	0.0764309
$467$	9.26543	0.428753	0.214377	−	0.976751i	$-0.431228\pi$
$467$	9.26543	0.428753	0.214377	+	0.976751i	$0.431228\pi$
$468$	0	0
$469$	−19.4133	−0.896422
$470$	2.01899	0.0931289
$471$	0	0
$472$	6.34643	0.292118
$473$	−7.46749	−0.343356
$474$	0	0
$475$	−0.491946	−0.0225720
$476$	6.78929	0.311187
$477$	0	0
$478$	−3.23194	−0.147825
$479$	−4.42880	−0.202357	−0.101178	−	0.994868i	$-0.532261\pi$
$479$	−4.42880	−0.202357	−0.101178	+	0.994868i	$0.532261\pi$
$480$	0	0
$481$	38.5625	1.75830
$482$	−2.12154	−0.0966337
$483$	0	0
$484$	−1.96769	−0.0894402
$485$	30.5930	1.38916
$486$	0	0
$487$	24.2935	1.10084	0.550422	−	0.834886i	$-0.314467\pi$
$487$	24.2935	1.10084	0.550422	+	0.834886i	$0.314467\pi$
$488$	−0.713244	−0.0322871
$489$	0	0
$490$	−1.21283	−0.0547899
$491$	42.5903	1.92207	0.961037	−	0.276420i	$-0.0891481\pi$
$491$	42.5903	1.92207	0.961037	+	0.276420i	$0.0891481\pi$
$492$	0	0
$493$	−5.58604	−0.251583
$494$	3.19585	0.143788
$495$	0	0
$496$	−3.71582	−0.166845
$497$	−33.0379	−1.48195
$498$	0	0
$499$	29.9642	1.34138	0.670690	−	0.741737i	$-0.265998\pi$
$499$	29.9642	1.34138	0.670690	+	0.741737i	$0.265998\pi$
$500$	−22.2945	−0.997040
$501$	0	0
$502$	3.06973	0.137009
$503$	−23.4580	−1.04594	−0.522970	−	0.852351i	$-0.675176\pi$
$503$	−23.4580	−1.04594	−0.522970	+	0.852351i	$0.675176\pi$
$504$	0	0
$505$	−7.68707	−0.342070
$506$	−0.344102	−0.0152972
$507$	0	0
$508$	−17.6084	−0.781247
$509$	−12.5052	−0.554284	−0.277142	−	0.960829i	$-0.589387\pi$
$509$	−12.5052	−0.554284	−0.277142	+	0.960829i	$0.589387\pi$
$510$	0	0
$511$	38.8900	1.72039
$512$	13.4673	0.595176
$513$	0	0
$514$	−3.41072	−0.150440
$515$	42.4230	1.86938
$516$	0	0
$517$	−5.09413	−0.224039
$518$	4.37677	0.192304
$519$	0	0
$520$	7.89976	0.346428
$521$	17.7110	0.775931	0.387966	−	0.921674i	$-0.373178\pi$
$521$	17.7110	0.775931	0.387966	+	0.921674i	$0.373178\pi$
$522$	0	0
$523$	−24.7263	−1.08121	−0.540603	−	0.841278i	$-0.681804\pi$
$523$	−24.7263	−1.08121	−0.540603	+	0.841278i	$0.681804\pi$
$524$	27.5249	1.20243
$525$	0	0
$526$	−2.63760	−0.115005
$527$	1.06175	0.0462505
$528$	0	0
$529$	−19.3359	−0.840690
$530$	−4.34462	−0.188718
$531$	0	0
$532$	−22.0866	−0.957574
$533$	14.0834	0.610019
$534$	0	0
$535$	6.23914	0.269742
$536$	4.36552	0.188562
$537$	0	0
$538$	3.04672	0.131354
$539$	3.06010	0.131808
$540$	0	0
$541$	−14.3455	−0.616759	−0.308380	−	0.951263i	$-0.599787\pi$
$541$	−14.3455	−0.616759	−0.308380	+	0.951263i	$0.599787\pi$
$542$	1.18632	0.0509570
$543$	0	0
$544$	−2.29631	−0.0984535
$545$	−16.4648	−0.705273
$546$	0	0
$547$	−13.1006	−0.560142	−0.280071	−	0.959979i	$-0.590358\pi$
$547$	−13.1006	−0.560142	−0.280071	+	0.959979i	$0.590358\pi$
$548$	38.9586	1.66423
$549$	0	0
$550$	−0.0249889	−0.00106553
$551$	18.1722	0.774162
$552$	0	0
$553$	49.7248	2.11451
$554$	−3.54650	−0.150676
$555$	0	0
$556$	8.36066	0.354571
$557$	−26.6862	−1.13073	−0.565365	−	0.824841i	$-0.691265\pi$
$557$	−26.6862	−1.13073	−0.565365	+	0.824841i	$0.691265\pi$
$558$	0	0
$559$	−37.5135	−1.58665
$560$	−26.6234	−1.12505
$561$	0	0
$562$	4.28399	0.180709
$563$	35.8990	1.51296	0.756482	−	0.654015i	$-0.226917\pi$
$563$	35.8990	1.51296	0.756482	+	0.654015i	$0.226917\pi$
$564$	0	0
$565$	−20.0133	−0.841965
$566$	−2.61114	−0.109754
$567$	0	0
$568$	7.42933	0.311728
$569$	−43.3017	−1.81530	−0.907650	−	0.419727i	$-0.862126\pi$
$569$	−43.3017	−1.81530	−0.907650	+	0.419727i	$0.862126\pi$
$570$	0	0
$571$	18.0647	0.755983	0.377992	−	0.925809i	$-0.376615\pi$
$571$	18.0647	0.755983	0.377992	+	0.925809i	$0.376615\pi$
$572$	−9.88482	−0.413305
$573$	0	0
$574$	1.59844	0.0667175
$575$	0.266092	0.0110968
$576$	0	0
$577$	40.3884	1.68139	0.840696	−	0.541507i	$-0.182146\pi$
$577$	40.3884	1.68139	0.840696	+	0.541507i	$0.182146\pi$
$578$	−2.84324	−0.118263
$579$	0	0
$580$	22.2769	0.924997
$581$	−34.1277	−1.41585
$582$	0	0
$583$	10.9620	0.453998
$584$	−8.74530	−0.361883
$585$	0	0
$586$	0.549392	0.0226952
$587$	−8.96421	−0.369992	−0.184996	−	0.982739i	$-0.559227\pi$
$587$	−8.96421	−0.369992	−0.184996	+	0.982739i	$0.559227\pi$
$588$	0	0
$589$	−3.45403	−0.142321
$590$	3.52659	0.145187
$591$	0	0
$592$	29.2248	1.20113
$593$	26.1692	1.07464	0.537321	−	0.843378i	$-0.319436\pi$
$593$	26.1692	1.07464	0.537321	+	0.843378i	$0.319436\pi$
$594$	0	0
$595$	7.60731	0.311869
$596$	8.20770	0.336201
$597$	0	0
$598$	−1.72862	−0.0706886
$599$	−25.8018	−1.05423	−0.527116	−	0.849794i	$-0.676726\pi$
$599$	−25.8018	−1.05423	−0.527116	+	0.849794i	$0.676726\pi$
$600$	0	0
$601$	−20.9671	−0.855267	−0.427633	−	0.903952i	$-0.640653\pi$
$601$	−20.9671	−0.855267	−0.427633	+	0.903952i	$0.640653\pi$
$602$	−4.25772	−0.173532
$603$	0	0
$604$	−35.6022	−1.44863
$605$	−2.20477	−0.0896365
$606$	0	0
$607$	15.8671	0.644025	0.322013	−	0.946735i	$-0.395641\pi$
$607$	15.8671	0.644025	0.322013	+	0.946735i	$0.395641\pi$
$608$	7.47023	0.302958
$609$	0	0
$610$	−0.396336	−0.0160472
$611$	−25.5908	−1.03529
$612$	0	0
$613$	−30.5257	−1.23292	−0.616462	−	0.787385i	$-0.711434\pi$
$613$	−30.5257	−1.23292	−0.616462	+	0.787385i	$0.711434\pi$
$614$	−3.29104	−0.132816
$615$	0	0
$616$	−2.26224	−0.0911484
$617$	−39.2025	−1.57823	−0.789116	−	0.614244i	$-0.789461\pi$
$617$	−39.2025	−1.57823	−0.789116	+	0.614244i	$0.789461\pi$
$618$	0	0
$619$	−5.80498	−0.233322	−0.116661	−	0.993172i	$-0.537219\pi$
$619$	−5.80498	−0.233322	−0.116661	+	0.993172i	$0.537219\pi$
$620$	−4.23421	−0.170050
$621$	0	0
$622$	3.74805	0.150283
$623$	45.1630	1.80942
$624$	0	0
$625$	−24.2856	−0.971425
$626$	−0.629043	−0.0251416
$627$	0	0
$628$	−38.5271	−1.53740
$629$	−8.35063	−0.332961
$630$	0	0
$631$	−10.9232	−0.434845	−0.217423	−	0.976078i	$-0.569765\pi$
$631$	−10.9232	−0.434845	−0.217423	+	0.976078i	$0.569765\pi$
$632$	−11.1818	−0.444787
$633$	0	0
$634$	3.31512	0.131660
$635$	−19.7300	−0.782961
$636$	0	0
$637$	15.3726	0.609086
$638$	0.923076	0.0365449
$639$	0	0
$640$	12.1754	0.481275
$641$	20.5925	0.813355	0.406677	−	0.913572i	$-0.366687\pi$
$641$	20.5925	0.813355	0.406677	+	0.913572i	$0.366687\pi$
$642$	0	0
$643$	−32.4513	−1.27975	−0.639877	−	0.768477i	$-0.721015\pi$
$643$	−32.4513	−1.27975	−0.639877	+	0.768477i	$0.721015\pi$
$644$	11.9465	0.470760
$645$	0	0
$646$	−0.692055	−0.0272285
$647$	−24.4762	−0.962258	−0.481129	−	0.876650i	$-0.659773\pi$
$647$	−24.4762	−0.962258	−0.481129	+	0.876650i	$0.659773\pi$
$648$	0	0
$649$	−8.89797	−0.349276
$650$	−0.125534	−0.00492384
$651$	0	0
$652$	−28.1800	−1.10362
$653$	−9.38845	−0.367399	−0.183699	−	0.982982i	$-0.558807\pi$
$653$	−9.38845	−0.367399	−0.183699	+	0.982982i	$0.558807\pi$
$654$	0	0
$655$	30.8413	1.20507
$656$	10.6732	0.416718
$657$	0	0
$658$	−2.90450	−0.113229
$659$	−5.44984	−0.212296	−0.106148	−	0.994350i	$-0.533852\pi$
$659$	−5.44984	−0.212296	−0.106148	+	0.994350i	$0.533852\pi$
$660$	0	0
$661$	5.23046	0.203441	0.101721	−	0.994813i	$-0.467565\pi$
$661$	5.23046	0.203441	0.101721	+	0.994813i	$0.467565\pi$
$662$	1.21929	0.0473890
$663$	0	0
$664$	7.67439	0.297824
$665$	−24.7477	−0.959675
$666$	0	0
$667$	−9.82929	−0.380592
$668$	0.350503	0.0135614
$669$	0	0
$670$	2.42584	0.0937182
$671$	1.00000	0.0386046
$672$	0	0
$673$	−50.7702	−1.95705	−0.978523	−	0.206136i	$-0.933911\pi$
$673$	−50.7702	−1.95705	−0.978523	+	0.206136i	$0.933911\pi$
$674$	6.46673	0.249089
$675$	0	0
$676$	−24.0773	−0.926049
$677$	38.0216	1.46129	0.730645	−	0.682757i	$-0.239219\pi$
$677$	38.0216	1.46129	0.730645	+	0.682757i	$0.239219\pi$
$678$	0	0
$679$	−44.0110	−1.68899
$680$	−1.71068	−0.0656015
$681$	0	0
$682$	−0.175451	−0.00671836
$683$	−20.3542	−0.778831	−0.389416	−	0.921062i	$-0.627323\pi$
$683$	−20.3542	−0.778831	−0.389416	+	0.921062i	$0.627323\pi$
$684$	0	0
$685$	43.6527	1.66788
$686$	−2.24640	−0.0857681
$687$	0	0
$688$	−28.4299	−1.08388
$689$	55.0683	2.09793
$690$	0	0
$691$	−2.62631	−0.0999094	−0.0499547	−	0.998751i	$-0.515908\pi$
$691$	−2.62631	−0.0999094	−0.0499547	+	0.998751i	$0.515908\pi$
$692$	0.993268	0.0377584
$693$	0	0
$694$	1.52951	0.0580593
$695$	9.36801	0.355349
$696$	0	0
$697$	−3.04973	−0.115517
$698$	2.50222	0.0947106
$699$	0	0
$700$	0.867566	0.0327909
$701$	−36.7376	−1.38756	−0.693780	−	0.720187i	$-0.744056\pi$
$701$	−36.7376	−1.38756	−0.693780	+	0.720187i	$0.744056\pi$
$702$	0	0
$703$	27.1658	1.02458
$704$	−7.23485	−0.272674
$705$	0	0
$706$	3.83978	0.144512
$707$	11.0586	0.415901
$708$	0	0
$709$	29.6385	1.11310	0.556549	−	0.830815i	$-0.312125\pi$
$709$	29.6385	1.11310	0.556549	+	0.830815i	$0.312125\pi$
$710$	4.12833	0.154934
$711$	0	0
$712$	−10.1559	−0.380610
$713$	1.86827	0.0699673
$714$	0	0
$715$	−11.0758	−0.414212
$716$	23.7142	0.886240
$717$	0	0
$718$	1.83072	0.0683220
$719$	−20.2891	−0.756655	−0.378327	−	0.925672i	$-0.623501\pi$
$719$	−20.2891	−0.756655	−0.378327	+	0.925672i	$0.623501\pi$
$720$	0	0
$721$	−61.0295	−2.27286
$722$	−1.16415	−0.0433251
$723$	0	0
$724$	−46.9145	−1.74356
$725$	−0.713810	−0.0265102
$726$	0	0
$727$	−12.1406	−0.450271	−0.225136	−	0.974327i	$-0.572283\pi$
$727$	−12.1406	−0.450271	−0.225136	+	0.974327i	$0.572283\pi$
$728$	−11.3646	−0.421199
$729$	0	0
$730$	−4.85959	−0.179862
$731$	8.12349	0.300458
$732$	0	0
$733$	3.41541	0.126151	0.0630755	−	0.998009i	$-0.479909\pi$
$733$	3.41541	0.126151	0.0630755	+	0.998009i	$0.479909\pi$
$734$	−1.38705	−0.0511971
$735$	0	0
$736$	−4.04062	−0.148939
$737$	−6.12066	−0.225457
$738$	0	0
$739$	11.0875	0.407862	0.203931	−	0.978985i	$-0.434628\pi$
$739$	11.0875	0.407862	0.203931	+	0.978985i	$0.434628\pi$
$740$	33.3019	1.22420
$741$	0	0
$742$	6.25015	0.229450
$743$	15.5926	0.572036	0.286018	−	0.958224i	$-0.407668\pi$
$743$	15.5926	0.572036	0.286018	+	0.958224i	$0.407668\pi$
$744$	0	0
$745$	9.19662	0.336938
$746$	−3.12774	−0.114515
$747$	0	0
$748$	2.14054	0.0782659
$749$	−8.97560	−0.327961
$750$	0	0
$751$	2.31000	0.0842932	0.0421466	−	0.999111i	$-0.486580\pi$
$751$	2.31000	0.0842932	0.0421466	+	0.999111i	$0.486580\pi$
$752$	−19.3941	−0.707231
$753$	0	0
$754$	4.63715	0.168875
$755$	−39.8918	−1.45181
$756$	0	0
$757$	23.7474	0.863116	0.431558	−	0.902085i	$-0.357964\pi$
$757$	23.7474	0.863116	0.431558	+	0.902085i	$0.357964\pi$
$758$	−0.415916	−0.0151067
$759$	0	0
$760$	5.56509	0.201867
$761$	7.80814	0.283045	0.141523	−	0.989935i	$-0.454800\pi$
$761$	7.80814	0.283045	0.141523	+	0.989935i	$0.454800\pi$
$762$	0	0
$763$	23.6862	0.857496
$764$	−51.1902	−1.85199
$765$	0	0
$766$	1.61648	0.0584059
$767$	−44.6997	−1.61401
$768$	0	0
$769$	−24.6045	−0.887261	−0.443630	−	0.896210i	$-0.646310\pi$
$769$	−24.6045	−0.887261	−0.443630	+	0.896210i	$0.646310\pi$
$770$	−1.25709	−0.0453022
$771$	0	0
$772$	7.67348	0.276175
$773$	0.312960	0.0112564	0.00562819	−	0.999984i	$-0.498208\pi$
$773$	0.312960	0.0112564	0.00562819	+	0.999984i	$0.498208\pi$
$774$	0	0
$775$	0.135675	0.00487359
$776$	9.89688	0.355277
$777$	0	0
$778$	1.37599	0.0493317
$779$	9.92122	0.355464
$780$	0	0
$781$	−10.4162	−0.372723
$782$	0.374330	0.0133860
$783$	0	0
$784$	11.6503	0.416081
$785$	−43.1691	−1.54077
$786$	0	0
$787$	9.32767	0.332495	0.166248	−	0.986084i	$-0.446835\pi$
$787$	9.32767	0.332495	0.166248	+	0.986084i	$0.446835\pi$
$788$	−43.2233	−1.53976
$789$	0	0
$790$	−6.21349	−0.221066
$791$	28.7910	1.02369
$792$	0	0
$793$	5.02358	0.178393
$794$	4.69608	0.166658
$795$	0	0
$796$	−2.24570	−0.0795968
$797$	−10.3564	−0.366842	−0.183421	−	0.983034i	$-0.558717\pi$
$797$	−10.3564	−0.366842	−0.183421	+	0.983034i	$0.558717\pi$
$798$	0	0
$799$	5.54163	0.196049
$800$	−0.293433	−0.0103744
$801$	0	0
$802$	4.91501	0.173555
$803$	12.2613	0.432692
$804$	0	0
$805$	13.3859	0.471793
$806$	−0.881391	−0.0310457
$807$	0	0
$808$	−2.48678	−0.0874845
$809$	21.7280	0.763917	0.381958	−	0.924180i	$-0.375250\pi$
$809$	21.7280	0.763917	0.381958	+	0.924180i	$0.375250\pi$
$810$	0	0
$811$	13.2505	0.465286	0.232643	−	0.972562i	$-0.425263\pi$
$811$	13.2505	0.465286	0.232643	+	0.972562i	$0.425263\pi$
$812$	−32.0474	−1.12464
$813$	0	0
$814$	1.37992	0.0483660
$815$	−31.5754	−1.10604
$816$	0	0
$817$	−26.4269	−0.924560
$818$	−4.36233	−0.152525
$819$	0	0
$820$	12.1622	0.424722
$821$	16.6433	0.580856	0.290428	−	0.956897i	$-0.406202\pi$
$821$	16.6433	0.580856	0.290428	+	0.956897i	$0.406202\pi$
$822$	0	0
$823$	−19.5756	−0.682364	−0.341182	−	0.939997i	$-0.610827\pi$
$823$	−19.5756	−0.682364	−0.341182	+	0.939997i	$0.610827\pi$
$824$	13.7239	0.478094
$825$	0	0
$826$	−5.07333	−0.176524
$827$	−20.1005	−0.698964	−0.349482	−	0.936943i	$-0.613642\pi$
$827$	−20.1005	−0.698964	−0.349482	+	0.936943i	$0.613642\pi$
$828$	0	0
$829$	55.2719	1.91967	0.959835	−	0.280564i	$-0.0905214\pi$
$829$	55.2719	1.91967	0.959835	+	0.280564i	$0.0905214\pi$
$830$	4.26451	0.148023
$831$	0	0
$832$	−36.3449	−1.26003
$833$	−3.32892	−0.115340
$834$	0	0
$835$	0.392734	0.0135911
$836$	−6.96349	−0.240837
$837$	0	0
$838$	−2.97926	−0.102917
$839$	9.09374	0.313951	0.156975	−	0.987603i	$-0.449826\pi$
$839$	9.09374	0.313951	0.156975	+	0.987603i	$0.449826\pi$
$840$	0	0
$841$	−2.63226	−0.0907676
$842$	0.0760589	0.00262116
$843$	0	0
$844$	−18.4994	−0.636777
$845$	−26.9783	−0.928081
$846$	0	0
$847$	3.17177	0.108983
$848$	41.7339	1.43315
$849$	0	0
$850$	0.0271841	0.000932407 0
$851$	−14.6939	−0.503700
$852$	0	0
$853$	−30.4639	−1.04307	−0.521533	−	0.853231i	$-0.674639\pi$
$853$	−30.4639	−1.04307	−0.521533	+	0.853231i	$0.674639\pi$
$854$	0.570167	0.0195107
$855$	0	0
$856$	2.01837	0.0689865
$857$	36.7917	1.25678	0.628391	−	0.777898i	$-0.283714\pi$
$857$	36.7917	1.25678	0.628391	+	0.777898i	$0.283714\pi$
$858$	0	0
$859$	−19.7020	−0.672225	−0.336112	−	0.941822i	$-0.609112\pi$
$859$	−19.7020	−0.672225	−0.336112	+	0.941822i	$0.609112\pi$
$860$	−32.3961	−1.10470
$861$	0	0
$862$	−4.37534	−0.149025
$863$	40.6000	1.38204	0.691020	−	0.722836i	$-0.257162\pi$
$863$	40.6000	1.38204	0.691020	+	0.722836i	$0.257162\pi$
$864$	0	0
$865$	1.11294	0.0378412
$866$	0.980794	0.0333287
$867$	0	0
$868$	6.09131	0.206753
$869$	15.6773	0.531817
$870$	0	0
$871$	−30.7476	−1.04184
$872$	−5.32638	−0.180374
$873$	0	0
$874$	−1.21775	−0.0411910
$875$	35.9371	1.21490
$876$	0	0
$877$	32.5425	1.09888	0.549441	−	0.835532i	$-0.314841\pi$
$877$	32.5425	1.09888	0.549441	+	0.835532i	$0.314841\pi$
$878$	1.02750	0.0346763
$879$	0	0
$880$	−8.39388	−0.282958
$881$	−9.99390	−0.336703	−0.168352	−	0.985727i	$-0.553844\pi$
$881$	−9.99390	−0.336703	−0.168352	+	0.985727i	$0.553844\pi$
$882$	0	0
$883$	−47.6765	−1.60444	−0.802221	−	0.597028i	$-0.796348\pi$
$883$	−47.6765	−1.60444	−0.802221	+	0.597028i	$0.796348\pi$
$884$	10.7532	0.361668
$885$	0	0
$886$	−2.73215	−0.0917883
$887$	32.5954	1.09445	0.547224	−	0.836986i	$-0.315685\pi$
$887$	32.5954	1.09445	0.547224	+	0.836986i	$0.315685\pi$
$888$	0	0
$889$	28.3835	0.951951
$890$	−5.64346	−0.189169
$891$	0	0
$892$	−4.77613	−0.159917
$893$	−18.0277	−0.603275
$894$	0	0
$895$	26.5714	0.888185
$896$	−17.5155	−0.585151
$897$	0	0
$898$	0.209365	0.00698659
$899$	−5.01177	−0.167152
$900$	0	0
$901$	−11.9249	−0.397277
$902$	0.503958	0.0167800
$903$	0	0
$904$	−6.47432	−0.215333
$905$	−52.5671	−1.74739
$906$	0	0
$907$	11.8687	0.394094	0.197047	−	0.980394i	$-0.436865\pi$
$907$	11.8687	0.394094	0.197047	+	0.980394i	$0.436865\pi$
$908$	−38.3558	−1.27288
$909$	0	0
$910$	−6.31507	−0.209342
$911$	22.8072	0.755636	0.377818	−	0.925880i	$-0.376674\pi$
$911$	22.8072	0.755636	0.377818	+	0.925880i	$0.376674\pi$
$912$	0	0
$913$	−10.7598	−0.356099
$914$	2.84662	0.0941579
$915$	0	0
$916$	−29.1776	−0.964056
$917$	−44.3682	−1.46517
$918$	0	0
$919$	55.9065	1.84419	0.922093	−	0.386969i	$-0.126478\pi$
$919$	55.9065	1.84419	0.922093	+	0.386969i	$0.126478\pi$
$920$	−3.01014	−0.0992413
$921$	0	0
$922$	−2.57864	−0.0849231
$923$	−52.3268	−1.72236
$924$	0	0
$925$	−1.06708	−0.0350854
$926$	4.42977	0.145571
$927$	0	0
$928$	10.8392	0.355816
$929$	42.5584	1.39630	0.698149	−	0.715953i	$-0.254007\pi$
$929$	42.5584	1.39630	0.698149	+	0.715953i	$0.254007\pi$
$930$	0	0
$931$	10.8295	0.354921
$932$	−18.0599	−0.591573
$933$	0	0
$934$	1.66559	0.0544996
$935$	2.39845	0.0784376
$936$	0	0
$937$	−18.2206	−0.595242	−0.297621	−	0.954684i	$-0.596193\pi$
$937$	−18.2206	−0.595242	−0.297621	+	0.954684i	$0.596193\pi$
$938$	−3.48980	−0.113946
$939$	0	0
$940$	−22.0998	−0.720815
$941$	30.8164	1.00458	0.502292	−	0.864698i	$-0.332490\pi$
$941$	30.8164	1.00458	0.502292	+	0.864698i	$0.332490\pi$
$942$	0	0
$943$	−5.36635	−0.174752
$944$	−33.8760	−1.10257
$945$	0	0
$946$	−1.34238	−0.0436446
$947$	52.0370	1.69098	0.845488	−	0.533994i	$-0.179310\pi$
$947$	52.0370	1.69098	0.845488	+	0.533994i	$0.179310\pi$
$948$	0	0
$949$	61.5956	1.99948
$950$	−0.0884339	−0.00286917
$951$	0	0
$952$	2.46098	0.0797607
$953$	19.4381	0.629662	0.314831	−	0.949148i	$-0.398052\pi$
$953$	19.4381	0.629662	0.314831	+	0.949148i	$0.398052\pi$
$954$	0	0
$955$	−57.3579	−1.85606
$956$	35.3767	1.14416
$957$	0	0
$958$	−0.796135	−0.0257220
$959$	−62.7985	−2.02787
$960$	0	0
$961$	−30.0474	−0.969271
$962$	6.93212	0.223500
$963$	0	0
$964$	23.2224	0.747942
$965$	8.59803	0.276781
$966$	0	0
$967$	−30.8565	−0.992280	−0.496140	−	0.868243i	$-0.665250\pi$
$967$	−30.8565	−0.992280	−0.496140	+	0.868243i	$0.665250\pi$
$968$	−0.713244	−0.0229245
$969$	0	0
$970$	5.49951	0.176578
$971$	5.96171	0.191320	0.0956601	−	0.995414i	$-0.469504\pi$
$971$	5.96171	0.191320	0.0956601	+	0.995414i	$0.469504\pi$
$972$	0	0
$973$	−13.4768	−0.432046
$974$	4.36708	0.139930
$975$	0	0
$976$	3.80716	0.121864
$977$	−4.76596	−0.152477	−0.0762383	−	0.997090i	$-0.524291\pi$
$977$	−4.76596	−0.152477	−0.0762383	+	0.997090i	$0.524291\pi$
$978$	0	0
$979$	14.2391	0.455083
$980$	13.2756	0.424073
$981$	0	0
$982$	7.65618	0.244318
$983$	−52.1600	−1.66365	−0.831823	−	0.555042i	$-0.812702\pi$
$983$	−52.1600	−1.66365	−0.831823	+	0.555042i	$0.812702\pi$
$984$	0	0
$985$	−48.4311	−1.54314
$986$	−1.00417	−0.0319791
$987$	0	0
$988$	−34.9817	−1.11291
$989$	14.2942	0.454530
$990$	0	0
$991$	20.1207	0.639156	0.319578	−	0.947560i	$-0.396459\pi$
$991$	20.1207	0.639156	0.319578	+	0.947560i	$0.396459\pi$
$992$	−2.06024	−0.0654126
$993$	0	0
$994$	−5.93900	−0.188374
$995$	−2.51628	−0.0797714
$996$	0	0
$997$	15.9577	0.505384	0.252692	−	0.967547i	$-0.418684\pi$
$997$	15.9577	0.505384	0.252692	+	0.967547i	$0.418684\pi$
$998$	5.38646	0.170505
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.j.1.8		14
3.2	odd	2		2013.2.a.h.1.7	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.h.1.7	✓	14	3.2	odd	2
6039.2.a.j.1.8		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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+0.179763 q^{2} -1.96769 q^{4} -2.20477 q^{5} +3.17177 q^{7} -0.713244 q^{8} +O(q^{10})\)	\(q+0.179763 q^{2} -1.96769 q^{4} -2.20477 q^{5} +3.17177 q^{7} -0.713244 q^{8} -0.396336 q^{10} +1.00000 q^{11} +5.02358 q^{13} +0.570167 q^{14} +3.80716 q^{16} -1.08785 q^{17} +3.53893 q^{19} +4.33828 q^{20} +0.179763 q^{22} -1.91419 q^{23} -0.139010 q^{25} +0.903056 q^{26} -6.24104 q^{28} +5.13495 q^{29} -0.976010 q^{31} +2.11088 q^{32} -0.195555 q^{34} -6.99300 q^{35} +7.67629 q^{37} +0.636169 q^{38} +1.57254 q^{40} +2.80345 q^{41} -7.46749 q^{43} -1.96769 q^{44} -0.344102 q^{46} -5.09413 q^{47} +3.06010 q^{49} -0.0249889 q^{50} -9.88482 q^{52} +10.9620 q^{53} -2.20477 q^{55} -2.26224 q^{56} +0.923076 q^{58} -8.89797 q^{59} +1.00000 q^{61} -0.175451 q^{62} -7.23485 q^{64} -11.0758 q^{65} -6.12066 q^{67} +2.14054 q^{68} -1.25709 q^{70} -10.4162 q^{71} +12.2613 q^{73} +1.37992 q^{74} -6.96349 q^{76} +3.17177 q^{77} +15.6773 q^{79} -8.39388 q^{80} +0.503958 q^{82} -10.7598 q^{83} +2.39845 q^{85} -1.34238 q^{86} -0.713244 q^{88} +14.2391 q^{89} +15.9336 q^{91} +3.76653 q^{92} -0.915737 q^{94} -7.80250 q^{95} -13.8759 q^{97} +0.550094 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) \) 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7} + 6 q^{10} + 14 q^{11} + q^{13} + 7 q^{14} + 17 q^{16} + 9 q^{17} + 22 q^{19} - 23 q^{20} + q^{22} - q^{23} + 25 q^{25} - 4 q^{26} + 37 q^{28} + 6 q^{29} + 9 q^{31} - 4 q^{32} + 8 q^{34} - 18 q^{35} + 18 q^{37} - 8 q^{38} + 16 q^{40} + 25 q^{41} + 25 q^{43} + 15 q^{44} + 20 q^{46} - 36 q^{47} + 25 q^{49} - 2 q^{50} - 13 q^{52} - q^{55} + 40 q^{56} + 33 q^{58} - 17 q^{59} + 14 q^{61} + 13 q^{62} - 6 q^{64} + 61 q^{65} + 22 q^{67} - 66 q^{68} + 44 q^{70} + 13 q^{71} + 20 q^{73} + 12 q^{74} + 49 q^{76} + 9 q^{77} + 31 q^{79} - 88 q^{80} + 2 q^{82} - 32 q^{83} + 2 q^{85} + 14 q^{86} + 21 q^{89} + 45 q^{91} + 14 q^{92} - 31 q^{94} - 23 q^{95} + 37 q^{97} + 38 q^{98}+O(q^{100}) \) 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7 + 6 * q^10 + 14 * q^11 + q^13 + 7 * q^14 + 17 * q^16 + 9 * q^17 + 22 * q^19 - 23 * q^20 + q^22 - q^23 + 25 * q^25 - 4 * q^26 + 37 * q^28 + 6 * q^29 + 9 * q^31 - 4 * q^32 + 8 * q^34 - 18 * q^35 + 18 * q^37 - 8 * q^38 + 16 * q^40 + 25 * q^41 + 25 * q^43 + 15 * q^44 + 20 * q^46 - 36 * q^47 + 25 * q^49 - 2 * q^50 - 13 * q^52 - q^55 + 40 * q^56 + 33 * q^58 - 17 * q^59 + 14 * q^61 + 13 * q^62 - 6 * q^64 + 61 * q^65 + 22 * q^67 - 66 * q^68 + 44 * q^70 + 13 * q^71 + 20 * q^73 + 12 * q^74 + 49 * q^76 + 9 * q^77 + 31 * q^79 - 88 * q^80 + 2 * q^82 - 32 * q^83 + 2 * q^85 + 14 * q^86 + 21 * q^89 + 45 * q^91 + 14 * q^92 - 31 * q^94 - 23 * q^95 + 37 * q^97 + 38 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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