LMFDB - Embedded newform 6039.2.a.j.1.12

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			1.12
Root			$1.93923$ 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)

$f(q)$	$=$	$q+1.93923 q^{2} +1.76061 q^{4} -3.07794 q^{5} -3.15900 q^{7} -0.464237 q^{8} +O(q^{10})$	\(q+1.93923 q^{2} +1.76061 q^{4} -3.07794 q^{5} -3.15900 q^{7} -0.464237 q^{8} -5.96883 q^{10} +1.00000 q^{11} -4.31016 q^{13} -6.12601 q^{14} -4.42148 q^{16} +3.81369 q^{17} -4.23281 q^{19} -5.41904 q^{20} +1.93923 q^{22} -6.48918 q^{23} +4.47371 q^{25} -8.35838 q^{26} -5.56175 q^{28} +3.27479 q^{29} +10.0842 q^{31} -7.64578 q^{32} +7.39562 q^{34} +9.72319 q^{35} -1.85262 q^{37} -8.20839 q^{38} +1.42889 q^{40} +10.7788 q^{41} +10.4003 q^{43} +1.76061 q^{44} -12.5840 q^{46} -10.3730 q^{47} +2.97925 q^{49} +8.67555 q^{50} -7.58850 q^{52} -8.28262 q^{53} -3.07794 q^{55} +1.46652 q^{56} +6.35056 q^{58} +10.0151 q^{59} +1.00000 q^{61} +19.5555 q^{62} -5.98396 q^{64} +13.2664 q^{65} -5.80576 q^{67} +6.71441 q^{68} +18.8555 q^{70} +3.40168 q^{71} +11.2828 q^{73} -3.59265 q^{74} -7.45232 q^{76} -3.15900 q^{77} +4.26886 q^{79} +13.6090 q^{80} +20.9026 q^{82} -17.4095 q^{83} -11.7383 q^{85} +20.1685 q^{86} -0.464237 q^{88} +9.75884 q^{89} +13.6158 q^{91} -11.4249 q^{92} -20.1156 q^{94} +13.0283 q^{95} +0.814992 q^{97} +5.77745 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.93923	1.37124	0.685621	−	0.727959i	$-0.259531\pi$
$2$	1.93923	1.37124	0.685621	+	0.727959i	$0.259531\pi$
$3$	0	0
$3$	0	0
$4$	1.76061	0.880304
$5$	−3.07794	−1.37650	−0.688248	−	0.725475i	$-0.741620\pi$
$5$	−3.07794	−1.37650	−0.688248	+	0.725475i	$0.741620\pi$
$6$	0	0
$7$	−3.15900	−1.19399	−0.596994	−	0.802246i	$-0.703638\pi$
$7$	−3.15900	−1.19399	−0.596994	+	0.802246i	$0.703638\pi$
$8$	−0.464237	−0.164132
$9$	0	0
$10$	−5.96883	−1.88751
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.31016	−1.19542	−0.597711	−	0.801711i	$-0.703923\pi$
$13$	−4.31016	−1.19542	−0.597711	+	0.801711i	$0.703923\pi$
$14$	−6.12601	−1.63725
$15$	0	0
$16$	−4.42148	−1.10537
$17$	3.81369	0.924956	0.462478	−	0.886631i	$-0.346960\pi$
$17$	3.81369	0.924956	0.462478	+	0.886631i	$0.346960\pi$
$18$	0	0
$19$	−4.23281	−0.971073	−0.485537	−	0.874216i	$-0.661376\pi$
$19$	−4.23281	−0.971073	−0.485537	+	0.874216i	$0.661376\pi$
$20$	−5.41904	−1.21173
$21$	0	0
$22$	1.93923	0.413445
$23$	−6.48918	−1.35309	−0.676544	−	0.736402i	$-0.736523\pi$
$23$	−6.48918	−1.35309	−0.676544	+	0.736402i	$0.736523\pi$
$24$	0	0
$25$	4.47371	0.894742
$26$	−8.35838	−1.63921
$27$	0	0
$28$	−5.56175	−1.05107
$29$	3.27479	0.608113	0.304056	−	0.952654i	$-0.401659\pi$
$29$	3.27479	0.608113	0.304056	+	0.952654i	$0.401659\pi$
$30$	0	0
$31$	10.0842	1.81117	0.905585	−	0.424164i	$-0.139432\pi$
$31$	10.0842	1.81117	0.905585	+	0.424164i	$0.139432\pi$
$32$	−7.64578	−1.35160
$33$	0	0
$34$	7.39562	1.26834
$35$	9.72319	1.64352
$36$	0	0
$37$	−1.85262	−0.304569	−0.152284	−	0.988337i	$-0.548663\pi$
$37$	−1.85262	−0.304569	−0.152284	+	0.988337i	$0.548663\pi$
$38$	−8.20839	−1.33158
$39$	0	0
$40$	1.42889	0.225928
$41$	10.7788	1.68337	0.841685	−	0.539969i	$-0.181564\pi$
$41$	10.7788	1.68337	0.841685	+	0.539969i	$0.181564\pi$
$42$	0	0
$43$	10.4003	1.58603	0.793013	−	0.609205i	$-0.208511\pi$
$43$	10.4003	1.58603	0.793013	+	0.609205i	$0.208511\pi$
$44$	1.76061	0.265422
$45$	0	0
$46$	−12.5840	−1.85541
$47$	−10.3730	−1.51305	−0.756527	−	0.653962i	$-0.773106\pi$
$47$	−10.3730	−1.51305	−0.756527	+	0.653962i	$0.773106\pi$
$48$	0	0
$49$	2.97925	0.425607
$50$	8.67555	1.22691
$51$	0	0
$52$	−7.58850	−1.05234
$53$	−8.28262	−1.13771	−0.568853	−	0.822439i	$-0.692613\pi$
$53$	−8.28262	−1.13771	−0.568853	+	0.822439i	$0.692613\pi$
$54$	0	0
$55$	−3.07794	−0.415029
$56$	1.46652	0.195972
$57$	0	0
$58$	6.35056	0.833869
$59$	10.0151	1.30385	0.651925	−	0.758283i	$-0.273962\pi$
$59$	10.0151	1.30385	0.651925	+	0.758283i	$0.273962\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	19.5555	2.48355
$63$	0	0
$64$	−5.98396	−0.747995
$65$	13.2664	1.64550
$66$	0	0
$67$	−5.80576	−0.709287	−0.354643	−	0.935002i	$-0.615398\pi$
$67$	−5.80576	−0.709287	−0.354643	+	0.935002i	$0.615398\pi$
$68$	6.71441	0.814242
$69$	0	0
$70$	18.8555	2.25366
$71$	3.40168	0.403705	0.201853	−	0.979416i	$-0.435304\pi$
$71$	3.40168	0.403705	0.201853	+	0.979416i	$0.435304\pi$
$72$	0	0
$73$	11.2828	1.32055	0.660276	−	0.751023i	$-0.270439\pi$
$73$	11.2828	1.32055	0.660276	+	0.751023i	$0.270439\pi$
$74$	−3.59265	−0.417638
$75$	0	0
$76$	−7.45232	−0.854840
$77$	−3.15900	−0.360001
$78$	0	0
$79$	4.26886	0.480284	0.240142	−	0.970738i	$-0.422806\pi$
$79$	4.26886	0.480284	0.240142	+	0.970738i	$0.422806\pi$
$80$	13.6090	1.52154
$81$	0	0
$82$	20.9026	2.30831
$83$	−17.4095	−1.91094	−0.955469	−	0.295092i	$-0.904650\pi$
$83$	−17.4095	−1.91094	−0.955469	+	0.295092i	$0.904650\pi$
$84$	0	0
$85$	−11.7383	−1.27320
$86$	20.1685	2.17483
$87$	0	0
$88$	−0.464237	−0.0494878
$89$	9.75884	1.03444	0.517218	−	0.855854i	$-0.326968\pi$
$89$	9.75884	1.03444	0.517218	+	0.855854i	$0.326968\pi$
$90$	0	0
$91$	13.6158	1.42732
$92$	−11.4249	−1.19113
$93$	0	0
$94$	−20.1156	−2.07476
$95$	13.0283	1.33668
$96$	0	0
$97$	0.814992	0.0827499	0.0413749	−	0.999144i	$-0.486826\pi$
$97$	0.814992	0.0827499	0.0413749	+	0.999144i	$0.486826\pi$
$98$	5.77745	0.583610
$99$	0	0
$100$	7.87645	0.787645
$101$	−8.64218	−0.859929	−0.429965	−	0.902846i	$-0.641474\pi$
$101$	−8.64218	−0.859929	−0.429965	+	0.902846i	$0.641474\pi$
$102$	0	0
$103$	1.53886	0.151629	0.0758143	−	0.997122i	$-0.475844\pi$
$103$	1.53886	0.151629	0.0758143	+	0.997122i	$0.475844\pi$
$104$	2.00093	0.196208
$105$	0	0
$106$	−16.0619	−1.56007
$107$	1.27301	0.123067	0.0615333	−	0.998105i	$-0.480401\pi$
$107$	1.27301	0.123067	0.0615333	+	0.998105i	$0.480401\pi$
$108$	0	0
$109$	10.8977	1.04381	0.521905	−	0.853004i	$-0.325222\pi$
$109$	10.8977	1.04381	0.521905	+	0.853004i	$0.325222\pi$
$110$	−5.96883	−0.569105
$111$	0	0
$112$	13.9674	1.31980
$113$	12.6330	1.18841	0.594206	−	0.804313i	$-0.297466\pi$
$113$	12.6330	1.18841	0.594206	+	0.804313i	$0.297466\pi$
$114$	0	0
$115$	19.9733	1.86252
$116$	5.76561	0.535324
$117$	0	0
$118$	19.4215	1.78789
$119$	−12.0474	−1.10439
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.93923	0.175570
$123$	0	0
$124$	17.7543	1.59438
$125$	1.61989	0.144887
$126$	0	0
$127$	12.4848	1.10784	0.553921	−	0.832569i	$-0.313131\pi$
$127$	12.4848	1.10784	0.553921	+	0.832569i	$0.313131\pi$
$128$	3.68729	0.325913
$129$	0	0
$130$	25.7266	2.25637
$131$	−0.0498953	−0.00435938	−0.00217969	−	0.999998i	$-0.500694\pi$
$131$	−0.0498953	−0.00435938	−0.00217969	+	0.999998i	$0.500694\pi$
$132$	0	0
$133$	13.3714	1.15945
$134$	−11.2587	−0.972604
$135$	0	0
$136$	−1.77045	−0.151815
$137$	−0.243742	−0.0208243	−0.0104121	−	0.999946i	$-0.503314\pi$
$137$	−0.243742	−0.0208243	−0.0104121	+	0.999946i	$0.503314\pi$
$138$	0	0
$139$	−21.1871	−1.79706	−0.898531	−	0.438910i	$-0.855365\pi$
$139$	−21.1871	−1.79706	−0.898531	+	0.438910i	$0.855365\pi$
$140$	17.1187	1.44680
$141$	0	0
$142$	6.59664	0.553578
$143$	−4.31016	−0.360434
$144$	0	0
$145$	−10.0796	−0.837065
$146$	21.8799	1.81080
$147$	0	0
$148$	−3.26174	−0.268113
$149$	−13.4964	−1.10567	−0.552834	−	0.833291i	$-0.686454\pi$
$149$	−13.4964	−1.10567	−0.552834	+	0.833291i	$0.686454\pi$
$150$	0	0
$151$	2.14503	0.174560	0.0872801	−	0.996184i	$-0.472182\pi$
$151$	2.14503	0.174560	0.0872801	+	0.996184i	$0.472182\pi$
$152$	1.96503	0.159385
$153$	0	0
$154$	−6.12601	−0.493648
$155$	−31.0385	−2.49307
$156$	0	0
$157$	−2.78006	−0.221873	−0.110936	−	0.993828i	$-0.535385\pi$
$157$	−2.78006	−0.221873	−0.110936	+	0.993828i	$0.535385\pi$
$158$	8.27830	0.658586
$159$	0	0
$160$	23.5332	1.86047
$161$	20.4993	1.61557
$162$	0	0
$163$	16.8129	1.31689	0.658444	−	0.752630i	$-0.271215\pi$
$163$	16.8129	1.31689	0.658444	+	0.752630i	$0.271215\pi$
$164$	18.9773	1.48188
$165$	0	0
$166$	−33.7609	−2.62036
$167$	−14.1771	−1.09706	−0.548529	−	0.836132i	$-0.684812\pi$
$167$	−14.1771	−1.09706	−0.548529	+	0.836132i	$0.684812\pi$
$168$	0	0
$169$	5.57746	0.429036
$170$	−22.7633	−1.74586
$171$	0	0
$172$	18.3108	1.39618
$173$	−19.2019	−1.45989	−0.729945	−	0.683505i	$-0.760455\pi$
$173$	−19.2019	−1.45989	−0.729945	+	0.683505i	$0.760455\pi$
$174$	0	0
$175$	−14.1324	−1.06831
$176$	−4.42148	−0.333281
$177$	0	0
$178$	18.9246	1.41846
$179$	6.27478	0.468999	0.234500	−	0.972116i	$-0.424655\pi$
$179$	6.27478	0.468999	0.234500	+	0.972116i	$0.424655\pi$
$180$	0	0
$181$	14.9701	1.11272	0.556359	−	0.830942i	$-0.312198\pi$
$181$	14.9701	1.11272	0.556359	+	0.830942i	$0.312198\pi$
$182$	26.4041	1.95720
$183$	0	0
$184$	3.01251	0.222085
$185$	5.70225	0.419238
$186$	0	0
$187$	3.81369	0.278885
$188$	−18.2627	−1.33195
$189$	0	0
$190$	25.2649	1.83291
$191$	1.11274	0.0805152	0.0402576	−	0.999189i	$-0.487182\pi$
$191$	1.11274	0.0805152	0.0402576	+	0.999189i	$0.487182\pi$
$192$	0	0
$193$	−16.4747	−1.18587	−0.592937	−	0.805249i	$-0.702032\pi$
$193$	−16.4747	−1.18587	−0.592937	+	0.805249i	$0.702032\pi$
$194$	1.58046	0.113470
$195$	0	0
$196$	5.24529	0.374664
$197$	−25.4239	−1.81138	−0.905690	−	0.423941i	$-0.860646\pi$
$197$	−25.4239	−1.81138	−0.905690	+	0.423941i	$0.860646\pi$
$198$	0	0
$199$	−8.05688	−0.571137	−0.285569	−	0.958358i	$-0.592182\pi$
$199$	−8.05688	−0.571137	−0.285569	+	0.958358i	$0.592182\pi$
$200$	−2.07686	−0.146856
$201$	0	0
$202$	−16.7592	−1.17917
$203$	−10.3450	−0.726079
$204$	0	0
$205$	−33.1766	−2.31715
$206$	2.98420	0.207919
$207$	0	0
$208$	19.0573	1.32138
$209$	−4.23281	−0.292790
$210$	0	0
$211$	21.8405	1.50356	0.751780	−	0.659414i	$-0.229196\pi$
$211$	21.8405	1.50356	0.751780	+	0.659414i	$0.229196\pi$
$212$	−14.5825	−1.00153
$213$	0	0
$214$	2.46866	0.168754
$215$	−32.0114	−2.18316
$216$	0	0
$217$	−31.8559	−2.16252
$218$	21.1331	1.43132
$219$	0	0
$220$	−5.41904	−0.365352
$221$	−16.4376	−1.10571
$222$	0	0
$223$	12.6761	0.848852	0.424426	−	0.905463i	$-0.360476\pi$
$223$	12.6761	0.848852	0.424426	+	0.905463i	$0.360476\pi$
$224$	24.1530	1.61379
$225$	0	0
$226$	24.4983	1.62960
$227$	−18.0164	−1.19579	−0.597897	−	0.801573i	$-0.703997\pi$
$227$	−18.0164	−1.19579	−0.597897	+	0.801573i	$0.703997\pi$
$228$	0	0
$229$	21.1854	1.39997	0.699984	−	0.714159i	$-0.253191\pi$
$229$	21.1854	1.39997	0.699984	+	0.714159i	$0.253191\pi$
$230$	38.7328	2.55397
$231$	0	0
$232$	−1.52028	−0.0998110
$233$	28.3068	1.85444	0.927221	−	0.374516i	$-0.122191\pi$
$233$	28.3068	1.85444	0.927221	+	0.374516i	$0.122191\pi$
$234$	0	0
$235$	31.9274	2.08271
$236$	17.6326	1.14778
$237$	0	0
$238$	−23.3627	−1.51438
$239$	−4.89918	−0.316902	−0.158451	−	0.987367i	$-0.550650\pi$
$239$	−4.89918	−0.316902	−0.158451	+	0.987367i	$0.550650\pi$
$240$	0	0
$241$	27.3485	1.76167	0.880836	−	0.473422i	$-0.156981\pi$
$241$	27.3485	1.76167	0.880836	+	0.473422i	$0.156981\pi$
$242$	1.93923	0.124658
$243$	0	0
$244$	1.76061	0.112711
$245$	−9.16995	−0.585847
$246$	0	0
$247$	18.2441	1.16084
$248$	−4.68144	−0.297272
$249$	0	0
$250$	3.14134	0.198676
$251$	−2.95590	−0.186575	−0.0932874	−	0.995639i	$-0.529738\pi$
$251$	−2.95590	−0.186575	−0.0932874	+	0.995639i	$0.529738\pi$
$252$	0	0
$253$	−6.48918	−0.407971
$254$	24.2108	1.51912
$255$	0	0
$256$	19.1184	1.19490
$257$	−8.57140	−0.534669	−0.267335	−	0.963604i	$-0.586143\pi$
$257$	−8.57140	−0.534669	−0.267335	+	0.963604i	$0.586143\pi$
$258$	0	0
$259$	5.85242	0.363652
$260$	23.3569	1.44854
$261$	0	0
$262$	−0.0967585	−0.00597776
$263$	1.84001	0.113460	0.0567300	−	0.998390i	$-0.481933\pi$
$263$	1.84001	0.113460	0.0567300	+	0.998390i	$0.481933\pi$
$264$	0	0
$265$	25.4934	1.56605
$266$	25.9303	1.58989
$267$	0	0
$268$	−10.2217	−0.624388
$269$	8.65480	0.527692	0.263846	−	0.964565i	$-0.415009\pi$
$269$	8.65480	0.527692	0.263846	+	0.964565i	$0.415009\pi$
$270$	0	0
$271$	18.4447	1.12044	0.560219	−	0.828344i	$-0.310717\pi$
$271$	18.4447	1.12044	0.560219	+	0.828344i	$0.310717\pi$
$272$	−16.8621	−1.02242
$273$	0	0
$274$	−0.472671	−0.0285551
$275$	4.47371	0.269775
$276$	0	0
$277$	8.09606	0.486445	0.243223	−	0.969971i	$-0.421795\pi$
$277$	8.09606	0.486445	0.243223	+	0.969971i	$0.421795\pi$
$278$	−41.0865	−2.46421
$279$	0	0
$280$	−4.51386	−0.269755
$281$	17.8764	1.06642	0.533208	−	0.845984i	$-0.320986\pi$
$281$	17.8764	1.06642	0.533208	+	0.845984i	$0.320986\pi$
$282$	0	0
$283$	17.7034	1.05236	0.526179	−	0.850374i	$-0.323624\pi$
$283$	17.7034	1.05236	0.526179	+	0.850374i	$0.323624\pi$
$284$	5.98903	0.355383
$285$	0	0
$286$	−8.35838	−0.494241
$287$	−34.0503	−2.00992
$288$	0	0
$289$	−2.45576	−0.144456
$290$	−19.5466	−1.14782
$291$	0	0
$292$	19.8646	1.16249
$293$	13.3362	0.779109	0.389554	−	0.921003i	$-0.372629\pi$
$293$	13.3362	0.779109	0.389554	+	0.921003i	$0.372629\pi$
$294$	0	0
$295$	−30.8258	−1.79474
$296$	0.860054	0.0499896
$297$	0	0
$298$	−26.1726	−1.51614
$299$	27.9694	1.61751
$300$	0	0
$301$	−32.8544	−1.89370
$302$	4.15971	0.239364
$303$	0	0
$304$	18.7153	1.07339
$305$	−3.07794	−0.176242
$306$	0	0
$307$	−32.1020	−1.83216	−0.916080	−	0.400996i	$-0.868664\pi$
$307$	−32.1020	−1.83216	−0.916080	+	0.400996i	$0.868664\pi$
$308$	−5.56175	−0.316910
$309$	0	0
$310$	−60.1907	−3.41860
$311$	−19.1427	−1.08548	−0.542742	−	0.839899i	$-0.682614\pi$
$311$	−19.1427	−1.08548	−0.542742	+	0.839899i	$0.682614\pi$
$312$	0	0
$313$	7.12592	0.402781	0.201390	−	0.979511i	$-0.435454\pi$
$313$	7.12592	0.402781	0.201390	+	0.979511i	$0.435454\pi$
$314$	−5.39117	−0.304241
$315$	0	0
$316$	7.51579	0.422796
$317$	30.7643	1.72789	0.863947	−	0.503584i	$-0.167985\pi$
$317$	30.7643	1.72789	0.863947	+	0.503584i	$0.167985\pi$
$318$	0	0
$319$	3.27479	0.183353
$320$	18.4183	1.02961
$321$	0	0
$322$	39.7528	2.21534
$323$	−16.1426	−0.898200
$324$	0	0
$325$	−19.2824	−1.06959
$326$	32.6040	1.80577
$327$	0	0
$328$	−5.00393	−0.276296
$329$	32.7682	1.80657
$330$	0	0
$331$	−1.20244	−0.0660920	−0.0330460	−	0.999454i	$-0.510521\pi$
$331$	−1.20244	−0.0660920	−0.0330460	+	0.999454i	$0.510521\pi$
$332$	−30.6512	−1.68221
$333$	0	0
$334$	−27.4926	−1.50433
$335$	17.8698	0.976331
$336$	0	0
$337$	−27.8849	−1.51898	−0.759492	−	0.650516i	$-0.774553\pi$
$337$	−27.8849	−1.51898	−0.759492	+	0.650516i	$0.774553\pi$
$338$	10.8160	0.588312
$339$	0	0
$340$	−20.6666	−1.12080
$341$	10.0842	0.546089
$342$	0	0
$343$	12.7015	0.685818
$344$	−4.82818	−0.260318
$345$	0	0
$346$	−37.2368	−2.00186
$347$	−8.66808	−0.465327	−0.232663	−	0.972557i	$-0.574744\pi$
$347$	−8.66808	−0.465327	−0.232663	+	0.972557i	$0.574744\pi$
$348$	0	0
$349$	−3.27364	−0.175234	−0.0876170	−	0.996154i	$-0.527925\pi$
$349$	−3.27364	−0.175234	−0.0876170	+	0.996154i	$0.527925\pi$
$350$	−27.4060	−1.46491
$351$	0	0
$352$	−7.64578	−0.407521
$353$	−23.4498	−1.24811	−0.624054	−	0.781381i	$-0.714515\pi$
$353$	−23.4498	−1.24811	−0.624054	+	0.781381i	$0.714515\pi$
$354$	0	0
$355$	−10.4702	−0.555699
$356$	17.1815	0.910617
$357$	0	0
$358$	12.1682	0.643111
$359$	26.9502	1.42238	0.711188	−	0.703001i	$-0.248157\pi$
$359$	26.9502	1.42238	0.711188	+	0.703001i	$0.248157\pi$
$360$	0	0
$361$	−1.08332	−0.0570167
$362$	29.0304	1.52580
$363$	0	0
$364$	23.9720	1.25648
$365$	−34.7278	−1.81774
$366$	0	0
$367$	−6.45575	−0.336987	−0.168494	−	0.985703i	$-0.553890\pi$
$367$	−6.45575	−0.336987	−0.168494	+	0.985703i	$0.553890\pi$
$368$	28.6918	1.49566
$369$	0	0
$370$	11.0580	0.574877
$371$	26.1648	1.35841
$372$	0	0
$373$	9.95114	0.515250	0.257625	−	0.966245i	$-0.417060\pi$
$373$	9.95114	0.515250	0.257625	+	0.966245i	$0.417060\pi$
$374$	7.39562	0.382418
$375$	0	0
$376$	4.81552	0.248341
$377$	−14.1148	−0.726952
$378$	0	0
$379$	−5.99448	−0.307916	−0.153958	−	0.988077i	$-0.549202\pi$
$379$	−5.99448	−0.307916	−0.153958	+	0.988077i	$0.549202\pi$
$380$	22.9378	1.17668
$381$	0	0
$382$	2.15786	0.110406
$383$	−2.43976	−0.124666	−0.0623329	−	0.998055i	$-0.519854\pi$
$383$	−2.43976	−0.124666	−0.0623329	+	0.998055i	$0.519854\pi$
$384$	0	0
$385$	9.72319	0.495540
$386$	−31.9482	−1.62612
$387$	0	0
$388$	1.43488	0.0728450
$389$	1.63009	0.0826491	0.0413245	−	0.999146i	$-0.486842\pi$
$389$	1.63009	0.0826491	0.0413245	+	0.999146i	$0.486842\pi$
$390$	0	0
$391$	−24.7477	−1.25155
$392$	−1.38308	−0.0698559
$393$	0	0
$394$	−49.3028	−2.48384
$395$	−13.1393	−0.661110
$396$	0	0
$397$	−29.1550	−1.46325	−0.731624	−	0.681709i	$-0.761237\pi$
$397$	−29.1550	−1.46325	−0.731624	+	0.681709i	$0.761237\pi$
$398$	−15.6241	−0.783167
$399$	0	0
$400$	−19.7804	−0.989020
$401$	28.8479	1.44059	0.720297	−	0.693666i	$-0.244005\pi$
$401$	28.8479	1.44059	0.720297	+	0.693666i	$0.244005\pi$
$402$	0	0
$403$	−43.4644	−2.16512
$404$	−15.2155	−0.756999
$405$	0	0
$406$	−20.0614	−0.995630
$407$	−1.85262	−0.0918310
$408$	0	0
$409$	19.7104	0.974619	0.487309	−	0.873229i	$-0.337978\pi$
$409$	19.7104	0.974619	0.487309	+	0.873229i	$0.337978\pi$
$410$	−64.3370	−3.17738
$411$	0	0
$412$	2.70933	0.133479
$413$	−31.6375	−1.55678
$414$	0	0
$415$	53.5853	2.63040
$416$	32.9545	1.61573
$417$	0	0
$418$	−8.20839	−0.401485
$419$	6.33286	0.309381	0.154690	−	0.987963i	$-0.450562\pi$
$419$	6.33286	0.309381	0.154690	+	0.987963i	$0.450562\pi$
$420$	0	0
$421$	30.9731	1.50954	0.754768	−	0.655992i	$-0.227750\pi$
$421$	30.9731	1.50954	0.754768	+	0.655992i	$0.227750\pi$
$422$	42.3537	2.06174
$423$	0	0
$424$	3.84510	0.186734
$425$	17.0613	0.827597
$426$	0	0
$427$	−3.15900	−0.152874
$428$	2.24127	0.108336
$429$	0	0
$430$	−62.0774	−2.99364
$431$	0.711680	0.0342804	0.0171402	−	0.999853i	$-0.494544\pi$
$431$	0.711680	0.0342804	0.0171402	+	0.999853i	$0.494544\pi$
$432$	0	0
$433$	38.6317	1.85652	0.928259	−	0.371934i	$-0.121305\pi$
$433$	38.6317	1.85652	0.928259	+	0.371934i	$0.121305\pi$
$434$	−61.7758	−2.96533
$435$	0	0
$436$	19.1866	0.918870
$437$	27.4675	1.31395
$438$	0	0
$439$	10.1209	0.483043	0.241522	−	0.970395i	$-0.422353\pi$
$439$	10.1209	0.483043	0.241522	+	0.970395i	$0.422353\pi$
$440$	1.42889	0.0681197
$441$	0	0
$442$	−31.8763	−1.51620
$443$	27.6987	1.31600	0.658002	−	0.753016i	$-0.271402\pi$
$443$	27.6987	1.31600	0.658002	+	0.753016i	$0.271402\pi$
$444$	0	0
$445$	−30.0371	−1.42390
$446$	24.5818	1.16398
$447$	0	0
$448$	18.9033	0.893097
$449$	−20.3574	−0.960727	−0.480364	−	0.877069i	$-0.659495\pi$
$449$	−20.3574	−0.960727	−0.480364	+	0.877069i	$0.659495\pi$
$450$	0	0
$451$	10.7788	0.507555
$452$	22.2418	1.04616
$453$	0	0
$454$	−34.9380	−1.63972
$455$	−41.9085	−1.96470
$456$	0	0
$457$	−29.6278	−1.38593	−0.692964	−	0.720972i	$-0.743696\pi$
$457$	−29.6278	−1.38593	−0.692964	+	0.720972i	$0.743696\pi$
$458$	41.0832	1.91969
$459$	0	0
$460$	35.1651	1.63958
$461$	−13.6161	−0.634163	−0.317081	−	0.948398i	$-0.602703\pi$
$461$	−13.6161	−0.634163	−0.317081	+	0.948398i	$0.602703\pi$
$462$	0	0
$463$	−16.1578	−0.750914	−0.375457	−	0.926840i	$-0.622514\pi$
$463$	−16.1578	−0.750914	−0.375457	+	0.926840i	$0.622514\pi$
$464$	−14.4794	−0.672189
$465$	0	0
$466$	54.8934	2.54289
$467$	12.4292	0.575155	0.287577	−	0.957757i	$-0.407150\pi$
$467$	12.4292	0.575155	0.287577	+	0.957757i	$0.407150\pi$
$468$	0	0
$469$	18.3404	0.846880
$470$	61.9145	2.85590
$471$	0	0
$472$	−4.64936	−0.214004
$473$	10.4003	0.478205
$474$	0	0
$475$	−18.9364	−0.868860
$476$	−21.2108	−0.972196
$477$	0	0
$478$	−9.50063	−0.434549
$479$	−29.1644	−1.33255	−0.666277	−	0.745704i	$-0.732113\pi$
$479$	−29.1644	−1.33255	−0.666277	+	0.745704i	$0.732113\pi$
$480$	0	0
$481$	7.98509	0.364089
$482$	53.0350	2.41568
$483$	0	0
$484$	1.76061	0.0800276
$485$	−2.50850	−0.113905
$486$	0	0
$487$	36.8143	1.66822	0.834108	−	0.551601i	$-0.185983\pi$
$487$	36.8143	1.66822	0.834108	+	0.551601i	$0.185983\pi$
$488$	−0.464237	−0.0210150
$489$	0	0
$490$	−17.7826	−0.803337
$491$	−38.5259	−1.73865	−0.869326	−	0.494239i	$-0.835447\pi$
$491$	−38.5259	−1.73865	−0.869326	+	0.494239i	$0.835447\pi$
$492$	0	0
$493$	12.4890	0.562477
$494$	35.3794	1.59180
$495$	0	0
$496$	−44.5869	−2.00201
$497$	−10.7459	−0.482019
$498$	0	0
$499$	1.68198	0.0752959	0.0376480	−	0.999291i	$-0.488013\pi$
$499$	1.68198	0.0752959	0.0376480	+	0.999291i	$0.488013\pi$
$500$	2.85199	0.127545
$501$	0	0
$502$	−5.73217	−0.255839
$503$	−20.1033	−0.896360	−0.448180	−	0.893943i	$-0.647928\pi$
$503$	−20.1033	−0.896360	−0.448180	+	0.893943i	$0.647928\pi$
$504$	0	0
$505$	26.6001	1.18369
$506$	−12.5840	−0.559427
$507$	0	0
$508$	21.9807	0.975238
$509$	−32.9375	−1.45993	−0.729964	−	0.683485i	$-0.760463\pi$
$509$	−32.9375	−1.45993	−0.729964	+	0.683485i	$0.760463\pi$
$510$	0	0
$511$	−35.6423	−1.57672
$512$	29.7004	1.31259
$513$	0	0
$514$	−16.6219	−0.733161
$515$	−4.73652	−0.208716
$516$	0	0
$517$	−10.3730	−0.456203
$518$	11.3492	0.498654
$519$	0	0
$520$	−6.15875	−0.270079
$521$	−6.18719	−0.271066	−0.135533	−	0.990773i	$-0.543275\pi$
$521$	−6.18719	−0.271066	−0.135533	+	0.990773i	$0.543275\pi$
$522$	0	0
$523$	19.2567	0.842035	0.421017	−	0.907053i	$-0.361673\pi$
$523$	19.2567	0.842035	0.421017	+	0.907053i	$0.361673\pi$
$524$	−0.0878461	−0.00383757
$525$	0	0
$526$	3.56820	0.155581
$527$	38.4579	1.67525
$528$	0	0
$529$	19.1095	0.830846
$530$	49.4376	2.14743
$531$	0	0
$532$	23.5418	1.02067
$533$	−46.4585	−2.01234
$534$	0	0
$535$	−3.91825	−0.169401
$536$	2.69525	0.116417
$537$	0	0
$538$	16.7836	0.723594
$539$	2.97925	0.128325
$540$	0	0
$541$	−32.4081	−1.39333	−0.696666	−	0.717395i	$-0.745334\pi$
$541$	−32.4081	−1.39333	−0.696666	+	0.717395i	$0.745334\pi$
$542$	35.7686	1.53639
$543$	0	0
$544$	−29.1586	−1.25017
$545$	−33.5425	−1.43680
$546$	0	0
$547$	−10.6214	−0.454140	−0.227070	−	0.973878i	$-0.572915\pi$
$547$	−10.6214	−0.454140	−0.227070	+	0.973878i	$0.572915\pi$
$548$	−0.429133	−0.0183317
$549$	0	0
$550$	8.67555	0.369927
$551$	−13.8615	−0.590522
$552$	0	0
$553$	−13.4853	−0.573454
$554$	15.7001	0.667034
$555$	0	0
$556$	−37.3021	−1.58196
$557$	33.3894	1.41475	0.707376	−	0.706837i	$-0.249879\pi$
$557$	33.3894	1.41475	0.707376	+	0.706837i	$0.249879\pi$
$558$	0	0
$559$	−44.8268	−1.89597
$560$	−42.9909	−1.81670
$561$	0	0
$562$	34.6664	1.46231
$563$	−8.93867	−0.376720	−0.188360	−	0.982100i	$-0.560317\pi$
$563$	−8.93867	−0.376720	−0.188360	+	0.982100i	$0.560317\pi$
$564$	0	0
$565$	−38.8836	−1.63585
$566$	34.3309	1.44304
$567$	0	0
$568$	−1.57918	−0.0662611
$569$	2.35475	0.0987163	0.0493582	−	0.998781i	$-0.484282\pi$
$569$	2.35475	0.0987163	0.0493582	+	0.998781i	$0.484282\pi$
$570$	0	0
$571$	4.24507	0.177651	0.0888253	−	0.996047i	$-0.471689\pi$
$571$	4.24507	0.177651	0.0888253	+	0.996047i	$0.471689\pi$
$572$	−7.58850	−0.317291
$573$	0	0
$574$	−66.0313	−2.75609
$575$	−29.0307	−1.21066
$576$	0	0
$577$	−8.98406	−0.374011	−0.187006	−	0.982359i	$-0.559878\pi$
$577$	−8.98406	−0.374011	−0.187006	+	0.982359i	$0.559878\pi$
$578$	−4.76227	−0.198084
$579$	0	0
$580$	−17.7462	−0.736871
$581$	54.9964	2.28164
$582$	0	0
$583$	−8.28262	−0.343031
$584$	−5.23789	−0.216745
$585$	0	0
$586$	25.8619	1.06835
$587$	−0.0674569	−0.00278424	−0.00139212	−	0.999999i	$-0.500443\pi$
$587$	−0.0674569	−0.00278424	−0.00139212	+	0.999999i	$0.500443\pi$
$588$	0	0
$589$	−42.6844	−1.75878
$590$	−59.7782	−2.46103
$591$	0	0
$592$	8.19132	0.336661
$593$	19.4214	0.797540	0.398770	−	0.917051i	$-0.369437\pi$
$593$	19.4214	0.797540	0.398770	+	0.917051i	$0.369437\pi$
$594$	0	0
$595$	37.0813	1.52018
$596$	−23.7619	−0.973324
$597$	0	0
$598$	54.2390	2.21800
$599$	19.6058	0.801070	0.400535	−	0.916282i	$-0.368824\pi$
$599$	19.6058	0.801070	0.400535	+	0.916282i	$0.368824\pi$
$600$	0	0
$601$	32.7244	1.33485	0.667427	−	0.744675i	$-0.267395\pi$
$601$	32.7244	1.33485	0.667427	+	0.744675i	$0.267395\pi$
$602$	−63.7122	−2.59672
$603$	0	0
$604$	3.77656	0.153666
$605$	−3.07794	−0.125136
$606$	0	0
$607$	−6.92562	−0.281102	−0.140551	−	0.990073i	$-0.544887\pi$
$607$	−6.92562	−0.281102	−0.140551	+	0.990073i	$0.544887\pi$
$608$	32.3631	1.31250
$609$	0	0
$610$	−5.96883	−0.241671
$611$	44.7092	1.80874
$612$	0	0
$613$	10.1296	0.409131	0.204566	−	0.978853i	$-0.434422\pi$
$613$	10.1296	0.409131	0.204566	+	0.978853i	$0.434422\pi$
$614$	−62.2532	−2.51233
$615$	0	0
$616$	1.46652	0.0590878
$617$	−3.90198	−0.157088	−0.0785439	−	0.996911i	$-0.525027\pi$
$617$	−3.90198	−0.157088	−0.0785439	+	0.996911i	$0.525027\pi$
$618$	0	0
$619$	8.08612	0.325009	0.162504	−	0.986708i	$-0.448043\pi$
$619$	8.08612	0.325009	0.162504	+	0.986708i	$0.448043\pi$
$620$	−54.6466	−2.19466
$621$	0	0
$622$	−37.1221	−1.48846
$623$	−30.8281	−1.23510
$624$	0	0
$625$	−27.3545	−1.09418
$626$	13.8188	0.552310
$627$	0	0
$628$	−4.89459	−0.195315
$629$	−7.06532	−0.281713
$630$	0	0
$631$	14.8852	0.592569	0.296284	−	0.955100i	$-0.404252\pi$
$631$	14.8852	0.592569	0.296284	+	0.955100i	$0.404252\pi$
$632$	−1.98176	−0.0788302
$633$	0	0
$634$	59.6589	2.36936
$635$	−38.4273	−1.52494
$636$	0	0
$637$	−12.8410	−0.508780
$638$	6.35056	0.251421
$639$	0	0
$640$	−11.3492	−0.448618
$641$	−6.49332	−0.256471	−0.128235	−	0.991744i	$-0.540931\pi$
$641$	−6.49332	−0.256471	−0.128235	+	0.991744i	$0.540931\pi$
$642$	0	0
$643$	−0.679815	−0.0268093	−0.0134046	−	0.999910i	$-0.504267\pi$
$643$	−0.679815	−0.0268093	−0.0134046	+	0.999910i	$0.504267\pi$
$644$	36.0912	1.42219
$645$	0	0
$646$	−31.3043	−1.23165
$647$	−9.53160	−0.374726	−0.187363	−	0.982291i	$-0.559994\pi$
$647$	−9.53160	−0.374726	−0.187363	+	0.982291i	$0.559994\pi$
$648$	0	0
$649$	10.0151	0.393126
$650$	−37.3930	−1.46667
$651$	0	0
$652$	29.6009	1.15926
$653$	15.7563	0.616591	0.308296	−	0.951291i	$-0.400241\pi$
$653$	15.7563	0.616591	0.308296	+	0.951291i	$0.400241\pi$
$654$	0	0
$655$	0.153575	0.00600066
$656$	−47.6583	−1.86075
$657$	0	0
$658$	63.5450	2.47724
$659$	−22.5633	−0.878941	−0.439470	−	0.898257i	$-0.644834\pi$
$659$	−22.5633	−0.878941	−0.439470	+	0.898257i	$0.644834\pi$
$660$	0	0
$661$	−47.5006	−1.84756	−0.923779	−	0.382927i	$-0.874916\pi$
$661$	−47.5006	−1.84756	−0.923779	+	0.382927i	$0.874916\pi$
$662$	−2.33180	−0.0906281
$663$	0	0
$664$	8.08211	0.313647
$665$	−41.1564	−1.59598
$666$	0	0
$667$	−21.2507	−0.822830
$668$	−24.9603	−0.965744
$669$	0	0
$670$	34.6536	1.33879
$671$	1.00000	0.0386046
$672$	0	0
$673$	6.74671	0.260067	0.130033	−	0.991510i	$-0.458492\pi$
$673$	6.74671	0.260067	0.130033	+	0.991510i	$0.458492\pi$
$674$	−54.0751	−2.08290
$675$	0	0
$676$	9.81973	0.377682
$677$	−31.7778	−1.22132	−0.610660	−	0.791893i	$-0.709096\pi$
$677$	−31.7778	−1.22132	−0.610660	+	0.791893i	$0.709096\pi$
$678$	0	0
$679$	−2.57456	−0.0988024
$680$	5.44935	0.208973
$681$	0	0
$682$	19.5555	0.748819
$683$	4.24491	0.162427	0.0812135	−	0.996697i	$-0.474120\pi$
$683$	4.24491	0.162427	0.0812135	+	0.996697i	$0.474120\pi$
$684$	0	0
$685$	0.750222	0.0286645
$686$	24.6312	0.940422
$687$	0	0
$688$	−45.9845	−1.75314
$689$	35.6994	1.36004
$690$	0	0
$691$	39.5060	1.50288	0.751440	−	0.659801i	$-0.229359\pi$
$691$	39.5060	1.50288	0.751440	+	0.659801i	$0.229359\pi$
$692$	−33.8069	−1.28515
$693$	0	0
$694$	−16.8094	−0.638076
$695$	65.2125	2.47365
$696$	0	0
$697$	41.1071	1.55704
$698$	−6.34834	−0.240288
$699$	0	0
$700$	−24.8817	−0.940438
$701$	2.69413	0.101756	0.0508779	−	0.998705i	$-0.483798\pi$
$701$	2.69413	0.101756	0.0508779	+	0.998705i	$0.483798\pi$
$702$	0	0
$703$	7.84179	0.295759
$704$	−5.98396	−0.225529
$705$	0	0
$706$	−45.4746	−1.71146
$707$	27.3006	1.02675
$708$	0	0
$709$	44.7755	1.68158	0.840790	−	0.541362i	$-0.182091\pi$
$709$	44.7755	1.68158	0.840790	+	0.541362i	$0.182091\pi$
$710$	−20.3041	−0.761997
$711$	0	0
$712$	−4.53041	−0.169784
$713$	−65.4380	−2.45067
$714$	0	0
$715$	13.2664	0.496135
$716$	11.0474	0.412862
$717$	0	0
$718$	52.2626	1.95042
$719$	28.5746	1.06565	0.532827	−	0.846224i	$-0.321130\pi$
$719$	28.5746	1.06565	0.532827	+	0.846224i	$0.321130\pi$
$720$	0	0
$721$	−4.86126	−0.181043
$722$	−2.10080	−0.0781836
$723$	0	0
$724$	26.3564	0.979529
$725$	14.6504	0.544104
$726$	0	0
$727$	−25.0251	−0.928130	−0.464065	−	0.885801i	$-0.653610\pi$
$727$	−25.0251	−0.928130	−0.464065	+	0.885801i	$0.653610\pi$
$728$	−6.32094	−0.234270
$729$	0	0
$730$	−67.3451	−2.49256
$731$	39.6634	1.46700
$732$	0	0
$733$	−0.528034	−0.0195034	−0.00975169	−	0.999952i	$-0.503104\pi$
$733$	−0.528034	−0.0195034	−0.00975169	+	0.999952i	$0.503104\pi$
$734$	−12.5192	−0.462091
$735$	0	0
$736$	49.6148	1.82883
$737$	−5.80576	−0.213858
$738$	0	0
$739$	20.9456	0.770496	0.385248	−	0.922813i	$-0.374116\pi$
$739$	20.9456	0.770496	0.385248	+	0.922813i	$0.374116\pi$
$740$	10.0394	0.369057
$741$	0	0
$742$	50.7395	1.86270
$743$	17.0344	0.624932	0.312466	−	0.949929i	$-0.398845\pi$
$743$	17.0344	0.624932	0.312466	+	0.949929i	$0.398845\pi$
$744$	0	0
$745$	41.5411	1.52195
$746$	19.2975	0.706533
$747$	0	0
$748$	6.71441	0.245503
$749$	−4.02143	−0.146940
$750$	0	0
$751$	−10.2079	−0.372491	−0.186246	−	0.982503i	$-0.559632\pi$
$751$	−10.2079	−0.372491	−0.186246	+	0.982503i	$0.559632\pi$
$752$	45.8639	1.67248
$753$	0	0
$754$	−27.3719	−0.996826
$755$	−6.60228	−0.240281
$756$	0	0
$757$	18.2567	0.663549	0.331775	−	0.943359i	$-0.392353\pi$
$757$	18.2567	0.663549	0.331775	+	0.943359i	$0.392353\pi$
$758$	−11.6247	−0.422227
$759$	0	0
$760$	−6.04823	−0.219392
$761$	−5.09224	−0.184594	−0.0922968	−	0.995732i	$-0.529421\pi$
$761$	−5.09224	−0.184594	−0.0922968	+	0.995732i	$0.529421\pi$
$762$	0	0
$763$	−34.4258	−1.24630
$764$	1.95910	0.0708779
$765$	0	0
$766$	−4.73125	−0.170947
$767$	−43.1665	−1.55865
$768$	0	0
$769$	−9.02962	−0.325617	−0.162808	−	0.986658i	$-0.552055\pi$
$769$	−9.02962	−0.325617	−0.162808	+	0.986658i	$0.552055\pi$
$770$	18.8555	0.679505
$771$	0	0
$772$	−29.0054	−1.04393
$773$	9.13122	0.328427	0.164214	−	0.986425i	$-0.447491\pi$
$773$	9.13122	0.328427	0.164214	+	0.986425i	$0.447491\pi$
$774$	0	0
$775$	45.1137	1.62053
$776$	−0.378349	−0.0135819
$777$	0	0
$778$	3.16113	0.113332
$779$	−45.6247	−1.63468
$780$	0	0
$781$	3.40168	0.121722
$782$	−47.9915	−1.71617
$783$	0	0
$784$	−13.1727	−0.470453
$785$	8.55685	0.305407
$786$	0	0
$787$	28.3647	1.01109	0.505546	−	0.862800i	$-0.331291\pi$
$787$	28.3647	1.01109	0.505546	+	0.862800i	$0.331291\pi$
$788$	−44.7615	−1.59456
$789$	0	0
$790$	−25.4801	−0.906541
$791$	−39.9076	−1.41895
$792$	0	0
$793$	−4.31016	−0.153058
$794$	−56.5382	−2.00647
$795$	0	0
$796$	−14.1850	−0.502774
$797$	−36.2460	−1.28390	−0.641950	−	0.766747i	$-0.721874\pi$
$797$	−36.2460	−1.28390	−0.641950	+	0.766747i	$0.721874\pi$
$798$	0	0
$799$	−39.5593	−1.39951
$800$	−34.2050	−1.20933
$801$	0	0
$802$	55.9426	1.97540
$803$	11.2828	0.398162
$804$	0	0
$805$	−63.0956	−2.22383
$806$	−84.2874	−2.96890
$807$	0	0
$808$	4.01202	0.141142
$809$	53.3004	1.87394	0.936971	−	0.349408i	$-0.113617\pi$
$809$	53.3004	1.87394	0.936971	+	0.349408i	$0.113617\pi$
$810$	0	0
$811$	13.8664	0.486914	0.243457	−	0.969912i	$-0.421719\pi$
$811$	13.8664	0.486914	0.243457	+	0.969912i	$0.421719\pi$
$812$	−18.2135	−0.639170
$813$	0	0
$814$	−3.59265	−0.125922
$815$	−51.7491	−1.81269
$816$	0	0
$817$	−44.0224	−1.54015
$818$	38.2231	1.33644
$819$	0	0
$820$	−58.4109	−2.03980
$821$	42.4160	1.48033	0.740164	−	0.672427i	$-0.234748\pi$
$821$	42.4160	1.48033	0.740164	+	0.672427i	$0.234748\pi$
$822$	0	0
$823$	35.8679	1.25028	0.625138	−	0.780514i	$-0.285043\pi$
$823$	35.8679	1.25028	0.625138	+	0.780514i	$0.285043\pi$
$824$	−0.714396	−0.0248871
$825$	0	0
$826$	−61.3524	−2.13472
$827$	7.27629	0.253022	0.126511	−	0.991965i	$-0.459622\pi$
$827$	7.27629	0.253022	0.126511	+	0.991965i	$0.459622\pi$
$828$	0	0
$829$	−21.2488	−0.738000	−0.369000	−	0.929429i	$-0.620300\pi$
$829$	−21.2488	−0.738000	−0.369000	+	0.929429i	$0.620300\pi$
$830$	103.914	3.60691
$831$	0	0
$832$	25.7918	0.894171
$833$	11.3619	0.393668
$834$	0	0
$835$	43.6363	1.51010
$836$	−7.45232	−0.257744
$837$	0	0
$838$	12.2809	0.424236
$839$	−27.7675	−0.958641	−0.479320	−	0.877640i	$-0.659117\pi$
$839$	−27.7675	−0.958641	−0.479320	+	0.877640i	$0.659117\pi$
$840$	0	0
$841$	−18.2758	−0.630199
$842$	60.0639	2.06994
$843$	0	0
$844$	38.4525	1.32359
$845$	−17.1671	−0.590566
$846$	0	0
$847$	−3.15900	−0.108544
$848$	36.6214	1.25758
$849$	0	0
$850$	33.0859	1.13484
$851$	12.0220	0.412108
$852$	0	0
$853$	−7.41186	−0.253777	−0.126889	−	0.991917i	$-0.540499\pi$
$853$	−7.41186	−0.253777	−0.126889	+	0.991917i	$0.540499\pi$
$854$	−6.12601	−0.209628
$855$	0	0
$856$	−0.590978	−0.0201992
$857$	13.6447	0.466095	0.233047	−	0.972465i	$-0.425130\pi$
$857$	13.6447	0.466095	0.233047	+	0.972465i	$0.425130\pi$
$858$	0	0
$859$	−39.3829	−1.34373	−0.671864	−	0.740675i	$-0.734506\pi$
$859$	−39.3829	−1.34373	−0.671864	+	0.740675i	$0.734506\pi$
$860$	−56.3595	−1.92184
$861$	0	0
$862$	1.38011	0.0470068
$863$	48.1420	1.63877	0.819387	−	0.573241i	$-0.194314\pi$
$863$	48.1420	1.63877	0.819387	+	0.573241i	$0.194314\pi$
$864$	0	0
$865$	59.1022	2.00953
$866$	74.9156	2.54574
$867$	0	0
$868$	−56.0857	−1.90367
$869$	4.26886	0.144811
$870$	0	0
$871$	25.0238	0.847898
$872$	−5.05911	−0.171323
$873$	0	0
$874$	53.2657	1.80174
$875$	−5.11722	−0.172994
$876$	0	0
$877$	37.6587	1.27164	0.635822	−	0.771836i	$-0.280661\pi$
$877$	37.6587	1.27164	0.635822	+	0.771836i	$0.280661\pi$
$878$	19.6267	0.662369
$879$	0	0
$880$	13.6090	0.458760
$881$	−32.6783	−1.10096	−0.550480	−	0.834848i	$-0.685555\pi$
$881$	−32.6783	−1.10096	−0.550480	+	0.834848i	$0.685555\pi$
$882$	0	0
$883$	−2.47331	−0.0832335	−0.0416167	−	0.999134i	$-0.513251\pi$
$883$	−2.47331	−0.0832335	−0.0416167	+	0.999134i	$0.513251\pi$
$884$	−28.9402	−0.973364
$885$	0	0
$886$	53.7141	1.80456
$887$	−2.57792	−0.0865580	−0.0432790	−	0.999063i	$-0.513780\pi$
$887$	−2.57792	−0.0865580	−0.0432790	+	0.999063i	$0.513780\pi$
$888$	0	0
$889$	−39.4393	−1.32275
$890$	−58.2489	−1.95251
$891$	0	0
$892$	22.3176	0.747248
$893$	43.9068	1.46929
$894$	0	0
$895$	−19.3134	−0.645576
$896$	−11.6481	−0.389136
$897$	0	0
$898$	−39.4777	−1.31739
$899$	33.0235	1.10140
$900$	0	0
$901$	−31.5874	−1.05233
$902$	20.9026	0.695981
$903$	0	0
$904$	−5.86470	−0.195057
$905$	−46.0770	−1.53165
$906$	0	0
$907$	8.34904	0.277225	0.138613	−	0.990347i	$-0.455736\pi$
$907$	8.34904	0.277225	0.138613	+	0.990347i	$0.455736\pi$
$908$	−31.7199	−1.05266
$909$	0	0
$910$	−81.2702	−2.69408
$911$	8.66605	0.287119	0.143560	−	0.989642i	$-0.454145\pi$
$911$	8.66605	0.287119	0.143560	+	0.989642i	$0.454145\pi$
$912$	0	0
$913$	−17.4095	−0.576169
$914$	−57.4550	−1.90044
$915$	0	0
$916$	37.2991	1.23240
$917$	0.157619	0.00520504
$918$	0	0
$919$	17.4019	0.574035	0.287017	−	0.957925i	$-0.407336\pi$
$919$	17.4019	0.574035	0.287017	+	0.957925i	$0.407336\pi$
$920$	−9.27234	−0.305700
$921$	0	0
$922$	−26.4046	−0.869591
$923$	−14.6618	−0.482599
$924$	0	0
$925$	−8.28809	−0.272511
$926$	−31.3336	−1.02969
$927$	0	0
$928$	−25.0383	−0.821922
$929$	−7.38998	−0.242458	−0.121229	−	0.992625i	$-0.538683\pi$
$929$	−7.38998	−0.242458	−0.121229	+	0.992625i	$0.538683\pi$
$930$	0	0
$931$	−12.6106	−0.413296
$932$	49.8372	1.63247
$933$	0	0
$934$	24.1031	0.788676
$935$	−11.7383	−0.383884
$936$	0	0
$937$	41.9703	1.37111	0.685555	−	0.728021i	$-0.259560\pi$
$937$	41.9703	1.37111	0.685555	+	0.728021i	$0.259560\pi$
$938$	35.5662	1.16128
$939$	0	0
$940$	56.2116	1.83342
$941$	51.7763	1.68786	0.843929	−	0.536454i	$-0.180237\pi$
$941$	51.7763	1.68786	0.843929	+	0.536454i	$0.180237\pi$
$942$	0	0
$943$	−69.9458	−2.27775
$944$	−44.2814	−1.44124
$945$	0	0
$946$	20.1685	0.655734
$947$	−22.0964	−0.718037	−0.359019	−	0.933330i	$-0.616889\pi$
$947$	−22.0964	−0.718037	−0.359019	+	0.933330i	$0.616889\pi$
$948$	0	0
$949$	−48.6307	−1.57862
$950$	−36.7219	−1.19142
$951$	0	0
$952$	5.59286	0.181266
$953$	36.4561	1.18093	0.590464	−	0.807064i	$-0.298945\pi$
$953$	36.4561	1.18093	0.590464	+	0.807064i	$0.298945\pi$
$954$	0	0
$955$	−3.42495	−0.110829
$956$	−8.62553	−0.278970
$957$	0	0
$958$	−56.5564	−1.82725
$959$	0.769979	0.0248639
$960$	0	0
$961$	70.6905	2.28034
$962$	15.4849	0.499253
$963$	0	0
$964$	48.1500	1.55081
$965$	50.7081	1.63235
$966$	0	0
$967$	1.95027	0.0627164	0.0313582	−	0.999508i	$-0.490017\pi$
$967$	1.95027	0.0627164	0.0313582	+	0.999508i	$0.490017\pi$
$968$	−0.464237	−0.0149211
$969$	0	0
$970$	−4.86455	−0.156191
$971$	40.8271	1.31020	0.655102	−	0.755540i	$-0.272626\pi$
$971$	40.8271	1.31020	0.655102	+	0.755540i	$0.272626\pi$
$972$	0	0
$973$	66.9298	2.14567
$974$	71.3914	2.28753
$975$	0	0
$976$	−4.42148	−0.141528
$977$	29.4296	0.941535	0.470767	−	0.882257i	$-0.343977\pi$
$977$	29.4296	0.941535	0.470767	+	0.882257i	$0.343977\pi$
$978$	0	0
$979$	9.75884	0.311894
$980$	−16.1447	−0.515723
$981$	0	0
$982$	−74.7106	−2.38411
$983$	−12.2766	−0.391562	−0.195781	−	0.980648i	$-0.562724\pi$
$983$	−12.2766	−0.391562	−0.195781	+	0.980648i	$0.562724\pi$
$984$	0	0
$985$	78.2533	2.49336
$986$	24.2191	0.771292
$987$	0	0
$988$	32.1207	1.02189
$989$	−67.4892	−2.14603
$990$	0	0
$991$	−30.0821	−0.955588	−0.477794	−	0.878472i	$-0.658563\pi$
$991$	−30.0821	−0.955588	−0.477794	+	0.878472i	$0.658563\pi$
$992$	−77.1014	−2.44797
$993$	0	0
$994$	−20.8387	−0.660965
$995$	24.7986	0.786168
$996$	0	0
$997$	28.9187	0.915866	0.457933	−	0.888987i	$-0.348590\pi$
$997$	28.9187	0.915866	0.457933	+	0.888987i	$0.348590\pi$
$998$	3.26175	0.103249
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.j.1.12		14
3.2	odd	2		2013.2.a.h.1.3	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.h.1.3	✓	14	3.2	odd	2
6039.2.a.j.1.12		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)

\(f(q)\)	\(=\)	\(q+1.93923 q^{2} +1.76061 q^{4} -3.07794 q^{5} -3.15900 q^{7} -0.464237 q^{8} +O(q^{10})\)	\(q+1.93923 q^{2} +1.76061 q^{4} -3.07794 q^{5} -3.15900 q^{7} -0.464237 q^{8} -5.96883 q^{10} +1.00000 q^{11} -4.31016 q^{13} -6.12601 q^{14} -4.42148 q^{16} +3.81369 q^{17} -4.23281 q^{19} -5.41904 q^{20} +1.93923 q^{22} -6.48918 q^{23} +4.47371 q^{25} -8.35838 q^{26} -5.56175 q^{28} +3.27479 q^{29} +10.0842 q^{31} -7.64578 q^{32} +7.39562 q^{34} +9.72319 q^{35} -1.85262 q^{37} -8.20839 q^{38} +1.42889 q^{40} +10.7788 q^{41} +10.4003 q^{43} +1.76061 q^{44} -12.5840 q^{46} -10.3730 q^{47} +2.97925 q^{49} +8.67555 q^{50} -7.58850 q^{52} -8.28262 q^{53} -3.07794 q^{55} +1.46652 q^{56} +6.35056 q^{58} +10.0151 q^{59} +1.00000 q^{61} +19.5555 q^{62} -5.98396 q^{64} +13.2664 q^{65} -5.80576 q^{67} +6.71441 q^{68} +18.8555 q^{70} +3.40168 q^{71} +11.2828 q^{73} -3.59265 q^{74} -7.45232 q^{76} -3.15900 q^{77} +4.26886 q^{79} +13.6090 q^{80} +20.9026 q^{82} -17.4095 q^{83} -11.7383 q^{85} +20.1685 q^{86} -0.464237 q^{88} +9.75884 q^{89} +13.6158 q^{91} -11.4249 q^{92} -20.1156 q^{94} +13.0283 q^{95} +0.814992 q^{97} +5.77745 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) \) 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

Related objects

Downloads

Learn more