LMFDB - Embedded newform 6039.2.a.i.1.13

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} + \cdots - 47 $ x^13 - 2x^12 - 19x^11 + 35x^10 + 136x^9 - 220x^8 - 469x^7 + 610x^6 + 841x^5 - 760x^4 - 742x^3 + 366x^2 + 236x - 47
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-2.59890$ 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)

$f(q)$	$=$	$q+2.59890 q^{2} +4.75428 q^{4} +3.50938 q^{5} +0.252288 q^{7} +7.15810 q^{8} +O(q^{10})$	\(q+2.59890 q^{2} +4.75428 q^{4} +3.50938 q^{5} +0.252288 q^{7} +7.15810 q^{8} +9.12053 q^{10} -1.00000 q^{11} +0.314950 q^{13} +0.655671 q^{14} +9.09463 q^{16} -3.13246 q^{17} +0.841231 q^{19} +16.6846 q^{20} -2.59890 q^{22} -1.45222 q^{23} +7.31576 q^{25} +0.818524 q^{26} +1.19945 q^{28} +6.76829 q^{29} -7.07730 q^{31} +9.31982 q^{32} -8.14095 q^{34} +0.885374 q^{35} +8.89628 q^{37} +2.18627 q^{38} +25.1205 q^{40} +8.43604 q^{41} +1.80409 q^{43} -4.75428 q^{44} -3.77418 q^{46} -2.30017 q^{47} -6.93635 q^{49} +19.0129 q^{50} +1.49736 q^{52} +4.41103 q^{53} -3.50938 q^{55} +1.80590 q^{56} +17.5901 q^{58} -4.48036 q^{59} -1.00000 q^{61} -18.3932 q^{62} +6.03203 q^{64} +1.10528 q^{65} -0.0414722 q^{67} -14.8926 q^{68} +2.30100 q^{70} +9.80832 q^{71} +8.90879 q^{73} +23.1205 q^{74} +3.99945 q^{76} -0.252288 q^{77} -8.51680 q^{79} +31.9165 q^{80} +21.9244 q^{82} -11.2151 q^{83} -10.9930 q^{85} +4.68865 q^{86} -7.15810 q^{88} -5.58487 q^{89} +0.0794581 q^{91} -6.90427 q^{92} -5.97790 q^{94} +2.95220 q^{95} +7.54000 q^{97} -18.0269 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * 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 $

$2$	2.59890	1.83770	0.918850	−	0.394607i	$-0.129119\pi$
$2$	2.59890	1.83770	0.918850	+	0.394607i	$0.129119\pi$
$3$	0	0
$3$	0	0
$4$	4.75428	2.37714
$5$	3.50938	1.56944	0.784721	−	0.619849i	$-0.212806\pi$
$5$	3.50938	1.56944	0.784721	+	0.619849i	$0.212806\pi$
$6$	0	0
$7$	0.252288	0.0953558	0.0476779	−	0.998863i	$-0.484818\pi$
$7$	0.252288	0.0953558	0.0476779	+	0.998863i	$0.484818\pi$
$8$	7.15810	2.53077
$9$	0	0
$10$	9.12053	2.88416
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.314950	0.0873514	0.0436757	−	0.999046i	$-0.486093\pi$
$13$	0.314950	0.0873514	0.0436757	+	0.999046i	$0.486093\pi$
$14$	0.655671	0.175235
$15$	0	0
$16$	9.09463	2.27366
$17$	−3.13246	−0.759733	−0.379866	−	0.925041i	$-0.624030\pi$
$17$	−3.13246	−0.759733	−0.379866	+	0.925041i	$0.624030\pi$
$18$	0	0
$19$	0.841231	0.192992	0.0964958	−	0.995333i	$-0.469237\pi$
$19$	0.841231	0.192992	0.0964958	+	0.995333i	$0.469237\pi$
$20$	16.6846	3.73079
$21$	0	0
$22$	−2.59890	−0.554087
$23$	−1.45222	−0.302809	−0.151405	−	0.988472i	$-0.548380\pi$
$23$	−1.45222	−0.302809	−0.151405	+	0.988472i	$0.548380\pi$
$24$	0	0
$25$	7.31576	1.46315
$26$	0.818524	0.160526
$27$	0	0
$28$	1.19945	0.226674
$29$	6.76829	1.25684	0.628420	−	0.777874i	$-0.283702\pi$
$29$	6.76829	1.25684	0.628420	+	0.777874i	$0.283702\pi$
$30$	0	0
$31$	−7.07730	−1.27112	−0.635560	−	0.772051i	$-0.719231\pi$
$31$	−7.07730	−1.27112	−0.635560	+	0.772051i	$0.719231\pi$
$32$	9.31982	1.64753
$33$	0	0
$34$	−8.14095	−1.39616
$35$	0.885374	0.149656
$36$	0	0
$37$	8.89628	1.46254	0.731270	−	0.682088i	$-0.238928\pi$
$37$	8.89628	1.46254	0.731270	+	0.682088i	$0.238928\pi$
$38$	2.18627	0.354661
$39$	0	0
$40$	25.1205	3.97190
$41$	8.43604	1.31749	0.658744	−	0.752367i	$-0.271088\pi$
$41$	8.43604	1.31749	0.658744	+	0.752367i	$0.271088\pi$
$42$	0	0
$43$	1.80409	0.275121	0.137561	−	0.990493i	$-0.456074\pi$
$43$	1.80409	0.275121	0.137561	+	0.990493i	$0.456074\pi$
$44$	−4.75428	−0.716735
$45$	0	0
$46$	−3.77418	−0.556473
$47$	−2.30017	−0.335514	−0.167757	−	0.985828i	$-0.553652\pi$
$47$	−2.30017	−0.335514	−0.167757	+	0.985828i	$0.553652\pi$
$48$	0	0
$49$	−6.93635	−0.990907
$50$	19.0129	2.68883
$51$	0	0
$52$	1.49736	0.207647
$53$	4.41103	0.605901	0.302951	−	0.953006i	$-0.402028\pi$
$53$	4.41103	0.605901	0.302951	+	0.953006i	$0.402028\pi$
$54$	0	0
$55$	−3.50938	−0.473205
$56$	1.80590	0.241324
$57$	0	0
$58$	17.5901	2.30969
$59$	−4.48036	−0.583293	−0.291647	−	0.956526i	$-0.594203\pi$
$59$	−4.48036	−0.583293	−0.291647	+	0.956526i	$0.594203\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−18.3932	−2.33594
$63$	0	0
$64$	6.03203	0.754004
$65$	1.10528	0.137093
$66$	0	0
$67$	−0.0414722	−0.00506664	−0.00253332	−	0.999997i	$-0.500806\pi$
$67$	−0.0414722	−0.00506664	−0.00253332	+	0.999997i	$0.500806\pi$
$68$	−14.8926	−1.80599
$69$	0	0
$70$	2.30100	0.275022
$71$	9.80832	1.16403	0.582017	−	0.813177i	$-0.302264\pi$
$71$	9.80832	1.16403	0.582017	+	0.813177i	$0.302264\pi$
$72$	0	0
$73$	8.90879	1.04269	0.521347	−	0.853345i	$-0.325430\pi$
$73$	8.90879	1.04269	0.521347	+	0.853345i	$0.325430\pi$
$74$	23.1205	2.68771
$75$	0	0
$76$	3.99945	0.458768
$77$	−0.252288	−0.0287509
$78$	0	0
$79$	−8.51680	−0.958215	−0.479108	−	0.877756i	$-0.659040\pi$
$79$	−8.51680	−0.958215	−0.479108	+	0.877756i	$0.659040\pi$
$80$	31.9165	3.56837
$81$	0	0
$82$	21.9244	2.42115
$83$	−11.2151	−1.23102	−0.615509	−	0.788130i	$-0.711050\pi$
$83$	−11.2151	−1.23102	−0.615509	+	0.788130i	$0.711050\pi$
$84$	0	0
$85$	−10.9930	−1.19236
$86$	4.68865	0.505590
$87$	0	0
$88$	−7.15810	−0.763056
$89$	−5.58487	−0.591995	−0.295997	−	0.955189i	$-0.595652\pi$
$89$	−5.58487	−0.591995	−0.295997	+	0.955189i	$0.595652\pi$
$90$	0	0
$91$	0.0794581	0.00832947
$92$	−6.90427	−0.719820
$93$	0	0
$94$	−5.97790	−0.616573
$95$	2.95220	0.302889
$96$	0	0
$97$	7.54000	0.765571	0.382786	−	0.923837i	$-0.374965\pi$
$97$	7.54000	0.765571	0.382786	+	0.923837i	$0.374965\pi$
$98$	−18.0269	−1.82099
$99$	0	0
$100$	34.7812	3.47812
$101$	5.85246	0.582342	0.291171	−	0.956671i	$-0.405955\pi$
$101$	5.85246	0.582342	0.291171	+	0.956671i	$0.405955\pi$
$102$	0	0
$103$	−17.3802	−1.71253	−0.856263	−	0.516541i	$-0.827219\pi$
$103$	−17.3802	−1.71253	−0.856263	+	0.516541i	$0.827219\pi$
$104$	2.25444	0.221066
$105$	0	0
$106$	11.4638	1.11346
$107$	7.20395	0.696432	0.348216	−	0.937414i	$-0.386788\pi$
$107$	7.20395	0.696432	0.348216	+	0.937414i	$0.386788\pi$
$108$	0	0
$109$	3.06602	0.293672	0.146836	−	0.989161i	$-0.453091\pi$
$109$	3.06602	0.293672	0.146836	+	0.989161i	$0.453091\pi$
$110$	−9.12053	−0.869608
$111$	0	0
$112$	2.29446	0.216806
$113$	12.0330	1.13197	0.565986	−	0.824415i	$-0.308496\pi$
$113$	12.0330	1.13197	0.565986	+	0.824415i	$0.308496\pi$
$114$	0	0
$115$	−5.09640	−0.475242
$116$	32.1784	2.98769
$117$	0	0
$118$	−11.6440	−1.07192
$119$	−0.790281	−0.0724450
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.59890	−0.235293
$123$	0	0
$124$	−33.6475	−3.02163
$125$	8.12687	0.726889
$126$	0	0
$127$	−7.56103	−0.670933	−0.335466	−	0.942052i	$-0.608894\pi$
$127$	−7.56103	−0.670933	−0.335466	+	0.942052i	$0.608894\pi$
$128$	−2.96300	−0.261895
$129$	0	0
$130$	2.87251	0.251936
$131$	−10.0905	−0.881613	−0.440807	−	0.897602i	$-0.645308\pi$
$131$	−10.0905	−0.881613	−0.440807	+	0.897602i	$0.645308\pi$
$132$	0	0
$133$	0.212232	0.0184029
$134$	−0.107782	−0.00931097
$135$	0	0
$136$	−22.4225	−1.92271
$137$	−11.2651	−0.962444	−0.481222	−	0.876599i	$-0.659807\pi$
$137$	−11.2651	−0.962444	−0.481222	+	0.876599i	$0.659807\pi$
$138$	0	0
$139$	10.2656	0.870720	0.435360	−	0.900257i	$-0.356621\pi$
$139$	10.2656	0.870720	0.435360	+	0.900257i	$0.356621\pi$
$140$	4.20932	0.355752
$141$	0	0
$142$	25.4908	2.13914
$143$	−0.314950	−0.0263374
$144$	0	0
$145$	23.7525	1.97254
$146$	23.1530	1.91616
$147$	0	0
$148$	42.2954	3.47666
$149$	−16.9755	−1.39069	−0.695345	−	0.718676i	$-0.744748\pi$
$149$	−16.9755	−1.39069	−0.695345	+	0.718676i	$0.744748\pi$
$150$	0	0
$151$	−10.4087	−0.847044	−0.423522	−	0.905886i	$-0.639206\pi$
$151$	−10.4087	−0.847044	−0.423522	+	0.905886i	$0.639206\pi$
$152$	6.02161	0.488417
$153$	0	0
$154$	−0.655671	−0.0528355
$155$	−24.8369	−1.99495
$156$	0	0
$157$	−15.3822	−1.22764	−0.613818	−	0.789448i	$-0.710367\pi$
$157$	−15.3822	−1.22764	−0.613818	+	0.789448i	$0.710367\pi$
$158$	−22.1343	−1.76091
$159$	0	0
$160$	32.7068	2.58570
$161$	−0.366378	−0.0288746
$162$	0	0
$163$	0.539069	0.0422232	0.0211116	−	0.999777i	$-0.493279\pi$
$163$	0.539069	0.0422232	0.0211116	+	0.999777i	$0.493279\pi$
$164$	40.1073	3.13185
$165$	0	0
$166$	−29.1470	−2.26224
$167$	−21.2985	−1.64813	−0.824063	−	0.566498i	$-0.808298\pi$
$167$	−21.2985	−1.64813	−0.824063	+	0.566498i	$0.808298\pi$
$168$	0	0
$169$	−12.9008	−0.992370
$170$	−28.5697	−2.19119
$171$	0	0
$172$	8.57715	0.654002
$173$	−20.9450	−1.59242	−0.796209	−	0.605022i	$-0.793164\pi$
$173$	−20.9450	−1.59242	−0.796209	+	0.605022i	$0.793164\pi$
$174$	0	0
$175$	1.84568	0.139520
$176$	−9.09463	−0.685533
$177$	0	0
$178$	−14.5145	−1.08791
$179$	2.59129	0.193682	0.0968409	−	0.995300i	$-0.469126\pi$
$179$	2.59129	0.193682	0.0968409	+	0.995300i	$0.469126\pi$
$180$	0	0
$181$	−10.8521	−0.806634	−0.403317	−	0.915060i	$-0.632143\pi$
$181$	−10.8521	−0.806634	−0.403317	+	0.915060i	$0.632143\pi$
$182$	0.206504	0.0153071
$183$	0	0
$184$	−10.3952	−0.766341
$185$	31.2204	2.29537
$186$	0	0
$187$	3.13246	0.229068
$188$	−10.9356	−0.797563
$189$	0	0
$190$	7.67247	0.556620
$191$	−14.8806	−1.07672	−0.538362	−	0.842714i	$-0.680957\pi$
$191$	−14.8806	−1.07672	−0.538362	+	0.842714i	$0.680957\pi$
$192$	0	0
$193$	−0.222935	−0.0160472	−0.00802362	−	0.999968i	$-0.502554\pi$
$193$	−0.222935	−0.0160472	−0.00802362	+	0.999968i	$0.502554\pi$
$194$	19.5957	1.40689
$195$	0	0
$196$	−32.9774	−2.35553
$197$	−2.16031	−0.153916	−0.0769580	−	0.997034i	$-0.524521\pi$
$197$	−2.16031	−0.153916	−0.0769580	+	0.997034i	$0.524521\pi$
$198$	0	0
$199$	24.6468	1.74716	0.873581	−	0.486678i	$-0.161792\pi$
$199$	24.6468	1.74716	0.873581	+	0.486678i	$0.161792\pi$
$200$	52.3669	3.70290
$201$	0	0
$202$	15.2100	1.07017
$203$	1.70756	0.119847
$204$	0	0
$205$	29.6053	2.06772
$206$	−45.1695	−3.14711
$207$	0	0
$208$	2.86435	0.198607
$209$	−0.841231	−0.0581891
$210$	0	0
$211$	12.6063	0.867851	0.433926	−	0.900949i	$-0.357128\pi$
$211$	12.6063	0.867851	0.433926	+	0.900949i	$0.357128\pi$
$212$	20.9713	1.44031
$213$	0	0
$214$	18.7223	1.27983
$215$	6.33124	0.431787
$216$	0	0
$217$	−1.78552	−0.121209
$218$	7.96828	0.539680
$219$	0	0
$220$	−16.6846	−1.12487
$221$	−0.986568	−0.0663637
$222$	0	0
$223$	10.5825	0.708657	0.354329	−	0.935121i	$-0.384709\pi$
$223$	10.5825	0.708657	0.354329	+	0.935121i	$0.384709\pi$
$224$	2.35128	0.157101
$225$	0	0
$226$	31.2726	2.08023
$227$	−9.60279	−0.637359	−0.318680	−	0.947862i	$-0.603239\pi$
$227$	−9.60279	−0.637359	−0.318680	+	0.947862i	$0.603239\pi$
$228$	0	0
$229$	4.99502	0.330080	0.165040	−	0.986287i	$-0.447225\pi$
$229$	4.99502	0.330080	0.165040	+	0.986287i	$0.447225\pi$
$230$	−13.2450	−0.873352
$231$	0	0
$232$	48.4481	3.18077
$233$	−18.4718	−1.21013	−0.605064	−	0.796177i	$-0.706853\pi$
$233$	−18.4718	−1.21013	−0.605064	+	0.796177i	$0.706853\pi$
$234$	0	0
$235$	−8.07216	−0.526569
$236$	−21.3009	−1.38657
$237$	0	0
$238$	−2.05386	−0.133132
$239$	−6.77945	−0.438526	−0.219263	−	0.975666i	$-0.570365\pi$
$239$	−6.77945	−0.438526	−0.219263	+	0.975666i	$0.570365\pi$
$240$	0	0
$241$	15.6701	1.00940	0.504701	−	0.863294i	$-0.331603\pi$
$241$	15.6701	1.00940	0.504701	+	0.863294i	$0.331603\pi$
$242$	2.59890	0.167064
$243$	0	0
$244$	−4.75428	−0.304362
$245$	−24.3423	−1.55517
$246$	0	0
$247$	0.264946	0.0168581
$248$	−50.6600	−3.21691
$249$	0	0
$250$	21.1209	1.33580
$251$	−15.5439	−0.981120	−0.490560	−	0.871407i	$-0.663208\pi$
$251$	−15.5439	−0.981120	−0.490560	+	0.871407i	$0.663208\pi$
$252$	0	0
$253$	1.45222	0.0913004
$254$	−19.6504	−1.23297
$255$	0	0
$256$	−19.7646	−1.23529
$257$	−17.3834	−1.08434	−0.542172	−	0.840267i	$-0.682398\pi$
$257$	−17.3834	−1.08434	−0.542172	+	0.840267i	$0.682398\pi$
$258$	0	0
$259$	2.24442	0.139462
$260$	5.25481	0.325890
$261$	0	0
$262$	−26.2243	−1.62014
$263$	13.4938	0.832062	0.416031	−	0.909350i	$-0.363421\pi$
$263$	13.4938	0.832062	0.416031	+	0.909350i	$0.363421\pi$
$264$	0	0
$265$	15.4800	0.950928
$266$	0.551571	0.0338190
$267$	0	0
$268$	−0.197171	−0.0120441
$269$	−17.3064	−1.05519	−0.527594	−	0.849497i	$-0.676906\pi$
$269$	−17.3064	−1.05519	−0.527594	+	0.849497i	$0.676906\pi$
$270$	0	0
$271$	−2.53256	−0.153842	−0.0769210	−	0.997037i	$-0.524509\pi$
$271$	−2.53256	−0.153842	−0.0769210	+	0.997037i	$0.524509\pi$
$272$	−28.4885	−1.72737
$273$	0	0
$274$	−29.2769	−1.76868
$275$	−7.31576	−0.441157
$276$	0	0
$277$	23.6178	1.41906	0.709529	−	0.704676i	$-0.248908\pi$
$277$	23.6178	1.41906	0.709529	+	0.704676i	$0.248908\pi$
$278$	26.6794	1.60012
$279$	0	0
$280$	6.33760	0.378744
$281$	−10.4991	−0.626323	−0.313162	−	0.949700i	$-0.601388\pi$
$281$	−10.4991	−0.626323	−0.313162	+	0.949700i	$0.601388\pi$
$282$	0	0
$283$	7.65586	0.455093	0.227547	−	0.973767i	$-0.426930\pi$
$283$	7.65586	0.455093	0.227547	+	0.973767i	$0.426930\pi$
$284$	46.6315	2.76707
$285$	0	0
$286$	−0.818524	−0.0484003
$287$	2.12831	0.125630
$288$	0	0
$289$	−7.18771	−0.422806
$290$	61.7304	3.62493
$291$	0	0
$292$	42.3549	2.47863
$293$	−13.3177	−0.778026	−0.389013	−	0.921232i	$-0.627184\pi$
$293$	−13.3177	−0.778026	−0.389013	+	0.921232i	$0.627184\pi$
$294$	0	0
$295$	−15.7233	−0.915445
$296$	63.6805	3.70135
$297$	0	0
$298$	−44.1177	−2.55567
$299$	−0.457378	−0.0264508
$300$	0	0
$301$	0.455150	0.0262344
$302$	−27.0510	−1.55661
$303$	0	0
$304$	7.65068	0.438797
$305$	−3.50938	−0.200947
$306$	0	0
$307$	−7.23744	−0.413063	−0.206531	−	0.978440i	$-0.566218\pi$
$307$	−7.23744	−0.413063	−0.206531	+	0.978440i	$0.566218\pi$
$308$	−1.19945	−0.0683449
$309$	0	0
$310$	−64.5487	−3.66612
$311$	8.38540	0.475492	0.237746	−	0.971327i	$-0.423591\pi$
$311$	8.38540	0.475492	0.237746	+	0.971327i	$0.423591\pi$
$312$	0	0
$313$	8.07210	0.456262	0.228131	−	0.973630i	$-0.426739\pi$
$313$	8.07210	0.456262	0.228131	+	0.973630i	$0.426739\pi$
$314$	−39.9769	−2.25603
$315$	0	0
$316$	−40.4913	−2.27781
$317$	26.3618	1.48063	0.740314	−	0.672261i	$-0.234677\pi$
$317$	26.3618	1.48063	0.740314	+	0.672261i	$0.234677\pi$
$318$	0	0
$319$	−6.76829	−0.378952
$320$	21.1687	1.18337
$321$	0	0
$322$	−0.952180	−0.0530629
$323$	−2.63512	−0.146622
$324$	0	0
$325$	2.30410	0.127808
$326$	1.40099	0.0775935
$327$	0	0
$328$	60.3860	3.33426
$329$	−0.580304	−0.0319932
$330$	0	0
$331$	20.6870	1.13706	0.568531	−	0.822662i	$-0.307512\pi$
$331$	20.6870	1.13706	0.568531	+	0.822662i	$0.307512\pi$
$332$	−53.3198	−2.92630
$333$	0	0
$334$	−55.3526	−3.02876
$335$	−0.145542	−0.00795180
$336$	0	0
$337$	2.66866	0.145371	0.0726856	−	0.997355i	$-0.476843\pi$
$337$	2.66866	0.145371	0.0726856	+	0.997355i	$0.476843\pi$
$338$	−33.5279	−1.82368
$339$	0	0
$340$	−52.2638	−2.83440
$341$	7.07730	0.383257
$342$	0	0
$343$	−3.51597	−0.189845
$344$	12.9139	0.696269
$345$	0	0
$346$	−54.4339	−2.92638
$347$	10.6683	0.572707	0.286354	−	0.958124i	$-0.407557\pi$
$347$	10.6683	0.572707	0.286354	+	0.958124i	$0.407557\pi$
$348$	0	0
$349$	25.9644	1.38984	0.694922	−	0.719085i	$-0.255439\pi$
$349$	25.9644	1.38984	0.694922	+	0.719085i	$0.255439\pi$
$350$	4.79673	0.256396
$351$	0	0
$352$	−9.31982	−0.496748
$353$	29.9585	1.59453	0.797266	−	0.603629i	$-0.206279\pi$
$353$	29.9585	1.59453	0.797266	+	0.603629i	$0.206279\pi$
$354$	0	0
$355$	34.4211	1.82688
$356$	−26.5520	−1.40725
$357$	0	0
$358$	6.73449	0.355929
$359$	−22.7413	−1.20024	−0.600121	−	0.799909i	$-0.704881\pi$
$359$	−22.7413	−1.20024	−0.600121	+	0.799909i	$0.704881\pi$
$360$	0	0
$361$	−18.2923	−0.962754
$362$	−28.2036	−1.48235
$363$	0	0
$364$	0.377766	0.0198003
$365$	31.2643	1.63645
$366$	0	0
$367$	−3.01744	−0.157509	−0.0787545	−	0.996894i	$-0.525094\pi$
$367$	−3.01744	−0.157509	−0.0787545	+	0.996894i	$0.525094\pi$
$368$	−13.2074	−0.688484
$369$	0	0
$370$	81.1388	4.21821
$371$	1.11285	0.0577762
$372$	0	0
$373$	−2.16528	−0.112114	−0.0560570	−	0.998428i	$-0.517853\pi$
$373$	−2.16528	−0.112114	−0.0560570	+	0.998428i	$0.517853\pi$
$374$	8.14095	0.420958
$375$	0	0
$376$	−16.4648	−0.849108
$377$	2.13167	0.109787
$378$	0	0
$379$	4.56064	0.234264	0.117132	−	0.993116i	$-0.462630\pi$
$379$	4.56064	0.234264	0.117132	+	0.993116i	$0.462630\pi$
$380$	14.0356	0.720010
$381$	0	0
$382$	−38.6732	−1.97869
$383$	21.2912	1.08793	0.543964	−	0.839109i	$-0.316923\pi$
$383$	21.2912	1.08793	0.543964	+	0.839109i	$0.316923\pi$
$384$	0	0
$385$	−0.885374	−0.0451228
$386$	−0.579386	−0.0294900
$387$	0	0
$388$	35.8473	1.81987
$389$	30.9732	1.57041	0.785203	−	0.619238i	$-0.212559\pi$
$389$	30.9732	1.57041	0.785203	+	0.619238i	$0.212559\pi$
$390$	0	0
$391$	4.54903	0.230054
$392$	−49.6511	−2.50776
$393$	0	0
$394$	−5.61444	−0.282851
$395$	−29.8887	−1.50386
$396$	0	0
$397$	23.5944	1.18417	0.592085	−	0.805875i	$-0.298305\pi$
$397$	23.5944	1.18417	0.592085	+	0.805875i	$0.298305\pi$
$398$	64.0545	3.21076
$399$	0	0
$400$	66.5341	3.32670
$401$	20.4153	1.01949	0.509745	−	0.860326i	$-0.329740\pi$
$401$	20.4153	1.01949	0.509745	+	0.860326i	$0.329740\pi$
$402$	0	0
$403$	−2.22900	−0.111034
$404$	27.8243	1.38431
$405$	0	0
$406$	4.43777	0.220243
$407$	−8.89628	−0.440972
$408$	0	0
$409$	−17.8248	−0.881379	−0.440689	−	0.897660i	$-0.645266\pi$
$409$	−17.8248	−0.881379	−0.440689	+	0.897660i	$0.645266\pi$
$410$	76.9411	3.79985
$411$	0	0
$412$	−82.6305	−4.07091
$413$	−1.13034	−0.0556204
$414$	0	0
$415$	−39.3581	−1.93201
$416$	2.93528	0.143914
$417$	0	0
$418$	−2.18627	−0.106934
$419$	5.42075	0.264821	0.132410	−	0.991195i	$-0.457728\pi$
$419$	5.42075	0.264821	0.132410	+	0.991195i	$0.457728\pi$
$420$	0	0
$421$	38.0113	1.85256	0.926279	−	0.376840i	$-0.122989\pi$
$421$	38.0113	1.85256	0.926279	+	0.376840i	$0.122989\pi$
$422$	32.7624	1.59485
$423$	0	0
$424$	31.5746	1.53340
$425$	−22.9163	−1.11160
$426$	0	0
$427$	−0.252288	−0.0122091
$428$	34.2496	1.65552
$429$	0	0
$430$	16.4543	0.793495
$431$	−14.5523	−0.700961	−0.350480	−	0.936570i	$-0.613982\pi$
$431$	−14.5523	−0.700961	−0.350480	+	0.936570i	$0.613982\pi$
$432$	0	0
$433$	−1.86310	−0.0895346	−0.0447673	−	0.998997i	$-0.514255\pi$
$433$	−1.86310	−0.0895346	−0.0447673	+	0.998997i	$0.514255\pi$
$434$	−4.64038	−0.222745
$435$	0	0
$436$	14.5767	0.698099
$437$	−1.22165	−0.0584396
$438$	0	0
$439$	0.643741	0.0307241	0.0153620	−	0.999882i	$-0.495110\pi$
$439$	0.643741	0.0307241	0.0153620	+	0.999882i	$0.495110\pi$
$440$	−25.1205	−1.19757
$441$	0	0
$442$	−2.56399	−0.121957
$443$	18.0271	0.856492	0.428246	−	0.903662i	$-0.359132\pi$
$443$	18.0271	0.856492	0.428246	+	0.903662i	$0.359132\pi$
$444$	0	0
$445$	−19.5994	−0.929102
$446$	27.5029	1.30230
$447$	0	0
$448$	1.52181	0.0718987
$449$	−27.3932	−1.29277	−0.646383	−	0.763013i	$-0.723719\pi$
$449$	−27.3932	−1.29277	−0.646383	+	0.763013i	$0.723719\pi$
$450$	0	0
$451$	−8.43604	−0.397237
$452$	57.2084	2.69086
$453$	0	0
$454$	−24.9567	−1.17128
$455$	0.278849	0.0130726
$456$	0	0
$457$	−25.1056	−1.17439	−0.587196	−	0.809445i	$-0.699768\pi$
$457$	−25.1056	−1.17439	−0.587196	+	0.809445i	$0.699768\pi$
$458$	12.9816	0.606589
$459$	0	0
$460$	−24.2297	−1.12972
$461$	−38.5232	−1.79421	−0.897103	−	0.441822i	$-0.854332\pi$
$461$	−38.5232	−1.79421	−0.897103	+	0.441822i	$0.854332\pi$
$462$	0	0
$463$	−3.29366	−0.153070	−0.0765348	−	0.997067i	$-0.524386\pi$
$463$	−3.29366	−0.153070	−0.0765348	+	0.997067i	$0.524386\pi$
$464$	61.5551	2.85762
$465$	0	0
$466$	−48.0064	−2.22385
$467$	33.9437	1.57073	0.785364	−	0.619035i	$-0.212476\pi$
$467$	33.9437	1.57073	0.785364	+	0.619035i	$0.212476\pi$
$468$	0	0
$469$	−0.0104629	−0.000483134 0
$470$	−20.9787	−0.967676
$471$	0	0
$472$	−32.0709	−1.47618
$473$	−1.80409	−0.0829522
$474$	0	0
$475$	6.15424	0.282376
$476$	−3.75722	−0.172212
$477$	0	0
$478$	−17.6191	−0.805879
$479$	−10.0038	−0.457084	−0.228542	−	0.973534i	$-0.573396\pi$
$479$	−10.0038	−0.457084	−0.228542	+	0.973534i	$0.573396\pi$
$480$	0	0
$481$	2.80188	0.127755
$482$	40.7251	1.85498
$483$	0	0
$484$	4.75428	0.216104
$485$	26.4607	1.20152
$486$	0	0
$487$	−10.7252	−0.486005	−0.243002	−	0.970026i	$-0.578132\pi$
$487$	−10.7252	−0.486005	−0.243002	+	0.970026i	$0.578132\pi$
$488$	−7.15810	−0.324032
$489$	0	0
$490$	−63.2632	−2.85794
$491$	−8.35754	−0.377170	−0.188585	−	0.982057i	$-0.560390\pi$
$491$	−8.35754	−0.377170	−0.188585	+	0.982057i	$0.560390\pi$
$492$	0	0
$493$	−21.2014	−0.954863
$494$	0.688567	0.0309801
$495$	0	0
$496$	−64.3654	−2.89009
$497$	2.47452	0.110997
$498$	0	0
$499$	11.9000	0.532715	0.266358	−	0.963874i	$-0.414180\pi$
$499$	11.9000	0.532715	0.266358	+	0.963874i	$0.414180\pi$
$500$	38.6374	1.72792
$501$	0	0
$502$	−40.3970	−1.80300
$503$	21.4919	0.958278	0.479139	−	0.877739i	$-0.340949\pi$
$503$	21.4919	0.958278	0.479139	+	0.877739i	$0.340949\pi$
$504$	0	0
$505$	20.5385	0.913952
$506$	3.77418	0.167783
$507$	0	0
$508$	−35.9473	−1.59490
$509$	37.0220	1.64097	0.820485	−	0.571668i	$-0.193703\pi$
$509$	37.0220	1.64097	0.820485	+	0.571668i	$0.193703\pi$
$510$	0	0
$511$	2.24758	0.0994270
$512$	−45.4402	−2.00819
$513$	0	0
$514$	−45.1776	−1.99270
$515$	−60.9939	−2.68771
$516$	0	0
$517$	2.30017	0.101161
$518$	5.83303	0.256289
$519$	0	0
$520$	7.91170	0.346951
$521$	−25.1196	−1.10051	−0.550256	−	0.834996i	$-0.685470\pi$
$521$	−25.1196	−1.10051	−0.550256	+	0.834996i	$0.685470\pi$
$522$	0	0
$523$	29.0802	1.27159	0.635794	−	0.771859i	$-0.280673\pi$
$523$	29.0802	1.27159	0.635794	+	0.771859i	$0.280673\pi$
$524$	−47.9732	−2.09572
$525$	0	0
$526$	35.0690	1.52908
$527$	22.1693	0.965712
$528$	0	0
$529$	−20.8911	−0.908307
$530$	40.2309	1.74752
$531$	0	0
$532$	1.00901	0.0437462
$533$	2.65693	0.115084
$534$	0	0
$535$	25.2814	1.09301
$536$	−0.296863	−0.0128225
$537$	0	0
$538$	−44.9775	−1.93912
$539$	6.93635	0.298770
$540$	0	0
$541$	−0.392586	−0.0168786	−0.00843929	−	0.999964i	$-0.502686\pi$
$541$	−0.392586	−0.0168786	−0.00843929	+	0.999964i	$0.502686\pi$
$542$	−6.58186	−0.282715
$543$	0	0
$544$	−29.1939	−1.25168
$545$	10.7598	0.460901
$546$	0	0
$547$	15.1084	0.645990	0.322995	−	0.946401i	$-0.395310\pi$
$547$	15.1084	0.645990	0.322995	+	0.946401i	$0.395310\pi$
$548$	−53.5575	−2.28786
$549$	0	0
$550$	−19.0129	−0.810713
$551$	5.69370	0.242560
$552$	0	0
$553$	−2.14869	−0.0913714
$554$	61.3804	2.60780
$555$	0	0
$556$	48.8057	2.06982
$557$	44.8252	1.89931	0.949653	−	0.313304i	$-0.101436\pi$
$557$	44.8252	1.89931	0.949653	+	0.313304i	$0.101436\pi$
$558$	0	0
$559$	0.568198	0.0240322
$560$	8.05215	0.340265
$561$	0	0
$562$	−27.2861	−1.15099
$563$	−18.8989	−0.796492	−0.398246	−	0.917279i	$-0.630381\pi$
$563$	−18.8989	−0.796492	−0.398246	+	0.917279i	$0.630381\pi$
$564$	0	0
$565$	42.2285	1.77657
$566$	19.8968	0.836325
$567$	0	0
$568$	70.2089	2.94590
$569$	−2.20824	−0.0925744	−0.0462872	−	0.998928i	$-0.514739\pi$
$569$	−2.20824	−0.0925744	−0.0462872	+	0.998928i	$0.514739\pi$
$570$	0	0
$571$	−12.1493	−0.508431	−0.254216	−	0.967148i	$-0.581817\pi$
$571$	−12.1493	−0.508431	−0.254216	+	0.967148i	$0.581817\pi$
$572$	−1.49736	−0.0626078
$573$	0	0
$574$	5.53126	0.230870
$575$	−10.6241	−0.443056
$576$	0	0
$577$	−11.7039	−0.487238	−0.243619	−	0.969871i	$-0.578335\pi$
$577$	−11.7039	−0.487238	−0.243619	+	0.969871i	$0.578335\pi$
$578$	−18.6801	−0.776991
$579$	0	0
$580$	112.926	4.68900
$581$	−2.82944	−0.117385
$582$	0	0
$583$	−4.41103	−0.182686
$584$	63.7700	2.63882
$585$	0	0
$586$	−34.6113	−1.42978
$587$	−32.9310	−1.35921	−0.679604	−	0.733580i	$-0.737848\pi$
$587$	−32.9310	−1.35921	−0.679604	+	0.733580i	$0.737848\pi$
$588$	0	0
$589$	−5.95364	−0.245315
$590$	−40.8632	−1.68231
$591$	0	0
$592$	80.9084	3.32531
$593$	−35.1041	−1.44155	−0.720776	−	0.693168i	$-0.756214\pi$
$593$	−35.1041	−1.44155	−0.720776	+	0.693168i	$0.756214\pi$
$594$	0	0
$595$	−2.77340	−0.113698
$596$	−80.7064	−3.30586
$597$	0	0
$598$	−1.18868	−0.0486087
$599$	35.7935	1.46248	0.731241	−	0.682119i	$-0.238941\pi$
$599$	35.7935	1.46248	0.731241	+	0.682119i	$0.238941\pi$
$600$	0	0
$601$	−8.14840	−0.332380	−0.166190	−	0.986094i	$-0.553147\pi$
$601$	−8.14840	−0.332380	−0.166190	+	0.986094i	$0.553147\pi$
$602$	1.18289	0.0482110
$603$	0	0
$604$	−49.4857	−2.01354
$605$	3.50938	0.142677
$606$	0	0
$607$	23.1424	0.939320	0.469660	−	0.882847i	$-0.344377\pi$
$607$	23.1424	0.939320	0.469660	+	0.882847i	$0.344377\pi$
$608$	7.84012	0.317959
$609$	0	0
$610$	−9.12053	−0.369279
$611$	−0.724437	−0.0293076
$612$	0	0
$613$	−4.11938	−0.166380	−0.0831901	−	0.996534i	$-0.526511\pi$
$613$	−4.11938	−0.166380	−0.0831901	+	0.996534i	$0.526511\pi$
$614$	−18.8094	−0.759085
$615$	0	0
$616$	−1.80590	−0.0727619
$617$	25.0455	1.00829	0.504146	−	0.863618i	$-0.331807\pi$
$617$	25.0455	1.00829	0.504146	+	0.863618i	$0.331807\pi$
$618$	0	0
$619$	42.8615	1.72275	0.861374	−	0.507972i	$-0.169605\pi$
$619$	42.8615	1.72275	0.861374	+	0.507972i	$0.169605\pi$
$620$	−118.082	−4.74228
$621$	0	0
$622$	21.7928	0.873812
$623$	−1.40899	−0.0564502
$624$	0	0
$625$	−8.05850	−0.322340
$626$	20.9786	0.838473
$627$	0	0
$628$	−73.1314	−2.91826
$629$	−27.8672	−1.11114
$630$	0	0
$631$	−13.0422	−0.519204	−0.259602	−	0.965716i	$-0.583591\pi$
$631$	−13.0422	−0.519204	−0.259602	+	0.965716i	$0.583591\pi$
$632$	−60.9641	−2.42502
$633$	0	0
$634$	68.5118	2.72095
$635$	−26.5345	−1.05299
$636$	0	0
$637$	−2.18460	−0.0865572
$638$	−17.5901	−0.696399
$639$	0	0
$640$	−10.3983	−0.411028
$641$	−41.7027	−1.64716	−0.823580	−	0.567201i	$-0.808026\pi$
$641$	−41.7027	−1.64716	−0.823580	+	0.567201i	$0.808026\pi$
$642$	0	0
$643$	18.6948	0.737249	0.368625	−	0.929578i	$-0.379829\pi$
$643$	18.6948	0.737249	0.368625	+	0.929578i	$0.379829\pi$
$644$	−1.74186	−0.0686391
$645$	0	0
$646$	−6.84841	−0.269447
$647$	−37.1905	−1.46211	−0.731055	−	0.682319i	$-0.760972\pi$
$647$	−37.1905	−1.46211	−0.731055	+	0.682319i	$0.760972\pi$
$648$	0	0
$649$	4.48036	0.175869
$650$	5.98812	0.234873
$651$	0	0
$652$	2.56289	0.100370
$653$	−35.5250	−1.39020	−0.695100	−	0.718914i	$-0.744640\pi$
$653$	−35.5250	−1.39020	−0.695100	+	0.718914i	$0.744640\pi$
$654$	0	0
$655$	−35.4115	−1.38364
$656$	76.7226	2.99551
$657$	0	0
$658$	−1.50815	−0.0587939
$659$	25.4043	0.989613	0.494806	−	0.869003i	$-0.335239\pi$
$659$	25.4043	0.989613	0.494806	+	0.869003i	$0.335239\pi$
$660$	0	0
$661$	−22.8515	−0.888819	−0.444409	−	0.895824i	$-0.646586\pi$
$661$	−22.8515	−0.888819	−0.444409	+	0.895824i	$0.646586\pi$
$662$	53.7635	2.08958
$663$	0	0
$664$	−80.2789	−3.11543
$665$	0.744804	0.0288823
$666$	0	0
$667$	−9.82906	−0.380583
$668$	−101.259	−3.91783
$669$	0	0
$670$	−0.378249	−0.0146130
$671$	1.00000	0.0386046
$672$	0	0
$673$	24.2548	0.934954	0.467477	−	0.884005i	$-0.345163\pi$
$673$	24.2548	0.934954	0.467477	+	0.884005i	$0.345163\pi$
$674$	6.93558	0.267148
$675$	0	0
$676$	−61.3341	−2.35900
$677$	17.7813	0.683390	0.341695	−	0.939811i	$-0.388999\pi$
$677$	17.7813	0.683390	0.341695	+	0.939811i	$0.388999\pi$
$678$	0	0
$679$	1.90225	0.0730017
$680$	−78.6889	−3.01758
$681$	0	0
$682$	18.3932	0.704312
$683$	−40.4477	−1.54769	−0.773844	−	0.633376i	$-0.781669\pi$
$683$	−40.4477	−1.54769	−0.773844	+	0.633376i	$0.781669\pi$
$684$	0	0
$685$	−39.5336	−1.51050
$686$	−9.13766	−0.348877
$687$	0	0
$688$	16.4075	0.625531
$689$	1.38925	0.0529264
$690$	0	0
$691$	49.4541	1.88132	0.940661	−	0.339349i	$-0.110207\pi$
$691$	49.4541	1.88132	0.940661	+	0.339349i	$0.110207\pi$
$692$	−99.5783	−3.78540
$693$	0	0
$694$	27.7260	1.05246
$695$	36.0260	1.36654
$696$	0	0
$697$	−26.4255	−1.00094
$698$	67.4789	2.55411
$699$	0	0
$700$	8.77486	0.331659
$701$	8.75971	0.330850	0.165425	−	0.986222i	$-0.447100\pi$
$701$	8.75971	0.330850	0.165425	+	0.986222i	$0.447100\pi$
$702$	0	0
$703$	7.48383	0.282258
$704$	−6.03203	−0.227341
$705$	0	0
$706$	77.8592	2.93027
$707$	1.47651	0.0555297
$708$	0	0
$709$	1.59624	0.0599480	0.0299740	−	0.999551i	$-0.490458\pi$
$709$	1.59624	0.0599480	0.0299740	+	0.999551i	$0.490458\pi$
$710$	89.4571	3.35726
$711$	0	0
$712$	−39.9770	−1.49820
$713$	10.2778	0.384907
$714$	0	0
$715$	−1.10528	−0.0413351
$716$	12.3197	0.460409
$717$	0	0
$718$	−59.1025	−2.20568
$719$	−24.2156	−0.903091	−0.451545	−	0.892248i	$-0.649127\pi$
$719$	−24.2156	−0.903091	−0.451545	+	0.892248i	$0.649127\pi$
$720$	0	0
$721$	−4.38482	−0.163299
$722$	−47.5399	−1.76925
$723$	0	0
$724$	−51.5942	−1.91748
$725$	49.5152	1.83895
$726$	0	0
$727$	6.33810	0.235067	0.117534	−	0.993069i	$-0.462501\pi$
$727$	6.33810	0.235067	0.117534	+	0.993069i	$0.462501\pi$
$728$	0.568769	0.0210800
$729$	0	0
$730$	81.2529	3.00730
$731$	−5.65124	−0.209019
$732$	0	0
$733$	−23.5758	−0.870790	−0.435395	−	0.900239i	$-0.643391\pi$
$733$	−23.5758	−0.870790	−0.435395	+	0.900239i	$0.643391\pi$
$734$	−7.84202	−0.289454
$735$	0	0
$736$	−13.5345	−0.498887
$737$	0.0414722	0.00152765
$738$	0	0
$739$	35.2908	1.29819	0.649096	−	0.760706i	$-0.275147\pi$
$739$	35.2908	1.29819	0.649096	+	0.760706i	$0.275147\pi$
$740$	148.431	5.45642
$741$	0	0
$742$	2.89218	0.106175
$743$	−37.4102	−1.37245	−0.686224	−	0.727390i	$-0.740733\pi$
$743$	−37.4102	−1.37245	−0.686224	+	0.727390i	$0.740733\pi$
$744$	0	0
$745$	−59.5736	−2.18261
$746$	−5.62735	−0.206032
$747$	0	0
$748$	14.8926	0.544527
$749$	1.81747	0.0664089
$750$	0	0
$751$	−4.73838	−0.172906	−0.0864530	−	0.996256i	$-0.527553\pi$
$751$	−4.73838	−0.172906	−0.0864530	+	0.996256i	$0.527553\pi$
$752$	−20.9191	−0.762843
$753$	0	0
$754$	5.54001	0.201755
$755$	−36.5279	−1.32939
$756$	0	0
$757$	6.11168	0.222133	0.111066	−	0.993813i	$-0.464573\pi$
$757$	6.11168	0.222133	0.111066	+	0.993813i	$0.464573\pi$
$758$	11.8526	0.430507
$759$	0	0
$760$	21.1321	0.766543
$761$	33.3959	1.21060	0.605300	−	0.795998i	$-0.293053\pi$
$761$	33.3959	1.21060	0.605300	+	0.795998i	$0.293053\pi$
$762$	0	0
$763$	0.773520	0.0280033
$764$	−70.7466	−2.55952
$765$	0	0
$766$	55.3336	1.99928
$767$	−1.41109	−0.0509515
$768$	0	0
$769$	−5.52203	−0.199130	−0.0995648	−	0.995031i	$-0.531745\pi$
$769$	−5.52203	−0.199130	−0.0995648	+	0.995031i	$0.531745\pi$
$770$	−2.30100	−0.0829222
$771$	0	0
$772$	−1.05990	−0.0381465
$773$	−35.4489	−1.27501	−0.637504	−	0.770447i	$-0.720033\pi$
$773$	−35.4489	−1.27501	−0.637504	+	0.770447i	$0.720033\pi$
$774$	0	0
$775$	−51.7758	−1.85984
$776$	53.9721	1.93749
$777$	0	0
$778$	80.4964	2.88593
$779$	7.09665	0.254264
$780$	0	0
$781$	−9.80832	−0.350969
$782$	11.8225	0.422770
$783$	0	0
$784$	−63.0835	−2.25298
$785$	−53.9821	−1.92670
$786$	0	0
$787$	18.2222	0.649551	0.324776	−	0.945791i	$-0.394711\pi$
$787$	18.2222	0.649551	0.324776	+	0.945791i	$0.394711\pi$
$788$	−10.2707	−0.365880
$789$	0	0
$790$	−77.6778	−2.76365
$791$	3.03579	0.107940
$792$	0	0
$793$	−0.314950	−0.0111842
$794$	61.3196	2.17615
$795$	0	0
$796$	117.178	4.15325
$797$	40.8082	1.44550	0.722751	−	0.691109i	$-0.242877\pi$
$797$	40.8082	1.44550	0.722751	+	0.691109i	$0.242877\pi$
$798$	0	0
$799$	7.20517	0.254901
$800$	68.1815	2.41058
$801$	0	0
$802$	53.0572	1.87352
$803$	−8.90879	−0.314384
$804$	0	0
$805$	−1.28576	−0.0453171
$806$	−5.79294	−0.204047
$807$	0	0
$808$	41.8925	1.47377
$809$	47.5619	1.67219	0.836094	−	0.548586i	$-0.184834\pi$
$809$	47.5619	1.67219	0.836094	+	0.548586i	$0.184834\pi$
$810$	0	0
$811$	−10.5129	−0.369157	−0.184578	−	0.982818i	$-0.559092\pi$
$811$	−10.5129	−0.369157	−0.184578	+	0.982818i	$0.559092\pi$
$812$	8.11821	0.284893
$813$	0	0
$814$	−23.1205	−0.810375
$815$	1.89180	0.0662668
$816$	0	0
$817$	1.51766	0.0530961
$818$	−46.3248	−1.61971
$819$	0	0
$820$	140.752	4.91526
$821$	−21.5569	−0.752339	−0.376170	−	0.926551i	$-0.622759\pi$
$821$	−21.5569	−0.752339	−0.376170	+	0.926551i	$0.622759\pi$
$822$	0	0
$823$	−5.18282	−0.180662	−0.0903309	−	0.995912i	$-0.528792\pi$
$823$	−5.18282	−0.180662	−0.0903309	+	0.995912i	$0.528792\pi$
$824$	−124.409	−4.33401
$825$	0	0
$826$	−2.93764	−0.102214
$827$	−8.46918	−0.294502	−0.147251	−	0.989099i	$-0.547043\pi$
$827$	−8.46918	−0.294502	−0.147251	+	0.989099i	$0.547043\pi$
$828$	0	0
$829$	23.7804	0.825927	0.412964	−	0.910748i	$-0.364494\pi$
$829$	23.7804	0.825927	0.412964	+	0.910748i	$0.364494\pi$
$830$	−102.288	−3.55046
$831$	0	0
$832$	1.89979	0.0658633
$833$	21.7278	0.752825
$834$	0	0
$835$	−74.7445	−2.58664
$836$	−3.99945	−0.138324
$837$	0	0
$838$	14.0880	0.486661
$839$	−0.710743	−0.0245376	−0.0122688	−	0.999925i	$-0.503905\pi$
$839$	−0.710743	−0.0245376	−0.0122688	+	0.999925i	$0.503905\pi$
$840$	0	0
$841$	16.8098	0.579647
$842$	98.7875	3.40444
$843$	0	0
$844$	59.9337	2.06300
$845$	−45.2738	−1.55747
$846$	0	0
$847$	0.252288	0.00866871
$848$	40.1167	1.37761
$849$	0	0
$850$	−59.5572	−2.04279
$851$	−12.9194	−0.442871
$852$	0	0
$853$	24.6568	0.844231	0.422116	−	0.906542i	$-0.361288\pi$
$853$	24.6568	0.844231	0.422116	+	0.906542i	$0.361288\pi$
$854$	−0.655671	−0.0224366
$855$	0	0
$856$	51.5666	1.76251
$857$	3.62125	0.123699	0.0618497	−	0.998085i	$-0.480300\pi$
$857$	3.62125	0.123699	0.0618497	+	0.998085i	$0.480300\pi$
$858$	0	0
$859$	14.1354	0.482295	0.241147	−	0.970488i	$-0.422476\pi$
$859$	14.1354	0.482295	0.241147	+	0.970488i	$0.422476\pi$
$860$	30.1005	1.02642
$861$	0	0
$862$	−37.8200	−1.28816
$863$	10.6439	0.362324	0.181162	−	0.983453i	$-0.442014\pi$
$863$	10.6439	0.362324	0.181162	+	0.983453i	$0.442014\pi$
$864$	0	0
$865$	−73.5039	−2.49921
$866$	−4.84200	−0.164538
$867$	0	0
$868$	−8.48885	−0.288130
$869$	8.51680	0.288913
$870$	0	0
$871$	−0.0130617	−0.000442578 0
$872$	21.9469	0.743216
$873$	0	0
$874$	−3.17496	−0.107395
$875$	2.05031	0.0693131
$876$	0	0
$877$	7.29331	0.246278	0.123139	−	0.992389i	$-0.460704\pi$
$877$	7.29331	0.246278	0.123139	+	0.992389i	$0.460704\pi$
$878$	1.67302	0.0564616
$879$	0	0
$880$	−31.9165	−1.07591
$881$	12.8614	0.433311	0.216655	−	0.976248i	$-0.430485\pi$
$881$	12.8614	0.433311	0.216655	+	0.976248i	$0.430485\pi$
$882$	0	0
$883$	21.2952	0.716639	0.358319	−	0.933599i	$-0.383350\pi$
$883$	21.2952	0.716639	0.358319	+	0.933599i	$0.383350\pi$
$884$	−4.69042	−0.157756
$885$	0	0
$886$	46.8505	1.57397
$887$	−57.3228	−1.92471	−0.962356	−	0.271793i	$-0.912383\pi$
$887$	−57.3228	−1.92471	−0.962356	+	0.271793i	$0.912383\pi$
$888$	0	0
$889$	−1.90756	−0.0639774
$890$	−50.9370	−1.70741
$891$	0	0
$892$	50.3122	1.68458
$893$	−1.93497	−0.0647513
$894$	0	0
$895$	9.09381	0.303973
$896$	−0.747528	−0.0249732
$897$	0	0
$898$	−71.1922	−2.37572
$899$	−47.9012	−1.59759
$900$	0	0
$901$	−13.8174	−0.460323
$902$	−21.9244	−0.730003
$903$	0	0
$904$	86.1337	2.86476
$905$	−38.0843	−1.26597
$906$	0	0
$907$	19.1793	0.636838	0.318419	−	0.947950i	$-0.396848\pi$
$907$	19.1793	0.636838	0.318419	+	0.947950i	$0.396848\pi$
$908$	−45.6543	−1.51509
$909$	0	0
$910$	0.724700	0.0240236
$911$	10.8028	0.357912	0.178956	−	0.983857i	$-0.442728\pi$
$911$	10.8028	0.357912	0.178956	+	0.983857i	$0.442728\pi$
$912$	0	0
$913$	11.2151	0.371166
$914$	−65.2470	−2.15818
$915$	0	0
$916$	23.7477	0.784648
$917$	−2.54572	−0.0840670
$918$	0	0
$919$	18.6484	0.615155	0.307578	−	0.951523i	$-0.400482\pi$
$919$	18.6484	0.615155	0.307578	+	0.951523i	$0.400482\pi$
$920$	−36.4806	−1.20273
$921$	0	0
$922$	−100.118	−3.29721
$923$	3.08913	0.101680
$924$	0	0
$925$	65.0830	2.13992
$926$	−8.55990	−0.281296
$927$	0	0
$928$	63.0793	2.07068
$929$	47.6052	1.56188	0.780938	−	0.624608i	$-0.214741\pi$
$929$	47.6052	1.56188	0.780938	+	0.624608i	$0.214741\pi$
$930$	0	0
$931$	−5.83507	−0.191237
$932$	−87.8201	−2.87664
$933$	0	0
$934$	88.2163	2.88652
$935$	10.9930	0.359509
$936$	0	0
$937$	9.10229	0.297359	0.148680	−	0.988885i	$-0.452498\pi$
$937$	9.10229	0.297359	0.148680	+	0.988885i	$0.452498\pi$
$938$	−0.0271921	−0.000887855 0
$939$	0	0
$940$	−38.3773	−1.25173
$941$	34.9570	1.13956	0.569782	−	0.821796i	$-0.307028\pi$
$941$	34.9570	1.13956	0.569782	+	0.821796i	$0.307028\pi$
$942$	0	0
$943$	−12.2510	−0.398947
$944$	−40.7472	−1.32621
$945$	0	0
$946$	−4.68865	−0.152441
$947$	19.1383	0.621910	0.310955	−	0.950425i	$-0.399351\pi$
$947$	19.1383	0.621910	0.310955	+	0.950425i	$0.399351\pi$
$948$	0	0
$949$	2.80582	0.0910809
$950$	15.9943	0.518922
$951$	0	0
$952$	−5.65691	−0.183342
$953$	−45.7828	−1.48305	−0.741525	−	0.670925i	$-0.765897\pi$
$953$	−45.7828	−1.48305	−0.741525	+	0.670925i	$0.765897\pi$
$954$	0	0
$955$	−52.2217	−1.68986
$956$	−32.2314	−1.04244
$957$	0	0
$958$	−25.9988	−0.839982
$959$	−2.84205	−0.0917747
$960$	0	0
$961$	19.0881	0.615746
$962$	7.28182	0.234775
$963$	0	0
$964$	74.5002	2.39949
$965$	−0.782365	−0.0251852
$966$	0	0
$967$	−9.33277	−0.300122	−0.150061	−	0.988677i	$-0.547947\pi$
$967$	−9.33277	−0.300122	−0.150061	+	0.988677i	$0.547947\pi$
$968$	7.15810	0.230070
$969$	0	0
$970$	68.7688	2.20803
$971$	−2.32628	−0.0746540	−0.0373270	−	0.999303i	$-0.511884\pi$
$971$	−2.32628	−0.0746540	−0.0373270	+	0.999303i	$0.511884\pi$
$972$	0	0
$973$	2.58989	0.0830282
$974$	−27.8737	−0.893131
$975$	0	0
$976$	−9.09463	−0.291112
$977$	38.7073	1.23836	0.619178	−	0.785251i	$-0.287466\pi$
$977$	38.7073	1.23836	0.619178	+	0.785251i	$0.287466\pi$
$978$	0	0
$979$	5.58487	0.178493
$980$	−115.730	−3.69686
$981$	0	0
$982$	−21.7204	−0.693126
$983$	43.4665	1.38637	0.693184	−	0.720761i	$-0.256207\pi$
$983$	43.4665	1.38637	0.693184	+	0.720761i	$0.256207\pi$
$984$	0	0
$985$	−7.58136	−0.241562
$986$	−55.1003	−1.75475
$987$	0	0
$988$	1.25963	0.0400740
$989$	−2.61994	−0.0833093
$990$	0	0
$991$	50.1426	1.59283	0.796417	−	0.604748i	$-0.206726\pi$
$991$	50.1426	1.59283	0.796417	+	0.604748i	$0.206726\pi$
$992$	−65.9591	−2.09420
$993$	0	0
$994$	6.43103	0.203980
$995$	86.4949	2.74207
$996$	0	0
$997$	−46.5501	−1.47426	−0.737128	−	0.675753i	$-0.763818\pi$
$997$	−46.5501	−1.47426	−0.737128	+	0.675753i	$0.763818\pi$
$998$	30.9268	0.978971
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.i.1.13		13
3.2	odd	2		2013.2.a.e.1.1	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.e.1.1	✓	13	3.2	odd	2
6039.2.a.i.1.13		13	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)

\(f(q)\)	\(=\)	\(q+2.59890 q^{2} +4.75428 q^{4} +3.50938 q^{5} +0.252288 q^{7} +7.15810 q^{8} +O(q^{10})\)	\(q+2.59890 q^{2} +4.75428 q^{4} +3.50938 q^{5} +0.252288 q^{7} +7.15810 q^{8} +9.12053 q^{10} -1.00000 q^{11} +0.314950 q^{13} +0.655671 q^{14} +9.09463 q^{16} -3.13246 q^{17} +0.841231 q^{19} +16.6846 q^{20} -2.59890 q^{22} -1.45222 q^{23} +7.31576 q^{25} +0.818524 q^{26} +1.19945 q^{28} +6.76829 q^{29} -7.07730 q^{31} +9.31982 q^{32} -8.14095 q^{34} +0.885374 q^{35} +8.89628 q^{37} +2.18627 q^{38} +25.1205 q^{40} +8.43604 q^{41} +1.80409 q^{43} -4.75428 q^{44} -3.77418 q^{46} -2.30017 q^{47} -6.93635 q^{49} +19.0129 q^{50} +1.49736 q^{52} +4.41103 q^{53} -3.50938 q^{55} +1.80590 q^{56} +17.5901 q^{58} -4.48036 q^{59} -1.00000 q^{61} -18.3932 q^{62} +6.03203 q^{64} +1.10528 q^{65} -0.0414722 q^{67} -14.8926 q^{68} +2.30100 q^{70} +9.80832 q^{71} +8.90879 q^{73} +23.1205 q^{74} +3.99945 q^{76} -0.252288 q^{77} -8.51680 q^{79} +31.9165 q^{80} +21.9244 q^{82} -11.2151 q^{83} -10.9930 q^{85} +4.68865 q^{86} -7.15810 q^{88} -5.58487 q^{89} +0.0794581 q^{91} -6.90427 q^{92} -5.97790 q^{94} +2.95220 q^{95} +7.54000 q^{97} -18.0269 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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