LMFDB - Embedded newform 6039.2.a.d.1.1

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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 4x^{10} - 6x^{9} + 37x^{8} - 2x^{7} - 109x^{6} + 55x^{5} + 115x^{4} - 76x^{3} - 29x^{2} + 14x + 3 $ x^11 - 4x^10 - 6x^9 + 37x^8 - 2x^7 - 109x^6 + 55x^5 + 115x^4 - 76x^3 - 29x^2 + 14x + 3
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			1.1
Root			$-2.14662$ 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)

$f(q)$	$=$	$q-2.14662 q^{2} +2.60797 q^{4} +4.24506 q^{5} +0.408872 q^{7} -1.30509 q^{8} +O(q^{10})$	\(q-2.14662 q^{2} +2.60797 q^{4} +4.24506 q^{5} +0.408872 q^{7} -1.30509 q^{8} -9.11254 q^{10} +1.00000 q^{11} -1.49968 q^{13} -0.877693 q^{14} -2.41442 q^{16} +7.27063 q^{17} -1.94900 q^{19} +11.0710 q^{20} -2.14662 q^{22} -0.493201 q^{23} +13.0206 q^{25} +3.21925 q^{26} +1.06633 q^{28} +2.03143 q^{29} -0.979706 q^{31} +7.79302 q^{32} -15.6073 q^{34} +1.73569 q^{35} -2.19668 q^{37} +4.18376 q^{38} -5.54019 q^{40} +3.81354 q^{41} +1.15052 q^{43} +2.60797 q^{44} +1.05872 q^{46} -6.83861 q^{47} -6.83282 q^{49} -27.9502 q^{50} -3.91113 q^{52} +4.05514 q^{53} +4.24506 q^{55} -0.533615 q^{56} -4.36071 q^{58} +5.72355 q^{59} +1.00000 q^{61} +2.10306 q^{62} -11.8998 q^{64} -6.36625 q^{65} -8.88123 q^{67} +18.9616 q^{68} -3.72586 q^{70} -7.77111 q^{71} +12.4479 q^{73} +4.71544 q^{74} -5.08294 q^{76} +0.408872 q^{77} +11.5347 q^{79} -10.2494 q^{80} -8.18623 q^{82} -7.10658 q^{83} +30.8643 q^{85} -2.46974 q^{86} -1.30509 q^{88} +8.88821 q^{89} -0.613179 q^{91} -1.28626 q^{92} +14.6799 q^{94} -8.27363 q^{95} +10.6085 q^{97} +14.6675 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8}+O(q^{10}) $ 11 * q + 4 * q^2 + 6 * q^4 + 13 * q^5 - 5 * q^7 + 9 * q^8	$ 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8} + 6 q^{10} + 11 q^{11} - 3 q^{13} + 9 q^{14} + 4 q^{16} + 7 q^{17} - 8 q^{19} + 25 q^{20} + 4 q^{22} + 15 q^{23} + 4 q^{25} + 2 q^{26} + 13 q^{28} + 8 q^{29} - 17 q^{31} + 27 q^{32} - 18 q^{34} + 2 q^{35} - 10 q^{37} + 30 q^{38} + 10 q^{40} + 25 q^{41} - 7 q^{43} + 6 q^{44} + 32 q^{46} + 30 q^{47} - 2 q^{49} - 11 q^{50} - 7 q^{52} + 18 q^{53} + 13 q^{55} + 20 q^{56} - 13 q^{58} + 43 q^{59} + 11 q^{61} - 7 q^{62} + 25 q^{64} + 27 q^{65} - 30 q^{67} - 10 q^{68} - 4 q^{70} + 7 q^{71} + 6 q^{73} + 44 q^{74} - 19 q^{76} - 5 q^{77} + 17 q^{79} + 22 q^{80} + 8 q^{82} + 34 q^{83} + 10 q^{85} - 2 q^{86} + 9 q^{88} + 41 q^{89} - 39 q^{91} + 32 q^{92} + 55 q^{94} + 9 q^{95} - 41 q^{97} + 29 q^{98}+O(q^{100}) $ 11 * q + 4 * q^2 + 6 * q^4 + 13 * q^5 - 5 * q^7 + 9 * q^8 + 6 * q^10 + 11 * q^11 - 3 * q^13 + 9 * q^14 + 4 * q^16 + 7 * q^17 - 8 * q^19 + 25 * q^20 + 4 * q^22 + 15 * q^23 + 4 * q^25 + 2 * q^26 + 13 * q^28 + 8 * q^29 - 17 * q^31 + 27 * q^32 - 18 * q^34 + 2 * q^35 - 10 * q^37 + 30 * q^38 + 10 * q^40 + 25 * q^41 - 7 * q^43 + 6 * q^44 + 32 * q^46 + 30 * q^47 - 2 * q^49 - 11 * q^50 - 7 * q^52 + 18 * q^53 + 13 * q^55 + 20 * q^56 - 13 * q^58 + 43 * q^59 + 11 * q^61 - 7 * q^62 + 25 * q^64 + 27 * q^65 - 30 * q^67 - 10 * q^68 - 4 * q^70 + 7 * q^71 + 6 * q^73 + 44 * q^74 - 19 * q^76 - 5 * q^77 + 17 * q^79 + 22 * q^80 + 8 * q^82 + 34 * q^83 + 10 * q^85 - 2 * q^86 + 9 * q^88 + 41 * q^89 - 39 * q^91 + 32 * q^92 + 55 * q^94 + 9 * q^95 - 41 * q^97 + 29 * 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 $

$2$	−2.14662	−1.51789	−0.758945	−	0.651155i	$-0.774285\pi$
$2$	−2.14662	−1.51789	−0.758945	+	0.651155i	$0.774285\pi$
$3$	0	0
$3$	0	0
$4$	2.60797	1.30399
$5$	4.24506	1.89845	0.949225	−	0.314597i	$-0.101869\pi$
$5$	4.24506	1.89845	0.949225	+	0.314597i	$0.101869\pi$
$6$	0	0
$7$	0.408872	0.154539	0.0772696	−	0.997010i	$-0.475380\pi$
$7$	0.408872	0.154539	0.0772696	+	0.997010i	$0.475380\pi$
$8$	−1.30509	−0.461419
$9$	0	0
$10$	−9.11254	−2.88164
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.49968	−0.415937	−0.207969	−	0.978136i	$-0.566685\pi$
$13$	−1.49968	−0.415937	−0.207969	+	0.978136i	$0.566685\pi$
$14$	−0.877693	−0.234573
$15$	0	0
$16$	−2.41442	−0.603605
$17$	7.27063	1.76339	0.881693	−	0.471824i	$-0.156404\pi$
$17$	7.27063	1.76339	0.881693	+	0.471824i	$0.156404\pi$
$18$	0	0
$19$	−1.94900	−0.447131	−0.223566	−	0.974689i	$-0.571770\pi$
$19$	−1.94900	−0.447131	−0.223566	+	0.974689i	$0.571770\pi$
$20$	11.0710	2.47555
$21$	0	0
$22$	−2.14662	−0.457661
$23$	−0.493201	−0.102840	−0.0514198	−	0.998677i	$-0.516375\pi$
$23$	−0.493201	−0.102840	−0.0514198	+	0.998677i	$0.516375\pi$
$24$	0	0
$25$	13.0206	2.60411
$26$	3.21925	0.631346
$27$	0	0
$28$	1.06633	0.201517
$29$	2.03143	0.377228	0.188614	−	0.982051i	$-0.439601\pi$
$29$	2.03143	0.377228	0.188614	+	0.982051i	$0.439601\pi$
$30$	0	0
$31$	−0.979706	−0.175960	−0.0879802	−	0.996122i	$-0.528041\pi$
$31$	−0.979706	−0.175960	−0.0879802	+	0.996122i	$0.528041\pi$
$32$	7.79302	1.37762
$33$	0	0
$34$	−15.6073	−2.67662
$35$	1.73569	0.293385
$36$	0	0
$37$	−2.19668	−0.361132	−0.180566	−	0.983563i	$-0.557793\pi$
$37$	−2.19668	−0.361132	−0.180566	+	0.983563i	$0.557793\pi$
$38$	4.18376	0.678696
$39$	0	0
$40$	−5.54019	−0.875980
$41$	3.81354	0.595575	0.297788	−	0.954632i	$-0.403751\pi$
$41$	3.81354	0.595575	0.297788	+	0.954632i	$0.403751\pi$
$42$	0	0
$43$	1.15052	0.175453	0.0877267	−	0.996145i	$-0.472040\pi$
$43$	1.15052	0.175453	0.0877267	+	0.996145i	$0.472040\pi$
$44$	2.60797	0.393167
$45$	0	0
$46$	1.05872	0.156099
$47$	−6.83861	−0.997513	−0.498757	−	0.866742i	$-0.666210\pi$
$47$	−6.83861	−0.997513	−0.498757	+	0.866742i	$0.666210\pi$
$48$	0	0
$49$	−6.83282	−0.976118
$50$	−27.9502	−3.95276
$51$	0	0
$52$	−3.91113	−0.542377
$53$	4.05514	0.557017	0.278508	−	0.960434i	$-0.410160\pi$
$53$	4.05514	0.557017	0.278508	+	0.960434i	$0.410160\pi$
$54$	0	0
$55$	4.24506	0.572404
$56$	−0.533615	−0.0713073
$57$	0	0
$58$	−4.36071	−0.572590
$59$	5.72355	0.745142	0.372571	−	0.928004i	$-0.378476\pi$
$59$	5.72355	0.745142	0.372571	+	0.928004i	$0.378476\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	2.10306	0.267088
$63$	0	0
$64$	−11.8998	−1.48748
$65$	−6.36625	−0.789636
$66$	0	0
$67$	−8.88123	−1.08502	−0.542508	−	0.840051i	$-0.682525\pi$
$67$	−8.88123	−1.08502	−0.542508	+	0.840051i	$0.682525\pi$
$68$	18.9616	2.29943
$69$	0	0
$70$	−3.72586	−0.445326
$71$	−7.77111	−0.922262	−0.461131	−	0.887332i	$-0.652556\pi$
$71$	−7.77111	−0.922262	−0.461131	+	0.887332i	$0.652556\pi$
$72$	0	0
$73$	12.4479	1.45691	0.728456	−	0.685093i	$-0.240238\pi$
$73$	12.4479	1.45691	0.728456	+	0.685093i	$0.240238\pi$
$74$	4.71544	0.548159
$75$	0	0
$76$	−5.08294	−0.583054
$77$	0.408872	0.0465953
$78$	0	0
$79$	11.5347	1.29775	0.648875	−	0.760895i	$-0.275240\pi$
$79$	11.5347	1.29775	0.648875	+	0.760895i	$0.275240\pi$
$80$	−10.2494	−1.14591
$81$	0	0
$82$	−8.18623	−0.904017
$83$	−7.10658	−0.780048	−0.390024	−	0.920805i	$-0.627533\pi$
$83$	−7.10658	−0.780048	−0.390024	+	0.920805i	$0.627533\pi$
$84$	0	0
$85$	30.8643	3.34770
$86$	−2.46974	−0.266319
$87$	0	0
$88$	−1.30509	−0.139123
$89$	8.88821	0.942148	0.471074	−	0.882094i	$-0.343866\pi$
$89$	8.88821	0.942148	0.471074	+	0.882094i	$0.343866\pi$
$90$	0	0
$91$	−0.613179	−0.0642786
$92$	−1.28626	−0.134102
$93$	0	0
$94$	14.6799	1.51411
$95$	−8.27363	−0.848857
$96$	0	0
$97$	10.6085	1.07713	0.538564	−	0.842584i	$-0.318967\pi$
$97$	10.6085	1.07713	0.538564	+	0.842584i	$0.318967\pi$
$98$	14.6675	1.48164
$99$	0	0
$100$	33.9573	3.39573
$101$	11.8586	1.17998	0.589989	−	0.807411i	$-0.299132\pi$
$101$	11.8586	1.17998	0.589989	+	0.807411i	$0.299132\pi$
$102$	0	0
$103$	−14.4134	−1.42019	−0.710096	−	0.704105i	$-0.751348\pi$
$103$	−14.4134	−1.42019	−0.710096	+	0.704105i	$0.751348\pi$
$104$	1.95722	0.191921
$105$	0	0
$106$	−8.70485	−0.845490
$107$	−4.56268	−0.441091	−0.220545	−	0.975377i	$-0.570784\pi$
$107$	−4.56268	−0.441091	−0.220545	+	0.975377i	$0.570784\pi$
$108$	0	0
$109$	−0.778710	−0.0745869	−0.0372934	−	0.999304i	$-0.511874\pi$
$109$	−0.778710	−0.0745869	−0.0372934	+	0.999304i	$0.511874\pi$
$110$	−9.11254	−0.868846
$111$	0	0
$112$	−0.987189	−0.0932806
$113$	13.3179	1.25285	0.626423	−	0.779483i	$-0.284518\pi$
$113$	13.3179	1.25285	0.626423	+	0.779483i	$0.284518\pi$
$114$	0	0
$115$	−2.09367	−0.195236
$116$	5.29793	0.491900
$117$	0	0
$118$	−12.2863	−1.13104
$119$	2.97276	0.272512
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.14662	−0.194346
$123$	0	0
$124$	−2.55505	−0.229450
$125$	34.0478	3.04533
$126$	0	0
$127$	19.0948	1.69439	0.847196	−	0.531280i	$-0.178289\pi$
$127$	19.0948	1.69439	0.847196	+	0.531280i	$0.178289\pi$
$128$	9.95831	0.880198
$129$	0	0
$130$	13.6659	1.19858
$131$	7.66992	0.670124	0.335062	−	0.942196i	$-0.391243\pi$
$131$	7.66992	0.670124	0.335062	+	0.942196i	$0.391243\pi$
$132$	0	0
$133$	−0.796893	−0.0690993
$134$	19.0646	1.64693
$135$	0	0
$136$	−9.48881	−0.813659
$137$	−13.0186	−1.11225	−0.556125	−	0.831098i	$-0.687713\pi$
$137$	−13.0186	−1.11225	−0.556125	+	0.831098i	$0.687713\pi$
$138$	0	0
$139$	−1.41250	−0.119806	−0.0599032	−	0.998204i	$-0.519079\pi$
$139$	−1.41250	−0.119806	−0.0599032	+	0.998204i	$0.519079\pi$
$140$	4.52663	0.382570
$141$	0	0
$142$	16.6816	1.39989
$143$	−1.49968	−0.125410
$144$	0	0
$145$	8.62356	0.716148
$146$	−26.7208	−2.21143
$147$	0	0
$148$	−5.72889	−0.470912
$149$	1.67851	0.137509	0.0687545	−	0.997634i	$-0.478097\pi$
$149$	1.67851	0.137509	0.0687545	+	0.997634i	$0.478097\pi$
$150$	0	0
$151$	0.421110	0.0342695	0.0171347	−	0.999853i	$-0.494546\pi$
$151$	0.421110	0.0342695	0.0171347	+	0.999853i	$0.494546\pi$
$152$	2.54362	0.206315
$153$	0	0
$154$	−0.877693	−0.0707265
$155$	−4.15892	−0.334052
$156$	0	0
$157$	16.2370	1.29585	0.647927	−	0.761703i	$-0.275636\pi$
$157$	16.2370	1.29585	0.647927	+	0.761703i	$0.275636\pi$
$158$	−24.7605	−1.96984
$159$	0	0
$160$	33.0819	2.61535
$161$	−0.201656	−0.0158928
$162$	0	0
$163$	2.34565	0.183726	0.0918628	−	0.995772i	$-0.470718\pi$
$163$	2.34565	0.183726	0.0918628	+	0.995772i	$0.470718\pi$
$164$	9.94562	0.776623
$165$	0	0
$166$	15.2551	1.18403
$167$	22.9327	1.77458	0.887292	−	0.461208i	$-0.152584\pi$
$167$	22.9327	1.77458	0.887292	+	0.461208i	$0.152584\pi$
$168$	0	0
$169$	−10.7510	−0.826996
$170$	−66.2538	−5.08144
$171$	0	0
$172$	3.00054	0.228789
$173$	−12.6639	−0.962819	−0.481409	−	0.876496i	$-0.659875\pi$
$173$	−12.6639	−0.962819	−0.481409	+	0.876496i	$0.659875\pi$
$174$	0	0
$175$	5.32375	0.402438
$176$	−2.41442	−0.181994
$177$	0	0
$178$	−19.0796	−1.43008
$179$	4.98340	0.372477	0.186238	−	0.982505i	$-0.440370\pi$
$179$	4.98340	0.372477	0.186238	+	0.982505i	$0.440370\pi$
$180$	0	0
$181$	−17.0746	−1.26915	−0.634574	−	0.772862i	$-0.718824\pi$
$181$	−17.0746	−1.26915	−0.634574	+	0.772862i	$0.718824\pi$
$182$	1.31626	0.0975678
$183$	0	0
$184$	0.643672	0.0474521
$185$	−9.32505	−0.685592
$186$	0	0
$187$	7.27063	0.531681
$188$	−17.8349	−1.30074
$189$	0	0
$190$	17.7603	1.28847
$191$	−18.3089	−1.32478	−0.662392	−	0.749157i	$-0.730459\pi$
$191$	−18.3089	−1.32478	−0.662392	+	0.749157i	$0.730459\pi$
$192$	0	0
$193$	−16.1105	−1.15966	−0.579830	−	0.814737i	$-0.696881\pi$
$193$	−16.1105	−1.15966	−0.579830	+	0.814737i	$0.696881\pi$
$194$	−22.7724	−1.63496
$195$	0	0
$196$	−17.8198	−1.27284
$197$	26.9193	1.91792	0.958961	−	0.283538i	$-0.0915083\pi$
$197$	26.9193	1.91792	0.958961	+	0.283538i	$0.0915083\pi$
$198$	0	0
$199$	8.60007	0.609643	0.304821	−	0.952410i	$-0.401403\pi$
$199$	8.60007	0.609643	0.304821	+	0.952410i	$0.401403\pi$
$200$	−16.9930	−1.20159
$201$	0	0
$202$	−25.4560	−1.79108
$203$	0.830597	0.0582965
$204$	0	0
$205$	16.1887	1.13067
$206$	30.9400	2.15569
$207$	0	0
$208$	3.62086	0.251062
$209$	−1.94900	−0.134815
$210$	0	0
$211$	20.2204	1.39203	0.696013	−	0.718029i	$-0.254956\pi$
$211$	20.2204	1.39203	0.696013	+	0.718029i	$0.254956\pi$
$212$	10.5757	0.726343
$213$	0	0
$214$	9.79433	0.669526
$215$	4.88405	0.333089
$216$	0	0
$217$	−0.400575	−0.0271928
$218$	1.67159	0.113215
$219$	0	0
$220$	11.0710	0.746408
$221$	−10.9036	−0.733458
$222$	0	0
$223$	21.9636	1.47079	0.735395	−	0.677639i	$-0.236997\pi$
$223$	21.9636	1.47079	0.735395	+	0.677639i	$0.236997\pi$
$224$	3.18635	0.212897
$225$	0	0
$226$	−28.5885	−1.90168
$227$	3.90659	0.259289	0.129645	−	0.991561i	$-0.458616\pi$
$227$	3.90659	0.259289	0.129645	+	0.991561i	$0.458616\pi$
$228$	0	0
$229$	−18.6339	−1.23136	−0.615681	−	0.787996i	$-0.711119\pi$
$229$	−18.6339	−1.23136	−0.615681	+	0.787996i	$0.711119\pi$
$230$	4.49432	0.296346
$231$	0	0
$232$	−2.65120	−0.174060
$233$	−18.9756	−1.24313	−0.621566	−	0.783362i	$-0.713503\pi$
$233$	−18.9756	−1.24313	−0.621566	+	0.783362i	$0.713503\pi$
$234$	0	0
$235$	−29.0303	−1.89373
$236$	14.9269	0.971656
$237$	0	0
$238$	−6.38138	−0.413643
$239$	−21.4310	−1.38626	−0.693128	−	0.720815i	$-0.743768\pi$
$239$	−21.4310	−1.38626	−0.693128	+	0.720815i	$0.743768\pi$
$240$	0	0
$241$	−8.66593	−0.558221	−0.279111	−	0.960259i	$-0.590040\pi$
$241$	−8.66593	−0.558221	−0.279111	+	0.960259i	$0.590040\pi$
$242$	−2.14662	−0.137990
$243$	0	0
$244$	2.60797	0.166958
$245$	−29.0058	−1.85311
$246$	0	0
$247$	2.92288	0.185979
$248$	1.27860	0.0811914
$249$	0	0
$250$	−73.0877	−4.62247
$251$	−1.04201	−0.0657709	−0.0328854	−	0.999459i	$-0.510470\pi$
$251$	−1.04201	−0.0657709	−0.0328854	+	0.999459i	$0.510470\pi$
$252$	0	0
$253$	−0.493201	−0.0310073
$254$	−40.9893	−2.57190
$255$	0	0
$256$	2.42291	0.151432
$257$	23.4146	1.46056	0.730282	−	0.683146i	$-0.239389\pi$
$257$	23.4146	1.46056	0.730282	+	0.683146i	$0.239389\pi$
$258$	0	0
$259$	−0.898162	−0.0558091
$260$	−16.6030	−1.02968
$261$	0	0
$262$	−16.4644	−1.01717
$263$	−15.3322	−0.945424	−0.472712	−	0.881217i	$-0.656725\pi$
$263$	−15.3322	−0.945424	−0.472712	+	0.881217i	$0.656725\pi$
$264$	0	0
$265$	17.2143	1.05747
$266$	1.71062	0.104885
$267$	0	0
$268$	−23.1620	−1.41485
$269$	−2.74704	−0.167490	−0.0837449	−	0.996487i	$-0.526688\pi$
$269$	−2.74704	−0.167490	−0.0837449	+	0.996487i	$0.526688\pi$
$270$	0	0
$271$	−18.7742	−1.14045	−0.570227	−	0.821487i	$-0.693145\pi$
$271$	−18.7742	−1.14045	−0.570227	+	0.821487i	$0.693145\pi$
$272$	−17.5543	−1.06439
$273$	0	0
$274$	27.9459	1.68827
$275$	13.0206	0.785170
$276$	0	0
$277$	−20.0716	−1.20598	−0.602991	−	0.797748i	$-0.706025\pi$
$277$	−20.0716	−1.20598	−0.602991	+	0.797748i	$0.706025\pi$
$278$	3.03209	0.181853
$279$	0	0
$280$	−2.26523	−0.135373
$281$	6.46762	0.385826	0.192913	−	0.981216i	$-0.438207\pi$
$281$	6.46762	0.385826	0.192913	+	0.981216i	$0.438207\pi$
$282$	0	0
$283$	3.72416	0.221378	0.110689	−	0.993855i	$-0.464694\pi$
$283$	3.72416	0.221378	0.110689	+	0.993855i	$0.464694\pi$
$284$	−20.2669	−1.20262
$285$	0	0
$286$	3.21925	0.190358
$287$	1.55925	0.0920398
$288$	0	0
$289$	35.8620	2.10953
$290$	−18.5115	−1.08703
$291$	0	0
$292$	32.4637	1.89979
$293$	28.0147	1.63664	0.818319	−	0.574764i	$-0.194906\pi$
$293$	28.0147	1.63664	0.818319	+	0.574764i	$0.194906\pi$
$294$	0	0
$295$	24.2968	1.41462
$296$	2.86686	0.166633
$297$	0	0
$298$	−3.60312	−0.208723
$299$	0.739646	0.0427748
$300$	0	0
$301$	0.470418	0.0271144
$302$	−0.903964	−0.0520173
$303$	0	0
$304$	4.70570	0.269891
$305$	4.24506	0.243072
$306$	0	0
$307$	4.62768	0.264116	0.132058	−	0.991242i	$-0.457842\pi$
$307$	4.62768	0.264116	0.132058	+	0.991242i	$0.457842\pi$
$308$	1.06633	0.0607597
$309$	0	0
$310$	8.92761	0.507054
$311$	4.99747	0.283380	0.141690	−	0.989911i	$-0.454746\pi$
$311$	4.99747	0.283380	0.141690	+	0.989911i	$0.454746\pi$
$312$	0	0
$313$	−27.0058	−1.52646	−0.763230	−	0.646127i	$-0.776388\pi$
$313$	−27.0058	−1.52646	−0.763230	+	0.646127i	$0.776388\pi$
$314$	−34.8546	−1.96696
$315$	0	0
$316$	30.0821	1.69225
$317$	−14.6041	−0.820248	−0.410124	−	0.912030i	$-0.634515\pi$
$317$	−14.6041	−0.820248	−0.410124	+	0.912030i	$0.634515\pi$
$318$	0	0
$319$	2.03143	0.113738
$320$	−50.5154	−2.82390
$321$	0	0
$322$	0.432880	0.0241234
$323$	−14.1705	−0.788465
$324$	0	0
$325$	−19.5267	−1.08315
$326$	−5.03522	−0.278875
$327$	0	0
$328$	−4.97701	−0.274810
$329$	−2.79612	−0.154155
$330$	0	0
$331$	−3.54858	−0.195048	−0.0975238	−	0.995233i	$-0.531092\pi$
$331$	−3.54858	−0.195048	−0.0975238	+	0.995233i	$0.531092\pi$
$332$	−18.5338	−1.01717
$333$	0	0
$334$	−49.2277	−2.69362
$335$	−37.7014	−2.05985
$336$	0	0
$337$	−16.7964	−0.914957	−0.457478	−	0.889221i	$-0.651247\pi$
$337$	−16.7964	−0.914957	−0.457478	+	0.889221i	$0.651247\pi$
$338$	23.0782	1.25529
$339$	0	0
$340$	80.4932	4.36536
$341$	−0.979706	−0.0530541
$342$	0	0
$343$	−5.65586	−0.305388
$344$	−1.50154	−0.0809574
$345$	0	0
$346$	27.1846	1.46145
$347$	−13.1793	−0.707504	−0.353752	−	0.935339i	$-0.615094\pi$
$347$	−13.1793	−0.707504	−0.353752	+	0.935339i	$0.615094\pi$
$348$	0	0
$349$	−4.22013	−0.225898	−0.112949	−	0.993601i	$-0.536030\pi$
$349$	−4.22013	−0.225898	−0.112949	+	0.993601i	$0.536030\pi$
$350$	−11.4281	−0.610856
$351$	0	0
$352$	7.79302	0.415369
$353$	12.3066	0.655013	0.327507	−	0.944849i	$-0.393792\pi$
$353$	12.3066	0.655013	0.327507	+	0.944849i	$0.393792\pi$
$354$	0	0
$355$	−32.9889	−1.75087
$356$	23.1802	1.22855
$357$	0	0
$358$	−10.6975	−0.565378
$359$	−16.4213	−0.866683	−0.433342	−	0.901230i	$-0.642666\pi$
$359$	−16.4213	−0.866683	−0.433342	+	0.901230i	$0.642666\pi$
$360$	0	0
$361$	−15.2014	−0.800074
$362$	36.6528	1.92643
$363$	0	0
$364$	−1.59915	−0.0838185
$365$	52.8419	2.76587
$366$	0	0
$367$	−13.6364	−0.711815	−0.355908	−	0.934521i	$-0.615828\pi$
$367$	−13.6364	−0.711815	−0.355908	+	0.934521i	$0.615828\pi$
$368$	1.19080	0.0620745
$369$	0	0
$370$	20.0173	1.04065
$371$	1.65804	0.0860809
$372$	0	0
$373$	8.15846	0.422429	0.211214	−	0.977440i	$-0.432258\pi$
$373$	8.15846	0.422429	0.211214	+	0.977440i	$0.432258\pi$
$374$	−15.6073	−0.807032
$375$	0	0
$376$	8.92499	0.460271
$377$	−3.04651	−0.156903
$378$	0	0
$379$	−4.29727	−0.220736	−0.110368	−	0.993891i	$-0.535203\pi$
$379$	−4.29727	−0.220736	−0.110368	+	0.993891i	$0.535203\pi$
$380$	−21.5774	−1.10690
$381$	0	0
$382$	39.3022	2.01088
$383$	13.0420	0.666413	0.333207	−	0.942854i	$-0.391869\pi$
$383$	13.0420	0.666413	0.333207	+	0.942854i	$0.391869\pi$
$384$	0	0
$385$	1.73569	0.0884589
$386$	34.5832	1.76024
$387$	0	0
$388$	27.6667	1.40456
$389$	−29.8866	−1.51531	−0.757655	−	0.652655i	$-0.773655\pi$
$389$	−29.8866	−1.51531	−0.757655	+	0.652655i	$0.773655\pi$
$390$	0	0
$391$	−3.58588	−0.181346
$392$	8.91744	0.450399
$393$	0	0
$394$	−57.7855	−2.91119
$395$	48.9654	2.46372
$396$	0	0
$397$	−18.4903	−0.928001	−0.464001	−	0.885835i	$-0.653586\pi$
$397$	−18.4903	−0.928001	−0.464001	+	0.885835i	$0.653586\pi$
$398$	−18.4611	−0.925370
$399$	0	0
$400$	−31.4371	−1.57186
$401$	38.8630	1.94073	0.970364	−	0.241648i	$-0.0776878\pi$
$401$	38.8630	1.94073	0.970364	+	0.241648i	$0.0776878\pi$
$402$	0	0
$403$	1.46925	0.0731885
$404$	30.9270	1.53868
$405$	0	0
$406$	−1.78298	−0.0884876
$407$	−2.19668	−0.108885
$408$	0	0
$409$	29.7500	1.47104	0.735522	−	0.677500i	$-0.236937\pi$
$409$	29.7500	1.47104	0.735522	+	0.677500i	$0.236937\pi$
$410$	−34.7511	−1.71623
$411$	0	0
$412$	−37.5897	−1.85191
$413$	2.34020	0.115154
$414$	0	0
$415$	−30.1679	−1.48088
$416$	−11.6871	−0.573005
$417$	0	0
$418$	4.18376	0.204634
$419$	13.0059	0.635377	0.317689	−	0.948195i	$-0.397093\pi$
$419$	13.0059	0.635377	0.317689	+	0.948195i	$0.397093\pi$
$420$	0	0
$421$	3.05401	0.148843	0.0744216	−	0.997227i	$-0.476289\pi$
$421$	3.05401	0.148843	0.0744216	+	0.997227i	$0.476289\pi$
$422$	−43.4054	−2.11294
$423$	0	0
$424$	−5.29232	−0.257018
$425$	94.6677	4.59206
$426$	0	0
$427$	0.408872	0.0197867
$428$	−11.8993	−0.575176
$429$	0	0
$430$	−10.4842	−0.505593
$431$	19.9060	0.958840	0.479420	−	0.877586i	$-0.340847\pi$
$431$	19.9060	0.958840	0.479420	+	0.877586i	$0.340847\pi$
$432$	0	0
$433$	−11.0275	−0.529947	−0.264973	−	0.964256i	$-0.585363\pi$
$433$	−11.0275	−0.529947	−0.264973	+	0.964256i	$0.585363\pi$
$434$	0.859882	0.0412756
$435$	0	0
$436$	−2.03085	−0.0972603
$437$	0.961250	0.0459828
$438$	0	0
$439$	10.0476	0.479544	0.239772	−	0.970829i	$-0.422927\pi$
$439$	10.0476	0.479544	0.239772	+	0.970829i	$0.422927\pi$
$440$	−5.54019	−0.264118
$441$	0	0
$442$	23.4059	1.11331
$443$	−1.69107	−0.0803452	−0.0401726	−	0.999193i	$-0.512791\pi$
$443$	−1.69107	−0.0803452	−0.0401726	+	0.999193i	$0.512791\pi$
$444$	0	0
$445$	37.7310	1.78862
$446$	−47.1474	−2.23250
$447$	0	0
$448$	−4.86550	−0.229873
$449$	39.0267	1.84179	0.920893	−	0.389815i	$-0.127461\pi$
$449$	39.0267	1.84179	0.920893	+	0.389815i	$0.127461\pi$
$450$	0	0
$451$	3.81354	0.179573
$452$	34.7328	1.63370
$453$	0	0
$454$	−8.38595	−0.393572
$455$	−2.60298	−0.122030
$456$	0	0
$457$	−39.2869	−1.83776	−0.918882	−	0.394533i	$-0.870906\pi$
$457$	−39.2869	−1.83776	−0.918882	+	0.394533i	$0.870906\pi$
$458$	39.9999	1.86907
$459$	0	0
$460$	−5.46024	−0.254585
$461$	−10.8819	−0.506822	−0.253411	−	0.967359i	$-0.581552\pi$
$461$	−10.8819	−0.506822	−0.253411	+	0.967359i	$0.581552\pi$
$462$	0	0
$463$	31.8665	1.48096	0.740482	−	0.672077i	$-0.234597\pi$
$463$	31.8665	1.48096	0.740482	+	0.672077i	$0.234597\pi$
$464$	−4.90473	−0.227696
$465$	0	0
$466$	40.7333	1.88694
$467$	−22.8435	−1.05707	−0.528535	−	0.848912i	$-0.677258\pi$
$467$	−22.8435	−1.05707	−0.528535	+	0.848912i	$0.677258\pi$
$468$	0	0
$469$	−3.63129	−0.167677
$470$	62.3170	2.87447
$471$	0	0
$472$	−7.46974	−0.343822
$473$	1.15052	0.0529012
$474$	0	0
$475$	−25.3771	−1.16438
$476$	7.75288	0.355352
$477$	0	0
$478$	46.0042	2.10418
$479$	−4.02432	−0.183876	−0.0919380	−	0.995765i	$-0.529306\pi$
$479$	−4.02432	−0.183876	−0.0919380	+	0.995765i	$0.529306\pi$
$480$	0	0
$481$	3.29433	0.150208
$482$	18.6024	0.847318
$483$	0	0
$484$	2.60797	0.118544
$485$	45.0337	2.04488
$486$	0	0
$487$	−21.8146	−0.988514	−0.494257	−	0.869316i	$-0.664560\pi$
$487$	−21.8146	−0.988514	−0.494257	+	0.869316i	$0.664560\pi$
$488$	−1.30509	−0.0590786
$489$	0	0
$490$	62.2644	2.81282
$491$	11.6277	0.524752	0.262376	−	0.964966i	$-0.415494\pi$
$491$	11.6277	0.524752	0.262376	+	0.964966i	$0.415494\pi$
$492$	0	0
$493$	14.7698	0.665198
$494$	−6.27432	−0.282295
$495$	0	0
$496$	2.36542	0.106211
$497$	−3.17739	−0.142526
$498$	0	0
$499$	14.2839	0.639435	0.319718	−	0.947513i	$-0.396412\pi$
$499$	14.2839	0.639435	0.319718	+	0.947513i	$0.396412\pi$
$500$	88.7959	3.97107
$501$	0	0
$502$	2.23679	0.0998329
$503$	−2.85657	−0.127368	−0.0636840	−	0.997970i	$-0.520285\pi$
$503$	−2.85657	−0.127368	−0.0636840	+	0.997970i	$0.520285\pi$
$504$	0	0
$505$	50.3407	2.24013
$506$	1.05872	0.0470657
$507$	0	0
$508$	49.7988	2.20947
$509$	−31.5352	−1.39777	−0.698886	−	0.715233i	$-0.746321\pi$
$509$	−31.5352	−1.39777	−0.698886	+	0.715233i	$0.746321\pi$
$510$	0	0
$511$	5.08958	0.225150
$512$	−25.1177	−1.11005
$513$	0	0
$514$	−50.2623	−2.21697
$515$	−61.1857	−2.69616
$516$	0	0
$517$	−6.83861	−0.300762
$518$	1.92801	0.0847120
$519$	0	0
$520$	8.30852	0.364353
$521$	23.7305	1.03965	0.519826	−	0.854272i	$-0.325997\pi$
$521$	23.7305	1.03965	0.519826	+	0.854272i	$0.325997\pi$
$522$	0	0
$523$	3.00257	0.131293	0.0656466	−	0.997843i	$-0.479089\pi$
$523$	3.00257	0.131293	0.0656466	+	0.997843i	$0.479089\pi$
$524$	20.0030	0.873834
$525$	0	0
$526$	32.9124	1.43505
$527$	−7.12308	−0.310286
$528$	0	0
$529$	−22.7568	−0.989424
$530$	−36.9526	−1.60512
$531$	0	0
$532$	−2.07827	−0.0901046
$533$	−5.71910	−0.247722
$534$	0	0
$535$	−19.3689	−0.837388
$536$	11.5908	0.500646
$537$	0	0
$538$	5.89684	0.254231
$539$	−6.83282	−0.294311
$540$	0	0
$541$	7.72541	0.332141	0.166071	−	0.986114i	$-0.446892\pi$
$541$	7.72541	0.332141	0.166071	+	0.986114i	$0.446892\pi$
$542$	40.3012	1.73108
$543$	0	0
$544$	56.6601	2.42928
$545$	−3.30567	−0.141599
$546$	0	0
$547$	−35.4778	−1.51692	−0.758462	−	0.651718i	$-0.774049\pi$
$547$	−35.4778	−1.51692	−0.758462	+	0.651718i	$0.774049\pi$
$548$	−33.9521	−1.45036
$549$	0	0
$550$	−27.9502	−1.19180
$551$	−3.95926	−0.168670
$552$	0	0
$553$	4.71621	0.200553
$554$	43.0860	1.83055
$555$	0	0
$556$	−3.68376	−0.156226
$557$	29.2868	1.24092	0.620462	−	0.784237i	$-0.286945\pi$
$557$	29.2868	1.24092	0.620462	+	0.784237i	$0.286945\pi$
$558$	0	0
$559$	−1.72542	−0.0729776
$560$	−4.19068	−0.177089
$561$	0	0
$562$	−13.8835	−0.585641
$563$	34.7716	1.46545	0.732723	−	0.680527i	$-0.238249\pi$
$563$	34.7716	1.46545	0.732723	+	0.680527i	$0.238249\pi$
$564$	0	0
$565$	56.5355	2.37847
$566$	−7.99436	−0.336028
$567$	0	0
$568$	10.1420	0.425549
$569$	18.4019	0.771448	0.385724	−	0.922614i	$-0.373952\pi$
$569$	18.4019	0.771448	0.385724	+	0.922614i	$0.373952\pi$
$570$	0	0
$571$	−22.7245	−0.950991	−0.475496	−	0.879718i	$-0.657731\pi$
$571$	−22.7245	−0.950991	−0.475496	+	0.879718i	$0.657731\pi$
$572$	−3.91113	−0.163533
$573$	0	0
$574$	−3.34712	−0.139706
$575$	−6.42176	−0.267806
$576$	0	0
$577$	6.70212	0.279013	0.139506	−	0.990221i	$-0.455448\pi$
$577$	6.70212	0.279013	0.139506	+	0.990221i	$0.455448\pi$
$578$	−76.9820	−3.20203
$579$	0	0
$580$	22.4900	0.933848
$581$	−2.90568	−0.120548
$582$	0	0
$583$	4.05514	0.167947
$584$	−16.2456	−0.672246
$585$	0	0
$586$	−60.1370	−2.48424
$587$	31.0245	1.28052	0.640259	−	0.768159i	$-0.278827\pi$
$587$	31.0245	1.28052	0.640259	+	0.768159i	$0.278827\pi$
$588$	0	0
$589$	1.90945	0.0786774
$590$	−52.1560	−2.14723
$591$	0	0
$592$	5.30371	0.217981
$593$	−25.8313	−1.06076	−0.530382	−	0.847759i	$-0.677952\pi$
$593$	−25.8313	−1.06076	−0.530382	+	0.847759i	$0.677952\pi$
$594$	0	0
$595$	12.6195	0.517351
$596$	4.37751	0.179310
$597$	0	0
$598$	−1.58774	−0.0649274
$599$	5.63372	0.230187	0.115094	−	0.993355i	$-0.463283\pi$
$599$	5.63372	0.230187	0.115094	+	0.993355i	$0.463283\pi$
$600$	0	0
$601$	44.2316	1.80425	0.902123	−	0.431480i	$-0.142008\pi$
$601$	44.2316	1.80425	0.902123	+	0.431480i	$0.142008\pi$
$602$	−1.00981	−0.0411567
$603$	0	0
$604$	1.09824	0.0446870
$605$	4.24506	0.172586
$606$	0	0
$607$	−0.679704	−0.0275883	−0.0137942	−	0.999905i	$-0.504391\pi$
$607$	−0.679704	−0.0275883	−0.0137942	+	0.999905i	$0.504391\pi$
$608$	−15.1886	−0.615979
$609$	0	0
$610$	−9.11254	−0.368956
$611$	10.2557	0.414903
$612$	0	0
$613$	−19.3145	−0.780105	−0.390053	−	0.920793i	$-0.627543\pi$
$613$	−19.3145	−0.780105	−0.390053	+	0.920793i	$0.627543\pi$
$614$	−9.93387	−0.400899
$615$	0	0
$616$	−0.533615	−0.0215000
$617$	8.36395	0.336720	0.168360	−	0.985726i	$-0.446153\pi$
$617$	8.36395	0.336720	0.168360	+	0.985726i	$0.446153\pi$
$618$	0	0
$619$	−17.2130	−0.691848	−0.345924	−	0.938263i	$-0.612434\pi$
$619$	−17.2130	−0.691848	−0.345924	+	0.938263i	$0.612434\pi$
$620$	−10.8463	−0.435600
$621$	0	0
$622$	−10.7277	−0.430140
$623$	3.63414	0.145599
$624$	0	0
$625$	79.4324	3.17730
$626$	57.9712	2.31700
$627$	0	0
$628$	42.3457	1.68978
$629$	−15.9712	−0.636815
$630$	0	0
$631$	35.8438	1.42692	0.713460	−	0.700695i	$-0.247127\pi$
$631$	35.8438	1.42692	0.713460	+	0.700695i	$0.247127\pi$
$632$	−15.0538	−0.598806
$633$	0	0
$634$	31.3494	1.24505
$635$	81.0588	3.21672
$636$	0	0
$637$	10.2471	0.406004
$638$	−4.36071	−0.172642
$639$	0	0
$640$	42.2737	1.67101
$641$	35.7794	1.41320	0.706600	−	0.707613i	$-0.250228\pi$
$641$	35.7794	1.41320	0.706600	+	0.707613i	$0.250228\pi$
$642$	0	0
$643$	−32.3830	−1.27706	−0.638530	−	0.769597i	$-0.720457\pi$
$643$	−32.3830	−1.27706	−0.638530	+	0.769597i	$0.720457\pi$
$644$	−0.525915	−0.0207239
$645$	0	0
$646$	30.4186	1.19680
$647$	−37.5425	−1.47595	−0.737974	−	0.674830i	$-0.764217\pi$
$647$	−37.5425	−1.47595	−0.737974	+	0.674830i	$0.764217\pi$
$648$	0	0
$649$	5.72355	0.224669
$650$	41.9164	1.64410
$651$	0	0
$652$	6.11740	0.239576
$653$	−32.8214	−1.28440	−0.642201	−	0.766537i	$-0.721978\pi$
$653$	−32.8214	−1.28440	−0.642201	+	0.766537i	$0.721978\pi$
$654$	0	0
$655$	32.5593	1.27220
$656$	−9.20749	−0.359492
$657$	0	0
$658$	6.00220	0.233990
$659$	39.7911	1.55004	0.775021	−	0.631935i	$-0.217739\pi$
$659$	39.7911	1.55004	0.775021	+	0.631935i	$0.217739\pi$
$660$	0	0
$661$	16.8264	0.654473	0.327236	−	0.944943i	$-0.393883\pi$
$661$	16.8264	0.654473	0.327236	+	0.944943i	$0.393883\pi$
$662$	7.61745	0.296060
$663$	0	0
$664$	9.27471	0.359929
$665$	−3.38286	−0.131182
$666$	0	0
$667$	−1.00191	−0.0387939
$668$	59.8078	2.31403
$669$	0	0
$670$	80.9305	3.12662
$671$	1.00000	0.0386046
$672$	0	0
$673$	12.2217	0.471113	0.235557	−	0.971861i	$-0.424309\pi$
$673$	12.2217	0.471113	0.235557	+	0.971861i	$0.424309\pi$
$674$	36.0554	1.38880
$675$	0	0
$676$	−28.0382	−1.07839
$677$	−15.0903	−0.579968	−0.289984	−	0.957031i	$-0.593650\pi$
$677$	−15.0903	−0.579968	−0.289984	+	0.957031i	$0.593650\pi$
$678$	0	0
$679$	4.33752	0.166459
$680$	−40.2806	−1.54469
$681$	0	0
$682$	2.10306	0.0805302
$683$	−13.9910	−0.535352	−0.267676	−	0.963509i	$-0.586256\pi$
$683$	−13.9910	−0.535352	−0.267676	+	0.963509i	$0.586256\pi$
$684$	0	0
$685$	−55.2646	−2.11155
$686$	12.1410	0.463545
$687$	0	0
$688$	−2.77785	−0.105904
$689$	−6.08143	−0.231684
$690$	0	0
$691$	31.1855	1.18635	0.593177	−	0.805072i	$-0.297874\pi$
$691$	31.1855	1.18635	0.593177	+	0.805072i	$0.297874\pi$
$692$	−33.0271	−1.25550
$693$	0	0
$694$	28.2910	1.07391
$695$	−5.99614	−0.227447
$696$	0	0
$697$	27.7268	1.05023
$698$	9.05900	0.342888
$699$	0	0
$700$	13.8842	0.524774
$701$	−47.7306	−1.80276	−0.901380	−	0.433029i	$-0.857445\pi$
$701$	−47.7306	−1.80276	−0.901380	+	0.433029i	$0.857445\pi$
$702$	0	0
$703$	4.28133	0.161474
$704$	−11.8998	−0.448491
$705$	0	0
$706$	−26.4175	−0.994237
$707$	4.84867	0.182353
$708$	0	0
$709$	−15.3797	−0.577596	−0.288798	−	0.957390i	$-0.593256\pi$
$709$	−15.3797	−0.577596	−0.288798	+	0.957390i	$0.593256\pi$
$710$	70.8146	2.65762
$711$	0	0
$712$	−11.5999	−0.434725
$713$	0.483193	0.0180957
$714$	0	0
$715$	−6.36625	−0.238084
$716$	12.9966	0.485705
$717$	0	0
$718$	35.2503	1.31553
$719$	42.1771	1.57294	0.786470	−	0.617629i	$-0.211907\pi$
$719$	42.1771	1.57294	0.786470	+	0.617629i	$0.211907\pi$
$720$	0	0
$721$	−5.89323	−0.219475
$722$	32.6316	1.21442
$723$	0	0
$724$	−44.5302	−1.65495
$725$	26.4504	0.982344
$726$	0	0
$727$	−25.1615	−0.933187	−0.466594	−	0.884472i	$-0.654519\pi$
$727$	−25.1615	−0.933187	−0.466594	+	0.884472i	$0.654519\pi$
$728$	0.800253	0.0296593
$729$	0	0
$730$	−113.432	−4.19829
$731$	8.36503	0.309392
$732$	0	0
$733$	0.148201	0.00547393	0.00273696	−	0.999996i	$-0.499129\pi$
$733$	0.148201	0.00547393	0.00273696	+	0.999996i	$0.499129\pi$
$734$	29.2722	1.08046
$735$	0	0
$736$	−3.84353	−0.141674
$737$	−8.88123	−0.327144
$738$	0	0
$739$	40.1687	1.47763	0.738815	−	0.673909i	$-0.235386\pi$
$739$	40.1687	1.47763	0.738815	+	0.673909i	$0.235386\pi$
$740$	−24.3195	−0.894003
$741$	0	0
$742$	−3.55917	−0.130661
$743$	−18.4479	−0.676789	−0.338394	−	0.941004i	$-0.609884\pi$
$743$	−18.4479	−0.676789	−0.338394	+	0.941004i	$0.609884\pi$
$744$	0	0
$745$	7.12538	0.261054
$746$	−17.5131	−0.641200
$747$	0	0
$748$	18.9616	0.693305
$749$	−1.86555	−0.0681658
$750$	0	0
$751$	25.7181	0.938466	0.469233	−	0.883075i	$-0.344531\pi$
$751$	25.7181	0.938466	0.469233	+	0.883075i	$0.344531\pi$
$752$	16.5113	0.602104
$753$	0	0
$754$	6.53969	0.238161
$755$	1.78764	0.0650589
$756$	0	0
$757$	−23.5628	−0.856406	−0.428203	−	0.903683i	$-0.640853\pi$
$757$	−23.5628	−0.856406	−0.428203	+	0.903683i	$0.640853\pi$
$758$	9.22460	0.335053
$759$	0	0
$760$	10.7978	0.391678
$761$	−24.6803	−0.894659	−0.447329	−	0.894369i	$-0.647625\pi$
$761$	−24.6803	−0.894659	−0.447329	+	0.894369i	$0.647625\pi$
$762$	0	0
$763$	−0.318393	−0.0115266
$764$	−47.7491	−1.72750
$765$	0	0
$766$	−27.9961	−1.01154
$767$	−8.58350	−0.309932
$768$	0	0
$769$	15.1950	0.547946	0.273973	−	0.961737i	$-0.411662\pi$
$769$	15.1950	0.547946	0.273973	+	0.961737i	$0.411662\pi$
$770$	−3.72586	−0.134271
$771$	0	0
$772$	−42.0158	−1.51218
$773$	15.5875	0.560642	0.280321	−	0.959906i	$-0.409559\pi$
$773$	15.5875	0.560642	0.280321	+	0.959906i	$0.409559\pi$
$774$	0	0
$775$	−12.7563	−0.458221
$776$	−13.8450	−0.497007
$777$	0	0
$778$	64.1551	2.30007
$779$	−7.43260	−0.266300
$780$	0	0
$781$	−7.77111	−0.278072
$782$	7.69753	0.275263
$783$	0	0
$784$	16.4973	0.589189
$785$	68.9271	2.46011
$786$	0	0
$787$	−30.9748	−1.10413	−0.552065	−	0.833801i	$-0.686160\pi$
$787$	−30.9748	−1.10413	−0.552065	+	0.833801i	$0.686160\pi$
$788$	70.2049	2.50095
$789$	0	0
$790$	−105.110	−3.73965
$791$	5.44534	0.193614
$792$	0	0
$793$	−1.49968	−0.0532553
$794$	39.6916	1.40860
$795$	0	0
$796$	22.4288	0.794966
$797$	−1.38010	−0.0488856	−0.0244428	−	0.999701i	$-0.507781\pi$
$797$	−1.38010	−0.0488856	−0.0244428	+	0.999701i	$0.507781\pi$
$798$	0	0
$799$	−49.7209	−1.75900
$800$	101.470	3.58749
$801$	0	0
$802$	−83.4242	−2.94581
$803$	12.4479	0.439275
$804$	0	0
$805$	−0.856045	−0.0301716
$806$	−3.15392	−0.111092
$807$	0	0
$808$	−15.4766	−0.544464
$809$	5.17587	0.181974	0.0909869	−	0.995852i	$-0.470998\pi$
$809$	5.17587	0.181974	0.0909869	+	0.995852i	$0.470998\pi$
$810$	0	0
$811$	43.3280	1.52145	0.760727	−	0.649072i	$-0.224843\pi$
$811$	43.3280	1.52145	0.760727	+	0.649072i	$0.224843\pi$
$812$	2.16618	0.0760179
$813$	0	0
$814$	4.71544	0.165276
$815$	9.95744	0.348794
$816$	0	0
$817$	−2.24237	−0.0784507
$818$	−63.8620	−2.23288
$819$	0	0
$820$	42.2198	1.47438
$821$	6.96270	0.243000	0.121500	−	0.992591i	$-0.461230\pi$
$821$	6.96270	0.243000	0.121500	+	0.992591i	$0.461230\pi$
$822$	0	0
$823$	−8.34495	−0.290887	−0.145443	−	0.989367i	$-0.546461\pi$
$823$	−8.34495	−0.290887	−0.145443	+	0.989367i	$0.546461\pi$
$824$	18.8107	0.655303
$825$	0	0
$826$	−5.02352	−0.174791
$827$	15.0449	0.523163	0.261582	−	0.965181i	$-0.415756\pi$
$827$	15.0449	0.523163	0.261582	+	0.965181i	$0.415756\pi$
$828$	0	0
$829$	31.1525	1.08197	0.540986	−	0.841032i	$-0.318051\pi$
$829$	31.1525	1.08197	0.540986	+	0.841032i	$0.318051\pi$
$830$	64.7589	2.24782
$831$	0	0
$832$	17.8459	0.618696
$833$	−49.6789	−1.72127
$834$	0	0
$835$	97.3507	3.36896
$836$	−5.08294	−0.175797
$837$	0	0
$838$	−27.9186	−0.964432
$839$	0.0930000	0.00321072	0.00160536	−	0.999999i	$-0.499489\pi$
$839$	0.0930000	0.00321072	0.00160536	+	0.999999i	$0.499489\pi$
$840$	0	0
$841$	−24.8733	−0.857699
$842$	−6.55579	−0.225928
$843$	0	0
$844$	52.7342	1.81518
$845$	−45.6385	−1.57001
$846$	0	0
$847$	0.408872	0.0140490
$848$	−9.79082	−0.336218
$849$	0	0
$850$	−203.215	−6.97023
$851$	1.08341	0.0371387
$852$	0	0
$853$	−16.3120	−0.558512	−0.279256	−	0.960217i	$-0.590088\pi$
$853$	−16.3120	−0.558512	−0.279256	+	0.960217i	$0.590088\pi$
$854$	−0.877693	−0.0300340
$855$	0	0
$856$	5.95470	0.203527
$857$	9.09800	0.310782	0.155391	−	0.987853i	$-0.450336\pi$
$857$	9.09800	0.310782	0.155391	+	0.987853i	$0.450336\pi$
$858$	0	0
$859$	32.5107	1.10925	0.554626	−	0.832100i	$-0.312861\pi$
$859$	32.5107	1.10925	0.554626	+	0.832100i	$0.312861\pi$
$860$	12.7375	0.434344
$861$	0	0
$862$	−42.7307	−1.45541
$863$	10.6104	0.361183	0.180591	−	0.983558i	$-0.442199\pi$
$863$	10.6104	0.361183	0.180591	+	0.983558i	$0.442199\pi$
$864$	0	0
$865$	−53.7591	−1.82786
$866$	23.6718	0.804401
$867$	0	0
$868$	−1.04469	−0.0354591
$869$	11.5347	0.391287
$870$	0	0
$871$	13.3190	0.451298
$872$	1.01629	0.0344158
$873$	0	0
$874$	−2.06344	−0.0697968
$875$	13.9212	0.470623
$876$	0	0
$877$	11.5441	0.389816	0.194908	−	0.980822i	$-0.437559\pi$
$877$	11.5441	0.389816	0.194908	+	0.980822i	$0.437559\pi$
$878$	−21.5683	−0.727895
$879$	0	0
$880$	−10.2494	−0.345506
$881$	22.0029	0.741296	0.370648	−	0.928773i	$-0.379136\pi$
$881$	22.0029	0.741296	0.370648	+	0.928773i	$0.379136\pi$
$882$	0	0
$883$	−53.7244	−1.80797	−0.903985	−	0.427564i	$-0.859372\pi$
$883$	−53.7244	−1.80797	−0.903985	+	0.427564i	$0.859372\pi$
$884$	−28.4364	−0.956419
$885$	0	0
$886$	3.63009	0.121955
$887$	11.1368	0.373936	0.186968	−	0.982366i	$-0.440134\pi$
$887$	11.1368	0.373936	0.186968	+	0.982366i	$0.440134\pi$
$888$	0	0
$889$	7.80735	0.261850
$890$	−80.9941	−2.71493
$891$	0	0
$892$	57.2804	1.91789
$893$	13.3284	0.446019
$894$	0	0
$895$	21.1548	0.707129
$896$	4.07168	0.136025
$897$	0	0
$898$	−83.7756	−2.79563
$899$	−1.99021	−0.0663772
$900$	0	0
$901$	29.4834	0.982235
$902$	−8.18623	−0.272571
$903$	0	0
$904$	−17.3811	−0.578087
$905$	−72.4830	−2.40941
$906$	0	0
$907$	−45.5629	−1.51289	−0.756446	−	0.654056i	$-0.773066\pi$
$907$	−45.5629	−1.51289	−0.756446	+	0.654056i	$0.773066\pi$
$908$	10.1883	0.338110
$909$	0	0
$910$	5.58762	0.185228
$911$	−8.76624	−0.290438	−0.145219	−	0.989400i	$-0.546389\pi$
$911$	−8.76624	−0.290438	−0.145219	+	0.989400i	$0.546389\pi$
$912$	0	0
$913$	−7.10658	−0.235193
$914$	84.3340	2.78952
$915$	0	0
$916$	−48.5967	−1.60568
$917$	3.13602	0.103561
$918$	0	0
$919$	16.3059	0.537881	0.268941	−	0.963157i	$-0.413326\pi$
$919$	16.3059	0.537881	0.268941	+	0.963157i	$0.413326\pi$
$920$	2.73243	0.0900855
$921$	0	0
$922$	23.3594	0.769299
$923$	11.6542	0.383603
$924$	0	0
$925$	−28.6020	−0.940429
$926$	−68.4053	−2.24794
$927$	0	0
$928$	15.8310	0.519678
$929$	22.5078	0.738458	0.369229	−	0.929338i	$-0.379622\pi$
$929$	22.5078	0.738458	0.369229	+	0.929338i	$0.379622\pi$
$930$	0	0
$931$	13.3172	0.436453
$932$	−49.4878	−1.62103
$933$	0	0
$934$	49.0362	1.60451
$935$	30.8643	1.00937
$936$	0	0
$937$	37.7483	1.23318	0.616592	−	0.787283i	$-0.288513\pi$
$937$	37.7483	1.23318	0.616592	+	0.787283i	$0.288513\pi$
$938$	7.79500	0.254516
$939$	0	0
$940$	−75.7103	−2.46940
$941$	12.5979	0.410681	0.205341	−	0.978691i	$-0.434170\pi$
$941$	12.5979	0.410681	0.205341	+	0.978691i	$0.434170\pi$
$942$	0	0
$943$	−1.88085	−0.0612487
$944$	−13.8190	−0.449771
$945$	0	0
$946$	−2.46974	−0.0802981
$947$	−37.1724	−1.20794	−0.603970	−	0.797007i	$-0.706415\pi$
$947$	−37.1724	−1.20794	−0.603970	+	0.797007i	$0.706415\pi$
$948$	0	0
$949$	−18.6678	−0.605984
$950$	54.4750	1.76740
$951$	0	0
$952$	−3.87971	−0.125742
$953$	−46.4784	−1.50558	−0.752791	−	0.658259i	$-0.771293\pi$
$953$	−46.4784	−1.50558	−0.752791	+	0.658259i	$0.771293\pi$
$954$	0	0
$955$	−77.7224	−2.51504
$956$	−55.8914	−1.80766
$957$	0	0
$958$	8.63869	0.279103
$959$	−5.32293	−0.171886
$960$	0	0
$961$	−30.0402	−0.969038
$962$	−7.07166	−0.228000
$963$	0	0
$964$	−22.6005	−0.727913
$965$	−68.3902	−2.20156
$966$	0	0
$967$	44.0531	1.41665	0.708327	−	0.705885i	$-0.249450\pi$
$967$	44.0531	1.41665	0.708327	+	0.705885i	$0.249450\pi$
$968$	−1.30509	−0.0419471
$969$	0	0
$970$	−96.6702	−3.10389
$971$	−26.6687	−0.855840	−0.427920	−	0.903817i	$-0.640753\pi$
$971$	−26.6687	−0.855840	−0.427920	+	0.903817i	$0.640753\pi$
$972$	0	0
$973$	−0.577531	−0.0185148
$974$	46.8276	1.50045
$975$	0	0
$976$	−2.41442	−0.0772837
$977$	0.328176	0.0104993	0.00524964	−	0.999986i	$-0.498329\pi$
$977$	0.328176	0.0104993	0.00524964	+	0.999986i	$0.498329\pi$
$978$	0	0
$979$	8.88821	0.284068
$980$	−75.6463	−2.41643
$981$	0	0
$982$	−24.9603	−0.796515
$983$	−12.0908	−0.385637	−0.192819	−	0.981234i	$-0.561763\pi$
$983$	−12.0908	−0.385637	−0.192819	+	0.981234i	$0.561763\pi$
$984$	0	0
$985$	114.274	3.64108
$986$	−31.7051	−1.00970
$987$	0	0
$988$	7.62280	0.242514
$989$	−0.567440	−0.0180436
$990$	0	0
$991$	−16.4953	−0.523990	−0.261995	−	0.965069i	$-0.584380\pi$
$991$	−16.4953	−0.523990	−0.261995	+	0.965069i	$0.584380\pi$
$992$	−7.63487	−0.242407
$993$	0	0
$994$	6.82066	0.216338
$995$	36.5078	1.15738
$996$	0	0
$997$	−39.6185	−1.25473	−0.627365	−	0.778725i	$-0.715867\pi$
$997$	−39.6185	−1.25473	−0.627365	+	0.778725i	$0.715867\pi$
$998$	−30.6621	−0.970592
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.d.1.1		11
3.2	odd	2		2013.2.a.a.1.11	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.a.1.11	✓	11	3.2	odd	2
6039.2.a.d.1.1		11	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)

\(f(q)\)	\(=\)	\(q-2.14662 q^{2} +2.60797 q^{4} +4.24506 q^{5} +0.408872 q^{7} -1.30509 q^{8} +O(q^{10})\)	\(q-2.14662 q^{2} +2.60797 q^{4} +4.24506 q^{5} +0.408872 q^{7} -1.30509 q^{8} -9.11254 q^{10} +1.00000 q^{11} -1.49968 q^{13} -0.877693 q^{14} -2.41442 q^{16} +7.27063 q^{17} -1.94900 q^{19} +11.0710 q^{20} -2.14662 q^{22} -0.493201 q^{23} +13.0206 q^{25} +3.21925 q^{26} +1.06633 q^{28} +2.03143 q^{29} -0.979706 q^{31} +7.79302 q^{32} -15.6073 q^{34} +1.73569 q^{35} -2.19668 q^{37} +4.18376 q^{38} -5.54019 q^{40} +3.81354 q^{41} +1.15052 q^{43} +2.60797 q^{44} +1.05872 q^{46} -6.83861 q^{47} -6.83282 q^{49} -27.9502 q^{50} -3.91113 q^{52} +4.05514 q^{53} +4.24506 q^{55} -0.533615 q^{56} -4.36071 q^{58} +5.72355 q^{59} +1.00000 q^{61} +2.10306 q^{62} -11.8998 q^{64} -6.36625 q^{65} -8.88123 q^{67} +18.9616 q^{68} -3.72586 q^{70} -7.77111 q^{71} +12.4479 q^{73} +4.71544 q^{74} -5.08294 q^{76} +0.408872 q^{77} +11.5347 q^{79} -10.2494 q^{80} -8.18623 q^{82} -7.10658 q^{83} +30.8643 q^{85} -2.46974 q^{86} -1.30509 q^{88} +8.88821 q^{89} -0.613179 q^{91} -1.28626 q^{92} +14.6799 q^{94} -8.27363 q^{95} +10.6085 q^{97} +14.6675 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8}+O(q^{10}) \) 11 * q + 4 * q^2 + 6 * q^4 + 13 * q^5 - 5 * q^7 + 9 * q^8	\( 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8} + 6 q^{10} + 11 q^{11} - 3 q^{13} + 9 q^{14} + 4 q^{16} + 7 q^{17} - 8 q^{19} + 25 q^{20} + 4 q^{22} + 15 q^{23} + 4 q^{25} + 2 q^{26} + 13 q^{28} + 8 q^{29} - 17 q^{31} + 27 q^{32} - 18 q^{34} + 2 q^{35} - 10 q^{37} + 30 q^{38} + 10 q^{40} + 25 q^{41} - 7 q^{43} + 6 q^{44} + 32 q^{46} + 30 q^{47} - 2 q^{49} - 11 q^{50} - 7 q^{52} + 18 q^{53} + 13 q^{55} + 20 q^{56} - 13 q^{58} + 43 q^{59} + 11 q^{61} - 7 q^{62} + 25 q^{64} + 27 q^{65} - 30 q^{67} - 10 q^{68} - 4 q^{70} + 7 q^{71} + 6 q^{73} + 44 q^{74} - 19 q^{76} - 5 q^{77} + 17 q^{79} + 22 q^{80} + 8 q^{82} + 34 q^{83} + 10 q^{85} - 2 q^{86} + 9 q^{88} + 41 q^{89} - 39 q^{91} + 32 q^{92} + 55 q^{94} + 9 q^{95} - 41 q^{97} + 29 q^{98}+O(q^{100}) \) 11 * q + 4 * q^2 + 6 * q^4 + 13 * q^5 - 5 * q^7 + 9 * q^8 + 6 * q^10 + 11 * q^11 - 3 * q^13 + 9 * q^14 + 4 * q^16 + 7 * q^17 - 8 * q^19 + 25 * q^20 + 4 * q^22 + 15 * q^23 + 4 * q^25 + 2 * q^26 + 13 * q^28 + 8 * q^29 - 17 * q^31 + 27 * q^32 - 18 * q^34 + 2 * q^35 - 10 * q^37 + 30 * q^38 + 10 * q^40 + 25 * q^41 - 7 * q^43 + 6 * q^44 + 32 * q^46 + 30 * q^47 - 2 * q^49 - 11 * q^50 - 7 * q^52 + 18 * q^53 + 13 * q^55 + 20 * q^56 - 13 * q^58 + 43 * q^59 + 11 * q^61 - 7 * q^62 + 25 * q^64 + 27 * q^65 - 30 * q^67 - 10 * q^68 - 4 * q^70 + 7 * q^71 + 6 * q^73 + 44 * q^74 - 19 * q^76 - 5 * q^77 + 17 * q^79 + 22 * q^80 + 8 * q^82 + 34 * q^83 + 10 * q^85 - 2 * q^86 + 9 * q^88 + 41 * q^89 - 39 * q^91 + 32 * q^92 + 55 * q^94 + 9 * q^95 - 41 * q^97 + 29 * q^98

Introduction

L-functions

Modular forms

Varieties

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