LMFDB - Embedded newform 6039.2.a.n.1.6

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:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-1.79702 q^{2} +1.22928 q^{4} +4.13243 q^{5} +4.22858 q^{7} +1.38500 q^{8} +O(q^{10})$	\(q-1.79702 q^{2} +1.22928 q^{4} +4.13243 q^{5} +4.22858 q^{7} +1.38500 q^{8} -7.42606 q^{10} -1.00000 q^{11} -5.95009 q^{13} -7.59885 q^{14} -4.94743 q^{16} -6.87502 q^{17} -0.114410 q^{19} +5.07992 q^{20} +1.79702 q^{22} -3.90616 q^{23} +12.0770 q^{25} +10.6924 q^{26} +5.19812 q^{28} -1.80774 q^{29} -8.12966 q^{31} +6.12063 q^{32} +12.3545 q^{34} +17.4743 q^{35} -4.76890 q^{37} +0.205597 q^{38} +5.72341 q^{40} +1.81748 q^{41} +0.000598963 q^{43} -1.22928 q^{44} +7.01944 q^{46} +6.87638 q^{47} +10.8809 q^{49} -21.7026 q^{50} -7.31433 q^{52} -10.2253 q^{53} -4.13243 q^{55} +5.85658 q^{56} +3.24855 q^{58} -6.58478 q^{59} -1.00000 q^{61} +14.6092 q^{62} -1.10404 q^{64} -24.5883 q^{65} -9.06126 q^{67} -8.45132 q^{68} -31.4017 q^{70} -12.7148 q^{71} +12.9759 q^{73} +8.56980 q^{74} -0.140642 q^{76} -4.22858 q^{77} +8.35873 q^{79} -20.4449 q^{80} -3.26605 q^{82} +2.71294 q^{83} -28.4105 q^{85} -0.00107635 q^{86} -1.38500 q^{88} -16.6064 q^{89} -25.1605 q^{91} -4.80176 q^{92} -12.3570 q^{94} -0.472791 q^{95} +17.1502 q^{97} -19.5532 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) $ 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8	$ 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8} - 25 q^{11} + 4 q^{13} - 18 q^{14} + 21 q^{16} - 20 q^{17} + 14 q^{19} - 12 q^{20} + 5 q^{22} - 20 q^{23} + 13 q^{25} - 16 q^{26} - 14 q^{28} - 28 q^{29} - 12 q^{31} - 35 q^{32} + 6 q^{34} - 10 q^{35} - 8 q^{37} - 32 q^{38} + 24 q^{40} - 26 q^{41} + 18 q^{43} - 25 q^{44} + 4 q^{46} - 12 q^{47} + 23 q^{49} - 43 q^{50} + 22 q^{52} - 36 q^{53} + 4 q^{55} - 26 q^{56} - 20 q^{58} - 46 q^{59} - 25 q^{61} + 14 q^{62} - 13 q^{64} - 60 q^{65} - 20 q^{67} - 44 q^{68} - 20 q^{70} - 52 q^{71} + 6 q^{73} - 32 q^{74} - 4 q^{77} + 26 q^{79} - 52 q^{80} + 6 q^{82} - 38 q^{83} - 4 q^{85} - 34 q^{86} + 15 q^{88} - 82 q^{89} - 58 q^{91} - 36 q^{92} + 16 q^{94} - 30 q^{95} - 35 q^{98}+O(q^{100}) $ 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8 - 25 * q^11 + 4 * q^13 - 18 * q^14 + 21 * q^16 - 20 * q^17 + 14 * q^19 - 12 * q^20 + 5 * q^22 - 20 * q^23 + 13 * q^25 - 16 * q^26 - 14 * q^28 - 28 * q^29 - 12 * q^31 - 35 * q^32 + 6 * q^34 - 10 * q^35 - 8 * q^37 - 32 * q^38 + 24 * q^40 - 26 * q^41 + 18 * q^43 - 25 * q^44 + 4 * q^46 - 12 * q^47 + 23 * q^49 - 43 * q^50 + 22 * q^52 - 36 * q^53 + 4 * q^55 - 26 * q^56 - 20 * q^58 - 46 * q^59 - 25 * q^61 + 14 * q^62 - 13 * q^64 - 60 * q^65 - 20 * q^67 - 44 * q^68 - 20 * q^70 - 52 * q^71 + 6 * q^73 - 32 * q^74 - 4 * q^77 + 26 * q^79 - 52 * q^80 + 6 * q^82 - 38 * q^83 - 4 * q^85 - 34 * q^86 + 15 * q^88 - 82 * q^89 - 58 * q^91 - 36 * q^92 + 16 * q^94 - 30 * q^95 - 35 * 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 $

$2$	−1.79702	−1.27068	−0.635342	−	0.772230i	$-0.719141\pi$
$2$	−1.79702	−1.27068	−0.635342	+	0.772230i	$0.719141\pi$
$3$	0	0
$3$	0	0
$4$	1.22928	0.614640
$5$	4.13243	1.84808	0.924039	−	0.382298i	$-0.124867\pi$
$5$	4.13243	1.84808	0.924039	+	0.382298i	$0.124867\pi$
$6$	0	0
$7$	4.22858	1.59825	0.799127	−	0.601162i	$-0.205295\pi$
$7$	4.22858	1.59825	0.799127	+	0.601162i	$0.205295\pi$
$8$	1.38500	0.489671
$9$	0	0
$10$	−7.42606	−2.34833
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.95009	−1.65026	−0.825130	−	0.564944i	$-0.808898\pi$
$13$	−5.95009	−1.65026	−0.825130	+	0.564944i	$0.808898\pi$
$14$	−7.59885	−2.03088
$15$	0	0
$16$	−4.94743	−1.23686
$17$	−6.87502	−1.66744	−0.833718	−	0.552190i	$-0.813792\pi$
$17$	−6.87502	−1.66744	−0.833718	+	0.552190i	$0.813792\pi$
$18$	0	0
$19$	−0.114410	−0.0262474	−0.0131237	−	0.999914i	$-0.504178\pi$
$19$	−0.114410	−0.0262474	−0.0131237	+	0.999914i	$0.504178\pi$
$20$	5.07992	1.13590
$21$	0	0
$22$	1.79702	0.383126
$23$	−3.90616	−0.814490	−0.407245	−	0.913319i	$-0.633510\pi$
$23$	−3.90616	−0.814490	−0.407245	+	0.913319i	$0.633510\pi$
$24$	0	0
$25$	12.0770	2.41539
$26$	10.6924	2.09696
$27$	0	0
$28$	5.19812	0.982351
$29$	−1.80774	−0.335689	−0.167845	−	0.985813i	$-0.553681\pi$
$29$	−1.80774	−0.335689	−0.167845	+	0.985813i	$0.553681\pi$
$30$	0	0
$31$	−8.12966	−1.46013	−0.730065	−	0.683378i	$-0.760510\pi$
$31$	−8.12966	−1.46013	−0.730065	+	0.683378i	$0.760510\pi$
$32$	6.12063	1.08199
$33$	0	0
$34$	12.3545	2.11879
$35$	17.4743	2.95370
$36$	0	0
$37$	−4.76890	−0.784002	−0.392001	−	0.919965i	$-0.628217\pi$
$37$	−4.76890	−0.784002	−0.392001	+	0.919965i	$0.628217\pi$
$38$	0.205597	0.0333522
$39$	0	0
$40$	5.72341	0.904950
$41$	1.81748	0.283843	0.141921	−	0.989878i	$-0.454672\pi$
$41$	1.81748	0.283843	0.141921	+	0.989878i	$0.454672\pi$
$42$	0	0
$43$	0.000598963 0	9.13411e−5 0	4.56705e−5	−	1.00000i	$-0.499985\pi$
$43$	0.000598963 0	9.13411e−5 0	4.56705e−5		1.00000i	$0.499985\pi$
$44$	−1.22928	−0.185321
$45$	0	0
$46$	7.01944	1.03496
$47$	6.87638	1.00302	0.501512	−	0.865151i	$-0.332777\pi$
$47$	6.87638	1.00302	0.501512	+	0.865151i	$0.332777\pi$
$48$	0	0
$49$	10.8809	1.55442
$50$	−21.7026	−3.06921
$51$	0	0
$52$	−7.31433	−1.01432
$53$	−10.2253	−1.40455	−0.702275	−	0.711906i	$-0.747832\pi$
$53$	−10.2253	−1.40455	−0.702275	+	0.711906i	$0.747832\pi$
$54$	0	0
$55$	−4.13243	−0.557217
$56$	5.85658	0.782618
$57$	0	0
$58$	3.24855	0.426555
$59$	−6.58478	−0.857266	−0.428633	−	0.903479i	$-0.641005\pi$
$59$	−6.58478	−0.857266	−0.428633	+	0.903479i	$0.641005\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	14.6092	1.85537
$63$	0	0
$64$	−1.10404	−0.138005
$65$	−24.5883	−3.04981
$66$	0	0
$67$	−9.06126	−1.10701	−0.553505	−	0.832846i	$-0.686710\pi$
$67$	−9.06126	−1.10701	−0.553505	+	0.832846i	$0.686710\pi$
$68$	−8.45132	−1.02487
$69$	0	0
$70$	−31.4017	−3.75322
$71$	−12.7148	−1.50897	−0.754485	−	0.656318i	$-0.772113\pi$
$71$	−12.7148	−1.50897	−0.754485	+	0.656318i	$0.772113\pi$
$72$	0	0
$73$	12.9759	1.51871	0.759357	−	0.650674i	$-0.225514\pi$
$73$	12.9759	1.51871	0.759357	+	0.650674i	$0.225514\pi$
$74$	8.56980	0.996219
$75$	0	0
$76$	−0.140642	−0.0161327
$77$	−4.22858	−0.481892
$78$	0	0
$79$	8.35873	0.940431	0.470215	−	0.882552i	$-0.344176\pi$
$79$	8.35873	0.940431	0.470215	+	0.882552i	$0.344176\pi$
$80$	−20.4449	−2.28581
$81$	0	0
$82$	−3.26605	−0.360675
$83$	2.71294	0.297784	0.148892	−	0.988853i	$-0.452429\pi$
$83$	2.71294	0.297784	0.148892	+	0.988853i	$0.452429\pi$
$84$	0	0
$85$	−28.4105	−3.08155
$86$	−0.00107635	−0.000116066 0
$87$	0	0
$88$	−1.38500	−0.147641
$89$	−16.6064	−1.76028	−0.880138	−	0.474718i	$-0.842550\pi$
$89$	−16.6064	−1.76028	−0.880138	+	0.474718i	$0.842550\pi$
$90$	0	0
$91$	−25.1605	−2.63753
$92$	−4.80176	−0.500618
$93$	0	0
$94$	−12.3570	−1.27453
$95$	−0.472791	−0.0485073
$96$	0	0
$97$	17.1502	1.74133	0.870667	−	0.491873i	$-0.163687\pi$
$97$	17.1502	1.74133	0.870667	+	0.491873i	$0.163687\pi$
$98$	−19.5532	−1.97517
$99$	0	0
$100$	14.8460	1.48460
$101$	−15.5825	−1.55051	−0.775256	−	0.631647i	$-0.782379\pi$
$101$	−15.5825	−1.55051	−0.775256	+	0.631647i	$0.782379\pi$
$102$	0	0
$103$	−12.1112	−1.19335	−0.596677	−	0.802482i	$-0.703513\pi$
$103$	−12.1112	−1.19335	−0.596677	+	0.802482i	$0.703513\pi$
$104$	−8.24087	−0.808084
$105$	0	0
$106$	18.3750	1.78474
$107$	1.87666	0.181424	0.0907120	−	0.995877i	$-0.471086\pi$
$107$	1.87666	0.181424	0.0907120	+	0.995877i	$0.471086\pi$
$108$	0	0
$109$	12.9578	1.24113	0.620567	−	0.784153i	$-0.286902\pi$
$109$	12.9578	1.24113	0.620567	+	0.784153i	$0.286902\pi$
$110$	7.42606	0.708047
$111$	0	0
$112$	−20.9206	−1.97681
$113$	1.15087	0.108265	0.0541325	−	0.998534i	$-0.482761\pi$
$113$	1.15087	0.108265	0.0541325	+	0.998534i	$0.482761\pi$
$114$	0	0
$115$	−16.1419	−1.50524
$116$	−2.22222	−0.206328
$117$	0	0
$118$	11.8330	1.08931
$119$	−29.0716	−2.66499
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.79702	0.162695
$123$	0	0
$124$	−9.99363	−0.897455
$125$	29.2451	2.61576
$126$	0	0
$127$	−14.4814	−1.28502	−0.642509	−	0.766278i	$-0.722107\pi$
$127$	−14.4814	−1.28502	−0.642509	+	0.766278i	$0.722107\pi$
$128$	−10.2573	−0.906624
$129$	0	0
$130$	44.1857	3.87535
$131$	3.43385	0.300017	0.150008	−	0.988685i	$-0.452070\pi$
$131$	3.43385	0.300017	0.150008	+	0.988685i	$0.452070\pi$
$132$	0	0
$133$	−0.483792	−0.0419501
$134$	16.2833	1.40666
$135$	0	0
$136$	−9.52188	−0.816495
$137$	−14.0878	−1.20360	−0.601799	−	0.798648i	$-0.705549\pi$
$137$	−14.0878	−1.20360	−0.601799	+	0.798648i	$0.705549\pi$
$138$	0	0
$139$	10.7447	0.911358	0.455679	−	0.890144i	$-0.349397\pi$
$139$	10.7447	0.911358	0.455679	+	0.890144i	$0.349397\pi$
$140$	21.4808	1.81546
$141$	0	0
$142$	22.8487	1.91742
$143$	5.95009	0.497572
$144$	0	0
$145$	−7.47037	−0.620380
$146$	−23.3179	−1.92981
$147$	0	0
$148$	−5.86231	−0.481879
$149$	−21.7408	−1.78108	−0.890539	−	0.454906i	$-0.849673\pi$
$149$	−21.7408	−1.78108	−0.890539	+	0.454906i	$0.849673\pi$
$150$	0	0
$151$	5.01304	0.407955	0.203978	−	0.978976i	$-0.434613\pi$
$151$	5.01304	0.407955	0.203978	+	0.978976i	$0.434613\pi$
$152$	−0.158457	−0.0128526
$153$	0	0
$154$	7.59885	0.612333
$155$	−33.5952	−2.69844
$156$	0	0
$157$	8.11572	0.647705	0.323853	−	0.946108i	$-0.395022\pi$
$157$	8.11572	0.647705	0.323853	+	0.946108i	$0.395022\pi$
$158$	−15.0208	−1.19499
$159$	0	0
$160$	25.2931	1.99959
$161$	−16.5175	−1.30176
$162$	0	0
$163$	−19.6507	−1.53916	−0.769579	−	0.638551i	$-0.779534\pi$
$163$	−19.6507	−1.53916	−0.769579	+	0.638551i	$0.779534\pi$
$164$	2.23419	0.174461
$165$	0	0
$166$	−4.87521	−0.378390
$167$	−9.43965	−0.730462	−0.365231	−	0.930917i	$-0.619010\pi$
$167$	−9.43965	−0.730462	−0.365231	+	0.930917i	$0.619010\pi$
$168$	0	0
$169$	22.4036	1.72336
$170$	51.0543	3.91568
$171$	0	0
$172$	0.000736294 0	5.61419e−5 0
$173$	14.8804	1.13134	0.565668	−	0.824633i	$-0.308619\pi$
$173$	14.8804	1.13134	0.565668	+	0.824633i	$0.308619\pi$
$174$	0	0
$175$	51.0685	3.86041
$176$	4.94743	0.372927
$177$	0	0
$178$	29.8421	2.23676
$179$	3.80389	0.284316	0.142158	−	0.989844i	$-0.454596\pi$
$179$	3.80389	0.284316	0.142158	+	0.989844i	$0.454596\pi$
$180$	0	0
$181$	−0.450976	−0.0335208	−0.0167604	−	0.999860i	$-0.505335\pi$
$181$	−0.450976	−0.0335208	−0.0167604	+	0.999860i	$0.505335\pi$
$182$	45.2139	3.35147
$183$	0	0
$184$	−5.41002	−0.398832
$185$	−19.7071	−1.44890
$186$	0	0
$187$	6.87502	0.502751
$188$	8.45300	0.616498
$189$	0	0
$190$	0.849614	0.0616375
$191$	−10.9248	−0.790494	−0.395247	−	0.918575i	$-0.629341\pi$
$191$	−10.9248	−0.790494	−0.395247	+	0.918575i	$0.629341\pi$
$192$	0	0
$193$	6.24835	0.449766	0.224883	−	0.974386i	$-0.427800\pi$
$193$	6.24835	0.449766	0.224883	+	0.974386i	$0.427800\pi$
$194$	−30.8192	−2.21269
$195$	0	0
$196$	13.3757	0.955407
$197$	11.3428	0.808143	0.404072	−	0.914727i	$-0.367595\pi$
$197$	11.3428	0.808143	0.404072	+	0.914727i	$0.367595\pi$
$198$	0	0
$199$	9.57964	0.679083	0.339541	−	0.940591i	$-0.389728\pi$
$199$	9.57964	0.679083	0.339541	+	0.940591i	$0.389728\pi$
$200$	16.7266	1.18275
$201$	0	0
$202$	28.0020	1.97021
$203$	−7.64419	−0.536517
$204$	0	0
$205$	7.51060	0.524563
$206$	21.7641	1.51638
$207$	0	0
$208$	29.4377	2.04114
$209$	0.114410	0.00791390
$210$	0	0
$211$	21.4885	1.47933	0.739663	−	0.672977i	$-0.234985\pi$
$211$	21.4885	1.47933	0.739663	+	0.672977i	$0.234985\pi$
$212$	−12.5697	−0.863293
$213$	0	0
$214$	−3.37240	−0.230533
$215$	0.00247517	0.000168805 0
$216$	0	0
$217$	−34.3769	−2.33366
$218$	−23.2855	−1.57709
$219$	0	0
$220$	−5.07992	−0.342488
$221$	40.9070	2.75170
$222$	0	0
$223$	−2.87993	−0.192854	−0.0964271	−	0.995340i	$-0.530741\pi$
$223$	−2.87993	−0.192854	−0.0964271	+	0.995340i	$0.530741\pi$
$224$	25.8816	1.72929
$225$	0	0
$226$	−2.06814	−0.137571
$227$	−25.7966	−1.71218	−0.856089	−	0.516828i	$-0.827113\pi$
$227$	−25.7966	−1.71218	−0.856089	+	0.516828i	$0.827113\pi$
$228$	0	0
$229$	−13.5014	−0.892200	−0.446100	−	0.894983i	$-0.647187\pi$
$229$	−13.5014	−0.892200	−0.446100	+	0.894983i	$0.647187\pi$
$230$	29.0073	1.91269
$231$	0	0
$232$	−2.50372	−0.164377
$233$	6.77150	0.443616	0.221808	−	0.975090i	$-0.428804\pi$
$233$	6.77150	0.443616	0.221808	+	0.975090i	$0.428804\pi$
$234$	0	0
$235$	28.4162	1.85367
$236$	−8.09455	−0.526910
$237$	0	0
$238$	52.2422	3.38636
$239$	5.72817	0.370525	0.185262	−	0.982689i	$-0.440686\pi$
$239$	5.72817	0.370525	0.185262	+	0.982689i	$0.440686\pi$
$240$	0	0
$241$	−19.3348	−1.24546	−0.622731	−	0.782436i	$-0.713977\pi$
$241$	−19.3348	−1.24546	−0.622731	+	0.782436i	$0.713977\pi$
$242$	−1.79702	−0.115517
$243$	0	0
$244$	−1.22928	−0.0786966
$245$	44.9646	2.87268
$246$	0	0
$247$	0.680750	0.0433151
$248$	−11.2596	−0.714983
$249$	0	0
$250$	−52.5540	−3.32381
$251$	24.5158	1.54742	0.773711	−	0.633539i	$-0.218398\pi$
$251$	24.5158	1.54742	0.773711	+	0.633539i	$0.218398\pi$
$252$	0	0
$253$	3.90616	0.245578
$254$	26.0234	1.63285
$255$	0	0
$256$	20.6406	1.29004
$257$	6.58361	0.410674	0.205337	−	0.978691i	$-0.434171\pi$
$257$	6.58361	0.410674	0.205337	+	0.978691i	$0.434171\pi$
$258$	0	0
$259$	−20.1657	−1.25303
$260$	−30.2260	−1.87454
$261$	0	0
$262$	−6.17069	−0.381227
$263$	−1.91765	−0.118247	−0.0591236	−	0.998251i	$-0.518831\pi$
$263$	−1.91765	−0.118247	−0.0591236	+	0.998251i	$0.518831\pi$
$264$	0	0
$265$	−42.2552	−2.59572
$266$	0.869383	0.0533053
$267$	0	0
$268$	−11.1388	−0.680412
$269$	−3.53464	−0.215511	−0.107755	−	0.994177i	$-0.534366\pi$
$269$	−3.53464	−0.215511	−0.107755	+	0.994177i	$0.534366\pi$
$270$	0	0
$271$	−12.8059	−0.777901	−0.388951	−	0.921259i	$-0.627162\pi$
$271$	−12.8059	−0.777901	−0.388951	+	0.921259i	$0.627162\pi$
$272$	34.0137	2.06238
$273$	0	0
$274$	25.3160	1.52939
$275$	−12.0770	−0.728269
$276$	0	0
$277$	13.0569	0.784510	0.392255	−	0.919856i	$-0.371695\pi$
$277$	13.0569	0.784510	0.392255	+	0.919856i	$0.371695\pi$
$278$	−19.3085	−1.15805
$279$	0	0
$280$	24.2019	1.44634
$281$	−23.0464	−1.37483	−0.687415	−	0.726264i	$-0.741255\pi$
$281$	−23.0464	−1.37483	−0.687415	+	0.726264i	$0.741255\pi$
$282$	0	0
$283$	13.7644	0.818207	0.409103	−	0.912488i	$-0.365842\pi$
$283$	13.7644	0.818207	0.409103	+	0.912488i	$0.365842\pi$
$284$	−15.6301	−0.927473
$285$	0	0
$286$	−10.6924	−0.632257
$287$	7.68536	0.453653
$288$	0	0
$289$	30.2658	1.78034
$290$	13.4244	0.788308
$291$	0	0
$292$	15.9510	0.933463
$293$	10.7562	0.628384	0.314192	−	0.949359i	$-0.398266\pi$
$293$	10.7562	0.628384	0.314192	+	0.949359i	$0.398266\pi$
$294$	0	0
$295$	−27.2112	−1.58429
$296$	−6.60491	−0.383903
$297$	0	0
$298$	39.0687	2.26319
$299$	23.2420	1.34412
$300$	0	0
$301$	0.00253277	0.000145986 0
$302$	−9.00853	−0.518383
$303$	0	0
$304$	0.566035	0.0324643
$305$	−4.13243	−0.236622
$306$	0	0
$307$	−21.0831	−1.20328	−0.601639	−	0.798768i	$-0.705485\pi$
$307$	−21.0831	−1.20328	−0.601639	+	0.798768i	$0.705485\pi$
$308$	−5.19812	−0.296190
$309$	0	0
$310$	60.3713	3.42886
$311$	5.47069	0.310214	0.155107	−	0.987898i	$-0.450428\pi$
$311$	5.47069	0.310214	0.155107	+	0.987898i	$0.450428\pi$
$312$	0	0
$313$	−23.5030	−1.32847	−0.664234	−	0.747524i	$-0.731242\pi$
$313$	−23.5030	−1.32847	−0.664234	+	0.747524i	$0.731242\pi$
$314$	−14.5841	−0.823029
$315$	0	0
$316$	10.2752	0.578027
$317$	15.5042	0.870802	0.435401	−	0.900237i	$-0.356607\pi$
$317$	15.5042	0.870802	0.435401	+	0.900237i	$0.356607\pi$
$318$	0	0
$319$	1.80774	0.101214
$320$	−4.56237	−0.255044
$321$	0	0
$322$	29.6823	1.65413
$323$	0.786570	0.0437659
$324$	0	0
$325$	−71.8591	−3.98603
$326$	35.3126	1.95579
$327$	0	0
$328$	2.51721	0.138989
$329$	29.0773	1.60309
$330$	0	0
$331$	−13.4645	−0.740077	−0.370038	−	0.929016i	$-0.620655\pi$
$331$	−13.4645	−0.740077	−0.370038	+	0.929016i	$0.620655\pi$
$332$	3.33497	0.183030
$333$	0	0
$334$	16.9632	0.928187
$335$	−37.4450	−2.04584
$336$	0	0
$337$	2.25582	0.122882	0.0614411	−	0.998111i	$-0.480430\pi$
$337$	2.25582	0.122882	0.0614411	+	0.998111i	$0.480430\pi$
$338$	−40.2597	−2.18984
$339$	0	0
$340$	−34.9245	−1.89405
$341$	8.12966	0.440246
$342$	0	0
$343$	16.4108	0.886098
$344$	0.000829563 0	4.47270e−5 0
$345$	0	0
$346$	−26.7404	−1.43757
$347$	−6.94172	−0.372651	−0.186325	−	0.982488i	$-0.559658\pi$
$347$	−6.94172	−0.372651	−0.186325	+	0.982488i	$0.559658\pi$
$348$	0	0
$349$	−19.2568	−1.03079	−0.515396	−	0.856952i	$-0.672355\pi$
$349$	−19.2568	−1.03079	−0.515396	+	0.856952i	$0.672355\pi$
$350$	−91.7711	−4.90537
$351$	0	0
$352$	−6.12063	−0.326231
$353$	−17.2457	−0.917895	−0.458947	−	0.888463i	$-0.651773\pi$
$353$	−17.2457	−0.917895	−0.458947	+	0.888463i	$0.651773\pi$
$354$	0	0
$355$	−52.5430	−2.78869
$356$	−20.4139	−1.08194
$357$	0	0
$358$	−6.83566	−0.361276
$359$	−7.01416	−0.370193	−0.185097	−	0.982720i	$-0.559260\pi$
$359$	−7.01416	−0.370193	−0.185097	+	0.982720i	$0.559260\pi$
$360$	0	0
$361$	−18.9869	−0.999311
$362$	0.810413	0.0425944
$363$	0	0
$364$	−30.9293	−1.62113
$365$	53.6220	2.80670
$366$	0	0
$367$	−0.793430	−0.0414167	−0.0207084	−	0.999786i	$-0.506592\pi$
$367$	−0.793430	−0.0414167	−0.0207084	+	0.999786i	$0.506592\pi$
$368$	19.3254	1.00741
$369$	0	0
$370$	35.4141	1.84109
$371$	−43.2384	−2.24483
$372$	0	0
$373$	21.5789	1.11731	0.558656	−	0.829399i	$-0.311317\pi$
$373$	21.5789	1.11731	0.558656	+	0.829399i	$0.311317\pi$
$374$	−12.3545	−0.638838
$375$	0	0
$376$	9.52377	0.491151
$377$	10.7562	0.553975
$378$	0	0
$379$	−0.364682	−0.0187325	−0.00936623	−	0.999956i	$-0.502981\pi$
$379$	−0.364682	−0.0187325	−0.00936623	+	0.999956i	$0.502981\pi$
$380$	−0.581193	−0.0298145
$381$	0	0
$382$	19.6322	1.00447
$383$	−5.90918	−0.301945	−0.150973	−	0.988538i	$-0.548241\pi$
$383$	−5.90918	−0.301945	−0.150973	+	0.988538i	$0.548241\pi$
$384$	0	0
$385$	−17.4743	−0.890574
$386$	−11.2284	−0.571511
$387$	0	0
$388$	21.0824	1.07029
$389$	0.0700961	0.00355401	0.00177701	−	0.999998i	$-0.499434\pi$
$389$	0.0700961	0.00355401	0.00177701	+	0.999998i	$0.499434\pi$
$390$	0	0
$391$	26.8549	1.35811
$392$	15.0700	0.761152
$393$	0	0
$394$	−20.3833	−1.02690
$395$	34.5419	1.73799
$396$	0	0
$397$	9.10974	0.457205	0.228602	−	0.973520i	$-0.426584\pi$
$397$	9.10974	0.457205	0.228602	+	0.973520i	$0.426584\pi$
$398$	−17.2148	−0.862900
$399$	0	0
$400$	−59.7500	−2.98750
$401$	−30.5240	−1.52429	−0.762147	−	0.647404i	$-0.775855\pi$
$401$	−30.5240	−1.52429	−0.762147	+	0.647404i	$0.775855\pi$
$402$	0	0
$403$	48.3722	2.40959
$404$	−19.1552	−0.953007
$405$	0	0
$406$	13.7368	0.681744
$407$	4.76890	0.236385
$408$	0	0
$409$	20.0857	0.993173	0.496587	−	0.867987i	$-0.334586\pi$
$409$	20.0857	0.993173	0.496587	+	0.867987i	$0.334586\pi$
$410$	−13.4967	−0.666555
$411$	0	0
$412$	−14.8881	−0.733483
$413$	−27.8443	−1.37013
$414$	0	0
$415$	11.2110	0.550329
$416$	−36.4184	−1.78556
$417$	0	0
$418$	−0.205597	−0.0100561
$419$	20.3405	0.993700	0.496850	−	0.867836i	$-0.334490\pi$
$419$	20.3405	0.993700	0.496850	+	0.867836i	$0.334490\pi$
$420$	0	0
$421$	13.0548	0.636254	0.318127	−	0.948048i	$-0.396946\pi$
$421$	13.0548	0.636254	0.318127	+	0.948048i	$0.396946\pi$
$422$	−38.6152	−1.87976
$423$	0	0
$424$	−14.1620	−0.687767
$425$	−83.0294	−4.02752
$426$	0	0
$427$	−4.22858	−0.204635
$428$	2.30695	0.111510
$429$	0	0
$430$	−0.00444794	−0.000214499 0
$431$	19.6386	0.945959	0.472979	−	0.881073i	$-0.343178\pi$
$431$	19.6386	0.945959	0.472979	+	0.881073i	$0.343178\pi$
$432$	0	0
$433$	0.252308	0.0121251	0.00606256	−	0.999982i	$-0.498070\pi$
$433$	0.252308	0.0121251	0.00606256	+	0.999982i	$0.498070\pi$
$434$	61.7761	2.96535
$435$	0	0
$436$	15.9288	0.762851
$437$	0.446903	0.0213783
$438$	0	0
$439$	−21.0439	−1.00437	−0.502186	−	0.864760i	$-0.667471\pi$
$439$	−21.0439	−1.00437	−0.502186	+	0.864760i	$0.667471\pi$
$440$	−5.72341	−0.272853
$441$	0	0
$442$	−73.5107	−3.49655
$443$	34.5338	1.64075	0.820376	−	0.571824i	$-0.193764\pi$
$443$	34.5338	1.64075	0.820376	+	0.571824i	$0.193764\pi$
$444$	0	0
$445$	−68.6248	−3.25313
$446$	5.17528	0.245057
$447$	0	0
$448$	−4.66853	−0.220567
$449$	−37.4052	−1.76526	−0.882630	−	0.470069i	$-0.844229\pi$
$449$	−37.4052	−1.76526	−0.882630	+	0.470069i	$0.844229\pi$
$450$	0	0
$451$	−1.81748	−0.0855818
$452$	1.41475	0.0665441
$453$	0	0
$454$	46.3570	2.17564
$455$	−103.974	−4.87437
$456$	0	0
$457$	−29.9860	−1.40269	−0.701344	−	0.712823i	$-0.747416\pi$
$457$	−29.9860	−1.40269	−0.701344	+	0.712823i	$0.747416\pi$
$458$	24.2623	1.13371
$459$	0	0
$460$	−19.8429	−0.925182
$461$	9.54911	0.444746	0.222373	−	0.974962i	$-0.428620\pi$
$461$	9.54911	0.444746	0.222373	+	0.974962i	$0.428620\pi$
$462$	0	0
$463$	20.3600	0.946209	0.473104	−	0.881006i	$-0.343133\pi$
$463$	20.3600	0.946209	0.473104	+	0.881006i	$0.343133\pi$
$464$	8.94368	0.415200
$465$	0	0
$466$	−12.1685	−0.563696
$467$	−15.9468	−0.737930	−0.368965	−	0.929443i	$-0.620288\pi$
$467$	−15.9468	−0.737930	−0.368965	+	0.929443i	$0.620288\pi$
$468$	0	0
$469$	−38.3163	−1.76928
$470$	−51.0644	−2.35543
$471$	0	0
$472$	−9.11991	−0.419778
$473$	−0.000598963 0	−2.75404e−5 0
$474$	0	0
$475$	−1.38172	−0.0633979
$476$	−35.7371	−1.63801
$477$	0	0
$478$	−10.2936	−0.470820
$479$	40.2343	1.83835	0.919176	−	0.393847i	$-0.128856\pi$
$479$	40.2343	1.83835	0.919176	+	0.393847i	$0.128856\pi$
$480$	0	0
$481$	28.3754	1.29381
$482$	34.7450	1.58259
$483$	0	0
$484$	1.22928	0.0558764
$485$	70.8718	3.21812
$486$	0	0
$487$	0.969646	0.0439388	0.0219694	−	0.999759i	$-0.493006\pi$
$487$	0.969646	0.0439388	0.0219694	+	0.999759i	$0.493006\pi$
$488$	−1.38500	−0.0626959
$489$	0	0
$490$	−80.8023	−3.65028
$491$	5.83295	0.263238	0.131619	−	0.991300i	$-0.457983\pi$
$491$	5.83295	0.263238	0.131619	+	0.991300i	$0.457983\pi$
$492$	0	0
$493$	12.4283	0.559741
$494$	−1.22332	−0.0550398
$495$	0	0
$496$	40.2209	1.80597
$497$	−53.7656	−2.41172
$498$	0	0
$499$	19.7460	0.883954	0.441977	−	0.897026i	$-0.354277\pi$
$499$	19.7460	0.883954	0.441977	+	0.897026i	$0.354277\pi$
$500$	35.9504	1.60775
$501$	0	0
$502$	−44.0553	−1.96629
$503$	14.4566	0.644588	0.322294	−	0.946640i	$-0.395546\pi$
$503$	14.4566	0.644588	0.322294	+	0.946640i	$0.395546\pi$
$504$	0	0
$505$	−64.3934	−2.86547
$506$	−7.01944	−0.312052
$507$	0	0
$508$	−17.8017	−0.789824
$509$	−3.47334	−0.153953	−0.0769766	−	0.997033i	$-0.524527\pi$
$509$	−3.47334	−0.153953	−0.0769766	+	0.997033i	$0.524527\pi$
$510$	0	0
$511$	54.8697	2.42729
$512$	−16.5771	−0.732609
$513$	0	0
$514$	−11.8309	−0.521838
$515$	−50.0487	−2.20541
$516$	0	0
$517$	−6.87638	−0.302423
$518$	36.2381	1.59221
$519$	0	0
$520$	−34.0548	−1.49340
$521$	−14.4838	−0.634546	−0.317273	−	0.948334i	$-0.602767\pi$
$521$	−14.4838	−0.634546	−0.317273	+	0.948334i	$0.602767\pi$
$522$	0	0
$523$	−27.4917	−1.20213	−0.601065	−	0.799200i	$-0.705257\pi$
$523$	−27.4917	−1.20213	−0.601065	+	0.799200i	$0.705257\pi$
$524$	4.22116	0.184402
$525$	0	0
$526$	3.44605	0.150255
$527$	55.8915	2.43467
$528$	0	0
$529$	−7.74194	−0.336606
$530$	75.9335	3.29834
$531$	0	0
$532$	−0.594716	−0.0257842
$533$	−10.8142	−0.468414
$534$	0	0
$535$	7.75518	0.335286
$536$	−12.5498	−0.542070
$537$	0	0
$538$	6.35182	0.273846
$539$	−10.8809	−0.468674
$540$	0	0
$541$	6.05357	0.260263	0.130132	−	0.991497i	$-0.458460\pi$
$541$	6.05357	0.260263	0.130132	+	0.991497i	$0.458460\pi$
$542$	23.0124	0.988467
$543$	0	0
$544$	−42.0795	−1.80414
$545$	53.5473	2.29371
$546$	0	0
$547$	32.9576	1.40916	0.704582	−	0.709622i	$-0.251134\pi$
$547$	32.9576	1.40916	0.704582	+	0.709622i	$0.251134\pi$
$548$	−17.3178	−0.739780
$549$	0	0
$550$	21.7026	0.925400
$551$	0.206824	0.00881098
$552$	0	0
$553$	35.3456	1.50305
$554$	−23.4634	−0.996866
$555$	0	0
$556$	13.2083	0.560157
$557$	−12.4314	−0.526735	−0.263367	−	0.964696i	$-0.584833\pi$
$557$	−12.4314	−0.526735	−0.263367	+	0.964696i	$0.584833\pi$
$558$	0	0
$559$	−0.00356389	−0.000150736 0
$560$	−86.4530	−3.65331
$561$	0	0
$562$	41.4148	1.74698
$563$	−5.86096	−0.247010	−0.123505	−	0.992344i	$-0.539414\pi$
$563$	−5.86096	−0.247010	−0.123505	+	0.992344i	$0.539414\pi$
$564$	0	0
$565$	4.75590	0.200082
$566$	−24.7348	−1.03968
$567$	0	0
$568$	−17.6100	−0.738898
$569$	35.8528	1.50303	0.751514	−	0.659718i	$-0.229324\pi$
$569$	35.8528	1.50303	0.751514	+	0.659718i	$0.229324\pi$
$570$	0	0
$571$	38.0482	1.59227	0.796134	−	0.605120i	$-0.206875\pi$
$571$	38.0482	1.59227	0.796134	+	0.605120i	$0.206875\pi$
$572$	7.31433	0.305828
$573$	0	0
$574$	−13.8107	−0.576450
$575$	−47.1745	−1.96731
$576$	0	0
$577$	14.4611	0.602024	0.301012	−	0.953620i	$-0.402676\pi$
$577$	14.4611	0.602024	0.301012	+	0.953620i	$0.402676\pi$
$578$	−54.3883	−2.26226
$579$	0	0
$580$	−9.18318	−0.381311
$581$	11.4719	0.475935
$582$	0	0
$583$	10.2253	0.423488
$584$	17.9716	0.743670
$585$	0	0
$586$	−19.3291	−0.798478
$587$	12.5605	0.518429	0.259214	−	0.965820i	$-0.416536\pi$
$587$	12.5605	0.518429	0.259214	+	0.965820i	$0.416536\pi$
$588$	0	0
$589$	0.930114	0.0383247
$590$	48.8990	2.01314
$591$	0	0
$592$	23.5938	0.969699
$593$	−21.6528	−0.889175	−0.444588	−	0.895735i	$-0.646650\pi$
$593$	−21.6528	−0.889175	−0.444588	+	0.895735i	$0.646650\pi$
$594$	0	0
$595$	−120.136	−4.92510
$596$	−26.7256	−1.09472
$597$	0	0
$598$	−41.7663	−1.70795
$599$	8.14091	0.332629	0.166314	−	0.986073i	$-0.446813\pi$
$599$	8.14091	0.332629	0.166314	+	0.986073i	$0.446813\pi$
$600$	0	0
$601$	−0.887153	−0.0361877	−0.0180939	−	0.999836i	$-0.505760\pi$
$601$	−0.887153	−0.0361877	−0.0180939	+	0.999836i	$0.505760\pi$
$602$	−0.00455143	−0.000185502 0
$603$	0	0
$604$	6.16243	0.250746
$605$	4.13243	0.168007
$606$	0	0
$607$	3.16751	0.128565	0.0642827	−	0.997932i	$-0.479524\pi$
$607$	3.16751	0.128565	0.0642827	+	0.997932i	$0.479524\pi$
$608$	−0.700261	−0.0283993
$609$	0	0
$610$	7.42606	0.300672
$611$	−40.9151	−1.65525
$612$	0	0
$613$	25.3632	1.02441	0.512205	−	0.858863i	$-0.328829\pi$
$613$	25.3632	1.02441	0.512205	+	0.858863i	$0.328829\pi$
$614$	37.8868	1.52899
$615$	0	0
$616$	−5.85658	−0.235968
$617$	8.47573	0.341220	0.170610	−	0.985339i	$-0.445426\pi$
$617$	8.47573	0.341220	0.170610	+	0.985339i	$0.445426\pi$
$618$	0	0
$619$	−16.5870	−0.666687	−0.333344	−	0.942805i	$-0.608177\pi$
$619$	−16.5870	−0.666687	−0.333344	+	0.942805i	$0.608177\pi$
$620$	−41.2980	−1.65857
$621$	0	0
$622$	−9.83093	−0.394184
$623$	−70.2216	−2.81337
$624$	0	0
$625$	60.4684	2.41874
$626$	42.2354	1.68807
$627$	0	0
$628$	9.97650	0.398106
$629$	32.7862	1.30727
$630$	0	0
$631$	14.9136	0.593700	0.296850	−	0.954924i	$-0.404064\pi$
$631$	14.9136	0.593700	0.296850	+	0.954924i	$0.404064\pi$
$632$	11.5768	0.460502
$633$	0	0
$634$	−27.8613	−1.10652
$635$	−59.8434	−2.37481
$636$	0	0
$637$	−64.7425	−2.56519
$638$	−3.24855	−0.128611
$639$	0	0
$640$	−42.3875	−1.67551
$641$	0.0827782	0.00326954	0.00163477	−	0.999999i	$-0.499480\pi$
$641$	0.0827782	0.00326954	0.00163477	+	0.999999i	$0.499480\pi$
$642$	0	0
$643$	19.8688	0.783547	0.391774	−	0.920062i	$-0.371862\pi$
$643$	19.8688	0.783547	0.391774	+	0.920062i	$0.371862\pi$
$644$	−20.3046	−0.800115
$645$	0	0
$646$	−1.41348	−0.0556127
$647$	−13.9341	−0.547807	−0.273904	−	0.961757i	$-0.588315\pi$
$647$	−13.9341	−0.547807	−0.273904	+	0.961757i	$0.588315\pi$
$648$	0	0
$649$	6.58478	0.258475
$650$	129.132	5.06498
$651$	0	0
$652$	−24.1562	−0.946029
$653$	−17.9089	−0.700830	−0.350415	−	0.936594i	$-0.613959\pi$
$653$	−17.9089	−0.700830	−0.350415	+	0.936594i	$0.613959\pi$
$654$	0	0
$655$	14.1901	0.554454
$656$	−8.99185	−0.351073
$657$	0	0
$658$	−52.2526	−2.03702
$659$	−7.08434	−0.275967	−0.137983	−	0.990435i	$-0.544062\pi$
$659$	−7.08434	−0.275967	−0.137983	+	0.990435i	$0.544062\pi$
$660$	0	0
$661$	42.3330	1.64656	0.823282	−	0.567633i	$-0.192141\pi$
$661$	42.3330	1.64656	0.823282	+	0.567633i	$0.192141\pi$
$662$	24.1960	0.940405
$663$	0	0
$664$	3.75742	0.145816
$665$	−1.99924	−0.0775270
$666$	0	0
$667$	7.06133	0.273416
$668$	−11.6040	−0.448971
$669$	0	0
$670$	67.2894	2.59962
$671$	1.00000	0.0386046
$672$	0	0
$673$	46.2241	1.78181	0.890904	−	0.454192i	$-0.150072\pi$
$673$	46.2241	1.78181	0.890904	+	0.454192i	$0.150072\pi$
$674$	−4.05375	−0.156145
$675$	0	0
$676$	27.5403	1.05924
$677$	−8.24391	−0.316839	−0.158420	−	0.987372i	$-0.550640\pi$
$677$	−8.24391	−0.316839	−0.158420	+	0.987372i	$0.550640\pi$
$678$	0	0
$679$	72.5208	2.78309
$680$	−39.3485	−1.50895
$681$	0	0
$682$	−14.6092	−0.559414
$683$	−38.3645	−1.46798	−0.733989	−	0.679162i	$-0.762343\pi$
$683$	−38.3645	−1.46798	−0.733989	+	0.679162i	$0.762343\pi$
$684$	0	0
$685$	−58.2166	−2.22434
$686$	−29.4905	−1.12595
$687$	0	0
$688$	−0.00296333	−0.000112976 0
$689$	60.8413	2.31787
$690$	0	0
$691$	−41.4049	−1.57512	−0.787558	−	0.616240i	$-0.788655\pi$
$691$	−41.4049	−1.57512	−0.787558	+	0.616240i	$0.788655\pi$
$692$	18.2922	0.695364
$693$	0	0
$694$	12.4744	0.473522
$695$	44.4019	1.68426
$696$	0	0
$697$	−12.4952	−0.473289
$698$	34.6049	1.30981
$699$	0	0
$700$	62.7775	2.37277
$701$	27.8921	1.05347	0.526735	−	0.850029i	$-0.323416\pi$
$701$	27.8921	1.05347	0.526735	+	0.850029i	$0.323416\pi$
$702$	0	0
$703$	0.545609	0.0205780
$704$	1.10404	0.0416101
$705$	0	0
$706$	30.9908	1.16636
$707$	−65.8917	−2.47811
$708$	0	0
$709$	−4.19632	−0.157596	−0.0787980	−	0.996891i	$-0.525108\pi$
$709$	−4.19632	−0.157596	−0.0787980	+	0.996891i	$0.525108\pi$
$710$	94.4208	3.54355
$711$	0	0
$712$	−22.9999	−0.861956
$713$	31.7557	1.18926
$714$	0	0
$715$	24.5883	0.919552
$716$	4.67605	0.174752
$717$	0	0
$718$	12.6046	0.470399
$719$	48.7664	1.81868	0.909339	−	0.416055i	$-0.136588\pi$
$719$	48.7664	1.81868	0.909339	+	0.416055i	$0.136588\pi$
$720$	0	0
$721$	−51.2133	−1.90728
$722$	34.1199	1.26981
$723$	0	0
$724$	−0.554376	−0.0206032
$725$	−21.8321	−0.810822
$726$	0	0
$727$	−32.3367	−1.19930	−0.599652	−	0.800261i	$-0.704694\pi$
$727$	−32.3367	−1.19930	−0.599652	+	0.800261i	$0.704694\pi$
$728$	−34.8472	−1.29152
$729$	0	0
$730$	−96.3598	−3.56643
$731$	−0.00411788	−0.000152305 0
$732$	0	0
$733$	−33.1274	−1.22359	−0.611795	−	0.791017i	$-0.709552\pi$
$733$	−33.1274	−1.22359	−0.611795	+	0.791017i	$0.709552\pi$
$734$	1.42581	0.0526276
$735$	0	0
$736$	−23.9082	−0.881266
$737$	9.06126	0.333776
$738$	0	0
$739$	12.1344	0.446372	0.223186	−	0.974776i	$-0.428354\pi$
$739$	12.1344	0.446372	0.223186	+	0.974776i	$0.428354\pi$
$740$	−24.2256	−0.890550
$741$	0	0
$742$	77.7003	2.85247
$743$	43.8579	1.60899	0.804496	−	0.593958i	$-0.202436\pi$
$743$	43.8579	1.60899	0.804496	+	0.593958i	$0.202436\pi$
$744$	0	0
$745$	−89.8425	−3.29157
$746$	−38.7777	−1.41975
$747$	0	0
$748$	8.45132	0.309011
$749$	7.93563	0.289962
$750$	0	0
$751$	−17.4328	−0.636130	−0.318065	−	0.948069i	$-0.603033\pi$
$751$	−17.4328	−0.636130	−0.318065	+	0.948069i	$0.603033\pi$
$752$	−34.0204	−1.24060
$753$	0	0
$754$	−19.3292	−0.703927
$755$	20.7160	0.753934
$756$	0	0
$757$	−10.6697	−0.387796	−0.193898	−	0.981022i	$-0.562113\pi$
$757$	−10.6697	−0.387796	−0.193898	+	0.981022i	$0.562113\pi$
$758$	0.655341	0.0238031
$759$	0	0
$760$	−0.654814	−0.0237526
$761$	−25.4388	−0.922154	−0.461077	−	0.887360i	$-0.652537\pi$
$761$	−25.4388	−0.922154	−0.461077	+	0.887360i	$0.652537\pi$
$762$	0	0
$763$	54.7932	1.98365
$764$	−13.4297	−0.485870
$765$	0	0
$766$	10.6189	0.383677
$767$	39.1801	1.41471
$768$	0	0
$769$	−33.2863	−1.20033	−0.600167	−	0.799875i	$-0.704899\pi$
$769$	−33.2863	−1.20033	−0.600167	+	0.799875i	$0.704899\pi$
$770$	31.4017	1.13164
$771$	0	0
$772$	7.68097	0.276444
$773$	−25.8479	−0.929686	−0.464843	−	0.885393i	$-0.653889\pi$
$773$	−25.8479	−0.929686	−0.464843	+	0.885393i	$0.653889\pi$
$774$	0	0
$775$	−98.1817	−3.52679
$776$	23.7529	0.852681
$777$	0	0
$778$	−0.125964	−0.00451603
$779$	−0.207938	−0.00745014
$780$	0	0
$781$	12.7148	0.454971
$782$	−48.2588	−1.72573
$783$	0	0
$784$	−53.8326	−1.92259
$785$	33.5376	1.19701
$786$	0	0
$787$	33.1813	1.18279	0.591393	−	0.806384i	$-0.298578\pi$
$787$	33.1813	1.18279	0.591393	+	0.806384i	$0.298578\pi$
$788$	13.9435	0.496717
$789$	0	0
$790$	−62.0724	−2.20844
$791$	4.86656	0.173035
$792$	0	0
$793$	5.95009	0.211294
$794$	−16.3704	−0.580963
$795$	0	0
$796$	11.7761	0.417392
$797$	35.7596	1.26667	0.633335	−	0.773878i	$-0.281686\pi$
$797$	35.7596	1.26667	0.633335	+	0.773878i	$0.281686\pi$
$798$	0	0
$799$	−47.2752	−1.67248
$800$	73.9187	2.61342
$801$	0	0
$802$	54.8522	1.93690
$803$	−12.9759	−0.457909
$804$	0	0
$805$	−68.2574	−2.40576
$806$	−86.9259	−3.06183
$807$	0	0
$808$	−21.5817	−0.759241
$809$	20.5071	0.720993	0.360496	−	0.932761i	$-0.382607\pi$
$809$	20.5071	0.720993	0.360496	+	0.932761i	$0.382607\pi$
$810$	0	0
$811$	−38.3261	−1.34581	−0.672906	−	0.739728i	$-0.734954\pi$
$811$	−38.3261	−1.34581	−0.672906	+	0.739728i	$0.734954\pi$
$812$	−9.39686	−0.329765
$813$	0	0
$814$	−8.56980	−0.300371
$815$	−81.2050	−2.84449
$816$	0	0
$817$	−6.85273e−5 0	−2.39747e−6 0
$818$	−36.0944	−1.26201
$819$	0	0
$820$	9.23264	0.322418
$821$	−11.9912	−0.418496	−0.209248	−	0.977863i	$-0.567102\pi$
$821$	−11.9912	−0.418496	−0.209248	+	0.977863i	$0.567102\pi$
$822$	0	0
$823$	−22.7456	−0.792861	−0.396430	−	0.918065i	$-0.629751\pi$
$823$	−22.7456	−0.792861	−0.396430	+	0.918065i	$0.629751\pi$
$824$	−16.7740	−0.584350
$825$	0	0
$826$	50.0368	1.74100
$827$	−20.5343	−0.714047	−0.357024	−	0.934095i	$-0.616208\pi$
$827$	−20.5343	−0.714047	−0.357024	+	0.934095i	$0.616208\pi$
$828$	0	0
$829$	9.48682	0.329491	0.164746	−	0.986336i	$-0.447320\pi$
$829$	9.48682	0.329491	0.164746	+	0.986336i	$0.447320\pi$
$830$	−20.1465	−0.699294
$831$	0	0
$832$	6.56915	0.227744
$833$	−74.8065	−2.59189
$834$	0	0
$835$	−39.0087	−1.34995
$836$	0.140642	0.00486420
$837$	0	0
$838$	−36.5523	−1.26268
$839$	−40.8237	−1.40939	−0.704695	−	0.709510i	$-0.748916\pi$
$839$	−40.8237	−1.40939	−0.704695	+	0.709510i	$0.748916\pi$
$840$	0	0
$841$	−25.7321	−0.887313
$842$	−23.4598	−0.808479
$843$	0	0
$844$	26.4153	0.909253
$845$	92.5814	3.18490
$846$	0	0
$847$	4.22858	0.145296
$848$	50.5888	1.73723
$849$	0	0
$850$	149.205	5.11770
$851$	18.6281	0.638561
$852$	0	0
$853$	14.6419	0.501329	0.250665	−	0.968074i	$-0.419351\pi$
$853$	14.6419	0.501329	0.250665	+	0.968074i	$0.419351\pi$
$854$	7.59885	0.260027
$855$	0	0
$856$	2.59918	0.0888380
$857$	3.14537	0.107444	0.0537219	−	0.998556i	$-0.482892\pi$
$857$	3.14537	0.107444	0.0537219	+	0.998556i	$0.482892\pi$
$858$	0	0
$859$	5.53158	0.188735	0.0943675	−	0.995537i	$-0.469917\pi$
$859$	5.53158	0.188735	0.0943675	+	0.995537i	$0.469917\pi$
$860$	0.00304268	0.000103755 0
$861$	0	0
$862$	−35.2910	−1.20202
$863$	21.2821	0.724452	0.362226	−	0.932090i	$-0.382017\pi$
$863$	21.2821	0.724452	0.362226	+	0.932090i	$0.382017\pi$
$864$	0	0
$865$	61.4922	2.09080
$866$	−0.453402	−0.0154072
$867$	0	0
$868$	−42.2589	−1.43436
$869$	−8.35873	−0.283551
$870$	0	0
$871$	53.9153	1.82685
$872$	17.9466	0.607747
$873$	0	0
$874$	−0.803093	−0.0271650
$875$	123.665	4.18065
$876$	0	0
$877$	−53.0646	−1.79186	−0.895931	−	0.444192i	$-0.853491\pi$
$877$	−53.0646	−1.79186	−0.895931	+	0.444192i	$0.853491\pi$
$878$	37.8163	1.27624
$879$	0	0
$880$	20.4449	0.689198
$881$	40.9268	1.37886	0.689430	−	0.724353i	$-0.257861\pi$
$881$	40.9268	1.37886	0.689430	+	0.724353i	$0.257861\pi$
$882$	0	0
$883$	18.5469	0.624153	0.312076	−	0.950057i	$-0.398975\pi$
$883$	18.5469	0.624153	0.312076	+	0.950057i	$0.398975\pi$
$884$	50.2862	1.69131
$885$	0	0
$886$	−62.0580	−2.08488
$887$	3.31658	0.111360	0.0556800	−	0.998449i	$-0.482267\pi$
$887$	3.31658	0.111360	0.0556800	+	0.998449i	$0.482267\pi$
$888$	0	0
$889$	−61.2359	−2.05379
$890$	123.320	4.13370
$891$	0	0
$892$	−3.54024	−0.118536
$893$	−0.786726	−0.0263268
$894$	0	0
$895$	15.7193	0.525438
$896$	−43.3738	−1.44902
$897$	0	0
$898$	67.2178	2.24309
$899$	14.6963	0.490150
$900$	0	0
$901$	70.2989	2.34200
$902$	3.26605	0.108747
$903$	0	0
$904$	1.59396	0.0530142
$905$	−1.86363	−0.0619491
$906$	0	0
$907$	47.6886	1.58347	0.791737	−	0.610862i	$-0.209177\pi$
$907$	47.6886	1.58347	0.791737	+	0.610862i	$0.209177\pi$
$908$	−31.7112	−1.05237
$909$	0	0
$910$	186.843	6.19379
$911$	10.3974	0.344480	0.172240	−	0.985055i	$-0.444900\pi$
$911$	10.3974	0.344480	0.172240	+	0.985055i	$0.444900\pi$
$912$	0	0
$913$	−2.71294	−0.0897853
$914$	53.8855	1.78237
$915$	0	0
$916$	−16.5971	−0.548382
$917$	14.5203	0.479503
$918$	0	0
$919$	−59.9521	−1.97764	−0.988818	−	0.149128i	$-0.952353\pi$
$919$	−59.9521	−1.97764	−0.988818	+	0.149128i	$0.952353\pi$
$920$	−22.3565	−0.737073
$921$	0	0
$922$	−17.1599	−0.565133
$923$	75.6543	2.49019
$924$	0	0
$925$	−57.5938	−1.89367
$926$	−36.5873	−1.20233
$927$	0	0
$928$	−11.0645	−0.363211
$929$	−3.13584	−0.102884	−0.0514418	−	0.998676i	$-0.516382\pi$
$929$	−3.13584	−0.102884	−0.0514418	+	0.998676i	$0.516382\pi$
$930$	0	0
$931$	−1.24488	−0.0407994
$932$	8.32407	0.272664
$933$	0	0
$934$	28.6567	0.937676
$935$	28.4105	0.929123
$936$	0	0
$937$	−55.4987	−1.81306	−0.906531	−	0.422139i	$-0.861279\pi$
$937$	−55.4987	−1.81306	−0.906531	+	0.422139i	$0.861279\pi$
$938$	68.8551	2.24820
$939$	0	0
$940$	34.9314	1.13934
$941$	0.0229290	0.000747463 0	0.000373731	−	1.00000i	$-0.499881\pi$
$941$	0.0229290	0.000747463 0	0.000373731		1.00000i	$0.499881\pi$
$942$	0	0
$943$	−7.09936	−0.231187
$944$	32.5778	1.06032
$945$	0	0
$946$	0.00107635	3.49951e−5 0
$947$	−11.9133	−0.387129	−0.193565	−	0.981088i	$-0.562005\pi$
$947$	−11.9133	−0.387129	−0.193565	+	0.981088i	$0.562005\pi$
$948$	0	0
$949$	−77.2078	−2.50627
$950$	2.48299	0.0805587
$951$	0	0
$952$	−40.2641	−1.30497
$953$	38.9528	1.26180	0.630902	−	0.775862i	$-0.282685\pi$
$953$	38.9528	1.26180	0.630902	+	0.775862i	$0.282685\pi$
$954$	0	0
$955$	−45.1462	−1.46090
$956$	7.04153	0.227739
$957$	0	0
$958$	−72.3018	−2.33597
$959$	−59.5712	−1.92366
$960$	0	0
$961$	35.0914	1.13198
$962$	−50.9911	−1.64402
$963$	0	0
$964$	−23.7678	−0.765511
$965$	25.8209	0.831203
$966$	0	0
$967$	−30.5683	−0.983009	−0.491504	−	0.870875i	$-0.663553\pi$
$967$	−30.5683	−0.983009	−0.491504	+	0.870875i	$0.663553\pi$
$968$	1.38500	0.0445155
$969$	0	0
$970$	−127.358	−4.08922
$971$	−13.5495	−0.434823	−0.217412	−	0.976080i	$-0.569761\pi$
$971$	−13.5495	−0.434823	−0.217412	+	0.976080i	$0.569761\pi$
$972$	0	0
$973$	45.4351	1.45658
$974$	−1.74247	−0.0558324
$975$	0	0
$976$	4.94743	0.158363
$977$	−47.4768	−1.51892	−0.759459	−	0.650555i	$-0.774536\pi$
$977$	−47.4768	−1.51892	−0.759459	+	0.650555i	$0.774536\pi$
$978$	0	0
$979$	16.6064	0.530743
$980$	55.2741	1.76567
$981$	0	0
$982$	−10.4819	−0.334492
$983$	−38.2094	−1.21869	−0.609345	−	0.792905i	$-0.708568\pi$
$983$	−38.2094	−1.21869	−0.609345	+	0.792905i	$0.708568\pi$
$984$	0	0
$985$	46.8734	1.49351
$986$	−22.3338	−0.711254
$987$	0	0
$988$	0.836832	0.0266232
$989$	−0.00233964	−7.43964e−5 0
$990$	0	0
$991$	−4.55647	−0.144741	−0.0723705	−	0.997378i	$-0.523056\pi$
$991$	−4.55647	−0.144741	−0.0723705	+	0.997378i	$0.523056\pi$
$992$	−49.7587	−1.57984
$993$	0	0
$994$	96.6178	3.06453
$995$	39.5872	1.25500
$996$	0	0
$997$	40.2658	1.27523	0.637616	−	0.770354i	$-0.279921\pi$
$997$	40.2658	1.27523	0.637616	+	0.770354i	$0.279921\pi$
$998$	−35.4840	−1.12323
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.n.1.6	✓	25
3.2	odd	2		6039.2.a.o.1.20	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.6	✓	25	1.1	even	1	trivial
6039.2.a.o.1.20	yes	25	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.79702 q^{2} +1.22928 q^{4} +4.13243 q^{5} +4.22858 q^{7} +1.38500 q^{8} +O(q^{10})\)	\(q-1.79702 q^{2} +1.22928 q^{4} +4.13243 q^{5} +4.22858 q^{7} +1.38500 q^{8} -7.42606 q^{10} -1.00000 q^{11} -5.95009 q^{13} -7.59885 q^{14} -4.94743 q^{16} -6.87502 q^{17} -0.114410 q^{19} +5.07992 q^{20} +1.79702 q^{22} -3.90616 q^{23} +12.0770 q^{25} +10.6924 q^{26} +5.19812 q^{28} -1.80774 q^{29} -8.12966 q^{31} +6.12063 q^{32} +12.3545 q^{34} +17.4743 q^{35} -4.76890 q^{37} +0.205597 q^{38} +5.72341 q^{40} +1.81748 q^{41} +0.000598963 q^{43} -1.22928 q^{44} +7.01944 q^{46} +6.87638 q^{47} +10.8809 q^{49} -21.7026 q^{50} -7.31433 q^{52} -10.2253 q^{53} -4.13243 q^{55} +5.85658 q^{56} +3.24855 q^{58} -6.58478 q^{59} -1.00000 q^{61} +14.6092 q^{62} -1.10404 q^{64} -24.5883 q^{65} -9.06126 q^{67} -8.45132 q^{68} -31.4017 q^{70} -12.7148 q^{71} +12.9759 q^{73} +8.56980 q^{74} -0.140642 q^{76} -4.22858 q^{77} +8.35873 q^{79} -20.4449 q^{80} -3.26605 q^{82} +2.71294 q^{83} -28.4105 q^{85} -0.00107635 q^{86} -1.38500 q^{88} -16.6064 q^{89} -25.1605 q^{91} -4.80176 q^{92} -12.3570 q^{94} -0.472791 q^{95} +17.1502 q^{97} -19.5532 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) \) 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8	\( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8} - 25 q^{11} + 4 q^{13} - 18 q^{14} + 21 q^{16} - 20 q^{17} + 14 q^{19} - 12 q^{20} + 5 q^{22} - 20 q^{23} + 13 q^{25} - 16 q^{26} - 14 q^{28} - 28 q^{29} - 12 q^{31} - 35 q^{32} + 6 q^{34} - 10 q^{35} - 8 q^{37} - 32 q^{38} + 24 q^{40} - 26 q^{41} + 18 q^{43} - 25 q^{44} + 4 q^{46} - 12 q^{47} + 23 q^{49} - 43 q^{50} + 22 q^{52} - 36 q^{53} + 4 q^{55} - 26 q^{56} - 20 q^{58} - 46 q^{59} - 25 q^{61} + 14 q^{62} - 13 q^{64} - 60 q^{65} - 20 q^{67} - 44 q^{68} - 20 q^{70} - 52 q^{71} + 6 q^{73} - 32 q^{74} - 4 q^{77} + 26 q^{79} - 52 q^{80} + 6 q^{82} - 38 q^{83} - 4 q^{85} - 34 q^{86} + 15 q^{88} - 82 q^{89} - 58 q^{91} - 36 q^{92} + 16 q^{94} - 30 q^{95} - 35 q^{98}+O(q^{100}) \) 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8 - 25 * q^11 + 4 * q^13 - 18 * q^14 + 21 * q^16 - 20 * q^17 + 14 * q^19 - 12 * q^20 + 5 * q^22 - 20 * q^23 + 13 * q^25 - 16 * q^26 - 14 * q^28 - 28 * q^29 - 12 * q^31 - 35 * q^32 + 6 * q^34 - 10 * q^35 - 8 * q^37 - 32 * q^38 + 24 * q^40 - 26 * q^41 + 18 * q^43 - 25 * q^44 + 4 * q^46 - 12 * q^47 + 23 * q^49 - 43 * q^50 + 22 * q^52 - 36 * q^53 + 4 * q^55 - 26 * q^56 - 20 * q^58 - 46 * q^59 - 25 * q^61 + 14 * q^62 - 13 * q^64 - 60 * q^65 - 20 * q^67 - 44 * q^68 - 20 * q^70 - 52 * q^71 + 6 * q^73 - 32 * q^74 - 4 * q^77 + 26 * q^79 - 52 * q^80 + 6 * q^82 - 38 * q^83 - 4 * q^85 - 34 * q^86 + 15 * q^88 - 82 * q^89 - 58 * q^91 - 36 * q^92 + 16 * q^94 - 30 * q^95 - 35 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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