LMFDB - Embedded newform 6039.2.a.c.1.7

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:	$11$
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} - 2x^{10} - 14x^{9} + 27x^{8} + 66x^{7} - 125x^{6} - 115x^{5} + 227x^{4} + 40x^{3} - 129x^{2} + 26x - 1 $ x^11 - 2x^10 - 14x^9 + 27x^8 + 66x^7 - 125x^6 - 115x^5 + 227x^4 + 40x^3 - 129x^2 + 26x - 1
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			1.7
Root			$0.942842$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.942842 q^{2} -1.11105 q^{4} -3.49194 q^{5} +1.81837 q^{7} -2.93323 q^{8} +O(q^{10})$	\(q+0.942842 q^{2} -1.11105 q^{4} -3.49194 q^{5} +1.81837 q^{7} -2.93323 q^{8} -3.29234 q^{10} -1.00000 q^{11} +0.0495663 q^{13} +1.71444 q^{14} -0.543473 q^{16} +7.74735 q^{17} -8.43210 q^{19} +3.87971 q^{20} -0.942842 q^{22} +2.63679 q^{23} +7.19362 q^{25} +0.0467332 q^{26} -2.02030 q^{28} +8.93168 q^{29} -6.37998 q^{31} +5.35405 q^{32} +7.30453 q^{34} -6.34964 q^{35} +8.12883 q^{37} -7.95014 q^{38} +10.2426 q^{40} -1.61240 q^{41} +1.65032 q^{43} +1.11105 q^{44} +2.48607 q^{46} +3.69954 q^{47} -3.69353 q^{49} +6.78245 q^{50} -0.0550706 q^{52} -1.04707 q^{53} +3.49194 q^{55} -5.33370 q^{56} +8.42117 q^{58} +8.66125 q^{59} +1.00000 q^{61} -6.01531 q^{62} +6.13497 q^{64} -0.173082 q^{65} -3.49836 q^{67} -8.60768 q^{68} -5.98670 q^{70} -13.4818 q^{71} -14.2256 q^{73} +7.66420 q^{74} +9.36848 q^{76} -1.81837 q^{77} -12.8071 q^{79} +1.89777 q^{80} -1.52024 q^{82} +3.54926 q^{83} -27.0532 q^{85} +1.55599 q^{86} +2.93323 q^{88} -3.49530 q^{89} +0.0901300 q^{91} -2.92960 q^{92} +3.48808 q^{94} +29.4444 q^{95} +2.90558 q^{97} -3.48241 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{2} + 10 q^{4} + q^{5} - 11 q^{7} + 3 q^{8}+O(q^{10}) $ 11 * q + 2 * q^2 + 10 * q^4 + q^5 - 11 * q^7 + 3 * q^8	$ 11 q + 2 q^{2} + 10 q^{4} + q^{5} - 11 q^{7} + 3 q^{8} - 8 q^{10} - 11 q^{11} - 13 q^{13} - 5 q^{14} + 4 q^{16} + 13 q^{17} - 12 q^{19} + 7 q^{20} - 2 q^{22} + 3 q^{23} + 12 q^{25} - 12 q^{26} - 13 q^{28} - 2 q^{29} + q^{31} + 23 q^{32} - 14 q^{34} + 4 q^{35} - 14 q^{37} + 8 q^{38} - 34 q^{40} - 3 q^{41} - 21 q^{43} - 10 q^{44} - 12 q^{46} + 16 q^{47} - 18 q^{49} + 13 q^{50} - 33 q^{52} - q^{55} - 16 q^{56} - 17 q^{58} - 3 q^{59} + 11 q^{61} + 21 q^{62} - 7 q^{64} + q^{65} - 24 q^{67} - 2 q^{68} + 4 q^{70} - 7 q^{71} - 42 q^{73} + 16 q^{74} - 13 q^{76} + 11 q^{77} - 11 q^{79} - 42 q^{80} - 38 q^{82} + 34 q^{83} - 14 q^{85} - 42 q^{86} - 3 q^{88} - 29 q^{89} + 9 q^{91} - 42 q^{92} - 33 q^{94} + 31 q^{95} - 45 q^{97} + 33 q^{98}+O(q^{100}) $ 11 * q + 2 * q^2 + 10 * q^4 + q^5 - 11 * q^7 + 3 * q^8 - 8 * q^10 - 11 * q^11 - 13 * q^13 - 5 * q^14 + 4 * q^16 + 13 * q^17 - 12 * q^19 + 7 * q^20 - 2 * q^22 + 3 * q^23 + 12 * q^25 - 12 * q^26 - 13 * q^28 - 2 * q^29 + q^31 + 23 * q^32 - 14 * q^34 + 4 * q^35 - 14 * q^37 + 8 * q^38 - 34 * q^40 - 3 * q^41 - 21 * q^43 - 10 * q^44 - 12 * q^46 + 16 * q^47 - 18 * q^49 + 13 * q^50 - 33 * q^52 - q^55 - 16 * q^56 - 17 * q^58 - 3 * q^59 + 11 * q^61 + 21 * q^62 - 7 * q^64 + q^65 - 24 * q^67 - 2 * q^68 + 4 * q^70 - 7 * q^71 - 42 * q^73 + 16 * q^74 - 13 * q^76 + 11 * q^77 - 11 * q^79 - 42 * q^80 - 38 * q^82 + 34 * q^83 - 14 * q^85 - 42 * q^86 - 3 * q^88 - 29 * q^89 + 9 * q^91 - 42 * q^92 - 33 * q^94 + 31 * q^95 - 45 * q^97 + 33 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.942842	0.666690	0.333345	−	0.942805i	$-0.391823\pi$
$2$	0.942842	0.666690	0.333345	+	0.942805i	$0.391823\pi$
$3$	0	0
$3$	0	0
$4$	−1.11105	−0.555524
$5$	−3.49194	−1.56164	−0.780821	−	0.624755i	$-0.785199\pi$
$5$	−3.49194	−1.56164	−0.780821	+	0.624755i	$0.785199\pi$
$6$	0	0
$7$	1.81837	0.687280	0.343640	−	0.939102i	$-0.388340\pi$
$7$	1.81837	0.687280	0.343640	+	0.939102i	$0.388340\pi$
$8$	−2.93323	−1.03705
$9$	0	0
$10$	−3.29234	−1.04113
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.0495663	0.0137472	0.00687361	−	0.999976i	$-0.497812\pi$
$13$	0.0495663	0.0137472	0.00687361	+	0.999976i	$0.497812\pi$
$14$	1.71444	0.458203
$15$	0	0
$16$	−0.543473	−0.135868
$17$	7.74735	1.87901	0.939504	−	0.342538i	$-0.111287\pi$
$17$	7.74735	1.87901	0.939504	+	0.342538i	$0.111287\pi$
$18$	0	0
$19$	−8.43210	−1.93446	−0.967228	−	0.253908i	$-0.918284\pi$
$19$	−8.43210	−1.93446	−0.967228	+	0.253908i	$0.918284\pi$
$20$	3.87971	0.867530
$21$	0	0
$22$	−0.942842	−0.201015
$23$	2.63679	0.549808	0.274904	−	0.961472i	$-0.411354\pi$
$23$	2.63679	0.549808	0.274904	+	0.961472i	$0.411354\pi$
$24$	0	0
$25$	7.19362	1.43872
$26$	0.0467332	0.00916514
$27$	0	0
$28$	−2.02030	−0.381801
$29$	8.93168	1.65857	0.829286	−	0.558825i	$-0.188747\pi$
$29$	8.93168	1.65857	0.829286	+	0.558825i	$0.188747\pi$
$30$	0	0
$31$	−6.37998	−1.14588	−0.572939	−	0.819598i	$-0.694197\pi$
$31$	−6.37998	−1.14588	−0.572939	+	0.819598i	$0.694197\pi$
$32$	5.35405	0.946471
$33$	0	0
$34$	7.30453	1.25272
$35$	−6.34964	−1.07328
$36$	0	0
$37$	8.12883	1.33637	0.668185	−	0.743995i	$-0.267071\pi$
$37$	8.12883	1.33637	0.668185	+	0.743995i	$0.267071\pi$
$38$	−7.95014	−1.28968
$39$	0	0
$40$	10.2426	1.61950
$41$	−1.61240	−0.251815	−0.125907	−	0.992042i	$-0.540184\pi$
$41$	−1.61240	−0.251815	−0.125907	+	0.992042i	$0.540184\pi$
$42$	0	0
$43$	1.65032	0.251671	0.125836	−	0.992051i	$-0.459839\pi$
$43$	1.65032	0.251671	0.125836	+	0.992051i	$0.459839\pi$
$44$	1.11105	0.167497
$45$	0	0
$46$	2.48607	0.366551
$47$	3.69954	0.539633	0.269816	−	0.962912i	$-0.413037\pi$
$47$	3.69954	0.539633	0.269816	+	0.962912i	$0.413037\pi$
$48$	0	0
$49$	−3.69353	−0.527647
$50$	6.78245	0.959183
$51$	0	0
$52$	−0.0550706	−0.00763692
$53$	−1.04707	−0.143826	−0.0719131	−	0.997411i	$-0.522910\pi$
$53$	−1.04707	−0.143826	−0.0719131	+	0.997411i	$0.522910\pi$
$54$	0	0
$55$	3.49194	0.470853
$56$	−5.33370	−0.712745
$57$	0	0
$58$	8.42117	1.10575
$59$	8.66125	1.12760	0.563800	−	0.825912i	$-0.309339\pi$
$59$	8.66125	1.12760	0.563800	+	0.825912i	$0.309339\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−6.01531	−0.763945
$63$	0	0
$64$	6.13497	0.766871
$65$	−0.173082	−0.0214682
$66$	0	0
$67$	−3.49836	−0.427392	−0.213696	−	0.976900i	$-0.568550\pi$
$67$	−3.49836	−0.427392	−0.213696	+	0.976900i	$0.568550\pi$
$68$	−8.60768	−1.04383
$69$	0	0
$70$	−5.98670	−0.715548
$71$	−13.4818	−1.59999	−0.799996	−	0.600006i	$-0.795165\pi$
$71$	−13.4818	−1.59999	−0.799996	+	0.600006i	$0.795165\pi$
$72$	0	0
$73$	−14.2256	−1.66498	−0.832488	−	0.554044i	$-0.813084\pi$
$73$	−14.2256	−1.66498	−0.832488	+	0.554044i	$0.813084\pi$
$74$	7.66420	0.890945
$75$	0	0
$76$	9.36848	1.07464
$77$	−1.81837	−0.207223
$78$	0	0
$79$	−12.8071	−1.44091	−0.720453	−	0.693504i	$-0.756066\pi$
$79$	−12.8071	−1.44091	−0.720453	+	0.693504i	$0.756066\pi$
$80$	1.89777	0.212177
$81$	0	0
$82$	−1.52024	−0.167882
$83$	3.54926	0.389582	0.194791	−	0.980845i	$-0.437597\pi$
$83$	3.54926	0.389582	0.194791	+	0.980845i	$0.437597\pi$
$84$	0	0
$85$	−27.0532	−2.93434
$86$	1.55599	0.167787
$87$	0	0
$88$	2.93323	0.312683
$89$	−3.49530	−0.370501	−0.185251	−	0.982691i	$-0.559310\pi$
$89$	−3.49530	−0.370501	−0.185251	+	0.982691i	$0.559310\pi$
$90$	0	0
$91$	0.0901300	0.00944819
$92$	−2.92960	−0.305432
$93$	0	0
$94$	3.48808	0.359768
$95$	29.4444	3.02093
$96$	0	0
$97$	2.90558	0.295017	0.147509	−	0.989061i	$-0.452875\pi$
$97$	2.90558	0.295017	0.147509	+	0.989061i	$0.452875\pi$
$98$	−3.48241	−0.351777
$99$	0	0
$100$	−7.99246	−0.799246
$101$	−4.01746	−0.399752	−0.199876	−	0.979821i	$-0.564054\pi$
$101$	−4.01746	−0.399752	−0.199876	+	0.979821i	$0.564054\pi$
$102$	0	0
$103$	−8.35920	−0.823657	−0.411828	−	0.911261i	$-0.635110\pi$
$103$	−8.35920	−0.823657	−0.411828	+	0.911261i	$0.635110\pi$
$104$	−0.145389	−0.0142566
$105$	0	0
$106$	−0.987223	−0.0958875
$107$	0.438376	0.0423794	0.0211897	−	0.999775i	$-0.493255\pi$
$107$	0.438376	0.0423794	0.0211897	+	0.999775i	$0.493255\pi$
$108$	0	0
$109$	−5.04229	−0.482964	−0.241482	−	0.970405i	$-0.577633\pi$
$109$	−5.04229	−0.482964	−0.241482	+	0.970405i	$0.577633\pi$
$110$	3.29234	0.313913
$111$	0	0
$112$	−0.988236	−0.0933795
$113$	−11.3993	−1.07235	−0.536177	−	0.844106i	$-0.680132\pi$
$113$	−11.3993	−1.07235	−0.536177	+	0.844106i	$0.680132\pi$
$114$	0	0
$115$	−9.20749	−0.858603
$116$	−9.92354	−0.921377
$117$	0	0
$118$	8.16619	0.751759
$119$	14.0876	1.29140
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.942842	0.0853609
$123$	0	0
$124$	7.08846	0.636563
$125$	−7.65998	−0.685129
$126$	0	0
$127$	16.6435	1.47688	0.738438	−	0.674321i	$-0.235564\pi$
$127$	16.6435	1.47688	0.738438	+	0.674321i	$0.235564\pi$
$128$	−4.92379	−0.435206
$129$	0	0
$130$	−0.163189	−0.0143127
$131$	15.0473	1.31469	0.657346	−	0.753589i	$-0.271679\pi$
$131$	15.0473	1.31469	0.657346	+	0.753589i	$0.271679\pi$
$132$	0	0
$133$	−15.3327	−1.32951
$134$	−3.29840	−0.284938
$135$	0	0
$136$	−22.7247	−1.94863
$137$	−5.65682	−0.483295	−0.241647	−	0.970364i	$-0.577688\pi$
$137$	−5.65682	−0.483295	−0.241647	+	0.970364i	$0.577688\pi$
$138$	0	0
$139$	21.1733	1.79590	0.897950	−	0.440098i	$-0.145056\pi$
$139$	21.1733	1.79590	0.897950	+	0.440098i	$0.145056\pi$
$140$	7.05476	0.596236
$141$	0	0
$142$	−12.7112	−1.06670
$143$	−0.0495663	−0.00414494
$144$	0	0
$145$	−31.1889	−2.59009
$146$	−13.4125	−1.11002
$147$	0	0
$148$	−9.03152	−0.742387
$149$	−21.2658	−1.74217	−0.871083	−	0.491137i	$-0.836582\pi$
$149$	−21.2658	−1.74217	−0.871083	+	0.491137i	$0.836582\pi$
$150$	0	0
$151$	−8.05881	−0.655817	−0.327908	−	0.944710i	$-0.606344\pi$
$151$	−8.05881	−0.655817	−0.327908	+	0.944710i	$0.606344\pi$
$152$	24.7333	2.00613
$153$	0	0
$154$	−1.71444	−0.138153
$155$	22.2785	1.78945
$156$	0	0
$157$	1.96933	0.157170	0.0785849	−	0.996907i	$-0.474960\pi$
$157$	1.96933	0.157170	0.0785849	+	0.996907i	$0.474960\pi$
$158$	−12.0750	−0.960638
$159$	0	0
$160$	−18.6960	−1.47805
$161$	4.79465	0.377872
$162$	0	0
$163$	−20.7663	−1.62654	−0.813269	−	0.581888i	$-0.802314\pi$
$163$	−20.7663	−1.62654	−0.813269	+	0.581888i	$0.802314\pi$
$164$	1.79146	0.139889
$165$	0	0
$166$	3.34639	0.259730
$167$	1.21018	0.0936465	0.0468233	−	0.998903i	$-0.485090\pi$
$167$	1.21018	0.0936465	0.0468233	+	0.998903i	$0.485090\pi$
$168$	0	0
$169$	−12.9975	−0.999811
$170$	−25.5069	−1.95629
$171$	0	0
$172$	−1.83359	−0.139810
$173$	2.13714	0.162483	0.0812417	−	0.996694i	$-0.474111\pi$
$173$	2.13714	0.162483	0.0812417	+	0.996694i	$0.474111\pi$
$174$	0	0
$175$	13.0807	0.988806
$176$	0.543473	0.0409658
$177$	0	0
$178$	−3.29552	−0.247009
$179$	−13.7854	−1.03037	−0.515184	−	0.857080i	$-0.672276\pi$
$179$	−13.7854	−1.03037	−0.515184	+	0.857080i	$0.672276\pi$
$180$	0	0
$181$	9.01267	0.669906	0.334953	−	0.942235i	$-0.391279\pi$
$181$	9.01267	0.669906	0.334953	+	0.942235i	$0.391279\pi$
$182$	0.0849783	0.00629901
$183$	0	0
$184$	−7.73429	−0.570180
$185$	−28.3853	−2.08693
$186$	0	0
$187$	−7.74735	−0.566542
$188$	−4.11036	−0.299779
$189$	0	0
$190$	27.7614	2.01402
$191$	6.79478	0.491653	0.245827	−	0.969314i	$-0.420941\pi$
$191$	6.79478	0.491653	0.245827	+	0.969314i	$0.420941\pi$
$192$	0	0
$193$	−6.80903	−0.490125	−0.245062	−	0.969507i	$-0.578808\pi$
$193$	−6.80903	−0.490125	−0.245062	+	0.969507i	$0.578808\pi$
$194$	2.73951	0.196685
$195$	0	0
$196$	4.10369	0.293121
$197$	−19.9506	−1.42142	−0.710712	−	0.703483i	$-0.751627\pi$
$197$	−19.9506	−1.42142	−0.710712	+	0.703483i	$0.751627\pi$
$198$	0	0
$199$	−7.46854	−0.529431	−0.264715	−	0.964327i	$-0.585278\pi$
$199$	−7.46854	−0.529431	−0.264715	+	0.964327i	$0.585278\pi$
$200$	−21.1005	−1.49203
$201$	0	0
$202$	−3.78783	−0.266511
$203$	16.2411	1.13990
$204$	0	0
$205$	5.63040	0.393245
$206$	−7.88141	−0.549124
$207$	0	0
$208$	−0.0269380	−0.00186781
$209$	8.43210	0.583261
$210$	0	0
$211$	23.9405	1.64813	0.824066	−	0.566493i	$-0.191700\pi$
$211$	23.9405	1.64813	0.824066	+	0.566493i	$0.191700\pi$
$212$	1.16335	0.0798990
$213$	0	0
$214$	0.413319	0.0282539
$215$	−5.76281	−0.393020
$216$	0	0
$217$	−11.6012	−0.787538
$218$	−4.75408	−0.321987
$219$	0	0
$220$	−3.87971	−0.261570
$221$	0.384007	0.0258311
$222$	0	0
$223$	10.8163	0.724311	0.362156	−	0.932118i	$-0.382041\pi$
$223$	10.8163	0.724311	0.362156	+	0.932118i	$0.382041\pi$
$224$	9.73564	0.650490
$225$	0	0
$226$	−10.7477	−0.714928
$227$	26.3046	1.74590	0.872950	−	0.487810i	$-0.162204\pi$
$227$	26.3046	1.74590	0.872950	+	0.487810i	$0.162204\pi$
$228$	0	0
$229$	−24.8377	−1.64132	−0.820661	−	0.571416i	$-0.806394\pi$
$229$	−24.8377	−1.64132	−0.820661	+	0.571416i	$0.806394\pi$
$230$	−8.68121	−0.572422
$231$	0	0
$232$	−26.1987	−1.72003
$233$	−15.2240	−0.997358	−0.498679	−	0.866787i	$-0.666181\pi$
$233$	−15.2240	−0.997358	−0.498679	+	0.866787i	$0.666181\pi$
$234$	0	0
$235$	−12.9185	−0.842713
$236$	−9.62307	−0.626409
$237$	0	0
$238$	13.2823	0.860966
$239$	0.265685	0.0171857	0.00859287	−	0.999963i	$-0.497265\pi$
$239$	0.265685	0.0171857	0.00859287	+	0.999963i	$0.497265\pi$
$240$	0	0
$241$	−23.8197	−1.53436	−0.767182	−	0.641429i	$-0.778342\pi$
$241$	−23.8197	−1.53436	−0.767182	+	0.641429i	$0.778342\pi$
$242$	0.942842	0.0606082
$243$	0	0
$244$	−1.11105	−0.0711276
$245$	12.8976	0.823995
$246$	0	0
$247$	−0.417948	−0.0265934
$248$	18.7139	1.18834
$249$	0	0
$250$	−7.22215	−0.456769
$251$	12.9390	0.816702	0.408351	−	0.912825i	$-0.366104\pi$
$251$	12.9390	0.816702	0.408351	+	0.912825i	$0.366104\pi$
$252$	0	0
$253$	−2.63679	−0.165773
$254$	15.6922	0.984618
$255$	0	0
$256$	−16.9123	−1.05702
$257$	14.2280	0.887519	0.443759	−	0.896146i	$-0.353644\pi$
$257$	14.2280	0.887519	0.443759	+	0.896146i	$0.353644\pi$
$258$	0	0
$259$	14.7812	0.918461
$260$	0.192303	0.0119261
$261$	0	0
$262$	14.1873	0.876491
$263$	−21.1376	−1.30340	−0.651699	−	0.758478i	$-0.725943\pi$
$263$	−21.1376	−1.30340	−0.651699	+	0.758478i	$0.725943\pi$
$264$	0	0
$265$	3.65631	0.224605
$266$	−14.4563	−0.886373
$267$	0	0
$268$	3.88684	0.237427
$269$	1.60031	0.0975728	0.0487864	−	0.998809i	$-0.484465\pi$
$269$	1.60031	0.0975728	0.0487864	+	0.998809i	$0.484465\pi$
$270$	0	0
$271$	−24.2839	−1.47514	−0.737571	−	0.675269i	$-0.764027\pi$
$271$	−24.2839	−1.47514	−0.737571	+	0.675269i	$0.764027\pi$
$272$	−4.21047	−0.255298
$273$	0	0
$274$	−5.33349	−0.322208
$275$	−7.19362	−0.433792
$276$	0	0
$277$	4.46699	0.268395	0.134198	−	0.990955i	$-0.457154\pi$
$277$	4.46699	0.268395	0.134198	+	0.990955i	$0.457154\pi$
$278$	19.9631	1.19731
$279$	0	0
$280$	18.6249	1.11305
$281$	−11.3532	−0.677276	−0.338638	−	0.940917i	$-0.609966\pi$
$281$	−11.3532	−0.677276	−0.338638	+	0.940917i	$0.609966\pi$
$282$	0	0
$283$	4.15294	0.246867	0.123433	−	0.992353i	$-0.460609\pi$
$283$	4.15294	0.246867	0.123433	+	0.992353i	$0.460609\pi$
$284$	14.9789	0.888834
$285$	0	0
$286$	−0.0467332	−0.00276339
$287$	−2.93195	−0.173067
$288$	0	0
$289$	43.0214	2.53067
$290$	−29.4062	−1.72679
$291$	0	0
$292$	15.8053	0.924934
$293$	−0.185221	−0.0108207	−0.00541037	−	0.999985i	$-0.501722\pi$
$293$	−0.185221	−0.0108207	−0.00541037	+	0.999985i	$0.501722\pi$
$294$	0	0
$295$	−30.2445	−1.76091
$296$	−23.8437	−1.38589
$297$	0	0
$298$	−20.0503	−1.16148
$299$	0.130696	0.00755833
$300$	0	0
$301$	3.00089	0.172969
$302$	−7.59818	−0.437226
$303$	0	0
$304$	4.58262	0.262831
$305$	−3.49194	−0.199948
$306$	0	0
$307$	−28.1720	−1.60786	−0.803931	−	0.594722i	$-0.797262\pi$
$307$	−28.1720	−1.60786	−0.803931	+	0.594722i	$0.797262\pi$
$308$	2.02030	0.115117
$309$	0	0
$310$	21.0051	1.19301
$311$	−15.7010	−0.890321	−0.445161	−	0.895451i	$-0.646853\pi$
$311$	−15.7010	−0.890321	−0.445161	+	0.895451i	$0.646853\pi$
$312$	0	0
$313$	6.44602	0.364351	0.182175	−	0.983266i	$-0.441686\pi$
$313$	6.44602	0.364351	0.182175	+	0.983266i	$0.441686\pi$
$314$	1.85677	0.104783
$315$	0	0
$316$	14.2293	0.800459
$317$	−14.6815	−0.824596	−0.412298	−	0.911049i	$-0.635274\pi$
$317$	−14.6815	−0.824596	−0.412298	+	0.911049i	$0.635274\pi$
$318$	0	0
$319$	−8.93168	−0.500078
$320$	−21.4229	−1.19758
$321$	0	0
$322$	4.52060	0.251923
$323$	−65.3264	−3.63486
$324$	0	0
$325$	0.356561	0.0197785
$326$	−19.5793	−1.08440
$327$	0	0
$328$	4.72954	0.261145
$329$	6.72713	0.370879
$330$	0	0
$331$	33.3092	1.83084	0.915420	−	0.402500i	$-0.131859\pi$
$331$	33.3092	1.83084	0.915420	+	0.402500i	$0.131859\pi$
$332$	−3.94340	−0.216422
$333$	0	0
$334$	1.14101	0.0624332
$335$	12.2160	0.667433
$336$	0	0
$337$	−19.9521	−1.08686	−0.543430	−	0.839454i	$-0.682875\pi$
$337$	−19.9521	−1.08686	−0.543430	+	0.839454i	$0.682875\pi$
$338$	−12.2546	−0.666564
$339$	0	0
$340$	30.0575	1.63010
$341$	6.37998	0.345495
$342$	0	0
$343$	−19.4448	−1.04992
$344$	−4.84076	−0.260996
$345$	0	0
$346$	2.01498	0.108326
$347$	−10.1816	−0.546574	−0.273287	−	0.961932i	$-0.588111\pi$
$347$	−10.1816	−0.546574	−0.273287	+	0.961932i	$0.588111\pi$
$348$	0	0
$349$	16.9472	0.907162	0.453581	−	0.891215i	$-0.350146\pi$
$349$	16.9472	0.907162	0.453581	+	0.891215i	$0.350146\pi$
$350$	12.3330	0.659227
$351$	0	0
$352$	−5.35405	−0.285372
$353$	−33.9199	−1.80537	−0.902686	−	0.430300i	$-0.858408\pi$
$353$	−33.9199	−1.80537	−0.902686	+	0.430300i	$0.858408\pi$
$354$	0	0
$355$	47.0775	2.49861
$356$	3.88345	0.205822
$357$	0	0
$358$	−12.9974	−0.686936
$359$	5.01922	0.264904	0.132452	−	0.991189i	$-0.457715\pi$
$359$	5.01922	0.264904	0.132452	+	0.991189i	$0.457715\pi$
$360$	0	0
$361$	52.1003	2.74212
$362$	8.49752	0.446620
$363$	0	0
$364$	−0.100139	−0.00524870
$365$	49.6747	2.60009
$366$	0	0
$367$	−18.9220	−0.987720	−0.493860	−	0.869541i	$-0.664414\pi$
$367$	−18.9220	−0.987720	−0.493860	+	0.869541i	$0.664414\pi$
$368$	−1.43302	−0.0747014
$369$	0	0
$370$	−26.7629	−1.39134
$371$	−1.90396	−0.0988489
$372$	0	0
$373$	−25.5470	−1.32277	−0.661386	−	0.750046i	$-0.730031\pi$
$373$	−25.5470	−1.32277	−0.661386	+	0.750046i	$0.730031\pi$
$374$	−7.30453	−0.377708
$375$	0	0
$376$	−10.8516	−0.559628
$377$	0.442711	0.0228008
$378$	0	0
$379$	−36.9463	−1.89781	−0.948903	−	0.315568i	$-0.897805\pi$
$379$	−36.9463	−1.89781	−0.948903	+	0.315568i	$0.897805\pi$
$380$	−32.7141	−1.67820
$381$	0	0
$382$	6.40641	0.327780
$383$	2.59368	0.132531	0.0662655	−	0.997802i	$-0.478892\pi$
$383$	2.59368	0.132531	0.0662655	+	0.997802i	$0.478892\pi$
$384$	0	0
$385$	6.34964	0.323607
$386$	−6.41984	−0.326761
$387$	0	0
$388$	−3.22825	−0.163889
$389$	9.79596	0.496675	0.248337	−	0.968674i	$-0.420116\pi$
$389$	9.79596	0.496675	0.248337	+	0.968674i	$0.420116\pi$
$390$	0	0
$391$	20.4281	1.03309
$392$	10.8340	0.547197
$393$	0	0
$394$	−18.8103	−0.947650
$395$	44.7214	2.25018
$396$	0	0
$397$	−7.80554	−0.391749	−0.195875	−	0.980629i	$-0.562755\pi$
$397$	−7.80554	−0.391749	−0.195875	+	0.980629i	$0.562755\pi$
$398$	−7.04166	−0.352966
$399$	0	0
$400$	−3.90954	−0.195477
$401$	−30.0943	−1.50284	−0.751419	−	0.659825i	$-0.770630\pi$
$401$	−30.0943	−1.50284	−0.751419	+	0.659825i	$0.770630\pi$
$402$	0	0
$403$	−0.316232	−0.0157526
$404$	4.46359	0.222072
$405$	0	0
$406$	15.3128	0.759962
$407$	−8.12883	−0.402931
$408$	0	0
$409$	2.92991	0.144875	0.0724374	−	0.997373i	$-0.476922\pi$
$409$	2.92991	0.144875	0.0724374	+	0.997373i	$0.476922\pi$
$410$	5.30858	0.262172
$411$	0	0
$412$	9.28748	0.457561
$413$	15.7494	0.774976
$414$	0	0
$415$	−12.3938	−0.608387
$416$	0.265380	0.0130113
$417$	0	0
$418$	7.95014	0.388854
$419$	−8.24319	−0.402706	−0.201353	−	0.979519i	$-0.564534\pi$
$419$	−8.24319	−0.402706	−0.201353	+	0.979519i	$0.564534\pi$
$420$	0	0
$421$	−0.713317	−0.0347650	−0.0173825	−	0.999849i	$-0.505533\pi$
$421$	−0.713317	−0.0347650	−0.0173825	+	0.999849i	$0.505533\pi$
$422$	22.5721	1.09879
$423$	0	0
$424$	3.07130	0.149155
$425$	55.7315	2.70337
$426$	0	0
$427$	1.81837	0.0879972
$428$	−0.487057	−0.0235428
$429$	0	0
$430$	−5.43342	−0.262023
$431$	38.3428	1.84691	0.923453	−	0.383711i	$-0.125354\pi$
$431$	38.3428	1.84691	0.923453	+	0.383711i	$0.125354\pi$
$432$	0	0
$433$	−3.98585	−0.191548	−0.0957739	−	0.995403i	$-0.530533\pi$
$433$	−3.98585	−0.191548	−0.0957739	+	0.995403i	$0.530533\pi$
$434$	−10.9381	−0.525044
$435$	0	0
$436$	5.60223	0.268298
$437$	−22.2336	−1.06358
$438$	0	0
$439$	31.2766	1.49275	0.746376	−	0.665524i	$-0.231792\pi$
$439$	31.2766	1.49275	0.746376	+	0.665524i	$0.231792\pi$
$440$	−10.2426	−0.488299
$441$	0	0
$442$	0.362058	0.0172214
$443$	12.3617	0.587321	0.293661	−	0.955910i	$-0.405126\pi$
$443$	12.3617	0.587321	0.293661	+	0.955910i	$0.405126\pi$
$444$	0	0
$445$	12.2054	0.578590
$446$	10.1980	0.482891
$447$	0	0
$448$	11.1556	0.527055
$449$	−2.46911	−0.116524	−0.0582622	−	0.998301i	$-0.518556\pi$
$449$	−2.46911	−0.116524	−0.0582622	+	0.998301i	$0.518556\pi$
$450$	0	0
$451$	1.61240	0.0759250
$452$	12.6652	0.595719
$453$	0	0
$454$	24.8011	1.16397
$455$	−0.314728	−0.0147547
$456$	0	0
$457$	−28.1649	−1.31750	−0.658748	−	0.752364i	$-0.728914\pi$
$457$	−28.1649	−1.31750	−0.658748	+	0.752364i	$0.728914\pi$
$458$	−23.4180	−1.09425
$459$	0	0
$460$	10.2300	0.476975
$461$	−33.1318	−1.54310	−0.771552	−	0.636167i	$-0.780519\pi$
$461$	−33.1318	−1.54310	−0.771552	+	0.636167i	$0.780519\pi$
$462$	0	0
$463$	1.89987	0.0882947	0.0441473	−	0.999025i	$-0.485943\pi$
$463$	1.89987	0.0882947	0.0441473	+	0.999025i	$0.485943\pi$
$464$	−4.85413	−0.225347
$465$	0	0
$466$	−14.3538	−0.664929
$467$	9.23493	0.427342	0.213671	−	0.976906i	$-0.431458\pi$
$467$	9.23493	0.427342	0.213671	+	0.976906i	$0.431458\pi$
$468$	0	0
$469$	−6.36131	−0.293738
$470$	−12.1801	−0.561828
$471$	0	0
$472$	−25.4054	−1.16938
$473$	−1.65032	−0.0758818
$474$	0	0
$475$	−60.6573	−2.78315
$476$	−15.6520	−0.717407
$477$	0	0
$478$	0.250499	0.0114576
$479$	27.1839	1.24207	0.621033	−	0.783785i	$-0.286713\pi$
$479$	27.1839	1.24207	0.621033	+	0.783785i	$0.286713\pi$
$480$	0	0
$481$	0.402916	0.0183714
$482$	−22.4582	−1.02295
$483$	0	0
$484$	−1.11105	−0.0505022
$485$	−10.1461	−0.460711
$486$	0	0
$487$	−33.9039	−1.53633	−0.768165	−	0.640252i	$-0.778830\pi$
$487$	−33.9039	−1.53633	−0.768165	+	0.640252i	$0.778830\pi$
$488$	−2.93323	−0.132781
$489$	0	0
$490$	12.1604	0.549349
$491$	−6.92051	−0.312318	−0.156159	−	0.987732i	$-0.549911\pi$
$491$	−6.92051	−0.312318	−0.156159	+	0.987732i	$0.549911\pi$
$492$	0	0
$493$	69.1969	3.11647
$494$	−0.394059	−0.0177296
$495$	0	0
$496$	3.46734	0.155688
$497$	−24.5149	−1.09964
$498$	0	0
$499$	25.0512	1.12145	0.560724	−	0.828003i	$-0.310523\pi$
$499$	25.0512	1.12145	0.560724	+	0.828003i	$0.310523\pi$
$500$	8.51061	0.380606
$501$	0	0
$502$	12.1994	0.544487
$503$	31.2220	1.39212	0.696061	−	0.717983i	$-0.254934\pi$
$503$	31.2220	1.39212	0.696061	+	0.717983i	$0.254934\pi$
$504$	0	0
$505$	14.0287	0.624269
$506$	−2.48607	−0.110519
$507$	0	0
$508$	−18.4918	−0.820441
$509$	18.6446	0.826408	0.413204	−	0.910639i	$-0.364410\pi$
$509$	18.6446	0.826408	0.413204	+	0.910639i	$0.364410\pi$
$510$	0	0
$511$	−25.8673	−1.14430
$512$	−6.09804	−0.269498
$513$	0	0
$514$	13.4148	0.591700
$515$	29.1898	1.28626
$516$	0	0
$517$	−3.69954	−0.162705
$518$	13.9364	0.612329
$519$	0	0
$520$	0.507690	0.0222637
$521$	−3.71320	−0.162678	−0.0813391	−	0.996686i	$-0.525920\pi$
$521$	−3.71320	−0.162678	−0.0813391	+	0.996686i	$0.525920\pi$
$522$	0	0
$523$	−17.0578	−0.745888	−0.372944	−	0.927854i	$-0.621652\pi$
$523$	−17.0578	−0.745888	−0.372944	+	0.927854i	$0.621652\pi$
$524$	−16.7183	−0.730343
$525$	0	0
$526$	−19.9294	−0.868963
$527$	−49.4279	−2.15311
$528$	0	0
$529$	−16.0474	−0.697711
$530$	3.44732	0.149742
$531$	0	0
$532$	17.0354	0.738577
$533$	−0.0799208	−0.00346175
$534$	0	0
$535$	−1.53078	−0.0661814
$536$	10.2615	0.443228
$537$	0	0
$538$	1.50884	0.0650508
$539$	3.69353	0.159091
$540$	0	0
$541$	−24.3002	−1.04475	−0.522373	−	0.852717i	$-0.674953\pi$
$541$	−24.3002	−1.04475	−0.522373	+	0.852717i	$0.674953\pi$
$542$	−22.8959	−0.983463
$543$	0	0
$544$	41.4797	1.77843
$545$	17.6074	0.754216
$546$	0	0
$547$	22.6753	0.969525	0.484762	−	0.874646i	$-0.338906\pi$
$547$	22.6753	0.969525	0.484762	+	0.874646i	$0.338906\pi$
$548$	6.28500	0.268482
$549$	0	0
$550$	−6.78245	−0.289205
$551$	−75.3129	−3.20844
$552$	0	0
$553$	−23.2880	−0.990306
$554$	4.21166	0.178936
$555$	0	0
$556$	−23.5246	−0.997666
$557$	13.8294	0.585972	0.292986	−	0.956117i	$-0.405351\pi$
$557$	13.8294	0.585972	0.292986	+	0.956117i	$0.405351\pi$
$558$	0	0
$559$	0.0818003	0.00345978
$560$	3.45086	0.145825
$561$	0	0
$562$	−10.7043	−0.451533
$563$	3.01616	0.127116	0.0635580	−	0.997978i	$-0.479755\pi$
$563$	3.01616	0.127116	0.0635580	+	0.997978i	$0.479755\pi$
$564$	0	0
$565$	39.8056	1.67463
$566$	3.91557	0.164584
$567$	0	0
$568$	39.5451	1.65928
$569$	−27.5304	−1.15413	−0.577067	−	0.816697i	$-0.695803\pi$
$569$	−27.5304	−1.15413	−0.577067	+	0.816697i	$0.695803\pi$
$570$	0	0
$571$	−11.8631	−0.496455	−0.248228	−	0.968702i	$-0.579848\pi$
$571$	−11.8631	−0.496455	−0.248228	+	0.968702i	$0.579848\pi$
$572$	0.0550706	0.00230262
$573$	0	0
$574$	−2.76436	−0.115382
$575$	18.9680	0.791022
$576$	0	0
$577$	36.4041	1.51552	0.757761	−	0.652532i	$-0.226293\pi$
$577$	36.4041	1.51552	0.757761	+	0.652532i	$0.226293\pi$
$578$	40.5624	1.68717
$579$	0	0
$580$	34.6524	1.43886
$581$	6.45387	0.267752
$582$	0	0
$583$	1.04707	0.0433652
$584$	41.7268	1.72667
$585$	0	0
$586$	−0.174634	−0.00721408
$587$	30.5450	1.26073	0.630365	−	0.776299i	$-0.282905\pi$
$587$	30.5450	1.26073	0.630365	+	0.776299i	$0.282905\pi$
$588$	0	0
$589$	53.7966	2.21665
$590$	−28.5158	−1.17398
$591$	0	0
$592$	−4.41780	−0.181570
$593$	24.2159	0.994426	0.497213	−	0.867628i	$-0.334357\pi$
$593$	24.2159	0.994426	0.497213	+	0.867628i	$0.334357\pi$
$594$	0	0
$595$	−49.1928	−2.01671
$596$	23.6274	0.967815
$597$	0	0
$598$	0.123225	0.00503906
$599$	23.9401	0.978165	0.489083	−	0.872237i	$-0.337332\pi$
$599$	23.9401	0.978165	0.489083	+	0.872237i	$0.337332\pi$
$600$	0	0
$601$	1.66406	0.0678783	0.0339392	−	0.999424i	$-0.489195\pi$
$601$	1.66406	0.0678783	0.0339392	+	0.999424i	$0.489195\pi$
$602$	2.82937	0.115316
$603$	0	0
$604$	8.95373	0.364322
$605$	−3.49194	−0.141967
$606$	0	0
$607$	−7.58111	−0.307708	−0.153854	−	0.988094i	$-0.549169\pi$
$607$	−7.58111	−0.307708	−0.153854	+	0.988094i	$0.549169\pi$
$608$	−45.1459	−1.83091
$609$	0	0
$610$	−3.29234	−0.133303
$611$	0.183372	0.00741845
$612$	0	0
$613$	−25.1894	−1.01739	−0.508695	−	0.860947i	$-0.669872\pi$
$613$	−25.1894	−1.01739	−0.508695	+	0.860947i	$0.669872\pi$
$614$	−26.5618	−1.07195
$615$	0	0
$616$	5.33370	0.214901
$617$	7.65279	0.308090	0.154045	−	0.988064i	$-0.450770\pi$
$617$	7.65279	0.308090	0.154045	+	0.988064i	$0.450770\pi$
$618$	0	0
$619$	27.5986	1.10928	0.554640	−	0.832091i	$-0.312856\pi$
$619$	27.5986	1.10928	0.554640	+	0.832091i	$0.312856\pi$
$620$	−24.7525	−0.994083
$621$	0	0
$622$	−14.8035	−0.593568
$623$	−6.35576	−0.254638
$624$	0	0
$625$	−9.21994	−0.368798
$626$	6.07758	0.242909
$627$	0	0
$628$	−2.18802	−0.0873116
$629$	62.9769	2.51105
$630$	0	0
$631$	12.6174	0.502289	0.251145	−	0.967950i	$-0.419193\pi$
$631$	12.6174	0.502289	0.251145	+	0.967950i	$0.419193\pi$
$632$	37.5660	1.49430
$633$	0	0
$634$	−13.8424	−0.549750
$635$	−58.1182	−2.30635
$636$	0	0
$637$	−0.183074	−0.00725367
$638$	−8.42117	−0.333397
$639$	0	0
$640$	17.1936	0.679635
$641$	−14.1833	−0.560205	−0.280103	−	0.959970i	$-0.590369\pi$
$641$	−14.1833	−0.560205	−0.280103	+	0.959970i	$0.590369\pi$
$642$	0	0
$643$	3.69437	0.145692	0.0728459	−	0.997343i	$-0.476792\pi$
$643$	3.69437	0.145692	0.0728459	+	0.997343i	$0.476792\pi$
$644$	−5.32710	−0.209917
$645$	0	0
$646$	−61.5925	−2.42332
$647$	−46.3072	−1.82052	−0.910262	−	0.414033i	$-0.864120\pi$
$647$	−46.3072	−1.82052	−0.910262	+	0.414033i	$0.864120\pi$
$648$	0	0
$649$	−8.66125	−0.339984
$650$	0.336181	0.0131861
$651$	0	0
$652$	23.0723	0.903582
$653$	−7.51510	−0.294088	−0.147044	−	0.989130i	$-0.546976\pi$
$653$	−7.51510	−0.294088	−0.147044	+	0.989130i	$0.546976\pi$
$654$	0	0
$655$	−52.5443	−2.05308
$656$	0.876297	0.0342136
$657$	0	0
$658$	6.34262	0.247261
$659$	−5.51263	−0.214742	−0.107371	−	0.994219i	$-0.534243\pi$
$659$	−5.51263	−0.214742	−0.107371	+	0.994219i	$0.534243\pi$
$660$	0	0
$661$	4.49436	0.174810	0.0874052	−	0.996173i	$-0.472143\pi$
$661$	4.49436	0.174810	0.0874052	+	0.996173i	$0.472143\pi$
$662$	31.4053	1.22060
$663$	0	0
$664$	−10.4108	−0.404017
$665$	53.5408	2.07622
$666$	0	0
$667$	23.5509	0.911896
$668$	−1.34457	−0.0520229
$669$	0	0
$670$	11.5178	0.444971
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	43.6433	1.68233	0.841164	−	0.540781i	$-0.181871\pi$
$673$	43.6433	1.68233	0.841164	+	0.540781i	$0.181871\pi$
$674$	−18.8117	−0.724599
$675$	0	0
$676$	14.4409	0.555419
$677$	−33.9134	−1.30340	−0.651699	−	0.758478i	$-0.725943\pi$
$677$	−33.9134	−1.30340	−0.651699	+	0.758478i	$0.725943\pi$
$678$	0	0
$679$	5.28343	0.202759
$680$	79.3533	3.04306
$681$	0	0
$682$	6.01531	0.230338
$683$	−24.0834	−0.921526	−0.460763	−	0.887523i	$-0.652424\pi$
$683$	−24.0834	−0.921526	−0.460763	+	0.887523i	$0.652424\pi$
$684$	0	0
$685$	19.7532	0.754733
$686$	−18.3334	−0.699972
$687$	0	0
$688$	−0.896904	−0.0341941
$689$	−0.0518994	−0.00197721
$690$	0	0
$691$	11.1661	0.424779	0.212390	−	0.977185i	$-0.431875\pi$
$691$	11.1661	0.424779	0.212390	+	0.977185i	$0.431875\pi$
$692$	−2.37446	−0.0902635
$693$	0	0
$694$	−9.59960	−0.364396
$695$	−73.9360	−2.80455
$696$	0	0
$697$	−12.4918	−0.473162
$698$	15.9785	0.604796
$699$	0	0
$700$	−14.5333	−0.549306
$701$	−43.1498	−1.62975	−0.814873	−	0.579639i	$-0.803194\pi$
$701$	−43.1498	−1.62975	−0.814873	+	0.579639i	$0.803194\pi$
$702$	0	0
$703$	−68.5431	−2.58515
$704$	−6.13497	−0.231220
$705$	0	0
$706$	−31.9811	−1.20362
$707$	−7.30523	−0.274741
$708$	0	0
$709$	−13.9514	−0.523956	−0.261978	−	0.965074i	$-0.584375\pi$
$709$	−13.9514	−0.523956	−0.261978	+	0.965074i	$0.584375\pi$
$710$	44.3866	1.66580
$711$	0	0
$712$	10.2525	0.384229
$713$	−16.8226	−0.630012
$714$	0	0
$715$	0.173082	0.00647291
$716$	15.3162	0.572395
$717$	0	0
$718$	4.73233	0.176609
$719$	46.8731	1.74807	0.874036	−	0.485862i	$-0.161494\pi$
$719$	46.8731	1.74807	0.874036	+	0.485862i	$0.161494\pi$
$720$	0	0
$721$	−15.2001	−0.566082
$722$	49.1224	1.82815
$723$	0	0
$724$	−10.0135	−0.372149
$725$	64.2511	2.38623
$726$	0	0
$727$	16.2896	0.604148	0.302074	−	0.953285i	$-0.402321\pi$
$727$	16.2896	0.604148	0.302074	+	0.953285i	$0.402321\pi$
$728$	−0.264372	−0.00979827
$729$	0	0
$730$	46.8354	1.73346
$731$	12.7856	0.472893
$732$	0	0
$733$	26.3688	0.973953	0.486976	−	0.873415i	$-0.338100\pi$
$733$	26.3688	0.973953	0.486976	+	0.873415i	$0.338100\pi$
$734$	−17.8405	−0.658503
$735$	0	0
$736$	14.1175	0.520377
$737$	3.49836	0.128864
$738$	0	0
$739$	−16.4740	−0.606007	−0.303004	−	0.952989i	$-0.597989\pi$
$739$	−16.4740	−0.606007	−0.303004	+	0.952989i	$0.597989\pi$
$740$	31.5375	1.15934
$741$	0	0
$742$	−1.79514	−0.0659016
$743$	31.6315	1.16045	0.580224	−	0.814457i	$-0.302965\pi$
$743$	31.6315	1.16045	0.580224	+	0.814457i	$0.302965\pi$
$744$	0	0
$745$	74.2589	2.72064
$746$	−24.0867	−0.881878
$747$	0	0
$748$	8.60768	0.314728
$749$	0.797130	0.0291265
$750$	0	0
$751$	40.4481	1.47597	0.737986	−	0.674816i	$-0.235777\pi$
$751$	40.4481	1.47597	0.737986	+	0.674816i	$0.235777\pi$
$752$	−2.01060	−0.0733190
$753$	0	0
$754$	0.417406	0.0152010
$755$	28.1408	1.02415
$756$	0	0
$757$	−10.5609	−0.383841	−0.191921	−	0.981410i	$-0.561472\pi$
$757$	−10.5609	−0.383841	−0.191921	+	0.981410i	$0.561472\pi$
$758$	−34.8346	−1.26525
$759$	0	0
$760$	−86.3670	−3.13286
$761$	21.2871	0.771655	0.385828	−	0.922571i	$-0.373916\pi$
$761$	21.2871	0.771655	0.385828	+	0.922571i	$0.373916\pi$
$762$	0	0
$763$	−9.16875	−0.331931
$764$	−7.54934	−0.273125
$765$	0	0
$766$	2.44543	0.0883571
$767$	0.429306	0.0155014
$768$	0	0
$769$	15.1444	0.546120	0.273060	−	0.961997i	$-0.411964\pi$
$769$	15.1444	0.546120	0.273060	+	0.961997i	$0.411964\pi$
$770$	5.98670	0.215746
$771$	0	0
$772$	7.56516	0.272276
$773$	−14.6618	−0.527348	−0.263674	−	0.964612i	$-0.584934\pi$
$773$	−14.6618	−0.527348	−0.263674	+	0.964612i	$0.584934\pi$
$774$	0	0
$775$	−45.8951	−1.64860
$776$	−8.52274	−0.305949
$777$	0	0
$778$	9.23604	0.331128
$779$	13.5959	0.487125
$780$	0	0
$781$	13.4818	0.482416
$782$	19.2605	0.688753
$783$	0	0
$784$	2.00733	0.0716904
$785$	−6.87678	−0.245443
$786$	0	0
$787$	−44.7202	−1.59410	−0.797052	−	0.603910i	$-0.793608\pi$
$787$	−44.7202	−1.59410	−0.797052	+	0.603910i	$0.793608\pi$
$788$	22.1661	0.789636
$789$	0	0
$790$	42.1652	1.50017
$791$	−20.7281	−0.737007
$792$	0	0
$793$	0.0495663	0.00176015
$794$	−7.35940	−0.261175
$795$	0	0
$796$	8.29792	0.294112
$797$	42.3541	1.50026	0.750130	−	0.661290i	$-0.229991\pi$
$797$	42.3541	1.50026	0.750130	+	0.661290i	$0.229991\pi$
$798$	0	0
$799$	28.6616	1.01397
$800$	38.5150	1.36171
$801$	0	0
$802$	−28.3742	−1.00193
$803$	14.2256	0.502009
$804$	0	0
$805$	−16.7426	−0.590100
$806$	−0.298157	−0.0105021
$807$	0	0
$808$	11.7841	0.414564
$809$	−19.8867	−0.699180	−0.349590	−	0.936903i	$-0.613679\pi$
$809$	−19.8867	−0.699180	−0.349590	+	0.936903i	$0.613679\pi$
$810$	0	0
$811$	−12.7950	−0.449292	−0.224646	−	0.974440i	$-0.572123\pi$
$811$	−12.7950	−0.449292	−0.224646	+	0.974440i	$0.572123\pi$
$812$	−18.0447	−0.633244
$813$	0	0
$814$	−7.66420	−0.268630
$815$	72.5144	2.54007
$816$	0	0
$817$	−13.9157	−0.486847
$818$	2.76244	0.0965866
$819$	0	0
$820$	−6.25565	−0.218457
$821$	19.3033	0.673690	0.336845	−	0.941560i	$-0.390640\pi$
$821$	19.3033	0.673690	0.336845	+	0.941560i	$0.390640\pi$
$822$	0	0
$823$	−21.2555	−0.740921	−0.370460	−	0.928848i	$-0.620800\pi$
$823$	−21.2555	−0.740921	−0.370460	+	0.928848i	$0.620800\pi$
$824$	24.5194	0.854175
$825$	0	0
$826$	14.8492	0.516669
$827$	5.50486	0.191423	0.0957114	−	0.995409i	$-0.469487\pi$
$827$	5.50486	0.191423	0.0957114	+	0.995409i	$0.469487\pi$
$828$	0	0
$829$	32.6065	1.13247	0.566235	−	0.824244i	$-0.308400\pi$
$829$	32.6065	1.13247	0.566235	+	0.824244i	$0.308400\pi$
$830$	−11.6854	−0.405605
$831$	0	0
$832$	0.304088	0.0105423
$833$	−28.6150	−0.991452
$834$	0	0
$835$	−4.22587	−0.146242
$836$	−9.36848	−0.324016
$837$	0	0
$838$	−7.77203	−0.268480
$839$	−10.2087	−0.352443	−0.176221	−	0.984351i	$-0.556387\pi$
$839$	−10.2087	−0.352443	−0.176221	+	0.984351i	$0.556387\pi$
$840$	0	0
$841$	50.7750	1.75086
$842$	−0.672546	−0.0231775
$843$	0	0
$844$	−26.5991	−0.915578
$845$	45.3866	1.56135
$846$	0	0
$847$	1.81837	0.0624800
$848$	0.569055	0.0195414
$849$	0	0
$850$	52.5460	1.80231
$851$	21.4340	0.734747
$852$	0	0
$853$	−40.7415	−1.39496	−0.697481	−	0.716603i	$-0.745696\pi$
$853$	−40.7415	−1.39496	−0.697481	+	0.716603i	$0.745696\pi$
$854$	1.71444	0.0586668
$855$	0	0
$856$	−1.28586	−0.0439496
$857$	−41.4107	−1.41456	−0.707282	−	0.706932i	$-0.750079\pi$
$857$	−41.4107	−1.41456	−0.707282	+	0.706932i	$0.750079\pi$
$858$	0	0
$859$	18.5889	0.634244	0.317122	−	0.948385i	$-0.397284\pi$
$859$	18.5889	0.634244	0.317122	+	0.948385i	$0.397284\pi$
$860$	6.40276	0.218332
$861$	0	0
$862$	36.1512	1.23131
$863$	30.9600	1.05389	0.526946	−	0.849899i	$-0.323337\pi$
$863$	30.9600	1.05389	0.526946	+	0.849899i	$0.323337\pi$
$864$	0	0
$865$	−7.46274	−0.253741
$866$	−3.75803	−0.127703
$867$	0	0
$868$	12.8895	0.437497
$869$	12.8071	0.434450
$870$	0	0
$871$	−0.173401	−0.00587545
$872$	14.7902	0.500859
$873$	0	0
$874$	−20.9628	−0.709078
$875$	−13.9287	−0.470875
$876$	0	0
$877$	21.9728	0.741969	0.370985	−	0.928639i	$-0.379020\pi$
$877$	21.9728	0.741969	0.370985	+	0.928639i	$0.379020\pi$
$878$	29.4889	0.995204
$879$	0	0
$880$	−1.89777	−0.0639739
$881$	2.24572	0.0756602	0.0378301	−	0.999284i	$-0.487955\pi$
$881$	2.24572	0.0756602	0.0378301	+	0.999284i	$0.487955\pi$
$882$	0	0
$883$	−17.8354	−0.600208	−0.300104	−	0.953907i	$-0.597021\pi$
$883$	−17.8354	−0.600208	−0.300104	+	0.953907i	$0.597021\pi$
$884$	−0.426651	−0.0143498
$885$	0	0
$886$	11.6551	0.391561
$887$	3.58048	0.120221	0.0601103	−	0.998192i	$-0.480855\pi$
$887$	3.58048	0.120221	0.0601103	+	0.998192i	$0.480855\pi$
$888$	0	0
$889$	30.2641	1.01503
$890$	11.5077	0.385740
$891$	0	0
$892$	−12.0174	−0.402372
$893$	−31.1949	−1.04390
$894$	0	0
$895$	48.1377	1.60907
$896$	−8.95328	−0.299108
$897$	0	0
$898$	−2.32798	−0.0776857
$899$	−56.9839	−1.90052
$900$	0	0
$901$	−8.11202	−0.270251
$902$	1.52024	0.0506185
$903$	0	0
$904$	33.4367	1.11209
$905$	−31.4717	−1.04615
$906$	0	0
$907$	−5.73095	−0.190293	−0.0951465	−	0.995463i	$-0.530332\pi$
$907$	−5.73095	−0.190293	−0.0951465	+	0.995463i	$0.530332\pi$
$908$	−29.2257	−0.969890
$909$	0	0
$910$	−0.296739	−0.00983680
$911$	8.76057	0.290251	0.145125	−	0.989413i	$-0.453641\pi$
$911$	8.76057	0.290251	0.145125	+	0.989413i	$0.453641\pi$
$912$	0	0
$913$	−3.54926	−0.117463
$914$	−26.5550	−0.878362
$915$	0	0
$916$	27.5959	0.911794
$917$	27.3616	0.903561
$918$	0	0
$919$	−43.6126	−1.43865	−0.719323	−	0.694676i	$-0.755548\pi$
$919$	−43.6126	−1.43865	−0.719323	+	0.694676i	$0.755548\pi$
$920$	27.0077	0.890416
$921$	0	0
$922$	−31.2381	−1.02877
$923$	−0.668242	−0.0219954
$924$	0	0
$925$	58.4757	1.92267
$926$	1.79128	0.0588652
$927$	0	0
$928$	47.8206	1.56979
$929$	−16.3467	−0.536317	−0.268159	−	0.963375i	$-0.586415\pi$
$929$	−16.3467	−0.536317	−0.268159	+	0.963375i	$0.586415\pi$
$930$	0	0
$931$	31.1442	1.02071
$932$	16.9146	0.554057
$933$	0	0
$934$	8.70708	0.284904
$935$	27.0532	0.884736
$936$	0	0
$937$	22.0142	0.719171	0.359586	−	0.933112i	$-0.382918\pi$
$937$	22.0142	0.719171	0.359586	+	0.933112i	$0.382918\pi$
$938$	−5.99771	−0.195832
$939$	0	0
$940$	14.3531	0.468148
$941$	40.6955	1.32663	0.663317	−	0.748338i	$-0.269148\pi$
$941$	40.6955	1.32663	0.663317	+	0.748338i	$0.269148\pi$
$942$	0	0
$943$	−4.25156	−0.138450
$944$	−4.70716	−0.153205
$945$	0	0
$946$	−1.55599	−0.0505896
$947$	6.43989	0.209268	0.104634	−	0.994511i	$-0.466633\pi$
$947$	6.43989	0.209268	0.104634	+	0.994511i	$0.466633\pi$
$948$	0	0
$949$	−0.705108	−0.0228888
$950$	−57.1903	−1.85550
$951$	0	0
$952$	−41.3220	−1.33925
$953$	8.62127	0.279270	0.139635	−	0.990203i	$-0.455407\pi$
$953$	8.62127	0.279270	0.139635	+	0.990203i	$0.455407\pi$
$954$	0	0
$955$	−23.7270	−0.767786
$956$	−0.295189	−0.00954710
$957$	0	0
$958$	25.6301	0.828073
$959$	−10.2862	−0.332159
$960$	0	0
$961$	9.70408	0.313035
$962$	0.379886	0.0122480
$963$	0	0
$964$	26.4649	0.852377
$965$	23.7767	0.765399
$966$	0	0
$967$	−10.5897	−0.340541	−0.170270	−	0.985397i	$-0.554464\pi$
$967$	−10.5897	−0.340541	−0.170270	+	0.985397i	$0.554464\pi$
$968$	−2.93323	−0.0942775
$969$	0	0
$970$	−9.56618	−0.307152
$971$	34.3252	1.10155	0.550774	−	0.834654i	$-0.314333\pi$
$971$	34.3252	1.10155	0.550774	+	0.834654i	$0.314333\pi$
$972$	0	0
$973$	38.5010	1.23429
$974$	−31.9660	−1.02426
$975$	0	0
$976$	−0.543473	−0.0173961
$977$	21.8729	0.699775	0.349887	−	0.936792i	$-0.386220\pi$
$977$	21.8729	0.699775	0.349887	+	0.936792i	$0.386220\pi$
$978$	0	0
$979$	3.49530	0.111710
$980$	−14.3298	−0.457749
$981$	0	0
$982$	−6.52495	−0.208220
$983$	20.9511	0.668236	0.334118	−	0.942531i	$-0.391562\pi$
$983$	20.9511	0.668236	0.334118	+	0.942531i	$0.391562\pi$
$984$	0	0
$985$	69.6664	2.21976
$986$	65.2417	2.07772
$987$	0	0
$988$	0.464361	0.0147733
$989$	4.35154	0.138371
$990$	0	0
$991$	30.6419	0.973373	0.486687	−	0.873577i	$-0.338205\pi$
$991$	30.6419	0.973373	0.486687	+	0.873577i	$0.338205\pi$
$992$	−34.1587	−1.08454
$993$	0	0
$994$	−23.1136	−0.733120
$995$	26.0797	0.826781
$996$	0	0
$997$	−47.6634	−1.50951	−0.754757	−	0.656004i	$-0.772245\pi$
$997$	−47.6634	−1.50951	−0.754757	+	0.656004i	$0.772245\pi$
$998$	23.6194	0.747658
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.c.1.7		11
3.2	odd	2		2013.2.a.b.1.5	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.b.1.5	✓	11	3.2	odd	2
6039.2.a.c.1.7		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)

\(f(q)\)	\(=\)	\(q+0.942842 q^{2} -1.11105 q^{4} -3.49194 q^{5} +1.81837 q^{7} -2.93323 q^{8} +O(q^{10})\)	\(q+0.942842 q^{2} -1.11105 q^{4} -3.49194 q^{5} +1.81837 q^{7} -2.93323 q^{8} -3.29234 q^{10} -1.00000 q^{11} +0.0495663 q^{13} +1.71444 q^{14} -0.543473 q^{16} +7.74735 q^{17} -8.43210 q^{19} +3.87971 q^{20} -0.942842 q^{22} +2.63679 q^{23} +7.19362 q^{25} +0.0467332 q^{26} -2.02030 q^{28} +8.93168 q^{29} -6.37998 q^{31} +5.35405 q^{32} +7.30453 q^{34} -6.34964 q^{35} +8.12883 q^{37} -7.95014 q^{38} +10.2426 q^{40} -1.61240 q^{41} +1.65032 q^{43} +1.11105 q^{44} +2.48607 q^{46} +3.69954 q^{47} -3.69353 q^{49} +6.78245 q^{50} -0.0550706 q^{52} -1.04707 q^{53} +3.49194 q^{55} -5.33370 q^{56} +8.42117 q^{58} +8.66125 q^{59} +1.00000 q^{61} -6.01531 q^{62} +6.13497 q^{64} -0.173082 q^{65} -3.49836 q^{67} -8.60768 q^{68} -5.98670 q^{70} -13.4818 q^{71} -14.2256 q^{73} +7.66420 q^{74} +9.36848 q^{76} -1.81837 q^{77} -12.8071 q^{79} +1.89777 q^{80} -1.52024 q^{82} +3.54926 q^{83} -27.0532 q^{85} +1.55599 q^{86} +2.93323 q^{88} -3.49530 q^{89} +0.0901300 q^{91} -2.92960 q^{92} +3.48808 q^{94} +29.4444 q^{95} +2.90558 q^{97} -3.48241 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{2} + 10 q^{4} + q^{5} - 11 q^{7} + 3 q^{8}+O(q^{10}) \) 11 * q + 2 * q^2 + 10 * q^4 + q^5 - 11 * q^7 + 3 * q^8	\( 11 q + 2 q^{2} + 10 q^{4} + q^{5} - 11 q^{7} + 3 q^{8} - 8 q^{10} - 11 q^{11} - 13 q^{13} - 5 q^{14} + 4 q^{16} + 13 q^{17} - 12 q^{19} + 7 q^{20} - 2 q^{22} + 3 q^{23} + 12 q^{25} - 12 q^{26} - 13 q^{28} - 2 q^{29} + q^{31} + 23 q^{32} - 14 q^{34} + 4 q^{35} - 14 q^{37} + 8 q^{38} - 34 q^{40} - 3 q^{41} - 21 q^{43} - 10 q^{44} - 12 q^{46} + 16 q^{47} - 18 q^{49} + 13 q^{50} - 33 q^{52} - q^{55} - 16 q^{56} - 17 q^{58} - 3 q^{59} + 11 q^{61} + 21 q^{62} - 7 q^{64} + q^{65} - 24 q^{67} - 2 q^{68} + 4 q^{70} - 7 q^{71} - 42 q^{73} + 16 q^{74} - 13 q^{76} + 11 q^{77} - 11 q^{79} - 42 q^{80} - 38 q^{82} + 34 q^{83} - 14 q^{85} - 42 q^{86} - 3 q^{88} - 29 q^{89} + 9 q^{91} - 42 q^{92} - 33 q^{94} + 31 q^{95} - 45 q^{97} + 33 q^{98}+O(q^{100}) \) 11 * q + 2 * q^2 + 10 * q^4 + q^5 - 11 * q^7 + 3 * q^8 - 8 * q^10 - 11 * q^11 - 13 * q^13 - 5 * q^14 + 4 * q^16 + 13 * q^17 - 12 * q^19 + 7 * q^20 - 2 * q^22 + 3 * q^23 + 12 * q^25 - 12 * q^26 - 13 * q^28 - 2 * q^29 + q^31 + 23 * q^32 - 14 * q^34 + 4 * q^35 - 14 * q^37 + 8 * q^38 - 34 * q^40 - 3 * q^41 - 21 * q^43 - 10 * q^44 - 12 * q^46 + 16 * q^47 - 18 * q^49 + 13 * q^50 - 33 * q^52 - q^55 - 16 * q^56 - 17 * q^58 - 3 * q^59 + 11 * q^61 + 21 * q^62 - 7 * q^64 + q^65 - 24 * q^67 - 2 * q^68 + 4 * q^70 - 7 * q^71 - 42 * q^73 + 16 * q^74 - 13 * q^76 + 11 * q^77 - 11 * q^79 - 42 * q^80 - 38 * q^82 + 34 * q^83 - 14 * q^85 - 42 * q^86 - 3 * q^88 - 29 * q^89 + 9 * q^91 - 42 * q^92 - 33 * q^94 + 31 * q^95 - 45 * q^97 + 33 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more