LMFDB - Embedded newform 6039.2.a.o.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:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-2.47686 q^{2} +4.13484 q^{4} +0.821809 q^{5} -3.73620 q^{7} -5.28771 q^{8} +O(q^{10})$	\(q-2.47686 q^{2} +4.13484 q^{4} +0.821809 q^{5} -3.73620 q^{7} -5.28771 q^{8} -2.03551 q^{10} +1.00000 q^{11} +6.75247 q^{13} +9.25405 q^{14} +4.82724 q^{16} +4.83541 q^{17} -7.06509 q^{19} +3.39805 q^{20} -2.47686 q^{22} -2.72642 q^{23} -4.32463 q^{25} -16.7249 q^{26} -15.4486 q^{28} +3.71896 q^{29} +3.15172 q^{31} -1.38099 q^{32} -11.9767 q^{34} -3.07044 q^{35} -3.39069 q^{37} +17.4992 q^{38} -4.34549 q^{40} +5.93952 q^{41} +5.77004 q^{43} +4.13484 q^{44} +6.75296 q^{46} +1.41937 q^{47} +6.95919 q^{49} +10.7115 q^{50} +27.9204 q^{52} +7.67983 q^{53} +0.821809 q^{55} +19.7559 q^{56} -9.21134 q^{58} -11.9817 q^{59} -1.00000 q^{61} -7.80638 q^{62} -6.23397 q^{64} +5.54924 q^{65} -8.35493 q^{67} +19.9937 q^{68} +7.60506 q^{70} +7.46667 q^{71} -6.87191 q^{73} +8.39828 q^{74} -29.2130 q^{76} -3.73620 q^{77} -0.687247 q^{79} +3.96707 q^{80} -14.7114 q^{82} +11.9993 q^{83} +3.97379 q^{85} -14.2916 q^{86} -5.28771 q^{88} +18.7090 q^{89} -25.2286 q^{91} -11.2733 q^{92} -3.51559 q^{94} -5.80615 q^{95} +14.3172 q^{97} -17.2370 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.47686	−1.75141	−0.875703	−	0.482851i	$-0.839601\pi$
$2$	−2.47686	−1.75141	−0.875703	+	0.482851i	$0.839601\pi$
$3$	0	0
$3$	0	0
$4$	4.13484	2.06742
$5$	0.821809	0.367524	0.183762	−	0.982971i	$-0.441172\pi$
$5$	0.821809	0.367524	0.183762	+	0.982971i	$0.441172\pi$
$6$	0	0
$7$	−3.73620	−1.41215	−0.706076	−	0.708137i	$-0.749536\pi$
$7$	−3.73620	−1.41215	−0.706076	+	0.708137i	$0.749536\pi$
$8$	−5.28771	−1.86949
$9$	0	0
$10$	−2.03551	−0.643684
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.75247	1.87280	0.936399	−	0.350936i	$-0.114137\pi$
$13$	6.75247	1.87280	0.936399	+	0.350936i	$0.114137\pi$
$14$	9.25405	2.47325
$15$	0	0
$16$	4.82724	1.20681
$17$	4.83541	1.17276	0.586380	−	0.810036i	$-0.300552\pi$
$17$	4.83541	1.17276	0.586380	+	0.810036i	$0.300552\pi$
$18$	0	0
$19$	−7.06509	−1.62084	−0.810421	−	0.585848i	$-0.800762\pi$
$19$	−7.06509	−1.62084	−0.810421	+	0.585848i	$0.800762\pi$
$20$	3.39805	0.759827
$21$	0	0
$22$	−2.47686	−0.528069
$23$	−2.72642	−0.568498	−0.284249	−	0.958751i	$-0.591744\pi$
$23$	−2.72642	−0.568498	−0.284249	+	0.958751i	$0.591744\pi$
$24$	0	0
$25$	−4.32463	−0.864926
$26$	−16.7249	−3.28003
$27$	0	0
$28$	−15.4486	−2.91951
$29$	3.71896	0.690593	0.345296	−	0.938494i	$-0.387778\pi$
$29$	3.71896	0.690593	0.345296	+	0.938494i	$0.387778\pi$
$30$	0	0
$31$	3.15172	0.566066	0.283033	−	0.959110i	$-0.408659\pi$
$31$	3.15172	0.566066	0.283033	+	0.959110i	$0.408659\pi$
$32$	−1.38099	−0.244126
$33$	0	0
$34$	−11.9767	−2.05398
$35$	−3.07044	−0.518999
$36$	0	0
$37$	−3.39069	−0.557427	−0.278713	−	0.960374i	$-0.589908\pi$
$37$	−3.39069	−0.557427	−0.278713	+	0.960374i	$0.589908\pi$
$38$	17.4992	2.83875
$39$	0	0
$40$	−4.34549	−0.687082
$41$	5.93952	0.927597	0.463799	−	0.885941i	$-0.346486\pi$
$41$	5.93952	0.927597	0.463799	+	0.885941i	$0.346486\pi$
$42$	0	0
$43$	5.77004	0.879923	0.439961	−	0.898017i	$-0.354992\pi$
$43$	5.77004	0.879923	0.439961	+	0.898017i	$0.354992\pi$
$44$	4.13484	0.623351
$45$	0	0
$46$	6.75296	0.995670
$47$	1.41937	0.207037	0.103518	−	0.994628i	$-0.466990\pi$
$47$	1.41937	0.207037	0.103518	+	0.994628i	$0.466990\pi$
$48$	0	0
$49$	6.95919	0.994171
$50$	10.7115	1.51484
$51$	0	0
$52$	27.9204	3.87186
$53$	7.67983	1.05491	0.527453	−	0.849584i	$-0.323147\pi$
$53$	7.67983	1.05491	0.527453	+	0.849584i	$0.323147\pi$
$54$	0	0
$55$	0.821809	0.110813
$56$	19.7559	2.64000
$57$	0	0
$58$	−9.21134	−1.20951
$59$	−11.9817	−1.55988	−0.779941	−	0.625853i	$-0.784751\pi$
$59$	−11.9817	−1.55988	−0.779941	+	0.625853i	$0.784751\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−7.80638	−0.991411
$63$	0	0
$64$	−6.23397	−0.779246
$65$	5.54924	0.688299
$66$	0	0
$67$	−8.35493	−1.02072	−0.510359	−	0.859962i	$-0.670487\pi$
$67$	−8.35493	−1.02072	−0.510359	+	0.859962i	$0.670487\pi$
$68$	19.9937	2.42459
$69$	0	0
$70$	7.60506	0.908979
$71$	7.46667	0.886130	0.443065	−	0.896489i	$-0.353891\pi$
$71$	7.46667	0.886130	0.443065	+	0.896489i	$0.353891\pi$
$72$	0	0
$73$	−6.87191	−0.804297	−0.402148	−	0.915575i	$-0.631736\pi$
$73$	−6.87191	−0.804297	−0.402148	+	0.915575i	$0.631736\pi$
$74$	8.39828	0.976280
$75$	0	0
$76$	−29.2130	−3.35096
$77$	−3.73620	−0.425780
$78$	0	0
$79$	−0.687247	−0.0773213	−0.0386607	−	0.999252i	$-0.512309\pi$
$79$	−0.687247	−0.0773213	−0.0386607	+	0.999252i	$0.512309\pi$
$80$	3.96707	0.443532
$81$	0	0
$82$	−14.7114	−1.62460
$83$	11.9993	1.31710	0.658550	−	0.752537i	$-0.271170\pi$
$83$	11.9993	1.31710	0.658550	+	0.752537i	$0.271170\pi$
$84$	0	0
$85$	3.97379	0.431018
$86$	−14.2916	−1.54110
$87$	0	0
$88$	−5.28771	−0.563672
$89$	18.7090	1.98315	0.991575	−	0.129533i	$-0.0413478\pi$
$89$	18.7090	1.98315	0.991575	+	0.129533i	$0.0413478\pi$
$90$	0	0
$91$	−25.2286	−2.64467
$92$	−11.2733	−1.17532
$93$	0	0
$94$	−3.51559	−0.362606
$95$	−5.80615	−0.595698
$96$	0	0
$97$	14.3172	1.45369	0.726846	−	0.686801i	$-0.240986\pi$
$97$	14.3172	1.45369	0.726846	+	0.686801i	$0.240986\pi$
$98$	−17.2370	−1.74120
$99$	0	0
$100$	−17.8817	−1.78817
$101$	−2.46806	−0.245581	−0.122791	−	0.992433i	$-0.539184\pi$
$101$	−2.46806	−0.245581	−0.122791	+	0.992433i	$0.539184\pi$
$102$	0	0
$103$	−8.78301	−0.865416	−0.432708	−	0.901534i	$-0.642442\pi$
$103$	−8.78301	−0.865416	−0.432708	+	0.901534i	$0.642442\pi$
$104$	−35.7051	−3.50117
$105$	0	0
$106$	−19.0219	−1.84757
$107$	−8.68217	−0.839337	−0.419669	−	0.907677i	$-0.637854\pi$
$107$	−8.68217	−0.839337	−0.419669	+	0.907677i	$0.637854\pi$
$108$	0	0
$109$	1.41823	0.135842	0.0679209	−	0.997691i	$-0.478363\pi$
$109$	1.41823	0.135842	0.0679209	+	0.997691i	$0.478363\pi$
$110$	−2.03551	−0.194078
$111$	0	0
$112$	−18.0355	−1.70420
$113$	−16.8145	−1.58177	−0.790887	−	0.611963i	$-0.790380\pi$
$113$	−16.8145	−1.58177	−0.790887	+	0.611963i	$0.790380\pi$
$114$	0	0
$115$	−2.24060	−0.208937
$116$	15.3773	1.42775
$117$	0	0
$118$	29.6770	2.73199
$119$	−18.0661	−1.65611
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.47686	0.224245
$123$	0	0
$124$	13.0319	1.17030
$125$	−7.66306	−0.685405
$126$	0	0
$127$	−6.19343	−0.549578	−0.274789	−	0.961505i	$-0.588608\pi$
$127$	−6.19343	−0.549578	−0.274789	+	0.961505i	$0.588608\pi$
$128$	18.2027	1.60890
$129$	0	0
$130$	−13.7447	−1.20549
$131$	−5.46443	−0.477430	−0.238715	−	0.971090i	$-0.576726\pi$
$131$	−5.46443	−0.477430	−0.238715	+	0.971090i	$0.576726\pi$
$132$	0	0
$133$	26.3966	2.28887
$134$	20.6940	1.78769
$135$	0	0
$136$	−25.5683	−2.19246
$137$	8.10373	0.692348	0.346174	−	0.938170i	$-0.387481\pi$
$137$	8.10373	0.692348	0.346174	+	0.938170i	$0.387481\pi$
$138$	0	0
$139$	7.21865	0.612278	0.306139	−	0.951987i	$-0.400963\pi$
$139$	7.21865	0.612278	0.306139	+	0.951987i	$0.400963\pi$
$140$	−12.6958	−1.07299
$141$	0	0
$142$	−18.4939	−1.55197
$143$	6.75247	0.564670
$144$	0	0
$145$	3.05627	0.253810
$146$	17.0208	1.40865
$147$	0	0
$148$	−14.0200	−1.15244
$149$	4.48415	0.367356	0.183678	−	0.982986i	$-0.441200\pi$
$149$	4.48415	0.367356	0.183678	+	0.982986i	$0.441200\pi$
$150$	0	0
$151$	17.9415	1.46006	0.730031	−	0.683414i	$-0.239506\pi$
$151$	17.9415	1.46006	0.730031	+	0.683414i	$0.239506\pi$
$152$	37.3581	3.03014
$153$	0	0
$154$	9.25405	0.745713
$155$	2.59011	0.208043
$156$	0	0
$157$	−8.12082	−0.648112	−0.324056	−	0.946038i	$-0.605047\pi$
$157$	−8.12082	−0.648112	−0.324056	+	0.946038i	$0.605047\pi$
$158$	1.70222	0.135421
$159$	0	0
$160$	−1.13491	−0.0897222
$161$	10.1864	0.802805
$162$	0	0
$163$	−21.4823	−1.68263	−0.841313	−	0.540548i	$-0.818217\pi$
$163$	−21.4823	−1.68263	−0.841313	+	0.540548i	$0.818217\pi$
$164$	24.5590	1.91773
$165$	0	0
$166$	−29.7207	−2.30678
$167$	9.29874	0.719558	0.359779	−	0.933038i	$-0.382852\pi$
$167$	9.29874	0.719558	0.359779	+	0.933038i	$0.382852\pi$
$168$	0	0
$169$	32.5959	2.50738
$170$	−9.84252	−0.754887
$171$	0	0
$172$	23.8582	1.81917
$173$	−9.00472	−0.684616	−0.342308	−	0.939588i	$-0.611209\pi$
$173$	−9.00472	−0.684616	−0.342308	+	0.939588i	$0.611209\pi$
$174$	0	0
$175$	16.1577	1.22141
$176$	4.82724	0.363867
$177$	0	0
$178$	−46.3396	−3.47330
$179$	−6.21995	−0.464901	−0.232451	−	0.972608i	$-0.574674\pi$
$179$	−6.21995	−0.464901	−0.232451	+	0.972608i	$0.574674\pi$
$180$	0	0
$181$	−17.2017	−1.27859	−0.639295	−	0.768962i	$-0.720774\pi$
$181$	−17.2017	−1.27859	−0.639295	+	0.768962i	$0.720774\pi$
$182$	62.4877	4.63190
$183$	0	0
$184$	14.4165	1.06280
$185$	−2.78650	−0.204868
$186$	0	0
$187$	4.83541	0.353601
$188$	5.86889	0.428032
$189$	0	0
$190$	14.3810	1.04331
$191$	18.5876	1.34495	0.672475	−	0.740120i	$-0.265231\pi$
$191$	18.5876	1.34495	0.672475	+	0.740120i	$0.265231\pi$
$192$	0	0
$193$	3.59810	0.258997	0.129499	−	0.991580i	$-0.458663\pi$
$193$	3.59810	0.258997	0.129499	+	0.991580i	$0.458663\pi$
$194$	−35.4617	−2.54600
$195$	0	0
$196$	28.7752	2.05537
$197$	6.69489	0.476991	0.238496	−	0.971144i	$-0.423346\pi$
$197$	6.69489	0.476991	0.238496	+	0.971144i	$0.423346\pi$
$198$	0	0
$199$	13.2642	0.940277	0.470138	−	0.882593i	$-0.344204\pi$
$199$	13.2642	0.940277	0.470138	+	0.882593i	$0.344204\pi$
$200$	22.8674	1.61697
$201$	0	0
$202$	6.11305	0.430113
$203$	−13.8948	−0.975222
$204$	0	0
$205$	4.88115	0.340914
$206$	21.7543	1.51569
$207$	0	0
$208$	32.5958	2.26011
$209$	−7.06509	−0.488702
$210$	0	0
$211$	25.6669	1.76698	0.883490	−	0.468450i	$-0.155188\pi$
$211$	25.6669	1.76698	0.883490	+	0.468450i	$0.155188\pi$
$212$	31.7549	2.18093
$213$	0	0
$214$	21.5045	1.47002
$215$	4.74187	0.323393
$216$	0	0
$217$	−11.7755	−0.799371
$218$	−3.51276	−0.237914
$219$	0	0
$220$	3.39805	0.229096
$221$	32.6510	2.19634
$222$	0	0
$223$	−9.49674	−0.635949	−0.317974	−	0.948099i	$-0.603003\pi$
$223$	−9.49674	−0.635949	−0.317974	+	0.948099i	$0.603003\pi$
$224$	5.15964	0.344743
$225$	0	0
$226$	41.6471	2.77033
$227$	5.20526	0.345485	0.172743	−	0.984967i	$-0.444737\pi$
$227$	5.20526	0.345485	0.172743	+	0.984967i	$0.444737\pi$
$228$	0	0
$229$	−2.64473	−0.174768	−0.0873842	−	0.996175i	$-0.527851\pi$
$229$	−2.64473	−0.174768	−0.0873842	+	0.996175i	$0.527851\pi$
$230$	5.54964	0.365933
$231$	0	0
$232$	−19.6648	−1.29106
$233$	−6.86751	−0.449905	−0.224953	−	0.974370i	$-0.572223\pi$
$233$	−6.86751	−0.449905	−0.224953	+	0.974370i	$0.572223\pi$
$234$	0	0
$235$	1.16645	0.0760910
$236$	−49.5424	−3.22493
$237$	0	0
$238$	44.7472	2.90053
$239$	13.7526	0.889583	0.444791	−	0.895634i	$-0.353278\pi$
$239$	13.7526	0.889583	0.444791	+	0.895634i	$0.353278\pi$
$240$	0	0
$241$	7.00545	0.451261	0.225630	−	0.974213i	$-0.427556\pi$
$241$	7.00545	0.451261	0.225630	+	0.974213i	$0.427556\pi$
$242$	−2.47686	−0.159219
$243$	0	0
$244$	−4.13484	−0.264706
$245$	5.71913	0.365382
$246$	0	0
$247$	−47.7068	−3.03551
$248$	−16.6654	−1.05825
$249$	0	0
$250$	18.9803	1.20042
$251$	11.5698	0.730281	0.365140	−	0.930952i	$-0.381021\pi$
$251$	11.5698	0.730281	0.365140	+	0.930952i	$0.381021\pi$
$252$	0	0
$253$	−2.72642	−0.171409
$254$	15.3403	0.962534
$255$	0	0
$256$	−32.6175	−2.03859
$257$	−18.8280	−1.17446	−0.587229	−	0.809421i	$-0.699781\pi$
$257$	−18.8280	−1.17446	−0.587229	+	0.809421i	$0.699781\pi$
$258$	0	0
$259$	12.6683	0.787171
$260$	22.9452	1.42300
$261$	0	0
$262$	13.5346	0.836173
$263$	−24.4782	−1.50939	−0.754697	−	0.656074i	$-0.772216\pi$
$263$	−24.4782	−1.50939	−0.754697	+	0.656074i	$0.772216\pi$
$264$	0	0
$265$	6.31135	0.387703
$266$	−65.3807	−4.00875
$267$	0	0
$268$	−34.5463	−2.11025
$269$	−17.0492	−1.03951	−0.519755	−	0.854315i	$-0.673977\pi$
$269$	−17.0492	−1.03951	−0.519755	+	0.854315i	$0.673977\pi$
$270$	0	0
$271$	0.362668	0.0220305	0.0110152	−	0.999939i	$-0.496494\pi$
$271$	0.362668	0.0220305	0.0110152	+	0.999939i	$0.496494\pi$
$272$	23.3417	1.41530
$273$	0	0
$274$	−20.0718	−1.21258
$275$	−4.32463	−0.260785
$276$	0	0
$277$	−20.0124	−1.20243	−0.601216	−	0.799087i	$-0.705317\pi$
$277$	−20.0124	−1.20243	−0.601216	+	0.799087i	$0.705317\pi$
$278$	−17.8796	−1.07235
$279$	0	0
$280$	16.2356	0.970263
$281$	2.95859	0.176494	0.0882472	−	0.996099i	$-0.471873\pi$
$281$	2.95859	0.176494	0.0882472	+	0.996099i	$0.471873\pi$
$282$	0	0
$283$	17.9188	1.06516	0.532580	−	0.846380i	$-0.321223\pi$
$283$	17.9188	1.06516	0.532580	+	0.846380i	$0.321223\pi$
$284$	30.8735	1.83200
$285$	0	0
$286$	−16.7249	−0.988966
$287$	−22.1912	−1.30991
$288$	0	0
$289$	6.38124	0.375367
$290$	−7.56996	−0.444523
$291$	0	0
$292$	−28.4143	−1.66282
$293$	18.2802	1.06794	0.533970	−	0.845504i	$-0.320700\pi$
$293$	18.2802	1.06794	0.533970	+	0.845504i	$0.320700\pi$
$294$	0	0
$295$	−9.84665	−0.573294
$296$	17.9290	1.04210
$297$	0	0
$298$	−11.1066	−0.643389
$299$	−18.4101	−1.06468
$300$	0	0
$301$	−21.5580	−1.24258
$302$	−44.4387	−2.55716
$303$	0	0
$304$	−34.1049	−1.95605
$305$	−0.821809	−0.0470566
$306$	0	0
$307$	18.9319	1.08050	0.540250	−	0.841504i	$-0.318330\pi$
$307$	18.9319	1.08050	0.540250	+	0.841504i	$0.318330\pi$
$308$	−15.4486	−0.880266
$309$	0	0
$310$	−6.41535	−0.364368
$311$	15.5714	0.882973	0.441487	−	0.897268i	$-0.354451\pi$
$311$	15.5714	0.882973	0.441487	+	0.897268i	$0.354451\pi$
$312$	0	0
$313$	14.8521	0.839488	0.419744	−	0.907643i	$-0.362120\pi$
$313$	14.8521	0.839488	0.419744	+	0.907643i	$0.362120\pi$
$314$	20.1142	1.13511
$315$	0	0
$316$	−2.84166	−0.159856
$317$	−8.59120	−0.482530	−0.241265	−	0.970459i	$-0.577562\pi$
$317$	−8.59120	−0.482530	−0.241265	+	0.970459i	$0.577562\pi$
$318$	0	0
$319$	3.71896	0.208222
$320$	−5.12313	−0.286392
$321$	0	0
$322$	−25.2304	−1.40604
$323$	−34.1626	−1.90086
$324$	0	0
$325$	−29.2019	−1.61983
$326$	53.2088	2.94696
$327$	0	0
$328$	−31.4065	−1.73413
$329$	−5.30306	−0.292367
$330$	0	0
$331$	−28.8345	−1.58489	−0.792444	−	0.609945i	$-0.791192\pi$
$331$	−28.8345	−1.58489	−0.792444	+	0.609945i	$0.791192\pi$
$332$	49.6154	2.72300
$333$	0	0
$334$	−23.0317	−1.26024
$335$	−6.86616	−0.375138
$336$	0	0
$337$	−25.5173	−1.39001	−0.695007	−	0.719003i	$-0.744599\pi$
$337$	−25.5173	−1.39001	−0.695007	+	0.719003i	$0.744599\pi$
$338$	−80.7355	−4.39143
$339$	0	0
$340$	16.4310	0.891095
$341$	3.15172	0.170675
$342$	0	0
$343$	0.152459	0.00823200
$344$	−30.5103	−1.64501
$345$	0	0
$346$	22.3035	1.19904
$347$	−27.3665	−1.46911	−0.734556	−	0.678548i	$-0.762610\pi$
$347$	−27.3665	−1.46911	−0.734556	+	0.678548i	$0.762610\pi$
$348$	0	0
$349$	27.7991	1.48805	0.744026	−	0.668150i	$-0.232914\pi$
$349$	27.7991	1.48805	0.744026	+	0.668150i	$0.232914\pi$
$350$	−40.0204	−2.13918
$351$	0	0
$352$	−1.38099	−0.0736068
$353$	8.58631	0.457003	0.228502	−	0.973544i	$-0.426617\pi$
$353$	8.58631	0.457003	0.228502	+	0.973544i	$0.426617\pi$
$354$	0	0
$355$	6.13617	0.325674
$356$	77.3588	4.10001
$357$	0	0
$358$	15.4060	0.814230
$359$	19.7629	1.04305	0.521524	−	0.853237i	$-0.325364\pi$
$359$	19.7629	1.04305	0.521524	+	0.853237i	$0.325364\pi$
$360$	0	0
$361$	30.9154	1.62713
$362$	42.6061	2.23933
$363$	0	0
$364$	−104.316	−5.46766
$365$	−5.64740	−0.295598
$366$	0	0
$367$	7.56024	0.394641	0.197321	−	0.980339i	$-0.436776\pi$
$367$	7.56024	0.394641	0.197321	+	0.980339i	$0.436776\pi$
$368$	−13.1611	−0.686069
$369$	0	0
$370$	6.90178	0.358806
$371$	−28.6934	−1.48969
$372$	0	0
$373$	33.5487	1.73709	0.868543	−	0.495613i	$-0.165057\pi$
$373$	33.5487	1.73709	0.868543	+	0.495613i	$0.165057\pi$
$374$	−11.9767	−0.619298
$375$	0	0
$376$	−7.50523	−0.387053
$377$	25.1122	1.29334
$378$	0	0
$379$	−22.1699	−1.13879	−0.569396	−	0.822063i	$-0.692823\pi$
$379$	−22.1699	−1.13879	−0.569396	+	0.822063i	$0.692823\pi$
$380$	−24.0075	−1.23156
$381$	0	0
$382$	−46.0389	−2.35555
$383$	−2.24937	−0.114938	−0.0574688	−	0.998347i	$-0.518303\pi$
$383$	−2.24937	−0.114938	−0.0574688	+	0.998347i	$0.518303\pi$
$384$	0	0
$385$	−3.07044	−0.156484
$386$	−8.91200	−0.453609
$387$	0	0
$388$	59.1994	3.00539
$389$	27.9029	1.41473	0.707366	−	0.706848i	$-0.249883\pi$
$389$	27.9029	1.41473	0.707366	+	0.706848i	$0.249883\pi$
$390$	0	0
$391$	−13.1834	−0.666712
$392$	−36.7982	−1.85859
$393$	0	0
$394$	−16.5823	−0.835405
$395$	−0.564786	−0.0284175
$396$	0	0
$397$	−17.5939	−0.883011	−0.441506	−	0.897259i	$-0.645555\pi$
$397$	−17.5939	−0.883011	−0.441506	+	0.897259i	$0.645555\pi$
$398$	−32.8537	−1.64681
$399$	0	0
$400$	−20.8760	−1.04380
$401$	34.5684	1.72626	0.863132	−	0.504978i	$-0.168499\pi$
$401$	34.5684	1.72626	0.863132	+	0.504978i	$0.168499\pi$
$402$	0	0
$403$	21.2819	1.06013
$404$	−10.2051	−0.507720
$405$	0	0
$406$	34.4154	1.70801
$407$	−3.39069	−0.168070
$408$	0	0
$409$	29.2598	1.44681	0.723403	−	0.690426i	$-0.242577\pi$
$409$	29.2598	1.44681	0.723403	+	0.690426i	$0.242577\pi$
$410$	−12.0899	−0.597079
$411$	0	0
$412$	−36.3164	−1.78918
$413$	44.7660	2.20279
$414$	0	0
$415$	9.86117	0.484066
$416$	−9.32507	−0.457199
$417$	0	0
$418$	17.4992	0.855916
$419$	−6.64861	−0.324806	−0.162403	−	0.986725i	$-0.551924\pi$
$419$	−6.64861	−0.324806	−0.162403	+	0.986725i	$0.551924\pi$
$420$	0	0
$421$	11.4807	0.559533	0.279766	−	0.960068i	$-0.409743\pi$
$421$	11.4807	0.559533	0.279766	+	0.960068i	$0.409743\pi$
$422$	−63.5733	−3.09470
$423$	0	0
$424$	−40.6087	−1.97213
$425$	−20.9114	−1.01435
$426$	0	0
$427$	3.73620	0.180807
$428$	−35.8994	−1.73526
$429$	0	0
$430$	−11.7450	−0.566392
$431$	−34.4051	−1.65724	−0.828618	−	0.559815i	$-0.810872\pi$
$431$	−34.4051	−1.65724	−0.828618	+	0.559815i	$0.810872\pi$
$432$	0	0
$433$	−0.705505	−0.0339044	−0.0169522	−	0.999856i	$-0.505396\pi$
$433$	−0.705505	−0.0339044	−0.0169522	+	0.999856i	$0.505396\pi$
$434$	29.1662	1.40002
$435$	0	0
$436$	5.86415	0.280842
$437$	19.2624	0.921445
$438$	0	0
$439$	21.0564	1.00497	0.502485	−	0.864586i	$-0.332419\pi$
$439$	21.0564	1.00497	0.502485	+	0.864586i	$0.332419\pi$
$440$	−4.34549	−0.207163
$441$	0	0
$442$	−80.8720	−3.84669
$443$	−15.9966	−0.760019	−0.380010	−	0.924983i	$-0.624079\pi$
$443$	−15.9966	−0.760019	−0.380010	+	0.924983i	$0.624079\pi$
$444$	0	0
$445$	15.3752	0.728855
$446$	23.5221	1.11380
$447$	0	0
$448$	23.2914	1.10041
$449$	3.10396	0.146485	0.0732425	−	0.997314i	$-0.476665\pi$
$449$	3.10396	0.146485	0.0732425	+	0.997314i	$0.476665\pi$
$450$	0	0
$451$	5.93952	0.279681
$452$	−69.5252	−3.27019
$453$	0	0
$454$	−12.8927	−0.605084
$455$	−20.7331	−0.971982
$456$	0	0
$457$	1.19343	0.0558264	0.0279132	−	0.999610i	$-0.491114\pi$
$457$	1.19343	0.0558264	0.0279132	+	0.999610i	$0.491114\pi$
$458$	6.55062	0.306090
$459$	0	0
$460$	−9.26451	−0.431960
$461$	23.9933	1.11748	0.558741	−	0.829342i	$-0.311285\pi$
$461$	23.9933	1.11748	0.558741	+	0.829342i	$0.311285\pi$
$462$	0	0
$463$	29.7230	1.38134	0.690672	−	0.723168i	$-0.257315\pi$
$463$	29.7230	1.38134	0.690672	+	0.723168i	$0.257315\pi$
$464$	17.9523	0.833415
$465$	0	0
$466$	17.0099	0.787967
$467$	−27.1088	−1.25445	−0.627223	−	0.778840i	$-0.715808\pi$
$467$	−27.1088	−1.25445	−0.627223	+	0.778840i	$0.715808\pi$
$468$	0	0
$469$	31.2157	1.44141
$470$	−2.88914	−0.133266
$471$	0	0
$472$	63.3557	2.91618
$473$	5.77004	0.265307
$474$	0	0
$475$	30.5539	1.40191
$476$	−74.7004	−3.42389
$477$	0	0
$478$	−34.0633	−1.55802
$479$	−18.2617	−0.834397	−0.417199	−	0.908815i	$-0.636988\pi$
$479$	−18.2617	−0.834397	−0.417199	+	0.908815i	$0.636988\pi$
$480$	0	0
$481$	−22.8956	−1.04395
$482$	−17.3515	−0.790340
$483$	0	0
$484$	4.13484	0.187947
$485$	11.7660	0.534266
$486$	0	0
$487$	27.2273	1.23379	0.616893	−	0.787047i	$-0.288391\pi$
$487$	27.2273	1.23379	0.616893	+	0.787047i	$0.288391\pi$
$488$	5.28771	0.239363
$489$	0	0
$490$	−14.1655	−0.639931
$491$	8.52648	0.384795	0.192397	−	0.981317i	$-0.438374\pi$
$491$	8.52648	0.384795	0.192397	+	0.981317i	$0.438374\pi$
$492$	0	0
$493$	17.9827	0.809900
$494$	118.163	5.31641
$495$	0	0
$496$	15.2141	0.683134
$497$	−27.8970	−1.25135
$498$	0	0
$499$	−4.80164	−0.214951	−0.107475	−	0.994208i	$-0.534277\pi$
$499$	−4.80164	−0.214951	−0.107475	+	0.994208i	$0.534277\pi$
$500$	−31.6856	−1.41702
$501$	0	0
$502$	−28.6569	−1.27902
$503$	19.4829	0.868700	0.434350	−	0.900744i	$-0.356978\pi$
$503$	19.4829	0.868700	0.434350	+	0.900744i	$0.356978\pi$
$504$	0	0
$505$	−2.02828	−0.0902571
$506$	6.75296	0.300206
$507$	0	0
$508$	−25.6089	−1.13621
$509$	13.6959	0.607059	0.303529	−	0.952822i	$-0.401835\pi$
$509$	13.6959	0.607059	0.303529	+	0.952822i	$0.401835\pi$
$510$	0	0
$511$	25.6749	1.13579
$512$	44.3838	1.96150
$513$	0	0
$514$	46.6343	2.05695
$515$	−7.21796	−0.318061
$516$	0	0
$517$	1.41937	0.0624240
$518$	−31.3777	−1.37866
$519$	0	0
$520$	−29.3428	−1.28677
$521$	−34.9033	−1.52914	−0.764570	−	0.644540i	$-0.777049\pi$
$521$	−34.9033	−1.52914	−0.764570	+	0.644540i	$0.777049\pi$
$522$	0	0
$523$	18.0407	0.788864	0.394432	−	0.918925i	$-0.370941\pi$
$523$	18.0407	0.788864	0.394432	+	0.918925i	$0.370941\pi$
$524$	−22.5946	−0.987048
$525$	0	0
$526$	60.6292	2.64356
$527$	15.2399	0.663860
$528$	0	0
$529$	−15.5666	−0.676810
$530$	−15.6323	−0.679025
$531$	0	0
$532$	109.146	4.73207
$533$	40.1065	1.73720
$534$	0	0
$535$	−7.13508	−0.308477
$536$	44.1785	1.90822
$537$	0	0
$538$	42.2286	1.82060
$539$	6.95919	0.299754
$540$	0	0
$541$	12.0043	0.516107	0.258053	−	0.966131i	$-0.416919\pi$
$541$	12.0043	0.516107	0.258053	+	0.966131i	$0.416919\pi$
$542$	−0.898278	−0.0385843
$543$	0	0
$544$	−6.67764	−0.286301
$545$	1.16551	0.0499251
$546$	0	0
$547$	−43.3280	−1.85257	−0.926286	−	0.376821i	$-0.877017\pi$
$547$	−43.3280	−1.85257	−0.926286	+	0.376821i	$0.877017\pi$
$548$	33.5077	1.43138
$549$	0	0
$550$	10.7115	0.456740
$551$	−26.2747	−1.11934
$552$	0	0
$553$	2.56769	0.109189
$554$	49.5681	2.10595
$555$	0	0
$556$	29.8480	1.26584
$557$	31.6275	1.34010	0.670050	−	0.742316i	$-0.266273\pi$
$557$	31.6275	1.34010	0.670050	+	0.742316i	$0.266273\pi$
$558$	0	0
$559$	38.9620	1.64792
$560$	−14.8218	−0.626334
$561$	0	0
$562$	−7.32801	−0.309113
$563$	−31.7420	−1.33777	−0.668883	−	0.743367i	$-0.733227\pi$
$563$	−31.7420	−1.33777	−0.668883	+	0.743367i	$0.733227\pi$
$564$	0	0
$565$	−13.8183	−0.581340
$566$	−44.3823	−1.86553
$567$	0	0
$568$	−39.4816	−1.65661
$569$	33.1158	1.38829	0.694143	−	0.719837i	$-0.255784\pi$
$569$	33.1158	1.38829	0.694143	+	0.719837i	$0.255784\pi$
$570$	0	0
$571$	16.8155	0.703708	0.351854	−	0.936055i	$-0.385551\pi$
$571$	16.8155	0.703708	0.351854	+	0.936055i	$0.385551\pi$
$572$	27.9204	1.16741
$573$	0	0
$574$	54.9646	2.29418
$575$	11.7908	0.491709
$576$	0	0
$577$	2.99960	0.124875	0.0624375	−	0.998049i	$-0.480113\pi$
$577$	2.99960	0.124875	0.0624375	+	0.998049i	$0.480113\pi$
$578$	−15.8054	−0.657420
$579$	0	0
$580$	12.6372	0.524731
$581$	−44.8320	−1.85994
$582$	0	0
$583$	7.67983	0.318066
$584$	36.3367	1.50362
$585$	0	0
$586$	−45.2775	−1.87039
$587$	44.8335	1.85048	0.925238	−	0.379387i	$-0.123865\pi$
$587$	44.8335	1.85048	0.925238	+	0.379387i	$0.123865\pi$
$588$	0	0
$589$	−22.2672	−0.917504
$590$	24.3888	1.00407
$591$	0	0
$592$	−16.3677	−0.672708
$593$	23.9589	0.983875	0.491938	−	0.870631i	$-0.336289\pi$
$593$	23.9589	0.983875	0.491938	+	0.870631i	$0.336289\pi$
$594$	0	0
$595$	−14.8469	−0.608662
$596$	18.5413	0.759480
$597$	0	0
$598$	45.5992	1.86469
$599$	−16.9350	−0.691946	−0.345973	−	0.938244i	$-0.612451\pi$
$599$	−16.9350	−0.691946	−0.345973	+	0.938244i	$0.612451\pi$
$600$	0	0
$601$	26.0786	1.06377	0.531884	−	0.846817i	$-0.321484\pi$
$601$	26.0786	1.06377	0.531884	+	0.846817i	$0.321484\pi$
$602$	53.3962	2.17627
$603$	0	0
$604$	74.1855	3.01856
$605$	0.821809	0.0334113
$606$	0	0
$607$	8.12229	0.329674	0.164837	−	0.986321i	$-0.447290\pi$
$607$	8.12229	0.329674	0.164837	+	0.986321i	$0.447290\pi$
$608$	9.75678	0.395690
$609$	0	0
$610$	2.03551	0.0824152
$611$	9.58428	0.387738
$612$	0	0
$613$	−11.4803	−0.463686	−0.231843	−	0.972753i	$-0.574476\pi$
$613$	−11.4803	−0.463686	−0.231843	+	0.972753i	$0.574476\pi$
$614$	−46.8917	−1.89239
$615$	0	0
$616$	19.7559	0.795990
$617$	34.7309	1.39821	0.699106	−	0.715018i	$-0.253581\pi$
$617$	34.7309	1.39821	0.699106	+	0.715018i	$0.253581\pi$
$618$	0	0
$619$	31.5034	1.26623	0.633113	−	0.774059i	$-0.281777\pi$
$619$	31.5034	1.26623	0.633113	+	0.774059i	$0.281777\pi$
$620$	10.7097	0.430112
$621$	0	0
$622$	−38.5682	−1.54644
$623$	−69.9006	−2.80051
$624$	0	0
$625$	15.3256	0.613023
$626$	−36.7865	−1.47028
$627$	0	0
$628$	−33.5783	−1.33992
$629$	−16.3954	−0.653728
$630$	0	0
$631$	−1.82599	−0.0726916	−0.0363458	−	0.999339i	$-0.511572\pi$
$631$	−1.82599	−0.0726916	−0.0363458	+	0.999339i	$0.511572\pi$
$632$	3.63396	0.144551
$633$	0	0
$634$	21.2792	0.845106
$635$	−5.08981	−0.201983
$636$	0	0
$637$	46.9918	1.86188
$638$	−9.21134	−0.364680
$639$	0	0
$640$	14.9591	0.591310
$641$	15.2017	0.600430	0.300215	−	0.953872i	$-0.402942\pi$
$641$	15.2017	0.600430	0.300215	+	0.953872i	$0.402942\pi$
$642$	0	0
$643$	16.8332	0.663836	0.331918	−	0.943308i	$-0.392304\pi$
$643$	16.8332	0.663836	0.331918	+	0.943308i	$0.392304\pi$
$644$	42.1194	1.65974
$645$	0	0
$646$	84.6161	3.32917
$647$	−1.00459	−0.0394945	−0.0197472	−	0.999805i	$-0.506286\pi$
$647$	−1.00459	−0.0394945	−0.0197472	+	0.999805i	$0.506286\pi$
$648$	0	0
$649$	−11.9817	−0.470322
$650$	72.3292	2.83698
$651$	0	0
$652$	−88.8261	−3.47870
$653$	9.51765	0.372454	0.186227	−	0.982507i	$-0.440374\pi$
$653$	9.51765	0.372454	0.186227	+	0.982507i	$0.440374\pi$
$654$	0	0
$655$	−4.49072	−0.175467
$656$	28.6715	1.11943
$657$	0	0
$658$	13.1350	0.512054
$659$	34.8312	1.35683	0.678416	−	0.734678i	$-0.262667\pi$
$659$	34.8312	1.35683	0.678416	+	0.734678i	$0.262667\pi$
$660$	0	0
$661$	−19.8721	−0.772936	−0.386468	−	0.922303i	$-0.626305\pi$
$661$	−19.8721	−0.772936	−0.386468	+	0.922303i	$0.626305\pi$
$662$	71.4191	2.77578
$663$	0	0
$664$	−63.4491	−2.46230
$665$	21.6929	0.841216
$666$	0	0
$667$	−10.1394	−0.392601
$668$	38.4488	1.48763
$669$	0	0
$670$	17.0065	0.657019
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−35.6038	−1.37243	−0.686213	−	0.727400i	$-0.740728\pi$
$673$	−35.6038	−1.37243	−0.686213	+	0.727400i	$0.740728\pi$
$674$	63.2028	2.43448
$675$	0	0
$676$	134.779	5.18380
$677$	14.7492	0.566858	0.283429	−	0.958993i	$-0.408528\pi$
$677$	14.7492	0.566858	0.283429	+	0.958993i	$0.408528\pi$
$678$	0	0
$679$	−53.4919	−2.05283
$680$	−21.0122	−0.805782
$681$	0	0
$682$	−7.80638	−0.298922
$683$	20.1926	0.772650	0.386325	−	0.922363i	$-0.373744\pi$
$683$	20.1926	0.772650	0.386325	+	0.922363i	$0.373744\pi$
$684$	0	0
$685$	6.65972	0.254455
$686$	−0.377619	−0.0144176
$687$	0	0
$688$	27.8534	1.06190
$689$	51.8578	1.97563
$690$	0	0
$691$	−14.5160	−0.552215	−0.276107	−	0.961127i	$-0.589045\pi$
$691$	−14.5160	−0.552215	−0.276107	+	0.961127i	$0.589045\pi$
$692$	−37.2331	−1.41539
$693$	0	0
$694$	67.7831	2.57301
$695$	5.93235	0.225027
$696$	0	0
$697$	28.7200	1.08785
$698$	−68.8546	−2.60618
$699$	0	0
$700$	66.8095	2.52516
$701$	5.40239	0.204045	0.102023	−	0.994782i	$-0.467469\pi$
$701$	5.40239	0.204045	0.102023	+	0.994782i	$0.467469\pi$
$702$	0	0
$703$	23.9555	0.903500
$704$	−6.23397	−0.234952
$705$	0	0
$706$	−21.2671	−0.800398
$707$	9.22118	0.346798
$708$	0	0
$709$	9.71469	0.364843	0.182421	−	0.983220i	$-0.441606\pi$
$709$	9.71469	0.364843	0.182421	+	0.983220i	$0.441606\pi$
$710$	−15.1984	−0.570388
$711$	0	0
$712$	−98.9278	−3.70748
$713$	−8.59292	−0.321807
$714$	0	0
$715$	5.54924	0.207530
$716$	−25.7185	−0.961147
$717$	0	0
$718$	−48.9500	−1.82680
$719$	−0.690169	−0.0257390	−0.0128695	−	0.999917i	$-0.504097\pi$
$719$	−0.690169	−0.0257390	−0.0128695	+	0.999917i	$0.504097\pi$
$720$	0	0
$721$	32.8151	1.22210
$722$	−76.5732	−2.84976
$723$	0	0
$724$	−71.1262	−2.64338
$725$	−16.0831	−0.597312
$726$	0	0
$727$	24.4586	0.907118	0.453559	−	0.891226i	$-0.350154\pi$
$727$	24.4586	0.907118	0.453559	+	0.891226i	$0.350154\pi$
$728$	133.401	4.94419
$729$	0	0
$730$	13.9878	0.517713
$731$	27.9005	1.03194
$732$	0	0
$733$	−6.43611	−0.237723	−0.118862	−	0.992911i	$-0.537924\pi$
$733$	−6.43611	−0.237723	−0.118862	+	0.992911i	$0.537924\pi$
$734$	−18.7257	−0.691177
$735$	0	0
$736$	3.76515	0.138785
$737$	−8.35493	−0.307758
$738$	0	0
$739$	33.7241	1.24056	0.620280	−	0.784380i	$-0.287019\pi$
$739$	33.7241	1.24056	0.620280	+	0.784380i	$0.287019\pi$
$740$	−11.5217	−0.423548
$741$	0	0
$742$	71.0695	2.60904
$743$	3.06877	0.112582	0.0562912	−	0.998414i	$-0.482072\pi$
$743$	3.06877	0.112582	0.0562912	+	0.998414i	$0.482072\pi$
$744$	0	0
$745$	3.68512	0.135012
$746$	−83.0955	−3.04234
$747$	0	0
$748$	19.9937	0.731041
$749$	32.4383	1.18527
$750$	0	0
$751$	9.88308	0.360639	0.180319	−	0.983608i	$-0.442287\pi$
$751$	9.88308	0.360639	0.180319	+	0.983608i	$0.442287\pi$
$752$	6.85166	0.249854
$753$	0	0
$754$	−62.1993	−2.26517
$755$	14.7445	0.536608
$756$	0	0
$757$	−13.1020	−0.476201	−0.238101	−	0.971241i	$-0.576525\pi$
$757$	−13.1020	−0.476201	−0.238101	+	0.971241i	$0.576525\pi$
$758$	54.9118	1.99449
$759$	0	0
$760$	30.7012	1.11365
$761$	−39.1774	−1.42018	−0.710090	−	0.704111i	$-0.751346\pi$
$761$	−39.1774	−1.42018	−0.710090	+	0.704111i	$0.751346\pi$
$762$	0	0
$763$	−5.29879	−0.191829
$764$	76.8567	2.78058
$765$	0	0
$766$	5.57138	0.201302
$767$	−80.9060	−2.92135
$768$	0	0
$769$	23.6144	0.851557	0.425778	−	0.904828i	$-0.360000\pi$
$769$	23.6144	0.851557	0.425778	+	0.904828i	$0.360000\pi$
$770$	7.60506	0.274067
$771$	0	0
$772$	14.8776	0.535456
$773$	15.1482	0.544844	0.272422	−	0.962178i	$-0.412175\pi$
$773$	15.1482	0.544844	0.272422	+	0.962178i	$0.412175\pi$
$774$	0	0
$775$	−13.6300	−0.489605
$776$	−75.7052	−2.71766
$777$	0	0
$778$	−69.1116	−2.47777
$779$	−41.9632	−1.50349
$780$	0	0
$781$	7.46667	0.267178
$782$	32.6534	1.16768
$783$	0	0
$784$	33.5937	1.19978
$785$	−6.67376	−0.238197
$786$	0	0
$787$	41.7588	1.48854	0.744270	−	0.667879i	$-0.232798\pi$
$787$	41.7588	1.48854	0.744270	+	0.667879i	$0.232798\pi$
$788$	27.6823	0.986142
$789$	0	0
$790$	1.39890	0.0497705
$791$	62.8223	2.23370
$792$	0	0
$793$	−6.75247	−0.239787
$794$	43.5776	1.54651
$795$	0	0
$796$	54.8455	1.94395
$797$	−34.4098	−1.21886	−0.609430	−	0.792840i	$-0.708602\pi$
$797$	−34.4098	−1.21886	−0.609430	+	0.792840i	$0.708602\pi$
$798$	0	0
$799$	6.86326	0.242805
$800$	5.97225	0.211151
$801$	0	0
$802$	−85.6212	−3.02339
$803$	−6.87191	−0.242505
$804$	0	0
$805$	8.37131	0.295050
$806$	−52.7124	−1.85671
$807$	0	0
$808$	13.0504	0.459112
$809$	0.858625	0.0301877	0.0150938	−	0.999886i	$-0.495195\pi$
$809$	0.858625	0.0301877	0.0150938	+	0.999886i	$0.495195\pi$
$810$	0	0
$811$	−34.7221	−1.21926	−0.609629	−	0.792687i	$-0.708681\pi$
$811$	−34.7221	−1.21926	−0.609629	+	0.792687i	$0.708681\pi$
$812$	−57.4527	−2.01619
$813$	0	0
$814$	8.39828	0.294360
$815$	−17.6544	−0.618406
$816$	0	0
$817$	−40.7658	−1.42622
$818$	−72.4725	−2.53394
$819$	0	0
$820$	20.1828	0.704814
$821$	44.9142	1.56751	0.783757	−	0.621067i	$-0.213301\pi$
$821$	44.9142	1.56751	0.783757	+	0.621067i	$0.213301\pi$
$822$	0	0
$823$	47.7347	1.66393	0.831964	−	0.554830i	$-0.187217\pi$
$823$	47.7347	1.66393	0.831964	+	0.554830i	$0.187217\pi$
$824$	46.4420	1.61788
$825$	0	0
$826$	−110.879	−3.85798
$827$	34.1487	1.18747	0.593733	−	0.804662i	$-0.297653\pi$
$827$	34.1487	1.18747	0.593733	+	0.804662i	$0.297653\pi$
$828$	0	0
$829$	−0.290820	−0.0101006	−0.00505029	−	0.999987i	$-0.501608\pi$
$829$	−0.290820	−0.0101006	−0.00505029	+	0.999987i	$0.501608\pi$
$830$	−24.4247	−0.847795
$831$	0	0
$832$	−42.0947	−1.45937
$833$	33.6506	1.16592
$834$	0	0
$835$	7.64178	0.264455
$836$	−29.2130	−1.01035
$837$	0	0
$838$	16.4677	0.568867
$839$	−23.2848	−0.803881	−0.401940	−	0.915666i	$-0.631664\pi$
$839$	−23.2848	−0.803881	−0.401940	+	0.915666i	$0.631664\pi$
$840$	0	0
$841$	−15.1694	−0.523081
$842$	−28.4360	−0.979969
$843$	0	0
$844$	106.128	3.65309
$845$	26.7876	0.921521
$846$	0	0
$847$	−3.73620	−0.128377
$848$	37.0724	1.27307
$849$	0	0
$850$	51.7946	1.77654
$851$	9.24446	0.316896
$852$	0	0
$853$	34.5641	1.18345	0.591726	−	0.806139i	$-0.298447\pi$
$853$	34.5641	1.18345	0.591726	+	0.806139i	$0.298447\pi$
$854$	−9.25405	−0.316667
$855$	0	0
$856$	45.9088	1.56913
$857$	33.0283	1.12823	0.564113	−	0.825698i	$-0.309218\pi$
$857$	33.0283	1.12823	0.564113	+	0.825698i	$0.309218\pi$
$858$	0	0
$859$	28.3164	0.966144	0.483072	−	0.875581i	$-0.339521\pi$
$859$	28.3164	0.966144	0.483072	+	0.875581i	$0.339521\pi$
$860$	19.6069	0.668589
$861$	0	0
$862$	85.2167	2.90249
$863$	33.4172	1.13753	0.568767	−	0.822499i	$-0.307421\pi$
$863$	33.4172	1.13753	0.568767	+	0.822499i	$0.307421\pi$
$864$	0	0
$865$	−7.40016	−0.251613
$866$	1.74744	0.0593804
$867$	0	0
$868$	−48.6897	−1.65264
$869$	−0.687247	−0.0233133
$870$	0	0
$871$	−56.4164	−1.91160
$872$	−7.49918	−0.253954
$873$	0	0
$874$	−47.7103	−1.61382
$875$	28.6307	0.967896
$876$	0	0
$877$	−26.7344	−0.902758	−0.451379	−	0.892332i	$-0.649068\pi$
$877$	−26.7344	−0.902758	−0.451379	+	0.892332i	$0.649068\pi$
$878$	−52.1539	−1.76011
$879$	0	0
$880$	3.96707	0.133730
$881$	10.4194	0.351037	0.175519	−	0.984476i	$-0.443840\pi$
$881$	10.4194	0.351037	0.175519	+	0.984476i	$0.443840\pi$
$882$	0	0
$883$	22.3583	0.752416	0.376208	−	0.926535i	$-0.377228\pi$
$883$	22.3583	0.752416	0.376208	+	0.926535i	$0.377228\pi$
$884$	135.007	4.54077
$885$	0	0
$886$	39.6212	1.33110
$887$	38.1506	1.28097	0.640486	−	0.767970i	$-0.278733\pi$
$887$	38.1506	1.28097	0.640486	+	0.767970i	$0.278733\pi$
$888$	0	0
$889$	23.1399	0.776087
$890$	−38.0823	−1.27652
$891$	0	0
$892$	−39.2675	−1.31477
$893$	−10.0280	−0.335574
$894$	0	0
$895$	−5.11161	−0.170862
$896$	−68.0088	−2.27201
$897$	0	0
$898$	−7.68808	−0.256555
$899$	11.7211	0.390921
$900$	0	0
$901$	37.1352	1.23715
$902$	−14.7114	−0.489835
$903$	0	0
$904$	88.9101	2.95711
$905$	−14.1365	−0.469912
$906$	0	0
$907$	−28.1034	−0.933159	−0.466579	−	0.884479i	$-0.654514\pi$
$907$	−28.1034	−0.933159	−0.466579	+	0.884479i	$0.654514\pi$
$908$	21.5229	0.714263
$909$	0	0
$910$	51.3530	1.70233
$911$	59.0151	1.95526	0.977629	−	0.210337i	$-0.0674562\pi$
$911$	59.0151	1.95526	0.977629	+	0.210337i	$0.0674562\pi$
$912$	0	0
$913$	11.9993	0.397120
$914$	−2.95597	−0.0977747
$915$	0	0
$916$	−10.9355	−0.361320
$917$	20.4162	0.674203
$918$	0	0
$919$	−36.5389	−1.20531	−0.602654	−	0.798003i	$-0.705890\pi$
$919$	−36.5389	−1.20531	−0.602654	+	0.798003i	$0.705890\pi$
$920$	11.8476	0.390604
$921$	0	0
$922$	−59.4282	−1.95716
$923$	50.4185	1.65954
$924$	0	0
$925$	14.6635	0.482133
$926$	−73.6197	−2.41929
$927$	0	0
$928$	−5.13582	−0.168592
$929$	38.0982	1.24996	0.624981	−	0.780640i	$-0.285107\pi$
$929$	38.0982	1.24996	0.624981	+	0.780640i	$0.285107\pi$
$930$	0	0
$931$	−49.1673	−1.61139
$932$	−28.3961	−0.930144
$933$	0	0
$934$	67.1447	2.19704
$935$	3.97379	0.129957
$936$	0	0
$937$	−42.5551	−1.39021	−0.695107	−	0.718906i	$-0.744643\pi$
$937$	−42.5551	−1.39021	−0.695107	+	0.718906i	$0.744643\pi$
$938$	−77.3170	−2.52449
$939$	0	0
$940$	4.82310	0.157312
$941$	−33.8142	−1.10231	−0.551156	−	0.834402i	$-0.685813\pi$
$941$	−33.8142	−1.10231	−0.551156	+	0.834402i	$0.685813\pi$
$942$	0	0
$943$	−16.1936	−0.527337
$944$	−57.8385	−1.88248
$945$	0	0
$946$	−14.2916	−0.464660
$947$	19.0565	0.619252	0.309626	−	0.950858i	$-0.399796\pi$
$947$	19.0565	0.619252	0.309626	+	0.950858i	$0.399796\pi$
$948$	0	0
$949$	−46.4024	−1.50629
$950$	−75.6777	−2.45531
$951$	0	0
$952$	95.5282	3.09609
$953$	−5.82791	−0.188784	−0.0943922	−	0.995535i	$-0.530091\pi$
$953$	−5.82791	−0.188784	−0.0943922	+	0.995535i	$0.530091\pi$
$954$	0	0
$955$	15.2754	0.494302
$956$	56.8649	1.83914
$957$	0	0
$958$	45.2316	1.46137
$959$	−30.2772	−0.977701
$960$	0	0
$961$	−21.0666	−0.679569
$962$	56.7092	1.82838
$963$	0	0
$964$	28.9664	0.932946
$965$	2.95695	0.0951877
$966$	0	0
$967$	−8.43040	−0.271104	−0.135552	−	0.990770i	$-0.543281\pi$
$967$	−8.43040	−0.271104	−0.135552	+	0.990770i	$0.543281\pi$
$968$	−5.28771	−0.169953
$969$	0	0
$970$	−29.1428	−0.935717
$971$	44.1597	1.41715	0.708576	−	0.705634i	$-0.249338\pi$
$971$	44.1597	1.41715	0.708576	+	0.705634i	$0.249338\pi$
$972$	0	0
$973$	−26.9703	−0.864629
$974$	−67.4382	−2.16086
$975$	0	0
$976$	−4.82724	−0.154516
$977$	−18.1660	−0.581183	−0.290591	−	0.956847i	$-0.593852\pi$
$977$	−18.1660	−0.581183	−0.290591	+	0.956847i	$0.593852\pi$
$978$	0	0
$979$	18.7090	0.597942
$980$	23.6477	0.755398
$981$	0	0
$982$	−21.1189	−0.673931
$983$	−1.72964	−0.0551668	−0.0275834	−	0.999620i	$-0.508781\pi$
$983$	−1.72964	−0.0551668	−0.0275834	+	0.999620i	$0.508781\pi$
$984$	0	0
$985$	5.50192	0.175306
$986$	−44.5407	−1.41846
$987$	0	0
$988$	−197.260	−6.27568
$989$	−15.7316	−0.500234
$990$	0	0
$991$	−7.83333	−0.248834	−0.124417	−	0.992230i	$-0.539706\pi$
$991$	−7.83333	−0.248834	−0.124417	+	0.992230i	$0.539706\pi$
$992$	−4.35248	−0.138191
$993$	0	0
$994$	69.0969	2.19162
$995$	10.9007	0.345574
$996$	0	0
$997$	−54.7739	−1.73471	−0.867353	−	0.497694i	$-0.834180\pi$
$997$	−54.7739	−1.73471	−0.867353	+	0.497694i	$0.834180\pi$
$998$	11.8930	0.376466
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.o.1.1	yes	25
3.2	odd	2		6039.2.a.n.1.25	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.25	✓	25	3.2	odd	2
6039.2.a.o.1.1	yes	25	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-2.47686 q^{2} +4.13484 q^{4} +0.821809 q^{5} -3.73620 q^{7} -5.28771 q^{8} +O(q^{10})\)	\(q-2.47686 q^{2} +4.13484 q^{4} +0.821809 q^{5} -3.73620 q^{7} -5.28771 q^{8} -2.03551 q^{10} +1.00000 q^{11} +6.75247 q^{13} +9.25405 q^{14} +4.82724 q^{16} +4.83541 q^{17} -7.06509 q^{19} +3.39805 q^{20} -2.47686 q^{22} -2.72642 q^{23} -4.32463 q^{25} -16.7249 q^{26} -15.4486 q^{28} +3.71896 q^{29} +3.15172 q^{31} -1.38099 q^{32} -11.9767 q^{34} -3.07044 q^{35} -3.39069 q^{37} +17.4992 q^{38} -4.34549 q^{40} +5.93952 q^{41} +5.77004 q^{43} +4.13484 q^{44} +6.75296 q^{46} +1.41937 q^{47} +6.95919 q^{49} +10.7115 q^{50} +27.9204 q^{52} +7.67983 q^{53} +0.821809 q^{55} +19.7559 q^{56} -9.21134 q^{58} -11.9817 q^{59} -1.00000 q^{61} -7.80638 q^{62} -6.23397 q^{64} +5.54924 q^{65} -8.35493 q^{67} +19.9937 q^{68} +7.60506 q^{70} +7.46667 q^{71} -6.87191 q^{73} +8.39828 q^{74} -29.2130 q^{76} -3.73620 q^{77} -0.687247 q^{79} +3.96707 q^{80} -14.7114 q^{82} +11.9993 q^{83} +3.97379 q^{85} -14.2916 q^{86} -5.28771 q^{88} +18.7090 q^{89} -25.2286 q^{91} -11.2733 q^{92} -3.51559 q^{94} -5.80615 q^{95} +14.3172 q^{97} -17.2370 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) \) 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

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