LMFDB - Embedded newform 6039.2.a.n.1.10

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			1.10
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)

$f(q)$	$=$	$q-0.926278 q^{2} -1.14201 q^{4} -3.01280 q^{5} +2.23888 q^{7} +2.91037 q^{8} +O(q^{10})$	\(q-0.926278 q^{2} -1.14201 q^{4} -3.01280 q^{5} +2.23888 q^{7} +2.91037 q^{8} +2.79069 q^{10} -1.00000 q^{11} -3.86374 q^{13} -2.07382 q^{14} -0.411796 q^{16} +0.440707 q^{17} +3.02117 q^{19} +3.44064 q^{20} +0.926278 q^{22} -5.93879 q^{23} +4.07696 q^{25} +3.57890 q^{26} -2.55682 q^{28} +8.56058 q^{29} -0.629158 q^{31} -5.43931 q^{32} -0.408217 q^{34} -6.74529 q^{35} +5.92820 q^{37} -2.79844 q^{38} -8.76837 q^{40} -9.04985 q^{41} +3.19468 q^{43} +1.14201 q^{44} +5.50097 q^{46} +2.54522 q^{47} -1.98743 q^{49} -3.77640 q^{50} +4.41243 q^{52} -8.76682 q^{53} +3.01280 q^{55} +6.51597 q^{56} -7.92947 q^{58} +7.36876 q^{59} -1.00000 q^{61} +0.582775 q^{62} +5.86191 q^{64} +11.6407 q^{65} +0.878261 q^{67} -0.503291 q^{68} +6.24801 q^{70} -5.06787 q^{71} +14.1572 q^{73} -5.49116 q^{74} -3.45021 q^{76} -2.23888 q^{77} +16.1085 q^{79} +1.24066 q^{80} +8.38268 q^{82} -7.46706 q^{83} -1.32776 q^{85} -2.95916 q^{86} -2.91037 q^{88} +1.43455 q^{89} -8.65044 q^{91} +6.78216 q^{92} -2.35758 q^{94} -9.10218 q^{95} -4.10138 q^{97} +1.84091 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.926278	−0.654977	−0.327489	−	0.944855i	$-0.606202\pi$
$2$	−0.926278	−0.654977	−0.327489	+	0.944855i	$0.606202\pi$
$3$	0	0
$3$	0	0
$4$	−1.14201	−0.571005
$5$	−3.01280	−1.34736	−0.673682	−	0.739021i	$-0.735288\pi$
$5$	−3.01280	−1.34736	−0.673682	+	0.739021i	$0.735288\pi$
$6$	0	0
$7$	2.23888	0.846216	0.423108	−	0.906079i	$-0.360939\pi$
$7$	2.23888	0.846216	0.423108	+	0.906079i	$0.360939\pi$
$8$	2.91037	1.02897
$9$	0	0
$10$	2.79069	0.882494
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.86374	−1.07161	−0.535804	−	0.844342i	$-0.679991\pi$
$13$	−3.86374	−1.07161	−0.535804	+	0.844342i	$0.679991\pi$
$14$	−2.07382	−0.554252
$15$	0	0
$16$	−0.411796	−0.102949
$17$	0.440707	0.106887	0.0534436	−	0.998571i	$-0.482980\pi$
$17$	0.440707	0.106887	0.0534436	+	0.998571i	$0.482980\pi$
$18$	0	0
$19$	3.02117	0.693104	0.346552	−	0.938031i	$-0.387352\pi$
$19$	3.02117	0.693104	0.346552	+	0.938031i	$0.387352\pi$
$20$	3.44064	0.769352
$21$	0	0
$22$	0.926278	0.197483
$23$	−5.93879	−1.23832	−0.619162	−	0.785263i	$-0.712528\pi$
$23$	−5.93879	−1.23832	−0.619162	+	0.785263i	$0.712528\pi$
$24$	0	0
$25$	4.07696	0.815392
$26$	3.57890	0.701879
$27$	0	0
$28$	−2.55682	−0.483193
$29$	8.56058	1.58966	0.794830	−	0.606833i	$-0.207560\pi$
$29$	8.56058	1.58966	0.794830	+	0.606833i	$0.207560\pi$
$30$	0	0
$31$	−0.629158	−0.113000	−0.0565001	−	0.998403i	$-0.517994\pi$
$31$	−0.629158	−0.113000	−0.0565001	+	0.998403i	$0.517994\pi$
$32$	−5.43931	−0.961543
$33$	0	0
$34$	−0.408217	−0.0700086
$35$	−6.74529	−1.14016
$36$	0	0
$37$	5.92820	0.974589	0.487295	−	0.873238i	$-0.337984\pi$
$37$	5.92820	0.974589	0.487295	+	0.873238i	$0.337984\pi$
$38$	−2.79844	−0.453968
$39$	0	0
$40$	−8.76837	−1.38640
$41$	−9.04985	−1.41335	−0.706675	−	0.707539i	$-0.749805\pi$
$41$	−9.04985	−1.41335	−0.706675	+	0.707539i	$0.749805\pi$
$42$	0	0
$43$	3.19468	0.487184	0.243592	−	0.969878i	$-0.421674\pi$
$43$	3.19468	0.487184	0.243592	+	0.969878i	$0.421674\pi$
$44$	1.14201	0.172164
$45$	0	0
$46$	5.50097	0.811074
$47$	2.54522	0.371259	0.185629	−	0.982620i	$-0.440568\pi$
$47$	2.54522	0.371259	0.185629	+	0.982620i	$0.440568\pi$
$48$	0	0
$49$	−1.98743	−0.283918
$50$	−3.77640	−0.534063
$51$	0	0
$52$	4.41243	0.611893
$53$	−8.76682	−1.20422	−0.602108	−	0.798415i	$-0.705672\pi$
$53$	−8.76682	−1.20422	−0.602108	+	0.798415i	$0.705672\pi$
$54$	0	0
$55$	3.01280	0.406246
$56$	6.51597	0.870733
$57$	0	0
$58$	−7.92947	−1.04119
$59$	7.36876	0.959331	0.479665	−	0.877451i	$-0.340758\pi$
$59$	7.36876	0.959331	0.479665	+	0.877451i	$0.340758\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0.582775	0.0740125
$63$	0	0
$64$	5.86191	0.732738
$65$	11.6407	1.44385
$66$	0	0
$67$	0.878261	0.107297	0.0536483	−	0.998560i	$-0.482915\pi$
$67$	0.878261	0.107297	0.0536483	+	0.998560i	$0.482915\pi$
$68$	−0.503291	−0.0610330
$69$	0	0
$70$	6.24801	0.746780
$71$	−5.06787	−0.601446	−0.300723	−	0.953712i	$-0.597228\pi$
$71$	−5.06787	−0.601446	−0.300723	+	0.953712i	$0.597228\pi$
$72$	0	0
$73$	14.1572	1.65698	0.828488	−	0.560006i	$-0.189201\pi$
$73$	14.1572	1.65698	0.828488	+	0.560006i	$0.189201\pi$
$74$	−5.49116	−0.638334
$75$	0	0
$76$	−3.45021	−0.395766
$77$	−2.23888	−0.255144
$78$	0	0
$79$	16.1085	1.81234	0.906172	−	0.422910i	$-0.138991\pi$
$79$	16.1085	1.81234	0.906172	+	0.422910i	$0.138991\pi$
$80$	1.24066	0.138710
$81$	0	0
$82$	8.38268	0.925712
$83$	−7.46706	−0.819617	−0.409808	−	0.912172i	$-0.634404\pi$
$83$	−7.46706	−0.819617	−0.409808	+	0.912172i	$0.634404\pi$
$84$	0	0
$85$	−1.32776	−0.144016
$86$	−2.95916	−0.319095
$87$	0	0
$88$	−2.91037	−0.310247
$89$	1.43455	0.152062	0.0760310	−	0.997105i	$-0.475775\pi$
$89$	1.43455	0.152062	0.0760310	+	0.997105i	$0.475775\pi$
$90$	0	0
$91$	−8.65044	−0.906812
$92$	6.78216	0.707089
$93$	0	0
$94$	−2.35758	−0.243166
$95$	−9.10218	−0.933864
$96$	0	0
$97$	−4.10138	−0.416432	−0.208216	−	0.978083i	$-0.566766\pi$
$97$	−4.10138	−0.416432	−0.208216	+	0.978083i	$0.566766\pi$
$98$	1.84091	0.185960
$99$	0	0
$100$	−4.65593	−0.465593
$101$	18.5286	1.84367	0.921834	−	0.387586i	$-0.126691\pi$
$101$	18.5286	1.84367	0.921834	+	0.387586i	$0.126691\pi$
$102$	0	0
$103$	14.9366	1.47175	0.735876	−	0.677117i	$-0.236771\pi$
$103$	14.9366	1.47175	0.735876	+	0.677117i	$0.236771\pi$
$104$	−11.2449	−1.10266
$105$	0	0
$106$	8.12051	0.788734
$107$	−8.48007	−0.819800	−0.409900	−	0.912131i	$-0.634436\pi$
$107$	−8.48007	−0.819800	−0.409900	+	0.912131i	$0.634436\pi$
$108$	0	0
$109$	−13.1193	−1.25661	−0.628303	−	0.777969i	$-0.716250\pi$
$109$	−13.1193	−1.25661	−0.628303	+	0.777969i	$0.716250\pi$
$110$	−2.79069	−0.266082
$111$	0	0
$112$	−0.921962	−0.0871172
$113$	3.07877	0.289627	0.144813	−	0.989459i	$-0.453742\pi$
$113$	3.07877	0.289627	0.144813	+	0.989459i	$0.453742\pi$
$114$	0	0
$115$	17.8924	1.66847
$116$	−9.77626	−0.907703
$117$	0	0
$118$	−6.82552	−0.628340
$119$	0.986688	0.0904496
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.926278	0.0838613
$123$	0	0
$124$	0.718504	0.0645236
$125$	2.78093	0.248734
$126$	0	0
$127$	18.1448	1.61009	0.805044	−	0.593215i	$-0.202142\pi$
$127$	18.1448	1.61009	0.805044	+	0.593215i	$0.202142\pi$
$128$	5.44887	0.481616
$129$	0	0
$130$	−10.7825	−0.945687
$131$	−11.2061	−0.979085	−0.489542	−	0.871979i	$-0.662836\pi$
$131$	−11.2061	−0.979085	−0.489542	+	0.871979i	$0.662836\pi$
$132$	0	0
$133$	6.76403	0.586516
$134$	−0.813513	−0.0702769
$135$	0	0
$136$	1.28262	0.109984
$137$	−14.3501	−1.22601	−0.613007	−	0.790077i	$-0.710040\pi$
$137$	−14.3501	−1.22601	−0.613007	+	0.790077i	$0.710040\pi$
$138$	0	0
$139$	−15.8060	−1.34064	−0.670322	−	0.742070i	$-0.733844\pi$
$139$	−15.8060	−1.34064	−0.670322	+	0.742070i	$0.733844\pi$
$140$	7.70318	0.651038
$141$	0	0
$142$	4.69426	0.393933
$143$	3.86374	0.323102
$144$	0	0
$145$	−25.7913	−2.14185
$146$	−13.1135	−1.08528
$147$	0	0
$148$	−6.77005	−0.556495
$149$	−2.26414	−0.185486	−0.0927428	−	0.995690i	$-0.529563\pi$
$149$	−2.26414	−0.185486	−0.0927428	+	0.995690i	$0.529563\pi$
$150$	0	0
$151$	−20.0595	−1.63242	−0.816210	−	0.577755i	$-0.803929\pi$
$151$	−20.0595	−1.63242	−0.816210	+	0.577755i	$0.803929\pi$
$152$	8.79274	0.713185
$153$	0	0
$154$	2.07382	0.167113
$155$	1.89553	0.152252
$156$	0	0
$157$	7.10967	0.567414	0.283707	−	0.958911i	$-0.408436\pi$
$157$	7.10967	0.567414	0.283707	+	0.958911i	$0.408436\pi$
$158$	−14.9209	−1.18704
$159$	0	0
$160$	16.3875	1.29555
$161$	−13.2962	−1.04789
$162$	0	0
$163$	−2.08082	−0.162982	−0.0814912	−	0.996674i	$-0.525968\pi$
$163$	−2.08082	−0.162982	−0.0814912	+	0.996674i	$0.525968\pi$
$164$	10.3350	0.807029
$165$	0	0
$166$	6.91658	0.536830
$167$	12.1974	0.943860	0.471930	−	0.881636i	$-0.343558\pi$
$167$	12.1974	0.943860	0.471930	+	0.881636i	$0.343558\pi$
$168$	0	0
$169$	1.92848	0.148344
$170$	1.22988	0.0943272
$171$	0	0
$172$	−3.64836	−0.278185
$173$	−20.5841	−1.56498	−0.782492	−	0.622661i	$-0.786052\pi$
$173$	−20.5841	−1.56498	−0.782492	+	0.622661i	$0.786052\pi$
$174$	0	0
$175$	9.12781	0.689998
$176$	0.411796	0.0310403
$177$	0	0
$178$	−1.32879	−0.0995972
$179$	24.2526	1.81272	0.906361	−	0.422504i	$-0.138849\pi$
$179$	24.2526	1.81272	0.906361	+	0.422504i	$0.138849\pi$
$180$	0	0
$181$	−4.46564	−0.331928	−0.165964	−	0.986132i	$-0.553074\pi$
$181$	−4.46564	−0.331928	−0.165964	+	0.986132i	$0.553074\pi$
$182$	8.01271	0.593941
$183$	0	0
$184$	−17.2841	−1.27420
$185$	−17.8605	−1.31313
$186$	0	0
$187$	−0.440707	−0.0322277
$188$	−2.90667	−0.211991
$189$	0	0
$190$	8.43115	0.611660
$191$	3.23006	0.233719	0.116860	−	0.993148i	$-0.462717\pi$
$191$	3.23006	0.233719	0.116860	+	0.993148i	$0.462717\pi$
$192$	0	0
$193$	25.4703	1.83339	0.916697	−	0.399584i	$-0.130845\pi$
$193$	25.4703	1.83339	0.916697	+	0.399584i	$0.130845\pi$
$194$	3.79902	0.272754
$195$	0	0
$196$	2.26966	0.162119
$197$	0.466820	0.0332595	0.0166298	−	0.999862i	$-0.494706\pi$
$197$	0.466820	0.0332595	0.0166298	+	0.999862i	$0.494706\pi$
$198$	0	0
$199$	−1.52097	−0.107819	−0.0539093	−	0.998546i	$-0.517168\pi$
$199$	−1.52097	−0.107819	−0.0539093	+	0.998546i	$0.517168\pi$
$200$	11.8655	0.839016
$201$	0	0
$202$	−17.1627	−1.20756
$203$	19.1661	1.34519
$204$	0	0
$205$	27.2654	1.90430
$206$	−13.8355	−0.963964
$207$	0	0
$208$	1.59107	0.110321
$209$	−3.02117	−0.208979
$210$	0	0
$211$	20.0214	1.37833	0.689166	−	0.724604i	$-0.257977\pi$
$211$	20.0214	1.37833	0.689166	+	0.724604i	$0.257977\pi$
$212$	10.0118	0.687612
$213$	0	0
$214$	7.85490	0.536950
$215$	−9.62494	−0.656415
$216$	0	0
$217$	−1.40861	−0.0956225
$218$	12.1522	0.823048
$219$	0	0
$220$	−3.44064	−0.231968
$221$	−1.70278	−0.114541
$222$	0	0
$223$	−0.191453	−0.0128207	−0.00641033	−	0.999979i	$-0.502040\pi$
$223$	−0.191453	−0.0128207	−0.00641033	+	0.999979i	$0.502040\pi$
$224$	−12.1779	−0.813673
$225$	0	0
$226$	−2.85180	−0.189699
$227$	−6.45425	−0.428383	−0.214192	−	0.976792i	$-0.568712\pi$
$227$	−6.45425	−0.428383	−0.214192	+	0.976792i	$0.568712\pi$
$228$	0	0
$229$	−18.0747	−1.19441	−0.597204	−	0.802090i	$-0.703722\pi$
$229$	−18.0747	−1.19441	−0.597204	+	0.802090i	$0.703722\pi$
$230$	−16.5733	−1.09281
$231$	0	0
$232$	24.9145	1.63572
$233$	−1.11848	−0.0732742	−0.0366371	−	0.999329i	$-0.511665\pi$
$233$	−1.11848	−0.0732742	−0.0366371	+	0.999329i	$0.511665\pi$
$234$	0	0
$235$	−7.66825	−0.500221
$236$	−8.41519	−0.547782
$237$	0	0
$238$	−0.913948	−0.0592424
$239$	−30.4855	−1.97194	−0.985971	−	0.166915i	$-0.946619\pi$
$239$	−30.4855	−1.97194	−0.985971	+	0.166915i	$0.946619\pi$
$240$	0	0
$241$	3.26629	0.210400	0.105200	−	0.994451i	$-0.466452\pi$
$241$	3.26629	0.210400	0.105200	+	0.994451i	$0.466452\pi$
$242$	−0.926278	−0.0595434
$243$	0	0
$244$	1.14201	0.0731096
$245$	5.98773	0.382542
$246$	0	0
$247$	−11.6730	−0.742736
$248$	−1.83108	−0.116274
$249$	0	0
$250$	−2.57592	−0.162915
$251$	5.35900	0.338257	0.169128	−	0.985594i	$-0.445905\pi$
$251$	5.35900	0.338257	0.169128	+	0.985594i	$0.445905\pi$
$252$	0	0
$253$	5.93879	0.373369
$254$	−16.8071	−1.05457
$255$	0	0
$256$	−16.7710	−1.04819
$257$	31.0072	1.93418	0.967089	−	0.254439i	$-0.0818908\pi$
$257$	31.0072	1.93418	0.967089	+	0.254439i	$0.0818908\pi$
$258$	0	0
$259$	13.2725	0.824713
$260$	−13.2938	−0.824443
$261$	0	0
$262$	10.3800	0.641278
$263$	−27.6139	−1.70275	−0.851374	−	0.524560i	$-0.824230\pi$
$263$	−27.6139	−1.70275	−0.851374	+	0.524560i	$0.824230\pi$
$264$	0	0
$265$	26.4127	1.62252
$266$	−6.26537	−0.384155
$267$	0	0
$268$	−1.00298	−0.0612669
$269$	17.1085	1.04312	0.521561	−	0.853214i	$-0.325350\pi$
$269$	17.1085	1.04312	0.521561	+	0.853214i	$0.325350\pi$
$270$	0	0
$271$	−17.2027	−1.04499	−0.522494	−	0.852643i	$-0.674998\pi$
$271$	−17.2027	−1.04499	−0.522494	+	0.852643i	$0.674998\pi$
$272$	−0.181481	−0.0110039
$273$	0	0
$274$	13.2922	0.803012
$275$	−4.07696	−0.245850
$276$	0	0
$277$	−7.75361	−0.465869	−0.232935	−	0.972492i	$-0.574833\pi$
$277$	−7.75361	−0.465869	−0.232935	+	0.972492i	$0.574833\pi$
$278$	14.6407	0.878092
$279$	0	0
$280$	−19.6313	−1.17320
$281$	−26.5733	−1.58523	−0.792615	−	0.609723i	$-0.791281\pi$
$281$	−26.5733	−1.58523	−0.792615	+	0.609723i	$0.791281\pi$
$282$	0	0
$283$	−10.8092	−0.642539	−0.321269	−	0.946988i	$-0.604109\pi$
$283$	−10.8092	−0.642539	−0.321269	+	0.946988i	$0.604109\pi$
$284$	5.78756	0.343428
$285$	0	0
$286$	−3.57890	−0.211625
$287$	−20.2615	−1.19600
$288$	0	0
$289$	−16.8058	−0.988575
$290$	23.8899	1.40286
$291$	0	0
$292$	−16.1677	−0.946141
$293$	−14.1325	−0.825631	−0.412815	−	0.910815i	$-0.635455\pi$
$293$	−14.1325	−0.825631	−0.412815	+	0.910815i	$0.635455\pi$
$294$	0	0
$295$	−22.2006	−1.29257
$296$	17.2533	1.00283
$297$	0	0
$298$	2.09722	0.121489
$299$	22.9459	1.32700
$300$	0	0
$301$	7.15250	0.412263
$302$	18.5807	1.06920
$303$	0	0
$304$	−1.24411	−0.0713545
$305$	3.01280	0.172512
$306$	0	0
$307$	−5.54844	−0.316666	−0.158333	−	0.987386i	$-0.550612\pi$
$307$	−5.54844	−0.316666	−0.158333	+	0.987386i	$0.550612\pi$
$308$	2.55682	0.145688
$309$	0	0
$310$	−1.75578	−0.0997219
$311$	−29.8511	−1.69270	−0.846350	−	0.532628i	$-0.821205\pi$
$311$	−29.8511	−1.69270	−0.846350	+	0.532628i	$0.821205\pi$
$312$	0	0
$313$	−3.31243	−0.187230	−0.0936149	−	0.995608i	$-0.529842\pi$
$313$	−3.31243	−0.187230	−0.0936149	+	0.995608i	$0.529842\pi$
$314$	−6.58553	−0.371643
$315$	0	0
$316$	−18.3960	−1.03486
$317$	−12.6163	−0.708604	−0.354302	−	0.935131i	$-0.615282\pi$
$317$	−12.6163	−0.708604	−0.354302	+	0.935131i	$0.615282\pi$
$318$	0	0
$319$	−8.56058	−0.479300
$320$	−17.6607	−0.987266
$321$	0	0
$322$	12.3160	0.686344
$323$	1.33145	0.0740839
$324$	0	0
$325$	−15.7523	−0.873781
$326$	1.92742	0.106750
$327$	0	0
$328$	−26.3384	−1.45430
$329$	5.69844	0.314165
$330$	0	0
$331$	−5.46906	−0.300607	−0.150303	−	0.988640i	$-0.548025\pi$
$331$	−5.46906	−0.300607	−0.150303	+	0.988640i	$0.548025\pi$
$332$	8.52746	0.468005
$333$	0	0
$334$	−11.2981	−0.618207
$335$	−2.64602	−0.144568
$336$	0	0
$337$	15.2084	0.828455	0.414228	−	0.910173i	$-0.364052\pi$
$337$	15.2084	0.828455	0.414228	+	0.910173i	$0.364052\pi$
$338$	−1.78630	−0.0971621
$339$	0	0
$340$	1.51632	0.0822338
$341$	0.629158	0.0340708
$342$	0	0
$343$	−20.1217	−1.08647
$344$	9.29772	0.501299
$345$	0	0
$346$	19.0666	1.02503
$347$	−7.97403	−0.428069	−0.214034	−	0.976826i	$-0.568660\pi$
$347$	−7.97403	−0.428069	−0.214034	+	0.976826i	$0.568660\pi$
$348$	0	0
$349$	6.05068	0.323885	0.161943	−	0.986800i	$-0.448224\pi$
$349$	6.05068	0.323885	0.161943	+	0.986800i	$0.448224\pi$
$350$	−8.45489	−0.451933
$351$	0	0
$352$	5.43931	0.289916
$353$	−14.6700	−0.780806	−0.390403	−	0.920644i	$-0.627664\pi$
$353$	−14.6700	−0.780806	−0.390403	+	0.920644i	$0.627664\pi$
$354$	0	0
$355$	15.2685	0.810367
$356$	−1.63827	−0.0868281
$357$	0	0
$358$	−22.4646	−1.18729
$359$	−13.1214	−0.692521	−0.346260	−	0.938138i	$-0.612549\pi$
$359$	−13.1214	−0.692521	−0.346260	+	0.938138i	$0.612549\pi$
$360$	0	0
$361$	−9.87253	−0.519607
$362$	4.13642	0.217406
$363$	0	0
$364$	9.87888	0.517794
$365$	−42.6529	−2.23255
$366$	0	0
$367$	−17.2726	−0.901621	−0.450811	−	0.892620i	$-0.648865\pi$
$367$	−17.2726	−0.901621	−0.450811	+	0.892620i	$0.648865\pi$
$368$	2.44557	0.127484
$369$	0	0
$370$	16.5438	0.860069
$371$	−19.6278	−1.01903
$372$	0	0
$373$	−27.6887	−1.43367	−0.716833	−	0.697244i	$-0.754409\pi$
$373$	−27.6887	−1.43367	−0.716833	+	0.697244i	$0.754409\pi$
$374$	0.408217	0.0211084
$375$	0	0
$376$	7.40755	0.382015
$377$	−33.0758	−1.70349
$378$	0	0
$379$	10.7589	0.552649	0.276325	−	0.961064i	$-0.410884\pi$
$379$	10.7589	0.552649	0.276325	+	0.961064i	$0.410884\pi$
$380$	10.3948	0.533241
$381$	0	0
$382$	−2.99194	−0.153081
$383$	−15.2418	−0.778821	−0.389411	−	0.921064i	$-0.627321\pi$
$383$	−15.2418	−0.778821	−0.389411	+	0.921064i	$0.627321\pi$
$384$	0	0
$385$	6.74529	0.343772
$386$	−23.5926	−1.20083
$387$	0	0
$388$	4.68381	0.237785
$389$	11.0535	0.560435	0.280218	−	0.959936i	$-0.409593\pi$
$389$	11.0535	0.560435	0.280218	+	0.959936i	$0.409593\pi$
$390$	0	0
$391$	−2.61727	−0.132361
$392$	−5.78416	−0.292144
$393$	0	0
$394$	−0.432405	−0.0217842
$395$	−48.5316	−2.44189
$396$	0	0
$397$	5.16172	0.259059	0.129530	−	0.991576i	$-0.458653\pi$
$397$	5.16172	0.259059	0.129530	+	0.991576i	$0.458653\pi$
$398$	1.40884	0.0706188
$399$	0	0
$400$	−1.67888	−0.0839439
$401$	−0.397368	−0.0198436	−0.00992180	−	0.999951i	$-0.503158\pi$
$401$	−0.397368	−0.0198436	−0.00992180	+	0.999951i	$0.503158\pi$
$402$	0	0
$403$	2.43090	0.121092
$404$	−21.1599	−1.05274
$405$	0	0
$406$	−17.7531	−0.881072
$407$	−5.92820	−0.293850
$408$	0	0
$409$	0.156659	0.00774629	0.00387315	−	0.999992i	$-0.498767\pi$
$409$	0.156659	0.00774629	0.00387315	+	0.999992i	$0.498767\pi$
$410$	−25.2553	−1.24727
$411$	0	0
$412$	−17.0578	−0.840377
$413$	16.4977	0.811801
$414$	0	0
$415$	22.4968	1.10432
$416$	21.0161	1.03040
$417$	0	0
$418$	2.79844	0.136876
$419$	−20.8281	−1.01752	−0.508759	−	0.860909i	$-0.669896\pi$
$419$	−20.8281	−1.01752	−0.508759	+	0.860909i	$0.669896\pi$
$420$	0	0
$421$	16.1238	0.785824	0.392912	−	0.919576i	$-0.371468\pi$
$421$	16.1238	0.785824	0.392912	+	0.919576i	$0.371468\pi$
$422$	−18.5454	−0.902776
$423$	0	0
$424$	−25.5147	−1.23910
$425$	1.79674	0.0871549
$426$	0	0
$427$	−2.23888	−0.108347
$428$	9.68432	0.468109
$429$	0	0
$430$	8.91536	0.429937
$431$	−13.1273	−0.632318	−0.316159	−	0.948706i	$-0.602393\pi$
$431$	−13.1273	−0.632318	−0.316159	+	0.948706i	$0.602393\pi$
$432$	0	0
$433$	8.39574	0.403474	0.201737	−	0.979440i	$-0.435341\pi$
$433$	8.39574	0.403474	0.201737	+	0.979440i	$0.435341\pi$
$434$	1.30476	0.0626306
$435$	0	0
$436$	14.9824	0.717528
$437$	−17.9421	−0.858288
$438$	0	0
$439$	−35.0080	−1.67084	−0.835421	−	0.549611i	$-0.814776\pi$
$439$	−35.0080	−1.67084	−0.835421	+	0.549611i	$0.814776\pi$
$440$	8.76837	0.418016
$441$	0	0
$442$	1.57724	0.0750218
$443$	2.26391	0.107561	0.0537807	−	0.998553i	$-0.482873\pi$
$443$	2.26391	0.107561	0.0537807	+	0.998553i	$0.482873\pi$
$444$	0	0
$445$	−4.32201	−0.204883
$446$	0.177339	0.00839724
$447$	0	0
$448$	13.1241	0.620055
$449$	7.72927	0.364767	0.182383	−	0.983228i	$-0.441619\pi$
$449$	7.72927	0.364767	0.182383	+	0.983228i	$0.441619\pi$
$450$	0	0
$451$	9.04985	0.426141
$452$	−3.51599	−0.165378
$453$	0	0
$454$	5.97843	0.280582
$455$	26.0620	1.22181
$456$	0	0
$457$	−35.3937	−1.65565	−0.827824	−	0.560988i	$-0.810421\pi$
$457$	−35.3937	−1.65565	−0.827824	+	0.560988i	$0.810421\pi$
$458$	16.7422	0.782310
$459$	0	0
$460$	−20.4333	−0.952707
$461$	16.8619	0.785339	0.392670	−	0.919680i	$-0.371552\pi$
$461$	16.8619	0.785339	0.392670	+	0.919680i	$0.371552\pi$
$462$	0	0
$463$	−1.11156	−0.0516586	−0.0258293	−	0.999666i	$-0.508223\pi$
$463$	−1.11156	−0.0516586	−0.0258293	+	0.999666i	$0.508223\pi$
$464$	−3.52521	−0.163654
$465$	0	0
$466$	1.03603	0.0479930
$467$	−0.720363	−0.0333344	−0.0166672	−	0.999861i	$-0.505306\pi$
$467$	−0.720363	−0.0333344	−0.0166672	+	0.999861i	$0.505306\pi$
$468$	0	0
$469$	1.96632	0.0907961
$470$	7.10293	0.327634
$471$	0	0
$472$	21.4458	0.987125
$473$	−3.19468	−0.146892
$474$	0	0
$475$	12.3172	0.565152
$476$	−1.12681	−0.0516471
$477$	0	0
$478$	28.2380	1.29158
$479$	3.40158	0.155422	0.0777111	−	0.996976i	$-0.475239\pi$
$479$	3.40158	0.155422	0.0777111	+	0.996976i	$0.475239\pi$
$480$	0	0
$481$	−22.9050	−1.04438
$482$	−3.02549	−0.137807
$483$	0	0
$484$	−1.14201	−0.0519095
$485$	12.3566	0.561086
$486$	0	0
$487$	−0.765544	−0.0346901	−0.0173451	−	0.999850i	$-0.505521\pi$
$487$	−0.765544	−0.0346901	−0.0173451	+	0.999850i	$0.505521\pi$
$488$	−2.91037	−0.131746
$489$	0	0
$490$	−5.54630	−0.250556
$491$	−15.6369	−0.705685	−0.352842	−	0.935683i	$-0.614785\pi$
$491$	−15.6369	−0.705685	−0.352842	+	0.935683i	$0.614785\pi$
$492$	0	0
$493$	3.77270	0.169914
$494$	10.8125	0.486475
$495$	0	0
$496$	0.259085	0.0116333
$497$	−11.3463	−0.508953
$498$	0	0
$499$	2.80238	0.125452	0.0627260	−	0.998031i	$-0.480021\pi$
$499$	2.80238	0.125452	0.0627260	+	0.998031i	$0.480021\pi$
$500$	−3.17585	−0.142028
$501$	0	0
$502$	−4.96392	−0.221551
$503$	−8.03193	−0.358126	−0.179063	−	0.983838i	$-0.557307\pi$
$503$	−8.03193	−0.358126	−0.179063	+	0.983838i	$0.557307\pi$
$504$	0	0
$505$	−55.8230	−2.48409
$506$	−5.50097	−0.244548
$507$	0	0
$508$	−20.7215	−0.919368
$509$	22.2477	0.986113	0.493056	−	0.869997i	$-0.335880\pi$
$509$	22.2477	0.986113	0.493056	+	0.869997i	$0.335880\pi$
$510$	0	0
$511$	31.6963	1.40216
$512$	4.63685	0.204922
$513$	0	0
$514$	−28.7213	−1.26684
$515$	−45.0011	−1.98299
$516$	0	0
$517$	−2.54522	−0.111939
$518$	−12.2940	−0.540169
$519$	0	0
$520$	33.8787	1.48568
$521$	−23.5663	−1.03246	−0.516229	−	0.856450i	$-0.672665\pi$
$521$	−23.5663	−1.03246	−0.516229	+	0.856450i	$0.672665\pi$
$522$	0	0
$523$	16.7929	0.734304	0.367152	−	0.930161i	$-0.380333\pi$
$523$	16.7929	0.734304	0.367152	+	0.930161i	$0.380333\pi$
$524$	12.7975	0.559062
$525$	0	0
$526$	25.5782	1.11526
$527$	−0.277274	−0.0120783
$528$	0	0
$529$	12.2693	0.533447
$530$	−24.4655	−1.06271
$531$	0	0
$532$	−7.72459	−0.334903
$533$	34.9663	1.51456
$534$	0	0
$535$	25.5488	1.10457
$536$	2.55607	0.110405
$537$	0	0
$538$	−15.8472	−0.683222
$539$	1.98743	0.0856046
$540$	0	0
$541$	−23.1415	−0.994932	−0.497466	−	0.867483i	$-0.665736\pi$
$541$	−23.1415	−0.994932	−0.497466	+	0.867483i	$0.665736\pi$
$542$	15.9345	0.684444
$543$	0	0
$544$	−2.39714	−0.102777
$545$	39.5260	1.69311
$546$	0	0
$547$	−19.6581	−0.840519	−0.420259	−	0.907404i	$-0.638061\pi$
$547$	−19.6581	−0.840519	−0.420259	+	0.907404i	$0.638061\pi$
$548$	16.3880	0.700060
$549$	0	0
$550$	3.77640	0.161026
$551$	25.8630	1.10180
$552$	0	0
$553$	36.0649	1.53363
$554$	7.18200	0.305134
$555$	0	0
$556$	18.0506	0.765514
$557$	−22.0819	−0.935641	−0.467820	−	0.883824i	$-0.654961\pi$
$557$	−22.0819	−0.935641	−0.467820	+	0.883824i	$0.654961\pi$
$558$	0	0
$559$	−12.3434	−0.522071
$560$	2.77769	0.117379
$561$	0	0
$562$	24.6143	1.03829
$563$	−46.8881	−1.97610	−0.988049	−	0.154141i	$-0.950739\pi$
$563$	−46.8881	−1.97610	−0.988049	+	0.154141i	$0.950739\pi$
$564$	0	0
$565$	−9.27572	−0.390233
$566$	10.0123	0.420848
$567$	0	0
$568$	−14.7494	−0.618871
$569$	−17.0707	−0.715640	−0.357820	−	0.933791i	$-0.616480\pi$
$569$	−17.0707	−0.715640	−0.357820	+	0.933791i	$0.616480\pi$
$570$	0	0
$571$	6.12409	0.256285	0.128143	−	0.991756i	$-0.459099\pi$
$571$	6.12409	0.256285	0.128143	+	0.991756i	$0.459099\pi$
$572$	−4.41243	−0.184493
$573$	0	0
$574$	18.7678	0.783352
$575$	−24.2122	−1.00972
$576$	0	0
$577$	−28.2191	−1.17478	−0.587388	−	0.809305i	$-0.699844\pi$
$577$	−28.2191	−1.17478	−0.587388	+	0.809305i	$0.699844\pi$
$578$	15.5668	0.647494
$579$	0	0
$580$	29.4539	1.22301
$581$	−16.7178	−0.693573
$582$	0	0
$583$	8.76682	0.363085
$584$	41.2028	1.70498
$585$	0	0
$586$	13.0906	0.540770
$587$	37.5954	1.55173	0.775865	−	0.630899i	$-0.217314\pi$
$587$	37.5954	1.55173	0.775865	+	0.630899i	$0.217314\pi$
$588$	0	0
$589$	−1.90079	−0.0783208
$590$	20.5639	0.846603
$591$	0	0
$592$	−2.44121	−0.100333
$593$	28.5914	1.17411	0.587053	−	0.809548i	$-0.300288\pi$
$593$	28.5914	1.17411	0.587053	+	0.809548i	$0.300288\pi$
$594$	0	0
$595$	−2.97269	−0.121869
$596$	2.58567	0.105913
$597$	0	0
$598$	−21.2543	−0.869154
$599$	35.0071	1.43035	0.715176	−	0.698945i	$-0.246347\pi$
$599$	35.0071	1.43035	0.715176	+	0.698945i	$0.246347\pi$
$600$	0	0
$601$	30.7967	1.25622	0.628111	−	0.778124i	$-0.283828\pi$
$601$	30.7967	1.25622	0.628111	+	0.778124i	$0.283828\pi$
$602$	−6.62520	−0.270023
$603$	0	0
$604$	22.9082	0.932120
$605$	−3.01280	−0.122488
$606$	0	0
$607$	−10.4644	−0.424735	−0.212368	−	0.977190i	$-0.568117\pi$
$607$	−10.4644	−0.424735	−0.212368	+	0.977190i	$0.568117\pi$
$608$	−16.4331	−0.666450
$609$	0	0
$610$	−2.79069	−0.112992
$611$	−9.83408	−0.397844
$612$	0	0
$613$	10.4359	0.421502	0.210751	−	0.977540i	$-0.432409\pi$
$613$	10.4359	0.421502	0.210751	+	0.977540i	$0.432409\pi$
$614$	5.13940	0.207409
$615$	0	0
$616$	−6.51597	−0.262536
$617$	−36.7076	−1.47779	−0.738896	−	0.673820i	$-0.764652\pi$
$617$	−36.7076	−1.47779	−0.738896	+	0.673820i	$0.764652\pi$
$618$	0	0
$619$	−40.5806	−1.63107	−0.815537	−	0.578705i	$-0.803558\pi$
$619$	−40.5806	−1.63107	−0.815537	+	0.578705i	$0.803558\pi$
$620$	−2.16471	−0.0869368
$621$	0	0
$622$	27.6504	1.10868
$623$	3.21178	0.128677
$624$	0	0
$625$	−28.7632	−1.15053
$626$	3.06823	0.122631
$627$	0	0
$628$	−8.11931	−0.323996
$629$	2.61260	0.104171
$630$	0	0
$631$	44.2181	1.76029	0.880146	−	0.474702i	$-0.157444\pi$
$631$	44.2181	1.76029	0.880146	+	0.474702i	$0.157444\pi$
$632$	46.8816	1.86485
$633$	0	0
$634$	11.6862	0.464120
$635$	−54.6665	−2.16938
$636$	0	0
$637$	7.67891	0.304249
$638$	7.92947	0.313931
$639$	0	0
$640$	−16.4163	−0.648913
$641$	−29.2319	−1.15459	−0.577295	−	0.816536i	$-0.695892\pi$
$641$	−29.2319	−1.15459	−0.577295	+	0.816536i	$0.695892\pi$
$642$	0	0
$643$	14.1519	0.558096	0.279048	−	0.960277i	$-0.409981\pi$
$643$	14.1519	0.558096	0.279048	+	0.960277i	$0.409981\pi$
$644$	15.1844	0.598350
$645$	0	0
$646$	−1.23329	−0.0485233
$647$	8.07912	0.317623	0.158811	−	0.987309i	$-0.449234\pi$
$647$	8.07912	0.317623	0.158811	+	0.987309i	$0.449234\pi$
$648$	0	0
$649$	−7.36876	−0.289249
$650$	14.5910	0.572307
$651$	0	0
$652$	2.37632	0.0930637
$653$	12.3179	0.482036	0.241018	−	0.970521i	$-0.422519\pi$
$653$	12.3179	0.482036	0.241018	+	0.970521i	$0.422519\pi$
$654$	0	0
$655$	33.7618	1.31918
$656$	3.72670	0.145503
$657$	0	0
$658$	−5.27834	−0.205771
$659$	−12.5261	−0.487948	−0.243974	−	0.969782i	$-0.578451\pi$
$659$	−12.5261	−0.487948	−0.243974	+	0.969782i	$0.578451\pi$
$660$	0	0
$661$	−44.2616	−1.72158	−0.860788	−	0.508963i	$-0.830029\pi$
$661$	−44.2616	−1.72158	−0.860788	+	0.508963i	$0.830029\pi$
$662$	5.06587	0.196891
$663$	0	0
$664$	−21.7319	−0.843363
$665$	−20.3787	−0.790251
$666$	0	0
$667$	−50.8395	−1.96851
$668$	−13.9295	−0.538948
$669$	0	0
$670$	2.45095	0.0946886
$671$	1.00000	0.0386046
$672$	0	0
$673$	−15.6213	−0.602156	−0.301078	−	0.953600i	$-0.597346\pi$
$673$	−15.6213	−0.602156	−0.301078	+	0.953600i	$0.597346\pi$
$674$	−14.0872	−0.542620
$675$	0	0
$676$	−2.20234	−0.0847053
$677$	28.2183	1.08452	0.542259	−	0.840211i	$-0.317569\pi$
$677$	28.2183	1.08452	0.542259	+	0.840211i	$0.317569\pi$
$678$	0	0
$679$	−9.18248	−0.352391
$680$	−3.86428	−0.148188
$681$	0	0
$682$	−0.582775	−0.0223156
$683$	21.5858	0.825956	0.412978	−	0.910741i	$-0.364489\pi$
$683$	21.5858	0.825956	0.412978	+	0.910741i	$0.364489\pi$
$684$	0	0
$685$	43.2341	1.65189
$686$	18.6383	0.711615
$687$	0	0
$688$	−1.31556	−0.0501552
$689$	33.8727	1.29045
$690$	0	0
$691$	−43.7411	−1.66399	−0.831995	−	0.554783i	$-0.812801\pi$
$691$	−43.7411	−1.66399	−0.831995	+	0.554783i	$0.812801\pi$
$692$	23.5073	0.893613
$693$	0	0
$694$	7.38617	0.280375
$695$	47.6202	1.80634
$696$	0	0
$697$	−3.98833	−0.151069
$698$	−5.60461	−0.212138
$699$	0	0
$700$	−10.4240	−0.393992
$701$	20.7357	0.783177	0.391589	−	0.920140i	$-0.371926\pi$
$701$	20.7357	0.783177	0.391589	+	0.920140i	$0.371926\pi$
$702$	0	0
$703$	17.9101	0.675492
$704$	−5.86191	−0.220929
$705$	0	0
$706$	13.5885	0.511411
$707$	41.4833	1.56014
$708$	0	0
$709$	38.3956	1.44198	0.720988	−	0.692948i	$-0.243688\pi$
$709$	38.3956	1.44198	0.720988	+	0.692948i	$0.243688\pi$
$710$	−14.1429	−0.530772
$711$	0	0
$712$	4.17508	0.156468
$713$	3.73644	0.139931
$714$	0	0
$715$	−11.6407	−0.435336
$716$	−27.6967	−1.03507
$717$	0	0
$718$	12.1541	0.453585
$719$	34.1955	1.27528	0.637639	−	0.770335i	$-0.279911\pi$
$719$	34.1955	1.27528	0.637639	+	0.770335i	$0.279911\pi$
$720$	0	0
$721$	33.4413	1.24542
$722$	9.14470	0.340331
$723$	0	0
$724$	5.09980	0.189533
$725$	34.9011	1.29620
$726$	0	0
$727$	−19.5867	−0.726431	−0.363216	−	0.931705i	$-0.618321\pi$
$727$	−19.5867	−0.726431	−0.363216	+	0.931705i	$0.618321\pi$
$728$	−25.1760	−0.933085
$729$	0	0
$730$	39.5084	1.46227
$731$	1.40792	0.0520737
$732$	0	0
$733$	29.4150	1.08647	0.543235	−	0.839581i	$-0.317199\pi$
$733$	29.4150	1.08647	0.543235	+	0.839581i	$0.317199\pi$
$734$	15.9992	0.590541
$735$	0	0
$736$	32.3029	1.19070
$737$	−0.878261	−0.0323511
$738$	0	0
$739$	−44.4363	−1.63462	−0.817309	−	0.576200i	$-0.804535\pi$
$739$	−44.4363	−1.63462	−0.817309	+	0.576200i	$0.804535\pi$
$740$	20.3968	0.749802
$741$	0	0
$742$	18.1808	0.667439
$743$	4.68567	0.171901	0.0859504	−	0.996299i	$-0.472607\pi$
$743$	4.68567	0.171901	0.0859504	+	0.996299i	$0.472607\pi$
$744$	0	0
$745$	6.82140	0.249917
$746$	25.6474	0.939019
$747$	0	0
$748$	0.503291	0.0184022
$749$	−18.9858	−0.693727
$750$	0	0
$751$	−31.1595	−1.13703	−0.568513	−	0.822675i	$-0.692481\pi$
$751$	−31.1595	−1.13703	−0.568513	+	0.822675i	$0.692481\pi$
$752$	−1.04811	−0.0382208
$753$	0	0
$754$	30.6374	1.11575
$755$	60.4353	2.19947
$756$	0	0
$757$	11.2022	0.407150	0.203575	−	0.979059i	$-0.434744\pi$
$757$	11.2022	0.407150	0.203575	+	0.979059i	$0.434744\pi$
$758$	−9.96576	−0.361973
$759$	0	0
$760$	−26.4908	−0.960921
$761$	24.4537	0.886447	0.443223	−	0.896411i	$-0.353835\pi$
$761$	24.4537	0.886447	0.443223	+	0.896411i	$0.353835\pi$
$762$	0	0
$763$	−29.3726	−1.06336
$764$	−3.68876	−0.133455
$765$	0	0
$766$	14.1182	0.510110
$767$	−28.4710	−1.02803
$768$	0	0
$769$	43.6514	1.57411	0.787055	−	0.616883i	$-0.211605\pi$
$769$	43.6514	1.57411	0.787055	+	0.616883i	$0.211605\pi$
$770$	−6.24801	−0.225163
$771$	0	0
$772$	−29.0873	−1.04688
$773$	32.9837	1.18634	0.593170	−	0.805077i	$-0.297876\pi$
$773$	32.9837	1.18634	0.593170	+	0.805077i	$0.297876\pi$
$774$	0	0
$775$	−2.56505	−0.0921394
$776$	−11.9365	−0.428497
$777$	0	0
$778$	−10.2386	−0.367072
$779$	−27.3411	−0.979598
$780$	0	0
$781$	5.06787	0.181343
$782$	2.42432	0.0866934
$783$	0	0
$784$	0.818416	0.0292292
$785$	−21.4200	−0.764513
$786$	0	0
$787$	−53.9059	−1.92154	−0.960769	−	0.277350i	$-0.910544\pi$
$787$	−53.9059	−1.92154	−0.960769	+	0.277350i	$0.910544\pi$
$788$	−0.533112	−0.0189913
$789$	0	0
$790$	44.9537	1.59938
$791$	6.89299	0.245087
$792$	0	0
$793$	3.86374	0.137205
$794$	−4.78118	−0.169678
$795$	0	0
$796$	1.73696	0.0615650
$797$	−14.7906	−0.523912	−0.261956	−	0.965080i	$-0.584367\pi$
$797$	−14.7906	−0.523912	−0.261956	+	0.965080i	$0.584367\pi$
$798$	0	0
$799$	1.12170	0.0396828
$800$	−22.1758	−0.784035
$801$	0	0
$802$	0.368073	0.0129971
$803$	−14.1572	−0.499597
$804$	0	0
$805$	40.0589	1.41189
$806$	−2.25169	−0.0793124
$807$	0	0
$808$	53.9252	1.89708
$809$	−5.63134	−0.197988	−0.0989938	−	0.995088i	$-0.531562\pi$
$809$	−5.63134	−0.197988	−0.0989938	+	0.995088i	$0.531562\pi$
$810$	0	0
$811$	56.9456	1.99963	0.999815	−	0.0192532i	$-0.00612885\pi$
$811$	56.9456	1.99963	0.999815	+	0.0192532i	$0.00612885\pi$
$812$	−21.8878	−0.768113
$813$	0	0
$814$	5.49116	0.192465
$815$	6.26909	0.219597
$816$	0	0
$817$	9.65168	0.337670
$818$	−0.145110	−0.00507365
$819$	0	0
$820$	−31.1373	−1.08736
$821$	0.746728	0.0260610	0.0130305	−	0.999915i	$-0.495852\pi$
$821$	0.746728	0.0260610	0.0130305	+	0.999915i	$0.495852\pi$
$822$	0	0
$823$	−17.1908	−0.599233	−0.299616	−	0.954060i	$-0.596859\pi$
$823$	−17.1908	−0.599233	−0.299616	+	0.954060i	$0.596859\pi$
$824$	43.4712	1.51439
$825$	0	0
$826$	−15.2815	−0.531711
$827$	32.3225	1.12396	0.561981	−	0.827150i	$-0.310039\pi$
$827$	32.3225	1.12396	0.561981	+	0.827150i	$0.310039\pi$
$828$	0	0
$829$	3.79141	0.131681	0.0658406	−	0.997830i	$-0.479027\pi$
$829$	3.79141	0.131681	0.0658406	+	0.997830i	$0.479027\pi$
$830$	−20.8383	−0.723307
$831$	0	0
$832$	−22.6489	−0.785208
$833$	−0.875874	−0.0303472
$834$	0	0
$835$	−36.7482	−1.27172
$836$	3.45021	0.119328
$837$	0	0
$838$	19.2926	0.666452
$839$	27.0204	0.932848	0.466424	−	0.884561i	$-0.345542\pi$
$839$	27.0204	0.932848	0.466424	+	0.884561i	$0.345542\pi$
$840$	0	0
$841$	44.2835	1.52702
$842$	−14.9351	−0.514697
$843$	0	0
$844$	−22.8646	−0.787034
$845$	−5.81011	−0.199874
$846$	0	0
$847$	2.23888	0.0769287
$848$	3.61014	0.123973
$849$	0	0
$850$	−1.66428	−0.0570845
$851$	−35.2063	−1.20686
$852$	0	0
$853$	45.9383	1.57290	0.786449	−	0.617655i	$-0.211917\pi$
$853$	45.9383	1.57290	0.786449	+	0.617655i	$0.211917\pi$
$854$	2.07382	0.0709647
$855$	0	0
$856$	−24.6802	−0.843551
$857$	−13.5754	−0.463726	−0.231863	−	0.972748i	$-0.574482\pi$
$857$	−13.5754	−0.463726	−0.231863	+	0.972748i	$0.574482\pi$
$858$	0	0
$859$	−3.97110	−0.135492	−0.0677462	−	0.997703i	$-0.521581\pi$
$859$	−3.97110	−0.135492	−0.0677462	+	0.997703i	$0.521581\pi$
$860$	10.9918	0.374816
$861$	0	0
$862$	12.1595	0.414154
$863$	−31.5985	−1.07563	−0.537813	−	0.843064i	$-0.680749\pi$
$863$	−31.5985	−1.07563	−0.537813	+	0.843064i	$0.680749\pi$
$864$	0	0
$865$	62.0159	2.10860
$866$	−7.77679	−0.264266
$867$	0	0
$868$	1.60864	0.0546009
$869$	−16.1085	−0.546442
$870$	0	0
$871$	−3.39337	−0.114980
$872$	−38.1822	−1.29301
$873$	0	0
$874$	16.6194	0.562159
$875$	6.22617	0.210483
$876$	0	0
$877$	−1.10744	−0.0373956	−0.0186978	−	0.999825i	$-0.505952\pi$
$877$	−1.10744	−0.0373956	−0.0186978	+	0.999825i	$0.505952\pi$
$878$	32.4271	1.09436
$879$	0	0
$880$	−1.24066	−0.0418226
$881$	−28.4348	−0.957994	−0.478997	−	0.877816i	$-0.659000\pi$
$881$	−28.4348	−0.957994	−0.478997	+	0.877816i	$0.659000\pi$
$882$	0	0
$883$	−13.7861	−0.463939	−0.231969	−	0.972723i	$-0.574517\pi$
$883$	−13.7861	−0.463939	−0.231969	+	0.972723i	$0.574517\pi$
$884$	1.94459	0.0654035
$885$	0	0
$886$	−2.09701	−0.0704503
$887$	34.6726	1.16419	0.582096	−	0.813120i	$-0.302233\pi$
$887$	34.6726	1.16419	0.582096	+	0.813120i	$0.302233\pi$
$888$	0	0
$889$	40.6239	1.36248
$890$	4.00339	0.134194
$891$	0	0
$892$	0.218641	0.00732065
$893$	7.68955	0.257321
$894$	0	0
$895$	−73.0681	−2.44240
$896$	12.1993	0.407551
$897$	0	0
$898$	−7.15945	−0.238914
$899$	−5.38595	−0.179632
$900$	0	0
$901$	−3.86360	−0.128715
$902$	−8.38268	−0.279113
$903$	0	0
$904$	8.96038	0.298018
$905$	13.4541	0.447228
$906$	0	0
$907$	−59.6041	−1.97912	−0.989561	−	0.144118i	$-0.953966\pi$
$907$	−59.6041	−1.97912	−0.989561	+	0.144118i	$0.953966\pi$
$908$	7.37081	0.244609
$909$	0	0
$910$	−24.1407	−0.800256
$911$	16.1094	0.533727	0.266864	−	0.963734i	$-0.414013\pi$
$911$	16.1094	0.533727	0.266864	+	0.963734i	$0.414013\pi$
$912$	0	0
$913$	7.46706	0.247124
$914$	32.7844	1.08441
$915$	0	0
$916$	20.6414	0.682012
$917$	−25.0892	−0.828517
$918$	0	0
$919$	−38.9602	−1.28518	−0.642589	−	0.766211i	$-0.722140\pi$
$919$	−38.9602	−1.28518	−0.642589	+	0.766211i	$0.722140\pi$
$920$	52.0736	1.71681
$921$	0	0
$922$	−15.6189	−0.514379
$923$	19.5809	0.644514
$924$	0	0
$925$	24.1690	0.794672
$926$	1.02961	0.0338352
$927$	0	0
$928$	−46.5636	−1.52853
$929$	−45.4860	−1.49235	−0.746174	−	0.665751i	$-0.768111\pi$
$929$	−45.4860	−1.49235	−0.746174	+	0.665751i	$0.768111\pi$
$930$	0	0
$931$	−6.00436	−0.196785
$932$	1.27732	0.0418399
$933$	0	0
$934$	0.667257	0.0218333
$935$	1.32776	0.0434224
$936$	0	0
$937$	−38.1653	−1.24681	−0.623404	−	0.781900i	$-0.714251\pi$
$937$	−38.1653	−1.24681	−0.623404	+	0.781900i	$0.714251\pi$
$938$	−1.82136	−0.0594694
$939$	0	0
$940$	8.75721	0.285629
$941$	58.9633	1.92215	0.961075	−	0.276288i	$-0.0891045\pi$
$941$	58.9633	1.92215	0.961075	+	0.276288i	$0.0891045\pi$
$942$	0	0
$943$	53.7452	1.75018
$944$	−3.03443	−0.0987623
$945$	0	0
$946$	2.95916	0.0962107
$947$	40.0700	1.30210	0.651051	−	0.759034i	$-0.274329\pi$
$947$	40.0700	1.30210	0.651051	+	0.759034i	$0.274329\pi$
$948$	0	0
$949$	−54.6998	−1.77563
$950$	−11.4091	−0.370162
$951$	0	0
$952$	2.87163	0.0930701
$953$	16.2023	0.524844	0.262422	−	0.964953i	$-0.415479\pi$
$953$	16.2023	0.524844	0.262422	+	0.964953i	$0.415479\pi$
$954$	0	0
$955$	−9.73153	−0.314905
$956$	34.8147	1.12599
$957$	0	0
$958$	−3.15081	−0.101798
$959$	−32.1282	−1.03747
$960$	0	0
$961$	−30.6042	−0.987231
$962$	21.2164	0.684044
$963$	0	0
$964$	−3.73013	−0.120139
$965$	−76.7369	−2.47025
$966$	0	0
$967$	−39.0480	−1.25570	−0.627850	−	0.778334i	$-0.716065\pi$
$967$	−39.0480	−1.25570	−0.627850	+	0.778334i	$0.716065\pi$
$968$	2.91037	0.0935430
$969$	0	0
$970$	−11.4457	−0.367499
$971$	0.673013	0.0215980	0.0107990	−	0.999942i	$-0.496563\pi$
$971$	0.673013	0.0215980	0.0107990	+	0.999942i	$0.496563\pi$
$972$	0	0
$973$	−35.3876	−1.13447
$974$	0.709106	0.0227212
$975$	0	0
$976$	0.411796	0.0131813
$977$	−6.63719	−0.212343	−0.106171	−	0.994348i	$-0.533859\pi$
$977$	−6.63719	−0.212343	−0.106171	+	0.994348i	$0.533859\pi$
$978$	0	0
$979$	−1.43455	−0.0458484
$980$	−6.83804	−0.218433
$981$	0	0
$982$	14.4841	0.462208
$983$	5.75805	0.183653	0.0918266	−	0.995775i	$-0.470729\pi$
$983$	5.75805	0.183653	0.0918266	+	0.995775i	$0.470729\pi$
$984$	0	0
$985$	−1.40643	−0.0448127
$986$	−3.49457	−0.111290
$987$	0	0
$988$	13.3307	0.424106
$989$	−18.9726	−0.603292
$990$	0	0
$991$	−8.34089	−0.264957	−0.132479	−	0.991186i	$-0.542294\pi$
$991$	−8.34089	−0.264957	−0.132479	+	0.991186i	$0.542294\pi$
$992$	3.42218	0.108654
$993$	0	0
$994$	10.5099	0.333353
$995$	4.58238	0.145271
$996$	0	0
$997$	−17.8266	−0.564574	−0.282287	−	0.959330i	$-0.591093\pi$
$997$	−17.8266	−0.564574	−0.282287	+	0.959330i	$0.591093\pi$
$998$	−2.59579	−0.0821682
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.n.1.10	✓	25
3.2	odd	2		6039.2.a.o.1.16	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.10	✓	25	1.1	even	1	trivial
6039.2.a.o.1.16	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)

\(f(q)\)	\(=\)	\(q-0.926278 q^{2} -1.14201 q^{4} -3.01280 q^{5} +2.23888 q^{7} +2.91037 q^{8} +O(q^{10})\)	\(q-0.926278 q^{2} -1.14201 q^{4} -3.01280 q^{5} +2.23888 q^{7} +2.91037 q^{8} +2.79069 q^{10} -1.00000 q^{11} -3.86374 q^{13} -2.07382 q^{14} -0.411796 q^{16} +0.440707 q^{17} +3.02117 q^{19} +3.44064 q^{20} +0.926278 q^{22} -5.93879 q^{23} +4.07696 q^{25} +3.57890 q^{26} -2.55682 q^{28} +8.56058 q^{29} -0.629158 q^{31} -5.43931 q^{32} -0.408217 q^{34} -6.74529 q^{35} +5.92820 q^{37} -2.79844 q^{38} -8.76837 q^{40} -9.04985 q^{41} +3.19468 q^{43} +1.14201 q^{44} +5.50097 q^{46} +2.54522 q^{47} -1.98743 q^{49} -3.77640 q^{50} +4.41243 q^{52} -8.76682 q^{53} +3.01280 q^{55} +6.51597 q^{56} -7.92947 q^{58} +7.36876 q^{59} -1.00000 q^{61} +0.582775 q^{62} +5.86191 q^{64} +11.6407 q^{65} +0.878261 q^{67} -0.503291 q^{68} +6.24801 q^{70} -5.06787 q^{71} +14.1572 q^{73} -5.49116 q^{74} -3.45021 q^{76} -2.23888 q^{77} +16.1085 q^{79} +1.24066 q^{80} +8.38268 q^{82} -7.46706 q^{83} -1.32776 q^{85} -2.95916 q^{86} -2.91037 q^{88} +1.43455 q^{89} -8.65044 q^{91} +6.78216 q^{92} -2.35758 q^{94} -9.10218 q^{95} -4.10138 q^{97} +1.84091 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) \) 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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