LMFDB - Embedded newform 6039.2.a.o.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:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.07776 q^{2} -0.838436 q^{4} +3.04682 q^{5} +3.44893 q^{7} +3.05915 q^{8} +O(q^{10})$	\(q-1.07776 q^{2} -0.838436 q^{4} +3.04682 q^{5} +3.44893 q^{7} +3.05915 q^{8} -3.28374 q^{10} +1.00000 q^{11} +2.71326 q^{13} -3.71712 q^{14} -1.62015 q^{16} +3.42003 q^{17} -1.81606 q^{19} -2.55457 q^{20} -1.07776 q^{22} -6.08551 q^{23} +4.28313 q^{25} -2.92424 q^{26} -2.89171 q^{28} +0.376637 q^{29} +4.93900 q^{31} -4.37216 q^{32} -3.68597 q^{34} +10.5083 q^{35} +3.50288 q^{37} +1.95727 q^{38} +9.32069 q^{40} +5.87719 q^{41} -8.07962 q^{43} -0.838436 q^{44} +6.55872 q^{46} +6.22226 q^{47} +4.89515 q^{49} -4.61618 q^{50} -2.27489 q^{52} +2.61126 q^{53} +3.04682 q^{55} +10.5508 q^{56} -0.405924 q^{58} -2.54527 q^{59} -1.00000 q^{61} -5.32305 q^{62} +7.95245 q^{64} +8.26682 q^{65} +7.44820 q^{67} -2.86748 q^{68} -11.3254 q^{70} +4.42352 q^{71} +9.20943 q^{73} -3.77526 q^{74} +1.52265 q^{76} +3.44893 q^{77} +5.23667 q^{79} -4.93632 q^{80} -6.33419 q^{82} -8.81625 q^{83} +10.4202 q^{85} +8.70788 q^{86} +3.05915 q^{88} +8.15358 q^{89} +9.35785 q^{91} +5.10231 q^{92} -6.70609 q^{94} -5.53320 q^{95} +1.34982 q^{97} -5.27579 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) $ 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8	$ 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8} + 25 q^{11} + 4 q^{13} + 18 q^{14} + 21 q^{16} + 20 q^{17} + 14 q^{19} + 12 q^{20} + 5 q^{22} + 20 q^{23} + 13 q^{25} + 16 q^{26} - 14 q^{28} + 28 q^{29} - 12 q^{31} + 35 q^{32} + 6 q^{34} + 10 q^{35} - 8 q^{37} + 32 q^{38} + 24 q^{40} + 26 q^{41} + 18 q^{43} + 25 q^{44} + 4 q^{46} + 12 q^{47} + 23 q^{49} + 43 q^{50} + 22 q^{52} + 36 q^{53} + 4 q^{55} + 26 q^{56} - 20 q^{58} + 46 q^{59} - 25 q^{61} - 14 q^{62} - 13 q^{64} + 60 q^{65} - 20 q^{67} + 44 q^{68} - 20 q^{70} + 52 q^{71} + 6 q^{73} + 32 q^{74} + 4 q^{77} + 26 q^{79} + 52 q^{80} + 6 q^{82} + 38 q^{83} - 4 q^{85} + 34 q^{86} + 15 q^{88} + 82 q^{89} - 58 q^{91} + 36 q^{92} + 16 q^{94} + 30 q^{95} + 35 q^{98}+O(q^{100}) $ 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8 + 25 * q^11 + 4 * q^13 + 18 * q^14 + 21 * q^16 + 20 * q^17 + 14 * q^19 + 12 * q^20 + 5 * q^22 + 20 * q^23 + 13 * q^25 + 16 * q^26 - 14 * q^28 + 28 * q^29 - 12 * q^31 + 35 * q^32 + 6 * q^34 + 10 * q^35 - 8 * q^37 + 32 * q^38 + 24 * q^40 + 26 * q^41 + 18 * q^43 + 25 * q^44 + 4 * q^46 + 12 * q^47 + 23 * q^49 + 43 * q^50 + 22 * q^52 + 36 * q^53 + 4 * q^55 + 26 * q^56 - 20 * q^58 + 46 * q^59 - 25 * q^61 - 14 * q^62 - 13 * q^64 + 60 * q^65 - 20 * q^67 + 44 * q^68 - 20 * q^70 + 52 * q^71 + 6 * q^73 + 32 * q^74 + 4 * q^77 + 26 * q^79 + 52 * q^80 + 6 * q^82 + 38 * q^83 - 4 * q^85 + 34 * q^86 + 15 * q^88 + 82 * q^89 - 58 * q^91 + 36 * q^92 + 16 * q^94 + 30 * q^95 + 35 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.07776	−0.762091	−0.381045	−	0.924556i	$-0.624436\pi$
$2$	−1.07776	−0.762091	−0.381045	+	0.924556i	$0.624436\pi$
$3$	0	0
$3$	0	0
$4$	−0.838436	−0.419218
$5$	3.04682	1.36258	0.681290	−	0.732013i	$-0.261419\pi$
$5$	3.04682	1.36258	0.681290	+	0.732013i	$0.261419\pi$
$6$	0	0
$7$	3.44893	1.30357	0.651787	−	0.758402i	$-0.274019\pi$
$7$	3.44893	1.30357	0.651787	+	0.758402i	$0.274019\pi$
$8$	3.05915	1.08157
$9$	0	0
$10$	−3.28374	−1.03841
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.71326	0.752523	0.376261	−	0.926514i	$-0.377209\pi$
$13$	2.71326	0.752523	0.376261	+	0.926514i	$0.377209\pi$
$14$	−3.71712	−0.993442
$15$	0	0
$16$	−1.62015	−0.405038
$17$	3.42003	0.829480	0.414740	−	0.909940i	$-0.363873\pi$
$17$	3.42003	0.829480	0.414740	+	0.909940i	$0.363873\pi$
$18$	0	0
$19$	−1.81606	−0.416632	−0.208316	−	0.978062i	$-0.566798\pi$
$19$	−1.81606	−0.416632	−0.208316	+	0.978062i	$0.566798\pi$
$20$	−2.55457	−0.571218
$21$	0	0
$22$	−1.07776	−0.229779
$23$	−6.08551	−1.26892	−0.634459	−	0.772957i	$-0.718777\pi$
$23$	−6.08551	−1.26892	−0.634459	+	0.772957i	$0.718777\pi$
$24$	0	0
$25$	4.28313	0.856626
$26$	−2.92424	−0.573491
$27$	0	0
$28$	−2.89171	−0.546482
$29$	0.376637	0.0699397	0.0349699	−	0.999388i	$-0.488866\pi$
$29$	0.376637	0.0699397	0.0349699	+	0.999388i	$0.488866\pi$
$30$	0	0
$31$	4.93900	0.887071	0.443536	−	0.896257i	$-0.353724\pi$
$31$	4.93900	0.887071	0.443536	+	0.896257i	$0.353724\pi$
$32$	−4.37216	−0.772897
$33$	0	0
$34$	−3.68597	−0.632139
$35$	10.5083	1.77623
$36$	0	0
$37$	3.50288	0.575870	0.287935	−	0.957650i	$-0.407031\pi$
$37$	3.50288	0.575870	0.287935	+	0.957650i	$0.407031\pi$
$38$	1.95727	0.317511
$39$	0	0
$40$	9.32069	1.47373
$41$	5.87719	0.917862	0.458931	−	0.888472i	$-0.348232\pi$
$41$	5.87719	0.917862	0.458931	+	0.888472i	$0.348232\pi$
$42$	0	0
$43$	−8.07962	−1.23213	−0.616065	−	0.787695i	$-0.711274\pi$
$43$	−8.07962	−1.23213	−0.616065	+	0.787695i	$0.711274\pi$
$44$	−0.838436	−0.126399
$45$	0	0
$46$	6.55872	0.967030
$47$	6.22226	0.907609	0.453805	−	0.891101i	$-0.350066\pi$
$47$	6.22226	0.907609	0.453805	+	0.891101i	$0.350066\pi$
$48$	0	0
$49$	4.89515	0.699307
$50$	−4.61618	−0.652827
$51$	0	0
$52$	−2.27489	−0.315471
$53$	2.61126	0.358684	0.179342	−	0.983787i	$-0.442603\pi$
$53$	2.61126	0.358684	0.179342	+	0.983787i	$0.442603\pi$
$54$	0	0
$55$	3.04682	0.410834
$56$	10.5508	1.40991
$57$	0	0
$58$	−0.405924	−0.0533004
$59$	−2.54527	−0.331365	−0.165683	−	0.986179i	$-0.552983\pi$
$59$	−2.54527	−0.331365	−0.165683	+	0.986179i	$0.552983\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−5.32305	−0.676028
$63$	0	0
$64$	7.95245	0.994056
$65$	8.26682	1.02537
$66$	0	0
$67$	7.44820	0.909942	0.454971	−	0.890506i	$-0.349650\pi$
$67$	7.44820	0.909942	0.454971	+	0.890506i	$0.349650\pi$
$68$	−2.86748	−0.347733
$69$	0	0
$70$	−11.3254	−1.35364
$71$	4.42352	0.524975	0.262488	−	0.964935i	$-0.415457\pi$
$71$	4.42352	0.524975	0.262488	+	0.964935i	$0.415457\pi$
$72$	0	0
$73$	9.20943	1.07788	0.538941	−	0.842344i	$-0.318825\pi$
$73$	9.20943	1.07788	0.538941	+	0.842344i	$0.318825\pi$
$74$	−3.77526	−0.438865
$75$	0	0
$76$	1.52265	0.174660
$77$	3.44893	0.393043
$78$	0	0
$79$	5.23667	0.589171	0.294585	−	0.955625i	$-0.404818\pi$
$79$	5.23667	0.589171	0.294585	+	0.955625i	$0.404818\pi$
$80$	−4.93632	−0.551898
$81$	0	0
$82$	−6.33419	−0.699494
$83$	−8.81625	−0.967710	−0.483855	−	0.875148i	$-0.660764\pi$
$83$	−8.81625	−0.967710	−0.483855	+	0.875148i	$0.660764\pi$
$84$	0	0
$85$	10.4202	1.13023
$86$	8.70788	0.938995
$87$	0	0
$88$	3.05915	0.326106
$89$	8.15358	0.864278	0.432139	−	0.901807i	$-0.357759\pi$
$89$	8.15358	0.864278	0.432139	+	0.901807i	$0.357759\pi$
$90$	0	0
$91$	9.35785	0.980970
$92$	5.10231	0.531953
$93$	0	0
$94$	−6.70609	−0.691681
$95$	−5.53320	−0.567695
$96$	0	0
$97$	1.34982	0.137054	0.0685269	−	0.997649i	$-0.478170\pi$
$97$	1.34982	0.137054	0.0685269	+	0.997649i	$0.478170\pi$
$98$	−5.27579	−0.532935
$99$	0	0
$100$	−3.59113	−0.359113
$101$	4.97047	0.494580	0.247290	−	0.968942i	$-0.420460\pi$
$101$	4.97047	0.494580	0.247290	+	0.968942i	$0.420460\pi$
$102$	0	0
$103$	−18.1042	−1.78386	−0.891929	−	0.452176i	$-0.850648\pi$
$103$	−18.1042	−1.78386	−0.891929	+	0.452176i	$0.850648\pi$
$104$	8.30027	0.813908
$105$	0	0
$106$	−2.81431	−0.273350
$107$	1.38867	0.134247	0.0671237	−	0.997745i	$-0.478618\pi$
$107$	1.38867	0.134247	0.0671237	+	0.997745i	$0.478618\pi$
$108$	0	0
$109$	−11.4775	−1.09935	−0.549673	−	0.835380i	$-0.685248\pi$
$109$	−11.4775	−1.09935	−0.549673	+	0.835380i	$0.685248\pi$
$110$	−3.28374	−0.313092
$111$	0	0
$112$	−5.58780	−0.527998
$113$	8.65339	0.814042	0.407021	−	0.913419i	$-0.366568\pi$
$113$	8.65339	0.814042	0.407021	+	0.913419i	$0.366568\pi$
$114$	0	0
$115$	−18.5415	−1.72900
$116$	−0.315786	−0.0293200
$117$	0	0
$118$	2.74318	0.252530
$119$	11.7955	1.08129
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.07776	0.0975757
$123$	0	0
$124$	−4.14104	−0.371876
$125$	−2.18417	−0.195358
$126$	0	0
$127$	−8.67839	−0.770082	−0.385041	−	0.922899i	$-0.625813\pi$
$127$	−8.67839	−0.770082	−0.385041	+	0.922899i	$0.625813\pi$
$128$	0.173509	0.0153362
$129$	0	0
$130$	−8.90964	−0.781427
$131$	3.54246	0.309506	0.154753	−	0.987953i	$-0.450542\pi$
$131$	3.54246	0.309506	0.154753	+	0.987953i	$0.450542\pi$
$132$	0	0
$133$	−6.26346	−0.543111
$134$	−8.02736	−0.693458
$135$	0	0
$136$	10.4624	0.897143
$137$	18.0703	1.54385	0.771924	−	0.635715i	$-0.219295\pi$
$137$	18.0703	1.54385	0.771924	+	0.635715i	$0.219295\pi$
$138$	0	0
$139$	−3.32448	−0.281979	−0.140989	−	0.990011i	$-0.545028\pi$
$139$	−3.32448	−0.281979	−0.140989	+	0.990011i	$0.545028\pi$
$140$	−8.81053	−0.744626
$141$	0	0
$142$	−4.76749	−0.400079
$143$	2.71326	0.226894
$144$	0	0
$145$	1.14755	0.0952986
$146$	−9.92554	−0.821444
$147$	0	0
$148$	−2.93694	−0.241415
$149$	−1.93908	−0.158856	−0.0794278	−	0.996841i	$-0.525309\pi$
$149$	−1.93908	−0.158856	−0.0794278	+	0.996841i	$0.525309\pi$
$150$	0	0
$151$	−8.63107	−0.702386	−0.351193	−	0.936303i	$-0.614224\pi$
$151$	−8.63107	−0.702386	−0.351193	+	0.936303i	$0.614224\pi$
$152$	−5.55559	−0.450618
$153$	0	0
$154$	−3.71712	−0.299534
$155$	15.0483	1.20871
$156$	0	0
$157$	−7.40501	−0.590984	−0.295492	−	0.955345i	$-0.595484\pi$
$157$	−7.40501	−0.590984	−0.295492	+	0.955345i	$0.595484\pi$
$158$	−5.64386	−0.449002
$159$	0	0
$160$	−13.3212	−1.05313
$161$	−20.9885	−1.65413
$162$	0	0
$163$	−4.33911	−0.339865	−0.169933	−	0.985456i	$-0.554355\pi$
$163$	−4.33911	−0.339865	−0.169933	+	0.985456i	$0.554355\pi$
$164$	−4.92764	−0.384784
$165$	0	0
$166$	9.50180	0.737482
$167$	12.1403	0.939441	0.469721	−	0.882815i	$-0.344355\pi$
$167$	12.1403	0.939441	0.469721	+	0.882815i	$0.344355\pi$
$168$	0	0
$169$	−5.63822	−0.433709
$170$	−11.2305	−0.861340
$171$	0	0
$172$	6.77424	0.516531
$173$	−4.34300	−0.330192	−0.165096	−	0.986278i	$-0.552793\pi$
$173$	−4.34300	−0.330192	−0.165096	+	0.986278i	$0.552793\pi$
$174$	0	0
$175$	14.7722	1.11668
$176$	−1.62015	−0.122124
$177$	0	0
$178$	−8.78759	−0.658658
$179$	6.46245	0.483026	0.241513	−	0.970398i	$-0.422356\pi$
$179$	6.46245	0.483026	0.241513	+	0.970398i	$0.422356\pi$
$180$	0	0
$181$	−14.7017	−1.09277	−0.546385	−	0.837534i	$-0.683996\pi$
$181$	−14.7017	−1.09277	−0.546385	+	0.837534i	$0.683996\pi$
$182$	−10.0855	−0.747588
$183$	0	0
$184$	−18.6165	−1.37243
$185$	10.6727	0.784669
$186$	0	0
$187$	3.42003	0.250098
$188$	−5.21696	−0.380486
$189$	0	0
$190$	5.96346	0.432635
$191$	24.4638	1.77014	0.885070	−	0.465459i	$-0.154111\pi$
$191$	24.4638	1.77014	0.885070	+	0.465459i	$0.154111\pi$
$192$	0	0
$193$	−2.45063	−0.176400	−0.0882001	−	0.996103i	$-0.528111\pi$
$193$	−2.45063	−0.176400	−0.0882001	+	0.996103i	$0.528111\pi$
$194$	−1.45478	−0.104447
$195$	0	0
$196$	−4.10427	−0.293162
$197$	9.82650	0.700109	0.350055	−	0.936729i	$-0.386163\pi$
$197$	9.82650	0.700109	0.350055	+	0.936729i	$0.386163\pi$
$198$	0	0
$199$	−18.7230	−1.32724	−0.663619	−	0.748071i	$-0.730980\pi$
$199$	−18.7230	−1.32724	−0.663619	+	0.748071i	$0.730980\pi$
$200$	13.1027	0.926503
$201$	0	0
$202$	−5.35697	−0.376915
$203$	1.29900	0.0911717
$204$	0	0
$205$	17.9067	1.25066
$206$	19.5119	1.35946
$207$	0	0
$208$	−4.39590	−0.304801
$209$	−1.81606	−0.125619
$210$	0	0
$211$	−12.2966	−0.846531	−0.423265	−	0.906006i	$-0.639116\pi$
$211$	−12.2966	−0.846531	−0.423265	+	0.906006i	$0.639116\pi$
$212$	−2.18938	−0.150367
$213$	0	0
$214$	−1.49665	−0.102309
$215$	−24.6172	−1.67888
$216$	0	0
$217$	17.0343	1.15636
$218$	12.3700	0.837801
$219$	0	0
$220$	−2.55457	−0.172229
$221$	9.27944	0.624203
$222$	0	0
$223$	−0.629713	−0.0421687	−0.0210844	−	0.999778i	$-0.506712\pi$
$223$	−0.629713	−0.0421687	−0.0210844	+	0.999778i	$0.506712\pi$
$224$	−15.0793	−1.00753
$225$	0	0
$226$	−9.32627	−0.620374
$227$	−21.8544	−1.45053	−0.725264	−	0.688471i	$-0.758282\pi$
$227$	−21.8544	−1.45053	−0.725264	+	0.688471i	$0.758282\pi$
$228$	0	0
$229$	21.6849	1.43298	0.716489	−	0.697598i	$-0.245748\pi$
$229$	21.6849	1.43298	0.716489	+	0.697598i	$0.245748\pi$
$230$	19.9832	1.31766
$231$	0	0
$232$	1.15219	0.0756449
$233$	13.8094	0.904681	0.452341	−	0.891845i	$-0.350589\pi$
$233$	13.8094	0.904681	0.452341	+	0.891845i	$0.350589\pi$
$234$	0	0
$235$	18.9581	1.23669
$236$	2.13404	0.138914
$237$	0	0
$238$	−12.7127	−0.824040
$239$	8.10196	0.524072	0.262036	−	0.965058i	$-0.415606\pi$
$239$	8.10196	0.524072	0.262036	+	0.965058i	$0.415606\pi$
$240$	0	0
$241$	−5.88395	−0.379018	−0.189509	−	0.981879i	$-0.560690\pi$
$241$	−5.88395	−0.379018	−0.189509	+	0.981879i	$0.560690\pi$
$242$	−1.07776	−0.0692810
$243$	0	0
$244$	0.838436	0.0536754
$245$	14.9147	0.952862
$246$	0	0
$247$	−4.92743	−0.313525
$248$	15.1091	0.959432
$249$	0	0
$250$	2.35401	0.148881
$251$	−18.2741	−1.15345	−0.576726	−	0.816938i	$-0.695670\pi$
$251$	−18.2741	−1.15345	−0.576726	+	0.816938i	$0.695670\pi$
$252$	0	0
$253$	−6.08551	−0.382593
$254$	9.35321	0.586872
$255$	0	0
$256$	−16.0919	−1.00574
$257$	−3.47117	−0.216526	−0.108263	−	0.994122i	$-0.534529\pi$
$257$	−3.47117	−0.216526	−0.108263	+	0.994122i	$0.534529\pi$
$258$	0	0
$259$	12.0812	0.750689
$260$	−6.93120	−0.429855
$261$	0	0
$262$	−3.81791	−0.235871
$263$	1.56304	0.0963814	0.0481907	−	0.998838i	$-0.484654\pi$
$263$	1.56304	0.0963814	0.0481907	+	0.998838i	$0.484654\pi$
$264$	0	0
$265$	7.95605	0.488737
$266$	6.75050	0.413900
$267$	0	0
$268$	−6.24483	−0.381464
$269$	−11.7725	−0.717782	−0.358891	−	0.933379i	$-0.616845\pi$
$269$	−11.7725	−0.717782	−0.358891	+	0.933379i	$0.616845\pi$
$270$	0	0
$271$	20.5162	1.24627	0.623135	−	0.782114i	$-0.285859\pi$
$271$	20.5162	1.24627	0.623135	+	0.782114i	$0.285859\pi$
$272$	−5.54098	−0.335971
$273$	0	0
$274$	−19.4754	−1.17655
$275$	4.28313	0.258283
$276$	0	0
$277$	−0.101873	−0.00612094	−0.00306047	−	0.999995i	$-0.500974\pi$
$277$	−0.101873	−0.00612094	−0.00306047	+	0.999995i	$0.500974\pi$
$278$	3.58299	0.214893
$279$	0	0
$280$	32.1464	1.92112
$281$	−14.3249	−0.854552	−0.427276	−	0.904121i	$-0.640527\pi$
$281$	−14.3249	−0.854552	−0.427276	+	0.904121i	$0.640527\pi$
$282$	0	0
$283$	−26.4737	−1.57370	−0.786850	−	0.617144i	$-0.788290\pi$
$283$	−26.4737	−1.57370	−0.786850	+	0.617144i	$0.788290\pi$
$284$	−3.70884	−0.220079
$285$	0	0
$286$	−2.92424	−0.172914
$287$	20.2700	1.19650
$288$	0	0
$289$	−5.30336	−0.311963
$290$	−1.23678	−0.0726261
$291$	0	0
$292$	−7.72151	−0.451867
$293$	7.46832	0.436304	0.218152	−	0.975915i	$-0.429997\pi$
$293$	7.46832	0.436304	0.218152	+	0.975915i	$0.429997\pi$
$294$	0	0
$295$	−7.75498	−0.451512
$296$	10.7158	0.622845
$297$	0	0
$298$	2.08986	0.121062
$299$	−16.5116	−0.954889
$300$	0	0
$301$	−27.8661	−1.60617
$302$	9.30221	0.535282
$303$	0	0
$304$	2.94229	0.168752
$305$	−3.04682	−0.174461
$306$	0	0
$307$	23.2988	1.32973	0.664867	−	0.746962i	$-0.268488\pi$
$307$	23.2988	1.32973	0.664867	+	0.746962i	$0.268488\pi$
$308$	−2.89171	−0.164770
$309$	0	0
$310$	−16.2184	−0.921143
$311$	17.3926	0.986244	0.493122	−	0.869960i	$-0.335856\pi$
$311$	17.3926	0.986244	0.493122	+	0.869960i	$0.335856\pi$
$312$	0	0
$313$	−1.69958	−0.0960659	−0.0480330	−	0.998846i	$-0.515295\pi$
$313$	−1.69958	−0.0960659	−0.0480330	+	0.998846i	$0.515295\pi$
$314$	7.98081	0.450383
$315$	0	0
$316$	−4.39061	−0.246991
$317$	−29.6094	−1.66303	−0.831515	−	0.555502i	$-0.812526\pi$
$317$	−29.6094	−1.66303	−0.831515	+	0.555502i	$0.812526\pi$
$318$	0	0
$319$	0.376637	0.0210876
$320$	24.2297	1.35448
$321$	0	0
$322$	22.6206	1.26060
$323$	−6.21098	−0.345588
$324$	0	0
$325$	11.6212	0.644631
$326$	4.67651	0.259008
$327$	0	0
$328$	17.9792	0.992735
$329$	21.4602	1.18314
$330$	0	0
$331$	−4.47943	−0.246211	−0.123106	−	0.992394i	$-0.539285\pi$
$331$	−4.47943	−0.246211	−0.123106	+	0.992394i	$0.539285\pi$
$332$	7.39186	0.405681
$333$	0	0
$334$	−13.0843	−0.715939
$335$	22.6933	1.23987
$336$	0	0
$337$	6.91513	0.376691	0.188345	−	0.982103i	$-0.439688\pi$
$337$	6.91513	0.376691	0.188345	+	0.982103i	$0.439688\pi$
$338$	6.07664	0.330526
$339$	0	0
$340$	−8.73670	−0.473814
$341$	4.93900	0.267462
$342$	0	0
$343$	−7.25949	−0.391976
$344$	−24.7168	−1.33264
$345$	0	0
$346$	4.68070	0.251636
$347$	18.5560	0.996140	0.498070	−	0.867137i	$-0.334042\pi$
$347$	18.5560	0.996140	0.498070	+	0.867137i	$0.334042\pi$
$348$	0	0
$349$	−6.98021	−0.373642	−0.186821	−	0.982394i	$-0.559818\pi$
$349$	−6.98021	−0.373642	−0.186821	+	0.982394i	$0.559818\pi$
$350$	−15.9209	−0.851008
$351$	0	0
$352$	−4.37216	−0.233037
$353$	25.0961	1.33573	0.667867	−	0.744281i	$-0.267208\pi$
$353$	25.0961	1.33573	0.667867	+	0.744281i	$0.267208\pi$
$354$	0	0
$355$	13.4777	0.715321
$356$	−6.83625	−0.362321
$357$	0	0
$358$	−6.96496	−0.368110
$359$	−26.6233	−1.40513	−0.702563	−	0.711621i	$-0.747961\pi$
$359$	−26.6233	−1.40513	−0.702563	+	0.711621i	$0.747961\pi$
$360$	0	0
$361$	−15.7019	−0.826418
$362$	15.8449	0.832790
$363$	0	0
$364$	−7.84596	−0.411240
$365$	28.0595	1.46870
$366$	0	0
$367$	2.88012	0.150341	0.0751705	−	0.997171i	$-0.476050\pi$
$367$	2.88012	0.150341	0.0751705	+	0.997171i	$0.476050\pi$
$368$	9.85947	0.513960
$369$	0	0
$370$	−11.5025	−0.597989
$371$	9.00607	0.467572
$372$	0	0
$373$	−5.86690	−0.303777	−0.151888	−	0.988398i	$-0.548535\pi$
$373$	−5.86690	−0.303777	−0.151888	+	0.988398i	$0.548535\pi$
$374$	−3.68597	−0.190597
$375$	0	0
$376$	19.0348	0.981646
$377$	1.02191	0.0526313
$378$	0	0
$379$	25.8822	1.32948	0.664739	−	0.747076i	$-0.268543\pi$
$379$	25.8822	1.32948	0.664739	+	0.747076i	$0.268543\pi$
$380$	4.63924	0.237988
$381$	0	0
$382$	−26.3661	−1.34901
$383$	−18.2813	−0.934133	−0.467066	−	0.884222i	$-0.654689\pi$
$383$	−18.2813	−0.934133	−0.467066	+	0.884222i	$0.654689\pi$
$384$	0	0
$385$	10.5083	0.535552
$386$	2.64119	0.134433
$387$	0	0
$388$	−1.13174	−0.0574554
$389$	6.60365	0.334818	0.167409	−	0.985887i	$-0.446460\pi$
$389$	6.60365	0.334818	0.167409	+	0.985887i	$0.446460\pi$
$390$	0	0
$391$	−20.8127	−1.05254
$392$	14.9750	0.756351
$393$	0	0
$394$	−10.5906	−0.533547
$395$	15.9552	0.802793
$396$	0	0
$397$	5.31584	0.266795	0.133397	−	0.991063i	$-0.457411\pi$
$397$	5.31584	0.266795	0.133397	+	0.991063i	$0.457411\pi$
$398$	20.1789	1.01148
$399$	0	0
$400$	−6.93933	−0.346967
$401$	−23.5518	−1.17612	−0.588060	−	0.808817i	$-0.700108\pi$
$401$	−23.5518	−1.17612	−0.588060	+	0.808817i	$0.700108\pi$
$402$	0	0
$403$	13.4008	0.667541
$404$	−4.16742	−0.207337
$405$	0	0
$406$	−1.40001	−0.0694811
$407$	3.50288	0.173631
$408$	0	0
$409$	29.8085	1.47393	0.736967	−	0.675929i	$-0.236258\pi$
$409$	29.8085	1.47393	0.736967	+	0.675929i	$0.236258\pi$
$410$	−19.2992	−0.953117
$411$	0	0
$412$	15.1792	0.747825
$413$	−8.77846	−0.431960
$414$	0	0
$415$	−26.8616	−1.31858
$416$	−11.8628	−0.581622
$417$	0	0
$418$	1.95727	0.0957333
$419$	−22.0112	−1.07532	−0.537660	−	0.843162i	$-0.680691\pi$
$419$	−22.0112	−1.07532	−0.537660	+	0.843162i	$0.680691\pi$
$420$	0	0
$421$	−2.64031	−0.128681	−0.0643404	−	0.997928i	$-0.520494\pi$
$421$	−2.64031	−0.128681	−0.0643404	+	0.997928i	$0.520494\pi$
$422$	13.2527	0.645133
$423$	0	0
$424$	7.98824	0.387943
$425$	14.6485	0.710554
$426$	0	0
$427$	−3.44893	−0.166906
$428$	−1.16431	−0.0562790
$429$	0	0
$430$	26.5314	1.27946
$431$	11.5029	0.554076	0.277038	−	0.960859i	$-0.410647\pi$
$431$	11.5029	0.554076	0.277038	+	0.960859i	$0.410647\pi$
$432$	0	0
$433$	19.3604	0.930403	0.465201	−	0.885205i	$-0.345982\pi$
$433$	19.3604	0.930403	0.465201	+	0.885205i	$0.345982\pi$
$434$	−18.3589	−0.881254
$435$	0	0
$436$	9.62316	0.460866
$437$	11.0516	0.528671
$438$	0	0
$439$	−13.8560	−0.661312	−0.330656	−	0.943751i	$-0.607270\pi$
$439$	−13.8560	−0.661312	−0.330656	+	0.943751i	$0.607270\pi$
$440$	9.32069	0.444346
$441$	0	0
$442$	−10.0010	−0.475699
$443$	13.1908	0.626714	0.313357	−	0.949635i	$-0.398546\pi$
$443$	13.1908	0.626714	0.313357	+	0.949635i	$0.398546\pi$
$444$	0	0
$445$	24.8425	1.17765
$446$	0.678679	0.0321364
$447$	0	0
$448$	27.4275	1.29583
$449$	10.4842	0.494782	0.247391	−	0.968916i	$-0.420427\pi$
$449$	10.4842	0.494782	0.247391	+	0.968916i	$0.420427\pi$
$450$	0	0
$451$	5.87719	0.276746
$452$	−7.25531	−0.341261
$453$	0	0
$454$	23.5538	1.10543
$455$	28.5117	1.33665
$456$	0	0
$457$	5.55221	0.259722	0.129861	−	0.991532i	$-0.458547\pi$
$457$	5.55221	0.259722	0.129861	+	0.991532i	$0.458547\pi$
$458$	−23.3711	−1.09206
$459$	0	0
$460$	15.5458	0.724829
$461$	−14.2444	−0.663430	−0.331715	−	0.943380i	$-0.607627\pi$
$461$	−14.2444	−0.663430	−0.331715	+	0.943380i	$0.607627\pi$
$462$	0	0
$463$	5.17058	0.240297	0.120149	−	0.992756i	$-0.461663\pi$
$463$	5.17058	0.240297	0.120149	+	0.992756i	$0.461663\pi$
$464$	−0.610210	−0.0283283
$465$	0	0
$466$	−14.8832	−0.689449
$467$	36.3845	1.68368	0.841838	−	0.539731i	$-0.181474\pi$
$467$	36.3845	1.68368	0.841838	+	0.539731i	$0.181474\pi$
$468$	0	0
$469$	25.6883	1.18618
$470$	−20.4323	−0.942471
$471$	0	0
$472$	−7.78635	−0.358396
$473$	−8.07962	−0.371501
$474$	0	0
$475$	−7.77841	−0.356898
$476$	−9.88975	−0.453296
$477$	0	0
$478$	−8.73196	−0.399390
$479$	37.5482	1.71562	0.857810	−	0.513967i	$-0.171825\pi$
$479$	37.5482	1.71562	0.857810	+	0.513967i	$0.171825\pi$
$480$	0	0
$481$	9.50422	0.433355
$482$	6.34148	0.288846
$483$	0	0
$484$	−0.838436	−0.0381107
$485$	4.11267	0.186747
$486$	0	0
$487$	11.4763	0.520042	0.260021	−	0.965603i	$-0.416270\pi$
$487$	11.4763	0.520042	0.260021	+	0.965603i	$0.416270\pi$
$488$	−3.05915	−0.138481
$489$	0	0
$490$	−16.0744	−0.726167
$491$	15.7233	0.709581	0.354791	−	0.934946i	$-0.384552\pi$
$491$	15.7233	0.709581	0.354791	+	0.934946i	$0.384552\pi$
$492$	0	0
$493$	1.28811	0.0580136
$494$	5.31058	0.238934
$495$	0	0
$496$	−8.00194	−0.359298
$497$	15.2564	0.684345
$498$	0	0
$499$	−31.2829	−1.40041	−0.700207	−	0.713940i	$-0.746909\pi$
$499$	−31.2829	−1.40041	−0.700207	+	0.713940i	$0.746909\pi$
$500$	1.83129	0.0818977
$501$	0	0
$502$	19.6951	0.879035
$503$	22.4470	1.00086	0.500431	−	0.865777i	$-0.333175\pi$
$503$	22.4470	1.00086	0.500431	+	0.865777i	$0.333175\pi$
$504$	0	0
$505$	15.1441	0.673905
$506$	6.55872	0.291570
$507$	0	0
$508$	7.27627	0.322832
$509$	7.38076	0.327146	0.163573	−	0.986531i	$-0.447698\pi$
$509$	7.38076	0.327146	0.163573	+	0.986531i	$0.447698\pi$
$510$	0	0
$511$	31.7627	1.40510
$512$	16.9962	0.751131
$513$	0	0
$514$	3.74108	0.165012
$515$	−55.1602	−2.43065
$516$	0	0
$517$	6.22226	0.273655
$518$	−13.0206	−0.572093
$519$	0	0
$520$	25.2894	1.10902
$521$	27.5979	1.20909	0.604543	−	0.796572i	$-0.293356\pi$
$521$	27.5979	1.20909	0.604543	+	0.796572i	$0.293356\pi$
$522$	0	0
$523$	21.3348	0.932904	0.466452	−	0.884547i	$-0.345532\pi$
$523$	21.3348	0.932904	0.466452	+	0.884547i	$0.345532\pi$
$524$	−2.97012	−0.129750
$525$	0	0
$526$	−1.68458	−0.0734513
$527$	16.8916	0.735808
$528$	0	0
$529$	14.0335	0.610151
$530$	−8.57471	−0.372462
$531$	0	0
$532$	5.25151	0.227682
$533$	15.9463	0.690712
$534$	0	0
$535$	4.23102	0.182923
$536$	22.7851	0.984168
$537$	0	0
$538$	12.6879	0.547015
$539$	4.89515	0.210849
$540$	0	0
$541$	2.69913	0.116045	0.0580224	−	0.998315i	$-0.481521\pi$
$541$	2.69913	0.116045	0.0580224	+	0.998315i	$0.481521\pi$
$542$	−22.1115	−0.949770
$543$	0	0
$544$	−14.9530	−0.641102
$545$	−34.9699	−1.49795
$546$	0	0
$547$	6.51857	0.278714	0.139357	−	0.990242i	$-0.455496\pi$
$547$	6.51857	0.278714	0.139357	+	0.990242i	$0.455496\pi$
$548$	−15.1508	−0.647208
$549$	0	0
$550$	−4.61618	−0.196835
$551$	−0.683994	−0.0291391
$552$	0	0
$553$	18.0609	0.768028
$554$	0.109794	0.00466471
$555$	0	0
$556$	2.78736	0.118210
$557$	−1.33512	−0.0565708	−0.0282854	−	0.999600i	$-0.509005\pi$
$557$	−1.33512	−0.0565708	−0.0282854	+	0.999600i	$0.509005\pi$
$558$	0	0
$559$	−21.9221	−0.927206
$560$	−17.0250	−0.719440
$561$	0	0
$562$	15.4388	0.651246
$563$	−9.26909	−0.390646	−0.195323	−	0.980739i	$-0.562575\pi$
$563$	−9.26909	−0.390646	−0.195323	+	0.980739i	$0.562575\pi$
$564$	0	0
$565$	26.3653	1.10920
$566$	28.5323	1.19930
$567$	0	0
$568$	13.5322	0.567799
$569$	−24.4489	−1.02495	−0.512475	−	0.858702i	$-0.671271\pi$
$569$	−24.4489	−1.02495	−0.512475	+	0.858702i	$0.671271\pi$
$570$	0	0
$571$	−13.5222	−0.565888	−0.282944	−	0.959136i	$-0.591311\pi$
$571$	−13.5222	−0.565888	−0.282944	+	0.959136i	$0.591311\pi$
$572$	−2.27489	−0.0951181
$573$	0	0
$574$	−21.8462	−0.911843
$575$	−26.0651	−1.08699
$576$	0	0
$577$	14.4217	0.600383	0.300192	−	0.953879i	$-0.402949\pi$
$577$	14.4217	0.600383	0.300192	+	0.953879i	$0.402949\pi$
$578$	5.71575	0.237744
$579$	0	0
$580$	−0.962144	−0.0399509
$581$	−30.4067	−1.26148
$582$	0	0
$583$	2.61126	0.108147
$584$	28.1730	1.16581
$585$	0	0
$586$	−8.04905	−0.332503
$587$	−32.9640	−1.36057	−0.680285	−	0.732947i	$-0.738144\pi$
$587$	−32.9640	−1.36057	−0.680285	+	0.732947i	$0.738144\pi$
$588$	0	0
$589$	−8.96951	−0.369582
$590$	8.35799	0.344093
$591$	0	0
$592$	−5.67520	−0.233249
$593$	44.9564	1.84614	0.923069	−	0.384633i	$-0.125672\pi$
$593$	44.9564	1.84614	0.923069	+	0.384633i	$0.125672\pi$
$594$	0	0
$595$	35.9387	1.47334
$596$	1.62579	0.0665951
$597$	0	0
$598$	17.7955	0.727712
$599$	−10.6539	−0.435306	−0.217653	−	0.976026i	$-0.569840\pi$
$599$	−10.6539	−0.435306	−0.217653	+	0.976026i	$0.569840\pi$
$600$	0	0
$601$	0.426911	0.0174140	0.00870702	−	0.999962i	$-0.497228\pi$
$601$	0.426911	0.0174140	0.00870702	+	0.999962i	$0.497228\pi$
$602$	30.0329	1.22405
$603$	0	0
$604$	7.23659	0.294453
$605$	3.04682	0.123871
$606$	0	0
$607$	−37.0498	−1.50381	−0.751903	−	0.659274i	$-0.770864\pi$
$607$	−37.0498	−1.50381	−0.751903	+	0.659274i	$0.770864\pi$
$608$	7.94010	0.322013
$609$	0	0
$610$	3.28374	0.132955
$611$	16.8826	0.682997
$612$	0	0
$613$	1.90829	0.0770750	0.0385375	−	0.999257i	$-0.487730\pi$
$613$	1.90829	0.0770750	0.0385375	+	0.999257i	$0.487730\pi$
$614$	−25.1105	−1.01338
$615$	0	0
$616$	10.5508	0.425104
$617$	36.7736	1.48045	0.740226	−	0.672359i	$-0.234719\pi$
$617$	36.7736	1.48045	0.740226	+	0.672359i	$0.234719\pi$
$618$	0	0
$619$	−25.8507	−1.03903	−0.519513	−	0.854462i	$-0.673887\pi$
$619$	−25.8507	−1.03903	−0.519513	+	0.854462i	$0.673887\pi$
$620$	−12.6170	−0.506711
$621$	0	0
$622$	−18.7450	−0.751608
$623$	28.1212	1.12665
$624$	0	0
$625$	−28.0704	−1.12282
$626$	1.83174	0.0732110
$627$	0	0
$628$	6.20862	0.247751
$629$	11.9800	0.477673
$630$	0	0
$631$	−44.0002	−1.75162	−0.875810	−	0.482657i	$-0.839672\pi$
$631$	−44.0002	−1.75162	−0.875810	+	0.482657i	$0.839672\pi$
$632$	16.0197	0.637231
$633$	0	0
$634$	31.9118	1.26738
$635$	−26.4415	−1.04930
$636$	0	0
$637$	13.2818	0.526244
$638$	−0.405924	−0.0160707
$639$	0	0
$640$	0.528651	0.0208968
$641$	−45.9375	−1.81442	−0.907211	−	0.420677i	$-0.861793\pi$
$641$	−45.9375	−1.81442	−0.907211	+	0.420677i	$0.861793\pi$
$642$	0	0
$643$	−10.9831	−0.433132	−0.216566	−	0.976268i	$-0.569486\pi$
$643$	−10.9831	−0.433132	−0.216566	+	0.976268i	$0.569486\pi$
$644$	17.5975	0.693440
$645$	0	0
$646$	6.69393	0.263369
$647$	−9.24470	−0.363447	−0.181723	−	0.983350i	$-0.558168\pi$
$647$	−9.24470	−0.363447	−0.181723	+	0.983350i	$0.558168\pi$
$648$	0	0
$649$	−2.54527	−0.0999104
$650$	−12.5249	−0.491267
$651$	0	0
$652$	3.63806	0.142478
$653$	−36.3448	−1.42228	−0.711140	−	0.703050i	$-0.751821\pi$
$653$	−36.3448	−1.42228	−0.711140	+	0.703050i	$0.751821\pi$
$654$	0	0
$655$	10.7932	0.421727
$656$	−9.52194	−0.371769
$657$	0	0
$658$	−23.1289	−0.901657
$659$	12.8729	0.501458	0.250729	−	0.968057i	$-0.419330\pi$
$659$	12.8729	0.501458	0.250729	+	0.968057i	$0.419330\pi$
$660$	0	0
$661$	24.6328	0.958105	0.479053	−	0.877786i	$-0.340980\pi$
$661$	24.6328	0.958105	0.479053	+	0.877786i	$0.340980\pi$
$662$	4.82774	0.187635
$663$	0	0
$664$	−26.9702	−1.04665
$665$	−19.0837	−0.740032
$666$	0	0
$667$	−2.29203	−0.0887478
$668$	−10.1788	−0.393831
$669$	0	0
$670$	−24.4579	−0.944893
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	28.8608	1.11250	0.556250	−	0.831015i	$-0.312240\pi$
$673$	28.8608	1.11250	0.556250	+	0.831015i	$0.312240\pi$
$674$	−7.45284	−0.287073
$675$	0	0
$676$	4.72729	0.181819
$677$	27.3298	1.05037	0.525184	−	0.850989i	$-0.323996\pi$
$677$	27.3298	1.05037	0.525184	+	0.850989i	$0.323996\pi$
$678$	0	0
$679$	4.65545	0.178660
$680$	31.8771	1.22243
$681$	0	0
$682$	−5.32305	−0.203830
$683$	−4.55326	−0.174226	−0.0871128	−	0.996198i	$-0.527764\pi$
$683$	−4.55326	−0.174226	−0.0871128	+	0.996198i	$0.527764\pi$
$684$	0	0
$685$	55.0569	2.10362
$686$	7.82398	0.298721
$687$	0	0
$688$	13.0902	0.499060
$689$	7.08503	0.269918
$690$	0	0
$691$	6.69036	0.254513	0.127257	−	0.991870i	$-0.459383\pi$
$691$	6.69036	0.254513	0.127257	+	0.991870i	$0.459383\pi$
$692$	3.64132	0.138422
$693$	0	0
$694$	−19.9989	−0.759149
$695$	−10.1291	−0.384219
$696$	0	0
$697$	20.1002	0.761348
$698$	7.52298	0.284749
$699$	0	0
$700$	−12.3856	−0.468131
$701$	−3.64599	−0.137707	−0.0688536	−	0.997627i	$-0.521934\pi$
$701$	−3.64599	−0.137707	−0.0688536	+	0.997627i	$0.521934\pi$
$702$	0	0
$703$	−6.36143	−0.239926
$704$	7.95245	0.299719
$705$	0	0
$706$	−27.0476	−1.01795
$707$	17.1428	0.644722
$708$	0	0
$709$	8.31591	0.312310	0.156155	−	0.987733i	$-0.450090\pi$
$709$	8.31591	0.312310	0.156155	+	0.987733i	$0.450090\pi$
$710$	−14.5257	−0.545140
$711$	0	0
$712$	24.9430	0.934779
$713$	−30.0564	−1.12562
$714$	0	0
$715$	8.26682	0.309162
$716$	−5.41835	−0.202493
$717$	0	0
$718$	28.6935	1.07083
$719$	−10.6667	−0.397800	−0.198900	−	0.980020i	$-0.563737\pi$
$719$	−10.6667	−0.397800	−0.198900	+	0.980020i	$0.563737\pi$
$720$	0	0
$721$	−62.4401	−2.32539
$722$	16.9229	0.629805
$723$	0	0
$724$	12.3264	0.458109
$725$	1.61319	0.0599122
$726$	0	0
$727$	−8.80084	−0.326405	−0.163203	−	0.986593i	$-0.552182\pi$
$727$	−8.80084	−0.326405	−0.163203	+	0.986593i	$0.552182\pi$
$728$	28.6271	1.06099
$729$	0	0
$730$	−30.2414	−1.11928
$731$	−27.6326	−1.02203
$732$	0	0
$733$	−4.17577	−0.154236	−0.0771179	−	0.997022i	$-0.524572\pi$
$733$	−4.17577	−0.154236	−0.0771179	+	0.997022i	$0.524572\pi$
$734$	−3.10407	−0.114573
$735$	0	0
$736$	26.6069	0.980742
$737$	7.44820	0.274358
$738$	0	0
$739$	54.2009	1.99381	0.996906	−	0.0786054i	$-0.0250467\pi$
$739$	54.2009	1.99381	0.996906	+	0.0786054i	$0.0250467\pi$
$740$	−8.94834	−0.328947
$741$	0	0
$742$	−9.70637	−0.356332
$743$	22.4243	0.822666	0.411333	−	0.911485i	$-0.365063\pi$
$743$	22.4243	0.822666	0.411333	+	0.911485i	$0.365063\pi$
$744$	0	0
$745$	−5.90803	−0.216453
$746$	6.32310	0.231505
$747$	0	0
$748$	−2.86748	−0.104845
$749$	4.78942	0.175002
$750$	0	0
$751$	−46.3856	−1.69263	−0.846316	−	0.532681i	$-0.821185\pi$
$751$	−46.3856	−1.69263	−0.846316	+	0.532681i	$0.821185\pi$
$752$	−10.0810	−0.367617
$753$	0	0
$754$	−1.10138	−0.0401098
$755$	−26.2973	−0.957058
$756$	0	0
$757$	−2.80886	−0.102090	−0.0510448	−	0.998696i	$-0.516255\pi$
$757$	−2.80886	−0.102090	−0.0510448	+	0.998696i	$0.516255\pi$
$758$	−27.8947	−1.01318
$759$	0	0
$760$	−16.9269	−0.614003
$761$	18.9731	0.687773	0.343887	−	0.939011i	$-0.388256\pi$
$761$	18.9731	0.687773	0.343887	+	0.939011i	$0.388256\pi$
$762$	0	0
$763$	−39.5852	−1.43308
$764$	−20.5113	−0.742074
$765$	0	0
$766$	19.7029	0.711894
$767$	−6.90597	−0.249360
$768$	0	0
$769$	27.9821	1.00906	0.504530	−	0.863394i	$-0.331666\pi$
$769$	27.9821	1.00906	0.504530	+	0.863394i	$0.331666\pi$
$770$	−11.3254	−0.408139
$771$	0	0
$772$	2.05470	0.0739501
$773$	−17.6472	−0.634727	−0.317363	−	0.948304i	$-0.602797\pi$
$773$	−17.6472	−0.634727	−0.317363	+	0.948304i	$0.602797\pi$
$774$	0	0
$775$	21.1544	0.759888
$776$	4.12931	0.148234
$777$	0	0
$778$	−7.11714	−0.255162
$779$	−10.6733	−0.382411
$780$	0	0
$781$	4.42352	0.158286
$782$	22.4310	0.802132
$783$	0	0
$784$	−7.93089	−0.283246
$785$	−22.5617	−0.805263
$786$	0	0
$787$	−50.6943	−1.80706	−0.903528	−	0.428528i	$-0.859032\pi$
$787$	−50.6943	−1.80706	−0.903528	+	0.428528i	$0.859032\pi$
$788$	−8.23889	−0.293498
$789$	0	0
$790$	−17.1959	−0.611801
$791$	29.8450	1.06116
$792$	0	0
$793$	−2.71326	−0.0963507
$794$	−5.72920	−0.203322
$795$	0	0
$796$	15.6980	0.556402
$797$	−13.5778	−0.480951	−0.240476	−	0.970655i	$-0.577303\pi$
$797$	−13.5778	−0.480951	−0.240476	+	0.970655i	$0.577303\pi$
$798$	0	0
$799$	21.2803	0.752844
$800$	−18.7266	−0.662084
$801$	0	0
$802$	25.3831	0.896310
$803$	9.20943	0.324994
$804$	0	0
$805$	−63.9484	−2.25388
$806$	−14.4428	−0.508727
$807$	0	0
$808$	15.2054	0.534924
$809$	−44.4156	−1.56157	−0.780785	−	0.624799i	$-0.785181\pi$
$809$	−44.4156	−1.56157	−0.780785	+	0.624799i	$0.785181\pi$
$810$	0	0
$811$	41.7978	1.46772	0.733859	−	0.679301i	$-0.237717\pi$
$811$	41.7978	1.46772	0.733859	+	0.679301i	$0.237717\pi$
$812$	−1.08913	−0.0382208
$813$	0	0
$814$	−3.77526	−0.132323
$815$	−13.2205	−0.463094
$816$	0	0
$817$	14.6730	0.513345
$818$	−32.1263	−1.12327
$819$	0	0
$820$	−15.0137	−0.524300
$821$	6.14459	0.214448	0.107224	−	0.994235i	$-0.465804\pi$
$821$	6.14459	0.214448	0.107224	+	0.994235i	$0.465804\pi$
$822$	0	0
$823$	−10.4844	−0.365464	−0.182732	−	0.983163i	$-0.558494\pi$
$823$	−10.4844	−0.365464	−0.182732	+	0.983163i	$0.558494\pi$
$824$	−55.3834	−1.92937
$825$	0	0
$826$	9.46106	0.329192
$827$	−1.30033	−0.0452168	−0.0226084	−	0.999744i	$-0.507197\pi$
$827$	−1.30033	−0.0452168	−0.0226084	+	0.999744i	$0.507197\pi$
$828$	0	0
$829$	51.8433	1.80059	0.900296	−	0.435279i	$-0.143350\pi$
$829$	51.8433	1.80059	0.900296	+	0.435279i	$0.143350\pi$
$830$	28.9503	1.00488
$831$	0	0
$832$	21.5770	0.748050
$833$	16.7416	0.580061
$834$	0	0
$835$	36.9892	1.28006
$836$	1.52265	0.0526618
$837$	0	0
$838$	23.7228	0.819491
$839$	−8.55057	−0.295198	−0.147599	−	0.989047i	$-0.547155\pi$
$839$	−8.55057	−0.295198	−0.147599	+	0.989047i	$0.547155\pi$
$840$	0	0
$841$	−28.8581	−0.995108
$842$	2.84562	0.0980665
$843$	0	0
$844$	10.3099	0.354881
$845$	−17.1787	−0.590964
$846$	0	0
$847$	3.44893	0.118507
$848$	−4.23065	−0.145281
$849$	0	0
$850$	−15.7875	−0.541507
$851$	−21.3168	−0.730731
$852$	0	0
$853$	−3.34288	−0.114458	−0.0572289	−	0.998361i	$-0.518226\pi$
$853$	−3.34288	−0.114458	−0.0572289	+	0.998361i	$0.518226\pi$
$854$	3.71712	0.127197
$855$	0	0
$856$	4.24814	0.145198
$857$	−39.8478	−1.36117	−0.680587	−	0.732667i	$-0.738275\pi$
$857$	−39.8478	−1.36117	−0.680587	+	0.732667i	$0.738275\pi$
$858$	0	0
$859$	−9.75337	−0.332781	−0.166390	−	0.986060i	$-0.553211\pi$
$859$	−9.75337	−0.332781	−0.166390	+	0.986060i	$0.553211\pi$
$860$	20.6399	0.703815
$861$	0	0
$862$	−12.3974	−0.422256
$863$	13.2225	0.450098	0.225049	−	0.974347i	$-0.427746\pi$
$863$	13.2225	0.450098	0.225049	+	0.974347i	$0.427746\pi$
$864$	0	0
$865$	−13.2323	−0.449913
$866$	−20.8659	−0.709051
$867$	0	0
$868$	−14.2822	−0.484768
$869$	5.23667	0.177642
$870$	0	0
$871$	20.2089	0.684752
$872$	−35.1114	−1.18902
$873$	0	0
$874$	−11.9110	−0.402896
$875$	−7.53307	−0.254664
$876$	0	0
$877$	−13.0314	−0.440039	−0.220020	−	0.975495i	$-0.570612\pi$
$877$	−13.0314	−0.440039	−0.220020	+	0.975495i	$0.570612\pi$
$878$	14.9335	0.503980
$879$	0	0
$880$	−4.93632	−0.166403
$881$	−16.8445	−0.567507	−0.283753	−	0.958897i	$-0.591580\pi$
$881$	−16.8445	−0.567507	−0.283753	+	0.958897i	$0.591580\pi$
$882$	0	0
$883$	−11.2257	−0.377775	−0.188888	−	0.981999i	$-0.560488\pi$
$883$	−11.2257	−0.377775	−0.188888	+	0.981999i	$0.560488\pi$
$884$	−7.78022	−0.261677
$885$	0	0
$886$	−14.2165	−0.477613
$887$	18.9067	0.634823	0.317412	−	0.948288i	$-0.397186\pi$
$887$	18.9067	0.634823	0.317412	+	0.948288i	$0.397186\pi$
$888$	0	0
$889$	−29.9312	−1.00386
$890$	−26.7742	−0.897475
$891$	0	0
$892$	0.527974	0.0176779
$893$	−11.3000	−0.378139
$894$	0	0
$895$	19.6899	0.658162
$896$	0.598421	0.0199918
$897$	0	0
$898$	−11.2995	−0.377068
$899$	1.86021	0.0620415
$900$	0	0
$901$	8.93061	0.297522
$902$	−6.33419	−0.210905
$903$	0	0
$904$	26.4720	0.880446
$905$	−44.7935	−1.48899
$906$	0	0
$907$	27.4334	0.910911	0.455456	−	0.890259i	$-0.349476\pi$
$907$	27.4334	0.910911	0.455456	+	0.890259i	$0.349476\pi$
$908$	18.3235	0.608088
$909$	0	0
$910$	−30.7288	−1.01865
$911$	15.5416	0.514916	0.257458	−	0.966289i	$-0.417115\pi$
$911$	15.5416	0.514916	0.257458	+	0.966289i	$0.417115\pi$
$912$	0	0
$913$	−8.81625	−0.291775
$914$	−5.98395	−0.197931
$915$	0	0
$916$	−18.1814	−0.600730
$917$	12.2177	0.403464
$918$	0	0
$919$	50.8268	1.67662	0.838310	−	0.545193i	$-0.183544\pi$
$919$	50.8268	1.67662	0.838310	+	0.545193i	$0.183544\pi$
$920$	−56.7212	−1.87004
$921$	0	0
$922$	15.3521	0.505593
$923$	12.0022	0.395056
$924$	0	0
$925$	15.0033	0.493305
$926$	−5.57264	−0.183128
$927$	0	0
$928$	−1.64672	−0.0540562
$929$	−8.78485	−0.288221	−0.144111	−	0.989562i	$-0.546032\pi$
$929$	−8.78485	−0.288221	−0.144111	+	0.989562i	$0.546032\pi$
$930$	0	0
$931$	−8.88987	−0.291354
$932$	−11.5783	−0.379259
$933$	0	0
$934$	−39.2138	−1.28311
$935$	10.4202	0.340778
$936$	0	0
$937$	56.6855	1.85184	0.925918	−	0.377725i	$-0.123293\pi$
$937$	56.6855	1.85184	0.925918	+	0.377725i	$0.123293\pi$
$938$	−27.6858	−0.903975
$939$	0	0
$940$	−15.8952	−0.518443
$941$	−41.0121	−1.33696	−0.668479	−	0.743731i	$-0.733054\pi$
$941$	−41.0121	−1.33696	−0.668479	+	0.743731i	$0.733054\pi$
$942$	0	0
$943$	−35.7657	−1.16469
$944$	4.12372	0.134216
$945$	0	0
$946$	8.70788	0.283118
$947$	−9.18122	−0.298349	−0.149175	−	0.988811i	$-0.547662\pi$
$947$	−9.18122	−0.298349	−0.149175	+	0.988811i	$0.547662\pi$
$948$	0	0
$949$	24.9876	0.811131
$950$	8.38325	0.271988
$951$	0	0
$952$	36.0841	1.16949
$953$	−17.0299	−0.551653	−0.275826	−	0.961207i	$-0.588951\pi$
$953$	−17.0299	−0.551653	−0.275826	+	0.961207i	$0.588951\pi$
$954$	0	0
$955$	74.5369	2.41196
$956$	−6.79297	−0.219700
$957$	0	0
$958$	−40.4679	−1.30746
$959$	62.3232	2.01252
$960$	0	0
$961$	−6.60625	−0.213105
$962$	−10.2433	−0.330256
$963$	0	0
$964$	4.93331	0.158891
$965$	−7.46663	−0.240359
$966$	0	0
$967$	−16.5606	−0.532555	−0.266277	−	0.963896i	$-0.585794\pi$
$967$	−16.5606	−0.532555	−0.266277	+	0.963896i	$0.585794\pi$
$968$	3.05915	0.0983248
$969$	0	0
$970$	−4.43247	−0.142318
$971$	−12.9086	−0.414256	−0.207128	−	0.978314i	$-0.566412\pi$
$971$	−12.9086	−0.414256	−0.207128	+	0.978314i	$0.566412\pi$
$972$	0	0
$973$	−11.4659	−0.367580
$974$	−12.3687	−0.396319
$975$	0	0
$976$	1.62015	0.0518599
$977$	6.73089	0.215340	0.107670	−	0.994187i	$-0.465661\pi$
$977$	6.73089	0.215340	0.107670	+	0.994187i	$0.465661\pi$
$978$	0	0
$979$	8.15358	0.260590
$980$	−12.5050	−0.399457
$981$	0	0
$982$	−16.9459	−0.540765
$983$	−3.89460	−0.124219	−0.0621093	−	0.998069i	$-0.519783\pi$
$983$	−3.89460	−0.124219	−0.0621093	+	0.998069i	$0.519783\pi$
$984$	0	0
$985$	29.9396	0.953955
$986$	−1.38827	−0.0442116
$987$	0	0
$988$	4.13134	0.131435
$989$	49.1686	1.56347
$990$	0	0
$991$	47.0696	1.49522	0.747608	−	0.664141i	$-0.231202\pi$
$991$	47.0696	1.49522	0.747608	+	0.664141i	$0.231202\pi$
$992$	−21.5941	−0.685614
$993$	0	0
$994$	−16.4428	−0.521533
$995$	−57.0457	−1.80847
$996$	0	0
$997$	−35.0684	−1.11063	−0.555314	−	0.831641i	$-0.687402\pi$
$997$	−35.0684	−1.11063	−0.555314	+	0.831641i	$0.687402\pi$
$998$	33.7154	1.06724
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.o.1.7	yes	25
3.2	odd	2		6039.2.a.n.1.19	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.19	✓	25	3.2	odd	2
6039.2.a.o.1.7	yes	25	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)

\(f(q)\)	\(=\)	\(q-1.07776 q^{2} -0.838436 q^{4} +3.04682 q^{5} +3.44893 q^{7} +3.05915 q^{8} +O(q^{10})\)	\(q-1.07776 q^{2} -0.838436 q^{4} +3.04682 q^{5} +3.44893 q^{7} +3.05915 q^{8} -3.28374 q^{10} +1.00000 q^{11} +2.71326 q^{13} -3.71712 q^{14} -1.62015 q^{16} +3.42003 q^{17} -1.81606 q^{19} -2.55457 q^{20} -1.07776 q^{22} -6.08551 q^{23} +4.28313 q^{25} -2.92424 q^{26} -2.89171 q^{28} +0.376637 q^{29} +4.93900 q^{31} -4.37216 q^{32} -3.68597 q^{34} +10.5083 q^{35} +3.50288 q^{37} +1.95727 q^{38} +9.32069 q^{40} +5.87719 q^{41} -8.07962 q^{43} -0.838436 q^{44} +6.55872 q^{46} +6.22226 q^{47} +4.89515 q^{49} -4.61618 q^{50} -2.27489 q^{52} +2.61126 q^{53} +3.04682 q^{55} +10.5508 q^{56} -0.405924 q^{58} -2.54527 q^{59} -1.00000 q^{61} -5.32305 q^{62} +7.95245 q^{64} +8.26682 q^{65} +7.44820 q^{67} -2.86748 q^{68} -11.3254 q^{70} +4.42352 q^{71} +9.20943 q^{73} -3.77526 q^{74} +1.52265 q^{76} +3.44893 q^{77} +5.23667 q^{79} -4.93632 q^{80} -6.33419 q^{82} -8.81625 q^{83} +10.4202 q^{85} +8.70788 q^{86} +3.05915 q^{88} +8.15358 q^{89} +9.35785 q^{91} +5.10231 q^{92} -6.70609 q^{94} -5.53320 q^{95} +1.34982 q^{97} -5.27579 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) \) 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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