LMFDB - Embedded newform 6004.2.a.e.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.98454 q^{3} +1.21429 q^{5} +4.02832 q^{7} +0.938380 q^{9} +O(q^{10})$	$q+1.98454 q^{3} +1.21429 q^{5} +4.02832 q^{7} +0.938380 q^{9} +2.10051 q^{11} -1.82614 q^{13} +2.40980 q^{15} -2.39722 q^{17} +1.00000 q^{19} +7.99434 q^{21} +0.824686 q^{23} -3.52550 q^{25} -4.09136 q^{27} +7.86118 q^{29} +1.02625 q^{31} +4.16854 q^{33} +4.89154 q^{35} -1.30416 q^{37} -3.62403 q^{39} +1.13073 q^{41} +12.7901 q^{43} +1.13946 q^{45} -3.49164 q^{47} +9.22735 q^{49} -4.75736 q^{51} -0.890313 q^{53} +2.55063 q^{55} +1.98454 q^{57} +2.48627 q^{59} +6.87706 q^{61} +3.78009 q^{63} -2.21746 q^{65} +8.07970 q^{67} +1.63662 q^{69} +3.03821 q^{71} -11.7230 q^{73} -6.99648 q^{75} +8.46154 q^{77} +1.00000 q^{79} -10.9346 q^{81} +16.1796 q^{83} -2.91092 q^{85} +15.6008 q^{87} +16.7784 q^{89} -7.35626 q^{91} +2.03663 q^{93} +1.21429 q^{95} -5.70676 q^{97} +1.97108 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * q^99

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	0
$2$	0	0
$3$	1.98454	1.14577	0.572886	−	0.819635i	$-0.305824\pi$
$3$	1.98454	1.14577	0.572886	+	0.819635i	$0.305824\pi$
$4$	0	0
$5$	1.21429	0.543047	0.271523	−	0.962432i	$-0.412473\pi$
$5$	1.21429	0.543047	0.271523	+	0.962432i	$0.412473\pi$
$6$	0	0
$7$	4.02832	1.52256	0.761281	−	0.648423i	$-0.224571\pi$
$7$	4.02832	1.52256	0.761281	+	0.648423i	$0.224571\pi$
$8$	0	0
$9$	0.938380	0.312793
$10$	0	0
$11$	2.10051	0.633329	0.316664	−	0.948538i	$-0.397437\pi$
$11$	2.10051	0.633329	0.316664	+	0.948538i	$0.397437\pi$
$12$	0	0
$13$	−1.82614	−0.506479	−0.253240	−	0.967404i	$-0.581496\pi$
$13$	−1.82614	−0.506479	−0.253240	+	0.967404i	$0.581496\pi$
$14$	0	0
$15$	2.40980	0.622208
$16$	0	0
$17$	−2.39722	−0.581411	−0.290705	−	0.956813i	$-0.593890\pi$
$17$	−2.39722	−0.581411	−0.290705	+	0.956813i	$0.593890\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	7.99434	1.74451
$22$	0	0
$23$	0.824686	0.171959	0.0859794	−	0.996297i	$-0.472598\pi$
$23$	0.824686	0.171959	0.0859794	+	0.996297i	$0.472598\pi$
$24$	0	0
$25$	−3.52550	−0.705100
$26$	0	0
$27$	−4.09136	−0.787382
$28$	0	0
$29$	7.86118	1.45978	0.729892	−	0.683562i	$-0.239570\pi$
$29$	7.86118	1.45978	0.729892	+	0.683562i	$0.239570\pi$
$30$	0	0
$31$	1.02625	0.184320	0.0921600	−	0.995744i	$-0.470623\pi$
$31$	1.02625	0.184320	0.0921600	+	0.995744i	$0.470623\pi$
$32$	0	0
$33$	4.16854	0.725650
$34$	0	0
$35$	4.89154	0.826822
$36$	0	0
$37$	−1.30416	−0.214403	−0.107201	−	0.994237i	$-0.534189\pi$
$37$	−1.30416	−0.214403	−0.107201	+	0.994237i	$0.534189\pi$
$38$	0	0
$39$	−3.62403	−0.580310
$40$	0	0
$41$	1.13073	0.176591	0.0882955	−	0.996094i	$-0.471858\pi$
$41$	1.13073	0.176591	0.0882955	+	0.996094i	$0.471858\pi$
$42$	0	0
$43$	12.7901	1.95047	0.975237	−	0.221164i	$-0.0709857\pi$
$43$	12.7901	1.95047	0.975237	+	0.221164i	$0.0709857\pi$
$44$	0	0
$45$	1.13946	0.169861
$46$	0	0
$47$	−3.49164	−0.509309	−0.254654	−	0.967032i	$-0.581962\pi$
$47$	−3.49164	−0.509309	−0.254654	+	0.967032i	$0.581962\pi$
$48$	0	0
$49$	9.22735	1.31819
$50$	0	0
$51$	−4.75736	−0.666164
$52$	0	0
$53$	−0.890313	−0.122294	−0.0611469	−	0.998129i	$-0.519476\pi$
$53$	−0.890313	−0.122294	−0.0611469	+	0.998129i	$0.519476\pi$
$54$	0	0
$55$	2.55063	0.343927
$56$	0	0
$57$	1.98454	0.262858
$58$	0	0
$59$	2.48627	0.323684	0.161842	−	0.986817i	$-0.448256\pi$
$59$	2.48627	0.323684	0.161842	+	0.986817i	$0.448256\pi$
$60$	0	0
$61$	6.87706	0.880518	0.440259	−	0.897871i	$-0.354887\pi$
$61$	6.87706	0.880518	0.440259	+	0.897871i	$0.354887\pi$
$62$	0	0
$63$	3.78009	0.476247
$64$	0	0
$65$	−2.21746	−0.275042
$66$	0	0
$67$	8.07970	0.987092	0.493546	−	0.869720i	$-0.335700\pi$
$67$	8.07970	0.987092	0.493546	+	0.869720i	$0.335700\pi$
$68$	0	0
$69$	1.63662	0.197026
$70$	0	0
$71$	3.03821	0.360570	0.180285	−	0.983614i	$-0.442298\pi$
$71$	3.03821	0.360570	0.180285	+	0.983614i	$0.442298\pi$
$72$	0	0
$73$	−11.7230	−1.37207	−0.686035	−	0.727568i	$-0.740650\pi$
$73$	−11.7230	−1.37207	−0.686035	+	0.727568i	$0.740650\pi$
$74$	0	0
$75$	−6.99648	−0.807884
$76$	0	0
$77$	8.46154	0.964282
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	−10.9346	−1.21495
$82$	0	0
$83$	16.1796	1.77595	0.887973	−	0.459895i	$-0.152113\pi$
$83$	16.1796	1.77595	0.887973	+	0.459895i	$0.152113\pi$
$84$	0	0
$85$	−2.91092	−0.315733
$86$	0	0
$87$	15.6008	1.67258
$88$	0	0
$89$	16.7784	1.77851	0.889256	−	0.457411i	$-0.151223\pi$
$89$	16.7784	1.77851	0.889256	+	0.457411i	$0.151223\pi$
$90$	0	0
$91$	−7.35626	−0.771146
$92$	0	0
$93$	2.03663	0.211189
$94$	0	0
$95$	1.21429	0.124583
$96$	0	0
$97$	−5.70676	−0.579433	−0.289717	−	0.957112i	$-0.593561\pi$
$97$	−5.70676	−0.579433	−0.289717	+	0.957112i	$0.593561\pi$
$98$	0	0
$99$	1.97108	0.198101
$100$	0	0
$101$	−6.84294	−0.680898	−0.340449	−	0.940263i	$-0.610579\pi$
$101$	−6.84294	−0.680898	−0.340449	+	0.940263i	$0.610579\pi$
$102$	0	0
$103$	17.0111	1.67616	0.838078	−	0.545551i	$-0.183679\pi$
$103$	17.0111	1.67616	0.838078	+	0.545551i	$0.183679\pi$
$104$	0	0
$105$	9.70744	0.947349
$106$	0	0
$107$	6.43276	0.621878	0.310939	−	0.950430i	$-0.399356\pi$
$107$	6.43276	0.621878	0.310939	+	0.950430i	$0.399356\pi$
$108$	0	0
$109$	7.41662	0.710383	0.355192	−	0.934794i	$-0.384416\pi$
$109$	7.41662	0.710383	0.355192	+	0.934794i	$0.384416\pi$
$110$	0	0
$111$	−2.58815	−0.245657
$112$	0	0
$113$	−15.5388	−1.46176	−0.730882	−	0.682504i	$-0.760891\pi$
$113$	−15.5388	−1.46176	−0.730882	+	0.682504i	$0.760891\pi$
$114$	0	0
$115$	1.00141	0.0933817
$116$	0	0
$117$	−1.71361	−0.158423
$118$	0	0
$119$	−9.65676	−0.885233
$120$	0	0
$121$	−6.58784	−0.598895
$122$	0	0
$123$	2.24398	0.202333
$124$	0	0
$125$	−10.3524	−0.925949
$126$	0	0
$127$	−18.3173	−1.62540	−0.812700	−	0.582683i	$-0.802003\pi$
$127$	−18.3173	−1.62540	−0.812700	+	0.582683i	$0.802003\pi$
$128$	0	0
$129$	25.3824	2.23480
$130$	0	0
$131$	6.03302	0.527108	0.263554	−	0.964645i	$-0.415105\pi$
$131$	6.03302	0.527108	0.263554	+	0.964645i	$0.415105\pi$
$132$	0	0
$133$	4.02832	0.349299
$134$	0	0
$135$	−4.96809	−0.427585
$136$	0	0
$137$	−1.68630	−0.144070	−0.0720351	−	0.997402i	$-0.522949\pi$
$137$	−1.68630	−0.144070	−0.0720351	+	0.997402i	$0.522949\pi$
$138$	0	0
$139$	20.3278	1.72418	0.862089	−	0.506757i	$-0.169156\pi$
$139$	20.3278	1.72418	0.862089	+	0.506757i	$0.169156\pi$
$140$	0	0
$141$	−6.92929	−0.583552
$142$	0	0
$143$	−3.83583	−0.320768
$144$	0	0
$145$	9.54575	0.792731
$146$	0	0
$147$	18.3120	1.51035
$148$	0	0
$149$	−12.5330	−1.02674	−0.513371	−	0.858167i	$-0.671604\pi$
$149$	−12.5330	−1.02674	−0.513371	+	0.858167i	$0.671604\pi$
$150$	0	0
$151$	−11.5700	−0.941554	−0.470777	−	0.882252i	$-0.656026\pi$
$151$	−11.5700	−0.941554	−0.470777	+	0.882252i	$0.656026\pi$
$152$	0	0
$153$	−2.24950	−0.181861
$154$	0	0
$155$	1.24617	0.100094
$156$	0	0
$157$	−14.5501	−1.16122	−0.580611	−	0.814181i	$-0.697186\pi$
$157$	−14.5501	−1.16122	−0.580611	+	0.814181i	$0.697186\pi$
$158$	0	0
$159$	−1.76686	−0.140121
$160$	0	0
$161$	3.32210	0.261818
$162$	0	0
$163$	12.1138	0.948827	0.474414	−	0.880302i	$-0.342660\pi$
$163$	12.1138	0.948827	0.474414	+	0.880302i	$0.342660\pi$
$164$	0	0
$165$	5.06182	0.394062
$166$	0	0
$167$	−0.301326	−0.0233173	−0.0116586	−	0.999932i	$-0.503711\pi$
$167$	−0.301326	−0.0233173	−0.0116586	+	0.999932i	$0.503711\pi$
$168$	0	0
$169$	−9.66522	−0.743479
$170$	0	0
$171$	0.938380	0.0717597
$172$	0	0
$173$	−20.0578	−1.52497	−0.762483	−	0.647008i	$-0.776020\pi$
$173$	−20.0578	−1.52497	−0.762483	+	0.647008i	$0.776020\pi$
$174$	0	0
$175$	−14.2018	−1.07356
$176$	0	0
$177$	4.93408	0.370868
$178$	0	0
$179$	15.8624	1.18562	0.592808	−	0.805344i	$-0.298019\pi$
$179$	15.8624	1.18562	0.592808	+	0.805344i	$0.298019\pi$
$180$	0	0
$181$	11.3590	0.844308	0.422154	−	0.906524i	$-0.361274\pi$
$181$	11.3590	0.844308	0.422154	+	0.906524i	$0.361274\pi$
$182$	0	0
$183$	13.6478	1.00887
$184$	0	0
$185$	−1.58363	−0.116431
$186$	0	0
$187$	−5.03539	−0.368224
$188$	0	0
$189$	−16.4813	−1.19884
$190$	0	0
$191$	14.7628	1.06820	0.534098	−	0.845423i	$-0.320651\pi$
$191$	14.7628	1.06820	0.534098	+	0.845423i	$0.320651\pi$
$192$	0	0
$193$	0.466797	0.0336008	0.0168004	−	0.999859i	$-0.494652\pi$
$193$	0.466797	0.0336008	0.0168004	+	0.999859i	$0.494652\pi$
$194$	0	0
$195$	−4.40063	−0.315135
$196$	0	0
$197$	−18.3371	−1.30647	−0.653233	−	0.757157i	$-0.726588\pi$
$197$	−18.3371	−1.30647	−0.653233	+	0.757157i	$0.726588\pi$
$198$	0	0
$199$	3.53634	0.250685	0.125342	−	0.992114i	$-0.459997\pi$
$199$	3.53634	0.250685	0.125342	+	0.992114i	$0.459997\pi$
$200$	0	0
$201$	16.0344	1.13098
$202$	0	0
$203$	31.6673	2.22261
$204$	0	0
$205$	1.37304	0.0958972
$206$	0	0
$207$	0.773868	0.0537875
$208$	0	0
$209$	2.10051	0.145296
$210$	0	0
$211$	−20.3274	−1.39939	−0.699697	−	0.714440i	$-0.746682\pi$
$211$	−20.3274	−1.39939	−0.699697	+	0.714440i	$0.746682\pi$
$212$	0	0
$213$	6.02944	0.413131
$214$	0	0
$215$	15.5309	1.05920
$216$	0	0
$217$	4.13406	0.280639
$218$	0	0
$219$	−23.2647	−1.57208
$220$	0	0
$221$	4.37765	0.294473
$222$	0	0
$223$	−13.7204	−0.918785	−0.459393	−	0.888233i	$-0.651933\pi$
$223$	−13.7204	−0.918785	−0.459393	+	0.888233i	$0.651933\pi$
$224$	0	0
$225$	−3.30826	−0.220551
$226$	0	0
$227$	5.02835	0.333743	0.166872	−	0.985979i	$-0.446633\pi$
$227$	5.02835	0.333743	0.166872	+	0.985979i	$0.446633\pi$
$228$	0	0
$229$	3.64139	0.240630	0.120315	−	0.992736i	$-0.461610\pi$
$229$	3.64139	0.240630	0.120315	+	0.992736i	$0.461610\pi$
$230$	0	0
$231$	16.7922	1.10485
$232$	0	0
$233$	−26.5549	−1.73967	−0.869833	−	0.493346i	$-0.835774\pi$
$233$	−26.5549	−1.73967	−0.869833	+	0.493346i	$0.835774\pi$
$234$	0	0
$235$	−4.23987	−0.276578
$236$	0	0
$237$	1.98454	0.128909
$238$	0	0
$239$	−18.3311	−1.18574	−0.592869	−	0.805299i	$-0.702005\pi$
$239$	−18.3311	−1.18574	−0.592869	+	0.805299i	$0.702005\pi$
$240$	0	0
$241$	−14.1402	−0.910847	−0.455424	−	0.890275i	$-0.650512\pi$
$241$	−14.1402	−0.910847	−0.455424	+	0.890275i	$0.650512\pi$
$242$	0	0
$243$	−9.42599	−0.604677
$244$	0	0
$245$	11.2047	0.715840
$246$	0	0
$247$	−1.82614	−0.116194
$248$	0	0
$249$	32.1091	2.03483
$250$	0	0
$251$	0.355841	0.0224605	0.0112302	−	0.999937i	$-0.496425\pi$
$251$	0.355841	0.0224605	0.0112302	+	0.999937i	$0.496425\pi$
$252$	0	0
$253$	1.73226	0.108906
$254$	0	0
$255$	−5.77682	−0.361758
$256$	0	0
$257$	−0.344878	−0.0215129	−0.0107564	−	0.999942i	$-0.503424\pi$
$257$	−0.344878	−0.0215129	−0.0107564	+	0.999942i	$0.503424\pi$
$258$	0	0
$259$	−5.25357	−0.326441
$260$	0	0
$261$	7.37677	0.456611
$262$	0	0
$263$	−28.1636	−1.73664	−0.868321	−	0.496003i	$-0.834801\pi$
$263$	−28.1636	−1.73664	−0.868321	+	0.496003i	$0.834801\pi$
$264$	0	0
$265$	−1.08110	−0.0664113
$266$	0	0
$267$	33.2974	2.03777
$268$	0	0
$269$	−21.3650	−1.30265	−0.651323	−	0.758801i	$-0.725785\pi$
$269$	−21.3650	−1.30265	−0.651323	+	0.758801i	$0.725785\pi$
$270$	0	0
$271$	21.4789	1.30475	0.652376	−	0.757896i	$-0.273773\pi$
$271$	21.4789	1.30475	0.652376	+	0.757896i	$0.273773\pi$
$272$	0	0
$273$	−14.5988	−0.883557
$274$	0	0
$275$	−7.40536	−0.446560
$276$	0	0
$277$	14.9680	0.899340	0.449670	−	0.893195i	$-0.351542\pi$
$277$	14.9680	0.899340	0.449670	+	0.893195i	$0.351542\pi$
$278$	0	0
$279$	0.963013	0.0576541
$280$	0	0
$281$	7.99239	0.476786	0.238393	−	0.971169i	$-0.423379\pi$
$281$	7.99239	0.476786	0.238393	+	0.971169i	$0.423379\pi$
$282$	0	0
$283$	−31.4051	−1.86684	−0.933419	−	0.358787i	$-0.883190\pi$
$283$	−31.4051	−1.86684	−0.933419	+	0.358787i	$0.883190\pi$
$284$	0	0
$285$	2.40980	0.142744
$286$	0	0
$287$	4.55496	0.268871
$288$	0	0
$289$	−11.2533	−0.661962
$290$	0	0
$291$	−11.3253	−0.663898
$292$	0	0
$293$	−8.97135	−0.524112	−0.262056	−	0.965053i	$-0.584400\pi$
$293$	−8.97135	−0.524112	−0.262056	+	0.965053i	$0.584400\pi$
$294$	0	0
$295$	3.01905	0.175776
$296$	0	0
$297$	−8.59396	−0.498672
$298$	0	0
$299$	−1.50599	−0.0870936
$300$	0	0
$301$	51.5226	2.96971
$302$	0	0
$303$	−13.5801	−0.780154
$304$	0	0
$305$	8.35075	0.478162
$306$	0	0
$307$	−18.7902	−1.07241	−0.536206	−	0.844087i	$-0.680143\pi$
$307$	−18.7902	−1.07241	−0.536206	+	0.844087i	$0.680143\pi$
$308$	0	0
$309$	33.7592	1.92049
$310$	0	0
$311$	14.3047	0.811144	0.405572	−	0.914063i	$-0.367072\pi$
$311$	14.3047	0.811144	0.405572	+	0.914063i	$0.367072\pi$
$312$	0	0
$313$	−22.5207	−1.27295	−0.636474	−	0.771298i	$-0.719608\pi$
$313$	−22.5207	−1.27295	−0.636474	+	0.771298i	$0.719608\pi$
$314$	0	0
$315$	4.59012	0.258624
$316$	0	0
$317$	32.8582	1.84550	0.922750	−	0.385399i	$-0.125936\pi$
$317$	32.8582	1.84550	0.922750	+	0.385399i	$0.125936\pi$
$318$	0	0
$319$	16.5125	0.924524
$320$	0	0
$321$	12.7660	0.712531
$322$	0	0
$323$	−2.39722	−0.133385
$324$	0	0
$325$	6.43805	0.357119
$326$	0	0
$327$	14.7185	0.813937
$328$	0	0
$329$	−14.0655	−0.775454
$330$	0	0
$331$	9.06609	0.498317	0.249159	−	0.968463i	$-0.419846\pi$
$331$	9.06609	0.498317	0.249159	+	0.968463i	$0.419846\pi$
$332$	0	0
$333$	−1.22380	−0.0670637
$334$	0	0
$335$	9.81109	0.536037
$336$	0	0
$337$	−6.38471	−0.347797	−0.173899	−	0.984764i	$-0.555636\pi$
$337$	−6.38471	−0.347797	−0.173899	+	0.984764i	$0.555636\pi$
$338$	0	0
$339$	−30.8372	−1.67485
$340$	0	0
$341$	2.15565	0.116735
$342$	0	0
$343$	8.97246	0.484467
$344$	0	0
$345$	1.98733	0.106994
$346$	0	0
$347$	2.00447	0.107605	0.0538027	−	0.998552i	$-0.482866\pi$
$347$	2.00447	0.107605	0.0538027	+	0.998552i	$0.482866\pi$
$348$	0	0
$349$	9.72437	0.520534	0.260267	−	0.965537i	$-0.416189\pi$
$349$	9.72437	0.520534	0.260267	+	0.965537i	$0.416189\pi$
$350$	0	0
$351$	7.47138	0.398793
$352$	0	0
$353$	−9.50638	−0.505973	−0.252987	−	0.967470i	$-0.581413\pi$
$353$	−9.50638	−0.505973	−0.252987	+	0.967470i	$0.581413\pi$
$354$	0	0
$355$	3.68927	0.195806
$356$	0	0
$357$	−19.1642	−1.01428
$358$	0	0
$359$	20.9577	1.10610	0.553052	−	0.833147i	$-0.313463\pi$
$359$	20.9577	1.10610	0.553052	+	0.833147i	$0.313463\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−13.0738	−0.686197
$364$	0	0
$365$	−14.2351	−0.745098
$366$	0	0
$367$	−8.20855	−0.428483	−0.214241	−	0.976781i	$-0.568728\pi$
$367$	−8.20855	−0.428483	−0.214241	+	0.976781i	$0.568728\pi$
$368$	0	0
$369$	1.06106	0.0552365
$370$	0	0
$371$	−3.58646	−0.186200
$372$	0	0
$373$	−9.04921	−0.468550	−0.234275	−	0.972170i	$-0.575272\pi$
$373$	−9.04921	−0.468550	−0.234275	+	0.972170i	$0.575272\pi$
$374$	0	0
$375$	−20.5448	−1.06093
$376$	0	0
$377$	−14.3556	−0.739351
$378$	0	0
$379$	−28.3302	−1.45523	−0.727613	−	0.685988i	$-0.759370\pi$
$379$	−28.3302	−1.45523	−0.727613	+	0.685988i	$0.759370\pi$
$380$	0	0
$381$	−36.3514	−1.86234
$382$	0	0
$383$	4.48662	0.229256	0.114628	−	0.993409i	$-0.463432\pi$
$383$	4.48662	0.229256	0.114628	+	0.993409i	$0.463432\pi$
$384$	0	0
$385$	10.2748	0.523650
$386$	0	0
$387$	12.0020	0.610095
$388$	0	0
$389$	29.9619	1.51913	0.759565	−	0.650431i	$-0.225412\pi$
$389$	29.9619	1.51913	0.759565	+	0.650431i	$0.225412\pi$
$390$	0	0
$391$	−1.97695	−0.0999787
$392$	0	0
$393$	11.9727	0.603945
$394$	0	0
$395$	1.21429	0.0610975
$396$	0	0
$397$	−2.87398	−0.144241	−0.0721204	−	0.997396i	$-0.522977\pi$
$397$	−2.87398	−0.144241	−0.0721204	+	0.997396i	$0.522977\pi$
$398$	0	0
$399$	7.99434	0.400218
$400$	0	0
$401$	15.5054	0.774304	0.387152	−	0.922016i	$-0.373459\pi$
$401$	15.5054	0.774304	0.387152	+	0.922016i	$0.373459\pi$
$402$	0	0
$403$	−1.87408	−0.0933543
$404$	0	0
$405$	−13.2777	−0.659777
$406$	0	0
$407$	−2.73941	−0.135787
$408$	0	0
$409$	26.5888	1.31473	0.657367	−	0.753571i	$-0.271670\pi$
$409$	26.5888	1.31473	0.657367	+	0.753571i	$0.271670\pi$
$410$	0	0
$411$	−3.34652	−0.165072
$412$	0	0
$413$	10.0155	0.492829
$414$	0	0
$415$	19.6468	0.964422
$416$	0	0
$417$	40.3411	1.97551
$418$	0	0
$419$	−14.1286	−0.690226	−0.345113	−	0.938561i	$-0.612159\pi$
$419$	−14.1286	−0.690226	−0.345113	+	0.938561i	$0.612159\pi$
$420$	0	0
$421$	−18.6284	−0.907893	−0.453947	−	0.891029i	$-0.649984\pi$
$421$	−18.6284	−0.907893	−0.453947	+	0.891029i	$0.649984\pi$
$422$	0	0
$423$	−3.27649	−0.159308
$424$	0	0
$425$	8.45140	0.409953
$426$	0	0
$427$	27.7030	1.34064
$428$	0	0
$429$	−7.61233	−0.367527
$430$	0	0
$431$	−2.38365	−0.114817	−0.0574083	−	0.998351i	$-0.518284\pi$
$431$	−2.38365	−0.114817	−0.0574083	+	0.998351i	$0.518284\pi$
$432$	0	0
$433$	−21.3129	−1.02423	−0.512115	−	0.858917i	$-0.671138\pi$
$433$	−21.3129	−1.02423	−0.512115	+	0.858917i	$0.671138\pi$
$434$	0	0
$435$	18.9439	0.908289
$436$	0	0
$437$	0.824686	0.0394501
$438$	0	0
$439$	0.555580	0.0265164	0.0132582	−	0.999912i	$-0.495780\pi$
$439$	0.555580	0.0265164	0.0132582	+	0.999912i	$0.495780\pi$
$440$	0	0
$441$	8.65875	0.412322
$442$	0	0
$443$	13.2088	0.627567	0.313783	−	0.949495i	$-0.398403\pi$
$443$	13.2088	0.627567	0.313783	+	0.949495i	$0.398403\pi$
$444$	0	0
$445$	20.3739	0.965815
$446$	0	0
$447$	−24.8722	−1.17641
$448$	0	0
$449$	−8.71816	−0.411436	−0.205718	−	0.978611i	$-0.565953\pi$
$449$	−8.71816	−0.411436	−0.205718	+	0.978611i	$0.565953\pi$
$450$	0	0
$451$	2.37512	0.111840
$452$	0	0
$453$	−22.9611	−1.07881
$454$	0	0
$455$	−8.93263	−0.418768
$456$	0	0
$457$	18.3193	0.856942	0.428471	−	0.903555i	$-0.359052\pi$
$457$	18.3193	0.856942	0.428471	+	0.903555i	$0.359052\pi$
$458$	0	0
$459$	9.80788	0.457793
$460$	0	0
$461$	−16.9839	−0.791017	−0.395509	−	0.918462i	$-0.629432\pi$
$461$	−16.9839	−0.791017	−0.395509	+	0.918462i	$0.629432\pi$
$462$	0	0
$463$	10.4685	0.486512	0.243256	−	0.969962i	$-0.421785\pi$
$463$	10.4685	0.486512	0.243256	+	0.969962i	$0.421785\pi$
$464$	0	0
$465$	2.47306	0.114685
$466$	0	0
$467$	21.6712	1.00283	0.501413	−	0.865208i	$-0.332814\pi$
$467$	21.6712	1.00283	0.501413	+	0.865208i	$0.332814\pi$
$468$	0	0
$469$	32.5476	1.50291
$470$	0	0
$471$	−28.8751	−1.33050
$472$	0	0
$473$	26.8658	1.23529
$474$	0	0
$475$	−3.52550	−0.161761
$476$	0	0
$477$	−0.835452	−0.0382527
$478$	0	0
$479$	−25.4733	−1.16390	−0.581952	−	0.813223i	$-0.697711\pi$
$479$	−25.4733	−1.16390	−0.581952	+	0.813223i	$0.697711\pi$
$480$	0	0
$481$	2.38158	0.108591
$482$	0	0
$483$	6.59282	0.299983
$484$	0	0
$485$	−6.92965	−0.314659
$486$	0	0
$487$	8.08850	0.366525	0.183262	−	0.983064i	$-0.441334\pi$
$487$	8.08850	0.366525	0.183262	+	0.983064i	$0.441334\pi$
$488$	0	0
$489$	24.0403	1.08714
$490$	0	0
$491$	22.9117	1.03399	0.516995	−	0.855988i	$-0.327051\pi$
$491$	22.9117	1.03399	0.516995	+	0.855988i	$0.327051\pi$
$492$	0	0
$493$	−18.8450	−0.848735
$494$	0	0
$495$	2.39346	0.107578
$496$	0	0
$497$	12.2389	0.548989
$498$	0	0
$499$	14.2094	0.636101	0.318050	−	0.948074i	$-0.396972\pi$
$499$	14.2094	0.636101	0.318050	+	0.948074i	$0.396972\pi$
$500$	0	0
$501$	−0.597992	−0.0267163
$502$	0	0
$503$	11.0210	0.491400	0.245700	−	0.969346i	$-0.420982\pi$
$503$	11.0210	0.491400	0.245700	+	0.969346i	$0.420982\pi$
$504$	0	0
$505$	−8.30931	−0.369760
$506$	0	0
$507$	−19.1810	−0.851857
$508$	0	0
$509$	33.5680	1.48787	0.743937	−	0.668250i	$-0.232956\pi$
$509$	33.5680	1.48787	0.743937	+	0.668250i	$0.232956\pi$
$510$	0	0
$511$	−47.2239	−2.08906
$512$	0	0
$513$	−4.09136	−0.180638
$514$	0	0
$515$	20.6564	0.910231
$516$	0	0
$517$	−7.33425	−0.322560
$518$	0	0
$519$	−39.8054	−1.74726
$520$	0	0
$521$	38.8799	1.70336	0.851680	−	0.524062i	$-0.175584\pi$
$521$	38.8799	1.70336	0.851680	+	0.524062i	$0.175584\pi$
$522$	0	0
$523$	−4.84376	−0.211803	−0.105901	−	0.994377i	$-0.533773\pi$
$523$	−4.84376	−0.211803	−0.105901	+	0.994377i	$0.533773\pi$
$524$	0	0
$525$	−28.1841	−1.23005
$526$	0	0
$527$	−2.46015	−0.107166
$528$	0	0
$529$	−22.3199	−0.970430
$530$	0	0
$531$	2.33306	0.101246
$532$	0	0
$533$	−2.06488	−0.0894397
$534$	0	0
$535$	7.81123	0.337709
$536$	0	0
$537$	31.4796	1.35844
$538$	0	0
$539$	19.3822	0.834849
$540$	0	0
$541$	−36.4090	−1.56535	−0.782673	−	0.622434i	$-0.786144\pi$
$541$	−36.4090	−1.56535	−0.782673	+	0.622434i	$0.786144\pi$
$542$	0	0
$543$	22.5423	0.967385
$544$	0	0
$545$	9.00592	0.385771
$546$	0	0
$547$	−11.4474	−0.489455	−0.244728	−	0.969592i	$-0.578699\pi$
$547$	−11.4474	−0.489455	−0.244728	+	0.969592i	$0.578699\pi$
$548$	0	0
$549$	6.45330	0.275420
$550$	0	0
$551$	7.86118	0.334898
$552$	0	0
$553$	4.02832	0.171302
$554$	0	0
$555$	−3.14277	−0.133403
$556$	0	0
$557$	34.2508	1.45125	0.725627	−	0.688088i	$-0.241550\pi$
$557$	34.2508	1.45125	0.725627	+	0.688088i	$0.241550\pi$
$558$	0	0
$559$	−23.3565	−0.987874
$560$	0	0
$561$	−9.99291	−0.421901
$562$	0	0
$563$	−22.5493	−0.950338	−0.475169	−	0.879895i	$-0.657613\pi$
$563$	−22.5493	−0.950338	−0.475169	+	0.879895i	$0.657613\pi$
$564$	0	0
$565$	−18.8686	−0.793806
$566$	0	0
$567$	−44.0480	−1.84984
$568$	0	0
$569$	−20.9590	−0.878648	−0.439324	−	0.898329i	$-0.644782\pi$
$569$	−20.9590	−0.878648	−0.439324	+	0.898329i	$0.644782\pi$
$570$	0	0
$571$	−4.45355	−0.186375	−0.0931876	−	0.995649i	$-0.529706\pi$
$571$	−4.45355	−0.186375	−0.0931876	+	0.995649i	$0.529706\pi$
$572$	0	0
$573$	29.2972	1.22391
$574$	0	0
$575$	−2.90743	−0.121248
$576$	0	0
$577$	−15.5420	−0.647020	−0.323510	−	0.946225i	$-0.604863\pi$
$577$	−15.5420	−0.647020	−0.323510	+	0.946225i	$0.604863\pi$
$578$	0	0
$579$	0.926374	0.0384988
$580$	0	0
$581$	65.1768	2.70399
$582$	0	0
$583$	−1.87011	−0.0774522
$584$	0	0
$585$	−2.08082	−0.0860313
$586$	0	0
$587$	−18.9080	−0.780417	−0.390208	−	0.920727i	$-0.627597\pi$
$587$	−18.9080	−0.780417	−0.390208	+	0.920727i	$0.627597\pi$
$588$	0	0
$589$	1.02625	0.0422859
$590$	0	0
$591$	−36.3907	−1.49691
$592$	0	0
$593$	−11.7089	−0.480827	−0.240414	−	0.970671i	$-0.577283\pi$
$593$	−11.7089	−0.480827	−0.240414	+	0.970671i	$0.577283\pi$
$594$	0	0
$595$	−11.7261	−0.480723
$596$	0	0
$597$	7.01800	0.287227
$598$	0	0
$599$	17.1576	0.701041	0.350520	−	0.936555i	$-0.386005\pi$
$599$	17.1576	0.701041	0.350520	+	0.936555i	$0.386005\pi$
$600$	0	0
$601$	−1.69784	−0.0692563	−0.0346282	−	0.999400i	$-0.511025\pi$
$601$	−1.69784	−0.0692563	−0.0346282	+	0.999400i	$0.511025\pi$
$602$	0	0
$603$	7.58182	0.308756
$604$	0	0
$605$	−7.99955	−0.325228
$606$	0	0
$607$	−5.91643	−0.240141	−0.120070	−	0.992765i	$-0.538312\pi$
$607$	−5.91643	−0.240141	−0.120070	+	0.992765i	$0.538312\pi$
$608$	0	0
$609$	62.8449	2.54661
$610$	0	0
$611$	6.37622	0.257954
$612$	0	0
$613$	−23.8056	−0.961498	−0.480749	−	0.876858i	$-0.659635\pi$
$613$	−23.8056	−0.961498	−0.480749	+	0.876858i	$0.659635\pi$
$614$	0	0
$615$	2.72484	0.109876
$616$	0	0
$617$	−47.1814	−1.89945	−0.949726	−	0.313082i	$-0.898638\pi$
$617$	−47.1814	−1.89945	−0.949726	+	0.313082i	$0.898638\pi$
$618$	0	0
$619$	−34.2036	−1.37476	−0.687380	−	0.726298i	$-0.741239\pi$
$619$	−34.2036	−1.37476	−0.687380	+	0.726298i	$0.741239\pi$
$620$	0	0
$621$	−3.37408	−0.135397
$622$	0	0
$623$	67.5889	2.70789
$624$	0	0
$625$	5.05667	0.202267
$626$	0	0
$627$	4.16854	0.166476
$628$	0	0
$629$	3.12636	0.124656
$630$	0	0
$631$	−19.6632	−0.782780	−0.391390	−	0.920225i	$-0.628006\pi$
$631$	−19.6632	−0.782780	−0.391390	+	0.920225i	$0.628006\pi$
$632$	0	0
$633$	−40.3404	−1.60339
$634$	0	0
$635$	−22.2425	−0.882668
$636$	0	0
$637$	−16.8504	−0.667637
$638$	0	0
$639$	2.85100	0.112784
$640$	0	0
$641$	29.7373	1.17455	0.587276	−	0.809387i	$-0.300200\pi$
$641$	29.7373	1.17455	0.587276	+	0.809387i	$0.300200\pi$
$642$	0	0
$643$	−38.8036	−1.53026	−0.765131	−	0.643874i	$-0.777326\pi$
$643$	−38.8036	−1.53026	−0.765131	+	0.643874i	$0.777326\pi$
$644$	0	0
$645$	30.8216	1.21360
$646$	0	0
$647$	19.4570	0.764934	0.382467	−	0.923969i	$-0.375075\pi$
$647$	19.4570	0.764934	0.382467	+	0.923969i	$0.375075\pi$
$648$	0	0
$649$	5.22244	0.204999
$650$	0	0
$651$	8.20420	0.321548
$652$	0	0
$653$	23.4540	0.917826	0.458913	−	0.888481i	$-0.348239\pi$
$653$	23.4540	0.917826	0.458913	+	0.888481i	$0.348239\pi$
$654$	0	0
$655$	7.32584	0.286244
$656$	0	0
$657$	−11.0006	−0.429174
$658$	0	0
$659$	−38.6883	−1.50708	−0.753541	−	0.657400i	$-0.771656\pi$
$659$	−38.6883	−1.50708	−0.753541	+	0.657400i	$0.771656\pi$
$660$	0	0
$661$	−22.3655	−0.869916	−0.434958	−	0.900451i	$-0.643237\pi$
$661$	−22.3655	−0.869916	−0.434958	+	0.900451i	$0.643237\pi$
$662$	0	0
$663$	8.68760	0.337398
$664$	0	0
$665$	4.89154	0.189686
$666$	0	0
$667$	6.48300	0.251023
$668$	0	0
$669$	−27.2286	−1.05272
$670$	0	0
$671$	14.4454	0.557657
$672$	0	0
$673$	4.44339	0.171280	0.0856400	−	0.996326i	$-0.472707\pi$
$673$	4.44339	0.171280	0.0856400	+	0.996326i	$0.472707\pi$
$674$	0	0
$675$	14.4241	0.555183
$676$	0	0
$677$	−21.7702	−0.836695	−0.418348	−	0.908287i	$-0.637391\pi$
$677$	−21.7702	−0.836695	−0.418348	+	0.908287i	$0.637391\pi$
$678$	0	0
$679$	−22.9886	−0.882222
$680$	0	0
$681$	9.97894	0.382394
$682$	0	0
$683$	−22.0382	−0.843267	−0.421633	−	0.906766i	$-0.638543\pi$
$683$	−22.0382	−0.843267	−0.421633	+	0.906766i	$0.638543\pi$
$684$	0	0
$685$	−2.04765	−0.0782368
$686$	0	0
$687$	7.22647	0.275707
$688$	0	0
$689$	1.62583	0.0619393
$690$	0	0
$691$	28.7137	1.09232	0.546160	−	0.837681i	$-0.316089\pi$
$691$	28.7137	1.09232	0.546160	+	0.837681i	$0.316089\pi$
$692$	0	0
$693$	7.94013	0.301621
$694$	0	0
$695$	24.6838	0.936309
$696$	0	0
$697$	−2.71062	−0.102672
$698$	0	0
$699$	−52.6991	−1.99326
$700$	0	0
$701$	−14.0839	−0.531941	−0.265971	−	0.963981i	$-0.585692\pi$
$701$	−14.0839	−0.531941	−0.265971	+	0.963981i	$0.585692\pi$
$702$	0	0
$703$	−1.30416	−0.0491874
$704$	0	0
$705$	−8.41416	−0.316896
$706$	0	0
$707$	−27.5656	−1.03671
$708$	0	0
$709$	−34.3520	−1.29012	−0.645059	−	0.764133i	$-0.723167\pi$
$709$	−34.3520	−1.29012	−0.645059	+	0.764133i	$0.723167\pi$
$710$	0	0
$711$	0.938380	0.0351920
$712$	0	0
$713$	0.846334	0.0316955
$714$	0	0
$715$	−4.65780	−0.174192
$716$	0	0
$717$	−36.3786	−1.35858
$718$	0	0
$719$	25.3434	0.945149	0.472574	−	0.881291i	$-0.343325\pi$
$719$	25.3434	0.945149	0.472574	+	0.881291i	$0.343325\pi$
$720$	0	0
$721$	68.5262	2.55205
$722$	0	0
$723$	−28.0616	−1.04362
$724$	0	0
$725$	−27.7146	−1.02929
$726$	0	0
$727$	−29.0013	−1.07560	−0.537799	−	0.843073i	$-0.680744\pi$
$727$	−29.0013	−1.07560	−0.537799	+	0.843073i	$0.680744\pi$
$728$	0	0
$729$	14.0975	0.522131
$730$	0	0
$731$	−30.6607	−1.13403
$732$	0	0
$733$	−51.9223	−1.91780	−0.958898	−	0.283752i	$-0.908421\pi$
$733$	−51.9223	−1.91780	−0.958898	+	0.283752i	$0.908421\pi$
$734$	0	0
$735$	22.2361	0.820189
$736$	0	0
$737$	16.9715	0.625154
$738$	0	0
$739$	38.9547	1.43297	0.716485	−	0.697602i	$-0.245750\pi$
$739$	38.9547	1.43297	0.716485	+	0.697602i	$0.245750\pi$
$740$	0	0
$741$	−3.62403	−0.133132
$742$	0	0
$743$	35.3476	1.29678	0.648388	−	0.761310i	$-0.275443\pi$
$743$	35.3476	1.29678	0.648388	+	0.761310i	$0.275443\pi$
$744$	0	0
$745$	−15.2187	−0.557569
$746$	0	0
$747$	15.1826	0.555504
$748$	0	0
$749$	25.9132	0.946848
$750$	0	0
$751$	17.4742	0.637644	0.318822	−	0.947815i	$-0.396713\pi$
$751$	17.4742	0.637644	0.318822	+	0.947815i	$0.396713\pi$
$752$	0	0
$753$	0.706179	0.0257346
$754$	0	0
$755$	−14.0493	−0.511308
$756$	0	0
$757$	29.6970	1.07936	0.539678	−	0.841871i	$-0.318546\pi$
$757$	29.6970	1.07936	0.539678	+	0.841871i	$0.318546\pi$
$758$	0	0
$759$	3.43774	0.124782
$760$	0	0
$761$	37.8801	1.37315	0.686577	−	0.727057i	$-0.259112\pi$
$761$	37.8801	1.37315	0.686577	+	0.727057i	$0.259112\pi$
$762$	0	0
$763$	29.8765	1.08160
$764$	0	0
$765$	−2.73154	−0.0987592
$766$	0	0
$767$	−4.54026	−0.163939
$768$	0	0
$769$	35.6471	1.28547	0.642734	−	0.766089i	$-0.277800\pi$
$769$	35.6471	1.28547	0.642734	+	0.766089i	$0.277800\pi$
$770$	0	0
$771$	−0.684422	−0.0246488
$772$	0	0
$773$	22.5396	0.810695	0.405347	−	0.914163i	$-0.367151\pi$
$773$	22.5396	0.810695	0.405347	+	0.914163i	$0.367151\pi$
$774$	0	0
$775$	−3.61805	−0.129964
$776$	0	0
$777$	−10.4259	−0.374027
$778$	0	0
$779$	1.13073	0.0405128
$780$	0	0
$781$	6.38181	0.228359
$782$	0	0
$783$	−32.1629	−1.14941
$784$	0	0
$785$	−17.6680	−0.630598
$786$	0	0
$787$	36.4421	1.29902	0.649511	−	0.760353i	$-0.274974\pi$
$787$	36.4421	1.29902	0.649511	+	0.760353i	$0.274974\pi$
$788$	0	0
$789$	−55.8916	−1.98980
$790$	0	0
$791$	−62.5951	−2.22563
$792$	0	0
$793$	−12.5585	−0.445964
$794$	0	0
$795$	−2.14548	−0.0760922
$796$	0	0
$797$	17.3589	0.614883	0.307441	−	0.951567i	$-0.400527\pi$
$797$	17.3589	0.614883	0.307441	+	0.951567i	$0.400527\pi$
$798$	0	0
$799$	8.37023	0.296118
$800$	0	0
$801$	15.7445	0.556306
$802$	0	0
$803$	−24.6243	−0.868972
$804$	0	0
$805$	4.03399	0.142179
$806$	0	0
$807$	−42.3995	−1.49253
$808$	0	0
$809$	−50.8807	−1.78887	−0.894435	−	0.447199i	$-0.852422\pi$
$809$	−50.8807	−1.78887	−0.894435	+	0.447199i	$0.852422\pi$
$810$	0	0
$811$	9.42632	0.331003	0.165501	−	0.986210i	$-0.447076\pi$
$811$	9.42632	0.331003	0.165501	+	0.986210i	$0.447076\pi$
$812$	0	0
$813$	42.6257	1.49495
$814$	0	0
$815$	14.7097	0.515257
$816$	0	0
$817$	12.7901	0.447469
$818$	0	0
$819$	−6.90297	−0.241209
$820$	0	0
$821$	−43.4452	−1.51625	−0.758124	−	0.652110i	$-0.773884\pi$
$821$	−43.4452	−1.51625	−0.758124	+	0.652110i	$0.773884\pi$
$822$	0	0
$823$	−7.87761	−0.274596	−0.137298	−	0.990530i	$-0.543842\pi$
$823$	−7.87761	−0.274596	−0.137298	+	0.990530i	$0.543842\pi$
$824$	0	0
$825$	−14.6962	−0.511656
$826$	0	0
$827$	38.1230	1.32566	0.662832	−	0.748768i	$-0.269354\pi$
$827$	38.1230	1.32566	0.662832	+	0.748768i	$0.269354\pi$
$828$	0	0
$829$	32.3938	1.12508	0.562541	−	0.826769i	$-0.309824\pi$
$829$	32.3938	1.12508	0.562541	+	0.826769i	$0.309824\pi$
$830$	0	0
$831$	29.7045	1.03044
$832$	0	0
$833$	−22.1200	−0.766411
$834$	0	0
$835$	−0.365897	−0.0126624
$836$	0	0
$837$	−4.19876	−0.145130
$838$	0	0
$839$	−10.2052	−0.352321	−0.176161	−	0.984361i	$-0.556368\pi$
$839$	−10.2052	−0.352321	−0.176161	+	0.984361i	$0.556368\pi$
$840$	0	0
$841$	32.7982	1.13097
$842$	0	0
$843$	15.8612	0.546288
$844$	0	0
$845$	−11.7364	−0.403744
$846$	0	0
$847$	−26.5379	−0.911854
$848$	0	0
$849$	−62.3245	−2.13897
$850$	0	0
$851$	−1.07552	−0.0368684
$852$	0	0
$853$	55.8963	1.91385	0.956926	−	0.290333i	$-0.0937660\pi$
$853$	55.8963	1.91385	0.956926	+	0.290333i	$0.0937660\pi$
$854$	0	0
$855$	1.13946	0.0389689
$856$	0	0
$857$	−6.44719	−0.220232	−0.110116	−	0.993919i	$-0.535122\pi$
$857$	−6.44719	−0.220232	−0.110116	+	0.993919i	$0.535122\pi$
$858$	0	0
$859$	−40.6206	−1.38596	−0.692979	−	0.720958i	$-0.743702\pi$
$859$	−40.6206	−1.38596	−0.692979	+	0.720958i	$0.743702\pi$
$860$	0	0
$861$	9.03947	0.308064
$862$	0	0
$863$	20.6333	0.702365	0.351183	−	0.936307i	$-0.385780\pi$
$863$	20.6333	0.702365	0.351183	+	0.936307i	$0.385780\pi$
$864$	0	0
$865$	−24.3560	−0.828128
$866$	0	0
$867$	−22.3327	−0.758457
$868$	0	0
$869$	2.10051	0.0712551
$870$	0	0
$871$	−14.7546	−0.499942
$872$	0	0
$873$	−5.35510	−0.181243
$874$	0	0
$875$	−41.7029	−1.40981
$876$	0	0
$877$	−1.71754	−0.0579973	−0.0289987	−	0.999579i	$-0.509232\pi$
$877$	−1.71754	−0.0579973	−0.0289987	+	0.999579i	$0.509232\pi$
$878$	0	0
$879$	−17.8040	−0.600513
$880$	0	0
$881$	53.6374	1.80709	0.903545	−	0.428494i	$-0.140956\pi$
$881$	53.6374	1.80709	0.903545	+	0.428494i	$0.140956\pi$
$882$	0	0
$883$	−21.8880	−0.736590	−0.368295	−	0.929709i	$-0.620058\pi$
$883$	−21.8880	−0.736590	−0.368295	+	0.929709i	$0.620058\pi$
$884$	0	0
$885$	5.99140	0.201399
$886$	0	0
$887$	−9.24892	−0.310548	−0.155274	−	0.987871i	$-0.549626\pi$
$887$	−9.24892	−0.310548	−0.155274	+	0.987871i	$0.549626\pi$
$888$	0	0
$889$	−73.7880	−2.47477
$890$	0	0
$891$	−22.9682	−0.769465
$892$	0	0
$893$	−3.49164	−0.116843
$894$	0	0
$895$	19.2616	0.643844
$896$	0	0
$897$	−2.98869	−0.0997894
$898$	0	0
$899$	8.06754	0.269068
$900$	0	0
$901$	2.13427	0.0711030
$902$	0	0
$903$	102.248	3.40262
$904$	0	0
$905$	13.7931	0.458499
$906$	0	0
$907$	44.5947	1.48074	0.740372	−	0.672198i	$-0.234649\pi$
$907$	44.5947	1.48074	0.740372	+	0.672198i	$0.234649\pi$
$908$	0	0
$909$	−6.42128	−0.212980
$910$	0	0
$911$	0.309455	0.0102527	0.00512636	−	0.999987i	$-0.498368\pi$
$911$	0.309455	0.0102527	0.00512636	+	0.999987i	$0.498368\pi$
$912$	0	0
$913$	33.9856	1.12476
$914$	0	0
$915$	16.5724	0.547865
$916$	0	0
$917$	24.3029	0.802554
$918$	0	0
$919$	8.53334	0.281489	0.140744	−	0.990046i	$-0.455050\pi$
$919$	8.53334	0.281489	0.140744	+	0.990046i	$0.455050\pi$
$920$	0	0
$921$	−37.2898	−1.22874
$922$	0	0
$923$	−5.54820	−0.182621
$924$	0	0
$925$	4.59782	0.151175
$926$	0	0
$927$	15.9629	0.524290
$928$	0	0
$929$	−58.4777	−1.91859	−0.959296	−	0.282404i	$-0.908868\pi$
$929$	−58.4777	−1.91859	−0.959296	+	0.282404i	$0.908868\pi$
$930$	0	0
$931$	9.22735	0.302414
$932$	0	0
$933$	28.3881	0.929386
$934$	0	0
$935$	−6.11442	−0.199963
$936$	0	0
$937$	−6.13026	−0.200267	−0.100133	−	0.994974i	$-0.531927\pi$
$937$	−6.13026	−0.200267	−0.100133	+	0.994974i	$0.531927\pi$
$938$	0	0
$939$	−44.6932	−1.45851
$940$	0	0
$941$	−11.8082	−0.384938	−0.192469	−	0.981303i	$-0.561649\pi$
$941$	−11.8082	−0.384938	−0.192469	+	0.981303i	$0.561649\pi$
$942$	0	0
$943$	0.932500	0.0303664
$944$	0	0
$945$	−20.0131	−0.651025
$946$	0	0
$947$	−0.115140	−0.00374154	−0.00187077	−	0.999998i	$-0.500595\pi$
$947$	−0.115140	−0.00374154	−0.00187077	+	0.999998i	$0.500595\pi$
$948$	0	0
$949$	21.4078	0.694925
$950$	0	0
$951$	65.2082	2.11452
$952$	0	0
$953$	−24.3970	−0.790295	−0.395147	−	0.918618i	$-0.629307\pi$
$953$	−24.3970	−0.790295	−0.395147	+	0.918618i	$0.629307\pi$
$954$	0	0
$955$	17.9263	0.580080
$956$	0	0
$957$	32.7697	1.05929
$958$	0	0
$959$	−6.79294	−0.219356
$960$	0	0
$961$	−29.9468	−0.966026
$962$	0	0
$963$	6.03637	0.194519
$964$	0	0
$965$	0.566826	0.0182468
$966$	0	0
$967$	−37.4739	−1.20508	−0.602540	−	0.798088i	$-0.705845\pi$
$967$	−37.4739	−1.20508	−0.602540	+	0.798088i	$0.705845\pi$
$968$	0	0
$969$	−4.75736	−0.152829
$970$	0	0
$971$	−1.24815	−0.0400550	−0.0200275	−	0.999799i	$-0.506375\pi$
$971$	−1.24815	−0.0400550	−0.0200275	+	0.999799i	$0.506375\pi$
$972$	0	0
$973$	81.8867	2.62517
$974$	0	0
$975$	12.7765	0.409177
$976$	0	0
$977$	55.8320	1.78622	0.893112	−	0.449834i	$-0.148517\pi$
$977$	55.8320	1.78622	0.893112	+	0.449834i	$0.148517\pi$
$978$	0	0
$979$	35.2433	1.12638
$980$	0	0
$981$	6.95960	0.222203
$982$	0	0
$983$	31.4280	1.00240	0.501198	−	0.865333i	$-0.332893\pi$
$983$	31.4280	1.00240	0.501198	+	0.865333i	$0.332893\pi$
$984$	0	0
$985$	−22.2666	−0.709472
$986$	0	0
$987$	−27.9134	−0.888493
$988$	0	0
$989$	10.5478	0.335401
$990$	0	0
$991$	11.7049	0.371818	0.185909	−	0.982567i	$-0.440477\pi$
$991$	11.7049	0.371818	0.185909	+	0.982567i	$0.440477\pi$
$992$	0	0
$993$	17.9920	0.570958
$994$	0	0
$995$	4.29414	0.136134
$996$	0	0
$997$	6.68121	0.211596	0.105798	−	0.994388i	$-0.466260\pi$
$997$	6.68121	0.211596	0.105798	+	0.994388i	$0.466260\pi$
$998$	0	0
$999$	5.33579	0.168817

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.e.1.16	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.e.1.16	✓	24	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.98454 q^{3} +1.21429 q^{5} +4.02832 q^{7} +0.938380 q^{9} +O(q^{10})\)	\(q+1.98454 q^{3} +1.21429 q^{5} +4.02832 q^{7} +0.938380 q^{9} +2.10051 q^{11} -1.82614 q^{13} +2.40980 q^{15} -2.39722 q^{17} +1.00000 q^{19} +7.99434 q^{21} +0.824686 q^{23} -3.52550 q^{25} -4.09136 q^{27} +7.86118 q^{29} +1.02625 q^{31} +4.16854 q^{33} +4.89154 q^{35} -1.30416 q^{37} -3.62403 q^{39} +1.13073 q^{41} +12.7901 q^{43} +1.13946 q^{45} -3.49164 q^{47} +9.22735 q^{49} -4.75736 q^{51} -0.890313 q^{53} +2.55063 q^{55} +1.98454 q^{57} +2.48627 q^{59} +6.87706 q^{61} +3.78009 q^{63} -2.21746 q^{65} +8.07970 q^{67} +1.63662 q^{69} +3.03821 q^{71} -11.7230 q^{73} -6.99648 q^{75} +8.46154 q^{77} +1.00000 q^{79} -10.9346 q^{81} +16.1796 q^{83} -2.91092 q^{85} +15.6008 q^{87} +16.7784 q^{89} -7.35626 q^{91} +2.03663 q^{93} +1.21429 q^{95} -5.70676 q^{97} +1.97108 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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