LMFDB - Embedded newform 6008.2.a.d.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, 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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$49$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.83267 q^{3} +0.130732 q^{5} -0.665399 q^{7} +0.358681 q^{9} +O(q^{10})$	$q-1.83267 q^{3} +0.130732 q^{5} -0.665399 q^{7} +0.358681 q^{9} -0.194392 q^{11} +1.97689 q^{13} -0.239589 q^{15} -5.28864 q^{17} +4.65240 q^{19} +1.21946 q^{21} -4.84381 q^{23} -4.98291 q^{25} +4.84067 q^{27} +8.97846 q^{29} +4.34791 q^{31} +0.356256 q^{33} -0.0869891 q^{35} -5.81028 q^{37} -3.62299 q^{39} -8.98662 q^{41} -6.81604 q^{43} +0.0468912 q^{45} +4.61972 q^{47} -6.55724 q^{49} +9.69233 q^{51} +10.3018 q^{53} -0.0254132 q^{55} -8.52632 q^{57} -13.4916 q^{59} +12.7233 q^{61} -0.238666 q^{63} +0.258443 q^{65} +6.68629 q^{67} +8.87712 q^{69} -5.25871 q^{71} -7.23447 q^{73} +9.13203 q^{75} +0.129348 q^{77} +10.5243 q^{79} -9.94739 q^{81} +6.00586 q^{83} -0.691396 q^{85} -16.4546 q^{87} -14.7725 q^{89} -1.31542 q^{91} -7.96828 q^{93} +0.608219 q^{95} +4.62962 q^{97} -0.0697245 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * 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
$3$	−1.83267	−1.05809	−0.529046	−	0.848593i	$-0.677450\pi$
$3$	−1.83267	−1.05809	−0.529046	+	0.848593i	$0.677450\pi$
$4$	0	0
$5$	0.130732	0.0584653	0.0292326	−	0.999573i	$-0.490694\pi$
$5$	0.130732	0.0584653	0.0292326	+	0.999573i	$0.490694\pi$
$6$	0	0
$7$	−0.665399	−0.251497	−0.125749	−	0.992062i	$-0.540133\pi$
$7$	−0.665399	−0.251497	−0.125749	+	0.992062i	$0.540133\pi$
$8$	0	0
$9$	0.358681	0.119560
$10$	0	0
$11$	−0.194392	−0.0586113	−0.0293056	−	0.999570i	$-0.509330\pi$
$11$	−0.194392	−0.0586113	−0.0293056	+	0.999570i	$0.509330\pi$
$12$	0	0
$13$	1.97689	0.548291	0.274145	−	0.961688i	$-0.411605\pi$
$13$	1.97689	0.548291	0.274145	+	0.961688i	$0.411605\pi$
$14$	0	0
$15$	−0.239589	−0.0618617
$16$	0	0
$17$	−5.28864	−1.28268	−0.641342	−	0.767255i	$-0.721622\pi$
$17$	−5.28864	−1.28268	−0.641342	+	0.767255i	$0.721622\pi$
$18$	0	0
$19$	4.65240	1.06733	0.533667	−	0.845694i	$-0.320813\pi$
$19$	4.65240	1.06733	0.533667	+	0.845694i	$0.320813\pi$
$20$	0	0
$21$	1.21946	0.266107
$22$	0	0
$23$	−4.84381	−1.01001	−0.505003	−	0.863118i	$-0.668508\pi$
$23$	−4.84381	−1.01001	−0.505003	+	0.863118i	$0.668508\pi$
$24$	0	0
$25$	−4.98291	−0.996582
$26$	0	0
$27$	4.84067	0.931587
$28$	0	0
$29$	8.97846	1.66726	0.833629	−	0.552324i	$-0.186259\pi$
$29$	8.97846	1.66726	0.833629	+	0.552324i	$0.186259\pi$
$30$	0	0
$31$	4.34791	0.780907	0.390454	−	0.920623i	$-0.372318\pi$
$31$	4.34791	0.780907	0.390454	+	0.920623i	$0.372318\pi$
$32$	0	0
$33$	0.356256	0.0620161
$34$	0	0
$35$	−0.0869891	−0.0147038
$36$	0	0
$37$	−5.81028	−0.955204	−0.477602	−	0.878576i	$-0.658494\pi$
$37$	−5.81028	−0.955204	−0.477602	+	0.878576i	$0.658494\pi$
$38$	0	0
$39$	−3.62299	−0.580142
$40$	0	0
$41$	−8.98662	−1.40347	−0.701737	−	0.712436i	$-0.747592\pi$
$41$	−8.98662	−1.40347	−0.701737	+	0.712436i	$0.747592\pi$
$42$	0	0
$43$	−6.81604	−1.03944	−0.519719	−	0.854338i	$-0.673963\pi$
$43$	−6.81604	−1.03944	−0.519719	+	0.854338i	$0.673963\pi$
$44$	0	0
$45$	0.0468912	0.00699012
$46$	0	0
$47$	4.61972	0.673856	0.336928	−	0.941530i	$-0.390612\pi$
$47$	4.61972	0.673856	0.336928	+	0.941530i	$0.390612\pi$
$48$	0	0
$49$	−6.55724	−0.936749
$50$	0	0
$51$	9.69233	1.35720
$52$	0	0
$53$	10.3018	1.41507	0.707534	−	0.706680i	$-0.249808\pi$
$53$	10.3018	1.41507	0.707534	+	0.706680i	$0.249808\pi$
$54$	0	0
$55$	−0.0254132	−0.00342672
$56$	0	0
$57$	−8.52632	−1.12934
$58$	0	0
$59$	−13.4916	−1.75645	−0.878227	−	0.478244i	$-0.841274\pi$
$59$	−13.4916	−1.75645	−0.878227	+	0.478244i	$0.841274\pi$
$60$	0	0
$61$	12.7233	1.62906	0.814529	−	0.580123i	$-0.196995\pi$
$61$	12.7233	1.62906	0.814529	+	0.580123i	$0.196995\pi$
$62$	0	0
$63$	−0.238666	−0.0300691
$64$	0	0
$65$	0.258443	0.0320560
$66$	0	0
$67$	6.68629	0.816860	0.408430	−	0.912790i	$-0.366076\pi$
$67$	6.68629	0.816860	0.408430	+	0.912790i	$0.366076\pi$
$68$	0	0
$69$	8.87712	1.06868
$70$	0	0
$71$	−5.25871	−0.624094	−0.312047	−	0.950067i	$-0.601015\pi$
$71$	−5.25871	−0.624094	−0.312047	+	0.950067i	$0.601015\pi$
$72$	0	0
$73$	−7.23447	−0.846731	−0.423366	−	0.905959i	$-0.639151\pi$
$73$	−7.23447	−0.846731	−0.423366	+	0.905959i	$0.639151\pi$
$74$	0	0
$75$	9.13203	1.05448
$76$	0	0
$77$	0.129348	0.0147406
$78$	0	0
$79$	10.5243	1.18408	0.592041	−	0.805908i	$-0.298322\pi$
$79$	10.5243	1.18408	0.592041	+	0.805908i	$0.298322\pi$
$80$	0	0
$81$	−9.94739	−1.10527
$82$	0	0
$83$	6.00586	0.659229	0.329614	−	0.944116i	$-0.393081\pi$
$83$	6.00586	0.659229	0.329614	+	0.944116i	$0.393081\pi$
$84$	0	0
$85$	−0.691396	−0.0749924
$86$	0	0
$87$	−16.4546	−1.76411
$88$	0	0
$89$	−14.7725	−1.56588	−0.782941	−	0.622096i	$-0.786281\pi$
$89$	−14.7725	−1.56588	−0.782941	+	0.622096i	$0.786281\pi$
$90$	0	0
$91$	−1.31542	−0.137894
$92$	0	0
$93$	−7.96828	−0.826273
$94$	0	0
$95$	0.608219	0.0624020
$96$	0	0
$97$	4.62962	0.470067	0.235034	−	0.971987i	$-0.424480\pi$
$97$	4.62962	0.470067	0.235034	+	0.971987i	$0.424480\pi$
$98$	0	0
$99$	−0.0697245	−0.00700758
$100$	0	0
$101$	−17.6604	−1.75728	−0.878638	−	0.477488i	$-0.841547\pi$
$101$	−17.6604	−1.75728	−0.878638	+	0.477488i	$0.841547\pi$
$102$	0	0
$103$	15.4001	1.51741	0.758707	−	0.651432i	$-0.225832\pi$
$103$	15.4001	1.51741	0.758707	+	0.651432i	$0.225832\pi$
$104$	0	0
$105$	0.159422	0.0155580
$106$	0	0
$107$	3.53767	0.342000	0.171000	−	0.985271i	$-0.445300\pi$
$107$	3.53767	0.342000	0.171000	+	0.985271i	$0.445300\pi$
$108$	0	0
$109$	−5.15734	−0.493983	−0.246992	−	0.969018i	$-0.579442\pi$
$109$	−5.15734	−0.493983	−0.246992	+	0.969018i	$0.579442\pi$
$110$	0	0
$111$	10.6483	1.01069
$112$	0	0
$113$	11.0840	1.04269	0.521347	−	0.853344i	$-0.325430\pi$
$113$	11.0840	1.04269	0.521347	+	0.853344i	$0.325430\pi$
$114$	0	0
$115$	−0.633243	−0.0590502
$116$	0	0
$117$	0.709073	0.0655538
$118$	0	0
$119$	3.51905	0.322591
$120$	0	0
$121$	−10.9622	−0.996565
$122$	0	0
$123$	16.4695	1.48501
$124$	0	0
$125$	−1.30509	−0.116731
$126$	0	0
$127$	7.57646	0.672302	0.336151	−	0.941808i	$-0.390875\pi$
$127$	7.57646	0.672302	0.336151	+	0.941808i	$0.390875\pi$
$128$	0	0
$129$	12.4916	1.09982
$130$	0	0
$131$	5.28503	0.461755	0.230878	−	0.972983i	$-0.425840\pi$
$131$	5.28503	0.461755	0.230878	+	0.972983i	$0.425840\pi$
$132$	0	0
$133$	−3.09570	−0.268432
$134$	0	0
$135$	0.632831	0.0544655
$136$	0	0
$137$	10.1982	0.871292	0.435646	−	0.900118i	$-0.356520\pi$
$137$	10.1982	0.871292	0.435646	+	0.900118i	$0.356520\pi$
$138$	0	0
$139$	−5.07106	−0.430121	−0.215061	−	0.976601i	$-0.568995\pi$
$139$	−5.07106	−0.430121	−0.215061	+	0.976601i	$0.568995\pi$
$140$	0	0
$141$	−8.46643	−0.713002
$142$	0	0
$143$	−0.384291	−0.0321360
$144$	0	0
$145$	1.17377	0.0974767
$146$	0	0
$147$	12.0173	0.991168
$148$	0	0
$149$	13.6577	1.11888	0.559439	−	0.828871i	$-0.311017\pi$
$149$	13.6577	1.11888	0.559439	+	0.828871i	$0.311017\pi$
$150$	0	0
$151$	1.82325	0.148374	0.0741868	−	0.997244i	$-0.476364\pi$
$151$	1.82325	0.148374	0.0741868	+	0.997244i	$0.476364\pi$
$152$	0	0
$153$	−1.89693	−0.153358
$154$	0	0
$155$	0.568412	0.0456560
$156$	0	0
$157$	−0.315654	−0.0251919	−0.0125960	−	0.999921i	$-0.504010\pi$
$157$	−0.315654	−0.0251919	−0.0125960	+	0.999921i	$0.504010\pi$
$158$	0	0
$159$	−18.8799	−1.49727
$160$	0	0
$161$	3.22307	0.254013
$162$	0	0
$163$	6.04146	0.473203	0.236602	−	0.971607i	$-0.423966\pi$
$163$	6.04146	0.473203	0.236602	+	0.971607i	$0.423966\pi$
$164$	0	0
$165$	0.0465741	0.00362579
$166$	0	0
$167$	21.3814	1.65454	0.827272	−	0.561801i	$-0.189891\pi$
$167$	21.3814	1.65454	0.827272	+	0.561801i	$0.189891\pi$
$168$	0	0
$169$	−9.09191	−0.699377
$170$	0	0
$171$	1.66873	0.127611
$172$	0	0
$173$	−6.52530	−0.496109	−0.248055	−	0.968746i	$-0.579791\pi$
$173$	−6.52530	−0.496109	−0.248055	+	0.968746i	$0.579791\pi$
$174$	0	0
$175$	3.31562	0.250637
$176$	0	0
$177$	24.7256	1.85849
$178$	0	0
$179$	−16.2022	−1.21101	−0.605503	−	0.795843i	$-0.707028\pi$
$179$	−16.2022	−1.21101	−0.605503	+	0.795843i	$0.707028\pi$
$180$	0	0
$181$	−15.0995	−1.12234	−0.561168	−	0.827702i	$-0.689648\pi$
$181$	−15.0995	−1.12234	−0.561168	+	0.827702i	$0.689648\pi$
$182$	0	0
$183$	−23.3177	−1.72369
$184$	0	0
$185$	−0.759591	−0.0558463
$186$	0	0
$187$	1.02807	0.0751797
$188$	0	0
$189$	−3.22097	−0.234291
$190$	0	0
$191$	9.89495	0.715973	0.357987	−	0.933727i	$-0.383463\pi$
$191$	9.89495	0.715973	0.357987	+	0.933727i	$0.383463\pi$
$192$	0	0
$193$	13.4039	0.964832	0.482416	−	0.875942i	$-0.339759\pi$
$193$	13.4039	0.964832	0.482416	+	0.875942i	$0.339759\pi$
$194$	0	0
$195$	−0.473641	−0.0339182
$196$	0	0
$197$	−26.4144	−1.88195	−0.940975	−	0.338475i	$-0.890089\pi$
$197$	−26.4144	−1.88195	−0.940975	+	0.338475i	$0.890089\pi$
$198$	0	0
$199$	3.12246	0.221345	0.110673	−	0.993857i	$-0.464700\pi$
$199$	3.12246	0.221345	0.110673	+	0.993857i	$0.464700\pi$
$200$	0	0
$201$	−12.2538	−0.864314
$202$	0	0
$203$	−5.97426	−0.419311
$204$	0	0
$205$	−1.17484	−0.0820545
$206$	0	0
$207$	−1.73738	−0.120757
$208$	0	0
$209$	−0.904388	−0.0625578
$210$	0	0
$211$	−3.11026	−0.214119	−0.107059	−	0.994253i	$-0.534144\pi$
$211$	−3.11026	−0.214119	−0.107059	+	0.994253i	$0.534144\pi$
$212$	0	0
$213$	9.63748	0.660349
$214$	0	0
$215$	−0.891077	−0.0607710
$216$	0	0
$217$	−2.89309	−0.196396
$218$	0	0
$219$	13.2584	0.895920
$220$	0	0
$221$	−10.4551	−0.703283
$222$	0	0
$223$	17.7967	1.19175	0.595876	−	0.803076i	$-0.296805\pi$
$223$	17.7967	1.19175	0.595876	+	0.803076i	$0.296805\pi$
$224$	0	0
$225$	−1.78727	−0.119152
$226$	0	0
$227$	−20.6918	−1.37336	−0.686681	−	0.726959i	$-0.740933\pi$
$227$	−20.6918	−1.37336	−0.686681	+	0.726959i	$0.740933\pi$
$228$	0	0
$229$	14.5042	0.958461	0.479231	−	0.877689i	$-0.340916\pi$
$229$	14.5042	0.958461	0.479231	+	0.877689i	$0.340916\pi$
$230$	0	0
$231$	−0.237052	−0.0155969
$232$	0	0
$233$	8.25667	0.540912	0.270456	−	0.962732i	$-0.412825\pi$
$233$	8.25667	0.540912	0.270456	+	0.962732i	$0.412825\pi$
$234$	0	0
$235$	0.603947	0.0393971
$236$	0	0
$237$	−19.2877	−1.25287
$238$	0	0
$239$	11.6277	0.752134	0.376067	−	0.926593i	$-0.377276\pi$
$239$	11.6277	0.752134	0.376067	+	0.926593i	$0.377276\pi$
$240$	0	0
$241$	14.0048	0.902126	0.451063	−	0.892492i	$-0.351045\pi$
$241$	14.0048	0.902126	0.451063	+	0.892492i	$0.351045\pi$
$242$	0	0
$243$	3.70829	0.237887
$244$	0	0
$245$	−0.857244	−0.0547673
$246$	0	0
$247$	9.19729	0.585210
$248$	0	0
$249$	−11.0068	−0.697525
$250$	0	0
$251$	30.7401	1.94030	0.970148	−	0.242513i	$-0.0779719\pi$
$251$	30.7401	1.94030	0.970148	+	0.242513i	$0.0779719\pi$
$252$	0	0
$253$	0.941597	0.0591977
$254$	0	0
$255$	1.26710	0.0793489
$256$	0	0
$257$	4.64590	0.289803	0.144901	−	0.989446i	$-0.453713\pi$
$257$	4.64590	0.289803	0.144901	+	0.989446i	$0.453713\pi$
$258$	0	0
$259$	3.86615	0.240231
$260$	0	0
$261$	3.22040	0.199338
$262$	0	0
$263$	11.8492	0.730656	0.365328	−	0.930879i	$-0.380957\pi$
$263$	11.8492	0.730656	0.365328	+	0.930879i	$0.380957\pi$
$264$	0	0
$265$	1.34678	0.0827323
$266$	0	0
$267$	27.0731	1.65685
$268$	0	0
$269$	14.5162	0.885068	0.442534	−	0.896752i	$-0.354080\pi$
$269$	14.5162	0.885068	0.442534	+	0.896752i	$0.354080\pi$
$270$	0	0
$271$	−16.1680	−0.982134	−0.491067	−	0.871122i	$-0.663393\pi$
$271$	−16.1680	−0.982134	−0.491067	+	0.871122i	$0.663393\pi$
$272$	0	0
$273$	2.41073	0.145904
$274$	0	0
$275$	0.968635	0.0584109
$276$	0	0
$277$	11.8283	0.710696	0.355348	−	0.934734i	$-0.384362\pi$
$277$	11.8283	0.710696	0.355348	+	0.934734i	$0.384362\pi$
$278$	0	0
$279$	1.55951	0.0933655
$280$	0	0
$281$	1.06946	0.0637986	0.0318993	−	0.999491i	$-0.489844\pi$
$281$	1.06946	0.0637986	0.0318993	+	0.999491i	$0.489844\pi$
$282$	0	0
$283$	24.1828	1.43752	0.718759	−	0.695260i	$-0.244711\pi$
$283$	24.1828	1.43752	0.718759	+	0.695260i	$0.244711\pi$
$284$	0	0
$285$	−1.11467	−0.0660271
$286$	0	0
$287$	5.97969	0.352970
$288$	0	0
$289$	10.9697	0.645276
$290$	0	0
$291$	−8.48458	−0.497375
$292$	0	0
$293$	−14.1275	−0.825340	−0.412670	−	0.910881i	$-0.635404\pi$
$293$	−14.1275	−0.825340	−0.412670	+	0.910881i	$0.635404\pi$
$294$	0	0
$295$	−1.76379	−0.102692
$296$	0	0
$297$	−0.940985	−0.0546015
$298$	0	0
$299$	−9.57569	−0.553776
$300$	0	0
$301$	4.53539	0.261415
$302$	0	0
$303$	32.3657	1.85936
$304$	0	0
$305$	1.66335	0.0952433
$306$	0	0
$307$	32.1243	1.83343	0.916715	−	0.399542i	$-0.130831\pi$
$307$	32.1243	1.83343	0.916715	+	0.399542i	$0.130831\pi$
$308$	0	0
$309$	−28.2232	−1.60556
$310$	0	0
$311$	0.674358	0.0382393	0.0191197	−	0.999817i	$-0.493914\pi$
$311$	0.674358	0.0382393	0.0191197	+	0.999817i	$0.493914\pi$
$312$	0	0
$313$	17.0142	0.961698	0.480849	−	0.876804i	$-0.340329\pi$
$313$	17.0142	0.961698	0.480849	+	0.876804i	$0.340329\pi$
$314$	0	0
$315$	−0.0312013	−0.00175800
$316$	0	0
$317$	35.2042	1.97727	0.988634	−	0.150344i	$-0.0480382\pi$
$317$	35.2042	1.97727	0.988634	+	0.150344i	$0.0480382\pi$
$318$	0	0
$319$	−1.74534	−0.0977201
$320$	0	0
$321$	−6.48338	−0.361867
$322$	0	0
$323$	−24.6049	−1.36905
$324$	0	0
$325$	−9.85066	−0.546416
$326$	0	0
$327$	9.45170	0.522680
$328$	0	0
$329$	−3.07396	−0.169473
$330$	0	0
$331$	9.96675	0.547822	0.273911	−	0.961755i	$-0.411683\pi$
$331$	9.96675	0.547822	0.273911	+	0.961755i	$0.411683\pi$
$332$	0	0
$333$	−2.08404	−0.114205
$334$	0	0
$335$	0.874114	0.0477579
$336$	0	0
$337$	13.9382	0.759262	0.379631	−	0.925138i	$-0.376051\pi$
$337$	13.9382	0.759262	0.379631	+	0.925138i	$0.376051\pi$
$338$	0	0
$339$	−20.3133	−1.10327
$340$	0	0
$341$	−0.845197	−0.0457700
$342$	0	0
$343$	9.02097	0.487087
$344$	0	0
$345$	1.16053	0.0624806
$346$	0	0
$347$	−8.93836	−0.479836	−0.239918	−	0.970793i	$-0.577121\pi$
$347$	−8.93836	−0.479836	−0.239918	+	0.970793i	$0.577121\pi$
$348$	0	0
$349$	24.4613	1.30938	0.654691	−	0.755897i	$-0.272799\pi$
$349$	24.4613	1.30938	0.654691	+	0.755897i	$0.272799\pi$
$350$	0	0
$351$	9.56947	0.510780
$352$	0	0
$353$	−0.247057	−0.0131495	−0.00657476	−	0.999978i	$-0.502093\pi$
$353$	−0.247057	−0.0131495	−0.00657476	+	0.999978i	$0.502093\pi$
$354$	0	0
$355$	−0.687483	−0.0364878
$356$	0	0
$357$	−6.44927	−0.341331
$358$	0	0
$359$	6.23078	0.328848	0.164424	−	0.986390i	$-0.447423\pi$
$359$	6.23078	0.328848	0.164424	+	0.986390i	$0.447423\pi$
$360$	0	0
$361$	2.64487	0.139203
$362$	0	0
$363$	20.0901	1.05446
$364$	0	0
$365$	−0.945779	−0.0495043
$366$	0	0
$367$	5.26507	0.274834	0.137417	−	0.990513i	$-0.456120\pi$
$367$	5.26507	0.274834	0.137417	+	0.990513i	$0.456120\pi$
$368$	0	0
$369$	−3.22333	−0.167800
$370$	0	0
$371$	−6.85483	−0.355885
$372$	0	0
$373$	33.0385	1.71067	0.855334	−	0.518077i	$-0.173352\pi$
$373$	33.0385	1.71067	0.855334	+	0.518077i	$0.173352\pi$
$374$	0	0
$375$	2.39180	0.123512
$376$	0	0
$377$	17.7494	0.914142
$378$	0	0
$379$	−17.5281	−0.900356	−0.450178	−	0.892939i	$-0.648640\pi$
$379$	−17.5281	−0.900356	−0.450178	+	0.892939i	$0.648640\pi$
$380$	0	0
$381$	−13.8851	−0.711358
$382$	0	0
$383$	7.76833	0.396943	0.198472	−	0.980107i	$-0.436402\pi$
$383$	7.76833	0.396943	0.198472	+	0.980107i	$0.436402\pi$
$384$	0	0
$385$	0.0169099	0.000861811 0
$386$	0	0
$387$	−2.44479	−0.124275
$388$	0	0
$389$	−7.68404	−0.389596	−0.194798	−	0.980843i	$-0.562405\pi$
$389$	−7.68404	−0.389596	−0.194798	+	0.980843i	$0.562405\pi$
$390$	0	0
$391$	25.6172	1.29552
$392$	0	0
$393$	−9.68572	−0.488580
$394$	0	0
$395$	1.37587	0.0692276
$396$	0	0
$397$	12.1284	0.608706	0.304353	−	0.952559i	$-0.401560\pi$
$397$	12.1284	0.608706	0.304353	+	0.952559i	$0.401560\pi$
$398$	0	0
$399$	5.67341	0.284026
$400$	0	0
$401$	−37.5974	−1.87752	−0.938762	−	0.344568i	$-0.888026\pi$
$401$	−37.5974	−1.87752	−0.938762	+	0.344568i	$0.888026\pi$
$402$	0	0
$403$	8.59534	0.428164
$404$	0	0
$405$	−1.30045	−0.0646196
$406$	0	0
$407$	1.12947	0.0559857
$408$	0	0
$409$	−0.128039	−0.00633111	−0.00316556	−	0.999995i	$-0.501008\pi$
$409$	−0.128039	−0.00633111	−0.00316556	+	0.999995i	$0.501008\pi$
$410$	0	0
$411$	−18.6900	−0.921908
$412$	0	0
$413$	8.97728	0.441743
$414$	0	0
$415$	0.785160	0.0385420
$416$	0	0
$417$	9.29357	0.455108
$418$	0	0
$419$	−9.22168	−0.450508	−0.225254	−	0.974300i	$-0.572321\pi$
$419$	−9.22168	−0.450508	−0.225254	+	0.974300i	$0.572321\pi$
$420$	0	0
$421$	29.7816	1.45147	0.725734	−	0.687975i	$-0.241500\pi$
$421$	29.7816	1.45147	0.725734	+	0.687975i	$0.241500\pi$
$422$	0	0
$423$	1.65701	0.0805664
$424$	0	0
$425$	26.3528	1.27830
$426$	0	0
$427$	−8.46610	−0.409703
$428$	0	0
$429$	0.704278	0.0340029
$430$	0	0
$431$	10.5867	0.509945	0.254972	−	0.966948i	$-0.417934\pi$
$431$	10.5867	0.509945	0.254972	+	0.966948i	$0.417934\pi$
$432$	0	0
$433$	−17.5282	−0.842349	−0.421175	−	0.906980i	$-0.638382\pi$
$433$	−17.5282	−0.842349	−0.421175	+	0.906980i	$0.638382\pi$
$434$	0	0
$435$	−2.15114	−0.103139
$436$	0	0
$437$	−22.5354	−1.07801
$438$	0	0
$439$	−30.7944	−1.46974	−0.734868	−	0.678210i	$-0.762756\pi$
$439$	−30.7944	−1.46974	−0.734868	+	0.678210i	$0.762756\pi$
$440$	0	0
$441$	−2.35196	−0.111998
$442$	0	0
$443$	−7.90445	−0.375552	−0.187776	−	0.982212i	$-0.560128\pi$
$443$	−7.90445	−0.375552	−0.187776	+	0.982212i	$0.560128\pi$
$444$	0	0
$445$	−1.93124	−0.0915497
$446$	0	0
$447$	−25.0300	−1.18388
$448$	0	0
$449$	−7.36018	−0.347348	−0.173674	−	0.984803i	$-0.555564\pi$
$449$	−7.36018	−0.347348	−0.173674	+	0.984803i	$0.555564\pi$
$450$	0	0
$451$	1.74692	0.0822594
$452$	0	0
$453$	−3.34141	−0.156993
$454$	0	0
$455$	−0.171968	−0.00806198
$456$	0	0
$457$	22.1642	1.03680	0.518399	−	0.855139i	$-0.326528\pi$
$457$	22.1642	1.03680	0.518399	+	0.855139i	$0.326528\pi$
$458$	0	0
$459$	−25.6005	−1.19493
$460$	0	0
$461$	−13.1989	−0.614733	−0.307366	−	0.951591i	$-0.599448\pi$
$461$	−13.1989	−0.614733	−0.307366	+	0.951591i	$0.599448\pi$
$462$	0	0
$463$	16.6745	0.774931	0.387466	−	0.921884i	$-0.373351\pi$
$463$	16.6745	0.774931	0.387466	+	0.921884i	$0.373351\pi$
$464$	0	0
$465$	−1.04171	−0.0483082
$466$	0	0
$467$	13.3937	0.619787	0.309893	−	0.950771i	$-0.399707\pi$
$467$	13.3937	0.619787	0.309893	+	0.950771i	$0.399707\pi$
$468$	0	0
$469$	−4.44905	−0.205438
$470$	0	0
$471$	0.578490	0.0266554
$472$	0	0
$473$	1.32498	0.0609227
$474$	0	0
$475$	−23.1825	−1.06369
$476$	0	0
$477$	3.69507	0.169186
$478$	0	0
$479$	21.5971	0.986798	0.493399	−	0.869803i	$-0.335754\pi$
$479$	21.5971	0.986798	0.493399	+	0.869803i	$0.335754\pi$
$480$	0	0
$481$	−11.4863	−0.523730
$482$	0	0
$483$	−5.90682	−0.268770
$484$	0	0
$485$	0.605241	0.0274826
$486$	0	0
$487$	2.59597	0.117634	0.0588172	−	0.998269i	$-0.481267\pi$
$487$	2.59597	0.117634	0.0588172	+	0.998269i	$0.481267\pi$
$488$	0	0
$489$	−11.0720	−0.500693
$490$	0	0
$491$	23.5376	1.06224	0.531118	−	0.847298i	$-0.321772\pi$
$491$	23.5376	1.06224	0.531118	+	0.847298i	$0.321772\pi$
$492$	0	0
$493$	−47.4838	−2.13856
$494$	0	0
$495$	−0.00911525	−0.000409700 0
$496$	0	0
$497$	3.49914	0.156958
$498$	0	0
$499$	24.7023	1.10583	0.552914	−	0.833238i	$-0.313516\pi$
$499$	24.7023	1.10583	0.552914	+	0.833238i	$0.313516\pi$
$500$	0	0
$501$	−39.1851	−1.75066
$502$	0	0
$503$	−10.8849	−0.485333	−0.242666	−	0.970110i	$-0.578022\pi$
$503$	−10.8849	−0.485333	−0.242666	+	0.970110i	$0.578022\pi$
$504$	0	0
$505$	−2.30879	−0.102740
$506$	0	0
$507$	16.6625	0.740006
$508$	0	0
$509$	6.58906	0.292055	0.146028	−	0.989281i	$-0.453351\pi$
$509$	6.58906	0.292055	0.146028	+	0.989281i	$0.453351\pi$
$510$	0	0
$511$	4.81381	0.212950
$512$	0	0
$513$	22.5207	0.994315
$514$	0	0
$515$	2.01329	0.0887160
$516$	0	0
$517$	−0.898035	−0.0394955
$518$	0	0
$519$	11.9587	0.524929
$520$	0	0
$521$	25.2629	1.10679	0.553394	−	0.832919i	$-0.313332\pi$
$521$	25.2629	1.10679	0.553394	+	0.832919i	$0.313332\pi$
$522$	0	0
$523$	−2.95466	−0.129198	−0.0645990	−	0.997911i	$-0.520577\pi$
$523$	−2.95466	−0.129198	−0.0645990	+	0.997911i	$0.520577\pi$
$524$	0	0
$525$	−6.07644	−0.265198
$526$	0	0
$527$	−22.9945	−1.00166
$528$	0	0
$529$	0.462541	0.0201105
$530$	0	0
$531$	−4.83917	−0.210002
$532$	0	0
$533$	−17.7656	−0.769512
$534$	0	0
$535$	0.462488	0.0199951
$536$	0	0
$537$	29.6932	1.28136
$538$	0	0
$539$	1.27467	0.0549040
$540$	0	0
$541$	31.8713	1.37026	0.685128	−	0.728423i	$-0.259746\pi$
$541$	31.8713	1.37026	0.685128	+	0.728423i	$0.259746\pi$
$542$	0	0
$543$	27.6724	1.18754
$544$	0	0
$545$	−0.674231	−0.0288809
$546$	0	0
$547$	−39.4371	−1.68621	−0.843105	−	0.537749i	$-0.819275\pi$
$547$	−39.4371	−1.68621	−0.843105	+	0.537749i	$0.819275\pi$
$548$	0	0
$549$	4.56362	0.194771
$550$	0	0
$551$	41.7714	1.77952
$552$	0	0
$553$	−7.00289	−0.297793
$554$	0	0
$555$	1.39208	0.0590905
$556$	0	0
$557$	30.3533	1.28611	0.643055	−	0.765820i	$-0.277667\pi$
$557$	30.3533	1.28611	0.643055	+	0.765820i	$0.277667\pi$
$558$	0	0
$559$	−13.4746	−0.569914
$560$	0	0
$561$	−1.88411	−0.0795471
$562$	0	0
$563$	38.7866	1.63466	0.817331	−	0.576169i	$-0.195453\pi$
$563$	38.7866	1.63466	0.817331	+	0.576169i	$0.195453\pi$
$564$	0	0
$565$	1.44904	0.0609614
$566$	0	0
$567$	6.61898	0.277971
$568$	0	0
$569$	−8.97210	−0.376130	−0.188065	−	0.982157i	$-0.560222\pi$
$569$	−8.97210	−0.376130	−0.188065	+	0.982157i	$0.560222\pi$
$570$	0	0
$571$	33.1013	1.38525	0.692623	−	0.721300i	$-0.256455\pi$
$571$	33.1013	1.38525	0.692623	+	0.721300i	$0.256455\pi$
$572$	0	0
$573$	−18.1342	−0.757566
$574$	0	0
$575$	24.1363	1.00655
$576$	0	0
$577$	−24.6608	−1.02664	−0.513321	−	0.858196i	$-0.671585\pi$
$577$	−24.6608	−1.02664	−0.513321	+	0.858196i	$0.671585\pi$
$578$	0	0
$579$	−24.5649	−1.02088
$580$	0	0
$581$	−3.99629	−0.165794
$582$	0	0
$583$	−2.00259	−0.0829389
$584$	0	0
$585$	0.0926987	0.00383262
$586$	0	0
$587$	4.60869	0.190221	0.0951105	−	0.995467i	$-0.469680\pi$
$587$	4.60869	0.190221	0.0951105	+	0.995467i	$0.469680\pi$
$588$	0	0
$589$	20.2282	0.833490
$590$	0	0
$591$	48.4090	1.99128
$592$	0	0
$593$	31.5563	1.29586	0.647930	−	0.761700i	$-0.275635\pi$
$593$	31.5563	1.29586	0.647930	+	0.761700i	$0.275635\pi$
$594$	0	0
$595$	0.460054	0.0188604
$596$	0	0
$597$	−5.72243	−0.234204
$598$	0	0
$599$	31.8145	1.29990	0.649952	−	0.759975i	$-0.274789\pi$
$599$	31.8145	1.29990	0.649952	+	0.759975i	$0.274789\pi$
$600$	0	0
$601$	−46.5964	−1.90071	−0.950354	−	0.311170i	$-0.899279\pi$
$601$	−46.5964	−1.90071	−0.950354	+	0.311170i	$0.899279\pi$
$602$	0	0
$603$	2.39824	0.0976641
$604$	0	0
$605$	−1.43311	−0.0582644
$606$	0	0
$607$	46.8926	1.90331	0.951657	−	0.307164i	$-0.0993801\pi$
$607$	46.8926	1.90331	0.951657	+	0.307164i	$0.0993801\pi$
$608$	0	0
$609$	10.9488	0.443670
$610$	0	0
$611$	9.13268	0.369469
$612$	0	0
$613$	−20.6339	−0.833397	−0.416698	−	0.909045i	$-0.636813\pi$
$613$	−20.6339	−0.833397	−0.416698	+	0.909045i	$0.636813\pi$
$614$	0	0
$615$	2.15310	0.0868212
$616$	0	0
$617$	11.0579	0.445175	0.222587	−	0.974913i	$-0.428550\pi$
$617$	11.0579	0.445175	0.222587	+	0.974913i	$0.428550\pi$
$618$	0	0
$619$	−24.2019	−0.972758	−0.486379	−	0.873748i	$-0.661683\pi$
$619$	−24.2019	−0.972758	−0.486379	+	0.873748i	$0.661683\pi$
$620$	0	0
$621$	−23.4473	−0.940908
$622$	0	0
$623$	9.82961	0.393815
$624$	0	0
$625$	24.7439	0.989757
$626$	0	0
$627$	1.65745	0.0661920
$628$	0	0
$629$	30.7285	1.22522
$630$	0	0
$631$	5.83838	0.232422	0.116211	−	0.993225i	$-0.462925\pi$
$631$	5.83838	0.232422	0.116211	+	0.993225i	$0.462925\pi$
$632$	0	0
$633$	5.70007	0.226558
$634$	0	0
$635$	0.990487	0.0393063
$636$	0	0
$637$	−12.9630	−0.513611
$638$	0	0
$639$	−1.88620	−0.0746169
$640$	0	0
$641$	7.66620	0.302797	0.151398	−	0.988473i	$-0.451622\pi$
$641$	7.66620	0.302797	0.151398	+	0.988473i	$0.451622\pi$
$642$	0	0
$643$	4.55440	0.179608	0.0898040	−	0.995959i	$-0.471376\pi$
$643$	4.55440	0.179608	0.0898040	+	0.995959i	$0.471376\pi$
$644$	0	0
$645$	1.63305	0.0643013
$646$	0	0
$647$	18.0768	0.710671	0.355335	−	0.934739i	$-0.384367\pi$
$647$	18.0768	0.710671	0.355335	+	0.934739i	$0.384367\pi$
$648$	0	0
$649$	2.62265	0.102948
$650$	0	0
$651$	5.30209	0.207805
$652$	0	0
$653$	−15.1503	−0.592877	−0.296439	−	0.955052i	$-0.595799\pi$
$653$	−15.1503	−0.592877	−0.296439	+	0.955052i	$0.595799\pi$
$654$	0	0
$655$	0.690924	0.0269966
$656$	0	0
$657$	−2.59487	−0.101235
$658$	0	0
$659$	25.2205	0.982452	0.491226	−	0.871032i	$-0.336549\pi$
$659$	25.2205	0.982452	0.491226	+	0.871032i	$0.336549\pi$
$660$	0	0
$661$	8.22722	0.320002	0.160001	−	0.987117i	$-0.448850\pi$
$661$	8.22722	0.320002	0.160001	+	0.987117i	$0.448850\pi$
$662$	0	0
$663$	19.1607	0.744139
$664$	0	0
$665$	−0.404708	−0.0156939
$666$	0	0
$667$	−43.4900	−1.68394
$668$	0	0
$669$	−32.6154	−1.26098
$670$	0	0
$671$	−2.47331	−0.0954811
$672$	0	0
$673$	−14.6843	−0.566037	−0.283018	−	0.959114i	$-0.591336\pi$
$673$	−14.6843	−0.566037	−0.283018	+	0.959114i	$0.591336\pi$
$674$	0	0
$675$	−24.1206	−0.928403
$676$	0	0
$677$	−42.2263	−1.62289	−0.811445	−	0.584429i	$-0.801319\pi$
$677$	−42.2263	−1.62289	−0.811445	+	0.584429i	$0.801319\pi$
$678$	0	0
$679$	−3.08055	−0.118221
$680$	0	0
$681$	37.9212	1.45314
$682$	0	0
$683$	−14.8272	−0.567346	−0.283673	−	0.958921i	$-0.591553\pi$
$683$	−14.8272	−0.567346	−0.283673	+	0.958921i	$0.591553\pi$
$684$	0	0
$685$	1.33324	0.0509403
$686$	0	0
$687$	−26.5813	−1.01414
$688$	0	0
$689$	20.3656	0.775868
$690$	0	0
$691$	−47.8949	−1.82201	−0.911005	−	0.412396i	$-0.864692\pi$
$691$	−47.8949	−1.82201	−0.911005	+	0.412396i	$0.864692\pi$
$692$	0	0
$693$	0.0463946	0.00176239
$694$	0	0
$695$	−0.662951	−0.0251472
$696$	0	0
$697$	47.5270	1.80021
$698$	0	0
$699$	−15.1318	−0.572336
$700$	0	0
$701$	−52.0331	−1.96526	−0.982631	−	0.185569i	$-0.940587\pi$
$701$	−52.0331	−1.96526	−0.982631	+	0.185569i	$0.940587\pi$
$702$	0	0
$703$	−27.0318	−1.01952
$704$	0	0
$705$	−1.10684	−0.0416858
$706$	0	0
$707$	11.7512	0.441950
$708$	0	0
$709$	−3.32991	−0.125057	−0.0625287	−	0.998043i	$-0.519916\pi$
$709$	−3.32991	−0.125057	−0.0625287	+	0.998043i	$0.519916\pi$
$710$	0	0
$711$	3.77488	0.141569
$712$	0	0
$713$	−21.0605	−0.788721
$714$	0	0
$715$	−0.0502392	−0.00187884
$716$	0	0
$717$	−21.3098	−0.795827
$718$	0	0
$719$	−38.1038	−1.42103	−0.710516	−	0.703681i	$-0.751538\pi$
$719$	−38.1038	−1.42103	−0.710516	+	0.703681i	$0.751538\pi$
$720$	0	0
$721$	−10.2472	−0.381625
$722$	0	0
$723$	−25.6661	−0.954533
$724$	0	0
$725$	−44.7389	−1.66156
$726$	0	0
$727$	5.05611	0.187521	0.0937604	−	0.995595i	$-0.470111\pi$
$727$	5.05611	0.187521	0.0937604	+	0.995595i	$0.470111\pi$
$728$	0	0
$729$	23.0461	0.853559
$730$	0	0
$731$	36.0476	1.33327
$732$	0	0
$733$	−6.80089	−0.251197	−0.125598	−	0.992081i	$-0.540085\pi$
$733$	−6.80089	−0.251197	−0.125598	+	0.992081i	$0.540085\pi$
$734$	0	0
$735$	1.57104	0.0579489
$736$	0	0
$737$	−1.29976	−0.0478772
$738$	0	0
$739$	−24.4617	−0.899839	−0.449919	−	0.893069i	$-0.648547\pi$
$739$	−24.4617	−0.899839	−0.449919	+	0.893069i	$0.648547\pi$
$740$	0	0
$741$	−16.8556	−0.619206
$742$	0	0
$743$	22.4989	0.825404	0.412702	−	0.910866i	$-0.364585\pi$
$743$	22.4989	0.825404	0.412702	+	0.910866i	$0.364585\pi$
$744$	0	0
$745$	1.78550	0.0654155
$746$	0	0
$747$	2.15419	0.0788176
$748$	0	0
$749$	−2.35396	−0.0860119
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	−56.3364	−2.05301
$754$	0	0
$755$	0.238357	0.00867470
$756$	0	0
$757$	16.3800	0.595342	0.297671	−	0.954669i	$-0.403790\pi$
$757$	16.3800	0.595342	0.297671	+	0.954669i	$0.403790\pi$
$758$	0	0
$759$	−1.72564	−0.0626366
$760$	0	0
$761$	36.8712	1.33658	0.668291	−	0.743900i	$-0.267026\pi$
$761$	36.8712	1.33658	0.668291	+	0.743900i	$0.267026\pi$
$762$	0	0
$763$	3.43169	0.124235
$764$	0	0
$765$	−0.247990	−0.00896611
$766$	0	0
$767$	−26.6714	−0.963048
$768$	0	0
$769$	36.5525	1.31812	0.659058	−	0.752092i	$-0.270955\pi$
$769$	36.5525	1.31812	0.659058	+	0.752092i	$0.270955\pi$
$770$	0	0
$771$	−8.51439	−0.306638
$772$	0	0
$773$	8.00250	0.287830	0.143915	−	0.989590i	$-0.454031\pi$
$773$	8.00250	0.287830	0.143915	+	0.989590i	$0.454031\pi$
$774$	0	0
$775$	−21.6652	−0.778238
$776$	0	0
$777$	−7.08539	−0.254187
$778$	0	0
$779$	−41.8094	−1.49798
$780$	0	0
$781$	1.02225	0.0365789
$782$	0	0
$783$	43.4617	1.55320
$784$	0	0
$785$	−0.0412662	−0.00147285
$786$	0	0
$787$	−42.6884	−1.52168	−0.760838	−	0.648942i	$-0.775212\pi$
$787$	−42.6884	−1.52168	−0.760838	+	0.648942i	$0.775212\pi$
$788$	0	0
$789$	−21.7158	−0.773102
$790$	0	0
$791$	−7.37528	−0.262235
$792$	0	0
$793$	25.1527	0.893197
$794$	0	0
$795$	−2.46821	−0.0875384
$796$	0	0
$797$	−22.3425	−0.791411	−0.395706	−	0.918377i	$-0.629500\pi$
$797$	−22.3425	−0.791411	−0.395706	+	0.918377i	$0.629500\pi$
$798$	0	0
$799$	−24.4320	−0.864344
$800$	0	0
$801$	−5.29862	−0.187217
$802$	0	0
$803$	1.40632	0.0496280
$804$	0	0
$805$	0.421359	0.0148510
$806$	0	0
$807$	−26.6034	−0.936484
$808$	0	0
$809$	−3.36015	−0.118137	−0.0590684	−	0.998254i	$-0.518813\pi$
$809$	−3.36015	−0.118137	−0.0590684	+	0.998254i	$0.518813\pi$
$810$	0	0
$811$	20.7242	0.727726	0.363863	−	0.931452i	$-0.381458\pi$
$811$	20.7242	0.727726	0.363863	+	0.931452i	$0.381458\pi$
$812$	0	0
$813$	29.6305	1.03919
$814$	0	0
$815$	0.789813	0.0276660
$816$	0	0
$817$	−31.7110	−1.10943
$818$	0	0
$819$	−0.471816	−0.0164866
$820$	0	0
$821$	−18.3354	−0.639911	−0.319956	−	0.947433i	$-0.603668\pi$
$821$	−18.3354	−0.639911	−0.319956	+	0.947433i	$0.603668\pi$
$822$	0	0
$823$	−23.1478	−0.806882	−0.403441	−	0.915006i	$-0.632186\pi$
$823$	−23.1478	−0.806882	−0.403441	+	0.915006i	$0.632186\pi$
$824$	0	0
$825$	−1.77519	−0.0618042
$826$	0	0
$827$	11.5900	0.403023	0.201511	−	0.979486i	$-0.435415\pi$
$827$	11.5900	0.403023	0.201511	+	0.979486i	$0.435415\pi$
$828$	0	0
$829$	−9.41376	−0.326953	−0.163477	−	0.986547i	$-0.552271\pi$
$829$	−9.41376	−0.326953	−0.163477	+	0.986547i	$0.552271\pi$
$830$	0	0
$831$	−21.6775	−0.751983
$832$	0	0
$833$	34.6789	1.20155
$834$	0	0
$835$	2.79524	0.0967334
$836$	0	0
$837$	21.0468	0.727483
$838$	0	0
$839$	0.460102	0.0158845	0.00794225	−	0.999968i	$-0.497472\pi$
$839$	0.460102	0.0158845	0.00794225	+	0.999968i	$0.497472\pi$
$840$	0	0
$841$	51.6128	1.77975
$842$	0	0
$843$	−1.95997	−0.0675048
$844$	0	0
$845$	−1.18861	−0.0408893
$846$	0	0
$847$	7.29424	0.250633
$848$	0	0
$849$	−44.3191	−1.52103
$850$	0	0
$851$	28.1439	0.964761
$852$	0	0
$853$	42.5659	1.45743	0.728715	−	0.684817i	$-0.240118\pi$
$853$	42.5659	1.45743	0.728715	+	0.684817i	$0.240118\pi$
$854$	0	0
$855$	0.218157	0.00746080
$856$	0	0
$857$	5.48824	0.187475	0.0937374	−	0.995597i	$-0.470119\pi$
$857$	5.48824	0.187475	0.0937374	+	0.995597i	$0.470119\pi$
$858$	0	0
$859$	12.4623	0.425207	0.212604	−	0.977139i	$-0.431806\pi$
$859$	12.4623	0.425207	0.212604	+	0.977139i	$0.431806\pi$
$860$	0	0
$861$	−10.9588	−0.373475
$862$	0	0
$863$	−57.0707	−1.94271	−0.971354	−	0.237635i	$-0.923628\pi$
$863$	−57.0707	−1.94271	−0.971354	+	0.237635i	$0.923628\pi$
$864$	0	0
$865$	−0.853067	−0.0290051
$866$	0	0
$867$	−20.1038	−0.682762
$868$	0	0
$869$	−2.04584	−0.0694005
$870$	0	0
$871$	13.2181	0.447877
$872$	0	0
$873$	1.66056	0.0562014
$874$	0	0
$875$	0.868404	0.0293574
$876$	0	0
$877$	32.6751	1.10336	0.551679	−	0.834056i	$-0.313987\pi$
$877$	32.6751	1.10336	0.551679	+	0.834056i	$0.313987\pi$
$878$	0	0
$879$	25.8911	0.873286
$880$	0	0
$881$	−43.0023	−1.44879	−0.724393	−	0.689388i	$-0.757880\pi$
$881$	−43.0023	−1.44879	−0.724393	+	0.689388i	$0.757880\pi$
$882$	0	0
$883$	40.0283	1.34706	0.673530	−	0.739160i	$-0.264777\pi$
$883$	40.0283	1.34706	0.673530	+	0.739160i	$0.264777\pi$
$884$	0	0
$885$	3.23244	0.108657
$886$	0	0
$887$	−3.67563	−0.123416	−0.0617079	−	0.998094i	$-0.519655\pi$
$887$	−3.67563	−0.123416	−0.0617079	+	0.998094i	$0.519655\pi$
$888$	0	0
$889$	−5.04136	−0.169082
$890$	0	0
$891$	1.93369	0.0647810
$892$	0	0
$893$	21.4928	0.719230
$894$	0	0
$895$	−2.11814	−0.0708017
$896$	0	0
$897$	17.5491	0.585947
$898$	0	0
$899$	39.0375	1.30197
$900$	0	0
$901$	−54.4827	−1.81508
$902$	0	0
$903$	−8.31187	−0.276602
$904$	0	0
$905$	−1.97399	−0.0656176
$906$	0	0
$907$	18.3684	0.609913	0.304956	−	0.952366i	$-0.401358\pi$
$907$	18.3684	0.609913	0.304956	+	0.952366i	$0.401358\pi$
$908$	0	0
$909$	−6.33445	−0.210101
$910$	0	0
$911$	−21.8625	−0.724338	−0.362169	−	0.932112i	$-0.617964\pi$
$911$	−21.8625	−0.724338	−0.362169	+	0.932112i	$0.617964\pi$
$912$	0	0
$913$	−1.16749	−0.0386382
$914$	0	0
$915$	−3.04838	−0.100776
$916$	0	0
$917$	−3.51665	−0.116130
$918$	0	0
$919$	25.2165	0.831817	0.415908	−	0.909406i	$-0.363464\pi$
$919$	25.2165	0.831817	0.415908	+	0.909406i	$0.363464\pi$
$920$	0	0
$921$	−58.8732	−1.93994
$922$	0	0
$923$	−10.3959	−0.342185
$924$	0	0
$925$	28.9521	0.951939
$926$	0	0
$927$	5.52371	0.181422
$928$	0	0
$929$	34.1401	1.12010	0.560051	−	0.828458i	$-0.310782\pi$
$929$	34.1401	1.12010	0.560051	+	0.828458i	$0.310782\pi$
$930$	0	0
$931$	−30.5070	−0.999825
$932$	0	0
$933$	−1.23588	−0.0404607
$934$	0	0
$935$	0.134401	0.00439540
$936$	0	0
$937$	−41.8920	−1.36855	−0.684276	−	0.729223i	$-0.739882\pi$
$937$	−41.8920	−1.36855	−0.684276	+	0.729223i	$0.739882\pi$
$938$	0	0
$939$	−31.1814	−1.01757
$940$	0	0
$941$	32.8495	1.07086	0.535432	−	0.844578i	$-0.320149\pi$
$941$	32.8495	1.07086	0.535432	+	0.844578i	$0.320149\pi$
$942$	0	0
$943$	43.5295	1.41752
$944$	0	0
$945$	−0.421085	−0.0136979
$946$	0	0
$947$	41.1594	1.33750	0.668751	−	0.743487i	$-0.266829\pi$
$947$	41.1594	1.33750	0.668751	+	0.743487i	$0.266829\pi$
$948$	0	0
$949$	−14.3018	−0.464255
$950$	0	0
$951$	−64.5178	−2.09213
$952$	0	0
$953$	−4.92982	−0.159693	−0.0798463	−	0.996807i	$-0.525443\pi$
$953$	−4.92982	−0.159693	−0.0798463	+	0.996807i	$0.525443\pi$
$954$	0	0
$955$	1.29359	0.0418596
$956$	0	0
$957$	3.19863	0.103397
$958$	0	0
$959$	−6.78588	−0.219127
$960$	0	0
$961$	−12.0957	−0.390184
$962$	0	0
$963$	1.26889	0.0408896
$964$	0	0
$965$	1.75232	0.0564092
$966$	0	0
$967$	−34.5330	−1.11051	−0.555253	−	0.831681i	$-0.687379\pi$
$967$	−34.5330	−1.11051	−0.555253	+	0.831681i	$0.687379\pi$
$968$	0	0
$969$	45.0926	1.44858
$970$	0	0
$971$	21.7410	0.697701	0.348850	−	0.937178i	$-0.386572\pi$
$971$	21.7410	0.697701	0.348850	+	0.937178i	$0.386572\pi$
$972$	0	0
$973$	3.37427	0.108174
$974$	0	0
$975$	18.0530	0.578159
$976$	0	0
$977$	−26.8806	−0.859985	−0.429993	−	0.902832i	$-0.641484\pi$
$977$	−26.8806	−0.859985	−0.429993	+	0.902832i	$0.641484\pi$
$978$	0	0
$979$	2.87165	0.0917783
$980$	0	0
$981$	−1.84984	−0.0590608
$982$	0	0
$983$	10.2716	0.327613	0.163807	−	0.986492i	$-0.447623\pi$
$983$	10.2716	0.327613	0.163807	+	0.986492i	$0.447623\pi$
$984$	0	0
$985$	−3.45322	−0.110029
$986$	0	0
$987$	5.63355	0.179318
$988$	0	0
$989$	33.0157	1.04984
$990$	0	0
$991$	−12.0132	−0.381612	−0.190806	−	0.981628i	$-0.561110\pi$
$991$	−12.0132	−0.381612	−0.190806	+	0.981628i	$0.561110\pi$
$992$	0	0
$993$	−18.2658	−0.579647
$994$	0	0
$995$	0.408206	0.0129410
$996$	0	0
$997$	17.3475	0.549400	0.274700	−	0.961530i	$-0.411421\pi$
$997$	17.3475	0.549400	0.274700	+	0.961530i	$0.411421\pi$
$998$	0	0
$999$	−28.1256	−0.889856

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.d.1.11	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.d.1.11	✓	49	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.83267 q^{3} +0.130732 q^{5} -0.665399 q^{7} +0.358681 q^{9} +O(q^{10})\)	\(q-1.83267 q^{3} +0.130732 q^{5} -0.665399 q^{7} +0.358681 q^{9} -0.194392 q^{11} +1.97689 q^{13} -0.239589 q^{15} -5.28864 q^{17} +4.65240 q^{19} +1.21946 q^{21} -4.84381 q^{23} -4.98291 q^{25} +4.84067 q^{27} +8.97846 q^{29} +4.34791 q^{31} +0.356256 q^{33} -0.0869891 q^{35} -5.81028 q^{37} -3.62299 q^{39} -8.98662 q^{41} -6.81604 q^{43} +0.0468912 q^{45} +4.61972 q^{47} -6.55724 q^{49} +9.69233 q^{51} +10.3018 q^{53} -0.0254132 q^{55} -8.52632 q^{57} -13.4916 q^{59} +12.7233 q^{61} -0.238666 q^{63} +0.258443 q^{65} +6.68629 q^{67} +8.87712 q^{69} -5.25871 q^{71} -7.23447 q^{73} +9.13203 q^{75} +0.129348 q^{77} +10.5243 q^{79} -9.94739 q^{81} +6.00586 q^{83} -0.691396 q^{85} -16.4546 q^{87} -14.7725 q^{89} -1.31542 q^{91} -7.96828 q^{93} +0.608219 q^{95} +4.62962 q^{97} -0.0697245 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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