LMFDB - Embedded newform 6008.2.a.e.1.22

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:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
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)

$f(q)$	$=$	$q-0.410037 q^{3} +0.180422 q^{5} -3.49245 q^{7} -2.83187 q^{9} +O(q^{10})$	$q-0.410037 q^{3} +0.180422 q^{5} -3.49245 q^{7} -2.83187 q^{9} -2.01646 q^{11} -5.49316 q^{13} -0.0739798 q^{15} -3.92356 q^{17} -4.57585 q^{19} +1.43203 q^{21} +0.277604 q^{23} -4.96745 q^{25} +2.39128 q^{27} -3.41534 q^{29} -5.41640 q^{31} +0.826824 q^{33} -0.630115 q^{35} -2.83207 q^{37} +2.25240 q^{39} +2.24221 q^{41} -0.371008 q^{43} -0.510932 q^{45} +3.31206 q^{47} +5.19719 q^{49} +1.60880 q^{51} +13.6411 q^{53} -0.363814 q^{55} +1.87627 q^{57} +0.754139 q^{59} -1.25383 q^{61} +9.89016 q^{63} -0.991087 q^{65} -4.39453 q^{67} -0.113828 q^{69} -5.03532 q^{71} -0.365337 q^{73} +2.03684 q^{75} +7.04238 q^{77} -0.408161 q^{79} +7.51509 q^{81} -7.26930 q^{83} -0.707897 q^{85} +1.40041 q^{87} +8.52128 q^{89} +19.1846 q^{91} +2.22093 q^{93} -0.825584 q^{95} +11.6354 q^{97} +5.71035 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * 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
$3$	−0.410037	−0.236735	−0.118368	−	0.992970i	$-0.537766\pi$
$3$	−0.410037	−0.236735	−0.118368	+	0.992970i	$0.537766\pi$
$4$	0	0
$5$	0.180422	0.0806872	0.0403436	−	0.999186i	$-0.487155\pi$
$5$	0.180422	0.0806872	0.0403436	+	0.999186i	$0.487155\pi$
$6$	0	0
$7$	−3.49245	−1.32002	−0.660010	−	0.751256i	$-0.729448\pi$
$7$	−3.49245	−1.32002	−0.660010	+	0.751256i	$0.729448\pi$
$8$	0	0
$9$	−2.83187	−0.943957
$10$	0	0
$11$	−2.01646	−0.607986	−0.303993	−	0.952674i	$-0.598320\pi$
$11$	−2.01646	−0.607986	−0.303993	+	0.952674i	$0.598320\pi$
$12$	0	0
$13$	−5.49316	−1.52353	−0.761764	−	0.647855i	$-0.775666\pi$
$13$	−5.49316	−1.52353	−0.761764	+	0.647855i	$0.775666\pi$
$14$	0	0
$15$	−0.0739798	−0.0191015
$16$	0	0
$17$	−3.92356	−0.951603	−0.475801	−	0.879553i	$-0.657842\pi$
$17$	−3.92356	−0.951603	−0.475801	+	0.879553i	$0.657842\pi$
$18$	0	0
$19$	−4.57585	−1.04977	−0.524886	−	0.851173i	$-0.675892\pi$
$19$	−4.57585	−1.04977	−0.524886	+	0.851173i	$0.675892\pi$
$20$	0	0
$21$	1.43203	0.312495
$22$	0	0
$23$	0.277604	0.0578844	0.0289422	−	0.999581i	$-0.490786\pi$
$23$	0.277604	0.0578844	0.0289422	+	0.999581i	$0.490786\pi$
$24$	0	0
$25$	−4.96745	−0.993490
$26$	0	0
$27$	2.39128	0.460203
$28$	0	0
$29$	−3.41534	−0.634212	−0.317106	−	0.948390i	$-0.602711\pi$
$29$	−3.41534	−0.634212	−0.317106	+	0.948390i	$0.602711\pi$
$30$	0	0
$31$	−5.41640	−0.972815	−0.486407	−	0.873732i	$-0.661693\pi$
$31$	−5.41640	−0.972815	−0.486407	+	0.873732i	$0.661693\pi$
$32$	0	0
$33$	0.826824	0.143932
$34$	0	0
$35$	−0.630115	−0.106509
$36$	0	0
$37$	−2.83207	−0.465590	−0.232795	−	0.972526i	$-0.574787\pi$
$37$	−2.83207	−0.465590	−0.232795	+	0.972526i	$0.574787\pi$
$38$	0	0
$39$	2.25240	0.360672
$40$	0	0
$41$	2.24221	0.350174	0.175087	−	0.984553i	$-0.443979\pi$
$41$	2.24221	0.350174	0.175087	+	0.984553i	$0.443979\pi$
$42$	0	0
$43$	−0.371008	−0.0565781	−0.0282891	−	0.999600i	$-0.509006\pi$
$43$	−0.371008	−0.0565781	−0.0282891	+	0.999600i	$0.509006\pi$
$44$	0	0
$45$	−0.510932	−0.0761652
$46$	0	0
$47$	3.31206	0.483114	0.241557	−	0.970387i	$-0.422342\pi$
$47$	3.31206	0.483114	0.241557	+	0.970387i	$0.422342\pi$
$48$	0	0
$49$	5.19719	0.742455
$50$	0	0
$51$	1.60880	0.225278
$52$	0	0
$53$	13.6411	1.87375	0.936877	−	0.349658i	$-0.113702\pi$
$53$	13.6411	1.87375	0.936877	+	0.349658i	$0.113702\pi$
$54$	0	0
$55$	−0.363814	−0.0490567
$56$	0	0
$57$	1.87627	0.248518
$58$	0	0
$59$	0.754139	0.0981805	0.0490902	−	0.998794i	$-0.484368\pi$
$59$	0.754139	0.0981805	0.0490902	+	0.998794i	$0.484368\pi$
$60$	0	0
$61$	−1.25383	−0.160537	−0.0802683	−	0.996773i	$-0.525578\pi$
$61$	−1.25383	−0.160537	−0.0802683	+	0.996773i	$0.525578\pi$
$62$	0	0
$63$	9.89016	1.24604
$64$	0	0
$65$	−0.991087	−0.122929
$66$	0	0
$67$	−4.39453	−0.536877	−0.268439	−	0.963297i	$-0.586508\pi$
$67$	−4.39453	−0.536877	−0.268439	+	0.963297i	$0.586508\pi$
$68$	0	0
$69$	−0.113828	−0.0137033
$70$	0	0
$71$	−5.03532	−0.597583	−0.298791	−	0.954318i	$-0.596584\pi$
$71$	−5.03532	−0.597583	−0.298791	+	0.954318i	$0.596584\pi$
$72$	0	0
$73$	−0.365337	−0.0427595	−0.0213797	−	0.999771i	$-0.506806\pi$
$73$	−0.365337	−0.0427595	−0.0213797	+	0.999771i	$0.506806\pi$
$74$	0	0
$75$	2.03684	0.235194
$76$	0	0
$77$	7.04238	0.802554
$78$	0	0
$79$	−0.408161	−0.0459216	−0.0229608	−	0.999736i	$-0.507309\pi$
$79$	−0.408161	−0.0459216	−0.0229608	+	0.999736i	$0.507309\pi$
$80$	0	0
$81$	7.51509	0.835010
$82$	0	0
$83$	−7.26930	−0.797909	−0.398955	−	0.916971i	$-0.630627\pi$
$83$	−7.26930	−0.797909	−0.398955	+	0.916971i	$0.630627\pi$
$84$	0	0
$85$	−0.707897	−0.0767822
$86$	0	0
$87$	1.40041	0.150140
$88$	0	0
$89$	8.52128	0.903254	0.451627	−	0.892207i	$-0.350844\pi$
$89$	8.52128	0.903254	0.451627	+	0.892207i	$0.350844\pi$
$90$	0	0
$91$	19.1846	2.01109
$92$	0	0
$93$	2.22093	0.230299
$94$	0	0
$95$	−0.825584	−0.0847031
$96$	0	0
$97$	11.6354	1.18139	0.590697	−	0.806893i	$-0.298853\pi$
$97$	11.6354	1.18139	0.590697	+	0.806893i	$0.298853\pi$
$98$	0	0
$99$	5.71035	0.573912
$100$	0	0
$101$	−6.23326	−0.620232	−0.310116	−	0.950699i	$-0.600368\pi$
$101$	−6.23326	−0.620232	−0.310116	+	0.950699i	$0.600368\pi$
$102$	0	0
$103$	−13.0353	−1.28440	−0.642202	−	0.766535i	$-0.721979\pi$
$103$	−13.0353	−1.28440	−0.642202	+	0.766535i	$0.721979\pi$
$104$	0	0
$105$	0.258370	0.0252144
$106$	0	0
$107$	−5.24662	−0.507210	−0.253605	−	0.967308i	$-0.581616\pi$
$107$	−5.24662	−0.507210	−0.253605	+	0.967308i	$0.581616\pi$
$108$	0	0
$109$	−0.624958	−0.0598602	−0.0299301	−	0.999552i	$-0.509528\pi$
$109$	−0.624958	−0.0598602	−0.0299301	+	0.999552i	$0.509528\pi$
$110$	0	0
$111$	1.16125	0.110221
$112$	0	0
$113$	4.57500	0.430380	0.215190	−	0.976572i	$-0.430963\pi$
$113$	4.57500	0.430380	0.215190	+	0.976572i	$0.430963\pi$
$114$	0	0
$115$	0.0500859	0.00467053
$116$	0	0
$117$	15.5559	1.43814
$118$	0	0
$119$	13.7028	1.25614
$120$	0	0
$121$	−6.93389	−0.630353
$122$	0	0
$123$	−0.919387	−0.0828984
$124$	0	0
$125$	−1.79835	−0.160849
$126$	0	0
$127$	−18.3317	−1.62668	−0.813339	−	0.581790i	$-0.802352\pi$
$127$	−18.3317	−1.62668	−0.813339	+	0.581790i	$0.802352\pi$
$128$	0	0
$129$	0.152127	0.0133940
$130$	0	0
$131$	−1.97073	−0.172183	−0.0860916	−	0.996287i	$-0.527438\pi$
$131$	−1.97073	−0.172183	−0.0860916	+	0.996287i	$0.527438\pi$
$132$	0	0
$133$	15.9809	1.38572
$134$	0	0
$135$	0.431440	0.0371325
$136$	0	0
$137$	4.92357	0.420649	0.210324	−	0.977632i	$-0.432548\pi$
$137$	4.92357	0.420649	0.210324	+	0.977632i	$0.432548\pi$
$138$	0	0
$139$	1.08412	0.0919537	0.0459768	−	0.998943i	$-0.485360\pi$
$139$	1.08412	0.0919537	0.0459768	+	0.998943i	$0.485360\pi$
$140$	0	0
$141$	−1.35807	−0.114370
$142$	0	0
$143$	11.0767	0.926283
$144$	0	0
$145$	−0.616202	−0.0511728
$146$	0	0
$147$	−2.13104	−0.175765
$148$	0	0
$149$	21.0672	1.72589	0.862946	−	0.505296i	$-0.168617\pi$
$149$	21.0672	1.72589	0.862946	+	0.505296i	$0.168617\pi$
$150$	0	0
$151$	−10.9313	−0.889579	−0.444790	−	0.895635i	$-0.646722\pi$
$151$	−10.9313	−0.889579	−0.444790	+	0.895635i	$0.646722\pi$
$152$	0	0
$153$	11.1110	0.898272
$154$	0	0
$155$	−0.977239	−0.0784937
$156$	0	0
$157$	−6.84675	−0.546430	−0.273215	−	0.961953i	$-0.588087\pi$
$157$	−6.84675	−0.546430	−0.273215	+	0.961953i	$0.588087\pi$
$158$	0	0
$159$	−5.59337	−0.443583
$160$	0	0
$161$	−0.969517	−0.0764086
$162$	0	0
$163$	−7.77039	−0.608624	−0.304312	−	0.952572i	$-0.598426\pi$
$163$	−7.77039	−0.608624	−0.304312	+	0.952572i	$0.598426\pi$
$164$	0	0
$165$	0.149177	0.0116134
$166$	0	0
$167$	5.91472	0.457695	0.228847	−	0.973462i	$-0.426504\pi$
$167$	5.91472	0.457695	0.228847	+	0.973462i	$0.426504\pi$
$168$	0	0
$169$	17.1748	1.32114
$170$	0	0
$171$	12.9582	0.990938
$172$	0	0
$173$	11.9324	0.907203	0.453602	−	0.891205i	$-0.350139\pi$
$173$	11.9324	0.907203	0.453602	+	0.891205i	$0.350139\pi$
$174$	0	0
$175$	17.3485	1.31143
$176$	0	0
$177$	−0.309225	−0.0232428
$178$	0	0
$179$	−5.53752	−0.413894	−0.206947	−	0.978352i	$-0.566353\pi$
$179$	−5.53752	−0.413894	−0.206947	+	0.978352i	$0.566353\pi$
$180$	0	0
$181$	5.89037	0.437828	0.218914	−	0.975744i	$-0.429749\pi$
$181$	5.89037	0.437828	0.218914	+	0.975744i	$0.429749\pi$
$182$	0	0
$183$	0.514117	0.0380046
$184$	0	0
$185$	−0.510969	−0.0375672
$186$	0	0
$187$	7.91170	0.578561
$188$	0	0
$189$	−8.35143	−0.607477
$190$	0	0
$191$	23.1837	1.67751	0.838756	−	0.544507i	$-0.183283\pi$
$191$	23.1837	1.67751	0.838756	+	0.544507i	$0.183283\pi$
$192$	0	0
$193$	−3.24077	−0.233276	−0.116638	−	0.993175i	$-0.537212\pi$
$193$	−3.24077	−0.233276	−0.116638	+	0.993175i	$0.537212\pi$
$194$	0	0
$195$	0.406383	0.0291017
$196$	0	0
$197$	−12.1180	−0.863371	−0.431686	−	0.902024i	$-0.642081\pi$
$197$	−12.1180	−0.863371	−0.431686	+	0.902024i	$0.642081\pi$
$198$	0	0
$199$	−6.10518	−0.432785	−0.216392	−	0.976306i	$-0.569429\pi$
$199$	−6.10518	−0.432785	−0.216392	+	0.976306i	$0.569429\pi$
$200$	0	0
$201$	1.80192	0.127098
$202$	0	0
$203$	11.9279	0.837173
$204$	0	0
$205$	0.404544	0.0282545
$206$	0	0
$207$	−0.786138	−0.0546404
$208$	0	0
$209$	9.22701	0.638246
$210$	0	0
$211$	−14.3433	−0.987435	−0.493718	−	0.869622i	$-0.664362\pi$
$211$	−14.3433	−0.987435	−0.493718	+	0.869622i	$0.664362\pi$
$212$	0	0
$213$	2.06467	0.141469
$214$	0	0
$215$	−0.0669380	−0.00456513
$216$	0	0
$217$	18.9165	1.28414
$218$	0	0
$219$	0.149802	0.0101227
$220$	0	0
$221$	21.5527	1.44979
$222$	0	0
$223$	0.0622068	0.00416568	0.00208284	−	0.999998i	$-0.499337\pi$
$223$	0.0622068	0.00416568	0.00208284	+	0.999998i	$0.499337\pi$
$224$	0	0
$225$	14.0672	0.937811
$226$	0	0
$227$	−23.2565	−1.54359	−0.771795	−	0.635871i	$-0.780641\pi$
$227$	−23.2565	−1.54359	−0.771795	+	0.635871i	$0.780641\pi$
$228$	0	0
$229$	−19.6069	−1.29566	−0.647830	−	0.761785i	$-0.724323\pi$
$229$	−19.6069	−1.29566	−0.647830	+	0.761785i	$0.724323\pi$
$230$	0	0
$231$	−2.88764	−0.189993
$232$	0	0
$233$	−17.1423	−1.12303	−0.561515	−	0.827466i	$-0.689781\pi$
$233$	−17.1423	−1.12303	−0.561515	+	0.827466i	$0.689781\pi$
$234$	0	0
$235$	0.597569	0.0389811
$236$	0	0
$237$	0.167361	0.0108713
$238$	0	0
$239$	−7.64249	−0.494351	−0.247176	−	0.968971i	$-0.579502\pi$
$239$	−7.64249	−0.494351	−0.247176	+	0.968971i	$0.579502\pi$
$240$	0	0
$241$	−6.82941	−0.439921	−0.219961	−	0.975509i	$-0.570593\pi$
$241$	−6.82941	−0.439921	−0.219961	+	0.975509i	$0.570593\pi$
$242$	0	0
$243$	−10.2553	−0.657879
$244$	0	0
$245$	0.937688	0.0599067
$246$	0	0
$247$	25.1358	1.59936
$248$	0	0
$249$	2.98068	0.188893
$250$	0	0
$251$	−20.3213	−1.28267	−0.641333	−	0.767262i	$-0.721618\pi$
$251$	−20.3213	−1.28267	−0.641333	+	0.767262i	$0.721618\pi$
$252$	0	0
$253$	−0.559777	−0.0351929
$254$	0	0
$255$	0.290264	0.0181770
$256$	0	0
$257$	11.2765	0.703411	0.351705	−	0.936111i	$-0.385602\pi$
$257$	11.2765	0.703411	0.351705	+	0.936111i	$0.385602\pi$
$258$	0	0
$259$	9.89087	0.614588
$260$	0	0
$261$	9.67179	0.598669
$262$	0	0
$263$	−27.2337	−1.67930	−0.839650	−	0.543127i	$-0.817240\pi$
$263$	−27.2337	−1.67930	−0.839650	+	0.543127i	$0.817240\pi$
$264$	0	0
$265$	2.46116	0.151188
$266$	0	0
$267$	−3.49404	−0.213832
$268$	0	0
$269$	11.4551	0.698432	0.349216	−	0.937042i	$-0.386448\pi$
$269$	11.4551	0.698432	0.349216	+	0.937042i	$0.386448\pi$
$270$	0	0
$271$	−14.7846	−0.898103	−0.449052	−	0.893506i	$-0.648238\pi$
$271$	−14.7846	−0.898103	−0.449052	+	0.893506i	$0.648238\pi$
$272$	0	0
$273$	−7.86638	−0.476095
$274$	0	0
$275$	10.0167	0.604028
$276$	0	0
$277$	−8.47888	−0.509446	−0.254723	−	0.967014i	$-0.581984\pi$
$277$	−8.47888	−0.509446	−0.254723	+	0.967014i	$0.581984\pi$
$278$	0	0
$279$	15.3385	0.918295
$280$	0	0
$281$	−11.7238	−0.699384	−0.349692	−	0.936865i	$-0.613714\pi$
$281$	−11.7238	−0.699384	−0.349692	+	0.936865i	$0.613714\pi$
$282$	0	0
$283$	−6.23273	−0.370497	−0.185249	−	0.982692i	$-0.559309\pi$
$283$	−6.23273	−0.370497	−0.185249	+	0.982692i	$0.559309\pi$
$284$	0	0
$285$	0.338520	0.0200522
$286$	0	0
$287$	−7.83078	−0.462237
$288$	0	0
$289$	−1.60569	−0.0944522
$290$	0	0
$291$	−4.77094	−0.279677
$292$	0	0
$293$	−11.9895	−0.700436	−0.350218	−	0.936668i	$-0.613893\pi$
$293$	−11.9895	−0.700436	−0.350218	+	0.936668i	$0.613893\pi$
$294$	0	0
$295$	0.136063	0.00792191
$296$	0	0
$297$	−4.82193	−0.279797
$298$	0	0
$299$	−1.52492	−0.0881885
$300$	0	0
$301$	1.29572	0.0746843
$302$	0	0
$303$	2.55587	0.146831
$304$	0	0
$305$	−0.226219	−0.0129533
$306$	0	0
$307$	11.1016	0.633601	0.316800	−	0.948492i	$-0.397391\pi$
$307$	11.1016	0.633601	0.316800	+	0.948492i	$0.397391\pi$
$308$	0	0
$309$	5.34495	0.304064
$310$	0	0
$311$	10.9242	0.619456	0.309728	−	0.950825i	$-0.399762\pi$
$311$	10.9242	0.619456	0.309728	+	0.950825i	$0.399762\pi$
$312$	0	0
$313$	−5.27447	−0.298131	−0.149065	−	0.988827i	$-0.547626\pi$
$313$	−5.27447	−0.298131	−0.149065	+	0.988827i	$0.547626\pi$
$314$	0	0
$315$	1.78440	0.100540
$316$	0	0
$317$	13.8124	0.775782	0.387891	−	0.921705i	$-0.373204\pi$
$317$	13.8124	0.775782	0.387891	+	0.921705i	$0.373204\pi$
$318$	0	0
$319$	6.88689	0.385592
$320$	0	0
$321$	2.15131	0.120074
$322$	0	0
$323$	17.9536	0.998965
$324$	0	0
$325$	27.2870	1.51361
$326$	0	0
$327$	0.256256	0.0141710
$328$	0	0
$329$	−11.5672	−0.637720
$330$	0	0
$331$	1.03867	0.0570904	0.0285452	−	0.999593i	$-0.490913\pi$
$331$	1.03867	0.0570904	0.0285452	+	0.999593i	$0.490913\pi$
$332$	0	0
$333$	8.02006	0.439497
$334$	0	0
$335$	−0.792871	−0.0433192
$336$	0	0
$337$	−9.88970	−0.538727	−0.269363	−	0.963039i	$-0.586813\pi$
$337$	−9.88970	−0.538727	−0.269363	+	0.963039i	$0.586813\pi$
$338$	0	0
$339$	−1.87592	−0.101886
$340$	0	0
$341$	10.9220	0.591458
$342$	0	0
$343$	6.29623	0.339964
$344$	0	0
$345$	−0.0205371	−0.00110568
$346$	0	0
$347$	19.3954	1.04120	0.520600	−	0.853801i	$-0.325708\pi$
$347$	19.3954	1.04120	0.520600	+	0.853801i	$0.325708\pi$
$348$	0	0
$349$	7.49915	0.401420	0.200710	−	0.979651i	$-0.435675\pi$
$349$	7.49915	0.401420	0.200710	+	0.979651i	$0.435675\pi$
$350$	0	0
$351$	−13.1357	−0.701132
$352$	0	0
$353$	9.08893	0.483755	0.241878	−	0.970307i	$-0.422237\pi$
$353$	9.08893	0.483755	0.241878	+	0.970307i	$0.422237\pi$
$354$	0	0
$355$	−0.908484	−0.0482173
$356$	0	0
$357$	−5.61867	−0.297371
$358$	0	0
$359$	0.102206	0.00539424	0.00269712	−	0.999996i	$-0.499141\pi$
$359$	0.102206	0.00539424	0.00269712	+	0.999996i	$0.499141\pi$
$360$	0	0
$361$	1.93837	0.102019
$362$	0	0
$363$	2.84315	0.149227
$364$	0	0
$365$	−0.0659149	−0.00345015
$366$	0	0
$367$	22.8350	1.19198	0.595988	−	0.802993i	$-0.296760\pi$
$367$	22.8350	1.19198	0.595988	+	0.802993i	$0.296760\pi$
$368$	0	0
$369$	−6.34963	−0.330549
$370$	0	0
$371$	−47.6410	−2.47340
$372$	0	0
$373$	7.10713	0.367993	0.183997	−	0.982927i	$-0.441096\pi$
$373$	7.10713	0.367993	0.183997	+	0.982927i	$0.441096\pi$
$374$	0	0
$375$	0.737389	0.0380786
$376$	0	0
$377$	18.7610	0.966240
$378$	0	0
$379$	13.3809	0.687329	0.343665	−	0.939092i	$-0.388332\pi$
$379$	13.3809	0.687329	0.343665	+	0.939092i	$0.388332\pi$
$380$	0	0
$381$	7.51669	0.385092
$382$	0	0
$383$	−28.7677	−1.46996	−0.734980	−	0.678089i	$-0.762808\pi$
$383$	−28.7677	−1.46996	−0.734980	+	0.678089i	$0.762808\pi$
$384$	0	0
$385$	1.27060	0.0647559
$386$	0	0
$387$	1.05065	0.0534073
$388$	0	0
$389$	17.2013	0.872142	0.436071	−	0.899912i	$-0.356370\pi$
$389$	17.2013	0.872142	0.436071	+	0.899912i	$0.356370\pi$
$390$	0	0
$391$	−1.08920	−0.0550830
$392$	0	0
$393$	0.808071	0.0407618
$394$	0	0
$395$	−0.0736412	−0.00370529
$396$	0	0
$397$	−13.9670	−0.700985	−0.350493	−	0.936565i	$-0.613986\pi$
$397$	−13.9670	−0.700985	−0.350493	+	0.936565i	$0.613986\pi$
$398$	0	0
$399$	−6.55276	−0.328048
$400$	0	0
$401$	−26.9062	−1.34363	−0.671815	−	0.740719i	$-0.734485\pi$
$401$	−26.9062	−1.34363	−0.671815	+	0.740719i	$0.734485\pi$
$402$	0	0
$403$	29.7532	1.48211
$404$	0	0
$405$	1.35589	0.0673747
$406$	0	0
$407$	5.71076	0.283072
$408$	0	0
$409$	23.1755	1.14596	0.572978	−	0.819571i	$-0.305788\pi$
$409$	23.1755	1.14596	0.572978	+	0.819571i	$0.305788\pi$
$410$	0	0
$411$	−2.01885	−0.0995823
$412$	0	0
$413$	−2.63379	−0.129600
$414$	0	0
$415$	−1.31154	−0.0643811
$416$	0	0
$417$	−0.444529	−0.0217687
$418$	0	0
$419$	−17.7164	−0.865502	−0.432751	−	0.901514i	$-0.642457\pi$
$419$	−17.7164	−0.865502	−0.432751	+	0.901514i	$0.642457\pi$
$420$	0	0
$421$	−23.2412	−1.13271	−0.566354	−	0.824162i	$-0.691647\pi$
$421$	−23.2412	−1.13271	−0.566354	+	0.824162i	$0.691647\pi$
$422$	0	0
$423$	−9.37933	−0.456039
$424$	0	0
$425$	19.4901	0.945407
$426$	0	0
$427$	4.37894	0.211912
$428$	0	0
$429$	−4.54187	−0.219284
$430$	0	0
$431$	19.8001	0.953737	0.476868	−	0.878975i	$-0.341772\pi$
$431$	19.8001	0.953737	0.476868	+	0.878975i	$0.341772\pi$
$432$	0	0
$433$	16.6762	0.801409	0.400705	−	0.916207i	$-0.368765\pi$
$433$	16.6762	0.801409	0.400705	+	0.916207i	$0.368765\pi$
$434$	0	0
$435$	0.252666	0.0121144
$436$	0	0
$437$	−1.27027	−0.0607654
$438$	0	0
$439$	−26.6952	−1.27409	−0.637046	−	0.770826i	$-0.719844\pi$
$439$	−26.6952	−1.27409	−0.637046	+	0.770826i	$0.719844\pi$
$440$	0	0
$441$	−14.7178	−0.700846
$442$	0	0
$443$	−10.8551	−0.515743	−0.257871	−	0.966179i	$-0.583021\pi$
$443$	−10.8551	−0.515743	−0.257871	+	0.966179i	$0.583021\pi$
$444$	0	0
$445$	1.53743	0.0728810
$446$	0	0
$447$	−8.63833	−0.408579
$448$	0	0
$449$	16.9569	0.800248	0.400124	−	0.916461i	$-0.368967\pi$
$449$	16.9569	0.800248	0.400124	+	0.916461i	$0.368967\pi$
$450$	0	0
$451$	−4.52132	−0.212901
$452$	0	0
$453$	4.48225	0.210595
$454$	0	0
$455$	3.46132	0.162269
$456$	0	0
$457$	−34.8495	−1.63019	−0.815095	−	0.579327i	$-0.803315\pi$
$457$	−34.8495	−1.63019	−0.815095	+	0.579327i	$0.803315\pi$
$458$	0	0
$459$	−9.38234	−0.437930
$460$	0	0
$461$	−12.7420	−0.593453	−0.296727	−	0.954962i	$-0.595895\pi$
$461$	−12.7420	−0.593453	−0.296727	+	0.954962i	$0.595895\pi$
$462$	0	0
$463$	13.9365	0.647684	0.323842	−	0.946111i	$-0.395025\pi$
$463$	13.9365	0.647684	0.323842	+	0.946111i	$0.395025\pi$
$464$	0	0
$465$	0.400704	0.0185822
$466$	0	0
$467$	29.2729	1.35459	0.677294	−	0.735712i	$-0.263152\pi$
$467$	29.2729	1.35459	0.677294	+	0.735712i	$0.263152\pi$
$468$	0	0
$469$	15.3477	0.708689
$470$	0	0
$471$	2.80742	0.129359
$472$	0	0
$473$	0.748122	0.0343987
$474$	0	0
$475$	22.7303	1.04294
$476$	0	0
$477$	−38.6299	−1.76874
$478$	0	0
$479$	1.99534	0.0911695	0.0455847	−	0.998960i	$-0.485485\pi$
$479$	1.99534	0.0911695	0.0455847	+	0.998960i	$0.485485\pi$
$480$	0	0
$481$	15.5570	0.709339
$482$	0	0
$483$	0.397538	0.0180886
$484$	0	0
$485$	2.09928	0.0953234
$486$	0	0
$487$	−0.566670	−0.0256783	−0.0128391	−	0.999918i	$-0.504087\pi$
$487$	−0.566670	−0.0256783	−0.0128391	+	0.999918i	$0.504087\pi$
$488$	0	0
$489$	3.18615	0.144083
$490$	0	0
$491$	−2.03998	−0.0920630	−0.0460315	−	0.998940i	$-0.514657\pi$
$491$	−2.03998	−0.0920630	−0.0460315	+	0.998940i	$0.514657\pi$
$492$	0	0
$493$	13.4003	0.603518
$494$	0	0
$495$	1.03027	0.0463074
$496$	0	0
$497$	17.5856	0.788822
$498$	0	0
$499$	−10.8936	−0.487665	−0.243832	−	0.969817i	$-0.578405\pi$
$499$	−10.8936	−0.487665	−0.243832	+	0.969817i	$0.578405\pi$
$500$	0	0
$501$	−2.42526	−0.108352
$502$	0	0
$503$	29.4660	1.31383	0.656913	−	0.753967i	$-0.271862\pi$
$503$	29.4660	1.31383	0.656913	+	0.753967i	$0.271862\pi$
$504$	0	0
$505$	−1.12462	−0.0500448
$506$	0	0
$507$	−7.04230	−0.312760
$508$	0	0
$509$	33.8480	1.50029	0.750144	−	0.661275i	$-0.229984\pi$
$509$	33.8480	1.50029	0.750144	+	0.661275i	$0.229984\pi$
$510$	0	0
$511$	1.27592	0.0564434
$512$	0	0
$513$	−10.9421	−0.483107
$514$	0	0
$515$	−2.35185	−0.103635
$516$	0	0
$517$	−6.67864	−0.293726
$518$	0	0
$519$	−4.89272	−0.214767
$520$	0	0
$521$	−0.842945	−0.0369301	−0.0184650	−	0.999830i	$-0.505878\pi$
$521$	−0.842945	−0.0369301	−0.0184650	+	0.999830i	$0.505878\pi$
$522$	0	0
$523$	17.5336	0.766689	0.383345	−	0.923605i	$-0.374772\pi$
$523$	17.5336	0.766689	0.383345	+	0.923605i	$0.374772\pi$
$524$	0	0
$525$	−7.11355	−0.310461
$526$	0	0
$527$	21.2516	0.925733
$528$	0	0
$529$	−22.9229	−0.996649
$530$	0	0
$531$	−2.13562	−0.0926781
$532$	0	0
$533$	−12.3168	−0.533499
$534$	0	0
$535$	−0.946607	−0.0409254
$536$	0	0
$537$	2.27059	0.0979831
$538$	0	0
$539$	−10.4799	−0.451402
$540$	0	0
$541$	12.6336	0.543161	0.271580	−	0.962416i	$-0.412454\pi$
$541$	12.6336	0.543161	0.271580	+	0.962416i	$0.412454\pi$
$542$	0	0
$543$	−2.41527	−0.103649
$544$	0	0
$545$	−0.112756	−0.00482995
$546$	0	0
$547$	−10.6721	−0.456308	−0.228154	−	0.973625i	$-0.573269\pi$
$547$	−10.6721	−0.456308	−0.228154	+	0.973625i	$0.573269\pi$
$548$	0	0
$549$	3.55069	0.151540
$550$	0	0
$551$	15.6281	0.665777
$552$	0	0
$553$	1.42548	0.0606175
$554$	0	0
$555$	0.209516	0.00889346
$556$	0	0
$557$	−15.5565	−0.659149	−0.329574	−	0.944130i	$-0.606905\pi$
$557$	−15.5565	−0.659149	−0.329574	+	0.944130i	$0.606905\pi$
$558$	0	0
$559$	2.03800	0.0861984
$560$	0	0
$561$	−3.24409	−0.136966
$562$	0	0
$563$	12.1477	0.511963	0.255981	−	0.966682i	$-0.417601\pi$
$563$	12.1477	0.511963	0.255981	+	0.966682i	$0.417601\pi$
$564$	0	0
$565$	0.825432	0.0347262
$566$	0	0
$567$	−26.2461	−1.10223
$568$	0	0
$569$	21.7486	0.911749	0.455875	−	0.890044i	$-0.349327\pi$
$569$	21.7486	0.911749	0.455875	+	0.890044i	$0.349327\pi$
$570$	0	0
$571$	−30.3672	−1.27083	−0.635414	−	0.772172i	$-0.719171\pi$
$571$	−30.3672	−1.27083	−0.635414	+	0.772172i	$0.719171\pi$
$572$	0	0
$573$	−9.50617	−0.397126
$574$	0	0
$575$	−1.37898	−0.0575076
$576$	0	0
$577$	−18.2687	−0.760538	−0.380269	−	0.924876i	$-0.624169\pi$
$577$	−18.2687	−0.760538	−0.380269	+	0.924876i	$0.624169\pi$
$578$	0	0
$579$	1.32884	0.0552245
$580$	0	0
$581$	25.3876	1.05326
$582$	0	0
$583$	−27.5068	−1.13922
$584$	0	0
$585$	2.80663	0.116040
$586$	0	0
$587$	18.5051	0.763787	0.381893	−	0.924206i	$-0.375272\pi$
$587$	18.5051	0.763787	0.381893	+	0.924206i	$0.375272\pi$
$588$	0	0
$589$	24.7846	1.02123
$590$	0	0
$591$	4.96883	0.204390
$592$	0	0
$593$	−8.07163	−0.331462	−0.165731	−	0.986171i	$-0.552998\pi$
$593$	−8.07163	−0.331462	−0.165731	+	0.986171i	$0.552998\pi$
$594$	0	0
$595$	2.47229	0.101354
$596$	0	0
$597$	2.50335	0.102455
$598$	0	0
$599$	−34.7015	−1.41787	−0.708933	−	0.705276i	$-0.750823\pi$
$599$	−34.7015	−1.41787	−0.708933	+	0.705276i	$0.750823\pi$
$600$	0	0
$601$	−19.9084	−0.812081	−0.406041	−	0.913855i	$-0.633091\pi$
$601$	−19.9084	−0.812081	−0.406041	+	0.913855i	$0.633091\pi$
$602$	0	0
$603$	12.4447	0.506789
$604$	0	0
$605$	−1.25103	−0.0508615
$606$	0	0
$607$	36.9716	1.50063	0.750316	−	0.661080i	$-0.229902\pi$
$607$	36.9716	1.50063	0.750316	+	0.661080i	$0.229902\pi$
$608$	0	0
$609$	−4.89087	−0.198188
$610$	0	0
$611$	−18.1937	−0.736038
$612$	0	0
$613$	19.7708	0.798537	0.399268	−	0.916834i	$-0.369264\pi$
$613$	19.7708	0.798537	0.399268	+	0.916834i	$0.369264\pi$
$614$	0	0
$615$	−0.165878	−0.00668884
$616$	0	0
$617$	−5.64097	−0.227097	−0.113548	−	0.993532i	$-0.536222\pi$
$617$	−5.64097	−0.227097	−0.113548	+	0.993532i	$0.536222\pi$
$618$	0	0
$619$	22.0900	0.887872	0.443936	−	0.896059i	$-0.353582\pi$
$619$	22.0900	0.887872	0.443936	+	0.896059i	$0.353582\pi$
$620$	0	0
$621$	0.663829	0.0266386
$622$	0	0
$623$	−29.7601	−1.19231
$624$	0	0
$625$	24.5128	0.980511
$626$	0	0
$627$	−3.78342	−0.151095
$628$	0	0
$629$	11.1118	0.443057
$630$	0	0
$631$	35.0590	1.39568	0.697839	−	0.716255i	$-0.254145\pi$
$631$	35.0590	1.39568	0.697839	+	0.716255i	$0.254145\pi$
$632$	0	0
$633$	5.88129	0.233760
$634$	0	0
$635$	−3.30745	−0.131252
$636$	0	0
$637$	−28.5490	−1.13115
$638$	0	0
$639$	14.2594	0.564092
$640$	0	0
$641$	−33.9762	−1.34198	−0.670990	−	0.741467i	$-0.734130\pi$
$641$	−33.9762	−1.34198	−0.670990	+	0.741467i	$0.734130\pi$
$642$	0	0
$643$	−26.4390	−1.04265	−0.521326	−	0.853358i	$-0.674562\pi$
$643$	−26.4390	−1.04265	−0.521326	+	0.853358i	$0.674562\pi$
$644$	0	0
$645$	0.0274471	0.00108073
$646$	0	0
$647$	50.0530	1.96779	0.983893	−	0.178757i	$-0.0572076\pi$
$647$	50.0530	1.96779	0.983893	+	0.178757i	$0.0572076\pi$
$648$	0	0
$649$	−1.52069	−0.0596923
$650$	0	0
$651$	−7.75647	−0.304000
$652$	0	0
$653$	7.65981	0.299751	0.149876	−	0.988705i	$-0.452113\pi$
$653$	7.65981	0.299751	0.149876	+	0.988705i	$0.452113\pi$
$654$	0	0
$655$	−0.355563	−0.0138930
$656$	0	0
$657$	1.03459	0.0403631
$658$	0	0
$659$	−13.0367	−0.507837	−0.253918	−	0.967226i	$-0.581719\pi$
$659$	−13.0367	−0.507837	−0.253918	+	0.967226i	$0.581719\pi$
$660$	0	0
$661$	−5.09055	−0.197999	−0.0989997	−	0.995087i	$-0.531564\pi$
$661$	−5.09055	−0.197999	−0.0989997	+	0.995087i	$0.531564\pi$
$662$	0	0
$663$	−8.83742	−0.343217
$664$	0	0
$665$	2.88331	0.111810
$666$	0	0
$667$	−0.948111	−0.0367110
$668$	0	0
$669$	−0.0255071	−0.000986162 0
$670$	0	0
$671$	2.52830	0.0976040
$672$	0	0
$673$	33.8236	1.30380	0.651901	−	0.758304i	$-0.273972\pi$
$673$	33.8236	1.30380	0.651901	+	0.758304i	$0.273972\pi$
$674$	0	0
$675$	−11.8786	−0.457206
$676$	0	0
$677$	23.8890	0.918130	0.459065	−	0.888403i	$-0.348184\pi$
$677$	23.8890	0.918130	0.459065	+	0.888403i	$0.348184\pi$
$678$	0	0
$679$	−40.6360	−1.55946
$680$	0	0
$681$	9.53604	0.365422
$682$	0	0
$683$	−1.24029	−0.0474583	−0.0237292	−	0.999718i	$-0.507554\pi$
$683$	−1.24029	−0.0474583	−0.0237292	+	0.999718i	$0.507554\pi$
$684$	0	0
$685$	0.888321	0.0339410
$686$	0	0
$687$	8.03955	0.306728
$688$	0	0
$689$	−74.9330	−2.85472
$690$	0	0
$691$	−33.5565	−1.27655	−0.638274	−	0.769809i	$-0.720351\pi$
$691$	−33.5565	−1.27655	−0.638274	+	0.769809i	$0.720351\pi$
$692$	0	0
$693$	−19.9431	−0.757576
$694$	0	0
$695$	0.195599	0.00741949
$696$	0	0
$697$	−8.79743	−0.333226
$698$	0	0
$699$	7.02899	0.265861
$700$	0	0
$701$	2.80272	0.105857	0.0529286	−	0.998598i	$-0.483144\pi$
$701$	2.80272	0.105857	0.0529286	+	0.998598i	$0.483144\pi$
$702$	0	0
$703$	12.9591	0.488763
$704$	0	0
$705$	−0.245026	−0.00922820
$706$	0	0
$707$	21.7693	0.818720
$708$	0	0
$709$	28.5481	1.07214	0.536072	−	0.844172i	$-0.319907\pi$
$709$	28.5481	1.07214	0.536072	+	0.844172i	$0.319907\pi$
$710$	0	0
$711$	1.15586	0.0433480
$712$	0	0
$713$	−1.50361	−0.0563108
$714$	0	0
$715$	1.99849	0.0747392
$716$	0	0
$717$	3.13370	0.117030
$718$	0	0
$719$	11.7166	0.436957	0.218479	−	0.975842i	$-0.429891\pi$
$719$	11.7166	0.436957	0.218479	+	0.975842i	$0.429891\pi$
$720$	0	0
$721$	45.5250	1.69544
$722$	0	0
$723$	2.80031	0.104145
$724$	0	0
$725$	16.9655	0.630083
$726$	0	0
$727$	10.2324	0.379497	0.189749	−	0.981833i	$-0.439233\pi$
$727$	10.2324	0.379497	0.189749	+	0.981833i	$0.439233\pi$
$728$	0	0
$729$	−18.3402	−0.679268
$730$	0	0
$731$	1.45567	0.0538399
$732$	0	0
$733$	25.4209	0.938941	0.469470	−	0.882948i	$-0.344445\pi$
$733$	25.4209	0.938941	0.469470	+	0.882948i	$0.344445\pi$
$734$	0	0
$735$	−0.384487	−0.0141820
$736$	0	0
$737$	8.86140	0.326414
$738$	0	0
$739$	−25.1764	−0.926129	−0.463064	−	0.886325i	$-0.653250\pi$
$739$	−25.1764	−0.926129	−0.463064	+	0.886325i	$0.653250\pi$
$740$	0	0
$741$	−10.3066	−0.378624
$742$	0	0
$743$	−42.3613	−1.55408	−0.777042	−	0.629449i	$-0.783281\pi$
$743$	−42.3613	−1.55408	−0.777042	+	0.629449i	$0.783281\pi$
$744$	0	0
$745$	3.80099	0.139257
$746$	0	0
$747$	20.5857	0.753192
$748$	0	0
$749$	18.3235	0.669528
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	8.33247	0.303652
$754$	0	0
$755$	−1.97225	−0.0717777
$756$	0	0
$757$	39.9426	1.45174	0.725869	−	0.687833i	$-0.241438\pi$
$757$	39.9426	1.45174	0.725869	+	0.687833i	$0.241438\pi$
$758$	0	0
$759$	0.229529	0.00833139
$760$	0	0
$761$	14.0575	0.509583	0.254792	−	0.966996i	$-0.417993\pi$
$761$	14.0575	0.509583	0.254792	+	0.966996i	$0.417993\pi$
$762$	0	0
$763$	2.18263	0.0790167
$764$	0	0
$765$	2.00467	0.0724791
$766$	0	0
$767$	−4.14260	−0.149581
$768$	0	0
$769$	0.0919031	0.00331411	0.00165705	−	0.999999i	$-0.499473\pi$
$769$	0.0919031	0.00331411	0.00165705	+	0.999999i	$0.499473\pi$
$770$	0	0
$771$	−4.62380	−0.166522
$772$	0	0
$773$	1.63172	0.0586888	0.0293444	−	0.999569i	$-0.490658\pi$
$773$	1.63172	0.0586888	0.0293444	+	0.999569i	$0.490658\pi$
$774$	0	0
$775$	26.9057	0.966481
$776$	0	0
$777$	−4.05562	−0.145495
$778$	0	0
$779$	−10.2600	−0.367602
$780$	0	0
$781$	10.1535	0.363322
$782$	0	0
$783$	−8.16703	−0.291866
$784$	0	0
$785$	−1.23531	−0.0440899
$786$	0	0
$787$	−17.5960	−0.627230	−0.313615	−	0.949550i	$-0.601540\pi$
$787$	−17.5960	−0.627230	−0.313615	+	0.949550i	$0.601540\pi$
$788$	0	0
$789$	11.1668	0.397549
$790$	0	0
$791$	−15.9780	−0.568111
$792$	0	0
$793$	6.88749	0.244582
$794$	0	0
$795$	−1.00917	−0.0357915
$796$	0	0
$797$	23.4818	0.831767	0.415884	−	0.909418i	$-0.363472\pi$
$797$	23.4818	0.831767	0.415884	+	0.909418i	$0.363472\pi$
$798$	0	0
$799$	−12.9951	−0.459733
$800$	0	0
$801$	−24.1311	−0.852632
$802$	0	0
$803$	0.736688	0.0259972
$804$	0	0
$805$	−0.174922	−0.00616520
$806$	0	0
$807$	−4.69703	−0.165343
$808$	0	0
$809$	−18.3873	−0.646463	−0.323232	−	0.946320i	$-0.604769\pi$
$809$	−18.3873	−0.646463	−0.323232	+	0.946320i	$0.604769\pi$
$810$	0	0
$811$	−36.4010	−1.27821	−0.639106	−	0.769119i	$-0.720695\pi$
$811$	−36.4010	−1.27821	−0.639106	+	0.769119i	$0.720695\pi$
$812$	0	0
$813$	6.06225	0.212613
$814$	0	0
$815$	−1.40195	−0.0491082
$816$	0	0
$817$	1.69767	0.0593941
$818$	0	0
$819$	−54.3282	−1.89838
$820$	0	0
$821$	40.5516	1.41526	0.707631	−	0.706582i	$-0.249764\pi$
$821$	40.5516	1.41526	0.707631	+	0.706582i	$0.249764\pi$
$822$	0	0
$823$	27.4855	0.958086	0.479043	−	0.877791i	$-0.340984\pi$
$823$	27.4855	0.958086	0.479043	+	0.877791i	$0.340984\pi$
$824$	0	0
$825$	−4.10720	−0.142994
$826$	0	0
$827$	−15.7910	−0.549105	−0.274553	−	0.961572i	$-0.588530\pi$
$827$	−15.7910	−0.549105	−0.274553	+	0.961572i	$0.588530\pi$
$828$	0	0
$829$	−17.9188	−0.622345	−0.311172	−	0.950354i	$-0.600722\pi$
$829$	−17.9188	−0.622345	−0.311172	+	0.950354i	$0.600722\pi$
$830$	0	0
$831$	3.47665	0.120604
$832$	0	0
$833$	−20.3915	−0.706523
$834$	0	0
$835$	1.06715	0.0369301
$836$	0	0
$837$	−12.9522	−0.447692
$838$	0	0
$839$	−43.4244	−1.49918	−0.749588	−	0.661905i	$-0.769748\pi$
$839$	−43.4244	−1.49918	−0.749588	+	0.661905i	$0.769748\pi$
$840$	0	0
$841$	−17.3355	−0.597775
$842$	0	0
$843$	4.80720	0.165569
$844$	0	0
$845$	3.09871	0.106599
$846$	0	0
$847$	24.2162	0.832080
$848$	0	0
$849$	2.55565	0.0877097
$850$	0	0
$851$	−0.786194	−0.0269504
$852$	0	0
$853$	−38.5267	−1.31913	−0.659564	−	0.751649i	$-0.729259\pi$
$853$	−38.5267	−1.31913	−0.659564	+	0.751649i	$0.729259\pi$
$854$	0	0
$855$	2.33795	0.0799561
$856$	0	0
$857$	−12.4762	−0.426180	−0.213090	−	0.977033i	$-0.568353\pi$
$857$	−12.4762	−0.426180	−0.213090	+	0.977033i	$0.568353\pi$
$858$	0	0
$859$	−0.909040	−0.0310160	−0.0155080	−	0.999880i	$-0.504937\pi$
$859$	−0.909040	−0.0310160	−0.0155080	+	0.999880i	$0.504937\pi$
$860$	0	0
$861$	3.21091	0.109428
$862$	0	0
$863$	27.1218	0.923237	0.461619	−	0.887079i	$-0.347269\pi$
$863$	27.1218	0.923237	0.461619	+	0.887079i	$0.347269\pi$
$864$	0	0
$865$	2.15287	0.0731997
$866$	0	0
$867$	0.658391	0.0223601
$868$	0	0
$869$	0.823040	0.0279197
$870$	0	0
$871$	24.1399	0.817948
$872$	0	0
$873$	−32.9499	−1.11518
$874$	0	0
$875$	6.28064	0.212324
$876$	0	0
$877$	47.2206	1.59453	0.797264	−	0.603631i	$-0.206280\pi$
$877$	47.2206	1.59453	0.797264	+	0.603631i	$0.206280\pi$
$878$	0	0
$879$	4.91616	0.165818
$880$	0	0
$881$	−8.51318	−0.286816	−0.143408	−	0.989664i	$-0.545806\pi$
$881$	−8.51318	−0.286816	−0.143408	+	0.989664i	$0.545806\pi$
$882$	0	0
$883$	2.18194	0.0734282	0.0367141	−	0.999326i	$-0.488311\pi$
$883$	2.18194	0.0734282	0.0367141	+	0.999326i	$0.488311\pi$
$884$	0	0
$885$	−0.0557910	−0.00187539
$886$	0	0
$887$	−15.4269	−0.517985	−0.258992	−	0.965879i	$-0.583390\pi$
$887$	−15.4269	−0.517985	−0.258992	+	0.965879i	$0.583390\pi$
$888$	0	0
$889$	64.0226	2.14725
$890$	0	0
$891$	−15.1539	−0.507674
$892$	0	0
$893$	−15.1555	−0.507159
$894$	0	0
$895$	−0.999091	−0.0333959
$896$	0	0
$897$	0.625275	0.0208773
$898$	0	0
$899$	18.4988	0.616971
$900$	0	0
$901$	−53.5218	−1.78307
$902$	0	0
$903$	−0.531295	−0.0176804
$904$	0	0
$905$	1.06275	0.0353271
$906$	0	0
$907$	10.1984	0.338631	0.169316	−	0.985562i	$-0.445844\pi$
$907$	10.1984	0.338631	0.169316	+	0.985562i	$0.445844\pi$
$908$	0	0
$909$	17.6518	0.585473
$910$	0	0
$911$	−40.1284	−1.32951	−0.664756	−	0.747061i	$-0.731464\pi$
$911$	−40.1284	−1.32951	−0.664756	+	0.747061i	$0.731464\pi$
$912$	0	0
$913$	14.6583	0.485117
$914$	0	0
$915$	0.0927581	0.00306649
$916$	0	0
$917$	6.88266	0.227285
$918$	0	0
$919$	1.22547	0.0404247	0.0202123	−	0.999796i	$-0.493566\pi$
$919$	1.22547	0.0404247	0.0202123	+	0.999796i	$0.493566\pi$
$920$	0	0
$921$	−4.55206	−0.149996
$922$	0	0
$923$	27.6598	0.910434
$924$	0	0
$925$	14.0682	0.462559
$926$	0	0
$927$	36.9142	1.21242
$928$	0	0
$929$	12.1194	0.397626	0.198813	−	0.980037i	$-0.436291\pi$
$929$	12.1194	0.397626	0.198813	+	0.980037i	$0.436291\pi$
$930$	0	0
$931$	−23.7815	−0.779408
$932$	0	0
$933$	−4.47934	−0.146647
$934$	0	0
$935$	1.42745	0.0466825
$936$	0	0
$937$	13.1678	0.430173	0.215086	−	0.976595i	$-0.430997\pi$
$937$	13.1678	0.430173	0.215086	+	0.976595i	$0.430997\pi$
$938$	0	0
$939$	2.16273	0.0705780
$940$	0	0
$941$	−4.69697	−0.153117	−0.0765584	−	0.997065i	$-0.524393\pi$
$941$	−4.69697	−0.153117	−0.0765584	+	0.997065i	$0.524393\pi$
$942$	0	0
$943$	0.622445	0.0202696
$944$	0	0
$945$	−1.50678	−0.0490156
$946$	0	0
$947$	−8.17089	−0.265518	−0.132759	−	0.991148i	$-0.542384\pi$
$947$	−8.17089	−0.265518	−0.132759	+	0.991148i	$0.542384\pi$
$948$	0	0
$949$	2.00686	0.0651453
$950$	0	0
$951$	−5.66360	−0.183655
$952$	0	0
$953$	1.04551	0.0338673	0.0169336	−	0.999857i	$-0.494610\pi$
$953$	1.04551	0.0338673	0.0169336	+	0.999857i	$0.494610\pi$
$954$	0	0
$955$	4.18285	0.135354
$956$	0	0
$957$	−2.82388	−0.0912831
$958$	0	0
$959$	−17.1953	−0.555265
$960$	0	0
$961$	−1.66257	−0.0536313
$962$	0	0
$963$	14.8577	0.478784
$964$	0	0
$965$	−0.584706	−0.0188224
$966$	0	0
$967$	6.74171	0.216799	0.108399	−	0.994107i	$-0.465427\pi$
$967$	6.74171	0.216799	0.108399	+	0.994107i	$0.465427\pi$
$968$	0	0
$969$	−7.36164	−0.236490
$970$	0	0
$971$	−17.3681	−0.557370	−0.278685	−	0.960383i	$-0.589898\pi$
$971$	−17.3681	−0.557370	−0.278685	+	0.960383i	$0.589898\pi$
$972$	0	0
$973$	−3.78622	−0.121381
$974$	0	0
$975$	−11.1887	−0.358324
$976$	0	0
$977$	−35.9777	−1.15103	−0.575514	−	0.817792i	$-0.695198\pi$
$977$	−35.9777	−1.15103	−0.575514	+	0.817792i	$0.695198\pi$
$978$	0	0
$979$	−17.1828	−0.549165
$980$	0	0
$981$	1.76980	0.0565054
$982$	0	0
$983$	−41.9631	−1.33842	−0.669208	−	0.743075i	$-0.733366\pi$
$983$	−41.9631	−1.33842	−0.669208	+	0.743075i	$0.733366\pi$
$984$	0	0
$985$	−2.18635	−0.0696630
$986$	0	0
$987$	4.74298	0.150971
$988$	0	0
$989$	−0.102993	−0.00327499
$990$	0	0
$991$	−30.5998	−0.972036	−0.486018	−	0.873949i	$-0.661551\pi$
$991$	−30.5998	−0.972036	−0.486018	+	0.873949i	$0.661551\pi$
$992$	0	0
$993$	−0.425893	−0.0135153
$994$	0	0
$995$	−1.10151	−0.0349202
$996$	0	0
$997$	62.5075	1.97963	0.989817	−	0.142348i	$-0.0454651\pi$
$997$	62.5075	1.97963	0.989817	+	0.142348i	$0.0454651\pi$
$998$	0	0
$999$	−6.77229	−0.214266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.22	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.22	✓	50	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)

\(f(q)\)	\(=\)	\(q-0.410037 q^{3} +0.180422 q^{5} -3.49245 q^{7} -2.83187 q^{9} +O(q^{10})\)	\(q-0.410037 q^{3} +0.180422 q^{5} -3.49245 q^{7} -2.83187 q^{9} -2.01646 q^{11} -5.49316 q^{13} -0.0739798 q^{15} -3.92356 q^{17} -4.57585 q^{19} +1.43203 q^{21} +0.277604 q^{23} -4.96745 q^{25} +2.39128 q^{27} -3.41534 q^{29} -5.41640 q^{31} +0.826824 q^{33} -0.630115 q^{35} -2.83207 q^{37} +2.25240 q^{39} +2.24221 q^{41} -0.371008 q^{43} -0.510932 q^{45} +3.31206 q^{47} +5.19719 q^{49} +1.60880 q^{51} +13.6411 q^{53} -0.363814 q^{55} +1.87627 q^{57} +0.754139 q^{59} -1.25383 q^{61} +9.89016 q^{63} -0.991087 q^{65} -4.39453 q^{67} -0.113828 q^{69} -5.03532 q^{71} -0.365337 q^{73} +2.03684 q^{75} +7.04238 q^{77} -0.408161 q^{79} +7.51509 q^{81} -7.26930 q^{83} -0.707897 q^{85} +1.40041 q^{87} +8.52128 q^{89} +19.1846 q^{91} +2.22093 q^{93} -0.825584 q^{95} +11.6354 q^{97} +5.71035 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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