LMFDB - Embedded newform 6008.2.a.e.1.40

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			1.40
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)

$f(q)$	$=$	$q+2.15655 q^{3} +1.11683 q^{5} -1.03539 q^{7} +1.65070 q^{9} +O(q^{10})$	$q+2.15655 q^{3} +1.11683 q^{5} -1.03539 q^{7} +1.65070 q^{9} +2.85069 q^{11} +3.74207 q^{13} +2.40850 q^{15} +5.19567 q^{17} +1.79315 q^{19} -2.23286 q^{21} +6.06395 q^{23} -3.75269 q^{25} -2.90984 q^{27} +0.462966 q^{29} +1.21315 q^{31} +6.14765 q^{33} -1.15635 q^{35} -7.34179 q^{37} +8.06994 q^{39} +6.52229 q^{41} +2.05942 q^{43} +1.84355 q^{45} -5.41764 q^{47} -5.92797 q^{49} +11.2047 q^{51} +2.63975 q^{53} +3.18374 q^{55} +3.86702 q^{57} -1.09961 q^{59} +14.7487 q^{61} -1.70911 q^{63} +4.17926 q^{65} -13.2027 q^{67} +13.0772 q^{69} -2.35417 q^{71} -6.45396 q^{73} -8.09284 q^{75} -2.95157 q^{77} +0.844569 q^{79} -11.2273 q^{81} +13.6825 q^{83} +5.80269 q^{85} +0.998407 q^{87} +0.307688 q^{89} -3.87449 q^{91} +2.61622 q^{93} +2.00265 q^{95} -5.44127 q^{97} +4.70562 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.15655	1.24508	0.622542	−	0.782587i	$-0.286100\pi$
$3$	2.15655	1.24508	0.622542	+	0.782587i	$0.286100\pi$
$4$	0	0
$5$	1.11683	0.499463	0.249731	−	0.968315i	$-0.419658\pi$
$5$	1.11683	0.499463	0.249731	+	0.968315i	$0.419658\pi$
$6$	0	0
$7$	−1.03539	−0.391340	−0.195670	−	0.980670i	$-0.562688\pi$
$7$	−1.03539	−0.391340	−0.195670	+	0.980670i	$0.562688\pi$
$8$	0	0
$9$	1.65070	0.550232
$10$	0	0
$11$	2.85069	0.859515	0.429758	−	0.902944i	$-0.358599\pi$
$11$	2.85069	0.859515	0.429758	+	0.902944i	$0.358599\pi$
$12$	0	0
$13$	3.74207	1.03786	0.518931	−	0.854816i	$-0.326330\pi$
$13$	3.74207	1.03786	0.518931	+	0.854816i	$0.326330\pi$
$14$	0	0
$15$	2.40850	0.621872
$16$	0	0
$17$	5.19567	1.26013	0.630067	−	0.776540i	$-0.283027\pi$
$17$	5.19567	1.26013	0.630067	+	0.776540i	$0.283027\pi$
$18$	0	0
$19$	1.79315	0.411377	0.205689	−	0.978617i	$-0.434057\pi$
$19$	1.79315	0.411377	0.205689	+	0.978617i	$0.434057\pi$
$20$	0	0
$21$	−2.23286	−0.487251
$22$	0	0
$23$	6.06395	1.26442	0.632211	−	0.774796i	$-0.282148\pi$
$23$	6.06395	1.26442	0.632211	+	0.774796i	$0.282148\pi$
$24$	0	0
$25$	−3.75269	−0.750537
$26$	0	0
$27$	−2.90984	−0.559998
$28$	0	0
$29$	0.462966	0.0859706	0.0429853	−	0.999076i	$-0.486313\pi$
$29$	0.462966	0.0859706	0.0429853	+	0.999076i	$0.486313\pi$
$30$	0	0
$31$	1.21315	0.217888	0.108944	−	0.994048i	$-0.465253\pi$
$31$	1.21315	0.217888	0.108944	+	0.994048i	$0.465253\pi$
$32$	0	0
$33$	6.14765	1.07017
$34$	0	0
$35$	−1.15635	−0.195460
$36$	0	0
$37$	−7.34179	−1.20698	−0.603492	−	0.797369i	$-0.706224\pi$
$37$	−7.34179	−1.20698	−0.603492	+	0.797369i	$0.706224\pi$
$38$	0	0
$39$	8.06994	1.29223
$40$	0	0
$41$	6.52229	1.01861	0.509305	−	0.860586i	$-0.329902\pi$
$41$	6.52229	1.01861	0.509305	+	0.860586i	$0.329902\pi$
$42$	0	0
$43$	2.05942	0.314059	0.157029	−	0.987594i	$-0.449808\pi$
$43$	2.05942	0.314059	0.157029	+	0.987594i	$0.449808\pi$
$44$	0	0
$45$	1.84355	0.274820
$46$	0	0
$47$	−5.41764	−0.790245	−0.395122	−	0.918629i	$-0.629298\pi$
$47$	−5.41764	−0.790245	−0.395122	+	0.918629i	$0.629298\pi$
$48$	0	0
$49$	−5.92797	−0.846853
$50$	0	0
$51$	11.2047	1.56897
$52$	0	0
$53$	2.63975	0.362597	0.181299	−	0.983428i	$-0.441970\pi$
$53$	2.63975	0.362597	0.181299	+	0.983428i	$0.441970\pi$
$54$	0	0
$55$	3.18374	0.429296
$56$	0	0
$57$	3.86702	0.512199
$58$	0	0
$59$	−1.09961	−0.143157	−0.0715787	−	0.997435i	$-0.522804\pi$
$59$	−1.09961	−0.143157	−0.0715787	+	0.997435i	$0.522804\pi$
$60$	0	0
$61$	14.7487	1.88837	0.944187	−	0.329411i	$-0.106850\pi$
$61$	14.7487	1.88837	0.944187	+	0.329411i	$0.106850\pi$
$62$	0	0
$63$	−1.70911	−0.215328
$64$	0	0
$65$	4.17926	0.518373
$66$	0	0
$67$	−13.2027	−1.61297	−0.806483	−	0.591257i	$-0.798632\pi$
$67$	−13.2027	−1.61297	−0.806483	+	0.591257i	$0.798632\pi$
$68$	0	0
$69$	13.0772	1.57431
$70$	0	0
$71$	−2.35417	−0.279389	−0.139694	−	0.990195i	$-0.544612\pi$
$71$	−2.35417	−0.279389	−0.139694	+	0.990195i	$0.544612\pi$
$72$	0	0
$73$	−6.45396	−0.755378	−0.377689	−	0.925932i	$-0.623281\pi$
$73$	−6.45396	−0.755378	−0.377689	+	0.925932i	$0.623281\pi$
$74$	0	0
$75$	−8.09284	−0.934481
$76$	0	0
$77$	−2.95157	−0.336363
$78$	0	0
$79$	0.844569	0.0950214	0.0475107	−	0.998871i	$-0.484871\pi$
$79$	0.844569	0.0950214	0.0475107	+	0.998871i	$0.484871\pi$
$80$	0	0
$81$	−11.2273	−1.24748
$82$	0	0
$83$	13.6825	1.50185	0.750925	−	0.660388i	$-0.229608\pi$
$83$	13.6825	1.50185	0.750925	+	0.660388i	$0.229608\pi$
$84$	0	0
$85$	5.80269	0.629390
$86$	0	0
$87$	0.998407	0.107040
$88$	0	0
$89$	0.307688	0.0326149	0.0163074	−	0.999867i	$-0.494809\pi$
$89$	0.307688	0.0326149	0.0163074	+	0.999867i	$0.494809\pi$
$90$	0	0
$91$	−3.87449	−0.406157
$92$	0	0
$93$	2.61622	0.271289
$94$	0	0
$95$	2.00265	0.205467
$96$	0	0
$97$	−5.44127	−0.552478	−0.276239	−	0.961089i	$-0.589088\pi$
$97$	−5.44127	−0.552478	−0.276239	+	0.961089i	$0.589088\pi$
$98$	0	0
$99$	4.70562	0.472933
$100$	0	0
$101$	−7.01883	−0.698400	−0.349200	−	0.937048i	$-0.613547\pi$
$101$	−7.01883	−0.698400	−0.349200	+	0.937048i	$0.613547\pi$
$102$	0	0
$103$	−5.77002	−0.568536	−0.284268	−	0.958745i	$-0.591751\pi$
$103$	−5.77002	−0.568536	−0.284268	+	0.958745i	$0.591751\pi$
$104$	0	0
$105$	−2.49373	−0.243363
$106$	0	0
$107$	0.369862	0.0357559	0.0178779	−	0.999840i	$-0.494309\pi$
$107$	0.369862	0.0357559	0.0178779	+	0.999840i	$0.494309\pi$
$108$	0	0
$109$	15.5269	1.48720	0.743602	−	0.668623i	$-0.233116\pi$
$109$	15.5269	1.48720	0.743602	+	0.668623i	$0.233116\pi$
$110$	0	0
$111$	−15.8329	−1.50279
$112$	0	0
$113$	8.41022	0.791167	0.395583	−	0.918430i	$-0.370542\pi$
$113$	8.41022	0.791167	0.395583	+	0.918430i	$0.370542\pi$
$114$	0	0
$115$	6.77242	0.631531
$116$	0	0
$117$	6.17702	0.571065
$118$	0	0
$119$	−5.37953	−0.493141
$120$	0	0
$121$	−2.87357	−0.261233
$122$	0	0
$123$	14.0656	1.26825
$124$	0	0
$125$	−9.77528	−0.874328
$126$	0	0
$127$	−10.5742	−0.938311	−0.469156	−	0.883116i	$-0.655442\pi$
$127$	−10.5742	−0.938311	−0.469156	+	0.883116i	$0.655442\pi$
$128$	0	0
$129$	4.44124	0.391029
$130$	0	0
$131$	5.63867	0.492653	0.246326	−	0.969187i	$-0.420776\pi$
$131$	5.63867	0.492653	0.246326	+	0.969187i	$0.420776\pi$
$132$	0	0
$133$	−1.85661	−0.160988
$134$	0	0
$135$	−3.24980	−0.279698
$136$	0	0
$137$	12.8379	1.09682	0.548408	−	0.836211i	$-0.315234\pi$
$137$	12.8379	1.09682	0.548408	+	0.836211i	$0.315234\pi$
$138$	0	0
$139$	−8.80162	−0.746544	−0.373272	−	0.927722i	$-0.621764\pi$
$139$	−8.80162	−0.746544	−0.373272	+	0.927722i	$0.621764\pi$
$140$	0	0
$141$	−11.6834	−0.983920
$142$	0	0
$143$	10.6675	0.892059
$144$	0	0
$145$	0.517055	0.0429391
$146$	0	0
$147$	−12.7840	−1.05440
$148$	0	0
$149$	−0.134806	−0.0110438	−0.00552188	−	0.999985i	$-0.501758\pi$
$149$	−0.134806	−0.0110438	−0.00552188	+	0.999985i	$0.501758\pi$
$150$	0	0
$151$	0.150825	0.0122740	0.00613699	−	0.999981i	$-0.498047\pi$
$151$	0.150825	0.0122740	0.00613699	+	0.999981i	$0.498047\pi$
$152$	0	0
$153$	8.57647	0.693367
$154$	0	0
$155$	1.35489	0.108827
$156$	0	0
$157$	21.8541	1.74415	0.872075	−	0.489371i	$-0.162774\pi$
$157$	21.8541	1.74415	0.872075	+	0.489371i	$0.162774\pi$
$158$	0	0
$159$	5.69274	0.451464
$160$	0	0
$161$	−6.27854	−0.494819
$162$	0	0
$163$	12.6530	0.991062	0.495531	−	0.868590i	$-0.334974\pi$
$163$	12.6530	0.991062	0.495531	+	0.868590i	$0.334974\pi$
$164$	0	0
$165$	6.86589	0.534509
$166$	0	0
$167$	22.3537	1.72978	0.864890	−	0.501962i	$-0.167388\pi$
$167$	22.3537	1.72978	0.864890	+	0.501962i	$0.167388\pi$
$168$	0	0
$169$	1.00306	0.0771586
$170$	0	0
$171$	2.95995	0.226353
$172$	0	0
$173$	20.5153	1.55975	0.779875	−	0.625935i	$-0.215283\pi$
$173$	20.5153	1.55975	0.779875	+	0.625935i	$0.215283\pi$
$174$	0	0
$175$	3.88549	0.293715
$176$	0	0
$177$	−2.37137	−0.178243
$178$	0	0
$179$	−14.2623	−1.06601	−0.533005	−	0.846112i	$-0.678937\pi$
$179$	−14.2623	−1.06601	−0.533005	+	0.846112i	$0.678937\pi$
$180$	0	0
$181$	−22.2037	−1.65039	−0.825194	−	0.564849i	$-0.808934\pi$
$181$	−22.2037	−1.65039	−0.825194	+	0.564849i	$0.808934\pi$
$182$	0	0
$183$	31.8062	2.35118
$184$	0	0
$185$	−8.19955	−0.602843
$186$	0	0
$187$	14.8112	1.08311
$188$	0	0
$189$	3.01281	0.219150
$190$	0	0
$191$	17.0531	1.23392	0.616959	−	0.786995i	$-0.288365\pi$
$191$	17.0531	1.23392	0.616959	+	0.786995i	$0.288365\pi$
$192$	0	0
$193$	1.05421	0.0758835	0.0379417	−	0.999280i	$-0.487920\pi$
$193$	1.05421	0.0758835	0.0379417	+	0.999280i	$0.487920\pi$
$194$	0	0
$195$	9.01277	0.645418
$196$	0	0
$197$	2.78465	0.198398	0.0991991	−	0.995068i	$-0.468372\pi$
$197$	2.78465	0.198398	0.0991991	+	0.995068i	$0.468372\pi$
$198$	0	0
$199$	7.43942	0.527367	0.263683	−	0.964609i	$-0.415063\pi$
$199$	7.43942	0.527367	0.263683	+	0.964609i	$0.415063\pi$
$200$	0	0
$201$	−28.4722	−2.00828
$202$	0	0
$203$	−0.479349	−0.0336437
$204$	0	0
$205$	7.28430	0.508758
$206$	0	0
$207$	10.0097	0.695725
$208$	0	0
$209$	5.11172	0.353585
$210$	0	0
$211$	−13.4294	−0.924517	−0.462259	−	0.886745i	$-0.652961\pi$
$211$	−13.4294	−0.924517	−0.462259	+	0.886745i	$0.652961\pi$
$212$	0	0
$213$	−5.07688	−0.347862
$214$	0	0
$215$	2.30003	0.156860
$216$	0	0
$217$	−1.25608	−0.0852684
$218$	0	0
$219$	−13.9183	−0.940509
$220$	0	0
$221$	19.4425	1.30785
$222$	0	0
$223$	5.26025	0.352252	0.176126	−	0.984368i	$-0.443643\pi$
$223$	5.26025	0.352252	0.176126	+	0.984368i	$0.443643\pi$
$224$	0	0
$225$	−6.19454	−0.412970
$226$	0	0
$227$	−12.4816	−0.828430	−0.414215	−	0.910179i	$-0.635944\pi$
$227$	−12.4816	−0.828430	−0.414215	+	0.910179i	$0.635944\pi$
$228$	0	0
$229$	3.73617	0.246893	0.123447	−	0.992351i	$-0.460605\pi$
$229$	3.73617	0.246893	0.123447	+	0.992351i	$0.460605\pi$
$230$	0	0
$231$	−6.36520	−0.418799
$232$	0	0
$233$	6.63095	0.434408	0.217204	−	0.976126i	$-0.430306\pi$
$233$	6.63095	0.434408	0.217204	+	0.976126i	$0.430306\pi$
$234$	0	0
$235$	−6.05060	−0.394698
$236$	0	0
$237$	1.82135	0.118310
$238$	0	0
$239$	15.1290	0.978617	0.489308	−	0.872111i	$-0.337249\pi$
$239$	15.1290	0.978617	0.489308	+	0.872111i	$0.337249\pi$
$240$	0	0
$241$	6.26415	0.403509	0.201755	−	0.979436i	$-0.435336\pi$
$241$	6.26415	0.403509	0.201755	+	0.979436i	$0.435336\pi$
$242$	0	0
$243$	−15.4827	−0.993214
$244$	0	0
$245$	−6.62055	−0.422971
$246$	0	0
$247$	6.71009	0.426953
$248$	0	0
$249$	29.5070	1.86993
$250$	0	0
$251$	6.24782	0.394359	0.197180	−	0.980367i	$-0.436822\pi$
$251$	6.24782	0.394359	0.197180	+	0.980367i	$0.436822\pi$
$252$	0	0
$253$	17.2864	1.08679
$254$	0	0
$255$	12.5138	0.783643
$256$	0	0
$257$	1.51441	0.0944666	0.0472333	−	0.998884i	$-0.484960\pi$
$257$	1.51441	0.0944666	0.0472333	+	0.998884i	$0.484960\pi$
$258$	0	0
$259$	7.60160	0.472341
$260$	0	0
$261$	0.764216	0.0473038
$262$	0	0
$263$	−10.6885	−0.659082	−0.329541	−	0.944141i	$-0.606894\pi$
$263$	−10.6885	−0.659082	−0.329541	+	0.944141i	$0.606894\pi$
$264$	0	0
$265$	2.94816	0.181104
$266$	0	0
$267$	0.663544	0.0406082
$268$	0	0
$269$	−15.0899	−0.920047	−0.460024	−	0.887907i	$-0.652159\pi$
$269$	−15.0899	−0.920047	−0.460024	+	0.887907i	$0.652159\pi$
$270$	0	0
$271$	3.21500	0.195297	0.0976486	−	0.995221i	$-0.468868\pi$
$271$	3.21500	0.195297	0.0976486	+	0.995221i	$0.468868\pi$
$272$	0	0
$273$	−8.35552	−0.505699
$274$	0	0
$275$	−10.6977	−0.645098
$276$	0	0
$277$	12.5164	0.752036	0.376018	−	0.926612i	$-0.377293\pi$
$277$	12.5164	0.752036	0.376018	+	0.926612i	$0.377293\pi$
$278$	0	0
$279$	2.00254	0.119889
$280$	0	0
$281$	12.8585	0.767072	0.383536	−	0.923526i	$-0.374706\pi$
$281$	12.8585	0.767072	0.383536	+	0.923526i	$0.374706\pi$
$282$	0	0
$283$	−8.81621	−0.524070	−0.262035	−	0.965058i	$-0.584394\pi$
$283$	−8.81621	−0.524070	−0.262035	+	0.965058i	$0.584394\pi$
$284$	0	0
$285$	4.31881	0.255824
$286$	0	0
$287$	−6.75310	−0.398623
$288$	0	0
$289$	9.99497	0.587940
$290$	0	0
$291$	−11.7344	−0.687881
$292$	0	0
$293$	−14.6682	−0.856924	−0.428462	−	0.903560i	$-0.640944\pi$
$293$	−14.6682	−0.856924	−0.428462	+	0.903560i	$0.640944\pi$
$294$	0	0
$295$	−1.22808	−0.0715017
$296$	0	0
$297$	−8.29504	−0.481327
$298$	0	0
$299$	22.6917	1.31230
$300$	0	0
$301$	−2.13230	−0.122904
$302$	0	0
$303$	−15.1364	−0.869566
$304$	0	0
$305$	16.4718	0.943172
$306$	0	0
$307$	−9.04718	−0.516350	−0.258175	−	0.966098i	$-0.583121\pi$
$307$	−9.04718	−0.516350	−0.258175	+	0.966098i	$0.583121\pi$
$308$	0	0
$309$	−12.4433	−0.707875
$310$	0	0
$311$	8.12467	0.460708	0.230354	−	0.973107i	$-0.426012\pi$
$311$	8.12467	0.460708	0.230354	+	0.973107i	$0.426012\pi$
$312$	0	0
$313$	7.65731	0.432817	0.216408	−	0.976303i	$-0.430566\pi$
$313$	7.65731	0.432817	0.216408	+	0.976303i	$0.430566\pi$
$314$	0	0
$315$	−1.90879	−0.107548
$316$	0	0
$317$	−25.2787	−1.41979	−0.709897	−	0.704306i	$-0.751258\pi$
$317$	−25.2787	−1.41979	−0.709897	+	0.704306i	$0.751258\pi$
$318$	0	0
$319$	1.31977	0.0738930
$320$	0	0
$321$	0.797624	0.0445190
$322$	0	0
$323$	9.31662	0.518391
$324$	0	0
$325$	−14.0428	−0.778954
$326$	0	0
$327$	33.4844	1.85169
$328$	0	0
$329$	5.60936	0.309254
$330$	0	0
$331$	−20.0214	−1.10047	−0.550237	−	0.835009i	$-0.685463\pi$
$331$	−20.0214	−1.10047	−0.550237	+	0.835009i	$0.685463\pi$
$332$	0	0
$333$	−12.1191	−0.664121
$334$	0	0
$335$	−14.7452	−0.805616
$336$	0	0
$337$	16.0377	0.873632	0.436816	−	0.899551i	$-0.356106\pi$
$337$	16.0377	0.873632	0.436816	+	0.899551i	$0.356106\pi$
$338$	0	0
$339$	18.1370	0.985068
$340$	0	0
$341$	3.45832	0.187278
$342$	0	0
$343$	13.3855	0.722747
$344$	0	0
$345$	14.6050	0.786309
$346$	0	0
$347$	−21.2603	−1.14131	−0.570656	−	0.821189i	$-0.693311\pi$
$347$	−21.2603	−1.14131	−0.570656	+	0.821189i	$0.693311\pi$
$348$	0	0
$349$	−17.9375	−0.960171	−0.480085	−	0.877222i	$-0.659394\pi$
$349$	−17.9375	−0.960171	−0.480085	+	0.877222i	$0.659394\pi$
$350$	0	0
$351$	−10.8888	−0.581201
$352$	0	0
$353$	4.12954	0.219793	0.109897	−	0.993943i	$-0.464948\pi$
$353$	4.12954	0.219793	0.109897	+	0.993943i	$0.464948\pi$
$354$	0	0
$355$	−2.62922	−0.139544
$356$	0	0
$357$	−11.6012	−0.614002
$358$	0	0
$359$	6.46751	0.341342	0.170671	−	0.985328i	$-0.445406\pi$
$359$	6.46751	0.341342	0.170671	+	0.985328i	$0.445406\pi$
$360$	0	0
$361$	−15.7846	−0.830769
$362$	0	0
$363$	−6.19698	−0.325257
$364$	0	0
$365$	−7.20799	−0.377283
$366$	0	0
$367$	−12.6490	−0.660270	−0.330135	−	0.943934i	$-0.607094\pi$
$367$	−12.6490	−0.660270	−0.330135	+	0.943934i	$0.607094\pi$
$368$	0	0
$369$	10.7663	0.560472
$370$	0	0
$371$	−2.73316	−0.141899
$372$	0	0
$373$	22.3923	1.15943	0.579714	−	0.814820i	$-0.303164\pi$
$373$	22.3923	1.15943	0.579714	+	0.814820i	$0.303164\pi$
$374$	0	0
$375$	−21.0809	−1.08861
$376$	0	0
$377$	1.73245	0.0892256
$378$	0	0
$379$	−10.4196	−0.535217	−0.267608	−	0.963528i	$-0.586233\pi$
$379$	−10.4196	−0.535217	−0.267608	+	0.963528i	$0.586233\pi$
$380$	0	0
$381$	−22.8038	−1.16828
$382$	0	0
$383$	−13.5146	−0.690563	−0.345281	−	0.938499i	$-0.612216\pi$
$383$	−13.5146	−0.690563	−0.345281	+	0.938499i	$0.612216\pi$
$384$	0	0
$385$	−3.29641	−0.168001
$386$	0	0
$387$	3.39948	0.172805
$388$	0	0
$389$	−16.3967	−0.831345	−0.415673	−	0.909514i	$-0.636454\pi$
$389$	−16.3967	−0.831345	−0.415673	+	0.909514i	$0.636454\pi$
$390$	0	0
$391$	31.5063	1.59334
$392$	0	0
$393$	12.1601	0.613394
$394$	0	0
$395$	0.943242	0.0474597
$396$	0	0
$397$	−7.08382	−0.355527	−0.177763	−	0.984073i	$-0.556886\pi$
$397$	−7.08382	−0.355527	−0.177763	+	0.984073i	$0.556886\pi$
$398$	0	0
$399$	−4.00386	−0.200444
$400$	0	0
$401$	11.3630	0.567440	0.283720	−	0.958907i	$-0.408431\pi$
$401$	11.3630	0.567440	0.283720	+	0.958907i	$0.408431\pi$
$402$	0	0
$403$	4.53969	0.226138
$404$	0	0
$405$	−12.5390	−0.623068
$406$	0	0
$407$	−20.9292	−1.03742
$408$	0	0
$409$	−32.2956	−1.59691	−0.798457	−	0.602052i	$-0.794350\pi$
$409$	−32.2956	−1.59691	−0.798457	+	0.602052i	$0.794350\pi$
$410$	0	0
$411$	27.6855	1.36563
$412$	0	0
$413$	1.13853	0.0560232
$414$	0	0
$415$	15.2811	0.750117
$416$	0	0
$417$	−18.9811	−0.929509
$418$	0	0
$419$	−12.7646	−0.623593	−0.311797	−	0.950149i	$-0.600931\pi$
$419$	−12.7646	−0.623593	−0.311797	+	0.950149i	$0.600931\pi$
$420$	0	0
$421$	12.7387	0.620845	0.310422	−	0.950599i	$-0.399530\pi$
$421$	12.7387	0.620845	0.310422	+	0.950599i	$0.399530\pi$
$422$	0	0
$423$	−8.94288	−0.434818
$424$	0	0
$425$	−19.4977	−0.945778
$426$	0	0
$427$	−15.2706	−0.738996
$428$	0	0
$429$	23.0049	1.11069
$430$	0	0
$431$	−14.0208	−0.675360	−0.337680	−	0.941261i	$-0.609642\pi$
$431$	−14.0208	−0.675360	−0.337680	+	0.941261i	$0.609642\pi$
$432$	0	0
$433$	−15.1063	−0.725963	−0.362982	−	0.931796i	$-0.618241\pi$
$433$	−15.1063	−0.725963	−0.362982	+	0.931796i	$0.618241\pi$
$434$	0	0
$435$	1.11505	0.0534627
$436$	0	0
$437$	10.8736	0.520154
$438$	0	0
$439$	2.57545	0.122919	0.0614597	−	0.998110i	$-0.480424\pi$
$439$	2.57545	0.122919	0.0614597	+	0.998110i	$0.480424\pi$
$440$	0	0
$441$	−9.78528	−0.465966
$442$	0	0
$443$	−4.14957	−0.197152	−0.0985760	−	0.995130i	$-0.531429\pi$
$443$	−4.14957	−0.197152	−0.0985760	+	0.995130i	$0.531429\pi$
$444$	0	0
$445$	0.343636	0.0162899
$446$	0	0
$447$	−0.290716	−0.0137504
$448$	0	0
$449$	11.7581	0.554900	0.277450	−	0.960740i	$-0.410511\pi$
$449$	11.7581	0.554900	0.277450	+	0.960740i	$0.410511\pi$
$450$	0	0
$451$	18.5930	0.875511
$452$	0	0
$453$	0.325262	0.0152821
$454$	0	0
$455$	−4.32716	−0.202860
$456$	0	0
$457$	5.84962	0.273634	0.136817	−	0.990596i	$-0.456313\pi$
$457$	5.84962	0.273634	0.136817	+	0.990596i	$0.456313\pi$
$458$	0	0
$459$	−15.1186	−0.705674
$460$	0	0
$461$	−21.1899	−0.986913	−0.493457	−	0.869770i	$-0.664267\pi$
$461$	−21.1899	−0.986913	−0.493457	+	0.869770i	$0.664267\pi$
$462$	0	0
$463$	−18.2654	−0.848864	−0.424432	−	0.905460i	$-0.639526\pi$
$463$	−18.2654	−0.848864	−0.424432	+	0.905460i	$0.639526\pi$
$464$	0	0
$465$	2.92188	0.135499
$466$	0	0
$467$	−31.3091	−1.44881	−0.724407	−	0.689373i	$-0.757886\pi$
$467$	−31.3091	−1.44881	−0.724407	+	0.689373i	$0.757886\pi$
$468$	0	0
$469$	13.6699	0.631218
$470$	0	0
$471$	47.1295	2.17161
$472$	0	0
$473$	5.87077	0.269938
$474$	0	0
$475$	−6.72914	−0.308754
$476$	0	0
$477$	4.35742	0.199513
$478$	0	0
$479$	15.9830	0.730281	0.365141	−	0.930952i	$-0.381021\pi$
$479$	15.9830	0.730281	0.365141	+	0.930952i	$0.381021\pi$
$480$	0	0
$481$	−27.4735	−1.25268
$482$	0	0
$483$	−13.5400	−0.616090
$484$	0	0
$485$	−6.07699	−0.275942
$486$	0	0
$487$	9.82136	0.445048	0.222524	−	0.974927i	$-0.428570\pi$
$487$	9.82136	0.445048	0.222524	+	0.974927i	$0.428570\pi$
$488$	0	0
$489$	27.2869	1.23395
$490$	0	0
$491$	−9.55488	−0.431206	−0.215603	−	0.976481i	$-0.569172\pi$
$491$	−9.55488	−0.431206	−0.215603	+	0.976481i	$0.569172\pi$
$492$	0	0
$493$	2.40542	0.108334
$494$	0	0
$495$	5.25539	0.236212
$496$	0	0
$497$	2.43748	0.109336
$498$	0	0
$499$	−9.73479	−0.435789	−0.217895	−	0.975972i	$-0.569919\pi$
$499$	−9.73479	−0.435789	−0.217895	+	0.975972i	$0.569919\pi$
$500$	0	0
$501$	48.2068	2.15372
$502$	0	0
$503$	−37.0775	−1.65320	−0.826602	−	0.562787i	$-0.809729\pi$
$503$	−37.0775	−1.65320	−0.826602	+	0.562787i	$0.809729\pi$
$504$	0	0
$505$	−7.83886	−0.348825
$506$	0	0
$507$	2.16315	0.0960689
$508$	0	0
$509$	35.9330	1.59270	0.796351	−	0.604834i	$-0.206761\pi$
$509$	35.9330	1.59270	0.796351	+	0.604834i	$0.206761\pi$
$510$	0	0
$511$	6.68235	0.295610
$512$	0	0
$513$	−5.21778	−0.230371
$514$	0	0
$515$	−6.44414	−0.283963
$516$	0	0
$517$	−15.4440	−0.679227
$518$	0	0
$519$	44.2422	1.94202
$520$	0	0
$521$	38.8629	1.70261	0.851307	−	0.524668i	$-0.175810\pi$
$521$	38.8629	1.70261	0.851307	+	0.524668i	$0.175810\pi$
$522$	0	0
$523$	12.7790	0.558786	0.279393	−	0.960177i	$-0.409867\pi$
$523$	12.7790	0.558786	0.279393	+	0.960177i	$0.409867\pi$
$524$	0	0
$525$	8.37924	0.365700
$526$	0	0
$527$	6.30313	0.274569
$528$	0	0
$529$	13.7715	0.598762
$530$	0	0
$531$	−1.81513	−0.0787698
$532$	0	0
$533$	24.4068	1.05718
$534$	0	0
$535$	0.413073	0.0178587
$536$	0	0
$537$	−30.7572	−1.32727
$538$	0	0
$539$	−16.8988	−0.727883
$540$	0	0
$541$	−14.9793	−0.644011	−0.322006	−	0.946738i	$-0.604357\pi$
$541$	−14.9793	−0.644011	−0.322006	+	0.946738i	$0.604357\pi$
$542$	0	0
$543$	−47.8833	−2.05487
$544$	0	0
$545$	17.3409	0.742802
$546$	0	0
$547$	−19.8845	−0.850201	−0.425101	−	0.905146i	$-0.639761\pi$
$547$	−19.8845	−0.850201	−0.425101	+	0.905146i	$0.639761\pi$
$548$	0	0
$549$	24.3456	1.03904
$550$	0	0
$551$	0.830168	0.0353663
$552$	0	0
$553$	−0.874457	−0.0371857
$554$	0	0
$555$	−17.6827	−0.750589
$556$	0	0
$557$	26.3583	1.11684	0.558418	−	0.829560i	$-0.311409\pi$
$557$	26.3583	1.11684	0.558418	+	0.829560i	$0.311409\pi$
$558$	0	0
$559$	7.70649	0.325950
$560$	0	0
$561$	31.9411	1.34856
$562$	0	0
$563$	−30.5426	−1.28722	−0.643609	−	0.765354i	$-0.722564\pi$
$563$	−30.5426	−1.28722	−0.643609	+	0.765354i	$0.722564\pi$
$564$	0	0
$565$	9.39280	0.395158
$566$	0	0
$567$	11.6246	0.488187
$568$	0	0
$569$	11.5650	0.484830	0.242415	−	0.970173i	$-0.422060\pi$
$569$	11.5650	0.484830	0.242415	+	0.970173i	$0.422060\pi$
$570$	0	0
$571$	−33.9047	−1.41887	−0.709433	−	0.704773i	$-0.751049\pi$
$571$	−33.9047	−1.41887	−0.709433	+	0.704773i	$0.751049\pi$
$572$	0	0
$573$	36.7758	1.53633
$574$	0	0
$575$	−22.7561	−0.948995
$576$	0	0
$577$	−45.5815	−1.89758	−0.948791	−	0.315904i	$-0.897692\pi$
$577$	−45.5815	−1.89758	−0.948791	+	0.315904i	$0.897692\pi$
$578$	0	0
$579$	2.27345	0.0944812
$580$	0	0
$581$	−14.1667	−0.587734
$582$	0	0
$583$	7.52510	0.311658
$584$	0	0
$585$	6.89869	0.285226
$586$	0	0
$587$	−47.1595	−1.94648	−0.973241	−	0.229787i	$-0.926197\pi$
$587$	−47.1595	−1.94648	−0.973241	+	0.229787i	$0.926197\pi$
$588$	0	0
$589$	2.17536	0.0896343
$590$	0	0
$591$	6.00523	0.247022
$592$	0	0
$593$	−13.5334	−0.555752	−0.277876	−	0.960617i	$-0.589630\pi$
$593$	−13.5334	−0.555752	−0.277876	+	0.960617i	$0.589630\pi$
$594$	0	0
$595$	−6.00804	−0.246305
$596$	0	0
$597$	16.0435	0.656615
$598$	0	0
$599$	−26.6936	−1.09067	−0.545336	−	0.838217i	$-0.683598\pi$
$599$	−26.6936	−1.09067	−0.545336	+	0.838217i	$0.683598\pi$
$600$	0	0
$601$	16.4173	0.669677	0.334838	−	0.942276i	$-0.391318\pi$
$601$	16.4173	0.669677	0.334838	+	0.942276i	$0.391318\pi$
$602$	0	0
$603$	−21.7936	−0.887506
$604$	0	0
$605$	−3.20929	−0.130476
$606$	0	0
$607$	−5.17188	−0.209920	−0.104960	−	0.994476i	$-0.533471\pi$
$607$	−5.17188	−0.209920	−0.104960	+	0.994476i	$0.533471\pi$
$608$	0	0
$609$	−1.03374	−0.0418892
$610$	0	0
$611$	−20.2732	−0.820165
$612$	0	0
$613$	47.5270	1.91960	0.959799	−	0.280689i	$-0.0905630\pi$
$613$	47.5270	1.91960	0.959799	+	0.280689i	$0.0905630\pi$
$614$	0	0
$615$	15.7089	0.633446
$616$	0	0
$617$	−46.9899	−1.89174	−0.945870	−	0.324545i	$-0.894789\pi$
$617$	−46.9899	−1.89174	−0.945870	+	0.324545i	$0.894789\pi$
$618$	0	0
$619$	17.3290	0.696512	0.348256	−	0.937399i	$-0.386774\pi$
$619$	17.3290	0.696512	0.348256	+	0.937399i	$0.386774\pi$
$620$	0	0
$621$	−17.6451	−0.708074
$622$	0	0
$623$	−0.318576	−0.0127635
$624$	0	0
$625$	7.84608	0.313843
$626$	0	0
$627$	11.0237	0.440243
$628$	0	0
$629$	−38.1455	−1.52096
$630$	0	0
$631$	29.5377	1.17588	0.587939	−	0.808905i	$-0.299940\pi$
$631$	29.5377	1.17588	0.587939	+	0.808905i	$0.299940\pi$
$632$	0	0
$633$	−28.9611	−1.15110
$634$	0	0
$635$	−11.8096	−0.468651
$636$	0	0
$637$	−22.1829	−0.878917
$638$	0	0
$639$	−3.88602	−0.153729
$640$	0	0
$641$	23.6635	0.934650	0.467325	−	0.884086i	$-0.345218\pi$
$641$	23.6635	0.934650	0.467325	+	0.884086i	$0.345218\pi$
$642$	0	0
$643$	25.1262	0.990880	0.495440	−	0.868642i	$-0.335007\pi$
$643$	25.1262	0.990880	0.495440	+	0.868642i	$0.335007\pi$
$644$	0	0
$645$	4.96011	0.195304
$646$	0	0
$647$	−25.1637	−0.989288	−0.494644	−	0.869096i	$-0.664702\pi$
$647$	−25.1637	−0.989288	−0.494644	+	0.869096i	$0.664702\pi$
$648$	0	0
$649$	−3.13465	−0.123046
$650$	0	0
$651$	−2.70880	−0.106166
$652$	0	0
$653$	−25.9591	−1.01586	−0.507929	−	0.861399i	$-0.669589\pi$
$653$	−25.9591	−1.01586	−0.507929	+	0.861399i	$0.669589\pi$
$654$	0	0
$655$	6.29745	0.246062
$656$	0	0
$657$	−10.6535	−0.415633
$658$	0	0
$659$	−5.37016	−0.209192	−0.104596	−	0.994515i	$-0.533355\pi$
$659$	−5.37016	−0.209192	−0.104596	+	0.994515i	$0.533355\pi$
$660$	0	0
$661$	16.7159	0.650174	0.325087	−	0.945684i	$-0.394606\pi$
$661$	16.7159	0.650174	0.325087	+	0.945684i	$0.394606\pi$
$662$	0	0
$663$	41.9288	1.62838
$664$	0	0
$665$	−2.07352	−0.0804076
$666$	0	0
$667$	2.80740	0.108703
$668$	0	0
$669$	11.3440	0.438583
$670$	0	0
$671$	42.0439	1.62309
$672$	0	0
$673$	−31.5558	−1.21639	−0.608193	−	0.793790i	$-0.708105\pi$
$673$	−31.5558	−1.21639	−0.608193	+	0.793790i	$0.708105\pi$
$674$	0	0
$675$	10.9197	0.420300
$676$	0	0
$677$	15.4499	0.593788	0.296894	−	0.954911i	$-0.404049\pi$
$677$	15.4499	0.593788	0.296894	+	0.954911i	$0.404049\pi$
$678$	0	0
$679$	5.63383	0.216207
$680$	0	0
$681$	−26.9171	−1.03146
$682$	0	0
$683$	42.1816	1.61404	0.807018	−	0.590527i	$-0.201080\pi$
$683$	42.1816	1.61404	0.807018	+	0.590527i	$0.201080\pi$
$684$	0	0
$685$	14.3378	0.547818
$686$	0	0
$687$	8.05724	0.307403
$688$	0	0
$689$	9.87811	0.376326
$690$	0	0
$691$	−35.8621	−1.36426	−0.682130	−	0.731231i	$-0.738946\pi$
$691$	−35.8621	−1.36426	−0.682130	+	0.731231i	$0.738946\pi$
$692$	0	0
$693$	−4.87215	−0.185078
$694$	0	0
$695$	−9.82993	−0.372871
$696$	0	0
$697$	33.8877	1.28359
$698$	0	0
$699$	14.3000	0.540874
$700$	0	0
$701$	27.2216	1.02815	0.514073	−	0.857747i	$-0.328136\pi$
$701$	27.2216	1.02815	0.514073	+	0.857747i	$0.328136\pi$
$702$	0	0
$703$	−13.1649	−0.496525
$704$	0	0
$705$	−13.0484	−0.491431
$706$	0	0
$707$	7.26722	0.273312
$708$	0	0
$709$	−31.7905	−1.19392	−0.596958	−	0.802272i	$-0.703624\pi$
$709$	−31.7905	−1.19392	−0.596958	+	0.802272i	$0.703624\pi$
$710$	0	0
$711$	1.39413	0.0522838
$712$	0	0
$713$	7.35649	0.275503
$714$	0	0
$715$	11.9138	0.445550
$716$	0	0
$717$	32.6265	1.21846
$718$	0	0
$719$	−15.8452	−0.590926	−0.295463	−	0.955354i	$-0.595474\pi$
$719$	−15.8452	−0.590926	−0.295463	+	0.955354i	$0.595474\pi$
$720$	0	0
$721$	5.97420	0.222491
$722$	0	0
$723$	13.5089	0.502402
$724$	0	0
$725$	−1.73736	−0.0645241
$726$	0	0
$727$	15.6071	0.578837	0.289418	−	0.957203i	$-0.406538\pi$
$727$	15.6071	0.578837	0.289418	+	0.957203i	$0.406538\pi$
$728$	0	0
$729$	0.292759	0.0108429
$730$	0	0
$731$	10.7001	0.395756
$732$	0	0
$733$	−16.5963	−0.612999	−0.306499	−	0.951871i	$-0.599158\pi$
$733$	−16.5963	−0.612999	−0.306499	+	0.951871i	$0.599158\pi$
$734$	0	0
$735$	−14.2775	−0.526635
$736$	0	0
$737$	−37.6368	−1.38637
$738$	0	0
$739$	−14.8369	−0.545785	−0.272892	−	0.962045i	$-0.587980\pi$
$739$	−14.8369	−0.545785	−0.272892	+	0.962045i	$0.587980\pi$
$740$	0	0
$741$	14.4706	0.531592
$742$	0	0
$743$	−30.2839	−1.11101	−0.555504	−	0.831514i	$-0.687475\pi$
$743$	−30.2839	−1.11101	−0.555504	+	0.831514i	$0.687475\pi$
$744$	0	0
$745$	−0.150556	−0.00551595
$746$	0	0
$747$	22.5856	0.826366
$748$	0	0
$749$	−0.382950	−0.0139927
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	13.4737	0.491010
$754$	0	0
$755$	0.168446	0.00613040
$756$	0	0
$757$	7.95776	0.289230	0.144615	−	0.989488i	$-0.453806\pi$
$757$	7.95776	0.289230	0.144615	+	0.989488i	$0.453806\pi$
$758$	0	0
$759$	37.2790	1.35314
$760$	0	0
$761$	−5.15733	−0.186953	−0.0934765	−	0.995621i	$-0.529798\pi$
$761$	−5.15733	−0.186953	−0.0934765	+	0.995621i	$0.529798\pi$
$762$	0	0
$763$	−16.0763	−0.582002
$764$	0	0
$765$	9.57848	0.346311
$766$	0	0
$767$	−4.11482	−0.148578
$768$	0	0
$769$	−30.1482	−1.08717	−0.543586	−	0.839353i	$-0.682934\pi$
$769$	−30.1482	−1.08717	−0.543586	+	0.839353i	$0.682934\pi$
$770$	0	0
$771$	3.26591	0.117619
$772$	0	0
$773$	44.6084	1.60445	0.802227	−	0.597019i	$-0.203648\pi$
$773$	44.6084	1.60445	0.802227	+	0.597019i	$0.203648\pi$
$774$	0	0
$775$	−4.55257	−0.163533
$776$	0	0
$777$	16.3932	0.588103
$778$	0	0
$779$	11.6955	0.419033
$780$	0	0
$781$	−6.71101	−0.240139
$782$	0	0
$783$	−1.34715	−0.0481434
$784$	0	0
$785$	24.4074	0.871138
$786$	0	0
$787$	−37.8061	−1.34764	−0.673821	−	0.738895i	$-0.735348\pi$
$787$	−37.8061	−1.34764	−0.673821	+	0.738895i	$0.735348\pi$
$788$	0	0
$789$	−23.0503	−0.820612
$790$	0	0
$791$	−8.70784	−0.309615
$792$	0	0
$793$	55.1905	1.95987
$794$	0	0
$795$	6.35784	0.225489
$796$	0	0
$797$	12.5860	0.445819	0.222910	−	0.974839i	$-0.428445\pi$
$797$	12.5860	0.445819	0.222910	+	0.974839i	$0.428445\pi$
$798$	0	0
$799$	−28.1483	−0.995815
$800$	0	0
$801$	0.507899	0.0179457
$802$	0	0
$803$	−18.3982	−0.649259
$804$	0	0
$805$	−7.01208	−0.247143
$806$	0	0
$807$	−32.5421	−1.14554
$808$	0	0
$809$	27.8387	0.978757	0.489378	−	0.872072i	$-0.337224\pi$
$809$	27.8387	0.978757	0.489378	+	0.872072i	$0.337224\pi$
$810$	0	0
$811$	47.6031	1.67157	0.835786	−	0.549056i	$-0.185013\pi$
$811$	47.6031	1.67157	0.835786	+	0.549056i	$0.185013\pi$
$812$	0	0
$813$	6.93329	0.243161
$814$	0	0
$815$	14.1313	0.494998
$816$	0	0
$817$	3.69285	0.129197
$818$	0	0
$819$	−6.39561	−0.223481
$820$	0	0
$821$	21.0081	0.733186	0.366593	−	0.930381i	$-0.380524\pi$
$821$	21.0081	0.733186	0.366593	+	0.930381i	$0.380524\pi$
$822$	0	0
$823$	16.9903	0.592245	0.296122	−	0.955150i	$-0.404306\pi$
$823$	16.9903	0.592245	0.296122	+	0.955150i	$0.404306\pi$
$824$	0	0
$825$	−23.0702	−0.803201
$826$	0	0
$827$	−0.312392	−0.0108629	−0.00543146	−	0.999985i	$-0.501729\pi$
$827$	−0.312392	−0.0108629	−0.00543146	+	0.999985i	$0.501729\pi$
$828$	0	0
$829$	−4.03882	−0.140274	−0.0701370	−	0.997537i	$-0.522344\pi$
$829$	−4.03882	−0.140274	−0.0701370	+	0.997537i	$0.522344\pi$
$830$	0	0
$831$	26.9921	0.936347
$832$	0	0
$833$	−30.7998	−1.06715
$834$	0	0
$835$	24.9653	0.863960
$836$	0	0
$837$	−3.53007	−0.122017
$838$	0	0
$839$	−47.1610	−1.62818	−0.814090	−	0.580739i	$-0.802764\pi$
$839$	−47.1610	−1.62818	−0.814090	+	0.580739i	$0.802764\pi$
$840$	0	0
$841$	−28.7857	−0.992609
$842$	0	0
$843$	27.7299	0.955069
$844$	0	0
$845$	1.12025	0.0385378
$846$	0	0
$847$	2.97526	0.102231
$848$	0	0
$849$	−19.0126	−0.652510
$850$	0	0
$851$	−44.5203	−1.52614
$852$	0	0
$853$	−36.4086	−1.24661	−0.623304	−	0.781980i	$-0.714210\pi$
$853$	−36.4086	−1.24661	−0.623304	+	0.781980i	$0.714210\pi$
$854$	0	0
$855$	3.30577	0.113055
$856$	0	0
$857$	23.3770	0.798544	0.399272	−	0.916832i	$-0.369263\pi$
$857$	23.3770	0.798544	0.399272	+	0.916832i	$0.369263\pi$
$858$	0	0
$859$	−23.3953	−0.798238	−0.399119	−	0.916899i	$-0.630684\pi$
$859$	−23.3953	−0.798238	−0.399119	+	0.916899i	$0.630684\pi$
$860$	0	0
$861$	−14.5634	−0.496319
$862$	0	0
$863$	9.88486	0.336485	0.168242	−	0.985746i	$-0.446191\pi$
$863$	9.88486	0.336485	0.168242	+	0.985746i	$0.446191\pi$
$864$	0	0
$865$	22.9122	0.779037
$866$	0	0
$867$	21.5546	0.732034
$868$	0	0
$869$	2.40760	0.0816724
$870$	0	0
$871$	−49.4054	−1.67404
$872$	0	0
$873$	−8.98189	−0.303991
$874$	0	0
$875$	10.1212	0.342159
$876$	0	0
$877$	36.0742	1.21814	0.609069	−	0.793117i	$-0.291543\pi$
$877$	36.0742	1.21814	0.609069	+	0.793117i	$0.291543\pi$
$878$	0	0
$879$	−31.6326	−1.06694
$880$	0	0
$881$	27.2176	0.916985	0.458493	−	0.888698i	$-0.348390\pi$
$881$	27.2176	0.916985	0.458493	+	0.888698i	$0.348390\pi$
$882$	0	0
$883$	8.59263	0.289165	0.144583	−	0.989493i	$-0.453816\pi$
$883$	8.59263	0.289165	0.144583	+	0.989493i	$0.453816\pi$
$884$	0	0
$885$	−2.64842	−0.0890256
$886$	0	0
$887$	−15.5178	−0.521036	−0.260518	−	0.965469i	$-0.583893\pi$
$887$	−15.5178	−0.521036	−0.260518	+	0.965469i	$0.583893\pi$
$888$	0	0
$889$	10.9484	0.367199
$890$	0	0
$891$	−32.0055	−1.07223
$892$	0	0
$893$	−9.71466	−0.325089
$894$	0	0
$895$	−15.9285	−0.532432
$896$	0	0
$897$	48.9358	1.63392
$898$	0	0
$899$	0.561647	0.0187320
$900$	0	0
$901$	13.7153	0.456921
$902$	0	0
$903$	−4.59840	−0.153025
$904$	0	0
$905$	−24.7978	−0.824307
$906$	0	0
$907$	−24.3134	−0.807313	−0.403657	−	0.914911i	$-0.632261\pi$
$907$	−24.3134	−0.807313	−0.403657	+	0.914911i	$0.632261\pi$
$908$	0	0
$909$	−11.5860	−0.384282
$910$	0	0
$911$	15.3373	0.508148	0.254074	−	0.967185i	$-0.418229\pi$
$911$	15.3373	0.508148	0.254074	+	0.967185i	$0.418229\pi$
$912$	0	0
$913$	39.0046	1.29086
$914$	0	0
$915$	35.5222	1.17433
$916$	0	0
$917$	−5.83821	−0.192795
$918$	0	0
$919$	−25.2258	−0.832121	−0.416061	−	0.909337i	$-0.636590\pi$
$919$	−25.2258	−0.832121	−0.416061	+	0.909337i	$0.636590\pi$
$920$	0	0
$921$	−19.5107	−0.642898
$922$	0	0
$923$	−8.80947	−0.289967
$924$	0	0
$925$	27.5514	0.905886
$926$	0	0
$927$	−9.52454	−0.312827
$928$	0	0
$929$	−27.5440	−0.903689	−0.451845	−	0.892097i	$-0.649234\pi$
$929$	−27.5440	−0.903689	−0.451845	+	0.892097i	$0.649234\pi$
$930$	0	0
$931$	−10.6298	−0.348376
$932$	0	0
$933$	17.5212	0.573620
$934$	0	0
$935$	16.5417	0.540970
$936$	0	0
$937$	33.9734	1.10986	0.554931	−	0.831897i	$-0.312745\pi$
$937$	33.9734	1.10986	0.554931	+	0.831897i	$0.312745\pi$
$938$	0	0
$939$	16.5133	0.538893
$940$	0	0
$941$	32.6469	1.06426	0.532129	−	0.846663i	$-0.321392\pi$
$941$	32.6469	1.06426	0.532129	+	0.846663i	$0.321392\pi$
$942$	0	0
$943$	39.5508	1.28795
$944$	0	0
$945$	3.36480	0.109457
$946$	0	0
$947$	−60.1595	−1.95492	−0.977461	−	0.211118i	$-0.932290\pi$
$947$	−60.1595	−1.95492	−0.977461	+	0.211118i	$0.932290\pi$
$948$	0	0
$949$	−24.1511	−0.783979
$950$	0	0
$951$	−54.5147	−1.76776
$952$	0	0
$953$	19.5815	0.634307	0.317154	−	0.948374i	$-0.397273\pi$
$953$	19.5815	0.634307	0.317154	+	0.948374i	$0.397273\pi$
$954$	0	0
$955$	19.0454	0.616296
$956$	0	0
$957$	2.84615	0.0920029
$958$	0	0
$959$	−13.2922	−0.429228
$960$	0	0
$961$	−29.5283	−0.952525
$962$	0	0
$963$	0.610529	0.0196740
$964$	0	0
$965$	1.17737	0.0379009
$966$	0	0
$967$	25.5911	0.822954	0.411477	−	0.911420i	$-0.365013\pi$
$967$	25.5911	0.822954	0.411477	+	0.911420i	$0.365013\pi$
$968$	0	0
$969$	20.0917	0.645440
$970$	0	0
$971$	29.3292	0.941220	0.470610	−	0.882341i	$-0.344034\pi$
$971$	29.3292	0.941220	0.470610	+	0.882341i	$0.344034\pi$
$972$	0	0
$973$	9.11309	0.292152
$974$	0	0
$975$	−30.2840	−0.969863
$976$	0	0
$977$	−27.0385	−0.865038	−0.432519	−	0.901625i	$-0.642375\pi$
$977$	−27.0385	−0.865038	−0.432519	+	0.901625i	$0.642375\pi$
$978$	0	0
$979$	0.877123	0.0280330
$980$	0	0
$981$	25.6301	0.818307
$982$	0	0
$983$	11.4927	0.366561	0.183281	−	0.983061i	$-0.441328\pi$
$983$	11.4927	0.366561	0.183281	+	0.983061i	$0.441328\pi$
$984$	0	0
$985$	3.10999	0.0990924
$986$	0	0
$987$	12.0969	0.385047
$988$	0	0
$989$	12.4882	0.397102
$990$	0	0
$991$	−50.0434	−1.58968	−0.794841	−	0.606818i	$-0.792446\pi$
$991$	−50.0434	−1.58968	−0.794841	+	0.606818i	$0.792446\pi$
$992$	0	0
$993$	−43.1770	−1.37018
$994$	0	0
$995$	8.30858	0.263400
$996$	0	0
$997$	52.7736	1.67136	0.835678	−	0.549219i	$-0.185075\pi$
$997$	52.7736	1.67136	0.835678	+	0.549219i	$0.185075\pi$
$998$	0	0
$999$	21.3634	0.675909

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.40	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.40	✓	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)

\(f(q)\)	\(=\)	\(q+2.15655 q^{3} +1.11683 q^{5} -1.03539 q^{7} +1.65070 q^{9} +O(q^{10})\)	\(q+2.15655 q^{3} +1.11683 q^{5} -1.03539 q^{7} +1.65070 q^{9} +2.85069 q^{11} +3.74207 q^{13} +2.40850 q^{15} +5.19567 q^{17} +1.79315 q^{19} -2.23286 q^{21} +6.06395 q^{23} -3.75269 q^{25} -2.90984 q^{27} +0.462966 q^{29} +1.21315 q^{31} +6.14765 q^{33} -1.15635 q^{35} -7.34179 q^{37} +8.06994 q^{39} +6.52229 q^{41} +2.05942 q^{43} +1.84355 q^{45} -5.41764 q^{47} -5.92797 q^{49} +11.2047 q^{51} +2.63975 q^{53} +3.18374 q^{55} +3.86702 q^{57} -1.09961 q^{59} +14.7487 q^{61} -1.70911 q^{63} +4.17926 q^{65} -13.2027 q^{67} +13.0772 q^{69} -2.35417 q^{71} -6.45396 q^{73} -8.09284 q^{75} -2.95157 q^{77} +0.844569 q^{79} -11.2273 q^{81} +13.6825 q^{83} +5.80269 q^{85} +0.998407 q^{87} +0.307688 q^{89} -3.87449 q^{91} +2.61622 q^{93} +2.00265 q^{95} -5.44127 q^{97} +4.70562 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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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