LMFDB - Embedded newform 6008.2.a.e.1.34

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			1.34
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)

$f(q)$	$=$	$q+1.38028 q^{3} -3.43256 q^{5} +3.30717 q^{7} -1.09484 q^{9} +O(q^{10})$	$q+1.38028 q^{3} -3.43256 q^{5} +3.30717 q^{7} -1.09484 q^{9} -0.715209 q^{11} +5.15563 q^{13} -4.73789 q^{15} -1.52677 q^{17} +7.44406 q^{19} +4.56481 q^{21} -3.93916 q^{23} +6.78248 q^{25} -5.65201 q^{27} -4.20561 q^{29} +6.78288 q^{31} -0.987187 q^{33} -11.3521 q^{35} -5.07948 q^{37} +7.11620 q^{39} +1.57167 q^{41} -1.14412 q^{43} +3.75809 q^{45} +4.06761 q^{47} +3.93737 q^{49} -2.10737 q^{51} +3.64581 q^{53} +2.45500 q^{55} +10.2749 q^{57} +12.5592 q^{59} -5.54003 q^{61} -3.62081 q^{63} -17.6970 q^{65} -2.31555 q^{67} -5.43713 q^{69} +6.03969 q^{71} -6.14485 q^{73} +9.36170 q^{75} -2.36532 q^{77} +12.9617 q^{79} -4.51682 q^{81} +8.06286 q^{83} +5.24074 q^{85} -5.80490 q^{87} -15.5246 q^{89} +17.0506 q^{91} +9.36225 q^{93} -25.5522 q^{95} +4.13304 q^{97} +0.783037 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.38028	0.796903	0.398452	−	0.917189i	$-0.369548\pi$
$3$	1.38028	0.796903	0.398452	+	0.917189i	$0.369548\pi$
$4$	0	0
$5$	−3.43256	−1.53509	−0.767544	−	0.640996i	$-0.778521\pi$
$5$	−3.43256	−1.53509	−0.767544	+	0.640996i	$0.778521\pi$
$6$	0	0
$7$	3.30717	1.24999	0.624996	−	0.780628i	$-0.285100\pi$
$7$	3.30717	1.24999	0.624996	+	0.780628i	$0.285100\pi$
$8$	0	0
$9$	−1.09484	−0.364945
$10$	0	0
$11$	−0.715209	−0.215644	−0.107822	−	0.994170i	$-0.534388\pi$
$11$	−0.715209	−0.215644	−0.107822	+	0.994170i	$0.534388\pi$
$12$	0	0
$13$	5.15563	1.42992	0.714958	−	0.699168i	$-0.246446\pi$
$13$	5.15563	1.42992	0.714958	+	0.699168i	$0.246446\pi$
$14$	0	0
$15$	−4.73789	−1.22332
$16$	0	0
$17$	−1.52677	−0.370297	−0.185148	−	0.982711i	$-0.559277\pi$
$17$	−1.52677	−0.370297	−0.185148	+	0.982711i	$0.559277\pi$
$18$	0	0
$19$	7.44406	1.70779	0.853893	−	0.520449i	$-0.174235\pi$
$19$	7.44406	1.70779	0.853893	+	0.520449i	$0.174235\pi$
$20$	0	0
$21$	4.56481	0.996123
$22$	0	0
$23$	−3.93916	−0.821372	−0.410686	−	0.911777i	$-0.634711\pi$
$23$	−3.93916	−0.821372	−0.410686	+	0.911777i	$0.634711\pi$
$24$	0	0
$25$	6.78248	1.35650
$26$	0	0
$27$	−5.65201	−1.08773
$28$	0	0
$29$	−4.20561	−0.780961	−0.390481	−	0.920611i	$-0.627691\pi$
$29$	−4.20561	−0.780961	−0.390481	+	0.920611i	$0.627691\pi$
$30$	0	0
$31$	6.78288	1.21824	0.609121	−	0.793078i	$-0.291523\pi$
$31$	6.78288	1.21824	0.609121	+	0.793078i	$0.291523\pi$
$32$	0	0
$33$	−0.987187	−0.171847
$34$	0	0
$35$	−11.3521	−1.91885
$36$	0	0
$37$	−5.07948	−0.835061	−0.417530	−	0.908663i	$-0.637104\pi$
$37$	−5.07948	−0.835061	−0.417530	+	0.908663i	$0.637104\pi$
$38$	0	0
$39$	7.11620	1.13950
$40$	0	0
$41$	1.57167	0.245453	0.122727	−	0.992441i	$-0.460836\pi$
$41$	1.57167	0.245453	0.122727	+	0.992441i	$0.460836\pi$
$42$	0	0
$43$	−1.14412	−0.174476	−0.0872381	−	0.996187i	$-0.527804\pi$
$43$	−1.14412	−0.174476	−0.0872381	+	0.996187i	$0.527804\pi$
$44$	0	0
$45$	3.75809	0.560223
$46$	0	0
$47$	4.06761	0.593322	0.296661	−	0.954983i	$-0.404127\pi$
$47$	4.06761	0.593322	0.296661	+	0.954983i	$0.404127\pi$
$48$	0	0
$49$	3.93737	0.562482
$50$	0	0
$51$	−2.10737	−0.295091
$52$	0	0
$53$	3.64581	0.500790	0.250395	−	0.968144i	$-0.419439\pi$
$53$	3.64581	0.500790	0.250395	+	0.968144i	$0.419439\pi$
$54$	0	0
$55$	2.45500	0.331032
$56$	0	0
$57$	10.2749	1.36094
$58$	0	0
$59$	12.5592	1.63507	0.817536	−	0.575877i	$-0.195339\pi$
$59$	12.5592	1.63507	0.817536	+	0.575877i	$0.195339\pi$
$60$	0	0
$61$	−5.54003	−0.709328	−0.354664	−	0.934994i	$-0.615405\pi$
$61$	−5.54003	−0.709328	−0.354664	+	0.934994i	$0.615405\pi$
$62$	0	0
$63$	−3.62081	−0.456179
$64$	0	0
$65$	−17.6970	−2.19505
$66$	0	0
$67$	−2.31555	−0.282890	−0.141445	−	0.989946i	$-0.545175\pi$
$67$	−2.31555	−0.282890	−0.141445	+	0.989946i	$0.545175\pi$
$68$	0	0
$69$	−5.43713	−0.654554
$70$	0	0
$71$	6.03969	0.716780	0.358390	−	0.933572i	$-0.383326\pi$
$71$	6.03969	0.716780	0.358390	+	0.933572i	$0.383326\pi$
$72$	0	0
$73$	−6.14485	−0.719201	−0.359600	−	0.933106i	$-0.617087\pi$
$73$	−6.14485	−0.719201	−0.359600	+	0.933106i	$0.617087\pi$
$74$	0	0
$75$	9.36170	1.08100
$76$	0	0
$77$	−2.36532	−0.269553
$78$	0	0
$79$	12.9617	1.45830	0.729152	−	0.684351i	$-0.239915\pi$
$79$	12.9617	1.45830	0.729152	+	0.684351i	$0.239915\pi$
$80$	0	0
$81$	−4.51682	−0.501869
$82$	0	0
$83$	8.06286	0.885014	0.442507	−	0.896765i	$-0.354089\pi$
$83$	8.06286	0.885014	0.442507	+	0.896765i	$0.354089\pi$
$84$	0	0
$85$	5.24074	0.568438
$86$	0	0
$87$	−5.80490	−0.622351
$88$	0	0
$89$	−15.5246	−1.64560	−0.822802	−	0.568328i	$-0.807590\pi$
$89$	−15.5246	−1.64560	−0.822802	+	0.568328i	$0.807590\pi$
$90$	0	0
$91$	17.0506	1.78738
$92$	0	0
$93$	9.36225	0.970820
$94$	0	0
$95$	−25.5522	−2.62160
$96$	0	0
$97$	4.13304	0.419647	0.209824	−	0.977739i	$-0.432711\pi$
$97$	4.13304	0.419647	0.209824	+	0.977739i	$0.432711\pi$
$98$	0	0
$99$	0.783037	0.0786982
$100$	0	0
$101$	−3.07921	−0.306393	−0.153197	−	0.988196i	$-0.548957\pi$
$101$	−3.07921	−0.306393	−0.153197	+	0.988196i	$0.548957\pi$
$102$	0	0
$103$	−12.9048	−1.27155	−0.635775	−	0.771874i	$-0.719319\pi$
$103$	−12.9048	−1.27155	−0.635775	+	0.771874i	$0.719319\pi$
$104$	0	0
$105$	−15.6690	−1.52914
$106$	0	0
$107$	18.1027	1.75006	0.875029	−	0.484071i	$-0.160842\pi$
$107$	18.1027	1.75006	0.875029	+	0.484071i	$0.160842\pi$
$108$	0	0
$109$	−1.37607	−0.131804	−0.0659018	−	0.997826i	$-0.520992\pi$
$109$	−1.37607	−0.131804	−0.0659018	+	0.997826i	$0.520992\pi$
$110$	0	0
$111$	−7.01108	−0.665463
$112$	0	0
$113$	17.9587	1.68941	0.844707	−	0.535228i	$-0.179774\pi$
$113$	17.9587	1.68941	0.844707	+	0.535228i	$0.179774\pi$
$114$	0	0
$115$	13.5214	1.26088
$116$	0	0
$117$	−5.64457	−0.521841
$118$	0	0
$119$	−5.04929	−0.462868
$120$	0	0
$121$	−10.4885	−0.953498
$122$	0	0
$123$	2.16934	0.195602
$124$	0	0
$125$	−6.11848	−0.547253
$126$	0	0
$127$	20.8271	1.84810	0.924052	−	0.382267i	$-0.124857\pi$
$127$	20.8271	1.84810	0.924052	+	0.382267i	$0.124857\pi$
$128$	0	0
$129$	−1.57920	−0.139041
$130$	0	0
$131$	6.81472	0.595404	0.297702	−	0.954659i	$-0.403780\pi$
$131$	6.81472	0.595404	0.297702	+	0.954659i	$0.403780\pi$
$132$	0	0
$133$	24.6188	2.13472
$134$	0	0
$135$	19.4009	1.66976
$136$	0	0
$137$	−13.4759	−1.15133	−0.575664	−	0.817686i	$-0.695256\pi$
$137$	−13.4759	−1.15133	−0.575664	+	0.817686i	$0.695256\pi$
$138$	0	0
$139$	−11.6302	−0.986460	−0.493230	−	0.869899i	$-0.664184\pi$
$139$	−11.6302	−0.986460	−0.493230	+	0.869899i	$0.664184\pi$
$140$	0	0
$141$	5.61443	0.472820
$142$	0	0
$143$	−3.68736	−0.308352
$144$	0	0
$145$	14.4360	1.19884
$146$	0	0
$147$	5.43466	0.448243
$148$	0	0
$149$	−15.7126	−1.28723	−0.643614	−	0.765350i	$-0.722566\pi$
$149$	−15.7126	−1.28723	−0.643614	+	0.765350i	$0.722566\pi$
$150$	0	0
$151$	6.13637	0.499371	0.249685	−	0.968327i	$-0.419673\pi$
$151$	6.13637	0.499371	0.249685	+	0.968327i	$0.419673\pi$
$152$	0	0
$153$	1.67157	0.135138
$154$	0	0
$155$	−23.2827	−1.87011
$156$	0	0
$157$	6.25511	0.499213	0.249606	−	0.968347i	$-0.419699\pi$
$157$	6.25511	0.499213	0.249606	+	0.968347i	$0.419699\pi$
$158$	0	0
$159$	5.03222	0.399081
$160$	0	0
$161$	−13.0275	−1.02671
$162$	0	0
$163$	24.7242	1.93655	0.968274	−	0.249891i	$-0.0803947\pi$
$163$	24.7242	1.93655	0.968274	+	0.249891i	$0.0803947\pi$
$164$	0	0
$165$	3.38858	0.263801
$166$	0	0
$167$	0.736630	0.0570021	0.0285011	−	0.999594i	$-0.490927\pi$
$167$	0.736630	0.0570021	0.0285011	+	0.999594i	$0.490927\pi$
$168$	0	0
$169$	13.5805	1.04466
$170$	0	0
$171$	−8.15003	−0.623249
$172$	0	0
$173$	12.0400	0.915387	0.457694	−	0.889110i	$-0.348676\pi$
$173$	12.0400	0.915387	0.457694	+	0.889110i	$0.348676\pi$
$174$	0	0
$175$	22.4308	1.69561
$176$	0	0
$177$	17.3352	1.30299
$178$	0	0
$179$	13.6240	1.01831	0.509154	−	0.860676i	$-0.329958\pi$
$179$	13.6240	1.01831	0.509154	+	0.860676i	$0.329958\pi$
$180$	0	0
$181$	11.7372	0.872421	0.436210	−	0.899845i	$-0.356320\pi$
$181$	11.7372	0.872421	0.436210	+	0.899845i	$0.356320\pi$
$182$	0	0
$183$	−7.64677	−0.565266
$184$	0	0
$185$	17.4356	1.28189
$186$	0	0
$187$	1.09196	0.0798521
$188$	0	0
$189$	−18.6921	−1.35965
$190$	0	0
$191$	−1.41855	−0.102643	−0.0513215	−	0.998682i	$-0.516343\pi$
$191$	−1.41855	−0.102643	−0.0513215	+	0.998682i	$0.516343\pi$
$192$	0	0
$193$	13.5267	0.973672	0.486836	−	0.873493i	$-0.338151\pi$
$193$	13.5267	0.973672	0.486836	+	0.873493i	$0.338151\pi$
$194$	0	0
$195$	−24.4268	−1.74924
$196$	0	0
$197$	−6.85311	−0.488264	−0.244132	−	0.969742i	$-0.578503\pi$
$197$	−6.85311	−0.488264	−0.244132	+	0.969742i	$0.578503\pi$
$198$	0	0
$199$	23.9050	1.69458	0.847291	−	0.531129i	$-0.178232\pi$
$199$	23.9050	1.69458	0.847291	+	0.531129i	$0.178232\pi$
$200$	0	0
$201$	−3.19610	−0.225436
$202$	0	0
$203$	−13.9087	−0.976196
$204$	0	0
$205$	−5.39484	−0.376792
$206$	0	0
$207$	4.31274	0.299756
$208$	0	0
$209$	−5.32406	−0.368273
$210$	0	0
$211$	11.0715	0.762192	0.381096	−	0.924536i	$-0.375547\pi$
$211$	11.0715	0.762192	0.381096	+	0.924536i	$0.375547\pi$
$212$	0	0
$213$	8.33645	0.571204
$214$	0	0
$215$	3.92725	0.267836
$216$	0	0
$217$	22.4321	1.52279
$218$	0	0
$219$	−8.48160	−0.573133
$220$	0	0
$221$	−7.87148	−0.529493
$222$	0	0
$223$	15.8141	1.05899	0.529494	−	0.848314i	$-0.322382\pi$
$223$	15.8141	1.05899	0.529494	+	0.848314i	$0.322382\pi$
$224$	0	0
$225$	−7.42571	−0.495047
$226$	0	0
$227$	−11.9752	−0.794822	−0.397411	−	0.917641i	$-0.630091\pi$
$227$	−11.9752	−0.794822	−0.397411	+	0.917641i	$0.630091\pi$
$228$	0	0
$229$	6.96277	0.460113	0.230056	−	0.973177i	$-0.426109\pi$
$229$	6.96277	0.460113	0.230056	+	0.973177i	$0.426109\pi$
$230$	0	0
$231$	−3.26479	−0.214808
$232$	0	0
$233$	−19.3936	−1.27052	−0.635258	−	0.772300i	$-0.719106\pi$
$233$	−19.3936	−1.27052	−0.635258	+	0.772300i	$0.719106\pi$
$234$	0	0
$235$	−13.9623	−0.910801
$236$	0	0
$237$	17.8907	1.16213
$238$	0	0
$239$	15.1021	0.976876	0.488438	−	0.872599i	$-0.337567\pi$
$239$	15.1021	0.976876	0.488438	+	0.872599i	$0.337567\pi$
$240$	0	0
$241$	−14.5806	−0.939222	−0.469611	−	0.882873i	$-0.655606\pi$
$241$	−14.5806	−0.939222	−0.469611	+	0.882873i	$0.655606\pi$
$242$	0	0
$243$	10.7216	0.687788
$244$	0	0
$245$	−13.5153	−0.863459
$246$	0	0
$247$	38.3789	2.44199
$248$	0	0
$249$	11.1290	0.705271
$250$	0	0
$251$	−4.72647	−0.298332	−0.149166	−	0.988812i	$-0.547659\pi$
$251$	−4.72647	−0.298332	−0.149166	+	0.988812i	$0.547659\pi$
$252$	0	0
$253$	2.81733	0.177124
$254$	0	0
$255$	7.23367	0.452990
$256$	0	0
$257$	0.0945381	0.00589713	0.00294856	−	0.999996i	$-0.499061\pi$
$257$	0.0945381	0.00589713	0.00294856	+	0.999996i	$0.499061\pi$
$258$	0	0
$259$	−16.7987	−1.04382
$260$	0	0
$261$	4.60445	0.285008
$262$	0	0
$263$	15.6136	0.962774	0.481387	−	0.876508i	$-0.340133\pi$
$263$	15.6136	0.962774	0.481387	+	0.876508i	$0.340133\pi$
$264$	0	0
$265$	−12.5145	−0.768757
$266$	0	0
$267$	−21.4282	−1.31139
$268$	0	0
$269$	12.0602	0.735323	0.367662	−	0.929960i	$-0.380158\pi$
$269$	12.0602	0.735323	0.367662	+	0.929960i	$0.380158\pi$
$270$	0	0
$271$	−4.82291	−0.292971	−0.146485	−	0.989213i	$-0.546796\pi$
$271$	−4.82291	−0.292971	−0.146485	+	0.989213i	$0.546796\pi$
$272$	0	0
$273$	23.5345	1.42437
$274$	0	0
$275$	−4.85089	−0.292520
$276$	0	0
$277$	−10.8417	−0.651415	−0.325707	−	0.945471i	$-0.605602\pi$
$277$	−10.8417	−0.651415	−0.325707	+	0.945471i	$0.605602\pi$
$278$	0	0
$279$	−7.42614	−0.444592
$280$	0	0
$281$	−11.5903	−0.691420	−0.345710	−	0.938341i	$-0.612362\pi$
$281$	−11.5903	−0.691420	−0.345710	+	0.938341i	$0.612362\pi$
$282$	0	0
$283$	−12.2298	−0.726989	−0.363494	−	0.931596i	$-0.618416\pi$
$283$	−12.2298	−0.726989	−0.363494	+	0.931596i	$0.618416\pi$
$284$	0	0
$285$	−35.2691	−2.08916
$286$	0	0
$287$	5.19777	0.306815
$288$	0	0
$289$	−14.6690	−0.862880
$290$	0	0
$291$	5.70475	0.334418
$292$	0	0
$293$	11.8915	0.694710	0.347355	−	0.937734i	$-0.387080\pi$
$293$	11.8915	0.694710	0.347355	+	0.937734i	$0.387080\pi$
$294$	0	0
$295$	−43.1103	−2.50998
$296$	0	0
$297$	4.04237	0.234562
$298$	0	0
$299$	−20.3089	−1.17449
$300$	0	0
$301$	−3.78379	−0.218094
$302$	0	0
$303$	−4.25017	−0.244166
$304$	0	0
$305$	19.0165	1.08888
$306$	0	0
$307$	−22.9784	−1.31144	−0.655722	−	0.755002i	$-0.727636\pi$
$307$	−22.9784	−1.31144	−0.655722	+	0.755002i	$0.727636\pi$
$308$	0	0
$309$	−17.8122	−1.01330
$310$	0	0
$311$	10.1728	0.576848	0.288424	−	0.957503i	$-0.406869\pi$
$311$	10.1728	0.576848	0.288424	+	0.957503i	$0.406869\pi$
$312$	0	0
$313$	10.0258	0.566692	0.283346	−	0.959018i	$-0.408556\pi$
$313$	10.0258	0.566692	0.283346	+	0.959018i	$0.408556\pi$
$314$	0	0
$315$	12.4287	0.700275
$316$	0	0
$317$	21.3486	1.19906	0.599528	−	0.800354i	$-0.295355\pi$
$317$	21.3486	1.19906	0.599528	+	0.800354i	$0.295355\pi$
$318$	0	0
$319$	3.00789	0.168409
$320$	0	0
$321$	24.9868	1.39463
$322$	0	0
$323$	−11.3654	−0.632387
$324$	0	0
$325$	34.9680	1.93967
$326$	0	0
$327$	−1.89936	−0.105035
$328$	0	0
$329$	13.4523	0.741648
$330$	0	0
$331$	23.7276	1.30419	0.652093	−	0.758139i	$-0.273891\pi$
$331$	23.7276	1.30419	0.652093	+	0.758139i	$0.273891\pi$
$332$	0	0
$333$	5.56120	0.304752
$334$	0	0
$335$	7.94827	0.434261
$336$	0	0
$337$	22.0819	1.20288	0.601438	−	0.798919i	$-0.294595\pi$
$337$	22.0819	1.20288	0.601438	+	0.798919i	$0.294595\pi$
$338$	0	0
$339$	24.7880	1.34630
$340$	0	0
$341$	−4.85118	−0.262706
$342$	0	0
$343$	−10.1286	−0.546895
$344$	0	0
$345$	18.6633	1.00480
$346$	0	0
$347$	23.7902	1.27712	0.638562	−	0.769570i	$-0.279529\pi$
$347$	23.7902	1.27712	0.638562	+	0.769570i	$0.279529\pi$
$348$	0	0
$349$	23.9757	1.28339	0.641695	−	0.766960i	$-0.278232\pi$
$349$	23.9757	1.28339	0.641695	+	0.766960i	$0.278232\pi$
$350$	0	0
$351$	−29.1397	−1.55536
$352$	0	0
$353$	1.63502	0.0870233	0.0435117	−	0.999053i	$-0.486145\pi$
$353$	1.63502	0.0870233	0.0435117	+	0.999053i	$0.486145\pi$
$354$	0	0
$355$	−20.7316	−1.10032
$356$	0	0
$357$	−6.96942	−0.368861
$358$	0	0
$359$	−12.8539	−0.678405	−0.339202	−	0.940713i	$-0.610157\pi$
$359$	−12.8539	−0.678405	−0.339202	+	0.940713i	$0.610157\pi$
$360$	0	0
$361$	36.4141	1.91653
$362$	0	0
$363$	−14.4770	−0.759845
$364$	0	0
$365$	21.0926	1.10404
$366$	0	0
$367$	27.2219	1.42097	0.710486	−	0.703711i	$-0.248475\pi$
$367$	27.2219	1.42097	0.710486	+	0.703711i	$0.248475\pi$
$368$	0	0
$369$	−1.72072	−0.0895770
$370$	0	0
$371$	12.0573	0.625984
$372$	0	0
$373$	16.3061	0.844298	0.422149	−	0.906527i	$-0.361276\pi$
$373$	16.3061	0.844298	0.422149	+	0.906527i	$0.361276\pi$
$374$	0	0
$375$	−8.44519	−0.436108
$376$	0	0
$377$	−21.6826	−1.11671
$378$	0	0
$379$	−6.96706	−0.357874	−0.178937	−	0.983861i	$-0.557266\pi$
$379$	−6.96706	−0.357874	−0.178937	+	0.983861i	$0.557266\pi$
$380$	0	0
$381$	28.7471	1.47276
$382$	0	0
$383$	−28.0321	−1.43238	−0.716188	−	0.697907i	$-0.754115\pi$
$383$	−28.0321	−1.43238	−0.716188	+	0.697907i	$0.754115\pi$
$384$	0	0
$385$	8.11910	0.413788
$386$	0	0
$387$	1.25262	0.0636743
$388$	0	0
$389$	25.1118	1.27322	0.636610	−	0.771186i	$-0.280336\pi$
$389$	25.1118	1.27322	0.636610	+	0.771186i	$0.280336\pi$
$390$	0	0
$391$	6.01420	0.304151
$392$	0	0
$393$	9.40619	0.474480
$394$	0	0
$395$	−44.4918	−2.23863
$396$	0	0
$397$	−13.0339	−0.654154	−0.327077	−	0.944998i	$-0.606064\pi$
$397$	−13.0339	−0.654154	−0.327077	+	0.944998i	$0.606064\pi$
$398$	0	0
$399$	33.9807	1.70116
$400$	0	0
$401$	−12.2412	−0.611296	−0.305648	−	0.952145i	$-0.598873\pi$
$401$	−12.2412	−0.611296	−0.305648	+	0.952145i	$0.598873\pi$
$402$	0	0
$403$	34.9700	1.74198
$404$	0	0
$405$	15.5043	0.770414
$406$	0	0
$407$	3.63289	0.180076
$408$	0	0
$409$	−5.12069	−0.253202	−0.126601	−	0.991954i	$-0.540407\pi$
$409$	−5.12069	−0.253202	−0.126601	+	0.991954i	$0.540407\pi$
$410$	0	0
$411$	−18.6005	−0.917497
$412$	0	0
$413$	41.5355	2.04383
$414$	0	0
$415$	−27.6763	−1.35857
$416$	0	0
$417$	−16.0529	−0.786113
$418$	0	0
$419$	−1.48489	−0.0725419	−0.0362709	−	0.999342i	$-0.511548\pi$
$419$	−1.48489	−0.0725419	−0.0362709	+	0.999342i	$0.511548\pi$
$420$	0	0
$421$	−1.32660	−0.0646545	−0.0323272	−	0.999477i	$-0.510292\pi$
$421$	−1.32660	−0.0646545	−0.0323272	+	0.999477i	$0.510292\pi$
$422$	0	0
$423$	−4.45337	−0.216530
$424$	0	0
$425$	−10.3553	−0.502306
$426$	0	0
$427$	−18.3218	−0.886655
$428$	0	0
$429$	−5.08957	−0.245727
$430$	0	0
$431$	1.11034	0.0534832	0.0267416	−	0.999642i	$-0.491487\pi$
$431$	1.11034	0.0534832	0.0267416	+	0.999642i	$0.491487\pi$
$432$	0	0
$433$	−18.7731	−0.902179	−0.451089	−	0.892479i	$-0.648964\pi$
$433$	−18.7731	−0.902179	−0.451089	+	0.892479i	$0.648964\pi$
$434$	0	0
$435$	19.9257	0.955363
$436$	0	0
$437$	−29.3234	−1.40273
$438$	0	0
$439$	−1.77248	−0.0845960	−0.0422980	−	0.999105i	$-0.513468\pi$
$439$	−1.77248	−0.0845960	−0.0422980	+	0.999105i	$0.513468\pi$
$440$	0	0
$441$	−4.31078	−0.205275
$442$	0	0
$443$	−23.5716	−1.11992	−0.559961	−	0.828519i	$-0.689184\pi$
$443$	−23.5716	−1.11992	−0.559961	+	0.828519i	$0.689184\pi$
$444$	0	0
$445$	53.2891	2.52615
$446$	0	0
$447$	−21.6878	−1.02580
$448$	0	0
$449$	11.3934	0.537688	0.268844	−	0.963184i	$-0.413358\pi$
$449$	11.3934	0.537688	0.268844	+	0.963184i	$0.413358\pi$
$450$	0	0
$451$	−1.12407	−0.0529304
$452$	0	0
$453$	8.46989	0.397950
$454$	0	0
$455$	−58.5271	−2.74379
$456$	0	0
$457$	−37.3769	−1.74842	−0.874209	−	0.485549i	$-0.838620\pi$
$457$	−37.3769	−1.74842	−0.874209	+	0.485549i	$0.838620\pi$
$458$	0	0
$459$	8.62933	0.402783
$460$	0	0
$461$	28.6343	1.33363	0.666817	−	0.745221i	$-0.267656\pi$
$461$	28.6343	1.33363	0.666817	+	0.745221i	$0.267656\pi$
$462$	0	0
$463$	1.79462	0.0834032	0.0417016	−	0.999130i	$-0.486722\pi$
$463$	1.79462	0.0834032	0.0417016	+	0.999130i	$0.486722\pi$
$464$	0	0
$465$	−32.1365	−1.49029
$466$	0	0
$467$	25.7225	1.19029	0.595147	−	0.803617i	$-0.297094\pi$
$467$	25.7225	1.19029	0.595147	+	0.803617i	$0.297094\pi$
$468$	0	0
$469$	−7.65792	−0.353610
$470$	0	0
$471$	8.63379	0.397824
$472$	0	0
$473$	0.818283	0.0376247
$474$	0	0
$475$	50.4892	2.31660
$476$	0	0
$477$	−3.99156	−0.182761
$478$	0	0
$479$	−24.5167	−1.12020	−0.560099	−	0.828426i	$-0.689237\pi$
$479$	−24.5167	−1.12020	−0.560099	+	0.828426i	$0.689237\pi$
$480$	0	0
$481$	−26.1879	−1.19407
$482$	0	0
$483$	−17.9815	−0.818188
$484$	0	0
$485$	−14.1869	−0.644195
$486$	0	0
$487$	1.10226	0.0499482	0.0249741	−	0.999688i	$-0.492050\pi$
$487$	1.10226	0.0499482	0.0249741	+	0.999688i	$0.492050\pi$
$488$	0	0
$489$	34.1262	1.54324
$490$	0	0
$491$	23.1395	1.04427	0.522136	−	0.852862i	$-0.325135\pi$
$491$	23.1395	1.04427	0.522136	+	0.852862i	$0.325135\pi$
$492$	0	0
$493$	6.42100	0.289187
$494$	0	0
$495$	−2.68782	−0.120809
$496$	0	0
$497$	19.9743	0.895969
$498$	0	0
$499$	−27.4595	−1.22926	−0.614628	−	0.788817i	$-0.710694\pi$
$499$	−27.4595	−1.22926	−0.614628	+	0.788817i	$0.710694\pi$
$500$	0	0
$501$	1.01675	0.0454252
$502$	0	0
$503$	15.2079	0.678086	0.339043	−	0.940771i	$-0.389897\pi$
$503$	15.2079	0.678086	0.339043	+	0.940771i	$0.389897\pi$
$504$	0	0
$505$	10.5696	0.470341
$506$	0	0
$507$	18.7449	0.832491
$508$	0	0
$509$	21.6588	0.960011	0.480005	−	0.877266i	$-0.340635\pi$
$509$	21.6588	0.960011	0.480005	+	0.877266i	$0.340635\pi$
$510$	0	0
$511$	−20.3221	−0.898996
$512$	0	0
$513$	−42.0739	−1.85761
$514$	0	0
$515$	44.2966	1.95194
$516$	0	0
$517$	−2.90919	−0.127946
$518$	0	0
$519$	16.6186	0.729475
$520$	0	0
$521$	−23.3182	−1.02159	−0.510795	−	0.859703i	$-0.670649\pi$
$521$	−23.3182	−1.02159	−0.510795	+	0.859703i	$0.670649\pi$
$522$	0	0
$523$	−11.9308	−0.521698	−0.260849	−	0.965380i	$-0.584002\pi$
$523$	−11.9308	−0.521698	−0.260849	+	0.965380i	$0.584002\pi$
$524$	0	0
$525$	30.9607	1.35124
$526$	0	0
$527$	−10.3559	−0.451111
$528$	0	0
$529$	−7.48300	−0.325348
$530$	0	0
$531$	−13.7503	−0.596712
$532$	0	0
$533$	8.10294	0.350977
$534$	0	0
$535$	−62.1388	−2.68649
$536$	0	0
$537$	18.8049	0.811492
$538$	0	0
$539$	−2.81604	−0.121296
$540$	0	0
$541$	−30.0986	−1.29404	−0.647021	−	0.762472i	$-0.723985\pi$
$541$	−30.0986	−1.29404	−0.647021	+	0.762472i	$0.723985\pi$
$542$	0	0
$543$	16.2006	0.695235
$544$	0	0
$545$	4.72344	0.202330
$546$	0	0
$547$	−11.1630	−0.477296	−0.238648	−	0.971106i	$-0.576704\pi$
$547$	−11.1630	−0.477296	−0.238648	+	0.971106i	$0.576704\pi$
$548$	0	0
$549$	6.06542	0.258866
$550$	0	0
$551$	−31.3068	−1.33371
$552$	0	0
$553$	42.8665	1.82287
$554$	0	0
$555$	24.0660	1.02154
$556$	0	0
$557$	−16.1492	−0.684264	−0.342132	−	0.939652i	$-0.611149\pi$
$557$	−16.1492	−0.684264	−0.342132	+	0.939652i	$0.611149\pi$
$558$	0	0
$559$	−5.89865	−0.249486
$560$	0	0
$561$	1.50721	0.0636344
$562$	0	0
$563$	2.34711	0.0989191	0.0494595	−	0.998776i	$-0.484250\pi$
$563$	2.34711	0.0989191	0.0494595	+	0.998776i	$0.484250\pi$
$564$	0	0
$565$	−61.6444	−2.59340
$566$	0	0
$567$	−14.9379	−0.627333
$568$	0	0
$569$	9.18458	0.385038	0.192519	−	0.981293i	$-0.438334\pi$
$569$	9.18458	0.385038	0.192519	+	0.981293i	$0.438334\pi$
$570$	0	0
$571$	−24.8042	−1.03802	−0.519012	−	0.854767i	$-0.673700\pi$
$571$	−24.8042	−1.03802	−0.519012	+	0.854767i	$0.673700\pi$
$572$	0	0
$573$	−1.95800	−0.0817965
$574$	0	0
$575$	−26.7173	−1.11419
$576$	0	0
$577$	−26.8202	−1.11654	−0.558269	−	0.829660i	$-0.688534\pi$
$577$	−26.8202	−1.11654	−0.558269	+	0.829660i	$0.688534\pi$
$578$	0	0
$579$	18.6706	0.775922
$580$	0	0
$581$	26.6653	1.10626
$582$	0	0
$583$	−2.60752	−0.107992
$584$	0	0
$585$	19.3753	0.801072
$586$	0	0
$587$	−36.2607	−1.49664	−0.748319	−	0.663339i	$-0.769139\pi$
$587$	−36.2607	−1.49664	−0.748319	+	0.663339i	$0.769139\pi$
$588$	0	0
$589$	50.4922	2.08049
$590$	0	0
$591$	−9.45918	−0.389099
$592$	0	0
$593$	−13.5126	−0.554895	−0.277448	−	0.960741i	$-0.589489\pi$
$593$	−13.5126	−0.554895	−0.277448	+	0.960741i	$0.589489\pi$
$594$	0	0
$595$	17.3320	0.710543
$596$	0	0
$597$	32.9956	1.35042
$598$	0	0
$599$	26.2268	1.07160	0.535800	−	0.844345i	$-0.320010\pi$
$599$	26.2268	1.07160	0.535800	+	0.844345i	$0.320010\pi$
$600$	0	0
$601$	−16.9506	−0.691428	−0.345714	−	0.938340i	$-0.612363\pi$
$601$	−16.9506	−0.691428	−0.345714	+	0.938340i	$0.612363\pi$
$602$	0	0
$603$	2.53515	0.103239
$604$	0	0
$605$	36.0023	1.46370
$606$	0	0
$607$	−37.2591	−1.51230	−0.756149	−	0.654399i	$-0.772922\pi$
$607$	−37.2591	−1.51230	−0.756149	+	0.654399i	$0.772922\pi$
$608$	0	0
$609$	−19.1978	−0.777934
$610$	0	0
$611$	20.9711	0.848400
$612$	0	0
$613$	−14.4194	−0.582393	−0.291197	−	0.956663i	$-0.594053\pi$
$613$	−14.4194	−0.582393	−0.291197	+	0.956663i	$0.594053\pi$
$614$	0	0
$615$	−7.44638	−0.300267
$616$	0	0
$617$	27.0114	1.08744	0.543719	−	0.839267i	$-0.317016\pi$
$617$	27.0114	1.08744	0.543719	+	0.839267i	$0.317016\pi$
$618$	0	0
$619$	18.4323	0.740855	0.370427	−	0.928861i	$-0.379211\pi$
$619$	18.4323	0.740855	0.370427	+	0.928861i	$0.379211\pi$
$620$	0	0
$621$	22.2642	0.893431
$622$	0	0
$623$	−51.3425	−2.05699
$624$	0	0
$625$	−12.9104	−0.516414
$626$	0	0
$627$	−7.34868	−0.293478
$628$	0	0
$629$	7.75520	0.309220
$630$	0	0
$631$	−2.25049	−0.0895906	−0.0447953	−	0.998996i	$-0.514264\pi$
$631$	−2.25049	−0.0895906	−0.0447953	+	0.998996i	$0.514264\pi$
$632$	0	0
$633$	15.2817	0.607393
$634$	0	0
$635$	−71.4902	−2.83700
$636$	0	0
$637$	20.2996	0.804301
$638$	0	0
$639$	−6.61247	−0.261585
$640$	0	0
$641$	16.7234	0.660533	0.330267	−	0.943888i	$-0.392861\pi$
$641$	16.7234	0.660533	0.330267	+	0.943888i	$0.392861\pi$
$642$	0	0
$643$	3.08542	0.121677	0.0608386	−	0.998148i	$-0.480622\pi$
$643$	3.08542	0.121677	0.0608386	+	0.998148i	$0.480622\pi$
$644$	0	0
$645$	5.42070	0.213440
$646$	0	0
$647$	−38.4215	−1.51050	−0.755252	−	0.655435i	$-0.772485\pi$
$647$	−38.4215	−1.51050	−0.755252	+	0.655435i	$0.772485\pi$
$648$	0	0
$649$	−8.98248	−0.352593
$650$	0	0
$651$	30.9626	1.21352
$652$	0	0
$653$	−3.35524	−0.131301	−0.0656503	−	0.997843i	$-0.520912\pi$
$653$	−3.35524	−0.131301	−0.0656503	+	0.997843i	$0.520912\pi$
$654$	0	0
$655$	−23.3919	−0.913998
$656$	0	0
$657$	6.72761	0.262469
$658$	0	0
$659$	−21.8296	−0.850359	−0.425179	−	0.905109i	$-0.639789\pi$
$659$	−21.8296	−0.850359	−0.425179	+	0.905109i	$0.639789\pi$
$660$	0	0
$661$	−35.8057	−1.39268	−0.696340	−	0.717712i	$-0.745189\pi$
$661$	−35.8057	−1.39268	−0.696340	+	0.717712i	$0.745189\pi$
$662$	0	0
$663$	−10.8648	−0.421954
$664$	0	0
$665$	−84.5055	−3.27698
$666$	0	0
$667$	16.5666	0.641460
$668$	0	0
$669$	21.8278	0.843910
$670$	0	0
$671$	3.96228	0.152962
$672$	0	0
$673$	−43.1693	−1.66406	−0.832028	−	0.554734i	$-0.812820\pi$
$673$	−43.1693	−1.66406	−0.832028	+	0.554734i	$0.812820\pi$
$674$	0	0
$675$	−38.3346	−1.47550
$676$	0	0
$677$	6.38497	0.245394	0.122697	−	0.992444i	$-0.460846\pi$
$677$	6.38497	0.245394	0.122697	+	0.992444i	$0.460846\pi$
$678$	0	0
$679$	13.6687	0.524556
$680$	0	0
$681$	−16.5291	−0.633396
$682$	0	0
$683$	−19.5133	−0.746656	−0.373328	−	0.927699i	$-0.621783\pi$
$683$	−19.5133	−0.746656	−0.373328	+	0.927699i	$0.621783\pi$
$684$	0	0
$685$	46.2570	1.76739
$686$	0	0
$687$	9.61055	0.366665
$688$	0	0
$689$	18.7964	0.716088
$690$	0	0
$691$	−16.9634	−0.645319	−0.322659	−	0.946515i	$-0.604577\pi$
$691$	−16.9634	−0.645319	−0.322659	+	0.946515i	$0.604577\pi$
$692$	0	0
$693$	2.58964	0.0983721
$694$	0	0
$695$	39.9214	1.51430
$696$	0	0
$697$	−2.39958	−0.0908905
$698$	0	0
$699$	−26.7685	−1.01248
$700$	0	0
$701$	−2.32280	−0.0877311	−0.0438655	−	0.999037i	$-0.513967\pi$
$701$	−2.32280	−0.0877311	−0.0438655	+	0.999037i	$0.513967\pi$
$702$	0	0
$703$	−37.8120	−1.42610
$704$	0	0
$705$	−19.2719	−0.725820
$706$	0	0
$707$	−10.1835	−0.382989
$708$	0	0
$709$	−36.4777	−1.36995	−0.684974	−	0.728568i	$-0.740186\pi$
$709$	−36.4777	−1.36995	−0.684974	+	0.728568i	$0.740186\pi$
$710$	0	0
$711$	−14.1909	−0.532202
$712$	0	0
$713$	−26.7189	−1.00063
$714$	0	0
$715$	12.6571	0.473348
$716$	0	0
$717$	20.8451	0.778475
$718$	0	0
$719$	−8.31384	−0.310054	−0.155027	−	0.987910i	$-0.549546\pi$
$719$	−8.31384	−0.310054	−0.155027	+	0.987910i	$0.549546\pi$
$720$	0	0
$721$	−42.6785	−1.58943
$722$	0	0
$723$	−20.1253	−0.748469
$724$	0	0
$725$	−28.5244	−1.05937
$726$	0	0
$727$	27.7570	1.02945	0.514725	−	0.857356i	$-0.327894\pi$
$727$	27.7570	1.02945	0.514725	+	0.857356i	$0.327894\pi$
$728$	0	0
$729$	28.3492	1.04997
$730$	0	0
$731$	1.74681	0.0646080
$732$	0	0
$733$	45.2015	1.66955	0.834777	−	0.550588i	$-0.185597\pi$
$733$	45.2015	1.66955	0.834777	+	0.550588i	$0.185597\pi$
$734$	0	0
$735$	−18.6548	−0.688093
$736$	0	0
$737$	1.65610	0.0610034
$738$	0	0
$739$	19.7131	0.725159	0.362580	−	0.931953i	$-0.381896\pi$
$739$	19.7131	0.725159	0.362580	+	0.931953i	$0.381896\pi$
$740$	0	0
$741$	52.9734	1.94603
$742$	0	0
$743$	2.33662	0.0857224	0.0428612	−	0.999081i	$-0.486353\pi$
$743$	2.33662	0.0857224	0.0428612	+	0.999081i	$0.486353\pi$
$744$	0	0
$745$	53.9346	1.97601
$746$	0	0
$747$	−8.82751	−0.322982
$748$	0	0
$749$	59.8688	2.18756
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−6.52384	−0.237742
$754$	0	0
$755$	−21.0635	−0.766579
$756$	0	0
$757$	−35.4457	−1.28830	−0.644148	−	0.764901i	$-0.722788\pi$
$757$	−35.4457	−1.28830	−0.644148	+	0.764901i	$0.722788\pi$
$758$	0	0
$759$	3.88869	0.141150
$760$	0	0
$761$	−26.7618	−0.970116	−0.485058	−	0.874482i	$-0.661201\pi$
$761$	−26.7618	−0.970116	−0.485058	+	0.874482i	$0.661201\pi$
$762$	0	0
$763$	−4.55090	−0.164753
$764$	0	0
$765$	−5.73775	−0.207449
$766$	0	0
$767$	64.7508	2.33801
$768$	0	0
$769$	1.18115	0.0425934	0.0212967	−	0.999773i	$-0.493221\pi$
$769$	1.18115	0.0425934	0.0212967	+	0.999773i	$0.493221\pi$
$770$	0	0
$771$	0.130489	0.00469944
$772$	0	0
$773$	6.27705	0.225770	0.112885	−	0.993608i	$-0.463991\pi$
$773$	6.27705	0.225770	0.112885	+	0.993608i	$0.463991\pi$
$774$	0	0
$775$	46.0048	1.65254
$776$	0	0
$777$	−23.1868	−0.831823
$778$	0	0
$779$	11.6996	0.419181
$780$	0	0
$781$	−4.31964	−0.154569
$782$	0	0
$783$	23.7701	0.849475
$784$	0	0
$785$	−21.4711	−0.766335
$786$	0	0
$787$	−12.1036	−0.431447	−0.215724	−	0.976454i	$-0.569211\pi$
$787$	−12.1036	−0.431447	−0.215724	+	0.976454i	$0.569211\pi$
$788$	0	0
$789$	21.5510	0.767237
$790$	0	0
$791$	59.3926	2.11176
$792$	0	0
$793$	−28.5623	−1.01428
$794$	0	0
$795$	−17.2734	−0.612625
$796$	0	0
$797$	−53.8903	−1.90889	−0.954446	−	0.298383i	$-0.903553\pi$
$797$	−53.8903	−1.90889	−0.954446	+	0.298383i	$0.903553\pi$
$798$	0	0
$799$	−6.21031	−0.219705
$800$	0	0
$801$	16.9969	0.600556
$802$	0	0
$803$	4.39486	0.155091
$804$	0	0
$805$	44.7176	1.57609
$806$	0	0
$807$	16.6464	0.585981
$808$	0	0
$809$	18.4573	0.648923	0.324461	−	0.945899i	$-0.394817\pi$
$809$	18.4573	0.648923	0.324461	+	0.945899i	$0.394817\pi$
$810$	0	0
$811$	49.7375	1.74652	0.873260	−	0.487254i	$-0.162002\pi$
$811$	49.7375	1.74652	0.873260	+	0.487254i	$0.162002\pi$
$812$	0	0
$813$	−6.65694	−0.233469
$814$	0	0
$815$	−84.8673	−2.97277
$816$	0	0
$817$	−8.51688	−0.297968
$818$	0	0
$819$	−18.6676	−0.652297
$820$	0	0
$821$	40.4080	1.41025	0.705125	−	0.709083i	$-0.250891\pi$
$821$	40.4080	1.41025	0.705125	+	0.709083i	$0.250891\pi$
$822$	0	0
$823$	28.5688	0.995845	0.497923	−	0.867221i	$-0.334096\pi$
$823$	28.5688	0.995845	0.497923	+	0.867221i	$0.334096\pi$
$824$	0	0
$825$	−6.69557	−0.233110
$826$	0	0
$827$	15.8368	0.550698	0.275349	−	0.961344i	$-0.411207\pi$
$827$	15.8368	0.550698	0.275349	+	0.961344i	$0.411207\pi$
$828$	0	0
$829$	27.5314	0.956204	0.478102	−	0.878304i	$-0.341325\pi$
$829$	27.5314	0.956204	0.478102	+	0.878304i	$0.341325\pi$
$830$	0	0
$831$	−14.9645	−0.519114
$832$	0	0
$833$	−6.01147	−0.208285
$834$	0	0
$835$	−2.52853	−0.0875033
$836$	0	0
$837$	−38.3369	−1.32512
$838$	0	0
$839$	−2.77127	−0.0956750	−0.0478375	−	0.998855i	$-0.515233\pi$
$839$	−2.77127	−0.0956750	−0.0478375	+	0.998855i	$0.515233\pi$
$840$	0	0
$841$	−11.3129	−0.390099
$842$	0	0
$843$	−15.9978	−0.550994
$844$	0	0
$845$	−46.6161	−1.60364
$846$	0	0
$847$	−34.6872	−1.19187
$848$	0	0
$849$	−16.8806	−0.579340
$850$	0	0
$851$	20.0089	0.685896
$852$	0	0
$853$	3.26328	0.111733	0.0558664	−	0.998438i	$-0.482208\pi$
$853$	3.26328	0.111733	0.0558664	+	0.998438i	$0.482208\pi$
$854$	0	0
$855$	27.9755	0.956742
$856$	0	0
$857$	2.70878	0.0925301	0.0462650	−	0.998929i	$-0.485268\pi$
$857$	2.70878	0.0925301	0.0462650	+	0.998929i	$0.485268\pi$
$858$	0	0
$859$	38.1510	1.30170	0.650848	−	0.759208i	$-0.274414\pi$
$859$	38.1510	1.30170	0.650848	+	0.759208i	$0.274414\pi$
$860$	0	0
$861$	7.17436	0.244502
$862$	0	0
$863$	−3.41001	−0.116078	−0.0580391	−	0.998314i	$-0.518485\pi$
$863$	−3.41001	−0.116078	−0.0580391	+	0.998314i	$0.518485\pi$
$864$	0	0
$865$	−41.3282	−1.40520
$866$	0	0
$867$	−20.2472	−0.687632
$868$	0	0
$869$	−9.27032	−0.314474
$870$	0	0
$871$	−11.9381	−0.404508
$872$	0	0
$873$	−4.52501	−0.153148
$874$	0	0
$875$	−20.2348	−0.684063
$876$	0	0
$877$	18.6335	0.629210	0.314605	−	0.949223i	$-0.398128\pi$
$877$	18.6335	0.629210	0.314605	+	0.949223i	$0.398128\pi$
$878$	0	0
$879$	16.4136	0.553617
$880$	0	0
$881$	37.7308	1.27118	0.635592	−	0.772025i	$-0.280756\pi$
$881$	37.7308	1.27118	0.635592	+	0.772025i	$0.280756\pi$
$882$	0	0
$883$	−12.9761	−0.436681	−0.218340	−	0.975873i	$-0.570064\pi$
$883$	−12.9761	−0.436681	−0.218340	+	0.975873i	$0.570064\pi$
$884$	0	0
$885$	−59.5042	−2.00021
$886$	0	0
$887$	−47.7085	−1.60190	−0.800948	−	0.598734i	$-0.795671\pi$
$887$	−47.7085	−1.60190	−0.800948	+	0.598734i	$0.795671\pi$
$888$	0	0
$889$	68.8786	2.31012
$890$	0	0
$891$	3.23047	0.108225
$892$	0	0
$893$	30.2795	1.01327
$894$	0	0
$895$	−46.7653	−1.56319
$896$	0	0
$897$	−28.0319	−0.935957
$898$	0	0
$899$	−28.5261	−0.951399
$900$	0	0
$901$	−5.56632	−0.185441
$902$	0	0
$903$	−5.22268	−0.173800
$904$	0	0
$905$	−40.2887	−1.33924
$906$	0	0
$907$	40.5414	1.34615	0.673077	−	0.739572i	$-0.264972\pi$
$907$	40.5414	1.34615	0.673077	+	0.739572i	$0.264972\pi$
$908$	0	0
$909$	3.37124	0.111817
$910$	0	0
$911$	−31.0323	−1.02815	−0.514073	−	0.857747i	$-0.671864\pi$
$911$	−31.0323	−1.02815	−0.514073	+	0.857747i	$0.671864\pi$
$912$	0	0
$913$	−5.76663	−0.190848
$914$	0	0
$915$	26.2480	0.867733
$916$	0	0
$917$	22.5374	0.744251
$918$	0	0
$919$	49.2321	1.62402	0.812009	−	0.583644i	$-0.198374\pi$
$919$	49.2321	1.62402	0.812009	+	0.583644i	$0.198374\pi$
$920$	0	0
$921$	−31.7165	−1.04509
$922$	0	0
$923$	31.1384	1.02493
$924$	0	0
$925$	−34.4515	−1.13276
$926$	0	0
$927$	14.1287	0.464047
$928$	0	0
$929$	−46.5816	−1.52829	−0.764146	−	0.645043i	$-0.776839\pi$
$929$	−46.5816	−1.52829	−0.764146	+	0.645043i	$0.776839\pi$
$930$	0	0
$931$	29.3100	0.960598
$932$	0	0
$933$	14.0413	0.459692
$934$	0	0
$935$	−3.74823	−0.122580
$936$	0	0
$937$	−35.6760	−1.16548	−0.582742	−	0.812657i	$-0.698020\pi$
$937$	−35.6760	−1.16548	−0.582742	+	0.812657i	$0.698020\pi$
$938$	0	0
$939$	13.8384	0.451598
$940$	0	0
$941$	42.4784	1.38476	0.692379	−	0.721534i	$-0.256563\pi$
$941$	42.4784	1.38476	0.692379	+	0.721534i	$0.256563\pi$
$942$	0	0
$943$	−6.19105	−0.201608
$944$	0	0
$945$	64.1619	2.08719
$946$	0	0
$947$	36.4930	1.18586	0.592932	−	0.805252i	$-0.297970\pi$
$947$	36.4930	1.18586	0.592932	+	0.805252i	$0.297970\pi$
$948$	0	0
$949$	−31.6806	−1.02840
$950$	0	0
$951$	29.4669	0.955531
$952$	0	0
$953$	41.3087	1.33812	0.669061	−	0.743208i	$-0.266697\pi$
$953$	41.3087	1.33812	0.669061	+	0.743208i	$0.266697\pi$
$954$	0	0
$955$	4.86928	0.157566
$956$	0	0
$957$	4.15172	0.134206
$958$	0	0
$959$	−44.5672	−1.43915
$960$	0	0
$961$	15.0075	0.484112
$962$	0	0
$963$	−19.8195	−0.638676
$964$	0	0
$965$	−46.4312	−1.49467
$966$	0	0
$967$	22.5504	0.725172	0.362586	−	0.931950i	$-0.381894\pi$
$967$	22.5504	0.725172	0.362586	+	0.931950i	$0.381894\pi$
$968$	0	0
$969$	−15.6874	−0.503951
$970$	0	0
$971$	−33.4442	−1.07328	−0.536638	−	0.843813i	$-0.680306\pi$
$971$	−33.4442	−1.07328	−0.536638	+	0.843813i	$0.680306\pi$
$972$	0	0
$973$	−38.4630	−1.23307
$974$	0	0
$975$	48.2655	1.54573
$976$	0	0
$977$	−35.1927	−1.12591	−0.562957	−	0.826486i	$-0.690336\pi$
$977$	−35.1927	−1.12591	−0.562957	+	0.826486i	$0.690336\pi$
$978$	0	0
$979$	11.1033	0.354864
$980$	0	0
$981$	1.50657	0.0481011
$982$	0	0
$983$	6.33888	0.202179	0.101089	−	0.994877i	$-0.467767\pi$
$983$	6.33888	0.202179	0.101089	+	0.994877i	$0.467767\pi$
$984$	0	0
$985$	23.5237	0.749528
$986$	0	0
$987$	18.5679	0.591021
$988$	0	0
$989$	4.50686	0.143310
$990$	0	0
$991$	−32.1283	−1.02059	−0.510294	−	0.860000i	$-0.670463\pi$
$991$	−32.1283	−1.02059	−0.510294	+	0.860000i	$0.670463\pi$
$992$	0	0
$993$	32.7506	1.03931
$994$	0	0
$995$	−82.0555	−2.60133
$996$	0	0
$997$	14.6952	0.465402	0.232701	−	0.972548i	$-0.425244\pi$
$997$	14.6952	0.465402	0.232701	+	0.972548i	$0.425244\pi$
$998$	0	0
$999$	28.7092	0.908320

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.34	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.34	✓	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)

\(f(q)\)	\(=\)	\(q+1.38028 q^{3} -3.43256 q^{5} +3.30717 q^{7} -1.09484 q^{9} +O(q^{10})\)	\(q+1.38028 q^{3} -3.43256 q^{5} +3.30717 q^{7} -1.09484 q^{9} -0.715209 q^{11} +5.15563 q^{13} -4.73789 q^{15} -1.52677 q^{17} +7.44406 q^{19} +4.56481 q^{21} -3.93916 q^{23} +6.78248 q^{25} -5.65201 q^{27} -4.20561 q^{29} +6.78288 q^{31} -0.987187 q^{33} -11.3521 q^{35} -5.07948 q^{37} +7.11620 q^{39} +1.57167 q^{41} -1.14412 q^{43} +3.75809 q^{45} +4.06761 q^{47} +3.93737 q^{49} -2.10737 q^{51} +3.64581 q^{53} +2.45500 q^{55} +10.2749 q^{57} +12.5592 q^{59} -5.54003 q^{61} -3.62081 q^{63} -17.6970 q^{65} -2.31555 q^{67} -5.43713 q^{69} +6.03969 q^{71} -6.14485 q^{73} +9.36170 q^{75} -2.36532 q^{77} +12.9617 q^{79} -4.51682 q^{81} +8.06286 q^{83} +5.24074 q^{85} -5.80490 q^{87} -15.5246 q^{89} +17.0506 q^{91} +9.36225 q^{93} -25.5522 q^{95} +4.13304 q^{97} +0.783037 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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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