LMFDB - Embedded newform 6008.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$49$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-3.18877 q^{3} +2.23846 q^{5} +2.57450 q^{7} +7.16824 q^{9} +O(q^{10})$	$q-3.18877 q^{3} +2.23846 q^{5} +2.57450 q^{7} +7.16824 q^{9} +2.62888 q^{11} -2.46272 q^{13} -7.13794 q^{15} -1.10159 q^{17} +5.55935 q^{19} -8.20949 q^{21} +1.46525 q^{23} +0.0107186 q^{25} -13.2915 q^{27} -6.66999 q^{29} +10.6095 q^{31} -8.38287 q^{33} +5.76293 q^{35} +9.34117 q^{37} +7.85303 q^{39} +2.42957 q^{41} +8.37014 q^{43} +16.0458 q^{45} +2.50946 q^{47} -0.371939 q^{49} +3.51270 q^{51} +9.61441 q^{53} +5.88464 q^{55} -17.7275 q^{57} -3.69008 q^{59} -8.77340 q^{61} +18.4546 q^{63} -5.51270 q^{65} +4.18460 q^{67} -4.67233 q^{69} -5.99275 q^{71} -6.74481 q^{73} -0.0341790 q^{75} +6.76805 q^{77} +2.05123 q^{79} +20.8789 q^{81} +4.68964 q^{83} -2.46586 q^{85} +21.2691 q^{87} +4.75441 q^{89} -6.34027 q^{91} -33.8314 q^{93} +12.4444 q^{95} +18.5657 q^{97} +18.8444 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.18877	−1.84104	−0.920518	−	0.390701i	$-0.872233\pi$
$3$	−3.18877	−1.84104	−0.920518	+	0.390701i	$0.872233\pi$
$4$	0	0
$5$	2.23846	1.00107	0.500536	−	0.865716i	$-0.333136\pi$
$5$	2.23846	1.00107	0.500536	+	0.865716i	$0.333136\pi$
$6$	0	0
$7$	2.57450	0.973070	0.486535	−	0.873661i	$-0.338261\pi$
$7$	2.57450	0.973070	0.486535	+	0.873661i	$0.338261\pi$
$8$	0	0
$9$	7.16824	2.38941
$10$	0	0
$11$	2.62888	0.792636	0.396318	−	0.918113i	$-0.370288\pi$
$11$	2.62888	0.792636	0.396318	+	0.918113i	$0.370288\pi$
$12$	0	0
$13$	−2.46272	−0.683035	−0.341517	−	0.939875i	$-0.610941\pi$
$13$	−2.46272	−0.683035	−0.341517	+	0.939875i	$0.610941\pi$
$14$	0	0
$15$	−7.13794	−1.84301
$16$	0	0
$17$	−1.10159	−0.267174	−0.133587	−	0.991037i	$-0.542650\pi$
$17$	−1.10159	−0.267174	−0.133587	+	0.991037i	$0.542650\pi$
$18$	0	0
$19$	5.55935	1.27540	0.637701	−	0.770284i	$-0.279886\pi$
$19$	5.55935	1.27540	0.637701	+	0.770284i	$0.279886\pi$
$20$	0	0
$21$	−8.20949	−1.79146
$22$	0	0
$23$	1.46525	0.305525	0.152762	−	0.988263i	$-0.451183\pi$
$23$	1.46525	0.305525	0.152762	+	0.988263i	$0.451183\pi$
$24$	0	0
$25$	0.0107186	0.00214371
$26$	0	0
$27$	−13.2915	−2.55796
$28$	0	0
$29$	−6.66999	−1.23859	−0.619293	−	0.785160i	$-0.712581\pi$
$29$	−6.66999	−1.23859	−0.619293	+	0.785160i	$0.712581\pi$
$30$	0	0
$31$	10.6095	1.90553	0.952766	−	0.303707i	$-0.0982243\pi$
$31$	10.6095	1.90553	0.952766	+	0.303707i	$0.0982243\pi$
$32$	0	0
$33$	−8.38287	−1.45927
$34$	0	0
$35$	5.76293	0.974113
$36$	0	0
$37$	9.34117	1.53568	0.767840	−	0.640642i	$-0.221332\pi$
$37$	9.34117	1.53568	0.767840	+	0.640642i	$0.221332\pi$
$38$	0	0
$39$	7.85303	1.25749
$40$	0	0
$41$	2.42957	0.379434	0.189717	−	0.981839i	$-0.439243\pi$
$41$	2.42957	0.379434	0.189717	+	0.981839i	$0.439243\pi$
$42$	0	0
$43$	8.37014	1.27643	0.638217	−	0.769856i	$-0.279672\pi$
$43$	8.37014	1.27643	0.638217	+	0.769856i	$0.279672\pi$
$44$	0	0
$45$	16.0458	2.39197
$46$	0	0
$47$	2.50946	0.366042	0.183021	−	0.983109i	$-0.441412\pi$
$47$	2.50946	0.366042	0.183021	+	0.983109i	$0.441412\pi$
$48$	0	0
$49$	−0.371939	−0.0531342
$50$	0	0
$51$	3.51270	0.491877
$52$	0	0
$53$	9.61441	1.32064	0.660320	−	0.750984i	$-0.270420\pi$
$53$	9.61441	1.32064	0.660320	+	0.750984i	$0.270420\pi$
$54$	0	0
$55$	5.88464	0.793485
$56$	0	0
$57$	−17.7275	−2.34806
$58$	0	0
$59$	−3.69008	−0.480407	−0.240204	−	0.970723i	$-0.577214\pi$
$59$	−3.69008	−0.480407	−0.240204	+	0.970723i	$0.577214\pi$
$60$	0	0
$61$	−8.77340	−1.12332	−0.561660	−	0.827368i	$-0.689837\pi$
$61$	−8.77340	−1.12332	−0.561660	+	0.827368i	$0.689837\pi$
$62$	0	0
$63$	18.4546	2.32507
$64$	0	0
$65$	−5.51270	−0.683767
$66$	0	0
$67$	4.18460	0.511230	0.255615	−	0.966779i	$-0.417722\pi$
$67$	4.18460	0.511230	0.255615	+	0.966779i	$0.417722\pi$
$68$	0	0
$69$	−4.67233	−0.562482
$70$	0	0
$71$	−5.99275	−0.711209	−0.355604	−	0.934637i	$-0.615725\pi$
$71$	−5.99275	−0.711209	−0.355604	+	0.934637i	$0.615725\pi$
$72$	0	0
$73$	−6.74481	−0.789420	−0.394710	−	0.918806i	$-0.629155\pi$
$73$	−6.74481	−0.789420	−0.394710	+	0.918806i	$0.629155\pi$
$74$	0	0
$75$	−0.0341790	−0.00394665
$76$	0	0
$77$	6.76805	0.771291
$78$	0	0
$79$	2.05123	0.230781	0.115390	−	0.993320i	$-0.463188\pi$
$79$	2.05123	0.230781	0.115390	+	0.993320i	$0.463188\pi$
$80$	0	0
$81$	20.8789	2.31988
$82$	0	0
$83$	4.68964	0.514755	0.257378	−	0.966311i	$-0.417142\pi$
$83$	4.68964	0.514755	0.257378	+	0.966311i	$0.417142\pi$
$84$	0	0
$85$	−2.46586	−0.267460
$86$	0	0
$87$	21.2691	2.28028
$88$	0	0
$89$	4.75441	0.503967	0.251983	−	0.967732i	$-0.418917\pi$
$89$	4.75441	0.503967	0.251983	+	0.967732i	$0.418917\pi$
$90$	0	0
$91$	−6.34027	−0.664641
$92$	0	0
$93$	−33.8314	−3.50815
$94$	0	0
$95$	12.4444	1.27677
$96$	0	0
$97$	18.5657	1.88507	0.942533	−	0.334113i	$-0.108437\pi$
$97$	18.5657	1.88507	0.942533	+	0.334113i	$0.108437\pi$
$98$	0	0
$99$	18.8444	1.89393
$100$	0	0
$101$	−1.21174	−0.120572	−0.0602861	−	0.998181i	$-0.519201\pi$
$101$	−1.21174	−0.120572	−0.0602861	+	0.998181i	$0.519201\pi$
$102$	0	0
$103$	6.20847	0.611738	0.305869	−	0.952074i	$-0.401053\pi$
$103$	6.20847	0.611738	0.305869	+	0.952074i	$0.401053\pi$
$104$	0	0
$105$	−18.3766	−1.79338
$106$	0	0
$107$	−19.3652	−1.87211	−0.936054	−	0.351856i	$-0.885551\pi$
$107$	−19.3652	−1.87211	−0.936054	+	0.351856i	$0.885551\pi$
$108$	0	0
$109$	−8.80581	−0.843444	−0.421722	−	0.906725i	$-0.638574\pi$
$109$	−8.80581	−0.843444	−0.421722	+	0.906725i	$0.638574\pi$
$110$	0	0
$111$	−29.7868	−2.82724
$112$	0	0
$113$	5.46967	0.514544	0.257272	−	0.966339i	$-0.417176\pi$
$113$	5.46967	0.514544	0.257272	+	0.966339i	$0.417176\pi$
$114$	0	0
$115$	3.27990	0.305852
$116$	0	0
$117$	−17.6533	−1.63205
$118$	0	0
$119$	−2.83604	−0.259979
$120$	0	0
$121$	−4.08901	−0.371728
$122$	0	0
$123$	−7.74732	−0.698552
$124$	0	0
$125$	−11.1683	−0.998925
$126$	0	0
$127$	15.3068	1.35826	0.679130	−	0.734018i	$-0.262357\pi$
$127$	15.3068	1.35826	0.679130	+	0.734018i	$0.262357\pi$
$128$	0	0
$129$	−26.6904	−2.34996
$130$	0	0
$131$	1.98264	0.173224	0.0866122	−	0.996242i	$-0.472396\pi$
$131$	1.98264	0.173224	0.0866122	+	0.996242i	$0.472396\pi$
$132$	0	0
$133$	14.3125	1.24106
$134$	0	0
$135$	−29.7526	−2.56070
$136$	0	0
$137$	−7.92658	−0.677214	−0.338607	−	0.940928i	$-0.609956\pi$
$137$	−7.92658	−0.677214	−0.338607	+	0.940928i	$0.609956\pi$
$138$	0	0
$139$	1.26291	0.107118	0.0535592	−	0.998565i	$-0.482943\pi$
$139$	1.26291	0.107118	0.0535592	+	0.998565i	$0.482943\pi$
$140$	0	0
$141$	−8.00207	−0.673896
$142$	0	0
$143$	−6.47418	−0.541398
$144$	0	0
$145$	−14.9305	−1.23991
$146$	0	0
$147$	1.18603	0.0978219
$148$	0	0
$149$	−18.0910	−1.48207	−0.741034	−	0.671467i	$-0.765664\pi$
$149$	−18.0910	−1.48207	−0.741034	+	0.671467i	$0.765664\pi$
$150$	0	0
$151$	−5.09991	−0.415025	−0.207512	−	0.978232i	$-0.566537\pi$
$151$	−5.09991	−0.415025	−0.207512	+	0.978232i	$0.566537\pi$
$152$	0	0
$153$	−7.89643	−0.638389
$154$	0	0
$155$	23.7491	1.90757
$156$	0	0
$157$	−17.2678	−1.37812	−0.689059	−	0.724706i	$-0.741976\pi$
$157$	−17.2678	−1.37812	−0.689059	+	0.724706i	$0.741976\pi$
$158$	0	0
$159$	−30.6581	−2.43135
$160$	0	0
$161$	3.77228	0.297297
$162$	0	0
$163$	−7.38285	−0.578269	−0.289135	−	0.957288i	$-0.593368\pi$
$163$	−7.38285	−0.578269	−0.289135	+	0.957288i	$0.593368\pi$
$164$	0	0
$165$	−18.7648	−1.46083
$166$	0	0
$167$	3.86891	0.299385	0.149693	−	0.988733i	$-0.452172\pi$
$167$	3.86891	0.299385	0.149693	+	0.988733i	$0.452172\pi$
$168$	0	0
$169$	−6.93502	−0.533463
$170$	0	0
$171$	39.8507	3.04746
$172$	0	0
$173$	−13.0078	−0.988964	−0.494482	−	0.869188i	$-0.664642\pi$
$173$	−13.0078	−0.988964	−0.494482	+	0.869188i	$0.664642\pi$
$174$	0	0
$175$	0.0275950	0.00208598
$176$	0	0
$177$	11.7668	0.884447
$178$	0	0
$179$	−22.0930	−1.65131	−0.825655	−	0.564175i	$-0.809194\pi$
$179$	−22.0930	−1.65131	−0.825655	+	0.564175i	$0.809194\pi$
$180$	0	0
$181$	10.5119	0.781343	0.390671	−	0.920530i	$-0.372243\pi$
$181$	10.5119	0.781343	0.390671	+	0.920530i	$0.372243\pi$
$182$	0	0
$183$	27.9763	2.06807
$184$	0	0
$185$	20.9099	1.53732
$186$	0	0
$187$	−2.89593	−0.211772
$188$	0	0
$189$	−34.2191	−2.48907
$190$	0	0
$191$	2.56162	0.185352	0.0926762	−	0.995696i	$-0.470458\pi$
$191$	2.56162	0.185352	0.0926762	+	0.995696i	$0.470458\pi$
$192$	0	0
$193$	17.8582	1.28546	0.642730	−	0.766092i	$-0.277802\pi$
$193$	17.8582	1.28546	0.642730	+	0.766092i	$0.277802\pi$
$194$	0	0
$195$	17.5787	1.25884
$196$	0	0
$197$	10.9182	0.777888	0.388944	−	0.921261i	$-0.372840\pi$
$197$	10.9182	0.777888	0.388944	+	0.921261i	$0.372840\pi$
$198$	0	0
$199$	−7.18328	−0.509209	−0.254605	−	0.967045i	$-0.581945\pi$
$199$	−7.18328	−0.509209	−0.254605	+	0.967045i	$0.581945\pi$
$200$	0	0
$201$	−13.3437	−0.941193
$202$	0	0
$203$	−17.1719	−1.20523
$204$	0	0
$205$	5.43850	0.379841
$206$	0	0
$207$	10.5032	0.730025
$208$	0	0
$209$	14.6148	1.01093
$210$	0	0
$211$	0.920786	0.0633895	0.0316948	−	0.999498i	$-0.489910\pi$
$211$	0.920786	0.0633895	0.0316948	+	0.999498i	$0.489910\pi$
$212$	0	0
$213$	19.1095	1.30936
$214$	0	0
$215$	18.7362	1.27780
$216$	0	0
$217$	27.3143	1.85422
$218$	0	0
$219$	21.5076	1.45335
$220$	0	0
$221$	2.71290	0.182489
$222$	0	0
$223$	20.8265	1.39465	0.697323	−	0.716757i	$-0.254374\pi$
$223$	20.8265	1.39465	0.697323	+	0.716757i	$0.254374\pi$
$224$	0	0
$225$	0.0768332	0.00512221
$226$	0	0
$227$	14.5999	0.969026	0.484513	−	0.874784i	$-0.338997\pi$
$227$	14.5999	0.969026	0.484513	+	0.874784i	$0.338997\pi$
$228$	0	0
$229$	−6.76933	−0.447330	−0.223665	−	0.974666i	$-0.571802\pi$
$229$	−6.76933	−0.447330	−0.223665	+	0.974666i	$0.571802\pi$
$230$	0	0
$231$	−21.5817	−1.41997
$232$	0	0
$233$	27.9654	1.83207	0.916037	−	0.401094i	$-0.131370\pi$
$233$	27.9654	1.83207	0.916037	+	0.401094i	$0.131370\pi$
$234$	0	0
$235$	5.61733	0.366434
$236$	0	0
$237$	−6.54088	−0.424876
$238$	0	0
$239$	−17.1378	−1.10855	−0.554275	−	0.832334i	$-0.687004\pi$
$239$	−17.1378	−1.10855	−0.554275	+	0.832334i	$0.687004\pi$
$240$	0	0
$241$	7.58462	0.488568	0.244284	−	0.969704i	$-0.421447\pi$
$241$	7.58462	0.488568	0.244284	+	0.969704i	$0.421447\pi$
$242$	0	0
$243$	−26.7034	−1.71302
$244$	0	0
$245$	−0.832572	−0.0531911
$246$	0	0
$247$	−13.6911	−0.871144
$248$	0	0
$249$	−14.9542	−0.947683
$250$	0	0
$251$	−4.53609	−0.286316	−0.143158	−	0.989700i	$-0.545726\pi$
$251$	−4.53609	−0.286316	−0.143158	+	0.989700i	$0.545726\pi$
$252$	0	0
$253$	3.85195	0.242170
$254$	0	0
$255$	7.86306	0.492404
$256$	0	0
$257$	9.67946	0.603788	0.301894	−	0.953341i	$-0.402381\pi$
$257$	9.67946	0.603788	0.301894	+	0.953341i	$0.402381\pi$
$258$	0	0
$259$	24.0489	1.49432
$260$	0	0
$261$	−47.8121	−2.95949
$262$	0	0
$263$	12.3016	0.758550	0.379275	−	0.925284i	$-0.376173\pi$
$263$	12.3016	0.758550	0.379275	+	0.925284i	$0.376173\pi$
$264$	0	0
$265$	21.5215	1.32206
$266$	0	0
$267$	−15.1607	−0.927821
$268$	0	0
$269$	1.04380	0.0636415	0.0318208	−	0.999494i	$-0.489869\pi$
$269$	1.04380	0.0636415	0.0318208	+	0.999494i	$0.489869\pi$
$270$	0	0
$271$	−6.36917	−0.386900	−0.193450	−	0.981110i	$-0.561968\pi$
$271$	−6.36917	−0.386900	−0.193450	+	0.981110i	$0.561968\pi$
$272$	0	0
$273$	20.2176	1.22363
$274$	0	0
$275$	0.0281778	0.00169918
$276$	0	0
$277$	−29.4417	−1.76898	−0.884491	−	0.466558i	$-0.845494\pi$
$277$	−29.4417	−1.76898	−0.884491	+	0.466558i	$0.845494\pi$
$278$	0	0
$279$	76.0518	4.55310
$280$	0	0
$281$	16.6561	0.993621	0.496810	−	0.867859i	$-0.334504\pi$
$281$	16.6561	0.993621	0.496810	+	0.867859i	$0.334504\pi$
$282$	0	0
$283$	4.53601	0.269638	0.134819	−	0.990870i	$-0.456955\pi$
$283$	4.53601	0.269638	0.134819	+	0.990870i	$0.456955\pi$
$284$	0	0
$285$	−39.6823	−2.35057
$286$	0	0
$287$	6.25492	0.369216
$288$	0	0
$289$	−15.7865	−0.928618
$290$	0	0
$291$	−59.2019	−3.47047
$292$	0	0
$293$	32.1459	1.87798	0.938991	−	0.343943i	$-0.111763\pi$
$293$	32.1459	1.87798	0.938991	+	0.343943i	$0.111763\pi$
$294$	0	0
$295$	−8.26011	−0.480922
$296$	0	0
$297$	−34.9418	−2.02753
$298$	0	0
$299$	−3.60849	−0.208684
$300$	0	0
$301$	21.5489	1.24206
$302$	0	0
$303$	3.86395	0.221978
$304$	0	0
$305$	−19.6389	−1.12452
$306$	0	0
$307$	5.44703	0.310879	0.155439	−	0.987845i	$-0.450321\pi$
$307$	5.44703	0.310879	0.155439	+	0.987845i	$0.450321\pi$
$308$	0	0
$309$	−19.7974	−1.12623
$310$	0	0
$311$	−3.46535	−0.196502	−0.0982509	−	0.995162i	$-0.531325\pi$
$311$	−3.46535	−0.196502	−0.0982509	+	0.995162i	$0.531325\pi$
$312$	0	0
$313$	31.3427	1.77159	0.885797	−	0.464074i	$-0.153613\pi$
$313$	31.3427	1.77159	0.885797	+	0.464074i	$0.153613\pi$
$314$	0	0
$315$	41.3100	2.32756
$316$	0	0
$317$	−6.81049	−0.382515	−0.191258	−	0.981540i	$-0.561257\pi$
$317$	−6.81049	−0.382515	−0.191258	+	0.981540i	$0.561257\pi$
$318$	0	0
$319$	−17.5346	−0.981748
$320$	0	0
$321$	61.7512	3.44662
$322$	0	0
$323$	−6.12410	−0.340754
$324$	0	0
$325$	−0.0263968	−0.00146423
$326$	0	0
$327$	28.0797	1.55281
$328$	0	0
$329$	6.46060	0.356185
$330$	0	0
$331$	−9.17158	−0.504115	−0.252058	−	0.967712i	$-0.581107\pi$
$331$	−9.17158	−0.504115	−0.252058	+	0.967712i	$0.581107\pi$
$332$	0	0
$333$	66.9597	3.66937
$334$	0	0
$335$	9.36707	0.511778
$336$	0	0
$337$	−10.4435	−0.568894	−0.284447	−	0.958692i	$-0.591810\pi$
$337$	−10.4435	−0.568894	−0.284447	+	0.958692i	$0.591810\pi$
$338$	0	0
$339$	−17.4415	−0.947293
$340$	0	0
$341$	27.8912	1.51039
$342$	0	0
$343$	−18.9791	−1.02477
$344$	0	0
$345$	−10.4588	−0.563085
$346$	0	0
$347$	6.35636	0.341227	0.170614	−	0.985338i	$-0.445425\pi$
$347$	6.35636	0.341227	0.170614	+	0.985338i	$0.445425\pi$
$348$	0	0
$349$	−2.25949	−0.120948	−0.0604738	−	0.998170i	$-0.519261\pi$
$349$	−2.25949	−0.120948	−0.0604738	+	0.998170i	$0.519261\pi$
$350$	0	0
$351$	32.7333	1.74717
$352$	0	0
$353$	−17.6193	−0.937779	−0.468890	−	0.883257i	$-0.655346\pi$
$353$	−17.6193	−0.937779	−0.468890	+	0.883257i	$0.655346\pi$
$354$	0	0
$355$	−13.4146	−0.711970
$356$	0	0
$357$	9.04346	0.478631
$358$	0	0
$359$	3.27691	0.172948	0.0864742	−	0.996254i	$-0.472440\pi$
$359$	3.27691	0.172948	0.0864742	+	0.996254i	$0.472440\pi$
$360$	0	0
$361$	11.9063	0.626648
$362$	0	0
$363$	13.0389	0.684365
$364$	0	0
$365$	−15.0980	−0.790265
$366$	0	0
$367$	5.04081	0.263128	0.131564	−	0.991308i	$-0.458000\pi$
$367$	5.04081	0.263128	0.131564	+	0.991308i	$0.458000\pi$
$368$	0	0
$369$	17.4157	0.906625
$370$	0	0
$371$	24.7523	1.28508
$372$	0	0
$373$	−17.1609	−0.888560	−0.444280	−	0.895888i	$-0.646540\pi$
$373$	−17.1609	−0.888560	−0.444280	+	0.895888i	$0.646540\pi$
$374$	0	0
$375$	35.6132	1.83906
$376$	0	0
$377$	16.4263	0.845998
$378$	0	0
$379$	35.7697	1.83737	0.918683	−	0.394995i	$-0.129254\pi$
$379$	35.7697	1.83737	0.918683	+	0.394995i	$0.129254\pi$
$380$	0	0
$381$	−48.8098	−2.50060
$382$	0	0
$383$	37.3773	1.90989	0.954945	−	0.296783i	$-0.0959138\pi$
$383$	37.3773	1.90989	0.954945	+	0.296783i	$0.0959138\pi$
$384$	0	0
$385$	15.1500	0.772117
$386$	0	0
$387$	59.9991	3.04993
$388$	0	0
$389$	15.4709	0.784405	0.392203	−	0.919879i	$-0.371713\pi$
$389$	15.4709	0.784405	0.392203	+	0.919879i	$0.371713\pi$
$390$	0	0
$391$	−1.61409	−0.0816283
$392$	0	0
$393$	−6.32219	−0.318912
$394$	0	0
$395$	4.59159	0.231028
$396$	0	0
$397$	11.9712	0.600818	0.300409	−	0.953811i	$-0.402877\pi$
$397$	11.9712	0.600818	0.300409	+	0.953811i	$0.402877\pi$
$398$	0	0
$399$	−45.6394	−2.28483
$400$	0	0
$401$	−1.03438	−0.0516546	−0.0258273	−	0.999666i	$-0.508222\pi$
$401$	−1.03438	−0.0516546	−0.0258273	+	0.999666i	$0.508222\pi$
$402$	0	0
$403$	−26.1283	−1.30154
$404$	0	0
$405$	46.7367	2.32236
$406$	0	0
$407$	24.5568	1.21723
$408$	0	0
$409$	6.85826	0.339119	0.169560	−	0.985520i	$-0.445765\pi$
$409$	6.85826	0.339119	0.169560	+	0.985520i	$0.445765\pi$
$410$	0	0
$411$	25.2760	1.24677
$412$	0	0
$413$	−9.50012	−0.467470
$414$	0	0
$415$	10.4976	0.515307
$416$	0	0
$417$	−4.02712	−0.197209
$418$	0	0
$419$	−19.1711	−0.936571	−0.468286	−	0.883577i	$-0.655128\pi$
$419$	−19.1711	−0.936571	−0.468286	+	0.883577i	$0.655128\pi$
$420$	0	0
$421$	12.4164	0.605138	0.302569	−	0.953127i	$-0.402156\pi$
$421$	12.4164	0.605138	0.302569	+	0.953127i	$0.402156\pi$
$422$	0	0
$423$	17.9884	0.874625
$424$	0	0
$425$	−0.0118074	−0.000572744 0
$426$	0	0
$427$	−22.5871	−1.09307
$428$	0	0
$429$	20.6446	0.996733
$430$	0	0
$431$	4.15904	0.200334	0.100167	−	0.994971i	$-0.468062\pi$
$431$	4.15904	0.200334	0.100167	+	0.994971i	$0.468062\pi$
$432$	0	0
$433$	−29.1035	−1.39862	−0.699312	−	0.714817i	$-0.746510\pi$
$433$	−29.1035	−1.39862	−0.699312	+	0.714817i	$0.746510\pi$
$434$	0	0
$435$	47.6100	2.28272
$436$	0	0
$437$	8.14581	0.389667
$438$	0	0
$439$	−17.4302	−0.831899	−0.415950	−	0.909388i	$-0.636551\pi$
$439$	−17.4302	−0.831899	−0.415950	+	0.909388i	$0.636551\pi$
$440$	0	0
$441$	−2.66615	−0.126959
$442$	0	0
$443$	−24.8039	−1.17847	−0.589235	−	0.807962i	$-0.700571\pi$
$443$	−24.8039	−1.17847	−0.589235	+	0.807962i	$0.700571\pi$
$444$	0	0
$445$	10.6426	0.504507
$446$	0	0
$447$	57.6878	2.72854
$448$	0	0
$449$	17.8192	0.840940	0.420470	−	0.907307i	$-0.361865\pi$
$449$	17.8192	0.840940	0.420470	+	0.907307i	$0.361865\pi$
$450$	0	0
$451$	6.38703	0.300753
$452$	0	0
$453$	16.2624	0.764076
$454$	0	0
$455$	−14.1925	−0.665353
$456$	0	0
$457$	−2.82186	−0.132001	−0.0660005	−	0.997820i	$-0.521024\pi$
$457$	−2.82186	−0.132001	−0.0660005	+	0.997820i	$0.521024\pi$
$458$	0	0
$459$	14.6418	0.683419
$460$	0	0
$461$	−3.10779	−0.144744	−0.0723721	−	0.997378i	$-0.523057\pi$
$461$	−3.10779	−0.144744	−0.0723721	+	0.997378i	$0.523057\pi$
$462$	0	0
$463$	−1.32805	−0.0617196	−0.0308598	−	0.999524i	$-0.509825\pi$
$463$	−1.32805	−0.0617196	−0.0308598	+	0.999524i	$0.509825\pi$
$464$	0	0
$465$	−75.7303	−3.51191
$466$	0	0
$467$	−4.95905	−0.229477	−0.114739	−	0.993396i	$-0.536603\pi$
$467$	−4.95905	−0.229477	−0.114739	+	0.993396i	$0.536603\pi$
$468$	0	0
$469$	10.7733	0.497463
$470$	0	0
$471$	55.0629	2.53716
$472$	0	0
$473$	22.0041	1.01175
$474$	0	0
$475$	0.0595882	0.00273409
$476$	0	0
$477$	68.9184	3.15556
$478$	0	0
$479$	−33.2214	−1.51793	−0.758963	−	0.651133i	$-0.774294\pi$
$479$	−33.2214	−1.51793	−0.758963	+	0.651133i	$0.774294\pi$
$480$	0	0
$481$	−23.0047	−1.04892
$482$	0	0
$483$	−12.0289	−0.547335
$484$	0	0
$485$	41.5588	1.88709
$486$	0	0
$487$	34.3941	1.55854	0.779272	−	0.626686i	$-0.215589\pi$
$487$	34.3941	1.55854	0.779272	+	0.626686i	$0.215589\pi$
$488$	0	0
$489$	23.5422	1.06461
$490$	0	0
$491$	27.9369	1.26077	0.630387	−	0.776281i	$-0.282896\pi$
$491$	27.9369	1.26077	0.630387	+	0.776281i	$0.282896\pi$
$492$	0	0
$493$	7.34757	0.330918
$494$	0	0
$495$	42.1825	1.89596
$496$	0	0
$497$	−15.4283	−0.692056
$498$	0	0
$499$	−26.1864	−1.17227	−0.586133	−	0.810215i	$-0.699350\pi$
$499$	−26.1864	−1.17227	−0.586133	+	0.810215i	$0.699350\pi$
$500$	0	0
$501$	−12.3371	−0.551179
$502$	0	0
$503$	26.6946	1.19025	0.595127	−	0.803631i	$-0.297102\pi$
$503$	26.6946	1.19025	0.595127	+	0.803631i	$0.297102\pi$
$504$	0	0
$505$	−2.71243	−0.120701
$506$	0	0
$507$	22.1142	0.982125
$508$	0	0
$509$	35.2091	1.56062	0.780309	−	0.625394i	$-0.215062\pi$
$509$	35.2091	1.56062	0.780309	+	0.625394i	$0.215062\pi$
$510$	0	0
$511$	−17.3645	−0.768161
$512$	0	0
$513$	−73.8922	−3.26242
$514$	0	0
$515$	13.8974	0.612394
$516$	0	0
$517$	6.59705	0.290138
$518$	0	0
$519$	41.4788	1.82072
$520$	0	0
$521$	−25.5618	−1.11988	−0.559941	−	0.828532i	$-0.689176\pi$
$521$	−25.5618	−1.11988	−0.559941	+	0.828532i	$0.689176\pi$
$522$	0	0
$523$	31.5712	1.38051	0.690256	−	0.723565i	$-0.257498\pi$
$523$	31.5712	1.38051	0.690256	+	0.723565i	$0.257498\pi$
$524$	0	0
$525$	−0.0879939	−0.00384037
$526$	0	0
$527$	−11.6873	−0.509108
$528$	0	0
$529$	−20.8531	−0.906655
$530$	0	0
$531$	−26.4514	−1.14789
$532$	0	0
$533$	−5.98333	−0.259167
$534$	0	0
$535$	−43.3484	−1.87411
$536$	0	0
$537$	70.4495	3.04012
$538$	0	0
$539$	−0.977782	−0.0421161
$540$	0	0
$541$	14.1107	0.606666	0.303333	−	0.952885i	$-0.401901\pi$
$541$	14.1107	0.606666	0.303333	+	0.952885i	$0.401901\pi$
$542$	0	0
$543$	−33.5200	−1.43848
$544$	0	0
$545$	−19.7115	−0.844347
$546$	0	0
$547$	−20.6835	−0.884361	−0.442181	−	0.896926i	$-0.645795\pi$
$547$	−20.6835	−0.884361	−0.442181	+	0.896926i	$0.645795\pi$
$548$	0	0
$549$	−62.8898	−2.68407
$550$	0	0
$551$	−37.0808	−1.57969
$552$	0	0
$553$	5.28088	0.224566
$554$	0	0
$555$	−66.6767	−2.83027
$556$	0	0
$557$	9.33742	0.395639	0.197820	−	0.980238i	$-0.436614\pi$
$557$	9.33742	0.395639	0.197820	+	0.980238i	$0.436614\pi$
$558$	0	0
$559$	−20.6133	−0.871849
$560$	0	0
$561$	9.23446	0.389879
$562$	0	0
$563$	−43.2892	−1.82442	−0.912210	−	0.409722i	$-0.865626\pi$
$563$	−43.2892	−1.82442	−0.912210	+	0.409722i	$0.865626\pi$
$564$	0	0
$565$	12.2437	0.515095
$566$	0	0
$567$	53.7528	2.25740
$568$	0	0
$569$	−21.9937	−0.922024	−0.461012	−	0.887394i	$-0.652513\pi$
$569$	−21.9937	−0.922024	−0.461012	+	0.887394i	$0.652513\pi$
$570$	0	0
$571$	4.04641	0.169337	0.0846684	−	0.996409i	$-0.473017\pi$
$571$	4.04641	0.169337	0.0846684	+	0.996409i	$0.473017\pi$
$572$	0	0
$573$	−8.16841	−0.341240
$574$	0	0
$575$	0.0157053	0.000654957 0
$576$	0	0
$577$	−13.1425	−0.547131	−0.273566	−	0.961853i	$-0.588203\pi$
$577$	−13.1425	−0.547131	−0.273566	+	0.961853i	$0.588203\pi$
$578$	0	0
$579$	−56.9456	−2.36658
$580$	0	0
$581$	12.0735	0.500893
$582$	0	0
$583$	25.2751	1.04679
$584$	0	0
$585$	−39.5164	−1.63380
$586$	0	0
$587$	20.4044	0.842182	0.421091	−	0.907018i	$-0.361647\pi$
$587$	20.4044	0.842182	0.421091	+	0.907018i	$0.361647\pi$
$588$	0	0
$589$	58.9821	2.43032
$590$	0	0
$591$	−34.8155	−1.43212
$592$	0	0
$593$	14.9489	0.613879	0.306940	−	0.951729i	$-0.400695\pi$
$593$	14.9489	0.613879	0.306940	+	0.951729i	$0.400695\pi$
$594$	0	0
$595$	−6.34836	−0.260258
$596$	0	0
$597$	22.9058	0.937473
$598$	0	0
$599$	−14.6804	−0.599823	−0.299912	−	0.953967i	$-0.596957\pi$
$599$	−14.6804	−0.599823	−0.299912	+	0.953967i	$0.596957\pi$
$600$	0	0
$601$	4.88210	0.199145	0.0995725	−	0.995030i	$-0.468252\pi$
$601$	4.88210	0.199145	0.0995725	+	0.995030i	$0.468252\pi$
$602$	0	0
$603$	29.9962	1.22154
$604$	0	0
$605$	−9.15310	−0.372126
$606$	0	0
$607$	−15.7612	−0.639728	−0.319864	−	0.947463i	$-0.603637\pi$
$607$	−15.7612	−0.639728	−0.319864	+	0.947463i	$0.603637\pi$
$608$	0	0
$609$	54.7572	2.21887
$610$	0	0
$611$	−6.18008	−0.250019
$612$	0	0
$613$	21.4094	0.864718	0.432359	−	0.901701i	$-0.357681\pi$
$613$	21.4094	0.864718	0.432359	+	0.901701i	$0.357681\pi$
$614$	0	0
$615$	−17.3421	−0.699301
$616$	0	0
$617$	10.9209	0.439659	0.219829	−	0.975538i	$-0.429450\pi$
$617$	10.9209	0.439659	0.219829	+	0.975538i	$0.429450\pi$
$618$	0	0
$619$	21.8057	0.876447	0.438223	−	0.898866i	$-0.355608\pi$
$619$	21.8057	0.876447	0.438223	+	0.898866i	$0.355608\pi$
$620$	0	0
$621$	−19.4754	−0.781520
$622$	0	0
$623$	12.2402	0.490395
$624$	0	0
$625$	−25.0535	−1.00214
$626$	0	0
$627$	−46.6033	−1.86116
$628$	0	0
$629$	−10.2901	−0.410294
$630$	0	0
$631$	38.7513	1.54266	0.771332	−	0.636433i	$-0.219591\pi$
$631$	38.7513	1.54266	0.771332	+	0.636433i	$0.219591\pi$
$632$	0	0
$633$	−2.93617	−0.116702
$634$	0	0
$635$	34.2637	1.35971
$636$	0	0
$637$	0.915981	0.0362925
$638$	0	0
$639$	−42.9574	−1.69937
$640$	0	0
$641$	35.8847	1.41736	0.708681	−	0.705529i	$-0.249291\pi$
$641$	35.8847	1.41736	0.708681	+	0.705529i	$0.249291\pi$
$642$	0	0
$643$	17.7011	0.698062	0.349031	−	0.937111i	$-0.386511\pi$
$643$	17.7011	0.698062	0.349031	+	0.937111i	$0.386511\pi$
$644$	0	0
$645$	−59.7455	−2.35248
$646$	0	0
$647$	30.5913	1.20267	0.601334	−	0.798998i	$-0.294636\pi$
$647$	30.5913	1.20267	0.601334	+	0.798998i	$0.294636\pi$
$648$	0	0
$649$	−9.70076	−0.380788
$650$	0	0
$651$	−87.0990	−3.41368
$652$	0	0
$653$	−25.5534	−0.999980	−0.499990	−	0.866031i	$-0.666663\pi$
$653$	−25.5534	−0.999980	−0.499990	+	0.866031i	$0.666663\pi$
$654$	0	0
$655$	4.43808	0.173410
$656$	0	0
$657$	−48.3484	−1.88625
$658$	0	0
$659$	−35.1780	−1.37034	−0.685171	−	0.728382i	$-0.740273\pi$
$659$	−35.1780	−1.37034	−0.685171	+	0.728382i	$0.740273\pi$
$660$	0	0
$661$	45.6359	1.77503	0.887516	−	0.460777i	$-0.152429\pi$
$661$	45.6359	1.77503	0.887516	+	0.460777i	$0.152429\pi$
$662$	0	0
$663$	−8.65079	−0.335969
$664$	0	0
$665$	32.0381	1.24238
$666$	0	0
$667$	−9.77318	−0.378419
$668$	0	0
$669$	−66.4109	−2.56759
$670$	0	0
$671$	−23.0642	−0.890383
$672$	0	0
$673$	−45.9091	−1.76966	−0.884832	−	0.465910i	$-0.845727\pi$
$673$	−45.9091	−1.76966	−0.884832	+	0.465910i	$0.845727\pi$
$674$	0	0
$675$	−0.142466	−0.00548352
$676$	0	0
$677$	−37.2828	−1.43290	−0.716448	−	0.697641i	$-0.754233\pi$
$677$	−37.2828	−1.43290	−0.716448	+	0.697641i	$0.754233\pi$
$678$	0	0
$679$	47.7976	1.83430
$680$	0	0
$681$	−46.5555	−1.78401
$682$	0	0
$683$	−30.0842	−1.15114	−0.575571	−	0.817752i	$-0.695220\pi$
$683$	−30.0842	−1.15114	−0.575571	+	0.817752i	$0.695220\pi$
$684$	0	0
$685$	−17.7434	−0.677939
$686$	0	0
$687$	21.5858	0.823551
$688$	0	0
$689$	−23.6776	−0.902044
$690$	0	0
$691$	46.9833	1.78733	0.893665	−	0.448734i	$-0.148125\pi$
$691$	46.9833	1.78733	0.893665	+	0.448734i	$0.148125\pi$
$692$	0	0
$693$	48.5150	1.84293
$694$	0	0
$695$	2.82697	0.107233
$696$	0	0
$697$	−2.67638	−0.101375
$698$	0	0
$699$	−89.1751	−3.37291
$700$	0	0
$701$	24.8857	0.939921	0.469960	−	0.882688i	$-0.344268\pi$
$701$	24.8857	0.939921	0.469960	+	0.882688i	$0.344268\pi$
$702$	0	0
$703$	51.9308	1.95861
$704$	0	0
$705$	−17.9124	−0.674618
$706$	0	0
$707$	−3.11962	−0.117325
$708$	0	0
$709$	−16.7074	−0.627461	−0.313730	−	0.949512i	$-0.601579\pi$
$709$	−16.7074	−0.627461	−0.313730	+	0.949512i	$0.601579\pi$
$710$	0	0
$711$	14.7037	0.551431
$712$	0	0
$713$	15.5456	0.582187
$714$	0	0
$715$	−14.4922	−0.541978
$716$	0	0
$717$	54.6483	2.04088
$718$	0	0
$719$	−20.9297	−0.780547	−0.390273	−	0.920699i	$-0.627619\pi$
$719$	−20.9297	−0.780547	−0.390273	+	0.920699i	$0.627619\pi$
$720$	0	0
$721$	15.9837	0.595264
$722$	0	0
$723$	−24.1856	−0.899472
$724$	0	0
$725$	−0.0714927	−0.00265517
$726$	0	0
$727$	40.8161	1.51378	0.756892	−	0.653540i	$-0.226717\pi$
$727$	40.8161	1.51378	0.756892	+	0.653540i	$0.226717\pi$
$728$	0	0
$729$	22.5141	0.833855
$730$	0	0
$731$	−9.22043	−0.341030
$732$	0	0
$733$	−48.1166	−1.77723	−0.888613	−	0.458657i	$-0.848331\pi$
$733$	−48.1166	−1.77723	−0.888613	+	0.458657i	$0.848331\pi$
$734$	0	0
$735$	2.65488	0.0979267
$736$	0	0
$737$	11.0008	0.405219
$738$	0	0
$739$	35.1025	1.29127	0.645633	−	0.763648i	$-0.276594\pi$
$739$	35.1025	1.29127	0.645633	+	0.763648i	$0.276594\pi$
$740$	0	0
$741$	43.6577	1.60381
$742$	0	0
$743$	4.11952	0.151131	0.0755653	−	0.997141i	$-0.475924\pi$
$743$	4.11952	0.151131	0.0755653	+	0.997141i	$0.475924\pi$
$744$	0	0
$745$	−40.4959	−1.48366
$746$	0	0
$747$	33.6165	1.22996
$748$	0	0
$749$	−49.8558	−1.82169
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	14.4645	0.527117
$754$	0	0
$755$	−11.4160	−0.415470
$756$	0	0
$757$	21.7736	0.791375	0.395688	−	0.918385i	$-0.370506\pi$
$757$	21.7736	0.791375	0.395688	+	0.918385i	$0.370506\pi$
$758$	0	0
$759$	−12.2830	−0.445844
$760$	0	0
$761$	37.0426	1.34279	0.671397	−	0.741097i	$-0.265694\pi$
$761$	37.0426	1.34279	0.671397	+	0.741097i	$0.265694\pi$
$762$	0	0
$763$	−22.6706	−0.820730
$764$	0	0
$765$	−17.6759	−0.639073
$766$	0	0
$767$	9.08762	0.328135
$768$	0	0
$769$	4.69845	0.169430	0.0847152	−	0.996405i	$-0.473002\pi$
$769$	4.69845	0.169430	0.0847152	+	0.996405i	$0.473002\pi$
$770$	0	0
$771$	−30.8655	−1.11160
$772$	0	0
$773$	−41.8840	−1.50646	−0.753231	−	0.657756i	$-0.771506\pi$
$773$	−41.8840	−1.50646	−0.753231	+	0.657756i	$0.771506\pi$
$774$	0	0
$775$	0.113719	0.00408491
$776$	0	0
$777$	−76.6863	−2.75110
$778$	0	0
$779$	13.5068	0.483931
$780$	0	0
$781$	−15.7542	−0.563730
$782$	0	0
$783$	88.6544	3.16825
$784$	0	0
$785$	−38.6533	−1.37959
$786$	0	0
$787$	19.4432	0.693076	0.346538	−	0.938036i	$-0.387357\pi$
$787$	19.4432	0.693076	0.346538	+	0.938036i	$0.387357\pi$
$788$	0	0
$789$	−39.2270	−1.39652
$790$	0	0
$791$	14.0817	0.500687
$792$	0	0
$793$	21.6064	0.767266
$794$	0	0
$795$	−68.6271	−2.43395
$796$	0	0
$797$	−36.1864	−1.28179	−0.640895	−	0.767629i	$-0.721437\pi$
$797$	−36.1864	−1.28179	−0.640895	+	0.767629i	$0.721437\pi$
$798$	0	0
$799$	−2.76438	−0.0977969
$800$	0	0
$801$	34.0808	1.20418
$802$	0	0
$803$	−17.7313	−0.625723
$804$	0	0
$805$	8.44411	0.297616
$806$	0	0
$807$	−3.32843	−0.117166
$808$	0	0
$809$	12.5071	0.439724	0.219862	−	0.975531i	$-0.429439\pi$
$809$	12.5071	0.439724	0.219862	+	0.975531i	$0.429439\pi$
$810$	0	0
$811$	−43.9774	−1.54426	−0.772129	−	0.635466i	$-0.780808\pi$
$811$	−43.9774	−1.54426	−0.772129	+	0.635466i	$0.780808\pi$
$812$	0	0
$813$	20.3098	0.712296
$814$	0	0
$815$	−16.5262	−0.578889
$816$	0	0
$817$	46.5325	1.62797
$818$	0	0
$819$	−45.4486	−1.58810
$820$	0	0
$821$	25.2717	0.881990	0.440995	−	0.897510i	$-0.354626\pi$
$821$	25.2717	0.881990	0.440995	+	0.897510i	$0.354626\pi$
$822$	0	0
$823$	41.4501	1.44486	0.722430	−	0.691444i	$-0.243025\pi$
$823$	41.4501	1.44486	0.722430	+	0.691444i	$0.243025\pi$
$824$	0	0
$825$	−0.0898524	−0.00312826
$826$	0	0
$827$	31.3487	1.09010	0.545050	−	0.838404i	$-0.316511\pi$
$827$	31.3487	1.09010	0.545050	+	0.838404i	$0.316511\pi$
$828$	0	0
$829$	12.7564	0.443046	0.221523	−	0.975155i	$-0.428897\pi$
$829$	12.7564	0.443046	0.221523	+	0.975155i	$0.428897\pi$
$830$	0	0
$831$	93.8828	3.25676
$832$	0	0
$833$	0.409723	0.0141961
$834$	0	0
$835$	8.66042	0.299706
$836$	0	0
$837$	−141.017	−4.87427
$838$	0	0
$839$	2.50293	0.0864106	0.0432053	−	0.999066i	$-0.486243\pi$
$839$	2.50293	0.0864106	0.0432053	+	0.999066i	$0.486243\pi$
$840$	0	0
$841$	15.4888	0.534096
$842$	0	0
$843$	−53.1125	−1.82929
$844$	0	0
$845$	−15.5238	−0.534035
$846$	0	0
$847$	−10.5272	−0.361718
$848$	0	0
$849$	−14.4643	−0.496413
$850$	0	0
$851$	13.6871	0.469188
$852$	0	0
$853$	18.5637	0.635611	0.317805	−	0.948156i	$-0.397054\pi$
$853$	18.5637	0.635611	0.317805	+	0.948156i	$0.397054\pi$
$854$	0	0
$855$	89.2043	3.05072
$856$	0	0
$857$	35.5990	1.21604	0.608020	−	0.793922i	$-0.291964\pi$
$857$	35.5990	1.21604	0.608020	+	0.793922i	$0.291964\pi$
$858$	0	0
$859$	8.62788	0.294379	0.147190	−	0.989108i	$-0.452977\pi$
$859$	8.62788	0.294379	0.147190	+	0.989108i	$0.452977\pi$
$860$	0	0
$861$	−19.9455	−0.679741
$862$	0	0
$863$	−9.58592	−0.326309	−0.163154	−	0.986601i	$-0.552167\pi$
$863$	−9.58592	−0.326309	−0.163154	+	0.986601i	$0.552167\pi$
$864$	0	0
$865$	−29.1175	−0.990024
$866$	0	0
$867$	50.3395	1.70962
$868$	0	0
$869$	5.39242	0.182925
$870$	0	0
$871$	−10.3055	−0.349188
$872$	0	0
$873$	133.084	4.50420
$874$	0	0
$875$	−28.7529	−0.972025
$876$	0	0
$877$	−20.6357	−0.696819	−0.348409	−	0.937342i	$-0.613278\pi$
$877$	−20.6357	−0.696819	−0.348409	+	0.937342i	$0.613278\pi$
$878$	0	0
$879$	−102.506	−3.45743
$880$	0	0
$881$	0.563989	0.0190013	0.00950064	−	0.999955i	$-0.496976\pi$
$881$	0.563989	0.0190013	0.00950064	+	0.999955i	$0.496976\pi$
$882$	0	0
$883$	−5.65169	−0.190194	−0.0950972	−	0.995468i	$-0.530316\pi$
$883$	−5.65169	−0.190194	−0.0950972	+	0.995468i	$0.530316\pi$
$884$	0	0
$885$	26.3396	0.885395
$886$	0	0
$887$	−21.6703	−0.727617	−0.363809	−	0.931474i	$-0.618524\pi$
$887$	−21.6703	−0.727617	−0.363809	+	0.931474i	$0.618524\pi$
$888$	0	0
$889$	39.4074	1.32168
$890$	0	0
$891$	54.8881	1.83882
$892$	0	0
$893$	13.9509	0.466850
$894$	0	0
$895$	−49.4544	−1.65308
$896$	0	0
$897$	11.5066	0.384195
$898$	0	0
$899$	−70.7656	−2.36017
$900$	0	0
$901$	−10.5911	−0.352841
$902$	0	0
$903$	−68.7145	−2.28668
$904$	0	0
$905$	23.5305	0.782180
$906$	0	0
$907$	−22.4865	−0.746652	−0.373326	−	0.927700i	$-0.621783\pi$
$907$	−22.4865	−0.746652	−0.373326	+	0.927700i	$0.621783\pi$
$908$	0	0
$909$	−8.68601	−0.288097
$910$	0	0
$911$	26.9169	0.891797	0.445898	−	0.895084i	$-0.352884\pi$
$911$	26.9169	0.891797	0.445898	+	0.895084i	$0.352884\pi$
$912$	0	0
$913$	12.3285	0.408013
$914$	0	0
$915$	62.6240	2.07029
$916$	0	0
$917$	5.10432	0.168560
$918$	0	0
$919$	−35.9235	−1.18501	−0.592504	−	0.805567i	$-0.701861\pi$
$919$	−35.9235	−1.18501	−0.592504	+	0.805567i	$0.701861\pi$
$920$	0	0
$921$	−17.3693	−0.572338
$922$	0	0
$923$	14.7584	0.485780
$924$	0	0
$925$	0.100124	0.00329206
$926$	0	0
$927$	44.5038	1.46170
$928$	0	0
$929$	7.82499	0.256730	0.128365	−	0.991727i	$-0.459027\pi$
$929$	7.82499	0.256730	0.128365	+	0.991727i	$0.459027\pi$
$930$	0	0
$931$	−2.06774	−0.0677674
$932$	0	0
$933$	11.0502	0.361767
$934$	0	0
$935$	−6.48244	−0.211999
$936$	0	0
$937$	18.2637	0.596648	0.298324	−	0.954465i	$-0.403572\pi$
$937$	18.2637	0.596648	0.298324	+	0.954465i	$0.403572\pi$
$938$	0	0
$939$	−99.9445	−3.26157
$940$	0	0
$941$	33.7453	1.10006	0.550032	−	0.835144i	$-0.314616\pi$
$941$	33.7453	1.10006	0.550032	+	0.835144i	$0.314616\pi$
$942$	0	0
$943$	3.55991	0.115927
$944$	0	0
$945$	−76.5982	−2.49174
$946$	0	0
$947$	−8.54030	−0.277523	−0.138761	−	0.990326i	$-0.544312\pi$
$947$	−8.54030	−0.277523	−0.138761	+	0.990326i	$0.544312\pi$
$948$	0	0
$949$	16.6105	0.539201
$950$	0	0
$951$	21.7171	0.704224
$952$	0	0
$953$	−34.5534	−1.11929	−0.559646	−	0.828731i	$-0.689063\pi$
$953$	−34.5534	−1.11929	−0.559646	+	0.828731i	$0.689063\pi$
$954$	0	0
$955$	5.73410	0.185551
$956$	0	0
$957$	55.9137	1.80743
$958$	0	0
$959$	−20.4070	−0.658977
$960$	0	0
$961$	81.5625	2.63105
$962$	0	0
$963$	−138.815	−4.47324
$964$	0	0
$965$	39.9749	1.28684
$966$	0	0
$967$	27.9987	0.900378	0.450189	−	0.892933i	$-0.351357\pi$
$967$	27.9987	0.900378	0.450189	+	0.892933i	$0.351357\pi$
$968$	0	0
$969$	19.5283	0.627340
$970$	0	0
$971$	13.9549	0.447833	0.223917	−	0.974608i	$-0.428116\pi$
$971$	13.9549	0.447833	0.223917	+	0.974608i	$0.428116\pi$
$972$	0	0
$973$	3.25136	0.104234
$974$	0	0
$975$	0.0841732	0.00269570
$976$	0	0
$977$	−59.3079	−1.89743	−0.948713	−	0.316138i	$-0.897614\pi$
$977$	−59.3079	−1.89743	−0.948713	+	0.316138i	$0.897614\pi$
$978$	0	0
$979$	12.4988	0.399462
$980$	0	0
$981$	−63.1221	−2.01533
$982$	0	0
$983$	−8.30963	−0.265036	−0.132518	−	0.991181i	$-0.542306\pi$
$983$	−8.30963	−0.265036	−0.132518	+	0.991181i	$0.542306\pi$
$984$	0	0
$985$	24.4399	0.778721
$986$	0	0
$987$	−20.6014	−0.655748
$988$	0	0
$989$	12.2643	0.389982
$990$	0	0
$991$	38.0164	1.20763	0.603815	−	0.797124i	$-0.293647\pi$
$991$	38.0164	1.20763	0.603815	+	0.797124i	$0.293647\pi$
$992$	0	0
$993$	29.2460	0.928094
$994$	0	0
$995$	−16.0795	−0.509755
$996$	0	0
$997$	−30.0599	−0.952006	−0.476003	−	0.879444i	$-0.657915\pi$
$997$	−30.0599	−0.952006	−0.476003	+	0.879444i	$0.657915\pi$
$998$	0	0
$999$	−124.159	−3.92820

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.d.1.1	✓	49	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-3.18877 q^{3} +2.23846 q^{5} +2.57450 q^{7} +7.16824 q^{9} +O(q^{10})\)	\(q-3.18877 q^{3} +2.23846 q^{5} +2.57450 q^{7} +7.16824 q^{9} +2.62888 q^{11} -2.46272 q^{13} -7.13794 q^{15} -1.10159 q^{17} +5.55935 q^{19} -8.20949 q^{21} +1.46525 q^{23} +0.0107186 q^{25} -13.2915 q^{27} -6.66999 q^{29} +10.6095 q^{31} -8.38287 q^{33} +5.76293 q^{35} +9.34117 q^{37} +7.85303 q^{39} +2.42957 q^{41} +8.37014 q^{43} +16.0458 q^{45} +2.50946 q^{47} -0.371939 q^{49} +3.51270 q^{51} +9.61441 q^{53} +5.88464 q^{55} -17.7275 q^{57} -3.69008 q^{59} -8.77340 q^{61} +18.4546 q^{63} -5.51270 q^{65} +4.18460 q^{67} -4.67233 q^{69} -5.99275 q^{71} -6.74481 q^{73} -0.0341790 q^{75} +6.76805 q^{77} +2.05123 q^{79} +20.8789 q^{81} +4.68964 q^{83} -2.46586 q^{85} +21.2691 q^{87} +4.75441 q^{89} -6.34027 q^{91} -33.8314 q^{93} +12.4444 q^{95} +18.5657 q^{97} +18.8444 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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