LMFDB - Embedded newform 6008.2.a.e.1.27

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			1.27
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.343448 q^{3} -0.515640 q^{5} -3.09823 q^{7} -2.88204 q^{9} +O(q^{10})$	$q+0.343448 q^{3} -0.515640 q^{5} -3.09823 q^{7} -2.88204 q^{9} +3.36617 q^{11} +6.89124 q^{13} -0.177095 q^{15} -2.74330 q^{17} +5.90448 q^{19} -1.06408 q^{21} +0.434326 q^{23} -4.73412 q^{25} -2.02018 q^{27} +0.715207 q^{29} -0.476381 q^{31} +1.15611 q^{33} +1.59757 q^{35} -10.3893 q^{37} +2.36678 q^{39} +5.09378 q^{41} -2.97309 q^{43} +1.48610 q^{45} +5.04407 q^{47} +2.59903 q^{49} -0.942183 q^{51} +8.94342 q^{53} -1.73573 q^{55} +2.02788 q^{57} -9.71744 q^{59} -9.74340 q^{61} +8.92924 q^{63} -3.55340 q^{65} +7.35306 q^{67} +0.149168 q^{69} -1.40295 q^{71} +16.7996 q^{73} -1.62592 q^{75} -10.4292 q^{77} -5.53188 q^{79} +7.95230 q^{81} -12.9852 q^{83} +1.41456 q^{85} +0.245636 q^{87} +0.906320 q^{89} -21.3506 q^{91} -0.163612 q^{93} -3.04458 q^{95} +0.281398 q^{97} -9.70146 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.343448	0.198290	0.0991449	−	0.995073i	$-0.468389\pi$
$3$	0.343448	0.198290	0.0991449	+	0.995073i	$0.468389\pi$
$4$	0	0
$5$	−0.515640	−0.230601	−0.115301	−	0.993331i	$-0.536783\pi$
$5$	−0.515640	−0.230601	−0.115301	+	0.993331i	$0.536783\pi$
$6$	0	0
$7$	−3.09823	−1.17102	−0.585511	−	0.810665i	$-0.699106\pi$
$7$	−3.09823	−1.17102	−0.585511	+	0.810665i	$0.699106\pi$
$8$	0	0
$9$	−2.88204	−0.960681
$10$	0	0
$11$	3.36617	1.01494	0.507470	−	0.861670i	$-0.330581\pi$
$11$	3.36617	1.01494	0.507470	+	0.861670i	$0.330581\pi$
$12$	0	0
$13$	6.89124	1.91128	0.955642	−	0.294529i	$-0.0951629\pi$
$13$	6.89124	1.91128	0.955642	+	0.294529i	$0.0951629\pi$
$14$	0	0
$15$	−0.177095	−0.0457259
$16$	0	0
$17$	−2.74330	−0.665349	−0.332675	−	0.943042i	$-0.607951\pi$
$17$	−2.74330	−0.665349	−0.332675	+	0.943042i	$0.607951\pi$
$18$	0	0
$19$	5.90448	1.35458	0.677290	−	0.735716i	$-0.263154\pi$
$19$	5.90448	1.35458	0.677290	+	0.735716i	$0.263154\pi$
$20$	0	0
$21$	−1.06408	−0.232202
$22$	0	0
$23$	0.434326	0.0905631	0.0452816	−	0.998974i	$-0.485581\pi$
$23$	0.434326	0.0905631	0.0452816	+	0.998974i	$0.485581\pi$
$24$	0	0
$25$	−4.73412	−0.946823
$26$	0	0
$27$	−2.02018	−0.388783
$28$	0	0
$29$	0.715207	0.132811	0.0664053	−	0.997793i	$-0.478847\pi$
$29$	0.715207	0.132811	0.0664053	+	0.997793i	$0.478847\pi$
$30$	0	0
$31$	−0.476381	−0.0855606	−0.0427803	−	0.999085i	$-0.513622\pi$
$31$	−0.476381	−0.0855606	−0.0427803	+	0.999085i	$0.513622\pi$
$32$	0	0
$33$	1.15611	0.201252
$34$	0	0
$35$	1.59757	0.270039
$36$	0	0
$37$	−10.3893	−1.70799	−0.853995	−	0.520281i	$-0.825827\pi$
$37$	−10.3893	−1.70799	−0.853995	+	0.520281i	$0.825827\pi$
$38$	0	0
$39$	2.36678	0.378988
$40$	0	0
$41$	5.09378	0.795516	0.397758	−	0.917490i	$-0.369788\pi$
$41$	5.09378	0.795516	0.397758	+	0.917490i	$0.369788\pi$
$42$	0	0
$43$	−2.97309	−0.453392	−0.226696	−	0.973966i	$-0.572792\pi$
$43$	−2.97309	−0.453392	−0.226696	+	0.973966i	$0.572792\pi$
$44$	0	0
$45$	1.48610	0.221534
$46$	0	0
$47$	5.04407	0.735753	0.367877	−	0.929875i	$-0.380085\pi$
$47$	5.04407	0.735753	0.367877	+	0.929875i	$0.380085\pi$
$48$	0	0
$49$	2.59903	0.371291
$50$	0	0
$51$	−0.942183	−0.131932
$52$	0	0
$53$	8.94342	1.22847	0.614237	−	0.789122i	$-0.289464\pi$
$53$	8.94342	1.22847	0.614237	+	0.789122i	$0.289464\pi$
$54$	0	0
$55$	−1.73573	−0.234046
$56$	0	0
$57$	2.02788	0.268600
$58$	0	0
$59$	−9.71744	−1.26510	−0.632552	−	0.774518i	$-0.717992\pi$
$59$	−9.71744	−1.26510	−0.632552	+	0.774518i	$0.717992\pi$
$60$	0	0
$61$	−9.74340	−1.24751	−0.623757	−	0.781618i	$-0.714395\pi$
$61$	−9.74340	−1.24751	−0.623757	+	0.781618i	$0.714395\pi$
$62$	0	0
$63$	8.92924	1.12498
$64$	0	0
$65$	−3.55340	−0.440744
$66$	0	0
$67$	7.35306	0.898319	0.449160	−	0.893452i	$-0.351723\pi$
$67$	7.35306	0.898319	0.449160	+	0.893452i	$0.351723\pi$
$68$	0	0
$69$	0.149168	0.0179578
$70$	0	0
$71$	−1.40295	−0.166500	−0.0832499	−	0.996529i	$-0.526530\pi$
$71$	−1.40295	−0.166500	−0.0832499	+	0.996529i	$0.526530\pi$
$72$	0	0
$73$	16.7996	1.96625	0.983124	−	0.182939i	$-0.0585612\pi$
$73$	16.7996	1.96625	0.983124	+	0.182939i	$0.0585612\pi$
$74$	0	0
$75$	−1.62592	−0.187745
$76$	0	0
$77$	−10.4292	−1.18852
$78$	0	0
$79$	−5.53188	−0.622385	−0.311192	−	0.950347i	$-0.600728\pi$
$79$	−5.53188	−0.622385	−0.311192	+	0.950347i	$0.600728\pi$
$80$	0	0
$81$	7.95230	0.883589
$82$	0	0
$83$	−12.9852	−1.42531	−0.712657	−	0.701512i	$-0.752509\pi$
$83$	−12.9852	−1.42531	−0.712657	+	0.701512i	$0.752509\pi$
$84$	0	0
$85$	1.41456	0.153430
$86$	0	0
$87$	0.245636	0.0263350
$88$	0	0
$89$	0.906320	0.0960697	0.0480349	−	0.998846i	$-0.484704\pi$
$89$	0.906320	0.0960697	0.0480349	+	0.998846i	$0.484704\pi$
$90$	0	0
$91$	−21.3506	−2.23816
$92$	0	0
$93$	−0.163612	−0.0169658
$94$	0	0
$95$	−3.04458	−0.312368
$96$	0	0
$97$	0.281398	0.0285716	0.0142858	−	0.999898i	$-0.495453\pi$
$97$	0.281398	0.0285716	0.0142858	+	0.999898i	$0.495453\pi$
$98$	0	0
$99$	−9.70146	−0.975033
$100$	0	0
$101$	18.4146	1.83232	0.916161	−	0.400810i	$-0.131271\pi$
$101$	18.4146	1.83232	0.916161	+	0.400810i	$0.131271\pi$
$102$	0	0
$103$	10.3375	1.01858	0.509291	−	0.860595i	$-0.329908\pi$
$103$	10.3375	1.01858	0.509291	+	0.860595i	$0.329908\pi$
$104$	0	0
$105$	0.548683	0.0535459
$106$	0	0
$107$	7.19350	0.695422	0.347711	−	0.937602i	$-0.386959\pi$
$107$	7.19350	0.695422	0.347711	+	0.937602i	$0.386959\pi$
$108$	0	0
$109$	8.15842	0.781435	0.390717	−	0.920511i	$-0.372227\pi$
$109$	8.15842	0.781435	0.390717	+	0.920511i	$0.372227\pi$
$110$	0	0
$111$	−3.56819	−0.338677
$112$	0	0
$113$	−3.72132	−0.350073	−0.175036	−	0.984562i	$-0.556004\pi$
$113$	−3.72132	−0.350073	−0.175036	+	0.984562i	$0.556004\pi$
$114$	0	0
$115$	−0.223956	−0.0208840
$116$	0	0
$117$	−19.8608	−1.83614
$118$	0	0
$119$	8.49939	0.779138
$120$	0	0
$121$	0.331129	0.0301026
$122$	0	0
$123$	1.74945	0.157743
$124$	0	0
$125$	5.01930	0.448940
$126$	0	0
$127$	−4.16513	−0.369596	−0.184798	−	0.982777i	$-0.559163\pi$
$127$	−4.16513	−0.369596	−0.184798	+	0.982777i	$0.559163\pi$
$128$	0	0
$129$	−1.02110	−0.0899030
$130$	0	0
$131$	−8.44647	−0.737972	−0.368986	−	0.929435i	$-0.620295\pi$
$131$	−8.44647	−0.737972	−0.368986	+	0.929435i	$0.620295\pi$
$132$	0	0
$133$	−18.2934	−1.58624
$134$	0	0
$135$	1.04168	0.0896538
$136$	0	0
$137$	8.24979	0.704827	0.352413	−	0.935844i	$-0.385361\pi$
$137$	8.24979	0.704827	0.352413	+	0.935844i	$0.385361\pi$
$138$	0	0
$139$	19.5669	1.65964	0.829822	−	0.558029i	$-0.188442\pi$
$139$	19.5669	1.65964	0.829822	+	0.558029i	$0.188442\pi$
$140$	0	0
$141$	1.73238	0.145892
$142$	0	0
$143$	23.1971	1.93984
$144$	0	0
$145$	−0.368789	−0.0306263
$146$	0	0
$147$	0.892633	0.0736231
$148$	0	0
$149$	5.72499	0.469009	0.234505	−	0.972115i	$-0.424653\pi$
$149$	5.72499	0.469009	0.234505	+	0.972115i	$0.424653\pi$
$150$	0	0
$151$	11.2188	0.912977	0.456488	−	0.889729i	$-0.349107\pi$
$151$	11.2188	0.912977	0.456488	+	0.889729i	$0.349107\pi$
$152$	0	0
$153$	7.90632	0.639188
$154$	0	0
$155$	0.245641	0.0197304
$156$	0	0
$157$	11.0698	0.883467	0.441733	−	0.897146i	$-0.354364\pi$
$157$	11.0698	0.883467	0.441733	+	0.897146i	$0.354364\pi$
$158$	0	0
$159$	3.07160	0.243594
$160$	0	0
$161$	−1.34564	−0.106051
$162$	0	0
$163$	−2.14185	−0.167763	−0.0838813	−	0.996476i	$-0.526732\pi$
$163$	−2.14185	−0.167763	−0.0838813	+	0.996476i	$0.526732\pi$
$164$	0	0
$165$	−0.596134	−0.0464090
$166$	0	0
$167$	−13.1466	−1.01731	−0.508657	−	0.860970i	$-0.669858\pi$
$167$	−13.1466	−1.01731	−0.508657	+	0.860970i	$0.669858\pi$
$168$	0	0
$169$	34.4891	2.65301
$170$	0	0
$171$	−17.0170	−1.30132
$172$	0	0
$173$	−5.61280	−0.426733	−0.213367	−	0.976972i	$-0.568443\pi$
$173$	−5.61280	−0.426733	−0.213367	+	0.976972i	$0.568443\pi$
$174$	0	0
$175$	14.6674	1.10875
$176$	0	0
$177$	−3.33744	−0.250857
$178$	0	0
$179$	4.12561	0.308363	0.154181	−	0.988043i	$-0.450726\pi$
$179$	4.12561	0.308363	0.154181	+	0.988043i	$0.450726\pi$
$180$	0	0
$181$	12.6748	0.942111	0.471056	−	0.882104i	$-0.343873\pi$
$181$	12.6748	0.942111	0.471056	+	0.882104i	$0.343873\pi$
$182$	0	0
$183$	−3.34635	−0.247369
$184$	0	0
$185$	5.35714	0.393865
$186$	0	0
$187$	−9.23444	−0.675289
$188$	0	0
$189$	6.25897	0.455273
$190$	0	0
$191$	14.2870	1.03377	0.516885	−	0.856055i	$-0.327091\pi$
$191$	14.2870	1.03377	0.516885	+	0.856055i	$0.327091\pi$
$192$	0	0
$193$	26.1431	1.88182	0.940911	−	0.338654i	$-0.109972\pi$
$193$	26.1431	1.88182	0.940911	+	0.338654i	$0.109972\pi$
$194$	0	0
$195$	−1.22041	−0.0873951
$196$	0	0
$197$	13.8668	0.987966	0.493983	−	0.869472i	$-0.335541\pi$
$197$	13.8668	0.987966	0.493983	+	0.869472i	$0.335541\pi$
$198$	0	0
$199$	24.3548	1.72647	0.863234	−	0.504804i	$-0.168435\pi$
$199$	24.3548	1.72647	0.863234	+	0.504804i	$0.168435\pi$
$200$	0	0
$201$	2.52539	0.178128
$202$	0	0
$203$	−2.21588	−0.155524
$204$	0	0
$205$	−2.62656	−0.183447
$206$	0	0
$207$	−1.25175	−0.0870023
$208$	0	0
$209$	19.8755	1.37482
$210$	0	0
$211$	−8.23438	−0.566878	−0.283439	−	0.958990i	$-0.591475\pi$
$211$	−8.23438	−0.566878	−0.283439	+	0.958990i	$0.591475\pi$
$212$	0	0
$213$	−0.481841	−0.0330152
$214$	0	0
$215$	1.53304	0.104553
$216$	0	0
$217$	1.47594	0.100193
$218$	0	0
$219$	5.76980	0.389887
$220$	0	0
$221$	−18.9048	−1.27167
$222$	0	0
$223$	−10.4987	−0.703044	−0.351522	−	0.936180i	$-0.614336\pi$
$223$	−10.4987	−0.703044	−0.351522	+	0.936180i	$0.614336\pi$
$224$	0	0
$225$	13.6439	0.909595
$226$	0	0
$227$	4.26952	0.283378	0.141689	−	0.989911i	$-0.454747\pi$
$227$	4.26952	0.283378	0.141689	+	0.989911i	$0.454747\pi$
$228$	0	0
$229$	−14.6811	−0.970156	−0.485078	−	0.874471i	$-0.661209\pi$
$229$	−14.6811	−0.970156	−0.485078	+	0.874471i	$0.661209\pi$
$230$	0	0
$231$	−3.58188	−0.235671
$232$	0	0
$233$	25.7022	1.68380	0.841902	−	0.539630i	$-0.181436\pi$
$233$	25.7022	1.68380	0.841902	+	0.539630i	$0.181436\pi$
$234$	0	0
$235$	−2.60092	−0.169665
$236$	0	0
$237$	−1.89991	−0.123413
$238$	0	0
$239$	−1.27590	−0.0825314	−0.0412657	−	0.999148i	$-0.513139\pi$
$239$	−1.27590	−0.0825314	−0.0412657	+	0.999148i	$0.513139\pi$
$240$	0	0
$241$	7.09166	0.456814	0.228407	−	0.973566i	$-0.426648\pi$
$241$	7.09166	0.456814	0.228407	+	0.973566i	$0.426648\pi$
$242$	0	0
$243$	8.79173	0.563990
$244$	0	0
$245$	−1.34017	−0.0856200
$246$	0	0
$247$	40.6892	2.58899
$248$	0	0
$249$	−4.45975	−0.282625
$250$	0	0
$251$	−0.986908	−0.0622931	−0.0311465	−	0.999515i	$-0.509916\pi$
$251$	−0.986908	−0.0622931	−0.0311465	+	0.999515i	$0.509916\pi$
$252$	0	0
$253$	1.46202	0.0919161
$254$	0	0
$255$	0.485827	0.0304237
$256$	0	0
$257$	−16.2595	−1.01424	−0.507120	−	0.861875i	$-0.669290\pi$
$257$	−16.2595	−1.01424	−0.507120	+	0.861875i	$0.669290\pi$
$258$	0	0
$259$	32.1885	2.00009
$260$	0	0
$261$	−2.06126	−0.127589
$262$	0	0
$263$	26.6080	1.64072	0.820360	−	0.571848i	$-0.193773\pi$
$263$	26.6080	1.64072	0.820360	+	0.571848i	$0.193773\pi$
$264$	0	0
$265$	−4.61158	−0.283287
$266$	0	0
$267$	0.311274	0.0190496
$268$	0	0
$269$	−7.21445	−0.439873	−0.219936	−	0.975514i	$-0.570585\pi$
$269$	−7.21445	−0.439873	−0.219936	+	0.975514i	$0.570585\pi$
$270$	0	0
$271$	−8.37271	−0.508606	−0.254303	−	0.967125i	$-0.581846\pi$
$271$	−8.37271	−0.508606	−0.254303	+	0.967125i	$0.581846\pi$
$272$	0	0
$273$	−7.33284	−0.443803
$274$	0	0
$275$	−15.9359	−0.960968
$276$	0	0
$277$	−2.26449	−0.136060	−0.0680301	−	0.997683i	$-0.521671\pi$
$277$	−2.26449	−0.136060	−0.0680301	+	0.997683i	$0.521671\pi$
$278$	0	0
$279$	1.37295	0.0821965
$280$	0	0
$281$	9.68208	0.577584	0.288792	−	0.957392i	$-0.406746\pi$
$281$	9.68208	0.577584	0.288792	+	0.957392i	$0.406746\pi$
$282$	0	0
$283$	−3.63538	−0.216101	−0.108051	−	0.994145i	$-0.534461\pi$
$283$	−3.63538	−0.216101	−0.108051	+	0.994145i	$0.534461\pi$
$284$	0	0
$285$	−1.04566	−0.0619394
$286$	0	0
$287$	−15.7817	−0.931566
$288$	0	0
$289$	−9.47428	−0.557311
$290$	0	0
$291$	0.0966455	0.00566546
$292$	0	0
$293$	26.6553	1.55722	0.778610	−	0.627508i	$-0.215925\pi$
$293$	26.6553	1.55722	0.778610	+	0.627508i	$0.215925\pi$
$294$	0	0
$295$	5.01070	0.291734
$296$	0	0
$297$	−6.80027	−0.394591
$298$	0	0
$299$	2.99304	0.173092
$300$	0	0
$301$	9.21132	0.530931
$302$	0	0
$303$	6.32446	0.363331
$304$	0	0
$305$	5.02408	0.287678
$306$	0	0
$307$	13.2265	0.754878	0.377439	−	0.926034i	$-0.376805\pi$
$307$	13.2265	0.754878	0.377439	+	0.926034i	$0.376805\pi$
$308$	0	0
$309$	3.55038	0.201974
$310$	0	0
$311$	−4.48783	−0.254481	−0.127241	−	0.991872i	$-0.540612\pi$
$311$	−4.48783	−0.254481	−0.127241	+	0.991872i	$0.540612\pi$
$312$	0	0
$313$	1.55093	0.0876638	0.0438319	−	0.999039i	$-0.486043\pi$
$313$	1.55093	0.0876638	0.0438319	+	0.999039i	$0.486043\pi$
$314$	0	0
$315$	−4.60427	−0.259421
$316$	0	0
$317$	25.0826	1.40878	0.704388	−	0.709815i	$-0.251221\pi$
$317$	25.0826	1.40878	0.704388	+	0.709815i	$0.251221\pi$
$318$	0	0
$319$	2.40751	0.134795
$320$	0	0
$321$	2.47059	0.137895
$322$	0	0
$323$	−16.1978	−0.901269
$324$	0	0
$325$	−32.6239	−1.80965
$326$	0	0
$327$	2.80199	0.154951
$328$	0	0
$329$	−15.6277	−0.861582
$330$	0	0
$331$	−9.79211	−0.538223	−0.269112	−	0.963109i	$-0.586730\pi$
$331$	−9.79211	−0.538223	−0.269112	+	0.963109i	$0.586730\pi$
$332$	0	0
$333$	29.9424	1.64083
$334$	0	0
$335$	−3.79153	−0.207153
$336$	0	0
$337$	−22.7208	−1.23768	−0.618842	−	0.785516i	$-0.712398\pi$
$337$	−22.7208	−1.23768	−0.618842	+	0.785516i	$0.712398\pi$
$338$	0	0
$339$	−1.27808	−0.0694159
$340$	0	0
$341$	−1.60358	−0.0868389
$342$	0	0
$343$	13.6352	0.736232
$344$	0	0
$345$	−0.0769171	−0.00414108
$346$	0	0
$347$	−14.8362	−0.796447	−0.398223	−	0.917288i	$-0.630373\pi$
$347$	−14.8362	−0.796447	−0.398223	+	0.917288i	$0.630373\pi$
$348$	0	0
$349$	3.21654	0.172177	0.0860887	−	0.996287i	$-0.472563\pi$
$349$	3.21654	0.172177	0.0860887	+	0.996287i	$0.472563\pi$
$350$	0	0
$351$	−13.9215	−0.743075
$352$	0	0
$353$	−21.8583	−1.16340	−0.581699	−	0.813404i	$-0.697612\pi$
$353$	−21.8583	−1.16340	−0.581699	+	0.813404i	$0.697612\pi$
$354$	0	0
$355$	0.723418	0.0383951
$356$	0	0
$357$	2.91910	0.154495
$358$	0	0
$359$	−17.1130	−0.903191	−0.451595	−	0.892223i	$-0.649145\pi$
$359$	−17.1130	−0.903191	−0.451595	+	0.892223i	$0.649145\pi$
$360$	0	0
$361$	15.8629	0.834889
$362$	0	0
$363$	0.113726	0.00596905
$364$	0	0
$365$	−8.66256	−0.453419
$366$	0	0
$367$	−27.0160	−1.41023	−0.705113	−	0.709095i	$-0.749104\pi$
$367$	−27.0160	−1.41023	−0.705113	+	0.709095i	$0.749104\pi$
$368$	0	0
$369$	−14.6805	−0.764237
$370$	0	0
$371$	−27.7088	−1.43857
$372$	0	0
$373$	26.3165	1.36262	0.681309	−	0.731996i	$-0.261411\pi$
$373$	26.3165	1.36262	0.681309	+	0.731996i	$0.261411\pi$
$374$	0	0
$375$	1.72387	0.0890202
$376$	0	0
$377$	4.92866	0.253839
$378$	0	0
$379$	7.25669	0.372751	0.186376	−	0.982479i	$-0.440326\pi$
$379$	7.25669	0.372751	0.186376	+	0.982479i	$0.440326\pi$
$380$	0	0
$381$	−1.43051	−0.0732871
$382$	0	0
$383$	−6.90410	−0.352783	−0.176392	−	0.984320i	$-0.556442\pi$
$383$	−6.90410	−0.352783	−0.176392	+	0.984320i	$0.556442\pi$
$384$	0	0
$385$	5.37770	0.274073
$386$	0	0
$387$	8.56857	0.435565
$388$	0	0
$389$	−3.64038	−0.184575	−0.0922873	−	0.995732i	$-0.529418\pi$
$389$	−3.64038	−0.184575	−0.0922873	+	0.995732i	$0.529418\pi$
$390$	0	0
$391$	−1.19149	−0.0602561
$392$	0	0
$393$	−2.90092	−0.146332
$394$	0	0
$395$	2.85246	0.143523
$396$	0	0
$397$	12.7633	0.640571	0.320286	−	0.947321i	$-0.396221\pi$
$397$	12.7633	0.640571	0.320286	+	0.947321i	$0.396221\pi$
$398$	0	0
$399$	−6.28285	−0.314536
$400$	0	0
$401$	7.23113	0.361106	0.180553	−	0.983565i	$-0.442211\pi$
$401$	7.23113	0.361106	0.180553	+	0.983565i	$0.442211\pi$
$402$	0	0
$403$	−3.28286	−0.163531
$404$	0	0
$405$	−4.10052	−0.203757
$406$	0	0
$407$	−34.9722	−1.73351
$408$	0	0
$409$	−13.5958	−0.672267	−0.336134	−	0.941814i	$-0.609119\pi$
$409$	−13.5958	−0.672267	−0.336134	+	0.941814i	$0.609119\pi$
$410$	0	0
$411$	2.83337	0.139760
$412$	0	0
$413$	30.1069	1.48146
$414$	0	0
$415$	6.69570	0.328679
$416$	0	0
$417$	6.72021	0.329090
$418$	0	0
$419$	−21.9840	−1.07399	−0.536995	−	0.843586i	$-0.680441\pi$
$419$	−21.9840	−1.07399	−0.536995	+	0.843586i	$0.680441\pi$
$420$	0	0
$421$	0.710762	0.0346404	0.0173202	−	0.999850i	$-0.494487\pi$
$421$	0.710762	0.0346404	0.0173202	+	0.999850i	$0.494487\pi$
$422$	0	0
$423$	−14.5372	−0.706824
$424$	0	0
$425$	12.9871	0.629968
$426$	0	0
$427$	30.1873	1.46087
$428$	0	0
$429$	7.96700	0.384650
$430$	0	0
$431$	24.9021	1.19949	0.599746	−	0.800190i	$-0.295268\pi$
$431$	24.9021	1.19949	0.599746	+	0.800190i	$0.295268\pi$
$432$	0	0
$433$	33.0635	1.58893	0.794465	−	0.607309i	$-0.207751\pi$
$433$	33.0635	1.58893	0.794465	+	0.607309i	$0.207751\pi$
$434$	0	0
$435$	−0.126660	−0.00607288
$436$	0	0
$437$	2.56447	0.122675
$438$	0	0
$439$	36.2075	1.72809	0.864044	−	0.503416i	$-0.167924\pi$
$439$	36.2075	1.72809	0.864044	+	0.503416i	$0.167924\pi$
$440$	0	0
$441$	−7.49053	−0.356692
$442$	0	0
$443$	−8.82326	−0.419206	−0.209603	−	0.977787i	$-0.567217\pi$
$443$	−8.82326	−0.419206	−0.209603	+	0.977787i	$0.567217\pi$
$444$	0	0
$445$	−0.467335	−0.0221538
$446$	0	0
$447$	1.96624	0.0929997
$448$	0	0
$449$	−5.88516	−0.277738	−0.138869	−	0.990311i	$-0.544347\pi$
$449$	−5.88516	−0.277738	−0.138869	+	0.990311i	$0.544347\pi$
$450$	0	0
$451$	17.1466	0.807400
$452$	0	0
$453$	3.85309	0.181034
$454$	0	0
$455$	11.0092	0.516121
$456$	0	0
$457$	8.79112	0.411231	0.205616	−	0.978633i	$-0.434080\pi$
$457$	8.79112	0.411231	0.205616	+	0.978633i	$0.434080\pi$
$458$	0	0
$459$	5.54196	0.258676
$460$	0	0
$461$	−10.3747	−0.483197	−0.241599	−	0.970376i	$-0.577672\pi$
$461$	−10.3747	−0.483197	−0.241599	+	0.970376i	$0.577672\pi$
$462$	0	0
$463$	−41.0615	−1.90829	−0.954146	−	0.299342i	$-0.903233\pi$
$463$	−41.0615	−1.90829	−0.954146	+	0.299342i	$0.903233\pi$
$464$	0	0
$465$	0.0843650	0.00391233
$466$	0	0
$467$	−32.4988	−1.50387	−0.751933	−	0.659239i	$-0.770879\pi$
$467$	−32.4988	−1.50387	−0.751933	+	0.659239i	$0.770879\pi$
$468$	0	0
$469$	−22.7815	−1.05195
$470$	0	0
$471$	3.80190	0.175182
$472$	0	0
$473$	−10.0079	−0.460165
$474$	0	0
$475$	−27.9525	−1.28255
$476$	0	0
$477$	−25.7753	−1.18017
$478$	0	0
$479$	−29.6681	−1.35557	−0.677784	−	0.735261i	$-0.737060\pi$
$479$	−29.6681	−1.35557	−0.677784	+	0.735261i	$0.737060\pi$
$480$	0	0
$481$	−71.5951	−3.26446
$482$	0	0
$483$	−0.462158	−0.0210289
$484$	0	0
$485$	−0.145100	−0.00658864
$486$	0	0
$487$	−1.70979	−0.0774781	−0.0387390	−	0.999249i	$-0.512334\pi$
$487$	−1.70979	−0.0774781	−0.0387390	+	0.999249i	$0.512334\pi$
$488$	0	0
$489$	−0.735614	−0.0332656
$490$	0	0
$491$	−43.4862	−1.96250	−0.981252	−	0.192732i	$-0.938265\pi$
$491$	−43.4862	−1.96250	−0.981252	+	0.192732i	$0.938265\pi$
$492$	0	0
$493$	−1.96203	−0.0883654
$494$	0	0
$495$	5.00246	0.224844
$496$	0	0
$497$	4.34667	0.194975
$498$	0	0
$499$	−2.03095	−0.0909178	−0.0454589	−	0.998966i	$-0.514475\pi$
$499$	−2.03095	−0.0909178	−0.0454589	+	0.998966i	$0.514475\pi$
$500$	0	0
$501$	−4.51517	−0.201723
$502$	0	0
$503$	39.7730	1.77339	0.886696	−	0.462353i	$-0.152995\pi$
$503$	39.7730	1.77339	0.886696	+	0.462353i	$0.152995\pi$
$504$	0	0
$505$	−9.49531	−0.422536
$506$	0	0
$507$	11.8452	0.526065
$508$	0	0
$509$	−24.3853	−1.08086	−0.540430	−	0.841389i	$-0.681738\pi$
$509$	−24.3853	−1.08086	−0.540430	+	0.841389i	$0.681738\pi$
$510$	0	0
$511$	−52.0491	−2.30252
$512$	0	0
$513$	−11.9281	−0.526638
$514$	0	0
$515$	−5.33041	−0.234886
$516$	0	0
$517$	16.9792	0.746745
$518$	0	0
$519$	−1.92771	−0.0846169
$520$	0	0
$521$	−5.08675	−0.222854	−0.111427	−	0.993773i	$-0.535542\pi$
$521$	−5.08675	−0.222854	−0.111427	+	0.993773i	$0.535542\pi$
$522$	0	0
$523$	20.8229	0.910522	0.455261	−	0.890358i	$-0.349546\pi$
$523$	20.8229	0.910522	0.455261	+	0.890358i	$0.349546\pi$
$524$	0	0
$525$	5.03748	0.219854
$526$	0	0
$527$	1.30686	0.0569277
$528$	0	0
$529$	−22.8114	−0.991798
$530$	0	0
$531$	28.0061	1.21536
$532$	0	0
$533$	35.1025	1.52046
$534$	0	0
$535$	−3.70926	−0.160365
$536$	0	0
$537$	1.41693	0.0611452
$538$	0	0
$539$	8.74880	0.376838
$540$	0	0
$541$	16.6226	0.714659	0.357330	−	0.933978i	$-0.383687\pi$
$541$	16.6226	0.714659	0.357330	+	0.933978i	$0.383687\pi$
$542$	0	0
$543$	4.35314	0.186811
$544$	0	0
$545$	−4.20681	−0.180200
$546$	0	0
$547$	16.4550	0.703564	0.351782	−	0.936082i	$-0.385576\pi$
$547$	16.4550	0.703564	0.351782	+	0.936082i	$0.385576\pi$
$548$	0	0
$549$	28.0809	1.19846
$550$	0	0
$551$	4.22292	0.179903
$552$	0	0
$553$	17.1390	0.728826
$554$	0	0
$555$	1.83990	0.0780993
$556$	0	0
$557$	14.8187	0.627889	0.313944	−	0.949441i	$-0.398349\pi$
$557$	14.8187	0.627889	0.313944	+	0.949441i	$0.398349\pi$
$558$	0	0
$559$	−20.4883	−0.866561
$560$	0	0
$561$	−3.17155	−0.133903
$562$	0	0
$563$	−16.0405	−0.676026	−0.338013	−	0.941141i	$-0.609755\pi$
$563$	−16.0405	−0.676026	−0.338013	+	0.941141i	$0.609755\pi$
$564$	0	0
$565$	1.91886	0.0807272
$566$	0	0
$567$	−24.6381	−1.03470
$568$	0	0
$569$	−0.969004	−0.0406228	−0.0203114	−	0.999794i	$-0.506466\pi$
$569$	−0.969004	−0.0406228	−0.0203114	+	0.999794i	$0.506466\pi$
$570$	0	0
$571$	−22.6264	−0.946887	−0.473444	−	0.880824i	$-0.656989\pi$
$571$	−22.6264	−0.946887	−0.473444	+	0.880824i	$0.656989\pi$
$572$	0	0
$573$	4.90684	0.204986
$574$	0	0
$575$	−2.05615	−0.0857473
$576$	0	0
$577$	−3.91281	−0.162893	−0.0814463	−	0.996678i	$-0.525954\pi$
$577$	−3.91281	−0.162893	−0.0814463	+	0.996678i	$0.525954\pi$
$578$	0	0
$579$	8.97880	0.373146
$580$	0	0
$581$	40.2313	1.66907
$582$	0	0
$583$	30.1051	1.24683
$584$	0	0
$585$	10.2410	0.423415
$586$	0	0
$587$	−20.2990	−0.837829	−0.418915	−	0.908026i	$-0.637589\pi$
$587$	−20.2990	−0.837829	−0.418915	+	0.908026i	$0.637589\pi$
$588$	0	0
$589$	−2.81278	−0.115899
$590$	0	0
$591$	4.76251	0.195904
$592$	0	0
$593$	33.9897	1.39579	0.697894	−	0.716201i	$-0.254121\pi$
$593$	33.9897	1.39579	0.697894	+	0.716201i	$0.254121\pi$
$594$	0	0
$595$	−4.38262	−0.179670
$596$	0	0
$597$	8.36462	0.342341
$598$	0	0
$599$	18.6637	0.762579	0.381290	−	0.924456i	$-0.375480\pi$
$599$	18.6637	0.762579	0.381290	+	0.924456i	$0.375480\pi$
$600$	0	0
$601$	−27.5926	−1.12552	−0.562762	−	0.826619i	$-0.690261\pi$
$601$	−27.5926	−1.12552	−0.562762	+	0.826619i	$0.690261\pi$
$602$	0	0
$603$	−21.1918	−0.862998
$604$	0	0
$605$	−0.170743	−0.00694170
$606$	0	0
$607$	36.2459	1.47117	0.735587	−	0.677430i	$-0.236906\pi$
$607$	36.2459	1.47117	0.735587	+	0.677430i	$0.236906\pi$
$608$	0	0
$609$	−0.761038	−0.0308388
$610$	0	0
$611$	34.7599	1.40623
$612$	0	0
$613$	−11.4554	−0.462679	−0.231339	−	0.972873i	$-0.574311\pi$
$613$	−11.4554	−0.462679	−0.231339	+	0.972873i	$0.574311\pi$
$614$	0	0
$615$	−0.902086	−0.0363756
$616$	0	0
$617$	14.9117	0.600324	0.300162	−	0.953888i	$-0.402959\pi$
$617$	14.9117	0.600324	0.300162	+	0.953888i	$0.402959\pi$
$618$	0	0
$619$	7.45267	0.299548	0.149774	−	0.988720i	$-0.452145\pi$
$619$	7.45267	0.299548	0.149774	+	0.988720i	$0.452145\pi$
$620$	0	0
$621$	−0.877414	−0.0352094
$622$	0	0
$623$	−2.80799	−0.112500
$624$	0	0
$625$	21.0824	0.843297
$626$	0	0
$627$	6.82620	0.272612
$628$	0	0
$629$	28.5010	1.13641
$630$	0	0
$631$	−20.7325	−0.825348	−0.412674	−	0.910879i	$-0.635405\pi$
$631$	−20.7325	−0.825348	−0.412674	+	0.910879i	$0.635405\pi$
$632$	0	0
$633$	−2.82808	−0.112406
$634$	0	0
$635$	2.14771	0.0852292
$636$	0	0
$637$	17.9106	0.709642
$638$	0	0
$639$	4.04337	0.159953
$640$	0	0
$641$	−18.3496	−0.724766	−0.362383	−	0.932029i	$-0.618037\pi$
$641$	−18.3496	−0.724766	−0.362383	+	0.932029i	$0.618037\pi$
$642$	0	0
$643$	−23.8401	−0.940160	−0.470080	−	0.882624i	$-0.655775\pi$
$643$	−23.8401	−0.940160	−0.470080	+	0.882624i	$0.655775\pi$
$644$	0	0
$645$	0.526521	0.0207317
$646$	0	0
$647$	15.2467	0.599411	0.299705	−	0.954032i	$-0.403112\pi$
$647$	15.2467	0.599411	0.299705	+	0.954032i	$0.403112\pi$
$648$	0	0
$649$	−32.7106	−1.28400
$650$	0	0
$651$	0.506909	0.0198673
$652$	0	0
$653$	39.3804	1.54107	0.770536	−	0.637396i	$-0.219989\pi$
$653$	39.3804	1.54107	0.770536	+	0.637396i	$0.219989\pi$
$654$	0	0
$655$	4.35534	0.170177
$656$	0	0
$657$	−48.4173	−1.88894
$658$	0	0
$659$	34.3768	1.33913	0.669566	−	0.742753i	$-0.266480\pi$
$659$	34.3768	1.33913	0.669566	+	0.742753i	$0.266480\pi$
$660$	0	0
$661$	−35.8195	−1.39322	−0.696608	−	0.717452i	$-0.745308\pi$
$661$	−35.8195	−1.39322	−0.696608	+	0.717452i	$0.745308\pi$
$662$	0	0
$663$	−6.49280	−0.252160
$664$	0	0
$665$	9.43283	0.365789
$666$	0	0
$667$	0.310633	0.0120277
$668$	0	0
$669$	−3.60575	−0.139406
$670$	0	0
$671$	−32.7980	−1.26615
$672$	0	0
$673$	38.7385	1.49326	0.746629	−	0.665240i	$-0.231671\pi$
$673$	38.7385	1.49326	0.746629	+	0.665240i	$0.231671\pi$
$674$	0	0
$675$	9.56375	0.368109
$676$	0	0
$677$	48.7473	1.87351	0.936756	−	0.349982i	$-0.113812\pi$
$677$	48.7473	1.87351	0.936756	+	0.349982i	$0.113812\pi$
$678$	0	0
$679$	−0.871835	−0.0334580
$680$	0	0
$681$	1.46636	0.0561910
$682$	0	0
$683$	−17.7784	−0.680273	−0.340136	−	0.940376i	$-0.610473\pi$
$683$	−17.7784	−0.680273	−0.340136	+	0.940376i	$0.610473\pi$
$684$	0	0
$685$	−4.25392	−0.162534
$686$	0	0
$687$	−5.04220	−0.192372
$688$	0	0
$689$	61.6312	2.34796
$690$	0	0
$691$	4.33759	0.165010	0.0825048	−	0.996591i	$-0.473708\pi$
$691$	4.33759	0.165010	0.0825048	+	0.996591i	$0.473708\pi$
$692$	0	0
$693$	30.0574	1.14178
$694$	0	0
$695$	−10.0895	−0.382716
$696$	0	0
$697$	−13.9738	−0.529295
$698$	0	0
$699$	8.82736	0.333881
$700$	0	0
$701$	−4.91232	−0.185536	−0.0927679	−	0.995688i	$-0.529571\pi$
$701$	−4.91232	−0.185536	−0.0927679	+	0.995688i	$0.529571\pi$
$702$	0	0
$703$	−61.3434	−2.31361
$704$	0	0
$705$	−0.893282	−0.0336429
$706$	0	0
$707$	−57.0527	−2.14569
$708$	0	0
$709$	13.2486	0.497560	0.248780	−	0.968560i	$-0.419970\pi$
$709$	13.2486	0.497560	0.248780	+	0.968560i	$0.419970\pi$
$710$	0	0
$711$	15.9431	0.597913
$712$	0	0
$713$	−0.206905	−0.00774864
$714$	0	0
$715$	−11.9613	−0.447329
$716$	0	0
$717$	−0.438207	−0.0163651
$718$	0	0
$719$	4.93680	0.184112	0.0920558	−	0.995754i	$-0.470656\pi$
$719$	4.93680	0.184112	0.0920558	+	0.995754i	$0.470656\pi$
$720$	0	0
$721$	−32.0279	−1.19278
$722$	0	0
$723$	2.43562	0.0905816
$724$	0	0
$725$	−3.38587	−0.125748
$726$	0	0
$727$	6.18466	0.229377	0.114688	−	0.993402i	$-0.463413\pi$
$727$	6.18466	0.229377	0.114688	+	0.993402i	$0.463413\pi$
$728$	0	0
$729$	−20.8374	−0.771756
$730$	0	0
$731$	8.15609	0.301664
$732$	0	0
$733$	−6.66526	−0.246187	−0.123093	−	0.992395i	$-0.539282\pi$
$733$	−6.66526	−0.246187	−0.123093	+	0.992395i	$0.539282\pi$
$734$	0	0
$735$	−0.460277	−0.0169776
$736$	0	0
$737$	24.7517	0.911740
$738$	0	0
$739$	19.2596	0.708476	0.354238	−	0.935155i	$-0.384740\pi$
$739$	19.2596	0.708476	0.354238	+	0.935155i	$0.384740\pi$
$740$	0	0
$741$	13.9746	0.513370
$742$	0	0
$743$	−15.7832	−0.579031	−0.289515	−	0.957173i	$-0.593494\pi$
$743$	−15.7832	−0.579031	−0.289515	+	0.957173i	$0.593494\pi$
$744$	0	0
$745$	−2.95203	−0.108154
$746$	0	0
$747$	37.4240	1.36927
$748$	0	0
$749$	−22.2871	−0.814354
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−0.338952	−0.0123521
$754$	0	0
$755$	−5.78489	−0.210534
$756$	0	0
$757$	24.7476	0.899466	0.449733	−	0.893163i	$-0.351519\pi$
$757$	24.7476	0.899466	0.449733	+	0.893163i	$0.351519\pi$
$758$	0	0
$759$	0.502126	0.0182260
$760$	0	0
$761$	21.3236	0.772979	0.386490	−	0.922294i	$-0.373688\pi$
$761$	21.3236	0.772979	0.386490	+	0.922294i	$0.373688\pi$
$762$	0	0
$763$	−25.2767	−0.915077
$764$	0	0
$765$	−4.07681	−0.147398
$766$	0	0
$767$	−66.9652	−2.41797
$768$	0	0
$769$	0.0881767	0.00317973	0.00158987	−	0.999999i	$-0.499494\pi$
$769$	0.0881767	0.00317973	0.00158987	+	0.999999i	$0.499494\pi$
$770$	0	0
$771$	−5.58430	−0.201114
$772$	0	0
$773$	45.5142	1.63703	0.818517	−	0.574483i	$-0.194797\pi$
$773$	45.5142	1.63703	0.818517	+	0.574483i	$0.194797\pi$
$774$	0	0
$775$	2.25525	0.0810108
$776$	0	0
$777$	11.0551	0.396598
$778$	0	0
$779$	30.0761	1.07759
$780$	0	0
$781$	−4.72258	−0.168987
$782$	0	0
$783$	−1.44484	−0.0516345
$784$	0	0
$785$	−5.70803	−0.203728
$786$	0	0
$787$	8.08193	0.288090	0.144045	−	0.989571i	$-0.453989\pi$
$787$	8.08193	0.288090	0.144045	+	0.989571i	$0.453989\pi$
$788$	0	0
$789$	9.13846	0.325338
$790$	0	0
$791$	11.5295	0.409943
$792$	0	0
$793$	−67.1441	−2.38436
$794$	0	0
$795$	−1.58384	−0.0561730
$796$	0	0
$797$	−7.26236	−0.257246	−0.128623	−	0.991694i	$-0.541056\pi$
$797$	−7.26236	−0.257246	−0.128623	+	0.991694i	$0.541056\pi$
$798$	0	0
$799$	−13.8374	−0.489533
$800$	0	0
$801$	−2.61205	−0.0922924
$802$	0	0
$803$	56.5505	1.99562
$804$	0	0
$805$	0.693866	0.0244556
$806$	0	0
$807$	−2.47779	−0.0872223
$808$	0	0
$809$	25.6438	0.901588	0.450794	−	0.892628i	$-0.351141\pi$
$809$	25.6438	0.901588	0.450794	+	0.892628i	$0.351141\pi$
$810$	0	0
$811$	−8.64096	−0.303425	−0.151712	−	0.988425i	$-0.548479\pi$
$811$	−8.64096	−0.303425	−0.151712	+	0.988425i	$0.548479\pi$
$812$	0	0
$813$	−2.87559	−0.100851
$814$	0	0
$815$	1.10442	0.0386862
$816$	0	0
$817$	−17.5545	−0.614156
$818$	0	0
$819$	61.5335	2.15015
$820$	0	0
$821$	18.9555	0.661552	0.330776	−	0.943709i	$-0.392690\pi$
$821$	18.9555	0.661552	0.330776	+	0.943709i	$0.392690\pi$
$822$	0	0
$823$	25.7522	0.897666	0.448833	−	0.893616i	$-0.351840\pi$
$823$	25.7522	0.897666	0.448833	+	0.893616i	$0.351840\pi$
$824$	0	0
$825$	−5.47314	−0.190550
$826$	0	0
$827$	1.61480	0.0561522	0.0280761	−	0.999606i	$-0.491062\pi$
$827$	1.61480	0.0561522	0.0280761	+	0.999606i	$0.491062\pi$
$828$	0	0
$829$	14.0914	0.489413	0.244707	−	0.969597i	$-0.421308\pi$
$829$	14.0914	0.489413	0.244707	+	0.969597i	$0.421308\pi$
$830$	0	0
$831$	−0.777736	−0.0269793
$832$	0	0
$833$	−7.12994	−0.247038
$834$	0	0
$835$	6.77890	0.234594
$836$	0	0
$837$	0.962375	0.0332645
$838$	0	0
$839$	33.3762	1.15228	0.576138	−	0.817353i	$-0.304559\pi$
$839$	33.3762	1.15228	0.576138	+	0.817353i	$0.304559\pi$
$840$	0	0
$841$	−28.4885	−0.982361
$842$	0	0
$843$	3.32529	0.114529
$844$	0	0
$845$	−17.7840	−0.611787
$846$	0	0
$847$	−1.02591	−0.0352508
$848$	0	0
$849$	−1.24856	−0.0428506
$850$	0	0
$851$	−4.51234	−0.154681
$852$	0	0
$853$	0.535872	0.0183479	0.00917396	−	0.999958i	$-0.497080\pi$
$853$	0.535872	0.0183479	0.00917396	+	0.999958i	$0.497080\pi$
$854$	0	0
$855$	8.77463	0.300086
$856$	0	0
$857$	−9.21396	−0.314743	−0.157371	−	0.987539i	$-0.550302\pi$
$857$	−9.21396	−0.314743	−0.157371	+	0.987539i	$0.550302\pi$
$858$	0	0
$859$	−45.3827	−1.54844	−0.774219	−	0.632918i	$-0.781857\pi$
$859$	−45.3827	−1.54844	−0.774219	+	0.632918i	$0.781857\pi$
$860$	0	0
$861$	−5.42020	−0.184720
$862$	0	0
$863$	−18.4663	−0.628601	−0.314300	−	0.949324i	$-0.601770\pi$
$863$	−18.4663	−0.628601	−0.314300	+	0.949324i	$0.601770\pi$
$864$	0	0
$865$	2.89418	0.0984052
$866$	0	0
$867$	−3.25392	−0.110509
$868$	0	0
$869$	−18.6213	−0.631683
$870$	0	0
$871$	50.6717	1.71694
$872$	0	0
$873$	−0.811000	−0.0274482
$874$	0	0
$875$	−15.5509	−0.525718
$876$	0	0
$877$	14.0271	0.473663	0.236832	−	0.971551i	$-0.423891\pi$
$877$	14.0271	0.473663	0.236832	+	0.971551i	$0.423891\pi$
$878$	0	0
$879$	9.15472	0.308781
$880$	0	0
$881$	−38.1317	−1.28469	−0.642345	−	0.766416i	$-0.722038\pi$
$881$	−38.1317	−1.28469	−0.642345	+	0.766416i	$0.722038\pi$
$882$	0	0
$883$	31.4041	1.05683	0.528416	−	0.848986i	$-0.322786\pi$
$883$	31.4041	1.05683	0.528416	+	0.848986i	$0.322786\pi$
$884$	0	0
$885$	1.72092	0.0578479
$886$	0	0
$887$	−21.1972	−0.711733	−0.355866	−	0.934537i	$-0.615814\pi$
$887$	−21.1972	−0.711733	−0.355866	+	0.934537i	$0.615814\pi$
$888$	0	0
$889$	12.9045	0.432804
$890$	0	0
$891$	26.7688	0.896790
$892$	0	0
$893$	29.7826	0.996637
$894$	0	0
$895$	−2.12733	−0.0711088
$896$	0	0
$897$	1.02795	0.0343224
$898$	0	0
$899$	−0.340711	−0.0113634
$900$	0	0
$901$	−24.5345	−0.817363
$902$	0	0
$903$	3.16361	0.105278
$904$	0	0
$905$	−6.53564	−0.217252
$906$	0	0
$907$	9.36000	0.310794	0.155397	−	0.987852i	$-0.450334\pi$
$907$	9.36000	0.310794	0.155397	+	0.987852i	$0.450334\pi$
$908$	0	0
$909$	−53.0717	−1.76028
$910$	0	0
$911$	3.56627	0.118156	0.0590778	−	0.998253i	$-0.481184\pi$
$911$	3.56627	0.118156	0.0590778	+	0.998253i	$0.481184\pi$
$912$	0	0
$913$	−43.7106	−1.44661
$914$	0	0
$915$	1.72551	0.0570437
$916$	0	0
$917$	26.1691	0.864181
$918$	0	0
$919$	52.7064	1.73862	0.869312	−	0.494264i	$-0.164562\pi$
$919$	52.7064	1.73862	0.869312	+	0.494264i	$0.164562\pi$
$920$	0	0
$921$	4.54263	0.149685
$922$	0	0
$923$	−9.66808	−0.318229
$924$	0	0
$925$	49.1842	1.61716
$926$	0	0
$927$	−29.7930	−0.978532
$928$	0	0
$929$	51.7441	1.69767	0.848835	−	0.528658i	$-0.177305\pi$
$929$	51.7441	1.69767	0.848835	+	0.528658i	$0.177305\pi$
$930$	0	0
$931$	15.3459	0.502943
$932$	0	0
$933$	−1.54134	−0.0504611
$934$	0	0
$935$	4.76165	0.155722
$936$	0	0
$937$	−20.7051	−0.676407	−0.338204	−	0.941073i	$-0.609819\pi$
$937$	−20.7051	−0.676407	−0.338204	+	0.941073i	$0.609819\pi$
$938$	0	0
$939$	0.532664	0.0173828
$940$	0	0
$941$	18.1156	0.590551	0.295276	−	0.955412i	$-0.404589\pi$
$941$	18.1156	0.590551	0.295276	+	0.955412i	$0.404589\pi$
$942$	0	0
$943$	2.21236	0.0720444
$944$	0	0
$945$	−3.22738	−0.104987
$946$	0	0
$947$	−42.5973	−1.38423	−0.692113	−	0.721789i	$-0.743320\pi$
$947$	−42.5973	−1.38423	−0.692113	+	0.721789i	$0.743320\pi$
$948$	0	0
$949$	115.770	3.75806
$950$	0	0
$951$	8.61456	0.279346
$952$	0	0
$953$	−37.6241	−1.21876	−0.609381	−	0.792877i	$-0.708582\pi$
$953$	−37.6241	−1.21876	−0.609381	+	0.792877i	$0.708582\pi$
$954$	0	0
$955$	−7.36694	−0.238389
$956$	0	0
$957$	0.826855	0.0267284
$958$	0	0
$959$	−25.5597	−0.825367
$960$	0	0
$961$	−30.7731	−0.992679
$962$	0	0
$963$	−20.7320	−0.668079
$964$	0	0
$965$	−13.4804	−0.433950
$966$	0	0
$967$	−47.9367	−1.54154	−0.770770	−	0.637114i	$-0.780128\pi$
$967$	−47.9367	−1.54154	−0.770770	+	0.637114i	$0.780128\pi$
$968$	0	0
$969$	−5.56310	−0.178712
$970$	0	0
$971$	−39.6857	−1.27357	−0.636787	−	0.771040i	$-0.719737\pi$
$971$	−39.6857	−1.27357	−0.636787	+	0.771040i	$0.719737\pi$
$972$	0	0
$973$	−60.6228	−1.94348
$974$	0	0
$975$	−11.2046	−0.358835
$976$	0	0
$977$	20.7512	0.663891	0.331945	−	0.943299i	$-0.392295\pi$
$977$	20.7512	0.663891	0.331945	+	0.943299i	$0.392295\pi$
$978$	0	0
$979$	3.05083	0.0975050
$980$	0	0
$981$	−23.5129	−0.750710
$982$	0	0
$983$	−41.4880	−1.32326	−0.661631	−	0.749830i	$-0.730135\pi$
$983$	−41.4880	−1.32326	−0.661631	+	0.749830i	$0.730135\pi$
$984$	0	0
$985$	−7.15026	−0.227826
$986$	0	0
$987$	−5.36730	−0.170843
$988$	0	0
$989$	−1.29129	−0.0410606
$990$	0	0
$991$	48.7393	1.54825	0.774127	−	0.633030i	$-0.218189\pi$
$991$	48.7393	1.54825	0.774127	+	0.633030i	$0.218189\pi$
$992$	0	0
$993$	−3.36308	−0.106724
$994$	0	0
$995$	−12.5583	−0.398126
$996$	0	0
$997$	1.19412	0.0378182	0.0189091	−	0.999821i	$-0.493981\pi$
$997$	1.19412	0.0378182	0.0189091	+	0.999821i	$0.493981\pi$
$998$	0	0
$999$	20.9882	0.664038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.27	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.27	✓	50	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.343448 q^{3} -0.515640 q^{5} -3.09823 q^{7} -2.88204 q^{9} +O(q^{10})\)	\(q+0.343448 q^{3} -0.515640 q^{5} -3.09823 q^{7} -2.88204 q^{9} +3.36617 q^{11} +6.89124 q^{13} -0.177095 q^{15} -2.74330 q^{17} +5.90448 q^{19} -1.06408 q^{21} +0.434326 q^{23} -4.73412 q^{25} -2.02018 q^{27} +0.715207 q^{29} -0.476381 q^{31} +1.15611 q^{33} +1.59757 q^{35} -10.3893 q^{37} +2.36678 q^{39} +5.09378 q^{41} -2.97309 q^{43} +1.48610 q^{45} +5.04407 q^{47} +2.59903 q^{49} -0.942183 q^{51} +8.94342 q^{53} -1.73573 q^{55} +2.02788 q^{57} -9.71744 q^{59} -9.74340 q^{61} +8.92924 q^{63} -3.55340 q^{65} +7.35306 q^{67} +0.149168 q^{69} -1.40295 q^{71} +16.7996 q^{73} -1.62592 q^{75} -10.4292 q^{77} -5.53188 q^{79} +7.95230 q^{81} -12.9852 q^{83} +1.41456 q^{85} +0.245636 q^{87} +0.906320 q^{89} -21.3506 q^{91} -0.163612 q^{93} -3.04458 q^{95} +0.281398 q^{97} -9.70146 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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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