LMFDB - Embedded newform 6028.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, 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("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62082 q^{3} -2.77332 q^{5} -0.589123 q^{7} +3.86872 q^{9} +O(q^{10})$	$q-2.62082 q^{3} -2.77332 q^{5} -0.589123 q^{7} +3.86872 q^{9} -1.00000 q^{11} -2.70553 q^{13} +7.26839 q^{15} +3.38950 q^{17} +1.58703 q^{19} +1.54399 q^{21} -5.18220 q^{23} +2.69131 q^{25} -2.27676 q^{27} +1.44822 q^{29} -5.38743 q^{31} +2.62082 q^{33} +1.63383 q^{35} -5.38980 q^{37} +7.09072 q^{39} -6.27962 q^{41} +7.59517 q^{43} -10.7292 q^{45} +2.89264 q^{47} -6.65293 q^{49} -8.88328 q^{51} -7.94047 q^{53} +2.77332 q^{55} -4.15932 q^{57} -4.68779 q^{59} -6.22310 q^{61} -2.27915 q^{63} +7.50330 q^{65} -7.91575 q^{67} +13.5816 q^{69} +7.48402 q^{71} -1.27951 q^{73} -7.05346 q^{75} +0.589123 q^{77} -8.96508 q^{79} -5.63918 q^{81} -12.5525 q^{83} -9.40017 q^{85} -3.79554 q^{87} -4.32970 q^{89} +1.59389 q^{91} +14.1195 q^{93} -4.40133 q^{95} +10.9230 q^{97} -3.86872 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) $ 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9	$ 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9} - 27 q^{11} + 2 q^{13} + 10 q^{15} + 17 q^{17} + 4 q^{19} - 4 q^{21} + 25 q^{23} + 21 q^{25} + 32 q^{27} - q^{29} + 8 q^{31} - 5 q^{33} + 30 q^{35} - 24 q^{37} + 10 q^{39} + 31 q^{41} + 17 q^{43} - 15 q^{45} + 27 q^{47} + 40 q^{49} + 30 q^{51} + 2 q^{55} + 13 q^{57} + 31 q^{59} + 2 q^{61} + 61 q^{63} + 2 q^{67} - 15 q^{69} + 40 q^{71} + 17 q^{73} + 30 q^{75} - 7 q^{77} + 33 q^{79} + 31 q^{81} + 66 q^{83} + 26 q^{85} + 43 q^{87} + 22 q^{89} + 18 q^{91} - 9 q^{93} + 95 q^{95} - 21 q^{97} - 28 q^{99}+O(q^{100}) $ 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9 - 27 * q^11 + 2 * q^13 + 10 * q^15 + 17 * q^17 + 4 * q^19 - 4 * q^21 + 25 * q^23 + 21 * q^25 + 32 * q^27 - q^29 + 8 * q^31 - 5 * q^33 + 30 * q^35 - 24 * q^37 + 10 * q^39 + 31 * q^41 + 17 * q^43 - 15 * q^45 + 27 * q^47 + 40 * q^49 + 30 * q^51 + 2 * q^55 + 13 * q^57 + 31 * q^59 + 2 * q^61 + 61 * q^63 + 2 * q^67 - 15 * q^69 + 40 * q^71 + 17 * q^73 + 30 * q^75 - 7 * q^77 + 33 * q^79 + 31 * q^81 + 66 * q^83 + 26 * q^85 + 43 * q^87 + 22 * q^89 + 18 * q^91 - 9 * q^93 + 95 * q^95 - 21 * q^97 - 28 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.62082	−1.51313	−0.756567	−	0.653917i	$-0.773125\pi$
$3$	−2.62082	−1.51313	−0.756567	+	0.653917i	$0.773125\pi$
$4$	0	0
$5$	−2.77332	−1.24027	−0.620134	−	0.784496i	$-0.712922\pi$
$5$	−2.77332	−1.24027	−0.620134	+	0.784496i	$0.712922\pi$
$6$	0	0
$7$	−0.589123	−0.222668	−0.111334	−	0.993783i	$-0.535512\pi$
$7$	−0.589123	−0.222668	−0.111334	+	0.993783i	$0.535512\pi$
$8$	0	0
$9$	3.86872	1.28957
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.70553	−0.750379	−0.375189	−	0.926948i	$-0.622422\pi$
$13$	−2.70553	−0.750379	−0.375189	+	0.926948i	$0.622422\pi$
$14$	0	0
$15$	7.26839	1.87669
$16$	0	0
$17$	3.38950	0.822074	0.411037	−	0.911619i	$-0.365167\pi$
$17$	3.38950	0.822074	0.411037	+	0.911619i	$0.365167\pi$
$18$	0	0
$19$	1.58703	0.364089	0.182044	−	0.983290i	$-0.441729\pi$
$19$	1.58703	0.364089	0.182044	+	0.983290i	$0.441729\pi$
$20$	0	0
$21$	1.54399	0.336926
$22$	0	0
$23$	−5.18220	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$23$	−5.18220	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$24$	0	0
$25$	2.69131	0.538263
$26$	0	0
$27$	−2.27676	−0.438162
$28$	0	0
$29$	1.44822	0.268928	0.134464	−	0.990918i	$-0.457069\pi$
$29$	1.44822	0.268928	0.134464	+	0.990918i	$0.457069\pi$
$30$	0	0
$31$	−5.38743	−0.967612	−0.483806	−	0.875175i	$-0.660746\pi$
$31$	−5.38743	−0.967612	−0.483806	+	0.875175i	$0.660746\pi$
$32$	0	0
$33$	2.62082	0.456227
$34$	0	0
$35$	1.63383	0.276167
$36$	0	0
$37$	−5.38980	−0.886078	−0.443039	−	0.896502i	$-0.646100\pi$
$37$	−5.38980	−0.886078	−0.443039	+	0.896502i	$0.646100\pi$
$38$	0	0
$39$	7.09072	1.13542
$40$	0	0
$41$	−6.27962	−0.980712	−0.490356	−	0.871522i	$-0.663133\pi$
$41$	−6.27962	−0.980712	−0.490356	+	0.871522i	$0.663133\pi$
$42$	0	0
$43$	7.59517	1.15825	0.579127	−	0.815238i	$-0.303394\pi$
$43$	7.59517	1.15825	0.579127	+	0.815238i	$0.303394\pi$
$44$	0	0
$45$	−10.7292	−1.59941
$46$	0	0
$47$	2.89264	0.421935	0.210967	−	0.977493i	$-0.432339\pi$
$47$	2.89264	0.421935	0.210967	+	0.977493i	$0.432339\pi$
$48$	0	0
$49$	−6.65293	−0.950419
$50$	0	0
$51$	−8.88328	−1.24391
$52$	0	0
$53$	−7.94047	−1.09071	−0.545354	−	0.838206i	$-0.683605\pi$
$53$	−7.94047	−1.09071	−0.545354	+	0.838206i	$0.683605\pi$
$54$	0	0
$55$	2.77332	0.373955
$56$	0	0
$57$	−4.15932	−0.550915
$58$	0	0
$59$	−4.68779	−0.610299	−0.305149	−	0.952304i	$-0.598706\pi$
$59$	−4.68779	−0.610299	−0.305149	+	0.952304i	$0.598706\pi$
$60$	0	0
$61$	−6.22310	−0.796787	−0.398393	−	0.917215i	$-0.630432\pi$
$61$	−6.22310	−0.796787	−0.398393	+	0.917215i	$0.630432\pi$
$62$	0	0
$63$	−2.27915	−0.287146
$64$	0	0
$65$	7.50330	0.930670
$66$	0	0
$67$	−7.91575	−0.967063	−0.483532	−	0.875327i	$-0.660646\pi$
$67$	−7.91575	−0.967063	−0.483532	+	0.875327i	$0.660646\pi$
$68$	0	0
$69$	13.5816	1.63504
$70$	0	0
$71$	7.48402	0.888190	0.444095	−	0.895980i	$-0.353525\pi$
$71$	7.48402	0.888190	0.444095	+	0.895980i	$0.353525\pi$
$72$	0	0
$73$	−1.27951	−0.149755	−0.0748775	−	0.997193i	$-0.523857\pi$
$73$	−1.27951	−0.149755	−0.0748775	+	0.997193i	$0.523857\pi$
$74$	0	0
$75$	−7.05346	−0.814464
$76$	0	0
$77$	0.589123	0.0671368
$78$	0	0
$79$	−8.96508	−1.00865	−0.504325	−	0.863514i	$-0.668259\pi$
$79$	−8.96508	−1.00865	−0.504325	+	0.863514i	$0.668259\pi$
$80$	0	0
$81$	−5.63918	−0.626575
$82$	0	0
$83$	−12.5525	−1.37781	−0.688907	−	0.724850i	$-0.741909\pi$
$83$	−12.5525	−1.37781	−0.688907	+	0.724850i	$0.741909\pi$
$84$	0	0
$85$	−9.40017	−1.01959
$86$	0	0
$87$	−3.79554	−0.406924
$88$	0	0
$89$	−4.32970	−0.458947	−0.229473	−	0.973315i	$-0.573700\pi$
$89$	−4.32970	−0.458947	−0.229473	+	0.973315i	$0.573700\pi$
$90$	0	0
$91$	1.59389	0.167085
$92$	0	0
$93$	14.1195	1.46413
$94$	0	0
$95$	−4.40133	−0.451567
$96$	0	0
$97$	10.9230	1.10906	0.554530	−	0.832163i	$-0.312898\pi$
$97$	10.9230	1.10906	0.554530	+	0.832163i	$0.312898\pi$
$98$	0	0
$99$	−3.86872	−0.388821
$100$	0	0
$101$	0.923831	0.0919246	0.0459623	−	0.998943i	$-0.485365\pi$
$101$	0.923831	0.0919246	0.0459623	+	0.998943i	$0.485365\pi$
$102$	0	0
$103$	5.89670	0.581019	0.290509	−	0.956872i	$-0.406175\pi$
$103$	5.89670	0.581019	0.290509	+	0.956872i	$0.406175\pi$
$104$	0	0
$105$	−4.28198	−0.417878
$106$	0	0
$107$	−11.0043	−1.06383	−0.531913	−	0.846799i	$-0.678527\pi$
$107$	−11.0043	−1.06383	−0.531913	+	0.846799i	$0.678527\pi$
$108$	0	0
$109$	−14.4429	−1.38338	−0.691691	−	0.722193i	$-0.743134\pi$
$109$	−14.4429	−1.38338	−0.691691	+	0.722193i	$0.743134\pi$
$110$	0	0
$111$	14.1257	1.34075
$112$	0	0
$113$	0.642098	0.0604035	0.0302018	−	0.999544i	$-0.490385\pi$
$113$	0.642098	0.0604035	0.0302018	+	0.999544i	$0.490385\pi$
$114$	0	0
$115$	14.3719	1.34019
$116$	0	0
$117$	−10.4669	−0.967668
$118$	0	0
$119$	−1.99683	−0.183049
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	16.4578	1.48395
$124$	0	0
$125$	6.40273	0.572677
$126$	0	0
$127$	−4.73944	−0.420557	−0.210279	−	0.977642i	$-0.567437\pi$
$127$	−4.73944	−0.420557	−0.210279	+	0.977642i	$0.567437\pi$
$128$	0	0
$129$	−19.9056	−1.75259
$130$	0	0
$131$	8.69563	0.759741	0.379870	−	0.925040i	$-0.375969\pi$
$131$	8.69563	0.759741	0.379870	+	0.925040i	$0.375969\pi$
$132$	0	0
$133$	−0.934954	−0.0810708
$134$	0	0
$135$	6.31418	0.543438
$136$	0	0
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	0	0
$139$	−7.41308	−0.628769	−0.314384	−	0.949296i	$-0.601798\pi$
$139$	−7.41308	−0.628769	−0.314384	+	0.949296i	$0.601798\pi$
$140$	0	0
$141$	−7.58110	−0.638444
$142$	0	0
$143$	2.70553	0.226248
$144$	0	0
$145$	−4.01639	−0.333543
$146$	0	0
$147$	17.4362	1.43811
$148$	0	0
$149$	−23.7920	−1.94912	−0.974559	−	0.224129i	$-0.928046\pi$
$149$	−23.7920	−1.94912	−0.974559	+	0.224129i	$0.928046\pi$
$150$	0	0
$151$	5.60377	0.456028	0.228014	−	0.973658i	$-0.426777\pi$
$151$	5.60377	0.456028	0.228014	+	0.973658i	$0.426777\pi$
$152$	0	0
$153$	13.1130	1.06012
$154$	0	0
$155$	14.9411	1.20010
$156$	0	0
$157$	−9.95166	−0.794229	−0.397114	−	0.917769i	$-0.629988\pi$
$157$	−9.95166	−0.794229	−0.397114	+	0.917769i	$0.629988\pi$
$158$	0	0
$159$	20.8106	1.65039
$160$	0	0
$161$	3.05296	0.240607
$162$	0	0
$163$	−5.24176	−0.410566	−0.205283	−	0.978703i	$-0.565812\pi$
$163$	−5.24176	−0.410566	−0.205283	+	0.978703i	$0.565812\pi$
$164$	0	0
$165$	−7.26839	−0.565843
$166$	0	0
$167$	10.3685	0.802342	0.401171	−	0.916003i	$-0.368603\pi$
$167$	10.3685	0.802342	0.401171	+	0.916003i	$0.368603\pi$
$168$	0	0
$169$	−5.68011	−0.436932
$170$	0	0
$171$	6.13975	0.469519
$172$	0	0
$173$	9.22949	0.701705	0.350853	−	0.936431i	$-0.385892\pi$
$173$	9.22949	0.701705	0.350853	+	0.936431i	$0.385892\pi$
$174$	0	0
$175$	−1.58552	−0.119854
$176$	0	0
$177$	12.2859	0.923463
$178$	0	0
$179$	−14.2163	−1.06258	−0.531288	−	0.847191i	$-0.678292\pi$
$179$	−14.2163	−1.06258	−0.531288	+	0.847191i	$0.678292\pi$
$180$	0	0
$181$	−6.37007	−0.473483	−0.236742	−	0.971573i	$-0.576079\pi$
$181$	−6.37007	−0.473483	−0.236742	+	0.971573i	$0.576079\pi$
$182$	0	0
$183$	16.3097	1.20564
$184$	0	0
$185$	14.9477	1.09897
$186$	0	0
$187$	−3.38950	−0.247865
$188$	0	0
$189$	1.34129	0.0975645
$190$	0	0
$191$	16.1552	1.16895	0.584475	−	0.811412i	$-0.301301\pi$
$191$	16.1552	1.16895	0.584475	+	0.811412i	$0.301301\pi$
$192$	0	0
$193$	−4.36580	−0.314257	−0.157128	−	0.987578i	$-0.550224\pi$
$193$	−4.36580	−0.314257	−0.157128	+	0.987578i	$0.550224\pi$
$194$	0	0
$195$	−19.6648	−1.40823
$196$	0	0
$197$	1.16170	0.0827676	0.0413838	−	0.999143i	$-0.486823\pi$
$197$	1.16170	0.0827676	0.0413838	+	0.999143i	$0.486823\pi$
$198$	0	0
$199$	17.9581	1.27302	0.636508	−	0.771270i	$-0.280378\pi$
$199$	17.9581	1.27302	0.636508	+	0.771270i	$0.280378\pi$
$200$	0	0
$201$	20.7458	1.46330
$202$	0	0
$203$	−0.853182	−0.0598816
$204$	0	0
$205$	17.4154	1.21635
$206$	0	0
$207$	−20.0485	−1.39346
$208$	0	0
$209$	−1.58703	−0.109777
$210$	0	0
$211$	26.0006	1.78995	0.894977	−	0.446113i	$-0.147192\pi$
$211$	26.0006	1.78995	0.894977	+	0.446113i	$0.147192\pi$
$212$	0	0
$213$	−19.6143	−1.34395
$214$	0	0
$215$	−21.0639	−1.43654
$216$	0	0
$217$	3.17386	0.215456
$218$	0	0
$219$	3.35337	0.226599
$220$	0	0
$221$	−9.17039	−0.616867
$222$	0	0
$223$	−15.1078	−1.01169	−0.505845	−	0.862624i	$-0.668819\pi$
$223$	−15.1078	−1.01169	−0.505845	+	0.862624i	$0.668819\pi$
$224$	0	0
$225$	10.4119	0.694129
$226$	0	0
$227$	22.4491	1.49000	0.744999	−	0.667066i	$-0.232450\pi$
$227$	22.4491	1.49000	0.744999	+	0.667066i	$0.232450\pi$
$228$	0	0
$229$	−19.6474	−1.29834	−0.649170	−	0.760644i	$-0.724883\pi$
$229$	−19.6474	−1.29834	−0.649170	+	0.760644i	$0.724883\pi$
$230$	0	0
$231$	−1.54399	−0.101587
$232$	0	0
$233$	16.3625	1.07194	0.535972	−	0.844236i	$-0.319945\pi$
$233$	16.3625	1.07194	0.535972	+	0.844236i	$0.319945\pi$
$234$	0	0
$235$	−8.02222	−0.523312
$236$	0	0
$237$	23.4959	1.52622
$238$	0	0
$239$	−9.70013	−0.627449	−0.313724	−	0.949514i	$-0.601577\pi$
$239$	−9.70013	−0.627449	−0.313724	+	0.949514i	$0.601577\pi$
$240$	0	0
$241$	16.7817	1.08101	0.540503	−	0.841342i	$-0.318234\pi$
$241$	16.7817	1.08101	0.540503	+	0.841342i	$0.318234\pi$
$242$	0	0
$243$	21.6096	1.38625
$244$	0	0
$245$	18.4507	1.17877
$246$	0	0
$247$	−4.29374	−0.273204
$248$	0	0
$249$	32.8978	2.08482
$250$	0	0
$251$	10.6036	0.669290	0.334645	−	0.942344i	$-0.391384\pi$
$251$	10.6036	0.669290	0.334645	+	0.942344i	$0.391384\pi$
$252$	0	0
$253$	5.18220	0.325802
$254$	0	0
$255$	24.6362	1.54278
$256$	0	0
$257$	25.3993	1.58437	0.792184	−	0.610282i	$-0.208944\pi$
$257$	25.3993	1.58437	0.792184	+	0.610282i	$0.208944\pi$
$258$	0	0
$259$	3.17526	0.197301
$260$	0	0
$261$	5.60276	0.346802
$262$	0	0
$263$	−2.85140	−0.175825	−0.0879125	−	0.996128i	$-0.528020\pi$
$263$	−2.85140	−0.175825	−0.0879125	+	0.996128i	$0.528020\pi$
$264$	0	0
$265$	22.0215	1.35277
$266$	0	0
$267$	11.3474	0.694448
$268$	0	0
$269$	−25.2429	−1.53908	−0.769542	−	0.638596i	$-0.779516\pi$
$269$	−25.2429	−1.53908	−0.769542	+	0.638596i	$0.779516\pi$
$270$	0	0
$271$	5.99921	0.364426	0.182213	−	0.983259i	$-0.441674\pi$
$271$	5.99921	0.364426	0.182213	+	0.983259i	$0.441674\pi$
$272$	0	0
$273$	−4.17731	−0.252822
$274$	0	0
$275$	−2.69131	−0.162292
$276$	0	0
$277$	12.7950	0.768776	0.384388	−	0.923172i	$-0.374413\pi$
$277$	12.7950	0.768776	0.384388	+	0.923172i	$0.374413\pi$
$278$	0	0
$279$	−20.8425	−1.24781
$280$	0	0
$281$	4.78687	0.285560	0.142780	−	0.989754i	$-0.454396\pi$
$281$	4.78687	0.285560	0.142780	+	0.989754i	$0.454396\pi$
$282$	0	0
$283$	−11.7647	−0.699338	−0.349669	−	0.936873i	$-0.613706\pi$
$283$	−11.7647	−0.699338	−0.349669	+	0.936873i	$0.613706\pi$
$284$	0	0
$285$	11.5351	0.683282
$286$	0	0
$287$	3.69947	0.218373
$288$	0	0
$289$	−5.51129	−0.324194
$290$	0	0
$291$	−28.6272	−1.67816
$292$	0	0
$293$	13.8129	0.806961	0.403480	−	0.914988i	$-0.367800\pi$
$293$	13.8129	0.806961	0.403480	+	0.914988i	$0.367800\pi$
$294$	0	0
$295$	13.0008	0.756934
$296$	0	0
$297$	2.27676	0.132111
$298$	0	0
$299$	14.0206	0.810832
$300$	0	0
$301$	−4.47449	−0.257906
$302$	0	0
$303$	−2.42120	−0.139094
$304$	0	0
$305$	17.2587	0.988229
$306$	0	0
$307$	−26.8512	−1.53248	−0.766240	−	0.642554i	$-0.777875\pi$
$307$	−26.8512	−1.53248	−0.766240	+	0.642554i	$0.777875\pi$
$308$	0	0
$309$	−15.4542	−0.879159
$310$	0	0
$311$	4.13562	0.234510	0.117255	−	0.993102i	$-0.462591\pi$
$311$	4.13562	0.234510	0.117255	+	0.993102i	$0.462591\pi$
$312$	0	0
$313$	9.56625	0.540716	0.270358	−	0.962760i	$-0.412858\pi$
$313$	9.56625	0.540716	0.270358	+	0.962760i	$0.412858\pi$
$314$	0	0
$315$	6.32082	0.356138
$316$	0	0
$317$	−14.8749	−0.835460	−0.417730	−	0.908571i	$-0.637174\pi$
$317$	−14.8749	−0.835460	−0.417730	+	0.908571i	$0.637174\pi$
$318$	0	0
$319$	−1.44822	−0.0810849
$320$	0	0
$321$	28.8403	1.60971
$322$	0	0
$323$	5.37922	0.299308
$324$	0	0
$325$	−7.28143	−0.403901
$326$	0	0
$327$	37.8524	2.09324
$328$	0	0
$329$	−1.70412	−0.0939513
$330$	0	0
$331$	10.5139	0.577897	0.288949	−	0.957345i	$-0.406694\pi$
$331$	10.5139	0.577897	0.288949	+	0.957345i	$0.406694\pi$
$332$	0	0
$333$	−20.8516	−1.14266
$334$	0	0
$335$	21.9529	1.19942
$336$	0	0
$337$	−0.647142	−0.0352520	−0.0176260	−	0.999845i	$-0.505611\pi$
$337$	−0.647142	−0.0352520	−0.0176260	+	0.999845i	$0.505611\pi$
$338$	0	0
$339$	−1.68283	−0.0913986
$340$	0	0
$341$	5.38743	0.291746
$342$	0	0
$343$	8.04326	0.434295
$344$	0	0
$345$	−37.6662	−2.02788
$346$	0	0
$347$	0.560573	0.0300931	0.0150466	−	0.999887i	$-0.495210\pi$
$347$	0.560573	0.0300931	0.0150466	+	0.999887i	$0.495210\pi$
$348$	0	0
$349$	−25.1822	−1.34797	−0.673985	−	0.738745i	$-0.735419\pi$
$349$	−25.1822	−1.34797	−0.673985	+	0.738745i	$0.735419\pi$
$350$	0	0
$351$	6.15983	0.328787
$352$	0	0
$353$	19.1196	1.01764	0.508818	−	0.860874i	$-0.330083\pi$
$353$	19.1196	1.01764	0.508818	+	0.860874i	$0.330083\pi$
$354$	0	0
$355$	−20.7556	−1.10159
$356$	0	0
$357$	5.23335	0.276978
$358$	0	0
$359$	−12.7589	−0.673390	−0.336695	−	0.941614i	$-0.609309\pi$
$359$	−12.7589	−0.673390	−0.336695	+	0.941614i	$0.609309\pi$
$360$	0	0
$361$	−16.4813	−0.867439
$362$	0	0
$363$	−2.62082	−0.137558
$364$	0	0
$365$	3.54849	0.185736
$366$	0	0
$367$	21.4120	1.11770	0.558849	−	0.829269i	$-0.311243\pi$
$367$	21.4120	1.11770	0.558849	+	0.829269i	$0.311243\pi$
$368$	0	0
$369$	−24.2941	−1.26470
$370$	0	0
$371$	4.67791	0.242865
$372$	0	0
$373$	8.44261	0.437142	0.218571	−	0.975821i	$-0.429861\pi$
$373$	8.44261	0.437142	0.218571	+	0.975821i	$0.429861\pi$
$374$	0	0
$375$	−16.7804	−0.866537
$376$	0	0
$377$	−3.91821	−0.201798
$378$	0	0
$379$	−3.30447	−0.169739	−0.0848696	−	0.996392i	$-0.527047\pi$
$379$	−3.30447	−0.169739	−0.0848696	+	0.996392i	$0.527047\pi$
$380$	0	0
$381$	12.4212	0.636359
$382$	0	0
$383$	−4.36915	−0.223253	−0.111626	−	0.993750i	$-0.535606\pi$
$383$	−4.36915	−0.223253	−0.111626	+	0.993750i	$0.535606\pi$
$384$	0	0
$385$	−1.63383	−0.0832676
$386$	0	0
$387$	29.3836	1.49365
$388$	0	0
$389$	−11.5966	−0.587970	−0.293985	−	0.955810i	$-0.594982\pi$
$389$	−11.5966	−0.587970	−0.293985	+	0.955810i	$0.594982\pi$
$390$	0	0
$391$	−17.5651	−0.888304
$392$	0	0
$393$	−22.7897	−1.14959
$394$	0	0
$395$	24.8630	1.25100
$396$	0	0
$397$	9.77694	0.490690	0.245345	−	0.969436i	$-0.421099\pi$
$397$	9.77694	0.490690	0.245345	+	0.969436i	$0.421099\pi$
$398$	0	0
$399$	2.45035	0.122671
$400$	0	0
$401$	−3.44749	−0.172159	−0.0860796	−	0.996288i	$-0.527434\pi$
$401$	−3.44749	−0.172159	−0.0860796	+	0.996288i	$0.527434\pi$
$402$	0	0
$403$	14.5759	0.726075
$404$	0	0
$405$	15.6393	0.777121
$406$	0	0
$407$	5.38980	0.267163
$408$	0	0
$409$	−6.33650	−0.313320	−0.156660	−	0.987653i	$-0.550073\pi$
$409$	−6.33650	−0.313320	−0.156660	+	0.987653i	$0.550073\pi$
$410$	0	0
$411$	2.62082	0.129276
$412$	0	0
$413$	2.76169	0.135894
$414$	0	0
$415$	34.8121	1.70886
$416$	0	0
$417$	19.4284	0.951411
$418$	0	0
$419$	24.1397	1.17930	0.589651	−	0.807658i	$-0.299265\pi$
$419$	24.1397	1.17930	0.589651	+	0.807658i	$0.299265\pi$
$420$	0	0
$421$	5.88611	0.286871	0.143436	−	0.989660i	$-0.454185\pi$
$421$	5.88611	0.286871	0.143436	+	0.989660i	$0.454185\pi$
$422$	0	0
$423$	11.1908	0.544116
$424$	0	0
$425$	9.12221	0.442492
$426$	0	0
$427$	3.66618	0.177419
$428$	0	0
$429$	−7.09072	−0.342343
$430$	0	0
$431$	8.83029	0.425340	0.212670	−	0.977124i	$-0.431784\pi$
$431$	8.83029	0.425340	0.212670	+	0.977124i	$0.431784\pi$
$432$	0	0
$433$	14.3523	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$433$	14.3523	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$434$	0	0
$435$	10.5262	0.504695
$436$	0	0
$437$	−8.22429	−0.393421
$438$	0	0
$439$	11.1510	0.532209	0.266104	−	0.963944i	$-0.414263\pi$
$439$	11.1510	0.532209	0.266104	+	0.963944i	$0.414263\pi$
$440$	0	0
$441$	−25.7383	−1.22563
$442$	0	0
$443$	18.3785	0.873188	0.436594	−	0.899659i	$-0.356185\pi$
$443$	18.3785	0.873188	0.436594	+	0.899659i	$0.356185\pi$
$444$	0	0
$445$	12.0076	0.569217
$446$	0	0
$447$	62.3547	2.94928
$448$	0	0
$449$	23.8390	1.12503	0.562517	−	0.826786i	$-0.309833\pi$
$449$	23.8390	1.12503	0.562517	+	0.826786i	$0.309833\pi$
$450$	0	0
$451$	6.27962	0.295696
$452$	0	0
$453$	−14.6865	−0.690032
$454$	0	0
$455$	−4.42037	−0.207230
$456$	0	0
$457$	−0.937551	−0.0438568	−0.0219284	−	0.999760i	$-0.506981\pi$
$457$	−0.937551	−0.0438568	−0.0219284	+	0.999760i	$0.506981\pi$
$458$	0	0
$459$	−7.71706	−0.360202
$460$	0	0
$461$	−33.1952	−1.54605	−0.773027	−	0.634374i	$-0.781258\pi$
$461$	−33.1952	−1.54605	−0.773027	+	0.634374i	$0.781258\pi$
$462$	0	0
$463$	32.8566	1.52698	0.763488	−	0.645823i	$-0.223485\pi$
$463$	32.8566	1.52698	0.763488	+	0.645823i	$0.223485\pi$
$464$	0	0
$465$	−39.1580	−1.81591
$466$	0	0
$467$	14.7524	0.682659	0.341330	−	0.939944i	$-0.389123\pi$
$467$	14.7524	0.682659	0.341330	+	0.939944i	$0.389123\pi$
$468$	0	0
$469$	4.66335	0.215334
$470$	0	0
$471$	26.0815	1.20177
$472$	0	0
$473$	−7.59517	−0.349226
$474$	0	0
$475$	4.27119	0.195975
$476$	0	0
$477$	−30.7194	−1.40655
$478$	0	0
$479$	16.1817	0.739360	0.369680	−	0.929159i	$-0.379467\pi$
$479$	16.1817	0.739360	0.369680	+	0.929159i	$0.379467\pi$
$480$	0	0
$481$	14.5823	0.664894
$482$	0	0
$483$	−8.00126	−0.364070
$484$	0	0
$485$	−30.2929	−1.37553
$486$	0	0
$487$	0.835607	0.0378650	0.0189325	−	0.999821i	$-0.493973\pi$
$487$	0.835607	0.0378650	0.0189325	+	0.999821i	$0.493973\pi$
$488$	0	0
$489$	13.7377	0.621242
$490$	0	0
$491$	13.0588	0.589336	0.294668	−	0.955600i	$-0.404791\pi$
$491$	13.0588	0.589336	0.294668	+	0.955600i	$0.404791\pi$
$492$	0	0
$493$	4.90875	0.221079
$494$	0	0
$495$	10.7292	0.482242
$496$	0	0
$497$	−4.40901	−0.197771
$498$	0	0
$499$	15.5107	0.694354	0.347177	−	0.937800i	$-0.387140\pi$
$499$	15.5107	0.694354	0.347177	+	0.937800i	$0.387140\pi$
$500$	0	0
$501$	−27.1741	−1.21405
$502$	0	0
$503$	1.85095	0.0825296	0.0412648	−	0.999148i	$-0.486861\pi$
$503$	1.85095	0.0825296	0.0412648	+	0.999148i	$0.486861\pi$
$504$	0	0
$505$	−2.56208	−0.114011
$506$	0	0
$507$	14.8866	0.661136
$508$	0	0
$509$	32.0281	1.41962	0.709811	−	0.704393i	$-0.248781\pi$
$509$	32.0281	1.41962	0.709811	+	0.704393i	$0.248781\pi$
$510$	0	0
$511$	0.753788	0.0333456
$512$	0	0
$513$	−3.61327	−0.159530
$514$	0	0
$515$	−16.3534	−0.720619
$516$	0	0
$517$	−2.89264	−0.127218
$518$	0	0
$519$	−24.1889	−1.06177
$520$	0	0
$521$	12.3261	0.540017	0.270009	−	0.962858i	$-0.412973\pi$
$521$	12.3261	0.540017	0.270009	+	0.962858i	$0.412973\pi$
$522$	0	0
$523$	−13.6513	−0.596929	−0.298465	−	0.954421i	$-0.596474\pi$
$523$	−13.6513	−0.596929	−0.298465	+	0.954421i	$0.596474\pi$
$524$	0	0
$525$	4.15536	0.181355
$526$	0	0
$527$	−18.2607	−0.795449
$528$	0	0
$529$	3.85520	0.167618
$530$	0	0
$531$	−18.1357	−0.787025
$532$	0	0
$533$	16.9897	0.735906
$534$	0	0
$535$	30.5185	1.31943
$536$	0	0
$537$	37.2584	1.60782
$538$	0	0
$539$	6.65293	0.286562
$540$	0	0
$541$	−5.89326	−0.253371	−0.126686	−	0.991943i	$-0.540434\pi$
$541$	−5.89326	−0.253371	−0.126686	+	0.991943i	$0.540434\pi$
$542$	0	0
$543$	16.6948	0.716443
$544$	0	0
$545$	40.0549	1.71576
$546$	0	0
$547$	−32.0552	−1.37058	−0.685291	−	0.728269i	$-0.740325\pi$
$547$	−32.0552	−1.37058	−0.685291	+	0.728269i	$0.740325\pi$
$548$	0	0
$549$	−24.0754	−1.02751
$550$	0	0
$551$	2.29837	0.0979137
$552$	0	0
$553$	5.28154	0.224594
$554$	0	0
$555$	−39.1752	−1.66289
$556$	0	0
$557$	−27.8259	−1.17902	−0.589510	−	0.807761i	$-0.700679\pi$
$557$	−27.8259	−1.17902	−0.589510	+	0.807761i	$0.700679\pi$
$558$	0	0
$559$	−20.5490	−0.869129
$560$	0	0
$561$	8.88328	0.375052
$562$	0	0
$563$	36.9874	1.55883	0.779416	−	0.626507i	$-0.215516\pi$
$563$	36.9874	1.55883	0.779416	+	0.626507i	$0.215516\pi$
$564$	0	0
$565$	−1.78075	−0.0749165
$566$	0	0
$567$	3.32217	0.139518
$568$	0	0
$569$	−42.6076	−1.78621	−0.893103	−	0.449853i	$-0.851476\pi$
$569$	−42.6076	−1.78621	−0.893103	+	0.449853i	$0.851476\pi$
$570$	0	0
$571$	−1.20912	−0.0506002	−0.0253001	−	0.999680i	$-0.508054\pi$
$571$	−1.20912	−0.0506002	−0.0253001	+	0.999680i	$0.508054\pi$
$572$	0	0
$573$	−42.3399	−1.76878
$574$	0	0
$575$	−13.9469	−0.581627
$576$	0	0
$577$	−33.8241	−1.40811	−0.704057	−	0.710143i	$-0.748630\pi$
$577$	−33.8241	−1.40811	−0.704057	+	0.710143i	$0.748630\pi$
$578$	0	0
$579$	11.4420	0.475512
$580$	0	0
$581$	7.39496	0.306795
$582$	0	0
$583$	7.94047	0.328861
$584$	0	0
$585$	29.0282	1.20017
$586$	0	0
$587$	26.5228	1.09471	0.547356	−	0.836900i	$-0.315634\pi$
$587$	26.5228	1.09471	0.547356	+	0.836900i	$0.315634\pi$
$588$	0	0
$589$	−8.55000	−0.352297
$590$	0	0
$591$	−3.04461	−0.125238
$592$	0	0
$593$	−21.7244	−0.892113	−0.446057	−	0.895005i	$-0.647172\pi$
$593$	−21.7244	−0.892113	−0.446057	+	0.895005i	$0.647172\pi$
$594$	0	0
$595$	5.53786	0.227030
$596$	0	0
$597$	−47.0650	−1.92624
$598$	0	0
$599$	12.0142	0.490887	0.245444	−	0.969411i	$-0.421066\pi$
$599$	12.0142	0.490887	0.245444	+	0.969411i	$0.421066\pi$
$600$	0	0
$601$	−13.3730	−0.545498	−0.272749	−	0.962085i	$-0.587933\pi$
$601$	−13.3730	−0.545498	−0.272749	+	0.962085i	$0.587933\pi$
$602$	0	0
$603$	−30.6238	−1.24710
$604$	0	0
$605$	−2.77332	−0.112752
$606$	0	0
$607$	25.9483	1.05321	0.526605	−	0.850110i	$-0.323465\pi$
$607$	25.9483	1.05321	0.526605	+	0.850110i	$0.323465\pi$
$608$	0	0
$609$	2.23604	0.0906089
$610$	0	0
$611$	−7.82612	−0.316611
$612$	0	0
$613$	−16.1536	−0.652437	−0.326219	−	0.945294i	$-0.605775\pi$
$613$	−16.1536	−0.652437	−0.326219	+	0.945294i	$0.605775\pi$
$614$	0	0
$615$	−45.6427	−1.84049
$616$	0	0
$617$	41.1636	1.65719	0.828593	−	0.559852i	$-0.189142\pi$
$617$	41.1636	1.65719	0.828593	+	0.559852i	$0.189142\pi$
$618$	0	0
$619$	22.1137	0.888824	0.444412	−	0.895822i	$-0.353413\pi$
$619$	22.1137	0.888824	0.444412	+	0.895822i	$0.353413\pi$
$620$	0	0
$621$	11.7986	0.473462
$622$	0	0
$623$	2.55072	0.102193
$624$	0	0
$625$	−31.2134	−1.24854
$626$	0	0
$627$	4.15932	0.166107
$628$	0	0
$629$	−18.2687	−0.728422
$630$	0	0
$631$	5.51375	0.219499	0.109750	−	0.993959i	$-0.464995\pi$
$631$	5.51375	0.219499	0.109750	+	0.993959i	$0.464995\pi$
$632$	0	0
$633$	−68.1429	−2.70844
$634$	0	0
$635$	13.1440	0.521603
$636$	0	0
$637$	17.9997	0.713174
$638$	0	0
$639$	28.9536	1.14539
$640$	0	0
$641$	46.8675	1.85116	0.925578	−	0.378558i	$-0.123580\pi$
$641$	46.8675	1.85116	0.925578	+	0.378558i	$0.123580\pi$
$642$	0	0
$643$	9.35097	0.368766	0.184383	−	0.982854i	$-0.440971\pi$
$643$	9.35097	0.368766	0.184383	+	0.982854i	$0.440971\pi$
$644$	0	0
$645$	55.2047	2.17368
$646$	0	0
$647$	−3.24533	−0.127587	−0.0637935	−	0.997963i	$-0.520320\pi$
$647$	−3.24533	−0.127587	−0.0637935	+	0.997963i	$0.520320\pi$
$648$	0	0
$649$	4.68779	0.184012
$650$	0	0
$651$	−8.31814	−0.326013
$652$	0	0
$653$	−21.8697	−0.855828	−0.427914	−	0.903819i	$-0.640751\pi$
$653$	−21.8697	−0.855828	−0.427914	+	0.903819i	$0.640751\pi$
$654$	0	0
$655$	−24.1158	−0.942282
$656$	0	0
$657$	−4.95006	−0.193120
$658$	0	0
$659$	4.42340	0.172311	0.0861555	−	0.996282i	$-0.472542\pi$
$659$	4.42340	0.172311	0.0861555	+	0.996282i	$0.472542\pi$
$660$	0	0
$661$	−24.5791	−0.956018	−0.478009	−	0.878355i	$-0.658641\pi$
$661$	−24.5791	−0.956018	−0.478009	+	0.878355i	$0.658641\pi$
$662$	0	0
$663$	24.0340	0.933402
$664$	0	0
$665$	2.59293	0.100549
$666$	0	0
$667$	−7.50498	−0.290594
$668$	0	0
$669$	39.5948	1.53082
$670$	0	0
$671$	6.22310	0.240240
$672$	0	0
$673$	14.0796	0.542727	0.271363	−	0.962477i	$-0.412525\pi$
$673$	14.0796	0.542727	0.271363	+	0.962477i	$0.412525\pi$
$674$	0	0
$675$	−6.12746	−0.235846
$676$	0	0
$677$	38.7560	1.48951	0.744757	−	0.667336i	$-0.232565\pi$
$677$	38.7560	1.48951	0.744757	+	0.667336i	$0.232565\pi$
$678$	0	0
$679$	−6.43498	−0.246952
$680$	0	0
$681$	−58.8351	−2.25456
$682$	0	0
$683$	−37.4096	−1.43144	−0.715718	−	0.698389i	$-0.753901\pi$
$683$	−37.4096	−1.43144	−0.715718	+	0.698389i	$0.753901\pi$
$684$	0	0
$685$	2.77332	0.105963
$686$	0	0
$687$	51.4925	1.96456
$688$	0	0
$689$	21.4832	0.818443
$690$	0	0
$691$	13.0897	0.497955	0.248978	−	0.968509i	$-0.419905\pi$
$691$	13.0897	0.497955	0.248978	+	0.968509i	$0.419905\pi$
$692$	0	0
$693$	2.27915	0.0865778
$694$	0	0
$695$	20.5588	0.779841
$696$	0	0
$697$	−21.2848	−0.806218
$698$	0	0
$699$	−42.8833	−1.62200
$700$	0	0
$701$	−36.4599	−1.37707	−0.688536	−	0.725202i	$-0.741747\pi$
$701$	−36.4599	−1.37707	−0.688536	+	0.725202i	$0.741747\pi$
$702$	0	0
$703$	−8.55376	−0.322611
$704$	0	0
$705$	21.0248	0.791841
$706$	0	0
$707$	−0.544250	−0.0204686
$708$	0	0
$709$	−31.5949	−1.18657	−0.593285	−	0.804992i	$-0.702169\pi$
$709$	−31.5949	−1.18657	−0.593285	+	0.804992i	$0.702169\pi$
$710$	0	0
$711$	−34.6834	−1.30073
$712$	0	0
$713$	27.9188	1.04557
$714$	0	0
$715$	−7.50330	−0.280608
$716$	0	0
$717$	25.4223	0.949414
$718$	0	0
$719$	32.1297	1.19824	0.599118	−	0.800661i	$-0.295518\pi$
$719$	32.1297	1.19824	0.599118	+	0.800661i	$0.295518\pi$
$720$	0	0
$721$	−3.47388	−0.129374
$722$	0	0
$723$	−43.9819	−1.63571
$724$	0	0
$725$	3.89762	0.144754
$726$	0	0
$727$	−32.9519	−1.22212	−0.611060	−	0.791584i	$-0.709257\pi$
$727$	−32.9519	−1.22212	−0.611060	+	0.791584i	$0.709257\pi$
$728$	0	0
$729$	−39.7173	−1.47101
$730$	0	0
$731$	25.7438	0.952170
$732$	0	0
$733$	−24.2869	−0.897058	−0.448529	−	0.893768i	$-0.648052\pi$
$733$	−24.2869	−0.897058	−0.448529	+	0.893768i	$0.648052\pi$
$734$	0	0
$735$	−48.3561	−1.78364
$736$	0	0
$737$	7.91575	0.291580
$738$	0	0
$739$	−36.3539	−1.33730	−0.668650	−	0.743577i	$-0.733127\pi$
$739$	−36.3539	−1.33730	−0.668650	+	0.743577i	$0.733127\pi$
$740$	0	0
$741$	11.2531	0.413395
$742$	0	0
$743$	5.40556	0.198311	0.0991554	−	0.995072i	$-0.468386\pi$
$743$	5.40556	0.198311	0.0991554	+	0.995072i	$0.468386\pi$
$744$	0	0
$745$	65.9830	2.41743
$746$	0	0
$747$	−48.5620	−1.77679
$748$	0	0
$749$	6.48289	0.236880
$750$	0	0
$751$	−49.9968	−1.82441	−0.912205	−	0.409735i	$-0.865621\pi$
$751$	−49.9968	−1.82441	−0.912205	+	0.409735i	$0.865621\pi$
$752$	0	0
$753$	−27.7900	−1.01273
$754$	0	0
$755$	−15.5411	−0.565597
$756$	0	0
$757$	−1.06590	−0.0387409	−0.0193705	−	0.999812i	$-0.506166\pi$
$757$	−1.06590	−0.0387409	−0.0193705	+	0.999812i	$0.506166\pi$
$758$	0	0
$759$	−13.5816	−0.492982
$760$	0	0
$761$	−39.2845	−1.42406	−0.712032	−	0.702147i	$-0.752225\pi$
$761$	−39.2845	−1.42406	−0.712032	+	0.702147i	$0.752225\pi$
$762$	0	0
$763$	8.50867	0.308035
$764$	0	0
$765$	−36.3666	−1.31484
$766$	0	0
$767$	12.6830	0.457955
$768$	0	0
$769$	24.3520	0.878154	0.439077	−	0.898449i	$-0.355305\pi$
$769$	24.3520	0.878154	0.439077	+	0.898449i	$0.355305\pi$
$770$	0	0
$771$	−66.5672	−2.39736
$772$	0	0
$773$	8.58622	0.308825	0.154412	−	0.988006i	$-0.450652\pi$
$773$	8.58622	0.308825	0.154412	+	0.988006i	$0.450652\pi$
$774$	0	0
$775$	−14.4993	−0.520830
$776$	0	0
$777$	−8.32179	−0.298543
$778$	0	0
$779$	−9.96593	−0.357066
$780$	0	0
$781$	−7.48402	−0.267799
$782$	0	0
$783$	−3.29725	−0.117834
$784$	0	0
$785$	27.5991	0.985056
$786$	0	0
$787$	−14.9290	−0.532160	−0.266080	−	0.963951i	$-0.585728\pi$
$787$	−14.9290	−0.532160	−0.266080	+	0.963951i	$0.585728\pi$
$788$	0	0
$789$	7.47302	0.266047
$790$	0	0
$791$	−0.378275	−0.0134499
$792$	0	0
$793$	16.8368	0.597892
$794$	0	0
$795$	−57.7144	−2.04692
$796$	0	0
$797$	−33.9939	−1.20412	−0.602062	−	0.798449i	$-0.705654\pi$
$797$	−33.9939	−1.20412	−0.602062	+	0.798449i	$0.705654\pi$
$798$	0	0
$799$	9.80460	0.346862
$800$	0	0
$801$	−16.7504	−0.591845
$802$	0	0
$803$	1.27951	0.0451529
$804$	0	0
$805$	−8.46683	−0.298417
$806$	0	0
$807$	66.1571	2.32884
$808$	0	0
$809$	−43.0080	−1.51208	−0.756041	−	0.654525i	$-0.772869\pi$
$809$	−43.0080	−1.51208	−0.756041	+	0.654525i	$0.772869\pi$
$810$	0	0
$811$	−34.8173	−1.22260	−0.611300	−	0.791399i	$-0.709353\pi$
$811$	−34.8173	−1.22260	−0.611300	+	0.791399i	$0.709353\pi$
$812$	0	0
$813$	−15.7229	−0.551425
$814$	0	0
$815$	14.5371	0.509212
$816$	0	0
$817$	12.0537	0.421707
$818$	0	0
$819$	6.16631	0.215468
$820$	0	0
$821$	7.12119	0.248531	0.124266	−	0.992249i	$-0.460343\pi$
$821$	7.12119	0.248531	0.124266	+	0.992249i	$0.460343\pi$
$822$	0	0
$823$	34.3003	1.19563	0.597817	−	0.801633i	$-0.296035\pi$
$823$	34.3003	1.19563	0.597817	+	0.801633i	$0.296035\pi$
$824$	0	0
$825$	7.05346	0.245570
$826$	0	0
$827$	47.2009	1.64134	0.820668	−	0.571405i	$-0.193602\pi$
$827$	47.2009	1.64134	0.820668	+	0.571405i	$0.193602\pi$
$828$	0	0
$829$	−16.4101	−0.569948	−0.284974	−	0.958535i	$-0.591985\pi$
$829$	−16.4101	−0.569948	−0.284974	+	0.958535i	$0.591985\pi$
$830$	0	0
$831$	−33.5334	−1.16326
$832$	0	0
$833$	−22.5501	−0.781315
$834$	0	0
$835$	−28.7553	−0.995118
$836$	0	0
$837$	12.2659	0.423970
$838$	0	0
$839$	−25.4750	−0.879496	−0.439748	−	0.898121i	$-0.644932\pi$
$839$	−25.4750	−0.879496	−0.439748	+	0.898121i	$0.644932\pi$
$840$	0	0
$841$	−26.9027	−0.927678
$842$	0	0
$843$	−12.5455	−0.432091
$844$	0	0
$845$	15.7528	0.541912
$846$	0	0
$847$	−0.589123	−0.0202425
$848$	0	0
$849$	30.8332	1.05819
$850$	0	0
$851$	27.9310	0.957464
$852$	0	0
$853$	13.4169	0.459385	0.229693	−	0.973263i	$-0.426228\pi$
$853$	13.4169	0.459385	0.229693	+	0.973263i	$0.426228\pi$
$854$	0	0
$855$	−17.0275	−0.582329
$856$	0	0
$857$	9.10571	0.311045	0.155522	−	0.987832i	$-0.450294\pi$
$857$	9.10571	0.311045	0.155522	+	0.987832i	$0.450294\pi$
$858$	0	0
$859$	−2.81722	−0.0961221	−0.0480611	−	0.998844i	$-0.515304\pi$
$859$	−2.81722	−0.0961221	−0.0480611	+	0.998844i	$0.515304\pi$
$860$	0	0
$861$	−9.69567	−0.330427
$862$	0	0
$863$	−10.5320	−0.358512	−0.179256	−	0.983802i	$-0.557369\pi$
$863$	−10.5320	−0.358512	−0.179256	+	0.983802i	$0.557369\pi$
$864$	0	0
$865$	−25.5963	−0.870302
$866$	0	0
$867$	14.4441	0.490548
$868$	0	0
$869$	8.96508	0.304119
$870$	0	0
$871$	21.4163	0.725664
$872$	0	0
$873$	42.2579	1.43021
$874$	0	0
$875$	−3.77200	−0.127517
$876$	0	0
$877$	−8.76571	−0.295997	−0.147999	−	0.988988i	$-0.547283\pi$
$877$	−8.76571	−0.295997	−0.147999	+	0.988988i	$0.547283\pi$
$878$	0	0
$879$	−36.2013	−1.22104
$880$	0	0
$881$	25.6693	0.864822	0.432411	−	0.901677i	$-0.357663\pi$
$881$	25.6693	0.864822	0.432411	+	0.901677i	$0.357663\pi$
$882$	0	0
$883$	19.5410	0.657608	0.328804	−	0.944398i	$-0.393354\pi$
$883$	19.5410	0.657608	0.328804	+	0.944398i	$0.393354\pi$
$884$	0	0
$885$	−34.0727	−1.14534
$886$	0	0
$887$	21.3872	0.718112	0.359056	−	0.933316i	$-0.383099\pi$
$887$	21.3872	0.718112	0.359056	+	0.933316i	$0.383099\pi$
$888$	0	0
$889$	2.79211	0.0936445
$890$	0	0
$891$	5.63918	0.188920
$892$	0	0
$893$	4.59069	0.153622
$894$	0	0
$895$	39.4264	1.31788
$896$	0	0
$897$	−36.7455	−1.22690
$898$	0	0
$899$	−7.80220	−0.260218
$900$	0	0
$901$	−26.9142	−0.896642
$902$	0	0
$903$	11.7269	0.390246
$904$	0	0
$905$	17.6662	0.587246
$906$	0	0
$907$	−35.7293	−1.18637	−0.593186	−	0.805066i	$-0.702130\pi$
$907$	−35.7293	−1.18637	−0.593186	+	0.805066i	$0.702130\pi$
$908$	0	0
$909$	3.57404	0.118543
$910$	0	0
$911$	−10.2419	−0.339328	−0.169664	−	0.985502i	$-0.554268\pi$
$911$	−10.2419	−0.339328	−0.169664	+	0.985502i	$0.554268\pi$
$912$	0	0
$913$	12.5525	0.415426
$914$	0	0
$915$	−45.2319	−1.49532
$916$	0	0
$917$	−5.12280	−0.169170
$918$	0	0
$919$	−34.8196	−1.14859	−0.574297	−	0.818647i	$-0.694725\pi$
$919$	−34.8196	−1.14859	−0.574297	+	0.818647i	$0.694725\pi$
$920$	0	0
$921$	70.3723	2.31885
$922$	0	0
$923$	−20.2482	−0.666479
$924$	0	0
$925$	−14.5057	−0.476943
$926$	0	0
$927$	22.8127	0.749266
$928$	0	0
$929$	33.1097	1.08629	0.543147	−	0.839638i	$-0.317233\pi$
$929$	33.1097	1.08629	0.543147	+	0.839638i	$0.317233\pi$
$930$	0	0
$931$	−10.5584	−0.346037
$932$	0	0
$933$	−10.8387	−0.354844
$934$	0	0
$935$	9.40017	0.307419
$936$	0	0
$937$	−0.243205	−0.00794516	−0.00397258	−	0.999992i	$-0.501265\pi$
$937$	−0.243205	−0.00794516	−0.00397258	+	0.999992i	$0.501265\pi$
$938$	0	0
$939$	−25.0715	−0.818176
$940$	0	0
$941$	37.2079	1.21294	0.606471	−	0.795106i	$-0.292585\pi$
$941$	37.2079	1.21294	0.606471	+	0.795106i	$0.292585\pi$
$942$	0	0
$943$	32.5423	1.05972
$944$	0	0
$945$	−3.71983	−0.121006
$946$	0	0
$947$	33.4046	1.08550	0.542751	−	0.839893i	$-0.317383\pi$
$947$	33.4046	1.08550	0.542751	+	0.839893i	$0.317383\pi$
$948$	0	0
$949$	3.46175	0.112373
$950$	0	0
$951$	38.9846	1.26416
$952$	0	0
$953$	−0.631378	−0.0204523	−0.0102262	−	0.999948i	$-0.503255\pi$
$953$	−0.631378	−0.0204523	−0.0102262	+	0.999948i	$0.503255\pi$
$954$	0	0
$955$	−44.8036	−1.44981
$956$	0	0
$957$	3.79554	0.122692
$958$	0	0
$959$	0.589123	0.0190238
$960$	0	0
$961$	−1.97555	−0.0637275
$962$	0	0
$963$	−42.5726	−1.37188
$964$	0	0
$965$	12.1078	0.389762
$966$	0	0
$967$	5.91823	0.190317	0.0951587	−	0.995462i	$-0.469664\pi$
$967$	5.91823	0.190317	0.0951587	+	0.995462i	$0.469664\pi$
$968$	0	0
$969$	−14.0980	−0.452893
$970$	0	0
$971$	−5.80747	−0.186371	−0.0931853	−	0.995649i	$-0.529705\pi$
$971$	−5.80747	−0.186371	−0.0931853	+	0.995649i	$0.529705\pi$
$972$	0	0
$973$	4.36722	0.140007
$974$	0	0
$975$	19.0833	0.611156
$976$	0	0
$977$	26.5313	0.848811	0.424406	−	0.905472i	$-0.360483\pi$
$977$	26.5313	0.848811	0.424406	+	0.905472i	$0.360483\pi$
$978$	0	0
$979$	4.32970	0.138378
$980$	0	0
$981$	−55.8757	−1.78397
$982$	0	0
$983$	−15.3363	−0.489152	−0.244576	−	0.969630i	$-0.578649\pi$
$983$	−15.3363	−0.489152	−0.244576	+	0.969630i	$0.578649\pi$
$984$	0	0
$985$	−3.22176	−0.102654
$986$	0	0
$987$	4.46620	0.142161
$988$	0	0
$989$	−39.3597	−1.25157
$990$	0	0
$991$	40.0441	1.27204	0.636022	−	0.771671i	$-0.280579\pi$
$991$	40.0441	1.27204	0.636022	+	0.771671i	$0.280579\pi$
$992$	0	0
$993$	−27.5551	−0.874436
$994$	0	0
$995$	−49.8036	−1.57888
$996$	0	0
$997$	1.50416	0.0476373	0.0238187	−	0.999716i	$-0.492418\pi$
$997$	1.50416	0.0476373	0.0238187	+	0.999716i	$0.492418\pi$
$998$	0	0
$999$	12.2713	0.388246

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.e.1.3	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.e.1.3	✓	27	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q-2.62082 q^{3} -2.77332 q^{5} -0.589123 q^{7} +3.86872 q^{9} +O(q^{10})\)	\(q-2.62082 q^{3} -2.77332 q^{5} -0.589123 q^{7} +3.86872 q^{9} -1.00000 q^{11} -2.70553 q^{13} +7.26839 q^{15} +3.38950 q^{17} +1.58703 q^{19} +1.54399 q^{21} -5.18220 q^{23} +2.69131 q^{25} -2.27676 q^{27} +1.44822 q^{29} -5.38743 q^{31} +2.62082 q^{33} +1.63383 q^{35} -5.38980 q^{37} +7.09072 q^{39} -6.27962 q^{41} +7.59517 q^{43} -10.7292 q^{45} +2.89264 q^{47} -6.65293 q^{49} -8.88328 q^{51} -7.94047 q^{53} +2.77332 q^{55} -4.15932 q^{57} -4.68779 q^{59} -6.22310 q^{61} -2.27915 q^{63} +7.50330 q^{65} -7.91575 q^{67} +13.5816 q^{69} +7.48402 q^{71} -1.27951 q^{73} -7.05346 q^{75} +0.589123 q^{77} -8.96508 q^{79} -5.63918 q^{81} -12.5525 q^{83} -9.40017 q^{85} -3.79554 q^{87} -4.32970 q^{89} +1.59389 q^{91} +14.1195 q^{93} -4.40133 q^{95} +10.9230 q^{97} -3.86872 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) \) 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9} - 27 q^{11} + 2 q^{13} + 10 q^{15} + 17 q^{17} + 4 q^{19} - 4 q^{21} + 25 q^{23} + 21 q^{25} + 32 q^{27} - q^{29} + 8 q^{31} - 5 q^{33} + 30 q^{35} - 24 q^{37} + 10 q^{39} + 31 q^{41} + 17 q^{43} - 15 q^{45} + 27 q^{47} + 40 q^{49} + 30 q^{51} + 2 q^{55} + 13 q^{57} + 31 q^{59} + 2 q^{61} + 61 q^{63} + 2 q^{67} - 15 q^{69} + 40 q^{71} + 17 q^{73} + 30 q^{75} - 7 q^{77} + 33 q^{79} + 31 q^{81} + 66 q^{83} + 26 q^{85} + 43 q^{87} + 22 q^{89} + 18 q^{91} - 9 q^{93} + 95 q^{95} - 21 q^{97} - 28 q^{99}+O(q^{100}) \) 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9 - 27 * q^11 + 2 * q^13 + 10 * q^15 + 17 * q^17 + 4 * q^19 - 4 * q^21 + 25 * q^23 + 21 * q^25 + 32 * q^27 - q^29 + 8 * q^31 - 5 * q^33 + 30 * q^35 - 24 * q^37 + 10 * q^39 + 31 * q^41 + 17 * q^43 - 15 * q^45 + 27 * q^47 + 40 * q^49 + 30 * q^51 + 2 * q^55 + 13 * q^57 + 31 * q^59 + 2 * q^61 + 61 * q^63 + 2 * q^67 - 15 * q^69 + 40 * q^71 + 17 * q^73 + 30 * q^75 - 7 * q^77 + 33 * q^79 + 31 * q^81 + 66 * q^83 + 26 * q^85 + 43 * q^87 + 22 * q^89 + 18 * q^91 - 9 * q^93 + 95 * q^95 - 21 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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