LMFDB - Embedded newform 6028.2.a.d.1.24

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:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.24
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)

$f(q)$	$=$	$q+2.42031 q^{3} +3.69219 q^{5} -4.84237 q^{7} +2.85788 q^{9} +O(q^{10})$	$q+2.42031 q^{3} +3.69219 q^{5} -4.84237 q^{7} +2.85788 q^{9} -1.00000 q^{11} -3.71443 q^{13} +8.93623 q^{15} -3.07979 q^{17} -2.32543 q^{19} -11.7200 q^{21} -3.41726 q^{23} +8.63227 q^{25} -0.343965 q^{27} +2.45629 q^{29} -8.99061 q^{31} -2.42031 q^{33} -17.8790 q^{35} +5.73842 q^{37} -8.99006 q^{39} -1.72339 q^{41} -8.48317 q^{43} +10.5518 q^{45} -3.91594 q^{47} +16.4486 q^{49} -7.45403 q^{51} +9.81865 q^{53} -3.69219 q^{55} -5.62826 q^{57} -5.78111 q^{59} -5.85383 q^{61} -13.8389 q^{63} -13.7144 q^{65} -11.7116 q^{67} -8.27081 q^{69} -7.66765 q^{71} -0.840484 q^{73} +20.8927 q^{75} +4.84237 q^{77} +12.1478 q^{79} -9.40615 q^{81} +10.0675 q^{83} -11.3712 q^{85} +5.94497 q^{87} +1.04737 q^{89} +17.9867 q^{91} -21.7600 q^{93} -8.58593 q^{95} +3.31954 q^{97} -2.85788 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9}+O(q^{10}) $ 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9	$ 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9} - 27 q^{11} - 6 q^{15} - 21 q^{17} - 3 q^{19} - 4 q^{21} - 44 q^{23} + 38 q^{25} - 18 q^{27} + q^{29} - 8 q^{31} + 6 q^{33} - 33 q^{35} + 11 q^{37} - 13 q^{39} - 19 q^{41} - 11 q^{43} + 17 q^{45} - 37 q^{47} + 41 q^{49} - 49 q^{51} - 12 q^{53} - q^{55} - 50 q^{57} - 14 q^{59} + 12 q^{61} - 53 q^{63} - 55 q^{65} - 5 q^{67} + 14 q^{69} - 67 q^{71} - 27 q^{73} - 70 q^{75} + 14 q^{77} - 31 q^{79} - 5 q^{81} - 55 q^{83} - 3 q^{85} - 31 q^{87} + 11 q^{89} - 11 q^{91} - 24 q^{93} - 47 q^{95} - q^{97} - 33 q^{99}+O(q^{100}) $ 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9 - 27 * q^11 - 6 * q^15 - 21 * q^17 - 3 * q^19 - 4 * q^21 - 44 * q^23 + 38 * q^25 - 18 * q^27 + q^29 - 8 * q^31 + 6 * q^33 - 33 * q^35 + 11 * q^37 - 13 * q^39 - 19 * q^41 - 11 * q^43 + 17 * q^45 - 37 * q^47 + 41 * q^49 - 49 * q^51 - 12 * q^53 - q^55 - 50 * q^57 - 14 * q^59 + 12 * q^61 - 53 * q^63 - 55 * q^65 - 5 * q^67 + 14 * q^69 - 67 * q^71 - 27 * q^73 - 70 * q^75 + 14 * q^77 - 31 * q^79 - 5 * q^81 - 55 * q^83 - 3 * q^85 - 31 * q^87 + 11 * q^89 - 11 * q^91 - 24 * q^93 - 47 * q^95 - q^97 - 33 * 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
$3$	2.42031	1.39736	0.698682	−	0.715432i	$-0.253770\pi$
$3$	2.42031	1.39736	0.698682	+	0.715432i	$0.253770\pi$
$4$	0	0
$5$	3.69219	1.65120	0.825599	−	0.564258i	$-0.190838\pi$
$5$	3.69219	1.65120	0.825599	+	0.564258i	$0.190838\pi$
$6$	0	0
$7$	−4.84237	−1.83025	−0.915123	−	0.403175i	$-0.867906\pi$
$7$	−4.84237	−1.83025	−0.915123	+	0.403175i	$0.867906\pi$
$8$	0	0
$9$	2.85788	0.952628
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.71443	−1.03020	−0.515099	−	0.857131i	$-0.672245\pi$
$13$	−3.71443	−1.03020	−0.515099	+	0.857131i	$0.672245\pi$
$14$	0	0
$15$	8.93623	2.30733
$16$	0	0
$17$	−3.07979	−0.746958	−0.373479	−	0.927639i	$-0.621835\pi$
$17$	−3.07979	−0.746958	−0.373479	+	0.927639i	$0.621835\pi$
$18$	0	0
$19$	−2.32543	−0.533491	−0.266745	−	0.963767i	$-0.585948\pi$
$19$	−2.32543	−0.533491	−0.266745	+	0.963767i	$0.585948\pi$
$20$	0	0
$21$	−11.7200	−2.55752
$22$	0	0
$23$	−3.41726	−0.712548	−0.356274	−	0.934382i	$-0.615953\pi$
$23$	−3.41726	−0.712548	−0.356274	+	0.934382i	$0.615953\pi$
$24$	0	0
$25$	8.63227	1.72645
$26$	0	0
$27$	−0.343965	−0.0661961
$28$	0	0
$29$	2.45629	0.456122	0.228061	−	0.973647i	$-0.426762\pi$
$29$	2.45629	0.456122	0.228061	+	0.973647i	$0.426762\pi$
$30$	0	0
$31$	−8.99061	−1.61476	−0.807381	−	0.590031i	$-0.799115\pi$
$31$	−8.99061	−1.61476	−0.807381	+	0.590031i	$0.799115\pi$
$32$	0	0
$33$	−2.42031	−0.421321
$34$	0	0
$35$	−17.8790	−3.02210
$36$	0	0
$37$	5.73842	0.943390	0.471695	−	0.881762i	$-0.343642\pi$
$37$	5.73842	0.943390	0.471695	+	0.881762i	$0.343642\pi$
$38$	0	0
$39$	−8.99006	−1.43956
$40$	0	0
$41$	−1.72339	−0.269148	−0.134574	−	0.990904i	$-0.542967\pi$
$41$	−1.72339	−0.269148	−0.134574	+	0.990904i	$0.542967\pi$
$42$	0	0
$43$	−8.48317	−1.29367	−0.646836	−	0.762629i	$-0.723908\pi$
$43$	−8.48317	−1.29367	−0.646836	+	0.762629i	$0.723908\pi$
$44$	0	0
$45$	10.5518	1.57298
$46$	0	0
$47$	−3.91594	−0.571199	−0.285599	−	0.958349i	$-0.592193\pi$
$47$	−3.91594	−0.571199	−0.285599	+	0.958349i	$0.592193\pi$
$48$	0	0
$49$	16.4486	2.34980
$50$	0	0
$51$	−7.45403	−1.04377
$52$	0	0
$53$	9.81865	1.34870	0.674348	−	0.738414i	$-0.264425\pi$
$53$	9.81865	1.34870	0.674348	+	0.738414i	$0.264425\pi$
$54$	0	0
$55$	−3.69219	−0.497855
$56$	0	0
$57$	−5.62826	−0.745481
$58$	0	0
$59$	−5.78111	−0.752636	−0.376318	−	0.926491i	$-0.622810\pi$
$59$	−5.78111	−0.752636	−0.376318	+	0.926491i	$0.622810\pi$
$60$	0	0
$61$	−5.85383	−0.749506	−0.374753	−	0.927125i	$-0.622273\pi$
$61$	−5.85383	−0.749506	−0.374753	+	0.927125i	$0.622273\pi$
$62$	0	0
$63$	−13.8389	−1.74354
$64$	0	0
$65$	−13.7144	−1.70106
$66$	0	0
$67$	−11.7116	−1.43079	−0.715397	−	0.698718i	$-0.753754\pi$
$67$	−11.7116	−1.43079	−0.715397	+	0.698718i	$0.753754\pi$
$68$	0	0
$69$	−8.27081	−0.995689
$70$	0	0
$71$	−7.66765	−0.909982	−0.454991	−	0.890496i	$-0.650358\pi$
$71$	−7.66765	−0.909982	−0.454991	+	0.890496i	$0.650358\pi$
$72$	0	0
$73$	−0.840484	−0.0983712	−0.0491856	−	0.998790i	$-0.515663\pi$
$73$	−0.840484	−0.0983712	−0.0491856	+	0.998790i	$0.515663\pi$
$74$	0	0
$75$	20.8927	2.41248
$76$	0	0
$77$	4.84237	0.551840
$78$	0	0
$79$	12.1478	1.36674	0.683368	−	0.730074i	$-0.260515\pi$
$79$	12.1478	1.36674	0.683368	+	0.730074i	$0.260515\pi$
$80$	0	0
$81$	−9.40615	−1.04513
$82$	0	0
$83$	10.0675	1.10505	0.552526	−	0.833496i	$-0.313664\pi$
$83$	10.0675	1.10505	0.552526	+	0.833496i	$0.313664\pi$
$84$	0	0
$85$	−11.3712	−1.23338
$86$	0	0
$87$	5.94497	0.637368
$88$	0	0
$89$	1.04737	0.111021	0.0555105	−	0.998458i	$-0.482321\pi$
$89$	1.04737	0.111021	0.0555105	+	0.998458i	$0.482321\pi$
$90$	0	0
$91$	17.9867	1.88551
$92$	0	0
$93$	−21.7600	−2.25641
$94$	0	0
$95$	−8.58593	−0.880898
$96$	0	0
$97$	3.31954	0.337048	0.168524	−	0.985698i	$-0.446100\pi$
$97$	3.31954	0.337048	0.168524	+	0.985698i	$0.446100\pi$
$98$	0	0
$99$	−2.85788	−0.287228
$100$	0	0
$101$	6.29939	0.626813	0.313406	−	0.949619i	$-0.398530\pi$
$101$	6.29939	0.626813	0.313406	+	0.949619i	$0.398530\pi$
$102$	0	0
$103$	12.6211	1.24360	0.621798	−	0.783178i	$-0.286403\pi$
$103$	12.6211	1.24360	0.621798	+	0.783178i	$0.286403\pi$
$104$	0	0
$105$	−43.2726	−4.22297
$106$	0	0
$107$	−2.07089	−0.200200	−0.100100	−	0.994977i	$-0.531916\pi$
$107$	−2.07089	−0.200200	−0.100100	+	0.994977i	$0.531916\pi$
$108$	0	0
$109$	−7.06497	−0.676702	−0.338351	−	0.941020i	$-0.609869\pi$
$109$	−7.06497	−0.676702	−0.338351	+	0.941020i	$0.609869\pi$
$110$	0	0
$111$	13.8887	1.31826
$112$	0	0
$113$	−7.99876	−0.752460	−0.376230	−	0.926526i	$-0.622780\pi$
$113$	−7.99876	−0.752460	−0.376230	+	0.926526i	$0.622780\pi$
$114$	0	0
$115$	−12.6172	−1.17656
$116$	0	0
$117$	−10.6154	−0.981395
$118$	0	0
$119$	14.9135	1.36712
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.17113	−0.376098
$124$	0	0
$125$	13.4110	1.19952
$126$	0	0
$127$	−14.8043	−1.31367	−0.656834	−	0.754035i	$-0.728105\pi$
$127$	−14.8043	−1.31367	−0.656834	+	0.754035i	$0.728105\pi$
$128$	0	0
$129$	−20.5319	−1.80773
$130$	0	0
$131$	9.98327	0.872242	0.436121	−	0.899888i	$-0.356352\pi$
$131$	9.98327	0.872242	0.436121	+	0.899888i	$0.356352\pi$
$132$	0	0
$133$	11.2606	0.976419
$134$	0	0
$135$	−1.26998	−0.109303
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	−8.78513	−0.745145	−0.372573	−	0.928003i	$-0.621524\pi$
$139$	−8.78513	−0.745145	−0.372573	+	0.928003i	$0.621524\pi$
$140$	0	0
$141$	−9.47778	−0.798173
$142$	0	0
$143$	3.71443	0.310616
$144$	0	0
$145$	9.06909	0.753147
$146$	0	0
$147$	39.8106	3.28353
$148$	0	0
$149$	1.65484	0.135569	0.0677847	−	0.997700i	$-0.478407\pi$
$149$	1.65484	0.135569	0.0677847	+	0.997700i	$0.478407\pi$
$150$	0	0
$151$	−9.64429	−0.784841	−0.392420	−	0.919786i	$-0.628362\pi$
$151$	−9.64429	−0.784841	−0.392420	+	0.919786i	$0.628362\pi$
$152$	0	0
$153$	−8.80168	−0.711574
$154$	0	0
$155$	−33.1950	−2.66629
$156$	0	0
$157$	−12.0918	−0.965032	−0.482516	−	0.875887i	$-0.660277\pi$
$157$	−12.0918	−0.965032	−0.482516	+	0.875887i	$0.660277\pi$
$158$	0	0
$159$	23.7642	1.88462
$160$	0	0
$161$	16.5476	1.30414
$162$	0	0
$163$	20.7196	1.62288	0.811442	−	0.584432i	$-0.198683\pi$
$163$	20.7196	1.62288	0.811442	+	0.584432i	$0.198683\pi$
$164$	0	0
$165$	−8.93623	−0.695685
$166$	0	0
$167$	12.0536	0.932737	0.466369	−	0.884590i	$-0.345562\pi$
$167$	12.0536	0.932737	0.466369	+	0.884590i	$0.345562\pi$
$168$	0	0
$169$	0.796993	0.0613072
$170$	0	0
$171$	−6.64581	−0.508218
$172$	0	0
$173$	0.354187	0.0269284	0.0134642	−	0.999909i	$-0.495714\pi$
$173$	0.354187	0.0269284	0.0134642	+	0.999909i	$0.495714\pi$
$174$	0	0
$175$	−41.8007	−3.15983
$176$	0	0
$177$	−13.9921	−1.05171
$178$	0	0
$179$	6.30496	0.471255	0.235628	−	0.971843i	$-0.424285\pi$
$179$	6.30496	0.471255	0.235628	+	0.971843i	$0.424285\pi$
$180$	0	0
$181$	−7.35448	−0.546654	−0.273327	−	0.961921i	$-0.588124\pi$
$181$	−7.35448	−0.546654	−0.273327	+	0.961921i	$0.588124\pi$
$182$	0	0
$183$	−14.1681	−1.04733
$184$	0	0
$185$	21.1873	1.55772
$186$	0	0
$187$	3.07979	0.225216
$188$	0	0
$189$	1.66561	0.121155
$190$	0	0
$191$	−4.62359	−0.334551	−0.167276	−	0.985910i	$-0.553497\pi$
$191$	−4.62359	−0.334551	−0.167276	+	0.985910i	$0.553497\pi$
$192$	0	0
$193$	−21.4483	−1.54388	−0.771940	−	0.635696i	$-0.780713\pi$
$193$	−21.4483	−1.54388	−0.771940	+	0.635696i	$0.780713\pi$
$194$	0	0
$195$	−33.1930	−2.37700
$196$	0	0
$197$	14.3876	1.02507	0.512537	−	0.858665i	$-0.328706\pi$
$197$	14.3876	1.02507	0.512537	+	0.858665i	$0.328706\pi$
$198$	0	0
$199$	2.93601	0.208128	0.104064	−	0.994571i	$-0.466815\pi$
$199$	2.93601	0.208128	0.104064	+	0.994571i	$0.466815\pi$
$200$	0	0
$201$	−28.3455	−1.99934
$202$	0	0
$203$	−11.8943	−0.834814
$204$	0	0
$205$	−6.36307	−0.444416
$206$	0	0
$207$	−9.76613	−0.678793
$208$	0	0
$209$	2.32543	0.160853
$210$	0	0
$211$	−17.4824	−1.20354	−0.601770	−	0.798669i	$-0.705538\pi$
$211$	−17.4824	−1.20354	−0.601770	+	0.798669i	$0.705538\pi$
$212$	0	0
$213$	−18.5581	−1.27158
$214$	0	0
$215$	−31.3215	−2.13611
$216$	0	0
$217$	43.5359	2.95541
$218$	0	0
$219$	−2.03423	−0.137460
$220$	0	0
$221$	11.4397	0.769515
$222$	0	0
$223$	−23.2037	−1.55384	−0.776918	−	0.629601i	$-0.783218\pi$
$223$	−23.2037	−1.55384	−0.776918	+	0.629601i	$0.783218\pi$
$224$	0	0
$225$	24.6700	1.64467
$226$	0	0
$227$	16.7984	1.11495	0.557474	−	0.830194i	$-0.311771\pi$
$227$	16.7984	1.11495	0.557474	+	0.830194i	$0.311771\pi$
$228$	0	0
$229$	17.0857	1.12906	0.564528	−	0.825414i	$-0.309058\pi$
$229$	17.0857	1.12906	0.564528	+	0.825414i	$0.309058\pi$
$230$	0	0
$231$	11.7200	0.771121
$232$	0	0
$233$	−25.2290	−1.65281	−0.826404	−	0.563077i	$-0.809617\pi$
$233$	−25.2290	−1.65281	−0.826404	+	0.563077i	$0.809617\pi$
$234$	0	0
$235$	−14.4584	−0.943162
$236$	0	0
$237$	29.4014	1.90983
$238$	0	0
$239$	22.6813	1.46713	0.733564	−	0.679620i	$-0.237855\pi$
$239$	22.6813	1.46713	0.733564	+	0.679620i	$0.237855\pi$
$240$	0	0
$241$	−27.0641	−1.74335	−0.871677	−	0.490080i	$-0.836967\pi$
$241$	−27.0641	−1.74335	−0.871677	+	0.490080i	$0.836967\pi$
$242$	0	0
$243$	−21.7339	−1.39423
$244$	0	0
$245$	60.7313	3.87998
$246$	0	0
$247$	8.63765	0.549601
$248$	0	0
$249$	24.3664	1.54416
$250$	0	0
$251$	−17.9522	−1.13313	−0.566565	−	0.824017i	$-0.691728\pi$
$251$	−17.9522	−1.13313	−0.566565	+	0.824017i	$0.691728\pi$
$252$	0	0
$253$	3.41726	0.214841
$254$	0	0
$255$	−27.5217	−1.72348
$256$	0	0
$257$	−10.7539	−0.670809	−0.335404	−	0.942074i	$-0.608873\pi$
$257$	−10.7539	−0.670809	−0.335404	+	0.942074i	$0.608873\pi$
$258$	0	0
$259$	−27.7876	−1.72663
$260$	0	0
$261$	7.01979	0.434514
$262$	0	0
$263$	7.51537	0.463417	0.231709	−	0.972785i	$-0.425568\pi$
$263$	7.51537	0.463417	0.231709	+	0.972785i	$0.425568\pi$
$264$	0	0
$265$	36.2523	2.22696
$266$	0	0
$267$	2.53495	0.155137
$268$	0	0
$269$	23.1507	1.41152	0.705760	−	0.708451i	$-0.250606\pi$
$269$	23.1507	1.41152	0.705760	+	0.708451i	$0.250606\pi$
$270$	0	0
$271$	11.0356	0.670366	0.335183	−	0.942153i	$-0.391202\pi$
$271$	11.0356	0.670366	0.335183	+	0.942153i	$0.391202\pi$
$272$	0	0
$273$	43.5332	2.63475
$274$	0	0
$275$	−8.63227	−0.520545
$276$	0	0
$277$	−19.8704	−1.19390	−0.596949	−	0.802279i	$-0.703621\pi$
$277$	−19.8704	−1.19390	−0.596949	+	0.802279i	$0.703621\pi$
$278$	0	0
$279$	−25.6941	−1.53827
$280$	0	0
$281$	−9.61116	−0.573354	−0.286677	−	0.958027i	$-0.592551\pi$
$281$	−9.61116	−0.573354	−0.286677	+	0.958027i	$0.592551\pi$
$282$	0	0
$283$	22.1122	1.31443	0.657216	−	0.753702i	$-0.271734\pi$
$283$	22.1122	1.31443	0.657216	+	0.753702i	$0.271734\pi$
$284$	0	0
$285$	−20.7806	−1.23094
$286$	0	0
$287$	8.34529	0.492607
$288$	0	0
$289$	−7.51490	−0.442053
$290$	0	0
$291$	8.03430	0.470979
$292$	0	0
$293$	31.6785	1.85068	0.925338	−	0.379144i	$-0.123782\pi$
$293$	31.6785	1.85068	0.925338	+	0.379144i	$0.123782\pi$
$294$	0	0
$295$	−21.3450	−1.24275
$296$	0	0
$297$	0.343965	0.0199589
$298$	0	0
$299$	12.6932	0.734065
$300$	0	0
$301$	41.0787	2.36774
$302$	0	0
$303$	15.2465	0.875886
$304$	0	0
$305$	−21.6135	−1.23758
$306$	0	0
$307$	24.8319	1.41723	0.708614	−	0.705596i	$-0.249321\pi$
$307$	24.8319	1.41723	0.708614	+	0.705596i	$0.249321\pi$
$308$	0	0
$309$	30.5470	1.73776
$310$	0	0
$311$	−10.0008	−0.567095	−0.283548	−	0.958958i	$-0.591511\pi$
$311$	−10.0008	−0.567095	−0.283548	+	0.958958i	$0.591511\pi$
$312$	0	0
$313$	19.7297	1.11519	0.557593	−	0.830114i	$-0.311725\pi$
$313$	19.7297	1.11519	0.557593	+	0.830114i	$0.311725\pi$
$314$	0	0
$315$	−51.0960	−2.87893
$316$	0	0
$317$	25.0370	1.40622	0.703110	−	0.711081i	$-0.251794\pi$
$317$	25.0370	1.40622	0.703110	+	0.711081i	$0.251794\pi$
$318$	0	0
$319$	−2.45629	−0.137526
$320$	0	0
$321$	−5.01219	−0.279753
$322$	0	0
$323$	7.16184	0.398495
$324$	0	0
$325$	−32.0640	−1.77859
$326$	0	0
$327$	−17.0994	−0.945599
$328$	0	0
$329$	18.9625	1.04543
$330$	0	0
$331$	13.6076	0.747942	0.373971	−	0.927440i	$-0.377996\pi$
$331$	13.6076	0.747942	0.373971	+	0.927440i	$0.377996\pi$
$332$	0	0
$333$	16.3997	0.898699
$334$	0	0
$335$	−43.2413	−2.36252
$336$	0	0
$337$	−21.3110	−1.16088	−0.580442	−	0.814302i	$-0.697120\pi$
$337$	−21.3110	−1.16088	−0.580442	+	0.814302i	$0.697120\pi$
$338$	0	0
$339$	−19.3595	−1.05146
$340$	0	0
$341$	8.99061	0.486869
$342$	0	0
$343$	−45.7536	−2.47046
$344$	0	0
$345$	−30.5374	−1.64408
$346$	0	0
$347$	1.56918	0.0842378	0.0421189	−	0.999113i	$-0.486589\pi$
$347$	1.56918	0.0842378	0.0421189	+	0.999113i	$0.486589\pi$
$348$	0	0
$349$	−5.87349	−0.314401	−0.157200	−	0.987567i	$-0.550247\pi$
$349$	−5.87349	−0.314401	−0.157200	+	0.987567i	$0.550247\pi$
$350$	0	0
$351$	1.27763	0.0681950
$352$	0	0
$353$	20.0883	1.06919	0.534596	−	0.845108i	$-0.320464\pi$
$353$	20.0883	1.06919	0.534596	+	0.845108i	$0.320464\pi$
$354$	0	0
$355$	−28.3104	−1.50256
$356$	0	0
$357$	36.0952	1.91036
$358$	0	0
$359$	33.0801	1.74590	0.872951	−	0.487809i	$-0.162204\pi$
$359$	33.0801	1.74590	0.872951	+	0.487809i	$0.162204\pi$
$360$	0	0
$361$	−13.5924	−0.715388
$362$	0	0
$363$	2.42031	0.127033
$364$	0	0
$365$	−3.10323	−0.162430
$366$	0	0
$367$	−1.49731	−0.0781590	−0.0390795	−	0.999236i	$-0.512443\pi$
$367$	−1.49731	−0.0781590	−0.0390795	+	0.999236i	$0.512443\pi$
$368$	0	0
$369$	−4.92524	−0.256398
$370$	0	0
$371$	−47.5456	−2.46844
$372$	0	0
$373$	−30.6078	−1.58481	−0.792405	−	0.609995i	$-0.791171\pi$
$373$	−30.6078	−1.58481	−0.792405	+	0.609995i	$0.791171\pi$
$374$	0	0
$375$	32.4588	1.67616
$376$	0	0
$377$	−9.12372	−0.469895
$378$	0	0
$379$	−26.9825	−1.38600	−0.692999	−	0.720938i	$-0.743711\pi$
$379$	−26.9825	−1.38600	−0.692999	+	0.720938i	$0.743711\pi$
$380$	0	0
$381$	−35.8309	−1.83567
$382$	0	0
$383$	−25.1895	−1.28712	−0.643561	−	0.765395i	$-0.722544\pi$
$383$	−25.1895	−1.28712	−0.643561	+	0.765395i	$0.722544\pi$
$384$	0	0
$385$	17.8790	0.911196
$386$	0	0
$387$	−24.2439	−1.23239
$388$	0	0
$389$	7.42498	0.376461	0.188231	−	0.982125i	$-0.439725\pi$
$389$	7.42498	0.376461	0.188231	+	0.982125i	$0.439725\pi$
$390$	0	0
$391$	10.5244	0.532244
$392$	0	0
$393$	24.1626	1.21884
$394$	0	0
$395$	44.8520	2.25675
$396$	0	0
$397$	−22.9023	−1.14944	−0.574718	−	0.818352i	$-0.694888\pi$
$397$	−22.9023	−1.14944	−0.574718	+	0.818352i	$0.694888\pi$
$398$	0	0
$399$	27.2541	1.36441
$400$	0	0
$401$	7.77754	0.388392	0.194196	−	0.980963i	$-0.437790\pi$
$401$	7.77754	0.388392	0.194196	+	0.980963i	$0.437790\pi$
$402$	0	0
$403$	33.3950	1.66352
$404$	0	0
$405$	−34.7293	−1.72571
$406$	0	0
$407$	−5.73842	−0.284443
$408$	0	0
$409$	27.0984	1.33993	0.669964	−	0.742393i	$-0.266309\pi$
$409$	27.0984	1.33993	0.669964	+	0.742393i	$0.266309\pi$
$410$	0	0
$411$	2.42031	0.119385
$412$	0	0
$413$	27.9943	1.37751
$414$	0	0
$415$	37.1711	1.82466
$416$	0	0
$417$	−21.2627	−1.04124
$418$	0	0
$419$	8.14413	0.397867	0.198933	−	0.980013i	$-0.436252\pi$
$419$	8.14413	0.397867	0.198933	+	0.980013i	$0.436252\pi$
$420$	0	0
$421$	−8.70466	−0.424239	−0.212120	−	0.977244i	$-0.568037\pi$
$421$	−8.70466	−0.424239	−0.212120	+	0.977244i	$0.568037\pi$
$422$	0	0
$423$	−11.1913	−0.544140
$424$	0	0
$425$	−26.5856	−1.28959
$426$	0	0
$427$	28.3464	1.37178
$428$	0	0
$429$	8.99006	0.434044
$430$	0	0
$431$	33.2517	1.60168	0.800840	−	0.598879i	$-0.204387\pi$
$431$	33.2517	1.60168	0.800840	+	0.598879i	$0.204387\pi$
$432$	0	0
$433$	−2.95413	−0.141966	−0.0709832	−	0.997478i	$-0.522614\pi$
$433$	−2.95413	−0.141966	−0.0709832	+	0.997478i	$0.522614\pi$
$434$	0	0
$435$	21.9500	1.05242
$436$	0	0
$437$	7.94660	0.380137
$438$	0	0
$439$	22.7745	1.08697	0.543483	−	0.839420i	$-0.317105\pi$
$439$	22.7745	1.08697	0.543483	+	0.839420i	$0.317105\pi$
$440$	0	0
$441$	47.0082	2.23848
$442$	0	0
$443$	26.2169	1.24560	0.622802	−	0.782379i	$-0.285994\pi$
$443$	26.2169	1.24560	0.622802	+	0.782379i	$0.285994\pi$
$444$	0	0
$445$	3.86709	0.183317
$446$	0	0
$447$	4.00521	0.189440
$448$	0	0
$449$	−25.9538	−1.22483	−0.612417	−	0.790535i	$-0.709803\pi$
$449$	−25.9538	−1.22483	−0.612417	+	0.790535i	$0.709803\pi$
$450$	0	0
$451$	1.72339	0.0811512
$452$	0	0
$453$	−23.3421	−1.09671
$454$	0	0
$455$	66.4102	3.11336
$456$	0	0
$457$	18.8087	0.879836	0.439918	−	0.898038i	$-0.355007\pi$
$457$	18.8087	0.879836	0.439918	+	0.898038i	$0.355007\pi$
$458$	0	0
$459$	1.05934	0.0494457
$460$	0	0
$461$	12.7761	0.595042	0.297521	−	0.954715i	$-0.403840\pi$
$461$	12.7761	0.595042	0.297521	+	0.954715i	$0.403840\pi$
$462$	0	0
$463$	23.4582	1.09019	0.545097	−	0.838373i	$-0.316493\pi$
$463$	23.4582	1.09019	0.545097	+	0.838373i	$0.316493\pi$
$464$	0	0
$465$	−80.3422	−3.72578
$466$	0	0
$467$	−17.8308	−0.825113	−0.412557	−	0.910932i	$-0.635364\pi$
$467$	−17.8308	−0.825113	−0.412557	+	0.910932i	$0.635364\pi$
$468$	0	0
$469$	56.7117	2.61870
$470$	0	0
$471$	−29.2659	−1.34850
$472$	0	0
$473$	8.48317	0.390057
$474$	0	0
$475$	−20.0737	−0.921047
$476$	0	0
$477$	28.0606	1.28481
$478$	0	0
$479$	24.0319	1.09805	0.549024	−	0.835807i	$-0.315000\pi$
$479$	24.0319	1.09805	0.549024	+	0.835807i	$0.315000\pi$
$480$	0	0
$481$	−21.3149	−0.971878
$482$	0	0
$483$	40.0504	1.82236
$484$	0	0
$485$	12.2564	0.556533
$486$	0	0
$487$	−4.21570	−0.191031	−0.0955157	−	0.995428i	$-0.530450\pi$
$487$	−4.21570	−0.191031	−0.0955157	+	0.995428i	$0.530450\pi$
$488$	0	0
$489$	50.1478	2.26776
$490$	0	0
$491$	19.1141	0.862606	0.431303	−	0.902207i	$-0.358054\pi$
$491$	19.1141	0.862606	0.431303	+	0.902207i	$0.358054\pi$
$492$	0	0
$493$	−7.56485	−0.340704
$494$	0	0
$495$	−10.5518	−0.474270
$496$	0	0
$497$	37.1296	1.66549
$498$	0	0
$499$	−26.1308	−1.16977	−0.584887	−	0.811115i	$-0.698861\pi$
$499$	−26.1308	−1.16977	−0.584887	+	0.811115i	$0.698861\pi$
$500$	0	0
$501$	29.1735	1.30337
$502$	0	0
$503$	−13.7040	−0.611031	−0.305515	−	0.952187i	$-0.598829\pi$
$503$	−13.7040	−0.611031	−0.305515	+	0.952187i	$0.598829\pi$
$504$	0	0
$505$	23.2585	1.03499
$506$	0	0
$507$	1.92897	0.0856685
$508$	0	0
$509$	19.0918	0.846227	0.423114	−	0.906077i	$-0.360937\pi$
$509$	19.0918	0.846227	0.423114	+	0.906077i	$0.360937\pi$
$510$	0	0
$511$	4.06994	0.180043
$512$	0	0
$513$	0.799867	0.0353150
$514$	0	0
$515$	46.5995	2.05342
$516$	0	0
$517$	3.91594	0.172223
$518$	0	0
$519$	0.857242	0.0376287
$520$	0	0
$521$	38.3446	1.67991	0.839953	−	0.542660i	$-0.182583\pi$
$521$	38.3446	1.67991	0.839953	+	0.542660i	$0.182583\pi$
$522$	0	0
$523$	−44.2993	−1.93707	−0.968536	−	0.248875i	$-0.919939\pi$
$523$	−44.2993	−1.93707	−0.968536	+	0.248875i	$0.919939\pi$
$524$	0	0
$525$	−101.170	−4.41544
$526$	0	0
$527$	27.6892	1.20616
$528$	0	0
$529$	−11.3223	−0.492276
$530$	0	0
$531$	−16.5217	−0.716982
$532$	0	0
$533$	6.40140	0.277276
$534$	0	0
$535$	−7.64611	−0.330570
$536$	0	0
$537$	15.2599	0.658515
$538$	0	0
$539$	−16.4486	−0.708491
$540$	0	0
$541$	−1.47486	−0.0634093	−0.0317047	−	0.999497i	$-0.510094\pi$
$541$	−1.47486	−0.0634093	−0.0317047	+	0.999497i	$0.510094\pi$
$542$	0	0
$543$	−17.8001	−0.763875
$544$	0	0
$545$	−26.0852	−1.11737
$546$	0	0
$547$	−5.50577	−0.235410	−0.117705	−	0.993049i	$-0.537554\pi$
$547$	−5.50577	−0.235410	−0.117705	+	0.993049i	$0.537554\pi$
$548$	0	0
$549$	−16.7296	−0.714001
$550$	0	0
$551$	−5.71193	−0.243337
$552$	0	0
$553$	−58.8242	−2.50146
$554$	0	0
$555$	51.2798	2.17671
$556$	0	0
$557$	−26.7539	−1.13360	−0.566800	−	0.823856i	$-0.691819\pi$
$557$	−26.7539	−1.13360	−0.566800	+	0.823856i	$0.691819\pi$
$558$	0	0
$559$	31.5102	1.33274
$560$	0	0
$561$	7.45403	0.314710
$562$	0	0
$563$	−29.6759	−1.25069	−0.625345	−	0.780348i	$-0.715042\pi$
$563$	−29.6759	−1.25069	−0.625345	+	0.780348i	$0.715042\pi$
$564$	0	0
$565$	−29.5330	−1.24246
$566$	0	0
$567$	45.5481	1.91284
$568$	0	0
$569$	−22.5794	−0.946577	−0.473289	−	0.880907i	$-0.656933\pi$
$569$	−22.5794	−0.946577	−0.473289	+	0.880907i	$0.656933\pi$
$570$	0	0
$571$	−6.58017	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$571$	−6.58017	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$572$	0	0
$573$	−11.1905	−0.467490
$574$	0	0
$575$	−29.4987	−1.23018
$576$	0	0
$577$	24.1732	1.00634	0.503172	−	0.864186i	$-0.332166\pi$
$577$	24.1732	1.00634	0.503172	+	0.864186i	$0.332166\pi$
$578$	0	0
$579$	−51.9114	−2.15736
$580$	0	0
$581$	−48.7506	−2.02252
$582$	0	0
$583$	−9.81865	−0.406647
$584$	0	0
$585$	−39.1941	−1.62048
$586$	0	0
$587$	−24.6960	−1.01931	−0.509657	−	0.860378i	$-0.670228\pi$
$587$	−24.6960	−1.01931	−0.509657	+	0.860378i	$0.670228\pi$
$588$	0	0
$589$	20.9070	0.861460
$590$	0	0
$591$	34.8224	1.43240
$592$	0	0
$593$	−18.3911	−0.755231	−0.377615	−	0.925963i	$-0.623256\pi$
$593$	−18.3911	−0.755231	−0.377615	+	0.925963i	$0.623256\pi$
$594$	0	0
$595$	55.0634	2.25738
$596$	0	0
$597$	7.10604	0.290831
$598$	0	0
$599$	9.32988	0.381208	0.190604	−	0.981667i	$-0.438955\pi$
$599$	9.32988	0.381208	0.190604	+	0.981667i	$0.438955\pi$
$600$	0	0
$601$	−12.5048	−0.510081	−0.255040	−	0.966930i	$-0.582089\pi$
$601$	−12.5048	−0.510081	−0.255040	+	0.966930i	$0.582089\pi$
$602$	0	0
$603$	−33.4703	−1.36301
$604$	0	0
$605$	3.69219	0.150109
$606$	0	0
$607$	−21.2512	−0.862559	−0.431279	−	0.902218i	$-0.641938\pi$
$607$	−21.2512	−0.862559	−0.431279	+	0.902218i	$0.641938\pi$
$608$	0	0
$609$	−28.7878	−1.16654
$610$	0	0
$611$	14.5455	0.588448
$612$	0	0
$613$	−5.26406	−0.212613	−0.106307	−	0.994333i	$-0.533903\pi$
$613$	−5.26406	−0.212613	−0.106307	+	0.994333i	$0.533903\pi$
$614$	0	0
$615$	−15.4006	−0.621012
$616$	0	0
$617$	25.6826	1.03394	0.516972	−	0.856002i	$-0.327059\pi$
$617$	25.6826	1.03394	0.516972	+	0.856002i	$0.327059\pi$
$618$	0	0
$619$	−40.1200	−1.61256	−0.806280	−	0.591535i	$-0.798522\pi$
$619$	−40.1200	−1.61256	−0.806280	+	0.591535i	$0.798522\pi$
$620$	0	0
$621$	1.17542	0.0471678
$622$	0	0
$623$	−5.07175	−0.203196
$624$	0	0
$625$	6.35470	0.254188
$626$	0	0
$627$	5.62826	0.224771
$628$	0	0
$629$	−17.6731	−0.704673
$630$	0	0
$631$	9.49021	0.377799	0.188900	−	0.981996i	$-0.439508\pi$
$631$	9.49021	0.377799	0.188900	+	0.981996i	$0.439508\pi$
$632$	0	0
$633$	−42.3128	−1.68178
$634$	0	0
$635$	−54.6602	−2.16912
$636$	0	0
$637$	−61.0971	−2.42076
$638$	0	0
$639$	−21.9132	−0.866874
$640$	0	0
$641$	2.53002	0.0999299	0.0499650	−	0.998751i	$-0.484089\pi$
$641$	2.53002	0.0999299	0.0499650	+	0.998751i	$0.484089\pi$
$642$	0	0
$643$	10.5267	0.415134	0.207567	−	0.978221i	$-0.433446\pi$
$643$	10.5267	0.415134	0.207567	+	0.978221i	$0.433446\pi$
$644$	0	0
$645$	−75.8076	−2.98492
$646$	0	0
$647$	−29.3130	−1.15241	−0.576206	−	0.817305i	$-0.695467\pi$
$647$	−29.3130	−1.15241	−0.576206	+	0.817305i	$0.695467\pi$
$648$	0	0
$649$	5.78111	0.226928
$650$	0	0
$651$	105.370	4.12978
$652$	0	0
$653$	39.9132	1.56192	0.780962	−	0.624579i	$-0.214729\pi$
$653$	39.9132	1.56192	0.780962	+	0.624579i	$0.214729\pi$
$654$	0	0
$655$	36.8601	1.44024
$656$	0	0
$657$	−2.40201	−0.0937112
$658$	0	0
$659$	−8.40244	−0.327313	−0.163656	−	0.986517i	$-0.552329\pi$
$659$	−8.40244	−0.327313	−0.163656	+	0.986517i	$0.552329\pi$
$660$	0	0
$661$	3.16414	0.123071	0.0615353	−	0.998105i	$-0.480400\pi$
$661$	3.16414	0.123071	0.0615353	+	0.998105i	$0.480400\pi$
$662$	0	0
$663$	27.6875	1.07529
$664$	0	0
$665$	41.5763	1.61226
$666$	0	0
$667$	−8.39378	−0.325008
$668$	0	0
$669$	−56.1601	−2.17128
$670$	0	0
$671$	5.85383	0.225985
$672$	0	0
$673$	−20.6676	−0.796678	−0.398339	−	0.917238i	$-0.630413\pi$
$673$	−20.6676	−0.796678	−0.398339	+	0.917238i	$0.630413\pi$
$674$	0	0
$675$	−2.96920	−0.114284
$676$	0	0
$677$	−26.8668	−1.03257	−0.516287	−	0.856416i	$-0.672686\pi$
$677$	−26.8668	−1.03257	−0.516287	+	0.856416i	$0.672686\pi$
$678$	0	0
$679$	−16.0744	−0.616881
$680$	0	0
$681$	40.6572	1.55799
$682$	0	0
$683$	−10.5003	−0.401783	−0.200892	−	0.979613i	$-0.564384\pi$
$683$	−10.5003	−0.401783	−0.200892	+	0.979613i	$0.564384\pi$
$684$	0	0
$685$	3.69219	0.141071
$686$	0	0
$687$	41.3527	1.57770
$688$	0	0
$689$	−36.4707	−1.38942
$690$	0	0
$691$	46.1120	1.75418	0.877091	−	0.480325i	$-0.159481\pi$
$691$	46.1120	1.75418	0.877091	+	0.480325i	$0.159481\pi$
$692$	0	0
$693$	13.8389	0.525698
$694$	0	0
$695$	−32.4364	−1.23038
$696$	0	0
$697$	5.30767	0.201042
$698$	0	0
$699$	−61.0620	−2.30958
$700$	0	0
$701$	5.82314	0.219937	0.109969	−	0.993935i	$-0.464925\pi$
$701$	5.82314	0.219937	0.109969	+	0.993935i	$0.464925\pi$
$702$	0	0
$703$	−13.3443	−0.503290
$704$	0	0
$705$	−34.9938	−1.31794
$706$	0	0
$707$	−30.5040	−1.14722
$708$	0	0
$709$	−7.41342	−0.278417	−0.139208	−	0.990263i	$-0.544456\pi$
$709$	−7.41342	−0.278417	−0.139208	+	0.990263i	$0.544456\pi$
$710$	0	0
$711$	34.7170	1.30199
$712$	0	0
$713$	30.7232	1.15059
$714$	0	0
$715$	13.7144	0.512889
$716$	0	0
$717$	54.8956	2.05011
$718$	0	0
$719$	−51.2767	−1.91230	−0.956150	−	0.292878i	$-0.905387\pi$
$719$	−51.2767	−1.91230	−0.956150	+	0.292878i	$0.905387\pi$
$720$	0	0
$721$	−61.1161	−2.27608
$722$	0	0
$723$	−65.5035	−2.43610
$724$	0	0
$725$	21.2033	0.787472
$726$	0	0
$727$	5.23050	0.193989	0.0969943	−	0.995285i	$-0.469077\pi$
$727$	5.23050	0.193989	0.0969943	+	0.995285i	$0.469077\pi$
$728$	0	0
$729$	−24.3842	−0.903118
$730$	0	0
$731$	26.1264	0.966319
$732$	0	0
$733$	11.2005	0.413700	0.206850	−	0.978373i	$-0.433679\pi$
$733$	11.2005	0.413700	0.206850	+	0.978373i	$0.433679\pi$
$734$	0	0
$735$	146.988	5.42175
$736$	0	0
$737$	11.7116	0.431401
$738$	0	0
$739$	−44.1930	−1.62566	−0.812832	−	0.582498i	$-0.802076\pi$
$739$	−44.1930	−1.62566	−0.812832	+	0.582498i	$0.802076\pi$
$740$	0	0
$741$	20.9058	0.767993
$742$	0	0
$743$	7.18667	0.263653	0.131827	−	0.991273i	$-0.457916\pi$
$743$	7.18667	0.263653	0.131827	+	0.991273i	$0.457916\pi$
$744$	0	0
$745$	6.10997	0.223852
$746$	0	0
$747$	28.7717	1.05270
$748$	0	0
$749$	10.0280	0.366416
$750$	0	0
$751$	21.4265	0.781863	0.390931	−	0.920420i	$-0.372153\pi$
$751$	21.4265	0.781863	0.390931	+	0.920420i	$0.372153\pi$
$752$	0	0
$753$	−43.4497	−1.58340
$754$	0	0
$755$	−35.6085	−1.29593
$756$	0	0
$757$	−33.8985	−1.23206	−0.616031	−	0.787722i	$-0.711261\pi$
$757$	−33.8985	−1.23206	−0.616031	+	0.787722i	$0.711261\pi$
$758$	0	0
$759$	8.27081	0.300212
$760$	0	0
$761$	20.2683	0.734726	0.367363	−	0.930078i	$-0.380261\pi$
$761$	20.2683	0.734726	0.367363	+	0.930078i	$0.380261\pi$
$762$	0	0
$763$	34.2113	1.23853
$764$	0	0
$765$	−32.4975	−1.17495
$766$	0	0
$767$	21.4735	0.775364
$768$	0	0
$769$	19.5174	0.703814	0.351907	−	0.936035i	$-0.385533\pi$
$769$	19.5174	0.703814	0.351907	+	0.936035i	$0.385533\pi$
$770$	0	0
$771$	−26.0277	−0.937364
$772$	0	0
$773$	−41.5928	−1.49599	−0.747995	−	0.663705i	$-0.768983\pi$
$773$	−41.5928	−1.49599	−0.747995	+	0.663705i	$0.768983\pi$
$774$	0	0
$775$	−77.6093	−2.78781
$776$	0	0
$777$	−67.2544	−2.41274
$778$	0	0
$779$	4.00762	0.143588
$780$	0	0
$781$	7.66765	0.274370
$782$	0	0
$783$	−0.844877	−0.0301934
$784$	0	0
$785$	−44.6453	−1.59346
$786$	0	0
$787$	0.543665	0.0193795	0.00968977	−	0.999953i	$-0.496916\pi$
$787$	0.543665	0.0193795	0.00968977	+	0.999953i	$0.496916\pi$
$788$	0	0
$789$	18.1895	0.647563
$790$	0	0
$791$	38.7330	1.37719
$792$	0	0
$793$	21.7437	0.772140
$794$	0	0
$795$	87.7418	3.11188
$796$	0	0
$797$	−29.3913	−1.04109	−0.520546	−	0.853833i	$-0.674272\pi$
$797$	−29.3913	−1.04109	−0.520546	+	0.853833i	$0.674272\pi$
$798$	0	0
$799$	12.0603	0.426662
$800$	0	0
$801$	2.99326	0.105762
$802$	0	0
$803$	0.840484	0.0296600
$804$	0	0
$805$	61.0971	2.15339
$806$	0	0
$807$	56.0317	1.97241
$808$	0	0
$809$	−15.1780	−0.533629	−0.266815	−	0.963748i	$-0.585971\pi$
$809$	−15.1780	−0.533629	−0.266815	+	0.963748i	$0.585971\pi$
$810$	0	0
$811$	−40.9989	−1.43967	−0.719834	−	0.694146i	$-0.755782\pi$
$811$	−40.9989	−1.43967	−0.719834	+	0.694146i	$0.755782\pi$
$812$	0	0
$813$	26.7096	0.936746
$814$	0	0
$815$	76.5007	2.67970
$816$	0	0
$817$	19.7270	0.690162
$818$	0	0
$819$	51.4038	1.79619
$820$	0	0
$821$	33.2894	1.16181	0.580904	−	0.813972i	$-0.302699\pi$
$821$	33.2894	1.16181	0.580904	+	0.813972i	$0.302699\pi$
$822$	0	0
$823$	−34.5983	−1.20602	−0.603010	−	0.797733i	$-0.706032\pi$
$823$	−34.5983	−1.20602	−0.603010	+	0.797733i	$0.706032\pi$
$824$	0	0
$825$	−20.8927	−0.727392
$826$	0	0
$827$	−20.4347	−0.710583	−0.355291	−	0.934756i	$-0.615618\pi$
$827$	−20.4347	−0.710583	−0.355291	+	0.934756i	$0.615618\pi$
$828$	0	0
$829$	−53.7948	−1.86837	−0.934185	−	0.356790i	$-0.883871\pi$
$829$	−53.7948	−1.86837	−0.934185	+	0.356790i	$0.883871\pi$
$830$	0	0
$831$	−48.0925	−1.66831
$832$	0	0
$833$	−50.6582	−1.75520
$834$	0	0
$835$	44.5043	1.54013
$836$	0	0
$837$	3.09245	0.106891
$838$	0	0
$839$	31.7442	1.09593	0.547965	−	0.836501i	$-0.315403\pi$
$839$	31.7442	1.09593	0.547965	+	0.836501i	$0.315403\pi$
$840$	0	0
$841$	−22.9666	−0.791953
$842$	0	0
$843$	−23.2620	−0.801184
$844$	0	0
$845$	2.94265	0.101230
$846$	0	0
$847$	−4.84237	−0.166386
$848$	0	0
$849$	53.5182	1.83674
$850$	0	0
$851$	−19.6097	−0.672210
$852$	0	0
$853$	−48.1288	−1.64790	−0.823950	−	0.566663i	$-0.808234\pi$
$853$	−48.1288	−1.64790	−0.823950	+	0.566663i	$0.808234\pi$
$854$	0	0
$855$	−24.5376	−0.839168
$856$	0	0
$857$	6.86107	0.234370	0.117185	−	0.993110i	$-0.462613\pi$
$857$	6.86107	0.234370	0.117185	+	0.993110i	$0.462613\pi$
$858$	0	0
$859$	−9.85956	−0.336404	−0.168202	−	0.985753i	$-0.553796\pi$
$859$	−9.85956	−0.336404	−0.168202	+	0.985753i	$0.553796\pi$
$860$	0	0
$861$	20.1982	0.688351
$862$	0	0
$863$	−34.0038	−1.15750	−0.578752	−	0.815504i	$-0.696460\pi$
$863$	−34.0038	−1.15750	−0.578752	+	0.815504i	$0.696460\pi$
$864$	0	0
$865$	1.30773	0.0444641
$866$	0	0
$867$	−18.1884	−0.617709
$868$	0	0
$869$	−12.1478	−0.412086
$870$	0	0
$871$	43.5018	1.47400
$872$	0	0
$873$	9.48685	0.321081
$874$	0	0
$875$	−64.9412	−2.19541
$876$	0	0
$877$	41.3754	1.39715	0.698574	−	0.715538i	$-0.253818\pi$
$877$	41.3754	1.39715	0.698574	+	0.715538i	$0.253818\pi$
$878$	0	0
$879$	76.6716	2.58607
$880$	0	0
$881$	−19.0676	−0.642403	−0.321202	−	0.947011i	$-0.604087\pi$
$881$	−19.0676	−0.642403	−0.321202	+	0.947011i	$0.604087\pi$
$882$	0	0
$883$	9.50375	0.319827	0.159913	−	0.987131i	$-0.448879\pi$
$883$	9.50375	0.319827	0.159913	+	0.987131i	$0.448879\pi$
$884$	0	0
$885$	−51.6613	−1.73658
$886$	0	0
$887$	−47.2992	−1.58815	−0.794076	−	0.607818i	$-0.792045\pi$
$887$	−47.2992	−1.58815	−0.794076	+	0.607818i	$0.792045\pi$
$888$	0	0
$889$	71.6879	2.40433
$890$	0	0
$891$	9.40615	0.315118
$892$	0	0
$893$	9.10625	0.304729
$894$	0	0
$895$	23.2791	0.778135
$896$	0	0
$897$	30.7214	1.02576
$898$	0	0
$899$	−22.0835	−0.736527
$900$	0	0
$901$	−30.2394	−1.00742
$902$	0	0
$903$	99.4230	3.30859
$904$	0	0
$905$	−27.1541	−0.902634
$906$	0	0
$907$	−41.4673	−1.37690	−0.688449	−	0.725284i	$-0.741708\pi$
$907$	−41.4673	−1.37690	−0.688449	+	0.725284i	$0.741708\pi$
$908$	0	0
$909$	18.0029	0.597119
$910$	0	0
$911$	−21.6451	−0.717133	−0.358566	−	0.933504i	$-0.616734\pi$
$911$	−21.6451	−0.717133	−0.358566	+	0.933504i	$0.616734\pi$
$912$	0	0
$913$	−10.0675	−0.333185
$914$	0	0
$915$	−52.3112	−1.72936
$916$	0	0
$917$	−48.3427	−1.59642
$918$	0	0
$919$	−5.50890	−0.181722	−0.0908608	−	0.995864i	$-0.528962\pi$
$919$	−5.50890	−0.181722	−0.0908608	+	0.995864i	$0.528962\pi$
$920$	0	0
$921$	60.1007	1.98039
$922$	0	0
$923$	28.4809	0.937462
$924$	0	0
$925$	49.5355	1.62872
$926$	0	0
$927$	36.0697	1.18468
$928$	0	0
$929$	23.7646	0.779691	0.389845	−	0.920880i	$-0.372528\pi$
$929$	23.7646	0.779691	0.389845	+	0.920880i	$0.372528\pi$
$930$	0	0
$931$	−38.2501	−1.25360
$932$	0	0
$933$	−24.2051	−0.792439
$934$	0	0
$935$	11.3712	0.371877
$936$	0	0
$937$	10.4587	0.341671	0.170836	−	0.985300i	$-0.445353\pi$
$937$	10.4587	0.341671	0.170836	+	0.985300i	$0.445353\pi$
$938$	0	0
$939$	47.7518	1.55832
$940$	0	0
$941$	10.5862	0.345101	0.172550	−	0.985001i	$-0.444799\pi$
$941$	10.5862	0.345101	0.172550	+	0.985001i	$0.444799\pi$
$942$	0	0
$943$	5.88926	0.191781
$944$	0	0
$945$	6.14973	0.200051
$946$	0	0
$947$	13.1429	0.427087	0.213544	−	0.976934i	$-0.431499\pi$
$947$	13.1429	0.427087	0.213544	+	0.976934i	$0.431499\pi$
$948$	0	0
$949$	3.12192	0.101342
$950$	0	0
$951$	60.5973	1.96500
$952$	0	0
$953$	5.10082	0.165232	0.0826160	−	0.996581i	$-0.473673\pi$
$953$	5.10082	0.165232	0.0826160	+	0.996581i	$0.473673\pi$
$954$	0	0
$955$	−17.0712	−0.552411
$956$	0	0
$957$	−5.94497	−0.192174
$958$	0	0
$959$	−4.84237	−0.156368
$960$	0	0
$961$	49.8311	1.60745
$962$	0	0
$963$	−5.91836	−0.190716
$964$	0	0
$965$	−79.1911	−2.54925
$966$	0	0
$967$	24.5959	0.790952	0.395476	−	0.918476i	$-0.370580\pi$
$967$	24.5959	0.790952	0.395476	+	0.918476i	$0.370580\pi$
$968$	0	0
$969$	17.3338	0.556843
$970$	0	0
$971$	60.1603	1.93064	0.965318	−	0.261077i	$-0.0840776\pi$
$971$	60.1603	1.93064	0.965318	+	0.261077i	$0.0840776\pi$
$972$	0	0
$973$	42.5409	1.36380
$974$	0	0
$975$	−77.6046	−2.48534
$976$	0	0
$977$	25.6318	0.820034	0.410017	−	0.912078i	$-0.365523\pi$
$977$	25.6318	0.820034	0.410017	+	0.912078i	$0.365523\pi$
$978$	0	0
$979$	−1.04737	−0.0334741
$980$	0	0
$981$	−20.1909	−0.644645
$982$	0	0
$983$	−28.3756	−0.905042	−0.452521	−	0.891754i	$-0.649475\pi$
$983$	−28.3756	−0.905042	−0.452521	+	0.891754i	$0.649475\pi$
$984$	0	0
$985$	53.1218	1.69260
$986$	0	0
$987$	45.8949	1.46085
$988$	0	0
$989$	28.9892	0.921803
$990$	0	0
$991$	55.8094	1.77285	0.886423	−	0.462877i	$-0.153183\pi$
$991$	55.8094	1.77285	0.886423	+	0.462877i	$0.153183\pi$
$992$	0	0
$993$	32.9346	1.04515
$994$	0	0
$995$	10.8403	0.343661
$996$	0	0
$997$	28.6889	0.908586	0.454293	−	0.890852i	$-0.349892\pi$
$997$	28.6889	0.908586	0.454293	+	0.890852i	$0.349892\pi$
$998$	0	0
$999$	−1.97381	−0.0624487

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.d.1.24	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.d.1.24	✓	27	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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q+2.42031 q^{3} +3.69219 q^{5} -4.84237 q^{7} +2.85788 q^{9} +O(q^{10})\)	\(q+2.42031 q^{3} +3.69219 q^{5} -4.84237 q^{7} +2.85788 q^{9} -1.00000 q^{11} -3.71443 q^{13} +8.93623 q^{15} -3.07979 q^{17} -2.32543 q^{19} -11.7200 q^{21} -3.41726 q^{23} +8.63227 q^{25} -0.343965 q^{27} +2.45629 q^{29} -8.99061 q^{31} -2.42031 q^{33} -17.8790 q^{35} +5.73842 q^{37} -8.99006 q^{39} -1.72339 q^{41} -8.48317 q^{43} +10.5518 q^{45} -3.91594 q^{47} +16.4486 q^{49} -7.45403 q^{51} +9.81865 q^{53} -3.69219 q^{55} -5.62826 q^{57} -5.78111 q^{59} -5.85383 q^{61} -13.8389 q^{63} -13.7144 q^{65} -11.7116 q^{67} -8.27081 q^{69} -7.66765 q^{71} -0.840484 q^{73} +20.8927 q^{75} +4.84237 q^{77} +12.1478 q^{79} -9.40615 q^{81} +10.0675 q^{83} -11.3712 q^{85} +5.94497 q^{87} +1.04737 q^{89} +17.9867 q^{91} -21.7600 q^{93} -8.58593 q^{95} +3.31954 q^{97} -2.85788 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9}+O(q^{10}) \) 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9	\( 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9} - 27 q^{11} - 6 q^{15} - 21 q^{17} - 3 q^{19} - 4 q^{21} - 44 q^{23} + 38 q^{25} - 18 q^{27} + q^{29} - 8 q^{31} + 6 q^{33} - 33 q^{35} + 11 q^{37} - 13 q^{39} - 19 q^{41} - 11 q^{43} + 17 q^{45} - 37 q^{47} + 41 q^{49} - 49 q^{51} - 12 q^{53} - q^{55} - 50 q^{57} - 14 q^{59} + 12 q^{61} - 53 q^{63} - 55 q^{65} - 5 q^{67} + 14 q^{69} - 67 q^{71} - 27 q^{73} - 70 q^{75} + 14 q^{77} - 31 q^{79} - 5 q^{81} - 55 q^{83} - 3 q^{85} - 31 q^{87} + 11 q^{89} - 11 q^{91} - 24 q^{93} - 47 q^{95} - q^{97} - 33 q^{99}+O(q^{100}) \) 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9 - 27 * q^11 - 6 * q^15 - 21 * q^17 - 3 * q^19 - 4 * q^21 - 44 * q^23 + 38 * q^25 - 18 * q^27 + q^29 - 8 * q^31 + 6 * q^33 - 33 * q^35 + 11 * q^37 - 13 * q^39 - 19 * q^41 - 11 * q^43 + 17 * q^45 - 37 * q^47 + 41 * q^49 - 49 * q^51 - 12 * q^53 - q^55 - 50 * q^57 - 14 * q^59 + 12 * q^61 - 53 * q^63 - 55 * q^65 - 5 * q^67 + 14 * q^69 - 67 * q^71 - 27 * q^73 - 70 * q^75 + 14 * q^77 - 31 * q^79 - 5 * q^81 - 55 * q^83 - 3 * q^85 - 31 * q^87 + 11 * q^89 - 11 * q^91 - 24 * q^93 - 47 * q^95 - q^97 - 33 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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