LMFDB - Embedded newform 6028.2.a.f.1.28

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:	$29$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.28
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)

$f(q)$	$=$	$q+3.34495 q^{3} -1.55264 q^{5} -5.16237 q^{7} +8.18871 q^{9} +O(q^{10})$	$q+3.34495 q^{3} -1.55264 q^{5} -5.16237 q^{7} +8.18871 q^{9} +1.00000 q^{11} -5.65191 q^{13} -5.19350 q^{15} +1.58279 q^{17} +0.0288425 q^{19} -17.2679 q^{21} +7.85691 q^{23} -2.58931 q^{25} +17.3560 q^{27} +2.65626 q^{29} +8.68610 q^{31} +3.34495 q^{33} +8.01530 q^{35} +9.07749 q^{37} -18.9054 q^{39} -1.21842 q^{41} -4.91964 q^{43} -12.7141 q^{45} -6.66585 q^{47} +19.6501 q^{49} +5.29437 q^{51} +0.215830 q^{53} -1.55264 q^{55} +0.0964767 q^{57} +0.410309 q^{59} +0.112356 q^{61} -42.2732 q^{63} +8.77537 q^{65} +10.7794 q^{67} +26.2810 q^{69} -14.8717 q^{71} +11.4343 q^{73} -8.66114 q^{75} -5.16237 q^{77} -10.2845 q^{79} +33.4889 q^{81} +10.3772 q^{83} -2.45751 q^{85} +8.88506 q^{87} -7.13668 q^{89} +29.1773 q^{91} +29.0546 q^{93} -0.0447819 q^{95} -3.42963 q^{97} +8.18871 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.34495	1.93121	0.965605	−	0.260014i	$-0.0837273\pi$
$3$	3.34495	1.93121	0.965605	+	0.260014i	$0.0837273\pi$
$4$	0	0
$5$	−1.55264	−0.694361	−0.347180	−	0.937798i	$-0.612861\pi$
$5$	−1.55264	−0.694361	−0.347180	+	0.937798i	$0.612861\pi$
$6$	0	0
$7$	−5.16237	−1.95119	−0.975597	−	0.219569i	$-0.929535\pi$
$7$	−5.16237	−1.95119	−0.975597	+	0.219569i	$0.929535\pi$
$8$	0	0
$9$	8.18871	2.72957
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.65191	−1.56756	−0.783779	−	0.621040i	$-0.786711\pi$
$13$	−5.65191	−1.56756	−0.783779	+	0.621040i	$0.786711\pi$
$14$	0	0
$15$	−5.19350	−1.34096
$16$	0	0
$17$	1.58279	0.383884	0.191942	−	0.981406i	$-0.438521\pi$
$17$	1.58279	0.383884	0.191942	+	0.981406i	$0.438521\pi$
$18$	0	0
$19$	0.0288425	0.00661692	0.00330846	−	0.999995i	$-0.498947\pi$
$19$	0.0288425	0.00661692	0.00330846	+	0.999995i	$0.498947\pi$
$20$	0	0
$21$	−17.2679	−3.76816
$22$	0	0
$23$	7.85691	1.63828	0.819140	−	0.573594i	$-0.194451\pi$
$23$	7.85691	1.63828	0.819140	+	0.573594i	$0.194451\pi$
$24$	0	0
$25$	−2.58931	−0.517863
$26$	0	0
$27$	17.3560	3.34016
$28$	0	0
$29$	2.65626	0.493255	0.246627	−	0.969110i	$-0.420678\pi$
$29$	2.65626	0.493255	0.246627	+	0.969110i	$0.420678\pi$
$30$	0	0
$31$	8.68610	1.56007	0.780035	−	0.625736i	$-0.215201\pi$
$31$	8.68610	1.56007	0.780035	+	0.625736i	$0.215201\pi$
$32$	0	0
$33$	3.34495	0.582282
$34$	0	0
$35$	8.01530	1.35483
$36$	0	0
$37$	9.07749	1.49233	0.746165	−	0.665761i	$-0.231893\pi$
$37$	9.07749	1.49233	0.746165	+	0.665761i	$0.231893\pi$
$38$	0	0
$39$	−18.9054	−3.02728
$40$	0	0
$41$	−1.21842	−0.190285	−0.0951423	−	0.995464i	$-0.530331\pi$
$41$	−1.21842	−0.190285	−0.0951423	+	0.995464i	$0.530331\pi$
$42$	0	0
$43$	−4.91964	−0.750238	−0.375119	−	0.926977i	$-0.622398\pi$
$43$	−4.91964	−0.750238	−0.375119	+	0.926977i	$0.622398\pi$
$44$	0	0
$45$	−12.7141	−1.89531
$46$	0	0
$47$	−6.66585	−0.972314	−0.486157	−	0.873872i	$-0.661602\pi$
$47$	−6.66585	−0.972314	−0.486157	+	0.873872i	$0.661602\pi$
$48$	0	0
$49$	19.6501	2.80716
$50$	0	0
$51$	5.29437	0.741361
$52$	0	0
$53$	0.215830	0.0296465	0.0148233	−	0.999890i	$-0.495281\pi$
$53$	0.215830	0.0296465	0.0148233	+	0.999890i	$0.495281\pi$
$54$	0	0
$55$	−1.55264	−0.209358
$56$	0	0
$57$	0.0964767	0.0127787
$58$	0	0
$59$	0.410309	0.0534177	0.0267089	−	0.999643i	$-0.491497\pi$
$59$	0.410309	0.0534177	0.0267089	+	0.999643i	$0.491497\pi$
$60$	0	0
$61$	0.112356	0.0143858	0.00719288	−	0.999974i	$-0.497710\pi$
$61$	0.112356	0.0143858	0.00719288	+	0.999974i	$0.497710\pi$
$62$	0	0
$63$	−42.2732	−5.32592
$64$	0	0
$65$	8.77537	1.08845
$66$	0	0
$67$	10.7794	1.31691	0.658454	−	0.752621i	$-0.271211\pi$
$67$	10.7794	1.31691	0.658454	+	0.752621i	$0.271211\pi$
$68$	0	0
$69$	26.2810	3.16386
$70$	0	0
$71$	−14.8717	−1.76494	−0.882472	−	0.470364i	$-0.844122\pi$
$71$	−14.8717	−1.76494	−0.882472	+	0.470364i	$0.844122\pi$
$72$	0	0
$73$	11.4343	1.33829	0.669143	−	0.743134i	$-0.266661\pi$
$73$	11.4343	1.33829	0.669143	+	0.743134i	$0.266661\pi$
$74$	0	0
$75$	−8.66114	−1.00010
$76$	0	0
$77$	−5.16237	−0.588307
$78$	0	0
$79$	−10.2845	−1.15710	−0.578548	−	0.815648i	$-0.696381\pi$
$79$	−10.2845	−1.15710	−0.578548	+	0.815648i	$0.696381\pi$
$80$	0	0
$81$	33.4889	3.72098
$82$	0	0
$83$	10.3772	1.13904	0.569522	−	0.821976i	$-0.307128\pi$
$83$	10.3772	1.13904	0.569522	+	0.821976i	$0.307128\pi$
$84$	0	0
$85$	−2.45751	−0.266554
$86$	0	0
$87$	8.88506	0.952579
$88$	0	0
$89$	−7.13668	−0.756487	−0.378243	−	0.925706i	$-0.623472\pi$
$89$	−7.13668	−0.756487	−0.378243	+	0.925706i	$0.623472\pi$
$90$	0	0
$91$	29.1773	3.05861
$92$	0	0
$93$	29.0546	3.01282
$94$	0	0
$95$	−0.0447819	−0.00459453
$96$	0	0
$97$	−3.42963	−0.348226	−0.174113	−	0.984726i	$-0.555706\pi$
$97$	−3.42963	−0.348226	−0.174113	+	0.984726i	$0.555706\pi$
$98$	0	0
$99$	8.18871	0.822996
$100$	0	0
$101$	14.8320	1.47584	0.737921	−	0.674888i	$-0.235808\pi$
$101$	14.8320	1.47584	0.737921	+	0.674888i	$0.235808\pi$
$102$	0	0
$103$	13.7961	1.35937	0.679686	−	0.733503i	$-0.262116\pi$
$103$	13.7961	1.35937	0.679686	+	0.733503i	$0.262116\pi$
$104$	0	0
$105$	26.8108	2.61647
$106$	0	0
$107$	18.1639	1.75597	0.877986	−	0.478687i	$-0.158887\pi$
$107$	18.1639	1.75597	0.877986	+	0.478687i	$0.158887\pi$
$108$	0	0
$109$	−5.12926	−0.491294	−0.245647	−	0.969359i	$-0.579000\pi$
$109$	−5.12926	−0.491294	−0.245647	+	0.969359i	$0.579000\pi$
$110$	0	0
$111$	30.3638	2.88200
$112$	0	0
$113$	2.88910	0.271784	0.135892	−	0.990724i	$-0.456610\pi$
$113$	2.88910	0.271784	0.135892	+	0.990724i	$0.456610\pi$
$114$	0	0
$115$	−12.1989	−1.13756
$116$	0	0
$117$	−46.2819	−4.27876
$118$	0	0
$119$	−8.17098	−0.749032
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.07555	−0.367480
$124$	0	0
$125$	11.7835	1.05394
$126$	0	0
$127$	10.6964	0.949154	0.474577	−	0.880214i	$-0.342601\pi$
$127$	10.6964	0.949154	0.474577	+	0.880214i	$0.342601\pi$
$128$	0	0
$129$	−16.4560	−1.44887
$130$	0	0
$131$	−7.36031	−0.643074	−0.321537	−	0.946897i	$-0.604199\pi$
$131$	−7.36031	−0.643074	−0.321537	+	0.946897i	$0.604199\pi$
$132$	0	0
$133$	−0.148896	−0.0129109
$134$	0	0
$135$	−26.9476	−2.31928
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	18.2393	1.54704	0.773521	−	0.633771i	$-0.218494\pi$
$139$	18.2393	1.54704	0.773521	+	0.633771i	$0.218494\pi$
$140$	0	0
$141$	−22.2969	−1.87774
$142$	0	0
$143$	−5.65191	−0.472637
$144$	0	0
$145$	−4.12421	−0.342497
$146$	0	0
$147$	65.7287	5.42121
$148$	0	0
$149$	−1.31177	−0.107464	−0.0537320	−	0.998555i	$-0.517112\pi$
$149$	−1.31177	−0.107464	−0.0537320	+	0.998555i	$0.517112\pi$
$150$	0	0
$151$	−11.7724	−0.958025	−0.479012	−	0.877808i	$-0.659005\pi$
$151$	−11.7724	−0.958025	−0.479012	+	0.877808i	$0.659005\pi$
$152$	0	0
$153$	12.9610	1.04784
$154$	0	0
$155$	−13.4864	−1.08325
$156$	0	0
$157$	0.170154	0.0135798	0.00678988	−	0.999977i	$-0.497839\pi$
$157$	0.170154	0.0135798	0.00678988	+	0.999977i	$0.497839\pi$
$158$	0	0
$159$	0.721941	0.0572537
$160$	0	0
$161$	−40.5603	−3.19660
$162$	0	0
$163$	−14.2409	−1.11544	−0.557718	−	0.830031i	$-0.688323\pi$
$163$	−14.2409	−1.11544	−0.557718	+	0.830031i	$0.688323\pi$
$164$	0	0
$165$	−5.19350	−0.404314
$166$	0	0
$167$	7.04922	0.545485	0.272743	−	0.962087i	$-0.412069\pi$
$167$	7.04922	0.545485	0.272743	+	0.962087i	$0.412069\pi$
$168$	0	0
$169$	18.9441	1.45724
$170$	0	0
$171$	0.236183	0.0180613
$172$	0	0
$173$	−1.60161	−0.121768	−0.0608841	−	0.998145i	$-0.519392\pi$
$173$	−1.60161	−0.121768	−0.0608841	+	0.998145i	$0.519392\pi$
$174$	0	0
$175$	13.3670	1.01045
$176$	0	0
$177$	1.37247	0.103161
$178$	0	0
$179$	−11.4102	−0.852840	−0.426420	−	0.904525i	$-0.640225\pi$
$179$	−11.4102	−0.852840	−0.426420	+	0.904525i	$0.640225\pi$
$180$	0	0
$181$	10.8371	0.805513	0.402756	−	0.915307i	$-0.368052\pi$
$181$	10.8371	0.805513	0.402756	+	0.915307i	$0.368052\pi$
$182$	0	0
$183$	0.375827	0.0277819
$184$	0	0
$185$	−14.0941	−1.03622
$186$	0	0
$187$	1.58279	0.115745
$188$	0	0
$189$	−89.5981	−6.51731
$190$	0	0
$191$	10.9522	0.792470	0.396235	−	0.918149i	$-0.370317\pi$
$191$	10.9522	0.792470	0.396235	+	0.918149i	$0.370317\pi$
$192$	0	0
$193$	20.0341	1.44208	0.721042	−	0.692891i	$-0.243663\pi$
$193$	20.0341	1.44208	0.721042	+	0.692891i	$0.243663\pi$
$194$	0	0
$195$	29.3532	2.10203
$196$	0	0
$197$	9.38771	0.668846	0.334423	−	0.942423i	$-0.391459\pi$
$197$	9.38771	0.668846	0.334423	+	0.942423i	$0.391459\pi$
$198$	0	0
$199$	7.08234	0.502053	0.251027	−	0.967980i	$-0.419232\pi$
$199$	7.08234	0.502053	0.251027	+	0.967980i	$0.419232\pi$
$200$	0	0
$201$	36.0564	2.54322
$202$	0	0
$203$	−13.7126	−0.962436
$204$	0	0
$205$	1.89176	0.132126
$206$	0	0
$207$	64.3380	4.47180
$208$	0	0
$209$	0.0288425	0.00199508
$210$	0	0
$211$	10.1641	0.699725	0.349863	−	0.936801i	$-0.386228\pi$
$211$	10.1641	0.699725	0.349863	+	0.936801i	$0.386228\pi$
$212$	0	0
$213$	−49.7451	−3.40848
$214$	0	0
$215$	7.63842	0.520936
$216$	0	0
$217$	−44.8409	−3.04400
$218$	0	0
$219$	38.2473	2.58451
$220$	0	0
$221$	−8.94581	−0.601761
$222$	0	0
$223$	−22.3372	−1.49581	−0.747904	−	0.663807i	$-0.768940\pi$
$223$	−22.3372	−1.49581	−0.747904	+	0.663807i	$0.768940\pi$
$224$	0	0
$225$	−21.2032	−1.41354
$226$	0	0
$227$	−7.10714	−0.471718	−0.235859	−	0.971787i	$-0.575790\pi$
$227$	−7.10714	−0.471718	−0.235859	+	0.971787i	$0.575790\pi$
$228$	0	0
$229$	11.2593	0.744035	0.372017	−	0.928226i	$-0.378666\pi$
$229$	11.2593	0.744035	0.372017	+	0.928226i	$0.378666\pi$
$230$	0	0
$231$	−17.2679	−1.13614
$232$	0	0
$233$	6.12199	0.401065	0.200533	−	0.979687i	$-0.435733\pi$
$233$	6.12199	0.401065	0.200533	+	0.979687i	$0.435733\pi$
$234$	0	0
$235$	10.3496	0.675137
$236$	0	0
$237$	−34.4012	−2.23460
$238$	0	0
$239$	−23.0878	−1.49343	−0.746713	−	0.665146i	$-0.768369\pi$
$239$	−23.0878	−1.49343	−0.746713	+	0.665146i	$0.768369\pi$
$240$	0	0
$241$	10.1021	0.650734	0.325367	−	0.945588i	$-0.394512\pi$
$241$	10.1021	0.650734	0.325367	+	0.945588i	$0.394512\pi$
$242$	0	0
$243$	59.9507	3.84584
$244$	0	0
$245$	−30.5095	−1.94918
$246$	0	0
$247$	−0.163015	−0.0103724
$248$	0	0
$249$	34.7112	2.19973
$250$	0	0
$251$	8.86799	0.559742	0.279871	−	0.960038i	$-0.409708\pi$
$251$	8.86799	0.559742	0.279871	+	0.960038i	$0.409708\pi$
$252$	0	0
$253$	7.85691	0.493960
$254$	0	0
$255$	−8.22025	−0.514772
$256$	0	0
$257$	−29.3088	−1.82824	−0.914118	−	0.405449i	$-0.867115\pi$
$257$	−29.3088	−1.82824	−0.914118	+	0.405449i	$0.867115\pi$
$258$	0	0
$259$	−46.8614	−2.91183
$260$	0	0
$261$	21.7513	1.34637
$262$	0	0
$263$	6.57149	0.405216	0.202608	−	0.979260i	$-0.435058\pi$
$263$	6.57149	0.405216	0.202608	+	0.979260i	$0.435058\pi$
$264$	0	0
$265$	−0.335106	−0.0205854
$266$	0	0
$267$	−23.8719	−1.46093
$268$	0	0
$269$	−27.0036	−1.64644	−0.823221	−	0.567721i	$-0.807825\pi$
$269$	−27.0036	−1.64644	−0.823221	+	0.567721i	$0.807825\pi$
$270$	0	0
$271$	24.4437	1.48485	0.742423	−	0.669931i	$-0.233676\pi$
$271$	24.4437	1.48485	0.742423	+	0.669931i	$0.233676\pi$
$272$	0	0
$273$	97.5966	5.90682
$274$	0	0
$275$	−2.58931	−0.156142
$276$	0	0
$277$	−5.82181	−0.349799	−0.174899	−	0.984586i	$-0.555960\pi$
$277$	−5.82181	−0.349799	−0.174899	+	0.984586i	$0.555960\pi$
$278$	0	0
$279$	71.1280	4.25832
$280$	0	0
$281$	31.0777	1.85394	0.926969	−	0.375139i	$-0.122405\pi$
$281$	31.0777	1.85394	0.926969	+	0.375139i	$0.122405\pi$
$282$	0	0
$283$	1.08568	0.0645370	0.0322685	−	0.999479i	$-0.489727\pi$
$283$	1.08568	0.0645370	0.0322685	+	0.999479i	$0.489727\pi$
$284$	0	0
$285$	−0.149793	−0.00887300
$286$	0	0
$287$	6.28992	0.371282
$288$	0	0
$289$	−14.4948	−0.852633
$290$	0	0
$291$	−11.4720	−0.672498
$292$	0	0
$293$	−19.1798	−1.12050	−0.560249	−	0.828324i	$-0.689295\pi$
$293$	−19.1798	−1.12050	−0.560249	+	0.828324i	$0.689295\pi$
$294$	0	0
$295$	−0.637062	−0.0370912
$296$	0	0
$297$	17.3560	1.00710
$298$	0	0
$299$	−44.4066	−2.56810
$300$	0	0
$301$	25.3970	1.46386
$302$	0	0
$303$	49.6124	2.85016
$304$	0	0
$305$	−0.174449	−0.00998891
$306$	0	0
$307$	5.84725	0.333720	0.166860	−	0.985981i	$-0.446637\pi$
$307$	5.84725	0.333720	0.166860	+	0.985981i	$0.446637\pi$
$308$	0	0
$309$	46.1474	2.62523
$310$	0	0
$311$	9.65805	0.547658	0.273829	−	0.961778i	$-0.411710\pi$
$311$	9.65805	0.547658	0.273829	+	0.961778i	$0.411710\pi$
$312$	0	0
$313$	−2.55891	−0.144638	−0.0723191	−	0.997382i	$-0.523040\pi$
$313$	−2.55891	−0.144638	−0.0723191	+	0.997382i	$0.523040\pi$
$314$	0	0
$315$	65.6350	3.69811
$316$	0	0
$317$	−13.5488	−0.760977	−0.380489	−	0.924786i	$-0.624244\pi$
$317$	−13.5488	−0.760977	−0.380489	+	0.924786i	$0.624244\pi$
$318$	0	0
$319$	2.65626	0.148722
$320$	0	0
$321$	60.7574	3.39115
$322$	0	0
$323$	0.0456517	0.00254013
$324$	0	0
$325$	14.6346	0.811780
$326$	0	0
$327$	−17.1571	−0.948791
$328$	0	0
$329$	34.4116	1.89717
$330$	0	0
$331$	−7.19108	−0.395258	−0.197629	−	0.980277i	$-0.563324\pi$
$331$	−7.19108	−0.395258	−0.197629	+	0.980277i	$0.563324\pi$
$332$	0	0
$333$	74.3329	4.07342
$334$	0	0
$335$	−16.7364	−0.914409
$336$	0	0
$337$	−1.09045	−0.0594008	−0.0297004	−	0.999559i	$-0.509455\pi$
$337$	−1.09045	−0.0594008	−0.0297004	+	0.999559i	$0.509455\pi$
$338$	0	0
$339$	9.66390	0.524871
$340$	0	0
$341$	8.68610	0.470379
$342$	0	0
$343$	−65.3046	−3.52612
$344$	0	0
$345$	−40.8049	−2.19686
$346$	0	0
$347$	2.84578	0.152769	0.0763847	−	0.997078i	$-0.475662\pi$
$347$	2.84578	0.152769	0.0763847	+	0.997078i	$0.475662\pi$
$348$	0	0
$349$	31.2200	1.67117	0.835584	−	0.549363i	$-0.185130\pi$
$349$	31.2200	1.67117	0.835584	+	0.549363i	$0.185130\pi$
$350$	0	0
$351$	−98.0946	−5.23590
$352$	0	0
$353$	12.0790	0.642901	0.321451	−	0.946926i	$-0.395830\pi$
$353$	12.0790	0.642901	0.321451	+	0.946926i	$0.395830\pi$
$354$	0	0
$355$	23.0903	1.22551
$356$	0	0
$357$	−27.3315	−1.44654
$358$	0	0
$359$	−10.3503	−0.546266	−0.273133	−	0.961976i	$-0.588060\pi$
$359$	−10.3503	−0.546266	−0.273133	+	0.961976i	$0.588060\pi$
$360$	0	0
$361$	−18.9992	−0.999956
$362$	0	0
$363$	3.34495	0.175565
$364$	0	0
$365$	−17.7534	−0.929254
$366$	0	0
$367$	−10.8672	−0.567264	−0.283632	−	0.958933i	$-0.591539\pi$
$367$	−10.8672	−0.567264	−0.283632	+	0.958933i	$0.591539\pi$
$368$	0	0
$369$	−9.97726	−0.519395
$370$	0	0
$371$	−1.11420	−0.0578461
$372$	0	0
$373$	18.1080	0.937598	0.468799	−	0.883305i	$-0.344687\pi$
$373$	18.1080	0.937598	0.468799	+	0.883305i	$0.344687\pi$
$374$	0	0
$375$	39.4151	2.03539
$376$	0	0
$377$	−15.0129	−0.773206
$378$	0	0
$379$	−25.3188	−1.30054	−0.650269	−	0.759704i	$-0.725344\pi$
$379$	−25.3188	−1.30054	−0.650269	+	0.759704i	$0.725344\pi$
$380$	0	0
$381$	35.7790	1.83302
$382$	0	0
$383$	8.23000	0.420533	0.210267	−	0.977644i	$-0.432567\pi$
$383$	8.23000	0.420533	0.210267	+	0.977644i	$0.432567\pi$
$384$	0	0
$385$	8.01530	0.408497
$386$	0	0
$387$	−40.2855	−2.04783
$388$	0	0
$389$	−9.63758	−0.488645	−0.244322	−	0.969694i	$-0.578566\pi$
$389$	−9.63758	−0.488645	−0.244322	+	0.969694i	$0.578566\pi$
$390$	0	0
$391$	12.4359	0.628909
$392$	0	0
$393$	−24.6199	−1.24191
$394$	0	0
$395$	15.9681	0.803443
$396$	0	0
$397$	0.325265	0.0163246	0.00816230	−	0.999967i	$-0.497402\pi$
$397$	0.325265	0.0163246	0.00816230	+	0.999967i	$0.497402\pi$
$398$	0	0
$399$	−0.498049	−0.0249336
$400$	0	0
$401$	16.0930	0.803646	0.401823	−	0.915717i	$-0.368377\pi$
$401$	16.0930	0.803646	0.401823	+	0.915717i	$0.368377\pi$
$402$	0	0
$403$	−49.0931	−2.44550
$404$	0	0
$405$	−51.9961	−2.58371
$406$	0	0
$407$	9.07749	0.449954
$408$	0	0
$409$	−14.1424	−0.699298	−0.349649	−	0.936881i	$-0.613699\pi$
$409$	−14.1424	−0.699298	−0.349649	+	0.936881i	$0.613699\pi$
$410$	0	0
$411$	3.34495	0.164994
$412$	0	0
$413$	−2.11817	−0.104228
$414$	0	0
$415$	−16.1120	−0.790908
$416$	0	0
$417$	61.0098	2.98766
$418$	0	0
$419$	8.02716	0.392152	0.196076	−	0.980589i	$-0.437180\pi$
$419$	8.02716	0.392152	0.196076	+	0.980589i	$0.437180\pi$
$420$	0	0
$421$	−3.62581	−0.176711	−0.0883556	−	0.996089i	$-0.528161\pi$
$421$	−3.62581	−0.176711	−0.0883556	+	0.996089i	$0.528161\pi$
$422$	0	0
$423$	−54.5847	−2.65400
$424$	0	0
$425$	−4.09835	−0.198799
$426$	0	0
$427$	−0.580026	−0.0280694
$428$	0	0
$429$	−18.9054	−0.912760
$430$	0	0
$431$	−1.45866	−0.0702614	−0.0351307	−	0.999383i	$-0.511185\pi$
$431$	−1.45866	−0.0702614	−0.0351307	+	0.999383i	$0.511185\pi$
$432$	0	0
$433$	−22.0758	−1.06090	−0.530449	−	0.847717i	$-0.677977\pi$
$433$	−22.0758	−1.06090	−0.530449	+	0.847717i	$0.677977\pi$
$434$	0	0
$435$	−13.7953	−0.661433
$436$	0	0
$437$	0.226613	0.0108404
$438$	0	0
$439$	−13.0621	−0.623420	−0.311710	−	0.950177i	$-0.600902\pi$
$439$	−13.0621	−0.623420	−0.311710	+	0.950177i	$0.600902\pi$
$440$	0	0
$441$	160.909	7.66234
$442$	0	0
$443$	−2.30388	−0.109461	−0.0547304	−	0.998501i	$-0.517430\pi$
$443$	−2.30388	−0.109461	−0.0547304	+	0.998501i	$0.517430\pi$
$444$	0	0
$445$	11.0807	0.525275
$446$	0	0
$447$	−4.38780	−0.207536
$448$	0	0
$449$	−36.5396	−1.72441	−0.862204	−	0.506560i	$-0.830917\pi$
$449$	−36.5396	−1.72441	−0.862204	+	0.506560i	$0.830917\pi$
$450$	0	0
$451$	−1.21842	−0.0573730
$452$	0	0
$453$	−39.3781	−1.85015
$454$	0	0
$455$	−45.3018	−2.12378
$456$	0	0
$457$	−21.1505	−0.989380	−0.494690	−	0.869069i	$-0.664718\pi$
$457$	−21.1505	−0.989380	−0.494690	+	0.869069i	$0.664718\pi$
$458$	0	0
$459$	27.4710	1.28224
$460$	0	0
$461$	−36.8396	−1.71579	−0.857896	−	0.513823i	$-0.828229\pi$
$461$	−36.8396	−1.71579	−0.857896	+	0.513823i	$0.828229\pi$
$462$	0	0
$463$	−11.1271	−0.517118	−0.258559	−	0.965995i	$-0.583248\pi$
$463$	−11.1271	−0.517118	−0.258559	+	0.965995i	$0.583248\pi$
$464$	0	0
$465$	−45.1113	−2.09199
$466$	0	0
$467$	−24.0319	−1.11207	−0.556033	−	0.831160i	$-0.687677\pi$
$467$	−24.0319	−1.11207	−0.556033	+	0.831160i	$0.687677\pi$
$468$	0	0
$469$	−55.6470	−2.56954
$470$	0	0
$471$	0.569157	0.0262254
$472$	0	0
$473$	−4.91964	−0.226205
$474$	0	0
$475$	−0.0746822	−0.00342666
$476$	0	0
$477$	1.76737	0.0809223
$478$	0	0
$479$	18.9831	0.867360	0.433680	−	0.901067i	$-0.357215\pi$
$479$	18.9831	0.867360	0.433680	+	0.901067i	$0.357215\pi$
$480$	0	0
$481$	−51.3052	−2.33931
$482$	0	0
$483$	−135.672	−6.17331
$484$	0	0
$485$	5.32497	0.241795
$486$	0	0
$487$	5.15551	0.233619	0.116809	−	0.993154i	$-0.462733\pi$
$487$	5.15551	0.233619	0.116809	+	0.993154i	$0.462733\pi$
$488$	0	0
$489$	−47.6352	−2.15414
$490$	0	0
$491$	23.1236	1.04355	0.521776	−	0.853083i	$-0.325270\pi$
$491$	23.1236	1.04355	0.521776	+	0.853083i	$0.325270\pi$
$492$	0	0
$493$	4.20431	0.189353
$494$	0	0
$495$	−12.7141	−0.571457
$496$	0	0
$497$	76.7732	3.44375
$498$	0	0
$499$	−37.3080	−1.67014	−0.835069	−	0.550146i	$-0.814572\pi$
$499$	−37.3080	−1.67014	−0.835069	+	0.550146i	$0.814572\pi$
$500$	0	0
$501$	23.5793	1.05345
$502$	0	0
$503$	28.9127	1.28915	0.644577	−	0.764540i	$-0.277034\pi$
$503$	28.9127	1.28915	0.644577	+	0.764540i	$0.277034\pi$
$504$	0	0
$505$	−23.0288	−1.02477
$506$	0	0
$507$	63.3671	2.81423
$508$	0	0
$509$	−12.1432	−0.538238	−0.269119	−	0.963107i	$-0.586732\pi$
$509$	−12.1432	−0.538238	−0.269119	+	0.963107i	$0.586732\pi$
$510$	0	0
$511$	−59.0282	−2.61126
$512$	0	0
$513$	0.500590	0.0221016
$514$	0	0
$515$	−21.4204	−0.943895
$516$	0	0
$517$	−6.66585	−0.293164
$518$	0	0
$519$	−5.35731	−0.235160
$520$	0	0
$521$	−3.38069	−0.148111	−0.0740554	−	0.997254i	$-0.523594\pi$
$521$	−3.38069	−0.148111	−0.0740554	+	0.997254i	$0.523594\pi$
$522$	0	0
$523$	−21.6513	−0.946747	−0.473373	−	0.880862i	$-0.656964\pi$
$523$	−21.6513	−0.946747	−0.473373	+	0.880862i	$0.656964\pi$
$524$	0	0
$525$	44.7120	1.95139
$526$	0	0
$527$	13.7483	0.598886
$528$	0	0
$529$	38.7311	1.68396
$530$	0	0
$531$	3.35990	0.145807
$532$	0	0
$533$	6.88638	0.298282
$534$	0	0
$535$	−28.2020	−1.21928
$536$	0	0
$537$	−38.1667	−1.64701
$538$	0	0
$539$	19.6501	0.846390
$540$	0	0
$541$	8.29931	0.356815	0.178408	−	0.983957i	$-0.442905\pi$
$541$	8.29931	0.356815	0.178408	+	0.983957i	$0.442905\pi$
$542$	0	0
$543$	36.2495	1.55561
$544$	0	0
$545$	7.96388	0.341135
$546$	0	0
$547$	−13.3177	−0.569425	−0.284712	−	0.958613i	$-0.591898\pi$
$547$	−13.3177	−0.569425	−0.284712	+	0.958613i	$0.591898\pi$
$548$	0	0
$549$	0.920054	0.0392669
$550$	0	0
$551$	0.0766131	0.00326383
$552$	0	0
$553$	53.0924	2.25772
$554$	0	0
$555$	−47.1440	−2.00115
$556$	0	0
$557$	−2.37943	−0.100820	−0.0504098	−	0.998729i	$-0.516053\pi$
$557$	−2.37943	−0.100820	−0.0504098	+	0.998729i	$0.516053\pi$
$558$	0	0
$559$	27.8054	1.17604
$560$	0	0
$561$	5.29437	0.223529
$562$	0	0
$563$	8.39342	0.353741	0.176870	−	0.984234i	$-0.443403\pi$
$563$	8.39342	0.353741	0.176870	+	0.984234i	$0.443403\pi$
$564$	0	0
$565$	−4.48573	−0.188716
$566$	0	0
$567$	−172.882	−7.26036
$568$	0	0
$569$	2.95710	0.123968	0.0619840	−	0.998077i	$-0.480257\pi$
$569$	2.95710	0.123968	0.0619840	+	0.998077i	$0.480257\pi$
$570$	0	0
$571$	−10.2092	−0.427240	−0.213620	−	0.976917i	$-0.568525\pi$
$571$	−10.2092	−0.427240	−0.213620	+	0.976917i	$0.568525\pi$
$572$	0	0
$573$	36.6344	1.53043
$574$	0	0
$575$	−20.3440	−0.848404
$576$	0	0
$577$	−6.08294	−0.253236	−0.126618	−	0.991952i	$-0.540412\pi$
$577$	−6.08294	−0.253236	−0.126618	+	0.991952i	$0.540412\pi$
$578$	0	0
$579$	67.0130	2.78497
$580$	0	0
$581$	−53.5709	−2.22250
$582$	0	0
$583$	0.215830	0.00893877
$584$	0	0
$585$	71.8590	2.97100
$586$	0	0
$587$	−45.4976	−1.87789	−0.938943	−	0.344071i	$-0.888194\pi$
$587$	−45.4976	−1.87789	−0.938943	+	0.344071i	$0.888194\pi$
$588$	0	0
$589$	0.250529	0.0103229
$590$	0	0
$591$	31.4014	1.29168
$592$	0	0
$593$	−28.1832	−1.15735	−0.578674	−	0.815559i	$-0.696429\pi$
$593$	−28.1832	−1.15735	−0.578674	+	0.815559i	$0.696429\pi$
$594$	0	0
$595$	12.6866	0.520099
$596$	0	0
$597$	23.6901	0.969570
$598$	0	0
$599$	33.1221	1.35333	0.676666	−	0.736290i	$-0.263424\pi$
$599$	33.1221	1.35333	0.676666	+	0.736290i	$0.263424\pi$
$600$	0	0
$601$	20.8491	0.850454	0.425227	−	0.905087i	$-0.360194\pi$
$601$	20.8491	0.850454	0.425227	+	0.905087i	$0.360194\pi$
$602$	0	0
$603$	88.2690	3.59459
$604$	0	0
$605$	−1.55264	−0.0631237
$606$	0	0
$607$	−32.5998	−1.32318	−0.661592	−	0.749864i	$-0.730119\pi$
$607$	−32.5998	−1.32318	−0.661592	+	0.749864i	$0.730119\pi$
$608$	0	0
$609$	−45.8680	−1.85867
$610$	0	0
$611$	37.6748	1.52416
$612$	0	0
$613$	−11.5721	−0.467393	−0.233696	−	0.972310i	$-0.575082\pi$
$613$	−11.5721	−0.467393	−0.233696	+	0.972310i	$0.575082\pi$
$614$	0	0
$615$	6.32785	0.255163
$616$	0	0
$617$	28.4365	1.14481	0.572404	−	0.819972i	$-0.306011\pi$
$617$	28.4365	1.14481	0.572404	+	0.819972i	$0.306011\pi$
$618$	0	0
$619$	8.65944	0.348052	0.174026	−	0.984741i	$-0.444322\pi$
$619$	8.65944	0.348052	0.174026	+	0.984741i	$0.444322\pi$
$620$	0	0
$621$	136.365	5.47212
$622$	0	0
$623$	36.8422	1.47605
$624$	0	0
$625$	−5.34887	−0.213955
$626$	0	0
$627$	0.0964767	0.00385291
$628$	0	0
$629$	14.3678	0.572882
$630$	0	0
$631$	25.1088	0.999566	0.499783	−	0.866151i	$-0.333413\pi$
$631$	25.1088	0.999566	0.499783	+	0.866151i	$0.333413\pi$
$632$	0	0
$633$	33.9984	1.35132
$634$	0	0
$635$	−16.6077	−0.659055
$636$	0	0
$637$	−111.061	−4.40038
$638$	0	0
$639$	−121.780	−4.81754
$640$	0	0
$641$	−20.2131	−0.798369	−0.399185	−	0.916871i	$-0.630707\pi$
$641$	−20.2131	−0.798369	−0.399185	+	0.916871i	$0.630707\pi$
$642$	0	0
$643$	−41.7932	−1.64816	−0.824081	−	0.566472i	$-0.808308\pi$
$643$	−41.7932	−1.64816	−0.824081	+	0.566472i	$0.808308\pi$
$644$	0	0
$645$	25.5502	1.00604
$646$	0	0
$647$	4.05220	0.159308	0.0796542	−	0.996823i	$-0.474618\pi$
$647$	4.05220	0.159308	0.0796542	+	0.996823i	$0.474618\pi$
$648$	0	0
$649$	0.410309	0.0161060
$650$	0	0
$651$	−149.991	−5.87860
$652$	0	0
$653$	−11.7240	−0.458794	−0.229397	−	0.973333i	$-0.573675\pi$
$653$	−11.7240	−0.458794	−0.229397	+	0.973333i	$0.573675\pi$
$654$	0	0
$655$	11.4279	0.446525
$656$	0	0
$657$	93.6324	3.65295
$658$	0	0
$659$	47.5163	1.85097	0.925486	−	0.378781i	$-0.123657\pi$
$659$	47.5163	1.85097	0.925486	+	0.378781i	$0.123657\pi$
$660$	0	0
$661$	2.61780	0.101821	0.0509103	−	0.998703i	$-0.483788\pi$
$661$	2.61780	0.101821	0.0509103	+	0.998703i	$0.483788\pi$
$662$	0	0
$663$	−29.9233	−1.16213
$664$	0	0
$665$	0.231181	0.00896481
$666$	0	0
$667$	20.8700	0.808089
$668$	0	0
$669$	−74.7168	−2.88872
$670$	0	0
$671$	0.112356	0.00433747
$672$	0	0
$673$	16.4990	0.635989	0.317995	−	0.948093i	$-0.396991\pi$
$673$	16.4990	0.635989	0.317995	+	0.948093i	$0.396991\pi$
$674$	0	0
$675$	−44.9401	−1.72975
$676$	0	0
$677$	−44.0371	−1.69248	−0.846242	−	0.532799i	$-0.821140\pi$
$677$	−44.0371	−1.69248	−0.846242	+	0.532799i	$0.821140\pi$
$678$	0	0
$679$	17.7050	0.679457
$680$	0	0
$681$	−23.7731	−0.910986
$682$	0	0
$683$	40.8497	1.56307	0.781535	−	0.623862i	$-0.214437\pi$
$683$	40.8497	1.56307	0.781535	+	0.623862i	$0.214437\pi$
$684$	0	0
$685$	−1.55264	−0.0593233
$686$	0	0
$687$	37.6618	1.43689
$688$	0	0
$689$	−1.21985	−0.0464727
$690$	0	0
$691$	44.9793	1.71109	0.855547	−	0.517725i	$-0.173221\pi$
$691$	44.9793	1.71109	0.855547	+	0.517725i	$0.173221\pi$
$692$	0	0
$693$	−42.2732	−1.60583
$694$	0	0
$695$	−28.3191	−1.07420
$696$	0	0
$697$	−1.92850	−0.0730473
$698$	0	0
$699$	20.4778	0.774541
$700$	0	0
$701$	4.26088	0.160931	0.0804656	−	0.996757i	$-0.474359\pi$
$701$	4.26088	0.160931	0.0804656	+	0.996757i	$0.474359\pi$
$702$	0	0
$703$	0.261817	0.00987462
$704$	0	0
$705$	34.6191	1.30383
$706$	0	0
$707$	−76.5684	−2.87965
$708$	0	0
$709$	−33.0538	−1.24136	−0.620681	−	0.784063i	$-0.713144\pi$
$709$	−33.0538	−1.24136	−0.620681	+	0.784063i	$0.713144\pi$
$710$	0	0
$711$	−84.2168	−3.15838
$712$	0	0
$713$	68.2459	2.55583
$714$	0	0
$715$	8.77537	0.328180
$716$	0	0
$717$	−77.2276	−2.88412
$718$	0	0
$719$	−6.13271	−0.228712	−0.114356	−	0.993440i	$-0.536480\pi$
$719$	−6.13271	−0.228712	−0.114356	+	0.993440i	$0.536480\pi$
$720$	0	0
$721$	−71.2207	−2.65240
$722$	0	0
$723$	33.7911	1.25670
$724$	0	0
$725$	−6.87789	−0.255438
$726$	0	0
$727$	−18.4492	−0.684245	−0.342122	−	0.939655i	$-0.611146\pi$
$727$	−18.4492	−0.684245	−0.342122	+	0.939655i	$0.611146\pi$
$728$	0	0
$729$	100.066	3.70613
$730$	0	0
$731$	−7.78678	−0.288004
$732$	0	0
$733$	14.6921	0.542665	0.271333	−	0.962486i	$-0.412536\pi$
$733$	14.6921	0.542665	0.271333	+	0.962486i	$0.412536\pi$
$734$	0	0
$735$	−102.053	−3.76428
$736$	0	0
$737$	10.7794	0.397063
$738$	0	0
$739$	−1.88905	−0.0694899	−0.0347449	−	0.999396i	$-0.511062\pi$
$739$	−1.88905	−0.0694899	−0.0347449	+	0.999396i	$0.511062\pi$
$740$	0	0
$741$	−0.545278	−0.0200313
$742$	0	0
$743$	50.1051	1.83818	0.919089	−	0.394050i	$-0.128926\pi$
$743$	50.1051	1.83818	0.919089	+	0.394050i	$0.128926\pi$
$744$	0	0
$745$	2.03670	0.0746188
$746$	0	0
$747$	84.9758	3.10910
$748$	0	0
$749$	−93.7689	−3.42624
$750$	0	0
$751$	−9.64307	−0.351881	−0.175940	−	0.984401i	$-0.556297\pi$
$751$	−9.64307	−0.351881	−0.175940	+	0.984401i	$0.556297\pi$
$752$	0	0
$753$	29.6630	1.08098
$754$	0	0
$755$	18.2783	0.665215
$756$	0	0
$757$	4.39008	0.159560	0.0797800	−	0.996812i	$-0.474578\pi$
$757$	4.39008	0.159560	0.0797800	+	0.996812i	$0.474578\pi$
$758$	0	0
$759$	26.2810	0.953940
$760$	0	0
$761$	−12.5591	−0.455267	−0.227633	−	0.973747i	$-0.573099\pi$
$761$	−12.5591	−0.455267	−0.227633	+	0.973747i	$0.573099\pi$
$762$	0	0
$763$	26.4791	0.958609
$764$	0	0
$765$	−20.1238	−0.727578
$766$	0	0
$767$	−2.31903	−0.0837354
$768$	0	0
$769$	13.7866	0.497157	0.248579	−	0.968612i	$-0.420037\pi$
$769$	13.7866	0.497157	0.248579	+	0.968612i	$0.420037\pi$
$770$	0	0
$771$	−98.0367	−3.53071
$772$	0	0
$773$	35.2512	1.26790	0.633949	−	0.773375i	$-0.281433\pi$
$773$	35.2512	1.26790	0.633949	+	0.773375i	$0.281433\pi$
$774$	0	0
$775$	−22.4911	−0.807902
$776$	0	0
$777$	−156.749	−5.62335
$778$	0	0
$779$	−0.0351421	−0.00125910
$780$	0	0
$781$	−14.8717	−0.532151
$782$	0	0
$783$	46.1020	1.64755
$784$	0	0
$785$	−0.264187	−0.00942925
$786$	0	0
$787$	49.6255	1.76896	0.884479	−	0.466580i	$-0.154514\pi$
$787$	49.6255	1.76896	0.884479	+	0.466580i	$0.154514\pi$
$788$	0	0
$789$	21.9813	0.782556
$790$	0	0
$791$	−14.9146	−0.530303
$792$	0	0
$793$	−0.635028	−0.0225505
$794$	0	0
$795$	−1.12091	−0.0397547
$796$	0	0
$797$	19.0069	0.673259	0.336630	−	0.941637i	$-0.390713\pi$
$797$	19.0069	0.673259	0.336630	+	0.941637i	$0.390713\pi$
$798$	0	0
$799$	−10.5507	−0.373256
$800$	0	0
$801$	−58.4402	−2.06488
$802$	0	0
$803$	11.4343	0.403508
$804$	0	0
$805$	62.9755	2.21959
$806$	0	0
$807$	−90.3259	−3.17962
$808$	0	0
$809$	−18.2870	−0.642936	−0.321468	−	0.946920i	$-0.604176\pi$
$809$	−18.2870	−0.642936	−0.321468	+	0.946920i	$0.604176\pi$
$810$	0	0
$811$	−32.1970	−1.13059	−0.565295	−	0.824889i	$-0.691238\pi$
$811$	−32.1970	−1.13059	−0.565295	+	0.824889i	$0.691238\pi$
$812$	0	0
$813$	81.7629	2.86755
$814$	0	0
$815$	22.1110	0.774515
$816$	0	0
$817$	−0.141895	−0.00496426
$818$	0	0
$819$	238.924	8.34869
$820$	0	0
$821$	−4.64969	−0.162275	−0.0811376	−	0.996703i	$-0.525855\pi$
$821$	−4.64969	−0.162275	−0.0811376	+	0.996703i	$0.525855\pi$
$822$	0	0
$823$	17.5603	0.612112	0.306056	−	0.952013i	$-0.400990\pi$
$823$	17.5603	0.612112	0.306056	+	0.952013i	$0.400990\pi$
$824$	0	0
$825$	−8.66114	−0.301542
$826$	0	0
$827$	−25.1376	−0.874120	−0.437060	−	0.899432i	$-0.643980\pi$
$827$	−25.1376	−0.874120	−0.437060	+	0.899432i	$0.643980\pi$
$828$	0	0
$829$	57.2925	1.98985	0.994925	−	0.100619i	$-0.0320823\pi$
$829$	57.2925	1.98985	0.994925	+	0.100619i	$0.0320823\pi$
$830$	0	0
$831$	−19.4737	−0.675535
$832$	0	0
$833$	31.1021	1.07762
$834$	0	0
$835$	−10.9449	−0.378764
$836$	0	0
$837$	150.756	5.21089
$838$	0	0
$839$	−30.1455	−1.04074	−0.520369	−	0.853942i	$-0.674206\pi$
$839$	−30.1455	−1.04074	−0.520369	+	0.853942i	$0.674206\pi$
$840$	0	0
$841$	−21.9443	−0.756700
$842$	0	0
$843$	103.953	3.58034
$844$	0	0
$845$	−29.4133	−1.01185
$846$	0	0
$847$	−5.16237	−0.177381
$848$	0	0
$849$	3.63155	0.124634
$850$	0	0
$851$	71.3210	2.44485
$852$	0	0
$853$	13.9724	0.478405	0.239202	−	0.970970i	$-0.423114\pi$
$853$	13.9724	0.478405	0.239202	+	0.970970i	$0.423114\pi$
$854$	0	0
$855$	−0.366706	−0.0125411
$856$	0	0
$857$	53.0189	1.81109	0.905546	−	0.424249i	$-0.139462\pi$
$857$	53.0189	1.81109	0.905546	+	0.424249i	$0.139462\pi$
$858$	0	0
$859$	−4.08731	−0.139457	−0.0697285	−	0.997566i	$-0.522213\pi$
$859$	−4.08731	−0.139457	−0.0697285	+	0.997566i	$0.522213\pi$
$860$	0	0
$861$	21.0395	0.717024
$862$	0	0
$863$	19.0361	0.647997	0.323998	−	0.946058i	$-0.394973\pi$
$863$	19.0361	0.647997	0.323998	+	0.946058i	$0.394973\pi$
$864$	0	0
$865$	2.48672	0.0845511
$866$	0	0
$867$	−48.4843	−1.64661
$868$	0	0
$869$	−10.2845	−0.348878
$870$	0	0
$871$	−60.9239	−2.06433
$872$	0	0
$873$	−28.0842	−0.950508
$874$	0	0
$875$	−60.8306	−2.05645
$876$	0	0
$877$	−50.3862	−1.70142	−0.850711	−	0.525634i	$-0.823828\pi$
$877$	−50.3862	−1.70142	−0.850711	+	0.525634i	$0.823828\pi$
$878$	0	0
$879$	−64.1557	−2.16392
$880$	0	0
$881$	−18.1561	−0.611695	−0.305847	−	0.952081i	$-0.598940\pi$
$881$	−18.1561	−0.611695	−0.305847	+	0.952081i	$0.598940\pi$
$882$	0	0
$883$	21.4421	0.721584	0.360792	−	0.932646i	$-0.382506\pi$
$883$	21.4421	0.721584	0.360792	+	0.932646i	$0.382506\pi$
$884$	0	0
$885$	−2.13094	−0.0716308
$886$	0	0
$887$	39.9835	1.34251	0.671257	−	0.741225i	$-0.265755\pi$
$887$	39.9835	1.34251	0.671257	+	0.741225i	$0.265755\pi$
$888$	0	0
$889$	−55.2189	−1.85198
$890$	0	0
$891$	33.4889	1.12192
$892$	0	0
$893$	−0.192259	−0.00643372
$894$	0	0
$895$	17.7159	0.592179
$896$	0	0
$897$	−148.538	−4.95954
$898$	0	0
$899$	23.0725	0.769512
$900$	0	0
$901$	0.341615	0.0113808
$902$	0	0
$903$	84.9518	2.82702
$904$	0	0
$905$	−16.8260	−0.559317
$906$	0	0
$907$	−10.5866	−0.351523	−0.175761	−	0.984433i	$-0.556239\pi$
$907$	−10.5866	−0.351523	−0.175761	+	0.984433i	$0.556239\pi$
$908$	0	0
$909$	121.455	4.02841
$910$	0	0
$911$	−19.3004	−0.639451	−0.319725	−	0.947510i	$-0.603591\pi$
$911$	−19.3004	−0.639451	−0.319725	+	0.947510i	$0.603591\pi$
$912$	0	0
$913$	10.3772	0.343435
$914$	0	0
$915$	−0.583523	−0.0192907
$916$	0	0
$917$	37.9967	1.25476
$918$	0	0
$919$	35.4265	1.16861	0.584306	−	0.811534i	$-0.301367\pi$
$919$	35.4265	1.16861	0.584306	+	0.811534i	$0.301367\pi$
$920$	0	0
$921$	19.5588	0.644483
$922$	0	0
$923$	84.0535	2.76665
$924$	0	0
$925$	−23.5045	−0.772822
$926$	0	0
$927$	112.972	3.71050
$928$	0	0
$929$	−35.3094	−1.15846	−0.579232	−	0.815163i	$-0.696648\pi$
$929$	−35.3094	−1.15846	−0.579232	+	0.815163i	$0.696648\pi$
$930$	0	0
$931$	0.566758	0.0185747
$932$	0	0
$933$	32.3057	1.05764
$934$	0	0
$935$	−2.45751	−0.0803691
$936$	0	0
$937$	10.2877	0.336083	0.168042	−	0.985780i	$-0.446256\pi$
$937$	10.2877	0.336083	0.168042	+	0.985780i	$0.446256\pi$
$938$	0	0
$939$	−8.55944	−0.279327
$940$	0	0
$941$	26.5130	0.864300	0.432150	−	0.901802i	$-0.357755\pi$
$941$	26.5130	0.864300	0.432150	+	0.901802i	$0.357755\pi$
$942$	0	0
$943$	−9.57299	−0.311739
$944$	0	0
$945$	139.113	4.52536
$946$	0	0
$947$	44.1358	1.43422	0.717110	−	0.696960i	$-0.245464\pi$
$947$	44.1358	1.43422	0.717110	+	0.696960i	$0.245464\pi$
$948$	0	0
$949$	−64.6258	−2.09784
$950$	0	0
$951$	−45.3202	−1.46961
$952$	0	0
$953$	−4.14673	−0.134326	−0.0671629	−	0.997742i	$-0.521395\pi$
$953$	−4.14673	−0.134326	−0.0671629	+	0.997742i	$0.521395\pi$
$954$	0	0
$955$	−17.0047	−0.550260
$956$	0	0
$957$	8.88506	0.287213
$958$	0	0
$959$	−5.16237	−0.166702
$960$	0	0
$961$	44.4484	1.43382
$962$	0	0
$963$	148.739	4.79305
$964$	0	0
$965$	−31.1057	−1.00133
$966$	0	0
$967$	23.0054	0.739804	0.369902	−	0.929071i	$-0.379391\pi$
$967$	23.0054	0.739804	0.369902	+	0.929071i	$0.379391\pi$
$968$	0	0
$969$	0.152703	0.00490552
$970$	0	0
$971$	−16.6214	−0.533405	−0.266703	−	0.963779i	$-0.585934\pi$
$971$	−16.6214	−0.533405	−0.266703	+	0.963779i	$0.585934\pi$
$972$	0	0
$973$	−94.1583	−3.01858
$974$	0	0
$975$	48.9520	1.56772
$976$	0	0
$977$	38.0057	1.21591	0.607955	−	0.793971i	$-0.291990\pi$
$977$	38.0057	1.21591	0.607955	+	0.793971i	$0.291990\pi$
$978$	0	0
$979$	−7.13668	−0.228089
$980$	0	0
$981$	−42.0020	−1.34102
$982$	0	0
$983$	58.8893	1.87828	0.939138	−	0.343539i	$-0.111626\pi$
$983$	58.8893	1.87828	0.939138	+	0.343539i	$0.111626\pi$
$984$	0	0
$985$	−14.5757	−0.464421
$986$	0	0
$987$	115.105	3.66384
$988$	0	0
$989$	−38.6532	−1.22910
$990$	0	0
$991$	35.1039	1.11511	0.557555	−	0.830140i	$-0.311739\pi$
$991$	35.1039	1.11511	0.557555	+	0.830140i	$0.311739\pi$
$992$	0	0
$993$	−24.0538	−0.763325
$994$	0	0
$995$	−10.9963	−0.348606
$996$	0	0
$997$	−19.7687	−0.626081	−0.313041	−	0.949740i	$-0.601348\pi$
$997$	−19.7687	−0.626081	−0.313041	+	0.949740i	$0.601348\pi$
$998$	0	0
$999$	157.549	4.98463

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.f.1.28	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.f.1.28	✓	29	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)

\(f(q)\)	\(=\)	\(q+3.34495 q^{3} -1.55264 q^{5} -5.16237 q^{7} +8.18871 q^{9} +O(q^{10})\)	\(q+3.34495 q^{3} -1.55264 q^{5} -5.16237 q^{7} +8.18871 q^{9} +1.00000 q^{11} -5.65191 q^{13} -5.19350 q^{15} +1.58279 q^{17} +0.0288425 q^{19} -17.2679 q^{21} +7.85691 q^{23} -2.58931 q^{25} +17.3560 q^{27} +2.65626 q^{29} +8.68610 q^{31} +3.34495 q^{33} +8.01530 q^{35} +9.07749 q^{37} -18.9054 q^{39} -1.21842 q^{41} -4.91964 q^{43} -12.7141 q^{45} -6.66585 q^{47} +19.6501 q^{49} +5.29437 q^{51} +0.215830 q^{53} -1.55264 q^{55} +0.0964767 q^{57} +0.410309 q^{59} +0.112356 q^{61} -42.2732 q^{63} +8.77537 q^{65} +10.7794 q^{67} +26.2810 q^{69} -14.8717 q^{71} +11.4343 q^{73} -8.66114 q^{75} -5.16237 q^{77} -10.2845 q^{79} +33.4889 q^{81} +10.3772 q^{83} -2.45751 q^{85} +8.88506 q^{87} -7.13668 q^{89} +29.1773 q^{91} +29.0546 q^{93} -0.0447819 q^{95} -3.42963 q^{97} +8.18871 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more