LMFDB - Embedded newform 6028.2.a.d.1.23

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			1.23
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)

$f(q)$	$=$	$q+2.26016 q^{3} +0.892342 q^{5} +1.42130 q^{7} +2.10834 q^{9} +O(q^{10})$	$q+2.26016 q^{3} +0.892342 q^{5} +1.42130 q^{7} +2.10834 q^{9} -1.00000 q^{11} -6.21682 q^{13} +2.01684 q^{15} -5.54664 q^{17} -1.00121 q^{19} +3.21237 q^{21} -6.61564 q^{23} -4.20373 q^{25} -2.01529 q^{27} -4.82780 q^{29} +5.24795 q^{31} -2.26016 q^{33} +1.26829 q^{35} +10.4755 q^{37} -14.0510 q^{39} +11.1076 q^{41} +5.50946 q^{43} +1.88136 q^{45} -3.63696 q^{47} -4.97990 q^{49} -12.5363 q^{51} -5.51616 q^{53} -0.892342 q^{55} -2.26289 q^{57} -5.15816 q^{59} -5.06560 q^{61} +2.99659 q^{63} -5.54753 q^{65} +1.30509 q^{67} -14.9524 q^{69} +2.98250 q^{71} -6.99685 q^{73} -9.50111 q^{75} -1.42130 q^{77} -10.8553 q^{79} -10.8799 q^{81} -1.67284 q^{83} -4.94950 q^{85} -10.9116 q^{87} -18.4423 q^{89} -8.83597 q^{91} +11.8612 q^{93} -0.893418 q^{95} -7.07466 q^{97} -2.10834 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.26016	1.30491	0.652453	−	0.757829i	$-0.273740\pi$
$3$	2.26016	1.30491	0.652453	+	0.757829i	$0.273740\pi$
$4$	0	0
$5$	0.892342	0.399068	0.199534	−	0.979891i	$-0.436057\pi$
$5$	0.892342	0.399068	0.199534	+	0.979891i	$0.436057\pi$
$6$	0	0
$7$	1.42130	0.537201	0.268601	−	0.963252i	$-0.413439\pi$
$7$	1.42130	0.537201	0.268601	+	0.963252i	$0.413439\pi$
$8$	0	0
$9$	2.10834	0.702781
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.21682	−1.72423	−0.862117	−	0.506709i	$-0.830862\pi$
$13$	−6.21682	−1.72423	−0.862117	+	0.506709i	$0.830862\pi$
$14$	0	0
$15$	2.01684	0.520746
$16$	0	0
$17$	−5.54664	−1.34526	−0.672628	−	0.739980i	$-0.734835\pi$
$17$	−5.54664	−1.34526	−0.672628	+	0.739980i	$0.734835\pi$
$18$	0	0
$19$	−1.00121	−0.229692	−0.114846	−	0.993383i	$-0.536637\pi$
$19$	−1.00121	−0.229692	−0.114846	+	0.993383i	$0.536637\pi$
$20$	0	0
$21$	3.21237	0.700997
$22$	0	0
$23$	−6.61564	−1.37946	−0.689728	−	0.724069i	$-0.742270\pi$
$23$	−6.61564	−1.37946	−0.689728	+	0.724069i	$0.742270\pi$
$24$	0	0
$25$	−4.20373	−0.840745
$26$	0	0
$27$	−2.01529	−0.387843
$28$	0	0
$29$	−4.82780	−0.896499	−0.448250	−	0.893908i	$-0.647952\pi$
$29$	−4.82780	−0.896499	−0.448250	+	0.893908i	$0.647952\pi$
$30$	0	0
$31$	5.24795	0.942560	0.471280	−	0.881984i	$-0.343792\pi$
$31$	5.24795	0.942560	0.471280	+	0.881984i	$0.343792\pi$
$32$	0	0
$33$	−2.26016	−0.393444
$34$	0	0
$35$	1.26829	0.214380
$36$	0	0
$37$	10.4755	1.72215	0.861077	−	0.508475i	$-0.169791\pi$
$37$	10.4755	1.72215	0.861077	+	0.508475i	$0.169791\pi$
$38$	0	0
$39$	−14.0510	−2.24997
$40$	0	0
$41$	11.1076	1.73472	0.867361	−	0.497680i	$-0.165815\pi$
$41$	11.1076	1.73472	0.867361	+	0.497680i	$0.165815\pi$
$42$	0	0
$43$	5.50946	0.840184	0.420092	−	0.907481i	$-0.361998\pi$
$43$	5.50946	0.840184	0.420092	+	0.907481i	$0.361998\pi$
$44$	0	0
$45$	1.88136	0.280457
$46$	0	0
$47$	−3.63696	−0.530506	−0.265253	−	0.964179i	$-0.585455\pi$
$47$	−3.63696	−0.530506	−0.265253	+	0.964179i	$0.585455\pi$
$48$	0	0
$49$	−4.97990	−0.711415
$50$	0	0
$51$	−12.5363	−1.75543
$52$	0	0
$53$	−5.51616	−0.757703	−0.378852	−	0.925457i	$-0.623681\pi$
$53$	−5.51616	−0.757703	−0.378852	+	0.925457i	$0.623681\pi$
$54$	0	0
$55$	−0.892342	−0.120323
$56$	0	0
$57$	−2.26289	−0.299727
$58$	0	0
$59$	−5.15816	−0.671536	−0.335768	−	0.941945i	$-0.608996\pi$
$59$	−5.15816	−0.671536	−0.335768	+	0.941945i	$0.608996\pi$
$60$	0	0
$61$	−5.06560	−0.648583	−0.324292	−	0.945957i	$-0.605126\pi$
$61$	−5.06560	−0.648583	−0.324292	+	0.945957i	$0.605126\pi$
$62$	0	0
$63$	2.99659	0.377535
$64$	0	0
$65$	−5.54753	−0.688086
$66$	0	0
$67$	1.30509	0.159442	0.0797210	−	0.996817i	$-0.474597\pi$
$67$	1.30509	0.159442	0.0797210	+	0.996817i	$0.474597\pi$
$68$	0	0
$69$	−14.9524	−1.80006
$70$	0	0
$71$	2.98250	0.353958	0.176979	−	0.984215i	$-0.443368\pi$
$71$	2.98250	0.353958	0.176979	+	0.984215i	$0.443368\pi$
$72$	0	0
$73$	−6.99685	−0.818919	−0.409460	−	0.912328i	$-0.634283\pi$
$73$	−6.99685	−0.818919	−0.409460	+	0.912328i	$0.634283\pi$
$74$	0	0
$75$	−9.50111	−1.09709
$76$	0	0
$77$	−1.42130	−0.161972
$78$	0	0
$79$	−10.8553	−1.22132	−0.610660	−	0.791893i	$-0.709096\pi$
$79$	−10.8553	−1.22132	−0.610660	+	0.791893i	$0.709096\pi$
$80$	0	0
$81$	−10.8799	−1.20888
$82$	0	0
$83$	−1.67284	−0.183618	−0.0918088	−	0.995777i	$-0.529265\pi$
$83$	−1.67284	−0.183618	−0.0918088	+	0.995777i	$0.529265\pi$
$84$	0	0
$85$	−4.94950	−0.536848
$86$	0	0
$87$	−10.9116	−1.16985
$88$	0	0
$89$	−18.4423	−1.95488	−0.977441	−	0.211210i	$-0.932260\pi$
$89$	−18.4423	−1.95488	−0.977441	+	0.211210i	$0.932260\pi$
$90$	0	0
$91$	−8.83597	−0.926261
$92$	0	0
$93$	11.8612	1.22995
$94$	0	0
$95$	−0.893418	−0.0916627
$96$	0	0
$97$	−7.07466	−0.718323	−0.359161	−	0.933275i	$-0.616937\pi$
$97$	−7.07466	−0.718323	−0.359161	+	0.933275i	$0.616937\pi$
$98$	0	0
$99$	−2.10834	−0.211897
$100$	0	0
$101$	−11.0751	−1.10201	−0.551007	−	0.834501i	$-0.685756\pi$
$101$	−11.0751	−1.10201	−0.551007	+	0.834501i	$0.685756\pi$
$102$	0	0
$103$	14.0796	1.38731	0.693653	−	0.720310i	$-0.256000\pi$
$103$	14.0796	1.38731	0.693653	+	0.720310i	$0.256000\pi$
$104$	0	0
$105$	2.86654	0.279745
$106$	0	0
$107$	13.1200	1.26835	0.634177	−	0.773188i	$-0.281339\pi$
$107$	13.1200	1.26835	0.634177	+	0.773188i	$0.281339\pi$
$108$	0	0
$109$	−6.57991	−0.630241	−0.315121	−	0.949052i	$-0.602045\pi$
$109$	−6.57991	−0.630241	−0.315121	+	0.949052i	$0.602045\pi$
$110$	0	0
$111$	23.6762	2.24725
$112$	0	0
$113$	12.9104	1.21451	0.607253	−	0.794508i	$-0.292271\pi$
$113$	12.9104	1.21451	0.607253	+	0.794508i	$0.292271\pi$
$114$	0	0
$115$	−5.90341	−0.550496
$116$	0	0
$117$	−13.1072	−1.21176
$118$	0	0
$119$	−7.88344	−0.722674
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	25.1051	2.26365
$124$	0	0
$125$	−8.21287	−0.734582
$126$	0	0
$127$	7.53457	0.668585	0.334293	−	0.942469i	$-0.391503\pi$
$127$	7.53457	0.668585	0.334293	+	0.942469i	$0.391503\pi$
$128$	0	0
$129$	12.4523	1.09636
$130$	0	0
$131$	−7.27599	−0.635706	−0.317853	−	0.948140i	$-0.602962\pi$
$131$	−7.27599	−0.635706	−0.317853	+	0.948140i	$0.602962\pi$
$132$	0	0
$133$	−1.42301	−0.123391
$134$	0	0
$135$	−1.79833	−0.154775
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	−7.58937	−0.643722	−0.321861	−	0.946787i	$-0.604308\pi$
$139$	−7.58937	−0.643722	−0.321861	+	0.946787i	$0.604308\pi$
$140$	0	0
$141$	−8.22014	−0.692260
$142$	0	0
$143$	6.21682	0.519876
$144$	0	0
$145$	−4.30805	−0.357764
$146$	0	0
$147$	−11.2554	−0.928330
$148$	0	0
$149$	−16.9733	−1.39051	−0.695255	−	0.718763i	$-0.744709\pi$
$149$	−16.9733	−1.39051	−0.695255	+	0.718763i	$0.744709\pi$
$150$	0	0
$151$	11.3474	0.923439	0.461720	−	0.887026i	$-0.347233\pi$
$151$	11.3474	0.923439	0.461720	+	0.887026i	$0.347233\pi$
$152$	0	0
$153$	−11.6942	−0.945421
$154$	0	0
$155$	4.68297	0.376145
$156$	0	0
$157$	23.2876	1.85855	0.929277	−	0.369383i	$-0.120431\pi$
$157$	23.2876	1.85855	0.929277	+	0.369383i	$0.120431\pi$
$158$	0	0
$159$	−12.4674	−0.988732
$160$	0	0
$161$	−9.40281	−0.741045
$162$	0	0
$163$	12.2824	0.962035	0.481018	−	0.876711i	$-0.340267\pi$
$163$	12.2824	0.962035	0.481018	+	0.876711i	$0.340267\pi$
$164$	0	0
$165$	−2.01684	−0.157011
$166$	0	0
$167$	−16.6473	−1.28821	−0.644105	−	0.764937i	$-0.722770\pi$
$167$	−16.6473	−1.28821	−0.644105	+	0.764937i	$0.722770\pi$
$168$	0	0
$169$	25.6488	1.97299
$170$	0	0
$171$	−2.11088	−0.161423
$172$	0	0
$173$	2.89567	0.220153	0.110077	−	0.993923i	$-0.464890\pi$
$173$	2.89567	0.220153	0.110077	+	0.993923i	$0.464890\pi$
$174$	0	0
$175$	−5.97476	−0.451649
$176$	0	0
$177$	−11.6583	−0.876291
$178$	0	0
$179$	19.6017	1.46510	0.732550	−	0.680713i	$-0.238330\pi$
$179$	19.6017	1.46510	0.732550	+	0.680713i	$0.238330\pi$
$180$	0	0
$181$	1.13092	0.0840608	0.0420304	−	0.999116i	$-0.486617\pi$
$181$	1.13092	0.0840608	0.0420304	+	0.999116i	$0.486617\pi$
$182$	0	0
$183$	−11.4491	−0.846341
$184$	0	0
$185$	9.34769	0.687256
$186$	0	0
$187$	5.54664	0.405610
$188$	0	0
$189$	−2.86433	−0.208350
$190$	0	0
$191$	26.9970	1.95344	0.976719	−	0.214524i	$-0.0688199\pi$
$191$	26.9970	1.95344	0.976719	+	0.214524i	$0.0688199\pi$
$192$	0	0
$193$	−8.51687	−0.613058	−0.306529	−	0.951861i	$-0.599168\pi$
$193$	−8.51687	−0.613058	−0.306529	+	0.951861i	$0.599168\pi$
$194$	0	0
$195$	−12.5383	−0.897888
$196$	0	0
$197$	−21.7856	−1.55216	−0.776079	−	0.630636i	$-0.782794\pi$
$197$	−21.7856	−1.55216	−0.776079	+	0.630636i	$0.782794\pi$
$198$	0	0
$199$	12.4982	0.885977	0.442989	−	0.896527i	$-0.353918\pi$
$199$	12.4982	0.885977	0.442989	+	0.896527i	$0.353918\pi$
$200$	0	0
$201$	2.94972	0.208057
$202$	0	0
$203$	−6.86175	−0.481600
$204$	0	0
$205$	9.91181	0.692271
$206$	0	0
$207$	−13.9480	−0.969456
$208$	0	0
$209$	1.00121	0.0692548
$210$	0	0
$211$	−16.4115	−1.12982	−0.564908	−	0.825154i	$-0.691088\pi$
$211$	−16.4115	−1.12982	−0.564908	+	0.825154i	$0.691088\pi$
$212$	0	0
$213$	6.74095	0.461882
$214$	0	0
$215$	4.91632	0.335290
$216$	0	0
$217$	7.45892	0.506344
$218$	0	0
$219$	−15.8140	−1.06861
$220$	0	0
$221$	34.4824	2.31954
$222$	0	0
$223$	−5.61461	−0.375982	−0.187991	−	0.982171i	$-0.560198\pi$
$223$	−5.61461	−0.375982	−0.187991	+	0.982171i	$0.560198\pi$
$224$	0	0
$225$	−8.86290	−0.590860
$226$	0	0
$227$	20.2329	1.34291	0.671453	−	0.741047i	$-0.265670\pi$
$227$	20.2329	1.34291	0.671453	+	0.741047i	$0.265670\pi$
$228$	0	0
$229$	9.22963	0.609911	0.304955	−	0.952367i	$-0.401358\pi$
$229$	9.22963	0.609911	0.304955	+	0.952367i	$0.401358\pi$
$230$	0	0
$231$	−3.21237	−0.211359
$232$	0	0
$233$	21.9949	1.44093	0.720467	−	0.693489i	$-0.243927\pi$
$233$	21.9949	1.44093	0.720467	+	0.693489i	$0.243927\pi$
$234$	0	0
$235$	−3.24542	−0.211708
$236$	0	0
$237$	−24.5348	−1.59371
$238$	0	0
$239$	−5.81086	−0.375873	−0.187936	−	0.982181i	$-0.560180\pi$
$239$	−5.81086	−0.375873	−0.187936	+	0.982181i	$0.560180\pi$
$240$	0	0
$241$	11.7372	0.756058	0.378029	−	0.925794i	$-0.376602\pi$
$241$	11.7372	0.756058	0.378029	+	0.925794i	$0.376602\pi$
$242$	0	0
$243$	−18.5445	−1.18963
$244$	0	0
$245$	−4.44378	−0.283903
$246$	0	0
$247$	6.22431	0.396043
$248$	0	0
$249$	−3.78089	−0.239604
$250$	0	0
$251$	4.50437	0.284313	0.142157	−	0.989844i	$-0.454596\pi$
$251$	4.50437	0.284313	0.142157	+	0.989844i	$0.454596\pi$
$252$	0	0
$253$	6.61564	0.415922
$254$	0	0
$255$	−11.1867	−0.700537
$256$	0	0
$257$	−30.7685	−1.91928	−0.959642	−	0.281225i	$-0.909259\pi$
$257$	−30.7685	−1.91928	−0.959642	+	0.281225i	$0.909259\pi$
$258$	0	0
$259$	14.8888	0.925143
$260$	0	0
$261$	−10.1787	−0.630043
$262$	0	0
$263$	−0.692675	−0.0427122	−0.0213561	−	0.999772i	$-0.506798\pi$
$263$	−0.692675	−0.0427122	−0.0213561	+	0.999772i	$0.506798\pi$
$264$	0	0
$265$	−4.92231	−0.302375
$266$	0	0
$267$	−41.6827	−2.55094
$268$	0	0
$269$	−18.8464	−1.14908	−0.574541	−	0.818476i	$-0.694819\pi$
$269$	−18.8464	−1.14908	−0.574541	+	0.818476i	$0.694819\pi$
$270$	0	0
$271$	0.0493418	0.00299730	0.00149865	−	0.999999i	$-0.499523\pi$
$271$	0.0493418	0.00299730	0.00149865	+	0.999999i	$0.499523\pi$
$272$	0	0
$273$	−19.9707	−1.20868
$274$	0	0
$275$	4.20373	0.253494
$276$	0	0
$277$	0.341489	0.0205181	0.0102591	−	0.999947i	$-0.496734\pi$
$277$	0.341489	0.0205181	0.0102591	+	0.999947i	$0.496734\pi$
$278$	0	0
$279$	11.0645	0.662413
$280$	0	0
$281$	−3.20849	−0.191402	−0.0957012	−	0.995410i	$-0.530509\pi$
$281$	−3.20849	−0.191402	−0.0957012	+	0.995410i	$0.530509\pi$
$282$	0	0
$283$	−30.1701	−1.79343	−0.896713	−	0.442613i	$-0.854051\pi$
$283$	−30.1701	−1.79343	−0.896713	+	0.442613i	$0.854051\pi$
$284$	0	0
$285$	−2.01927	−0.119611
$286$	0	0
$287$	15.7873	0.931895
$288$	0	0
$289$	13.7652	0.809716
$290$	0	0
$291$	−15.9899	−0.937344
$292$	0	0
$293$	22.0946	1.29078	0.645390	−	0.763854i	$-0.276695\pi$
$293$	22.0946	1.29078	0.645390	+	0.763854i	$0.276695\pi$
$294$	0	0
$295$	−4.60285	−0.267988
$296$	0	0
$297$	2.01529	0.116939
$298$	0	0
$299$	41.1282	2.37851
$300$	0	0
$301$	7.83060	0.451348
$302$	0	0
$303$	−25.0316	−1.43803
$304$	0	0
$305$	−4.52025	−0.258829
$306$	0	0
$307$	−27.0847	−1.54580	−0.772901	−	0.634526i	$-0.781195\pi$
$307$	−27.0847	−1.54580	−0.772901	+	0.634526i	$0.781195\pi$
$308$	0	0
$309$	31.8222	1.81030
$310$	0	0
$311$	21.4957	1.21891	0.609455	−	0.792821i	$-0.291388\pi$
$311$	21.4957	1.21891	0.609455	+	0.792821i	$0.291388\pi$
$312$	0	0
$313$	14.4578	0.817202	0.408601	−	0.912713i	$-0.366017\pi$
$313$	14.4578	0.817202	0.408601	+	0.912713i	$0.366017\pi$
$314$	0	0
$315$	2.67398	0.150662
$316$	0	0
$317$	25.1973	1.41522	0.707610	−	0.706604i	$-0.249774\pi$
$317$	25.1973	1.41522	0.707610	+	0.706604i	$0.249774\pi$
$318$	0	0
$319$	4.82780	0.270305
$320$	0	0
$321$	29.6532	1.65508
$322$	0	0
$323$	5.55332	0.308995
$324$	0	0
$325$	26.1338	1.44964
$326$	0	0
$327$	−14.8717	−0.822406
$328$	0	0
$329$	−5.16922	−0.284988
$330$	0	0
$331$	0.939069	0.0516159	0.0258079	−	0.999667i	$-0.491784\pi$
$331$	0.939069	0.0516159	0.0258079	+	0.999667i	$0.491784\pi$
$332$	0	0
$333$	22.0859	1.21030
$334$	0	0
$335$	1.16459	0.0636281
$336$	0	0
$337$	−30.0175	−1.63516	−0.817580	−	0.575815i	$-0.804685\pi$
$337$	−30.0175	−1.63516	−0.817580	+	0.575815i	$0.804685\pi$
$338$	0	0
$339$	29.1796	1.58482
$340$	0	0
$341$	−5.24795	−0.284193
$342$	0	0
$343$	−17.0270	−0.919374
$344$	0	0
$345$	−13.3427	−0.718346
$346$	0	0
$347$	5.03004	0.270026	0.135013	−	0.990844i	$-0.456892\pi$
$347$	5.03004	0.270026	0.135013	+	0.990844i	$0.456892\pi$
$348$	0	0
$349$	−33.6435	−1.80090	−0.900448	−	0.434963i	$-0.856762\pi$
$349$	−33.6435	−1.80090	−0.900448	+	0.434963i	$0.856762\pi$
$350$	0	0
$351$	12.5287	0.668732
$352$	0	0
$353$	4.95891	0.263936	0.131968	−	0.991254i	$-0.457870\pi$
$353$	4.95891	0.263936	0.131968	+	0.991254i	$0.457870\pi$
$354$	0	0
$355$	2.66141	0.141253
$356$	0	0
$357$	−17.8179	−0.943022
$358$	0	0
$359$	−22.4309	−1.18386	−0.591928	−	0.805991i	$-0.701633\pi$
$359$	−22.4309	−1.18386	−0.591928	+	0.805991i	$0.701633\pi$
$360$	0	0
$361$	−17.9976	−0.947241
$362$	0	0
$363$	2.26016	0.118628
$364$	0	0
$365$	−6.24358	−0.326804
$366$	0	0
$367$	−17.3428	−0.905286	−0.452643	−	0.891692i	$-0.649519\pi$
$367$	−17.3428	−0.905286	−0.452643	+	0.891692i	$0.649519\pi$
$368$	0	0
$369$	23.4187	1.21913
$370$	0	0
$371$	−7.84013	−0.407039
$372$	0	0
$373$	32.0405	1.65900	0.829498	−	0.558510i	$-0.188627\pi$
$373$	32.0405	1.65900	0.829498	+	0.558510i	$0.188627\pi$
$374$	0	0
$375$	−18.5624	−0.958561
$376$	0	0
$377$	30.0135	1.54577
$378$	0	0
$379$	29.5697	1.51889	0.759446	−	0.650571i	$-0.225470\pi$
$379$	29.5697	1.51889	0.759446	+	0.650571i	$0.225470\pi$
$380$	0	0
$381$	17.0294	0.872441
$382$	0	0
$383$	−7.26134	−0.371037	−0.185518	−	0.982641i	$-0.559396\pi$
$383$	−7.26134	−0.371037	−0.185518	+	0.982641i	$0.559396\pi$
$384$	0	0
$385$	−1.26829	−0.0646379
$386$	0	0
$387$	11.6158	0.590466
$388$	0	0
$389$	5.07140	0.257130	0.128565	−	0.991701i	$-0.458963\pi$
$389$	5.07140	0.257130	0.128565	+	0.991701i	$0.458963\pi$
$390$	0	0
$391$	36.6945	1.85572
$392$	0	0
$393$	−16.4449	−0.829537
$394$	0	0
$395$	−9.68667	−0.487390
$396$	0	0
$397$	−17.4937	−0.877985	−0.438993	−	0.898491i	$-0.644665\pi$
$397$	−17.4937	−0.877985	−0.438993	+	0.898491i	$0.644665\pi$
$398$	0	0
$399$	−3.21625	−0.161014
$400$	0	0
$401$	−18.1500	−0.906368	−0.453184	−	0.891417i	$-0.649712\pi$
$401$	−18.1500	−0.906368	−0.453184	+	0.891417i	$0.649712\pi$
$402$	0	0
$403$	−32.6256	−1.62519
$404$	0	0
$405$	−9.70861	−0.482425
$406$	0	0
$407$	−10.4755	−0.519249
$408$	0	0
$409$	16.4450	0.813152	0.406576	−	0.913617i	$-0.366723\pi$
$409$	16.4450	0.813152	0.406576	+	0.913617i	$0.366723\pi$
$410$	0	0
$411$	2.26016	0.111486
$412$	0	0
$413$	−7.33130	−0.360750
$414$	0	0
$415$	−1.49274	−0.0732759
$416$	0	0
$417$	−17.1532	−0.839997
$418$	0	0
$419$	−34.8869	−1.70434	−0.852168	−	0.523268i	$-0.824713\pi$
$419$	−34.8869	−1.70434	−0.852168	+	0.523268i	$0.824713\pi$
$420$	0	0
$421$	−9.19876	−0.448320	−0.224160	−	0.974552i	$-0.571964\pi$
$421$	−9.19876	−0.448320	−0.224160	+	0.974552i	$0.571964\pi$
$422$	0	0
$423$	−7.66797	−0.372829
$424$	0	0
$425$	23.3165	1.13102
$426$	0	0
$427$	−7.19974	−0.348420
$428$	0	0
$429$	14.0510	0.678390
$430$	0	0
$431$	21.5026	1.03574	0.517871	−	0.855459i	$-0.326725\pi$
$431$	21.5026	1.03574	0.517871	+	0.855459i	$0.326725\pi$
$432$	0	0
$433$	24.0229	1.15447	0.577235	−	0.816578i	$-0.304132\pi$
$433$	24.0229	1.15447	0.577235	+	0.816578i	$0.304132\pi$
$434$	0	0
$435$	−9.73689	−0.466848
$436$	0	0
$437$	6.62361	0.316850
$438$	0	0
$439$	4.61213	0.220125	0.110063	−	0.993925i	$-0.464895\pi$
$439$	4.61213	0.220125	0.110063	+	0.993925i	$0.464895\pi$
$440$	0	0
$441$	−10.4993	−0.499969
$442$	0	0
$443$	−27.7314	−1.31756	−0.658779	−	0.752336i	$-0.728927\pi$
$443$	−27.7314	−1.31756	−0.658779	+	0.752336i	$0.728927\pi$
$444$	0	0
$445$	−16.4569	−0.780130
$446$	0	0
$447$	−38.3625	−1.81449
$448$	0	0
$449$	−39.0269	−1.84179	−0.920897	−	0.389807i	$-0.872542\pi$
$449$	−39.0269	−1.84179	−0.920897	+	0.389807i	$0.872542\pi$
$450$	0	0
$451$	−11.1076	−0.523038
$452$	0	0
$453$	25.6470	1.20500
$454$	0	0
$455$	−7.88471	−0.369641
$456$	0	0
$457$	−38.4515	−1.79868	−0.899342	−	0.437246i	$-0.855954\pi$
$457$	−38.4515	−1.79868	−0.899342	+	0.437246i	$0.855954\pi$
$458$	0	0
$459$	11.1781	0.521748
$460$	0	0
$461$	3.46829	0.161535	0.0807673	−	0.996733i	$-0.474263\pi$
$461$	3.46829	0.161535	0.0807673	+	0.996733i	$0.474263\pi$
$462$	0	0
$463$	1.98226	0.0921233	0.0460616	−	0.998939i	$-0.485333\pi$
$463$	1.98226	0.0921233	0.0460616	+	0.998939i	$0.485333\pi$
$464$	0	0
$465$	10.5843	0.490834
$466$	0	0
$467$	−6.76846	−0.313207	−0.156603	−	0.987662i	$-0.550054\pi$
$467$	−6.76846	−0.313207	−0.156603	+	0.987662i	$0.550054\pi$
$468$	0	0
$469$	1.85492	0.0856525
$470$	0	0
$471$	52.6339	2.42524
$472$	0	0
$473$	−5.50946	−0.253325
$474$	0	0
$475$	4.20879	0.193113
$476$	0	0
$477$	−11.6300	−0.532500
$478$	0	0
$479$	23.0319	1.05235	0.526177	−	0.850375i	$-0.323625\pi$
$479$	23.0319	1.05235	0.526177	+	0.850375i	$0.323625\pi$
$480$	0	0
$481$	−65.1240	−2.96940
$482$	0	0
$483$	−21.2519	−0.966995
$484$	0	0
$485$	−6.31302	−0.286659
$486$	0	0
$487$	−7.21076	−0.326751	−0.163375	−	0.986564i	$-0.552238\pi$
$487$	−7.21076	−0.326751	−0.163375	+	0.986564i	$0.552238\pi$
$488$	0	0
$489$	27.7603	1.25537
$490$	0	0
$491$	−1.32406	−0.0597541	−0.0298770	−	0.999554i	$-0.509512\pi$
$491$	−1.32406	−0.0597541	−0.0298770	+	0.999554i	$0.509512\pi$
$492$	0	0
$493$	26.7780	1.20602
$494$	0	0
$495$	−1.88136	−0.0845610
$496$	0	0
$497$	4.23903	0.190147
$498$	0	0
$499$	−10.7761	−0.482406	−0.241203	−	0.970475i	$-0.577542\pi$
$499$	−10.7761	−0.482406	−0.241203	+	0.970475i	$0.577542\pi$
$500$	0	0
$501$	−37.6257	−1.68099
$502$	0	0
$503$	−7.51560	−0.335104	−0.167552	−	0.985863i	$-0.553586\pi$
$503$	−7.51560	−0.335104	−0.167552	+	0.985863i	$0.553586\pi$
$504$	0	0
$505$	−9.88278	−0.439778
$506$	0	0
$507$	57.9705	2.57456
$508$	0	0
$509$	32.0716	1.42155	0.710774	−	0.703421i	$-0.248345\pi$
$509$	32.0716	1.42155	0.710774	+	0.703421i	$0.248345\pi$
$510$	0	0
$511$	−9.94463	−0.439924
$512$	0	0
$513$	2.01772	0.0890844
$514$	0	0
$515$	12.5638	0.553629
$516$	0	0
$517$	3.63696	0.159953
$518$	0	0
$519$	6.54468	0.287280
$520$	0	0
$521$	−12.7803	−0.559914	−0.279957	−	0.960013i	$-0.590320\pi$
$521$	−12.7803	−0.559914	−0.279957	+	0.960013i	$0.590320\pi$
$522$	0	0
$523$	18.1645	0.794279	0.397140	−	0.917758i	$-0.370003\pi$
$523$	18.1645	0.794279	0.397140	+	0.917758i	$0.370003\pi$
$524$	0	0
$525$	−13.5039	−0.589360
$526$	0	0
$527$	−29.1085	−1.26799
$528$	0	0
$529$	20.7667	0.902898
$530$	0	0
$531$	−10.8752	−0.471943
$532$	0	0
$533$	−69.0541	−2.99107
$534$	0	0
$535$	11.7075	0.506159
$536$	0	0
$537$	44.3031	1.91182
$538$	0	0
$539$	4.97990	0.214500
$540$	0	0
$541$	−1.24715	−0.0536193	−0.0268096	−	0.999641i	$-0.508535\pi$
$541$	−1.24715	−0.0536193	−0.0268096	+	0.999641i	$0.508535\pi$
$542$	0	0
$543$	2.55607	0.109691
$544$	0	0
$545$	−5.87154	−0.251509
$546$	0	0
$547$	27.6745	1.18328	0.591639	−	0.806203i	$-0.298481\pi$
$547$	27.6745	1.18328	0.591639	+	0.806203i	$0.298481\pi$
$548$	0	0
$549$	−10.6800	−0.455812
$550$	0	0
$551$	4.83361	0.205919
$552$	0	0
$553$	−15.4287	−0.656095
$554$	0	0
$555$	21.1273	0.896805
$556$	0	0
$557$	24.1592	1.02366	0.511830	−	0.859087i	$-0.328968\pi$
$557$	24.1592	1.02366	0.511830	+	0.859087i	$0.328968\pi$
$558$	0	0
$559$	−34.2513	−1.44868
$560$	0	0
$561$	12.5363	0.529283
$562$	0	0
$563$	−23.7588	−1.00131	−0.500656	−	0.865646i	$-0.666908\pi$
$563$	−23.7588	−1.00131	−0.500656	+	0.865646i	$0.666908\pi$
$564$	0	0
$565$	11.5205	0.484670
$566$	0	0
$567$	−15.4636	−0.649412
$568$	0	0
$569$	20.2077	0.847152	0.423576	−	0.905861i	$-0.360775\pi$
$569$	20.2077	0.847152	0.423576	+	0.905861i	$0.360775\pi$
$570$	0	0
$571$	40.6878	1.70273	0.851366	−	0.524571i	$-0.175774\pi$
$571$	40.6878	1.70273	0.851366	+	0.524571i	$0.175774\pi$
$572$	0	0
$573$	61.0178	2.54905
$574$	0	0
$575$	27.8103	1.15977
$576$	0	0
$577$	20.8293	0.867134	0.433567	−	0.901121i	$-0.357255\pi$
$577$	20.8293	0.867134	0.433567	+	0.901121i	$0.357255\pi$
$578$	0	0
$579$	−19.2495	−0.799983
$580$	0	0
$581$	−2.37760	−0.0986397
$582$	0	0
$583$	5.51616	0.228456
$584$	0	0
$585$	−11.6961	−0.483574
$586$	0	0
$587$	36.0234	1.48685	0.743423	−	0.668821i	$-0.233201\pi$
$587$	36.0234	1.48685	0.743423	+	0.668821i	$0.233201\pi$
$588$	0	0
$589$	−5.25428	−0.216499
$590$	0	0
$591$	−49.2390	−2.02542
$592$	0	0
$593$	−28.4960	−1.17019	−0.585096	−	0.810964i	$-0.698943\pi$
$593$	−28.4960	−1.17019	−0.585096	+	0.810964i	$0.698943\pi$
$594$	0	0
$595$	−7.03473	−0.288396
$596$	0	0
$597$	28.2481	1.15612
$598$	0	0
$599$	−14.3623	−0.586827	−0.293414	−	0.955986i	$-0.594791\pi$
$599$	−14.3623	−0.586827	−0.293414	+	0.955986i	$0.594791\pi$
$600$	0	0
$601$	6.76297	0.275867	0.137934	−	0.990441i	$-0.455954\pi$
$601$	6.76297	0.275867	0.137934	+	0.990441i	$0.455954\pi$
$602$	0	0
$603$	2.75158	0.112053
$604$	0	0
$605$	0.892342	0.0362789
$606$	0	0
$607$	−7.94783	−0.322592	−0.161296	−	0.986906i	$-0.551567\pi$
$607$	−7.94783	−0.322592	−0.161296	+	0.986906i	$0.551567\pi$
$608$	0	0
$609$	−15.5087	−0.628444
$610$	0	0
$611$	22.6103	0.914716
$612$	0	0
$613$	−21.3108	−0.860735	−0.430368	−	0.902654i	$-0.641616\pi$
$613$	−21.3108	−0.860735	−0.430368	+	0.902654i	$0.641616\pi$
$614$	0	0
$615$	22.4023	0.903349
$616$	0	0
$617$	−6.83350	−0.275106	−0.137553	−	0.990494i	$-0.543924\pi$
$617$	−6.83350	−0.275106	−0.137553	+	0.990494i	$0.543924\pi$
$618$	0	0
$619$	19.9812	0.803111	0.401556	−	0.915835i	$-0.368470\pi$
$619$	19.9812	0.803111	0.401556	+	0.915835i	$0.368470\pi$
$620$	0	0
$621$	13.3324	0.535012
$622$	0	0
$623$	−26.2121	−1.05016
$624$	0	0
$625$	13.6899	0.547597
$626$	0	0
$627$	2.26289	0.0903710
$628$	0	0
$629$	−58.1035	−2.31674
$630$	0	0
$631$	41.4503	1.65011	0.825056	−	0.565052i	$-0.191144\pi$
$631$	41.4503	1.65011	0.825056	+	0.565052i	$0.191144\pi$
$632$	0	0
$633$	−37.0928	−1.47430
$634$	0	0
$635$	6.72342	0.266811
$636$	0	0
$637$	30.9591	1.22665
$638$	0	0
$639$	6.28814	0.248755
$640$	0	0
$641$	9.10718	0.359712	0.179856	−	0.983693i	$-0.442437\pi$
$641$	9.10718	0.359712	0.179856	+	0.983693i	$0.442437\pi$
$642$	0	0
$643$	−43.6329	−1.72071	−0.860357	−	0.509692i	$-0.829759\pi$
$643$	−43.6329	−1.72071	−0.860357	+	0.509692i	$0.829759\pi$
$644$	0	0
$645$	11.1117	0.437523
$646$	0	0
$647$	−7.55592	−0.297054	−0.148527	−	0.988908i	$-0.547453\pi$
$647$	−7.55592	−0.297054	−0.148527	+	0.988908i	$0.547453\pi$
$648$	0	0
$649$	5.15816	0.202476
$650$	0	0
$651$	16.8584	0.660732
$652$	0	0
$653$	3.35827	0.131419	0.0657096	−	0.997839i	$-0.479069\pi$
$653$	3.35827	0.131419	0.0657096	+	0.997839i	$0.479069\pi$
$654$	0	0
$655$	−6.49267	−0.253690
$656$	0	0
$657$	−14.7518	−0.575521
$658$	0	0
$659$	38.9277	1.51641	0.758204	−	0.652017i	$-0.226077\pi$
$659$	38.9277	1.51641	0.758204	+	0.652017i	$0.226077\pi$
$660$	0	0
$661$	1.26536	0.0492168	0.0246084	−	0.999697i	$-0.492166\pi$
$661$	1.26536	0.0492168	0.0246084	+	0.999697i	$0.492166\pi$
$662$	0	0
$663$	77.9359	3.02678
$664$	0	0
$665$	−1.26982	−0.0492413
$666$	0	0
$667$	31.9389	1.23668
$668$	0	0
$669$	−12.6900	−0.490622
$670$	0	0
$671$	5.06560	0.195555
$672$	0	0
$673$	0.470899	0.0181518	0.00907591	−	0.999959i	$-0.497111\pi$
$673$	0.470899	0.0181518	0.00907591	+	0.999959i	$0.497111\pi$
$674$	0	0
$675$	8.47173	0.326077
$676$	0	0
$677$	−37.1977	−1.42962	−0.714812	−	0.699317i	$-0.753488\pi$
$677$	−37.1977	−1.42962	−0.714812	+	0.699317i	$0.753488\pi$
$678$	0	0
$679$	−10.0552	−0.385884
$680$	0	0
$681$	45.7297	1.75237
$682$	0	0
$683$	−3.46422	−0.132555	−0.0662773	−	0.997801i	$-0.521112\pi$
$683$	−3.46422	−0.132555	−0.0662773	+	0.997801i	$0.521112\pi$
$684$	0	0
$685$	0.892342	0.0340946
$686$	0	0
$687$	20.8605	0.795877
$688$	0	0
$689$	34.2930	1.30646
$690$	0	0
$691$	−27.2707	−1.03743	−0.518713	−	0.854948i	$-0.673589\pi$
$691$	−27.2707	−1.03743	−0.518713	+	0.854948i	$0.673589\pi$
$692$	0	0
$693$	−2.99659	−0.113831
$694$	0	0
$695$	−6.77232	−0.256889
$696$	0	0
$697$	−61.6100	−2.33365
$698$	0	0
$699$	49.7121	1.88029
$700$	0	0
$701$	−11.8758	−0.448541	−0.224271	−	0.974527i	$-0.572000\pi$
$701$	−11.8758	−0.448541	−0.224271	+	0.974527i	$0.572000\pi$
$702$	0	0
$703$	−10.4881	−0.395565
$704$	0	0
$705$	−7.33518	−0.276259
$706$	0	0
$707$	−15.7411	−0.592003
$708$	0	0
$709$	38.4227	1.44299	0.721497	−	0.692417i	$-0.243454\pi$
$709$	38.4227	1.44299	0.721497	+	0.692417i	$0.243454\pi$
$710$	0	0
$711$	−22.8868	−0.858321
$712$	0	0
$713$	−34.7185	−1.30022
$714$	0	0
$715$	5.54753	0.207466
$716$	0	0
$717$	−13.1335	−0.490479
$718$	0	0
$719$	−48.8439	−1.82157	−0.910784	−	0.412882i	$-0.864522\pi$
$719$	−48.8439	−1.82157	−0.910784	+	0.412882i	$0.864522\pi$
$720$	0	0
$721$	20.0114	0.745262
$722$	0	0
$723$	26.5279	0.986585
$724$	0	0
$725$	20.2947	0.753727
$726$	0	0
$727$	39.6110	1.46909	0.734545	−	0.678560i	$-0.237396\pi$
$727$	39.6110	1.46909	0.734545	+	0.678560i	$0.237396\pi$
$728$	0	0
$729$	−9.27395	−0.343479
$730$	0	0
$731$	−30.5590	−1.13026
$732$	0	0
$733$	17.5157	0.646958	0.323479	−	0.946235i	$-0.395147\pi$
$733$	17.5157	0.646958	0.323479	+	0.946235i	$0.395147\pi$
$734$	0	0
$735$	−10.0437	−0.370466
$736$	0	0
$737$	−1.30509	−0.0480736
$738$	0	0
$739$	−44.6047	−1.64081	−0.820406	−	0.571782i	$-0.806252\pi$
$739$	−44.6047	−1.64081	−0.820406	+	0.571782i	$0.806252\pi$
$740$	0	0
$741$	14.0680	0.516799
$742$	0	0
$743$	−5.34527	−0.196099	−0.0980495	−	0.995182i	$-0.531260\pi$
$743$	−5.34527	−0.196099	−0.0980495	+	0.995182i	$0.531260\pi$
$744$	0	0
$745$	−15.1460	−0.554908
$746$	0	0
$747$	−3.52691	−0.129043
$748$	0	0
$749$	18.6474	0.681361
$750$	0	0
$751$	−13.6581	−0.498392	−0.249196	−	0.968453i	$-0.580166\pi$
$751$	−13.6581	−0.498392	−0.249196	+	0.968453i	$0.580166\pi$
$752$	0	0
$753$	10.1806	0.371002
$754$	0	0
$755$	10.1258	0.368515
$756$	0	0
$757$	39.6526	1.44120	0.720599	−	0.693353i	$-0.243867\pi$
$757$	39.6526	1.44120	0.720599	+	0.693353i	$0.243867\pi$
$758$	0	0
$759$	14.9524	0.542739
$760$	0	0
$761$	31.7653	1.15149	0.575746	−	0.817628i	$-0.304712\pi$
$761$	31.7653	1.15149	0.575746	+	0.817628i	$0.304712\pi$
$762$	0	0
$763$	−9.35204	−0.338566
$764$	0	0
$765$	−10.4352	−0.377287
$766$	0	0
$767$	32.0674	1.15789
$768$	0	0
$769$	44.8982	1.61907	0.809535	−	0.587072i	$-0.199719\pi$
$769$	44.8982	1.61907	0.809535	+	0.587072i	$0.199719\pi$
$770$	0	0
$771$	−69.5418	−2.50449
$772$	0	0
$773$	13.1905	0.474431	0.237215	−	0.971457i	$-0.423765\pi$
$773$	13.1905	0.474431	0.237215	+	0.971457i	$0.423765\pi$
$774$	0	0
$775$	−22.0609	−0.792453
$776$	0	0
$777$	33.6511	1.20723
$778$	0	0
$779$	−11.1210	−0.398452
$780$	0	0
$781$	−2.98250	−0.106722
$782$	0	0
$783$	9.72941	0.347701
$784$	0	0
$785$	20.7805	0.741689
$786$	0	0
$787$	−44.8107	−1.59733	−0.798665	−	0.601776i	$-0.794460\pi$
$787$	−44.8107	−1.59733	−0.798665	+	0.601776i	$0.794460\pi$
$788$	0	0
$789$	−1.56556	−0.0557354
$790$	0	0
$791$	18.3495	0.652434
$792$	0	0
$793$	31.4919	1.11831
$794$	0	0
$795$	−11.1252	−0.394571
$796$	0	0
$797$	16.8132	0.595554	0.297777	−	0.954635i	$-0.403755\pi$
$797$	16.8132	0.595554	0.297777	+	0.954635i	$0.403755\pi$
$798$	0	0
$799$	20.1729	0.713666
$800$	0	0
$801$	−38.8827	−1.37385
$802$	0	0
$803$	6.99685	0.246913
$804$	0	0
$805$	−8.39053	−0.295727
$806$	0	0
$807$	−42.5959	−1.49945
$808$	0	0
$809$	−30.1946	−1.06159	−0.530794	−	0.847501i	$-0.678106\pi$
$809$	−30.1946	−1.06159	−0.530794	+	0.847501i	$0.678106\pi$
$810$	0	0
$811$	−0.366592	−0.0128728	−0.00643639	−	0.999979i	$-0.502049\pi$
$811$	−0.366592	−0.0128728	−0.00643639	+	0.999979i	$0.502049\pi$
$812$	0	0
$813$	0.111521	0.00391120
$814$	0	0
$815$	10.9601	0.383917
$816$	0	0
$817$	−5.51610	−0.192984
$818$	0	0
$819$	−18.6293	−0.650959
$820$	0	0
$821$	−7.20944	−0.251611	−0.125806	−	0.992055i	$-0.540152\pi$
$821$	−7.20944	−0.251611	−0.125806	+	0.992055i	$0.540152\pi$
$822$	0	0
$823$	−1.38828	−0.0483925	−0.0241962	−	0.999707i	$-0.507703\pi$
$823$	−1.38828	−0.0483925	−0.0241962	+	0.999707i	$0.507703\pi$
$824$	0	0
$825$	9.50111	0.330786
$826$	0	0
$827$	34.9555	1.21552	0.607761	−	0.794120i	$-0.292068\pi$
$827$	34.9555	1.21552	0.607761	+	0.794120i	$0.292068\pi$
$828$	0	0
$829$	51.4642	1.78743	0.893713	−	0.448640i	$-0.148091\pi$
$829$	51.4642	1.78743	0.893713	+	0.448640i	$0.148091\pi$
$830$	0	0
$831$	0.771822	0.0267742
$832$	0	0
$833$	27.6217	0.957035
$834$	0	0
$835$	−14.8551	−0.514083
$836$	0	0
$837$	−10.5761	−0.365565
$838$	0	0
$839$	−1.96133	−0.0677126	−0.0338563	−	0.999427i	$-0.510779\pi$
$839$	−1.96133	−0.0677126	−0.0338563	+	0.999427i	$0.510779\pi$
$840$	0	0
$841$	−5.69240	−0.196289
$842$	0	0
$843$	−7.25171	−0.249762
$844$	0	0
$845$	22.8875	0.787355
$846$	0	0
$847$	1.42130	0.0488365
$848$	0	0
$849$	−68.1893	−2.34025
$850$	0	0
$851$	−69.3018	−2.37563
$852$	0	0
$853$	−49.9774	−1.71119	−0.855597	−	0.517643i	$-0.826810\pi$
$853$	−49.9774	−1.71119	−0.855597	+	0.517643i	$0.826810\pi$
$854$	0	0
$855$	−1.88363	−0.0644188
$856$	0	0
$857$	8.56273	0.292497	0.146249	−	0.989248i	$-0.453280\pi$
$857$	8.56273	0.292497	0.146249	+	0.989248i	$0.453280\pi$
$858$	0	0
$859$	−33.7351	−1.15103	−0.575514	−	0.817792i	$-0.695198\pi$
$859$	−33.7351	−1.15103	−0.575514	+	0.817792i	$0.695198\pi$
$860$	0	0
$861$	35.6819	1.21604
$862$	0	0
$863$	−19.0609	−0.648839	−0.324420	−	0.945913i	$-0.605169\pi$
$863$	−19.0609	−0.648839	−0.324420	+	0.945913i	$0.605169\pi$
$864$	0	0
$865$	2.58392	0.0878561
$866$	0	0
$867$	31.1115	1.05660
$868$	0	0
$869$	10.8553	0.368242
$870$	0	0
$871$	−8.11350	−0.274915
$872$	0	0
$873$	−14.9158	−0.504824
$874$	0	0
$875$	−11.6730	−0.394618
$876$	0	0
$877$	33.5262	1.13210	0.566050	−	0.824371i	$-0.308471\pi$
$877$	33.5262	1.13210	0.566050	+	0.824371i	$0.308471\pi$
$878$	0	0
$879$	49.9374	1.68435
$880$	0	0
$881$	33.4840	1.12810	0.564051	−	0.825740i	$-0.309242\pi$
$881$	33.4840	1.12810	0.564051	+	0.825740i	$0.309242\pi$
$882$	0	0
$883$	−7.58156	−0.255140	−0.127570	−	0.991830i	$-0.540718\pi$
$883$	−7.58156	−0.255140	−0.127570	+	0.991830i	$0.540718\pi$
$884$	0	0
$885$	−10.4032	−0.349700
$886$	0	0
$887$	13.3590	0.448550	0.224275	−	0.974526i	$-0.427999\pi$
$887$	13.3590	0.448550	0.224275	+	0.974526i	$0.427999\pi$
$888$	0	0
$889$	10.7089	0.359165
$890$	0	0
$891$	10.8799	0.364491
$892$	0	0
$893$	3.64135	0.121853
$894$	0	0
$895$	17.4914	0.584674
$896$	0	0
$897$	92.9565	3.10373
$898$	0	0
$899$	−25.3360	−0.845004
$900$	0	0
$901$	30.5961	1.01931
$902$	0	0
$903$	17.6984	0.588967
$904$	0	0
$905$	1.00917	0.0335459
$906$	0	0
$907$	−10.3037	−0.342127	−0.171064	−	0.985260i	$-0.554720\pi$
$907$	−10.3037	−0.342127	−0.171064	+	0.985260i	$0.554720\pi$
$908$	0	0
$909$	−23.3501	−0.774475
$910$	0	0
$911$	−28.8754	−0.956685	−0.478342	−	0.878173i	$-0.658762\pi$
$911$	−28.8754	−0.956685	−0.478342	+	0.878173i	$0.658762\pi$
$912$	0	0
$913$	1.67284	0.0553628
$914$	0	0
$915$	−10.2165	−0.337747
$916$	0	0
$917$	−10.3414	−0.341502
$918$	0	0
$919$	−0.0753325	−0.00248499	−0.00124250	−	0.999999i	$-0.500395\pi$
$919$	−0.0753325	−0.00248499	−0.00124250	+	0.999999i	$0.500395\pi$
$920$	0	0
$921$	−61.2158	−2.01713
$922$	0	0
$923$	−18.5417	−0.610307
$924$	0	0
$925$	−44.0359	−1.44789
$926$	0	0
$927$	29.6847	0.974972
$928$	0	0
$929$	−33.9110	−1.11258	−0.556292	−	0.830987i	$-0.687776\pi$
$929$	−33.9110	−1.11258	−0.556292	+	0.830987i	$0.687776\pi$
$930$	0	0
$931$	4.98590	0.163406
$932$	0	0
$933$	48.5838	1.59056
$934$	0	0
$935$	4.94950	0.161866
$936$	0	0
$937$	−22.1684	−0.724209	−0.362105	−	0.932137i	$-0.617942\pi$
$937$	−22.1684	−0.724209	−0.362105	+	0.932137i	$0.617942\pi$
$938$	0	0
$939$	32.6770	1.06637
$940$	0	0
$941$	−50.3163	−1.64027	−0.820133	−	0.572173i	$-0.806100\pi$
$941$	−50.3163	−1.64027	−0.820133	+	0.572173i	$0.806100\pi$
$942$	0	0
$943$	−73.4841	−2.39297
$944$	0	0
$945$	−2.55597	−0.0831456
$946$	0	0
$947$	5.97665	0.194215	0.0971074	−	0.995274i	$-0.469041\pi$
$947$	5.97665	0.194215	0.0971074	+	0.995274i	$0.469041\pi$
$948$	0	0
$949$	43.4981	1.41201
$950$	0	0
$951$	56.9500	1.84673
$952$	0	0
$953$	−42.9545	−1.39143	−0.695717	−	0.718316i	$-0.744913\pi$
$953$	−42.9545	−1.39143	−0.695717	+	0.718316i	$0.744913\pi$
$954$	0	0
$955$	24.0906	0.779554
$956$	0	0
$957$	10.9116	0.352722
$958$	0	0
$959$	1.42130	0.0458962
$960$	0	0
$961$	−3.45900	−0.111581
$962$	0	0
$963$	27.6614	0.891375
$964$	0	0
$965$	−7.59996	−0.244651
$966$	0	0
$967$	−20.6287	−0.663375	−0.331687	−	0.943389i	$-0.607618\pi$
$967$	−20.6287	−0.663375	−0.331687	+	0.943389i	$0.607618\pi$
$968$	0	0
$969$	12.5514	0.403210
$970$	0	0
$971$	−27.5489	−0.884086	−0.442043	−	0.896994i	$-0.645746\pi$
$971$	−27.5489	−0.884086	−0.442043	+	0.896994i	$0.645746\pi$
$972$	0	0
$973$	−10.7868	−0.345808
$974$	0	0
$975$	59.0667	1.89165
$976$	0	0
$977$	−47.6128	−1.52327	−0.761634	−	0.648008i	$-0.775602\pi$
$977$	−47.6128	−1.52327	−0.761634	+	0.648008i	$0.775602\pi$
$978$	0	0
$979$	18.4423	0.589419
$980$	0	0
$981$	−13.8727	−0.442922
$982$	0	0
$983$	−50.9913	−1.62637	−0.813184	−	0.582006i	$-0.802268\pi$
$983$	−50.9913	−1.62637	−0.813184	+	0.582006i	$0.802268\pi$
$984$	0	0
$985$	−19.4402	−0.619416
$986$	0	0
$987$	−11.6833	−0.371883
$988$	0	0
$989$	−36.4486	−1.15900
$990$	0	0
$991$	43.9094	1.39483	0.697414	−	0.716669i	$-0.254334\pi$
$991$	43.9094	1.39483	0.697414	+	0.716669i	$0.254334\pi$
$992$	0	0
$993$	2.12245	0.0673539
$994$	0	0
$995$	11.1527	0.353565
$996$	0	0
$997$	−28.6427	−0.907123	−0.453562	−	0.891225i	$-0.649847\pi$
$997$	−28.6427	−0.907123	−0.453562	+	0.891225i	$0.649847\pi$
$998$	0	0
$999$	−21.1111	−0.667925

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.d.1.23	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.d.1.23	✓	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)

\(f(q)\)	\(=\)	\(q+2.26016 q^{3} +0.892342 q^{5} +1.42130 q^{7} +2.10834 q^{9} +O(q^{10})\)	\(q+2.26016 q^{3} +0.892342 q^{5} +1.42130 q^{7} +2.10834 q^{9} -1.00000 q^{11} -6.21682 q^{13} +2.01684 q^{15} -5.54664 q^{17} -1.00121 q^{19} +3.21237 q^{21} -6.61564 q^{23} -4.20373 q^{25} -2.01529 q^{27} -4.82780 q^{29} +5.24795 q^{31} -2.26016 q^{33} +1.26829 q^{35} +10.4755 q^{37} -14.0510 q^{39} +11.1076 q^{41} +5.50946 q^{43} +1.88136 q^{45} -3.63696 q^{47} -4.97990 q^{49} -12.5363 q^{51} -5.51616 q^{53} -0.892342 q^{55} -2.26289 q^{57} -5.15816 q^{59} -5.06560 q^{61} +2.99659 q^{63} -5.54753 q^{65} +1.30509 q^{67} -14.9524 q^{69} +2.98250 q^{71} -6.99685 q^{73} -9.50111 q^{75} -1.42130 q^{77} -10.8553 q^{79} -10.8799 q^{81} -1.67284 q^{83} -4.94950 q^{85} -10.9116 q^{87} -18.4423 q^{89} -8.83597 q^{91} +11.8612 q^{93} -0.893418 q^{95} -7.07466 q^{97} -2.10834 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9}+O(q^{10}) \) 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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