LMFDB - Embedded newform 6028.2.a.e.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.1338223384$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+1.41110 q^{3} -1.10550 q^{5} -3.91271 q^{7} -1.00881 q^{9} +O(q^{10})$	$q+1.41110 q^{3} -1.10550 q^{5} -3.91271 q^{7} -1.00881 q^{9} -1.00000 q^{11} +2.55958 q^{13} -1.55997 q^{15} -2.06299 q^{17} +4.47552 q^{19} -5.52121 q^{21} +2.63846 q^{23} -3.77787 q^{25} -5.65681 q^{27} -7.02933 q^{29} -2.21152 q^{31} -1.41110 q^{33} +4.32550 q^{35} +7.48870 q^{37} +3.61181 q^{39} +10.6937 q^{41} -5.52171 q^{43} +1.11524 q^{45} +3.99508 q^{47} +8.30932 q^{49} -2.91107 q^{51} -10.4184 q^{53} +1.10550 q^{55} +6.31539 q^{57} -5.63881 q^{59} +1.62988 q^{61} +3.94718 q^{63} -2.82962 q^{65} +2.07899 q^{67} +3.72311 q^{69} +13.8306 q^{71} +8.54047 q^{73} -5.33094 q^{75} +3.91271 q^{77} -1.97116 q^{79} -4.95588 q^{81} +13.0889 q^{83} +2.28063 q^{85} -9.91906 q^{87} +14.3027 q^{89} -10.0149 q^{91} -3.12067 q^{93} -4.94769 q^{95} +1.25896 q^{97} +1.00881 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) $ 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9	$ 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9} - 27 q^{11} + 2 q^{13} + 10 q^{15} + 17 q^{17} + 4 q^{19} - 4 q^{21} + 25 q^{23} + 21 q^{25} + 32 q^{27} - q^{29} + 8 q^{31} - 5 q^{33} + 30 q^{35} - 24 q^{37} + 10 q^{39} + 31 q^{41} + 17 q^{43} - 15 q^{45} + 27 q^{47} + 40 q^{49} + 30 q^{51} + 2 q^{55} + 13 q^{57} + 31 q^{59} + 2 q^{61} + 61 q^{63} + 2 q^{67} - 15 q^{69} + 40 q^{71} + 17 q^{73} + 30 q^{75} - 7 q^{77} + 33 q^{79} + 31 q^{81} + 66 q^{83} + 26 q^{85} + 43 q^{87} + 22 q^{89} + 18 q^{91} - 9 q^{93} + 95 q^{95} - 21 q^{97} - 28 q^{99}+O(q^{100}) $ 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9 - 27 * q^11 + 2 * q^13 + 10 * q^15 + 17 * q^17 + 4 * q^19 - 4 * q^21 + 25 * q^23 + 21 * q^25 + 32 * q^27 - q^29 + 8 * q^31 - 5 * q^33 + 30 * q^35 - 24 * q^37 + 10 * q^39 + 31 * q^41 + 17 * q^43 - 15 * q^45 + 27 * q^47 + 40 * q^49 + 30 * q^51 + 2 * q^55 + 13 * q^57 + 31 * q^59 + 2 * q^61 + 61 * q^63 + 2 * q^67 - 15 * q^69 + 40 * q^71 + 17 * q^73 + 30 * q^75 - 7 * q^77 + 33 * q^79 + 31 * q^81 + 66 * q^83 + 26 * q^85 + 43 * q^87 + 22 * q^89 + 18 * q^91 - 9 * q^93 + 95 * q^95 - 21 * q^97 - 28 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.41110	0.814697	0.407348	−	0.913273i	$-0.366454\pi$
$3$	1.41110	0.814697	0.407348	+	0.913273i	$0.366454\pi$
$4$	0	0
$5$	−1.10550	−0.494395	−0.247197	−	0.968965i	$-0.579510\pi$
$5$	−1.10550	−0.494395	−0.247197	+	0.968965i	$0.579510\pi$
$6$	0	0
$7$	−3.91271	−1.47887	−0.739433	−	0.673230i	$-0.764906\pi$
$7$	−3.91271	−1.47887	−0.739433	+	0.673230i	$0.764906\pi$
$8$	0	0
$9$	−1.00881	−0.336270
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.55958	0.709900	0.354950	−	0.934885i	$-0.384498\pi$
$13$	2.55958	0.709900	0.354950	+	0.934885i	$0.384498\pi$
$14$	0	0
$15$	−1.55997	−0.402782
$16$	0	0
$17$	−2.06299	−0.500348	−0.250174	−	0.968201i	$-0.580488\pi$
$17$	−2.06299	−0.500348	−0.250174	+	0.968201i	$0.580488\pi$
$18$	0	0
$19$	4.47552	1.02676	0.513378	−	0.858163i	$-0.328394\pi$
$19$	4.47552	1.02676	0.513378	+	0.858163i	$0.328394\pi$
$20$	0	0
$21$	−5.52121	−1.20483
$22$	0	0
$23$	2.63846	0.550156	0.275078	−	0.961422i	$-0.411296\pi$
$23$	2.63846	0.550156	0.275078	+	0.961422i	$0.411296\pi$
$24$	0	0
$25$	−3.77787	−0.755574
$26$	0	0
$27$	−5.65681	−1.08865
$28$	0	0
$29$	−7.02933	−1.30531	−0.652657	−	0.757653i	$-0.726346\pi$
$29$	−7.02933	−1.30531	−0.652657	+	0.757653i	$0.726346\pi$
$30$	0	0
$31$	−2.21152	−0.397201	−0.198601	−	0.980081i	$-0.563640\pi$
$31$	−2.21152	−0.397201	−0.198601	+	0.980081i	$0.563640\pi$
$32$	0	0
$33$	−1.41110	−0.245640
$34$	0	0
$35$	4.32550	0.731143
$36$	0	0
$37$	7.48870	1.23113	0.615567	−	0.788084i	$-0.288927\pi$
$37$	7.48870	1.23113	0.615567	+	0.788084i	$0.288927\pi$
$38$	0	0
$39$	3.61181	0.578353
$40$	0	0
$41$	10.6937	1.67007	0.835037	−	0.550194i	$-0.185446\pi$
$41$	10.6937	1.67007	0.835037	+	0.550194i	$0.185446\pi$
$42$	0	0
$43$	−5.52171	−0.842054	−0.421027	−	0.907048i	$-0.638330\pi$
$43$	−5.52171	−0.842054	−0.421027	+	0.907048i	$0.638330\pi$
$44$	0	0
$45$	1.11524	0.166250
$46$	0	0
$47$	3.99508	0.582742	0.291371	−	0.956610i	$-0.405889\pi$
$47$	3.99508	0.582742	0.291371	+	0.956610i	$0.405889\pi$
$48$	0	0
$49$	8.30932	1.18705
$50$	0	0
$51$	−2.91107	−0.407632
$52$	0	0
$53$	−10.4184	−1.43107	−0.715535	−	0.698577i	$-0.753817\pi$
$53$	−10.4184	−1.43107	−0.715535	+	0.698577i	$0.753817\pi$
$54$	0	0
$55$	1.10550	0.149066
$56$	0	0
$57$	6.31539	0.836494
$58$	0	0
$59$	−5.63881	−0.734111	−0.367055	−	0.930199i	$-0.619634\pi$
$59$	−5.63881	−0.734111	−0.367055	+	0.930199i	$0.619634\pi$
$60$	0	0
$61$	1.62988	0.208685	0.104343	−	0.994541i	$-0.466726\pi$
$61$	1.62988	0.208685	0.104343	+	0.994541i	$0.466726\pi$
$62$	0	0
$63$	3.94718	0.497298
$64$	0	0
$65$	−2.82962	−0.350971
$66$	0	0
$67$	2.07899	0.253989	0.126994	−	0.991903i	$-0.459467\pi$
$67$	2.07899	0.253989	0.126994	+	0.991903i	$0.459467\pi$
$68$	0	0
$69$	3.72311	0.448210
$70$	0	0
$71$	13.8306	1.64139	0.820695	−	0.571367i	$-0.193587\pi$
$71$	13.8306	1.64139	0.820695	+	0.571367i	$0.193587\pi$
$72$	0	0
$73$	8.54047	0.999586	0.499793	−	0.866145i	$-0.333409\pi$
$73$	8.54047	0.999586	0.499793	+	0.866145i	$0.333409\pi$
$74$	0	0
$75$	−5.33094	−0.615564
$76$	0	0
$77$	3.91271	0.445895
$78$	0	0
$79$	−1.97116	−0.221773	−0.110886	−	0.993833i	$-0.535369\pi$
$79$	−1.97116	−0.221773	−0.110886	+	0.993833i	$0.535369\pi$
$80$	0	0
$81$	−4.95588	−0.550653
$82$	0	0
$83$	13.0889	1.43670	0.718348	−	0.695684i	$-0.244899\pi$
$83$	13.0889	1.43670	0.718348	+	0.695684i	$0.244899\pi$
$84$	0	0
$85$	2.28063	0.247369
$86$	0	0
$87$	−9.91906	−1.06344
$88$	0	0
$89$	14.3027	1.51608	0.758041	−	0.652207i	$-0.226157\pi$
$89$	14.3027	1.51608	0.758041	+	0.652207i	$0.226157\pi$
$90$	0	0
$91$	−10.0149	−1.04985
$92$	0	0
$93$	−3.12067	−0.323598
$94$	0	0
$95$	−4.94769	−0.507622
$96$	0	0
$97$	1.25896	0.127828	0.0639141	−	0.997955i	$-0.479642\pi$
$97$	1.25896	0.127828	0.0639141	+	0.997955i	$0.479642\pi$
$98$	0	0
$99$	1.00881	0.101389
$100$	0	0
$101$	7.33365	0.729726	0.364863	−	0.931061i	$-0.381116\pi$
$101$	7.33365	0.729726	0.364863	+	0.931061i	$0.381116\pi$
$102$	0	0
$103$	−2.61736	−0.257896	−0.128948	−	0.991651i	$-0.541160\pi$
$103$	−2.61736	−0.257896	−0.128948	+	0.991651i	$0.541160\pi$
$104$	0	0
$105$	6.10370	0.595660
$106$	0	0
$107$	−7.31696	−0.707357	−0.353679	−	0.935367i	$-0.615069\pi$
$107$	−7.31696	−0.707357	−0.353679	+	0.935367i	$0.615069\pi$
$108$	0	0
$109$	−19.6123	−1.87852	−0.939259	−	0.343209i	$-0.888486\pi$
$109$	−19.6123	−1.87852	−0.939259	+	0.343209i	$0.888486\pi$
$110$	0	0
$111$	10.5673	1.00300
$112$	0	0
$113$	7.85941	0.739351	0.369675	−	0.929161i	$-0.379469\pi$
$113$	7.85941	0.739351	0.369675	+	0.929161i	$0.379469\pi$
$114$	0	0
$115$	−2.91681	−0.271994
$116$	0	0
$117$	−2.58213	−0.238718
$118$	0	0
$119$	8.07188	0.739948
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	15.0898	1.36060
$124$	0	0
$125$	9.70393	0.867946
$126$	0	0
$127$	12.7049	1.12737	0.563687	−	0.825988i	$-0.309382\pi$
$127$	12.7049	1.12737	0.563687	+	0.825988i	$0.309382\pi$
$128$	0	0
$129$	−7.79167	−0.686018
$130$	0	0
$131$	−8.57406	−0.749120	−0.374560	−	0.927203i	$-0.622206\pi$
$131$	−8.57406	−0.749120	−0.374560	+	0.927203i	$0.622206\pi$
$132$	0	0
$133$	−17.5114	−1.51843
$134$	0	0
$135$	6.25361	0.538225
$136$	0	0
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	0	0
$139$	7.52191	0.638000	0.319000	−	0.947755i	$-0.396653\pi$
$139$	7.52191	0.638000	0.319000	+	0.947755i	$0.396653\pi$
$140$	0	0
$141$	5.63743	0.474758
$142$	0	0
$143$	−2.55958	−0.214043
$144$	0	0
$145$	7.77093	0.645340
$146$	0	0
$147$	11.7252	0.967082
$148$	0	0
$149$	12.2649	1.00478	0.502391	−	0.864641i	$-0.332454\pi$
$149$	12.2649	1.00478	0.502391	+	0.864641i	$0.332454\pi$
$150$	0	0
$151$	1.12895	0.0918726	0.0459363	−	0.998944i	$-0.485373\pi$
$151$	1.12895	0.0918726	0.0459363	+	0.998944i	$0.485373\pi$
$152$	0	0
$153$	2.08116	0.168252
$154$	0	0
$155$	2.44484	0.196374
$156$	0	0
$157$	1.21664	0.0970983	0.0485491	−	0.998821i	$-0.484540\pi$
$157$	1.21664	0.0970983	0.0485491	+	0.998821i	$0.484540\pi$
$158$	0	0
$159$	−14.7013	−1.16589
$160$	0	0
$161$	−10.3235	−0.813607
$162$	0	0
$163$	18.4791	1.44739	0.723696	−	0.690119i	$-0.242442\pi$
$163$	18.4791	1.44739	0.723696	+	0.690119i	$0.242442\pi$
$164$	0	0
$165$	1.55997	0.121443
$166$	0	0
$167$	0.126250	0.00976952	0.00488476	−	0.999988i	$-0.498445\pi$
$167$	0.126250	0.00976952	0.00488476	+	0.999988i	$0.498445\pi$
$168$	0	0
$169$	−6.44854	−0.496042
$170$	0	0
$171$	−4.51494	−0.345266
$172$	0	0
$173$	−17.7185	−1.34712	−0.673558	−	0.739135i	$-0.735235\pi$
$173$	−17.7185	−1.34712	−0.673558	+	0.739135i	$0.735235\pi$
$174$	0	0
$175$	14.7817	1.11739
$176$	0	0
$177$	−7.95690	−0.598078
$178$	0	0
$179$	18.4705	1.38055	0.690274	−	0.723548i	$-0.257490\pi$
$179$	18.4705	1.38055	0.690274	+	0.723548i	$0.257490\pi$
$180$	0	0
$181$	11.2870	0.838953	0.419476	−	0.907766i	$-0.362214\pi$
$181$	11.2870	0.838953	0.419476	+	0.907766i	$0.362214\pi$
$182$	0	0
$183$	2.29992	0.170015
$184$	0	0
$185$	−8.27876	−0.608666
$186$	0	0
$187$	2.06299	0.150861
$188$	0	0
$189$	22.1335	1.60997
$190$	0	0
$191$	4.46902	0.323367	0.161684	−	0.986843i	$-0.448308\pi$
$191$	4.46902	0.323367	0.161684	+	0.986843i	$0.448308\pi$
$192$	0	0
$193$	11.2491	0.809730	0.404865	−	0.914376i	$-0.367318\pi$
$193$	11.2491	0.809730	0.404865	+	0.914376i	$0.367318\pi$
$194$	0	0
$195$	−3.99286	−0.285935
$196$	0	0
$197$	18.9380	1.34928	0.674640	−	0.738147i	$-0.264299\pi$
$197$	18.9380	1.34928	0.674640	+	0.738147i	$0.264299\pi$
$198$	0	0
$199$	−11.8953	−0.843233	−0.421617	−	0.906774i	$-0.638537\pi$
$199$	−11.8953	−0.843233	−0.421617	+	0.906774i	$0.638537\pi$
$200$	0	0
$201$	2.93365	0.206924
$202$	0	0
$203$	27.5038	1.93039
$204$	0	0
$205$	−11.8219	−0.825675
$206$	0	0
$207$	−2.66170	−0.185001
$208$	0	0
$209$	−4.47552	−0.309578
$210$	0	0
$211$	19.9053	1.37034	0.685168	−	0.728385i	$-0.259729\pi$
$211$	19.9053	1.37034	0.685168	+	0.728385i	$0.259729\pi$
$212$	0	0
$213$	19.5163	1.33723
$214$	0	0
$215$	6.10425	0.416307
$216$	0	0
$217$	8.65305	0.587407
$218$	0	0
$219$	12.0514	0.814359
$220$	0	0
$221$	−5.28039	−0.355197
$222$	0	0
$223$	−10.6492	−0.713120	−0.356560	−	0.934272i	$-0.616050\pi$
$223$	−10.6492	−0.713120	−0.356560	+	0.934272i	$0.616050\pi$
$224$	0	0
$225$	3.81115	0.254077
$226$	0	0
$227$	17.1551	1.13863	0.569314	−	0.822120i	$-0.307209\pi$
$227$	17.1551	1.13863	0.569314	+	0.822120i	$0.307209\pi$
$228$	0	0
$229$	19.5738	1.29347	0.646735	−	0.762715i	$-0.276134\pi$
$229$	19.5738	1.29347	0.646735	+	0.762715i	$0.276134\pi$
$230$	0	0
$231$	5.52121	0.363269
$232$	0	0
$233$	16.3413	1.07055	0.535276	−	0.844677i	$-0.320208\pi$
$233$	16.3413	1.07055	0.535276	+	0.844677i	$0.320208\pi$
$234$	0	0
$235$	−4.41655	−0.288104
$236$	0	0
$237$	−2.78149	−0.180677
$238$	0	0
$239$	21.9845	1.42206	0.711030	−	0.703161i	$-0.248229\pi$
$239$	21.9845	1.42206	0.711030	+	0.703161i	$0.248229\pi$
$240$	0	0
$241$	−8.87023	−0.571382	−0.285691	−	0.958322i	$-0.592223\pi$
$241$	−8.87023	−0.571382	−0.285691	+	0.958322i	$0.592223\pi$
$242$	0	0
$243$	9.97722	0.640039
$244$	0	0
$245$	−9.18595	−0.586869
$246$	0	0
$247$	11.4555	0.728894
$248$	0	0
$249$	18.4697	1.17047
$250$	0	0
$251$	1.76172	0.111199	0.0555994	−	0.998453i	$-0.482293\pi$
$251$	1.76172	0.111199	0.0555994	+	0.998453i	$0.482293\pi$
$252$	0	0
$253$	−2.63846	−0.165878
$254$	0	0
$255$	3.21819	0.201531
$256$	0	0
$257$	−6.89931	−0.430367	−0.215183	−	0.976574i	$-0.569035\pi$
$257$	−6.89931	−0.430367	−0.215183	+	0.976574i	$0.569035\pi$
$258$	0	0
$259$	−29.3011	−1.82068
$260$	0	0
$261$	7.09125	0.438937
$262$	0	0
$263$	25.9518	1.60026	0.800129	−	0.599828i	$-0.204764\pi$
$263$	25.9518	1.60026	0.800129	+	0.599828i	$0.204764\pi$
$264$	0	0
$265$	11.5175	0.707514
$266$	0	0
$267$	20.1825	1.23515
$268$	0	0
$269$	−31.4322	−1.91645	−0.958227	−	0.286009i	$-0.907671\pi$
$269$	−31.4322	−1.91645	−0.958227	+	0.286009i	$0.907671\pi$
$270$	0	0
$271$	−24.9623	−1.51635	−0.758177	−	0.652049i	$-0.773910\pi$
$271$	−24.9623	−1.51635	−0.758177	+	0.652049i	$0.773910\pi$
$272$	0	0
$273$	−14.1320	−0.855307
$274$	0	0
$275$	3.77787	0.227814
$276$	0	0
$277$	26.4964	1.59201	0.796007	−	0.605287i	$-0.206942\pi$
$277$	26.4964	1.59201	0.796007	+	0.605287i	$0.206942\pi$
$278$	0	0
$279$	2.23100	0.133567
$280$	0	0
$281$	−9.37437	−0.559228	−0.279614	−	0.960112i	$-0.590207\pi$
$281$	−9.37437	−0.559228	−0.279614	+	0.960112i	$0.590207\pi$
$282$	0	0
$283$	10.6135	0.630907	0.315453	−	0.948941i	$-0.397843\pi$
$283$	10.6135	0.630907	0.315453	+	0.948941i	$0.397843\pi$
$284$	0	0
$285$	−6.98166	−0.413558
$286$	0	0
$287$	−41.8413	−2.46982
$288$	0	0
$289$	−12.7441	−0.749652
$290$	0	0
$291$	1.77652	0.104141
$292$	0	0
$293$	1.99392	0.116486	0.0582429	−	0.998302i	$-0.481450\pi$
$293$	1.99392	0.116486	0.0582429	+	0.998302i	$0.481450\pi$
$294$	0	0
$295$	6.23371	0.362940
$296$	0	0
$297$	5.65681	0.328242
$298$	0	0
$299$	6.75334	0.390556
$300$	0	0
$301$	21.6049	1.24528
$302$	0	0
$303$	10.3485	0.594505
$304$	0	0
$305$	−1.80184	−0.103173
$306$	0	0
$307$	−22.9822	−1.31166	−0.655832	−	0.754907i	$-0.727682\pi$
$307$	−22.9822	−1.31166	−0.655832	+	0.754907i	$0.727682\pi$
$308$	0	0
$309$	−3.69335	−0.210107
$310$	0	0
$311$	−19.4169	−1.10103	−0.550514	−	0.834826i	$-0.685568\pi$
$311$	−19.4169	−1.10103	−0.550514	+	0.834826i	$0.685568\pi$
$312$	0	0
$313$	−1.32503	−0.0748954	−0.0374477	−	0.999299i	$-0.511923\pi$
$313$	−1.32503	−0.0748954	−0.0374477	+	0.999299i	$0.511923\pi$
$314$	0	0
$315$	−4.36360	−0.245861
$316$	0	0
$317$	−30.1601	−1.69396	−0.846980	−	0.531625i	$-0.821582\pi$
$317$	−30.1601	−1.69396	−0.846980	+	0.531625i	$0.821582\pi$
$318$	0	0
$319$	7.02933	0.393567
$320$	0	0
$321$	−10.3249	−0.576282
$322$	0	0
$323$	−9.23295	−0.513735
$324$	0	0
$325$	−9.66977	−0.536382
$326$	0	0
$327$	−27.6748	−1.53042
$328$	0	0
$329$	−15.6316	−0.861797
$330$	0	0
$331$	21.2630	1.16872	0.584359	−	0.811495i	$-0.301346\pi$
$331$	21.2630	1.16872	0.584359	+	0.811495i	$0.301346\pi$
$332$	0	0
$333$	−7.55467	−0.413993
$334$	0	0
$335$	−2.29832	−0.125571
$336$	0	0
$337$	−7.69562	−0.419207	−0.209604	−	0.977786i	$-0.567217\pi$
$337$	−7.69562	−0.419207	−0.209604	+	0.977786i	$0.567217\pi$
$338$	0	0
$339$	11.0904	0.602347
$340$	0	0
$341$	2.21152	0.119761
$342$	0	0
$343$	−5.12299	−0.276615
$344$	0	0
$345$	−4.11590	−0.221593
$346$	0	0
$347$	−0.783821	−0.0420777	−0.0210389	−	0.999779i	$-0.506697\pi$
$347$	−0.783821	−0.0420777	−0.0210389	+	0.999779i	$0.506697\pi$
$348$	0	0
$349$	15.7986	0.845681	0.422840	−	0.906204i	$-0.361033\pi$
$349$	15.7986	0.845681	0.422840	+	0.906204i	$0.361033\pi$
$350$	0	0
$351$	−14.4791	−0.772836
$352$	0	0
$353$	−22.8481	−1.21608	−0.608041	−	0.793905i	$-0.708044\pi$
$353$	−22.8481	−1.21608	−0.608041	+	0.793905i	$0.708044\pi$
$354$	0	0
$355$	−15.2897	−0.811494
$356$	0	0
$357$	11.3902	0.602833
$358$	0	0
$359$	−1.91771	−0.101213	−0.0506066	−	0.998719i	$-0.516115\pi$
$359$	−1.91771	−0.101213	−0.0506066	+	0.998719i	$0.516115\pi$
$360$	0	0
$361$	1.03029	0.0542259
$362$	0	0
$363$	1.41110	0.0740633
$364$	0	0
$365$	−9.44149	−0.494190
$366$	0	0
$367$	−17.9816	−0.938630	−0.469315	−	0.883031i	$-0.655499\pi$
$367$	−17.9816	−0.938630	−0.469315	+	0.883031i	$0.655499\pi$
$368$	0	0
$369$	−10.7879	−0.561595
$370$	0	0
$371$	40.7640	2.11636
$372$	0	0
$373$	−5.59585	−0.289742	−0.144871	−	0.989451i	$-0.546277\pi$
$373$	−5.59585	−0.289742	−0.144871	+	0.989451i	$0.546277\pi$
$374$	0	0
$375$	13.6932	0.707113
$376$	0	0
$377$	−17.9922	−0.926643
$378$	0	0
$379$	20.7283	1.06474	0.532369	−	0.846512i	$-0.321302\pi$
$379$	20.7283	1.06474	0.532369	+	0.846512i	$0.321302\pi$
$380$	0	0
$381$	17.9278	0.918468
$382$	0	0
$383$	3.86536	0.197511	0.0987554	−	0.995112i	$-0.468514\pi$
$383$	3.86536	0.197511	0.0987554	+	0.995112i	$0.468514\pi$
$384$	0	0
$385$	−4.32550	−0.220448
$386$	0	0
$387$	5.57035	0.283157
$388$	0	0
$389$	−0.725270	−0.0367727	−0.0183863	−	0.999831i	$-0.505853\pi$
$389$	−0.725270	−0.0367727	−0.0183863	+	0.999831i	$0.505853\pi$
$390$	0	0
$391$	−5.44311	−0.275270
$392$	0	0
$393$	−12.0988	−0.610305
$394$	0	0
$395$	2.17911	0.109643
$396$	0	0
$397$	−29.0556	−1.45826	−0.729130	−	0.684375i	$-0.760075\pi$
$397$	−29.0556	−1.45826	−0.729130	+	0.684375i	$0.760075\pi$
$398$	0	0
$399$	−24.7103	−1.23706
$400$	0	0
$401$	8.06370	0.402682	0.201341	−	0.979521i	$-0.435470\pi$
$401$	8.06370	0.402682	0.201341	+	0.979521i	$0.435470\pi$
$402$	0	0
$403$	−5.66057	−0.281973
$404$	0	0
$405$	5.47872	0.272240
$406$	0	0
$407$	−7.48870	−0.371201
$408$	0	0
$409$	16.2964	0.805807	0.402903	−	0.915243i	$-0.368001\pi$
$409$	16.2964	0.805807	0.402903	+	0.915243i	$0.368001\pi$
$410$	0	0
$411$	−1.41110	−0.0696042
$412$	0	0
$413$	22.0630	1.08565
$414$	0	0
$415$	−14.4698	−0.710294
$416$	0	0
$417$	10.6141	0.519776
$418$	0	0
$419$	8.69526	0.424791	0.212396	−	0.977184i	$-0.431873\pi$
$419$	8.69526	0.424791	0.212396	+	0.977184i	$0.431873\pi$
$420$	0	0
$421$	7.45578	0.363373	0.181686	−	0.983357i	$-0.441844\pi$
$421$	7.45578	0.363373	0.181686	+	0.983357i	$0.441844\pi$
$422$	0	0
$423$	−4.03027	−0.195958
$424$	0	0
$425$	7.79370	0.378050
$426$	0	0
$427$	−6.37727	−0.308618
$428$	0	0
$429$	−3.61181	−0.174380
$430$	0	0
$431$	−16.2704	−0.783717	−0.391858	−	0.920026i	$-0.628168\pi$
$431$	−16.2704	−0.783717	−0.391858	+	0.920026i	$0.628168\pi$
$432$	0	0
$433$	27.6642	1.32945	0.664727	−	0.747086i	$-0.268548\pi$
$433$	27.6642	1.32945	0.664727	+	0.747086i	$0.268548\pi$
$434$	0	0
$435$	10.9655	0.525757
$436$	0	0
$437$	11.8085	0.564876
$438$	0	0
$439$	9.20284	0.439228	0.219614	−	0.975587i	$-0.429520\pi$
$439$	9.20284	0.439228	0.219614	+	0.975587i	$0.429520\pi$
$440$	0	0
$441$	−8.38251	−0.399167
$442$	0	0
$443$	36.3797	1.72845	0.864225	−	0.503105i	$-0.167809\pi$
$443$	36.3797	1.72845	0.864225	+	0.503105i	$0.167809\pi$
$444$	0	0
$445$	−15.8116	−0.749542
$446$	0	0
$447$	17.3070	0.818592
$448$	0	0
$449$	2.61650	0.123480	0.0617400	−	0.998092i	$-0.480335\pi$
$449$	2.61650	0.123480	0.0617400	+	0.998092i	$0.480335\pi$
$450$	0	0
$451$	−10.6937	−0.503546
$452$	0	0
$453$	1.59306	0.0748483
$454$	0	0
$455$	11.0715	0.519039
$456$	0	0
$457$	25.5801	1.19659	0.598293	−	0.801278i	$-0.295846\pi$
$457$	25.5801	1.19659	0.598293	+	0.801278i	$0.295846\pi$
$458$	0	0
$459$	11.6699	0.544706
$460$	0	0
$461$	−25.2324	−1.17519	−0.587594	−	0.809156i	$-0.699925\pi$
$461$	−25.2324	−1.17519	−0.587594	+	0.809156i	$0.699925\pi$
$462$	0	0
$463$	−26.1087	−1.21337	−0.606687	−	0.794941i	$-0.707502\pi$
$463$	−26.1087	−1.21337	−0.606687	+	0.794941i	$0.707502\pi$
$464$	0	0
$465$	3.44990	0.159985
$466$	0	0
$467$	−28.3254	−1.31075	−0.655373	−	0.755306i	$-0.727488\pi$
$467$	−28.3254	−1.31075	−0.655373	+	0.755306i	$0.727488\pi$
$468$	0	0
$469$	−8.13448	−0.375615
$470$	0	0
$471$	1.71679	0.0791056
$472$	0	0
$473$	5.52171	0.253889
$474$	0	0
$475$	−16.9079	−0.775789
$476$	0	0
$477$	10.5101	0.481225
$478$	0	0
$479$	9.26419	0.423292	0.211646	−	0.977346i	$-0.432118\pi$
$479$	9.26419	0.423292	0.211646	+	0.977346i	$0.432118\pi$
$480$	0	0
$481$	19.1679	0.873983
$482$	0	0
$483$	−14.5675	−0.662843
$484$	0	0
$485$	−1.39178	−0.0631976
$486$	0	0
$487$	−4.65289	−0.210842	−0.105421	−	0.994428i	$-0.533619\pi$
$487$	−4.65289	−0.210842	−0.105421	+	0.994428i	$0.533619\pi$
$488$	0	0
$489$	26.0757	1.17919
$490$	0	0
$491$	13.3985	0.604666	0.302333	−	0.953202i	$-0.402235\pi$
$491$	13.3985	0.604666	0.302333	+	0.953202i	$0.402235\pi$
$492$	0	0
$493$	14.5014	0.653112
$494$	0	0
$495$	−1.11524	−0.0501262
$496$	0	0
$497$	−54.1152	−2.42740
$498$	0	0
$499$	−25.4469	−1.13916	−0.569579	−	0.821937i	$-0.692894\pi$
$499$	−25.4469	−1.13916	−0.569579	+	0.821937i	$0.692894\pi$
$500$	0	0
$501$	0.178151	0.00795920
$502$	0	0
$503$	7.56322	0.337227	0.168614	−	0.985682i	$-0.446071\pi$
$503$	7.56322	0.337227	0.168614	+	0.985682i	$0.446071\pi$
$504$	0	0
$505$	−8.10735	−0.360772
$506$	0	0
$507$	−9.09951	−0.404123
$508$	0	0
$509$	20.1898	0.894896	0.447448	−	0.894310i	$-0.352333\pi$
$509$	20.1898	0.894896	0.447448	+	0.894310i	$0.352333\pi$
$510$	0	0
$511$	−33.4164	−1.47825
$512$	0	0
$513$	−25.3172	−1.11778
$514$	0	0
$515$	2.89349	0.127503
$516$	0	0
$517$	−3.99508	−0.175703
$518$	0	0
$519$	−25.0026	−1.09749
$520$	0	0
$521$	43.9442	1.92523	0.962615	−	0.270874i	$-0.0873128\pi$
$521$	43.9442	1.92523	0.962615	+	0.270874i	$0.0873128\pi$
$522$	0	0
$523$	−34.2823	−1.49906	−0.749531	−	0.661970i	$-0.769721\pi$
$523$	−34.2823	−1.49906	−0.749531	+	0.661970i	$0.769721\pi$
$524$	0	0
$525$	20.8584	0.910336
$526$	0	0
$527$	4.56235	0.198739
$528$	0	0
$529$	−16.0385	−0.697328
$530$	0	0
$531$	5.68848	0.246859
$532$	0	0
$533$	27.3714	1.18559
$534$	0	0
$535$	8.08890	0.349714
$536$	0	0
$537$	26.0636	1.12473
$538$	0	0
$539$	−8.30932	−0.357908
$540$	0	0
$541$	−28.6755	−1.23285	−0.616427	−	0.787412i	$-0.711421\pi$
$541$	−28.6755	−1.23285	−0.616427	+	0.787412i	$0.711421\pi$
$542$	0	0
$543$	15.9270	0.683492
$544$	0	0
$545$	21.6814	0.928729
$546$	0	0
$547$	−30.8595	−1.31946	−0.659729	−	0.751504i	$-0.729329\pi$
$547$	−30.8595	−1.31946	−0.659729	+	0.751504i	$0.729329\pi$
$548$	0	0
$549$	−1.64424	−0.0701745
$550$	0	0
$551$	−31.4599	−1.34024
$552$	0	0
$553$	7.71257	0.327972
$554$	0	0
$555$	−11.6821	−0.495878
$556$	0	0
$557$	30.9639	1.31198	0.655992	−	0.754768i	$-0.272251\pi$
$557$	30.9639	1.31198	0.655992	+	0.754768i	$0.272251\pi$
$558$	0	0
$559$	−14.1333	−0.597774
$560$	0	0
$561$	2.91107	0.122906
$562$	0	0
$563$	36.0945	1.52120	0.760600	−	0.649220i	$-0.224905\pi$
$563$	36.0945	1.52120	0.760600	+	0.649220i	$0.224905\pi$
$564$	0	0
$565$	−8.68857	−0.365531
$566$	0	0
$567$	19.3909	0.814343
$568$	0	0
$569$	−19.2686	−0.807780	−0.403890	−	0.914808i	$-0.632342\pi$
$569$	−19.2686	−0.807780	−0.403890	+	0.914808i	$0.632342\pi$
$570$	0	0
$571$	−7.83056	−0.327699	−0.163849	−	0.986485i	$-0.552391\pi$
$571$	−7.83056	−0.327699	−0.163849	+	0.986485i	$0.552391\pi$
$572$	0	0
$573$	6.30622	0.263446
$574$	0	0
$575$	−9.96774	−0.415684
$576$	0	0
$577$	−14.8794	−0.619436	−0.309718	−	0.950828i	$-0.600235\pi$
$577$	−14.8794	−0.619436	−0.309718	+	0.950828i	$0.600235\pi$
$578$	0	0
$579$	15.8736	0.659684
$580$	0	0
$581$	−51.2132	−2.12468
$582$	0	0
$583$	10.4184	0.431484
$584$	0	0
$585$	2.85454	0.118021
$586$	0	0
$587$	−37.9369	−1.56582	−0.782911	−	0.622134i	$-0.786266\pi$
$587$	−37.9369	−1.56582	−0.782911	+	0.622134i	$0.786266\pi$
$588$	0	0
$589$	−9.89771	−0.407828
$590$	0	0
$591$	26.7234	1.09925
$592$	0	0
$593$	18.2467	0.749300	0.374650	−	0.927166i	$-0.377763\pi$
$593$	18.2467	0.749300	0.374650	+	0.927166i	$0.377763\pi$
$594$	0	0
$595$	−8.92346	−0.365826
$596$	0	0
$597$	−16.7854	−0.686979
$598$	0	0
$599$	−11.9945	−0.490082	−0.245041	−	0.969513i	$-0.578801\pi$
$599$	−11.9945	−0.490082	−0.245041	+	0.969513i	$0.578801\pi$
$600$	0	0
$601$	−25.5365	−1.04166	−0.520828	−	0.853662i	$-0.674377\pi$
$601$	−25.5365	−1.04166	−0.520828	+	0.853662i	$0.674377\pi$
$602$	0	0
$603$	−2.09730	−0.0854087
$604$	0	0
$605$	−1.10550	−0.0449450
$606$	0	0
$607$	−11.7867	−0.478406	−0.239203	−	0.970970i	$-0.576886\pi$
$607$	−11.7867	−0.478406	−0.239203	+	0.970970i	$0.576886\pi$
$608$	0	0
$609$	38.8104	1.57268
$610$	0	0
$611$	10.2257	0.413688
$612$	0	0
$613$	28.8587	1.16559	0.582795	−	0.812619i	$-0.301959\pi$
$613$	28.8587	1.16559	0.582795	+	0.812619i	$0.301959\pi$
$614$	0	0
$615$	−16.6818	−0.672675
$616$	0	0
$617$	−17.9041	−0.720792	−0.360396	−	0.932799i	$-0.617358\pi$
$617$	−17.9041	−0.720792	−0.360396	+	0.932799i	$0.617358\pi$
$618$	0	0
$619$	13.7081	0.550975	0.275487	−	0.961305i	$-0.411161\pi$
$619$	13.7081	0.550975	0.275487	+	0.961305i	$0.411161\pi$
$620$	0	0
$621$	−14.9253	−0.598930
$622$	0	0
$623$	−55.9623	−2.24208
$624$	0	0
$625$	8.16165	0.326466
$626$	0	0
$627$	−6.31539	−0.252212
$628$	0	0
$629$	−15.4491	−0.615996
$630$	0	0
$631$	−25.0510	−0.997264	−0.498632	−	0.866814i	$-0.666164\pi$
$631$	−25.0510	−0.997264	−0.498632	+	0.866814i	$0.666164\pi$
$632$	0	0
$633$	28.0882	1.11641
$634$	0	0
$635$	−14.0452	−0.557368
$636$	0	0
$637$	21.2684	0.842684
$638$	0	0
$639$	−13.9524	−0.551949
$640$	0	0
$641$	−23.8694	−0.942787	−0.471393	−	0.881923i	$-0.656249\pi$
$641$	−23.8694	−0.942787	−0.471393	+	0.881923i	$0.656249\pi$
$642$	0	0
$643$	25.7585	1.01581	0.507907	−	0.861412i	$-0.330419\pi$
$643$	25.7585	1.01581	0.507907	+	0.861412i	$0.330419\pi$
$644$	0	0
$645$	8.61369	0.339164
$646$	0	0
$647$	4.08913	0.160760	0.0803802	−	0.996764i	$-0.474387\pi$
$647$	4.08913	0.160760	0.0803802	+	0.996764i	$0.474387\pi$
$648$	0	0
$649$	5.63881	0.221343
$650$	0	0
$651$	12.2103	0.478559
$652$	0	0
$653$	−14.4213	−0.564350	−0.282175	−	0.959363i	$-0.591056\pi$
$653$	−14.4213	−0.564350	−0.282175	+	0.959363i	$0.591056\pi$
$654$	0	0
$655$	9.47863	0.370361
$656$	0	0
$657$	−8.61570	−0.336130
$658$	0	0
$659$	13.8343	0.538907	0.269454	−	0.963013i	$-0.413157\pi$
$659$	13.8343	0.538907	0.269454	+	0.963013i	$0.413157\pi$
$660$	0	0
$661$	−38.9401	−1.51459	−0.757296	−	0.653072i	$-0.773480\pi$
$661$	−38.9401	−1.51459	−0.757296	+	0.653072i	$0.773480\pi$
$662$	0	0
$663$	−7.45113	−0.289378
$664$	0	0
$665$	19.3589	0.750705
$666$	0	0
$667$	−18.5466	−0.718127
$668$	0	0
$669$	−15.0270	−0.580977
$670$	0	0
$671$	−1.62988	−0.0629210
$672$	0	0
$673$	26.7917	1.03274	0.516372	−	0.856364i	$-0.327282\pi$
$673$	26.7917	1.03274	0.516372	+	0.856364i	$0.327282\pi$
$674$	0	0
$675$	21.3707	0.822559
$676$	0	0
$677$	18.7420	0.720314	0.360157	−	0.932892i	$-0.382723\pi$
$677$	18.7420	0.720314	0.360157	+	0.932892i	$0.382723\pi$
$678$	0	0
$679$	−4.92596	−0.189041
$680$	0	0
$681$	24.2076	0.927636
$682$	0	0
$683$	7.55097	0.288930	0.144465	−	0.989510i	$-0.453854\pi$
$683$	7.55097	0.288930	0.144465	+	0.989510i	$0.453854\pi$
$684$	0	0
$685$	1.10550	0.0422390
$686$	0	0
$687$	27.6204	1.05379
$688$	0	0
$689$	−26.6666	−1.01592
$690$	0	0
$691$	3.10764	0.118220	0.0591100	−	0.998251i	$-0.481174\pi$
$691$	3.10764	0.118220	0.0591100	+	0.998251i	$0.481174\pi$
$692$	0	0
$693$	−3.94718	−0.149941
$694$	0	0
$695$	−8.31547	−0.315424
$696$	0	0
$697$	−22.0610	−0.835618
$698$	0	0
$699$	23.0591	0.872175
$700$	0	0
$701$	−18.3751	−0.694017	−0.347009	−	0.937862i	$-0.612803\pi$
$701$	−18.3751	−0.694017	−0.347009	+	0.937862i	$0.612803\pi$
$702$	0	0
$703$	33.5158	1.26407
$704$	0	0
$705$	−6.23218	−0.234718
$706$	0	0
$707$	−28.6945	−1.07917
$708$	0	0
$709$	2.92181	0.109731	0.0548655	−	0.998494i	$-0.482527\pi$
$709$	2.92181	0.109731	0.0548655	+	0.998494i	$0.482527\pi$
$710$	0	0
$711$	1.98852	0.0745753
$712$	0	0
$713$	−5.83500	−0.218523
$714$	0	0
$715$	2.82962	0.105822
$716$	0	0
$717$	31.0223	1.15855
$718$	0	0
$719$	14.8195	0.552674	0.276337	−	0.961061i	$-0.410879\pi$
$719$	14.8195	0.552674	0.276337	+	0.961061i	$0.410879\pi$
$720$	0	0
$721$	10.2410	0.381394
$722$	0	0
$723$	−12.5167	−0.465503
$724$	0	0
$725$	26.5559	0.986262
$726$	0	0
$727$	−17.5437	−0.650659	−0.325330	−	0.945601i	$-0.605475\pi$
$727$	−17.5437	−0.650659	−0.325330	+	0.945601i	$0.605475\pi$
$728$	0	0
$729$	28.9464	1.07209
$730$	0	0
$731$	11.3912	0.421320
$732$	0	0
$733$	32.7917	1.21119	0.605594	−	0.795774i	$-0.292936\pi$
$733$	32.7917	1.21119	0.605594	+	0.795774i	$0.292936\pi$
$734$	0	0
$735$	−12.9623	−0.478120
$736$	0	0
$737$	−2.07899	−0.0765805
$738$	0	0
$739$	33.3459	1.22665	0.613324	−	0.789831i	$-0.289832\pi$
$739$	33.3459	1.22665	0.613324	+	0.789831i	$0.289832\pi$
$740$	0	0
$741$	16.1648	0.593827
$742$	0	0
$743$	−43.2636	−1.58719	−0.793593	−	0.608449i	$-0.791792\pi$
$743$	−43.2636	−1.58719	−0.793593	+	0.608449i	$0.791792\pi$
$744$	0	0
$745$	−13.5589	−0.496759
$746$	0	0
$747$	−13.2042	−0.483117
$748$	0	0
$749$	28.6292	1.04609
$750$	0	0
$751$	31.2138	1.13901	0.569504	−	0.821989i	$-0.307135\pi$
$751$	31.2138	1.13901	0.569504	+	0.821989i	$0.307135\pi$
$752$	0	0
$753$	2.48596	0.0905933
$754$	0	0
$755$	−1.24805	−0.0454213
$756$	0	0
$757$	−39.4727	−1.43466	−0.717331	−	0.696733i	$-0.754636\pi$
$757$	−39.4727	−1.43466	−0.717331	+	0.696733i	$0.754636\pi$
$758$	0	0
$759$	−3.72311	−0.135140
$760$	0	0
$761$	−42.1007	−1.52615	−0.763074	−	0.646311i	$-0.776311\pi$
$761$	−42.1007	−1.52615	−0.763074	+	0.646311i	$0.776311\pi$
$762$	0	0
$763$	76.7373	2.77808
$764$	0	0
$765$	−2.30072	−0.0831828
$766$	0	0
$767$	−14.4330	−0.521145
$768$	0	0
$769$	−6.88336	−0.248220	−0.124110	−	0.992268i	$-0.539608\pi$
$769$	−6.88336	−0.248220	−0.124110	+	0.992268i	$0.539608\pi$
$770$	0	0
$771$	−9.73558	−0.350619
$772$	0	0
$773$	−18.1777	−0.653805	−0.326902	−	0.945058i	$-0.606005\pi$
$773$	−18.1777	−0.653805	−0.326902	+	0.945058i	$0.606005\pi$
$774$	0	0
$775$	8.35484	0.300115
$776$	0	0
$777$	−41.3467	−1.48330
$778$	0	0
$779$	47.8598	1.71476
$780$	0	0
$781$	−13.8306	−0.494898
$782$	0	0
$783$	39.7636	1.42104
$784$	0	0
$785$	−1.34499	−0.0480049
$786$	0	0
$787$	28.9454	1.03179	0.515897	−	0.856651i	$-0.327459\pi$
$787$	28.9454	1.03179	0.515897	+	0.856651i	$0.327459\pi$
$788$	0	0
$789$	36.6205	1.30372
$790$	0	0
$791$	−30.7516	−1.09340
$792$	0	0
$793$	4.17182	0.148146
$794$	0	0
$795$	16.2523	0.576409
$796$	0	0
$797$	−39.0792	−1.38426	−0.692128	−	0.721775i	$-0.743327\pi$
$797$	−39.0792	−1.38426	−0.692128	+	0.721775i	$0.743327\pi$
$798$	0	0
$799$	−8.24180	−0.291574
$800$	0	0
$801$	−14.4287	−0.509812
$802$	0	0
$803$	−8.54047	−0.301387
$804$	0	0
$805$	11.4126	0.402243
$806$	0	0
$807$	−44.3538	−1.56133
$808$	0	0
$809$	4.71982	0.165940	0.0829700	−	0.996552i	$-0.473559\pi$
$809$	4.71982	0.165940	0.0829700	+	0.996552i	$0.473559\pi$
$810$	0	0
$811$	28.5860	1.00379	0.501894	−	0.864929i	$-0.332637\pi$
$811$	28.5860	1.00379	0.501894	+	0.864929i	$0.332637\pi$
$812$	0	0
$813$	−35.2242	−1.23537
$814$	0	0
$815$	−20.4286	−0.715583
$816$	0	0
$817$	−24.7126	−0.864583
$818$	0	0
$819$	10.1031	0.353032
$820$	0	0
$821$	1.77226	0.0618523	0.0309261	−	0.999522i	$-0.490154\pi$
$821$	1.77226	0.0618523	0.0309261	+	0.999522i	$0.490154\pi$
$822$	0	0
$823$	49.1570	1.71351	0.856753	−	0.515727i	$-0.172478\pi$
$823$	49.1570	1.71351	0.856753	+	0.515727i	$0.172478\pi$
$824$	0	0
$825$	5.33094	0.185599
$826$	0	0
$827$	38.8452	1.35078	0.675390	−	0.737460i	$-0.263975\pi$
$827$	38.8452	1.35078	0.675390	+	0.737460i	$0.263975\pi$
$828$	0	0
$829$	−18.3535	−0.637445	−0.318722	−	0.947848i	$-0.603254\pi$
$829$	−18.3535	−0.637445	−0.318722	+	0.947848i	$0.603254\pi$
$830$	0	0
$831$	37.3890	1.29701
$832$	0	0
$833$	−17.1420	−0.593936
$834$	0	0
$835$	−0.139569	−0.00483000
$836$	0	0
$837$	12.5102	0.432415
$838$	0	0
$839$	35.1271	1.21272	0.606360	−	0.795190i	$-0.292629\pi$
$839$	35.1271	1.21272	0.606360	+	0.795190i	$0.292629\pi$
$840$	0	0
$841$	20.4115	0.703846
$842$	0	0
$843$	−13.2281	−0.455601
$844$	0	0
$845$	7.12886	0.245240
$846$	0	0
$847$	−3.91271	−0.134442
$848$	0	0
$849$	14.9767	0.513998
$850$	0	0
$851$	19.7586	0.677316
$852$	0	0
$853$	−27.1971	−0.931210	−0.465605	−	0.884993i	$-0.654163\pi$
$853$	−27.1971	−0.931210	−0.465605	+	0.884993i	$0.654163\pi$
$854$	0	0
$855$	4.99127	0.170698
$856$	0	0
$857$	14.5319	0.496400	0.248200	−	0.968709i	$-0.420161\pi$
$857$	14.5319	0.496400	0.248200	+	0.968709i	$0.420161\pi$
$858$	0	0
$859$	12.0294	0.410439	0.205220	−	0.978716i	$-0.434209\pi$
$859$	12.0294	0.410439	0.205220	+	0.978716i	$0.434209\pi$
$860$	0	0
$861$	−59.0421	−2.01215
$862$	0	0
$863$	26.3003	0.895272	0.447636	−	0.894216i	$-0.352266\pi$
$863$	26.3003	0.895272	0.447636	+	0.894216i	$0.352266\pi$
$864$	0	0
$865$	19.5878	0.666007
$866$	0	0
$867$	−17.9831	−0.610739
$868$	0	0
$869$	1.97116	0.0668669
$870$	0	0
$871$	5.32134	0.180307
$872$	0	0
$873$	−1.27005	−0.0429847
$874$	0	0
$875$	−37.9687	−1.28358
$876$	0	0
$877$	23.4759	0.792726	0.396363	−	0.918094i	$-0.370272\pi$
$877$	23.4759	0.792726	0.396363	+	0.918094i	$0.370272\pi$
$878$	0	0
$879$	2.81361	0.0949006
$880$	0	0
$881$	49.1042	1.65436	0.827181	−	0.561935i	$-0.189943\pi$
$881$	49.1042	1.65436	0.827181	+	0.561935i	$0.189943\pi$
$882$	0	0
$883$	−6.14740	−0.206876	−0.103438	−	0.994636i	$-0.532984\pi$
$883$	−6.14740	−0.206876	−0.103438	+	0.994636i	$0.532984\pi$
$884$	0	0
$885$	8.79636	0.295686
$886$	0	0
$887$	37.9399	1.27390	0.636949	−	0.770906i	$-0.280196\pi$
$887$	37.9399	1.27390	0.636949	+	0.770906i	$0.280196\pi$
$888$	0	0
$889$	−49.7105	−1.66724
$890$	0	0
$891$	4.95588	0.166028
$892$	0	0
$893$	17.8800	0.598333
$894$	0	0
$895$	−20.4191	−0.682536
$896$	0	0
$897$	9.52961	0.318185
$898$	0	0
$899$	15.5455	0.518472
$900$	0	0
$901$	21.4929	0.716034
$902$	0	0
$903$	30.4866	1.01453
$904$	0	0
$905$	−12.4777	−0.414774
$906$	0	0
$907$	−28.5356	−0.947511	−0.473755	−	0.880657i	$-0.657102\pi$
$907$	−28.5356	−0.947511	−0.473755	+	0.880657i	$0.657102\pi$
$908$	0	0
$909$	−7.39825	−0.245384
$910$	0	0
$911$	−6.38947	−0.211693	−0.105846	−	0.994383i	$-0.533755\pi$
$911$	−6.38947	−0.211693	−0.105846	+	0.994383i	$0.533755\pi$
$912$	0	0
$913$	−13.0889	−0.433180
$914$	0	0
$915$	−2.54257	−0.0840546
$916$	0	0
$917$	33.5478	1.10785
$918$	0	0
$919$	32.2671	1.06439	0.532197	−	0.846620i	$-0.321366\pi$
$919$	32.2671	1.06439	0.532197	+	0.846620i	$0.321366\pi$
$920$	0	0
$921$	−32.4301	−1.06861
$922$	0	0
$923$	35.4005	1.16522
$924$	0	0
$925$	−28.2913	−0.930214
$926$	0	0
$927$	2.64042	0.0867227
$928$	0	0
$929$	18.8779	0.619363	0.309682	−	0.950840i	$-0.399778\pi$
$929$	18.8779	0.619363	0.309682	+	0.950840i	$0.399778\pi$
$930$	0	0
$931$	37.1885	1.21880
$932$	0	0
$933$	−27.3990	−0.897004
$934$	0	0
$935$	−2.28063	−0.0745847
$936$	0	0
$937$	4.68065	0.152910	0.0764551	−	0.997073i	$-0.475640\pi$
$937$	4.68065	0.152910	0.0764551	+	0.997073i	$0.475640\pi$
$938$	0	0
$939$	−1.86975	−0.0610170
$940$	0	0
$941$	15.6193	0.509173	0.254587	−	0.967050i	$-0.418061\pi$
$941$	15.6193	0.509173	0.254587	+	0.967050i	$0.418061\pi$
$942$	0	0
$943$	28.2148	0.918801
$944$	0	0
$945$	−24.4686	−0.795962
$946$	0	0
$947$	−20.4280	−0.663822	−0.331911	−	0.943311i	$-0.607693\pi$
$947$	−20.4280	−0.663822	−0.331911	+	0.943311i	$0.607693\pi$
$948$	0	0
$949$	21.8600	0.709606
$950$	0	0
$951$	−42.5588	−1.38006
$952$	0	0
$953$	18.5026	0.599357	0.299679	−	0.954040i	$-0.403121\pi$
$953$	18.5026	0.599357	0.299679	+	0.954040i	$0.403121\pi$
$954$	0	0
$955$	−4.94050	−0.159871
$956$	0	0
$957$	9.91906	0.320638
$958$	0	0
$959$	3.91271	0.126348
$960$	0	0
$961$	−26.1092	−0.842231
$962$	0	0
$963$	7.38141	0.237863
$964$	0	0
$965$	−12.4359	−0.400326
$966$	0	0
$967$	−21.6697	−0.696849	−0.348425	−	0.937337i	$-0.613283\pi$
$967$	−21.6697	−0.696849	−0.348425	+	0.937337i	$0.613283\pi$
$968$	0	0
$969$	−13.0286	−0.418538
$970$	0	0
$971$	−38.9339	−1.24945	−0.624724	−	0.780846i	$-0.714788\pi$
$971$	−38.9339	−1.24945	−0.624724	+	0.780846i	$0.714788\pi$
$972$	0	0
$973$	−29.4311	−0.943517
$974$	0	0
$975$	−13.6450	−0.436989
$976$	0	0
$977$	1.63997	0.0524673	0.0262336	−	0.999656i	$-0.491649\pi$
$977$	1.63997	0.0524673	0.0262336	+	0.999656i	$0.491649\pi$
$978$	0	0
$979$	−14.3027	−0.457116
$980$	0	0
$981$	19.7851	0.631688
$982$	0	0
$983$	24.2774	0.774328	0.387164	−	0.922011i	$-0.373455\pi$
$983$	24.2774	0.774328	0.387164	+	0.922011i	$0.373455\pi$
$984$	0	0
$985$	−20.9360	−0.667076
$986$	0	0
$987$	−22.0577	−0.702103
$988$	0	0
$989$	−14.5688	−0.463261
$990$	0	0
$991$	−45.6160	−1.44904	−0.724519	−	0.689255i	$-0.757938\pi$
$991$	−45.6160	−1.44904	−0.724519	+	0.689255i	$0.757938\pi$
$992$	0	0
$993$	30.0041	0.952151
$994$	0	0
$995$	13.1502	0.416890
$996$	0	0
$997$	−20.2588	−0.641603	−0.320801	−	0.947146i	$-0.603952\pi$
$997$	−20.2588	−0.641603	−0.320801	+	0.947146i	$0.603952\pi$
$998$	0	0
$999$	−42.3622	−1.34028

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.e.1.19	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.e.1.19	✓	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)

\(f(q)\)	\(=\)	\(q+1.41110 q^{3} -1.10550 q^{5} -3.91271 q^{7} -1.00881 q^{9} +O(q^{10})\)	\(q+1.41110 q^{3} -1.10550 q^{5} -3.91271 q^{7} -1.00881 q^{9} -1.00000 q^{11} +2.55958 q^{13} -1.55997 q^{15} -2.06299 q^{17} +4.47552 q^{19} -5.52121 q^{21} +2.63846 q^{23} -3.77787 q^{25} -5.65681 q^{27} -7.02933 q^{29} -2.21152 q^{31} -1.41110 q^{33} +4.32550 q^{35} +7.48870 q^{37} +3.61181 q^{39} +10.6937 q^{41} -5.52171 q^{43} +1.11524 q^{45} +3.99508 q^{47} +8.30932 q^{49} -2.91107 q^{51} -10.4184 q^{53} +1.10550 q^{55} +6.31539 q^{57} -5.63881 q^{59} +1.62988 q^{61} +3.94718 q^{63} -2.82962 q^{65} +2.07899 q^{67} +3.72311 q^{69} +13.8306 q^{71} +8.54047 q^{73} -5.33094 q^{75} +3.91271 q^{77} -1.97116 q^{79} -4.95588 q^{81} +13.0889 q^{83} +2.28063 q^{85} -9.91906 q^{87} +14.3027 q^{89} -10.0149 q^{91} -3.12067 q^{93} -4.94769 q^{95} +1.25896 q^{97} +1.00881 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) \) 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9} - 27 q^{11} + 2 q^{13} + 10 q^{15} + 17 q^{17} + 4 q^{19} - 4 q^{21} + 25 q^{23} + 21 q^{25} + 32 q^{27} - q^{29} + 8 q^{31} - 5 q^{33} + 30 q^{35} - 24 q^{37} + 10 q^{39} + 31 q^{41} + 17 q^{43} - 15 q^{45} + 27 q^{47} + 40 q^{49} + 30 q^{51} + 2 q^{55} + 13 q^{57} + 31 q^{59} + 2 q^{61} + 61 q^{63} + 2 q^{67} - 15 q^{69} + 40 q^{71} + 17 q^{73} + 30 q^{75} - 7 q^{77} + 33 q^{79} + 31 q^{81} + 66 q^{83} + 26 q^{85} + 43 q^{87} + 22 q^{89} + 18 q^{91} - 9 q^{93} + 95 q^{95} - 21 q^{97} - 28 q^{99}+O(q^{100}) \) 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9 - 27 * q^11 + 2 * q^13 + 10 * q^15 + 17 * q^17 + 4 * q^19 - 4 * q^21 + 25 * q^23 + 21 * q^25 + 32 * q^27 - q^29 + 8 * q^31 - 5 * q^33 + 30 * q^35 - 24 * q^37 + 10 * q^39 + 31 * q^41 + 17 * q^43 - 15 * q^45 + 27 * q^47 + 40 * q^49 + 30 * q^51 + 2 * q^55 + 13 * q^57 + 31 * q^59 + 2 * q^61 + 61 * q^63 + 2 * q^67 - 15 * q^69 + 40 * q^71 + 17 * q^73 + 30 * q^75 - 7 * q^77 + 33 * q^79 + 31 * q^81 + 66 * q^83 + 26 * q^85 + 43 * q^87 + 22 * q^89 + 18 * q^91 - 9 * q^93 + 95 * q^95 - 21 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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