LMFDB - Embedded newform 2028.2.a.m.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2028,2,Mod(1,2028)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2028.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2028, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2028.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,4,0,0,0,0,0,4,0,0,0,0,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$16.1936615299$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{43})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 23x^{2} + 100 $ x^4 - 23*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.14474$ of defining polynomial
Character	$\chi$	$=$	2028.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.14474 q^{5} +2.41269 q^{7} +1.00000 q^{9} -3.46410 q^{11} +4.14474 q^{15} -5.17891 q^{17} +3.46410 q^{19} +2.41269 q^{21} +2.00000 q^{23} +12.1789 q^{25} +1.00000 q^{27} +3.17891 q^{29} +1.05141 q^{31} -3.46410 q^{33} +10.0000 q^{35} -7.60885 q^{37} -0.680643 q^{41} +12.1789 q^{43} +4.14474 q^{45} +10.3923 q^{47} -1.17891 q^{49} -5.17891 q^{51} +1.17891 q^{53} -14.3578 q^{55} +3.46410 q^{57} -11.7536 q^{59} +5.00000 q^{61} +2.41269 q^{63} -2.41269 q^{67} +2.00000 q^{69} +3.46410 q^{71} -14.8469 q^{73} +12.1789 q^{75} -8.35782 q^{77} +1.82109 q^{79} +1.00000 q^{81} -1.36129 q^{83} -21.4653 q^{85} +3.17891 q^{87} -6.92820 q^{89} +1.05141 q^{93} +14.3578 q^{95} -17.6304 q^{97} -3.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9} + 2 q^{17} + 8 q^{23} + 26 q^{25} + 4 q^{27} - 10 q^{29} + 40 q^{35} + 26 q^{43} + 18 q^{49} + 2 q^{51} - 18 q^{53} - 12 q^{55} + 20 q^{61} + 8 q^{69} + 26 q^{75} + 12 q^{77} + 30 q^{79}+ \cdots + 12 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 + 2 * q^17 + 8 * q^23 + 26 * q^25 + 4 * q^27 - 10 * q^29 + 40 * q^35 + 26 * q^43 + 18 * q^49 + 2 * q^51 - 18 * q^53 - 12 * q^55 + 20 * q^61 + 8 * q^69 + 26 * q^75 + 12 * q^77 + 30 * q^79 + 4 * q^81 - 10 * q^87 + 12 * q^95

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	4.14474	1.85359	0.926793	−	0.375572i	$-0.122554\pi$
$5$	4.14474	1.85359	0.926793	+	0.375572i	$0.122554\pi$
$6$	0	0
$7$	2.41269	0.911913	0.455956	−	0.890002i	$-0.349297\pi$
$7$	2.41269	0.911913	0.455956	+	0.890002i	$0.349297\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	4.14474	1.07017
$16$	0	0
$17$	−5.17891	−1.25607	−0.628035	−	0.778185i	$-0.716140\pi$
$17$	−5.17891	−1.25607	−0.628035	+	0.778185i	$0.716140\pi$
$18$	0	0
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	0	0
$21$	2.41269	0.526493
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	12.1789	2.43578
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.17891	0.590308	0.295154	−	0.955450i	$-0.404629\pi$
$29$	3.17891	0.590308	0.295154	+	0.955450i	$0.404629\pi$
$30$	0	0
$31$	1.05141	0.188838	0.0944192	−	0.995533i	$-0.469901\pi$
$31$	1.05141	0.188838	0.0944192	+	0.995533i	$0.469901\pi$
$32$	0	0
$33$	−3.46410	−0.603023
$34$	0	0
$35$	10.0000	1.69031
$36$	0	0
$37$	−7.60885	−1.25089	−0.625443	−	0.780270i	$-0.715082\pi$
$37$	−7.60885	−1.25089	−0.625443	+	0.780270i	$0.715082\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.680643	−0.106299	−0.0531493	−	0.998587i	$-0.516926\pi$
$41$	−0.680643	−0.106299	−0.0531493	+	0.998587i	$0.516926\pi$
$42$	0	0
$43$	12.1789	1.85727	0.928633	−	0.371000i	$-0.120985\pi$
$43$	12.1789	1.85727	0.928633	+	0.371000i	$0.120985\pi$
$44$	0	0
$45$	4.14474	0.617862
$46$	0	0
$47$	10.3923	1.51587	0.757937	−	0.652328i	$-0.226208\pi$
$47$	10.3923	1.51587	0.757937	+	0.652328i	$0.226208\pi$
$48$	0	0
$49$	−1.17891	−0.168415
$50$	0	0
$51$	−5.17891	−0.725192
$52$	0	0
$53$	1.17891	0.161936	0.0809678	−	0.996717i	$-0.474199\pi$
$53$	1.17891	0.161936	0.0809678	+	0.996717i	$0.474199\pi$
$54$	0	0
$55$	−14.3578	−1.93601
$56$	0	0
$57$	3.46410	0.458831
$58$	0	0
$59$	−11.7536	−1.53019	−0.765094	−	0.643919i	$-0.777307\pi$
$59$	−11.7536	−1.53019	−0.765094	+	0.643919i	$0.777307\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	0	0
$63$	2.41269	0.303971
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.41269	−0.294757	−0.147379	−	0.989080i	$-0.547084\pi$
$67$	−2.41269	−0.294757	−0.147379	+	0.989080i	$0.547084\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	3.46410	0.411113	0.205557	−	0.978645i	$-0.434100\pi$
$71$	3.46410	0.411113	0.205557	+	0.978645i	$0.434100\pi$
$72$	0	0
$73$	−14.8469	−1.73770	−0.868851	−	0.495074i	$-0.835141\pi$
$73$	−14.8469	−1.73770	−0.868851	+	0.495074i	$0.835141\pi$
$74$	0	0
$75$	12.1789	1.40630
$76$	0	0
$77$	−8.35782	−0.952462
$78$	0	0
$79$	1.82109	0.204889	0.102444	−	0.994739i	$-0.467334\pi$
$79$	1.82109	0.204889	0.102444	+	0.994739i	$0.467334\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.36129	−0.149421	−0.0747103	−	0.997205i	$-0.523803\pi$
$83$	−1.36129	−0.149421	−0.0747103	+	0.997205i	$0.523803\pi$
$84$	0	0
$85$	−21.4653	−2.32823
$86$	0	0
$87$	3.17891	0.340815
$88$	0	0
$89$	−6.92820	−0.734388	−0.367194	−	0.930144i	$-0.619682\pi$
$89$	−6.92820	−0.734388	−0.367194	+	0.930144i	$0.619682\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.05141	0.109026
$94$	0	0
$95$	14.3578	1.47308
$96$	0	0
$97$	−17.6304	−1.79009	−0.895047	−	0.445971i	$-0.852858\pi$
$97$	−17.6304	−1.79009	−0.895047	+	0.445971i	$0.852858\pi$
$98$	0	0
$99$	−3.46410	−0.348155
$100$	0	0
$101$	−0.821092	−0.0817017	−0.0408508	−	0.999165i	$-0.513007\pi$
$101$	−0.821092	−0.0817017	−0.0408508	+	0.999165i	$0.513007\pi$
$102$	0	0
$103$	8.17891	0.805892	0.402946	−	0.915224i	$-0.367986\pi$
$103$	8.17891	0.805892	0.402946	+	0.915224i	$0.367986\pi$
$104$	0	0
$105$	10.0000	0.975900
$106$	0	0
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	0	0
$109$	1.05141	0.100707	0.0503533	−	0.998731i	$-0.483965\pi$
$109$	1.05141	0.100707	0.0503533	+	0.998731i	$0.483965\pi$
$110$	0	0
$111$	−7.60885	−0.722200
$112$	0	0
$113$	0.821092	0.0772418	0.0386209	−	0.999254i	$-0.487704\pi$
$113$	0.821092	0.0772418	0.0386209	+	0.999254i	$0.487704\pi$
$114$	0	0
$115$	8.28949	0.772999
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.4951	−1.14543
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.680643	−0.0613715
$124$	0	0
$125$	29.7547	2.66135
$126$	0	0
$127$	14.1789	1.25817	0.629087	−	0.777335i	$-0.283429\pi$
$127$	14.1789	1.25817	0.629087	+	0.777335i	$0.283429\pi$
$128$	0	0
$129$	12.1789	1.07229
$130$	0	0
$131$	−16.3578	−1.42919	−0.714595	−	0.699539i	$-0.753389\pi$
$131$	−16.3578	−1.42919	−0.714595	+	0.699539i	$0.753389\pi$
$132$	0	0
$133$	8.35782	0.724715
$134$	0	0
$135$	4.14474	0.356723
$136$	0	0
$137$	−6.24756	−0.533765	−0.266883	−	0.963729i	$-0.585994\pi$
$137$	−6.24756	−0.533765	−0.266883	+	0.963729i	$0.585994\pi$
$138$	0	0
$139$	5.82109	0.493739	0.246869	−	0.969049i	$-0.420598\pi$
$139$	5.82109	0.493739	0.246869	+	0.969049i	$0.420598\pi$
$140$	0	0
$141$	10.3923	0.875190
$142$	0	0
$143$	0	0
$144$	0	0
$145$	13.1758	1.09419
$146$	0	0
$147$	−1.17891	−0.0972347
$148$	0	0
$149$	0.680643	0.0557605	0.0278802	−	0.999611i	$-0.491124\pi$
$149$	0.680643	0.0557605	0.0278802	+	0.999611i	$0.491124\pi$
$150$	0	0
$151$	6.18667	0.503464	0.251732	−	0.967797i	$-0.419000\pi$
$151$	6.18667	0.503464	0.251732	+	0.967797i	$0.419000\pi$
$152$	0	0
$153$	−5.17891	−0.418690
$154$	0	0
$155$	4.35782	0.350028
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	0	0
$159$	1.17891	0.0934935
$160$	0	0
$161$	4.82539	0.380294
$162$	0	0
$163$	−1.05141	−0.0823526	−0.0411763	−	0.999152i	$-0.513111\pi$
$163$	−1.05141	−0.0823526	−0.0411763	+	0.999152i	$0.513111\pi$
$164$	0	0
$165$	−14.3578	−1.11775
$166$	0	0
$167$	−6.92820	−0.536120	−0.268060	−	0.963402i	$-0.586383\pi$
$167$	−6.92820	−0.536120	−0.268060	+	0.963402i	$0.586383\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	3.46410	0.264906
$172$	0	0
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	29.3840	2.22122
$176$	0	0
$177$	−11.7536	−0.883454
$178$	0	0
$179$	−14.3578	−1.07315	−0.536577	−	0.843851i	$-0.680283\pi$
$179$	−14.3578	−1.07315	−0.536577	+	0.843851i	$0.680283\pi$
$180$	0	0
$181$	−13.5367	−1.00618	−0.503088	−	0.864235i	$-0.667803\pi$
$181$	−13.5367	−1.00618	−0.503088	+	0.864235i	$0.667803\pi$
$182$	0	0
$183$	5.00000	0.369611
$184$	0	0
$185$	−31.5367	−2.31863
$186$	0	0
$187$	17.9403	1.31192
$188$	0	0
$189$	2.41269	0.175498
$190$	0	0
$191$	20.3578	1.47304	0.736520	−	0.676416i	$-0.236468\pi$
$191$	20.3578	1.47304	0.736520	+	0.676416i	$0.236468\pi$
$192$	0	0
$193$	0.370765	0.0266882	0.0133441	−	0.999911i	$-0.495752\pi$
$193$	0.370765	0.0266882	0.0133441	+	0.999911i	$0.495752\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.5790	1.18120	0.590602	−	0.806963i	$-0.298890\pi$
$197$	16.5790	1.18120	0.590602	+	0.806963i	$0.298890\pi$
$198$	0	0
$199$	−12.5367	−0.888705	−0.444352	−	0.895852i	$-0.646566\pi$
$199$	−12.5367	−0.888705	−0.444352	+	0.895852i	$0.646566\pi$
$200$	0	0
$201$	−2.41269	−0.170178
$202$	0	0
$203$	7.66973	0.538310
$204$	0	0
$205$	−2.82109	−0.197034
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	18.5367	1.27612	0.638060	−	0.769986i	$-0.279737\pi$
$211$	18.5367	1.27612	0.638060	+	0.769986i	$0.279737\pi$
$212$	0	0
$213$	3.46410	0.237356
$214$	0	0
$215$	50.4785	3.44260
$216$	0	0
$217$	2.53673	0.172204
$218$	0	0
$219$	−14.8469	−1.00326
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−2.10282	−0.140815	−0.0704075	−	0.997518i	$-0.522430\pi$
$223$	−2.10282	−0.140815	−0.0704075	+	0.997518i	$0.522430\pi$
$224$	0	0
$225$	12.1789	0.811927
$226$	0	0
$227$	−22.1459	−1.46987	−0.734937	−	0.678135i	$-0.762788\pi$
$227$	−22.1459	−1.46987	−0.734937	+	0.678135i	$0.762788\pi$
$228$	0	0
$229$	16.5790	1.09557	0.547785	−	0.836619i	$-0.315471\pi$
$229$	16.5790	1.09557	0.547785	+	0.836619i	$0.315471\pi$
$230$	0	0
$231$	−8.35782	−0.549904
$232$	0	0
$233$	22.7156	1.48815	0.744075	−	0.668096i	$-0.232890\pi$
$233$	22.7156	1.48815	0.744075	+	0.668096i	$0.232890\pi$
$234$	0	0
$235$	43.0735	2.80980
$236$	0	0
$237$	1.82109	0.118293
$238$	0	0
$239$	−15.2177	−0.984351	−0.492175	−	0.870496i	$-0.663798\pi$
$239$	−15.2177	−0.984351	−0.492175	+	0.870496i	$0.663798\pi$
$240$	0	0
$241$	−0.680643	−0.0438441	−0.0219220	−	0.999760i	$-0.506979\pi$
$241$	−0.680643	−0.0438441	−0.0219220	+	0.999760i	$0.506979\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.88627	−0.312173
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.36129	−0.0862680
$250$	0	0
$251$	−24.7156	−1.56004	−0.780018	−	0.625756i	$-0.784790\pi$
$251$	−24.7156	−1.56004	−0.780018	+	0.625756i	$0.784790\pi$
$252$	0	0
$253$	−6.92820	−0.435572
$254$	0	0
$255$	−21.4653	−1.34421
$256$	0	0
$257$	−10.8211	−0.675001	−0.337501	−	0.941325i	$-0.609581\pi$
$257$	−10.8211	−0.675001	−0.337501	+	0.941325i	$0.609581\pi$
$258$	0	0
$259$	−18.3578	−1.14070
$260$	0	0
$261$	3.17891	0.196769
$262$	0	0
$263$	9.64218	0.594562	0.297281	−	0.954790i	$-0.403920\pi$
$263$	9.64218	0.594562	0.297281	+	0.954790i	$0.403920\pi$
$264$	0	0
$265$	4.88627	0.300161
$266$	0	0
$267$	−6.92820	−0.423999
$268$	0	0
$269$	10.3578	0.631527	0.315764	−	0.948838i	$-0.397739\pi$
$269$	10.3578	0.631527	0.315764	+	0.948838i	$0.397739\pi$
$270$	0	0
$271$	7.97961	0.484727	0.242363	−	0.970186i	$-0.422077\pi$
$271$	7.97961	0.484727	0.242363	+	0.970186i	$0.422077\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−42.1890	−2.54409
$276$	0	0
$277$	−11.1789	−0.671676	−0.335838	−	0.941920i	$-0.609019\pi$
$277$	−11.1789	−0.671676	−0.335838	+	0.941920i	$0.609019\pi$
$278$	0	0
$279$	1.05141	0.0629461
$280$	0	0
$281$	−18.0012	−1.07386	−0.536929	−	0.843627i	$-0.680416\pi$
$281$	−18.0012	−1.07386	−0.536929	+	0.843627i	$0.680416\pi$
$282$	0	0
$283$	15.8211	0.940466	0.470233	−	0.882542i	$-0.344170\pi$
$283$	15.8211	0.940466	0.470233	+	0.882542i	$0.344170\pi$
$284$	0	0
$285$	14.3578	0.850484
$286$	0	0
$287$	−1.64218	−0.0969350
$288$	0	0
$289$	9.82109	0.577711
$290$	0	0
$291$	−17.6304	−1.03351
$292$	0	0
$293$	0.0608864	0.00355702	0.00177851	−	0.999998i	$-0.499434\pi$
$293$	0.0608864	0.00355702	0.00177851	+	0.999998i	$0.499434\pi$
$294$	0	0
$295$	−48.7156	−2.83633
$296$	0	0
$297$	−3.46410	−0.201008
$298$	0	0
$299$	0	0
$300$	0	0
$301$	29.3840	1.69366
$302$	0	0
$303$	−0.821092	−0.0471705
$304$	0	0
$305$	20.7237	1.18664
$306$	0	0
$307$	−4.51551	−0.257714	−0.128857	−	0.991663i	$-0.541131\pi$
$307$	−4.51551	−0.257714	−0.128857	+	0.991663i	$0.541131\pi$
$308$	0	0
$309$	8.17891	0.465282
$310$	0	0
$311$	−34.3578	−1.94825	−0.974127	−	0.226003i	$-0.927434\pi$
$311$	−34.3578	−1.94825	−0.974127	+	0.226003i	$0.927434\pi$
$312$	0	0
$313$	−28.8945	−1.63322	−0.816608	−	0.577193i	$-0.804148\pi$
$313$	−28.8945	−1.63322	−0.816608	+	0.577193i	$0.804148\pi$
$314$	0	0
$315$	10.0000	0.563436
$316$	0	0
$317$	−15.8983	−0.892939	−0.446470	−	0.894799i	$-0.647319\pi$
$317$	−15.8983	−0.892939	−0.446470	+	0.894799i	$0.647319\pi$
$318$	0	0
$319$	−11.0121	−0.616557
$320$	0	0
$321$	−10.0000	−0.558146
$322$	0	0
$323$	−17.9403	−0.998223
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.05141	0.0581430
$328$	0	0
$329$	25.0735	1.38234
$330$	0	0
$331$	26.6614	1.46544	0.732722	−	0.680528i	$-0.238250\pi$
$331$	26.6614	1.46544	0.732722	+	0.680528i	$0.238250\pi$
$332$	0	0
$333$	−7.60885	−0.416962
$334$	0	0
$335$	−10.0000	−0.546358
$336$	0	0
$337$	2.64218	0.143929	0.0719644	−	0.997407i	$-0.477073\pi$
$337$	2.64218	0.143929	0.0719644	+	0.997407i	$0.477073\pi$
$338$	0	0
$339$	0.821092	0.0445956
$340$	0	0
$341$	−3.64218	−0.197235
$342$	0	0
$343$	−19.7332	−1.06549
$344$	0	0
$345$	8.28949	0.446291
$346$	0	0
$347$	−2.35782	−0.126574	−0.0632871	−	0.997995i	$-0.520158\pi$
$347$	−2.35782	−0.126574	−0.0632871	+	0.997995i	$0.520158\pi$
$348$	0	0
$349$	22.4558	1.20203	0.601015	−	0.799238i	$-0.294763\pi$
$349$	22.4558	1.20203	0.601015	+	0.799238i	$0.294763\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−13.7955	−0.734261	−0.367131	−	0.930169i	$-0.619660\pi$
$353$	−13.7955	−0.734261	−0.367131	+	0.930169i	$0.619660\pi$
$354$	0	0
$355$	14.3578	0.762034
$356$	0	0
$357$	−12.4951	−0.661312
$358$	0	0
$359$	5.56692	0.293811	0.146905	−	0.989151i	$-0.453069\pi$
$359$	5.56692	0.293811	0.146905	+	0.989151i	$0.453069\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−61.5367	−3.22098
$366$	0	0
$367$	24.1789	1.26213	0.631064	−	0.775730i	$-0.282618\pi$
$367$	24.1789	1.26213	0.631064	+	0.775730i	$0.282618\pi$
$368$	0	0
$369$	−0.680643	−0.0354329
$370$	0	0
$371$	2.84434	0.147671
$372$	0	0
$373$	11.3578	0.588085	0.294043	−	0.955792i	$-0.404999\pi$
$373$	11.3578	0.588085	0.294043	+	0.955792i	$0.404999\pi$
$374$	0	0
$375$	29.7547	1.53653
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−12.8050	−0.657749	−0.328874	−	0.944374i	$-0.606669\pi$
$379$	−12.8050	−0.657749	−0.328874	+	0.944374i	$0.606669\pi$
$380$	0	0
$381$	14.1789	0.726407
$382$	0	0
$383$	−33.1580	−1.69429	−0.847146	−	0.531360i	$-0.821681\pi$
$383$	−33.1580	−1.69429	−0.847146	+	0.531360i	$0.821681\pi$
$384$	0	0
$385$	−34.6410	−1.76547
$386$	0	0
$387$	12.1789	0.619089
$388$	0	0
$389$	−27.5367	−1.39617	−0.698084	−	0.716016i	$-0.745964\pi$
$389$	−27.5367	−1.39617	−0.698084	+	0.716016i	$0.745964\pi$
$390$	0	0
$391$	−10.3578	−0.523817
$392$	0	0
$393$	−16.3578	−0.825143
$394$	0	0
$395$	7.54796	0.379779
$396$	0	0
$397$	14.9078	0.748202	0.374101	−	0.927388i	$-0.377951\pi$
$397$	14.9078	0.748202	0.374101	+	0.927388i	$0.377951\pi$
$398$	0	0
$399$	8.35782	0.418414
$400$	0	0
$401$	8.97013	0.447947	0.223974	−	0.974595i	$-0.428097\pi$
$401$	8.97013	0.447947	0.223974	+	0.974595i	$0.428097\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	4.14474	0.205954
$406$	0	0
$407$	26.3578	1.30651
$408$	0	0
$409$	−25.2392	−1.24800	−0.624000	−	0.781424i	$-0.714493\pi$
$409$	−25.2392	−1.24800	−0.624000	+	0.781424i	$0.714493\pi$
$410$	0	0
$411$	−6.24756	−0.308169
$412$	0	0
$413$	−28.3578	−1.39540
$414$	0	0
$415$	−5.64218	−0.276964
$416$	0	0
$417$	5.82109	0.285060
$418$	0	0
$419$	32.7156	1.59826	0.799132	−	0.601156i	$-0.205293\pi$
$419$	32.7156	1.59826	0.799132	+	0.601156i	$0.205293\pi$
$420$	0	0
$421$	−5.93768	−0.289385	−0.144692	−	0.989477i	$-0.546219\pi$
$421$	−5.93768	−0.289385	−0.144692	+	0.989477i	$0.546219\pi$
$422$	0	0
$423$	10.3923	0.505291
$424$	0	0
$425$	−63.0735	−3.05951
$426$	0	0
$427$	12.0635	0.583792
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.56692	−0.268149	−0.134074	−	0.990971i	$-0.542806\pi$
$431$	−5.56692	−0.268149	−0.134074	+	0.990971i	$0.542806\pi$
$432$	0	0
$433$	25.3578	1.21862	0.609309	−	0.792933i	$-0.291447\pi$
$433$	25.3578	1.21862	0.609309	+	0.792933i	$0.291447\pi$
$434$	0	0
$435$	13.1758	0.631730
$436$	0	0
$437$	6.92820	0.331421
$438$	0	0
$439$	32.8945	1.56997	0.784985	−	0.619514i	$-0.212670\pi$
$439$	32.8945	1.56997	0.784985	+	0.619514i	$0.212670\pi$
$440$	0	0
$441$	−1.17891	−0.0561385
$442$	0	0
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	0	0
$445$	−28.7156	−1.36125
$446$	0	0
$447$	0.680643	0.0321933
$448$	0	0
$449$	28.3326	1.33710	0.668548	−	0.743669i	$-0.266916\pi$
$449$	28.3326	1.33710	0.668548	+	0.743669i	$0.266916\pi$
$450$	0	0
$451$	2.35782	0.111025
$452$	0	0
$453$	6.18667	0.290675
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.4138	0.954919	0.477460	−	0.878654i	$-0.341558\pi$
$457$	20.4138	0.954919	0.477460	+	0.878654i	$0.341558\pi$
$458$	0	0
$459$	−5.17891	−0.241731
$460$	0	0
$461$	4.88627	0.227576	0.113788	−	0.993505i	$-0.463701\pi$
$461$	4.88627	0.227576	0.113788	+	0.993505i	$0.463701\pi$
$462$	0	0
$463$	−21.0945	−0.980344	−0.490172	−	0.871626i	$-0.663066\pi$
$463$	−21.0945	−0.980344	−0.490172	+	0.871626i	$0.663066\pi$
$464$	0	0
$465$	4.35782	0.202089
$466$	0	0
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	−5.82109	−0.268793
$470$	0	0
$471$	7.00000	0.322543
$472$	0	0
$473$	−42.1890	−1.93985
$474$	0	0
$475$	42.1890	1.93576
$476$	0	0
$477$	1.17891	0.0539785
$478$	0	0
$479$	6.92820	0.316558	0.158279	−	0.987394i	$-0.449406\pi$
$479$	6.92820	0.316558	0.158279	+	0.987394i	$0.449406\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	4.82539	0.219563
$484$	0	0
$485$	−73.0735	−3.31809
$486$	0	0
$487$	−24.2487	−1.09881	−0.549407	−	0.835555i	$-0.685146\pi$
$487$	−24.2487	−1.09881	−0.549407	+	0.835555i	$0.685146\pi$
$488$	0	0
$489$	−1.05141	−0.0475463
$490$	0	0
$491$	26.7156	1.20566	0.602830	−	0.797870i	$-0.294040\pi$
$491$	26.7156	1.20566	0.602830	+	0.797870i	$0.294040\pi$
$492$	0	0
$493$	−16.4633	−0.741469
$494$	0	0
$495$	−14.3578	−0.645336
$496$	0	0
$497$	8.35782	0.374899
$498$	0	0
$499$	−11.7536	−0.526163	−0.263081	−	0.964774i	$-0.584739\pi$
$499$	−11.7536	−0.526163	−0.263081	+	0.964774i	$0.584739\pi$
$500$	0	0
$501$	−6.92820	−0.309529
$502$	0	0
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	−3.40322	−0.151441
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.50603	0.244051	0.122025	−	0.992527i	$-0.461061\pi$
$509$	5.50603	0.244051	0.122025	+	0.992527i	$0.461061\pi$
$510$	0	0
$511$	−35.8211	−1.58463
$512$	0	0
$513$	3.46410	0.152944
$514$	0	0
$515$	33.8995	1.49379
$516$	0	0
$517$	−36.0000	−1.58328
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	17.8945	0.783974	0.391987	−	0.919971i	$-0.371788\pi$
$521$	17.8945	0.783974	0.391987	+	0.919971i	$0.371788\pi$
$522$	0	0
$523$	1.64218	0.0718077	0.0359038	−	0.999355i	$-0.488569\pi$
$523$	1.64218	0.0718077	0.0359038	+	0.999355i	$0.488569\pi$
$524$	0	0
$525$	29.3840	1.28242
$526$	0	0
$527$	−5.44514	−0.237194
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−11.7536	−0.510062
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−41.4474	−1.79193
$536$	0	0
$537$	−14.3578	−0.619586
$538$	0	0
$539$	4.08386	0.175904
$540$	0	0
$541$	−10.0215	−0.430860	−0.215430	−	0.976519i	$-0.569115\pi$
$541$	−10.0215	−0.430860	−0.215430	+	0.976519i	$0.569115\pi$
$542$	0	0
$543$	−13.5367	−0.580916
$544$	0	0
$545$	4.35782	0.186668
$546$	0	0
$547$	−20.5367	−0.878087	−0.439043	−	0.898466i	$-0.644683\pi$
$547$	−20.5367	−0.878087	−0.439043	+	0.898466i	$0.644683\pi$
$548$	0	0
$549$	5.00000	0.213395
$550$	0	0
$551$	11.0121	0.469130
$552$	0	0
$553$	4.39374	0.186841
$554$	0	0
$555$	−31.5367	−1.33866
$556$	0	0
$557$	−26.9104	−1.14023	−0.570115	−	0.821565i	$-0.693101\pi$
$557$	−26.9104	−1.14023	−0.570115	+	0.821565i	$0.693101\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	17.9403	0.757439
$562$	0	0
$563$	−4.35782	−0.183660	−0.0918300	−	0.995775i	$-0.529272\pi$
$563$	−4.35782	−0.183660	−0.0918300	+	0.995775i	$0.529272\pi$
$564$	0	0
$565$	3.40322	0.143174
$566$	0	0
$567$	2.41269	0.101324
$568$	0	0
$569$	−14.7156	−0.616911	−0.308456	−	0.951239i	$-0.599812\pi$
$569$	−14.7156	−0.616911	−0.308456	+	0.951239i	$0.599812\pi$
$570$	0	0
$571$	30.3578	1.27044	0.635218	−	0.772333i	$-0.280910\pi$
$571$	30.3578	1.27044	0.635218	+	0.772333i	$0.280910\pi$
$572$	0	0
$573$	20.3578	0.850460
$574$	0	0
$575$	24.3578	1.01579
$576$	0	0
$577$	10.4532	0.435172	0.217586	−	0.976041i	$-0.430182\pi$
$577$	10.4532	0.435172	0.217586	+	0.976041i	$0.430182\pi$
$578$	0	0
$579$	0.370765	0.0154085
$580$	0	0
$581$	−3.28437	−0.136258
$582$	0	0
$583$	−4.08386	−0.169136
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.03102	−0.372750	−0.186375	−	0.982479i	$-0.559674\pi$
$587$	−9.03102	−0.372750	−0.186375	+	0.982479i	$0.559674\pi$
$588$	0	0
$589$	3.64218	0.150074
$590$	0	0
$591$	16.5790	0.681968
$592$	0	0
$593$	−0.0608864	−0.00250030	−0.00125015	−	0.999999i	$-0.500398\pi$
$593$	−0.0608864	−0.00250030	−0.00125015	+	0.999999i	$0.500398\pi$
$594$	0	0
$595$	−51.7891	−2.12315
$596$	0	0
$597$	−12.5367	−0.513094
$598$	0	0
$599$	32.3578	1.32210	0.661052	−	0.750340i	$-0.270110\pi$
$599$	32.3578	1.32210	0.661052	+	0.750340i	$0.270110\pi$
$600$	0	0
$601$	11.1789	0.455997	0.227999	−	0.973661i	$-0.426782\pi$
$601$	11.1789	0.455997	0.227999	+	0.973661i	$0.426782\pi$
$602$	0	0
$603$	−2.41269	−0.0982525
$604$	0	0
$605$	4.14474	0.168508
$606$	0	0
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	0	0
$609$	7.66973	0.310793
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−28.7033	−1.15932	−0.579658	−	0.814860i	$-0.696814\pi$
$613$	−28.7033	−1.15932	−0.579658	+	0.814860i	$0.696814\pi$
$614$	0	0
$615$	−2.82109	−0.113757
$616$	0	0
$617$	4.14474	0.166861	0.0834306	−	0.996514i	$-0.473412\pi$
$617$	4.14474	0.166861	0.0834306	+	0.996514i	$0.473412\pi$
$618$	0	0
$619$	−22.4558	−0.902574	−0.451287	−	0.892379i	$-0.649035\pi$
$619$	−22.4558	−0.902574	−0.451287	+	0.892379i	$0.649035\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	−16.7156	−0.669698
$624$	0	0
$625$	62.4313	2.49725
$626$	0	0
$627$	−12.0000	−0.479234
$628$	0	0
$629$	39.4055	1.57120
$630$	0	0
$631$	12.8050	0.509759	0.254879	−	0.966973i	$-0.417964\pi$
$631$	12.8050	0.509759	0.254879	+	0.966973i	$0.417964\pi$
$632$	0	0
$633$	18.5367	0.736769
$634$	0	0
$635$	58.7680	2.33214
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.46410	0.137038
$640$	0	0
$641$	−33.5367	−1.32462	−0.662310	−	0.749229i	$-0.730424\pi$
$641$	−33.5367	−1.32462	−0.662310	+	0.749229i	$0.730424\pi$
$642$	0	0
$643$	−3.77398	−0.148831	−0.0744156	−	0.997227i	$-0.523709\pi$
$643$	−3.77398	−0.148831	−0.0744156	+	0.997227i	$0.523709\pi$
$644$	0	0
$645$	50.4785	1.98759
$646$	0	0
$647$	−36.3578	−1.42937	−0.714687	−	0.699445i	$-0.753431\pi$
$647$	−36.3578	−1.42937	−0.714687	+	0.699445i	$0.753431\pi$
$648$	0	0
$649$	40.7156	1.59823
$650$	0	0
$651$	2.53673	0.0994221
$652$	0	0
$653$	−22.3578	−0.874929	−0.437464	−	0.899236i	$-0.644123\pi$
$653$	−22.3578	−0.874929	−0.437464	+	0.899236i	$0.644123\pi$
$654$	0	0
$655$	−67.7990	−2.64913
$656$	0	0
$657$	−14.8469	−0.579234
$658$	0	0
$659$	36.7156	1.43024	0.715119	−	0.699003i	$-0.246372\pi$
$659$	36.7156	1.43024	0.715119	+	0.699003i	$0.246372\pi$
$660$	0	0
$661$	−39.0956	−1.52064	−0.760322	−	0.649546i	$-0.774959\pi$
$661$	−39.0956	−1.52064	−0.760322	+	0.649546i	$0.774959\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	34.6410	1.34332
$666$	0	0
$667$	6.35782	0.246176
$668$	0	0
$669$	−2.10282	−0.0812995
$670$	0	0
$671$	−17.3205	−0.668651
$672$	0	0
$673$	−17.3578	−0.669095	−0.334547	−	0.942379i	$-0.608583\pi$
$673$	−17.3578	−0.669095	−0.334547	+	0.942379i	$0.608583\pi$
$674$	0	0
$675$	12.1789	0.468766
$676$	0	0
$677$	−26.3578	−1.01301	−0.506507	−	0.862236i	$-0.669063\pi$
$677$	−26.3578	−1.01301	−0.506507	+	0.862236i	$0.669063\pi$
$678$	0	0
$679$	−42.5367	−1.63241
$680$	0	0
$681$	−22.1459	−0.848633
$682$	0	0
$683$	21.4044	0.819015	0.409508	−	0.912307i	$-0.365700\pi$
$683$	21.4044	0.819015	0.409508	+	0.912307i	$0.365700\pi$
$684$	0	0
$685$	−25.8945	−0.989380
$686$	0	0
$687$	16.5790	0.632527
$688$	0	0
$689$	0	0
$690$	0	0
$691$	37.6735	1.43317	0.716583	−	0.697502i	$-0.245705\pi$
$691$	37.6735	1.43317	0.716583	+	0.697502i	$0.245705\pi$
$692$	0	0
$693$	−8.35782	−0.317487
$694$	0	0
$695$	24.1269	0.915187
$696$	0	0
$697$	3.52499	0.133518
$698$	0	0
$699$	22.7156	0.859184
$700$	0	0
$701$	−25.6422	−0.968492	−0.484246	−	0.874932i	$-0.660906\pi$
$701$	−25.6422	−0.968492	−0.484246	+	0.874932i	$0.660906\pi$
$702$	0	0
$703$	−26.3578	−0.994104
$704$	0	0
$705$	43.0735	1.62224
$706$	0	0
$707$	−1.98104	−0.0745048
$708$	0	0
$709$	29.4449	1.10583	0.552913	−	0.833239i	$-0.313516\pi$
$709$	29.4449	1.10583	0.552913	+	0.833239i	$0.313516\pi$
$710$	0	0
$711$	1.82109	0.0682963
$712$	0	0
$713$	2.10282	0.0787511
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15.2177	−0.568315
$718$	0	0
$719$	7.64218	0.285005	0.142503	−	0.989794i	$-0.454485\pi$
$719$	7.64218	0.285005	0.142503	+	0.989794i	$0.454485\pi$
$720$	0	0
$721$	19.7332	0.734903
$722$	0	0
$723$	−0.680643	−0.0253134
$724$	0	0
$725$	38.7156	1.43786
$726$	0	0
$727$	16.5367	0.613313	0.306657	−	0.951820i	$-0.400790\pi$
$727$	16.5367	0.613313	0.306657	+	0.951820i	$0.400790\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−63.0735	−2.33286
$732$	0	0
$733$	41.8182	1.54459	0.772295	−	0.635264i	$-0.219109\pi$
$733$	41.8182	1.54459	0.772295	+	0.635264i	$0.219109\pi$
$734$	0	0
$735$	−4.88627	−0.180233
$736$	0	0
$737$	8.35782	0.307864
$738$	0	0
$739$	2.10282	0.0773533	0.0386767	−	0.999252i	$-0.487686\pi$
$739$	2.10282	0.0773533	0.0386767	+	0.999252i	$0.487686\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	11.7536	0.431197	0.215599	−	0.976482i	$-0.430830\pi$
$743$	11.7536	0.431197	0.215599	+	0.976482i	$0.430830\pi$
$744$	0	0
$745$	2.82109	0.103357
$746$	0	0
$747$	−1.36129	−0.0498069
$748$	0	0
$749$	−24.1269	−0.881579
$750$	0	0
$751$	42.3578	1.54566	0.772829	−	0.634614i	$-0.218841\pi$
$751$	42.3578	1.54566	0.772829	+	0.634614i	$0.218841\pi$
$752$	0	0
$753$	−24.7156	−0.900688
$754$	0	0
$755$	25.6422	0.933215
$756$	0	0
$757$	6.71563	0.244084	0.122042	−	0.992525i	$-0.461056\pi$
$757$	6.71563	0.244084	0.122042	+	0.992525i	$0.461056\pi$
$758$	0	0
$759$	−6.92820	−0.251478
$760$	0	0
$761$	−48.4974	−1.75803	−0.879015	−	0.476794i	$-0.841799\pi$
$761$	−48.4974	−1.75803	−0.879015	+	0.476794i	$0.841799\pi$
$762$	0	0
$763$	2.53673	0.0918356
$764$	0	0
$765$	−21.4653	−0.776078
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−6.92820	−0.249837	−0.124919	−	0.992167i	$-0.539867\pi$
$769$	−6.92820	−0.249837	−0.124919	+	0.992167i	$0.539867\pi$
$770$	0	0
$771$	−10.8211	−0.389712
$772$	0	0
$773$	−7.54796	−0.271481	−0.135741	−	0.990744i	$-0.543341\pi$
$773$	−7.54796	−0.271481	−0.135741	+	0.990744i	$0.543341\pi$
$774$	0	0
$775$	12.8050	0.459969
$776$	0	0
$777$	−18.3578	−0.658583
$778$	0	0
$779$	−2.35782	−0.0844775
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	3.17891	0.113605
$784$	0	0
$785$	29.0132	1.03553
$786$	0	0
$787$	−40.5178	−1.44430	−0.722152	−	0.691734i	$-0.756847\pi$
$787$	−40.5178	−1.44430	−0.722152	+	0.691734i	$0.756847\pi$
$788$	0	0
$789$	9.64218	0.343271
$790$	0	0
$791$	1.98104	0.0704378
$792$	0	0
$793$	0	0
$794$	0	0
$795$	4.88627	0.173298
$796$	0	0
$797$	51.0735	1.80911	0.904557	−	0.426352i	$-0.140202\pi$
$797$	51.0735	1.80911	0.904557	+	0.426352i	$0.140202\pi$
$798$	0	0
$799$	−53.8208	−1.90404
$800$	0	0
$801$	−6.92820	−0.244796
$802$	0	0
$803$	51.4313	1.81497
$804$	0	0
$805$	20.0000	0.704907
$806$	0	0
$807$	10.3578	0.364612
$808$	0	0
$809$	12.4633	0.438185	0.219093	−	0.975704i	$-0.429690\pi$
$809$	12.4633	0.438185	0.219093	+	0.975704i	$0.429690\pi$
$810$	0	0
$811$	−5.87680	−0.206362	−0.103181	−	0.994663i	$-0.532902\pi$
$811$	−5.87680	−0.206362	−0.103181	+	0.994663i	$0.532902\pi$
$812$	0	0
$813$	7.97961	0.279857
$814$	0	0
$815$	−4.35782	−0.152648
$816$	0	0
$817$	42.1890	1.47601
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.54796	−0.263426	−0.131713	−	0.991288i	$-0.542048\pi$
$821$	−7.54796	−0.263426	−0.131713	+	0.991288i	$0.542048\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	0	0
$825$	−42.1890	−1.46883
$826$	0	0
$827$	42.8087	1.48861	0.744303	−	0.667842i	$-0.232782\pi$
$827$	42.8087	1.48861	0.744303	+	0.667842i	$0.232782\pi$
$828$	0	0
$829$	−0.642183	−0.0223039	−0.0111520	−	0.999938i	$-0.503550\pi$
$829$	−0.642183	−0.0223039	−0.0111520	+	0.999938i	$0.503550\pi$
$830$	0	0
$831$	−11.1789	−0.387792
$832$	0	0
$833$	6.10546	0.211542
$834$	0	0
$835$	−28.7156	−0.993745
$836$	0	0
$837$	1.05141	0.0363420
$838$	0	0
$839$	−2.10282	−0.0725973	−0.0362986	−	0.999341i	$-0.511557\pi$
$839$	−2.10282	−0.0725973	−0.0362986	+	0.999341i	$0.511557\pi$
$840$	0	0
$841$	−18.8945	−0.651536
$842$	0	0
$843$	−18.0012	−0.619993
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.41269	0.0829011
$848$	0	0
$849$	15.8211	0.542978
$850$	0	0
$851$	−15.2177	−0.521656
$852$	0	0
$853$	37.7344	1.29200	0.646000	−	0.763338i	$-0.276441\pi$
$853$	37.7344	1.29200	0.646000	+	0.763338i	$0.276441\pi$
$854$	0	0
$855$	14.3578	0.491027
$856$	0	0
$857$	8.82109	0.301323	0.150661	−	0.988585i	$-0.451860\pi$
$857$	8.82109	0.301323	0.150661	+	0.988585i	$0.451860\pi$
$858$	0	0
$859$	−25.2524	−0.861599	−0.430800	−	0.902448i	$-0.641768\pi$
$859$	−25.2524	−0.861599	−0.430800	+	0.902448i	$0.641768\pi$
$860$	0	0
$861$	−1.64218	−0.0559655
$862$	0	0
$863$	−6.18667	−0.210597	−0.105298	−	0.994441i	$-0.533580\pi$
$863$	−6.18667	−0.210597	−0.105298	+	0.994441i	$0.533580\pi$
$864$	0	0
$865$	−74.6054	−2.53666
$866$	0	0
$867$	9.82109	0.333542
$868$	0	0
$869$	−6.30845	−0.213999
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−17.6304	−0.596698
$874$	0	0
$875$	71.7891	2.42691
$876$	0	0
$877$	−6.24756	−0.210965	−0.105483	−	0.994421i	$-0.533639\pi$
$877$	−6.24756	−0.210965	−0.105483	+	0.994421i	$0.533639\pi$
$878$	0	0
$879$	0.0608864	0.00205365
$880$	0	0
$881$	49.1789	1.65688	0.828440	−	0.560078i	$-0.189229\pi$
$881$	49.1789	1.65688	0.828440	+	0.560078i	$0.189229\pi$
$882$	0	0
$883$	−38.8945	−1.30891	−0.654453	−	0.756103i	$-0.727101\pi$
$883$	−38.8945	−1.30891	−0.654453	+	0.756103i	$0.727101\pi$
$884$	0	0
$885$	−48.7156	−1.63756
$886$	0	0
$887$	44.3578	1.48939	0.744695	−	0.667405i	$-0.232595\pi$
$887$	44.3578	1.48939	0.744695	+	0.667405i	$0.232595\pi$
$888$	0	0
$889$	34.2094	1.14735
$890$	0	0
$891$	−3.46410	−0.116052
$892$	0	0
$893$	36.0000	1.20469
$894$	0	0
$895$	−59.5095	−1.98918
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.34233	0.111473
$900$	0	0
$901$	−6.10546	−0.203402
$902$	0	0
$903$	29.3840	0.977838
$904$	0	0
$905$	−56.1063	−1.86504
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	−0.821092	−0.0272339
$910$	0	0
$911$	−40.7156	−1.34897	−0.674485	−	0.738289i	$-0.735634\pi$
$911$	−40.7156	−1.34897	−0.674485	+	0.738289i	$0.735634\pi$
$912$	0	0
$913$	4.71563	0.156065
$914$	0	0
$915$	20.7237	0.685105
$916$	0	0
$917$	−39.4664	−1.30330
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	−4.51551	−0.148791
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−92.6674	−3.04689
$926$	0	0
$927$	8.17891	0.268631
$928$	0	0
$929$	11.8145	0.387620	0.193810	−	0.981039i	$-0.437915\pi$
$929$	11.8145	0.387620	0.193810	+	0.981039i	$0.437915\pi$
$930$	0	0
$931$	−4.08386	−0.133843
$932$	0	0
$933$	−34.3578	−1.12482
$934$	0	0
$935$	74.3578	2.43176
$936$	0	0
$937$	−7.89454	−0.257903	−0.128952	−	0.991651i	$-0.541161\pi$
$937$	−7.89454	−0.257903	−0.128952	+	0.991651i	$0.541161\pi$
$938$	0	0
$939$	−28.8945	−0.942938
$940$	0	0
$941$	−54.5623	−1.77868	−0.889340	−	0.457246i	$-0.848836\pi$
$941$	−54.5623	−1.77868	−0.889340	+	0.457246i	$0.848836\pi$
$942$	0	0
$943$	−1.36129	−0.0443296
$944$	0	0
$945$	10.0000	0.325300
$946$	0	0
$947$	4.20563	0.136665	0.0683323	−	0.997663i	$-0.478232\pi$
$947$	4.20563	0.136665	0.0683323	+	0.997663i	$0.478232\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−15.8983	−0.515539
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	84.3780	2.73041
$956$	0	0
$957$	−11.0121	−0.355969
$958$	0	0
$959$	−15.0735	−0.486747
$960$	0	0
$961$	−29.8945	−0.964340
$962$	0	0
$963$	−10.0000	−0.322245
$964$	0	0
$965$	1.53673	0.0494689
$966$	0	0
$967$	61.6123	1.98132	0.990659	−	0.136363i	$-0.0435412\pi$
$967$	61.6123	1.98132	0.990659	+	0.136363i	$0.0435412\pi$
$968$	0	0
$969$	−17.9403	−0.576324
$970$	0	0
$971$	−40.7156	−1.30663	−0.653313	−	0.757088i	$-0.726622\pi$
$971$	−40.7156	−1.30663	−0.653313	+	0.757088i	$0.726622\pi$
$972$	0	0
$973$	14.0445	0.450246
$974$	0	0
$975$	0	0
$976$	0	0
$977$	36.6829	1.17359	0.586796	−	0.809735i	$-0.300389\pi$
$977$	36.6829	1.17359	0.586796	+	0.809735i	$0.300389\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	1.05141	0.0335689
$982$	0	0
$983$	14.4762	0.461718	0.230859	−	0.972987i	$-0.425846\pi$
$983$	14.4762	0.461718	0.230859	+	0.972987i	$0.425846\pi$
$984$	0	0
$985$	68.7156	2.18946
$986$	0	0
$987$	25.0735	0.798097
$988$	0	0
$989$	24.3578	0.774534
$990$	0	0
$991$	23.0735	0.732952	0.366476	−	0.930427i	$-0.380564\pi$
$991$	23.0735	0.732952	0.366476	+	0.930427i	$0.380564\pi$
$992$	0	0
$993$	26.6614	0.846074
$994$	0	0
$995$	−51.9615	−1.64729
$996$	0	0
$997$	12.6422	0.400382	0.200191	−	0.979757i	$-0.435844\pi$
$997$	12.6422	0.400382	0.200191	+	0.979757i	$0.435844\pi$
$998$	0	0
$999$	−7.60885	−0.240733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.2.a.m.1.4		4
3.2	odd	2		6084.2.a.bd.1.1		4
4.3	odd	2		8112.2.a.cr.1.4		4
13.2	odd	12		156.2.q.b.121.1	yes	4
13.3	even	3		2028.2.i.n.529.4		8
13.4	even	6		2028.2.i.n.2005.1		8
13.5	odd	4		2028.2.b.e.337.1		4
13.6	odd	12		2028.2.q.f.361.1		4
13.7	odd	12		156.2.q.b.49.2	✓	4
13.8	odd	4		2028.2.b.e.337.4		4
13.9	even	3		2028.2.i.n.2005.4		8
13.10	even	6		2028.2.i.n.529.1		8
13.11	odd	12		2028.2.q.f.1837.2		4
13.12	even	2	inner	2028.2.a.m.1.1		4
39.2	even	12		468.2.t.d.433.2		4
39.5	even	4		6084.2.b.o.4393.4		4
39.8	even	4		6084.2.b.o.4393.1		4
39.20	even	12		468.2.t.d.361.1		4
39.38	odd	2		6084.2.a.bd.1.4		4
52.7	even	12		624.2.bv.f.49.2		4
52.15	even	12		624.2.bv.f.433.1		4
52.51	odd	2		8112.2.a.cr.1.1		4
65.2	even	12		3900.2.bw.j.2149.4		8
65.7	even	12		3900.2.bw.j.49.1		8
65.28	even	12		3900.2.bw.j.2149.1		8
65.33	even	12		3900.2.bw.j.49.4		8
65.54	odd	12		3900.2.cd.i.901.1		4
65.59	odd	12		3900.2.cd.i.2701.1		4
156.59	odd	12		1872.2.by.j.1297.1		4
156.119	odd	12		1872.2.by.j.433.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.q.b.49.2	✓	4	13.7	odd	12
156.2.q.b.121.1	yes	4	13.2	odd	12
468.2.t.d.361.1		4	39.20	even	12
468.2.t.d.433.2		4	39.2	even	12
624.2.bv.f.49.2		4	52.7	even	12
624.2.bv.f.433.1		4	52.15	even	12
1872.2.by.j.433.2		4	156.119	odd	12
1872.2.by.j.1297.1		4	156.59	odd	12
2028.2.a.m.1.1		4	13.12	even	2	inner
2028.2.a.m.1.4		4	1.1	even	1	trivial
2028.2.b.e.337.1		4	13.5	odd	4
2028.2.b.e.337.4		4	13.8	odd	4
2028.2.i.n.529.1		8	13.10	even	6
2028.2.i.n.529.4		8	13.3	even	3
2028.2.i.n.2005.1		8	13.4	even	6
2028.2.i.n.2005.4		8	13.9	even	3
2028.2.q.f.361.1		4	13.6	odd	12
2028.2.q.f.1837.2		4	13.11	odd	12
3900.2.bw.j.49.1		8	65.7	even	12
3900.2.bw.j.49.4		8	65.33	even	12
3900.2.bw.j.2149.1		8	65.28	even	12
3900.2.bw.j.2149.4		8	65.2	even	12
3900.2.cd.i.901.1		4	65.54	odd	12
3900.2.cd.i.2701.1		4	65.59	odd	12
6084.2.a.bd.1.1		4	3.2	odd	2
6084.2.a.bd.1.4		4	39.38	odd	2
6084.2.b.o.4393.1		4	39.8	even	4
6084.2.b.o.4393.4		4	39.5	even	4
8112.2.a.cr.1.1		4	52.51	odd	2
8112.2.a.cr.1.4		4	4.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2028.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.14474 q^{5} +2.41269 q^{7} +1.00000 q^{9} -3.46410 q^{11} +4.14474 q^{15} -5.17891 q^{17} +3.46410 q^{19} +2.41269 q^{21} +2.00000 q^{23} +12.1789 q^{25} +1.00000 q^{27} +3.17891 q^{29} +1.05141 q^{31} -3.46410 q^{33} +10.0000 q^{35} -7.60885 q^{37} -0.680643 q^{41} +12.1789 q^{43} +4.14474 q^{45} +10.3923 q^{47} -1.17891 q^{49} -5.17891 q^{51} +1.17891 q^{53} -14.3578 q^{55} +3.46410 q^{57} -11.7536 q^{59} +5.00000 q^{61} +2.41269 q^{63} -2.41269 q^{67} +2.00000 q^{69} +3.46410 q^{71} -14.8469 q^{73} +12.1789 q^{75} -8.35782 q^{77} +1.82109 q^{79} +1.00000 q^{81} -1.36129 q^{83} -21.4653 q^{85} +3.17891 q^{87} -6.92820 q^{89} +1.05141 q^{93} +14.3578 q^{95} -17.6304 q^{97} -3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9} + 2 q^{17} + 8 q^{23} + 26 q^{25} + 4 q^{27} - 10 q^{29} + 40 q^{35} + 26 q^{43} + 18 q^{49} + 2 q^{51} - 18 q^{53} - 12 q^{55} + 20 q^{61} + 8 q^{69} + 26 q^{75} + 12 q^{77} + 30 q^{79}+ \cdots + 12 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 + 2 * q^17 + 8 * q^23 + 26 * q^25 + 4 * q^27 - 10 * q^29 + 40 * q^35 + 26 * q^43 + 18 * q^49 + 2 * q^51 - 18 * q^53 - 12 * q^55 + 20 * q^61 + 8 * q^69 + 26 * q^75 + 12 * q^77 + 30 * q^79 + 4 * q^81 - 10 * q^87 + 12 * q^95

Introduction

L-functions

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