LMFDB - Embedded newform 2028.4.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2028,4,Mod(1,2028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2028, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2028.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2028.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:	$119.655873492$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 312x^{4} + 1528x^{3} + 8737x^{2} - 35142x + 6054 $ x^6 - 2x^5 - 312x^4 + 1528x^3 + 8737x^2 - 35142*x + 6054
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.17021$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -11.8987 q^{5} -22.2285 q^{7} +9.00000 q^{9} +56.6439 q^{11} +35.6961 q^{15} +41.1241 q^{17} +114.832 q^{19} +66.6855 q^{21} -67.7051 q^{23} +16.5792 q^{25} -27.0000 q^{27} -134.204 q^{29} -265.864 q^{31} -169.932 q^{33} +264.490 q^{35} +69.3690 q^{37} -114.408 q^{41} +307.434 q^{43} -107.088 q^{45} -518.083 q^{47} +151.106 q^{49} -123.372 q^{51} +304.436 q^{53} -673.989 q^{55} -344.497 q^{57} -702.573 q^{59} -649.365 q^{61} -200.056 q^{63} -719.352 q^{67} +203.115 q^{69} -903.626 q^{71} +562.264 q^{73} -49.7376 q^{75} -1259.11 q^{77} +530.771 q^{79} +81.0000 q^{81} +986.923 q^{83} -489.324 q^{85} +402.613 q^{87} +313.761 q^{89} +797.591 q^{93} -1366.35 q^{95} +139.088 q^{97} +509.795 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 18 q^{3} + 54 q^{9} + 64 q^{17} + 208 q^{23} + 166 q^{25} - 162 q^{27} - 440 q^{29} + 176 q^{35} + 466 q^{43} + 76 q^{49} - 192 q^{51} + 732 q^{53} - 1404 q^{55} - 1190 q^{61} - 624 q^{69} - 498 q^{75}+ \cdots - 180 q^{95}+O(q^{100}) $ 6 * q - 18 * q^3 + 54 * q^9 + 64 * q^17 + 208 * q^23 + 166 * q^25 - 162 * q^27 - 440 * q^29 + 176 * q^35 + 466 * q^43 + 76 * q^49 - 192 * q^51 + 732 * q^53 - 1404 * q^55 - 1190 * q^61 - 624 * q^69 - 498 * q^75 - 2076 * q^77 + 2222 * q^79 + 486 * q^81 + 1320 * q^87 - 180 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−11.8987	−1.06425	−0.532126	−	0.846665i	$-0.678607\pi$
$5$	−11.8987	−1.06425	−0.532126	+	0.846665i	$0.678607\pi$
$6$	0	0
$7$	−22.2285	−1.20023	−0.600113	−	0.799915i	$-0.704878\pi$
$7$	−22.2285	−1.20023	−0.600113	+	0.799915i	$0.704878\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	56.6439	1.55262	0.776308	−	0.630354i	$-0.217090\pi$
$11$	56.6439	1.55262	0.776308	+	0.630354i	$0.217090\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	35.6961	0.614447
$16$	0	0
$17$	41.1241	0.586710	0.293355	−	0.956004i	$-0.405228\pi$
$17$	41.1241	0.586710	0.293355	+	0.956004i	$0.405228\pi$
$18$	0	0
$19$	114.832	1.38654	0.693271	−	0.720677i	$-0.256169\pi$
$19$	114.832	1.38654	0.693271	+	0.720677i	$0.256169\pi$
$20$	0	0
$21$	66.6855	0.692951
$22$	0	0
$23$	−67.7051	−0.613804	−0.306902	−	0.951741i	$-0.599292\pi$
$23$	−67.7051	−0.613804	−0.306902	+	0.951741i	$0.599292\pi$
$24$	0	0
$25$	16.5792	0.132634
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−134.204	−0.859348	−0.429674	−	0.902984i	$-0.641372\pi$
$29$	−134.204	−0.859348	−0.429674	+	0.902984i	$0.641372\pi$
$30$	0	0
$31$	−265.864	−1.54034	−0.770170	−	0.637839i	$-0.779829\pi$
$31$	−265.864	−1.54034	−0.770170	+	0.637839i	$0.779829\pi$
$32$	0	0
$33$	−169.932	−0.896403
$34$	0	0
$35$	264.490	1.27734
$36$	0	0
$37$	69.3690	0.308221	0.154111	−	0.988054i	$-0.450749\pi$
$37$	69.3690	0.308221	0.154111	+	0.988054i	$0.450749\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−114.408	−0.435795	−0.217897	−	0.975972i	$-0.569920\pi$
$41$	−114.408	−0.435795	−0.217897	+	0.975972i	$0.569920\pi$
$42$	0	0
$43$	307.434	1.09031	0.545154	−	0.838336i	$-0.316471\pi$
$43$	307.434	1.09031	0.545154	+	0.838336i	$0.316471\pi$
$44$	0	0
$45$	−107.088	−0.354751
$46$	0	0
$47$	−518.083	−1.60788	−0.803938	−	0.594713i	$-0.797266\pi$
$47$	−518.083	−1.60788	−0.803938	+	0.594713i	$0.797266\pi$
$48$	0	0
$49$	151.106	0.440542
$50$	0	0
$51$	−123.372	−0.338737
$52$	0	0
$53$	304.436	0.789010	0.394505	−	0.918894i	$-0.370916\pi$
$53$	304.436	0.789010	0.394505	+	0.918894i	$0.370916\pi$
$54$	0	0
$55$	−673.989	−1.65237
$56$	0	0
$57$	−344.497	−0.800521
$58$	0	0
$59$	−702.573	−1.55029	−0.775145	−	0.631783i	$-0.782323\pi$
$59$	−702.573	−1.55029	−0.775145	+	0.631783i	$0.782323\pi$
$60$	0	0
$61$	−649.365	−1.36300	−0.681498	−	0.731820i	$-0.738671\pi$
$61$	−649.365	−1.36300	−0.681498	+	0.731820i	$0.738671\pi$
$62$	0	0
$63$	−200.056	−0.400075
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−719.352	−1.31168	−0.655842	−	0.754898i	$-0.727686\pi$
$67$	−719.352	−1.31168	−0.655842	+	0.754898i	$0.727686\pi$
$68$	0	0
$69$	203.115	0.354380
$70$	0	0
$71$	−903.626	−1.51043	−0.755216	−	0.655476i	$-0.772468\pi$
$71$	−903.626	−1.51043	−0.755216	+	0.655476i	$0.772468\pi$
$72$	0	0
$73$	562.264	0.901479	0.450740	−	0.892655i	$-0.351160\pi$
$73$	562.264	0.901479	0.450740	+	0.892655i	$0.351160\pi$
$74$	0	0
$75$	−49.7376	−0.0765761
$76$	0	0
$77$	−1259.11	−1.86349
$78$	0	0
$79$	530.771	0.755904	0.377952	−	0.925825i	$-0.376628\pi$
$79$	530.771	0.755904	0.377952	+	0.925825i	$0.376628\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	986.923	1.30517	0.652583	−	0.757717i	$-0.273685\pi$
$83$	986.923	1.30517	0.652583	+	0.757717i	$0.273685\pi$
$84$	0	0
$85$	−489.324	−0.624408
$86$	0	0
$87$	402.613	0.496145
$88$	0	0
$89$	313.761	0.373692	0.186846	−	0.982389i	$-0.440174\pi$
$89$	313.761	0.373692	0.186846	+	0.982389i	$0.440174\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	797.591	0.889316
$94$	0	0
$95$	−1366.35	−1.47563
$96$	0	0
$97$	139.088	0.145591	0.0727953	−	0.997347i	$-0.476808\pi$
$97$	139.088	0.145591	0.0727953	+	0.997347i	$0.476808\pi$
$98$	0	0
$99$	509.795	0.517538
$100$	0	0
$101$	1051.31	1.03574	0.517868	−	0.855460i	$-0.326726\pi$
$101$	1051.31	1.03574	0.517868	+	0.855460i	$0.326726\pi$
$102$	0	0
$103$	1861.88	1.78113	0.890565	−	0.454855i	$-0.150309\pi$
$103$	1861.88	1.78113	0.890565	+	0.454855i	$0.150309\pi$
$104$	0	0
$105$	−793.471	−0.737474
$106$	0	0
$107$	1355.91	1.22505	0.612527	−	0.790449i	$-0.290153\pi$
$107$	1355.91	1.22505	0.612527	+	0.790449i	$0.290153\pi$
$108$	0	0
$109$	343.720	0.302041	0.151020	−	0.988531i	$-0.451744\pi$
$109$	343.720	0.302041	0.151020	+	0.988531i	$0.451744\pi$
$110$	0	0
$111$	−208.107	−0.177952
$112$	0	0
$113$	98.2882	0.0818246	0.0409123	−	0.999163i	$-0.486974\pi$
$113$	98.2882	0.0818246	0.0409123	+	0.999163i	$0.486974\pi$
$114$	0	0
$115$	805.603	0.653242
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−914.128	−0.704184
$120$	0	0
$121$	1877.53	1.41061
$122$	0	0
$123$	343.225	0.251606
$124$	0	0
$125$	1290.07	0.923097
$126$	0	0
$127$	−1243.22	−0.868646	−0.434323	−	0.900757i	$-0.643012\pi$
$127$	−1243.22	−0.868646	−0.434323	+	0.900757i	$0.643012\pi$
$128$	0	0
$129$	−922.302	−0.629490
$130$	0	0
$131$	316.587	0.211148	0.105574	−	0.994411i	$-0.466332\pi$
$131$	316.587	0.211148	0.105574	+	0.994411i	$0.466332\pi$
$132$	0	0
$133$	−2552.55	−1.66416
$134$	0	0
$135$	321.265	0.204816
$136$	0	0
$137$	−2469.80	−1.54021	−0.770107	−	0.637914i	$-0.779797\pi$
$137$	−2469.80	−1.54021	−0.770107	+	0.637914i	$0.779797\pi$
$138$	0	0
$139$	1387.07	0.846402	0.423201	−	0.906036i	$-0.360906\pi$
$139$	1387.07	0.846402	0.423201	+	0.906036i	$0.360906\pi$
$140$	0	0
$141$	1554.25	0.928307
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1596.86	0.914564
$146$	0	0
$147$	−453.317	−0.254347
$148$	0	0
$149$	554.229	0.304726	0.152363	−	0.988325i	$-0.451312\pi$
$149$	554.229	0.304726	0.152363	+	0.988325i	$0.451312\pi$
$150$	0	0
$151$	578.030	0.311519	0.155760	−	0.987795i	$-0.450217\pi$
$151$	578.030	0.311519	0.155760	+	0.987795i	$0.450217\pi$
$152$	0	0
$153$	370.117	0.195570
$154$	0	0
$155$	3163.43	1.63931
$156$	0	0
$157$	−2525.44	−1.28377	−0.641886	−	0.766800i	$-0.721848\pi$
$157$	−2525.44	−1.28377	−0.641886	+	0.766800i	$0.721848\pi$
$158$	0	0
$159$	−913.309	−0.455535
$160$	0	0
$161$	1504.98	0.736703
$162$	0	0
$163$	2132.18	1.02457	0.512286	−	0.858815i	$-0.328799\pi$
$163$	2132.18	1.02457	0.512286	+	0.858815i	$0.328799\pi$
$164$	0	0
$165$	2021.97	0.953999
$166$	0	0
$167$	−2253.36	−1.04413	−0.522066	−	0.852905i	$-0.674839\pi$
$167$	−2253.36	−1.04413	−0.522066	+	0.852905i	$0.674839\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	1033.49	0.462181
$172$	0	0
$173$	507.586	0.223069	0.111535	−	0.993761i	$-0.464423\pi$
$173$	507.586	0.223069	0.111535	+	0.993761i	$0.464423\pi$
$174$	0	0
$175$	−368.531	−0.159190
$176$	0	0
$177$	2107.72	0.895061
$178$	0	0
$179$	674.607	0.281690	0.140845	−	0.990032i	$-0.455018\pi$
$179$	674.607	0.281690	0.140845	+	0.990032i	$0.455018\pi$
$180$	0	0
$181$	3291.89	1.35185	0.675923	−	0.736973i	$-0.263745\pi$
$181$	3291.89	1.35185	0.675923	+	0.736973i	$0.263745\pi$
$182$	0	0
$183$	1948.10	0.786926
$184$	0	0
$185$	−825.401	−0.328025
$186$	0	0
$187$	2329.43	0.910935
$188$	0	0
$189$	600.169	0.230984
$190$	0	0
$191$	−885.021	−0.335277	−0.167638	−	0.985849i	$-0.553614\pi$
$191$	−885.021	−0.335277	−0.167638	+	0.985849i	$0.553614\pi$
$192$	0	0
$193$	−3802.41	−1.41815	−0.709076	−	0.705132i	$-0.750888\pi$
$193$	−3802.41	−1.41815	−0.709076	+	0.705132i	$0.750888\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3022.98	1.09329	0.546646	−	0.837363i	$-0.315904\pi$
$197$	3022.98	1.09329	0.546646	+	0.837363i	$0.315904\pi$
$198$	0	0
$199$	−2228.02	−0.793669	−0.396834	−	0.917890i	$-0.629891\pi$
$199$	−2228.02	−0.793669	−0.396834	+	0.917890i	$0.629891\pi$
$200$	0	0
$201$	2158.06	0.757301
$202$	0	0
$203$	2983.16	1.03141
$204$	0	0
$205$	1361.31	0.463796
$206$	0	0
$207$	−609.346	−0.204601
$208$	0	0
$209$	6504.54	2.15277
$210$	0	0
$211$	−1833.53	−0.598226	−0.299113	−	0.954218i	$-0.596691\pi$
$211$	−1833.53	−0.598226	−0.299113	+	0.954218i	$0.596691\pi$
$212$	0	0
$213$	2710.88	0.872048
$214$	0	0
$215$	−3658.07	−1.16036
$216$	0	0
$217$	5909.75	1.84876
$218$	0	0
$219$	−1686.79	−0.520469
$220$	0	0
$221$	0	0
$222$	0	0
$223$	171.520	0.0515061	0.0257530	−	0.999668i	$-0.491802\pi$
$223$	171.520	0.0515061	0.0257530	+	0.999668i	$0.491802\pi$
$224$	0	0
$225$	149.213	0.0442112
$226$	0	0
$227$	4348.78	1.27154	0.635768	−	0.771881i	$-0.280684\pi$
$227$	4348.78	1.27154	0.635768	+	0.771881i	$0.280684\pi$
$228$	0	0
$229$	−4321.88	−1.24715	−0.623577	−	0.781762i	$-0.714321\pi$
$229$	−4321.88	−1.24715	−0.623577	+	0.781762i	$0.714321\pi$
$230$	0	0
$231$	3777.32	1.07589
$232$	0	0
$233$	−1159.14	−0.325913	−0.162957	−	0.986633i	$-0.552103\pi$
$233$	−1159.14	−0.325913	−0.162957	+	0.986633i	$0.552103\pi$
$234$	0	0
$235$	6164.52	1.71119
$236$	0	0
$237$	−1592.31	−0.436421
$238$	0	0
$239$	1503.63	0.406954	0.203477	−	0.979080i	$-0.434776\pi$
$239$	1503.63	0.406954	0.203477	+	0.979080i	$0.434776\pi$
$240$	0	0
$241$	−6200.53	−1.65731	−0.828655	−	0.559760i	$-0.810893\pi$
$241$	−6200.53	−1.65731	−0.828655	+	0.559760i	$0.810893\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−1797.96	−0.468847
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−2960.77	−0.753538
$250$	0	0
$251$	3034.10	0.762991	0.381495	−	0.924371i	$-0.375409\pi$
$251$	3034.10	0.762991	0.381495	+	0.924371i	$0.375409\pi$
$252$	0	0
$253$	−3835.08	−0.953001
$254$	0	0
$255$	1467.97	0.360502
$256$	0	0
$257$	−927.678	−0.225163	−0.112582	−	0.993642i	$-0.535912\pi$
$257$	−927.678	−0.225163	−0.112582	+	0.993642i	$0.535912\pi$
$258$	0	0
$259$	−1541.97	−0.369935
$260$	0	0
$261$	−1207.84	−0.286449
$262$	0	0
$263$	6872.55	1.61133	0.805665	−	0.592371i	$-0.201808\pi$
$263$	6872.55	1.61133	0.805665	+	0.592371i	$0.201808\pi$
$264$	0	0
$265$	−3622.40	−0.839706
$266$	0	0
$267$	−941.282	−0.215751
$268$	0	0
$269$	5601.41	1.26961	0.634803	−	0.772674i	$-0.281081\pi$
$269$	5601.41	1.26961	0.634803	+	0.772674i	$0.281081\pi$
$270$	0	0
$271$	−8154.97	−1.82797	−0.913984	−	0.405750i	$-0.867010\pi$
$271$	−8154.97	−1.82797	−0.913984	+	0.405750i	$0.867010\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	939.110	0.205929
$276$	0	0
$277$	1862.11	0.403911	0.201956	−	0.979395i	$-0.435270\pi$
$277$	1862.11	0.403911	0.201956	+	0.979395i	$0.435270\pi$
$278$	0	0
$279$	−2392.77	−0.513447
$280$	0	0
$281$	−1050.87	−0.223095	−0.111548	−	0.993759i	$-0.535581\pi$
$281$	−1050.87	−0.223095	−0.111548	+	0.993759i	$0.535581\pi$
$282$	0	0
$283$	−1378.57	−0.289566	−0.144783	−	0.989463i	$-0.546248\pi$
$283$	−1378.57	−0.289566	−0.144783	+	0.989463i	$0.546248\pi$
$284$	0	0
$285$	4099.06	0.851956
$286$	0	0
$287$	2543.12	0.523052
$288$	0	0
$289$	−3221.80	−0.655771
$290$	0	0
$291$	−417.265	−0.0840567
$292$	0	0
$293$	−5146.84	−1.02622	−0.513109	−	0.858324i	$-0.671506\pi$
$293$	−5146.84	−1.02622	−0.513109	+	0.858324i	$0.671506\pi$
$294$	0	0
$295$	8359.70	1.64990
$296$	0	0
$297$	−1529.38	−0.298801
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−6833.80	−1.30862
$302$	0	0
$303$	−3153.93	−0.597983
$304$	0	0
$305$	7726.61	1.45057
$306$	0	0
$307$	2186.04	0.406396	0.203198	−	0.979138i	$-0.434866\pi$
$307$	2186.04	0.406396	0.203198	+	0.979138i	$0.434866\pi$
$308$	0	0
$309$	−5585.64	−1.02834
$310$	0	0
$311$	6096.03	1.11149	0.555746	−	0.831352i	$-0.312433\pi$
$311$	6096.03	1.11149	0.555746	+	0.831352i	$0.312433\pi$
$312$	0	0
$313$	−9073.90	−1.63862	−0.819309	−	0.573352i	$-0.805643\pi$
$313$	−9073.90	−1.63862	−0.819309	+	0.573352i	$0.805643\pi$
$314$	0	0
$315$	2380.41	0.425781
$316$	0	0
$317$	6755.86	1.19699	0.598497	−	0.801125i	$-0.295765\pi$
$317$	6755.86	1.19699	0.598497	+	0.801125i	$0.295765\pi$
$318$	0	0
$319$	−7601.84	−1.33424
$320$	0	0
$321$	−4067.73	−0.707286
$322$	0	0
$323$	4722.38	0.813499
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1031.16	−0.174383
$328$	0	0
$329$	11516.2	1.92981
$330$	0	0
$331$	3967.10	0.658766	0.329383	−	0.944196i	$-0.393159\pi$
$331$	3967.10	0.658766	0.329383	+	0.944196i	$0.393159\pi$
$332$	0	0
$333$	624.321	0.102740
$334$	0	0
$335$	8559.35	1.39596
$336$	0	0
$337$	8962.13	1.44866	0.724330	−	0.689454i	$-0.242149\pi$
$337$	8962.13	1.44866	0.724330	+	0.689454i	$0.242149\pi$
$338$	0	0
$339$	−294.865	−0.0472414
$340$	0	0
$341$	−15059.5	−2.39155
$342$	0	0
$343$	4265.52	0.671476
$344$	0	0
$345$	−2416.81	−0.377150
$346$	0	0
$347$	5102.66	0.789410	0.394705	−	0.918808i	$-0.370847\pi$
$347$	5102.66	0.789410	0.394705	+	0.918808i	$0.370847\pi$
$348$	0	0
$349$	−5246.39	−0.804679	−0.402340	−	0.915490i	$-0.631803\pi$
$349$	−5246.39	−0.804679	−0.402340	+	0.915490i	$0.631803\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9406.59	1.41831	0.709153	−	0.705054i	$-0.249077\pi$
$353$	9406.59	1.41831	0.709153	+	0.705054i	$0.249077\pi$
$354$	0	0
$355$	10752.0	1.60748
$356$	0	0
$357$	2742.38	0.406561
$358$	0	0
$359$	−3949.68	−0.580657	−0.290329	−	0.956927i	$-0.593765\pi$
$359$	−3949.68	−0.580657	−0.290329	+	0.956927i	$0.593765\pi$
$360$	0	0
$361$	6327.43	0.922501
$362$	0	0
$363$	−5632.58	−0.814418
$364$	0	0
$365$	−6690.21	−0.959402
$366$	0	0
$367$	11264.2	1.60214	0.801072	−	0.598569i	$-0.204264\pi$
$367$	11264.2	1.60214	0.801072	+	0.598569i	$0.204264\pi$
$368$	0	0
$369$	−1029.68	−0.145265
$370$	0	0
$371$	−6767.16	−0.946991
$372$	0	0
$373$	−582.141	−0.0808100	−0.0404050	−	0.999183i	$-0.512865\pi$
$373$	−582.141	−0.0808100	−0.0404050	+	0.999183i	$0.512865\pi$
$374$	0	0
$375$	−3870.20	−0.532950
$376$	0	0
$377$	0	0
$378$	0	0
$379$	2233.34	0.302688	0.151344	−	0.988481i	$-0.451640\pi$
$379$	2233.34	0.302688	0.151344	+	0.988481i	$0.451640\pi$
$380$	0	0
$381$	3729.66	0.501513
$382$	0	0
$383$	−4925.73	−0.657162	−0.328581	−	0.944476i	$-0.606570\pi$
$383$	−4925.73	−0.657162	−0.328581	+	0.944476i	$0.606570\pi$
$384$	0	0
$385$	14981.7	1.98322
$386$	0	0
$387$	2766.91	0.363436
$388$	0	0
$389$	3460.45	0.451032	0.225516	−	0.974239i	$-0.427593\pi$
$389$	3460.45	0.451032	0.225516	+	0.974239i	$0.427593\pi$
$390$	0	0
$391$	−2784.31	−0.360125
$392$	0	0
$393$	−949.761	−0.121906
$394$	0	0
$395$	−6315.49	−0.804473
$396$	0	0
$397$	−1056.16	−0.133520	−0.0667598	−	0.997769i	$-0.521266\pi$
$397$	−1056.16	−0.133520	−0.0667598	+	0.997769i	$0.521266\pi$
$398$	0	0
$399$	7657.64	0.960806
$400$	0	0
$401$	2515.15	0.313218	0.156609	−	0.987661i	$-0.449944\pi$
$401$	2515.15	0.313218	0.156609	+	0.987661i	$0.449944\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−963.795	−0.118250
$406$	0	0
$407$	3929.33	0.478549
$408$	0	0
$409$	1138.26	0.137612	0.0688061	−	0.997630i	$-0.478081\pi$
$409$	1138.26	0.137612	0.0688061	+	0.997630i	$0.478081\pi$
$410$	0	0
$411$	7409.40	0.889243
$412$	0	0
$413$	15617.1	1.86070
$414$	0	0
$415$	−11743.1	−1.38903
$416$	0	0
$417$	−4161.22	−0.488670
$418$	0	0
$419$	−15640.8	−1.82364	−0.911818	−	0.410595i	$-0.865321\pi$
$419$	−15640.8	−1.82364	−0.911818	+	0.410595i	$0.865321\pi$
$420$	0	0
$421$	10804.7	1.25080	0.625400	−	0.780304i	$-0.284936\pi$
$421$	10804.7	1.25080	0.625400	+	0.780304i	$0.284936\pi$
$422$	0	0
$423$	−4662.75	−0.535959
$424$	0	0
$425$	681.805	0.0778175
$426$	0	0
$427$	14434.4	1.63590
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4791.62	0.535508	0.267754	−	0.963487i	$-0.413719\pi$
$431$	4791.62	0.535508	0.267754	+	0.963487i	$0.413719\pi$
$432$	0	0
$433$	−3228.28	−0.358294	−0.179147	−	0.983822i	$-0.557334\pi$
$433$	−3228.28	−0.358294	−0.179147	+	0.983822i	$0.557334\pi$
$434$	0	0
$435$	−4790.57	−0.528024
$436$	0	0
$437$	−7774.72	−0.851065
$438$	0	0
$439$	8614.39	0.936543	0.468272	−	0.883585i	$-0.344877\pi$
$439$	8614.39	0.936543	0.468272	+	0.883585i	$0.344877\pi$
$440$	0	0
$441$	1359.95	0.146847
$442$	0	0
$443$	2008.57	0.215418	0.107709	−	0.994182i	$-0.465649\pi$
$443$	2008.57	0.215418	0.107709	+	0.994182i	$0.465649\pi$
$444$	0	0
$445$	−3733.35	−0.397702
$446$	0	0
$447$	−1662.69	−0.175934
$448$	0	0
$449$	7212.48	0.758080	0.379040	−	0.925380i	$-0.376254\pi$
$449$	7212.48	0.758080	0.379040	+	0.925380i	$0.376254\pi$
$450$	0	0
$451$	−6480.53	−0.676621
$452$	0	0
$453$	−1734.09	−0.179856
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7473.18	0.764947	0.382473	−	0.923966i	$-0.375072\pi$
$457$	7473.18	0.764947	0.382473	+	0.923966i	$0.375072\pi$
$458$	0	0
$459$	−1110.35	−0.112912
$460$	0	0
$461$	−14910.9	−1.50644	−0.753222	−	0.657766i	$-0.771501\pi$
$461$	−14910.9	−1.50644	−0.753222	+	0.657766i	$0.771501\pi$
$462$	0	0
$463$	12351.1	1.23975	0.619875	−	0.784701i	$-0.287183\pi$
$463$	12351.1	1.23975	0.619875	+	0.784701i	$0.287183\pi$
$464$	0	0
$465$	−9490.30	−0.946456
$466$	0	0
$467$	16740.9	1.65884	0.829420	−	0.558626i	$-0.188671\pi$
$467$	16740.9	1.65884	0.829420	+	0.558626i	$0.188671\pi$
$468$	0	0
$469$	15990.1	1.57432
$470$	0	0
$471$	7576.33	0.741186
$472$	0	0
$473$	17414.3	1.69283
$474$	0	0
$475$	1903.83	0.183902
$476$	0	0
$477$	2739.93	0.263003
$478$	0	0
$479$	−5697.16	−0.543444	−0.271722	−	0.962376i	$-0.587593\pi$
$479$	−5697.16	−0.543444	−0.271722	+	0.962376i	$0.587593\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−4514.95	−0.425336
$484$	0	0
$485$	−1654.97	−0.154945
$486$	0	0
$487$	9766.53	0.908756	0.454378	−	0.890809i	$-0.349862\pi$
$487$	9766.53	0.908756	0.454378	+	0.890809i	$0.349862\pi$
$488$	0	0
$489$	−6396.54	−0.591537
$490$	0	0
$491$	12179.0	1.11941	0.559704	−	0.828692i	$-0.310915\pi$
$491$	12179.0	1.11941	0.559704	+	0.828692i	$0.310915\pi$
$492$	0	0
$493$	−5519.03	−0.504188
$494$	0	0
$495$	−6065.90	−0.550792
$496$	0	0
$497$	20086.2	1.81286
$498$	0	0
$499$	555.913	0.0498719	0.0249360	−	0.999689i	$-0.492062\pi$
$499$	555.913	0.0498719	0.0249360	+	0.999689i	$0.492062\pi$
$500$	0	0
$501$	6760.08	0.602830
$502$	0	0
$503$	10017.9	0.888025	0.444013	−	0.896021i	$-0.353555\pi$
$503$	10017.9	0.888025	0.444013	+	0.896021i	$0.353555\pi$
$504$	0	0
$505$	−12509.2	−1.10229
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10873.6	0.946882	0.473441	−	0.880826i	$-0.343012\pi$
$509$	10873.6	0.946882	0.473441	+	0.880826i	$0.343012\pi$
$510$	0	0
$511$	−12498.3	−1.08198
$512$	0	0
$513$	−3100.47	−0.266840
$514$	0	0
$515$	−22154.0	−1.89557
$516$	0	0
$517$	−29346.2	−2.49641
$518$	0	0
$519$	−1522.76	−0.128789
$520$	0	0
$521$	19951.6	1.67773	0.838863	−	0.544343i	$-0.183221\pi$
$521$	19951.6	1.67773	0.838863	+	0.544343i	$0.183221\pi$
$522$	0	0
$523$	−12801.9	−1.07034	−0.535171	−	0.844744i	$-0.679753\pi$
$523$	−12801.9	−1.07034	−0.535171	+	0.844744i	$0.679753\pi$
$524$	0	0
$525$	1105.59	0.0919085
$526$	0	0
$527$	−10933.4	−0.903733
$528$	0	0
$529$	−7583.02	−0.623245
$530$	0	0
$531$	−6323.15	−0.516764
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−16133.6	−1.30377
$536$	0	0
$537$	−2023.82	−0.162634
$538$	0	0
$539$	8559.21	0.683991
$540$	0	0
$541$	1263.22	0.100388	0.0501941	−	0.998739i	$-0.484016\pi$
$541$	1263.22	0.100388	0.0501941	+	0.998739i	$0.484016\pi$
$542$	0	0
$543$	−9875.66	−0.780488
$544$	0	0
$545$	−4089.83	−0.321448
$546$	0	0
$547$	1998.46	0.156212	0.0781061	−	0.996945i	$-0.475113\pi$
$547$	1998.46	0.156212	0.0781061	+	0.996945i	$0.475113\pi$
$548$	0	0
$549$	−5844.29	−0.454332
$550$	0	0
$551$	−15411.0	−1.19152
$552$	0	0
$553$	−11798.2	−0.907255
$554$	0	0
$555$	2476.20	0.189386
$556$	0	0
$557$	7438.01	0.565815	0.282907	−	0.959147i	$-0.408701\pi$
$557$	7438.01	0.565815	0.282907	+	0.959147i	$0.408701\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−6988.29	−0.525928
$562$	0	0
$563$	6553.59	0.490588	0.245294	−	0.969449i	$-0.421116\pi$
$563$	6553.59	0.490588	0.245294	+	0.969449i	$0.421116\pi$
$564$	0	0
$565$	−1169.50	−0.0870820
$566$	0	0
$567$	−1800.51	−0.133358
$568$	0	0
$569$	−20975.8	−1.54543	−0.772717	−	0.634751i	$-0.781103\pi$
$569$	−20975.8	−1.54543	−0.772717	+	0.634751i	$0.781103\pi$
$570$	0	0
$571$	14419.5	1.05681	0.528404	−	0.848993i	$-0.322791\pi$
$571$	14419.5	1.05681	0.528404	+	0.848993i	$0.322791\pi$
$572$	0	0
$573$	2655.06	0.193572
$574$	0	0
$575$	−1122.50	−0.0814110
$576$	0	0
$577$	3857.12	0.278291	0.139146	−	0.990272i	$-0.455564\pi$
$577$	3857.12	0.278291	0.139146	+	0.990272i	$0.455564\pi$
$578$	0	0
$579$	11407.2	0.818771
$580$	0	0
$581$	−21937.8	−1.56649
$582$	0	0
$583$	17244.5	1.22503
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20470.6	−1.43937	−0.719686	−	0.694300i	$-0.755714\pi$
$587$	−20470.6	−1.43937	−0.719686	+	0.694300i	$0.755714\pi$
$588$	0	0
$589$	−30529.7	−2.13575
$590$	0	0
$591$	−9068.95	−0.631213
$592$	0	0
$593$	15080.7	1.04433	0.522167	−	0.852843i	$-0.325124\pi$
$593$	15080.7	1.04433	0.522167	+	0.852843i	$0.325124\pi$
$594$	0	0
$595$	10876.9	0.749430
$596$	0	0
$597$	6684.06	0.458225
$598$	0	0
$599$	21083.9	1.43817	0.719087	−	0.694920i	$-0.244560\pi$
$599$	21083.9	1.43817	0.719087	+	0.694920i	$0.244560\pi$
$600$	0	0
$601$	7834.64	0.531749	0.265875	−	0.964008i	$-0.414339\pi$
$601$	7834.64	0.531749	0.265875	+	0.964008i	$0.414339\pi$
$602$	0	0
$603$	−6474.17	−0.437228
$604$	0	0
$605$	−22340.1	−1.50125
$606$	0	0
$607$	9609.53	0.642568	0.321284	−	0.946983i	$-0.395886\pi$
$607$	9609.53	0.642568	0.321284	+	0.946983i	$0.395886\pi$
$608$	0	0
$609$	−8949.47	−0.595486
$610$	0	0
$611$	0	0
$612$	0	0
$613$	3849.01	0.253605	0.126803	−	0.991928i	$-0.459528\pi$
$613$	3849.01	0.253605	0.126803	+	0.991928i	$0.459528\pi$
$614$	0	0
$615$	−4083.93	−0.267773
$616$	0	0
$617$	23068.6	1.50520	0.752598	−	0.658480i	$-0.228800\pi$
$617$	23068.6	1.50520	0.752598	+	0.658480i	$0.228800\pi$
$618$	0	0
$619$	29594.4	1.92165	0.960823	−	0.277162i	$-0.0893940\pi$
$619$	29594.4	1.92165	0.960823	+	0.277162i	$0.0893940\pi$
$620$	0	0
$621$	1828.04	0.118127
$622$	0	0
$623$	−6974.43	−0.448514
$624$	0	0
$625$	−17422.5	−1.11504
$626$	0	0
$627$	−19513.6	−1.24290
$628$	0	0
$629$	2852.74	0.180837
$630$	0	0
$631$	16121.1	1.01707	0.508535	−	0.861041i	$-0.330187\pi$
$631$	16121.1	1.01707	0.508535	+	0.861041i	$0.330187\pi$
$632$	0	0
$633$	5500.60	0.345386
$634$	0	0
$635$	14792.7	0.924458
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−8132.63	−0.503477
$640$	0	0
$641$	−23080.8	−1.42221	−0.711104	−	0.703087i	$-0.751804\pi$
$641$	−23080.8	−1.42221	−0.711104	+	0.703087i	$0.751804\pi$
$642$	0	0
$643$	17620.9	1.08071	0.540357	−	0.841436i	$-0.318289\pi$
$643$	17620.9	1.08071	0.540357	+	0.841436i	$0.318289\pi$
$644$	0	0
$645$	10974.2	0.669936
$646$	0	0
$647$	−22838.3	−1.38774	−0.693868	−	0.720103i	$-0.744095\pi$
$647$	−22838.3	−1.38774	−0.693868	+	0.720103i	$0.744095\pi$
$648$	0	0
$649$	−39796.4	−2.40700
$650$	0	0
$651$	−17729.2	−1.06738
$652$	0	0
$653$	16996.1	1.01854	0.509270	−	0.860607i	$-0.329915\pi$
$653$	16996.1	1.01854	0.509270	+	0.860607i	$0.329915\pi$
$654$	0	0
$655$	−3766.97	−0.224714
$656$	0	0
$657$	5060.37	0.300493
$658$	0	0
$659$	−14758.6	−0.872404	−0.436202	−	0.899849i	$-0.643677\pi$
$659$	−14758.6	−0.872404	−0.436202	+	0.899849i	$0.643677\pi$
$660$	0	0
$661$	−21461.1	−1.26285	−0.631424	−	0.775438i	$-0.717529\pi$
$661$	−21461.1	−1.26285	−0.631424	+	0.775438i	$0.717529\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	30372.0	1.77109
$666$	0	0
$667$	9086.31	0.527471
$668$	0	0
$669$	−514.561	−0.0297370
$670$	0	0
$671$	−36782.6	−2.11621
$672$	0	0
$673$	8501.30	0.486926	0.243463	−	0.969910i	$-0.421717\pi$
$673$	8501.30	0.486926	0.243463	+	0.969910i	$0.421717\pi$
$674$	0	0
$675$	−447.638	−0.0255254
$676$	0	0
$677$	18229.8	1.03490	0.517452	−	0.855712i	$-0.326881\pi$
$677$	18229.8	1.03490	0.517452	+	0.855712i	$0.326881\pi$
$678$	0	0
$679$	−3091.72	−0.174741
$680$	0	0
$681$	−13046.3	−0.734121
$682$	0	0
$683$	33290.6	1.86505	0.932526	−	0.361104i	$-0.117600\pi$
$683$	33290.6	1.86505	0.932526	+	0.361104i	$0.117600\pi$
$684$	0	0
$685$	29387.4	1.63918
$686$	0	0
$687$	12965.7	0.720044
$688$	0	0
$689$	0	0
$690$	0	0
$691$	3827.46	0.210714	0.105357	−	0.994434i	$-0.466401\pi$
$691$	3827.46	0.210714	0.105357	+	0.994434i	$0.466401\pi$
$692$	0	0
$693$	−11332.0	−0.621163
$694$	0	0
$695$	−16504.4	−0.900786
$696$	0	0
$697$	−4704.95	−0.255685
$698$	0	0
$699$	3477.42	0.188166
$700$	0	0
$701$	24647.4	1.32799	0.663993	−	0.747739i	$-0.268860\pi$
$701$	24647.4	1.32799	0.663993	+	0.747739i	$0.268860\pi$
$702$	0	0
$703$	7965.79	0.427362
$704$	0	0
$705$	−18493.5	−0.987954
$706$	0	0
$707$	−23369.1	−1.24312
$708$	0	0
$709$	16195.3	0.857868	0.428934	−	0.903336i	$-0.358889\pi$
$709$	16195.3	0.857868	0.428934	+	0.903336i	$0.358889\pi$
$710$	0	0
$711$	4776.94	0.251968
$712$	0	0
$713$	18000.3	0.945466
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−4510.90	−0.234955
$718$	0	0
$719$	29905.6	1.55117	0.775584	−	0.631245i	$-0.217456\pi$
$719$	29905.6	1.55117	0.775584	+	0.631245i	$0.217456\pi$
$720$	0	0
$721$	−41386.8	−2.13776
$722$	0	0
$723$	18601.6	0.956848
$724$	0	0
$725$	−2225.00	−0.113978
$726$	0	0
$727$	−5867.26	−0.299318	−0.149659	−	0.988738i	$-0.547818\pi$
$727$	−5867.26	−0.299318	−0.149659	+	0.988738i	$0.547818\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	12643.0	0.639695
$732$	0	0
$733$	−25782.9	−1.29920	−0.649600	−	0.760276i	$-0.725064\pi$
$733$	−25782.9	−1.29920	−0.649600	+	0.760276i	$0.725064\pi$
$734$	0	0
$735$	5393.89	0.270689
$736$	0	0
$737$	−40746.9	−2.03654
$738$	0	0
$739$	4225.39	0.210329	0.105165	−	0.994455i	$-0.466463\pi$
$739$	4225.39	0.210329	0.105165	+	0.994455i	$0.466463\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9329.49	0.460654	0.230327	−	0.973113i	$-0.426020\pi$
$743$	9329.49	0.460654	0.230327	+	0.973113i	$0.426020\pi$
$744$	0	0
$745$	−6594.61	−0.324306
$746$	0	0
$747$	8882.30	0.435056
$748$	0	0
$749$	−30139.9	−1.47034
$750$	0	0
$751$	7179.23	0.348834	0.174417	−	0.984672i	$-0.444196\pi$
$751$	7179.23	0.348834	0.174417	+	0.984672i	$0.444196\pi$
$752$	0	0
$753$	−9102.30	−0.440513
$754$	0	0
$755$	−6877.81	−0.331535
$756$	0	0
$757$	−24207.8	−1.16228	−0.581141	−	0.813803i	$-0.697393\pi$
$757$	−24207.8	−1.16228	−0.581141	+	0.813803i	$0.697393\pi$
$758$	0	0
$759$	11505.2	0.550215
$760$	0	0
$761$	−23052.2	−1.09809	−0.549043	−	0.835794i	$-0.685008\pi$
$761$	−23052.2	−1.09809	−0.549043	+	0.835794i	$0.685008\pi$
$762$	0	0
$763$	−7640.39	−0.362517
$764$	0	0
$765$	−4403.92	−0.208136
$766$	0	0
$767$	0	0
$768$	0	0
$769$	14336.2	0.672272	0.336136	−	0.941813i	$-0.390880\pi$
$769$	14336.2	0.672272	0.336136	+	0.941813i	$0.390880\pi$
$770$	0	0
$771$	2783.03	0.129998
$772$	0	0
$773$	−1396.21	−0.0649651	−0.0324826	−	0.999472i	$-0.510341\pi$
$773$	−1396.21	−0.0649651	−0.0324826	+	0.999472i	$0.510341\pi$
$774$	0	0
$775$	−4407.81	−0.204301
$776$	0	0
$777$	4625.90	0.213582
$778$	0	0
$779$	−13137.8	−0.604248
$780$	0	0
$781$	−51184.8	−2.34512
$782$	0	0
$783$	3623.51	0.165382
$784$	0	0
$785$	30049.5	1.36626
$786$	0	0
$787$	43829.8	1.98521	0.992607	−	0.121372i	$-0.0387293\pi$
$787$	43829.8	1.98521	0.992607	+	0.121372i	$0.0387293\pi$
$788$	0	0
$789$	−20617.7	−0.930302
$790$	0	0
$791$	−2184.80	−0.0982079
$792$	0	0
$793$	0	0
$794$	0	0
$795$	10867.2	0.484805
$796$	0	0
$797$	32540.1	1.44621	0.723106	−	0.690737i	$-0.242714\pi$
$797$	32540.1	1.44621	0.723106	+	0.690737i	$0.242714\pi$
$798$	0	0
$799$	−21305.7	−0.943357
$800$	0	0
$801$	2823.85	0.124564
$802$	0	0
$803$	31848.8	1.39965
$804$	0	0
$805$	−17907.3	−0.784038
$806$	0	0
$807$	−16804.2	−0.733007
$808$	0	0
$809$	18307.8	0.795632	0.397816	−	0.917465i	$-0.369768\pi$
$809$	18307.8	0.795632	0.397816	+	0.917465i	$0.369768\pi$
$810$	0	0
$811$	−38732.4	−1.67704	−0.838519	−	0.544872i	$-0.816578\pi$
$811$	−38732.4	−1.67704	−0.838519	+	0.544872i	$0.816578\pi$
$812$	0	0
$813$	24464.9	1.05538
$814$	0	0
$815$	−25370.2	−1.09040
$816$	0	0
$817$	35303.3	1.51176
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−44477.6	−1.89072	−0.945358	−	0.326033i	$-0.894288\pi$
$821$	−44477.6	−1.89072	−0.945358	+	0.326033i	$0.894288\pi$
$822$	0	0
$823$	2329.38	0.0986599	0.0493299	−	0.998783i	$-0.484291\pi$
$823$	2329.38	0.0986599	0.0493299	+	0.998783i	$0.484291\pi$
$824$	0	0
$825$	−2817.33	−0.118893
$826$	0	0
$827$	−32012.1	−1.34604	−0.673018	−	0.739626i	$-0.735002\pi$
$827$	−32012.1	−1.34604	−0.673018	+	0.739626i	$0.735002\pi$
$828$	0	0
$829$	−43281.7	−1.81331	−0.906656	−	0.421871i	$-0.861373\pi$
$829$	−43281.7	−1.81331	−0.906656	+	0.421871i	$0.861373\pi$
$830$	0	0
$831$	−5586.34	−0.233198
$832$	0	0
$833$	6214.09	0.258470
$834$	0	0
$835$	26812.1	1.11122
$836$	0	0
$837$	7178.32	0.296439
$838$	0	0
$839$	1434.56	0.0590303	0.0295152	−	0.999564i	$-0.490604\pi$
$839$	1434.56	0.0590303	0.0295152	+	0.999564i	$0.490604\pi$
$840$	0	0
$841$	−6378.23	−0.261521
$842$	0	0
$843$	3152.61	0.128804
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−41734.6	−1.69305
$848$	0	0
$849$	4135.70	0.167181
$850$	0	0
$851$	−4696.63	−0.189187
$852$	0	0
$853$	−28862.3	−1.15853	−0.579265	−	0.815139i	$-0.696660\pi$
$853$	−28862.3	−1.15853	−0.579265	+	0.815139i	$0.696660\pi$
$854$	0	0
$855$	−12297.2	−0.491877
$856$	0	0
$857$	8629.75	0.343975	0.171988	−	0.985099i	$-0.444981\pi$
$857$	8629.75	0.343975	0.171988	+	0.985099i	$0.444981\pi$
$858$	0	0
$859$	−29542.9	−1.17345	−0.586724	−	0.809787i	$-0.699583\pi$
$859$	−29542.9	−1.17345	−0.586724	+	0.809787i	$0.699583\pi$
$860$	0	0
$861$	−7629.37	−0.301984
$862$	0	0
$863$	18901.7	0.745565	0.372782	−	0.927919i	$-0.378404\pi$
$863$	18901.7	0.745565	0.372782	+	0.927919i	$0.378404\pi$
$864$	0	0
$865$	−6039.61	−0.237402
$866$	0	0
$867$	9665.41	0.378610
$868$	0	0
$869$	30064.9	1.17363
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1251.80	0.0485302
$874$	0	0
$875$	−28676.2	−1.10792
$876$	0	0
$877$	−5779.22	−0.222520	−0.111260	−	0.993791i	$-0.535489\pi$
$877$	−5779.22	−0.222520	−0.111260	+	0.993791i	$0.535489\pi$
$878$	0	0
$879$	15440.5	0.592487
$880$	0	0
$881$	32932.7	1.25940	0.629700	−	0.776839i	$-0.283178\pi$
$881$	32932.7	1.25940	0.629700	+	0.776839i	$0.283178\pi$
$882$	0	0
$883$	−34060.5	−1.29811	−0.649053	−	0.760743i	$-0.724835\pi$
$883$	−34060.5	−1.29811	−0.649053	+	0.760743i	$0.724835\pi$
$884$	0	0
$885$	−25079.1	−0.952571
$886$	0	0
$887$	43728.7	1.65532	0.827658	−	0.561233i	$-0.189673\pi$
$887$	43728.7	1.65532	0.827658	+	0.561233i	$0.189673\pi$
$888$	0	0
$889$	27634.9	1.04257
$890$	0	0
$891$	4588.15	0.172513
$892$	0	0
$893$	−59492.6	−2.22939
$894$	0	0
$895$	−8026.95	−0.299789
$896$	0	0
$897$	0	0
$898$	0	0
$899$	35680.0	1.32369
$900$	0	0
$901$	12519.7	0.462920
$902$	0	0
$903$	20501.4	0.755530
$904$	0	0
$905$	−39169.2	−1.43870
$906$	0	0
$907$	−6302.24	−0.230719	−0.115360	−	0.993324i	$-0.536802\pi$
$907$	−6302.24	−0.230719	−0.115360	+	0.993324i	$0.536802\pi$
$908$	0	0
$909$	9461.80	0.345246
$910$	0	0
$911$	39635.4	1.44147	0.720736	−	0.693210i	$-0.243804\pi$
$911$	39635.4	1.44147	0.720736	+	0.693210i	$0.243804\pi$
$912$	0	0
$913$	55903.1	2.02642
$914$	0	0
$915$	−23179.8	−0.837488
$916$	0	0
$917$	−7037.25	−0.253425
$918$	0	0
$919$	−27293.3	−0.979676	−0.489838	−	0.871814i	$-0.662944\pi$
$919$	−27293.3	−0.979676	−0.489838	+	0.871814i	$0.662944\pi$
$920$	0	0
$921$	−6558.11	−0.234633
$922$	0	0
$923$	0	0
$924$	0	0
$925$	1150.08	0.0408805
$926$	0	0
$927$	16756.9	0.593710
$928$	0	0
$929$	−3165.12	−0.111781	−0.0558903	−	0.998437i	$-0.517800\pi$
$929$	−3165.12	−0.111781	−0.0558903	+	0.998437i	$0.517800\pi$
$930$	0	0
$931$	17351.8	0.610830
$932$	0	0
$933$	−18288.1	−0.641720
$934$	0	0
$935$	−27717.2	−0.969465
$936$	0	0
$937$	−25319.5	−0.882766	−0.441383	−	0.897319i	$-0.645512\pi$
$937$	−25319.5	−0.882766	−0.441383	+	0.897319i	$0.645512\pi$
$938$	0	0
$939$	27221.7	0.946057
$940$	0	0
$941$	−29219.6	−1.01226	−0.506128	−	0.862458i	$-0.668924\pi$
$941$	−29219.6	−1.01226	−0.506128	+	0.862458i	$0.668924\pi$
$942$	0	0
$943$	7746.03	0.267492
$944$	0	0
$945$	−7141.24	−0.245825
$946$	0	0
$947$	22104.8	0.758512	0.379256	−	0.925292i	$-0.376180\pi$
$947$	22104.8	0.758512	0.379256	+	0.925292i	$0.376180\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−20267.6	−0.691084
$952$	0	0
$953$	−50838.7	−1.72805	−0.864023	−	0.503453i	$-0.832063\pi$
$953$	−50838.7	−1.72805	−0.864023	+	0.503453i	$0.832063\pi$
$954$	0	0
$955$	10530.6	0.356819
$956$	0	0
$957$	22805.5	0.770322
$958$	0	0
$959$	54900.0	1.84860
$960$	0	0
$961$	40892.5	1.37265
$962$	0	0
$963$	12203.2	0.408352
$964$	0	0
$965$	45243.8	1.50927
$966$	0	0
$967$	−16456.6	−0.547267	−0.273634	−	0.961834i	$-0.588226\pi$
$967$	−16456.6	−0.547267	−0.273634	+	0.961834i	$0.588226\pi$
$968$	0	0
$969$	−14167.1	−0.469674
$970$	0	0
$971$	20427.8	0.675138	0.337569	−	0.941301i	$-0.390395\pi$
$971$	20427.8	0.675138	0.337569	+	0.941301i	$0.390395\pi$
$972$	0	0
$973$	−30832.5	−1.01587
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14359.7	−0.470223	−0.235111	−	0.971968i	$-0.575546\pi$
$977$	−14359.7	−0.470223	−0.235111	+	0.971968i	$0.575546\pi$
$978$	0	0
$979$	17772.6	0.580199
$980$	0	0
$981$	3093.48	0.100680
$982$	0	0
$983$	−5478.74	−0.177767	−0.0888834	−	0.996042i	$-0.528330\pi$
$983$	−5478.74	−0.177767	−0.0888834	+	0.996042i	$0.528330\pi$
$984$	0	0
$985$	−35969.6	−1.16354
$986$	0	0
$987$	−34548.6	−1.11418
$988$	0	0
$989$	−20814.9	−0.669235
$990$	0	0
$991$	−30372.3	−0.973572	−0.486786	−	0.873521i	$-0.661831\pi$
$991$	−30372.3	−0.973572	−0.486786	+	0.873521i	$0.661831\pi$
$992$	0	0
$993$	−11901.3	−0.380339
$994$	0	0
$995$	26510.5	0.844664
$996$	0	0
$997$	−20728.3	−0.658447	−0.329224	−	0.944252i	$-0.606787\pi$
$997$	−20728.3	−0.658447	−0.329224	+	0.944252i	$0.606787\pi$
$998$	0	0
$999$	−1872.96	−0.0593172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.4.a.o.1.2		6
13.2	odd	12		156.4.q.c.121.3	yes	6
13.5	odd	4		2028.4.b.h.337.5		6
13.7	odd	12		156.4.q.c.49.1	✓	6
13.8	odd	4		2028.4.b.h.337.2		6
13.12	even	2	inner	2028.4.a.o.1.5		6
39.2	even	12		468.4.t.f.433.1		6
39.20	even	12		468.4.t.f.361.3		6
52.7	even	12		624.4.bv.f.49.1		6
52.15	even	12		624.4.bv.f.433.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.4.q.c.49.1	✓	6	13.7	odd	12
156.4.q.c.121.3	yes	6	13.2	odd	12
468.4.t.f.361.3		6	39.20	even	12
468.4.t.f.433.1		6	39.2	even	12
624.4.bv.f.49.1		6	52.7	even	12
624.4.bv.f.433.3		6	52.15	even	12
2028.4.a.o.1.2		6	1.1	even	1	trivial
2028.4.a.o.1.5		6	13.12	even	2	inner
2028.4.b.h.337.2		6	13.8	odd	4
2028.4.b.h.337.5		6	13.5	odd	4

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:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2028.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -11.8987 q^{5} -22.2285 q^{7} +9.00000 q^{9} +56.6439 q^{11} +35.6961 q^{15} +41.1241 q^{17} +114.832 q^{19} +66.6855 q^{21} -67.7051 q^{23} +16.5792 q^{25} -27.0000 q^{27} -134.204 q^{29} -265.864 q^{31} -169.932 q^{33} +264.490 q^{35} +69.3690 q^{37} -114.408 q^{41} +307.434 q^{43} -107.088 q^{45} -518.083 q^{47} +151.106 q^{49} -123.372 q^{51} +304.436 q^{53} -673.989 q^{55} -344.497 q^{57} -702.573 q^{59} -649.365 q^{61} -200.056 q^{63} -719.352 q^{67} +203.115 q^{69} -903.626 q^{71} +562.264 q^{73} -49.7376 q^{75} -1259.11 q^{77} +530.771 q^{79} +81.0000 q^{81} +986.923 q^{83} -489.324 q^{85} +402.613 q^{87} +313.761 q^{89} +797.591 q^{93} -1366.35 q^{95} +139.088 q^{97} +509.795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 18 q^{3} + 54 q^{9} + 64 q^{17} + 208 q^{23} + 166 q^{25} - 162 q^{27} - 440 q^{29} + 176 q^{35} + 466 q^{43} + 76 q^{49} - 192 q^{51} + 732 q^{53} - 1404 q^{55} - 1190 q^{61} - 624 q^{69} - 498 q^{75}+ \cdots - 180 q^{95}+O(q^{100}) \) 6 * q - 18 * q^3 + 54 * q^9 + 64 * q^17 + 208 * q^23 + 166 * q^25 - 162 * q^27 - 440 * q^29 + 176 * q^35 + 466 * q^43 + 76 * q^49 - 192 * q^51 + 732 * q^53 - 1404 * q^55 - 1190 * q^61 - 624 * q^69 - 498 * q^75 - 2076 * q^77 + 2222 * q^79 + 486 * q^81 + 1320 * q^87 - 180 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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