LMFDB - Embedded newform 2028.4.a.s.1.3

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 700 x^{7} + 159 x^{6} + 144359 x^{5} + 120897 x^{4} - 9003831 x^{3} - 4147432 x^{2} + \cdots - 143123539 $ x^9 - x^8 - 700x^7 + 159x^6 + 144359x^5 + 120897x^4 - 9003831x^3 - 4147432x^2 + 141805568*x - 143123539
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 13^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$19.0051$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} -8.06618 q^{5} +19.7548 q^{7} +9.00000 q^{9} -34.8446 q^{11} -24.1985 q^{15} -12.3447 q^{17} -35.0327 q^{19} +59.2643 q^{21} -167.807 q^{23} -59.9368 q^{25} +27.0000 q^{27} +126.543 q^{29} +333.168 q^{31} -104.534 q^{33} -159.345 q^{35} -38.8774 q^{37} +175.361 q^{41} +146.306 q^{43} -72.5956 q^{45} +414.946 q^{47} +47.2508 q^{49} -37.0342 q^{51} -76.7920 q^{53} +281.063 q^{55} -105.098 q^{57} +71.9387 q^{59} +499.860 q^{61} +177.793 q^{63} +387.011 q^{67} -503.420 q^{69} -600.863 q^{71} +92.9877 q^{73} -179.810 q^{75} -688.346 q^{77} +1102.98 q^{79} +81.0000 q^{81} +221.478 q^{83} +99.5749 q^{85} +379.630 q^{87} +49.3746 q^{89} +999.504 q^{93} +282.580 q^{95} +548.221 q^{97} -313.601 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 27 q^{3} + 11 q^{5} + 37 q^{7} + 81 q^{9} + 61 q^{11} + 33 q^{15} + 104 q^{17} + 328 q^{19} + 111 q^{21} + 54 q^{23} + 376 q^{25} + 243 q^{27} + 61 q^{29} + 333 q^{31} + 183 q^{33} - 338 q^{35} + 144 q^{37}+ \cdots + 549 q^{99}+O(q^{100}) $ 9 * q + 27 * q^3 + 11 * q^5 + 37 * q^7 + 81 * q^9 + 61 * q^11 + 33 * q^15 + 104 * q^17 + 328 * q^19 + 111 * q^21 + 54 * q^23 + 376 * q^25 + 243 * q^27 + 61 * q^29 + 333 * q^31 + 183 * q^33 - 338 * q^35 + 144 * q^37 + 304 * q^41 - 248 * q^43 + 99 * q^45 + 1034 * q^47 + 1142 * q^49 + 312 * q^51 - 737 * q^53 - 394 * q^55 + 984 * q^57 + 1115 * q^59 + 440 * q^61 + 333 * q^63 + 2046 * q^67 + 162 * q^69 + 1548 * q^71 + 2407 * q^73 + 1128 * q^75 - 354 * q^77 - 1361 * q^79 + 729 * q^81 + 3365 * q^83 + 1000 * q^85 + 183 * q^87 + 1412 * q^89 + 999 * q^93 - 1532 * q^95 + 3443 * q^97 + 549 * q^99

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
$5$	−8.06618	−0.721461	−0.360730	−	0.932670i	$-0.617473\pi$
$5$	−8.06618	−0.721461	−0.360730	+	0.932670i	$0.617473\pi$
$6$	0	0
$7$	19.7548	1.06666	0.533329	−	0.845908i	$-0.320941\pi$
$7$	19.7548	1.06666	0.533329	+	0.845908i	$0.320941\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−34.8446	−0.955094	−0.477547	−	0.878606i	$-0.658474\pi$
$11$	−34.8446	−0.955094	−0.477547	+	0.878606i	$0.658474\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−24.1985	−0.416536
$16$	0	0
$17$	−12.3447	−0.176120	−0.0880600	−	0.996115i	$-0.528067\pi$
$17$	−12.3447	−0.176120	−0.0880600	+	0.996115i	$0.528067\pi$
$18$	0	0
$19$	−35.0327	−0.423003	−0.211501	−	0.977378i	$-0.567835\pi$
$19$	−35.0327	−0.423003	−0.211501	+	0.977378i	$0.567835\pi$
$20$	0	0
$21$	59.2643	0.615835
$22$	0	0
$23$	−167.807	−1.52131	−0.760655	−	0.649157i	$-0.775122\pi$
$23$	−167.807	−1.52131	−0.760655	+	0.649157i	$0.775122\pi$
$24$	0	0
$25$	−59.9368	−0.479494
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	126.543	0.810293	0.405147	−	0.914252i	$-0.367220\pi$
$29$	126.543	0.810293	0.405147	+	0.914252i	$0.367220\pi$
$30$	0	0
$31$	333.168	1.93028	0.965141	−	0.261731i	$-0.0842934\pi$
$31$	333.168	1.93028	0.965141	+	0.261731i	$0.0842934\pi$
$32$	0	0
$33$	−104.534	−0.551424
$34$	0	0
$35$	−159.345	−0.769551
$36$	0	0
$37$	−38.8774	−0.172741	−0.0863703	−	0.996263i	$-0.527527\pi$
$37$	−38.8774	−0.172741	−0.0863703	+	0.996263i	$0.527527\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	175.361	0.667969	0.333984	−	0.942579i	$-0.391607\pi$
$41$	175.361	0.667969	0.333984	+	0.942579i	$0.391607\pi$
$42$	0	0
$43$	146.306	0.518872	0.259436	−	0.965760i	$-0.416463\pi$
$43$	146.306	0.518872	0.259436	+	0.965760i	$0.416463\pi$
$44$	0	0
$45$	−72.5956	−0.240487
$46$	0	0
$47$	414.946	1.28779	0.643895	−	0.765114i	$-0.277317\pi$
$47$	414.946	1.28779	0.643895	+	0.765114i	$0.277317\pi$
$48$	0	0
$49$	47.2508	0.137757
$50$	0	0
$51$	−37.0342	−0.101683
$52$	0	0
$53$	−76.7920	−0.199023	−0.0995113	−	0.995036i	$-0.531728\pi$
$53$	−76.7920	−0.199023	−0.0995113	+	0.995036i	$0.531728\pi$
$54$	0	0
$55$	281.063	0.689063
$56$	0	0
$57$	−105.098	−0.244221
$58$	0	0
$59$	71.9387	0.158739	0.0793697	−	0.996845i	$-0.474709\pi$
$59$	71.9387	0.158739	0.0793697	+	0.996845i	$0.474709\pi$
$60$	0	0
$61$	499.860	1.04919	0.524594	−	0.851353i	$-0.324217\pi$
$61$	499.860	1.04919	0.524594	+	0.851353i	$0.324217\pi$
$62$	0	0
$63$	177.793	0.355552
$64$	0	0
$65$	0	0
$66$	0	0
$67$	387.011	0.705685	0.352843	−	0.935683i	$-0.385215\pi$
$67$	387.011	0.705685	0.352843	+	0.935683i	$0.385215\pi$
$68$	0	0
$69$	−503.420	−0.878328
$70$	0	0
$71$	−600.863	−1.00436	−0.502178	−	0.864764i	$-0.667468\pi$
$71$	−600.863	−1.00436	−0.502178	+	0.864764i	$0.667468\pi$
$72$	0	0
$73$	92.9877	0.149088	0.0745438	−	0.997218i	$-0.476250\pi$
$73$	92.9877	0.149088	0.0745438	+	0.997218i	$0.476250\pi$
$74$	0	0
$75$	−179.810	−0.276836
$76$	0	0
$77$	−688.346	−1.01876
$78$	0	0
$79$	1102.98	1.57082	0.785412	−	0.618973i	$-0.212451\pi$
$79$	1102.98	1.57082	0.785412	+	0.618973i	$0.212451\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	221.478	0.292896	0.146448	−	0.989218i	$-0.453216\pi$
$83$	221.478	0.292896	0.146448	+	0.989218i	$0.453216\pi$
$84$	0	0
$85$	99.5749	0.127064
$86$	0	0
$87$	379.630	0.467823
$88$	0	0
$89$	49.3746	0.0588056	0.0294028	−	0.999568i	$-0.490639\pi$
$89$	49.3746	0.0588056	0.0294028	+	0.999568i	$0.490639\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	999.504	1.11445
$94$	0	0
$95$	282.580	0.305180
$96$	0	0
$97$	548.221	0.573849	0.286925	−	0.957953i	$-0.407367\pi$
$97$	548.221	0.573849	0.286925	+	0.957953i	$0.407367\pi$
$98$	0	0
$99$	−313.601	−0.318365
$100$	0	0
$101$	−160.729	−0.158348	−0.0791739	−	0.996861i	$-0.525228\pi$
$101$	−160.729	−0.158348	−0.0791739	+	0.996861i	$0.525228\pi$
$102$	0	0
$103$	−1747.70	−1.67190	−0.835952	−	0.548803i	$-0.815084\pi$
$103$	−1747.70	−1.67190	−0.835952	+	0.548803i	$0.815084\pi$
$104$	0	0
$105$	−478.036	−0.444301
$106$	0	0
$107$	−541.397	−0.489148	−0.244574	−	0.969631i	$-0.578648\pi$
$107$	−541.397	−0.489148	−0.244574	+	0.969631i	$0.578648\pi$
$108$	0	0
$109$	−1038.56	−0.912624	−0.456312	−	0.889820i	$-0.650830\pi$
$109$	−1038.56	−0.912624	−0.456312	+	0.889820i	$0.650830\pi$
$110$	0	0
$111$	−116.632	−0.0997318
$112$	0	0
$113$	−244.303	−0.203381	−0.101691	−	0.994816i	$-0.532425\pi$
$113$	−244.303	−0.203381	−0.101691	+	0.994816i	$0.532425\pi$
$114$	0	0
$115$	1353.56	1.09757
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−243.868	−0.187860
$120$	0	0
$121$	−116.856	−0.0877956
$122$	0	0
$123$	526.082	0.385652
$124$	0	0
$125$	1491.73	1.06740
$126$	0	0
$127$	1926.10	1.34578	0.672888	−	0.739745i	$-0.265054\pi$
$127$	1926.10	1.34578	0.672888	+	0.739745i	$0.265054\pi$
$128$	0	0
$129$	438.919	0.299571
$130$	0	0
$131$	−2558.01	−1.70607	−0.853033	−	0.521856i	$-0.825240\pi$
$131$	−2558.01	−1.70607	−0.853033	+	0.521856i	$0.825240\pi$
$132$	0	0
$133$	−692.062	−0.451199
$134$	0	0
$135$	−217.787	−0.138845
$136$	0	0
$137$	2840.46	1.77136	0.885681	−	0.464294i	$-0.153692\pi$
$137$	2840.46	1.77136	0.885681	+	0.464294i	$0.153692\pi$
$138$	0	0
$139$	649.111	0.396093	0.198046	−	0.980193i	$-0.436540\pi$
$139$	649.111	0.396093	0.198046	+	0.980193i	$0.436540\pi$
$140$	0	0
$141$	1244.84	0.743506
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1020.72	−0.584595
$146$	0	0
$147$	141.752	0.0795342
$148$	0	0
$149$	2155.21	1.18498	0.592490	−	0.805578i	$-0.298145\pi$
$149$	2155.21	1.18498	0.592490	+	0.805578i	$0.298145\pi$
$150$	0	0
$151$	3145.77	1.69536	0.847679	−	0.530510i	$-0.178000\pi$
$151$	3145.77	1.69536	0.847679	+	0.530510i	$0.178000\pi$
$152$	0	0
$153$	−111.103	−0.0587067
$154$	0	0
$155$	−2687.39	−1.39262
$156$	0	0
$157$	1592.67	0.809611	0.404806	−	0.914403i	$-0.367339\pi$
$157$	1592.67	0.809611	0.404806	+	0.914403i	$0.367339\pi$
$158$	0	0
$159$	−230.376	−0.114906
$160$	0	0
$161$	−3314.98	−1.62272
$162$	0	0
$163$	1411.04	0.678043	0.339021	−	0.940779i	$-0.389904\pi$
$163$	1411.04	0.678043	0.339021	+	0.940779i	$0.389904\pi$
$164$	0	0
$165$	843.188	0.397831
$166$	0	0
$167$	−766.584	−0.355210	−0.177605	−	0.984102i	$-0.556835\pi$
$167$	−766.584	−0.355210	−0.177605	+	0.984102i	$0.556835\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−315.294	−0.141001
$172$	0	0
$173$	1835.96	0.806853	0.403426	−	0.915012i	$-0.367819\pi$
$173$	1835.96	0.806853	0.403426	+	0.915012i	$0.367819\pi$
$174$	0	0
$175$	−1184.04	−0.511456
$176$	0	0
$177$	215.816	0.0916482
$178$	0	0
$179$	−4126.42	−1.72304	−0.861518	−	0.507727i	$-0.830486\pi$
$179$	−4126.42	−1.72304	−0.861518	+	0.507727i	$0.830486\pi$
$180$	0	0
$181$	1919.56	0.788285	0.394143	−	0.919049i	$-0.371042\pi$
$181$	1919.56	0.788285	0.394143	+	0.919049i	$0.371042\pi$
$182$	0	0
$183$	1499.58	0.605749
$184$	0	0
$185$	313.592	0.124626
$186$	0	0
$187$	430.147	0.168211
$188$	0	0
$189$	533.379	0.205278
$190$	0	0
$191$	−309.598	−0.117286	−0.0586432	−	0.998279i	$-0.518677\pi$
$191$	−309.598	−0.117286	−0.0586432	+	0.998279i	$0.518677\pi$
$192$	0	0
$193$	2143.97	0.799619	0.399809	−	0.916598i	$-0.369076\pi$
$193$	2143.97	0.799619	0.399809	+	0.916598i	$0.369076\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1371.77	0.496113	0.248056	−	0.968746i	$-0.420208\pi$
$197$	1371.77	0.496113	0.248056	+	0.968746i	$0.420208\pi$
$198$	0	0
$199$	3713.31	1.32276	0.661381	−	0.750050i	$-0.269971\pi$
$199$	3713.31	1.32276	0.661381	+	0.750050i	$0.269971\pi$
$200$	0	0
$201$	1161.03	0.407428
$202$	0	0
$203$	2499.83	0.864305
$204$	0	0
$205$	−1414.49	−0.481914
$206$	0	0
$207$	−1510.26	−0.507103
$208$	0	0
$209$	1220.70	0.404007
$210$	0	0
$211$	−1355.67	−0.442313	−0.221156	−	0.975238i	$-0.570983\pi$
$211$	−1355.67	−0.442313	−0.221156	+	0.975238i	$0.570983\pi$
$212$	0	0
$213$	−1802.59	−0.579865
$214$	0	0
$215$	−1180.13	−0.374346
$216$	0	0
$217$	6581.65	2.05895
$218$	0	0
$219$	278.963	0.0860757
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−749.668	−0.225119	−0.112559	−	0.993645i	$-0.535905\pi$
$223$	−749.668	−0.225119	−0.112559	+	0.993645i	$0.535905\pi$
$224$	0	0
$225$	−539.431	−0.159831
$226$	0	0
$227$	4210.36	1.23106	0.615532	−	0.788112i	$-0.288941\pi$
$227$	4210.36	1.23106	0.615532	+	0.788112i	$0.288941\pi$
$228$	0	0
$229$	3948.11	1.13930	0.569648	−	0.821889i	$-0.307080\pi$
$229$	3948.11	1.13930	0.569648	+	0.821889i	$0.307080\pi$
$230$	0	0
$231$	−2065.04	−0.588180
$232$	0	0
$233$	0.632659	0.000177884 0	8.89418e−5	−	1.00000i	$-0.499972\pi$
$233$	0.632659	0.000177884 0	8.89418e−5		1.00000i	$0.499972\pi$
$234$	0	0
$235$	−3347.03	−0.929090
$236$	0	0
$237$	3308.94	0.906916
$238$	0	0
$239$	−2403.88	−0.650604	−0.325302	−	0.945610i	$-0.605466\pi$
$239$	−2403.88	−0.650604	−0.325302	+	0.945610i	$0.605466\pi$
$240$	0	0
$241$	−3001.74	−0.802321	−0.401160	−	0.916008i	$-0.631393\pi$
$241$	−3001.74	−0.802321	−0.401160	+	0.916008i	$0.631393\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−381.133	−0.0993865
$246$	0	0
$247$	0	0
$248$	0	0
$249$	664.435	0.169104
$250$	0	0
$251$	4612.16	1.15983	0.579914	−	0.814677i	$-0.303086\pi$
$251$	4612.16	1.15983	0.579914	+	0.814677i	$0.303086\pi$
$252$	0	0
$253$	5847.15	1.45299
$254$	0	0
$255$	298.725	0.0733603
$256$	0	0
$257$	3103.00	0.753150	0.376575	−	0.926386i	$-0.377102\pi$
$257$	3103.00	0.753150	0.376575	+	0.926386i	$0.377102\pi$
$258$	0	0
$259$	−768.013	−0.184255
$260$	0	0
$261$	1138.89	0.270098
$262$	0	0
$263$	2997.48	0.702786	0.351393	−	0.936228i	$-0.385708\pi$
$263$	2997.48	0.702786	0.351393	+	0.936228i	$0.385708\pi$
$264$	0	0
$265$	619.418	0.143587
$266$	0	0
$267$	148.124	0.0339514
$268$	0	0
$269$	8704.75	1.97300	0.986501	−	0.163753i	$-0.0523599\pi$
$269$	8704.75	1.97300	0.986501	+	0.163753i	$0.0523599\pi$
$270$	0	0
$271$	6798.60	1.52393	0.761966	−	0.647618i	$-0.224235\pi$
$271$	6798.60	1.52393	0.761966	+	0.647618i	$0.224235\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2088.47	0.457962
$276$	0	0
$277$	1943.73	0.421615	0.210807	−	0.977528i	$-0.432391\pi$
$277$	1943.73	0.421615	0.210807	+	0.977528i	$0.432391\pi$
$278$	0	0
$279$	2998.51	0.643427
$280$	0	0
$281$	5673.88	1.20454	0.602269	−	0.798293i	$-0.294263\pi$
$281$	5673.88	1.20454	0.602269	+	0.798293i	$0.294263\pi$
$282$	0	0
$283$	−8188.55	−1.72000	−0.859998	−	0.510298i	$-0.829535\pi$
$283$	−8188.55	−1.72000	−0.859998	+	0.510298i	$0.829535\pi$
$284$	0	0
$285$	847.740	0.176196
$286$	0	0
$287$	3464.21	0.712494
$288$	0	0
$289$	−4760.61	−0.968982
$290$	0	0
$291$	1644.66	0.331312
$292$	0	0
$293$	−8469.31	−1.68868	−0.844339	−	0.535810i	$-0.820006\pi$
$293$	−8469.31	−1.68868	−0.844339	+	0.535810i	$0.820006\pi$
$294$	0	0
$295$	−580.271	−0.114524
$296$	0	0
$297$	−940.803	−0.183808
$298$	0	0
$299$	0	0
$300$	0	0
$301$	2890.25	0.553458
$302$	0	0
$303$	−482.187	−0.0914222
$304$	0	0
$305$	−4031.96	−0.756948
$306$	0	0
$307$	8182.72	1.52121	0.760607	−	0.649213i	$-0.224902\pi$
$307$	8182.72	1.52121	0.760607	+	0.649213i	$0.224902\pi$
$308$	0	0
$309$	−5243.10	−0.965274
$310$	0	0
$311$	−8779.90	−1.60084	−0.800422	−	0.599437i	$-0.795391\pi$
$311$	−8779.90	−1.60084	−0.800422	+	0.599437i	$0.795391\pi$
$312$	0	0
$313$	5717.11	1.03243	0.516214	−	0.856460i	$-0.327341\pi$
$313$	5717.11	1.03243	0.516214	+	0.856460i	$0.327341\pi$
$314$	0	0
$315$	−1434.11	−0.256517
$316$	0	0
$317$	−3265.29	−0.578539	−0.289270	−	0.957248i	$-0.593412\pi$
$317$	−3265.29	−0.578539	−0.289270	+	0.957248i	$0.593412\pi$
$318$	0	0
$319$	−4409.35	−0.773906
$320$	0	0
$321$	−1624.19	−0.282410
$322$	0	0
$323$	432.470	0.0744992
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3115.68	−0.526904
$328$	0	0
$329$	8197.17	1.37363
$330$	0	0
$331$	−1432.62	−0.237897	−0.118949	−	0.992900i	$-0.537952\pi$
$331$	−1432.62	−0.237897	−0.118949	+	0.992900i	$0.537952\pi$
$332$	0	0
$333$	−349.896	−0.0575802
$334$	0	0
$335$	−3121.70	−0.509124
$336$	0	0
$337$	−8973.26	−1.45046	−0.725229	−	0.688507i	$-0.758266\pi$
$337$	−8973.26	−1.45046	−0.725229	+	0.688507i	$0.758266\pi$
$338$	0	0
$339$	−732.908	−0.117422
$340$	0	0
$341$	−11609.1	−1.84360
$342$	0	0
$343$	−5842.46	−0.919717
$344$	0	0
$345$	4060.68	0.633680
$346$	0	0
$347$	4336.82	0.670931	0.335465	−	0.942053i	$-0.391106\pi$
$347$	4336.82	0.670931	0.335465	+	0.942053i	$0.391106\pi$
$348$	0	0
$349$	−2392.94	−0.367024	−0.183512	−	0.983017i	$-0.558747\pi$
$349$	−2392.94	−0.367024	−0.183512	+	0.983017i	$0.558747\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5965.15	0.899413	0.449706	−	0.893176i	$-0.351529\pi$
$353$	5965.15	0.899413	0.449706	+	0.893176i	$0.351529\pi$
$354$	0	0
$355$	4846.67	0.724604
$356$	0	0
$357$	−731.603	−0.108461
$358$	0	0
$359$	−7862.03	−1.15583	−0.577914	−	0.816098i	$-0.696133\pi$
$359$	−7862.03	−1.15583	−0.577914	+	0.816098i	$0.696133\pi$
$360$	0	0
$361$	−5631.71	−0.821069
$362$	0	0
$363$	−350.568	−0.0506888
$364$	0	0
$365$	−750.056	−0.107561
$366$	0	0
$367$	3571.24	0.507949	0.253974	−	0.967211i	$-0.418262\pi$
$367$	3571.24	0.507949	0.253974	+	0.967211i	$0.418262\pi$
$368$	0	0
$369$	1578.25	0.222656
$370$	0	0
$371$	−1517.01	−0.212289
$372$	0	0
$373$	−5821.64	−0.808131	−0.404066	−	0.914730i	$-0.632403\pi$
$373$	−5821.64	−0.808131	−0.404066	+	0.914730i	$0.632403\pi$
$374$	0	0
$375$	4475.20	0.616262
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−7847.38	−1.06357	−0.531785	−	0.846879i	$-0.678479\pi$
$379$	−7847.38	−1.06357	−0.531785	+	0.846879i	$0.678479\pi$
$380$	0	0
$381$	5778.29	0.776984
$382$	0	0
$383$	12512.9	1.66940	0.834702	−	0.550702i	$-0.185640\pi$
$383$	12512.9	1.66940	0.834702	+	0.550702i	$0.185640\pi$
$384$	0	0
$385$	5552.32	0.734994
$386$	0	0
$387$	1316.76	0.172957
$388$	0	0
$389$	−6990.56	−0.911145	−0.455573	−	0.890199i	$-0.650565\pi$
$389$	−6990.56	−0.911145	−0.455573	+	0.890199i	$0.650565\pi$
$390$	0	0
$391$	2071.53	0.267933
$392$	0	0
$393$	−7674.04	−0.984998
$394$	0	0
$395$	−8896.85	−1.13329
$396$	0	0
$397$	13647.2	1.72527	0.862637	−	0.505823i	$-0.168811\pi$
$397$	13647.2	1.72527	0.862637	+	0.505823i	$0.168811\pi$
$398$	0	0
$399$	−2076.19	−0.260500
$400$	0	0
$401$	5889.14	0.733390	0.366695	−	0.930341i	$-0.380489\pi$
$401$	5889.14	0.733390	0.366695	+	0.930341i	$0.380489\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−653.361	−0.0801623
$406$	0	0
$407$	1354.67	0.164983
$408$	0	0
$409$	6467.05	0.781846	0.390923	−	0.920423i	$-0.372156\pi$
$409$	6467.05	0.781846	0.390923	+	0.920423i	$0.372156\pi$
$410$	0	0
$411$	8521.37	1.02270
$412$	0	0
$413$	1421.13	0.169320
$414$	0	0
$415$	−1786.48	−0.211313
$416$	0	0
$417$	1947.33	0.228684
$418$	0	0
$419$	3387.71	0.394989	0.197494	−	0.980304i	$-0.436720\pi$
$419$	3387.71	0.394989	0.197494	+	0.980304i	$0.436720\pi$
$420$	0	0
$421$	−634.791	−0.0734865	−0.0367432	−	0.999325i	$-0.511698\pi$
$421$	−634.791	−0.0734865	−0.0367432	+	0.999325i	$0.511698\pi$
$422$	0	0
$423$	3734.52	0.429263
$424$	0	0
$425$	739.904	0.0844485
$426$	0	0
$427$	9874.61	1.11912
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11254.3	1.25778	0.628888	−	0.777496i	$-0.283510\pi$
$431$	11254.3	1.25778	0.628888	+	0.777496i	$0.283510\pi$
$432$	0	0
$433$	−3544.92	−0.393436	−0.196718	−	0.980460i	$-0.563028\pi$
$433$	−3544.92	−0.393436	−0.196718	+	0.980460i	$0.563028\pi$
$434$	0	0
$435$	−3062.16	−0.337516
$436$	0	0
$437$	5878.72	0.643518
$438$	0	0
$439$	−2754.60	−0.299475	−0.149738	−	0.988726i	$-0.547843\pi$
$439$	−2754.60	−0.299475	−0.149738	+	0.988726i	$0.547843\pi$
$440$	0	0
$441$	425.257	0.0459191
$442$	0	0
$443$	−15296.5	−1.64054	−0.820271	−	0.571975i	$-0.806177\pi$
$443$	−15296.5	−1.64054	−0.820271	+	0.571975i	$0.806177\pi$
$444$	0	0
$445$	−398.265	−0.0424260
$446$	0	0
$447$	6465.64	0.684148
$448$	0	0
$449$	−9453.61	−0.993638	−0.496819	−	0.867854i	$-0.665499\pi$
$449$	−9453.61	−0.993638	−0.496819	+	0.867854i	$0.665499\pi$
$450$	0	0
$451$	−6110.37	−0.637973
$452$	0	0
$453$	9437.30	0.978815
$454$	0	0
$455$	0	0
$456$	0	0
$457$	462.480	0.0473389	0.0236695	−	0.999720i	$-0.492465\pi$
$457$	462.480	0.0473389	0.0236695	+	0.999720i	$0.492465\pi$
$458$	0	0
$459$	−333.308	−0.0338943
$460$	0	0
$461$	−9022.26	−0.911515	−0.455758	−	0.890104i	$-0.650632\pi$
$461$	−9022.26	−0.911515	−0.455758	+	0.890104i	$0.650632\pi$
$462$	0	0
$463$	−6656.08	−0.668109	−0.334054	−	0.942554i	$-0.608417\pi$
$463$	−6656.08	−0.668109	−0.334054	+	0.942554i	$0.608417\pi$
$464$	0	0
$465$	−8062.18	−0.804031
$466$	0	0
$467$	−11824.1	−1.17164	−0.585818	−	0.810442i	$-0.699227\pi$
$467$	−11824.1	−1.17164	−0.585818	+	0.810442i	$0.699227\pi$
$468$	0	0
$469$	7645.31	0.752724
$470$	0	0
$471$	4778.01	0.467429
$472$	0	0
$473$	−5097.98	−0.495571
$474$	0	0
$475$	2099.75	0.202827
$476$	0	0
$477$	−691.128	−0.0663409
$478$	0	0
$479$	−5116.57	−0.488063	−0.244031	−	0.969767i	$-0.578470\pi$
$479$	−5116.57	−0.488063	−0.244031	+	0.969767i	$0.578470\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−9944.95	−0.936875
$484$	0	0
$485$	−4422.05	−0.414010
$486$	0	0
$487$	8481.62	0.789197	0.394598	−	0.918854i	$-0.370884\pi$
$487$	8481.62	0.789197	0.394598	+	0.918854i	$0.370884\pi$
$488$	0	0
$489$	4233.11	0.391468
$490$	0	0
$491$	18962.0	1.74286	0.871430	−	0.490520i	$-0.163193\pi$
$491$	18962.0	1.74286	0.871430	+	0.490520i	$0.163193\pi$
$492$	0	0
$493$	−1562.15	−0.142709
$494$	0	0
$495$	2529.56	0.229688
$496$	0	0
$497$	−11869.9	−1.07130
$498$	0	0
$499$	−657.460	−0.0589818	−0.0294909	−	0.999565i	$-0.509389\pi$
$499$	−657.460	−0.0589818	−0.0294909	+	0.999565i	$0.509389\pi$
$500$	0	0
$501$	−2299.75	−0.205080
$502$	0	0
$503$	17719.0	1.57068	0.785341	−	0.619063i	$-0.212487\pi$
$503$	17719.0	1.57068	0.785341	+	0.619063i	$0.212487\pi$
$504$	0	0
$505$	1296.47	0.114242
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1703.48	0.148341	0.0741703	−	0.997246i	$-0.476369\pi$
$509$	1703.48	0.148341	0.0741703	+	0.997246i	$0.476369\pi$
$510$	0	0
$511$	1836.95	0.159025
$512$	0	0
$513$	−945.882	−0.0814069
$514$	0	0
$515$	14097.3	1.20621
$516$	0	0
$517$	−14458.6	−1.22996
$518$	0	0
$519$	5507.88	0.465837
$520$	0	0
$521$	15954.8	1.34163	0.670817	−	0.741623i	$-0.265943\pi$
$521$	15954.8	1.34163	0.670817	+	0.741623i	$0.265943\pi$
$522$	0	0
$523$	3169.79	0.265020	0.132510	−	0.991182i	$-0.457696\pi$
$523$	3169.79	0.265020	0.132510	+	0.991182i	$0.457696\pi$
$524$	0	0
$525$	−3552.11	−0.295289
$526$	0	0
$527$	−4112.87	−0.339961
$528$	0	0
$529$	15992.1	1.31438
$530$	0	0
$531$	647.449	0.0529131
$532$	0	0
$533$	0	0
$534$	0	0
$535$	4367.01	0.352901
$536$	0	0
$537$	−12379.3	−0.994795
$538$	0	0
$539$	−1646.43	−0.131571
$540$	0	0
$541$	−17117.5	−1.36033	−0.680166	−	0.733058i	$-0.738092\pi$
$541$	−17117.5	−1.36033	−0.680166	+	0.733058i	$0.738092\pi$
$542$	0	0
$543$	5758.67	0.455117
$544$	0	0
$545$	8377.21	0.658423
$546$	0	0
$547$	−6769.75	−0.529166	−0.264583	−	0.964363i	$-0.585234\pi$
$547$	−6769.75	−0.529166	−0.264583	+	0.964363i	$0.585234\pi$
$548$	0	0
$549$	4498.74	0.349729
$550$	0	0
$551$	−4433.15	−0.342756
$552$	0	0
$553$	21789.1	1.67553
$554$	0	0
$555$	940.776	0.0719526
$556$	0	0
$557$	−21507.3	−1.63607	−0.818036	−	0.575167i	$-0.804937\pi$
$557$	−21507.3	−1.63607	−0.818036	+	0.575167i	$0.804937\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1290.44	0.0971168
$562$	0	0
$563$	5873.05	0.439644	0.219822	−	0.975540i	$-0.429452\pi$
$563$	5873.05	0.439644	0.219822	+	0.975540i	$0.429452\pi$
$564$	0	0
$565$	1970.59	0.146732
$566$	0	0
$567$	1600.14	0.118517
$568$	0	0
$569$	19169.9	1.41238	0.706189	−	0.708023i	$-0.250413\pi$
$569$	19169.9	1.41238	0.706189	+	0.708023i	$0.250413\pi$
$570$	0	0
$571$	16025.4	1.17450	0.587251	−	0.809405i	$-0.300210\pi$
$571$	16025.4	1.17450	0.587251	+	0.809405i	$0.300210\pi$
$572$	0	0
$573$	−928.794	−0.0677154
$574$	0	0
$575$	10057.8	0.729459
$576$	0	0
$577$	5087.97	0.367097	0.183549	−	0.983011i	$-0.441242\pi$
$577$	5087.97	0.367097	0.183549	+	0.983011i	$0.441242\pi$
$578$	0	0
$579$	6431.91	0.461660
$580$	0	0
$581$	4375.25	0.312420
$582$	0	0
$583$	2675.79	0.190085
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5047.27	−0.354894	−0.177447	−	0.984130i	$-0.556784\pi$
$587$	−5047.27	−0.354894	−0.177447	+	0.984130i	$0.556784\pi$
$588$	0	0
$589$	−11671.8	−0.816514
$590$	0	0
$591$	4115.30	0.286431
$592$	0	0
$593$	6135.16	0.424858	0.212429	−	0.977177i	$-0.431863\pi$
$593$	6135.16	0.424858	0.212429	+	0.977177i	$0.431863\pi$
$594$	0	0
$595$	1967.08	0.135533
$596$	0	0
$597$	11139.9	0.763697
$598$	0	0
$599$	−20513.8	−1.39928	−0.699642	−	0.714494i	$-0.746657\pi$
$599$	−20513.8	−1.39928	−0.699642	+	0.714494i	$0.746657\pi$
$600$	0	0
$601$	26100.9	1.77151	0.885754	−	0.464155i	$-0.153642\pi$
$601$	26100.9	1.77151	0.885754	+	0.464155i	$0.153642\pi$
$602$	0	0
$603$	3483.10	0.235228
$604$	0	0
$605$	942.581	0.0633411
$606$	0	0
$607$	11846.4	0.792145	0.396072	−	0.918219i	$-0.370373\pi$
$607$	11846.4	0.792145	0.396072	+	0.918219i	$0.370373\pi$
$608$	0	0
$609$	7499.50	0.499007
$610$	0	0
$611$	0	0
$612$	0	0
$613$	17738.7	1.16878	0.584389	−	0.811474i	$-0.301334\pi$
$613$	17738.7	1.16878	0.584389	+	0.811474i	$0.301334\pi$
$614$	0	0
$615$	−4243.47	−0.278233
$616$	0	0
$617$	−18888.0	−1.23242	−0.616210	−	0.787582i	$-0.711333\pi$
$617$	−18888.0	−1.23242	−0.616210	+	0.787582i	$0.711333\pi$
$618$	0	0
$619$	18402.9	1.19495	0.597474	−	0.801888i	$-0.296171\pi$
$619$	18402.9	1.19495	0.597474	+	0.801888i	$0.296171\pi$
$620$	0	0
$621$	−4530.78	−0.292776
$622$	0	0
$623$	975.384	0.0627254
$624$	0	0
$625$	−4540.49	−0.290591
$626$	0	0
$627$	3662.10	0.233254
$628$	0	0
$629$	479.931	0.0304231
$630$	0	0
$631$	−4368.45	−0.275603	−0.137801	−	0.990460i	$-0.544004\pi$
$631$	−4368.45	−0.275603	−0.137801	+	0.990460i	$0.544004\pi$
$632$	0	0
$633$	−4067.00	−0.255369
$634$	0	0
$635$	−15536.2	−0.970924
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−5407.76	−0.334785
$640$	0	0
$641$	26554.6	1.63626	0.818130	−	0.575034i	$-0.195011\pi$
$641$	26554.6	1.63626	0.818130	+	0.575034i	$0.195011\pi$
$642$	0	0
$643$	11827.1	0.725374	0.362687	−	0.931911i	$-0.381859\pi$
$643$	11827.1	0.725374	0.362687	+	0.931911i	$0.381859\pi$
$644$	0	0
$645$	−3540.40	−0.216129
$646$	0	0
$647$	22800.5	1.38544	0.692721	−	0.721206i	$-0.256412\pi$
$647$	22800.5	1.38544	0.692721	+	0.721206i	$0.256412\pi$
$648$	0	0
$649$	−2506.67	−0.151611
$650$	0	0
$651$	19745.0	1.18873
$652$	0	0
$653$	8882.03	0.532283	0.266141	−	0.963934i	$-0.414251\pi$
$653$	8882.03	0.532283	0.266141	+	0.963934i	$0.414251\pi$
$654$	0	0
$655$	20633.4	1.23086
$656$	0	0
$657$	836.890	0.0496959
$658$	0	0
$659$	−27773.8	−1.64175	−0.820875	−	0.571108i	$-0.806514\pi$
$659$	−27773.8	−1.64175	−0.820875	+	0.571108i	$0.806514\pi$
$660$	0	0
$661$	−9585.52	−0.564044	−0.282022	−	0.959408i	$-0.591005\pi$
$661$	−9585.52	−0.564044	−0.282022	+	0.959408i	$0.591005\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5582.30	0.325522
$666$	0	0
$667$	−21234.8	−1.23271
$668$	0	0
$669$	−2249.00	−0.129972
$670$	0	0
$671$	−17417.4	−1.00207
$672$	0	0
$673$	−19945.8	−1.14243	−0.571215	−	0.820800i	$-0.693528\pi$
$673$	−19945.8	−1.14243	−0.571215	+	0.820800i	$0.693528\pi$
$674$	0	0
$675$	−1618.29	−0.0922787
$676$	0	0
$677$	−26567.3	−1.50822	−0.754109	−	0.656749i	$-0.771931\pi$
$677$	−26567.3	−1.50822	−0.754109	+	0.656749i	$0.771931\pi$
$678$	0	0
$679$	10830.0	0.612100
$680$	0	0
$681$	12631.1	0.710755
$682$	0	0
$683$	−11543.7	−0.646718	−0.323359	−	0.946276i	$-0.604812\pi$
$683$	−11543.7	−0.646718	−0.323359	+	0.946276i	$0.604812\pi$
$684$	0	0
$685$	−22911.6	−1.27797
$686$	0	0
$687$	11844.3	0.657773
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−11795.5	−0.649378	−0.324689	−	0.945821i	$-0.605260\pi$
$691$	−11795.5	−0.649378	−0.324689	+	0.945821i	$0.605260\pi$
$692$	0	0
$693$	−6195.12	−0.339586
$694$	0	0
$695$	−5235.85	−0.285766
$696$	0	0
$697$	−2164.78	−0.117643
$698$	0	0
$699$	1.89798	0.000102701 0
$700$	0	0
$701$	−24573.6	−1.32401	−0.662006	−	0.749499i	$-0.730294\pi$
$701$	−24573.6	−1.32401	−0.662006	+	0.749499i	$0.730294\pi$
$702$	0	0
$703$	1361.98	0.0730697
$704$	0	0
$705$	−10041.1	−0.536410
$706$	0	0
$707$	−3175.16	−0.168903
$708$	0	0
$709$	693.306	0.0367245	0.0183622	−	0.999831i	$-0.494155\pi$
$709$	693.306	0.0367245	0.0183622	+	0.999831i	$0.494155\pi$
$710$	0	0
$711$	9926.83	0.523608
$712$	0	0
$713$	−55907.8	−2.93656
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7211.65	−0.375626
$718$	0	0
$719$	1906.40	0.0988826	0.0494413	−	0.998777i	$-0.484256\pi$
$719$	1906.40	0.0988826	0.0494413	+	0.998777i	$0.484256\pi$
$720$	0	0
$721$	−34525.4	−1.78335
$722$	0	0
$723$	−9005.23	−0.463220
$724$	0	0
$725$	−7584.60	−0.388531
$726$	0	0
$727$	30753.4	1.56889	0.784444	−	0.620199i	$-0.212948\pi$
$727$	30753.4	1.56889	0.784444	+	0.620199i	$0.212948\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−1806.11	−0.0913838
$732$	0	0
$733$	−20777.2	−1.04696	−0.523482	−	0.852037i	$-0.675367\pi$
$733$	−20777.2	−1.04696	−0.523482	+	0.852037i	$0.675367\pi$
$734$	0	0
$735$	−1143.40	−0.0573808
$736$	0	0
$737$	−13485.2	−0.673996
$738$	0	0
$739$	30233.4	1.50494	0.752472	−	0.658625i	$-0.228862\pi$
$739$	30233.4	1.50494	0.752472	+	0.658625i	$0.228862\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14761.6	0.728868	0.364434	−	0.931229i	$-0.381262\pi$
$743$	14761.6	0.728868	0.364434	+	0.931229i	$0.381262\pi$
$744$	0	0
$745$	−17384.3	−0.854916
$746$	0	0
$747$	1993.30	0.0976321
$748$	0	0
$749$	−10695.2	−0.521754
$750$	0	0
$751$	37659.9	1.82987	0.914933	−	0.403607i	$-0.132244\pi$
$751$	37659.9	1.82987	0.914933	+	0.403607i	$0.132244\pi$
$752$	0	0
$753$	13836.5	0.669628
$754$	0	0
$755$	−25374.3	−1.22313
$756$	0	0
$757$	−35506.0	−1.70474	−0.852371	−	0.522938i	$-0.824836\pi$
$757$	−35506.0	−1.70474	−0.852371	+	0.522938i	$0.824836\pi$
$758$	0	0
$759$	17541.5	0.838886
$760$	0	0
$761$	23032.0	1.09712	0.548561	−	0.836110i	$-0.315176\pi$
$761$	23032.0	1.09712	0.548561	+	0.836110i	$0.315176\pi$
$762$	0	0
$763$	−20516.5	−0.973457
$764$	0	0
$765$	896.174	0.0423546
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−30946.3	−1.45117	−0.725587	−	0.688131i	$-0.758432\pi$
$769$	−30946.3	−1.45117	−0.725587	+	0.688131i	$0.758432\pi$
$770$	0	0
$771$	9308.99	0.434831
$772$	0	0
$773$	−20957.7	−0.975156	−0.487578	−	0.873079i	$-0.662120\pi$
$773$	−20957.7	−0.975156	−0.487578	+	0.873079i	$0.662120\pi$
$774$	0	0
$775$	−19969.0	−0.925559
$776$	0	0
$777$	−2304.04	−0.106380
$778$	0	0
$779$	−6143.35	−0.282553
$780$	0	0
$781$	20936.8	0.959254
$782$	0	0
$783$	3416.67	0.155941
$784$	0	0
$785$	−12846.8	−0.584103
$786$	0	0
$787$	−13418.7	−0.607783	−0.303891	−	0.952707i	$-0.598286\pi$
$787$	−13418.7	−0.607783	−0.303891	+	0.952707i	$0.598286\pi$
$788$	0	0
$789$	8992.44	0.405753
$790$	0	0
$791$	−4826.14	−0.216938
$792$	0	0
$793$	0	0
$794$	0	0
$795$	1858.25	0.0829000
$796$	0	0
$797$	−7004.96	−0.311328	−0.155664	−	0.987810i	$-0.549752\pi$
$797$	−7004.96	−0.311328	−0.155664	+	0.987810i	$0.549752\pi$
$798$	0	0
$799$	−5122.41	−0.226806
$800$	0	0
$801$	444.372	0.0196019
$802$	0	0
$803$	−3240.12	−0.142393
$804$	0	0
$805$	26739.2	1.17073
$806$	0	0
$807$	26114.2	1.13911
$808$	0	0
$809$	−14787.5	−0.642645	−0.321322	−	0.946970i	$-0.604127\pi$
$809$	−14787.5	−0.642645	−0.321322	+	0.946970i	$0.604127\pi$
$810$	0	0
$811$	19156.7	0.829450	0.414725	−	0.909947i	$-0.363878\pi$
$811$	19156.7	0.829450	0.414725	+	0.909947i	$0.363878\pi$
$812$	0	0
$813$	20395.8	0.879842
$814$	0	0
$815$	−11381.7	−0.489181
$816$	0	0
$817$	−5125.50	−0.219484
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17292.8	−0.735108	−0.367554	−	0.930002i	$-0.619805\pi$
$821$	−17292.8	−0.735108	−0.367554	+	0.930002i	$0.619805\pi$
$822$	0	0
$823$	−41414.6	−1.75410	−0.877048	−	0.480403i	$-0.840491\pi$
$823$	−41414.6	−1.75410	−0.877048	+	0.480403i	$0.840491\pi$
$824$	0	0
$825$	6265.41	0.264404
$826$	0	0
$827$	−39076.5	−1.64308	−0.821538	−	0.570153i	$-0.806884\pi$
$827$	−39076.5	−1.64308	−0.821538	+	0.570153i	$0.806884\pi$
$828$	0	0
$829$	549.901	0.0230384	0.0115192	−	0.999934i	$-0.496333\pi$
$829$	549.901	0.0230384	0.0115192	+	0.999934i	$0.496333\pi$
$830$	0	0
$831$	5831.18	0.243419
$832$	0	0
$833$	−583.299	−0.0242618
$834$	0	0
$835$	6183.40	0.256270
$836$	0	0
$837$	8995.53	0.371483
$838$	0	0
$839$	−12455.0	−0.512507	−0.256253	−	0.966610i	$-0.582488\pi$
$839$	−12455.0	−0.512507	−0.256253	+	0.966610i	$0.582488\pi$
$840$	0	0
$841$	−8375.79	−0.343425
$842$	0	0
$843$	17021.6	0.695440
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2308.46	−0.0936478
$848$	0	0
$849$	−24565.6	−0.993040
$850$	0	0
$851$	6523.88	0.262792
$852$	0	0
$853$	−44057.5	−1.76846	−0.884232	−	0.467047i	$-0.845318\pi$
$853$	−44057.5	−1.76846	−0.884232	+	0.467047i	$0.845318\pi$
$854$	0	0
$855$	2543.22	0.101727
$856$	0	0
$857$	−14600.8	−0.581978	−0.290989	−	0.956726i	$-0.593984\pi$
$857$	−14600.8	−0.581978	−0.290989	+	0.956726i	$0.593984\pi$
$858$	0	0
$859$	−3371.92	−0.133933	−0.0669664	−	0.997755i	$-0.521332\pi$
$859$	−3371.92	−0.133933	−0.0669664	+	0.997755i	$0.521332\pi$
$860$	0	0
$861$	10392.6	0.411358
$862$	0	0
$863$	30606.4	1.20725	0.603623	−	0.797270i	$-0.293723\pi$
$863$	30606.4	1.20725	0.603623	+	0.797270i	$0.293723\pi$
$864$	0	0
$865$	−14809.2	−0.582113
$866$	0	0
$867$	−14281.8	−0.559442
$868$	0	0
$869$	−38432.9	−1.50028
$870$	0	0
$871$	0	0
$872$	0	0
$873$	4933.99	0.191283
$874$	0	0
$875$	29468.8	1.13855
$876$	0	0
$877$	−7920.58	−0.304970	−0.152485	−	0.988306i	$-0.548728\pi$
$877$	−7920.58	−0.304970	−0.152485	+	0.988306i	$0.548728\pi$
$878$	0	0
$879$	−25407.9	−0.974958
$880$	0	0
$881$	42779.1	1.63594	0.817972	−	0.575259i	$-0.195099\pi$
$881$	42779.1	1.63594	0.817972	+	0.575259i	$0.195099\pi$
$882$	0	0
$883$	−28199.6	−1.07474	−0.537368	−	0.843348i	$-0.680581\pi$
$883$	−28199.6	−1.07474	−0.537368	+	0.843348i	$0.680581\pi$
$884$	0	0
$885$	−1740.81	−0.0661206
$886$	0	0
$887$	−21813.1	−0.825717	−0.412859	−	0.910795i	$-0.635470\pi$
$887$	−21813.1	−0.825717	−0.412859	+	0.910795i	$0.635470\pi$
$888$	0	0
$889$	38049.6	1.43548
$890$	0	0
$891$	−2822.41	−0.106122
$892$	0	0
$893$	−14536.7	−0.544738
$894$	0	0
$895$	33284.5	1.24310
$896$	0	0
$897$	0	0
$898$	0	0
$899$	42160.2	1.56409
$900$	0	0
$901$	947.978	0.0350519
$902$	0	0
$903$	8670.74	0.319539
$904$	0	0
$905$	−15483.5	−0.568717
$906$	0	0
$907$	−15282.8	−0.559488	−0.279744	−	0.960075i	$-0.590250\pi$
$907$	−15282.8	−0.559488	−0.279744	+	0.960075i	$0.590250\pi$
$908$	0	0
$909$	−1446.56	−0.0527826
$910$	0	0
$911$	−3870.85	−0.140776	−0.0703880	−	0.997520i	$-0.522424\pi$
$911$	−3870.85	−0.140776	−0.0703880	+	0.997520i	$0.522424\pi$
$912$	0	0
$913$	−7717.31	−0.279743
$914$	0	0
$915$	−12095.9	−0.437024
$916$	0	0
$917$	−50533.0	−1.81979
$918$	0	0
$919$	−39980.2	−1.43507	−0.717533	−	0.696524i	$-0.754729\pi$
$919$	−39980.2	−1.43507	−0.717533	+	0.696524i	$0.754729\pi$
$920$	0	0
$921$	24548.2	0.878273
$922$	0	0
$923$	0	0
$924$	0	0
$925$	2330.18	0.0828281
$926$	0	0
$927$	−15729.3	−0.557301
$928$	0	0
$929$	−21680.0	−0.765661	−0.382830	−	0.923819i	$-0.625051\pi$
$929$	−21680.0	−0.765661	−0.382830	+	0.923819i	$0.625051\pi$
$930$	0	0
$931$	−1655.32	−0.0582717
$932$	0	0
$933$	−26339.7	−0.924247
$934$	0	0
$935$	−3469.65	−0.121358
$936$	0	0
$937$	27340.5	0.953228	0.476614	−	0.879113i	$-0.341864\pi$
$937$	27340.5	0.953228	0.476614	+	0.879113i	$0.341864\pi$
$938$	0	0
$939$	17151.3	0.596073
$940$	0	0
$941$	−44214.3	−1.53172	−0.765859	−	0.643009i	$-0.777686\pi$
$941$	−44214.3	−1.53172	−0.765859	+	0.643009i	$0.777686\pi$
$942$	0	0
$943$	−29426.7	−1.01619
$944$	0	0
$945$	−4302.33	−0.148100
$946$	0	0
$947$	−46558.7	−1.59763	−0.798814	−	0.601579i	$-0.794539\pi$
$947$	−46558.7	−1.59763	−0.798814	+	0.601579i	$0.794539\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−9795.87	−0.334020
$952$	0	0
$953$	−24497.2	−0.832680	−0.416340	−	0.909209i	$-0.636687\pi$
$953$	−24497.2	−0.832680	−0.416340	+	0.909209i	$0.636687\pi$
$954$	0	0
$955$	2497.27	0.0846176
$956$	0	0
$957$	−13228.0	−0.446815
$958$	0	0
$959$	56112.6	1.88944
$960$	0	0
$961$	81209.9	2.72599
$962$	0	0
$963$	−4872.58	−0.163049
$964$	0	0
$965$	−17293.7	−0.576894
$966$	0	0
$967$	30838.8	1.02555	0.512777	−	0.858522i	$-0.328617\pi$
$967$	30838.8	1.02555	0.512777	+	0.858522i	$0.328617\pi$
$968$	0	0
$969$	1297.41	0.0430121
$970$	0	0
$971$	−39031.9	−1.29000	−0.645002	−	0.764181i	$-0.723143\pi$
$971$	−39031.9	−1.29000	−0.645002	+	0.764181i	$0.723143\pi$
$972$	0	0
$973$	12823.0	0.422495
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−38599.7	−1.26398	−0.631992	−	0.774975i	$-0.717762\pi$
$977$	−38599.7	−1.26398	−0.631992	+	0.774975i	$0.717762\pi$
$978$	0	0
$979$	−1720.44	−0.0561649
$980$	0	0
$981$	−9347.04	−0.304208
$982$	0	0
$983$	−31291.4	−1.01530	−0.507651	−	0.861563i	$-0.669486\pi$
$983$	−31291.4	−1.01530	−0.507651	+	0.861563i	$0.669486\pi$
$984$	0	0
$985$	−11064.9	−0.357926
$986$	0	0
$987$	24591.5	0.793066
$988$	0	0
$989$	−24551.2	−0.789365
$990$	0	0
$991$	−7301.25	−0.234038	−0.117019	−	0.993130i	$-0.537334\pi$
$991$	−7301.25	−0.234038	−0.117019	+	0.993130i	$0.537334\pi$
$992$	0	0
$993$	−4297.86	−0.137350
$994$	0	0
$995$	−29952.2	−0.954322
$996$	0	0
$997$	47982.5	1.52419	0.762097	−	0.647462i	$-0.224170\pi$
$997$	47982.5	1.52419	0.762097	+	0.647462i	$0.224170\pi$
$998$	0	0
$999$	−1049.69	−0.0332439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.4.a.s.1.3	yes	9
13.5	odd	4		2028.4.b.l.337.12		18
13.8	odd	4		2028.4.b.l.337.7		18
13.12	even	2		2028.4.a.r.1.7	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2028.4.a.r.1.7	✓	9	13.12	even	2
2028.4.a.s.1.3	yes	9	1.1	even	1	trivial
2028.4.b.l.337.7		18	13.8	odd	4
2028.4.b.l.337.12		18	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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -8.06618 q^{5} +19.7548 q^{7} +9.00000 q^{9} -34.8446 q^{11} -24.1985 q^{15} -12.3447 q^{17} -35.0327 q^{19} +59.2643 q^{21} -167.807 q^{23} -59.9368 q^{25} +27.0000 q^{27} +126.543 q^{29} +333.168 q^{31} -104.534 q^{33} -159.345 q^{35} -38.8774 q^{37} +175.361 q^{41} +146.306 q^{43} -72.5956 q^{45} +414.946 q^{47} +47.2508 q^{49} -37.0342 q^{51} -76.7920 q^{53} +281.063 q^{55} -105.098 q^{57} +71.9387 q^{59} +499.860 q^{61} +177.793 q^{63} +387.011 q^{67} -503.420 q^{69} -600.863 q^{71} +92.9877 q^{73} -179.810 q^{75} -688.346 q^{77} +1102.98 q^{79} +81.0000 q^{81} +221.478 q^{83} +99.5749 q^{85} +379.630 q^{87} +49.3746 q^{89} +999.504 q^{93} +282.580 q^{95} +548.221 q^{97} -313.601 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 27 q^{3} + 11 q^{5} + 37 q^{7} + 81 q^{9} + 61 q^{11} + 33 q^{15} + 104 q^{17} + 328 q^{19} + 111 q^{21} + 54 q^{23} + 376 q^{25} + 243 q^{27} + 61 q^{29} + 333 q^{31} + 183 q^{33} - 338 q^{35} + 144 q^{37}+ \cdots + 549 q^{99}+O(q^{100}) \) 9 * q + 27 * q^3 + 11 * q^5 + 37 * q^7 + 81 * q^9 + 61 * q^11 + 33 * q^15 + 104 * q^17 + 328 * q^19 + 111 * q^21 + 54 * q^23 + 376 * q^25 + 243 * q^27 + 61 * q^29 + 333 * q^31 + 183 * q^33 - 338 * q^35 + 144 * q^37 + 304 * q^41 - 248 * q^43 + 99 * q^45 + 1034 * q^47 + 1142 * q^49 + 312 * q^51 - 737 * q^53 - 394 * q^55 + 984 * q^57 + 1115 * q^59 + 440 * q^61 + 333 * q^63 + 2046 * q^67 + 162 * q^69 + 1548 * q^71 + 2407 * q^73 + 1128 * q^75 - 354 * q^77 - 1361 * q^79 + 729 * q^81 + 3365 * q^83 + 1000 * q^85 + 183 * q^87 + 1412 * q^89 + 999 * q^93 - 1532 * q^95 + 3443 * q^97 + 549 * q^99

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