LMFDB - Embedded newform 2028.4.a.n.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:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 122x^{2} - 594x - 689 $ x^4 - 122x^2 - 594x - 689
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$13.0917$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +5.67342 q^{5} -19.5100 q^{7} +9.00000 q^{9} -5.57917 q^{11} +17.0203 q^{15} -105.422 q^{17} -135.422 q^{19} -58.5299 q^{21} +45.5398 q^{23} -92.8123 q^{25} +27.0000 q^{27} +226.357 q^{29} +31.4566 q^{31} -16.7375 q^{33} -110.688 q^{35} +90.2951 q^{37} -72.6290 q^{41} +375.453 q^{43} +51.0608 q^{45} +637.171 q^{47} +37.6383 q^{49} -316.267 q^{51} +578.656 q^{53} -31.6530 q^{55} -406.266 q^{57} -76.0867 q^{59} +756.520 q^{61} -175.590 q^{63} -235.444 q^{67} +136.619 q^{69} +314.806 q^{71} -407.548 q^{73} -278.437 q^{75} +108.849 q^{77} +185.914 q^{79} +81.0000 q^{81} +694.924 q^{83} -598.106 q^{85} +679.072 q^{87} -13.7380 q^{89} +94.3698 q^{93} -768.307 q^{95} +652.689 q^{97} -50.2125 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 7 q^{5} + 11 q^{7} + 36 q^{9} - 20 q^{11} + 21 q^{15} + 7 q^{17} + 18 q^{19} + 33 q^{21} - 60 q^{23} + 67 q^{25} + 108 q^{27} + 75 q^{29} + 87 q^{31} - 60 q^{33} + 556 q^{35} - 51 q^{37}+ \cdots - 180 q^{99}+O(q^{100}) $ 4 * q + 12 * q^3 + 7 * q^5 + 11 * q^7 + 36 * q^9 - 20 * q^11 + 21 * q^15 + 7 * q^17 + 18 * q^19 + 33 * q^21 - 60 * q^23 + 67 * q^25 + 108 * q^27 + 75 * q^29 + 87 * q^31 - 60 * q^33 + 556 * q^35 - 51 * q^37 + 271 * q^41 - q^43 + 63 * q^45 - 32 * q^47 + 153 * q^49 + 21 * q^51 + 757 * q^53 + 626 * q^55 + 54 * q^57 - 610 * q^59 + 1166 * q^61 + 99 * q^63 + 1191 * q^67 - 180 * q^69 + 980 * q^71 + 482 * q^73 + 201 * q^75 + 406 * q^77 + 2225 * q^79 + 324 * q^81 - 1518 * q^83 - 2603 * q^85 + 225 * q^87 + 1078 * q^89 + 261 * q^93 + 154 * q^95 + 2527 * q^97 - 180 * 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
$5$	5.67342	0.507446	0.253723	−	0.967277i	$-0.418345\pi$
$5$	5.67342	0.507446	0.253723	+	0.967277i	$0.418345\pi$
$6$	0	0
$7$	−19.5100	−1.05344	−0.526719	−	0.850039i	$-0.676578\pi$
$7$	−19.5100	−1.05344	−0.526719	+	0.850039i	$0.676578\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−5.57917	−0.152926	−0.0764629	−	0.997072i	$-0.524363\pi$
$11$	−5.57917	−0.152926	−0.0764629	+	0.997072i	$0.524363\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	17.0203	0.292974
$16$	0	0
$17$	−105.422	−1.50404	−0.752021	−	0.659140i	$-0.770921\pi$
$17$	−105.422	−1.50404	−0.752021	+	0.659140i	$0.770921\pi$
$18$	0	0
$19$	−135.422	−1.63516	−0.817578	−	0.575818i	$-0.804684\pi$
$19$	−135.422	−1.63516	−0.817578	+	0.575818i	$0.804684\pi$
$20$	0	0
$21$	−58.5299	−0.608203
$22$	0	0
$23$	45.5398	0.412856	0.206428	−	0.978462i	$-0.433816\pi$
$23$	45.5398	0.412856	0.206428	+	0.978462i	$0.433816\pi$
$24$	0	0
$25$	−92.8123	−0.742498
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	226.357	1.44943	0.724715	−	0.689048i	$-0.241971\pi$
$29$	226.357	1.44943	0.724715	+	0.689048i	$0.241971\pi$
$30$	0	0
$31$	31.4566	0.182251	0.0911254	−	0.995839i	$-0.470954\pi$
$31$	31.4566	0.182251	0.0911254	+	0.995839i	$0.470954\pi$
$32$	0	0
$33$	−16.7375	−0.0882917
$34$	0	0
$35$	−110.688	−0.534564
$36$	0	0
$37$	90.2951	0.401201	0.200600	−	0.979673i	$-0.435711\pi$
$37$	90.2951	0.401201	0.200600	+	0.979673i	$0.435711\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−72.6290	−0.276652	−0.138326	−	0.990387i	$-0.544172\pi$
$41$	−72.6290	−0.276652	−0.138326	+	0.990387i	$0.544172\pi$
$42$	0	0
$43$	375.453	1.33154	0.665768	−	0.746158i	$-0.268104\pi$
$43$	375.453	1.33154	0.665768	+	0.746158i	$0.268104\pi$
$44$	0	0
$45$	51.0608	0.169149
$46$	0	0
$47$	637.171	1.97747	0.988734	−	0.149682i	$-0.0478251\pi$
$47$	637.171	1.97747	0.988734	+	0.149682i	$0.0478251\pi$
$48$	0	0
$49$	37.6383	0.109733
$50$	0	0
$51$	−316.267	−0.868358
$52$	0	0
$53$	578.656	1.49971	0.749854	−	0.661603i	$-0.230123\pi$
$53$	578.656	1.49971	0.749854	+	0.661603i	$0.230123\pi$
$54$	0	0
$55$	−31.6530	−0.0776016
$56$	0	0
$57$	−406.266	−0.944057
$58$	0	0
$59$	−76.0867	−0.167892	−0.0839461	−	0.996470i	$-0.526752\pi$
$59$	−76.0867	−0.167892	−0.0839461	+	0.996470i	$0.526752\pi$
$60$	0	0
$61$	756.520	1.58791	0.793954	−	0.607978i	$-0.208019\pi$
$61$	756.520	1.58791	0.793954	+	0.607978i	$0.208019\pi$
$62$	0	0
$63$	−175.590	−0.351146
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−235.444	−0.429315	−0.214657	−	0.976689i	$-0.568864\pi$
$67$	−235.444	−0.429315	−0.214657	+	0.976689i	$0.568864\pi$
$68$	0	0
$69$	136.619	0.238363
$70$	0	0
$71$	314.806	0.526206	0.263103	−	0.964768i	$-0.415254\pi$
$71$	314.806	0.526206	0.263103	+	0.964768i	$0.415254\pi$
$72$	0	0
$73$	−407.548	−0.653423	−0.326711	−	0.945124i	$-0.605941\pi$
$73$	−407.548	−0.653423	−0.326711	+	0.945124i	$0.605941\pi$
$74$	0	0
$75$	−278.437	−0.428681
$76$	0	0
$77$	108.849	0.161098
$78$	0	0
$79$	185.914	0.264772	0.132386	−	0.991198i	$-0.457736\pi$
$79$	185.914	0.264772	0.132386	+	0.991198i	$0.457736\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	694.924	0.919010	0.459505	−	0.888175i	$-0.348027\pi$
$83$	694.924	0.919010	0.459505	+	0.888175i	$0.348027\pi$
$84$	0	0
$85$	−598.106	−0.763220
$86$	0	0
$87$	679.072	0.836829
$88$	0	0
$89$	−13.7380	−0.0163620	−0.00818102	−	0.999967i	$-0.502604\pi$
$89$	−13.7380	−0.0163620	−0.00818102	+	0.999967i	$0.502604\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	94.3698	0.105223
$94$	0	0
$95$	−768.307	−0.829754
$96$	0	0
$97$	652.689	0.683201	0.341601	−	0.939845i	$-0.389031\pi$
$97$	652.689	0.683201	0.341601	+	0.939845i	$0.389031\pi$
$98$	0	0
$99$	−50.2125	−0.0509752
$100$	0	0
$101$	383.876	0.378189	0.189095	−	0.981959i	$-0.439445\pi$
$101$	383.876	0.378189	0.189095	+	0.981959i	$0.439445\pi$
$102$	0	0
$103$	970.879	0.928772	0.464386	−	0.885633i	$-0.346275\pi$
$103$	970.879	0.928772	0.464386	+	0.885633i	$0.346275\pi$
$104$	0	0
$105$	−332.065	−0.308630
$106$	0	0
$107$	−902.438	−0.815346	−0.407673	−	0.913128i	$-0.633660\pi$
$107$	−902.438	−0.815346	−0.407673	+	0.913128i	$0.633660\pi$
$108$	0	0
$109$	−630.574	−0.554111	−0.277055	−	0.960854i	$-0.589359\pi$
$109$	−630.574	−0.554111	−0.277055	+	0.960854i	$0.589359\pi$
$110$	0	0
$111$	270.885	0.231633
$112$	0	0
$113$	−516.418	−0.429916	−0.214958	−	0.976623i	$-0.568961\pi$
$113$	−516.418	−0.429916	−0.214958	+	0.976623i	$0.568961\pi$
$114$	0	0
$115$	258.366	0.209503
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2056.79	1.58441
$120$	0	0
$121$	−1299.87	−0.976614
$122$	0	0
$123$	−217.887	−0.159725
$124$	0	0
$125$	−1235.74	−0.884224
$126$	0	0
$127$	−165.162	−0.115400	−0.0576999	−	0.998334i	$-0.518377\pi$
$127$	−165.162	−0.115400	−0.0576999	+	0.998334i	$0.518377\pi$
$128$	0	0
$129$	1126.36	0.768763
$130$	0	0
$131$	2676.77	1.78527	0.892635	−	0.450779i	$-0.148854\pi$
$131$	2676.77	1.78527	0.892635	+	0.450779i	$0.148854\pi$
$132$	0	0
$133$	2642.08	1.72254
$134$	0	0
$135$	153.182	0.0976581
$136$	0	0
$137$	−1752.84	−1.09310	−0.546552	−	0.837425i	$-0.684060\pi$
$137$	−1752.84	−1.09310	−0.546552	+	0.837425i	$0.684060\pi$
$138$	0	0
$139$	1180.33	0.720249	0.360124	−	0.932904i	$-0.382734\pi$
$139$	1180.33	0.720249	0.360124	+	0.932904i	$0.382734\pi$
$140$	0	0
$141$	1911.51	1.14169
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1284.22	0.735508
$146$	0	0
$147$	112.915	0.0633542
$148$	0	0
$149$	3391.51	1.86472	0.932361	−	0.361528i	$-0.117745\pi$
$149$	3391.51	1.86472	0.932361	+	0.361528i	$0.117745\pi$
$150$	0	0
$151$	−51.1288	−0.0275550	−0.0137775	−	0.999905i	$-0.504386\pi$
$151$	−51.1288	−0.0275550	−0.0137775	+	0.999905i	$0.504386\pi$
$152$	0	0
$153$	−948.802	−0.501347
$154$	0	0
$155$	178.467	0.0924825
$156$	0	0
$157$	−107.766	−0.0547812	−0.0273906	−	0.999625i	$-0.508720\pi$
$157$	−107.766	−0.0547812	−0.0273906	+	0.999625i	$0.508720\pi$
$158$	0	0
$159$	1735.97	0.865857
$160$	0	0
$161$	−888.479	−0.434919
$162$	0	0
$163$	−1486.47	−0.714291	−0.357146	−	0.934049i	$-0.616250\pi$
$163$	−1486.47	−0.714291	−0.357146	+	0.934049i	$0.616250\pi$
$164$	0	0
$165$	−94.9590	−0.0448033
$166$	0	0
$167$	2556.10	1.18441	0.592206	−	0.805787i	$-0.298257\pi$
$167$	2556.10	1.18441	0.592206	+	0.805787i	$0.298257\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−1218.80	−0.545052
$172$	0	0
$173$	3528.35	1.55061	0.775306	−	0.631586i	$-0.217596\pi$
$173$	3528.35	1.55061	0.775306	+	0.631586i	$0.217596\pi$
$174$	0	0
$175$	1810.76	0.782176
$176$	0	0
$177$	−228.260	−0.0969326
$178$	0	0
$179$	2396.76	1.00080	0.500398	−	0.865795i	$-0.333187\pi$
$179$	2396.76	1.00080	0.500398	+	0.865795i	$0.333187\pi$
$180$	0	0
$181$	1646.29	0.676064	0.338032	−	0.941135i	$-0.390239\pi$
$181$	1646.29	0.676064	0.338032	+	0.941135i	$0.390239\pi$
$182$	0	0
$183$	2269.56	0.916779
$184$	0	0
$185$	512.282	0.203588
$186$	0	0
$187$	588.170	0.230007
$188$	0	0
$189$	−526.769	−0.202734
$190$	0	0
$191$	827.484	0.313480	0.156740	−	0.987640i	$-0.449902\pi$
$191$	827.484	0.313480	0.156740	+	0.987640i	$0.449902\pi$
$192$	0	0
$193$	−3398.39	−1.26747	−0.633735	−	0.773551i	$-0.718479\pi$
$193$	−3398.39	−1.26747	−0.633735	+	0.773551i	$0.718479\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5316.91	−1.92292	−0.961458	−	0.274953i	$-0.911338\pi$
$197$	−5316.91	−1.92292	−0.961458	+	0.274953i	$0.911338\pi$
$198$	0	0
$199$	−3181.55	−1.13334	−0.566669	−	0.823946i	$-0.691768\pi$
$199$	−3181.55	−1.13334	−0.566669	+	0.823946i	$0.691768\pi$
$200$	0	0
$201$	−706.333	−0.247865
$202$	0	0
$203$	−4416.22	−1.52689
$204$	0	0
$205$	−412.055	−0.140386
$206$	0	0
$207$	409.858	0.137619
$208$	0	0
$209$	755.543	0.250057
$210$	0	0
$211$	670.453	0.218748	0.109374	−	0.994001i	$-0.465115\pi$
$211$	670.453	0.218748	0.109374	+	0.994001i	$0.465115\pi$
$212$	0	0
$213$	944.419	0.303805
$214$	0	0
$215$	2130.11	0.675684
$216$	0	0
$217$	−613.717	−0.191990
$218$	0	0
$219$	−1222.64	−0.377254
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−5036.49	−1.51241	−0.756207	−	0.654332i	$-0.772950\pi$
$223$	−5036.49	−1.51241	−0.756207	+	0.654332i	$0.772950\pi$
$224$	0	0
$225$	−835.310	−0.247499
$226$	0	0
$227$	5037.54	1.47292	0.736461	−	0.676480i	$-0.236496\pi$
$227$	5037.54	1.47292	0.736461	+	0.676480i	$0.236496\pi$
$228$	0	0
$229$	−4661.22	−1.34508	−0.672538	−	0.740063i	$-0.734796\pi$
$229$	−4661.22	−1.34508	−0.672538	+	0.740063i	$0.734796\pi$
$230$	0	0
$231$	326.548	0.0930099
$232$	0	0
$233$	132.855	0.0373545	0.0186772	−	0.999826i	$-0.494055\pi$
$233$	132.855	0.0373545	0.0186772	+	0.999826i	$0.494055\pi$
$234$	0	0
$235$	3614.94	1.00346
$236$	0	0
$237$	557.742	0.152866
$238$	0	0
$239$	−1398.04	−0.378376	−0.189188	−	0.981941i	$-0.560586\pi$
$239$	−1398.04	−0.378376	−0.189188	+	0.981941i	$0.560586\pi$
$240$	0	0
$241$	−2555.48	−0.683041	−0.341520	−	0.939874i	$-0.610942\pi$
$241$	−2555.48	−0.683041	−0.341520	+	0.939874i	$0.610942\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	213.538	0.0556834
$246$	0	0
$247$	0	0
$248$	0	0
$249$	2084.77	0.530590
$250$	0	0
$251$	−2282.17	−0.573901	−0.286951	−	0.957945i	$-0.592642\pi$
$251$	−2282.17	−0.573901	−0.286951	+	0.957945i	$0.592642\pi$
$252$	0	0
$253$	−254.074	−0.0631364
$254$	0	0
$255$	−1794.32	−0.440645
$256$	0	0
$257$	−5114.63	−1.24141	−0.620703	−	0.784045i	$-0.713153\pi$
$257$	−5114.63	−1.24141	−0.620703	+	0.784045i	$0.713153\pi$
$258$	0	0
$259$	−1761.65	−0.422640
$260$	0	0
$261$	2037.22	0.483144
$262$	0	0
$263$	5983.53	1.40289	0.701445	−	0.712724i	$-0.252539\pi$
$263$	5983.53	1.40289	0.701445	+	0.712724i	$0.252539\pi$
$264$	0	0
$265$	3282.96	0.761022
$266$	0	0
$267$	−41.2139	−0.00944663
$268$	0	0
$269$	−1653.52	−0.374784	−0.187392	−	0.982285i	$-0.560003\pi$
$269$	−1653.52	−0.374784	−0.187392	+	0.982285i	$0.560003\pi$
$270$	0	0
$271$	6373.97	1.42875	0.714375	−	0.699763i	$-0.246711\pi$
$271$	6373.97	1.42875	0.714375	+	0.699763i	$0.246711\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	517.815	0.113547
$276$	0	0
$277$	1836.90	0.398444	0.199222	−	0.979954i	$-0.436159\pi$
$277$	1836.90	0.398444	0.199222	+	0.979954i	$0.436159\pi$
$278$	0	0
$279$	283.109	0.0607503
$280$	0	0
$281$	4214.79	0.894781	0.447390	−	0.894339i	$-0.352353\pi$
$281$	4214.79	0.894781	0.447390	+	0.894339i	$0.352353\pi$
$282$	0	0
$283$	5015.35	1.05347	0.526735	−	0.850030i	$-0.323416\pi$
$283$	5015.35	1.05347	0.526735	+	0.850030i	$0.323416\pi$
$284$	0	0
$285$	−2304.92	−0.479059
$286$	0	0
$287$	1416.99	0.291436
$288$	0	0
$289$	6200.89	1.26214
$290$	0	0
$291$	1958.07	0.394446
$292$	0	0
$293$	−954.946	−0.190405	−0.0952023	−	0.995458i	$-0.530350\pi$
$293$	−954.946	−0.190405	−0.0952023	+	0.995458i	$0.530350\pi$
$294$	0	0
$295$	−431.672	−0.0851963
$296$	0	0
$297$	−150.638	−0.0294306
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−7325.08	−1.40269
$302$	0	0
$303$	1151.63	0.218348
$304$	0	0
$305$	4292.06	0.805779
$306$	0	0
$307$	−7207.06	−1.33983	−0.669916	−	0.742437i	$-0.733670\pi$
$307$	−7207.06	−1.33983	−0.669916	+	0.742437i	$0.733670\pi$
$308$	0	0
$309$	2912.64	0.536227
$310$	0	0
$311$	1176.61	0.214532	0.107266	−	0.994230i	$-0.465790\pi$
$311$	1176.61	0.214532	0.107266	+	0.994230i	$0.465790\pi$
$312$	0	0
$313$	9578.32	1.72971	0.864854	−	0.502023i	$-0.167411\pi$
$313$	9578.32	1.72971	0.864854	+	0.502023i	$0.167411\pi$
$314$	0	0
$315$	−996.194	−0.178188
$316$	0	0
$317$	−7243.53	−1.28340	−0.641699	−	0.766957i	$-0.721770\pi$
$317$	−7243.53	−1.28340	−0.641699	+	0.766957i	$0.721770\pi$
$318$	0	0
$319$	−1262.89	−0.221655
$320$	0	0
$321$	−2707.31	−0.470740
$322$	0	0
$323$	14276.5	2.45934
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1891.72	−0.319916
$328$	0	0
$329$	−12431.2	−2.08314
$330$	0	0
$331$	5134.89	0.852687	0.426343	−	0.904561i	$-0.359802\pi$
$331$	5134.89	0.852687	0.426343	+	0.904561i	$0.359802\pi$
$332$	0	0
$333$	812.656	0.133734
$334$	0	0
$335$	−1335.78	−0.217854
$336$	0	0
$337$	3989.68	0.644901	0.322451	−	0.946586i	$-0.395493\pi$
$337$	3989.68	0.644901	0.322451	+	0.946586i	$0.395493\pi$
$338$	0	0
$339$	−1549.25	−0.248212
$340$	0	0
$341$	−175.502	−0.0278708
$342$	0	0
$343$	5957.59	0.937842
$344$	0	0
$345$	775.099	0.120956
$346$	0	0
$347$	−5874.10	−0.908755	−0.454378	−	0.890809i	$-0.650138\pi$
$347$	−5874.10	−0.908755	−0.454378	+	0.890809i	$0.650138\pi$
$348$	0	0
$349$	10093.5	1.54811	0.774057	−	0.633116i	$-0.218224\pi$
$349$	10093.5	1.54811	0.774057	+	0.633116i	$0.218224\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6440.42	−0.971074	−0.485537	−	0.874216i	$-0.661376\pi$
$353$	−6440.42	−0.971074	−0.485537	+	0.874216i	$0.661376\pi$
$354$	0	0
$355$	1786.03	0.267022
$356$	0	0
$357$	6170.36	0.914762
$358$	0	0
$359$	−3177.72	−0.467170	−0.233585	−	0.972336i	$-0.575046\pi$
$359$	−3177.72	−0.467170	−0.233585	+	0.972336i	$0.575046\pi$
$360$	0	0
$361$	11480.1	1.67373
$362$	0	0
$363$	−3899.62	−0.563848
$364$	0	0
$365$	−2312.19	−0.331577
$366$	0	0
$367$	338.142	0.0480950	0.0240475	−	0.999711i	$-0.492345\pi$
$367$	338.142	0.0480950	0.0240475	+	0.999711i	$0.492345\pi$
$368$	0	0
$369$	−653.661	−0.0922174
$370$	0	0
$371$	−11289.6	−1.57985
$372$	0	0
$373$	9786.58	1.35852	0.679262	−	0.733896i	$-0.262300\pi$
$373$	9786.58	1.35852	0.679262	+	0.733896i	$0.262300\pi$
$374$	0	0
$375$	−3707.22	−0.510507
$376$	0	0
$377$	0	0
$378$	0	0
$379$	8952.02	1.21328	0.606642	−	0.794975i	$-0.292516\pi$
$379$	8952.02	1.21328	0.606642	+	0.794975i	$0.292516\pi$
$380$	0	0
$381$	−495.487	−0.0666261
$382$	0	0
$383$	−7233.32	−0.965027	−0.482514	−	0.875889i	$-0.660276\pi$
$383$	−7233.32	−0.965027	−0.482514	+	0.875889i	$0.660276\pi$
$384$	0	0
$385$	617.548	0.0817485
$386$	0	0
$387$	3379.08	0.443846
$388$	0	0
$389$	10733.4	1.39898	0.699491	−	0.714641i	$-0.253410\pi$
$389$	10733.4	1.39898	0.699491	+	0.714641i	$0.253410\pi$
$390$	0	0
$391$	−4800.91	−0.620953
$392$	0	0
$393$	8030.31	1.03073
$394$	0	0
$395$	1054.77	0.134357
$396$	0	0
$397$	7180.49	0.907754	0.453877	−	0.891064i	$-0.350041\pi$
$397$	7180.49	0.907754	0.453877	+	0.891064i	$0.350041\pi$
$398$	0	0
$399$	7926.24	0.994506
$400$	0	0
$401$	613.503	0.0764013	0.0382006	−	0.999270i	$-0.487837\pi$
$401$	613.503	0.0764013	0.0382006	+	0.999270i	$0.487837\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	459.547	0.0563829
$406$	0	0
$407$	−503.772	−0.0613539
$408$	0	0
$409$	12572.5	1.51997	0.759985	−	0.649941i	$-0.225206\pi$
$409$	12572.5	1.51997	0.759985	+	0.649941i	$0.225206\pi$
$410$	0	0
$411$	−5258.52	−0.631104
$412$	0	0
$413$	1484.45	0.176864
$414$	0	0
$415$	3942.60	0.466348
$416$	0	0
$417$	3541.00	0.415836
$418$	0	0
$419$	−9682.06	−1.12888	−0.564439	−	0.825475i	$-0.690907\pi$
$419$	−9682.06	−1.12888	−0.564439	+	0.825475i	$0.690907\pi$
$420$	0	0
$421$	−3930.59	−0.455025	−0.227512	−	0.973775i	$-0.573059\pi$
$421$	−3930.59	−0.455025	−0.227512	+	0.973775i	$0.573059\pi$
$422$	0	0
$423$	5734.54	0.659156
$424$	0	0
$425$	9784.49	1.11675
$426$	0	0
$427$	−14759.7	−1.67276
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−938.551	−0.104892	−0.0524460	−	0.998624i	$-0.516702\pi$
$431$	−938.551	−0.104892	−0.0524460	+	0.998624i	$0.516702\pi$
$432$	0	0
$433$	−7096.84	−0.787650	−0.393825	−	0.919186i	$-0.628848\pi$
$433$	−7096.84	−0.787650	−0.393825	+	0.919186i	$0.628848\pi$
$434$	0	0
$435$	3852.66	0.424646
$436$	0	0
$437$	−6167.09	−0.675084
$438$	0	0
$439$	−14182.8	−1.54194	−0.770968	−	0.636874i	$-0.780227\pi$
$439$	−14182.8	−1.54194	−0.770968	+	0.636874i	$0.780227\pi$
$440$	0	0
$441$	338.745	0.0365775
$442$	0	0
$443$	12640.7	1.35571	0.677855	−	0.735195i	$-0.262910\pi$
$443$	12640.7	1.35571	0.677855	+	0.735195i	$0.262910\pi$
$444$	0	0
$445$	−77.9413	−0.00830286
$446$	0	0
$447$	10174.5	1.07660
$448$	0	0
$449$	−17483.1	−1.83759	−0.918793	−	0.394739i	$-0.870835\pi$
$449$	−17483.1	−1.83759	−0.918793	+	0.394739i	$0.870835\pi$
$450$	0	0
$451$	405.209	0.0423072
$452$	0	0
$453$	−153.386	−0.0159089
$454$	0	0
$455$	0	0
$456$	0	0
$457$	553.725	0.0566787	0.0283394	−	0.999598i	$-0.490978\pi$
$457$	553.725	0.0566787	0.0283394	+	0.999598i	$0.490978\pi$
$458$	0	0
$459$	−2846.41	−0.289453
$460$	0	0
$461$	9239.42	0.933455	0.466727	−	0.884401i	$-0.345433\pi$
$461$	9239.42	0.933455	0.466727	+	0.884401i	$0.345433\pi$
$462$	0	0
$463$	−5386.78	−0.540701	−0.270351	−	0.962762i	$-0.587140\pi$
$463$	−5386.78	−0.540701	−0.270351	+	0.962762i	$0.587140\pi$
$464$	0	0
$465$	535.400	0.0533948
$466$	0	0
$467$	10755.4	1.06574	0.532871	−	0.846197i	$-0.321113\pi$
$467$	10755.4	1.06574	0.532871	+	0.846197i	$0.321113\pi$
$468$	0	0
$469$	4593.51	0.452257
$470$	0	0
$471$	−323.297	−0.0316279
$472$	0	0
$473$	−2094.72	−0.203626
$474$	0	0
$475$	12568.8	1.21410
$476$	0	0
$477$	5207.91	0.499903
$478$	0	0
$479$	11689.1	1.11500	0.557502	−	0.830176i	$-0.311760\pi$
$479$	11689.1	1.11500	0.557502	+	0.830176i	$0.311760\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−2665.44	−0.251100
$484$	0	0
$485$	3702.98	0.346688
$486$	0	0
$487$	537.670	0.0500291	0.0250145	−	0.999687i	$-0.492037\pi$
$487$	537.670	0.0500291	0.0250145	+	0.999687i	$0.492037\pi$
$488$	0	0
$489$	−4459.42	−0.412396
$490$	0	0
$491$	−14170.4	−1.30244	−0.651222	−	0.758887i	$-0.725744\pi$
$491$	−14170.4	−1.30244	−0.651222	+	0.758887i	$0.725744\pi$
$492$	0	0
$493$	−23863.1	−2.18000
$494$	0	0
$495$	−284.877	−0.0258672
$496$	0	0
$497$	−6141.86	−0.554326
$498$	0	0
$499$	11824.1	1.06076	0.530382	−	0.847759i	$-0.322049\pi$
$499$	11824.1	1.06076	0.530382	+	0.847759i	$0.322049\pi$
$500$	0	0
$501$	7668.29	0.683820
$502$	0	0
$503$	−7771.20	−0.688868	−0.344434	−	0.938810i	$-0.611929\pi$
$503$	−7771.20	−0.688868	−0.344434	+	0.938810i	$0.611929\pi$
$504$	0	0
$505$	2177.89	0.191911
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11205.7	0.975807	0.487903	−	0.872898i	$-0.337762\pi$
$509$	11205.7	0.975807	0.487903	+	0.872898i	$0.337762\pi$
$510$	0	0
$511$	7951.24	0.688341
$512$	0	0
$513$	−3656.40	−0.314686
$514$	0	0
$515$	5508.21	0.471302
$516$	0	0
$517$	−3554.89	−0.302406
$518$	0	0
$519$	10585.1	0.895246
$520$	0	0
$521$	−10763.7	−0.905118	−0.452559	−	0.891734i	$-0.649489\pi$
$521$	−10763.7	−0.905118	−0.452559	+	0.891734i	$0.649489\pi$
$522$	0	0
$523$	21206.0	1.77299	0.886496	−	0.462736i	$-0.153132\pi$
$523$	21206.0	1.77299	0.886496	+	0.462736i	$0.153132\pi$
$524$	0	0
$525$	5432.29	0.451590
$526$	0	0
$527$	−3316.23	−0.274113
$528$	0	0
$529$	−10093.1	−0.829550
$530$	0	0
$531$	−684.780	−0.0559641
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−5119.91	−0.413744
$536$	0	0
$537$	7190.29	0.577810
$538$	0	0
$539$	−209.990	−0.0167809
$540$	0	0
$541$	8378.58	0.665847	0.332924	−	0.942954i	$-0.391965\pi$
$541$	8378.58	0.665847	0.332924	+	0.942954i	$0.391965\pi$
$542$	0	0
$543$	4938.86	0.390325
$544$	0	0
$545$	−3577.52	−0.281182
$546$	0	0
$547$	17579.7	1.37414	0.687069	−	0.726592i	$-0.258897\pi$
$547$	17579.7	1.37414	0.687069	+	0.726592i	$0.258897\pi$
$548$	0	0
$549$	6808.68	0.529303
$550$	0	0
$551$	−30653.8	−2.37004
$552$	0	0
$553$	−3627.18	−0.278921
$554$	0	0
$555$	1536.85	0.117542
$556$	0	0
$557$	−2969.36	−0.225881	−0.112941	−	0.993602i	$-0.536027\pi$
$557$	−2969.36	−0.225881	−0.112941	+	0.993602i	$0.536027\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1764.51	0.132794
$562$	0	0
$563$	9124.37	0.683031	0.341516	−	0.939876i	$-0.389060\pi$
$563$	9124.37	0.683031	0.341516	+	0.939876i	$0.389060\pi$
$564$	0	0
$565$	−2929.86	−0.218159
$566$	0	0
$567$	−1580.31	−0.117049
$568$	0	0
$569$	16345.4	1.20428	0.602140	−	0.798390i	$-0.294315\pi$
$569$	16345.4	1.20428	0.602140	+	0.798390i	$0.294315\pi$
$570$	0	0
$571$	9870.90	0.723440	0.361720	−	0.932287i	$-0.382190\pi$
$571$	9870.90	0.723440	0.361720	+	0.932287i	$0.382190\pi$
$572$	0	0
$573$	2482.45	0.180988
$574$	0	0
$575$	−4226.65	−0.306545
$576$	0	0
$577$	−11297.7	−0.815126	−0.407563	−	0.913177i	$-0.633621\pi$
$577$	−11297.7	−0.815126	−0.407563	+	0.913177i	$0.633621\pi$
$578$	0	0
$579$	−10195.2	−0.731774
$580$	0	0
$581$	−13557.9	−0.968120
$582$	0	0
$583$	−3228.42	−0.229344
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21225.6	−1.49246	−0.746230	−	0.665688i	$-0.768138\pi$
$587$	−21225.6	−1.49246	−0.746230	+	0.665688i	$0.768138\pi$
$588$	0	0
$589$	−4259.92	−0.298008
$590$	0	0
$591$	−15950.7	−1.11020
$592$	0	0
$593$	21708.9	1.50333	0.751666	−	0.659544i	$-0.229250\pi$
$593$	21708.9	1.50333	0.751666	+	0.659544i	$0.229250\pi$
$594$	0	0
$595$	11669.0	0.804006
$596$	0	0
$597$	−9544.65	−0.654332
$598$	0	0
$599$	1041.35	0.0710324	0.0355162	−	0.999369i	$-0.488692\pi$
$599$	1041.35	0.0710324	0.0355162	+	0.999369i	$0.488692\pi$
$600$	0	0
$601$	12939.4	0.878217	0.439108	−	0.898434i	$-0.355294\pi$
$601$	12939.4	0.878217	0.439108	+	0.898434i	$0.355294\pi$
$602$	0	0
$603$	−2119.00	−0.143105
$604$	0	0
$605$	−7374.73	−0.495579
$606$	0	0
$607$	19887.2	1.32981	0.664906	−	0.746927i	$-0.268472\pi$
$607$	19887.2	1.32981	0.664906	+	0.746927i	$0.268472\pi$
$608$	0	0
$609$	−13248.7	−0.881548
$610$	0	0
$611$	0	0
$612$	0	0
$613$	20379.6	1.34278	0.671390	−	0.741104i	$-0.265698\pi$
$613$	20379.6	1.34278	0.671390	+	0.741104i	$0.265698\pi$
$614$	0	0
$615$	−1236.17	−0.0810520
$616$	0	0
$617$	5067.56	0.330652	0.165326	−	0.986239i	$-0.447132\pi$
$617$	5067.56	0.330652	0.165326	+	0.986239i	$0.447132\pi$
$618$	0	0
$619$	−6642.99	−0.431348	−0.215674	−	0.976465i	$-0.569195\pi$
$619$	−6642.99	−0.431348	−0.215674	+	0.976465i	$0.569195\pi$
$620$	0	0
$621$	1229.57	0.0794542
$622$	0	0
$623$	268.027	0.0172364
$624$	0	0
$625$	4590.65	0.293801
$626$	0	0
$627$	2266.63	0.144371
$628$	0	0
$629$	−9519.13	−0.603422
$630$	0	0
$631$	−27636.8	−1.74359	−0.871793	−	0.489875i	$-0.837042\pi$
$631$	−27636.8	−1.74359	−0.871793	+	0.489875i	$0.837042\pi$
$632$	0	0
$633$	2011.36	0.126294
$634$	0	0
$635$	−937.035	−0.0585592
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2833.26	0.175402
$640$	0	0
$641$	−10576.5	−0.651712	−0.325856	−	0.945419i	$-0.605653\pi$
$641$	−10576.5	−0.651712	−0.325856	+	0.945419i	$0.605653\pi$
$642$	0	0
$643$	−16052.3	−0.984510	−0.492255	−	0.870451i	$-0.663827\pi$
$643$	−16052.3	−0.984510	−0.492255	+	0.870451i	$0.663827\pi$
$644$	0	0
$645$	6390.32	0.390106
$646$	0	0
$647$	−12877.4	−0.782476	−0.391238	−	0.920289i	$-0.627953\pi$
$647$	−12877.4	−0.782476	−0.391238	+	0.920289i	$0.627953\pi$
$648$	0	0
$649$	424.500	0.0256750
$650$	0	0
$651$	−1841.15	−0.110845
$652$	0	0
$653$	−7674.89	−0.459941	−0.229971	−	0.973198i	$-0.573863\pi$
$653$	−7674.89	−0.459941	−0.229971	+	0.973198i	$0.573863\pi$
$654$	0	0
$655$	15186.5	0.905929
$656$	0	0
$657$	−3667.93	−0.217808
$658$	0	0
$659$	−18616.6	−1.10045	−0.550227	−	0.835015i	$-0.685459\pi$
$659$	−18616.6	−1.10045	−0.550227	+	0.835015i	$0.685459\pi$
$660$	0	0
$661$	−21300.6	−1.25340	−0.626699	−	0.779261i	$-0.715595\pi$
$661$	−21300.6	−1.25340	−0.626699	+	0.779261i	$0.715595\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	14989.6	0.874095
$666$	0	0
$667$	10308.3	0.598407
$668$	0	0
$669$	−15109.5	−0.873193
$670$	0	0
$671$	−4220.75	−0.242832
$672$	0	0
$673$	−12231.5	−0.700581	−0.350291	−	0.936641i	$-0.613917\pi$
$673$	−12231.5	−0.700581	−0.350291	+	0.936641i	$0.613917\pi$
$674$	0	0
$675$	−2505.93	−0.142894
$676$	0	0
$677$	−33485.1	−1.90094	−0.950471	−	0.310813i	$-0.899399\pi$
$677$	−33485.1	−1.90094	−0.950471	+	0.310813i	$0.899399\pi$
$678$	0	0
$679$	−12733.9	−0.719710
$680$	0	0
$681$	15112.6	0.850392
$682$	0	0
$683$	20867.1	1.16904	0.584522	−	0.811378i	$-0.301282\pi$
$683$	20867.1	1.16904	0.584522	+	0.811378i	$0.301282\pi$
$684$	0	0
$685$	−9944.61	−0.554692
$686$	0	0
$687$	−13983.7	−0.776580
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−5683.55	−0.312898	−0.156449	−	0.987686i	$-0.550005\pi$
$691$	−5683.55	−0.312898	−0.156449	+	0.987686i	$0.550005\pi$
$692$	0	0
$693$	979.644	0.0536993
$694$	0	0
$695$	6696.53	0.365488
$696$	0	0
$697$	7656.73	0.416096
$698$	0	0
$699$	398.564	0.0215666
$700$	0	0
$701$	−14124.9	−0.761039	−0.380519	−	0.924773i	$-0.624255\pi$
$701$	−14124.9	−0.761039	−0.380519	+	0.924773i	$0.624255\pi$
$702$	0	0
$703$	−12228.0	−0.656025
$704$	0	0
$705$	10844.8	0.579347
$706$	0	0
$707$	−7489.41	−0.398399
$708$	0	0
$709$	19631.9	1.03990	0.519952	−	0.854195i	$-0.325950\pi$
$709$	19631.9	1.03990	0.519952	+	0.854195i	$0.325950\pi$
$710$	0	0
$711$	1673.23	0.0882573
$712$	0	0
$713$	1432.53	0.0752434
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−4194.12	−0.218455
$718$	0	0
$719$	10623.1	0.551007	0.275503	−	0.961300i	$-0.411155\pi$
$719$	10623.1	0.551007	0.275503	+	0.961300i	$0.411155\pi$
$720$	0	0
$721$	−18941.8	−0.978405
$722$	0	0
$723$	−7666.44	−0.394354
$724$	0	0
$725$	−21008.7	−1.07620
$726$	0	0
$727$	−3361.06	−0.171464	−0.0857322	−	0.996318i	$-0.527323\pi$
$727$	−3361.06	−0.171464	−0.0857322	+	0.996318i	$0.527323\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−39581.2	−2.00269
$732$	0	0
$733$	−8833.43	−0.445116	−0.222558	−	0.974919i	$-0.571441\pi$
$733$	−8833.43	−0.445116	−0.222558	+	0.974919i	$0.571441\pi$
$734$	0	0
$735$	640.614	0.0321488
$736$	0	0
$737$	1313.58	0.0656533
$738$	0	0
$739$	28857.3	1.43645	0.718223	−	0.695813i	$-0.244956\pi$
$739$	28857.3	1.43645	0.718223	+	0.695813i	$0.244956\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13543.9	0.668743	0.334371	−	0.942441i	$-0.391476\pi$
$743$	13543.9	0.668743	0.334371	+	0.942441i	$0.391476\pi$
$744$	0	0
$745$	19241.5	0.946247
$746$	0	0
$747$	6254.31	0.306337
$748$	0	0
$749$	17606.5	0.858916
$750$	0	0
$751$	−7394.96	−0.359316	−0.179658	−	0.983729i	$-0.557499\pi$
$751$	−7394.96	−0.359316	−0.179658	+	0.983729i	$0.557499\pi$
$752$	0	0
$753$	−6846.51	−0.331342
$754$	0	0
$755$	−290.075	−0.0139827
$756$	0	0
$757$	−23785.1	−1.14198	−0.570992	−	0.820955i	$-0.693441\pi$
$757$	−23785.1	−1.14198	−0.570992	+	0.820955i	$0.693441\pi$
$758$	0	0
$759$	−762.222	−0.0364518
$760$	0	0
$761$	986.867	0.0470091	0.0235045	−	0.999724i	$-0.492518\pi$
$761$	986.867	0.0470091	0.0235045	+	0.999724i	$0.492518\pi$
$762$	0	0
$763$	12302.5	0.583722
$764$	0	0
$765$	−5382.96	−0.254407
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−12859.5	−0.603022	−0.301511	−	0.953463i	$-0.597491\pi$
$769$	−12859.5	−0.603022	−0.301511	+	0.953463i	$0.597491\pi$
$770$	0	0
$771$	−15343.9	−0.716727
$772$	0	0
$773$	33339.1	1.55126	0.775630	−	0.631188i	$-0.217433\pi$
$773$	33339.1	1.55126	0.775630	+	0.631188i	$0.217433\pi$
$774$	0	0
$775$	−2919.56	−0.135321
$776$	0	0
$777$	−5284.96	−0.244011
$778$	0	0
$779$	9835.57	0.452369
$780$	0	0
$781$	−1756.36	−0.0804705
$782$	0	0
$783$	6111.65	0.278943
$784$	0	0
$785$	−611.401	−0.0277985
$786$	0	0
$787$	40581.2	1.83807	0.919037	−	0.394171i	$-0.128968\pi$
$787$	40581.2	1.83807	0.919037	+	0.394171i	$0.128968\pi$
$788$	0	0
$789$	17950.6	0.809959
$790$	0	0
$791$	10075.3	0.452890
$792$	0	0
$793$	0	0
$794$	0	0
$795$	9848.89	0.439376
$796$	0	0
$797$	−22164.9	−0.985094	−0.492547	−	0.870286i	$-0.663934\pi$
$797$	−22164.9	−0.985094	−0.492547	+	0.870286i	$0.663934\pi$
$798$	0	0
$799$	−67172.2	−2.97419
$800$	0	0
$801$	−123.642	−0.00545401
$802$	0	0
$803$	2273.78	0.0999252
$804$	0	0
$805$	−5040.72	−0.220698
$806$	0	0
$807$	−4960.56	−0.216382
$808$	0	0
$809$	−40740.1	−1.77052	−0.885258	−	0.465101i	$-0.846018\pi$
$809$	−40740.1	−1.77052	−0.885258	+	0.465101i	$0.846018\pi$
$810$	0	0
$811$	32854.7	1.42255	0.711273	−	0.702916i	$-0.248119\pi$
$811$	32854.7	1.42255	0.711273	+	0.702916i	$0.248119\pi$
$812$	0	0
$813$	19121.9	0.824889
$814$	0	0
$815$	−8433.38	−0.362465
$816$	0	0
$817$	−50844.7	−2.17727
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13255.6	0.563489	0.281745	−	0.959489i	$-0.409087\pi$
$821$	13255.6	0.563489	0.281745	+	0.959489i	$0.409087\pi$
$822$	0	0
$823$	28691.8	1.21523	0.607615	−	0.794232i	$-0.292126\pi$
$823$	28691.8	1.21523	0.607615	+	0.794232i	$0.292126\pi$
$824$	0	0
$825$	1553.45	0.0655564
$826$	0	0
$827$	41617.9	1.74994	0.874968	−	0.484181i	$-0.160883\pi$
$827$	41617.9	1.74994	0.874968	+	0.484181i	$0.160883\pi$
$828$	0	0
$829$	5580.51	0.233798	0.116899	−	0.993144i	$-0.462705\pi$
$829$	5580.51	0.233798	0.116899	+	0.993144i	$0.462705\pi$
$830$	0	0
$831$	5510.71	0.230042
$832$	0	0
$833$	−3967.92	−0.165042
$834$	0	0
$835$	14501.8	0.601025
$836$	0	0
$837$	849.328	0.0350742
$838$	0	0
$839$	6766.94	0.278451	0.139226	−	0.990261i	$-0.455539\pi$
$839$	6766.94	0.278451	0.139226	+	0.990261i	$0.455539\pi$
$840$	0	0
$841$	26848.6	1.10085
$842$	0	0
$843$	12644.4	0.516602
$844$	0	0
$845$	0	0
$846$	0	0
$847$	25360.5	1.02880
$848$	0	0
$849$	15046.1	0.608221
$850$	0	0
$851$	4112.02	0.165638
$852$	0	0
$853$	23869.0	0.958102	0.479051	−	0.877787i	$-0.340981\pi$
$853$	23869.0	0.958102	0.479051	+	0.877787i	$0.340981\pi$
$854$	0	0
$855$	−6914.76	−0.276585
$856$	0	0
$857$	−11507.3	−0.458670	−0.229335	−	0.973348i	$-0.573655\pi$
$857$	−11507.3	−0.458670	−0.229335	+	0.973348i	$0.573655\pi$
$858$	0	0
$859$	−40591.0	−1.61228	−0.806139	−	0.591726i	$-0.798447\pi$
$859$	−40591.0	−1.61228	−0.806139	+	0.591726i	$0.798447\pi$
$860$	0	0
$861$	4250.96	0.168261
$862$	0	0
$863$	−24143.3	−0.952315	−0.476158	−	0.879360i	$-0.657971\pi$
$863$	−24143.3	−0.952315	−0.476158	+	0.879360i	$0.657971\pi$
$864$	0	0
$865$	20017.8	0.786852
$866$	0	0
$867$	18602.7	0.728696
$868$	0	0
$869$	−1037.25	−0.0404904
$870$	0	0
$871$	0	0
$872$	0	0
$873$	5874.20	0.227734
$874$	0	0
$875$	24109.3	0.931476
$876$	0	0
$877$	17759.9	0.683817	0.341909	−	0.939733i	$-0.388927\pi$
$877$	17759.9	0.683817	0.341909	+	0.939733i	$0.388927\pi$
$878$	0	0
$879$	−2864.84	−0.109930
$880$	0	0
$881$	27527.3	1.05269	0.526344	−	0.850272i	$-0.323562\pi$
$881$	27527.3	1.05269	0.526344	+	0.850272i	$0.323562\pi$
$882$	0	0
$883$	14882.4	0.567193	0.283596	−	0.958944i	$-0.408472\pi$
$883$	14882.4	0.567193	0.283596	+	0.958944i	$0.408472\pi$
$884$	0	0
$885$	−1295.02	−0.0491881
$886$	0	0
$887$	−44353.7	−1.67897	−0.839487	−	0.543380i	$-0.817144\pi$
$887$	−44353.7	−1.67897	−0.839487	+	0.543380i	$0.817144\pi$
$888$	0	0
$889$	3222.31	0.121567
$890$	0	0
$891$	−451.913	−0.0169917
$892$	0	0
$893$	−86287.1	−3.23347
$894$	0	0
$895$	13597.9	0.507851
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7120.43	0.264160
$900$	0	0
$901$	−61003.4	−2.25562
$902$	0	0
$903$	−21975.2	−0.809845
$904$	0	0
$905$	9340.08	0.343066
$906$	0	0
$907$	46291.2	1.69468	0.847340	−	0.531050i	$-0.178202\pi$
$907$	46291.2	1.69468	0.847340	+	0.531050i	$0.178202\pi$
$908$	0	0
$909$	3454.88	0.126063
$910$	0	0
$911$	−48843.4	−1.77635	−0.888175	−	0.459506i	$-0.848027\pi$
$911$	−48843.4	−1.77635	−0.888175	+	0.459506i	$0.848027\pi$
$912$	0	0
$913$	−3877.10	−0.140540
$914$	0	0
$915$	12876.2	0.465216
$916$	0	0
$917$	−52223.7	−1.88067
$918$	0	0
$919$	12769.7	0.458361	0.229181	−	0.973384i	$-0.426395\pi$
$919$	12769.7	0.458361	0.229181	+	0.973384i	$0.426395\pi$
$920$	0	0
$921$	−21621.2	−0.773552
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−8380.49	−0.297891
$926$	0	0
$927$	8737.91	0.309591
$928$	0	0
$929$	−36189.8	−1.27809	−0.639046	−	0.769168i	$-0.720671\pi$
$929$	−36189.8	−1.27809	−0.639046	+	0.769168i	$0.720671\pi$
$930$	0	0
$931$	−5097.05	−0.179430
$932$	0	0
$933$	3529.83	0.123860
$934$	0	0
$935$	3336.94	0.116716
$936$	0	0
$937$	−51638.7	−1.80039	−0.900193	−	0.435491i	$-0.856575\pi$
$937$	−51638.7	−1.80039	−0.900193	+	0.435491i	$0.856575\pi$
$938$	0	0
$939$	28735.0	0.998647
$940$	0	0
$941$	−17015.4	−0.589466	−0.294733	−	0.955580i	$-0.595231\pi$
$941$	−17015.4	−0.589466	−0.294733	+	0.955580i	$0.595231\pi$
$942$	0	0
$943$	−3307.51	−0.114218
$944$	0	0
$945$	−2988.58	−0.102877
$946$	0	0
$947$	27797.2	0.953843	0.476921	−	0.878946i	$-0.341753\pi$
$947$	27797.2	0.953843	0.476921	+	0.878946i	$0.341753\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−21730.6	−0.740970
$952$	0	0
$953$	−32522.5	−1.10546	−0.552732	−	0.833359i	$-0.686415\pi$
$953$	−32522.5	−1.10546	−0.552732	+	0.833359i	$0.686415\pi$
$954$	0	0
$955$	4694.67	0.159074
$956$	0	0
$957$	−3788.66	−0.127973
$958$	0	0
$959$	34197.9	1.15152
$960$	0	0
$961$	−28801.5	−0.966785
$962$	0	0
$963$	−8121.94	−0.271782
$964$	0	0
$965$	−19280.5	−0.643173
$966$	0	0
$967$	15192.7	0.505237	0.252619	−	0.967566i	$-0.418708\pi$
$967$	15192.7	0.505237	0.252619	+	0.967566i	$0.418708\pi$
$968$	0	0
$969$	42829.6	1.41990
$970$	0	0
$971$	4065.98	0.134381	0.0671903	−	0.997740i	$-0.478597\pi$
$971$	4065.98	0.134381	0.0671903	+	0.997740i	$0.478597\pi$
$972$	0	0
$973$	−23028.2	−0.758738
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15468.9	−0.506546	−0.253273	−	0.967395i	$-0.581507\pi$
$977$	−15468.9	−0.506546	−0.253273	+	0.967395i	$0.581507\pi$
$978$	0	0
$979$	76.6464	0.00250218
$980$	0	0
$981$	−5675.17	−0.184704
$982$	0	0
$983$	15997.7	0.519073	0.259537	−	0.965733i	$-0.416430\pi$
$983$	15997.7	0.519073	0.259537	+	0.965733i	$0.416430\pi$
$984$	0	0
$985$	−30165.1	−0.975777
$986$	0	0
$987$	−37293.6	−1.20270
$988$	0	0
$989$	17098.1	0.549733
$990$	0	0
$991$	−6027.96	−0.193223	−0.0966117	−	0.995322i	$-0.530800\pi$
$991$	−6027.96	−0.193223	−0.0966117	+	0.995322i	$0.530800\pi$
$992$	0	0
$993$	15404.7	0.492299
$994$	0	0
$995$	−18050.3	−0.575108
$996$	0	0
$997$	−21539.6	−0.684219	−0.342110	−	0.939660i	$-0.611141\pi$
$997$	−21539.6	−0.684219	−0.342110	+	0.939660i	$0.611141\pi$
$998$	0	0
$999$	2437.97	0.0772111

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.4.a.n.1.3		4
13.3	even	3		156.4.i.a.61.3	✓	8
13.5	odd	4		2028.4.b.j.337.3		8
13.8	odd	4		2028.4.b.j.337.6		8
13.9	even	3		156.4.i.a.133.3	yes	8
13.12	even	2		2028.4.a.m.1.2		4
39.29	odd	6		468.4.l.d.217.2		8
39.35	odd	6		468.4.l.d.289.2		8
52.3	odd	6		624.4.q.l.529.3		8
52.35	odd	6		624.4.q.l.289.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.4.i.a.61.3	✓	8	13.3	even	3
156.4.i.a.133.3	yes	8	13.9	even	3
468.4.l.d.217.2		8	39.29	odd	6
468.4.l.d.289.2		8	39.35	odd	6
624.4.q.l.289.3		8	52.35	odd	6
624.4.q.l.529.3		8	52.3	odd	6
2028.4.a.m.1.2		4	13.12	even	2
2028.4.a.n.1.3		4	1.1	even	1	trivial
2028.4.b.j.337.3		8	13.5	odd	4
2028.4.b.j.337.6		8	13.8	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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +5.67342 q^{5} -19.5100 q^{7} +9.00000 q^{9} -5.57917 q^{11} +17.0203 q^{15} -105.422 q^{17} -135.422 q^{19} -58.5299 q^{21} +45.5398 q^{23} -92.8123 q^{25} +27.0000 q^{27} +226.357 q^{29} +31.4566 q^{31} -16.7375 q^{33} -110.688 q^{35} +90.2951 q^{37} -72.6290 q^{41} +375.453 q^{43} +51.0608 q^{45} +637.171 q^{47} +37.6383 q^{49} -316.267 q^{51} +578.656 q^{53} -31.6530 q^{55} -406.266 q^{57} -76.0867 q^{59} +756.520 q^{61} -175.590 q^{63} -235.444 q^{67} +136.619 q^{69} +314.806 q^{71} -407.548 q^{73} -278.437 q^{75} +108.849 q^{77} +185.914 q^{79} +81.0000 q^{81} +694.924 q^{83} -598.106 q^{85} +679.072 q^{87} -13.7380 q^{89} +94.3698 q^{93} -768.307 q^{95} +652.689 q^{97} -50.2125 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 7 q^{5} + 11 q^{7} + 36 q^{9} - 20 q^{11} + 21 q^{15} + 7 q^{17} + 18 q^{19} + 33 q^{21} - 60 q^{23} + 67 q^{25} + 108 q^{27} + 75 q^{29} + 87 q^{31} - 60 q^{33} + 556 q^{35} - 51 q^{37}+ \cdots - 180 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 + 7 * q^5 + 11 * q^7 + 36 * q^9 - 20 * q^11 + 21 * q^15 + 7 * q^17 + 18 * q^19 + 33 * q^21 - 60 * q^23 + 67 * q^25 + 108 * q^27 + 75 * q^29 + 87 * q^31 - 60 * q^33 + 556 * q^35 - 51 * q^37 + 271 * q^41 - q^43 + 63 * q^45 - 32 * q^47 + 153 * q^49 + 21 * q^51 + 757 * q^53 + 626 * q^55 + 54 * q^57 - 610 * q^59 + 1166 * q^61 + 99 * q^63 + 1191 * q^67 - 180 * q^69 + 980 * q^71 + 482 * q^73 + 201 * q^75 + 406 * q^77 + 2225 * q^79 + 324 * q^81 - 1518 * q^83 - 2603 * q^85 + 225 * q^87 + 1078 * q^89 + 261 * q^93 + 154 * q^95 + 2527 * q^97 - 180 * q^99

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