LMFDB - Embedded newform 4900.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4900,2,Mod(1,4900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.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:	$39.1266969904$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 16 $ x^4 - 11*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.31342$ of defining polynomial
Character	$\chi$	$=$	4900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +2.27492 q^{11} +6.09095 q^{13} -4.77753 q^{17} +4.27492 q^{19} +0.894797 q^{23} +5.19615 q^{27} -3.27492 q^{29} +4.27492 q^{31} -3.94027 q^{33} -5.61478 q^{37} -10.5498 q^{39} +11.2749 q^{41} -6.50958 q^{43} -2.15068 q^{47} +8.27492 q^{51} +7.40437 q^{53} -7.40437 q^{57} -4.27492 q^{59} -1.54983 q^{61} -13.9140 q^{67} -1.54983 q^{69} +10.5498 q^{71} -2.15068 q^{73} -0.274917 q^{79} -9.00000 q^{81} -5.67232 q^{83} +5.67232 q^{87} +7.00000 q^{89} -7.40437 q^{93} +6.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{11} + 2 q^{19} + 2 q^{29} + 2 q^{31} - 12 q^{39} + 30 q^{41} + 18 q^{51} - 2 q^{59} + 24 q^{61} + 24 q^{69} + 12 q^{71} + 14 q^{79} - 36 q^{81} + 28 q^{89}+O(q^{100}) $ 4 * q - 6 * q^11 + 2 * q^19 + 2 * q^29 + 2 * q^31 - 12 * q^39 + 30 * q^41 + 18 * q^51 - 2 * q^59 + 24 * q^61 + 24 * q^69 + 12 * q^71 + 14 * q^79 - 36 * q^81 + 28 * q^89

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.73205	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	−1.73205	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.27492	0.685913	0.342957	−	0.939351i	$-0.388572\pi$
$11$	2.27492	0.685913	0.342957	+	0.939351i	$0.388572\pi$
$12$	0	0
$13$	6.09095	1.68933	0.844663	−	0.535299i	$-0.179801\pi$
$13$	6.09095	1.68933	0.844663	+	0.535299i	$0.179801\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.77753	−1.15872	−0.579360	−	0.815072i	$-0.696697\pi$
$17$	−4.77753	−1.15872	−0.579360	+	0.815072i	$0.696697\pi$
$18$	0	0
$19$	4.27492	0.980733	0.490367	−	0.871516i	$-0.336863\pi$
$19$	4.27492	0.980733	0.490367	+	0.871516i	$0.336863\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.894797	0.186578	0.0932891	−	0.995639i	$-0.470262\pi$
$23$	0.894797	0.186578	0.0932891	+	0.995639i	$0.470262\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.19615	1.00000
$28$	0	0
$29$	−3.27492	−0.608137	−0.304068	−	0.952650i	$-0.598345\pi$
$29$	−3.27492	−0.608137	−0.304068	+	0.952650i	$0.598345\pi$
$30$	0	0
$31$	4.27492	0.767798	0.383899	−	0.923375i	$-0.374581\pi$
$31$	4.27492	0.767798	0.383899	+	0.923375i	$0.374581\pi$
$32$	0	0
$33$	−3.94027	−0.685913
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.61478	−0.923064	−0.461532	−	0.887124i	$-0.652700\pi$
$37$	−5.61478	−0.923064	−0.461532	+	0.887124i	$0.652700\pi$
$38$	0	0
$39$	−10.5498	−1.68933
$40$	0	0
$41$	11.2749	1.76085	0.880423	−	0.474189i	$-0.157259\pi$
$41$	11.2749	1.76085	0.880423	+	0.474189i	$0.157259\pi$
$42$	0	0
$43$	−6.50958	−0.992701	−0.496351	−	0.868122i	$-0.665327\pi$
$43$	−6.50958	−0.992701	−0.496351	+	0.868122i	$0.665327\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.15068	−0.313709	−0.156854	−	0.987622i	$-0.550135\pi$
$47$	−2.15068	−0.313709	−0.156854	+	0.987622i	$0.550135\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	8.27492	1.15872
$52$	0	0
$53$	7.40437	1.01707	0.508534	−	0.861042i	$-0.330187\pi$
$53$	7.40437	1.01707	0.508534	+	0.861042i	$0.330187\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−7.40437	−0.980733
$58$	0	0
$59$	−4.27492	−0.556547	−0.278273	−	0.960502i	$-0.589762\pi$
$59$	−4.27492	−0.556547	−0.278273	+	0.960502i	$0.589762\pi$
$60$	0	0
$61$	−1.54983	−0.198436	−0.0992180	−	0.995066i	$-0.531634\pi$
$61$	−1.54983	−0.198436	−0.0992180	+	0.995066i	$0.531634\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13.9140	−1.69986	−0.849930	−	0.526896i	$-0.823356\pi$
$67$	−13.9140	−1.69986	−0.849930	+	0.526896i	$0.823356\pi$
$68$	0	0
$69$	−1.54983	−0.186578
$70$	0	0
$71$	10.5498	1.25204	0.626018	−	0.779809i	$-0.284684\pi$
$71$	10.5498	1.25204	0.626018	+	0.779809i	$0.284684\pi$
$72$	0	0
$73$	−2.15068	−0.251718	−0.125859	−	0.992048i	$-0.540169\pi$
$73$	−2.15068	−0.251718	−0.125859	+	0.992048i	$0.540169\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.274917	−0.0309306	−0.0154653	−	0.999880i	$-0.504923\pi$
$79$	−0.274917	−0.0309306	−0.0154653	+	0.999880i	$0.504923\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	−5.67232	−0.622618	−0.311309	−	0.950309i	$-0.600767\pi$
$83$	−5.67232	−0.622618	−0.311309	+	0.950309i	$0.600767\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.67232	0.608137
$88$	0	0
$89$	7.00000	0.741999	0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000	0.741999	0.370999	+	0.928633i	$0.379015\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−7.40437	−0.767798
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.92820	0.703452	0.351726	−	0.936103i	$-0.385595\pi$
$97$	6.92820	0.703452	0.351726	+	0.936103i	$0.385595\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.54983	−0.154214	−0.0771071	−	0.997023i	$-0.524568\pi$
$101$	−1.54983	−0.154214	−0.0771071	+	0.997023i	$0.524568\pi$
$102$	0	0
$103$	−2.56930	−0.253161	−0.126581	−	0.991956i	$-0.540400\pi$
$103$	−2.56930	−0.253161	−0.126581	+	0.991956i	$0.540400\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.9140	−1.34511	−0.672556	−	0.740046i	$-0.734804\pi$
$107$	−13.9140	−1.34511	−0.672556	+	0.740046i	$0.734804\pi$
$108$	0	0
$109$	3.54983	0.340012	0.170006	−	0.985443i	$-0.445621\pi$
$109$	3.54983	0.340012	0.170006	+	0.985443i	$0.445621\pi$
$110$	0	0
$111$	9.72508	0.923064
$112$	0	0
$113$	13.0192	1.22474	0.612369	−	0.790572i	$-0.290217\pi$
$113$	13.0192	1.22474	0.612369	+	0.790572i	$0.290217\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.82475	−0.529523
$122$	0	0
$123$	−19.5287	−1.76085
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.78959	0.158801	0.0794004	−	0.996843i	$-0.474699\pi$
$127$	1.78959	0.158801	0.0794004	+	0.996843i	$0.474699\pi$
$128$	0	0
$129$	11.2749	0.992701
$130$	0	0
$131$	18.2749	1.59669	0.798343	−	0.602202i	$-0.205710\pi$
$131$	18.2749	1.59669	0.798343	+	0.602202i	$0.205710\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.19397	0.785494	0.392747	−	0.919647i	$-0.371525\pi$
$137$	9.19397	0.785494	0.392747	+	0.919647i	$0.371525\pi$
$138$	0	0
$139$	17.0997	1.45037	0.725187	−	0.688551i	$-0.241753\pi$
$139$	17.0997	1.45037	0.725187	+	0.688551i	$0.241753\pi$
$140$	0	0
$141$	3.72508	0.313709
$142$	0	0
$143$	13.8564	1.15873
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.54983	0.618507	0.309253	−	0.950980i	$-0.399921\pi$
$149$	7.54983	0.618507	0.309253	+	0.950980i	$0.399921\pi$
$150$	0	0
$151$	−20.2749	−1.64995	−0.824975	−	0.565170i	$-0.808811\pi$
$151$	−20.2749	−1.64995	−0.824975	+	0.565170i	$0.808811\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.8685	−0.867399	−0.433699	−	0.901058i	$-0.642792\pi$
$157$	−10.8685	−0.867399	−0.433699	+	0.901058i	$0.642792\pi$
$158$	0	0
$159$	−12.8248	−1.01707
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−7.40437	−0.579955	−0.289978	−	0.957033i	$-0.593648\pi$
$163$	−7.40437	−0.579955	−0.289978	+	0.957033i	$0.593648\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.6005	−0.975058	−0.487529	−	0.873107i	$-0.662102\pi$
$167$	−12.6005	−0.975058	−0.487529	+	0.873107i	$0.662102\pi$
$168$	0	0
$169$	24.0997	1.85382
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.7490	1.42546	0.712731	−	0.701438i	$-0.247458\pi$
$173$	18.7490	1.42546	0.712731	+	0.701438i	$0.247458\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.40437	0.556547
$178$	0	0
$179$	0.274917	0.0205483	0.0102741	−	0.999947i	$-0.496730\pi$
$179$	0.274917	0.0205483	0.0102741	+	0.999947i	$0.496730\pi$
$180$	0	0
$181$	16.7251	1.24317	0.621583	−	0.783348i	$-0.286490\pi$
$181$	16.7251	1.24317	0.621583	+	0.783348i	$0.286490\pi$
$182$	0	0
$183$	2.68439	0.198436
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−10.8685	−0.794782
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.8248	1.65154	0.825771	−	0.564006i	$-0.190741\pi$
$191$	22.8248	1.65154	0.825771	+	0.564006i	$0.190741\pi$
$192$	0	0
$193$	−9.19397	−0.661796	−0.330898	−	0.943666i	$-0.607352\pi$
$193$	−9.19397	−0.661796	−0.330898	+	0.943666i	$0.607352\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	26.0383	1.85515	0.927576	−	0.373634i	$-0.121888\pi$
$197$	26.0383	1.85515	0.927576	+	0.373634i	$0.121888\pi$
$198$	0	0
$199$	−9.72508	−0.689393	−0.344696	−	0.938714i	$-0.612018\pi$
$199$	−9.72508	−0.689393	−0.344696	+	0.938714i	$0.612018\pi$
$200$	0	0
$201$	24.0997	1.69986
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.72508	0.672698
$210$	0	0
$211$	−19.6495	−1.35273	−0.676364	−	0.736568i	$-0.736445\pi$
$211$	−19.6495	−1.35273	−0.676364	+	0.736568i	$0.736445\pi$
$212$	0	0
$213$	−18.2728	−1.25204
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3.72508	0.251718
$220$	0	0
$221$	−29.0997	−1.95746
$222$	0	0
$223$	−8.71780	−0.583787	−0.291893	−	0.956451i	$-0.594285\pi$
$223$	−8.71780	−0.583787	−0.291893	+	0.956451i	$0.594285\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.45203	0.428236	0.214118	−	0.976808i	$-0.431312\pi$
$227$	6.45203	0.428236	0.214118	+	0.976808i	$0.431312\pi$
$228$	0	0
$229$	4.27492	0.282494	0.141247	−	0.989974i	$-0.454889\pi$
$229$	4.27492	0.282494	0.141247	+	0.989974i	$0.454889\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.6339	1.22075	0.610375	−	0.792113i	$-0.291019\pi$
$233$	18.6339	1.22075	0.610375	+	0.792113i	$0.291019\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.476171	0.0309306
$238$	0	0
$239$	−14.5498	−0.941151	−0.470575	−	0.882360i	$-0.655954\pi$
$239$	−14.5498	−0.941151	−0.470575	+	0.882360i	$0.655954\pi$
$240$	0	0
$241$	12.8248	0.826115	0.413057	−	0.910705i	$-0.364461\pi$
$241$	12.8248	0.826115	0.413057	+	0.910705i	$0.364461\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	26.0383	1.65678
$248$	0	0
$249$	9.82475	0.622618
$250$	0	0
$251$	5.45017	0.344011	0.172006	−	0.985096i	$-0.444975\pi$
$251$	5.45017	0.344011	0.172006	+	0.985096i	$0.444975\pi$
$252$	0	0
$253$	2.03559	0.127976
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.7249	1.54230	0.771148	−	0.636656i	$-0.219683\pi$
$257$	24.7249	1.54230	0.771148	+	0.636656i	$0.219683\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	26.9331	1.66077	0.830383	−	0.557193i	$-0.188122\pi$
$263$	26.9331	1.66077	0.830383	+	0.557193i	$0.188122\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−12.1244	−0.741999
$268$	0	0
$269$	29.5498	1.80169	0.900843	−	0.434146i	$-0.142950\pi$
$269$	29.5498	1.80169	0.900843	+	0.434146i	$0.142950\pi$
$270$	0	0
$271$	12.8248	0.779048	0.389524	−	0.921016i	$-0.372639\pi$
$271$	12.8248	0.779048	0.389524	+	0.921016i	$0.372639\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.6339	1.11960	0.559802	−	0.828626i	$-0.310877\pi$
$277$	18.6339	1.11960	0.559802	+	0.828626i	$0.310877\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	19.5863	1.16428	0.582142	−	0.813087i	$-0.302215\pi$
$283$	19.5863	1.16428	0.582142	+	0.813087i	$0.302215\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	5.82475	0.342632
$290$	0	0
$291$	−12.0000	−0.703452
$292$	0	0
$293$	−6.92820	−0.404750	−0.202375	−	0.979308i	$-0.564866\pi$
$293$	−6.92820	−0.404750	−0.202375	+	0.979308i	$0.564866\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	11.8208	0.685913
$298$	0	0
$299$	5.45017	0.315191
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.68439	0.154214
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.99782	0.228167	0.114084	−	0.993471i	$-0.463607\pi$
$307$	3.99782	0.228167	0.114084	+	0.993471i	$0.463607\pi$
$308$	0	0
$309$	4.45017	0.253161
$310$	0	0
$311$	−12.8248	−0.727225	−0.363612	−	0.931550i	$-0.618457\pi$
$311$	−12.8248	−0.727225	−0.363612	+	0.931550i	$0.618457\pi$
$312$	0	0
$313$	−14.4477	−0.816630	−0.408315	−	0.912841i	$-0.633884\pi$
$313$	−14.4477	−0.816630	−0.408315	+	0.912841i	$0.633884\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.82518	0.214844	0.107422	−	0.994214i	$-0.465740\pi$
$317$	3.82518	0.214844	0.107422	+	0.994214i	$0.465740\pi$
$318$	0	0
$319$	−7.45017	−0.417129
$320$	0	0
$321$	24.0997	1.34511
$322$	0	0
$323$	−20.4235	−1.13640
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−6.14849	−0.340012
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.82475	−0.265192	−0.132596	−	0.991170i	$-0.542331\pi$
$331$	−4.82475	−0.265192	−0.132596	+	0.991170i	$0.542331\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−13.0192	−0.709198	−0.354599	−	0.935018i	$-0.615383\pi$
$337$	−13.0192	−0.709198	−0.354599	+	0.935018i	$0.615383\pi$
$338$	0	0
$339$	−22.5498	−1.22474
$340$	0	0
$341$	9.72508	0.526643
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.1244	0.650870	0.325435	−	0.945564i	$-0.394489\pi$
$347$	12.1244	0.650870	0.325435	+	0.945564i	$0.394489\pi$
$348$	0	0
$349$	11.2749	0.603532	0.301766	−	0.953382i	$-0.402424\pi$
$349$	11.2749	0.603532	0.301766	+	0.953382i	$0.402424\pi$
$350$	0	0
$351$	31.6495	1.68933
$352$	0	0
$353$	−21.2608	−1.13160	−0.565799	−	0.824543i	$-0.691432\pi$
$353$	−21.2608	−1.13160	−0.565799	+	0.824543i	$0.691432\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.37459	−0.0725479	−0.0362739	−	0.999342i	$-0.511549\pi$
$359$	−1.37459	−0.0725479	−0.0362739	+	0.999342i	$0.511549\pi$
$360$	0	0
$361$	−0.725083	−0.0381623
$362$	0	0
$363$	10.0888	0.529523
$364$	0	0
$365$	0	0
$366$	0	0
$367$	14.7512	0.770007	0.385003	−	0.922915i	$-0.374200\pi$
$367$	14.7512	0.770007	0.385003	+	0.922915i	$0.374200\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	5.61478	0.290722	0.145361	−	0.989379i	$-0.453566\pi$
$373$	5.61478	0.290722	0.145361	+	0.989379i	$0.453566\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−19.9474	−1.02734
$378$	0	0
$379$	23.6495	1.21479	0.607397	−	0.794399i	$-0.292214\pi$
$379$	23.6495	1.21479	0.607397	+	0.794399i	$0.292214\pi$
$380$	0	0
$381$	−3.09967	−0.158801
$382$	0	0
$383$	−20.0049	−1.02220	−0.511101	−	0.859520i	$-0.670762\pi$
$383$	−20.0049	−1.02220	−0.511101	+	0.859520i	$0.670762\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.37459	−0.272502	−0.136251	−	0.990674i	$-0.543505\pi$
$389$	−5.37459	−0.272502	−0.136251	+	0.990674i	$0.543505\pi$
$390$	0	0
$391$	−4.27492	−0.216192
$392$	0	0
$393$	−31.6531	−1.59669
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−15.1698	−0.761352	−0.380676	−	0.924708i	$-0.624309\pi$
$397$	−15.1698	−0.761352	−0.380676	+	0.924708i	$0.624309\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	0	0
$403$	26.0383	1.29706
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.7732	−0.633142
$408$	0	0
$409$	−10.0997	−0.499396	−0.249698	−	0.968324i	$-0.580331\pi$
$409$	−10.0997	−0.499396	−0.249698	+	0.968324i	$0.580331\pi$
$410$	0	0
$411$	−15.9244	−0.785494
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−29.6175	−1.45037
$418$	0	0
$419$	−17.0997	−0.835373	−0.417687	−	0.908591i	$-0.637159\pi$
$419$	−17.0997	−0.835373	−0.417687	+	0.908591i	$0.637159\pi$
$420$	0	0
$421$	3.27492	0.159610	0.0798048	−	0.996811i	$-0.474570\pi$
$421$	3.27492	0.159610	0.0798048	+	0.996811i	$0.474570\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−24.0000	−1.15873
$430$	0	0
$431$	19.3746	0.933241	0.466620	−	0.884458i	$-0.345471\pi$
$431$	19.3746	0.933241	0.466620	+	0.884458i	$0.345471\pi$
$432$	0	0
$433$	26.8756	1.29156	0.645778	−	0.763525i	$-0.276533\pi$
$433$	26.8756	1.29156	0.645778	+	0.763525i	$0.276533\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.82518	0.182983
$438$	0	0
$439$	−1.17525	−0.0560915	−0.0280458	−	0.999607i	$-0.508928\pi$
$439$	−1.17525	−0.0560915	−0.0280458	+	0.999607i	$0.508928\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.1244	−0.576046	−0.288023	−	0.957624i	$-0.592998\pi$
$443$	−12.1244	−0.576046	−0.288023	+	0.957624i	$0.592998\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−13.0767	−0.618507
$448$	0	0
$449$	−25.8248	−1.21875	−0.609373	−	0.792884i	$-0.708579\pi$
$449$	−25.8248	−1.21875	−0.609373	+	0.792884i	$0.708579\pi$
$450$	0	0
$451$	25.6495	1.20779
$452$	0	0
$453$	35.1172	1.64995
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.4235	0.955372	0.477686	−	0.878531i	$-0.341476\pi$
$457$	20.4235	0.955372	0.477686	+	0.878531i	$0.341476\pi$
$458$	0	0
$459$	−24.8248	−1.15872
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	6.50958	0.302526	0.151263	−	0.988494i	$-0.451666\pi$
$463$	6.50958	0.302526	0.151263	+	0.988494i	$0.451666\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.1676	0.886973	0.443486	−	0.896281i	$-0.353741\pi$
$467$	19.1676	0.886973	0.443486	+	0.896281i	$0.353741\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	18.8248	0.867399
$472$	0	0
$473$	−14.8087	−0.680907
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12.8248	0.585978	0.292989	−	0.956116i	$-0.405350\pi$
$479$	12.8248	0.585978	0.292989	+	0.956116i	$0.405350\pi$
$480$	0	0
$481$	−34.1993	−1.55936
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	33.4427	1.51543	0.757716	−	0.652584i	$-0.226315\pi$
$487$	33.4427	1.51543	0.757716	+	0.652584i	$0.226315\pi$
$488$	0	0
$489$	12.8248	0.579955
$490$	0	0
$491$	−13.4502	−0.606997	−0.303499	−	0.952832i	$-0.598155\pi$
$491$	−13.4502	−0.606997	−0.303499	+	0.952832i	$0.598155\pi$
$492$	0	0
$493$	15.6460	0.704660
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	39.3746	1.76265	0.881324	−	0.472512i	$-0.156653\pi$
$499$	39.3746	1.76265	0.881324	+	0.472512i	$0.156653\pi$
$500$	0	0
$501$	21.8248	0.975058
$502$	0	0
$503$	16.1797	0.721418	0.360709	−	0.932678i	$-0.382535\pi$
$503$	16.1797	0.721418	0.360709	+	0.932678i	$0.382535\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−41.7419	−1.85382
$508$	0	0
$509$	29.5498	1.30977	0.654887	−	0.755727i	$-0.272716\pi$
$509$	29.5498	1.30977	0.654887	+	0.755727i	$0.272716\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	22.2131	0.980733
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.89261	−0.215177
$518$	0	0
$519$	−32.4743	−1.42546
$520$	0	0
$521$	12.8248	0.561863	0.280931	−	0.959728i	$-0.409357\pi$
$521$	12.8248	0.561863	0.280931	+	0.959728i	$0.409357\pi$
$522$	0	0
$523$	−11.7057	−0.511856	−0.255928	−	0.966696i	$-0.582381\pi$
$523$	−11.7057	−0.511856	−0.255928	+	0.966696i	$0.582381\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20.4235	−0.889663
$528$	0	0
$529$	−22.1993	−0.965189
$530$	0	0
$531$	0	0
$532$	0	0
$533$	68.6750	2.97464
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−0.476171	−0.0205483
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−2.45017	−0.105341	−0.0526704	−	0.998612i	$-0.516773\pi$
$541$	−2.45017	−0.105341	−0.0526704	+	0.998612i	$0.516773\pi$
$542$	0	0
$543$	−28.9687	−1.24317
$544$	0	0
$545$	0	0
$546$	0	0
$547$	36.1271	1.54468	0.772341	−	0.635208i	$-0.219086\pi$
$547$	36.1271	1.54468	0.772341	+	0.635208i	$0.219086\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−14.0000	−0.596420
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.61478	0.237906	0.118953	−	0.992900i	$-0.462046\pi$
$557$	5.61478	0.237906	0.118953	+	0.992900i	$0.462046\pi$
$558$	0	0
$559$	−39.6495	−1.67700
$560$	0	0
$561$	18.8248	0.794782
$562$	0	0
$563$	−12.2394	−0.515831	−0.257916	−	0.966167i	$-0.583036\pi$
$563$	−12.2394	−0.515831	−0.257916	+	0.966167i	$0.583036\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−29.3746	−1.23145	−0.615723	−	0.787962i	$-0.711136\pi$
$569$	−29.3746	−1.23145	−0.615723	+	0.787962i	$0.711136\pi$
$570$	0	0
$571$	−0.274917	−0.0115049	−0.00575246	−	0.999983i	$-0.501831\pi$
$571$	−0.274917	−0.0115049	−0.00575246	+	0.999983i	$0.501831\pi$
$572$	0	0
$573$	−39.5336	−1.65154
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.24163	0.343103	0.171552	−	0.985175i	$-0.445122\pi$
$577$	8.24163	0.343103	0.171552	+	0.985175i	$0.445122\pi$
$578$	0	0
$579$	15.9244	0.661796
$580$	0	0
$581$	0	0
$582$	0	0
$583$	16.8443	0.697621
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.9715	0.576665	0.288333	−	0.957530i	$-0.406899\pi$
$587$	13.9715	0.576665	0.288333	+	0.957530i	$0.406899\pi$
$588$	0	0
$589$	18.2749	0.753005
$590$	0	0
$591$	−45.0997	−1.85515
$592$	0	0
$593$	−20.3084	−0.833968	−0.416984	−	0.908914i	$-0.636913\pi$
$593$	−20.3084	−0.833968	−0.416984	+	0.908914i	$0.636913\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.8443	0.689393
$598$	0	0
$599$	−2.27492	−0.0929506	−0.0464753	−	0.998919i	$-0.514799\pi$
$599$	−2.27492	−0.0929506	−0.0464753	+	0.998919i	$0.514799\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.1868	1.30642	0.653211	−	0.757176i	$-0.273421\pi$
$607$	32.1868	1.30642	0.653211	+	0.757176i	$0.273421\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13.0997	−0.529956
$612$	0	0
$613$	37.0219	1.49530	0.747650	−	0.664093i	$-0.231182\pi$
$613$	37.0219	1.49530	0.747650	+	0.664093i	$0.231182\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.57919	0.144093	0.0720464	−	0.997401i	$-0.477047\pi$
$617$	3.57919	0.144093	0.0720464	+	0.997401i	$0.477047\pi$
$618$	0	0
$619$	−43.9244	−1.76547	−0.882736	−	0.469870i	$-0.844301\pi$
$619$	−43.9244	−1.76547	−0.882736	+	0.469870i	$0.844301\pi$
$620$	0	0
$621$	4.64950	0.186578
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−16.8443	−0.672698
$628$	0	0
$629$	26.8248	1.06957
$630$	0	0
$631$	2.90033	0.115460	0.0577302	−	0.998332i	$-0.481614\pi$
$631$	2.90033	0.115460	0.0577302	+	0.998332i	$0.481614\pi$
$632$	0	0
$633$	34.0339	1.35273
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	28.0997	1.10987	0.554935	−	0.831894i	$-0.312743\pi$
$641$	28.0997	1.10987	0.554935	+	0.831894i	$0.312743\pi$
$642$	0	0
$643$	−38.3353	−1.51180	−0.755898	−	0.654689i	$-0.772800\pi$
$643$	−38.3353	−1.51180	−0.755898	+	0.654689i	$0.772800\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.779710	−0.0306535	−0.0153268	−	0.999883i	$-0.504879\pi$
$647$	−0.779710	−0.0306535	−0.0153268	+	0.999883i	$0.504879\pi$
$648$	0	0
$649$	−9.72508	−0.381743
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−37.0219	−1.44878	−0.724389	−	0.689392i	$-0.757878\pi$
$653$	−37.0219	−1.44878	−0.724389	+	0.689392i	$0.757878\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−25.4502	−0.991398	−0.495699	−	0.868494i	$-0.665088\pi$
$659$	−25.4502	−0.991398	−0.495699	+	0.868494i	$0.665088\pi$
$660$	0	0
$661$	15.5498	0.604818	0.302409	−	0.953178i	$-0.402209\pi$
$661$	15.5498	0.604818	0.302409	+	0.953178i	$0.402209\pi$
$662$	0	0
$663$	50.4021	1.95746
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.93039	−0.113465
$668$	0	0
$669$	15.0997	0.583787
$670$	0	0
$671$	−3.52575	−0.136110
$672$	0	0
$673$	3.57919	0.137968	0.0689838	−	0.997618i	$-0.478024\pi$
$673$	3.57919	0.137968	0.0689838	+	0.997618i	$0.478024\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−24.6098	−0.945831	−0.472916	−	0.881108i	$-0.656798\pi$
$677$	−24.6098	−0.945831	−0.472916	+	0.881108i	$0.656798\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−11.1752	−0.428236
$682$	0	0
$683$	−15.7035	−0.600879	−0.300440	−	0.953801i	$-0.597133\pi$
$683$	−15.7035	−0.600879	−0.300440	+	0.953801i	$0.597133\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−7.40437	−0.282494
$688$	0	0
$689$	45.0997	1.71816
$690$	0	0
$691$	7.37459	0.280542	0.140271	−	0.990113i	$-0.455203\pi$
$691$	7.37459	0.280542	0.140271	+	0.990113i	$0.455203\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−53.8662	−2.04033
$698$	0	0
$699$	−32.2749	−1.22075
$700$	0	0
$701$	−13.8248	−0.522154	−0.261077	−	0.965318i	$-0.584078\pi$
$701$	−13.8248	−0.522154	−0.261077	+	0.965318i	$0.584078\pi$
$702$	0	0
$703$	−24.0027	−0.905280
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−25.5498	−0.959544	−0.479772	−	0.877393i	$-0.659281\pi$
$709$	−25.5498	−0.959544	−0.479772	+	0.877393i	$0.659281\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.82518	0.143254
$714$	0	0
$715$	0	0
$716$	0	0
$717$	25.2011	0.941151
$718$	0	0
$719$	−7.37459	−0.275026	−0.137513	−	0.990500i	$-0.543911\pi$
$719$	−7.37459	−0.275026	−0.137513	+	0.990500i	$0.543911\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−22.2131	−0.826115
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.6915	0.693228	0.346614	−	0.938008i	$-0.387331\pi$
$727$	18.6915	0.693228	0.346614	+	0.938008i	$0.387331\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	31.0997	1.15026
$732$	0	0
$733$	−33.3276	−1.23098	−0.615491	−	0.788144i	$-0.711042\pi$
$733$	−33.3276	−1.23098	−0.615491	+	0.788144i	$0.711042\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−31.6531	−1.16596
$738$	0	0
$739$	31.9244	1.17436	0.587179	−	0.809457i	$-0.300238\pi$
$739$	31.9244	1.17436	0.587179	+	0.809457i	$0.300238\pi$
$740$	0	0
$741$	−45.0997	−1.65678
$742$	0	0
$743$	−19.5287	−0.716440	−0.358220	−	0.933637i	$-0.616616\pi$
$743$	−19.5287	−0.716440	−0.358220	+	0.933637i	$0.616616\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−22.2749	−0.812823	−0.406412	−	0.913690i	$-0.633220\pi$
$751$	−22.2749	−0.812823	−0.406412	+	0.913690i	$0.633220\pi$
$752$	0	0
$753$	−9.43996	−0.344011
$754$	0	0
$755$	0	0
$756$	0	0
$757$	9.43996	0.343101	0.171551	−	0.985175i	$-0.445122\pi$
$757$	9.43996	0.343101	0.171551	+	0.985175i	$0.445122\pi$
$758$	0	0
$759$	−3.52575	−0.127976
$760$	0	0
$761$	29.9244	1.08476	0.542380	−	0.840133i	$-0.317523\pi$
$761$	29.9244	1.08476	0.542380	+	0.840133i	$0.317523\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−26.0383	−0.940189
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	−42.8248	−1.54230
$772$	0	0
$773$	27.2366	0.979634	0.489817	−	0.871825i	$-0.337064\pi$
$773$	27.2366	0.979634	0.489817	+	0.871825i	$0.337064\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	48.1993	1.72692
$780$	0	0
$781$	24.0000	0.858788
$782$	0	0
$783$	−17.0170	−0.608137
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.73205	0.0617409	0.0308705	−	0.999523i	$-0.490172\pi$
$787$	1.73205	0.0617409	0.0308705	+	0.999523i	$0.490172\pi$
$788$	0	0
$789$	−46.6495	−1.66077
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9.43996	−0.335223
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29.3873	1.04095	0.520476	−	0.853876i	$-0.325754\pi$
$797$	29.3873	1.04095	0.520476	+	0.853876i	$0.325754\pi$
$798$	0	0
$799$	10.2749	0.363500
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.89261	−0.172657
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−51.1818	−1.80169
$808$	0	0
$809$	43.1993	1.51881	0.759404	−	0.650619i	$-0.225491\pi$
$809$	43.1993	1.51881	0.759404	+	0.650619i	$0.225491\pi$
$810$	0	0
$811$	−22.5498	−0.791832	−0.395916	−	0.918287i	$-0.629573\pi$
$811$	−22.5498	−0.791832	−0.395916	+	0.918287i	$0.629573\pi$
$812$	0	0
$813$	−22.2131	−0.779048
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−27.8279	−0.973575
$818$	0	0
$819$	0	0
$820$	0	0
$821$	17.3746	0.606377	0.303189	−	0.952931i	$-0.401949\pi$
$821$	17.3746	0.606377	0.303189	+	0.952931i	$0.401949\pi$
$822$	0	0
$823$	−32.3019	−1.12597	−0.562987	−	0.826466i	$-0.690348\pi$
$823$	−32.3019	−1.12597	−0.562987	+	0.826466i	$0.690348\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−45.5670	−1.58452	−0.792261	−	0.610183i	$-0.791096\pi$
$827$	−45.5670	−1.58452	−0.792261	+	0.610183i	$0.791096\pi$
$828$	0	0
$829$	−1.92442	−0.0668379	−0.0334189	−	0.999441i	$-0.510640\pi$
$829$	−1.92442	−0.0668379	−0.0334189	+	0.999441i	$0.510640\pi$
$830$	0	0
$831$	−32.2749	−1.11960
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	22.2131	0.767798
$838$	0	0
$839$	−10.9003	−0.376321	−0.188161	−	0.982138i	$-0.560253\pi$
$839$	−10.9003	−0.376321	−0.188161	+	0.982138i	$0.560253\pi$
$840$	0	0
$841$	−18.2749	−0.630170
$842$	0	0
$843$	−10.3923	−0.357930
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−33.9244	−1.16428
$850$	0	0
$851$	−5.02409	−0.172224
$852$	0	0
$853$	30.4547	1.04275	0.521375	−	0.853327i	$-0.325419\pi$
$853$	30.4547	1.04275	0.521375	+	0.853327i	$0.325419\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−33.3276	−1.13845	−0.569224	−	0.822182i	$-0.692756\pi$
$857$	−33.3276	−1.13845	−0.569224	+	0.822182i	$0.692756\pi$
$858$	0	0
$859$	−35.3746	−1.20697	−0.603483	−	0.797376i	$-0.706221\pi$
$859$	−35.3746	−1.20697	−0.603483	+	0.797376i	$0.706221\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.1435	−0.855895	−0.427947	−	0.903804i	$-0.640763\pi$
$863$	−25.1435	−0.855895	−0.427947	+	0.903804i	$0.640763\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−10.0888	−0.342632
$868$	0	0
$869$	−0.625414	−0.0212157
$870$	0	0
$871$	−84.7492	−2.87162
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−44.6722	−1.50847	−0.754237	−	0.656602i	$-0.771993\pi$
$877$	−44.6722	−1.50847	−0.754237	+	0.656602i	$0.771993\pi$
$878$	0	0
$879$	12.0000	0.404750
$880$	0	0
$881$	−40.0241	−1.34845	−0.674223	−	0.738528i	$-0.735521\pi$
$881$	−40.0241	−1.34845	−0.674223	+	0.738528i	$0.735521\pi$
$882$	0	0
$883$	20.6695	0.695585	0.347792	−	0.937572i	$-0.386931\pi$
$883$	20.6695	0.695585	0.347792	+	0.937572i	$0.386931\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39.2301	−1.31722	−0.658609	−	0.752486i	$-0.728855\pi$
$887$	−39.2301	−1.31722	−0.658609	+	0.752486i	$0.728855\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−20.4743	−0.685913
$892$	0	0
$893$	−9.19397	−0.307664
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−9.43996	−0.315191
$898$	0	0
$899$	−14.0000	−0.466926
$900$	0	0
$901$	−35.3746	−1.17850
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	45.3210	1.50486	0.752430	−	0.658672i	$-0.228881\pi$
$907$	45.3210	1.50486	0.752430	+	0.658672i	$0.228881\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.09967	0.168960	0.0844798	−	0.996425i	$-0.473077\pi$
$911$	5.09967	0.168960	0.0844798	+	0.996425i	$0.473077\pi$
$912$	0	0
$913$	−12.9041	−0.427062
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	5.92442	0.195429	0.0977143	−	0.995215i	$-0.468847\pi$
$919$	5.92442	0.195429	0.0977143	+	0.995215i	$0.468847\pi$
$920$	0	0
$921$	−6.92442	−0.228167
$922$	0	0
$923$	64.2585	2.11509
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17.9003	0.587291	0.293645	−	0.955914i	$-0.405132\pi$
$929$	17.9003	0.587291	0.293645	+	0.955914i	$0.405132\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	22.2131	0.727225
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10.5074	0.343262	0.171631	−	0.985161i	$-0.445096\pi$
$937$	10.5074	0.343262	0.171631	+	0.985161i	$0.445096\pi$
$938$	0	0
$939$	25.0241	0.816630
$940$	0	0
$941$	−4.27492	−0.139358	−0.0696792	−	0.997569i	$-0.522198\pi$
$941$	−4.27492	−0.139358	−0.0696792	+	0.997569i	$0.522198\pi$
$942$	0	0
$943$	10.0888	0.328535
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.7419	1.35643	0.678214	−	0.734865i	$-0.262754\pi$
$947$	41.7419	1.35643	0.678214	+	0.734865i	$0.262754\pi$
$948$	0	0
$949$	−13.0997	−0.425233
$950$	0	0
$951$	−6.62541	−0.214844
$952$	0	0
$953$	−29.6175	−0.959405	−0.479702	−	0.877431i	$-0.659255\pi$
$953$	−29.6175	−0.959405	−0.479702	+	0.877431i	$0.659255\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	12.9041	0.417129
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−12.7251	−0.410487
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−6.50958	−0.209334	−0.104667	−	0.994507i	$-0.533378\pi$
$967$	−6.50958	−0.209334	−0.104667	+	0.994507i	$0.533378\pi$
$968$	0	0
$969$	35.3746	1.13640
$970$	0	0
$971$	−15.9244	−0.511039	−0.255519	−	0.966804i	$-0.582246\pi$
$971$	−15.9244	−0.511039	−0.255519	+	0.966804i	$0.582246\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16.8443	−0.538898	−0.269449	−	0.963015i	$-0.586842\pi$
$977$	−16.8443	−0.538898	−0.269449	+	0.963015i	$0.586842\pi$
$978$	0	0
$979$	15.9244	0.508947
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−37.2103	−1.18682	−0.593412	−	0.804899i	$-0.702220\pi$
$983$	−37.2103	−1.18682	−0.593412	+	0.804899i	$0.702220\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.82475	−0.185216
$990$	0	0
$991$	−34.4743	−1.09511	−0.547555	−	0.836769i	$-0.684441\pi$
$991$	−34.4743	−1.09511	−0.547555	+	0.836769i	$0.684441\pi$
$992$	0	0
$993$	8.35671	0.265192
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−4.77753	−0.151306	−0.0756529	−	0.997134i	$-0.524104\pi$
$997$	−4.77753	−0.151306	−0.0756529	+	0.997134i	$0.524104\pi$
$998$	0	0
$999$	−29.1752	−0.923064

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.a.bf.1.2		4
5.2	odd	4		980.2.e.c.589.3		4
5.3	odd	4		980.2.e.c.589.1		4
5.4	even	2	inner	4900.2.a.bf.1.4		4
7.3	odd	6		700.2.i.f.401.1		8
7.5	odd	6		700.2.i.f.501.1		8
7.6	odd	2		4900.2.a.be.1.4		4
35.2	odd	12		980.2.q.b.949.2		4
35.3	even	12		140.2.q.b.9.2	yes	4
35.12	even	12		140.2.q.b.109.1	yes	4
35.13	even	4		980.2.e.f.589.4		4
35.17	even	12		140.2.q.a.9.1	✓	4
35.18	odd	12		980.2.q.b.569.1		4
35.19	odd	6		700.2.i.f.501.4		8
35.23	odd	12		980.2.q.g.949.2		4
35.24	odd	6		700.2.i.f.401.4		8
35.27	even	4		980.2.e.f.589.2		4
35.32	odd	12		980.2.q.g.569.2		4
35.33	even	12		140.2.q.a.109.1	yes	4
35.34	odd	2		4900.2.a.be.1.2		4
105.17	odd	12		1260.2.bm.a.289.2		4
105.38	odd	12		1260.2.bm.b.289.1		4
105.47	odd	12		1260.2.bm.b.109.2		4
105.68	odd	12		1260.2.bm.a.109.2		4
140.3	odd	12		560.2.bw.a.289.2		4
140.47	odd	12		560.2.bw.a.529.1		4
140.87	odd	12		560.2.bw.e.289.1		4
140.103	odd	12		560.2.bw.e.529.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.q.a.9.1	✓	4	35.17	even	12
140.2.q.a.109.1	yes	4	35.33	even	12
140.2.q.b.9.2	yes	4	35.3	even	12
140.2.q.b.109.1	yes	4	35.12	even	12
560.2.bw.a.289.2		4	140.3	odd	12
560.2.bw.a.529.1		4	140.47	odd	12
560.2.bw.e.289.1		4	140.87	odd	12
560.2.bw.e.529.1		4	140.103	odd	12
700.2.i.f.401.1		8	7.3	odd	6
700.2.i.f.401.4		8	35.24	odd	6
700.2.i.f.501.1		8	7.5	odd	6
700.2.i.f.501.4		8	35.19	odd	6
980.2.e.c.589.1		4	5.3	odd	4
980.2.e.c.589.3		4	5.2	odd	4
980.2.e.f.589.2		4	35.27	even	4
980.2.e.f.589.4		4	35.13	even	4
980.2.q.b.569.1		4	35.18	odd	12
980.2.q.b.949.2		4	35.2	odd	12
980.2.q.g.569.2		4	35.32	odd	12
980.2.q.g.949.2		4	35.23	odd	12
1260.2.bm.a.109.2		4	105.68	odd	12
1260.2.bm.a.289.2		4	105.17	odd	12
1260.2.bm.b.109.2		4	105.47	odd	12
1260.2.bm.b.289.1		4	105.38	odd	12
4900.2.a.be.1.2		4	35.34	odd	2
4900.2.a.be.1.4		4	7.6	odd	2
4900.2.a.bf.1.2		4	1.1	even	1	trivial
4900.2.a.bf.1.4		4	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +2.27492 q^{11} +6.09095 q^{13} -4.77753 q^{17} +4.27492 q^{19} +0.894797 q^{23} +5.19615 q^{27} -3.27492 q^{29} +4.27492 q^{31} -3.94027 q^{33} -5.61478 q^{37} -10.5498 q^{39} +11.2749 q^{41} -6.50958 q^{43} -2.15068 q^{47} +8.27492 q^{51} +7.40437 q^{53} -7.40437 q^{57} -4.27492 q^{59} -1.54983 q^{61} -13.9140 q^{67} -1.54983 q^{69} +10.5498 q^{71} -2.15068 q^{73} -0.274917 q^{79} -9.00000 q^{81} -5.67232 q^{83} +5.67232 q^{87} +7.00000 q^{89} -7.40437 q^{93} +6.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{11} + 2 q^{19} + 2 q^{29} + 2 q^{31} - 12 q^{39} + 30 q^{41} + 18 q^{51} - 2 q^{59} + 24 q^{61} + 24 q^{69} + 12 q^{71} + 14 q^{79} - 36 q^{81} + 28 q^{89}+O(q^{100}) \) 4 * q - 6 * q^11 + 2 * q^19 + 2 * q^29 + 2 * q^31 - 12 * q^39 + 30 * q^41 + 18 * q^51 - 2 * q^59 + 24 * q^61 + 24 * q^69 + 12 * q^71 + 14 * q^79 - 36 * q^81 + 28 * q^89

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