LMFDB - Embedded newform 980.2.c.b.979.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,2,Mod(979,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

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

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

chi := DirichletCharacter("980.979");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$7.82533939809$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3317760000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 $ x^8 - 4x^6 + 7x^4 - 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			979.1
Root			$-1.01575 - 1.40294i$ of defining polynomial
Character	$\chi$	$=$	980.979
Dual form			980.2.c.b.979.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -2.80588i q^{3} +2.00000 q^{4} +2.23607i q^{5} +3.96812i q^{6} -2.82843 q^{8} -4.87298 q^{9} +O(q^{10})$	$q-1.41421 q^{2} -2.80588i q^{3} +2.00000 q^{4} +2.23607i q^{5} +3.96812i q^{6} -2.82843 q^{8} -4.87298 q^{9} -3.16228i q^{10} -5.61177i q^{12} +6.27415 q^{15} +4.00000 q^{16} +6.89144 q^{18} +4.47214i q^{20} -8.92295 q^{23} +7.93624i q^{24} -5.00000 q^{25} +5.25537i q^{27} -10.7460 q^{29} -8.87298 q^{30} -5.65685 q^{32} -9.74597 q^{36} -6.32456i q^{40} -8.15624i q^{41} -3.62535 q^{43} -10.8963i q^{45} +12.6190 q^{46} +9.48683i q^{47} -11.2235i q^{48} +7.07107 q^{50} -7.43222i q^{54} +15.1971 q^{58} +12.5483 q^{60} +0.219999i q^{61} +8.00000 q^{64} -11.5717 q^{67} +25.0367i q^{69} +13.7829 q^{72} +14.0294i q^{75} +8.94427i q^{80} +0.127017 q^{81} +11.5347i q^{82} -8.72878i q^{83} +5.12702 q^{86} +30.1519i q^{87} +3.74812i q^{89} +15.4097i q^{90} -17.8459 q^{92} -13.4164i q^{94} +15.8725i q^{96} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} - 8 q^{9}+O(q^{10}) $ 8 * q + 16 * q^4 - 8 * q^9	$ 8 q + 16 q^{4} - 8 q^{9} + 32 q^{16} - 40 q^{25} - 24 q^{29} - 40 q^{30} - 16 q^{36} + 8 q^{46} + 64 q^{64} + 32 q^{81} + 72 q^{86}+O(q^{100}) $ 8 * q + 16 * q^4 - 8 * q^9 + 32 * q^16 - 40 * q^25 - 24 * q^29 - 40 * q^30 - 16 * q^36 + 8 * q^46 + 64 * q^64 + 32 * q^81 + 72 * q^86

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	−1.41421	−1.00000
$2$	−1.41421	−1.00000
$3$	− 2.80588i	− 1.61998i	−0.586445	−	0.809989i	$-0.699473\pi$
$3$	− 2.80588i	− 1.61998i	0.586445	−	0.809989i	$-0.300527\pi$
$4$	2.00000	1.00000
$5$	2.23607i	1.00000i
$5$	2.23607i	1.00000i
$6$	3.96812i	1.61998i
$7$	0	0
$7$	0	0
$8$	−2.82843	−1.00000
$9$	−4.87298	−1.62433
$10$	− 3.16228i	− 1.00000i
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	− 5.61177i	− 1.61998i
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	6.27415	1.61998
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	6.89144	1.62433
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	4.47214i	1.00000i
$21$	0	0
$22$	0	0
$23$	−8.92295	−1.86056	−0.930281	−	0.366847i	$-0.880437\pi$
$23$	−8.92295	−1.86056	−0.930281	+	0.366847i	$0.880437\pi$
$24$	7.93624i	1.61998i
$25$	−5.00000	−1.00000
$26$	0	0
$27$	5.25537i	1.01140i
$28$	0	0
$29$	−10.7460	−1.99548	−0.997738	−	0.0672232i	$-0.978586\pi$
$29$	−10.7460	−1.99548	−0.997738	+	0.0672232i	$0.978586\pi$
$30$	−8.87298	−1.61998
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−5.65685	−1.00000
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−9.74597	−1.62433
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	− 6.32456i	− 1.00000i
$41$	− 8.15624i	− 1.27379i	−0.770950	−	0.636895i	$-0.780218\pi$
$41$	− 8.15624i	− 1.27379i	0.770950	−	0.636895i	$-0.219782\pi$
$42$	0	0
$43$	−3.62535	−0.552860	−0.276430	−	0.961034i	$-0.589151\pi$
$43$	−3.62535	−0.552860	−0.276430	+	0.961034i	$0.589151\pi$
$44$	0	0
$45$	− 10.8963i	− 1.62433i
$46$	12.6190	1.86056
$47$	9.48683i	1.38380i	0.721995	+	0.691898i	$0.243225\pi$
$47$	9.48683i	1.38380i	−0.721995	+	0.691898i	$0.756775\pi$
$48$	− 11.2235i	− 1.61998i
$49$	0	0
$50$	7.07107	1.00000
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	− 7.43222i	− 1.01140i
$55$	0	0
$56$	0	0
$57$	0	0
$58$	15.1971	1.99548
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	12.5483	1.61998
$61$	0.219999i	0.0281680i	0.999901	+	0.0140840i	$0.00448323\pi$
$61$	0.219999i	0.0281680i	−0.999901	+	0.0140840i	$0.995517\pi$
$62$	0	0
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	−11.5717	−1.41371	−0.706857	−	0.707357i	$-0.749887\pi$
$67$	−11.5717	−1.41371	−0.706857	+	0.707357i	$0.749887\pi$
$68$	0	0
$69$	25.0367i	3.01407i
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	13.7829	1.62433
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	14.0294i	1.61998i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	8.94427i	1.00000i
$81$	0.127017	0.0141130
$82$	11.5347i	1.27379i
$83$	− 8.72878i	− 0.958108i	−0.877785	−	0.479054i	$-0.840980\pi$
$83$	− 8.72878i	− 0.958108i	0.877785	−	0.479054i	$-0.159020\pi$
$84$	0	0
$85$	0	0
$86$	5.12702	0.552860
$87$	30.1519i	3.23263i
$88$	0	0
$89$	3.74812i	0.397300i	0.980071	+	0.198650i	$0.0636557\pi$
$89$	3.74812i	0.397300i	−0.980071	+	0.198650i	$0.936344\pi$
$90$	15.4097i	1.62433i
$91$	0	0
$92$	−17.8459	−1.86056
$93$	0	0
$94$	− 13.4164i	− 1.38380i
$95$	0	0
$96$	15.8725i	1.61998i
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	0	0
$100$	−10.0000	−1.00000
$101$	20.0606i	1.99610i	0.0623905	+	0.998052i	$0.480128\pi$
$101$	20.0606i	1.99610i	−0.0623905	+	0.998052i	$0.519872\pi$
$102$	0	0
$103$	− 3.11701i	− 0.307128i	−0.988139	−	0.153564i	$-0.950925\pi$
$103$	− 3.11701i	− 0.307128i	0.988139	−	0.153564i	$-0.0490751\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.976550	0.0944066	0.0472033	−	0.998885i	$-0.484969\pi$
$107$	0.976550	0.0944066	0.0472033	+	0.998885i	$0.484969\pi$
$108$	10.5107i	1.01140i
$109$	19.6190	1.87915	0.939577	−	0.342337i	$-0.111218\pi$
$109$	19.6190	1.87915	0.939577	+	0.342337i	$0.111218\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	− 19.9523i	− 1.86056i
$116$	−21.4919	−1.99548
$117$	0	0
$118$	0	0
$119$	0	0
$120$	−17.7460	−1.61998
$121$	11.0000	1.00000
$122$	− 0.311126i	− 0.0281680i
$123$	−22.8855	−2.06351
$124$	0	0
$125$	− 11.1803i	− 1.00000i
$126$	0	0
$127$	4.24264	0.376473	0.188237	−	0.982124i	$-0.439723\pi$
$127$	4.24264	0.376473	0.188237	+	0.982124i	$0.439723\pi$
$128$	−11.3137	−1.00000
$129$	10.1723i	0.895622i
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	16.3649	1.41371
$135$	−11.7514	−1.01140
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	− 35.4073i	− 3.01407i
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	26.6190	2.24172
$142$	0	0
$143$	0	0
$144$	−19.4919	−1.62433
$145$	− 24.0287i	− 1.99548i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.8730	−1.30037	−0.650183	−	0.759778i	$-0.725308\pi$
$149$	−15.8730	−1.30037	−0.650183	+	0.759778i	$0.725308\pi$
$150$	− 19.8406i	− 1.61998i
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	− 12.6491i	− 1.00000i
$161$	0	0
$162$	−0.179629	−0.0141130
$163$	−12.7279	−0.996928	−0.498464	−	0.866910i	$-0.666102\pi$
$163$	−12.7279	−0.996928	−0.498464	+	0.866910i	$0.666102\pi$
$164$	− 16.3125i	− 1.27379i
$165$	0	0
$166$	12.3444i	0.958108i
$167$	25.5641i	1.97821i	0.147219	+	0.989104i	$0.452968\pi$
$167$	25.5641i	1.97821i	−0.147219	+	0.989104i	$0.547032\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	−7.25070	−0.552860
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	− 42.6413i	− 3.23263i
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 5.30064i	− 0.397300i
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	− 21.7926i	− 1.62433i
$181$	− 11.6844i	− 0.868491i	−0.900794	−	0.434246i	$-0.857015\pi$
$181$	− 11.6844i	− 0.868491i	0.900794	−	0.434246i	$-0.142985\pi$
$182$	0	0
$183$	0.617292	0.0456316
$184$	25.2379	1.86056
$185$	0	0
$186$	0	0
$187$	0	0
$188$	18.9737i	1.38380i
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	− 22.4471i	− 1.61998i
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	14.1421	1.00000
$201$	32.4690i	2.29018i
$202$	− 28.3700i	− 1.99610i
$203$	0	0
$204$	0	0
$205$	18.2379	1.27379
$206$	4.40812i	0.307128i
$207$	43.4814	3.02216
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	−1.38105	−0.0944066
$215$	− 8.10653i	− 0.552860i
$216$	− 14.8644i	− 1.01140i
$217$	0	0
$218$	−27.7454	−1.87915
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 3.16228i	− 0.211762i	−0.994379	−	0.105881i	$-0.966234\pi$
$223$	− 3.16228i	− 0.211762i	0.994379	−	0.105881i	$-0.0337662\pi$
$224$	0	0
$225$	24.3649	1.62433
$226$	0	0
$227$	− 28.4605i	− 1.88899i	−0.328526	−	0.944495i	$-0.606552\pi$
$227$	− 28.4605i	− 1.88899i	0.328526	−	0.944495i	$-0.393448\pi$
$228$	0	0
$229$	− 26.8328i	− 1.77316i	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	− 26.8328i	− 1.77316i	0.462573	−	0.886581i	$-0.346926\pi$
$230$	28.2168i	1.86056i
$231$	0	0
$232$	30.3942	1.99548
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	−21.2132	−1.38380
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	25.0966	1.61998
$241$	− 13.4164i	− 0.864227i	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	− 13.4164i	− 0.864227i	0.901819	−	0.432113i	$-0.142232\pi$
$242$	−15.5563	−1.00000
$243$	15.4097i	0.988534i
$244$	0.439999i	0.0281680i
$245$	0	0
$246$	32.3649	2.06351
$247$	0	0
$248$	0	0
$249$	−24.4919	−1.55211
$250$	15.8114i	1.00000i
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	−6.00000	−0.376473
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	− 14.3858i	− 0.895622i
$259$	0	0
$260$	0	0
$261$	52.3649	3.24131
$262$	0	0
$263$	16.8693	1.04021	0.520104	−	0.854103i	$-0.325893\pi$
$263$	16.8693	1.04021	0.520104	+	0.854103i	$0.325893\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	10.5168	0.643617
$268$	−23.1435	−1.41371
$269$	− 31.9649i	− 1.94894i	−0.224523	−	0.974469i	$-0.572083\pi$
$269$	− 31.9649i	− 1.94894i	0.224523	−	0.974469i	$-0.427917\pi$
$270$	16.6190	1.01140
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	50.0735i	3.01407i
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	−37.6449	−2.24172
$283$	15.8114i	0.939889i	0.882696	+	0.469945i	$0.155726\pi$
$283$	15.8114i	0.939889i	−0.882696	+	0.469945i	$0.844274\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	27.5658	1.62433
$289$	−17.0000	−1.00000
$290$	33.9817i	1.99548i
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	22.4478	1.30037
$299$	0	0
$300$	28.0588i	1.61998i
$301$	0	0
$302$	0	0
$303$	56.2877	3.23364
$304$	0	0
$305$	−0.491933	−0.0281680
$306$	0	0
$307$	− 13.7183i	− 0.782944i	−0.920190	−	0.391472i	$-0.871966\pi$
$307$	− 13.7183i	− 0.782944i	0.920190	−	0.391472i	$-0.128034\pi$
$308$	0	0
$309$	−8.74597	−0.497541
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	17.8885i	1.00000i
$321$	− 2.74009i	− 0.152937i
$322$	0	0
$323$	0	0
$324$	0.254033	0.0141130
$325$	0	0
$326$	18.0000	0.996928
$327$	− 55.0485i	− 3.04419i
$328$	23.0693i	1.27379i
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	− 17.4576i	− 0.958108i
$333$	0	0
$334$	− 36.1531i	− 1.97821i
$335$	− 25.8752i	− 1.41371i
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	18.3848	1.00000
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	10.2540	0.552860
$345$	−55.9839	−3.01407
$346$	0	0
$347$	36.6683	1.96846	0.984230	−	0.176896i	$-0.0566056\pi$
$347$	36.6683	1.96846	0.984230	+	0.176896i	$0.0566056\pi$
$348$	60.3039i	3.23263i
$349$	− 35.9331i	− 1.92345i	−0.274011	−	0.961727i	$-0.588351\pi$
$349$	− 35.9331i	− 1.92345i	0.274011	−	0.961727i	$-0.411649\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	7.49624i	0.397300i
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	30.8195i	1.62433i
$361$	−19.0000	−1.00000
$362$	16.5242i	0.868491i
$363$	− 30.8647i	− 1.61998i
$364$	0	0
$365$	0	0
$366$	−0.872983	−0.0456316
$367$	31.4870i	1.64361i	0.569771	+	0.821803i	$0.307032\pi$
$367$	31.4870i	1.64361i	−0.569771	+	0.821803i	$0.692968\pi$
$368$	−35.6918	−1.86056
$369$	39.7452i	2.06905i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−31.3707	−1.61998
$376$	− 26.8328i	− 1.38380i
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	− 11.9044i	− 0.609879i
$382$	0	0
$383$	9.03990i	0.461917i	0.972964	+	0.230959i	$0.0741862\pi$
$383$	9.03990i	0.461917i	−0.972964	+	0.230959i	$0.925814\pi$
$384$	31.7450i	1.61998i
$385$	0	0
$386$	0	0
$387$	17.6663	0.898027
$388$	0	0
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	−20.0000	−1.00000
$401$	−21.9839	−1.09782	−0.548911	−	0.835881i	$-0.684957\pi$
$401$	−21.9839	−1.09782	−0.548911	+	0.835881i	$0.684957\pi$
$402$	− 45.9181i	− 2.29018i
$403$	0	0
$404$	40.1212i	1.99610i
$405$	0.284018i	0.0141130i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	23.5887i	1.16639i	0.812333	+	0.583193i	$0.198197\pi$
$409$	23.5887i	1.16639i	−0.812333	+	0.583193i	$0.801803\pi$
$410$	−25.7923	−1.27379
$411$	0	0
$412$	− 6.23402i	− 0.307128i
$413$	0	0
$414$	−61.4919	−3.02216
$415$	19.5181	0.958108
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−30.8569	−1.50387	−0.751935	−	0.659237i	$-0.770879\pi$
$421$	−30.8569	−1.50387	−0.751935	+	0.659237i	$0.770879\pi$
$422$	0	0
$423$	− 46.2292i	− 2.24774i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	1.95310	0.0944066
$429$	0	0
$430$	11.4644i	0.552860i
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	21.0215i	1.01140i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	−67.4218	−3.23263
$436$	39.2379	1.87915
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−28.7219	−1.36462	−0.682310	−	0.731063i	$-0.739025\pi$
$443$	−28.7219	−1.36462	−0.682310	+	0.731063i	$0.739025\pi$
$444$	0	0
$445$	−8.38105	−0.397300
$446$	4.47214i	0.211762i
$447$	44.5377i	2.10656i
$448$	0	0
$449$	37.3649	1.76336	0.881680	−	0.471848i	$-0.156413\pi$
$449$	37.3649	1.76336	0.881680	+	0.471848i	$0.156413\pi$
$450$	−34.4572	−1.62433
$451$	0	0
$452$	0	0
$453$	0	0
$454$	40.2492i	1.88899i
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	37.9473i	1.77316i
$459$	0	0
$460$	− 39.9046i	− 1.86056i
$461$	8.94427i	0.416576i	0.978068	+	0.208288i	$0.0667892\pi$
$461$	8.94427i	0.416576i	−0.978068	+	0.208288i	$0.933211\pi$
$462$	0	0
$463$	−41.9659	−1.95032	−0.975161	−	0.221497i	$-0.928906\pi$
$463$	−41.9659	−1.95032	−0.975161	+	0.221497i	$0.928906\pi$
$464$	−42.9839	−1.99548
$465$	0	0
$466$	0	0
$467$	− 42.3994i	− 1.96201i	−0.193984	−	0.981005i	$-0.562141\pi$
$467$	− 42.3994i	− 1.96201i	0.193984	−	0.981005i	$-0.437859\pi$
$468$	0	0
$469$	0	0
$470$	30.0000	1.38380
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−35.4919	−1.61998
$481$	0	0
$482$	18.9737i	0.864227i
$483$	0	0
$484$	22.0000	1.00000
$485$	0	0
$486$	− 21.7926i	− 0.988534i
$487$	38.1838	1.73027	0.865136	−	0.501538i	$-0.167232\pi$
$487$	38.1838	1.73027	0.865136	+	0.501538i	$0.167232\pi$
$488$	− 0.622252i	− 0.0281680i
$489$	35.7131i	1.61500i
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−45.7709	−2.06351
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	34.6368	1.55211
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 22.3607i	− 1.00000i
$501$	71.7298	3.20465
$502$	0	0
$503$	42.7105i	1.90437i	0.305526	+	0.952184i	$0.401168\pi$
$503$	42.7105i	1.90437i	−0.305526	+	0.952184i	$0.598832\pi$
$504$	0	0
$505$	−44.8569	−1.99610
$506$	0	0
$507$	36.4765i	1.61998i
$508$	8.48528	0.376473
$509$	− 27.5568i	− 1.22144i	−0.791849	−	0.610718i	$-0.790881\pi$
$509$	− 27.5568i	− 1.22144i	0.791849	−	0.610718i	$-0.209119\pi$
$510$	0	0
$511$	0	0
$512$	−22.6274	−1.00000
$513$	0	0
$514$	0	0
$515$	6.96985	0.307128
$516$	20.3446i	0.895622i
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17.8885i	0.783711i	0.920027	+	0.391856i	$0.128167\pi$
$521$	17.8885i	0.783711i	−0.920027	+	0.391856i	$0.871833\pi$
$522$	−74.0552	−3.24131
$523$	− 34.7851i	− 1.52104i	−0.649312	−	0.760522i	$-0.724943\pi$
$523$	− 34.7851i	− 1.52104i	0.649312	−	0.760522i	$-0.275057\pi$
$524$	0	0
$525$	0	0
$526$	−23.8569	−1.04021
$527$	0	0
$528$	0	0
$529$	56.6190	2.46169
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	−14.8730	−0.643617
$535$	2.18363i	0.0944066i
$536$	32.7298	1.41371
$537$	0	0
$538$	45.2053i	1.94894i
$539$	0	0
$540$	−23.5027	−1.01140
$541$	4.23790	0.182202	0.0911008	−	0.995842i	$-0.470961\pi$
$541$	4.23790	0.182202	0.0911008	+	0.995842i	$0.470961\pi$
$542$	0	0
$543$	−32.7849	−1.40694
$544$	0	0
$545$	43.8693i	1.87915i
$546$	0	0
$547$	26.0731	1.11481	0.557403	−	0.830242i	$-0.311798\pi$
$547$	26.0731	1.11481	0.557403	+	0.830242i	$0.311798\pi$
$548$	0	0
$549$	− 1.07205i	− 0.0457541i
$550$	0	0
$551$	0	0
$552$	− 70.8146i	− 3.01407i
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	16.9706	0.715860
$563$	− 24.9418i	− 1.05117i	−0.850740	−	0.525586i	$-0.823846\pi$
$563$	− 24.9418i	− 1.05117i	0.850740	−	0.525586i	$-0.176154\pi$
$564$	53.2379	2.24172
$565$	0	0
$566$	− 22.3607i	− 0.939889i
$567$	0	0
$568$	0	0
$569$	36.0000	1.50920	0.754599	−	0.656186i	$-0.227831\pi$
$569$	36.0000	1.50920	0.754599	+	0.656186i	$0.227831\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	44.6147	1.86056
$576$	−38.9839	−1.62433
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	24.0416	1.00000
$579$	0	0
$580$	− 48.0574i	− 1.99548i
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 47.4342i	− 1.95782i	−0.204298	−	0.978909i	$-0.565491\pi$
$587$	− 47.4342i	− 1.95782i	0.204298	−	0.978909i	$-0.434509\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	−31.7460	−1.30037
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	− 39.6812i	− 1.61998i
$601$	40.2492i	1.64180i	0.571072	+	0.820900i	$0.306528\pi$
$601$	40.2492i	1.64180i	−0.571072	+	0.820900i	$0.693472\pi$
$602$	0	0
$603$	56.3889	2.29633
$604$	0	0
$605$	24.5967i	1.00000i
$606$	−79.6028	−3.23364
$607$	48.3223i	1.96134i	0.195667	+	0.980670i	$0.437313\pi$
$607$	48.3223i	1.96134i	−0.195667	+	0.980670i	$0.562687\pi$
$608$	0	0
$609$	0	0
$610$	0.695699	0.0281680
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	19.4006i	0.782944i
$615$	− 51.1734i	− 2.06351i
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	12.3687	0.497541
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	− 46.8934i	− 1.88177i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9.48683i	0.376473i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	− 25.2982i	− 1.00000i
$641$	−48.6028	−1.91970	−0.959848	−	0.280521i	$-0.909493\pi$
$641$	−48.6028	−1.91970	−0.959848	+	0.280521i	$0.909493\pi$
$642$	3.87507i	0.152937i
$643$	41.1096i	1.62120i	0.585597	+	0.810602i	$0.300860\pi$
$643$	41.1096i	1.62120i	−0.585597	+	0.810602i	$0.699140\pi$
$644$	0	0
$645$	−22.7460	−0.895622
$646$	0	0
$647$	− 7.79540i	− 0.306469i	−0.988190	−	0.153234i	$-0.951031\pi$
$647$	− 7.79540i	− 0.306469i	0.988190	−	0.153234i	$-0.0489689\pi$
$648$	−0.359257	−0.0141130
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−25.4558	−0.996928
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	77.8503i	3.04419i
$655$	0	0
$656$	− 32.6249i	− 1.27379i
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	47.8374i	1.86066i	0.366723	+	0.930330i	$0.380480\pi$
$661$	47.8374i	1.86066i	−0.366723	+	0.930330i	$0.619520\pi$
$662$	0	0
$663$	0	0
$664$	24.6887i	0.958108i
$665$	0	0
$666$	0	0
$667$	95.8857	3.71271
$668$	51.1282i	1.97821i
$669$	−8.87298	−0.343049
$670$	36.5931i	1.41371i
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	− 26.2769i	− 1.01140i
$676$	−26.0000	−1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−79.8569	−3.06012
$682$	0	0
$683$	46.5678	1.78187	0.890934	−	0.454132i	$-0.150051\pi$
$683$	46.5678	1.78187	0.890934	+	0.454132i	$0.150051\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−75.2898	−2.87248
$688$	−14.5014	−0.552860
$689$	0	0
$690$	79.1731	3.01407
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	−51.8569	−1.96846
$695$	0	0
$696$	− 85.2825i	− 3.23263i
$697$	0	0
$698$	50.8170i	1.92345i
$699$	0	0
$700$	0	0
$701$	−4.63508	−0.175065	−0.0875323	−	0.996162i	$-0.527898\pi$
$701$	−4.63508	−0.175065	−0.0875323	+	0.996162i	$0.527898\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	59.5218i	2.24172i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	46.2379	1.73650	0.868250	−	0.496126i	$-0.165245\pi$
$709$	46.2379	1.73650	0.868250	+	0.496126i	$0.165245\pi$
$710$	0	0
$711$	0	0
$712$	− 10.6013i	− 0.397300i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	− 43.5853i	− 1.62433i
$721$	0	0
$722$	26.8701	1.00000
$723$	−37.6449	−1.40003
$724$	− 23.3687i	− 0.868491i
$725$	53.7298	1.99548
$726$	43.6493i	1.61998i
$727$	30.5536i	1.13317i	0.824003	+	0.566585i	$0.191736\pi$
$727$	30.5536i	1.13317i	−0.824003	+	0.566585i	$0.808264\pi$
$728$	0	0
$729$	43.6190	1.61552
$730$	0	0
$731$	0	0
$732$	1.23458	0.0456316
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	− 44.5293i	− 1.64361i
$735$	0	0
$736$	50.4758	1.86056
$737$	0	0
$738$	− 56.2082i	− 2.06905i
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	54.5142	1.99993	0.999966	−	0.00819813i	$-0.00260958\pi$
$743$	54.5142	1.99993	0.999966	+	0.00819813i	$0.00260958\pi$
$744$	0	0
$745$	− 35.4931i	− 1.30037i
$746$	0	0
$747$	42.5352i	1.55628i
$748$	0	0
$749$	0	0
$750$	44.3649	1.61998
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	37.9473i	1.38380i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	35.7771i	1.29692i	0.761249	+	0.648459i	$0.224586\pi$
$761$	35.7771i	1.29692i	−0.761249	+	0.648459i	$0.775414\pi$
$762$	16.8353i	0.609879i
$763$	0	0
$764$	0	0
$765$	0	0
$766$	− 12.7844i	− 0.461917i
$767$	0	0
$768$	− 44.8941i	− 1.61998i
$769$	− 53.6656i	− 1.93523i	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	− 53.6656i	− 1.93523i	0.252426	−	0.967616i	$-0.418771\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	−24.9839	−0.898027
$775$	0	0
$776$	0	0
$777$	0	0
$778$	33.9411	1.21685
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 56.4741i	− 2.01822i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	53.6229i	1.91145i	0.294260	+	0.955725i	$0.404927\pi$
$787$	53.6229i	1.91145i	−0.294260	+	0.955725i	$0.595073\pi$
$788$	0	0
$789$	− 47.3334i	− 1.68511i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	28.2843	1.00000
$801$	− 18.2645i	− 0.645345i
$802$	31.0899	1.09782
$803$	0	0
$804$	64.9379i	2.29018i
$805$	0	0
$806$	0	0
$807$	−89.6899	−3.15724
$808$	− 56.7399i	− 1.99610i
$809$	−42.4919	−1.49394	−0.746968	−	0.664860i	$-0.768491\pi$
$809$	−42.4919	−1.49394	−0.746968	+	0.664860i	$0.768491\pi$
$810$	− 0.401662i	− 0.0141130i
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 28.4605i	− 0.996928i
$816$	0	0
$817$	0	0
$818$	− 33.3595i	− 1.16639i
$819$	0	0
$820$	36.4758	1.27379
$821$	−48.0000	−1.67521	−0.837606	−	0.546275i	$-0.816045\pi$
$821$	−48.0000	−1.67521	−0.837606	+	0.546275i	$0.816045\pi$
$822$	0	0
$823$	−41.2702	−1.43859	−0.719295	−	0.694705i	$-0.755535\pi$
$823$	−41.2702	−1.43859	−0.719295	+	0.694705i	$0.755535\pi$
$824$	8.81623i	0.307128i
$825$	0	0
$826$	0	0
$827$	24.8157	0.862928	0.431464	−	0.902130i	$-0.357997\pi$
$827$	24.8157	0.862928	0.431464	+	0.902130i	$0.357997\pi$
$828$	86.9627	3.02216
$829$	− 13.4164i	− 0.465971i	−0.972480	−	0.232986i	$-0.925151\pi$
$829$	− 13.4164i	− 0.465971i	0.972480	−	0.232986i	$-0.0748495\pi$
$830$	−27.6028	−0.958108
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−57.1630	−1.97821
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	86.4758	2.98192
$842$	43.6382	1.50387
$843$	33.6706i	1.15968i
$844$	0	0
$845$	− 29.0689i	− 1.00000i
$846$	65.3779i	2.24774i
$847$	0	0
$848$	0	0
$849$	44.3649	1.52260
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	−2.76210	−0.0944066
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	− 16.2131i	− 0.552860i
$861$	0	0
$862$	0	0
$863$	20.7755	0.707208	0.353604	−	0.935395i	$-0.384956\pi$
$863$	20.7755	0.707208	0.353604	+	0.935395i	$0.384956\pi$
$864$	− 29.7289i	− 1.01140i
$865$	0	0
$866$	0	0
$867$	47.7000i	1.61998i
$868$	0	0
$869$	0	0
$870$	95.3488	3.23263
$871$	0	0
$872$	−55.4908	−1.87915
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 39.4612i	− 1.32948i	−0.747074	−	0.664741i	$-0.768542\pi$
$881$	− 39.4612i	− 1.32948i	0.747074	−	0.664741i	$-0.231458\pi$
$882$	0	0
$883$	−55.1543	−1.85609	−0.928045	−	0.372467i	$-0.878512\pi$
$883$	−55.1543	−1.85609	−0.928045	+	0.372467i	$0.878512\pi$
$884$	0	0
$885$	0	0
$886$	40.6190	1.36462
$887$	− 59.5458i	− 1.99935i	−0.0254417	−	0.999676i	$-0.508099\pi$
$887$	− 59.5458i	− 1.99935i	0.0254417	−	0.999676i	$-0.491901\pi$
$888$	0	0
$889$	0	0
$890$	11.8526	0.397300
$891$	0	0
$892$	− 6.32456i	− 0.211762i
$893$	0	0
$894$	− 62.9859i	− 2.10656i
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−52.8420	−1.76336
$899$	0	0
$900$	48.7298	1.62433
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	26.1270	0.868491
$906$	0	0
$907$	−49.9123	−1.65731	−0.828656	−	0.559759i	$-0.810894\pi$
$907$	−49.9123	−1.65731	−0.828656	+	0.559759i	$0.810894\pi$
$908$	− 56.9210i	− 1.88899i
$909$	− 97.7549i	− 3.24233i
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	1.38031i	0.0456316i
$916$	− 53.6656i	− 1.77316i
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	56.4337i	1.86056i
$921$	−38.4919	−1.26835
$922$	− 12.6491i	− 0.416576i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	59.3488	1.95032
$927$	15.1891i	0.498877i
$928$	60.7884	1.99548
$929$	55.7737i	1.82987i	0.403596	+	0.914937i	$0.367760\pi$
$929$	55.7737i	1.82987i	−0.403596	+	0.914937i	$0.632240\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	59.9618i	1.96201i
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	−42.4264	−1.38380
$941$	− 44.7214i	− 1.45787i	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	− 44.7214i	− 1.45787i	0.684580	−	0.728937i	$-0.259985\pi$
$942$	0	0
$943$	72.7777i	2.36997i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−38.6214	−1.25503	−0.627514	−	0.778605i	$-0.715927\pi$
$947$	−38.6214	−1.25503	−0.627514	+	0.778605i	$0.715927\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	50.1932	1.61998
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−4.75871	−0.153347
$964$	− 26.8328i	− 0.864227i
$965$	0	0
$966$	0	0
$967$	−12.2674	−0.394494	−0.197247	−	0.980354i	$-0.563200\pi$
$967$	−12.2674	−0.394494	−0.197247	+	0.980354i	$0.563200\pi$
$968$	−31.1127	−1.00000
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	30.8195i	0.988534i
$973$	0	0
$974$	−54.0000	−1.73027
$975$	0	0
$976$	0.879997i	0.0281680i
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	− 50.5059i	− 1.61500i
$979$	0	0
$980$	0	0
$981$	−95.6028	−3.05236
$982$	0	0
$983$	− 59.2347i	− 1.88929i	−0.328090	−	0.944646i	$-0.606405\pi$
$983$	− 59.2347i	− 1.88929i	0.328090	−	0.944646i	$-0.393595\pi$
$984$	64.7298	2.06351
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	32.3488	1.02863
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	−48.9839	−1.55211
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.c.b.979.1		8
4.3	odd	2	inner	980.2.c.b.979.8		8
5.4	even	2	inner	980.2.c.b.979.8		8
7.2	even	3		980.2.s.a.619.4		8
7.3	odd	6		980.2.s.a.19.4		8
7.4	even	3		140.2.s.a.19.3	yes	8
7.5	odd	6		140.2.s.a.59.3	yes	8
7.6	odd	2	inner	980.2.c.b.979.4		8
20.19	odd	2	CM	980.2.c.b.979.1		8
28.3	even	6		980.2.s.a.19.1		8
28.11	odd	6		140.2.s.a.19.2	✓	8
28.19	even	6		140.2.s.a.59.2	yes	8
28.23	odd	6		980.2.s.a.619.1		8
28.27	even	2	inner	980.2.c.b.979.5		8
35.4	even	6		140.2.s.a.19.2	✓	8
35.9	even	6		980.2.s.a.619.1		8
35.12	even	12		700.2.p.b.451.1		8
35.18	odd	12		700.2.p.b.551.1		8
35.19	odd	6		140.2.s.a.59.2	yes	8
35.24	odd	6		980.2.s.a.19.1		8
35.32	odd	12		700.2.p.b.551.4		8
35.33	even	12		700.2.p.b.451.4		8
35.34	odd	2	inner	980.2.c.b.979.5		8
140.19	even	6		140.2.s.a.59.3	yes	8
140.39	odd	6		140.2.s.a.19.3	yes	8
140.47	odd	12		700.2.p.b.451.4		8
140.59	even	6		980.2.s.a.19.4		8
140.67	even	12		700.2.p.b.551.1		8
140.79	odd	6		980.2.s.a.619.4		8
140.103	odd	12		700.2.p.b.451.1		8
140.123	even	12		700.2.p.b.551.4		8
140.139	even	2	inner	980.2.c.b.979.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.s.a.19.2	✓	8	28.11	odd	6
140.2.s.a.19.2	✓	8	35.4	even	6
140.2.s.a.19.3	yes	8	7.4	even	3
140.2.s.a.19.3	yes	8	140.39	odd	6
140.2.s.a.59.2	yes	8	28.19	even	6
140.2.s.a.59.2	yes	8	35.19	odd	6
140.2.s.a.59.3	yes	8	7.5	odd	6
140.2.s.a.59.3	yes	8	140.19	even	6
700.2.p.b.451.1		8	35.12	even	12
700.2.p.b.451.1		8	140.103	odd	12
700.2.p.b.451.4		8	35.33	even	12
700.2.p.b.451.4		8	140.47	odd	12
700.2.p.b.551.1		8	35.18	odd	12
700.2.p.b.551.1		8	140.67	even	12
700.2.p.b.551.4		8	35.32	odd	12
700.2.p.b.551.4		8	140.123	even	12
980.2.c.b.979.1		8	1.1	even	1	trivial
980.2.c.b.979.1		8	20.19	odd	2	CM
980.2.c.b.979.4		8	7.6	odd	2	inner
980.2.c.b.979.4		8	140.139	even	2	inner
980.2.c.b.979.5		8	28.27	even	2	inner
980.2.c.b.979.5		8	35.34	odd	2	inner
980.2.c.b.979.8		8	4.3	odd	2	inner
980.2.c.b.979.8		8	5.4	even	2	inner
980.2.s.a.19.1		8	28.3	even	6
980.2.s.a.19.1		8	35.24	odd	6
980.2.s.a.19.4		8	7.3	odd	6
980.2.s.a.19.4		8	140.59	even	6
980.2.s.a.619.1		8	28.23	odd	6
980.2.s.a.619.1		8	35.9	even	6
980.2.s.a.619.4		8	7.2	even	3
980.2.s.a.619.4		8	140.79	odd	6

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:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -2.80588i q^{3} +2.00000 q^{4} +2.23607i q^{5} +3.96812i q^{6} -2.82843 q^{8} -4.87298 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -2.80588i q^{3} +2.00000 q^{4} +2.23607i q^{5} +3.96812i q^{6} -2.82843 q^{8} -4.87298 q^{9} -3.16228i q^{10} -5.61177i q^{12} +6.27415 q^{15} +4.00000 q^{16} +6.89144 q^{18} +4.47214i q^{20} -8.92295 q^{23} +7.93624i q^{24} -5.00000 q^{25} +5.25537i q^{27} -10.7460 q^{29} -8.87298 q^{30} -5.65685 q^{32} -9.74597 q^{36} -6.32456i q^{40} -8.15624i q^{41} -3.62535 q^{43} -10.8963i q^{45} +12.6190 q^{46} +9.48683i q^{47} -11.2235i q^{48} +7.07107 q^{50} -7.43222i q^{54} +15.1971 q^{58} +12.5483 q^{60} +0.219999i q^{61} +8.00000 q^{64} -11.5717 q^{67} +25.0367i q^{69} +13.7829 q^{72} +14.0294i q^{75} +8.94427i q^{80} +0.127017 q^{81} +11.5347i q^{82} -8.72878i q^{83} +5.12702 q^{86} +30.1519i q^{87} +3.74812i q^{89} +15.4097i q^{90} -17.8459 q^{92} -13.4164i q^{94} +15.8725i q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} - 8 q^{9}+O(q^{10}) \) 8 * q + 16 * q^4 - 8 * q^9	\( 8 q + 16 q^{4} - 8 q^{9} + 32 q^{16} - 40 q^{25} - 24 q^{29} - 40 q^{30} - 16 q^{36} + 8 q^{46} + 64 q^{64} + 32 q^{81} + 72 q^{86}+O(q^{100}) \) 8 * q + 16 * q^4 - 8 * q^9 + 32 * q^16 - 40 * q^25 - 24 * q^29 - 40 * q^30 - 16 * q^36 + 8 * q^46 + 64 * q^64 + 32 * q^81 + 72 * q^86

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