LMFDB - Embedded newform 980.2.c.b.979.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [980,2,Mod(979,980)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [8,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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			979.6
Root			$-1.72286 - 0.178197i$ of defining polynomial
Character	$\chi$	$=$	980.979
Dual form			980.2.c.b.979.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} -0.356394i q^{3} +2.00000 q^{4} -2.23607i q^{5} -0.504017i q^{6} +2.82843 q^{8} +2.87298 q^{9} -3.16228i q^{10} -0.712788i q^{12} -0.796921 q^{15} +4.00000 q^{16} +4.06301 q^{18} -4.47214i q^{20} -7.50873 q^{23} -1.00803i q^{24} -5.00000 q^{25} -2.09310i q^{27} +4.74597 q^{29} -1.12702 q^{30} +5.65685 q^{32} +5.74597 q^{36} -6.32456i q^{40} -12.6284i q^{41} +9.10257 q^{43} -6.42419i q^{45} -10.6190 q^{46} +9.48683i q^{47} -1.42558i q^{48} -7.07107 q^{50} -2.96008i q^{54} +6.71181 q^{58} -1.59384 q^{60} +13.6364i q^{61} +8.00000 q^{64} -15.8144 q^{67} +2.67607i q^{69} +8.12602 q^{72} +1.78197i q^{75} -8.94427i q^{80} +7.87298 q^{81} -17.8592i q^{82} +18.2156i q^{83} +12.8730 q^{86} -1.69143i q^{87} -14.1404i q^{89} -9.08517i q^{90} -15.0175 q^{92} +13.4164i q^{94} -2.01607i q^{96} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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

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
$3$	− 0.356394i	− 0.205764i	−0.994694	−	0.102882i	$-0.967194\pi$
$3$	− 0.356394i	− 0.205764i	0.994694	−	0.102882i	$-0.0328064\pi$
$4$	2.00000	1.00000
$5$	− 2.23607i	− 1.00000i
$5$	− 2.23607i	− 1.00000i
$6$	− 0.504017i	− 0.205764i
$7$	0	0
$7$	0	0
$8$	2.82843	1.00000
$9$	2.87298	0.957661
$10$	− 3.16228i	− 1.00000i
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	− 0.712788i	− 0.205764i
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−0.796921	−0.205764
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	4.06301	0.957661
$19$	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
$23$	−7.50873	−1.56568	−0.782839	−	0.622224i	$-0.786229\pi$
$23$	−7.50873	−1.56568	−0.782839	+	0.622224i	$0.786229\pi$
$24$	− 1.00803i	− 0.205764i
$25$	−5.00000	−1.00000
$26$	0	0
$27$	− 2.09310i	− 0.402816i
$28$	0	0
$29$	4.74597	0.881304	0.440652	−	0.897678i	$-0.354747\pi$
$29$	4.74597	0.881304	0.440652	+	0.897678i	$0.354747\pi$
$30$	−1.12702	−0.205764
$31$	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
$36$	5.74597	0.957661
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	− 6.32456i	− 1.00000i
$41$	− 12.6284i	− 1.97222i	−0.166092	−	0.986110i	$-0.553115\pi$
$41$	− 12.6284i	− 1.97222i	0.166092	−	0.986110i	$-0.446885\pi$
$42$	0	0
$43$	9.10257	1.38813	0.694065	−	0.719913i	$-0.255818\pi$
$43$	9.10257	1.38813	0.694065	+	0.719913i	$0.255818\pi$
$44$	0	0
$45$	− 6.42419i	− 0.957661i
$46$	−10.6190	−1.56568
$47$	9.48683i	1.38380i	0.721995	+	0.691898i	$0.243225\pi$
$47$	9.48683i	1.38380i	−0.721995	+	0.691898i	$0.756775\pi$
$48$	− 1.42558i	− 0.205764i
$49$	0	0
$50$	−7.07107	−1.00000
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	− 2.96008i	− 0.402816i
$55$	0	0
$56$	0	0
$57$	0	0
$58$	6.71181	0.881304
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−1.59384	−0.205764
$61$	13.6364i	1.74596i	0.487753	+	0.872982i	$0.337817\pi$
$61$	13.6364i	1.74596i	−0.487753	+	0.872982i	$0.662183\pi$
$62$	0	0
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	−15.8144	−1.93203	−0.966017	−	0.258478i	$-0.916779\pi$
$67$	−15.8144	−1.93203	−0.966017	+	0.258478i	$0.916779\pi$
$68$	0	0
$69$	2.67607i	0.322161i
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	8.12602	0.957661
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	1.78197i	0.205764i
$76$	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
$81$	7.87298	0.874776
$82$	− 17.8592i	− 1.97222i
$83$	18.2156i	1.99942i	0.0240199	+	0.999711i	$0.492353\pi$
$83$	18.2156i	1.99942i	−0.0240199	+	0.999711i	$0.507647\pi$
$84$	0	0
$85$	0	0
$86$	12.8730	1.38813
$87$	− 1.69143i	− 0.181341i
$88$	0	0
$89$	− 14.1404i	− 1.49888i	−0.662071	−	0.749441i	$-0.730322\pi$
$89$	− 14.1404i	− 1.49888i	0.662071	−	0.749441i	$-0.269678\pi$
$90$	− 9.08517i	− 0.957661i
$91$	0	0
$92$	−15.0175	−1.56568
$93$	0	0
$94$	13.4164i	1.38380i
$95$	0	0
$96$	− 2.01607i	− 0.205764i
$97$	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
$101$	11.1163i	1.10612i	0.833143	+	0.553058i	$0.186539\pi$
$101$	11.1163i	1.10612i	−0.833143	+	0.553058i	$0.813461\pi$
$102$	0	0
$103$	18.9284i	1.86507i	0.361079	+	0.932535i	$0.382408\pi$
$103$	18.9284i	1.86507i	−0.361079	+	0.932535i	$0.617592\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.4082	−1.68292	−0.841458	−	0.540322i	$-0.818302\pi$
$107$	−17.4082	−1.68292	−0.841458	+	0.540322i	$0.818302\pi$
$108$	− 4.18619i	− 0.402816i
$109$	−3.61895	−0.346633	−0.173316	−	0.984866i	$-0.555448\pi$
$109$	−3.61895	−0.346633	−0.173316	+	0.984866i	$0.555448\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	16.7900i	1.56568i
$116$	9.49193	0.881304
$117$	0	0
$118$	0	0
$119$	0	0
$120$	−2.25403	−0.205764
$121$	11.0000	1.00000
$122$	19.2848i	1.74596i
$123$	−4.50068	−0.405812
$124$	0	0
$125$	11.1803i	1.00000i
$126$	0	0
$127$	−4.24264	−0.376473	−0.188237	−	0.982124i	$-0.560277\pi$
$127$	−4.24264	−0.376473	−0.188237	+	0.982124i	$0.560277\pi$
$128$	11.3137	1.00000
$129$	− 3.24410i	− 0.285627i
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	−22.3649	−1.93203
$135$	−4.68030	−0.402816
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	3.78453i	0.322161i
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	3.38105	0.284736
$142$	0	0
$143$	0	0
$144$	11.4919	0.957661
$145$	− 10.6123i	− 0.881304i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.12702	−0.665791	−0.332896	−	0.942964i	$-0.608026\pi$
$149$	−8.12702	−0.665791	−0.332896	+	0.942964i	$0.608026\pi$
$150$	2.52009i	0.205764i
$151$	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
$162$	11.1341	0.874776
$163$	12.7279	0.996928	0.498464	−	0.866910i	$-0.333898\pi$
$163$	12.7279	0.996928	0.498464	+	0.866910i	$0.333898\pi$
$164$	− 25.2567i	− 1.97222i
$165$	0	0
$166$	25.7608i	1.99942i
$167$	− 16.0772i	− 1.24409i	−0.782980	−	0.622047i	$-0.786301\pi$
$167$	− 16.0772i	− 1.24409i	0.782980	−	0.622047i	$-0.213699\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	18.2051	1.38813
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	− 2.39205i	− 0.181341i
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 19.9976i	− 1.49888i
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	− 12.8484i	− 0.957661i
$181$	15.1485i	1.12598i	0.826465	+	0.562988i	$0.190348\pi$
$181$	15.1485i	1.12598i	−0.826465	+	0.562988i	$0.809652\pi$
$182$	0	0
$183$	4.85993	0.359257
$184$	−21.2379	−1.56568
$185$	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			$\pi$
$192$	− 2.85115i	− 0.205764i
$193$	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
$201$	5.63615i	0.397543i
$202$	15.7209i	1.10612i
$203$	0	0
$204$	0	0
$205$	−28.2379	−1.97222
$206$	26.7688i	1.86507i
$207$	−21.5725	−1.49939
$208$	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
$214$	−24.6190	−1.68292
$215$	− 20.3540i	− 1.38813i
$216$	− 5.92017i	− 0.402816i
$217$	0	0
$218$	−5.11797	−0.346633
$219$	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
$225$	−14.3649	−0.957661
$226$	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	$0.346926\pi$
$229$	26.8328i	1.77316i	−0.462573	+	0.886581i	$0.653074\pi$
$230$	23.7447i	1.56568i
$231$	0	0
$232$	13.4236	0.881304
$233$	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			$\pi$
$240$	−3.18768	−0.205764
$241$	13.4164i	0.864227i	0.901819	+	0.432113i	$0.142232\pi$
$241$	13.4164i	0.864227i	−0.901819	+	0.432113i	$0.857768\pi$
$242$	15.5563	1.00000
$243$	− 9.08517i	− 0.582814i
$244$	27.2728i	1.74596i
$245$	0	0
$246$	−6.36492	−0.405812
$247$	0	0
$248$	0	0
$249$	6.49193	0.411410
$250$	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	$0.5\pi$
$258$	− 4.58785i	− 0.285627i
$259$	0	0
$260$	0	0
$261$	13.6351	0.843990
$262$	0	0
$263$	32.4257	1.99945	0.999727	−	0.0233719i	$-0.00744017\pi$
$263$	32.4257	1.99945	0.999727	+	0.0233719i	$0.00744017\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−5.03956	−0.308416
$268$	−31.6288	−1.93203
$269$	− 9.60427i	− 0.585583i	−0.956176	−	0.292791i	$-0.905416\pi$
$269$	− 9.60427i	− 0.585583i	0.956176	−	0.292791i	$-0.0945841\pi$
$270$	−6.61895	−0.402816
$271$	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
$276$	5.35213i	0.322161i
$277$	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	$0.616517\pi$
$282$	4.78153	0.284736
$283$	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
$288$	16.2520	0.957661
$289$	−17.0000	−1.00000
$290$	− 15.0081i	− 0.881304i
$291$	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
$298$	−11.4933	−0.665791
$299$	0	0
$300$	3.56394i	0.205764i
$301$	0	0
$302$	0	0
$303$	3.96179	0.227599
$304$	0	0
$305$	30.4919	1.74596
$306$	0	0
$307$	− 21.0668i	− 1.20234i	−0.799120	−	0.601172i	$-0.794701\pi$
$307$	− 21.0668i	− 1.20234i	0.799120	−	0.601172i	$-0.205299\pi$
$308$	0	0
$309$	6.74597	0.383765
$310$	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
$321$	6.20419i	0.346284i
$322$	0	0
$323$	0	0
$324$	15.7460	0.874776
$325$	0	0
$326$	18.0000	0.996928
$327$	1.28977i	0.0713246i
$328$	− 35.7184i	− 1.97222i
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	36.4312i	1.99942i
$333$	0	0
$334$	− 22.7367i	− 1.24409i
$335$	35.3620i	1.93203i
$336$	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
$344$	25.7460	1.38813
$345$	5.98387	0.322161
$346$	0	0
$347$	12.6267	0.677837	0.338918	−	0.940816i	$-0.389939\pi$
$347$	12.6267	0.677837	0.338918	+	0.940816i	$0.389939\pi$
$348$	− 3.38287i	− 0.181341i
$349$	− 9.10025i	− 0.487125i	−0.969885	−	0.243563i	$-0.921684\pi$
$349$	− 9.10025i	− 0.487125i	0.969885	−	0.243563i	$-0.0783162\pi$
$350$	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
$356$	− 28.2808i	− 1.49888i
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	− 18.1703i	− 0.957661i
$361$	−19.0000	−1.00000
$362$	21.4232i	1.12598i
$363$	− 3.92033i	− 0.205764i
$364$	0	0
$365$	0	0
$366$	6.87298	0.359257
$367$	− 34.6493i	− 1.80868i	−0.426817	−	0.904338i	$-0.640365\pi$
$367$	− 34.6493i	− 1.80868i	0.426817	−	0.904338i	$-0.359635\pi$
$368$	−30.0349	−1.56568
$369$	− 36.2811i	− 1.88872i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	3.98461	0.205764
$376$	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
$381$	1.51205i	0.0774647i
$382$	0	0
$383$	− 37.5004i	− 1.91618i	−0.286466	−	0.958091i	$-0.592480\pi$
$383$	− 37.5004i	− 1.91618i	0.286466	−	0.958091i	$-0.407520\pi$
$384$	− 4.03214i	− 0.205764i
$385$	0	0
$386$	0	0
$387$	26.1515	1.32936
$388$	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
$401$	39.9839	1.99670	0.998350	−	0.0574304i	$-0.0182907\pi$
$401$	39.9839	1.99670	0.998350	+	0.0574304i	$0.0182907\pi$
$402$	7.97072i	0.397543i
$403$	0	0
$404$	22.2326i	1.10612i
$405$	− 17.6045i	− 0.874776i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 16.6605i	− 0.823809i	−0.911227	−	0.411905i	$-0.864864\pi$
$409$	− 16.6605i	− 0.823809i	0.911227	−	0.411905i	$-0.135136\pi$
$410$	−39.9344	−1.97222
$411$	0	0
$412$	37.8568i	1.86507i
$413$	0	0
$414$	−30.5081	−1.49939
$415$	40.7313	1.99942
$416$	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
$421$	38.8569	1.89377	0.946883	−	0.321577i	$-0.104213\pi$
$421$	38.8569	1.89377	0.946883	+	0.321577i	$0.104213\pi$
$422$	0	0
$423$	27.2555i	1.32521i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−34.8165	−1.68292
$429$	0	0
$430$	− 28.7849i	− 1.38813i
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	− 8.37238i	− 0.402816i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	−3.78216	−0.181341
$436$	−7.23790	−0.346633
$437$	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
$443$	12.2903	0.583928	0.291964	−	0.956429i	$-0.405691\pi$
$443$	12.2903	0.583928	0.291964	+	0.956429i	$0.405691\pi$
$444$	0	0
$445$	−31.6190	−1.49888
$446$	− 4.47214i	− 0.211762i
$447$	2.89642i	0.136996i
$448$	0	0
$449$	−1.36492	−0.0644144	−0.0322072	−	0.999481i	$-0.510254\pi$
$449$	−1.36492	−0.0644144	−0.0322072	+	0.999481i	$0.510254\pi$
$450$	−20.3151	−0.957661
$451$	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
$460$	33.5801i	1.56568i
$461$	− 8.94427i	− 0.416576i	−0.978068	−	0.208288i	$-0.933211\pi$
$461$	− 8.94427i	− 0.416576i	0.978068	−	0.208288i	$-0.0667892\pi$
$462$	0	0
$463$	−29.2380	−1.35881	−0.679403	−	0.733766i	$-0.737761\pi$
$463$	−29.2380	−1.35881	−0.679403	+	0.733766i	$0.737761\pi$
$464$	18.9839	0.881304
$465$	0	0
$466$	0	0
$467$	13.9389i	0.645015i	0.946567	+	0.322507i	$0.104526\pi$
$467$	13.9389i	0.645015i	−0.946567	+	0.322507i	$0.895474\pi$
$468$	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	$0.5\pi$
$480$	−4.50807	−0.205764
$481$	0	0
$482$	18.9737i	0.864227i
$483$	0	0
$484$	22.0000	1.00000
$485$	0	0
$486$	− 12.8484i	− 0.582814i
$487$	−38.1838	−1.73027	−0.865136	−	0.501538i	$-0.832768\pi$
$487$	−38.1838	−1.73027	−0.865136	+	0.501538i	$0.832768\pi$
$488$	38.5696i	1.74596i
$489$	− 4.53615i	− 0.205132i
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−9.00135	−0.405812
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	9.18098	0.411410
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	22.3607i	1.00000i
$501$	−5.72983	−0.255990
$502$	0	0
$503$	− 33.2237i	− 1.48137i	−0.671852	−	0.740685i	$-0.734501\pi$
$503$	− 33.2237i	− 1.48137i	0.671852	−	0.740685i	$-0.265499\pi$
$504$	0	0
$505$	24.8569	1.10612
$506$	0	0
$507$	4.63312i	0.205764i
$508$	−8.48528	−0.376473
$509$	17.1645i	0.760804i	0.924821	+	0.380402i	$0.124214\pi$
$509$	17.1645i	0.760804i	−0.924821	+	0.380402i	$0.875786\pi$
$510$	0	0
$511$	0	0
$512$	22.6274	1.00000
$513$	0	0
$514$	0	0
$515$	42.3252	1.86507
$516$	− 6.48820i	− 0.285627i
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 17.8885i	− 0.783711i	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	− 17.8885i	− 0.783711i	0.920027	−	0.391856i	$-0.128167\pi$
$522$	19.2829	0.843990
$523$	− 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
$526$	45.8569	1.99945
$527$	0	0
$528$	0	0
$529$	33.3810	1.45135
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	−7.12702	−0.308416
$535$	38.9260i	1.68292i
$536$	−44.7298	−1.93203
$537$	0	0
$538$	− 13.5825i	− 0.585583i
$539$	0	0
$540$	−9.36061	−0.402816
$541$	−42.2379	−1.81595	−0.907975	−	0.419025i	$-0.862372\pi$
$541$	−42.2379	−1.81595	−0.907975	+	0.419025i	$0.862372\pi$
$542$	0	0
$543$	5.39882	0.231686
$544$	0	0
$545$	8.09222i	0.346633i
$546$	0	0
$547$	−20.5959	−0.880618	−0.440309	−	0.897846i	$-0.645131\pi$
$547$	−20.5959	−0.880618	−0.440309	+	0.897846i	$0.645131\pi$
$548$	0	0
$549$	39.1772i	1.67204i
$550$	0	0
$551$	0	0
$552$	7.56906i	0.322161i
$553$	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
$563$	− 22.4923i	− 0.947939i	−0.880541	−	0.473970i	$-0.842821\pi$
$563$	− 22.4923i	− 0.947939i	0.880541	−	0.473970i	$-0.157179\pi$
$564$	6.76210	0.284736
$565$	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
$575$	37.5437	1.56568
$576$	22.9839	0.957661
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	−24.0416	−1.00000
$579$	0	0
$580$	− 21.2246i	− 0.881304i
$581$	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
$596$	−16.2540	−0.665791
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	5.04017i	0.205764i
$601$	− 40.2492i	− 1.64180i	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	− 40.2492i	− 1.64180i	0.571072	−	0.820900i	$-0.306528\pi$
$602$	0	0
$603$	−45.4345	−1.85023
$604$	0	0
$605$	− 24.5967i	− 1.00000i
$606$	5.60282	0.227599
$607$	− 32.5109i	− 1.31958i	−0.751452	−	0.659788i	$-0.770646\pi$
$607$	− 32.5109i	− 1.31958i	0.751452	−	0.659788i	$-0.229354\pi$
$608$	0	0
$609$	0	0
$610$	43.1221	1.74596
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	− 29.7929i	− 1.20234i
$615$	10.0638i	0.405812i
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	9.54024	0.383765
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	15.7165i	0.630681i
$622$	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
$641$	36.6028	1.44572	0.722862	−	0.690992i	$-0.242826\pi$
$641$	36.6028	1.44572	0.722862	+	0.690992i	$0.242826\pi$
$642$	8.77405i	0.346284i
$643$	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
$645$	−7.25403	−0.285627
$646$	0	0
$647$	− 39.6388i	− 1.55836i	−0.626800	−	0.779180i	$-0.715636\pi$
$647$	− 39.6388i	− 1.55836i	0.626800	−	0.779180i	$-0.284364\pi$
$648$	22.2682	0.874776
$649$	0	0
$650$	0	0
$651$	0	0
$652$	25.4558	0.996928
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	1.82401i	0.0713246i
$655$	0	0
$656$	− 50.5135i	− 1.97222i
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	7.58820i	0.295147i	0.989051	+	0.147573i	$0.0471463\pi$
$661$	7.58820i	0.295147i	−0.989051	+	0.147573i	$0.952854\pi$
$662$	0	0
$663$	0	0
$664$	51.5215i	1.99942i
$665$	0	0
$666$	0	0
$667$	−35.6362	−1.37984
$668$	− 32.1545i	− 1.24409i
$669$	−1.12702	−0.0435730
$670$	50.0095i	1.93203i
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	10.4655i	0.402816i
$676$	−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
$681$	−10.1431	−0.388686
$682$	0	0
$683$	2.72720	0.104354	0.0521768	−	0.998638i	$-0.483384\pi$
$683$	2.72720	0.104354	0.0521768	+	0.998638i	$0.483384\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.56305	0.364853
$688$	36.4103	1.38813
$689$	0	0
$690$	8.46247	0.322161
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	17.8569	0.677837
$695$	0	0
$696$	− 4.78410i	− 0.181341i
$697$	0	0
$698$	− 12.8697i	− 0.487125i
$699$	0	0
$700$	0	0
$701$	−43.3649	−1.63787	−0.818935	−	0.573886i	$-0.805435\pi$
$701$	−43.3649	−1.63787	−0.818935	+	0.573886i	$0.805435\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	− 7.56026i	− 0.284736i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−0.237900	−0.00893452	−0.00446726	−	0.999990i	$-0.501422\pi$
$709$	−0.237900	−0.00893452	−0.00446726	+	0.999990i	$0.501422\pi$
$710$	0	0
$711$	0	0
$712$	− 39.9952i	− 1.49888i
$713$	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	$0.5\pi$
$720$	− 25.6967i	− 0.957661i
$721$	0	0
$722$	−26.8701	−1.00000
$723$	4.78153	0.177827
$724$	30.2969i	1.12598i
$725$	−23.7298	−0.881304
$726$	− 5.54419i	− 0.205764i
$727$	23.2051i	0.860630i	0.902679	+	0.430315i	$0.141598\pi$
$727$	23.2051i	0.860630i	−0.902679	+	0.430315i	$0.858402\pi$
$728$	0	0
$729$	20.3810	0.754854
$730$	0	0
$731$	0	0
$732$	9.71987	0.359257
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	− 49.0014i	− 1.80868i
$735$	0	0
$736$	−42.4758	−1.56568
$737$	0	0
$738$	− 51.3092i	− 1.88872i
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.6442	1.01417	0.507083	−	0.861897i	$-0.330724\pi$
$743$	27.6442	1.01417	0.507083	+	0.861897i	$0.330724\pi$
$744$	0	0
$745$	18.1726i	0.665791i
$746$	0	0
$747$	52.3331i	1.91477i
$748$	0	0
$749$	0	0
$750$	5.63508	0.205764
$751$	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	$-0.775414\pi$
$761$	− 35.7771i	− 1.29692i	0.761249	−	0.648459i	$-0.224586\pi$
$762$	2.13836i	0.0774647i
$763$	0	0
$764$	0	0
$765$	0	0
$766$	− 53.0336i	− 1.91618i
$767$	0	0
$768$	− 5.70230i	− 0.205764i
$769$	53.6656i	1.93523i	0.252426	+	0.967616i	$0.418771\pi$
$769$	53.6656i	1.93523i	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	36.9839	1.32936
$775$	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
$783$	− 9.93376i	− 0.355004i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 12.5133i	− 0.446051i	−0.974813	−	0.223026i	$-0.928407\pi$
$787$	− 12.5133i	− 0.446051i	0.974813	−	0.223026i	$-0.0715934\pi$
$788$	0	0
$789$	− 11.5563i	− 0.411416i
$790$	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
$801$	− 40.6252i	− 1.43542i
$802$	56.5457	1.99670
$803$	0	0
$804$	11.2723i	0.397543i
$805$	0	0
$806$	0	0
$807$	−3.42290	−0.120492
$808$	31.4417i	1.10612i
$809$	−11.5081	−0.404602	−0.202301	−	0.979323i	$-0.564842\pi$
$809$	−11.5081	−0.404602	−0.202301	+	0.979323i	$0.564842\pi$
$810$	− 24.8966i	− 0.874776i
$811$	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
$818$	− 23.5615i	− 0.823809i
$819$	0	0
$820$	−56.4758	−1.97222
$821$	−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
$823$	13.8841	0.483969	0.241985	−	0.970280i	$-0.422202\pi$
$823$	13.8841	0.483969	0.241985	+	0.970280i	$0.422202\pi$
$824$	53.5376i	1.86507i
$825$	0	0
$826$	0	0
$827$	57.3426	1.99400	0.997000	−	0.0774065i	$-0.0246639\pi$
$827$	57.3426	1.99400	0.997000	+	0.0774065i	$0.0246639\pi$
$828$	−43.1449	−1.49939
$829$	13.4164i	0.465971i	0.972480	+	0.232986i	$0.0748495\pi$
$829$	13.4164i	0.465971i	−0.972480	+	0.232986i	$0.925151\pi$
$830$	57.6028	1.99942
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−35.9498	−1.24409
$836$	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
$841$	−6.47580	−0.223303
$842$	54.9519	1.89377
$843$	4.27673i	0.147298i
$844$	0	0
$845$	29.0689i	1.00000i
$846$	38.5451i	1.32521i
$847$	0	0
$848$	0	0
$849$	5.63508	0.193396
$850$	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
$856$	−49.2379	−1.68292
$857$	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	$0.5\pi$
$860$	− 40.7079i	− 1.38813i
$861$	0	0
$862$	0	0
$863$	−37.2072	−1.26655	−0.633274	−	0.773928i	$-0.718289\pi$
$863$	−37.2072	−1.26655	−0.633274	+	0.773928i	$0.718289\pi$
$864$	− 11.8403i	− 0.402816i
$865$	0	0
$866$	0	0
$867$	6.05870i	0.205764i
$868$	0	0
$869$	0	0
$870$	−5.34878	−0.181341
$871$	0	0
$872$	−10.2359	−0.346633
$873$	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
$881$	18.6766i	0.629230i	0.949219	+	0.314615i	$0.101875\pi$
$881$	18.6766i	0.629230i	−0.949219	+	0.314615i	$0.898125\pi$
$882$	0	0
$883$	55.1543	1.85609	0.928045	−	0.372467i	$-0.121488\pi$
$883$	55.1543	1.85609	0.928045	+	0.372467i	$0.121488\pi$
$884$	0	0
$885$	0	0
$886$	17.3810	0.583928
$887$	31.0853i	1.04374i	0.853024	+	0.521871i	$0.174766\pi$
$887$	31.0853i	1.04374i	−0.853024	+	0.521871i	$0.825234\pi$
$888$	0	0
$889$	0	0
$890$	−44.7159	−1.49888
$891$	0	0
$892$	− 6.32456i	− 0.211762i
$893$	0	0
$894$	4.09616i	0.136996i
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−1.93028	−0.0644144
$899$	0	0
$900$	−28.7298	−0.957661
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	33.8730	1.12598
$906$	0	0
$907$	−54.1550	−1.79819	−0.899093	−	0.437758i	$-0.855773\pi$
$907$	−54.1550	−1.79819	−0.899093	+	0.437758i	$0.855773\pi$
$908$	− 56.9210i	− 1.88899i
$909$	31.9370i	1.05928i
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 10.8671i	− 0.359257i
$916$	53.6656i	1.77316i
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	47.4894i	1.56568i
$921$	−7.50807	−0.247399
$922$	− 12.6491i	− 0.416576i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−41.3488	−1.35881
$927$	54.3810i	1.78611i
$928$	26.8472	0.881304
$929$	6.58017i	0.215888i	0.994157	+	0.107944i	$0.0344268\pi$
$929$	6.58017i	0.215888i	−0.994157	+	0.107944i	$0.965573\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	19.7126i	0.645015i
$935$	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	$0.259985\pi$
$941$	44.7214i	1.45787i	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	94.8231i	3.08786i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.1898	0.721070	0.360535	−	0.932746i	$-0.382594\pi$
$947$	22.1898	0.721070	0.360535	+	0.932746i	$0.382594\pi$
$948$	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
$960$	−6.37537	−0.205764
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−50.0135	−1.61166
$964$	26.8328i	0.864227i
$965$	0	0
$966$	0	0
$967$	−58.9365	−1.89527	−0.947635	−	0.319356i	$-0.896533\pi$
$967$	−58.9365	−1.89527	−0.947635	+	0.319356i	$0.896533\pi$
$968$	31.1127	1.00000
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	− 18.1703i	− 0.582814i
$973$	0	0
$974$	−54.0000	−1.73027
$975$	0	0
$976$	54.5456i	1.74596i
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	− 6.41509i	− 0.205132i
$979$	0	0
$980$	0	0
$981$	−10.3972	−0.331957
$982$	0	0
$983$	11.8005i	0.376378i	0.982133	+	0.188189i	$0.0602618\pi$
$983$	11.8005i	0.376378i	−0.982133	+	0.188189i	$0.939738\pi$
$984$	−12.7298	−0.405812
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−68.3488	−2.17336
$990$	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
$996$	12.9839	0.411410
$997$	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.979.6		8
4.3	odd	2	inner	980.2.c.b.979.3		8
5.4	even	2	inner	980.2.c.b.979.3		8
7.2	even	3		980.2.s.a.619.2		8
7.3	odd	6		980.2.s.a.19.2		8
7.4	even	3		140.2.s.a.19.1	✓	8
7.5	odd	6		140.2.s.a.59.1	yes	8
7.6	odd	2	inner	980.2.c.b.979.7		8
20.19	odd	2	CM	980.2.c.b.979.6		8
28.3	even	6		980.2.s.a.19.3		8
28.11	odd	6		140.2.s.a.19.4	yes	8
28.19	even	6		140.2.s.a.59.4	yes	8
28.23	odd	6		980.2.s.a.619.3		8
28.27	even	2	inner	980.2.c.b.979.2		8
35.4	even	6		140.2.s.a.19.4	yes	8
35.9	even	6		980.2.s.a.619.3		8
35.12	even	12		700.2.p.b.451.3		8
35.18	odd	12		700.2.p.b.551.3		8
35.19	odd	6		140.2.s.a.59.4	yes	8
35.24	odd	6		980.2.s.a.19.3		8
35.32	odd	12		700.2.p.b.551.2		8
35.33	even	12		700.2.p.b.451.2		8
35.34	odd	2	inner	980.2.c.b.979.2		8
140.19	even	6		140.2.s.a.59.1	yes	8
140.39	odd	6		140.2.s.a.19.1	✓	8
140.47	odd	12		700.2.p.b.451.2		8
140.59	even	6		980.2.s.a.19.2		8
140.67	even	12		700.2.p.b.551.3		8
140.79	odd	6		980.2.s.a.619.2		8
140.103	odd	12		700.2.p.b.451.3		8
140.123	even	12		700.2.p.b.551.2		8
140.139	even	2	inner	980.2.c.b.979.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.s.a.19.1	✓	8	7.4	even	3
140.2.s.a.19.1	✓	8	140.39	odd	6
140.2.s.a.19.4	yes	8	28.11	odd	6
140.2.s.a.19.4	yes	8	35.4	even	6
140.2.s.a.59.1	yes	8	7.5	odd	6
140.2.s.a.59.1	yes	8	140.19	even	6
140.2.s.a.59.4	yes	8	28.19	even	6
140.2.s.a.59.4	yes	8	35.19	odd	6
700.2.p.b.451.2		8	35.33	even	12
700.2.p.b.451.2		8	140.47	odd	12
700.2.p.b.451.3		8	35.12	even	12
700.2.p.b.451.3		8	140.103	odd	12
700.2.p.b.551.2		8	35.32	odd	12
700.2.p.b.551.2		8	140.123	even	12
700.2.p.b.551.3		8	35.18	odd	12
700.2.p.b.551.3		8	140.67	even	12
980.2.c.b.979.2		8	28.27	even	2	inner
980.2.c.b.979.2		8	35.34	odd	2	inner
980.2.c.b.979.3		8	4.3	odd	2	inner
980.2.c.b.979.3		8	5.4	even	2	inner
980.2.c.b.979.6		8	1.1	even	1	trivial
980.2.c.b.979.6		8	20.19	odd	2	CM
980.2.c.b.979.7		8	7.6	odd	2	inner
980.2.c.b.979.7		8	140.139	even	2	inner
980.2.s.a.19.2		8	7.3	odd	6
980.2.s.a.19.2		8	140.59	even	6
980.2.s.a.19.3		8	28.3	even	6
980.2.s.a.19.3		8	35.24	odd	6
980.2.s.a.619.2		8	7.2	even	3
980.2.s.a.619.2		8	140.79	odd	6
980.2.s.a.619.3		8	28.23	odd	6
980.2.s.a.619.3		8	35.9	even	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -0.356394i q^{3} +2.00000 q^{4} -2.23607i q^{5} -0.504017i q^{6} +2.82843 q^{8} +2.87298 q^{9} -3.16228i q^{10} -0.712788i q^{12} -0.796921 q^{15} +4.00000 q^{16} +4.06301 q^{18} -4.47214i q^{20} -7.50873 q^{23} -1.00803i q^{24} -5.00000 q^{25} -2.09310i q^{27} +4.74597 q^{29} -1.12702 q^{30} +5.65685 q^{32} +5.74597 q^{36} -6.32456i q^{40} -12.6284i q^{41} +9.10257 q^{43} -6.42419i q^{45} -10.6190 q^{46} +9.48683i q^{47} -1.42558i q^{48} -7.07107 q^{50} -2.96008i q^{54} +6.71181 q^{58} -1.59384 q^{60} +13.6364i q^{61} +8.00000 q^{64} -15.8144 q^{67} +2.67607i q^{69} +8.12602 q^{72} +1.78197i q^{75} -8.94427i q^{80} +7.87298 q^{81} -17.8592i q^{82} +18.2156i q^{83} +12.8730 q^{86} -1.69143i q^{87} -14.1404i q^{89} -9.08517i q^{90} -15.0175 q^{92} +13.4164i q^{94} -2.01607i q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} - 8 q^{9} + 32 q^{16} - 40 q^{25} - 24 q^{29} - 40 q^{30} - 16 q^{36} + 8 q^{46} + 64 q^{64} + 32 q^{81} + 72 q^{86}+O(q^{100}) \) 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

Introduction

L-functions

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