LMFDB - Embedded newform 980.2.c.a.979.4

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:	$\Q(\zeta_{16})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			979.4
Root			$-0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	980.979
Dual form			980.2.c.a.979.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +(2.23044 - 0.158513i) q^{5} +2.82843i q^{8} +3.00000 q^{9} +(-0.224171 - 3.15432i) q^{10} -4.01254 q^{13} +4.00000 q^{16} +4.90923 q^{17} -4.24264i q^{18} +(-4.46088 + 0.317025i) q^{20} +(4.94975 - 0.707107i) q^{25} +5.67459i q^{26} +9.89949 q^{29} -5.65685i q^{32} -6.94269i q^{34} -6.00000 q^{36} -7.07107i q^{37} +(0.448342 + 6.30864i) q^{40} +12.3003i q^{41} +(6.69133 - 0.475538i) q^{45} +(-1.00000 - 7.00000i) q^{50} +8.02509 q^{52} -14.0000i q^{53} -14.0000i q^{58} -7.25972i q^{61} -8.00000 q^{64} +(-8.94975 + 0.636039i) q^{65} -9.81845 q^{68} +8.48528i q^{72} -12.4860 q^{73} -10.0000 q^{74} +(8.92177 - 0.634051i) q^{80} +9.00000 q^{81} +17.3952 q^{82} +(10.9497 - 0.778175i) q^{85} -18.6089i q^{89} +(-0.672512 - 9.46297i) q^{90} -14.2793 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} + 24 q^{9} + 32 q^{16} - 48 q^{36} - 8 q^{50} - 64 q^{64} - 32 q^{65} - 80 q^{74} + 72 q^{81} + 48 q^{85}+O(q^{100}) $ 8 * q - 16 * q^4 + 24 * q^9 + 32 * q^16 - 48 * q^36 - 8 * q^50 - 64 * q^64 - 32 * q^65 - 80 * q^74 + 72 * q^81 + 48 * q^85

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 $

$2$		−	1.41421i		−	1.00000i
$2$		−	1.41421i		−	1.00000i
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	−2.00000			−1.00000
$5$	2.23044	−	0.158513i	0.997484	−	0.0708890i
$5$	2.23044	−	0.158513i	0.997484	−	0.0708890i
$6$		0			0
$7$		0			0
$7$		0			0
$8$			2.82843i			1.00000i
$9$	3.00000			1.00000
$10$	−0.224171	−	3.15432i	−0.0708890	−	0.997484i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−4.01254			−1.11288			−0.556440	−	0.830888i	$-0.687833\pi$
$13$	−4.01254			−1.11288			−0.556440	+	0.830888i	$0.687833\pi$
$14$		0			0
$15$		0			0
$16$	4.00000			1.00000
$17$	4.90923			1.19066			0.595331	−	0.803480i	$-0.297021\pi$
$17$	4.90923			1.19066			0.595331	+	0.803480i	$0.297021\pi$
$18$		−	4.24264i		−	1.00000i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−4.46088	+	0.317025i	−0.997484	+	0.0708890i
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	4.94975	−	0.707107i	0.989949	−	0.141421i
$26$			5.67459i			1.11288i
$27$		0			0
$28$		0			0
$29$	9.89949			1.83829			0.919145	−	0.393919i	$-0.128881\pi$
$29$	9.89949			1.83829			0.919145	+	0.393919i	$0.128881\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		−	5.65685i		−	1.00000i
$33$		0			0
$34$		−	6.94269i		−	1.19066i
$35$		0			0
$36$	−6.00000			−1.00000
$37$		−	7.07107i		−	1.16248i	−0.813733	−	0.581238i	$-0.802568\pi$
$37$		−	7.07107i		−	1.16248i	0.813733	−	0.581238i	$-0.197432\pi$
$38$		0			0
$39$		0			0
$40$	0.448342	+	6.30864i	0.0708890	+	0.997484i
$41$			12.3003i			1.92098i	0.278317	+	0.960489i	$0.410223\pi$
$41$			12.3003i			1.92098i	−0.278317	+	0.960489i	$0.589777\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	6.69133	−	0.475538i	0.997484	−	0.0708890i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$		0			0
$50$	−1.00000	−	7.00000i	−0.141421	−	0.989949i
$51$		0			0
$52$	8.02509			1.11288
$53$		−	14.0000i		−	1.92305i	−0.274721	−	0.961524i	$-0.588586\pi$
$53$		−	14.0000i		−	1.92305i	0.274721	−	0.961524i	$-0.411414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		−	14.0000i		−	1.83829i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		−	7.25972i		−	0.929512i	−0.885439	−	0.464756i	$-0.846142\pi$
$61$		−	7.25972i		−	0.929512i	0.885439	−	0.464756i	$-0.153858\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$	−8.94975	+	0.636039i	−1.11008	+	0.0788909i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−9.81845			−1.19066
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$			8.48528i			1.00000i
$73$	−12.4860			−1.46137			−0.730686	−	0.682713i	$-0.760800\pi$
$73$	−12.4860			−1.46137			−0.730686	+	0.682713i	$0.760800\pi$
$74$	−10.0000			−1.16248
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	8.92177	−	0.634051i	0.997484	−	0.0708890i
$81$	9.00000			1.00000
$82$	17.3952			1.92098
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	10.9497	−	0.778175i	1.18767	−	0.0844049i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	18.6089i		−	1.97254i	−0.165140	−	0.986270i	$-0.552808\pi$
$89$		−	18.6089i		−	1.97254i	0.165140	−	0.986270i	$-0.447192\pi$
$90$	−0.672512	−	9.46297i	−0.0708890	−	0.997484i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−14.2793			−1.44985			−0.724924	−	0.688829i	$-0.758125\pi$
$97$	−14.2793			−1.44985			−0.724924	+	0.688829i	$0.758125\pi$
$98$		0			0
$99$		0			0
$100$	−9.89949	+	1.41421i	−0.989949	+	0.141421i
$101$			19.2430i			1.91475i	0.288855	+	0.957373i	$0.406726\pi$
$101$			19.2430i			1.91475i	−0.288855	+	0.957373i	$0.593274\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		−	11.3492i		−	1.11288i
$105$		0			0
$106$	−19.7990			−1.92305
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	9.89949			0.948200			0.474100	−	0.880471i	$-0.342774\pi$
$109$	9.89949			0.948200			0.474100	+	0.880471i	$0.342774\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			14.0000i			1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$			14.0000i			1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$		0			0
$116$	−19.7990			−1.83829
$117$	−12.0376			−1.11288
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	−10.2668			−0.929512
$123$		0			0
$124$		0			0
$125$	10.9280	−	2.36176i	0.977434	−	0.211242i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$			11.3137i			1.00000i
$129$		0			0
$130$	0.899495	+	12.6569i	0.0788909	+	1.11008i
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$			13.8854i			1.19066i
$137$			21.2132i			1.81237i	0.422885	+	0.906183i	$0.361017\pi$
$137$			21.2132i			1.81237i	−0.422885	+	0.906183i	$0.638983\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	12.0000			1.00000
$145$	22.0803	−	1.56920i	1.83367	−	0.130315i
$146$			17.6578i			1.46137i
$147$		0			0
$148$			14.1421i			1.16248i
$149$	−20.0000			−1.63846			−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000			−1.63846			−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	14.7277			1.19066
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2.66752			−0.212891			−0.106446	−	0.994319i	$-0.533947\pi$
$157$	−2.66752			−0.212891			−0.106446	+	0.994319i	$0.533947\pi$
$158$		0			0
$159$		0			0
$160$	−0.896683	−	12.6173i	−0.0708890	−	0.997484i
$161$		0			0
$162$		−	12.7279i		−	1.00000i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		−	24.6005i		−	1.92098i
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	3.10051			0.238500
$170$	−1.10051	−	15.4853i	−0.0844049	−	1.18767i
$171$		0			0
$172$		0			0
$173$	−6.25425			−0.475502			−0.237751	−	0.971326i	$-0.576410\pi$
$173$	−6.25425			−0.475502			−0.237751	+	0.971326i	$0.576410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−26.3170			−1.97254
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	−13.3827	+	0.951076i	−0.997484	+	0.0708890i
$181$			24.2835i			1.80498i	0.430713	+	0.902489i	$0.358262\pi$
$181$			24.2835i			1.80498i	−0.430713	+	0.902489i	$0.641738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.12085	−	15.7716i	−0.0824068	−	1.15955i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$			14.0000i			1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$			14.0000i			1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$			20.1940i			1.44985i
$195$		0			0
$196$		0			0
$197$		−	28.0000i		−	1.99492i	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$		−	28.0000i		−	1.99492i	0.0712470	−	0.997459i	$-0.477302\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	2.00000	+	14.0000i	0.141421	+	0.989949i
$201$		0			0
$202$	27.2137			1.91475
$203$		0			0
$204$		0			0
$205$	1.94975	+	27.4350i	0.136176	+	1.91615i
$206$		0			0
$207$		0			0
$208$	−16.0502			−1.11288
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$			28.0000i			1.92305i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		−	14.0000i		−	0.948200i
$219$		0			0
$220$		0			0
$221$	−19.6985			−1.32506
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	14.8492	−	2.12132i	0.989949	−	0.141421i
$226$	19.7990			1.31701
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$			26.1857i			1.73040i	0.501430	+	0.865198i	$0.332808\pi$
$229$			26.1857i			1.73040i	−0.501430	+	0.865198i	$0.667192\pi$
$230$		0			0
$231$		0			0
$232$			28.0000i			1.83829i
$233$			7.07107i			0.463241i	0.972806	+	0.231621i	$0.0744028\pi$
$233$			7.07107i			0.463241i	−0.972806	+	0.231621i	$0.925597\pi$
$234$			17.0238i			1.11288i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		−	4.08947i		−	0.263426i	−0.991288	−	0.131713i	$-0.957952\pi$
$241$		−	4.08947i		−	0.263426i	0.991288	−	0.131713i	$-0.0420477\pi$
$242$		−	15.5563i		−	1.00000i
$243$		0			0
$244$			14.5194i			0.929512i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−3.34003	−	15.4546i	−0.211242	−	0.977434i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−30.3295			−1.89190			−0.945951	−	0.324308i	$-0.894869\pi$
$257$	−30.3295			−1.89190			−0.945951	+	0.324308i	$0.894869\pi$
$258$		0			0
$259$		0			0
$260$	17.8995	−	1.27208i	1.11008	−	0.0788909i
$261$	29.6985			1.83829
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$	−2.21918	−	31.2262i	−0.136323	−	1.91821i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	8.52782i		−	0.519950i	−0.965615	−	0.259975i	$-0.916286\pi$
$269$		−	8.52782i		−	0.519950i	0.965615	−	0.259975i	$-0.0837144\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	19.6369			1.19066
$273$		0			0
$274$	30.0000			1.81237
$275$		0			0
$276$		0			0
$277$			28.0000i			1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$			28.0000i			1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−29.6985			−1.77166			−0.885832	−	0.464007i	$-0.846411\pi$
$281$	−29.6985			−1.77166			−0.885832	+	0.464007i	$0.846411\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		−	16.9706i		−	1.00000i
$289$	7.10051			0.417677
$290$	−2.21918	−	31.2262i	−0.130315	−	1.83367i
$291$		0			0
$292$	24.9719			1.46137
$293$	29.8812			1.74568			0.872838	−	0.488010i	$-0.162277\pi$
$293$	29.8812			1.74568			0.872838	+	0.488010i	$0.162277\pi$
$294$		0			0
$295$		0			0
$296$	20.0000			1.16248
$297$		0			0
$298$			28.2843i			1.63846i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.15076	−	16.1924i	−0.0658922	−	0.927173i
$306$		−	20.8281i		−	1.19066i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	32.1229			1.81569			0.907846	−	0.419303i	$-0.137726\pi$
$313$	32.1229			1.81569			0.907846	+	0.419303i	$0.137726\pi$
$314$			3.77244i			0.212891i
$315$		0			0
$316$		0			0
$317$		−	28.0000i		−	1.57264i	−0.617822	−	0.786318i	$-0.711985\pi$
$317$		−	28.0000i		−	1.57264i	0.617822	−	0.786318i	$-0.288015\pi$
$318$		0			0
$319$		0			0
$320$	−17.8435	+	1.26810i	−0.997484	+	0.0708890i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−18.0000			−1.00000
$325$	−19.8611	+	2.83730i	−1.10169	+	0.157385i
$326$		0			0
$327$		0			0
$328$	−34.7904			−1.92098
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		−	21.2132i		−	1.16248i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			35.3553i			1.92593i	0.269630	+	0.962964i	$0.413099\pi$
$337$			35.3553i			1.92593i	−0.269630	+	0.962964i	$0.586901\pi$
$338$		−	4.38478i		−	0.238500i
$339$		0			0
$340$	−21.8995	+	1.55635i	−1.18767	+	0.0844049i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$			8.84485i			0.475502i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		−	23.0154i		−	1.23199i	−0.787752	−	0.615993i	$-0.788755\pi$
$349$		−	23.0154i		−	1.23199i	0.787752	−	0.615993i	$-0.211245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.77084			0.0942521			0.0471260	−	0.998889i	$-0.484994\pi$
$353$	1.77084			0.0942521			0.0471260	+	0.998889i	$0.484994\pi$
$354$		0			0
$355$		0			0
$356$			37.2178i			1.97254i
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$	1.34502	+	18.9259i	0.0708890	+	0.997484i
$361$	−19.0000			−1.00000
$362$	34.3421			1.80498
$363$		0			0
$364$		0			0
$365$	−27.8492	+	1.97918i	−1.45770	+	0.103595i
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$			36.9008i			1.92098i
$370$	−22.3044	+	1.58513i	−1.15955	+	0.0824068i
$371$		0			0
$372$		0			0
$373$			14.0000i			0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$			14.0000i			0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−39.7222			−2.04579
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$	19.7990			1.00774
$387$		0			0
$388$	28.5587			1.44985
$389$	−9.89949			−0.501924			−0.250962	−	0.967997i	$-0.580747\pi$
$389$	−9.89949			−0.501924			−0.250962	+	0.967997i	$0.580747\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−39.5980			−1.99492
$395$		0			0
$396$		0			0
$397$	−39.6996			−1.99247			−0.996233	−	0.0867112i	$-0.972364\pi$
$397$	−39.6996			−1.99247			−0.996233	+	0.0867112i	$0.972364\pi$
$398$		0			0
$399$		0			0
$400$	19.7990	−	2.82843i	0.989949	−	0.141421i
$401$	−29.6985			−1.48307			−0.741536	−	0.670913i	$-0.765902\pi$
$401$	−29.6985			−1.48307			−0.741536	+	0.670913i	$0.765902\pi$
$402$		0			0
$403$		0			0
$404$		−	38.4859i		−	1.91475i
$405$	20.0740	−	1.42661i	0.997484	−	0.0708890i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	9.76406i		−	0.482802i	−0.970425	−	0.241401i	$-0.922393\pi$
$409$		−	9.76406i		−	0.482802i	0.970425	−	0.241401i	$-0.0776069\pi$
$410$	38.7990	−	2.75736i	1.91615	−	0.136176i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$			22.6984i			1.11288i
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−30.0000			−1.46211			−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000			−1.46211			−0.731055	+	0.682318i	$0.760972\pi$
$422$		0			0
$423$		0			0
$424$	39.5980			1.92305
$425$	24.2994	−	3.47135i	1.17870	−	0.168385i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−40.5963			−1.95093			−0.975467	−	0.220146i	$-0.929347\pi$
$433$	−40.5963			−1.95093			−0.975467	+	0.220146i	$0.929347\pi$
$434$		0			0
$435$		0			0
$436$	−19.7990			−0.948200
$437$		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
$442$			27.8579i			1.32506i
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$	−2.94975	−	41.5061i	−0.139831	−	1.96758i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−40.0000			−1.88772			−0.943858	−	0.330350i	$-0.892833\pi$
$449$	−40.0000			−1.88772			−0.943858	+	0.330350i	$0.892833\pi$
$450$	−3.00000	−	21.0000i	−0.141421	−	0.989949i
$451$		0			0
$452$		−	28.0000i		−	1.31701i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	42.0000i		−	1.96468i	−0.187112	−	0.982339i	$-0.559913\pi$
$457$		−	42.0000i		−	1.96468i	0.187112	−	0.982339i	$-0.440087\pi$
$458$	37.0321			1.73040
$459$		0			0
$460$		0			0
$461$		−	27.4538i		−	1.27865i	−0.768937	−	0.639324i	$-0.779214\pi$
$461$		−	27.4538i		−	1.27865i	0.768937	−	0.639324i	$-0.220786\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$	39.5980			1.83829
$465$		0			0
$466$	10.0000			0.463241
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	24.0753			1.11288
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	42.0000i		−	1.92305i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$			28.3730i			1.29370i
$482$	−5.78338			−0.263426
$483$		0			0
$484$	−22.0000			−1.00000
$485$	−31.8492	+	2.26346i	−1.44620	+	0.102778i
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	20.5336			0.929512
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	48.5989			2.18878
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	−21.8561	+	4.72352i	−0.977434	+	0.211242i
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	3.05025	+	42.9203i	0.135734	+	1.90993i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	44.4775i		−	1.97143i	−0.168416	−	0.985716i	$-0.553865\pi$
$509$		−	44.4775i		−	1.97143i	0.168416	−	0.985716i	$-0.446135\pi$
$510$		0			0
$511$		0			0
$512$		−	22.6274i		−	1.00000i
$513$		0			0
$514$			42.8924i			1.89190i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−1.79899	−	25.3137i	−0.0788909	−	1.11008i
$521$			35.6327i			1.56110i	0.625096	+	0.780548i	$0.285060\pi$
$521$			35.6327i			1.56110i	−0.625096	+	0.780548i	$0.714940\pi$
$522$		−	42.0000i		−	1.83829i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$	−44.1605	+	3.13839i	−1.91821	+	0.136323i
$531$		0			0
$532$		0			0
$533$		−	49.3553i		−	2.13782i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−12.0602			−0.519950
$539$		0			0
$540$		0			0
$541$	20.0000			0.859867			0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000			0.859867			0.429934	+	0.902861i	$0.358537\pi$
$542$		0			0
$543$		0			0
$544$		−	27.7708i		−	1.19066i
$545$	22.0803	−	1.56920i	0.945814	−	0.0672169i
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		−	42.4264i		−	1.81237i
$549$		−	21.7792i		−	0.929512i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	39.5980			1.68236
$555$		0			0
$556$		0			0
$557$		−	28.0000i		−	1.18640i	−0.805056	−	0.593199i	$-0.797865\pi$
$557$		−	28.0000i		−	1.18640i	0.805056	−	0.593199i	$-0.202135\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$			42.0000i			1.77166i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$	2.21918	+	31.2262i	0.0933615	+	1.31370i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	9.89949			0.415008			0.207504	−	0.978234i	$-0.433466\pi$
$569$	9.89949			0.415008			0.207504	+	0.978234i	$0.433466\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−24.0000			−1.00000
$577$	16.5210			0.687780			0.343890	−	0.939010i	$-0.388255\pi$
$577$	16.5210			0.687780			0.343890	+	0.939010i	$0.388255\pi$
$578$		−	10.0416i		−	0.417677i
$579$		0			0
$580$	−44.1605	+	3.13839i	−1.83367	+	0.130315i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		−	35.3157i		−	1.46137i
$585$	−26.8492	+	1.90812i	−1.11008	+	0.0788909i
$586$		−	42.2584i		−	1.74568i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		−	28.2843i		−	1.16248i
$593$	48.6214			1.99664			0.998321	−	0.0579298i	$-0.0184500\pi$
$593$	48.6214			1.99664			0.998321	+	0.0579298i	$0.0184500\pi$
$594$		0			0
$595$		0			0
$596$	40.0000			1.63846
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$		−	9.13001i		−	0.372421i	−0.982510	−	0.186210i	$-0.940379\pi$
$601$		−	9.13001i		−	0.372421i	0.982510	−	0.186210i	$-0.0596206\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	24.5349	−	1.74364i	0.997484	−	0.0708890i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$	−22.8995	+	1.62742i	−0.927173	+	0.0658922i
$611$		0			0
$612$	−29.4554			−1.19066
$613$			1.41421i			0.0571195i	0.999592	+	0.0285598i	$0.00909209\pi$
$613$			1.41421i			0.0571195i	−0.999592	+	0.0285598i	$0.990908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			4.24264i			0.170802i	0.996347	+	0.0854011i	$0.0272172\pi$
$617$			4.24264i			0.170802i	−0.996347	+	0.0854011i	$0.972783\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	24.0000	−	7.00000i	0.960000	−	0.280000i
$626$		−	45.4286i		−	1.81569i
$627$		0			0
$628$	5.33504			0.212891
$629$		−	34.7135i		−	1.38412i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	−39.5980			−1.57264
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	1.79337	+	25.2346i	0.0708890	+	0.997484i
$641$	29.6985			1.17302			0.586510	−	0.809942i	$-0.300502\pi$
$641$	29.6985			1.17302			0.586510	+	0.809942i	$0.300502\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$			25.4558i			1.00000i
$649$		0			0
$650$	4.01254	+	28.0878i	0.157385	+	1.10169i
$651$		0			0
$652$		0			0
$653$			12.7279i			0.498082i	0.968493	+	0.249041i	$0.0801154\pi$
$653$			12.7279i			0.498082i	−0.968493	+	0.249041i	$0.919885\pi$
$654$		0			0
$655$		0			0
$656$			49.2011i			1.92098i
$657$	−37.4579			−1.46137
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		−	50.7862i		−	1.97535i	−0.156508	−	0.987677i	$-0.550024\pi$
$661$		−	50.7862i		−	1.97535i	0.156508	−	0.987677i	$-0.449976\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−30.0000			−1.16248
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			15.5563i			0.599653i	0.953994	+	0.299827i	$0.0969288\pi$
$673$			15.5563i			0.599653i	−0.953994	+	0.299827i	$0.903071\pi$
$674$	50.0000			1.92593
$675$		0			0
$676$	−6.20101			−0.238500
$677$	−47.2764			−1.81698			−0.908489	−	0.417908i	$-0.862763\pi$
$677$	−47.2764			−1.81698			−0.908489	+	0.417908i	$0.862763\pi$
$678$		0			0
$679$		0			0
$680$	2.20101	+	30.9706i	0.0844049	+	1.18767i
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$	3.36256	+	47.3148i	0.128477	+	1.80781i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			56.1756i			2.14012i
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	12.5085			0.475502
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$			60.3848i			2.28724i
$698$	−32.5487			−1.23199
$699$		0			0
$700$		0			0
$701$	29.6985			1.12170			0.560848	−	0.827919i	$-0.310475\pi$
$701$	29.6985			1.12170			0.560848	+	0.827919i	$0.310475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		−	2.50434i		−	0.0942521i
$707$		0			0
$708$		0			0
$709$	9.89949			0.371783			0.185892	−	0.982570i	$-0.440483\pi$
$709$	9.89949			0.371783			0.185892	+	0.982570i	$0.440483\pi$
$710$		0			0
$711$		0			0
$712$	52.6339			1.97254
$713$		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	$0.5\pi$
$720$	26.7653	−	1.90215i	0.997484	−	0.0708890i
$721$		0			0
$722$			26.8701i			1.00000i
$723$		0			0
$724$		−	48.5670i		−	1.80498i
$725$	49.0000	−	7.00000i	1.81981	−	0.259973i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$	2.79899	+	39.3848i	0.103595	+	1.45770i
$731$		0			0
$732$		0			0
$733$	16.9694			0.626779			0.313389	−	0.949625i	$-0.398536\pi$
$733$	16.9694			0.626779			0.313389	+	0.949625i	$0.398536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	52.1856			1.92098
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	2.24171	+	31.5432i	0.0824068	+	1.15955i
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	−44.6088	+	3.17025i	−1.63434	+	0.116149i
$746$	19.7990			0.724893
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$			56.1756i			2.04579i
$755$		0			0
$756$		0			0
$757$			24.0416i			0.873808i	0.899508	+	0.436904i	$0.143925\pi$
$757$			24.0416i			0.873808i	−0.899508	+	0.436904i	$0.856075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			22.4132i			0.812478i	0.913767	+	0.406239i	$0.133160\pi$
$761$			22.4132i			0.812478i	−0.913767	+	0.406239i	$0.866840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	32.8492	−	2.33452i	1.18767	−	0.0844049i
$766$		0			0
$767$		0			0
$768$		0			0
$769$			41.3073i			1.48958i	0.667300	+	0.744789i	$0.267450\pi$
$769$			41.3073i			1.48958i	−0.667300	+	0.744789i	$0.732550\pi$
$770$		0			0
$771$		0			0
$772$		−	28.0000i		−	1.00774i
$773$	−27.6395			−0.994122			−0.497061	−	0.867715i	$-0.665588\pi$
$773$	−27.6395			−0.994122			−0.497061	+	0.867715i	$0.665588\pi$
$774$		0			0
$775$		0			0
$776$		−	40.3881i		−	1.44985i
$777$		0			0
$778$			14.0000i			0.501924i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5.94975	+	0.422836i	−0.212356	+	0.0150916i
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$			56.0000i			1.99492i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$			29.1299i			1.03443i
$794$			56.1437i			1.99247i
$795$		0			0
$796$		0			0
$797$	0.425811			0.0150830			0.00754150	−	0.999972i	$-0.497599\pi$
$797$	0.425811			0.0150830			0.00754150	+	0.999972i	$0.497599\pi$
$798$		0			0
$799$		0			0
$800$	−4.00000	−	28.0000i	−0.141421	−	0.989949i
$801$		−	55.8267i		−	1.97254i
$802$			42.0000i			1.48307i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−54.4273			−1.91475
$809$	−10.0000			−0.351581			−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000			−0.351581			−0.175791	+	0.984428i	$0.556248\pi$
$810$	−2.01754	−	28.3889i	−0.0708890	−	0.997484i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−13.8085			−0.482802
$819$		0			0
$820$	−3.89949	−	54.8701i	−0.136176	−	1.91615i
$821$	−50.0000			−1.74501			−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000			−1.74501			−0.872506	+	0.488603i	$0.837507\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$			2.18731i			0.0759686i	0.999278	+	0.0379843i	$0.0120937\pi$
$829$			2.18731i			0.0759686i	−0.999278	+	0.0379843i	$0.987906\pi$
$830$		0			0
$831$		0			0
$832$	32.1003			1.11288
$833$		0			0
$834$		0			0
$835$		0			0
$836$		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
$841$	69.0000			2.37931
$842$			42.4264i			1.46211i
$843$		0			0
$844$		0			0
$845$	6.91550	−	0.491469i	0.237900	−	0.0169071i
$846$		0			0
$847$		0			0
$848$		−	56.0000i		−	1.92305i
$849$		0			0
$850$	−4.90923	−	34.3646i	−0.168385	−	1.17870i
$851$		0			0
$852$		0			0
$853$	50.8631			1.74152			0.870760	−	0.491709i	$-0.163628\pi$
$853$	50.8631			1.74152			0.870760	+	0.491709i	$0.163628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−56.6465			−1.93501			−0.967503	−	0.252858i	$-0.918630\pi$
$857$	−56.6465			−1.93501			−0.967503	+	0.252858i	$0.918630\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−13.9497	+	0.991378i	−0.474306	+	0.0337079i
$866$			57.4118i			1.95093i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$			28.0000i			0.948200i
$873$	−42.8380			−1.44985
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	32.5269i		−	1.09836i	−0.835705	−	0.549178i	$-0.814941\pi$
$877$		−	32.5269i		−	1.09836i	0.835705	−	0.549178i	$-0.185059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			10.4300i			0.351395i	0.984444	+	0.175697i	$0.0562180\pi$
$881$			10.4300i			0.351395i	−0.984444	+	0.175697i	$0.943782\pi$
$882$		0			0
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$	39.3970			1.32506
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$	−58.6985	+	4.17157i	−1.96758	+	0.139831i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$			56.5685i			1.88772i
$899$		0			0
$900$	−29.6985	+	4.24264i	−0.989949	+	0.141421i
$901$		−	68.7292i		−	2.28970i
$902$		0			0
$903$		0			0
$904$	−39.5980			−1.31701
$905$	3.84924	+	54.1630i	0.127953	+	1.80044i
$906$		0			0
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$			57.7289i			1.91475i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	−59.3970			−1.96468
$915$		0			0
$916$		−	52.3713i		−	1.73040i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$	−38.8255			−1.27865
$923$		0			0
$924$		0			0
$925$	−5.00000	−	35.0000i	−0.164399	−	1.15079i
$926$		0			0
$927$		0			0
$928$		−	56.0000i		−	1.83829i
$929$		−	54.5586i		−	1.79001i	−0.446056	−	0.895005i	$-0.647172\pi$
$929$		−	54.5586i		−	1.79001i	0.446056	−	0.895005i	$-0.352828\pi$
$930$		0			0
$931$		0			0
$932$		−	14.1421i		−	0.463241i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		−	34.0476i		−	1.11288i
$937$	58.8882			1.92379			0.961897	−	0.273413i	$-0.0881527\pi$
$937$	58.8882			1.92379			0.961897	+	0.273413i	$0.0881527\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			40.6732i			1.32591i	0.748660	+	0.662955i	$0.230698\pi$
$941$			40.6732i			1.32591i	−0.748660	+	0.662955i	$0.769302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$	50.1005			1.62633
$950$		0			0
$951$		0			0
$952$		0			0
$953$			56.0000i			1.81402i	0.421111	+	0.907009i	$0.361640\pi$
$953$			56.0000i			1.81402i	−0.421111	+	0.907009i	$0.638360\pi$
$954$	−59.3970			−1.92305
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$	40.1254			1.29370
$963$		0			0
$964$			8.17893i			0.263426i
$965$	2.21918	+	31.2262i	0.0714378	+	1.00521i
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$			31.1127i			1.00000i
$969$		0			0
$970$	3.20101	+	45.0416i	0.102778	+	1.44620i
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		−	29.0389i		−	0.929512i
$977$			38.1838i			1.22161i	0.791782	+	0.610803i	$0.209153\pi$
$977$			38.1838i			1.22161i	−0.791782	+	0.610803i	$0.790847\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	29.6985			0.948200
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$	−4.43835	−	62.4524i	−0.141418	−	1.98990i
$986$		−	68.7292i		−	2.18878i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		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
$996$		0			0
$997$	34.8129			1.10254			0.551268	−	0.834328i	$-0.314144\pi$
$997$	34.8129			1.10254			0.551268	+	0.834328i	$0.314144\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.c.a.979.4	yes	8
4.3	odd	2	CM	980.2.c.a.979.4	yes	8
5.4	even	2	inner	980.2.c.a.979.5	yes	8
7.2	even	3		980.2.s.c.619.2		16
7.3	odd	6		980.2.s.c.19.7		16
7.4	even	3		980.2.s.c.19.6		16
7.5	odd	6		980.2.s.c.619.3		16
7.6	odd	2	inner	980.2.c.a.979.1	✓	8
20.19	odd	2	inner	980.2.c.a.979.5	yes	8
28.3	even	6		980.2.s.c.19.7		16
28.11	odd	6		980.2.s.c.19.6		16
28.19	even	6		980.2.s.c.619.3		16
28.23	odd	6		980.2.s.c.619.2		16
28.27	even	2	inner	980.2.c.a.979.1	✓	8
35.4	even	6		980.2.s.c.19.3		16
35.9	even	6		980.2.s.c.619.7		16
35.19	odd	6		980.2.s.c.619.6		16
35.24	odd	6		980.2.s.c.19.2		16
35.34	odd	2	inner	980.2.c.a.979.8	yes	8
140.19	even	6		980.2.s.c.619.6		16
140.39	odd	6		980.2.s.c.19.3		16
140.59	even	6		980.2.s.c.19.2		16
140.79	odd	6		980.2.s.c.619.7		16
140.139	even	2	inner	980.2.c.a.979.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.c.a.979.1	✓	8	7.6	odd	2	inner
980.2.c.a.979.1	✓	8	28.27	even	2	inner
980.2.c.a.979.4	yes	8	1.1	even	1	trivial
980.2.c.a.979.4	yes	8	4.3	odd	2	CM
980.2.c.a.979.5	yes	8	5.4	even	2	inner
980.2.c.a.979.5	yes	8	20.19	odd	2	inner
980.2.c.a.979.8	yes	8	35.34	odd	2	inner
980.2.c.a.979.8	yes	8	140.139	even	2	inner
980.2.s.c.19.2		16	35.24	odd	6
980.2.s.c.19.2		16	140.59	even	6
980.2.s.c.19.3		16	35.4	even	6
980.2.s.c.19.3		16	140.39	odd	6
980.2.s.c.19.6		16	7.4	even	3
980.2.s.c.19.6		16	28.11	odd	6
980.2.s.c.19.7		16	7.3	odd	6
980.2.s.c.19.7		16	28.3	even	6
980.2.s.c.619.2		16	7.2	even	3
980.2.s.c.619.2		16	28.23	odd	6
980.2.s.c.619.3		16	7.5	odd	6
980.2.s.c.619.3		16	28.19	even	6
980.2.s.c.619.6		16	35.19	odd	6
980.2.s.c.619.6		16	140.19	even	6
980.2.s.c.619.7		16	35.9	even	6
980.2.s.c.619.7		16	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 $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +(2.23044 - 0.158513i) q^{5} +2.82843i q^{8} +3.00000 q^{9} +(-0.224171 - 3.15432i) q^{10} -4.01254 q^{13} +4.00000 q^{16} +4.90923 q^{17} -4.24264i q^{18} +(-4.46088 + 0.317025i) q^{20} +(4.94975 - 0.707107i) q^{25} +5.67459i q^{26} +9.89949 q^{29} -5.65685i q^{32} -6.94269i q^{34} -6.00000 q^{36} -7.07107i q^{37} +(0.448342 + 6.30864i) q^{40} +12.3003i q^{41} +(6.69133 - 0.475538i) q^{45} +(-1.00000 - 7.00000i) q^{50} +8.02509 q^{52} -14.0000i q^{53} -14.0000i q^{58} -7.25972i q^{61} -8.00000 q^{64} +(-8.94975 + 0.636039i) q^{65} -9.81845 q^{68} +8.48528i q^{72} -12.4860 q^{73} -10.0000 q^{74} +(8.92177 - 0.634051i) q^{80} +9.00000 q^{81} +17.3952 q^{82} +(10.9497 - 0.778175i) q^{85} -18.6089i q^{89} +(-0.672512 - 9.46297i) q^{90} -14.2793 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 24 q^{9} + 32 q^{16} - 48 q^{36} - 8 q^{50} - 64 q^{64} - 32 q^{65} - 80 q^{74} + 72 q^{81} + 48 q^{85}+O(q^{100}) \) 8 * q - 16 * q^4 + 24 * q^9 + 32 * q^16 - 48 * q^36 - 8 * q^50 - 64 * q^64 - 32 * q^65 - 80 * q^74 + 72 * q^81 + 48 * q^85

Introduction

L-functions

Modular forms

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