LMFDB - Embedded newform 980.2.c.a.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:	$\Q(\zeta_{16})$
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			979.6
Root			$-0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	980.979
Dual form			980.2.c.a.979.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} +(-0.158513 - 2.23044i) q^{5} -2.82843i q^{8} +3.00000 q^{9} +(3.15432 - 0.224171i) q^{10} -5.99162 q^{13} +4.00000 q^{16} -6.62567 q^{17} +4.24264i q^{18} +(0.317025 + 4.46088i) q^{20} +(-4.94975 + 0.707107i) q^{25} -8.47343i q^{26} -9.89949 q^{29} +5.65685i q^{32} -9.37011i q^{34} -6.00000 q^{36} +7.07107i q^{37} +(-6.30864 + 0.448342i) q^{40} -3.56420i q^{41} +(-0.475538 - 6.69133i) q^{45} +(-1.00000 - 7.00000i) q^{50} +11.9832 q^{52} -14.0000i q^{53} -14.0000i q^{58} -13.8310i q^{61} -8.00000 q^{64} +(0.949747 + 13.3640i) q^{65} +13.2513 q^{68} -8.48528i q^{72} -11.6662 q^{73} -10.0000 q^{74} +(-0.634051 - 8.92177i) q^{80} +9.00000 q^{81} +5.04054 q^{82} +(1.05025 + 14.7782i) q^{85} +3.11586i q^{89} +(9.46297 - 0.672512i) q^{90} +13.5684 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.158513	−	2.23044i	−0.0708890	−	0.997484i
$5$	−0.158513	−	2.23044i	−0.0708890	−	0.997484i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		−	2.82843i		−	1.00000i
$9$	3.00000			1.00000
$10$	3.15432	−	0.224171i	0.997484	−	0.0708890i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−5.99162			−1.66178			−0.830888	−	0.556440i	$-0.812167\pi$
$13$	−5.99162			−1.66178			−0.830888	+	0.556440i	$0.812167\pi$
$14$		0			0
$15$		0			0
$16$	4.00000			1.00000
$17$	−6.62567			−1.60696			−0.803480	−	0.595331i	$-0.797021\pi$
$17$	−6.62567			−1.60696			−0.803480	+	0.595331i	$0.797021\pi$
$18$			4.24264i			1.00000i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	0.317025	+	4.46088i	0.0708890	+	0.997484i
$21$		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
$26$		−	8.47343i		−	1.66178i
$27$		0			0
$28$		0			0
$29$	−9.89949			−1.83829			−0.919145	−	0.393919i	$-0.871119\pi$
$29$	−9.89949			−1.83829			−0.919145	+	0.393919i	$0.871119\pi$
$30$		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
$34$		−	9.37011i		−	1.60696i
$35$		0			0
$36$	−6.00000			−1.00000
$37$			7.07107i			1.16248i	0.813733	+	0.581238i	$0.197432\pi$
$37$			7.07107i			1.16248i	−0.813733	+	0.581238i	$0.802568\pi$
$38$		0			0
$39$		0			0
$40$	−6.30864	+	0.448342i	−0.997484	+	0.0708890i
$41$		−	3.56420i		−	0.556635i	−0.960489	−	0.278317i	$-0.910223\pi$
$41$		−	3.56420i		−	0.556635i	0.960489	−	0.278317i	$-0.0897767\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	−0.475538	−	6.69133i	−0.0708890	−	0.997484i
$46$		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
$52$	11.9832			1.66178
$53$		−	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
$61$		−	13.8310i		−	1.77088i	−0.464756	−	0.885439i	$-0.653858\pi$
$61$		−	13.8310i		−	1.77088i	0.464756	−	0.885439i	$-0.346142\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$	0.949747	+	13.3640i	0.117802	+	1.65760i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	13.2513			1.60696
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		−	8.48528i		−	1.00000i
$73$	−11.6662			−1.36543			−0.682713	−	0.730686i	$-0.739200\pi$
$73$	−11.6662			−1.36543			−0.682713	+	0.730686i	$0.739200\pi$
$74$	−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			$\pi$
$80$	−0.634051	−	8.92177i	−0.0708890	−	0.997484i
$81$	9.00000			1.00000
$82$	5.04054			0.556635
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	1.05025	+	14.7782i	0.113916	+	1.60292i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			3.11586i			0.330281i	0.986270	+	0.165140i	$0.0528077\pi$
$89$			3.11586i			0.330281i	−0.986270	+	0.165140i	$0.947192\pi$
$90$	9.46297	−	0.672512i	0.997484	−	0.0708890i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	13.5684			1.37766			0.688829	−	0.724924i	$-0.258125\pi$
$97$	13.5684			1.37766			0.688829	+	0.724924i	$0.258125\pi$
$98$		0			0
$99$		0			0
$100$	9.89949	−	1.41421i	0.989949	−	0.141421i
$101$			5.80591i			0.577710i	0.957373	+	0.288855i	$0.0932745\pi$
$101$			5.80591i			0.577710i	−0.957373	+	0.288855i	$0.906726\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$			16.9469i			1.66178i
$105$		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	$-0.657226\pi$
$109$	−9.89949			−0.948200			−0.474100	+	0.880471i	$0.657226\pi$
$110$		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
$117$	−17.9749			−1.66178
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	19.5600			1.77088
$123$		0			0
$124$		0			0
$125$	2.36176	+	10.9280i	0.211242	+	0.977434i
$126$		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
$130$	−18.8995	+	1.34315i	−1.65760	+	0.117802i
$131$		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
$136$			18.7402i			1.60696i
$137$		−	21.2132i		−	1.81237i	−0.422885	−	0.906183i	$-0.638983\pi$
$137$		−	21.2132i		−	1.81237i	0.422885	−	0.906183i	$-0.361017\pi$
$138$		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
$145$	1.56920	+	22.0803i	0.130315	+	1.83367i
$146$		−	16.4985i		−	1.36543i
$147$		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
$153$	−19.8770			−1.60696
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−24.9176			−1.98864			−0.994319	−	0.106446i	$-0.966053\pi$
$157$	−24.9176			−1.98864			−0.994319	+	0.106446i	$0.966053\pi$
$158$		0			0
$159$		0			0
$160$	12.6173	−	0.896683i	0.997484	−	0.0708890i
$161$		0			0
$162$			12.7279i			1.00000i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$			7.12840i			0.556635i
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	22.8995			1.76150
$170$	−20.8995	+	1.48528i	−1.60292	+	0.113916i
$171$		0			0
$172$		0			0
$173$	25.5516			1.94265			0.971326	−	0.237751i	$-0.0764102\pi$
$173$	25.5516			1.94265			0.971326	+	0.237751i	$0.0764102\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−4.40649			−0.330281
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	0.951076	+	13.3827i	0.0708890	+	0.997484i
$181$		−	11.5893i		−	0.861425i	−0.902489	−	0.430713i	$-0.858262\pi$
$181$		−	11.5893i		−	0.861425i	0.902489	−	0.430713i	$-0.141738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	15.7716	−	1.12085i	1.15955	−	0.0824068i
$186$		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	$0.831909\pi$
$194$			19.1886i			1.37766i
$195$		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
$202$	−8.21080			−0.577710
$203$		0			0
$204$		0			0
$205$	−7.94975	+	0.564971i	−0.555234	+	0.0394593i
$206$		0			0
$207$		0			0
$208$	−23.9665			−1.66178
$209$		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
$221$	39.6985			2.67041
$222$		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
$229$			15.1760i			1.00286i	0.865198	+	0.501430i	$0.167192\pi$
$229$			15.1760i			1.00286i	−0.865198	+	0.501430i	$0.832808\pi$
$230$		0			0
$231$		0			0
$232$			28.0000i			1.83829i
$233$		−	7.07107i		−	0.463241i	−0.972806	−	0.231621i	$-0.925597\pi$
$233$		−	7.07107i		−	0.463241i	0.972806	−	0.231621i	$-0.0744028\pi$
$234$		−	25.4203i		−	1.66178i
$235$		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
$241$			30.7779i			1.98258i	0.131713	+	0.991288i	$0.457952\pi$
$241$			30.7779i			1.98258i	−0.131713	+	0.991288i	$0.542048\pi$
$242$			15.5563i			1.00000i
$243$		0			0
$244$			27.6620i			1.77088i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−15.4546	+	3.34003i	−0.977434	+	0.211242i
$251$		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
$257$	−10.3981			−0.648616			−0.324308	−	0.945951i	$-0.605131\pi$
$257$	−10.3981			−0.648616			−0.324308	+	0.945951i	$0.605131\pi$
$258$		0			0
$259$		0			0
$260$	−1.89949	−	26.7279i	−0.117802	−	1.65760i
$261$	−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
$265$	−31.2262	+	2.21918i	−1.91821	+	0.136323i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	31.6745i		−	1.93123i	−0.259975	−	0.965615i	$-0.583714\pi$
$269$		−	31.6745i		−	1.93123i	0.259975	−	0.965615i	$-0.416286\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	−26.5027			−1.60696
$273$		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	$-0.153589\pi$
$281$	29.6985			1.77166			0.885832	+	0.464007i	$0.153589\pi$
$282$		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
$289$	26.8995			1.58232
$290$	−31.2262	+	2.21918i	−1.83367	+	0.130315i
$291$		0			0
$292$	23.3324			1.36543
$293$	16.7068			0.976019			0.488010	−	0.872838i	$-0.337723\pi$
$293$	16.7068			0.976019			0.488010	+	0.872838i	$0.337723\pi$
$294$		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
$305$	−30.8492	+	2.19239i	−1.76642	+	0.125536i
$306$		−	28.1103i		−	1.60696i
$307$		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
$313$	−14.8365			−0.838607			−0.419303	−	0.907846i	$-0.637726\pi$
$313$	−14.8365			−0.838607			−0.419303	+	0.907846i	$0.637726\pi$
$314$		−	35.2387i		−	1.98864i
$315$		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
$320$	1.26810	+	17.8435i	0.0708890	+	0.997484i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−18.0000			−1.00000
$325$	29.6570	−	4.23671i	1.64507	−	0.235011i
$326$		0			0
$327$		0			0
$328$	−10.0811			−0.556635
$329$		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	$-0.586901\pi$
$337$		−	35.3553i		−	1.92593i	0.269630	−	0.962964i	$-0.413099\pi$
$338$			32.3848i			1.76150i
$339$		0			0
$340$	−2.10051	−	29.5563i	−0.113916	−	1.60292i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$			36.1354i			1.94265i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$			29.4328i			1.57550i	0.615993	+	0.787752i	$0.288755\pi$
$349$			29.4328i			1.57550i	−0.615993	+	0.787752i	$0.711245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	37.5348			1.99778			0.998889	−	0.0471260i	$-0.0150062\pi$
$353$	37.5348			1.99778			0.998889	+	0.0471260i	$0.0150062\pi$
$354$		0			0
$355$		0			0
$356$		−	6.23172i		−	0.330281i
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$	−18.9259	+	1.34502i	−0.997484	+	0.0708890i
$361$	−19.0000			−1.00000
$362$	16.3897			0.861425
$363$		0			0
$364$		0			0
$365$	1.84924	+	26.0208i	0.0967938	+	1.36199i
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		−	10.6926i		−	0.556635i
$370$	1.58513	+	22.3044i	0.0824068	+	1.15955i
$371$		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
$377$	59.3140			3.05483
$378$		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
$388$	−27.1367			−1.37766
$389$	9.89949			0.501924			0.250962	−	0.967997i	$-0.419253\pi$
$389$	9.89949			0.501924			0.250962	+	0.967997i	$0.419253\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	39.5980			1.99492
$395$		0			0
$396$		0			0
$397$	−3.45542			−0.173422			−0.0867112	−	0.996233i	$-0.527636\pi$
$397$	−3.45542			−0.173422			−0.0867112	+	0.996233i	$0.527636\pi$
$398$		0			0
$399$		0			0
$400$	−19.7990	+	2.82843i	−0.989949	+	0.141421i
$401$	29.6985			1.48307			0.741536	−	0.670913i	$-0.234098\pi$
$401$	29.6985			1.48307			0.741536	+	0.670913i	$0.234098\pi$
$402$		0			0
$403$		0			0
$404$		−	11.6118i		−	0.577710i
$405$	−1.42661	−	20.0740i	−0.0708890	−	0.997484i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			39.2513i			1.94085i	0.241401	+	0.970425i	$0.422393\pi$
$409$			39.2513i			1.94085i	−0.241401	+	0.970425i	$0.577607\pi$
$410$	−0.798990	−	11.2426i	−0.0394593	−	0.555234i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		−	33.8937i		−	1.66178i
$417$		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
$425$	32.7954	−	4.68506i	1.59081	−	0.227259i
$426$		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
$433$	9.16187			0.440291			0.220146	−	0.975467i	$-0.429347\pi$
$433$	9.16187			0.440291			0.220146	+	0.975467i	$0.429347\pi$
$434$		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
$442$			56.1421i			2.67041i
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$	6.94975	−	0.493903i	0.329450	−	0.0234133i
$446$		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	$-0.440087\pi$
$458$	−21.4621			−1.00286
$459$		0			0
$460$		0			0
$461$		−	33.0196i		−	1.53787i	−0.639324	−	0.768937i	$-0.720786\pi$
$461$		−	33.0196i		−	1.53787i	0.639324	−	0.768937i	$-0.279214\pi$
$462$		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			$\pi$
$468$	35.9497			1.66178
$469$		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
$481$		−	42.3671i		−	1.93178i
$482$	−43.5265			−1.98258
$483$		0			0
$484$	−22.0000			−1.00000
$485$	−2.15076	−	30.2635i	−0.0976609	−	1.37419i
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	−39.1200			−1.77088
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	65.5908			2.95406
$494$		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			$\pi$
$500$	−4.72352	−	21.8561i	−0.211242	−	0.977434i
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	12.9497	−	0.920310i	0.576256	−	0.0409533i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	7.59928i		−	0.336832i	−0.985716	−	0.168416i	$-0.946135\pi$
$509$		−	7.59928i		−	0.336832i	0.985716	−	0.168416i	$-0.0538652\pi$
$510$		0			0
$511$		0			0
$512$			22.6274i			1.00000i
$513$		0			0
$514$		−	14.7051i		−	0.648616i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	37.7990	−	2.68629i	1.65760	−	0.117802i
$521$		−	28.5361i		−	1.25019i	−0.780548	−	0.625096i	$-0.785060\pi$
$521$		−	28.5361i		−	1.25019i	0.780548	−	0.625096i	$-0.214940\pi$
$522$		−	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
$530$	−3.13839	−	44.1605i	−0.136323	−	1.91821i
$531$		0			0
$532$		0			0
$533$			21.3553i			0.925002i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	44.7946			1.93123
$539$		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
$544$		−	37.4804i		−	1.60696i
$545$	1.56920	+	22.0803i	0.0672169	+	0.945814i
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$			42.4264i			1.81237i
$549$		−	41.4930i		−	1.77088i
$550$		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
$565$	31.2262	−	2.21918i	1.31370	−	0.0933615i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9.89949			−0.415008			−0.207504	−	0.978234i	$-0.566534\pi$
$569$	−9.89949			−0.415008			−0.207504	+	0.978234i	$0.566534\pi$
$570$		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
$577$	−45.1116			−1.87802			−0.939010	−	0.343890i	$-0.888255\pi$
$577$	−45.1116			−1.87802			−0.939010	+	0.343890i	$0.888255\pi$
$578$			38.0416i			1.58232i
$579$		0			0
$580$	−3.13839	−	44.1605i	−0.130315	−	1.83367i
$581$		0			0
$582$		0			0
$583$		0			0
$584$			32.9970i			1.36543i
$585$	2.84924	+	40.0919i	0.117802	+	1.65760i
$586$			23.6269i			0.976019i
$587$		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
$593$	2.82137			0.115860			0.0579298	−	0.998321i	$-0.481550\pi$
$593$	2.82137			0.115860			0.0579298	+	0.998321i	$0.481550\pi$
$594$		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
$601$			48.1731i			1.96502i	0.186210	+	0.982510i	$0.440379\pi$
$601$			48.1731i			1.96502i	−0.186210	+	0.982510i	$0.559621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.74364	−	24.5349i	−0.0708890	−	0.997484i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$	−3.10051	−	43.6274i	−0.125536	−	1.76642i
$611$		0			0
$612$	39.7540			1.60696
$613$		−	1.41421i		−	0.0571195i	−0.999592	−	0.0285598i	$-0.990908\pi$
$613$		−	1.41421i		−	0.0571195i	0.999592	−	0.0285598i	$-0.00909209\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	4.24264i		−	0.170802i	−0.996347	−	0.0854011i	$-0.972783\pi$
$617$		−	4.24264i		−	0.170802i	0.996347	−	0.0854011i	$-0.0272172\pi$
$618$		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
$626$		−	20.9819i		−	0.838607i
$627$		0			0
$628$	49.8351			1.98864
$629$		−	46.8506i		−	1.86805i
$630$		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
$640$	−25.2346	+	1.79337i	−0.997484	+	0.0708890i
$641$	−29.6985			−1.17302			−0.586510	−	0.809942i	$-0.699498\pi$
$641$	−29.6985			−1.17302			−0.586510	+	0.809942i	$0.699498\pi$
$642$		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
$650$	5.99162	+	41.9413i	0.235011	+	1.64507i
$651$		0			0
$652$		0			0
$653$		−	12.7279i		−	0.498082i	−0.968493	−	0.249041i	$-0.919885\pi$
$653$		−	12.7279i		−	0.498082i	0.968493	−	0.249041i	$-0.0801154\pi$
$654$		0			0
$655$		0			0
$656$		−	14.2568i		−	0.556635i
$657$	−34.9986			−1.36543
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		−	8.04762i		−	0.313016i	−0.987677	−	0.156508i	$-0.949976\pi$
$661$		−	8.04762i		−	0.313016i	0.987677	−	0.156508i	$-0.0500237\pi$
$662$		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	$-0.903071\pi$
$673$		−	15.5563i		−	0.599653i	0.953994	−	0.299827i	$-0.0969288\pi$
$674$	50.0000			1.92593
$675$		0			0
$676$	−45.7990			−1.76150
$677$	−21.7473			−0.835817			−0.417908	−	0.908489i	$-0.637237\pi$
$677$	−21.7473			−0.835817			−0.417908	+	0.908489i	$0.637237\pi$
$678$		0			0
$679$		0			0
$680$	41.7990	−	2.97056i	1.60292	−	0.113916i
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$	−47.3148	+	3.36256i	−1.80781	+	0.128477i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			83.8827i			3.19567i
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	−51.1032			−1.94265
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$			23.6152i			0.894490i
$698$	−41.6243			−1.57550
$699$		0			0
$700$		0			0
$701$	−29.6985			−1.12170			−0.560848	−	0.827919i	$-0.689525\pi$
$701$	−29.6985			−1.12170			−0.560848	+	0.827919i	$0.689525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$			53.0823i			1.99778i
$707$		0			0
$708$		0			0
$709$	−9.89949			−0.371783			−0.185892	−	0.982570i	$-0.559517\pi$
$709$	−9.89949			−0.371783			−0.185892	+	0.982570i	$0.559517\pi$
$710$		0			0
$711$		0			0
$712$	8.81298			0.330281
$713$		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	$0.5\pi$
$720$	−1.90215	−	26.7653i	−0.0708890	−	0.997484i
$721$		0			0
$722$		−	26.8701i		−	1.00000i
$723$		0			0
$724$			23.1786i			0.861425i
$725$	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
$730$	−36.7990	+	2.61522i	−1.36199	+	0.0967938i
$731$		0			0
$732$		0			0
$733$	−51.4202			−1.89925			−0.949625	−	0.313389i	$-0.898536\pi$
$733$	−51.4202			−1.89925			−0.949625	+	0.313389i	$0.898536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	15.1216			0.556635
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	−31.5432	+	2.24171i	−1.15955	+	0.0824068i
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	3.17025	+	44.6088i	0.116149	+	1.63434i
$746$	−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
$754$			83.8827i			3.05483i
$755$		0			0
$756$		0			0
$757$		−	24.0416i		−	0.873808i	−0.899508	−	0.436904i	$-0.856075\pi$
$757$		−	24.0416i		−	0.873808i	0.899508	−	0.436904i	$-0.143925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			50.4148i			1.82753i	0.406239	+	0.913767i	$0.366840\pi$
$761$			50.4148i			1.82753i	−0.406239	+	0.913767i	$0.633160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	3.15076	+	44.3345i	0.113916	+	1.60292i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	37.0096i		−	1.33460i	−0.744789	−	0.667300i	$-0.767450\pi$
$769$		−	37.0096i		−	1.33460i	0.744789	−	0.667300i	$-0.232550\pi$
$770$		0			0
$771$		0			0
$772$		−	28.0000i		−	1.00774i
$773$	−48.2500			−1.73543			−0.867715	−	0.497061i	$-0.834412\pi$
$773$	−48.2500			−1.73543			−0.867715	+	0.497061i	$0.834412\pi$
$774$		0			0
$775$		0			0
$776$		−	38.3771i		−	1.37766i
$777$		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
$785$	3.94975	+	55.5772i	0.140973	+	1.98363i
$786$		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
$793$			82.8701i			2.94280i
$794$		−	4.88670i		−	0.173422i
$795$		0			0
$796$		0			0
$797$	56.4608			1.99994			0.999972	−	0.00754150i	$-0.00240056\pi$
$797$	56.4608			1.99994			0.999972	+	0.00754150i	$0.00240056\pi$
$798$		0			0
$799$		0			0
$800$	−4.00000	−	28.0000i	−0.141421	−	0.989949i
$801$			9.34758i			0.330281i
$802$			42.0000i			1.48307i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	16.4216			0.577710
$809$	−10.0000			−0.351581			−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000			−0.351581			−0.175791	+	0.984428i	$0.556248\pi$
$810$	28.3889	−	2.01754i	0.997484	−	0.0708890i
$811$		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
$818$	−55.5097			−1.94085
$819$		0			0
$820$	15.8995	−	1.12994i	0.555234	−	0.0394593i
$821$	−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
$829$		−	57.5432i		−	1.99856i	−0.0379843	−	0.999278i	$-0.512094\pi$
$829$		−	57.5432i		−	1.99856i	0.0379843	−	0.999278i	$-0.487906\pi$
$830$		0			0
$831$		0			0
$832$	47.9329			1.66178
$833$		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
$845$	−3.62986	−	51.0760i	−0.124871	−	1.75707i
$846$		0			0
$847$		0			0
$848$		−	56.0000i		−	1.92305i
$849$		0			0
$850$	6.62567	+	46.3797i	0.227259	+	1.59081i
$851$		0			0
$852$		0			0
$853$	−28.7219			−0.983418			−0.491709	−	0.870760i	$-0.663628\pi$
$853$	−28.7219			−0.983418			−0.491709	+	0.870760i	$0.663628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−14.8046			−0.505716			−0.252858	−	0.967503i	$-0.581370\pi$
$857$	−14.8046			−0.505716			−0.252858	+	0.967503i	$0.581370\pi$
$858$		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
$865$	−4.05025	−	56.9914i	−0.137713	−	1.93776i
$866$			12.9568i			0.440291i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$			28.0000i			0.948200i
$873$	40.7051			1.37766
$874$		0			0
$875$		0			0
$876$		0			0
$877$			32.5269i			1.09836i	0.835705	+	0.549178i	$0.185059\pi$
$877$			32.5269i			1.09836i	−0.835705	+	0.549178i	$0.814941\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			58.4398i			1.96889i	0.175697	+	0.984444i	$0.443782\pi$
$881$			58.4398i			1.96889i	−0.175697	+	0.984444i	$0.556218\pi$
$882$		0			0
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$	−79.3970			−2.67041
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$	0.698485	+	9.82843i	0.0234133	+	0.329450i
$891$		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
$901$			92.7594i			3.09026i
$902$		0			0
$903$		0			0
$904$	39.5980			1.31701
$905$	−25.8492	+	1.83705i	−0.859258	+	0.0610656i
$906$		0			0
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$			17.4177i			0.577710i
$910$		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
$916$		−	30.3520i		−	1.00286i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$	46.6967			1.53787
$923$		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
$929$			27.1911i			0.892112i	0.895005	+	0.446056i	$0.147172\pi$
$929$			27.1911i			0.892112i	−0.895005	+	0.446056i	$0.852828\pi$
$930$		0			0
$931$		0			0
$932$			14.1421i			0.463241i
$933$		0			0
$934$		0			0
$935$		0			0
$936$			50.8406i			1.66178i
$937$	−16.7386			−0.546827			−0.273413	−	0.961897i	$-0.588153\pi$
$937$	−16.7386			−0.546827			−0.273413	+	0.961897i	$0.588153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	45.9313i		−	1.49732i	−0.662955	−	0.748660i	$-0.730698\pi$
$941$		−	45.9313i		−	1.49732i	0.662955	−	0.748660i	$-0.269302\pi$
$942$		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
$949$	69.8995			2.26903
$950$		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
$962$	59.9162			1.93178
$963$		0			0
$964$		−	61.5557i		−	1.98258i
$965$	31.2262	−	2.21918i	1.00521	−	0.0714378i
$966$		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
$970$	42.7990	−	3.04163i	1.37419	−	0.0976609i
$971$		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
$976$		−	55.3240i		−	1.77088i
$977$		−	38.1838i		−	1.22161i	−0.791782	−	0.610803i	$-0.790847\pi$
$977$		−	38.1838i		−	1.22161i	0.791782	−	0.610803i	$-0.209153\pi$
$978$		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
$985$	−62.4524	+	4.43835i	−1.98990	+	0.141418i
$986$			92.7594i			2.95406i
$987$		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
$997$	−52.6883			−1.66866			−0.834328	−	0.551268i	$-0.814144\pi$
$997$	−52.6883			−1.66866			−0.834328	+	0.551268i	$0.814144\pi$
$998$		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.979.6	yes	8
4.3	odd	2	CM	980.2.c.a.979.6	yes	8
5.4	even	2	inner	980.2.c.a.979.3	yes	8
7.2	even	3		980.2.s.c.619.5		16
7.3	odd	6		980.2.s.c.19.1		16
7.4	even	3		980.2.s.c.19.4		16
7.5	odd	6		980.2.s.c.619.8		16
7.6	odd	2	inner	980.2.c.a.979.7	yes	8
20.19	odd	2	inner	980.2.c.a.979.3	yes	8
28.3	even	6		980.2.s.c.19.1		16
28.11	odd	6		980.2.s.c.19.4		16
28.19	even	6		980.2.s.c.619.8		16
28.23	odd	6		980.2.s.c.619.5		16
28.27	even	2	inner	980.2.c.a.979.7	yes	8
35.4	even	6		980.2.s.c.19.8		16
35.9	even	6		980.2.s.c.619.1		16
35.19	odd	6		980.2.s.c.619.4		16
35.24	odd	6		980.2.s.c.19.5		16
35.34	odd	2	inner	980.2.c.a.979.2	✓	8
140.19	even	6		980.2.s.c.619.4		16
140.39	odd	6		980.2.s.c.19.8		16
140.59	even	6		980.2.s.c.19.5		16
140.79	odd	6		980.2.s.c.619.1		16
140.139	even	2	inner	980.2.c.a.979.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.c.a.979.2	✓	8	35.34	odd	2	inner
980.2.c.a.979.2	✓	8	140.139	even	2	inner
980.2.c.a.979.3	yes	8	5.4	even	2	inner
980.2.c.a.979.3	yes	8	20.19	odd	2	inner
980.2.c.a.979.6	yes	8	1.1	even	1	trivial
980.2.c.a.979.6	yes	8	4.3	odd	2	CM
980.2.c.a.979.7	yes	8	7.6	odd	2	inner
980.2.c.a.979.7	yes	8	28.27	even	2	inner
980.2.s.c.19.1		16	7.3	odd	6
980.2.s.c.19.1		16	28.3	even	6
980.2.s.c.19.4		16	7.4	even	3
980.2.s.c.19.4		16	28.11	odd	6
980.2.s.c.19.5		16	35.24	odd	6
980.2.s.c.19.5		16	140.59	even	6
980.2.s.c.19.8		16	35.4	even	6
980.2.s.c.19.8		16	140.39	odd	6
980.2.s.c.619.1		16	35.9	even	6
980.2.s.c.619.1		16	140.79	odd	6
980.2.s.c.619.4		16	35.19	odd	6
980.2.s.c.619.4		16	140.19	even	6
980.2.s.c.619.5		16	7.2	even	3
980.2.s.c.619.5		16	28.23	odd	6
980.2.s.c.619.8		16	7.5	odd	6
980.2.s.c.619.8		16	28.19	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)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +(-0.158513 - 2.23044i) q^{5} -2.82843i q^{8} +3.00000 q^{9} +(3.15432 - 0.224171i) q^{10} -5.99162 q^{13} +4.00000 q^{16} -6.62567 q^{17} +4.24264i q^{18} +(0.317025 + 4.46088i) q^{20} +(-4.94975 + 0.707107i) q^{25} -8.47343i q^{26} -9.89949 q^{29} +5.65685i q^{32} -9.37011i q^{34} -6.00000 q^{36} +7.07107i q^{37} +(-6.30864 + 0.448342i) q^{40} -3.56420i q^{41} +(-0.475538 - 6.69133i) q^{45} +(-1.00000 - 7.00000i) q^{50} +11.9832 q^{52} -14.0000i q^{53} -14.0000i q^{58} -13.8310i q^{61} -8.00000 q^{64} +(0.949747 + 13.3640i) q^{65} +13.2513 q^{68} -8.48528i q^{72} -11.6662 q^{73} -10.0000 q^{74} +(-0.634051 - 8.92177i) q^{80} +9.00000 q^{81} +5.04054 q^{82} +(1.05025 + 14.7782i) q^{85} +3.11586i q^{89} +(9.46297 - 0.672512i) q^{90} +13.5684 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 24 q^{9} + 32 q^{16} - 48 q^{36} - 8 q^{50} - 64 q^{64} - 32 q^{65} - 80 q^{74} + 72 q^{81} + 48 q^{85}+O(q^{100}) \) 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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