LMFDB - Embedded newform 980.2.k.f.883.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,2,Mod(687,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

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

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

chi := DirichletCharacter("980.687");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$980 = 2^{2} \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	980.k (of order $4$ , degree $2$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.82533939809$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 1$ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			883.1
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	980.883
Dual form			980.2.k.f.687.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-2.12132 - 0.707107i) q^{5} +(-2.00000 - 2.00000i) q^{8} +3.00000i q^{9} +(-2.82843 + 1.41421i) q^{10} +(-4.24264 - 4.24264i) q^{13} -4.00000 q^{16} +(-5.65685 + 5.65685i) q^{17} +(3.00000 + 3.00000i) q^{18} +(-1.41421 + 4.24264i) q^{20} +(4.00000 + 3.00000i) q^{25} -8.48528 q^{26} -10.0000i q^{29} +(-4.00000 + 4.00000i) q^{32} +11.3137i q^{34} +6.00000 q^{36} +(-7.00000 + 7.00000i) q^{37} +(2.82843 + 5.65685i) q^{40} +1.41421 q^{41} +(2.12132 - 6.36396i) q^{45} +(7.00000 - 1.00000i) q^{50} +(-8.48528 + 8.48528i) q^{52} +(-5.00000 - 5.00000i) q^{53} +(-10.0000 - 10.0000i) q^{58} -1.41421 q^{61} +8.00000i q^{64} +(6.00000 + 12.0000i) q^{65} +(11.3137 + 11.3137i) q^{68} +(6.00000 - 6.00000i) q^{72} +(4.24264 + 4.24264i) q^{73} +14.0000i q^{74} +(8.48528 + 2.82843i) q^{80} -9.00000 q^{81} +(1.41421 - 1.41421i) q^{82} +(16.0000 - 8.00000i) q^{85} -18.3848i q^{89} +(-4.24264 - 8.48528i) q^{90} +(5.65685 - 5.65685i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 4 q^{2} - 8 q^{8} - 16 q^{16} + 12 q^{18} + 16 q^{25} - 16 q^{32} + 24 q^{36} - 28 q^{37} + 28 q^{50} - 20 q^{53} - 40 q^{58} + 24 q^{65} + 24 q^{72} - 36 q^{81} + 64 q^{85}+O(q^{100})$ 4 * q + 4 * q^2 - 8 * q^8 - 16 * q^16 + 12 * q^18 + 16 * q^25 - 16 * q^32 + 24 * q^36 - 28 * q^37 + 28 * q^50 - 20 * q^53 - 40 * q^58 + 24 * q^65 + 24 * q^72 - 36 * q^81 + 64 * q^85

Character values

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

$n$	$101$	$197$	$491$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

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

)

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

$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		−	2.00000i		−	1.00000i
$5$	−2.12132	−	0.707107i	−0.948683	−	0.316228i
$5$	−2.12132	−	0.707107i	−0.948683	−	0.316228i
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$			3.00000i			1.00000i
$10$	−2.82843	+	1.41421i	−0.894427	+	0.447214i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−4.24264	−	4.24264i	−1.17670	−	1.17670i	−0.980581	−	0.196116i	$-0.937167\pi$
$13$	−4.24264	−	4.24264i	−1.17670	−	1.17670i	−0.196116	−	0.980581i	$-0.562833\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	−5.65685	+	5.65685i	−1.37199	+	1.37199i	−0.514496	+	0.857493i	$0.672021\pi$
$17$	−5.65685	+	5.65685i	−1.37199	+	1.37199i	−0.857493	+	0.514496i	$0.827979\pi$
$18$	3.00000	+	3.00000i	0.707107	+	0.707107i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−1.41421	+	4.24264i	−0.316228	+	0.948683i
$21$		0			0
$22$		0			0
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$	4.00000	+	3.00000i	0.800000	+	0.600000i
$26$	−8.48528			−1.66410
$27$		0			0
$28$		0			0
$29$		−	10.0000i		−	1.85695i	−0.371391	−	0.928477i	$-0.621119\pi$
$29$		−	10.0000i		−	1.85695i	0.371391	−	0.928477i	$-0.378881\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−4.00000	+	4.00000i	−0.707107	+	0.707107i
$33$		0			0
$34$			11.3137i			1.94029i
$35$		0			0
$36$	6.00000			1.00000
$37$	−7.00000	+	7.00000i	−1.15079	+	1.15079i	−0.164399	+	0.986394i	$0.552568\pi$
$37$	−7.00000	+	7.00000i	−1.15079	+	1.15079i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		0			0
$40$	2.82843	+	5.65685i	0.447214	+	0.894427i
$41$	1.41421			0.220863			0.110432	−	0.993884i	$-0.464777\pi$
$41$	1.41421			0.220863			0.110432	+	0.993884i	$0.464777\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	2.12132	−	6.36396i	0.316228	−	0.948683i
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$		0			0
$50$	7.00000	−	1.00000i	0.989949	−	0.141421i
$51$		0			0
$52$	−8.48528	+	8.48528i	−1.17670	+	1.17670i
$53$	−5.00000	−	5.00000i	−0.686803	−	0.686803i	0.274721	−	0.961524i	$-0.411414\pi$
$53$	−5.00000	−	5.00000i	−0.686803	−	0.686803i	−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−10.0000	−	10.0000i	−1.31306	−	1.31306i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−1.41421			−0.181071			−0.0905357	−	0.995893i	$-0.528858\pi$
$61$	−1.41421			−0.181071			−0.0905357	+	0.995893i	$0.528858\pi$
$62$		0			0
$63$		0			0
$64$			8.00000i			1.00000i
$65$	6.00000	+	12.0000i	0.744208	+	1.48842i
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	11.3137	+	11.3137i	1.37199	+	1.37199i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	6.00000	−	6.00000i	0.707107	−	0.707107i
$73$	4.24264	+	4.24264i	0.496564	+	0.496564i	0.910366	−	0.413803i	$-0.135800\pi$
$73$	4.24264	+	4.24264i	0.496564	+	0.496564i	−0.413803	+	0.910366i	$0.635800\pi$
$74$			14.0000i			1.62747i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	8.48528	+	2.82843i	0.948683	+	0.316228i
$81$	−9.00000			−1.00000
$82$	1.41421	−	1.41421i	0.156174	−	0.156174i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	16.0000	−	8.00000i	1.73544	−	0.867722i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	18.3848i		−	1.94878i	−0.224860	−	0.974391i	$-0.572192\pi$
$89$		−	18.3848i		−	1.94878i	0.224860	−	0.974391i	$-0.427808\pi$
$90$	−4.24264	−	8.48528i	−0.447214	−	0.894427i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	5.65685	−	5.65685i	0.574367	−	0.574367i	−0.358979	−	0.933346i	$-0.616875\pi$
$97$	5.65685	−	5.65685i	0.574367	−	0.574367i	0.933346	+	0.358979i	$0.116875\pi$
$98$		0			0
$99$		0			0
$100$	6.00000	−	8.00000i	0.600000	−	0.800000i
$101$	−12.7279			−1.26648			−0.633238	−	0.773957i	$-0.718274\pi$
$101$	−12.7279			−1.26648			−0.633238	+	0.773957i	$0.718274\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$			16.9706i			1.66410i
$105$		0			0
$106$	−10.0000			−0.971286
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		−	20.0000i		−	1.91565i	−0.287348	−	0.957826i	$-0.592774\pi$
$109$		−	20.0000i		−	1.91565i	0.287348	−	0.957826i	$-0.407226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−15.0000	−	15.0000i	−1.41108	−	1.41108i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−15.0000	−	15.0000i	−1.41108	−	1.41108i	−0.658505	−	0.752577i	$-0.728811\pi$
$114$		0			0
$115$		0			0
$116$	−20.0000			−1.85695
$117$	12.7279	−	12.7279i	1.17670	−	1.17670i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	−1.41421	+	1.41421i	−0.128037	+	0.128037i
$123$		0			0
$124$		0			0
$125$	−6.36396	−	9.19239i	−0.569210	−	0.822192i
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	8.00000	+	8.00000i	0.707107	+	0.707107i
$129$		0			0
$130$	18.0000	+	6.00000i	1.57870	+	0.526235i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	22.6274			1.94029
$137$	−7.00000	+	7.00000i	−0.598050	+	0.598050i	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−7.00000	+	7.00000i	−0.598050	+	0.598050i	0.341743	+	0.939793i	$0.388983\pi$
$138$		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
$144$		−	12.0000i		−	1.00000i
$145$	−7.07107	+	21.2132i	−0.587220	+	1.76166i
$146$	8.48528			0.702247
$147$		0			0
$148$	14.0000	+	14.0000i	1.15079	+	1.15079i
$149$			14.0000i			1.14692i	0.819232	+	0.573462i	$0.194400\pi$
$149$			14.0000i			1.14692i	−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
$153$	−16.9706	−	16.9706i	−1.37199	−	1.37199i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	15.5563	−	15.5563i	1.24153	−	1.24153i	0.282166	−	0.959366i	$-0.408947\pi$
$157$	15.5563	−	15.5563i	1.24153	−	1.24153i	0.959366	−	0.282166i	$-0.0910530\pi$
$158$		0			0
$159$		0			0
$160$	11.3137	−	5.65685i	0.894427	−	0.447214i
$161$		0			0
$162$	−9.00000	+	9.00000i	−0.707107	+	0.707107i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		−	2.82843i		−	0.220863i
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$			23.0000i			1.76923i
$170$	8.00000	−	24.0000i	0.613572	−	1.84072i
$171$		0			0
$172$		0			0
$173$	2.82843	+	2.82843i	0.215041	+	0.215041i	0.806405	−	0.591364i	$-0.201410\pi$
$173$	2.82843	+	2.82843i	0.215041	+	0.215041i	−0.591364	+	0.806405i	$0.701410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−18.3848	−	18.3848i	−1.37800	−	1.37800i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−12.7279	−	4.24264i	−0.948683	−	0.316228i
$181$	26.8701			1.99724			0.998618	−	0.0525588i	$-0.0167377\pi$
$181$	26.8701			1.99724			0.998618	+	0.0525588i	$0.0167377\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	19.7990	−	9.89949i	1.45565	−	0.727825i
$186$		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
$193$	5.00000	+	5.00000i	0.359908	+	0.359908i	0.863779	−	0.503871i	$-0.168091\pi$
$193$	5.00000	+	5.00000i	0.359908	+	0.359908i	−0.503871	+	0.863779i	$0.668091\pi$
$194$		−	11.3137i		−	0.812277i
$195$		0			0
$196$		0			0
$197$	−15.0000	+	15.0000i	−1.06871	+	1.06871i	−0.0712470	+	0.997459i	$0.522698\pi$
$197$	−15.0000	+	15.0000i	−1.06871	+	1.06871i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		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
$202$	−12.7279	+	12.7279i	−0.895533	+	0.895533i
$203$		0			0
$204$		0			0
$205$	−3.00000	−	1.00000i	−0.209529	−	0.0698430i
$206$		0			0
$207$		0			0
$208$	16.9706	+	16.9706i	1.17670	+	1.17670i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−10.0000	+	10.0000i	−0.686803	+	0.686803i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−20.0000	−	20.0000i	−1.35457	−	1.35457i
$219$		0			0
$220$		0			0
$221$	48.0000			3.22883
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$	−9.00000	+	12.0000i	−0.600000	+	0.800000i
$226$	−30.0000			−1.99557
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		−	24.0416i		−	1.58872i	−0.607450	−	0.794358i	$-0.707808\pi$
$229$		−	24.0416i		−	1.58872i	0.607450	−	0.794358i	$-0.292192\pi$
$230$		0			0
$231$		0			0
$232$	−20.0000	+	20.0000i	−1.31306	+	1.31306i
$233$	−21.0000	−	21.0000i	−1.37576	−	1.37576i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−21.0000	−	21.0000i	−1.37576	−	1.37576i	−0.524097	−	0.851658i	$-0.675597\pi$
$234$		−	25.4558i		−	1.66410i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−15.5563			−1.00207			−0.501036	−	0.865426i	$-0.667048\pi$
$241$	−15.5563			−1.00207			−0.501036	+	0.865426i	$0.667048\pi$
$242$	11.0000	−	11.0000i	0.707107	−	0.707107i
$243$		0			0
$244$			2.82843i			0.181071i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−15.5563	−	2.82843i	−0.983870	−	0.178885i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−22.6274	+	22.6274i	−1.41146	+	1.41146i	−0.661622	+	0.749838i	$0.730131\pi$
$257$	−22.6274	+	22.6274i	−1.41146	+	1.41146i	−0.749838	+	0.661622i	$0.769869\pi$
$258$		0			0
$259$		0			0
$260$	24.0000	−	12.0000i	1.48842	−	0.744208i
$261$	30.0000			1.85695
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$	7.07107	+	14.1421i	0.434372	+	0.868744i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			32.5269i			1.98320i	0.129339	+	0.991600i	$0.458714\pi$
$269$			32.5269i			1.98320i	−0.129339	+	0.991600i	$0.541286\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	22.6274	−	22.6274i	1.37199	−	1.37199i
$273$		0			0
$274$			14.0000i			0.845771i
$275$		0			0
$276$		0			0
$277$	−5.00000	+	5.00000i	−0.300421	+	0.300421i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−5.00000	+	5.00000i	−0.300421	+	0.300421i	0.540758	+	0.841178i	$0.318138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−12.0000	−	12.0000i	−0.707107	−	0.707107i
$289$		−	47.0000i		−	2.76471i
$290$	14.1421	+	28.2843i	0.830455	+	1.66091i
$291$		0			0
$292$	8.48528	−	8.48528i	0.496564	−	0.496564i
$293$	−2.82843	−	2.82843i	−0.165238	−	0.165238i	0.619644	−	0.784883i	$-0.287277\pi$
$293$	−2.82843	−	2.82843i	−0.165238	−	0.165238i	−0.784883	+	0.619644i	$0.787277\pi$
$294$		0			0
$295$		0			0
$296$	28.0000			1.62747
$297$		0			0
$298$	14.0000	+	14.0000i	0.810998	+	0.810998i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.00000	+	1.00000i	0.171780	+	0.0572598i
$306$	−33.9411			−1.94029
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	18.3848	+	18.3848i	1.03917	+	1.03917i	0.999201	+	0.0399680i	$0.0127256\pi$
$313$	18.3848	+	18.3848i	1.03917	+	1.03917i	0.0399680	+	0.999201i	$0.487274\pi$
$314$		−	31.1127i		−	1.75579i
$315$		0			0
$316$		0			0
$317$	−25.0000	+	25.0000i	−1.40414	+	1.40414i	−0.617822	+	0.786318i	$0.711985\pi$
$317$	−25.0000	+	25.0000i	−1.40414	+	1.40414i	−0.786318	+	0.617822i	$0.788015\pi$
$318$		0			0
$319$		0			0
$320$	5.65685	−	16.9706i	0.316228	−	0.948683i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			18.0000i			1.00000i
$325$	−4.24264	−	29.6985i	−0.235339	−	1.64738i
$326$		0			0
$327$		0			0
$328$	−2.82843	−	2.82843i	−0.156174	−	0.156174i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−21.0000	−	21.0000i	−1.15079	−	1.15079i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−7.00000	+	7.00000i	−0.381314	+	0.381314i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−7.00000	+	7.00000i	−0.381314	+	0.381314i	0.490261	+	0.871576i	$0.336901\pi$
$338$	23.0000	+	23.0000i	1.25104	+	1.25104i
$339$		0			0
$340$	−16.0000	−	32.0000i	−0.867722	−	1.73544i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	5.65685			0.304114
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$			18.3848i			0.984115i	0.870563	+	0.492057i	$0.163755\pi$
$349$			18.3848i			0.984115i	−0.870563	+	0.492057i	$0.836245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−11.3137	−	11.3137i	−0.602168	−	0.602168i	0.338719	−	0.940887i	$-0.390006\pi$
$353$	−11.3137	−	11.3137i	−0.602168	−	0.602168i	−0.940887	+	0.338719i	$0.890006\pi$
$354$		0			0
$355$		0			0
$356$	−36.7696			−1.94878
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−16.9706	+	8.48528i	−0.894427	+	0.447214i
$361$	−19.0000			−1.00000
$362$	26.8701	−	26.8701i	1.41226	−	1.41226i
$363$		0			0
$364$		0			0
$365$	−6.00000	−	12.0000i	−0.314054	−	0.628109i
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$			4.24264i			0.220863i
$370$	9.89949	−	29.6985i	0.514650	−	1.54395i
$371$		0			0
$372$		0			0
$373$	−25.0000	−	25.0000i	−1.29445	−	1.29445i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	−25.0000	−	25.0000i	−1.29445	−	1.29445i	−0.362446	−	0.932005i	$-0.618058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−42.4264	+	42.4264i	−2.18507	+	2.18507i
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$	10.0000			0.508987
$387$		0			0
$388$	−11.3137	−	11.3137i	−0.574367	−	0.574367i
$389$			20.0000i			1.01404i	0.861934	+	0.507020i	$0.169253\pi$
$389$			20.0000i			1.01404i	−0.861934	+	0.507020i	$0.830747\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$			30.0000i			1.51138i
$395$		0			0
$396$		0			0
$397$	8.48528	−	8.48528i	0.425864	−	0.425864i	−0.461353	−	0.887217i	$-0.652636\pi$
$397$	8.48528	−	8.48528i	0.425864	−	0.425864i	0.887217	+	0.461353i	$0.152636\pi$
$398$		0			0
$399$		0			0
$400$	−16.0000	−	12.0000i	−0.800000	−	0.600000i
$401$	40.0000			1.99750			0.998752	−	0.0499376i	$-0.0159023\pi$
$401$	40.0000			1.99750			0.998752	+	0.0499376i	$0.0159023\pi$
$402$		0			0
$403$		0			0
$404$			25.4558i			1.26648i
$405$	19.0919	+	6.36396i	0.948683	+	0.316228i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			24.0416i			1.18878i	0.804176	+	0.594391i	$0.202607\pi$
$409$			24.0416i			1.18878i	−0.804176	+	0.594391i	$0.797393\pi$
$410$	−4.00000	+	2.00000i	−0.197546	+	0.0987730i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	33.9411			1.66410
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	28.0000			1.36464			0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000			1.36464			0.682318	+	0.731055i	$0.260972\pi$
$422$		0			0
$423$		0			0
$424$			20.0000i			0.971286i
$425$	−39.5980	+	5.65685i	−1.92078	+	0.274398i
$426$		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
$433$	24.0416	+	24.0416i	1.15537	+	1.15537i	0.985460	+	0.169907i	$0.0543467\pi$
$433$	24.0416	+	24.0416i	1.15537	+	1.15537i	0.169907	+	0.985460i	$0.445653\pi$
$434$		0			0
$435$		0			0
$436$	−40.0000			−1.91565
$437$		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
$442$	48.0000	−	48.0000i	2.28313	−	2.28313i
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	−13.0000	+	39.0000i	−0.616259	+	1.84878i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			14.0000i			0.660701i	0.943858	+	0.330350i	$0.107167\pi$
$449$			14.0000i			0.660701i	−0.943858	+	0.330350i	$0.892833\pi$
$450$	3.00000	+	21.0000i	0.141421	+	0.989949i
$451$		0			0
$452$	−30.0000	+	30.0000i	−1.41108	+	1.41108i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	25.0000	−	25.0000i	1.16945	−	1.16945i	0.187112	−	0.982339i	$-0.440087\pi$
$457$	25.0000	−	25.0000i	1.16945	−	1.16945i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	−24.0416	−	24.0416i	−1.12339	−	1.12339i
$459$		0			0
$460$		0			0
$461$	12.7279			0.592798			0.296399	−	0.955064i	$-0.404214\pi$
$461$	12.7279			0.592798			0.296399	+	0.955064i	$0.404214\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$			40.0000i			1.85695i
$465$		0			0
$466$	−42.0000			−1.94561
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	−25.4558	−	25.4558i	−1.17670	−	1.17670i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	15.0000	−	15.0000i	0.686803	−	0.686803i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	59.3970			2.70827
$482$	−15.5563	+	15.5563i	−0.708572	+	0.708572i
$483$		0			0
$484$		−	22.0000i		−	1.00000i
$485$	−16.0000	+	8.00000i	−0.726523	+	0.363261i
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$	2.82843	+	2.82843i	0.128037	+	0.128037i
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	56.5685	+	56.5685i	2.54772	+	2.54772i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−18.3848	+	12.7279i	−0.822192	+	0.569210i
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$	27.0000	+	9.00000i	1.20148	+	0.400495i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			38.1838i			1.69247i	0.532813	+	0.846233i	$0.321135\pi$
$509$			38.1838i			1.69247i	−0.532813	+	0.846233i	$0.678865\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	−	16.0000i	0.707107	−	0.707107i
$513$		0			0
$514$			45.2548i			1.99611i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	12.0000	−	36.0000i	0.526235	−	1.57870i
$521$	−43.8406			−1.92069			−0.960346	−	0.278810i	$-0.910060\pi$
$521$	−43.8406			−1.92069			−0.960346	+	0.278810i	$0.910060\pi$
$522$	30.0000	−	30.0000i	1.31306	−	1.31306i
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			23.0000i			1.00000i
$530$	21.2132	+	7.07107i	0.921443	+	0.307148i
$531$		0			0
$532$		0			0
$533$	−6.00000	−	6.00000i	−0.259889	−	0.259889i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	32.5269	+	32.5269i	1.40233	+	1.40233i
$539$		0			0
$540$		0			0
$541$	−42.0000			−1.80572			−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000			−1.80572			−0.902861	+	0.429934i	$0.858537\pi$
$542$		0			0
$543$		0			0
$544$		−	45.2548i		−	1.94029i
$545$	−14.1421	+	42.4264i	−0.605783	+	1.81735i
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$	14.0000	+	14.0000i	0.598050	+	0.598050i
$549$		−	4.24264i		−	0.181071i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$			10.0000i			0.424859i
$555$		0			0
$556$		0			0
$557$	5.00000	−	5.00000i	0.211857	−	0.211857i	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	5.00000	−	5.00000i	0.211857	−	0.211857i	0.805056	+	0.593199i	$0.202135\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	10.0000	−	10.0000i	0.421825	−	0.421825i
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$	21.2132	+	42.4264i	0.892446	+	1.78489i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	40.0000i		−	1.67689i	−0.544988	−	0.838444i	$-0.683466\pi$
$569$		−	40.0000i		−	1.67689i	0.544988	−	0.838444i	$-0.316534\pi$
$570$		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
$577$	−1.41421	+	1.41421i	−0.0588745	+	0.0588745i	−0.735931	−	0.677057i	$-0.763255\pi$
$577$	−1.41421	+	1.41421i	−0.0588745	+	0.0588745i	0.677057	+	0.735931i	$0.263255\pi$
$578$	−47.0000	−	47.0000i	−1.95494	−	1.95494i
$579$		0			0
$580$	42.4264	+	14.1421i	1.76166	+	0.587220i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		−	16.9706i		−	0.702247i
$585$	−36.0000	+	18.0000i	−1.48842	+	0.744208i
$586$	−5.65685			−0.233682
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	28.0000	−	28.0000i	1.15079	−	1.15079i
$593$	−32.5269	−	32.5269i	−1.33572	−	1.33572i	−0.900159	−	0.435561i	$-0.856550\pi$
$593$	−32.5269	−	32.5269i	−1.33572	−	1.33572i	−0.435561	−	0.900159i	$-0.643450\pi$
$594$		0			0
$595$		0			0
$596$	28.0000			1.14692
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−41.0122			−1.67292			−0.836461	−	0.548026i	$-0.815379\pi$
$601$	−41.0122			−1.67292			−0.836461	+	0.548026i	$0.815379\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−23.3345	−	7.77817i	−0.948683	−	0.316228i
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$	4.00000	−	2.00000i	0.161955	−	0.0809776i
$611$		0			0
$612$	−33.9411	+	33.9411i	−1.37199	+	1.37199i
$613$	1.00000	+	1.00000i	0.0403896	+	0.0403896i	0.727013	−	0.686624i	$-0.240908\pi$
$613$	1.00000	+	1.00000i	0.0403896	+	0.0403896i	−0.686624	+	0.727013i	$0.740908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.00000	+	3.00000i	−0.120775	+	0.120775i	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−3.00000	+	3.00000i	−0.120775	+	0.120775i	0.644136	+	0.764911i	$0.277217\pi$
$618$		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
$625$	7.00000	+	24.0000i	0.280000	+	0.960000i
$626$	36.7696			1.46961
$627$		0			0
$628$	−31.1127	−	31.1127i	−1.24153	−	1.24153i
$629$		−	79.1960i		−	3.15775i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$			50.0000i			1.98575i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−11.3137	−	22.6274i	−0.447214	−	0.894427i
$641$	50.0000			1.97488			0.987441	−	0.157991i	$-0.0505015\pi$
$641$	50.0000			1.97488			0.987441	+	0.157991i	$0.0505015\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	18.0000	+	18.0000i	0.707107	+	0.707107i
$649$		0			0
$650$	−33.9411	−	25.4558i	−1.33128	−	0.998460i
$651$		0			0
$652$		0			0
$653$	−9.00000	−	9.00000i	−0.352197	−	0.352197i	0.508729	−	0.860927i	$-0.330115\pi$
$653$	−9.00000	−	9.00000i	−0.352197	−	0.352197i	−0.860927	+	0.508729i	$0.830115\pi$
$654$		0			0
$655$		0			0
$656$	−5.65685			−0.220863
$657$	−12.7279	+	12.7279i	−0.496564	+	0.496564i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−26.8701			−1.04512			−0.522562	−	0.852601i	$-0.675024\pi$
$661$	−26.8701			−1.04512			−0.522562	+	0.852601i	$0.675024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−42.0000			−1.62747
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	11.0000	+	11.0000i	0.424019	+	0.424019i	0.886585	−	0.462566i	$-0.153071\pi$
$673$	11.0000	+	11.0000i	0.424019	+	0.424019i	−0.462566	+	0.886585i	$0.653071\pi$
$674$			14.0000i			0.539260i
$675$		0			0
$676$	46.0000			1.76923
$677$	1.41421	−	1.41421i	0.0543526	−	0.0543526i	−0.679408	−	0.733761i	$-0.737763\pi$
$677$	1.41421	−	1.41421i	0.0543526	−	0.0543526i	0.733761	+	0.679408i	$0.237763\pi$
$678$		0			0
$679$		0			0
$680$	−48.0000	−	16.0000i	−1.84072	−	0.613572i
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$	19.7990	−	9.89949i	0.756481	−	0.378240i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			42.4264i			1.61632i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	5.65685	−	5.65685i	0.215041	−	0.215041i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−8.00000	+	8.00000i	−0.303022	+	0.303022i
$698$	18.3848	+	18.3848i	0.695874	+	0.695874i
$699$		0			0
$700$		0			0
$701$	−10.0000			−0.377695			−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000			−0.377695			−0.188847	+	0.982006i	$0.560475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−22.6274			−0.851594
$707$		0			0
$708$		0			0
$709$			30.0000i			1.12667i	0.826227	+	0.563337i	$0.190483\pi$
$709$			30.0000i			1.12667i	−0.826227	+	0.563337i	$0.809517\pi$
$710$		0			0
$711$		0			0
$712$	−36.7696	+	36.7696i	−1.37800	+	1.37800i
$713$		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	$0.5\pi$
$720$	−8.48528	+	25.4558i	−0.316228	+	0.948683i
$721$		0			0
$722$	−19.0000	+	19.0000i	−0.707107	+	0.707107i
$723$		0			0
$724$		−	53.7401i		−	1.99724i
$725$	30.0000	−	40.0000i	1.11417	−	1.48556i
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$	−18.0000	−	6.00000i	−0.666210	−	0.222070i
$731$		0			0
$732$		0			0
$733$	−38.1838	−	38.1838i	−1.41035	−	1.41035i	−0.757410	−	0.652940i	$-0.773536\pi$
$733$	−38.1838	−	38.1838i	−1.41035	−	1.41035i	−0.652940	−	0.757410i	$-0.726464\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	4.24264	+	4.24264i	0.156174	+	0.156174i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−19.7990	−	39.5980i	−0.727825	−	1.45565i
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$	9.89949	−	29.6985i	0.362689	−	1.08807i
$746$	−50.0000			−1.83063
$747$		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
$754$			84.8528i			3.09016i
$755$		0			0
$756$		0			0
$757$	17.0000	−	17.0000i	0.617876	−	0.617876i	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	17.0000	−	17.0000i	0.617876	−	0.617876i	0.944986	+	0.327111i	$0.106075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−55.1543			−1.99934			−0.999671	−	0.0256326i	$-0.991840\pi$
$761$	−55.1543			−1.99934			−0.999671	+	0.0256326i	$0.991840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	24.0000	+	48.0000i	0.867722	+	1.73544i
$766$		0			0
$767$		0			0
$768$		0			0
$769$			52.3259i			1.88692i	0.331486	+	0.943460i	$0.392450\pi$
$769$			52.3259i			1.88692i	−0.331486	+	0.943460i	$0.607550\pi$
$770$		0			0
$771$		0			0
$772$	10.0000	−	10.0000i	0.359908	−	0.359908i
$773$	−24.0416	−	24.0416i	−0.864717	−	0.864717i	0.127164	−	0.991882i	$-0.459412\pi$
$773$	−24.0416	−	24.0416i	−0.864717	−	0.864717i	−0.991882	+	0.127164i	$0.959412\pi$
$774$		0			0
$775$		0			0
$776$	−22.6274			−0.812277
$777$		0			0
$778$	20.0000	+	20.0000i	0.717035	+	0.717035i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−44.0000	+	22.0000i	−1.57043	+	0.785214i
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	30.0000	+	30.0000i	1.06871	+	1.06871i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	6.00000	+	6.00000i	0.213066	+	0.213066i
$794$		−	16.9706i		−	0.602263i
$795$		0			0
$796$		0			0
$797$	36.7696	−	36.7696i	1.30244	−	1.30244i	0.375705	−	0.926739i	$-0.377401\pi$
$797$	36.7696	−	36.7696i	1.30244	−	1.30244i	0.926739	−	0.375705i	$-0.122599\pi$
$798$		0			0
$799$		0			0
$800$	−28.0000	+	4.00000i	−0.989949	+	0.141421i
$801$	55.1543			1.94878
$802$	40.0000	−	40.0000i	1.41245	−	1.41245i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	25.4558	+	25.4558i	0.895533	+	0.895533i
$809$		−	56.0000i		−	1.96886i	−0.175791	−	0.984428i	$-0.556248\pi$
$809$		−	56.0000i		−	1.96886i	0.175791	−	0.984428i	$-0.443752\pi$
$810$	25.4558	−	12.7279i	0.894427	−	0.447214i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	24.0416	+	24.0416i	0.840596	+	0.840596i
$819$		0			0
$820$	−2.00000	+	6.00000i	−0.0698430	+	0.209529i
$821$	28.0000			0.977207			0.488603	−	0.872506i	$-0.337507\pi$
$821$	28.0000			0.977207			0.488603	+	0.872506i	$0.337507\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$		−	52.3259i		−	1.81735i	−0.417500	−	0.908677i	$-0.637094\pi$
$829$		−	52.3259i		−	1.81735i	0.417500	−	0.908677i	$-0.362906\pi$
$830$		0			0
$831$		0			0
$832$	33.9411	−	33.9411i	1.17670	−	1.17670i
$833$		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
$841$	−71.0000			−2.44828
$842$	28.0000	−	28.0000i	0.964944	−	0.964944i
$843$		0			0
$844$		0			0
$845$	16.2635	−	48.7904i	0.559480	−	1.67844i
$846$		0			0
$847$		0			0
$848$	20.0000	+	20.0000i	0.686803	+	0.686803i
$849$		0			0
$850$	−33.9411	+	45.2548i	−1.16417	+	1.55223i
$851$		0			0
$852$		0			0
$853$	−25.4558	−	25.4558i	−0.871592	−	0.871592i	0.121054	−	0.992646i	$-0.461372\pi$
$853$	−25.4558	−	25.4558i	−0.871592	−	0.871592i	−0.992646	+	0.121054i	$0.961372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	41.0122	−	41.0122i	1.40095	−	1.40095i	0.603858	−	0.797092i	$-0.293630\pi$
$857$	41.0122	−	41.0122i	1.40095	−	1.40095i	0.797092	−	0.603858i	$-0.206370\pi$
$858$		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		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$	−4.00000	−	8.00000i	−0.136004	−	0.272008i
$866$	48.0833			1.63394
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−40.0000	+	40.0000i	−1.35457	+	1.35457i
$873$	16.9706	+	16.9706i	0.574367	+	0.574367i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−23.0000	+	23.0000i	−0.776655	+	0.776655i	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−23.0000	+	23.0000i	−0.776655	+	0.776655i	0.202606	+	0.979260i	$0.435059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12.7279			0.428815			0.214407	−	0.976744i	$-0.431218\pi$
$881$	12.7279			0.428815			0.214407	+	0.976744i	$0.431218\pi$
$882$		0			0
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		−	96.0000i		−	3.22883i
$885$		0			0
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		0			0
$890$	26.0000	+	52.0000i	0.871522	+	1.74304i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	14.0000	+	14.0000i	0.467186	+	0.467186i
$899$		0			0
$900$	24.0000	+	18.0000i	0.800000	+	0.600000i
$901$	56.5685			1.88457
$902$		0			0
$903$		0			0
$904$			60.0000i			1.99557i
$905$	−57.0000	−	19.0000i	−1.89474	−	0.631581i
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$		−	38.1838i		−	1.26648i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		−	50.0000i		−	1.65385i
$915$		0			0
$916$	−48.0833			−1.58872
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$	12.7279	−	12.7279i	0.419172	−	0.419172i
$923$		0			0
$924$		0			0
$925$	−49.0000	+	7.00000i	−1.61111	+	0.230159i
$926$		0			0
$927$		0			0
$928$	40.0000	+	40.0000i	1.31306	+	1.31306i
$929$			4.24264i			0.139197i	0.997575	+	0.0695983i	$0.0221717\pi$
$929$			4.24264i			0.139197i	−0.997575	+	0.0695983i	$0.977828\pi$
$930$		0			0
$931$		0			0
$932$	−42.0000	+	42.0000i	−1.37576	+	1.37576i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−50.9117			−1.66410
$937$	33.9411	−	33.9411i	1.10881	−	1.10881i	0.115501	−	0.993307i	$-0.463153\pi$
$937$	33.9411	−	33.9411i	1.10881	−	1.10881i	0.993307	−	0.115501i	$-0.0368473\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−26.8701			−0.875939			−0.437969	−	0.898990i	$-0.644302\pi$
$941$	−26.8701			−0.875939			−0.437969	+	0.898990i	$0.644302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$		−	36.0000i		−	1.16861i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	15.0000	+	15.0000i	0.485898	+	0.485898i	0.907009	−	0.421111i	$-0.138360\pi$
$953$	15.0000	+	15.0000i	0.485898	+	0.485898i	−0.421111	+	0.907009i	$0.638360\pi$
$954$		−	30.0000i		−	0.971286i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$	59.3970	−	59.3970i	1.91504	−	1.91504i
$963$		0			0
$964$			31.1127i			1.00207i
$965$	−7.07107	−	14.1421i	−0.227626	−	0.455251i
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	−22.0000	−	22.0000i	−0.707107	−	0.707107i
$969$		0			0
$970$	−8.00000	+	24.0000i	−0.256865	+	0.770594i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	5.65685			0.181071
$977$	27.0000	−	27.0000i	0.863807	−	0.863807i	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	27.0000	−	27.0000i	0.863807	−	0.863807i	0.991778	+	0.127971i	$0.0408466\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	60.0000			1.91565
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$	42.4264	−	21.2132i	1.35182	−	0.675909i
$986$	113.137			3.60302
$987$		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
$997$	−8.48528	+	8.48528i	−0.268732	+	0.268732i	−0.828589	−	0.559857i	$-0.810856\pi$
$997$	−8.48528	+	8.48528i	−0.268732	+	0.268732i	0.559857	+	0.828589i	$0.310856\pi$
$998$		0			0
$999$		0			0

Display

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a_n

instead) (See

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a_n

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.k.f.883.1	yes	4
4.3	odd	2	CM	980.2.k.f.883.1	yes	4
5.2	odd	4	inner	980.2.k.f.687.1	✓	4
7.2	even	3		980.2.x.g.263.2		8
7.3	odd	6		980.2.x.g.863.1		8
7.4	even	3		980.2.x.g.863.2		8
7.5	odd	6		980.2.x.g.263.1		8
7.6	odd	2	inner	980.2.k.f.883.2	yes	4
20.7	even	4	inner	980.2.k.f.687.1	✓	4
28.3	even	6		980.2.x.g.863.1		8
28.11	odd	6		980.2.x.g.863.2		8
28.19	even	6		980.2.x.g.263.1		8
28.23	odd	6		980.2.x.g.263.2		8
28.27	even	2	inner	980.2.k.f.883.2	yes	4
35.2	odd	12		980.2.x.g.67.2		8
35.12	even	12		980.2.x.g.67.1		8
35.17	even	12		980.2.x.g.667.1		8
35.27	even	4	inner	980.2.k.f.687.2	yes	4
35.32	odd	12		980.2.x.g.667.2		8
140.27	odd	4	inner	980.2.k.f.687.2	yes	4
140.47	odd	12		980.2.x.g.67.1		8
140.67	even	12		980.2.x.g.667.2		8
140.87	odd	12		980.2.x.g.667.1		8
140.107	even	12		980.2.x.g.67.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.k.f.687.1	✓	4	5.2	odd	4	inner
980.2.k.f.687.1	✓	4	20.7	even	4	inner
980.2.k.f.687.2	yes	4	35.27	even	4	inner
980.2.k.f.687.2	yes	4	140.27	odd	4	inner
980.2.k.f.883.1	yes	4	1.1	even	1	trivial
980.2.k.f.883.1	yes	4	4.3	odd	2	CM
980.2.k.f.883.2	yes	4	7.6	odd	2	inner
980.2.k.f.883.2	yes	4	28.27	even	2	inner
980.2.x.g.67.1		8	35.12	even	12
980.2.x.g.67.1		8	140.47	odd	12
980.2.x.g.67.2		8	35.2	odd	12
980.2.x.g.67.2		8	140.107	even	12
980.2.x.g.263.1		8	7.5	odd	6
980.2.x.g.263.1		8	28.19	even	6
980.2.x.g.263.2		8	7.2	even	3
980.2.x.g.263.2		8	28.23	odd	6
980.2.x.g.667.1		8	35.17	even	12
980.2.x.g.667.1		8	140.87	odd	12
980.2.x.g.667.2		8	35.32	odd	12
980.2.x.g.667.2		8	140.67	even	12
980.2.x.g.863.1		8	7.3	odd	6
980.2.x.g.863.1		8	28.3	even	6
980.2.x.g.863.2		8	7.4	even	3
980.2.x.g.863.2		8	28.11	odd	6

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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	980.2.k.f.883.1

Level	$980$
Weight	$2$
Character	980.883
Analytic conductor	$7.825$
Analytic rank	$0$
Dimension	$4$
CM discriminant	-4
Inner twists	$8$