Embedded newform 354.2.g.a.101.7

• Label: 354.2.g.a.101.7
• Level: $354$
• Weight: $2$
• Character: 354.101
• Analytic conductor: $2.827$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	354.g (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.82670423155\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(10\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			101.7
Character	\(\chi\)	\(=\)	354.101
Dual form			354.2.g.a.347.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.561187 + 0.827689i) q^{2} +(1.17407 - 1.27341i) q^{3} +(-0.370138 - 0.928977i) q^{4} +(1.38516 - 2.99397i) q^{5} +(0.395111 + 1.68638i) q^{6} +(0.533222 + 1.92049i) q^{7} +(0.976621 + 0.214970i) q^{8} +(-0.243123 - 2.99013i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 10 q^{2} + q^{3} - 10 q^{4} + q^{6} + 2 q^{7} - 10 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/354\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{11}{58}\right)\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.561187	+	0.827689i	−0.396819	+	0.585265i
							\(3\)	1.17407	−	1.27341i	0.677849	−	0.735201i
\(5\)	1.38516	−	2.99397i	0.619461	−	1.33894i								−0.303050	−	0.952975i	\(-0.598005\pi\)
							0.922512	−	0.385969i	\(-0.126133\pi\)
\(7\)	0.533222	+	1.92049i	0.201539	+	0.725878i	0.993352	+	0.115115i	\(0.0367235\pi\)
							−0.791813	+	0.610763i	\(0.790863\pi\)
\(11\)	−1.75038	−	1.65805i	−0.527760	−	0.499921i	0.376784	−	0.926301i	\(-0.377030\pi\)
							−0.904544	+	0.426380i	\(0.859788\pi\)
\(13\)	−0.135152	+	1.24270i	−0.0374844	+	0.344663i	0.960345	+	0.278814i	\(0.0899412\pi\)
							−0.997830	+	0.0658493i	\(0.979024\pi\)
\(17\)	1.40811	+	0.390960i	0.341517	+	0.0948218i	0.434052	−	0.900888i	\(-0.357083\pi\)
							−0.0925351	+	0.995709i	\(0.529497\pi\)
\(19\)	−0.190281	−	0.144648i	−0.0436536	−	0.0331846i	0.583120	−	0.812386i	\(-0.301832\pi\)
							−0.626773	+	0.779202i	\(0.715625\pi\)
\(23\)	−2.16833	−	4.08991i	−0.452128	−	0.852805i	−0.999869	−	0.0161567i	\(-0.994857\pi\)
							0.547741	−	0.836648i	\(-0.315488\pi\)
\(29\)	3.20337	−	2.17194i	0.594851	−	0.403319i	−0.226277	−	0.974063i	\(-0.572655\pi\)
							0.821128	+	0.570744i	\(0.193345\pi\)
\(31\)	0.378750	+	0.498236i	0.0680254	+	0.0894859i	0.828847	−	0.559475i	\(-0.188997\pi\)
							−0.760822	+	0.648961i	\(0.775204\pi\)
\(37\)	2.13771	+	9.71172i	0.351438	+	1.59660i	0.738308	+	0.674464i	\(0.235625\pi\)
							−0.386870	+	0.922134i	\(0.626444\pi\)
\(41\)	10.6407	+	5.64136i	1.66180	+	0.881033i	0.990124	+	0.140191i	\(0.0447717\pi\)
							0.671680	+	0.740842i	\(0.265573\pi\)
\(43\)	3.54938	+	3.74703i	0.541275	+	0.571417i	0.938050	−	0.346499i	\(-0.112630\pi\)
							−0.396776	+	0.917916i	\(0.629871\pi\)
\(47\)	−6.50675	+	3.01034i	−0.949107	+	0.439104i	−0.832470	−	0.554071i	\(-0.813074\pi\)
							−0.116638	+	0.993175i	\(0.537212\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.a.101.7	✓	280
3.2	odd	2		354.2.g.b.101.4	yes	280
59.52	odd	58		354.2.g.b.347.4	yes	280
177.170	even	58	inner	354.2.g.a.347.7	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.101.7	✓	280	1.1	even	1	trivial
354.2.g.a.347.7	yes	280	177.170	even	58	inner
354.2.g.b.101.4	yes	280	3.2	odd	2
354.2.g.b.347.4	yes	280	59.52	odd	58