Embedded newform 354.3.f.a.13.1

• Label: 354.3.f.a.13.1
• Level: $354$
• Weight: $3$
• Character: 354.13
• Analytic conductor: $9.646$
• Analytic rank: $0$
• Dimension: $560$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.64580135835\)
Analytic rank:	\(0\)
Dimension:	\(560\)
Relative dimension:	\(20\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			13.1
Character	\(\chi\)	\(=\)	354.13
Dual form			354.3.f.a.109.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07786 + 0.915542i) q^{2} +(-1.37887 + 1.04819i) q^{3} +(0.323564 - 1.97365i) q^{4} +(-4.07963 + 7.69500i) q^{5} +(0.526568 - 2.39222i) q^{6} +(-11.4872 + 1.24931i) q^{7} +(1.45821 + 2.42356i) q^{8} +(0.802585 - 2.89065i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} - 8 q^{7} - 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{45}{58}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07786	+	0.915542i	−0.538930	+	0.457771i
							\(3\)	−1.37887	+	1.04819i	−0.459625	+	0.349397i
\(5\)	−4.07963	+	7.69500i	−0.815926	+	1.53900i								0.0261598	+	0.999658i	\(0.491672\pi\)
							−0.842086	+	0.539343i	\(0.818673\pi\)
\(7\)	−11.4872	+	1.24931i	−1.64103	+	0.178473i	−0.881561	−	0.472070i	\(-0.843507\pi\)
							−0.759470	+	0.650543i	\(0.774541\pi\)
\(11\)	−4.82995	−	14.3348i	−0.439086	−	1.30316i	−0.907648	−	0.419731i	\(-0.862124\pi\)
							0.468562	−	0.883431i	\(-0.344772\pi\)
\(13\)	−6.63882	+	1.84326i	−0.510678	+	0.141789i	−0.513303	−	0.858207i	\(-0.671578\pi\)
							0.00262479	+	0.999997i	\(0.499165\pi\)
\(17\)	27.0530	+	2.94219i	1.59135	+	0.173070i	0.860494	−	0.509460i	\(-0.170155\pi\)
							0.730857	+	0.682530i	\(0.239121\pi\)
\(19\)	1.77434	+	32.7257i	0.0933862	+	1.72241i	0.552062	+	0.833803i	\(0.313841\pi\)
							−0.458676	+	0.888604i	\(0.651676\pi\)
\(23\)	−5.53997	+	11.9744i	−0.240868	+	0.520628i	−0.989876	−	0.141933i	\(-0.954668\pi\)
							0.749008	+	0.662561i	\(0.230530\pi\)
\(29\)	16.1390	−	19.0003i	0.556516	−	0.655182i	−0.409910	−	0.912126i	\(-0.634440\pi\)
							0.966426	+	0.256944i	\(0.0827157\pi\)
\(31\)	−14.1766	−	0.768635i	−0.457311	−	0.0247947i	−0.175955	−	0.984398i	\(-0.556301\pi\)
							−0.281356	+	0.959604i	\(0.590784\pi\)
\(37\)	26.2706	−	43.6620i	0.710015	−	1.18005i	−0.266339	−	0.963879i	\(-0.585814\pi\)
							0.976355	−	0.216175i	\(-0.0693583\pi\)
\(41\)	33.7436	−	15.6115i	0.823016	−	0.380768i	0.0372607	−	0.999306i	\(-0.488137\pi\)
							0.785755	+	0.618538i	\(0.212275\pi\)
\(43\)	−15.1840	+	45.0644i	−0.353116	+	1.04801i	0.613040	+	0.790052i	\(0.289946\pi\)
							−0.966156	+	0.257958i	\(0.916950\pi\)
\(47\)	−44.9910	+	23.8527i	−0.957255	+	0.507505i	−0.872349	−	0.488884i	\(-0.837404\pi\)
							−0.0849067	+	0.996389i	\(0.527059\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.f.a.13.1	✓	560
59.50	odd	58	inner	354.3.f.a.109.1	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.f.a.13.1	✓	560	1.1	even	1	trivial
354.3.f.a.109.1	yes	560	59.50	odd	58	inner