Embedded newform 354.3.f.a.13.12

• Label: 354.3.f.a.13.12
• 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			13.12
Character	\(\chi\)	\(=\)	354.13
Dual form			354.3.f.a.109.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.07786 - 0.915542i) q^{2} +(-1.37887 + 1.04819i) q^{3} +(0.323564 - 1.97365i) q^{4} +(-1.81071 + 3.41535i) q^{5} +(-0.526568 + 2.39222i) q^{6} +(0.390182 - 0.0424348i) q^{7} +(-1.45821 - 2.42356i) q^{8} +(0.802585 - 2.89065i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} - 8 q^{7} - 60 q^{9}+O(q^{10}) \)

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
\(5\)	−1.81071	+	3.41535i	−0.362141	+	0.683071i								−0.995900	−	0.0904600i	\(-0.971166\pi\)
							0.633759	+	0.773531i	\(0.281511\pi\)
\(7\)	0.390182	−	0.0424348i	0.0557403	−	0.00606212i	−0.0802064	−	0.996778i	\(-0.525558\pi\)
							0.135947	+	0.990716i	\(0.456592\pi\)
\(11\)	1.68129	+	4.98990i	0.152845	+	0.453627i	0.996532	−	0.0832076i	\(-0.0265165\pi\)
							−0.843688	+	0.536835i	\(0.819620\pi\)
\(13\)	−15.6450	+	4.34381i	−1.20346	+	0.334140i	−0.810679	−	0.585491i	\(-0.800902\pi\)
							−0.392783	+	0.919631i	\(0.628488\pi\)
\(17\)	−20.1745	−	2.19411i	−1.18674	−	0.129065i	−0.506620	−	0.862170i	\(-0.669105\pi\)
							−0.680116	+	0.733104i	\(0.738071\pi\)
\(19\)	0.211253	+	3.89633i	0.0111186	+	0.205070i	0.998814	+	0.0486823i	\(0.0155022\pi\)
							−0.987696	+	0.156388i	\(0.950015\pi\)
\(23\)	−9.86188	+	21.3161i	−0.428778	+	0.926787i	0.565970	+	0.824426i	\(0.308502\pi\)
							−0.994747	+	0.102361i	\(0.967360\pi\)
\(29\)	−30.1613	+	35.5086i	−1.04004	+	1.22443i	−0.0660605	+	0.997816i	\(0.521043\pi\)
							−0.973984	+	0.226619i	\(0.927233\pi\)
\(31\)	−22.4329	−	1.21628i	−0.723641	−	0.0392347i	−0.311361	−	0.950292i	\(-0.600785\pi\)
							−0.412280	+	0.911057i	\(0.635268\pi\)
\(37\)	10.9271	−	18.1610i	0.295327	−	0.490837i	−0.673296	−	0.739373i	\(-0.735122\pi\)
							0.968623	+	0.248536i	\(0.0799496\pi\)
\(41\)	53.6412	−	24.8171i	1.30832	−	0.605294i	0.363109	−	0.931747i	\(-0.381715\pi\)
							0.945214	+	0.326452i	\(0.105853\pi\)
\(43\)	0.666146	−	1.97705i	0.0154918	−	0.0459779i	−0.939628	−	0.342197i	\(-0.888829\pi\)
							0.955120	+	0.296219i	\(0.0957257\pi\)
\(47\)	−78.6999	+	41.7240i	−1.67447	+	0.887745i	−0.687622	+	0.726069i	\(0.741345\pi\)
							−0.986844	+	0.161676i	\(0.948310\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.f.a.13.12	✓	560
59.50	odd	58	inner	354.3.f.a.109.12	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.f.a.13.12	✓	560	1.1	even	1	trivial
354.3.f.a.109.12	yes	560	59.50	odd	58	inner