Embedded newform 349.2.e.a.227.9

• Label: 349.2.e.a.227.9
• Level: $349$
• Weight: $2$
• Character: 349.227
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $58$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(58\)
Relative dimension:	\(29\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			227.9
Character	\(\chi\)	\(=\)	349.227
Dual form			349.2.e.a.123.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06897 - 0.617172i) q^{2} +(0.625735 + 1.08381i) q^{3} +(-0.238198 - 0.412572i) q^{4} +(0.733047 + 1.26968i) q^{5} -1.54474i q^{6} +(-0.833145 - 0.481016i) q^{7} +3.05672i q^{8} +(0.716910 - 1.24172i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q - 3 q^{2} + 27 q^{4} + 2 q^{5} - 29 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/349\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06897	−	0.617172i	−0.755878	−	0.436406i	0.0719360	−	0.997409i	\(-0.477082\pi\)
							−0.827814	+	0.561003i	\(0.810416\pi\)
\(3\)	0.625735	+	1.08381i	0.361269	+	0.625735i	0.988170	−	0.153363i	\(-0.0490104\pi\)
							−0.626901	+	0.779099i	\(0.715677\pi\)
\(5\)	0.733047	+	1.26968i	0.327829	+	0.567816i	0.982081	−	0.188461i	\(-0.0603497\pi\)
							−0.654252	+	0.756277i	\(0.727016\pi\)
\(7\)	−0.833145	−	0.481016i	−0.314899	−	0.181807i	0.334218	−	0.942496i	\(-0.391528\pi\)
							−0.649117	+	0.760689i	\(0.724861\pi\)
\(11\)		−	0.842874i		−	0.254136i	−0.991894	−	0.127068i	\(-0.959443\pi\)
							0.991894	−	0.127068i	\(-0.0405566\pi\)
\(13\)	5.14146	+	2.96842i	1.42598	+	0.823292i	0.996801	−	0.0799257i	\(-0.0254683\pi\)
							0.429183	+	0.903218i	\(0.358802\pi\)
\(17\)	−2.44219			−0.592318			−0.296159	−	0.955139i	\(-0.595706\pi\)
							−0.296159	+	0.955139i	\(0.595706\pi\)
\(19\)	3.80272	+	6.58650i	0.872403	+	1.51105i	0.859504	+	0.511129i	\(0.170773\pi\)
							0.0128987	+	0.999917i	\(0.495894\pi\)
\(23\)	0.0324754	−	0.0562491i	0.00677159	−	0.0117287i	−0.862620	−	0.505853i	\(-0.831178\pi\)
							0.869391	+	0.494124i	\(0.164511\pi\)
\(29\)	3.19627	+	5.53610i	0.593532	+	1.02803i	0.993752	+	0.111609i	\(0.0356003\pi\)
							−0.400220	+	0.916419i	\(0.631066\pi\)
\(31\)	−6.39813			−1.14914			−0.574569	−	0.818456i	\(-0.694830\pi\)
							−0.574569	+	0.818456i	\(0.694830\pi\)
\(37\)	8.13346			1.33713			0.668566	−	0.743653i	\(-0.266908\pi\)
							0.668566	+	0.743653i	\(0.266908\pi\)
\(41\)	−2.84700			−0.444627			−0.222314	−	0.974975i	\(-0.571361\pi\)
							−0.222314	+	0.974975i	\(0.571361\pi\)
\(43\)	6.80445	−	3.92855i	1.03767	−	0.599099i	0.118497	−	0.992954i	\(-0.462192\pi\)
							0.919172	+	0.393856i	\(0.128859\pi\)
\(47\)			10.6006i			1.54626i	0.634250	+	0.773128i	\(0.281309\pi\)
							−0.634250	+	0.773128i	\(0.718691\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.e.a.227.9	yes	58
349.123	even	6	inner	349.2.e.a.123.9	✓	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.e.a.123.9	✓	58	349.123	even	6	inner
349.2.e.a.227.9	yes	58	1.1	even	1	trivial