Embedded newform 349.2.e.a.123.9

• Label: 349.2.e.a.123.9
• Level: $349$
• Weight: $2$
• Character: 349.123
• 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			123.9
Character	\(\chi\)	\(=\)	349.123
Dual form			349.2.e.a.227.9

$q$-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 \(\left(\mathbb{Z}/349\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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	−0.827814	−	0.561003i	\(-0.810416\pi\)
							0.0719360	+	0.997409i	\(0.477082\pi\)
\(3\)	0.625735	−	1.08381i	0.361269	−	0.625735i	−0.626901	−	0.779099i	\(-0.715677\pi\)
							0.988170	+	0.153363i	\(0.0490104\pi\)
\(5\)	0.733047	−	1.26968i	0.327829	−	0.567816i	−0.654252	−	0.756277i	\(-0.727016\pi\)
							0.982081	+	0.188461i	\(0.0603497\pi\)
\(7\)	−0.833145	+	0.481016i	−0.314899	+	0.181807i	−0.649117	−	0.760689i	\(-0.724861\pi\)
							0.334218	+	0.942496i	\(0.391528\pi\)
\(11\)			0.842874i			0.254136i	0.991894	+	0.127068i	\(0.0405566\pi\)
							−0.991894	+	0.127068i	\(0.959443\pi\)
\(13\)	5.14146	−	2.96842i	1.42598	−	0.823292i	0.429183	−	0.903218i	\(-0.358802\pi\)
							0.996801	+	0.0799257i	\(0.0254683\pi\)
\(17\)	−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	0.0128987	−	0.999917i	\(-0.495894\pi\)
							0.859504	−	0.511129i	\(-0.170773\pi\)
\(23\)	0.0324754	+	0.0562491i	0.00677159	+	0.0117287i	0.869391	−	0.494124i	\(-0.164511\pi\)
							−0.862620	+	0.505853i	\(0.831178\pi\)
\(29\)	3.19627	−	5.53610i	0.593532	−	1.02803i	−0.400220	−	0.916419i	\(-0.631066\pi\)
							0.993752	−	0.111609i	\(-0.0356003\pi\)
\(31\)	−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	0.919172	−	0.393856i	\(-0.128859\pi\)
							0.118497	+	0.992954i	\(0.462192\pi\)
\(47\)		−	10.6006i		−	1.54626i	−0.634250	−	0.773128i	\(-0.718691\pi\)
							0.634250	−	0.773128i	\(-0.281309\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.e.a.123.9	✓	58	1.1	even	1	trivial
349.2.e.a.227.9	yes	58	349.227	even	6	inner