Embedded newform 349.2.e.a.123.10

• Label: 349.2.e.a.123.10
• 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			123.10
Character	\(\chi\)	\(=\)	349.123
Dual form			349.2.e.a.227.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.03662 + 0.598491i) q^{2} +(0.00513702 - 0.00889758i) q^{3} +(-0.283617 + 0.491238i) q^{4} +(-0.193021 + 0.334323i) q^{5} +0.0122978i q^{6} +(3.23720 - 1.86900i) q^{7} -3.07293i q^{8} +(1.49995 + 2.59798i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q - 3 q^{2} + 27 q^{4} + 2 q^{5} - 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.03662	+	0.598491i	−0.732999	+	0.423197i	−0.819518	−	0.573053i	\(-0.805759\pi\)
							0.0865193	+	0.996250i	\(0.472426\pi\)
\(3\)	0.00513702	−	0.00889758i	0.00296586	−	0.00513702i	−0.864539	−	0.502566i	\(-0.832389\pi\)
							0.867505	+	0.497429i	\(0.165723\pi\)
\(5\)	−0.193021	+	0.334323i	−0.0863217	+	0.149514i	−0.905954	−	0.423377i	\(-0.860845\pi\)
							0.819632	+	0.572890i	\(0.194178\pi\)
\(7\)	3.23720	−	1.86900i	1.22355	−	0.706415i	0.257875	−	0.966178i	\(-0.416978\pi\)
							0.965672	+	0.259763i	\(0.0836444\pi\)
\(11\)		−	4.48490i		−	1.35225i	−0.736788	−	0.676124i	\(-0.763658\pi\)
							0.736788	−	0.676124i	\(-0.236342\pi\)
\(13\)	−1.35467	+	0.782122i	−0.375719	+	0.216921i	−0.675954	−	0.736944i	\(-0.736268\pi\)
							0.300235	+	0.953865i	\(0.402935\pi\)
\(17\)	6.40967			1.55457			0.777287	−	0.629146i	\(-0.216595\pi\)
							0.777287	+	0.629146i	\(0.216595\pi\)
\(19\)	−2.90902	+	5.03856i	−0.667374	+	1.15593i	0.311262	+	0.950324i	\(0.399248\pi\)
							−0.978636	+	0.205602i	\(0.934085\pi\)
\(23\)	2.27143	+	3.93424i	0.473626	+	0.820345i	0.999544	−	0.0301904i	\(-0.00961138\pi\)
							−0.525918	+	0.850535i	\(0.676278\pi\)
\(29\)	3.46755	−	6.00597i	0.643908	−	1.11528i	−0.340645	−	0.940192i	\(-0.610645\pi\)
							0.984553	−	0.175089i	\(-0.0560213\pi\)
\(31\)	−2.15649			−0.387317			−0.193659	−	0.981069i	\(-0.562035\pi\)
							−0.193659	+	0.981069i	\(0.562035\pi\)
\(37\)	4.45227			0.731949			0.365975	−	0.930625i	\(-0.380736\pi\)
							0.365975	+	0.930625i	\(0.380736\pi\)
\(41\)	5.24919			0.819785			0.409893	−	0.912134i	\(-0.365566\pi\)
							0.409893	+	0.912134i	\(0.365566\pi\)
\(43\)	−7.50306	−	4.33190i	−1.14421	−	0.660608i	−0.196738	−	0.980456i	\(-0.563035\pi\)
							−0.947469	+	0.319848i	\(0.896368\pi\)
\(47\)			2.22995i			0.325272i	0.986686	+	0.162636i	\(0.0519996\pi\)
							−0.986686	+	0.162636i	\(0.948000\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.e.a.123.10	✓	58
349.227	even	6	inner	349.2.e.a.227.10	yes	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.e.a.123.10	✓	58	1.1	even	1	trivial
349.2.e.a.227.10	yes	58	349.227	even	6	inner