Embedded newform 196.2.p.a.103.5

• Label: 196.2.p.a.103.5
• Level: $196$
• Weight: $2$
• Character: 196.103
• Analytic conductor: $1.565$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	196.p (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.56506787962\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(26\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			103.5
Character	\(\chi\)	\(=\)	196.103
Dual form			196.2.p.a.59.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20490 - 0.740415i) q^{2} +(1.89744 + 1.29365i) q^{3} +(0.903573 + 1.78425i) q^{4} +(0.459106 + 0.0344052i) q^{5} +(-1.32839 - 2.96362i) q^{6} +(-0.101996 - 2.64378i) q^{7} +(0.232371 - 2.81887i) q^{8} +(0.830721 + 2.11664i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 13 q^{2} - 13 q^{4} - 22 q^{5} - 14 q^{6} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\)	\(99\)	\(101\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{29}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20490	−	0.740415i	−0.851994	−	0.523552i
							\(3\)	1.89744	+	1.29365i	1.09549	+	0.746891i	0.969488	−	0.245139i	\(-0.0788336\pi\)
0.126000	+	0.992030i	\(0.459786\pi\)
\(5\)	0.459106	+	0.0344052i	0.205318	+	0.0153865i	0.176992	−	0.984212i	\(-0.443363\pi\)
							0.0283261	+	0.999599i	\(0.490982\pi\)
\(7\)	−0.101996	−	2.64378i	−0.0385507	−	0.999257i
							\(11\)	5.59412	+	2.19553i	1.68669	+	0.661978i	0.998008	−	0.0630850i	\(-0.0200939\pi\)
0.688683	+	0.725063i	\(0.258189\pi\)
\(13\)	−4.33422	+	3.45643i	−1.20210	+	0.958641i	−0.999786	−	0.0206804i	\(-0.993417\pi\)
							−0.202311	+	0.979321i	\(0.564845\pi\)
\(17\)	2.11293	+	2.27720i	0.512461	+	0.552302i	0.935278	−	0.353913i	\(-0.115149\pi\)
							−0.422817	+	0.906215i	\(0.638959\pi\)
\(19\)	3.06297	−	5.30522i	0.702693	−	1.21710i	−0.264824	−	0.964297i	\(-0.585314\pi\)
							0.967518	−	0.252804i	\(-0.0813527\pi\)
\(23\)	−3.35196	+	3.61255i	−0.698932	+	0.753269i	−0.978363	−	0.206897i	\(-0.933664\pi\)
							0.279431	+	0.960166i	\(0.409854\pi\)
\(29\)	0.395863	+	1.73439i	0.0735100	+	0.322068i	0.998291	−	0.0584303i	\(-0.0186095\pi\)
							−0.924782	+	0.380499i	\(0.875752\pi\)
\(31\)	−1.92532	−	3.33475i	−0.345798	−	0.598939i	0.639701	−	0.768624i	\(-0.279058\pi\)
							−0.985498	+	0.169685i	\(0.945725\pi\)
\(37\)	−2.78661	−	0.859554i	−0.458115	−	0.141310i	0.0571041	−	0.998368i	\(-0.481813\pi\)
							−0.515219	+	0.857058i	\(0.672290\pi\)
\(41\)	−4.16086	−	8.64012i	−0.649818	−	1.34936i	−0.922028	−	0.387123i	\(-0.873469\pi\)
							0.272210	−	0.962238i	\(-0.412245\pi\)
\(43\)	−0.134323	+	0.278925i	−0.0204841	+	0.0425357i	−0.910960	−	0.412495i	\(-0.864657\pi\)
							0.890476	+	0.455031i	\(0.150372\pi\)
\(47\)	−3.46750	+	0.522642i	−0.505787	+	0.0762351i	−0.396979	−	0.917828i	\(-0.629941\pi\)
							−0.108808	+	0.994063i	\(0.534703\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.2.p.a.103.5	yes	312
4.3	odd	2	inner	196.2.p.a.103.8	yes	312
49.10	odd	42	inner	196.2.p.a.59.8	yes	312
196.59	even	42	inner	196.2.p.a.59.5	✓	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.p.a.59.5	✓	312	196.59	even	42	inner
196.2.p.a.59.8	yes	312	49.10	odd	42	inner
196.2.p.a.103.5	yes	312	1.1	even	1	trivial
196.2.p.a.103.8	yes	312	4.3	odd	2	inner