Embedded newform 196.2.p.a.103.17

• Label: 196.2.p.a.103.17
• 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			103.17
Character	\(\chi\)	\(=\)	196.103
Dual form			196.2.p.a.59.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.368520 + 1.36535i) q^{2} +(-0.268248 - 0.182888i) q^{3} +(-1.72839 + 1.00632i) q^{4} +(-3.87544 - 0.290424i) q^{5} +(0.150853 - 0.433651i) q^{6} +(-1.56212 + 2.13536i) q^{7} +(-2.01093 - 1.98901i) q^{8} +(-1.05751 - 2.69450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 13 q^{2} - 13 q^{4} - 22 q^{5} - 14 q^{6} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.368520	+	1.36535i	0.260583	+	0.965451i
							\(3\)	−0.268248	−	0.182888i	−0.154873	−	0.105591i	0.483404	−	0.875397i	\(-0.339400\pi\)
−0.638277	+	0.769807i	\(0.720353\pi\)
\(5\)	−3.87544	−	0.290424i	−1.73315	−	0.129882i	−0.829598	−	0.558361i	\(-0.811431\pi\)
							−0.903552	+	0.428479i	\(0.859050\pi\)
\(7\)	−1.56212	+	2.13536i	−0.590426	+	0.807091i
							\(11\)	3.83683	+	1.50584i	1.15685	+	0.454029i	0.864673	−	0.502336i	\(-0.167526\pi\)
0.292175	+	0.956365i	\(0.405621\pi\)
\(13\)	−3.48169	+	2.77655i	−0.965646	+	0.770077i	−0.973216	−	0.229894i	\(-0.926162\pi\)
							0.00756928	+	0.999971i	\(0.497591\pi\)
\(17\)	1.72962	+	1.86409i	0.419495	+	0.452108i	0.907074	−	0.420971i	\(-0.138311\pi\)
							−0.487579	+	0.873079i	\(0.662120\pi\)
\(19\)	−2.67127	+	4.62677i	−0.612831	+	1.06145i	0.377930	+	0.925834i	\(0.376636\pi\)
							−0.990761	+	0.135620i	\(0.956697\pi\)
\(23\)	−1.95374	+	2.10563i	−0.407383	+	0.439054i	−0.903100	−	0.429430i	\(-0.858715\pi\)
							0.495718	+	0.868484i	\(0.334905\pi\)
\(29\)	−0.966524	−	4.23462i	−0.179479	−	0.786349i	−0.981871	−	0.189551i	\(-0.939297\pi\)
							0.802392	−	0.596798i	\(-0.203561\pi\)
\(31\)	−1.62402	−	2.81289i	−0.291683	−	0.505210i	0.682525	−	0.730863i	\(-0.260882\pi\)
							−0.974208	+	0.225652i	\(0.927549\pi\)
\(37\)	−2.52615	−	0.779213i	−0.415296	−	0.128102i	0.0800638	−	0.996790i	\(-0.474488\pi\)
							−0.495360	+	0.868688i	\(0.664964\pi\)
\(41\)	0.309311	+	0.642291i	0.0483063	+	0.100309i	0.923725	−	0.383056i	\(-0.125129\pi\)
							−0.875419	+	0.483365i	\(0.839414\pi\)
\(43\)	−3.62349	+	7.52426i	−0.552578	+	1.14744i	0.418399	+	0.908263i	\(0.362591\pi\)
							−0.970976	+	0.239176i	\(0.923123\pi\)
\(47\)	5.43584	−	0.819322i	0.792899	−	0.119510i	0.259911	−	0.965633i	\(-0.416307\pi\)
							0.532989	+	0.846122i	\(0.321069\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.2.p.a.103.17	yes	312
4.3	odd	2	inner	196.2.p.a.103.14	yes	312
49.10	odd	42	inner	196.2.p.a.59.14	✓	312
196.59	even	42	inner	196.2.p.a.59.17	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.p.a.59.14	✓	312	49.10	odd	42	inner
196.2.p.a.59.17	yes	312	196.59	even	42	inner
196.2.p.a.103.14	yes	312	4.3	odd	2	inner
196.2.p.a.103.17	yes	312	1.1	even	1	trivial