Embedded newform 196.2.p.a.59.17

• Label: 196.2.p.a.59.17
• Level: $196$
• Weight: $2$
• Character: 196.59
• 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			59.17
Character	\(\chi\)	\(=\)	196.59
Dual form			196.2.p.a.103.17

$q$-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 \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\)	\(99\)	\(101\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{42}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.p.a.59.14	✓	312	4.3	odd	2	inner
196.2.p.a.59.17	yes	312	1.1	even	1	trivial
196.2.p.a.103.14	yes	312	49.5	odd	42	inner
196.2.p.a.103.17	yes	312	196.103	even	42	inner