Embedded newform 196.2.p.a.103.1

• Label: 196.2.p.a.103.1
• Level: $196$
• Weight: $2$
• Character: 196.103
• Analytic conductor: $1.565$
• Analytic rank: $0$
• Dimension: $312$
• 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			103.1
Character	\(\chi\)	\(=\)	196.103
Dual form			196.2.p.a.59.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41417 - 0.0104853i) q^{2} +(-1.15179 - 0.785275i) q^{3} +(1.99978 + 0.0296562i) q^{4} +(2.72463 + 0.204183i) q^{5} +(1.62059 + 1.12259i) q^{6} +(-1.69335 + 2.03287i) q^{7} +(-2.82773 - 0.0629074i) q^{8} +(-0.386067 - 0.983684i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 13 q^{2} - 13 q^{4} - 22 q^{5} - 14 q^{6} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41417	−	0.0104853i	−0.999973	−	0.00741425i
							\(3\)	−1.15179	−	0.785275i	−0.664984	−	0.453379i	0.183203	−	0.983075i	\(-0.441353\pi\)
−0.848187	+	0.529697i	\(0.822306\pi\)
\(5\)	2.72463	+	0.204183i	1.21849	+	0.0913134i	0.668408	−	0.743795i	\(-0.266976\pi\)
							0.550085	+	0.835109i	\(0.314595\pi\)
\(7\)	−1.69335	+	2.03287i	−0.640027	+	0.768353i
							\(11\)	4.00653	+	1.57245i	1.20801	+	0.474110i	0.881998	−	0.471254i	\(-0.156198\pi\)
0.326016	+	0.945364i	\(0.394294\pi\)
\(13\)	2.14265	−	1.70870i	0.594264	−	0.473909i	−0.279576	−	0.960124i	\(-0.590194\pi\)
							0.873840	+	0.486214i	\(0.161622\pi\)
\(17\)	2.62031	+	2.82402i	0.635518	+	0.684926i	0.965898	−	0.258922i	\(-0.0833674\pi\)
							−0.330380	+	0.943848i	\(0.607177\pi\)
\(19\)	3.33091	−	5.76930i	0.764162	−	1.32357i	−0.176526	−	0.984296i	\(-0.556486\pi\)
							0.940688	−	0.339272i	\(-0.110181\pi\)
\(23\)	3.71540	−	4.00425i	0.774714	−	0.834943i	−0.215206	−	0.976569i	\(-0.569042\pi\)
							0.989920	+	0.141625i	\(0.0452328\pi\)
\(29\)	0.688317	+	3.01572i	0.127817	+	0.560004i	0.997763	+	0.0668560i	\(0.0212968\pi\)
							−0.869945	+	0.493148i	\(0.835846\pi\)
\(31\)	−1.69507	−	2.93595i	−0.304444	−	0.527312i	0.672694	−	0.739921i	\(-0.265137\pi\)
							−0.977137	+	0.212609i	\(0.931804\pi\)
\(37\)	−7.82638	−	2.41412i	−1.28665	−	0.396879i	−0.425374	−	0.905018i	\(-0.639857\pi\)
							−0.861275	+	0.508139i	\(0.830334\pi\)
\(41\)	3.34988	+	6.95609i	0.523163	+	1.08636i	0.980401	+	0.197015i	\(0.0631247\pi\)
							−0.457237	+	0.889345i	\(0.651161\pi\)
\(43\)	−1.60718	+	3.33735i	−0.245093	+	0.508940i	−0.986832	−	0.161748i	\(-0.948287\pi\)
							0.741739	+	0.670688i	\(0.234001\pi\)
\(47\)	8.45195	−	1.27393i	1.23284	−	0.185821i	0.499883	−	0.866093i	\(-0.333376\pi\)
							0.732960	+	0.680271i	\(0.238138\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.2.p.a.103.1	yes	312
4.3	odd	2	inner	196.2.p.a.103.4	yes	312
49.10	odd	42	inner	196.2.p.a.59.4	yes	312
196.59	even	42	inner	196.2.p.a.59.1	✓	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.p.a.59.1	✓	312	196.59	even	42	inner
196.2.p.a.59.4	yes	312	49.10	odd	42	inner
196.2.p.a.103.1	yes	312	1.1	even	1	trivial
196.2.p.a.103.4	yes	312	4.3	odd	2	inner