Embedded newform 196.2.p.a.103.12

• Label: 196.2.p.a.103.12
• 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			103.12
Character	\(\chi\)	\(=\)	196.103
Dual form			196.2.p.a.59.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.167781 - 1.40423i) q^{2} +(-1.97899 - 1.34925i) q^{3} +(-1.94370 + 0.471205i) q^{4} +(-0.0159493 - 0.00119524i) q^{5} +(-1.56262 + 3.00533i) q^{6} +(-2.59622 + 0.509531i) q^{7} +(0.987794 + 2.65033i) q^{8} +(0.999898 + 2.54770i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 13 q^{2} - 13 q^{4} - 22 q^{5} - 14 q^{6} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.167781	−	1.40423i	−0.118639	−	0.992937i
							\(3\)	−1.97899	−	1.34925i	−1.14257	−	0.778992i	−0.164318	−	0.986407i	\(-0.552542\pi\)
−0.978253	+	0.207416i	\(0.933495\pi\)
\(5\)	−0.0159493	−	0.00119524i	−0.00713275	−	0.000534526i	0.0711632	−	0.997465i	\(-0.477329\pi\)
							−0.0782960	+	0.996930i	\(0.524948\pi\)
\(7\)	−2.59622	+	0.509531i	−0.981280	+	0.192585i
							\(11\)	1.65591	+	0.649897i	0.499276	+	0.195951i	0.601597	−	0.798800i	\(-0.294532\pi\)
−0.102321	+	0.994751i	\(0.532627\pi\)
\(13\)	−2.07204	+	1.65240i	−0.574681	+	0.458293i	−0.867195	−	0.497968i	\(-0.834080\pi\)
							0.292514	+	0.956261i	\(0.405508\pi\)
\(17\)	−3.29762	−	3.55399i	−0.799790	−	0.861968i	0.193079	−	0.981183i	\(-0.438153\pi\)
							−0.992868	+	0.119215i	\(0.961962\pi\)
\(19\)	−0.682883	+	1.18279i	−0.156664	+	0.271350i	−0.933664	−	0.358151i	\(-0.883407\pi\)
							0.777000	+	0.629501i	\(0.216741\pi\)
\(23\)	−3.75388	+	4.04572i	−0.782738	+	0.843591i	−0.990914	−	0.134500i	\(-0.957057\pi\)
							0.208175	+	0.978092i	\(0.433248\pi\)
\(29\)	0.487147	+	2.13433i	0.0904609	+	0.396335i	0.999806	−	0.0197080i	\(-0.00627365\pi\)
							−0.909345	+	0.416043i	\(0.863417\pi\)
\(31\)	−4.96153	−	8.59362i	−0.891116	−	1.54346i	−0.838539	−	0.544842i	\(-0.816590\pi\)
							−0.0525777	−	0.998617i	\(-0.516744\pi\)
\(37\)	1.78032	+	0.549155i	0.292682	+	0.0902805i	0.437619	−	0.899160i	\(-0.355822\pi\)
							−0.144937	+	0.989441i	\(0.546298\pi\)
\(41\)	−1.78963	−	3.71620i	−0.279493	−	0.580373i	0.713211	−	0.700949i	\(-0.247240\pi\)
							−0.992704	+	0.120576i	\(0.961526\pi\)
\(43\)	4.01716	−	8.34173i	0.612612	−	1.27210i	−0.331804	−	0.943348i	\(-0.607657\pi\)
							0.944416	−	0.328753i	\(-0.106628\pi\)
\(47\)	−9.62933	+	1.45139i	−1.40458	+	0.211707i	−0.807213	−	0.590260i	\(-0.799025\pi\)
							−0.597369	+	0.801967i	\(0.703787\pi\)