Embedded newform 196.2.j.a.83.15

• Label: 196.2.j.a.83.15
• Level: $196$
• Weight: $2$
• Character: 196.83
• Analytic conductor: $1.565$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	196.j (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.56506787962\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(26\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			83.15
Character	\(\chi\)	\(=\)	196.83
Dual form			196.2.j.a.111.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.145336 + 1.40673i) q^{2} +(-0.0665173 + 0.0834100i) q^{3} +(-1.95776 + 0.408895i) q^{4} +(1.11698 + 0.890758i) q^{5} +(-0.127002 - 0.0814491i) q^{6} +(-1.34404 + 2.27894i) q^{7} +(-0.859735 - 2.69460i) q^{8} +(0.665030 + 2.91369i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 5 q^{2} - 5 q^{4} - 14 q^{5} - 7 q^{6} - 11 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(99\)	\(101\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.145336	+	1.40673i	0.102768	+	0.994705i
							\(3\)	−0.0665173	+	0.0834100i	−0.0384038	+	0.0481568i	−0.800663	−	0.599116i	\(-0.795519\pi\)
0.762259	+	0.647272i	\(0.224090\pi\)
\(5\)	1.11698	+	0.890758i	0.499527	+	0.398359i	0.840582	−	0.541684i	\(-0.182213\pi\)
							−0.341056	+	0.940043i	\(0.610784\pi\)
\(7\)	−1.34404	+	2.27894i	−0.507999	+	0.861358i
							\(11\)	−1.57408	−	0.359273i	−0.474603	−	0.108325i	−0.0214728	−	0.999769i	\(-0.506836\pi\)
−0.453130	+	0.891444i	\(0.649693\pi\)
\(13\)	0.776378	+	0.177203i	0.215329	+	0.0491474i	0.328824	−	0.944391i	\(-0.393348\pi\)
							−0.113496	+	0.993539i	\(0.536205\pi\)
\(17\)	1.17894	+	2.44810i	0.285936	+	0.593753i	0.993620	−	0.112776i	\(-0.0359743\pi\)
							−0.707684	+	0.706529i	\(0.750260\pi\)
\(19\)	−0.719271			−0.165012			−0.0825061	−	0.996591i	\(-0.526292\pi\)
							−0.0825061	+	0.996591i	\(0.526292\pi\)
\(23\)	1.19976	−	2.49133i	0.250167	−	0.519478i	−0.737633	−	0.675202i	\(-0.764057\pi\)
							0.987801	+	0.155724i	\(0.0497709\pi\)
\(29\)	3.80797	−	1.83382i	0.707122	−	0.340532i	−0.0455189	−	0.998963i	\(-0.514494\pi\)
							0.752641	+	0.658432i	\(0.228780\pi\)
\(31\)	7.03624			1.26375			0.631873	−	0.775072i	\(-0.282286\pi\)
							0.631873	+	0.775072i	\(0.282286\pi\)
\(37\)	4.28735	−	2.06468i	0.704837	−	0.339432i	−0.0468956	−	0.998900i	\(-0.514933\pi\)
							0.751732	+	0.659468i	\(0.229219\pi\)
\(41\)	−0.875154	−	0.697912i	−0.136676	−	0.108996i	0.552765	−	0.833337i	\(-0.313573\pi\)
							−0.689442	+	0.724341i	\(0.742144\pi\)
\(43\)	4.71379	−	3.75912i	0.718847	−	0.573261i	−0.194276	−	0.980947i	\(-0.562236\pi\)
							0.913122	+	0.407686i	\(0.133664\pi\)
\(47\)	−2.47552	+	10.8460i	−0.361092	+	1.58205i	0.389338	+	0.921095i	\(0.372704\pi\)
							−0.750430	+	0.660950i	\(0.770153\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.2.j.a.83.15	yes	156
4.3	odd	2	inner	196.2.j.a.83.9	✓	156
49.13	odd	14	inner	196.2.j.a.111.9	yes	156
196.111	even	14	inner	196.2.j.a.111.15	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.j.a.83.9	✓	156	4.3	odd	2	inner
196.2.j.a.83.15	yes	156	1.1	even	1	trivial
196.2.j.a.111.9	yes	156	49.13	odd	14	inner
196.2.j.a.111.15	yes	156	196.111	even	14	inner