Embedded newform 196.2.j.a.111.9

• Label: 196.2.j.a.111.9
• Level: $196$
• Weight: $2$
• Character: 196.111
• 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			111.9
Character	\(\chi\)	\(=\)	196.111
Dual form			196.2.j.a.83.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.741298 - 1.20436i) q^{2} +(0.0665173 + 0.0834100i) q^{3} +(-0.900953 + 1.78558i) q^{4} +(1.11698 - 0.890758i) q^{5} +(0.0511463 - 0.141942i) q^{6} +(1.34404 + 2.27894i) q^{7} +(2.81835 - 0.238575i) q^{8} +(0.665030 - 2.91369i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 5 q^{2} - 5 q^{4} - 14 q^{5} - 7 q^{6} - 11 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(99\)	\(101\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.741298	−	1.20436i	−0.524177	−	0.851609i
							\(3\)	0.0665173	+	0.0834100i	0.0384038	+	0.0481568i	0.800663	−	0.599116i	\(-0.204481\pi\)
−0.762259	+	0.647272i	\(0.775910\pi\)
\(5\)	1.11698	−	0.890758i	0.499527	−	0.398359i	−0.341056	−	0.940043i	\(-0.610784\pi\)
							0.840582	+	0.541684i	\(0.182213\pi\)
\(7\)	1.34404	+	2.27894i	0.507999	+	0.861358i
							\(11\)	1.57408	−	0.359273i	0.474603	−	0.108325i	0.0214728	−	0.999769i	\(-0.493164\pi\)
0.453130	+	0.891444i	\(0.350307\pi\)
\(13\)	0.776378	−	0.177203i	0.215329	−	0.0491474i	−0.113496	−	0.993539i	\(-0.536205\pi\)
							0.328824	+	0.944391i	\(0.393348\pi\)
\(17\)	1.17894	−	2.44810i	0.285936	−	0.593753i	−0.707684	−	0.706529i	\(-0.750260\pi\)
							0.993620	+	0.112776i	\(0.0359743\pi\)
\(19\)	0.719271			0.165012			0.0825061	−	0.996591i	\(-0.473708\pi\)
							0.0825061	+	0.996591i	\(0.473708\pi\)
\(23\)	−1.19976	−	2.49133i	−0.250167	−	0.519478i	0.737633	−	0.675202i	\(-0.235943\pi\)
							−0.987801	+	0.155724i	\(0.950229\pi\)
\(29\)	3.80797	+	1.83382i	0.707122	+	0.340532i	0.752641	−	0.658432i	\(-0.228780\pi\)
							−0.0455189	+	0.998963i	\(0.514494\pi\)
\(31\)	−7.03624			−1.26375			−0.631873	−	0.775072i	\(-0.717714\pi\)
							−0.631873	+	0.775072i	\(0.717714\pi\)
\(37\)	4.28735	+	2.06468i	0.704837	+	0.339432i	0.751732	−	0.659468i	\(-0.229219\pi\)
							−0.0468956	+	0.998900i	\(0.514933\pi\)
\(41\)	−0.875154	+	0.697912i	−0.136676	+	0.108996i	−0.689442	−	0.724341i	\(-0.742144\pi\)
							0.552765	+	0.833337i	\(0.313573\pi\)
\(43\)	−4.71379	−	3.75912i	−0.718847	−	0.573261i	0.194276	−	0.980947i	\(-0.437764\pi\)
							−0.913122	+	0.407686i	\(0.866336\pi\)
\(47\)	2.47552	+	10.8460i	0.361092	+	1.58205i	0.750430	+	0.660950i	\(0.229847\pi\)
							−0.389338	+	0.921095i	\(0.627296\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.j.a.83.9	✓	156	196.83	even	14	inner
196.2.j.a.83.15	yes	156	49.34	odd	14	inner
196.2.j.a.111.9	yes	156	1.1	even	1	trivial
196.2.j.a.111.15	yes	156	4.3	odd	2	inner