Embedded newform 196.3.g.j.67.2

• Label: 196.3.g.j.67.2
• Level: $196$
• Weight: $3$
• Character: 196.67
• Analytic conductor: $5.341$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	196.g (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.34061318146\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	12.0.1728283481971641.1
Defining polynomial:	x^12 - 2*x^11 + 4*x^10 - 6*x^9 + 6*x^8 - 8*x^7 + 9*x^6 - 16*x^5 + 24*x^4 - 48*x^3 + 64*x^2 - 64*x + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{12} \)
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			67.2
Root			\(0.341867 + 1.37227i\) of defining polynomial
Character	\(\chi\)	\(=\)	196.67
Dual form			196.3.g.j.79.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.64441 + 1.13838i) q^{2} +(-1.35124 - 0.780139i) q^{3} +(1.40817 - 3.74394i) q^{4} +(1.71871 + 2.97689i) q^{5} +(3.11009 - 0.255361i) q^{6} +(1.94643 + 7.75960i) q^{8} +(-3.28276 - 5.68592i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} - q^{4} - 4 q^{5} - 12 q^{6} - 26 q^{8} + 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(99\)	\(101\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.64441	+	1.13838i	−0.822205	+	0.569192i
							\(3\)	−1.35124	−	0.780139i	−0.450414	−	0.260046i	0.257591	−	0.966254i	\(-0.417071\pi\)
−0.708005	+	0.706208i	\(0.750405\pi\)
\(5\)	1.71871	+	2.97689i	0.343742	+	0.595379i	0.985124	−	0.171843i	\(-0.0549721\pi\)
							−0.641382	+	0.767221i	\(0.721639\pi\)
\(7\)		0			0
							\(11\)	−7.34727	−	4.24195i	−0.667933	−	0.385631i	0.127360	−	0.991857i	\(-0.459350\pi\)
−0.795293	+	0.606225i	\(0.792683\pi\)
\(13\)	18.5685			1.42834			0.714172	−	0.699970i	\(-0.246803\pi\)
							0.714172	+	0.699970i	\(0.246803\pi\)
\(17\)	−4.43742	+	7.68584i	−0.261025	+	0.452108i	−0.966514	−	0.256612i	\(-0.917394\pi\)
							0.705490	+	0.708720i	\(0.250727\pi\)
\(19\)	26.2686	−	15.1662i	1.38256	−	0.798221i	0.390097	−	0.920774i	\(-0.372441\pi\)
							0.992462	+	0.122553i	\(0.0391080\pi\)
\(23\)	22.9751	−	13.2647i	0.998917	−	0.576725i	0.0909892	−	0.995852i	\(-0.470997\pi\)
							0.907928	+	0.419127i	\(0.137664\pi\)
\(29\)	18.6245			0.642225			0.321112	−	0.947041i	\(-0.395943\pi\)
							0.321112	+	0.947041i	\(0.395943\pi\)
\(31\)	35.7273	+	20.6272i	1.15249	+	0.665393i	0.949494	−	0.313786i	\(-0.101597\pi\)
							0.203000	+	0.979179i	\(0.434931\pi\)
\(37\)	1.74673	+	3.02542i	0.0472089	+	0.0817682i	0.888664	−	0.458558i	\(-0.151634\pi\)
							−0.841455	+	0.540327i	\(0.818301\pi\)
\(41\)	−37.7556			−0.920868			−0.460434	−	0.887694i	\(-0.652306\pi\)
							−0.460434	+	0.887694i	\(0.652306\pi\)
\(43\)		−	50.8159i		−	1.18177i	−0.806757	−	0.590883i	\(-0.798780\pi\)
							0.806757	−	0.590883i	\(-0.201220\pi\)
\(47\)	45.0169	−	25.9905i	0.957806	−	0.552990i	0.0623089	−	0.998057i	\(-0.480154\pi\)
							0.895497	+	0.445067i	\(0.146820\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.3.g.j.67.2		12
4.3	odd	2	inner	196.3.g.j.67.6		12
7.2	even	3	inner	196.3.g.j.79.6		12
7.3	odd	6		28.3.c.a.15.3	✓	6
7.4	even	3		196.3.c.g.99.3		6
7.5	odd	6		196.3.g.k.79.6		12
7.6	odd	2		196.3.g.k.67.2		12
21.17	even	6		252.3.g.a.127.4		6
28.3	even	6		28.3.c.a.15.4	yes	6
28.11	odd	6		196.3.c.g.99.4		6
28.19	even	6		196.3.g.k.79.2		12
28.23	odd	6	inner	196.3.g.j.79.2		12
28.27	even	2		196.3.g.k.67.6		12
56.3	even	6		448.3.d.d.127.3		6
56.45	odd	6		448.3.d.d.127.4		6
84.59	odd	6		252.3.g.a.127.3		6
112.3	even	12		1792.3.g.g.127.5		12
112.45	odd	12		1792.3.g.g.127.7		12
112.59	even	12		1792.3.g.g.127.8		12
112.101	odd	12		1792.3.g.g.127.6		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.3.c.a.15.3	✓	6	7.3	odd	6
28.3.c.a.15.4	yes	6	28.3	even	6
196.3.c.g.99.3		6	7.4	even	3
196.3.c.g.99.4		6	28.11	odd	6
196.3.g.j.67.2		12	1.1	even	1	trivial
196.3.g.j.67.6		12	4.3	odd	2	inner
196.3.g.j.79.2		12	28.23	odd	6	inner
196.3.g.j.79.6		12	7.2	even	3	inner
196.3.g.k.67.2		12	7.6	odd	2
196.3.g.k.67.6		12	28.27	even	2
196.3.g.k.79.2		12	28.19	even	6
196.3.g.k.79.6		12	7.5	odd	6
252.3.g.a.127.3		6	84.59	odd	6
252.3.g.a.127.4		6	21.17	even	6
448.3.d.d.127.3		6	56.3	even	6
448.3.d.d.127.4		6	56.45	odd	6
1792.3.g.g.127.5		12	112.3	even	12
1792.3.g.g.127.6		12	112.101	odd	12
1792.3.g.g.127.7		12	112.45	odd	12
1792.3.g.g.127.8		12	112.59	even	12