Embedded newform 196.3.g.j.67.4

• Label: 196.3.g.j.67.4
• 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			67.4
Root			\(1.40501 + 0.161055i\) of defining polynomial
Character	\(\chi\)	\(=\)	196.67
Dual form			196.3.g.j.79.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.489450 + 1.93919i) q^{2} +(3.93608 + 2.27250i) q^{3} +(-3.52088 + 1.89827i) q^{4} +(0.683969 + 1.18467i) q^{5} +(-2.48028 + 8.74507i) q^{6} +(-5.40439 - 5.89853i) q^{8} +(5.82850 + 10.0953i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} - q^{4} - 4 q^{5} - 12 q^{6} - 26 q^{8} + 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.489450	+	1.93919i	0.244725	+	0.969593i
							\(3\)	3.93608	+	2.27250i	1.31203	+	0.757499i	0.982432	−	0.186623i	\(-0.0597542\pi\)
0.329596	+	0.944122i	\(0.393088\pi\)
\(5\)	0.683969	+	1.18467i	0.136794	+	0.236934i	0.926281	−	0.376833i	\(-0.122987\pi\)
							−0.789487	+	0.613767i	\(0.789654\pi\)
\(7\)		0			0
							\(11\)	13.2565	+	7.65363i	1.20513	+	0.695785i	0.961692	−	0.274131i	\(-0.0883900\pi\)
0.243442	+	0.969915i	\(0.421723\pi\)
\(13\)	−19.9460			−1.53431			−0.767156	−	0.641461i	\(-0.778329\pi\)
							−0.767156	+	0.641461i	\(0.778329\pi\)
\(17\)	−2.36794	+	4.10139i	−0.139290	+	0.241258i	−0.927228	−	0.374497i	\(-0.877816\pi\)
							0.787938	+	0.615755i	\(0.211149\pi\)
\(19\)	11.4977	−	6.63823i	0.605145	−	0.349380i	−0.165918	−	0.986140i	\(-0.553059\pi\)
							0.771063	+	0.636759i	\(0.219725\pi\)
\(23\)	12.9460	−	7.47437i	0.562869	−	0.324973i	−0.191427	−	0.981507i	\(-0.561312\pi\)
							0.754296	+	0.656534i	\(0.227978\pi\)
\(29\)	6.20763			0.214056			0.107028	−	0.994256i	\(-0.465867\pi\)
							0.107028	+	0.994256i	\(0.465867\pi\)
\(31\)	−16.0548	−	9.26926i	−0.517898	−	0.299008i	0.218176	−	0.975909i	\(-0.429989\pi\)
							−0.736074	+	0.676901i	\(0.763323\pi\)
\(37\)	13.7608	+	23.8344i	0.371914	+	0.644173i	0.989860	−	0.142047i	\(-0.0453684\pi\)
							−0.617946	+	0.786220i	\(0.712035\pi\)
\(41\)	11.1064			0.270887			0.135443	−	0.990785i	\(-0.456754\pi\)
							0.135443	+	0.990785i	\(0.456754\pi\)
\(43\)		−	27.0248i		−	0.628483i	−0.949343	−	0.314241i	\(-0.898250\pi\)
							0.949343	−	0.314241i	\(-0.101750\pi\)
\(47\)	−26.8235	+	15.4865i	−0.570712	+	0.329501i	−0.757434	−	0.652912i	\(-0.773547\pi\)
							0.186722	+	0.982413i	\(0.440214\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.3.g.j.67.4		12
4.3	odd	2	inner	196.3.g.j.67.5		12
7.2	even	3	inner	196.3.g.j.79.5		12
7.3	odd	6		28.3.c.a.15.1	✓	6
7.4	even	3		196.3.c.g.99.1		6
7.5	odd	6		196.3.g.k.79.5		12
7.6	odd	2		196.3.g.k.67.4		12
21.17	even	6		252.3.g.a.127.6		6
28.3	even	6		28.3.c.a.15.2	yes	6
28.11	odd	6		196.3.c.g.99.2		6
28.19	even	6		196.3.g.k.79.4		12
28.23	odd	6	inner	196.3.g.j.79.4		12
28.27	even	2		196.3.g.k.67.5		12
56.3	even	6		448.3.d.d.127.6		6
56.45	odd	6		448.3.d.d.127.1		6
84.59	odd	6		252.3.g.a.127.5		6
112.3	even	12		1792.3.g.g.127.11		12
112.45	odd	12		1792.3.g.g.127.1		12
112.59	even	12		1792.3.g.g.127.2		12
112.101	odd	12		1792.3.g.g.127.12		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.3.c.a.15.1	✓	6	7.3	odd	6
28.3.c.a.15.2	yes	6	28.3	even	6
196.3.c.g.99.1		6	7.4	even	3
196.3.c.g.99.2		6	28.11	odd	6
196.3.g.j.67.4		12	1.1	even	1	trivial
196.3.g.j.67.5		12	4.3	odd	2	inner
196.3.g.j.79.4		12	28.23	odd	6	inner
196.3.g.j.79.5		12	7.2	even	3	inner
196.3.g.k.67.4		12	7.6	odd	2
196.3.g.k.67.5		12	28.27	even	2
196.3.g.k.79.4		12	28.19	even	6
196.3.g.k.79.5		12	7.5	odd	6
252.3.g.a.127.5		6	84.59	odd	6
252.3.g.a.127.6		6	21.17	even	6
448.3.d.d.127.1		6	56.45	odd	6
448.3.d.d.127.6		6	56.3	even	6
1792.3.g.g.127.1		12	112.45	odd	12
1792.3.g.g.127.2		12	112.59	even	12
1792.3.g.g.127.11		12	112.3	even	12
1792.3.g.g.127.12		12	112.101	odd	12