Embedded newform 196.3.g.j.67.1

• Label: 196.3.g.j.67.1
• 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			67.1
Root			\(0.0455406 - 1.41348i\) of defining polynomial
Character	\(\chi\)	\(=\)	196.67
Dual form			196.3.g.j.79.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.84709 - 0.766985i) q^{2} +(2.58484 + 1.49236i) q^{3} +(2.82347 + 2.83338i) q^{4} +(-3.40268 - 5.89361i) q^{5} +(-3.62981 - 4.73905i) q^{6} +(-3.04204 - 7.39905i) q^{8} +(-0.0457311 - 0.0792086i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} - q^{4} - 4 q^{5} - 12 q^{6} - 26 q^{8} + 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.84709	−	0.766985i	−0.923544	−	0.383492i
							\(3\)	2.58484	+	1.49236i	0.861614	+	0.497453i	0.864552	−	0.502543i	\(-0.167602\pi\)
−0.00293865	+	0.999996i	\(0.500935\pi\)
\(5\)	−3.40268	−	5.89361i	−0.680536	−	1.17872i	−0.974818	−	0.223004i	\(-0.928414\pi\)
							0.294282	−	0.955719i	\(-0.404920\pi\)
\(7\)		0			0
							\(11\)	−12.4211	−	7.17132i	−1.12919	−	0.651938i	−0.185459	−	0.982652i	\(-0.559377\pi\)
−0.943731	+	0.330714i	\(0.892710\pi\)
\(13\)	−4.62243			−0.355572			−0.177786	−	0.984069i	\(-0.556893\pi\)
							−0.177786	+	0.984069i	\(0.556893\pi\)
\(17\)	5.80536	−	10.0552i	0.341492	−	0.591481i	−0.643218	−	0.765683i	\(-0.722401\pi\)
							0.984710	+	0.174202i	\(0.0557346\pi\)
\(19\)	19.4361	−	11.2214i	1.02295	−	0.590601i	0.107994	−	0.994152i	\(-0.465557\pi\)
							0.914957	+	0.403550i	\(0.132224\pi\)
\(23\)	−0.739535	+	0.426971i	−0.0321537	+	0.0185639i	−0.515991	−	0.856594i	\(-0.672576\pi\)
							0.483837	+	0.875158i	\(0.339243\pi\)
\(29\)	−42.8321			−1.47697			−0.738485	−	0.674270i	\(-0.764459\pi\)
							−0.738485	+	0.674270i	\(0.764459\pi\)
\(31\)	1.34219	+	0.774912i	0.0432963	+	0.0249972i	0.521492	−	0.853256i	\(-0.325376\pi\)
							−0.478196	+	0.878253i	\(0.658709\pi\)
\(37\)	−22.5075	−	38.9842i	−0.608312	−	1.05363i	−0.991519	−	0.129964i	\(-0.958514\pi\)
							0.383207	−	0.923663i	\(-0.374820\pi\)
\(41\)	36.6492			0.893883			0.446942	−	0.894563i	\(-0.352513\pi\)
							0.446942	+	0.894563i	\(0.352513\pi\)
\(43\)			27.9894i			0.650916i	0.945557	+	0.325458i	\(0.105518\pi\)
							−0.945557	+	0.325458i	\(0.894482\pi\)
\(47\)	36.5237	−	21.0870i	0.777100	−	0.448659i	−0.0583012	−	0.998299i	\(-0.518568\pi\)
							0.835402	+	0.549640i	\(0.185235\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.3.g.j.67.1		12
4.3	odd	2	inner	196.3.g.j.67.3		12
7.2	even	3	inner	196.3.g.j.79.3		12
7.3	odd	6		28.3.c.a.15.5	✓	6
7.4	even	3		196.3.c.g.99.5		6
7.5	odd	6		196.3.g.k.79.3		12
7.6	odd	2		196.3.g.k.67.1		12
21.17	even	6		252.3.g.a.127.2		6
28.3	even	6		28.3.c.a.15.6	yes	6
28.11	odd	6		196.3.c.g.99.6		6
28.19	even	6		196.3.g.k.79.1		12
28.23	odd	6	inner	196.3.g.j.79.1		12
28.27	even	2		196.3.g.k.67.3		12
56.3	even	6		448.3.d.d.127.5		6
56.45	odd	6		448.3.d.d.127.2		6
84.59	odd	6		252.3.g.a.127.1		6
112.3	even	12		1792.3.g.g.127.10		12
112.45	odd	12		1792.3.g.g.127.4		12
112.59	even	12		1792.3.g.g.127.3		12
112.101	odd	12		1792.3.g.g.127.9		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.3.c.a.15.5	✓	6	7.3	odd	6
28.3.c.a.15.6	yes	6	28.3	even	6
196.3.c.g.99.5		6	7.4	even	3
196.3.c.g.99.6		6	28.11	odd	6
196.3.g.j.67.1		12	1.1	even	1	trivial
196.3.g.j.67.3		12	4.3	odd	2	inner
196.3.g.j.79.1		12	28.23	odd	6	inner
196.3.g.j.79.3		12	7.2	even	3	inner
196.3.g.k.67.1		12	7.6	odd	2
196.3.g.k.67.3		12	28.27	even	2
196.3.g.k.79.1		12	28.19	even	6
196.3.g.k.79.3		12	7.5	odd	6
252.3.g.a.127.1		6	84.59	odd	6
252.3.g.a.127.2		6	21.17	even	6
448.3.d.d.127.2		6	56.45	odd	6
448.3.d.d.127.5		6	56.3	even	6
1792.3.g.g.127.3		12	112.59	even	12
1792.3.g.g.127.4		12	112.45	odd	12
1792.3.g.g.127.9		12	112.101	odd	12
1792.3.g.g.127.10		12	112.3	even	12