Embedded newform 7.12.c.a.4.6

• Label: 7.12.c.a.4.6
• Level: $7$
• Weight: $12$
• Character: 7.4
• Analytic conductor: $5.378$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	7.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.37840226392\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - x^11 + 1846*x^10 + 9475*x^9 + 2735534*x^8 + 11305015*x^7 + 1247863105*x^6 + 8299724444*x^5 + 456730004116*x^4 + 1967575859568*x^3 + 7281700309008*x^2 + 5939576563968*x + 4089842896896
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 2^{20}\cdot 3^{3}\cdot 7^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			4.6
Root			\(-17.5066 - 30.3223i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.4
Dual form			7.12.c.a.2.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(37.0132 + 64.1087i) q^{2} +(-82.0189 + 142.061i) q^{3} +(-1715.95 + 2972.11i) q^{4} +(1670.28 + 2893.01i) q^{5} -12143.1 q^{6} +(-26385.1 - 35793.2i) q^{7} -102445. q^{8} +(75119.3 + 130110. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 22 q^{2} - 244 q^{3} - 2556 q^{4} - 8782 q^{5} + 38140 q^{6} - 504 q^{7} + 97008 q^{8} - 172348 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	37.0132	+	64.1087i	0.817883	+	1.41661i	0.907239	+	0.420615i	\(0.138186\pi\)
							−0.0893562	+	0.996000i	\(0.528481\pi\)
\(3\)	−82.0189	+	142.061i	−0.194871	+	0.337526i	−0.946858	−	0.321651i	\(-0.895762\pi\)
							0.751987	+	0.659178i	\(0.229095\pi\)
\(5\)	1670.28	+	2893.01i	0.239031	+	0.414014i	0.960436	−	0.278499i	\(-0.0898369\pi\)
							−0.721406	+	0.692513i	\(0.756504\pi\)
\(7\)	−26385.1	−	35793.2i	−0.593362	−	0.804936i
							\(11\)	−178457.	+	309097.i	−0.334098	+	0.578675i	−0.983311	−	0.181931i	\(-0.941765\pi\)
0.649213	+	0.760607i	\(0.275098\pi\)
\(13\)	2.06258e6			1.54072			0.770359	−	0.637611i	\(-0.220077\pi\)
							0.770359	+	0.637611i	\(0.220077\pi\)
\(17\)	1.53495e6	−	2.65861e6i	0.262196	−	0.454136i	−0.704630	−	0.709575i	\(-0.748887\pi\)
							0.966825	+	0.255439i	\(0.0822201\pi\)
\(19\)	6.51852e6	+	1.12904e7i	0.603955	+	1.04608i	0.992216	+	0.124531i	\(0.0397425\pi\)
							−0.388261	+	0.921549i	\(0.626924\pi\)
\(23\)	−2.24585e7	−	3.88993e7i	−0.727576	−	1.26020i	−0.957905	−	0.287086i	\(-0.907313\pi\)
							0.230329	−	0.973113i	\(-0.426020\pi\)
\(29\)	−1.16423e8			−1.05402			−0.527012	−	0.849858i	\(-0.676688\pi\)
							−0.527012	+	0.849858i	\(0.676688\pi\)
\(31\)	1.09920e8	−	1.90386e8i	0.689581	−	1.19439i	−0.282392	−	0.959299i	\(-0.591128\pi\)
							0.971973	−	0.235091i	\(-0.0755387\pi\)
\(37\)	−3.99465e7	−	6.91894e7i	−0.0947042	−	0.164032i	0.814781	−	0.579769i	\(-0.196857\pi\)
							−0.909485	+	0.415737i	\(0.863524\pi\)
\(41\)	9.01434e8			1.21513			0.607565	−	0.794270i	\(-0.292146\pi\)
							0.607565	+	0.794270i	\(0.292146\pi\)
\(43\)	5.79780e8			0.601432			0.300716	−	0.953714i	\(-0.402774\pi\)
							0.300716	+	0.953714i	\(0.402774\pi\)
\(47\)	−5.34035e8	−	9.24976e8i	−0.339650	−	0.588291i	0.644717	−	0.764422i	\(-0.276975\pi\)
							−0.984367	+	0.176130i	\(0.943642\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.12.c.a.4.6	yes	12
3.2	odd	2		63.12.e.b.46.1		12
4.3	odd	2		112.12.i.c.81.4		12
7.2	even	3	inner	7.12.c.a.2.6	✓	12
7.3	odd	6		49.12.a.f.1.1		6
7.4	even	3		49.12.a.g.1.1		6
7.5	odd	6		49.12.c.i.30.6		12
7.6	odd	2		49.12.c.i.18.6		12
21.2	odd	6		63.12.e.b.37.1		12
28.23	odd	6		112.12.i.c.65.4		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.12.c.a.2.6	✓	12	7.2	even	3	inner
7.12.c.a.4.6	yes	12	1.1	even	1	trivial
49.12.a.f.1.1		6	7.3	odd	6
49.12.a.g.1.1		6	7.4	even	3
49.12.c.i.18.6		12	7.6	odd	2
49.12.c.i.30.6		12	7.5	odd	6
63.12.e.b.37.1		12	21.2	odd	6
63.12.e.b.46.1		12	3.2	odd	2
112.12.i.c.65.4		12	28.23	odd	6
112.12.i.c.81.4		12	4.3	odd	2