Embedded newform 7.12.c.a.4.4

• Label: 7.12.c.a.4.4
• 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			4.4
Root			\(-1.81957 - 3.15159i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.4
Dual form			7.12.c.a.2.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.63915 + 9.76729i) q^{2} +(-23.0866 + 39.9871i) q^{3} +(960.400 - 1663.46i) q^{4} +(4304.93 + 7456.37i) q^{5} -520.754 q^{6} +(44340.4 + 3354.68i) q^{7} +44761.3 q^{8} +(87507.5 + 151567. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 22 q^{2} - 244 q^{3} - 2556 q^{4} - 8782 q^{5} + 38140 q^{6} - 504 q^{7} + 97008 q^{8} - 172348 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.63915	+	9.76729i	0.124609	+	0.215829i	0.921580	−	0.388189i	\(-0.126899\pi\)
							−0.796971	+	0.604017i	\(0.793566\pi\)
\(3\)	−23.0866	+	39.9871i	−0.0548520	+	0.0950064i	−0.892148	−	0.451744i	\(-0.850802\pi\)
							0.837296	+	0.546750i	\(0.184135\pi\)
\(5\)	4304.93	+	7456.37i	0.616072	+	1.06707i	0.990195	+	0.139689i	\(0.0446103\pi\)
							−0.374123	+	0.927379i	\(0.622056\pi\)
\(7\)	44340.4	+	3354.68i	0.997150	+	0.0754417i
							\(11\)	316770.	−	548662.i	0.593041	−	1.02718i	−0.400779	−	0.916175i	\(-0.631261\pi\)
0.993820	−	0.111002i	\(-0.0354061\pi\)
\(13\)	−2.31023e6			−1.72570			−0.862852	−	0.505456i	\(-0.831324\pi\)
							−0.862852	+	0.505456i	\(0.831324\pi\)
\(17\)	541013.	−	937063.i	0.0924142	−	0.160066i	−0.816112	−	0.577893i	\(-0.803875\pi\)
							0.908526	+	0.417827i	\(0.137208\pi\)
\(19\)	3.46533e6	+	6.00213e6i	0.321070	+	0.556110i	0.980709	−	0.195473i	\(-0.0626242\pi\)
							−0.659639	+	0.751583i	\(0.729291\pi\)
\(23\)	33074.7	+	57287.1i	0.00107150	+	0.00185590i	0.866561	−	0.499072i	\(-0.166326\pi\)
							−0.865489	+	0.500928i	\(0.832992\pi\)
\(29\)	−1.55800e8			−1.41051			−0.705257	−	0.708952i	\(-0.749168\pi\)
							−0.705257	+	0.708952i	\(0.749168\pi\)
\(31\)	−4.72954e7	+	8.19180e7i	−0.296708	+	0.513913i	−0.975381	−	0.220528i	\(-0.929222\pi\)
							0.678673	+	0.734441i	\(0.262555\pi\)
\(37\)	−1.78046e8	−	3.08384e8i	−0.422106	−	0.731109i	0.574039	−	0.818828i	\(-0.305376\pi\)
							−0.996145	+	0.0877186i	\(0.972042\pi\)
\(41\)	7.58734e8			1.02277			0.511385	−	0.859352i	\(-0.329132\pi\)
							0.511385	+	0.859352i	\(0.329132\pi\)
\(43\)	−1.10138e9			−1.14251			−0.571255	−	0.820772i	\(-0.693543\pi\)
							−0.571255	+	0.820772i	\(0.693543\pi\)
\(47\)	−1.12802e9	−	1.95379e9i	−0.717430	−	1.24263i	−0.962015	−	0.272998i	\(-0.911985\pi\)
							0.244585	−	0.969628i	\(-0.421348\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.12.c.a.4.4	yes	12
3.2	odd	2		63.12.e.b.46.3		12
4.3	odd	2		112.12.i.c.81.3		12
7.2	even	3	inner	7.12.c.a.2.4	✓	12
7.3	odd	6		49.12.a.f.1.3		6
7.4	even	3		49.12.a.g.1.3		6
7.5	odd	6		49.12.c.i.30.4		12
7.6	odd	2		49.12.c.i.18.4		12
21.2	odd	6		63.12.e.b.37.3		12
28.23	odd	6		112.12.i.c.65.3		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.12.c.a.2.4	✓	12	7.2	even	3	inner
7.12.c.a.4.4	yes	12	1.1	even	1	trivial
49.12.a.f.1.3		6	7.3	odd	6
49.12.a.g.1.3		6	7.4	even	3
49.12.c.i.18.4		12	7.6	odd	2
49.12.c.i.30.4		12	7.5	odd	6
63.12.e.b.37.3		12	21.2	odd	6
63.12.e.b.46.3		12	3.2	odd	2
112.12.i.c.65.3		12	28.23	odd	6
112.12.i.c.81.3		12	4.3	odd	2