Embedded newform 12.13.d.a.7.2

• Label: 12.13.d.a.7.2
• Level: $12$
• Weight: $13$
• Character: 12.7
• Analytic conductor: $10.968$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	12.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9679258073\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - 3*x^11 + 1570*x^10 - 4077*x^9 + 1884069*x^8 - 3551868*x^7 + 881574992*x^6 + 2167220880*x^5 + 304613421264*x^4 + 27792635520*x^3 + 5806366144000*x^2 + 1120796928000*x + 104882177440000
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{57}\cdot 3^{25} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			7.2
Root			\(14.6572 - 25.3870i\) of defining polynomial
Character	\(\chi\)	\(=\)	12.7
Dual form			12.13.d.a.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-62.3881 + 14.2733i) q^{2} +420.888i q^{3} +(3688.55 - 1780.96i) q^{4} -21622.0 q^{5} +(-6007.45 - 26258.4i) q^{6} +155240. i q^{7} +(-204701. + 163759. i) q^{8} -177147. q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 90 q^{2} - 4692 q^{4} - 10296 q^{5} - 48114 q^{6} - 648000 q^{8} - 2125764 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(7\)
\(\chi(n)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−62.3881	+	14.2733i	−0.974814	+	0.223020i
							\(3\)			420.888i			0.577350i
\(5\)	−21622.0			−1.38381										−0.691904	−	0.721989i	\(-0.743228\pi\)
							−0.691904	+	0.721989i	\(0.743228\pi\)
\(7\)			155240.i			1.31952i	0.751477	+	0.659759i	\(0.229342\pi\)
							−0.751477	+	0.659759i	\(0.770658\pi\)
\(11\)		−	3.12388e6i		−	1.76335i	−0.471860	−	0.881673i	\(-0.656417\pi\)
							0.471860	−	0.881673i	\(-0.343583\pi\)
\(13\)	1.10272e6			0.228458			0.114229	−	0.993454i	\(-0.463560\pi\)
							0.114229	+	0.993454i	\(0.463560\pi\)
\(17\)	3.10754e7			1.28743			0.643714	−	0.765267i	\(-0.277393\pi\)
							0.643714	+	0.765267i	\(0.277393\pi\)
\(19\)		−	2.82999e7i		−	0.601537i	−0.953697	−	0.300769i	\(-0.902757\pi\)
							0.953697	−	0.300769i	\(-0.0972432\pi\)
\(23\)		−	1.30643e8i		−	0.882508i	−0.897382	−	0.441254i	\(-0.854534\pi\)
							0.897382	−	0.441254i	\(-0.145466\pi\)
\(29\)	−7.45018e8			−1.25250			−0.626252	−	0.779621i	\(-0.715412\pi\)
							−0.626252	+	0.779621i	\(0.715412\pi\)
\(31\)		−	3.42331e8i		−	0.385724i	−0.981226	−	0.192862i	\(-0.938223\pi\)
							0.981226	−	0.192862i	\(-0.0617769\pi\)
\(37\)	1.27314e9			0.496211			0.248105	−	0.968733i	\(-0.420192\pi\)
							0.248105	+	0.968733i	\(0.420192\pi\)
\(41\)	−8.57107e8			−0.180440			−0.0902199	−	0.995922i	\(-0.528757\pi\)
							−0.0902199	+	0.995922i	\(0.528757\pi\)
\(43\)		−	6.63854e9i		−	1.05018i	−0.851048	−	0.525088i	\(-0.824033\pi\)
							0.851048	−	0.525088i	\(-0.175967\pi\)
\(47\)			7.75614e9i			0.719546i	0.933040	+	0.359773i	\(0.117146\pi\)
							−0.933040	+	0.359773i	\(0.882854\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.13.d.a.7.2	yes	12
3.2	odd	2		36.13.d.d.19.11		12
4.3	odd	2	inner	12.13.d.a.7.1	✓	12
8.3	odd	2		192.13.g.e.127.11		12
8.5	even	2		192.13.g.e.127.5		12
12.11	even	2		36.13.d.d.19.12		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.13.d.a.7.1	✓	12	4.3	odd	2	inner
12.13.d.a.7.2	yes	12	1.1	even	1	trivial
36.13.d.d.19.11		12	3.2	odd	2
36.13.d.d.19.12		12	12.11	even	2
192.13.g.e.127.5		12	8.5	even	2
192.13.g.e.127.11		12	8.3	odd	2