Embedded newform 12.13.d.a.7.8

• Label: 12.13.d.a.7.8
• 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			7.8
Root			\(-16.4016 - 28.4085i\) of defining polynomial
Character	\(\chi\)	\(=\)	12.7
Dual form			12.13.d.a.7.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(18.1780 + 61.3642i) q^{2} -420.888i q^{3} +(-3435.12 + 2230.95i) q^{4} -204.488 q^{5} +(25827.5 - 7650.89i) q^{6} -121613. i q^{7} +(-199344. - 170239. i) q^{8} -177147. q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 90 q^{2} - 4692 q^{4} - 10296 q^{5} - 48114 q^{6} - 648000 q^{8} - 2125764 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	18.1780	+	61.3642i	0.284031	+	0.958815i
							\(3\)		−	420.888i		−	0.577350i
\(5\)	−204.488			−0.0130872										−0.00654361	−	0.999979i	\(-0.502083\pi\)
							−0.00654361	+	0.999979i	\(0.502083\pi\)
\(7\)		−	121613.i		−	1.03369i	−0.856079	−	0.516845i	\(-0.827106\pi\)
							0.856079	−	0.516845i	\(-0.172894\pi\)
\(11\)		−	2.28583e6i		−	1.29029i	−0.764059	−	0.645146i	\(-0.776797\pi\)
							0.764059	−	0.645146i	\(-0.223203\pi\)
\(13\)	6.03292e6			1.24988			0.624939	−	0.780674i	\(-0.285124\pi\)
							0.624939	+	0.780674i	\(0.285124\pi\)
\(17\)	−1.11018e7			−0.459938			−0.229969	−	0.973198i	\(-0.573862\pi\)
							−0.229969	+	0.973198i	\(0.573862\pi\)
\(19\)		−	5.58358e7i		−	1.18684i	−0.804894	−	0.593419i	\(-0.797778\pi\)
							0.804894	−	0.593419i	\(-0.202222\pi\)
\(23\)			7.22448e7i			0.488022i	0.969772	+	0.244011i	\(0.0784633\pi\)
							−0.969772	+	0.244011i	\(0.921537\pi\)
\(29\)	−7.70727e8			−1.29572			−0.647862	−	0.761757i	\(-0.724337\pi\)
							−0.647862	+	0.761757i	\(0.724337\pi\)
\(31\)		−	3.21570e8i		−	0.362330i	−0.983453	−	0.181165i	\(-0.942013\pi\)
							0.983453	−	0.181165i	\(-0.0579869\pi\)
\(37\)	−3.35054e9			−1.30589			−0.652943	−	0.757407i	\(-0.726466\pi\)
							−0.652943	+	0.757407i	\(0.726466\pi\)
\(41\)	6.76783e9			1.42478			0.712388	−	0.701786i	\(-0.247614\pi\)
							0.712388	+	0.701786i	\(0.247614\pi\)
\(43\)		−	3.68199e9i		−	0.582467i	−0.956652	−	0.291234i	\(-0.905934\pi\)
							0.956652	−	0.291234i	\(-0.0940657\pi\)
\(47\)			7.64797e9i			0.709511i	0.934959	+	0.354755i	\(0.115436\pi\)
							−0.934959	+	0.354755i	\(0.884564\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.13.d.a.7.8	yes	12
3.2	odd	2		36.13.d.d.19.5		12
4.3	odd	2	inner	12.13.d.a.7.7	✓	12
8.3	odd	2		192.13.g.e.127.4		12
8.5	even	2		192.13.g.e.127.10		12
12.11	even	2		36.13.d.d.19.6		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.13.d.a.7.7	✓	12	4.3	odd	2	inner
12.13.d.a.7.8	yes	12	1.1	even	1	trivial
36.13.d.d.19.5		12	3.2	odd	2
36.13.d.d.19.6		12	12.11	even	2
192.13.g.e.127.4		12	8.3	odd	2
192.13.g.e.127.10		12	8.5	even	2