Embedded newform 12.11.d.a.7.8

• Label: 12.11.d.a.7.8
• Level: $12$
• Weight: $11$
• Character: 12.7
• Analytic conductor: $7.624$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	12.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.62428703208\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	x^10 - 2*x^9 + 198*x^8 - 1066*x^7 + 30608*x^6 - 147714*x^5 + 2092829*x^4 - 9587938*x^3 + 101463432*x^2 - 344403128*x + 1648684816
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{39}\cdot 3^{18} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			7.8
Root			\(-6.13017 + 10.6178i\) of defining polynomial
Character	\(\chi\)	\(=\)	12.7
Dual form			12.11.d.a.7.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(20.5505 + 24.5291i) q^{2} -140.296i q^{3} +(-179.355 + 1008.17i) q^{4} -4874.35 q^{5} +(3441.34 - 2883.15i) q^{6} +22628.8i q^{7} +(-28415.4 + 16319.0i) q^{8} -19683.0 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 22 q^{2} - 644 q^{4} + 3116 q^{5} + 2430 q^{6} + 20176 q^{8} - 196830 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	20.5505	+	24.5291i	0.642203	+	0.766535i
							\(3\)		−	140.296i		−	0.577350i
\(5\)	−4874.35			−1.55979										−0.779896	−	0.625909i	\(-0.784728\pi\)
							−0.779896	+	0.625909i	\(0.784728\pi\)
\(7\)			22628.8i			1.34639i	0.739463	+	0.673197i	\(0.235079\pi\)
							−0.739463	+	0.673197i	\(0.764921\pi\)
\(11\)		−	77310.1i		−	0.480035i	−0.970769	−	0.240017i	\(-0.922847\pi\)
							0.970769	−	0.240017i	\(-0.0771532\pi\)
\(13\)	−319786.			−0.861276			−0.430638	−	0.902525i	\(-0.641711\pi\)
							−0.430638	+	0.902525i	\(0.641711\pi\)
\(17\)	2.42820e6			1.71017			0.855085	−	0.518487i	\(-0.173505\pi\)
							0.855085	+	0.518487i	\(0.173505\pi\)
\(19\)			1.56392e6i			0.631605i	0.948825	+	0.315802i	\(0.102274\pi\)
							−0.948825	+	0.315802i	\(0.897726\pi\)
\(23\)			2.60831e6i			0.405247i	0.979257	+	0.202623i	\(0.0649467\pi\)
							−0.979257	+	0.202623i	\(0.935053\pi\)
\(29\)	−9.86234e6			−0.480828			−0.240414	−	0.970670i	\(-0.577283\pi\)
							−0.240414	+	0.970670i	\(0.577283\pi\)
\(31\)			4.14393e7i			1.44745i	0.690087	+	0.723726i	\(0.257572\pi\)
							−0.690087	+	0.723726i	\(0.742428\pi\)
\(37\)	−7.55735e7			−1.08983			−0.544917	−	0.838490i	\(-0.683439\pi\)
							−0.544917	+	0.838490i	\(0.683439\pi\)
\(41\)	9.41061e6			0.0812266			0.0406133	−	0.999175i	\(-0.487069\pi\)
							0.0406133	+	0.999175i	\(0.487069\pi\)
\(43\)		−	8.08340e6i		−	0.0549860i	−0.999622	−	0.0274930i	\(-0.991248\pi\)
							0.999622	−	0.0274930i	\(-0.00875239\pi\)
\(47\)		−	1.70311e8i		−	0.742598i	−0.928513	−	0.371299i	\(-0.878913\pi\)
							0.928513	−	0.371299i	\(-0.121087\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.11.d.a.7.8	yes	10
3.2	odd	2		36.11.d.e.19.3		10
4.3	odd	2	inner	12.11.d.a.7.7	✓	10
8.3	odd	2		192.11.g.e.127.5		10
8.5	even	2		192.11.g.e.127.10		10
12.11	even	2		36.11.d.e.19.4		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.11.d.a.7.7	✓	10	4.3	odd	2	inner
12.11.d.a.7.8	yes	10	1.1	even	1	trivial
36.11.d.e.19.3		10	3.2	odd	2
36.11.d.e.19.4		10	12.11	even	2
192.11.g.e.127.5		10	8.3	odd	2
192.11.g.e.127.10		10	8.5	even	2