Embedded newform 12.8.b.b.11.1

• Label: 12.8.b.b.11.1
• Level: $12$
• Weight: $8$
• Character: 12.11
• Analytic conductor: $3.749$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	12.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.74862030581\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial:	\( x^{8} + 41x^{6} + 1008x^{4} + 41984x^{2} + 1048576 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{20}\cdot 3^{5} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			11.1
Root			\(4.51977 - 3.40172i\) of defining polynomial
Character	\(\chi\)	\(=\)	12.11
Dual form			12.8.b.b.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.03954 - 6.80344i) q^{2} +(46.2722 + 6.77368i) q^{3} +(35.4264 + 123.000i) q^{4} -426.666i q^{5} +(-372.195 - 376.041i) q^{6} -780.788i q^{7} +(516.584 - 1352.88i) q^{8} +(2095.23 + 626.867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 328 q^{4} - 1512 q^{6} + 2088 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.03954	−	6.80344i	−0.798990	−	0.601345i
							\(3\)	46.2722	+	6.77368i	0.989455	+	0.144844i
\(5\)		−	426.666i		−	1.52649i								−0.646112	−	0.763243i	\(-0.723606\pi\)
							0.646112	−	0.763243i	\(-0.276394\pi\)
\(7\)		−	780.788i		−	0.860380i	−0.902738	−	0.430190i	\(-0.858447\pi\)
							0.902738	−	0.430190i	\(-0.141553\pi\)
\(11\)	−1923.23			−0.435670			−0.217835	−	0.975986i	\(-0.569899\pi\)
							−0.217835	+	0.975986i	\(0.569899\pi\)
\(13\)	8028.82			1.01356			0.506781	−	0.862075i	\(-0.330835\pi\)
							0.506781	+	0.862075i	\(0.330835\pi\)
\(17\)			8167.93i			0.403219i	0.979466	+	0.201609i	\(0.0646171\pi\)
							−0.979466	+	0.201609i	\(0.935383\pi\)
\(19\)			30292.2i			1.01319i	0.862183	+	0.506597i	\(0.169097\pi\)
							−0.862183	+	0.506597i	\(0.830903\pi\)
\(23\)	−25899.2			−0.443852			−0.221926	−	0.975064i	\(-0.571234\pi\)
							−0.221926	+	0.975064i	\(0.571234\pi\)
\(29\)			109562.i			0.834193i	0.908862	+	0.417096i	\(0.136952\pi\)
							−0.908862	+	0.417096i	\(0.863048\pi\)
\(31\)			36004.5i			0.217065i	0.994093	+	0.108533i	\(0.0346152\pi\)
							−0.994093	+	0.108533i	\(0.965385\pi\)
\(37\)	318880.			1.03496			0.517478	−	0.855697i	\(-0.326871\pi\)
							0.517478	+	0.855697i	\(0.326871\pi\)
\(41\)			64088.3i			0.145223i	0.997360	+	0.0726115i	\(0.0231333\pi\)
							−0.997360	+	0.0726115i	\(0.976867\pi\)
\(43\)		−	794243.i		−	1.52340i	−0.647930	−	0.761700i	\(-0.724365\pi\)
							0.647930	−	0.761700i	\(-0.275635\pi\)
\(47\)	−60806.6			−0.0854296			−0.0427148	−	0.999087i	\(-0.513601\pi\)
							−0.0427148	+	0.999087i	\(0.513601\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.8.b.b.11.1	✓	8
3.2	odd	2	inner	12.8.b.b.11.8	yes	8
4.3	odd	2	inner	12.8.b.b.11.7	yes	8
8.3	odd	2		192.8.c.e.191.8		8
8.5	even	2		192.8.c.e.191.1		8
12.11	even	2	inner	12.8.b.b.11.2	yes	8
24.5	odd	2		192.8.c.e.191.7		8
24.11	even	2		192.8.c.e.191.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.8.b.b.11.1	✓	8	1.1	even	1	trivial
12.8.b.b.11.2	yes	8	12.11	even	2	inner
12.8.b.b.11.7	yes	8	4.3	odd	2	inner
12.8.b.b.11.8	yes	8	3.2	odd	2	inner
192.8.c.e.191.1		8	8.5	even	2
192.8.c.e.191.2		8	24.11	even	2
192.8.c.e.191.7		8	24.5	odd	2
192.8.c.e.191.8		8	8.3	odd	2