Embedded newform 12.8.b.a.11.3

• Label: 12.8.b.a.11.3
• Level: $12$
• Weight: $8$
• Character: 12.11
• Analytic conductor: $3.749$
• Analytic rank: $0$
• Dimension: $4$
• 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:	\(4\)
Coefficient field:	\(\Q(\sqrt{3}, \sqrt{-5})\)
Defining polynomial:	\( x^{4} + x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{5}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			11.3
Root			\(0.866025 - 1.11803i\) of defining polynomial
Character	\(\chi\)	\(=\)	12.11
Dual form			12.8.b.a.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.3923 - 4.47214i) q^{2} +(31.1769 + 34.8569i) q^{3} +(88.0000 - 92.9516i) q^{4} +205.718i q^{5} +(479.885 + 222.816i) q^{6} -999.230i q^{7} +(498.831 - 1359.53i) q^{8} +(-243.000 + 2173.46i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 352 q^{4} + 1296 q^{6} - 972 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.3923	−	4.47214i	0.918559	−	0.395285i
							\(3\)	31.1769	+	34.8569i	0.666667	+	0.745356i
\(5\)			205.718i			0.736000i								0.929826	+	0.368000i	\(0.119957\pi\)
							−0.929826	+	0.368000i	\(0.880043\pi\)
\(7\)		−	999.230i		−	1.10109i	−0.834806	−	0.550544i	\(-0.814420\pi\)
							0.834806	−	0.550544i	\(-0.185580\pi\)
\(11\)	−1974.54			−0.447292			−0.223646	−	0.974670i	\(-0.571796\pi\)
							−0.223646	+	0.974670i	\(0.571796\pi\)
\(13\)	−12730.0			−1.60704			−0.803520	−	0.595278i	\(-0.797042\pi\)
							−0.803520	+	0.595278i	\(0.797042\pi\)
\(17\)			18854.5i			0.930774i	0.885107	+	0.465387i	\(0.154085\pi\)
							−0.885107	+	0.465387i	\(0.845915\pi\)
\(19\)		−	23911.8i		−	0.799788i	−0.916561	−	0.399894i	\(-0.869047\pi\)
							0.916561	−	0.399894i	\(-0.130953\pi\)
\(23\)	72372.0			1.24029			0.620145	−	0.784487i	\(-0.287074\pi\)
							0.620145	+	0.784487i	\(0.287074\pi\)
\(29\)		−	106106.i		−	0.807879i	−0.914786	−	0.403940i	\(-0.867641\pi\)
							0.914786	−	0.403940i	\(-0.132359\pi\)
\(31\)			90372.2i			0.544840i	0.962178	+	0.272420i	\(0.0878240\pi\)
							−0.962178	+	0.272420i	\(0.912176\pi\)
\(37\)	43310.0			0.140567			0.0702833	−	0.997527i	\(-0.477610\pi\)
							0.0702833	+	0.997527i	\(0.477610\pi\)
\(41\)			785790.i			1.78059i	0.455388	+	0.890293i	\(0.349501\pi\)
							−0.455388	+	0.890293i	\(0.650499\pi\)
\(43\)			70805.9i			0.135809i	0.997692	+	0.0679047i	\(0.0216314\pi\)
							−0.997692	+	0.0679047i	\(0.978369\pi\)
\(47\)	563762.			0.792051			0.396025	−	0.918240i	\(-0.370389\pi\)
							0.396025	+	0.918240i	\(0.370389\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.8.b.a.11.3	yes	4
3.2	odd	2	inner	12.8.b.a.11.2	yes	4
4.3	odd	2	inner	12.8.b.a.11.1	✓	4
8.3	odd	2		192.8.c.b.191.4		4
8.5	even	2		192.8.c.b.191.1		4
12.11	even	2	inner	12.8.b.a.11.4	yes	4
24.5	odd	2		192.8.c.b.191.3		4
24.11	even	2		192.8.c.b.191.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.8.b.a.11.1	✓	4	4.3	odd	2	inner
12.8.b.a.11.2	yes	4	3.2	odd	2	inner
12.8.b.a.11.3	yes	4	1.1	even	1	trivial
12.8.b.a.11.4	yes	4	12.11	even	2	inner
192.8.c.b.191.1		4	8.5	even	2
192.8.c.b.191.2		4	24.11	even	2
192.8.c.b.191.3		4	24.5	odd	2
192.8.c.b.191.4		4	8.3	odd	2