Embedded newform 12.14.b.a.11.2

• Label: 12.14.b.a.11.2
• Level: $12$
• Weight: $14$
• Character: 12.11
• Analytic conductor: $12.868$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	12.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8677114742\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			11.2
Character	\(\chi\)	\(=\)	12.11
Dual form			12.14.b.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-89.8019 + 11.2966i) q^{2} +(-433.314 - 1185.99i) q^{3} +(7936.77 - 2028.92i) q^{4} +24994.7i q^{5} +(52310.1 + 101609. i) q^{6} +71320.8i q^{7} +(-689818. + 271859. i) q^{8} +(-1.21880e6 + 1.02781e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 8376 q^{4} - 135720 q^{6} - 327240 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−89.8019	+	11.2966i	−0.992181	+	0.124811i
							\(3\)	−433.314	−	1185.99i	−0.343174	−	0.939272i
\(5\)			24994.7i			0.715390i								0.933839	+	0.357695i	\(0.116437\pi\)
							−0.933839	+	0.357695i	\(0.883563\pi\)
\(7\)			71320.8i			0.229128i	0.993416	+	0.114564i	\(0.0365472\pi\)
							−0.993416	+	0.114564i	\(0.963453\pi\)
\(11\)	4.95247e6			0.842888			0.421444	−	0.906855i	\(-0.361523\pi\)
							0.421444	+	0.906855i	\(0.361523\pi\)
\(13\)	1.35351e7			0.777732			0.388866	−	0.921294i	\(-0.372867\pi\)
							0.388866	+	0.921294i	\(0.372867\pi\)
\(17\)		−	5.28848e7i		−	0.531390i	−0.964057	−	0.265695i	\(-0.914399\pi\)
							0.964057	−	0.265695i	\(-0.0856014\pi\)
\(19\)		−	3.46946e8i		−	1.69186i	−0.533298	−	0.845928i	\(-0.679047\pi\)
							0.533298	−	0.845928i	\(-0.320953\pi\)
\(23\)	−4.73538e8			−0.666996			−0.333498	−	0.942751i	\(-0.608229\pi\)
							−0.333498	+	0.942751i	\(0.608229\pi\)
\(29\)		−	6.10944e9i		−	1.90728i	−0.300952	−	0.953639i	\(-0.597304\pi\)
							0.300952	−	0.953639i	\(-0.402696\pi\)
\(31\)			6.26028e9i			1.26690i	0.773784	+	0.633450i	\(0.218362\pi\)
							−0.773784	+	0.633450i	\(0.781638\pi\)
\(37\)	1.95426e10			1.25219			0.626097	−	0.779745i	\(-0.284651\pi\)
							0.626097	+	0.779745i	\(0.284651\pi\)
\(41\)		−	3.71309e10i		−	1.22079i	−0.792098	−	0.610394i	\(-0.791011\pi\)
							0.792098	−	0.610394i	\(-0.208989\pi\)
\(43\)		−	2.19497e10i		−	0.529521i	−0.964314	−	0.264761i	\(-0.914707\pi\)
							0.964314	−	0.264761i	\(-0.0852930\pi\)
\(47\)	9.42331e10			1.27517			0.637584	−	0.770381i	\(-0.279934\pi\)
							0.637584	+	0.770381i	\(0.279934\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.14.b.a.11.2	yes	24
3.2	odd	2	inner	12.14.b.a.11.23	yes	24
4.3	odd	2	inner	12.14.b.a.11.24	yes	24
12.11	even	2	inner	12.14.b.a.11.1	✓	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.14.b.a.11.1	✓	24	12.11	even	2	inner
12.14.b.a.11.2	yes	24	1.1	even	1	trivial
12.14.b.a.11.23	yes	24	3.2	odd	2	inner
12.14.b.a.11.24	yes	24	4.3	odd	2	inner