Embedded newform 12.16.b.a.11.20

• Label: 12.16.b.a.11.20
• Level: $12$
• Weight: $16$
• Character: 12.11
• Analytic conductor: $17.123$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	12.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.1232206120\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			11.20
Character	\(\chi\)	\(=\)	12.11
Dual form			12.16.b.a.11.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(83.3106 + 160.709i) q^{2} +(-3555.47 - 1306.74i) q^{3} +(-18886.7 + 26777.5i) q^{4} -93360.9i q^{5} +(-86203.8 - 680260. i) q^{6} -1.14542e6i q^{7} +(-5.87684e6 - 804409. i) q^{8} +(1.09338e7 + 9.29212e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 26968 q^{4} + 823656 q^{6} - 3812052 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	83.3106	+	160.709i	0.460230	+	0.887800i
							\(3\)	−3555.47	−	1306.74i	−0.938614	−	0.344968i
\(5\)		−	93360.9i		−	0.534429i								−0.963637	−	0.267214i	\(-0.913897\pi\)
							0.963637	−	0.267214i	\(-0.0861032\pi\)
\(7\)		−	1.14542e6i		−	0.525688i	−0.964838	−	0.262844i	\(-0.915339\pi\)
							0.964838	−	0.262844i	\(-0.0846605\pi\)
\(11\)	1.80841e6			0.0279803			0.0139901	−	0.999902i	\(-0.495547\pi\)
							0.0139901	+	0.999902i	\(0.495547\pi\)
\(13\)	3.17574e8			1.40369			0.701843	−	0.712331i	\(-0.252361\pi\)
							0.701843	+	0.712331i	\(0.252361\pi\)
\(17\)			2.67956e9i			1.58378i	0.610661	+	0.791892i	\(0.290904\pi\)
							−0.610661	+	0.791892i	\(0.709096\pi\)
\(19\)		−	2.07292e9i		−	0.532024i	−0.963970	−	0.266012i	\(-0.914294\pi\)
							0.963970	−	0.266012i	\(-0.0857060\pi\)
\(23\)	2.43757e10			1.49279			0.746393	−	0.665505i	\(-0.231784\pi\)
							0.746393	+	0.665505i	\(0.231784\pi\)
\(29\)		−	4.77985e9i		−	0.0514552i	−0.999669	−	0.0257276i	\(-0.991810\pi\)
							0.999669	−	0.0257276i	\(-0.00819025\pi\)
\(31\)		−	5.30495e10i		−	0.346313i	−0.984894	−	0.173156i	\(-0.944603\pi\)
							0.984894	−	0.173156i	\(-0.0553966\pi\)
\(37\)	4.06413e11			0.703808			0.351904	−	0.936036i	\(-0.385534\pi\)
							0.351904	+	0.936036i	\(0.385534\pi\)
\(41\)		−	1.36874e12i		−	1.09760i	−0.835954	−	0.548799i	\(-0.815085\pi\)
							0.835954	−	0.548799i	\(-0.184915\pi\)
\(43\)		−	2.85013e12i		−	1.59901i	−0.600661	−	0.799504i	\(-0.705096\pi\)
							0.600661	−	0.799504i	\(-0.294904\pi\)
\(47\)	−7.27503e11			−0.209460			−0.104730	−	0.994501i	\(-0.533398\pi\)
							−0.104730	+	0.994501i	\(0.533398\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.16.b.a.11.20	yes	28
3.2	odd	2	inner	12.16.b.a.11.9	✓	28
4.3	odd	2	inner	12.16.b.a.11.10	yes	28
12.11	even	2	inner	12.16.b.a.11.19	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.16.b.a.11.9	✓	28	3.2	odd	2	inner
12.16.b.a.11.10	yes	28	4.3	odd	2	inner
12.16.b.a.11.19	yes	28	12.11	even	2	inner
12.16.b.a.11.20	yes	28	1.1	even	1	trivial