Embedded newform 12.16.b.a.11.28

• Label: 12.16.b.a.11.28
• 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			11.28
Character	\(\chi\)	\(=\)	12.11
Dual form			12.16.b.a.11.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(172.746 + 54.1000i) q^{2} +(-3105.31 + 2169.32i) q^{3} +(26914.4 + 18691.1i) q^{4} +192307. i q^{5} +(-653790. + 206744. i) q^{6} -366926. i q^{7} +(3.63816e6 + 4.68489e6i) q^{8} +(4.93700e6 - 1.34728e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 26968 q^{4} + 823656 q^{6} - 3812052 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	172.746	+	54.1000i	0.954296	+	0.298863i
							\(3\)	−3105.31	+	2169.32i	−0.819777	+	0.572683i
\(5\)			192307.i			1.10083i								0.834891	+	0.550415i	\(0.185530\pi\)
							−0.834891	+	0.550415i	\(0.814470\pi\)
\(7\)		−	366926.i		−	0.168401i	−0.996449	−	0.0842003i	\(-0.973166\pi\)
							0.996449	−	0.0842003i	\(-0.0268335\pi\)
\(11\)	−5.82623e7			−0.901452			−0.450726	−	0.892662i	\(-0.648835\pi\)
							−0.450726	+	0.892662i	\(0.648835\pi\)
\(13\)	−2.65726e8			−1.17452			−0.587259	−	0.809399i	\(-0.699793\pi\)
							−0.587259	+	0.809399i	\(0.699793\pi\)
\(17\)			1.79306e9i			1.05981i	0.848058	+	0.529904i	\(0.177772\pi\)
							−0.848058	+	0.529904i	\(0.822228\pi\)
\(19\)		−	4.75795e9i		−	1.22115i	−0.791960	−	0.610573i	\(-0.790939\pi\)
							0.791960	−	0.610573i	\(-0.209061\pi\)
\(23\)	−1.92879e10			−1.18121			−0.590604	−	0.806961i	\(-0.701111\pi\)
							−0.590604	+	0.806961i	\(0.701111\pi\)
\(29\)			1.16647e11i			1.25571i	0.778332	+	0.627853i	\(0.216066\pi\)
							−0.778332	+	0.627853i	\(0.783934\pi\)
\(31\)		−	1.69225e11i		−	1.10472i	−0.833606	−	0.552360i	\(-0.813727\pi\)
							0.833606	−	0.552360i	\(-0.186273\pi\)
\(37\)	7.59020e11			1.31444			0.657219	−	0.753699i	\(-0.271733\pi\)
							0.657219	+	0.753699i	\(0.271733\pi\)
\(41\)			1.15439e12i			0.925705i	0.886435	+	0.462852i	\(0.153174\pi\)
							−0.886435	+	0.462852i	\(0.846826\pi\)
\(43\)			1.91686e12i			1.07542i	0.843130	+	0.537709i	\(0.180710\pi\)
							−0.843130	+	0.537709i	\(0.819290\pi\)
\(47\)	1.35211e12			0.389293			0.194647	−	0.980873i	\(-0.437644\pi\)
							0.194647	+	0.980873i	\(0.437644\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.16.b.a.11.28	yes	28
3.2	odd	2	inner	12.16.b.a.11.1	✓	28
4.3	odd	2	inner	12.16.b.a.11.2	yes	28
12.11	even	2	inner	12.16.b.a.11.27	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.16.b.a.11.1	✓	28	3.2	odd	2	inner
12.16.b.a.11.2	yes	28	4.3	odd	2	inner
12.16.b.a.11.27	yes	28	12.11	even	2	inner
12.16.b.a.11.28	yes	28	1.1	even	1	trivial