Embedded newform 12.16.b.a.11.5

• Label: 12.16.b.a.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	12.11
Dual form			12.16.b.a.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-166.441 - 71.1726i) q^{2} +(-8.61090 - 3787.99i) q^{3} +(22636.9 + 23692.0i) q^{4} -99785.5i q^{5} +(-268168. + 631087. i) q^{6} +1.78364e6i q^{7} +(-2.08148e6 - 5.55444e6i) q^{8} +(-1.43488e7 + 65235.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 26968 q^{4} + 823656 q^{6} - 3812052 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−166.441	−	71.1726i	−0.919463	−	0.393177i
							\(3\)	−8.61090	−	3787.99i	−0.00227321	−	0.999997i
\(5\)		−	99785.5i		−	0.571205i								−0.958348	−	0.285603i	\(-0.907806\pi\)
							0.958348	−	0.285603i	\(-0.0921938\pi\)
\(7\)			1.78364e6i			0.818601i	0.912400	+	0.409300i	\(0.134227\pi\)
							−0.912400	+	0.409300i	\(0.865773\pi\)
\(11\)	−1.16077e8			−1.79597			−0.897987	−	0.440022i	\(-0.854971\pi\)
							−0.897987	+	0.440022i	\(0.854971\pi\)
\(13\)	2.21330e8			0.978282			0.489141	−	0.872205i	\(-0.337310\pi\)
							0.489141	+	0.872205i	\(0.337310\pi\)
\(17\)			1.04886e9i			0.619941i	0.950746	+	0.309971i	\(0.100319\pi\)
							−0.950746	+	0.309971i	\(0.899681\pi\)
\(19\)			4.40175e9i			1.12973i	0.825185	+	0.564863i	\(0.191071\pi\)
							−0.825185	+	0.564863i	\(0.808929\pi\)
\(23\)	1.18058e10			0.722998			0.361499	−	0.932373i	\(-0.382265\pi\)
							0.361499	+	0.932373i	\(0.382265\pi\)
\(29\)		−	3.11714e10i		−	0.335561i	−0.985824	−	0.167780i	\(-0.946340\pi\)
							0.985824	−	0.167780i	\(-0.0536599\pi\)
\(31\)		−	8.47798e10i		−	0.553452i	−0.960949	−	0.276726i	\(-0.910751\pi\)
							0.960949	−	0.276726i	\(-0.0892494\pi\)
\(37\)	−5.63984e11			−0.976684			−0.488342	−	0.872652i	\(-0.662398\pi\)
							−0.488342	+	0.872652i	\(0.662398\pi\)
\(41\)			1.67904e12i			1.34642i	0.739450	+	0.673211i	\(0.235085\pi\)
							−0.739450	+	0.673211i	\(0.764915\pi\)
\(43\)			6.70438e11i			0.376136i	0.982156	+	0.188068i	\(0.0602226\pi\)
							−0.982156	+	0.188068i	\(0.939777\pi\)
\(47\)	4.09939e12			1.18028			0.590141	−	0.807300i	\(-0.299072\pi\)
							0.590141	+	0.807300i	\(0.299072\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.16.b.a.11.5	✓	28
3.2	odd	2	inner	12.16.b.a.11.24	yes	28
4.3	odd	2	inner	12.16.b.a.11.23	yes	28
12.11	even	2	inner	12.16.b.a.11.6	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.16.b.a.11.5	✓	28	1.1	even	1	trivial
12.16.b.a.11.6	yes	28	12.11	even	2	inner
12.16.b.a.11.23	yes	28	4.3	odd	2	inner
12.16.b.a.11.24	yes	28	3.2	odd	2	inner