Embedded newform 12.20.b.a.11.10

• Label: 12.20.b.a.11.10
• Level: $12$
• Weight: $20$
• Character: 12.11
• Analytic conductor: $27.458$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	12.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(27.4580035868\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			11.10
Character	\(\chi\)	\(=\)	12.11
Dual form			12.20.b.a.11.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-519.517 + 504.371i) q^{2} +(-19784.5 + 27763.9i) q^{3} +(15507.7 - 524059. i) q^{4} +1.34492e6i q^{5} +(-3.72492e6 - 2.44026e7i) q^{6} +1.72563e8i q^{7} +(2.56263e8 + 2.80079e8i) q^{8} +(-3.79408e8 - 1.09859e9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 47880 q^{4} + 21771144 q^{6} - 809473644 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−519.517	+	504.371i	−0.717488	+	0.696571i
							\(3\)	−19784.5	+	27763.9i	−0.580328	+	0.814383i
\(5\)			1.34492e6i			0.307950i								0.988075	+	0.153975i	\(0.0492076\pi\)
							−0.988075	+	0.153975i	\(0.950792\pi\)
\(7\)			1.72563e8i			1.61627i	0.588994	+	0.808137i	\(0.299524\pi\)
							−0.588994	+	0.808137i	\(0.700476\pi\)
\(11\)	−1.32949e10			−1.70002			−0.850009	−	0.526768i	\(-0.823404\pi\)
							−0.850009	+	0.526768i	\(0.823404\pi\)
\(13\)	−2.84987e10			−0.745355			−0.372678	−	0.927961i	\(-0.621560\pi\)
							−0.372678	+	0.927961i	\(0.621560\pi\)
\(17\)		−	1.96504e11i		−	0.401889i	−0.979603	−	0.200945i	\(-0.935599\pi\)
							0.979603	−	0.200945i	\(-0.0644011\pi\)
\(19\)			2.26261e12i			1.60861i	0.594219	+	0.804304i	\(0.297461\pi\)
							−0.594219	+	0.804304i	\(0.702539\pi\)
\(23\)	−9.34021e12			−1.08129			−0.540645	−	0.841251i	\(-0.681820\pi\)
							−0.540645	+	0.841251i	\(0.681820\pi\)
\(29\)			1.66276e13i			0.212838i	0.994321	+	0.106419i	\(0.0339384\pi\)
							−0.994321	+	0.106419i	\(0.966062\pi\)
\(31\)		−	4.90603e13i		−	0.333268i	−0.986019	−	0.166634i	\(-0.946710\pi\)
							0.986019	−	0.166634i	\(-0.0532899\pi\)
\(37\)	9.66977e14			1.22321			0.611603	−	0.791165i	\(-0.290525\pi\)
							0.611603	+	0.791165i	\(0.290525\pi\)
\(41\)		−	1.16915e15i		−	0.557730i	−0.960330	−	0.278865i	\(-0.910042\pi\)
							0.960330	−	0.278865i	\(-0.0899582\pi\)
\(43\)			3.53338e14i			0.107211i	0.998562	+	0.0536055i	\(0.0170714\pi\)
							−0.998562	+	0.0536055i	\(0.982929\pi\)
\(47\)	2.94330e15			0.383624			0.191812	−	0.981432i	\(-0.438564\pi\)
							0.191812	+	0.981432i	\(0.438564\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.20.b.a.11.10	yes	36
3.2	odd	2	inner	12.20.b.a.11.27	yes	36
4.3	odd	2	inner	12.20.b.a.11.28	yes	36
12.11	even	2	inner	12.20.b.a.11.9	✓	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.20.b.a.11.9	✓	36	12.11	even	2	inner
12.20.b.a.11.10	yes	36	1.1	even	1	trivial
12.20.b.a.11.27	yes	36	3.2	odd	2	inner
12.20.b.a.11.28	yes	36	4.3	odd	2	inner