Embedded newform 16.12.e.a.13.9

• Label: 16.12.e.a.13.9
• Level: $16$
• Weight: $12$
• Character: 16.13
• Analytic conductor: $12.293$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.2934908890\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(21\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.9
Character	\(\chi\)	\(=\)	16.13
Dual form			16.12.e.a.5.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.87277 + 44.7299i) q^{2} +(-291.167 - 291.167i) q^{3} +(-1953.53 - 614.837i) q^{4} +(-9753.88 + 9753.88i) q^{5} +(15025.0 - 11022.8i) q^{6} +16447.0i q^{7} +(40927.8 - 83155.6i) q^{8} -7590.38i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.87277	+	44.7299i	−0.151868	+	0.988401i
							\(3\)	−291.167	−	291.167i	−0.691792	−	0.691792i	0.270834	−	0.962626i	\(-0.412701\pi\)
−0.962626	+	0.270834i	\(0.912701\pi\)
\(5\)	−9753.88	+	9753.88i	−1.39586	+	1.39586i	−0.584384	+	0.811477i	\(0.698664\pi\)
							−0.811477	+	0.584384i	\(0.801336\pi\)
\(7\)			16447.0i			0.369868i	0.982751	+	0.184934i	\(0.0592071\pi\)
							−0.982751	+	0.184934i	\(0.940793\pi\)
\(11\)	477540.	−	477540.i	0.894026	−	0.894026i	−0.100873	−	0.994899i	\(-0.532164\pi\)
							0.994899	+	0.100873i	\(0.0321637\pi\)
\(13\)	665864.	+	665864.i	0.497390	+	0.497390i	0.910625	−	0.413235i	\(-0.135601\pi\)
							−0.413235	+	0.910625i	\(0.635601\pi\)
\(17\)	−5.80462e6			−0.991527			−0.495764	−	0.868457i	\(-0.665112\pi\)
							−0.495764	+	0.868457i	\(0.665112\pi\)
\(19\)	1.91371e6	+	1.91371e6i	0.177309	+	0.177309i	0.790182	−	0.612873i	\(-0.209986\pi\)
							−0.612873	+	0.790182i	\(0.709986\pi\)
\(23\)		−	1.46860e7i		−	0.475775i	−0.971293	−	0.237888i	\(-0.923545\pi\)
							0.971293	−	0.237888i	\(-0.0764550\pi\)
\(29\)	5.83665e7	+	5.83665e7i	0.528414	+	0.528414i	0.920099	−	0.391685i	\(-0.128108\pi\)
							−0.391685	+	0.920099i	\(0.628108\pi\)
\(31\)	8.50845e7			0.533778			0.266889	−	0.963727i	\(-0.414004\pi\)
							0.266889	+	0.963727i	\(0.414004\pi\)
\(37\)	2.52847e7	−	2.52847e7i	0.0599443	−	0.0599443i	−0.676499	−	0.736443i	\(-0.736504\pi\)
							0.736443	+	0.676499i	\(0.236504\pi\)
\(41\)		−	9.24887e8i		−	1.24674i	−0.781925	−	0.623372i	\(-0.785762\pi\)
							0.781925	−	0.623372i	\(-0.214238\pi\)
\(43\)	1.00305e9	−	1.00305e9i	1.04051	−	1.04051i	0.0413672	−	0.999144i	\(-0.486829\pi\)
							0.999144	−	0.0413672i	\(-0.0131714\pi\)
\(47\)	−2.86182e9			−1.82014			−0.910068	−	0.414459i	\(-0.863971\pi\)
							−0.910068	+	0.414459i	\(0.863971\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.13.9	yes	42
4.3	odd	2		64.12.e.a.17.16		42
8.3	odd	2		128.12.e.a.33.6		42
8.5	even	2		128.12.e.b.33.16		42
16.3	odd	4		128.12.e.a.97.6		42
16.5	even	4	inner	16.12.e.a.5.9	✓	42
16.11	odd	4		64.12.e.a.49.16		42
16.13	even	4		128.12.e.b.97.16		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.9	✓	42	16.5	even	4	inner
16.12.e.a.13.9	yes	42	1.1	even	1	trivial
64.12.e.a.17.16		42	4.3	odd	2
64.12.e.a.49.16		42	16.11	odd	4
128.12.e.a.33.6		42	8.3	odd	2
128.12.e.a.97.6		42	16.3	odd	4
128.12.e.b.33.16		42	8.5	even	2
128.12.e.b.97.16		42	16.13	even	4