Embedded newform 16.12.e.a.13.20

• Label: 16.12.e.a.13.20
• 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			13.20
Character	\(\chi\)	\(=\)	16.13
Dual form			16.12.e.a.5.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(44.1073 + 10.1265i) q^{2} +(-179.701 - 179.701i) q^{3} +(1842.91 + 893.305i) q^{4} +(2961.29 - 2961.29i) q^{5} +(-6106.38 - 9745.87i) q^{6} +5179.04i q^{7} +(72239.7 + 58063.5i) q^{8} -112562. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	44.1073	+	10.1265i	0.974643	+	0.223766i
							\(3\)	−179.701	−	179.701i	−0.426956	−	0.426956i	0.460634	−	0.887590i	\(-0.347622\pi\)
−0.887590	+	0.460634i	\(0.847622\pi\)
\(5\)	2961.29	−	2961.29i	0.423785	−	0.423785i	−0.462719	−	0.886505i	\(-0.653126\pi\)
							0.886505	+	0.462719i	\(0.153126\pi\)
\(7\)			5179.04i			0.116469i	0.998303	+	0.0582345i	\(0.0185471\pi\)
							−0.998303	+	0.0582345i	\(0.981453\pi\)
\(11\)	694803.	−	694803.i	1.30077	−	1.30077i	0.372903	−	0.927870i	\(-0.378362\pi\)
							0.927870	−	0.372903i	\(-0.121638\pi\)
\(13\)	524615.	+	524615.i	0.391879	+	0.391879i	0.875357	−	0.483478i	\(-0.160627\pi\)
							−0.483478	+	0.875357i	\(0.660627\pi\)
\(17\)	1.82572e6			0.311864			0.155932	−	0.987768i	\(-0.450162\pi\)
							0.155932	+	0.987768i	\(0.450162\pi\)
\(19\)	−601185.	−	601185.i	−0.0557011	−	0.0557011i	0.678708	−	0.734409i	\(-0.262540\pi\)
							−0.734409	+	0.678708i	\(0.762540\pi\)
\(23\)		−	2.32829e7i		−	0.754284i	−0.926156	−	0.377142i	\(-0.876907\pi\)
							0.926156	−	0.377142i	\(-0.123093\pi\)
\(29\)	189495.	+	189495.i	0.00171557	+	0.00171557i	0.707964	−	0.706248i	\(-0.249614\pi\)
							−0.706248	+	0.707964i	\(0.749614\pi\)
\(31\)	−2.48770e8			−1.56066			−0.780330	−	0.625367i	\(-0.784949\pi\)
							−0.780330	+	0.625367i	\(0.784949\pi\)
\(37\)	−4.30934e8	+	4.30934e8i	−1.02165	+	1.02165i	−0.0218884	+	0.999760i	\(0.506968\pi\)
							−0.999760	+	0.0218884i	\(0.993032\pi\)
\(41\)			1.02672e8i			0.138401i	0.997603	+	0.0692007i	\(0.0220449\pi\)
							−0.997603	+	0.0692007i	\(0.977955\pi\)
\(43\)	−6.91645e7	+	6.91645e7i	−0.0717475	+	0.0717475i	−0.742070	−	0.670322i	\(-0.766156\pi\)
							0.670322	+	0.742070i	\(0.266156\pi\)
\(47\)	2.50575e8			0.159367			0.0796836	−	0.996820i	\(-0.474609\pi\)
							0.0796836	+	0.996820i	\(0.474609\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.13.20	yes	42
4.3	odd	2		64.12.e.a.17.14		42
8.3	odd	2		128.12.e.a.33.8		42
8.5	even	2		128.12.e.b.33.14		42
16.3	odd	4		128.12.e.a.97.8		42
16.5	even	4	inner	16.12.e.a.5.20	✓	42
16.11	odd	4		64.12.e.a.49.14		42
16.13	even	4		128.12.e.b.97.14		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.20	✓	42	16.5	even	4	inner
16.12.e.a.13.20	yes	42	1.1	even	1	trivial
64.12.e.a.17.14		42	4.3	odd	2
64.12.e.a.49.14		42	16.11	odd	4
128.12.e.a.33.8		42	8.3	odd	2
128.12.e.a.97.8		42	16.3	odd	4
128.12.e.b.33.14		42	8.5	even	2
128.12.e.b.97.14		42	16.13	even	4