Embedded newform 16.12.e.a.13.12

• Label: 16.12.e.a.13.12
• 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			13.12
Character	\(\chi\)	\(=\)	16.13
Dual form			16.12.e.a.5.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.445844 - 45.2526i) q^{2} +(-296.568 - 296.568i) q^{3} +(-2047.60 + 40.3512i) q^{4} +(-2867.98 + 2867.98i) q^{5} +(-13288.2 + 13552.7i) q^{6} +8380.90i q^{7} +(2738.91 + 92641.4i) q^{8} -1242.24i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.445844	−	45.2526i	−0.00985185	−	0.999951i
							\(3\)	−296.568	−	296.568i	−0.704623	−	0.704623i	0.260776	−	0.965399i	\(-0.416022\pi\)
−0.965399	+	0.260776i	\(0.916022\pi\)
\(5\)	−2867.98	+	2867.98i	−0.410432	+	0.410432i	−0.881889	−	0.471457i	\(-0.843728\pi\)
							0.471457	+	0.881889i	\(0.343728\pi\)
\(7\)			8380.90i			0.188474i	0.995550	+	0.0942370i	\(0.0300411\pi\)
							−0.995550	+	0.0942370i	\(0.969959\pi\)
\(11\)	84515.5	−	84515.5i	0.158226	−	0.158226i	−0.623554	−	0.781780i	\(-0.714312\pi\)
							0.781780	+	0.623554i	\(0.214312\pi\)
\(13\)	−385486.	−	385486.i	−0.287952	−	0.287952i	0.548318	−	0.836270i	\(-0.315268\pi\)
							−0.836270	+	0.548318i	\(0.815268\pi\)
\(17\)	1.56551e6			0.267416			0.133708	−	0.991021i	\(-0.457311\pi\)
							0.133708	+	0.991021i	\(0.457311\pi\)
\(19\)	9.59253e6	+	9.59253e6i	0.888768	+	0.888768i	0.994405	−	0.105637i	\(-0.0336882\pi\)
							−0.105637	+	0.994405i	\(0.533688\pi\)
\(23\)			3.88214e7i			1.25767i	0.777537	+	0.628837i	\(0.216469\pi\)
							−0.777537	+	0.628837i	\(0.783531\pi\)
\(29\)	−8.81048e7	−	8.81048e7i	−0.797647	−	0.797647i	0.185077	−	0.982724i	\(-0.440746\pi\)
							−0.982724	+	0.185077i	\(0.940746\pi\)
\(31\)	−7.23333e7			−0.453784			−0.226892	−	0.973920i	\(-0.572856\pi\)
							−0.226892	+	0.973920i	\(0.572856\pi\)
\(37\)	−9.98585e7	+	9.98585e7i	−0.236742	+	0.236742i	−0.815500	−	0.578758i	\(-0.803538\pi\)
							0.578758	+	0.815500i	\(0.303538\pi\)
\(41\)			1.13304e9i			1.52733i	0.645611	+	0.763667i	\(0.276603\pi\)
							−0.645611	+	0.763667i	\(0.723397\pi\)
\(43\)	−1.05585e9	+	1.05585e9i	−1.09528	+	1.09528i	−0.100329	+	0.994954i	\(0.531990\pi\)
							−0.994954	+	0.100329i	\(0.968010\pi\)
\(47\)	1.33450e9			0.848750			0.424375	−	0.905486i	\(-0.360494\pi\)
							0.424375	+	0.905486i	\(0.360494\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.13.12	yes	42
4.3	odd	2		64.12.e.a.17.17		42
8.3	odd	2		128.12.e.a.33.5		42
8.5	even	2		128.12.e.b.33.17		42
16.3	odd	4		128.12.e.a.97.5		42
16.5	even	4	inner	16.12.e.a.5.12	✓	42
16.11	odd	4		64.12.e.a.49.17		42
16.13	even	4		128.12.e.b.97.17		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.12	✓	42	16.5	even	4	inner
16.12.e.a.13.12	yes	42	1.1	even	1	trivial
64.12.e.a.17.17		42	4.3	odd	2
64.12.e.a.49.17		42	16.11	odd	4
128.12.e.a.33.5		42	8.3	odd	2
128.12.e.a.97.5		42	16.3	odd	4
128.12.e.b.33.17		42	8.5	even	2
128.12.e.b.97.17		42	16.13	even	4