Embedded newform 16.12.e.a.13.8

• Label: 16.12.e.a.13.8
• 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			13.8
Character	\(\chi\)	\(=\)	16.13
Dual form			16.12.e.a.5.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-26.9679 - 36.3419i) q^{2} +(-199.508 - 199.508i) q^{3} +(-593.467 + 1960.13i) q^{4} +(8937.54 - 8937.54i) q^{5} +(-1870.19 + 12630.8i) q^{6} -55653.3i q^{7} +(87239.3 - 31292.8i) q^{8} -97540.0i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−26.9679	−	36.3419i	−0.595912	−	0.803050i
							\(3\)	−199.508	−	199.508i	−0.474017	−	0.474017i	0.429195	−	0.903212i	\(-0.358797\pi\)
−0.903212	+	0.429195i	\(0.858797\pi\)
\(5\)	8937.54	−	8937.54i	1.27904	−	1.27904i	0.337830	−	0.941207i	\(-0.390307\pi\)
							0.941207	−	0.337830i	\(-0.109693\pi\)
\(7\)		−	55653.3i		−	1.25156i	−0.780000	−	0.625780i	\(-0.784781\pi\)
							0.780000	−	0.625780i	\(-0.215219\pi\)
\(11\)	−146129.	+	146129.i	−0.273575	+	0.273575i	−0.830538	−	0.556963i	\(-0.811967\pi\)
							0.556963	+	0.830538i	\(0.311967\pi\)
\(13\)	1.33136e6	+	1.33136e6i	0.994502	+	0.994502i	0.999985	−	0.00548268i	\(-0.00174520\pi\)
							−0.00548268	+	0.999985i	\(0.501745\pi\)
\(17\)	−2.92311e6			−0.499317			−0.249659	−	0.968334i	\(-0.580318\pi\)
							−0.249659	+	0.968334i	\(0.580318\pi\)
\(19\)	−1.11550e7	−	1.11550e7i	−1.03354	−	1.03354i	−0.999418	−	0.0341197i	\(-0.989137\pi\)
							−0.0341197	−	0.999418i	\(-0.510863\pi\)
\(23\)			2.12560e7i			0.688619i	0.938856	+	0.344309i	\(0.111887\pi\)
							−0.938856	+	0.344309i	\(0.888113\pi\)
\(29\)	−1.31742e7	−	1.31742e7i	−0.119271	−	0.119271i	0.644952	−	0.764223i	\(-0.276877\pi\)
							−0.764223	+	0.644952i	\(0.776877\pi\)
\(31\)	1.84262e8			1.15597			0.577986	−	0.816047i	\(-0.303839\pi\)
							0.577986	+	0.816047i	\(0.303839\pi\)
\(37\)	6.58484e7	−	6.58484e7i	0.156112	−	0.156112i	−0.624729	−	0.780841i	\(-0.714791\pi\)
							0.780841	+	0.624729i	\(0.214791\pi\)
\(41\)			7.48317e8i			1.00873i	0.863491	+	0.504364i	\(0.168273\pi\)
							−0.863491	+	0.504364i	\(0.831727\pi\)
\(43\)	−3.10180e8	+	3.10180e8i	−0.321764	+	0.321764i	−0.849443	−	0.527680i	\(-0.823062\pi\)
							0.527680	+	0.849443i	\(0.323062\pi\)
\(47\)	−1.85656e9			−1.18078			−0.590392	−	0.807117i	\(-0.701027\pi\)
							−0.590392	+	0.807117i	\(0.701027\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.13.8	yes	42
4.3	odd	2		64.12.e.a.17.15		42
8.3	odd	2		128.12.e.a.33.7		42
8.5	even	2		128.12.e.b.33.15		42
16.3	odd	4		128.12.e.a.97.7		42
16.5	even	4	inner	16.12.e.a.5.8	✓	42
16.11	odd	4		64.12.e.a.49.15		42
16.13	even	4		128.12.e.b.97.15		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.8	✓	42	16.5	even	4	inner
16.12.e.a.13.8	yes	42	1.1	even	1	trivial
64.12.e.a.17.15		42	4.3	odd	2
64.12.e.a.49.15		42	16.11	odd	4
128.12.e.a.33.7		42	8.3	odd	2
128.12.e.a.97.7		42	16.3	odd	4
128.12.e.b.33.15		42	8.5	even	2
128.12.e.b.97.15		42	16.13	even	4