Embedded newform 16.12.e.a.5.17

• Label: 16.12.e.a.5.17
• Level: $16$
• Weight: $12$
• Character: 16.5
• 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			5.17
Character	\(\chi\)	\(=\)	16.5
Dual form			16.12.e.a.13.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(33.6326 - 30.2795i) q^{2} +(-145.072 + 145.072i) q^{3} +(214.299 - 2036.76i) q^{4} +(-4652.81 - 4652.81i) q^{5} +(-486.430 + 9271.83i) q^{6} +43253.0i q^{7} +(-54464.6 - 74990.3i) q^{8} +135056. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	33.6326	−	30.2795i	0.743182	−	0.669090i
							\(3\)	−145.072	+	145.072i	−0.344679	+	0.344679i	−0.858123	−	0.513444i	\(-0.828369\pi\)
0.513444	+	0.858123i	\(0.328369\pi\)
\(5\)	−4652.81	−	4652.81i	−0.665856	−	0.665856i	0.290898	−	0.956754i	\(-0.406046\pi\)
							−0.956754	+	0.290898i	\(0.906046\pi\)
\(7\)			43253.0i			0.972696i	0.873765	+	0.486348i	\(0.161671\pi\)
							−0.873765	+	0.486348i	\(0.838329\pi\)
\(11\)	−685114.	−	685114.i	−1.28263	−	1.28263i	−0.939163	−	0.343472i	\(-0.888397\pi\)
							−0.343472	−	0.939163i	\(-0.611603\pi\)
\(13\)	−1.40072e6	+	1.40072e6i	−1.04632	+	1.04632i	−0.0474458	+	0.998874i	\(0.515108\pi\)
							−0.998874	+	0.0474458i	\(0.984892\pi\)
\(17\)	3.84591e6			0.656946			0.328473	−	0.944513i	\(-0.393466\pi\)
							0.328473	+	0.944513i	\(0.393466\pi\)
\(19\)	−7.81673e6	+	7.81673e6i	−0.724236	+	0.724236i	−0.969465	−	0.245229i	\(-0.921137\pi\)
							0.245229	+	0.969465i	\(0.421137\pi\)
\(23\)		−	1.99742e7i		−	0.647094i	−0.946212	−	0.323547i	\(-0.895125\pi\)
							0.946212	−	0.323547i	\(-0.104875\pi\)
\(29\)	1.18663e8	−	1.18663e8i	1.07430	−	1.07430i	0.0772929	−	0.997008i	\(-0.475372\pi\)
							0.997008	−	0.0772929i	\(-0.0246277\pi\)
\(31\)	5.32181e7			0.333864			0.166932	−	0.985968i	\(-0.446614\pi\)
							0.166932	+	0.985968i	\(0.446614\pi\)
\(37\)	−3.70312e8	−	3.70312e8i	−0.877928	−	0.877928i	0.115392	−	0.993320i	\(-0.463188\pi\)
							−0.993320	+	0.115392i	\(0.963188\pi\)
\(41\)		−	8.83045e7i		−	0.119034i	−0.998227	−	0.0595171i	\(-0.981044\pi\)
							0.998227	−	0.0595171i	\(-0.0189561\pi\)
\(43\)	1.81785e8	+	1.81785e8i	0.188573	+	0.188573i	0.795079	−	0.606506i	\(-0.207429\pi\)
							−0.606506	+	0.795079i	\(0.707429\pi\)
\(47\)	−1.92709e9			−1.22564			−0.612821	−	0.790222i	\(-0.709965\pi\)
							−0.612821	+	0.790222i	\(0.709965\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.5.17	✓	42
4.3	odd	2		64.12.e.a.49.13		42
8.3	odd	2		128.12.e.a.97.9		42
8.5	even	2		128.12.e.b.97.13		42
16.3	odd	4		64.12.e.a.17.13		42
16.5	even	4		128.12.e.b.33.13		42
16.11	odd	4		128.12.e.a.33.9		42
16.13	even	4	inner	16.12.e.a.13.17	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.17	✓	42	1.1	even	1	trivial
16.12.e.a.13.17	yes	42	16.13	even	4	inner
64.12.e.a.17.13		42	16.3	odd	4
64.12.e.a.49.13		42	4.3	odd	2
128.12.e.a.33.9		42	16.11	odd	4
128.12.e.a.97.9		42	8.3	odd	2
128.12.e.b.33.13		42	16.5	even	4
128.12.e.b.97.13		42	8.5	even	2