Embedded newform 16.12.e.a.5.3

• Label: 16.12.e.a.5.3
• 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			5.3
Character	\(\chi\)	\(=\)	16.5
Dual form			16.12.e.a.13.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-42.3872 + 15.8533i) q^{2} +(-435.014 + 435.014i) q^{3} +(1545.35 - 1343.95i) q^{4} +(-1517.34 - 1517.34i) q^{5} +(11542.6 - 25335.4i) q^{6} -59722.7i q^{7} +(-44197.0 + 81465.1i) q^{8} -201326. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−42.3872	+	15.8533i	−0.936634	+	0.350311i
							\(3\)	−435.014	+	435.014i	−1.03356	+	1.03356i	−0.0341434	+	0.999417i	\(0.510870\pi\)
−0.999417	+	0.0341434i	\(0.989130\pi\)
\(5\)	−1517.34	−	1517.34i	−0.217144	−	0.217144i	0.590150	−	0.807294i	\(-0.299069\pi\)
							−0.807294	+	0.590150i	\(0.799069\pi\)
\(7\)		−	59722.7i		−	1.34307i	−0.740971	−	0.671537i	\(-0.765634\pi\)
							0.740971	−	0.671537i	\(-0.234366\pi\)
\(11\)	56196.6	+	56196.6i	0.105208	+	0.105208i	0.757752	−	0.652543i	\(-0.226298\pi\)
							−0.652543	+	0.757752i	\(0.726298\pi\)
\(13\)	−315796.	+	315796.i	−0.235895	+	0.235895i	−0.815148	−	0.579253i	\(-0.803344\pi\)
							0.579253	+	0.815148i	\(0.303344\pi\)
\(17\)	8.11170e6			1.38562			0.692808	−	0.721122i	\(-0.256373\pi\)
							0.692808	+	0.721122i	\(0.256373\pi\)
\(19\)	−1.33811e7	+	1.33811e7i	−1.23978	+	1.23978i	−0.279695	+	0.960089i	\(0.590233\pi\)
							−0.960089	+	0.279695i	\(0.909767\pi\)
\(23\)			2.84613e7i			0.922043i	0.887389	+	0.461022i	\(0.152517\pi\)
							−0.887389	+	0.461022i	\(0.847483\pi\)
\(29\)	7.59695e7	−	7.59695e7i	0.687781	−	0.687781i	−0.273960	−	0.961741i	\(-0.588334\pi\)
							0.961741	+	0.273960i	\(0.0883335\pi\)
\(31\)	5.67278e7			0.355882			0.177941	−	0.984041i	\(-0.443056\pi\)
							0.177941	+	0.984041i	\(0.443056\pi\)
\(37\)	3.75680e8	+	3.75680e8i	0.890654	+	0.890654i	0.994585	−	0.103931i	\(-0.0331421\pi\)
							−0.103931	+	0.994585i	\(0.533142\pi\)
\(41\)		−	1.51696e7i		−	0.0204486i	−0.999948	−	0.0102243i	\(-0.996745\pi\)
							0.999948	−	0.0102243i	\(-0.00325455\pi\)
\(43\)	1.08339e9	+	1.08339e9i	1.12385	+	1.12385i	0.991158	+	0.132689i	\(0.0423610\pi\)
							0.132689	+	0.991158i	\(0.457639\pi\)
\(47\)	2.79798e9			1.77954			0.889768	−	0.456414i	\(-0.150866\pi\)
							0.889768	+	0.456414i	\(0.150866\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.5.3	✓	42
4.3	odd	2		64.12.e.a.49.18		42
8.3	odd	2		128.12.e.a.97.4		42
8.5	even	2		128.12.e.b.97.18		42
16.3	odd	4		64.12.e.a.17.18		42
16.5	even	4		128.12.e.b.33.18		42
16.11	odd	4		128.12.e.a.33.4		42
16.13	even	4	inner	16.12.e.a.13.3	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.3	✓	42	1.1	even	1	trivial
16.12.e.a.13.3	yes	42	16.13	even	4	inner
64.12.e.a.17.18		42	16.3	odd	4
64.12.e.a.49.18		42	4.3	odd	2
128.12.e.a.33.4		42	16.11	odd	4
128.12.e.a.97.4		42	8.3	odd	2
128.12.e.b.33.18		42	16.5	even	4
128.12.e.b.97.18		42	8.5	even	2