Embedded newform 16.12.e.a.13.4

• Label: 16.12.e.a.13.4
• 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			13.4
Character	\(\chi\)	\(=\)	16.13
Dual form			16.12.e.a.5.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-36.4599 + 26.8082i) q^{2} +(-475.976 - 475.976i) q^{3} +(610.643 - 1954.84i) q^{4} +(2805.14 - 2805.14i) q^{5} +(30114.1 + 4593.97i) q^{6} -76033.6i q^{7} +(30141.9 + 87643.6i) q^{8} +275960. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−36.4599	+	26.8082i	−0.805657	+	0.592383i
							\(3\)	−475.976	−	475.976i	−1.13089	−	1.13089i	−0.990030	−	0.140855i	\(-0.955015\pi\)
−0.140855	−	0.990030i	\(-0.544985\pi\)
\(5\)	2805.14	−	2805.14i	0.401439	−	0.401439i	−0.477301	−	0.878740i	\(-0.658385\pi\)
							0.878740	+	0.477301i	\(0.158385\pi\)
\(7\)		−	76033.6i		−	1.70988i	−0.518725	−	0.854941i	\(-0.673593\pi\)
							0.518725	−	0.854941i	\(-0.326407\pi\)
\(11\)	409752.	−	409752.i	0.767116	−	0.767116i	−0.210482	−	0.977598i	\(-0.567503\pi\)
							0.977598	+	0.210482i	\(0.0675032\pi\)
\(13\)	−1.80631e6	−	1.80631e6i	−1.34929	−	1.34929i	−0.886435	−	0.462853i	\(-0.846826\pi\)
							−0.462853	−	0.886435i	\(-0.653174\pi\)
\(17\)	−1.50853e6			−0.257683			−0.128842	−	0.991665i	\(-0.541126\pi\)
							−0.128842	+	0.991665i	\(0.541126\pi\)
\(19\)	5.45825e6	+	5.45825e6i	0.505718	+	0.505718i	0.913209	−	0.407491i	\(-0.133596\pi\)
							−0.407491	+	0.913209i	\(0.633596\pi\)
\(23\)		−	2.28696e6i		−	0.0740893i	−0.999314	−	0.0370446i	\(-0.988206\pi\)
							0.999314	−	0.0370446i	\(-0.0117944\pi\)
\(29\)	4.38084e7	+	4.38084e7i	0.396615	+	0.396615i	0.877037	−	0.480422i	\(-0.159517\pi\)
							−0.480422	+	0.877037i	\(0.659517\pi\)
\(31\)	2.76655e7			0.173560			0.0867798	−	0.996228i	\(-0.472342\pi\)
							0.0867798	+	0.996228i	\(0.472342\pi\)
\(37\)	2.40566e8	−	2.40566e8i	0.570327	−	0.570327i	−0.361893	−	0.932220i	\(-0.617869\pi\)
							0.932220	+	0.361893i	\(0.117869\pi\)
\(41\)			1.08336e8i			0.146037i	0.997331	+	0.0730184i	\(0.0232632\pi\)
							−0.997331	+	0.0730184i	\(0.976737\pi\)
\(43\)	1.91015e8	−	1.91015e8i	0.198149	−	0.198149i	−0.601057	−	0.799206i	\(-0.705254\pi\)
							0.799206	+	0.601057i	\(0.205254\pi\)
\(47\)	−1.72497e9			−1.09709			−0.548547	−	0.836120i	\(-0.684819\pi\)
							−0.548547	+	0.836120i	\(0.684819\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.13.4	yes	42
4.3	odd	2		64.12.e.a.17.19		42
8.3	odd	2		128.12.e.a.33.3		42
8.5	even	2		128.12.e.b.33.19		42
16.3	odd	4		128.12.e.a.97.3		42
16.5	even	4	inner	16.12.e.a.5.4	✓	42
16.11	odd	4		64.12.e.a.49.19		42
16.13	even	4		128.12.e.b.97.19		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.4	✓	42	16.5	even	4	inner
16.12.e.a.13.4	yes	42	1.1	even	1	trivial
64.12.e.a.17.19		42	4.3	odd	2
64.12.e.a.49.19		42	16.11	odd	4
128.12.e.a.33.3		42	8.3	odd	2
128.12.e.a.97.3		42	16.3	odd	4
128.12.e.b.33.19		42	8.5	even	2
128.12.e.b.97.19		42	16.13	even	4