Embedded newform 16.12.e.a.13.16

• Label: 16.12.e.a.13.16
• 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			13.16
Character	\(\chi\)	\(=\)	16.13
Dual form			16.12.e.a.5.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(30.2883 - 33.6246i) q^{2} +(-56.4628 - 56.4628i) q^{3} +(-213.234 - 2036.87i) q^{4} +(6260.45 - 6260.45i) q^{5} +(-3608.71 + 188.377i) q^{6} +32944.0i q^{7} +(-74947.5 - 54523.5i) q^{8} -170771. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	30.2883	−	33.6246i	0.669284	−	0.743007i
							\(3\)	−56.4628	−	56.4628i	−0.134151	−	0.134151i	0.636842	−	0.770994i	\(-0.280240\pi\)
−0.770994	+	0.636842i	\(0.780240\pi\)
\(5\)	6260.45	−	6260.45i	0.895922	−	0.895922i	−0.0991500	−	0.995072i	\(-0.531612\pi\)
							0.995072	+	0.0991500i	\(0.0316124\pi\)
\(7\)			32944.0i			0.740861i	0.928860	+	0.370431i	\(0.120790\pi\)
							−0.928860	+	0.370431i	\(0.879210\pi\)
\(11\)	−151744.	+	151744.i	−0.284088	+	0.284088i	−0.834737	−	0.550649i	\(-0.814380\pi\)
							0.550649	+	0.834737i	\(0.314380\pi\)
\(13\)	−1.05140e6	−	1.05140e6i	−0.785380	−	0.785380i	0.195353	−	0.980733i	\(-0.437415\pi\)
							−0.980733	+	0.195353i	\(0.937415\pi\)
\(17\)	−1.95797e6			−0.334455			−0.167228	−	0.985918i	\(-0.553481\pi\)
							−0.167228	+	0.985918i	\(0.553481\pi\)
\(19\)	5.25340e6	+	5.25340e6i	0.486738	+	0.486738i	0.907275	−	0.420537i	\(-0.138158\pi\)
							−0.420537	+	0.907275i	\(0.638158\pi\)
\(23\)		−	4.62502e7i		−	1.49834i	−0.662377	−	0.749170i	\(-0.730452\pi\)
							0.662377	−	0.749170i	\(-0.269548\pi\)
\(29\)	1.16274e8	+	1.16274e8i	1.05267	+	1.05267i	0.998533	+	0.0541405i	\(0.0172419\pi\)
							0.0541405	+	0.998533i	\(0.482758\pi\)
\(31\)	2.86231e8			1.79567			0.897836	−	0.440331i	\(-0.145139\pi\)
							0.897836	+	0.440331i	\(0.145139\pi\)
\(37\)	3.33901e8	−	3.33901e8i	0.791606	−	0.791606i	−0.190150	−	0.981755i	\(-0.560897\pi\)
							0.981755	+	0.190150i	\(0.0608973\pi\)
\(41\)		−	1.31100e8i		−	0.176722i	−0.996089	−	0.0883610i	\(-0.971837\pi\)
							0.996089	−	0.0883610i	\(-0.0281629\pi\)
\(43\)	5.60465e8	−	5.60465e8i	0.581396	−	0.581396i	−0.353891	−	0.935287i	\(-0.615141\pi\)
							0.935287	+	0.353891i	\(0.115141\pi\)
\(47\)	1.10264e8			0.0701288			0.0350644	−	0.999385i	\(-0.488836\pi\)
							0.0350644	+	0.999385i	\(0.488836\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.13.16	yes	42
4.3	odd	2		64.12.e.a.17.12		42
8.3	odd	2		128.12.e.a.33.10		42
8.5	even	2		128.12.e.b.33.12		42
16.3	odd	4		128.12.e.a.97.10		42
16.5	even	4	inner	16.12.e.a.5.16	✓	42
16.11	odd	4		64.12.e.a.49.12		42
16.13	even	4		128.12.e.b.97.12		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.16	✓	42	16.5	even	4	inner
16.12.e.a.13.16	yes	42	1.1	even	1	trivial
64.12.e.a.17.12		42	4.3	odd	2
64.12.e.a.49.12		42	16.11	odd	4
128.12.e.a.33.10		42	8.3	odd	2
128.12.e.a.97.10		42	16.3	odd	4
128.12.e.b.33.12		42	8.5	even	2
128.12.e.b.97.12		42	16.13	even	4