Embedded newform 16.14.e.a.5.12

• Label: 16.14.e.a.5.12
• Level: $16$
• Weight: $14$
• Character: 16.5
• Analytic conductor: $17.157$
• Analytic rank: $0$
• Dimension: $50$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.1569486323\)
Analytic rank:	\(0\)
Dimension:	\(50\)
Relative dimension:	\(25\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.12
Character	\(\chi\)	\(=\)	16.5
Dual form			16.14.e.a.13.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-18.7534 - 88.5455i) q^{2} +(1610.25 - 1610.25i) q^{3} +(-7488.62 + 3321.06i) q^{4} +(39780.2 + 39780.2i) q^{5} +(-172778. - 112383. i) q^{6} -439933. i q^{7} +(434502. + 600803. i) q^{8} -3.59146e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 2 q^{2} - 2 q^{3} + 360 q^{4} - 2 q^{5} - 255056 q^{6} - 1076876 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−18.7534	−	88.5455i	−0.207198	−	0.978299i
							\(3\)	1610.25	−	1610.25i	1.27528	−	1.27528i	0.331994	−	0.943281i	\(-0.392279\pi\)
0.943281	−	0.331994i	\(-0.107721\pi\)
\(5\)	39780.2	+	39780.2i	1.13857	+	1.13857i	0.988706	+	0.149869i	\(0.0478851\pi\)
							0.149869	+	0.988706i	\(0.452115\pi\)
\(7\)		−	439933.i		−	1.41335i	−0.707539	−	0.706674i	\(-0.750195\pi\)
							0.707539	−	0.706674i	\(-0.249805\pi\)
\(11\)	−2.46114e6	−	2.46114e6i	−0.418875	−	0.418875i	0.465941	−	0.884816i	\(-0.345716\pi\)
							−0.884816	+	0.465941i	\(0.845716\pi\)
\(13\)	5.83461e6	−	5.83461e6i	0.335259	−	0.335259i	−0.519321	−	0.854579i	\(-0.673815\pi\)
							0.854579	+	0.519321i	\(0.173815\pi\)
\(17\)	−6.96279e7			−0.699625			−0.349812	−	0.936820i	\(-0.613755\pi\)
							−0.349812	+	0.936820i	\(0.613755\pi\)
\(19\)	1.89520e7	−	1.89520e7i	0.0924179	−	0.0924179i	−0.659386	−	0.751804i	\(-0.729184\pi\)
							0.751804	+	0.659386i	\(0.229184\pi\)
\(23\)			1.01724e8i			0.143282i	0.997430	+	0.0716411i	\(0.0228236\pi\)
							−0.997430	+	0.0716411i	\(0.977176\pi\)
\(29\)	−1.53387e9	+	1.53387e9i	−0.478852	+	0.478852i	−0.904765	−	0.425912i	\(-0.859953\pi\)
							0.425912	+	0.904765i	\(0.359953\pi\)
\(31\)	6.07081e9			1.22856			0.614279	−	0.789089i	\(-0.289447\pi\)
							0.614279	+	0.789089i	\(0.289447\pi\)
\(37\)	2.43167e9	+	2.43167e9i	0.155809	+	0.155809i	0.780707	−	0.624897i	\(-0.214859\pi\)
							−0.624897	+	0.780707i	\(0.714859\pi\)
\(41\)			2.21934e10i			0.729674i	0.931071	+	0.364837i	\(0.118875\pi\)
							−0.931071	+	0.364837i	\(0.881125\pi\)
\(43\)	1.85698e10	+	1.85698e10i	0.447985	+	0.447985i	0.894684	−	0.446699i	\(-0.147401\pi\)
							−0.446699	+	0.894684i	\(0.647401\pi\)
\(47\)	−2.26074e9			−0.0305924			−0.0152962	−	0.999883i	\(-0.504869\pi\)
							−0.0152962	+	0.999883i	\(0.504869\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.14.e.a.5.12	✓	50
4.3	odd	2		64.14.e.a.49.2		50
8.3	odd	2		128.14.e.a.97.24		50
8.5	even	2		128.14.e.b.97.2		50
16.3	odd	4		64.14.e.a.17.2		50
16.5	even	4		128.14.e.b.33.2		50
16.11	odd	4		128.14.e.a.33.24		50
16.13	even	4	inner	16.14.e.a.13.12	yes	50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.14.e.a.5.12	✓	50	1.1	even	1	trivial
16.14.e.a.13.12	yes	50	16.13	even	4	inner
64.14.e.a.17.2		50	16.3	odd	4
64.14.e.a.49.2		50	4.3	odd	2
128.14.e.a.33.24		50	16.11	odd	4
128.14.e.a.97.24		50	8.3	odd	2
128.14.e.b.33.2		50	16.5	even	4
128.14.e.b.97.2		50	8.5	even	2