Embedded newform 816.2.bm.c.251.13

• Label: 816.2.bm.c.251.13
• Level: $816$
• Weight: $2$
• Character: 816.251
• Analytic conductor: $6.516$
• Analytic rank: $0$
• Dimension: $272$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	816.bm (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.51579280494\)
Analytic rank:	\(0\)
Dimension:	\(272\)
Relative dimension:	\(136\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			251.13
Character	\(\chi\)	\(=\)	816.251
Dual form			816.2.bm.c.803.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36342 - 0.375628i) q^{2} +(-0.839781 - 1.51485i) q^{3} +(1.71781 + 1.02427i) q^{4} -2.27638 q^{5} +(0.575951 + 2.38082i) q^{6} +(1.09870 - 1.09870i) q^{7} +(-1.95734 - 2.04177i) q^{8} +(-1.58954 + 2.54428i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 8 q^{4} - 10 q^{6} - 8 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/816\mathbb{Z}\right)^\times\).

\(n\)	\(241\)	\(511\)	\(545\)	\(613\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36342	−	0.375628i	−0.964081	−	0.265609i
							\(3\)	−0.839781	−	1.51485i	−0.484848	−	0.874599i
\(5\)	−2.27638			−1.01803										−0.509014	−	0.860758i	\(-0.669990\pi\)
							−0.509014	+	0.860758i	\(0.669990\pi\)
\(7\)	1.09870	−	1.09870i	0.415269	−	0.415269i	−0.468301	−	0.883569i	\(-0.655134\pi\)
							0.883569	+	0.468301i	\(0.155134\pi\)
\(11\)	3.54638			1.06927			0.534637	−	0.845082i	\(-0.320448\pi\)
							0.534637	+	0.845082i	\(0.320448\pi\)
\(13\)	1.70155	−	1.70155i	0.471925	−	0.471925i	−0.430612	−	0.902537i	\(-0.641702\pi\)
							0.902537	+	0.430612i	\(0.141702\pi\)
\(17\)	3.29445	−	2.47924i	0.799021	−	0.601303i
							\(19\)	−0.256413	+	0.256413i	−0.0588252	+	0.0588252i	−0.735907	−	0.677082i	\(-0.763244\pi\)
0.677082	+	0.735907i	\(0.263244\pi\)
\(23\)	0.152024	+	0.152024i	0.0316992	+	0.0316992i	0.722779	−	0.691080i	\(-0.242865\pi\)
							−0.691080	+	0.722779i	\(0.742865\pi\)
\(29\)		−	7.58036i		−	1.40764i	−0.710379	−	0.703819i	\(-0.751477\pi\)
							0.710379	−	0.703819i	\(-0.248523\pi\)
\(31\)	−0.659429	+	0.659429i	−0.118437	+	0.118437i	−0.763841	−	0.645404i	\(-0.776689\pi\)
							0.645404	+	0.763841i	\(0.276689\pi\)
\(37\)		−	2.36013i		−	0.388002i	−0.981001	−	0.194001i	\(-0.937853\pi\)
							0.981001	−	0.194001i	\(-0.0621466\pi\)
\(41\)	−1.51412	+	1.51412i	−0.236466	+	0.236466i	−0.815385	−	0.578919i	\(-0.803475\pi\)
							0.578919	+	0.815385i	\(0.303475\pi\)
\(43\)	−2.12482	−	2.12482i	−0.324032	−	0.324032i	0.526280	−	0.850312i	\(-0.323587\pi\)
							−0.850312	+	0.526280i	\(0.823587\pi\)
\(47\)	2.83423			0.413415			0.206707	−	0.978403i	\(-0.433725\pi\)
							0.206707	+	0.978403i	\(0.433725\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	816.2.bm.c.251.13	yes	272
3.2	odd	2	inner	816.2.bm.c.251.124	yes	272
16.3	odd	4		816.2.r.c.659.13	yes	272
17.4	even	4		816.2.r.c.395.124	yes	272
48.35	even	4		816.2.r.c.659.124	yes	272
51.38	odd	4		816.2.r.c.395.13	✓	272
272.259	odd	4	inner	816.2.bm.c.803.124	yes	272
816.803	even	4	inner	816.2.bm.c.803.13	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
816.2.r.c.395.13	✓	272	51.38	odd	4
816.2.r.c.395.124	yes	272	17.4	even	4
816.2.r.c.659.13	yes	272	16.3	odd	4
816.2.r.c.659.124	yes	272	48.35	even	4
816.2.bm.c.251.13	yes	272	1.1	even	1	trivial
816.2.bm.c.251.124	yes	272	3.2	odd	2	inner
816.2.bm.c.803.13	yes	272	816.803	even	4	inner
816.2.bm.c.803.124	yes	272	272.259	odd	4	inner