Embedded newform 260.2.p.d.103.16

• Label: 260.2.p.d.103.16
• Level: $260$
• Weight: $2$
• Character: 260.103
• Analytic conductor: $2.076$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	260.p (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.07611045255\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			103.16
Character	\(\chi\)	\(=\)	260.103
Dual form			260.2.p.d.207.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0324895 - 1.41384i) q^{2} +(2.32949 - 2.32949i) q^{3} +(-1.99789 + 0.0918700i) q^{4} +(2.23599 + 0.0191440i) q^{5} +(-3.36921 - 3.21784i) q^{6} +(-1.97121 + 1.97121i) q^{7} +(0.194800 + 2.82171i) q^{8} -7.85303i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/260\mathbb{Z}\right)^\times\).

\(n\)	\(41\)	\(131\)	\(157\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0324895	−	1.41384i	−0.0229736	−	0.999736i
							\(3\)	2.32949	−	2.32949i	1.34493	−	1.34493i	0.453856	−	0.891075i	\(-0.350048\pi\)
0.891075	−	0.453856i	\(-0.149952\pi\)
\(5\)	2.23599	+	0.0191440i	0.999963	+	0.00856145i
							\(7\)	−1.97121	+	1.97121i	−0.745047	+	0.745047i	−0.973544	−	0.228498i	\(-0.926619\pi\)
0.228498	+	0.973544i	\(0.426619\pi\)
\(11\)	−2.76564			−0.833871			−0.416935	−	0.908936i	\(-0.636896\pi\)
							−0.416935	+	0.908936i	\(0.636896\pi\)
\(13\)	3.56433	+	0.543628i	0.988568	+	0.150775i
							\(17\)	−1.79865	+	1.79865i	−0.436237	+	0.436237i	−0.890744	−	0.454506i	\(-0.849816\pi\)
0.454506	+	0.890744i	\(0.349816\pi\)
\(19\)			2.06756i			0.474330i	0.971469	+	0.237165i	\(0.0762182\pi\)
							−0.971469	+	0.237165i	\(0.923782\pi\)
\(23\)	−1.07900	+	1.07900i	−0.224988	+	0.224988i	−0.810595	−	0.585607i	\(-0.800856\pi\)
							0.585607	+	0.810595i	\(0.300856\pi\)
\(29\)			7.19247i			1.33561i	0.744337	+	0.667804i	\(0.232766\pi\)
							−0.744337	+	0.667804i	\(0.767234\pi\)
\(31\)	2.08767			0.374957			0.187478	−	0.982269i	\(-0.439969\pi\)
							0.187478	+	0.982269i	\(0.439969\pi\)
\(37\)	−2.10529	−	2.10529i	−0.346108	−	0.346108i	0.512549	−	0.858658i	\(-0.328701\pi\)
							−0.858658	+	0.512549i	\(0.828701\pi\)
\(41\)		−	2.07353i		−	0.323831i	−0.986805	−	0.161915i	\(-0.948233\pi\)
							0.986805	−	0.161915i	\(-0.0517671\pi\)
\(43\)	−1.37442	+	1.37442i	−0.209596	+	0.209596i	−0.804096	−	0.594500i	\(-0.797350\pi\)
							0.594500	+	0.804096i	\(0.297350\pi\)
\(47\)	3.03509	−	3.03509i	0.442713	−	0.442713i	−0.450210	−	0.892923i	\(-0.648651\pi\)
							0.892923	+	0.450210i	\(0.148651\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	260.2.p.d.103.16	yes	64
4.3	odd	2	inner	260.2.p.d.103.1	✓	64
5.2	odd	4	inner	260.2.p.d.207.32	yes	64
13.12	even	2	inner	260.2.p.d.103.17	yes	64
20.7	even	4	inner	260.2.p.d.207.17	yes	64
52.51	odd	2	inner	260.2.p.d.103.32	yes	64
65.12	odd	4	inner	260.2.p.d.207.1	yes	64
260.207	even	4	inner	260.2.p.d.207.16	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.p.d.103.1	✓	64	4.3	odd	2	inner
260.2.p.d.103.16	yes	64	1.1	even	1	trivial
260.2.p.d.103.17	yes	64	13.12	even	2	inner
260.2.p.d.103.32	yes	64	52.51	odd	2	inner
260.2.p.d.207.1	yes	64	65.12	odd	4	inner
260.2.p.d.207.16	yes	64	260.207	even	4	inner
260.2.p.d.207.17	yes	64	20.7	even	4	inner
260.2.p.d.207.32	yes	64	5.2	odd	4	inner