Embedded newform 260.2.p.d.103.15

• Label: 260.2.p.d.103.15
• 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			103.15
Character	\(\chi\)	\(=\)	260.103
Dual form			260.2.p.d.207.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.358430 - 1.36804i) q^{2} +(-1.80624 + 1.80624i) q^{3} +(-1.74306 + 0.980692i) q^{4} +(1.12997 - 1.92955i) q^{5} +(3.11841 + 1.82359i) q^{6} +(0.163803 - 0.163803i) q^{7} +(1.96639 + 2.03306i) q^{8} -3.52499i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.358430	−	1.36804i	−0.253448	−	0.967349i
							\(3\)	−1.80624	+	1.80624i	−1.04283	+	1.04283i	−0.0437910	+	0.999041i	\(0.513944\pi\)
−0.999041	+	0.0437910i	\(0.986056\pi\)
\(5\)	1.12997	−	1.92955i	0.505339	−	0.862921i
							\(7\)	0.163803	−	0.163803i	0.0619118	−	0.0619118i	−0.675473	−	0.737385i	\(-0.736060\pi\)
0.737385	+	0.675473i	\(0.236060\pi\)
\(11\)	−5.50310			−1.65925			−0.829623	−	0.558324i	\(-0.811445\pi\)
							−0.829623	+	0.558324i	\(0.811445\pi\)
\(13\)	−3.60119	+	0.177198i	−0.998792	+	0.0491458i
							\(17\)	−5.33812	+	5.33812i	−1.29468	+	1.29468i	−0.362829	+	0.931856i	\(0.618189\pi\)
−0.931856	+	0.362829i	\(0.881811\pi\)
\(19\)		−	3.95414i		−	0.907143i	−0.891220	−	0.453572i	\(-0.850150\pi\)
							0.891220	−	0.453572i	\(-0.149850\pi\)
\(23\)	−3.90248	+	3.90248i	−0.813722	+	0.813722i	−0.985190	−	0.171467i	\(-0.945149\pi\)
							0.171467	+	0.985190i	\(0.445149\pi\)
\(29\)		−	2.85390i		−	0.529955i	−0.964255	−	0.264978i	\(-0.914635\pi\)
							0.964255	−	0.264978i	\(-0.0853646\pi\)
\(31\)	4.84261			0.869759			0.434880	−	0.900489i	\(-0.356791\pi\)
							0.434880	+	0.900489i	\(0.356791\pi\)
\(37\)	−1.81836	−	1.81836i	−0.298937	−	0.298937i	0.541660	−	0.840597i	\(-0.317796\pi\)
							−0.840597	+	0.541660i	\(0.817796\pi\)
\(41\)		−	1.07125i		−	0.167301i	−0.996495	−	0.0836505i	\(-0.973342\pi\)
							0.996495	−	0.0836505i	\(-0.0266579\pi\)
\(43\)	2.39102	−	2.39102i	0.364627	−	0.364627i	−0.500886	−	0.865513i	\(-0.666992\pi\)
							0.865513	+	0.500886i	\(0.166992\pi\)
\(47\)	−2.94373	+	2.94373i	−0.429388	+	0.429388i	−0.888420	−	0.459032i	\(-0.848196\pi\)
							0.459032	+	0.888420i	\(0.348196\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	260.2.p.d.103.15	yes	64
4.3	odd	2	inner	260.2.p.d.103.2	✓	64
5.2	odd	4	inner	260.2.p.d.207.31	yes	64
13.12	even	2	inner	260.2.p.d.103.18	yes	64
20.7	even	4	inner	260.2.p.d.207.18	yes	64
52.51	odd	2	inner	260.2.p.d.103.31	yes	64
65.12	odd	4	inner	260.2.p.d.207.2	yes	64
260.207	even	4	inner	260.2.p.d.207.15	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.p.d.103.2	✓	64	4.3	odd	2	inner
260.2.p.d.103.15	yes	64	1.1	even	1	trivial
260.2.p.d.103.18	yes	64	13.12	even	2	inner
260.2.p.d.103.31	yes	64	52.51	odd	2	inner
260.2.p.d.207.2	yes	64	65.12	odd	4	inner
260.2.p.d.207.15	yes	64	260.207	even	4	inner
260.2.p.d.207.18	yes	64	20.7	even	4	inner
260.2.p.d.207.31	yes	64	5.2	odd	4	inner