Embedded newform 656.2.o.a.573.13

• Label: 656.2.o.a.573.13
• Level: $656$
• Weight: $2$
• Character: 656.573
• Analytic conductor: $5.238$
• Analytic rank: $0$
• Dimension: $164$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 656 = 2^{4} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	656.o (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.23818637260\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(82\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			573.13
Character	\(\chi\)	\(=\)	656.573
Dual form			656.2.o.a.245.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23625 - 0.686792i) q^{2} +(-0.0704098 + 0.0704098i) q^{3} +(1.05663 + 1.69809i) q^{4} +(-2.73754 + 2.73754i) q^{5} +(0.135401 - 0.0386873i) q^{6} -4.71049 q^{7} +(-0.140026 - 2.82496i) q^{8} +2.99008i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 4 q^{2} - 4 q^{4} - 4 q^{5} - 16 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/656\mathbb{Z}\right)^\times\).

\(n\)	\(129\)	\(165\)	\(575\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23625	−	0.686792i	−0.874162	−	0.485635i
							\(3\)	−0.0704098	+	0.0704098i	−0.0406511	+	0.0406511i	−0.727140	−	0.686489i	\(-0.759151\pi\)
0.686489	+	0.727140i	\(0.259151\pi\)
\(5\)	−2.73754	+	2.73754i	−1.22427	+	1.22427i	−0.258165	+	0.966101i	\(0.583118\pi\)
							−0.966101	+	0.258165i	\(0.916882\pi\)
\(7\)	−4.71049			−1.78040			−0.890198	−	0.455574i	\(-0.849434\pi\)
							−0.890198	+	0.455574i	\(0.849434\pi\)
\(11\)	1.43872	+	1.43872i	0.433791	+	0.433791i	0.889916	−	0.456125i	\(-0.150763\pi\)
							−0.456125	+	0.889916i	\(0.650763\pi\)
\(13\)	2.35845	−	2.35845i	0.654117	−	0.654117i	−0.299865	−	0.953982i	\(-0.596942\pi\)
							0.953982	+	0.299865i	\(0.0969415\pi\)
\(17\)		−	3.04660i		−	0.738909i	−0.929249	−	0.369455i	\(-0.879545\pi\)
							0.929249	−	0.369455i	\(-0.120455\pi\)
\(19\)	−3.59911	+	3.59911i	−0.825693	+	0.825693i	−0.986918	−	0.161225i	\(-0.948456\pi\)
							0.161225	+	0.986918i	\(0.448456\pi\)
\(23\)		−	3.02686i		−	0.631143i	−0.948902	−	0.315572i	\(-0.897804\pi\)
							0.948902	−	0.315572i	\(-0.102196\pi\)
\(29\)	6.96246	−	6.96246i	1.29290	−	1.29290i	0.359909	−	0.932988i	\(-0.382808\pi\)
							0.932988	−	0.359909i	\(-0.117192\pi\)
\(31\)	−5.91537			−1.06243			−0.531215	−	0.847237i	\(-0.678265\pi\)
							−0.531215	+	0.847237i	\(0.678265\pi\)
\(37\)	−1.93409	+	1.93409i	−0.317962	+	0.317962i	−0.847984	−	0.530022i	\(-0.822184\pi\)
							0.530022	+	0.847984i	\(0.322184\pi\)
\(41\)	5.86009	+	2.58056i	0.915193	+	0.403016i
							\(43\)	−3.26637	+	3.26637i	−0.498116	+	0.498116i	−0.910851	−	0.412735i	\(-0.864574\pi\)
0.412735	+	0.910851i	\(0.364574\pi\)
\(47\)		−	1.15228i		−	0.168078i	−0.996462	−	0.0840388i	\(-0.973218\pi\)
							0.996462	−	0.0840388i	\(-0.0267820\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	656.2.o.a.573.13	yes	164
16.5	even	4	inner	656.2.o.a.245.14	yes	164
41.40	even	2	inner	656.2.o.a.573.14	yes	164
656.245	even	4	inner	656.2.o.a.245.13	✓	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
656.2.o.a.245.13	✓	164	656.245	even	4	inner
656.2.o.a.245.14	yes	164	16.5	even	4	inner
656.2.o.a.573.13	yes	164	1.1	even	1	trivial
656.2.o.a.573.14	yes	164	41.40	even	2	inner