Embedded newform 656.2.o.a.245.2

• Label: 656.2.o.a.245.2
• Level: $656$
• Weight: $2$
• Character: 656.245
• Analytic conductor: $5.238$
• Analytic rank: $0$
• Dimension: $164$
• CM: no
• 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			245.2
Character	\(\chi\)	\(=\)	656.245
Dual form			656.2.o.a.573.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41162 + 0.0856576i) q^{2} +(2.26252 + 2.26252i) q^{3} +(1.98533 - 0.241831i) q^{4} +(-1.97321 - 1.97321i) q^{5} +(-3.38762 - 3.00001i) q^{6} -2.05040 q^{7} +(-2.78180 + 0.511432i) q^{8} +7.23800i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 4 q^{2} - 4 q^{4} - 4 q^{5} - 16 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41162	+	0.0856576i	−0.998164	+	0.0605691i
							\(3\)	2.26252	+	2.26252i	1.30627	+	1.30627i	0.924088	+	0.382179i	\(0.124826\pi\)
0.382179	+	0.924088i	\(0.375174\pi\)
\(5\)	−1.97321	−	1.97321i	−0.882446	−	0.882446i	0.111336	−	0.993783i	\(-0.464487\pi\)
							−0.993783	+	0.111336i	\(0.964487\pi\)
\(7\)	−2.05040			−0.774977			−0.387488	−	0.921875i	\(-0.626657\pi\)
							−0.387488	+	0.921875i	\(0.626657\pi\)
\(11\)	−3.02726	+	3.02726i	−0.912755	+	0.912755i	−0.996488	−	0.0837335i	\(-0.973316\pi\)
							0.0837335	+	0.996488i	\(0.473316\pi\)
\(13\)	−3.75983	−	3.75983i	−1.04279	−	1.04279i	−0.999043	−	0.0437464i	\(-0.986071\pi\)
							−0.0437464	−	0.999043i	\(-0.513929\pi\)
\(17\)		−	1.85643i		−	0.450250i	−0.974330	−	0.225125i	\(-0.927721\pi\)
							0.974330	−	0.225125i	\(-0.0722791\pi\)
\(19\)	−4.54034	−	4.54034i	−1.04162	−	1.04162i	−0.999095	−	0.0425294i	\(-0.986458\pi\)
							−0.0425294	−	0.999095i	\(-0.513542\pi\)
\(23\)			8.60390i			1.79404i	0.441993	+	0.897018i	\(0.354272\pi\)
							−0.441993	+	0.897018i	\(0.645728\pi\)
\(29\)	−0.660365	−	0.660365i	−0.122627	−	0.122627i	0.643130	−	0.765757i	\(-0.277635\pi\)
							−0.765757	+	0.643130i	\(0.777635\pi\)
\(31\)	4.09217			0.734976			0.367488	−	0.930028i	\(-0.380218\pi\)
							0.367488	+	0.930028i	\(0.380218\pi\)
\(37\)	−4.46637	−	4.46637i	−0.734266	−	0.734266i	0.237196	−	0.971462i	\(-0.423772\pi\)
							−0.971462	+	0.237196i	\(0.923772\pi\)
\(41\)	−5.19828	+	3.73870i	−0.811835	+	0.583887i
							\(43\)	3.31521	+	3.31521i	0.505566	+	0.505566i	0.913162	−	0.407597i	\(-0.133633\pi\)
−0.407597	+	0.913162i	\(0.633633\pi\)
\(47\)			4.72653i			0.689435i	0.938707	+	0.344717i	\(0.112025\pi\)
							−0.938707	+	0.344717i	\(0.887975\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	656.2.o.a.245.2	yes	164
16.13	even	4	inner	656.2.o.a.573.1	yes	164
41.40	even	2	inner	656.2.o.a.245.1	✓	164
656.573	even	4	inner	656.2.o.a.573.2	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
656.2.o.a.245.1	✓	164	41.40	even	2	inner
656.2.o.a.245.2	yes	164	1.1	even	1	trivial
656.2.o.a.573.1	yes	164	16.13	even	4	inner
656.2.o.a.573.2	yes	164	656.573	even	4	inner