Embedded newform 656.2.o.a.245.8

• Label: 656.2.o.a.245.8
• 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			245.8
Character	\(\chi\)	\(=\)	656.245
Dual form			656.2.o.a.573.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37799 + 0.318038i) q^{2} +(1.64581 + 1.64581i) q^{3} +(1.79770 - 0.876504i) q^{4} +(1.05875 + 1.05875i) q^{5} +(-2.79134 - 1.74448i) q^{6} -0.937357 q^{7} +(-2.19845 + 1.77955i) q^{8} +2.41740i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 4 q^{2} - 4 q^{4} - 4 q^{5} - 16 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37799	+	0.318038i	−0.974385	+	0.224887i
							\(3\)	1.64581	+	1.64581i	0.950210	+	0.950210i	0.998818	−	0.0486078i	\(-0.0154785\pi\)
−0.0486078	+	0.998818i	\(0.515478\pi\)
\(5\)	1.05875	+	1.05875i	0.473487	+	0.473487i	0.903041	−	0.429554i	\(-0.141329\pi\)
							−0.429554	+	0.903041i	\(0.641329\pi\)
\(7\)	−0.937357			−0.354288			−0.177144	−	0.984185i	\(-0.556686\pi\)
							−0.177144	+	0.984185i	\(0.556686\pi\)
\(11\)	3.74605	−	3.74605i	1.12948	−	1.12948i	0.139216	−	0.990262i	\(-0.455542\pi\)
							0.990262	−	0.139216i	\(-0.0444582\pi\)
\(13\)	−0.659366	−	0.659366i	−0.182875	−	0.182875i	0.609732	−	0.792607i	\(-0.291277\pi\)
							−0.792607	+	0.609732i	\(0.791277\pi\)
\(17\)			5.56372i			1.34940i	0.738092	+	0.674700i	\(0.235727\pi\)
							−0.738092	+	0.674700i	\(0.764273\pi\)
\(19\)	4.91609	+	4.91609i	1.12783	+	1.12783i	0.990530	+	0.137298i	\(0.0438418\pi\)
							0.137298	+	0.990530i	\(0.456158\pi\)
\(23\)			7.92682i			1.65286i	0.563042	+	0.826428i	\(0.309631\pi\)
							−0.563042	+	0.826428i	\(0.690369\pi\)
\(29\)	−0.225032	−	0.225032i	−0.0417873	−	0.0417873i	0.685904	−	0.727692i	\(-0.259407\pi\)
							−0.727692	+	0.685904i	\(0.759407\pi\)
\(31\)	−6.94370			−1.24713			−0.623563	−	0.781773i	\(-0.714316\pi\)
							−0.623563	+	0.781773i	\(0.714316\pi\)
\(37\)	6.02825	+	6.02825i	0.991037	+	0.991037i	0.999960	−	0.00892271i	\(-0.00284022\pi\)
							−0.00892271	+	0.999960i	\(0.502840\pi\)
\(41\)	3.49691	−	5.36392i	0.546125	−	0.837704i
							\(43\)	−6.60609	−	6.60609i	−1.00742	−	1.00742i	−0.999972	−	0.00744625i	\(-0.997630\pi\)
−0.00744625	−	0.999972i	\(-0.502370\pi\)
\(47\)			2.03944i			0.297482i	0.988876	+	0.148741i	\(0.0475221\pi\)
							−0.988876	+	0.148741i	\(0.952478\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	656.2.o.a.245.8	yes	164
16.13	even	4	inner	656.2.o.a.573.7	yes	164
41.40	even	2	inner	656.2.o.a.245.7	✓	164
656.573	even	4	inner	656.2.o.a.573.8	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
656.2.o.a.245.7	✓	164	41.40	even	2	inner
656.2.o.a.245.8	yes	164	1.1	even	1	trivial
656.2.o.a.573.7	yes	164	16.13	even	4	inner
656.2.o.a.573.8	yes	164	656.573	even	4	inner