Embedded newform 164.2.o.b.135.1

• Label: 164.2.o.b.135.1
• Level: $164$
• Weight: $2$
• Character: 164.135
• Analytic conductor: $1.310$
• Analytic rank: $0$
• Dimension: $288$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 164 = 2^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	164.o (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.30954659315\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(18\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			135.1
Character	\(\chi\)	\(=\)	164.135
Dual form			164.2.o.b.147.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41391 + 0.0293664i) q^{2} +(0.353906 - 0.854404i) q^{3} +(1.99828 - 0.0830427i) q^{4} +(-3.50074 + 1.78372i) q^{5} +(-0.475300 + 1.21844i) q^{6} +(-2.34559 + 1.43738i) q^{7} +(-2.82294 + 0.176097i) q^{8} +(1.51656 + 1.51656i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{27}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41391	+	0.0293664i	−0.999784	+	0.0207652i
							\(3\)	0.353906	−	0.854404i	0.204328	−	0.493290i	−0.788184	−	0.615439i	\(-0.788979\pi\)
0.992512	+	0.122149i	\(0.0389786\pi\)
\(5\)	−3.50074	+	1.78372i	−1.56558	+	0.797702i	−0.999642	−	0.0267476i	\(-0.991485\pi\)
							−0.565936	+	0.824449i	\(0.691485\pi\)
\(7\)	−2.34559	+	1.43738i	−0.886549	+	0.543278i	−0.889612	−	0.456718i	\(-0.849025\pi\)
							0.00306302	+	0.999995i	\(0.499025\pi\)
\(11\)	−0.158678	+	2.01620i	−0.0478433	+	0.607907i	0.925529	+	0.378677i	\(0.123621\pi\)
							−0.973372	+	0.229230i	\(0.926379\pi\)
\(13\)	−0.0579378	+	0.0139096i	−0.0160691	+	0.00385784i	−0.241472	−	0.970408i	\(-0.577630\pi\)
							0.225403	+	0.974266i	\(0.427630\pi\)
\(17\)	−2.54830	−	2.98367i	−0.618053	−	0.723647i	0.359779	−	0.933038i	\(-0.382852\pi\)
							−0.977832	+	0.209390i	\(0.932852\pi\)
\(19\)	−1.87210	+	7.79786i	−0.429489	+	1.78895i	0.166589	+	0.986026i	\(0.446725\pi\)
							−0.596079	+	0.802926i	\(0.703275\pi\)
\(23\)	−2.22493	+	1.61651i	−0.463931	+	0.337065i	−0.795071	−	0.606516i	\(-0.792567\pi\)
							0.331141	+	0.943581i	\(0.392567\pi\)
\(29\)	−3.97267	+	4.65140i	−0.737706	+	0.863743i	−0.994716	−	0.102666i	\(-0.967263\pi\)
							0.257010	+	0.966409i	\(0.417263\pi\)
\(31\)	−2.29609	−	7.06665i	−0.412390	−	1.26921i	−0.914564	−	0.404441i	\(-0.867466\pi\)
							0.502174	−	0.864767i	\(-0.332534\pi\)
\(37\)	0.859166	−	2.64424i	0.141246	−	0.434711i	−0.855263	−	0.518194i	\(-0.826605\pi\)
							0.996509	+	0.0834833i	\(0.0266045\pi\)
\(41\)	4.45739	−	4.59692i	0.696128	−	0.717918i
							\(43\)	3.59693	+	0.569698i	0.548527	+	0.0868782i	0.424546	−	0.905406i	\(-0.360434\pi\)
0.123981	+	0.992285i	\(0.460434\pi\)
\(47\)	4.73066	+	2.89895i	0.690037	+	0.422855i	0.822829	−	0.568288i	\(-0.192394\pi\)
							−0.132792	+	0.991144i	\(0.542394\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.135.1	✓	288
4.3	odd	2	inner	164.2.o.b.135.17	yes	288
41.24	odd	40	inner	164.2.o.b.147.17	yes	288
164.147	even	40	inner	164.2.o.b.147.1	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.135.1	✓	288	1.1	even	1	trivial
164.2.o.b.135.17	yes	288	4.3	odd	2	inner
164.2.o.b.147.1	yes	288	164.147	even	40	inner
164.2.o.b.147.17	yes	288	41.24	odd	40	inner