Embedded newform 164.2.o.b.135.3

• Label: 164.2.o.b.135.3
• 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			135.3
Character	\(\chi\)	\(=\)	164.135
Dual form			164.2.o.b.147.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23324 + 0.692177i) q^{2} +(1.26929 - 3.06434i) q^{3} +(1.04178 - 1.70725i) q^{4} +(2.22680 - 1.13461i) q^{5} +(0.555720 + 4.65765i) q^{6} +(-1.69374 + 1.03793i) q^{7} +(-0.103055 + 2.82655i) q^{8} +(-5.65776 - 5.65776i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23324	+	0.692177i	−0.872035	+	0.489443i
							\(3\)	1.26929	−	3.06434i	0.732826	−	1.76920i	0.0998835	−	0.994999i	\(-0.468153\pi\)
0.632942	−	0.774199i	\(-0.281847\pi\)
\(5\)	2.22680	−	1.13461i	0.995857	−	0.507415i	0.121445	−	0.992598i	\(-0.461247\pi\)
							0.874412	+	0.485184i	\(0.161247\pi\)
\(7\)	−1.69374	+	1.03793i	−0.640174	+	0.392299i	−0.804370	−	0.594129i	\(-0.797497\pi\)
							0.164196	+	0.986428i	\(0.447497\pi\)
\(11\)	−0.211159	+	2.68304i	−0.0636670	+	0.808966i	0.879340	+	0.476194i	\(0.157984\pi\)
							−0.943007	+	0.332772i	\(0.892016\pi\)
\(13\)	0.213068	−	0.0511530i	0.0590943	−	0.0141873i	−0.203790	−	0.979015i	\(-0.565326\pi\)
							0.262885	+	0.964827i	\(0.415326\pi\)
\(17\)	2.71207	+	3.17543i	0.657774	+	0.770154i	0.984580	−	0.174937i	\(-0.0559722\pi\)
							−0.326806	+	0.945092i	\(0.605972\pi\)
\(19\)	−0.907341	+	3.77935i	−0.208158	+	0.867042i	0.766302	+	0.642481i	\(0.222095\pi\)
							−0.974460	+	0.224561i	\(0.927905\pi\)
\(23\)	5.66038	−	4.11251i	1.18027	−	0.857517i	0.188068	−	0.982156i	\(-0.439777\pi\)
							0.992202	+	0.124639i	\(0.0397773\pi\)
\(29\)	3.78327	−	4.42964i	0.702535	−	0.822563i	−0.288314	−	0.957536i	\(-0.593095\pi\)
							0.990849	+	0.134973i	\(0.0430946\pi\)
\(31\)	−0.955574	−	2.94095i	−0.171626	−	0.528211i	0.827837	−	0.560968i	\(-0.189571\pi\)
							−0.999463	+	0.0327575i	\(0.989571\pi\)
\(37\)	−0.213779	+	0.657943i	−0.0351450	+	0.108165i	−0.967090	−	0.254434i	\(-0.918111\pi\)
							0.931945	+	0.362600i	\(0.118111\pi\)
\(41\)	−5.96458	−	2.32891i	−0.931511	−	0.363714i
							\(43\)	−4.94314	−	0.782917i	−0.753822	−	0.119394i	−0.232315	−	0.972641i	\(-0.574630\pi\)
−0.521507	+	0.853247i	\(0.674630\pi\)
\(47\)	8.85613	+	5.42704i	1.29180	+	0.791616i	0.986988	−	0.160797i	\(-0.0514063\pi\)
							0.304812	+	0.952413i	\(0.401406\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.135.3	✓	288
4.3	odd	2	inner	164.2.o.b.135.14	yes	288
41.24	odd	40	inner	164.2.o.b.147.14	yes	288
164.147	even	40	inner	164.2.o.b.147.3	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.135.3	✓	288	1.1	even	1	trivial
164.2.o.b.135.14	yes	288	4.3	odd	2	inner
164.2.o.b.147.3	yes	288	164.147	even	40	inner
164.2.o.b.147.14	yes	288	41.24	odd	40	inner