Embedded newform 164.2.o.b.135.18

• Label: 164.2.o.b.135.18
• Level: $164$
• Weight: $2$
• Character: 164.135
• Analytic conductor: $1.310$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• 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			135.18
Character	\(\chi\)	\(=\)	164.135
Dual form			164.2.o.b.147.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.35658 - 0.399600i) q^{2} +(-0.663681 + 1.60227i) q^{3} +(1.68064 - 1.08418i) q^{4} +(1.22577 - 0.624560i) q^{5} +(-0.260072 + 2.43882i) q^{6} +(-1.33847 + 0.820213i) q^{7} +(1.84669 - 2.14237i) q^{8} +(-0.00546890 - 0.00546890i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.35658	−	0.399600i	0.959250	−	0.282560i
							\(3\)	−0.663681	+	1.60227i	−0.383176	+	0.925070i	0.608171	+	0.793806i	\(0.291903\pi\)
−0.991347	+	0.131264i	\(0.958097\pi\)
\(5\)	1.22577	−	0.624560i	0.548180	−	0.279312i	−0.157886	−	0.987457i	\(-0.550468\pi\)
							0.706066	+	0.708146i	\(0.250468\pi\)
\(7\)	−1.33847	+	0.820213i	−0.505893	+	0.310012i	−0.751923	−	0.659251i	\(-0.770874\pi\)
							0.246030	+	0.969262i	\(0.420874\pi\)
\(11\)	−0.0312438	+	0.396991i	−0.00942037	+	0.119697i	−0.999829	−	0.0184690i	\(-0.994121\pi\)
							0.990409	+	0.138166i	\(0.0441208\pi\)
\(13\)	−1.80991	+	0.434521i	−0.501979	+	0.120515i	−0.476524	−	0.879161i	\(-0.658104\pi\)
							−0.0254552	+	0.999676i	\(0.508104\pi\)
\(17\)	−3.70473	−	4.33768i	−0.898529	−	1.05204i	−0.998372	−	0.0570395i	\(-0.981834\pi\)
							0.0998425	−	0.995003i	\(-0.468166\pi\)
\(19\)	0.0591700	−	0.246461i	0.0135745	−	0.0565420i	−0.965172	−	0.261618i	\(-0.915744\pi\)
							0.978746	+	0.205076i	\(0.0657441\pi\)
\(23\)	−0.527413	+	0.383188i	−0.109973	+	0.0799002i	−0.641413	−	0.767196i	\(-0.721651\pi\)
							0.531439	+	0.847096i	\(0.321651\pi\)
\(29\)	1.35306	−	1.58423i	0.251257	−	0.294184i	−0.620415	−	0.784274i	\(-0.713036\pi\)
							0.871672	+	0.490089i	\(0.163036\pi\)
\(31\)	−3.22828	−	9.93563i	−0.579817	−	1.78449i	−0.619158	−	0.785266i	\(-0.712526\pi\)
							0.0393412	−	0.999226i	\(-0.487474\pi\)
\(37\)	−0.115544	+	0.355607i	−0.0189953	+	0.0584614i	−0.960105	−	0.279640i	\(-0.909785\pi\)
							0.941110	+	0.338102i	\(0.109785\pi\)
\(41\)	0.0405979	+	6.40300i	0.00634033	+	0.999980i
							\(43\)	2.48784	+	0.394034i	0.379391	+	0.0600897i	0.343218	−	0.939256i	\(-0.388483\pi\)
0.0361735	+	0.999346i	\(0.488483\pi\)
\(47\)	−2.26558	−	1.38835i	−0.330468	−	0.202511i	0.347359	−	0.937732i	\(-0.387078\pi\)
							−0.677828	+	0.735221i	\(0.737078\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.135.18	yes	288
4.3	odd	2	inner	164.2.o.b.135.4	✓	288
41.24	odd	40	inner	164.2.o.b.147.4	yes	288
164.147	even	40	inner	164.2.o.b.147.18	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.135.4	✓	288	4.3	odd	2	inner
164.2.o.b.135.18	yes	288	1.1	even	1	trivial
164.2.o.b.147.4	yes	288	41.24	odd	40	inner
164.2.o.b.147.18	yes	288	164.147	even	40	inner