Embedded newform 164.2.o.b.11.5

• Label: 164.2.o.b.11.5
• Level: $164$
• Weight: $2$
• Character: 164.11
• 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			11.5
Character	\(\chi\)	\(=\)	164.11
Dual form			164.2.o.b.15.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00048 + 0.999519i) q^{2} +(1.13045 - 2.72914i) q^{3} +(0.00192156 - 2.00000i) q^{4} +(-0.430197 + 0.844309i) q^{5} +(1.59684 + 3.86035i) q^{6} +(0.0257188 + 0.107126i) q^{7} +(1.99712 + 2.00288i) q^{8} +(-4.04896 - 4.04896i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00048	+	0.999519i	−0.707446	+	0.706767i
							\(3\)	1.13045	−	2.72914i	0.652663	−	1.57567i	−0.156235	−	0.987720i	\(-0.549936\pi\)
0.808899	−	0.587948i	\(-0.200064\pi\)
\(5\)	−0.430197	+	0.844309i	−0.192390	+	0.377587i	−0.966970	−	0.254889i	\(-0.917961\pi\)
							0.774580	+	0.632475i	\(0.217961\pi\)
\(7\)	0.0257188	+	0.107126i	0.00972079	+	0.0404900i	0.977020	−	0.213150i	\(-0.0683722\pi\)
							−0.967299	+	0.253640i	\(0.918372\pi\)
\(11\)	3.23203	−	3.78422i	0.974494	−	1.14099i	−0.0154999	−	0.999880i	\(-0.504934\pi\)
							0.989994	−	0.141107i	\(-0.0450660\pi\)
\(13\)	−2.01034	−	3.28058i	−0.557569	−	0.909870i	−0.999915	−	0.0130254i	\(-0.995854\pi\)
							0.442346	−	0.896844i	\(-0.354146\pi\)
\(17\)	0.108842	+	1.38297i	0.0263981	+	0.335420i	0.995908	+	0.0903689i	\(0.0288046\pi\)
							−0.969510	+	0.245051i	\(0.921195\pi\)
\(19\)	−3.34801	−	2.05166i	−0.768085	−	0.470683i	0.0824541	−	0.996595i	\(-0.473724\pi\)
							−0.850539	+	0.525912i	\(0.823724\pi\)
\(23\)	6.86039	+	4.98437i	1.43049	+	1.03931i	0.989925	+	0.141592i	\(0.0452220\pi\)
							0.440565	+	0.897720i	\(0.354778\pi\)
\(29\)	−0.498000	+	6.32769i	−0.0924762	+	1.17502i	0.759166	+	0.650898i	\(0.225607\pi\)
							−0.851642	+	0.524124i	\(0.824393\pi\)
\(31\)	−0.564960	+	1.73877i	−0.101470	+	0.312292i	−0.988886	−	0.148678i	\(-0.952498\pi\)
							0.887416	+	0.460970i	\(0.152498\pi\)
\(37\)	0.756379	+	2.32790i	0.124348	+	0.382704i	0.993782	−	0.111346i	\(-0.0355161\pi\)
							−0.869434	+	0.494050i	\(0.835516\pi\)
\(41\)	−0.811348	+	6.35151i	−0.126711	+	0.991940i
							\(43\)	−0.666293	−	4.20681i	−0.101609	−	0.641532i	−0.984955	−	0.172811i	\(-0.944715\pi\)
0.883346	−	0.468721i	\(-0.155285\pi\)
\(47\)	0.398882	−	1.66146i	0.0581828	−	0.242349i	−0.935699	−	0.352800i	\(-0.885230\pi\)
							0.993882	+	0.110451i	\(0.0352296\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.11.5	yes	288
4.3	odd	2	inner	164.2.o.b.11.2	✓	288
41.15	odd	40	inner	164.2.o.b.15.2	yes	288
164.15	even	40	inner	164.2.o.b.15.5	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.11.2	✓	288	4.3	odd	2	inner
164.2.o.b.11.5	yes	288	1.1	even	1	trivial
164.2.o.b.15.2	yes	288	41.15	odd	40	inner
164.2.o.b.15.5	yes	288	164.15	even	40	inner