Embedded newform 164.2.o.b.11.9

• Label: 164.2.o.b.11.9
• Level: $164$
• Weight: $2$
• Character: 164.11
• 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			11.9
Character	\(\chi\)	\(=\)	164.11
Dual form			164.2.o.b.15.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.00473161 - 1.41421i) q^{2} +(-0.494050 + 1.19274i) q^{3} +(-1.99996 + 0.0133829i) q^{4} +(-1.78033 + 3.49410i) q^{5} +(1.68912 + 0.693045i) q^{6} +(-0.375912 - 1.56578i) q^{7} +(0.0283892 + 2.82828i) q^{8} +(0.942771 + 0.942771i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.00473161	−	1.41421i	−0.00334575	−	0.999994i
							\(3\)	−0.494050	+	1.19274i	−0.285240	+	0.688630i	−0.999942	−	0.0107991i	\(-0.996562\pi\)
0.714702	+	0.699429i	\(0.246562\pi\)
\(5\)	−1.78033	+	3.49410i	−0.796190	+	1.56261i	0.0302218	+	0.999543i	\(0.490379\pi\)
							−0.826411	+	0.563067i	\(0.809621\pi\)
\(7\)	−0.375912	−	1.56578i	−0.142081	−	0.591811i	−0.997293	−	0.0735319i	\(-0.976573\pi\)
							0.855212	−	0.518279i	\(-0.173427\pi\)
\(11\)	−2.53026	+	2.96255i	−0.762902	+	0.893243i	−0.996875	−	0.0789999i	\(-0.974827\pi\)
							0.233973	+	0.972243i	\(0.424827\pi\)
\(13\)	1.25641	+	2.05027i	0.348465	+	0.568643i	0.978071	−	0.208271i	\(-0.0667837\pi\)
							−0.629606	+	0.776914i	\(0.716784\pi\)
\(17\)	−0.283888	−	3.60714i	−0.0688530	−	0.874860i	−0.930250	−	0.366927i	\(-0.880410\pi\)
							0.861397	−	0.507933i	\(-0.169590\pi\)
\(19\)	−3.76422	−	2.30672i	−0.863571	−	0.529197i	0.0187614	−	0.999824i	\(-0.494028\pi\)
							−0.882332	+	0.470627i	\(0.844028\pi\)
\(23\)	7.12703	+	5.17809i	1.48609	+	1.07971i	0.975530	+	0.219865i	\(0.0705615\pi\)
							0.510559	+	0.859843i	\(0.329438\pi\)
\(29\)	0.227410	−	2.88952i	0.0422290	−	0.536571i	−0.939025	−	0.343850i	\(-0.888269\pi\)
							0.981254	−	0.192721i	\(-0.0617312\pi\)
\(31\)	0.405218	−	1.24713i	0.0727792	−	0.223991i	−0.908050	−	0.418863i	\(-0.862429\pi\)
							0.980829	+	0.194871i	\(0.0624289\pi\)
\(37\)	2.54198	+	7.82340i	0.417898	+	1.28616i	0.909633	+	0.415413i	\(0.136363\pi\)
							−0.491735	+	0.870745i	\(0.663637\pi\)
\(41\)	3.29581	+	5.48977i	0.514719	+	0.857359i
							\(43\)	−0.293420	−	1.85258i	−0.0447461	−	0.282516i	0.955162	−	0.296084i	\(-0.0956809\pi\)
−0.999908	+	0.0135683i	\(0.995681\pi\)
\(47\)	−1.22771	+	5.11377i	−0.179080	+	0.745920i	0.808460	+	0.588551i	\(0.200301\pi\)
							−0.987540	+	0.157369i	\(0.949699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.11.9	✓	288
4.3	odd	2	inner	164.2.o.b.11.12	yes	288
41.15	odd	40	inner	164.2.o.b.15.12	yes	288
164.15	even	40	inner	164.2.o.b.15.9	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.11.9	✓	288	1.1	even	1	trivial
164.2.o.b.11.12	yes	288	4.3	odd	2	inner
164.2.o.b.15.9	yes	288	164.15	even	40	inner
164.2.o.b.15.12	yes	288	41.15	odd	40	inner