Embedded newform 20.8.e.b.3.2

• Label: 20.8.e.b.3.2
• Level: $20$
• Weight: $8$
• Character: 20.3
• Analytic conductor: $6.248$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	20.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.24770050968\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.2
Character	\(\chi\)	\(=\)	20.3
Dual form			20.8.e.b.7.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.1445 - 1.94961i) q^{2} +(-31.8653 + 31.8653i) q^{3} +(120.398 + 43.4547i) q^{4} +(92.5871 + 263.728i) q^{5} +(417.246 - 292.996i) q^{6} +(-789.211 - 789.211i) q^{7} +(-1257.05 - 719.009i) q^{8} +156.211i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 14 q^{2} - 560 q^{5} + 104 q^{6} - 3052 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\)	\(11\)	\(17\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.1445	−	1.94961i	−0.985041	−	0.172323i
							\(3\)	−31.8653	+	31.8653i	−0.681386	+	0.681386i	−0.960312	−	0.278927i	\(-0.910021\pi\)
0.278927	+	0.960312i	\(0.410021\pi\)
\(5\)	92.5871	+	263.728i	0.331250	+	0.943543i
							\(7\)	−789.211	−	789.211i	−0.869661	−	0.869661i	0.122773	−	0.992435i	\(-0.460821\pi\)
−0.992435	+	0.122773i	\(0.960821\pi\)
\(11\)		−	4795.88i		−	1.08641i	−0.839600	−	0.543205i	\(-0.817211\pi\)
							0.839600	−	0.543205i	\(-0.182789\pi\)
\(13\)	−5751.69	−	5751.69i	−0.726095	−	0.726095i	0.243744	−	0.969840i	\(-0.421624\pi\)
							−0.969840	+	0.243744i	\(0.921624\pi\)
\(17\)	17067.3	−	17067.3i	0.842545	−	0.842545i	−0.146644	−	0.989189i	\(-0.546847\pi\)
							0.989189	+	0.146644i	\(0.0468473\pi\)
\(19\)	−8191.73			−0.273992			−0.136996	−	0.990572i	\(-0.543745\pi\)
							−0.136996	+	0.990572i	\(0.543745\pi\)
\(23\)	−29160.0	+	29160.0i	−0.499735	+	0.499735i	−0.911356	−	0.411620i	\(-0.864963\pi\)
							0.411620	+	0.911356i	\(0.364963\pi\)
\(29\)		−	236423.i		−	1.80010i	−0.435790	−	0.900049i	\(-0.643531\pi\)
							0.435790	−	0.900049i	\(-0.356469\pi\)
\(31\)			97046.0i			0.585075i	0.956254	+	0.292538i	\(0.0944997\pi\)
							−0.956254	+	0.292538i	\(0.905500\pi\)
\(37\)	−209979.	+	209979.i	−0.681505	+	0.681505i	−0.960339	−	0.278834i	\(-0.910052\pi\)
							0.278834	+	0.960339i	\(0.410052\pi\)
\(41\)	−265802.			−0.602303			−0.301152	−	0.953576i	\(-0.597371\pi\)
							−0.301152	+	0.953576i	\(0.597371\pi\)
\(43\)	−298328.	+	298328.i	−0.572208	+	0.572208i	−0.932745	−	0.360537i	\(-0.882593\pi\)
							0.360537	+	0.932745i	\(0.382593\pi\)
\(47\)	354810.	+	354810.i	0.498486	+	0.498486i	0.910967	−	0.412480i	\(-0.135337\pi\)
							−0.412480	+	0.910967i	\(0.635337\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.8.e.b.3.2	✓	36
4.3	odd	2	inner	20.8.e.b.3.10	yes	36
5.2	odd	4	inner	20.8.e.b.7.10	yes	36
5.3	odd	4		100.8.e.e.7.9		36
5.4	even	2		100.8.e.e.43.17		36
20.3	even	4		100.8.e.e.7.17		36
20.7	even	4	inner	20.8.e.b.7.2	yes	36
20.19	odd	2		100.8.e.e.43.9		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.8.e.b.3.2	✓	36	1.1	even	1	trivial
20.8.e.b.3.10	yes	36	4.3	odd	2	inner
20.8.e.b.7.2	yes	36	20.7	even	4	inner
20.8.e.b.7.10	yes	36	5.2	odd	4	inner
100.8.e.e.7.9		36	5.3	odd	4
100.8.e.e.7.17		36	20.3	even	4
100.8.e.e.43.9		36	20.19	odd	2
100.8.e.e.43.17		36	5.4	even	2