Embedded newform 20.6.e.b.3.5

• Label: 20.6.e.b.3.5
• Level: $20$
• Weight: $6$
• Character: 20.3
• Analytic conductor: $3.208$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	20.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.20767639626\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.5
Character	\(\chi\)	\(=\)	20.3
Dual form			20.6.e.b.7.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.41367 + 5.11607i) q^{2} +(16.4802 - 16.4802i) q^{3} +(-20.3484 - 24.6970i) q^{4} +(21.8249 - 51.4653i) q^{5} +(44.5361 + 124.092i) q^{6} +(78.4623 + 78.4623i) q^{7} +(175.466 - 44.4934i) q^{8} -300.193i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 10 q^{2} - 80 q^{5} - 184 q^{6} + 500 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.41367	+	5.11607i	−0.426681	+	0.904402i
							\(3\)	16.4802	−	16.4802i	1.05720	−	1.05720i	0.0589430	−	0.998261i	\(-0.481227\pi\)
0.998261	−	0.0589430i	\(-0.0187730\pi\)
\(5\)	21.8249	−	51.4653i	0.390416	−	0.920639i
							\(7\)	78.4623	+	78.4623i	0.605224	+	0.605224i	0.941694	−	0.336470i	\(-0.109233\pi\)
−0.336470	+	0.941694i	\(0.609233\pi\)
\(11\)			208.153i			0.518683i	0.965786	+	0.259341i	\(0.0835054\pi\)
							−0.965786	+	0.259341i	\(0.916495\pi\)
\(13\)	−401.203	−	401.203i	−0.658425	−	0.658425i	0.296582	−	0.955007i	\(-0.404153\pi\)
							−0.955007	+	0.296582i	\(0.904153\pi\)
\(17\)	−515.793	+	515.793i	−0.432865	+	0.432865i	−0.889602	−	0.456737i	\(-0.849018\pi\)
							0.456737	+	0.889602i	\(0.349018\pi\)
\(19\)	2467.80			1.56829			0.784145	−	0.620578i	\(-0.213102\pi\)
							0.784145	+	0.620578i	\(0.213102\pi\)
\(23\)	−2925.63	+	2925.63i	−1.15319	+	1.15319i	−0.167276	+	0.985910i	\(0.553497\pi\)
							−0.985910	+	0.167276i	\(0.946503\pi\)
\(29\)			6391.35i			1.41123i	0.708596	+	0.705614i	\(0.249329\pi\)
							−0.708596	+	0.705614i	\(0.750671\pi\)
\(31\)			556.567i			0.104019i	0.998647	+	0.0520095i	\(0.0165626\pi\)
							−0.998647	+	0.0520095i	\(0.983437\pi\)
\(37\)	−1963.77	+	1963.77i	−0.235823	+	0.235823i	−0.815118	−	0.579295i	\(-0.803328\pi\)
							0.579295	+	0.815118i	\(0.303328\pi\)
\(41\)	6139.41			0.570383			0.285192	−	0.958471i	\(-0.407943\pi\)
							0.285192	+	0.958471i	\(0.407943\pi\)
\(43\)	7375.44	−	7375.44i	0.608298	−	0.608298i	−0.334203	−	0.942501i	\(-0.608467\pi\)
							0.942501	+	0.334203i	\(0.108467\pi\)
\(47\)	825.227	+	825.227i	0.0544915	+	0.0544915i	0.733827	−	0.679336i	\(-0.237732\pi\)
							−0.679336	+	0.733827i	\(0.737732\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.6.e.b.3.5	yes	24
4.3	odd	2	inner	20.6.e.b.3.2	✓	24
5.2	odd	4	inner	20.6.e.b.7.2	yes	24
5.3	odd	4		100.6.e.e.7.11		24
5.4	even	2		100.6.e.e.43.8		24
20.3	even	4		100.6.e.e.7.8		24
20.7	even	4	inner	20.6.e.b.7.5	yes	24
20.19	odd	2		100.6.e.e.43.11		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.6.e.b.3.2	✓	24	4.3	odd	2	inner
20.6.e.b.3.5	yes	24	1.1	even	1	trivial
20.6.e.b.7.2	yes	24	5.2	odd	4	inner
20.6.e.b.7.5	yes	24	20.7	even	4	inner
100.6.e.e.7.8		24	20.3	even	4
100.6.e.e.7.11		24	5.3	odd	4
100.6.e.e.43.8		24	5.4	even	2
100.6.e.e.43.11		24	20.19	odd	2