Embedded newform 20.10.e.b.3.7

• Label: 20.10.e.b.3.7
• Level: $20$
• Weight: $10$
• Character: 20.3
• Analytic conductor: $10.301$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	20.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3007167233\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.7
Character	\(\chi\)	\(=\)	20.3
Dual form			20.10.e.b.7.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.96719 - 20.3139i) q^{2} +(181.251 - 181.251i) q^{3} +(-313.310 + 404.945i) q^{4} +(-997.734 - 978.597i) q^{5} +(-5488.49 - 1875.36i) q^{6} +(-1096.38 - 1096.38i) q^{7} +(11348.8 + 2328.39i) q^{8} -46021.0i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 30 q^{2} - 1440 q^{5} + 6152 q^{6} - 17100 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.96719	−	20.3139i	−0.440492	−	0.897757i
							\(3\)	181.251	−	181.251i	1.29192	−	1.29192i	0.358321	−	0.933599i	\(-0.383349\pi\)
0.933599	−	0.358321i	\(-0.116651\pi\)
\(5\)	−997.734	−	978.597i	−0.713920	−	0.700227i
							\(7\)	−1096.38	−	1096.38i	−0.172592	−	0.172592i	0.615525	−	0.788117i	\(-0.288944\pi\)
−0.788117	+	0.615525i	\(0.788944\pi\)
\(11\)			24135.6i			0.497041i	0.968627	+	0.248520i	\(0.0799443\pi\)
							−0.968627	+	0.248520i	\(0.920056\pi\)
\(13\)	−34010.0	−	34010.0i	−0.330264	−	0.330264i	0.522422	−	0.852687i	\(-0.325028\pi\)
							−0.852687	+	0.522422i	\(0.825028\pi\)
\(17\)	−348629.	+	348629.i	−1.01238	+	1.01238i	−0.0124561	+	0.999922i	\(0.503965\pi\)
							−0.999922	+	0.0124561i	\(0.996035\pi\)
\(19\)	764214.			1.34531			0.672657	−	0.739954i	\(-0.265153\pi\)
							0.672657	+	0.739954i	\(0.265153\pi\)
\(23\)	27449.8	−	27449.8i	0.0204533	−	0.0204533i	−0.696806	−	0.717259i	\(-0.745396\pi\)
							0.717259	+	0.696806i	\(0.245396\pi\)
\(29\)		−	5.01486e6i		−	1.31664i	−0.752737	−	0.658321i	\(-0.771267\pi\)
							0.752737	−	0.658321i	\(-0.228733\pi\)
\(31\)		−	7.76401e6i		−	1.50993i	−0.655762	−	0.754967i	\(-0.727653\pi\)
							0.655762	−	0.754967i	\(-0.272347\pi\)
\(37\)	3.31907e6	−	3.31907e6i	0.291144	−	0.291144i	−0.546388	−	0.837532i	\(-0.683998\pi\)
							0.837532	+	0.546388i	\(0.183998\pi\)
\(41\)	−9.42386e6			−0.520837			−0.260418	−	0.965496i	\(-0.583860\pi\)
							−0.260418	+	0.965496i	\(0.583860\pi\)
\(43\)	1.86665e7	−	1.86665e7i	0.832634	−	0.832634i	−0.155242	−	0.987876i	\(-0.549616\pi\)
							0.987876	+	0.155242i	\(0.0496158\pi\)
\(47\)	−1.31878e7	−	1.31878e7i	−0.394213	−	0.394213i	0.481973	−	0.876186i	\(-0.339920\pi\)
							−0.876186	+	0.481973i	\(0.839920\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.10.e.b.3.7	✓	48
4.3	odd	2	inner	20.10.e.b.3.20	yes	48
5.2	odd	4	inner	20.10.e.b.7.20	yes	48
5.3	odd	4		100.10.e.e.7.5		48
5.4	even	2		100.10.e.e.43.18		48
20.3	even	4		100.10.e.e.7.18		48
20.7	even	4	inner	20.10.e.b.7.7	yes	48
20.19	odd	2		100.10.e.e.43.5		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.10.e.b.3.7	✓	48	1.1	even	1	trivial
20.10.e.b.3.20	yes	48	4.3	odd	2	inner
20.10.e.b.7.7	yes	48	20.7	even	4	inner
20.10.e.b.7.20	yes	48	5.2	odd	4	inner
100.10.e.e.7.5		48	5.3	odd	4
100.10.e.e.7.18		48	20.3	even	4
100.10.e.e.43.5		48	20.19	odd	2
100.10.e.e.43.18		48	5.4	even	2