Newform orbit 1020.2.s.a

• Label: 1020.2.s.a
• Level: $1020$
• Weight: $2$
• Character orbit: 1020.s
• Analytic conductor: $8.145$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1020.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.14474100617\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 8 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
353.1		0		−1.71576	−	0.237011i		0		−0.244881	+	2.22262i		0				3.72738i		0		2.88765	+	0.813305i		0
353.2		0		−1.71240	+	0.260150i		0		0.131237	−	2.23221i		0			−	2.09334i		0		2.86464	−	0.890962i		0
353.3		0		−1.68197	+	0.413492i		0		2.22895	+	0.178237i		0				0.402897i		0		2.65805	−	1.39096i		0
353.4		0		−1.63716	−	0.565439i		0		1.21902	+	1.87457i		0			−	2.47580i		0		2.36056	+	1.85142i		0
353.5		0		−1.59708	−	0.670321i		0		−1.46551	−	1.68887i		0			−	1.08017i		0		2.10134	+	2.14112i		0
353.6		0		−1.59499	+	0.675275i		0		−2.07492	+	0.833484i		0			−	3.80262i		0		2.08801	−	2.15412i		0
353.7		0		−1.25453	+	1.19421i		0		−1.58050	+	1.58177i		0				0.890151i		0		0.147702	−	2.99636i		0
353.8		0		−1.24592	−	1.20320i		0		2.08847	−	0.798923i		0			−	0.0977078i		0		0.104609	+	2.99818i		0
353.9		0		−1.16152	−	1.28486i		0		−2.23397	+	0.0967453i		0			−	1.31829i		0		−0.301733	+	2.98479i		0
353.10		0		−1.14742	+	1.29747i		0		2.18542	+	0.473212i		0				4.56623i		0		−0.366853	−	2.97749i		0
353.11		0		−1.09796	+	1.33959i		0		−1.55650	−	1.60540i		0				1.62877i		0		−0.588988	−	2.94161i		0
353.12		0		−1.09282	−	1.34378i		0		1.09049	−	1.95213i		0				4.41760i		0		−0.611506	+	2.93702i		0
353.13		0		−0.899531	−	1.48015i		0		−1.91305	+	1.15768i		0				0.446571i		0		−1.38169	+	2.66288i		0
353.14		0		−0.473940	+	1.66595i		0		0.993178	−	2.00340i		0				2.59004i		0		−2.55076	−	1.57912i		0
353.15		0		−0.451969	+	1.67204i		0		2.23595	−	0.0229198i		0			−	4.01673i		0		−2.59145	−	1.51142i		0
353.16		0		−0.217767	+	1.71831i		0		0.571668	+	2.16176i		0			−	1.96685i		0		−2.90515	−	0.748382i		0
353.17		0		−0.192771	−	1.72129i		0		0.482536	−	2.18338i		0			−	4.76163i		0		−2.92568	+	0.663629i		0
353.18		0		−0.0749995	−	1.73043i		0		1.43943	+	1.71115i		0				2.94350i		0		−2.98875	+	0.259562i		0
353.19		0		0.0749995	−	1.73043i		0		−1.43943	−	1.71115i		0				2.94350i		0		−2.98875	−	0.259562i		0
353.20		0		0.192771	−	1.72129i		0		−0.482536	+	2.18338i		0			−	4.76163i		0		−2.92568	−	0.663629i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
85.f	odd	4	1	inner
255.k	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1020.2.s.a	✓	72
3.b	odd	2	1	inner	1020.2.s.a	✓	72
5.c	odd	4	1		1020.2.bn.a	yes	72
15.e	even	4	1		1020.2.bn.a	yes	72
17.c	even	4	1		1020.2.bn.a	yes	72
51.f	odd	4	1		1020.2.bn.a	yes	72
85.f	odd	4	1	inner	1020.2.s.a	✓	72
255.k	even	4	1	inner	1020.2.s.a	✓	72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1020.2.s.a	✓	72	1.a	even	1	1	trivial
1020.2.s.a	✓	72	3.b	odd	2	1	inner
1020.2.s.a	✓	72	85.f	odd	4	1	inner
1020.2.s.a	✓	72	255.k	even	4	1	inner
1020.2.bn.a	yes	72	5.c	odd	4	1
1020.2.bn.a	yes	72	15.e	even	4	1
1020.2.bn.a	yes	72	17.c	even	4	1
1020.2.bn.a	yes	72	51.f	odd	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1020, [\chi])\).