Newform orbit 1600.3.b.g

• Label: 1600.3.b.g
• Level: $1600$
• Weight: $3$
• Character orbit: 1600.b
• Analytic conductor: $43.597$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(43.5968422976\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-15}) \)
Defining polynomial:	\( x^{2} - x + 4 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta q^{3} - 2 \beta q^{7} - 6 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\)	\(577\)	\(901\)	\(1151\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1151.1

0.500000	+	1.93649i
0.500000	−	1.93649i

0

−

3.87298i

0

0

0

−

7.74597i

0

−6.00000

0

1151.2

0

3.87298i

0

0

0

7.74597i

0

−6.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1600.3.b.g		2
4.b	odd	2	1	inner	1600.3.b.g		2
5.b	even	2	1		1600.3.b.h		2
5.c	odd	4	2		1600.3.h.i		4
8.b	even	2	1		400.3.b.d	yes	2
8.d	odd	2	1		400.3.b.d	yes	2
20.d	odd	2	1		1600.3.b.h		2
20.e	even	4	2		1600.3.h.i		4
24.f	even	2	1		3600.3.e.v		2
24.h	odd	2	1		3600.3.e.v		2
40.e	odd	2	1		400.3.b.c	✓	2
40.f	even	2	1		400.3.b.c	✓	2
40.i	odd	4	2		400.3.h.c		4
40.k	even	4	2		400.3.h.c		4
120.i	odd	2	1		3600.3.e.i		2
120.m	even	2	1		3600.3.e.i		2
120.q	odd	4	2		3600.3.j.b		4
120.w	even	4	2		3600.3.j.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
400.3.b.c	✓	2	40.e	odd	2	1
400.3.b.c	✓	2	40.f	even	2	1
400.3.b.d	yes	2	8.b	even	2	1
400.3.b.d	yes	2	8.d	odd	2	1
400.3.h.c		4	40.i	odd	4	2
400.3.h.c		4	40.k	even	4	2
1600.3.b.g		2	1.a	even	1	1	trivial
1600.3.b.g		2	4.b	odd	2	1	inner
1600.3.b.h		2	5.b	even	2	1
1600.3.b.h		2	20.d	odd	2	1
1600.3.h.i		4	5.c	odd	4	2
1600.3.h.i		4	20.e	even	4	2
3600.3.e.i		2	120.i	odd	2	1
3600.3.e.i		2	120.m	even	2	1
3600.3.e.v		2	24.f	even	2	1
3600.3.e.v		2	24.h	odd	2	1
3600.3.j.b		4	120.q	odd	4	2
3600.3.j.b		4	120.w	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{2} + 15 \)

\( T_{7}^{2} + 60 \)

\( T_{13} + 20 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 15 \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 60 \)
$11$	\( T^{2} + 375 \)