Newform orbit 50.4.b.a

• Label: 50.4.b.a
• Level: $50$
• Weight: $4$
• Character orbit: 50.b
• Analytic conductor: $2.950$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.95009550029\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} + 4 \beta q^{3} - 4 q^{4} - 16 q^{6} - 2 \beta q^{7} - 4 \beta q^{8} - 37 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 32 q^{6} - 74 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\)	\(27\)
\(\chi(n)\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

49.1

	−	1.00000i
		1.00000i

−

2.00000i

−

8.00000i

−4.00000

0

−16.0000

4.00000i

8.00000i

−37.0000

0

49.2

2.00000i

8.00000i

−4.00000

0

−16.0000

−

4.00000i

−

8.00000i

−37.0000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	50.4.b.a		2
3.b	odd	2	1		450.4.c.d		2
4.b	odd	2	1		400.4.c.c		2
5.b	even	2	1	inner	50.4.b.a		2
5.c	odd	4	1		10.4.a.a	✓	1
5.c	odd	4	1		50.4.a.c		1
15.d	odd	2	1		450.4.c.d		2
15.e	even	4	1		90.4.a.a		1
15.e	even	4	1		450.4.a.q		1
20.d	odd	2	1		400.4.c.c		2
20.e	even	4	1		80.4.a.f		1
20.e	even	4	1		400.4.a.b		1
35.f	even	4	1		490.4.a.o		1
35.f	even	4	1		2450.4.a.b		1
35.k	even	12	2		490.4.e.a		2
35.l	odd	12	2		490.4.e.i		2
40.i	odd	4	1		320.4.a.m		1
40.i	odd	4	1		1600.4.a.d		1
40.k	even	4	1		320.4.a.b		1
40.k	even	4	1		1600.4.a.bx		1
45.k	odd	12	2		810.4.e.c		2
45.l	even	12	2		810.4.e.w		2
55.e	even	4	1		1210.4.a.b		1
60.l	odd	4	1		720.4.a.j		1
65.h	odd	4	1		1690.4.a.a		1
80.i	odd	4	1		1280.4.d.j		2
80.j	even	4	1		1280.4.d.g		2
80.s	even	4	1		1280.4.d.g		2
80.t	odd	4	1		1280.4.d.j		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
10.4.a.a	✓	1	5.c	odd	4	1
50.4.a.c		1	5.c	odd	4	1
50.4.b.a		2	1.a	even	1	1	trivial
50.4.b.a		2	5.b	even	2	1	inner
80.4.a.f		1	20.e	even	4	1
90.4.a.a		1	15.e	even	4	1
320.4.a.b		1	40.k	even	4	1
320.4.a.m		1	40.i	odd	4	1
400.4.a.b		1	20.e	even	4	1
400.4.c.c		2	4.b	odd	2	1
400.4.c.c		2	20.d	odd	2	1
450.4.a.q		1	15.e	even	4	1
450.4.c.d		2	3.b	odd	2	1
450.4.c.d		2	15.d	odd	2	1
490.4.a.o		1	35.f	even	4	1
490.4.e.a		2	35.k	even	12	2
490.4.e.i		2	35.l	odd	12	2
720.4.a.j		1	60.l	odd	4	1
810.4.e.c		2	45.k	odd	12	2
810.4.e.w		2	45.l	even	12	2
1210.4.a.b		1	55.e	even	4	1
1280.4.d.g		2	80.j	even	4	1
1280.4.d.g		2	80.s	even	4	1
1280.4.d.j		2	80.i	odd	4	1
1280.4.d.j		2	80.t	odd	4	1
1600.4.a.d		1	40.i	odd	4	1
1600.4.a.bx		1	40.k	even	4	1
1690.4.a.a		1	65.h	odd	4	1
2450.4.a.b		1	35.f	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 64 \) acting on \(S_{4}^{\mathrm{new}}(50, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 4 \)
$3$	\( T^{2} + 64 \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 16 \)
$11$	\( (T - 12)^{2} \)