Newform orbit 3900.2.j.e

• Label: 3900.2.j.e
• Level: $3900$
• Weight: $2$
• Character orbit: 3900.j
• Analytic conductor: $31.142$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3900.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(31.1416567883\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 156)
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.

\(f(q)\)	\(=\)	\( q + i q^{3} + 4 q^{7} - q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{7} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times\).

\(n\)	\(301\)	\(1301\)	\(1951\)	\(3277\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

649.1

	−	1.00000i
		1.00000i

0

−

1.00000i

0

0

0

4.00000

0

−1.00000

0

649.2

0

1.00000i

0

0

0

4.00000

0

−1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
65.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3900.2.j.e		2
5.b	even	2	1		3900.2.j.b		2
5.c	odd	4	1		156.2.b.b	✓	2
5.c	odd	4	1		3900.2.c.a		2
13.b	even	2	1		3900.2.j.b		2
15.e	even	4	1		468.2.b.c		2
20.e	even	4	1		624.2.c.d		2
35.f	even	4	1		7644.2.e.b		2
40.i	odd	4	1		2496.2.c.b		2
40.k	even	4	1		2496.2.c.i		2
60.l	odd	4	1		1872.2.c.h		2
65.d	even	2	1	inner	3900.2.j.e		2
65.f	even	4	1		2028.2.a.d		1
65.h	odd	4	1		156.2.b.b	✓	2
65.h	odd	4	1		3900.2.c.a		2
65.k	even	4	1		2028.2.a.f		1
65.o	even	12	2		2028.2.i.d		2
65.q	odd	12	2		2028.2.q.e		4
65.r	odd	12	2		2028.2.q.e		4
65.t	even	12	2		2028.2.i.a		2
195.j	odd	4	1		6084.2.a.d		1
195.s	even	4	1		468.2.b.c		2
195.u	odd	4	1		6084.2.a.n		1
260.l	odd	4	1		8112.2.a.d		1
260.p	even	4	1		624.2.c.d		2
260.s	odd	4	1		8112.2.a.l		1
455.s	even	4	1		7644.2.e.b		2
520.bc	even	4	1		2496.2.c.i		2
520.bg	odd	4	1		2496.2.c.b		2
780.w	odd	4	1		1872.2.c.h		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.2.b.b	✓	2	5.c	odd	4	1
156.2.b.b	✓	2	65.h	odd	4	1
468.2.b.c		2	15.e	even	4	1
468.2.b.c		2	195.s	even	4	1
624.2.c.d		2	20.e	even	4	1
624.2.c.d		2	260.p	even	4	1
1872.2.c.h		2	60.l	odd	4	1
1872.2.c.h		2	780.w	odd	4	1
2028.2.a.d		1	65.f	even	4	1
2028.2.a.f		1	65.k	even	4	1
2028.2.i.a		2	65.t	even	12	2
2028.2.i.d		2	65.o	even	12	2
2028.2.q.e		4	65.q	odd	12	2
2028.2.q.e		4	65.r	odd	12	2
2496.2.c.b		2	40.i	odd	4	1
2496.2.c.b		2	520.bg	odd	4	1
2496.2.c.i		2	40.k	even	4	1
2496.2.c.i		2	520.bc	even	4	1
3900.2.c.a		2	5.c	odd	4	1
3900.2.c.a		2	65.h	odd	4	1
3900.2.j.b		2	5.b	even	2	1
3900.2.j.b		2	13.b	even	2	1
3900.2.j.e		2	1.a	even	1	1	trivial
3900.2.j.e		2	65.d	even	2	1	inner
6084.2.a.d		1	195.j	odd	4	1
6084.2.a.n		1	195.u	odd	4	1
7644.2.e.b		2	35.f	even	4	1
7644.2.e.b		2	455.s	even	4	1
8112.2.a.d		1	260.l	odd	4	1
8112.2.a.l		1	260.s	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3900, [\chi])\):

\( T_{7} - 4 \)

\( T_{11}^{2} + 36 \)

\( T_{37} - 8 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 1 \)
$5$	\( T^{2} \)
$7$	\( (T - 4)^{2} \)
$11$	\( T^{2} + 36 \)