Newform orbit 1984.2.a.r

• Label: 1984.2.a.r
• Level: $1984$
• Weight: $2$
• Character orbit: 1984.a
• Self dual: yes
• Analytic conductor: $15.842$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1984 = 2^{6} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1984.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(15.8423197610\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta + 1) q^{3} - q^{5} + (\beta - 2) q^{7} + (2 \beta + 3) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 6 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−0.618034
1.61803

0

−1.23607

0

−1.00000

0

−4.23607

0

−1.47214

0

1.2

0

3.23607

0

−1.00000

0

0.236068

0

7.47214

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(31\)	\(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1984.2.a.r		2
4.b	odd	2	1		1984.2.a.n		2
8.b	even	2	1		31.2.a.a	✓	2
8.d	odd	2	1		496.2.a.i		2
24.f	even	2	1		4464.2.a.bf		2
24.h	odd	2	1		279.2.a.a		2
40.f	even	2	1		775.2.a.d		2
40.i	odd	4	2		775.2.b.d		4
56.h	odd	2	1		1519.2.a.a		2
88.b	odd	2	1		3751.2.a.b		2
104.e	even	2	1		5239.2.a.f		2
120.i	odd	2	1		6975.2.a.y		2
136.h	even	2	1		8959.2.a.b		2
248.g	odd	2	1		961.2.a.f		2
248.l	odd	6	2		961.2.c.c		4
248.p	even	6	2		961.2.c.e		4
248.r	odd	10	2		961.2.d.a		4
248.r	odd	10	2		961.2.d.g		4
248.u	even	10	2		961.2.d.c		4
248.u	even	10	2		961.2.d.d		4
248.bc	even	30	4		961.2.g.a		8
248.bc	even	30	4		961.2.g.h		8
248.bf	odd	30	4		961.2.g.d		8
248.bf	odd	30	4		961.2.g.e		8
744.o	even	2	1		8649.2.a.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
31.2.a.a	✓	2	8.b	even	2	1
279.2.a.a		2	24.h	odd	2	1
496.2.a.i		2	8.d	odd	2	1
775.2.a.d		2	40.f	even	2	1
775.2.b.d		4	40.i	odd	4	2
961.2.a.f		2	248.g	odd	2	1
961.2.c.c		4	248.l	odd	6	2
961.2.c.e		4	248.p	even	6	2
961.2.d.a		4	248.r	odd	10	2
961.2.d.c		4	248.u	even	10	2
961.2.d.d		4	248.u	even	10	2
961.2.d.g		4	248.r	odd	10	2
961.2.g.a		8	248.bc	even	30	4
961.2.g.d		8	248.bf	odd	30	4
961.2.g.e		8	248.bf	odd	30	4
961.2.g.h		8	248.bc	even	30	4
1519.2.a.a		2	56.h	odd	2	1
1984.2.a.n		2	4.b	odd	2	1
1984.2.a.r		2	1.a	even	1	1	trivial
3751.2.a.b		2	88.b	odd	2	1
4464.2.a.bf		2	24.f	even	2	1
5239.2.a.f		2	104.e	even	2	1
6975.2.a.y		2	120.i	odd	2	1
8649.2.a.c		2	744.o	even	2	1
8959.2.a.b		2	136.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1984))\):

\( T_{3}^{2} - 2T_{3} - 4 \)

\( T_{5} + 1 \)

\( T_{7}^{2} + 4T_{7} - 1 \)

\( T_{19}^{2} - 5 \)

Hecke characteristic polynomials

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