Newform orbit 5000.2.a.c

• Label: 5000.2.a.c
• Level: $5000$
• Weight: $2$
• Character orbit: 5000.a
• Self dual: yes
• Analytic conductor: $39.925$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$5000 = 2^{3} \cdot 5^{4}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5000.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$39.9252010106$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{5})$
Defining polynomial:	$x^{2} - x - 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \frac{1}{2}(1 + \sqrt{5})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (-2*b + 1) * q^3 + (b + 1) * q^7 + 2 * q^9 + (2*b - 2) * q^11 + (-b + 5) * q^13 + (2*b + 2) * q^17 + (b + 6) * q^19 + (-3*b - 1) * q^21 + (4*b - 1) * q^23 + (2*b - 1) * q^27 + (-b + 6) * q^29 + (-6*b + 5) * q^31 + (2*b - 6) * q^33 + (-2*b - 5) * q^37 + (-9*b + 7) * q^39 + (6*b - 4) * q^41 + (-b + 6) * q^43 + (-3*b + 5) * q^47 + (3*b - 5) * q^49 + (-6*b - 2) * q^51 + (8*b - 5) * q^53 + (-13*b + 4) * q^57 + (-b + 4) * q^59 + (-10*b + 3) * q^61 + (2*b + 2) * q^63 + (6*b + 4) * q^67 + (-2*b - 9) * q^69 + (-b - 7) * q^71 + 11 * q^73 + 2*b * q^77 + (-7*b + 6) * q^79 - 11 * q^81 + (4*b - 11) * q^83 + (-11*b + 8) * q^87 + 4 * q^89 + (3*b + 4) * q^91 + (-4*b + 17) * q^93 + (-15*b + 9) * q^97 + (4*b - 4) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 3 * q^7 + 4 * q^9 - 2 * q^11 + 9 * q^13 + 6 * q^17 + 13 * q^19 - 5 * q^21 + 2 * q^23 + 11 * q^29 + 4 * q^31 - 10 * q^33 - 12 * q^37 + 5 * q^39 - 2 * q^41 + 11 * q^43 + 7 * q^47 - 7 * q^49 - 10 * q^51 - 2 * q^53 - 5 * q^57 + 7 * q^59 - 4 * q^61 + 6 * q^63 + 14 * q^67 - 20 * q^69 - 15 * q^71 + 22 * q^73 + 2 * q^77 + 5 * q^79 - 22 * q^81 - 18 * q^83 + 5 * q^87 + 8 * q^89 + 11 * q^91 + 30 * q^93 + 3 * q^97 - 4 * q^99

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

1.61803
−0.618034

0

−2.23607

0

0

0

2.61803

0

2.00000

0

1.2

0

2.23607

0

0

0

0.381966

0

2.00000

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$+1$

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	5000.2.a.c		2
4.b	odd	2	1		10000.2.a.g		2
5.b	even	2	1		5000.2.a.a		2
20.d	odd	2	1		10000.2.a.i		2
25.d	even	5	2		200.2.m.a	✓	4
25.e	even	10	2		1000.2.m.a		4
25.f	odd	20	4		1000.2.q.a		8
100.j	odd	10	2		400.2.u.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.2.m.a	✓	4	25.d	even	5	2
400.2.u.a		4	100.j	odd	10	2
1000.2.m.a		4	25.e	even	10	2
1000.2.q.a		8	25.f	odd	20	4
5000.2.a.a		2	5.b	even	2	1
5000.2.a.c		2	1.a	even	1	1	trivial
10000.2.a.g		2	4.b	odd	2	1
10000.2.a.i		2	20.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(5000))$:

$T_{3}^{2} - 5$

$T_{7}^{2} - 3T_{7} + 1$

$T_{11}^{2} + 2T_{11} - 4$

Hecke characteristic polynomials

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