Properties

Label 2500.1.b.a
Level $2500$
Weight $1$
Character orbit 2500.b
Self dual yes
Analytic conductor $1.248$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -4
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2500 = 2^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2500.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.24766253158\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 100)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.6250000.1

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + q^{4} - q^{8} + q^{9} + ( 1 - \beta ) q^{13} + q^{16} + \beta q^{17} - q^{18} + ( -1 + \beta ) q^{26} -\beta q^{29} - q^{32} -\beta q^{34} + q^{36} + \beta q^{37} + ( -1 + \beta ) q^{41} + q^{49} + ( 1 - \beta ) q^{52} + ( 1 - \beta ) q^{53} + \beta q^{58} + ( -1 + \beta ) q^{61} + q^{64} + \beta q^{68} - q^{72} + ( 1 - \beta ) q^{73} -\beta q^{74} + q^{81} + ( 1 - \beta ) q^{82} -\beta q^{89} + \beta q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} + 2q^{9} + q^{13} + 2q^{16} + q^{17} - 2q^{18} - q^{26} - q^{29} - 2q^{32} - q^{34} + 2q^{36} + q^{37} - q^{41} + 2q^{49} + q^{52} + q^{53} + q^{58} - q^{61} + 2q^{64} + q^{68} - 2q^{72} + q^{73} - q^{74} + 2q^{81} + q^{82} - q^{89} + q^{97} - 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2500\mathbb{Z}\right)^\times\).

\(n\) \(1251\) \(1877\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1251.1
1.61803
−0.618034
−1.00000 0 1.00000 0 0 0 −1.00000 1.00000 0
1251.2 −1.00000 0 1.00000 0 0 0 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2500.1.b.a 2
4.b odd 2 1 CM 2500.1.b.a 2
5.b even 2 1 2500.1.b.b 2
5.c odd 4 2 2500.1.d.a 4
20.d odd 2 1 2500.1.b.b 2
20.e even 4 2 2500.1.d.a 4
25.d even 5 2 500.1.j.a 4
25.d even 5 2 2500.1.j.b 4
25.e even 10 2 100.1.j.a 4
25.e even 10 2 2500.1.j.a 4
25.f odd 20 4 500.1.h.a 8
25.f odd 20 4 2500.1.h.e 8
75.h odd 10 2 900.1.x.a 4
100.h odd 10 2 100.1.j.a 4
100.h odd 10 2 2500.1.j.a 4
100.j odd 10 2 500.1.j.a 4
100.j odd 10 2 2500.1.j.b 4
100.l even 20 4 500.1.h.a 8
100.l even 20 4 2500.1.h.e 8
200.o even 10 2 1600.1.bh.a 4
200.s odd 10 2 1600.1.bh.a 4
300.r even 10 2 900.1.x.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.1.j.a 4 25.e even 10 2
100.1.j.a 4 100.h odd 10 2
500.1.h.a 8 25.f odd 20 4
500.1.h.a 8 100.l even 20 4
500.1.j.a 4 25.d even 5 2
500.1.j.a 4 100.j odd 10 2
900.1.x.a 4 75.h odd 10 2
900.1.x.a 4 300.r even 10 2
1600.1.bh.a 4 200.o even 10 2
1600.1.bh.a 4 200.s odd 10 2
2500.1.b.a 2 1.a even 1 1 trivial
2500.1.b.a 2 4.b odd 2 1 CM
2500.1.b.b 2 5.b even 2 1
2500.1.b.b 2 20.d odd 2 1
2500.1.d.a 4 5.c odd 4 2
2500.1.d.a 4 20.e even 4 2
2500.1.h.e 8 25.f odd 20 4
2500.1.h.e 8 100.l even 20 4
2500.1.j.a 4 25.e even 10 2
2500.1.j.a 4 100.h odd 10 2
2500.1.j.b 4 25.d even 5 2
2500.1.j.b 4 100.j odd 10 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} - T_{13} - 1 \) acting on \(S_{1}^{\mathrm{new}}(2500, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( -1 - T + T^{2} \)
$17$ \( -1 - T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( -1 + T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( -1 - T + T^{2} \)
$41$ \( -1 + T + T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( -1 - T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( -1 + T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( -1 - T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( -1 + T + T^{2} \)
$97$ \( -1 - T + T^{2} \)
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