Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 2500 = 2^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2500.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.24766253158\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - q^{4} -\beta_{3} q^{8} - q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} - q^{4} -\beta_{3} q^{8} - q^{9} -\beta_{1} q^{13} + q^{16} + ( -\beta_{1} - \beta_{3} ) q^{17} -\beta_{3} q^{18} + \beta_{2} q^{26} + ( 1 + \beta_{2} ) q^{29} + \beta_{3} q^{32} + ( 1 + \beta_{2} ) q^{34} + q^{36} + ( -\beta_{1} - \beta_{3} ) q^{37} + \beta_{2} q^{41} - q^{49} + \beta_{1} q^{52} -\beta_{1} q^{53} + ( \beta_{1} + \beta_{3} ) q^{58} + \beta_{2} q^{61} - q^{64} + ( \beta_{1} + \beta_{3} ) q^{68} + \beta_{3} q^{72} -\beta_{1} q^{73} + ( 1 + \beta_{2} ) q^{74} + q^{81} + \beta_{1} q^{82} + ( 1 + \beta_{2} ) q^{89} + ( -\beta_{1} - \beta_{3} ) q^{97} -\beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{9} + 4q^{16} - 2q^{26} + 2q^{29} + 2q^{34} + 4q^{36} - 2q^{41} - 4q^{49} - 2q^{61} - 4q^{64} + 2q^{74} + 4q^{81} + 2q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 2 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2499.1
1.61803i
0.618034i
0.618034i
1.61803i
1.00000i 0 −1.00000 0 0 0 1.00000i −1.00000 0
2499.2 1.00000i 0 −1.00000 0 0 0 1.00000i −1.00000 0
2499.3 1.00000i 0 −1.00000 0 0 0 1.00000i −1.00000 0
2499.4 1.00000i 0 −1.00000 0 0 0 1.00000i −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 \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

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

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2500, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( 1 + 3 T^{2} + T^{4} \)
$17$ \( 1 + 3 T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( -1 - T + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( 1 + 3 T^{2} + T^{4} \)
$41$ \( ( -1 + T + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( 1 + 3 T^{2} + T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( -1 + T + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( 1 + 3 T^{2} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( -1 - T + T^{2} )^{2} \)
$97$ \( 1 + 3 T^{2} + T^{4} \)
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