Properties

Label 2500.1.j.b
Level $2500$
Weight $1$
Character orbit 2500.j
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.j (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.24766253158\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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

The \(q\)-expansion and trace form are shown below.

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

Character values

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

\(n\) \(1251\) \(1877\)
\(\chi(n)\) \(-1\) \(-\zeta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−0.309017 + 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 0.951057i
0.809017 0.587785i 0 0.309017 0.951057i 0 0 0 −0.309017 0.951057i −0.809017 0.587785i 0
751.1 −0.309017 0.951057i 0 −0.809017 + 0.587785i 0 0 0 0.809017 + 0.587785i 0.309017 0.951057i 0
1751.1 −0.309017 + 0.951057i 0 −0.809017 0.587785i 0 0 0 0.809017 0.587785i 0.309017 + 0.951057i 0
2251.1 0.809017 + 0.587785i 0 0.309017 + 0.951057i 0 0 0 −0.309017 + 0.951057i −0.809017 + 0.587785i 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 \(\Q(\sqrt{-1}) \)
25.d even 5 1 inner
100.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2500.1.j.b 4
4.b odd 2 1 CM 2500.1.j.b 4
5.b even 2 1 2500.1.j.a 4
5.c odd 4 2 2500.1.h.e 8
20.d odd 2 1 2500.1.j.a 4
20.e even 4 2 2500.1.h.e 8
25.d even 5 2 500.1.j.a 4
25.d even 5 1 2500.1.b.a 2
25.d even 5 1 inner 2500.1.j.b 4
25.e even 10 2 100.1.j.a 4
25.e even 10 1 2500.1.b.b 2
25.e even 10 1 2500.1.j.a 4
25.f odd 20 4 500.1.h.a 8
25.f odd 20 2 2500.1.d.a 4
25.f odd 20 2 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 1 2500.1.b.b 2
100.h odd 10 1 2500.1.j.a 4
100.j odd 10 2 500.1.j.a 4
100.j odd 10 1 2500.1.b.a 2
100.j odd 10 1 inner 2500.1.j.b 4
100.l even 20 4 500.1.h.a 8
100.l even 20 2 2500.1.d.a 4
100.l even 20 2 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 25.d even 5 1
2500.1.b.a 2 100.j odd 10 1
2500.1.b.b 2 25.e even 10 1
2500.1.b.b 2 100.h odd 10 1
2500.1.d.a 4 25.f odd 20 2
2500.1.d.a 4 100.l even 20 2
2500.1.h.e 8 5.c odd 4 2
2500.1.h.e 8 20.e even 4 2
2500.1.h.e 8 25.f odd 20 2
2500.1.h.e 8 100.l even 20 2
2500.1.j.a 4 5.b even 2 1
2500.1.j.a 4 20.d odd 2 1
2500.1.j.a 4 25.e even 10 1
2500.1.j.a 4 100.h odd 10 1
2500.1.j.b 4 1.a even 1 1 trivial
2500.1.j.b 4 4.b odd 2 1 CM
2500.1.j.b 4 25.d even 5 1 inner
2500.1.j.b 4 100.j odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2500, [\chi])\):

\( T_{3} \)
\( T_{13}^{4} + 3 T_{13}^{3} + 4 T_{13}^{2} + 2 T_{13} + 1 \)

Hecke characteristic polynomials

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