Properties

Label 2500.1.j.d
Level $2500$
Weight $1$
Character orbit 2500.j
Analytic conductor $1.248$
Analytic rank $0$
Dimension $8$
Projective image $D_{5}$
CM discriminant -20
Inner twists $8$

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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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 500)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.250000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{20} q^{2} + ( \zeta_{20}^{5} + \zeta_{20}^{9} ) q^{3} + \zeta_{20}^{2} q^{4} + ( 1 - \zeta_{20}^{6} ) q^{6} + ( -\zeta_{20} - \zeta_{20}^{9} ) q^{7} -\zeta_{20}^{3} q^{8} + ( -1 - \zeta_{20}^{4} - \zeta_{20}^{8} ) q^{9} +O(q^{10})\) \( q -\zeta_{20} q^{2} + ( \zeta_{20}^{5} + \zeta_{20}^{9} ) q^{3} + \zeta_{20}^{2} q^{4} + ( 1 - \zeta_{20}^{6} ) q^{6} + ( -\zeta_{20} - \zeta_{20}^{9} ) q^{7} -\zeta_{20}^{3} q^{8} + ( -1 - \zeta_{20}^{4} - \zeta_{20}^{8} ) q^{9} + ( -\zeta_{20} + \zeta_{20}^{7} ) q^{12} + ( -1 + \zeta_{20}^{2} ) q^{14} + \zeta_{20}^{4} q^{16} + ( \zeta_{20} + \zeta_{20}^{5} + \zeta_{20}^{9} ) q^{18} + ( 1 + \zeta_{20}^{4} - \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{21} + ( \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{23} + ( \zeta_{20}^{2} - \zeta_{20}^{8} ) q^{24} + ( \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{27} + ( \zeta_{20} - \zeta_{20}^{3} ) q^{28} + ( -1 - \zeta_{20}^{4} ) q^{29} -\zeta_{20}^{5} q^{32} + ( 1 - \zeta_{20}^{2} - \zeta_{20}^{6} ) q^{36} + ( -\zeta_{20}^{2} - \zeta_{20}^{6} ) q^{41} + ( -\zeta_{20} - \zeta_{20}^{5} + \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{42} + ( -\zeta_{20}^{3} - \zeta_{20}^{7} ) q^{43} + ( -\zeta_{20}^{6} + \zeta_{20}^{8} ) q^{46} + ( -\zeta_{20}^{5} - \zeta_{20}^{9} ) q^{47} + ( -\zeta_{20}^{3} + \zeta_{20}^{9} ) q^{48} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{8} ) q^{49} + ( -1 - \zeta_{20}^{4} + \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{54} + ( -\zeta_{20}^{2} + \zeta_{20}^{4} ) q^{56} + ( \zeta_{20} + \zeta_{20}^{5} ) q^{58} + ( 1 - \zeta_{20}^{2} ) q^{61} + ( \zeta_{20} - \zeta_{20}^{3} + \zeta_{20}^{5} - \zeta_{20}^{7} + 2 \zeta_{20}^{9} ) q^{63} + \zeta_{20}^{6} q^{64} + 2 \zeta_{20}^{3} q^{67} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{69} + ( -\zeta_{20} + \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{72} + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{81} + ( \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{82} + ( -\zeta_{20}^{7} + \zeta_{20}^{9} ) q^{83} + ( -1 + \zeta_{20}^{2} + \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{84} + ( \zeta_{20}^{4} + \zeta_{20}^{8} ) q^{86} + ( \zeta_{20}^{3} - \zeta_{20}^{5} - 2 \zeta_{20}^{9} ) q^{87} + ( -1 + \zeta_{20}^{2} ) q^{89} + ( \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{92} + ( -1 + \zeta_{20}^{6} ) q^{94} + ( 1 + \zeta_{20}^{4} ) q^{96} + ( \zeta_{20} - \zeta_{20}^{3} + \zeta_{20}^{9} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} + 6 q^{6} - 4 q^{9} + O(q^{10}) \) \( 8 q + 2 q^{4} + 6 q^{6} - 4 q^{9} - 6 q^{14} - 2 q^{16} + 2 q^{21} + 4 q^{24} - 6 q^{29} + 4 q^{36} - 4 q^{41} - 4 q^{46} - 4 q^{49} - 2 q^{54} - 4 q^{56} + 6 q^{61} + 2 q^{64} - 2 q^{69} - 2 q^{84} - 4 q^{86} - 6 q^{89} - 6 q^{94} + 6 q^{96} + O(q^{100}) \)

Character values

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

\(n\) \(1251\) \(1877\)
\(\chi(n)\) \(-1\) \(-\zeta_{20}^{2}\)

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
0.587785 + 0.809017i
−0.587785 0.809017i
0.951057 0.309017i
−0.951057 + 0.309017i
0.951057 + 0.309017i
−0.951057 0.309017i
0.587785 0.809017i
−0.587785 + 0.809017i
−0.587785 0.809017i −0.587785 0.190983i −0.309017 + 0.951057i 0 0.190983 + 0.587785i 1.61803i 0.951057 0.309017i −0.500000 0.363271i 0
251.2 0.587785 + 0.809017i 0.587785 + 0.190983i −0.309017 + 0.951057i 0 0.190983 + 0.587785i 1.61803i −0.951057 + 0.309017i −0.500000 0.363271i 0
751.1 −0.951057 + 0.309017i −0.951057 1.30902i 0.809017 0.587785i 0 1.30902 + 0.951057i 0.618034i −0.587785 + 0.809017i −0.500000 + 1.53884i 0
751.2 0.951057 0.309017i 0.951057 + 1.30902i 0.809017 0.587785i 0 1.30902 + 0.951057i 0.618034i 0.587785 0.809017i −0.500000 + 1.53884i 0
1751.1 −0.951057 0.309017i −0.951057 + 1.30902i 0.809017 + 0.587785i 0 1.30902 0.951057i 0.618034i −0.587785 0.809017i −0.500000 1.53884i 0
1751.2 0.951057 + 0.309017i 0.951057 1.30902i 0.809017 + 0.587785i 0 1.30902 0.951057i 0.618034i 0.587785 + 0.809017i −0.500000 1.53884i 0
2251.1 −0.587785 + 0.809017i −0.587785 + 0.190983i −0.309017 0.951057i 0 0.190983 0.587785i 1.61803i 0.951057 + 0.309017i −0.500000 + 0.363271i 0
2251.2 0.587785 0.809017i 0.587785 0.190983i −0.309017 0.951057i 0 0.190983 0.587785i 1.61803i −0.951057 0.309017i −0.500000 + 0.363271i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2251.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner
100.h odd 10 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.d 8
4.b odd 2 1 inner 2500.1.j.d 8
5.b even 2 1 inner 2500.1.j.d 8
5.c odd 4 1 2500.1.h.b 4
5.c odd 4 1 2500.1.h.c 4
20.d odd 2 1 CM 2500.1.j.d 8
20.e even 4 1 2500.1.h.b 4
20.e even 4 1 2500.1.h.c 4
25.d even 5 1 500.1.b.a 4
25.d even 5 2 2500.1.j.c 8
25.d even 5 1 inner 2500.1.j.d 8
25.e even 10 1 500.1.b.a 4
25.e even 10 2 2500.1.j.c 8
25.e even 10 1 inner 2500.1.j.d 8
25.f odd 20 1 500.1.d.a 2
25.f odd 20 1 500.1.d.b 2
25.f odd 20 2 2500.1.h.a 4
25.f odd 20 1 2500.1.h.b 4
25.f odd 20 1 2500.1.h.c 4
25.f odd 20 2 2500.1.h.d 4
100.h odd 10 1 500.1.b.a 4
100.h odd 10 2 2500.1.j.c 8
100.h odd 10 1 inner 2500.1.j.d 8
100.j odd 10 1 500.1.b.a 4
100.j odd 10 2 2500.1.j.c 8
100.j odd 10 1 inner 2500.1.j.d 8
100.l even 20 1 500.1.d.a 2
100.l even 20 1 500.1.d.b 2
100.l even 20 2 2500.1.h.a 4
100.l even 20 1 2500.1.h.b 4
100.l even 20 1 2500.1.h.c 4
100.l even 20 2 2500.1.h.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.1.b.a 4 25.d even 5 1
500.1.b.a 4 25.e even 10 1
500.1.b.a 4 100.h odd 10 1
500.1.b.a 4 100.j odd 10 1
500.1.d.a 2 25.f odd 20 1
500.1.d.a 2 100.l even 20 1
500.1.d.b 2 25.f odd 20 1
500.1.d.b 2 100.l even 20 1
2500.1.h.a 4 25.f odd 20 2
2500.1.h.a 4 100.l even 20 2
2500.1.h.b 4 5.c odd 4 1
2500.1.h.b 4 20.e even 4 1
2500.1.h.b 4 25.f odd 20 1
2500.1.h.b 4 100.l even 20 1
2500.1.h.c 4 5.c odd 4 1
2500.1.h.c 4 20.e even 4 1
2500.1.h.c 4 25.f odd 20 1
2500.1.h.c 4 100.l even 20 1
2500.1.h.d 4 25.f odd 20 2
2500.1.h.d 4 100.l even 20 2
2500.1.j.c 8 25.d even 5 2
2500.1.j.c 8 25.e even 10 2
2500.1.j.c 8 100.h odd 10 2
2500.1.j.c 8 100.j odd 10 2
2500.1.j.d 8 1.a even 1 1 trivial
2500.1.j.d 8 4.b odd 2 1 inner
2500.1.j.d 8 5.b even 2 1 inner
2500.1.j.d 8 20.d odd 2 1 CM
2500.1.j.d 8 25.d even 5 1 inner
2500.1.j.d 8 25.e even 10 1 inner
2500.1.j.d 8 100.h odd 10 1 inner
2500.1.j.d 8 100.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 \(S_{1}^{\mathrm{new}}(2500, [\chi])\):

\( T_{3}^{8} + T_{3}^{6} + 6 T_{3}^{4} - 4 T_{3}^{2} + 1 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$3$ \( 1 - 4 T^{2} + 6 T^{4} + T^{6} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 1 + 3 T^{2} + T^{4} )^{2} \)
$11$ \( T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( 1 - 4 T^{2} + 6 T^{4} + T^{6} + T^{8} \)
$29$ \( ( 1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( ( 1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$43$ \( ( 1 + 3 T^{2} + T^{4} )^{2} \)
$47$ \( 1 - 4 T^{2} + 6 T^{4} + T^{6} + T^{8} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( ( 1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$67$ \( 256 - 64 T^{2} + 16 T^{4} - 4 T^{6} + T^{8} \)
$71$ \( T^{8} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( 1 + T^{2} + 6 T^{4} - 4 T^{6} + T^{8} \)
$89$ \( ( 1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$97$ \( T^{8} \)
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