Newspace parameters
Level: | \( N \) | \(=\) | \( 2500 = 2^{2} \cdot 5^{4} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2500.h (of order \(10\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.24766253158\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{10})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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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 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.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2500\mathbb{Z}\right)^\times\).
\(n\) | \(1251\) | \(1877\) |
\(\chi(n)\) | \(-1\) | \(\zeta_{10}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
499.1 |
|
−0.309017 | − | 0.951057i | −1.30902 | + | 0.951057i | −0.809017 | + | 0.587785i | 0 | 1.30902 | + | 0.951057i | −0.618034 | 0.809017 | + | 0.587785i | 0.500000 | − | 1.53884i | 0 | ||||||||||||||||||
999.1 | 0.809017 | − | 0.587785i | −0.190983 | + | 0.587785i | 0.309017 | − | 0.951057i | 0 | 0.190983 | + | 0.587785i | 1.61803 | −0.309017 | − | 0.951057i | 0.500000 | + | 0.363271i | 0 | |||||||||||||||||||
1499.1 | 0.809017 | + | 0.587785i | −0.190983 | − | 0.587785i | 0.309017 | + | 0.951057i | 0 | 0.190983 | − | 0.587785i | 1.61803 | −0.309017 | + | 0.951057i | 0.500000 | − | 0.363271i | 0 | |||||||||||||||||||
1999.1 | −0.309017 | + | 0.951057i | −1.30902 | − | 0.951057i | −0.809017 | − | 0.587785i | 0 | 1.30902 | − | 0.951057i | −0.618034 | 0.809017 | − | 0.587785i | 0.500000 | + | 1.53884i | 0 | |||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
20.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-5}) \) |
25.d | even | 5 | 1 | inner |
100.h | 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.h.c | 4 | |
4.b | odd | 2 | 1 | 2500.1.h.b | 4 | ||
5.b | even | 2 | 1 | 2500.1.h.b | 4 | ||
5.c | odd | 4 | 2 | 2500.1.j.d | 8 | ||
20.d | odd | 2 | 1 | CM | 2500.1.h.c | 4 | |
20.e | even | 4 | 2 | 2500.1.j.d | 8 | ||
25.d | even | 5 | 1 | 500.1.d.a | 2 | ||
25.d | even | 5 | 1 | inner | 2500.1.h.c | 4 | |
25.d | even | 5 | 2 | 2500.1.h.d | 4 | ||
25.e | even | 10 | 1 | 500.1.d.b | 2 | ||
25.e | even | 10 | 2 | 2500.1.h.a | 4 | ||
25.e | even | 10 | 1 | 2500.1.h.b | 4 | ||
25.f | odd | 20 | 2 | 500.1.b.a | ✓ | 4 | |
25.f | odd | 20 | 4 | 2500.1.j.c | 8 | ||
25.f | odd | 20 | 2 | 2500.1.j.d | 8 | ||
100.h | odd | 10 | 1 | 500.1.d.a | 2 | ||
100.h | odd | 10 | 1 | inner | 2500.1.h.c | 4 | |
100.h | odd | 10 | 2 | 2500.1.h.d | 4 | ||
100.j | odd | 10 | 1 | 500.1.d.b | 2 | ||
100.j | odd | 10 | 2 | 2500.1.h.a | 4 | ||
100.j | odd | 10 | 1 | 2500.1.h.b | 4 | ||
100.l | even | 20 | 2 | 500.1.b.a | ✓ | 4 | |
100.l | even | 20 | 4 | 2500.1.j.c | 8 | ||
100.l | even | 20 | 2 | 2500.1.j.d | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
500.1.b.a | ✓ | 4 | 25.f | odd | 20 | 2 | |
500.1.b.a | ✓ | 4 | 100.l | even | 20 | 2 | |
500.1.d.a | 2 | 25.d | even | 5 | 1 | ||
500.1.d.a | 2 | 100.h | odd | 10 | 1 | ||
500.1.d.b | 2 | 25.e | even | 10 | 1 | ||
500.1.d.b | 2 | 100.j | odd | 10 | 1 | ||
2500.1.h.a | 4 | 25.e | even | 10 | 2 | ||
2500.1.h.a | 4 | 100.j | odd | 10 | 2 | ||
2500.1.h.b | 4 | 4.b | odd | 2 | 1 | ||
2500.1.h.b | 4 | 5.b | even | 2 | 1 | ||
2500.1.h.b | 4 | 25.e | even | 10 | 1 | ||
2500.1.h.b | 4 | 100.j | odd | 10 | 1 | ||
2500.1.h.c | 4 | 1.a | even | 1 | 1 | trivial | |
2500.1.h.c | 4 | 20.d | odd | 2 | 1 | CM | |
2500.1.h.c | 4 | 25.d | even | 5 | 1 | inner | |
2500.1.h.c | 4 | 100.h | odd | 10 | 1 | inner | |
2500.1.h.d | 4 | 25.d | even | 5 | 2 | ||
2500.1.h.d | 4 | 100.h | odd | 10 | 2 | ||
2500.1.j.c | 8 | 25.f | odd | 20 | 4 | ||
2500.1.j.c | 8 | 100.l | even | 20 | 4 | ||
2500.1.j.d | 8 | 5.c | odd | 4 | 2 | ||
2500.1.j.d | 8 | 20.e | even | 4 | 2 | ||
2500.1.j.d | 8 | 25.f | odd | 20 | 2 | ||
2500.1.j.d | 8 | 100.l | even | 20 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 3T_{3}^{3} + 4T_{3}^{2} + 2T_{3} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2500, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$3$
\( T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1 \)
$5$
\( T^{4} \)
$7$
\( (T^{2} - T - 1)^{2} \)
$11$
\( T^{4} \)
$13$
\( T^{4} \)
$17$
\( T^{4} \)
$19$
\( T^{4} \)
$23$
\( T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1 \)
$29$
\( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \)
$31$
\( T^{4} \)
$37$
\( T^{4} \)
$41$
\( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \)
$43$
\( (T^{2} - T - 1)^{2} \)
$47$
\( T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1 \)
$53$
\( T^{4} \)
$59$
\( T^{4} \)
$61$
\( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \)
$67$
\( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \)
$71$
\( T^{4} \)
$73$
\( T^{4} \)
$79$
\( T^{4} \)
$83$
\( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \)
$89$
\( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \)
$97$
\( T^{4} \)
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