Newspace parameters
Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 25.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.47504775014\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
24.1 |
|
− | 1.00000i | 7.00000i | 7.00000 | 0 | 7.00000 | − | 6.00000i | − | 15.0000i | −22.0000 | 0 | |||||||||||||||||||||
24.2 | 1.00000i | − | 7.00000i | 7.00000 | 0 | 7.00000 | 6.00000i | 15.0000i | −22.0000 | 0 | ||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 25.4.b.b | 2 | |
3.b | odd | 2 | 1 | 225.4.b.f | 2 | ||
4.b | odd | 2 | 1 | 400.4.c.e | 2 | ||
5.b | even | 2 | 1 | inner | 25.4.b.b | 2 | |
5.c | odd | 4 | 1 | 25.4.a.a | ✓ | 1 | |
5.c | odd | 4 | 1 | 25.4.a.b | yes | 1 | |
15.d | odd | 2 | 1 | 225.4.b.f | 2 | ||
15.e | even | 4 | 1 | 225.4.a.c | 1 | ||
15.e | even | 4 | 1 | 225.4.a.e | 1 | ||
20.d | odd | 2 | 1 | 400.4.c.e | 2 | ||
20.e | even | 4 | 1 | 400.4.a.c | 1 | ||
20.e | even | 4 | 1 | 400.4.a.s | 1 | ||
35.f | even | 4 | 1 | 1225.4.a.h | 1 | ||
35.f | even | 4 | 1 | 1225.4.a.i | 1 | ||
40.i | odd | 4 | 1 | 1600.4.a.i | 1 | ||
40.i | odd | 4 | 1 | 1600.4.a.bt | 1 | ||
40.k | even | 4 | 1 | 1600.4.a.h | 1 | ||
40.k | even | 4 | 1 | 1600.4.a.bs | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
25.4.a.a | ✓ | 1 | 5.c | odd | 4 | 1 | |
25.4.a.b | yes | 1 | 5.c | odd | 4 | 1 | |
25.4.b.b | 2 | 1.a | even | 1 | 1 | trivial | |
25.4.b.b | 2 | 5.b | even | 2 | 1 | inner | |
225.4.a.c | 1 | 15.e | even | 4 | 1 | ||
225.4.a.e | 1 | 15.e | even | 4 | 1 | ||
225.4.b.f | 2 | 3.b | odd | 2 | 1 | ||
225.4.b.f | 2 | 15.d | odd | 2 | 1 | ||
400.4.a.c | 1 | 20.e | even | 4 | 1 | ||
400.4.a.s | 1 | 20.e | even | 4 | 1 | ||
400.4.c.e | 2 | 4.b | odd | 2 | 1 | ||
400.4.c.e | 2 | 20.d | odd | 2 | 1 | ||
1225.4.a.h | 1 | 35.f | even | 4 | 1 | ||
1225.4.a.i | 1 | 35.f | even | 4 | 1 | ||
1600.4.a.h | 1 | 40.k | even | 4 | 1 | ||
1600.4.a.i | 1 | 40.i | odd | 4 | 1 | ||
1600.4.a.bs | 1 | 40.k | even | 4 | 1 | ||
1600.4.a.bt | 1 | 40.i | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 1 \)
acting on \(S_{4}^{\mathrm{new}}(25, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 1 \)
$3$
\( T^{2} + 49 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 36 \)
$11$
\( (T + 43)^{2} \)
$13$
\( T^{2} + 784 \)
$17$
\( T^{2} + 8281 \)
$19$
\( (T - 35)^{2} \)
$23$
\( T^{2} + 26244 \)
$29$
\( (T + 160)^{2} \)
$31$
\( (T - 42)^{2} \)
$37$
\( T^{2} + 98596 \)
$41$
\( (T + 203)^{2} \)
$43$
\( T^{2} + 8464 \)
$47$
\( T^{2} + 38416 \)
$53$
\( T^{2} + 6724 \)
$59$
\( (T - 280)^{2} \)
$61$
\( (T + 518)^{2} \)
$67$
\( T^{2} + 19881 \)
$71$
\( (T - 412)^{2} \)
$73$
\( T^{2} + 582169 \)
$79$
\( (T + 510)^{2} \)
$83$
\( T^{2} + 603729 \)
$89$
\( (T - 945)^{2} \)
$97$
\( T^{2} + 1552516 \)
show more
show less