Newspace parameters
Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 25.c (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.58424907710\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(i, \sqrt{21})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 11x^{2} + 25 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 2^{3} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 11x^{2} + 25 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{3} + 6\nu ) / 5 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{3} + 10\nu^{2} + 16\nu + 55 ) / 5 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} - 10\nu^{2} + 16\nu - 55 ) / 5 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + \beta_{2} - 2\beta_1 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{3} + \beta_{2} - 22 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( -3\beta_{3} - 3\beta_{2} + 16\beta_1 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 |
|
−4.58258 | − | 4.58258i | −4.58258 | + | 4.58258i | 26.0000i | 0 | 42.0000 | 41.2432 | + | 41.2432i | 45.8258 | − | 45.8258i | 39.0000i | 0 | ||||||||||||||||||||||
7.2 | 4.58258 | + | 4.58258i | 4.58258 | − | 4.58258i | 26.0000i | 0 | 42.0000 | −41.2432 | − | 41.2432i | −45.8258 | + | 45.8258i | 39.0000i | 0 | |||||||||||||||||||||||
18.1 | −4.58258 | + | 4.58258i | −4.58258 | − | 4.58258i | − | 26.0000i | 0 | 42.0000 | 41.2432 | − | 41.2432i | 45.8258 | + | 45.8258i | − | 39.0000i | 0 | |||||||||||||||||||||
18.2 | 4.58258 | − | 4.58258i | 4.58258 | + | 4.58258i | − | 26.0000i | 0 | 42.0000 | −41.2432 | + | 41.2432i | −45.8258 | − | 45.8258i | − | 39.0000i | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 25.5.c.c | ✓ | 4 |
3.b | odd | 2 | 1 | 225.5.g.j | 4 | ||
4.b | odd | 2 | 1 | 400.5.p.l | 4 | ||
5.b | even | 2 | 1 | inner | 25.5.c.c | ✓ | 4 |
5.c | odd | 4 | 2 | inner | 25.5.c.c | ✓ | 4 |
15.d | odd | 2 | 1 | 225.5.g.j | 4 | ||
15.e | even | 4 | 2 | 225.5.g.j | 4 | ||
20.d | odd | 2 | 1 | 400.5.p.l | 4 | ||
20.e | even | 4 | 2 | 400.5.p.l | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
25.5.c.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
25.5.c.c | ✓ | 4 | 5.b | even | 2 | 1 | inner |
25.5.c.c | ✓ | 4 | 5.c | odd | 4 | 2 | inner |
225.5.g.j | 4 | 3.b | odd | 2 | 1 | ||
225.5.g.j | 4 | 15.d | odd | 2 | 1 | ||
225.5.g.j | 4 | 15.e | even | 4 | 2 | ||
400.5.p.l | 4 | 4.b | odd | 2 | 1 | ||
400.5.p.l | 4 | 20.d | odd | 2 | 1 | ||
400.5.p.l | 4 | 20.e | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 1764 \)
acting on \(S_{5}^{\mathrm{new}}(25, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 1764 \)
$3$
\( T^{4} + 1764 \)
$5$
\( T^{4} \)
$7$
\( T^{4} + 11573604 \)
$11$
\( (T + 108)^{4} \)
$13$
\( T^{4} + 2962842624 \)
$17$
\( T^{4} + 451584 \)
$19$
\( (T^{2} + 19600)^{2} \)
$23$
\( T^{4} + 68707942884 \)
$29$
\( (T^{2} + 656100)^{2} \)
$31$
\( (T + 728)^{4} \)
$37$
\( T^{4} + 758487711744 \)
$41$
\( (T - 1512)^{4} \)
$43$
\( T^{4} + 11573604 \)
$47$
\( T^{4} + 4080909924 \)
$53$
\( T^{4} + \cdots + 368370941912064 \)
$59$
\( (T^{2} + 14288400)^{2} \)
$61$
\( (T - 4592)^{4} \)
$67$
\( T^{4} + \cdots + 498205702352484 \)
$71$
\( (T - 432)^{4} \)
$73$
\( T^{4} + 68\!\cdots\!84 \)
$79$
\( (T^{2} + 78145600)^{2} \)
$83$
\( T^{4} + 1438948991844 \)
$89$
\( (T^{2} + 175032900)^{2} \)
$97$
\( T^{4} + 17\!\cdots\!24 \)
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