Newspace parameters
Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 75.f (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.04360198270\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(i, \sqrt{6})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{4} + 9 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 1 \) |
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 \( x^{4} + 9 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{2} ) / 3 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} ) / 3 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( 3\beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( 3\beta_{3} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
\(n\) | \(26\) | \(52\) |
\(\chi(n)\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 |
|
−2.44949 | − | 2.44949i | −1.22474 | + | 1.22474i | 8.00000i | 0 | 6.00000 | 6.12372 | + | 6.12372i | 9.79796 | − | 9.79796i | − | 3.00000i | 0 | |||||||||||||||||||||
7.2 | 2.44949 | + | 2.44949i | 1.22474 | − | 1.22474i | 8.00000i | 0 | 6.00000 | −6.12372 | − | 6.12372i | −9.79796 | + | 9.79796i | − | 3.00000i | 0 | ||||||||||||||||||||||
43.1 | −2.44949 | + | 2.44949i | −1.22474 | − | 1.22474i | − | 8.00000i | 0 | 6.00000 | 6.12372 | − | 6.12372i | 9.79796 | + | 9.79796i | 3.00000i | 0 | ||||||||||||||||||||||
43.2 | 2.44949 | − | 2.44949i | 1.22474 | + | 1.22474i | − | 8.00000i | 0 | 6.00000 | −6.12372 | + | 6.12372i | −9.79796 | − | 9.79796i | 3.00000i | 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 | 75.3.f.b | ✓ | 4 |
3.b | odd | 2 | 1 | 225.3.g.b | 4 | ||
4.b | odd | 2 | 1 | 1200.3.bg.g | 4 | ||
5.b | even | 2 | 1 | inner | 75.3.f.b | ✓ | 4 |
5.c | odd | 4 | 2 | inner | 75.3.f.b | ✓ | 4 |
15.d | odd | 2 | 1 | 225.3.g.b | 4 | ||
15.e | even | 4 | 2 | 225.3.g.b | 4 | ||
20.d | odd | 2 | 1 | 1200.3.bg.g | 4 | ||
20.e | even | 4 | 2 | 1200.3.bg.g | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.3.f.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
75.3.f.b | ✓ | 4 | 5.b | even | 2 | 1 | inner |
75.3.f.b | ✓ | 4 | 5.c | odd | 4 | 2 | inner |
225.3.g.b | 4 | 3.b | odd | 2 | 1 | ||
225.3.g.b | 4 | 15.d | odd | 2 | 1 | ||
225.3.g.b | 4 | 15.e | even | 4 | 2 | ||
1200.3.bg.g | 4 | 4.b | odd | 2 | 1 | ||
1200.3.bg.g | 4 | 20.d | odd | 2 | 1 | ||
1200.3.bg.g | 4 | 20.e | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 144 \)
acting on \(S_{3}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 144 \)
$3$
\( T^{4} + 9 \)
$5$
\( T^{4} \)
$7$
\( T^{4} + 5625 \)
$11$
\( (T - 6)^{4} \)
$13$
\( T^{4} + 729 \)
$17$
\( T^{4} + 345744 \)
$19$
\( (T^{2} + 529)^{2} \)
$23$
\( T^{4} + 90000 \)
$29$
\( (T^{2} + 36)^{2} \)
$31$
\( (T - 25)^{4} \)
$37$
\( T^{4} + 1440000 \)
$41$
\( (T + 60)^{4} \)
$43$
\( T^{4} + 51883209 \)
$47$
\( T^{4} + 11664 \)
$53$
\( T^{4} + 1440000 \)
$59$
\( (T^{2} + 324)^{2} \)
$61$
\( (T + 37)^{4} \)
$67$
\( T^{4} + 1750329 \)
$71$
\( (T - 132)^{4} \)
$73$
\( T^{4} + 1440000 \)
$79$
\( (T^{2} + 100)^{2} \)
$83$
\( T^{4} + 144 \)
$89$
\( (T^{2} + 17424)^{2} \)
$97$
\( T^{4} + 1172889 \)
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