Newspace parameters
| Level: | \( N \) | \(=\) | \( 1875 = 3 \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1875.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(14.9719503790\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{5})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 3x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 75) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 3x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 2\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).
| \(n\) | \(626\) | \(1252\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1249.1 |
|
− | 1.00000i | 1.00000i | 1.00000 | 0 | 1.00000 | − | 4.47214i | − | 3.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||
| 1249.2 | − | 1.00000i | 1.00000i | 1.00000 | 0 | 1.00000 | 4.47214i | − | 3.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||
| 1249.3 | 1.00000i | − | 1.00000i | 1.00000 | 0 | 1.00000 | − | 4.47214i | 3.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||
| 1249.4 | 1.00000i | − | 1.00000i | 1.00000 | 0 | 1.00000 | 4.47214i | 3.00000i | −1.00000 | 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 | 1875.2.b.b | 4 | |
| 5.b | even | 2 | 1 | inner | 1875.2.b.b | 4 | |
| 5.c | odd | 4 | 1 | 1875.2.a.a | 2 | ||
| 5.c | odd | 4 | 1 | 1875.2.a.d | 2 | ||
| 15.e | even | 4 | 1 | 5625.2.a.a | 2 | ||
| 15.e | even | 4 | 1 | 5625.2.a.h | 2 | ||
| 25.d | even | 5 | 2 | 375.2.i.a | 8 | ||
| 25.e | even | 10 | 2 | 375.2.i.a | 8 | ||
| 25.f | odd | 20 | 2 | 75.2.g.a | ✓ | 4 | |
| 25.f | odd | 20 | 2 | 375.2.g.a | 4 | ||
| 75.l | even | 20 | 2 | 225.2.h.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.2.g.a | ✓ | 4 | 25.f | odd | 20 | 2 | |
| 225.2.h.a | 4 | 75.l | even | 20 | 2 | ||
| 375.2.g.a | 4 | 25.f | odd | 20 | 2 | ||
| 375.2.i.a | 8 | 25.d | even | 5 | 2 | ||
| 375.2.i.a | 8 | 25.e | even | 10 | 2 | ||
| 1875.2.a.a | 2 | 5.c | odd | 4 | 1 | ||
| 1875.2.a.d | 2 | 5.c | odd | 4 | 1 | ||
| 1875.2.b.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 1875.2.b.b | 4 | 5.b | even | 2 | 1 | inner | |
| 5625.2.a.a | 2 | 15.e | even | 4 | 1 | ||
| 5625.2.a.h | 2 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1875, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
|
| $3$ |
\( (T^{2} + 1)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 20)^{2} \)
|
| $11$ |
\( (T^{2} - 2 T - 4)^{2} \)
|
| $13$ |
\( T^{4} + 43T^{2} + 361 \)
|
| $17$ |
\( T^{4} + 23T^{2} + 121 \)
|
| $19$ |
\( (T^{2} - 2 T - 4)^{2} \)
|
| $23$ |
\( (T^{2} + 20)^{2} \)
|
| $29$ |
\( (T^{2} + 11 T + 29)^{2} \)
|
| $31$ |
\( (T^{2} - 10 T + 20)^{2} \)
|
| $37$ |
\( T^{4} + 75T^{2} + 625 \)
|
| $41$ |
\( (T^{2} + 5 T + 5)^{2} \)
|
| $43$ |
\( T^{4} + 92T^{2} + 1936 \)
|
| $47$ |
\( T^{4} + 28T^{2} + 16 \)
|
| $53$ |
\( T^{4} + 15T^{2} + 25 \)
|
| $59$ |
\( (T - 4)^{4} \)
|
| $61$ |
\( (T^{2} - T - 1)^{2} \)
|
| $67$ |
\( T^{4} + 28T^{2} + 16 \)
|
| $71$ |
\( (T^{2} + 6 T + 4)^{2} \)
|
| $73$ |
\( T^{4} + 75T^{2} + 625 \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} + 168T^{2} + 1936 \)
|
| $89$ |
\( (T^{2} + 13 T + 41)^{2} \)
|
| $97$ |
\( T^{4} + 83T^{2} + 361 \)
|
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