Newspace parameters
| Level: | \( N \) | \(=\) | \( 375 = 3 \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 375.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.99439007580\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 3x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 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 \(\left(\mathbb{Z}/375\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(251\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 124.1 |
|
− | 2.61803i | − | 1.00000i | −4.85410 | 0 | −2.61803 | − | 1.38197i | 7.47214i | −1.00000 | 0 | |||||||||||||||||||||||||||
| 124.2 | − | 0.381966i | − | 1.00000i | 1.85410 | 0 | −0.381966 | − | 3.61803i | − | 1.47214i | −1.00000 | 0 | |||||||||||||||||||||||||||
| 124.3 | 0.381966i | 1.00000i | 1.85410 | 0 | −0.381966 | 3.61803i | 1.47214i | −1.00000 | 0 | |||||||||||||||||||||||||||||||
| 124.4 | 2.61803i | 1.00000i | −4.85410 | 0 | −2.61803 | 1.38197i | − | 7.47214i | −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 | 375.2.b.a | 4 | |
| 3.b | odd | 2 | 1 | 1125.2.b.b | 4 | ||
| 4.b | odd | 2 | 1 | 6000.2.f.n | 4 | ||
| 5.b | even | 2 | 1 | inner | 375.2.b.a | 4 | |
| 5.c | odd | 4 | 1 | 375.2.a.a | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 375.2.a.d | yes | 2 | |
| 15.d | odd | 2 | 1 | 1125.2.b.b | 4 | ||
| 15.e | even | 4 | 1 | 1125.2.a.a | 2 | ||
| 15.e | even | 4 | 1 | 1125.2.a.f | 2 | ||
| 20.d | odd | 2 | 1 | 6000.2.f.n | 4 | ||
| 20.e | even | 4 | 1 | 6000.2.a.m | 2 | ||
| 20.e | even | 4 | 1 | 6000.2.a.q | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 375.2.a.a | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 375.2.a.d | yes | 2 | 5.c | odd | 4 | 1 | |
| 375.2.b.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 375.2.b.a | 4 | 5.b | even | 2 | 1 | inner | |
| 1125.2.a.a | 2 | 15.e | even | 4 | 1 | ||
| 1125.2.a.f | 2 | 15.e | even | 4 | 1 | ||
| 1125.2.b.b | 4 | 3.b | odd | 2 | 1 | ||
| 1125.2.b.b | 4 | 15.d | odd | 2 | 1 | ||
| 6000.2.a.m | 2 | 20.e | even | 4 | 1 | ||
| 6000.2.a.q | 2 | 20.e | even | 4 | 1 | ||
| 6000.2.f.n | 4 | 4.b | odd | 2 | 1 | ||
| 6000.2.f.n | 4 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 7T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(375, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 7T^{2} + 1 \)
$3$
\( (T^{2} + 1)^{2} \)
$5$
\( T^{4} \)
$7$
\( T^{4} + 15T^{2} + 25 \)
$11$
\( (T^{2} + 8 T + 11)^{2} \)
$13$
\( (T^{2} + 9)^{2} \)
$17$
\( T^{4} + 43T^{2} + 361 \)
$19$
\( (T - 1)^{4} \)
$23$
\( (T^{2} + 45)^{2} \)
$29$
\( (T^{2} - 4 T - 1)^{2} \)
$31$
\( (T^{2} - 5 T - 25)^{2} \)
$37$
\( (T^{2} + 25)^{2} \)
$41$
\( (T^{2} + 15 T + 45)^{2} \)
$43$
\( T^{4} + 27T^{2} + 121 \)
$47$
\( T^{4} + 258 T^{2} + 14641 \)
$53$
\( T^{4} + 15T^{2} + 25 \)
$59$
\( (T^{2} - 3 T - 149)^{2} \)
$61$
\( (T^{2} - T - 31)^{2} \)
$67$
\( (T^{2} + 64)^{2} \)
$71$
\( (T^{2} - 9 T + 9)^{2} \)
$73$
\( T^{4} + 335 T^{2} + 21025 \)
$79$
\( (T^{2} + 10 T + 20)^{2} \)
$83$
\( T^{4} + 123T^{2} + 3721 \)
$89$
\( (T^{2} - 2 T - 19)^{2} \)
$97$
\( T^{4} + 203 T^{2} + 10201 \)
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