Newspace parameters
Level: | \( N \) | \(=\) | \( 375 = 3 \cdot 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 375.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(2.99439007580\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{5}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x - 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
0.381966 | −1.00000 | −1.85410 | 0 | −0.381966 | 3.61803 | −1.47214 | 1.00000 | 0 | ||||||||||||||||||||||||
1.2 | 2.61803 | −1.00000 | 4.85410 | 0 | −2.61803 | 1.38197 | 7.47214 | 1.00000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(5\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 375.2.a.d | yes | 2 |
3.b | odd | 2 | 1 | 1125.2.a.a | 2 | ||
4.b | odd | 2 | 1 | 6000.2.a.q | 2 | ||
5.b | even | 2 | 1 | 375.2.a.a | ✓ | 2 | |
5.c | odd | 4 | 2 | 375.2.b.a | 4 | ||
15.d | odd | 2 | 1 | 1125.2.a.f | 2 | ||
15.e | even | 4 | 2 | 1125.2.b.b | 4 | ||
20.d | odd | 2 | 1 | 6000.2.a.m | 2 | ||
20.e | even | 4 | 2 | 6000.2.f.n | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
375.2.a.a | ✓ | 2 | 5.b | even | 2 | 1 | |
375.2.a.d | yes | 2 | 1.a | even | 1 | 1 | trivial |
375.2.b.a | 4 | 5.c | odd | 4 | 2 | ||
1125.2.a.a | 2 | 3.b | odd | 2 | 1 | ||
1125.2.a.f | 2 | 15.d | odd | 2 | 1 | ||
1125.2.b.b | 4 | 15.e | even | 4 | 2 | ||
6000.2.a.m | 2 | 20.d | odd | 2 | 1 | ||
6000.2.a.q | 2 | 4.b | odd | 2 | 1 | ||
6000.2.f.n | 4 | 20.e | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 3T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(375))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 3T + 1 \)
$3$
\( (T + 1)^{2} \)
$5$
\( T^{2} \)
$7$
\( T^{2} - 5T + 5 \)
$11$
\( T^{2} + 8T + 11 \)
$13$
\( (T - 3)^{2} \)
$17$
\( T^{2} - 9T + 19 \)
$19$
\( (T + 1)^{2} \)
$23$
\( T^{2} - 45 \)
$29$
\( T^{2} + 4T - 1 \)
$31$
\( T^{2} - 5T - 25 \)
$37$
\( (T - 5)^{2} \)
$41$
\( T^{2} + 15T + 45 \)
$43$
\( T^{2} + 7T + 11 \)
$47$
\( T^{2} + 4T - 121 \)
$53$
\( T^{2} + 5T + 5 \)
$59$
\( T^{2} + 3T - 149 \)
$61$
\( T^{2} - T - 31 \)
$67$
\( (T + 8)^{2} \)
$71$
\( T^{2} - 9T + 9 \)
$73$
\( T^{2} - 25T + 145 \)
$79$
\( T^{2} - 10T + 20 \)
$83$
\( T^{2} + T - 61 \)
$89$
\( T^{2} + 2T - 19 \)
$97$
\( T^{2} + T - 101 \)
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