Newspace parameters
Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 12 \) |
Character orbit: | \([\chi]\) | \(=\) | 75.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(57.6257385420\) |
Analytic rank: | \(1\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 3) |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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 |
|
−78.0000 | 243.000 | 4036.00 | 0 | −18954.0 | 27760.0 | −155064. | 59049.0 | 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 | 75.12.a.a | 1 | |
3.b | odd | 2 | 1 | 225.12.a.f | 1 | ||
5.b | even | 2 | 1 | 3.12.a.a | ✓ | 1 | |
5.c | odd | 4 | 2 | 75.12.b.a | 2 | ||
15.d | odd | 2 | 1 | 9.12.a.a | 1 | ||
15.e | even | 4 | 2 | 225.12.b.a | 2 | ||
20.d | odd | 2 | 1 | 48.12.a.f | 1 | ||
35.c | odd | 2 | 1 | 147.12.a.c | 1 | ||
40.e | odd | 2 | 1 | 192.12.a.g | 1 | ||
40.f | even | 2 | 1 | 192.12.a.q | 1 | ||
45.h | odd | 6 | 2 | 81.12.c.e | 2 | ||
45.j | even | 6 | 2 | 81.12.c.a | 2 | ||
60.h | even | 2 | 1 | 144.12.a.l | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3.12.a.a | ✓ | 1 | 5.b | even | 2 | 1 | |
9.12.a.a | 1 | 15.d | odd | 2 | 1 | ||
48.12.a.f | 1 | 20.d | odd | 2 | 1 | ||
75.12.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
75.12.b.a | 2 | 5.c | odd | 4 | 2 | ||
81.12.c.a | 2 | 45.j | even | 6 | 2 | ||
81.12.c.e | 2 | 45.h | odd | 6 | 2 | ||
144.12.a.l | 1 | 60.h | even | 2 | 1 | ||
147.12.a.c | 1 | 35.c | odd | 2 | 1 | ||
192.12.a.g | 1 | 40.e | odd | 2 | 1 | ||
192.12.a.q | 1 | 40.f | even | 2 | 1 | ||
225.12.a.f | 1 | 3.b | odd | 2 | 1 | ||
225.12.b.a | 2 | 15.e | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 78 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 78 \)
$3$
\( T - 243 \)
$5$
\( T \)
$7$
\( T - 27760 \)
$11$
\( T - 637836 \)
$13$
\( T + 766214 \)
$17$
\( T + 3084354 \)
$19$
\( T + 19511404 \)
$23$
\( T + 15312360 \)
$29$
\( T - 10751262 \)
$31$
\( T + 50937400 \)
$37$
\( T + 664740830 \)
$41$
\( T - 898833450 \)
$43$
\( T - 957947188 \)
$47$
\( T - 1555741344 \)
$53$
\( T + 3792417030 \)
$59$
\( T - 555306924 \)
$61$
\( T - 4950420998 \)
$67$
\( T + 5292399284 \)
$71$
\( T + 14831086248 \)
$73$
\( T + 13971005210 \)
$79$
\( T - 3720542360 \)
$83$
\( T + 8768454036 \)
$89$
\( T + 25472769174 \)
$97$
\( T - 39092494846 \)
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