Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(12.0287864860\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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 |
|
−7.00000 | −9.00000 | 17.0000 | 0 | 63.0000 | −12.0000 | 105.000 | 81.0000 | 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 | 75.6.a.a | 1 | |
| 3.b | odd | 2 | 1 | 225.6.a.h | 1 | ||
| 5.b | even | 2 | 1 | 15.6.a.b | ✓ | 1 | |
| 5.c | odd | 4 | 2 | 75.6.b.a | 2 | ||
| 15.d | odd | 2 | 1 | 45.6.a.a | 1 | ||
| 15.e | even | 4 | 2 | 225.6.b.a | 2 | ||
| 20.d | odd | 2 | 1 | 240.6.a.b | 1 | ||
| 35.c | odd | 2 | 1 | 735.6.a.b | 1 | ||
| 40.e | odd | 2 | 1 | 960.6.a.x | 1 | ||
| 40.f | even | 2 | 1 | 960.6.a.k | 1 | ||
| 60.h | even | 2 | 1 | 720.6.a.q | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.6.a.b | ✓ | 1 | 5.b | even | 2 | 1 | |
| 45.6.a.a | 1 | 15.d | odd | 2 | 1 | ||
| 75.6.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
| 75.6.b.a | 2 | 5.c | odd | 4 | 2 | ||
| 225.6.a.h | 1 | 3.b | odd | 2 | 1 | ||
| 225.6.b.a | 2 | 15.e | even | 4 | 2 | ||
| 240.6.a.b | 1 | 20.d | odd | 2 | 1 | ||
| 720.6.a.q | 1 | 60.h | even | 2 | 1 | ||
| 735.6.a.b | 1 | 35.c | odd | 2 | 1 | ||
| 960.6.a.k | 1 | 40.f | even | 2 | 1 | ||
| 960.6.a.x | 1 | 40.e | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 7 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T + 7 \)
|
| $3$ |
\( T + 9 \)
|
| $5$ |
\( T \)
|
| $7$ |
\( T + 12 \)
|
| $11$ |
\( T - 112 \)
|
| $13$ |
\( T - 974 \)
|
| $17$ |
\( T + 2182 \)
|
| $19$ |
\( T - 1420 \)
|
| $23$ |
\( T + 3216 \)
|
| $29$ |
\( T + 4150 \)
|
| $31$ |
\( T + 5688 \)
|
| $37$ |
\( T + 6482 \)
|
| $41$ |
\( T - 5402 \)
|
| $43$ |
\( T - 21764 \)
|
| $47$ |
\( T - 368 \)
|
| $53$ |
\( T + 12586 \)
|
| $59$ |
\( T + 25520 \)
|
| $61$ |
\( T - 11782 \)
|
| $67$ |
\( T - 13188 \)
|
| $71$ |
\( T + 35968 \)
|
| $73$ |
\( T + 73186 \)
|
| $79$ |
\( T + 52440 \)
|
| $83$ |
\( T + 69036 \)
|
| $89$ |
\( T + 33870 \)
|
| $97$ |
\( T + 143042 \)
|
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