Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(23.4288769113\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.717484.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 244x - 1016 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2\cdot 5 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 244x - 1016 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} - 9\nu - 160 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 21\nu - 169 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta _1 + 3 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 9\beta_{2} + 42\beta _1 + 1627 ) / 10 \)
|
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 |
|
−14.0237 | 27.0000 | 68.6631 | 0 | −378.639 | −1077.72 | 832.121 | 729.000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 1.67926 | 27.0000 | −125.180 | 0 | 45.3401 | 1415.20 | −425.156 | 729.000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 18.3444 | 27.0000 | 208.517 | 0 | 495.299 | −254.472 | 1477.04 | 729.000 | 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 | 75.8.a.h | yes | 3 |
| 3.b | odd | 2 | 1 | 225.8.a.x | 3 | ||
| 5.b | even | 2 | 1 | 75.8.a.g | ✓ | 3 | |
| 5.c | odd | 4 | 2 | 75.8.b.f | 6 | ||
| 15.d | odd | 2 | 1 | 225.8.a.y | 3 | ||
| 15.e | even | 4 | 2 | 225.8.b.q | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.8.a.g | ✓ | 3 | 5.b | even | 2 | 1 | |
| 75.8.a.h | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 75.8.b.f | 6 | 5.c | odd | 4 | 2 | ||
| 225.8.a.x | 3 | 3.b | odd | 2 | 1 | ||
| 225.8.a.y | 3 | 15.d | odd | 2 | 1 | ||
| 225.8.b.q | 6 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 6T_{2}^{2} - 250T_{2} + 432 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 6 T^{2} + \cdots + 432 \)
|
| $3$ |
\( (T - 27)^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} - 83 T^{2} + \cdots - 388118385 \)
|
| $11$ |
\( T^{3} + \cdots + 102949073496 \)
|
| $13$ |
\( T^{3} + \cdots + 13216978519 \)
|
| $17$ |
\( T^{3} + \cdots + 22129700848056 \)
|
| $19$ |
\( T^{3} + \cdots + 19077937228225 \)
|
| $23$ |
\( T^{3} + \cdots - 83778646871160 \)
|
| $29$ |
\( T^{3} + \cdots + 853069492719000 \)
|
| $31$ |
\( T^{3} + \cdots + 857899428441975 \)
|
| $37$ |
\( T^{3} + \cdots - 14\!\cdots\!80 \)
|
| $41$ |
\( T^{3} + \cdots + 30\!\cdots\!20 \)
|
| $43$ |
\( T^{3} + \cdots - 16\!\cdots\!33 \)
|
| $47$ |
\( T^{3} + \cdots - 52\!\cdots\!56 \)
|
| $53$ |
\( T^{3} + \cdots + 89\!\cdots\!60 \)
|
| $59$ |
\( T^{3} + \cdots + 15\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 14\!\cdots\!87 \)
|
| $67$ |
\( T^{3} + \cdots + 17\!\cdots\!31 \)
|
| $71$ |
\( T^{3} + \cdots - 21\!\cdots\!08 \)
|
| $73$ |
\( T^{3} + \cdots + 45\!\cdots\!00 \)
|
| $79$ |
\( T^{3} + \cdots + 18\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 87\!\cdots\!36 \)
|
| $89$ |
\( T^{3} + \cdots + 19\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 75\!\cdots\!29 \)
|
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