Newspace parameters
| Level: | \( N \) | \(=\) | \( 675 = 3^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 675.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(39.8262892539\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.3173728.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - 29x^{2} + 118 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 29x^{2} + 118 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 21\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 15 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 21\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−4.90965 | 0 | 16.1047 | 0 | 0 | 2.10469 | −39.7912 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −2.21254 | 0 | −3.10469 | 0 | 0 | −17.1047 | 24.5695 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 2.21254 | 0 | −3.10469 | 0 | 0 | −17.1047 | −24.5695 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 4.90965 | 0 | 16.1047 | 0 | 0 | 2.10469 | 39.7912 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 675.4.a.w | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 675.4.a.w | ✓ | 4 |
| 5.b | even | 2 | 1 | 675.4.a.x | yes | 4 | |
| 5.c | odd | 4 | 2 | 675.4.b.o | 8 | ||
| 15.d | odd | 2 | 1 | 675.4.a.x | yes | 4 | |
| 15.e | even | 4 | 2 | 675.4.b.o | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 675.4.a.w | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 675.4.a.w | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 675.4.a.x | yes | 4 | 5.b | even | 2 | 1 | |
| 675.4.a.x | yes | 4 | 15.d | odd | 2 | 1 | |
| 675.4.b.o | 8 | 5.c | odd | 4 | 2 | ||
| 675.4.b.o | 8 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(675))\):
|
\( T_{2}^{4} - 29T_{2}^{2} + 118 \)
|
|
\( T_{7}^{2} + 15T_{7} - 36 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 29T^{2} + 118 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 15 T - 36)^{2} \)
|
| $11$ |
\( T^{4} - 6092 T^{2} + 7257472 \)
|
| $13$ |
\( (T^{2} + 11 T - 4490)^{2} \)
|
| $17$ |
\( T^{4} - 11468 T^{2} + 15980032 \)
|
| $19$ |
\( (T^{2} - 173 T + 5176)^{2} \)
|
| $23$ |
\( T^{4} - 47664 T^{2} + 244684800 \)
|
| $29$ |
\( T^{4} - 56492 T^{2} + 26288512 \)
|
| $31$ |
\( (T^{2} - 372 T + 31275)^{2} \)
|
| $37$ |
\( (T^{2} - 38 T - 1115)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 12373196800 \)
|
| $43$ |
\( (T^{2} + 140 T - 24989)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 6419834368 \)
|
| $53$ |
\( T^{4} - 89228 T^{2} + 475783552 \)
|
| $59$ |
\( T^{4} - 1856 T^{2} + 483328 \)
|
| $61$ |
\( (T^{2} - 89 T - 516926)^{2} \)
|
| $67$ |
\( (T^{2} + 69 T - 319932)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 8294391808 \)
|
| $73$ |
\( (T^{2} + 518 T + 65605)^{2} \)
|
| $79$ |
\( (T^{2} - 1038 T - 108495)^{2} \)
|
| $83$ |
\( T^{4} - 64332 T^{2} + 611712 \)
|
| $89$ |
\( T^{4} + \cdots + 4761627379200 \)
|
| $97$ |
\( (T^{2} - 1025 T - 678386)^{2} \)
|
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