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: | 4.4.467024.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - 26x^{2} + 101 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 135) |
| 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
\( x^{4} - 26x^{2} + 101 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{3} - 45\nu ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{3} + 69\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} - 15 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( 4\beta_{3} + 15 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 15\beta_{2} + 23\beta_1 ) / 6 \)
|
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 |
|
−3.56155 | 0 | 4.68466 | 0 | 0 | −6.06288 | 11.8078 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −3.56155 | 0 | 4.68466 | 0 | 0 | 6.06288 | 11.8078 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0.561553 | 0 | −7.68466 | 0 | 0 | −29.8369 | −8.80776 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0.561553 | 0 | −7.68466 | 0 | 0 | 29.8369 | −8.80776 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 15.d | 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.t | 4 | |
| 3.b | odd | 2 | 1 | 675.4.a.ba | 4 | ||
| 5.b | even | 2 | 1 | 675.4.a.ba | 4 | ||
| 5.c | odd | 4 | 2 | 135.4.b.b | ✓ | 8 | |
| 15.d | odd | 2 | 1 | inner | 675.4.a.t | 4 | |
| 15.e | even | 4 | 2 | 135.4.b.b | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.4.b.b | ✓ | 8 | 5.c | odd | 4 | 2 | |
| 135.4.b.b | ✓ | 8 | 15.e | even | 4 | 2 | |
| 675.4.a.t | 4 | 1.a | even | 1 | 1 | trivial | |
| 675.4.a.t | 4 | 15.d | odd | 2 | 1 | inner | |
| 675.4.a.ba | 4 | 3.b | odd | 2 | 1 | ||
| 675.4.a.ba | 4 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(675))\):
|
\( T_{2}^{2} + 3T_{2} - 2 \)
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|
\( T_{7}^{4} - 927T_{7}^{2} + 32724 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 3 T - 2)^{2} \)
|
| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} - 927 T^{2} + 32724 \)
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| $11$ |
\( T^{4} - 6903 T^{2} + 11813364 \)
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| $13$ |
\( T^{4} - 5436 T^{2} + 2094336 \)
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| $17$ |
\( (T^{2} - 33 T - 956)^{2} \)
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| $19$ |
\( (T^{2} + 55 T - 1118)^{2} \)
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| $23$ |
\( (T^{2} - 3 T - 12854)^{2} \)
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| $29$ |
\( T^{4} - 6876 T^{2} + 8377344 \)
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| $31$ |
\( (T^{2} + 134 T - 25499)^{2} \)
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| $37$ |
\( T^{4} - 26892 T^{2} + 169641216 \)
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| $41$ |
\( T^{4} + \cdots + 6509327184 \)
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| $43$ |
\( T^{4} + \cdots + 3738520656 \)
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| $47$ |
\( (T^{2} - 558 T + 40288)^{2} \)
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| $53$ |
\( (T^{2} - 912 T + 191599)^{2} \)
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| $59$ |
\( T^{4} + \cdots + 87371640144 \)
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| $61$ |
\( (T^{2} + 899 T + 202012)^{2} \)
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| $67$ |
\( T^{4} + \cdots + 197389073664 \)
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| $71$ |
\( T^{4} + \cdots + 28669365504 \)
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| $73$ |
\( T^{4} + \cdots + 27039219444 \)
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| $79$ |
\( (T^{2} - 341 T + 18016)^{2} \)
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| $83$ |
\( (T^{2} - 192 T - 1184609)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 171008948304 \)
|
| $97$ |
\( T^{4} + \cdots + 801172005696 \)
|
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