Newspace parameters
| Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 225.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(70.2866307339\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{65}, \sqrt{85})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 46x^{2} - 115x - 35 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{2}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 45) |
| 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
\( x^{4} - x^{3} - 46x^{2} - 115x - 35 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4\nu^{3} + 20\nu^{2} + 116\nu - 10 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( -20\nu^{3} + 40\nu^{2} + 1000\nu + 1240 \)
|
| \(\beta_{3}\) | \(=\) |
\( 60\nu^{3} - 240\nu^{2} - 1920\nu - 1200 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 30\beta _1 + 60 ) / 240 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{3} + 3\beta_{2} + 270\beta _1 + 5580 ) / 240 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 16\beta_{3} + 11\beta_{2} + 510\beta _1 + 7260 ) / 60 \)
|
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 |
|
−16.1245 | 0 | 132.000 | 0 | 0 | −891.964 | −64.4981 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −16.1245 | 0 | 132.000 | 0 | 0 | 891.964 | −64.4981 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 16.1245 | 0 | 132.000 | 0 | 0 | −891.964 | 64.4981 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 16.1245 | 0 | 132.000 | 0 | 0 | 891.964 | 64.4981 | 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 |
| 5.b | even | 2 | 1 | inner |
| 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 | 225.8.a.ba | 4 | |
| 3.b | odd | 2 | 1 | inner | 225.8.a.ba | 4 | |
| 5.b | even | 2 | 1 | inner | 225.8.a.ba | 4 | |
| 5.c | odd | 4 | 2 | 45.8.b.c | ✓ | 4 | |
| 15.d | odd | 2 | 1 | inner | 225.8.a.ba | 4 | |
| 15.e | even | 4 | 2 | 45.8.b.c | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.8.b.c | ✓ | 4 | 5.c | odd | 4 | 2 | |
| 45.8.b.c | ✓ | 4 | 15.e | even | 4 | 2 | |
| 225.8.a.ba | 4 | 1.a | even | 1 | 1 | trivial | |
| 225.8.a.ba | 4 | 3.b | odd | 2 | 1 | inner | |
| 225.8.a.ba | 4 | 5.b | even | 2 | 1 | inner | |
| 225.8.a.ba | 4 | 15.d | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(225))\):
|
\( T_{2}^{2} - 260 \)
|
|
\( T_{7}^{2} - 795600 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 260)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} - 795600)^{2} \)
|
| $11$ |
\( (T^{2} - 14994000)^{2} \)
|
| $13$ |
\( (T^{2} - 96267600)^{2} \)
|
| $17$ |
\( (T^{2} - 119165540)^{2} \)
|
| $19$ |
\( (T + 14924)^{4} \)
|
| $23$ |
\( (T^{2} - 50960)^{2} \)
|
| $29$ |
\( (T^{2} - 18668754000)^{2} \)
|
| $31$ |
\( (T + 244192)^{4} \)
|
| $37$ |
\( (T^{2} - 261218552400)^{2} \)
|
| $41$ |
\( (T^{2} - 379744776000)^{2} \)
|
| $43$ |
\( (T^{2} - 430973337600)^{2} \)
|
| $47$ |
\( (T^{2} - 1745086063760)^{2} \)
|
| $53$ |
\( (T^{2} - 1223085907940)^{2} \)
|
| $59$ |
\( (T^{2} - 3849849954000)^{2} \)
|
| $61$ |
\( (T - 1762442)^{4} \)
|
| $67$ |
\( (T^{2} - 987336417600)^{2} \)
|
| $71$ |
\( (T^{2} - 206759304000)^{2} \)
|
| $73$ |
\( (T^{2} - 16456432262400)^{2} \)
|
| $79$ |
\( (T + 2168216)^{4} \)
|
| $83$ |
\( (T^{2} - 560059173440)^{2} \)
|
| $89$ |
\( (T^{2} - 28200960000000)^{2} \)
|
| $97$ |
\( (T^{2} - 127797447585600)^{2} \)
|
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