Newspace parameters
| Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 825.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(48.6765757547\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{17})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 9x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 165) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 9x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{2} - 5\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).
| \(n\) | \(376\) | \(551\) | \(727\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 199.1 |
|
− | 2.56155i | − | 3.00000i | 1.43845 | 0 | −7.68466 | 6.24621i | − | 24.1771i | −9.00000 | 0 | |||||||||||||||||||||||||||
| 199.2 | − | 1.56155i | 3.00000i | 5.56155 | 0 | 4.68466 | 10.2462i | − | 21.1771i | −9.00000 | 0 | |||||||||||||||||||||||||||||
| 199.3 | 1.56155i | − | 3.00000i | 5.56155 | 0 | 4.68466 | − | 10.2462i | 21.1771i | −9.00000 | 0 | |||||||||||||||||||||||||||||
| 199.4 | 2.56155i | 3.00000i | 1.43845 | 0 | −7.68466 | − | 6.24621i | 24.1771i | −9.00000 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 825.4.c.j | 4 | |
| 5.b | even | 2 | 1 | inner | 825.4.c.j | 4 | |
| 5.c | odd | 4 | 1 | 165.4.a.c | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 825.4.a.m | 2 | ||
| 15.e | even | 4 | 1 | 495.4.a.d | 2 | ||
| 15.e | even | 4 | 1 | 2475.4.a.n | 2 | ||
| 55.e | even | 4 | 1 | 1815.4.a.n | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.4.a.c | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 495.4.a.d | 2 | 15.e | even | 4 | 1 | ||
| 825.4.a.m | 2 | 5.c | odd | 4 | 1 | ||
| 825.4.c.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 825.4.c.j | 4 | 5.b | even | 2 | 1 | inner | |
| 1815.4.a.n | 2 | 55.e | even | 4 | 1 | ||
| 2475.4.a.n | 2 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(825, [\chi])\):
|
\( T_{2}^{4} + 9T_{2}^{2} + 16 \)
|
|
\( T_{7}^{4} + 144T_{7}^{2} + 4096 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 9T^{2} + 16 \)
|
| $3$ |
\( (T^{2} + 9)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 144T^{2} + 4096 \)
|
| $11$ |
\( (T + 11)^{4} \)
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| $13$ |
\( T^{4} + 4084 T^{2} + 4032064 \)
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| $17$ |
\( T^{4} + 16584 T^{2} + 66650896 \)
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| $19$ |
\( (T^{2} - 170 T + 5168)^{2} \)
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| $23$ |
\( T^{4} + 38288 T^{2} + 131239936 \)
|
| $29$ |
\( (T^{2} - 158 T + 1328)^{2} \)
|
| $31$ |
\( (T^{2} - 60 T + 288)^{2} \)
|
| $37$ |
\( T^{4} + 175816 T^{2} + 350288656 \)
|
| $41$ |
\( (T^{2} - 38 T - 100432)^{2} \)
|
| $43$ |
\( T^{4} + 263824 T^{2} + 1478656 \)
|
| $47$ |
\( T^{4} + \cdots + 34500833536 \)
|
| $53$ |
\( T^{4} + \cdots + 11571735184 \)
|
| $59$ |
\( (T^{2} + 248 T - 229424)^{2} \)
|
| $61$ |
\( (T^{2} - 72 T - 561812)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 18850191616 \)
|
| $71$ |
\( (T^{2} - 2060 T + 1052672)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 2002741504 \)
|
| $79$ |
\( (T^{2} + 642 T - 294912)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 563736678976 \)
|
| $89$ |
\( (T^{2} + 244 T - 54748)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 595175218576 \)
|
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