Newspace parameters
| Level: | \( N \) | \(=\) | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 450.g (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(46.5164833877\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 30) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + 3\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 3\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 307.1 |
|
−2.00000 | − | 2.00000i | 0 | 8.00000i | 0 | 0 | −44.4393 | − | 44.4393i | 16.0000 | − | 16.0000i | 0 | 0 | ||||||||||||||||||||||||
| 307.2 | −2.00000 | − | 2.00000i | 0 | 8.00000i | 0 | 0 | 58.4393 | + | 58.4393i | 16.0000 | − | 16.0000i | 0 | 0 | |||||||||||||||||||||||||
| 343.1 | −2.00000 | + | 2.00000i | 0 | − | 8.00000i | 0 | 0 | −44.4393 | + | 44.4393i | 16.0000 | + | 16.0000i | 0 | 0 | ||||||||||||||||||||||||
| 343.2 | −2.00000 | + | 2.00000i | 0 | − | 8.00000i | 0 | 0 | 58.4393 | − | 58.4393i | 16.0000 | + | 16.0000i | 0 | 0 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 450.5.g.f | 4 | |
| 3.b | odd | 2 | 1 | 150.5.f.e | 4 | ||
| 5.b | even | 2 | 1 | 90.5.g.e | 4 | ||
| 5.c | odd | 4 | 1 | 90.5.g.e | 4 | ||
| 5.c | odd | 4 | 1 | inner | 450.5.g.f | 4 | |
| 15.d | odd | 2 | 1 | 30.5.f.a | ✓ | 4 | |
| 15.e | even | 4 | 1 | 30.5.f.a | ✓ | 4 | |
| 15.e | even | 4 | 1 | 150.5.f.e | 4 | ||
| 60.h | even | 2 | 1 | 240.5.bg.b | 4 | ||
| 60.l | odd | 4 | 1 | 240.5.bg.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 30.5.f.a | ✓ | 4 | 15.d | odd | 2 | 1 | |
| 30.5.f.a | ✓ | 4 | 15.e | even | 4 | 1 | |
| 90.5.g.e | 4 | 5.b | even | 2 | 1 | ||
| 90.5.g.e | 4 | 5.c | odd | 4 | 1 | ||
| 150.5.f.e | 4 | 3.b | odd | 2 | 1 | ||
| 150.5.f.e | 4 | 15.e | even | 4 | 1 | ||
| 240.5.bg.b | 4 | 60.h | even | 2 | 1 | ||
| 240.5.bg.b | 4 | 60.l | odd | 4 | 1 | ||
| 450.5.g.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 450.5.g.f | 4 | 5.c | odd | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(450, [\chi])\):
|
\( T_{7}^{4} - 28T_{7}^{3} + 392T_{7}^{2} + 145432T_{7} + 26977636 \)
|
|
\( T_{11}^{2} + 232T_{11} + 10810 \)
|
|
\( T_{17}^{4} - 392T_{17}^{3} + 76832T_{17}^{2} + 14866208T_{17} + 1438229776 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4 T + 8)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 28 T^{3} + \cdots + 26977636 \)
|
| $11$ |
\( (T^{2} + 232 T + 10810)^{2} \)
|
| $13$ |
\( T^{4} - 336 T^{3} + \cdots + 618815376 \)
|
| $17$ |
\( T^{4} + \cdots + 1438229776 \)
|
| $19$ |
\( T^{4} + \cdots + 24945043600 \)
|
| $23$ |
\( T^{4} + \cdots + 4830250000 \)
|
| $29$ |
\( T^{4} + \cdots + 60069108100 \)
|
| $31$ |
\( (T^{2} + 280 T - 276104)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 307479122064 \)
|
| $41$ |
\( (T^{2} + 196 T - 2193812)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 273914063424 \)
|
| $47$ |
\( T^{4} + \cdots + 100134845427600 \)
|
| $53$ |
\( T^{4} + \cdots + 4407010106944 \)
|
| $59$ |
\( T^{4} + \cdots + 812612102500 \)
|
| $61$ |
\( (T^{2} + 2268 T - 8149140)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 502668237829696 \)
|
| $71$ |
\( (T^{2} + 9032 T + 19716880)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 21\!\cdots\!16 \)
|
| $79$ |
\( T^{4} + \cdots + 12490004174400 \)
|
| $83$ |
\( T^{4} + \cdots + 12\!\cdots\!56 \)
|
| $89$ |
\( T^{4} + \cdots + 50\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 41\!\cdots\!04 \)
|
| show more | |
| show less |