Newspace parameters
Level: | \( N \) | \(=\) | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 450.f (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(26.5508595026\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | yes |
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
\(\beta_{1}\) | \(=\) | \( \zeta_{8}^{2} \) |
\(\beta_{2}\) | \(=\) | \( \zeta_{8}^{3} + \zeta_{8} \) |
\(\beta_{3}\) | \(=\) | \( -\zeta_{8}^{3} + \zeta_{8} \) |
\(\zeta_{8}\) | \(=\) | \( ( \beta_{3} + \beta_{2} ) / 2 \) |
\(\zeta_{8}^{2}\) | \(=\) | \( \beta_1 \) |
\(\zeta_{8}^{3}\) | \(=\) | \( ( -\beta_{3} + \beta_{2} ) / 2 \) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
107.1 |
|
−1.41421 | + | 1.41421i | 0 | − | 4.00000i | 0 | 0 | −9.00000 | − | 9.00000i | 5.65685 | + | 5.65685i | 0 | 0 | |||||||||||||||||||||||
107.2 | 1.41421 | − | 1.41421i | 0 | − | 4.00000i | 0 | 0 | −9.00000 | − | 9.00000i | −5.65685 | − | 5.65685i | 0 | 0 | ||||||||||||||||||||||||
143.1 | −1.41421 | − | 1.41421i | 0 | 4.00000i | 0 | 0 | −9.00000 | + | 9.00000i | 5.65685 | − | 5.65685i | 0 | 0 | |||||||||||||||||||||||||
143.2 | 1.41421 | + | 1.41421i | 0 | 4.00000i | 0 | 0 | −9.00000 | + | 9.00000i | −5.65685 | + | 5.65685i | 0 | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 450.4.f.a | ✓ | 4 |
3.b | odd | 2 | 1 | inner | 450.4.f.a | ✓ | 4 |
5.b | even | 2 | 1 | 450.4.f.c | yes | 4 | |
5.c | odd | 4 | 1 | inner | 450.4.f.a | ✓ | 4 |
5.c | odd | 4 | 1 | 450.4.f.c | yes | 4 | |
15.d | odd | 2 | 1 | 450.4.f.c | yes | 4 | |
15.e | even | 4 | 1 | inner | 450.4.f.a | ✓ | 4 |
15.e | even | 4 | 1 | 450.4.f.c | yes | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
450.4.f.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
450.4.f.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
450.4.f.a | ✓ | 4 | 5.c | odd | 4 | 1 | inner |
450.4.f.a | ✓ | 4 | 15.e | even | 4 | 1 | inner |
450.4.f.c | yes | 4 | 5.b | even | 2 | 1 | |
450.4.f.c | yes | 4 | 5.c | odd | 4 | 1 | |
450.4.f.c | yes | 4 | 15.d | odd | 2 | 1 | |
450.4.f.c | yes | 4 | 15.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 18T_{7} + 162 \)
acting on \(S_{4}^{\mathrm{new}}(450, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 16 \)
$3$
\( T^{4} \)
$5$
\( T^{4} \)
$7$
\( (T^{2} + 18 T + 162)^{2} \)
$11$
\( (T^{2} + 1458)^{2} \)
$13$
\( (T^{2} - 18 T + 162)^{2} \)
$17$
\( T^{4} + 37015056 \)
$19$
\( (T^{2} + 1444)^{2} \)
$23$
\( T^{4} + 3111696 \)
$29$
\( (T^{2} - 5832)^{2} \)
$31$
\( (T + 236)^{4} \)
$37$
\( (T^{2} + 558 T + 155682)^{2} \)
$41$
\( (T^{2} + 118098)^{2} \)
$43$
\( (T^{2} + 720 T + 259200)^{2} \)
$47$
\( T^{4} + 16796160000 \)
$53$
\( T^{4} + 61505984016 \)
$59$
\( (T^{2} - 118098)^{2} \)
$61$
\( (T - 110)^{4} \)
$67$
\( (T^{2} + 1404 T + 985608)^{2} \)
$71$
\( (T^{2} + 5832)^{2} \)
$73$
\( (T^{2} + 756 T + 285768)^{2} \)
$79$
\( (T^{2} + 73984)^{2} \)
$83$
\( T^{4} + 49787136 \)
$89$
\( (T^{2} - 2217618)^{2} \)
$97$
\( (T^{2} + 1368 T + 935712)^{2} \)
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