Newspace parameters
| Level: | \( N \) | \(=\) | \( 4 = 2^{2} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.62951444024\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-39}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 10 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-39}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−10.0000 | − | 12.4900i | − | 99.9200i | −56.0000 | + | 249.800i | 610.000 | −1248.00 | + | 999.200i | − | 1398.88i | 3680.00 | − | 1798.56i | −3423.00 | −6100.00 | − | 7618.90i | ||||||||||||
| 3.2 | −10.0000 | + | 12.4900i | 99.9200i | −56.0000 | − | 249.800i | 610.000 | −1248.00 | − | 999.200i | 1398.88i | 3680.00 | + | 1798.56i | −3423.00 | −6100.00 | + | 7618.90i | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4.9.b.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 36.9.d.b | 2 | ||
| 4.b | odd | 2 | 1 | inner | 4.9.b.b | ✓ | 2 |
| 5.b | even | 2 | 1 | 100.9.b.c | 2 | ||
| 5.c | odd | 4 | 2 | 100.9.d.b | 4 | ||
| 8.b | even | 2 | 1 | 64.9.c.b | 2 | ||
| 8.d | odd | 2 | 1 | 64.9.c.b | 2 | ||
| 12.b | even | 2 | 1 | 36.9.d.b | 2 | ||
| 16.e | even | 4 | 2 | 256.9.d.e | 4 | ||
| 16.f | odd | 4 | 2 | 256.9.d.e | 4 | ||
| 20.d | odd | 2 | 1 | 100.9.b.c | 2 | ||
| 20.e | even | 4 | 2 | 100.9.d.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4.9.b.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 4.9.b.b | ✓ | 2 | 4.b | odd | 2 | 1 | inner |
| 36.9.d.b | 2 | 3.b | odd | 2 | 1 | ||
| 36.9.d.b | 2 | 12.b | even | 2 | 1 | ||
| 64.9.c.b | 2 | 8.b | even | 2 | 1 | ||
| 64.9.c.b | 2 | 8.d | odd | 2 | 1 | ||
| 100.9.b.c | 2 | 5.b | even | 2 | 1 | ||
| 100.9.b.c | 2 | 20.d | odd | 2 | 1 | ||
| 100.9.d.b | 4 | 5.c | odd | 4 | 2 | ||
| 100.9.d.b | 4 | 20.e | even | 4 | 2 | ||
| 256.9.d.e | 4 | 16.e | even | 4 | 2 | ||
| 256.9.d.e | 4 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 9984 \)
acting on \(S_{9}^{\mathrm{new}}(4, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 20T + 256 \)
$3$
\( T^{2} + 9984 \)
$5$
\( (T - 610)^{2} \)
$7$
\( T^{2} + 1956864 \)
$11$
\( T^{2} + 341702400 \)
$13$
\( (T + 5470)^{2} \)
$17$
\( (T - 73090)^{2} \)
$19$
\( T^{2} + 379641600 \)
$23$
\( T^{2} + 56268585984 \)
$29$
\( (T + 128222)^{2} \)
$31$
\( T^{2} + 4616601600 \)
$37$
\( (T + 3472030)^{2} \)
$41$
\( (T - 2146882)^{2} \)
$43$
\( T^{2} + 35142983526144 \)
$47$
\( T^{2} + 58160324112384 \)
$53$
\( (T - 824290)^{2} \)
$59$
\( T^{2} + 13879469510400 \)
$61$
\( (T + 14746078)^{2} \)
$67$
\( T^{2} + \cdots + 232766284318464 \)
$71$
\( T^{2} + 1430516505600 \)
$73$
\( (T + 5725630)^{2} \)
$79$
\( T^{2} + 12\!\cdots\!00 \)
$83$
\( T^{2} + 26\!\cdots\!24 \)
$89$
\( (T + 83324222)^{2} \)
$97$
\( (T - 120619010)^{2} \)
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