Newspace parameters
| Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 912.o (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(24.8502001097\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 57) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(229\) | \(305\) | \(799\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 721.1 |
|
0 | − | 1.73205i | 0 | 4.00000 | 0 | 10.0000 | 0 | −3.00000 | 0 | |||||||||||||||||||||||
| 721.2 | 0 | 1.73205i | 0 | 4.00000 | 0 | 10.0000 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 912.3.o.a | 2 | |
| 3.b | odd | 2 | 1 | 2736.3.o.e | 2 | ||
| 4.b | odd | 2 | 1 | 57.3.c.a | ✓ | 2 | |
| 12.b | even | 2 | 1 | 171.3.c.c | 2 | ||
| 19.b | odd | 2 | 1 | inner | 912.3.o.a | 2 | |
| 57.d | even | 2 | 1 | 2736.3.o.e | 2 | ||
| 76.d | even | 2 | 1 | 57.3.c.a | ✓ | 2 | |
| 228.b | odd | 2 | 1 | 171.3.c.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.3.c.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 57.3.c.a | ✓ | 2 | 76.d | even | 2 | 1 | |
| 171.3.c.c | 2 | 12.b | even | 2 | 1 | ||
| 171.3.c.c | 2 | 228.b | odd | 2 | 1 | ||
| 912.3.o.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 912.3.o.a | 2 | 19.b | odd | 2 | 1 | inner | |
| 2736.3.o.e | 2 | 3.b | odd | 2 | 1 | ||
| 2736.3.o.e | 2 | 57.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 4 \)
acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 3 \)
$5$
\( (T - 4)^{2} \)
$7$
\( (T - 10)^{2} \)
$11$
\( (T + 10)^{2} \)
$13$
\( T^{2} + 588 \)
$17$
\( (T - 10)^{2} \)
$19$
\( (T + 19)^{2} \)
$23$
\( (T - 20)^{2} \)
$29$
\( T^{2} + 1200 \)
$31$
\( T^{2} + 300 \)
$37$
\( T^{2} + 108 \)
$41$
\( T^{2} + 1200 \)
$43$
\( (T - 10)^{2} \)
$47$
\( (T - 80)^{2} \)
$53$
\( T^{2} + 1728 \)
$59$
\( T^{2} + 1200 \)
$61$
\( (T + 10)^{2} \)
$67$
\( T^{2} + 5808 \)
$71$
\( T^{2} + 10800 \)
$73$
\( (T + 10)^{2} \)
$79$
\( T^{2} + 300 \)
$83$
\( (T + 70)^{2} \)
$89$
\( T^{2} + 10800 \)
$97$
\( T^{2} + 5808 \)
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