Newspace parameters
| Level: | \( N \) | \(=\) | \( 230 = 2 \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 230.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.83655924649\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1101.1 |
| Defining polynomial: |
\( x^{3} - x^{2} - 9x + 12 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 9x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 7 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | −3.11903 | 1.00000 | −1.00000 | −3.11903 | 4.50973 | 1.00000 | 6.72833 | −1.00000 | |||||||||||||||||||||||||||
| 1.2 | 1.00000 | 1.43163 | 1.00000 | −1.00000 | 1.43163 | 3.08719 | 1.00000 | −0.950444 | −1.00000 | ||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 2.68740 | 1.00000 | −1.00000 | 2.68740 | −4.59692 | 1.00000 | 4.22212 | −1.00000 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(23\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 230.2.a.d | ✓ | 3 |
| 3.b | odd | 2 | 1 | 2070.2.a.z | 3 | ||
| 4.b | odd | 2 | 1 | 1840.2.a.r | 3 | ||
| 5.b | even | 2 | 1 | 1150.2.a.q | 3 | ||
| 5.c | odd | 4 | 2 | 1150.2.b.j | 6 | ||
| 8.b | even | 2 | 1 | 7360.2.a.bz | 3 | ||
| 8.d | odd | 2 | 1 | 7360.2.a.ce | 3 | ||
| 20.d | odd | 2 | 1 | 9200.2.a.cf | 3 | ||
| 23.b | odd | 2 | 1 | 5290.2.a.r | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 230.2.a.d | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1150.2.a.q | 3 | 5.b | even | 2 | 1 | ||
| 1150.2.b.j | 6 | 5.c | odd | 4 | 2 | ||
| 1840.2.a.r | 3 | 4.b | odd | 2 | 1 | ||
| 2070.2.a.z | 3 | 3.b | odd | 2 | 1 | ||
| 5290.2.a.r | 3 | 23.b | odd | 2 | 1 | ||
| 7360.2.a.bz | 3 | 8.b | even | 2 | 1 | ||
| 7360.2.a.ce | 3 | 8.d | odd | 2 | 1 | ||
| 9200.2.a.cf | 3 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - T_{3}^{2} - 9T_{3} + 12 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(230))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 1)^{3} \)
$3$
\( T^{3} - T^{2} - 9T + 12 \)
$5$
\( (T + 1)^{3} \)
$7$
\( T^{3} - 3 T^{2} - 21 T + 64 \)
$11$
\( T^{3} - 3 T^{2} - 39 T + 144 \)
$13$
\( T^{3} + T^{2} - 15 T - 18 \)
$17$
\( T^{3} + 7 T^{2} + 7 T - 18 \)
$19$
\( T^{3} - 3 T^{2} - 21 T + 64 \)
$23$
\( (T + 1)^{3} \)
$29$
\( T^{3} + 4 T^{2} - 32 T + 24 \)
$31$
\( T^{3} + 5 T^{2} - 7 T - 8 \)
$37$
\( T^{3} + 2 T^{2} - 40 T - 32 \)
$41$
\( T^{3} - T^{2} - 59 T + 186 \)
$43$
\( (T - 8)^{3} \)
$47$
\( T^{3} + 14 T^{2} + 4 T - 288 \)
$53$
\( (T + 6)^{3} \)
$59$
\( T^{3} - 14 T^{2} + 28 T + 144 \)
$61$
\( T^{3} - T^{2} - 157 T + 526 \)
$67$
\( T^{3} - 8 T^{2} - 144 T + 384 \)
$71$
\( T^{3} - 11 T^{2} + 31 T - 24 \)
$73$
\( T^{3} + 8 T^{2} - 40 T - 248 \)
$79$
\( T^{3} + 4 T^{2} - 240 T - 1152 \)
$83$
\( T^{3} - 8 T^{2} - 20 T + 96 \)
$89$
\( T^{3} - 18 T^{2} - 48 T + 1152 \)
$97$
\( T^{3} + 33 T^{2} + 279 T + 166 \)
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