Newspace parameters
| Level: | \( N \) | \(=\) | \( 230 = 2 \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 230.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(36.8882785570\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.27980.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 47x - 106 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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
\( x^{3} - 47x - 106 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4\nu - 31 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} + 4\beta _1 + 93 ) / 3 \)
|
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 |
|
−4.00000 | −22.3561 | 16.0000 | 25.0000 | 89.4244 | −110.123 | −64.0000 | 256.795 | −100.000 | |||||||||||||||||||||||||||
| 1.2 | −4.00000 | −10.5273 | 16.0000 | 25.0000 | 42.1092 | 156.388 | −64.0000 | −132.176 | −100.000 | ||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 6.88339 | 16.0000 | 25.0000 | −27.5336 | −45.2649 | −64.0000 | −195.619 | −100.000 | ||||||||||||||||||||||||||||
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 | 230.6.a.c | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 230.6.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 26T_{3}^{2} + 9T_{3} - 1620 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(230))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{3} \)
|
| $3$ |
\( T^{3} + 26 T^{2} + \cdots - 1620 \)
|
| $5$ |
\( (T - 25)^{3} \)
|
| $7$ |
\( T^{3} - T^{2} + \cdots - 779544 \)
|
| $11$ |
\( T^{3} - 178 T^{2} + \cdots + 21004632 \)
|
| $13$ |
\( T^{3} + 236 T^{2} + \cdots - 16482042 \)
|
| $17$ |
\( T^{3} + 1683 T^{2} + \cdots + 73501360 \)
|
| $19$ |
\( T^{3} + 194 T^{2} + \cdots - 841047768 \)
|
| $23$ |
\( (T - 529)^{3} \)
|
| $29$ |
\( T^{3} - 6901 T^{2} + \cdots + 809182077 \)
|
| $31$ |
\( T^{3} + \cdots + 30813892995 \)
|
| $37$ |
\( T^{3} + \cdots + 881126317804 \)
|
| $41$ |
\( T^{3} + \cdots + 1050649591 \)
|
| $43$ |
\( T^{3} + \cdots - 1900451896320 \)
|
| $47$ |
\( T^{3} + \cdots - 98633296700 \)
|
| $53$ |
\( T^{3} + \cdots + 1549341371484 \)
|
| $59$ |
\( T^{3} + \cdots - 22979507348832 \)
|
| $61$ |
\( T^{3} + \cdots - 4638543486768 \)
|
| $67$ |
\( T^{3} + \cdots + 4102954370460 \)
|
| $71$ |
\( T^{3} + \cdots - 1521596424555 \)
|
| $73$ |
\( T^{3} + \cdots + 4522640531874 \)
|
| $79$ |
\( T^{3} + \cdots - 11537777158848 \)
|
| $83$ |
\( T^{3} + \cdots - 70485885180896 \)
|
| $89$ |
\( T^{3} + \cdots - 650267509666176 \)
|
| $97$ |
\( T^{3} + \cdots - 42022925072144 \)
|
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