Newspace parameters
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 40 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(28.9018654169\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 3876249523x - 18467420411022 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{6}\cdot 5\cdot 7\cdot 13 \) |
| 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
\( x^{3} - 3876249523x - 18467420411022 \)
:
| \(\beta_{1}\) | \(=\) |
\( 18\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 324\nu^{2} - 2315430\nu - 837269896968 \)
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| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 128635\beta _1 + 837269896968 ) / 324 \)
|
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 |
|
−983454. | −1.16226e9 | 4.17427e11 | 1.57969e13 | 1.14303e15 | 5.39162e15 | 1.30140e17 | 1.35085e18 | −1.55355e19 | |||||||||||||||||||||||||||
| 1.2 | 264126. | −1.16226e9 | −4.79993e11 | −6.30487e13 | −3.06983e14 | −4.59086e16 | −2.71983e17 | 1.35085e18 | −1.66528e19 | ||||||||||||||||||||||||||||
| 1.3 | 1.25290e6 | −1.16226e9 | 1.02001e12 | −6.13018e12 | −1.45620e15 | 3.89397e16 | 5.89182e17 | 1.35085e18 | −7.68052e18 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
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 | 3.40.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 9.40.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.40.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 9.40.a.c | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 533574T_{2}^{2} - 1161004440960T_{2} + 325448454458769408 \)
acting on \(S_{40}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots + 32\!\cdots\!08 \)
|
| $3$ |
\( (T + 1162261467)^{3} \)
|
| $5$ |
\( T^{3} + \cdots - 61\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots + 96\!\cdots\!40 \)
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| $11$ |
\( T^{3} + \cdots + 53\!\cdots\!36 \)
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| $13$ |
\( T^{3} + \cdots + 16\!\cdots\!36 \)
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| $17$ |
\( T^{3} + \cdots + 77\!\cdots\!44 \)
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| $19$ |
\( T^{3} + \cdots - 28\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots + 14\!\cdots\!40 \)
|
| $29$ |
\( T^{3} + \cdots - 12\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 65\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots + 53\!\cdots\!80 \)
|
| $41$ |
\( T^{3} + \cdots - 72\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots + 39\!\cdots\!68 \)
|
| $47$ |
\( T^{3} + \cdots + 42\!\cdots\!16 \)
|
| $53$ |
\( T^{3} + \cdots - 61\!\cdots\!80 \)
|
| $59$ |
\( T^{3} + \cdots - 56\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 27\!\cdots\!52 \)
|
| $67$ |
\( T^{3} + \cdots - 88\!\cdots\!76 \)
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| $71$ |
\( T^{3} + \cdots + 61\!\cdots\!12 \)
|
| $73$ |
\( T^{3} + \cdots - 42\!\cdots\!60 \)
|
| $79$ |
\( T^{3} + \cdots - 23\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 76\!\cdots\!76 \)
|
| $89$ |
\( T^{3} + \cdots + 15\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 15\!\cdots\!56 \)
|
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