Newspace parameters
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 68 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(56.8580703860\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{3} - x^{2} - 195874426031875504236526x - 11619230000023993068059089554543524 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{7}\cdot 5^{2}\cdot 7\cdot 11 \) |
| 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
\( x^{3} - x^{2} - 195874426031875504236526x - 11619230000023993068059089554543524 \)
:
| \(\beta_{1}\) | \(=\) |
\( 34560\nu - 11520 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 32768\nu^{2} - 2915692819806720\nu - 4278942128140692450941731328 ) / 2699271 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 11520 ) / 34560 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 24293439\beta_{2} + 759295005158\beta _1 + 38510479153274979136935002112 ) / 294912 \)
|
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 |
|
8.58993e9 | −1.70268e16 | 7.37870e19 | −2.72411e22 | −1.46259e26 | −2.32072e28 | 6.33825e29 | 1.97201e32 | −2.34000e32 | |||||||||||||||||||||||||||
| 1.2 | 8.58993e9 | 1.29493e15 | 7.37870e19 | 3.81293e23 | 1.11233e25 | −1.02948e27 | 6.33825e29 | −9.10326e31 | 3.27529e33 | ||||||||||||||||||||||||||||
| 1.3 | 8.58993e9 | 1.33494e16 | 7.37870e19 | −2.84057e23 | 1.14671e26 | −2.26934e28 | 6.33825e29 | 8.54976e31 | −2.44003e33 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
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 | 2.68.a.b | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.68.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + \cdots + 29\!\cdots\!92 \)
acting on \(S_{68}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8589934592)^{3} \)
|
| $3$ |
\( T^{3} + \cdots + 29\!\cdots\!92 \)
|
| $5$ |
\( T^{3} + \cdots - 29\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots + 54\!\cdots\!44 \)
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| $11$ |
\( T^{3} + \cdots - 46\!\cdots\!68 \)
|
| $13$ |
\( T^{3} + \cdots + 10\!\cdots\!72 \)
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| $17$ |
\( T^{3} + \cdots + 29\!\cdots\!44 \)
|
| $19$ |
\( T^{3} + \cdots + 26\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots + 55\!\cdots\!92 \)
|
| $29$ |
\( T^{3} + \cdots - 10\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 17\!\cdots\!32 \)
|
| $37$ |
\( T^{3} + \cdots + 42\!\cdots\!84 \)
|
| $41$ |
\( T^{3} + \cdots + 11\!\cdots\!72 \)
|
| $43$ |
\( T^{3} + \cdots - 53\!\cdots\!48 \)
|
| $47$ |
\( T^{3} + \cdots + 17\!\cdots\!24 \)
|
| $53$ |
\( T^{3} + \cdots - 34\!\cdots\!28 \)
|
| $59$ |
\( T^{3} + \cdots - 15\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 47\!\cdots\!72 \)
|
| $67$ |
\( T^{3} + \cdots - 14\!\cdots\!16 \)
|
| $71$ |
\( T^{3} + \cdots + 54\!\cdots\!32 \)
|
| $73$ |
\( T^{3} + \cdots - 63\!\cdots\!28 \)
|
| $79$ |
\( T^{3} + \cdots - 12\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 39\!\cdots\!12 \)
|
| $89$ |
\( T^{3} + \cdots - 14\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 38\!\cdots\!36 \)
|
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