Newspace parameters
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 74 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(67.4967947474\) |
| 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: |
\( x^{3} - x^{2} - 413501459186944860372404680x - 2966140105783309949999568694815716833028 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{23}\cdot 3^{7}\cdot 5^{3} \) |
| 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
\( x^{3} - x^{2} - 413501459186944860372404680x - 2966140105783309949999568694815716833028 \)
:
| \(\beta_{1}\) | \(=\) |
\( 17280\nu - 5760 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 40960\nu^{2} - 440723106595950720\nu - 11291346512198027412866931825280 ) / 633843 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 5760 ) / 17280 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 5704587\beta_{2} + 229543284685391\beta _1 + 101622118609783568885122174279680 ) / 368640 \)
|
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 |
|
−6.87195e10 | −3.50368e17 | 4.72237e21 | −5.44340e25 | 2.40771e28 | −1.20967e31 | −3.24519e32 | 5.51727e34 | 3.74068e36 | |||||||||||||||||||||||||||
| 1.2 | −6.87195e10 | −2.54256e17 | 4.72237e21 | 3.27596e25 | 1.74723e28 | 8.99592e30 | −3.24519e32 | −2.93926e33 | −2.25122e36 | ||||||||||||||||||||||||||||
| 1.3 | −6.87195e10 | 3.00585e17 | 4.72237e21 | −1.10375e25 | −2.06560e28 | 7.83423e29 | −3.24519e32 | 2.27660e34 | 7.58489e35 | ||||||||||||||||||||||||||||
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 | 2.74.a.a | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.74.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + \cdots - 26\!\cdots\!76 \)
acting on \(S_{74}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 68719476736)^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 26\!\cdots\!76 \)
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| $5$ |
\( T^{3} + \cdots - 19\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots + 85\!\cdots\!72 \)
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| $11$ |
\( T^{3} + \cdots + 22\!\cdots\!92 \)
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| $13$ |
\( T^{3} + \cdots + 58\!\cdots\!64 \)
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| $17$ |
\( T^{3} + \cdots + 13\!\cdots\!52 \)
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| $19$ |
\( T^{3} + \cdots - 13\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots + 24\!\cdots\!64 \)
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| $29$ |
\( T^{3} + \cdots + 28\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 70\!\cdots\!32 \)
|
| $37$ |
\( T^{3} + \cdots + 55\!\cdots\!52 \)
|
| $41$ |
\( T^{3} + \cdots - 83\!\cdots\!68 \)
|
| $43$ |
\( T^{3} + \cdots - 12\!\cdots\!56 \)
|
| $47$ |
\( T^{3} + \cdots - 15\!\cdots\!28 \)
|
| $53$ |
\( T^{3} + \cdots + 23\!\cdots\!44 \)
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| $59$ |
\( T^{3} + \cdots + 63\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 24\!\cdots\!28 \)
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| $67$ |
\( T^{3} + \cdots - 33\!\cdots\!08 \)
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| $71$ |
\( T^{3} + \cdots + 46\!\cdots\!12 \)
|
| $73$ |
\( T^{3} + \cdots - 50\!\cdots\!16 \)
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| $79$ |
\( T^{3} + \cdots + 17\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 12\!\cdots\!36 \)
|
| $89$ |
\( T^{3} + \cdots - 20\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots + 12\!\cdots\!12 \)
|
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