Newspace parameters
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 42 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(31.9415011369\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 196497525461x^{2} + 10360343667016365x + 6095744045744274504000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{13}\cdot 3^{10}\cdot 5^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 196497525461x^{2} + 10360343667016365x + 6095744045744274504000 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 36\nu^{2} + 2847108\nu - 3536956170084 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 27\nu^{3} + 6696918\nu^{2} - 3100608741285\nu - 448170152274909880 ) / 5656 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 474518\beta _1 + 3536955695566 ) / 36 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 11312\beta_{3} - 372051\beta_{2} + 1210081143513\beta _1 - 419586565404959011 ) / 54 \)
|
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 |
|
−2.00342e6 | 3.48678e9 | 1.81468e12 | −3.43221e14 | −6.98551e15 | −2.11939e17 | 7.69998e17 | 1.21577e19 | 6.87617e20 | ||||||||||||||||||||||||||||||
| 1.2 | −1.59943e6 | 3.48678e9 | 3.59152e11 | 3.20915e14 | −5.57687e15 | 2.35480e17 | 2.94274e18 | 1.21577e19 | −5.13282e20 | |||||||||||||||||||||||||||||||
| 1.3 | 949201. | 3.48678e9 | −1.29804e12 | −1.14706e14 | 3.30966e15 | −7.22302e16 | −3.31942e18 | 1.21577e19 | −1.08879e20 | |||||||||||||||||||||||||||||||
| 1.4 | 2.58383e6 | 3.48678e9 | 4.47715e12 | 2.55548e14 | 9.00926e15 | 1.98945e17 | 5.88630e18 | 1.21577e19 | 6.60294e20 | |||||||||||||||||||||||||||||||
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 | 3.42.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 9.42.a.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.42.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 9.42.a.c | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 69822T_{2}^{3} - 7072082749728T_{2}^{2} - 2484749039974539264T_{2} + 7858870206246628249042944 \)
acting on \(S_{42}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 78\!\cdots\!44 \)
|
| $3$ |
\( (T - 3486784401)^{4} \)
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| $5$ |
\( T^{4} + \cdots + 32\!\cdots\!00 \)
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| $7$ |
\( T^{4} + \cdots + 71\!\cdots\!00 \)
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| $11$ |
\( T^{4} + \cdots + 63\!\cdots\!84 \)
|
| $13$ |
\( T^{4} + \cdots + 96\!\cdots\!44 \)
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| $17$ |
\( T^{4} + \cdots - 26\!\cdots\!44 \)
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| $19$ |
\( T^{4} + \cdots - 25\!\cdots\!00 \)
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| $23$ |
\( T^{4} + \cdots - 50\!\cdots\!00 \)
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| $29$ |
\( T^{4} + \cdots - 79\!\cdots\!00 \)
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| $31$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 20\!\cdots\!00 \)
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| $41$ |
\( T^{4} + \cdots - 33\!\cdots\!00 \)
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| $43$ |
\( T^{4} + \cdots - 62\!\cdots\!04 \)
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| $47$ |
\( T^{4} + \cdots - 16\!\cdots\!44 \)
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| $53$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
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| $59$ |
\( T^{4} + \cdots - 18\!\cdots\!00 \)
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| $61$ |
\( T^{4} + \cdots - 65\!\cdots\!64 \)
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| $67$ |
\( T^{4} + \cdots + 22\!\cdots\!76 \)
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| $71$ |
\( T^{4} + \cdots + 42\!\cdots\!36 \)
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| $73$ |
\( T^{4} + \cdots - 32\!\cdots\!00 \)
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| $79$ |
\( T^{4} + \cdots - 21\!\cdots\!00 \)
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| $83$ |
\( T^{4} + \cdots - 10\!\cdots\!76 \)
|
| $89$ |
\( T^{4} + \cdots + 66\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 31\!\cdots\!64 \)
|
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