Newspace parameters
| Level: | \( N \) | \(=\) | \( 100 = 2^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 20 \) |
| Character orbit: | \([\chi]\) | \(=\) | 100.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(228.816696556\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - 2x^{3} - 214954323x^{2} - 341671644076x + 8077617181385444 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{19}\cdot 3^{3}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 20) |
| 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
\( x^{4} - 2x^{3} - 214954323x^{2} - 341671644076x + 8077617181385444 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 2 \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 9844\nu^{2} - 116307691\nu + 801487176952 ) / 63 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 10180\nu^{2} + 115505659\nu - 837599102536 ) / 21 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 4 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 3\beta_{2} + 9548\beta _1 + 1719634600 ) / 16 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 2461\beta_{3} + 7635\beta_{2} + 139805319\beta _1 + 1026304658174 ) / 4 \)
|
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 |
|
0 | −55590.7 | 0 | 0 | 0 | −1.74271e8 | 0 | 1.92806e9 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −22420.4 | 0 | 0 | 0 | 8.55004e7 | 0 | −6.59585e8 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 40429.9 | 0 | 0 | 0 | −2.02472e8 | 0 | 4.72312e8 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 40661.2 | 0 | 0 | 0 | 1.43021e8 | 0 | 4.91076e8 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
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 | 100.20.a.c | 4 | |
| 5.b | even | 2 | 1 | 20.20.a.b | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 100.20.c.c | 8 | ||
| 20.d | odd | 2 | 1 | 80.20.a.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.20.a.b | ✓ | 4 | 5.b | even | 2 | 1 | |
| 80.20.a.h | 4 | 20.d | odd | 2 | 1 | ||
| 100.20.a.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 100.20.c.c | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 3080T_{3}^{3} - 3435711792T_{3}^{2} + 27175390721280T_{3} + 2048939287964476416 \)
acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(100))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} + \cdots + 20\!\cdots\!16 \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + \cdots + 43\!\cdots\!76 \)
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| $11$ |
\( T^{4} + \cdots + 38\!\cdots\!00 \)
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| $13$ |
\( T^{4} + \cdots - 38\!\cdots\!24 \)
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| $17$ |
\( T^{4} + \cdots + 25\!\cdots\!56 \)
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| $19$ |
\( T^{4} + \cdots + 73\!\cdots\!96 \)
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| $23$ |
\( T^{4} + \cdots + 43\!\cdots\!96 \)
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| $29$ |
\( T^{4} + \cdots + 39\!\cdots\!36 \)
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| $31$ |
\( T^{4} + \cdots + 29\!\cdots\!36 \)
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| $37$ |
\( T^{4} + \cdots - 48\!\cdots\!04 \)
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| $41$ |
\( T^{4} + \cdots - 13\!\cdots\!84 \)
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| $43$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
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| $47$ |
\( T^{4} + \cdots + 25\!\cdots\!16 \)
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| $53$ |
\( T^{4} + \cdots - 54\!\cdots\!64 \)
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| $59$ |
\( T^{4} + \cdots - 13\!\cdots\!64 \)
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| $61$ |
\( T^{4} + \cdots - 12\!\cdots\!04 \)
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| $67$ |
\( T^{4} + \cdots - 33\!\cdots\!64 \)
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| $71$ |
\( T^{4} + \cdots - 46\!\cdots\!64 \)
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| $73$ |
\( T^{4} + \cdots + 29\!\cdots\!96 \)
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| $79$ |
\( T^{4} + \cdots + 78\!\cdots\!56 \)
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| $83$ |
\( T^{4} + \cdots + 28\!\cdots\!76 \)
|
| $89$ |
\( T^{4} + \cdots - 82\!\cdots\!04 \)
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| $97$ |
\( T^{4} + \cdots + 65\!\cdots\!56 \)
|
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