Newspace parameters
| Level: | \( N \) | \(=\) | \( 100 = 2^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 100.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(76.8343180560\) |
| 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: |
\( x^{4} - 132543x^{2} - 1742450x + 702883056 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{2}\cdot 5^{2} \) |
| 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
\( x^{4} - 132543x^{2} - 1742450x + 702883056 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{3} - 380\nu^{2} - 967936\nu + 14728470 ) / 1377 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 16\nu^{3} + 4748\nu^{2} - 2043278\nu - 335566482 ) / 1377 \)
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| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 2\beta_{2} + 39\beta _1 + 265086 ) / 4 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 95\beta_{3} + 1187\beta_{2} + 487673\beta _1 + 10454700 ) / 8 \)
|
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 | −731.676 | 0 | 0 | 0 | −9112.28 | 0 | 358202. | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −140.231 | 0 | 0 | 0 | 31815.9 | 0 | −157482. | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 158.973 | 0 | 0 | 0 | −64162.5 | 0 | −151875. | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 692.934 | 0 | 0 | 0 | 30018.8 | 0 | 303011. | 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 | 100.12.a.d | ✓ | 4 |
| 5.b | even | 2 | 1 | 100.12.a.e | yes | 4 | |
| 5.c | odd | 4 | 2 | 100.12.c.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 100.12.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 100.12.a.e | yes | 4 | 5.b | even | 2 | 1 | |
| 100.12.c.d | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 20T_{3}^{3} - 530022T_{3}^{2} + 8638380T_{3} + 11302573221 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(100))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} + \cdots + 11302573221 \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + \cdots + 55\!\cdots\!76 \)
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| $11$ |
\( T^{4} + \cdots - 59\!\cdots\!75 \)
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| $13$ |
\( T^{4} + \cdots + 41\!\cdots\!36 \)
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| $17$ |
\( T^{4} + \cdots + 17\!\cdots\!01 \)
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| $19$ |
\( T^{4} + \cdots + 11\!\cdots\!41 \)
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| $23$ |
\( T^{4} + \cdots + 17\!\cdots\!36 \)
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| $29$ |
\( T^{4} + \cdots + 46\!\cdots\!16 \)
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| $31$ |
\( T^{4} + \cdots - 11\!\cdots\!44 \)
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| $37$ |
\( T^{4} + \cdots + 15\!\cdots\!56 \)
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| $41$ |
\( T^{4} + \cdots + 10\!\cdots\!01 \)
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| $43$ |
\( T^{4} + \cdots - 15\!\cdots\!00 \)
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| $47$ |
\( T^{4} + \cdots + 50\!\cdots\!96 \)
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| $53$ |
\( T^{4} + \cdots + 27\!\cdots\!16 \)
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| $59$ |
\( T^{4} + \cdots - 16\!\cdots\!04 \)
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| $61$ |
\( T^{4} + \cdots + 72\!\cdots\!36 \)
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| $67$ |
\( T^{4} + \cdots - 53\!\cdots\!19 \)
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| $71$ |
\( T^{4} + \cdots - 61\!\cdots\!84 \)
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| $73$ |
\( T^{4} + \cdots + 82\!\cdots\!61 \)
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| $79$ |
\( T^{4} + \cdots + 65\!\cdots\!36 \)
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| $83$ |
\( T^{4} + \cdots - 64\!\cdots\!79 \)
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| $89$ |
\( T^{4} + \cdots - 67\!\cdots\!39 \)
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| $97$ |
\( T^{4} + \cdots - 36\!\cdots\!64 \)
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