Newspace parameters
| Level: | \( N \) | \(=\) | \( 100 = 2^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 100.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(107.230928952\) |
| 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: |
\( x^{4} - 1054047x^{2} - 183513850x + 142150675896 \)
|
| 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
\( x^{4} - 1054047x^{2} - 183513850x + 142150675896 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{3} + 968\nu^{2} - 6527812\nu - 1611241848 ) / 1557 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{3} - 5260\nu^{2} - 4902304\nu + 1671060510 ) / 1557 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 522\beta _1 + 2108094 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 242\beta_{3} + 1315\beta_{2} + 3137582\beta _1 + 1101083100 ) / 8 \)
|
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 | −1882.41 | 0 | 0 | 0 | −20675.3 | 0 | 1.94915e6 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −385.938 | 0 | 0 | 0 | 41599.5 | 0 | −1.44537e6 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1438.55 | 0 | 0 | 0 | −504334. | 0 | 475107. | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 1689.80 | 0 | 0 | 0 | 563490. | 0 | 1.26110e6 | 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 | 100.14.a.e | yes | 4 |
| 5.b | even | 2 | 1 | 100.14.a.d | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 100.14.c.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 100.14.a.d | ✓ | 4 | 5.b | even | 2 | 1 | |
| 100.14.a.e | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 100.14.c.d | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 860T_{3}^{3} - 3938838T_{3}^{2} + 3241318140T_{3} + 1766010452661 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(100))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 1766010452661 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 24\!\cdots\!96 \)
|
| $11$ |
\( T^{4} + \cdots + 85\!\cdots\!25 \)
|
| $13$ |
\( T^{4} + \cdots - 81\!\cdots\!64 \)
|
| $17$ |
\( T^{4} + \cdots + 37\!\cdots\!81 \)
|
| $19$ |
\( T^{4} + \cdots + 54\!\cdots\!81 \)
|
| $23$ |
\( T^{4} + \cdots + 11\!\cdots\!56 \)
|
| $29$ |
\( T^{4} + \cdots - 21\!\cdots\!64 \)
|
| $31$ |
\( T^{4} + \cdots + 69\!\cdots\!76 \)
|
| $37$ |
\( T^{4} + \cdots + 18\!\cdots\!96 \)
|
| $41$ |
\( T^{4} + \cdots + 28\!\cdots\!21 \)
|
| $43$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 16\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots + 39\!\cdots\!36 \)
|
| $59$ |
\( T^{4} + \cdots - 20\!\cdots\!24 \)
|
| $61$ |
\( T^{4} + \cdots + 12\!\cdots\!16 \)
|
| $67$ |
\( T^{4} + \cdots + 10\!\cdots\!81 \)
|
| $71$ |
\( T^{4} + \cdots + 44\!\cdots\!56 \)
|
| $73$ |
\( T^{4} + \cdots + 16\!\cdots\!41 \)
|
| $79$ |
\( T^{4} + \cdots - 66\!\cdots\!04 \)
|
| $83$ |
\( T^{4} + \cdots - 19\!\cdots\!59 \)
|
| $89$ |
\( T^{4} + \cdots + 10\!\cdots\!81 \)
|
| $97$ |
\( T^{4} + \cdots - 46\!\cdots\!44 \)
|
| show more | |
| show less |