Newspace parameters
| Level: | \( N \) | \(=\) | \( 32 = 2^{5} \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 32.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(45.6619216320\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.2657491200.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - 624x^{2} - 4680x + 4329 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{27}\cdot 3^{5}\cdot 5^{2} \) |
| 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
\( x^{4} - 624x^{2} - 4680x + 4329 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -96\nu^{3} - 464\nu^{2} + 113952\nu + 481728 ) / 209 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7680\nu^{3} - 230400\nu^{2} - 2695680\nu + 44928000 ) / 209 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 385056\nu^{3} - 2419216\nu^{2} - 200242272\nu - 596751168 ) / 209 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 16\beta_{2} + 2731\beta_1 ) / 737280 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 40\beta_{2} + 811\beta _1 + 9584640 ) / 30720 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 357\beta_{3} - 4784\beta_{2} + 514167\beta _1 + 862617600 ) / 245760 \)
|
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 | −5730.06 | 0 | 298114. | 0 | 3.55222e6 | 0 | 1.84847e7 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −2757.10 | 0 | −108854. | 0 | −2.79703e6 | 0 | −6.74730e6 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 2757.10 | 0 | −108854. | 0 | 2.79703e6 | 0 | −6.74730e6 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 5730.06 | 0 | 298114. | 0 | −3.55222e6 | 0 | 1.84847e7 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 32.16.a.d | ✓ | 4 |
| 4.b | odd | 2 | 1 | inner | 32.16.a.d | ✓ | 4 |
| 8.b | even | 2 | 1 | 64.16.a.p | 4 | ||
| 8.d | odd | 2 | 1 | 64.16.a.p | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 32.16.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 32.16.a.d | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 64.16.a.p | 4 | 8.b | even | 2 | 1 | ||
| 64.16.a.p | 4 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 40435200T_{3}^{2} + 249588051148800 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(32))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 249588051148800 \)
|
| $5$ |
\( (T^{2} - 189260 T - 32450807900)^{2} \)
|
| $7$ |
\( T^{4} + \cdots + 98\!\cdots\!00 \)
|
| $11$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
| $13$ |
\( (T^{2} + \cdots - 39\!\cdots\!96)^{2} \)
|
| $17$ |
\( (T^{2} + \cdots - 63\!\cdots\!84)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + \cdots + 60\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
|
| $37$ |
\( (T^{2} + \cdots + 30\!\cdots\!56)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots - 47\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 48\!\cdots\!00 \)
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| $47$ |
\( T^{4} + \cdots + 51\!\cdots\!00 \)
|
| $53$ |
\( (T^{2} + \cdots + 31\!\cdots\!24)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 52\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + \cdots + 41\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots + 44\!\cdots\!00 \)
|
| $73$ |
\( (T^{2} + \cdots - 16\!\cdots\!76)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 91\!\cdots\!00 \)
|
| $89$ |
\( (T^{2} + \cdots - 15\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots + 96\!\cdots\!56)^{2} \)
|
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