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: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 10174x^{2} - 369720x - 3191805 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{27}\cdot 3\cdot 5 \) |
| 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
\( x^{4} - 10174x^{2} - 369720x - 3191805 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -32\nu^{3} - 688\nu^{2} + 394944\nu + 12373136 ) / 1169 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -6592\nu^{3} + 157536\nu^{2} + 64514176\nu + 1026510048 ) / 1503 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 113728\nu^{3} - 2343072\nu^{2} - 1010996608\nu - 19616429856 ) / 3507 \)
|
| \(\nu\) | \(=\) |
\( ( 7\beta_{3} + 48\beta_{2} + 602\beta_1 ) / 245760 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 197\beta_{3} + 1968\beta_{2} - 81938\beta _1 + 625090560 ) / 122880 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 77923\beta_{3} + 507792\beta_{2} + 1975298\beta _1 + 68146790400 ) / 245760 \)
|
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 | −6640.54 | 0 | 136415. | 0 | −3.99282e6 | 0 | 2.97478e7 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −2610.67 | 0 | −298458. | 0 | 2.33016e6 | 0 | −7.53332e6 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 634.991 | 0 | 134146. | 0 | 1.14384e6 | 0 | −1.39457e7 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 5704.21 | 0 | −70382.7 | 0 | −2.23695e6 | 0 | 1.81891e7 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
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 | 32.16.a.c | ✓ | 4 |
| 4.b | odd | 2 | 1 | 32.16.a.e | yes | 4 | |
| 8.b | even | 2 | 1 | 64.16.a.q | 4 | ||
| 8.d | odd | 2 | 1 | 64.16.a.o | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 32.16.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 32.16.a.e | yes | 4 | 4.b | odd | 2 | 1 | |
| 64.16.a.o | 4 | 8.d | odd | 2 | 1 | ||
| 64.16.a.q | 4 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 2912T_{3}^{3} - 37686912T_{3}^{2} - 76388935680T_{3} + 62793980448768 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(32))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 62793980448768 \)
|
| $5$ |
\( T^{4} + \cdots + 38\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots + 23\!\cdots\!16 \)
|
| $11$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
|
| $13$ |
\( T^{4} + \cdots + 31\!\cdots\!00 \)
|
| $17$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots - 44\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 11\!\cdots\!44 \)
|
| $29$ |
\( T^{4} + \cdots + 76\!\cdots\!52 \)
|
| $31$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots - 55\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots - 22\!\cdots\!48 \)
|
| $47$ |
\( T^{4} + \cdots - 41\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots - 32\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots - 20\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 40\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots - 10\!\cdots\!92 \)
|
| $71$ |
\( T^{4} + \cdots - 39\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots - 30\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 66\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 27\!\cdots\!40 \)
|
| $89$ |
\( T^{4} + \cdots + 15\!\cdots\!24 \)
|
| $97$ |
\( T^{4} + \cdots + 76\!\cdots\!00 \)
|
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