Newspace parameters
| Level: | \( N \) | \(=\) | \( 62 = 2 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 62.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(19.3678715800\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 2617x^{3} - 17755x^{2} + 1742092x + 24429360 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 2617x^{3} - 17755x^{2} + 1742092x + 24429360 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 20\nu^{3} + 2237\nu^{2} - 25424\nu - 1219152 ) / 1352 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{4} - 126\nu^{3} - 10847\nu^{2} + 158580\nu + 6055824 ) / 2704 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 20\nu^{3} - 2068\nu^{2} + 23903\nu + 1042716 ) / 338 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{4} + 8\beta_{2} + 9\beta _1 + 1044 \)
|
| \(\nu^{3}\) | \(=\) |
\( 26\beta_{4} - 104\beta_{3} - 156\beta_{2} + 1327\beta _1 + 12036 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4994\beta_{4} - 2080\beta_{3} + 13424\beta_{2} + 21249\beta _1 + 1356996 \)
|
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 |
|
−8.00000 | −69.3126 | 64.0000 | 413.350 | 554.501 | −1175.22 | −512.000 | 2617.23 | −3306.80 | |||||||||||||||||||||||||||||||||
| 1.2 | −8.00000 | −40.7973 | 64.0000 | −417.873 | 326.379 | −620.990 | −512.000 | −522.579 | 3342.99 | ||||||||||||||||||||||||||||||||||
| 1.3 | −8.00000 | −23.4184 | 64.0000 | 328.445 | 187.347 | 1228.01 | −512.000 | −1638.58 | −2627.56 | ||||||||||||||||||||||||||||||||||
| 1.4 | −8.00000 | 77.5058 | 64.0000 | −471.579 | −620.047 | −401.066 | −512.000 | 3820.15 | 3772.63 | ||||||||||||||||||||||||||||||||||
| 1.5 | −8.00000 | 84.0225 | 64.0000 | 299.658 | −672.180 | 837.264 | −512.000 | 4872.77 | −2397.26 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(31\) | \( +1 \) |
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 | 62.8.a.c | ✓ | 5 |
| 3.b | odd | 2 | 1 | 558.8.a.m | 5 | ||
| 4.b | odd | 2 | 1 | 496.8.a.c | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 62.8.a.c | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 496.8.a.c | 5 | 4.b | odd | 2 | 1 | ||
| 558.8.a.m | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 28T_{3}^{4} - 9650T_{3}^{3} + 62508T_{3}^{2} + 24510792T_{3} + 431251200 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(62))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{5} \)
|
| $3$ |
\( T^{5} - 28 T^{4} + \cdots + 431251200 \)
|
| $5$ |
\( T^{5} + \cdots - 8016877035000 \)
|
| $7$ |
\( T^{5} + \cdots + 300940831353600 \)
|
| $11$ |
\( T^{5} + \cdots + 68\!\cdots\!20 \)
|
| $13$ |
\( T^{5} + \cdots - 71\!\cdots\!40 \)
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| $17$ |
\( T^{5} + \cdots + 14\!\cdots\!24 \)
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| $19$ |
\( T^{5} + \cdots + 97\!\cdots\!68 \)
|
| $23$ |
\( T^{5} + \cdots + 82\!\cdots\!56 \)
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| $29$ |
\( T^{5} + \cdots - 50\!\cdots\!64 \)
|
| $31$ |
\( (T + 29791)^{5} \)
|
| $37$ |
\( T^{5} + \cdots - 13\!\cdots\!40 \)
|
| $41$ |
\( T^{5} + \cdots - 91\!\cdots\!72 \)
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| $43$ |
\( T^{5} + \cdots + 70\!\cdots\!12 \)
|
| $47$ |
\( T^{5} + \cdots + 29\!\cdots\!32 \)
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| $53$ |
\( T^{5} + \cdots - 75\!\cdots\!16 \)
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| $59$ |
\( T^{5} + \cdots + 26\!\cdots\!68 \)
|
| $61$ |
\( T^{5} + \cdots - 11\!\cdots\!36 \)
|
| $67$ |
\( T^{5} + \cdots - 20\!\cdots\!20 \)
|
| $71$ |
\( T^{5} + \cdots - 13\!\cdots\!52 \)
|
| $73$ |
\( T^{5} + \cdots - 16\!\cdots\!40 \)
|
| $79$ |
\( T^{5} + \cdots - 14\!\cdots\!60 \)
|
| $83$ |
\( T^{5} + \cdots - 20\!\cdots\!20 \)
|
| $89$ |
\( T^{5} + \cdots + 19\!\cdots\!32 \)
|
| $97$ |
\( T^{5} + \cdots + 12\!\cdots\!00 \)
|
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