Newspace parameters
| Level: | \( N \) | \(=\) | \( 52 = 2^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 52.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(26.7818634805\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 5049x^{3} - 87027x^{2} + 2867800x + 48215700 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{4} \) |
| 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
\( x^{5} - x^{4} - 5049x^{3} - 87027x^{2} + 2867800x + 48215700 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 113\nu^{4} - 3558\nu^{3} - 405502\nu^{2} + 2053669\nu + 113831770 ) / 59735 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -87\nu^{4} + 7497\nu^{3} + 307443\nu^{2} - 20196261\nu - 337166700 ) / 119470 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -168\nu^{4} + 2118\nu^{3} + 785247\nu^{2} + 4578036\nu - 363924080 ) / 59735 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{4} + 4\beta_{3} + 6\beta_{2} + 80\beta _1 + 18212 ) / 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{4} + 230\beta_{3} + 93\beta_{2} + 11818\beta _1 + 501976 ) / 9 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 10860\beta_{4} + 21596\beta_{3} + 29217\beta_{2} + 604669\beta _1 + 72038815 ) / 9 \)
|
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 | −179.797 | 0 | 2632.50 | 0 | −5669.07 | 0 | 12643.9 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −129.115 | 0 | −1993.22 | 0 | −7554.87 | 0 | −3012.30 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −77.4651 | 0 | 313.346 | 0 | 12252.5 | 0 | −13682.2 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 43.6557 | 0 | −1118.78 | 0 | −2400.89 | 0 | −17777.2 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 195.721 | 0 | 1887.16 | 0 | 6689.32 | 0 | 18623.8 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 52.10.a.b | ✓ | 5 |
| 4.b | odd | 2 | 1 | 208.10.a.i | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 52.10.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 208.10.a.i | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + 147T_{3}^{4} - 36801T_{3}^{3} - 6185619T_{3}^{2} - 27656640T_{3} + 15365376000 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(52))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + \cdots + 15365376000 \)
|
| $5$ |
\( T^{5} + \cdots - 34\!\cdots\!00 \)
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| $7$ |
\( T^{5} + \cdots + 84\!\cdots\!44 \)
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| $11$ |
\( T^{5} + \cdots - 10\!\cdots\!84 \)
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| $13$ |
\( (T + 28561)^{5} \)
|
| $17$ |
\( T^{5} + \cdots + 23\!\cdots\!32 \)
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| $19$ |
\( T^{5} + \cdots + 46\!\cdots\!20 \)
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| $23$ |
\( T^{5} + \cdots - 27\!\cdots\!84 \)
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| $29$ |
\( T^{5} + \cdots - 20\!\cdots\!40 \)
|
| $31$ |
\( T^{5} + \cdots + 65\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots - 24\!\cdots\!00 \)
|
| $41$ |
\( T^{5} + \cdots + 10\!\cdots\!72 \)
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| $43$ |
\( T^{5} + \cdots - 63\!\cdots\!68 \)
|
| $47$ |
\( T^{5} + \cdots - 32\!\cdots\!48 \)
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| $53$ |
\( T^{5} + \cdots + 58\!\cdots\!04 \)
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| $59$ |
\( T^{5} + \cdots + 31\!\cdots\!00 \)
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| $61$ |
\( T^{5} + \cdots - 12\!\cdots\!04 \)
|
| $67$ |
\( T^{5} + \cdots - 18\!\cdots\!00 \)
|
| $71$ |
\( T^{5} + \cdots + 16\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots + 91\!\cdots\!20 \)
|
| $79$ |
\( T^{5} + \cdots + 38\!\cdots\!32 \)
|
| $83$ |
\( T^{5} + \cdots + 57\!\cdots\!48 \)
|
| $89$ |
\( T^{5} + \cdots - 97\!\cdots\!80 \)
|
| $97$ |
\( T^{5} + \cdots - 25\!\cdots\!76 \)
|
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