Newspace parameters
| Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 325.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(19.1756207519\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{5} - x^{4} - 31x^{3} + 37x^{2} + 188x - 130 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{5} - x^{4} - 31x^{3} + 37x^{2} + 188x - 130 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 4\nu^{3} - 17\nu^{2} - 60\nu + 8 ) / 6 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} - 4\nu^{3} + 23\nu^{2} + 60\nu - 86 ) / 6 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} - 2\nu^{3} + 21\nu^{2} + 22\nu - 46 ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 13 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 2\beta_{3} + \beta_{2} + 19\beta _1 - 7 \)
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| \(\nu^{4}\) | \(=\) |
\( -4\beta_{4} + 25\beta_{3} + 19\beta_{2} - 16\beta _1 + 241 \)
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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 |
|
−4.98185 | 3.91681 | 16.8188 | 0 | −19.5129 | 32.4585 | −43.9339 | −11.6586 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.56487 | −1.74206 | −1.42145 | 0 | 4.46816 | 1.22645 | 24.1648 | −23.9652 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.653783 | 5.54502 | −7.57257 | 0 | 3.62524 | −7.58659 | −10.1811 | 3.74725 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 3.83244 | 1.29302 | 6.68760 | 0 | 4.95541 | −18.4527 | −5.02971 | −25.3281 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.06049 | −8.01278 | 8.48761 | 0 | −32.5359 | 18.3544 | 1.97995 | 37.2047 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 325.4.a.k | yes | 5 |
| 5.b | even | 2 | 1 | 325.4.a.i | ✓ | 5 | |
| 5.c | odd | 4 | 2 | 325.4.b.h | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 325.4.a.i | ✓ | 5 | 5.b | even | 2 | 1 | |
| 325.4.a.k | yes | 5 | 1.a | even | 1 | 1 | trivial |
| 325.4.b.h | 10 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(325))\):
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\( T_{2}^{5} - T_{2}^{4} - 31T_{2}^{3} + 37T_{2}^{2} + 188T_{2} - 130 \)
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\( T_{3}^{5} - T_{3}^{4} - 57T_{3}^{3} + 153T_{3}^{2} + 200T_{3} - 392 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - T^{4} + \cdots - 130 \)
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| $3$ |
\( T^{5} - T^{4} + \cdots - 392 \)
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| $5$ |
\( T^{5} \)
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| $7$ |
\( T^{5} - 26 T^{4} + \cdots - 102288 \)
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| $11$ |
\( T^{5} + 77 T^{4} + \cdots + 25116068 \)
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| $13$ |
\( (T + 13)^{5} \)
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| $17$ |
\( T^{5} + 73 T^{4} + \cdots + 19828508 \)
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| $19$ |
\( T^{5} + \cdots - 4913016192 \)
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| $23$ |
\( T^{5} + \cdots - 1505910528 \)
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| $29$ |
\( T^{5} + \cdots - 2717874652 \)
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| $31$ |
\( T^{5} + \cdots - 19810902888 \)
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| $37$ |
\( T^{5} + \cdots + 455049091584 \)
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| $41$ |
\( T^{5} + \cdots - 12725013504 \)
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| $43$ |
\( T^{5} + \cdots + 11730477712 \)
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| $47$ |
\( T^{5} + \cdots + 74990464640 \)
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| $53$ |
\( T^{5} + \cdots + 256822186914 \)
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| $59$ |
\( T^{5} + \cdots - 34612631436096 \)
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| $61$ |
\( T^{5} + \cdots + 19214667161816 \)
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| $67$ |
\( T^{5} + \cdots - 4619162534916 \)
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| $71$ |
\( T^{5} + \cdots + 6904899526656 \)
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| $73$ |
\( T^{5} + \cdots + 64468435248 \)
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| $79$ |
\( T^{5} + \cdots - 213811065504 \)
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| $83$ |
\( T^{5} + \cdots + 14144978104992 \)
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| $89$ |
\( T^{5} + \cdots - 77883058038240 \)
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| $97$ |
\( T^{5} + \cdots + 28787199835520 \)
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