Newspace parameters
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(26.8077322380\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 17722x^{2} + 125608x + 10385664 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5^{2} \) |
| 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
\( x^{4} - x^{3} - 17722x^{2} + 125608x + 10385664 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 3\nu^{2} - 16570\nu + 58416 ) / 228 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 9\nu - 8864 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 9\beta _1 + 8864 \)
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| \(\nu^{3}\) | \(=\) |
\( -3\beta_{3} + 228\beta_{2} + 16597\beta _1 - 85008 \)
|
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 |
|
−117.966 | −235.672 | 5724.04 | 0 | 27801.3 | 227749. | 291136. | −1.53878e6 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −5.16335 | 1952.42 | −8165.34 | 0 | −10081.0 | −455997. | 84458.7 | 2.21763e6 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 44.7600 | −1474.96 | −6188.54 | 0 | −66019.0 | −143529. | −643673. | 581170. | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 143.370 | 618.204 | 12362.8 | 0 | 88631.7 | 474578. | 597973. | −1.21215e6 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
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 | 25.14.a.d | yes | 4 |
| 5.b | even | 2 | 1 | 25.14.a.c | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 25.14.b.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 25.14.a.c | ✓ | 4 | 5.b | even | 2 | 1 | |
| 25.14.a.d | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 25.14.b.c | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 65T_{2}^{3} - 16138T_{2}^{2} + 675560T_{2} + 3908736 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(25))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 65 T^{3} + \cdots + 3908736 \)
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| $3$ |
\( T^{4} + \cdots + 419557862781 \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + \cdots + 70\!\cdots\!96 \)
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| $11$ |
\( T^{4} + \cdots + 60\!\cdots\!01 \)
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| $13$ |
\( T^{4} + \cdots + 31\!\cdots\!16 \)
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| $17$ |
\( T^{4} + \cdots - 86\!\cdots\!59 \)
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| $19$ |
\( T^{4} + \cdots + 91\!\cdots\!25 \)
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| $23$ |
\( T^{4} + \cdots + 72\!\cdots\!76 \)
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| $29$ |
\( T^{4} + \cdots + 20\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 30\!\cdots\!04 \)
|
| $37$ |
\( T^{4} + \cdots - 27\!\cdots\!44 \)
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| $41$ |
\( T^{4} + \cdots + 25\!\cdots\!81 \)
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| $43$ |
\( T^{4} + \cdots - 51\!\cdots\!04 \)
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| $47$ |
\( T^{4} + \cdots - 73\!\cdots\!24 \)
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| $53$ |
\( T^{4} + \cdots + 14\!\cdots\!56 \)
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| $59$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
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| $61$ |
\( T^{4} + \cdots + 47\!\cdots\!76 \)
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| $67$ |
\( T^{4} + \cdots + 30\!\cdots\!41 \)
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| $71$ |
\( T^{4} + \cdots - 91\!\cdots\!64 \)
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| $73$ |
\( T^{4} + \cdots + 14\!\cdots\!01 \)
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| $79$ |
\( T^{4} + \cdots - 19\!\cdots\!00 \)
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| $83$ |
\( T^{4} + \cdots - 43\!\cdots\!39 \)
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| $89$ |
\( T^{4} + \cdots + 17\!\cdots\!25 \)
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| $97$ |
\( T^{4} + \cdots - 10\!\cdots\!24 \)
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