Newspace parameters
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(69.8693360718\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{7} - 2 x^{6} - 11176187 x^{5} + 2419003474 x^{4} + 32443001467723 x^{3} + \cdots + 99\!\cdots\!78 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{13}\cdot 3^{3}\cdot 5^{10}\cdot 7^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2 x^{6} - 11176187 x^{5} + 2419003474 x^{4} + 32443001467723 x^{3} + \cdots + 99\!\cdots\!78 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 6348641 \nu^{6} - 4587100821 \nu^{5} + 68045569787032 \nu^{4} + \cdots + 10\!\cdots\!42 ) / 26\!\cdots\!04 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6348641 \nu^{6} + 4587100821 \nu^{5} - 68045569787032 \nu^{4} + \cdots - 12\!\cdots\!62 ) / 88\!\cdots\!68 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1001601967 \nu^{6} + 515952494523 \nu^{5} + \cdots - 10\!\cdots\!18 ) / 13\!\cdots\!52 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 270384493 \nu^{6} + 164545131537 \nu^{5} + \cdots - 42\!\cdots\!98 ) / 80\!\cdots\!72 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 41158122925 \nu^{6} + 24310943050833 \nu^{5} + \cdots - 64\!\cdots\!34 ) / 66\!\cdots\!76 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} - 323\beta _1 + 3193290 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - 2\beta_{5} - 3\beta_{4} - 672\beta_{3} - 5310\beta_{2} + 5371708\beta _1 - 1028672004 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 1239 \beta_{6} + 1870 \beta_{5} + 18689 \beta_{4} + 7655043 \beta_{3} + 23173635 \beta_{2} + \cdots + 17147364838012 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8557208 \beta_{6} - 19237552 \beta_{5} - 37544088 \beta_{4} - 9227449458 \beta_{3} + \cdots - 13\!\cdots\!26 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 14344521930 \beta_{6} + 23706455060 \beta_{5} + 212083675918 \beta_{4} + 55697602153169 \beta_{3} + \cdots + 10\!\cdots\!38 \)
|
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 |
|
−2433.48 | 81248.2 | 3.82468e6 | 0 | −1.97716e8 | 9.23572e8 | −4.20390e9 | −3.85908e9 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2133.11 | −148627. | 2.45299e6 | 0 | 3.17038e8 | −1.25395e9 | −7.59041e8 | 1.16297e10 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −745.447 | 122603. | −1.54146e6 | 0 | −9.13943e7 | −1.12071e9 | 2.71239e9 | 4.57120e9 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −724.148 | −45482.7 | −1.57276e6 | 0 | 3.29362e7 | 8.64838e8 | 2.65756e9 | −8.39168e9 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1029.06 | −120096. | −1.03819e6 | 0 | −1.23586e8 | −2.19729e8 | −3.22645e9 | 3.96271e9 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1542.62 | 110812. | 282527. | 0 | 1.70941e8 | 3.07836e8 | −2.79928e9 | 1.81900e9 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2727.50 | −50838.6 | 5.34213e6 | 0 | −1.38663e8 | −3.19642e8 | 8.85070e9 | −7.87579e9 | 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 | 25.22.a.d | ✓ | 7 |
| 5.b | even | 2 | 1 | 25.22.a.e | yes | 7 | |
| 5.c | odd | 4 | 2 | 25.22.b.d | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 25.22.a.d | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 25.22.a.e | yes | 7 | 5.b | even | 2 | 1 | |
| 25.22.b.d | 14 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 737 T_{2}^{6} - 10943402 T_{2}^{5} - 8245654024 T_{2}^{4} + 30199145968768 T_{2}^{3} + \cdots - 12\!\cdots\!48 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(25))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + \cdots - 12\!\cdots\!48 \)
|
| $3$ |
\( T^{7} + \cdots - 45\!\cdots\!01 \)
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| $5$ |
\( T^{7} \)
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| $7$ |
\( T^{7} + \cdots - 24\!\cdots\!48 \)
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| $11$ |
\( T^{7} + \cdots - 22\!\cdots\!83 \)
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| $13$ |
\( T^{7} + \cdots + 87\!\cdots\!84 \)
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| $17$ |
\( T^{7} + \cdots - 27\!\cdots\!23 \)
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| $19$ |
\( T^{7} + \cdots + 88\!\cdots\!25 \)
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| $23$ |
\( T^{7} + \cdots + 73\!\cdots\!44 \)
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| $29$ |
\( T^{7} + \cdots - 47\!\cdots\!00 \)
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| $31$ |
\( T^{7} + \cdots - 13\!\cdots\!68 \)
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| $37$ |
\( T^{7} + \cdots - 11\!\cdots\!48 \)
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| $41$ |
\( T^{7} + \cdots - 69\!\cdots\!73 \)
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| $43$ |
\( T^{7} + \cdots + 15\!\cdots\!64 \)
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| $47$ |
\( T^{7} + \cdots - 37\!\cdots\!48 \)
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| $53$ |
\( T^{7} + \cdots - 65\!\cdots\!76 \)
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| $59$ |
\( T^{7} + \cdots + 38\!\cdots\!00 \)
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| $61$ |
\( T^{7} + \cdots - 64\!\cdots\!08 \)
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| $67$ |
\( T^{7} + \cdots + 14\!\cdots\!77 \)
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| $71$ |
\( T^{7} + \cdots - 75\!\cdots\!88 \)
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| $73$ |
\( T^{7} + \cdots + 50\!\cdots\!19 \)
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| $79$ |
\( T^{7} + \cdots - 65\!\cdots\!00 \)
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| $83$ |
\( T^{7} + \cdots + 16\!\cdots\!29 \)
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| $89$ |
\( T^{7} + \cdots + 60\!\cdots\!25 \)
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| $97$ |
\( T^{7} + \cdots + 49\!\cdots\!52 \)
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