Newspace parameters
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(69.8693360718\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
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|
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| Defining polynomial: |
\( x^{6} + 109119x^{4} + 2971017489x^{2} + 8028699582016 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{17}\cdot 3^{2}\cdot 5^{8} \) |
| Twist minimal: | no (minimal twist has level 5) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 109119x^{4} + 2971017489x^{2} + 8028699582016 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -25\nu^{5} - 73565375\nu^{3} - 3939163981225\nu ) / 818329229028 \)
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| \(\beta_{2}\) | \(=\) |
\( ( -1240\nu^{4} - 73644440\nu^{2} - 213169166449 ) / 2117907 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -160\nu^{4} - 15575360\nu^{2} - 248392733551 ) / 2117907 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -21247\nu^{5} - 1904760977\nu^{3} - 45524851845919\nu ) / 1500270253218 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -12882337\nu^{5} - 1173868254887\nu^{3} - 19995109011893329\nu ) / 9001621519308 \)
|
| \(\nu\) | \(=\) |
\( ( 5\beta_{5} - 505\beta_{4} - 121\beta_1 ) / 4800 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -279\beta_{3} + 36\beta_{2} - 29098295 ) / 800 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -136195\beta_{5} + 13815095\beta_{4} - 24240193\beta_1 ) / 2400 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 16569999\beta_{3} - 3504456\beta_{2} + 1590638957315 ) / 800 \)
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| \(\nu^{5}\) | \(=\) |
\( ( 13706103605\beta_{5} - 1733899126105\beta_{4} + 4605452745143\beta_1 ) / 4800 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 24.1 |
|
− | 1965.76i | 94506.8i | −1.76707e6 | 0 | 1.85778e8 | 1.12850e9i | − | 6.48862e8i | 1.52881e9 | 0 | ||||||||||||||||||||||||||||||||||
| 24.2 | − | 1592.71i | − | 9397.92i | −439587. | 0 | −1.49682e7 | 1.90393e8i | − | 2.64003e9i | 1.03720e10 | 0 | ||||||||||||||||||||||||||||||||||
| 24.3 | − | 938.952i | − | 156099.i | 1.21552e6 | 0 | −1.46569e8 | − | 2.53688e8i | − | 3.11044e9i | −1.39065e10 | 0 | |||||||||||||||||||||||||||||||||
| 24.4 | 938.952i | 156099.i | 1.21552e6 | 0 | −1.46569e8 | 2.53688e8i | 3.11044e9i | −1.39065e10 | 0 | |||||||||||||||||||||||||||||||||||||
| 24.5 | 1592.71i | 9397.92i | −439587. | 0 | −1.49682e7 | − | 1.90393e8i | 2.64003e9i | 1.03720e10 | 0 | ||||||||||||||||||||||||||||||||||||
| 24.6 | 1965.76i | − | 94506.8i | −1.76707e6 | 0 | 1.85778e8 | − | 1.12850e9i | 6.48862e8i | 1.52881e9 | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 25.22.b.b | 6 | |
| 5.b | even | 2 | 1 | inner | 25.22.b.b | 6 | |
| 5.c | odd | 4 | 1 | 5.22.a.a | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 25.22.a.b | 3 | ||
| 15.e | even | 4 | 1 | 45.22.a.d | 3 | ||
| 20.e | even | 4 | 1 | 80.22.a.e | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.22.a.a | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 25.22.a.b | 3 | 5.c | odd | 4 | 1 | ||
| 25.22.b.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 25.22.b.b | 6 | 5.b | even | 2 | 1 | inner | |
| 45.22.a.d | 3 | 15.e | even | 4 | 1 | ||
| 80.22.a.e | 3 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 7282592T_{2}^{4} + 15445807122688T_{2}^{2} + 8642204661756985344 \)
acting on \(S_{22}^{\mathrm{new}}(25, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 86\!\cdots\!44 \)
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| $3$ |
\( T^{6} + \cdots + 19\!\cdots\!76 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} + \cdots + 29\!\cdots\!44 \)
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| $11$ |
\( (T^{3} + \cdots + 36\!\cdots\!72)^{2} \)
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| $13$ |
\( T^{6} + \cdots + 31\!\cdots\!96 \)
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| $17$ |
\( T^{6} + \cdots + 33\!\cdots\!44 \)
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| $19$ |
\( (T^{3} + \cdots - 30\!\cdots\!00)^{2} \)
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| $23$ |
\( T^{6} + \cdots + 39\!\cdots\!16 \)
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| $29$ |
\( (T^{3} + \cdots - 85\!\cdots\!00)^{2} \)
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| $31$ |
\( (T^{3} + \cdots - 17\!\cdots\!68)^{2} \)
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| $37$ |
\( T^{6} + \cdots + 53\!\cdots\!44 \)
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| $41$ |
\( (T^{3} + \cdots - 96\!\cdots\!88)^{2} \)
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| $43$ |
\( T^{6} + \cdots + 12\!\cdots\!56 \)
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| $47$ |
\( T^{6} + \cdots + 78\!\cdots\!44 \)
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| $53$ |
\( T^{6} + \cdots + 76\!\cdots\!76 \)
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| $59$ |
\( (T^{3} + \cdots + 25\!\cdots\!00)^{2} \)
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| $61$ |
\( (T^{3} + \cdots + 39\!\cdots\!72)^{2} \)
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| $67$ |
\( T^{6} + \cdots + 11\!\cdots\!44 \)
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| $71$ |
\( (T^{3} + \cdots - 55\!\cdots\!48)^{2} \)
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| $73$ |
\( T^{6} + \cdots + 20\!\cdots\!16 \)
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| $79$ |
\( (T^{3} + \cdots - 60\!\cdots\!00)^{2} \)
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| $83$ |
\( T^{6} + \cdots + 16\!\cdots\!36 \)
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| $89$ |
\( (T^{3} + \cdots - 77\!\cdots\!00)^{2} \)
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| $97$ |
\( T^{6} + \cdots + 78\!\cdots\!44 \)
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