Newspace parameters
| Level: | \( N \) | \(=\) | \( 7225 = 5^{2} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7225.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(57.6919154604\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.1397493.1 |
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| Defining polynomial: |
\( x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1445) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 2\nu + 2 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - \nu^{2} + 6\nu - 1 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 3\nu^{3} + 9\nu^{2} + 4\nu - 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 4\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} + 7\beta_{2} + 7\beta _1 + 9 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} + 12\beta_{3} + 18\beta_{2} + 20\beta _1 + 18 \)
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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 |
|
−1.68091 | 0.0689290 | 0.825466 | 0 | −0.115864 | 3.48244 | 1.97429 | −2.99525 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.05432 | −2.14741 | −0.888399 | 0 | 2.26407 | 4.45595 | 3.04531 | 1.61136 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.415466 | 2.09856 | −1.82739 | 0 | 0.871883 | 2.70116 | −1.59015 | 1.40397 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.801527 | 0.931071 | −1.35755 | 0 | 0.746278 | −1.48244 | −2.69117 | −2.13311 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.11662 | −1.09856 | 2.48009 | 0 | −2.32525 | −0.701156 | 1.01617 | −1.79316 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.40162 | 3.14741 | 3.76778 | 0 | 7.55888 | −2.45595 | 4.24555 | 6.90618 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(17\) | \( -1 \) |
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 | 7225.2.a.bi | 6 | |
| 5.b | even | 2 | 1 | 1445.2.a.l | ✓ | 6 | |
| 17.b | even | 2 | 1 | 7225.2.a.bh | 6 | ||
| 85.c | even | 2 | 1 | 1445.2.a.m | yes | 6 | |
| 85.j | even | 4 | 2 | 1445.2.d.h | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1445.2.a.l | ✓ | 6 | 5.b | even | 2 | 1 | |
| 1445.2.a.m | yes | 6 | 85.c | even | 2 | 1 | |
| 1445.2.d.h | 12 | 85.j | even | 4 | 2 | ||
| 7225.2.a.bh | 6 | 17.b | even | 2 | 1 | ||
| 7225.2.a.bi | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7225))\):
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\( T_{2}^{6} - 3T_{2}^{5} - 3T_{2}^{4} + 12T_{2}^{3} - 9T_{2} + 3 \)
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\( T_{3}^{6} - 3T_{3}^{5} - 6T_{3}^{4} + 17T_{3}^{3} + 6T_{3}^{2} - 15T_{3} + 1 \)
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\( T_{7}^{6} - 6T_{7}^{5} - 6T_{7}^{4} + 64T_{7}^{3} + 15T_{7}^{2} - 174T_{7} - 107 \)
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\( T_{11}^{6} - 6T_{11}^{5} - 21T_{11}^{4} + 114T_{11}^{3} + 207T_{11}^{2} - 495T_{11} - 807 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 3 \)
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| $3$ |
\( T^{6} - 3 T^{5} + \cdots + 1 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} - 6 T^{5} + \cdots - 107 \)
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| $11$ |
\( T^{6} - 6 T^{5} + \cdots - 807 \)
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| $13$ |
\( T^{6} - 9 T^{5} + \cdots + 433 \)
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| $17$ |
\( T^{6} \)
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| $19$ |
\( T^{6} + 21 T^{5} + \cdots - 89 \)
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| $23$ |
\( T^{6} - 18 T^{5} + \cdots - 4833 \)
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| $29$ |
\( T^{6} - 18 T^{5} + \cdots + 16473 \)
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| $31$ |
\( T^{6} + 6 T^{5} + \cdots + 2809 \)
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| $37$ |
\( T^{6} + 9 T^{5} + \cdots - 71927 \)
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| $41$ |
\( T^{6} - 3 T^{5} + \cdots - 21327 \)
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| $43$ |
\( T^{6} - 39 T^{5} + \cdots - 2771 \)
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| $47$ |
\( T^{6} + 3 T^{5} + \cdots + 3 \)
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| $53$ |
\( T^{6} - 24 T^{5} + \cdots + 1791 \)
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| $59$ |
\( T^{6} + 12 T^{5} + \cdots + 182037 \)
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| $61$ |
\( T^{6} - 27 T^{5} + \cdots + 15139 \)
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| $67$ |
\( T^{6} - 30 T^{5} + \cdots - 2753 \)
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| $71$ |
\( T^{6} + 18 T^{5} + \cdots + 1839 \)
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| $73$ |
\( T^{6} + 9 T^{5} + \cdots + 739 \)
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| $79$ |
\( T^{6} - 3 T^{5} + \cdots - 719 \)
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| $83$ |
\( T^{6} - 21 T^{5} + \cdots + 7617 \)
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| $89$ |
\( T^{6} - 6 T^{5} + \cdots - 19269 \)
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| $97$ |
\( T^{6} + 18 T^{5} + \cdots + 156421 \)
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