Newspace parameters
| Level: | \( N \) | \(=\) | \( 875 = 5^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 875.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(6.98691017686\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.3438125.1 |
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 6x^{4} + 12x^{3} + 6x^{2} - 15x + 5 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 6x^{4} + 12x^{3} + 6x^{2} - 15x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 6\nu^{3} + 5\nu^{2} + 7\nu - 5 \)
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| \(\beta_{3}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 7\nu^{3} - 5\nu^{2} - 11\nu + 5 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 6\nu^{2} + 11\nu - 7 \)
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| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 7\nu^{3} - 11\nu^{2} - 12\nu + 10 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 6\beta_{4} + 5\beta_{3} + \beta _1 + 7 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 6\beta_{3} + 7\beta_{2} + 18\beta _1 + 2 \)
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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 |
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−2.07183 | 1.41684 | 2.29250 | 0 | −2.93546 | 1.00000 | −0.606002 | −0.992564 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.40202 | 0.0555810 | −0.0343509 | 0 | −0.0779254 | 1.00000 | 2.85219 | −2.99691 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.525197 | 2.78976 | −1.72417 | 0 | 1.46517 | 1.00000 | −1.95592 | 4.78278 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.753136 | −0.885510 | −1.43279 | 0 | −0.666910 | 1.00000 | −2.58536 | −2.21587 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.87682 | −2.46338 | 1.52245 | 0 | −4.62332 | 1.00000 | −0.896272 | 3.06823 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.31870 | 2.08670 | 3.37636 | 0 | 4.83844 | 1.00000 | 3.19136 | 1.35433 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(7\) | \( -1 \) |
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 | 875.2.a.g | yes | 6 |
| 3.b | odd | 2 | 1 | 7875.2.a.r | 6 | ||
| 5.b | even | 2 | 1 | 875.2.a.f | ✓ | 6 | |
| 5.c | odd | 4 | 2 | 875.2.b.d | 12 | ||
| 7.b | odd | 2 | 1 | 6125.2.a.u | 6 | ||
| 15.d | odd | 2 | 1 | 7875.2.a.s | 6 | ||
| 35.c | odd | 2 | 1 | 6125.2.a.r | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 875.2.a.f | ✓ | 6 | 5.b | even | 2 | 1 | |
| 875.2.a.g | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 875.2.b.d | 12 | 5.c | odd | 4 | 2 | ||
| 6125.2.a.r | 6 | 35.c | odd | 2 | 1 | ||
| 6125.2.a.u | 6 | 7.b | odd | 2 | 1 | ||
| 7875.2.a.r | 6 | 3.b | odd | 2 | 1 | ||
| 7875.2.a.s | 6 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(875))\):
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\( T_{2}^{6} - 2T_{2}^{5} - 6T_{2}^{4} + 12T_{2}^{3} + 6T_{2}^{2} - 15T_{2} + 5 \)
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\( T_{3}^{6} - 3T_{3}^{5} - 6T_{3}^{4} + 21T_{3}^{3} - T_{3}^{2} - 18T_{3} + 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots + 5 \)
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| $3$ |
\( T^{6} - 3 T^{5} + \cdots + 1 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( (T - 1)^{6} \)
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| $11$ |
\( T^{6} + 3 T^{5} + \cdots + 19 \)
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| $13$ |
\( T^{6} - 16 T^{5} + \cdots - 5 \)
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| $17$ |
\( T^{6} - 8 T^{5} + \cdots + 779 \)
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| $19$ |
\( T^{6} - 9 T^{5} + \cdots + 5095 \)
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| $23$ |
\( T^{6} - 9 T^{5} + \cdots - 1709 \)
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| $29$ |
\( T^{6} + 9 T^{5} + \cdots - 125 \)
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| $31$ |
\( T^{6} - 15 T^{5} + \cdots + 1091 \)
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| $37$ |
\( T^{6} - 31 T^{5} + \cdots - 9311 \)
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| $41$ |
\( T^{6} - 9 T^{5} + \cdots - 49571 \)
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| $43$ |
\( T^{6} - 26 T^{5} + \cdots + 155 \)
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| $47$ |
\( T^{6} - 119 T^{4} + \cdots - 269 \)
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| $53$ |
\( T^{6} - 5 T^{5} + \cdots + 1019 \)
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| $59$ |
\( T^{6} - 5 T^{5} + \cdots + 1475 \)
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| $61$ |
\( T^{6} + T^{5} + \cdots - 188971 \)
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| $67$ |
\( T^{6} - 34 T^{5} + \cdots - 118829 \)
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| $71$ |
\( T^{6} + 7 T^{5} + \cdots + 7505 \)
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| $73$ |
\( T^{6} - 10 T^{5} + \cdots - 543911 \)
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| $79$ |
\( T^{6} + 22 T^{5} + \cdots + 3025 \)
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| $83$ |
\( T^{6} + 25 T^{5} + \cdots - 1021 \)
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| $89$ |
\( T^{6} + 4 T^{5} + \cdots + 19445 \)
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| $97$ |
\( T^{6} - 40 T^{5} + \cdots - 59 \)
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