Newspace parameters
| Level: | \( N \) | \(=\) | \( 2175 = 3 \cdot 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2175.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(128.329154262\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{6} - x^{5} - 31x^{4} + 9x^{3} + 230x^{2} + 32x - 128 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 435) |
| 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
\( x^{6} - x^{5} - 31x^{4} + 9x^{3} + 230x^{2} + 32x - 128 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 17\nu^{2} + 18\nu + 16 ) / 8 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 15\nu + 12 ) / 2 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} + 25\nu^{2} - 26\nu - 96 ) / 8 \)
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| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 25\nu^{3} + 26\nu^{2} + 128\nu - 24 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{2} + \beta _1 + 10 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + 17\beta _1 + 8 \)
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| \(\nu^{4}\) | \(=\) |
\( 21\beta_{4} + 4\beta_{3} + 29\beta_{2} + 33\beta _1 + 170 \)
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| \(\nu^{5}\) | \(=\) |
\( 8\beta_{5} + 66\beta_{4} + 58\beta_{3} + 82\beta_{2} + 337\beta _1 + 304 \)
|
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 |
|
−3.96002 | 3.00000 | 7.68179 | 0 | −11.8801 | −2.19133 | 1.26012 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3.16884 | 3.00000 | 2.04152 | 0 | −9.50651 | 6.36645 | 18.8814 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.886494 | 3.00000 | −7.21413 | 0 | −2.65948 | −17.9759 | 13.4872 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.691666 | 3.00000 | −7.52160 | 0 | 2.07500 | 14.5720 | −10.7358 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.33404 | 3.00000 | 3.11579 | 0 | 10.0021 | −1.26363 | −16.2841 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.98965 | 3.00000 | 16.8966 | 0 | 14.9690 | −22.5076 | 44.3911 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(29\) | \( +1 \) |
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 | 2175.4.a.l | 6 | |
| 5.b | even | 2 | 1 | 435.4.a.g | ✓ | 6 | |
| 15.d | odd | 2 | 1 | 1305.4.a.i | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 435.4.a.g | ✓ | 6 | 5.b | even | 2 | 1 | |
| 1305.4.a.i | 6 | 15.d | odd | 2 | 1 | ||
| 2175.4.a.l | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2175))\):
|
\( T_{2}^{6} - T_{2}^{5} - 31T_{2}^{4} + 9T_{2}^{3} + 230T_{2}^{2} + 32T_{2} - 128 \)
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\( T_{7}^{6} + 23T_{7}^{5} - 280T_{7}^{4} - 5872T_{7}^{3} + 20272T_{7}^{2} + 116624T_{7} + 103936 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} + \cdots - 128 \)
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| $3$ |
\( (T - 3)^{6} \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} + 23 T^{5} + \cdots + 103936 \)
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| $11$ |
\( T^{6} + \cdots + 4447644416 \)
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| $13$ |
\( T^{6} - 83 T^{5} + \cdots - 755576576 \)
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| $17$ |
\( T^{6} + \cdots + 35167804928 \)
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| $19$ |
\( T^{6} + \cdots + 70167660032 \)
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| $23$ |
\( T^{6} + \cdots - 17557826816 \)
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| $29$ |
\( (T + 29)^{6} \)
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| $31$ |
\( T^{6} + \cdots + 679337811968 \)
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| $37$ |
\( T^{6} + \cdots - 11015322431488 \)
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| $41$ |
\( T^{6} + \cdots - 2844920728576 \)
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| $43$ |
\( T^{6} + \cdots + 1852550152192 \)
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| $47$ |
\( T^{6} + \cdots - 36309886959616 \)
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| $53$ |
\( T^{6} + \cdots + 32\!\cdots\!00 \)
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| $59$ |
\( T^{6} + \cdots - 31\!\cdots\!44 \)
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| $61$ |
\( T^{6} + \cdots - 10\!\cdots\!00 \)
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| $67$ |
\( T^{6} + \cdots + 82\!\cdots\!64 \)
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| $71$ |
\( T^{6} + \cdots - 99\!\cdots\!48 \)
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| $73$ |
\( T^{6} + \cdots + 136347629123072 \)
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| $79$ |
\( T^{6} + \cdots - 859390658797568 \)
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| $83$ |
\( T^{6} + \cdots - 17\!\cdots\!52 \)
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| $89$ |
\( T^{6} + \cdots - 34\!\cdots\!76 \)
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| $97$ |
\( T^{6} + \cdots - 45\!\cdots\!08 \)
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