Newspace parameters
| Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 405.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(23.8957735523\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 44x^{5} + 74x^{4} + 479x^{3} - 460x^{2} - 1200x + 288 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3^{5} \) |
| Twist minimal: | no (minimal twist has level 45) |
| 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
\( x^{7} - 2x^{6} - 44x^{5} + 74x^{4} + 479x^{3} - 460x^{2} - 1200x + 288 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 23\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 39\nu^{4} + 35\nu^{3} + 310\nu^{2} - 114\nu - 240 ) / 12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 4\nu^{5} + 42\nu^{4} - 140\nu^{3} - 397\nu^{2} + 756\nu + 672 ) / 24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + 2\nu^{5} - 48\nu^{4} - 82\nu^{3} + 595\nu^{2} + 732\nu - 1104 ) / 24 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 23\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{6} + 2\beta_{5} + 2\beta_{4} - \beta_{3} + 31\beta_{2} - 6\beta _1 + 294 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{6} + 6\beta_{5} + 2\beta_{4} + 36\beta_{3} - 2\beta_{2} + 597\beta _1 - 26 \)
|
| \(\nu^{6}\) | \(=\) |
\( -76\beta_{6} + 84\beta_{5} + 92\beta_{4} - 38\beta_{3} + 897\beta_{2} - 328\beta _1 + 7615 \)
|
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 |
|
−5.31551 | 0 | 20.2546 | −5.00000 | 0 | −13.4337 | −65.1396 | 0 | 26.5775 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.26178 | 0 | 10.1628 | −5.00000 | 0 | 30.7639 | −9.21718 | 0 | 21.3089 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.19444 | 0 | −3.18442 | −5.00000 | 0 | 2.76605 | 24.5436 | 0 | 10.9722 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.225250 | 0 | −7.94926 | −5.00000 | 0 | −31.1940 | 3.59257 | 0 | 1.12625 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.57021 | 0 | −5.53444 | −5.00000 | 0 | 34.2398 | −21.2519 | 0 | −7.85104 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.04174 | 0 | 1.25219 | −5.00000 | 0 | −13.7122 | −20.5251 | 0 | −15.2087 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.38503 | 0 | 20.9986 | −5.00000 | 0 | 12.5702 | 69.9976 | 0 | −26.9252 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
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 | 405.4.a.m | 7 | |
| 3.b | odd | 2 | 1 | 405.4.a.n | 7 | ||
| 5.b | even | 2 | 1 | 2025.4.a.bb | 7 | ||
| 9.c | even | 3 | 2 | 45.4.e.c | ✓ | 14 | |
| 9.d | odd | 6 | 2 | 135.4.e.c | 14 | ||
| 15.d | odd | 2 | 1 | 2025.4.a.ba | 7 | ||
| 45.j | even | 6 | 2 | 225.4.e.d | 14 | ||
| 45.k | odd | 12 | 4 | 225.4.k.d | 28 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.4.e.c | ✓ | 14 | 9.c | even | 3 | 2 | |
| 135.4.e.c | 14 | 9.d | odd | 6 | 2 | ||
| 225.4.e.d | 14 | 45.j | even | 6 | 2 | ||
| 225.4.k.d | 28 | 45.k | odd | 12 | 4 | ||
| 405.4.a.m | 7 | 1.a | even | 1 | 1 | trivial | |
| 405.4.a.n | 7 | 3.b | odd | 2 | 1 | ||
| 2025.4.a.ba | 7 | 15.d | odd | 2 | 1 | ||
| 2025.4.a.bb | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 2T_{2}^{6} - 44T_{2}^{5} - 74T_{2}^{4} + 479T_{2}^{3} + 460T_{2}^{2} - 1200T_{2} - 288 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 2 T^{6} + \cdots - 288 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( (T + 5)^{7} \)
|
| $7$ |
\( T^{7} - 22 T^{6} + \cdots + 210450744 \)
|
| $11$ |
\( T^{7} + \cdots + 20019444768 \)
|
| $13$ |
\( T^{7} + \cdots + 159234392576 \)
|
| $17$ |
\( T^{7} + \cdots - 60588009792 \)
|
| $19$ |
\( T^{7} + \cdots - 1377989598400 \)
|
| $23$ |
\( T^{7} + \cdots + 4061042235738 \)
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| $29$ |
\( T^{7} + \cdots + 12128150971026 \)
|
| $31$ |
\( T^{7} + \cdots + 29481260805504 \)
|
| $37$ |
\( T^{7} + \cdots - 83646911884544 \)
|
| $41$ |
\( T^{7} + \cdots - 15\!\cdots\!61 \)
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| $43$ |
\( T^{7} + \cdots - 13\!\cdots\!84 \)
|
| $47$ |
\( T^{7} + \cdots - 65\!\cdots\!72 \)
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| $53$ |
\( T^{7} + \cdots - 11\!\cdots\!92 \)
|
| $59$ |
\( T^{7} + \cdots + 59\!\cdots\!08 \)
|
| $61$ |
\( T^{7} + \cdots + 12\!\cdots\!46 \)
|
| $67$ |
\( T^{7} + \cdots + 29\!\cdots\!03 \)
|
| $71$ |
\( T^{7} + \cdots - 32\!\cdots\!28 \)
|
| $73$ |
\( T^{7} + \cdots + 82\!\cdots\!12 \)
|
| $79$ |
\( T^{7} + \cdots + 16\!\cdots\!08 \)
|
| $83$ |
\( T^{7} + \cdots + 66\!\cdots\!72 \)
|
| $89$ |
\( T^{7} + \cdots - 32\!\cdots\!50 \)
|
| $97$ |
\( T^{7} + \cdots + 47\!\cdots\!76 \)
|
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