Newspace parameters
| Level: | \( N \) | \(=\) | \( 735 = 3 \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 735.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(43.3664038542\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{5} - 2x^{4} - 30x^{3} + 22x^{2} + 153x - 84 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 105) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 30x^{3} + 22x^{2} + 153x - 84 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 12 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu^{2} - 19\nu + 22 \)
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| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 23\nu^{2} + 29\nu + 54 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 12 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + 25\beta _1 + 14 \)
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| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 3\beta_{3} + 32\beta_{2} + 92\beta _1 + 264 \)
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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 |
|
−4.80647 | −3.00000 | 15.1022 | −5.00000 | 14.4194 | 0 | −34.1365 | 9.00000 | 24.0324 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.44043 | −3.00000 | −5.92517 | −5.00000 | 4.32128 | 0 | 20.0582 | 9.00000 | 7.20214 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.461250 | −3.00000 | −7.78725 | −5.00000 | −1.38375 | 0 | −7.28187 | 9.00000 | −2.30625 | ||||||||||||||||||||||||||||||||||
| 1.4 | 3.67994 | −3.00000 | 5.54199 | −5.00000 | −11.0398 | 0 | −9.04535 | 9.00000 | −18.3997 | ||||||||||||||||||||||||||||||||||
| 1.5 | 5.10571 | −3.00000 | 18.0682 | −5.00000 | −15.3171 | 0 | 51.4055 | 9.00000 | −25.5285 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( -1 \) |
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 | 735.4.a.z | 5 | |
| 3.b | odd | 2 | 1 | 2205.4.a.bs | 5 | ||
| 7.b | odd | 2 | 1 | 735.4.a.ba | 5 | ||
| 7.d | odd | 6 | 2 | 105.4.i.d | ✓ | 10 | |
| 21.c | even | 2 | 1 | 2205.4.a.br | 5 | ||
| 21.g | even | 6 | 2 | 315.4.j.h | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.4.i.d | ✓ | 10 | 7.d | odd | 6 | 2 | |
| 315.4.j.h | 10 | 21.g | even | 6 | 2 | ||
| 735.4.a.z | 5 | 1.a | even | 1 | 1 | trivial | |
| 735.4.a.ba | 5 | 7.b | odd | 2 | 1 | ||
| 2205.4.a.br | 5 | 21.c | even | 2 | 1 | ||
| 2205.4.a.bs | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(735))\):
|
\( T_{2}^{5} - 3T_{2}^{4} - 28T_{2}^{3} + 70T_{2}^{2} + 104T_{2} - 60 \)
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\( T_{11}^{5} - 43T_{11}^{4} - 3788T_{11}^{3} + 74940T_{11}^{2} + 4542452T_{11} + 35201580 \)
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\( T_{13}^{5} + 123T_{13}^{4} - 1074T_{13}^{3} - 566590T_{13}^{2} - 14427235T_{13} + 125988247 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 3 T^{4} + \cdots - 60 \)
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| $3$ |
\( (T + 3)^{5} \)
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| $5$ |
\( (T + 5)^{5} \)
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| $7$ |
\( T^{5} \)
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| $11$ |
\( T^{5} - 43 T^{4} + \cdots + 35201580 \)
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| $13$ |
\( T^{5} + 123 T^{4} + \cdots + 125988247 \)
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| $17$ |
\( T^{5} + 124 T^{4} + \cdots - 20514480 \)
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| $19$ |
\( T^{5} + 37 T^{4} + \cdots + 235295225 \)
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| $23$ |
\( T^{5} - 77 T^{4} + \cdots + 152403300 \)
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| $29$ |
\( T^{5} + \cdots - 15483733056 \)
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| $31$ |
\( T^{5} + \cdots + 69079322070 \)
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| $37$ |
\( T^{5} + \cdots - 212593667013 \)
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| $41$ |
\( T^{5} + \cdots + 914820763500 \)
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| $43$ |
\( T^{5} + \cdots + 45414054020 \)
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| $47$ |
\( T^{5} + \cdots - 439521150192 \)
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| $53$ |
\( T^{5} + \cdots + 12693539474880 \)
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| $59$ |
\( T^{5} + \cdots + 1309411612752 \)
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| $61$ |
\( T^{5} + \cdots + 16475195520000 \)
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| $67$ |
\( T^{5} + \cdots - 74697164857140 \)
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| $71$ |
\( T^{5} + \cdots + 12244368636072 \)
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| $73$ |
\( T^{5} + \cdots - 65528941041926 \)
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| $79$ |
\( T^{5} + \cdots + 1149495333516 \)
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| $83$ |
\( T^{5} + \cdots - 217219935694608 \)
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| $89$ |
\( T^{5} + \cdots + 8415535021416 \)
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| $97$ |
\( T^{5} + \cdots - 207081920604160 \)
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