Newspace parameters
| Level: | \( N \) | \(=\) | \( 1935 = 3^{2} \cdot 5 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1935.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(15.4510527911\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.230224.1 |
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|
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| Defining polynomial: |
\( x^{5} - 2x^{4} - 4x^{3} + 6x^{2} + 3x - 2 \)
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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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 4x^{3} + 6x^{2} + 3x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 4\nu + 1 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 5\beta_{2} + 7\beta _1 + 7 \)
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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 |
|
−2.66240 | 0 | 5.08835 | 1.00000 | 0 | 2.42596 | −8.22242 | 0 | −2.66240 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.75737 | 0 | 1.08833 | 1.00000 | 0 | −0.669031 | 1.60213 | 0 | −1.75737 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.575055 | 0 | −1.66931 | 1.00000 | 0 | −2.24437 | 2.11006 | 0 | −0.575055 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0.495924 | 0 | −1.75406 | 1.00000 | 0 | −1.25814 | −1.86173 | 0 | 0.495924 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.49889 | 0 | 0.246683 | 1.00000 | 0 | 1.74558 | −2.62804 | 0 | 1.49889 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(43\) | \( -1 \) |
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 | 1935.2.a.s | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1935.2.a.x | yes | 5 | |
| 5.b | even | 2 | 1 | 9675.2.a.ci | 5 | ||
| 15.d | odd | 2 | 1 | 9675.2.a.bx | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1935.2.a.s | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1935.2.a.x | yes | 5 | 3.b | odd | 2 | 1 | |
| 9675.2.a.bx | 5 | 15.d | odd | 2 | 1 | ||
| 9675.2.a.ci | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1935))\):
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\( T_{2}^{5} + 3T_{2}^{4} - 2T_{2}^{3} - 8T_{2}^{2} + 2 \)
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\( T_{7}^{5} - 8T_{7}^{3} - 2T_{7}^{2} + 14T_{7} + 8 \)
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\( T_{11}^{5} + 6T_{11}^{4} - 8T_{11}^{3} - 60T_{11}^{2} + 7T_{11} + 128 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 3 T^{4} + \cdots + 2 \)
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| $3$ |
\( T^{5} \)
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| $5$ |
\( (T - 1)^{5} \)
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| $7$ |
\( T^{5} - 8 T^{3} + \cdots + 8 \)
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| $11$ |
\( T^{5} + 6 T^{4} + \cdots + 128 \)
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| $13$ |
\( T^{5} + 2 T^{4} + \cdots + 2 \)
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| $17$ |
\( T^{5} + 6 T^{4} + \cdots + 94 \)
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| $19$ |
\( T^{5} + 8 T^{4} + \cdots - 404 \)
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| $23$ |
\( T^{5} + 6 T^{4} + \cdots - 94 \)
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| $29$ |
\( T^{5} + 20 T^{4} + \cdots - 16 \)
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| $31$ |
\( T^{5} + 8 T^{4} + \cdots + 244 \)
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| $37$ |
\( T^{5} - 2 T^{4} + \cdots - 1952 \)
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| $41$ |
\( T^{5} + 24 T^{4} + \cdots - 1424 \)
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| $43$ |
\( (T - 1)^{5} \)
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| $47$ |
\( T^{5} + 6 T^{4} + \cdots + 1016 \)
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| $53$ |
\( T^{5} + 14 T^{4} + \cdots - 3842 \)
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| $59$ |
\( T^{5} + 4 T^{4} + \cdots + 86144 \)
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| $61$ |
\( T^{5} + 16 T^{4} + \cdots + 30992 \)
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| $67$ |
\( T^{5} - 2 T^{4} + \cdots - 52 \)
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| $71$ |
\( T^{5} + 14 T^{4} + \cdots + 23008 \)
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| $73$ |
\( T^{5} - 8 T^{4} + \cdots - 13568 \)
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| $79$ |
\( T^{5} - 324 T^{3} + \cdots + 81968 \)
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| $83$ |
\( T^{5} + 10 T^{4} + \cdots - 122 \)
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| $89$ |
\( T^{5} + 18 T^{4} + \cdots - 2428 \)
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| $97$ |
\( T^{5} - 2 T^{4} + \cdots + 614 \)
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