Newspace parameters
| Level: | \( N \) | \(=\) | \( 3703 = 7 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3703.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(29.5686038685\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.70601.1 |
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| Defining polynomial: |
\( x^{5} - x^{4} - 5x^{3} + 2x^{2} + 3x - 1 \)
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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
\( x^{5} - x^{4} - 5x^{3} + 2x^{2} + 3x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 4\nu^{2} + \nu \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 2\nu + 2 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} - 2\nu + 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 2\beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 6\beta_{3} + 6\beta_{2} + 8\beta _1 + 9 \)
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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 |
|
−2.36697 | 3.00897 | 3.60257 | 4.35089 | −7.12215 | −1.00000 | −3.79324 | 6.05389 | −10.2985 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.13472 | −2.15334 | 2.55703 | −0.0441003 | 4.59679 | −1.00000 | −1.18911 | 1.63689 | 0.0941419 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.128035 | 1.79027 | −1.98361 | −0.908442 | −0.229217 | −1.00000 | 0.510042 | 0.205061 | 0.116313 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0.886926 | −0.427462 | −1.21336 | 4.02647 | −0.379127 | −1.00000 | −2.85001 | −2.81728 | 3.57118 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.74280 | −2.21843 | 1.03737 | −1.42482 | −3.86629 | −1.00000 | −1.67768 | 1.92143 | −2.48318 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( +1 \) |
| \(23\) | \( -1 \) |
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 | 3703.2.a.e | yes | 5 |
| 23.b | odd | 2 | 1 | 3703.2.a.d | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3703.2.a.d | ✓ | 5 | 23.b | odd | 2 | 1 | |
| 3703.2.a.e | yes | 5 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3703))\):
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\( T_{2}^{5} + 2T_{2}^{4} - 5T_{2}^{3} - 7T_{2}^{2} + 7T_{2} + 1 \)
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\( T_{5}^{5} - 6T_{5}^{4} - T_{5}^{3} + 30T_{5}^{2} + 24T_{5} + 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 2 T^{4} + \cdots + 1 \)
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| $3$ |
\( T^{5} - 11 T^{3} + \cdots + 11 \)
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| $5$ |
\( T^{5} - 6 T^{4} + \cdots + 1 \)
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| $7$ |
\( (T + 1)^{5} \)
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| $11$ |
\( T^{5} - T^{4} + \cdots - 53 \)
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| $13$ |
\( T^{5} - 10 T^{4} + \cdots - 23 \)
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| $17$ |
\( T^{5} - T^{4} - 31 T^{3} + \cdots - 9 \)
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| $19$ |
\( T^{5} - 7 T^{4} + \cdots - 199 \)
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| $23$ |
\( T^{5} \)
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| $29$ |
\( T^{5} + 2 T^{4} + \cdots + 199 \)
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| $31$ |
\( T^{5} - T^{4} - 40 T^{3} + \cdots + 9 \)
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| $37$ |
\( T^{5} - 2 T^{4} + \cdots - 211 \)
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| $41$ |
\( T^{5} + 14 T^{4} + \cdots + 1 \)
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| $43$ |
\( T^{5} - 15 T^{4} + \cdots + 4549 \)
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| $47$ |
\( T^{5} + 3 T^{4} + \cdots + 26737 \)
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| $53$ |
\( T^{5} - 21 T^{4} + \cdots - 7567 \)
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| $59$ |
\( T^{5} - 11 T^{4} + \cdots - 583 \)
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| $61$ |
\( T^{5} - 16 T^{4} + \cdots - 114273 \)
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| $67$ |
\( T^{5} + T^{4} + \cdots - 62971 \)
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| $71$ |
\( T^{5} + 3 T^{4} + \cdots + 77 \)
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| $73$ |
\( T^{5} - 9 T^{4} + \cdots - 2673 \)
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| $79$ |
\( T^{5} - 12 T^{4} + \cdots - 13853 \)
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| $83$ |
\( T^{5} - 18 T^{4} + \cdots + 22957 \)
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| $89$ |
\( T^{5} - 25 T^{4} + \cdots + 4489 \)
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| $97$ |
\( T^{5} - 133 T^{3} + \cdots - 567 \)
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