Newspace parameters
| Level: | \( N \) | \(=\) | \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5733.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(45.7782354788\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.46162368.1 |
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| Defining polynomial: |
\( x^{6} - 8x^{4} + 17x^{2} - 7 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 819) |
| 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
\( x^{6} - 8x^{4} + 17x^{2} - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 3 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 6\nu^{3} + 7\nu \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 5\beta_{2} + 12 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 6\beta_{3} + 17\beta_1 \)
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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 |
|
−2.16786 | 0 | 2.69963 | −1.51670 | 0 | 0 | −1.51670 | 0 | 3.28799 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.66159 | 0 | 0.760877 | 2.05891 | 0 | 0 | 2.05891 | 0 | −3.42107 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.734503 | 0 | −1.46050 | 2.54175 | 0 | 0 | 2.54175 | 0 | −1.86693 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.734503 | 0 | −1.46050 | −2.54175 | 0 | 0 | −2.54175 | 0 | −1.86693 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.66159 | 0 | 0.760877 | −2.05891 | 0 | 0 | −2.05891 | 0 | −3.42107 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.16786 | 0 | 2.69963 | 1.51670 | 0 | 0 | 1.51670 | 0 | 3.28799 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(13\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5733.2.a.bs | 6 | |
| 3.b | odd | 2 | 1 | inner | 5733.2.a.bs | 6 | |
| 7.b | odd | 2 | 1 | 5733.2.a.bt | 6 | ||
| 7.c | even | 3 | 2 | 819.2.j.i | ✓ | 12 | |
| 21.c | even | 2 | 1 | 5733.2.a.bt | 6 | ||
| 21.h | odd | 6 | 2 | 819.2.j.i | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 819.2.j.i | ✓ | 12 | 7.c | even | 3 | 2 | |
| 819.2.j.i | ✓ | 12 | 21.h | odd | 6 | 2 | |
| 5733.2.a.bs | 6 | 1.a | even | 1 | 1 | trivial | |
| 5733.2.a.bs | 6 | 3.b | odd | 2 | 1 | inner | |
| 5733.2.a.bt | 6 | 7.b | odd | 2 | 1 | ||
| 5733.2.a.bt | 6 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5733))\):
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\( T_{2}^{6} - 8T_{2}^{4} + 17T_{2}^{2} - 7 \)
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\( T_{5}^{6} - 13T_{5}^{4} + 52T_{5}^{2} - 63 \)
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\( T_{11}^{6} - 33T_{11}^{4} + 306T_{11}^{2} - 567 \)
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\( T_{17}^{6} - 83T_{17}^{4} + 1676T_{17}^{2} - 5887 \)
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\( T_{19}^{3} - 5T_{19}^{2} - 9T_{19} + 21 \)
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\( T_{31}^{3} - 4T_{31}^{2} - 59T_{31} - 21 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 8 T^{4} + \cdots - 7 \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} - 13 T^{4} + \cdots - 63 \)
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| $7$ |
\( T^{6} \)
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| $11$ |
\( T^{6} - 33 T^{4} + \cdots - 567 \)
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| $13$ |
\( (T + 1)^{6} \)
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| $17$ |
\( T^{6} - 83 T^{4} + \cdots - 5887 \)
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| $19$ |
\( (T^{3} - 5 T^{2} - 9 T + 21)^{2} \)
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| $23$ |
\( T^{6} - 46 T^{4} + \cdots - 63 \)
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| $29$ |
\( T^{6} - 164 T^{4} + \cdots - 135247 \)
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| $31$ |
\( (T^{3} - 4 T^{2} - 59 T - 21)^{2} \)
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| $37$ |
\( (T^{3} + 5 T^{2} + \cdots - 197)^{2} \)
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| $41$ |
\( T^{6} - 124 T^{4} + \cdots - 3087 \)
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| $43$ |
\( (T^{3} + 20 T^{2} + \cdots + 79)^{2} \)
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| $47$ |
\( T^{6} - 32 T^{4} + \cdots - 448 \)
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| $53$ |
\( T^{6} - 241 T^{4} + \cdots - 413343 \)
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| $59$ |
\( T^{6} - 257 T^{4} + \cdots - 80143 \)
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| $61$ |
\( (T^{3} + T^{2} - 150 T - 693)^{2} \)
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| $67$ |
\( (T^{3} + 33 T^{2} + \cdots + 1233)^{2} \)
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| $71$ |
\( T^{6} - 365 T^{4} + \cdots - 942823 \)
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| $73$ |
\( (T^{3} + 14 T^{2} + \cdots - 413)^{2} \)
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| $79$ |
\( (T^{3} + 10 T^{2} + \cdots - 1125)^{2} \)
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| $83$ |
\( T^{6} - 468 T^{4} + \cdots - 3720087 \)
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| $89$ |
\( T^{6} - 265 T^{4} + \cdots - 219303 \)
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| $97$ |
\( (T^{3} - 11 T^{2} + \cdots + 1631)^{2} \)
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