Newspace parameters
| Level: | \( N \) | \(=\) | \( 800 = 2^{5} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 800.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(47.2015280046\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.2106005.1 |
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|
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| Defining polynomial: |
\( x^{4} - 2x^{3} - 24x^{2} + 25x - 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 24x^{2} + 25x - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
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| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} - 4\nu - 50 \)
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| \(\beta_{3}\) | \(=\) |
\( 8\nu^{3} - 12\nu^{2} - 198\nu + 101 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 52 ) / 4 \)
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| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 3\beta_{2} + 105\beta _1 + 154 ) / 8 \)
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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 |
|
0 | −10.0971 | 0 | 0 | 0 | −10.5923 | 0 | 74.9510 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.221457 | 0 | 0 | 0 | −22.7992 | 0 | −26.9510 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.221457 | 0 | 0 | 0 | 22.7992 | 0 | −26.9510 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 10.0971 | 0 | 0 | 0 | 10.5923 | 0 | 74.9510 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 800.4.a.bb | yes | 4 |
| 4.b | odd | 2 | 1 | inner | 800.4.a.bb | yes | 4 |
| 5.b | even | 2 | 1 | 800.4.a.ba | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 800.4.c.o | 8 | ||
| 8.b | even | 2 | 1 | 1600.4.a.cw | 4 | ||
| 8.d | odd | 2 | 1 | 1600.4.a.cw | 4 | ||
| 20.d | odd | 2 | 1 | 800.4.a.ba | ✓ | 4 | |
| 20.e | even | 4 | 2 | 800.4.c.o | 8 | ||
| 40.e | odd | 2 | 1 | 1600.4.a.cx | 4 | ||
| 40.f | even | 2 | 1 | 1600.4.a.cx | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 800.4.a.ba | ✓ | 4 | 5.b | even | 2 | 1 | |
| 800.4.a.ba | ✓ | 4 | 20.d | odd | 2 | 1 | |
| 800.4.a.bb | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 800.4.a.bb | yes | 4 | 4.b | odd | 2 | 1 | inner |
| 800.4.c.o | 8 | 5.c | odd | 4 | 2 | ||
| 800.4.c.o | 8 | 20.e | even | 4 | 2 | ||
| 1600.4.a.cw | 4 | 8.b | even | 2 | 1 | ||
| 1600.4.a.cw | 4 | 8.d | odd | 2 | 1 | ||
| 1600.4.a.cx | 4 | 40.e | odd | 2 | 1 | ||
| 1600.4.a.cx | 4 | 40.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(800))\):
|
\( T_{3}^{4} - 102T_{3}^{2} + 5 \)
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\( T_{11}^{4} - 5982T_{11}^{2} + 6762845 \)
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\( T_{13}^{2} - 36T_{13} - 2272 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} - 102T^{2} + 5 \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} - 632 T^{2} + 58320 \)
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| $11$ |
\( T^{4} - 5982 T^{2} + 6762845 \)
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| $13$ |
\( (T^{2} - 36 T - 2272)^{2} \)
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| $17$ |
\( (T^{2} + 30 T - 2371)^{2} \)
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| $19$ |
\( T^{4} - 4390 T^{2} + 4753125 \)
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| $23$ |
\( T^{4} - 46392 T^{2} + 523059920 \)
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| $29$ |
\( (T^{2} + 36 T - 23040)^{2} \)
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| $31$ |
\( T^{4} + \cdots + 1457948880 \)
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| $37$ |
\( (T^{2} - 464 T + 30460)^{2} \)
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| $41$ |
\( (T^{2} - 362 T - 8775)^{2} \)
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| $43$ |
\( T^{4} + \cdots + 1179648000 \)
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| $47$ |
\( T^{4} + \cdots + 5516513280 \)
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| $53$ |
\( (T + 222)^{4} \)
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| $59$ |
\( T^{4} + \cdots + 83483873280 \)
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| $61$ |
\( (T^{2} - 888 T + 194540)^{2} \)
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| $67$ |
\( T^{4} + \cdots + 75081483405 \)
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| $71$ |
\( T^{4} - 428432 T^{2} + 7787520 \)
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| $73$ |
\( (T^{2} + 530 T - 680019)^{2} \)
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| $79$ |
\( T^{4} + \cdots + 37300611920 \)
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| $83$ |
\( T^{4} + \cdots + 842120280125 \)
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| $89$ |
\( (T^{2} - 450 T - 14275)^{2} \)
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| $97$ |
\( (T^{2} + 252 T - 243724)^{2} \)
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