Newspace parameters
| Level: | \( N \) | \(=\) | \( 775 = 5^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 775.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(45.7264802544\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
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| Defining polynomial: |
\( x^{5} - 2x^{4} - 32x^{3} + 19x^{2} + 228x + 172 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 31) |
| 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
\( x^{5} - 2x^{4} - 32x^{3} + 19x^{2} + 228x + 172 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 16\nu^{3} - 56\nu^{2} - 365\nu + 106 ) / 104 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{4} + 24\nu^{3} + 72\nu^{2} - 359\nu - 10 ) / 104 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -11\nu^{4} + 32\nu^{3} + 304\nu^{2} - 457\nu - 1582 ) / 104 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 2\beta_{3} + \beta_{2} + \beta _1 + 14 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} - 3\beta_{3} + 7\beta_{2} + 23\beta _1 + 23 \)
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| \(\nu^{4}\) | \(=\) |
\( 24\beta_{4} - 64\beta_{3} + 48\beta_{2} + 53\beta _1 + 310 \)
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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 |
|
−5.18275 | 2.52950 | 18.8609 | 0 | −13.1098 | −6.14402 | −56.2891 | −20.6016 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −3.15912 | 0.0377000 | 1.98006 | 0 | −0.119099 | 3.86558 | 19.0177 | −26.9986 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.92951 | −8.71716 | −4.27701 | 0 | 16.8198 | 5.68067 | 23.6886 | 48.9889 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.63578 | −5.22185 | −1.05267 | 0 | −13.7637 | −25.9132 | −23.8608 | 0.267763 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.63560 | 7.37181 | 13.4888 | 0 | 34.1727 | 13.5109 | 25.4436 | 27.3436 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(31\) | \( +1 \) |
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 | 775.4.a.f | 5 | |
| 5.b | even | 2 | 1 | 31.4.a.b | ✓ | 5 | |
| 15.d | odd | 2 | 1 | 279.4.a.h | 5 | ||
| 20.d | odd | 2 | 1 | 496.4.a.i | 5 | ||
| 35.c | odd | 2 | 1 | 1519.4.a.c | 5 | ||
| 40.e | odd | 2 | 1 | 1984.4.a.s | 5 | ||
| 40.f | even | 2 | 1 | 1984.4.a.r | 5 | ||
| 155.c | odd | 2 | 1 | 961.4.a.e | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 31.4.a.b | ✓ | 5 | 5.b | even | 2 | 1 | |
| 279.4.a.h | 5 | 15.d | odd | 2 | 1 | ||
| 496.4.a.i | 5 | 20.d | odd | 2 | 1 | ||
| 775.4.a.f | 5 | 1.a | even | 1 | 1 | trivial | |
| 961.4.a.e | 5 | 155.c | odd | 2 | 1 | ||
| 1519.4.a.c | 5 | 35.c | odd | 2 | 1 | ||
| 1984.4.a.r | 5 | 40.f | even | 2 | 1 | ||
| 1984.4.a.s | 5 | 40.e | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(775))\):
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\( T_{2}^{5} + 3T_{2}^{4} - 30T_{2}^{3} - 79T_{2}^{2} + 167T_{2} + 386 \)
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\( T_{3}^{5} + 4T_{3}^{4} - 74T_{3}^{3} - 188T_{3}^{2} + 856T_{3} - 32 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 3 T^{4} + \cdots + 386 \)
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| $3$ |
\( T^{5} + 4 T^{4} + \cdots - 32 \)
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| $5$ |
\( T^{5} \)
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| $7$ |
\( T^{5} + 9 T^{4} + \cdots - 47236 \)
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| $11$ |
\( T^{5} - 88 T^{4} + \cdots - 76793648 \)
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| $13$ |
\( T^{5} - 28 T^{4} + \cdots - 85935616 \)
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| $17$ |
\( T^{5} + 138 T^{4} + \cdots - 845793728 \)
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| $19$ |
\( T^{5} + 43 T^{4} + \cdots - 885299824 \)
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| $23$ |
\( T^{5} + \cdots + 1477525504 \)
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| $29$ |
\( T^{5} + \cdots + 6492808496 \)
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| $31$ |
\( (T + 31)^{5} \)
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| $37$ |
\( T^{5} + \cdots + 315180705232 \)
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| $41$ |
\( T^{5} + \cdots + 19192433688 \)
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| $43$ |
\( T^{5} + \cdots + 154176970896 \)
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| $47$ |
\( T^{5} + \cdots - 197408306432 \)
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| $53$ |
\( T^{5} + \cdots - 79406336128 \)
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| $59$ |
\( T^{5} + \cdots + 19804492743336 \)
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| $61$ |
\( T^{5} + \cdots + 922927740352 \)
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| $67$ |
\( T^{5} + \cdots + 22262005628928 \)
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| $71$ |
\( T^{5} + \cdots - 69573929276736 \)
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| $73$ |
\( T^{5} + \cdots + 1852644259168 \)
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| $79$ |
\( T^{5} + \cdots - 59985571648 \)
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| $83$ |
\( T^{5} + \cdots - 657431598704 \)
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| $89$ |
\( T^{5} + \cdots + 68090734165536 \)
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| $97$ |
\( T^{5} + \cdots + 31148274036888 \)
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