Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7803,2,Mod(1,7803)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7803, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7803.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7803 = 3^{3} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7803.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(62.3072686972\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.160801.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 5x^{3} + 4x^{2} + 3x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
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
\( x^{5} - x^{4} - 5x^{3} + 4x^{2} + 3x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 4\nu^{2} + 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 8 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{4} + 5\beta_{3} + 5\beta_{2} - 2\beta _1 + 5 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{4} + 4\beta_{3} - \beta_{2} + 2\beta _1 + 11 \)
|
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.43116 | 0 | 3.91053 | 1.77433 | 0 | 1.33114 | −4.64481 | 0 | −4.31367 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.530605 | 0 | −1.71846 | 1.60484 | 0 | −1.82457 | 1.97303 | 0 | −0.851535 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.40662 | 0 | −0.0214328 | 0.314752 | 0 | 4.28611 | −2.84338 | 0 | 0.442735 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.80078 | 0 | 1.24280 | −3.01952 | 0 | −3.07394 | −1.36355 | 0 | −5.43749 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.75437 | 0 | 5.58656 | 3.32561 | 0 | 0.281253 | 9.87870 | 0 | 9.15996 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(17\) | \(1\) |
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 | 7803.2.a.bm | yes | 5 |
| 3.b | odd | 2 | 1 | 7803.2.a.bj | ✓ | 5 | |
| 17.b | even | 2 | 1 | 7803.2.a.bl | yes | 5 | |
| 51.c | odd | 2 | 1 | 7803.2.a.bk | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7803.2.a.bj | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 7803.2.a.bk | yes | 5 | 51.c | odd | 2 | 1 | |
| 7803.2.a.bl | yes | 5 | 17.b | even | 2 | 1 | |
| 7803.2.a.bm | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7803))\):
|
\( T_{2}^{5} - 3T_{2}^{4} - 5T_{2}^{3} + 19T_{2}^{2} - 6T_{2} - 9 \)
|
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\( T_{5}^{5} - 4T_{5}^{4} - 5T_{5}^{3} + 35T_{5}^{2} - 39T_{5} + 9 \)
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\( T_{7}^{5} - T_{7}^{4} - 16T_{7}^{3} + T_{7}^{2} + 33T_{7} - 9 \)
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\( T_{11}^{5} + 8T_{11}^{4} - 4T_{11}^{3} - 147T_{11}^{2} - 270T_{11} - 27 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 3 T^{4} + \cdots - 9 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} - 4 T^{4} + \cdots + 9 \)
|
| $7$ |
\( T^{5} - T^{4} - 16 T^{3} + \cdots - 9 \)
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| $11$ |
\( T^{5} + 8 T^{4} + \cdots - 27 \)
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| $13$ |
\( T^{5} - T^{4} + \cdots - 107 \)
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| $17$ |
\( T^{5} \)
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| $19$ |
\( T^{5} + 6 T^{4} + \cdots + 9 \)
|
| $23$ |
\( T^{5} + 8 T^{4} + \cdots + 171 \)
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| $29$ |
\( T^{5} - 12 T^{4} + \cdots + 1377 \)
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| $31$ |
\( T^{5} - 9 T^{4} + \cdots + 83 \)
|
| $37$ |
\( T^{5} + 6 T^{4} + \cdots - 1 \)
|
| $41$ |
\( T^{5} + 3 T^{4} + \cdots + 837 \)
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| $43$ |
\( T^{5} + 13 T^{4} + \cdots + 81 \)
|
| $47$ |
\( T^{5} - 7 T^{4} + \cdots - 31401 \)
|
| $53$ |
\( T^{5} - 26 T^{4} + \cdots - 4563 \)
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| $59$ |
\( T^{5} + 9 T^{4} + \cdots + 92259 \)
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| $61$ |
\( T^{5} - 11 T^{4} + \cdots + 1663 \)
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| $67$ |
\( T^{5} + 14 T^{4} + \cdots - 991 \)
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| $71$ |
\( T^{5} - 22 T^{4} + \cdots + 22752 \)
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| $73$ |
\( T^{5} + 5 T^{4} + \cdots + 46449 \)
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| $79$ |
\( T^{5} + 43 T^{4} + \cdots - 13473 \)
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| $83$ |
\( T^{5} - 6 T^{4} + \cdots + 60867 \)
|
| $89$ |
\( T^{5} - 8 T^{4} + \cdots - 13203 \)
|
| $97$ |
\( T^{5} - 23 T^{4} + \cdots + 1411 \)
|
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