Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7254,2,Mod(1,7254)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7254, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("7254.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7254 = 2 \cdot 3^{2} \cdot 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7254.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: | \(57.9234816262\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.476177744.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 16x^{4} + 7x^{3} + 60x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 16x^{4} + 7x^{3} + 60x^{2} - x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 23\nu^{3} + 12\nu^{2} + 107\nu - 48 ) / 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 16\nu^{3} - 2\nu^{2} + 51\nu + 1 ) / 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{5} + 7\nu^{4} + 73\nu^{3} - 53\nu^{2} - 241\nu + 23 ) / 14 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -6\nu^{5} + 7\nu^{4} + 96\nu^{3} - 51\nu^{2} - 362\nu + 1 ) / 14 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 5 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} - 2\beta_{4} + \beta_{3} + 10\beta _1 + 3 \)
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| \(\nu^{4}\) | \(=\) |
\( 11\beta_{5} - 9\beta_{4} + 6\beta_{3} + 9\beta_{2} + 17\beta _1 + 44 \)
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| \(\nu^{5}\) | \(=\) |
\( 34\beta_{5} - 34\beta_{4} + 23\beta_{3} + 2\beta_{2} + 111\beta _1 + 57 \)
|
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 |
|
1.00000 | 0 | 1.00000 | −4.16137 | 0 | 2.80063 | 1.00000 | 0 | −4.16137 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | −3.93198 | 0 | −4.19915 | 1.00000 | 0 | −3.93198 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | −0.778810 | 0 | 2.23043 | 1.00000 | 0 | −0.778810 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0 | 1.00000 | 0.255424 | 0 | −3.78719 | 1.00000 | 0 | 0.255424 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0 | 1.00000 | 1.09762 | 0 | −2.13650 | 1.00000 | 0 | 1.09762 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 0 | 1.00000 | 2.51912 | 0 | −1.90822 | 1.00000 | 0 | 2.51912 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(13\) | \( -1 \) |
| \(31\) | \( +1 \) |
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 | 7254.2.a.bm | yes | 6 |
| 3.b | odd | 2 | 1 | 7254.2.a.bl | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7254.2.a.bl | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 7254.2.a.bm | yes | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7254))\):
|
\( T_{5}^{6} + 5T_{5}^{5} - 8T_{5}^{4} - 43T_{5}^{3} + 28T_{5}^{2} + 31T_{5} - 9 \)
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\( T_{7}^{6} + 7T_{7}^{5} - 2T_{7}^{4} - 91T_{7}^{3} - 96T_{7}^{2} + 279T_{7} + 405 \)
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\( T_{11}^{6} + T_{11}^{5} - 18T_{11}^{4} - 19T_{11}^{3} + 74T_{11}^{2} + 103T_{11} + 19 \)
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\( T_{17}^{6} + 13T_{17}^{5} + 6T_{17}^{4} - 547T_{17}^{3} - 2526T_{17}^{2} - 3213T_{17} + 243 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} + 5 T^{5} + \cdots - 9 \)
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| $7$ |
\( T^{6} + 7 T^{5} + \cdots + 405 \)
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| $11$ |
\( T^{6} + T^{5} + \cdots + 19 \)
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| $13$ |
\( (T - 1)^{6} \)
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| $17$ |
\( T^{6} + 13 T^{5} + \cdots + 243 \)
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| $19$ |
\( T^{6} + 7 T^{5} + \cdots - 113 \)
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| $23$ |
\( T^{6} - 88 T^{4} + \cdots - 1324 \)
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| $29$ |
\( T^{6} + 5 T^{5} + \cdots - 2349 \)
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| $31$ |
\( (T + 1)^{6} \)
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| $37$ |
\( T^{6} - 11 T^{5} + \cdots + 37971 \)
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| $41$ |
\( T^{6} + 15 T^{5} + \cdots + 2095 \)
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| $43$ |
\( T^{6} + 13 T^{5} + \cdots - 983 \)
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| $47$ |
\( T^{6} + 16 T^{5} + \cdots - 70964 \)
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| $53$ |
\( T^{6} + 8 T^{5} + \cdots - 499760 \)
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| $59$ |
\( T^{6} + 8 T^{5} + \cdots - 5204 \)
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| $61$ |
\( T^{6} - 274 T^{4} + \cdots + 5788 \)
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| $67$ |
\( T^{6} + 9 T^{5} + \cdots + 45379 \)
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| $71$ |
\( T^{6} + 18 T^{5} + \cdots + 433012 \)
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| $73$ |
\( T^{6} - T^{5} + \cdots + 8047 \)
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| $79$ |
\( T^{6} + 3 T^{5} + \cdots + 19 \)
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| $83$ |
\( T^{6} + 33 T^{5} + \cdots - 626475 \)
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| $89$ |
\( T^{6} + 8 T^{5} + \cdots - 323372 \)
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| $97$ |
\( T^{6} - 2 T^{5} + \cdots - 38476 \)
|
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