Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5312,2,Mod(1,5312)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5312, 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("5312.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5312 = 2^{6} \cdot 83 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5312.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: | \(42.4165335537\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.905177.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 7x^{4} + 9x^{3} + 7x^{2} - 9x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 664) |
| 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
\( x^{6} - x^{5} - 7x^{4} + 9x^{3} + 7x^{2} - 9x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + \nu^{2} - 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} + \nu^{4} - 6\nu^{3} - 3\nu^{2} + 8\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} + \nu^{4} - 6\nu^{3} - 3\nu^{2} + 6\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} + 7\nu^{3} - 3\nu^{2} - 9\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( -4\nu^{5} - \nu^{4} + 26\nu^{3} - 3\nu^{2} - 27\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 3\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 8 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{5} + 3\beta_{4} - 10\beta_{3} + 9\beta_{2} + 3\beta _1 + 4 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 7\beta_{5} - 17\beta_{4} + 20\beta_{3} - 9\beta_{2} + 3\beta _1 + 26 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -5\beta_{5} + 13\beta_{4} - 29\beta_{3} + 24\beta_{2} + 9\beta _1 - 1 ) / 2 \)
|
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 |
|
0 | −3.08520 | 0 | 0.890084 | 0 | −1.29942 | 0 | 6.51848 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.35090 | 0 | −2.49396 | 0 | 2.71769 | 0 | 2.52674 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.341119 | 0 | −2.49396 | 0 | 2.22131 | 0 | −2.88364 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.242220 | 0 | 3.60388 | 0 | −4.97458 | 0 | −2.94133 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.728308 | 0 | 0.890084 | 0 | 4.21127 | 0 | −2.46957 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.29114 | 0 | 3.60388 | 0 | 2.12373 | 0 | 2.24931 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(83\) | \(1\) |
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 | 5312.2.a.bm | 6 | |
| 4.b | odd | 2 | 1 | 5312.2.a.bp | 6 | ||
| 8.b | even | 2 | 1 | 1328.2.a.m | 6 | ||
| 8.d | odd | 2 | 1 | 664.2.a.e | ✓ | 6 | |
| 24.f | even | 2 | 1 | 5976.2.a.p | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 664.2.a.e | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 1328.2.a.m | 6 | 8.b | even | 2 | 1 | ||
| 5312.2.a.bm | 6 | 1.a | even | 1 | 1 | trivial | |
| 5312.2.a.bp | 6 | 4.b | odd | 2 | 1 | ||
| 5976.2.a.p | 6 | 24.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5312))\):
|
\( T_{3}^{6} + 3T_{3}^{5} - 6T_{3}^{4} - 17T_{3}^{3} + 4T_{3}^{2} + 6T_{3} + 1 \)
|
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\( T_{5}^{3} - 2T_{5}^{2} - 8T_{5} + 8 \)
|
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\( T_{7}^{6} - 5T_{7}^{5} - 18T_{7}^{4} + 135T_{7}^{3} - 164T_{7}^{2} - 194T_{7} + 349 \)
|
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\( T_{11}^{6} + 11T_{11}^{5} + 26T_{11}^{4} - 89T_{11}^{3} - 496T_{11}^{2} - 762T_{11} - 379 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 3 T^{5} + \cdots + 1 \)
|
| $5$ |
\( (T^{3} - 2 T^{2} - 8 T + 8)^{2} \)
|
| $7$ |
\( T^{6} - 5 T^{5} + \cdots + 349 \)
|
| $11$ |
\( T^{6} + 11 T^{5} + \cdots - 379 \)
|
| $13$ |
\( T^{6} - 4 T^{5} + \cdots - 64 \)
|
| $17$ |
\( T^{6} + 5 T^{5} + \cdots + 1637 \)
|
| $19$ |
\( T^{6} + 10 T^{5} + \cdots - 832 \)
|
| $23$ |
\( T^{6} - 11 T^{5} + \cdots + 15296 \)
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| $29$ |
\( T^{6} - 17 T^{5} + \cdots + 16421 \)
|
| $31$ |
\( T^{6} - 13 T^{5} + \cdots - 83 \)
|
| $37$ |
\( T^{6} - 5 T^{5} + \cdots - 12083 \)
|
| $41$ |
\( T^{6} + 9 T^{5} + \cdots + 2344 \)
|
| $43$ |
\( T^{6} + 20 T^{5} + \cdots + 42496 \)
|
| $47$ |
\( T^{6} - 6 T^{5} + \cdots - 4544 \)
|
| $53$ |
\( T^{6} + 2 T^{5} + \cdots + 5312 \)
|
| $59$ |
\( T^{6} + 21 T^{5} + \cdots + 696241 \)
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| $61$ |
\( T^{6} - 5 T^{5} + \cdots + 3569 \)
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| $67$ |
\( T^{6} + 16 T^{5} + \cdots - 16064 \)
|
| $71$ |
\( T^{6} - 10 T^{5} + \cdots - 41152 \)
|
| $73$ |
\( T^{6} - 44 T^{5} + \cdots - 188096 \)
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| $79$ |
\( T^{6} - 2 T^{5} + \cdots - 140608 \)
|
| $83$ |
\( (T + 1)^{6} \)
|
| $89$ |
\( T^{6} + 2 T^{5} + \cdots + 20992 \)
|
| $97$ |
\( T^{6} - 46 T^{5} + \cdots + 14848 \)
|
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