Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2312,2,Mod(1,2312)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2312, 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("2312.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2312 = 2^{3} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2312.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: | \(18.4614129473\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.3418281.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 9x^{4} - 4x^{3} + 18x^{2} + 12x - 1 \)
|
| 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
\( x^{6} - 9x^{4} - 4x^{3} + 18x^{2} + 12x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 4\nu + 1 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 4 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 8\nu^{3} + 3\nu^{2} + 14\nu + 2 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 7\beta_{3} + 2\beta_{2} + 8\beta _1 + 16 \)
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| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} + 2\beta_{4} + 12\beta_{3} + 18\beta_{2} + 31\beta _1 + 21 \)
|
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 | −2.06104 | 0 | 4.30892 | 0 | −1.72816 | 0 | 1.24788 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.60714 | 0 | 2.19003 | 0 | 4.55776 | 0 | −0.417107 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.857616 | 0 | −0.406879 | 0 | −0.172637 | 0 | −2.26449 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.0750494 | 0 | −2.06942 | 0 | −2.86317 | 0 | −2.99437 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.71374 | 0 | −0.776830 | 0 | −1.03061 | 0 | −0.0630875 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.73700 | 0 | 2.75417 | 0 | 4.23681 | 0 | 4.49118 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(17\) | \(-1\) |
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 | 2312.2.a.v | yes | 6 |
| 4.b | odd | 2 | 1 | 4624.2.a.bs | 6 | ||
| 17.b | even | 2 | 1 | 2312.2.a.u | ✓ | 6 | |
| 17.c | even | 4 | 2 | 2312.2.b.o | 12 | ||
| 68.d | odd | 2 | 1 | 4624.2.a.br | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2312.2.a.u | ✓ | 6 | 17.b | even | 2 | 1 | |
| 2312.2.a.v | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 2312.2.b.o | 12 | 17.c | even | 4 | 2 | ||
| 4624.2.a.br | 6 | 68.d | odd | 2 | 1 | ||
| 4624.2.a.bs | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2312))\):
|
\( T_{3}^{6} - 9T_{3}^{4} - 4T_{3}^{3} + 18T_{3}^{2} + 12T_{3} - 1 \)
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\( T_{5}^{6} - 6T_{5}^{5} + 38T_{5}^{3} - 15T_{5}^{2} - 54T_{5} - 17 \)
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\( T_{7}^{6} - 3T_{7}^{5} - 21T_{7}^{4} + 25T_{7}^{3} + 147T_{7}^{2} + 123T_{7} + 17 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} - 9 T^{4} + \cdots - 1 \)
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| $5$ |
\( T^{6} - 6 T^{5} + \cdots - 17 \)
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| $7$ |
\( T^{6} - 3 T^{5} + \cdots + 17 \)
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| $11$ |
\( T^{6} + 6 T^{5} + \cdots - 1864 \)
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| $13$ |
\( T^{6} + 6 T^{5} + \cdots + 199 \)
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| $17$ |
\( T^{6} \)
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| $19$ |
\( T^{6} - 6 T^{5} + \cdots + 1061 \)
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| $23$ |
\( T^{6} - 9 T^{5} + \cdots - 1 \)
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| $29$ |
\( T^{6} - 27 T^{5} + \cdots + 1009 \)
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| $31$ |
\( T^{6} - 9 T^{5} + \cdots + 11411 \)
|
| $37$ |
\( T^{6} - 15 T^{5} + \cdots + 35117 \)
|
| $41$ |
\( T^{6} - 21 T^{5} + \cdots - 10727 \)
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| $43$ |
\( T^{6} - 87 T^{4} + \cdots - 7883 \)
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| $47$ |
\( T^{6} - 3 T^{5} + \cdots - 2125 \)
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| $53$ |
\( T^{6} - 33 T^{5} + \cdots - 233000 \)
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| $59$ |
\( T^{6} - 93 T^{4} + \cdots - 1063 \)
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| $61$ |
\( T^{6} - 18 T^{5} + \cdots + 6803 \)
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| $67$ |
\( T^{6} + 9 T^{5} + \cdots - 36117 \)
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| $71$ |
\( T^{6} - 6 T^{5} + \cdots - 36109 \)
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| $73$ |
\( T^{6} - 27 T^{5} + \cdots - 12329 \)
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| $79$ |
\( T^{6} - 33 T^{5} + \cdots + 63208 \)
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| $83$ |
\( T^{6} + 12 T^{5} + \cdots + 73387 \)
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| $89$ |
\( T^{6} - 3 T^{5} + \cdots + 11917 \)
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| $97$ |
\( T^{6} - 51 T^{5} + \cdots - 197000 \)
|
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