Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2015,1,Mod(2014,2015)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2015.2014");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2015 = 5 \cdot 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2015.h (of order \(2\), degree \(1\), minimal) |
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: | \(1.00561600046\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{26})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 5x^{4} + 4x^{3} + 6x^{2} - 3x - 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{13}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
$q$-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 \(\nu = \zeta_{26} + \zeta_{26}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + 5\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 4\beta_{2} + 6 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 5\beta_{3} + 10\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2015\mathbb{Z}\right)^\times\).
| \(n\) | \(716\) | \(807\) | \(1861\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2014.1 |
|
−1.94188 | 1.49702 | 2.77091 | 1.00000 | −2.90704 | 1.13613 | −3.43891 | 1.24107 | −1.94188 | ||||||||||||||||||||||||||||||||||||
| 2014.2 | −1.49702 | −1.13613 | 1.24107 | 1.00000 | 1.70081 | −1.94188 | −0.360892 | 0.290790 | −1.49702 | |||||||||||||||||||||||||||||||||||||
| 2014.3 | −0.709210 | −1.77091 | −0.497021 | 1.00000 | 1.25595 | 0.241073 | 1.06170 | 2.13613 | −0.709210 | |||||||||||||||||||||||||||||||||||||
| 2014.4 | 0.241073 | 0.709210 | −0.941884 | 1.00000 | 0.170972 | 1.77091 | −0.468136 | −0.497021 | 0.241073 | |||||||||||||||||||||||||||||||||||||
| 2014.5 | 1.13613 | 1.94188 | 0.290790 | 1.00000 | 2.20623 | −1.49702 | −0.805754 | 2.77091 | 1.13613 | |||||||||||||||||||||||||||||||||||||
| 2014.6 | 1.77091 | −0.241073 | 2.13613 | 1.00000 | −0.426920 | −0.709210 | 2.01199 | −0.941884 | 1.77091 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 2015.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-2015}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2015.1.h.c | yes | 6 |
| 5.b | even | 2 | 1 | 2015.1.h.d | yes | 6 | |
| 13.b | even | 2 | 1 | 2015.1.h.e | yes | 6 | |
| 31.b | odd | 2 | 1 | 2015.1.h.b | ✓ | 6 | |
| 65.d | even | 2 | 1 | 2015.1.h.b | ✓ | 6 | |
| 155.c | odd | 2 | 1 | 2015.1.h.e | yes | 6 | |
| 403.b | odd | 2 | 1 | 2015.1.h.d | yes | 6 | |
| 2015.h | odd | 2 | 1 | CM | 2015.1.h.c | yes | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2015.1.h.b | ✓ | 6 | 31.b | odd | 2 | 1 | |
| 2015.1.h.b | ✓ | 6 | 65.d | even | 2 | 1 | |
| 2015.1.h.c | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 2015.1.h.c | yes | 6 | 2015.h | odd | 2 | 1 | CM |
| 2015.1.h.d | yes | 6 | 5.b | even | 2 | 1 | |
| 2015.1.h.d | yes | 6 | 403.b | odd | 2 | 1 | |
| 2015.1.h.e | yes | 6 | 13.b | even | 2 | 1 | |
| 2015.1.h.e | yes | 6 | 155.c | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2015, [\chi])\):
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\( T_{2}^{6} + T_{2}^{5} - 5T_{2}^{4} - 4T_{2}^{3} + 6T_{2}^{2} + 3T_{2} - 1 \)
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\( T_{3}^{6} - T_{3}^{5} - 5T_{3}^{4} + 4T_{3}^{3} + 6T_{3}^{2} - 3T_{3} - 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 5 T^{4} + \cdots - 1 \)
|
| $3$ |
\( T^{6} - T^{5} - 5 T^{4} + \cdots - 1 \)
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| $5$ |
\( (T - 1)^{6} \)
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| $7$ |
\( T^{6} + T^{5} - 5 T^{4} + \cdots - 1 \)
|
| $11$ |
\( T^{6} - T^{5} - 5 T^{4} + \cdots - 1 \)
|
| $13$ |
\( (T + 1)^{6} \)
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| $17$ |
\( T^{6} - T^{5} - 5 T^{4} + \cdots - 1 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} - T^{5} - 5 T^{4} + \cdots - 1 \)
|
| $29$ |
\( T^{6} \)
|
| $31$ |
\( (T + 1)^{6} \)
|
| $37$ |
\( T^{6} \)
|
| $41$ |
\( T^{6} \)
|
| $43$ |
\( T^{6} - T^{5} - 5 T^{4} + \cdots - 1 \)
|
| $47$ |
\( T^{6} + T^{5} - 5 T^{4} + \cdots - 1 \)
|
| $53$ |
\( T^{6} - T^{5} - 5 T^{4} + \cdots - 1 \)
|
| $59$ |
\( T^{6} \)
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| $61$ |
\( T^{6} \)
|
| $67$ |
\( T^{6} + T^{5} - 5 T^{4} + \cdots - 1 \)
|
| $71$ |
\( T^{6} \)
|
| $73$ |
\( T^{6} \)
|
| $79$ |
\( T^{6} \)
|
| $83$ |
\( T^{6} \)
|
| $89$ |
\( T^{6} - T^{5} - 5 T^{4} + \cdots - 1 \)
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| $97$ |
\( T^{6} + T^{5} - 5 T^{4} + \cdots - 1 \)
|
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