Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2014,2,Mod(1,2014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2014, 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("2014.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2014 = 2 \cdot 19 \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2014.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: | \(16.0818709671\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 13x^{6} - x^{5} + 50x^{4} + 21x^{3} - 61x^{2} - 52x - 10 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 13x^{6} - x^{5} + 50x^{4} + 21x^{3} - 61x^{2} - 52x - 10 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - 10\nu^{3} + \nu^{2} + 20\nu + 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{5} + 10\nu^{3} - 20\nu - 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 10\nu^{4} + \nu^{3} + 21\nu^{2} + 4\nu - 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{7} - \nu^{6} - 11\nu^{5} + 11\nu^{4} + 30\nu^{3} - 16\nu^{2} - 29\nu - 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{7} + \nu^{6} + 12\nu^{5} - 11\nu^{4} - 39\nu^{3} + 18\nu^{2} + 43\nu + 9 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{7} + \nu^{6} + 12\nu^{5} - 10\nu^{4} - 39\nu^{3} + 10\nu^{2} + 46\nu + 18 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{3} - 2\beta_{2} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - \beta_{6} + 8\beta_{3} + 8\beta_{2} - 3\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10\beta_{6} + 10\beta_{5} - 11\beta_{3} - 20\beta_{2} + 40\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10\beta_{7} - 11\beta_{6} - \beta_{5} + \beta_{4} + 60\beta_{3} + 61\beta_{2} - 40\beta _1 + 90 \)
|
| \(\nu^{7}\) | \(=\) |
\( -\beta_{7} + 80\beta_{6} + 80\beta_{5} + \beta_{4} - 103\beta_{3} - 171\beta_{2} + 282\beta _1 - 39 \)
|
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 | −2.40245 | 1.00000 | −2.70776 | 2.40245 | 0.173176 | −1.00000 | 2.77176 | 2.70776 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.14669 | 1.00000 | 1.85730 | 2.14669 | 3.24667 | −1.00000 | 1.60827 | −1.85730 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.89675 | 1.00000 | 0.769767 | 1.89675 | −3.09528 | −1.00000 | 0.597654 | −0.769767 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.303994 | 1.00000 | 0.384085 | −0.303994 | 3.38802 | −1.00000 | −2.90759 | −0.384085 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0.701197 | 1.00000 | 0.553365 | −0.701197 | −0.877068 | −1.00000 | −2.50832 | −0.553365 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 1.21468 | 1.00000 | 3.22718 | −1.21468 | −1.77598 | −1.00000 | −1.52455 | −3.22718 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 1.39426 | 1.00000 | −2.95656 | −1.39426 | 1.90796 | −1.00000 | −1.05603 | 2.95656 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 2.83175 | 1.00000 | −0.127379 | −2.83175 | −2.96750 | −1.00000 | 5.01881 | 0.127379 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(19\) | \( +1 \) |
| \(53\) | \( +1 \) |
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 | 2014.2.a.g | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2014.2.a.g | ✓ | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2014))\):
|
\( T_{3}^{8} - 13T_{3}^{6} + T_{3}^{5} + 50T_{3}^{4} - 21T_{3}^{3} - 61T_{3}^{2} + 52T_{3} - 10 \)
|
|
\( T_{7}^{8} - 24T_{7}^{6} - 6T_{7}^{5} + 179T_{7}^{4} + 84T_{7}^{3} - 391T_{7}^{2} - 236T_{7} + 52 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( T^{8} - 13 T^{6} + \cdots - 10 \)
|
| $5$ |
\( T^{8} - T^{7} - 15 T^{6} + \cdots - 1 \)
|
| $7$ |
\( T^{8} - 24 T^{6} + \cdots + 52 \)
|
| $11$ |
\( T^{8} + 6 T^{7} + \cdots + 1087 \)
|
| $13$ |
\( T^{8} + 5 T^{7} + \cdots - 1250 \)
|
| $17$ |
\( T^{8} + 9 T^{7} + \cdots + 7715 \)
|
| $19$ |
\( (T + 1)^{8} \)
|
| $23$ |
\( T^{8} + T^{7} + \cdots - 1225 \)
|
| $29$ |
\( T^{8} + 23 T^{7} + \cdots + 79750 \)
|
| $31$ |
\( T^{8} + 6 T^{7} + \cdots + 91843 \)
|
| $37$ |
\( T^{8} - T^{7} + \cdots + 3940 \)
|
| $41$ |
\( T^{8} + 3 T^{7} + \cdots - 67790 \)
|
| $43$ |
\( T^{8} - 9 T^{7} + \cdots - 16415 \)
|
| $47$ |
\( T^{8} + 13 T^{7} + \cdots - 290 \)
|
| $53$ |
\( (T + 1)^{8} \)
|
| $59$ |
\( T^{8} + 9 T^{7} + \cdots - 293473 \)
|
| $61$ |
\( T^{8} - 113 T^{6} + \cdots + 1108 \)
|
| $67$ |
\( T^{8} - 9 T^{7} + \cdots - 84152 \)
|
| $71$ |
\( T^{8} + 31 T^{7} + \cdots - 1298680 \)
|
| $73$ |
\( T^{8} - 5 T^{7} + \cdots + 119984 \)
|
| $79$ |
\( T^{8} - 19 T^{7} + \cdots - 3087173 \)
|
| $83$ |
\( T^{8} - 11 T^{7} + \cdots - 2009954 \)
|
| $89$ |
\( T^{8} + 41 T^{7} + \cdots + 14035633 \)
|
| $97$ |
\( T^{8} + 24 T^{7} + \cdots + 1027951 \)
|
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