Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1250,4,Mod(1,1250)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1250, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1250.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1250 = 2 \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1250.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: | \(73.7523875072\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 145x^{4} + 120x^{3} + 5125x^{2} - 2431x - 1069 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 5 \) |
| 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
\( x^{6} - x^{5} - 145x^{4} + 120x^{3} + 5125x^{2} - 2431x - 1069 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{5} + 20\nu^{4} + 495\nu^{3} - 1605\nu^{2} - 19800\nu + 413 ) / 9360 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 17\nu^{5} + 60\nu^{4} - 1245\nu^{3} - 4425\nu^{2} - 6880\nu - 3207 ) / 9360 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -19\nu^{5} + 40\nu^{4} + 2355\nu^{3} - 8085\nu^{2} - 70540\nu + 246409 ) / 9360 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -53\nu^{5} - 80\nu^{4} + 7185\nu^{3} + 5445\nu^{2} - 234620\nu + 16483 ) / 4680 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{4} - \beta_{3} + 7\beta_{2} - \beta _1 + 52 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} + 10\beta_{3} - 14\beta_{2} + 78\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -18\beta_{5} - 156\beta_{4} - 114\beta_{3} + 978\beta_{2} - 93\beta _1 + 4088 \)
|
| \(\nu^{5}\) | \(=\) |
\( 210\beta_{5} + 30\beta_{4} + 1425\beta_{3} - 2655\beta_{2} + 6185\beta _1 - 924 \)
|
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 |
|
−2.00000 | −8.57516 | 4.00000 | 0 | 17.1503 | −1.39161 | −8.00000 | 46.5333 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −5.66784 | 4.00000 | 0 | 11.3357 | −16.6100 | −8.00000 | 5.12438 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0.660070 | 4.00000 | 0 | −1.32014 | −15.8349 | −8.00000 | −26.5643 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 1.86712 | 4.00000 | 0 | −3.73425 | 30.0531 | −8.00000 | −23.5138 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 9.67902 | 4.00000 | 0 | −19.3580 | 20.9347 | −8.00000 | 66.6834 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 10.0368 | 4.00000 | 0 | −20.0736 | −23.1512 | −8.00000 | 73.7370 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
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 | 1250.4.a.c | ✓ | 6 |
| 5.b | even | 2 | 1 | 1250.4.a.f | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1250.4.a.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1250.4.a.f | yes | 6 | 5.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 8T_{3}^{5} - 120T_{3}^{4} + 760T_{3}^{3} + 3480T_{3}^{2} - 11408T_{3} + 5819 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1250))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{6} \)
|
| $3$ |
\( T^{6} - 8 T^{5} + \cdots + 5819 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 6 T^{5} + \cdots + 5331276 \)
|
| $11$ |
\( T^{6} - 107 T^{5} + \cdots + 138384 \)
|
| $13$ |
\( T^{6} + \cdots + 7545700404 \)
|
| $17$ |
\( T^{6} + \cdots + 142258923171 \)
|
| $19$ |
\( T^{6} + \cdots - 92597273100 \)
|
| $23$ |
\( T^{6} + \cdots - 252583777296 \)
|
| $29$ |
\( T^{6} + \cdots - 365288720400 \)
|
| $31$ |
\( T^{6} + \cdots - 6475098886576 \)
|
| $37$ |
\( T^{6} + \cdots - 18880670882564 \)
|
| $41$ |
\( T^{6} + \cdots + 5327780215104 \)
|
| $43$ |
\( T^{6} + \cdots + 11\!\cdots\!24 \)
|
| $47$ |
\( T^{6} + \cdots - 34910336855664 \)
|
| $53$ |
\( T^{6} + \cdots - 55\!\cdots\!36 \)
|
| $59$ |
\( T^{6} + \cdots - 64\!\cdots\!25 \)
|
| $61$ |
\( T^{6} + \cdots - 205699969593756 \)
|
| $67$ |
\( T^{6} + \cdots - 23165951433309 \)
|
| $71$ |
\( T^{6} + \cdots + 11933041882644 \)
|
| $73$ |
\( T^{6} + \cdots - 39\!\cdots\!16 \)
|
| $79$ |
\( T^{6} + \cdots + 38\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 33\!\cdots\!44 \)
|
| $89$ |
\( T^{6} + \cdots - 21\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 85\!\cdots\!39 \)
|
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