Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1110,4,Mod(1,1110)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1110, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1110.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1110.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: | \(65.4921201064\) |
| 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: |
\( x^{6} - x^{5} - 652x^{4} - 636x^{3} + 100007x^{2} + 107093x - 3177844 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{6} - x^{5} - 652x^{4} - 636x^{3} + 100007x^{2} + 107093x - 3177844 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 33\nu^{5} + 1736\nu^{4} - 118068\nu^{3} + 475888\nu^{2} + 27293815\nu - 182219036 ) / 6257200 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 43\nu^{5} + 1314\nu^{4} - 25858\nu^{3} - 545070\nu^{2} + 1384239\nu + 12126756 ) / 1251440 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -83\nu^{5} + 374\nu^{4} + 48093\nu^{3} - 63998\nu^{2} - 4509410\nu - 1102724 ) / 782150 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 83\nu^{5} - 374\nu^{4} - 48093\nu^{3} + 63998\nu^{2} + 6073710\nu + 320574 ) / 782150 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2127\nu^{5} + 30316\nu^{4} + 926192\nu^{3} - 12328172\nu^{2} - 61091685\nu + 922164484 ) / 12514400 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{5} - \beta_{4} + 3\beta_{3} + 2\beta_{2} + 3\beta _1 + 220 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -48\beta_{5} + 249\beta_{4} + 349\beta_{3} + 80\beta_{2} - 56\beta _1 + 1521 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -782\beta_{5} + 10\beta_{4} + 1844\beta_{3} + 1622\beta_{2} + 1133\beta _1 + 77497 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -31776\beta_{5} + 91581\beta_{4} + 141037\beta_{3} + 57888\beta_{2} - 26864\beta _1 + 1159557 ) / 2 \)
|
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 |
|
−2.00000 | 3.00000 | 4.00000 | 5.00000 | −6.00000 | −36.4453 | −8.00000 | 9.00000 | −10.0000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 3.00000 | 4.00000 | 5.00000 | −6.00000 | −9.65508 | −8.00000 | 9.00000 | −10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 3.00000 | 4.00000 | 5.00000 | −6.00000 | −7.78261 | −8.00000 | 9.00000 | −10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 3.00000 | 4.00000 | 5.00000 | −6.00000 | 1.57196 | −8.00000 | 9.00000 | −10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 3.00000 | 4.00000 | 5.00000 | −6.00000 | 22.8758 | −8.00000 | 9.00000 | −10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 3.00000 | 4.00000 | 5.00000 | −6.00000 | 31.4353 | −8.00000 | 9.00000 | −10.0000 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(37\) | \(1\) |
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 | 1110.4.a.r | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1110.4.a.r | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} - 2T_{7}^{5} - 1496T_{7}^{4} + 5242T_{7}^{3} + 357767T_{7}^{2} + 1399784T_{7} - 3095696 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1110))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{6} \)
|
| $3$ |
\( (T - 3)^{6} \)
|
| $5$ |
\( (T - 5)^{6} \)
|
| $7$ |
\( T^{6} - 2 T^{5} + \cdots - 3095696 \)
|
| $11$ |
\( T^{6} - 53 T^{5} + \cdots + 964051800 \)
|
| $13$ |
\( T^{6} + \cdots + 25854488760 \)
|
| $17$ |
\( T^{6} + \cdots + 222651767100 \)
|
| $19$ |
\( T^{6} + \cdots - 1046737200 \)
|
| $23$ |
\( T^{6} + \cdots - 1649046784000 \)
|
| $29$ |
\( T^{6} + \cdots - 7023930761304 \)
|
| $31$ |
\( T^{6} + \cdots + 223677560160 \)
|
| $37$ |
\( (T + 37)^{6} \)
|
| $41$ |
\( T^{6} + \cdots - 22149606105840 \)
|
| $43$ |
\( T^{6} + \cdots + 91784225637376 \)
|
| $47$ |
\( T^{6} + \cdots + 36\!\cdots\!20 \)
|
| $53$ |
\( T^{6} + \cdots + 314127716294464 \)
|
| $59$ |
\( T^{6} + \cdots + 5222628026880 \)
|
| $61$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots + 169181357552256 \)
|
| $71$ |
\( T^{6} + \cdots + 99\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots - 15\!\cdots\!64 \)
|
| $79$ |
\( T^{6} + \cdots + 52\!\cdots\!80 \)
|
| $83$ |
\( T^{6} + \cdots + 11\!\cdots\!52 \)
|
| $89$ |
\( T^{6} + \cdots - 28\!\cdots\!52 \)
|
| $97$ |
\( T^{6} + \cdots - 71\!\cdots\!60 \)
|
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