Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1035,4,Mod(1,1035)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1035, 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("1035.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1035 = 3^{2} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1035.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: | \(61.0669768559\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 115) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 25\nu^{2} - 11\nu + 98 ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 3\nu^{3} + 17\nu^{2} - 41\nu - 42 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 7\nu^{3} + 25\nu^{2} - 109\nu - 162 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + 2\beta_{2} + 15\beta _1 + 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( 23\beta_{4} - 25\beta_{3} - 11\beta_{2} + 21\beta _1 + 169 \)
|
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 |
|
−5.60878 | 0 | 23.4584 | 5.00000 | 0 | 11.4426 | −86.7031 | 0 | −28.0439 | |||||||||||||||||||||||||||||||||
| 1.2 | −4.41740 | 0 | 11.5134 | 5.00000 | 0 | −8.97260 | −15.5200 | 0 | −22.0870 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.404957 | 0 | −7.83601 | 5.00000 | 0 | 13.7888 | 6.41290 | 0 | −2.02479 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.49214 | 0 | −5.77352 | 5.00000 | 0 | 4.33445 | −20.5520 | 0 | 7.46070 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.93900 | 0 | 0.637693 | 5.00000 | 0 | −23.5932 | −21.6378 | 0 | 14.6950 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 1035.4.a.k | 5 | |
| 3.b | odd | 2 | 1 | 115.4.a.e | ✓ | 5 | |
| 12.b | even | 2 | 1 | 1840.4.a.n | 5 | ||
| 15.d | odd | 2 | 1 | 575.4.a.j | 5 | ||
| 15.e | even | 4 | 2 | 575.4.b.i | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 115.4.a.e | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 575.4.a.j | 5 | 15.d | odd | 2 | 1 | ||
| 575.4.b.i | 10 | 15.e | even | 4 | 2 | ||
| 1035.4.a.k | 5 | 1.a | even | 1 | 1 | trivial | |
| 1840.4.a.n | 5 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1035))\):
|
\( T_{2}^{5} + 6T_{2}^{4} - 13T_{2}^{3} - 72T_{2}^{2} + 82T_{2} + 44 \)
|
|
\( T_{7}^{5} + 3T_{7}^{4} - 484T_{7}^{3} + 1757T_{7}^{2} + 34281T_{7} - 144774 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 6 T^{4} + \cdots + 44 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( (T - 5)^{5} \)
|
| $7$ |
\( T^{5} + 3 T^{4} + \cdots - 144774 \)
|
| $11$ |
\( T^{5} + 23 T^{4} + \cdots + 74136848 \)
|
| $13$ |
\( T^{5} - 132 T^{4} + \cdots + 1550116 \)
|
| $17$ |
\( T^{5} + \cdots + 1039045340 \)
|
| $19$ |
\( T^{5} + 161 T^{4} + \cdots - 801280 \)
|
| $23$ |
\( (T - 23)^{5} \)
|
| $29$ |
\( T^{5} + \cdots + 6149898500 \)
|
| $31$ |
\( T^{5} - 32 T^{4} + \cdots - 438072447 \)
|
| $37$ |
\( T^{5} + \cdots + 1590700778176 \)
|
| $41$ |
\( T^{5} + \cdots - 114116030755 \)
|
| $43$ |
\( T^{5} + \cdots + 504784881664 \)
|
| $47$ |
\( T^{5} + \cdots - 117787714816 \)
|
| $53$ |
\( T^{5} + \cdots - 5720332226904 \)
|
| $59$ |
\( T^{5} + \cdots - 24279649927232 \)
|
| $61$ |
\( T^{5} + \cdots - 34095834816896 \)
|
| $67$ |
\( T^{5} + \cdots + 5644442112 \)
|
| $71$ |
\( T^{5} + \cdots - 15638892903635 \)
|
| $73$ |
\( T^{5} + \cdots - 100895881632176 \)
|
| $79$ |
\( T^{5} + \cdots - 90481602379776 \)
|
| $83$ |
\( T^{5} + \cdots - 18307318870176 \)
|
| $89$ |
\( T^{5} + \cdots + 115104799418880 \)
|
| $97$ |
\( T^{5} + \cdots - 480989167569272 \)
|
| show more | |
| show less |