Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1911,4,Mod(1,1911)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1911, 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("1911.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1911.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: | \(112.752650021\) |
| 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: |
\( x^{6} - x^{5} - 31x^{4} + 33x^{3} + 220x^{2} - 154x - 160 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 273) |
| 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
\( x^{6} - x^{5} - 31x^{4} + 33x^{3} + 220x^{2} - 154x - 160 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + \nu^{2} - 16\nu - 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 2\nu^{3} - 19\nu^{2} - 24\nu + 32 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 2\nu^{4} - 23\nu^{3} - 28\nu^{2} + 96\nu + 22 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 16\beta _1 - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} - 2\beta_{3} + 21\beta_{2} - 8\beta _1 + 185 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} - 4\beta_{4} + 27\beta_{3} - 37\beta_{2} + 288\beta _1 - 176 \)
|
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 |
|
−3.71660 | −3.00000 | 5.81310 | 9.49795 | 11.1498 | 0 | 8.12784 | 9.00000 | −35.3000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.82577 | −3.00000 | −4.66658 | −12.8060 | 5.47730 | 0 | 23.1262 | 9.00000 | 23.3807 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.401623 | −3.00000 | −7.83870 | 21.2272 | −1.20487 | 0 | −6.36119 | 9.00000 | 8.52535 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.31034 | −3.00000 | −2.66232 | −12.6220 | −6.93103 | 0 | −24.6336 | 9.00000 | −29.1612 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 4.78080 | −3.00000 | 14.8561 | −11.7374 | −14.3424 | 0 | 32.7775 | 9.00000 | −56.1143 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 5.04960 | −3.00000 | 17.4984 | 9.44025 | −15.1488 | 0 | 47.9632 | 9.00000 | 47.6694 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 1911.4.a.q | 6 | |
| 7.b | odd | 2 | 1 | 273.4.a.j | ✓ | 6 | |
| 21.c | even | 2 | 1 | 819.4.a.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.4.a.j | ✓ | 6 | 7.b | odd | 2 | 1 | |
| 819.4.a.k | 6 | 21.c | even | 2 | 1 | ||
| 1911.4.a.q | 6 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1911))\):
|
\( T_{2}^{6} - 7T_{2}^{5} - 11T_{2}^{4} + 127T_{2}^{3} - 40T_{2}^{2} - 382T_{2} + 152 \)
|
|
\( T_{5}^{6} - 3T_{5}^{5} - 541T_{5}^{4} - 213T_{5}^{3} + 79276T_{5}^{2} + 57096T_{5} - 3610944 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 7 T^{5} + \cdots + 152 \)
|
| $3$ |
\( (T + 3)^{6} \)
|
| $5$ |
\( T^{6} - 3 T^{5} + \cdots - 3610944 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 1729985760 \)
|
| $13$ |
\( (T - 13)^{6} \)
|
| $17$ |
\( T^{6} + \cdots - 22635225088 \)
|
| $19$ |
\( T^{6} + \cdots + 64439481600 \)
|
| $23$ |
\( T^{6} + \cdots - 229413349744 \)
|
| $29$ |
\( T^{6} + \cdots - 5254293222040 \)
|
| $31$ |
\( T^{6} + \cdots + 33684663296 \)
|
| $37$ |
\( T^{6} + \cdots - 9792596611840 \)
|
| $41$ |
\( T^{6} + \cdots - 1489927781888 \)
|
| $43$ |
\( T^{6} + \cdots + 54297743174336 \)
|
| $47$ |
\( T^{6} + \cdots - 19\!\cdots\!08 \)
|
| $53$ |
\( T^{6} + \cdots + 13\!\cdots\!68 \)
|
| $59$ |
\( T^{6} + \cdots + 40\!\cdots\!12 \)
|
| $61$ |
\( T^{6} + \cdots + 758169639717984 \)
|
| $67$ |
\( T^{6} + \cdots + 801332561341440 \)
|
| $71$ |
\( T^{6} + \cdots - 152699829808640 \)
|
| $73$ |
\( T^{6} + \cdots + 21\!\cdots\!88 \)
|
| $79$ |
\( T^{6} + \cdots - 10\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 33\!\cdots\!28 \)
|
| $89$ |
\( T^{6} + \cdots + 21\!\cdots\!24 \)
|
| $97$ |
\( T^{6} + \cdots + 26\!\cdots\!28 \)
|
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