Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6032,2,Mod(1,6032)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6032, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("6032.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6032 = 2^{4} \cdot 13 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6032.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: | \(48.1657624992\) |
| Analytic rank: | \(0\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 5x^{10} - 9x^{9} + 65x^{8} + 19x^{7} - 298x^{6} + 17x^{5} + 541x^{4} - 60x^{3} - 287x^{2} + 16x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 3016) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 5x^{10} - 9x^{9} + 65x^{8} + 19x^{7} - 298x^{6} + 17x^{5} + 541x^{4} - 60x^{3} - 287x^{2} + 16x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 27 \nu^{10} - 186 \nu^{9} + 493 \nu^{8} - 202 \nu^{7} - 4619 \nu^{6} + 11193 \nu^{5} + 8936 \nu^{4} + \cdots + 636 ) / 2308 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 44 \nu^{10} + 239 \nu^{9} - 376 \nu^{8} - 547 \nu^{7} + 6416 \nu^{6} - 8175 \nu^{5} - 23773 \nu^{4} + \cdots - 780 ) / 2308 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 53 \nu^{10} + 301 \nu^{9} + 229 \nu^{8} - 3557 \nu^{7} + 1801 \nu^{6} + 15790 \nu^{5} + \cdots - 5608 ) / 2308 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 92 \nu^{10} - 185 \nu^{9} - 1312 \nu^{8} + 829 \nu^{7} + 8196 \nu^{6} + 4609 \nu^{5} - 22785 \nu^{4} + \cdots + 2680 ) / 2308 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 59 \nu^{10} + 150 \nu^{9} + 1017 \nu^{8} - 2294 \nu^{7} - 5507 \nu^{6} + 11187 \nu^{5} + \cdots + 790 ) / 1154 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 125 \nu^{10} - 797 \nu^{9} - 453 \nu^{8} + 9173 \nu^{7} - 1809 \nu^{6} - 39026 \nu^{5} + 2113 \nu^{4} + \cdots - 11160 ) / 2308 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 154 \nu^{10} - 1125 \nu^{9} + 162 \nu^{8} + 12589 \nu^{7} - 12070 \nu^{6} - 48417 \nu^{5} + \cdots + 3884 ) / 2308 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 373 \nu^{10} - 1672 \nu^{9} - 3789 \nu^{8} + 21016 \nu^{7} + 11755 \nu^{6} - 90147 \nu^{5} + \cdots + 7504 ) / 2308 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{10} + \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + 7\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{10} + 3\beta_{9} - 2\beta_{7} + 4\beta_{6} - 4\beta_{5} + 3\beta_{4} - \beta_{3} + 8\beta_{2} + 14\beta _1 + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 17 \beta_{10} + 17 \beta_{9} + \beta_{8} - 13 \beta_{7} + 21 \beta_{6} - 18 \beta_{5} + 15 \beta_{4} + \cdots + 27 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 61 \beta_{10} + 59 \beta_{9} + 6 \beta_{8} - 40 \beta_{7} + 81 \beta_{6} - 72 \beta_{5} + 52 \beta_{4} + \cdots + 128 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 253 \beta_{10} + 246 \beta_{9} + 29 \beta_{8} - 173 \beta_{7} + 329 \beta_{6} - 274 \beta_{5} + \cdots + 311 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 924 \beta_{10} + 880 \beta_{9} + 128 \beta_{8} - 597 \beta_{7} + 1227 \beta_{6} - 1040 \beta_{5} + \cdots + 1190 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 3493 \beta_{10} + 3330 \beta_{9} + 510 \beta_{8} - 2286 \beta_{7} + 4612 \beta_{6} - 3802 \beta_{5} + \cdots + 3548 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 12651 \beta_{10} + 11937 \beta_{9} + 2005 \beta_{8} - 8105 \beta_{7} + 16828 \beta_{6} - 13955 \beta_{5} + \cdots + 12646 \)
|
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 |
|
0 | −2.53326 | 0 | −3.09469 | 0 | −0.871610 | 0 | 3.41739 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.67951 | 0 | 1.05669 | 0 | 1.62834 | 0 | −0.179232 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.59661 | 0 | −0.374330 | 0 | 3.78268 | 0 | −0.450834 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.863310 | 0 | −1.89826 | 0 | −2.78049 | 0 | −2.25470 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.0666645 | 0 | 3.14719 | 0 | −2.33769 | 0 | −2.99556 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.721074 | 0 | 0.0376400 | 0 | 1.18778 | 0 | −2.48005 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.22404 | 0 | −3.27866 | 0 | −0.601962 | 0 | −1.50173 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.82025 | 0 | 1.95827 | 0 | 3.06441 | 0 | 0.313322 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.73315 | 0 | 3.52761 | 0 | 2.06502 | 0 | 4.47010 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.98153 | 0 | 0.681937 | 0 | −4.55922 | 0 | 5.88949 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 3.12599 | 0 | −3.76339 | 0 | 2.42274 | 0 | 6.77179 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(13\) | \(-1\) |
| \(29\) | \(1\) |
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 | 6032.2.a.bc | 11 | |
| 4.b | odd | 2 | 1 | 3016.2.a.i | ✓ | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3016.2.a.i | ✓ | 11 | 4.b | odd | 2 | 1 | |
| 6032.2.a.bc | 11 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6032))\):
|
\( T_{3}^{11} - 6 T_{3}^{10} - 4 T_{3}^{9} + 76 T_{3}^{8} - 55 T_{3}^{7} - 311 T_{3}^{6} + 336 T_{3}^{5} + \cdots - 16 \)
|
|
\( T_{5}^{11} + 2 T_{5}^{10} - 31 T_{5}^{9} - 51 T_{5}^{8} + 334 T_{5}^{7} + 388 T_{5}^{6} - 1470 T_{5}^{5} + \cdots + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} \)
|
| $3$ |
\( T^{11} - 6 T^{10} + \cdots - 16 \)
|
| $5$ |
\( T^{11} + 2 T^{10} + \cdots + 16 \)
|
| $7$ |
\( T^{11} - 3 T^{10} + \cdots + 1744 \)
|
| $11$ |
\( T^{11} - 13 T^{10} + \cdots + 7584 \)
|
| $13$ |
\( (T - 1)^{11} \)
|
| $17$ |
\( T^{11} + 6 T^{10} + \cdots + 2640 \)
|
| $19$ |
\( T^{11} - 12 T^{10} + \cdots - 3678240 \)
|
| $23$ |
\( T^{11} - 13 T^{10} + \cdots - 1406464 \)
|
| $29$ |
\( (T + 1)^{11} \)
|
| $31$ |
\( T^{11} - 11 T^{10} + \cdots - 209856 \)
|
| $37$ |
\( T^{11} - 11 T^{10} + \cdots - 171776 \)
|
| $41$ |
\( T^{11} + 9 T^{10} + \cdots + 677376 \)
|
| $43$ |
\( T^{11} - 30 T^{10} + \cdots - 557360 \)
|
| $47$ |
\( T^{11} - T^{10} + \cdots + 35247504 \)
|
| $53$ |
\( T^{11} + 9 T^{10} + \cdots + 39040 \)
|
| $59$ |
\( T^{11} + \cdots - 1474957024 \)
|
| $61$ |
\( T^{11} + \cdots + 250784160 \)
|
| $67$ |
\( T^{11} - 25 T^{10} + \cdots + 19008 \)
|
| $71$ |
\( T^{11} - 26 T^{10} + \cdots - 52036272 \)
|
| $73$ |
\( T^{11} - 10 T^{10} + \cdots - 81483776 \)
|
| $79$ |
\( T^{11} - 14 T^{10} + \cdots + 49535776 \)
|
| $83$ |
\( T^{11} + \cdots - 54112692576 \)
|
| $89$ |
\( T^{11} + 11 T^{10} + \cdots + 18115584 \)
|
| $97$ |
\( T^{11} + \cdots - 108307456 \)
|
| show more | |
| show less |