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: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 2x^{8} - 19x^{7} + 32x^{6} + 118x^{5} - 131x^{4} - 312x^{3} + 96x^{2} + 336x + 128 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 1508) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 2x^{8} - 19x^{7} + 32x^{6} + 118x^{5} - 131x^{4} - 312x^{3} + 96x^{2} + 336x + 128 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{8} - 2\nu^{7} - 19\nu^{6} + 32\nu^{5} + 114\nu^{4} - 135\nu^{3} - 260\nu^{2} + 148\nu + 196 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{8} + 3\nu^{7} + 16\nu^{6} - 48\nu^{5} - 70\nu^{4} + 201\nu^{3} + 110\nu^{2} - 207\nu - 122 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{8} + 8\nu^{7} + 51\nu^{6} - 130\nu^{5} - 256\nu^{4} + 565\nu^{3} + 504\nu^{2} - 644\nu - 500 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 7\nu^{8} - 20\nu^{7} - 115\nu^{6} + 322\nu^{5} + 534\nu^{4} - 1369\nu^{3} - 924\nu^{2} + 1470\nu + 964 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -23\nu^{8} + 66\nu^{7} + 381\nu^{6} - 1068\nu^{5} - 1810\nu^{4} + 4593\nu^{3} + 3308\nu^{2} - 5068\nu - 3500 ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -27\nu^{8} + 78\nu^{7} + 445\nu^{6} - 1260\nu^{5} - 2090\nu^{4} + 5401\nu^{3} + 3752\nu^{2} - 5928\nu - 4012 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - \beta_{7} - \beta_{4} - \beta_{2} + 8\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} - \beta_{5} - 2\beta_{4} + 10\beta_{2} - \beta _1 + 38 \)
|
| \(\nu^{5}\) | \(=\) |
\( 14\beta_{8} - 14\beta_{7} + \beta_{6} - 11\beta_{4} - 2\beta_{3} - 12\beta_{2} + 72\beta _1 + 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} - 14\beta_{6} - 17\beta_{5} - 30\beta_{4} - 3\beta_{3} + 94\beta_{2} - 16\beta _1 + 330 \)
|
| \(\nu^{7}\) | \(=\) |
\( 158\beta_{8} - 155\beta_{7} + 18\beta_{6} - 7\beta_{5} - 111\beta_{4} - 37\beta_{3} - 134\beta_{2} + 679\beta _1 + 24 \)
|
| \(\nu^{8}\) | \(=\) |
\( 3 \beta_{8} + 22 \beta_{7} - 148 \beta_{6} - 223 \beta_{5} - 347 \beta_{4} - 63 \beta_{3} + 887 \beta_{2} + \cdots + 3033 \)
|
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 | −3.11842 | 0 | 2.65076 | 0 | 2.96737 | 0 | 6.72456 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.89990 | 0 | −4.20458 | 0 | 1.66583 | 0 | 5.40941 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.10347 | 0 | 2.88841 | 0 | −2.25318 | 0 | 1.42460 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.68531 | 0 | −0.142366 | 0 | −3.78735 | 0 | −0.159743 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.776734 | 0 | 3.73497 | 0 | −1.91773 | 0 | −2.39668 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.843132 | 0 | −1.88618 | 0 | 0.0856454 | 0 | −2.28913 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.911584 | 0 | −3.29353 | 0 | 5.14114 | 0 | −2.16901 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.11833 | 0 | −3.00093 | 0 | −3.40688 | 0 | 1.48731 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 3.15732 | 0 | 1.25346 | 0 | 2.50515 | 0 | 6.96869 | 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.w | 9 | |
| 4.b | odd | 2 | 1 | 1508.2.a.c | ✓ | 9 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1508.2.a.c | ✓ | 9 | 4.b | odd | 2 | 1 | |
| 6032.2.a.w | 9 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6032))\):
|
\( T_{3}^{9} + 2T_{3}^{8} - 19T_{3}^{7} - 32T_{3}^{6} + 118T_{3}^{5} + 131T_{3}^{4} - 312T_{3}^{3} - 96T_{3}^{2} + 336T_{3} - 128 \)
|
|
\( T_{5}^{9} + 2T_{5}^{8} - 34T_{5}^{7} - 55T_{5}^{6} + 398T_{5}^{5} + 477T_{5}^{4} - 1856T_{5}^{3} - 1352T_{5}^{2} + 2656T_{5} + 400 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} \)
|
| $3$ |
\( T^{9} + 2 T^{8} + \cdots - 128 \)
|
| $5$ |
\( T^{9} + 2 T^{8} + \cdots + 400 \)
|
| $7$ |
\( T^{9} - T^{8} + \cdots - 304 \)
|
| $11$ |
\( T^{9} + 7 T^{8} + \cdots + 1552 \)
|
| $13$ |
\( (T + 1)^{9} \)
|
| $17$ |
\( T^{9} - 6 T^{8} + \cdots - 520112 \)
|
| $19$ |
\( T^{9} + 12 T^{8} + \cdots - 16 \)
|
| $23$ |
\( T^{9} + 7 T^{8} + \cdots + 959488 \)
|
| $29$ |
\( (T + 1)^{9} \)
|
| $31$ |
\( T^{9} - 21 T^{8} + \cdots - 764816 \)
|
| $37$ |
\( T^{9} - 29 T^{8} + \cdots + 7140400 \)
|
| $41$ |
\( T^{9} - 17 T^{8} + \cdots - 20078288 \)
|
| $43$ |
\( T^{9} + 22 T^{8} + \cdots + 14912 \)
|
| $47$ |
\( T^{9} - 9 T^{8} + \cdots - 23888 \)
|
| $53$ |
\( T^{9} - 11 T^{8} + \cdots - 2642704 \)
|
| $59$ |
\( T^{9} + 13 T^{8} + \cdots + 210863536 \)
|
| $61$ |
\( T^{9} - 17 T^{8} + \cdots + 131291312 \)
|
| $67$ |
\( T^{9} + 23 T^{8} + \cdots + 72387856 \)
|
| $71$ |
\( T^{9} + 2 T^{8} + \cdots + 16475344 \)
|
| $73$ |
\( T^{9} - 20 T^{8} + \cdots + 178288 \)
|
| $79$ |
\( T^{9} - 18 T^{8} + \cdots - 41147456 \)
|
| $83$ |
\( T^{9} - 16 T^{8} + \cdots - 42985744 \)
|
| $89$ |
\( T^{9} + 7 T^{8} + \cdots - 83748272 \)
|
| $97$ |
\( T^{9} - 14 T^{8} + \cdots + 303289360 \)
|
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