Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6288,2,Mod(1,6288)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6288, 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("6288.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6288 = 2^{4} \cdot 3 \cdot 131 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6288.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: | \(50.2099327910\) |
| 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: |
\( x^{11} - 4 x^{10} - 29 x^{9} + 91 x^{8} + 332 x^{7} - 641 x^{6} - 1746 x^{5} + 1461 x^{4} + 3531 x^{3} + \cdots + 32 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 3144) |
| 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
\( x^{11} - 4 x^{10} - 29 x^{9} + 91 x^{8} + 332 x^{7} - 641 x^{6} - 1746 x^{5} + 1461 x^{4} + 3531 x^{3} + \cdots + 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 9 \nu^{10} + 47 \nu^{9} + 168 \nu^{8} - 811 \nu^{7} - 1499 \nu^{6} + 3754 \nu^{5} + 9732 \nu^{4} + \cdots + 5792 ) / 768 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 271 \nu^{10} + 1649 \nu^{9} + 4456 \nu^{8} - 34125 \nu^{7} - 19573 \nu^{6} + 218054 \nu^{5} + \cdots - 7072 ) / 768 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 503 \nu^{10} + 3129 \nu^{9} + 7688 \nu^{8} - 63141 \nu^{7} - 27517 \nu^{6} + 389078 \nu^{5} + \cdots - 9632 ) / 768 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 827 \nu^{10} + 5005 \nu^{9} + 13752 \nu^{8} - 103713 \nu^{7} - 62417 \nu^{6} + 663102 \nu^{5} + \cdots - 16928 ) / 768 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1151 \nu^{10} - 6977 \nu^{9} - 18984 \nu^{8} + 143837 \nu^{7} + 85317 \nu^{6} - 912806 \nu^{5} + \cdots + 16288 ) / 768 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1891 \nu^{10} + 11461 \nu^{9} + 31224 \nu^{8} - 236441 \nu^{7} - 140649 \nu^{6} + 1502158 \nu^{5} + \cdots - 27168 ) / 768 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1661 \nu^{10} - 10067 \nu^{9} - 27416 \nu^{8} + 207607 \nu^{7} + 123519 \nu^{6} - 1318386 \nu^{5} + \cdots + 25056 ) / 384 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 4187 \nu^{10} - 25389 \nu^{9} - 69048 \nu^{8} + 523649 \nu^{7} + 309745 \nu^{6} - 3325822 \nu^{5} + \cdots + 68896 ) / 768 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 4763 \nu^{10} - 28885 \nu^{9} - 78504 \nu^{8} + 595537 \nu^{7} + 352073 \nu^{6} - 3780606 \nu^{5} + \cdots + 74784 ) / 768 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{8} - \beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} + \beta _1 + 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{10} + \beta_{9} - \beta_{7} + 3\beta_{6} + \beta_{5} - 3\beta_{3} - \beta_{2} + 12\beta _1 + 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 2 \beta_{10} + 2 \beta_{9} - 14 \beta_{8} - 10 \beta_{7} + 23 \beta_{6} + 4 \beta_{5} + \beta_{4} + \cdots + 91 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 41 \beta_{10} + 16 \beta_{9} - 7 \beta_{8} - 21 \beta_{7} + 97 \beta_{6} + 27 \beta_{5} + 2 \beta_{4} + \cdots + 223 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 82 \beta_{10} + 27 \beta_{9} - 186 \beta_{8} - 122 \beta_{7} + 551 \beta_{6} + 110 \beta_{5} + \cdots + 1530 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 769 \beta_{10} + 159 \beta_{9} - 190 \beta_{8} - 331 \beta_{7} + 2551 \beta_{6} + 575 \beta_{5} + \cdots + 5272 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 2321 \beta_{10} + 68 \beta_{9} - 2432 \beta_{8} - 1553 \beta_{7} + 13214 \beta_{6} + 2439 \beta_{5} + \cdots + 29588 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 15266 \beta_{10} + 157 \beta_{9} - 3547 \beta_{8} - 4469 \beta_{7} + 63024 \beta_{6} + 11820 \beta_{5} + \cdots + 120279 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 58013 \beta_{10} - 8572 \beta_{9} - 31099 \beta_{8} - 18301 \beta_{7} + 314673 \beta_{6} + \cdots + 617757 \)
|
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 | −1.00000 | 0 | −3.82180 | 0 | −4.03134 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | −2.89977 | 0 | 0.0863246 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | −1.70800 | 0 | −1.32713 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | −1.05180 | 0 | 3.69094 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | 0.479979 | 0 | −1.67109 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.00000 | 0 | 0.964920 | 0 | −3.74988 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −1.00000 | 0 | 1.58496 | 0 | −4.99782 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | −1.00000 | 0 | 3.05414 | 0 | −0.886442 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | −1.00000 | 0 | 3.06612 | 0 | 3.64577 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | −1.00000 | 0 | 3.08015 | 0 | 2.08595 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | −1.00000 | 0 | 4.25110 | 0 | −2.84528 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(131\) | \( -1 \) |
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 | 6288.2.a.bo | 11 | |
| 4.b | odd | 2 | 1 | 3144.2.a.n | ✓ | 11 | |
| 12.b | even | 2 | 1 | 9432.2.a.v | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3144.2.a.n | ✓ | 11 | 4.b | odd | 2 | 1 | |
| 6288.2.a.bo | 11 | 1.a | even | 1 | 1 | trivial | |
| 9432.2.a.v | 11 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6288))\):
|
\( T_{5}^{11} - 7 T_{5}^{10} - 14 T_{5}^{9} + 185 T_{5}^{8} - 134 T_{5}^{7} - 1417 T_{5}^{6} + 2276 T_{5}^{5} + \cdots - 1792 \)
|
|
\( T_{7}^{11} + 10 T_{7}^{10} - 278 T_{7}^{8} - 666 T_{7}^{7} + 1892 T_{7}^{6} + 8133 T_{7}^{5} + \cdots + 1024 \)
|
|
\( T_{17}^{11} - 7 T_{17}^{10} - 83 T_{17}^{9} + 540 T_{17}^{8} + 2382 T_{17}^{7} - 12950 T_{17}^{6} + \cdots + 50976 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} \)
|
| $3$ |
\( (T + 1)^{11} \)
|
| $5$ |
\( T^{11} - 7 T^{10} + \cdots - 1792 \)
|
| $7$ |
\( T^{11} + 10 T^{10} + \cdots + 1024 \)
|
| $11$ |
\( T^{11} + 7 T^{10} + \cdots + 832 \)
|
| $13$ |
\( T^{11} - 15 T^{10} + \cdots - 76832 \)
|
| $17$ |
\( T^{11} - 7 T^{10} + \cdots + 50976 \)
|
| $19$ |
\( T^{11} + 6 T^{10} + \cdots - 1567744 \)
|
| $23$ |
\( T^{11} + 7 T^{10} + \cdots + 1465344 \)
|
| $29$ |
\( T^{11} - 9 T^{10} + \cdots + 2686192 \)
|
| $31$ |
\( T^{11} + 20 T^{10} + \cdots + 1195488 \)
|
| $37$ |
\( T^{11} - 18 T^{10} + \cdots + 6397632 \)
|
| $41$ |
\( T^{11} + \cdots + 366775808 \)
|
| $43$ |
\( T^{11} + 3 T^{10} + \cdots + 84856832 \)
|
| $47$ |
\( T^{11} + 10 T^{10} + \cdots - 3530752 \)
|
| $53$ |
\( T^{11} - 15 T^{10} + \cdots - 75478144 \)
|
| $59$ |
\( T^{11} + 12 T^{10} + \cdots - 13082624 \)
|
| $61$ |
\( T^{11} + \cdots - 2558608128 \)
|
| $67$ |
\( T^{11} + 8 T^{10} + \cdots - 67203072 \)
|
| $71$ |
\( T^{11} + \cdots + 7921696768 \)
|
| $73$ |
\( T^{11} + \cdots - 2074784128 \)
|
| $79$ |
\( T^{11} + 18 T^{10} + \cdots + 29225984 \)
|
| $83$ |
\( T^{11} - 20 T^{10} + \cdots + 987264 \)
|
| $89$ |
\( T^{11} - 9 T^{10} + \cdots - 16641472 \)
|
| $97$ |
\( T^{11} - 34 T^{10} + \cdots + 4580352 \)
|
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