Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6256,2,Mod(1,6256)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6256, 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("6256.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6256 = 2^{4} \cdot 17 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6256.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: | \(49.9544115045\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{11} - 5 x^{10} - 15 x^{9} + 103 x^{8} + 20 x^{7} - 672 x^{6} + 436 x^{5} + 1524 x^{4} - 1512 x^{3} + \cdots - 224 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 3128) |
| 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
\( x^{11} - 5 x^{10} - 15 x^{9} + 103 x^{8} + 20 x^{7} - 672 x^{6} + 436 x^{5} + 1524 x^{4} - 1512 x^{3} + \cdots - 224 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3 \nu^{10} - 8 \nu^{9} - 58 \nu^{8} + 168 \nu^{7} + 333 \nu^{6} - 1120 \nu^{5} - 478 \nu^{4} + \cdots + 1104 ) / 136 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 3 \nu^{10} + 8 \nu^{9} + 58 \nu^{8} - 168 \nu^{7} - 333 \nu^{6} + 1120 \nu^{5} + 546 \nu^{4} + \cdots - 16 ) / 136 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 2 \nu^{10} + 11 \nu^{9} + 33 \nu^{8} - 231 \nu^{7} - 103 \nu^{6} + 1574 \nu^{5} - 418 \nu^{4} + \cdots - 1144 ) / 136 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 4 \nu^{10} + 5 \nu^{9} + 83 \nu^{8} - 88 \nu^{7} - 546 \nu^{6} + 411 \nu^{5} + 1255 \nu^{4} + \cdots + 160 ) / 68 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 13 \nu^{10} - 29 \nu^{9} - 257 \nu^{8} + 575 \nu^{7} + 1528 \nu^{6} - 3516 \nu^{5} - 2638 \nu^{4} + \cdots + 704 ) / 136 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 29 \nu^{10} - 66 \nu^{9} - 606 \nu^{8} + 1352 \nu^{7} + 4035 \nu^{6} - 8798 \nu^{5} - 9358 \nu^{4} + \cdots + 2240 ) / 136 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 41 \nu^{10} + 64 \nu^{9} + 872 \nu^{8} - 1276 \nu^{7} - 5979 \nu^{6} + 8008 \nu^{5} + 14636 \nu^{4} + \cdots - 2032 ) / 136 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 45 \nu^{10} + 86 \nu^{9} + 938 \nu^{8} - 1738 \nu^{7} - 6185 \nu^{6} + 11088 \nu^{5} + 13800 \nu^{4} + \cdots - 4184 ) / 136 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 7\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 2\beta_{3} + 11\beta_{2} + 39 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{10} + 2\beta_{9} + 11\beta_{7} + 11\beta_{6} + 15\beta_{5} + 11\beta_{4} + 59\beta _1 + 13 \)
|
| \(\nu^{6}\) | \(=\) |
\( 6 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} + 28 \beta_{4} + 32 \beta_{3} + \cdots + 345 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 36 \beta_{10} + 34 \beta_{9} - 2 \beta_{8} + 105 \beta_{7} + 111 \beta_{6} + 177 \beta_{5} + \cdots + 135 \)
|
| \(\nu^{8}\) | \(=\) |
\( 116 \beta_{10} - 80 \beta_{9} + 32 \beta_{8} + 80 \beta_{7} + 40 \beta_{6} - 8 \beta_{5} + 306 \beta_{4} + \cdots + 3191 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 474 \beta_{10} + 426 \beta_{9} - 52 \beta_{8} + 987 \beta_{7} + 1115 \beta_{6} + 1935 \beta_{5} + \cdots + 1297 \)
|
| \(\nu^{10}\) | \(=\) |
\( 1582 \beta_{10} - 1124 \beta_{9} + 370 \beta_{8} + 1100 \beta_{7} + 554 \beta_{6} - 168 \beta_{5} + \cdots + 30121 \)
|
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 | −3.12015 | 0 | 2.45465 | 0 | −1.86509 | 0 | 6.73535 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.09870 | 0 | 1.35621 | 0 | −0.153784 | 0 | 6.60194 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.95192 | 0 | −2.99463 | 0 | 4.69465 | 0 | 5.71383 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.92558 | 0 | −3.69437 | 0 | −4.24297 | 0 | 0.707845 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.30531 | 0 | −0.555529 | 0 | −2.65676 | 0 | −1.29616 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.900372 | 0 | 3.47174 | 0 | 1.22572 | 0 | −2.18933 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −0.225199 | 0 | −4.02372 | 0 | 2.20509 | 0 | −2.94929 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.19833 | 0 | 0.300101 | 0 | 3.62545 | 0 | −1.56401 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.56097 | 0 | 0.0826655 | 0 | 1.33742 | 0 | −0.563387 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.59358 | 0 | −2.10107 | 0 | −4.00921 | 0 | 3.72665 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 3.17436 | 0 | −4.29603 | 0 | −0.160499 | 0 | 7.07655 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(17\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 6256.2.a.bf | 11 | |
| 4.b | odd | 2 | 1 | 3128.2.a.i | ✓ | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3128.2.a.i | ✓ | 11 | 4.b | odd | 2 | 1 | |
| 6256.2.a.bf | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6256))\):
|
\( T_{3}^{11} + 5 T_{3}^{10} - 15 T_{3}^{9} - 103 T_{3}^{8} + 20 T_{3}^{7} + 672 T_{3}^{6} + 436 T_{3}^{5} + \cdots + 224 \)
|
|
\( T_{5}^{11} + 10 T_{5}^{10} + 9 T_{5}^{9} - 186 T_{5}^{8} - 488 T_{5}^{7} + 814 T_{5}^{6} + 3257 T_{5}^{5} + \cdots - 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} \)
|
| $3$ |
\( T^{11} + 5 T^{10} + \cdots + 224 \)
|
| $5$ |
\( T^{11} + 10 T^{10} + \cdots - 64 \)
|
| $7$ |
\( T^{11} - 44 T^{9} + \cdots - 128 \)
|
| $11$ |
\( T^{11} + 10 T^{10} + \cdots - 68864 \)
|
| $13$ |
\( T^{11} - T^{10} + \cdots + 4592 \)
|
| $17$ |
\( (T - 1)^{11} \)
|
| $19$ |
\( T^{11} + 22 T^{10} + \cdots + 8225344 \)
|
| $23$ |
\( (T - 1)^{11} \)
|
| $29$ |
\( T^{11} + 19 T^{10} + \cdots + 38081536 \)
|
| $31$ |
\( T^{11} - 3 T^{10} + \cdots - 284672 \)
|
| $37$ |
\( T^{11} + 2 T^{10} + \cdots - 539648 \)
|
| $41$ |
\( T^{11} - 9 T^{10} + \cdots - 15284224 \)
|
| $43$ |
\( T^{11} + 32 T^{10} + \cdots + 53048576 \)
|
| $47$ |
\( T^{11} + 3 T^{10} + \cdots + 25274816 \)
|
| $53$ |
\( T^{11} + 8 T^{10} + \cdots - 295936 \)
|
| $59$ |
\( T^{11} + 10 T^{10} + \cdots - 6480896 \)
|
| $61$ |
\( T^{11} - 2 T^{10} + \cdots - 4918624 \)
|
| $67$ |
\( T^{11} + \cdots - 344598976 \)
|
| $71$ |
\( T^{11} + \cdots - 307268096 \)
|
| $73$ |
\( T^{11} + \cdots - 182455424 \)
|
| $79$ |
\( T^{11} + \cdots - 261880384 \)
|
| $83$ |
\( T^{11} + \cdots + 172446464 \)
|
| $89$ |
\( T^{11} + \cdots - 538715648 \)
|
| $97$ |
\( T^{11} + \cdots - 144680128 \)
|
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