Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [656,6,Mod(1,656)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(656, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("656.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 656 = 2^{4} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 656.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: | \(105.211785797\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 6 x^{11} - 2016 x^{10} + 9442 x^{9} + 1419186 x^{8} - 5164836 x^{7} - 410141644 x^{6} + \cdots + 2432900188800 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{17}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 328) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 6 x^{11} - 2016 x^{10} + 9442 x^{9} + 1419186 x^{8} - 5164836 x^{7} - 410141644 x^{6} + \cdots + 2432900188800 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 10\!\cdots\!59 \nu^{11} + \cdots - 30\!\cdots\!80 ) / 13\!\cdots\!80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 81\!\cdots\!29 \nu^{11} + \cdots - 75\!\cdots\!60 ) / 13\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 82\!\cdots\!31 \nu^{11} + \cdots - 24\!\cdots\!20 ) / 13\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\!\cdots\!95 \nu^{11} + \cdots + 30\!\cdots\!00 ) / 14\!\cdots\!20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 82\!\cdots\!31 \nu^{11} + \cdots - 22\!\cdots\!00 ) / 66\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 17\!\cdots\!15 \nu^{11} + \cdots - 17\!\cdots\!60 ) / 13\!\cdots\!80 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 58\!\cdots\!34 \nu^{11} + \cdots - 19\!\cdots\!60 ) / 44\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 31\!\cdots\!41 \nu^{11} + \cdots - 54\!\cdots\!20 ) / 13\!\cdots\!80 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 41\!\cdots\!59 \nu^{11} + \cdots - 33\!\cdots\!40 ) / 13\!\cdots\!80 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 45\!\cdots\!66 \nu^{11} + \cdots - 49\!\cdots\!00 ) / 13\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + 2\beta_{4} + \beta _1 + 338 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 6 \beta_{11} + 5 \beta_{10} + \beta_{9} - \beta_{8} - 5 \beta_{7} + 6 \beta_{6} + 4 \beta_{5} + \cdots + 375 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 7 \beta_{11} + 35 \beta_{10} - 48 \beta_{9} - 11 \beta_{8} - 31 \beta_{7} + 851 \beta_{6} + \cdots + 208304 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 5072 \beta_{11} + 5857 \beta_{10} + 1239 \beta_{9} - 1498 \beta_{8} - 4946 \beta_{7} + 5471 \beta_{6} + \cdots + 659859 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 8353 \beta_{11} + 40969 \beta_{10} - 60359 \beta_{9} - 99 \beta_{8} - 12102 \beta_{7} + \cdots + 146680241 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 3781116 \beta_{11} + 5681082 \beta_{10} + 881391 \beta_{9} - 1527645 \beta_{8} - 4273752 \beta_{7} + \cdots + 757497426 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 9097191 \beta_{11} + 42521691 \beta_{10} - 63759990 \beta_{9} + 4952820 \beta_{8} + \cdots + 109730534813 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2756288964 \beta_{11} + 5139758288 \beta_{10} + 432592720 \beta_{9} - 1358942500 \beta_{8} + \cdots + 748167086682 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 8811360448 \beta_{11} + 43428577508 \beta_{10} - 62920903188 \beta_{9} + 6212264344 \beta_{8} + \cdots + 84795471062804 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 2011526684870 \beta_{11} + 4509252766954 \beta_{10} + 58691737938 \beta_{9} - 1138337599114 \beta_{8} + \cdots + 690443474361138 \)
|
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 | −30.0715 | 0 | 12.9990 | 0 | 220.926 | 0 | 661.298 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −25.0399 | 0 | 60.5551 | 0 | −127.423 | 0 | 383.997 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −22.3942 | 0 | 80.4301 | 0 | −102.619 | 0 | 258.500 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −15.2554 | 0 | −45.6172 | 0 | −233.995 | 0 | −10.2719 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −6.19566 | 0 | −91.6695 | 0 | 209.419 | 0 | −204.614 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −3.29301 | 0 | 32.1612 | 0 | 115.349 | 0 | −232.156 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.0780745 | 0 | −68.3466 | 0 | −38.8811 | 0 | −242.994 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 4.48742 | 0 | 85.1953 | 0 | −60.8288 | 0 | −222.863 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 11.3498 | 0 | 14.2594 | 0 | 192.101 | 0 | −114.182 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 15.9624 | 0 | −40.1737 | 0 | −72.5976 | 0 | 11.7982 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 25.1688 | 0 | −4.90545 | 0 | 35.6544 | 0 | 390.468 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 27.2033 | 0 | 21.1124 | 0 | −209.105 | 0 | 497.019 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(41\) | \( +1 \) |
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 | 656.6.a.j | 12 | |
| 4.b | odd | 2 | 1 | 328.6.a.b | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 328.6.a.b | ✓ | 12 | 4.b | odd | 2 | 1 | |
| 656.6.a.j | 12 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 18 T_{3}^{11} - 1884 T_{3}^{10} - 30438 T_{3}^{9} + 1226262 T_{3}^{8} + 16960092 T_{3}^{7} + \cdots + 228089520000 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(656))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + \cdots + 228089520000 \)
|
| $5$ |
\( T^{12} + \cdots - 29\!\cdots\!72 \)
|
| $7$ |
\( T^{12} + \cdots - 40\!\cdots\!36 \)
|
| $11$ |
\( T^{12} + \cdots - 81\!\cdots\!00 \)
|
| $13$ |
\( T^{12} + \cdots - 56\!\cdots\!00 \)
|
| $17$ |
\( T^{12} + \cdots - 56\!\cdots\!56 \)
|
| $19$ |
\( T^{12} + \cdots - 72\!\cdots\!00 \)
|
| $23$ |
\( T^{12} + \cdots + 13\!\cdots\!00 \)
|
| $29$ |
\( T^{12} + \cdots - 76\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots + 26\!\cdots\!00 \)
|
| $37$ |
\( T^{12} + \cdots + 46\!\cdots\!80 \)
|
| $41$ |
\( (T + 1681)^{12} \)
|
| $43$ |
\( T^{12} + \cdots + 98\!\cdots\!68 \)
|
| $47$ |
\( T^{12} + \cdots + 20\!\cdots\!40 \)
|
| $53$ |
\( T^{12} + \cdots + 75\!\cdots\!40 \)
|
| $59$ |
\( T^{12} + \cdots - 53\!\cdots\!40 \)
|
| $61$ |
\( T^{12} + \cdots + 14\!\cdots\!00 \)
|
| $67$ |
\( T^{12} + \cdots + 38\!\cdots\!88 \)
|
| $71$ |
\( T^{12} + \cdots - 22\!\cdots\!76 \)
|
| $73$ |
\( T^{12} + \cdots - 50\!\cdots\!20 \)
|
| $79$ |
\( T^{12} + \cdots + 77\!\cdots\!80 \)
|
| $83$ |
\( T^{12} + \cdots - 10\!\cdots\!16 \)
|
| $89$ |
\( T^{12} + \cdots + 10\!\cdots\!72 \)
|
| $97$ |
\( T^{12} + \cdots + 15\!\cdots\!80 \)
|
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