Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [240,9,Mod(31,240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(240, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("240.31");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 240.e (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(97.7708664147\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{12} - x^{11} + 67817 x^{10} - 5043842 x^{9} + 3585199422 x^{8} - 247197333458 x^{7} + \cdots + 19\!\cdots\!76 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{36}\cdot 3^{18}\cdot 5^{12} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{12} - x^{11} + 67817 x^{10} - 5043842 x^{9} + 3585199422 x^{8} - 247197333458 x^{7} + \cdots + 19\!\cdots\!76 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 60\!\cdots\!39 \nu^{11} + \cdots + 41\!\cdots\!72 ) / 89\!\cdots\!92 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 49\!\cdots\!75 \nu^{11} + \cdots - 25\!\cdots\!00 ) / 14\!\cdots\!92 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 58\!\cdots\!83 \nu^{11} + \cdots - 18\!\cdots\!60 ) / 11\!\cdots\!12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 14\!\cdots\!95 \nu^{11} + \cdots + 77\!\cdots\!60 ) / 22\!\cdots\!24 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 48\!\cdots\!67 \nu^{11} + \cdots + 14\!\cdots\!36 ) / 30\!\cdots\!96 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 12\!\cdots\!81 \nu^{11} + \cdots - 55\!\cdots\!20 ) / 51\!\cdots\!32 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 16\!\cdots\!29 \nu^{11} + \cdots + 84\!\cdots\!80 ) / 57\!\cdots\!56 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 42\!\cdots\!07 \nu^{11} + \cdots - 21\!\cdots\!40 ) / 11\!\cdots\!12 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 11\!\cdots\!93 \nu^{11} + \cdots + 65\!\cdots\!08 ) / 10\!\cdots\!64 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 55\!\cdots\!03 \nu^{11} + \cdots + 28\!\cdots\!80 ) / 22\!\cdots\!24 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 28\!\cdots\!81 \nu^{11} + \cdots - 14\!\cdots\!60 ) / 10\!\cdots\!64 \)
|
| \(\nu\) | \(=\) |
\( ( 38 \beta_{11} - 27 \beta_{10} + 179 \beta_{9} + 54 \beta_{8} + 1062 \beta_{7} + 464 \beta_{6} + \cdots + 27000 ) / 324000 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 187 \beta_{11} + 8127 \beta_{10} + 7379 \beta_{9} - 12879 \beta_{8} - 83992 \beta_{7} + \cdots - 1220697000 ) / 108000 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 380858 \beta_{11} - 5948489 \beta_{9} - 22003924 \beta_{6} + 5542253 \beta_{5} + \cdots + 201529026000 ) / 162000 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 39949551 \beta_{11} - 544864671 \beta_{10} + 385066467 \beta_{9} + 352234467 \beta_{8} + \cdots - 46103423481000 ) / 108000 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 17424780158 \beta_{11} + 324969178743 \beta_{10} + 255270058111 \beta_{9} + \cdots - 12\!\cdots\!00 ) / 324000 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 1050110311625 \beta_{11} - 6355882582375 \beta_{9} - 13989402663375 \beta_{6} + \cdots + 65\!\cdots\!00 ) / 18000 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 18\!\cdots\!86 \beta_{11} + \cdots - 72\!\cdots\!00 ) / 324000 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 19\!\cdots\!51 \beta_{11} + \cdots - 90\!\cdots\!00 ) / 108000 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 12\!\cdots\!26 \beta_{11} + \cdots + 40\!\cdots\!00 ) / 162000 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 11\!\cdots\!03 \beta_{11} + \cdots - 43\!\cdots\!00 ) / 108000 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 72\!\cdots\!34 \beta_{11} + \cdots - 22\!\cdots\!00 ) / 324000 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(161\) | \(181\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | − | 46.7654i | 0 | −279.508 | 0 | − | 1883.05i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.2 | 0 | − | 46.7654i | 0 | −279.508 | 0 | 869.720i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.3 | 0 | − | 46.7654i | 0 | −279.508 | 0 | 2440.89i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.4 | 0 | − | 46.7654i | 0 | 279.508 | 0 | − | 2706.52i | 0 | −2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.5 | 0 | − | 46.7654i | 0 | 279.508 | 0 | 1536.92i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.6 | 0 | − | 46.7654i | 0 | 279.508 | 0 | 4099.87i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.7 | 0 | 46.7654i | 0 | −279.508 | 0 | − | 2440.89i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.8 | 0 | 46.7654i | 0 | −279.508 | 0 | − | 869.720i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.9 | 0 | 46.7654i | 0 | −279.508 | 0 | 1883.05i | 0 | −2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.10 | 0 | 46.7654i | 0 | 279.508 | 0 | − | 4099.87i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.11 | 0 | 46.7654i | 0 | 279.508 | 0 | − | 1536.92i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.12 | 0 | 46.7654i | 0 | 279.508 | 0 | 2706.52i | 0 | −2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 240.9.e.b | ✓ | 12 |
| 4.b | odd | 2 | 1 | inner | 240.9.e.b | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 240.9.e.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 240.9.e.b | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{12} + 36756528 T_{7}^{10} + 480310019919360 T_{7}^{8} + \cdots + 46\!\cdots\!84 \)
acting on \(S_{9}^{\mathrm{new}}(240, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( (T^{2} + 2187)^{6} \)
|
| $5$ |
\( (T^{2} - 78125)^{6} \)
|
| $7$ |
\( T^{12} + \cdots + 46\!\cdots\!84 \)
|
| $11$ |
\( T^{12} + \cdots + 32\!\cdots\!04 \)
|
| $13$ |
\( (T^{6} + \cdots - 34\!\cdots\!84)^{2} \)
|
| $17$ |
\( (T^{6} + \cdots + 18\!\cdots\!36)^{2} \)
|
| $19$ |
\( T^{12} + \cdots + 33\!\cdots\!64 \)
|
| $23$ |
\( T^{12} + \cdots + 47\!\cdots\!64 \)
|
| $29$ |
\( (T^{6} + \cdots - 27\!\cdots\!16)^{2} \)
|
| $31$ |
\( T^{12} + \cdots + 15\!\cdots\!00 \)
|
| $37$ |
\( (T^{6} + \cdots + 85\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{6} + \cdots + 18\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{12} + \cdots + 10\!\cdots\!84 \)
|
| $47$ |
\( T^{12} + \cdots + 14\!\cdots\!44 \)
|
| $53$ |
\( (T^{6} + \cdots + 13\!\cdots\!24)^{2} \)
|
| $59$ |
\( T^{12} + \cdots + 77\!\cdots\!24 \)
|
| $61$ |
\( (T^{6} + \cdots - 71\!\cdots\!44)^{2} \)
|
| $67$ |
\( T^{12} + \cdots + 84\!\cdots\!24 \)
|
| $71$ |
\( T^{12} + \cdots + 16\!\cdots\!84 \)
|
| $73$ |
\( (T^{6} + \cdots + 24\!\cdots\!36)^{2} \)
|
| $79$ |
\( T^{12} + \cdots + 14\!\cdots\!84 \)
|
| $83$ |
\( T^{12} + \cdots + 56\!\cdots\!24 \)
|
| $89$ |
\( (T^{6} + \cdots + 30\!\cdots\!16)^{2} \)
|
| $97$ |
\( (T^{6} + \cdots + 30\!\cdots\!04)^{2} \)
|
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