Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [260,10,Mod(1,260)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(260, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("260.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 260 = 2^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 260.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: | \(133.909317403\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 97268 x^{6} - 2071232 x^{5} + 2607329518 x^{4} + 97363112320 x^{3} - 14477077261188 x^{2} + \cdots + 84\!\cdots\!25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{16}\cdot 3\cdot 5 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 97268 x^{6} - 2071232 x^{5} + 2607329518 x^{4} + 97363112320 x^{3} - 14477077261188 x^{2} + \cdots + 84\!\cdots\!25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 32\nu - 24317 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 4962521352439 \nu^{7} + \cdots - 27\!\cdots\!13 ) / 22\!\cdots\!92 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 30074755108856 \nu^{7} + \cdots + 85\!\cdots\!79 ) / 11\!\cdots\!96 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 21590932197127 \nu^{7} + \cdots - 15\!\cdots\!57 ) / 17\!\cdots\!84 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 288252451269899 \nu^{7} + \cdots - 94\!\cdots\!69 ) / 22\!\cdots\!92 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 24963653005597 \nu^{7} + \cdots - 16\!\cdots\!25 ) / 17\!\cdots\!84 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 32\beta _1 + 24317 \)
|
| \(\nu^{3}\) | \(=\) |
\( -49\beta_{7} + 48\beta_{6} + 9\beta_{5} + 23\beta_{4} + 186\beta_{3} + 26\beta_{2} + 43650\beta _1 + 776712 \)
|
| \(\nu^{4}\) | \(=\) |
\( 1204 \beta_{7} + 3162 \beta_{6} - 6708 \beta_{5} - 10112 \beta_{4} - 5562 \beta_{3} + 57806 \beta_{2} + \cdots + 1061601197 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 3037370 \beta_{7} + 4031574 \beta_{6} - 560898 \beta_{5} + 1666642 \beta_{4} + 16378914 \beta_{3} + \cdots + 65063426160 \)
|
| \(\nu^{6}\) | \(=\) |
\( 78548900 \beta_{7} + 367384002 \beta_{6} - 643372692 \beta_{5} - 797522248 \beta_{4} + \cdots + 52323952035665 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 168264911535 \beta_{7} + 270635036382 \beta_{6} - 87476542965 \beta_{5} + 75479339181 \beta_{4} + \cdots + 42\!\cdots\!28 \)
|
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 | −244.228 | 0 | 625.000 | 0 | 10178.1 | 0 | 39964.5 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −203.691 | 0 | 625.000 | 0 | −3761.46 | 0 | 21806.9 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −62.0100 | 0 | 625.000 | 0 | 8181.50 | 0 | −15837.8 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −24.6590 | 0 | 625.000 | 0 | −9338.77 | 0 | −19074.9 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 26.1366 | 0 | 625.000 | 0 | −1157.02 | 0 | −18999.9 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 138.181 | 0 | 625.000 | 0 | 4766.66 | 0 | −588.929 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 138.967 | 0 | 625.000 | 0 | 3251.70 | 0 | −371.116 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 223.303 | 0 | 625.000 | 0 | −7312.70 | 0 | 30181.2 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(13\) | \(1\) |
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 | 260.10.a.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 260.10.a.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 8 T_{3}^{7} - 97240 T_{3}^{6} + 1487680 T_{3}^{5} + 2616226728 T_{3}^{4} + \cdots + 85\!\cdots\!00 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(260))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 85\!\cdots\!00 \)
|
| $5$ |
\( (T - 625)^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 38\!\cdots\!44 \)
|
| $11$ |
\( T^{8} + \cdots + 53\!\cdots\!12 \)
|
| $13$ |
\( (T + 28561)^{8} \)
|
| $17$ |
\( T^{8} + \cdots + 15\!\cdots\!76 \)
|
| $19$ |
\( T^{8} + \cdots - 42\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots - 27\!\cdots\!44 \)
|
| $29$ |
\( T^{8} + \cdots - 51\!\cdots\!24 \)
|
| $31$ |
\( T^{8} + \cdots + 26\!\cdots\!20 \)
|
| $37$ |
\( T^{8} + \cdots - 29\!\cdots\!00 \)
|
| $41$ |
\( T^{8} + \cdots - 30\!\cdots\!04 \)
|
| $43$ |
\( T^{8} + \cdots - 39\!\cdots\!72 \)
|
| $47$ |
\( T^{8} + \cdots + 55\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots - 92\!\cdots\!24 \)
|
| $59$ |
\( T^{8} + \cdots - 24\!\cdots\!60 \)
|
| $61$ |
\( T^{8} + \cdots - 29\!\cdots\!72 \)
|
| $67$ |
\( T^{8} + \cdots - 61\!\cdots\!80 \)
|
| $71$ |
\( T^{8} + \cdots - 97\!\cdots\!20 \)
|
| $73$ |
\( T^{8} + \cdots + 22\!\cdots\!84 \)
|
| $79$ |
\( T^{8} + \cdots + 17\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 30\!\cdots\!00 \)
|
| $89$ |
\( T^{8} + \cdots + 86\!\cdots\!04 \)
|
| $97$ |
\( T^{8} + \cdots + 49\!\cdots\!32 \)
|
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