Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4925,2,Mod(1,4925)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4925, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4925.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4925 = 5^{2} \cdot 197 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4925.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: | \(39.3263229955\) |
| Analytic rank: | \(1\) |
| Dimension: | \(10\) |
| Coefficient field: | 10.10.21886214112361.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2x^{9} - 9x^{8} + 16x^{7} + 26x^{6} - 38x^{5} - 27x^{4} + 32x^{3} + 6x^{2} - 7x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 985) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2x^{9} - 9x^{8} + 16x^{7} + 26x^{6} - 38x^{5} - 27x^{4} + 32x^{3} + 6x^{2} - 7x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{9} - 2\nu^{8} - 8\nu^{7} + 14\nu^{6} + 19\nu^{5} - 25\nu^{4} - 15\nu^{3} + 13\nu^{2} + 2\nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{9} + 2\nu^{8} + 9\nu^{7} - 16\nu^{6} - 26\nu^{5} + 38\nu^{4} + 27\nu^{3} - 31\nu^{2} - 7\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{9} - 2\nu^{8} - 9\nu^{7} + 16\nu^{6} + 26\nu^{5} - 38\nu^{4} - 27\nu^{3} + 32\nu^{2} + 6\nu - 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{9} - \nu^{8} - 11\nu^{7} + 8\nu^{6} + 40\nu^{5} - 19\nu^{4} - 52\nu^{3} + 17\nu^{2} + 18\nu - 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( -3\nu^{9} + 5\nu^{8} + 29\nu^{7} - 39\nu^{6} - 93\nu^{5} + 87\nu^{4} + 112\nu^{3} - 64\nu^{2} - 38\nu + 10 \)
|
| \(\beta_{7}\) | \(=\) |
\( 4\nu^{9} - 7\nu^{8} - 38\nu^{7} + 55\nu^{6} + 120\nu^{5} - 125\nu^{4} - 146\nu^{3} + 95\nu^{2} + 54\nu - 16 \)
|
| \(\beta_{8}\) | \(=\) |
\( -4\nu^{9} + 7\nu^{8} + 38\nu^{7} - 55\nu^{6} - 120\nu^{5} + 126\nu^{4} + 145\nu^{3} - 100\nu^{2} - 51\nu + 19 \)
|
| \(\beta_{9}\) | \(=\) |
\( -8\nu^{9} + 14\nu^{8} + 75\nu^{7} - 108\nu^{6} - 232\nu^{5} + 238\nu^{4} + 273\nu^{3} - 177\nu^{2} - 96\nu + 31 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{8} - \beta_{6} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} + \beta_{7} - \beta_{6} + 5\beta_{4} + 5\beta_{3} + \beta_{2} + 7\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{9} - 7\beta_{8} + \beta_{7} - 6\beta_{6} + \beta_{3} + 7\beta_{2} + 26\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9\beta_{9} - \beta_{8} + 9\beta_{7} - 7\beta_{6} + \beta_{5} + 25\beta_{4} + 24\beta_{3} + 9\beta_{2} + 43\beta _1 + 30 \)
|
| \(\nu^{7}\) | \(=\) |
\( 42 \beta_{9} - 39 \beta_{8} + 12 \beta_{7} - 31 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 9 \beta_{3} + \cdots + 17 \)
|
| \(\nu^{8}\) | \(=\) |
\( 64 \beta_{9} - 13 \beta_{8} + 63 \beta_{7} - 40 \beta_{6} + 13 \beta_{5} + 125 \beta_{4} + 116 \beta_{3} + \cdots + 141 \)
|
| \(\nu^{9}\) | \(=\) |
\( 245 \beta_{9} - 206 \beta_{8} + 102 \beta_{7} - 156 \beta_{6} + 28 \beta_{5} + 36 \beta_{4} + 61 \beta_{3} + \cdots + 111 \)
|
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 |
|
−2.12691 | 2.07716 | 2.52376 | 0 | −4.41794 | 2.06572 | −1.11399 | 1.31459 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.39693 | −0.475994 | −0.0485756 | 0 | 0.664932 | 3.39303 | 2.86172 | −2.77343 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.23959 | 2.93211 | −0.463416 | 0 | −3.63462 | −2.16985 | 3.05363 | 5.59727 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.638386 | −1.45932 | −1.59246 | 0 | 0.931607 | 2.37873 | 2.29338 | −0.870397 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.231201 | 0.400645 | −1.94655 | 0 | 0.0926295 | −1.70963 | −0.912444 | −2.83948 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.285611 | 1.26844 | −1.91843 | 0 | 0.362280 | 2.73433 | −1.11915 | −1.39107 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.888606 | −1.66962 | −1.21038 | 0 | −1.48364 | −1.49037 | −2.85276 | −0.212359 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.33777 | 1.51312 | −0.210373 | 0 | 2.02421 | 1.25982 | −2.95697 | −0.710457 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.24278 | −1.69515 | 3.03007 | 0 | −3.80186 | 3.86416 | 2.31023 | −0.126460 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.41585 | 0.108610 | 3.83635 | 0 | 0.262386 | −4.32596 | 4.43635 | −2.98820 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(197\) | \( +1 \) |
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 | 4925.2.a.j | 10 | |
| 5.b | even | 2 | 1 | 985.2.a.e | ✓ | 10 | |
| 15.d | odd | 2 | 1 | 8865.2.a.t | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 985.2.a.e | ✓ | 10 | 5.b | even | 2 | 1 | |
| 4925.2.a.j | 10 | 1.a | even | 1 | 1 | trivial | |
| 8865.2.a.t | 10 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4925))\):
|
\( T_{2}^{10} - 2T_{2}^{9} - 9T_{2}^{8} + 16T_{2}^{7} + 26T_{2}^{6} - 38T_{2}^{5} - 27T_{2}^{4} + 32T_{2}^{3} + 6T_{2}^{2} - 7T_{2} + 1 \)
|
|
\( T_{3}^{10} - 3T_{3}^{9} - 8T_{3}^{8} + 24T_{3}^{7} + 22T_{3}^{6} - 64T_{3}^{5} - 21T_{3}^{4} + 61T_{3}^{3} + T_{3}^{2} - 10T_{3} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 2 T^{9} + \cdots + 1 \)
|
| $3$ |
\( T^{10} - 3 T^{9} + \cdots + 1 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} - 6 T^{9} + \cdots + 5308 \)
|
| $11$ |
\( T^{10} + 11 T^{9} + \cdots + 876 \)
|
| $13$ |
\( T^{10} - 5 T^{9} + \cdots + 17477 \)
|
| $17$ |
\( T^{10} - 4 T^{9} + \cdots - 137988 \)
|
| $19$ |
\( T^{10} + 28 T^{9} + \cdots - 25811 \)
|
| $23$ |
\( T^{10} - 24 T^{9} + \cdots - 2075712 \)
|
| $29$ |
\( T^{10} + 16 T^{9} + \cdots + 98764 \)
|
| $31$ |
\( T^{10} + 34 T^{9} + \cdots - 73173232 \)
|
| $37$ |
\( T^{10} + 5 T^{9} + \cdots + 2426304 \)
|
| $41$ |
\( T^{10} + 15 T^{9} + \cdots + 337193 \)
|
| $43$ |
\( T^{10} - 5 T^{9} + \cdots + 2284 \)
|
| $47$ |
\( T^{10} - 14 T^{9} + \cdots + 28474764 \)
|
| $53$ |
\( T^{10} - 15 T^{9} + \cdots + 317244 \)
|
| $59$ |
\( T^{10} + 49 T^{9} + \cdots + 12406847 \)
|
| $61$ |
\( T^{10} + \cdots - 1693803719 \)
|
| $67$ |
\( T^{10} - 2 T^{9} + \cdots - 6547 \)
|
| $71$ |
\( T^{10} + \cdots + 150197316 \)
|
| $73$ |
\( T^{10} + 18 T^{9} + \cdots + 506257 \)
|
| $79$ |
\( T^{10} + 11 T^{9} + \cdots + 74241116 \)
|
| $83$ |
\( T^{10} + 4 T^{9} + \cdots + 132 \)
|
| $89$ |
\( T^{10} + 5 T^{9} + \cdots - 10810148 \)
|
| $97$ |
\( T^{10} + \cdots - 2022756524 \)
|
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