Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4235,2,Mod(1,4235)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4235, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4235.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4235 = 5 \cdot 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4235.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: | \(33.8166452560\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2x^{9} - 14x^{8} + 26x^{7} + 67x^{6} - 110x^{5} - 132x^{4} + 168x^{3} + 94x^{2} - 54x - 11 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2x^{9} - 14x^{8} + 26x^{7} + 67x^{6} - 110x^{5} - 132x^{4} + 168x^{3} + 94x^{2} - 54x - 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{8} - \nu^{7} - 12\nu^{6} + 9\nu^{5} + 43\nu^{4} - 19\nu^{3} - 49\nu^{2} + 3\nu + 7 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{9} + \nu^{8} + 12\nu^{7} - 9\nu^{6} - 43\nu^{5} + 19\nu^{4} + 49\nu^{3} - 6\nu^{2} - 7\nu + 9 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{9} - 15\nu^{7} + 73\nu^{5} - 3\nu^{4} - 124\nu^{3} + 11\nu^{2} + 36\nu - 6 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{9} - 14\nu^{7} - 3\nu^{6} + 64\nu^{5} + 27\nu^{4} - 108\nu^{3} - 61\nu^{2} + 44\nu + 14 ) / 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -4\nu^{9} + \nu^{8} + 57\nu^{7} - 3\nu^{6} - 265\nu^{5} - 35\nu^{4} + 442\nu^{3} + 120\nu^{2} - 145\nu - 24 ) / 3 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -4\nu^{9} + 58\nu^{7} + 9\nu^{6} - 274\nu^{5} - 78\nu^{4} + 458\nu^{3} + 169\nu^{2} - 136\nu - 28 ) / 3 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 6\nu^{9} - \nu^{8} - 85\nu^{7} - 3\nu^{6} + 390\nu^{5} + 89\nu^{4} - 628\nu^{3} - 242\nu^{2} + 167\nu + 49 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} + \beta_{7} - \beta_{3} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} + \beta_{8} + \beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 7\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{9} - 10\beta_{8} + 9\beta_{7} + 2\beta_{6} - 10\beta_{3} + 18\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10\beta_{9} + 9\beta_{8} + 12\beta_{7} + 22\beta_{6} + 12\beta_{5} + 10\beta_{4} - 12\beta_{3} + 47\beta_{2} + 77 \)
|
| \(\nu^{7}\) | \(=\) |
\( -9\beta_{9} - 77\beta_{8} + 71\beta_{7} + 27\beta_{6} + 3\beta_{5} - 80\beta_{3} + 3\beta_{2} + 90\beta _1 + 72 \)
|
| \(\nu^{8}\) | \(=\) |
\( 77 \beta_{9} + 59 \beta_{8} + 110 \beta_{7} + 187 \beta_{6} + 104 \beta_{5} + 77 \beta_{4} - 107 \beta_{3} + \cdots + 472 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 59 \beta_{9} - 546 \beta_{8} + 535 \beta_{7} + 265 \beta_{6} + 51 \beta_{5} + 3 \beta_{4} + \cdots + 562 \)
|
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 |
|
−2.51238 | −2.89922 | 4.31207 | 1.00000 | 7.28396 | −1.00000 | −5.80880 | 5.40550 | −2.51238 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.83371 | −1.24345 | 1.36250 | 1.00000 | 2.28012 | −1.00000 | 1.16899 | −1.45384 | −1.83371 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.71475 | 0.493317 | 0.940360 | 1.00000 | −0.845915 | −1.00000 | 1.81702 | −2.75664 | −1.71475 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.784066 | 1.73981 | −1.38524 | 1.00000 | −1.36412 | −1.00000 | 2.65425 | 0.0269309 | −0.784066 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.170316 | 0.667096 | −1.97099 | 1.00000 | −0.113617 | −1.00000 | 0.676324 | −2.55498 | −0.170316 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.547729 | −3.41697 | −1.69999 | 1.00000 | −1.87158 | −1.00000 | −2.02659 | 8.67570 | 0.547729 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.81029 | 2.40633 | 1.27715 | 1.00000 | 4.35616 | −1.00000 | −1.30857 | 2.79044 | 1.81029 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.85511 | −2.66752 | 1.44142 | 1.00000 | −4.94854 | −1.00000 | −1.03623 | 4.11568 | 1.85511 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.09068 | −1.17179 | 2.37096 | 1.00000 | −2.44983 | −1.00000 | 0.775558 | −1.62692 | 2.09068 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.71142 | 2.09240 | 5.35177 | 1.00000 | 5.67336 | −1.00000 | 9.08805 | 1.37813 | 2.71142 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(1\) |
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 | 4235.2.a.bk | yes | 10 |
| 11.b | odd | 2 | 1 | 4235.2.a.bi | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4235.2.a.bi | ✓ | 10 | 11.b | odd | 2 | 1 | |
| 4235.2.a.bk | yes | 10 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):
|
\( T_{2}^{10} - 2T_{2}^{9} - 14T_{2}^{8} + 26T_{2}^{7} + 67T_{2}^{6} - 110T_{2}^{5} - 132T_{2}^{4} + 168T_{2}^{3} + 94T_{2}^{2} - 54T_{2} - 11 \)
|
|
\( T_{3}^{10} + 4 T_{3}^{9} - 14 T_{3}^{8} - 58 T_{3}^{7} + 73 T_{3}^{6} + 282 T_{3}^{5} - 179 T_{3}^{4} + \cdots - 111 \)
|
|
\( T_{13}^{10} - 18 T_{13}^{9} + 102 T_{13}^{8} - 60 T_{13}^{7} - 1342 T_{13}^{6} + 4330 T_{13}^{5} + \cdots + 2497 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 2 T^{9} + \cdots - 11 \)
|
| $3$ |
\( T^{10} + 4 T^{9} + \cdots - 111 \)
|
| $5$ |
\( (T - 1)^{10} \)
|
| $7$ |
\( (T + 1)^{10} \)
|
| $11$ |
\( T^{10} \)
|
| $13$ |
\( T^{10} - 18 T^{9} + \cdots + 2497 \)
|
| $17$ |
\( T^{10} - 4 T^{9} + \cdots - 120531 \)
|
| $19$ |
\( T^{10} - 14 T^{9} + \cdots - 362363 \)
|
| $23$ |
\( T^{10} + 4 T^{9} + \cdots - 9867 \)
|
| $29$ |
\( T^{10} - 36 T^{9} + \cdots + 50990181 \)
|
| $31$ |
\( T^{10} + 18 T^{9} + \cdots - 443223 \)
|
| $37$ |
\( T^{10} - 4 T^{9} + \cdots + 221353 \)
|
| $41$ |
\( T^{10} - 38 T^{9} + \cdots + 1406725 \)
|
| $43$ |
\( T^{10} - 6 T^{9} + \cdots + 80793693 \)
|
| $47$ |
\( T^{10} + 18 T^{9} + \cdots - 2805575 \)
|
| $53$ |
\( T^{10} + 26 T^{9} + \cdots - 4676351 \)
|
| $59$ |
\( T^{10} + 14 T^{9} + \cdots + 76565913 \)
|
| $61$ |
\( T^{10} - 60 T^{9} + \cdots - 8879159 \)
|
| $67$ |
\( T^{10} + 10 T^{9} + \cdots - 43621883 \)
|
| $71$ |
\( T^{10} - 318 T^{8} + \cdots + 9317697 \)
|
| $73$ |
\( T^{10} + \cdots - 2443752719 \)
|
| $79$ |
\( T^{10} + \cdots - 524411063 \)
|
| $83$ |
\( T^{10} - 2 T^{9} + \cdots - 206723 \)
|
| $89$ |
\( T^{10} - 2 T^{9} + \cdots + 3891537 \)
|
| $97$ |
\( T^{10} + \cdots - 20760073331 \)
|
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