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: |
\( x^{10} - 2x^{9} - 14x^{8} + 26x^{7} + 63x^{6} - 106x^{5} - 96x^{4} + 140x^{3} + 38x^{2} - 38x - 11 \)
|
| 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
\( x^{10} - 2x^{9} - 14x^{8} + 26x^{7} + 63x^{6} - 106x^{5} - 96x^{4} + 140x^{3} + 38x^{2} - 38x - 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{9} + \nu^{8} + 17\nu^{7} - 9\nu^{6} - 95\nu^{5} + 6\nu^{4} + 188\nu^{3} + 67\nu^{2} - 97\nu - 44 ) / 11 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{9} + 7\nu^{8} + 23\nu^{7} - 92\nu^{6} - 65\nu^{5} + 381\nu^{4} - 33\nu^{3} - 511\nu^{2} + 157\nu + 121 ) / 11 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{9} + 7\nu^{8} + 57\nu^{7} - 87\nu^{6} - 261\nu^{5} + 329\nu^{4} + 390\nu^{3} - 372\nu^{2} - 85\nu + 44 ) / 11 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2\nu^{9} - 8\nu^{8} - 23\nu^{7} + 109\nu^{6} + 62\nu^{5} - 468\nu^{4} + 62\nu^{3} + 640\nu^{2} - 225\nu - 143 ) / 11 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -4\nu^{9} + 8\nu^{8} + 57\nu^{7} - 104\nu^{6} - 258\nu^{5} + 416\nu^{4} + 372\nu^{3} - 501\nu^{2} - 72\nu + 66 ) / 11 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 9 \nu^{9} - 20 \nu^{8} - 120 \nu^{7} + 257 \nu^{6} + 492 \nu^{5} - 1022 \nu^{4} - 570 \nu^{3} + \cdots - 198 ) / 11 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 12 \nu^{9} - 26 \nu^{8} - 160 \nu^{7} + 335 \nu^{6} + 647 \nu^{5} - 1334 \nu^{4} - 684 \nu^{3} + \cdots - 286 ) / 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} + 7\beta_{2} + \beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{9} + 2\beta_{8} + 9\beta_{7} + 9\beta_{6} - 8\beta_{5} + 10\beta_{4} + 2\beta_{2} + 29\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10\beta_{8} + 9\beta_{7} + 11\beta_{6} + 10\beta_{5} + 22\beta_{4} - 8\beta_{3} + 47\beta_{2} + 13\beta _1 + 96 \)
|
| \(\nu^{7}\) | \(=\) |
\( -10\beta_{9} + 24\beta_{8} + 69\beta_{7} + 66\beta_{6} - 52\beta_{5} + 80\beta_{4} + 25\beta_{2} + 178\beta _1 + 19 \)
|
| \(\nu^{8}\) | \(=\) |
\( 3 \beta_{9} + 77 \beta_{8} + 68 \beta_{7} + 91 \beta_{6} + 78 \beta_{5} + 188 \beta_{4} - 49 \beta_{3} + \cdots + 602 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 72 \beta_{9} + 211 \beta_{8} + 499 \beta_{7} + 453 \beta_{6} - 318 \beta_{5} + 600 \beta_{4} + \cdots + 219 \)
|
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 |
|
−2.46970 | −0.581915 | 4.09943 | −1.00000 | 1.43716 | 1.00000 | −5.18496 | −2.66137 | 2.46970 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.23949 | 1.46605 | 3.01534 | −1.00000 | −3.28322 | 1.00000 | −2.27384 | −0.850686 | 2.23949 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.40864 | 2.97205 | −0.0157228 | −1.00000 | −4.18655 | 1.00000 | 2.83944 | 5.83306 | 1.40864 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.416850 | 0.0737733 | −1.82624 | −1.00000 | −0.0307524 | 1.00000 | 1.59497 | −2.99456 | 0.416850 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.330750 | −2.01275 | −1.89060 | −1.00000 | 0.665719 | 1.00000 | 1.28682 | 1.05118 | 0.330750 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.799913 | 2.12674 | −1.36014 | −1.00000 | 1.70120 | 1.00000 | −2.68782 | 1.52301 | −0.799913 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.04772 | −2.73241 | −0.902293 | −1.00000 | −2.86278 | 1.00000 | −3.04078 | 4.46605 | −1.04772 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.78795 | 0.315097 | 1.19678 | −1.00000 | 0.563380 | 1.00000 | −1.43612 | −2.90071 | −1.78795 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.55273 | 2.87748 | 4.51641 | −1.00000 | 7.34543 | 1.00000 | 6.42372 | 5.27991 | −2.55273 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.67713 | −0.504113 | 5.16704 | −1.00000 | −1.34958 | 1.00000 | 8.47858 | −2.74587 | −2.67713 | |||||||||||||||||||||||||||||||||||||||||||||||||
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.bl | yes | 10 |
| 11.b | odd | 2 | 1 | 4235.2.a.bj | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4235.2.a.bj | ✓ | 10 | 11.b | odd | 2 | 1 | |
| 4235.2.a.bl | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):
|
\( T_{2}^{10} - 2T_{2}^{9} - 14T_{2}^{8} + 26T_{2}^{7} + 63T_{2}^{6} - 106T_{2}^{5} - 96T_{2}^{4} + 140T_{2}^{3} + 38T_{2}^{2} - 38T_{2} - 11 \)
|
|
\( T_{3}^{10} - 4T_{3}^{9} - 10T_{3}^{8} + 50T_{3}^{7} + 13T_{3}^{6} - 166T_{3}^{5} + 45T_{3}^{4} + 120T_{3}^{3} - 3T_{3}^{2} - 14T_{3} + 1 \)
|
|
\( T_{13}^{10} - 26 T_{13}^{9} + 246 T_{13}^{8} - 896 T_{13}^{7} - 206 T_{13}^{6} + 7894 T_{13}^{5} + \cdots + 14641 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 2 T^{9} + \cdots - 11 \)
|
| $3$ |
\( T^{10} - 4 T^{9} + \cdots + 1 \)
|
| $5$ |
\( (T + 1)^{10} \)
|
| $7$ |
\( (T - 1)^{10} \)
|
| $11$ |
\( T^{10} \)
|
| $13$ |
\( T^{10} - 26 T^{9} + \cdots + 14641 \)
|
| $17$ |
\( T^{10} - 4 T^{9} + \cdots - 138923 \)
|
| $19$ |
\( T^{10} - 6 T^{9} + \cdots - 78107 \)
|
| $23$ |
\( T^{10} + 8 T^{9} + \cdots + 18117 \)
|
| $29$ |
\( T^{10} + 4 T^{9} + \cdots - 4331 \)
|
| $31$ |
\( T^{10} - 18 T^{9} + \cdots + 2855497 \)
|
| $37$ |
\( T^{10} + 16 T^{9} + \cdots + 96009 \)
|
| $41$ |
\( T^{10} + 30 T^{9} + \cdots - 912483 \)
|
| $43$ |
\( T^{10} - 22 T^{9} + \cdots - 890747 \)
|
| $47$ |
\( T^{10} - 14 T^{9} + \cdots - 20631 \)
|
| $53$ |
\( T^{10} + 10 T^{9} + \cdots + 13631577 \)
|
| $59$ |
\( T^{10} - 22 T^{9} + \cdots - 2783 \)
|
| $61$ |
\( T^{10} - 12 T^{9} + \cdots - 3681183 \)
|
| $67$ |
\( T^{10} + 14 T^{9} + \cdots - 685883 \)
|
| $71$ |
\( T^{10} + 8 T^{9} + \cdots + 13539537 \)
|
| $73$ |
\( T^{10} - 30 T^{9} + \cdots - 53820239 \)
|
| $79$ |
\( T^{10} - 8 T^{9} + \cdots - 27573743 \)
|
| $83$ |
\( T^{10} + \cdots + 4234914213 \)
|
| $89$ |
\( T^{10} + \cdots + 721845289 \)
|
| $97$ |
\( T^{10} + \cdots + 261366237 \)
|
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