Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2014,2,Mod(1,2014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2014, 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("2014.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2014 = 2 \cdot 19 \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2014.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: | \(16.0818709671\) |
| Analytic rank: | \(0\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 4 x^{10} - 17 x^{9} + 77 x^{8} + 82 x^{7} - 483 x^{6} - 95 x^{5} + 1166 x^{4} - 84 x^{3} + \cdots + 224 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 4 x^{10} - 17 x^{9} + 77 x^{8} + 82 x^{7} - 483 x^{6} - 95 x^{5} + 1166 x^{4} - 84 x^{3} + \cdots + 224 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 63 \nu^{10} - 686 \nu^{9} + 1979 \nu^{8} + 14943 \nu^{7} - 19140 \nu^{6} - 108027 \nu^{5} + \cdots - 8016 ) / 8704 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 43 \nu^{10} + 1466 \nu^{9} - 169 \nu^{8} - 30005 \nu^{7} + 7660 \nu^{6} + 201065 \nu^{5} + \cdots + 94832 ) / 8704 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 333 \nu^{10} - 726 \nu^{9} - 7041 \nu^{8} + 13651 \nu^{7} + 52364 \nu^{6} - 82111 \nu^{5} + \cdots - 109328 ) / 8704 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 743 \nu^{10} + 2306 \nu^{9} + 14083 \nu^{8} - 43129 \nu^{7} - 88228 \nu^{6} + 254141 \nu^{5} + \cdots + 124720 ) / 8704 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 13 \nu^{10} + 19 \nu^{9} + 281 \nu^{8} - 340 \nu^{7} - 2103 \nu^{6} + 1873 \nu^{5} + 6578 \nu^{4} + \cdots + 2672 ) / 136 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 27 \nu^{10} + 80 \nu^{9} + 547 \nu^{8} - 1547 \nu^{7} - 3914 \nu^{6} + 9657 \nu^{5} + 13169 \nu^{4} + \cdots + 9776 ) / 272 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 485 \nu^{10} + 1126 \nu^{9} + 9657 \nu^{8} - 20859 \nu^{7} - 64556 \nu^{6} + 121607 \nu^{5} + \cdots + 73616 ) / 4352 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 485 \nu^{10} - 1126 \nu^{9} - 9657 \nu^{8} + 20859 \nu^{7} + 64556 \nu^{6} - 121607 \nu^{5} + \cdots - 91024 ) / 4352 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 735 \nu^{10} + 1970 \nu^{9} + 14747 \nu^{8} - 37281 \nu^{7} - 101988 \nu^{6} + 225893 \nu^{5} + \cdots + 171952 ) / 4352 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} + \beta_{8} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{10} + 9\beta_{9} + 11\beta_{8} - \beta_{6} - \beta_{5} - 3\beta_{4} + \beta_{3} + \beta_{2} + 10\beta _1 + 28 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3 \beta_{10} - 2 \beta_{9} + 12 \beta_{8} - 3 \beta_{7} - 2 \beta_{6} - 16 \beta_{5} + 10 \beta_{4} + \cdots - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 14 \beta_{10} + 78 \beta_{9} + 111 \beta_{8} - 5 \beta_{7} - 13 \beta_{6} - 18 \beta_{5} - 46 \beta_{4} + \cdots + 229 \)
|
| \(\nu^{7}\) | \(=\) |
\( 48 \beta_{10} - 32 \beta_{9} + 126 \beta_{8} - 53 \beta_{7} - 31 \beta_{6} - 189 \beta_{5} + 89 \beta_{4} + \cdots - 21 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 151 \beta_{10} + 697 \beta_{9} + 1095 \beta_{8} - 102 \beta_{7} - 125 \beta_{6} - 235 \beta_{5} + \cdots + 2014 \)
|
| \(\nu^{9}\) | \(=\) |
\( 577 \beta_{10} - 377 \beta_{9} + 1281 \beta_{8} - 702 \beta_{7} - 345 \beta_{6} - 2025 \beta_{5} + \cdots - 256 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 1485 \beta_{10} + 6398 \beta_{9} + 10710 \beta_{8} - 1468 \beta_{7} - 1097 \beta_{6} - 2723 \beta_{5} + \cdots + 18486 \)
|
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 |
|
1.00000 | −3.08426 | 1.00000 | −1.55975 | −3.08426 | 4.31153 | 1.00000 | 6.51266 | −1.55975 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.27434 | 1.00000 | 2.04339 | −2.27434 | −2.32452 | 1.00000 | 2.17261 | 2.04339 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.72690 | 1.00000 | −3.74019 | −1.72690 | −3.61735 | 1.00000 | −0.0178120 | −3.74019 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.905698 | 1.00000 | −0.944154 | −0.905698 | −3.23289 | 1.00000 | −2.17971 | −0.944154 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −0.542887 | 1.00000 | 3.79688 | −0.542887 | 1.24864 | 1.00000 | −2.70527 | 3.79688 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 0.813818 | 1.00000 | −0.744419 | 0.813818 | 4.53613 | 1.00000 | −2.33770 | −0.744419 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 0.880691 | 1.00000 | 1.44069 | 0.880691 | 2.79112 | 1.00000 | −2.22438 | 1.44069 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 2.29803 | 1.00000 | 3.26670 | 2.29803 | 0.687689 | 1.00000 | 2.28095 | 3.26670 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 2.61734 | 1.00000 | −2.97566 | 2.61734 | −1.15105 | 1.00000 | 3.85049 | −2.97566 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.00000 | 2.73837 | 1.00000 | 1.10916 | 2.73837 | 0.340266 | 1.00000 | 4.49869 | 1.10916 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.00000 | 3.18583 | 1.00000 | 1.30734 | 3.18583 | 3.41044 | 1.00000 | 7.14949 | 1.30734 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(19\) | \(1\) |
| \(53\) | \(1\) |
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 | 2014.2.a.k | ✓ | 11 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2014.2.a.k | ✓ | 11 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2014))\):
|
\( T_{3}^{11} - 4 T_{3}^{10} - 17 T_{3}^{9} + 77 T_{3}^{8} + 82 T_{3}^{7} - 483 T_{3}^{6} - 95 T_{3}^{5} + \cdots + 224 \)
|
|
\( T_{7}^{11} - 7 T_{7}^{10} - 21 T_{7}^{9} + 214 T_{7}^{8} + 32 T_{7}^{7} - 2153 T_{7}^{6} + 1486 T_{7}^{5} + \cdots - 1702 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{11} \)
|
| $3$ |
\( T^{11} - 4 T^{10} + \cdots + 224 \)
|
| $5$ |
\( T^{11} - 3 T^{10} + \cdots + 646 \)
|
| $7$ |
\( T^{11} - 7 T^{10} + \cdots - 1702 \)
|
| $11$ |
\( T^{11} - 7 T^{10} + \cdots + 2736 \)
|
| $13$ |
\( T^{11} - 4 T^{10} + \cdots + 1604 \)
|
| $17$ |
\( T^{11} + 3 T^{10} + \cdots - 28373 \)
|
| $19$ |
\( (T + 1)^{11} \)
|
| $23$ |
\( T^{11} - 20 T^{10} + \cdots - 3541984 \)
|
| $29$ |
\( T^{11} - 22 T^{10} + \cdots - 70258 \)
|
| $31$ |
\( T^{11} - 10 T^{10} + \cdots + 27842483 \)
|
| $37$ |
\( T^{11} - 18 T^{10} + \cdots + 28731734 \)
|
| $41$ |
\( T^{11} - 12 T^{10} + \cdots - 1363444 \)
|
| $43$ |
\( T^{11} - 14 T^{10} + \cdots - 2446976 \)
|
| $47$ |
\( T^{11} + 6 T^{10} + \cdots + 2744 \)
|
| $53$ |
\( (T + 1)^{11} \)
|
| $59$ |
\( T^{11} + 11 T^{10} + \cdots - 17811257 \)
|
| $61$ |
\( T^{11} + 9 T^{10} + \cdots - 33017202 \)
|
| $67$ |
\( T^{11} - 25 T^{10} + \cdots - 42042496 \)
|
| $71$ |
\( T^{11} + 2 T^{10} + \cdots - 21103468 \)
|
| $73$ |
\( T^{11} + 7 T^{10} + \cdots + 58854272 \)
|
| $79$ |
\( T^{11} + \cdots - 5263906288 \)
|
| $83$ |
\( T^{11} + \cdots - 490126778 \)
|
| $89$ |
\( T^{11} + 8 T^{10} + \cdots + 5731632 \)
|
| $97$ |
\( T^{11} + 15 T^{10} + \cdots - 14511088 \)
|
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