Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1911,4,Mod(1,1911)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1911, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1911.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1911.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: | \(112.752650021\) |
| 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: |
\( x^{11} - x^{10} - 75 x^{9} + 59 x^{8} + 1943 x^{7} - 1079 x^{6} - 20355 x^{5} + 7943 x^{4} + 75790 x^{3} + \cdots - 9072 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{11} - x^{10} - 75 x^{9} + 59 x^{8} + 1943 x^{7} - 1079 x^{6} - 20355 x^{5} + 7943 x^{4} + 75790 x^{3} + \cdots - 9072 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 23\nu - 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 81337 \nu^{10} - 60662 \nu^{9} + 6066649 \nu^{8} + 5635052 \nu^{7} - 152618091 \nu^{6} + \cdots - 77930576 ) / 4860608 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 255557 \nu^{10} + 226318 \nu^{9} - 18859557 \nu^{8} - 20288876 \nu^{7} + 466936399 \nu^{6} + \cdots + 1697676624 ) / 14581824 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 470329 \nu^{10} - 262886 \nu^{9} + 34707129 \nu^{8} + 26227132 \nu^{7} - 861554123 \nu^{6} + \cdots - 839946384 ) / 14581824 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 239243 \nu^{10} - 150418 \nu^{9} + 17654763 \nu^{8} + 14548628 \nu^{7} - 438008929 \nu^{6} + \cdots - 900073968 ) / 7290912 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 503785 \nu^{10} + 374294 \nu^{9} - 37309257 \nu^{8} - 34798636 \nu^{7} + 928629371 \nu^{6} + \cdots + 3226406736 ) / 14581824 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 993817 \nu^{10} + 719462 \nu^{9} - 73278393 \nu^{8} - 67279612 \nu^{7} + 1813729355 \nu^{6} + \cdots + 5032257168 ) / 14581824 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 1058419 \nu^{10} + 764386 \nu^{9} - 77998323 \nu^{8} - 72329780 \nu^{7} + 1927703961 \nu^{6} + \cdots + 6013661104 ) / 4860608 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 23\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + 31\beta_{2} + 3\beta _1 + 332 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{9} + \beta_{8} + 4\beta_{7} - 4\beta_{6} + 3\beta_{5} + \beta_{4} + 39\beta_{3} - 3\beta_{2} + 608\beta _1 + 90 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 60 \beta_{9} + 41 \beta_{8} - 55 \beta_{7} - 10 \beta_{6} + 76 \beta_{5} + 47 \beta_{4} - 44 \beta_{3} + \cdots + 8924 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 6 \beta_{10} - 6 \beta_{9} + 57 \beta_{8} + 231 \beta_{7} - 196 \beta_{6} + 106 \beta_{5} + \cdots + 2440 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 2463 \beta_{9} + 1328 \beta_{8} - 2262 \beta_{7} - 536 \beta_{6} + 3421 \beta_{5} + 1762 \beta_{4} + \cdots + 252746 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 434 \beta_{10} + 1209 \beta_{9} + 2192 \beta_{8} + 9685 \beta_{7} - 7240 \beta_{6} + 2945 \beta_{5} + \cdots + 48428 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 92 \beta_{10} - 88124 \beta_{9} + 40116 \beta_{8} - 82756 \beta_{7} - 21122 \beta_{6} + \cdots + 7351418 \)
|
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 |
|
−5.61453 | −3.00000 | 23.5229 | 9.01096 | 16.8436 | 0 | −87.1538 | 9.00000 | −50.5923 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.58807 | −3.00000 | 13.0504 | −3.67322 | 13.7642 | 0 | −23.1714 | 9.00000 | 16.8530 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.26282 | −3.00000 | 2.64599 | −8.19795 | 9.78846 | 0 | 17.4692 | 9.00000 | 26.7484 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.87392 | −3.00000 | 0.259427 | 17.0146 | 8.62177 | 0 | 22.2458 | 9.00000 | −48.8986 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.599393 | −3.00000 | −7.64073 | −21.8962 | 1.79818 | 0 | 9.37494 | 9.00000 | 13.1244 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.185628 | −3.00000 | −7.96554 | 10.0705 | 0.556883 | 0 | 2.96365 | 9.00000 | −1.86936 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.72226 | −3.00000 | −5.03380 | 14.2155 | −5.16679 | 0 | −22.4477 | 9.00000 | 24.4828 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.72931 | −3.00000 | −5.00948 | −3.74940 | −5.18793 | 0 | −22.4974 | 9.00000 | −6.48387 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 3.92588 | −3.00000 | 7.41251 | −0.183968 | −11.7776 | 0 | −2.30643 | 9.00000 | −0.722237 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 5.30132 | −3.00000 | 20.1040 | −13.9345 | −15.9040 | 0 | 64.1674 | 9.00000 | −73.8714 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 5.44558 | −3.00000 | 21.6543 | 19.3238 | −16.3367 | 0 | 74.3558 | 9.00000 | 105.229 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 1911.4.a.x | ✓ | 11 |
| 7.b | odd | 2 | 1 | 1911.4.a.y | yes | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1911.4.a.x | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
| 1911.4.a.y | yes | 11 | 7.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1911))\):
|
\( T_{2}^{11} - T_{2}^{10} - 75 T_{2}^{9} + 59 T_{2}^{8} + 1943 T_{2}^{7} - 1079 T_{2}^{6} - 20355 T_{2}^{5} + \cdots - 9072 \)
|
|
\( T_{5}^{11} - 18 T_{5}^{10} - 746 T_{5}^{9} + 14402 T_{5}^{8} + 148615 T_{5}^{7} - 3192700 T_{5}^{6} + \cdots - 2687897200 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - T^{10} + \cdots - 9072 \)
|
| $3$ |
\( (T + 3)^{11} \)
|
| $5$ |
\( T^{11} + \cdots - 2687897200 \)
|
| $7$ |
\( T^{11} \)
|
| $11$ |
\( T^{11} + \cdots - 487032853531520 \)
|
| $13$ |
\( (T - 13)^{11} \)
|
| $17$ |
\( T^{11} + \cdots - 47\!\cdots\!00 \)
|
| $19$ |
\( T^{11} + \cdots - 20\!\cdots\!12 \)
|
| $23$ |
\( T^{11} + \cdots - 18\!\cdots\!92 \)
|
| $29$ |
\( T^{11} + \cdots - 58\!\cdots\!56 \)
|
| $31$ |
\( T^{11} + \cdots - 12\!\cdots\!24 \)
|
| $37$ |
\( T^{11} + \cdots - 35\!\cdots\!00 \)
|
| $41$ |
\( T^{11} + \cdots + 23\!\cdots\!32 \)
|
| $43$ |
\( T^{11} + \cdots - 21\!\cdots\!64 \)
|
| $47$ |
\( T^{11} + \cdots + 32\!\cdots\!80 \)
|
| $53$ |
\( T^{11} + \cdots + 18\!\cdots\!04 \)
|
| $59$ |
\( T^{11} + \cdots - 24\!\cdots\!88 \)
|
| $61$ |
\( T^{11} + \cdots - 22\!\cdots\!96 \)
|
| $67$ |
\( T^{11} + \cdots + 65\!\cdots\!00 \)
|
| $71$ |
\( T^{11} + \cdots - 89\!\cdots\!04 \)
|
| $73$ |
\( T^{11} + \cdots - 47\!\cdots\!00 \)
|
| $79$ |
\( T^{11} + \cdots - 22\!\cdots\!92 \)
|
| $83$ |
\( T^{11} + \cdots + 10\!\cdots\!36 \)
|
| $89$ |
\( T^{11} + \cdots + 78\!\cdots\!84 \)
|
| $97$ |
\( T^{11} + \cdots + 85\!\cdots\!60 \)
|
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