Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [11,18,Mod(1,11)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11.1");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 11 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 11.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: | \(20.1544296079\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 142182x^{4} - 2828860x^{3} + 4365765216x^{2} - 37243791360x - 26396402886656 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{6}\cdot 11 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 142182x^{4} - 2828860x^{3} + 4365765216x^{2} - 37243791360x - 26396402886656 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 241 \nu^{5} + 38570 \nu^{4} - 32713946 \nu^{3} - 5277529280 \nu^{2} + 630900419504 \nu + 60488231279104 ) / 1199297232 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 647 \nu^{5} - 4810 \nu^{4} + 77082838 \nu^{3} + 4055795416 \nu^{2} - 1289276569120 \nu - 45895022347232 ) / 399765744 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 515 \nu^{5} + 62674 \nu^{4} - 62047942 \nu^{3} - 9335159080 \nu^{2} + 809609895664 \nu + 101672949614816 ) / 399765744 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 137 \nu^{5} - 12052 \nu^{4} + 14666998 \nu^{3} + 2828346388 \nu^{2} - 112940916976 \nu - 58484954744384 ) / 199882872 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - 3\beta_{2} + 59\beta _1 + 189574 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{5} + 147\beta_{4} + 28\beta_{3} - 693\beta_{2} + 80555\beta _1 + 2828426 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 51971\beta_{5} + 67203\beta_{4} + 11144\beta_{3} - 163809\beta_{2} + 7689309\beta _1 + 7656006050 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( 1790956\beta_{5} + 10074056\beta_{4} + 1008644\beta_{3} - 45374172\beta_{2} + 3866145158\beta _1 + 366185741464 \)
|
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 |
|
−680.229 | 19468.5 | 331640. | −363455. | −1.32430e7 | −8.96484e6 | −1.36432e8 | 2.49882e8 | 2.47233e8 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −270.129 | 12021.5 | −58102.1 | 724855. | −3.24736e6 | −2.07151e7 | 5.11015e7 | 1.53762e7 | −1.95805e8 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −226.629 | −17761.4 | −79711.3 | 301021. | 4.02525e6 | 9.77210e6 | 4.77696e7 | 1.86327e8 | −6.82201e7 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 158.415 | 4404.21 | −105977. | 157655. | 697694. | 1.69997e7 | −3.75521e7 | −1.09743e8 | 2.49750e7 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 451.956 | 8582.51 | 73191.9 | −1.00486e6 | 3.87891e6 | −1.88093e7 | −2.61592e7 | −5.54807e7 | −4.54152e8 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 566.617 | −14850.3 | 189982. | 532773. | −8.41443e6 | −9.59720e6 | 3.33796e7 | 9.13916e7 | 3.01878e8 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(1\) |
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 | 11.18.a.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 99.18.a.a | 6 | ||
| 11.b | odd | 2 | 1 | 121.18.a.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.18.a.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 99.18.a.a | 6 | 3.b | odd | 2 | 1 | ||
| 121.18.a.c | 6 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 568728T_{2}^{4} + 22630880T_{2}^{3} + 69852243456T_{2}^{2} + 1191801323520T_{2} - 1689369784745984 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(11))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 16\!\cdots\!84 \)
|
| $3$ |
\( T^{6} + \cdots + 23\!\cdots\!00 \)
|
| $5$ |
\( T^{6} + \cdots + 66\!\cdots\!50 \)
|
| $7$ |
\( T^{6} + \cdots + 55\!\cdots\!00 \)
|
| $11$ |
\( (T + 214358881)^{6} \)
|
| $13$ |
\( T^{6} + \cdots + 74\!\cdots\!00 \)
|
| $17$ |
\( T^{6} + \cdots - 39\!\cdots\!64 \)
|
| $19$ |
\( T^{6} + \cdots - 12\!\cdots\!44 \)
|
| $23$ |
\( T^{6} + \cdots - 28\!\cdots\!44 \)
|
| $29$ |
\( T^{6} + \cdots + 31\!\cdots\!76 \)
|
| $31$ |
\( T^{6} + \cdots - 86\!\cdots\!04 \)
|
| $37$ |
\( T^{6} + \cdots + 56\!\cdots\!34 \)
|
| $41$ |
\( T^{6} + \cdots - 10\!\cdots\!36 \)
|
| $43$ |
\( T^{6} + \cdots - 14\!\cdots\!64 \)
|
| $47$ |
\( T^{6} + \cdots + 20\!\cdots\!36 \)
|
| $53$ |
\( T^{6} + \cdots + 23\!\cdots\!64 \)
|
| $59$ |
\( T^{6} + \cdots - 14\!\cdots\!36 \)
|
| $61$ |
\( T^{6} + \cdots - 32\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots + 30\!\cdots\!04 \)
|
| $71$ |
\( T^{6} + \cdots - 18\!\cdots\!84 \)
|
| $73$ |
\( T^{6} + \cdots + 71\!\cdots\!96 \)
|
| $79$ |
\( T^{6} + \cdots - 82\!\cdots\!36 \)
|
| $83$ |
\( T^{6} + \cdots + 46\!\cdots\!24 \)
|
| $89$ |
\( T^{6} + \cdots + 17\!\cdots\!50 \)
|
| $97$ |
\( T^{6} + \cdots + 12\!\cdots\!50 \)
|
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