Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [11,12,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, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 11 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| 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: | \(8.45177498616\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 8126x^{3} - 2800x^{2} + 12829304x - 12233152 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 8126x^{3} - 2800x^{2} + 12829304x - 12233152 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{4} - 26\nu^{3} - 27582\nu^{2} - 1424\nu + 9058736 ) / 36792 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 170\nu^{3} - 12174\nu^{2} - 856312\nu + 21799264 ) / 36792 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 49\nu^{3} + 6918\nu^{2} - 219854\nu - 6782434 ) / 9198 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{4} - 3\beta_{3} + 3\beta_{2} + 2\beta _1 + 3251 \)
|
| \(\nu^{3}\) | \(=\) |
\( 114\beta_{4} + 96\beta_{3} + 72\beta_{2} + 4962\beta _1 + 9454 \)
|
| \(\nu^{4}\) | \(=\) |
\( 17142\beta_{4} - 16050\beta_{3} + 24282\beta_{2} + 37120\beta _1 + 16171230 \)
|
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 |
|
−69.0436 | 647.024 | 2719.01 | −6754.48 | −44672.8 | 16893.2 | −46329.2 | 241493. | 466354. | |||||||||||||||||||||||||||||||||
| 1.2 | −42.0793 | −546.334 | −277.330 | −10764.7 | 22989.4 | −57275.4 | 97848.3 | 121334. | 452970. | ||||||||||||||||||||||||||||||||||
| 1.3 | 6.95428 | −537.992 | −1999.64 | 4785.14 | −3741.35 | 76669.9 | −28148.4 | 112289. | 33277.2 | ||||||||||||||||||||||||||||||||||
| 1.4 | 50.7205 | 620.128 | 524.564 | 6454.63 | 31453.2 | −16519.2 | −77269.4 | 207411. | 327382. | ||||||||||||||||||||||||||||||||||
| 1.5 | 85.4482 | −22.8250 | 5253.39 | −2118.63 | −1950.35 | 59271.3 | 273895. | −176626. | −181033. | ||||||||||||||||||||||||||||||||||
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 | 11.12.a.b | ✓ | 5 |
| 3.b | odd | 2 | 1 | 99.12.a.c | 5 | ||
| 4.b | odd | 2 | 1 | 176.12.a.h | 5 | ||
| 5.b | even | 2 | 1 | 275.12.a.b | 5 | ||
| 11.b | odd | 2 | 1 | 121.12.a.d | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.12.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 99.12.a.c | 5 | 3.b | odd | 2 | 1 | ||
| 121.12.a.d | 5 | 11.b | odd | 2 | 1 | ||
| 176.12.a.h | 5 | 4.b | odd | 2 | 1 | ||
| 275.12.a.b | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 32T_{2}^{4} - 7718T_{2}^{3} + 140876T_{2}^{2} + 11993504T_{2} - 87564928 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(11))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 32 T^{4} + \cdots - 87564928 \)
|
| $3$ |
\( T^{5} + \cdots + 2691820257660 \)
|
| $5$ |
\( T^{5} + \cdots + 47\!\cdots\!50 \)
|
| $7$ |
\( T^{5} + \cdots - 72\!\cdots\!60 \)
|
| $11$ |
\( (T + 161051)^{5} \)
|
| $13$ |
\( T^{5} + \cdots + 97\!\cdots\!00 \)
|
| $17$ |
\( T^{5} + \cdots - 12\!\cdots\!08 \)
|
| $19$ |
\( T^{5} + \cdots + 24\!\cdots\!96 \)
|
| $23$ |
\( T^{5} + \cdots + 38\!\cdots\!12 \)
|
| $29$ |
\( T^{5} + \cdots + 20\!\cdots\!16 \)
|
| $31$ |
\( T^{5} + \cdots - 27\!\cdots\!48 \)
|
| $37$ |
\( T^{5} + \cdots - 46\!\cdots\!34 \)
|
| $41$ |
\( T^{5} + \cdots - 24\!\cdots\!72 \)
|
| $43$ |
\( T^{5} + \cdots + 21\!\cdots\!52 \)
|
| $47$ |
\( T^{5} + \cdots + 10\!\cdots\!84 \)
|
| $53$ |
\( T^{5} + \cdots - 30\!\cdots\!16 \)
|
| $59$ |
\( T^{5} + \cdots - 17\!\cdots\!04 \)
|
| $61$ |
\( T^{5} + \cdots - 69\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots - 66\!\cdots\!68 \)
|
| $71$ |
\( T^{5} + \cdots - 23\!\cdots\!32 \)
|
| $73$ |
\( T^{5} + \cdots - 30\!\cdots\!28 \)
|
| $79$ |
\( T^{5} + \cdots + 16\!\cdots\!68 \)
|
| $83$ |
\( T^{5} + \cdots + 21\!\cdots\!12 \)
|
| $89$ |
\( T^{5} + \cdots - 68\!\cdots\!30 \)
|
| $97$ |
\( T^{5} + \cdots + 83\!\cdots\!50 \)
|
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