Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2001,2,Mod(1,2001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, 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("2001.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2001 = 3 \cdot 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2001.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: | \(15.9780654445\) |
| 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: |
\( x^{11} - 2 x^{10} - 18 x^{9} + 30 x^{8} + 124 x^{7} - 152 x^{6} - 408 x^{5} + 285 x^{4} + 634 x^{3} + \cdots - 108 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{11} - 2 x^{10} - 18 x^{9} + 30 x^{8} + 124 x^{7} - 152 x^{6} - 408 x^{5} + 285 x^{4} + 634 x^{3} + \cdots - 108 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - \nu^{10} + 5 \nu^{9} + 9 \nu^{8} - 63 \nu^{7} - 13 \nu^{6} + 251 \nu^{5} - 27 \nu^{4} - 360 \nu^{3} + \cdots + 48 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 2 \nu^{10} + 7 \nu^{9} + 21 \nu^{8} - 87 \nu^{7} - 59 \nu^{6} + 343 \nu^{5} + 63 \nu^{4} - 498 \nu^{3} + \cdots + 153 ) / 9 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - \nu^{10} - \nu^{9} + 24 \nu^{8} + 6 \nu^{7} - 187 \nu^{6} + 23 \nu^{5} + 567 \nu^{4} - 150 \nu^{3} + \cdots + 162 ) / 9 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{8} - 2\nu^{7} - 12\nu^{6} + 22\nu^{5} + 44\nu^{4} - 68\nu^{3} - 57\nu^{2} + 52\nu + 31 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 4 \nu^{10} + 14 \nu^{9} + 42 \nu^{8} - 174 \nu^{7} - 109 \nu^{6} + 686 \nu^{5} + 18 \nu^{4} + \cdots + 135 ) / 9 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 2 \nu^{10} - 7 \nu^{9} - 30 \nu^{8} + 105 \nu^{7} + 167 \nu^{6} - 541 \nu^{5} - 450 \nu^{4} + 1101 \nu^{3} + \cdots - 369 ) / 9 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - \nu^{10} + 5 \nu^{9} + 15 \nu^{8} - 75 \nu^{7} - 91 \nu^{6} + 389 \nu^{5} + 303 \nu^{4} - 822 \nu^{3} + \cdots + 354 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} + \beta_{7} + \beta_{5} + \beta_{3} + 7\beta_{2} + \beta _1 + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} - \beta_{4} + 10\beta_{3} + 2\beta_{2} + 29\beta _1 + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12\beta_{9} + \beta_{8} + 12\beta_{7} + 10\beta_{5} + 12\beta_{3} + 47\beta_{2} + 14\beta _1 + 135 \)
|
| \(\nu^{7}\) | \(=\) |
\( 12 \beta_{10} + 16 \beta_{9} + 12 \beta_{8} + 3 \beta_{7} + \beta_{6} - 10 \beta_{5} - 10 \beta_{4} + \cdots + 89 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2 \beta_{10} + 110 \beta_{9} + 14 \beta_{8} + 107 \beta_{7} + 2 \beta_{6} + 78 \beta_{5} + 2 \beta_{4} + \cdots + 875 \)
|
| \(\nu^{9}\) | \(=\) |
\( 105 \beta_{10} + 172 \beta_{9} + 106 \beta_{8} + 53 \beta_{7} + 15 \beta_{6} - 73 \beta_{5} - 71 \beta_{4} + \cdots + 752 \)
|
| \(\nu^{10}\) | \(=\) |
\( 38 \beta_{10} + 910 \beta_{9} + 138 \beta_{8} + 856 \beta_{7} + 30 \beta_{6} + 559 \beta_{5} + \cdots + 5861 \)
|
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 |
|
−2.51124 | −1.00000 | 4.30634 | 2.07440 | 2.51124 | −0.329384 | −5.79178 | 1.00000 | −5.20933 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.96728 | −1.00000 | 1.87020 | −3.90206 | 1.96728 | −0.839519 | 0.255353 | 1.00000 | 7.67646 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.94502 | −1.00000 | 1.78310 | 0.890641 | 1.94502 | −3.69089 | 0.421884 | 1.00000 | −1.73231 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.05971 | −1.00000 | −0.877023 | −1.30384 | 1.05971 | 0.720797 | 3.04880 | 1.00000 | 1.38169 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.661934 | −1.00000 | −1.56184 | −1.16115 | 0.661934 | 4.80000 | 2.35770 | 1.00000 | 0.768607 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.467085 | −1.00000 | −1.78183 | −0.0105419 | 0.467085 | −1.85912 | 1.76644 | 1.00000 | 0.00492395 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.17662 | −1.00000 | −0.615555 | 2.85352 | −1.17662 | 3.62966 | −3.07753 | 1.00000 | 3.35752 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.76484 | −1.00000 | 1.11465 | −2.54815 | −1.76484 | −4.97592 | −1.56250 | 1.00000 | −4.49707 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.24285 | −1.00000 | 3.03039 | 3.33222 | −2.24285 | 0.336371 | 2.31101 | 1.00000 | 7.47368 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.70316 | −1.00000 | 5.30708 | 2.79988 | −2.70316 | 0.880738 | 8.93955 | 1.00000 | 7.56852 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.72479 | −1.00000 | 5.42450 | −1.02492 | −2.72479 | 4.32726 | 9.33107 | 1.00000 | −2.79269 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(23\) | \(-1\) |
| \(29\) | \(1\) |
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 | 2001.2.a.l | ✓ | 11 |
| 3.b | odd | 2 | 1 | 6003.2.a.m | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2001.2.a.l | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
| 6003.2.a.m | 11 | 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2001))\):
|
\( T_{2}^{11} - 2 T_{2}^{10} - 18 T_{2}^{9} + 30 T_{2}^{8} + 124 T_{2}^{7} - 152 T_{2}^{6} - 408 T_{2}^{5} + \cdots - 108 \)
|
|
\( T_{5}^{11} - 2 T_{5}^{10} - 27 T_{5}^{9} + 53 T_{5}^{8} + 233 T_{5}^{7} - 387 T_{5}^{6} - 862 T_{5}^{5} + \cdots - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 2 T^{10} + \cdots - 108 \)
|
| $3$ |
\( (T + 1)^{11} \)
|
| $5$ |
\( T^{11} - 2 T^{10} + \cdots - 8 \)
|
| $7$ |
\( T^{11} - 3 T^{10} + \cdots + 152 \)
|
| $11$ |
\( T^{11} - 11 T^{10} + \cdots - 5372 \)
|
| $13$ |
\( T^{11} + 5 T^{10} + \cdots - 5683 \)
|
| $17$ |
\( T^{11} - 15 T^{10} + \cdots - 14592 \)
|
| $19$ |
\( T^{11} + 6 T^{10} + \cdots - 49408 \)
|
| $23$ |
\( (T - 1)^{11} \)
|
| $29$ |
\( (T + 1)^{11} \)
|
| $31$ |
\( T^{11} - 35 T^{10} + \cdots + 756944 \)
|
| $37$ |
\( T^{11} + 28 T^{10} + \cdots + 759412 \)
|
| $41$ |
\( T^{11} - 10 T^{10} + \cdots - 5506272 \)
|
| $43$ |
\( T^{11} + 6 T^{10} + \cdots + 7091488 \)
|
| $47$ |
\( T^{11} + \cdots - 958100256 \)
|
| $53$ |
\( T^{11} + 7 T^{10} + \cdots - 80314464 \)
|
| $59$ |
\( T^{11} + 20 T^{10} + \cdots + 88926336 \)
|
| $61$ |
\( T^{11} + 20 T^{10} + \cdots - 40052104 \)
|
| $67$ |
\( T^{11} + 39 T^{10} + \cdots - 26096464 \)
|
| $71$ |
\( T^{11} - 49 T^{10} + \cdots + 66188592 \)
|
| $73$ |
\( T^{11} + \cdots - 42266597504 \)
|
| $79$ |
\( T^{11} + \cdots - 360776992 \)
|
| $83$ |
\( T^{11} - 13 T^{10} + \cdots - 15116544 \)
|
| $89$ |
\( T^{11} - 34 T^{10} + \cdots + 18922248 \)
|
| $97$ |
\( T^{11} + 11 T^{10} + \cdots - 580896 \)
|
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