Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2259,2,Mod(1,2259)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2259, 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("2259.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2259 = 3^{2} \cdot 251 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2259.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: | \(18.0382058166\) |
| 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: |
\( x^{11} - 3x^{10} - 13x^{9} + 42x^{8} + 49x^{7} - 187x^{6} - 46x^{5} + 289x^{4} + 5x^{3} - 135x^{2} - 14x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 753) |
| 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
\( x^{11} - 3x^{10} - 13x^{9} + 42x^{8} + 49x^{7} - 187x^{6} - 46x^{5} + 289x^{4} + 5x^{3} - 135x^{2} - 14x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 35 \nu^{10} + 56 \nu^{9} - 677 \nu^{8} - 903 \nu^{7} + 4450 \nu^{6} + 4769 \nu^{5} - 11239 \nu^{4} + \cdots - 4 ) / 218 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 27 \nu^{10} + 44 \nu^{9} + 379 \nu^{8} - 546 \nu^{7} - 1611 \nu^{6} + 1933 \nu^{5} + 1828 \nu^{4} + \cdots - 626 ) / 109 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 70 \nu^{10} + 106 \nu^{9} + 1027 \nu^{8} - 1355 \nu^{7} - 4867 \nu^{6} + 5177 \nu^{5} + 8199 \nu^{4} + \cdots + 335 ) / 109 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 175 \nu^{10} - 156 \nu^{9} - 2731 \nu^{8} + 1807 \nu^{7} + 14184 \nu^{6} - 5585 \nu^{5} - 27637 \nu^{4} + \cdots - 892 ) / 218 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 98 \nu^{10} - 83 \nu^{9} - 1525 \nu^{8} + 916 \nu^{7} + 7882 \nu^{6} - 2430 \nu^{5} - 15250 \nu^{4} + \cdots - 469 ) / 109 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 167 \nu^{10} - 256 \nu^{9} - 2433 \nu^{8} + 3256 \nu^{7} + 11345 \nu^{6} - 12178 \nu^{5} - 18335 \nu^{4} + \cdots - 371 ) / 109 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 333 \nu^{10} + 470 \nu^{9} + 4965 \nu^{8} - 5971 \nu^{7} - 24120 \nu^{6} + 22387 \nu^{5} + \cdots + 2598 ) / 218 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 177 \nu^{10} - 240 \nu^{9} - 2642 \nu^{8} + 2998 \nu^{7} + 12850 \nu^{6} - 10831 \nu^{5} - 22605 \nu^{4} + \cdots - 450 ) / 109 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{10} + \beta_{9} - \beta_{7} + \beta_{6} + 8\beta_{2} + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + 9 \beta_{6} + 10 \beta_{5} + \beta_{4} - 8 \beta_{3} + \cdots + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11 \beta_{10} + 12 \beta_{9} + \beta_{8} - 10 \beta_{7} + 10 \beta_{6} - \beta_{5} + 3 \beta_{4} + \cdots + 98 \)
|
| \(\nu^{7}\) | \(=\) |
\( 10 \beta_{10} + 13 \beta_{9} + 15 \beta_{8} - 15 \beta_{7} + 71 \beta_{6} + 79 \beta_{5} + 14 \beta_{4} + \cdots + 78 \)
|
| \(\nu^{8}\) | \(=\) |
\( 89 \beta_{10} + 111 \beta_{9} + 21 \beta_{8} - 81 \beta_{7} + 82 \beta_{6} - 16 \beta_{5} + 44 \beta_{4} + \cdots + 641 \)
|
| \(\nu^{9}\) | \(=\) |
\( 73 \beta_{10} + 131 \beta_{9} + 163 \beta_{8} - 152 \beta_{7} + 536 \beta_{6} + 576 \beta_{5} + \cdots + 658 \)
|
| \(\nu^{10}\) | \(=\) |
\( 649 \beta_{10} + 932 \beta_{9} + 269 \beta_{8} - 624 \beta_{7} + 645 \beta_{6} - 167 \beta_{5} + \cdots + 4361 \)
|
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 |
|
−2.58611 | 0 | 4.68798 | 1.25930 | 0 | 2.84296 | −6.95140 | 0 | −3.25670 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.20693 | 0 | 2.87052 | −1.99323 | 0 | 1.70614 | −1.92118 | 0 | 4.39892 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.28512 | 0 | −0.348478 | 4.01542 | 0 | 2.40078 | 3.01807 | 0 | −5.16028 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.637017 | 0 | −1.59421 | 0.973989 | 0 | −3.71761 | 2.28957 | 0 | −0.620448 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.360305 | 0 | −1.87018 | −2.80870 | 0 | 5.09207 | 1.39445 | 0 | 1.01199 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.204671 | 0 | −1.95811 | 3.25766 | 0 | 0.256166 | −0.810112 | 0 | 0.666750 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.05153 | 0 | −0.894277 | −3.78251 | 0 | 2.52264 | −3.04343 | 0 | −3.97743 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.80337 | 0 | 1.25215 | −1.60438 | 0 | −0.0400657 | −1.34865 | 0 | −2.89329 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.81978 | 0 | 1.31161 | 2.24591 | 0 | 4.40269 | −1.25271 | 0 | 4.08707 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.45111 | 0 | 4.00796 | 3.21500 | 0 | −1.09934 | 4.92173 | 0 | 7.88033 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.74500 | 0 | 5.53503 | −0.778475 | 0 | 4.63357 | 9.70368 | 0 | −2.13691 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(251\) | \( +1 \) |
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 | 2259.2.a.j | 11 | |
| 3.b | odd | 2 | 1 | 753.2.a.i | ✓ | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 753.2.a.i | ✓ | 11 | 3.b | odd | 2 | 1 | |
| 2259.2.a.j | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2259))\):
|
\( T_{2}^{11} - 3 T_{2}^{10} - 13 T_{2}^{9} + 42 T_{2}^{8} + 49 T_{2}^{7} - 187 T_{2}^{6} - 46 T_{2}^{5} + \cdots + 8 \)
|
|
\( T_{5}^{11} - 4 T_{5}^{10} - 29 T_{5}^{9} + 118 T_{5}^{8} + 277 T_{5}^{7} - 1158 T_{5}^{6} - 1066 T_{5}^{5} + \cdots + 3064 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 3 T^{10} + \cdots + 8 \)
|
| $3$ |
\( T^{11} \)
|
| $5$ |
\( T^{11} - 4 T^{10} + \cdots + 3064 \)
|
| $7$ |
\( T^{11} - 19 T^{10} + \cdots + 128 \)
|
| $11$ |
\( T^{11} - 5 T^{10} + \cdots + 256 \)
|
| $13$ |
\( T^{11} - 15 T^{10} + \cdots + 8824 \)
|
| $17$ |
\( T^{11} + 4 T^{10} + \cdots + 479200 \)
|
| $19$ |
\( T^{11} - 17 T^{10} + \cdots + 3232 \)
|
| $23$ |
\( T^{11} + 15 T^{10} + \cdots + 60944 \)
|
| $29$ |
\( T^{11} - 25 T^{10} + \cdots - 868408 \)
|
| $31$ |
\( T^{11} - 19 T^{10} + \cdots + 37811776 \)
|
| $37$ |
\( T^{11} - 16 T^{10} + \cdots - 410944 \)
|
| $41$ |
\( T^{11} - 6 T^{10} + \cdots + 19647968 \)
|
| $43$ |
\( T^{11} - 19 T^{10} + \cdots - 11628992 \)
|
| $47$ |
\( T^{11} + 23 T^{10} + \cdots - 8417792 \)
|
| $53$ |
\( T^{11} - 8 T^{10} + \cdots + 470384 \)
|
| $59$ |
\( T^{11} + \cdots + 130176640 \)
|
| $61$ |
\( T^{11} - 33 T^{10} + \cdots + 14257280 \)
|
| $67$ |
\( T^{11} - 14 T^{10} + \cdots + 77718656 \)
|
| $71$ |
\( T^{11} + T^{10} + \cdots + 7212544 \)
|
| $73$ |
\( T^{11} + \cdots - 152086174 \)
|
| $79$ |
\( T^{11} + \cdots - 97261754624 \)
|
| $83$ |
\( T^{11} + \cdots - 478971676 \)
|
| $89$ |
\( T^{11} + \cdots + 971206624 \)
|
| $97$ |
\( T^{11} + \cdots - 57304811264 \)
|
| show more | |
| show less |