Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3211,2,Mod(1,3211)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3211, 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("3211.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3211 = 13^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3211.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: | \(25.6399640890\) |
| 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: |
\( x^{11} - 2 x^{10} - 15 x^{9} + 26 x^{8} + 87 x^{7} - 121 x^{6} - 242 x^{5} + 236 x^{4} + 320 x^{3} + \cdots - 25 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 247) |
| 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
\( x^{11} - 2 x^{10} - 15 x^{9} + 26 x^{8} + 87 x^{7} - 121 x^{6} - 242 x^{5} + 236 x^{4} + 320 x^{3} + \cdots - 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 8\nu^{3} - 2\nu^{2} + 14\nu + 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{10} - 2\nu^{9} - 10\nu^{8} + 16\nu^{7} + 32\nu^{6} - 26\nu^{5} - 32\nu^{4} - 39\nu^{3} - 10\nu^{2} + 79\nu + 25 ) / 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 2 \nu^{10} + 4 \nu^{9} + 25 \nu^{8} - 37 \nu^{7} - 119 \nu^{6} + 92 \nu^{5} + 259 \nu^{4} - 12 \nu^{3} + \cdots - 5 ) / 5 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 2 \nu^{10} + 4 \nu^{9} + 25 \nu^{8} - 42 \nu^{7} - 114 \nu^{6} + 142 \nu^{5} + 224 \nu^{4} - 162 \nu^{3} + \cdots + 5 ) / 5 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 2 \nu^{10} + 4 \nu^{9} + 25 \nu^{8} - 42 \nu^{7} - 114 \nu^{6} + 142 \nu^{5} + 229 \nu^{4} + \cdots + 35 ) / 5 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{10} - 2 \nu^{9} - 15 \nu^{8} + 26 \nu^{7} + 87 \nu^{6} - 121 \nu^{5} - 237 \nu^{4} + 236 \nu^{3} + \cdots - 90 ) / 5 \)
|
| \(\beta_{10}\) | \(=\) |
\( \nu^{9} - 2\nu^{8} - 12\nu^{7} + 20\nu^{6} + 52\nu^{5} - 61\nu^{4} - 97\nu^{3} + 51\nu^{2} + 66\nu + 14 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} - \beta_{7} + \beta_{3} + 7\beta_{2} + \beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{4} + 8\beta_{3} + 10\beta_{2} + 18\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{9} + 9\beta_{8} - 8\beta_{7} + \beta_{5} + \beta_{4} + 10\beta_{3} + 44\beta_{2} + 11\beta _1 + 65 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{9} + 2\beta_{8} - 2\beta_{7} + \beta_{6} + \beta_{5} + 11\beta_{4} + 53\beta_{3} + 76\beta_{2} + 90\beta _1 + 69 \)
|
| \(\nu^{8}\) | \(=\) |
\( 12 \beta_{9} + 62 \beta_{8} - 51 \beta_{7} + 2 \beta_{6} + 14 \beta_{5} + 14 \beta_{4} + 78 \beta_{3} + \cdots + 355 \)
|
| \(\nu^{9}\) | \(=\) |
\( \beta_{10} + 16 \beta_{9} + 29 \beta_{8} - 27 \beta_{7} + 16 \beta_{6} + 20 \beta_{5} + 88 \beta_{4} + \cdots + 493 \)
|
| \(\nu^{10}\) | \(=\) |
\( 2 \beta_{10} + 104 \beta_{9} + 390 \beta_{8} - 308 \beta_{7} + 36 \beta_{6} + 137 \beta_{5} + \cdots + 2046 \)
|
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 |
|
−2.18269 | −1.54276 | 2.76413 | −1.58940 | 3.36737 | −2.06904 | −1.66785 | −0.619884 | 3.46916 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.83652 | 3.00179 | 1.37279 | 2.41636 | −5.51283 | 3.95523 | 1.15188 | 6.01073 | −4.43768 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.76782 | −0.159541 | 1.12519 | 0.550251 | 0.282041 | −0.0619072 | 1.54650 | −2.97455 | −0.972745 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.39632 | −2.66850 | −0.0502913 | 4.01572 | 3.72608 | −3.87097 | 2.86286 | 4.12088 | −5.60722 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.452472 | 1.96623 | −1.79527 | −2.35676 | −0.889666 | 1.84702 | 1.71725 | 0.866070 | 1.06637 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.217574 | −1.53678 | −1.95266 | −0.0541001 | 0.334364 | 1.79354 | 0.859996 | −0.638299 | 0.0117708 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.37910 | 0.582785 | −0.0980881 | −0.697466 | 0.803717 | −3.74407 | −2.89347 | −2.66036 | −0.961874 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.57958 | 1.48605 | 0.495064 | 3.41059 | 2.34733 | 4.37592 | −2.37716 | −0.791660 | 5.38730 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.85853 | −2.38357 | 1.45413 | −0.371051 | −4.42993 | 0.0566105 | −1.01452 | 2.68139 | −0.689608 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.47675 | 2.88757 | 4.13428 | 2.81800 | 7.15180 | −3.96536 | 5.28609 | 5.33809 | 6.97948 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.55944 | −1.63328 | 4.55072 | 1.85785 | −4.18027 | 3.68303 | 6.52842 | −0.332407 | 4.75506 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(1\) |
| \(19\) | \(-1\) |
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 | 3211.2.a.k | 11 | |
| 13.b | even | 2 | 1 | 3211.2.a.j | 11 | ||
| 13.e | even | 6 | 2 | 247.2.g.b | ✓ | 22 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 247.2.g.b | ✓ | 22 | 13.e | even | 6 | 2 | |
| 3211.2.a.j | 11 | 13.b | even | 2 | 1 | ||
| 3211.2.a.k | 11 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} - 2 T_{2}^{10} - 15 T_{2}^{9} + 26 T_{2}^{8} + 87 T_{2}^{7} - 121 T_{2}^{6} - 242 T_{2}^{5} + \cdots - 25 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3211))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 2 T^{10} + \cdots - 25 \)
|
| $3$ |
\( T^{11} - 22 T^{9} + \cdots - 58 \)
|
| $5$ |
\( T^{11} - 10 T^{10} + \cdots + 5 \)
|
| $7$ |
\( T^{11} - 2 T^{10} + \cdots + 88 \)
|
| $11$ |
\( T^{11} - 7 T^{10} + \cdots - 37714 \)
|
| $13$ |
\( T^{11} \)
|
| $17$ |
\( T^{11} + T^{10} + \cdots - 42203 \)
|
| $19$ |
\( (T - 1)^{11} \)
|
| $23$ |
\( T^{11} - 7 T^{10} + \cdots + 1619596 \)
|
| $29$ |
\( T^{11} + 12 T^{10} + \cdots + 11579 \)
|
| $31$ |
\( T^{11} - 13 T^{10} + \cdots - 20788504 \)
|
| $37$ |
\( T^{11} + \cdots - 292064923 \)
|
| $41$ |
\( T^{11} + \cdots - 1180912023 \)
|
| $43$ |
\( T^{11} - 20 T^{10} + \cdots + 4162496 \)
|
| $47$ |
\( T^{11} + \cdots - 125055362 \)
|
| $53$ |
\( T^{11} + 10 T^{10} + \cdots - 476303 \)
|
| $59$ |
\( T^{11} - 24 T^{10} + \cdots + 1348326 \)
|
| $61$ |
\( T^{11} - 289 T^{9} + \cdots - 14124875 \)
|
| $67$ |
\( T^{11} + 17 T^{10} + \cdots - 96249430 \)
|
| $71$ |
\( T^{11} + \cdots + 872463250 \)
|
| $73$ |
\( T^{11} + \cdots + 200069375 \)
|
| $79$ |
\( T^{11} - 12 T^{10} + \cdots + 2097370 \)
|
| $83$ |
\( T^{11} + 14 T^{10} + \cdots - 25465034 \)
|
| $89$ |
\( T^{11} - 7 T^{10} + \cdots - 10598 \)
|
| $97$ |
\( T^{11} + 14 T^{10} + \cdots - 18135050 \)
|
| show more | |
| show less |