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: | \(1\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 3x^{9} - 10x^{8} + 33x^{7} + 27x^{6} - 108x^{5} - 21x^{4} + 115x^{3} + 15x^{2} - 32x - 8 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 3x^{9} - 10x^{8} + 33x^{7} + 27x^{6} - 108x^{5} - 21x^{4} + 115x^{3} + 15x^{2} - 32x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{9} + \nu^{8} - 12\nu^{7} - 15\nu^{6} + 45\nu^{5} + 72\nu^{4} - 57\nu^{3} - 113\nu^{2} + 19\nu + 32 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{9} + \nu^{8} + 24\nu^{7} - 6\nu^{6} - 90\nu^{5} - 9\nu^{4} + 117\nu^{3} + 58\nu^{2} - 44\nu - 25 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5\nu^{9} - 7\nu^{8} - 54\nu^{7} + 69\nu^{6} + 165\nu^{5} - 180\nu^{4} - 129\nu^{3} + 113\nu^{2} - \nu - 8 ) / 6 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{9} - \nu^{8} + 15\nu^{7} + 12\nu^{6} - 75\nu^{5} - 45\nu^{4} + 138\nu^{3} + 59\nu^{2} - 70\nu - 26 ) / 3 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 5\nu^{9} - 13\nu^{8} - 48\nu^{7} + 135\nu^{6} + 111\nu^{5} - 390\nu^{4} - 15\nu^{3} + 305\nu^{2} - 43\nu - 44 ) / 6 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -\nu^{9} - \nu^{8} + 14\nu^{7} + 13\nu^{6} - 65\nu^{5} - 52\nu^{4} + 111\nu^{3} + 63\nu^{2} - 53\nu - 14 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} - \beta_{7} + \beta_{4} + 7\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + 7\beta_{3} - \beta_{2} + 27\beta _1 - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9\beta_{9} - 10\beta_{7} + \beta_{6} + 2\beta_{5} + 10\beta_{4} - \beta_{3} + 44\beta_{2} - 2\beta _1 + 73 \)
|
| \(\nu^{7}\) | \(=\) |
\( -10\beta_{9} + 10\beta_{8} - 9\beta_{6} + 12\beta_{5} + 13\beta_{4} + 42\beta_{3} - 11\beta_{2} + 150\beta _1 - 11 \)
|
| \(\nu^{8}\) | \(=\) |
\( 63\beta_{9} - 75\beta_{7} + 12\beta_{6} + 25\beta_{5} + 79\beta_{4} - 13\beta_{3} + 269\beta_{2} - 27\beta _1 + 401 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 75 \beta_{9} + 75 \beta_{8} - 3 \beta_{7} - 60 \beta_{6} + 104 \beta_{5} + 116 \beta_{4} + 244 \beta_{3} + \cdots - 94 \)
|
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 |
|
−2.47400 | −0.334323 | 4.12068 | −4.17391 | 0.827117 | −3.19720 | −5.24656 | −2.88823 | 10.3262 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.23001 | 2.74658 | 2.97295 | −0.788394 | −6.12491 | 0.297601 | −2.16969 | 4.54371 | 1.75813 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.17435 | −1.98019 | 2.72778 | 2.14453 | 4.30561 | −2.64934 | −1.58245 | 0.921144 | −4.66295 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.42669 | −1.66474 | 0.0354422 | 0.890041 | 2.37507 | 2.70731 | 2.80281 | −0.228641 | −1.26981 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.734167 | 1.94327 | −1.46100 | −1.17991 | −1.42668 | −3.07063 | 2.54095 | 0.776288 | 0.866248 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.304242 | −0.748590 | −1.90744 | −3.22734 | −0.227753 | 1.80333 | −1.18881 | −2.43961 | −0.981894 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.467372 | −3.16886 | −1.78156 | 1.07035 | −1.48104 | 4.47869 | −1.76740 | 7.04170 | 0.500252 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.01556 | 0.473201 | −0.968639 | 1.73777 | 0.480564 | −1.02472 | −3.01483 | −2.77608 | 1.76481 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.79296 | 2.17716 | 1.21469 | −4.05149 | 3.90355 | 2.54048 | −1.40803 | 1.74003 | −7.26414 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.45908 | 0.556495 | 4.04710 | −0.421658 | 1.36847 | −3.88552 | 5.03399 | −2.69031 | −1.03689 | |||||||||||||||||||||||||||||||||||||||||||||||||
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.g | 10 | |
| 13.b | even | 2 | 1 | 3211.2.a.h | 10 | ||
| 13.d | odd | 4 | 2 | 247.2.c.a | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 247.2.c.a | ✓ | 20 | 13.d | odd | 4 | 2 | |
| 3211.2.a.g | 10 | 1.a | even | 1 | 1 | trivial | |
| 3211.2.a.h | 10 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 3T_{2}^{9} - 10T_{2}^{8} - 33T_{2}^{7} + 27T_{2}^{6} + 108T_{2}^{5} - 21T_{2}^{4} - 115T_{2}^{3} + 15T_{2}^{2} + 32T_{2} - 8 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3211))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 3 T^{9} + \cdots - 8 \)
|
| $3$ |
\( T^{10} - 17 T^{8} + \cdots - 8 \)
|
| $5$ |
\( T^{10} + 8 T^{9} + \cdots + 76 \)
|
| $7$ |
\( T^{10} + 2 T^{9} + \cdots - 1712 \)
|
| $11$ |
\( T^{10} + 10 T^{9} + \cdots - 25468 \)
|
| $13$ |
\( T^{10} \)
|
| $17$ |
\( T^{10} - 64 T^{8} + \cdots + 13733 \)
|
| $19$ |
\( (T - 1)^{10} \)
|
| $23$ |
\( T^{10} + 4 T^{9} + \cdots - 2764 \)
|
| $29$ |
\( T^{10} - 6 T^{9} + \cdots + 11512 \)
|
| $31$ |
\( T^{10} + 8 T^{9} + \cdots + 2818 \)
|
| $37$ |
\( T^{10} - 4 T^{9} + \cdots + 887594 \)
|
| $41$ |
\( T^{10} + 28 T^{9} + \cdots - 45180 \)
|
| $43$ |
\( T^{10} - 12 T^{9} + \cdots - 409 \)
|
| $47$ |
\( T^{10} + 12 T^{9} + \cdots + 64052 \)
|
| $53$ |
\( T^{10} + 10 T^{9} + \cdots - 4170880 \)
|
| $59$ |
\( T^{10} + 16 T^{9} + \cdots + 359806 \)
|
| $61$ |
\( T^{10} + \cdots - 166286272 \)
|
| $67$ |
\( T^{10} + 24 T^{9} + \cdots + 11522234 \)
|
| $71$ |
\( T^{10} + 4 T^{9} + \cdots - 23351648 \)
|
| $73$ |
\( T^{10} + \cdots - 421249372 \)
|
| $79$ |
\( T^{10} + \cdots + 294923656 \)
|
| $83$ |
\( T^{10} + 34 T^{9} + \cdots + 1012592 \)
|
| $89$ |
\( T^{10} + \cdots - 329211448 \)
|
| $97$ |
\( T^{10} + 2 T^{9} + \cdots + 13338112 \)
|
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