Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3751,2,Mod(1,3751)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3751, 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("3751.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3751 = 11^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3751.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: | \(29.9518857982\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - x^{8} - 11x^{7} + 10x^{6} + 39x^{5} - 33x^{4} - 49x^{3} + 37x^{2} + 12x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - x^{8} - 11x^{7} + 10x^{6} + 39x^{5} - 33x^{4} - 49x^{3} + 37x^{2} + 12x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{8} + \nu^{7} - 9\nu^{6} - 8\nu^{5} + 19\nu^{4} + 9\nu^{3} - 3\nu^{2} + 11\nu - 6 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{8} - \nu^{7} - 27\nu^{6} + 4\nu^{5} + 65\nu^{4} + 3\nu^{3} - 37\nu^{2} - 19\nu - 2 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{8} - \nu^{7} - 9\nu^{6} + 8\nu^{5} + 23\nu^{4} - 19\nu^{3} - 19\nu^{2} + 11\nu + 6 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{8} + \nu^{7} - 11\nu^{6} - 10\nu^{5} + 37\nu^{4} + 25\nu^{3} - 41\nu^{2} - 13\nu + 4 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{8} + \nu^{7} + 31\nu^{6} - 8\nu^{5} - 97\nu^{4} + 25\nu^{3} + 97\nu^{2} - 29\nu - 10 ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 7\nu^{8} + 3\nu^{7} - 71\nu^{6} - 32\nu^{5} + 213\nu^{4} + 75\nu^{3} - 205\nu^{2} - 31\nu + 22 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - 2\beta_{6} - \beta_{5} - \beta_{3} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 6\beta_{2} + \beta _1 + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{8} - 16\beta_{6} - 7\beta_{5} - \beta_{4} - 7\beta_{3} + \beta_{2} + 11\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9\beta_{8} + 9\beta_{7} - 10\beta_{6} + 8\beta_{5} - 8\beta_{4} - 8\beta_{3} + 34\beta_{2} + 10\beta _1 + 55 \)
|
| \(\nu^{7}\) | \(=\) |
\( 52\beta_{8} + 2\beta_{7} - 102\beta_{6} - 41\beta_{5} - 10\beta_{4} - 42\beta_{3} + 12\beta_{2} + 48\beta _1 + 14 \)
|
| \(\nu^{8}\) | \(=\) |
\( 65\beta_{8} + 60\beta_{7} - 79\beta_{6} + 47\beta_{5} - 51\beta_{4} - 54\beta_{3} + 191\beta_{2} + 73\beta _1 + 276 \)
|
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 |
|
−2.45579 | −1.84654 | 4.03088 | 1.80050 | 4.53471 | 3.16816 | −4.98741 | 0.409722 | −4.42164 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.65560 | 0.578647 | 0.741010 | −0.297920 | −0.958007 | −3.18823 | 2.08438 | −2.66517 | 0.493236 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.54477 | 2.45385 | 0.386326 | −1.69106 | −3.79065 | 1.44886 | 2.49276 | 3.02139 | 2.61230 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.07686 | −2.35826 | −0.840377 | 1.33817 | 2.53951 | −1.16104 | 3.05868 | 2.56137 | −1.44102 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.227281 | −0.343934 | −1.94834 | −3.65931 | 0.0781697 | −2.11746 | 0.897383 | −2.88171 | 0.831690 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.431056 | 0.668658 | −1.81419 | 2.44004 | 0.288229 | −0.513254 | −1.64413 | −2.55290 | 1.05179 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.53410 | −2.80606 | 0.353466 | −3.71138 | −4.30478 | 0.756534 | −2.52595 | 4.87399 | −5.69363 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.76236 | 0.168782 | 1.10591 | −2.09917 | 0.297454 | 3.22613 | −1.57570 | −2.97151 | −3.69949 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.23278 | 1.48486 | 2.98531 | −2.11988 | 3.31536 | −1.61970 | 2.19999 | −0.795194 | −4.73323 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(-1\) |
| \(31\) | \(-1\) |
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 | 3751.2.a.i | ✓ | 9 |
| 11.b | odd | 2 | 1 | 3751.2.a.j | yes | 9 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3751.2.a.i | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 3751.2.a.j | yes | 9 | 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3751))\):
|
\( T_{2}^{9} + T_{2}^{8} - 11T_{2}^{7} - 10T_{2}^{6} + 39T_{2}^{5} + 33T_{2}^{4} - 49T_{2}^{3} - 37T_{2}^{2} + 12T_{2} + 4 \)
|
|
\( T_{3}^{9} + 2T_{3}^{8} - 11T_{3}^{7} - 17T_{3}^{6} + 38T_{3}^{5} + 29T_{3}^{4} - 50T_{3}^{3} + 7T_{3}^{2} + 6T_{3} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + T^{8} - 11 T^{7} + \cdots + 4 \)
|
| $3$ |
\( T^{9} + 2 T^{8} + \cdots - 1 \)
|
| $5$ |
\( T^{9} + 8 T^{8} + \cdots - 179 \)
|
| $7$ |
\( T^{9} - 21 T^{7} + \cdots + 73 \)
|
| $11$ |
\( T^{9} \)
|
| $13$ |
\( T^{9} - 4 T^{8} + \cdots - 43 \)
|
| $17$ |
\( T^{9} - 8 T^{8} + \cdots + 5939 \)
|
| $19$ |
\( T^{9} - 7 T^{8} + \cdots - 9983 \)
|
| $23$ |
\( T^{9} + 7 T^{8} + \cdots + 12347 \)
|
| $29$ |
\( T^{9} + 7 T^{8} + \cdots + 11889 \)
|
| $31$ |
\( (T - 1)^{9} \)
|
| $37$ |
\( T^{9} + 7 T^{8} + \cdots + 240181 \)
|
| $41$ |
\( T^{9} - 15 T^{8} + \cdots - 70916 \)
|
| $43$ |
\( T^{9} + 16 T^{8} + \cdots + 158933 \)
|
| $47$ |
\( T^{9} - 207 T^{7} + \cdots - 691777 \)
|
| $53$ |
\( T^{9} + 16 T^{8} + \cdots - 7546543 \)
|
| $59$ |
\( T^{9} + 29 T^{8} + \cdots + 8203843 \)
|
| $61$ |
\( T^{9} + 5 T^{8} + \cdots + 41716 \)
|
| $67$ |
\( T^{9} - 13 T^{8} + \cdots - 262877 \)
|
| $71$ |
\( T^{9} + 41 T^{8} + \cdots + 39059567 \)
|
| $73$ |
\( T^{9} + 31 T^{8} + \cdots - 1973519 \)
|
| $79$ |
\( T^{9} - 2 T^{8} + \cdots - 22805597 \)
|
| $83$ |
\( T^{9} - 4 T^{8} + \cdots + 973139 \)
|
| $89$ |
\( T^{9} + 27 T^{8} + \cdots - 19843 \)
|
| $97$ |
\( T^{9} + 22 T^{8} + \cdots - 266873 \)
|
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