Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2738,2,Mod(1,2738)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2738, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2738.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2738 = 2 \cdot 37^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2738.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: | \(21.8630400734\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\Q(\zeta_{38})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - x^{8} - 8x^{7} + 7x^{6} + 21x^{5} - 15x^{4} - 20x^{3} + 10x^{2} + 5x - 1 \)
|
| 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 \(\nu = \zeta_{38} + \zeta_{38}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + 5\nu \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 6\nu^{4} + 9\nu^{2} - 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 7\nu^{5} + 14\nu^{3} - 7\nu \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{8} - 8\nu^{6} + 20\nu^{4} - 16\nu^{2} + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 4\beta_{2} + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 5\beta_{3} + 10\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 6\beta_{4} + 15\beta_{2} + 20 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{7} + 7\beta_{5} + 21\beta_{3} + 35\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{8} + 8\beta_{6} + 28\beta_{4} + 56\beta_{2} + 70 \)
|
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 |
|
1.00000 | −3.38261 | 1.00000 | −2.72648 | −3.38261 | −0.152624 | 1.00000 | 8.44202 | −2.72648 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.72648 | 1.00000 | −0.0314505 | −2.72648 | 1.42593 | 1.00000 | 4.43367 | −0.0314505 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −2.58487 | 1.00000 | −3.38261 | −2.58487 | −3.95019 | 1.00000 | 3.68154 | −3.38261 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.819334 | 1.00000 | −0.605558 | −0.819334 | 0.239042 | 1.00000 | −2.32869 | −0.605558 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −0.739333 | 1.00000 | −2.58487 | −0.739333 | −0.650935 | 1.00000 | −2.45339 | −2.58487 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | −0.605558 | 1.00000 | 2.32729 | −0.605558 | 0.184773 | 1.00000 | −2.63330 | 2.32729 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | −0.0314505 | 1.00000 | 1.56234 | −0.0314505 | −4.80486 | 1.00000 | −2.99901 | 1.56234 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 1.56234 | 1.00000 | −0.819334 | 1.56234 | −1.93137 | 1.00000 | −0.559099 | −0.819334 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 2.32729 | 1.00000 | −0.739333 | 2.32729 | −4.35976 | 1.00000 | 2.41626 | −0.739333 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(37\) | \(-1\) |
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 | 2738.2.a.v | yes | 9 |
| 37.b | even | 2 | 1 | 2738.2.a.u | ✓ | 9 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2738.2.a.u | ✓ | 9 | 37.b | even | 2 | 1 | |
| 2738.2.a.v | yes | 9 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2738))\):
|
\( T_{3}^{9} + 7T_{3}^{8} + 7T_{3}^{7} - 45T_{3}^{6} - 100T_{3}^{5} + 13T_{3}^{4} + 170T_{3}^{3} + 139T_{3}^{2} + 36T_{3} + 1 \)
|
|
\( T_{5}^{9} + 7T_{5}^{8} + 7T_{5}^{7} - 45T_{5}^{6} - 100T_{5}^{5} + 13T_{5}^{4} + 170T_{5}^{3} + 139T_{5}^{2} + 36T_{5} + 1 \)
|
|
\( T_{7}^{9} + 14T_{7}^{8} + 66T_{7}^{7} + 96T_{7}^{6} - 99T_{7}^{5} - 287T_{7}^{4} - 64T_{7}^{3} + 46T_{7}^{2} + T_{7} - 1 \)
|
|
\( T_{13}^{9} + 3 T_{13}^{8} - 53 T_{13}^{7} - 170 T_{13}^{6} + 732 T_{13}^{5} + 2334 T_{13}^{4} - 2040 T_{13}^{3} - 5321 T_{13}^{2} - 46 T_{13} + 379 \)
|
|
\( T_{17}^{9} + 10 T_{17}^{8} - 40 T_{17}^{7} - 540 T_{17}^{6} + 202 T_{17}^{5} + 8595 T_{17}^{4} + 5594 T_{17}^{3} - 37295 T_{17}^{2} - 52545 T_{17} - 12769 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{9} \)
|
| $3$ |
\( T^{9} + 7 T^{8} + 7 T^{7} - 45 T^{6} + \cdots + 1 \)
|
| $5$ |
\( T^{9} + 7 T^{8} + 7 T^{7} - 45 T^{6} + \cdots + 1 \)
|
| $7$ |
\( T^{9} + 14 T^{8} + 66 T^{7} + 96 T^{6} + \cdots - 1 \)
|
| $11$ |
\( T^{9} + T^{8} - 46 T^{7} - 83 T^{6} + \cdots + 1901 \)
|
| $13$ |
\( T^{9} + 3 T^{8} - 53 T^{7} - 170 T^{6} + \cdots + 379 \)
|
| $17$ |
\( T^{9} + 10 T^{8} - 40 T^{7} + \cdots - 12769 \)
|
| $19$ |
\( T^{9} + 9 T^{8} - 59 T^{7} + \cdots - 1481 \)
|
| $23$ |
\( T^{9} - 3 T^{8} - 129 T^{7} + \cdots - 379 \)
|
| $29$ |
\( T^{9} + 12 T^{8} - 69 T^{7} + \cdots + 154697 \)
|
| $31$ |
\( T^{9} + 2 T^{8} - 89 T^{7} + \cdots + 1747 \)
|
| $37$ |
\( T^{9} \)
|
| $41$ |
\( T^{9} + 16 T^{8} - 15 T^{7} + \cdots + 60041 \)
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| $43$ |
\( T^{9} - 6 T^{8} - 193 T^{7} + \cdots + 289103 \)
|
| $47$ |
\( T^{9} + 15 T^{8} - 185 T^{7} + \cdots + 9187069 \)
|
| $53$ |
\( T^{9} + 7 T^{8} - 297 T^{7} + \cdots + 47107043 \)
|
| $59$ |
\( T^{9} + T^{8} - 312 T^{7} + \cdots - 565249 \)
|
| $61$ |
\( T^{9} + 7 T^{8} - 297 T^{7} + \cdots - 47151197 \)
|
| $67$ |
\( T^{9} + 66 T^{8} + 1727 T^{7} + \cdots - 30468247 \)
|
| $71$ |
\( T^{9} + 15 T^{8} - 128 T^{7} + \cdots - 453683 \)
|
| $73$ |
\( T^{9} + 50 T^{8} + \cdots - 393016331 \)
|
| $79$ |
\( T^{9} - 29 T^{8} + 36 T^{7} + \cdots + 11315299 \)
|
| $83$ |
\( T^{9} + 43 T^{8} + 427 T^{7} + \cdots - 99198581 \)
|
| $89$ |
\( T^{9} + 6 T^{8} - 345 T^{7} + \cdots + 11018443 \)
|
| $97$ |
\( T^{9} + 50 T^{8} + 881 T^{7} + \cdots - 18092827 \)
|
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