Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3375,2,Mod(1,3375)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3375, 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("3375.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3375 = 3^{3} \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3375.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: | \(26.9495106822\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 10x^{6} + 9x^{5} + 28x^{4} - 19x^{3} - 24x^{2} + 10x + 5 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 10x^{6} + 9x^{5} + 28x^{4} - 19x^{3} - 24x^{2} + 10x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{7} - 4\nu^{6} + 21\nu^{5} + 32\nu^{4} - 64\nu^{3} - 50\nu^{2} + 41\nu - 1 ) / 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{6} - 17\nu^{5} - 16\nu^{4} + 84\nu^{3} + 25\nu^{2} - 105\nu - 6 ) / 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{7} + 4\nu^{6} - 21\nu^{5} - 32\nu^{4} + 64\nu^{3} + 63\nu^{2} - 41\nu - 25 ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{7} + 5\nu^{6} + 42\nu^{5} - 40\nu^{4} - 115\nu^{3} + 69\nu^{2} + 56\nu - 28 ) / 13 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -2\nu^{7} - 4\nu^{6} + 21\nu^{5} + 45\nu^{4} - 64\nu^{3} - 141\nu^{2} + 54\nu + 77 ) / 13 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5\nu^{7} + 3\nu^{6} + 46\nu^{5} - 24\nu^{4} - 108\nu^{3} + 31\nu^{2} + 57\nu - 9 ) / 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 7\beta_{4} + 6\beta_{2} - \beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{7} + 8\beta_{5} - 8\beta_{4} + 6\beta_{3} - \beta_{2} + 27\beta _1 - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} + 8\beta_{6} + 44\beta_{4} - \beta_{3} + 33\beta_{2} - 11\beta _1 + 40 \)
|
| \(\nu^{7}\) | \(=\) |
\( -54\beta_{7} + 52\beta_{5} - 53\beta_{4} + 33\beta_{3} - 12\beta_{2} + 150\beta _1 - 13 \)
|
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 |
|
−2.36500 | 0 | 3.59322 | 0 | 0 | −3.25043 | −3.76796 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.02353 | 0 | 2.09468 | 0 | 0 | −0.428486 | −0.191579 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.13333 | 0 | −0.715564 | 0 | 0 | 4.74885 | 3.07763 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.765547 | 0 | −1.41394 | 0 | 0 | −4.41277 | 2.61353 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.333287 | 0 | −1.88892 | 0 | 0 | −2.03477 | −1.29612 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.950864 | 0 | −1.09586 | 0 | 0 | 1.05631 | −2.94374 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.54701 | 0 | 0.393232 | 0 | 0 | −0.463644 | −2.48568 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.45625 | 0 | 4.03315 | 0 | 0 | 2.78495 | 4.99392 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
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 | 3375.2.a.h | ✓ | 8 |
| 3.b | odd | 2 | 1 | 3375.2.a.l | yes | 8 | |
| 5.b | even | 2 | 1 | 3375.2.a.m | yes | 8 | |
| 15.d | odd | 2 | 1 | 3375.2.a.i | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3375.2.a.h | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 3375.2.a.i | yes | 8 | 15.d | odd | 2 | 1 | |
| 3375.2.a.l | yes | 8 | 3.b | odd | 2 | 1 | |
| 3375.2.a.m | yes | 8 | 5.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3375))\):
|
\( T_{2}^{8} + T_{2}^{7} - 10T_{2}^{6} - 9T_{2}^{5} + 28T_{2}^{4} + 19T_{2}^{3} - 24T_{2}^{2} - 10T_{2} + 5 \)
|
|
\( T_{7}^{8} + 2T_{7}^{7} - 31T_{7}^{6} - 65T_{7}^{5} + 209T_{7}^{4} + 414T_{7}^{3} - 180T_{7}^{2} - 324T_{7} - 81 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + T^{7} - 10 T^{6} + \cdots + 5 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 2 T^{7} + \cdots - 81 \)
|
| $11$ |
\( T^{8} - 9 T^{7} + \cdots + 405 \)
|
| $13$ |
\( T^{8} - 2 T^{7} + \cdots + 2025 \)
|
| $17$ |
\( T^{8} + 4 T^{7} + \cdots - 5 \)
|
| $19$ |
\( T^{8} + 8 T^{7} + \cdots - 15941 \)
|
| $23$ |
\( T^{8} - 3 T^{7} + \cdots + 21125 \)
|
| $29$ |
\( T^{8} - 21 T^{7} + \cdots + 486405 \)
|
| $31$ |
\( T^{8} + 10 T^{7} + \cdots + 38725 \)
|
| $37$ |
\( T^{8} + T^{7} + \cdots - 413505 \)
|
| $41$ |
\( T^{8} - 45 T^{7} + \cdots - 1225125 \)
|
| $43$ |
\( T^{8} + 14 T^{7} + \cdots + 110079 \)
|
| $47$ |
\( T^{8} - 2 T^{7} + \cdots + 44105 \)
|
| $53$ |
\( T^{8} + 6 T^{7} + \cdots + 25355 \)
|
| $59$ |
\( T^{8} - 26 T^{7} + \cdots + 20739645 \)
|
| $61$ |
\( T^{8} + 10 T^{7} + \cdots + 1414189 \)
|
| $67$ |
\( T^{8} - 20 T^{7} + \cdots - 2563245 \)
|
| $71$ |
\( T^{8} - 21 T^{7} + \cdots - 20768805 \)
|
| $73$ |
\( T^{8} - 5 T^{7} + \cdots + 6459345 \)
|
| $79$ |
\( T^{8} + 12 T^{7} + \cdots - 1518025 \)
|
| $83$ |
\( T^{8} + 17 T^{7} + \cdots + 853505 \)
|
| $89$ |
\( T^{8} - 64 T^{7} + \cdots + 3655125 \)
|
| $97$ |
\( T^{8} + 36 T^{7} + \cdots + 33129 \)
|
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