Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4025,2,Mod(1,4025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4025, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("4025.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4025 = 5^{2} \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4025.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: | \(32.1397868136\) |
| 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: |
\( x^{8} - 3x^{7} - 6x^{6} + 19x^{5} + 12x^{4} - 34x^{3} - 12x^{2} + 17x + 7 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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
\( x^{8} - 3x^{7} - 6x^{6} + 19x^{5} + 12x^{4} - 34x^{3} - 12x^{2} + 17x + 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 2\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 7\nu^{4} + 11\nu^{3} + 15\nu^{2} - 13\nu - 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{7} + 3\nu^{6} + 5\nu^{5} - 17\nu^{4} - 5\nu^{3} + 23\nu^{2} - 2\nu - 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 3\nu^{6} - 5\nu^{5} + 17\nu^{4} + 6\nu^{3} - 24\nu^{2} - \nu + 6 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{7} + 3\nu^{6} + 6\nu^{5} - 18\nu^{4} - 11\nu^{3} + 26\nu^{2} + 5\nu - 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{3} + 6\beta_{2} + 7\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + 7\beta_{6} + 6\beta_{5} + \beta_{3} + 9\beta_{2} + 21\beta _1 + 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{7} + 10\beta_{6} + 8\beta_{5} + \beta_{4} + 9\beta_{3} + 34\beta_{2} + 45\beta _1 + 33 \)
|
| \(\nu^{7}\) | \(=\) |
\( 11\beta_{7} + 43\beta_{6} + 31\beta_{5} + 3\beta_{4} + 15\beta_{3} + 63\beta_{2} + 122\beta _1 + 56 \)
|
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 |
|
−1.86556 | 1.47264 | 1.48032 | 0 | −2.74730 | −1.00000 | 0.969501 | −0.831331 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.34053 | −2.02792 | −0.202970 | 0 | 2.71850 | −1.00000 | 2.95316 | 1.11246 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.673262 | 0.897707 | −1.54672 | 0 | −0.604392 | −1.00000 | 2.38787 | −2.19412 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.468649 | 2.11571 | −1.78037 | 0 | −0.991526 | −1.00000 | 1.77166 | 1.47624 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.07669 | 0.452781 | −0.840734 | 0 | 0.487505 | −1.00000 | −3.05860 | −2.79499 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.38128 | −1.77230 | −0.0920619 | 0 | −2.44804 | −1.00000 | −2.88973 | 0.141045 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.33027 | −1.65734 | 3.43017 | 0 | −3.86204 | −1.00000 | 3.33268 | −0.253237 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.55976 | 2.51872 | 4.55237 | 0 | 6.44731 | −1.00000 | 6.53345 | 3.34393 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
| \(23\) | \(1\) |
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 | 4025.2.a.w | yes | 8 |
| 5.b | even | 2 | 1 | 4025.2.a.s | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4025.2.a.s | ✓ | 8 | 5.b | even | 2 | 1 | |
| 4025.2.a.w | yes | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):
|
\( T_{2}^{8} - 3T_{2}^{7} - 6T_{2}^{6} + 19T_{2}^{5} + 12T_{2}^{4} - 34T_{2}^{3} - 12T_{2}^{2} + 17T_{2} + 7 \)
|
|
\( T_{3}^{8} - 2T_{3}^{7} - 10T_{3}^{6} + 19T_{3}^{5} + 31T_{3}^{4} - 59T_{3}^{3} - 23T_{3}^{2} + 61T_{3} - 19 \)
|
|
\( T_{11}^{8} + 5T_{11}^{7} - 18T_{11}^{6} - 97T_{11}^{5} + 61T_{11}^{4} + 444T_{11}^{3} + 98T_{11}^{2} - 384T_{11} - 71 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 3 T^{7} + \cdots + 7 \)
|
| $3$ |
\( T^{8} - 2 T^{7} + \cdots - 19 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T + 1)^{8} \)
|
| $11$ |
\( T^{8} + 5 T^{7} + \cdots - 71 \)
|
| $13$ |
\( T^{8} - 9 T^{7} + \cdots - 111 \)
|
| $17$ |
\( T^{8} + T^{7} - 33 T^{6} + \cdots - 7 \)
|
| $19$ |
\( T^{8} + 4 T^{7} + \cdots - 243 \)
|
| $23$ |
\( (T + 1)^{8} \)
|
| $29$ |
\( T^{8} + 5 T^{7} + \cdots + 159651 \)
|
| $31$ |
\( T^{8} + T^{7} + \cdots - 195513 \)
|
| $37$ |
\( T^{8} - 18 T^{7} + \cdots + 3375 \)
|
| $41$ |
\( T^{8} + T^{7} + \cdots - 13569 \)
|
| $43$ |
\( T^{8} - 20 T^{7} + \cdots - 650071 \)
|
| $47$ |
\( T^{8} - 10 T^{7} + \cdots + 352131 \)
|
| $53$ |
\( T^{8} - 11 T^{7} + \cdots + 1118241 \)
|
| $59$ |
\( T^{8} - 20 T^{7} + \cdots - 71079 \)
|
| $61$ |
\( T^{8} + 6 T^{7} + \cdots + 5400047 \)
|
| $67$ |
\( T^{8} - 23 T^{7} + \cdots - 21960131 \)
|
| $71$ |
\( T^{8} - 3 T^{7} + \cdots - 461939 \)
|
| $73$ |
\( T^{8} + 8 T^{7} + \cdots + 7171111 \)
|
| $79$ |
\( T^{8} - 4 T^{7} + \cdots + 243 \)
|
| $83$ |
\( T^{8} - 4 T^{7} + \cdots + 18949233 \)
|
| $89$ |
\( T^{8} + 17 T^{7} + \cdots + 3313 \)
|
| $97$ |
\( T^{8} - 41 T^{7} + \cdots - 12386987 \)
|
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