Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7400,2,Mod(1,7400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7400, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("7400.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7400 = 2^{3} \cdot 5^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7400.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: | \(59.0892974957\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{8} - x^{7} - 12x^{6} + 8x^{5} + 39x^{4} - 14x^{3} - 36x^{2} + 11x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{8} - x^{7} - 12x^{6} + 8x^{5} + 39x^{4} - 14x^{3} - 36x^{2} + 11x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{7} - 2\nu^{6} - 22\nu^{5} + 13\nu^{4} + 58\nu^{3} - 5\nu^{2} - 30\nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{7} + \nu^{6} - 39\nu^{5} - 21\nu^{4} + 132\nu^{3} + 81\nu^{2} - 90\nu - 30 \)
|
| \(\beta_{4}\) | \(=\) |
\( -5\nu^{7} + 2\nu^{6} + 59\nu^{5} - \nu^{4} - 174\nu^{3} - 62\nu^{2} + 99\nu + 34 \)
|
| \(\beta_{5}\) | \(=\) |
\( 4\nu^{7} - 50\nu^{5} - 15\nu^{4} + 161\nu^{3} + 82\nu^{2} - 104\nu - 34 \)
|
| \(\beta_{6}\) | \(=\) |
\( -7\nu^{7} + 2\nu^{6} + 84\nu^{5} + 6\nu^{4} - 254\nu^{3} - 99\nu^{2} + 149\nu + 47 \)
|
| \(\beta_{7}\) | \(=\) |
\( -8\nu^{7} + \nu^{6} + 98\nu^{5} + 20\nu^{4} - 306\nu^{3} - 142\nu^{2} + 189\nu + 62 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - \beta_{4} + \beta_{3} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + 2\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{7} - 2\beta_{5} - 7\beta_{4} + 9\beta_{3} + \beta_{2} + 2\beta _1 + 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( -10\beta_{7} + 19\beta_{6} - 14\beta_{5} - 11\beta_{4} + 12\beta_{3} + 9\beta_{2} + 42\beta _1 + 21 \)
|
| \(\nu^{6}\) | \(=\) |
\( 45\beta_{7} + 6\beta_{6} - 30\beta_{5} - 54\beta_{4} + 76\beta_{3} + 12\beta_{2} + 27\beta _1 + 72 \)
|
| \(\nu^{7}\) | \(=\) |
\( -79\beta_{7} + 157\beta_{6} - 142\beta_{5} - 103\beta_{4} + 123\beta_{3} + 76\beta_{2} + 317\beta _1 + 187 \)
|
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 |
|
0 | −2.95803 | 0 | 0 | 0 | −0.939702 | 0 | 5.74994 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.07336 | 0 | 0 | 0 | −2.31363 | 0 | 1.29882 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.819833 | 0 | 0 | 0 | 3.46626 | 0 | −2.32787 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.800175 | 0 | 0 | 0 | −4.46764 | 0 | −2.35972 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.305014 | 0 | 0 | 0 | 0.258050 | 0 | −2.90697 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.31809 | 0 | 0 | 0 | 2.07329 | 0 | −1.26265 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.42472 | 0 | 0 | 0 | −0.242089 | 0 | −0.970174 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.60358 | 0 | 0 | 0 | −1.83454 | 0 | 3.77862 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(37\) | \( -1 \) |
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 | 7400.2.a.u | ✓ | 8 |
| 5.b | even | 2 | 1 | 7400.2.a.v | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7400.2.a.u | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 7400.2.a.v | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7400))\):
|
\( T_{3}^{8} + T_{3}^{7} - 12T_{3}^{6} - 8T_{3}^{5} + 39T_{3}^{4} + 14T_{3}^{3} - 36T_{3}^{2} - 11T_{3} + 6 \)
|
|
\( T_{7}^{8} + 4T_{7}^{7} - 15T_{7}^{6} - 62T_{7}^{5} + 18T_{7}^{4} + 195T_{7}^{3} + 124T_{7}^{2} - 14T_{7} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + T^{7} - 12 T^{6} + \cdots + 6 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 4 T^{7} + \cdots - 8 \)
|
| $11$ |
\( T^{8} - 48 T^{6} + \cdots - 288 \)
|
| $13$ |
\( T^{8} + 9 T^{7} + \cdots - 1068 \)
|
| $17$ |
\( T^{8} + 4 T^{7} + \cdots - 7528 \)
|
| $19$ |
\( T^{8} - 4 T^{7} + \cdots - 1641 \)
|
| $23$ |
\( T^{8} + 2 T^{7} + \cdots - 41788 \)
|
| $29$ |
\( T^{8} + 5 T^{7} + \cdots + 1264 \)
|
| $31$ |
\( T^{8} - 11 T^{7} + \cdots - 54888 \)
|
| $37$ |
\( (T - 1)^{8} \)
|
| $41$ |
\( T^{8} - 104 T^{6} + \cdots + 14499 \)
|
| $43$ |
\( T^{8} + 4 T^{7} + \cdots - 931052 \)
|
| $47$ |
\( T^{8} + 14 T^{7} + \cdots + 255168 \)
|
| $53$ |
\( T^{8} + 11 T^{7} + \cdots - 65412 \)
|
| $59$ |
\( T^{8} + 15 T^{7} + \cdots + 585268 \)
|
| $61$ |
\( T^{8} + 13 T^{7} + \cdots - 24587796 \)
|
| $67$ |
\( T^{8} + 19 T^{7} + \cdots + 13278 \)
|
| $71$ |
\( T^{8} - 40 T^{7} + \cdots - 22260148 \)
|
| $73$ |
\( T^{8} + 31 T^{7} + \cdots + 2429664 \)
|
| $79$ |
\( T^{8} - 2 T^{7} + \cdots + 298156 \)
|
| $83$ |
\( T^{8} + 4 T^{7} + \cdots + 35726136 \)
|
| $89$ |
\( T^{8} + 12 T^{7} + \cdots - 380742 \)
|
| $97$ |
\( T^{8} + 11 T^{7} + \cdots + 2651024 \)
|
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