Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4450,2,Mod(1,4450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4450, 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("4450.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4450 = 2 \cdot 5^{2} \cdot 89 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4450.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: | \(35.5334288995\) |
| 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: |
\( x^{8} - 4x^{7} - 4x^{6} + 26x^{5} - 3x^{4} - 42x^{3} + 20x^{2} + 4x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 890) |
| 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
\( x^{8} - 4x^{7} - 4x^{6} + 26x^{5} - 3x^{4} - 42x^{3} + 20x^{2} + 4x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{6} + 7\nu^{5} - 19\nu^{4} - 16\nu^{3} + 26\nu^{2} + 8\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 19\nu^{4} + 18\nu^{3} - 30\nu^{2} - 14\nu + 6 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{7} + 4\nu^{6} + 4\nu^{5} - 25\nu^{4} + 39\nu^{2} - 10\nu - 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -2\nu^{7} + 7\nu^{6} + 12\nu^{5} - 47\nu^{4} - 20\nu^{3} + 78\nu^{2} - 12 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2\nu^{7} - 7\nu^{6} - 12\nu^{5} + 49\nu^{4} + 16\nu^{3} - 86\nu^{2} + 10\nu + 12 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} - \beta_{3} + \beta_{2} + 2\beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} - \beta_{3} + 2\beta_{2} + 7\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} - 6\beta_{4} - 6\beta_{3} + 8\beta_{2} + 17\beta _1 + 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{7} + 5\beta_{6} - \beta_{5} - 9\beta_{4} - 11\beta_{3} + 19\beta_{2} + 51\beta _1 + 16 \)
|
| \(\nu^{6}\) | \(=\) |
\( 15\beta_{7} + 21\beta_{6} - 2\beta_{5} - 34\beta_{4} - 42\beta_{3} + 60\beta_{2} + 135\beta _1 + 70 \)
|
| \(\nu^{7}\) | \(=\) |
\( 47\beta_{7} + 79\beta_{6} - 13\beta_{5} - 61\beta_{4} - 101\beta_{3} + 155\beta_{2} + 387\beta _1 + 168 \)
|
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 | −2.85072 | 1.00000 | 0 | −2.85072 | 4.52360 | 1.00000 | 5.12662 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.67435 | 1.00000 | 0 | −2.67435 | −3.29849 | 1.00000 | 4.15217 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.31885 | 1.00000 | 0 | −1.31885 | 0.332784 | 1.00000 | −1.26063 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.680413 | 1.00000 | 0 | −0.680413 | −1.50187 | 1.00000 | −2.53704 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −0.421420 | 1.00000 | 0 | −0.421420 | 0.408024 | 1.00000 | −2.82241 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 0.461608 | 1.00000 | 0 | 0.461608 | 0.960393 | 1.00000 | −2.78692 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 1.57103 | 1.00000 | 0 | 1.57103 | 0.668798 | 1.00000 | −0.531874 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 1.91313 | 1.00000 | 0 | 1.91313 | −4.09323 | 1.00000 | 0.660064 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(89\) | \( +1 \) |
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 | 4450.2.a.bj | 8 | |
| 5.b | even | 2 | 1 | 4450.2.a.bi | 8 | ||
| 5.c | odd | 4 | 2 | 890.2.b.a | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 890.2.b.a | ✓ | 16 | 5.c | odd | 4 | 2 | |
| 4450.2.a.bi | 8 | 5.b | even | 2 | 1 | ||
| 4450.2.a.bj | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4450))\):
|
\( T_{3}^{8} + 4T_{3}^{7} - 4T_{3}^{6} - 26T_{3}^{5} - 3T_{3}^{4} + 42T_{3}^{3} + 20T_{3}^{2} - 8T_{3} - 4 \)
|
|
\( T_{7}^{8} + 2T_{7}^{7} - 24T_{7}^{6} - 46T_{7}^{5} + 90T_{7}^{4} + 48T_{7}^{3} - 120T_{7}^{2} + 56T_{7} - 8 \)
|
|
\( T_{11}^{8} + 4T_{11}^{7} - 30T_{11}^{6} - 124T_{11}^{5} + 165T_{11}^{4} + 700T_{11}^{3} - 320T_{11}^{2} - 932T_{11} + 28 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} + 4 T^{7} + \cdots - 4 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 2 T^{7} + \cdots - 8 \)
|
| $11$ |
\( T^{8} + 4 T^{7} + \cdots + 28 \)
|
| $13$ |
\( T^{8} + 4 T^{7} + \cdots + 648 \)
|
| $17$ |
\( T^{8} + 14 T^{7} + \cdots + 35312 \)
|
| $19$ |
\( T^{8} + 8 T^{7} + \cdots - 64 \)
|
| $23$ |
\( T^{8} + 24 T^{7} + \cdots + 15104 \)
|
| $29$ |
\( T^{8} + 10 T^{7} + \cdots - 311752 \)
|
| $31$ |
\( T^{8} - 94 T^{6} + \cdots + 200 \)
|
| $37$ |
\( T^{8} + 14 T^{7} + \cdots + 81928 \)
|
| $41$ |
\( T^{8} - 124 T^{6} + \cdots + 44288 \)
|
| $43$ |
\( T^{8} + 4 T^{7} + \cdots + 64708 \)
|
| $47$ |
\( T^{8} + 28 T^{7} + \cdots + 4448 \)
|
| $53$ |
\( T^{8} + 42 T^{7} + \cdots + 888184 \)
|
| $59$ |
\( T^{8} + 4 T^{7} + \cdots - 6891008 \)
|
| $61$ |
\( T^{8} + 2 T^{7} + \cdots + 2243950 \)
|
| $67$ |
\( T^{8} + 18 T^{7} + \cdots - 112 \)
|
| $71$ |
\( T^{8} - 20 T^{7} + \cdots + 774400 \)
|
| $73$ |
\( T^{8} + 18 T^{7} + \cdots - 8668736 \)
|
| $79$ |
\( T^{8} - 2 T^{7} + \cdots + 259328 \)
|
| $83$ |
\( T^{8} + 20 T^{7} + \cdots + 1417028 \)
|
| $89$ |
\( (T + 1)^{8} \)
|
| $97$ |
\( T^{8} + 16 T^{7} + \cdots + 31181888 \)
|
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