Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9800,2,Mod(1,9800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9800, 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("9800.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9800.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: | \(78.2533939809\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.43449.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 7x^{2} + 2x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1400) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 7x^{2} + 2x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 7\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 8\beta _1 + 3 \)
|
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 | −3.31553 | 0 | 0 | 0 | 0 | 0 | 7.99276 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.26536 | 0 | 0 | 0 | 0 | 0 | −1.39886 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.465834 | 0 | 0 | 0 | 0 | 0 | −2.78300 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.04673 | 0 | 0 | 0 | 0 | 0 | 1.18910 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
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 | 9800.2.a.cj | 4 | |
| 5.b | even | 2 | 1 | 9800.2.a.ct | 4 | ||
| 7.b | odd | 2 | 1 | 9800.2.a.cu | 4 | ||
| 7.c | even | 3 | 2 | 1400.2.q.m | yes | 8 | |
| 35.c | odd | 2 | 1 | 9800.2.a.ck | 4 | ||
| 35.j | even | 6 | 2 | 1400.2.q.l | ✓ | 8 | |
| 35.l | odd | 12 | 4 | 1400.2.bh.j | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1400.2.q.l | ✓ | 8 | 35.j | even | 6 | 2 | |
| 1400.2.q.m | yes | 8 | 7.c | even | 3 | 2 | |
| 1400.2.bh.j | 16 | 35.l | odd | 12 | 4 | ||
| 9800.2.a.cj | 4 | 1.a | even | 1 | 1 | trivial | |
| 9800.2.a.ck | 4 | 35.c | odd | 2 | 1 | ||
| 9800.2.a.ct | 4 | 5.b | even | 2 | 1 | ||
| 9800.2.a.cu | 4 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):
|
\( T_{3}^{4} + 3T_{3}^{3} - 4T_{3}^{2} - 11T_{3} - 4 \)
|
|
\( T_{11}^{4} - 37T_{11}^{2} - 49T_{11} + 134 \)
|
|
\( T_{13}^{4} - 4T_{13}^{3} - 23T_{13}^{2} + 105T_{13} - 84 \)
|
|
\( T_{19}^{4} + 10T_{19}^{3} - 13T_{19}^{2} - 359T_{19} - 824 \)
|
|
\( T_{23}^{4} - 79T_{23}^{2} + 91T_{23} + 953 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 3 T^{3} + \cdots - 4 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 37 T^{2} + \cdots + 134 \)
|
| $13$ |
\( T^{4} - 4 T^{3} + \cdots - 84 \)
|
| $17$ |
\( T^{4} - 7 T^{3} + \cdots + 5 \)
|
| $19$ |
\( T^{4} + 10 T^{3} + \cdots - 824 \)
|
| $23$ |
\( T^{4} - 79 T^{2} + \cdots + 953 \)
|
| $29$ |
\( T^{4} - 2 T^{3} + \cdots - 222 \)
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| $31$ |
\( T^{4} + 14 T^{3} + \cdots - 25 \)
|
| $37$ |
\( T^{4} - 2 T^{3} + \cdots - 92 \)
|
| $41$ |
\( T^{4} + 4 T^{3} + \cdots + 3319 \)
|
| $43$ |
\( T^{4} + 15 T^{3} + \cdots - 1956 \)
|
| $47$ |
\( T^{4} - 15 T^{3} + \cdots - 3015 \)
|
| $53$ |
\( T^{4} - 10 T^{3} + \cdots - 32 \)
|
| $59$ |
\( T^{4} + T^{3} + \cdots + 70 \)
|
| $61$ |
\( T^{4} + 25 T^{3} + \cdots - 2130 \)
|
| $67$ |
\( T^{4} - 4 T^{3} + \cdots - 1472 \)
|
| $71$ |
\( T^{4} + 20 T^{3} + \cdots - 1035 \)
|
| $73$ |
\( T^{4} - 2 T^{3} + \cdots + 1152 \)
|
| $79$ |
\( T^{4} - 2 T^{3} + \cdots - 149 \)
|
| $83$ |
\( T^{4} + 26 T^{3} + \cdots - 2826 \)
|
| $89$ |
\( T^{4} + 19 T^{3} + \cdots - 1437 \)
|
| $97$ |
\( T^{4} - 6 T^{3} + \cdots - 27 \)
|
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