Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4100,2,Mod(1,4100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4100, 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("4100.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4100.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: | \(32.7386648287\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 13x^{5} + 7x^{4} + 49x^{3} - 4x^{2} - 37x - 11 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 13x^{5} + 7x^{4} + 49x^{3} - 4x^{2} - 37x - 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 6\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 7\nu^{2} + 4\nu + 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 3\nu^{5} - 10\nu^{4} + 24\nu^{3} + 31\nu^{2} - 39\nu - 19 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2\nu^{6} - 3\nu^{5} - 23\nu^{4} + 24\nu^{3} + 74\nu^{2} - 39\nu - 38 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 7\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 8\beta_{2} + 10\beta _1 + 25 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} - 2\beta_{5} + \beta_{4} + 9\beta_{3} + 12\beta_{2} + 49\beta _1 + 25 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3\beta_{6} - 3\beta_{5} + 13\beta_{4} + 13\beta_{3} + 61\beta_{2} + 87\beta _1 + 172 \)
|
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 | −2.88961 | 0 | 0 | 0 | 4.40706 | 0 | 5.34984 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.56508 | 0 | 0 | 0 | −2.92431 | 0 | 3.57966 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.12349 | 0 | 0 | 0 | 2.98128 | 0 | −1.73778 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.377545 | 0 | 0 | 0 | 0.441953 | 0 | −2.85746 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.687910 | 0 | 0 | 0 | −4.33809 | 0 | −2.52678 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.18506 | 0 | 0 | 0 | 1.94794 | 0 | 1.77449 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.32767 | 0 | 0 | 0 | −2.51583 | 0 | 2.41803 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(41\) | \(1\) |
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 | 4100.2.a.h | ✓ | 7 |
| 5.b | even | 2 | 1 | 4100.2.a.i | yes | 7 | |
| 5.c | odd | 4 | 2 | 4100.2.d.f | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4100.2.a.h | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 4100.2.a.i | yes | 7 | 5.b | even | 2 | 1 | |
| 4100.2.d.f | 14 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} + T_{3}^{6} - 13T_{3}^{5} - 7T_{3}^{4} + 49T_{3}^{3} + 4T_{3}^{2} - 37T_{3} + 11 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4100))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} + T^{6} + \cdots + 11 \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( T^{7} - 33 T^{5} + \cdots + 361 \)
|
| $11$ |
\( T^{7} + 3 T^{6} + \cdots + 192 \)
|
| $13$ |
\( T^{7} + 7 T^{6} + \cdots + 571 \)
|
| $17$ |
\( T^{7} + 5 T^{6} + \cdots + 228 \)
|
| $19$ |
\( T^{7} - 3 T^{6} + \cdots + 14227 \)
|
| $23$ |
\( T^{7} + 4 T^{6} + \cdots - 27 \)
|
| $29$ |
\( T^{7} + T^{6} + \cdots - 19707 \)
|
| $31$ |
\( T^{7} + 4 T^{6} + \cdots - 81 \)
|
| $37$ |
\( T^{7} + 24 T^{6} + \cdots + 67491 \)
|
| $41$ |
\( (T + 1)^{7} \)
|
| $43$ |
\( T^{7} + 16 T^{6} + \cdots - 1004 \)
|
| $47$ |
\( T^{7} + 10 T^{6} + \cdots - 2187 \)
|
| $53$ |
\( T^{7} + 8 T^{6} + \cdots + 24681 \)
|
| $59$ |
\( T^{7} - 20 T^{6} + \cdots - 421761 \)
|
| $61$ |
\( T^{7} + 10 T^{6} + \cdots + 124757 \)
|
| $67$ |
\( T^{7} + 30 T^{6} + \cdots + 111231 \)
|
| $71$ |
\( T^{7} + 23 T^{6} + \cdots - 34413 \)
|
| $73$ |
\( T^{7} + 17 T^{6} + \cdots - 2355399 \)
|
| $79$ |
\( T^{7} - 11 T^{6} + \cdots - 325089 \)
|
| $83$ |
\( T^{7} - 4 T^{6} + \cdots + 3468 \)
|
| $89$ |
\( T^{7} - 8 T^{6} + \cdots + 938961 \)
|
| $97$ |
\( T^{7} + 47 T^{6} + \cdots + 159777 \)
|
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