Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5175,2,Mod(1,5175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5175, 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("5175.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5175.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: | \(41.3225830460\) |
| 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: |
\( x^{7} - 10x^{5} + 28x^{3} - 4x^{2} - 21x + 8 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 10x^{5} + 28x^{3} - 4x^{2} - 21x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} - \nu + 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 5\nu^{2} + 9\nu - 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 9\nu^{4} - \nu^{3} + 20\nu^{2} + 2\nu - 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 6\beta_{2} + \beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 7\beta_{3} + 8\beta_{2} + 20\beta _1 + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 9\beta_{4} + \beta_{3} + 35\beta_{2} + 11\beta _1 + 65 \)
|
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 |
|
−2.26478 | 0 | 3.12921 | 0 | 0 | 4.68661 | −2.55740 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.91486 | 0 | 1.66669 | 0 | 0 | 0.0284418 | 0.638246 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.21340 | 0 | −0.527654 | 0 | 0 | −4.59472 | 3.06706 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.461927 | 0 | −1.78662 | 0 | 0 | −1.03748 | −1.74915 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.811488 | 0 | −1.34149 | 0 | 0 | 0.370088 | −2.71158 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.62725 | 0 | 0.647939 | 0 | 0 | 1.84808 | −2.20014 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.49237 | 0 | 4.21193 | 0 | 0 | −2.30103 | 5.51296 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 5175.2.a.cc | ✓ | 7 |
| 3.b | odd | 2 | 1 | 5175.2.a.cd | yes | 7 | |
| 5.b | even | 2 | 1 | 5175.2.a.ce | yes | 7 | |
| 15.d | odd | 2 | 1 | 5175.2.a.cf | yes | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5175.2.a.cc | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 5175.2.a.cd | yes | 7 | 3.b | odd | 2 | 1 | |
| 5175.2.a.ce | yes | 7 | 5.b | even | 2 | 1 | |
| 5175.2.a.cf | yes | 7 | 15.d | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5175))\):
|
\( T_{2}^{7} - 10T_{2}^{5} + 28T_{2}^{3} - 4T_{2}^{2} - 21T_{2} + 8 \)
|
|
\( T_{7}^{7} + T_{7}^{6} - 26T_{7}^{5} - 26T_{7}^{4} + 96T_{7}^{3} + 62T_{7}^{2} - 37T_{7} + 1 \)
|
|
\( T_{11}^{7} + 6T_{11}^{6} - 20T_{11}^{5} - 196T_{11}^{4} - 340T_{11}^{3} + 92T_{11}^{2} + 433T_{11} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 10 T^{5} + \cdots + 8 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( T^{7} + T^{6} - 26 T^{5} + \cdots + 1 \)
|
| $11$ |
\( T^{7} + 6 T^{6} + \cdots + 16 \)
|
| $13$ |
\( T^{7} + 7 T^{6} + \cdots - 325 \)
|
| $17$ |
\( T^{7} - 2 T^{6} + \cdots + 1728 \)
|
| $19$ |
\( T^{7} + T^{6} + \cdots + 16 \)
|
| $23$ |
\( (T - 1)^{7} \)
|
| $29$ |
\( T^{7} + 20 T^{6} + \cdots + 6784 \)
|
| $31$ |
\( T^{7} + 5 T^{6} + \cdots - 191056 \)
|
| $37$ |
\( T^{7} - 8 T^{6} + \cdots - 25416 \)
|
| $41$ |
\( T^{7} + 36 T^{6} + \cdots - 30896 \)
|
| $43$ |
\( T^{7} - 3 T^{6} + \cdots + 729 \)
|
| $47$ |
\( T^{7} + 4 T^{6} + \cdots + 1536 \)
|
| $53$ |
\( T^{7} - 4 T^{6} + \cdots + 9216 \)
|
| $59$ |
\( T^{7} + 12 T^{6} + \cdots + 1664 \)
|
| $61$ |
\( T^{7} - 9 T^{6} + \cdots - 233204 \)
|
| $67$ |
\( T^{7} - 5 T^{6} + \cdots - 50076 \)
|
| $71$ |
\( T^{7} + 20 T^{6} + \cdots + 281216 \)
|
| $73$ |
\( T^{7} - 4 T^{6} + \cdots - 24894 \)
|
| $79$ |
\( T^{7} + 6 T^{6} + \cdots - 874068 \)
|
| $83$ |
\( T^{7} - 4 T^{6} + \cdots - 8762736 \)
|
| $89$ |
\( T^{7} + 42 T^{6} + \cdots - 12549568 \)
|
| $97$ |
\( T^{7} + T^{6} + \cdots - 1258300 \)
|
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