Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2300,2,Mod(1749,2300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2300, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2300.1749");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2300.c (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(18.3655924649\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.6594624.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 9x^{4} + 18x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 9x^{4} + 18x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 5\nu^{2} + 1 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 8\nu^{2} + 10 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 8\nu^{3} + 13\nu ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} + 19\nu^{3} + 41\nu ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 2\beta_{4} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{3} + 8\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{5} + 19\beta_{4} + 27\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).
| \(n\) | \(277\) | \(1151\) | \(1201\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1749.1 |
|
0 | − | 2.69963i | 0 | 0 | 0 | − | 3.39926i | 0 | −4.28799 | 0 | ||||||||||||||||||||||||||||||||||
| 1749.2 | 0 | − | 1.46050i | 0 | 0 | 0 | − | 4.92101i | 0 | 0.866926 | 0 | |||||||||||||||||||||||||||||||||||
| 1749.3 | 0 | − | 0.760877i | 0 | 0 | 0 | 0.478247i | 0 | 2.42107 | 0 | ||||||||||||||||||||||||||||||||||||
| 1749.4 | 0 | 0.760877i | 0 | 0 | 0 | − | 0.478247i | 0 | 2.42107 | 0 | ||||||||||||||||||||||||||||||||||||
| 1749.5 | 0 | 1.46050i | 0 | 0 | 0 | 4.92101i | 0 | 0.866926 | 0 | |||||||||||||||||||||||||||||||||||||
| 1749.6 | 0 | 2.69963i | 0 | 0 | 0 | 3.39926i | 0 | −4.28799 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2300.2.c.i | 6 | |
| 5.b | even | 2 | 1 | inner | 2300.2.c.i | 6 | |
| 5.c | odd | 4 | 1 | 2300.2.a.j | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 2300.2.a.k | yes | 3 | |
| 20.e | even | 4 | 1 | 9200.2.a.cc | 3 | ||
| 20.e | even | 4 | 1 | 9200.2.a.cg | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2300.2.a.j | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 2300.2.a.k | yes | 3 | 5.c | odd | 4 | 1 | |
| 2300.2.c.i | 6 | 1.a | even | 1 | 1 | trivial | |
| 2300.2.c.i | 6 | 5.b | even | 2 | 1 | inner | |
| 9200.2.a.cc | 3 | 20.e | even | 4 | 1 | ||
| 9200.2.a.cg | 3 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2300, [\chi])\):
|
\( T_{3}^{6} + 10T_{3}^{4} + 21T_{3}^{2} + 9 \)
|
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\( T_{7}^{6} + 36T_{7}^{4} + 288T_{7}^{2} + 64 \)
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\( T_{11}^{3} + 4T_{11}^{2} - 12T_{11} - 24 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} + 10 T^{4} + \cdots + 9 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} + 36 T^{4} + \cdots + 64 \)
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| $11$ |
\( (T^{3} + 4 T^{2} - 12 T - 24)^{2} \)
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| $13$ |
\( T^{6} + 46 T^{4} + \cdots + 729 \)
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| $17$ |
\( T^{6} + 56 T^{4} + \cdots + 5184 \)
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| $19$ |
\( (T^{3} - 6 T^{2} - 24 T + 56)^{2} \)
|
| $23$ |
\( (T^{2} + 1)^{3} \)
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| $29$ |
\( (T^{3} - 8 T^{2} + \cdots + 257)^{2} \)
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| $31$ |
\( (T^{3} + 20 T^{2} + \cdots + 127)^{2} \)
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| $37$ |
\( T^{6} + 36 T^{4} + \cdots + 64 \)
|
| $41$ |
\( (T^{3} - 4 T^{2} + \cdots + 321)^{2} \)
|
| $43$ |
\( T^{6} + 104 T^{4} + \cdots + 28224 \)
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| $47$ |
\( T^{6} + 174 T^{4} + \cdots + 16129 \)
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| $53$ |
\( T^{6} + 232 T^{4} + \cdots + 5184 \)
|
| $59$ |
\( (T^{3} - 9 T^{2} + \cdots + 721)^{2} \)
|
| $61$ |
\( (T^{3} + 22 T^{2} + \cdots - 72)^{2} \)
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| $67$ |
\( T^{6} + 520 T^{4} + \cdots + 1382976 \)
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| $71$ |
\( (T^{3} + 30 T^{2} + \cdots + 911)^{2} \)
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| $73$ |
\( T^{6} + 126 T^{4} + \cdots + 5929 \)
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| $79$ |
\( (T^{3} - 20 T^{2} + \cdots - 72)^{2} \)
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| $83$ |
\( T^{6} + 72 T^{4} + \cdots + 64 \)
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| $89$ |
\( (T^{3} + 6 T^{2} + \cdots + 216)^{2} \)
|
| $97$ |
\( T^{6} + 120 T^{4} + \cdots + 64 \)
|
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