Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6300,2,Mod(3149,6300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6300, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6300.3149");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6300.f (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: | \(50.3057532734\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 252) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( 2\zeta_{24}^{6} \)
|
| \(\beta_{2}\) | \(=\) |
\( -2\zeta_{24}^{6} + 4\zeta_{24}^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( -2\zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \)
|
| \(\beta_{4}\) | \(=\) |
\( 4\zeta_{24}^{4} - 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -3\zeta_{24}^{5} - 3\zeta_{24}^{3} + 3\zeta_{24} \)
|
| \(\beta_{6}\) | \(=\) |
\( -3\zeta_{24}^{5} + 3\zeta_{24}^{3} + 3\zeta_{24} \)
|
| \(\beta_{7}\) | \(=\) |
\( 4\zeta_{24}^{7} + 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} \)
|
| \(\zeta_{24}\) | \(=\) |
\( ( 3\beta_{7} + 2\beta_{6} + 2\beta_{5} + 6\beta_{3} + 3\beta_1 ) / 24 \)
|
| \(\zeta_{24}^{2}\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 4 \)
|
| \(\zeta_{24}^{3}\) | \(=\) |
\( ( \beta_{6} - \beta_{5} ) / 6 \)
|
| \(\zeta_{24}^{4}\) | \(=\) |
\( ( \beta_{4} + 2 ) / 4 \)
|
| \(\zeta_{24}^{5}\) | \(=\) |
\( ( 3\beta_{7} - 2\beta_{6} - 2\beta_{5} + 6\beta_{3} + 3\beta_1 ) / 24 \)
|
| \(\zeta_{24}^{6}\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\zeta_{24}^{7}\) | \(=\) |
\( ( 3\beta_{7} + 2\beta_{6} - 2\beta_{5} - 6\beta_{3} - 3\beta_1 ) / 24 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times\).
| \(n\) | \(2801\) | \(3151\) | \(3277\) | \(3601\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3149.1 |
|
0 | 0 | 0 | 0 | 0 | −2.44949 | − | 1.00000i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 3149.2 | 0 | 0 | 0 | 0 | 0 | −2.44949 | − | 1.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 3149.3 | 0 | 0 | 0 | 0 | 0 | −2.44949 | + | 1.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 3149.4 | 0 | 0 | 0 | 0 | 0 | −2.44949 | + | 1.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 3149.5 | 0 | 0 | 0 | 0 | 0 | 2.44949 | − | 1.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 3149.6 | 0 | 0 | 0 | 0 | 0 | 2.44949 | − | 1.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 3149.7 | 0 | 0 | 0 | 0 | 0 | 2.44949 | + | 1.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 3149.8 | 0 | 0 | 0 | 0 | 0 | 2.44949 | + | 1.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
| 35.c | odd | 2 | 1 | inner |
| 105.g | even | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6300, [\chi])\):
|
\( T_{11}^{2} + 18 \)
|
|
\( T_{41}^{2} - 12 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} \)
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| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{4} - 10 T^{2} + 49)^{2} \)
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| $11$ |
\( (T^{2} + 18)^{4} \)
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| $13$ |
\( (T^{2} - 24)^{4} \)
|
| $17$ |
\( (T^{2} + 12)^{4} \)
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| $19$ |
\( (T^{2} + 24)^{4} \)
|
| $23$ |
\( (T^{2} - 18)^{4} \)
|
| $29$ |
\( (T^{2} + 18)^{4} \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( (T^{2} + 64)^{4} \)
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| $41$ |
\( (T^{2} - 12)^{4} \)
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| $43$ |
\( (T^{2} + 4)^{4} \)
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| $47$ |
\( (T^{2} + 48)^{4} \)
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| $53$ |
\( (T^{2} - 162)^{4} \)
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| $59$ |
\( (T^{2} - 192)^{4} \)
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| $61$ |
\( (T^{2} + 96)^{4} \)
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| $67$ |
\( (T^{2} + 64)^{4} \)
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| $71$ |
\( (T^{2} + 18)^{4} \)
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| $73$ |
\( (T^{2} - 24)^{4} \)
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| $79$ |
\( (T - 4)^{8} \)
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| $83$ |
\( (T^{2} + 48)^{4} \)
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| $89$ |
\( (T^{2} - 108)^{4} \)
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| $97$ |
\( (T^{2} - 24)^{4} \)
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