Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [490,2,Mod(71,490)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(490, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("490.71");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 490.k (of order \(7\), degree \(6\), 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: | \(3.91266969904\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{14})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{7}]$ |
$q$-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 primitive root of unity \(\zeta_{14}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) |
| \(\chi(n)\) | \(-\zeta_{14}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 71.1 |
|
−0.900969 | + | 0.433884i | 0.678448 | − | 2.97247i | 0.623490 | − | 0.781831i | −0.222521 | + | 0.974928i | 0.678448 | + | 2.97247i | 2.57942 | + | 0.588735i | −0.222521 | + | 0.974928i | −5.67241 | − | 2.73169i | −0.222521 | − | 0.974928i | ||||||||||||||||||
| 141.1 | 0.623490 | − | 0.781831i | −1.52446 | − | 0.734141i | −0.222521 | − | 0.974928i | −0.900969 | − | 0.433884i | −1.52446 | + | 0.734141i | −1.14795 | + | 2.38374i | −0.900969 | − | 0.433884i | −0.0854576 | − | 0.107160i | −0.900969 | + | 0.433884i | |||||||||||||||||||
| 211.1 | −0.222521 | + | 0.974928i | 0.846011 | − | 1.06086i | −0.900969 | − | 0.433884i | 0.623490 | − | 0.781831i | 0.846011 | + | 1.06086i | 2.06853 | + | 1.64960i | 0.623490 | − | 0.781831i | 0.257865 | + | 1.12978i | 0.623490 | + | 0.781831i | |||||||||||||||||||
| 281.1 | −0.222521 | − | 0.974928i | 0.846011 | + | 1.06086i | −0.900969 | + | 0.433884i | 0.623490 | + | 0.781831i | 0.846011 | − | 1.06086i | 2.06853 | − | 1.64960i | 0.623490 | + | 0.781831i | 0.257865 | − | 1.12978i | 0.623490 | − | 0.781831i | |||||||||||||||||||
| 351.1 | 0.623490 | + | 0.781831i | −1.52446 | + | 0.734141i | −0.222521 | + | 0.974928i | −0.900969 | + | 0.433884i | −1.52446 | − | 0.734141i | −1.14795 | − | 2.38374i | −0.900969 | + | 0.433884i | −0.0854576 | + | 0.107160i | −0.900969 | − | 0.433884i | |||||||||||||||||||
| 421.1 | −0.900969 | − | 0.433884i | 0.678448 | + | 2.97247i | 0.623490 | + | 0.781831i | −0.222521 | − | 0.974928i | 0.678448 | − | 2.97247i | 2.57942 | − | 0.588735i | −0.222521 | − | 0.974928i | −5.67241 | + | 2.73169i | −0.222521 | + | 0.974928i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 49.e | even | 7 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 490.2.k.a | ✓ | 6 |
| 49.e | even | 7 | 1 | inner | 490.2.k.a | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 490.2.k.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 490.2.k.a | ✓ | 6 | 49.e | even | 7 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\):
|
\( T_{3}^{6} + 7T_{3}^{4} + 14T_{3}^{3} + 49 \)
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\( T_{11}^{6} - 10T_{11}^{5} + 30T_{11}^{4} + T_{11}^{3} + 88T_{11}^{2} + 65T_{11} + 169 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + T^{4} + \cdots + 1 \)
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| $3$ |
\( T^{6} + 7 T^{4} + \cdots + 49 \)
|
| $5$ |
\( T^{6} + T^{5} + T^{4} + \cdots + 1 \)
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| $7$ |
\( T^{6} - 7 T^{5} + \cdots + 343 \)
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| $11$ |
\( T^{6} - 10 T^{5} + \cdots + 169 \)
|
| $13$ |
\( T^{6} - T^{5} + T^{4} + \cdots + 1 \)
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| $17$ |
\( T^{6} - 2 T^{5} + \cdots + 169 \)
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| $19$ |
\( (T^{3} - 7 T - 7)^{2} \)
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| $23$ |
\( T^{6} + 7 T^{5} + \cdots + 49 \)
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| $29$ |
\( T^{6} - T^{5} + \cdots + 841 \)
|
| $31$ |
\( (T^{3} + 3 T^{2} - 46 T + 1)^{2} \)
|
| $37$ |
\( T^{6} + 4 T^{5} + \cdots + 1 \)
|
| $41$ |
\( T^{6} - 2 T^{5} + \cdots + 1 \)
|
| $43$ |
\( T^{6} - 10 T^{5} + \cdots + 6889 \)
|
| $47$ |
\( T^{6} + 34 T^{5} + \cdots + 57121 \)
|
| $53$ |
\( T^{6} + 33 T^{5} + \cdots + 16129 \)
|
| $59$ |
\( T^{6} + 22 T^{5} + \cdots + 169 \)
|
| $61$ |
\( T^{6} - 7 T^{5} + \cdots + 90601 \)
|
| $67$ |
\( (T^{3} + 15 T^{2} + \cdots - 3347)^{2} \)
|
| $71$ |
\( T^{6} - 9 T^{5} + \cdots + 841 \)
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| $73$ |
\( T^{6} + 12 T^{5} + \cdots + 1 \)
|
| $79$ |
\( (T^{3} + 10 T^{2} + \cdots - 349)^{2} \)
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| $83$ |
\( T^{6} - 15 T^{5} + \cdots + 9409 \)
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| $89$ |
\( T^{6} + 11 T^{5} + \cdots + 841 \)
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| $97$ |
\( (T^{3} + 27 T^{2} + \cdots + 197)^{2} \)
|
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