Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [33,6,Mod(1,33)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 33.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: | \(5.29266605383\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{313}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 78 \)
|
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{313})\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−8.34590 | −9.00000 | 37.6541 | −107.459 | 75.1131 | −26.6918 | −47.1885 | 81.0000 | 896.843 | ||||||||||||||||||||||||
| 1.2 | 9.34590 | −9.00000 | 55.3459 | 69.4590 | −84.1131 | 8.69181 | 218.189 | 81.0000 | 649.157 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(11\) | \(-1\) |
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 | 33.6.a.d | ✓ | 2 |
| 3.b | odd | 2 | 1 | 99.6.a.e | 2 | ||
| 4.b | odd | 2 | 1 | 528.6.a.q | 2 | ||
| 5.b | even | 2 | 1 | 825.6.a.d | 2 | ||
| 11.b | odd | 2 | 1 | 363.6.a.g | 2 | ||
| 33.d | even | 2 | 1 | 1089.6.a.o | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.6.a.d | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 99.6.a.e | 2 | 3.b | odd | 2 | 1 | ||
| 363.6.a.g | 2 | 11.b | odd | 2 | 1 | ||
| 528.6.a.q | 2 | 4.b | odd | 2 | 1 | ||
| 825.6.a.d | 2 | 5.b | even | 2 | 1 | ||
| 1089.6.a.o | 2 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 78 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(33))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - T - 78 \)
|
| $3$ |
\( (T + 9)^{2} \)
|
| $5$ |
\( T^{2} + 38T - 7464 \)
|
| $7$ |
\( T^{2} + 18T - 232 \)
|
| $11$ |
\( (T - 121)^{2} \)
|
| $13$ |
\( T^{2} + 66T - 878128 \)
|
| $17$ |
\( T^{2} + 920T + 210348 \)
|
| $19$ |
\( T^{2} + 2932 T + 2147904 \)
|
| $23$ |
\( T^{2} - 5246 T + 6827232 \)
|
| $29$ |
\( T^{2} + 12600 T + 38326572 \)
|
| $31$ |
\( T^{2} - 9936 T - 4245184 \)
|
| $37$ |
\( T^{2} - 5996 T + 975204 \)
|
| $41$ |
\( T^{2} - 24244 T + 105471636 \)
|
| $43$ |
\( T^{2} - 20360 T - 16684800 \)
|
| $47$ |
\( T^{2} + 5806 T - 38936064 \)
|
| $53$ |
\( T^{2} - 40770 T + 387938808 \)
|
| $59$ |
\( T^{2} - 18212 T - 974471136 \)
|
| $61$ |
\( T^{2} + 11398 T - 557566776 \)
|
| $67$ |
\( T^{2} + \cdots + 1022090128 \)
|
| $71$ |
\( T^{2} - 61446 T + 190949616 \)
|
| $73$ |
\( T^{2} - 53412 T + 705197636 \)
|
| $79$ |
\( T^{2} - 17122 T - 281721704 \)
|
| $83$ |
\( T^{2} + 14304 T - 922809744 \)
|
| $89$ |
\( T^{2} + 58140 T + 129501828 \)
|
| $97$ |
\( T^{2} + \cdots + 8284064476 \)
|
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