Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,5,Mod(19,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.19");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.d (of order \(6\), degree \(2\), 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: | \(5.06512819111\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{22})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 22x^{2} + 484 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 7) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 22x^{2} + 484 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 22 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 22 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 22\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 22\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
−3.34521 | + | 5.79407i | −8.53562 | + | 4.92804i | −14.3808 | − | 24.9083i | −6.57125 | − | 3.79391i | − | 65.9413i | 0 | 85.3808 | 8.07125 | − | 13.9798i | 43.9644 | − | 25.3828i | |||||||||||||||||
| 19.2 | 1.34521 | − | 2.32997i | 5.53562 | − | 3.19599i | 4.38083 | + | 7.58782i | 21.5712 | + | 12.4542i | − | 17.1971i | 0 | 66.6192 | −20.0712 | + | 34.7644i | 58.0356 | − | 33.5069i | ||||||||||||||||||
| 31.1 | −3.34521 | − | 5.79407i | −8.53562 | − | 4.92804i | −14.3808 | + | 24.9083i | −6.57125 | + | 3.79391i | 65.9413i | 0 | 85.3808 | 8.07125 | + | 13.9798i | 43.9644 | + | 25.3828i | |||||||||||||||||||
| 31.2 | 1.34521 | + | 2.32997i | 5.53562 | + | 3.19599i | 4.38083 | − | 7.58782i | 21.5712 | − | 12.4542i | 17.1971i | 0 | 66.6192 | −20.0712 | − | 34.7644i | 58.0356 | + | 33.5069i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 49.5.d.b | 4 | |
| 7.b | odd | 2 | 1 | 7.5.d.a | ✓ | 4 | |
| 7.c | even | 3 | 1 | 7.5.d.a | ✓ | 4 | |
| 7.c | even | 3 | 1 | 49.5.b.a | 4 | ||
| 7.d | odd | 6 | 1 | 49.5.b.a | 4 | ||
| 7.d | odd | 6 | 1 | inner | 49.5.d.b | 4 | |
| 21.c | even | 2 | 1 | 63.5.m.d | 4 | ||
| 21.g | even | 6 | 1 | 441.5.d.d | 4 | ||
| 21.h | odd | 6 | 1 | 63.5.m.d | 4 | ||
| 21.h | odd | 6 | 1 | 441.5.d.d | 4 | ||
| 28.d | even | 2 | 1 | 112.5.s.a | 4 | ||
| 28.f | even | 6 | 1 | 784.5.c.c | 4 | ||
| 28.g | odd | 6 | 1 | 112.5.s.a | 4 | ||
| 28.g | odd | 6 | 1 | 784.5.c.c | 4 | ||
| 35.c | odd | 2 | 1 | 175.5.i.a | 4 | ||
| 35.f | even | 4 | 2 | 175.5.j.a | 8 | ||
| 35.j | even | 6 | 1 | 175.5.i.a | 4 | ||
| 35.l | odd | 12 | 2 | 175.5.j.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.5.d.a | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 7.5.d.a | ✓ | 4 | 7.c | even | 3 | 1 | |
| 49.5.b.a | 4 | 7.c | even | 3 | 1 | ||
| 49.5.b.a | 4 | 7.d | odd | 6 | 1 | ||
| 49.5.d.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 49.5.d.b | 4 | 7.d | odd | 6 | 1 | inner | |
| 63.5.m.d | 4 | 21.c | even | 2 | 1 | ||
| 63.5.m.d | 4 | 21.h | odd | 6 | 1 | ||
| 112.5.s.a | 4 | 28.d | even | 2 | 1 | ||
| 112.5.s.a | 4 | 28.g | odd | 6 | 1 | ||
| 175.5.i.a | 4 | 35.c | odd | 2 | 1 | ||
| 175.5.i.a | 4 | 35.j | even | 6 | 1 | ||
| 175.5.j.a | 8 | 35.f | even | 4 | 2 | ||
| 175.5.j.a | 8 | 35.l | odd | 12 | 2 | ||
| 441.5.d.d | 4 | 21.g | even | 6 | 1 | ||
| 441.5.d.d | 4 | 21.h | odd | 6 | 1 | ||
| 784.5.c.c | 4 | 28.f | even | 6 | 1 | ||
| 784.5.c.c | 4 | 28.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 4T_{2}^{3} + 34T_{2}^{2} - 72T_{2} + 324 \)
acting on \(S_{5}^{\mathrm{new}}(49, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 4 T^{3} + \cdots + 324 \)
|
| $3$ |
\( T^{4} + 6 T^{3} + \cdots + 3969 \)
|
| $5$ |
\( T^{4} - 30 T^{3} + \cdots + 35721 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 58 T^{3} + \cdots + 30437289 \)
|
| $13$ |
\( T^{4} + 55272 T^{2} + 3111696 \)
|
| $17$ |
\( T^{4} + \cdots + 5076990009 \)
|
| $19$ |
\( T^{4} + \cdots + 3136784049 \)
|
| $23$ |
\( T^{4} - 290 T^{3} + \cdots + 254625849 \)
|
| $29$ |
\( (T^{2} + 1088 T + 188136)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 1071889573041 \)
|
| $37$ |
\( T^{4} + \cdots + 48279954529 \)
|
| $41$ |
\( T^{4} + \cdots + 3218392944144 \)
|
| $43$ |
\( (T^{2} - 1236 T - 2313076)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 4451285577249 \)
|
| $53$ |
\( T^{4} + \cdots + 6460874330625 \)
|
| $59$ |
\( T^{4} + \cdots + 163173619767249 \)
|
| $61$ |
\( T^{4} + \cdots + 14401820070729 \)
|
| $67$ |
\( T^{4} + \cdots + 151890252361 \)
|
| $71$ |
\( (T^{2} + 5204 T + 5524236)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 6851102086521 \)
|
| $79$ |
\( T^{4} + \cdots + 188584385155609 \)
|
| $83$ |
\( T^{4} + \cdots + 115179601694976 \)
|
| $89$ |
\( T^{4} + \cdots + 75\!\cdots\!81 \)
|
| $97$ |
\( T^{4} + \cdots + 11\!\cdots\!84 \)
|
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