Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,5,Mod(48,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.48");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.b (of order \(2\), degree \(1\), 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: | \(4\) |
| 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: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 22x^{2} + 484 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} ) / 22 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} + 44\nu - 22 ) / 22 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{2} + 77 ) / 11 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 7\beta_{2} - 7\beta_1 ) / 14 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 11\beta_{3} - 77 ) / 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( 22\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 48.1 |
|
−2.69042 | − | 6.39199i | −8.76166 | 24.9083i | 17.1971i | 0 | 66.6192 | 40.1425 | − | 67.0138i | ||||||||||||||||||||||||||||
| 48.2 | −2.69042 | 6.39199i | −8.76166 | − | 24.9083i | − | 17.1971i | 0 | 66.6192 | 40.1425 | 67.0138i | |||||||||||||||||||||||||||||
| 48.3 | 6.69042 | − | 9.85609i | 28.7617 | 7.58782i | − | 65.9413i | 0 | 85.3808 | −16.1425 | 50.7657i | |||||||||||||||||||||||||||||
| 48.4 | 6.69042 | 9.85609i | 28.7617 | − | 7.58782i | 65.9413i | 0 | 85.3808 | −16.1425 | − | 50.7657i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 49.5.b.a | 4 | |
| 3.b | odd | 2 | 1 | 441.5.d.d | 4 | ||
| 4.b | odd | 2 | 1 | 784.5.c.c | 4 | ||
| 7.b | odd | 2 | 1 | inner | 49.5.b.a | 4 | |
| 7.c | even | 3 | 1 | 7.5.d.a | ✓ | 4 | |
| 7.c | even | 3 | 1 | 49.5.d.b | 4 | ||
| 7.d | odd | 6 | 1 | 7.5.d.a | ✓ | 4 | |
| 7.d | odd | 6 | 1 | 49.5.d.b | 4 | ||
| 21.c | even | 2 | 1 | 441.5.d.d | 4 | ||
| 21.g | even | 6 | 1 | 63.5.m.d | 4 | ||
| 21.h | odd | 6 | 1 | 63.5.m.d | 4 | ||
| 28.d | even | 2 | 1 | 784.5.c.c | 4 | ||
| 28.f | even | 6 | 1 | 112.5.s.a | 4 | ||
| 28.g | odd | 6 | 1 | 112.5.s.a | 4 | ||
| 35.i | odd | 6 | 1 | 175.5.i.a | 4 | ||
| 35.j | even | 6 | 1 | 175.5.i.a | 4 | ||
| 35.k | even | 12 | 2 | 175.5.j.a | 8 | ||
| 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.c | even | 3 | 1 | |
| 7.5.d.a | ✓ | 4 | 7.d | odd | 6 | 1 | |
| 49.5.b.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 49.5.b.a | 4 | 7.b | odd | 2 | 1 | inner | |
| 49.5.d.b | 4 | 7.c | even | 3 | 1 | ||
| 49.5.d.b | 4 | 7.d | odd | 6 | 1 | ||
| 63.5.m.d | 4 | 21.g | even | 6 | 1 | ||
| 63.5.m.d | 4 | 21.h | odd | 6 | 1 | ||
| 112.5.s.a | 4 | 28.f | even | 6 | 1 | ||
| 112.5.s.a | 4 | 28.g | odd | 6 | 1 | ||
| 175.5.i.a | 4 | 35.i | odd | 6 | 1 | ||
| 175.5.i.a | 4 | 35.j | even | 6 | 1 | ||
| 175.5.j.a | 8 | 35.k | even | 12 | 2 | ||
| 175.5.j.a | 8 | 35.l | odd | 12 | 2 | ||
| 441.5.d.d | 4 | 3.b | odd | 2 | 1 | ||
| 441.5.d.d | 4 | 21.c | even | 2 | 1 | ||
| 784.5.c.c | 4 | 4.b | odd | 2 | 1 | ||
| 784.5.c.c | 4 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 4T_{2} - 18 \)
acting on \(S_{5}^{\mathrm{new}}(49, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 4 T - 18)^{2} \)
|
| $3$ |
\( T^{4} + 138T^{2} + 3969 \)
|
| $5$ |
\( T^{4} + 678 T^{2} + 35721 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 58 T - 5517)^{2} \)
|
| $13$ |
\( T^{4} + 55272 T^{2} + 3111696 \)
|
| $17$ |
\( T^{4} + \cdots + 5076990009 \)
|
| $19$ |
\( T^{4} + \cdots + 3136784049 \)
|
| $23$ |
\( (T^{2} + 290 T - 15957)^{2} \)
|
| $29$ |
\( (T^{2} + 1088 T + 188136)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 1071889573041 \)
|
| $37$ |
\( (T^{2} - 270 T - 219727)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 3218392944144 \)
|
| $43$ |
\( (T^{2} - 1236 T - 2313076)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 4451285577249 \)
|
| $53$ |
\( (T^{2} - 4510 T + 2541825)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 163173619767249 \)
|
| $61$ |
\( T^{4} + \cdots + 14401820070729 \)
|
| $67$ |
\( (T^{2} - 1318 T + 389731)^{2} \)
|
| $71$ |
\( (T^{2} + 5204 T + 5524236)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 6851102086521 \)
|
| $79$ |
\( (T^{2} - 8110 T + 13732603)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 115179601694976 \)
|
| $89$ |
\( T^{4} + \cdots + 75\!\cdots\!81 \)
|
| $97$ |
\( T^{4} + \cdots + 11\!\cdots\!84 \)
|
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