Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [845,2,Mod(146,845)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(845, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("845.146");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 845 = 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 845.e (of order \(3\), 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: | \(6.74735897080\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 65) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{12}^{3} + \zeta_{12} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 3 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).
| \(n\) | \(171\) | \(677\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 146.1 |
|
−0.866025 | − | 1.50000i | −1.36603 | − | 2.36603i | −0.500000 | + | 0.866025i | 1.00000 | −2.36603 | + | 4.09808i | 1.00000 | − | 1.73205i | −1.73205 | −2.23205 | + | 3.86603i | −0.866025 | − | 1.50000i | ||||||||||||||||
| 146.2 | 0.866025 | + | 1.50000i | 0.366025 | + | 0.633975i | −0.500000 | + | 0.866025i | 1.00000 | −0.633975 | + | 1.09808i | 1.00000 | − | 1.73205i | 1.73205 | 1.23205 | − | 2.13397i | 0.866025 | + | 1.50000i | |||||||||||||||||
| 191.1 | −0.866025 | + | 1.50000i | −1.36603 | + | 2.36603i | −0.500000 | − | 0.866025i | 1.00000 | −2.36603 | − | 4.09808i | 1.00000 | + | 1.73205i | −1.73205 | −2.23205 | − | 3.86603i | −0.866025 | + | 1.50000i | |||||||||||||||||
| 191.2 | 0.866025 | − | 1.50000i | 0.366025 | − | 0.633975i | −0.500000 | − | 0.866025i | 1.00000 | −0.633975 | − | 1.09808i | 1.00000 | + | 1.73205i | 1.73205 | 1.23205 | + | 2.13397i | 0.866025 | − | 1.50000i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 845.2.e.f | 4 | |
| 13.b | even | 2 | 1 | 845.2.e.e | 4 | ||
| 13.c | even | 3 | 1 | 845.2.a.d | 2 | ||
| 13.c | even | 3 | 1 | inner | 845.2.e.f | 4 | |
| 13.d | odd | 4 | 1 | 845.2.m.a | 4 | ||
| 13.d | odd | 4 | 1 | 845.2.m.c | 4 | ||
| 13.e | even | 6 | 1 | 65.2.a.c | ✓ | 2 | |
| 13.e | even | 6 | 1 | 845.2.e.e | 4 | ||
| 13.f | odd | 12 | 2 | 845.2.c.e | 4 | ||
| 13.f | odd | 12 | 1 | 845.2.m.a | 4 | ||
| 13.f | odd | 12 | 1 | 845.2.m.c | 4 | ||
| 39.h | odd | 6 | 1 | 585.2.a.k | 2 | ||
| 39.i | odd | 6 | 1 | 7605.2.a.be | 2 | ||
| 52.i | odd | 6 | 1 | 1040.2.a.h | 2 | ||
| 65.l | even | 6 | 1 | 325.2.a.g | 2 | ||
| 65.n | even | 6 | 1 | 4225.2.a.w | 2 | ||
| 65.r | odd | 12 | 2 | 325.2.b.e | 4 | ||
| 91.t | odd | 6 | 1 | 3185.2.a.k | 2 | ||
| 104.p | odd | 6 | 1 | 4160.2.a.bj | 2 | ||
| 104.s | even | 6 | 1 | 4160.2.a.y | 2 | ||
| 143.i | odd | 6 | 1 | 7865.2.a.h | 2 | ||
| 156.r | even | 6 | 1 | 9360.2.a.cm | 2 | ||
| 195.y | odd | 6 | 1 | 2925.2.a.z | 2 | ||
| 195.bf | even | 12 | 2 | 2925.2.c.v | 4 | ||
| 260.w | odd | 6 | 1 | 5200.2.a.ca | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.2.a.c | ✓ | 2 | 13.e | even | 6 | 1 | |
| 325.2.a.g | 2 | 65.l | even | 6 | 1 | ||
| 325.2.b.e | 4 | 65.r | odd | 12 | 2 | ||
| 585.2.a.k | 2 | 39.h | odd | 6 | 1 | ||
| 845.2.a.d | 2 | 13.c | even | 3 | 1 | ||
| 845.2.c.e | 4 | 13.f | odd | 12 | 2 | ||
| 845.2.e.e | 4 | 13.b | even | 2 | 1 | ||
| 845.2.e.e | 4 | 13.e | even | 6 | 1 | ||
| 845.2.e.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 845.2.e.f | 4 | 13.c | even | 3 | 1 | inner | |
| 845.2.m.a | 4 | 13.d | odd | 4 | 1 | ||
| 845.2.m.a | 4 | 13.f | odd | 12 | 1 | ||
| 845.2.m.c | 4 | 13.d | odd | 4 | 1 | ||
| 845.2.m.c | 4 | 13.f | odd | 12 | 1 | ||
| 1040.2.a.h | 2 | 52.i | odd | 6 | 1 | ||
| 2925.2.a.z | 2 | 195.y | odd | 6 | 1 | ||
| 2925.2.c.v | 4 | 195.bf | even | 12 | 2 | ||
| 3185.2.a.k | 2 | 91.t | odd | 6 | 1 | ||
| 4160.2.a.y | 2 | 104.s | even | 6 | 1 | ||
| 4160.2.a.bj | 2 | 104.p | odd | 6 | 1 | ||
| 4225.2.a.w | 2 | 65.n | even | 6 | 1 | ||
| 5200.2.a.ca | 2 | 260.w | odd | 6 | 1 | ||
| 7605.2.a.be | 2 | 39.i | odd | 6 | 1 | ||
| 7865.2.a.h | 2 | 143.i | odd | 6 | 1 | ||
| 9360.2.a.cm | 2 | 156.r | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\):
|
\( T_{2}^{4} + 3T_{2}^{2} + 9 \)
|
|
\( T_{7}^{2} - 2T_{7} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 3T^{2} + 9 \)
|
| $3$ |
\( T^{4} + 2 T^{3} + \cdots + 4 \)
|
| $5$ |
\( (T - 1)^{4} \)
|
| $7$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
| $11$ |
\( T^{4} + 6 T^{3} + \cdots + 36 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + 12T^{2} + 144 \)
|
| $19$ |
\( T^{4} + 2 T^{3} + \cdots + 676 \)
|
| $23$ |
\( T^{4} + 6 T^{3} + \cdots + 36 \)
|
| $29$ |
\( T^{4} - 12 T^{3} + \cdots + 576 \)
|
| $31$ |
\( (T^{2} + 10 T - 2)^{2} \)
|
| $37$ |
\( (T^{2} + 4 T + 16)^{2} \)
|
| $41$ |
\( T^{4} + 12T^{2} + 144 \)
|
| $43$ |
\( T^{4} + 10 T^{3} + \cdots + 4 \)
|
| $47$ |
\( (T + 6)^{4} \)
|
| $53$ |
\( (T^{2} - 108)^{2} \)
|
| $59$ |
\( T^{4} + 6 T^{3} + \cdots + 19044 \)
|
| $61$ |
\( T^{4} + 4 T^{3} + \cdots + 10816 \)
|
| $67$ |
\( T^{4} + 8 T^{3} + \cdots + 8464 \)
|
| $71$ |
\( T^{4} - 6 T^{3} + \cdots + 36 \)
|
| $73$ |
\( (T - 4)^{4} \)
|
| $79$ |
\( (T^{2} - 4 T - 104)^{2} \)
|
| $83$ |
\( (T - 6)^{4} \)
|
| $89$ |
\( T^{4} + 12 T^{3} + \cdots + 144 \)
|
| $97$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
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