Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3380,2,Mod(2029,3380)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3380, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3380.2029");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3380.c (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: | \(26.9894358832\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.839056.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 6x^{4} + 8x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 260) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 6x^{4} + 8x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{3} - 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + 4\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{4} + 8\nu^{2} + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} + 5\nu^{3} + \nu^{2} + 4\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} - 5\nu^{3} + \nu^{2} - 4\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{2} - 2\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -4\beta_{5} - 4\beta_{4} + \beta_{3} + 13 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} + 6\beta_{2} + 16\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).
| \(n\) | \(677\) | \(1691\) | \(1861\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2029.1 |
|
0 | − | 2.97840i | 0 | 0.254102 | − | 2.22158i | 0 | 0.335751i | 0 | −5.87086 | 0 | |||||||||||||||||||||||||||||||||
| 2029.2 | 0 | − | 1.44042i | 0 | 1.86081 | + | 1.23992i | 0 | 0.694243i | 0 | 0.925197 | 0 | ||||||||||||||||||||||||||||||||||
| 2029.3 | 0 | − | 0.233093i | 0 | −2.11491 | − | 0.726062i | 0 | 4.29014i | 0 | 2.94567 | 0 | ||||||||||||||||||||||||||||||||||
| 2029.4 | 0 | 0.233093i | 0 | −2.11491 | + | 0.726062i | 0 | − | 4.29014i | 0 | 2.94567 | 0 | ||||||||||||||||||||||||||||||||||
| 2029.5 | 0 | 1.44042i | 0 | 1.86081 | − | 1.23992i | 0 | − | 0.694243i | 0 | 0.925197 | 0 | ||||||||||||||||||||||||||||||||||
| 2029.6 | 0 | 2.97840i | 0 | 0.254102 | + | 2.22158i | 0 | − | 0.335751i | 0 | −5.87086 | 0 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3380.2.c.a | 6 | |
| 5.b | even | 2 | 1 | inner | 3380.2.c.a | 6 | |
| 13.b | even | 2 | 1 | 3380.2.c.b | 6 | ||
| 13.c | even | 3 | 2 | 260.2.ba.a | ✓ | 12 | |
| 13.d | odd | 4 | 2 | 3380.2.d.c | 12 | ||
| 39.i | odd | 6 | 2 | 2340.2.de.a | 12 | ||
| 52.j | odd | 6 | 2 | 1040.2.dh.c | 12 | ||
| 65.d | even | 2 | 1 | 3380.2.c.b | 6 | ||
| 65.g | odd | 4 | 2 | 3380.2.d.c | 12 | ||
| 65.n | even | 6 | 2 | 260.2.ba.a | ✓ | 12 | |
| 65.q | odd | 12 | 4 | 1300.2.i.i | 12 | ||
| 195.x | odd | 6 | 2 | 2340.2.de.a | 12 | ||
| 260.v | odd | 6 | 2 | 1040.2.dh.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 260.2.ba.a | ✓ | 12 | 13.c | even | 3 | 2 | |
| 260.2.ba.a | ✓ | 12 | 65.n | even | 6 | 2 | |
| 1040.2.dh.c | 12 | 52.j | odd | 6 | 2 | ||
| 1040.2.dh.c | 12 | 260.v | odd | 6 | 2 | ||
| 1300.2.i.i | 12 | 65.q | odd | 12 | 4 | ||
| 2340.2.de.a | 12 | 39.i | odd | 6 | 2 | ||
| 2340.2.de.a | 12 | 195.x | odd | 6 | 2 | ||
| 3380.2.c.a | 6 | 1.a | even | 1 | 1 | trivial | |
| 3380.2.c.a | 6 | 5.b | even | 2 | 1 | inner | |
| 3380.2.c.b | 6 | 13.b | even | 2 | 1 | ||
| 3380.2.c.b | 6 | 65.d | even | 2 | 1 | ||
| 3380.2.d.c | 12 | 13.d | odd | 4 | 2 | ||
| 3380.2.d.c | 12 | 65.g | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3380, [\chi])\):
|
\( T_{3}^{6} + 11T_{3}^{4} + 19T_{3}^{2} + 1 \)
|
|
\( T_{11} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 11 T^{4} + \cdots + 1 \)
|
| $5$ |
\( T^{6} - T^{4} + \cdots + 125 \)
|
| $7$ |
\( T^{6} + 19 T^{4} + \cdots + 1 \)
|
| $11$ |
\( (T + 1)^{6} \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} + 51 T^{4} + \cdots + 2209 \)
|
| $19$ |
\( (T^{3} + 5 T^{2} - 17 T - 53)^{2} \)
|
| $23$ |
\( T^{6} + 59 T^{4} + \cdots + 2401 \)
|
| $29$ |
\( (T^{3} - 5 T^{2} - 17 T + 53)^{2} \)
|
| $31$ |
\( (T^{3} + 6 T^{2} - 4 T - 16)^{2} \)
|
| $37$ |
\( T^{6} + 107 T^{4} + \cdots + 24649 \)
|
| $41$ |
\( (T^{3} + T^{2} - 77 T + 179)^{2} \)
|
| $43$ |
\( T^{6} + 99 T^{4} + \cdots + 12769 \)
|
| $47$ |
\( T^{6} + 204 T^{4} + \cdots + 87616 \)
|
| $53$ |
\( T^{6} + 80 T^{4} + \cdots + 16384 \)
|
| $59$ |
\( (T^{3} - T^{2} - 125 T + 109)^{2} \)
|
| $61$ |
\( (T^{3} + 11 T^{2} + \cdots - 227)^{2} \)
|
| $67$ |
\( T^{6} + 299 T^{4} + \cdots + 351649 \)
|
| $71$ |
\( (T^{3} - 7 T^{2} + \cdots - 137)^{2} \)
|
| $73$ |
\( T^{6} + 324 T^{4} + \cdots + 746496 \)
|
| $79$ |
\( (T^{3} - 2 T^{2} + \cdots + 512)^{2} \)
|
| $83$ |
\( T^{6} + 128 T^{4} + \cdots + 3136 \)
|
| $89$ |
\( (T^{3} + T^{2} - 125 T - 109)^{2} \)
|
| $97$ |
\( T^{6} + 299 T^{4} + \cdots + 67081 \)
|
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