Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6900,2,Mod(6349,6900)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6900, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6900.6349");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6900.f (of order \(2\), degree \(1\), 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: | \(55.0967773947\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.158155776.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 1380) |
| 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
\( x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 11\nu^{5} - 34\nu^{4} + 121\nu^{3} + 132\nu^{2} + 66\nu + 1203 ) / 681 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -29\nu^{5} + 69\nu^{4} - 92\nu^{3} - 575\nu^{2} - 2217\nu - 819 ) / 681 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 34\nu^{5} - 167\nu^{4} + 374\nu^{3} + 408\nu^{2} + 204\nu - 2163 ) / 681 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 40\nu^{5} - 103\nu^{4} + 213\nu^{3} + 707\nu^{2} + 3645\nu + 1341 ) / 681 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -123\nu^{5} + 277\nu^{4} - 218\nu^{3} - 3292\nu^{2} - 8229\nu - 3051 ) / 681 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{2} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + 2\beta_{4} + 7\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{5} + 13\beta_{4} - 2\beta_{3} + 29\beta_{2} + 13\beta _1 - 29 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -11\beta_{3} + 34\beta _1 - 95 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 46\beta_{5} - 197\beta_{4} - 46\beta_{3} - 493\beta_{2} + 197\beta _1 - 493 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times\).
| \(n\) | \(277\) | \(1201\) | \(3451\) | \(4601\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 6349.1 |
|
0 | − | 1.00000i | 0 | 0 | 0 | − | 3.73549i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 6349.2 | 0 | − | 1.00000i | 0 | 0 | 0 | 1.52644i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 6349.3 | 0 | − | 1.00000i | 0 | 0 | 0 | 4.20905i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 6349.4 | 0 | 1.00000i | 0 | 0 | 0 | − | 4.20905i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 6349.5 | 0 | 1.00000i | 0 | 0 | 0 | − | 1.52644i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 6349.6 | 0 | 1.00000i | 0 | 0 | 0 | 3.73549i | 0 | −1.00000 | 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 | 6900.2.f.r | 6 | |
| 5.b | even | 2 | 1 | inner | 6900.2.f.r | 6 | |
| 5.c | odd | 4 | 1 | 1380.2.a.j | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 6900.2.a.x | 3 | ||
| 15.e | even | 4 | 1 | 4140.2.a.s | 3 | ||
| 20.e | even | 4 | 1 | 5520.2.a.bv | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1380.2.a.j | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 4140.2.a.s | 3 | 15.e | even | 4 | 1 | ||
| 5520.2.a.bv | 3 | 20.e | even | 4 | 1 | ||
| 6900.2.a.x | 3 | 5.c | odd | 4 | 1 | ||
| 6900.2.f.r | 6 | 1.a | even | 1 | 1 | trivial | |
| 6900.2.f.r | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6900, [\chi])\):
|
\( T_{7}^{6} + 34T_{7}^{4} + 321T_{7}^{2} + 576 \)
|
|
\( T_{11}^{3} - 4T_{11}^{2} - 14T_{11} + 48 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 1)^{3} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 34 T^{4} + \cdots + 576 \)
|
| $11$ |
\( (T^{3} - 4 T^{2} - 14 T + 48)^{2} \)
|
| $13$ |
\( T^{6} + 40 T^{4} + \cdots + 144 \)
|
| $17$ |
\( T^{6} + 42 T^{4} + \cdots + 324 \)
|
| $19$ |
\( (T^{3} + 10 T^{2} + \cdots - 52)^{2} \)
|
| $23$ |
\( (T^{2} + 1)^{3} \)
|
| $29$ |
\( (T^{3} - 21 T + 18)^{2} \)
|
| $31$ |
\( (T^{3} - 10 T^{2} + 17 T + 4)^{2} \)
|
| $37$ |
\( T^{6} + 130 T^{4} + \cdots + 11664 \)
|
| $41$ |
\( (T^{3} - 8 T^{2} + \cdots + 582)^{2} \)
|
| $43$ |
\( T^{6} + 216 T^{4} + \cdots + 92416 \)
|
| $47$ |
\( T^{6} + 116 T^{4} + \cdots + 46656 \)
|
| $53$ |
\( T^{6} + 266 T^{4} + \cdots + 338724 \)
|
| $59$ |
\( (T^{3} - 21 T - 18)^{2} \)
|
| $61$ |
\( (T^{3} - 10 T^{2} + \cdots + 292)^{2} \)
|
| $67$ |
\( T^{6} + 66 T^{4} + \cdots + 16 \)
|
| $71$ |
\( (T^{3} + 8 T^{2} + \cdots - 582)^{2} \)
|
| $73$ |
\( T^{6} + 456 T^{4} + \cdots + 2226064 \)
|
| $79$ |
\( (T^{3} + 14 T^{2} - 96)^{2} \)
|
| $83$ |
\( T^{6} + 134 T^{4} + \cdots + 51984 \)
|
| $89$ |
\( (T^{3} + 2 T^{2} + \cdots - 912)^{2} \)
|
| $97$ |
\( T^{6} + 576 T^{4} + \cdots + 5456896 \)
|
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