Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1850,2,Mod(149,1850)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1850, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1850.149");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1850 = 2 \cdot 5^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1850.b (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: | \(14.7723243739\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.37161216.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 15x^{4} + 51x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| 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
\( x^{6} + 15x^{4} + 51x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 8\nu^{2} + 1 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 14\nu^{2} + 31 ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 14\nu^{3} + 43\nu ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} + 31\nu^{3} + 110\nu ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 4\beta_{4} - 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -8\beta_{3} + 14\beta_{2} + 39 \)
|
| \(\nu^{5}\) | \(=\) |
\( -14\beta_{5} + 62\beta_{4} + 69\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).
| \(n\) | \(1001\) | \(1777\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 149.1 |
|
− | 1.00000i | − | 2.27307i | −1.00000 | 0 | −2.27307 | 1.27307i | 1.00000i | −2.16686 | 0 | ||||||||||||||||||||||||||||||||||
| 149.2 | − | 1.00000i | 0.140435i | −1.00000 | 0 | 0.140435 | − | 1.14044i | 1.00000i | 2.98028 | 0 | |||||||||||||||||||||||||||||||||||
| 149.3 | − | 1.00000i | 3.13264i | −1.00000 | 0 | 3.13264 | − | 4.13264i | 1.00000i | −6.81342 | 0 | |||||||||||||||||||||||||||||||||||
| 149.4 | 1.00000i | − | 3.13264i | −1.00000 | 0 | 3.13264 | 4.13264i | − | 1.00000i | −6.81342 | 0 | |||||||||||||||||||||||||||||||||||
| 149.5 | 1.00000i | − | 0.140435i | −1.00000 | 0 | 0.140435 | 1.14044i | − | 1.00000i | 2.98028 | 0 | |||||||||||||||||||||||||||||||||||
| 149.6 | 1.00000i | 2.27307i | −1.00000 | 0 | −2.27307 | − | 1.27307i | − | 1.00000i | −2.16686 | 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 | 1850.2.b.p | 6 | |
| 5.b | even | 2 | 1 | inner | 1850.2.b.p | 6 | |
| 5.c | odd | 4 | 1 | 1850.2.a.y | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 1850.2.a.bc | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1850.2.a.y | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 1850.2.a.bc | yes | 3 | 5.c | odd | 4 | 1 | |
| 1850.2.b.p | 6 | 1.a | even | 1 | 1 | trivial | |
| 1850.2.b.p | 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 \(S_{2}^{\mathrm{new}}(1850, [\chi])\):
|
\( T_{3}^{6} + 15T_{3}^{4} + 51T_{3}^{2} + 1 \)
|
|
\( T_{7}^{6} + 20T_{7}^{4} + 52T_{7}^{2} + 36 \)
|
|
\( T_{13}^{6} + 36T_{13}^{4} + 228T_{13}^{2} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{3} \)
|
| $3$ |
\( T^{6} + 15 T^{4} + 51 T^{2} + 1 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 20 T^{4} + 52 T^{2} + 36 \)
|
| $11$ |
\( (T^{3} + 5 T^{2} - 9 T - 51)^{2} \)
|
| $13$ |
\( T^{6} + 36 T^{4} + 228 T^{2} + 4 \)
|
| $17$ |
\( T^{6} + 31 T^{4} + 123 T^{2} + 9 \)
|
| $19$ |
\( (T^{3} - T^{2} - 25 T - 11)^{2} \)
|
| $23$ |
\( T^{6} + 64 T^{4} + 960 T^{2} + \cdots + 2304 \)
|
| $29$ |
\( (T^{3} - 14 T^{2} + 36 T + 24)^{2} \)
|
| $31$ |
\( (T^{3} - 6 T^{2} - 36 T + 214)^{2} \)
|
| $37$ |
\( (T^{2} + 1)^{3} \)
|
| $41$ |
\( (T^{3} - 19 T^{2} + 51 T + 399)^{2} \)
|
| $43$ |
\( T^{6} + 248 T^{4} + 18064 T^{2} + \cdots + 318096 \)
|
| $47$ |
\( T^{6} \)
|
| $53$ |
\( T^{6} + 204 T^{4} + 9648 T^{2} + \cdots + 46656 \)
|
| $59$ |
\( (T^{3} - 8 T^{2} - 60 T - 84)^{2} \)
|
| $61$ |
\( (T^{3} - 12 T^{2} - 18 T + 238)^{2} \)
|
| $67$ |
\( T^{6} + 159 T^{4} + 5523 T^{2} + \cdots + 289 \)
|
| $71$ |
\( (T^{3} - 8 T^{2} - 54 T + 378)^{2} \)
|
| $73$ |
\( T^{6} + 107 T^{4} + 1195 T^{2} + \cdots + 81 \)
|
| $79$ |
\( (T^{3} + 2 T^{2} - 100 T + 88)^{2} \)
|
| $83$ |
\( T^{6} + 211 T^{4} + 3231 T^{2} + \cdots + 3969 \)
|
| $89$ |
\( (T^{3} + T^{2} - 189 T - 147)^{2} \)
|
| $97$ |
\( T^{6} + 204 T^{4} + 240 T^{2} + \cdots + 64 \)
|
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