Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1150,4,Mod(599,1150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1150.599");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1150 = 2 \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1150.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: | \(67.8521965066\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 91x^{4} + 2145x^{2} + 3600 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 230) |
| 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
\( x^{6} + 91x^{4} + 2145x^{2} + 3600 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 31\nu^{3} - 615\nu ) / 900 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 46\nu^{2} + 60 ) / 15 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 61\nu^{2} + 525 ) / 15 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 29\nu^{5} + 1799\nu^{3} + 21765\nu ) / 900 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} - 31 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 29\beta_{2} - 44\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -46\beta_{4} + 61\beta_{3} + 1366 \)
|
| \(\nu^{5}\) | \(=\) |
\( -31\beta_{5} + 1799\beta_{2} + 1979\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).
| \(n\) | \(51\) | \(277\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 599.1 |
|
− | 2.00000i | − | 6.84916i | −4.00000 | 0 | −13.6983 | − | 28.6094i | 8.00000i | −19.9110 | 0 | |||||||||||||||||||||||||||||||||
| 599.2 | − | 2.00000i | 1.34735i | −4.00000 | 0 | 2.69469 | 32.8794i | 8.00000i | 25.1847 | 0 | ||||||||||||||||||||||||||||||||||||
| 599.3 | − | 2.00000i | 6.50182i | −4.00000 | 0 | 13.0036 | 2.73001i | 8.00000i | −15.2736 | 0 | ||||||||||||||||||||||||||||||||||||
| 599.4 | 2.00000i | − | 6.50182i | −4.00000 | 0 | 13.0036 | − | 2.73001i | − | 8.00000i | −15.2736 | 0 | ||||||||||||||||||||||||||||||||||
| 599.5 | 2.00000i | − | 1.34735i | −4.00000 | 0 | 2.69469 | − | 32.8794i | − | 8.00000i | 25.1847 | 0 | ||||||||||||||||||||||||||||||||||
| 599.6 | 2.00000i | 6.84916i | −4.00000 | 0 | −13.6983 | 28.6094i | − | 8.00000i | −19.9110 | 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 | 1150.4.b.l | 6 | |
| 5.b | even | 2 | 1 | inner | 1150.4.b.l | 6 | |
| 5.c | odd | 4 | 1 | 230.4.a.g | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 1150.4.a.m | 3 | ||
| 15.e | even | 4 | 1 | 2070.4.a.ba | 3 | ||
| 20.e | even | 4 | 1 | 1840.4.a.j | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 230.4.a.g | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 1150.4.a.m | 3 | 5.c | odd | 4 | 1 | ||
| 1150.4.b.l | 6 | 1.a | even | 1 | 1 | trivial | |
| 1150.4.b.l | 6 | 5.b | even | 2 | 1 | inner | |
| 1840.4.a.j | 3 | 20.e | even | 4 | 1 | ||
| 2070.4.a.ba | 3 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1150, [\chi])\):
|
\( T_{3}^{6} + 91T_{3}^{4} + 2145T_{3}^{2} + 3600 \)
|
|
\( T_{7}^{6} + 1907T_{7}^{4} + 898993T_{7}^{2} + 6594624 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{3} \)
|
| $3$ |
\( T^{6} + 91 T^{4} + \cdots + 3600 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 1907 T^{4} + \cdots + 6594624 \)
|
| $11$ |
\( (T^{3} - 27 T^{2} + \cdots + 89388)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 75755956644 \)
|
| $17$ |
\( T^{6} + \cdots + 9321902500 \)
|
| $19$ |
\( (T^{3} - 185 T^{2} + \cdots - 37596)^{2} \)
|
| $23$ |
\( (T^{2} + 529)^{3} \)
|
| $29$ |
\( (T^{3} + 344 T^{2} + \cdots - 291288)^{2} \)
|
| $31$ |
\( (T^{3} - 397 T^{2} + \cdots + 1547080)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 10\!\cdots\!96 \)
|
| $41$ |
\( (T^{3} + 575 T^{2} + \cdots - 50953878)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 103264618086400 \)
|
| $47$ |
\( T^{6} + \cdots + 10137856000000 \)
|
| $53$ |
\( T^{6} + \cdots + 80\!\cdots\!16 \)
|
| $59$ |
\( (T^{3} + 142 T^{2} + \cdots + 9906704)^{2} \)
|
| $61$ |
\( (T^{3} + 49 T^{2} + \cdots - 23572158)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 12\!\cdots\!00 \)
|
| $71$ |
\( (T^{3} + 471 T^{2} + \cdots - 75603760)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 45\!\cdots\!16 \)
|
| $79$ |
\( (T^{3} - 860 T^{2} + \cdots + 8296704)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 11\!\cdots\!64 \)
|
| $89$ |
\( (T^{3} - 90 T^{2} + \cdots + 1109897568)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 47\!\cdots\!24 \)
|
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