Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,3,Mod(43,350)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(350, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.43");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 350.f (of order \(4\), degree \(2\), 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: | \(9.53680925261\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.12745506816.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 23x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 23x^{4} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 19\nu ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 29\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + 24\nu^{2} ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{4} + 23 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{6} - 22\nu^{2} \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -4\nu^{7} - 91\nu^{3} ) / 5 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 6\nu^{7} + 139\nu^{3} ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 5\beta_{3} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} + 3\beta_{6} \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5\beta_{4} - 23 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -19\beta_{2} + 29\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -12\beta_{5} - 55\beta_{3} \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -91\beta_{7} - 139\beta_{6} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 43.1 |
|
1.00000 | − | 1.00000i | −4.09557 | − | 4.09557i | − | 2.00000i | 0 | −8.19115 | −1.87083 | + | 1.87083i | −2.00000 | − | 2.00000i | 24.5474i | 0 | |||||||||||||||||||||||||||||||||
| 43.2 | 1.00000 | − | 1.00000i | −1.64608 | − | 1.64608i | − | 2.00000i | 0 | −3.29217 | −1.87083 | + | 1.87083i | −2.00000 | − | 2.00000i | − | 3.58082i | 0 | |||||||||||||||||||||||||||||||||
| 43.3 | 1.00000 | − | 1.00000i | −0.353916 | − | 0.353916i | − | 2.00000i | 0 | −0.707832 | 1.87083 | − | 1.87083i | −2.00000 | − | 2.00000i | − | 8.74949i | 0 | |||||||||||||||||||||||||||||||||
| 43.4 | 1.00000 | − | 1.00000i | 2.09557 | + | 2.09557i | − | 2.00000i | 0 | 4.19115 | 1.87083 | − | 1.87083i | −2.00000 | − | 2.00000i | − | 0.217143i | 0 | |||||||||||||||||||||||||||||||||
| 57.1 | 1.00000 | + | 1.00000i | −4.09557 | + | 4.09557i | 2.00000i | 0 | −8.19115 | −1.87083 | − | 1.87083i | −2.00000 | + | 2.00000i | − | 24.5474i | 0 | ||||||||||||||||||||||||||||||||||
| 57.2 | 1.00000 | + | 1.00000i | −1.64608 | + | 1.64608i | 2.00000i | 0 | −3.29217 | −1.87083 | − | 1.87083i | −2.00000 | + | 2.00000i | 3.58082i | 0 | |||||||||||||||||||||||||||||||||||
| 57.3 | 1.00000 | + | 1.00000i | −0.353916 | + | 0.353916i | 2.00000i | 0 | −0.707832 | 1.87083 | + | 1.87083i | −2.00000 | + | 2.00000i | 8.74949i | 0 | |||||||||||||||||||||||||||||||||||
| 57.4 | 1.00000 | + | 1.00000i | 2.09557 | − | 2.09557i | 2.00000i | 0 | 4.19115 | 1.87083 | + | 1.87083i | −2.00000 | + | 2.00000i | 0.217143i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 350.3.f.f | yes | 8 |
| 5.b | even | 2 | 1 | 350.3.f.e | ✓ | 8 | |
| 5.c | odd | 4 | 1 | 350.3.f.e | ✓ | 8 | |
| 5.c | odd | 4 | 1 | inner | 350.3.f.f | yes | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 350.3.f.e | ✓ | 8 | 5.b | even | 2 | 1 | |
| 350.3.f.e | ✓ | 8 | 5.c | odd | 4 | 1 | |
| 350.3.f.f | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 350.3.f.f | yes | 8 | 5.c | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 8T_{3}^{7} + 32T_{3}^{6} + 104T_{3}^{4} + 672T_{3}^{3} + 2048T_{3}^{2} + 1280T_{3} + 400 \)
acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 2)^{4} \)
|
| $3$ |
\( T^{8} + 8 T^{7} + \cdots + 400 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{4} + 49)^{2} \)
|
| $11$ |
\( (T^{4} - 4 T^{3} + \cdots - 215)^{2} \)
|
| $13$ |
\( T^{8} + 16 T^{7} + \cdots + 29584 \)
|
| $17$ |
\( T^{8} + 16 T^{7} + \cdots + 121176064 \)
|
| $19$ |
\( (T^{4} + 616 T^{2} + 19600)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 434775390625 \)
|
| $29$ |
\( T^{8} + \cdots + 14637370225 \)
|
| $31$ |
\( (T^{4} - 12 T^{3} + \cdots + 21676)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 1426950625 \)
|
| $41$ |
\( (T^{4} + 4 T^{3} + \cdots - 429236)^{2} \)
|
| $43$ |
\( T^{8} - 88 T^{7} + \cdots + 134351281 \)
|
| $47$ |
\( T^{8} + \cdots + 1870854955264 \)
|
| $53$ |
\( T^{8} + \cdots + 6039719117056 \)
|
| $59$ |
\( T^{8} + \cdots + 1282103290000 \)
|
| $61$ |
\( (T^{4} + 4 T^{3} + \cdots + 244780)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 23\!\cdots\!09 \)
|
| $71$ |
\( (T^{4} - 12 T^{3} + \cdots + 43192465)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 245376560250000 \)
|
| $79$ |
\( T^{8} + \cdots + 15\!\cdots\!25 \)
|
| $83$ |
\( T^{8} + \cdots + 599515812640144 \)
|
| $89$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
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