Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3850,2,Mod(1849,3850)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3850, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3850.1849");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3850.c (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: | \(30.7424047782\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 770) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{3} \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\zeta_{12}^{2} - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( ( \beta_{2} + 1 ) / 2 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times\).
| \(n\) | \(1751\) | \(2201\) | \(2927\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1849.1 |
|
− | 1.00000i | − | 2.73205i | −1.00000 | 0 | −2.73205 | 1.00000i | 1.00000i | −4.46410 | 0 | ||||||||||||||||||||||||||||
| 1849.2 | − | 1.00000i | 0.732051i | −1.00000 | 0 | 0.732051 | 1.00000i | 1.00000i | 2.46410 | 0 | ||||||||||||||||||||||||||||||
| 1849.3 | 1.00000i | − | 0.732051i | −1.00000 | 0 | 0.732051 | − | 1.00000i | − | 1.00000i | 2.46410 | 0 | ||||||||||||||||||||||||||||
| 1849.4 | 1.00000i | 2.73205i | −1.00000 | 0 | −2.73205 | − | 1.00000i | − | 1.00000i | −4.46410 | 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 | 3850.2.c.s | 4 | |
| 5.b | even | 2 | 1 | inner | 3850.2.c.s | 4 | |
| 5.c | odd | 4 | 1 | 770.2.a.h | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 3850.2.a.bm | 2 | ||
| 15.e | even | 4 | 1 | 6930.2.a.ca | 2 | ||
| 20.e | even | 4 | 1 | 6160.2.a.v | 2 | ||
| 35.f | even | 4 | 1 | 5390.2.a.bk | 2 | ||
| 55.e | even | 4 | 1 | 8470.2.a.ce | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 770.2.a.h | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 3850.2.a.bm | 2 | 5.c | odd | 4 | 1 | ||
| 3850.2.c.s | 4 | 1.a | even | 1 | 1 | trivial | |
| 3850.2.c.s | 4 | 5.b | even | 2 | 1 | inner | |
| 5390.2.a.bk | 2 | 35.f | even | 4 | 1 | ||
| 6160.2.a.v | 2 | 20.e | even | 4 | 1 | ||
| 6930.2.a.ca | 2 | 15.e | even | 4 | 1 | ||
| 8470.2.a.ce | 2 | 55.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 \(S_{2}^{\mathrm{new}}(3850, [\chi])\):
|
\( T_{3}^{4} + 8T_{3}^{2} + 4 \)
|
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\( T_{13}^{4} + 32T_{13}^{2} + 64 \)
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\( T_{17}^{2} + 12 \)
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\( T_{19}^{2} - 2T_{19} - 2 \)
|
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\( T_{37}^{4} + 56T_{37}^{2} + 484 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
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| $3$ |
\( T^{4} + 8T^{2} + 4 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 1)^{2} \)
|
| $11$ |
\( (T - 1)^{4} \)
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| $13$ |
\( T^{4} + 32T^{2} + 64 \)
|
| $17$ |
\( (T^{2} + 12)^{2} \)
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| $19$ |
\( (T^{2} - 2 T - 2)^{2} \)
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| $23$ |
\( T^{4} + 24T^{2} + 36 \)
|
| $29$ |
\( (T^{2} - 6 T + 6)^{2} \)
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| $31$ |
\( (T^{2} - 4 T - 44)^{2} \)
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| $37$ |
\( T^{4} + 56T^{2} + 484 \)
|
| $41$ |
\( (T^{2} + 6 T + 6)^{2} \)
|
| $43$ |
\( T^{4} + 104T^{2} + 1936 \)
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| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} + 24T^{2} + 36 \)
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| $59$ |
\( (T^{2} - 192)^{2} \)
|
| $61$ |
\( (T - 2)^{4} \)
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| $67$ |
\( T^{4} + 128T^{2} + 1024 \)
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| $71$ |
\( (T^{2} - 12 T + 24)^{2} \)
|
| $73$ |
\( T^{4} + 152T^{2} + 2704 \)
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| $79$ |
\( (T^{2} + 10 T + 22)^{2} \)
|
| $83$ |
\( T^{4} + 288T^{2} + 5184 \)
|
| $89$ |
\( (T^{2} - 24 T + 132)^{2} \)
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| $97$ |
\( T^{4} + 296T^{2} + 8836 \)
|
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