Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [190,4,Mod(1,190)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(190, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("190.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 190 = 2 \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 190.a (trivial) |
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: | yes |
| Analytic conductor: | \(11.2103629011\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{313}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 78 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{313})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.00000 | −8.34590 | 4.00000 | 5.00000 | 16.6918 | −22.3459 | −8.00000 | 42.6541 | −10.0000 | ||||||||||||||||||||||||
| 1.2 | −2.00000 | 9.34590 | 4.00000 | 5.00000 | −18.6918 | −4.65410 | −8.00000 | 60.3459 | −10.0000 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(19\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 190.4.a.d | ✓ | 2 |
| 3.b | odd | 2 | 1 | 1710.4.a.p | 2 | ||
| 4.b | odd | 2 | 1 | 1520.4.a.k | 2 | ||
| 5.b | even | 2 | 1 | 950.4.a.i | 2 | ||
| 5.c | odd | 4 | 2 | 950.4.b.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 190.4.a.d | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 950.4.a.i | 2 | 5.b | even | 2 | 1 | ||
| 950.4.b.f | 4 | 5.c | odd | 4 | 2 | ||
| 1520.4.a.k | 2 | 4.b | odd | 2 | 1 | ||
| 1710.4.a.p | 2 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - T_{3} - 78 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(190))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{2} \)
|
| $3$ |
\( T^{2} - T - 78 \)
|
| $5$ |
\( (T - 5)^{2} \)
|
| $7$ |
\( T^{2} + 27T + 104 \)
|
| $11$ |
\( T^{2} + 52T - 576 \)
|
| $13$ |
\( T^{2} + 17T - 632 \)
|
| $17$ |
\( T^{2} - 75T + 702 \)
|
| $19$ |
\( (T - 19)^{2} \)
|
| $23$ |
\( T^{2} + 203T + 6468 \)
|
| $29$ |
\( T^{2} - 183T + 2034 \)
|
| $31$ |
\( T^{2} + 18T - 52816 \)
|
| $37$ |
\( T^{2} - 78T - 36352 \)
|
| $41$ |
\( (T - 390)^{2} \)
|
| $43$ |
\( T^{2} - 338T + 28248 \)
|
| $47$ |
\( T^{2} - 24T - 44928 \)
|
| $53$ |
\( T^{2} + T - 219804 \)
|
| $59$ |
\( T^{2} + 687T - 40464 \)
|
| $61$ |
\( T^{2} + 1222 T + 357984 \)
|
| $67$ |
\( T^{2} + 43T - 350802 \)
|
| $71$ |
\( T^{2} - 1500 T + 461088 \)
|
| $73$ |
\( T^{2} + 983T + 122554 \)
|
| $79$ |
\( T^{2} - 750T + 140312 \)
|
| $83$ |
\( T^{2} - 1010 T - 323712 \)
|
| $89$ |
\( T^{2} - 164T - 599244 \)
|
| $97$ |
\( T^{2} + 954T - 523984 \)
|
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