Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [105,6,Mod(1,105)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(105, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("105.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 105.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: | \(16.8403010804\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(\beta = 4\sqrt{2}\). 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 |
|
−7.65685 | −9.00000 | 26.6274 | 25.0000 | 68.9117 | −49.0000 | 41.1371 | 81.0000 | −191.421 | ||||||||||||||||||||||||
| 1.2 | 3.65685 | −9.00000 | −18.6274 | 25.0000 | −32.9117 | −49.0000 | −185.137 | 81.0000 | 91.4214 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
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 | 105.6.a.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 315.6.a.f | 2 | ||
| 5.b | even | 2 | 1 | 525.6.a.h | 2 | ||
| 5.c | odd | 4 | 2 | 525.6.d.h | 4 | ||
| 7.b | odd | 2 | 1 | 735.6.a.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.6.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 315.6.a.f | 2 | 3.b | odd | 2 | 1 | ||
| 525.6.a.h | 2 | 5.b | even | 2 | 1 | ||
| 525.6.d.h | 4 | 5.c | odd | 4 | 2 | ||
| 735.6.a.e | 2 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 4T_{2} - 28 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(105))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 4T - 28 \)
|
| $3$ |
\( (T + 9)^{2} \)
|
| $5$ |
\( (T - 25)^{2} \)
|
| $7$ |
\( (T + 49)^{2} \)
|
| $11$ |
\( T^{2} + 176T - 312256 \)
|
| $13$ |
\( T^{2} - 692T + 114308 \)
|
| $17$ |
\( T^{2} + 428 T - 1603132 \)
|
| $19$ |
\( T^{2} + 1824 T + 724096 \)
|
| $23$ |
\( T^{2} - 8032 T + 15079904 \)
|
| $29$ |
\( T^{2} + 2948 T - 8333852 \)
|
| $31$ |
\( T^{2} - 14760 T + 51973488 \)
|
| $37$ |
\( T^{2} - 156 T - 176563548 \)
|
| $41$ |
\( T^{2} + 5980 T - 85728700 \)
|
| $43$ |
\( T^{2} - 25672 T + 124406768 \)
|
| $47$ |
\( T^{2} - 21904 T + 112481056 \)
|
| $53$ |
\( T^{2} + 53948 T + 726392324 \)
|
| $59$ |
\( T^{2} - 26296 T - 882188144 \)
|
| $61$ |
\( T^{2} + 16788 T - 251308476 \)
|
| $67$ |
\( T^{2} - 16264 T - 4247824 \)
|
| $71$ |
\( T^{2} - 22264 T + 123338224 \)
|
| $73$ |
\( T^{2} - 28684 T - 157081084 \)
|
| $79$ |
\( T^{2} + \cdots - 2543446976 \)
|
| $83$ |
\( T^{2} + \cdots - 1894968176 \)
|
| $89$ |
\( T^{2} + \cdots - 8699912828 \)
|
| $97$ |
\( T^{2} + \cdots + 14224705732 \)
|
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