Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [70,6,Mod(1,70)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(70, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("70.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 70.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: | \(11.2268673869\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{3369}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 842 \)
|
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{3369})\). 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 |
|
4.00000 | −27.5215 | 16.0000 | −25.0000 | −110.086 | −49.0000 | 64.0000 | 514.435 | −100.000 | ||||||||||||||||||||||||
| 1.2 | 4.00000 | 30.5215 | 16.0000 | −25.0000 | 122.086 | −49.0000 | 64.0000 | 688.565 | −100.000 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( +1 \) |
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 | 70.6.a.g | ✓ | 2 |
| 3.b | odd | 2 | 1 | 630.6.a.u | 2 | ||
| 4.b | odd | 2 | 1 | 560.6.a.m | 2 | ||
| 5.b | even | 2 | 1 | 350.6.a.q | 2 | ||
| 5.c | odd | 4 | 2 | 350.6.c.j | 4 | ||
| 7.b | odd | 2 | 1 | 490.6.a.v | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.6.a.g | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 350.6.a.q | 2 | 5.b | even | 2 | 1 | ||
| 350.6.c.j | 4 | 5.c | odd | 4 | 2 | ||
| 490.6.a.v | 2 | 7.b | odd | 2 | 1 | ||
| 560.6.a.m | 2 | 4.b | odd | 2 | 1 | ||
| 630.6.a.u | 2 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 3T_{3} - 840 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(70))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{2} \)
|
| $3$ |
\( T^{2} - 3T - 840 \)
|
| $5$ |
\( (T + 25)^{2} \)
|
| $7$ |
\( (T + 49)^{2} \)
|
| $11$ |
\( T^{2} - 959T + 222340 \)
|
| $13$ |
\( T^{2} + 393T - 150894 \)
|
| $17$ |
\( T^{2} + 2231 T + 1176118 \)
|
| $19$ |
\( T^{2} - 3342 T + 2761920 \)
|
| $23$ |
\( T^{2} + 450 T - 18900000 \)
|
| $29$ |
\( T^{2} + 4515 T - 19531926 \)
|
| $31$ |
\( T^{2} + 2036 T - 34014752 \)
|
| $37$ |
\( T^{2} + 1928 T - 5013620 \)
|
| $41$ |
\( T^{2} - 19318 T + 26317192 \)
|
| $43$ |
\( T^{2} + 14146 T + 36655768 \)
|
| $47$ |
\( T^{2} + 19745 T + 85936696 \)
|
| $53$ |
\( T^{2} - 26378 T + 75436792 \)
|
| $59$ |
\( T^{2} + 11104 T - 529992512 \)
|
| $61$ |
\( T^{2} + \cdots - 1197584192 \)
|
| $67$ |
\( T^{2} - 65208 T - 54732528 \)
|
| $71$ |
\( T^{2} - 17104 T - 307209920 \)
|
| $73$ |
\( T^{2} + \cdots - 3004645004 \)
|
| $79$ |
\( T^{2} - 72245 T + 662931856 \)
|
| $83$ |
\( T^{2} - 99676 T + 527636608 \)
|
| $89$ |
\( T^{2} + \cdots - 5491535720 \)
|
| $97$ |
\( T^{2} + \cdots - 4737795650 \)
|
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