Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [70,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("70.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(21.8669517839\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{1401}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 350 \)
|
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{1401})\). 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 |
|
−8.00000 | 3.78503 | 64.0000 | 125.000 | −30.2803 | −343.000 | −512.000 | −2172.67 | −1000.00 | ||||||||||||||||||||||||
| 1.2 | −8.00000 | 41.2150 | 64.0000 | 125.000 | −329.720 | −343.000 | −512.000 | −488.326 | −1000.00 | |||||||||||||||||||||||||
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 | 70.8.a.f | ✓ | 2 |
| 4.b | odd | 2 | 1 | 560.8.a.c | 2 | ||
| 5.b | even | 2 | 1 | 350.8.a.m | 2 | ||
| 5.c | odd | 4 | 2 | 350.8.c.e | 4 | ||
| 7.b | odd | 2 | 1 | 490.8.a.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.8.a.f | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 350.8.a.m | 2 | 5.b | even | 2 | 1 | ||
| 350.8.c.e | 4 | 5.c | odd | 4 | 2 | ||
| 490.8.a.f | 2 | 7.b | odd | 2 | 1 | ||
| 560.8.a.c | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 45T_{3} + 156 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(70))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{2} \)
|
| $3$ |
\( T^{2} - 45T + 156 \)
|
| $5$ |
\( (T - 125)^{2} \)
|
| $7$ |
\( (T + 343)^{2} \)
|
| $11$ |
\( T^{2} + 2845 T - 45666884 \)
|
| $13$ |
\( T^{2} - 753 T - 47548638 \)
|
| $17$ |
\( T^{2} + 6413 T - 762098414 \)
|
| $19$ |
\( T^{2} - 2502 T - 39401640 \)
|
| $23$ |
\( T^{2} + \cdots - 6348384720 \)
|
| $29$ |
\( T^{2} + \cdots - 6678017910 \)
|
| $31$ |
\( T^{2} + \cdots + 9262600192 \)
|
| $37$ |
\( T^{2} + \cdots + 150540514516 \)
|
| $41$ |
\( T^{2} + \cdots + 210810680032 \)
|
| $43$ |
\( T^{2} + \cdots - 36921167192 \)
|
| $47$ |
\( T^{2} + \cdots + 7157227168 \)
|
| $53$ |
\( T^{2} + \cdots - 2892684029072 \)
|
| $59$ |
\( T^{2} + \cdots + 115168186000 \)
|
| $61$ |
\( T^{2} + \cdots - 182619457760 \)
|
| $67$ |
\( T^{2} + \cdots + 5774798275776 \)
|
| $71$ |
\( T^{2} + \cdots + 76517476672 \)
|
| $73$ |
\( T^{2} + \cdots + 1669337367340 \)
|
| $79$ |
\( T^{2} + \cdots - 12471302443760 \)
|
| $83$ |
\( T^{2} + \cdots + 53719607334736 \)
|
| $89$ |
\( T^{2} + \cdots - 76016036072600 \)
|
| $97$ |
\( T^{2} + \cdots - 46589336442950 \)
|
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