Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [68,6,Mod(1,68)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(68, 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("68.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 68 = 2^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 68.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: | \(10.9060997473\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{13}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(\beta = 4\sqrt{13}\). 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 |
|
0 | −18.4222 | 0 | 18.8444 | 0 | 222.644 | 0 | 96.3776 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 10.4222 | 0 | −38.8444 | 0 | −94.6443 | 0 | −134.378 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(17\) | \(-1\) |
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 | 68.6.a.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 612.6.a.a | 2 | ||
| 4.b | odd | 2 | 1 | 272.6.a.h | 2 | ||
| 8.b | even | 2 | 1 | 1088.6.a.p | 2 | ||
| 8.d | odd | 2 | 1 | 1088.6.a.k | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 68.6.a.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 272.6.a.h | 2 | 4.b | odd | 2 | 1 | ||
| 612.6.a.a | 2 | 3.b | odd | 2 | 1 | ||
| 1088.6.a.k | 2 | 8.d | odd | 2 | 1 | ||
| 1088.6.a.p | 2 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 8T_{3} - 192 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(68))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 8T - 192 \)
|
| $5$ |
\( T^{2} + 20T - 732 \)
|
| $7$ |
\( T^{2} - 128T - 21072 \)
|
| $11$ |
\( T^{2} + 856T + 158016 \)
|
| $13$ |
\( T^{2} + 996T + 35012 \)
|
| $17$ |
\( (T - 289)^{2} \)
|
| $19$ |
\( T^{2} + 3096 T + 2095952 \)
|
| $23$ |
\( T^{2} - 1824 T - 4426704 \)
|
| $29$ |
\( T^{2} + 4900 T + 2906628 \)
|
| $31$ |
\( T^{2} + 2672 T + 1783024 \)
|
| $37$ |
\( T^{2} + 8724 T - 65638108 \)
|
| $41$ |
\( T^{2} + 13292 T - 25801884 \)
|
| $43$ |
\( T^{2} - 16664 T + 37133136 \)
|
| $47$ |
\( T^{2} - 9120 T - 361663488 \)
|
| $53$ |
\( T^{2} - 972 T - 4825692 \)
|
| $59$ |
\( T^{2} - 59832 T + 604365264 \)
|
| $61$ |
\( T^{2} - 10972 T - 49343996 \)
|
| $67$ |
\( T^{2} + 14104 T - 11823984 \)
|
| $71$ |
\( T^{2} - 51744 T + 441348912 \)
|
| $73$ |
\( T^{2} + \cdots - 1349543836 \)
|
| $79$ |
\( T^{2} + 35184 T - 520027408 \)
|
| $83$ |
\( T^{2} + \cdots - 3827008176 \)
|
| $89$ |
\( T^{2} + \cdots - 1032743196 \)
|
| $97$ |
\( T^{2} + \cdots + 4080403076 \)
|
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