Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [68,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("68.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 68 = 2^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(4.01212988039\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1524.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 7x + 1 \)
|
| 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{3} - x^{2} - 7x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} - 2\nu - 9 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} + \beta _1 + 19 ) / 4 \)
|
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 | −8.24143 | 0 | 9.43826 | 0 | 9.92665 | 0 | 40.9211 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 5.36156 | 0 | −2.53055 | 0 | 32.2301 | 0 | 1.74633 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 6.87987 | 0 | 19.0923 | 0 | −34.1567 | 0 | 20.3326 | 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 | 68.4.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 612.4.a.g | 3 | ||
| 4.b | odd | 2 | 1 | 272.4.a.i | 3 | ||
| 5.b | even | 2 | 1 | 1700.4.a.d | 3 | ||
| 5.c | odd | 4 | 2 | 1700.4.e.d | 6 | ||
| 8.b | even | 2 | 1 | 1088.4.a.t | 3 | ||
| 8.d | odd | 2 | 1 | 1088.4.a.w | 3 | ||
| 12.b | even | 2 | 1 | 2448.4.a.ba | 3 | ||
| 17.b | even | 2 | 1 | 1156.4.a.g | 3 | ||
| 17.c | even | 4 | 2 | 1156.4.b.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 68.4.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 272.4.a.i | 3 | 4.b | odd | 2 | 1 | ||
| 612.4.a.g | 3 | 3.b | odd | 2 | 1 | ||
| 1088.4.a.t | 3 | 8.b | even | 2 | 1 | ||
| 1088.4.a.w | 3 | 8.d | odd | 2 | 1 | ||
| 1156.4.a.g | 3 | 17.b | even | 2 | 1 | ||
| 1156.4.b.e | 6 | 17.c | even | 4 | 2 | ||
| 1700.4.a.d | 3 | 5.b | even | 2 | 1 | ||
| 1700.4.e.d | 6 | 5.c | odd | 4 | 2 | ||
| 2448.4.a.ba | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 4T_{3}^{2} - 64T_{3} + 304 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(68))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 4 T^{2} + \cdots + 304 \)
|
| $5$ |
\( T^{3} - 26 T^{2} + \cdots + 456 \)
|
| $7$ |
\( T^{3} - 8 T^{2} + \cdots + 10928 \)
|
| $11$ |
\( T^{3} - 60 T^{2} + \cdots + 7056 \)
|
| $13$ |
\( T^{3} - 82 T^{2} + \cdots + 19752 \)
|
| $17$ |
\( (T - 17)^{3} \)
|
| $19$ |
\( T^{3} + 124 T^{2} + \cdots - 695104 \)
|
| $23$ |
\( T^{3} + 120 T^{2} + \cdots - 253584 \)
|
| $29$ |
\( T^{3} + 46 T^{2} + \cdots - 1157016 \)
|
| $31$ |
\( T^{3} + 272 T^{2} + \cdots - 10158768 \)
|
| $37$ |
\( T^{3} - 186 T^{2} + \cdots - 1352504 \)
|
| $41$ |
\( T^{3} + 82 T^{2} + \cdots - 5658408 \)
|
| $43$ |
\( T^{3} - 300 T^{2} + \cdots + 10758208 \)
|
| $47$ |
\( T^{3} + 224 T^{2} + \cdots - 16309248 \)
|
| $53$ |
\( T^{3} - 202 T^{2} + \cdots + 3872712 \)
|
| $59$ |
\( T^{3} + 612 T^{2} + \cdots - 35759808 \)
|
| $61$ |
\( T^{3} - 786 T^{2} + \cdots + 3268072 \)
|
| $67$ |
\( T^{3} - 916 T^{2} + \cdots + 27387456 \)
|
| $71$ |
\( T^{3} + 1272 T^{2} + \cdots - 541975536 \)
|
| $73$ |
\( T^{3} + 306 T^{2} + \cdots - 6817448 \)
|
| $79$ |
\( T^{3} - 1568 T^{2} + \cdots + 180633008 \)
|
| $83$ |
\( T^{3} - 948 T^{2} + \cdots + 470015424 \)
|
| $89$ |
\( T^{3} - 478 T^{2} + \cdots - 505869288 \)
|
| $97$ |
\( T^{3} - 1318 T^{2} + \cdots + 137263992 \)
|
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