Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [137,2,Mod(1,137)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(137, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("137.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 137 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 137.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: | \(1.09395050769\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 10x^{5} + 28x^{3} - 3x^{2} - 19x + 7 \)
|
| 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 a basis \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 10x^{5} + 28x^{3} - 3x^{2} - 19x + 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 9\nu^{4} - 9\nu^{3} + 21\nu^{2} + 16\nu - 11 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 11\nu^{4} - 9\nu^{3} + 33\nu^{2} + 18\nu - 21 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{6} + 9\nu^{4} + \nu^{3} - 21\nu^{2} - 3\nu + 9 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} + \nu^{5} - 10\nu^{4} - 8\nu^{3} + 26\nu^{2} + 13\nu - 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 6\beta_{2} + \beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{6} + \beta_{5} - 8\beta_{4} - 6\beta_{3} + 8\beta_{2} + 19\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} - \beta_{5} - 10\beta_{4} + 8\beta_{3} + 34\beta_{2} + 10\beta _1 + 63 \)
|
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 |
|
−2.44249 | 1.77399 | 3.96576 | 0.640882 | −4.33295 | 0.952315 | −4.80136 | 0.147029 | −1.56535 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.76852 | −2.27749 | 1.12767 | −2.86766 | 4.02778 | 2.98969 | 1.54274 | 2.18694 | 5.07151 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.658869 | −0.372025 | −1.56589 | 2.88542 | 0.245115 | 1.80738 | 2.34946 | −2.86160 | −1.90111 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.487445 | 3.00378 | −1.76240 | −1.93572 | −1.46417 | 5.15852 | 1.83396 | 6.02267 | 0.943556 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.14107 | 1.78799 | −0.697961 | 1.22468 | 2.04022 | −1.35493 | −3.07856 | 0.196906 | 1.39745 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.95911 | −1.85245 | 1.83811 | 1.84548 | −3.62915 | 4.79800 | −0.317154 | 0.431564 | 3.61550 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.25715 | 0.936207 | 3.09471 | −3.79309 | 2.11315 | 0.649019 | 2.47092 | −2.12352 | −8.56156 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(137\) | \(-1\) |
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 | 137.2.a.b | ✓ | 7 |
| 3.b | odd | 2 | 1 | 1233.2.a.f | 7 | ||
| 4.b | odd | 2 | 1 | 2192.2.a.l | 7 | ||
| 5.b | even | 2 | 1 | 3425.2.a.c | 7 | ||
| 7.b | odd | 2 | 1 | 6713.2.a.d | 7 | ||
| 8.b | even | 2 | 1 | 8768.2.a.z | 7 | ||
| 8.d | odd | 2 | 1 | 8768.2.a.ba | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 137.2.a.b | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1233.2.a.f | 7 | 3.b | odd | 2 | 1 | ||
| 2192.2.a.l | 7 | 4.b | odd | 2 | 1 | ||
| 3425.2.a.c | 7 | 5.b | even | 2 | 1 | ||
| 6713.2.a.d | 7 | 7.b | odd | 2 | 1 | ||
| 8768.2.a.z | 7 | 8.b | even | 2 | 1 | ||
| 8768.2.a.ba | 7 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 10T_{2}^{5} + 28T_{2}^{3} + 3T_{2}^{2} - 19T_{2} - 7 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(137))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 10 T^{5} + \cdots - 7 \)
|
| $3$ |
\( T^{7} - 3 T^{6} + \cdots + 14 \)
|
| $5$ |
\( T^{7} + 2 T^{6} + \cdots + 88 \)
|
| $7$ |
\( T^{7} - 15 T^{6} + \cdots + 112 \)
|
| $11$ |
\( T^{7} + 3 T^{6} + \cdots + 16 \)
|
| $13$ |
\( T^{7} - 12 T^{6} + \cdots + 488 \)
|
| $17$ |
\( T^{7} + 6 T^{6} + \cdots - 4 \)
|
| $19$ |
\( T^{7} - 10 T^{6} + \cdots - 16 \)
|
| $23$ |
\( T^{7} + 3 T^{6} + \cdots - 11606 \)
|
| $29$ |
\( T^{7} + 9 T^{6} + \cdots + 7576 \)
|
| $31$ |
\( T^{7} - 13 T^{6} + \cdots + 98 \)
|
| $37$ |
\( T^{7} + 2 T^{6} + \cdots + 2332 \)
|
| $41$ |
\( T^{7} + T^{6} + \cdots + 7256 \)
|
| $43$ |
\( T^{7} - 7 T^{6} + \cdots - 12146 \)
|
| $47$ |
\( T^{7} - 15 T^{6} + \cdots + 9634 \)
|
| $53$ |
\( T^{7} + 8 T^{6} + \cdots + 15464 \)
|
| $59$ |
\( T^{7} + 6 T^{6} + \cdots + 232768 \)
|
| $61$ |
\( T^{7} - T^{6} + \cdots - 7285532 \)
|
| $67$ |
\( T^{7} - 24 T^{6} + \cdots + 184654 \)
|
| $71$ |
\( T^{7} - 16 T^{6} + \cdots + 221696 \)
|
| $73$ |
\( T^{7} + T^{6} + \cdots + 298312 \)
|
| $79$ |
\( T^{7} - 15 T^{6} + \cdots - 185806 \)
|
| $83$ |
\( T^{7} - 21 T^{6} + \cdots - 86338 \)
|
| $89$ |
\( T^{7} + 8 T^{6} + \cdots + 31528168 \)
|
| $97$ |
\( T^{7} - T^{6} + \cdots - 51016 \)
|
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