Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [354,6,Mod(1,354)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(354, 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("354.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 354 = 2 \cdot 3 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 354.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: | \(56.7758722138\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 290x^{3} - 616x^{2} + 4720x + 11900 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 290x^{3} - 616x^{2} + 4720x + 11900 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -29\nu^{4} + 189\nu^{3} + 7600\nu^{2} - 16386\nu - 96440 ) / 4500 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 29\nu^{4} - 189\nu^{3} - 7600\nu^{2} + 25386\nu + 93440 ) / 1500 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -47\nu^{4} + 177\nu^{3} + 12925\nu^{2} - 6798\nu - 159545 ) / 1125 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -89\nu^{4} + 399\nu^{3} + 24850\nu^{2} - 33426\nu - 374540 ) / 2250 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 10\beta_{4} - 7\beta_{3} + 8\beta_{2} + 8\beta _1 + 345 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 47\beta_{4} - 59\beta_{3} + 114\beta_{2} + 436\beta _1 + 1699 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2927\beta_{4} - 2219\beta_{3} + 2557\beta_{2} + 3625\beta _1 + 90945 ) / 3 \)
|
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 | 9.00000 | 16.0000 | −83.8793 | −36.0000 | −13.6774 | −64.0000 | 81.0000 | 335.517 | |||||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | 9.00000 | 16.0000 | −8.96408 | −36.0000 | −187.945 | −64.0000 | 81.0000 | 35.8563 | ||||||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 9.00000 | 16.0000 | −3.84671 | −36.0000 | 56.0281 | −64.0000 | 81.0000 | 15.3869 | ||||||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | 9.00000 | 16.0000 | 19.3501 | −36.0000 | 148.650 | −64.0000 | 81.0000 | −77.4004 | ||||||||||||||||||||||||||||||||||
| 1.5 | −4.00000 | 9.00000 | 16.0000 | 67.3400 | −36.0000 | −165.056 | −64.0000 | 81.0000 | −269.360 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(59\) | \(-1\) |
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 | 354.6.a.c | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1062.6.a.f | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 354.6.a.c | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1062.6.a.f | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} + 10T_{5}^{4} - 5970T_{5}^{3} + 32740T_{5}^{2} + 1194385T_{5} + 3768834 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(354))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{5} \)
|
| $3$ |
\( (T - 9)^{5} \)
|
| $5$ |
\( T^{5} + 10 T^{4} + \cdots + 3768834 \)
|
| $7$ |
\( T^{5} + \cdots + 3533772744 \)
|
| $11$ |
\( T^{5} + \cdots + 23908389964 \)
|
| $13$ |
\( T^{5} + \cdots + 23599739013786 \)
|
| $17$ |
\( T^{5} + \cdots - 20047268190894 \)
|
| $19$ |
\( T^{5} + \cdots - 10\!\cdots\!52 \)
|
| $23$ |
\( T^{5} + \cdots - 45\!\cdots\!72 \)
|
| $29$ |
\( T^{5} + \cdots + 42\!\cdots\!42 \)
|
| $31$ |
\( T^{5} + \cdots - 10\!\cdots\!30 \)
|
| $37$ |
\( T^{5} + \cdots + 23\!\cdots\!78 \)
|
| $41$ |
\( T^{5} + \cdots + 11\!\cdots\!42 \)
|
| $43$ |
\( T^{5} + \cdots + 13\!\cdots\!50 \)
|
| $47$ |
\( T^{5} + \cdots - 14\!\cdots\!08 \)
|
| $53$ |
\( T^{5} + \cdots + 45\!\cdots\!34 \)
|
| $59$ |
\( (T - 3481)^{5} \)
|
| $61$ |
\( T^{5} + \cdots - 15\!\cdots\!74 \)
|
| $67$ |
\( T^{5} + \cdots - 34\!\cdots\!06 \)
|
| $71$ |
\( T^{5} + \cdots + 40\!\cdots\!60 \)
|
| $73$ |
\( T^{5} + \cdots + 38\!\cdots\!94 \)
|
| $79$ |
\( T^{5} + \cdots - 86\!\cdots\!80 \)
|
| $83$ |
\( T^{5} + \cdots + 26\!\cdots\!36 \)
|
| $89$ |
\( T^{5} + \cdots - 80\!\cdots\!48 \)
|
| $97$ |
\( T^{5} + \cdots - 32\!\cdots\!34 \)
|
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