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: |
\( x^{5} - x^{4} - 835x^{3} + 14269x^{2} - 82497x + 143433 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{5} - x^{4} - 835x^{3} + 14269x^{2} - 82497x + 143433 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -14\nu^{4} - 115\nu^{3} + 10397\nu^{2} - 107621\nu + 298110 ) / 1089 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 29\nu^{4} + 316\nu^{3} - 20681\nu^{2} + 165290\nu - 264522 ) / 1089 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -70\nu^{4} - 575\nu^{3} + 53074\nu^{2} - 510880\nu + 1120290 ) / 1089 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 5\beta_{2} - 25\beta _1 + 340 \)
|
| \(\nu^{3}\) | \(=\) |
\( -11\beta_{4} + 14\beta_{3} + 84\beta_{2} + 1016\beta _1 - 8278 \)
|
| \(\nu^{4}\) | \(=\) |
\( 833\beta_{4} - 115\beta_{3} - 4481\beta_{2} - 34599\beta _1 + 341790 \)
|
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 | −100.786 | −36.0000 | −35.1732 | 64.0000 | 81.0000 | −403.143 | |||||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −9.00000 | 16.0000 | −21.6596 | −36.0000 | −77.1916 | 64.0000 | 81.0000 | −86.6382 | ||||||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | −9.00000 | 16.0000 | −9.50125 | −36.0000 | −69.2606 | 64.0000 | 81.0000 | −38.0050 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | −9.00000 | 16.0000 | 36.7026 | −36.0000 | −14.5171 | 64.0000 | 81.0000 | 146.810 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.00000 | −9.00000 | 16.0000 | 71.2441 | −36.0000 | 93.1425 | 64.0000 | 81.0000 | 284.976 | ||||||||||||||||||||||||||||||||||
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.d | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1062.6.a.d | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 354.6.a.d | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1062.6.a.d | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} + 24T_{5}^{4} - 8282T_{5}^{3} + 4532T_{5}^{2} + 6511281T_{5} + 54234444 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(354))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{5} \)
|
| $3$ |
\( (T + 9)^{5} \)
|
| $5$ |
\( T^{5} + 24 T^{4} + \cdots + 54234444 \)
|
| $7$ |
\( T^{5} + 103 T^{4} + \cdots - 254270063 \)
|
| $11$ |
\( T^{5} + \cdots + 5061448630155 \)
|
| $13$ |
\( T^{5} + \cdots + 73775492505089 \)
|
| $17$ |
\( T^{5} + \cdots + 913694892344575 \)
|
| $19$ |
\( T^{5} + \cdots - 39\!\cdots\!72 \)
|
| $23$ |
\( T^{5} + \cdots - 17\!\cdots\!50 \)
|
| $29$ |
\( T^{5} + \cdots + 26\!\cdots\!70 \)
|
| $31$ |
\( T^{5} + \cdots + 29\!\cdots\!60 \)
|
| $37$ |
\( T^{5} + \cdots - 11\!\cdots\!39 \)
|
| $41$ |
\( T^{5} + \cdots - 10\!\cdots\!95 \)
|
| $43$ |
\( T^{5} + \cdots - 36\!\cdots\!25 \)
|
| $47$ |
\( T^{5} + \cdots + 71\!\cdots\!80 \)
|
| $53$ |
\( T^{5} + \cdots + 31\!\cdots\!48 \)
|
| $59$ |
\( (T - 3481)^{5} \)
|
| $61$ |
\( T^{5} + \cdots - 10\!\cdots\!50 \)
|
| $67$ |
\( T^{5} + \cdots + 61\!\cdots\!08 \)
|
| $71$ |
\( T^{5} + \cdots - 25\!\cdots\!85 \)
|
| $73$ |
\( T^{5} + \cdots - 23\!\cdots\!64 \)
|
| $79$ |
\( T^{5} + \cdots - 60\!\cdots\!93 \)
|
| $83$ |
\( T^{5} + \cdots - 21\!\cdots\!01 \)
|
| $89$ |
\( T^{5} + \cdots + 74\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 33\!\cdots\!00 \)
|
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