Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [342,10,Mod(1,342)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(342, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("342.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 342.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: | \(176.142255968\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 34433x^{2} - 2723303x - 48270488 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 38) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 34433x^{2} - 2723303x - 48270488 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 27\nu^{3} - 2208\nu^{2} - 758211\nu - 17640024 ) / 20632 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -17\nu^{3} + 244\nu^{2} + 698613\nu + 30780440 ) / 20632 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -17\nu^{3} + 244\nu^{2} + 574821\nu + 30821704 ) / 20632 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -193\beta_{3} + 85\beta_{2} - 68\beta _1 + 103370 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -43865\beta_{3} + 35033\beta_{2} - 976\beta _1 + 12429538 ) / 6 \)
|
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 |
|
16.0000 | 0 | 256.000 | −2126.71 | 0 | 11469.0 | 4096.00 | 0 | −34027.4 | ||||||||||||||||||||||||||||||
| 1.2 | 16.0000 | 0 | 256.000 | −1263.95 | 0 | −3487.42 | 4096.00 | 0 | −20223.2 | |||||||||||||||||||||||||||||||
| 1.3 | 16.0000 | 0 | 256.000 | 2367.11 | 0 | 5859.40 | 4096.00 | 0 | 37873.7 | |||||||||||||||||||||||||||||||
| 1.4 | 16.0000 | 0 | 256.000 | 2418.56 | 0 | −1533.95 | 4096.00 | 0 | 38696.9 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 342.10.a.l | 4 | |
| 3.b | odd | 2 | 1 | 38.10.a.d | ✓ | 4 | |
| 12.b | even | 2 | 1 | 304.10.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.10.a.d | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 304.10.a.e | 4 | 12.b | even | 2 | 1 | ||
| 342.10.a.l | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 1395T_{5}^{3} - 7813530T_{5}^{2} + 6547343400T_{5} + 15389064288000 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(342))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 16)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 15389064288000 \)
|
| $7$ |
\( T^{4} + \cdots + 359494671206216 \)
|
| $11$ |
\( T^{4} + \cdots + 11\!\cdots\!24 \)
|
| $13$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
| $17$ |
\( T^{4} + \cdots + 27\!\cdots\!62 \)
|
| $19$ |
\( (T - 130321)^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 22\!\cdots\!08 \)
|
| $29$ |
\( T^{4} + \cdots - 34\!\cdots\!76 \)
|
| $31$ |
\( T^{4} + \cdots - 14\!\cdots\!96 \)
|
| $37$ |
\( T^{4} + \cdots - 58\!\cdots\!72 \)
|
| $41$ |
\( T^{4} + \cdots - 36\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots - 19\!\cdots\!12 \)
|
| $47$ |
\( T^{4} + \cdots + 52\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots - 20\!\cdots\!64 \)
|
| $59$ |
\( T^{4} + \cdots - 27\!\cdots\!04 \)
|
| $61$ |
\( T^{4} + \cdots + 43\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots - 18\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots - 32\!\cdots\!32 \)
|
| $73$ |
\( T^{4} + \cdots + 35\!\cdots\!34 \)
|
| $79$ |
\( T^{4} + \cdots + 29\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 94\!\cdots\!68 \)
|
| $89$ |
\( T^{4} + \cdots + 54\!\cdots\!84 \)
|
| $97$ |
\( T^{4} + \cdots + 37\!\cdots\!00 \)
|
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