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: | \(1\) |
| 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: |
\( x^{5} - x^{4} - 807833x^{3} + 190065997x^{2} + 44147599480x - 1741493366700 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 114) |
| 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
\( x^{5} - x^{4} - 807833x^{3} + 190065997x^{2} + 44147599480x - 1741493366700 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2257\nu^{4} + 6553857\nu^{3} + 2499767368\nu^{2} - 2751736979650\nu - 572122913857680 ) / 75307750105 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -24397\nu^{4} + 655428\nu^{3} + 18009022337\nu^{2} - 5491311849740\nu - 240604500501460 ) / 43033000060 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -141\nu^{4} - 44544\nu^{3} + 95850153\nu^{2} - 1203263112\nu - 4225246388180 ) / 61689740 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 25\beta_{4} - 89\beta_{3} + 223\beta_{2} - 907\beta _1 + 2907937 ) / 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -15745\beta_{4} + 61469\beta_{3} - 37978\beta_{2} + 2049685\beta _1 - 1021195699 ) / 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 18031145\beta_{4} - 79920133\beta_{3} + 163590611\beta_{2} - 1289695367\beta _1 + 2029670356721 ) / 9 \)
|
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 | −2503.42 | 0 | −12222.3 | −4096.00 | 0 | 40054.8 | |||||||||||||||||||||||||||||||||
| 1.2 | −16.0000 | 0 | 256.000 | −1989.33 | 0 | 5894.00 | −4096.00 | 0 | 31829.2 | ||||||||||||||||||||||||||||||||||
| 1.3 | −16.0000 | 0 | 256.000 | −683.894 | 0 | 9096.50 | −4096.00 | 0 | 10942.3 | ||||||||||||||||||||||||||||||||||
| 1.4 | −16.0000 | 0 | 256.000 | −71.5294 | 0 | 1741.51 | −4096.00 | 0 | 1144.47 | ||||||||||||||||||||||||||||||||||
| 1.5 | −16.0000 | 0 | 256.000 | 2350.17 | 0 | −2655.70 | −4096.00 | 0 | −37602.8 | ||||||||||||||||||||||||||||||||||
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.m | 5 | |
| 3.b | odd | 2 | 1 | 114.10.a.h | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.10.a.h | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 342.10.a.m | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} + 2898T_{5}^{4} - 3911139T_{5}^{3} - 15813555480T_{5}^{2} - 9114492063780T_{5} - 572550055329600 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(342))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{5} \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + \cdots - 572550055329600 \)
|
| $7$ |
\( T^{5} + \cdots - 30\!\cdots\!72 \)
|
| $11$ |
\( T^{5} + \cdots + 12\!\cdots\!00 \)
|
| $13$ |
\( T^{5} + \cdots - 38\!\cdots\!44 \)
|
| $17$ |
\( T^{5} + \cdots - 18\!\cdots\!12 \)
|
| $19$ |
\( (T - 130321)^{5} \)
|
| $23$ |
\( T^{5} + \cdots - 11\!\cdots\!00 \)
|
| $29$ |
\( T^{5} + \cdots + 86\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots - 10\!\cdots\!92 \)
|
| $37$ |
\( T^{5} + \cdots - 21\!\cdots\!72 \)
|
| $41$ |
\( T^{5} + \cdots + 48\!\cdots\!68 \)
|
| $43$ |
\( T^{5} + \cdots - 36\!\cdots\!16 \)
|
| $47$ |
\( T^{5} + \cdots + 70\!\cdots\!00 \)
|
| $53$ |
\( T^{5} + \cdots - 50\!\cdots\!96 \)
|
| $59$ |
\( T^{5} + \cdots + 23\!\cdots\!20 \)
|
| $61$ |
\( T^{5} + \cdots + 45\!\cdots\!32 \)
|
| $67$ |
\( T^{5} + \cdots - 17\!\cdots\!92 \)
|
| $71$ |
\( T^{5} + \cdots - 11\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots - 76\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots - 36\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 13\!\cdots\!80 \)
|
| $89$ |
\( T^{5} + \cdots - 19\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 42\!\cdots\!00 \)
|
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