Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [338,10,Mod(1,338)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("338.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 338 = 2 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 338.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: | \(174.082112623\) |
| 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: |
\( x^{5} - x^{4} - 60825x^{3} - 355103x^{2} + 717146696x - 24400132848 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 26) |
| 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
\( x^{5} - x^{4} - 60825x^{3} - 355103x^{2} + 717146696x - 24400132848 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 78\nu^{3} + 37069\nu^{2} - 1272330\nu - 14622936 ) / 17040 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -11\nu^{4} + 2278\nu^{3} + 288479\nu^{2} - 65543050\nu + 2397414024 ) / 460080 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 23\nu^{4} - 2149\nu^{3} - 937787\nu^{2} + 43530685\nu + 2494737468 ) / 460080 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -4\beta_{4} - \beta_{3} - 3\beta_{2} + 12\beta _1 + 24326 \)
|
| \(\nu^{3}\) | \(=\) |
\( -336\beta_{4} + 240\beta_{3} - 384\beta_{2} + 37309\beta _1 + 241788 \)
|
| \(\nu^{4}\) | \(=\) |
\( -174484\beta_{4} - 18349\beta_{3} - 158199\beta_{2} + 2082600\beta _1 + 905977022 \)
|
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 | −162.362 | 256.000 | −767.912 | −2597.79 | 2969.95 | 4096.00 | 6678.30 | −12286.6 | |||||||||||||||||||||||||||||||||
| 1.2 | 16.0000 | −157.164 | 256.000 | 2410.25 | −2514.62 | −9056.75 | 4096.00 | 5017.43 | 38563.9 | ||||||||||||||||||||||||||||||||||
| 1.3 | 16.0000 | 56.1848 | 256.000 | 839.868 | 898.956 | 627.345 | 4096.00 | −16526.3 | 13437.9 | ||||||||||||||||||||||||||||||||||
| 1.4 | 16.0000 | 103.398 | 256.000 | −646.596 | 1654.36 | 6855.66 | 4096.00 | −8991.91 | −10345.5 | ||||||||||||||||||||||||||||||||||
| 1.5 | 16.0000 | 240.943 | 256.000 | −622.606 | 3855.09 | −10111.2 | 4096.00 | 38370.4 | −9961.70 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( -1 \) |
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 | 338.10.a.j | 5 | |
| 13.b | even | 2 | 1 | 338.10.a.g | 5 | ||
| 13.d | odd | 4 | 2 | 26.10.b.a | ✓ | 10 | |
| 39.f | even | 4 | 2 | 234.10.b.a | 10 | ||
| 52.f | even | 4 | 2 | 208.10.f.a | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.10.b.a | ✓ | 10 | 13.d | odd | 4 | 2 | |
| 208.10.f.a | 10 | 52.f | even | 4 | 2 | ||
| 234.10.b.a | 10 | 39.f | even | 4 | 2 | ||
| 338.10.a.g | 5 | 13.b | even | 2 | 1 | ||
| 338.10.a.j | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(338))\):
|
\( T_{3}^{5} - 81T_{3}^{4} - 58201T_{3}^{3} + 2522001T_{3}^{2} + 682140456T_{3} - 35717361264 \)
|
|
\( T_{5}^{5} - 1213T_{5}^{4} - 3219353T_{5}^{3} - 43240635T_{5}^{2} + 1783123495200T_{5} + 625791896442000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 16)^{5} \)
|
| $3$ |
\( T^{5} + \cdots - 35717361264 \)
|
| $5$ |
\( T^{5} + \cdots + 625791896442000 \)
|
| $7$ |
\( T^{5} + \cdots - 11\!\cdots\!20 \)
|
| $11$ |
\( T^{5} + \cdots + 23\!\cdots\!00 \)
|
| $13$ |
\( T^{5} \)
|
| $17$ |
\( T^{5} + \cdots - 44\!\cdots\!76 \)
|
| $19$ |
\( T^{5} + \cdots - 30\!\cdots\!80 \)
|
| $23$ |
\( T^{5} + \cdots - 24\!\cdots\!00 \)
|
| $29$ |
\( T^{5} + \cdots + 24\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots + 11\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots - 21\!\cdots\!24 \)
|
| $41$ |
\( T^{5} + \cdots - 55\!\cdots\!00 \)
|
| $43$ |
\( T^{5} + \cdots + 17\!\cdots\!00 \)
|
| $47$ |
\( T^{5} + \cdots - 79\!\cdots\!96 \)
|
| $53$ |
\( T^{5} + \cdots + 84\!\cdots\!00 \)
|
| $59$ |
\( T^{5} + \cdots + 61\!\cdots\!48 \)
|
| $61$ |
\( T^{5} + \cdots - 16\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots + 62\!\cdots\!92 \)
|
| $71$ |
\( T^{5} + \cdots + 12\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots + 13\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots + 87\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 22\!\cdots\!08 \)
|
| $89$ |
\( T^{5} + \cdots + 87\!\cdots\!08 \)
|
| $97$ |
\( T^{5} + \cdots + 59\!\cdots\!76 \)
|
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