Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [342,5,Mod(37,342)]
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, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("342.37");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 342.d (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(35.3525273747\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 450x^{6} + 68229x^{4} + 4001228x^{2} + 77475204 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 38) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 450x^{6} + 68229x^{4} + 4001228x^{2} + 77475204 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} - 450\nu^{5} - 59427\nu^{3} - 770894\nu ) / 1249884 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + 450\nu^{5} + 59427\nu^{3} + 2020778\nu ) / 624942 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 2484\nu^{5} + 1017351\nu^{3} + 84215350\nu ) / 2499768 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 225\nu^{2} + 8802 ) / 71 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} - 352\nu^{4} - 33685\nu^{2} - 733602 ) / 3408 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + 352\nu^{4} + 37093\nu^{2} + 1115298 ) / 3408 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 151\nu^{7} + 55236\nu^{5} + 5973951\nu^{3} + 176424854\nu ) / 833256 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} - 112 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} + 13\beta_{3} - 257\beta_{2} - 294\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -225\beta_{6} - 225\beta_{5} + 71\beta_{4} + 16398 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -367\beta_{7} - 3067\beta_{3} + 65353\beta_{2} + 49114\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 45515\beta_{6} + 42107\beta_{5} - 24992\beta_{4} - 2732978 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 105723\beta_{7} + 607599\beta_{3} - 14907005\beta_{2} - 8671318\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(325\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
− | 2.82843i | 0 | −8.00000 | −41.6240 | 0 | −62.4342 | 22.6274i | 0 | 117.730i | |||||||||||||||||||||||||||||||||||||||||
| 37.2 | − | 2.82843i | 0 | −8.00000 | −26.7027 | 0 | 51.4469 | 22.6274i | 0 | 75.5266i | ||||||||||||||||||||||||||||||||||||||||||
| 37.3 | − | 2.82843i | 0 | −8.00000 | 26.2296 | 0 | −86.0910 | 22.6274i | 0 | − | 74.1886i | |||||||||||||||||||||||||||||||||||||||||
| 37.4 | − | 2.82843i | 0 | −8.00000 | 33.0971 | 0 | 16.0783 | 22.6274i | 0 | − | 93.6127i | |||||||||||||||||||||||||||||||||||||||||
| 37.5 | 2.82843i | 0 | −8.00000 | −41.6240 | 0 | −62.4342 | − | 22.6274i | 0 | − | 117.730i | |||||||||||||||||||||||||||||||||||||||||
| 37.6 | 2.82843i | 0 | −8.00000 | −26.7027 | 0 | 51.4469 | − | 22.6274i | 0 | − | 75.5266i | |||||||||||||||||||||||||||||||||||||||||
| 37.7 | 2.82843i | 0 | −8.00000 | 26.2296 | 0 | −86.0910 | − | 22.6274i | 0 | 74.1886i | ||||||||||||||||||||||||||||||||||||||||||
| 37.8 | 2.82843i | 0 | −8.00000 | 33.0971 | 0 | 16.0783 | − | 22.6274i | 0 | 93.6127i | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 342.5.d.a | 8 | |
| 3.b | odd | 2 | 1 | 38.5.b.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 304.5.e.e | 8 | ||
| 19.b | odd | 2 | 1 | inner | 342.5.d.a | 8 | |
| 57.d | even | 2 | 1 | 38.5.b.a | ✓ | 8 | |
| 228.b | odd | 2 | 1 | 304.5.e.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.5.b.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 38.5.b.a | ✓ | 8 | 57.d | even | 2 | 1 | |
| 304.5.e.e | 8 | 12.b | even | 2 | 1 | ||
| 304.5.e.e | 8 | 228.b | odd | 2 | 1 | ||
| 342.5.d.a | 8 | 1.a | even | 1 | 1 | trivial | |
| 342.5.d.a | 8 | 19.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 9T_{5}^{3} - 2074T_{5}^{2} - 6624T_{5} + 964896 \)
acting on \(S_{5}^{\mathrm{new}}(342, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} + 9 T^{3} + \cdots + 964896)^{2} \)
|
| $7$ |
\( (T^{4} + 81 T^{3} + \cdots + 4446118)^{2} \)
|
| $11$ |
\( (T^{4} - 3 T^{3} + \cdots + 76301016)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 57\!\cdots\!36 \)
|
| $17$ |
\( (T^{4} + 255 T^{3} + \cdots + 403423998)^{2} \)
|
| $19$ |
\( T^{8} + \cdots + 28\!\cdots\!81 \)
|
| $23$ |
\( (T^{4} - 198 T^{3} + \cdots + 18350234964)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 26\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots + 13\!\cdots\!16 \)
|
| $37$ |
\( T^{8} + \cdots + 61\!\cdots\!36 \)
|
| $41$ |
\( T^{8} + \cdots + 14\!\cdots\!04 \)
|
| $43$ |
\( (T^{4} + 4327 T^{3} + \cdots + 407751532960)^{2} \)
|
| $47$ |
\( (T^{4} + 1605 T^{3} + \cdots - 98774187816)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 82\!\cdots\!84 \)
|
| $59$ |
\( T^{8} + \cdots + 20\!\cdots\!00 \)
|
| $61$ |
\( (T^{4} + \cdots + 195962902247296)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 13\!\cdots\!96 \)
|
| $71$ |
\( T^{8} + \cdots + 22\!\cdots\!56 \)
|
| $73$ |
\( (T^{4} + \cdots - 15\!\cdots\!50)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 50\!\cdots\!76 \)
|
| $83$ |
\( (T^{4} + \cdots + 57645106800768)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 70\!\cdots\!96 \)
|
| $97$ |
\( T^{8} + \cdots + 11\!\cdots\!56 \)
|
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