Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [136,4,Mod(33,136)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(136, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("136.33");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 136.b (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: | \(8.02425976078\) |
| 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: |
\( x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{13} \) |
| Twist minimal: | yes |
| 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
\( x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{7} - 284\nu^{5} - 2159\nu^{3} - 3402\nu ) / 172 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{6} + 327\nu^{4} + 6158\nu^{2} + 17420 ) / 172 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 59\nu^{7} + 5485\nu^{5} + 33588\nu^{3} + 47900\nu ) / 1032 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -41\nu^{6} - 3781\nu^{4} - 20634\nu^{2} - 22844 ) / 172 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -109\nu^{7} - 9989\nu^{5} - 48702\nu^{3} - 15160\nu ) / 1032 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 45\nu^{6} + 4217\nu^{4} + 28386\nu^{2} + 35292 ) / 172 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{7} - 3\beta_{5} + 4\beta_{3} - 188 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{6} + 23\beta_{4} + 33\beta_{2} - 78\beta_1 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 277\beta_{7} + 279\beta_{5} - 342\beta_{3} + 14856 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -619\beta_{6} - 2075\beta_{4} - 3053\beta_{2} + 6708\beta_1 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -24035\beta_{7} - 24253\beta_{5} + 29526\beta_{3} - 1279856 ) / 8 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 53561\beta_{6} + 179881\beta_{4} + 265039\beta_{2} - 580024\beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).
| \(n\) | \(69\) | \(103\) | \(105\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 33.1 |
|
0 | − | 9.26065i | 0 | 16.4090i | 0 | 34.5010i | 0 | −58.7597 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 33.2 | 0 | − | 8.52350i | 0 | − | 18.2701i | 0 | − | 13.8757i | 0 | −45.6501 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 33.3 | 0 | − | 2.95309i | 0 | − | 4.89575i | 0 | 4.46235i | 0 | 18.2793 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 33.4 | 0 | − | 2.62097i | 0 | 11.5318i | 0 | − | 23.0485i | 0 | 20.1305 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 33.5 | 0 | 2.62097i | 0 | − | 11.5318i | 0 | 23.0485i | 0 | 20.1305 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 33.6 | 0 | 2.95309i | 0 | 4.89575i | 0 | − | 4.46235i | 0 | 18.2793 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 33.7 | 0 | 8.52350i | 0 | 18.2701i | 0 | 13.8757i | 0 | −45.6501 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 33.8 | 0 | 9.26065i | 0 | − | 16.4090i | 0 | − | 34.5010i | 0 | −58.7597 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 136.4.b.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1224.4.c.e | 8 | ||
| 4.b | odd | 2 | 1 | 272.4.b.f | 8 | ||
| 17.b | even | 2 | 1 | inner | 136.4.b.b | ✓ | 8 |
| 17.c | even | 4 | 2 | 2312.4.a.k | 8 | ||
| 51.c | odd | 2 | 1 | 1224.4.c.e | 8 | ||
| 68.d | odd | 2 | 1 | 272.4.b.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 136.4.b.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 136.4.b.b | ✓ | 8 | 17.b | even | 2 | 1 | inner |
| 272.4.b.f | 8 | 4.b | odd | 2 | 1 | ||
| 272.4.b.f | 8 | 68.d | odd | 2 | 1 | ||
| 1224.4.c.e | 8 | 3.b | odd | 2 | 1 | ||
| 1224.4.c.e | 8 | 51.c | odd | 2 | 1 | ||
| 2312.4.a.k | 8 | 17.c | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 174T_{3}^{6} + 8760T_{3}^{4} + 106624T_{3}^{2} + 373248 \)
acting on \(S_{4}^{\mathrm{new}}(136, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 174 T^{6} + \cdots + 373248 \)
|
| $5$ |
\( T^{8} + 760 T^{6} + \cdots + 286466048 \)
|
| $7$ |
\( T^{8} + \cdots + 2424307712 \)
|
| $11$ |
\( T^{8} + \cdots + 1063158272 \)
|
| $13$ |
\( (T^{4} - 22 T^{3} + \cdots + 8525216)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 582622237229761 \)
|
| $19$ |
\( (T^{4} - 24 T^{3} + \cdots + 44946176)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 2935871578112 \)
|
| $29$ |
\( T^{8} + \cdots + 14\!\cdots\!52 \)
|
| $31$ |
\( T^{8} + \cdots + 32\!\cdots\!52 \)
|
| $37$ |
\( T^{8} + \cdots + 54\!\cdots\!12 \)
|
| $41$ |
\( T^{8} + \cdots + 55\!\cdots\!32 \)
|
| $43$ |
\( (T^{4} - 4 T^{3} + \cdots + 112195072)^{2} \)
|
| $47$ |
\( (T^{4} - 156 T^{3} + \cdots + 134217728)^{2} \)
|
| $53$ |
\( (T^{4} - 236 T^{3} + \cdots - 14761769616)^{2} \)
|
| $59$ |
\( (T^{4} + 36 T^{3} + \cdots + 70465536)^{2} \)
|
| $61$ |
\( T^{8} + \cdots + 18\!\cdots\!28 \)
|
| $67$ |
\( (T^{4} + 312 T^{3} + \cdots + 30967766784)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 14\!\cdots\!28 \)
|
| $73$ |
\( T^{8} + \cdots + 21\!\cdots\!68 \)
|
| $79$ |
\( T^{8} + \cdots + 17\!\cdots\!72 \)
|
| $83$ |
\( (T^{4} - 1236 T^{3} + \cdots - 54225864704)^{2} \)
|
| $89$ |
\( (T^{4} - 34 T^{3} + \cdots + 21605388512)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 13\!\cdots\!92 \)
|
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