Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [154,2,Mod(15,154)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(154, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("154.15");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 154 = 2 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 154.f (of order \(5\), degree \(4\), 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: | \(1.22969619113\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.58140625.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 406\nu^{7} - 714\nu^{6} + 747\nu^{5} - 1896\nu^{4} + 2103\nu^{3} + 4949\nu^{2} + 1065\nu - 7800 ) / 1355 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 420\nu^{7} - 776\nu^{6} + 698\nu^{5} - 1924\nu^{4} + 2297\nu^{3} + 5129\nu^{2} + 1055\nu - 10265 ) / 1355 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 728\nu^{7} - 1327\nu^{6} + 1246\nu^{5} - 3353\nu^{4} + 3584\nu^{3} + 8547\nu^{2} + 2190\nu - 15715 ) / 1355 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -857\nu^{7} + 1666\nu^{6} - 1743\nu^{5} + 4424\nu^{4} - 4907\nu^{3} - 9470\nu^{2} - 2485\nu + 18200 ) / 1355 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 891\nu^{7} - 1623\nu^{6} + 1624\nu^{5} - 4492\nu^{4} + 4991\nu^{3} + 9520\nu^{2} + 3235\nu - 18960 ) / 1355 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 955\nu^{7} - 1829\nu^{6} + 1942\nu^{5} - 4891\nu^{4} + 5723\nu^{3} + 9646\nu^{2} + 2415\nu - 20550 ) / 1355 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{5} - 3\beta_{4} + 4\beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{7} - 5\beta_{6} + 4\beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + 4\beta _1 + 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{7} - 6\beta_{6} - 7\beta_{3} + 11\beta_{2} + 4\beta _1 - 13 \)
|
| \(\nu^{6}\) | \(=\) |
\( 7\beta_{6} + 7\beta_{5} - 8\beta_{4} - 7\beta_{3} + 21\beta_{2} - 2\beta _1 - 21 \)
|
| \(\nu^{7}\) | \(=\) |
\( 23\beta_{7} + 38\beta_{5} + 23\beta_{4} - 23\beta_{3} + 12\beta_{2} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/154\mathbb{Z}\right)^\times\).
| \(n\) | \(45\) | \(57\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 15.1 |
|
0.809017 | + | 0.587785i | −1.02988 | + | 3.16963i | 0.309017 | + | 0.951057i | 2.47539 | − | 1.79848i | −2.69625 | + | 1.95894i | 0.309017 | + | 0.951057i | −0.309017 | + | 0.951057i | −6.55888 | − | 4.76530i | 3.05975 | ||||||||||||||||||||||||||
| 15.2 | 0.809017 | + | 0.587785i | 0.220859 | − | 0.679734i | 0.309017 | + | 0.951057i | 0.451659 | − | 0.328150i | 0.578217 | − | 0.420099i | 0.309017 | + | 0.951057i | −0.309017 | + | 0.951057i | 2.01379 | + | 1.46310i | 0.558282 | |||||||||||||||||||||||||||
| 71.1 | −0.309017 | − | 0.951057i | −1.59089 | − | 1.15585i | −0.809017 | + | 0.587785i | −1.29224 | + | 3.97711i | −0.607666 | + | 1.87020i | −0.809017 | + | 0.587785i | 0.809017 | + | 0.587785i | 0.267892 | + | 0.824488i | 4.18178 | |||||||||||||||||||||||||||
| 71.2 | −0.309017 | − | 0.951057i | 1.89991 | + | 1.38036i | −0.809017 | + | 0.587785i | 0.865190 | − | 2.66278i | 0.725700 | − | 2.23347i | −0.809017 | + | 0.587785i | 0.809017 | + | 0.587785i | 0.777193 | + | 2.39195i | −2.79981 | |||||||||||||||||||||||||||
| 113.1 | 0.809017 | − | 0.587785i | −1.02988 | − | 3.16963i | 0.309017 | − | 0.951057i | 2.47539 | + | 1.79848i | −2.69625 | − | 1.95894i | 0.309017 | − | 0.951057i | −0.309017 | − | 0.951057i | −6.55888 | + | 4.76530i | 3.05975 | |||||||||||||||||||||||||||
| 113.2 | 0.809017 | − | 0.587785i | 0.220859 | + | 0.679734i | 0.309017 | − | 0.951057i | 0.451659 | + | 0.328150i | 0.578217 | + | 0.420099i | 0.309017 | − | 0.951057i | −0.309017 | − | 0.951057i | 2.01379 | − | 1.46310i | 0.558282 | |||||||||||||||||||||||||||
| 141.1 | −0.309017 | + | 0.951057i | −1.59089 | + | 1.15585i | −0.809017 | − | 0.587785i | −1.29224 | − | 3.97711i | −0.607666 | − | 1.87020i | −0.809017 | − | 0.587785i | 0.809017 | − | 0.587785i | 0.267892 | − | 0.824488i | 4.18178 | |||||||||||||||||||||||||||
| 141.2 | −0.309017 | + | 0.951057i | 1.89991 | − | 1.38036i | −0.809017 | − | 0.587785i | 0.865190 | + | 2.66278i | 0.725700 | + | 2.23347i | −0.809017 | − | 0.587785i | 0.809017 | − | 0.587785i | 0.777193 | − | 2.39195i | −2.79981 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 154.2.f.e | ✓ | 8 |
| 11.c | even | 5 | 1 | inner | 154.2.f.e | ✓ | 8 |
| 11.c | even | 5 | 1 | 1694.2.a.x | 4 | ||
| 11.d | odd | 10 | 1 | 1694.2.a.z | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 154.2.f.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 154.2.f.e | ✓ | 8 | 11.c | even | 5 | 1 | inner |
| 1694.2.a.x | 4 | 11.c | even | 5 | 1 | ||
| 1694.2.a.z | 4 | 11.d | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(154, [\chi])\):
|
\( T_{3}^{8} + T_{3}^{7} + 7T_{3}^{6} - 12T_{3}^{5} + 5T_{3}^{4} + 72T_{3}^{3} + 202T_{3}^{2} - 66T_{3} + 121 \)
|
|
\( T_{5}^{8} - 5T_{5}^{7} + 30T_{5}^{6} - 130T_{5}^{5} + 485T_{5}^{4} - 1150T_{5}^{3} + 2100T_{5}^{2} - 1400T_{5} + 400 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{8} + T^{7} + \cdots + 121 \)
|
| $5$ |
\( T^{8} - 5 T^{7} + \cdots + 400 \)
|
| $7$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $11$ |
\( T^{8} + 5 T^{7} + \cdots + 14641 \)
|
| $13$ |
\( T^{8} - 3 T^{7} + \cdots + 80656 \)
|
| $17$ |
\( T^{8} + 7 T^{7} + \cdots + 6241 \)
|
| $19$ |
\( T^{8} + 14 T^{7} + \cdots + 121 \)
|
| $23$ |
\( (T^{4} + 10 T^{3} + \cdots - 20)^{2} \)
|
| $29$ |
\( T^{8} + 23 T^{7} + \cdots + 4096 \)
|
| $31$ |
\( T^{8} - 3 T^{7} + \cdots + 16 \)
|
| $37$ |
\( T^{8} - 6 T^{7} + \cdots + 4145296 \)
|
| $41$ |
\( T^{8} + 2 T^{7} + \cdots + 256 \)
|
| $43$ |
\( (T^{4} + 8 T^{3} - 21 T^{2} + \cdots - 29)^{2} \)
|
| $47$ |
\( T^{8} - 34 T^{7} + \cdots + 839056 \)
|
| $53$ |
\( T^{8} - 13 T^{7} + \cdots + 4096 \)
|
| $59$ |
\( T^{8} + 3 T^{7} + \cdots + 44102881 \)
|
| $61$ |
\( T^{8} + 16 T^{7} + \cdots + 246016 \)
|
| $67$ |
\( (T^{4} - 9 T^{3} + \cdots + 341)^{2} \)
|
| $71$ |
\( T^{8} - 45 T^{7} + \cdots + 15366400 \)
|
| $73$ |
\( T^{8} - T^{7} + \cdots + 52113961 \)
|
| $79$ |
\( T^{8} + 21 T^{7} + \cdots + 38416 \)
|
| $83$ |
\( T^{8} - 12 T^{7} + \cdots + 11881 \)
|
| $89$ |
\( (T^{4} - 6 T^{3} + \cdots - 649)^{2} \)
|
| $97$ |
\( T^{8} - 5 T^{7} + \cdots + 2016400 \)
|
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