Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [330,2,Mod(31,330)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(330, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("330.31");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 330.m (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: | \(2.63506326670\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.2769390625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 2x^{6} + 5x^{5} + 21x^{4} + 75x^{3} - 198x^{2} - 87x + 841 \)
|
| 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
\( x^{8} - x^{7} - 2x^{6} + 5x^{5} + 21x^{4} + 75x^{3} - 198x^{2} - 87x + 841 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3622 \nu^{7} - 168690 \nu^{6} + 210024 \nu^{5} - 119321 \nu^{4} + 621001 \nu^{3} + \cdots + 20083080 ) / 11583209 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4915 \nu^{7} - 90523 \nu^{6} + 209410 \nu^{5} + 136080 \nu^{4} + 66153 \nu^{3} - 2218842 \nu^{2} + \cdots + 23823587 ) / 11583209 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 908\nu^{7} - 3211\nu^{6} + 11167\nu^{5} + 3050\nu^{4} + 96671\nu^{3} - 39405\nu^{2} - 93476\nu + 949895 ) / 399421 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 31462 \nu^{7} + 137254 \nu^{6} - 28223 \nu^{5} + 415469 \nu^{4} - 1154253 \nu^{3} + \cdots - 7703821 ) / 11583209 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 3648 \nu^{7} + 3143 \nu^{6} - 19751 \nu^{5} + 17019 \nu^{4} - 91661 \nu^{3} - 187200 \nu^{2} + \cdots - 912398 ) / 399421 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5692 \nu^{7} - 7492 \nu^{6} + 4739 \nu^{5} - 18791 \nu^{4} + 84213 \nu^{3} + 337956 \nu^{2} + \cdots + 105038 ) / 399421 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + 5\beta_{5} - 6\beta_{3} + 6\beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + 5\beta_{4} - 6\beta_{3} + \beta_{2} - \beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{7} + 11\beta_{6} + 21\beta_{5} + 11\beta_{4} + 10\beta_{3} - 9\beta _1 - 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{7} - 20\beta_{6} + 21\beta_{5} - 8\beta_{4} + 21\beta_{2} - 47 \)
|
| \(\nu^{6}\) | \(=\) |
\( -29\beta_{7} - 29\beta_{6} - 60\beta_{5} + 60\beta_{3} - 128\beta_{2} - 59\beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( 9\beta_{7} - 324\beta_{5} - 31\beta_{4} + 557\beta_{3} - 557\beta_{2} + 31\beta _1 - 324 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/330\mathbb{Z}\right)^\times\).
| \(n\) | \(67\) | \(211\) | \(221\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
−0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | −0.809017 | − | 0.587785i | −0.309017 | − | 0.951057i | −0.309017 | − | 0.951057i | −1.27873 | − | 0.929049i | 0.809017 | − | 0.587785i | 0.309017 | − | 0.951057i | 1.00000 | ||||||||||||||||||||||||||
| 31.2 | −0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | −0.809017 | − | 0.587785i | −0.309017 | − | 0.951057i | −0.309017 | − | 0.951057i | 2.58774 | + | 1.88011i | 0.809017 | − | 0.587785i | 0.309017 | − | 0.951057i | 1.00000 | |||||||||||||||||||||||||||
| 91.1 | 0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | 0.309017 | − | 0.951057i | 0.809017 | + | 0.587785i | 0.809017 | + | 0.587785i | −1.38778 | + | 4.27116i | −0.309017 | − | 0.951057i | −0.809017 | + | 0.587785i | 1.00000 | |||||||||||||||||||||||||||
| 91.2 | 0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | 0.309017 | − | 0.951057i | 0.809017 | + | 0.587785i | 0.809017 | + | 0.587785i | 1.57877 | − | 4.85894i | −0.309017 | − | 0.951057i | −0.809017 | + | 0.587785i | 1.00000 | |||||||||||||||||||||||||||
| 181.1 | −0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | −0.809017 | + | 0.587785i | −0.309017 | + | 0.951057i | −0.309017 | + | 0.951057i | −1.27873 | + | 0.929049i | 0.809017 | + | 0.587785i | 0.309017 | + | 0.951057i | 1.00000 | |||||||||||||||||||||||||||
| 181.2 | −0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | −0.809017 | + | 0.587785i | −0.309017 | + | 0.951057i | −0.309017 | + | 0.951057i | 2.58774 | − | 1.88011i | 0.809017 | + | 0.587785i | 0.309017 | + | 0.951057i | 1.00000 | |||||||||||||||||||||||||||
| 301.1 | 0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | 0.309017 | + | 0.951057i | 0.809017 | − | 0.587785i | 0.809017 | − | 0.587785i | −1.38778 | − | 4.27116i | −0.309017 | + | 0.951057i | −0.809017 | − | 0.587785i | 1.00000 | |||||||||||||||||||||||||||
| 301.2 | 0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | 0.309017 | + | 0.951057i | 0.809017 | − | 0.587785i | 0.809017 | − | 0.587785i | 1.57877 | + | 4.85894i | −0.309017 | + | 0.951057i | −0.809017 | − | 0.587785i | 1.00000 | |||||||||||||||||||||||||||
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 | 330.2.m.f | ✓ | 8 |
| 3.b | odd | 2 | 1 | 990.2.n.i | 8 | ||
| 11.c | even | 5 | 1 | inner | 330.2.m.f | ✓ | 8 |
| 11.c | even | 5 | 1 | 3630.2.a.bq | 4 | ||
| 11.d | odd | 10 | 1 | 3630.2.a.bs | 4 | ||
| 33.h | odd | 10 | 1 | 990.2.n.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 330.2.m.f | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 330.2.m.f | ✓ | 8 | 11.c | even | 5 | 1 | inner |
| 990.2.n.i | 8 | 3.b | odd | 2 | 1 | ||
| 990.2.n.i | 8 | 33.h | odd | 10 | 1 | ||
| 3630.2.a.bq | 4 | 11.c | even | 5 | 1 | ||
| 3630.2.a.bs | 4 | 11.d | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} - 3T_{7}^{7} + 38T_{7}^{6} - 76T_{7}^{5} + 505T_{7}^{4} - 896T_{7}^{3} + 808T_{7}^{2} + 7192T_{7} + 13456 \)
acting on \(S_{2}^{\mathrm{new}}(330, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $5$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $7$ |
\( T^{8} - 3 T^{7} + \cdots + 13456 \)
|
| $11$ |
\( T^{8} - 10 T^{7} + \cdots + 14641 \)
|
| $13$ |
\( T^{8} - T^{7} + \cdots + 14641 \)
|
| $17$ |
\( T^{8} - 3 T^{7} + \cdots + 256 \)
|
| $19$ |
\( T^{8} + 12 T^{7} + \cdots + 16 \)
|
| $23$ |
\( (T^{4} - 65 T^{2} + \cdots + 55)^{2} \)
|
| $29$ |
\( T^{8} - 27 T^{7} + \cdots + 524176 \)
|
| $31$ |
\( T^{8} + 6 T^{7} + \cdots + 26896 \)
|
| $37$ |
\( T^{8} + 4 T^{7} + \cdots + 3031081 \)
|
| $41$ |
\( T^{8} + 23 T^{7} + \cdots + 3254416 \)
|
| $43$ |
\( (T^{4} + T^{3} - 39 T^{2} + \cdots - 4)^{2} \)
|
| $47$ |
\( T^{8} - 5 T^{7} + \cdots + 555025 \)
|
| $53$ |
\( T^{8} + 18 T^{7} + \cdots + 1936 \)
|
| $59$ |
\( (T^{4} + 20 T^{3} + \cdots + 3025)^{2} \)
|
| $61$ |
\( T^{8} + 20 T^{7} + \cdots + 1638400 \)
|
| $67$ |
\( (T^{4} - 9 T^{3} - 89 T^{2} + \cdots + 76)^{2} \)
|
| $71$ |
\( T^{8} + 10 T^{7} + \cdots + 1638400 \)
|
| $73$ |
\( (T^{4} - 18 T^{3} + \cdots + 5776)^{2} \)
|
| $79$ |
\( T^{8} + 11 T^{7} + \cdots + 82373776 \)
|
| $83$ |
\( (T^{4} - 8 T^{3} + \cdots + 4096)^{2} \)
|
| $89$ |
\( (T^{4} + 23 T^{3} + \cdots - 6724)^{2} \)
|
| $97$ |
\( T^{8} - 16 T^{7} + \cdots + 1290496 \)
|
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