Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [117,3,Mod(53,117)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(117, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("117.53");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 117.c (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: | \(3.18801909302\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.1574161678336.15 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 20x^{6} + 106x^{4} + 164x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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
\( x^{8} + 20x^{6} + 106x^{4} + 164x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 16\nu^{4} + 47\nu^{2} + 16 ) / 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 16\nu^{4} - 37\nu^{2} + 34 ) / 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - 16\nu^{5} - 37\nu^{3} + 64\nu ) / 10 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{6} + 37\nu^{4} + 154\nu^{2} + 97 ) / 10 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 21\nu^{5} + 117\nu^{3} + 161\nu ) / 10 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2\nu^{7} + 37\nu^{5} + 164\nu^{3} + 197\nu ) / 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} + \beta_{4} - 10\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} - 12\beta_{3} - 16\beta_{2} + 47 \)
|
| \(\nu^{5}\) | \(=\) |
\( -16\beta_{7} + 18\beta_{6} - 14\beta_{4} + 115\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -32\beta_{5} + 145\beta_{3} + 219\beta_{2} - 533 \)
|
| \(\nu^{7}\) | \(=\) |
\( 219\beta_{7} - 251\beta_{6} + 177\beta_{4} - 1406\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).
| \(n\) | \(28\) | \(92\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 53.1 |
|
− | 3.55413i | 0 | −8.63185 | 0.118383i | 0 | −12.0700 | 16.4622i | 0 | 0.420750 | |||||||||||||||||||||||||||||||||||||||||
| 53.2 | − | 2.13992i | 0 | −0.579250 | − | 3.56642i | 0 | 0.858882 | − | 7.32013i | 0 | −7.63185 | ||||||||||||||||||||||||||||||||||||||||
| 53.3 | − | 1.65286i | 0 | 1.26805 | 2.99060i | 0 | 9.10296 | − | 8.70736i | 0 | 4.94305 | |||||||||||||||||||||||||||||||||||||||||
| 53.4 | − | 0.238646i | 0 | 3.94305 | 9.50384i | 0 | −5.89186 | − | 1.89558i | 0 | 2.26805 | |||||||||||||||||||||||||||||||||||||||||
| 53.5 | 0.238646i | 0 | 3.94305 | − | 9.50384i | 0 | −5.89186 | 1.89558i | 0 | 2.26805 | ||||||||||||||||||||||||||||||||||||||||||
| 53.6 | 1.65286i | 0 | 1.26805 | − | 2.99060i | 0 | 9.10296 | 8.70736i | 0 | 4.94305 | ||||||||||||||||||||||||||||||||||||||||||
| 53.7 | 2.13992i | 0 | −0.579250 | 3.56642i | 0 | 0.858882 | 7.32013i | 0 | −7.63185 | |||||||||||||||||||||||||||||||||||||||||||
| 53.8 | 3.55413i | 0 | −8.63185 | − | 0.118383i | 0 | −12.0700 | − | 16.4622i | 0 | 0.420750 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 117.3.c.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 117.3.c.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | 1872.3.f.d | 8 | ||
| 12.b | even | 2 | 1 | 1872.3.f.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 117.3.c.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 117.3.c.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 1872.3.f.d | 8 | 4.b | odd | 2 | 1 | ||
| 1872.3.f.d | 8 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(117, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 20 T^{6} + \cdots + 9 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 112 T^{6} + \cdots + 144 \)
|
| $7$ |
\( (T^{4} + 8 T^{3} + \cdots + 556)^{2} \)
|
| $11$ |
\( T^{8} + 192 T^{6} + \cdots + 24336 \)
|
| $13$ |
\( (T^{2} - 13)^{4} \)
|
| $17$ |
\( T^{8} + \cdots + 2850278544 \)
|
| $19$ |
\( (T^{4} - 32 T^{3} + \cdots - 93636)^{2} \)
|
| $23$ |
\( T^{8} + 784 T^{6} + \cdots + 77158656 \)
|
| $29$ |
\( T^{8} + \cdots + 5576504976 \)
|
| $31$ |
\( (T^{4} + 20 T^{3} + \cdots + 10204)^{2} \)
|
| $37$ |
\( (T^{4} + 52 T^{3} + \cdots - 3636032)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 6043750227216 \)
|
| $43$ |
\( (T^{4} - 64 T^{3} + \cdots + 184848)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 125511484176 \)
|
| $53$ |
\( T^{8} + \cdots + 19225523934864 \)
|
| $59$ |
\( T^{8} + \cdots + 101008530090000 \)
|
| $61$ |
\( (T^{4} + 56 T^{3} + \cdots - 2361584)^{2} \)
|
| $67$ |
\( (T^{4} - 92 T^{3} + \cdots - 9538292)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 40397566504464 \)
|
| $73$ |
\( (T^{4} + 236 T^{3} + \cdots - 286784)^{2} \)
|
| $79$ |
\( (T^{4} - 40 T^{3} + \cdots + 11328448)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 18793993721616 \)
|
| $89$ |
\( T^{8} + \cdots + 28996544904336 \)
|
| $97$ |
\( (T^{4} - 164 T^{3} + \cdots + 2153152)^{2} \)
|
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