Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [117,3,Mod(73,117)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(117, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("117.73");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 117.j (of order \(4\), degree \(2\), 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: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.897122304.10 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -12\nu^{7} - 34\nu^{5} - 170\nu^{3} - 1846\nu ) / 663 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 15\nu^{7} - 68\nu^{5} + 323\nu^{3} + 871\nu ) / 663 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 8\nu^{4} + 38\nu^{2} - 52 ) / 39 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 20\nu^{7} - 238\nu^{5} + 1462\nu^{3} - 2522\nu ) / 663 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{6} + 200 ) / 51 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -25\nu^{7} + 187\nu^{5} - 1054\nu^{3} + 1716\nu ) / 663 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 32\nu^{6} - 204\nu^{4} + 1632\nu^{2} - 2002 ) / 663 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{6} + \beta_{4} + 2\beta_{2} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{7} - \beta_{5} - 6\beta_{3} + 8 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{6} + 5\beta_{4} - 4\beta_{2} - 5\beta_1 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 19\beta_{7} + 8\beta_{5} - 48\beta_{3} - 38 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -22\beta_{6} - 8\beta_{4} - 54\beta_{2} - 35\beta_1 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 51\beta_{5} - 200 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -302\beta_{6} - 202\beta_{4} - 98\beta_{2} - 51\beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).
| \(n\) | \(28\) | \(92\) |
| \(\chi(n)\) | \(\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 73.1 |
|
−2.05719 | − | 2.05719i | 0 | 4.46410i | −3.01194 | − | 3.01194i | 0 | −2.46410 | + | 2.46410i | 0.954747 | − | 0.954747i | 0 | 12.3923i | ||||||||||||||||||||||||||||||||||
| 73.2 | −0.876327 | − | 0.876327i | 0 | − | 2.46410i | 4.78834 | + | 4.78834i | 0 | 4.46410 | − | 4.46410i | −5.66467 | + | 5.66467i | 0 | − | 8.39230i | |||||||||||||||||||||||||||||||||
| 73.3 | 0.876327 | + | 0.876327i | 0 | − | 2.46410i | −4.78834 | − | 4.78834i | 0 | 4.46410 | − | 4.46410i | 5.66467 | − | 5.66467i | 0 | − | 8.39230i | |||||||||||||||||||||||||||||||||
| 73.4 | 2.05719 | + | 2.05719i | 0 | 4.46410i | 3.01194 | + | 3.01194i | 0 | −2.46410 | + | 2.46410i | −0.954747 | + | 0.954747i | 0 | 12.3923i | |||||||||||||||||||||||||||||||||||
| 109.1 | −2.05719 | + | 2.05719i | 0 | − | 4.46410i | −3.01194 | + | 3.01194i | 0 | −2.46410 | − | 2.46410i | 0.954747 | + | 0.954747i | 0 | − | 12.3923i | |||||||||||||||||||||||||||||||||
| 109.2 | −0.876327 | + | 0.876327i | 0 | 2.46410i | 4.78834 | − | 4.78834i | 0 | 4.46410 | + | 4.46410i | −5.66467 | − | 5.66467i | 0 | 8.39230i | |||||||||||||||||||||||||||||||||||
| 109.3 | 0.876327 | − | 0.876327i | 0 | 2.46410i | −4.78834 | + | 4.78834i | 0 | 4.46410 | + | 4.46410i | 5.66467 | + | 5.66467i | 0 | 8.39230i | |||||||||||||||||||||||||||||||||||
| 109.4 | 2.05719 | − | 2.05719i | 0 | − | 4.46410i | 3.01194 | − | 3.01194i | 0 | −2.46410 | − | 2.46410i | −0.954747 | − | 0.954747i | 0 | − | 12.3923i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 13.d | odd | 4 | 1 | inner |
| 39.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 117.3.j.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 117.3.j.c | ✓ | 8 |
| 13.d | odd | 4 | 1 | inner | 117.3.j.c | ✓ | 8 |
| 39.f | even | 4 | 1 | inner | 117.3.j.c | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 117.3.j.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 117.3.j.c | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 117.3.j.c | ✓ | 8 | 13.d | odd | 4 | 1 | inner |
| 117.3.j.c | ✓ | 8 | 39.f | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 74T_{2}^{4} + 169 \)
acting on \(S_{3}^{\mathrm{new}}(117, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 74T^{4} + 169 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 2432 T^{4} + 692224 \)
|
| $7$ |
\( (T^{4} - 4 T^{3} + \cdots + 484)^{2} \)
|
| $11$ |
\( T^{8} + 2432 T^{4} + 692224 \)
|
| $13$ |
\( (T^{4} + 8 T^{3} + \cdots + 28561)^{2} \)
|
| $17$ |
\( (T^{4} + 1152 T^{2} + 269568)^{2} \)
|
| $19$ |
\( (T^{2} - 14 T + 98)^{4} \)
|
| $23$ |
\( (T^{4} + 1536 T^{2} + 479232)^{2} \)
|
| $29$ |
\( (T^{4} - 1280 T^{2} + 402688)^{2} \)
|
| $31$ |
\( (T^{4} + 68 T^{3} + \cdots + 357604)^{2} \)
|
| $37$ |
\( (T^{4} + 68 T^{3} + \cdots + 131044)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 8387928100864 \)
|
| $43$ |
\( (T^{4} + 7296 T^{2} + 6230016)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 90736445722624 \)
|
| $53$ |
\( (T^{4} - 5504 T^{2} + 1760512)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 17590575468544 \)
|
| $61$ |
\( (T^{2} + 8 T - 416)^{4} \)
|
| $67$ |
\( (T^{4} + 20 T^{3} + \cdots + 2116)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 97713066840064 \)
|
| $73$ |
\( (T^{4} - 76 T^{3} + \cdots + 49758916)^{2} \)
|
| $79$ |
\( (T^{2} + 200 T + 9568)^{4} \)
|
| $83$ |
\( T^{8} + \cdots + 3377832300544 \)
|
| $89$ |
\( T^{8} + \cdots + 40\!\cdots\!84 \)
|
| $97$ |
\( (T^{4} - 244 T^{3} + \cdots + 53963716)^{2} \)
|
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