Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [169,6,Mod(168,169)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("169.168");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 169.b (of order \(2\), degree \(1\), not 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: | \(27.1048655484\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 201x^{4} + 10512x^{2} + 65536 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 13) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 201x^{4} + 10512x^{2} + 65536 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 55\nu^{3} + 15344\nu ) / 19968 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 101\nu^{2} + 256 ) / 156 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 153\nu^{2} + 3792 ) / 208 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 153\nu^{3} + 4832\nu ) / 832 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 4\beta_{4} - 3\beta_{3} - 68 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{5} + 96\beta_{2} - 97\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -404\beta_{4} + 459\beta_{3} + 6612 \)
|
| \(\nu^{5}\) | \(=\) |
\( 220\beta_{5} - 14688\beta_{2} + 10009\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 168.1 |
|
− | 10.9678i | −15.7989 | −88.2923 | 51.6056i | 173.279i | − | 75.3967i | 617.402i | 6.60562 | 565.999 | ||||||||||||||||||||||||||||||||||
| 168.2 | − | 8.64858i | 10.2870 | −42.7979 | 92.1784i | − | 88.9676i | − | 2.86088i | 93.3863i | −137.178 | 797.212 | ||||||||||||||||||||||||||||||||||
| 168.3 | − | 4.68079i | 13.5120 | 10.0902 | − | 15.4272i | − | 63.2466i | 12.5359i | − | 197.015i | −60.4272 | −72.2115 | |||||||||||||||||||||||||||||||||
| 168.4 | 4.68079i | 13.5120 | 10.0902 | 15.4272i | 63.2466i | − | 12.5359i | 197.015i | −60.4272 | −72.2115 | ||||||||||||||||||||||||||||||||||||
| 168.5 | 8.64858i | 10.2870 | −42.7979 | − | 92.1784i | 88.9676i | 2.86088i | − | 93.3863i | −137.178 | 797.212 | |||||||||||||||||||||||||||||||||||
| 168.6 | 10.9678i | −15.7989 | −88.2923 | − | 51.6056i | − | 173.279i | 75.3967i | − | 617.402i | 6.60562 | 565.999 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 169.6.b.b | 6 | |
| 13.b | even | 2 | 1 | inner | 169.6.b.b | 6 | |
| 13.d | odd | 4 | 1 | 13.6.a.b | ✓ | 3 | |
| 13.d | odd | 4 | 1 | 169.6.a.b | 3 | ||
| 39.f | even | 4 | 1 | 117.6.a.d | 3 | ||
| 52.f | even | 4 | 1 | 208.6.a.j | 3 | ||
| 65.f | even | 4 | 1 | 325.6.b.c | 6 | ||
| 65.g | odd | 4 | 1 | 325.6.a.c | 3 | ||
| 65.k | even | 4 | 1 | 325.6.b.c | 6 | ||
| 91.i | even | 4 | 1 | 637.6.a.b | 3 | ||
| 104.j | odd | 4 | 1 | 832.6.a.s | 3 | ||
| 104.m | even | 4 | 1 | 832.6.a.t | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.6.a.b | ✓ | 3 | 13.d | odd | 4 | 1 | |
| 117.6.a.d | 3 | 39.f | even | 4 | 1 | ||
| 169.6.a.b | 3 | 13.d | odd | 4 | 1 | ||
| 169.6.b.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 169.6.b.b | 6 | 13.b | even | 2 | 1 | inner | |
| 208.6.a.j | 3 | 52.f | even | 4 | 1 | ||
| 325.6.a.c | 3 | 65.g | odd | 4 | 1 | ||
| 325.6.b.c | 6 | 65.f | even | 4 | 1 | ||
| 325.6.b.c | 6 | 65.k | even | 4 | 1 | ||
| 637.6.a.b | 3 | 91.i | even | 4 | 1 | ||
| 832.6.a.s | 3 | 104.j | odd | 4 | 1 | ||
| 832.6.a.t | 3 | 104.m | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 217T_{2}^{4} + 13272T_{2}^{2} + 197136 \)
acting on \(S_{6}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 217 T^{4} + \cdots + 197136 \)
|
| $3$ |
\( (T^{3} - 8 T^{2} + \cdots + 2196)^{2} \)
|
| $5$ |
\( T^{6} + \cdots + 5385504996 \)
|
| $7$ |
\( T^{6} + 5850 T^{4} + \cdots + 7311616 \)
|
| $11$ |
\( T^{6} + \cdots + 15\!\cdots\!36 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( (T^{3} + 908 T^{2} + \cdots - 77884638)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 20\!\cdots\!44 \)
|
| $23$ |
\( (T^{3} + 3624 T^{2} + \cdots - 5045833728)^{2} \)
|
| $29$ |
\( (T^{3} + 8758 T^{2} + \cdots - 221025174456)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 25\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 21\!\cdots\!76 \)
|
| $41$ |
\( T^{6} + \cdots + 86\!\cdots\!04 \)
|
| $43$ |
\( (T^{3} + 2032 T^{2} + \cdots - 281385762060)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 48\!\cdots\!56 \)
|
| $53$ |
\( (T^{3} + \cdots - 4415410372608)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 37\!\cdots\!84 \)
|
| $61$ |
\( (T^{3} + \cdots - 18650455523968)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 43\!\cdots\!96 \)
|
| $71$ |
\( T^{6} + \cdots + 13\!\cdots\!96 \)
|
| $73$ |
\( T^{6} + \cdots + 31\!\cdots\!64 \)
|
| $79$ |
\( (T^{3} + \cdots - 2044988893184)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 32\!\cdots\!24 \)
|
| $89$ |
\( T^{6} + \cdots + 83\!\cdots\!44 \)
|
| $97$ |
\( T^{6} + \cdots + 14\!\cdots\!56 \)
|
| show more | |
| show less |