Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [624,2,Mod(191,624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(624, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("624.191");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 624.bu (of order \(6\), 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: | \(4.98266508613\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 6.0.954288.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 5\nu^{4} + \nu^{3} + 9\nu^{2} - 6\nu - 45 ) / 27 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 9 ) / 27 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 2\nu^{4} - 2\nu^{3} - 6\nu - 18 ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} - 24\nu - 72 ) / 27 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} - 2\beta_{4} - 4\beta_{3} + 2\beta_{2} - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{5} + 2\beta_{4} + \beta_{3} + 4\beta_{2} + 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} + \beta_{4} - 10\beta_{3} - 4\beta_{2} + 6\beta _1 + 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).
| \(n\) | \(79\) | \(145\) | \(209\) | \(469\) |
| \(\chi(n)\) | \(-1\) | \(-1 - \beta_{3}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 191.1 |
|
0 | −1.25707 | + | 1.19154i | 0 | 2.71781i | 0 | 4.41751 | − | 2.55045i | 0 | 0.160442 | − | 2.99571i | 0 | ||||||||||||||||||||||||||||||
| 191.2 | 0 | −0.285997 | − | 1.70828i | 0 | 3.93569i | 0 | 0.449585 | − | 0.259568i | 0 | −2.83641 | + | 0.977122i | 0 | |||||||||||||||||||||||||||||||
| 191.3 | 0 | 1.04307 | + | 1.38276i | 0 | − | 1.45735i | 0 | 1.13290 | − | 0.654083i | 0 | −0.824030 | + | 2.88461i | 0 | ||||||||||||||||||||||||||||||
| 575.1 | 0 | −1.25707 | − | 1.19154i | 0 | − | 2.71781i | 0 | 4.41751 | + | 2.55045i | 0 | 0.160442 | + | 2.99571i | 0 | ||||||||||||||||||||||||||||||
| 575.2 | 0 | −0.285997 | + | 1.70828i | 0 | − | 3.93569i | 0 | 0.449585 | + | 0.259568i | 0 | −2.83641 | − | 0.977122i | 0 | ||||||||||||||||||||||||||||||
| 575.3 | 0 | 1.04307 | − | 1.38276i | 0 | 1.45735i | 0 | 1.13290 | + | 0.654083i | 0 | −0.824030 | − | 2.88461i | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 156.p | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 624.2.bu.l | yes | 6 |
| 3.b | odd | 2 | 1 | 624.2.bu.n | yes | 6 | |
| 4.b | odd | 2 | 1 | 624.2.bu.m | yes | 6 | |
| 12.b | even | 2 | 1 | 624.2.bu.k | ✓ | 6 | |
| 13.c | even | 3 | 1 | 624.2.bu.k | ✓ | 6 | |
| 39.i | odd | 6 | 1 | 624.2.bu.m | yes | 6 | |
| 52.j | odd | 6 | 1 | 624.2.bu.n | yes | 6 | |
| 156.p | even | 6 | 1 | inner | 624.2.bu.l | yes | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 624.2.bu.k | ✓ | 6 | 12.b | even | 2 | 1 | |
| 624.2.bu.k | ✓ | 6 | 13.c | even | 3 | 1 | |
| 624.2.bu.l | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 624.2.bu.l | yes | 6 | 156.p | even | 6 | 1 | inner |
| 624.2.bu.m | yes | 6 | 4.b | odd | 2 | 1 | |
| 624.2.bu.m | yes | 6 | 39.i | odd | 6 | 1 | |
| 624.2.bu.n | yes | 6 | 3.b | odd | 2 | 1 | |
| 624.2.bu.n | yes | 6 | 52.j | odd | 6 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\):
|
\( T_{5}^{6} + 25T_{5}^{4} + 163T_{5}^{2} + 243 \)
|
|
\( T_{7}^{6} - 12T_{7}^{5} + 58T_{7}^{4} - 120T_{7}^{3} + 124T_{7}^{2} - 60T_{7} + 12 \)
|
|
\( T_{11}^{6} - 10T_{11}^{5} + 78T_{11}^{4} - 232T_{11}^{3} + 544T_{11}^{2} + 132T_{11} + 36 \)
|
|
\( T_{17}^{6} + 3T_{17}^{5} - 12T_{17}^{4} - 45T_{17}^{3} + 198T_{17}^{2} + 405T_{17} + 243 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + T^{5} + \cdots + 27 \)
|
| $5$ |
\( T^{6} + 25 T^{4} + \cdots + 243 \)
|
| $7$ |
\( T^{6} - 12 T^{5} + \cdots + 12 \)
|
| $11$ |
\( T^{6} - 10 T^{5} + \cdots + 36 \)
|
| $13$ |
\( T^{6} + T^{5} + \cdots + 2197 \)
|
| $17$ |
\( T^{6} + 3 T^{5} + \cdots + 243 \)
|
| $19$ |
\( T^{6} + 6 T^{5} + \cdots + 108 \)
|
| $23$ |
\( T^{6} + 4 T^{5} + \cdots + 60516 \)
|
| $29$ |
\( (T^{2} - 9 T + 27)^{3} \)
|
| $31$ |
\( T^{6} + 88 T^{4} + \cdots + 13872 \)
|
| $37$ |
\( T^{6} + 9 T^{5} + \cdots + 26569 \)
|
| $41$ |
\( T^{6} + 15 T^{5} + \cdots + 27 \)
|
| $43$ |
\( T^{6} - 6 T^{5} + \cdots + 108 \)
|
| $47$ |
\( (T^{3} - 2 T^{2} + \cdots + 102)^{2} \)
|
| $53$ |
\( T^{6} + 217 T^{4} + \cdots + 41067 \)
|
| $59$ |
\( (T^{2} - 6 T + 36)^{3} \)
|
| $61$ |
\( T^{6} - 7 T^{5} + \cdots + 8649 \)
|
| $67$ |
\( T^{6} + 24 T^{5} + \cdots + 972 \)
|
| $71$ |
\( T^{6} + 26 T^{5} + \cdots + 311364 \)
|
| $73$ |
\( (T^{3} - 3 T^{2} - 21 T + 59)^{2} \)
|
| $79$ |
\( T^{6} + 256 T^{4} + \cdots + 442368 \)
|
| $83$ |
\( (T^{3} - 6 T^{2} - 90 T + 54)^{2} \)
|
| $89$ |
\( T^{6} - 24 T^{5} + \cdots + 62208 \)
|
| $97$ |
\( T^{6} + 4 T^{5} + \cdots + 242064 \)
|
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