Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [684,3,Mod(37,684)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(684, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("684.37");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 684.h (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: | \(18.6376500822\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.219615408.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 22x^{4} - 39x^{3} + 112x^{2} - 93x + 39 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 228) |
| 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
\( x^{6} - 3x^{5} + 22x^{4} - 39x^{3} + 112x^{2} - 93x + 39 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} - 32\nu^{3} + 43\nu^{2} + 16\nu - 15 ) / 19 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} + 9\nu^{2} - 8\nu - 7 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} - 13\nu^{2} + 12\nu - 17 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 8\nu^{5} - 20\nu^{4} + 128\nu^{3} - 172\nu^{2} + 392\nu - 168 ) / 19 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 8\nu^{5} - 20\nu^{4} + 204\nu^{3} - 286\nu^{2} + 1114\nu - 510 ) / 19 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 4\beta _1 + 12 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - 12\beta_{3} - 12\beta_{2} + 4\beta _1 - 132 ) / 24 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{5} - 7\beta_{4} - 9\beta_{3} - 9\beta_{2} - 16\beta _1 - 102 ) / 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 12\beta_{5} - 29\beta_{4} + 72\beta_{3} + 120\beta_{2} - 68\beta _1 + 1044 ) / 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -66\beta_{5} + 181\beta_{4} + 210\beta_{3} + 330\beta_{2} + 232\beta _1 + 2952 ) / 24 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(343\) | \(533\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
0 | 0 | 0 | −5.83539 | 0 | 5.24085 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 37.2 | 0 | 0 | 0 | −5.83539 | 0 | 5.24085 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 37.3 | 0 | 0 | 0 | −0.556406 | 0 | −9.52587 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 37.4 | 0 | 0 | 0 | −0.556406 | 0 | −9.52587 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 37.5 | 0 | 0 | 0 | 7.39180 | 0 | 3.28502 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 37.6 | 0 | 0 | 0 | 7.39180 | 0 | 3.28502 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 684.3.h.e | 6 | |
| 3.b | odd | 2 | 1 | 228.3.h.a | ✓ | 6 | |
| 4.b | odd | 2 | 1 | 2736.3.o.m | 6 | ||
| 12.b | even | 2 | 1 | 912.3.o.c | 6 | ||
| 19.b | odd | 2 | 1 | inner | 684.3.h.e | 6 | |
| 57.d | even | 2 | 1 | 228.3.h.a | ✓ | 6 | |
| 76.d | even | 2 | 1 | 2736.3.o.m | 6 | ||
| 228.b | odd | 2 | 1 | 912.3.o.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 228.3.h.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 228.3.h.a | ✓ | 6 | 57.d | even | 2 | 1 | |
| 684.3.h.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 684.3.h.e | 6 | 19.b | odd | 2 | 1 | inner | |
| 912.3.o.c | 6 | 12.b | even | 2 | 1 | ||
| 912.3.o.c | 6 | 228.b | odd | 2 | 1 | ||
| 2736.3.o.m | 6 | 4.b | odd | 2 | 1 | ||
| 2736.3.o.m | 6 | 76.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - T_{5}^{2} - 44T_{5} - 24 \)
acting on \(S_{3}^{\mathrm{new}}(684, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{3} - T^{2} - 44 T - 24)^{2} \)
|
| $7$ |
\( (T^{3} + T^{2} - 64 T + 164)^{2} \)
|
| $11$ |
\( (T^{3} - 13 T^{2} + \cdots + 12)^{2} \)
|
| $13$ |
\( T^{6} + 792 T^{4} + \cdots + 2495232 \)
|
| $17$ |
\( (T^{3} - 25 T^{2} + \cdots + 108)^{2} \)
|
| $19$ |
\( T^{6} + 10 T^{5} + \cdots + 47045881 \)
|
| $23$ |
\( (T^{3} + 14 T^{2} + \cdots - 5136)^{2} \)
|
| $29$ |
\( T^{6} + 3168 T^{4} + \cdots + 322486272 \)
|
| $31$ |
\( T^{6} + 3192 T^{4} + \cdots + 62208 \)
|
| $37$ |
\( T^{6} + 5976 T^{4} + \cdots + 832267008 \)
|
| $41$ |
\( T^{6} + \cdots + 29104386048 \)
|
| $43$ |
\( (T^{3} + 105 T^{2} + \cdots + 35764)^{2} \)
|
| $47$ |
\( (T^{3} + 11 T^{2} + \cdots - 25272)^{2} \)
|
| $53$ |
\( T^{6} + 4272 T^{4} + \cdots + 63700992 \)
|
| $59$ |
\( T^{6} + \cdots + 207873257472 \)
|
| $61$ |
\( (T^{3} - 107 T^{2} + \cdots + 104276)^{2} \)
|
| $67$ |
\( T^{6} + 4320 T^{4} + \cdots + 511377408 \)
|
| $71$ |
\( T^{6} + \cdots + 48612814848 \)
|
| $73$ |
\( (T^{3} - 51 T^{2} + \cdots + 115492)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 60723965952 \)
|
| $83$ |
\( (T^{3} - 202 T^{2} + \cdots - 259992)^{2} \)
|
| $89$ |
\( T^{6} + 3408 T^{4} + \cdots + 956510208 \)
|
| $97$ |
\( T^{6} + 8400 T^{4} + \cdots + 168210432 \)
|
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