Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1336,2,Mod(1,1336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1336, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1336.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1336 = 2^{3} \cdot 167 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1336.a (trivial) |
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: | yes |
| Analytic conductor: | \(10.6680137100\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - x^{8} - 13x^{7} + 8x^{6} + 56x^{5} - 15x^{4} - 81x^{3} + 2x^{2} + 13x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - x^{8} - 13x^{7} + 8x^{6} + 56x^{5} - 15x^{4} - 81x^{3} + 2x^{2} + 13x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{8} - 2\nu^{7} - 10\nu^{6} + 16\nu^{5} + 32\nu^{4} - 31\nu^{3} - 34\nu^{2} + 3\nu + 4 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{8} + 2\nu^{7} + 10\nu^{6} - 16\nu^{5} - 31\nu^{4} + 31\nu^{3} + 29\nu^{2} - 5\nu - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{8} + 5\nu^{7} + 18\nu^{6} - 42\nu^{5} - 46\nu^{4} + 92\nu^{3} + 27\nu^{2} - 34\nu - 1 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{8} - 9\nu^{7} - 22\nu^{6} + 72\nu^{5} + 30\nu^{4} - 137\nu^{3} + 31\nu^{2} + 5\nu - 1 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{8} + 9\nu^{7} + 22\nu^{6} - 72\nu^{5} - 30\nu^{4} + 139\nu^{3} - 33\nu^{2} - 13\nu + 5 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( 2\nu^{8} - 5\nu^{7} - 18\nu^{6} + 41\nu^{5} + 47\nu^{4} - 84\nu^{3} - 31\nu^{2} + 19\nu + 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -5\nu^{8} + 12\nu^{7} + 46\nu^{6} - 98\nu^{5} - 126\nu^{4} + 199\nu^{3} + 98\nu^{2} - 41\nu - 14 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} + \beta_{7} + \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{8} + 5\beta_{7} + \beta_{3} + 7\beta_{2} + 2\beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{8} + 8\beta_{7} + 8\beta_{6} + 8\beta_{5} - 2\beta_{4} + \beta_{3} + 11\beta_{2} + 19\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 29\beta_{8} + 26\beta_{7} + \beta_{6} + 2\beta_{5} - 3\beta_{4} + 9\beta_{3} + 50\beta_{2} + 21\beta _1 + 75 \)
|
| \(\nu^{7}\) | \(=\) |
\( 69\beta_{8} + 53\beta_{7} + 52\beta_{6} + 54\beta_{5} - 24\beta_{4} + 10\beta_{3} + 99\beta_{2} + 104\beta _1 + 94 \)
|
| \(\nu^{8}\) | \(=\) |
\( 189 \beta_{8} + 143 \beta_{7} + 17 \beta_{6} + 31 \beta_{5} - 46 \beta_{4} + 62 \beta_{3} + 365 \beta_{2} + \cdots + 443 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −2.70492 | 0 | −4.24392 | 0 | 1.27743 | 0 | 4.31658 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.37109 | 0 | 0.438298 | 0 | 3.46684 | 0 | 2.62208 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.63574 | 0 | −0.530251 | 0 | −0.325283 | 0 | −0.324343 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.460631 | 0 | 0.528623 | 0 | −1.88578 | 0 | −2.78782 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.0812095 | 0 | 2.29286 | 0 | 0.862825 | 0 | −2.99341 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.374925 | 0 | −2.61380 | 0 | 4.69526 | 0 | −2.85943 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.74816 | 0 | 1.48670 | 0 | −4.05717 | 0 | 0.0560672 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.76335 | 0 | −2.14065 | 0 | −2.31472 | 0 | 0.109415 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.20474 | 0 | −3.21786 | 0 | 0.280589 | 0 | 1.86086 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(167\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1336.2.a.c | ✓ | 9 |
| 4.b | odd | 2 | 1 | 2672.2.a.m | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1336.2.a.c | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 2672.2.a.m | 9 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{9} + T_{3}^{8} - 13T_{3}^{7} - 8T_{3}^{6} + 56T_{3}^{5} + 15T_{3}^{4} - 81T_{3}^{3} - 2T_{3}^{2} + 13T_{3} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1336))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} \)
|
| $3$ |
\( T^{9} + T^{8} - 13 T^{7} + \cdots - 1 \)
|
| $5$ |
\( T^{9} + 8 T^{8} + \cdots + 32 \)
|
| $7$ |
\( T^{9} - 2 T^{8} + \cdots - 29 \)
|
| $11$ |
\( T^{9} + 10 T^{8} + \cdots - 832 \)
|
| $13$ |
\( T^{9} + 13 T^{8} + \cdots + 50336 \)
|
| $17$ |
\( T^{9} + 8 T^{8} + \cdots + 38944 \)
|
| $19$ |
\( T^{9} + T^{8} + \cdots + 117452 \)
|
| $23$ |
\( T^{9} + 3 T^{8} + \cdots + 69728 \)
|
| $29$ |
\( T^{9} + 25 T^{8} + \cdots - 6539 \)
|
| $31$ |
\( T^{9} + T^{8} + \cdots + 558284 \)
|
| $37$ |
\( T^{9} + 35 T^{8} + \cdots + 955168 \)
|
| $41$ |
\( T^{9} + 16 T^{8} + \cdots - 6588896 \)
|
| $43$ |
\( T^{9} - 9 T^{8} + \cdots + 219424 \)
|
| $47$ |
\( T^{9} + T^{8} + \cdots - 35524 \)
|
| $53$ |
\( T^{9} + 29 T^{8} + \cdots + 3456 \)
|
| $59$ |
\( T^{9} + 14 T^{8} + \cdots + 44620928 \)
|
| $61$ |
\( T^{9} + 28 T^{8} + \cdots - 176929261 \)
|
| $67$ |
\( T^{9} - 19 T^{8} + \cdots + 12451104 \)
|
| $71$ |
\( T^{9} + 9 T^{8} + \cdots - 635392 \)
|
| $73$ |
\( T^{9} - 497 T^{7} + \cdots + 231346336 \)
|
| $79$ |
\( T^{9} + 18 T^{8} + \cdots + 292448 \)
|
| $83$ |
\( T^{9} + 13 T^{8} + \cdots + 128672 \)
|
| $89$ |
\( T^{9} + 21 T^{8} + \cdots - 2651312 \)
|
| $97$ |
\( T^{9} - 2 T^{8} + \cdots + 5812 \)
|
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