Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6936,2,Mod(1,6936)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6936, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6936.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6936 = 2^{3} \cdot 3 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6936.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: | \(55.3842388420\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.77984694272.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 10x^{6} + 36x^{5} + 39x^{4} - 96x^{3} - 56x^{2} + 64x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 408) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 10x^{6} + 36x^{5} + 39x^{4} - 96x^{3} - 56x^{2} + 64x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - 16\nu^{6} + 46\nu^{5} + 96\nu^{4} - 365\nu^{3} - 68\nu^{2} + 624\nu - 216 ) / 68 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{7} - 15\nu^{6} + 24\nu^{5} + 56\nu^{4} - 186\nu^{3} + 17\nu^{2} + 296\nu - 92 ) / 34 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 9\nu^{7} - 42\nu^{6} - 62\nu^{5} + 388\nu^{4} - 21\nu^{3} - 850\nu^{2} + 380\nu + 164 ) / 68 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13\nu^{7} - 72\nu^{6} - 14\nu^{5} + 500\nu^{4} - 393\nu^{3} - 748\nu^{2} + 904\nu - 292 ) / 68 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -27\nu^{7} + 160\nu^{6} - 18\nu^{5} - 1028\nu^{4} + 811\nu^{3} + 1632\nu^{2} - 1412\nu + 120 ) / 68 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 24\nu^{7} - 129\nu^{6} - 52\nu^{5} + 876\nu^{4} - 294\nu^{3} - 1479\nu^{2} + 628\nu - 16 ) / 34 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} - \beta_{3} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} - 2\beta_{6} + 3\beta_{5} - 3\beta_{4} - 4\beta_{3} - 2\beta_{2} + 7\beta _1 + 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{7} - 9\beta_{6} + 16\beta_{5} - 14\beta_{4} - 19\beta_{3} - 9\beta_{2} + 19\beta _1 + 36 \)
|
| \(\nu^{5}\) | \(=\) |
\( -25\beta_{7} - 45\beta_{6} + 63\beta_{5} - 54\beta_{4} - 75\beta_{3} - 48\beta_{2} + 85\beta _1 + 113 \)
|
| \(\nu^{6}\) | \(=\) |
\( -108\beta_{7} - 188\beta_{6} + 275\beta_{5} - 223\beta_{4} - 313\beta_{3} - 208\beta_{2} + 315\beta _1 + 492 \)
|
| \(\nu^{7}\) | \(=\) |
\( -463\beta_{7} - 804\beta_{6} + 1129\beta_{5} - 903\beta_{4} - 1262\beta_{3} - 918\beta_{2} + 1305\beta _1 + 1896 \)
|
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 | −1.00000 | 0 | −2.88319 | 0 | −3.28909 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | −2.71472 | 0 | 2.40302 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | −2.58886 | 0 | −2.17045 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | −0.850926 | 0 | 1.73366 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | −0.464863 | 0 | −1.75417 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.00000 | 0 | 0.940008 | 0 | 4.35003 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −1.00000 | 0 | 1.44964 | 0 | 0.0271141 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | −1.00000 | 0 | 3.11290 | 0 | −1.30012 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(17\) | \( -1 \) |
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 | 6936.2.a.bl | 8 | |
| 17.b | even | 2 | 1 | 6936.2.a.bo | 8 | ||
| 17.e | odd | 16 | 2 | 408.2.ba.a | ✓ | 16 | |
| 51.i | even | 16 | 2 | 1224.2.bq.e | 16 | ||
| 68.i | even | 16 | 2 | 816.2.bq.f | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 408.2.ba.a | ✓ | 16 | 17.e | odd | 16 | 2 | |
| 816.2.bq.f | 16 | 68.i | even | 16 | 2 | ||
| 1224.2.bq.e | 16 | 51.i | even | 16 | 2 | ||
| 6936.2.a.bl | 8 | 1.a | even | 1 | 1 | trivial | |
| 6936.2.a.bo | 8 | 17.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6936))\):
|
\( T_{5}^{8} + 4T_{5}^{7} - 10T_{5}^{6} - 52T_{5}^{5} - T_{5}^{4} + 136T_{5}^{3} + 44T_{5}^{2} - 80T_{5} - 34 \)
|
|
\( T_{7}^{8} - 24T_{7}^{6} - 16T_{7}^{5} + 150T_{7}^{4} + 144T_{7}^{3} - 264T_{7}^{2} - 288T_{7} + 8 \)
|
|
\( T_{11}^{8} + 4T_{11}^{7} - 30T_{11}^{6} - 140T_{11}^{5} + 97T_{11}^{4} + 1032T_{11}^{3} + 1256T_{11}^{2} + 160T_{11} - 272 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( T^{8} + 4 T^{7} + \cdots - 34 \)
|
| $7$ |
\( T^{8} - 24 T^{6} + \cdots + 8 \)
|
| $11$ |
\( T^{8} + 4 T^{7} + \cdots - 272 \)
|
| $13$ |
\( T^{8} + 4 T^{7} + \cdots - 388 \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( T^{8} - 4 T^{7} + \cdots - 8 \)
|
| $23$ |
\( T^{8} - 4 T^{7} + \cdots + 68 \)
|
| $29$ |
\( T^{8} - 108 T^{6} + \cdots - 16648 \)
|
| $31$ |
\( T^{8} + 8 T^{7} + \cdots + 8672 \)
|
| $37$ |
\( T^{8} + 16 T^{7} + \cdots + 92732 \)
|
| $41$ |
\( T^{8} + 20 T^{7} + \cdots + 2594 \)
|
| $43$ |
\( T^{8} - 28 T^{7} + \cdots + 69088 \)
|
| $47$ |
\( T^{8} - 24 T^{7} + \cdots - 184192 \)
|
| $53$ |
\( T^{8} - 8 T^{7} + \cdots + 378944 \)
|
| $59$ |
\( T^{8} - 16 T^{7} + \cdots - 5413136 \)
|
| $61$ |
\( T^{8} + 24 T^{7} + \cdots - 17532484 \)
|
| $67$ |
\( T^{8} - 316 T^{6} + \cdots + 2236384 \)
|
| $71$ |
\( T^{8} - 248 T^{6} + \cdots - 65008 \)
|
| $73$ |
\( T^{8} + 40 T^{7} + \cdots + 360928 \)
|
| $79$ |
\( T^{8} + 8 T^{7} + \cdots + 5030408 \)
|
| $83$ |
\( T^{8} - 32 T^{7} + \cdots + 893056 \)
|
| $89$ |
\( T^{8} - 16 T^{7} + \cdots - 6919288 \)
|
| $97$ |
\( T^{8} + 48 T^{7} + \cdots + 18526736 \)
|
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