Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [164,4,Mod(1,164)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(164, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("164.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 164 = 2^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 164.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: | \(9.67631324094\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 89x^{5} - 72x^{4} + 931x^{3} + 936x^{2} - 1419x - 936 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 89x^{5} - 72x^{4} + 931x^{3} + 936x^{2} - 1419x - 936 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 262\nu^{6} + 397\nu^{5} - 24028\nu^{4} - 49354\nu^{3} + 287178\nu^{2} + 674061\nu - 845676 ) / 105732 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -262\nu^{6} - 397\nu^{5} + 24028\nu^{4} + 49354\nu^{3} - 287178\nu^{2} - 251133\nu + 845676 ) / 105732 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -326\nu^{6} + 2869\nu^{5} + 23306\nu^{4} - 223636\nu^{3} - 27084\nu^{2} + 1939371\nu - 324684 ) / 105732 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 140\nu^{6} - 169\nu^{5} - 11270\nu^{4} + 2056\nu^{3} + 49650\nu^{2} + 120405\nu + 115110 ) / 17622 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -160\nu^{6} + 333\nu^{5} + 13859\nu^{4} - 16755\nu^{3} - 140797\nu^{2} + 72180\nu + 216690 ) / 5874 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3644\nu^{6} - 4231\nu^{5} - 316838\nu^{4} + 103192\nu^{3} + 3027666\nu^{2} - 156309\nu - 2841768 ) / 105732 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -4\beta_{6} - 3\beta_{5} + 2\beta_{4} - \beta_{3} - 15\beta_{2} + 107 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{6} + 10\beta_{5} - 10\beta_{4} - 26\beta_{3} + 21\beta_{2} + 75\beta _1 + 156 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -304\beta_{6} - 195\beta_{5} + 200\beta_{4} - 145\beta_{3} - 1203\beta_{2} + 60\beta _1 + 7373 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 136\beta_{6} + 760\beta_{5} - 652\beta_{4} - 2264\beta_{3} + 69\beta_{2} + 5805\beta _1 + 18804 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -22948\beta_{6} - 13863\beta_{5} + 15254\beta_{4} - 13669\beta_{3} - 92607\beta_{2} + 9876\beta _1 + 572699 ) / 4 \)
|
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 | −7.49632 | 0 | −18.6419 | 0 | −28.7226 | 0 | 29.1948 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −7.40882 | 0 | 6.76355 | 0 | −12.4395 | 0 | 27.8906 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.74615 | 0 | 21.8374 | 0 | −1.53250 | 0 | −23.9509 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.42181 | 0 | −15.9990 | 0 | 26.3525 | 0 | −24.9784 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 5.21746 | 0 | 2.54130 | 0 | 17.3104 | 0 | 0.221919 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 7.29517 | 0 | 14.3758 | 0 | 21.9585 | 0 | 26.2195 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 9.56047 | 0 | −0.877058 | 0 | −22.9268 | 0 | 64.4026 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(41\) | \(-1\) |
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 | 164.4.a.b | ✓ | 7 |
| 3.b | odd | 2 | 1 | 1476.4.a.h | 7 | ||
| 4.b | odd | 2 | 1 | 656.4.a.k | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 164.4.a.b | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 656.4.a.k | 7 | 4.b | odd | 2 | 1 | ||
| 1476.4.a.h | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} - 4T_{3}^{6} - 136T_{3}^{5} + 376T_{3}^{4} + 5456T_{3}^{3} - 7760T_{3}^{2} - 55748T_{3} - 50176 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(164))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 4 T^{6} + \cdots - 50176 \)
|
| $5$ |
\( T^{7} - 10 T^{6} + \cdots + 1411488 \)
|
| $7$ |
\( T^{7} - 1492 T^{5} + \cdots - 125748688 \)
|
| $11$ |
\( T^{7} + \cdots - 215871892128 \)
|
| $13$ |
\( T^{7} + \cdots + 15867990144 \)
|
| $17$ |
\( T^{7} + \cdots - 114289328256 \)
|
| $19$ |
\( T^{7} + \cdots + 73498533216 \)
|
| $23$ |
\( T^{7} + \cdots - 15223815020544 \)
|
| $29$ |
\( T^{7} + \cdots - 782179446748032 \)
|
| $31$ |
\( T^{7} + \cdots + 1198969929728 \)
|
| $37$ |
\( T^{7} + \cdots + 483348569362848 \)
|
| $41$ |
\( (T - 41)^{7} \)
|
| $43$ |
\( T^{7} + \cdots - 86\!\cdots\!24 \)
|
| $47$ |
\( T^{7} + \cdots - 14\!\cdots\!92 \)
|
| $53$ |
\( T^{7} + \cdots - 43\!\cdots\!56 \)
|
| $59$ |
\( T^{7} + \cdots + 81\!\cdots\!56 \)
|
| $61$ |
\( T^{7} + \cdots - 51\!\cdots\!72 \)
|
| $67$ |
\( T^{7} + \cdots - 11\!\cdots\!64 \)
|
| $71$ |
\( T^{7} + \cdots + 19\!\cdots\!76 \)
|
| $73$ |
\( T^{7} + \cdots + 30\!\cdots\!48 \)
|
| $79$ |
\( T^{7} + \cdots + 23\!\cdots\!64 \)
|
| $83$ |
\( T^{7} + \cdots + 41\!\cdots\!24 \)
|
| $89$ |
\( T^{7} + \cdots - 79\!\cdots\!32 \)
|
| $97$ |
\( T^{7} + \cdots + 25\!\cdots\!16 \)
|
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