Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [129,4,Mod(1,129)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(129, 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("129.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 129 = 3 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 129.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: | \(7.61124639074\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 59x^{6} + 43x^{5} + 1106x^{4} - 514x^{3} - 6880x^{2} + 864x + 6912 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 59x^{6} + 43x^{5} + 1106x^{4} - 514x^{3} - 6880x^{2} + 864x + 6912 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 4\nu^{4} - 27\nu^{3} + 92\nu^{2} + 74\nu - 144 ) / 24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + 3\nu^{6} - 47\nu^{5} - 81\nu^{4} + 526\nu^{3} - 138\nu^{2} - 776\nu + 3648 ) / 384 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} + 7\nu^{6} + 141\nu^{5} - 253\nu^{4} - 1834\nu^{3} + 2302\nu^{2} + 5144\nu - 4032 ) / 384 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} - 3\nu^{6} + 63\nu^{5} + 113\nu^{4} - 1150\nu^{3} - 790\nu^{2} + 5608\nu - 768 ) / 192 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - 13\nu^{6} - 47\nu^{5} + 607\nu^{4} + 782\nu^{3} - 7594\nu^{2} - 4744\nu + 16704 ) / 384 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 15 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{3} + 2\beta_{2} + 22\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} + 2\beta_{5} + 4\beta_{4} + 29\beta_{2} + 6\beta _1 + 331 \)
|
| \(\nu^{5}\) | \(=\) |
\( 35\beta_{7} - 27\beta_{6} + 35\beta_{5} + 16\beta_{4} + 78\beta_{3} + 78\beta_{2} + 544\beta _1 + 223 \)
|
| \(\nu^{6}\) | \(=\) |
\( 78\beta_{7} - 16\beta_{6} + 102\beta_{5} + 196\beta_{4} + 32\beta_{3} + 813\beta_{2} + 362\beta _1 + 8139 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1047\beta_{7} - 695\beta_{6} + 975\beta_{5} + 872\beta_{4} + 2518\beta_{3} + 2662\beta_{2} + 14172\beta _1 + 8667 \)
|
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 |
|
−5.06934 | 3.00000 | 17.6982 | −14.9596 | −15.2080 | 27.1292 | −49.1634 | 9.00000 | 75.8352 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.52533 | 3.00000 | 12.4786 | 18.7202 | −13.5760 | 22.6994 | −20.2671 | 9.00000 | −84.7151 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.88828 | 3.00000 | 0.342149 | −19.9221 | −8.66483 | −9.47486 | 22.1180 | 9.00000 | 57.5405 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.07486 | 3.00000 | −6.84467 | 13.4069 | −3.22458 | −6.64627 | 15.9560 | 9.00000 | −14.4106 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.14150 | 3.00000 | −6.69699 | −5.11237 | 3.42449 | 33.4335 | −16.7765 | 9.00000 | −5.83575 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.52419 | 3.00000 | 4.41994 | 17.7278 | 10.5726 | −8.91575 | −12.6168 | 9.00000 | 62.4764 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.36475 | 3.00000 | 11.0511 | 2.14600 | 13.0943 | 15.9941 | 13.3171 | 9.00000 | 9.36677 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 5.52736 | 3.00000 | 22.5517 | −8.00696 | 16.5821 | −22.2193 | 80.4327 | 9.00000 | −44.2574 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(43\) | \(-1\) |
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 | 129.4.a.f | ✓ | 8 |
| 3.b | odd | 2 | 1 | 387.4.a.i | 8 | ||
| 4.b | odd | 2 | 1 | 2064.4.a.ba | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 129.4.a.f | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 387.4.a.i | 8 | 3.b | odd | 2 | 1 | ||
| 2064.4.a.ba | 8 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - T_{2}^{7} - 59T_{2}^{6} + 43T_{2}^{5} + 1106T_{2}^{4} - 514T_{2}^{3} - 6880T_{2}^{2} + 864T_{2} + 6912 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(129))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{7} + \cdots + 6912 \)
|
| $3$ |
\( (T - 3)^{8} \)
|
| $5$ |
\( T^{8} - 4 T^{7} + \cdots + 116485248 \)
|
| $7$ |
\( T^{8} + \cdots + 4108009472 \)
|
| $11$ |
\( T^{8} + \cdots - 130355566056 \)
|
| $13$ |
\( T^{8} + \cdots + 27308921476 \)
|
| $17$ |
\( T^{8} + \cdots + 13478793512448 \)
|
| $19$ |
\( T^{8} + \cdots + 45992688497984 \)
|
| $23$ |
\( T^{8} + \cdots + 22\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots - 23\!\cdots\!28 \)
|
| $31$ |
\( T^{8} + \cdots - 32\!\cdots\!44 \)
|
| $37$ |
\( T^{8} + \cdots - 23\!\cdots\!24 \)
|
| $41$ |
\( T^{8} + \cdots - 71089654136832 \)
|
| $43$ |
\( (T - 43)^{8} \)
|
| $47$ |
\( T^{8} + \cdots - 25\!\cdots\!44 \)
|
| $53$ |
\( T^{8} + \cdots + 61\!\cdots\!08 \)
|
| $59$ |
\( T^{8} + \cdots - 61\!\cdots\!24 \)
|
| $61$ |
\( T^{8} + \cdots + 12\!\cdots\!52 \)
|
| $67$ |
\( T^{8} + \cdots + 42\!\cdots\!84 \)
|
| $71$ |
\( T^{8} + \cdots - 92\!\cdots\!12 \)
|
| $73$ |
\( T^{8} + \cdots + 14\!\cdots\!52 \)
|
| $79$ |
\( T^{8} + \cdots - 13\!\cdots\!44 \)
|
| $83$ |
\( T^{8} + \cdots + 42\!\cdots\!68 \)
|
| $89$ |
\( T^{8} + \cdots - 62\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 50\!\cdots\!52 \)
|
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