Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,12,Mod(1,13)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 13 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 13.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: | \(9.98846134727\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 13546x^{4} + 130998x^{3} + 49403509x^{2} - 776207317x - 22123683244 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 13546x^{4} + 130998x^{3} + 49403509x^{2} - 776207317x - 22123683244 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -1925\nu^{5} - 86014\nu^{4} + 15916032\nu^{3} + 462431442\nu^{2} - 26906396267\nu - 114096339268 ) / 502193280 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4549 \nu^{5} + 297830 \nu^{4} - 39656088 \nu^{3} - 2035001754 \nu^{2} + 79116342019 \nu + 1739047527524 ) / 1004386560 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5015 \nu^{5} + 381698 \nu^{4} - 44872104 \nu^{3} - 2440298814 \nu^{2} + 100337716769 \nu + 1234012729676 ) / 1004386560 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1745\nu^{5} + 170366\nu^{4} - 12096168\nu^{3} - 1100201058\nu^{2} + 15895857863\nu + 715844708372 ) / 251096640 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{4} - 5\beta_{3} - 2\beta_{2} - 13\beta _1 + 4517 \)
|
| \(\nu^{3}\) | \(=\) |
\( 58\beta_{5} - 166\beta_{4} + 50\beta_{3} - 52\beta_{2} + 6187\beta _1 - 59786 \)
|
| \(\nu^{4}\) | \(=\) |
\( 1254\beta_{5} + 26301\beta_{4} - 38115\beta_{3} - 8502\beta_{2} - 160011\beta _1 + 28173671 \)
|
| \(\nu^{5}\) | \(=\) |
\( 423516\beta_{5} - 1827024\beta_{4} + 915360\beta_{3} - 791376\beta_{2} + 41203985\beta _1 - 727365996 \)
|
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 |
|
−80.6176 | 827.494 | 4451.20 | 7679.45 | −66710.6 | −6800.32 | −193740. | 507599. | −619099. | ||||||||||||||||||||||||||||||||||||
| 1.2 | −70.8045 | −19.0788 | 2965.27 | −11885.7 | 1350.86 | −62220.3 | −64947.1 | −176783. | 841561. | |||||||||||||||||||||||||||||||||||||
| 1.3 | −6.24050 | −573.204 | −2009.06 | −3613.82 | 3577.08 | 14401.4 | 25318.0 | 151416. | 22552.1 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 45.5093 | 469.277 | 23.0963 | 4406.29 | 21356.5 | 63013.3 | −92151.9 | 43074.3 | 200527. | |||||||||||||||||||||||||||||||||||||
| 1.5 | 82.0961 | −696.954 | 4691.77 | 6795.70 | −57217.2 | 67467.4 | 217044. | 308598. | 557901. | |||||||||||||||||||||||||||||||||||||
| 1.6 | 85.0571 | 468.466 | 5186.72 | −69.9037 | 39846.4 | −80037.3 | 266970. | 42313.3 | −5945.81 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(1\) |
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 | 13.12.a.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 117.12.a.d | 6 | ||
| 4.b | odd | 2 | 1 | 208.12.a.h | 6 | ||
| 13.b | even | 2 | 1 | 169.12.a.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.12.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 117.12.a.d | 6 | 3.b | odd | 2 | 1 | ||
| 169.12.a.c | 6 | 13.b | even | 2 | 1 | ||
| 208.12.a.h | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 55T_{2}^{5} - 12286T_{2}^{4} + 603264T_{2}^{3} + 39388912T_{2}^{2} - 1594524928T_{2} - 11319915520 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(13))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 11319915520 \)
|
| $3$ |
\( T^{6} + \cdots - 13\!\cdots\!12 \)
|
| $5$ |
\( T^{6} + \cdots - 69\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots - 20\!\cdots\!96 \)
|
| $11$ |
\( T^{6} + \cdots - 18\!\cdots\!92 \)
|
| $13$ |
\( (T + 371293)^{6} \)
|
| $17$ |
\( T^{6} + \cdots - 14\!\cdots\!04 \)
|
| $19$ |
\( T^{6} + \cdots - 20\!\cdots\!84 \)
|
| $23$ |
\( T^{6} + \cdots + 21\!\cdots\!88 \)
|
| $29$ |
\( T^{6} + \cdots + 82\!\cdots\!12 \)
|
| $31$ |
\( T^{6} + \cdots - 54\!\cdots\!60 \)
|
| $37$ |
\( T^{6} + \cdots - 36\!\cdots\!32 \)
|
| $41$ |
\( T^{6} + \cdots + 89\!\cdots\!16 \)
|
| $43$ |
\( T^{6} + \cdots + 38\!\cdots\!60 \)
|
| $47$ |
\( T^{6} + \cdots + 46\!\cdots\!20 \)
|
| $53$ |
\( T^{6} + \cdots + 24\!\cdots\!72 \)
|
| $59$ |
\( T^{6} + \cdots - 28\!\cdots\!04 \)
|
| $61$ |
\( T^{6} + \cdots + 32\!\cdots\!92 \)
|
| $67$ |
\( T^{6} + \cdots - 12\!\cdots\!84 \)
|
| $71$ |
\( T^{6} + \cdots - 32\!\cdots\!16 \)
|
| $73$ |
\( T^{6} + \cdots - 73\!\cdots\!68 \)
|
| $79$ |
\( T^{6} + \cdots - 25\!\cdots\!96 \)
|
| $83$ |
\( T^{6} + \cdots - 50\!\cdots\!72 \)
|
| $89$ |
\( T^{6} + \cdots + 28\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 13\!\cdots\!00 \)
|
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