Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,14,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, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 13 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| 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: | \(13.9400207637\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{6} - x^{5} - 34578x^{4} - 245250x^{3} + 301528653x^{2} + 1141403491x - 761768771036 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 34578x^{4} - 245250x^{3} + 301528653x^{2} + 1141403491x - 761768771036 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 455\nu^{5} + 20106\nu^{4} - 13155416\nu^{3} - 753334486\nu^{2} + 58401824033\nu + 3416956810460 ) / 187891968 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 455\nu^{5} + 20106\nu^{4} - 13155416\nu^{3} - 690703830\nu^{2} + 57650256161\nu + 2695201130716 ) / 62630656 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 133\nu^{5} + 9641\nu^{4} - 3962109\nu^{3} - 317091025\nu^{2} + 18008741056\nu + 1387523313920 ) / 3914416 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 5889 \nu^{5} - 483910 \nu^{4} + 172729160 \nu^{3} + 15794382618 \nu^{2} + \cdots - 70012689899972 ) / 187891968 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 3\beta_{2} + 12\beta _1 + 11524 \)
|
| \(\nu^{3}\) | \(=\) |
\( -42\beta_{5} - 52\beta_{4} + 64\beta_{3} - 6\beta_{2} + 17173\beta _1 + 137038 \)
|
| \(\nu^{4}\) | \(=\) |
\( -1302\beta_{5} - 572\beta_{4} + 27725\beta_{3} - 92001\beta_{2} + 592192\beta _1 + 197610402 \)
|
| \(\nu^{5}\) | \(=\) |
\( -1156812\beta_{5} - 1478200\beta_{4} + 2280972\beta_{3} - 662136\beta_{2} + 361867113\beta _1 + 6800235080 \)
|
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 |
|
−151.869 | −2279.81 | 14872.1 | −53923.4 | 346232. | 358132. | −1.01450e6 | 3.60323e6 | 8.18928e6 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −104.576 | 1639.96 | 2744.04 | −27927.9 | −171499. | 85412.5 | 569723. | 1.09513e6 | 2.92058e6 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −79.8369 | −1213.97 | −1818.06 | 39770.7 | 96919.6 | 108654. | 799173. | −120602. | −3.17517e6 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 60.8052 | 1156.50 | −4494.73 | −18981.4 | 70321.3 | −435876. | −771419. | −256825. | −1.15417e6 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 62.5789 | −399.859 | −4275.88 | 29322.8 | −25022.8 | 173095. | −780226. | −1.43444e6 | 1.83499e6 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 147.897 | −1090.82 | 13681.5 | −45012.7 | −161328. | −17976.6 | 811887. | −404445. | −6.65725e6 | |||||||||||||||||||||||||||||||||||||
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 | 13.14.a.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 117.14.a.c | 6 | ||
| 13.b | even | 2 | 1 | 169.14.a.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.14.a.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 117.14.a.c | 6 | 3.b | odd | 2 | 1 | ||
| 169.14.a.a | 6 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 65T_{2}^{5} - 32818T_{2}^{4} - 1741272T_{2}^{3} + 268538080T_{2}^{2} + 7502807936T_{2} - 713559439360 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(13))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 713559439360 \)
|
| $3$ |
\( T^{6} + \cdots + 22\!\cdots\!44 \)
|
| $5$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots + 45\!\cdots\!08 \)
|
| $11$ |
\( T^{6} + \cdots + 11\!\cdots\!04 \)
|
| $13$ |
\( (T + 4826809)^{6} \)
|
| $17$ |
\( T^{6} + \cdots + 17\!\cdots\!32 \)
|
| $19$ |
\( T^{6} + \cdots + 51\!\cdots\!20 \)
|
| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!16 \)
|
| $29$ |
\( T^{6} + \cdots - 13\!\cdots\!60 \)
|
| $31$ |
\( T^{6} + \cdots + 13\!\cdots\!16 \)
|
| $37$ |
\( T^{6} + \cdots - 22\!\cdots\!44 \)
|
| $41$ |
\( T^{6} + \cdots - 56\!\cdots\!92 \)
|
| $43$ |
\( T^{6} + \cdots - 83\!\cdots\!24 \)
|
| $47$ |
\( T^{6} + \cdots + 14\!\cdots\!80 \)
|
| $53$ |
\( T^{6} + \cdots - 18\!\cdots\!08 \)
|
| $59$ |
\( T^{6} + \cdots - 48\!\cdots\!20 \)
|
| $61$ |
\( T^{6} + \cdots - 28\!\cdots\!32 \)
|
| $67$ |
\( T^{6} + \cdots + 24\!\cdots\!76 \)
|
| $71$ |
\( T^{6} + \cdots - 17\!\cdots\!72 \)
|
| $73$ |
\( T^{6} + \cdots - 26\!\cdots\!56 \)
|
| $79$ |
\( T^{6} + \cdots - 12\!\cdots\!80 \)
|
| $83$ |
\( T^{6} + \cdots - 46\!\cdots\!64 \)
|
| $89$ |
\( T^{6} + \cdots - 38\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 33\!\cdots\!80 \)
|
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