Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,16,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, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 13 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| 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: | \(18.5501556630\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{7} - 2 x^{6} - 172505 x^{5} + 9594016 x^{4} + 7133274223 x^{3} - 680104282298 x^{2} + \cdots + 23\!\cdots\!32 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{2}\cdot 5 \) |
| 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
\( x^{7} - 2 x^{6} - 172505 x^{5} + 9594016 x^{4} + 7133274223 x^{3} - 680104282298 x^{2} + \cdots + 23\!\cdots\!32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 41041543 \nu^{6} + 14123740235 \nu^{5} - 5785749609114 \nu^{4} + \cdots - 14\!\cdots\!64 ) / 41\!\cdots\!80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 25853207 \nu^{6} - 3848269423 \nu^{5} + 3853554298314 \nu^{4} + 319114537852750 \nu^{3} + \cdots - 57\!\cdots\!20 ) / 20\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 25853207 \nu^{6} + 3848269423 \nu^{5} - 3853554298314 \nu^{4} - 319114537852750 \nu^{3} + \cdots - 45\!\cdots\!20 ) / 20\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 23661403 \nu^{6} - 5201947975 \nu^{5} + 3187138376114 \nu^{4} + 513969319377406 \nu^{3} + \cdots + 15\!\cdots\!64 ) / 92\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 463704677 \nu^{6} + 73231607569 \nu^{5} - 67845749579310 \nu^{4} + \cdots - 65\!\cdots\!08 ) / 13\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - 81\beta _1 + 49311 \)
|
| \(\nu^{3}\) | \(=\) |
\( -48\beta_{6} - 130\beta_{5} + 964\beta_{4} - 97\beta_{3} - 47\beta_{2} + 89264\beta _1 - 3989668 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7520 \beta_{6} - 13742 \beta_{5} - 88231 \beta_{4} + 122694 \beta_{3} - 24809 \beta_{2} + \cdots + 4419484419 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 7861216 \beta_{6} - 19879132 \beta_{5} + 152277092 \beta_{4} - 19231490 \beta_{3} + \cdots - 666852880464 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1698563264 \beta_{6} - 693924636 \beta_{5} - 28227325375 \beta_{4} + 14843004603 \beta_{3} + \cdots + 474171031456999 \)
|
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 |
|
−348.737 | 4816.93 | 88849.8 | −293140. | −1.67985e6 | 2.09132e6 | −1.95578e7 | 8.85394e6 | 1.02229e8 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −219.106 | −3377.01 | 15239.6 | −113774. | 739926. | 714392. | 3.84058e6 | −2.94468e6 | 2.49287e7 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −208.673 | 5348.59 | 10776.4 | 82594.0 | −1.11610e6 | −3.96367e6 | 4.58906e6 | 1.42585e7 | −1.72351e7 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −98.0235 | −6818.66 | −23159.4 | 151115. | 668388. | 1.60747e6 | 5.48220e6 | 3.21452e7 | −1.48128e7 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 14.8141 | 1585.00 | −32548.5 | 102776. | 23480.3 | 1.31907e6 | −967607. | −1.18367e7 | 1.52254e6 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 200.146 | 2715.74 | 7290.37 | −170608. | 543544. | −1.73233e6 | −5.09924e6 | −6.97367e6 | −3.41464e7 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 314.580 | −5297.58 | 66192.8 | 71533.9 | −1.66652e6 | −3.72135e6 | 1.05148e7 | 1.37155e7 | 2.25031e7 | ||||||||||||||||||||||||||||||||||||||||
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 | 13.16.a.a | ✓ | 7 |
| 3.b | odd | 2 | 1 | 117.16.a.c | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.16.a.a | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 117.16.a.c | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 345 T_{2}^{6} - 121496 T_{2}^{5} - 47667996 T_{2}^{4} + 1317476032 T_{2}^{3} + \cdots - 14\!\cdots\!00 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(13))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + \cdots - 14\!\cdots\!00 \)
|
| $3$ |
\( T^{7} + \cdots + 13\!\cdots\!24 \)
|
| $5$ |
\( T^{7} + \cdots + 52\!\cdots\!00 \)
|
| $7$ |
\( T^{7} + \cdots + 80\!\cdots\!68 \)
|
| $11$ |
\( T^{7} + \cdots - 10\!\cdots\!52 \)
|
| $13$ |
\( (T - 62748517)^{7} \)
|
| $17$ |
\( T^{7} + \cdots - 23\!\cdots\!28 \)
|
| $19$ |
\( T^{7} + \cdots - 13\!\cdots\!04 \)
|
| $23$ |
\( T^{7} + \cdots + 40\!\cdots\!72 \)
|
| $29$ |
\( T^{7} + \cdots - 46\!\cdots\!76 \)
|
| $31$ |
\( T^{7} + \cdots - 10\!\cdots\!20 \)
|
| $37$ |
\( T^{7} + \cdots + 99\!\cdots\!52 \)
|
| $41$ |
\( T^{7} + \cdots + 82\!\cdots\!64 \)
|
| $43$ |
\( T^{7} + \cdots + 55\!\cdots\!20 \)
|
| $47$ |
\( T^{7} + \cdots - 12\!\cdots\!00 \)
|
| $53$ |
\( T^{7} + \cdots + 11\!\cdots\!76 \)
|
| $59$ |
\( T^{7} + \cdots + 32\!\cdots\!04 \)
|
| $61$ |
\( T^{7} + \cdots - 67\!\cdots\!88 \)
|
| $67$ |
\( T^{7} + \cdots - 20\!\cdots\!92 \)
|
| $71$ |
\( T^{7} + \cdots + 63\!\cdots\!64 \)
|
| $73$ |
\( T^{7} + \cdots + 36\!\cdots\!96 \)
|
| $79$ |
\( T^{7} + \cdots - 91\!\cdots\!76 \)
|
| $83$ |
\( T^{7} + \cdots + 10\!\cdots\!16 \)
|
| $89$ |
\( T^{7} + \cdots + 90\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots + 34\!\cdots\!00 \)
|
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