Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [23,6,Mod(1,23)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(23, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("23.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 23.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: | \(3.68882785570\) |
| 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: |
\( x^{6} - 2x^{5} - 149x^{4} + 215x^{3} + 6182x^{2} - 4625x - 79150 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2\cdot 5 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 149x^{4} + 215x^{3} + 6182x^{2} - 4625x - 79150 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 29\nu^{5} - 23\nu^{4} - 3610\nu^{3} + 1253\nu^{2} + 85533\nu + 21670 ) / 1236 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 27\nu^{5} + 7\nu^{4} - 3290\nu^{3} - 1689\nu^{2} + 75571\nu + 65666 ) / 1236 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 27\nu^{5} + 7\nu^{4} - 3290\nu^{3} - 2925\nu^{2} + 75571\nu + 127466 ) / 1236 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -11\nu^{5} + 165\nu^{4} + 1142\nu^{3} - 19271\nu^{2} - 16475\nu + 412546 ) / 1236 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 50 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{5} + 5\beta_{4} + 6\beta_{3} - 11\beta_{2} + 62\beta _1 + 26 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{5} - 113\beta_{4} + 129\beta_{3} - 13\beta_{2} - 12\beta _1 + 3359 \)
|
| \(\nu^{5}\) | \(=\) |
\( -245\beta_{5} + 576\beta_{4} + 806\beta_{3} - 1337\beta_{2} + 4759\beta _1 + 2993 \)
|
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 |
|
−8.76471 | −19.7842 | 44.8202 | −82.0499 | 173.403 | 213.509 | −112.365 | 148.416 | 719.144 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −5.03655 | −19.9612 | −6.63314 | 108.846 | 100.536 | −33.8609 | 194.578 | 155.451 | −548.208 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −4.64307 | 22.5726 | −10.4419 | 9.65155 | −104.806 | 199.091 | 197.061 | 266.522 | −44.8128 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 5.81709 | 5.42840 | 1.83858 | 83.3054 | 31.5775 | 84.5782 | −175.452 | −213.532 | 484.596 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 6.29078 | 29.7824 | 7.57397 | −45.2266 | 187.355 | −206.967 | −153.659 | 643.991 | −284.511 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 10.3365 | −2.03792 | 74.8423 | −32.5264 | −21.0649 | 43.6495 | 442.838 | −238.847 | −336.208 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(23\) | \(-1\) |
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 | 23.6.a.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 207.6.a.g | 6 | ||
| 4.b | odd | 2 | 1 | 368.6.a.h | 6 | ||
| 5.b | even | 2 | 1 | 575.6.a.c | 6 | ||
| 23.b | odd | 2 | 1 | 529.6.a.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 23.6.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 207.6.a.g | 6 | 3.b | odd | 2 | 1 | ||
| 368.6.a.h | 6 | 4.b | odd | 2 | 1 | ||
| 529.6.a.c | 6 | 23.b | odd | 2 | 1 | ||
| 575.6.a.c | 6 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 4T_{2}^{5} - 144T_{2}^{4} + 381T_{2}^{3} + 5928T_{2}^{2} - 7784T_{2} - 77528 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(23))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 4 T^{5} + \cdots - 77528 \)
|
| $3$ |
\( T^{6} - 16 T^{5} + \cdots - 2937024 \)
|
| $5$ |
\( T^{6} + \cdots - 10563094144 \)
|
| $7$ |
\( T^{6} + \cdots + 1099779388928 \)
|
| $11$ |
\( T^{6} + \cdots + 97362234604672 \)
|
| $13$ |
\( T^{6} + \cdots - 83\!\cdots\!88 \)
|
| $17$ |
\( T^{6} + \cdots - 22\!\cdots\!32 \)
|
| $19$ |
\( T^{6} + \cdots - 18\!\cdots\!48 \)
|
| $23$ |
\( (T - 529)^{6} \)
|
| $29$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 76\!\cdots\!36 \)
|
| $37$ |
\( T^{6} + \cdots - 24\!\cdots\!28 \)
|
| $41$ |
\( T^{6} + \cdots - 63\!\cdots\!16 \)
|
| $43$ |
\( T^{6} + \cdots - 11\!\cdots\!28 \)
|
| $47$ |
\( T^{6} + \cdots - 16\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots - 60\!\cdots\!96 \)
|
| $59$ |
\( T^{6} + \cdots - 35\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots - 14\!\cdots\!32 \)
|
| $67$ |
\( T^{6} + \cdots - 47\!\cdots\!72 \)
|
| $71$ |
\( T^{6} + \cdots + 63\!\cdots\!32 \)
|
| $73$ |
\( T^{6} + \cdots + 30\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots + 19\!\cdots\!92 \)
|
| $83$ |
\( T^{6} + \cdots + 75\!\cdots\!36 \)
|
| $89$ |
\( T^{6} + \cdots - 71\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 29\!\cdots\!00 \)
|
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