Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [69,8,Mod(1,69)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("69.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 69.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: | \(21.5545667584\) |
| 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: |
\( x^{6} - 2x^{5} - 466x^{4} + 540x^{3} + 48973x^{2} - 77282x - 1061812 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2} \) |
| 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
\( x^{6} - 2x^{5} - 466x^{4} + 540x^{3} + 48973x^{2} - 77282x - 1061812 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 156 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -45\nu^{5} - 175\nu^{4} + 18759\nu^{3} + 89711\nu^{2} - 1226918\nu - 4103988 ) / 10624 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} - 17\nu^{4} + 878\nu^{3} + 7143\nu^{2} - 61166\nu - 310308 ) / 664 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 43\nu^{5} + 241\nu^{4} - 17217\nu^{3} - 117345\nu^{2} + 957754\nu + 5819212 ) / 10624 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 156 \)
|
| \(\nu^{3}\) | \(=\) |
\( 10\beta_{5} + 5\beta_{4} + 6\beta_{3} + 6\beta_{2} + 258\beta _1 + 113 \)
|
| \(\nu^{4}\) | \(=\) |
\( 48\beta_{5} - 48\beta_{4} + 80\beta_{3} + 371\beta_{2} + 861\beta _1 + 40056 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3982\beta_{5} + 2271\beta_{4} + 1954\beta_{3} + 3052\beta_{2} + 78932\beta _1 + 111131 \)
|
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 |
|
−19.9294 | 27.0000 | 269.181 | 466.102 | −538.094 | −1376.23 | −2813.66 | 729.000 | −9289.15 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −9.60678 | 27.0000 | −35.7097 | 26.4395 | −259.383 | −248.118 | 1572.72 | 729.000 | −253.998 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −8.53998 | 27.0000 | −55.0688 | −446.195 | −230.579 | 1770.31 | 1563.40 | 729.000 | 3810.49 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 3.35108 | 27.0000 | −116.770 | 0.849066 | 90.4791 | 405.736 | −820.245 | 729.000 | 2.84529 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 10.6010 | 27.0000 | −15.6190 | 100.131 | 286.227 | −1216.42 | −1522.50 | 729.000 | 1061.49 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 16.1241 | 27.0000 | 131.987 | −519.327 | 435.351 | −439.281 | 64.2798 | 729.000 | −8373.68 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(23\) | \(1\) |
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 | 69.8.a.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 207.8.a.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.8.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 207.8.a.c | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 8T_{2}^{5} - 441T_{2}^{4} - 2364T_{2}^{3} + 44592T_{2}^{2} + 171760T_{2} - 936560 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(69))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 8 T^{5} + \cdots - 936560 \)
|
| $3$ |
\( (T - 27)^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 242778320000 \)
|
| $7$ |
\( T^{6} + \cdots + 13\!\cdots\!60 \)
|
| $11$ |
\( T^{6} + \cdots + 22\!\cdots\!00 \)
|
| $13$ |
\( T^{6} + \cdots + 20\!\cdots\!40 \)
|
| $17$ |
\( T^{6} + \cdots + 48\!\cdots\!00 \)
|
| $19$ |
\( T^{6} + \cdots + 23\!\cdots\!60 \)
|
| $23$ |
\( (T + 12167)^{6} \)
|
| $29$ |
\( T^{6} + \cdots + 30\!\cdots\!80 \)
|
| $31$ |
\( T^{6} + \cdots - 34\!\cdots\!72 \)
|
| $37$ |
\( T^{6} + \cdots + 67\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots - 86\!\cdots\!64 \)
|
| $43$ |
\( T^{6} + \cdots - 85\!\cdots\!12 \)
|
| $47$ |
\( T^{6} + \cdots - 28\!\cdots\!20 \)
|
| $53$ |
\( T^{6} + \cdots - 44\!\cdots\!80 \)
|
| $59$ |
\( T^{6} + \cdots + 30\!\cdots\!68 \)
|
| $61$ |
\( T^{6} + \cdots - 23\!\cdots\!80 \)
|
| $67$ |
\( T^{6} + \cdots + 41\!\cdots\!92 \)
|
| $71$ |
\( T^{6} + \cdots + 26\!\cdots\!44 \)
|
| $73$ |
\( T^{6} + \cdots - 42\!\cdots\!88 \)
|
| $79$ |
\( T^{6} + \cdots + 84\!\cdots\!32 \)
|
| $83$ |
\( T^{6} + \cdots + 69\!\cdots\!40 \)
|
| $89$ |
\( T^{6} + \cdots - 10\!\cdots\!44 \)
|
| $97$ |
\( T^{6} + \cdots - 11\!\cdots\!08 \)
|
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