Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [230,6,Mod(1,230)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(230, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("230.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 230 = 2 \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 230.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: | \(36.8882785570\) |
| 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: |
\( x^{6} - x^{5} - 1168x^{4} - 2857x^{3} + 297325x^{2} + 680040x - 8930700 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{6} - x^{5} - 1168x^{4} - 2857x^{3} + 297325x^{2} + 680040x - 8930700 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 51\nu^{5} + 1939\nu^{4} - 51558\nu^{3} - 1670407\nu^{2} + 1317025\nu + 92904390 ) / 1380040 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 49\nu^{5} - 843\nu^{4} - 57654\nu^{3} + 889995\nu^{2} + 14649059\nu - 166695750 ) / 414012 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -643\nu^{5} + 2613\nu^{4} + 731214\nu^{3} + 251391\nu^{2} - 171142265\nu - 222262350 ) / 4140120 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 416\nu^{5} - 7861\nu^{4} - 319078\nu^{3} + 4409948\nu^{2} + 34179795\nu - 336035700 ) / 1035030 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 + 389 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{5} + 22\beta_{4} + \beta_{3} + 24\beta_{2} + 653\beta _1 + 1916 \)
|
| \(\nu^{4}\) | \(=\) |
\( -18\beta_{5} + 856\beta_{4} + 766\beta_{3} + 1340\beta_{2} + 7597\beta _1 + 258320 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6750\beta_{5} + 22449\beta_{4} + 4641\beta_{3} + 33129\beta_{2} + 509251\beta _1 + 3035031 \)
|
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 |
|
4.00000 | −29.1755 | 16.0000 | 25.0000 | −116.702 | 213.932 | 64.0000 | 608.209 | 100.000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −14.9517 | 16.0000 | 25.0000 | −59.8068 | 52.9827 | 64.0000 | −19.4468 | 100.000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | −2.70878 | 16.0000 | 25.0000 | −10.8351 | −158.180 | 64.0000 | −235.663 | 100.000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | 9.33442 | 16.0000 | 25.0000 | 37.3377 | 234.539 | 64.0000 | −155.869 | 100.000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 4.00000 | 21.8456 | 16.0000 | 25.0000 | 87.3823 | 7.44621 | 64.0000 | 234.229 | 100.000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.00000 | 26.6560 | 16.0000 | 25.0000 | 106.624 | 15.2806 | 64.0000 | 467.540 | 100.000 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 230.6.a.h | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 230.6.a.h | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 11T_{3}^{5} - 1118T_{3}^{4} + 12081T_{3}^{3} + 252311T_{3}^{2} - 1797792T_{3} - 6422832 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(230))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{6} \)
|
| $3$ |
\( T^{6} - 11 T^{5} + \cdots - 6422832 \)
|
| $5$ |
\( (T - 25)^{6} \)
|
| $7$ |
\( T^{6} + \cdots - 47846477632 \)
|
| $11$ |
\( T^{6} + \cdots + 21\!\cdots\!80 \)
|
| $13$ |
\( T^{6} + \cdots + 478562359504036 \)
|
| $17$ |
\( T^{6} + \cdots - 27\!\cdots\!60 \)
|
| $19$ |
\( T^{6} + \cdots - 19\!\cdots\!40 \)
|
| $23$ |
\( (T - 529)^{6} \)
|
| $29$ |
\( T^{6} + \cdots + 66\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots - 75\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots - 80\!\cdots\!96 \)
|
| $41$ |
\( T^{6} + \cdots + 98\!\cdots\!86 \)
|
| $43$ |
\( T^{6} + \cdots - 10\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots + 17\!\cdots\!24 \)
|
| $53$ |
\( T^{6} + \cdots + 16\!\cdots\!32 \)
|
| $59$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots - 20\!\cdots\!76 \)
|
| $67$ |
\( T^{6} + \cdots - 68\!\cdots\!60 \)
|
| $71$ |
\( T^{6} + \cdots - 87\!\cdots\!48 \)
|
| $73$ |
\( T^{6} + \cdots + 87\!\cdots\!76 \)
|
| $79$ |
\( T^{6} + \cdots - 97\!\cdots\!80 \)
|
| $83$ |
\( T^{6} + \cdots + 48\!\cdots\!76 \)
|
| $89$ |
\( T^{6} + \cdots + 17\!\cdots\!20 \)
|
| $97$ |
\( T^{6} + \cdots - 23\!\cdots\!24 \)
|
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