Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [950,6,Mod(1,950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(950, 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("950.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 950.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: | \(152.364628822\) |
| 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: |
\( x^{6} - 2x^{5} - 942x^{4} + 2966x^{3} + 153479x^{2} - 629136x - 3379725 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{6} - 2x^{5} - 942x^{4} + 2966x^{3} + 153479x^{2} - 629136x - 3379725 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -266\nu^{5} - 1985\nu^{4} + 218502\nu^{3} + 1548593\nu^{2} - 18636073\nu - 137033550 ) / 1272375 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 23\nu^{5} + 80\nu^{4} - 19956\nu^{3} - 46829\nu^{2} + 2266894\nu + 1305900 ) / 43875 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -523\nu^{5} - 4080\nu^{4} + 455831\nu^{3} + 2873829\nu^{2} - 53724769\nu - 162745650 ) / 424125 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -1267\nu^{5} - 10695\nu^{4} + 1082924\nu^{3} + 7451841\nu^{2} - 117444601\nu - 408081600 ) / 424125 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + 2\beta_{4} + \beta_{3} + 5\beta_{2} - 2\beta _1 + 314 \)
|
| \(\nu^{3}\) | \(=\) |
\( -12\beta_{5} + 39\beta_{4} + 9\beta_{3} - 36\beta_{2} + 625\beta _1 - 726 \)
|
| \(\nu^{4}\) | \(=\) |
\( -811\beta_{5} + 1448\beta_{4} + 367\beta_{3} + 3968\beta_{2} - 1997\beta _1 + 191732 \)
|
| \(\nu^{5}\) | \(=\) |
\( -9627\beta_{5} + 32874\beta_{4} + 10476\beta_{3} - 34857\beta_{2} + 446596\beta _1 - 714270 \)
|
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 | −25.7561 | 16.0000 | 0 | 103.024 | 146.454 | −64.0000 | 420.376 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | −11.9399 | 16.0000 | 0 | 47.7598 | −38.6441 | −64.0000 | −100.438 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | −1.19220 | 16.0000 | 0 | 4.76880 | 158.276 | −64.0000 | −241.579 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | 11.7802 | 16.0000 | 0 | −47.1208 | 65.2750 | −64.0000 | −104.227 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −4.00000 | 12.5249 | 16.0000 | 0 | −50.0997 | −93.1772 | −64.0000 | −86.1264 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −4.00000 | 28.5831 | 16.0000 | 0 | −114.332 | −184.184 | −64.0000 | 573.994 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 950.6.a.o | ✓ | 6 |
| 5.b | even | 2 | 1 | 950.6.a.q | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 950.6.a.o | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 950.6.a.q | yes | 6 | 5.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 14T_{3}^{5} - 862T_{3}^{4} + 10262T_{3}^{3} + 113475T_{3}^{2} - 1177668T_{3} - 1546209 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(950))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{6} \)
|
| $3$ |
\( T^{6} - 14 T^{5} + \cdots - 1546209 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots - 1003478642185 \)
|
| $11$ |
\( T^{6} + \cdots + 98966745393075 \)
|
| $13$ |
\( T^{6} + \cdots - 695337118649475 \)
|
| $17$ |
\( T^{6} + \cdots - 55\!\cdots\!43 \)
|
| $19$ |
\( (T - 361)^{6} \)
|
| $23$ |
\( T^{6} + \cdots + 30\!\cdots\!71 \)
|
| $29$ |
\( T^{6} + \cdots - 27\!\cdots\!55 \)
|
| $31$ |
\( T^{6} + \cdots - 87\!\cdots\!57 \)
|
| $37$ |
\( T^{6} + \cdots - 62\!\cdots\!15 \)
|
| $41$ |
\( T^{6} + \cdots - 25\!\cdots\!75 \)
|
| $43$ |
\( T^{6} + \cdots - 34\!\cdots\!33 \)
|
| $47$ |
\( T^{6} + \cdots - 18\!\cdots\!25 \)
|
| $53$ |
\( T^{6} + \cdots + 59\!\cdots\!33 \)
|
| $59$ |
\( T^{6} + \cdots + 12\!\cdots\!20 \)
|
| $61$ |
\( T^{6} + \cdots + 91\!\cdots\!45 \)
|
| $67$ |
\( T^{6} + \cdots + 23\!\cdots\!35 \)
|
| $71$ |
\( T^{6} + \cdots + 18\!\cdots\!75 \)
|
| $73$ |
\( T^{6} + \cdots + 10\!\cdots\!61 \)
|
| $79$ |
\( T^{6} + \cdots + 23\!\cdots\!75 \)
|
| $83$ |
\( T^{6} + \cdots + 20\!\cdots\!95 \)
|
| $89$ |
\( T^{6} + \cdots - 15\!\cdots\!35 \)
|
| $97$ |
\( T^{6} + \cdots + 21\!\cdots\!25 \)
|
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