Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [21,22,Mod(1,21)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.1");
S:= CuspForms(chi, 22);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.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: | \(58.6902423003\) |
| 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: |
\( x^{6} - 3 x^{5} - 10018974 x^{4} - 2448700774 x^{3} + 18685028969301 x^{2} + \cdots + 19\!\cdots\!24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{8}\cdot 5^{2}\cdot 7^{3} \) |
| 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
\( x^{6} - 3 x^{5} - 10018974 x^{4} - 2448700774 x^{3} + 18685028969301 x^{2} + \cdots + 19\!\cdots\!24 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 371\nu - 3339474 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 771 \nu^{5} + 107730 \nu^{4} + 7682816740 \nu^{3} + 899757774470 \nu^{2} + \cdots - 14\!\cdots\!60 ) / 13920429056 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1855 \nu^{5} - 139078 \nu^{4} + 18649615668 \nu^{3} + 5223749240190 \nu^{2} + \cdots - 55\!\cdots\!68 ) / 4503668224 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 7829 \nu^{5} + 247598 \nu^{4} + 81554732684 \nu^{3} + 13805629476282 \nu^{2} + \cdots - 16\!\cdots\!28 ) / 9570294976 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 371\beta _1 + 3339474 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{5} - 3\beta_{4} - 22\beta_{3} + 574\beta_{2} + 6290567\beta _1 + 1236233569 \)
|
| \(\nu^{4}\) | \(=\) |
\( 1243\beta_{5} - 12551\beta_{4} + 74978\beta_{3} + 7918399\beta_{2} + 5176520321\beta _1 + 21005629946603 \)
|
| \(\nu^{5}\) | \(=\) |
\( 30067910 \beta_{5} - 31647950 \beta_{4} - 226802876 \beta_{3} + 7993182500 \beta_{2} + \cdots + 17\!\cdots\!70 \)
|
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 |
|
−2599.37 | 59049.0 | 4.65955e6 | 3.76002e7 | −1.53490e8 | −2.82475e8 | −6.66062e9 | 3.48678e9 | −9.77367e10 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1197.21 | 59049.0 | −663830. | 252969. | −7.06943e7 | −2.82475e8 | 3.30549e9 | 3.48678e9 | −3.02858e8 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 389.967 | 59049.0 | −1.94508e6 | −4.03104e7 | 2.30272e7 | −2.82475e8 | −1.57634e9 | 3.48678e9 | −1.57197e10 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 458.105 | 59049.0 | −1.88729e6 | 2.95608e7 | 2.70507e7 | −2.82475e8 | −1.82529e9 | 3.48678e9 | 1.35420e10 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2012.74 | 59049.0 | 1.95398e6 | −1.92932e7 | 1.18850e8 | −2.82475e8 | −2.88160e8 | 3.48678e9 | −3.88323e10 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2834.76 | 59049.0 | 5.93874e6 | 2.52284e7 | 1.67390e8 | −2.82475e8 | 1.08900e10 | 3.48678e9 | 7.15164e10 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
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 | 21.22.a.d | ✓ | 6 |
| 3.b | odd | 2 | 1 | 63.22.a.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.22.a.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 63.22.a.f | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 1899 T_{2}^{5} - 8516394 T_{2}^{4} + 14518674216 T_{2}^{3} + 10466049399936 T_{2}^{2} + \cdots + 31\!\cdots\!20 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(21))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 31\!\cdots\!20 \)
|
| $3$ |
\( (T - 59049)^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 55\!\cdots\!00 \)
|
| $7$ |
\( (T + 282475249)^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 15\!\cdots\!24 \)
|
| $13$ |
\( T^{6} + \cdots + 76\!\cdots\!80 \)
|
| $17$ |
\( T^{6} + \cdots - 39\!\cdots\!76 \)
|
| $19$ |
\( T^{6} + \cdots + 10\!\cdots\!84 \)
|
| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots - 46\!\cdots\!36 \)
|
| $31$ |
\( T^{6} + \cdots - 22\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 48\!\cdots\!64 \)
|
| $41$ |
\( T^{6} + \cdots + 30\!\cdots\!04 \)
|
| $43$ |
\( T^{6} + \cdots - 59\!\cdots\!80 \)
|
| $47$ |
\( T^{6} + \cdots + 77\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots - 37\!\cdots\!56 \)
|
| $59$ |
\( T^{6} + \cdots + 90\!\cdots\!20 \)
|
| $61$ |
\( T^{6} + \cdots - 19\!\cdots\!16 \)
|
| $67$ |
\( T^{6} + \cdots - 12\!\cdots\!64 \)
|
| $71$ |
\( T^{6} + \cdots - 97\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots + 60\!\cdots\!04 \)
|
| $79$ |
\( T^{6} + \cdots - 21\!\cdots\!24 \)
|
| $83$ |
\( T^{6} + \cdots + 76\!\cdots\!24 \)
|
| $89$ |
\( T^{6} + \cdots - 29\!\cdots\!76 \)
|
| $97$ |
\( T^{6} + \cdots + 12\!\cdots\!76 \)
|
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