Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [252,12,Mod(37,252)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(252, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("252.37");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 252.k (of order \(3\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(193.622481501\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Relative dimension: | \(7\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - 6 x^{13} + 198245134 x^{12} + 414863096508 x^{11} + \cdots + 37\!\cdots\!56 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{30}\cdot 3^{12}\cdot 7^{7} \) |
| Twist minimal: | no (minimal twist has level 84) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - 6 x^{13} + 198245134 x^{12} + 414863096508 x^{11} + \cdots + 37\!\cdots\!56 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 50\!\cdots\!47 \nu^{13} + \cdots - 56\!\cdots\!12 ) / 64\!\cdots\!36 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 50\!\cdots\!47 \nu^{13} + \cdots - 82\!\cdots\!24 ) / 64\!\cdots\!36 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 50\!\cdots\!66 \nu^{13} + \cdots - 19\!\cdots\!44 ) / 64\!\cdots\!36 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 84\!\cdots\!04 \nu^{13} + \cdots + 70\!\cdots\!36 ) / 45\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 45\!\cdots\!94 \nu^{13} + \cdots - 55\!\cdots\!00 ) / 14\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 73\!\cdots\!44 \nu^{13} + \cdots - 98\!\cdots\!64 ) / 22\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 49\!\cdots\!62 \nu^{13} + \cdots - 65\!\cdots\!52 ) / 65\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 42\!\cdots\!63 \nu^{13} + \cdots + 42\!\cdots\!28 ) / 22\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 16\!\cdots\!31 \nu^{13} + \cdots - 20\!\cdots\!28 ) / 45\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 19\!\cdots\!99 \nu^{13} + \cdots + 23\!\cdots\!00 ) / 50\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 23\!\cdots\!09 \nu^{13} + \cdots - 38\!\cdots\!72 ) / 31\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 19\!\cdots\!41 \nu^{13} + \cdots - 43\!\cdots\!36 ) / 15\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 94\!\cdots\!21 \nu^{13} + \cdots + 95\!\cdots\!36 ) / 73\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} + \beta _1 + 1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 11 \beta_{13} + 11 \beta_{12} - 38 \beta_{11} + 58 \beta_{9} - 20 \beta_{8} - 288 \beta_{7} - 38 \beta_{6} + \cdots + 96 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 9727 \beta_{12} - 131125 \beta_{10} + 541963 \beta_{9} + 720314 \beta_{7} - 1727168 \beta_{6} + \cdots - 89165142443 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 1192000460 \beta_{13} + 3652634255 \beta_{11} - 2346232175 \beta_{10} + 9942804330 \beta_{9} + \cdots - 41\!\cdots\!84 \)
|
| \(\nu^{5}\) | \(=\) |
\( 1426989509488 \beta_{13} + 1426989509488 \beta_{12} - 7368234223504 \beta_{11} - 16799038997286 \beta_{9} + \cdots - 9430804773782 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 97\!\cdots\!23 \beta_{12} + \cdots + 34\!\cdots\!03 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 18\!\cdots\!89 \beta_{13} + \cdots + 14\!\cdots\!60 \)
|
| \(\nu^{8}\) | \(=\) |
\( 70\!\cdots\!94 \beta_{13} + \cdots + 38\!\cdots\!09 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 22\!\cdots\!90 \beta_{12} + \cdots - 15\!\cdots\!04 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 46\!\cdots\!93 \beta_{13} + \cdots - 27\!\cdots\!57 \)
|
| \(\nu^{11}\) | \(=\) |
\( 27\!\cdots\!75 \beta_{13} + \cdots + 91\!\cdots\!75 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 26\!\cdots\!88 \beta_{12} + \cdots + 25\!\cdots\!05 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 31\!\cdots\!64 \beta_{13} + \cdots + 17\!\cdots\!12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
0 | 0 | 0 | −4987.65 | + | 8638.86i | 0 | 6340.58 | − | 44012.8i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.2 | 0 | 0 | 0 | −4369.65 | + | 7568.45i | 0 | −2593.70 | + | 44391.4i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.3 | 0 | 0 | 0 | −3193.56 | + | 5531.40i | 0 | 42025.8 | − | 14531.2i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.4 | 0 | 0 | 0 | −2050.38 | + | 3551.36i | 0 | −44391.9 | − | 2586.27i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.5 | 0 | 0 | 0 | 2471.06 | − | 4280.00i | 0 | −15790.8 | + | 41569.0i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.6 | 0 | 0 | 0 | 3945.47 | − | 6833.75i | 0 | 39151.4 | + | 21082.9i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.7 | 0 | 0 | 0 | 4575.70 | − | 7925.35i | 0 | −7241.01 | − | 43873.6i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.1 | 0 | 0 | 0 | −4987.65 | − | 8638.86i | 0 | 6340.58 | + | 44012.8i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.2 | 0 | 0 | 0 | −4369.65 | − | 7568.45i | 0 | −2593.70 | − | 44391.4i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.3 | 0 | 0 | 0 | −3193.56 | − | 5531.40i | 0 | 42025.8 | + | 14531.2i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.4 | 0 | 0 | 0 | −2050.38 | − | 3551.36i | 0 | −44391.9 | + | 2586.27i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.5 | 0 | 0 | 0 | 2471.06 | + | 4280.00i | 0 | −15790.8 | − | 41569.0i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.6 | 0 | 0 | 0 | 3945.47 | + | 6833.75i | 0 | 39151.4 | − | 21082.9i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.7 | 0 | 0 | 0 | 4575.70 | + | 7925.35i | 0 | −7241.01 | + | 43873.6i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 252.12.k.b | 14 | |
| 3.b | odd | 2 | 1 | 84.12.i.a | ✓ | 14 | |
| 7.c | even | 3 | 1 | inner | 252.12.k.b | 14 | |
| 21.h | odd | 6 | 1 | 84.12.i.a | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.12.i.a | ✓ | 14 | 3.b | odd | 2 | 1 | |
| 84.12.i.a | ✓ | 14 | 21.h | odd | 6 | 1 | |
| 252.12.k.b | 14 | 1.a | even | 1 | 1 | trivial | |
| 252.12.k.b | 14 | 7.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{14} + 7218 T_{5}^{13} + 228016270 T_{5}^{12} + 1113050335212 T_{5}^{11} + \cdots + 66\!\cdots\!00 \)
acting on \(S_{12}^{\mathrm{new}}(252, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} \)
|
| $3$ |
\( T^{14} \)
|
| $5$ |
\( T^{14} + \cdots + 66\!\cdots\!00 \)
|
| $7$ |
\( T^{14} + \cdots + 11\!\cdots\!07 \)
|
| $11$ |
\( T^{14} + \cdots + 16\!\cdots\!36 \)
|
| $13$ |
\( (T^{7} + \cdots + 14\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{14} + \cdots + 12\!\cdots\!84 \)
|
| $19$ |
\( T^{14} + \cdots + 10\!\cdots\!44 \)
|
| $23$ |
\( T^{14} + \cdots + 91\!\cdots\!44 \)
|
| $29$ |
\( (T^{7} + \cdots - 49\!\cdots\!20)^{2} \)
|
| $31$ |
\( T^{14} + \cdots + 62\!\cdots\!25 \)
|
| $37$ |
\( T^{14} + \cdots + 74\!\cdots\!00 \)
|
| $41$ |
\( (T^{7} + \cdots + 13\!\cdots\!64)^{2} \)
|
| $43$ |
\( (T^{7} + \cdots - 65\!\cdots\!40)^{2} \)
|
| $47$ |
\( T^{14} + \cdots + 10\!\cdots\!00 \)
|
| $53$ |
\( T^{14} + \cdots + 12\!\cdots\!76 \)
|
| $59$ |
\( T^{14} + \cdots + 37\!\cdots\!00 \)
|
| $61$ |
\( T^{14} + \cdots + 26\!\cdots\!16 \)
|
| $67$ |
\( T^{14} + \cdots + 94\!\cdots\!76 \)
|
| $71$ |
\( (T^{7} + \cdots + 61\!\cdots\!00)^{2} \)
|
| $73$ |
\( T^{14} + \cdots + 14\!\cdots\!00 \)
|
| $79$ |
\( T^{14} + \cdots + 20\!\cdots\!09 \)
|
| $83$ |
\( (T^{7} + \cdots + 46\!\cdots\!88)^{2} \)
|
| $89$ |
\( T^{14} + \cdots + 31\!\cdots\!00 \)
|
| $97$ |
\( (T^{7} + \cdots + 12\!\cdots\!00)^{2} \)
|
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