Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [252,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("252.37");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(129.789030713\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 3 x^{11} + 378960 x^{10} + 17798539 x^{9} + 135620837364 x^{8} + 3533018734497 x^{7} + \cdots + 11\!\cdots\!24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{20}\cdot 3^{10}\cdot 7^{5} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 3 x^{11} + 378960 x^{10} + 17798539 x^{9} + 135620837364 x^{8} + 3533018734497 x^{7} + \cdots + 11\!\cdots\!24 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 16\!\cdots\!35 \nu^{11} + \cdots - 17\!\cdots\!44 ) / 13\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 10\!\cdots\!65 \nu^{11} + \cdots - 11\!\cdots\!72 ) / 51\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 52\!\cdots\!31 \nu^{11} + \cdots + 24\!\cdots\!80 ) / 23\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 63\!\cdots\!48 \nu^{11} + \cdots - 44\!\cdots\!80 ) / 34\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 78\!\cdots\!39 \nu^{11} + \cdots + 67\!\cdots\!24 ) / 30\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 60\!\cdots\!53 \nu^{11} + \cdots + 94\!\cdots\!44 ) / 19\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 16\!\cdots\!73 \nu^{11} + \cdots + 28\!\cdots\!96 ) / 30\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 48\!\cdots\!35 \nu^{11} + \cdots - 14\!\cdots\!84 ) / 85\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 23\!\cdots\!37 \nu^{11} + \cdots - 24\!\cdots\!32 ) / 30\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 23\!\cdots\!83 \nu^{11} + \cdots + 25\!\cdots\!44 ) / 77\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 70\!\cdots\!61 \nu^{11} + \cdots - 38\!\cdots\!00 ) / 23\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} + 10 \beta_{8} - \beta_{7} + 2 \beta_{6} + 17 \beta_{5} + \cdots + 395 ) / 756 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 269 \beta_{10} + 12266 \beta_{8} + 1174 \beta_{7} + 8168 \beta_{6} - 24560 \beta_{5} + \cdots + 190987816 \beta_1 ) / 1512 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 165627 \beta_{11} - 8442 \beta_{9} - 186668 \beta_{8} + 549290 \beta_{6} - 1107022 \beta_{5} + \cdots - 796696644 ) / 168 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 20061595 \beta_{11} - 20061595 \beta_{10} - 108053852 \beta_{9} - 769875430 \beta_{8} + \cdots - 16095444264428 ) / 378 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 265126805851 \beta_{10} - 952225263437 \beta_{8} - 27749520394 \beta_{7} - 1226550604844 \beta_{6} + \cdots - 20\!\cdots\!76 \beta_1 ) / 756 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 247187605819 \beta_{11} + 2474631265142 \beta_{9} - 8717541101750 \beta_{8} - 24633162172848 \beta_{6} + \cdots + 36\!\cdots\!88 ) / 24 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 18\!\cdots\!63 \beta_{11} + \cdots + 20\!\cdots\!56 ) / 1512 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 96\!\cdots\!29 \beta_{10} + \cdots + 10\!\cdots\!58 \beta_1 ) / 189 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 37\!\cdots\!13 \beta_{11} + \cdots - 51\!\cdots\!60 ) / 84 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 14\!\cdots\!03 \beta_{11} + \cdots - 29\!\cdots\!64 ) / 1512 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 23\!\cdots\!31 \beta_{10} + \cdots - 40\!\cdots\!48 \beta_1 ) / 1512 \)
|
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 | −1257.17 | + | 2177.48i | 0 | −4051.17 | + | 4893.02i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.2 | 0 | 0 | 0 | −635.631 | + | 1100.94i | 0 | 464.037 | − | 6335.48i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.3 | 0 | 0 | 0 | −219.636 | + | 380.421i | 0 | 5903.02 | + | 2346.90i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.4 | 0 | 0 | 0 | 390.858 | − | 676.987i | 0 | −6278.11 | − | 969.005i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.5 | 0 | 0 | 0 | 815.188 | − | 1411.95i | 0 | −5488.12 | + | 3199.08i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.6 | 0 | 0 | 0 | 1107.89 | − | 1918.92i | 0 | 3132.34 | − | 5526.49i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.1 | 0 | 0 | 0 | −1257.17 | − | 2177.48i | 0 | −4051.17 | − | 4893.02i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.2 | 0 | 0 | 0 | −635.631 | − | 1100.94i | 0 | 464.037 | + | 6335.48i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.3 | 0 | 0 | 0 | −219.636 | − | 380.421i | 0 | 5903.02 | − | 2346.90i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.4 | 0 | 0 | 0 | 390.858 | + | 676.987i | 0 | −6278.11 | + | 969.005i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.5 | 0 | 0 | 0 | 815.188 | + | 1411.95i | 0 | −5488.12 | − | 3199.08i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.6 | 0 | 0 | 0 | 1107.89 | + | 1918.92i | 0 | 3132.34 | + | 5526.49i | 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 | 252.10.k.d | 12 | |
| 3.b | odd | 2 | 1 | 84.10.i.b | ✓ | 12 | |
| 7.c | even | 3 | 1 | inner | 252.10.k.d | 12 | |
| 21.h | odd | 6 | 1 | 84.10.i.b | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.10.i.b | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 84.10.i.b | ✓ | 12 | 21.h | odd | 6 | 1 | |
| 252.10.k.d | 12 | 1.a | even | 1 | 1 | trivial | |
| 252.10.k.d | 12 | 7.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(252, [\chi])\):
|
\( T_{5}^{12} - 403 T_{5}^{11} + 8236100 T_{5}^{10} - 4860275053 T_{5}^{9} + 53084029532900 T_{5}^{8} + \cdots + 15\!\cdots\!00 \)
|
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\( T_{13}^{6} - 9147 T_{13}^{5} - 31769189337 T_{13}^{4} - 657282566644625 T_{13}^{3} + \cdots + 15\!\cdots\!00 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} + \cdots + 15\!\cdots\!00 \)
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| $7$ |
\( T^{12} + \cdots + 43\!\cdots\!49 \)
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| $11$ |
\( T^{12} + \cdots + 47\!\cdots\!16 \)
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| $13$ |
\( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \)
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| $17$ |
\( T^{12} + \cdots + 15\!\cdots\!44 \)
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| $19$ |
\( T^{12} + \cdots + 44\!\cdots\!16 \)
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| $23$ |
\( T^{12} + \cdots + 54\!\cdots\!24 \)
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| $29$ |
\( (T^{6} + \cdots - 92\!\cdots\!20)^{2} \)
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| $31$ |
\( T^{12} + \cdots + 56\!\cdots\!25 \)
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| $37$ |
\( T^{12} + \cdots + 67\!\cdots\!00 \)
|
| $41$ |
\( (T^{6} + \cdots - 25\!\cdots\!28)^{2} \)
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| $43$ |
\( (T^{6} + \cdots - 12\!\cdots\!80)^{2} \)
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| $47$ |
\( T^{12} + \cdots + 16\!\cdots\!00 \)
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| $53$ |
\( T^{12} + \cdots + 25\!\cdots\!84 \)
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| $59$ |
\( T^{12} + \cdots + 23\!\cdots\!00 \)
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| $61$ |
\( T^{12} + \cdots + 32\!\cdots\!56 \)
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| $67$ |
\( T^{12} + \cdots + 40\!\cdots\!24 \)
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| $71$ |
\( (T^{6} + \cdots + 18\!\cdots\!00)^{2} \)
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| $73$ |
\( T^{12} + \cdots + 24\!\cdots\!00 \)
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| $79$ |
\( T^{12} + \cdots + 30\!\cdots\!41 \)
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| $83$ |
\( (T^{6} + \cdots - 12\!\cdots\!48)^{2} \)
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| $89$ |
\( T^{12} + \cdots + 15\!\cdots\!00 \)
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| $97$ |
\( (T^{6} + \cdots + 94\!\cdots\!00)^{2} \)
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