Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [126,9,Mod(19,126)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(126, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 5]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("126.19");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 126.n (of order \(6\), 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: | \(51.3297048677\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 2 x^{11} + 1771 x^{10} + 26038 x^{9} + 2442597 x^{8} + 26522276 x^{7} + 1175865280 x^{6} + \cdots + 36\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{10}\cdot 7^{3} \) |
| Twist minimal: | no (minimal twist has level 14) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{12} - 2 x^{11} + 1771 x^{10} + 26038 x^{9} + 2442597 x^{8} + 26522276 x^{7} + 1175865280 x^{6} + \cdots + 36\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 24\!\cdots\!14 \nu^{11} + \cdots - 26\!\cdots\!50 ) / 12\!\cdots\!50 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 20\!\cdots\!84 \nu^{11} + \cdots + 15\!\cdots\!00 ) / 14\!\cdots\!75 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 41\!\cdots\!32 \nu^{11} + \cdots + 11\!\cdots\!00 ) / 22\!\cdots\!75 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 43\!\cdots\!71 \nu^{11} + \cdots - 13\!\cdots\!00 ) / 22\!\cdots\!75 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 22\!\cdots\!98 \nu^{11} + \cdots - 52\!\cdots\!00 ) / 11\!\cdots\!50 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 14\!\cdots\!01 \nu^{11} + \cdots + 27\!\cdots\!00 ) / 49\!\cdots\!50 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 13\!\cdots\!92 \nu^{11} + \cdots + 93\!\cdots\!00 ) / 44\!\cdots\!50 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 70\!\cdots\!12 \nu^{11} + \cdots + 18\!\cdots\!50 ) / 22\!\cdots\!75 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 60\!\cdots\!17 \nu^{11} + \cdots - 17\!\cdots\!00 ) / 14\!\cdots\!50 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 28\!\cdots\!94 \nu^{11} + \cdots - 31\!\cdots\!50 ) / 44\!\cdots\!50 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 13\!\cdots\!67 \nu^{11} + \cdots - 23\!\cdots\!50 ) / 29\!\cdots\!50 \)
|
| \(\nu\) | \(=\) |
\( ( - \beta_{11} - 16 \beta_{10} - \beta_{8} - 13 \beta_{7} + 25 \beta_{6} - 9 \beta_{5} + 4 \beta_{4} + \cdots + 343 ) / 1008 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 18 \beta_{10} - 49 \beta_{9} - 221 \beta_{8} - 139 \beta_{7} + 161 \beta_{6} + 18 \beta_{5} + \cdots - 18 ) / 1008 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1719 \beta_{11} + 12399 \beta_{10} - 1719 \beta_{9} + 10197 \beta_{8} - 2670 \beta_{7} + \cdots - 7463217 ) / 1008 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 38339 \beta_{11} + 180779 \beta_{10} + 422426 \beta_{8} + 152291 \beta_{7} - 338667 \beta_{6} + \cdots - 313762235 ) / 504 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 1740162 \beta_{10} + 1425607 \beta_{9} + 4226891 \beta_{8} + 10126417 \beta_{7} - 19778119 \beta_{6} + \cdots + 1740162 ) / 504 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 8880085 \beta_{11} - 41399221 \beta_{10} + 8880085 \beta_{9} - 60656063 \beta_{8} + \cdots + 64028213221 ) / 72 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 2441634696 \beta_{11} - 15870609651 \beta_{10} - 25238396103 \beta_{8} - 13184814660 \beta_{7} + \cdots + 16283258667438 ) / 504 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 25493071377 \beta_{10} - 25897143496 \beta_{9} - 101211458282 \beta_{8} - 153362500381 \beta_{7} + \cdots - 25493071377 ) / 126 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 4181874760289 \beta_{11} + 21831088563341 \beta_{10} - 4181874760289 \beta_{9} + 28694475501481 \beta_{8} + \cdots - 28\!\cdots\!74 ) / 504 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 87304514607604 \beta_{11} + 528704325307279 \beta_{10} + 934094766761179 \beta_{8} + \cdots - 60\!\cdots\!10 ) / 252 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 74\!\cdots\!49 \beta_{10} + \cdots + 74\!\cdots\!49 ) / 504 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
−5.65685 | + | 9.79796i | 0 | −64.0000 | − | 110.851i | −928.109 | − | 535.844i | 0 | 2212.86 | + | 931.703i | 1448.15 | 0 | 10500.4 | − | 6062.38i | ||||||||||||||||||||||||||||||||||||||||||||
| 19.2 | −5.65685 | + | 9.79796i | 0 | −64.0000 | − | 110.851i | −372.872 | − | 215.278i | 0 | −1545.60 | + | 1837.37i | 1448.15 | 0 | 4218.56 | − | 2435.59i | |||||||||||||||||||||||||||||||||||||||||||||
| 19.3 | −5.65685 | + | 9.79796i | 0 | −64.0000 | − | 110.851i | 492.158 | + | 284.147i | 0 | −1740.97 | − | 1653.43i | 1448.15 | 0 | −5568.13 | + | 3214.76i | |||||||||||||||||||||||||||||||||||||||||||||
| 19.4 | 5.65685 | − | 9.79796i | 0 | −64.0000 | − | 110.851i | −753.641 | − | 435.115i | 0 | −1836.02 | + | 1547.20i | −1448.15 | 0 | −8526.48 | + | 4922.77i | |||||||||||||||||||||||||||||||||||||||||||||
| 19.5 | 5.65685 | − | 9.79796i | 0 | −64.0000 | − | 110.851i | −32.9005 | − | 18.9951i | 0 | 1399.84 | − | 1950.71i | −1448.15 | 0 | −372.227 | + | 214.905i | |||||||||||||||||||||||||||||||||||||||||||||
| 19.6 | 5.65685 | − | 9.79796i | 0 | −64.0000 | − | 110.851i | 758.365 | + | 437.842i | 0 | 855.887 | − | 2243.27i | −1448.15 | 0 | 8579.92 | − | 4953.62i | |||||||||||||||||||||||||||||||||||||||||||||
| 73.1 | −5.65685 | − | 9.79796i | 0 | −64.0000 | + | 110.851i | −928.109 | + | 535.844i | 0 | 2212.86 | − | 931.703i | 1448.15 | 0 | 10500.4 | + | 6062.38i | |||||||||||||||||||||||||||||||||||||||||||||
| 73.2 | −5.65685 | − | 9.79796i | 0 | −64.0000 | + | 110.851i | −372.872 | + | 215.278i | 0 | −1545.60 | − | 1837.37i | 1448.15 | 0 | 4218.56 | + | 2435.59i | |||||||||||||||||||||||||||||||||||||||||||||
| 73.3 | −5.65685 | − | 9.79796i | 0 | −64.0000 | + | 110.851i | 492.158 | − | 284.147i | 0 | −1740.97 | + | 1653.43i | 1448.15 | 0 | −5568.13 | − | 3214.76i | |||||||||||||||||||||||||||||||||||||||||||||
| 73.4 | 5.65685 | + | 9.79796i | 0 | −64.0000 | + | 110.851i | −753.641 | + | 435.115i | 0 | −1836.02 | − | 1547.20i | −1448.15 | 0 | −8526.48 | − | 4922.77i | |||||||||||||||||||||||||||||||||||||||||||||
| 73.5 | 5.65685 | + | 9.79796i | 0 | −64.0000 | + | 110.851i | −32.9005 | + | 18.9951i | 0 | 1399.84 | + | 1950.71i | −1448.15 | 0 | −372.227 | − | 214.905i | |||||||||||||||||||||||||||||||||||||||||||||
| 73.6 | 5.65685 | + | 9.79796i | 0 | −64.0000 | + | 110.851i | 758.365 | − | 437.842i | 0 | 855.887 | + | 2243.27i | −1448.15 | 0 | 8579.92 | + | 4953.62i | |||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 126.9.n.b | 12 | |
| 3.b | odd | 2 | 1 | 14.9.d.a | ✓ | 12 | |
| 7.d | odd | 6 | 1 | inner | 126.9.n.b | 12 | |
| 12.b | even | 2 | 1 | 112.9.s.c | 12 | ||
| 21.c | even | 2 | 1 | 98.9.d.b | 12 | ||
| 21.g | even | 6 | 1 | 14.9.d.a | ✓ | 12 | |
| 21.g | even | 6 | 1 | 98.9.b.c | 12 | ||
| 21.h | odd | 6 | 1 | 98.9.b.c | 12 | ||
| 21.h | odd | 6 | 1 | 98.9.d.b | 12 | ||
| 84.j | odd | 6 | 1 | 112.9.s.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.9.d.a | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 14.9.d.a | ✓ | 12 | 21.g | even | 6 | 1 | |
| 98.9.b.c | 12 | 21.g | even | 6 | 1 | ||
| 98.9.b.c | 12 | 21.h | odd | 6 | 1 | ||
| 98.9.d.b | 12 | 21.c | even | 2 | 1 | ||
| 98.9.d.b | 12 | 21.h | odd | 6 | 1 | ||
| 112.9.s.c | 12 | 12.b | even | 2 | 1 | ||
| 112.9.s.c | 12 | 84.j | odd | 6 | 1 | ||
| 126.9.n.b | 12 | 1.a | even | 1 | 1 | trivial | |
| 126.9.n.b | 12 | 7.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} + 1674 T_{5}^{11} - 190071 T_{5}^{10} - 1881848862 T_{5}^{9} + 18486646794 T_{5}^{8} + \cdots + 57\!\cdots\!25 \)
acting on \(S_{9}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 128 T^{2} + 16384)^{3} \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} + \cdots + 57\!\cdots\!25 \)
|
| $7$ |
\( T^{12} + \cdots + 36\!\cdots\!01 \)
|
| $11$ |
\( T^{12} + \cdots + 48\!\cdots\!81 \)
|
| $13$ |
\( T^{12} + \cdots + 35\!\cdots\!24 \)
|
| $17$ |
\( T^{12} + \cdots + 30\!\cdots\!01 \)
|
| $19$ |
\( T^{12} + \cdots + 94\!\cdots\!25 \)
|
| $23$ |
\( T^{12} + \cdots + 25\!\cdots\!41 \)
|
| $29$ |
\( (T^{6} + \cdots - 79\!\cdots\!12)^{2} \)
|
| $31$ |
\( T^{12} + \cdots + 95\!\cdots\!29 \)
|
| $37$ |
\( T^{12} + \cdots + 20\!\cdots\!21 \)
|
| $41$ |
\( T^{12} + \cdots + 50\!\cdots\!96 \)
|
| $43$ |
\( (T^{6} + \cdots - 32\!\cdots\!64)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 85\!\cdots\!49 \)
|
| $53$ |
\( T^{12} + \cdots + 10\!\cdots\!41 \)
|
| $59$ |
\( T^{12} + \cdots + 28\!\cdots\!01 \)
|
| $61$ |
\( T^{12} + \cdots + 13\!\cdots\!69 \)
|
| $67$ |
\( T^{12} + \cdots + 23\!\cdots\!89 \)
|
| $71$ |
\( (T^{6} + \cdots - 21\!\cdots\!64)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 26\!\cdots\!49 \)
|
| $79$ |
\( T^{12} + \cdots + 92\!\cdots\!25 \)
|
| $83$ |
\( T^{12} + \cdots + 17\!\cdots\!84 \)
|
| $89$ |
\( T^{12} + \cdots + 13\!\cdots\!21 \)
|
| $97$ |
\( T^{12} + \cdots + 52\!\cdots\!16 \)
|
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