Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [126,16,Mod(37,126)]
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, 2]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("126.37");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 126.g (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: | \(179.793816426\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 4 x^{19} - 375468372017 x^{18} + \cdots + 69\!\cdots\!07 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | multiple of \( 2^{32}\cdot 3^{37}\cdot 5^{6}\cdot 7^{21} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 4 x^{19} - 375468372017 x^{18} + \cdots + 69\!\cdots\!07 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 17\!\cdots\!25 \nu^{19} + \cdots - 27\!\cdots\!96 ) / 50\!\cdots\!92 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 17\!\cdots\!25 \nu^{19} + \cdots + 27\!\cdots\!04 ) / 50\!\cdots\!92 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 59\!\cdots\!06 \nu^{19} + \cdots - 12\!\cdots\!79 ) / 50\!\cdots\!92 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 24\!\cdots\!79 \nu^{19} + \cdots - 35\!\cdots\!08 ) / 12\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 35\!\cdots\!12 \nu^{19} + \cdots + 11\!\cdots\!95 ) / 12\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 41\!\cdots\!10 \nu^{19} + \cdots + 62\!\cdots\!61 ) / 12\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 99\!\cdots\!30 \nu^{19} + \cdots + 10\!\cdots\!86 ) / 15\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 13\!\cdots\!78 \nu^{19} + \cdots + 33\!\cdots\!47 ) / 12\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13\!\cdots\!66 \nu^{19} + \cdots - 33\!\cdots\!35 ) / 12\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 65\!\cdots\!91 \nu^{19} + \cdots + 29\!\cdots\!14 ) / 36\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 85\!\cdots\!01 \nu^{19} + \cdots + 58\!\cdots\!69 ) / 40\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 28\!\cdots\!68 \nu^{19} + \cdots - 55\!\cdots\!37 ) / 12\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 19\!\cdots\!67 \nu^{19} + \cdots - 20\!\cdots\!85 ) / 40\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 29\!\cdots\!15 \nu^{19} + \cdots + 25\!\cdots\!21 ) / 55\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 41\!\cdots\!44 \nu^{19} + \cdots + 89\!\cdots\!69 ) / 43\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 31\!\cdots\!81 \nu^{19} + \cdots - 39\!\cdots\!68 ) / 12\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 74\!\cdots\!51 \nu^{19} + \cdots + 11\!\cdots\!58 ) / 17\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 10\!\cdots\!15 \nu^{19} + \cdots + 21\!\cdots\!53 ) / 20\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 42\!\cdots\!09 \nu^{19} + \cdots + 66\!\cdots\!87 ) / 12\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} + \beta _1 + 1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 57 \beta_{13} + 3 \beta_{12} + 4 \beta_{11} - 201 \beta_{10} - 7 \beta_{9} - 103 \beta_{8} + \cdots + 37546835439 \)
|
| \(\nu^{3}\) | \(=\) |
\( 9 \beta_{19} + 12 \beta_{18} - 12 \beta_{17} + 21 \beta_{16} + 216 \beta_{15} - 309 \beta_{14} + \cdots - 313956954447954 \)
|
| \(\nu^{4}\) | \(=\) |
\( 467386 \beta_{19} + 1274136 \beta_{18} + 3441668 \beta_{17} + 1667946 \beta_{16} - 82983692 \beta_{15} + \cdots + 26\!\cdots\!87 \)
|
| \(\nu^{5}\) | \(=\) |
\( 1230924376670 \beta_{19} + 1437530291845 \beta_{18} - 3041842613490 \beta_{17} + 2985079187370 \beta_{16} + \cdots - 31\!\cdots\!36 \)
|
| \(\nu^{6}\) | \(=\) |
\( 56\!\cdots\!51 \beta_{19} + \cdots + 20\!\cdots\!24 \)
|
| \(\nu^{7}\) | \(=\) |
\( 13\!\cdots\!29 \beta_{19} + \cdots - 38\!\cdots\!13 \)
|
| \(\nu^{8}\) | \(=\) |
\( 60\!\cdots\!84 \beta_{19} + \cdots + 16\!\cdots\!12 \)
|
| \(\nu^{9}\) | \(=\) |
\( 14\!\cdots\!22 \beta_{19} + \cdots - 45\!\cdots\!47 \)
|
| \(\nu^{10}\) | \(=\) |
\( 58\!\cdots\!83 \beta_{19} + \cdots + 13\!\cdots\!51 \)
|
| \(\nu^{11}\) | \(=\) |
\( 14\!\cdots\!32 \beta_{19} + \cdots - 50\!\cdots\!73 \)
|
| \(\nu^{12}\) | \(=\) |
\( 53\!\cdots\!10 \beta_{19} + \cdots + 10\!\cdots\!11 \)
|
| \(\nu^{13}\) | \(=\) |
\( 13\!\cdots\!02 \beta_{19} + \cdots - 53\!\cdots\!23 \)
|
| \(\nu^{14}\) | \(=\) |
\( 43\!\cdots\!80 \beta_{19} + \cdots + 88\!\cdots\!69 \)
|
| \(\nu^{15}\) | \(=\) |
\( 12\!\cdots\!83 \beta_{19} + \cdots - 54\!\cdots\!68 \)
|
| \(\nu^{16}\) | \(=\) |
\( 33\!\cdots\!44 \beta_{19} + \cdots + 72\!\cdots\!11 \)
|
| \(\nu^{17}\) | \(=\) |
\( 12\!\cdots\!50 \beta_{19} + \cdots - 54\!\cdots\!54 \)
|
| \(\nu^{18}\) | \(=\) |
\( 21\!\cdots\!17 \beta_{19} + \cdots + 60\!\cdots\!98 \)
|
| \(\nu^{19}\) | \(=\) |
\( 11\!\cdots\!83 \beta_{19} + \cdots - 53\!\cdots\!17 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
−64.0000 | + | 110.851i | 0 | −8192.00 | − | 14189.0i | −137181. | + | 237604.i | 0 | −1.83741e6 | − | 1.17111e6i | 2.09715e6 | 0 | −1.75592e7 | − | 3.04133e7i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.2 | −64.0000 | + | 110.851i | 0 | −8192.00 | − | 14189.0i | −132990. | + | 230345.i | 0 | −202696. | + | 2.16944e6i | 2.09715e6 | 0 | −1.70227e7 | − | 2.94842e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.3 | −64.0000 | + | 110.851i | 0 | −8192.00 | − | 14189.0i | −65858.8 | + | 114071.i | 0 | 2.04465e6 | − | 752966.i | 2.09715e6 | 0 | −8.42993e6 | − | 1.46011e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.4 | −64.0000 | + | 110.851i | 0 | −8192.00 | − | 14189.0i | −26199.1 | + | 45378.2i | 0 | 280189. | + | 2.16080e6i | 2.09715e6 | 0 | −3.35349e6 | − | 5.80841e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.5 | −64.0000 | + | 110.851i | 0 | −8192.00 | − | 14189.0i | −17601.2 | + | 30486.1i | 0 | 639819. | − | 2.08283e6i | 2.09715e6 | 0 | −2.25295e6 | − | 3.90222e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.6 | −64.0000 | + | 110.851i | 0 | −8192.00 | − | 14189.0i | −5915.17 | + | 10245.4i | 0 | −1.80650e6 | − | 1.21824e6i | 2.09715e6 | 0 | −757142. | − | 1.31141e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.7 | −64.0000 | + | 110.851i | 0 | −8192.00 | − | 14189.0i | 68351.9 | − | 118389.i | 0 | 1.77611e6 | + | 1.26214e6i | 2.09715e6 | 0 | 8.74905e6 | + | 1.51538e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.8 | −64.0000 | + | 110.851i | 0 | −8192.00 | − | 14189.0i | 71087.4 | − | 123127.i | 0 | −2.10062e6 | + | 578739.i | 2.09715e6 | 0 | 9.09919e6 | + | 1.57603e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.9 | −64.0000 | + | 110.851i | 0 | −8192.00 | − | 14189.0i | 137490. | − | 238139.i | 0 | 255504. | − | 2.16386e6i | 2.09715e6 | 0 | 1.75987e7 | + | 3.04819e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.10 | −64.0000 | + | 110.851i | 0 | −8192.00 | − | 14189.0i | 153507. | − | 265882.i | 0 | −1.24378e6 | + | 1.78902e6i | 2.09715e6 | 0 | 1.96489e7 | + | 3.40330e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.1 | −64.0000 | − | 110.851i | 0 | −8192.00 | + | 14189.0i | −137181. | − | 237604.i | 0 | −1.83741e6 | + | 1.17111e6i | 2.09715e6 | 0 | −1.75592e7 | + | 3.04133e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.2 | −64.0000 | − | 110.851i | 0 | −8192.00 | + | 14189.0i | −132990. | − | 230345.i | 0 | −202696. | − | 2.16944e6i | 2.09715e6 | 0 | −1.70227e7 | + | 2.94842e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.3 | −64.0000 | − | 110.851i | 0 | −8192.00 | + | 14189.0i | −65858.8 | − | 114071.i | 0 | 2.04465e6 | + | 752966.i | 2.09715e6 | 0 | −8.42993e6 | + | 1.46011e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.4 | −64.0000 | − | 110.851i | 0 | −8192.00 | + | 14189.0i | −26199.1 | − | 45378.2i | 0 | 280189. | − | 2.16080e6i | 2.09715e6 | 0 | −3.35349e6 | + | 5.80841e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.5 | −64.0000 | − | 110.851i | 0 | −8192.00 | + | 14189.0i | −17601.2 | − | 30486.1i | 0 | 639819. | + | 2.08283e6i | 2.09715e6 | 0 | −2.25295e6 | + | 3.90222e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.6 | −64.0000 | − | 110.851i | 0 | −8192.00 | + | 14189.0i | −5915.17 | − | 10245.4i | 0 | −1.80650e6 | + | 1.21824e6i | 2.09715e6 | 0 | −757142. | + | 1.31141e6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.7 | −64.0000 | − | 110.851i | 0 | −8192.00 | + | 14189.0i | 68351.9 | + | 118389.i | 0 | 1.77611e6 | − | 1.26214e6i | 2.09715e6 | 0 | 8.74905e6 | − | 1.51538e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.8 | −64.0000 | − | 110.851i | 0 | −8192.00 | + | 14189.0i | 71087.4 | + | 123127.i | 0 | −2.10062e6 | − | 578739.i | 2.09715e6 | 0 | 9.09919e6 | − | 1.57603e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.9 | −64.0000 | − | 110.851i | 0 | −8192.00 | + | 14189.0i | 137490. | + | 238139.i | 0 | 255504. | + | 2.16386e6i | 2.09715e6 | 0 | 1.75987e7 | − | 3.04819e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.10 | −64.0000 | − | 110.851i | 0 | −8192.00 | + | 14189.0i | 153507. | + | 265882.i | 0 | −1.24378e6 | − | 1.78902e6i | 2.09715e6 | 0 | 1.96489e7 | − | 3.40330e7i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 126.16.g.g | ✓ | 20 |
| 3.b | odd | 2 | 1 | 126.16.g.h | yes | 20 | |
| 7.c | even | 3 | 1 | inner | 126.16.g.g | ✓ | 20 |
| 21.h | odd | 6 | 1 | 126.16.g.h | yes | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 126.16.g.g | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 126.16.g.g | ✓ | 20 | 7.c | even | 3 | 1 | inner |
| 126.16.g.h | yes | 20 | 3.b | odd | 2 | 1 | |
| 126.16.g.h | yes | 20 | 21.h | odd | 6 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{20} - 89383 T_{5}^{19} + 192128312395 T_{5}^{18} + \cdots + 11\!\cdots\!00 \)
acting on \(S_{16}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 128 T + 16384)^{10} \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} + \cdots + 11\!\cdots\!00 \)
|
| $7$ |
\( T^{20} + \cdots + 58\!\cdots\!49 \)
|
| $11$ |
\( T^{20} + \cdots + 27\!\cdots\!16 \)
|
| $13$ |
\( (T^{10} + \cdots - 61\!\cdots\!28)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 16\!\cdots\!16 \)
|
| $19$ |
\( T^{20} + \cdots + 96\!\cdots\!04 \)
|
| $23$ |
\( T^{20} + \cdots + 43\!\cdots\!36 \)
|
| $29$ |
\( (T^{10} + \cdots - 76\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 10\!\cdots\!49 \)
|
| $37$ |
\( T^{20} + \cdots + 35\!\cdots\!00 \)
|
| $41$ |
\( (T^{10} + \cdots + 15\!\cdots\!00)^{2} \)
|
| $43$ |
\( (T^{10} + \cdots - 15\!\cdots\!88)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 59\!\cdots\!96 \)
|
| $53$ |
\( T^{20} + \cdots + 19\!\cdots\!96 \)
|
| $59$ |
\( T^{20} + \cdots + 31\!\cdots\!04 \)
|
| $61$ |
\( T^{20} + \cdots + 57\!\cdots\!76 \)
|
| $67$ |
\( T^{20} + \cdots + 11\!\cdots\!44 \)
|
| $71$ |
\( (T^{10} + \cdots - 20\!\cdots\!00)^{2} \)
|
| $73$ |
\( T^{20} + \cdots + 14\!\cdots\!16 \)
|
| $79$ |
\( T^{20} + \cdots + 18\!\cdots\!25 \)
|
| $83$ |
\( (T^{10} + \cdots - 73\!\cdots\!12)^{2} \)
|
| $89$ |
\( T^{20} + \cdots + 42\!\cdots\!84 \)
|
| $97$ |
\( (T^{10} + \cdots + 51\!\cdots\!04)^{2} \)
|
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