Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [126,11,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, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("126.19");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| 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: | \(80.0550138369\) |
| 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: |
\( x^{12} + 7168 x^{10} - 191104 x^{9} + 39872585 x^{8} - 837614684 x^{7} + 83400850488 x^{6} + \cdots + 16\!\cdots\!56 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{6}\cdot 7^{4} \) |
| 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
\( x^{12} + 7168 x^{10} - 191104 x^{9} + 39872585 x^{8} - 837614684 x^{7} + 83400850488 x^{6} + \cdots + 16\!\cdots\!56 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 59\!\cdots\!91 \nu^{11} + \cdots + 54\!\cdots\!88 ) / 74\!\cdots\!80 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 28\!\cdots\!47 \nu^{11} + \cdots + 24\!\cdots\!60 ) / 30\!\cdots\!15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 36\!\cdots\!01 \nu^{11} + \cdots - 54\!\cdots\!20 ) / 25\!\cdots\!90 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 27\!\cdots\!15 \nu^{11} + \cdots - 73\!\cdots\!60 ) / 13\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 51\!\cdots\!17 \nu^{11} + \cdots + 73\!\cdots\!32 ) / 74\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11\!\cdots\!18 \nu^{11} + \cdots + 22\!\cdots\!76 ) / 13\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 14\!\cdots\!47 \nu^{11} + \cdots + 18\!\cdots\!56 ) / 12\!\cdots\!60 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 54\!\cdots\!25 \nu^{11} + \cdots - 16\!\cdots\!44 ) / 40\!\cdots\!20 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13\!\cdots\!45 \nu^{11} + \cdots + 10\!\cdots\!68 ) / 40\!\cdots\!20 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 42\!\cdots\!78 \nu^{11} + \cdots + 45\!\cdots\!80 ) / 12\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 65\!\cdots\!65 \nu^{11} + \cdots + 39\!\cdots\!64 ) / 12\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{11} + \beta_{10} + 17\beta_{6} - \beta_{3} - 4\beta_{2} ) / 168 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 64 \beta_{11} + 64 \beta_{10} - 20 \beta_{9} - 10 \beta_{8} - 477 \beta_{7} + 581 \beta_{4} + \cdots - 802816 ) / 336 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4429 \beta_{11} - 8858 \beta_{10} + 856 \beta_{9} - 856 \beta_{8} - 64861 \beta_{6} - 7472 \beta_{5} + \cdots + 8027224 ) / 168 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 995464 \beta_{11} + 497732 \beta_{10} + 190930 \beta_{9} + 381860 \beta_{8} + 2682089 \beta_{7} + \cdots - 190930 ) / 336 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 23896209 \beta_{11} + 23896209 \beta_{10} - 15725860 \beta_{9} - 7862930 \beta_{8} + \cdots - 74510630320 ) / 168 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 3372257800 \beta_{11} - 6744515600 \beta_{10} + 1417094874 \beta_{9} - 1417094874 \beta_{8} + \cdots + 16448907117082 ) / 336 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 286644095922 \beta_{11} + 143322047961 \beta_{10} + 54869325196 \beta_{9} + 109738650392 \beta_{8} + \cdots - 54869325196 ) / 168 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 21921601335180 \beta_{11} + 21921601335180 \beta_{10} - 18485762966292 \beta_{9} + \cdots - 96\!\cdots\!12 ) / 336 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 893649686832565 \beta_{11} + \cdots + 35\!\cdots\!26 ) / 168 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 28\!\cdots\!16 \beta_{11} + \cdots - 58\!\cdots\!66 ) / 336 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 56\!\cdots\!77 \beta_{11} + \cdots - 22\!\cdots\!16 ) / 168 \)
|
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 |
|
−11.3137 | + | 19.5959i | 0 | −256.000 | − | 443.405i | −4505.30 | − | 2601.14i | 0 | 7983.44 | − | 14789.9i | 11585.2 | 0 | 101943. | − | 58857.1i | ||||||||||||||||||||||||||||||||||||||||||||
| 19.2 | −11.3137 | + | 19.5959i | 0 | −256.000 | − | 443.405i | −32.1973 | − | 18.5891i | 0 | −11036.6 | + | 12675.5i | 11585.2 | 0 | 728.542 | − | 420.624i | |||||||||||||||||||||||||||||||||||||||||||||
| 19.3 | −11.3137 | + | 19.5959i | 0 | −256.000 | − | 443.405i | 4757.26 | + | 2746.61i | 0 | 15953.8 | + | 5286.95i | 11585.2 | 0 | −107645. | + | 62148.6i | |||||||||||||||||||||||||||||||||||||||||||||
| 19.4 | 11.3137 | − | 19.5959i | 0 | −256.000 | − | 443.405i | −1336.07 | − | 771.378i | 0 | 14974.8 | − | 7630.84i | −11585.2 | 0 | −30231.7 | + | 17454.3i | |||||||||||||||||||||||||||||||||||||||||||||
| 19.5 | 11.3137 | − | 19.5959i | 0 | −256.000 | − | 443.405i | 1221.89 | + | 705.456i | 0 | −16732.6 | + | 1579.95i | −11585.2 | 0 | 27648.1 | − | 15962.6i | |||||||||||||||||||||||||||||||||||||||||||||
| 19.6 | 11.3137 | − | 19.5959i | 0 | −256.000 | − | 443.405i | 3227.42 | + | 1863.35i | 0 | 4145.11 | + | 16287.8i | −11585.2 | 0 | 73028.2 | − | 42162.9i | |||||||||||||||||||||||||||||||||||||||||||||
| 73.1 | −11.3137 | − | 19.5959i | 0 | −256.000 | + | 443.405i | −4505.30 | + | 2601.14i | 0 | 7983.44 | + | 14789.9i | 11585.2 | 0 | 101943. | + | 58857.1i | |||||||||||||||||||||||||||||||||||||||||||||
| 73.2 | −11.3137 | − | 19.5959i | 0 | −256.000 | + | 443.405i | −32.1973 | + | 18.5891i | 0 | −11036.6 | − | 12675.5i | 11585.2 | 0 | 728.542 | + | 420.624i | |||||||||||||||||||||||||||||||||||||||||||||
| 73.3 | −11.3137 | − | 19.5959i | 0 | −256.000 | + | 443.405i | 4757.26 | − | 2746.61i | 0 | 15953.8 | − | 5286.95i | 11585.2 | 0 | −107645. | − | 62148.6i | |||||||||||||||||||||||||||||||||||||||||||||
| 73.4 | 11.3137 | + | 19.5959i | 0 | −256.000 | + | 443.405i | −1336.07 | + | 771.378i | 0 | 14974.8 | + | 7630.84i | −11585.2 | 0 | −30231.7 | − | 17454.3i | |||||||||||||||||||||||||||||||||||||||||||||
| 73.5 | 11.3137 | + | 19.5959i | 0 | −256.000 | + | 443.405i | 1221.89 | − | 705.456i | 0 | −16732.6 | − | 1579.95i | −11585.2 | 0 | 27648.1 | + | 15962.6i | |||||||||||||||||||||||||||||||||||||||||||||
| 73.6 | 11.3137 | + | 19.5959i | 0 | −256.000 | + | 443.405i | 3227.42 | − | 1863.35i | 0 | 4145.11 | − | 16287.8i | −11585.2 | 0 | 73028.2 | + | 42162.9i | |||||||||||||||||||||||||||||||||||||||||||||
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 | 126.11.n.b | 12 | |
| 3.b | odd | 2 | 1 | 14.11.d.a | ✓ | 12 | |
| 7.d | odd | 6 | 1 | inner | 126.11.n.b | 12 | |
| 12.b | even | 2 | 1 | 112.11.s.a | 12 | ||
| 21.c | even | 2 | 1 | 98.11.d.a | 12 | ||
| 21.g | even | 6 | 1 | 14.11.d.a | ✓ | 12 | |
| 21.g | even | 6 | 1 | 98.11.b.c | 12 | ||
| 21.h | odd | 6 | 1 | 98.11.b.c | 12 | ||
| 21.h | odd | 6 | 1 | 98.11.d.a | 12 | ||
| 84.j | odd | 6 | 1 | 112.11.s.a | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.11.d.a | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 14.11.d.a | ✓ | 12 | 21.g | even | 6 | 1 | |
| 98.11.b.c | 12 | 21.g | even | 6 | 1 | ||
| 98.11.b.c | 12 | 21.h | odd | 6 | 1 | ||
| 98.11.d.a | 12 | 21.c | even | 2 | 1 | ||
| 98.11.d.a | 12 | 21.h | odd | 6 | 1 | ||
| 112.11.s.a | 12 | 12.b | even | 2 | 1 | ||
| 112.11.s.a | 12 | 84.j | odd | 6 | 1 | ||
| 126.11.n.b | 12 | 1.a | even | 1 | 1 | trivial | |
| 126.11.n.b | 12 | 7.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} - 6666 T_{5}^{11} - 15531999 T_{5}^{10} + 202272110766 T_{5}^{9} + 417455844050826 T_{5}^{8} + \cdots + 74\!\cdots\!25 \)
acting on \(S_{11}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 512 T^{2} + 262144)^{3} \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} + \cdots + 74\!\cdots\!25 \)
|
| $7$ |
\( T^{12} + \cdots + 50\!\cdots\!01 \)
|
| $11$ |
\( T^{12} + \cdots + 29\!\cdots\!09 \)
|
| $13$ |
\( T^{12} + \cdots + 32\!\cdots\!16 \)
|
| $17$ |
\( T^{12} + \cdots + 63\!\cdots\!41 \)
|
| $19$ |
\( T^{12} + \cdots + 30\!\cdots\!25 \)
|
| $23$ |
\( T^{12} + \cdots + 17\!\cdots\!09 \)
|
| $29$ |
\( (T^{6} + \cdots + 13\!\cdots\!88)^{2} \)
|
| $31$ |
\( T^{12} + \cdots + 34\!\cdots\!69 \)
|
| $37$ |
\( T^{12} + \cdots + 38\!\cdots\!61 \)
|
| $41$ |
\( T^{12} + \cdots + 52\!\cdots\!24 \)
|
| $43$ |
\( (T^{6} + \cdots + 12\!\cdots\!72)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 17\!\cdots\!69 \)
|
| $53$ |
\( T^{12} + \cdots + 82\!\cdots\!09 \)
|
| $59$ |
\( T^{12} + \cdots + 14\!\cdots\!29 \)
|
| $61$ |
\( T^{12} + \cdots + 36\!\cdots\!09 \)
|
| $67$ |
\( T^{12} + \cdots + 23\!\cdots\!81 \)
|
| $71$ |
\( (T^{6} + \cdots - 12\!\cdots\!24)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 20\!\cdots\!81 \)
|
| $79$ |
\( T^{12} + \cdots + 28\!\cdots\!25 \)
|
| $83$ |
\( T^{12} + \cdots + 14\!\cdots\!56 \)
|
| $89$ |
\( T^{12} + \cdots + 18\!\cdots\!01 \)
|
| $97$ |
\( T^{12} + \cdots + 26\!\cdots\!76 \)
|
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