Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [162,11,Mod(161,162)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(162, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("162.161");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 162.b (of order \(2\), degree \(1\), 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: | \(102.927874933\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 4 x^{11} - 112022 x^{10} - 4403900 x^{9} + 4228288142 x^{8} + 340360803064 x^{7} + \cdots + 61\!\cdots\!28 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{15}\cdot 3^{44} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{12} - 4 x^{11} - 112022 x^{10} - 4403900 x^{9} + 4228288142 x^{8} + 340360803064 x^{7} + \cdots + 61\!\cdots\!28 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 18\!\cdots\!59 \nu^{11} + \cdots + 34\!\cdots\!28 ) / 14\!\cdots\!80 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 59\!\cdots\!36 \nu^{11} + \cdots - 15\!\cdots\!72 ) / 11\!\cdots\!60 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 84\!\cdots\!89 \nu^{11} + \cdots + 12\!\cdots\!08 ) / 11\!\cdots\!60 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 10\!\cdots\!14 \nu^{11} + \cdots - 15\!\cdots\!68 ) / 36\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\!\cdots\!93 \nu^{11} + \cdots + 17\!\cdots\!16 ) / 38\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 31\!\cdots\!66 \nu^{11} + \cdots + 51\!\cdots\!32 ) / 66\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 12\!\cdots\!83 \nu^{11} + \cdots + 43\!\cdots\!76 ) / 11\!\cdots\!60 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 89\!\cdots\!23 \nu^{11} + \cdots + 12\!\cdots\!16 ) / 73\!\cdots\!44 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 55\!\cdots\!68 \nu^{11} + \cdots + 84\!\cdots\!96 ) / 34\!\cdots\!20 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 90\!\cdots\!89 \nu^{11} + \cdots - 13\!\cdots\!28 ) / 11\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 63\!\cdots\!62 \nu^{11} + \cdots + 10\!\cdots\!04 ) / 66\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( ( 9\beta_{10} + 8\beta_{9} + 46\beta_{5} + 135\beta_{4} + 1161\beta_{3} + 9\beta_{2} - 26248\beta _1 + 17496 ) / 52488 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 36 \beta_{11} + 342 \beta_{10} + 220 \beta_{9} + 144 \beta_{8} + 270 \beta_{7} + 3780 \beta_{6} + \cdots + 980038440 ) / 52488 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 1305 \beta_{11} + 165648 \beta_{10} + 174043 \beta_{9} - 40302 \beta_{8} + 21735 \beta_{7} + \cdots + 21222688824 ) / 17496 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 1500138 \beta_{11} + 26786574 \beta_{10} + 36255262 \beta_{9} + 2974068 \beta_{8} + \cdots + 18069729846768 ) / 26244 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 214617735 \beta_{11} + 12242524968 \beta_{10} + 18571412789 \beta_{9} - 4362475590 \beta_{8} + \cdots + 20\!\cdots\!08 ) / 26244 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 17342767020 \beta_{11} + 303208472049 \beta_{10} + 550561444052 \beta_{9} - 19313574180 \beta_{8} + \cdots + 13\!\cdots\!56 ) / 4374 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 9027395472501 \beta_{11} + 303016097987631 \beta_{10} + 621766221287171 \beta_{9} + \cdots + 59\!\cdots\!88 ) / 13122 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 16\!\cdots\!00 \beta_{11} + \cdots + 96\!\cdots\!36 ) / 6561 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 36\!\cdots\!49 \beta_{11} + \cdots + 18\!\cdots\!08 ) / 729 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 10\!\cdots\!70 \beta_{11} + \cdots + 47\!\cdots\!60 ) / 6561 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 22\!\cdots\!25 \beta_{11} + \cdots + 92\!\cdots\!44 ) / 6561 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 161.1 |
|
− | 22.6274i | 0 | −512.000 | − | 5369.27i | 0 | 9924.74 | 11585.2i | 0 | −121493. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.2 | − | 22.6274i | 0 | −512.000 | − | 2549.46i | 0 | −4700.73 | 11585.2i | 0 | −57687.7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.3 | − | 22.6274i | 0 | −512.000 | − | 83.0202i | 0 | −19035.6 | 11585.2i | 0 | −1878.53 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.4 | − | 22.6274i | 0 | −512.000 | 1361.54i | 0 | −19432.3 | 11585.2i | 0 | 30808.1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.5 | − | 22.6274i | 0 | −512.000 | 4194.31i | 0 | 15954.7 | 11585.2i | 0 | 94906.3 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.6 | − | 22.6274i | 0 | −512.000 | 5971.54i | 0 | 19209.2 | 11585.2i | 0 | 135121. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.7 | 22.6274i | 0 | −512.000 | − | 5971.54i | 0 | 19209.2 | − | 11585.2i | 0 | 135121. | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.8 | 22.6274i | 0 | −512.000 | − | 4194.31i | 0 | 15954.7 | − | 11585.2i | 0 | 94906.3 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.9 | 22.6274i | 0 | −512.000 | − | 1361.54i | 0 | −19432.3 | − | 11585.2i | 0 | 30808.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.10 | 22.6274i | 0 | −512.000 | 83.0202i | 0 | −19035.6 | − | 11585.2i | 0 | −1878.53 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.11 | 22.6274i | 0 | −512.000 | 2549.46i | 0 | −4700.73 | − | 11585.2i | 0 | −57687.7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.12 | 22.6274i | 0 | −512.000 | 5369.27i | 0 | 9924.74 | − | 11585.2i | 0 | −121493. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 162.11.b.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | inner | 162.11.b.b | ✓ | 12 |
| 9.c | even | 3 | 2 | 162.11.d.g | 24 | ||
| 9.d | odd | 6 | 2 | 162.11.d.g | 24 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 162.11.b.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 162.11.b.b | ✓ | 12 | 3.b | odd | 2 | 1 | inner |
| 162.11.d.g | 24 | 9.c | even | 3 | 2 | ||
| 162.11.d.g | 24 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} + 90441036 T_{5}^{10} + \cdots + 15\!\cdots\!25 \)
acting on \(S_{11}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 512)^{6} \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} + \cdots + 15\!\cdots\!25 \)
|
| $7$ |
\( (T^{6} + \cdots - 52\!\cdots\!44)^{2} \)
|
| $11$ |
\( T^{12} + \cdots + 61\!\cdots\!36 \)
|
| $13$ |
\( (T^{6} + \cdots - 28\!\cdots\!31)^{2} \)
|
| $17$ |
\( T^{12} + \cdots + 24\!\cdots\!25 \)
|
| $19$ |
\( (T^{6} + \cdots + 11\!\cdots\!32)^{2} \)
|
| $23$ |
\( T^{12} + \cdots + 26\!\cdots\!00 \)
|
| $29$ |
\( T^{12} + \cdots + 16\!\cdots\!81 \)
|
| $31$ |
\( (T^{6} + \cdots - 23\!\cdots\!96)^{2} \)
|
| $37$ |
\( (T^{6} + \cdots + 62\!\cdots\!73)^{2} \)
|
| $41$ |
\( T^{12} + \cdots + 39\!\cdots\!96 \)
|
| $43$ |
\( (T^{6} + \cdots + 47\!\cdots\!12)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 11\!\cdots\!64 \)
|
| $53$ |
\( T^{12} + \cdots + 23\!\cdots\!84 \)
|
| $59$ |
\( T^{12} + \cdots + 45\!\cdots\!36 \)
|
| $61$ |
\( (T^{6} + \cdots - 56\!\cdots\!59)^{2} \)
|
| $67$ |
\( (T^{6} + \cdots + 31\!\cdots\!92)^{2} \)
|
| $71$ |
\( T^{12} + \cdots + 27\!\cdots\!96 \)
|
| $73$ |
\( (T^{6} + \cdots - 64\!\cdots\!43)^{2} \)
|
| $79$ |
\( (T^{6} + \cdots - 21\!\cdots\!76)^{2} \)
|
| $83$ |
\( T^{12} + \cdots + 74\!\cdots\!76 \)
|
| $89$ |
\( T^{12} + \cdots + 14\!\cdots\!69 \)
|
| $97$ |
\( (T^{6} + \cdots + 22\!\cdots\!52)^{2} \)
|
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