Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [161,2,Mod(20,161)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(161, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("161.20");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 161 = 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 161.k (of order \(22\), degree \(10\), 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: | \(1.28559147254\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{22})\) |
| Coefficient field: | 20.0.1570539027548129147161113769.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - x^{19} - x^{18} + 3 x^{17} - x^{16} - 5 x^{15} + 7 x^{14} + 3 x^{13} - 17 x^{12} + 11 x^{11} + \cdots + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{22}]$ |
$q$-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
\( x^{20} - x^{19} - x^{18} + 3 x^{17} - x^{16} - 5 x^{15} + 7 x^{14} + 3 x^{13} - 17 x^{12} + 11 x^{11} + \cdots + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{15} + 93\nu^{4} ) / 368 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{14} + 91\nu^{3} ) / 184 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{13} - 91\nu^{2} ) / 92 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{16} - 93\nu^{5} ) / 736 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{13} + \nu^{2} ) / 92 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{11} - 45 ) / 23 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -\nu^{12} - 45\nu ) / 46 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -\nu^{17} + 93\nu^{6} ) / 736 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -3\nu^{19} - 457\nu^{8} ) / 5888 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 3\nu^{18} + 457\nu^{7} ) / 2944 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 3 \nu^{19} - 3 \nu^{18} + 9 \nu^{17} - 3 \nu^{16} - 15 \nu^{15} + 21 \nu^{14} + 9 \nu^{13} + \cdots + 3072 ) / 5888 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 3\nu^{15} + 89\nu^{4} ) / 368 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( -5\nu^{16} - 271\nu^{5} ) / 736 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 11 \nu^{19} - 11 \nu^{18} - 11 \nu^{17} + 33 \nu^{16} - 11 \nu^{15} - 55 \nu^{14} + 77 \nu^{13} + \cdots - 5632 ) / 11776 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 7\nu^{18} + 85\nu^{7} ) / 2944 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 17 \nu^{19} + 17 \nu^{18} - 51 \nu^{17} + 17 \nu^{16} + 85 \nu^{15} - 119 \nu^{14} - 51 \nu^{13} + \cdots - 17408 ) / 11776 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - \nu^{19} + \nu^{18} + \nu^{17} - 3 \nu^{16} + \nu^{15} + 5 \nu^{14} - 7 \nu^{13} - 3 \nu^{12} + \cdots + 512 ) / 512 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( -17\nu^{19} - 627\nu^{8} ) / 5888 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{19} - \beta_{18} + \beta_{16} + \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \cdots - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{13} + 3\beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{14} - 5\beta_{5} \)
|
| \(\nu^{6}\) | \(=\) |
\( - 2 \beta_{17} + 2 \beta_{16} + 2 \beta_{15} + 2 \beta_{13} - 2 \beta_{10} + 5 \beta_{9} - 2 \beta_{8} + \cdots - 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( -3\beta_{16} + 7\beta_{11} \)
|
| \(\nu^{8}\) | \(=\) |
\( 3\beta_{19} - 17\beta_{10} \)
|
| \(\nu^{9}\) | \(=\) |
\( -6\beta_{17} - 17\beta_{12} \)
|
| \(\nu^{10}\) | \(=\) |
\( -11\beta_{18} - 23\beta_{15} \)
|
| \(\nu^{11}\) | \(=\) |
\( -23\beta_{7} - 45 \)
|
| \(\nu^{12}\) | \(=\) |
\( -46\beta_{8} - 45\beta_1 \)
|
| \(\nu^{13}\) | \(=\) |
\( -91\beta_{6} - \beta_{4} \)
|
| \(\nu^{14}\) | \(=\) |
\( - 91 \beta_{19} + 91 \beta_{18} - 91 \beta_{16} - 91 \beta_{15} - 91 \beta_{14} - 91 \beta_{13} + \cdots + 91 \)
|
| \(\nu^{15}\) | \(=\) |
\( 93\beta_{13} - 89\beta_{2} \)
|
| \(\nu^{16}\) | \(=\) |
\( -93\beta_{14} + 271\beta_{5} \)
|
| \(\nu^{17}\) | \(=\) |
\( - 186 \beta_{17} + 186 \beta_{16} + 186 \beta_{15} + 186 \beta_{13} - 186 \beta_{10} - 271 \beta_{9} + \cdots - 186 \)
|
| \(\nu^{18}\) | \(=\) |
\( 457\beta_{16} - 85\beta_{11} \)
|
| \(\nu^{19}\) | \(=\) |
\( -457\beta_{19} + 627\beta_{10} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/161\mathbb{Z}\right)^\times\).
| \(n\) | \(24\) | \(120\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{17}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 20.1 |
|
−0.242631 | − | 1.68753i | 0 | −0.869915 | + | 0.255430i | 0 | 0 | −1.43040 | − | 2.22575i | −0.774356 | − | 1.69560i | −1.96458 | − | 2.26725i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 20.2 | −0.0304694 | − | 0.211919i | 0 | 1.87500 | − | 0.550551i | 0 | 0 | 1.43040 | + | 2.22575i | −0.351682 | − | 0.770077i | −1.96458 | − | 2.26725i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.1 | −2.37408 | + | 1.52573i | 0 | 2.47758 | − | 5.42514i | 0 | 0 | −1.99953 | + | 1.73260i | 1.59208 | + | 11.0731i | −2.87848 | − | 0.845198i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.2 | 1.67514 | − | 1.07655i | 0 | 0.816314 | − | 1.78748i | 0 | 0 | 1.99953 | − | 1.73260i | 0.00990234 | + | 0.0688723i | −2.87848 | − | 0.845198i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 76.1 | −1.80816 | + | 2.08673i | 0 | −0.800361 | − | 5.56663i | 0 | 0 | −0.745394 | − | 2.53858i | 8.41759 | + | 5.40966i | 1.24625 | − | 2.72890i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 76.2 | 1.62177 | − | 1.87162i | 0 | −0.588199 | − | 4.09102i | 0 | 0 | 0.745394 | + | 2.53858i | −4.44400 | − | 2.85599i | 1.24625 | − | 2.72890i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.1 | −0.565283 | + | 0.165982i | 0 | −1.39051 | + | 0.893628i | 0 | 0 | 2.40666 | + | 1.09908i | 1.40933 | − | 1.62645i | −0.426945 | + | 2.96946i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.2 | 2.17964 | − | 0.639999i | 0 | 2.65871 | − | 1.70865i | 0 | 0 | −2.40666 | − | 1.09908i | 1.72625 | − | 1.99220i | −0.426945 | + | 2.96946i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 90.1 | −2.37408 | − | 1.52573i | 0 | 2.47758 | + | 5.42514i | 0 | 0 | −1.99953 | − | 1.73260i | 1.59208 | − | 11.0731i | −2.87848 | + | 0.845198i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 90.2 | 1.67514 | + | 1.07655i | 0 | 0.816314 | + | 1.78748i | 0 | 0 | 1.99953 | + | 1.73260i | 0.00990234 | − | 0.0688723i | −2.87848 | + | 0.845198i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.1 | −0.565283 | − | 0.165982i | 0 | −1.39051 | − | 0.893628i | 0 | 0 | 2.40666 | − | 1.09908i | 1.40933 | + | 1.62645i | −0.426945 | − | 2.96946i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.2 | 2.17964 | + | 0.639999i | 0 | 2.65871 | + | 1.70865i | 0 | 0 | −2.40666 | + | 1.09908i | 1.72625 | + | 1.99220i | −0.426945 | − | 2.96946i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 111.1 | −0.558594 | − | 1.22315i | 0 | 0.125653 | − | 0.145011i | 0 | 0 | 2.61882 | − | 0.376530i | −2.82795 | − | 0.830361i | 2.52376 | − | 1.62192i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 111.2 | 1.10267 | + | 2.41451i | 0 | −3.30427 | + | 3.81334i | 0 | 0 | −2.61882 | + | 0.376530i | −7.75715 | − | 2.27771i | 2.52376 | − | 1.62192i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 125.1 | −1.80816 | − | 2.08673i | 0 | −0.800361 | + | 5.56663i | 0 | 0 | −0.745394 | + | 2.53858i | 8.41759 | − | 5.40966i | 1.24625 | + | 2.72890i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 125.2 | 1.62177 | + | 1.87162i | 0 | −0.588199 | + | 4.09102i | 0 | 0 | 0.745394 | − | 2.53858i | −4.44400 | + | 2.85599i | 1.24625 | + | 2.72890i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 132.1 | −0.558594 | + | 1.22315i | 0 | 0.125653 | + | 0.145011i | 0 | 0 | 2.61882 | + | 0.376530i | −2.82795 | + | 0.830361i | 2.52376 | + | 1.62192i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 132.2 | 1.10267 | − | 2.41451i | 0 | −3.30427 | − | 3.81334i | 0 | 0 | −2.61882 | − | 0.376530i | −7.75715 | + | 2.27771i | 2.52376 | + | 1.62192i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 153.1 | −0.242631 | + | 1.68753i | 0 | −0.869915 | − | 0.255430i | 0 | 0 | −1.43040 | + | 2.22575i | −0.774356 | + | 1.69560i | −1.96458 | + | 2.26725i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 153.2 | −0.0304694 | + | 0.211919i | 0 | 1.87500 | + | 0.550551i | 0 | 0 | 1.43040 | − | 2.22575i | −0.351682 | + | 0.770077i | −1.96458 | + | 2.26725i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 23.d | odd | 22 | 1 | inner |
| 161.k | even | 22 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 161.2.k.a | ✓ | 20 |
| 7.b | odd | 2 | 1 | CM | 161.2.k.a | ✓ | 20 |
| 23.d | odd | 22 | 1 | inner | 161.2.k.a | ✓ | 20 |
| 161.k | even | 22 | 1 | inner | 161.2.k.a | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 161.2.k.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 161.2.k.a | ✓ | 20 | 7.b | odd | 2 | 1 | CM |
| 161.2.k.a | ✓ | 20 | 23.d | odd | 22 | 1 | inner |
| 161.2.k.a | ✓ | 20 | 161.k | even | 22 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} - 2 T_{2}^{19} + 3 T_{2}^{18} - 4 T_{2}^{17} + 71 T_{2}^{16} - 248 T_{2}^{15} + 425 T_{2}^{14} + \cdots + 4489 \)
acting on \(S_{2}^{\mathrm{new}}(161, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - 2 T^{19} + \cdots + 4489 \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} \)
|
| $7$ |
\( T^{20} + \cdots + 282475249 \)
|
| $11$ |
\( T^{20} + \cdots + 18282014521 \)
|
| $13$ |
\( T^{20} \)
|
| $17$ |
\( T^{20} \)
|
| $19$ |
\( T^{20} \)
|
| $23$ |
\( T^{20} + \cdots + 41426511213649 \)
|
| $29$ |
\( T^{20} + \cdots + 95\!\cdots\!01 \)
|
| $31$ |
\( T^{20} \)
|
| $37$ |
\( T^{20} + \cdots + 42\!\cdots\!49 \)
|
| $41$ |
\( T^{20} \)
|
| $43$ |
\( T^{20} + \cdots + 13\!\cdots\!89 \)
|
| $47$ |
\( T^{20} \)
|
| $53$ |
\( T^{20} + \cdots + 68\!\cdots\!49 \)
|
| $59$ |
\( T^{20} \)
|
| $61$ |
\( T^{20} \)
|
| $67$ |
\( T^{20} + \cdots + 16\!\cdots\!09 \)
|
| $71$ |
\( T^{20} + \cdots + 31\!\cdots\!81 \)
|
| $73$ |
\( T^{20} \)
|
| $79$ |
\( T^{20} + \cdots + 19\!\cdots\!61 \)
|
| $83$ |
\( T^{20} \)
|
| $89$ |
\( T^{20} \)
|
| $97$ |
\( T^{20} \)
|
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