Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [161,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("161.20");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 161 = 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(9.49930751092\) |
| 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 \(\left(\mathbb{Z}/161\mathbb{Z}\right)^\times\).
| \(n\) | \(24\) | \(120\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{5}\) |
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 |
|
0.576670 | + | 4.01083i | 0 | −8.07824 | + | 2.37198i | 0 | 0 | −10.0128 | − | 15.5802i | −0.705764 | − | 1.54541i | −17.6812 | − | 20.4052i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 20.2 | 0.788831 | + | 5.48644i | 0 | −21.8028 | + | 6.40189i | 0 | 0 | 10.0128 | + | 15.5802i | −33.9016 | − | 74.2343i | −17.6812 | − | 20.4052i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.1 | −0.277265 | + | 0.178187i | 0 | −3.27820 | + | 7.17825i | 0 | 0 | −13.9967 | + | 12.1282i | −0.745382 | − | 5.18425i | −25.9063 | − | 7.60678i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34.2 | 3.77196 | − | 2.42409i | 0 | 5.02813 | − | 11.0101i | 0 | 0 | 13.9967 | − | 12.1282i | −2.61869 | − | 18.2134i | −25.9063 | − | 7.60678i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 76.1 | −1.24898 | + | 1.44140i | 0 | 0.620835 | + | 4.31801i | 0 | 0 | −5.21776 | − | 17.7701i | −19.8352 | − | 12.7473i | 11.2162 | − | 24.5601i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 76.2 | 2.18095 | − | 2.51695i | 0 | −0.439971 | − | 3.06007i | 0 | 0 | 5.21776 | + | 17.7701i | 13.7521 | + | 8.83793i | 11.2162 | − | 24.5601i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.1 | −5.40834 | + | 1.58803i | 0 | 19.9983 | − | 12.8521i | 0 | 0 | 16.8466 | + | 7.69359i | −58.2183 | + | 67.1875i | −3.84250 | + | 26.7252i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.2 | −2.66342 | + | 0.782052i | 0 | −0.247804 | + | 0.159254i | 0 | 0 | −16.8466 | − | 7.69359i | 15.0779 | − | 17.4008i | −3.84250 | + | 26.7252i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 90.1 | −0.277265 | − | 0.178187i | 0 | −3.27820 | − | 7.17825i | 0 | 0 | −13.9967 | − | 12.1282i | −0.745382 | + | 5.18425i | −25.9063 | + | 7.60678i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 90.2 | 3.77196 | + | 2.42409i | 0 | 5.02813 | + | 11.0101i | 0 | 0 | 13.9967 | + | 12.1282i | −2.61869 | + | 18.2134i | −25.9063 | + | 7.60678i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.1 | −5.40834 | − | 1.58803i | 0 | 19.9983 | + | 12.8521i | 0 | 0 | 16.8466 | − | 7.69359i | −58.2183 | − | 67.1875i | −3.84250 | − | 26.7252i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.2 | −2.66342 | − | 0.782052i | 0 | −0.247804 | − | 0.159254i | 0 | 0 | −16.8466 | + | 7.69359i | 15.0779 | + | 17.4008i | −3.84250 | − | 26.7252i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 111.1 | −2.19083 | − | 4.79724i | 0 | −12.9749 | + | 14.9739i | 0 | 0 | 18.3317 | − | 2.63571i | 59.7775 | + | 17.5523i | 22.7138 | − | 14.5973i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 111.2 | −0.529562 | − | 1.15958i | 0 | 4.17470 | − | 4.81786i | 0 | 0 | −18.3317 | + | 2.63571i | −17.5826 | − | 5.16271i | 22.7138 | − | 14.5973i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 125.1 | −1.24898 | − | 1.44140i | 0 | 0.620835 | − | 4.31801i | 0 | 0 | −5.21776 | + | 17.7701i | −19.8352 | + | 12.7473i | 11.2162 | + | 24.5601i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 125.2 | 2.18095 | + | 2.51695i | 0 | −0.439971 | + | 3.06007i | 0 | 0 | 5.21776 | − | 17.7701i | 13.7521 | − | 8.83793i | 11.2162 | + | 24.5601i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 132.1 | −2.19083 | + | 4.79724i | 0 | −12.9749 | − | 14.9739i | 0 | 0 | 18.3317 | + | 2.63571i | 59.7775 | − | 17.5523i | 22.7138 | + | 14.5973i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 132.2 | −0.529562 | + | 1.15958i | 0 | 4.17470 | + | 4.81786i | 0 | 0 | −18.3317 | − | 2.63571i | −17.5826 | + | 5.16271i | 22.7138 | + | 14.5973i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 153.1 | 0.576670 | − | 4.01083i | 0 | −8.07824 | − | 2.37198i | 0 | 0 | −10.0128 | + | 15.5802i | −0.705764 | + | 1.54541i | −17.6812 | + | 20.4052i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 153.2 | 0.788831 | − | 5.48644i | 0 | −21.8028 | − | 6.40189i | 0 | 0 | 10.0128 | − | 15.5802i | −33.9016 | + | 74.2343i | −17.6812 | + | 20.4052i | 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 | 161.4.k.a | ✓ | 20 |
| 7.b | odd | 2 | 1 | CM | 161.4.k.a | ✓ | 20 |
| 23.d | odd | 22 | 1 | inner | 161.4.k.a | ✓ | 20 |
| 161.k | even | 22 | 1 | inner | 161.4.k.a | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 161.4.k.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 161.4.k.a | ✓ | 20 | 7.b | odd | 2 | 1 | CM |
| 161.4.k.a | ✓ | 20 | 23.d | odd | 22 | 1 | inner |
| 161.4.k.a | ✓ | 20 | 161.k | even | 22 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} + 10 T_{2}^{19} + 75 T_{2}^{18} + 500 T_{2}^{17} + 2333 T_{2}^{16} + 10390 T_{2}^{15} + \cdots + 491819329 \)
acting on \(S_{4}^{\mathrm{new}}(161, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots + 491819329 \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} \)
|
| $7$ |
\( T^{20} + \cdots + 22\!\cdots\!49 \)
|
| $11$ |
\( T^{20} + \cdots + 86\!\cdots\!21 \)
|
| $13$ |
\( T^{20} \)
|
| $17$ |
\( T^{20} \)
|
| $19$ |
\( T^{20} \)
|
| $23$ |
\( T^{20} + \cdots + 71\!\cdots\!49 \)
|
| $29$ |
\( T^{20} + \cdots + 37\!\cdots\!01 \)
|
| $31$ |
\( T^{20} \)
|
| $37$ |
\( T^{20} + \cdots + 13\!\cdots\!49 \)
|
| $41$ |
\( T^{20} \)
|
| $43$ |
\( T^{20} + \cdots + 92\!\cdots\!49 \)
|
| $47$ |
\( T^{20} \)
|
| $53$ |
\( T^{20} + \cdots + 12\!\cdots\!49 \)
|
| $59$ |
\( T^{20} \)
|
| $61$ |
\( T^{20} \)
|
| $67$ |
\( T^{20} + \cdots + 91\!\cdots\!49 \)
|
| $71$ |
\( T^{20} + \cdots + 20\!\cdots\!01 \)
|
| $73$ |
\( T^{20} \)
|
| $79$ |
\( T^{20} + \cdots + 26\!\cdots\!21 \)
|
| $83$ |
\( T^{20} \)
|
| $89$ |
\( T^{20} \)
|
| $97$ |
\( T^{20} \)
|
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