Newspace parameters
Level: | \( N \) | \(=\) | \( 161 = 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 161.k (of order \(22\), degree \(10\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.28559147254\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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.
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
20.1 | −0.347190 | − | 2.41476i | −0.922369 | − | 0.421232i | −3.79153 | + | 1.11329i | −1.48641 | + | 1.71541i | −0.696936 | + | 2.37354i | −1.44308 | + | 2.21755i | 1.97783 | + | 4.33085i | −1.29125 | − | 1.49019i | 4.65837 | + | 2.99375i |
20.2 | −0.347190 | − | 2.41476i | 0.922369 | + | 0.421232i | −3.79153 | + | 1.11329i | 1.48641 | − | 1.71541i | 0.696936 | − | 2.37354i | 2.61663 | − | 0.391471i | 1.97783 | + | 4.33085i | −1.29125 | − | 1.49019i | −4.65837 | − | 2.99375i |
20.3 | −0.198303 | − | 1.37923i | −2.35718 | − | 1.07649i | 0.0560394 | − | 0.0164547i | 2.67181 | − | 3.08344i | −1.01729 | + | 3.46456i | −1.66841 | + | 2.05339i | −1.19150 | − | 2.60901i | 2.43290 | + | 2.80772i | −4.78259 | − | 3.07359i |
20.4 | −0.198303 | − | 1.37923i | 2.35718 | + | 1.07649i | 0.0560394 | − | 0.0164547i | −2.67181 | + | 3.08344i | 1.01729 | − | 3.46456i | 2.56091 | − | 0.664634i | −1.19150 | − | 2.60901i | 2.43290 | + | 2.80772i | 4.78259 | + | 3.07359i |
20.5 | −0.0343259 | − | 0.238742i | −2.02155 | − | 0.923209i | 1.86317 | − | 0.547075i | −0.355896 | + | 0.410726i | −0.151017 | + | 0.514318i | 0.655104 | − | 2.56336i | −0.394959 | − | 0.864839i | 1.26975 | + | 1.46537i | 0.110274 | + | 0.0708688i |
20.6 | −0.0343259 | − | 0.238742i | 2.02155 | + | 0.923209i | 1.86317 | − | 0.547075i | 0.355896 | − | 0.410726i | 0.151017 | − | 0.514318i | −2.60386 | − | 0.468957i | −0.394959 | − | 0.864839i | 1.26975 | + | 1.46537i | −0.110274 | − | 0.0708688i |
20.7 | 0.187562 | + | 1.30452i | −1.36235 | − | 0.622165i | 0.252392 | − | 0.0741091i | 1.67205 | − | 1.92964i | 0.556102 | − | 1.89391i | 0.444888 | − | 2.60808i | 1.23900 | + | 2.71302i | −0.495672 | − | 0.572036i | 2.83087 | + | 1.81929i |
20.8 | 0.187562 | + | 1.30452i | 1.36235 | + | 0.622165i | 0.252392 | − | 0.0741091i | −1.67205 | + | 1.92964i | −0.556102 | + | 1.89391i | −2.55720 | − | 0.678751i | 1.23900 | + | 2.71302i | −0.495672 | − | 0.572036i | −2.83087 | − | 1.81929i |
20.9 | 0.259711 | + | 1.80633i | −0.739551 | − | 0.337742i | −1.27639 | + | 0.374782i | −2.45337 | + | 2.83134i | 0.418003 | − | 1.42359i | 2.48097 | + | 0.919137i | 0.507712 | + | 1.11173i | −1.53172 | − | 1.76769i | −5.75151 | − | 3.69627i |
20.10 | 0.259711 | + | 1.80633i | 0.739551 | + | 0.337742i | −1.27639 | + | 0.374782i | 2.45337 | − | 2.83134i | −0.418003 | + | 1.42359i | −0.194554 | + | 2.63859i | 0.507712 | + | 1.11173i | −1.53172 | − | 1.76769i | 5.75151 | + | 3.69627i |
20.11 | 0.365139 | + | 2.53960i | −2.08739 | − | 0.953277i | −4.39725 | + | 1.29115i | −0.106291 | + | 0.122667i | 1.65876 | − | 5.64920i | −2.64064 | − | 0.164315i | −2.75294 | − | 6.02810i | 1.48386 | + | 1.71246i | −0.350335 | − | 0.225147i |
20.12 | 0.365139 | + | 2.53960i | 2.08739 | + | 0.953277i | −4.39725 | + | 1.29115i | 0.106291 | − | 0.122667i | −1.65876 | + | 5.64920i | 0.947497 | − | 2.47027i | −2.75294 | − | 6.02810i | 1.48386 | + | 1.71246i | 0.350335 | + | 0.225147i |
34.1 | −1.68017 | + | 1.07978i | −0.664315 | − | 0.0955140i | 0.826210 | − | 1.80915i | −3.17412 | + | 0.932004i | 1.21929 | − | 0.556832i | 2.63780 | + | 0.204993i | −0.00316123 | − | 0.0219868i | −2.44629 | − | 0.718295i | 4.32668 | − | 4.99326i |
34.2 | −1.68017 | + | 1.07978i | 0.664315 | + | 0.0955140i | 0.826210 | − | 1.80915i | 3.17412 | − | 0.932004i | −1.21929 | + | 0.556832i | 0.172491 | − | 2.64012i | −0.00316123 | − | 0.0219868i | −2.44629 | − | 0.718295i | −4.32668 | + | 4.99326i |
34.3 | −1.53131 | + | 0.984115i | −3.07833 | − | 0.442597i | 0.545606 | − | 1.19471i | −0.648243 | + | 0.190341i | 5.14945 | − | 2.35168i | −2.63540 | + | 0.233843i | −0.177865 | − | 1.23708i | 6.40174 | + | 1.87972i | 0.805344 | − | 0.929417i |
34.4 | −1.53131 | + | 0.984115i | 3.07833 | + | 0.442597i | 0.545606 | − | 1.19471i | 0.648243 | − | 0.190341i | −5.14945 | + | 2.35168i | −0.606519 | + | 2.57529i | −0.177865 | − | 1.23708i | 6.40174 | + | 1.87972i | −0.805344 | + | 0.929417i |
34.5 | −0.494143 | + | 0.317566i | −1.18392 | − | 0.170222i | −0.687501 | + | 1.50542i | −0.804140 | + | 0.236117i | 0.639083 | − | 0.291860i | −1.03103 | − | 2.43659i | −0.305534 | − | 2.12504i | −1.50578 | − | 0.442138i | 0.322377 | − | 0.372043i |
34.6 | −0.494143 | + | 0.317566i | 1.18392 | + | 0.170222i | −0.687501 | + | 1.50542i | 0.804140 | − | 0.236117i | −0.639083 | + | 0.291860i | 2.26506 | + | 1.36730i | −0.305534 | − | 2.12504i | −1.50578 | − | 0.442138i | −0.322377 | + | 0.372043i |
34.7 | 0.430824 | − | 0.276874i | −2.93167 | − | 0.421510i | −0.721880 | + | 1.58070i | 3.09475 | − | 0.908700i | −1.37974 | + | 0.630106i | 2.40804 | − | 1.09607i | 0.272415 | + | 1.89469i | 5.53854 | + | 1.62626i | 1.08170 | − | 1.24835i |
34.8 | 0.430824 | − | 0.276874i | 2.93167 | + | 0.421510i | −0.721880 | + | 1.58070i | −3.09475 | + | 0.908700i | 1.37974 | − | 0.630106i | 1.42761 | − | 2.22754i | 0.272415 | + | 1.89469i | 5.53854 | + | 1.62626i | −1.08170 | + | 1.24835i |
See next 80 embeddings (of 120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
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.b | ✓ | 120 |
7.b | odd | 2 | 1 | inner | 161.2.k.b | ✓ | 120 |
23.d | odd | 22 | 1 | inner | 161.2.k.b | ✓ | 120 |
161.k | even | 22 | 1 | inner | 161.2.k.b | ✓ | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
161.2.k.b | ✓ | 120 | 1.a | even | 1 | 1 | trivial |
161.2.k.b | ✓ | 120 | 7.b | odd | 2 | 1 | inner |
161.2.k.b | ✓ | 120 | 23.d | odd | 22 | 1 | inner |
161.2.k.b | ✓ | 120 | 161.k | even | 22 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{60} + 11 T_{2}^{59} + 74 T_{2}^{58} + 372 T_{2}^{57} + 1533 T_{2}^{56} + 5465 T_{2}^{55} + 17463 T_{2}^{54} + 51343 T_{2}^{53} + 142150 T_{2}^{52} + 374850 T_{2}^{51} + 948807 T_{2}^{50} + 2306112 T_{2}^{49} + \cdots + 1849 \)
acting on \(S_{2}^{\mathrm{new}}(161, [\chi])\).