Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [170,2,Mod(23,170)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(170, base_ring=CyclotomicField(16))
chi = DirichletCharacter(H, H._module([12, 15]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("170.23");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 170 = 2 \cdot 5 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 170.r (of order \(16\), degree \(8\), 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: | \(1.35745683436\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{16})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{16}]$ |
$q$-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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | 0.923880 | − | 0.382683i | −2.47936 | − | 1.65666i | 0.707107 | − | 0.707107i | −2.20283 | + | 0.384106i | −2.92460 | − | 0.581740i | −2.11600 | − | 0.420899i | 0.382683 | − | 0.923880i | 2.25467 | + | 5.44325i | −1.88816 | + | 1.19785i |
23.2 | 0.923880 | − | 0.382683i | −0.628452 | − | 0.419918i | 0.707107 | − | 0.707107i | −0.448594 | − | 2.19061i | −0.741309 | − | 0.147456i | 1.27154 | + | 0.252926i | 0.382683 | − | 0.923880i | −0.929430 | − | 2.24384i | −1.25276 | − | 1.85219i |
23.3 | 0.923880 | − | 0.382683i | −0.432293 | − | 0.288849i | 0.707107 | − | 0.707107i | 0.601626 | + | 2.15361i | −0.509925 | − | 0.101430i | 3.10821 | + | 0.618262i | 0.382683 | − | 0.923880i | −1.04461 | − | 2.52190i | 1.37998 | + | 1.75945i |
23.4 | 0.923880 | − | 0.382683i | 1.41027 | + | 0.942312i | 0.707107 | − | 0.707107i | 2.21673 | − | 0.293448i | 1.66353 | + | 0.330896i | −5.02568 | − | 0.999670i | 0.382683 | − | 0.923880i | −0.0471409 | − | 0.113808i | 1.93569 | − | 1.11942i |
23.5 | 0.923880 | − | 0.382683i | 2.12983 | + | 1.42311i | 0.707107 | − | 0.707107i | −2.23146 | − | 0.143452i | 2.51231 | + | 0.499730i | 0.0411526 | + | 0.00818577i | 0.382683 | − | 0.923880i | 1.36290 | + | 3.29034i | −2.11650 | + | 0.721411i |
37.1 | 0.923880 | + | 0.382683i | −2.47936 | + | 1.65666i | 0.707107 | + | 0.707107i | −2.20283 | − | 0.384106i | −2.92460 | + | 0.581740i | −2.11600 | + | 0.420899i | 0.382683 | + | 0.923880i | 2.25467 | − | 5.44325i | −1.88816 | − | 1.19785i |
37.2 | 0.923880 | + | 0.382683i | −0.628452 | + | 0.419918i | 0.707107 | + | 0.707107i | −0.448594 | + | 2.19061i | −0.741309 | + | 0.147456i | 1.27154 | − | 0.252926i | 0.382683 | + | 0.923880i | −0.929430 | + | 2.24384i | −1.25276 | + | 1.85219i |
37.3 | 0.923880 | + | 0.382683i | −0.432293 | + | 0.288849i | 0.707107 | + | 0.707107i | 0.601626 | − | 2.15361i | −0.509925 | + | 0.101430i | 3.10821 | − | 0.618262i | 0.382683 | + | 0.923880i | −1.04461 | + | 2.52190i | 1.37998 | − | 1.75945i |
37.4 | 0.923880 | + | 0.382683i | 1.41027 | − | 0.942312i | 0.707107 | + | 0.707107i | 2.21673 | + | 0.293448i | 1.66353 | − | 0.330896i | −5.02568 | + | 0.999670i | 0.382683 | + | 0.923880i | −0.0471409 | + | 0.113808i | 1.93569 | + | 1.11942i |
37.5 | 0.923880 | + | 0.382683i | 2.12983 | − | 1.42311i | 0.707107 | + | 0.707107i | −2.23146 | + | 0.143452i | 2.51231 | − | 0.499730i | 0.0411526 | − | 0.00818577i | 0.382683 | + | 0.923880i | 1.36290 | − | 3.29034i | −2.11650 | − | 0.721411i |
97.1 | 0.382683 | − | 0.923880i | −0.592739 | + | 2.97990i | −0.707107 | − | 0.707107i | −1.33977 | + | 1.79026i | 2.52624 | + | 1.68798i | −0.842302 | − | 0.562808i | −0.923880 | + | 0.382683i | −5.75684 | − | 2.38456i | 1.14127 | + | 1.92289i |
97.2 | 0.382683 | − | 0.923880i | −0.364682 | + | 1.83338i | −0.707107 | − | 0.707107i | 0.389869 | − | 2.20182i | 1.55427 | + | 1.03853i | 3.99085 | + | 2.66660i | −0.923880 | + | 0.382683i | −0.456659 | − | 0.189154i | −1.88502 | − | 1.20279i |
97.3 | 0.382683 | − | 0.923880i | 0.0178485 | − | 0.0897304i | −0.707107 | − | 0.707107i | −0.936619 | − | 2.03045i | −0.0760698 | − | 0.0508282i | −3.86692 | − | 2.58379i | −0.923880 | + | 0.382683i | 2.76391 | + | 1.14485i | −2.23432 | + | 0.0883019i |
97.4 | 0.382683 | − | 0.923880i | 0.372511 | − | 1.87274i | −0.707107 | − | 0.707107i | 2.21955 | + | 0.271292i | −1.58763 | − | 1.06082i | −0.930986 | − | 0.622065i | −0.923880 | + | 0.382683i | −0.596748 | − | 0.247181i | 1.10003 | − | 1.94678i |
97.5 | 0.382683 | − | 0.923880i | 0.567062 | − | 2.85081i | −0.707107 | − | 0.707107i | −2.18819 | − | 0.460262i | −2.41680 | − | 1.61486i | 3.60477 | + | 2.40863i | −0.923880 | + | 0.382683i | −5.03395 | − | 2.08513i | −1.26261 | + | 1.84549i |
107.1 | −0.382683 | + | 0.923880i | −2.87914 | − | 0.572697i | −0.707107 | − | 0.707107i | 2.22112 | − | 0.258133i | 1.63090 | − | 2.44082i | −0.222469 | + | 0.332949i | 0.923880 | − | 0.382683i | 5.18983 | + | 2.14970i | −0.611501 | + | 2.15083i |
107.2 | −0.382683 | + | 0.923880i | −1.27477 | − | 0.253568i | −0.707107 | − | 0.707107i | −1.36286 | + | 1.77274i | 0.722100 | − | 1.08070i | −0.128721 | + | 0.192645i | 0.923880 | − | 0.382683i | −1.21089 | − | 0.501569i | −1.11625 | − | 1.93752i |
107.3 | −0.382683 | + | 0.923880i | −0.706293 | − | 0.140490i | −0.707107 | − | 0.707107i | −0.971395 | − | 2.01405i | 0.400083 | − | 0.598766i | 1.70679 | − | 2.55439i | 0.923880 | − | 0.382683i | −2.29253 | − | 0.949595i | 2.23248 | − | 0.126709i |
107.4 | −0.382683 | + | 0.923880i | 1.51925 | + | 0.302197i | −0.707107 | − | 0.707107i | 1.28645 | + | 1.82895i | −0.860584 | + | 1.28795i | 0.134383 | − | 0.201119i | 0.923880 | − | 0.382683i | −0.554854 | − | 0.229828i | −2.18203 | + | 0.488612i |
107.5 | −0.382683 | + | 0.923880i | 3.34096 | + | 0.664558i | −0.707107 | − | 0.707107i | −0.732362 | − | 2.11273i | −1.89250 | + | 2.83233i | −0.616963 | + | 0.923351i | 0.923880 | − | 0.382683i | 7.94873 | + | 3.29247i | 2.23217 | + | 0.131894i |
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
85.r | even | 16 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 170.2.r.b | yes | 40 |
5.b | even | 2 | 1 | 850.2.v.d | 40 | ||
5.c | odd | 4 | 1 | 170.2.o.b | ✓ | 40 | |
5.c | odd | 4 | 1 | 850.2.s.d | 40 | ||
17.e | odd | 16 | 1 | 170.2.o.b | ✓ | 40 | |
85.o | even | 16 | 1 | 850.2.v.d | 40 | ||
85.p | odd | 16 | 1 | 850.2.s.d | 40 | ||
85.r | even | 16 | 1 | inner | 170.2.r.b | yes | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
170.2.o.b | ✓ | 40 | 5.c | odd | 4 | 1 | |
170.2.o.b | ✓ | 40 | 17.e | odd | 16 | 1 | |
170.2.r.b | yes | 40 | 1.a | even | 1 | 1 | trivial |
170.2.r.b | yes | 40 | 85.r | even | 16 | 1 | inner |
850.2.s.d | 40 | 5.c | odd | 4 | 1 | ||
850.2.s.d | 40 | 85.p | odd | 16 | 1 | ||
850.2.v.d | 40 | 5.b | even | 2 | 1 | ||
850.2.v.d | 40 | 85.o | even | 16 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} - 8 T_{3}^{37} - 48 T_{3}^{36} - 88 T_{3}^{35} - 72 T_{3}^{34} - 408 T_{3}^{33} + \cdots + 524288 \) acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\).