↑

Properties

Label 368.2.s.b
Level $368$
Weight $2$
Character orbit 368.s
Analytic conductor $2.938$
Analytic rank $0$
Dimension $80$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [368,2,Mod(15,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 0, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.15");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 368.s (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: \(2.93849479438\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(8\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{9} + 12 q^{13} + 22 q^{17} + 66 q^{21} + 36 q^{25} + 34 q^{29} + 22 q^{33} + 12 q^{41} - 56 q^{49} - 66 q^{57} - 88 q^{61} - 154 q^{65} - 66 q^{69} - 16 q^{73} - 158 q^{77} - 248 q^{81} - 116 q^{85}+ \cdots - 22 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 0 −2.62148 + 1.19719i 0 −2.71500 + 2.35256i 0 −2.67155 − 1.71690i 0 3.47433 − 4.00959i 0
15.2 0 −1.72174 + 0.786292i 0 1.28247 − 1.11127i 0 0.445021 + 0.285998i 0 0.381548 − 0.440330i 0
15.3 0 −1.64422 + 0.750889i 0 2.87391 − 2.49026i 0 −1.97206 − 1.26737i 0 0.175032 − 0.201998i 0
15.4 0 −0.106877 + 0.0488093i 0 −1.32017 + 1.14393i 0 2.88624 + 1.85487i 0 −1.95554 + 2.25682i 0
15.5 0 0.106877 − 0.0488093i 0 −1.32017 + 1.14393i 0 −2.88624 − 1.85487i 0 −1.95554 + 2.25682i 0
15.6 0 1.64422 − 0.750889i 0 2.87391 − 2.49026i 0 1.97206 + 1.26737i 0 0.175032 − 0.201998i 0
15.7 0 1.72174 − 0.786292i 0 1.28247 − 1.11127i 0 −0.445021 − 0.285998i 0 0.381548 − 0.440330i 0
15.8 0 2.62148 − 1.19719i 0 −2.71500 + 2.35256i 0 2.67155 + 1.71690i 0 3.47433 − 4.00959i 0
63.1 0 −0.906661 + 3.08781i 0 0.0798308 + 0.124219i 0 −0.190741 + 1.32663i 0 −6.18874 − 3.97726i 0
63.2 0 −0.641049 + 2.18321i 0 −2.06240 − 3.20915i 0 −0.0632963 + 0.440236i 0 −1.83172 − 1.17717i 0
63.3 0 −0.364424 + 1.24112i 0 1.34269 + 2.08927i 0 0.274488 − 1.90910i 0 1.11620 + 0.717337i 0
63.4 0 −0.237705 + 0.809548i 0 −0.0390617 − 0.0607812i 0 0.668464 − 4.64927i 0 1.92490 + 1.23706i 0
63.5 0 0.237705 − 0.809548i 0 −0.0390617 − 0.0607812i 0 −0.668464 + 4.64927i 0 1.92490 + 1.23706i 0
63.6 0 0.364424 − 1.24112i 0 1.34269 + 2.08927i 0 −0.274488 + 1.90910i 0 1.11620 + 0.717337i 0
63.7 0 0.641049 − 2.18321i 0 −2.06240 − 3.20915i 0 0.0632963 − 0.440236i 0 −1.83172 − 1.17717i 0
63.8 0 0.906661 − 3.08781i 0 0.0798308 + 0.124219i 0 0.190741 − 1.32663i 0 −6.18874 − 3.97726i 0
79.1 0 −1.79190 + 2.78825i 0 −2.35821 − 1.07696i 0 −0.203995 − 0.0598984i 0 −3.31718 − 7.26362i 0
79.2 0 −0.732741 + 1.14017i 0 −2.77841 − 1.26886i 0 3.21084 + 0.942788i 0 0.483174 + 1.05800i 0
79.3 0 −0.656051 + 1.02084i 0 3.34986 + 1.52983i 0 −1.03868 − 0.304984i 0 0.634544 + 1.38946i 0
79.4 0 −0.0817633 + 0.127226i 0 −0.571708 − 0.261090i 0 1.43010 + 0.419915i 0 1.23674 + 2.70809i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.d odd 22 1 inner
92.h even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.2.s.b ✓ 80
4.b odd 2 1 inner 368.2.s.b ✓ 80
23.d odd 22 1 inner 368.2.s.b ✓ 80
92.h even 22 1 inner 368.2.s.b ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
368.2.s.b ✓ 80 1.a even 1 1 trivial
368.2.s.b ✓ 80 4.b odd 2 1 inner
368.2.s.b ✓ 80 23.d odd 22 1 inner
368.2.s.b ✓ 80 92.h even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{80} - 14 T_{3}^{78} + 205 T_{3}^{76} - 2425 T_{3}^{74} + 23638 T_{3}^{72} - 256257 T_{3}^{70} + \cdots + 279841 \) acting on \(S_{2}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display