Properties

Label 920.2.v.a
Level $920$
Weight $2$
Character orbit 920.v
Analytic conductor $7.346$
Analytic rank $0$
Dimension $264$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [920,2,Mod(323,920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(920, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("920.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 920.v (of order \(4\), degree \(2\), 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: \(7.34623698596\)
Analytic rank: \(0\)
Dimension: \(264\)
Relative dimension: \(132\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 264 q - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 264 q - 12 q^{8} + 16 q^{10} - 16 q^{12} - 16 q^{16} + 8 q^{17} - 28 q^{18} - 16 q^{22} - 8 q^{25} - 48 q^{26} - 12 q^{28} - 24 q^{30} + 40 q^{32} - 48 q^{35} + 32 q^{36} + 12 q^{38} + 16 q^{40} + 40 q^{42} - 64 q^{43} - 28 q^{48} - 40 q^{50} + 16 q^{52} - 40 q^{56} - 20 q^{58} - 100 q^{60} - 60 q^{62} - 8 q^{65} - 72 q^{66} + 48 q^{67} + 72 q^{68} - 60 q^{70} + 80 q^{72} - 40 q^{73} + 112 q^{75} + 48 q^{76} + 28 q^{78} - 60 q^{80} - 264 q^{81} + 44 q^{82} + 56 q^{86} - 68 q^{88} + 64 q^{91} + 16 q^{96} - 40 q^{97} + 16 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1 −1.41404 0.0224370i −1.94106 1.94106i 1.99899 + 0.0634534i 1.41681 + 1.72992i 2.70118 + 2.78828i −1.58571 1.58571i −2.82522 0.134577i 4.53543i −1.96461 2.47796i
323.2 −1.41346 0.0460389i −0.830909 0.830909i 1.99576 + 0.130149i −2.06164 0.865819i 1.13621 + 1.21271i −3.35989 3.35989i −2.81494 0.275843i 1.61918i 2.87419 + 1.31872i
323.3 −1.41336 0.0490551i 1.66418 + 1.66418i 1.99519 + 0.138665i 0.502259 + 2.17893i −2.27046 2.43373i 2.07765 + 2.07765i −2.81312 0.293859i 2.53901i −0.602987 3.10426i
323.4 −1.40971 0.112742i 2.42474 + 2.42474i 1.97458 + 0.317866i −1.74367 + 1.39986i −3.14481 3.69155i −2.58602 2.58602i −2.74775 0.670717i 8.75870i 2.61590 1.77682i
323.5 −1.40871 0.124592i 1.11995 + 1.11995i 1.96895 + 0.351030i 2.04565 0.902959i −1.43816 1.71723i 2.08392 + 2.08392i −2.72996 0.739818i 0.491408i −2.99423 + 1.01714i
323.6 −1.40681 0.144563i 1.37916 + 1.37916i 1.95820 + 0.406744i −0.158443 2.23045i −1.74084 2.13959i −0.711602 0.711602i −2.69601 0.855293i 0.804167i −0.0995414 + 3.16071i
323.7 −1.39818 + 0.212355i −1.45207 1.45207i 1.90981 0.593822i 2.18123 0.492171i 2.33860 + 1.72190i 2.01979 + 2.01979i −2.54416 + 1.23583i 1.21700i −2.94524 + 1.15134i
323.8 −1.39655 0.222850i −1.16052 1.16052i 1.90068 + 0.622439i −0.00501944 2.23606i 1.36209 + 1.87933i 0.119412 + 0.119412i −2.51567 1.29283i 0.306410i −0.491296 + 3.12388i
323.9 −1.38863 + 0.267761i −0.700567 0.700567i 1.85661 0.743644i −0.735317 2.11171i 1.16042 + 0.785246i 3.06000 + 3.06000i −2.37903 + 1.52978i 2.01841i 1.58652 + 2.73550i
323.10 −1.37226 0.341893i −0.562073 0.562073i 1.76622 + 0.938334i −0.819638 + 2.08043i 0.579144 + 0.963482i 0.0338316 + 0.0338316i −2.10291 1.89150i 2.36815i 1.83604 2.57467i
323.11 −1.36777 + 0.359447i 0.488759 + 0.488759i 1.74160 0.983282i 1.73112 1.41535i −0.844193 0.492828i −2.86140 2.86140i −2.02867 + 1.97092i 2.52223i −1.85903 + 2.55812i
323.12 −1.36421 + 0.372744i −2.34634 2.34634i 1.72212 1.01700i −2.20858 + 0.349547i 4.07547 + 2.32631i 1.42785 + 1.42785i −1.97025 + 2.02931i 8.01059i 2.88267 1.30009i
323.13 −1.36028 0.386815i 0.210032 + 0.210032i 1.70075 + 1.05236i −2.13300 + 0.671043i −0.204460 0.366946i 1.84573 + 1.84573i −1.90644 2.08938i 2.91177i 3.16106 0.0877327i
323.14 −1.33716 + 0.460449i −1.13667 1.13667i 1.57597 1.23139i −1.32806 + 1.79896i 2.04328 + 0.996526i 0.174949 + 0.174949i −1.54033 + 2.37221i 0.415965i 0.947486 3.01700i
323.15 −1.29776 0.561986i 1.12321 + 1.12321i 1.36834 + 1.45864i −1.82601 1.29062i −0.826426 2.08888i −1.78171 1.78171i −0.956042 2.66195i 0.476796i 1.64440 + 2.70110i
323.16 −1.28220 0.596623i −0.209415 0.209415i 1.28808 + 1.52998i 2.22712 0.199831i 0.143571 + 0.393455i 0.716662 + 0.716662i −0.738761 2.73024i 2.91229i −2.97484 1.07253i
323.17 −1.27843 + 0.604658i 0.523906 + 0.523906i 1.26878 1.54603i −1.86638 1.23152i −0.986562 0.352994i 1.76116 + 1.76116i −0.687226 + 2.74367i 2.45105i 3.13068 + 0.445889i
323.18 −1.27201 + 0.618053i 2.18353 + 2.18353i 1.23602 1.57234i 2.23354 + 0.106286i −4.12701 1.42794i −0.781712 0.781712i −0.600445 + 2.76396i 6.53562i −2.90678 + 1.24525i
323.19 −1.26157 0.639090i −1.73633 1.73633i 1.18313 + 1.61252i 1.09980 + 1.94690i 1.08083 + 3.30017i 3.40058 + 3.40058i −0.462060 2.79043i 3.02966i −0.143237 3.15903i
323.20 −1.25601 + 0.649962i 1.82618 + 1.82618i 1.15510 1.63271i −1.45456 1.69831i −3.48065 1.10675i −0.606946 0.606946i −0.389612 + 2.80146i 3.66990i 2.93077 + 1.18768i
See next 80 embeddings (of 264 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.132
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 920.2.v.a 264
5.c odd 4 1 inner 920.2.v.a 264
8.d odd 2 1 inner 920.2.v.a 264
40.k even 4 1 inner 920.2.v.a 264
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.v.a 264 1.a even 1 1 trivial
920.2.v.a 264 5.c odd 4 1 inner
920.2.v.a 264 8.d odd 2 1 inner
920.2.v.a 264 40.k even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(920, [\chi])\).