Properties

Label 342.2.w.a
Level $342$
Weight $2$
Character orbit 342.w
Analytic conductor $2.731$
Analytic rank $0$
Dimension $54$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [342,2,Mod(43,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([12, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.w (of order \(9\), degree \(6\), 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: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(9\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 54 q - 24 q^{7} + 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q - 24 q^{7} + 27 q^{8} - 3 q^{13} + 3 q^{14} + 9 q^{17} + 6 q^{18} + 15 q^{19} + 3 q^{22} - 12 q^{23} - 6 q^{25} - 27 q^{27} - 3 q^{28} + 9 q^{29} - 57 q^{33} + 18 q^{34} - 18 q^{35} - 6 q^{37} - 3 q^{38} - 6 q^{39} - 3 q^{41} + 6 q^{43} + 6 q^{44} + 54 q^{45} - 18 q^{47} + 6 q^{49} - 3 q^{50} - 15 q^{51} - 3 q^{52} - 24 q^{53} - 54 q^{54} - 36 q^{55} - 12 q^{56} + 12 q^{57} + 18 q^{58} + 69 q^{59} - 27 q^{60} + 84 q^{61} - 15 q^{62} + 39 q^{63} - 27 q^{64} + 24 q^{65} + 21 q^{66} - 30 q^{67} - 6 q^{68} + 42 q^{69} + 18 q^{70} - 60 q^{71} - 45 q^{73} + 3 q^{74} - 9 q^{78} - 18 q^{79} - 36 q^{81} + 3 q^{82} - 24 q^{83} - 21 q^{84} - 72 q^{85} + 3 q^{86} + 66 q^{87} + 9 q^{89} - 15 q^{91} + 6 q^{92} - 3 q^{93} + 6 q^{94} - 30 q^{95} - 78 q^{97} + 12 q^{98} - 51 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −0.173648 + 0.984808i −1.61970 + 0.613651i −0.939693 0.342020i −0.648788 + 3.67946i −0.323070 1.70165i −3.85338 0.500000 0.866025i 2.24687 1.98786i −3.51090 1.27786i
43.2 −0.173648 + 0.984808i −1.52941 + 0.812965i −0.939693 0.342020i 0.272361 1.54463i −0.535036 1.64734i 2.24074 0.500000 0.866025i 1.67817 2.48671i 1.47387 + 0.536446i
43.3 −0.173648 + 0.984808i −1.30786 1.13556i −0.939693 0.342020i −0.124608 + 0.706685i 1.34542 1.09080i 1.23071 0.500000 0.866025i 0.421002 + 2.97031i −0.674311 0.245429i
43.4 −0.173648 + 0.984808i −0.224346 1.71746i −0.939693 0.342020i −0.440347 + 2.49733i 1.73033 + 0.0772962i −1.32843 0.500000 0.866025i −2.89934 + 0.770610i −2.38293 0.867314i
43.5 −0.173648 + 0.984808i −0.0104059 + 1.73202i −0.939693 0.342020i −0.0597169 + 0.338671i −1.70390 0.311010i −2.70759 0.500000 0.866025i −2.99978 0.0360466i −0.323156 0.117619i
43.6 −0.173648 + 0.984808i 0.792978 1.53987i −0.939693 0.342020i 0.336337 1.90746i 1.37877 + 1.04833i −4.43581 0.500000 0.866025i −1.74237 2.44216i 1.82008 + 0.662455i
43.7 −0.173648 + 0.984808i 1.00018 + 1.41409i −0.939693 0.342020i −0.232829 + 1.32044i −1.56628 + 0.739433i 3.81041 0.500000 0.866025i −0.999276 + 2.82868i −1.25995 0.458584i
43.8 −0.173648 + 0.984808i 1.39081 1.03230i −0.939693 0.342020i 0.183383 1.04002i 0.775102 + 1.54894i 2.95323 0.500000 0.866025i 0.868725 2.87147i 0.992372 + 0.361194i
43.9 −0.173648 + 0.984808i 1.50775 + 0.852462i −0.939693 0.342020i 0.714207 4.05047i −1.10133 + 1.33681i −0.377794 0.500000 0.866025i 1.54662 + 2.57060i 3.86492 + 1.40671i
85.1 −0.766044 + 0.642788i −1.73088 0.0636567i 0.173648 0.984808i 0.451077 0.378498i 1.36685 1.06382i −1.49179 0.500000 + 0.866025i 2.99190 + 0.220364i −0.102251 + 0.579893i
85.2 −0.766044 + 0.642788i −1.13333 1.30979i 0.173648 0.984808i 1.81802 1.52550i 1.71010 + 0.274870i 1.50605 0.500000 + 0.866025i −0.431123 + 2.96886i −0.412111 + 2.33720i
85.3 −0.766044 + 0.642788i −1.01451 + 1.40384i 0.173648 0.984808i 1.60183 1.34410i −0.125210 1.72752i −0.685971 0.500000 + 0.866025i −0.941533 2.84842i −0.363106 + 2.05928i
85.4 −0.766044 + 0.642788i −0.916667 + 1.46960i 0.173648 0.984808i −2.20943 + 1.85393i −0.242432 1.71500i 1.38926 0.500000 + 0.866025i −1.31944 2.69427i 0.500836 2.84038i
85.5 −0.766044 + 0.642788i −0.379894 1.68988i 0.173648 0.984808i −2.05175 + 1.72162i 1.37725 + 1.05033i 0.931774 0.500000 + 0.866025i −2.71136 + 1.28395i 0.465093 2.63767i
85.6 −0.766044 + 0.642788i 0.705559 + 1.58183i 0.173648 0.984808i 0.287887 0.241566i −1.55727 0.758228i −3.26780 0.500000 + 0.866025i −2.00437 + 2.23215i −0.0652586 + 0.370100i
85.7 −0.766044 + 0.642788i 1.10517 1.33364i 0.173648 0.984808i −0.765245 + 0.642117i 0.0106356 + 1.73202i −5.16688 0.500000 + 0.866025i −0.557188 2.94780i 0.173467 0.983780i
85.8 −0.766044 + 0.642788i 1.67515 0.440302i 0.173648 0.984808i −1.49025 + 1.25047i −1.00022 + 1.41406i 2.53958 0.500000 + 0.866025i 2.61227 1.47515i 0.337812 1.91582i
85.9 −0.766044 + 0.642788i 1.68940 + 0.382000i 0.173648 0.984808i 2.35785 1.97847i −1.53970 + 0.793297i −1.63361 0.500000 + 0.866025i 2.70815 + 1.29070i −0.534481 + 3.03119i
139.1 0.939693 + 0.342020i −1.65424 0.513314i 0.766044 + 0.642788i −1.17751 0.428579i −1.37891 1.04814i 2.38962 0.500000 + 0.866025i 2.47302 + 1.69829i −0.959915 0.805465i
139.2 0.939693 + 0.342020i −1.63454 0.572959i 0.766044 + 0.642788i −1.12406 0.409125i −1.34000 1.09745i −3.90541 0.500000 + 0.866025i 2.34344 + 1.87305i −0.916343 0.768903i
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.w even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.w.a yes 54
9.c even 3 1 342.2.v.a 54
19.e even 9 1 342.2.v.a 54
171.w even 9 1 inner 342.2.w.a yes 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.v.a 54 9.c even 3 1
342.2.v.a 54 19.e even 9 1
342.2.w.a yes 54 1.a even 1 1 trivial
342.2.w.a yes 54 171.w even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{54} + 3 T_{5}^{52} + 6 T_{5}^{51} - 114 T_{5}^{50} - 105 T_{5}^{49} + 2256 T_{5}^{48} + 246 T_{5}^{47} + 22224 T_{5}^{46} - 33923 T_{5}^{45} - 137706 T_{5}^{44} - 84216 T_{5}^{43} + 4974390 T_{5}^{42} + \cdots + 387420489 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display