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Properties

Label 392.2.y.b
Level $392$
Weight $2$
Character orbit 392.y
Analytic conductor $3.130$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,2,Mod(9,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.y (of order \(21\), degree \(12\), 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: \(3.13013575923\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(7\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 10 q^{3} - q^{5} - 4 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q + 10 q^{3} - q^{5} - 4 q^{7} - 11 q^{9} + q^{11} + 10 q^{13} + 9 q^{15} + 3 q^{17} - 23 q^{19} - 10 q^{21} - q^{23} + 6 q^{25} - 32 q^{27} - 13 q^{29} - 29 q^{31} + 11 q^{33} + 30 q^{35} + 48 q^{37} + 51 q^{39} + 20 q^{41} - 14 q^{43} + 22 q^{45} - 34 q^{47} - 2 q^{49} + 3 q^{51} - 40 q^{53} - 9 q^{55} - 10 q^{57} + 38 q^{59} + 10 q^{61} + 9 q^{63} - 2 q^{65} - 37 q^{67} - 87 q^{69} + 5 q^{71} + 21 q^{73} - 12 q^{75} + 71 q^{77} + 19 q^{79} - 70 q^{81} - 55 q^{83} + 29 q^{85} - 165 q^{87} - 65 q^{89} - 15 q^{91} - 120 q^{93} + 7 q^{95} + 170 q^{97} + 110 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −2.81692 − 0.424582i 0 −0.149546 − 0.381038i 0 1.43463 + 2.22302i 0 4.88804 + 1.50776i 0
9.2 0 −2.23111 − 0.336286i 0 0.360771 + 0.919230i 0 −2.49326 − 0.885256i 0 1.99807 + 0.616322i 0
9.3 0 −0.694696 − 0.104709i 0 1.58953 + 4.05005i 0 2.54411 − 0.726282i 0 −2.39508 − 0.738784i 0
9.4 0 −0.506175 − 0.0762936i 0 −1.22985 − 3.13360i 0 −1.39551 + 2.24779i 0 −2.61633 − 0.807030i 0
9.5 0 0.263081 + 0.0396530i 0 −0.479382 − 1.22144i 0 −0.536348 − 2.59082i 0 −2.79908 − 0.863402i 0
9.6 0 2.07937 + 0.313415i 0 −0.306595 − 0.781192i 0 2.45929 + 0.975662i 0 1.35883 + 0.419144i 0
9.7 0 2.65131 + 0.399621i 0 0.913293 + 2.32703i 0 −2.63728 − 0.211582i 0 4.00305 + 1.23478i 0
25.1 0 −0.874484 − 2.22815i 0 3.44089 + 0.518630i 0 2.60911 − 0.438784i 0 −2.00077 + 1.85644i 0
25.2 0 −0.532847 − 1.35767i 0 1.09613 + 0.165214i 0 −2.44164 − 1.01903i 0 0.639808 − 0.593655i 0
25.3 0 −0.165318 − 0.421223i 0 −3.13260 − 0.472164i 0 2.54891 + 0.709281i 0 2.04906 − 1.90125i 0
25.4 0 0.242438 + 0.617721i 0 −0.793056 − 0.119534i 0 −0.905081 + 2.48613i 0 1.87635 − 1.74100i 0
25.5 0 0.610218 + 1.55481i 0 0.653119 + 0.0984419i 0 1.50007 − 2.17940i 0 0.154088 − 0.142973i 0
25.6 0 1.03636 + 2.64060i 0 3.74928 + 0.565113i 0 −0.477915 + 2.60223i 0 −3.69958 + 3.43271i 0
25.7 0 1.13684 + 2.89661i 0 −3.56403 − 0.537191i 0 −1.91804 − 1.82239i 0 −4.89880 + 4.54543i 0
65.1 0 −0.192917 − 2.57430i 0 0.277188 − 0.188984i 0 −2.63686 − 0.216760i 0 −3.62334 + 0.546130i 0
65.2 0 −0.167965 − 2.24134i 0 1.72106 − 1.17340i 0 2.19081 − 1.48335i 0 −2.02891 + 0.305809i 0
65.3 0 −0.138470 − 1.84775i 0 −3.02338 + 2.06131i 0 0.184852 + 2.63929i 0 −0.428529 + 0.0645904i 0
65.4 0 0.0603171 + 0.804876i 0 1.77912 − 1.21299i 0 0.287907 + 2.63004i 0 2.32231 − 0.350031i 0
65.5 0 0.0762273 + 1.01718i 0 −1.98810 + 1.35546i 0 −2.53769 − 0.748408i 0 1.93764 − 0.292053i 0
65.6 0 0.145655 + 1.94364i 0 1.90959 − 1.30194i 0 0.0445909 − 2.64538i 0 −0.790013 + 0.119075i 0
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.2.y.b ✓ 84
4.b odd 2 1 784.2.bg.e 84
49.g even 21 1 inner 392.2.y.b ✓ 84
196.o odd 42 1 784.2.bg.e 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.y.b ✓ 84 1.a even 1 1 trivial
392.2.y.b ✓ 84 49.g even 21 1 inner
784.2.bg.e 84 4.b odd 2 1
784.2.bg.e 84 196.o odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{84} - 10 T_{3}^{83} + 45 T_{3}^{82} - 94 T_{3}^{81} - 84 T_{3}^{80} + 1196 T_{3}^{79} + \cdots + 7127918329 \) acting on \(S_{2}^{\mathrm{new}}(392, [\chi])\). Copy content Toggle raw display