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Properties

Label 392.2.y.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 8 q^{3} - q^{5} + 4 q^{7} + 41 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 8 q^{3} - q^{5} + 4 q^{7} + 41 q^{9} + q^{11} - 2 q^{13} + q^{15} - 5 q^{17} - 13 q^{19} + 12 q^{21} - 5 q^{23} + 2 q^{25} + 4 q^{27} - 21 q^{29} + 23 q^{31} - 17 q^{33} - 22 q^{35} - 60 q^{37} - 61 q^{39} + 4 q^{41} + 22 q^{43} + 30 q^{45} + 82 q^{47} - 2 q^{49} + 3 q^{51} + 38 q^{53} + 69 q^{55} - 2 q^{57} - 6 q^{59} - 26 q^{61} - 37 q^{63} + 2 q^{65} + 43 q^{67} + 7 q^{69} - 21 q^{71} - 7 q^{73} - 80 q^{75} - 85 q^{77} - 25 q^{79} - 10 q^{81} - 17 q^{83} + 23 q^{85} + 47 q^{87} + 7 q^{89} - 3 q^{91} - 50 q^{93} - 129 q^{95} - 150 q^{97} - 94 q^{99}+O(q^{100}) \) 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 −3.25332 − 0.490359i 0 −1.54958 − 3.94826i 0 −1.43068 − 2.22557i 0 7.47690 + 2.30632i 0
9.2 0 −1.78890 − 0.269634i 0 1.38358 + 3.52530i 0 −2.47527 − 0.934375i 0 0.260753 + 0.0804317i 0
9.3 0 −1.51640 − 0.228560i 0 −0.147325 − 0.375377i 0 0.926271 + 2.47831i 0 −0.619491 − 0.191088i 0
9.4 0 −0.739962 − 0.111531i 0 −0.0672440 − 0.171335i 0 2.12593 − 1.57494i 0 −2.33161 − 0.719207i 0
9.5 0 1.51578 + 0.228466i 0 −1.31786 − 3.35786i 0 −2.27390 + 1.35254i 0 −0.621339 − 0.191658i 0
9.6 0 1.89105 + 0.285030i 0 0.866493 + 2.20779i 0 0.189580 + 2.63895i 0 0.628115 + 0.193748i 0
9.7 0 3.16923 + 0.477685i 0 −0.596962 − 1.52103i 0 1.36565 − 2.26605i 0 6.94914 + 2.14352i 0
25.1 0 −1.07048 − 2.72753i 0 −0.873295 − 0.131628i 0 2.12690 − 1.57362i 0 −4.09433 + 3.79898i 0
25.2 0 −0.633206 − 1.61338i 0 −2.32382 − 0.350259i 0 −2.63088 + 0.280091i 0 −0.00290016 + 0.00269096i 0
25.3 0 −0.600191 − 1.52926i 0 0.454269 + 0.0684701i 0 0.704180 + 2.55032i 0 0.220740 − 0.204817i 0
25.4 0 0.0317720 + 0.0809538i 0 2.96234 + 0.446501i 0 1.79423 + 1.94442i 0 2.19361 − 2.03537i 0
25.5 0 0.247413 + 0.630398i 0 −2.43262 − 0.366659i 0 −0.00945262 − 2.64573i 0 1.86297 − 1.72858i 0
25.6 0 0.281171 + 0.716412i 0 3.30995 + 0.498894i 0 −1.64091 − 2.07543i 0 1.76497 − 1.63765i 0
25.7 0 1.02100 + 2.60146i 0 −0.568892 − 0.0857466i 0 2.62933 + 0.294332i 0 −3.52599 + 3.27164i 0
65.1 0 −0.173239 − 2.31172i 0 −3.33484 + 2.27366i 0 1.12122 − 2.39643i 0 −2.34753 + 0.353833i 0
65.2 0 −0.118310 − 1.57873i 0 1.24081 − 0.845971i 0 2.07415 + 1.64252i 0 0.488092 − 0.0735680i 0
65.3 0 −0.0337386 − 0.450211i 0 3.66755 − 2.50050i 0 −1.88168 − 1.85991i 0 2.76494 − 0.416748i 0
65.4 0 −0.0247290 − 0.329985i 0 −1.09829 + 0.748802i 0 −2.39796 − 1.11795i 0 2.85821 − 0.430806i 0
65.5 0 0.100350 + 1.33908i 0 −0.499334 + 0.340440i 0 2.50671 − 0.846419i 0 1.18344 − 0.178375i 0
65.6 0 0.137858 + 1.83959i 0 −2.15672 + 1.47042i 0 −0.660063 + 2.56209i 0 −0.398603 + 0.0600797i 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.a ✓ 84
4.b odd 2 1 784.2.bg.f 84
49.g even 21 1 inner 392.2.y.a ✓ 84
196.o odd 42 1 784.2.bg.f 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.y.a ✓ 84 1.a even 1 1 trivial
392.2.y.a ✓ 84 49.g even 21 1 inner
784.2.bg.f 84 4.b odd 2 1
784.2.bg.f 84 196.o odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{84} + 8 T_{3}^{83} + T_{3}^{82} - 168 T_{3}^{81} - 426 T_{3}^{80} + 1016 T_{3}^{79} + \cdots + 782376841 \) acting on \(S_{2}^{\mathrm{new}}(392, [\chi])\). Copy content Toggle raw display