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Properties

Label 392.2.z.a
Level $392$
Weight $2$
Character orbit 392.z
Analytic conductor $3.130$
Analytic rank $0$
Dimension $648$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,2,Mod(37,392)]
 
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, 21, 32]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.z (of order \(42\), 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: \(648\)
Relative dimension: \(54\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9} - 6 q^{10} - 47 q^{12} - 33 q^{14} - 8 q^{15} - 17 q^{16} - 26 q^{17} - 8 q^{18} - 22 q^{20} - 18 q^{22} - 26 q^{23} - 74 q^{24} - 72 q^{25}+ \cdots + 151 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.41305 − 0.0573510i 0.882385 + 0.950985i 1.99342 + 0.162080i 0.913727 − 2.96223i −1.19231 − 1.39439i 2.15945 + 1.52866i −2.80751 − 0.343351i 0.0984214 − 1.31334i −1.46103 + 4.13337i
37.2 −1.41088 − 0.0969984i −1.16192 − 1.25225i 1.98118 + 0.273707i −0.978911 + 3.17355i 1.51787 + 1.87949i −0.0759419 + 2.64466i −2.76867 − 0.578340i 0.00611242 − 0.0815645i 1.68896 − 4.38256i
37.3 −1.40118 + 0.191593i −2.22424 − 2.39716i 1.92658 − 0.536910i 0.266970 − 0.865496i 3.57583 + 2.93269i 2.06958 + 1.64828i −2.59661 + 1.12142i −0.574944 + 7.67209i −0.208249 + 1.26386i
37.4 −1.39792 + 0.214038i −0.469951 − 0.506487i 1.90838 − 0.598418i 0.273835 − 0.887753i 0.765363 + 0.607442i 0.528093 − 2.59251i −2.53968 + 1.24501i 0.188515 − 2.51556i −0.192788 + 1.29962i
37.5 −1.37458 − 0.332474i −0.288166 − 0.310569i 1.77892 + 0.914022i −0.486553 + 1.57737i 0.292850 + 0.522709i −2.63963 − 0.179924i −2.14138 − 1.84784i 0.210777 − 2.81262i 1.19324 − 2.00645i
37.6 −1.37066 + 0.348278i 1.48945 + 1.60524i 1.75740 − 0.954740i −1.25029 + 4.05335i −2.60059 − 1.68150i 2.55536 − 0.685667i −2.07629 + 1.92069i −0.134159 + 1.79023i 0.302031 − 5.99121i
37.7 −1.34654 − 0.432240i 1.94533 + 2.09656i 1.62634 + 1.16406i 0.117162 − 0.379831i −1.71324 − 3.66395i −1.64975 + 2.06841i −1.68677 − 2.27042i −0.387093 + 5.16539i −0.321942 + 0.460815i
37.8 −1.26332 + 0.635636i 1.08411 + 1.16839i 1.19193 − 1.60602i −0.529378 + 1.71620i −2.11224 − 0.786948i −2.34752 + 1.22032i −0.484943 + 2.78654i 0.0343439 − 0.458288i −0.422109 − 2.50460i
37.9 −1.24570 − 0.669503i −1.67046 − 1.80033i 1.10353 + 1.66800i 0.687205 − 2.22786i 0.875568 + 3.36105i −0.555114 − 2.58686i −0.257941 − 2.81664i −0.226554 + 3.02315i −2.34761 + 2.31516i
37.10 −1.23135 + 0.695533i 1.73400 + 1.86880i 1.03247 − 1.71290i 0.880547 − 2.85466i −3.43498 − 1.09511i −1.33854 − 2.28217i −0.0799587 + 2.82730i −0.261493 + 3.48939i 0.901247 + 4.12755i
37.11 −1.19828 − 0.751090i 0.581041 + 0.626213i 0.871727 + 1.80003i −0.147712 + 0.478871i −0.225904 − 1.18679i 2.54532 − 0.722043i 0.307412 − 2.81167i 0.169656 − 2.26390i 0.536675 − 0.462874i
37.12 −1.09732 + 0.892129i −1.73400 − 1.86880i 0.408212 − 1.95790i −0.880547 + 2.85466i 3.56996 + 0.503723i −1.33854 − 2.28217i 1.29876 + 2.51261i −0.261493 + 3.48939i −1.58049 − 3.91803i
37.13 −1.07880 − 0.914431i −1.29408 − 1.39469i 0.327631 + 1.97298i 0.498626 − 1.61651i 0.120712 + 2.68794i −0.699420 + 2.55163i 1.45071 − 2.42805i −0.0463173 + 0.618061i −2.01610 + 1.28793i
37.14 −1.05324 + 0.943763i −1.08411 − 1.16839i 0.218622 − 1.98802i 0.529378 − 1.71620i 2.24451 + 0.207452i −2.34752 + 1.22032i 1.64596 + 2.30018i 0.0343439 − 0.458288i 1.06213 + 2.30718i
37.15 −0.824960 + 1.14867i −1.48945 − 1.60524i −0.638881 − 1.89521i 1.25029 − 4.05335i 3.07263 − 0.386620i 2.55536 − 0.685667i 2.70402 + 0.829613i −0.134159 + 1.79023i 3.62452 + 4.78003i
37.16 −0.804026 − 1.16342i 0.264186 + 0.284725i −0.707086 + 1.87084i −1.09949 + 3.56447i 0.118842 − 0.536285i −1.95202 − 1.78595i 2.74508 − 0.681563i 0.212916 − 2.84117i 5.03100 − 1.58676i
37.17 −0.767597 − 1.18777i −1.83996 − 1.98301i −0.821590 + 1.82346i −0.819891 + 2.65802i −0.943005 + 3.70760i 2.51957 − 0.807328i 2.79649 − 0.423820i −0.322667 + 4.30570i 3.78646 − 1.06645i
37.18 −0.766412 − 1.18853i 2.16919 + 2.33784i −0.825224 + 1.82181i 0.134837 − 0.437132i 1.11610 − 4.36991i 0.938839 − 2.47358i 2.79775 − 0.415453i −0.535880 + 7.15083i −0.622888 + 0.174765i
37.19 −0.709961 + 1.22309i 0.469951 + 0.506487i −0.991911 − 1.73670i −0.273835 + 0.887753i −0.953127 + 0.215208i 0.528093 − 2.59251i 2.82836 + 0.0197876i 0.188515 − 2.51556i −0.891391 − 0.965196i
37.20 −0.690255 + 1.23432i 2.22424 + 2.39716i −1.04710 − 1.70399i −0.266970 + 0.865496i −4.49415 + 1.09077i 2.06958 + 1.64828i 2.82604 − 0.116261i −0.574944 + 7.67209i −0.884022 − 0.926940i
See next 80 embeddings (of 648 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
49.g even 21 1 inner
392.z even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.2.z.a ✓ 648
8.b even 2 1 inner 392.2.z.a ✓ 648
49.g even 21 1 inner 392.2.z.a ✓ 648
392.z even 42 1 inner 392.2.z.a ✓ 648
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.z.a ✓ 648 1.a even 1 1 trivial
392.2.z.a ✓ 648 8.b even 2 1 inner
392.2.z.a ✓ 648 49.g even 21 1 inner
392.2.z.a ✓ 648 392.z even 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(392, [\chi])\).