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Properties

Label 237.2.h.b
Level $237$
Weight $2$
Character orbit 237.h
Analytic conductor $1.892$
Analytic rank $0$
Dimension $48$
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] = [237,2,Mod(56,237)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(237, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("237.56");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 237 = 3 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 237.h (of order \(6\), 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: \(1.89245452790\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 20 q^{4} - 3 q^{6} - 12 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 20 q^{4} - 3 q^{6} - 12 q^{7} - 2 q^{9} - 4 q^{10} + 4 q^{13} + 40 q^{18} + 8 q^{19} - 22 q^{21} + 20 q^{22} - 6 q^{24} + 28 q^{25} - 96 q^{28} - 33 q^{30} + 8 q^{31} + 18 q^{34} - 5 q^{36} + 12 q^{37} - 21 q^{39} - 26 q^{40} - 31 q^{42} - 30 q^{43} - 10 q^{45} - 56 q^{46} + 54 q^{48} + 16 q^{49} - 20 q^{51} + 8 q^{52} - 87 q^{54} - 76 q^{55} - 3 q^{60} + 30 q^{63} + 68 q^{64} - 60 q^{66} + 32 q^{67} - 36 q^{70} + 59 q^{72} + 16 q^{73} - 3 q^{75} + 32 q^{76} + 44 q^{79} - 2 q^{81} - 60 q^{82} + 45 q^{84} + 36 q^{85} + 38 q^{87} + 10 q^{88} + 16 q^{90} - 116 q^{97} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
56.1 −2.25698 + 1.30307i −0.886418 + 1.48804i 2.39596 − 4.14992i 2.14014 + 1.23561i 0.0616114 − 4.51353i −2.68612 − 1.55083i 7.27611i −1.42852 − 2.63805i −6.44031
56.2 −2.06164 + 1.19029i −0.674961 − 1.59513i 1.83357 − 3.17583i −0.970832 − 0.560510i 3.29018 + 2.48518i −4.20545 − 2.42802i 3.96875i −2.08886 + 2.15330i 2.66867
56.3 −1.93376 + 1.11646i 1.30217 + 1.14209i 1.49295 − 2.58587i 0.247146 + 0.142690i −3.79317 − 0.754708i 1.43511 + 0.828563i 2.20145i 0.391278 + 2.97437i −0.637229
56.4 −1.84859 + 1.06728i −0.994418 + 1.41814i 1.27819 − 2.21389i −3.28782 − 1.89822i 0.324708 − 3.68289i 1.21010 + 0.698653i 1.18763i −1.02227 − 2.82046i 8.10378
56.5 −1.80444 + 1.04179i 0.630090 − 1.61338i 1.17067 − 2.02766i 3.21590 + 1.85670i 0.543846 + 3.56767i 1.01485 + 0.585926i 0.711204i −2.20597 − 2.03315i −7.73720
56.6 −1.37191 + 0.792075i 1.70059 − 0.328625i 0.254766 − 0.441268i −2.02484 − 1.16904i −2.07277 + 1.79784i −3.29894 − 1.90464i − 2.36112i 2.78401 − 1.11771i 3.70387
56.7 −1.22810 + 0.709042i −1.73019 + 0.0801841i 0.00548220 − 0.00949545i −1.09115 − 0.629977i 2.06799 − 1.32525i −0.488213 − 0.281870i − 2.82062i 2.98714 − 0.277468i 1.78672
56.8 −1.02892 + 0.594049i −1.02535 − 1.39594i −0.294212 + 0.509591i 0.0549157 + 0.0317056i 1.88426 + 0.827209i 0.837864 + 0.483741i − 3.07530i −0.897313 + 2.86266i −0.0753386
56.9 −0.714364 + 0.412438i 1.32515 − 1.11533i −0.659789 + 1.14279i −0.638817 − 0.368821i −0.486637 + 1.34330i 1.89389 + 1.09344i − 2.73824i 0.512069 − 2.95597i 0.608464
56.10 −0.663589 + 0.383123i −0.452380 + 1.67193i −0.706433 + 1.22358i 1.83069 + 1.05695i −0.340361 − 1.28279i 4.38254 + 2.53026i − 2.61510i −2.59070 − 1.51270i −1.61977
56.11 −0.526436 + 0.303938i 1.55139 + 0.770193i −0.815243 + 1.41204i 3.35843 + 1.93899i −1.05080 + 0.0660685i −1.80930 − 1.04460i − 2.20689i 1.81361 + 2.38973i −2.35734
56.12 −0.257162 + 0.148472i 0.511253 + 1.65488i −0.955912 + 1.65569i −2.90383 − 1.67653i −0.377179 − 0.349664i −1.28634 − 0.742667i − 1.16160i −2.47724 + 1.69212i 0.995673
56.13 0.257162 − 0.148472i −1.17754 − 1.27020i −0.955912 + 1.65569i 2.90383 + 1.67653i −0.491408 − 0.151814i −1.28634 − 0.742667i 1.16160i −0.226802 + 2.99141i 0.995673
56.14 0.526436 − 0.303938i 0.108687 − 1.72864i −0.815243 + 1.41204i −3.35843 − 1.93899i −0.468182 − 0.943052i −1.80930 − 1.04460i 2.20689i −2.97637 − 0.375762i −2.35734
56.15 0.663589 − 0.383123i −1.67412 − 0.444192i −0.706433 + 1.22358i −1.83069 − 1.05695i −1.28111 + 0.346635i 4.38254 + 2.53026i 2.61510i 2.60539 + 1.48727i −1.61977
56.16 0.714364 − 0.412438i 1.62848 − 0.589952i −0.659789 + 1.14279i 0.638817 + 0.368821i 0.920011 − 1.09309i 1.89389 + 1.09344i 2.73824i 2.30391 − 1.92145i 0.608464
56.17 1.02892 − 0.594049i 0.696247 + 1.58595i −0.294212 + 0.509591i −0.0549157 − 0.0317056i 1.65852 + 1.21822i 0.837864 + 0.483741i 3.07530i −2.03048 + 2.20843i −0.0753386
56.18 1.22810 − 0.709042i −0.934538 + 1.45830i 0.00548220 − 0.00949545i 1.09115 + 0.629977i −0.113708 + 2.45356i −0.488213 − 0.281870i 2.82062i −1.25328 − 2.72567i 1.78672
56.19 1.37191 − 0.792075i 1.13489 − 1.30844i 0.254766 − 0.441268i 2.02484 + 1.16904i 0.520591 − 2.69399i −3.29894 − 1.90464i 2.36112i −0.424039 − 2.96988i 3.70387
56.20 1.80444 − 1.04179i 1.71227 + 0.261014i 1.17067 − 2.02766i −3.21590 − 1.85670i 3.36161 − 1.31285i 1.01485 + 0.585926i − 0.711204i 2.86374 + 0.893855i −7.73720
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 56.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
79.d odd 6 1 inner
237.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 237.2.h.b ✓ 48
3.b odd 2 1 inner 237.2.h.b ✓ 48
79.d odd 6 1 inner 237.2.h.b ✓ 48
237.h even 6 1 inner 237.2.h.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
237.2.h.b ✓ 48 1.a even 1 1 trivial
237.2.h.b ✓ 48 3.b odd 2 1 inner
237.2.h.b ✓ 48 79.d odd 6 1 inner
237.2.h.b ✓ 48 237.h even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 34 T_{2}^{46} + 656 T_{2}^{44} - 8654 T_{2}^{42} + 86308 T_{2}^{40} - 677792 T_{2}^{38} + \cdots + 123904 \) acting on \(S_{2}^{\mathrm{new}}(237, [\chi])\). Copy content Toggle raw display