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Properties

Label 931.2.x.h
Level $931$
Weight $2$
Character orbit 931.x
Analytic conductor $7.434$
Analytic rank $0$
Dimension $66$
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] = [931,2,Mod(226,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([6, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.226");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.x (of order \(9\), degree \(6\), not 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: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 133)
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.

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 3 q^{2} + 3 q^{3} - 3 q^{4} + 3 q^{5} + 12 q^{6} - 6 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q - 3 q^{2} + 3 q^{3} - 3 q^{4} + 3 q^{5} + 12 q^{6} - 6 q^{8} - 3 q^{9} - 15 q^{10} + 54 q^{12} + 12 q^{13} - 9 q^{15} + 9 q^{16} - 18 q^{18} - 6 q^{19} - 12 q^{22} - 3 q^{23} + 36 q^{24} - 3 q^{25} + 18 q^{26} - 30 q^{27} + 6 q^{29} + 3 q^{30} - 39 q^{31} + 6 q^{32} + 39 q^{33} + 24 q^{34} + 51 q^{36} + 6 q^{37} + 72 q^{38} - 24 q^{39} - 57 q^{40} + 30 q^{41} + 12 q^{43} - 48 q^{44} + 3 q^{45} - 12 q^{46} + 21 q^{47} + 21 q^{48} + 27 q^{50} - 3 q^{51} + 9 q^{52} - 9 q^{53} - 93 q^{54} + 27 q^{55} + 6 q^{57} - 42 q^{58} - 6 q^{59} + 48 q^{60} - 18 q^{61} + 27 q^{62} + 24 q^{64} + 21 q^{65} - 15 q^{66} - 18 q^{67} + 15 q^{68} - 42 q^{69} + 30 q^{71} + 48 q^{72} + 78 q^{73} + 21 q^{74} - 57 q^{75} - 42 q^{76} - 51 q^{78} - 81 q^{79} + 3 q^{80} + 48 q^{81} - 117 q^{82} + 18 q^{83} - 24 q^{85} - 66 q^{86} - 18 q^{87} - 45 q^{88} + 3 q^{89} + 48 q^{90} + 12 q^{92} + 33 q^{93} - 72 q^{94} + 93 q^{95} - 69 q^{96} - 3 q^{97} - 156 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1 −1.84748 − 1.55022i −0.311187 + 1.76483i 0.662701 + 3.75836i −0.419137 + 2.37705i 3.31078 − 2.77808i 0 2.19024 − 3.79361i −0.198712 − 0.0723253i 4.45928 − 3.74178i
226.2 −1.65958 − 1.39255i −0.0295780 + 0.167745i 0.467705 + 2.65249i 0.497961 − 2.82408i 0.282681 − 0.237198i 0 0.751108 − 1.30096i 2.79181 + 1.01614i −4.75908 + 3.99334i
226.3 −1.32143 − 1.10881i 0.241842 − 1.37156i 0.169415 + 0.960798i 0.0482084 − 0.273403i −1.84037 + 1.54425i 0 −0.883527 + 1.53031i 0.996401 + 0.362660i −0.366855 + 0.307828i
226.4 −0.500702 − 0.420139i −0.399846 + 2.26764i −0.273111 − 1.54889i 0.345735 − 1.96076i 1.15293 − 0.967420i 0 −1.16762 + 2.02238i −2.16323 − 0.787352i −0.996902 + 0.836500i
226.5 −0.220197 − 0.184767i 0.170748 − 0.968357i −0.332949 − 1.88825i −0.571402 + 3.24058i −0.216518 + 0.181680i 0 −0.563017 + 0.975174i 1.91052 + 0.695371i 0.724573 − 0.607989i
226.6 −0.157206 − 0.131911i 0.521576 − 2.95801i −0.339983 − 1.92814i 0.268492 − 1.52269i −0.472188 + 0.396213i 0 −0.406113 + 0.703409i −5.65868 − 2.05959i −0.243069 + 0.203959i
226.7 0.544758 + 0.457106i −0.00744360 + 0.0422148i −0.259481 − 1.47159i −0.146796 + 0.832520i −0.0233516 + 0.0195943i 0 1.24245 − 2.15199i 2.81735 + 1.02543i −0.460518 + 0.386420i
226.8 0.883971 + 0.741739i −0.461457 + 2.61705i −0.116070 − 0.658264i 0.201700 − 1.14390i −2.34908 + 1.97111i 0 1.53960 − 2.66666i −3.81693 − 1.38925i 1.02677 − 0.861562i
226.9 1.09342 + 0.917489i 0.181471 − 1.02917i 0.00648632 + 0.0367857i 0.712385 − 4.04013i 1.14268 − 0.958822i 0 1.40070 − 2.42608i 1.79281 + 0.652529i 4.48571 − 3.76396i
226.10 1.58782 + 1.33234i 0.536631 − 3.04339i 0.398749 + 2.26142i −0.478609 + 2.71432i 4.90690 − 4.11738i 0 −0.307082 + 0.531882i −6.15516 − 2.24029i −4.37635 + 3.67219i
226.11 1.86266 + 1.56296i −0.187857 + 1.06539i 0.679371 + 3.85291i −0.305833 + 1.73446i −2.01508 + 1.69085i 0 −2.32496 + 4.02695i 1.71931 + 0.625778i −3.28056 + 2.75271i
557.1 −2.36821 − 0.861957i −0.195003 + 0.163627i 3.33334 + 2.79701i −0.210808 + 0.176889i 0.602848 − 0.219419i 0 −2.96296 − 5.13199i −0.509692 + 2.89061i 0.651706 − 0.237202i
557.2 −2.16980 − 0.789743i 2.30919 − 1.93764i 2.55226 + 2.14160i 2.42092 − 2.03140i −6.54072 + 2.38063i 0 −1.53753 − 2.66307i 1.05696 − 5.99433i −6.85720 + 2.49582i
557.3 −1.79446 − 0.653129i −1.74767 + 1.46647i 1.26141 + 1.05845i −0.573481 + 0.481208i 4.09392 − 1.49006i 0 0.337378 + 0.584356i 0.382876 − 2.17140i 1.34338 − 0.488950i
557.4 −1.22377 − 0.445416i 0.801776 − 0.672770i −0.232870 − 0.195401i −0.236563 + 0.198500i −1.28085 + 0.466192i 0 1.50025 + 2.59852i −0.330719 + 1.87560i 0.377915 − 0.137550i
557.5 −0.660908 − 0.240551i 1.78112 − 1.49454i −1.15315 − 0.967611i −2.13448 + 1.79104i −1.53667 + 0.559302i 0 1.23269 + 2.13509i 0.417802 − 2.36947i 1.84153 − 0.670263i
557.6 0.117851 + 0.0428943i −0.749394 + 0.628816i −1.52004 − 1.27546i −1.40837 + 1.18176i −0.115290 + 0.0419620i 0 −0.249843 − 0.432741i −0.354763 + 2.01196i −0.216669 + 0.0788612i
557.7 0.330333 + 0.120231i −2.06604 + 1.73361i −1.43742 − 1.20614i 0.373709 − 0.313579i −0.890915 + 0.324266i 0 −0.681346 − 1.18013i 0.742158 − 4.20899i 0.161151 − 0.0586540i
557.8 0.740065 + 0.269362i 1.61643 − 1.35635i −1.05695 − 0.886885i 2.46031 − 2.06445i 1.56161 − 0.568381i 0 −1.33088 − 2.30515i 0.252230 − 1.43046i 2.37688 − 0.865112i
557.9 1.62363 + 0.590953i 0.517804 − 0.434489i 0.754859 + 0.633402i −2.84083 + 2.38374i 1.09748 − 0.399451i 0 −0.876530 − 1.51819i −0.441604 + 2.50446i −6.02112 + 2.19151i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 226.11
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.x.h 66
7.b odd 2 1 133.2.w.a yes 66
7.c even 3 1 931.2.v.h 66
7.c even 3 1 931.2.w.e 66
7.d odd 6 1 133.2.u.a ✓ 66
7.d odd 6 1 931.2.w.f 66
19.e even 9 1 931.2.v.h 66
133.u even 9 1 931.2.w.e 66
133.w even 9 1 inner 931.2.x.h 66
133.x odd 18 1 931.2.w.f 66
133.y odd 18 1 133.2.u.a ✓ 66
133.z odd 18 1 133.2.w.a yes 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.u.a ✓ 66 7.d odd 6 1
133.2.u.a ✓ 66 133.y odd 18 1
133.2.w.a yes 66 7.b odd 2 1
133.2.w.a yes 66 133.z odd 18 1
931.2.v.h 66 7.c even 3 1
931.2.v.h 66 19.e even 9 1
931.2.w.e 66 7.c even 3 1
931.2.w.e 66 133.u even 9 1
931.2.w.f 66 7.d odd 6 1
931.2.w.f 66 133.x odd 18 1
931.2.x.h 66 1.a even 1 1 trivial
931.2.x.h 66 133.w even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\):

\( T_{2}^{66} + 3 T_{2}^{65} + 6 T_{2}^{64} + 11 T_{2}^{63} + 6 T_{2}^{62} - 18 T_{2}^{61} + 309 T_{2}^{60} + \cdots + 29241 \) Copy content Toggle raw display
\( T_{3}^{66} - 3 T_{3}^{65} + 6 T_{3}^{64} + 19 T_{3}^{63} - 78 T_{3}^{62} + 144 T_{3}^{61} + 1206 T_{3}^{60} + \cdots + 361 \) Copy content Toggle raw display