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Properties

Label 667.2.h.a
Level $667$
Weight $2$
Character orbit 667.h
Analytic conductor $5.326$
Analytic rank $0$
Dimension $280$
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] = [667,2,Mod(59,667)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(667, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([14, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("667.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 667 = 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 667.h (of order \(11\), degree \(10\), 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: \(5.32602181482\)
Analytic rank: \(0\)
Dimension: \(280\)
Relative dimension: \(28\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 280 q - 2 q^{3} - 32 q^{4} - 8 q^{5} - 4 q^{6} - 10 q^{7} - 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 280 q - 2 q^{3} - 32 q^{4} - 8 q^{5} - 4 q^{6} - 10 q^{7} - 34 q^{9} + 29 q^{10} - 4 q^{11} - 34 q^{12} - 4 q^{13} - 16 q^{14} - 19 q^{15} - 40 q^{16} - 3 q^{17} + 2 q^{18} - 42 q^{19} - 22 q^{20} - 20 q^{21} + 82 q^{22} - 16 q^{23} - 44 q^{24} - 44 q^{25} - 22 q^{26} - 32 q^{27} - 4 q^{28} - 28 q^{29} - 44 q^{30} - 5 q^{31} - 60 q^{32} + 83 q^{33} + 62 q^{34} - 12 q^{35} - 120 q^{36} + 16 q^{37} - 34 q^{38} - 36 q^{39} - 22 q^{40} - 20 q^{41} + 50 q^{42} - 68 q^{43} + 14 q^{44} + 154 q^{45} + 16 q^{46} - 6 q^{47} - 118 q^{48} - 82 q^{49} - 142 q^{50} - 104 q^{51} + 74 q^{52} - 36 q^{53} + 46 q^{54} - 54 q^{55} + 184 q^{56} + 158 q^{57} + 14 q^{59} + 2 q^{60} - 24 q^{61} - 58 q^{62} - 104 q^{63} - 28 q^{64} + 3 q^{65} - 202 q^{66} + 30 q^{67} + 210 q^{68} + 293 q^{69} - 72 q^{70} + 7 q^{71} - 22 q^{72} - 72 q^{73} - 150 q^{74} + 55 q^{75} - 53 q^{76} - 58 q^{77} + 24 q^{78} + 68 q^{79} + 262 q^{80} + 110 q^{81} - 100 q^{82} - 64 q^{83} + 42 q^{84} - 42 q^{85} - 43 q^{86} - 2 q^{87} - 162 q^{88} - 112 q^{89} + 303 q^{90} + 572 q^{91} + 177 q^{92} - 164 q^{93} - 47 q^{94} - 85 q^{95} - 71 q^{96} - 85 q^{97} - 96 q^{98} - 115 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −1.73377 − 2.00088i 1.00284 + 0.644487i −0.712926 + 4.95851i 1.46921 − 3.21711i −0.449158 − 3.12396i −2.16980 − 0.637111i 6.70292 − 4.30770i −0.655917 − 1.43626i −8.98433 + 2.63804i
59.2 −1.68030 − 1.93917i 0.999107 + 0.642087i −0.652342 + 4.53714i −0.459105 + 1.00530i −0.433684 − 3.01634i −1.96459 − 0.576857i 5.57729 − 3.58431i −0.660306 − 1.44587i 2.72089 − 0.798924i
59.3 −1.46528 − 1.69103i −0.877186 − 0.563733i −0.427887 + 2.97602i −1.07610 + 2.35633i 0.332038 + 2.30937i −0.281406 − 0.0826282i 1.89481 − 1.21772i −0.794585 − 1.73990i 5.56140 − 1.63297i
59.4 −1.45585 − 1.68014i −2.57490 − 1.65479i −0.418743 + 2.91242i −1.48023 + 3.24126i 0.968392 + 6.73531i 2.28934 + 0.672211i 1.76245 − 1.13266i 2.64554 + 5.79293i 7.60076 − 2.23179i
59.5 −1.44817 − 1.67128i −1.44613 − 0.929373i −0.411343 + 2.86096i 0.525542 − 1.15078i 0.541007 + 3.76278i 2.08673 + 0.612719i 1.65642 − 1.06452i −0.0186792 − 0.0409017i −2.68434 + 0.788194i
59.6 −1.34129 − 1.54793i 2.73239 + 1.75600i −0.312403 + 2.17281i 1.37126 − 3.00265i −0.946762 − 6.58487i 1.19410 + 0.350621i 0.336268 − 0.216107i 3.13619 + 6.86729i −6.48715 + 1.90480i
59.7 −1.27384 − 1.47009i −1.47392 − 0.947230i −0.253868 + 1.76569i 1.52619 − 3.34190i 0.485024 + 3.37342i −4.44514 − 1.30521i −0.353720 + 0.227322i 0.0289469 + 0.0633850i −6.85702 + 2.01340i
59.8 −1.05550 − 1.21812i −0.162024 − 0.104126i −0.0850894 + 0.591810i 1.28649 − 2.81702i 0.0441787 + 0.307270i 2.24398 + 0.658892i −1.90116 + 1.22180i −1.23084 − 2.69515i −4.78936 + 1.40628i
59.9 −0.915048 − 1.05602i 2.24564 + 1.44319i 0.00676073 − 0.0470219i −0.363341 + 0.795607i −0.530835 − 3.69204i −0.833567 − 0.244757i −2.40684 + 1.54678i 1.71388 + 3.75288i 1.17265 − 0.344322i
59.10 −0.774959 − 0.894351i −1.66447 − 1.06969i 0.0853285 − 0.593473i −1.24672 + 2.72993i 0.333219 + 2.31759i −3.77699 − 1.10902i −2.58797 + 1.66319i 0.379988 + 0.832058i 3.40767 − 1.00058i
59.11 −0.599528 − 0.691893i 1.14225 + 0.734081i 0.165349 − 1.15002i −0.0418407 + 0.0916183i −0.176908 − 1.23042i −1.18047 − 0.346617i −2.43517 + 1.56499i −0.480379 − 1.05188i 0.0884747 − 0.0259785i
59.12 −0.241727 − 0.278967i −1.13970 − 0.732442i 0.265239 − 1.84478i 0.404841 − 0.886477i 0.0711689 + 0.494991i −2.42285 − 0.711414i −1.19981 + 0.771068i −0.483795 − 1.05936i −0.345159 + 0.101348i
59.13 −0.0551784 − 0.0636793i 1.26736 + 0.814485i 0.283619 − 1.97262i −0.506128 + 1.10827i −0.0180653 − 0.125647i 0.324786 + 0.0953658i −0.283032 + 0.181894i −0.303421 − 0.664400i 0.0985009 − 0.0289225i
59.14 0.0599302 + 0.0691631i 1.25978 + 0.809612i 0.283438 − 1.97135i −1.70763 + 3.73918i 0.0195036 + 0.135651i −4.55532 − 1.33756i 0.307308 − 0.197495i −0.314669 − 0.689030i −0.360952 + 0.105985i
59.15 0.0991171 + 0.114387i −0.327330 − 0.210362i 0.281369 − 1.95697i 0.699688 − 1.53210i −0.00838124 − 0.0582928i 3.78734 + 1.11206i 0.506398 − 0.325442i −1.18335 − 2.59118i 0.244604 − 0.0718222i
59.16 0.321350 + 0.370857i −2.04178 − 1.31217i 0.250360 − 1.74129i 0.882795 − 1.93305i −0.169497 − 1.17888i −1.89916 − 0.557644i 1.55185 − 0.997316i 1.20083 + 2.62945i 1.00057 − 0.293794i
59.17 0.446927 + 0.515781i 2.47295 + 1.58927i 0.218343 − 1.51861i 0.811027 − 1.77590i 0.285514 + 1.98579i −2.13197 − 0.626002i 2.02913 − 1.30404i 2.34347 + 5.13148i 1.27845 − 0.375386i
59.18 0.463843 + 0.535303i −2.67555 − 1.71947i 0.213230 − 1.48305i −0.455817 + 0.998101i −0.320596 − 2.22980i 3.73821 + 1.09764i 2.08452 − 1.33964i 2.95574 + 6.47217i −0.745714 + 0.218962i
59.19 0.596360 + 0.688236i 1.70141 + 1.09343i 0.166606 − 1.15877i −1.08646 + 2.37901i 0.262115 + 1.82305i 4.47682 + 1.31451i 2.42907 − 1.56107i 0.452966 + 0.991856i −2.28524 + 0.671007i
59.20 0.677230 + 0.781565i 0.376136 + 0.241728i 0.132426 − 0.921046i 1.46226 − 3.20191i 0.0658045 + 0.457680i 0.0514865 + 0.0151178i 2.54952 − 1.63848i −1.16320 − 2.54705i 3.49279 − 1.02557i
See next 80 embeddings (of 280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 667.2.h.a ✓ 280
23.c even 11 1 inner 667.2.h.a ✓ 280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
667.2.h.a ✓ 280 1.a even 1 1 trivial
667.2.h.a ✓ 280 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{280} + 44 T_{2}^{278} + 1082 T_{2}^{276} + 12 T_{2}^{275} + 19458 T_{2}^{274} + \cdots + 209440549587721 \) acting on \(S_{2}^{\mathrm{new}}(667, [\chi])\). Copy content Toggle raw display