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Properties

Label 277.2.e.c
Level $277$
Weight $2$
Character orbit 277.e
Analytic conductor $2.212$
Analytic rank $0$
Dimension $32$
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] = [277,2,Mod(117,277)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(277, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("277.117");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 277 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 277.e (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: \(2.21185613599\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 3 q^{3} - 26 q^{4} - 12 q^{5} + 12 q^{6} - 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 3 q^{3} - 26 q^{4} - 12 q^{5} + 12 q^{6} - 39 q^{9} - 10 q^{10} - 3 q^{11} - 16 q^{12} + 10 q^{13} - 36 q^{14} + 50 q^{16} - 6 q^{17} - 39 q^{18} - 18 q^{19} - 33 q^{20} + 10 q^{21} + 31 q^{22} - 14 q^{23} + 27 q^{24} + 16 q^{25} + 60 q^{27} - 2 q^{28} + 6 q^{29} + 54 q^{30} + 12 q^{31} - 17 q^{34} - 3 q^{35} + 49 q^{36} + 16 q^{39} + 17 q^{40} - 24 q^{41} - 36 q^{43} + 24 q^{44} - 18 q^{45} - 63 q^{46} + 26 q^{47} + 49 q^{48} - 16 q^{49} - 45 q^{50} - 22 q^{52} - 33 q^{53} - 43 q^{55} + 72 q^{56} + 11 q^{57} - 27 q^{58} - 42 q^{59} + 19 q^{62} + 33 q^{63} - 80 q^{64} + 42 q^{65} + 8 q^{66} + 3 q^{67} - 12 q^{68} - 18 q^{69} + 35 q^{70} - 10 q^{71} + 72 q^{72} + 50 q^{74} + 49 q^{75} - 54 q^{76} + 87 q^{77} - 135 q^{78} - 32 q^{79} + 198 q^{80} - 32 q^{81} + 22 q^{83} - 176 q^{84} - 34 q^{85} + 6 q^{86} - 70 q^{87} - 103 q^{88} + 36 q^{89} - 31 q^{90} + 65 q^{91} + 29 q^{92} + 66 q^{93} + 39 q^{94} - 15 q^{95} - 12 q^{96} - 51 q^{97} + 15 q^{98} + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1 − 2.59988i 1.09681 + 1.89974i −4.75935 1.77631 + 1.02555i 4.93908 − 2.85158i 0.567669 − 0.983232i 7.17397i −0.906005 + 1.56925i 2.66631 − 4.61818i
117.2 − 2.08094i 0.618230 + 1.07081i −2.33031 −1.80837 − 1.04406i 2.22828 − 1.28650i 0.670834 − 1.16192i 0.687348i 0.735584 − 1.27407i −2.17263 + 3.76311i
117.3 − 2.04368i −1.48475 − 2.57166i −2.17663 −1.65041 − 0.952867i −5.25565 + 3.03435i 2.33777 − 4.04913i 0.360975i −2.90896 + 5.03846i −1.94736 + 3.37292i
117.4 − 1.42595i 0.885664 + 1.53402i −0.0333270 1.47815 + 0.853411i 2.18743 − 1.26291i −2.63576 + 4.56528i − 2.80437i −0.0688014 + 0.119168i 1.21692 − 2.10777i
117.5 − 1.40679i 0.230387 + 0.399042i 0.0209441 −3.09389 − 1.78626i 0.561368 − 0.324106i 0.0179612 − 0.0311097i − 2.84304i 1.39384 − 2.41421i −2.51289 + 4.35245i
117.6 − 1.38615i 1.51043 + 2.61614i 0.0785944 −0.392857 − 0.226816i 3.62636 − 2.09368i 1.72417 − 2.98635i − 2.88124i −3.06279 + 5.30491i −0.314401 + 0.544558i
117.7 − 0.800028i −1.13468 − 1.96533i 1.35995 −2.31339 − 1.33564i −1.57232 + 0.907779i −1.44644 + 2.50530i − 2.68806i −1.07502 + 1.86198i −1.06855 + 1.85078i
117.8 − 0.614642i −0.878689 − 1.52193i 1.62222 0.874680 + 0.504997i −0.935444 + 0.540079i 0.293564 − 0.508468i − 2.22637i −0.0441870 + 0.0765342i 0.310392 − 0.537615i
117.9 − 0.0293514i 0.782301 + 1.35498i 1.99914 1.79108 + 1.03408i 0.0397707 − 0.0229616i 0.347093 − 0.601183i − 0.117380i 0.276011 − 0.478065i 0.0303517 − 0.0525708i
117.10 0.253687i 1.50949 + 2.61450i 1.93564 −2.02559 − 1.16947i −0.663265 + 0.382936i −1.15961 + 2.00851i 0.998420i −3.05709 + 5.29504i 0.296680 − 0.513864i
117.11 0.456674i 0.294438 + 0.509981i 1.79145 −2.79688 − 1.61478i −0.232895 + 0.134462i 1.12546 − 1.94935i 1.73146i 1.32661 − 2.29776i 0.737428 − 1.27726i
117.12 0.773922i −1.60768 − 2.78458i 1.40104 3.15193 + 1.81977i 2.15505 − 1.24422i 0.838503 − 1.45233i 2.63214i −3.66925 + 6.35533i −1.40836 + 2.43934i
117.13 1.56173i −1.64359 − 2.84678i −0.438996 −2.62385 − 1.51488i 4.44590 − 2.56684i −0.443045 + 0.767376i 2.43786i −3.90277 + 6.75979i 2.36583 − 4.09775i
117.14 2.20242i 0.947819 + 1.64167i −2.85067 −2.27410 − 1.31295i −3.61565 + 2.08750i −1.37055 + 2.37386i − 1.87353i −0.296722 + 0.513937i 2.89168 − 5.00854i
117.15 2.66932i −1.14545 − 1.98398i −5.12525 2.13384 + 1.23197i 5.29586 − 3.05757i −2.40585 + 4.16705i − 8.34228i −1.12411 + 1.94702i −3.28853 + 5.69589i
117.16 2.73760i 1.51927 + 2.63145i −5.49445 1.77336 + 1.02385i −7.20386 + 4.15915i 1.53823 − 2.66429i − 9.56642i −3.11636 + 5.39769i −2.80289 + 4.85475i
161.1 − 2.73760i 1.51927 − 2.63145i −5.49445 1.77336 − 1.02385i −7.20386 − 4.15915i 1.53823 + 2.66429i 9.56642i −3.11636 − 5.39769i −2.80289 − 4.85475i
161.2 − 2.66932i −1.14545 + 1.98398i −5.12525 2.13384 − 1.23197i 5.29586 + 3.05757i −2.40585 − 4.16705i 8.34228i −1.12411 − 1.94702i −3.28853 − 5.69589i
161.3 − 2.20242i 0.947819 − 1.64167i −2.85067 −2.27410 + 1.31295i −3.61565 − 2.08750i −1.37055 − 2.37386i 1.87353i −0.296722 − 0.513937i 2.89168 + 5.00854i
161.4 − 1.56173i −1.64359 + 2.84678i −0.438996 −2.62385 + 1.51488i 4.44590 + 2.56684i −0.443045 − 0.767376i − 2.43786i −3.90277 − 6.75979i 2.36583 + 4.09775i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
277.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 277.2.e.c ✓ 32
277.e even 6 1 inner 277.2.e.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
277.2.e.c ✓ 32 1.a even 1 1 trivial
277.2.e.c ✓ 32 277.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 45 T_{2}^{30} + 897 T_{2}^{28} + 10456 T_{2}^{26} + 79278 T_{2}^{24} + 411592 T_{2}^{22} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(277, [\chi])\). Copy content Toggle raw display