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Properties

Label 177.3.b.a
Level $177$
Weight $3$
Character orbit 177.b
Analytic conductor $4.823$
Analytic rank $0$
Dimension $38$
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] = [177,3,Mod(119,177)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(177, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.119");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 177.b (of order \(2\), degree \(1\), 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: \(4.82290067918\)
Analytic rank: \(0\)
Dimension: \(38\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 76 q^{4} - 8 q^{6} - 12 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q - 76 q^{4} - 8 q^{6} - 12 q^{7} + 20 q^{9} + 36 q^{10} - 4 q^{13} - 17 q^{15} + 100 q^{16} - 2 q^{18} - 28 q^{19} - 11 q^{21} + 84 q^{22} - 6 q^{24} - 166 q^{25} + 3 q^{27} + 12 q^{28} + 102 q^{30} - 40 q^{31} - 46 q^{33} - 148 q^{34} - 96 q^{36} + 112 q^{37} + 62 q^{39} - 56 q^{40} + 14 q^{42} + 164 q^{43} + 55 q^{45} - 4 q^{46} - 124 q^{48} + 242 q^{49} + 52 q^{51} + 8 q^{52} + 18 q^{54} - 228 q^{55} - 147 q^{57} - 80 q^{58} + 128 q^{60} + 12 q^{61} + 86 q^{63} + 48 q^{64} - 24 q^{66} + 124 q^{67} - 240 q^{69} + 148 q^{70} + 166 q^{72} - 192 q^{73} - 78 q^{75} - 304 q^{76} + 244 q^{78} + 64 q^{79} - 156 q^{81} - 180 q^{82} + 300 q^{84} - 52 q^{85} - 83 q^{87} - 96 q^{88} - 376 q^{90} - 332 q^{91} + 454 q^{93} + 768 q^{94} - 722 q^{96} + 416 q^{97} + 494 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
119.1 − 3.77988i 2.45428 − 1.72526i −10.2875 − 1.62741i −6.52128 − 9.27688i −7.78423 23.7661i 3.04695 − 8.46854i −6.15142
119.2 − 3.62092i −1.98632 + 2.24823i −9.11103 2.81281i 8.14066 + 7.19229i 8.99888 18.5066i −1.10909 − 8.93140i 10.1850
119.3 − 3.55815i −1.69290 − 2.47671i −8.66046 6.82324i −8.81252 + 6.02359i 0.148532 16.5826i −3.26820 + 8.38564i 24.2782
119.4 − 3.49731i −2.91161 − 0.722867i −8.23119 − 8.21287i −2.52809 + 10.1828i −3.56525 14.7978i 7.95493 + 4.20941i −28.7230
119.5 − 3.18759i 2.94308 + 0.581607i −6.16073 2.91835i 1.85393 − 9.38134i 12.1837 6.88752i 8.32347 + 3.42344i 9.30252
119.6 − 3.09739i 1.87116 + 2.34495i −5.59384 8.72041i 7.26322 − 5.79571i −11.2236 4.93675i −1.99754 + 8.77552i 27.0105
119.7 − 2.97620i −0.474775 + 2.96219i −4.85777 − 3.00594i 8.81608 + 1.41303i −3.67852 2.55289i −8.54918 − 2.81275i −8.94628
119.8 − 2.79604i 0.651071 − 2.92850i −3.81783 − 4.04341i −8.18820 − 1.82042i 6.95609 − 0.509352i −8.15221 − 3.81332i −11.3055
119.9 − 2.36527i 2.69227 + 1.32352i −1.59452 − 8.24482i 3.13049 − 6.36794i −0.755379 − 5.68962i 5.49658 + 7.12654i −19.5013
119.10 − 2.12813i −2.98150 + 0.332673i −0.528939 5.15735i 0.707972 + 6.34502i 2.69104 − 7.38687i 8.77866 − 1.98373i 10.9755
119.11 − 1.90736i −2.89185 + 0.798268i 0.361959 0.0951742i 1.52259 + 5.51580i −10.1724 − 8.31985i 7.72554 − 4.61693i 0.181532
119.12 − 1.89638i 0.180746 − 2.99455i 0.403751 2.43864i −5.67880 − 0.342762i −8.99368 − 8.35118i −8.93466 − 1.08250i 4.62459
119.13 − 1.50450i 2.07476 − 2.16688i 1.73647 9.35091i −3.26008 − 3.12148i 7.09053 − 8.63054i −0.390753 − 8.99151i 14.0685
119.14 − 1.50430i 2.93642 − 0.614364i 1.73708 1.04714i −0.924188 − 4.41725i −6.70636 − 8.63029i 8.24511 − 3.60806i 1.57522
119.15 − 1.30718i −2.48886 − 1.67498i 2.29128 − 3.80837i −2.18951 + 3.25339i 12.3524 − 8.22384i 3.38885 + 8.33761i −4.97823
119.16 − 1.19271i 0.625113 + 2.93415i 2.57744 0.192711i 3.49960 − 0.745581i 6.00162 − 7.84499i −8.21847 + 3.66835i 0.229849
119.17 − 0.472796i −1.33002 − 2.68906i 3.77646 − 7.63899i −1.27138 + 0.628829i −9.20830 − 3.67668i −5.46208 + 7.15302i −3.61168
119.18 − 0.201372i −2.12758 + 2.11504i 3.95945 − 6.21609i 0.425911 + 0.428435i −0.661651 − 1.60281i 0.0531837 − 8.99984i −1.25175
119.19 − 0.00789729i 2.45651 + 1.72208i 3.99994 4.78355i 0.0135997 − 0.0193998i 0.326558 − 0.0631778i 3.06891 + 8.46060i 0.0377771
119.20 0.00789729i 2.45651 − 1.72208i 3.99994 − 4.78355i 0.0135997 + 0.0193998i 0.326558 0.0631778i 3.06891 − 8.46060i 0.0377771
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 119.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.3.b.a ✓ 38
3.b odd 2 1 inner 177.3.b.a ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.3.b.a ✓ 38 1.a even 1 1 trivial
177.3.b.a ✓ 38 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(177, [\chi])\).