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 $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 177.b (of order \(2\), degree \(1\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 38q - 76q^{4} - 8q^{6} - 12q^{7} + 20q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 38q - 76q^{4} - 8q^{6} - 12q^{7} + 20q^{9} + 36q^{10} - 4q^{13} - 17q^{15} + 100q^{16} - 2q^{18} - 28q^{19} - 11q^{21} + 84q^{22} - 6q^{24} - 166q^{25} + 3q^{27} + 12q^{28} + 102q^{30} - 40q^{31} - 46q^{33} - 148q^{34} - 96q^{36} + 112q^{37} + 62q^{39} - 56q^{40} + 14q^{42} + 164q^{43} + 55q^{45} - 4q^{46} - 124q^{48} + 242q^{49} + 52q^{51} + 8q^{52} + 18q^{54} - 228q^{55} - 147q^{57} - 80q^{58} + 128q^{60} + 12q^{61} + 86q^{63} + 48q^{64} - 24q^{66} + 124q^{67} - 240q^{69} + 148q^{70} + 166q^{72} - 192q^{73} - 78q^{75} - 304q^{76} + 244q^{78} + 64q^{79} - 156q^{81} - 180q^{82} + 300q^{84} - 52q^{85} - 83q^{87} - 96q^{88} - 376q^{90} - 332q^{91} + 454q^{93} + 768q^{94} - 722q^{96} + 416q^{97} + 494q^{99} + O(q^{100}) \)

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.

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 \(S_{3}^{\mathrm{new}}(177, [\chi])\).