Properties

Label 177.9.c.a
Level $177$
Weight $9$
Character orbit 177.c
Analytic conductor $72.106$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 177.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(72.1060139808\)
Analytic rank: \(0\)
Dimension: \(80\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80q - 10240q^{4} + 160q^{7} + 174960q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 80q - 10240q^{4} + 160q^{7} + 174960q^{9} - 22680q^{12} - 59616q^{15} + 1199848q^{16} - 10608q^{17} - 27516q^{19} - 146436q^{20} - 974696q^{22} + 5718040q^{25} - 797484q^{26} - 3133000q^{28} + 1725924q^{29} + 4318800q^{35} - 22394880q^{36} - 732180q^{41} + 22752084q^{46} + 8703936q^{48} + 55899176q^{49} - 10373832q^{51} - 39265944q^{53} - 11408040q^{57} - 33575112q^{59} - 18034488q^{60} + 13038600q^{62} + 349920q^{63} - 241654260q^{64} - 35711928q^{66} + 36772608q^{68} - 235272660q^{71} - 63050712q^{74} + 74363184q^{75} + 9454680q^{76} - 10865988q^{78} + 17252580q^{79} + 318203976q^{80} + 382637520q^{81} - 20743128q^{84} - 27245820q^{85} + 105666984q^{86} + 29437992q^{87} + 82079788q^{88} + 121215992q^{94} - 690837276q^{95} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1 31.6572i 46.7654 −746.176 733.576 1480.46i −1197.79 15517.6i 2187.00 23222.9i
58.2 31.4468i −46.7654 −732.900 37.7655 1470.62i 1285.94 14997.0i 2187.00 1187.60i
58.3 29.8918i 46.7654 −637.520 −247.630 1397.90i 4057.23 11404.3i 2187.00 7402.10i
58.4 29.4383i −46.7654 −610.615 −616.312 1376.69i −3233.41 10439.3i 2187.00 18143.2i
58.5 29.1592i 46.7654 −594.257 −890.286 1363.64i −1275.54 9863.29i 2187.00 25960.0i
58.6 28.5404i −46.7654 −558.553 1019.37 1334.70i 1132.75 8634.97i 2187.00 29093.1i
58.7 28.1432i −46.7654 −536.039 −406.960 1316.13i 2744.16 7881.18i 2187.00 11453.1i
58.8 27.0356i 46.7654 −474.926 21.6634 1264.33i −141.547 5918.81i 2187.00 585.684i
58.9 25.6810i 46.7654 −403.515 951.178 1200.98i 4182.78 3788.34i 2187.00 24427.2i
58.10 25.6289i 46.7654 −400.842 −239.734 1198.55i −1755.35 3712.14i 2187.00 6144.13i
58.11 25.1007i −46.7654 −374.043 458.555 1173.84i −1157.18 2962.95i 2187.00 11510.0i
58.12 24.0665i −46.7654 −323.199 −984.641 1125.48i −1144.76 1617.24i 2187.00 23696.9i
58.13 23.8187i 46.7654 −311.333 925.940 1113.89i −3176.58 1317.95i 2187.00 22054.7i
58.14 23.2103i −46.7654 −282.716 684.951 1085.44i −2372.31 620.092i 2187.00 15897.9i
58.15 20.7900i 46.7654 −176.226 −602.288 972.254i 1280.77 1658.51i 2187.00 12521.6i
58.16 20.7097i 46.7654 −172.891 −840.056 968.496i −4543.72 1721.16i 2187.00 17397.3i
58.17 20.4347i −46.7654 −161.576 −84.1426 955.636i 4222.23 1929.52i 2187.00 1719.43i
58.18 20.4085i −46.7654 −160.507 −745.364 954.411i −9.58116 1948.88i 2187.00 15211.8i
58.19 19.3849i 46.7654 −119.774 778.298 906.541i 1420.14 2640.73i 2187.00 15087.2i
58.20 19.2811i −46.7654 −115.759 525.267 901.686i 658.593 2703.99i 2187.00 10127.7i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.80
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.9.c.a 80
59.b odd 2 1 inner 177.9.c.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.9.c.a 80 1.a even 1 1 trivial
177.9.c.a 80 59.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(177, [\chi])\).