Properties

Label 177.12.a.b
Level $177$
Weight $12$
Character orbit 177.a
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $27$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.996742959\)
Analytic rank: \(1\)
Dimension: \(27\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} - 383719q^{10} - 1816556q^{11} + 6352506q^{12} - 3951804q^{13} - 6207867q^{14} - 4176684q^{15} + 28295194q^{16} - 17723275q^{17} - 7558272q^{18} - 19573013q^{19} - 48468099q^{20} - 30758697q^{21} - 1729910q^{22} - 88593797q^{23} - 86458671q^{24} + 345714963q^{25} - 6676346q^{26} + 387420489q^{27} + 126954286q^{28} - 276632427q^{29} - 93243717q^{30} - 357680917q^{31} - 859842334q^{32} - 441423108q^{33} + 232730000q^{34} - 510315139q^{35} + 1543658958q^{36} - 660238257q^{37} - 2067286961q^{38} - 960288372q^{39} - 3388951110q^{40} - 1671147569q^{41} - 1508511681q^{42} - 1883107790q^{43} - 3895687630q^{44} - 1014934212q^{45} - 1720344243q^{46} - 5818572501q^{47} + 6875732142q^{48} - 18858180q^{49} - 21474519647q^{50} - 4306755825q^{51} - 42214560062q^{52} - 11444513368q^{53} - 1836660096q^{54} - 24401486484q^{55} - 50583585764q^{56} - 4756242159q^{57} - 45017395090q^{58} - 19302956073q^{59} - 11777748057q^{60} + 408637955q^{61} - 28543084070q^{62} - 7474363371q^{63} + 33067284293q^{64} - 21656714730q^{65} - 420368130q^{66} - 49803132690q^{67} - 16500749319q^{68} - 21528292671q^{69} - 45808890782q^{70} - 34127492216q^{71} - 21009457053q^{72} - 55734362153q^{73} - 40367816298q^{74} + 84008736009q^{75} - 14840406404q^{76} - 99723443615q^{77} - 1622352078q^{78} - 76484916442q^{79} + 93882788915q^{80} + 94143178827q^{81} + 52951239205q^{82} - 140433865655q^{83} + 30849891498q^{84} + 34329063335q^{85} + 175223869508q^{86} - 67221679761q^{87} + 268823645069q^{88} - 1191878597q^{89} - 22658223231q^{90} + 201632581559q^{91} - 206501888812q^{92} - 86916462831q^{93} + 319770144384q^{94} - 81387074885q^{95} - 208941687162q^{96} - 144896178730q^{97} + 135739195260q^{98} - 107265815244q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −88.8229 243.000 5841.51 11395.4 −21584.0 23008.9 −336951. 59049.0 −1.01218e6
1.2 −86.7335 243.000 5474.69 −11727.0 −21076.2 24236.6 −297209. 59049.0 1.01712e6
1.3 −78.8162 243.000 4163.99 6894.11 −19152.3 41129.0 −166774. 59049.0 −543367.
1.4 −71.6191 243.000 3081.30 −9710.01 −17403.4 37645.0 −74003.9 59049.0 695422.
1.5 −68.7593 243.000 2679.84 7324.59 −16708.5 −63928.8 −43445.2 59049.0 −503634.
1.6 −63.4822 243.000 1982.00 −1374.17 −15426.2 −16112.8 4190.11 59049.0 87235.4
1.7 −54.1540 243.000 884.652 −9497.19 −13159.4 −21091.6 62999.9 59049.0 514310.
1.8 −47.2991 243.000 189.204 4204.59 −11493.7 8235.98 87919.4 59049.0 −198873.
1.9 −45.9608 243.000 64.3958 −11937.5 −11168.5 −79874.8 91168.1 59049.0 548659.
1.10 −45.6492 243.000 35.8459 −2292.40 −11092.7 55379.8 91853.1 59049.0 104646.
1.11 −19.9033 243.000 −1651.86 11169.4 −4836.50 −27660.5 73639.4 59049.0 −222307.
1.12 −18.7578 243.000 −1696.14 −6326.10 −4558.15 −59668.2 70231.9 59049.0 118664.
1.13 −17.2441 243.000 −1750.64 7994.91 −4190.32 1325.62 65504.2 59049.0 −137865.
1.14 −4.91496 243.000 −2023.84 −1117.68 −1194.33 −21416.7 20012.9 59049.0 5493.36
1.15 −2.16187 243.000 −2043.33 654.400 −525.335 79533.7 8844.92 59049.0 −1414.73
1.16 10.6976 243.000 −1933.56 12608.0 2599.53 −75685.5 −42593.3 59049.0 134876.
1.17 13.5218 243.000 −1865.16 −12002.6 3285.79 −19689.8 −52912.9 59049.0 −162297.
1.18 32.2053 243.000 −1010.82 −5509.44 7825.90 −36737.1 −98510.2 59049.0 −177433.
1.19 33.0193 243.000 −957.729 −9842.24 8023.68 66020.7 −99246.9 59049.0 −324984.
1.20 35.0313 243.000 −820.805 1509.53 8512.61 260.533 −100498. 59049.0 52880.9
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(59\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.12.a.b 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.12.a.b 27 1.a even 1 1 trivial