Properties

Label 177.12.a.c
Level $177$
Weight $12$
Character orbit 177.a
Self dual yes
Analytic conductor $135.997$
Analytic rank $0$
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: \(0\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 27q - 46q^{2} - 6561q^{3} + 26142q^{4} - 2442q^{5} + 11178q^{6} + 170093q^{7} - 19341q^{8} + 1594323q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 27q - 46q^{2} - 6561q^{3} + 26142q^{4} - 2442q^{5} + 11178q^{6} + 170093q^{7} - 19341q^{8} + 1594323q^{9} + 140249q^{10} + 256992q^{11} - 6352506q^{12} + 2436978q^{13} + 5233061q^{14} + 593406q^{15} + 28295194q^{16} - 4565351q^{17} - 2716254q^{18} + 33607699q^{19} - 19208463q^{20} - 41332599q^{21} + 79735622q^{22} + 43966161q^{23} + 4699863q^{24} + 406675819q^{25} + 42605404q^{26} - 387420489q^{27} + 635747682q^{28} - 107217773q^{29} - 34080507q^{30} + 570926627q^{31} + 526569236q^{32} - 62449056q^{33} + 129790240q^{34} + 134356079q^{35} + 1543658958q^{36} - 107121371q^{37} + 208302581q^{38} - 592185654q^{39} - 958762162q^{40} - 1935967559q^{41} - 1271633823q^{42} + 1725943824q^{43} + 196885756q^{44} - 144197658q^{45} - 13265966407q^{46} + 1801256065q^{47} - 6875732142q^{48} + 10484289252q^{49} - 10067682271q^{50} + 1109380293q^{51} - 882697024q^{52} - 6214238922q^{53} + 660049722q^{54} + 4460552366q^{55} + 28328012310q^{56} - 8166670857q^{57} + 12220116750q^{58} - 19302956073q^{59} + 4667656509q^{60} + 13167821039q^{61} - 1162130230q^{62} + 10043821557q^{63} - 5337557395q^{64} - 16849896006q^{65} - 19375756146q^{66} - 16856763152q^{67} - 36171071977q^{68} - 10683777123q^{69} - 120177261588q^{70} - 5198545690q^{71} - 1142066709q^{72} - 25075321857q^{73} - 182979651978q^{74} - 98822224017q^{75} - 3501293988q^{76} - 42787697701q^{77} - 10353113172q^{78} + 6850314702q^{79} - 261464428159q^{80} + 94143178827q^{81} - 148881516273q^{82} + 30908370899q^{83} - 154486686726q^{84} - 49419624969q^{85} - 220725475224q^{86} + 26053918839q^{87} - 53091280787q^{88} + 28988060121q^{89} + 8281563201q^{90} + 97120614047q^{91} + 45374597708q^{92} - 138735170361q^{93} + 208966927220q^{94} - 125253904969q^{95} - 127956324348q^{96} + 367722840268q^{97} - 48265639912q^{98} + 15175120608q^{99} + O(q^{100}) \)

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 −83.3973 −243.000 4907.11 −4959.60 20265.5 14303.6 −238442. 59049.0 413617.
1.2 −81.8884 −243.000 4657.72 −12268.5 19898.9 −65634.7 −213706. 59049.0 1.00465e6
1.3 −79.2286 −243.000 4229.17 9672.04 19252.5 84867.6 −172811. 59049.0 −766302.
1.4 −70.9666 −243.000 2988.26 3032.24 17244.9 −62069.7 −66727.2 59049.0 −215188.
1.5 −66.1318 −243.000 2325.41 6882.52 16070.0 7668.92 −18345.7 59049.0 −455153.
1.6 −64.1897 −243.000 2072.32 −6307.39 15598.1 84626.9 −1560.87 59049.0 404869.
1.7 −58.2273 −243.000 1342.41 11765.2 14149.2 24587.2 41084.4 59049.0 −685053.
1.8 −41.9561 −243.000 −287.686 −4799.69 10195.3 −40967.8 97996.3 59049.0 201376.
1.9 −41.5867 −243.000 −318.543 9657.75 10105.6 −11578.9 98416.8 59049.0 −401634.
1.10 −38.5064 −243.000 −565.255 −13229.8 9357.06 −8923.95 100627. 59049.0 509431.
1.11 −11.8207 −243.000 −1908.27 −8832.11 2872.42 19217.9 46765.7 59049.0 104401.
1.12 −10.6257 −243.000 −1935.10 −9082.46 2582.03 57618.4 42323.0 59049.0 96507.1
1.13 −8.53597 −243.000 −1975.14 −453.475 2074.24 −60294.7 34341.4 59049.0 3870.85
1.14 −5.14065 −243.000 −2021.57 3789.43 1249.18 −36302.1 20920.3 59049.0 −19480.2
1.15 −3.32183 −243.000 −2036.97 10388.6 807.204 36007.4 13569.5 59049.0 −34509.2
1.16 7.89171 −243.000 −1985.72 −1405.99 −1917.69 25105.4 −31832.9 59049.0 −11095.6
1.17 18.0234 −243.000 −1723.16 −3879.65 −4379.69 32120.2 −67969.1 59049.0 −69924.5
1.18 24.9877 −243.000 −1423.61 11346.7 −6072.02 −46106.7 −86747.8 59049.0 283528.
1.19 41.1468 −243.000 −354.944 −4337.67 −9998.66 64289.5 −98873.4 59049.0 −178481.
1.20 43.4861 −243.000 −156.959 1833.80 −10567.1 33856.7 −95885.1 59049.0 79745.0
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.c 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.12.a.c 27 1.a even 1 1 trivial