Properties

Label 177.12.a.a
Level $177$
Weight $12$
Character orbit 177.a
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $26$
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: \(26\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} - 859751q^{10} + 579094q^{11} - 5606010q^{12} - 2018538q^{13} + 4157413q^{14} - 925344q^{15} + 20190274q^{16} - 13084493q^{17} - 4605822q^{18} + 9917231q^{19} + 10165633q^{20} + 24013017q^{21} - 89820518q^{22} - 63513223q^{23} + 28587735q^{24} + 218986852q^{25} - 77999532q^{26} - 373071582q^{27} - 444601862q^{28} + 81530981q^{29} + 208919493q^{30} - 408861231q^{31} - 26253128q^{32} - 140719842q^{33} - 508910076q^{34} - 75731421q^{35} + 1362260430q^{36} - 802381301q^{37} + 732704675q^{38} + 490504734q^{39} - 646130800q^{40} - 1354472849q^{41} - 1010251359q^{42} + 282952194q^{43} + 1846047996q^{44} + 224858592q^{45} + 9629305849q^{46} - 1196794197q^{47} - 4906236582q^{48} + 10889725683q^{49} - 6236232091q^{50} + 3179531799q^{51} - 1968200812q^{52} - 8276044236q^{53} + 1119214746q^{54} - 6672895076q^{55} + 2579741342q^{56} - 2409887133q^{57} - 9401656060q^{58} + 18588031774q^{59} - 2470248819q^{60} - 21181559029q^{61} - 6117706514q^{62} - 5835163131q^{63} + 42975855037q^{64} + 25680681860q^{65} + 21826385874q^{66} + 26234163394q^{67} + 19707344091q^{68} + 15433713189q^{69} + 129203099090q^{70} + 52088830406q^{71} - 6946819605q^{72} + 20943384867q^{73} + 41969200146q^{74} - 53213805036q^{75} + 223987219368q^{76} + 94604773153q^{77} + 18953886276q^{78} + 68965662774q^{79} + 218947784293q^{80} + 90656394426q^{81} + 11938614923q^{82} + 17947446393q^{83} + 108038252466q^{84} - 52849386709q^{85} + 384986147852q^{86} - 19812028383q^{87} - 49061112607q^{88} + 38570593981q^{89} - 50767436799q^{90} - 226268806999q^{91} - 79559686310q^{92} + 99353279133q^{93} - 16709400108q^{94} - 252795831501q^{95} + 6379510104q^{96} - 186894587836q^{97} - 252443311612q^{98} + 34194921606q^{99} + O(q^{100}) \)

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 −87.0020 −243.000 5521.36 −497.667 21141.5 10880.7 −302189. 59049.0 43298.0
1.2 −86.0142 −243.000 5350.44 12869.1 20901.4 −72810.3 −284057. 59049.0 −1.10693e6
1.3 −70.0902 −243.000 2864.64 3425.74 17031.9 −19416.7 −57238.6 59049.0 −240111.
1.4 −68.1728 −243.000 2599.53 −12352.0 16566.0 47034.4 −37599.4 59049.0 842068.
1.5 −66.9000 −243.000 2427.61 −5686.53 16256.7 26687.5 −25395.7 59049.0 380429.
1.6 −60.8905 −243.000 1659.66 −4960.99 14796.4 −87058.0 23646.5 59049.0 302077.
1.7 −49.4794 −243.000 400.209 6608.87 12023.5 −17933.1 81531.7 59049.0 −327003.
1.8 −43.6565 −243.000 −142.110 7124.43 10608.5 50312.1 95612.5 59049.0 −311028.
1.9 −42.1259 −243.000 −273.404 −6338.85 10236.6 −22948.3 97791.4 59049.0 267030.
1.10 −35.6107 −243.000 −779.875 −1849.70 8653.41 64152.0 100703. 59049.0 65869.2
1.11 −26.3367 −243.000 −1354.38 12487.3 6399.81 −48376.8 89607.3 59049.0 −328874.
1.12 −16.5456 −243.000 −1774.24 2844.36 4020.58 41460.7 63241.3 59049.0 −47061.6
1.13 −10.8994 −243.000 −1929.20 3570.42 2648.56 −66491.5 43349.2 59049.0 −38915.5
1.14 9.61747 −243.000 −1955.50 −8978.47 −2337.04 1935.43 −38503.6 59049.0 −86350.1
1.15 10.8639 −243.000 −1929.97 8237.50 −2639.94 85962.3 −43216.5 59049.0 89491.8
1.16 16.4601 −243.000 −1777.06 −11170.6 −3999.81 −64730.0 −62961.0 59049.0 −183870.
1.17 20.9819 −243.000 −1607.76 5799.05 −5098.61 22173.1 −76704.8 59049.0 121675.
1.18 27.2209 −243.000 −1307.02 −8750.00 −6614.68 −68225.0 −91326.7 59049.0 −238183.
1.19 38.1800 −243.000 −590.285 118.373 −9277.75 −24998.6 −100730. 59049.0 4519.50
1.20 40.1266 −243.000 −437.857 10858.4 −9750.76 −2158.10 −99749.0 59049.0 435712.
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.26
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.a 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.12.a.a 26 1.a even 1 1 trivial