Properties

Label 177.7.c.a
Level $177$
Weight $7$
Character orbit 177.c
Analytic conductor $40.720$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 177.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.7195728007\)
Analytic rank: \(0\)
Dimension: \(60\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 60q - 1920q^{4} - 408q^{7} + 14580q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 60q - 1920q^{4} - 408q^{7} + 14580q^{9} - 1944q^{12} - 4536q^{15} + 56616q^{16} + 8480q^{17} + 11376q^{19} + 40796q^{20} - 8232q^{22} + 197940q^{25} + 147252q^{26} + 71640q^{28} + 63456q^{29} - 364432q^{35} - 466560q^{36} + 99632q^{41} - 470316q^{46} + 171072q^{48} + 1737420q^{49} + 60912q^{51} + 92240q^{53} + 186624q^{57} + 917264q^{59} + 1063368q^{60} - 115768q^{62} - 99144q^{63} - 1107444q^{64} + 1172232q^{66} - 4247232q^{68} + 1498048q^{71} + 1161448q^{74} - 1477440q^{75} - 1045320q^{76} - 1060452q^{78} - 90600q^{79} + 77096q^{80} + 3542940q^{81} - 2225880q^{84} - 693408q^{85} - 1567768q^{86} + 1821528q^{87} + 62892q^{88} + 5268696q^{94} + 296128q^{95} + O(q^{100}) \)

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 15.3575i 15.5885 −171.853 −142.367 239.400i 584.638 1656.35i 243.000 2186.40i
58.2 15.1313i 15.5885 −164.955 −92.6753 235.873i −595.836 1527.58i 243.000 1402.29i
58.3 14.9223i −15.5885 −158.675 237.797 232.615i −568.491 1412.76i 243.000 3548.48i
58.4 14.6202i −15.5885 −149.749 −98.4329 227.906i −154.279 1253.67i 243.000 1439.11i
58.5 13.8068i 15.5885 −126.629 76.4303 215.227i −5.71721 864.707i 243.000 1055.26i
58.6 13.7128i −15.5885 −124.040 78.2520 213.761i 452.461 823.310i 243.000 1073.05i
58.7 13.2670i 15.5885 −112.012 130.160 206.811i 304.309 636.976i 243.000 1726.83i
58.8 13.2493i −15.5885 −111.544 −224.166 206.536i 460.347 629.930i 243.000 2970.05i
58.9 12.5406i −15.5885 −93.2673 48.4743 195.489i −222.016 367.030i 243.000 607.898i
58.10 11.8923i 15.5885 −77.4273 54.6144 185.383i −559.219 159.682i 243.000 649.492i
58.11 11.3092i 15.5885 −63.8971 −168.463 176.292i 92.9299 1.16321i 243.000 1905.18i
58.12 10.4738i −15.5885 −45.7013 65.8108 163.271i 49.8494 191.658i 243.000 689.291i
58.13 10.1526i −15.5885 −39.0745 −104.301 158.263i −654.718 253.057i 243.000 1058.93i
58.14 9.91595i 15.5885 −34.3260 −70.2821 154.574i 315.675 294.246i 243.000 696.914i
58.15 8.95127i 15.5885 −16.1253 −71.7905 139.537i 97.3010 428.540i 243.000 642.617i
58.16 8.91504i 15.5885 −15.4779 175.637 138.972i −256.162 432.577i 243.000 1565.81i
58.17 8.09269i −15.5885 −1.49168 −181.057 126.153i 502.973 505.861i 243.000 1465.24i
58.18 7.10973i −15.5885 13.4517 −5.28151 110.830i −92.4639 550.661i 243.000 37.5501i
58.19 7.02990i −15.5885 14.5805 242.375 109.585i 243.138 552.413i 243.000 1703.87i
58.20 6.65433i 15.5885 19.7199 168.519 103.731i 568.245 557.100i 243.000 1121.38i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.60
Significant digits:
Format:

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 177.7.c.a 60
59.b odd 2 1 inner 177.7.c.a 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.7.c.a 60 1.a even 1 1 trivial
177.7.c.a 60 59.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(177, [\chi])\).