Properties

Label 177.11.c.a
Level $177$
Weight $11$
Character orbit 177.c
Analytic conductor $112.458$
Analytic rank $0$
Dimension $100$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 177.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(112.458233723\)
Analytic rank: \(0\)
Dimension: \(100\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 100q - 51200q^{4} + 18392q^{7} + 1968300q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 100q - 51200q^{4} + 18392q^{7} + 1968300q^{9} + 620136q^{12} + 662904q^{15} + 27925160q^{16} - 2053136q^{17} - 5169828q^{19} - 1076324q^{20} - 1829224q^{22} + 215378180q^{25} - 22082700q^{26} - 102921320q^{28} - 112503588q^{29} - 76491392q^{35} - 1007769600q^{36} + 473464516q^{41} + 1215405588q^{46} - 1676272320q^{48} + 1975297276q^{49} + 733970808q^{51} + 3267506728q^{53} + 591502824q^{57} + 508142200q^{59} + 1264196808q^{60} - 6538206968q^{62} + 362009736q^{63} - 10324137972q^{64} - 2764346616q^{66} + 9997685952q^{68} + 14908523204q^{71} + 4863508712q^{74} + 1890481680q^{75} + 2044437240q^{76} - 758396196q^{78} - 3599839500q^{79} - 23217941144q^{80} + 38742048900q^{81} - 13094894808q^{84} + 23360564412q^{85} + 12186923752q^{86} + 7965322272q^{87} + 32415437996q^{88} - 22098322280q^{94} + 7834510028q^{95} + O(q^{100}) \)

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 63.4704i −140.296 −3004.49 −5840.70 8904.65i 20957.7 125703.i 19683.0 370712.i
58.2 63.4607i −140.296 −3003.25 3537.19 8903.28i −15274.6 125605.i 19683.0 224473.i
58.3 59.8859i 140.296 −2562.32 3689.76 8401.76i 4188.49 92123.7i 19683.0 220965.i
58.4 59.5712i 140.296 −2524.73 −4763.75 8357.61i 8815.37 89400.5i 19683.0 283782.i
58.5 58.9981i −140.296 −2456.77 878.334 8277.20i −5578.77 84530.8i 19683.0 51820.0i
58.6 58.2736i 140.296 −2371.81 −453.456 8175.56i 16053.8 78542.0i 19683.0 26424.5i
58.7 56.3537i −140.296 −2151.73 3579.32 7906.20i 7609.79 63551.9i 19683.0 201708.i
58.8 55.7243i 140.296 −2081.20 5222.72 7817.90i −1720.05 58911.7i 19683.0 291033.i
58.9 55.6543i 140.296 −2073.41 −1651.80 7808.09i −6427.48 58404.0i 19683.0 91929.7i
58.10 54.8156i −140.296 −1980.75 −4481.58 7690.41i −27371.2 52444.5i 19683.0 245660.i
58.11 53.2620i −140.296 −1812.84 3919.09 7472.46i 33417.1 42015.4i 19683.0 208739.i
58.12 51.4146i −140.296 −1619.46 −2372.07 7213.27i 15485.2 30615.5i 19683.0 121959.i
58.13 50.8928i 140.296 −1566.07 2107.07 7140.06i −26451.4 27587.7i 19683.0 107235.i
58.14 50.8045i −140.296 −1557.09 737.455 7127.67i −21294.5 27083.5i 19683.0 37466.0i
58.15 48.0671i −140.296 −1286.44 −1212.73 6743.62i −4240.63 12614.8i 19683.0 58292.2i
58.16 47.5424i 140.296 −1236.28 −4974.64 6670.02i 12474.7 10092.4i 19683.0 236507.i
58.17 46.8090i 140.296 −1167.08 −1767.61 6567.12i 23434.9 6697.50i 19683.0 82740.0i
58.18 46.3046i 140.296 −1120.11 3858.39 6496.35i 22459.5 4450.45i 19683.0 178661.i
58.19 45.0342i 140.296 −1004.08 −5626.54 6318.12i −22190.3 897.122i 19683.0 253387.i
58.20 44.5197i 140.296 −958.005 −404.139 6245.94i −20369.6 2938.10i 19683.0 17992.1i
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.100
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 177.11.c.a 100
59.b odd 2 1 inner 177.11.c.a 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.11.c.a 100 1.a even 1 1 trivial
177.11.c.a 100 59.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(177, [\chi])\).