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Properties

Label 177.10.a.c
Level $177$
Weight $10$
Character orbit 177.a
Self dual yes
Analytic conductor $91.161$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [177,10,Mod(1,177)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(177, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(91.1613430010\)
Analytic rank: \(0\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 36 q^{2} - 1782 q^{3} + 5718 q^{4} + 808 q^{5} - 2916 q^{6} + 21249 q^{7} + 9435 q^{8} + 144342 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 36 q^{2} - 1782 q^{3} + 5718 q^{4} + 808 q^{5} - 2916 q^{6} + 21249 q^{7} + 9435 q^{8} + 144342 q^{9} + 68441 q^{10} - 68033 q^{11} - 463158 q^{12} + 283817 q^{13} + 80285 q^{14} - 65448 q^{15} + 1067674 q^{16} + 436893 q^{17} + 236196 q^{18} + 1207580 q^{19} + 4209677 q^{20} - 1721169 q^{21} + 5460442 q^{22} + 2421966 q^{23} - 764235 q^{24} + 7441842 q^{25} - 2736526 q^{26} - 11691702 q^{27} + 4095246 q^{28} - 2320594 q^{29} - 5543721 q^{30} - 3178024 q^{31} - 20786874 q^{32} + 5510673 q^{33} - 13809336 q^{34} - 2630800 q^{35} + 37515798 q^{36} + 3981807 q^{37} - 24156377 q^{38} - 22989177 q^{39} - 29544450 q^{40} - 885225 q^{41} - 6503085 q^{42} + 12360835 q^{43} - 117711882 q^{44} + 5301288 q^{45} + 161066949 q^{46} + 75901252 q^{47} - 86481594 q^{48} + 170907951 q^{49} - 61318927 q^{50} - 35388333 q^{51} - 100762 q^{52} - 34790192 q^{53} - 19131876 q^{54} + 151773316 q^{55} - 417630344 q^{56} - 97813980 q^{57} - 432929294 q^{58} + 266581942 q^{59} - 340983837 q^{60} - 290555332 q^{61} + 158267098 q^{62} + 139414689 q^{63} - 131794443 q^{64} - 650690086 q^{65} - 442295802 q^{66} + 86645184 q^{67} + 62738541 q^{68} - 196179246 q^{69} + 429714610 q^{70} - 36567631 q^{71} + 61903035 q^{72} + 907807228 q^{73} - 171827242 q^{74} - 602789202 q^{75} + 1744504396 q^{76} - 310688725 q^{77} + 221658606 q^{78} + 2508604687 q^{79} + 3509441927 q^{80} + 947027862 q^{81} + 1759214793 q^{82} + 2185672083 q^{83} - 331714926 q^{84} + 2868860198 q^{85} + 2397001564 q^{86} + 187968114 q^{87} + 7683735877 q^{88} + 1320145942 q^{89} + 449041401 q^{90} + 3894639897 q^{91} + 3505964640 q^{92} + 257419944 q^{93} + 5406355552 q^{94} + 3093659122 q^{95} + 1683736794 q^{96} + 3904552980 q^{97} + 6137683116 q^{98} - 446364513 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −43.0478 −81.0000 1341.12 1406.99 3486.87 9392.26 −35691.7 6561.00 −60567.9
1.2 −38.2133 −81.0000 948.259 −2363.53 3095.28 4562.96 −16670.9 6561.00 90318.2
1.3 −36.7798 −81.0000 840.750 2305.05 2979.16 −1509.72 −12091.4 6561.00 −84779.2
1.4 −36.1747 −81.0000 796.607 898.892 2930.15 −4674.69 −10295.5 6561.00 −32517.1
1.5 −27.1769 −81.0000 226.586 −1812.37 2201.33 −9882.92 7756.68 6561.00 49254.6
1.6 −20.5236 −81.0000 −90.7810 −622.721 1662.41 1394.83 12371.2 6561.00 12780.5
1.7 −17.2166 −81.0000 −215.588 470.596 1394.55 −5413.27 12526.6 6561.00 −8102.07
1.8 −13.7066 −81.0000 −324.128 −1287.82 1110.24 3218.96 11460.5 6561.00 17651.6
1.9 −9.19311 −81.0000 −427.487 −1961.85 744.642 11918.6 8636.80 6561.00 18035.5
1.10 −7.89550 −81.0000 −449.661 876.449 639.536 −1202.42 7592.80 6561.00 −6920.01
1.11 −3.04761 −81.0000 −502.712 −901.977 246.857 1315.27 3092.45 6561.00 2748.88
1.12 1.33644 −81.0000 −510.214 2090.38 −108.251 107.167 −1366.13 6561.00 2793.66
1.13 13.0004 −81.0000 −342.991 −5.38377 −1053.03 −6261.56 −11115.2 6561.00 −69.9909
1.14 17.2254 −81.0000 −215.286 27.4764 −1395.26 8666.20 −12527.8 6561.00 473.292
1.15 18.6113 −81.0000 −165.619 1492.75 −1507.52 11801.9 −12611.4 6561.00 27782.1
1.16 25.6716 −81.0000 147.031 −2019.49 −2079.40 7281.90 −9369.35 6561.00 −51843.5
1.17 26.7529 −81.0000 203.719 −302.298 −2166.99 −5739.31 −8247.43 6561.00 −8087.35
1.18 31.5568 −81.0000 483.832 1956.26 −2556.10 −4902.75 −888.901 6561.00 61733.3
1.19 34.5007 −81.0000 678.300 −2125.49 −2794.56 −11083.2 5737.48 6561.00 −73331.0
1.20 38.7622 −81.0000 990.507 −782.368 −3139.74 8972.92 18548.0 6561.00 −30326.3
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
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 177.10.a.c ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.10.a.c ✓ 22 1.a even 1 1 trivial