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Properties

Label 177.10.a.b
Level $177$
Weight $10$
Character orbit 177.a
Self dual yes
Analytic conductor $91.161$
Analytic rank $1$
Dimension $21$
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: \(1\)
Dimension: \(21\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 20 q^{2} - 1701 q^{3} + 4950 q^{4} + 2058 q^{5} - 1620 q^{6} - 17167 q^{7} - 2853 q^{8} + 137781 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 20 q^{2} - 1701 q^{3} + 4950 q^{4} + 2058 q^{5} - 1620 q^{6} - 17167 q^{7} - 2853 q^{8} + 137781 q^{9} - 31559 q^{10} - 38751 q^{11} - 400950 q^{12} - 58915 q^{13} + 3453 q^{14} - 166698 q^{15} + 1655714 q^{16} - 64233 q^{17} + 131220 q^{18} - 1937236 q^{19} - 1065507 q^{20} + 1390527 q^{21} - 5386882 q^{22} - 1838574 q^{23} + 231093 q^{24} + 4565755 q^{25} - 839702 q^{26} - 11160261 q^{27} - 4471034 q^{28} + 15658544 q^{29} + 2556279 q^{30} - 14282802 q^{31} - 2205286 q^{32} + 3138831 q^{33} + 19005532 q^{34} - 8633300 q^{35} + 32476950 q^{36} + 7531195 q^{37} + 26649773 q^{38} + 4772115 q^{39} + 17775672 q^{40} + 18338245 q^{41} - 279693 q^{42} - 22480305 q^{43} - 80230922 q^{44} + 13502538 q^{45} - 83894107 q^{46} - 110397260 q^{47} - 134112834 q^{48} + 130653638 q^{49} + 65575693 q^{50} + 5202873 q^{51} + 177908014 q^{52} + 145498338 q^{53} - 10628820 q^{54} + 86448944 q^{55} + 354387888 q^{56} + 156916116 q^{57} + 115508368 q^{58} - 254464581 q^{59} + 86306067 q^{60} + 287595506 q^{61} + 819899030 q^{62} - 112632687 q^{63} + 822446413 q^{64} + 77238206 q^{65} + 436337442 q^{66} - 392860610 q^{67} + 167325073 q^{68} + 148924494 q^{69} - 424902116 q^{70} - 248960491 q^{71} - 18718533 q^{72} - 758406074 q^{73} - 923266846 q^{74} - 369826155 q^{75} - 2312747568 q^{76} - 878126795 q^{77} + 68015862 q^{78} - 1925801029 q^{79} - 1898919861 q^{80} + 903981141 q^{81} - 3249102191 q^{82} - 1650336307 q^{83} + 362153754 q^{84} - 2342480762 q^{85} - 3609864952 q^{86} - 1268342064 q^{87} - 5987792887 q^{88} - 574997526 q^{89} - 207058599 q^{90} - 4481387117 q^{91} - 5317166770 q^{92} + 1156906962 q^{93} - 5360726568 q^{94} - 2789231462 q^{95} + 178628166 q^{96} - 4651540898 q^{97} - 5566652976 q^{98} - 254245311 q^{99}+O(q^{100}) \) 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 −44.6602 −81.0000 1482.53 −714.289 3617.48 −10082.7 −43344.2 6561.00 31900.3
1.2 −39.7720 −81.0000 1069.81 1152.50 3221.53 −1754.07 −22185.4 6561.00 −45837.2
1.3 −36.0452 −81.0000 787.255 −242.182 2919.66 7611.77 −9921.60 6561.00 8729.49
1.4 −28.8751 −81.0000 321.769 −2145.37 2338.88 2495.81 5492.93 6561.00 61947.6
1.5 −24.7118 −81.0000 98.6742 583.392 2001.66 10055.1 10214.0 6561.00 −14416.7
1.6 −23.3610 −81.0000 33.7344 477.585 1892.24 1570.96 11172.7 6561.00 −11156.8
1.7 −22.9203 −81.0000 13.3420 1892.34 1856.55 −10585.6 11429.4 6561.00 −43373.0
1.8 −12.5454 −81.0000 −354.612 2378.79 1016.18 4679.29 10872.0 6561.00 −29843.0
1.9 −4.43662 −81.0000 −492.316 339.476 359.366 −12342.0 4455.77 6561.00 −1506.13
1.10 −4.11702 −81.0000 −495.050 −2185.60 333.479 −4537.74 4146.05 6561.00 8998.14
1.11 2.64725 −81.0000 −504.992 1268.04 −214.427 7400.58 −2692.23 6561.00 3356.80
1.12 7.87274 −81.0000 −450.020 −1394.70 −637.692 −8519.00 −7573.74 6561.00 −10980.1
1.13 11.8993 −81.0000 −370.407 −236.774 −963.842 4185.56 −10500.0 6561.00 −2817.44
1.14 15.7099 −81.0000 −265.198 −2481.38 −1272.50 1632.53 −12209.7 6561.00 −38982.3
1.15 16.0816 −81.0000 −253.381 2048.36 −1302.61 −8880.78 −12308.6 6561.00 32940.9
1.16 24.6766 −81.0000 96.9337 1351.78 −1998.80 −4831.46 −10242.4 6561.00 33357.2
1.17 29.8017 −81.0000 376.142 2386.14 −2413.94 4536.93 −4048.81 6561.00 71111.0
1.18 33.2747 −81.0000 595.206 −1222.14 −2695.25 7343.10 2768.67 6561.00 −40666.2
1.19 34.8201 −81.0000 700.437 −913.273 −2820.43 −1305.26 6561.40 6561.00 −31800.2
1.20 41.8735 −81.0000 1241.39 372.530 −3391.75 −8199.03 30542.0 6561.00 15599.1
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
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.b ✓ 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.10.a.b ✓ 21 1.a even 1 1 trivial