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Properties

Label 177.5.b.a
Level $177$
Weight $5$
Character orbit 177.b
Analytic conductor $18.296$
Analytic rank $0$
Dimension $78$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [177,5,Mod(119,177)]
 
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([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.119");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 177.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(18.2964834658\)
Analytic rank: \(0\)
Dimension: \(78\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 78 q - 612 q^{4} + 64 q^{6} + 76 q^{7} - 100 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 78 q - 612 q^{4} + 64 q^{6} + 76 q^{7} - 100 q^{9} - 156 q^{10} - 4 q^{13} + 13 q^{15} + 4948 q^{16} + 22 q^{18} + 812 q^{19} - 173 q^{21} - 1644 q^{22} - 678 q^{24} - 8238 q^{25} + 777 q^{27} - 3764 q^{28} + 1374 q^{30} + 4664 q^{31} - 1042 q^{33} + 3244 q^{34} + 3648 q^{36} - 3960 q^{37} - 7078 q^{39} - 1576 q^{40} + 4934 q^{42} - 1492 q^{43} - 2063 q^{45} - 2036 q^{46} - 2620 q^{48} + 24274 q^{49} + 7300 q^{51} + 8408 q^{52} - 14766 q^{54} + 9780 q^{55} + 6939 q^{57} - 3856 q^{58} + 4712 q^{60} - 212 q^{61} - 7438 q^{63} - 45760 q^{64} + 3048 q^{66} - 12972 q^{67} + 21672 q^{69} + 5828 q^{70} - 866 q^{72} - 5240 q^{73} - 20922 q^{75} + 12368 q^{76} - 16508 q^{78} - 14976 q^{79} + 25524 q^{81} - 14484 q^{82} + 9540 q^{84} + 11572 q^{85} + 5695 q^{87} + 62160 q^{88} - 31672 q^{90} + 8284 q^{91} - 9590 q^{93} - 10992 q^{94} + 34102 q^{96} - 55000 q^{97} - 14254 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
119.1 − 7.85862i 1.36439 + 8.89598i −45.7580 33.8616i 69.9102 − 10.7222i 13.8137 233.857i −77.2769 + 24.2751i 266.105
119.2 − 7.73459i −8.52815 − 2.87588i −43.8239 − 0.620129i −22.2437 + 65.9618i 71.8012 215.207i 64.4587 + 49.0518i −4.79645
119.3 − 7.49534i −6.13251 + 6.58728i −40.1801 − 24.9668i 49.3739 + 45.9653i −52.3496 181.238i −5.78454 − 80.7932i −187.134
119.4 − 7.42214i 2.94091 − 8.50594i −39.0882 22.5647i −63.1323 − 21.8278i 11.4871 171.364i −63.7021 − 50.0304i 167.478
119.5 − 7.37175i 8.47141 − 3.03895i −38.3427 − 40.5786i −22.4024 − 62.4491i 39.1148 164.705i 62.5296 − 51.4883i −299.136
119.6 − 7.08630i −1.58963 − 8.85850i −34.2156 − 8.65783i −62.7740 + 11.2646i −70.0243 129.081i −75.9462 + 28.1634i −61.3520
119.7 − 6.87418i 8.98827 + 0.459381i −31.2543 23.0718i 3.15787 − 61.7870i 23.0938 104.861i 80.5779 + 8.25808i 158.600
119.8 − 6.80672i 7.51401 + 4.95376i −30.3314 − 9.67205i 33.7188 − 51.1457i −60.1761 97.5501i 31.9206 + 74.4451i −65.8349
119.9 − 6.47467i −8.82514 − 1.76549i −25.9214 26.7513i −11.4310 + 57.1399i −48.5383 64.2378i 74.7661 + 31.1614i 173.206
119.10 − 6.42743i 2.05646 + 8.76190i −25.3118 − 39.0573i 56.3165 − 13.2178i 65.2885 59.8512i −72.5419 + 36.0371i −251.038
119.11 − 6.25305i −4.60229 − 7.73427i −23.1007 − 36.4421i −48.3628 + 28.7783i 14.4138 44.4009i −38.6379 + 71.1907i −227.875
119.12 − 5.88235i −8.00524 + 4.11293i −18.6020 − 15.2405i 24.1937 + 47.0896i 49.3247 15.3061i 47.1676 − 65.8500i −89.6502
119.13 − 5.74797i −1.10979 + 8.93131i −17.0392 1.93242i 51.3369 + 6.37903i −31.0983 5.97306i −78.5367 − 19.8237i 11.1075
119.14 − 5.57893i −6.24762 + 6.47821i −15.1244 38.3444i 36.1415 + 34.8550i 12.9947 − 4.88479i −2.93443 − 80.9468i 213.920
119.15 − 5.52059i 4.22607 + 7.94609i −14.4769 14.8774i 43.8671 − 23.3304i 83.7732 − 8.40835i −45.2807 + 67.1614i 82.1318
119.16 − 5.41049i 7.81854 − 4.45762i −13.2734 41.9044i −24.1179 − 42.3021i −64.0047 − 14.7525i 41.2592 − 69.7042i 226.723
119.17 − 5.25412i 4.74448 − 7.64787i −11.6058 − 7.04330i −40.1829 − 24.9281i 35.8552 − 23.0875i −35.9798 − 72.5703i −37.0064
119.18 − 5.04906i −3.85410 − 8.13301i −9.49305 33.9624i −41.0641 + 19.4596i 88.4932 − 32.8540i −51.2919 + 62.6909i 171.478
119.19 − 4.62880i 7.14239 − 5.47598i −5.42582 − 33.7032i −25.3472 − 33.0607i −90.1786 − 48.9458i 21.0273 − 78.2231i −156.006
119.20 − 4.27932i −6.98370 − 5.67697i −2.31259 − 6.15778i −24.2936 + 29.8855i −22.1503 − 58.5728i 16.5440 + 79.2925i −26.3511
See all 78 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 119.78
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.5.b.a ✓ 78
3.b odd 2 1 inner 177.5.b.a ✓ 78
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.5.b.a ✓ 78 1.a even 1 1 trivial
177.5.b.a ✓ 78 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(177, [\chi])\).