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Properties

Label 177.4.e.b
Level $177$
Weight $4$
Character orbit 177.e
Analytic conductor $10.443$
Analytic rank $0$
Dimension $420$
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] = [177,4,Mod(4,177)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(177, base_ring=CyclotomicField(58))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 177.e (of order \(29\), degree \(28\), 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: \(10.4433380710\)
Analytic rank: \(0\)
Dimension: \(420\)
Relative dimension: \(15\) over \(\Q(\zeta_{29})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{29}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9} - 754 q^{10} + 64 q^{11} - 168 q^{12} + 90 q^{13} + 1024 q^{14} - 42 q^{15} - 172 q^{16} + 96 q^{17} + 18 q^{18} + 16 q^{19} - 178 q^{20} - 243 q^{21} - 604 q^{22} + 48 q^{23} + 198 q^{24} - 173 q^{25} + 156 q^{26} - 405 q^{27} + 84 q^{28} - 50 q^{29} + 174 q^{30} + 222 q^{31} - 45 q^{32} + 192 q^{33} - 64 q^{34} - 132 q^{35} - 504 q^{36} + 30 q^{37} - 108 q^{38} + 270 q^{39} + 6984 q^{40} + 400 q^{41} + 114 q^{42} + 1220 q^{43} + 1180 q^{44} - 126 q^{45} + 4938 q^{46} + 370 q^{47} - 516 q^{48} + 771 q^{49} + 1048 q^{50} + 288 q^{51} - 1964 q^{52} - 822 q^{53} + 54 q^{54} - 2342 q^{55} - 13824 q^{56} + 48 q^{57} - 13150 q^{58} - 3423 q^{59} - 534 q^{60} - 2182 q^{61} + 845 q^{62} + 54 q^{63} - 8018 q^{64} + 4140 q^{65} - 1812 q^{66} + 506 q^{67} + 4826 q^{68} - 3684 q^{69} + 6882 q^{70} + 5874 q^{71} + 594 q^{72} + 206 q^{73} + 11124 q^{74} - 867 q^{75} + 16940 q^{76} + 9940 q^{77} + 468 q^{78} - 1762 q^{79} - 28954 q^{80} - 1215 q^{81} + 3770 q^{82} - 1216 q^{83} - 8013 q^{84} - 420 q^{85} - 2256 q^{86} + 5940 q^{87} + 13458 q^{88} - 1580 q^{89} + 522 q^{90} - 12204 q^{91} + 4562 q^{92} + 666 q^{93} + 176 q^{94} - 28456 q^{95} + 2040 q^{96} + 2574 q^{97} + 50690 q^{98} + 576 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −5.48057 − 0.596048i 1.94216 − 2.28649i 21.8684 + 4.81360i 0.769914 + 0.585273i −12.0070 + 11.3736i 4.65584 − 11.6853i −75.1876 − 25.3337i −1.45604 − 8.88144i −3.87071 − 3.66653i
4.2 −4.38291 − 0.476671i 1.94216 − 2.28649i 11.1698 + 2.45865i −11.3321 − 8.61446i −9.60221 + 9.09570i −7.94992 + 19.9528i −14.3603 − 4.83855i −1.45604 − 8.88144i 45.5615 + 43.1581i
4.3 −3.97824 − 0.432660i 1.94216 − 2.28649i 7.82627 + 1.72269i 6.73681 + 5.12119i −8.71565 + 8.25591i −6.58976 + 16.5390i −0.0516573 − 0.0174054i −1.45604 − 8.88144i −24.5850 − 23.2881i
4.4 −3.05964 − 0.332755i 1.94216 − 2.28649i 1.43768 + 0.316458i 12.7861 + 9.71975i −6.70314 + 6.34955i 6.84669 − 17.1839i 19.0391 + 6.41502i −1.45604 − 8.88144i −35.8865 − 33.9935i
4.5 −2.96757 − 0.322743i 1.94216 − 2.28649i 0.889340 + 0.195759i −11.1726 − 8.49315i −6.50144 + 6.15849i 12.7555 − 32.0138i 20.0545 + 6.75714i −1.45604 − 8.88144i 30.4142 + 28.8099i
4.6 −1.68795 − 0.183576i 1.94216 − 2.28649i −4.99749 − 1.10003i 3.49616 + 2.65771i −3.69801 + 3.50294i 0.0781424 − 0.196123i 21.1058 + 7.11136i −1.45604 − 8.88144i −5.41345 − 5.12789i
4.7 −0.488701 − 0.0531494i 1.94216 − 2.28649i −7.57696 − 1.66782i −7.57411 − 5.75769i −1.07066 + 1.01418i −3.71388 + 9.32114i 7.34103 + 2.47348i −1.45604 − 8.88144i 3.39546 + 3.21635i
4.8 0.253308 + 0.0275489i 1.94216 − 2.28649i −7.74956 − 1.70581i 7.78313 + 5.91658i 0.554954 − 0.525681i −0.440801 + 1.10633i −3.84774 − 1.29645i −1.45604 − 8.88144i 1.80853 + 1.71313i
4.9 1.24705 + 0.135624i 1.94216 − 2.28649i −6.27623 − 1.38150i 16.4405 + 12.4978i 2.73207 − 2.58795i −11.4144 + 28.6480i −17.1493 − 5.77826i −1.45604 − 8.88144i 18.8071 + 17.8150i
4.10 2.00109 + 0.217632i 1.94216 − 2.28649i −3.85597 − 0.848763i −3.87099 − 2.94265i 4.38404 − 4.15279i 5.75041 − 14.4324i −22.7916 − 7.67938i −1.45604 − 8.88144i −7.10578 − 6.73095i
4.11 2.91566 + 0.317097i 1.94216 − 2.28649i 0.587538 + 0.129327i −2.37530 − 1.80566i 6.38771 − 6.05076i 4.97289 − 12.4810i −20.5625 − 6.92833i −1.45604 − 8.88144i −6.35299 − 6.01788i
4.12 3.13622 + 0.341085i 1.94216 − 2.28649i 1.90660 + 0.419674i −14.9247 − 11.3455i 6.87093 − 6.50849i −10.7338 + 26.9398i −18.0802 − 6.09194i −1.45604 − 8.88144i −42.9375 − 40.6726i
4.13 4.20930 + 0.457789i 1.94216 − 2.28649i 9.69564 + 2.13417i 11.3095 + 8.59723i 9.22185 − 8.73540i 5.73956 − 14.4052i 7.73503 + 2.60623i −1.45604 − 8.88144i 43.6692 + 41.3657i
4.14 5.06804 + 0.551183i 1.94216 − 2.28649i 17.5683 + 3.86707i −15.9287 − 12.1087i 11.1032 − 10.5175i 8.66226 − 21.7406i 48.2567 + 16.2596i −1.45604 − 8.88144i −74.0531 − 70.1468i
4.15 5.20320 + 0.565882i 1.94216 − 2.28649i 18.9401 + 4.16902i 3.73106 + 2.83628i 11.3993 − 10.7980i −7.57844 + 19.0204i 56.5105 + 19.0406i −1.45604 − 8.88144i 17.8084 + 16.8690i
7.1 −2.86839 − 4.23056i 0.162417 − 2.99560i −6.70886 + 16.8380i 19.1429 − 8.85647i −13.1389 + 7.90544i −6.91419 + 24.9026i 50.5433 − 11.1254i −8.94724 − 0.973071i −92.3773 − 55.5815i
7.2 −2.57971 − 3.80479i 0.162417 − 2.99560i −4.86039 + 12.1987i −1.87335 + 0.866706i −11.8166 + 7.10982i 0.688121 − 2.47839i 23.0364 − 5.07071i −8.94724 − 0.973071i 8.13034 + 4.89186i
7.3 −2.23273 − 3.29303i 0.162417 − 2.99560i −2.89786 + 7.27309i 3.07784 − 1.42396i −10.2272 + 6.15353i 3.58463 − 12.9107i −0.663884 + 0.146132i −8.94724 − 0.973071i −11.5611 − 6.95609i
7.4 −2.02689 − 2.98944i 0.162417 − 2.99560i −1.86736 + 4.68671i −15.6898 + 7.25888i −9.28436 + 5.58622i −2.35079 + 8.46678i −10.4232 + 2.29432i −8.94724 − 0.973071i 53.5015 + 32.1908i
7.5 −1.28887 − 1.90094i 0.162417 − 2.99560i 1.00873 − 2.53171i 8.34074 − 3.85884i −5.90378 + 3.55218i 3.61604 − 13.0238i −24.0566 + 5.29526i −8.94724 − 0.973071i −18.0855 − 10.8817i
See next 80 embeddings (of 420 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.c even 29 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.4.e.b ✓ 420
59.c even 29 1 inner 177.4.e.b ✓ 420
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.4.e.b ✓ 420 1.a even 1 1 trivial
177.4.e.b ✓ 420 59.c even 29 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{420} - 2 T_{2}^{419} + 90 T_{2}^{418} - 210 T_{2}^{417} + 4973 T_{2}^{416} + \cdots + 72\!\cdots\!36 \) acting on \(S_{4}^{\mathrm{new}}(177, [\chi])\). Copy content Toggle raw display