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Properties

Label 177.4.e.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 420 q - 2 q^{2} + 45 q^{3} - 64 q^{4} + 14 q^{5} + 6 q^{6} + 6 q^{7} + 18 q^{8} - 135 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 420 q - 2 q^{2} + 45 q^{3} - 64 q^{4} + 14 q^{5} + 6 q^{6} + 6 q^{7} + 18 q^{8} - 135 q^{9} + 854 q^{10} + 32 q^{11} + 192 q^{12} - 22 q^{13} - 884 q^{14} - 42 q^{15} - 268 q^{16} - 56 q^{17} - 18 q^{18} - 112 q^{19} - 250 q^{20} + 243 q^{21} + 108 q^{22} + 16 q^{23} - 54 q^{24} - 197 q^{25} - 328 q^{26} + 405 q^{27} + 236 q^{28} - 54 q^{29} - 126 q^{30} + 66 q^{31} + 857 q^{32} - 96 q^{33} + 112 q^{34} + 532 q^{35} - 576 q^{36} + 110 q^{37} + 56 q^{38} + 66 q^{39} - 6044 q^{40} + 304 q^{41} - 306 q^{42} - 324 q^{43} - 568 q^{44} + 126 q^{45} + 5842 q^{46} + 4082 q^{47} + 804 q^{48} + 739 q^{49} + 576 q^{50} + 168 q^{51} - 1372 q^{52} - 1498 q^{53} + 54 q^{54} - 4422 q^{55} - 12036 q^{56} + 336 q^{57} + 6298 q^{58} - 3305 q^{59} + 750 q^{60} - 1718 q^{61} - 4333 q^{62} + 54 q^{63} - 6250 q^{64} - 5884 q^{65} - 324 q^{66} + 382 q^{67} + 734 q^{68} + 3780 q^{69} + 8082 q^{70} + 8638 q^{71} + 162 q^{72} + 8894 q^{73} + 12428 q^{74} + 243 q^{75} - 15468 q^{76} - 9916 q^{77} + 984 q^{78} - 578 q^{79} + 24570 q^{80} - 1215 q^{81} - 1382 q^{82} - 32 q^{83} + 7557 q^{84} + 708 q^{85} + 2296 q^{86} - 5928 q^{87} - 17646 q^{88} - 2284 q^{89} + 378 q^{90} + 11124 q^{91} + 762 q^{92} - 198 q^{93} - 1232 q^{94} + 24184 q^{95} - 396 q^{96} - 658 q^{97} - 4356 q^{98} + 288 q^{99}+O(q^{100}) \) 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 −5.26239 − 0.572319i −1.94216 + 2.28649i 19.5522 + 4.30376i −16.6552 − 12.6610i 11.5290 − 10.9208i −1.92301 + 4.82639i −60.2975 − 20.3166i −1.45604 − 8.88144i 80.4000 + 76.1589i
4.2 −4.95061 − 0.538411i −1.94216 + 2.28649i 16.4057 + 3.61116i 10.7888 + 8.20142i 10.8459 − 10.2738i 11.1308 − 27.9361i −41.5207 − 13.9900i −1.45604 − 8.88144i −48.9953 − 46.4108i
4.3 −4.04316 − 0.439720i −1.94216 + 2.28649i 8.34080 + 1.83595i 2.28067 + 1.73372i 8.85787 − 8.39062i 2.30882 − 5.79471i −2.08305 − 0.701862i −1.45604 − 8.88144i −8.45878 − 8.01258i
4.4 −3.43911 − 0.374026i −1.94216 + 2.28649i 3.87463 + 0.852870i −6.75733 − 5.13679i 7.53451 − 7.13706i 2.82453 − 7.08903i 13.2201 + 4.45438i −1.45604 − 8.88144i 21.3179 + 20.1934i
4.5 −2.99643 − 0.325881i −1.94216 + 2.28649i 1.05942 + 0.233196i −3.11311 − 2.36653i 6.56466 − 6.21838i −9.33995 + 23.4415i 19.7521 + 6.65525i −1.45604 − 8.88144i 8.55701 + 8.10563i
4.6 −2.03313 − 0.221116i −1.94216 + 2.28649i −3.72826 − 0.820651i 16.1269 + 12.2594i 4.45423 − 4.21927i −2.36180 + 5.92767i 22.9030 + 7.71693i −1.45604 − 8.88144i −30.0773 − 28.4907i
4.7 −0.689793 − 0.0750196i −1.94216 + 2.28649i −7.34278 − 1.61627i −13.2529 − 10.0746i 1.51122 − 1.43150i 3.00216 − 7.53485i 10.2041 + 3.43815i −1.45604 − 8.88144i 8.38595 + 7.94360i
4.8 0.115030 + 0.0125102i −1.94216 + 2.28649i −7.79989 − 1.71689i −3.58691 − 2.72670i −0.252010 + 0.238717i −13.1064 + 32.8946i −1.75295 − 0.590637i −1.45604 − 8.88144i −0.378490 − 0.358525i
4.9 0.857263 + 0.0932330i −1.94216 + 2.28649i −7.08676 − 1.55991i 6.24712 + 4.74894i −1.87812 + 1.77905i 0.937924 − 2.35401i −12.4672 − 4.20069i −1.45604 − 8.88144i 4.91267 + 4.65353i
4.10 1.07661 + 0.117088i −1.94216 + 2.28649i −6.66759 − 1.46765i 6.87670 + 5.22753i −2.35866 + 2.23424i 7.63111 − 19.1526i −15.2167 − 5.12709i −1.45604 − 8.88144i 6.79142 + 6.43318i
4.11 2.45810 + 0.267335i −1.94216 + 2.28649i −1.84216 − 0.405491i 7.30295 + 5.55156i −5.38528 + 5.10121i −2.79914 + 7.02531i −23.1651 − 7.80525i −1.45604 − 8.88144i 16.4673 + 15.5986i
4.12 3.41158 + 0.371031i −1.94216 + 2.28649i 3.68824 + 0.811842i −7.61387 − 5.78791i −7.47419 + 7.07992i 7.80642 − 19.5926i −13.7350 − 4.62785i −1.45604 − 8.88144i −23.8278 − 22.5709i
4.13 3.96175 + 0.430866i −1.94216 + 2.28649i 7.69682 + 1.69420i −13.8169 − 10.5033i −8.67951 + 8.22167i −7.10268 + 17.8264i −0.449126 − 0.151328i −1.45604 − 8.88144i −50.2135 − 47.5647i
4.14 4.44507 + 0.483430i −1.94216 + 2.28649i 11.7120 + 2.57800i 8.85655 + 6.73258i −9.73839 + 9.22469i −9.70013 + 24.3455i 16.9165 + 5.69982i −1.45604 − 8.88144i 36.1133 + 34.2083i
4.15 5.10094 + 0.554760i −1.94216 + 2.28649i 17.8988 + 3.93983i 10.4418 + 7.93765i −11.1753 + 10.5858i 11.7316 − 29.4442i 50.2158 + 16.9197i −1.45604 − 8.88144i 48.8595 + 46.2821i
7.1 −2.99760 − 4.42113i −0.162417 + 2.99560i −7.59966 + 19.0737i −18.0173 + 8.33567i 13.7308 − 8.26154i −1.41944 + 5.11237i 65.3749 − 14.3901i −8.94724 − 0.973071i 90.8616 + 54.6696i
7.2 −2.77836 − 4.09778i −0.162417 + 2.99560i −6.11139 + 15.3384i 11.4034 − 5.27577i 12.7266 − 7.65732i −0.571992 + 2.06013i 41.1522 − 9.05829i −8.94724 − 0.973071i −53.3017 − 32.0706i
7.3 −2.23303 − 3.29348i −0.162417 + 2.99560i −2.89945 + 7.27707i −6.22504 + 2.88001i 10.2286 − 6.15436i −3.74407 + 13.4849i −0.647313 + 0.142484i −8.94724 − 0.973071i 23.3860 + 14.0709i
7.4 −1.93104 − 2.84807i −0.162417 + 2.99560i −1.42147 + 3.56762i −12.2161 + 5.65175i 8.84530 − 5.32204i 8.68646 − 31.2858i −13.9786 + 3.07691i −8.94724 − 0.973071i 39.6862 + 23.8784i
7.5 −1.59837 − 2.35742i −0.162417 + 2.99560i −0.0415309 + 0.104235i 14.5110 − 6.71353i 7.32148 − 4.40519i 0.937125 − 3.37522i −21.9407 + 4.82952i −8.94724 − 0.973071i −39.0206 − 23.4779i
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.a ✓ 420
59.c even 29 1 inner 177.4.e.a ✓ 420
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.4.e.a ✓ 420 1.a even 1 1 trivial
177.4.e.a ✓ 420 59.c even 29 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{420} + 2 T_{2}^{419} + 94 T_{2}^{418} + 190 T_{2}^{417} + 5357 T_{2}^{416} + \cdots + 25\!\cdots\!36 \) acting on \(S_{4}^{\mathrm{new}}(177, [\chi])\). Copy content Toggle raw display