# Properties

 Label 177.2.e.a Level $177$ Weight $2$ Character orbit 177.e Analytic conductor $1.413$ Analytic rank $0$ Dimension $140$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$177 = 3 \cdot 59$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 177.e (of order $$29$$, degree $$28$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.41335211578$$ Analytic rank: $$0$$ Dimension: $$140$$ Relative dimension: $$5$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$140q - q^{2} - 5q^{3} - 9q^{4} - 2q^{5} - q^{6} - 2q^{7} - 9q^{8} - 5q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$140q - q^{2} - 5q^{3} - 9q^{4} - 2q^{5} - q^{6} - 2q^{7} - 9q^{8} - 5q^{9} + 88q^{10} - 14q^{11} - 9q^{12} - 12q^{13} - q^{14} - 2q^{15} - 41q^{16} - 16q^{17} - q^{18} - 10q^{19} - 32q^{20} + 27q^{21} - 26q^{22} - 22q^{23} - 9q^{24} + 27q^{25} - 56q^{26} - 5q^{27} - 50q^{28} - 24q^{29} - 28q^{30} - 24q^{31} + 106q^{32} - 14q^{33} - 54q^{34} - 70q^{35} - 9q^{36} - 28q^{37} - 80q^{38} - 12q^{39} - 50q^{40} - 40q^{41} - 30q^{42} + 4q^{43} - 104q^{44} - 2q^{45} - 28q^{46} + 31q^{47} - 41q^{48} - q^{49} + 39q^{50} - 16q^{51} + 62q^{52} + 4q^{53} - q^{54} + 5q^{55} + 96q^{56} - 10q^{57} + 128q^{58} - q^{59} - 32q^{60} - 16q^{61} + 223q^{62} - 2q^{63} + 97q^{64} + 121q^{65} - 26q^{66} - 12q^{67} + 10q^{68} + 36q^{69} - 2q^{70} - 22q^{71} - 9q^{72} + 179q^{73} - 38q^{74} - 31q^{75} + 112q^{76} - 62q^{77} - 56q^{78} - 84q^{79} + 204q^{80} - 5q^{81} - 152q^{82} - 88q^{83} + 95q^{84} - 118q^{85} - 118q^{86} + 34q^{87} + 18q^{88} - 86q^{89} - 28q^{90} + 78q^{91} - 174q^{92} - 24q^{93} - 164q^{94} + 218q^{95} - 39q^{96} - 84q^{97} + 129q^{98} - 14q^{99} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −2.29327 0.249408i 0.647386 0.762162i 3.24363 + 0.713978i −1.11360 0.846538i −1.67472 + 1.58638i 0.605271 1.51911i −2.88838 0.973208i −0.161782 0.986827i 2.34265 + 2.21908i
4.2 −1.38483 0.150610i 0.647386 0.762162i −0.0581619 0.0128024i 1.49641 + 1.13754i −1.01131 + 0.957965i −1.93379 + 4.85346i 2.71878 + 0.916062i −0.161782 0.986827i −1.90096 1.80068i
4.3 0.247015 + 0.0268645i 0.647386 0.762162i −1.89295 0.416669i −2.60143 1.97756i 0.180389 0.170874i 0.505225 1.26802i −0.927322 0.312451i −0.161782 0.986827i −0.589467 0.558372i
4.4 0.283755 + 0.0308602i 0.647386 0.762162i −1.87368 0.412427i 2.97558 + 2.26198i 0.207220 0.196289i 1.21093 3.03920i −1.05991 0.357126i −0.161782 0.986827i 0.774530 + 0.733674i
4.5 2.15319 + 0.234174i 0.647386 0.762162i 2.62816 + 0.578503i −1.34628 1.02342i 1.57243 1.48948i −0.734379 + 1.84315i 1.41845 + 0.477932i −0.161782 0.986827i −2.65915 2.51888i
7.1 −1.39179 2.05274i 0.0541389 0.998533i −1.53638 + 3.85601i −1.61683 + 0.748026i −2.12508 + 1.27862i −0.975578 + 3.51371i 5.20950 1.14670i −0.994138 0.108119i 3.78579 + 2.27784i
7.2 −0.408490 0.602477i 0.0541389 0.998533i 0.544161 1.36574i −2.21596 + 1.02521i −0.623709 + 0.375273i 0.762154 2.74503i −2.46688 + 0.543002i −0.994138 0.108119i 1.52286 + 0.916275i
7.3 −0.372741 0.549752i 0.0541389 0.998533i 0.576985 1.44812i 2.31671 1.07182i −0.569126 + 0.342432i −0.243434 + 0.876769i −2.30852 + 0.508144i −0.994138 0.108119i −1.45277 0.874104i
7.4 0.579343 + 0.854467i 0.0541389 0.998533i 0.345801 0.867894i 0.231383 0.107049i 0.884579 0.532233i −0.661330 + 2.38190i 2.95836 0.651184i −0.994138 0.108119i 0.225520 + 0.135691i
7.5 1.03249 + 1.52281i 0.0541389 0.998533i −0.512640 + 1.28663i 1.77030 0.819027i 1.57648 0.948534i 0.611139 2.20113i 1.10505 0.243241i −0.994138 0.108119i 3.07504 + 1.85019i
16.1 −2.19928 0.484098i −0.161782 0.986827i 2.78734 + 1.28956i 0.0503762 + 0.181439i −0.121917 + 2.24863i 2.06658 + 1.95757i −1.92037 1.45983i −0.947653 + 0.319302i −0.0229572 0.423421i
16.2 −1.29523 0.285102i −0.161782 0.986827i −0.218809 0.101232i 0.106655 + 0.384136i −0.0718011 + 1.32429i −2.67256 2.53159i 2.36617 + 1.79871i −0.947653 + 0.319302i −0.0286247 0.527952i
16.3 0.547213 + 0.120451i −0.161782 0.986827i −1.53022 0.707954i −0.382080 1.37613i 0.0303347 0.559491i −1.74470 1.65266i −1.64420 1.24989i −0.947653 + 0.319302i −0.0433236 0.799056i
16.4 1.94094 + 0.427233i −0.161782 0.986827i 1.76956 + 0.818686i 0.704369 + 2.53691i 0.107596 1.98449i 0.381927 + 0.361781i −0.0794753 0.0604155i −0.947653 + 0.319302i 0.283286 + 5.22491i
16.5 1.98298 + 0.436487i −0.161782 0.986827i 1.92655 + 0.891316i −0.867769 3.12542i 0.109927 2.02748i 2.78359 + 2.63676i 0.198401 + 0.150821i −0.947653 + 0.319302i −0.356563 6.57642i
19.1 −1.07727 + 2.03195i −0.725995 + 0.687699i −1.84595 2.72257i −3.23491 + 0.712057i −0.615278 2.21603i 1.31099 0.996592i 2.94796 0.320610i 0.0541389 0.998533i 2.03802 7.34027i
19.2 −0.454038 + 0.856407i −0.725995 + 0.687699i 0.595092 + 0.877695i 3.60206 0.792874i −0.259321 0.933989i 1.56865 1.19245i −2.94914 + 0.320738i 0.0541389 0.998533i −0.956451 + 3.44483i
19.3 0.0292574 0.0551852i −0.725995 + 0.687699i 1.12018 + 1.65215i −2.41343 + 0.531236i 0.0167101 + 0.0601845i −2.91820 + 2.21836i 0.248138 0.0269866i 0.0541389 0.998533i −0.0412942 + 0.148728i
19.4 0.655840 1.23705i −0.725995 + 0.687699i 0.0222185 + 0.0327698i 2.32817 0.512469i 0.374579 + 1.34911i −1.36308 + 1.03619i 2.83899 0.308758i 0.0541389 0.998533i 0.892959 3.21615i
19.5 1.31462 2.47964i −0.725995 + 0.687699i −3.29802 4.86421i 1.27307 0.280224i 0.750838 + 2.70428i 1.14406 0.869689i −10.8169 + 1.17641i 0.0541389 0.998533i 0.978753 3.52515i
See next 80 embeddings (of 140 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 175.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 177.2.e.a 140
3.b odd 2 1 531.2.i.c 140
59.c even 29 1 inner 177.2.e.a 140
177.h odd 58 1 531.2.i.c 140

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.e.a 140 1.a even 1 1 trivial
177.2.e.a 140 59.c even 29 1 inner
531.2.i.c 140 3.b odd 2 1
531.2.i.c 140 177.h odd 58 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$17\!\cdots\!76$$$$T_{2}^{109} +$$$$15\!\cdots\!73$$$$T_{2}^{108} -$$$$54\!\cdots\!81$$$$T_{2}^{107} +$$$$26\!\cdots\!18$$$$T_{2}^{106} +$$$$19\!\cdots\!24$$$$T_{2}^{105} -$$$$44\!\cdots\!52$$$$T_{2}^{104} +$$$$95\!\cdots\!64$$$$T_{2}^{103} -$$$$52\!\cdots\!48$$$$T_{2}^{102} +$$$$85\!\cdots\!70$$$$T_{2}^{101} +$$$$21\!\cdots\!05$$$$T_{2}^{100} +$$$$40\!\cdots\!45$$$$T_{2}^{99} +$$$$13\!\cdots\!06$$$$T_{2}^{98} -$$$$26\!\cdots\!97$$$$T_{2}^{97} +$$$$21\!\cdots\!47$$$$T_{2}^{96} +$$$$57\!\cdots\!05$$$$T_{2}^{95} +$$$$84\!\cdots\!26$$$$T_{2}^{94} +$$$$28\!\cdots\!67$$$$T_{2}^{93} +$$$$35\!\cdots\!42$$$$T_{2}^{92} -$$$$33\!\cdots\!80$$$$T_{2}^{91} +$$$$43\!\cdots\!99$$$$T_{2}^{90} -$$$$18\!\cdots\!41$$$$T_{2}^{89} +$$$$11\!\cdots\!44$$$$T_{2}^{88} +$$$$66\!\cdots\!53$$$$T_{2}^{87} -$$$$48\!\cdots\!26$$$$T_{2}^{86} -$$$$46\!\cdots\!87$$$$T_{2}^{85} +$$$$48\!\cdots\!56$$$$T_{2}^{84} +$$$$83\!\cdots\!66$$$$T_{2}^{83} +$$$$62\!\cdots\!14$$$$T_{2}^{82} -$$$$44\!\cdots\!26$$$$T_{2}^{81} +$$$$91\!\cdots\!50$$$$T_{2}^{80} +$$$$49\!\cdots\!08$$$$T_{2}^{79} +$$$$13\!\cdots\!30$$$$T_{2}^{78} -$$$$89\!\cdots\!48$$$$T_{2}^{77} +$$$$16\!\cdots\!22$$$$T_{2}^{76} +$$$$16\!\cdots\!85$$$$T_{2}^{75} +$$$$71\!\cdots\!71$$$$T_{2}^{74} +$$$$86\!\cdots\!21$$$$T_{2}^{73} +$$$$13\!\cdots\!80$$$$T_{2}^{72} +$$$$23\!\cdots\!50$$$$T_{2}^{71} +$$$$11\!\cdots\!94$$$$T_{2}^{70} +$$$$19\!\cdots\!59$$$$T_{2}^{69} +$$$$43\!\cdots\!96$$$$T_{2}^{68} +$$$$99\!\cdots\!74$$$$T_{2}^{67} +$$$$16\!\cdots\!48$$$$T_{2}^{66} +$$$$30\!\cdots\!27$$$$T_{2}^{65} +$$$$55\!\cdots\!80$$$$T_{2}^{64} +$$$$90\!\cdots\!16$$$$T_{2}^{63} +$$$$14\!\cdots\!36$$$$T_{2}^{62} +$$$$22\!\cdots\!43$$$$T_{2}^{61} +$$$$31\!\cdots\!29$$$$T_{2}^{60} +$$$$39\!\cdots\!90$$$$T_{2}^{59} +$$$$49\!\cdots\!38$$$$T_{2}^{58} +$$$$53\!\cdots\!38$$$$T_{2}^{57} +$$$$51\!\cdots\!10$$$$T_{2}^{56} +$$$$47\!\cdots\!65$$$$T_{2}^{55} +$$$$48\!\cdots\!67$$$$T_{2}^{54} +$$$$38\!\cdots\!07$$$$T_{2}^{53} +$$$$46\!\cdots\!46$$$$T_{2}^{52} +$$$$53\!\cdots\!75$$$$T_{2}^{51} +$$$$57\!\cdots\!11$$$$T_{2}^{50} +$$$$56\!\cdots\!93$$$$T_{2}^{49} +$$$$59\!\cdots\!87$$$$T_{2}^{48} +$$$$42\!\cdots\!58$$$$T_{2}^{47} +$$$$25\!\cdots\!24$$$$T_{2}^{46} +$$$$13\!\cdots\!04$$$$T_{2}^{45} +$$$$10\!\cdots\!55$$$$T_{2}^{44} -$$$$16\!\cdots\!60$$$$T_{2}^{43} +$$$$38\!\cdots\!51$$$$T_{2}^{42} -$$$$40\!\cdots\!88$$$$T_{2}^{41} +$$$$98\!\cdots\!00$$$$T_{2}^{40} -$$$$97\!\cdots\!56$$$$T_{2}^{39} +$$$$16\!\cdots\!63$$$$T_{2}^{38} -$$$$19\!\cdots\!51$$$$T_{2}^{37} +$$$$25\!\cdots\!35$$$$T_{2}^{36} -$$$$31\!\cdots\!78$$$$T_{2}^{35} +$$$$34\!\cdots\!38$$$$T_{2}^{34} -$$$$35\!\cdots\!25$$$$T_{2}^{33} +$$$$33\!\cdots\!11$$$$T_{2}^{32} -$$$$29\!\cdots\!36$$$$T_{2}^{31} +$$$$24\!\cdots\!65$$$$T_{2}^{30} -$$$$19\!\cdots\!67$$$$T_{2}^{29} +$$$$15\!\cdots\!93$$$$T_{2}^{28} -$$$$12\!\cdots\!74$$$$T_{2}^{27} +$$$$93\!\cdots\!61$$$$T_{2}^{26} -$$$$71\!\cdots\!64$$$$T_{2}^{25} +$$$$53\!\cdots\!00$$$$T_{2}^{24} -$$$$39\!\cdots\!03$$$$T_{2}^{23} +$$$$28\!\cdots\!93$$$$T_{2}^{22} -$$$$19\!\cdots\!77$$$$T_{2}^{21} +$$$$12\!\cdots\!47$$$$T_{2}^{20} -$$$$72\!\cdots\!14$$$$T_{2}^{19} +$$$$40\!\cdots\!84$$$$T_{2}^{18} -$$$$20\!\cdots\!56$$$$T_{2}^{17} +$$$$93\!\cdots\!05$$$$T_{2}^{16} -$$$$38\!\cdots\!68$$$$T_{2}^{15} +$$$$14\!\cdots\!80$$$$T_{2}^{14} -$$$$46\!\cdots\!77$$$$T_{2}^{13} +$$$$13\!\cdots\!12$$$$T_{2}^{12} -$$$$35\!\cdots\!97$$$$T_{2}^{11} +$$$$79\!\cdots\!45$$$$T_{2}^{10} -$$$$15\!\cdots\!98$$$$T_{2}^{9} +$$$$24\!\cdots\!23$$$$T_{2}^{8} -$$$$33\!\cdots\!67$$$$T_{2}^{7} +$$$$37\!\cdots\!26$$$$T_{2}^{6} -$$$$33\!\cdots\!63$$$$T_{2}^{5} +$$$$24\!\cdots\!88$$$$T_{2}^{4} -$$$$13\!\cdots\!02$$$$T_{2}^{3} +$$$$57\!\cdots\!02$$$$T_{2}^{2} -$$$$16\!\cdots\!73$$$$T_{2} + 23694752761$$">$$T_{2}^{140} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(177, [\chi])$$.