Properties

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

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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.

\(\operatorname{Tr}(f)(q) = \) \( 140q + q^{2} + 5q^{3} - q^{4} + 2q^{5} - q^{6} - 2q^{7} - 3q^{8} - 5q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 140q + q^{2} + 5q^{3} - q^{4} + 2q^{5} - q^{6} - 2q^{7} - 3q^{8} - 5q^{9} - 116q^{10} + 2q^{11} + q^{12} + 4q^{13} - 43q^{14} - 2q^{15} + 7q^{16} + q^{18} - 2q^{19} + 4q^{20} - 27q^{21} + 6q^{22} + 6q^{23} + 3q^{24} - 57q^{25} + 12q^{26} + 5q^{27} - 10q^{28} - 4q^{29} - 12q^{31} - 150q^{32} - 2q^{33} - 2q^{34} + 6q^{35} - q^{36} + 12q^{37} - 12q^{38} - 4q^{39} - 66q^{40} - 4q^{41} + 14q^{42} - 60q^{43} + 20q^{44} + 2q^{45} + 76q^{46} - 25q^{47} - 7q^{48} + 31q^{49} + 137q^{50} + 118q^{52} + 48q^{53} - q^{54} + 93q^{55} + 228q^{56} + 2q^{57} - 120q^{58} + 57q^{59} - 4q^{60} + 72q^{61} - 179q^{62} - 2q^{63} + 249q^{64} - 39q^{65} - 6q^{66} + 40q^{67} + 94q^{68} - 64q^{69} + 94q^{70} + 30q^{71} - 3q^{72} - 205q^{73} + 66q^{74} - q^{75} - 216q^{76} - 46q^{77} - 12q^{78} + 4q^{79} - 356q^{80} - 5q^{81} - 28q^{82} + 4q^{83} - 135q^{84} + 50q^{85} - 18q^{86} - 54q^{87} - 162q^{88} + 26q^{89} - 198q^{91} + 10q^{92} + 12q^{93} - 4q^{94} - 326q^{95} + 5q^{96} - 20q^{97} - 143q^{98} + 2q^{99} + O(q^{100}) \)

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 −2.53795 0.276019i −0.647386 + 0.762162i 4.41176 + 0.971102i −0.0920748 0.0699934i 1.85340 1.75564i 0.00524223 0.0131570i −6.09023 2.05204i −0.161782 0.986827i 0.214362 + 0.203054i
4.2 −0.756070 0.0822276i −0.647386 + 0.762162i −1.38836 0.305601i −0.128718 0.0978487i 0.552140 0.523015i 1.18304 2.96921i 2.46600 + 0.830893i −0.161782 0.986827i 0.0892739 + 0.0845647i
4.3 −0.233737 0.0254205i −0.647386 + 0.762162i −1.89925 0.418057i 1.40408 + 1.06736i 0.170693 0.161689i −0.853505 + 2.14214i 0.878915 + 0.296141i −0.161782 0.986827i −0.301054 0.285174i
4.4 1.81180 + 0.197046i −0.647386 + 0.762162i 1.29057 + 0.284075i 1.77642 + 1.35040i −1.32312 + 1.25332i −0.458287 + 1.15021i −1.17189 0.394857i −0.161782 0.986827i 2.95244 + 2.79670i
4.5 2.71009 + 0.294740i −0.647386 + 0.762162i 5.30448 + 1.16760i −2.37039 1.80192i −1.97912 + 1.87472i −0.223242 + 0.560295i 8.86474 + 2.98688i −0.161782 0.986827i −5.89286 5.58202i
7.1 −1.50851 2.22489i −0.0541389 + 0.998533i −1.93425 + 4.85460i 0.686980 0.317831i 2.30330 1.38585i −0.0286938 + 0.103346i 8.46834 1.86402i −0.994138 0.108119i −1.74346 1.04900i
7.2 −0.492250 0.726014i −0.0541389 + 0.998533i 0.455490 1.14319i −3.82749 + 1.77078i 0.751599 0.452222i −1.35341 + 4.87453i −2.76749 + 0.609171i −0.994138 0.108119i 3.16970 + 1.90714i
7.3 0.0649074 + 0.0957312i −0.0541389 + 0.998533i 0.735325 1.84553i 1.25197 0.579222i −0.0991048 + 0.0596294i 0.330724 1.19116i 0.450316 0.0991221i −0.994138 0.108119i 0.136712 + 0.0822566i
7.4 0.962498 + 1.41958i −0.0541389 + 0.998533i −0.348523 + 0.874727i 0.131897 0.0610220i −1.46960 + 0.884232i −0.721672 + 2.59923i 1.77283 0.390229i −0.994138 0.108119i 0.213576 + 0.128504i
7.5 1.53455 + 2.26329i −0.0541389 + 0.998533i −2.02735 + 5.08827i 1.27104 0.588045i −2.34304 + 1.40976i 1.26600 4.55972i −9.28622 + 2.04405i −0.994138 0.108119i 3.28138 + 1.97434i
16.1 −2.69139 0.592420i 0.161782 + 0.986827i 5.07747 + 2.34909i 0.982969 + 3.54033i 0.149197 2.75178i −0.252139 0.238838i −7.88603 5.99480i −0.947653 + 0.319302i −0.548189 10.1108i
16.2 −1.21818 0.268143i 0.161782 + 0.986827i −0.403079 0.186484i 0.0748466 + 0.269573i 0.0675300 1.24552i 0.579545 + 0.548975i 2.42703 + 1.84498i −0.947653 + 0.319302i −0.0188929 0.348459i
16.3 −0.0555609 0.0122299i 0.161782 + 0.986827i −1.81221 0.838419i −1.03314 3.72102i 0.00308001 0.0568075i −1.19345 1.13050i 0.181015 + 0.137604i −0.947653 + 0.319302i 0.0118943 + 0.219378i
16.4 0.821942 + 0.180923i 0.161782 + 0.986827i −1.17230 0.542362i 0.558556 + 2.01174i −0.0455643 + 0.840384i 2.86481 + 2.71369i −2.20545 1.67654i −0.947653 + 0.319302i 0.0951310 + 1.75459i
16.5 2.16657 + 0.476899i 0.161782 + 0.986827i 2.65145 + 1.22669i −0.194787 0.701560i −0.120104 + 2.21519i −1.18393 1.12147i 1.62739 + 1.23711i −0.947653 + 0.319302i −0.0874475 1.61287i
19.1 −1.07530 + 2.02823i 0.725995 0.687699i −1.83506 2.70652i 1.52678 0.336069i 0.614149 + 2.21197i 3.14104 2.38776i 2.89830 0.315209i 0.0541389 0.998533i −0.960116 + 3.45803i
19.2 −0.715981 + 1.35048i 0.725995 0.687699i −0.188804 0.278465i −3.13782 + 0.690687i 0.408928 + 1.47283i −2.11659 + 1.60899i −2.52792 + 0.274928i 0.0541389 0.998533i 1.31386 4.73210i
19.3 −0.166156 + 0.313404i 0.725995 0.687699i 1.05176 + 1.55123i 2.73415 0.601833i 0.0948990 + 0.341795i −2.80376 + 2.13136i −1.36621 + 0.148584i 0.0541389 0.998533i −0.265680 + 0.956892i
19.4 0.390110 0.735825i 0.725995 0.687699i 0.733121 + 1.08127i −0.577946 + 0.127216i −0.222808 0.802484i 0.804132 0.611285i 2.73754 0.297726i 0.0541389 0.998533i −0.131854 + 0.474895i
19.5 1.09892 2.07278i 0.725995 0.687699i −1.96641 2.90024i −2.10012 + 0.462272i −0.627639 2.26055i 0.717585 0.545494i −3.50786 + 0.381503i 0.0541389 0.998533i −1.34967 + 4.86109i
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:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(17\!\cdots\!30\)\( T_{2}^{109} + \)\(23\!\cdots\!65\)\( T_{2}^{108} - \)\(67\!\cdots\!43\)\( T_{2}^{107} + \)\(29\!\cdots\!78\)\( T_{2}^{106} - \)\(75\!\cdots\!58\)\( T_{2}^{105} + \)\(32\!\cdots\!24\)\( T_{2}^{104} - \)\(63\!\cdots\!52\)\( T_{2}^{103} + \)\(24\!\cdots\!38\)\( T_{2}^{102} - \)\(41\!\cdots\!34\)\( T_{2}^{101} + \)\(13\!\cdots\!37\)\( T_{2}^{100} - \)\(32\!\cdots\!69\)\( T_{2}^{99} + \)\(93\!\cdots\!84\)\( T_{2}^{98} - \)\(17\!\cdots\!43\)\( T_{2}^{97} + \)\(54\!\cdots\!71\)\( T_{2}^{96} - \)\(78\!\cdots\!63\)\( T_{2}^{95} + \)\(22\!\cdots\!72\)\( T_{2}^{94} - \)\(53\!\cdots\!93\)\( T_{2}^{93} + \)\(10\!\cdots\!14\)\( T_{2}^{92} - \)\(25\!\cdots\!94\)\( T_{2}^{91} + \)\(48\!\cdots\!37\)\( T_{2}^{90} - \)\(88\!\cdots\!77\)\( T_{2}^{89} + \)\(20\!\cdots\!42\)\( T_{2}^{88} - \)\(35\!\cdots\!17\)\( T_{2}^{87} + \)\(84\!\cdots\!60\)\( T_{2}^{86} - \)\(15\!\cdots\!29\)\( T_{2}^{85} + \)\(31\!\cdots\!52\)\( T_{2}^{84} - \)\(57\!\cdots\!22\)\( T_{2}^{83} + \)\(96\!\cdots\!80\)\( T_{2}^{82} - \)\(20\!\cdots\!00\)\( T_{2}^{81} + \)\(34\!\cdots\!54\)\( T_{2}^{80} - \)\(70\!\cdots\!42\)\( T_{2}^{79} + \)\(12\!\cdots\!58\)\( T_{2}^{78} - \)\(20\!\cdots\!62\)\( T_{2}^{77} + \)\(32\!\cdots\!46\)\( T_{2}^{76} - \)\(48\!\cdots\!09\)\( T_{2}^{75} + \)\(84\!\cdots\!47\)\( T_{2}^{74} - \)\(12\!\cdots\!19\)\( T_{2}^{73} + \)\(20\!\cdots\!82\)\( T_{2}^{72} - \)\(27\!\cdots\!04\)\( T_{2}^{71} + \)\(39\!\cdots\!68\)\( T_{2}^{70} - \)\(42\!\cdots\!91\)\( T_{2}^{69} + \)\(70\!\cdots\!46\)\( T_{2}^{68} - \)\(64\!\cdots\!82\)\( T_{2}^{67} + \)\(13\!\cdots\!54\)\( T_{2}^{66} - \)\(11\!\cdots\!83\)\( T_{2}^{65} + \)\(21\!\cdots\!46\)\( T_{2}^{64} - \)\(33\!\cdots\!68\)\( T_{2}^{63} + \)\(21\!\cdots\!94\)\( T_{2}^{62} + \)\(58\!\cdots\!79\)\( T_{2}^{61} + \)\(61\!\cdots\!93\)\( T_{2}^{60} - \)\(75\!\cdots\!02\)\( T_{2}^{59} + \)\(21\!\cdots\!30\)\( T_{2}^{58} - \)\(19\!\cdots\!18\)\( T_{2}^{57} + \)\(32\!\cdots\!12\)\( T_{2}^{56} + \)\(11\!\cdots\!95\)\( T_{2}^{55} - \)\(15\!\cdots\!57\)\( T_{2}^{54} + \)\(69\!\cdots\!21\)\( T_{2}^{53} + \)\(30\!\cdots\!74\)\( T_{2}^{52} - \)\(10\!\cdots\!79\)\( T_{2}^{51} + \)\(20\!\cdots\!47\)\( T_{2}^{50} + \)\(27\!\cdots\!87\)\( T_{2}^{49} - \)\(62\!\cdots\!85\)\( T_{2}^{48} + \)\(99\!\cdots\!66\)\( T_{2}^{47} + \)\(15\!\cdots\!64\)\( T_{2}^{46} - \)\(13\!\cdots\!84\)\( T_{2}^{45} + \)\(44\!\cdots\!55\)\( T_{2}^{44} - \)\(62\!\cdots\!36\)\( T_{2}^{43} + \)\(38\!\cdots\!13\)\( T_{2}^{42} - \)\(19\!\cdots\!70\)\( T_{2}^{41} - \)\(16\!\cdots\!30\)\( T_{2}^{40} + \)\(14\!\cdots\!64\)\( T_{2}^{39} + \)\(14\!\cdots\!45\)\( T_{2}^{38} + \)\(12\!\cdots\!95\)\( T_{2}^{37} - \)\(21\!\cdots\!89\)\( T_{2}^{36} + \)\(77\!\cdots\!48\)\( T_{2}^{35} + \)\(11\!\cdots\!98\)\( T_{2}^{34} - \)\(26\!\cdots\!55\)\( T_{2}^{33} - \)\(45\!\cdots\!07\)\( T_{2}^{32} + \)\(76\!\cdots\!14\)\( T_{2}^{31} + \)\(38\!\cdots\!71\)\( T_{2}^{30} + \)\(37\!\cdots\!79\)\( T_{2}^{29} - \)\(94\!\cdots\!75\)\( T_{2}^{28} - \)\(37\!\cdots\!62\)\( T_{2}^{27} + \)\(54\!\cdots\!37\)\( T_{2}^{26} + \)\(42\!\cdots\!42\)\( T_{2}^{25} + \)\(24\!\cdots\!46\)\( T_{2}^{24} - \)\(13\!\cdots\!47\)\( T_{2}^{23} - \)\(13\!\cdots\!77\)\( T_{2}^{22} + \)\(10\!\cdots\!55\)\( T_{2}^{21} + \)\(12\!\cdots\!01\)\( T_{2}^{20} + \)\(77\!\cdots\!44\)\( T_{2}^{19} + \)\(34\!\cdots\!50\)\( T_{2}^{18} + \)\(11\!\cdots\!28\)\( T_{2}^{17} + \)\(27\!\cdots\!79\)\( T_{2}^{16} + \)\(51\!\cdots\!06\)\( T_{2}^{15} + \)\(84\!\cdots\!92\)\( T_{2}^{14} + \)\(17\!\cdots\!19\)\( T_{2}^{13} + \)\(48\!\cdots\!20\)\( T_{2}^{12} + \)\(12\!\cdots\!29\)\( T_{2}^{11} + \)\(26\!\cdots\!19\)\( T_{2}^{10} + \)\(45\!\cdots\!66\)\( T_{2}^{9} + \)\(63\!\cdots\!17\)\( T_{2}^{8} + \)\(76\!\cdots\!75\)\( T_{2}^{7} + \)\(91\!\cdots\!76\)\( T_{2}^{6} + \)\(90\!\cdots\!65\)\( T_{2}^{5} + \)\(80\!\cdots\!60\)\( T_{2}^{4} + \)\(54\!\cdots\!72\)\( T_{2}^{3} + 302475667716 T_{2}^{2} + 14275010109 T_{2} + 335952241 \)">\(T_{2}^{140} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(177, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database