Properties

Label 177.3.h.a
Level $177$
Weight $3$
Character orbit 177.h
Analytic conductor $4.823$
Analytic rank $0$
Dimension $1064$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 177.h (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.82290067918\)
Analytic rank: \(0\)
Dimension: \(1064\)
Relative dimension: \(38\) over \(\Q(\zeta_{58})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{58}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1064q - 29q^{3} + 18q^{4} - 21q^{6} - 46q^{7} - 49q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 1064q - 29q^{3} + 18q^{4} - 21q^{6} - 46q^{7} - 49q^{9} - 94q^{10} - 29q^{12} - 54q^{13} - 12q^{15} - 158q^{16} - 27q^{18} - 30q^{19} - 18q^{21} - 142q^{22} - 23q^{24} + 108q^{25} - 32q^{27} - 70q^{28} - 131q^{30} - 18q^{31} + 17q^{33} + 90q^{34} + 67q^{36} - 170q^{37} - 91q^{39} - 2q^{40} - 43q^{42} - 222q^{43} - 461q^{45} - 54q^{46} - 1645q^{48} - 300q^{49} - 893q^{51} - 66q^{52} - 859q^{54} + 170q^{55} - 27q^{57} - 36q^{58} + 510q^{60} - 70q^{61} + 610q^{63} - 106q^{64} + 1619q^{66} - 182q^{67} + 1487q^{69} - 206q^{70} + 2241q^{72} + 134q^{73} + 542q^{75} + 246q^{76} - 273q^{78} - 122q^{79} + 127q^{81} + 122q^{82} - 329q^{84} - 6q^{85} + 54q^{87} + 38q^{88} + 347q^{90} + 274q^{91} - 483q^{93} - 826q^{94} + 693q^{96} - 474q^{97} - 523q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.23800 + 3.67424i 2.23781 1.99805i −8.78304 6.67669i 2.27487 + 0.906393i 4.57091 + 10.6958i 0.939802 0.434799i 22.5687 15.3019i 1.01559 8.94251i −6.14659 + 7.23632i
5.2 −1.21095 + 3.59398i −2.84830 + 0.941900i −8.26594 6.28360i 1.37514 + 0.547907i 0.0639885 11.3773i 8.82700 4.08381i 20.0368 13.5853i 7.22565 5.36563i −3.63441 + 4.27875i
5.3 −1.15362 + 3.42383i 0.846084 + 2.87822i −7.20737 5.47890i −6.99239 2.78602i −10.8306 0.423531i 0.665705 0.307988i 15.1117 10.2460i −7.56829 + 4.87043i 17.6054 20.7267i
5.4 −1.00783 + 2.99114i −2.99774 0.116447i −4.74683 3.60844i −2.80678 1.11832i 3.36953 8.84930i −12.2974 + 5.68939i 5.12741 3.47647i 8.97288 + 0.698155i 6.17382 7.26839i
5.5 −0.999960 + 2.96778i −1.56678 2.55836i −4.62340 3.51462i 4.69371 + 1.87015i 9.15935 2.09161i −2.98209 + 1.37966i 4.68548 3.17683i −4.09038 + 8.01678i −10.2437 + 12.0598i
5.6 −0.964344 + 2.86207i −0.263179 + 2.98843i −4.07712 3.09934i 5.65262 + 2.25221i −8.29931 3.63511i −4.32408 + 2.00053i 2.80324 1.90065i −8.86147 1.57299i −11.8970 + 14.0063i
5.7 −0.912087 + 2.70698i 2.96240 0.473468i −3.31146 2.51731i −6.32194 2.51889i −1.42030 + 8.45101i −6.08665 + 2.81599i 0.377434 0.255907i 8.55166 2.80521i 12.5848 14.8159i
5.8 −0.892372 + 2.64847i 0.713097 2.91402i −3.03367 2.30614i −2.87136 1.14405i 7.08133 + 4.48900i 2.33765 1.08151i −0.437890 + 0.296897i −7.98299 4.15595i 5.59230 6.58377i
5.9 −0.869361 + 2.58017i 2.78047 + 1.12649i −2.71713 2.06551i 3.05628 + 1.21773i −5.32377 + 6.19477i 9.38908 4.34385i −1.32265 + 0.896782i 6.46204 + 6.26435i −5.79898 + 6.82709i
5.10 −0.792761 + 2.35283i −2.47356 1.69749i −1.72297 1.30976i −7.18639 2.86332i 5.95485 4.47417i 10.9013 5.04348i −3.77238 + 2.55774i 3.23704 + 8.39771i 12.4340 14.6384i
5.11 −0.561287 + 1.66584i 2.68858 1.33099i 0.724387 + 0.550665i 7.94437 + 3.16532i 0.708151 + 5.22582i −11.0072 + 5.09247i −7.14376 + 4.84359i 5.45694 7.15694i −9.73200 + 11.4574i
5.12 −0.551931 + 1.63807i −1.84525 + 2.36538i 0.805720 + 0.612492i −3.83005 1.52603i −2.85622 4.32819i −0.768529 + 0.355559i −7.17084 + 4.86195i −2.19007 8.72947i 4.61367 5.43163i
5.13 −0.528130 + 1.56743i −2.80203 + 1.07173i 1.00644 + 0.765078i 6.70916 + 2.67318i −0.200026 4.95802i 6.64296 3.07336i −7.20679 + 4.88632i 6.70279 6.00604i −7.73334 + 9.10439i
5.14 −0.386111 + 1.14594i 1.94700 + 2.28237i 2.02029 + 1.53578i −1.21804 0.485310i −3.36720 + 1.34988i −3.89579 + 1.80238i −6.54344 + 4.43656i −1.41841 + 8.88753i 1.02643 1.20841i
5.15 −0.322077 + 0.955892i −0.589478 2.94152i 2.37438 + 1.80495i 5.46847 + 2.17884i 3.00163 + 0.383919i 3.90866 1.80834i −5.82961 + 3.95258i −8.30503 + 3.46792i −3.84400 + 4.52551i
5.16 −0.225119 + 0.668129i 0.0215190 2.99992i 2.78865 + 2.11988i −5.48264 2.18448i 1.99949 + 0.689717i −11.1057 + 5.13804i −4.37833 + 2.96858i −8.99907 0.129111i 2.69377 3.17135i
5.17 −0.221328 + 0.656877i −2.72451 1.25582i 2.80187 + 2.12993i −2.12339 0.846038i 1.42792 1.51172i −1.48778 + 0.688319i −4.31412 + 2.92505i 5.84586 + 6.84295i 1.02571 1.20756i
5.18 −0.192085 + 0.570088i 2.47232 1.69930i 2.89627 + 2.20169i 1.33026 + 0.530023i 0.493857 + 1.73585i 7.94838 3.67731i −3.80316 + 2.57861i 3.22474 8.40244i −0.557682 + 0.656554i
5.19 −0.0718854 + 0.213348i 2.81983 1.02398i 3.14402 + 2.39002i −7.60675 3.03081i 0.0157589 + 0.675216i 4.35627 2.01543i −1.48128 + 1.00433i 6.90294 5.77490i 1.19343 1.40502i
5.20 0.0718854 0.213348i 0.416138 + 2.97100i 3.14402 + 2.39002i 7.60675 + 3.03081i 0.663771 + 0.124789i 4.35627 2.01543i 1.48128 1.00433i −8.65366 + 2.47269i 1.19343 1.40502i
See next 80 embeddings (of 1064 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 167.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
59.c even 29 1 inner
177.h odd 58 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.3.h.a 1064
3.b odd 2 1 inner 177.3.h.a 1064
59.c even 29 1 inner 177.3.h.a 1064
177.h odd 58 1 inner 177.3.h.a 1064
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.3.h.a 1064 1.a even 1 1 trivial
177.3.h.a 1064 3.b odd 2 1 inner
177.3.h.a 1064 59.c even 29 1 inner
177.3.h.a 1064 177.h odd 58 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(177, [\chi])\).