Properties

Label 124.4.p.a
Level $124$
Weight $4$
Character orbit 124.p
Analytic conductor $7.316$
Analytic rank $0$
Dimension $368$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 124.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.31623684071\)
Analytic rank: \(0\)
Dimension: \(368\)
Relative dimension: \(46\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 368 q - 6 q^{2} + 6 q^{4} - 8 q^{5} - 9 q^{6} - 57 q^{8} + 360 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 368 q - 6 q^{2} + 6 q^{4} - 8 q^{5} - 9 q^{6} - 57 q^{8} + 360 q^{9} + 6 q^{10} - 283 q^{12} - 122 q^{13} + 120 q^{14} - 82 q^{16} - 14 q^{17} - 13 q^{18} + 157 q^{20} + 286 q^{21} + 99 q^{22} - 88 q^{24} - 3976 q^{25} - 3 q^{26} - 232 q^{28} - 20 q^{29} + 934 q^{32} - 144 q^{33} - 506 q^{34} + 155 q^{36} + 732 q^{37} + 38 q^{38} + 513 q^{40} - 18 q^{41} + 2209 q^{42} - 1433 q^{44} + 3738 q^{45} + 110 q^{46} + 3212 q^{48} - 1828 q^{49} + 4017 q^{50} + 3351 q^{52} + 10 q^{53} - 560 q^{54} - 214 q^{56} + 732 q^{57} - 1955 q^{58} - 9885 q^{60} - 3603 q^{62} + 399 q^{64} + 1236 q^{65} - 3808 q^{66} - 6702 q^{68} - 1128 q^{69} + 434 q^{70} + 10533 q^{72} - 986 q^{73} - 137 q^{74} + 5398 q^{76} - 20 q^{77} + 1059 q^{78} - 10 q^{80} + 2466 q^{81} + 2174 q^{82} - 1400 q^{84} + 1230 q^{85} - 3810 q^{86} - 1335 q^{88} + 1680 q^{89} - 781 q^{90} + 5770 q^{93} - 3968 q^{94} - 9770 q^{96} - 7784 q^{97} + 6746 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −2.82362 0.164797i −0.134736 + 1.28193i 7.94568 + 0.930647i −0.0229181 + 0.0396953i 0.591703 3.59749i 1.94572 + 9.15391i −22.2822 3.93722i 24.7848 + 5.26817i 0.0712537 0.108308i
3.2 −2.81278 + 0.297076i −0.949976 + 9.03842i 7.82349 1.67122i 6.63863 11.4984i −0.0130205 25.7053i −2.08859 9.82603i −21.5093 + 7.02495i −54.3805 11.5589i −15.2571 + 34.3148i
3.3 −2.80456 0.366691i 0.244891 2.32998i 7.73108 + 2.05681i −8.79653 + 15.2360i −1.54119 + 6.44476i −4.62413 21.7548i −20.9280 8.60336i 21.0412 + 4.47243i 30.2573 39.5047i
3.4 −2.80221 + 0.384244i 0.781206 7.43268i 7.70471 2.15346i −1.74753 + 3.02682i 0.666864 + 21.1281i 6.94622 + 32.6794i −20.7627 + 8.99494i −28.2244 5.99928i 3.73391 9.15324i
3.5 −2.70584 + 0.823655i 0.572364 5.44568i 6.64318 4.45737i 9.37792 16.2430i 2.93664 + 15.2066i −6.65861 31.3263i −14.3041 + 17.5326i −2.91787 0.620213i −11.9965 + 51.6753i
3.6 −2.62585 + 1.05116i −0.804966 + 7.65874i 5.79014 5.52035i −9.96666 + 17.2628i −5.93681 20.9568i 3.05831 + 14.3882i −9.40128 + 20.5819i −31.5984 6.71644i 8.02507 55.8059i
3.7 −2.50811 1.30742i 0.215453 2.04990i 4.58128 + 6.55834i 10.3394 17.9083i −3.22047 + 4.85970i 3.46899 + 16.3203i −2.91584 22.4388i 22.2543 + 4.73030i −49.3462 + 31.3982i
3.8 −2.46001 1.39582i −0.479513 + 4.56226i 4.10335 + 6.86750i 0.676684 1.17205i 7.54773 10.5539i −5.59922 26.3423i −0.508472 22.6217i 5.82568 + 1.23829i −3.30063 + 1.93873i
3.9 −2.41499 1.47235i 0.915780 8.71306i 3.66440 + 7.11141i 0.646337 1.11949i −15.0402 + 19.6937i −1.72916 8.13508i 1.62096 22.5693i −48.6688 10.3449i −3.20918 + 1.75193i
3.10 −2.30351 + 1.64129i −0.315000 + 2.99703i 2.61231 7.56147i 0.474237 0.821402i −4.19340 7.42069i −1.56190 7.34815i 6.39311 + 21.7055i 17.5270 + 3.72549i 0.255753 + 2.67047i
3.11 −2.27279 + 1.68358i 0.315000 2.99703i 2.33112 7.65284i 0.474237 0.821402i 4.32981 + 7.34193i 1.56190 + 7.34815i 7.58603 + 21.3179i 17.5270 + 3.72549i 0.305057 + 2.66529i
3.12 −2.25611 1.70586i −0.718973 + 6.84057i 2.18011 + 7.69722i −2.50613 + 4.34075i 13.2911 14.2066i 4.92655 + 23.1776i 8.21177 21.0848i −19.8664 4.22274i 13.0588 5.51813i
3.13 −1.81114 + 2.17250i 0.804966 7.65874i −1.43954 7.86942i −9.96666 + 17.2628i 15.1807 + 15.6198i −3.05831 14.3882i 19.7036 + 11.1252i −31.5984 6.71644i −19.4524 52.9179i
3.14 −1.61949 + 2.31889i −0.572364 + 5.44568i −2.75448 7.51085i 9.37792 16.2430i −11.7010 10.1465i 6.65861 + 31.3263i 21.8777 + 5.77646i −2.91787 0.620213i 22.4783 + 48.0518i
3.15 −1.48223 2.40894i 0.430178 4.09287i −3.60602 + 7.14119i −4.68227 + 8.10993i −10.4971 + 5.03029i 1.43654 + 6.75841i 22.5477 1.89817i 9.84342 + 2.09228i 26.4765 0.741420i
3.16 −1.23137 + 2.54632i −0.781206 + 7.43268i −4.96747 6.27090i −1.74753 + 3.02682i −17.9640 11.1416i −6.94622 32.6794i 22.0845 4.92697i −28.2244 5.99928i −5.55538 8.17690i
3.17 −1.20752 2.55771i −0.00192290 + 0.0182952i −5.08377 + 6.17699i 6.27992 10.8771i 0.0491158 0.0171736i −1.11632 5.25185i 21.9377 + 5.54396i 26.4097 + 5.61355i −35.4037 2.92782i
3.18 −1.15173 + 2.58331i 0.949976 9.03842i −5.34702 5.95058i 6.63863 11.4984i 22.2549 + 12.8639i 2.08859 + 9.82603i 21.5306 6.95955i −54.3805 11.5589i 22.0582 + 30.3928i
3.19 −1.00949 2.64215i −0.650852 + 6.19245i −5.96188 + 5.33442i −8.39771 + 14.5453i 17.0184 4.53154i −3.41539 16.0681i 20.1127 + 10.3671i −11.5128 2.44712i 46.9081 + 7.50475i
3.20 −0.737328 2.73063i −0.901200 + 8.57435i −6.91270 + 4.02674i 8.05240 13.9472i 24.0779 3.86126i −1.43706 6.76081i 16.0925 + 15.9070i −46.2973 9.84079i −44.0218 11.7045i
See next 80 embeddings (of 368 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 115.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
31.h odd 30 1 inner
124.p even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.4.p.a 368
4.b odd 2 1 inner 124.4.p.a 368
31.h odd 30 1 inner 124.4.p.a 368
124.p even 30 1 inner 124.4.p.a 368
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.p.a 368 1.a even 1 1 trivial
124.4.p.a 368 4.b odd 2 1 inner
124.4.p.a 368 31.h odd 30 1 inner
124.4.p.a 368 124.p even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(124, [\chi])\).