Properties

Label 138.5.g.a
Level $138$
Weight $5$
Character orbit 138.g
Analytic conductor $14.265$
Analytic rank $0$
Dimension $320$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 138.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.2650549056\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(32\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q + 20 q^{3} + 256 q^{4} - 64 q^{6} + 172 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q + 20 q^{3} + 256 q^{4} - 64 q^{6} + 172 q^{9} - 160 q^{12} - 200 q^{13} + 2202 q^{15} - 2048 q^{16} - 3392 q^{18} + 520 q^{19} - 982 q^{21} + 640 q^{22} + 512 q^{24} + 1680 q^{25} + 8786 q^{27} - 2752 q^{30} + 5368 q^{31} - 12050 q^{33} - 640 q^{34} - 1376 q^{36} - 8936 q^{37} + 132 q^{39} + 1984 q^{42} - 11704 q^{43} - 10396 q^{45} + 2112 q^{46} + 1280 q^{48} + 36752 q^{49} + 10060 q^{51} + 1600 q^{52} - 28896 q^{54} - 36788 q^{55} - 6074 q^{57} + 10880 q^{58} + 15648 q^{60} + 44536 q^{61} + 61826 q^{63} + 16384 q^{64} + 25600 q^{66} + 21628 q^{67} - 21406 q^{69} - 45568 q^{70} - 23552 q^{72} - 14212 q^{73} - 65058 q^{75} - 4160 q^{76} - 31840 q^{78} - 29024 q^{79} - 43964 q^{81} + 23808 q^{82} + 17184 q^{84} + 75252 q^{85} - 34892 q^{87} - 5120 q^{88} - 1088 q^{90} + 12840 q^{91} + 14988 q^{93} - 23040 q^{94} - 4096 q^{96} + 158244 q^{97} + 86652 q^{99}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −2.57283 1.17497i −8.98762 + 0.471881i 5.23889 + 6.04600i 23.8357 + 37.0891i 23.6780 + 9.34613i −6.99678 + 48.6637i −6.37488 21.7108i 80.5547 8.48217i −17.7466 123.430i
29.2 −2.57283 1.17497i −8.69255 + 2.33227i 5.23889 + 6.04600i −23.5192 36.5966i 25.1048 + 4.21297i 6.58071 45.7699i −6.37488 21.7108i 70.1210 40.5468i 17.5109 + 121.791i
29.3 −2.57283 1.17497i −7.85108 4.40006i 5.23889 + 6.04600i 0.208147 + 0.323883i 15.0295 + 20.5454i 0.671936 4.67342i −6.37488 21.7108i 42.2789 + 69.0905i −0.154973 1.07786i
29.4 −2.57283 1.17497i −7.78039 + 4.52388i 5.23889 + 6.04600i −1.76067 2.73965i 25.3330 2.49742i 1.63820 11.3939i −6.37488 21.7108i 40.0690 70.3951i 1.31088 + 9.11739i
29.5 −2.57283 1.17497i −6.25980 6.46644i 5.23889 + 6.04600i 3.94347 + 6.13615i 8.50750 + 23.9921i 4.85333 33.7557i −6.37488 21.7108i −2.62980 + 80.9573i −2.93606 20.4207i
29.6 −2.57283 1.17497i −4.43333 + 7.83234i 5.23889 + 6.04600i −6.99614 10.8862i 20.6090 14.9422i −12.6718 + 88.1344i −6.37488 21.7108i −41.6912 69.4467i 5.20889 + 36.2286i
29.7 −2.57283 1.17497i −3.79819 + 8.15927i 5.23889 + 6.04600i 10.9857 + 17.0941i 19.3590 16.5296i 6.94620 48.3119i −6.37488 21.7108i −52.1475 61.9810i −8.17929 56.8882i
29.8 −2.57283 1.17497i −1.98653 8.77802i 5.23889 + 6.04600i −19.3206 30.0634i −5.20294 + 24.9185i −7.22090 + 50.2225i −6.37488 21.7108i −73.1074 + 34.8755i 14.3849 + 100.049i
29.9 −2.57283 1.17497i 0.378009 8.99206i 5.23889 + 6.04600i 21.0690 + 32.7840i −11.5380 + 22.6909i −4.03906 + 28.0923i −6.37488 21.7108i −80.7142 6.79816i −15.6867 109.103i
29.10 −2.57283 1.17497i 3.35111 + 8.35285i 5.23889 + 6.04600i −4.10247 6.38357i 1.19252 25.4279i 6.03242 41.9564i −6.37488 21.7108i −58.5401 + 55.9827i 3.05444 + 21.2441i
29.11 −2.57283 1.17497i 5.31677 7.26168i 5.23889 + 6.04600i 10.0489 + 15.6363i −22.2114 + 12.4360i −0.294426 + 2.04778i −6.37488 21.7108i −24.4639 77.2173i −7.48175 52.0367i
29.12 −2.57283 1.17497i 6.96735 + 5.69702i 5.23889 + 6.04600i −16.5641 25.7742i −11.2320 22.8439i −5.23680 + 36.4227i −6.37488 21.7108i 16.0880 + 79.3863i 12.3326 + 85.7749i
29.13 −2.57283 1.17497i 7.44767 5.05294i 5.23889 + 6.04600i −5.01473 7.80307i −25.0986 + 4.24953i −6.24276 + 43.4194i −6.37488 21.7108i 29.9357 75.2652i 3.73365 + 25.9681i
29.14 −2.57283 1.17497i 8.46706 3.05105i 5.23889 + 6.04600i −23.2603 36.1937i −25.3692 2.09873i 8.08316 56.2196i −6.37488 21.7108i 62.3822 51.6668i 17.3181 + 120.450i
29.15 −2.57283 1.17497i 8.82038 + 1.78913i 5.23889 + 6.04600i 8.08738 + 12.5842i −20.5911 14.9668i −4.63055 + 32.2062i −6.37488 21.7108i 74.5980 + 31.5616i −6.02136 41.8795i
29.16 −2.57283 1.17497i 8.99975 0.0668439i 5.23889 + 6.04600i 22.3598 + 34.7925i −23.2334 10.4025i 12.5271 87.1280i −6.37488 21.7108i 80.9911 1.20316i −16.6477 115.787i
29.17 2.57283 + 1.17497i −8.98360 0.543047i 5.23889 + 6.04600i −0.208147 0.323883i −22.4752 11.9526i 0.671936 4.67342i 6.37488 + 21.7108i 80.4102 + 9.75704i −0.154973 1.07786i
29.18 2.57283 + 1.17497i −8.76210 + 2.05562i 5.23889 + 6.04600i −3.94347 6.13615i −24.9587 5.00647i 4.85333 33.7557i 6.37488 + 21.7108i 72.5489 36.0230i −2.93606 20.4207i
29.19 2.57283 + 1.17497i −7.30575 5.25605i 5.23889 + 6.04600i −23.8357 37.0891i −12.6207 22.1069i −6.99678 + 48.6637i 6.37488 + 21.7108i 25.7480 + 76.7987i −17.7466 123.430i
29.20 2.57283 + 1.17497i −6.41693 + 6.31055i 5.23889 + 6.04600i 19.3206 + 30.0634i −23.9244 + 8.69625i −7.22090 + 50.2225i 6.37488 + 21.7108i 1.35397 80.9887i 14.3849 + 100.049i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.c even 11 1 inner
69.h odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.5.g.a 320
3.b odd 2 1 inner 138.5.g.a 320
23.c even 11 1 inner 138.5.g.a 320
69.h odd 22 1 inner 138.5.g.a 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.5.g.a 320 1.a even 1 1 trivial
138.5.g.a 320 3.b odd 2 1 inner
138.5.g.a 320 23.c even 11 1 inner
138.5.g.a 320 69.h odd 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(138, [\chi])\).