Properties

Label 38.5.f.a
Level $38$
Weight $5$
Character orbit 38.f
Analytic conductor $3.928$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 38.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.92805859719\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36q + 12q^{3} + 48q^{6} + 90q^{7} - 84q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q + 12q^{3} + 48q^{6} + 90q^{7} - 84q^{9} + 90q^{11} - 432q^{12} - 330q^{13} + 576q^{14} + 2658q^{15} - 522q^{17} - 1236q^{19} - 864q^{20} - 1998q^{21} - 1344q^{22} - 1440q^{23} - 384q^{24} - 3276q^{25} + 576q^{26} + 1692q^{27} + 1056q^{28} + 4050q^{29} + 2808q^{31} + 2910q^{33} + 1536q^{34} + 1422q^{35} + 672q^{36} + 1872q^{38} - 9024q^{39} - 3060q^{41} - 3456q^{42} - 1218q^{43} - 5760q^{44} - 2592q^{45} - 2880q^{46} + 990q^{47} + 1536q^{48} - 9696q^{49} + 27648q^{50} + 35784q^{51} - 1488q^{52} + 17082q^{53} + 2160q^{54} + 4290q^{55} + 204q^{57} - 5376q^{58} - 11142q^{59} - 11856q^{60} - 35928q^{61} - 7200q^{62} - 58254q^{63} + 9216q^{64} - 34290q^{65} - 32928q^{66} + 44322q^{67} - 4752q^{68} + 21762q^{69} + 12864q^{70} + 40428q^{71} + 10752q^{72} + 6936q^{73} - 4032q^{74} - 576q^{76} - 36648q^{77} - 11040q^{78} - 29298q^{79} + 24324q^{81} - 14592q^{82} + 23958q^{83} + 36720q^{84} + 2772q^{85} + 30816q^{86} + 14256q^{87} + 26064q^{89} + 14784q^{90} + 100044q^{91} + 9360q^{92} + 26400q^{93} - 38646q^{95} - 55746q^{97} - 64512q^{98} - 69846q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.81808 + 2.16670i −3.47598 9.55018i −1.38919 7.87846i −6.10286 + 34.6110i 27.0120 + 9.83156i 33.4829 + 57.9941i 19.5959 + 11.3137i −17.0739 + 14.3267i −63.8963 76.1486i
3.2 −1.81808 + 2.16670i −1.13346 3.11415i −1.38919 7.87846i −0.544921 + 3.09040i 8.80815 + 3.20590i −41.9404 72.6430i 19.5959 + 11.3137i 53.6364 45.0063i −5.70527 6.79927i
3.3 −1.81808 + 2.16670i 0.470754 + 1.29339i −1.38919 7.87846i 8.21061 46.5647i −3.65825 1.33149i 33.5016 + 58.0265i 19.5959 + 11.3137i 60.5984 50.8481i 85.9642 + 102.448i
3.4 1.81808 2.16670i −5.55388 15.2592i −1.38919 7.87846i −3.70832 + 21.0309i −43.1594 15.7087i −2.05572 3.56062i −19.5959 11.3137i −139.947 + 117.429i 38.8257 + 46.2706i
3.5 1.81808 2.16670i −0.486419 1.33642i −1.38919 7.87846i 3.87248 21.9619i −3.77998 1.37580i −12.4808 21.6173i −19.5959 11.3137i 60.5002 50.7657i −40.5444 48.3190i
3.6 1.81808 2.16670i 4.86583 + 13.3688i −1.38919 7.87846i 1.39867 7.93225i 37.8126 + 13.7627i 43.6122 + 75.5386i −19.5959 11.3137i −92.9980 + 78.0346i −14.6439 17.4519i
13.1 −1.81808 2.16670i −3.47598 + 9.55018i −1.38919 + 7.87846i −6.10286 34.6110i 27.0120 9.83156i 33.4829 57.9941i 19.5959 11.3137i −17.0739 14.3267i −63.8963 + 76.1486i
13.2 −1.81808 2.16670i −1.13346 + 3.11415i −1.38919 + 7.87846i −0.544921 3.09040i 8.80815 3.20590i −41.9404 + 72.6430i 19.5959 11.3137i 53.6364 + 45.0063i −5.70527 + 6.79927i
13.3 −1.81808 2.16670i 0.470754 1.29339i −1.38919 + 7.87846i 8.21061 + 46.5647i −3.65825 + 1.33149i 33.5016 58.0265i 19.5959 11.3137i 60.5984 + 50.8481i 85.9642 102.448i
13.4 1.81808 + 2.16670i −5.55388 + 15.2592i −1.38919 + 7.87846i −3.70832 21.0309i −43.1594 + 15.7087i −2.05572 + 3.56062i −19.5959 + 11.3137i −139.947 117.429i 38.8257 46.2706i
13.5 1.81808 + 2.16670i −0.486419 + 1.33642i −1.38919 + 7.87846i 3.87248 + 21.9619i −3.77998 + 1.37580i −12.4808 + 21.6173i −19.5959 + 11.3137i 60.5002 + 50.7657i −40.5444 + 48.3190i
13.6 1.81808 + 2.16670i 4.86583 13.3688i −1.38919 + 7.87846i 1.39867 + 7.93225i 37.8126 13.7627i 43.6122 75.5386i −19.5959 + 11.3137i −92.9980 78.0346i −14.6439 + 17.4519i
15.1 −0.967379 + 2.65785i −10.7009 + 1.88685i −6.12836 5.14230i 8.30053 6.96497i 5.33682 30.2666i 31.7080 54.9198i 19.5959 11.3137i 34.8331 12.6782i 10.4821 + 28.7993i
15.2 −0.967379 + 2.65785i 1.72776 0.304651i −6.12836 5.14230i −25.8893 + 21.7237i −0.861682 + 4.88684i −9.12940 + 15.8126i 19.5959 11.3137i −73.2228 + 26.6509i −32.6936 89.8249i
15.3 −0.967379 + 2.65785i 10.2027 1.79900i −6.12836 5.14230i 24.4832 20.5438i −5.08835 + 28.8575i 6.11419 10.5901i 19.5959 11.3137i 24.7428 9.00564i 30.9179 + 84.9463i
15.4 0.967379 2.65785i −7.41178 + 1.30690i −6.12836 5.14230i −15.9705 + 13.4008i −3.69646 + 20.9637i −18.4009 + 31.8712i −19.5959 + 11.3137i −22.8886 + 8.33076i 20.1679 + 55.4109i
15.5 0.967379 2.65785i 3.41796 0.602679i −6.12836 5.14230i 10.5708 8.86998i 1.70463 9.66745i 40.2843 69.7745i −19.5959 + 11.3137i −64.7959 + 23.5838i −13.3491 36.6763i
15.6 0.967379 2.65785i 15.6933 2.76715i −6.12836 5.14230i 12.2941 10.3159i 7.82668 44.3873i −44.4015 + 76.9057i −19.5959 + 11.3137i 162.507 59.1476i −15.5252 42.6552i
21.1 −2.78546 0.491151i −9.25425 + 11.0288i 7.51754 + 2.73616i 5.57659 2.02971i 31.1941 26.1750i −33.2954 57.6693i −19.5959 11.3137i −21.9275 124.357i −16.5303 + 2.91473i
21.2 −2.78546 0.491151i 0.294342 0.350783i 7.51754 + 2.73616i −18.4019 + 6.69775i −0.992163 + 0.832524i 16.3449 + 28.3102i −19.5959 11.3137i 14.0291 + 79.5629i 54.5474 9.61818i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.5.f.a 36
19.f odd 18 1 inner 38.5.f.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.5.f.a 36 1.a even 1 1 trivial
38.5.f.a 36 19.f odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database