Properties

Label 38.9.d.a
Level $38$
Weight $9$
Character orbit 38.d
Analytic conductor $15.480$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 38.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.4803871823\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24q + 168q^{3} + 1536q^{4} - 558q^{5} + 896q^{6} + 11992q^{7} + 18448q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 168q^{3} + 1536q^{4} - 558q^{5} + 896q^{6} + 11992q^{7} + 18448q^{9} + 33516q^{11} + 60870q^{13} - 71424q^{14} - 351930q^{15} - 196608q^{16} + 3546q^{17} - 727738q^{19} - 142848q^{20} - 674568q^{21} - 681600q^{22} + 726q^{23} - 114688q^{24} - 277874q^{25} - 960000q^{26} + 767488q^{28} - 254286q^{29} - 71168q^{30} - 2796150q^{33} - 2740224q^{34} - 690672q^{35} - 2361344q^{36} - 6829056q^{38} - 1599924q^{39} - 5153976q^{41} + 4153600q^{42} + 5692338q^{43} + 2145024q^{44} + 42653840q^{45} + 11113938q^{47} - 2752512q^{48} + 17359944q^{49} + 62169786q^{51} + 7791360q^{52} + 10399050q^{53} - 24215680q^{54} + 40181572q^{55} - 27798558q^{57} - 10126848q^{58} - 58809492q^{59} - 45047040q^{60} - 45012614q^{61} + 5892096q^{62} + 62488724q^{63} - 50331648q^{64} + 45107968q^{66} - 93677268q^{67} + 907776q^{68} - 291840q^{70} + 5993046q^{71} + 11698176q^{72} + 23359860q^{73} - 31474944q^{74} - 1303552q^{76} + 35611680q^{77} + 94099200q^{78} - 93221166q^{79} - 9142272q^{80} - 115510580q^{81} + 6074880q^{82} - 194283156q^{83} - 15300674q^{85} + 58611456q^{86} + 89779380q^{87} - 49833126q^{89} - 63377664q^{90} - 101275308q^{91} - 92928q^{92} - 157549992q^{93} - 87570126q^{95} - 29360128q^{96} - 178814556q^{97} - 88409088q^{98} - 54444376q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −9.79796 5.65685i −127.651 73.6994i 64.0000 + 110.851i 381.154 660.178i 833.814 + 1444.21i 2763.12 1448.15i 7582.71 + 13133.6i −7469.06 + 4312.26i
27.2 −9.79796 5.65685i −57.0846 32.9578i 64.0000 + 110.851i −412.457 + 714.397i 372.875 + 645.838i 2843.86 1448.15i −1108.07 1919.23i 8082.48 4666.42i
27.3 −9.79796 5.65685i −31.2989 18.0704i 64.0000 + 110.851i 117.295 203.161i 204.444 + 354.107i −867.592 1448.15i −2627.42 4550.82i −2298.50 + 1327.04i
27.4 −9.79796 5.65685i 58.9458 + 34.0324i 64.0000 + 110.851i −530.398 + 918.676i −385.033 666.896i −2153.04 1448.15i −964.092 1669.86i 10393.6 6000.77i
27.5 −9.79796 5.65685i 65.0070 + 37.5318i 64.0000 + 110.851i 393.618 681.767i −424.624 735.470i −1847.62 1448.15i −463.230 802.337i −7713.31 + 4453.28i
27.6 −9.79796 5.65685i 99.7890 + 57.6132i 64.0000 + 110.851i −88.7120 + 153.654i −651.819 1128.98i 4081.69 1448.15i 3358.06 + 5816.33i 1738.39 1003.66i
27.7 9.79796 + 5.65685i −107.051 61.8057i 64.0000 + 110.851i 333.136 577.008i −699.252 1211.14i −377.866 1448.15i 4359.40 + 7550.70i 6528.10 3769.00i
27.8 9.79796 + 5.65685i −36.3650 20.9954i 64.0000 + 110.851i 106.719 184.842i −237.535 411.423i 680.248 1448.15i −2398.89 4155.00i 2091.25 1207.38i
27.9 9.79796 + 5.65685i −3.23455 1.86747i 64.0000 + 110.851i −499.152 + 864.556i −21.1280 36.5948i 3106.83 1448.15i −3273.53 5669.91i −9781.33 + 5647.26i
27.10 9.79796 + 5.65685i 30.3133 + 17.5014i 64.0000 + 110.851i −189.461 + 328.157i 198.006 + 342.956i −4482.18 1448.15i −2667.90 4620.94i −3712.67 + 2143.51i
27.11 9.79796 + 5.65685i 61.4665 + 35.4877i 64.0000 + 110.851i 219.968 380.996i 401.497 + 695.414i 2502.16 1448.15i −761.746 1319.38i 4310.48 2488.66i
27.12 9.79796 + 5.65685i 131.163 + 75.7272i 64.0000 + 110.851i −110.709 + 191.754i 856.755 + 1483.94i −253.604 1448.15i 8188.71 + 14183.3i −2169.45 + 1252.53i
31.1 −9.79796 + 5.65685i −127.651 + 73.6994i 64.0000 110.851i 381.154 + 660.178i 833.814 1444.21i 2763.12 1448.15i 7582.71 13133.6i −7469.06 4312.26i
31.2 −9.79796 + 5.65685i −57.0846 + 32.9578i 64.0000 110.851i −412.457 714.397i 372.875 645.838i 2843.86 1448.15i −1108.07 + 1919.23i 8082.48 + 4666.42i
31.3 −9.79796 + 5.65685i −31.2989 + 18.0704i 64.0000 110.851i 117.295 + 203.161i 204.444 354.107i −867.592 1448.15i −2627.42 + 4550.82i −2298.50 1327.04i
31.4 −9.79796 + 5.65685i 58.9458 34.0324i 64.0000 110.851i −530.398 918.676i −385.033 + 666.896i −2153.04 1448.15i −964.092 + 1669.86i 10393.6 + 6000.77i
31.5 −9.79796 + 5.65685i 65.0070 37.5318i 64.0000 110.851i 393.618 + 681.767i −424.624 + 735.470i −1847.62 1448.15i −463.230 + 802.337i −7713.31 4453.28i
31.6 −9.79796 + 5.65685i 99.7890 57.6132i 64.0000 110.851i −88.7120 153.654i −651.819 + 1128.98i 4081.69 1448.15i 3358.06 5816.33i 1738.39 + 1003.66i
31.7 9.79796 5.65685i −107.051 + 61.8057i 64.0000 110.851i 333.136 + 577.008i −699.252 + 1211.14i −377.866 1448.15i 4359.40 7550.70i 6528.10 + 3769.00i
31.8 9.79796 5.65685i −36.3650 + 20.9954i 64.0000 110.851i 106.719 + 184.842i −237.535 + 411.423i 680.248 1448.15i −2398.89 + 4155.00i 2091.25 + 1207.38i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.9.d.a 24
19.d odd 6 1 inner 38.9.d.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.9.d.a 24 1.a even 1 1 trivial
38.9.d.a 24 19.d odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database