Properties

Label 38.8.e.b
Level $38$
Weight $8$
Character orbit 38.e
Analytic conductor $11.871$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.8706309684\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36q - 84q^{3} + 672q^{6} - 2988q^{7} + 9216q^{8} + 1428q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q - 84q^{3} + 672q^{6} - 2988q^{7} + 9216q^{8} + 1428q^{9} + 1800q^{11} - 38334q^{13} + 20832q^{14} + 111228q^{15} - 86574q^{17} - 285888q^{18} - 160152q^{19} + 48384q^{20} - 71442q^{21} + 129984q^{22} + 142920q^{23} + 43008q^{24} - 514068q^{25} - 77856q^{26} + 159492q^{27} + 204288q^{28} - 626034q^{29} + 82656q^{31} + 1576494q^{33} + 85632q^{34} + 581760q^{35} + 91392q^{36} + 1090848q^{37} - 398496q^{38} - 282624q^{39} + 450780q^{41} - 898128q^{42} + 3066264q^{43} - 1039872q^{44} - 1208484q^{45} + 166080q^{46} - 1923516q^{47} + 688128q^{48} - 3667500q^{49} + 4263456q^{50} + 5070996q^{51} + 440448q^{52} - 4486374q^{53} + 1382688q^{54} - 5711508q^{55} - 3059712q^{56} - 5167644q^{57} + 1718304q^{58} - 5420364q^{59} - 3484416q^{60} + 20074776q^{61} + 5592864q^{62} - 4282596q^{63} - 4718592q^{64} + 10490790q^{65} - 6939360q^{66} + 5235372q^{67} - 171648q^{68} - 11912298q^{69} - 4654080q^{70} + 10980792q^{71} + 1462272q^{72} - 11218440q^{73} - 443088q^{74} - 8817984q^{75} + 470016q^{76} + 44477160q^{77} - 11046144q^{78} - 30314100q^{79} - 40768188q^{81} - 3606240q^{82} + 4370208q^{83} + 14171520q^{84} - 20507004q^{85} + 35895744q^{86} + 14888064q^{87} - 921600q^{88} + 9695220q^{89} + 34283424q^{90} - 24159408q^{91} - 3868416q^{92} + 4014432q^{93} - 46331520q^{94} - 16025892q^{95} - 47992770q^{97} - 32609808q^{98} + 62837148q^{99} + O(q^{100}) \)

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 7.51754 2.73616i −13.9835 + 79.3043i 49.0268 41.1384i −397.617 333.640i 111.868 + 634.435i 170.178 + 294.757i 256.000 443.405i −4038.53 1469.91i −3902.00 1420.21i
5.2 7.51754 2.73616i −11.7125 + 66.4251i 49.0268 41.1384i 272.185 + 228.391i 93.7003 + 531.401i −194.092 336.178i 256.000 443.405i −2220.00 808.013i 2671.08 + 972.193i
5.3 7.51754 2.73616i −3.06967 + 17.4090i 49.0268 41.1384i −9.97721 8.37187i 24.5574 + 139.272i 364.735 + 631.739i 256.000 443.405i 1761.46 + 641.118i −97.9109 35.6366i
5.4 7.51754 2.73616i −0.665685 + 3.77529i 49.0268 41.1384i 10.1638 + 8.52843i 5.32548 + 30.2023i −800.791 1387.01i 256.000 443.405i 2041.30 + 742.972i 99.7418 + 36.3031i
5.5 7.51754 2.73616i 8.60073 48.7771i 49.0268 41.1384i 379.177 + 318.167i −68.8058 390.217i 612.206 + 1060.37i 256.000 443.405i −250.129 91.0394i 3721.03 + 1354.35i
5.6 7.51754 2.73616i 11.6928 66.3132i 49.0268 41.1384i −157.410 132.083i −93.5425 530.506i −83.9701 145.440i 256.000 443.405i −2205.61 802.777i −1544.74 562.238i
9.1 −1.38919 7.87846i −66.5349 + 55.8294i −60.1403 + 21.8893i −312.664 113.800i 532.279 + 446.635i −817.968 + 1416.76i 256.000 + 443.405i 930.200 5275.43i −462.223 + 2621.40i
9.2 −1.38919 7.87846i −33.6098 + 28.2020i −60.1403 + 21.8893i 202.712 + 73.7810i 268.878 + 225.616i 234.915 406.885i 256.000 + 443.405i −45.5015 + 258.052i 299.677 1699.55i
9.3 −1.38919 7.87846i −12.0854 + 10.1408i −60.1403 + 21.8893i −307.410 111.888i 96.6830 + 81.1267i 444.097 769.199i 256.000 + 443.405i −336.549 + 1908.66i −454.457 + 2577.35i
9.4 −1.38919 7.87846i 16.0986 13.5083i −60.1403 + 21.8893i 358.329 + 130.421i −128.789 108.067i −386.454 + 669.359i 256.000 + 443.405i −303.078 + 1718.84i 529.732 3004.26i
9.5 −1.38919 7.87846i 37.7512 31.6770i −60.1403 + 21.8893i −192.429 70.0383i −302.010 253.416i −814.928 + 1411.50i 256.000 + 443.405i 41.9510 237.916i −284.475 + 1613.34i
9.6 −1.38919 7.87846i 65.8294 55.2375i −60.1403 + 21.8893i 133.061 + 48.4302i −526.636 441.900i 558.915 968.069i 256.000 + 443.405i 902.570 5118.73i 196.709 1115.59i
17.1 −1.38919 + 7.87846i −66.5349 55.8294i −60.1403 21.8893i −312.664 + 113.800i 532.279 446.635i −817.968 1416.76i 256.000 443.405i 930.200 + 5275.43i −462.223 2621.40i
17.2 −1.38919 + 7.87846i −33.6098 28.2020i −60.1403 21.8893i 202.712 73.7810i 268.878 225.616i 234.915 + 406.885i 256.000 443.405i −45.5015 258.052i 299.677 + 1699.55i
17.3 −1.38919 + 7.87846i −12.0854 10.1408i −60.1403 21.8893i −307.410 + 111.888i 96.6830 81.1267i 444.097 + 769.199i 256.000 443.405i −336.549 1908.66i −454.457 2577.35i
17.4 −1.38919 + 7.87846i 16.0986 + 13.5083i −60.1403 21.8893i 358.329 130.421i −128.789 + 108.067i −386.454 669.359i 256.000 443.405i −303.078 1718.84i 529.732 + 3004.26i
17.5 −1.38919 + 7.87846i 37.7512 + 31.6770i −60.1403 21.8893i −192.429 + 70.0383i −302.010 + 253.416i −814.928 1411.50i 256.000 443.405i 41.9510 + 237.916i −284.475 1613.34i
17.6 −1.38919 + 7.87846i 65.8294 + 55.2375i −60.1403 21.8893i 133.061 48.4302i −526.636 + 441.900i 558.915 + 968.069i 256.000 443.405i 902.570 + 5118.73i 196.709 + 1115.59i
23.1 7.51754 + 2.73616i −13.9835 79.3043i 49.0268 + 41.1384i −397.617 + 333.640i 111.868 634.435i 170.178 294.757i 256.000 + 443.405i −4038.53 + 1469.91i −3902.00 + 1420.21i
23.2 7.51754 + 2.73616i −11.7125 66.4251i 49.0268 + 41.1384i 272.185 228.391i 93.7003 531.401i −194.092 + 336.178i 256.000 + 443.405i −2220.00 + 808.013i 2671.08 972.193i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.8.e.b 36
19.e even 9 1 inner 38.8.e.b 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.8.e.b 36 1.a even 1 1 trivial
38.8.e.b 36 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{36} + \cdots\) acting on \(S_{8}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database