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Properties

Label 38.10.e.b
Level $38$
Weight $10$
Character orbit 38.e
Analytic conductor $19.571$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.5713617742\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48q + 33q^{3} + 528q^{6} - 25809q^{7} - 98304q^{8} + 7953q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q + 33q^{3} + 528q^{6} - 25809q^{7} - 98304q^{8} + 7953q^{9} - 55221q^{11} - 62208q^{12} + 21777q^{13} + 28464q^{14} - 280011q^{15} + 1055055q^{17} + 6083232q^{18} + 2408220q^{19} + 523776q^{20} - 5105673q^{21} - 2513280q^{22} - 456708q^{23} + 135168q^{24} + 19417530q^{25} - 1077120q^{26} - 11729661q^{27} - 30720q^{28} + 4679889q^{29} + 4879158q^{31} - 12360834q^{33} + 9206400q^{34} - 47475759q^{35} + 2035968q^{36} + 119843604q^{37} - 39471312q^{38} + 16552668q^{39} + 60082911q^{41} + 59187024q^{42} + 109939059q^{43} - 40212480q^{44} + 3067236q^{45} - 4919424q^{46} + 198627789q^{47} - 4325376q^{48} - 182954115q^{49} - 229688016q^{50} - 228228021q^{51} - 49495296q^{52} - 370799859q^{53} + 111760416q^{54} - 60369543q^{55} + 211427328q^{56} + 128158266q^{57} + 166196064q^{58} - 382689396q^{59} - 36777216q^{60} - 850085550q^{61} + 42954480q^{62} - 803298357q^{63} - 402653184q^{64} + 261122181q^{65} + 516819888q^{66} + 1146400764q^{67} + 125611008q^{68} - 178115667q^{69} - 759612144q^{70} + 363378456q^{71} - 65150976q^{72} + 1407478230q^{73} - 12153360q^{74} + 1715917770q^{75} - 128288256q^{76} - 193134048q^{77} - 1199414208q^{78} - 2098405449q^{79} + 3082805319q^{81} + 961326576q^{82} + 358330707q^{83} + 502249728q^{84} - 1961566422q^{85} - 588638592q^{86} - 1222269090q^{87} - 226185216q^{88} - 368108670q^{89} + 1740707616q^{90} - 3296437032q^{91} - 819648768q^{92} + 6597152658q^{93} + 2953278144q^{94} - 881990403q^{95} + 509607936q^{96} - 3752161206q^{97} - 458055696q^{98} + 6061589742q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 −15.0351 + 5.47232i −40.3563 + 228.872i 196.107 − 164.554i 931.504 + 781.625i −645.701 − 3661.95i −2748.77 − 4761.01i −2048.00 + 3547.24i −32257.8 − 11740.9i −18282.5 − 6654.30i
5.2 −15.0351 + 5.47232i −25.7718 + 146.159i 196.107 − 164.554i −400.333 − 335.920i −412.348 − 2338.54i 3638.56 + 6302.17i −2048.00 + 3547.24i −2202.31 − 801.574i 7857.31 + 2859.83i
5.3 −15.0351 + 5.47232i −13.8002 + 78.2646i 196.107 − 164.554i 1219.82 + 1023.55i −220.802 − 1252.23i 851.666 + 1475.13i −2048.00 + 3547.24i 12561.1 + 4571.86i −23941.4 − 8713.94i
5.4 −15.0351 + 5.47232i −11.5377 + 65.4333i 196.107 − 164.554i −2027.32 − 1701.12i −184.602 − 1046.93i −2305.11 − 3992.57i −2048.00 + 3547.24i 14347.6 + 5222.09i 39790.0 + 14482.4i
5.5 −15.0351 + 5.47232i 7.21020 − 40.8911i 196.107 − 164.554i 488.978 + 410.301i 115.363 + 654.257i −4413.16 − 7643.82i −2048.00 + 3547.24i 16875.9 + 6142.32i −9597.12 − 3493.06i
5.6 −15.0351 + 5.47232i 19.6083 − 111.204i 196.107 − 164.554i −757.043 − 635.234i 313.732 + 1779.26i 3786.65 + 6558.68i −2048.00 + 3547.24i 6514.11 + 2370.94i 14858.4 + 5408.02i
5.7 −15.0351 + 5.47232i 34.1990 − 193.952i 196.107 − 164.554i 1838.42 + 1542.62i 547.184 + 3103.23i 977.539 + 1693.15i −2048.00 + 3547.24i −17951.8 − 6533.94i −36082.6 − 13133.0i
5.8 −15.0351 + 5.47232i 48.1038 − 272.810i 196.107 − 164.554i −1032.82 − 866.635i 769.661 + 4364.97i −4220.81 − 7310.67i −2048.00 + 3547.24i −53615.6 − 19514.5i 20271.0 + 7378.03i
9.1 2.77837 + 15.7569i −172.140 + 144.442i −240.561 + 87.5572i 432.938 + 157.576i −2754.24 − 2311.08i 4035.63 − 6989.91i −2048.00 − 3547.24i 5350.58 − 30344.6i −1280.06 + 7259.57i
9.2 2.77837 + 15.7569i −127.396 + 106.898i −240.561 + 87.5572i 2138.47 + 778.340i −2038.34 − 1710.37i −5339.17 + 9247.71i −2048.00 − 3547.24i 1384.68 − 7852.94i −6322.78 + 35858.3i
9.3 2.77837 + 15.7569i −127.296 + 106.814i −240.561 + 87.5572i −1972.56 − 717.951i −2036.73 − 1709.02i −5187.34 + 8984.74i −2048.00 − 3547.24i 1377.11 − 7809.96i 5832.22 − 33076.1i
9.4 2.77837 + 15.7569i 15.1406 − 12.7045i −240.561 + 87.5572i −8.62619 − 3.13968i 242.250 + 203.272i 80.6742 − 139.732i −2048.00 − 3547.24i −3350.08 + 18999.3i 25.5049 − 144.645i
9.5 2.77837 + 15.7569i 30.0038 − 25.1762i −240.561 + 87.5572i −2542.70 − 925.467i 480.061 + 402.819i 3969.67 − 6875.67i −2048.00 − 3547.24i −3151.53 + 17873.2i 7517.95 − 42636.4i
9.6 2.77837 + 15.7569i 99.1452 − 83.1927i −240.561 + 87.5572i 750.555 + 273.180i 1586.32 + 1331.08i −2034.77 + 3524.32i −2048.00 − 3547.24i −509.171 + 2887.65i −2219.15 + 12585.4i
9.7 2.77837 + 15.7569i 147.173 − 123.493i −240.561 + 87.5572i 2321.00 + 844.777i 2354.77 + 1975.89i 2792.61 − 4836.95i −2048.00 − 3547.24i 2991.52 − 16965.8i −6862.47 + 38919.0i
9.8 2.77837 + 15.7569i 194.492 − 163.198i −240.561 + 87.5572i −1439.52 − 523.944i 3111.88 + 2611.17i −2289.54 + 3965.60i −2048.00 − 3547.24i 7775.61 − 44097.7i 4256.21 − 24138.2i
17.1 2.77837 − 15.7569i −172.140 − 144.442i −240.561 − 87.5572i 432.938 − 157.576i −2754.24 + 2311.08i 4035.63 + 6989.91i −2048.00 + 3547.24i 5350.58 + 30344.6i −1280.06 − 7259.57i
17.2 2.77837 − 15.7569i −127.396 − 106.898i −240.561 − 87.5572i 2138.47 − 778.340i −2038.34 + 1710.37i −5339.17 − 9247.71i −2048.00 + 3547.24i 1384.68 + 7852.94i −6322.78 − 35858.3i
17.3 2.77837 − 15.7569i −127.296 − 106.814i −240.561 − 87.5572i −1972.56 + 717.951i −2036.73 + 1709.02i −5187.34 − 8984.74i −2048.00 + 3547.24i 1377.11 + 7809.96i 5832.22 + 33076.1i
17.4 2.77837 − 15.7569i 15.1406 + 12.7045i −240.561 − 87.5572i −8.62619 + 3.13968i 242.250 − 203.272i 80.6742 + 139.732i −2048.00 + 3547.24i −3350.08 − 18999.3i 25.5049 + 144.645i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.8
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 38.10.e.b ✓ 48
19.e even 9 1 inner 38.10.e.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.e.b ✓ 48 1.a even 1 1 trivial
38.10.e.b ✓ 48 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(33\!\cdots\!05\)\( T_{3}^{42} + \)\(83\!\cdots\!25\)\( T_{3}^{41} - \)\(12\!\cdots\!36\)\( T_{3}^{40} - \)\(35\!\cdots\!25\)\( T_{3}^{39} + \)\(17\!\cdots\!53\)\( T_{3}^{38} + \)\(43\!\cdots\!96\)\( T_{3}^{37} + \)\(71\!\cdots\!36\)\( T_{3}^{36} + \)\(18\!\cdots\!36\)\( T_{3}^{35} - \)\(33\!\cdots\!03\)\( T_{3}^{34} + \)\(98\!\cdots\!59\)\( T_{3}^{33} + \)\(28\!\cdots\!31\)\( T_{3}^{32} - \)\(11\!\cdots\!50\)\( T_{3}^{31} + \)\(27\!\cdots\!73\)\( T_{3}^{30} + \)\(18\!\cdots\!55\)\( T_{3}^{29} - \)\(36\!\cdots\!57\)\( T_{3}^{28} - \)\(82\!\cdots\!89\)\( T_{3}^{27} + \)\(87\!\cdots\!16\)\( T_{3}^{26} + \)\(49\!\cdots\!44\)\( T_{3}^{25} + \)\(44\!\cdots\!12\)\( T_{3}^{24} - \)\(25\!\cdots\!75\)\( T_{3}^{23} - \)\(31\!\cdots\!10\)\( T_{3}^{22} - \)\(48\!\cdots\!61\)\( T_{3}^{21} + \)\(14\!\cdots\!16\)\( T_{3}^{20} - \)\(55\!\cdots\!57\)\( T_{3}^{19} + \)\(12\!\cdots\!49\)\( T_{3}^{18} + \)\(93\!\cdots\!18\)\( T_{3}^{17} - \)\(15\!\cdots\!92\)\( T_{3}^{16} - \)\(42\!\cdots\!79\)\( T_{3}^{15} + \)\(22\!\cdots\!36\)\( T_{3}^{14} + \)\(39\!\cdots\!62\)\( T_{3}^{13} + \)\(45\!\cdots\!60\)\( T_{3}^{12} + \)\(36\!\cdots\!40\)\( T_{3}^{11} + \)\(14\!\cdots\!97\)\( T_{3}^{10} + \)\(65\!\cdots\!33\)\( T_{3}^{9} + \)\(78\!\cdots\!62\)\( T_{3}^{8} + \)\(76\!\cdots\!98\)\( T_{3}^{7} + \)\(39\!\cdots\!58\)\( T_{3}^{6} + \)\(53\!\cdots\!42\)\( T_{3}^{5} + \)\(38\!\cdots\!12\)\( T_{3}^{4} - \)\(11\!\cdots\!23\)\( T_{3}^{3} - \)\(60\!\cdots\!98\)\( T_{3}^{2} + \)\(21\!\cdots\!28\)\( T_{3} + \)\(34\!\cdots\!21\)\( \)">\(T_{3}^{48} - \cdots\) acting on \(S_{10}^{\mathrm{new}}(38, [\chi])\).