Properties

Label 38.10.e.a
Level 38
Weight 10
Character orbit 38.e
Analytic conductor 19.571
Analytic rank 0
Dimension 42
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: \(42\)
Relative dimension: \(7\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 42q - 252q^{3} + 4032q^{6} + 14373q^{7} + 86016q^{8} + 12852q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 42q - 252q^{3} + 4032q^{6} + 14373q^{7} + 86016q^{8} + 12852q^{9} + 32625q^{11} - 62208q^{12} + 351237q^{13} + 165264q^{14} - 834165q^{15} + 1075035q^{17} - 3364608q^{18} + 696762q^{19} - 1396224q^{20} - 3148065q^{21} - 1462656q^{22} - 4882824q^{23} + 1032192q^{24} - 483270q^{25} - 70176q^{26} + 2570400q^{27} + 1519104q^{28} + 13670499q^{29} - 2263968q^{31} - 5777733q^{33} - 9526080q^{34} + 43019613q^{35} + 3290112q^{36} - 50105184q^{37} - 692160q^{38} - 89820600q^{39} - 94033860q^{41} - 43526160q^{42} + 30524217q^{43} + 23402496q^{44} + 39789630q^{45} - 18312576q^{46} + 81865791q^{47} + 33030144q^{48} + 40421580q^{49} - 16761600q^{50} - 28132314q^{51} - 91666176q^{52} + 126049077q^{53} - 106214544q^{54} + 553870071q^{55} + 117743616q^{56} + 384197394q^{57} - 63148512q^{58} + 135312639q^{59} - 129611520q^{60} + 132517452q^{61} - 419324112q^{62} + 577860735q^{63} - 352321536q^{64} - 184218951q^{65} + 413765760q^{66} + 229749621q^{67} + 152743680q^{68} - 517946637q^{69} - 688313808q^{70} - 2107934406q^{71} + 105283584q^{72} - 26577420q^{73} + 322027248q^{74} + 941894730q^{75} - 107163648q^{76} + 620006124q^{77} + 122597280q^{78} - 624601023q^{79} - 1037387736q^{81} + 1504541760q^{82} + 907729401q^{83} - 604998912q^{84} + 1462808898q^{85} + 222950208q^{86} + 63648732q^{87} - 133632000q^{88} - 847590846q^{89} - 4481693280q^{90} + 957045924q^{91} + 648163584q^{92} + 2438307564q^{93} + 2205393024q^{94} - 208077987q^{95} - 509607936q^{96} + 535342599q^{97} + 1861154832q^{98} - 11757151095q^{99} + O(q^{100}) \)

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 15.0351 5.47232i −40.8678 + 231.773i 196.107 164.554i 1498.95 + 1257.77i 653.886 + 3708.37i 4614.74 + 7992.96i 2048.00 3547.24i −33552.6 12212.2i 29419.8 + 10707.9i
5.2 15.0351 5.47232i −29.8720 + 169.412i 196.107 164.554i −424.097 355.860i 477.952 + 2710.60i −4175.68 7232.49i 2048.00 3547.24i −9312.26 3389.39i −8323.71 3029.58i
5.3 15.0351 5.47232i −19.9921 + 113.381i 196.107 164.554i −800.078 671.345i 319.873 + 1814.09i 928.665 + 1608.49i 2048.00 3547.24i 6040.49 + 2198.56i −15703.1 5715.45i
5.4 15.0351 5.47232i 5.37236 30.4682i 196.107 164.554i −860.596 722.125i −85.9578 487.491i 6057.39 + 10491.7i 2048.00 3547.24i 17596.5 + 6404.61i −16890.8 6147.76i
5.5 15.0351 5.47232i 10.6794 60.5660i 196.107 164.554i 1283.65 + 1077.11i −170.871 969.056i −13.1166 22.7186i 2048.00 3547.24i 14941.8 + 5438.36i 25194.1 + 9169.91i
5.6 15.0351 5.47232i 26.9513 152.849i 196.107 164.554i −1711.99 1436.53i −431.221 2445.58i −910.825 1577.60i 2048.00 3547.24i −4140.35 1506.96i −33601.1 12229.8i
5.7 15.0351 5.47232i 34.3807 194.983i 196.107 164.554i 317.823 + 266.685i −550.092 3119.72i −2691.74 4662.22i 2048.00 3547.24i −18340.3 6675.31i 6237.88 + 2270.40i
9.1 −2.77837 15.7569i −167.700 + 140.717i −240.561 + 87.5572i 1637.73 + 596.085i 2683.20 + 2251.47i 1264.29 2189.82i 2048.00 + 3547.24i 4904.09 27812.5i 4842.24 27461.7i
9.2 −2.77837 15.7569i −110.851 + 93.0147i −240.561 + 87.5572i −143.544 52.2458i 1773.61 + 1488.24i −1900.39 + 3291.57i 2048.00 + 3547.24i 218.206 1237.51i −424.414 + 2406.97i
9.3 −2.77837 15.7569i −69.9035 + 58.6560i −240.561 + 87.5572i −954.597 347.445i 1118.46 + 938.495i −2075.12 + 3594.21i 2048.00 + 3547.24i −1971.95 + 11183.5i −2822.44 + 16006.8i
9.4 −2.77837 15.7569i 41.1535 34.5319i −240.561 + 87.5572i −479.910 174.673i −658.456 552.510i 5188.46 8986.67i 2048.00 + 3547.24i −2916.76 + 16541.8i −1418.94 + 8047.21i
9.5 −2.77837 15.7569i 74.8003 62.7649i −240.561 + 87.5572i 1601.32 + 582.833i −1196.81 1004.24i 2055.81 3560.77i 2048.00 + 3547.24i −1762.26 + 9994.30i 4734.59 26851.2i
9.6 −2.77837 15.7569i 117.543 98.6302i −240.561 + 87.5572i −1860.80 677.275i −1880.69 1578.08i −1980.53 + 3430.38i 2048.00 + 3547.24i 670.500 3802.60i −5501.78 + 31202.1i
9.7 −2.77837 15.7569i 199.355 167.278i −240.561 + 87.5572i 1053.98 + 383.617i −3189.67 2676.45i −1724.05 + 2986.15i 2048.00 + 3547.24i 8342.29 47311.5i 3116.28 17673.3i
17.1 −2.77837 + 15.7569i −167.700 140.717i −240.561 87.5572i 1637.73 596.085i 2683.20 2251.47i 1264.29 + 2189.82i 2048.00 3547.24i 4904.09 + 27812.5i 4842.24 + 27461.7i
17.2 −2.77837 + 15.7569i −110.851 93.0147i −240.561 87.5572i −143.544 + 52.2458i 1773.61 1488.24i −1900.39 3291.57i 2048.00 3547.24i 218.206 + 1237.51i −424.414 2406.97i
17.3 −2.77837 + 15.7569i −69.9035 58.6560i −240.561 87.5572i −954.597 + 347.445i 1118.46 938.495i −2075.12 3594.21i 2048.00 3547.24i −1971.95 11183.5i −2822.44 16006.8i
17.4 −2.77837 + 15.7569i 41.1535 + 34.5319i −240.561 87.5572i −479.910 + 174.673i −658.456 + 552.510i 5188.46 + 8986.67i 2048.00 3547.24i −2916.76 16541.8i −1418.94 8047.21i
17.5 −2.77837 + 15.7569i 74.8003 + 62.7649i −240.561 87.5572i 1601.32 582.833i −1196.81 + 1004.24i 2055.81 + 3560.77i 2048.00 3547.24i −1762.26 9994.30i 4734.59 + 26851.2i
17.6 −2.77837 + 15.7569i 117.543 + 98.6302i −240.561 87.5572i −1860.80 + 677.275i −1880.69 + 1578.08i −1980.53 3430.38i 2048.00 3547.24i 670.500 + 3802.60i −5501.78 31202.1i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.7
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.a 42
19.e even 9 1 inner 38.10.e.a 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.e.a 42 1.a even 1 1 trivial
38.10.e.a 42 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{42} + \cdots\) acting on \(S_{10}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database