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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [38,10,Mod(5,38)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(38, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([16]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("38.5");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 5.7
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 \( T_{3}^{42} + 252 T_{3}^{41} + 25326 T_{3}^{40} + 3379662 T_{3}^{39} + 1211060970 T_{3}^{38} + \cdots + 64\!\cdots\!09 \) acting on \(S_{10}^{\mathrm{new}}(38, [\chi])\). Copy content Toggle raw display