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Properties

Label 38.8.e
Level $38$
Weight $8$
Character orbit 38.e
Rep. character $\chi_{38}(5,\cdot)$
Character field $\Q(\zeta_{9})$
Dimension $66$
Newform subspaces $2$
Sturm bound $40$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 38.e (of order \(9\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{9})\)
Newform subspaces: \( 2 \)
Sturm bound: \(40\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(38, [\chi])\).

Total New Old
Modular forms 222 66 156
Cusp forms 198 66 132
Eisenstein series 24 0 24

Trace form

\( 66 q - 39 q^{3} + 1032 q^{6} - 1182 q^{7} + 1536 q^{8} - 1677 q^{9} + O(q^{10}) \) \( 66 q - 39 q^{3} + 1032 q^{6} - 1182 q^{7} + 1536 q^{8} - 1677 q^{9} + 11586 q^{11} - 49464 q^{13} + 26976 q^{14} + 109410 q^{15} - 151824 q^{17} - 256848 q^{18} - 54246 q^{19} + 768 q^{20} + 101412 q^{21} + 232320 q^{22} + 243924 q^{23} + 66048 q^{24} - 646824 q^{25} - 169968 q^{26} + 203151 q^{27} + 300672 q^{28} - 899946 q^{29} + 141366 q^{31} + 1169529 q^{33} + 170592 q^{34} + 937860 q^{35} - 107328 q^{36} - 1508004 q^{37} - 617832 q^{38} + 1487220 q^{39} + 494157 q^{41} - 977184 q^{42} + 4192962 q^{43} - 221184 q^{44} - 2121030 q^{45} - 364800 q^{46} - 1335210 q^{47} + 319488 q^{48} - 6445785 q^{49} + 5928744 q^{50} + 9043041 q^{51} + 10368 q^{52} + 2034492 q^{53} + 896040 q^{54} - 7378326 q^{55} - 4909056 q^{56} - 12115488 q^{57} - 2854848 q^{58} - 6390831 q^{59} + 1150080 q^{60} + 16037106 q^{61} + 6348816 q^{62} + 3663876 q^{63} - 8650752 q^{64} + 11988480 q^{65} - 440088 q^{66} + 418347 q^{67} + 1300416 q^{68} - 1612422 q^{69} - 1805280 q^{70} - 14190498 q^{71} + 4641792 q^{72} - 9870690 q^{73} - 3567024 q^{74} + 2053248 q^{76} + 33000396 q^{77} - 11977104 q^{78} - 19536114 q^{79} - 46133223 q^{81} - 3259224 q^{82} + 14115522 q^{83} + 15048576 q^{84} + 8751552 q^{85} + 26557536 q^{86} + 40486866 q^{87} + 4088832 q^{88} - 10658520 q^{89} + 4171296 q^{90} - 4410684 q^{91} - 17966208 q^{92} - 14723742 q^{93} - 33954912 q^{94} - 36453096 q^{95} - 70997799 q^{97} - 17740800 q^{98} + 17017377 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(38, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
38.8.e.a 38.e 19.e $30$ $11.871$ None \(0\) \(45\) \(0\) \(1806\) $\mathrm{SU}(2)[C_{9}]$
38.8.e.b 38.e 19.e $36$ $11.871$ None \(0\) \(-84\) \(0\) \(-2988\) $\mathrm{SU}(2)[C_{9}]$

Decomposition of \(S_{8}^{\mathrm{old}}(38, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(38, [\chi]) \cong \) \(S_{8}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 2}\)