Properties

Label 38.10.a
Level $38$
Weight $10$
Character orbit 38.a
Rep. character $\chi_{38}(1,\cdot)$
Character field $\Q$
Dimension $13$
Newform subspaces $5$
Sturm bound $50$
Trace bound $3$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(50\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(38))\).

Total New Old
Modular forms 47 13 34
Cusp forms 43 13 30
Eisenstein series 4 0 4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(19\)FrickeDim
\(+\)\(+\)$+$\(3\)
\(+\)\(-\)$-$\(4\)
\(-\)\(+\)$-$\(4\)
\(-\)\(-\)$+$\(2\)
Plus space\(+\)\(5\)
Minus space\(-\)\(8\)

Trace form

\( 13 q - 16 q^{2} + 296 q^{3} + 3328 q^{4} - 2308 q^{5} + 1952 q^{6} + 5944 q^{7} - 4096 q^{8} + 75239 q^{9} + O(q^{10}) \) \( 13 q - 16 q^{2} + 296 q^{3} + 3328 q^{4} - 2308 q^{5} + 1952 q^{6} + 5944 q^{7} - 4096 q^{8} + 75239 q^{9} - 7840 q^{10} + 12554 q^{11} + 75776 q^{12} + 43522 q^{13} + 127424 q^{14} - 179392 q^{15} + 851968 q^{16} - 642222 q^{17} - 58064 q^{18} - 130321 q^{19} - 590848 q^{20} + 1565376 q^{21} + 1745856 q^{22} + 3201686 q^{23} + 499712 q^{24} + 4285001 q^{25} - 8220992 q^{26} + 8524160 q^{27} + 1521664 q^{28} + 7376494 q^{29} + 11703488 q^{30} + 17897460 q^{31} - 1048576 q^{32} - 46155212 q^{33} + 23043808 q^{34} + 7874562 q^{35} + 19261184 q^{36} - 4660078 q^{37} - 6255408 q^{38} + 101912994 q^{39} - 2007040 q^{40} - 16660946 q^{41} - 24989216 q^{42} - 43079262 q^{43} + 3213824 q^{44} - 101525240 q^{45} - 15514560 q^{46} + 52264094 q^{47} + 19398656 q^{48} + 37038761 q^{49} - 90002160 q^{50} - 23820580 q^{51} + 11141632 q^{52} + 96950450 q^{53} - 45272032 q^{54} - 132700946 q^{55} + 32620544 q^{56} - 21112002 q^{57} + 235968832 q^{58} - 358398768 q^{59} - 45924352 q^{60} - 229861552 q^{61} - 166077504 q^{62} - 595784674 q^{63} + 218103808 q^{64} + 78516376 q^{65} + 336624960 q^{66} - 222504220 q^{67} - 164408832 q^{68} - 191261588 q^{69} + 166765632 q^{70} + 57336224 q^{71} - 14864384 q^{72} - 438580250 q^{73} - 657503904 q^{74} + 1236573180 q^{75} - 33362176 q^{76} + 713884098 q^{77} - 14188480 q^{78} - 1270558956 q^{79} - 151257088 q^{80} + 1574745341 q^{81} - 225638176 q^{82} + 1130392872 q^{83} + 400736256 q^{84} + 59139426 q^{85} - 219498240 q^{86} - 1166432318 q^{87} + 446939136 q^{88} + 1297806654 q^{89} + 1927922080 q^{90} + 2158403728 q^{91} + 819631616 q^{92} + 500285756 q^{93} - 37863552 q^{94} - 653168852 q^{95} + 127926272 q^{96} + 2433642478 q^{97} - 727871248 q^{98} + 1414521614 q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(38))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 19
38.10.a.a 38.a 1.a $1$ $19.571$ \(\Q\) None \(16\) \(-119\) \(-684\) \(9149\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}-119q^{3}+2^{8}q^{4}-684q^{5}+\cdots\)
38.10.a.b 38.a 1.a $1$ $19.571$ \(\Q\) None \(16\) \(102\) \(-1581\) \(-4865\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+102q^{3}+2^{8}q^{4}-1581q^{5}+\cdots\)
38.10.a.c 38.a 1.a $3$ $19.571$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-48\) \(3\) \(486\) \(-13317\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(1+\beta _{1})q^{3}+2^{8}q^{4}+(162+\cdots)q^{5}+\cdots\)
38.10.a.d 38.a 1.a $4$ $19.571$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-64\) \(84\) \(-1395\) \(12307\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(21+\beta _{1})q^{3}+2^{8}q^{4}+(-350+\cdots)q^{5}+\cdots\)
38.10.a.e 38.a 1.a $4$ $19.571$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(64\) \(226\) \(866\) \(2670\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(57+\beta _{2})q^{3}+2^{8}q^{4}+(218+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(38))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(38)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)