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Properties

Label 230.6.a
Level $230$
Weight $6$
Character orbit 230.a
Rep. character $\chi_{230}(1,\cdot)$
Character field $\Q$
Dimension $34$
Newform subspaces $9$
Sturm bound $216$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(216\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(230))\).

Total New Old
Modular forms 184 34 150
Cusp forms 176 34 142
Eisenstein series 8 0 8

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

\(2\)\(5\)\(23\)FrickeDim
\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(-\)$-$\(5\)
\(+\)\(-\)\(+\)$-$\(5\)
\(+\)\(-\)\(-\)$+$\(3\)
\(-\)\(+\)\(+\)$-$\(6\)
\(-\)\(+\)\(-\)$+$\(3\)
\(-\)\(-\)\(+\)$+$\(3\)
\(-\)\(-\)\(-\)$-$\(6\)
Plus space\(+\)\(12\)
Minus space\(-\)\(22\)

Trace form

\( 34 q + 8 q^{2} - 8 q^{3} + 544 q^{4} + 16 q^{6} + 624 q^{7} + 128 q^{8} + 2502 q^{9} + O(q^{10}) \) \( 34 q + 8 q^{2} - 8 q^{3} + 544 q^{4} + 16 q^{6} + 624 q^{7} + 128 q^{8} + 2502 q^{9} + 156 q^{11} - 128 q^{12} + 580 q^{13} + 352 q^{14} - 2200 q^{15} + 8704 q^{16} - 3172 q^{17} + 5800 q^{18} + 4428 q^{19} + 1568 q^{21} - 8560 q^{22} + 256 q^{24} + 21250 q^{25} + 11488 q^{27} + 9984 q^{28} + 812 q^{29} + 20244 q^{31} + 2048 q^{32} + 12056 q^{33} - 15312 q^{34} + 1800 q^{35} + 40032 q^{36} + 39232 q^{37} + 23408 q^{38} + 36232 q^{39} - 1112 q^{41} - 55308 q^{43} + 2496 q^{44} + 27576 q^{47} - 2048 q^{48} + 133894 q^{49} + 5000 q^{50} + 18120 q^{51} + 9280 q^{52} - 11792 q^{53} + 6208 q^{54} + 5632 q^{56} - 21472 q^{57} + 13456 q^{58} - 41904 q^{59} - 35200 q^{60} + 33624 q^{61} + 116928 q^{62} + 286712 q^{63} + 139264 q^{64} + 87800 q^{65} + 106144 q^{66} - 97492 q^{67} - 50752 q^{68} + 19600 q^{70} - 106420 q^{71} + 92800 q^{72} - 7076 q^{73} + 54016 q^{74} - 5000 q^{75} + 70848 q^{76} + 194632 q^{77} + 60448 q^{78} + 457704 q^{79} + 553522 q^{81} + 200528 q^{82} + 284428 q^{83} + 25088 q^{84} + 66200 q^{85} + 12112 q^{86} + 55784 q^{87} - 136960 q^{88} + 75100 q^{89} + 27968 q^{91} + 125912 q^{93} + 132832 q^{94} - 75800 q^{95} + 4096 q^{96} + 178892 q^{97} - 449592 q^{98} + 776372 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(230))\) 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 5 23
230.6.a.a 230.a 1.a $1$ $36.888$ \(\Q\) None \(-4\) \(8\) \(-25\) \(199\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+8q^{3}+2^{4}q^{4}-5^{2}q^{5}-2^{5}q^{6}+\cdots\)
230.6.a.b 230.a 1.a $2$ $36.888$ \(\Q(\sqrt{2}) \) None \(-8\) \(6\) \(-50\) \(-164\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+(3+6\beta )q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
230.6.a.c 230.a 1.a $3$ $36.888$ 3.3.27980.1 None \(-12\) \(-26\) \(75\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+(-9+\beta _{2})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)
230.6.a.d 230.a 1.a $3$ $36.888$ 3.3.27980.1 None \(12\) \(-34\) \(75\) \(-121\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+(-11-\beta _{2})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)
230.6.a.e 230.a 1.a $3$ $36.888$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(12\) \(6\) \(-75\) \(5\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+(2-\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
230.6.a.f 230.a 1.a $5$ $36.888$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-20\) \(1\) \(125\) \(102\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}+5^{2}q^{5}-4\beta _{1}q^{6}+\cdots\)
230.6.a.g 230.a 1.a $5$ $36.888$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-20\) \(5\) \(-125\) \(130\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+(1+\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
230.6.a.h 230.a 1.a $6$ $36.888$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(24\) \(11\) \(150\) \(366\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+(2-\beta _{1})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)
230.6.a.i 230.a 1.a $6$ $36.888$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(24\) \(15\) \(-150\) \(106\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+(2+\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(230))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(230)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 2}\)