Properties

Label 231.4.a
Level $231$
Weight $4$
Character orbit 231.a
Rep. character $\chi_{231}(1,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $12$
Sturm bound $128$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 231.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(128\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(231))\).

Total New Old
Modular forms 100 28 72
Cusp forms 92 28 64
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.

\(3\)\(7\)\(11\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(5\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(5\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(5\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(20\)
Minus space\(-\)\(8\)

Trace form

\( 28 q + 8 q^{2} + 72 q^{4} + 32 q^{5} + 24 q^{6} + 96 q^{8} + 252 q^{9} + O(q^{10}) \) \( 28 q + 8 q^{2} + 72 q^{4} + 32 q^{5} + 24 q^{6} + 96 q^{8} + 252 q^{9} + 200 q^{10} + 288 q^{13} - 120 q^{15} + 328 q^{16} - 304 q^{17} + 72 q^{18} - 128 q^{19} + 416 q^{20} - 88 q^{22} - 312 q^{23} + 648 q^{24} + 836 q^{25} - 224 q^{26} + 720 q^{29} + 568 q^{31} + 896 q^{32} + 736 q^{34} + 112 q^{35} + 648 q^{36} + 88 q^{37} - 336 q^{38} + 1800 q^{40} + 1392 q^{41} - 168 q^{42} - 280 q^{43} + 288 q^{45} + 608 q^{46} + 840 q^{47} + 1372 q^{49} + 1400 q^{50} + 1368 q^{51} + 2720 q^{52} + 2256 q^{53} + 216 q^{54} + 52 q^{58} - 928 q^{59} - 252 q^{60} - 368 q^{61} - 3056 q^{62} - 2076 q^{64} + 960 q^{65} - 1832 q^{67} - 4976 q^{68} - 1536 q^{69} - 812 q^{70} - 1784 q^{71} + 864 q^{72} - 672 q^{73} - 4416 q^{74} - 1056 q^{75} - 3504 q^{76} - 60 q^{78} + 1040 q^{79} - 2712 q^{80} + 2268 q^{81} - 3368 q^{82} - 1616 q^{83} + 2240 q^{85} - 2040 q^{86} - 864 q^{87} - 660 q^{88} + 2536 q^{89} + 1800 q^{90} - 616 q^{91} - 4712 q^{92} - 936 q^{93} - 4280 q^{94} - 4256 q^{95} + 1008 q^{96} + 2568 q^{97} + 392 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(231))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 7 11
231.4.a.a 231.a 1.a $1$ $13.629$ \(\Q\) None 231.4.a.a \(-3\) \(-3\) \(-4\) \(-7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}-3q^{3}+q^{4}-4q^{5}+9q^{6}+\cdots\)
231.4.a.b 231.a 1.a $1$ $13.629$ \(\Q\) None 231.4.a.b \(-2\) \(3\) \(1\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}-4q^{4}+q^{5}-6q^{6}+\cdots\)
231.4.a.c 231.a 1.a $1$ $13.629$ \(\Q\) None 231.4.a.c \(2\) \(-3\) \(11\) \(-7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}-4q^{4}+11q^{5}-6q^{6}+\cdots\)
231.4.a.d 231.a 1.a $1$ $13.629$ \(\Q\) None 231.4.a.d \(3\) \(3\) \(-14\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}+3q^{3}+q^{4}-14q^{5}+9q^{6}+\cdots\)
231.4.a.e 231.a 1.a $1$ $13.629$ \(\Q\) None 231.4.a.e \(5\) \(3\) \(-6\) \(7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+5q^{2}+3q^{3}+17q^{4}-6q^{5}+15q^{6}+\cdots\)
231.4.a.f 231.a 1.a $2$ $13.629$ \(\Q(\sqrt{17}) \) None 231.4.a.f \(-3\) \(6\) \(-19\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+3q^{3}+(-3+3\beta )q^{4}+\cdots\)
231.4.a.g 231.a 1.a $2$ $13.629$ \(\Q(\sqrt{17}) \) None 231.4.a.g \(-3\) \(6\) \(25\) \(14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+3q^{3}+(-3+3\beta )q^{4}+\cdots\)
231.4.a.h 231.a 1.a $2$ $13.629$ \(\Q(\sqrt{17}) \) None 231.4.a.h \(3\) \(-6\) \(1\) \(14\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-3q^{3}+(-3+3\beta )q^{4}+\cdots\)
231.4.a.i 231.a 1.a $2$ $13.629$ \(\Q(\sqrt{37}) \) None 231.4.a.i \(3\) \(6\) \(2\) \(14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+3q^{3}+(2+3\beta )q^{4}+(-1+\cdots)q^{5}+\cdots\)
231.4.a.j 231.a 1.a $5$ $13.629$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 231.4.a.j \(-1\) \(-15\) \(7\) \(-35\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{2})q^{4}+(2-2\beta _{1}+\cdots)q^{5}+\cdots\)
231.4.a.k 231.a 1.a $5$ $13.629$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 231.4.a.k \(-1\) \(-15\) \(21\) \(35\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
231.4.a.l 231.a 1.a $5$ $13.629$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 231.4.a.l \(5\) \(15\) \(7\) \(-35\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(231))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(231)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 2}\)