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Properties

Label 242.4.a
Level $242$
Weight $4$
Character orbit 242.a
Rep. character $\chi_{242}(1,\cdot)$
Character field $\Q$
Dimension $27$
Newform subspaces $15$
Sturm bound $132$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 242.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(132\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(242))\).

Total New Old
Modular forms 111 27 84
Cusp forms 87 27 60
Eisenstein series 24 0 24

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

\(2\)\(11\)FrickeDim
\(+\)\(+\)$+$\(7\)
\(+\)\(-\)$-$\(6\)
\(-\)\(+\)$-$\(5\)
\(-\)\(-\)$+$\(9\)
Plus space\(+\)\(16\)
Minus space\(-\)\(11\)

Trace form

\( 27 q + 2 q^{2} + 6 q^{3} + 108 q^{4} - 8 q^{6} + 4 q^{7} + 8 q^{8} + 253 q^{9} + O(q^{10}) \) \( 27 q + 2 q^{2} + 6 q^{3} + 108 q^{4} - 8 q^{6} + 4 q^{7} + 8 q^{8} + 253 q^{9} - 4 q^{10} + 24 q^{12} + 138 q^{13} + 32 q^{14} - 256 q^{15} + 432 q^{16} - 126 q^{17} + 74 q^{18} + 12 q^{19} + 140 q^{21} - 64 q^{23} - 32 q^{24} + 703 q^{25} - 192 q^{26} + 204 q^{27} + 16 q^{28} - 142 q^{29} + 384 q^{30} + 32 q^{31} + 32 q^{32} + 204 q^{34} + 348 q^{35} + 1012 q^{36} + 240 q^{37} - 540 q^{38} - 288 q^{39} - 16 q^{40} - 122 q^{41} - 216 q^{42} - 88 q^{43} + 112 q^{45} - 784 q^{46} - 684 q^{47} + 96 q^{48} + 2239 q^{49} + 846 q^{50} - 884 q^{51} + 552 q^{52} - 640 q^{53} - 128 q^{54} + 128 q^{56} + 176 q^{57} + 424 q^{58} + 578 q^{59} - 1024 q^{60} + 458 q^{61} + 640 q^{62} - 656 q^{63} + 1728 q^{64} - 716 q^{65} - 494 q^{67} - 504 q^{68} - 236 q^{69} - 384 q^{70} + 2060 q^{71} + 296 q^{72} - 770 q^{73} + 1180 q^{74} + 1506 q^{75} + 48 q^{76} + 1232 q^{78} + 508 q^{79} + 2907 q^{81} - 1940 q^{82} - 1616 q^{83} + 560 q^{84} - 2568 q^{85} - 204 q^{86} + 392 q^{87} - 894 q^{89} - 1300 q^{90} - 6104 q^{91} - 256 q^{92} - 5736 q^{93} + 560 q^{94} + 1480 q^{95} - 128 q^{96} - 6614 q^{97} - 366 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(242))\) 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 11
242.4.a.a 242.a 1.a $1$ $14.278$ \(\Q\) None \(-2\) \(1\) \(-3\) \(10\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+4q^{4}-3q^{5}-2q^{6}+\cdots\)
242.4.a.b 242.a 1.a $1$ $14.278$ \(\Q\) None \(-2\) \(4\) \(3\) \(-8\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{3}+4q^{4}+3q^{5}-8q^{6}+\cdots\)
242.4.a.c 242.a 1.a $1$ $14.278$ \(\Q\) None \(-2\) \(5\) \(-15\) \(36\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+5q^{3}+4q^{4}-15q^{5}-10q^{6}+\cdots\)
242.4.a.d 242.a 1.a $1$ $14.278$ \(\Q\) None \(2\) \(-7\) \(-19\) \(-14\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-7q^{3}+4q^{4}-19q^{5}-14q^{6}+\cdots\)
242.4.a.e 242.a 1.a $1$ $14.278$ \(\Q\) None \(2\) \(4\) \(3\) \(8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{3}+4q^{4}+3q^{5}+8q^{6}+\cdots\)
242.4.a.f 242.a 1.a $1$ $14.278$ \(\Q\) None \(2\) \(4\) \(14\) \(8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{3}+4q^{4}+14q^{5}+8q^{6}+\cdots\)
242.4.a.g 242.a 1.a $1$ $14.278$ \(\Q\) None \(2\) \(5\) \(-15\) \(-36\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+5q^{3}+4q^{4}-15q^{5}+10q^{6}+\cdots\)
242.4.a.h 242.a 1.a $2$ $14.278$ \(\Q(\sqrt{5}) \) None \(-4\) \(-13\) \(-1\) \(-5\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-5-3\beta )q^{3}+4q^{4}-\beta q^{5}+\cdots\)
242.4.a.i 242.a 1.a $2$ $14.278$ \(\Q(\sqrt{3}) \) None \(-4\) \(-2\) \(-12\) \(6\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1+\beta )q^{3}+4q^{4}+(-6+\cdots)q^{5}+\cdots\)
242.4.a.j 242.a 1.a $2$ $14.278$ \(\Q(\sqrt{37}) \) None \(-4\) \(6\) \(4\) \(-42\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(3+\beta )q^{3}+4q^{4}+(2-3\beta )q^{5}+\cdots\)
242.4.a.k 242.a 1.a $2$ $14.278$ \(\Q(\sqrt{5}) \) None \(4\) \(-13\) \(-1\) \(5\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-5-3\beta )q^{3}+4q^{4}-\beta q^{5}+\cdots\)
242.4.a.l 242.a 1.a $2$ $14.278$ \(\Q(\sqrt{3}) \) None \(4\) \(-2\) \(-12\) \(-6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1+\beta )q^{3}+4q^{4}+(-6+\cdots)q^{5}+\cdots\)
242.4.a.m 242.a 1.a $2$ $14.278$ \(\Q(\sqrt{37}) \) None \(4\) \(6\) \(4\) \(42\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(3+\beta )q^{3}+4q^{4}+(2-3\beta )q^{5}+\cdots\)
242.4.a.n 242.a 1.a $4$ $14.278$ 4.4.978025.2 None \(-8\) \(4\) \(25\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(6-\beta _{2}+\cdots)q^{5}+\cdots\)
242.4.a.o 242.a 1.a $4$ $14.278$ 4.4.978025.2 None \(8\) \(4\) \(25\) \(3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(6-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(242))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(242)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 2}\)