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Properties

Label 251.2.a
Level $251$
Weight $2$
Character orbit 251.a
Rep. character $\chi_{251}(1,\cdot)$
Character field $\Q$
Dimension $21$
Newform subspaces $2$
Sturm bound $42$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 251.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(42\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(251))\).

Total New Old
Modular forms 22 22 0
Cusp forms 21 21 0
Eisenstein series 1 1 0

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

\(251\)Dim
\(+\)\(4\)
\(-\)\(17\)

Trace form

\( 21 q - 2 q^{3} + 24 q^{4} + 2 q^{6} + 6 q^{8} + 21 q^{9} + O(q^{10}) \) \( 21 q - 2 q^{3} + 24 q^{4} + 2 q^{6} + 6 q^{8} + 21 q^{9} + 6 q^{10} - 4 q^{11} - 8 q^{12} + 10 q^{13} - 8 q^{14} - 12 q^{15} + 34 q^{16} - 10 q^{18} + 4 q^{19} - 10 q^{20} + 18 q^{21} + 8 q^{22} + 2 q^{23} - 24 q^{24} + 25 q^{25} + 2 q^{26} - 14 q^{27} - 6 q^{28} + 16 q^{29} - 32 q^{30} + 10 q^{31} + 22 q^{32} - 16 q^{33} - 24 q^{34} - 10 q^{35} + 28 q^{36} + 14 q^{37} - 30 q^{38} + 26 q^{39} - 2 q^{40} - 40 q^{42} + 4 q^{43} - 44 q^{44} + 4 q^{45} + 2 q^{46} - 8 q^{47} - 76 q^{48} + 23 q^{49} - 12 q^{50} + 6 q^{53} - 58 q^{54} - 14 q^{55} - 56 q^{56} - 8 q^{57} - 40 q^{58} - 14 q^{59} - 104 q^{60} + 38 q^{61} - 72 q^{62} - 27 q^{63} + 62 q^{64} - 13 q^{65} - 38 q^{66} + 32 q^{67} - 18 q^{68} + 25 q^{69} - 16 q^{70} - 36 q^{71} + 8 q^{72} + 6 q^{73} + 16 q^{74} - 7 q^{75} + 40 q^{76} + 8 q^{77} - 14 q^{78} + 12 q^{79} - 32 q^{80} + 21 q^{81} + 22 q^{82} - q^{83} + 54 q^{84} + 50 q^{85} + 26 q^{86} + 20 q^{87} + 22 q^{88} + 16 q^{89} + 74 q^{90} - 4 q^{91} + 26 q^{92} + 5 q^{93} + 18 q^{94} + 4 q^{95} - 26 q^{96} - 4 q^{97} + 14 q^{98} - 34 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(251))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 251
251.2.a.a 251.a 1.a $4$ $2.004$ 4.4.725.1 None \(-2\) \(-2\) \(-3\) \(-3\) $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{3})q^{2}-\beta _{2}q^{3}-\beta _{3}q^{4}+(-1+\cdots)q^{5}+\cdots\)
251.2.a.b 251.a 1.a $17$ $2.004$ \(\mathbb{Q}[x]/(x^{17} - \cdots)\) None \(2\) \(0\) \(3\) \(3\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(2+\beta _{2})q^{4}-\beta _{7}q^{5}+\cdots\)