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Properties

Label 2450.2.a
Level $2450$
Weight $2$
Character orbit 2450.a
Rep. character $\chi_{2450}(1,\cdot)$
Character field $\Q$
Dimension $64$
Newform subspaces $47$
Sturm bound $840$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 47 \)
Sturm bound: \(840\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(3\), \(11\), \(13\), \(17\), \(19\), \(23\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2450))\).

Total New Old
Modular forms 468 64 404
Cusp forms 373 64 309
Eisenstein series 95 0 95

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

\(2\)\(5\)\(7\)FrickeDim
\(+\)\(+\)\(+\)$+$\(5\)
\(+\)\(+\)\(-\)$-$\(10\)
\(+\)\(-\)\(+\)$-$\(9\)
\(+\)\(-\)\(-\)$+$\(8\)
\(-\)\(+\)\(+\)$-$\(11\)
\(-\)\(+\)\(-\)$+$\(5\)
\(-\)\(-\)\(+\)$+$\(5\)
\(-\)\(-\)\(-\)$-$\(11\)
Plus space\(+\)\(23\)
Minus space\(-\)\(41\)

Trace form

\( 64 q - 2 q^{3} + 64 q^{4} + 4 q^{6} + 66 q^{9} + O(q^{10}) \) \( 64 q - 2 q^{3} + 64 q^{4} + 4 q^{6} + 66 q^{9} + 10 q^{11} - 2 q^{12} - 10 q^{13} + 64 q^{16} + 8 q^{17} - 8 q^{19} + 8 q^{22} + 8 q^{23} + 4 q^{24} + 10 q^{26} + 4 q^{27} + 24 q^{29} - 8 q^{31} + 2 q^{34} + 66 q^{36} + 24 q^{37} - 2 q^{38} + 24 q^{39} + 18 q^{41} + 40 q^{43} + 10 q^{44} + 4 q^{46} - 4 q^{47} - 2 q^{48} + 2 q^{51} - 10 q^{52} - 20 q^{53} + 22 q^{54} + 24 q^{57} - 12 q^{58} - 2 q^{59} + 2 q^{61} + 12 q^{62} + 64 q^{64} + 6 q^{66} + 16 q^{67} + 8 q^{68} + 36 q^{69} + 32 q^{71} + 4 q^{73} - 24 q^{74} - 8 q^{76} - 16 q^{78} + 36 q^{79} + 96 q^{81} - 4 q^{82} + 2 q^{83} + 8 q^{86} + 12 q^{87} + 8 q^{88} + 58 q^{89} + 8 q^{92} + 40 q^{93} + 28 q^{94} + 4 q^{96} - 8 q^{97} + 148 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2450))\) 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 7
2450.2.a.a 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(-3\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}+q^{4}+3q^{6}-q^{8}+6q^{9}+\cdots\)
2450.2.a.b 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(-3\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}+q^{4}+3q^{6}-q^{8}+6q^{9}+\cdots\)
2450.2.a.c 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(-3\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}+q^{4}+3q^{6}-q^{8}+6q^{9}+\cdots\)
2450.2.a.d 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots\)
2450.2.a.e 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(-2\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots\)
2450.2.a.f 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots\)
2450.2.a.g 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}-2q^{9}+\cdots\)
2450.2.a.h 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(0\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{8}-3q^{9}-2q^{11}+q^{16}+\cdots\)
2450.2.a.i 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{8}-3q^{9}-2q^{11}+q^{16}+\cdots\)
2450.2.a.j 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{8}-3q^{9}+3q^{11}-5q^{13}+\cdots\)
2450.2.a.k 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(0\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{8}-3q^{9}+3q^{11}+5q^{13}+\cdots\)
2450.2.a.l 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(0\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{8}-3q^{9}+4q^{11}-6q^{13}+\cdots\)
2450.2.a.m 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}-2q^{9}+\cdots\)
2450.2.a.n 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{8}+q^{9}+\cdots\)
2450.2.a.o 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{8}+q^{9}+\cdots\)
2450.2.a.p 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(2\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{8}+q^{9}+\cdots\)
2450.2.a.q 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(3\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}-3q^{6}-q^{8}+6q^{9}+\cdots\)
2450.2.a.r 2450.a 1.a $1$ $19.563$ \(\Q\) None \(-1\) \(3\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}-3q^{6}-q^{8}+6q^{9}+\cdots\)
2450.2.a.s 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(-3\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}+q^{4}-3q^{6}+q^{8}+6q^{9}+\cdots\)
2450.2.a.t 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(-2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{8}+q^{9}+\cdots\)
2450.2.a.u 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(-2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{8}+q^{9}+\cdots\)
2450.2.a.v 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(-2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{8}+q^{9}+\cdots\)
2450.2.a.w 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}-2q^{9}+\cdots\)
2450.2.a.x 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}-2q^{9}+\cdots\)
2450.2.a.y 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{8}-3q^{9}-2q^{11}+q^{16}+\cdots\)
2450.2.a.z 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{8}-3q^{9}-2q^{11}+q^{16}+\cdots\)
2450.2.a.ba 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{8}-3q^{9}+3q^{11}-5q^{13}+\cdots\)
2450.2.a.bb 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{8}-3q^{9}+3q^{11}+5q^{13}+\cdots\)
2450.2.a.bc 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}-2q^{9}+\cdots\)
2450.2.a.bd 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}-2q^{9}+\cdots\)
2450.2.a.be 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{8}+q^{9}+\cdots\)
2450.2.a.bf 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{8}+q^{9}+\cdots\)
2450.2.a.bg 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(3\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}+q^{4}+3q^{6}+q^{8}+6q^{9}+\cdots\)
2450.2.a.bh 2450.a 1.a $1$ $19.563$ \(\Q\) None \(1\) \(3\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}+q^{4}+3q^{6}+q^{8}+6q^{9}+\cdots\)
2450.2.a.bi 2450.a 1.a $2$ $19.563$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{8}-3q^{9}-4q^{11}+3\beta q^{13}+\cdots\)
2450.2.a.bj 2450.a 1.a $2$ $19.563$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{8}-q^{9}+\cdots\)
2450.2.a.bk 2450.a 1.a $2$ $19.563$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{8}-q^{9}+\cdots\)
2450.2.a.bl 2450.a 1.a $2$ $19.563$ \(\Q(\sqrt{6}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{8}+3q^{9}+\cdots\)
2450.2.a.bm 2450.a 1.a $2$ $19.563$ \(\Q(\sqrt{7}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{8}+4q^{9}+\cdots\)
2450.2.a.bn 2450.a 1.a $2$ $19.563$ \(\Q(\sqrt{2}) \) None \(2\) \(-4\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-2+\beta )q^{3}+q^{4}+(-2+\beta )q^{6}+\cdots\)
2450.2.a.bo 2450.a 1.a $2$ $19.563$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{8}-3q^{9}-4q^{11}+3\beta q^{13}+\cdots\)
2450.2.a.bp 2450.a 1.a $2$ $19.563$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+q^{8}-q^{9}+\cdots\)
2450.2.a.bq 2450.a 1.a $2$ $19.563$ \(\Q(\sqrt{6}) \) None \(2\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+q^{8}+3q^{9}+\cdots\)
2450.2.a.br 2450.a 1.a $2$ $19.563$ \(\Q(\sqrt{7}) \) None \(2\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+q^{8}+4q^{9}+\cdots\)
2450.2.a.bs 2450.a 1.a $2$ $19.563$ \(\Q(\sqrt{2}) \) None \(2\) \(4\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(2+\beta )q^{3}+q^{4}+(2+\beta )q^{6}+\cdots\)
2450.2.a.bt 2450.a 1.a $4$ $19.563$ \(\Q(\sqrt{2}, \sqrt{11})\) None \(-4\) \(0\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}-q^{8}+\cdots\)
2450.2.a.bu 2450.a 1.a $4$ $19.563$ \(\Q(\sqrt{2}, \sqrt{11})\) None \(4\) \(0\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2450))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2450)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(490))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\)\(^{\oplus 2}\)