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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2450))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 7
2450.2.a.a \(1\) \(19.563\) \(\Q\) None \(-1\) \(-3\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}-3q^{3}+q^{4}+3q^{6}-q^{8}+6q^{9}+\cdots\)
2450.2.a.b \(1\) \(19.563\) \(\Q\) None \(-1\) \(-3\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-q^{2}-3q^{3}+q^{4}+3q^{6}-q^{8}+6q^{9}+\cdots\)
2450.2.a.c \(1\) \(19.563\) \(\Q\) None \(-1\) \(-3\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q-q^{2}-3q^{3}+q^{4}+3q^{6}-q^{8}+6q^{9}+\cdots\)
2450.2.a.d \(1\) \(19.563\) \(\Q\) None \(-1\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots\)
2450.2.a.e \(1\) \(19.563\) \(\Q\) None \(-1\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots\)
2450.2.a.f \(1\) \(19.563\) \(\Q\) None \(-1\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots\)
2450.2.a.g \(1\) \(19.563\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}-2q^{9}+\cdots\)
2450.2.a.h \(1\) \(19.563\) \(\Q\) None \(-1\) \(0\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}-q^{8}-3q^{9}-2q^{11}+q^{16}+\cdots\)
2450.2.a.i \(1\) \(19.563\) \(\Q\) None \(-1\) \(0\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}-q^{8}-3q^{9}-2q^{11}+q^{16}+\cdots\)
2450.2.a.j \(1\) \(19.563\) \(\Q\) None \(-1\) \(0\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}-q^{8}-3q^{9}+3q^{11}-5q^{13}+\cdots\)
2450.2.a.k \(1\) \(19.563\) \(\Q\) None \(-1\) \(0\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}-q^{8}-3q^{9}+3q^{11}+5q^{13}+\cdots\)
2450.2.a.l \(1\) \(19.563\) \(\Q\) None \(-1\) \(0\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}-q^{8}-3q^{9}+4q^{11}-6q^{13}+\cdots\)
2450.2.a.m \(1\) \(19.563\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}-2q^{9}+\cdots\)
2450.2.a.n \(1\) \(19.563\) \(\Q\) None \(-1\) \(2\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{8}+q^{9}+\cdots\)
2450.2.a.o \(1\) \(19.563\) \(\Q\) None \(-1\) \(2\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{8}+q^{9}+\cdots\)
2450.2.a.p \(1\) \(19.563\) \(\Q\) None \(-1\) \(2\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{8}+q^{9}+\cdots\)
2450.2.a.q \(1\) \(19.563\) \(\Q\) None \(-1\) \(3\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}+3q^{3}+q^{4}-3q^{6}-q^{8}+6q^{9}+\cdots\)
2450.2.a.r \(1\) \(19.563\) \(\Q\) None \(-1\) \(3\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q-q^{2}+3q^{3}+q^{4}-3q^{6}-q^{8}+6q^{9}+\cdots\)
2450.2.a.s \(1\) \(19.563\) \(\Q\) None \(1\) \(-3\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+q^{2}-3q^{3}+q^{4}-3q^{6}+q^{8}+6q^{9}+\cdots\)
2450.2.a.t \(1\) \(19.563\) \(\Q\) None \(1\) \(-2\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{8}+q^{9}+\cdots\)
2450.2.a.u \(1\) \(19.563\) \(\Q\) None \(1\) \(-2\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{8}+q^{9}+\cdots\)
2450.2.a.v \(1\) \(19.563\) \(\Q\) None \(1\) \(-2\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{8}+q^{9}+\cdots\)
2450.2.a.w \(1\) \(19.563\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}-2q^{9}+\cdots\)
2450.2.a.x \(1\) \(19.563\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}-2q^{9}+\cdots\)
2450.2.a.y \(1\) \(19.563\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+q^{8}-3q^{9}-2q^{11}+q^{16}+\cdots\)
2450.2.a.z \(1\) \(19.563\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+q^{8}-3q^{9}-2q^{11}+q^{16}+\cdots\)
2450.2.a.ba \(1\) \(19.563\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+q^{8}-3q^{9}+3q^{11}-5q^{13}+\cdots\)
2450.2.a.bb \(1\) \(19.563\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+q^{8}-3q^{9}+3q^{11}+5q^{13}+\cdots\)
2450.2.a.bc \(1\) \(19.563\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}-2q^{9}+\cdots\)
2450.2.a.bd \(1\) \(19.563\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}-2q^{9}+\cdots\)
2450.2.a.be \(1\) \(19.563\) \(\Q\) None \(1\) \(2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{8}+q^{9}+\cdots\)
2450.2.a.bf \(1\) \(19.563\) \(\Q\) None \(1\) \(2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{8}+q^{9}+\cdots\)
2450.2.a.bg \(1\) \(19.563\) \(\Q\) None \(1\) \(3\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+q^{2}+3q^{3}+q^{4}+3q^{6}+q^{8}+6q^{9}+\cdots\)
2450.2.a.bh \(1\) \(19.563\) \(\Q\) None \(1\) \(3\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+q^{2}+3q^{3}+q^{4}+3q^{6}+q^{8}+6q^{9}+\cdots\)
2450.2.a.bi \(2\) \(19.563\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}-q^{8}-3q^{9}-4q^{11}+3\beta q^{13}+\cdots\)
2450.2.a.bj \(2\) \(19.563\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{8}-q^{9}+\cdots\)
2450.2.a.bk \(2\) \(19.563\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{8}-q^{9}+\cdots\)
2450.2.a.bl \(2\) \(19.563\) \(\Q(\sqrt{6}) \) None \(-2\) \(0\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{8}+3q^{9}+\cdots\)
2450.2.a.bm \(2\) \(19.563\) \(\Q(\sqrt{7}) \) None \(-2\) \(0\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{8}+4q^{9}+\cdots\)
2450.2.a.bn \(2\) \(19.563\) \(\Q(\sqrt{2}) \) None \(2\) \(-4\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+q^{2}+(-2+\beta )q^{3}+q^{4}+(-2+\beta )q^{6}+\cdots\)
2450.2.a.bo \(2\) \(19.563\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+q^{8}-3q^{9}-4q^{11}+3\beta q^{13}+\cdots\)
2450.2.a.bp \(2\) \(19.563\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+q^{8}-q^{9}+\cdots\)
2450.2.a.bq \(2\) \(19.563\) \(\Q(\sqrt{6}) \) None \(2\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+q^{8}+3q^{9}+\cdots\)
2450.2.a.br \(2\) \(19.563\) \(\Q(\sqrt{7}) \) None \(2\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+q^{8}+4q^{9}+\cdots\)
2450.2.a.bs \(2\) \(19.563\) \(\Q(\sqrt{2}) \) None \(2\) \(4\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+q^{2}+(2+\beta )q^{3}+q^{4}+(2+\beta )q^{6}+\cdots\)
2450.2.a.bt \(4\) \(19.563\) \(\Q(\sqrt{2}, \sqrt{11})\) None \(-4\) \(0\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}-q^{8}+\cdots\)
2450.2.a.bu \(4\) \(19.563\) \(\Q(\sqrt{2}, \sqrt{11})\) None \(4\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+q^{8}+\cdots\)

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

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