Properties

Label 630.2.a
Level 630
Weight 2
Character orbit a
Rep. character \(\chi_{630}(1,\cdot)\)
Character field \(\Q\)
Dimension 10
Newforms 10
Sturm bound 288
Trace bound 13

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Defining parameters

Level: \( N \) = \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 630.a (trivial)
Character field: \(\Q\)
Newforms: \( 10 \)
Sturm bound: \(288\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\), \(19\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(630))\).

Total New Old
Modular forms 160 10 150
Cusp forms 129 10 119
Eisenstein series 31 0 31

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

\(2\)\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(8\)

Trace form

\(10q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 10q^{4} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 10q^{4} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 8q^{13} \) \(\mathstrut +\mathstrut 10q^{16} \) \(\mathstrut +\mathstrut 12q^{17} \) \(\mathstrut +\mathstrut 20q^{19} \) \(\mathstrut +\mathstrut 16q^{22} \) \(\mathstrut +\mathstrut 24q^{23} \) \(\mathstrut +\mathstrut 10q^{25} \) \(\mathstrut +\mathstrut 8q^{26} \) \(\mathstrut -\mathstrut 20q^{29} \) \(\mathstrut -\mathstrut 8q^{31} \) \(\mathstrut -\mathstrut 2q^{32} \) \(\mathstrut +\mathstrut 4q^{34} \) \(\mathstrut -\mathstrut 2q^{35} \) \(\mathstrut -\mathstrut 4q^{37} \) \(\mathstrut +\mathstrut 12q^{38} \) \(\mathstrut +\mathstrut 4q^{41} \) \(\mathstrut -\mathstrut 8q^{46} \) \(\mathstrut +\mathstrut 8q^{47} \) \(\mathstrut +\mathstrut 10q^{49} \) \(\mathstrut -\mathstrut 2q^{50} \) \(\mathstrut +\mathstrut 8q^{52} \) \(\mathstrut -\mathstrut 4q^{53} \) \(\mathstrut -\mathstrut 8q^{55} \) \(\mathstrut -\mathstrut 12q^{58} \) \(\mathstrut +\mathstrut 12q^{59} \) \(\mathstrut -\mathstrut 16q^{61} \) \(\mathstrut -\mathstrut 8q^{62} \) \(\mathstrut +\mathstrut 10q^{64} \) \(\mathstrut -\mathstrut 4q^{65} \) \(\mathstrut +\mathstrut 8q^{67} \) \(\mathstrut +\mathstrut 12q^{68} \) \(\mathstrut +\mathstrut 2q^{70} \) \(\mathstrut +\mathstrut 16q^{71} \) \(\mathstrut -\mathstrut 20q^{73} \) \(\mathstrut -\mathstrut 4q^{74} \) \(\mathstrut +\mathstrut 20q^{76} \) \(\mathstrut -\mathstrut 8q^{77} \) \(\mathstrut -\mathstrut 8q^{79} \) \(\mathstrut -\mathstrut 20q^{82} \) \(\mathstrut -\mathstrut 28q^{83} \) \(\mathstrut +\mathstrut 16q^{85} \) \(\mathstrut +\mathstrut 8q^{86} \) \(\mathstrut +\mathstrut 16q^{88} \) \(\mathstrut -\mathstrut 36q^{89} \) \(\mathstrut +\mathstrut 4q^{91} \) \(\mathstrut +\mathstrut 24q^{92} \) \(\mathstrut -\mathstrut 24q^{94} \) \(\mathstrut -\mathstrut 12q^{95} \) \(\mathstrut +\mathstrut 4q^{97} \) \(\mathstrut -\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(630))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5 7
630.2.a.a \(1\) \(5.031\) \(\Q\) None \(-1\) \(0\) \(-1\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}+q^{10}+\cdots\)
630.2.a.b \(1\) \(5.031\) \(\Q\) None \(-1\) \(0\) \(-1\) \(1\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}+q^{10}+\cdots\)
630.2.a.c \(1\) \(5.031\) \(\Q\) None \(-1\) \(0\) \(-1\) \(1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}+q^{10}+\cdots\)
630.2.a.d \(1\) \(5.031\) \(\Q\) None \(-1\) \(0\) \(1\) \(-1\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{4}+q^{5}-q^{7}-q^{8}-q^{10}+\cdots\)
630.2.a.e \(1\) \(5.031\) \(\Q\) None \(-1\) \(0\) \(1\) \(-1\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}+q^{5}-q^{7}-q^{8}-q^{10}+\cdots\)
630.2.a.f \(1\) \(5.031\) \(\Q\) None \(-1\) \(0\) \(1\) \(1\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{4}+q^{5}+q^{7}-q^{8}-q^{10}+\cdots\)
630.2.a.g \(1\) \(5.031\) \(\Q\) None \(1\) \(0\) \(-1\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+q^{4}-q^{5}-q^{7}+q^{8}-q^{10}+\cdots\)
630.2.a.h \(1\) \(5.031\) \(\Q\) None \(1\) \(0\) \(-1\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}-q^{5}+q^{7}+q^{8}-q^{10}+\cdots\)
630.2.a.i \(1\) \(5.031\) \(\Q\) None \(1\) \(0\) \(1\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}+q^{10}+\cdots\)
630.2.a.j \(1\) \(5.031\) \(\Q\) None \(1\) \(0\) \(1\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}+q^{4}+q^{5}+q^{7}+q^{8}+q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(630))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(630)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 2}\)