Properties

Label 630.4.a
Level $630$
Weight $4$
Character orbit 630.a
Rep. character $\chi_{630}(1,\cdot)$
Character field $\Q$
Dimension $30$
Newform subspaces $27$
Sturm bound $576$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 630.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 27 \)
Sturm bound: \(576\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(630))\).

Total New Old
Modular forms 448 30 418
Cusp forms 416 30 386
Eisenstein series 32 0 32

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
\(+\)\(+\)\(+\)\(+\)$+$\(2\)
\(+\)\(+\)\(+\)\(-\)$-$\(1\)
\(+\)\(+\)\(-\)\(+\)$-$\(1\)
\(+\)\(+\)\(-\)\(-\)$+$\(2\)
\(+\)\(-\)\(+\)\(+\)$-$\(2\)
\(+\)\(-\)\(+\)\(-\)$+$\(2\)
\(+\)\(-\)\(-\)\(+\)$+$\(2\)
\(+\)\(-\)\(-\)\(-\)$-$\(2\)
\(-\)\(+\)\(+\)\(+\)$-$\(1\)
\(-\)\(+\)\(+\)\(-\)$+$\(2\)
\(-\)\(+\)\(-\)\(+\)$+$\(2\)
\(-\)\(+\)\(-\)\(-\)$-$\(1\)
\(-\)\(-\)\(+\)\(+\)$+$\(3\)
\(-\)\(-\)\(+\)\(-\)$-$\(2\)
\(-\)\(-\)\(-\)\(+\)$-$\(2\)
\(-\)\(-\)\(-\)\(-\)$+$\(3\)
Plus space\(+\)\(18\)
Minus space\(-\)\(12\)

Trace form

\( 30 q + 4 q^{2} + 120 q^{4} + 16 q^{8} + O(q^{10}) \) \( 30 q + 4 q^{2} + 120 q^{4} + 16 q^{8} - 172 q^{11} - 32 q^{13} + 480 q^{16} - 140 q^{17} - 260 q^{19} - 160 q^{22} + 72 q^{23} + 750 q^{25} - 80 q^{26} + 432 q^{29} + 1136 q^{31} + 64 q^{32} + 504 q^{34} + 70 q^{35} + 396 q^{37} - 488 q^{38} + 652 q^{41} + 456 q^{43} - 688 q^{44} + 16 q^{46} - 224 q^{47} + 1470 q^{49} + 100 q^{50} - 128 q^{52} + 2324 q^{53} - 360 q^{55} + 1272 q^{58} - 244 q^{59} - 496 q^{61} + 560 q^{62} + 1920 q^{64} - 440 q^{65} + 3088 q^{67} - 560 q^{68} - 140 q^{70} + 64 q^{71} + 332 q^{73} + 296 q^{74} - 1040 q^{76} + 616 q^{77} + 172 q^{79} + 2856 q^{82} + 316 q^{83} - 1060 q^{85} + 2064 q^{86} - 640 q^{88} + 1532 q^{89} - 1344 q^{91} + 288 q^{92} + 560 q^{94} - 2060 q^{95} + 716 q^{97} + 196 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(630))\) 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 3 5 7
630.4.a.a 630.a 1.a $1$ $37.171$ \(\Q\) None \(-2\) \(0\) \(-5\) \(-7\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}-7q^{7}-8q^{8}+\cdots\)
630.4.a.b 630.a 1.a $1$ $37.171$ \(\Q\) None \(-2\) \(0\) \(-5\) \(-7\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}-7q^{7}-8q^{8}+\cdots\)
630.4.a.c 630.a 1.a $1$ $37.171$ \(\Q\) None \(-2\) \(0\) \(-5\) \(7\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}+7q^{7}-8q^{8}+\cdots\)
630.4.a.d 630.a 1.a $1$ $37.171$ \(\Q\) None \(-2\) \(0\) \(5\) \(-7\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}-7q^{7}-8q^{8}+\cdots\)
630.4.a.e 630.a 1.a $1$ $37.171$ \(\Q\) None \(-2\) \(0\) \(5\) \(-7\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}-7q^{7}-8q^{8}+\cdots\)
630.4.a.f 630.a 1.a $1$ $37.171$ \(\Q\) None \(-2\) \(0\) \(5\) \(-7\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}-7q^{7}-8q^{8}+\cdots\)
630.4.a.g 630.a 1.a $1$ $37.171$ \(\Q\) None \(-2\) \(0\) \(5\) \(7\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}+7q^{7}-8q^{8}+\cdots\)
630.4.a.h 630.a 1.a $1$ $37.171$ \(\Q\) None \(-2\) \(0\) \(5\) \(7\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}+7q^{7}-8q^{8}+\cdots\)
630.4.a.i 630.a 1.a $1$ $37.171$ \(\Q\) None \(-2\) \(0\) \(5\) \(7\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}+7q^{7}-8q^{8}+\cdots\)
630.4.a.j 630.a 1.a $1$ $37.171$ \(\Q\) None \(-2\) \(0\) \(5\) \(7\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}+7q^{7}-8q^{8}+\cdots\)
630.4.a.k 630.a 1.a $1$ $37.171$ \(\Q\) None \(2\) \(0\) \(-5\) \(-7\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}-7q^{7}+8q^{8}+\cdots\)
630.4.a.l 630.a 1.a $1$ $37.171$ \(\Q\) None \(2\) \(0\) \(-5\) \(-7\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}-7q^{7}+8q^{8}+\cdots\)
630.4.a.m 630.a 1.a $1$ $37.171$ \(\Q\) None \(2\) \(0\) \(-5\) \(-7\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}-7q^{7}+8q^{8}+\cdots\)
630.4.a.n 630.a 1.a $1$ $37.171$ \(\Q\) None \(2\) \(0\) \(-5\) \(-7\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}-7q^{7}+8q^{8}+\cdots\)
630.4.a.o 630.a 1.a $1$ $37.171$ \(\Q\) None \(2\) \(0\) \(-5\) \(7\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.p 630.a 1.a $1$ $37.171$ \(\Q\) None \(2\) \(0\) \(-5\) \(7\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.q 630.a 1.a $1$ $37.171$ \(\Q\) None \(2\) \(0\) \(-5\) \(7\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.r 630.a 1.a $1$ $37.171$ \(\Q\) None \(2\) \(0\) \(-5\) \(7\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.s 630.a 1.a $1$ $37.171$ \(\Q\) None \(2\) \(0\) \(5\) \(-7\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}-7q^{7}+8q^{8}+\cdots\)
630.4.a.t 630.a 1.a $1$ $37.171$ \(\Q\) None \(2\) \(0\) \(5\) \(-7\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}-7q^{7}+8q^{8}+\cdots\)
630.4.a.u 630.a 1.a $1$ $37.171$ \(\Q\) None \(2\) \(0\) \(5\) \(7\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.v 630.a 1.a $1$ $37.171$ \(\Q\) None \(2\) \(0\) \(5\) \(7\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.w 630.a 1.a $1$ $37.171$ \(\Q\) None \(2\) \(0\) \(5\) \(7\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.x 630.a 1.a $1$ $37.171$ \(\Q\) None \(2\) \(0\) \(5\) \(7\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.y 630.a 1.a $2$ $37.171$ \(\Q(\sqrt{3649}) \) None \(-4\) \(0\) \(-10\) \(-14\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}-7q^{7}-8q^{8}+\cdots\)
630.4.a.z 630.a 1.a $2$ $37.171$ \(\Q(\sqrt{106}) \) None \(-4\) \(0\) \(-10\) \(14\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}+7q^{7}-8q^{8}+\cdots\)
630.4.a.ba 630.a 1.a $2$ $37.171$ \(\Q(\sqrt{3649}) \) None \(4\) \(0\) \(10\) \(-14\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}-7q^{7}+8q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(630))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(630)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 2}\)