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Properties

Label 630.6.a
Level $630$
Weight $6$
Character orbit 630.a
Rep. character $\chi_{630}(1,\cdot)$
Character field $\Q$
Dimension $50$
Newform subspaces $31$
Sturm bound $864$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 630.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 31 \)
Sturm bound: \(864\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(630))\).

Total New Old
Modular forms 736 50 686
Cusp forms 704 50 654
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
\(+\)\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(2\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(4\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(22\)
Minus space\(-\)\(28\)

Trace form

\( 50 q - 8 q^{2} + 800 q^{4} - 128 q^{8} + O(q^{10}) \) \( 50 q - 8 q^{2} + 800 q^{4} - 128 q^{8} + 176 q^{11} - 808 q^{13} + 12800 q^{16} + 28 q^{17} + 1220 q^{19} - 1856 q^{22} - 6024 q^{23} + 31250 q^{25} - 2528 q^{26} - 12924 q^{29} - 27560 q^{31} - 2048 q^{32} - 13552 q^{34} - 2450 q^{35} + 9484 q^{37} - 29072 q^{38} + 19060 q^{41} + 23280 q^{43} + 2816 q^{44} + 7200 q^{46} + 42952 q^{47} + 120050 q^{49} - 5000 q^{50} - 12928 q^{52} - 67876 q^{53} + 27800 q^{55} - 7728 q^{58} + 110876 q^{59} + 24624 q^{61} + 8288 q^{62} + 204800 q^{64} - 77900 q^{65} + 103832 q^{67} + 448 q^{68} + 9800 q^{70} + 82912 q^{71} + 179020 q^{73} - 8848 q^{74} + 19520 q^{76} - 47432 q^{77} - 63336 q^{79} + 188720 q^{82} - 320972 q^{83} + 46600 q^{85} + 93600 q^{86} - 29696 q^{88} + 27932 q^{89} + 22148 q^{91} - 96384 q^{92} - 147616 q^{94} + 33700 q^{95} + 13396 q^{97} - 19208 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(630))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 7
630.6.a.a 630.a 1.a $1$ $101.042$ \(\Q\) None 210.6.a.i \(-4\) \(0\) \(-25\) \(-49\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}-5^{2}q^{5}-7^{2}q^{7}-2^{6}q^{8}+\cdots\)
630.6.a.b 630.a 1.a $1$ $101.042$ \(\Q\) None 70.6.a.e \(-4\) \(0\) \(-25\) \(-49\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}-5^{2}q^{5}-7^{2}q^{7}-2^{6}q^{8}+\cdots\)
630.6.a.c 630.a 1.a $1$ $101.042$ \(\Q\) None 210.6.a.g \(-4\) \(0\) \(25\) \(-49\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}-7^{2}q^{7}-2^{6}q^{8}+\cdots\)
630.6.a.d 630.a 1.a $1$ $101.042$ \(\Q\) None 210.6.a.j \(-4\) \(0\) \(25\) \(-49\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}-7^{2}q^{7}-2^{6}q^{8}+\cdots\)
630.6.a.e 630.a 1.a $1$ $101.042$ \(\Q\) None 70.6.a.f \(-4\) \(0\) \(25\) \(49\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}+7^{2}q^{7}-2^{6}q^{8}+\cdots\)
630.6.a.f 630.a 1.a $1$ $101.042$ \(\Q\) None 210.6.a.h \(-4\) \(0\) \(25\) \(49\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}+7^{2}q^{7}-2^{6}q^{8}+\cdots\)
630.6.a.g 630.a 1.a $1$ $101.042$ \(\Q\) None 210.6.a.b \(4\) \(0\) \(-25\) \(-49\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}-7^{2}q^{7}+2^{6}q^{8}+\cdots\)
630.6.a.h 630.a 1.a $1$ $101.042$ \(\Q\) None 210.6.a.f \(4\) \(0\) \(-25\) \(-49\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}-7^{2}q^{7}+2^{6}q^{8}+\cdots\)
630.6.a.i 630.a 1.a $1$ $101.042$ \(\Q\) None 70.6.a.b \(4\) \(0\) \(-25\) \(-49\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}-7^{2}q^{7}+2^{6}q^{8}+\cdots\)
630.6.a.j 630.a 1.a $1$ $101.042$ \(\Q\) None 70.6.a.a \(4\) \(0\) \(-25\) \(49\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}+7^{2}q^{7}+2^{6}q^{8}+\cdots\)
630.6.a.k 630.a 1.a $1$ $101.042$ \(\Q\) None 210.6.a.c \(4\) \(0\) \(-25\) \(49\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}+7^{2}q^{7}+2^{6}q^{8}+\cdots\)
630.6.a.l 630.a 1.a $1$ $101.042$ \(\Q\) None 70.6.a.d \(4\) \(0\) \(25\) \(-49\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}-7^{2}q^{7}+2^{6}q^{8}+\cdots\)
630.6.a.m 630.a 1.a $1$ $101.042$ \(\Q\) None 210.6.a.d \(4\) \(0\) \(25\) \(-49\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}-7^{2}q^{7}+2^{6}q^{8}+\cdots\)
630.6.a.n 630.a 1.a $1$ $101.042$ \(\Q\) None 70.6.a.c \(4\) \(0\) \(25\) \(49\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}+7^{2}q^{7}+2^{6}q^{8}+\cdots\)
630.6.a.o 630.a 1.a $1$ $101.042$ \(\Q\) None 210.6.a.a \(4\) \(0\) \(25\) \(49\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}+7^{2}q^{7}+2^{6}q^{8}+\cdots\)
630.6.a.p 630.a 1.a $1$ $101.042$ \(\Q\) None 210.6.a.e \(4\) \(0\) \(25\) \(49\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}+7^{2}q^{7}+2^{6}q^{8}+\cdots\)
630.6.a.q 630.a 1.a $2$ $101.042$ \(\Q(\sqrt{62689}) \) None 210.6.a.o \(-8\) \(0\) \(-50\) \(-98\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}-5^{2}q^{5}-7^{2}q^{7}-2^{6}q^{8}+\cdots\)
630.6.a.r 630.a 1.a $2$ $101.042$ \(\Q(\sqrt{889}) \) None 630.6.a.r \(-8\) \(0\) \(-50\) \(-98\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}-5^{2}q^{5}-7^{2}q^{7}-2^{6}q^{8}+\cdots\)
630.6.a.s 630.a 1.a $2$ $101.042$ \(\Q(\sqrt{1129}) \) None 70.6.a.h \(-8\) \(0\) \(-50\) \(98\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}-5^{2}q^{5}+7^{2}q^{7}-2^{6}q^{8}+\cdots\)
630.6.a.t 630.a 1.a $2$ $101.042$ \(\Q(\sqrt{176089}) \) None 210.6.a.m \(-8\) \(0\) \(-50\) \(98\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}-5^{2}q^{5}+7^{2}q^{7}-2^{6}q^{8}+\cdots\)
630.6.a.u 630.a 1.a $2$ $101.042$ \(\Q(\sqrt{3369}) \) None 70.6.a.g \(-8\) \(0\) \(50\) \(-98\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}-7^{2}q^{7}-2^{6}q^{8}+\cdots\)
630.6.a.v 630.a 1.a $2$ $101.042$ \(\Q(\sqrt{27169}) \) None 210.6.a.n \(-8\) \(0\) \(50\) \(98\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}+7^{2}q^{7}-2^{6}q^{8}+\cdots\)
630.6.a.w 630.a 1.a $2$ $101.042$ \(\Q(\sqrt{1129}) \) None 630.6.a.w \(-8\) \(0\) \(50\) \(98\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}+7^{2}q^{7}-2^{6}q^{8}+\cdots\)
630.6.a.x 630.a 1.a $2$ $101.042$ \(\Q(\sqrt{1066}) \) None 210.6.a.l \(8\) \(0\) \(-50\) \(98\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}+7^{2}q^{7}+2^{6}q^{8}+\cdots\)
630.6.a.y 630.a 1.a $2$ $101.042$ \(\Q(\sqrt{1129}) \) None 630.6.a.w \(8\) \(0\) \(-50\) \(98\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}+7^{2}q^{7}+2^{6}q^{8}+\cdots\)
630.6.a.z 630.a 1.a $2$ $101.042$ \(\Q(\sqrt{116209}) \) None 210.6.a.k \(8\) \(0\) \(50\) \(-98\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}-7^{2}q^{7}+2^{6}q^{8}+\cdots\)
630.6.a.ba 630.a 1.a $2$ $101.042$ \(\Q(\sqrt{889}) \) None 630.6.a.r \(8\) \(0\) \(50\) \(-98\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}-7^{2}q^{7}+2^{6}q^{8}+\cdots\)
630.6.a.bb 630.a 1.a $3$ $101.042$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 630.6.a.bb \(-12\) \(0\) \(-75\) \(147\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}-5^{2}q^{5}+7^{2}q^{7}-2^{6}q^{8}+\cdots\)
630.6.a.bc 630.a 1.a $3$ $101.042$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 630.6.a.bc \(-12\) \(0\) \(75\) \(-147\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}-7^{2}q^{7}-2^{6}q^{8}+\cdots\)
630.6.a.bd 630.a 1.a $3$ $101.042$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 630.6.a.bc \(12\) \(0\) \(-75\) \(-147\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}-7^{2}q^{7}+2^{6}q^{8}+\cdots\)
630.6.a.be 630.a 1.a $3$ $101.042$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 630.6.a.bb \(12\) \(0\) \(75\) \(147\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}+7^{2}q^{7}+2^{6}q^{8}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(630))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(630)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 16}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 2}\)