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Properties

Label 3570.2.a
Level $3570$
Weight $2$
Character orbit 3570.a
Rep. character $\chi_{3570}(1,\cdot)$
Character field $\Q$
Dimension $63$
Newform subspaces $40$
Sturm bound $1728$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 3570 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3570.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 40 \)
Sturm bound: \(1728\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(11\), \(13\), \(19\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3570))\).

Total New Old
Modular forms 880 63 817
Cusp forms 849 63 786
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.

\(2\)\(3\)\(5\)\(7\)\(17\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(+\)\(+\)\(-\)\(+\)$-$\(3\)
\(+\)\(+\)\(+\)\(-\)\(-\)$+$\(1\)
\(+\)\(+\)\(-\)\(+\)\(+\)$-$\(1\)
\(+\)\(+\)\(-\)\(+\)\(-\)$+$\(3\)
\(+\)\(+\)\(-\)\(-\)\(+\)$+$\(2\)
\(+\)\(+\)\(-\)\(-\)\(-\)$-$\(2\)
\(+\)\(-\)\(+\)\(+\)\(+\)$-$\(2\)
\(+\)\(-\)\(+\)\(+\)\(-\)$+$\(2\)
\(+\)\(-\)\(+\)\(-\)\(+\)$+$\(3\)
\(+\)\(-\)\(+\)\(-\)\(-\)$-$\(1\)
\(+\)\(-\)\(-\)\(+\)\(+\)$+$\(2\)
\(+\)\(-\)\(-\)\(+\)\(-\)$-$\(2\)
\(+\)\(-\)\(-\)\(-\)\(+\)$-$\(2\)
\(+\)\(-\)\(-\)\(-\)\(-\)$+$\(2\)
\(-\)\(+\)\(+\)\(+\)\(+\)$-$\(3\)
\(-\)\(+\)\(+\)\(-\)\(+\)$+$\(2\)
\(-\)\(+\)\(+\)\(-\)\(-\)$-$\(3\)
\(-\)\(+\)\(-\)\(+\)\(+\)$+$\(2\)
\(-\)\(+\)\(-\)\(+\)\(-\)$-$\(3\)
\(-\)\(+\)\(-\)\(-\)\(+\)$-$\(2\)
\(-\)\(+\)\(-\)\(-\)\(-\)$+$\(1\)
\(-\)\(-\)\(+\)\(+\)\(+\)$+$\(2\)
\(-\)\(-\)\(+\)\(+\)\(-\)$-$\(3\)
\(-\)\(-\)\(+\)\(-\)\(+\)$-$\(2\)
\(-\)\(-\)\(+\)\(-\)\(-\)$+$\(1\)
\(-\)\(-\)\(-\)\(+\)\(+\)$-$\(3\)
\(-\)\(-\)\(-\)\(-\)\(-\)$-$\(4\)
Plus space\(+\)\(24\)
Minus space\(-\)\(39\)

Trace form

\( 63 q - q^{2} - q^{3} + 63 q^{4} - q^{5} - q^{6} - q^{7} - q^{8} + 63 q^{9} + O(q^{10}) \) \( 63 q - q^{2} - q^{3} + 63 q^{4} - q^{5} - q^{6} - q^{7} - q^{8} + 63 q^{9} - q^{10} - 12 q^{11} - q^{12} - 14 q^{13} - q^{14} - q^{15} + 63 q^{16} - q^{17} - q^{18} - 20 q^{19} - q^{20} - q^{21} + 4 q^{22} + 8 q^{23} - q^{24} + 63 q^{25} + 18 q^{26} - q^{27} - q^{28} - 30 q^{29} - q^{30} - q^{32} - 12 q^{33} - q^{34} - q^{35} + 63 q^{36} - 22 q^{37} + 44 q^{38} - 14 q^{39} - q^{40} + 22 q^{41} - q^{42} - 12 q^{43} - 12 q^{44} - q^{45} + 8 q^{46} + 16 q^{47} - q^{48} + 63 q^{49} - q^{50} - q^{51} - 14 q^{52} + 10 q^{53} - q^{54} - 12 q^{55} - q^{56} - 4 q^{57} + 50 q^{58} + 4 q^{59} - q^{60} + 34 q^{61} + 32 q^{62} - q^{63} + 63 q^{64} + 18 q^{65} + 20 q^{66} - 4 q^{67} - q^{68} + 8 q^{69} - q^{70} + 24 q^{71} - q^{72} + 22 q^{73} - 6 q^{74} - q^{75} - 20 q^{76} - 12 q^{77} + 2 q^{78} + 16 q^{79} - q^{80} + 63 q^{81} + 22 q^{82} - 20 q^{83} - q^{84} + 7 q^{85} + 36 q^{86} + 34 q^{87} + 4 q^{88} - 10 q^{89} - q^{90} - 14 q^{91} + 8 q^{92} + 32 q^{94} - 20 q^{95} - q^{96} - 2 q^{97} - q^{98} - 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3570))\) 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 17
3570.2.a.a 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(-1\) $+$ $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
3570.2.a.b 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(-1\) $+$ $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
3570.2.a.c 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(1\) $+$ $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
3570.2.a.d 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(-1\) \(1\) \(-1\) $+$ $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
3570.2.a.e 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(-1\) \(1\) \(-1\) $+$ $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
3570.2.a.f 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(-1\) \(1\) \(-1\) $+$ $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
3570.2.a.g 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(-1\) \(1\) \(-1\) $+$ $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
3570.2.a.h 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(-1\) \(1\) \(1\) $+$ $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
3570.2.a.i 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(-1\) \(1\) \(1\) $+$ $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
3570.2.a.j 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(1\) \(-1\) \(1\) $+$ $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
3570.2.a.k 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(1\) \(-1\) \(1\) $+$ $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
3570.2.a.l 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(1\) \(1\) \(-1\) $+$ $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
3570.2.a.m 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(1\) \(1\) \(-1\) $+$ $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
3570.2.a.n 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(1\) \(1\) \(1\) $+$ $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
3570.2.a.o 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(1\) \(1\) \(1\) $+$ $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
3570.2.a.p 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(1\) \(1\) \(1\) $+$ $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
3570.2.a.q 3570.a 1.a $1$ $28.507$ \(\Q\) None \(-1\) \(1\) \(1\) \(1\) $+$ $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
3570.2.a.r 3570.a 1.a $1$ $28.507$ \(\Q\) None \(1\) \(-1\) \(-1\) \(1\) $-$ $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
3570.2.a.s 3570.a 1.a $1$ $28.507$ \(\Q\) None \(1\) \(-1\) \(-1\) \(1\) $-$ $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
3570.2.a.t 3570.a 1.a $1$ $28.507$ \(\Q\) None \(1\) \(-1\) \(1\) \(-1\) $-$ $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
3570.2.a.u 3570.a 1.a $1$ $28.507$ \(\Q\) None \(1\) \(-1\) \(1\) \(1\) $-$ $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
3570.2.a.v 3570.a 1.a $1$ $28.507$ \(\Q\) None \(1\) \(1\) \(-1\) \(1\) $-$ $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
3570.2.a.w 3570.a 1.a $1$ $28.507$ \(\Q\) None \(1\) \(1\) \(-1\) \(1\) $-$ $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
3570.2.a.x 3570.a 1.a $1$ $28.507$ \(\Q\) None \(1\) \(1\) \(-1\) \(1\) $-$ $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
3570.2.a.y 3570.a 1.a $2$ $28.507$ \(\Q(\sqrt{17}) \) None \(-2\) \(-2\) \(-2\) \(-2\) $+$ $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
3570.2.a.z 3570.a 1.a $2$ $28.507$ \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(2\) \(2\) $+$ $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
3570.2.a.ba 3570.a 1.a $2$ $28.507$ \(\Q(\sqrt{5}) \) None \(-2\) \(2\) \(-2\) \(-2\) $+$ $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
3570.2.a.bb 3570.a 1.a $2$ $28.507$ \(\Q(\sqrt{17}) \) None \(-2\) \(2\) \(-2\) \(-2\) $+$ $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
3570.2.a.bc 3570.a 1.a $2$ $28.507$ \(\Q(\sqrt{17}) \) None \(-2\) \(2\) \(-2\) \(2\) $+$ $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
3570.2.a.bd 3570.a 1.a $2$ $28.507$ \(\Q(\sqrt{13}) \) None \(-2\) \(2\) \(2\) \(-2\) $+$ $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
3570.2.a.be 3570.a 1.a $2$ $28.507$ \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(2\) \(-2\) $-$ $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
3570.2.a.bf 3570.a 1.a $2$ $28.507$ \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(2\) \(-2\) $-$ $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
3570.2.a.bg 3570.a 1.a $2$ $28.507$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(2\) \(2\) $-$ $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
3570.2.a.bh 3570.a 1.a $2$ $28.507$ \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(-2\) \(-2\) $-$ $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
3570.2.a.bi 3570.a 1.a $3$ $28.507$ 3.3.229.1 None \(-3\) \(-3\) \(-3\) \(3\) $+$ $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
3570.2.a.bj 3570.a 1.a $3$ $28.507$ 3.3.473.1 None \(3\) \(-3\) \(-3\) \(-3\) $-$ $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
3570.2.a.bk 3570.a 1.a $3$ $28.507$ 3.3.321.1 None \(3\) \(-3\) \(-3\) \(3\) $-$ $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
3570.2.a.bl 3570.a 1.a $3$ $28.507$ 3.3.229.1 None \(3\) \(3\) \(-3\) \(-3\) $-$ $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
3570.2.a.bm 3570.a 1.a $3$ $28.507$ 3.3.169.1 None \(3\) \(3\) \(3\) \(-3\) $-$ $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
3570.2.a.bn 3570.a 1.a $4$ $28.507$ 4.4.2777.1 None \(4\) \(4\) \(4\) \(4\) $-$ $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3570))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3570)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(119))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(170))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(238))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(255))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(357))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(510))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(595))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(714))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1190))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1785))\)\(^{\oplus 2}\)