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Properties

Label 1870.2.a
Level $1870$
Weight $2$
Character orbit 1870.a
Rep. character $\chi_{1870}(1,\cdot)$
Character field $\Q$
Dimension $49$
Newform subspaces $22$
Sturm bound $648$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1870 = 2 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1870.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 22 \)
Sturm bound: \(648\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1870))\).

Total New Old
Modular forms 332 49 283
Cusp forms 317 49 268
Eisenstein series 15 0 15

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

\(2\)\(5\)\(11\)\(17\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(5\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(2\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(21\)
Minus space\(-\)\(28\)

Trace form

\( 49 q + q^{2} - 4 q^{3} + 49 q^{4} - 3 q^{5} - 4 q^{6} - 24 q^{7} + q^{8} + 37 q^{9} + O(q^{10}) \) \( 49 q + q^{2} - 4 q^{3} + 49 q^{4} - 3 q^{5} - 4 q^{6} - 24 q^{7} + q^{8} + 37 q^{9} + q^{10} + q^{11} - 4 q^{12} - 18 q^{13} + 4 q^{15} + 49 q^{16} + q^{17} + 13 q^{18} - 12 q^{19} - 3 q^{20} - 3 q^{22} + 16 q^{23} - 4 q^{24} + 49 q^{25} + 6 q^{26} + 8 q^{27} - 24 q^{28} + 22 q^{29} - 4 q^{30} - 8 q^{31} + q^{32} + 4 q^{33} + q^{34} + 8 q^{35} + 37 q^{36} - 34 q^{37} - 4 q^{38} + 8 q^{39} + q^{40} + 2 q^{41} + 32 q^{42} - 36 q^{43} + q^{44} - 7 q^{45} - 16 q^{46} + 16 q^{47} - 4 q^{48} + 9 q^{49} + q^{50} + 4 q^{51} - 18 q^{52} + 14 q^{53} + 8 q^{54} + q^{55} + 32 q^{57} + 14 q^{58} + 20 q^{59} + 4 q^{60} - 74 q^{61} - 8 q^{63} + 49 q^{64} - 2 q^{65} + 4 q^{66} - 100 q^{67} + q^{68} + 48 q^{69} - 16 q^{70} + 32 q^{71} + 13 q^{72} - 54 q^{73} + 22 q^{74} - 4 q^{75} - 12 q^{76} + 40 q^{78} - 64 q^{79} - 3 q^{80} + 41 q^{81} - 6 q^{82} - 12 q^{83} + q^{85} - 20 q^{86} + 24 q^{87} - 3 q^{88} + 50 q^{89} + 13 q^{90} - 32 q^{91} + 16 q^{92} + 48 q^{93} - 24 q^{94} + 20 q^{95} - 4 q^{96} - 6 q^{97} - 7 q^{98} + 13 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1870))\) 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 5 11 17
1870.2.a.a 1870.a 1.a $1$ $14.932$ \(\Q\) None 1870.2.a.a \(-1\) \(-3\) \(-1\) \(0\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}+q^{4}-q^{5}+3q^{6}-q^{8}+\cdots\)
1870.2.a.b 1870.a 1.a $1$ $14.932$ \(\Q\) None 1870.2.a.b \(-1\) \(-1\) \(-1\) \(-2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
1870.2.a.c 1870.a 1.a $1$ $14.932$ \(\Q\) None 1870.2.a.c \(-1\) \(0\) \(-1\) \(0\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-q^{8}-3q^{9}+q^{10}+\cdots\)
1870.2.a.d 1870.a 1.a $1$ $14.932$ \(\Q\) None 1870.2.a.d \(-1\) \(0\) \(1\) \(-4\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-3q^{9}+\cdots\)
1870.2.a.e 1870.a 1.a $1$ $14.932$ \(\Q\) None 1870.2.a.e \(-1\) \(1\) \(-1\) \(-4\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-4q^{7}+\cdots\)
1870.2.a.f 1870.a 1.a $1$ $14.932$ \(\Q\) None 1870.2.a.f \(-1\) \(1\) \(-1\) \(0\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{8}+\cdots\)
1870.2.a.g 1870.a 1.a $1$ $14.932$ \(\Q\) None 1870.2.a.g \(1\) \(-3\) \(1\) \(-4\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}+q^{4}+q^{5}-3q^{6}-4q^{7}+\cdots\)
1870.2.a.h 1870.a 1.a $1$ $14.932$ \(\Q\) None 1870.2.a.h \(1\) \(-1\) \(1\) \(-2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-2q^{7}+\cdots\)
1870.2.a.i 1870.a 1.a $1$ $14.932$ \(\Q\) None 1870.2.a.i \(1\) \(1\) \(1\) \(4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+4q^{7}+\cdots\)
1870.2.a.j 1870.a 1.a $2$ $14.932$ \(\Q(\sqrt{2}) \) None 1870.2.a.j \(-2\) \(-2\) \(2\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta )q^{3}+q^{4}+q^{5}+(1+\cdots)q^{6}+\cdots\)
1870.2.a.k 1870.a 1.a $2$ $14.932$ \(\Q(\sqrt{3}) \) None 1870.2.a.k \(-2\) \(0\) \(2\) \(2\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}+q^{5}-\beta q^{6}+(1+\cdots)q^{7}+\cdots\)
1870.2.a.l 1870.a 1.a $2$ $14.932$ \(\Q(\sqrt{2}) \) None 1870.2.a.l \(2\) \(-2\) \(-2\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta )q^{3}+q^{4}-q^{5}+(-1+\cdots)q^{6}+\cdots\)
1870.2.a.m 1870.a 1.a $2$ $14.932$ \(\Q(\sqrt{17}) \) None 1870.2.a.m \(2\) \(1\) \(2\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+q^{5}+\beta q^{6}+q^{8}+\cdots\)
1870.2.a.n 1870.a 1.a $3$ $14.932$ 3.3.564.1 None 1870.2.a.n \(-3\) \(1\) \(-3\) \(2\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
1870.2.a.o 1870.a 1.a $3$ $14.932$ 3.3.148.1 None 1870.2.a.o \(-3\) \(1\) \(3\) \(-4\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)
1870.2.a.p 1870.a 1.a $3$ $14.932$ 3.3.756.1 None 1870.2.a.p \(-3\) \(3\) \(3\) \(6\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
1870.2.a.q 1870.a 1.a $3$ $14.932$ 3.3.148.1 None 1870.2.a.q \(3\) \(-3\) \(3\) \(-6\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{2})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
1870.2.a.r 1870.a 1.a $3$ $14.932$ 3.3.940.1 None 1870.2.a.r \(3\) \(0\) \(-3\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
1870.2.a.s 1870.a 1.a $4$ $14.932$ 4.4.23252.1 None 1870.2.a.s \(4\) \(-3\) \(-4\) \(-4\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}-q^{5}+(-1+\cdots)q^{6}+\cdots\)
1870.2.a.t 1870.a 1.a $4$ $14.932$ 4.4.17428.1 None 1870.2.a.t \(4\) \(3\) \(-4\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}-q^{5}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
1870.2.a.u 1870.a 1.a $4$ $14.932$ 4.4.25492.1 None 1870.2.a.u \(4\) \(3\) \(4\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta _{2})q^{3}+q^{4}+q^{5}+(1+\beta _{2}+\cdots)q^{6}+\cdots\)
1870.2.a.v 1870.a 1.a $5$ $14.932$ 5.5.3102944.1 None 1870.2.a.v \(-5\) \(-1\) \(-5\) \(-8\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{3}q^{3}+q^{4}-q^{5}-\beta _{3}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1870))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1870)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(170))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(187))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(374))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(935))\)\(^{\oplus 2}\)