↑

Properties

Label 1785.2.a
Level $1785$
Weight $2$
Character orbit 1785.a
Rep. character $\chi_{1785}(1,\cdot)$
Character field $\Q$
Dimension $65$
Newform subspaces $30$
Sturm bound $576$
Trace bound $7$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 1785 = 3 \cdot 5 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1785.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 30 \)
Sturm bound: \(576\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1785))\).

Total New Old
Modular forms 296 65 231
Cusp forms 281 65 216
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.

\(3\)\(5\)\(7\)\(17\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(5\)
\(+\)\(+\)\(+\)\(-\)$-$\(5\)
\(+\)\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(+\)\(-\)\(-\)$+$\(4\)
\(+\)\(-\)\(+\)\(+\)$-$\(5\)
\(+\)\(-\)\(+\)\(-\)$+$\(3\)
\(+\)\(-\)\(-\)\(+\)$+$\(4\)
\(+\)\(-\)\(-\)\(-\)$-$\(4\)
\(-\)\(+\)\(+\)\(+\)$-$\(5\)
\(-\)\(+\)\(+\)\(-\)$+$\(3\)
\(-\)\(+\)\(-\)\(+\)$+$\(2\)
\(-\)\(+\)\(-\)\(-\)$-$\(6\)
\(-\)\(-\)\(+\)\(+\)$+$\(3\)
\(-\)\(-\)\(+\)\(-\)$-$\(7\)
\(-\)\(-\)\(-\)\(+\)$-$\(6\)
\(-\)\(-\)\(-\)\(-\)$+$\(1\)
Plus space\(+\)\(25\)
Minus space\(-\)\(40\)

Trace form

\( 65 q - 5 q^{2} + q^{3} + 63 q^{4} + q^{5} + 3 q^{6} - 7 q^{7} - 9 q^{8} + 65 q^{9} + O(q^{10}) \) \( 65 q - 5 q^{2} + q^{3} + 63 q^{4} + q^{5} + 3 q^{6} - 7 q^{7} - 9 q^{8} + 65 q^{9} + 3 q^{10} - 4 q^{11} + 7 q^{12} + 14 q^{13} - 5 q^{14} + q^{15} + 71 q^{16} + q^{17} - 5 q^{18} + 20 q^{19} + 7 q^{20} + q^{21} + 4 q^{22} - 24 q^{23} + 15 q^{24} + 65 q^{25} + 10 q^{26} + q^{27} - 17 q^{28} - 2 q^{29} + 3 q^{30} + 7 q^{32} + 12 q^{33} + 3 q^{34} + q^{35} + 63 q^{36} - 10 q^{37} - 4 q^{38} + 14 q^{39} + 15 q^{40} - 6 q^{41} + 3 q^{42} + 12 q^{43} + 36 q^{44} + q^{45} + 24 q^{46} + 31 q^{48} + 65 q^{49} - 5 q^{50} + q^{51} + 98 q^{52} - 26 q^{53} + 3 q^{54} + 12 q^{55} - 9 q^{56} + 4 q^{57} - 6 q^{58} + 12 q^{59} + 7 q^{60} - 2 q^{61} + 32 q^{62} - 7 q^{63} + 135 q^{64} - 2 q^{65} + 4 q^{66} + 4 q^{67} + 7 q^{68} + 8 q^{69} + 3 q^{70} - 8 q^{71} - 9 q^{72} - 6 q^{73} + 2 q^{74} + q^{75} - 4 q^{76} - 20 q^{77} - 54 q^{78} - 33 q^{80} + 65 q^{81} - 18 q^{82} + 36 q^{83} + 7 q^{84} - 7 q^{85} - 28 q^{86} - 18 q^{87} + 4 q^{88} - 38 q^{89} + 3 q^{90} + 14 q^{91} + 24 q^{92} + 20 q^{95} - 17 q^{96} - 30 q^{97} - 5 q^{98} - 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1785))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 7 17
1785.2.a.a 1785.a 1.a $1$ $14.253$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(1\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
1785.2.a.b 1785.a 1.a $1$ $14.253$ \(\Q\) None \(-1\) \(-1\) \(1\) \(1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
1785.2.a.c 1785.a 1.a $1$ $14.253$ \(\Q\) None \(-1\) \(1\) \(-1\) \(1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
1785.2.a.d 1785.a 1.a $1$ $14.253$ \(\Q\) None \(-1\) \(1\) \(1\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
1785.2.a.e 1785.a 1.a $1$ $14.253$ \(\Q\) None \(-1\) \(1\) \(1\) \(1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
1785.2.a.f 1785.a 1.a $1$ $14.253$ \(\Q\) None \(0\) \(-1\) \(-1\) \(1\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}-q^{5}+q^{7}+q^{9}+2q^{11}+\cdots\)
1785.2.a.g 1785.a 1.a $1$ $14.253$ \(\Q\) None \(0\) \(-1\) \(1\) \(-1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}+q^{5}-q^{7}+q^{9}-2q^{11}+\cdots\)
1785.2.a.h 1785.a 1.a $1$ $14.253$ \(\Q\) None \(0\) \(1\) \(-1\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-q^{5}+q^{7}+q^{9}-6q^{11}+\cdots\)
1785.2.a.i 1785.a 1.a $1$ $14.253$ \(\Q\) None \(1\) \(-1\) \(-1\) \(-1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
1785.2.a.j 1785.a 1.a $1$ $14.253$ \(\Q\) None \(1\) \(-1\) \(-1\) \(1\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
1785.2.a.k 1785.a 1.a $1$ $14.253$ \(\Q\) None \(1\) \(-1\) \(1\) \(-1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
1785.2.a.l 1785.a 1.a $1$ $14.253$ \(\Q\) None \(1\) \(-1\) \(1\) \(1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
1785.2.a.m 1785.a 1.a $1$ $14.253$ \(\Q\) None \(1\) \(1\) \(-1\) \(-1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
1785.2.a.n 1785.a 1.a $1$ $14.253$ \(\Q\) None \(1\) \(1\) \(-1\) \(1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
1785.2.a.o 1785.a 1.a $1$ $14.253$ \(\Q\) None \(1\) \(1\) \(1\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
1785.2.a.p 1785.a 1.a $2$ $14.253$ \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(-2\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}-q^{5}+\cdots\)
1785.2.a.q 1785.a 1.a $2$ $14.253$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(2\) \(-2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{3}+q^{4}+q^{5}-\beta q^{6}-q^{7}+\cdots\)
1785.2.a.r 1785.a 1.a $2$ $14.253$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(2\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+3q^{4}+q^{5}+\beta q^{6}+q^{7}+\cdots\)
1785.2.a.s 1785.a 1.a $2$ $14.253$ \(\Q(\sqrt{17}) \) None \(1\) \(2\) \(-2\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+(2+\beta )q^{4}-q^{5}+\beta q^{6}+\cdots\)
1785.2.a.t 1785.a 1.a $2$ $14.253$ \(\Q(\sqrt{17}) \) None \(1\) \(2\) \(2\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+(2+\beta )q^{4}+q^{5}+\beta q^{6}+\cdots\)
1785.2.a.u 1785.a 1.a $3$ $14.253$ 3.3.148.1 None \(-3\) \(-3\) \(-3\) \(3\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1785.2.a.v 1785.a 1.a $3$ $14.253$ 3.3.148.1 None \(-1\) \(3\) \(-3\) \(-3\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}-q^{5}+\cdots\)
1785.2.a.w 1785.a 1.a $3$ $14.253$ 3.3.148.1 None \(-1\) \(3\) \(3\) \(-3\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+q^{5}+\cdots\)
1785.2.a.x 1785.a 1.a $3$ $14.253$ 3.3.564.1 None \(-1\) \(3\) \(3\) \(3\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)
1785.2.a.y 1785.a 1.a $4$ $14.253$ 4.4.8468.1 None \(-2\) \(-4\) \(4\) \(-4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}-q^{3}+(2-\beta _{1})q^{4}+q^{5}-\beta _{2}q^{6}+\cdots\)
1785.2.a.z 1785.a 1.a $4$ $14.253$ 4.4.8468.1 None \(-1\) \(-4\) \(4\) \(4\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)
1785.2.a.ba 1785.a 1.a $4$ $14.253$ 4.4.11348.1 None \(2\) \(-4\) \(-4\) \(-4\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{2}-q^{3}+(2+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1785.2.a.bb 1785.a 1.a $5$ $14.253$ 5.5.246832.1 None \(-2\) \(-5\) \(-5\) \(-5\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{4})q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
1785.2.a.bc 1785.a 1.a $5$ $14.253$ 5.5.674848.1 None \(-1\) \(5\) \(-5\) \(5\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
1785.2.a.bd 1785.a 1.a $6$ $14.253$ 6.6.17749184.1 None \(3\) \(6\) \(6\) \(-6\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+q^{3}+(2+\beta _{4})q^{4}+q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1785))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1785)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(119))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(255))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(357))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(595))\)\(^{\oplus 2}\)