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Properties

Label 1617.2.a
Level $1617$
Weight $2$
Character orbit 1617.a
Rep. character $\chi_{1617}(1,\cdot)$
Character field $\Q$
Dimension $68$
Newform subspaces $28$
Sturm bound $448$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1617.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 28 \)
Sturm bound: \(448\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(2\), \(5\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1617))\).

Total New Old
Modular forms 240 68 172
Cusp forms 209 68 141
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.

\(3\)\(7\)\(11\)FrickeDim
\(+\)\(+\)\(+\)$+$\(7\)
\(+\)\(+\)\(-\)$-$\(9\)
\(+\)\(-\)\(+\)$-$\(10\)
\(+\)\(-\)\(-\)$+$\(7\)
\(-\)\(+\)\(+\)$-$\(8\)
\(-\)\(+\)\(-\)$+$\(6\)
\(-\)\(-\)\(+\)$+$\(9\)
\(-\)\(-\)\(-\)$-$\(12\)
Plus space\(+\)\(29\)
Minus space\(-\)\(39\)

Trace form

\( 68 q - 2 q^{2} + 2 q^{3} + 64 q^{4} - 8 q^{5} - 4 q^{6} + 6 q^{8} + 68 q^{9} + O(q^{10}) \) \( 68 q - 2 q^{2} + 2 q^{3} + 64 q^{4} - 8 q^{5} - 4 q^{6} + 6 q^{8} + 68 q^{9} + 4 q^{10} + 6 q^{12} - 4 q^{15} + 64 q^{16} - 20 q^{17} - 2 q^{18} - 4 q^{19} - 2 q^{22} - 12 q^{24} + 60 q^{25} + 12 q^{26} + 2 q^{27} - 4 q^{29} + 16 q^{30} + 16 q^{31} + 30 q^{32} + 2 q^{33} + 16 q^{34} + 64 q^{36} - 28 q^{37} + 36 q^{38} - 16 q^{39} + 36 q^{40} - 28 q^{41} + 8 q^{43} + 8 q^{44} - 8 q^{45} + 32 q^{46} + 16 q^{47} + 30 q^{48} - 14 q^{50} + 56 q^{52} - 72 q^{53} - 4 q^{54} + 8 q^{55} - 8 q^{57} - 8 q^{58} + 24 q^{59} + 20 q^{60} - 32 q^{61} + 24 q^{62} + 56 q^{64} - 112 q^{65} + 4 q^{66} + 4 q^{67} - 28 q^{68} - 8 q^{69} + 8 q^{71} + 6 q^{72} - 24 q^{73} - 124 q^{74} + 14 q^{75} - 4 q^{76} + 32 q^{78} - 16 q^{79} - 8 q^{80} + 68 q^{81} + 16 q^{82} - 8 q^{83} - 80 q^{85} + 4 q^{86} + 6 q^{88} - 56 q^{89} + 4 q^{90} + 32 q^{92} - 28 q^{93} - 32 q^{94} + 72 q^{95} - 28 q^{96} - 16 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1617))\) 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 7 11
1617.2.a.a 1617.a 1.a $1$ $12.912$ \(\Q\) None \(-2\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+q^{9}-q^{11}+\cdots\)
1617.2.a.b 1617.a 1.a $1$ $12.912$ \(\Q\) None \(-2\) \(1\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+2q^{4}-2q^{6}+q^{9}-q^{11}+\cdots\)
1617.2.a.c 1617.a 1.a $1$ $12.912$ \(\Q\) None \(-1\) \(-1\) \(4\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}+4q^{5}+q^{6}+3q^{8}+\cdots\)
1617.2.a.d 1617.a 1.a $1$ $12.912$ \(\Q\) None \(-1\) \(1\) \(-4\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}-4q^{5}-q^{6}+3q^{8}+\cdots\)
1617.2.a.e 1617.a 1.a $1$ $12.912$ \(\Q\) None \(-1\) \(1\) \(2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}+2q^{5}-q^{6}+3q^{8}+\cdots\)
1617.2.a.f 1617.a 1.a $1$ $12.912$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-q^{6}-3q^{8}+q^{9}+\cdots\)
1617.2.a.g 1617.a 1.a $1$ $12.912$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-q^{6}-3q^{8}+q^{9}+\cdots\)
1617.2.a.h 1617.a 1.a $1$ $12.912$ \(\Q\) None \(1\) \(1\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}+q^{9}+\cdots\)
1617.2.a.i 1617.a 1.a $1$ $12.912$ \(\Q\) None \(1\) \(1\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}+q^{9}+\cdots\)
1617.2.a.j 1617.a 1.a $1$ $12.912$ \(\Q\) None \(1\) \(1\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}+2q^{5}+q^{6}-3q^{8}+\cdots\)
1617.2.a.k 1617.a 1.a $2$ $12.912$ \(\Q(\sqrt{5}) \) None \(-3\) \(-2\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}-q^{3}+3\beta q^{4}+(1-2\beta )q^{5}+\cdots\)
1617.2.a.l 1617.a 1.a $2$ $12.912$ \(\Q(\sqrt{5}) \) None \(-3\) \(2\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+q^{3}+3\beta q^{4}+(-1+\cdots)q^{5}+\cdots\)
1617.2.a.m 1617.a 1.a $2$ $12.912$ \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(4\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}+2q^{5}+\cdots\)
1617.2.a.n 1617.a 1.a $2$ $12.912$ \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(-4\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}-2q^{5}+\cdots\)
1617.2.a.o 1617.a 1.a $2$ $12.912$ \(\Q(\sqrt{21}) \) None \(-1\) \(2\) \(-6\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+q^{3}+(3+\beta )q^{4}-3q^{5}-\beta q^{6}+\cdots\)
1617.2.a.p 1617.a 1.a $2$ $12.912$ \(\Q(\sqrt{5}) \) None \(1\) \(-2\) \(-2\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}-q^{5}-\beta q^{6}+\cdots\)
1617.2.a.q 1617.a 1.a $3$ $12.912$ 3.3.229.1 None \(0\) \(-3\) \(-4\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1617.2.a.r 1617.a 1.a $3$ $12.912$ 3.3.229.1 None \(0\) \(3\) \(4\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
1617.2.a.s 1617.a 1.a $3$ $12.912$ 3.3.837.1 None \(0\) \(3\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1617.2.a.t 1617.a 1.a $3$ $12.912$ 3.3.229.1 None \(2\) \(-3\) \(-4\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{2}-q^{3}+(2+\beta _{1})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1617.2.a.u 1617.a 1.a $4$ $12.912$ 4.4.2624.1 None \(-2\) \(-4\) \(8\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-q^{3}+\beta _{2}q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
1617.2.a.v 1617.a 1.a $4$ $12.912$ 4.4.2624.1 None \(-2\) \(4\) \(-8\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+q^{3}+\beta _{2}q^{4}+(-1+\cdots)q^{5}+\cdots\)
1617.2.a.w 1617.a 1.a $4$ $12.912$ 4.4.2624.1 None \(2\) \(-4\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-q^{3}+\beta _{2}q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots\)
1617.2.a.x 1617.a 1.a $4$ $12.912$ 4.4.11344.1 None \(2\) \(-4\) \(-4\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1617.2.a.y 1617.a 1.a $4$ $12.912$ 4.4.2624.1 None \(2\) \(4\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+q^{3}+\beta _{2}q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
1617.2.a.z 1617.a 1.a $4$ $12.912$ 4.4.11344.1 None \(2\) \(4\) \(4\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
1617.2.a.ba 1617.a 1.a $5$ $12.912$ 5.5.3676752.1 None \(2\) \(-5\) \(-4\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(-1+\beta _{4})q^{5}+\cdots\)
1617.2.a.bb 1617.a 1.a $5$ $12.912$ 5.5.3676752.1 None \(2\) \(5\) \(4\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+(1-\beta _{4})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1617))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1617)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(231))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(539))\)\(^{\oplus 2}\)