Properties

Label 1859.2.a
Level $1859$
Weight $2$
Character orbit 1859.a
Rep. character $\chi_{1859}(1,\cdot)$
Character field $\Q$
Dimension $128$
Newform subspaces $20$
Sturm bound $364$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1859.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 20 \)
Sturm bound: \(364\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1859))\).

Total New Old
Modular forms 196 128 68
Cusp forms 169 128 41
Eisenstein series 27 0 27

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

\(11\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(26\)
\(+\)\(-\)\(-\)\(37\)
\(-\)\(+\)\(-\)\(40\)
\(-\)\(-\)\(+\)\(25\)
Plus space\(+\)\(51\)
Minus space\(-\)\(77\)

Trace form

\( 128 q - q^{2} - q^{3} + 129 q^{4} - q^{5} + 2 q^{6} - 6 q^{7} - 3 q^{8} + 129 q^{9} + O(q^{10}) \) \( 128 q - q^{2} - q^{3} + 129 q^{4} - q^{5} + 2 q^{6} - 6 q^{7} - 3 q^{8} + 129 q^{9} + 4 q^{10} + 2 q^{11} + 2 q^{12} + 12 q^{14} + 7 q^{15} + 127 q^{16} + 5 q^{18} + 16 q^{20} + 10 q^{21} - q^{22} - 17 q^{23} + 36 q^{24} + 109 q^{25} - 19 q^{27} - 8 q^{28} + 6 q^{29} + 34 q^{30} + 3 q^{31} + 13 q^{32} + 3 q^{33} + 30 q^{34} + 14 q^{35} + 147 q^{36} - 19 q^{37} + 4 q^{38} + 14 q^{40} - 6 q^{41} - 24 q^{42} - 18 q^{43} + q^{44} - 2 q^{46} + 8 q^{47} - 8 q^{48} + 120 q^{49} - 33 q^{50} - 2 q^{51} + 4 q^{53} - 26 q^{54} - q^{55} - 4 q^{56} - 52 q^{57} - 18 q^{58} + q^{59} + 2 q^{60} - 14 q^{61} - 38 q^{62} - 4 q^{63} + 103 q^{64} - 14 q^{66} - 3 q^{67} - 10 q^{68} + 13 q^{69} - 8 q^{70} + 41 q^{71} + q^{72} - 42 q^{73} - 56 q^{74} + 12 q^{75} + 36 q^{76} - 2 q^{77} - 14 q^{79} - 22 q^{80} + 136 q^{81} - 54 q^{82} + 34 q^{83} - 48 q^{84} - 38 q^{85} - 4 q^{86} + 12 q^{87} - 15 q^{88} + 5 q^{89} - 30 q^{90} - 122 q^{92} + 5 q^{93} - 24 q^{94} + 32 q^{95} + 72 q^{96} - 29 q^{97} + 11 q^{98} + 11 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1859))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 11 13
1859.2.a.a 1859.a 1.a $1$ $14.844$ \(\Q\) None \(0\) \(-1\) \(1\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}+q^{5}+2q^{7}-2q^{9}+q^{11}+\cdots\)
1859.2.a.b 1859.a 1.a $1$ $14.844$ \(\Q\) None \(2\) \(-1\) \(-1\) \(2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+2q^{4}-q^{5}-2q^{6}+2q^{7}+\cdots\)
1859.2.a.c 1859.a 1.a $2$ $14.844$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(-6\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1-\beta )q^{3}+q^{4}-\beta q^{5}+\cdots\)
1859.2.a.d 1859.a 1.a $2$ $14.844$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{3}+q^{4}-\beta q^{5}+\cdots\)
1859.2.a.e 1859.a 1.a $3$ $14.844$ 3.3.756.1 None \(-3\) \(0\) \(-3\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}-q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
1859.2.a.f 1859.a 1.a $3$ $14.844$ 3.3.361.1 None \(-1\) \(1\) \(-1\) \(5\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+\beta _{2}q^{3}+(2+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1859.2.a.g 1859.a 1.a $3$ $14.844$ 3.3.361.1 None \(1\) \(1\) \(1\) \(-5\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1}-\beta _{2})q^{2}+\beta _{1}q^{3}+(2+\beta _{1}+\cdots)q^{4}+\cdots\)
1859.2.a.h 1859.a 1.a $3$ $14.844$ 3.3.756.1 None \(3\) \(0\) \(3\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}-q^{4}+(1+\beta _{2})q^{5}+\beta _{1}q^{6}+\cdots\)
1859.2.a.i 1859.a 1.a $4$ $14.844$ 4.4.1957.1 None \(-3\) \(0\) \(0\) \(-6\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1}-\beta _{2})q^{2}+(-\beta _{2}-\beta _{3})q^{3}+\cdots\)
1859.2.a.j 1859.a 1.a $6$ $14.844$ 6.6.84512328.1 None \(-1\) \(-2\) \(-4\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1859.2.a.k 1859.a 1.a $6$ $14.844$ 6.6.28561300.1 None \(0\) \(1\) \(-6\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+(-\beta _{2}-\beta _{4})q^{3}+(1+\beta _{2}+\cdots)q^{4}+\cdots\)
1859.2.a.l 1859.a 1.a $6$ $14.844$ 6.6.28561300.1 None \(0\) \(1\) \(6\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(-\beta _{2}-\beta _{4})q^{3}+(1+\beta _{2}+\cdots)q^{4}+\cdots\)
1859.2.a.m 1859.a 1.a $6$ $14.844$ 6.6.194616205.1 None \(0\) \(3\) \(-1\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1859.2.a.n 1859.a 1.a $6$ $14.844$ 6.6.84512328.1 None \(1\) \(-2\) \(4\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1859.2.a.o 1859.a 1.a $8$ $14.844$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-2\) \(0\) \(-8\) \(-14\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+\beta _{6}q^{3}+(\beta _{1}+\beta _{2}+\beta _{4}-\beta _{5}+\cdots)q^{4}+\cdots\)
1859.2.a.p 1859.a 1.a $8$ $14.844$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(2\) \(0\) \(8\) \(14\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{6}q^{3}+(\beta _{1}+\beta _{2}+\beta _{4}-\beta _{5}+\cdots)q^{4}+\cdots\)
1859.2.a.q 1859.a 1.a $9$ $14.844$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-5\) \(-4\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1-\beta _{1}+\beta _{3})q^{3}+(1-\beta _{5}+\cdots)q^{4}+\cdots\)
1859.2.a.r 1859.a 1.a $9$ $14.844$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-5\) \(4\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-1-\beta _{1}+\beta _{3})q^{3}+(1-\beta _{5}+\cdots)q^{4}+\cdots\)
1859.2.a.s 1859.a 1.a $21$ $14.844$ None \(0\) \(6\) \(-7\) \(-1\) $+$ $-$ $\mathrm{SU}(2)$
1859.2.a.t 1859.a 1.a $21$ $14.844$ None \(0\) \(6\) \(7\) \(1\) $-$ $+$ $\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1859))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1859)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(143))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)