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Properties

Label 6358.2.a
Level $6358$
Weight $2$
Character orbit 6358.a
Rep. character $\chi_{6358}(1,\cdot)$
Character field $\Q$
Dimension $225$
Newform subspaces $43$
Sturm bound $1836$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 6358 = 2 \cdot 11 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6358.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 43 \)
Sturm bound: \(1836\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6358))\).

Total New Old
Modular forms 954 225 729
Cusp forms 883 225 658
Eisenstein series 71 0 71

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

\(2\)\(11\)\(17\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(27\)
\(+\)\(+\)\(-\)\(-\)\(29\)
\(+\)\(-\)\(+\)\(-\)\(27\)
\(+\)\(-\)\(-\)\(+\)\(29\)
\(-\)\(+\)\(+\)\(-\)\(32\)
\(-\)\(+\)\(-\)\(+\)\(25\)
\(-\)\(-\)\(+\)\(+\)\(23\)
\(-\)\(-\)\(-\)\(-\)\(33\)
Plus space\(+\)\(104\)
Minus space\(-\)\(121\)

Trace form

\( 225 q + q^{2} - 4 q^{3} + 225 q^{4} - 2 q^{5} - 4 q^{6} - 8 q^{7} + q^{8} + 217 q^{9} + O(q^{10}) \) \( 225 q + q^{2} - 4 q^{3} + 225 q^{4} - 2 q^{5} - 4 q^{6} - 8 q^{7} + q^{8} + 217 q^{9} + 6 q^{10} - q^{11} - 4 q^{12} - 2 q^{13} - 8 q^{14} + 225 q^{16} - 3 q^{18} - 12 q^{19} - 2 q^{20} - 8 q^{21} - q^{22} - 16 q^{23} - 4 q^{24} + 211 q^{25} + 10 q^{26} + 8 q^{27} - 8 q^{28} + 6 q^{29} - 16 q^{31} + q^{32} + 32 q^{35} + 217 q^{36} - 10 q^{37} + 8 q^{38} + 6 q^{40} + 18 q^{41} + 32 q^{42} - 12 q^{43} - q^{44} + 38 q^{45} - 8 q^{46} + 32 q^{47} - 4 q^{48} + 209 q^{49} + 15 q^{50} - 2 q^{52} + 18 q^{53} + 8 q^{54} - 6 q^{55} - 8 q^{56} + 40 q^{57} + 22 q^{58} + 32 q^{59} - 42 q^{61} + 16 q^{62} + 32 q^{63} + 225 q^{64} + 36 q^{65} + 4 q^{66} - 24 q^{67} + 40 q^{69} + 24 q^{70} + 24 q^{71} - 3 q^{72} + 10 q^{73} + 30 q^{74} + 68 q^{75} - 12 q^{76} + 8 q^{77} + 32 q^{78} - 32 q^{79} - 2 q^{80} + 233 q^{81} + 18 q^{82} + 52 q^{83} - 8 q^{84} + 8 q^{86} + 88 q^{87} - q^{88} + 42 q^{89} + 78 q^{90} - 16 q^{92} + 96 q^{93} + 48 q^{95} - 4 q^{96} + 10 q^{97} + 25 q^{98} - 13 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6358))\) 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 11 17
6358.2.a.a 6358.a 1.a $1$ $50.769$ \(\Q\) None 6358.2.a.a \(-1\) \(-3\) \(3\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}+q^{4}+3q^{5}+3q^{6}-2q^{7}+\cdots\)
6358.2.a.b 6358.a 1.a $1$ $50.769$ \(\Q\) None 6358.2.a.b \(-1\) \(-2\) \(0\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+2q^{6}+2q^{7}-q^{8}+\cdots\)
6358.2.a.c 6358.a 1.a $1$ $50.769$ \(\Q\) None 6358.2.a.c \(-1\) \(-1\) \(-3\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-3q^{5}+q^{6}-2q^{7}+\cdots\)
6358.2.a.d 6358.a 1.a $1$ $50.769$ \(\Q\) None 374.2.a.a \(-1\) \(0\) \(0\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2q^{7}-q^{8}-3q^{9}+q^{11}+\cdots\)
6358.2.a.e 6358.a 1.a $1$ $50.769$ \(\Q\) None 6358.2.a.c \(-1\) \(1\) \(3\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+3q^{5}-q^{6}+2q^{7}+\cdots\)
6358.2.a.f 6358.a 1.a $1$ $50.769$ \(\Q\) None 6358.2.a.b \(-1\) \(2\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-2q^{7}-q^{8}+\cdots\)
6358.2.a.g 6358.a 1.a $1$ $50.769$ \(\Q\) None 6358.2.a.a \(-1\) \(3\) \(-3\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}-3q^{5}-3q^{6}+2q^{7}+\cdots\)
6358.2.a.h 6358.a 1.a $1$ $50.769$ \(\Q\) None 6358.2.a.h \(1\) \(-2\) \(0\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-2q^{6}+2q^{7}+q^{8}+\cdots\)
6358.2.a.i 6358.a 1.a $1$ $50.769$ \(\Q\) None 6358.2.a.h \(1\) \(2\) \(0\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}-2q^{7}+q^{8}+\cdots\)
6358.2.a.j 6358.a 1.a $2$ $50.769$ \(\Q(\sqrt{2}) \) None 6358.2.a.j \(-2\) \(-4\) \(4\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-2+\beta )q^{3}+q^{4}+(2-\beta )q^{5}+\cdots\)
6358.2.a.k 6358.a 1.a $2$ $50.769$ \(\Q(\sqrt{2}) \) None 6358.2.a.j \(-2\) \(4\) \(-4\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(2+\beta )q^{3}+q^{4}+(-2-\beta )q^{5}+\cdots\)
6358.2.a.l 6358.a 1.a $2$ $50.769$ \(\Q(\sqrt{2}) \) None 6358.2.a.l \(2\) \(-2\) \(-2\) \(4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-1-2\beta )q^{5}+\cdots\)
6358.2.a.m 6358.a 1.a $2$ $50.769$ \(\Q(\sqrt{2}) \) None 6358.2.a.l \(2\) \(2\) \(2\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1-2\beta )q^{5}+\cdots\)
6358.2.a.n 6358.a 1.a $3$ $50.769$ 3.3.148.1 None 6358.2.a.n \(-3\) \(-3\) \(1\) \(4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots\)
6358.2.a.o 6358.a 1.a $3$ $50.769$ 3.3.785.1 None 374.2.a.b \(-3\) \(1\) \(1\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{2}q^{5}-\beta _{1}q^{6}+\cdots\)
6358.2.a.p 6358.a 1.a $3$ $50.769$ 3.3.148.1 None 6358.2.a.n \(-3\) \(3\) \(-1\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta _{2})q^{3}+q^{4}+(-\beta _{1}+\beta _{2})q^{5}+\cdots\)
6358.2.a.q 6358.a 1.a $3$ $50.769$ \(\Q(\zeta_{18})^+\) None 374.2.b.a \(3\) \(-3\) \(-3\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta _{1})q^{3}+q^{4}+(-1-2\beta _{1}+\cdots)q^{5}+\cdots\)
6358.2.a.r 6358.a 1.a $3$ $50.769$ 3.3.257.1 None 374.2.a.c \(3\) \(-3\) \(1\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(\beta _{1}+\beta _{2})q^{5}+\cdots\)
6358.2.a.s 6358.a 1.a $3$ $50.769$ \(\Q(\zeta_{18})^+\) None 374.2.b.a \(3\) \(3\) \(3\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta _{1})q^{3}+q^{4}+(1+2\beta _{1}+\cdots)q^{5}+\cdots\)
6358.2.a.t 6358.a 1.a $4$ $50.769$ 4.4.55585.1 None 374.2.a.d \(-4\) \(-1\) \(-5\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
6358.2.a.u 6358.a 1.a $4$ $50.769$ 4.4.10273.1 None 374.2.b.b \(-4\) \(-1\) \(-3\) \(3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{2}q^{3}+q^{4}+(-1+\beta _{1}-\beta _{3})q^{5}+\cdots\)
6358.2.a.v 6358.a 1.a $4$ $50.769$ 4.4.10273.1 None 374.2.b.b \(-4\) \(1\) \(3\) \(-3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{2}q^{3}+q^{4}+(1-\beta _{1}+\beta _{3})q^{5}+\cdots\)
6358.2.a.w 6358.a 1.a $4$ $50.769$ \(\Q(\zeta_{16})^+\) None 374.2.h.a \(4\) \(-8\) \(-8\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}+(-2-\beta _{2}-\beta _{3})q^{5}+\cdots\)
6358.2.a.x 6358.a 1.a $4$ $50.769$ 4.4.17417.1 None 374.2.a.e \(4\) \(-1\) \(1\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{3}q^{3}+q^{4}+(1+\beta _{2}+\beta _{3})q^{5}+\cdots\)
6358.2.a.y 6358.a 1.a $4$ $50.769$ \(\Q(\zeta_{16})^+\) None 374.2.h.a \(4\) \(8\) \(8\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+(2+\beta _{2}-\beta _{3})q^{5}+\cdots\)
6358.2.a.z 6358.a 1.a $5$ $50.769$ 5.5.4403552.1 None 6358.2.a.z \(5\) \(-1\) \(-1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{2}q^{5}-\beta _{1}q^{6}+\cdots\)
6358.2.a.ba 6358.a 1.a $5$ $50.769$ 5.5.4403552.1 None 6358.2.a.z \(5\) \(1\) \(1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{2}q^{5}+\beta _{1}q^{6}+\cdots\)
6358.2.a.bb 6358.a 1.a $6$ $50.769$ 6.6.26427904.1 None 374.2.f.a \(-6\) \(-2\) \(-6\) \(2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+(-1+\beta _{3}+\beta _{4}+\cdots)q^{5}+\cdots\)
6358.2.a.bc 6358.a 1.a $6$ $50.769$ 6.6.26427904.1 None 374.2.f.a \(-6\) \(2\) \(6\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+(1-\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots\)
6358.2.a.bd 6358.a 1.a $6$ $50.769$ 6.6.1292517.1 None 6358.2.a.bd \(6\) \(0\) \(-3\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+(-\beta _{1}-\beta _{2})q^{5}+\cdots\)
6358.2.a.be 6358.a 1.a $6$ $50.769$ 6.6.1292517.1 None 6358.2.a.bd \(6\) \(0\) \(3\) \(3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+(\beta _{1}+\beta _{2})q^{5}+\cdots\)
6358.2.a.bf 6358.a 1.a $8$ $50.769$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 374.2.f.b \(8\) \(-6\) \(-6\) \(-10\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1+\beta _{5}+\cdots)q^{5}+\cdots\)
6358.2.a.bg 6358.a 1.a $8$ $50.769$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 374.2.f.b \(8\) \(6\) \(6\) \(10\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+(1-\beta _{5})q^{5}+\cdots\)
6358.2.a.bh 6358.a 1.a $9$ $50.769$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 6358.2.a.bh \(-9\) \(0\) \(-3\) \(3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-\beta _{1}+\beta _{2})q^{3}+q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots\)
6358.2.a.bi 6358.a 1.a $9$ $50.769$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 6358.2.a.bi \(-9\) \(0\) \(-3\) \(9\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{2}q^{5}+\beta _{1}q^{6}+\cdots\)
6358.2.a.bj 6358.a 1.a $9$ $50.769$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 6358.2.a.bi \(-9\) \(0\) \(3\) \(-9\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{2}q^{5}-\beta _{1}q^{6}+\cdots\)
6358.2.a.bk 6358.a 1.a $9$ $50.769$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 6358.2.a.bh \(-9\) \(0\) \(3\) \(-3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(\beta _{1}-\beta _{2})q^{3}+q^{4}+(\beta _{2}+\beta _{8})q^{5}+\cdots\)
6358.2.a.bl 6358.a 1.a $12$ $50.769$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 374.2.h.b \(12\) \(-4\) \(-4\) \(-12\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{6}q^{3}+q^{4}+(-\beta _{1}+\beta _{9})q^{5}+\cdots\)
6358.2.a.bm 6358.a 1.a $12$ $50.769$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 6358.2.a.bm \(12\) \(0\) \(-3\) \(-9\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{11}q^{5}+\beta _{1}q^{6}+\cdots\)
6358.2.a.bn 6358.a 1.a $12$ $50.769$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 6358.2.a.bm \(12\) \(0\) \(3\) \(9\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{11}q^{5}-\beta _{1}q^{6}+\cdots\)
6358.2.a.bo 6358.a 1.a $12$ $50.769$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 374.2.h.b \(12\) \(4\) \(4\) \(12\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{6}q^{3}+q^{4}+(\beta _{1}-\beta _{9})q^{5}+\cdots\)
6358.2.a.bp 6358.a 1.a $16$ $50.769$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 374.2.h.c \(-16\) \(-4\) \(-12\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+(-1-\beta _{13})q^{5}+\cdots\)
6358.2.a.bq 6358.a 1.a $16$ $50.769$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 374.2.h.c \(-16\) \(4\) \(12\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+(1+\beta _{13})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6358))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6358)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(187))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(289))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(374))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(578))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3179))\)\(^{\oplus 2}\)