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Properties

Label 6069.2.a
Level $6069$
Weight $2$
Character orbit 6069.a
Rep. character $\chi_{6069}(1,\cdot)$
Character field $\Q$
Dimension $272$
Newform subspaces $39$
Sturm bound $1632$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 6069 = 3 \cdot 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6069.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 39 \)
Sturm bound: \(1632\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(5\), \(11\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6069))\).

Total New Old
Modular forms 852 272 580
Cusp forms 781 272 509
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.

\(3\)\(7\)\(17\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(33\)
\(+\)\(+\)\(-\)\(-\)\(36\)
\(+\)\(-\)\(+\)\(-\)\(31\)
\(+\)\(-\)\(-\)\(+\)\(36\)
\(-\)\(+\)\(+\)\(-\)\(40\)
\(-\)\(+\)\(-\)\(+\)\(28\)
\(-\)\(-\)\(+\)\(+\)\(24\)
\(-\)\(-\)\(-\)\(-\)\(44\)
Plus space\(+\)\(121\)
Minus space\(-\)\(151\)

Trace form

\( 272 q + 276 q^{4} + 8 q^{5} + 4 q^{6} - 2 q^{7} + 272 q^{9} + O(q^{10}) \) \( 272 q + 276 q^{4} + 8 q^{5} + 4 q^{6} - 2 q^{7} + 272 q^{9} - 8 q^{12} + 16 q^{13} - 2 q^{14} + 8 q^{15} + 276 q^{16} + 16 q^{19} + 2 q^{21} - 32 q^{22} - 8 q^{23} + 12 q^{24} + 296 q^{25} - 16 q^{26} - 6 q^{28} + 8 q^{29} + 16 q^{31} + 8 q^{33} - 4 q^{35} + 276 q^{36} + 8 q^{37} + 32 q^{38} + 8 q^{40} + 16 q^{41} + 2 q^{42} - 8 q^{43} + 40 q^{44} + 8 q^{45} + 8 q^{46} - 8 q^{47} + 272 q^{49} - 8 q^{50} + 80 q^{52} - 24 q^{53} + 4 q^{54} - 8 q^{55} - 18 q^{56} + 8 q^{57} + 16 q^{58} - 8 q^{59} + 16 q^{60} + 24 q^{61} - 8 q^{62} - 2 q^{63} + 284 q^{64} + 32 q^{65} + 8 q^{66} + 32 q^{67} + 16 q^{69} - 4 q^{70} - 8 q^{71} + 16 q^{73} + 8 q^{74} - 32 q^{78} + 40 q^{80} + 272 q^{81} + 8 q^{82} - 32 q^{83} + 6 q^{84} + 8 q^{87} - 72 q^{88} + 40 q^{89} - 4 q^{91} + 24 q^{92} + 16 q^{93} + 32 q^{94} - 72 q^{95} + 28 q^{96} - 8 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6069))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 7 17
6069.2.a.a 6069.a 1.a $1$ $48.461$ \(\Q\) None 357.2.a.a \(-2\) \(-1\) \(3\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+2q^{4}+3q^{5}+2q^{6}+\cdots\)
6069.2.a.b 6069.a 1.a $1$ $48.461$ \(\Q\) None 21.2.a.a \(-1\) \(-1\) \(2\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}+2q^{5}+q^{6}+q^{7}+\cdots\)
6069.2.a.c 6069.a 1.a $1$ $48.461$ \(\Q\) None 357.2.a.c \(0\) \(1\) \(-1\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-q^{5}-q^{7}+q^{9}+5q^{11}+\cdots\)
6069.2.a.d 6069.a 1.a $1$ $48.461$ \(\Q\) None 357.2.a.b \(0\) \(1\) \(-1\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-q^{5}+q^{7}+q^{9}-3q^{11}+\cdots\)
6069.2.a.e 6069.a 1.a $1$ $48.461$ \(\Q\) None 357.2.a.d \(2\) \(-1\) \(-1\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+2q^{4}-q^{5}-2q^{6}+q^{7}+\cdots\)
6069.2.a.f 6069.a 1.a $2$ $48.461$ \(\Q(\sqrt{3}) \) None 357.2.a.e \(-2\) \(-2\) \(4\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}-q^{3}+(2-2\beta )q^{4}+(2+\cdots)q^{5}+\cdots\)
6069.2.a.g 6069.a 1.a $2$ $48.461$ \(\Q(\sqrt{2}) \) None 357.2.a.f \(0\) \(2\) \(2\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+(1+\beta )q^{5}+\beta q^{6}+q^{7}+\cdots\)
6069.2.a.h 6069.a 1.a $2$ $48.461$ \(\Q(\sqrt{13}) \) None 6069.2.a.h \(1\) \(-2\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{3}+(1+\beta )q^{4}-\beta q^{6}-q^{7}+\cdots\)
6069.2.a.i 6069.a 1.a $2$ $48.461$ \(\Q(\sqrt{13}) \) None 6069.2.a.h \(1\) \(2\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+(1+\beta )q^{4}+\beta q^{6}+q^{7}+\cdots\)
6069.2.a.j 6069.a 1.a $3$ $48.461$ \(\Q(\zeta_{14})^+\) None 6069.2.a.j \(-2\) \(-3\) \(-2\) \(-3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
6069.2.a.k 6069.a 1.a $3$ $48.461$ 3.3.148.1 None 357.2.f.a \(-2\) \(-3\) \(3\) \(-3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-q^{3}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
6069.2.a.l 6069.a 1.a $3$ $48.461$ \(\Q(\zeta_{14})^+\) None 6069.2.a.j \(-2\) \(3\) \(2\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
6069.2.a.m 6069.a 1.a $3$ $48.461$ 3.3.148.1 None 357.2.f.a \(-2\) \(3\) \(-3\) \(3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+q^{3}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
6069.2.a.n 6069.a 1.a $3$ $48.461$ \(\Q(\zeta_{18})^+\) None 6069.2.a.n \(0\) \(-3\) \(0\) \(3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots\)
6069.2.a.o 6069.a 1.a $3$ $48.461$ 3.3.257.1 None 6069.2.a.o \(0\) \(-3\) \(2\) \(-3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}-q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
6069.2.a.p 6069.a 1.a $3$ $48.461$ \(\Q(\zeta_{18})^+\) None 6069.2.a.n \(0\) \(3\) \(0\) \(-3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(-\beta _{1}+\beta _{2})q^{5}+\cdots\)
6069.2.a.q 6069.a 1.a $3$ $48.461$ 3.3.257.1 None 6069.2.a.o \(0\) \(3\) \(-2\) \(3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
6069.2.a.r 6069.a 1.a $3$ $48.461$ 3.3.316.1 None 357.2.a.g \(1\) \(-3\) \(-2\) \(-3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
6069.2.a.s 6069.a 1.a $4$ $48.461$ 4.4.7232.1 None 357.2.a.h \(2\) \(4\) \(2\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+q^{3}+(1-\beta _{2}-\beta _{3})q^{4}-\beta _{1}q^{5}+\cdots\)
6069.2.a.t 6069.a 1.a $5$ $48.461$ 5.5.1669781.1 None 6069.2.a.t \(-1\) \(-5\) \(-4\) \(5\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
6069.2.a.u 6069.a 1.a $5$ $48.461$ 5.5.1669781.1 None 6069.2.a.t \(-1\) \(5\) \(4\) \(-5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots\)
6069.2.a.v 6069.a 1.a $5$ $48.461$ 5.5.1502576.1 None 357.2.f.b \(0\) \(-5\) \(3\) \(5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(1+\beta _{2}+\cdots)q^{5}+\cdots\)
6069.2.a.w 6069.a 1.a $5$ $48.461$ 5.5.1502576.1 None 357.2.f.b \(0\) \(5\) \(-3\) \(-5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+(-1-\beta _{2}+\cdots)q^{5}+\cdots\)
6069.2.a.x 6069.a 1.a $6$ $48.461$ 6.6.2803712.1 None 357.2.k.a \(0\) \(-6\) \(6\) \(-6\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}+(1+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
6069.2.a.y 6069.a 1.a $6$ $48.461$ 6.6.2803712.1 None 357.2.k.a \(0\) \(6\) \(-6\) \(6\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
6069.2.a.z 6069.a 1.a $9$ $48.461$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 6069.2.a.z \(0\) \(-9\) \(3\) \(-9\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1-\beta _{3}+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)
6069.2.a.ba 6069.a 1.a $9$ $48.461$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 6069.2.a.ba \(0\) \(-9\) \(3\) \(9\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1-\beta _{4}+\beta _{5})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
6069.2.a.bb 6069.a 1.a $9$ $48.461$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 6069.2.a.ba \(0\) \(9\) \(-3\) \(-9\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1-\beta _{4}+\beta _{5})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
6069.2.a.bc 6069.a 1.a $9$ $48.461$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 6069.2.a.z \(0\) \(9\) \(-3\) \(9\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1-\beta _{3}+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)
6069.2.a.bd 6069.a 1.a $10$ $48.461$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 357.2.k.b \(4\) \(-10\) \(6\) \(10\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
6069.2.a.be 6069.a 1.a $10$ $48.461$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 357.2.k.b \(4\) \(10\) \(-6\) \(-10\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
6069.2.a.bf 6069.a 1.a $15$ $48.461$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None 6069.2.a.bf \(0\) \(-15\) \(-9\) \(-15\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{12}+\cdots)q^{5}+\cdots\)
6069.2.a.bg 6069.a 1.a $15$ $48.461$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None 6069.2.a.bg \(0\) \(-15\) \(-9\) \(15\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{13}+\cdots)q^{5}+\cdots\)
6069.2.a.bh 6069.a 1.a $15$ $48.461$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None 6069.2.a.bg \(0\) \(15\) \(9\) \(-15\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{13}+\cdots)q^{5}+\cdots\)
6069.2.a.bi 6069.a 1.a $15$ $48.461$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None 6069.2.a.bf \(0\) \(15\) \(9\) \(15\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{12}+\cdots)q^{5}+\cdots\)
6069.2.a.bj 6069.a 1.a $16$ $48.461$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 357.2.u.a \(-4\) \(-16\) \(-4\) \(16\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{14}q^{5}+\cdots\)
6069.2.a.bk 6069.a 1.a $16$ $48.461$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 357.2.u.a \(-4\) \(16\) \(4\) \(-16\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{14}q^{5}+\cdots\)
6069.2.a.bl 6069.a 1.a $24$ $48.461$ None 357.2.u.b \(4\) \(-24\) \(-4\) \(-24\) $+$ $+$ $-$ $\mathrm{SU}(2)$
6069.2.a.bm 6069.a 1.a $24$ $48.461$ None 357.2.u.b \(4\) \(24\) \(4\) \(24\) $-$ $-$ $-$ $\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6069))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6069)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(119))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(289))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(357))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(867))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2023))\)\(^{\oplus 2}\)