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Properties

Label 6909.2.a
Level $6909$
Weight $2$
Character orbit 6909.a
Rep. character $\chi_{6909}(1,\cdot)$
Character field $\Q$
Dimension $314$
Newform subspaces $38$
Sturm bound $1792$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 6909 = 3 \cdot 7^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6909.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 38 \)
Sturm bound: \(1792\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6909))\).

Total New Old
Modular forms 912 314 598
Cusp forms 881 314 567
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.

\(3\)\(7\)\(47\)FrickeDim
\(+\)\(+\)\(+\)$+$\(34\)
\(+\)\(+\)\(-\)$-$\(42\)
\(+\)\(-\)\(+\)$-$\(43\)
\(+\)\(-\)\(-\)$+$\(37\)
\(-\)\(+\)\(+\)$-$\(41\)
\(-\)\(+\)\(-\)$+$\(33\)
\(-\)\(-\)\(+\)$+$\(39\)
\(-\)\(-\)\(-\)$-$\(45\)
Plus space\(+\)\(143\)
Minus space\(-\)\(171\)

Trace form

\( 314 q + 2 q^{2} + 2 q^{3} + 318 q^{4} - 8 q^{5} + 2 q^{6} + 6 q^{8} + 314 q^{9} + O(q^{10}) \) \( 314 q + 2 q^{2} + 2 q^{3} + 318 q^{4} - 8 q^{5} + 2 q^{6} + 6 q^{8} + 314 q^{9} + 8 q^{11} + 6 q^{12} - 4 q^{13} + 342 q^{16} - 16 q^{17} + 2 q^{18} + 4 q^{19} - 24 q^{20} + 8 q^{23} + 6 q^{24} + 314 q^{25} + 4 q^{26} + 2 q^{27} + 8 q^{29} + 24 q^{30} + 20 q^{31} + 14 q^{32} + 8 q^{33} + 28 q^{34} + 318 q^{36} - 4 q^{37} + 28 q^{38} - 16 q^{39} + 16 q^{40} - 24 q^{41} + 8 q^{43} + 64 q^{44} - 8 q^{45} + 64 q^{46} + 14 q^{48} - 54 q^{50} + 8 q^{51} - 12 q^{52} - 64 q^{53} + 2 q^{54} + 12 q^{55} - 48 q^{58} + 52 q^{59} + 16 q^{60} + 12 q^{61} - 28 q^{62} + 374 q^{64} - 76 q^{65} - 16 q^{66} - 8 q^{67} + 4 q^{68} + 72 q^{71} + 6 q^{72} - 12 q^{73} - 24 q^{74} + 6 q^{75} - 44 q^{76} - 12 q^{78} - 8 q^{79} - 48 q^{80} + 314 q^{81} - 48 q^{82} - 12 q^{83} - 48 q^{85} - 116 q^{86} - 16 q^{87} - 112 q^{88} - 44 q^{89} - 200 q^{92} + 8 q^{93} + 8 q^{94} - 32 q^{95} + 14 q^{96} + 64 q^{97} + 8 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6909))\) 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 47
6909.2.a.a 6909.a 1.a $1$ $55.169$ \(\Q\) None \(-2\) \(-1\) \(3\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+2q^{4}+3q^{5}+2q^{6}+\cdots\)
6909.2.a.b 6909.a 1.a $1$ $55.169$ \(\Q\) None \(-1\) \(-1\) \(-2\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}-2q^{5}+q^{6}+3q^{8}+\cdots\)
6909.2.a.c 6909.a 1.a $1$ $55.169$ \(\Q\) None \(-1\) \(-1\) \(2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}+2q^{5}+q^{6}+3q^{8}+\cdots\)
6909.2.a.d 6909.a 1.a $1$ $55.169$ \(\Q\) None \(-1\) \(1\) \(-2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}-2q^{5}-q^{6}+3q^{8}+\cdots\)
6909.2.a.e 6909.a 1.a $1$ $55.169$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}-q^{6}+3q^{8}+q^{9}+\cdots\)
6909.2.a.f 6909.a 1.a $1$ $55.169$ \(\Q\) None \(0\) \(-1\) \(2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}+2q^{5}+q^{9}-2q^{11}+\cdots\)
6909.2.a.g 6909.a 1.a $1$ $55.169$ \(\Q\) None \(0\) \(1\) \(-2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-2q^{5}+q^{9}-2q^{11}+\cdots\)
6909.2.a.h 6909.a 1.a $1$ $55.169$ \(\Q\) None \(0\) \(1\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}+q^{5}+q^{9}-3q^{11}-2q^{12}+\cdots\)
6909.2.a.i 6909.a 1.a $1$ $55.169$ \(\Q\) None \(1\) \(1\) \(-4\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}-4q^{5}+q^{6}-3q^{8}+\cdots\)
6909.2.a.j 6909.a 1.a $1$ $55.169$ \(\Q\) None \(2\) \(-1\) \(-4\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+2q^{4}-4q^{5}-2q^{6}+\cdots\)
6909.2.a.k 6909.a 1.a $1$ $55.169$ \(\Q\) None \(2\) \(-1\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+2q^{4}+q^{5}-2q^{6}+q^{9}+\cdots\)
6909.2.a.l 6909.a 1.a $1$ $55.169$ \(\Q\) None \(2\) \(1\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+q^{3}+2q^{4}+2q^{6}+q^{9}+3q^{11}+\cdots\)
6909.2.a.m 6909.a 1.a $2$ $55.169$ \(\Q(\sqrt{17}) \) None \(-1\) \(2\) \(-1\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+q^{3}+(2+\beta )q^{4}+(-1+\beta )q^{5}+\cdots\)
6909.2.a.n 6909.a 1.a $2$ $55.169$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+q^{3}+3q^{4}+(1-\beta )q^{5}-\beta q^{6}+\cdots\)
6909.2.a.o 6909.a 1.a $3$ $55.169$ 3.3.169.1 None \(-1\) \(-3\) \(1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
6909.2.a.p 6909.a 1.a $3$ $55.169$ \(\Q(\zeta_{14})^+\) None \(-1\) \(3\) \(1\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}-\beta _{2}q^{5}-\beta _{1}q^{6}+\cdots\)
6909.2.a.q 6909.a 1.a $3$ $55.169$ \(\Q(\zeta_{14})^+\) None \(1\) \(3\) \(3\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(2-2\beta _{1}+\beta _{2})q^{5}+\cdots\)
6909.2.a.r 6909.a 1.a $4$ $55.169$ 4.4.2777.1 None \(-2\) \(-4\) \(7\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{3}q^{2}-q^{3}+(-\beta _{2}+\beta _{3})q^{4}+(2+\cdots)q^{5}+\cdots\)
6909.2.a.s 6909.a 1.a $4$ $55.169$ 4.4.141889.1 None \(-1\) \(4\) \(-1\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-\beta _{2}q^{5}+\cdots\)
6909.2.a.t 6909.a 1.a $5$ $55.169$ 5.5.486337.1 None \(0\) \(-5\) \(-5\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2}+\beta _{3})q^{4}+(-1+\cdots)q^{5}+\cdots\)
6909.2.a.u 6909.a 1.a $9$ $55.169$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(3\) \(-9\) \(-7\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{5}q^{2}-q^{3}+(1-\beta _{8})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
6909.2.a.v 6909.a 1.a $9$ $55.169$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(3\) \(9\) \(-3\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-\beta _{5}q^{5}+\cdots\)
6909.2.a.w 6909.a 1.a $10$ $55.169$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-2\) \(-10\) \(6\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{7}+\cdots)q^{5}+\cdots\)
6909.2.a.x 6909.a 1.a $10$ $55.169$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-2\) \(10\) \(-6\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(-1-\beta _{7}+\cdots)q^{5}+\cdots\)
6909.2.a.y 6909.a 1.a $14$ $55.169$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(2\) \(-14\) \(-2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{9}q^{5}+\cdots\)
6909.2.a.z 6909.a 1.a $14$ $55.169$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(2\) \(14\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{9}q^{5}+\cdots\)
6909.2.a.ba 6909.a 1.a $15$ $55.169$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(-6\) \(-15\) \(5\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\beta _{6}q^{5}+\cdots\)
6909.2.a.bb 6909.a 1.a $15$ $55.169$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(-6\) \(15\) \(-5\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-\beta _{6}q^{5}+\cdots\)
6909.2.a.bc 6909.a 1.a $15$ $55.169$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(-4\) \(-15\) \(3\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{13}q^{5}+\cdots\)
6909.2.a.bd 6909.a 1.a $15$ $55.169$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(-4\) \(15\) \(-3\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{13}q^{5}+\cdots\)
6909.2.a.be 6909.a 1.a $15$ $55.169$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(2\) \(-15\) \(-7\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{8}q^{5}+\cdots\)
6909.2.a.bf 6909.a 1.a $15$ $55.169$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(2\) \(15\) \(7\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{8}q^{5}+\cdots\)
6909.2.a.bg 6909.a 1.a $16$ $55.169$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(4\) \(-16\) \(-7\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{9}q^{5}+\cdots\)
6909.2.a.bh 6909.a 1.a $16$ $55.169$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(4\) \(16\) \(7\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{9}q^{5}+\cdots\)
6909.2.a.bi 6909.a 1.a $18$ $55.169$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(-2\) \(-18\) \(8\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{13}q^{5}+\cdots\)
6909.2.a.bj 6909.a 1.a $18$ $55.169$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(-2\) \(18\) \(-8\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{13}q^{5}+\cdots\)
6909.2.a.bk 6909.a 1.a $26$ $55.169$ None \(6\) \(-26\) \(-8\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$
6909.2.a.bl 6909.a 1.a $26$ $55.169$ None \(6\) \(26\) \(8\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6909))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6909)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(47))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(141))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(329))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(987))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2303))\)\(^{\oplus 2}\)