Properties

Label 6840.2.a
Level $6840$
Weight $2$
Character orbit 6840.a
Rep. character $\chi_{6840}(1,\cdot)$
Character field $\Q$
Dimension $90$
Newform subspaces $45$
Sturm bound $2880$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6840.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 45 \)
Sturm bound: \(2880\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6840))\).

Total New Old
Modular forms 1472 90 1382
Cusp forms 1409 90 1319
Eisenstein series 63 0 63

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

\(2\)\(3\)\(5\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(5\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(5\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(4\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(6\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(6\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(8\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(6\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(8\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(6\)
Plus space\(+\)\(42\)
Minus space\(-\)\(48\)

Trace form

\( 90 q + 2 q^{5} + O(q^{10}) \) \( 90 q + 2 q^{5} + 8 q^{17} + 90 q^{25} - 12 q^{29} + 8 q^{31} - 12 q^{35} + 4 q^{41} + 28 q^{43} + 36 q^{47} + 78 q^{49} + 16 q^{53} + 8 q^{55} + 16 q^{59} - 4 q^{61} + 8 q^{67} + 8 q^{71} - 48 q^{77} + 32 q^{79} - 60 q^{83} - 4 q^{85} - 36 q^{89} + 16 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6840))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 19
6840.2.a.a 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{7}-2q^{11}+6q^{17}-q^{19}+\cdots\)
6840.2.a.b 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{7}+2q^{11}-4q^{13}+2q^{17}+\cdots\)
6840.2.a.c 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-q^{7}-4q^{11}+q^{13}+7q^{17}+\cdots\)
6840.2.a.d 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-4q^{11}-2q^{13}-2q^{17}-q^{19}+\cdots\)
6840.2.a.e 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-4q^{11}+2q^{13}-6q^{17}-q^{19}+\cdots\)
6840.2.a.f 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{11}-q^{19}+4q^{23}+q^{25}+\cdots\)
6840.2.a.g 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{13}+6q^{17}+q^{19}-4q^{23}+\cdots\)
6840.2.a.h 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{11}-6q^{13}+6q^{17}-q^{19}+\cdots\)
6840.2.a.i 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{11}+4q^{13}+2q^{17}+q^{19}+\cdots\)
6840.2.a.j 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+2q^{7}-6q^{11}+4q^{13}+2q^{17}+\cdots\)
6840.2.a.k 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+2q^{7}-2q^{13}-4q^{17}+q^{19}+\cdots\)
6840.2.a.l 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{7}-4q^{11}-4q^{13}+2q^{17}+\cdots\)
6840.2.a.m 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{7}+2q^{13}-2q^{17}-q^{19}+\cdots\)
6840.2.a.n 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{7}+2q^{11}-4q^{13}-4q^{17}+\cdots\)
6840.2.a.o 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{7}+4q^{11}-6q^{17}-q^{19}+\cdots\)
6840.2.a.p 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(-1\) \(4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{7}+4q^{11}+2q^{13}+6q^{17}+\cdots\)
6840.2.a.q 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(1\) \(-2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}+2q^{11}-6q^{17}-q^{19}+\cdots\)
6840.2.a.r 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(1\) \(-2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}+4q^{11}-2q^{13}+q^{19}+\cdots\)
6840.2.a.s 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+2q^{11}-q^{19}-4q^{23}+q^{25}+\cdots\)
6840.2.a.t 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+4q^{11}-2q^{13}+2q^{17}+q^{19}+\cdots\)
6840.2.a.u 6840.a 1.a $1$ $54.618$ \(\Q\) None \(0\) \(0\) \(1\) \(4\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+4q^{7}-2q^{11}-4q^{13}+4q^{17}+\cdots\)
6840.2.a.v 6840.a 1.a $2$ $54.618$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-2\) \(-6\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+(-3+\beta )q^{7}+(1+3\beta )q^{11}+\cdots\)
6840.2.a.w 6840.a 1.a $2$ $54.618$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(-2\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(-1+\beta )q^{7}+(3-\beta )q^{11}+(1+\cdots)q^{13}+\cdots\)
6840.2.a.x 6840.a 1.a $2$ $54.618$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(2\) \(-4\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+(-2+\beta )q^{7}-\beta q^{11}-3\beta q^{13}+\cdots\)
6840.2.a.y 6840.a 1.a $2$ $54.618$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(2\) \(0\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{11}+(1+\beta )q^{13}+(-4+2\beta )q^{17}+\cdots\)
6840.2.a.z 6840.a 1.a $2$ $54.618$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(2\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+2\beta q^{7}+(-2-2\beta )q^{11}+(2+\cdots)q^{13}+\cdots\)
6840.2.a.ba 6840.a 1.a $2$ $54.618$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(2\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+\beta q^{7}+\beta q^{11}+(-2-\beta )q^{13}+\cdots\)
6840.2.a.bb 6840.a 1.a $2$ $54.618$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(2\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+\beta q^{7}+(2-\beta )q^{11}+(2-\beta )q^{13}+\cdots\)
6840.2.a.bc 6840.a 1.a $2$ $54.618$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(2\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+(1+\beta )q^{7}+(1+\beta )q^{11}+(-1+\cdots)q^{13}+\cdots\)
6840.2.a.bd 6840.a 1.a $2$ $54.618$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(2\) \(6\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(3+\beta )q^{7}+(-1+\beta )q^{11}+(3+\cdots)q^{13}+\cdots\)
6840.2.a.be 6840.a 1.a $3$ $54.618$ 3.3.148.1 None \(0\) \(0\) \(-3\) \(-2\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(-2\beta _{1}-\beta _{2})q^{7}+(1-\beta _{1})q^{11}+\cdots\)
6840.2.a.bf 6840.a 1.a $3$ $54.618$ 3.3.316.1 None \(0\) \(0\) \(-3\) \(-2\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(-1+\beta _{1})q^{7}+(1+\beta _{2})q^{11}+\cdots\)
6840.2.a.bg 6840.a 1.a $3$ $54.618$ 3.3.229.1 None \(0\) \(0\) \(-3\) \(-1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+\beta _{1}q^{7}+(2+\beta _{2})q^{13}+(-2+\cdots)q^{17}+\cdots\)
6840.2.a.bh 6840.a 1.a $3$ $54.618$ 3.3.1772.1 None \(0\) \(0\) \(-3\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+\beta _{2}q^{7}+(-2-\beta _{2})q^{11}+(1+\cdots)q^{13}+\cdots\)
6840.2.a.bi 6840.a 1.a $3$ $54.618$ 3.3.2804.1 None \(0\) \(0\) \(-3\) \(2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(1-\beta _{1})q^{7}+(-2-\beta _{2})q^{11}+\cdots\)
6840.2.a.bj 6840.a 1.a $3$ $54.618$ 3.3.568.1 None \(0\) \(0\) \(3\) \(-5\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+(-2-\beta _{2})q^{7}+(2-2\beta _{1})q^{11}+\cdots\)
6840.2.a.bk 6840.a 1.a $3$ $54.618$ 3.3.568.1 None \(0\) \(0\) \(3\) \(-2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(-1-\beta _{2})q^{7}+(-1+\beta _{2})q^{11}+\cdots\)
6840.2.a.bl 6840.a 1.a $3$ $54.618$ 3.3.148.1 None \(0\) \(0\) \(3\) \(-2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+(-2\beta _{1}-\beta _{2})q^{7}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
6840.2.a.bm 6840.a 1.a $3$ $54.618$ 3.3.316.1 None \(0\) \(0\) \(3\) \(-1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(-1+2\beta _{1}-\beta _{2})q^{7}-2\beta _{2}q^{11}+\cdots\)
6840.2.a.bn 6840.a 1.a $3$ $54.618$ 3.3.1016.1 None \(0\) \(0\) \(3\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+\beta _{2}q^{7}+(-1+\beta _{1}+\beta _{2})q^{11}+\cdots\)
6840.2.a.bo 6840.a 1.a $3$ $54.618$ 3.3.2804.1 None \(0\) \(0\) \(3\) \(2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+(1-\beta _{1})q^{7}+(2+\beta _{2})q^{11}+(3+\cdots)q^{13}+\cdots\)
6840.2.a.bp 6840.a 1.a $4$ $54.618$ 4.4.20308.1 None \(0\) \(0\) \(-4\) \(-4\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+(-1+\beta _{1})q^{7}+(1-\beta _{1}-\beta _{3})q^{11}+\cdots\)
6840.2.a.bq 6840.a 1.a $4$ $54.618$ 4.4.20308.1 None \(0\) \(0\) \(4\) \(-4\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(-1+\beta _{1})q^{7}+(-1+\beta _{1}+\beta _{3})q^{11}+\cdots\)
6840.2.a.br 6840.a 1.a $5$ $54.618$ 5.5.4636128.1 None \(0\) \(0\) \(-5\) \(2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+\beta _{4}q^{7}-\beta _{3}q^{11}+(\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)
6840.2.a.bs 6840.a 1.a $5$ $54.618$ 5.5.4636128.1 None \(0\) \(0\) \(5\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+\beta _{4}q^{7}+\beta _{3}q^{11}+(\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6840))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6840)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(285))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(342))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(380))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(456))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(570))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(684))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(760))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(855))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1140))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1368))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1710))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2280))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3420))\)\(^{\oplus 2}\)