Properties

Label 6897.2.a
Level $6897$
Weight $2$
Character orbit 6897.a
Rep. character $\chi_{6897}(1,\cdot)$
Character field $\Q$
Dimension $326$
Newform subspaces $43$
Sturm bound $1760$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 6897 = 3 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6897.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 43 \)
Sturm bound: \(1760\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(2\), \(5\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6897))\).

Total New Old
Modular forms 904 326 578
Cusp forms 857 326 531
Eisenstein series 47 0 47

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

\(3\)\(11\)\(19\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(41\)
\(+\)\(+\)\(-\)\(-\)\(37\)
\(+\)\(-\)\(+\)\(-\)\(40\)
\(+\)\(-\)\(-\)\(+\)\(45\)
\(-\)\(+\)\(+\)\(-\)\(45\)
\(-\)\(+\)\(-\)\(+\)\(33\)
\(-\)\(-\)\(+\)\(+\)\(35\)
\(-\)\(-\)\(-\)\(-\)\(50\)
Plus space\(+\)\(154\)
Minus space\(-\)\(172\)

Trace form

\( 326 q + 2 q^{2} + 328 q^{4} - 6 q^{5} + 2 q^{6} + 10 q^{7} + 6 q^{8} + 326 q^{9} + O(q^{10}) \) \( 326 q + 2 q^{2} + 328 q^{4} - 6 q^{5} + 2 q^{6} + 10 q^{7} + 6 q^{8} + 326 q^{9} + 8 q^{10} + 8 q^{12} - 4 q^{13} + 20 q^{14} + 4 q^{15} + 356 q^{16} - 10 q^{17} + 2 q^{18} + 4 q^{19} + 4 q^{20} + 20 q^{23} + 18 q^{24} + 320 q^{25} + 4 q^{26} + 28 q^{28} - 8 q^{29} - 4 q^{30} + 12 q^{31} + 14 q^{32} + 8 q^{34} + 14 q^{35} + 328 q^{36} + 24 q^{37} + 16 q^{39} - 12 q^{40} - 20 q^{41} - 8 q^{42} + 26 q^{43} - 6 q^{45} + 12 q^{46} + 34 q^{47} + 340 q^{49} - 38 q^{50} + 4 q^{51} - 40 q^{52} + 2 q^{54} + 48 q^{56} + 2 q^{57} - 32 q^{58} + 32 q^{59} - 8 q^{60} + 2 q^{61} - 32 q^{62} + 10 q^{63} + 412 q^{64} - 36 q^{65} + 16 q^{67} + 20 q^{68} + 4 q^{69} + 36 q^{70} + 32 q^{71} + 6 q^{72} - 2 q^{73} + 44 q^{74} + 24 q^{75} + 10 q^{76} + 4 q^{78} + 24 q^{79} + 16 q^{80} + 326 q^{81} + 24 q^{82} - 4 q^{83} + 8 q^{84} - 30 q^{85} + 100 q^{86} + 4 q^{87} - 8 q^{89} + 8 q^{90} + 68 q^{91} + 4 q^{92} - 8 q^{93} + 16 q^{94} + 2 q^{95} + 18 q^{96} + 12 q^{97} + 78 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6897))\) 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 11 19
6897.2.a.a 6897.a 1.a $1$ $55.073$ \(\Q\) None 57.2.a.c \(-1\) \(1\) \(-2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}-2q^{5}-q^{6}+3q^{8}+\cdots\)
6897.2.a.b 6897.a 1.a $1$ $55.073$ \(\Q\) None 627.2.a.a \(0\) \(1\) \(0\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-2q^{7}+q^{9}-2q^{12}+\cdots\)
6897.2.a.c 6897.a 1.a $1$ $55.073$ \(\Q\) None 6897.2.a.c \(0\) \(1\) \(2\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}+2q^{5}+q^{9}-2q^{12}+\cdots\)
6897.2.a.d 6897.a 1.a $1$ $55.073$ \(\Q\) None 6897.2.a.c \(0\) \(1\) \(2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}+2q^{5}+q^{9}-2q^{12}+\cdots\)
6897.2.a.e 6897.a 1.a $1$ $55.073$ \(\Q\) None 627.2.a.b \(0\) \(1\) \(4\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}+4q^{5}-2q^{7}+q^{9}-2q^{12}+\cdots\)
6897.2.a.f 6897.a 1.a $1$ $55.073$ \(\Q\) None 57.2.a.a \(2\) \(-1\) \(-3\) \(5\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+2q^{4}-3q^{5}-2q^{6}+\cdots\)
6897.2.a.g 6897.a 1.a $1$ $55.073$ \(\Q\) None 57.2.a.b \(2\) \(1\) \(1\) \(-3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+q^{3}+2q^{4}+q^{5}+2q^{6}-3q^{7}+\cdots\)
6897.2.a.h 6897.a 1.a $2$ $55.073$ \(\Q(\sqrt{3}) \) None 6897.2.a.h \(0\) \(2\) \(-4\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+q^{4}+(-2-\beta )q^{5}+\beta q^{6}+\cdots\)
6897.2.a.i 6897.a 1.a $2$ $55.073$ \(\Q(\sqrt{3}) \) None 6897.2.a.h \(0\) \(2\) \(-4\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+q^{4}+(-2+\beta )q^{5}+\beta q^{6}+\cdots\)
6897.2.a.j 6897.a 1.a $2$ $55.073$ \(\Q(\sqrt{3}) \) None 6897.2.a.j \(0\) \(2\) \(0\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+q^{4}-\beta q^{5}+\beta q^{6}+(-1+\cdots)q^{7}+\cdots\)
6897.2.a.k 6897.a 1.a $2$ $55.073$ \(\Q(\sqrt{3}) \) None 6897.2.a.j \(0\) \(2\) \(0\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+q^{4}+\beta q^{5}+\beta q^{6}+(1+\cdots)q^{7}+\cdots\)
6897.2.a.l 6897.a 1.a $3$ $55.073$ \(\Q(\zeta_{14})^+\) None 627.2.a.f \(-2\) \(-3\) \(-3\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
6897.2.a.m 6897.a 1.a $3$ $55.073$ 3.3.169.1 None 627.2.a.g \(-2\) \(3\) \(5\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
6897.2.a.n 6897.a 1.a $3$ $55.073$ 3.3.169.1 None 627.2.a.c \(2\) \(-3\) \(1\) \(5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
6897.2.a.o 6897.a 1.a $3$ $55.073$ \(\Q(\zeta_{14})^+\) None 627.2.a.d \(2\) \(3\) \(-7\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+q^{3}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
6897.2.a.p 6897.a 1.a $3$ $55.073$ 3.3.321.1 None 627.2.a.e \(2\) \(3\) \(-3\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
6897.2.a.q 6897.a 1.a $4$ $55.073$ 4.4.23377.1 None 627.2.a.h \(-1\) \(4\) \(3\) \(5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+(1+\beta _{2}+\cdots)q^{5}+\cdots\)
6897.2.a.r 6897.a 1.a $5$ $55.073$ 5.5.2179633.1 None 627.2.a.i \(-1\) \(-5\) \(3\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{4})q^{5}+\cdots\)
6897.2.a.s 6897.a 1.a $5$ $55.073$ 5.5.1920025.1 None 627.2.a.j \(-1\) \(-5\) \(7\) \(-7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(1-\beta _{4})q^{5}+\cdots\)
6897.2.a.t 6897.a 1.a $6$ $55.073$ 6.6.65858461.1 None 6897.2.a.t \(-3\) \(6\) \(-2\) \(-12\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+q^{3}+(1+\beta _{4}+\beta _{5})q^{4}+\cdots\)
6897.2.a.u 6897.a 1.a $6$ $55.073$ 6.6.157752128.1 None 6897.2.a.u \(-2\) \(6\) \(-4\) \(-10\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
6897.2.a.v 6897.a 1.a $6$ $55.073$ 6.6.162030669.1 None 6897.2.a.v \(-1\) \(6\) \(2\) \(-12\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2}+\beta _{4})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
6897.2.a.w 6897.a 1.a $6$ $55.073$ 6.6.162030669.1 None 6897.2.a.v \(1\) \(6\) \(2\) \(12\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2}+\beta _{4})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
6897.2.a.x 6897.a 1.a $6$ $55.073$ 6.6.157752128.1 None 6897.2.a.u \(2\) \(6\) \(-4\) \(10\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
6897.2.a.y 6897.a 1.a $6$ $55.073$ 6.6.65858461.1 None 6897.2.a.t \(3\) \(6\) \(-2\) \(12\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+q^{3}+(1+\beta _{4}+\beta _{5})q^{4}+\cdots\)
6897.2.a.z 6897.a 1.a $7$ $55.073$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 6897.2.a.z \(-2\) \(-7\) \(-2\) \(-6\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
6897.2.a.ba 6897.a 1.a $7$ $55.073$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 6897.2.a.z \(2\) \(-7\) \(-2\) \(6\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
6897.2.a.bb 6897.a 1.a $8$ $55.073$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 6897.2.a.bb \(-3\) \(-8\) \(-2\) \(-6\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\beta _{5}q^{5}+\cdots\)
6897.2.a.bc 6897.a 1.a $8$ $55.073$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 6897.2.a.bc \(-1\) \(-8\) \(-2\) \(-6\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1-\beta _{6}+\beta _{7})q^{4}-\beta _{4}q^{5}+\cdots\)
6897.2.a.bd 6897.a 1.a $8$ $55.073$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 6897.2.a.bc \(1\) \(-8\) \(-2\) \(6\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1-\beta _{6}+\beta _{7})q^{4}-\beta _{4}q^{5}+\cdots\)
6897.2.a.be 6897.a 1.a $8$ $55.073$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 6897.2.a.bb \(3\) \(-8\) \(-2\) \(6\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\beta _{5}q^{5}+\cdots\)
6897.2.a.bf 6897.a 1.a $12$ $55.073$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 627.2.j.a \(-1\) \(12\) \(-6\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}-\beta _{8}q^{5}-\beta _{1}q^{6}+\cdots\)
6897.2.a.bg 6897.a 1.a $12$ $55.073$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 627.2.j.a \(1\) \(12\) \(-6\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}-\beta _{8}q^{5}+\beta _{1}q^{6}+\cdots\)
6897.2.a.bh 6897.a 1.a $14$ $55.073$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None 6897.2.a.bh \(-4\) \(-14\) \(2\) \(-12\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{6}q^{5}+\cdots\)
6897.2.a.bi 6897.a 1.a $14$ $55.073$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None 6897.2.a.bi \(-4\) \(14\) \(2\) \(-20\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-\beta _{10}q^{5}+\cdots\)
6897.2.a.bj 6897.a 1.a $14$ $55.073$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None 6897.2.a.bh \(4\) \(-14\) \(2\) \(12\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{6}q^{5}+\cdots\)
6897.2.a.bk 6897.a 1.a $14$ $55.073$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None 6897.2.a.bi \(4\) \(14\) \(2\) \(20\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-\beta _{10}q^{5}+\cdots\)
6897.2.a.bl 6897.a 1.a $16$ $55.073$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 627.2.j.b \(-3\) \(-16\) \(8\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{9}q^{5}+\cdots\)
6897.2.a.bm 6897.a 1.a $16$ $55.073$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 627.2.j.b \(3\) \(-16\) \(8\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{9}q^{5}+\cdots\)
6897.2.a.bn 6897.a 1.a $20$ $55.073$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None 627.2.j.c \(-3\) \(-20\) \(-9\) \(3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{7}q^{5}+\cdots\)
6897.2.a.bo 6897.a 1.a $20$ $55.073$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None 627.2.j.c \(3\) \(-20\) \(-9\) \(-3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{7}q^{5}+\cdots\)
6897.2.a.bp 6897.a 1.a $24$ $55.073$ None 627.2.j.d \(-1\) \(24\) \(9\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$
6897.2.a.bq 6897.a 1.a $24$ $55.073$ None 627.2.j.d \(1\) \(24\) \(9\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6897))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6897)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(209))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(363))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(627))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2299))\)\(^{\oplus 2}\)