Properties

Label 6900.2.f
Level $6900$
Weight $2$
Character orbit 6900.f
Rep. character $\chi_{6900}(6349,\cdot)$
Character field $\Q$
Dimension $68$
Newform subspaces $19$
Sturm bound $2880$
Trace bound $21$

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Defining parameters

Level: \( N \) \(=\) \( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6900.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 19 \)
Sturm bound: \(2880\)
Trace bound: \(21\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(6900, [\chi])\).

Total New Old
Modular forms 1476 68 1408
Cusp forms 1404 68 1336
Eisenstein series 72 0 72

Trace form

\( 68 q - 68 q^{9} + O(q^{10}) \) \( 68 q - 68 q^{9} + 4 q^{19} - 4 q^{21} - 32 q^{29} + 12 q^{31} + 4 q^{39} + 8 q^{41} - 96 q^{49} + 16 q^{59} + 4 q^{61} - 8 q^{69} - 40 q^{71} + 16 q^{79} + 68 q^{81} + 32 q^{89} - 20 q^{91} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(6900, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
6900.2.f.a 6900.f 5.b $2$ $55.097$ \(\Q(\sqrt{-1}) \) None 6900.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}+3iq^{7}-q^{9}-5q^{11}-iq^{13}+\cdots\)
6900.2.f.b 6900.f 5.b $2$ $55.097$ \(\Q(\sqrt{-1}) \) None 1380.2.a.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+3iq^{7}-q^{9}-2q^{11}-2iq^{13}+\cdots\)
6900.2.f.c 6900.f 5.b $2$ $55.097$ \(\Q(\sqrt{-1}) \) None 1380.2.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-q^{9}+6iq^{13}+2iq^{17}-6q^{19}+\cdots\)
6900.2.f.d 6900.f 5.b $2$ $55.097$ \(\Q(\sqrt{-1}) \) None 1380.2.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+4iq^{7}-q^{9}+2iq^{13}-6iq^{17}+\cdots\)
6900.2.f.e 6900.f 5.b $2$ $55.097$ \(\Q(\sqrt{-1}) \) None 1380.2.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+iq^{7}-q^{9}-4iq^{13}+3iq^{17}+\cdots\)
6900.2.f.f 6900.f 5.b $2$ $55.097$ \(\Q(\sqrt{-1}) \) None 1380.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}+5iq^{7}-q^{9}+4iq^{13}+3iq^{17}+\cdots\)
6900.2.f.g 6900.f 5.b $2$ $55.097$ \(\Q(\sqrt{-1}) \) None 6900.2.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+3iq^{7}-q^{9}+q^{11}+iq^{13}+\cdots\)
6900.2.f.h 6900.f 5.b $4$ $55.097$ \(\Q(i, \sqrt{6})\) None 1380.2.a.f \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}-\beta _{1}q^{7}-q^{9}+(-2+\beta _{3})q^{11}+\cdots\)
6900.2.f.i 6900.f 5.b $4$ $55.097$ \(\Q(\zeta_{12})\) None 1380.2.a.h \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{12}q^{3}+(\zeta_{12}-2\zeta_{12}^{2})q^{7}-q^{9}+\cdots\)
6900.2.f.j 6900.f 5.b $4$ $55.097$ \(\Q(i, \sqrt{73})\) None 6900.2.a.n \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}-\beta _{2}q^{7}-q^{9}+(-1+\beta _{3})q^{11}+\cdots\)
6900.2.f.k 6900.f 5.b $4$ $55.097$ \(\Q(i, \sqrt{10})\) None 276.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+(2\beta _{1}-\beta _{2})q^{7}-q^{9}-4\beta _{1}q^{13}+\cdots\)
6900.2.f.l 6900.f 5.b $4$ $55.097$ \(\Q(\zeta_{8})\) None 276.2.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}q^{3}-\zeta_{8}^{2}q^{7}-q^{9}-4\zeta_{8}^{3}q^{11}+\cdots\)
6900.2.f.m 6900.f 5.b $4$ $55.097$ \(\Q(i, \sqrt{13})\) None 6900.2.a.k \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}+(\beta _{1}+3\beta _{2})q^{7}-q^{9}+\beta _{3}q^{11}+\cdots\)
6900.2.f.n 6900.f 5.b $4$ $55.097$ \(\Q(i, \sqrt{21})\) None 6900.2.a.o \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}+(\beta _{1}-\beta _{2})q^{7}-q^{9}+\beta _{3}q^{11}+\cdots\)
6900.2.f.o 6900.f 5.b $4$ $55.097$ \(\Q(i, \sqrt{15})\) None 1380.2.a.i \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+3\beta _{1}q^{7}-q^{9}+(1+\beta _{2})q^{11}+\cdots\)
6900.2.f.p 6900.f 5.b $4$ $55.097$ \(\Q(i, \sqrt{17})\) None 1380.2.a.g \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}+\beta _{1}q^{7}-q^{9}+(2-2\beta _{3})q^{11}+\cdots\)
6900.2.f.q 6900.f 5.b $4$ $55.097$ \(\Q(i, \sqrt{13})\) None 6900.2.a.l \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}+(\beta _{1}+3\beta _{2})q^{7}-q^{9}+(2+\beta _{3})q^{11}+\cdots\)
6900.2.f.r 6900.f 5.b $6$ $55.097$ 6.0.158155776.1 None 1380.2.a.j \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{3}+(-\beta _{2}-\beta _{5})q^{7}-q^{9}+(1+\cdots)q^{11}+\cdots\)
6900.2.f.s 6900.f 5.b $8$ $55.097$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 6900.2.a.ba \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{4}q^{3}+(-\beta _{4}+\beta _{5})q^{7}-q^{9}+\beta _{2}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(6900, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(6900, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(115, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(230, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(345, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(460, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(575, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(690, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1150, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1380, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1725, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2300, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(3450, [\chi])\)\(^{\oplus 2}\)