Properties

Label 3249.2.a
Level $3249$
Weight $2$
Character orbit 3249.a
Rep. character $\chi_{3249}(1,\cdot)$
Character field $\Q$
Dimension $133$
Newform subspaces $40$
Sturm bound $760$
Trace bound $91$

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Defining parameters

Level: \( N \) \(=\) \( 3249 = 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3249.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 40 \)
Sturm bound: \(760\)
Trace bound: \(91\)
Distinguishing \(T_p\): \(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3249))\).

Total New Old
Modular forms 420 150 270
Cusp forms 341 133 208
Eisenstein series 79 17 62

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

\(3\)\(19\)FrickeDim
\(+\)\(+\)\(+\)\(23\)
\(+\)\(-\)\(-\)\(33\)
\(-\)\(+\)\(-\)\(41\)
\(-\)\(-\)\(+\)\(36\)
Plus space\(+\)\(59\)
Minus space\(-\)\(74\)

Trace form

\( 133 q - 3 q^{2} + 121 q^{4} + 2 q^{7} - 3 q^{8} + O(q^{10}) \) \( 133 q - 3 q^{2} + 121 q^{4} + 2 q^{7} - 3 q^{8} + 10 q^{10} + 2 q^{11} - 6 q^{13} + 4 q^{14} + 97 q^{16} - 6 q^{17} + 14 q^{20} + 20 q^{22} + 8 q^{23} + 93 q^{25} + 6 q^{26} + 34 q^{28} - 4 q^{29} + 4 q^{31} + 21 q^{32} + 22 q^{34} + 20 q^{35} - 20 q^{37} + 66 q^{40} - 16 q^{41} + 18 q^{43} - 14 q^{44} + 8 q^{46} - 2 q^{47} + 75 q^{49} - q^{50} - 14 q^{52} + 10 q^{53} + 8 q^{55} - 36 q^{58} - 26 q^{59} - 20 q^{61} + 22 q^{62} + 31 q^{64} - 36 q^{65} + 8 q^{67} + 26 q^{68} - 24 q^{70} + 6 q^{71} + 2 q^{73} - 26 q^{74} + 12 q^{77} - 8 q^{79} + 60 q^{80} + 24 q^{82} + 44 q^{83} - 28 q^{85} + 12 q^{88} + 14 q^{89} + 20 q^{91} + 84 q^{92} - 24 q^{94} + 10 q^{97} - 47 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3249))\) 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 19
3249.2.a.a 3249.a 1.a $1$ $25.943$ \(\Q\) None 57.2.a.b \(-2\) \(0\) \(-1\) \(3\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-q^{5}+3q^{7}+2q^{10}+\cdots\)
3249.2.a.b 3249.a 1.a $1$ $25.943$ \(\Q\) None 57.2.a.a \(-2\) \(0\) \(3\) \(-5\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}+3q^{5}-5q^{7}-6q^{10}+\cdots\)
3249.2.a.c 3249.a 1.a $1$ $25.943$ \(\Q\) None 57.2.e.a \(-1\) \(0\) \(0\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+q^{7}+3q^{8}+2q^{11}+5q^{13}+\cdots\)
3249.2.a.d 3249.a 1.a $1$ $25.943$ \(\Q\) None 19.2.a.a \(0\) \(0\) \(-3\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}-3q^{5}-q^{7}-3q^{11}+4q^{13}+\cdots\)
3249.2.a.e 3249.a 1.a $1$ $25.943$ \(\Q\) \(\Q(\sqrt{-19}) \) 361.2.a.a \(0\) \(0\) \(1\) \(3\) $-$ $+$ $N(\mathrm{U}(1))$ \(q-2q^{4}+q^{5}+3q^{7}+5q^{11}+4q^{16}+\cdots\)
3249.2.a.f 3249.a 1.a $1$ $25.943$ \(\Q\) None 57.2.e.a \(1\) \(0\) \(0\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+q^{7}-3q^{8}+2q^{11}-5q^{13}+\cdots\)
3249.2.a.g 3249.a 1.a $1$ $25.943$ \(\Q\) None 57.2.a.c \(1\) \(0\) \(2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+2q^{5}-3q^{8}+2q^{10}+\cdots\)
3249.2.a.h 3249.a 1.a $2$ $25.943$ \(\Q(\sqrt{5}) \) None 1083.2.a.f \(-3\) \(0\) \(-1\) \(-5\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+3\beta q^{4}-\beta q^{5}+(-2+\cdots)q^{7}+\cdots\)
3249.2.a.i 3249.a 1.a $2$ $25.943$ \(\Q(\sqrt{5}) \) None 361.2.a.c \(-1\) \(0\) \(-2\) \(6\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}-2\beta q^{5}+3q^{7}+\cdots\)
3249.2.a.j 3249.a 1.a $2$ $25.943$ \(\Q(\sqrt{5}) \) None 1083.2.a.h \(-1\) \(0\) \(1\) \(-5\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}+(-1+3\beta )q^{5}+\cdots\)
3249.2.a.k 3249.a 1.a $2$ $25.943$ \(\Q(\sqrt{17}) \) None 1083.2.a.g \(-1\) \(0\) \(3\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(2+\beta )q^{4}+(1+\beta )q^{5}+(-1+\cdots)q^{7}+\cdots\)
3249.2.a.l 3249.a 1.a $2$ $25.943$ \(\Q(\sqrt{19}) \) \(\Q(\sqrt{-19}) \) 3249.2.a.l \(0\) \(0\) \(0\) \(-6\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-2q^{4}+\beta q^{5}-3q^{7}-\beta q^{11}+4q^{16}+\cdots\)
3249.2.a.m 3249.a 1.a $2$ $25.943$ \(\Q(\sqrt{5}) \) None 361.2.a.d \(0\) \(0\) \(-1\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-2\beta )q^{2}+3q^{4}+(-1+\beta )q^{5}+\cdots\)
3249.2.a.n 3249.a 1.a $2$ $25.943$ \(\Q(\sqrt{5}) \) None 361.2.a.d \(0\) \(0\) \(-1\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-2\beta )q^{2}+3q^{4}-\beta q^{5}+(-2+2\beta )q^{7}+\cdots\)
3249.2.a.o 3249.a 1.a $2$ $25.943$ \(\Q(\sqrt{5}) \) None 361.2.a.c \(1\) \(0\) \(-2\) \(6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}-2\beta q^{5}+3q^{7}+\cdots\)
3249.2.a.p 3249.a 1.a $2$ $25.943$ \(\Q(\sqrt{5}) \) None 1083.2.a.h \(1\) \(0\) \(1\) \(-5\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}+(-1+3\beta )q^{5}+\cdots\)
3249.2.a.q 3249.a 1.a $2$ $25.943$ \(\Q(\sqrt{17}) \) None 1083.2.a.g \(1\) \(0\) \(3\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(2+\beta )q^{4}+(1+\beta )q^{5}+(-1+\cdots)q^{7}+\cdots\)
3249.2.a.r 3249.a 1.a $2$ $25.943$ \(\Q(\sqrt{5}) \) None 1083.2.a.f \(3\) \(0\) \(-1\) \(-5\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+3\beta q^{4}-\beta q^{5}+(-2-\beta )q^{7}+\cdots\)
3249.2.a.s 3249.a 1.a $3$ $25.943$ \(\Q(\zeta_{18})^+\) None 19.2.e.a \(-3\) \(0\) \(3\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
3249.2.a.t 3249.a 1.a $3$ $25.943$ 3.3.564.1 None 57.2.e.b \(-1\) \(0\) \(-2\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
3249.2.a.u 3249.a 1.a $3$ $25.943$ \(\Q(\zeta_{18})^+\) \(\Q(\sqrt{-3}) \) 171.2.u.b \(0\) \(0\) \(0\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-2q^{4}+(-3\beta _{1}+\beta _{2})q^{7}+(4\beta _{1}-\beta _{2})q^{13}+\cdots\)
3249.2.a.v 3249.a 1.a $3$ $25.943$ \(\Q(\zeta_{18})^+\) \(\Q(\sqrt{-3}) \) 171.2.u.b \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-2q^{4}+(-3\beta _{1}+\beta _{2})q^{7}+(-4\beta _{1}+\cdots)q^{13}+\cdots\)
3249.2.a.w 3249.a 1.a $3$ $25.943$ \(\Q(\zeta_{18})^+\) None 57.2.i.a \(0\) \(0\) \(3\) \(-3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+\beta _{2}q^{4}+(1-\beta _{1})q^{5}+(-1+\cdots)q^{7}+\cdots\)
3249.2.a.x 3249.a 1.a $3$ $25.943$ \(\Q(\zeta_{18})^+\) None 57.2.i.a \(0\) \(0\) \(3\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(1-\beta _{1})q^{5}+(-1+\cdots)q^{7}+\cdots\)
3249.2.a.y 3249.a 1.a $3$ $25.943$ 3.3.564.1 None 57.2.e.b \(1\) \(0\) \(-2\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
3249.2.a.z 3249.a 1.a $3$ $25.943$ \(\Q(\zeta_{18})^+\) None 19.2.e.a \(3\) \(0\) \(3\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
3249.2.a.ba 3249.a 1.a $4$ $25.943$ \(\Q(\zeta_{20})^+\) None 3249.2.a.ba \(0\) \(0\) \(-2\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
3249.2.a.bb 3249.a 1.a $4$ $25.943$ \(\Q(\zeta_{20})^+\) None 3249.2.a.ba \(0\) \(0\) \(2\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{2})q^{5}+\cdots\)
3249.2.a.bc 3249.a 1.a $4$ $25.943$ \(\Q(\zeta_{20})^+\) None 361.2.a.i \(0\) \(0\) \(4\) \(-8\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(2+2\beta _{2})q^{5}+\cdots\)
3249.2.a.bd 3249.a 1.a $4$ $25.943$ 4.4.27648.1 None 171.2.f.c \(0\) \(0\) \(0\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{3}q^{5}+(-1+\cdots)q^{7}+\cdots\)
3249.2.a.be 3249.a 1.a $4$ $25.943$ 4.4.27648.1 None 171.2.f.c \(0\) \(0\) \(0\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{3}q^{5}+(-1+\cdots)q^{7}+\cdots\)
3249.2.a.bf 3249.a 1.a $4$ $25.943$ 4.4.13068.1 None 171.2.a.e \(0\) \(0\) \(0\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2+\beta _{3})q^{4}+(-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
3249.2.a.bg 3249.a 1.a $6$ $25.943$ 6.6.6357609.1 None 57.2.i.b \(0\) \(0\) \(-9\) \(9\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{3}+\beta _{4}+\beta _{5})q^{4}+(-2+\cdots)q^{5}+\cdots\)
3249.2.a.bh 3249.a 1.a $6$ $25.943$ 6.6.6357609.1 None 57.2.i.b \(0\) \(0\) \(-9\) \(9\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{3}+\beta _{4}+\beta _{5})q^{4}+(-2+\cdots)q^{5}+\cdots\)
3249.2.a.bi 3249.a 1.a $6$ $25.943$ 6.6.21415104.1 None 171.2.u.d \(0\) \(0\) \(0\) \(-6\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
3249.2.a.bj 3249.a 1.a $6$ $25.943$ 6.6.21415104.1 None 171.2.u.d \(0\) \(0\) \(0\) \(-6\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{1}+\beta _{3})q^{5}+\cdots\)
3249.2.a.bk 3249.a 1.a $8$ $25.943$ 8.8.9764000000.1 None 1083.2.a.r \(-4\) \(0\) \(2\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(1-\beta _{2}+\beta _{4}+\beta _{5}+\cdots)q^{4}+\cdots\)
3249.2.a.bl 3249.a 1.a $8$ $25.943$ 8.8.\(\cdots\).1 None 3249.2.a.bl \(0\) \(0\) \(0\) \(14\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}+\beta _{7})q^{5}+\cdots\)
3249.2.a.bm 3249.a 1.a $8$ $25.943$ 8.8.\(\cdots\).1 None 3249.2.a.bl \(0\) \(0\) \(0\) \(14\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(\beta _{1}-\beta _{7})q^{5}+\cdots\)
3249.2.a.bn 3249.a 1.a $8$ $25.943$ 8.8.9764000000.1 None 1083.2.a.r \(4\) \(0\) \(2\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(1-\beta _{2}+\beta _{4}+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3249))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3249)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(361))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1083))\)\(^{\oplus 2}\)