Properties

Label 3648.2.a
Level $3648$
Weight $2$
Character orbit 3648.a
Rep. character $\chi_{3648}(1,\cdot)$
Character field $\Q$
Dimension $72$
Newform subspaces $52$
Sturm bound $1280$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3648.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 52 \)
Sturm bound: \(1280\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\), \(23\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3648))\).

Total New Old
Modular forms 664 72 592
Cusp forms 617 72 545
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.

\(2\)\(3\)\(19\)FrickeDim
\(+\)\(+\)\(+\)$+$\(9\)
\(+\)\(+\)\(-\)$-$\(8\)
\(+\)\(-\)\(+\)$-$\(11\)
\(+\)\(-\)\(-\)$+$\(6\)
\(-\)\(+\)\(+\)$-$\(9\)
\(-\)\(+\)\(-\)$+$\(10\)
\(-\)\(-\)\(+\)$+$\(7\)
\(-\)\(-\)\(-\)$-$\(12\)
Plus space\(+\)\(32\)
Minus space\(-\)\(40\)

Trace form

\( 72 q + 72 q^{9} + O(q^{10}) \) \( 72 q + 72 q^{9} + 16 q^{17} + 88 q^{25} + 32 q^{29} + 32 q^{37} + 16 q^{41} + 72 q^{49} + 32 q^{53} - 16 q^{73} - 16 q^{77} + 72 q^{81} - 48 q^{85} - 16 q^{89} + 48 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3648))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 19
3648.2.a.a 3648.a 1.a $1$ $29.129$ \(\Q\) None 456.2.a.b \(0\) \(-1\) \(-4\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{5}-4q^{7}+q^{9}-4q^{11}+\cdots\)
3648.2.a.b 3648.a 1.a $1$ $29.129$ \(\Q\) None 1824.2.a.f \(0\) \(-1\) \(-3\) \(3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{5}+3q^{7}+q^{9}-3q^{11}+\cdots\)
3648.2.a.c 3648.a 1.a $1$ $29.129$ \(\Q\) None 114.2.a.b \(0\) \(-1\) \(-2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+q^{9}-4q^{11}-2q^{13}+\cdots\)
3648.2.a.d 3648.a 1.a $1$ $29.129$ \(\Q\) None 456.2.a.d \(0\) \(-1\) \(-2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+q^{9}-2q^{13}+2q^{15}+\cdots\)
3648.2.a.e 3648.a 1.a $1$ $29.129$ \(\Q\) None 228.2.a.b \(0\) \(-1\) \(-2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+q^{9}+2q^{11}-2q^{13}+\cdots\)
3648.2.a.f 3648.a 1.a $1$ $29.129$ \(\Q\) None 1824.2.a.e \(0\) \(-1\) \(-2\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+4q^{7}+q^{9}+6q^{11}+\cdots\)
3648.2.a.g 3648.a 1.a $1$ $29.129$ \(\Q\) None 456.2.a.a \(0\) \(-1\) \(-1\) \(3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+3q^{7}+q^{9}-5q^{11}+2q^{13}+\cdots\)
3648.2.a.h 3648.a 1.a $1$ $29.129$ \(\Q\) None 57.2.a.b \(0\) \(-1\) \(-1\) \(3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+3q^{7}+q^{9}+3q^{11}+6q^{13}+\cdots\)
3648.2.a.i 3648.a 1.a $1$ $29.129$ \(\Q\) None 114.2.a.c \(0\) \(-1\) \(0\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{7}+q^{9}+4q^{13}+6q^{17}+\cdots\)
3648.2.a.j 3648.a 1.a $1$ $29.129$ \(\Q\) None 114.2.a.a \(0\) \(-1\) \(0\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{7}+q^{9}+4q^{11}-2q^{17}+\cdots\)
3648.2.a.k 3648.a 1.a $1$ $29.129$ \(\Q\) None 1824.2.a.d \(0\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{9}-2q^{17}-q^{19}+6q^{23}+\cdots\)
3648.2.a.l 3648.a 1.a $1$ $29.129$ \(\Q\) None 1824.2.a.c \(0\) \(-1\) \(1\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-q^{7}+q^{9}+5q^{11}-4q^{13}+\cdots\)
3648.2.a.m 3648.a 1.a $1$ $29.129$ \(\Q\) None 1824.2.a.b \(0\) \(-1\) \(1\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{7}+q^{9}+3q^{11}-q^{15}+\cdots\)
3648.2.a.n 3648.a 1.a $1$ $29.129$ \(\Q\) None 1824.2.a.a \(0\) \(-1\) \(2\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}-4q^{7}+q^{9}-4q^{11}+\cdots\)
3648.2.a.o 3648.a 1.a $1$ $29.129$ \(\Q\) None 57.2.a.c \(0\) \(-1\) \(2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}+q^{9}-6q^{13}-2q^{15}+\cdots\)
3648.2.a.p 3648.a 1.a $1$ $29.129$ \(\Q\) None 456.2.a.c \(0\) \(-1\) \(3\) \(-3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{5}-3q^{7}+q^{9}+q^{11}+2q^{13}+\cdots\)
3648.2.a.q 3648.a 1.a $1$ $29.129$ \(\Q\) None 228.2.a.a \(0\) \(-1\) \(3\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{5}-q^{7}+q^{9}-5q^{11}+6q^{13}+\cdots\)
3648.2.a.r 3648.a 1.a $1$ $29.129$ \(\Q\) None 57.2.a.a \(0\) \(-1\) \(3\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{5}+5q^{7}+q^{9}+q^{11}-2q^{13}+\cdots\)
3648.2.a.s 3648.a 1.a $1$ $29.129$ \(\Q\) None 456.2.a.b \(0\) \(1\) \(-4\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{5}+4q^{7}+q^{9}+4q^{11}+\cdots\)
3648.2.a.t 3648.a 1.a $1$ $29.129$ \(\Q\) None 1824.2.a.f \(0\) \(1\) \(-3\) \(-3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{5}-3q^{7}+q^{9}+3q^{11}+\cdots\)
3648.2.a.u 3648.a 1.a $1$ $29.129$ \(\Q\) None 1824.2.a.e \(0\) \(1\) \(-2\) \(-4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}-4q^{7}+q^{9}-6q^{11}+\cdots\)
3648.2.a.v 3648.a 1.a $1$ $29.129$ \(\Q\) None 228.2.a.b \(0\) \(1\) \(-2\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}+q^{9}-2q^{11}-2q^{13}+\cdots\)
3648.2.a.w 3648.a 1.a $1$ $29.129$ \(\Q\) None 456.2.a.d \(0\) \(1\) \(-2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}+q^{9}-2q^{13}-2q^{15}+\cdots\)
3648.2.a.x 3648.a 1.a $1$ $29.129$ \(\Q\) None 114.2.a.b \(0\) \(1\) \(-2\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}+q^{9}+4q^{11}-2q^{13}+\cdots\)
3648.2.a.y 3648.a 1.a $1$ $29.129$ \(\Q\) None 57.2.a.b \(0\) \(1\) \(-1\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-3q^{7}+q^{9}-3q^{11}+6q^{13}+\cdots\)
3648.2.a.z 3648.a 1.a $1$ $29.129$ \(\Q\) None 456.2.a.a \(0\) \(1\) \(-1\) \(-3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-3q^{7}+q^{9}+5q^{11}+2q^{13}+\cdots\)
3648.2.a.ba 3648.a 1.a $1$ $29.129$ \(\Q\) None 1824.2.a.d \(0\) \(1\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{9}-2q^{17}+q^{19}-6q^{23}+\cdots\)
3648.2.a.bb 3648.a 1.a $1$ $29.129$ \(\Q\) None 114.2.a.a \(0\) \(1\) \(0\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{7}+q^{9}-4q^{11}-2q^{17}+\cdots\)
3648.2.a.bc 3648.a 1.a $1$ $29.129$ \(\Q\) None 114.2.a.c \(0\) \(1\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{7}+q^{9}+4q^{13}+6q^{17}+\cdots\)
3648.2.a.bd 3648.a 1.a $1$ $29.129$ \(\Q\) None 1824.2.a.b \(0\) \(1\) \(1\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-q^{7}+q^{9}-3q^{11}+q^{15}+\cdots\)
3648.2.a.be 3648.a 1.a $1$ $29.129$ \(\Q\) None 1824.2.a.c \(0\) \(1\) \(1\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+q^{7}+q^{9}-5q^{11}-4q^{13}+\cdots\)
3648.2.a.bf 3648.a 1.a $1$ $29.129$ \(\Q\) None 57.2.a.c \(0\) \(1\) \(2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+q^{9}-6q^{13}+2q^{15}+\cdots\)
3648.2.a.bg 3648.a 1.a $1$ $29.129$ \(\Q\) None 1824.2.a.a \(0\) \(1\) \(2\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+4q^{7}+q^{9}+4q^{11}+\cdots\)
3648.2.a.bh 3648.a 1.a $1$ $29.129$ \(\Q\) None 57.2.a.a \(0\) \(1\) \(3\) \(-5\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{5}-5q^{7}+q^{9}-q^{11}-2q^{13}+\cdots\)
3648.2.a.bi 3648.a 1.a $1$ $29.129$ \(\Q\) None 228.2.a.a \(0\) \(1\) \(3\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{5}+q^{7}+q^{9}+5q^{11}+6q^{13}+\cdots\)
3648.2.a.bj 3648.a 1.a $1$ $29.129$ \(\Q\) None 456.2.a.c \(0\) \(1\) \(3\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{5}+3q^{7}+q^{9}-q^{11}+2q^{13}+\cdots\)
3648.2.a.bk 3648.a 1.a $2$ $29.129$ \(\Q(\sqrt{33}) \) None 228.2.a.c \(0\) \(-2\) \(-3\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1-\beta )q^{5}+(1-\beta )q^{7}+q^{9}+\cdots\)
3648.2.a.bl 3648.a 1.a $2$ $29.129$ \(\Q(\sqrt{17}) \) None 456.2.a.f \(0\) \(-2\) \(-1\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-\beta q^{5}+\beta q^{7}+q^{9}+(-4+\beta )q^{11}+\cdots\)
3648.2.a.bm 3648.a 1.a $2$ $29.129$ \(\Q(\sqrt{33}) \) None 1824.2.a.o \(0\) \(-2\) \(1\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta q^{5}+\beta q^{7}+q^{9}+(4-\beta )q^{11}+\cdots\)
3648.2.a.bn 3648.a 1.a $2$ $29.129$ \(\Q(\sqrt{41}) \) None 456.2.a.e \(0\) \(-2\) \(1\) \(3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta q^{5}+(2-\beta )q^{7}+q^{9}+(2-\beta )q^{11}+\cdots\)
3648.2.a.bo 3648.a 1.a $2$ $29.129$ \(\Q(\sqrt{17}) \) None 1824.2.a.n \(0\) \(-2\) \(3\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(1+\beta )q^{5}+(-1+\beta )q^{7}+q^{9}+\cdots\)
3648.2.a.bp 3648.a 1.a $2$ $29.129$ \(\Q(\sqrt{17}) \) None 1824.2.a.m \(0\) \(-2\) \(3\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(1+\beta )q^{5}+(1-\beta )q^{7}+q^{9}+\cdots\)
3648.2.a.bq 3648.a 1.a $2$ $29.129$ \(\Q(\sqrt{33}) \) None 228.2.a.c \(0\) \(2\) \(-3\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1-\beta )q^{5}+(-1+\beta )q^{7}+\cdots\)
3648.2.a.br 3648.a 1.a $2$ $29.129$ \(\Q(\sqrt{17}) \) None 456.2.a.f \(0\) \(2\) \(-1\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-\beta q^{5}-\beta q^{7}+q^{9}+(4-\beta )q^{11}+\cdots\)
3648.2.a.bs 3648.a 1.a $2$ $29.129$ \(\Q(\sqrt{41}) \) None 456.2.a.e \(0\) \(2\) \(1\) \(-3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta q^{5}+(-2+\beta )q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
3648.2.a.bt 3648.a 1.a $2$ $29.129$ \(\Q(\sqrt{33}) \) None 1824.2.a.o \(0\) \(2\) \(1\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta q^{5}-\beta q^{7}+q^{9}+(-4+\beta )q^{11}+\cdots\)
3648.2.a.bu 3648.a 1.a $2$ $29.129$ \(\Q(\sqrt{17}) \) None 1824.2.a.m \(0\) \(2\) \(3\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(1+\beta )q^{5}+(-1+\beta )q^{7}+q^{9}+\cdots\)
3648.2.a.bv 3648.a 1.a $2$ $29.129$ \(\Q(\sqrt{17}) \) None 1824.2.a.n \(0\) \(2\) \(3\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(1+\beta )q^{5}+(1-\beta )q^{7}+q^{9}+\cdots\)
3648.2.a.bw 3648.a 1.a $3$ $29.129$ 3.3.229.1 None 1824.2.a.t \(0\) \(-3\) \(-5\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(-2-\beta _{2})q^{5}-\beta _{1}q^{7}+q^{9}+\cdots\)
3648.2.a.bx 3648.a 1.a $3$ $29.129$ 3.3.469.1 None 1824.2.a.s \(0\) \(-3\) \(3\) \(-5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(1-\beta _{1})q^{5}+(-2+\beta _{2})q^{7}+\cdots\)
3648.2.a.by 3648.a 1.a $3$ $29.129$ 3.3.229.1 None 1824.2.a.t \(0\) \(3\) \(-5\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(-2-\beta _{2})q^{5}+\beta _{1}q^{7}+q^{9}+\cdots\)
3648.2.a.bz 3648.a 1.a $3$ $29.129$ 3.3.469.1 None 1824.2.a.s \(0\) \(3\) \(3\) \(5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(1-\beta _{1})q^{5}+(2-\beta _{2})q^{7}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3648))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3648)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 14}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(304))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(456))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(608))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(912))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1216))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1824))\)\(^{\oplus 2}\)