Properties

Label 3136.2.a
Level $3136$
Weight $2$
Character orbit 3136.a
Rep. character $\chi_{3136}(1,\cdot)$
Character field $\Q$
Dimension $77$
Newform subspaces $52$
Sturm bound $896$
Trace bound $25$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3136.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 52 \)
Sturm bound: \(896\)
Trace bound: \(25\)
Distinguishing \(T_p\): \(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3136))\).

Total New Old
Modular forms 496 87 409
Cusp forms 401 77 324
Eisenstein series 95 10 85

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

\(2\)\(7\)FrickeDim
\(+\)\(+\)\(+\)\(17\)
\(+\)\(-\)\(-\)\(21\)
\(-\)\(+\)\(-\)\(21\)
\(-\)\(-\)\(+\)\(18\)
Plus space\(+\)\(35\)
Minus space\(-\)\(42\)

Trace form

\( 77 q - 2 q^{5} + 69 q^{9} + O(q^{10}) \) \( 77 q - 2 q^{5} + 69 q^{9} + 6 q^{13} + 6 q^{17} + 63 q^{25} + 2 q^{29} + 16 q^{33} - 2 q^{37} + 14 q^{41} - 42 q^{45} - 34 q^{53} + 4 q^{57} + 22 q^{61} + 12 q^{65} - 16 q^{69} - 2 q^{73} + 37 q^{81} + 40 q^{85} - 18 q^{89} + 36 q^{93} - 10 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3136))\) 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 7
3136.2.a.a 3136.a 1.a $1$ $25.041$ \(\Q\) None 56.2.i.a \(0\) \(-3\) \(-1\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-q^{5}+6q^{9}-q^{11}+2q^{13}+\cdots\)
3136.2.a.b 3136.a 1.a $1$ $25.041$ \(\Q\) None 56.2.i.a \(0\) \(-3\) \(1\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+q^{5}+6q^{9}+q^{11}-2q^{13}+\cdots\)
3136.2.a.c 3136.a 1.a $1$ $25.041$ \(\Q\) None 56.2.a.b \(0\) \(-2\) \(-4\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-4q^{5}+q^{9}+8q^{15}+2q^{17}+\cdots\)
3136.2.a.d 3136.a 1.a $1$ $25.041$ \(\Q\) None 1568.2.a.a \(0\) \(-2\) \(-2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-2q^{5}+q^{9}-4q^{11}-6q^{13}+\cdots\)
3136.2.a.e 3136.a 1.a $1$ $25.041$ \(\Q\) None 14.2.a.a \(0\) \(-2\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}-4q^{13}-6q^{17}+2q^{19}+\cdots\)
3136.2.a.f 3136.a 1.a $1$ $25.041$ \(\Q\) None 224.2.a.a \(0\) \(-2\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}+4q^{11}-4q^{13}+2q^{17}+\cdots\)
3136.2.a.g 3136.a 1.a $1$ $25.041$ \(\Q\) None 1568.2.a.a \(0\) \(-2\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+2q^{5}+q^{9}+4q^{11}+6q^{13}+\cdots\)
3136.2.a.h 3136.a 1.a $1$ $25.041$ \(\Q\) None 28.2.e.a \(0\) \(-1\) \(-3\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{5}-2q^{9}+3q^{11}-2q^{13}+\cdots\)
3136.2.a.i 3136.a 1.a $1$ $25.041$ \(\Q\) None 56.2.i.b \(0\) \(-1\) \(-1\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{9}-3q^{11}-6q^{13}+\cdots\)
3136.2.a.j 3136.a 1.a $1$ $25.041$ \(\Q\) None 56.2.i.b \(0\) \(-1\) \(1\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-2q^{9}+3q^{11}+6q^{13}+\cdots\)
3136.2.a.k 3136.a 1.a $1$ $25.041$ \(\Q\) None 28.2.e.a \(0\) \(-1\) \(3\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{5}-2q^{9}-3q^{11}+2q^{13}+\cdots\)
3136.2.a.l 3136.a 1.a $1$ $25.041$ \(\Q\) \(\Q(\sqrt{-1}) \) 1568.2.a.d \(0\) \(0\) \(-4\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-4q^{5}-3q^{9}+4q^{13}+8q^{17}+11q^{25}+\cdots\)
3136.2.a.m 3136.a 1.a $1$ $25.041$ \(\Q\) \(\Q(\sqrt{-1}) \) 32.2.a.a \(0\) \(0\) \(-2\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-2q^{5}-3q^{9}+6q^{13}-2q^{17}-q^{25}+\cdots\)
3136.2.a.n 3136.a 1.a $1$ $25.041$ \(\Q\) \(\Q(\sqrt{-7}) \) 49.2.a.a \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-3q^{9}-4q^{11}+8q^{23}-5q^{25}-2q^{29}+\cdots\)
3136.2.a.o 3136.a 1.a $1$ $25.041$ \(\Q\) \(\Q(\sqrt{-7}) \) 49.2.a.a \(0\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-3q^{9}+4q^{11}-8q^{23}-5q^{25}-2q^{29}+\cdots\)
3136.2.a.p 3136.a 1.a $1$ $25.041$ \(\Q\) None 56.2.a.a \(0\) \(0\) \(2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}-3q^{9}-4q^{11}+2q^{13}+6q^{17}+\cdots\)
3136.2.a.q 3136.a 1.a $1$ $25.041$ \(\Q\) None 56.2.a.a \(0\) \(0\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}-3q^{9}+4q^{11}+2q^{13}+6q^{17}+\cdots\)
3136.2.a.r 3136.a 1.a $1$ $25.041$ \(\Q\) \(\Q(\sqrt{-1}) \) 1568.2.a.d \(0\) \(0\) \(4\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q+4q^{5}-3q^{9}-4q^{13}-8q^{17}+11q^{25}+\cdots\)
3136.2.a.s 3136.a 1.a $1$ $25.041$ \(\Q\) None 28.2.e.a \(0\) \(1\) \(-3\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{5}-2q^{9}-3q^{11}-2q^{13}+\cdots\)
3136.2.a.t 3136.a 1.a $1$ $25.041$ \(\Q\) None 56.2.i.b \(0\) \(1\) \(-1\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{9}+3q^{11}-6q^{13}+\cdots\)
3136.2.a.u 3136.a 1.a $1$ $25.041$ \(\Q\) None 56.2.i.b \(0\) \(1\) \(1\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-2q^{9}-3q^{11}+6q^{13}+\cdots\)
3136.2.a.v 3136.a 1.a $1$ $25.041$ \(\Q\) None 28.2.e.a \(0\) \(1\) \(3\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{5}-2q^{9}+3q^{11}+2q^{13}+\cdots\)
3136.2.a.w 3136.a 1.a $1$ $25.041$ \(\Q\) None 56.2.a.b \(0\) \(2\) \(-4\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-4q^{5}+q^{9}-8q^{15}+2q^{17}+\cdots\)
3136.2.a.x 3136.a 1.a $1$ $25.041$ \(\Q\) None 1568.2.a.a \(0\) \(2\) \(-2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-2q^{5}+q^{9}+4q^{11}-6q^{13}+\cdots\)
3136.2.a.y 3136.a 1.a $1$ $25.041$ \(\Q\) None 224.2.a.a \(0\) \(2\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}-4q^{11}-4q^{13}+2q^{17}+\cdots\)
3136.2.a.z 3136.a 1.a $1$ $25.041$ \(\Q\) None 14.2.a.a \(0\) \(2\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}-4q^{13}-6q^{17}-2q^{19}+\cdots\)
3136.2.a.ba 3136.a 1.a $1$ $25.041$ \(\Q\) None 1568.2.a.a \(0\) \(2\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+2q^{5}+q^{9}-4q^{11}+6q^{13}+\cdots\)
3136.2.a.bb 3136.a 1.a $1$ $25.041$ \(\Q\) None 56.2.i.a \(0\) \(3\) \(-1\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-q^{5}+6q^{9}+q^{11}+2q^{13}+\cdots\)
3136.2.a.bc 3136.a 1.a $1$ $25.041$ \(\Q\) None 56.2.i.a \(0\) \(3\) \(1\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{5}+6q^{9}-q^{11}-2q^{13}+\cdots\)
3136.2.a.bd 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) None 224.2.i.a \(0\) \(-2\) \(-2\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+(-1+2\beta )q^{5}-2\beta q^{9}+\cdots\)
3136.2.a.be 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) None 224.2.i.a \(0\) \(-2\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+(1-2\beta )q^{5}-2\beta q^{9}+\cdots\)
3136.2.a.bf 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{5}) \) None 224.2.a.c \(0\) \(-2\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+(1-\beta )q^{5}+(3+2\beta )q^{9}+\cdots\)
3136.2.a.bg 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{7}) \) None 224.2.i.c \(0\) \(0\) \(-6\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-3q^{5}+4q^{9}+\beta q^{11}-4q^{13}+\cdots\)
3136.2.a.bh 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{3}) \) None 224.2.i.b \(0\) \(0\) \(-2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{5}+3\beta q^{11}-\beta q^{15}+5q^{17}+\cdots\)
3136.2.a.bi 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) 1568.2.a.o \(0\) \(0\) \(0\) \(0\) $-$ $+$ $N(\mathrm{U}(1))$ \(q+\beta q^{5}-3q^{9}-\beta q^{13}+5\beta q^{17}-3q^{25}+\cdots\)
3136.2.a.bj 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) 1568.2.a.p \(0\) \(0\) \(0\) \(0\) $-$ $+$ $N(\mathrm{U}(1))$ \(q-3\beta q^{5}-3q^{9}-5\beta q^{13}-3\beta q^{17}+\cdots\)
3136.2.a.bk 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) None 392.2.a.g \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+2\beta q^{5}-q^{9}-6q^{11}-4\beta q^{13}+\cdots\)
3136.2.a.bl 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) None 1568.2.a.q \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{9}-2q^{11}+2\beta q^{13}-3\beta q^{17}+\cdots\)
3136.2.a.bm 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) None 98.2.a.b \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+2\beta q^{5}-q^{9}-2q^{11}+4q^{15}+\cdots\)
3136.2.a.bn 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) None 98.2.a.b \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-2\beta q^{5}-q^{9}+2q^{11}-4q^{15}+\cdots\)
3136.2.a.bo 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) None 1568.2.a.q \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{9}+2q^{11}-2\beta q^{13}+3\beta q^{17}+\cdots\)
3136.2.a.bp 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) None 392.2.a.g \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-2\beta q^{5}-q^{9}+6q^{11}+4\beta q^{13}+\cdots\)
3136.2.a.bq 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) None 392.2.a.h \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-\beta q^{5}+5q^{9}-4q^{11}+\beta q^{13}+\cdots\)
3136.2.a.br 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) None 196.2.a.c \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+2\beta q^{3}-\beta q^{5}+5q^{9}-4q^{11}-3\beta q^{13}+\cdots\)
3136.2.a.bs 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) None 196.2.a.c \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2\beta q^{3}+\beta q^{5}+5q^{9}+4q^{11}+3\beta q^{13}+\cdots\)
3136.2.a.bt 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) None 392.2.a.h \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+\beta q^{5}+5q^{9}+4q^{11}-\beta q^{13}+\cdots\)
3136.2.a.bu 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{3}) \) None 224.2.i.b \(0\) \(0\) \(2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+q^{5}-3\beta q^{11}+\beta q^{15}-5q^{17}+\cdots\)
3136.2.a.bv 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{7}) \) None 224.2.i.c \(0\) \(0\) \(6\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+3q^{5}+4q^{9}-\beta q^{11}+4q^{13}+\cdots\)
3136.2.a.bw 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) None 224.2.i.a \(0\) \(2\) \(-2\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(-1-2\beta )q^{5}+2\beta q^{9}+\cdots\)
3136.2.a.bx 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{2}) \) None 224.2.i.a \(0\) \(2\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(1+2\beta )q^{5}+2\beta q^{9}+(1+\cdots)q^{11}+\cdots\)
3136.2.a.by 3136.a 1.a $2$ $25.041$ \(\Q(\sqrt{5}) \) None 224.2.a.c \(0\) \(2\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(1-\beta )q^{5}+(3+2\beta )q^{9}+\cdots\)
3136.2.a.bz 3136.a 1.a $4$ $25.041$ \(\Q(\sqrt{2}, \sqrt{5})\) None 1568.2.a.x \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+2\beta _{1}q^{5}+7q^{9}-\beta _{3}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3136))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3136)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(448))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(784))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1568))\)\(^{\oplus 2}\)