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 $25$

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

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

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