Properties

Label 1400.2.a
Level $1400$
Weight $2$
Character orbit 1400.a
Rep. character $\chi_{1400}(1,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $20$
Sturm bound $480$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 20 \)
Sturm bound: \(480\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1400))\).

Total New Old
Modular forms 264 28 236
Cusp forms 217 28 189
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.

\(2\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(5\)
\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(11\)
Minus space\(-\)\(17\)

Trace form

\( 28 q + 2 q^{3} + 20 q^{9} + O(q^{10}) \) \( 28 q + 2 q^{3} + 20 q^{9} + 8 q^{11} - 2 q^{13} - 4 q^{17} + 6 q^{19} - 2 q^{21} + 20 q^{27} + 12 q^{29} - 4 q^{31} - 24 q^{33} + 12 q^{37} + 16 q^{41} + 4 q^{43} - 4 q^{47} + 28 q^{49} + 56 q^{51} + 24 q^{53} + 12 q^{57} + 18 q^{59} + 38 q^{61} - 4 q^{63} + 32 q^{67} + 8 q^{69} - 8 q^{71} - 8 q^{73} + 4 q^{77} - 16 q^{79} - 26 q^{83} + 12 q^{87} + 4 q^{89} + 10 q^{91} + 32 q^{93} - 36 q^{97} + 44 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\) 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 5 7
1400.2.a.a 1400.a 1.a $1$ $11.179$ \(\Q\) None \(0\) \(-2\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-q^{7}+q^{9}+2q^{17}-2q^{19}+\cdots\)
1400.2.a.b 1400.a 1.a $1$ $11.179$ \(\Q\) None \(0\) \(-2\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-q^{7}+q^{9}+5q^{11}-8q^{17}+\cdots\)
1400.2.a.c 1400.a 1.a $1$ $11.179$ \(\Q\) None \(0\) \(-2\) \(0\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{7}+q^{9}+q^{11}-4q^{13}+\cdots\)
1400.2.a.d 1400.a 1.a $1$ $11.179$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}-2q^{9}-q^{11}+q^{13}+3q^{17}+\cdots\)
1400.2.a.e 1400.a 1.a $1$ $11.179$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}-2q^{9}-q^{11}+6q^{13}+\cdots\)
1400.2.a.f 1400.a 1.a $1$ $11.179$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{7}-3q^{9}+q^{11}+2q^{13}+4q^{17}+\cdots\)
1400.2.a.g 1400.a 1.a $1$ $11.179$ \(\Q\) None \(0\) \(0\) \(0\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{7}-3q^{9}-4q^{11}-2q^{13}+6q^{17}+\cdots\)
1400.2.a.h 1400.a 1.a $1$ $11.179$ \(\Q\) None \(0\) \(0\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{7}-3q^{9}+q^{11}-2q^{13}-4q^{17}+\cdots\)
1400.2.a.i 1400.a 1.a $1$ $11.179$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}-2q^{9}-q^{11}-6q^{13}+\cdots\)
1400.2.a.j 1400.a 1.a $1$ $11.179$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}-2q^{9}-q^{11}-q^{13}-3q^{17}+\cdots\)
1400.2.a.k 1400.a 1.a $1$ $11.179$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}-2q^{9}-5q^{11}-q^{13}+\cdots\)
1400.2.a.l 1400.a 1.a $1$ $11.179$ \(\Q\) None \(0\) \(2\) \(0\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{7}+q^{9}+q^{11}+4q^{13}+\cdots\)
1400.2.a.m 1400.a 1.a $1$ $11.179$ \(\Q\) None \(0\) \(2\) \(0\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{7}+q^{9}+5q^{11}+8q^{17}+\cdots\)
1400.2.a.n 1400.a 1.a $1$ $11.179$ \(\Q\) None \(0\) \(3\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-q^{7}+6q^{9}-5q^{11}+5q^{13}+\cdots\)
1400.2.a.o 1400.a 1.a $2$ $11.179$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}-q^{7}+(1+\beta )q^{9}+(-3+2\beta )q^{11}+\cdots\)
1400.2.a.p 1400.a 1.a $2$ $11.179$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}-q^{7}+(1+\beta )q^{9}-\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
1400.2.a.q 1400.a 1.a $2$ $11.179$ \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+q^{7}+(1+\beta )q^{9}+(-3+2\beta )q^{11}+\cdots\)
1400.2.a.r 1400.a 1.a $2$ $11.179$ \(\Q(\sqrt{33}) \) None \(0\) \(1\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+q^{7}+(5+\beta )q^{9}+(4-\beta )q^{11}+\cdots\)
1400.2.a.s 1400.a 1.a $3$ $11.179$ 3.3.568.1 None \(0\) \(-1\) \(0\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots\)
1400.2.a.t 1400.a 1.a $3$ $11.179$ 3.3.568.1 None \(0\) \(1\) \(0\) \(-3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1400))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1400)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 2}\)