Properties

Label 400.4.a
Level $400$
Weight $4$
Character orbit 400.a
Rep. character $\chi_{400}(1,\cdot)$
Character field $\Q$
Dimension $27$
Newform subspaces $24$
Sturm bound $240$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(240\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(400))\).

Total New Old
Modular forms 198 30 168
Cusp forms 162 27 135
Eisenstein series 36 3 33

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

\(2\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(7\)
\(-\)\(+\)\(-\)\(6\)
\(-\)\(-\)\(+\)\(7\)
Plus space\(+\)\(14\)
Minus space\(-\)\(13\)

Trace form

\( 27 q + 2 q^{3} - 26 q^{7} + 207 q^{9} + O(q^{10}) \) \( 27 q + 2 q^{3} - 26 q^{7} + 207 q^{9} - 20 q^{11} - 22 q^{13} + 26 q^{17} - 136 q^{19} + 92 q^{21} - 30 q^{23} + 236 q^{27} + 2 q^{29} + 84 q^{31} + 208 q^{33} + 170 q^{37} - 100 q^{39} + 214 q^{41} + 378 q^{43} + 430 q^{47} + 911 q^{49} - 896 q^{51} - 526 q^{53} + 728 q^{57} + 292 q^{59} - 754 q^{61} - 2170 q^{63} - 1186 q^{67} - 1372 q^{69} + 2132 q^{71} - 1198 q^{73} - 368 q^{77} + 784 q^{79} + 279 q^{81} + 3178 q^{83} + 2748 q^{87} - 454 q^{89} - 2828 q^{91} + 1608 q^{93} - 766 q^{97} + 2752 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(400))\) 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
400.4.a.a 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(-9\) \(0\) \(-26\) $+$ $-$ $\mathrm{SU}(2)$ \(q-9q^{3}-26q^{7}+54q^{9}+59q^{11}+\cdots\)
400.4.a.b 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(-8\) \(0\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{3}-4q^{7}+37q^{9}-12q^{11}+58q^{13}+\cdots\)
400.4.a.c 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(-7\) \(0\) \(-6\) $-$ $+$ $\mathrm{SU}(2)$ \(q-7q^{3}-6q^{7}+22q^{9}+43q^{11}-28q^{13}+\cdots\)
400.4.a.d 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(-7\) \(0\) \(34\) $-$ $-$ $\mathrm{SU}(2)$ \(q-7q^{3}+34q^{7}+22q^{9}-3^{3}q^{11}+\cdots\)
400.4.a.e 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(-6\) \(0\) \(-34\) $+$ $+$ $\mathrm{SU}(2)$ \(q-6q^{3}-34q^{7}+9q^{9}-2^{4}q^{11}-58q^{13}+\cdots\)
400.4.a.f 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(-5\) \(0\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{3}-2q^{7}-2q^{9}-39q^{11}+84q^{13}+\cdots\)
400.4.a.g 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(-4\) \(0\) \(24\) $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{3}+24q^{7}-11q^{9}+44q^{11}+\cdots\)
400.4.a.h 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(-2\) \(0\) \(-26\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-26q^{7}-23q^{9}+28q^{11}+\cdots\)
400.4.a.i 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(-1\) \(0\) \(-26\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-26q^{7}-26q^{9}-45q^{11}+44q^{13}+\cdots\)
400.4.a.j 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(-1\) \(0\) \(6\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+6q^{7}-26q^{9}+19q^{11}+12q^{13}+\cdots\)
400.4.a.k 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(1\) \(0\) \(-6\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-6q^{7}-26q^{9}+19q^{11}-12q^{13}+\cdots\)
400.4.a.l 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(1\) \(0\) \(26\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+26q^{7}-26q^{9}-45q^{11}-44q^{13}+\cdots\)
400.4.a.m 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(2\) \(0\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+6q^{7}-23q^{9}-2^{5}q^{11}+38q^{13}+\cdots\)
400.4.a.n 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(2\) \(0\) \(26\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+26q^{7}-23q^{9}+28q^{11}+\cdots\)
400.4.a.o 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(4\) \(0\) \(-16\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{3}-2^{4}q^{7}-11q^{9}+60q^{11}+\cdots\)
400.4.a.p 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(4\) \(0\) \(16\) $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{3}+2^{4}q^{7}-11q^{9}-6^{2}q^{11}+\cdots\)
400.4.a.q 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(5\) \(0\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{3}+2q^{7}-2q^{9}-39q^{11}-84q^{13}+\cdots\)
400.4.a.r 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(7\) \(0\) \(-34\) $-$ $+$ $\mathrm{SU}(2)$ \(q+7q^{3}-34q^{7}+22q^{9}-3^{3}q^{11}+\cdots\)
400.4.a.s 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(7\) \(0\) \(6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+7q^{3}+6q^{7}+22q^{9}+43q^{11}+28q^{13}+\cdots\)
400.4.a.t 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(9\) \(0\) \(26\) $+$ $+$ $\mathrm{SU}(2)$ \(q+9q^{3}+26q^{7}+54q^{9}+59q^{11}+\cdots\)
400.4.a.u 400.a 1.a $1$ $23.601$ \(\Q\) None \(0\) \(10\) \(0\) \(-18\) $+$ $+$ $\mathrm{SU}(2)$ \(q+10q^{3}-18q^{7}+73q^{9}+2^{4}q^{11}+\cdots\)
400.4.a.v 400.a 1.a $2$ $23.601$ \(\Q(\sqrt{6}) \) None \(0\) \(-4\) \(0\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{3}+(-2-3\beta )q^{7}+(1-4\beta )q^{9}+\cdots\)
400.4.a.w 400.a 1.a $2$ $23.601$ \(\Q(\sqrt{19}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+\beta q^{7}+7^{2}q^{9}-20q^{11}+6\beta q^{13}+\cdots\)
400.4.a.x 400.a 1.a $2$ $23.601$ \(\Q(\sqrt{6}) \) None \(0\) \(4\) \(0\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{3}+(2-3\beta )q^{7}+(1+4\beta )q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(400))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(400)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)