↑

Properties

Label 350.4.a
Level $350$
Weight $4$
Character orbit 350.a
Rep. character $\chi_{350}(1,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $24$
Sturm bound $240$
Trace bound $11$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(240\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(350))\).

Total New Old
Modular forms 192 28 164
Cusp forms 168 28 140
Eisenstein series 24 0 24

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
\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(-\)\(+\)$-$\(3\)
\(+\)\(-\)\(-\)$+$\(5\)
\(-\)\(+\)\(+\)$-$\(4\)
\(-\)\(+\)\(-\)$+$\(5\)
\(-\)\(-\)\(+\)$+$\(4\)
\(-\)\(-\)\(-\)$-$\(2\)
Plus space\(+\)\(17\)
Minus space\(-\)\(11\)

Trace form

\( 28 q + 4 q^{2} - 10 q^{3} + 112 q^{4} - 20 q^{6} + 16 q^{8} + 276 q^{9} + O(q^{10}) \) \( 28 q + 4 q^{2} - 10 q^{3} + 112 q^{4} - 20 q^{6} + 16 q^{8} + 276 q^{9} - 40 q^{11} - 40 q^{12} - 42 q^{13} - 28 q^{14} + 448 q^{16} + 28 q^{17} + 44 q^{18} + 178 q^{19} + 70 q^{21} + 8 q^{22} + 160 q^{23} - 80 q^{24} + 444 q^{26} + 44 q^{27} + 444 q^{29} - 268 q^{31} + 64 q^{32} + 744 q^{33} - 24 q^{34} + 1104 q^{36} + 188 q^{37} - 204 q^{38} - 272 q^{39} - 848 q^{41} + 84 q^{42} + 764 q^{43} - 160 q^{44} + 336 q^{46} - 1420 q^{47} - 160 q^{48} + 1372 q^{49} + 920 q^{51} - 168 q^{52} - 136 q^{53} + 1600 q^{54} - 112 q^{56} - 948 q^{57} + 848 q^{58} + 502 q^{59} - 2242 q^{61} + 360 q^{62} + 644 q^{63} + 1792 q^{64} - 456 q^{66} - 304 q^{67} + 112 q^{68} + 4776 q^{69} + 2872 q^{71} + 176 q^{72} - 968 q^{73} - 3344 q^{74} + 712 q^{76} + 84 q^{77} + 1264 q^{78} + 144 q^{79} + 7968 q^{81} - 1248 q^{82} - 3630 q^{83} + 280 q^{84} - 3832 q^{86} + 2500 q^{87} + 32 q^{88} - 3100 q^{89} - 1106 q^{91} + 640 q^{92} + 3200 q^{93} + 1592 q^{94} - 320 q^{96} + 4156 q^{97} + 196 q^{98} - 4780 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(350))\) 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
350.4.a.a 350.a 1.a $1$ $20.651$ \(\Q\) None \(-2\) \(-8\) \(0\) \(-7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-8q^{3}+4q^{4}+2^{4}q^{6}-7q^{7}+\cdots\)
350.4.a.b 350.a 1.a $1$ $20.651$ \(\Q\) None \(-2\) \(-7\) \(0\) \(-7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-7q^{3}+4q^{4}+14q^{6}-7q^{7}+\cdots\)
350.4.a.c 350.a 1.a $1$ $20.651$ \(\Q\) None \(-2\) \(-5\) \(0\) \(7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-5q^{3}+4q^{4}+10q^{6}+7q^{7}+\cdots\)
350.4.a.d 350.a 1.a $1$ $20.651$ \(\Q\) None \(-2\) \(-1\) \(0\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+4q^{4}+2q^{6}+7q^{7}+\cdots\)
350.4.a.e 350.a 1.a $1$ $20.651$ \(\Q\) None \(-2\) \(2\) \(0\) \(-7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{3}+4q^{4}-4q^{6}-7q^{7}+\cdots\)
350.4.a.f 350.a 1.a $1$ $20.651$ \(\Q\) None \(-2\) \(2\) \(0\) \(-7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{3}+4q^{4}-4q^{6}-7q^{7}+\cdots\)
350.4.a.g 350.a 1.a $1$ $20.651$ \(\Q\) None \(-2\) \(3\) \(0\) \(-7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
350.4.a.h 350.a 1.a $1$ $20.651$ \(\Q\) None \(-2\) \(4\) \(0\) \(7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{3}+4q^{4}-8q^{6}+7q^{7}+\cdots\)
350.4.a.i 350.a 1.a $1$ $20.651$ \(\Q\) None \(-2\) \(7\) \(0\) \(-7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+7q^{3}+4q^{4}-14q^{6}-7q^{7}+\cdots\)
350.4.a.j 350.a 1.a $1$ $20.651$ \(\Q\) None \(-2\) \(10\) \(0\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+10q^{3}+4q^{4}-20q^{6}+7q^{7}+\cdots\)
350.4.a.k 350.a 1.a $1$ $20.651$ \(\Q\) None \(2\) \(-10\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-10q^{3}+4q^{4}-20q^{6}-7q^{7}+\cdots\)
350.4.a.l 350.a 1.a $1$ $20.651$ \(\Q\) None \(2\) \(-8\) \(0\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-8q^{3}+4q^{4}-2^{4}q^{6}+7q^{7}+\cdots\)
350.4.a.m 350.a 1.a $1$ $20.651$ \(\Q\) None \(2\) \(-7\) \(0\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-7q^{3}+4q^{4}-14q^{6}+7q^{7}+\cdots\)
350.4.a.n 350.a 1.a $1$ $20.651$ \(\Q\) None \(2\) \(-4\) \(0\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-4q^{3}+4q^{4}-8q^{6}-7q^{7}+\cdots\)
350.4.a.o 350.a 1.a $1$ $20.651$ \(\Q\) None \(2\) \(-4\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-4q^{3}+4q^{4}-8q^{6}-7q^{7}+\cdots\)
350.4.a.p 350.a 1.a $1$ $20.651$ \(\Q\) None \(2\) \(-3\) \(0\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
350.4.a.q 350.a 1.a $1$ $20.651$ \(\Q\) None \(2\) \(-2\) \(0\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-2q^{3}+4q^{4}-4q^{6}+7q^{7}+\cdots\)
350.4.a.r 350.a 1.a $1$ $20.651$ \(\Q\) None \(2\) \(1\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+q^{3}+4q^{4}+2q^{6}-7q^{7}+\cdots\)
350.4.a.s 350.a 1.a $1$ $20.651$ \(\Q\) None \(2\) \(1\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+q^{3}+4q^{4}+2q^{6}-7q^{7}+\cdots\)
350.4.a.t 350.a 1.a $1$ $20.651$ \(\Q\) None \(2\) \(3\) \(0\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
350.4.a.u 350.a 1.a $1$ $20.651$ \(\Q\) None \(2\) \(8\) \(0\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+8q^{3}+4q^{4}+2^{4}q^{6}+7q^{7}+\cdots\)
350.4.a.v 350.a 1.a $1$ $20.651$ \(\Q\) None \(2\) \(8\) \(0\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+8q^{3}+4q^{4}+2^{4}q^{6}+7q^{7}+\cdots\)
350.4.a.w 350.a 1.a $3$ $20.651$ 3.3.51960.1 None \(-6\) \(-7\) \(0\) \(21\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-2-\beta _{1})q^{3}+4q^{4}+(4+2\beta _{1}+\cdots)q^{6}+\cdots\)
350.4.a.x 350.a 1.a $3$ $20.651$ 3.3.51960.1 None \(6\) \(7\) \(0\) \(-21\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(2+\beta _{1})q^{3}+4q^{4}+(4+2\beta _{1}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(350))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(350)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)