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Properties

Label 1050.4.a
Level $1050$
Weight $4$
Character orbit 1050.a
Rep. character $\chi_{1050}(1,\cdot)$
Character field $\Q$
Dimension $58$
Newform subspaces $39$
Sturm bound $960$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1050.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 39 \)
Sturm bound: \(960\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1050))\).

Total New Old
Modular forms 744 58 686
Cusp forms 696 58 638
Eisenstein series 48 0 48

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

\(2\)\(3\)\(5\)\(7\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(4\)
\(+\)\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(+\)\(-\)\(+\)$-$\(4\)
\(+\)\(+\)\(-\)\(-\)$+$\(4\)
\(+\)\(-\)\(+\)\(+\)$-$\(3\)
\(+\)\(-\)\(+\)\(-\)$+$\(4\)
\(+\)\(-\)\(-\)\(+\)$+$\(4\)
\(+\)\(-\)\(-\)\(-\)$-$\(4\)
\(-\)\(+\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(+\)\(-\)$+$\(4\)
\(-\)\(+\)\(-\)\(+\)$+$\(5\)
\(-\)\(+\)\(-\)\(-\)$-$\(3\)
\(-\)\(-\)\(+\)\(+\)$+$\(4\)
\(-\)\(-\)\(+\)\(-\)$-$\(2\)
\(-\)\(-\)\(-\)\(+\)$-$\(3\)
\(-\)\(-\)\(-\)\(-\)$+$\(5\)
Plus space\(+\)\(34\)
Minus space\(-\)\(24\)

Trace form

\( 58 q - 4 q^{2} + 232 q^{4} - 16 q^{8} + 522 q^{9} + O(q^{10}) \) \( 58 q - 4 q^{2} + 232 q^{4} - 16 q^{8} + 522 q^{9} - 80 q^{11} + 12 q^{13} + 928 q^{16} - 132 q^{17} - 36 q^{18} - 128 q^{19} + 42 q^{21} - 16 q^{22} + 248 q^{23} - 888 q^{26} - 340 q^{29} + 416 q^{31} - 64 q^{32} - 576 q^{33} + 72 q^{34} + 2088 q^{36} - 148 q^{37} - 64 q^{38} + 24 q^{39} + 692 q^{41} - 84 q^{42} - 1480 q^{43} - 320 q^{44} - 192 q^{46} + 1280 q^{47} + 2842 q^{49} + 384 q^{51} + 48 q^{52} + 1476 q^{53} + 192 q^{57} - 184 q^{58} + 2224 q^{59} + 4100 q^{61} - 96 q^{62} + 3712 q^{64} - 864 q^{66} + 1592 q^{67} - 528 q^{68} - 1728 q^{69} + 2120 q^{71} - 144 q^{72} + 2764 q^{73} + 3992 q^{74} - 512 q^{76} + 448 q^{77} - 48 q^{78} - 1248 q^{79} + 4698 q^{81} - 1704 q^{82} + 3376 q^{83} + 168 q^{84} + 3728 q^{86} + 192 q^{87} - 64 q^{88} + 8100 q^{89} + 784 q^{91} + 992 q^{92} + 456 q^{93} - 928 q^{94} - 116 q^{97} - 196 q^{98} - 720 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1050))\) 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 3 5 7
1050.4.a.a 1050.a 1.a $1$ $61.952$ \(\Q\) None \(-2\) \(-3\) \(0\) \(-7\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.b 1050.a 1.a $1$ $61.952$ \(\Q\) None \(-2\) \(-3\) \(0\) \(7\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.c 1050.a 1.a $1$ $61.952$ \(\Q\) None \(-2\) \(-3\) \(0\) \(7\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.d 1050.a 1.a $1$ $61.952$ \(\Q\) None \(-2\) \(-3\) \(0\) \(7\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.e 1050.a 1.a $1$ $61.952$ \(\Q\) None \(-2\) \(-3\) \(0\) \(7\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.f 1050.a 1.a $1$ $61.952$ \(\Q\) None \(-2\) \(-3\) \(0\) \(7\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.g 1050.a 1.a $1$ $61.952$ \(\Q\) None \(-2\) \(3\) \(0\) \(-7\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.h 1050.a 1.a $1$ $61.952$ \(\Q\) None \(-2\) \(3\) \(0\) \(-7\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.i 1050.a 1.a $1$ $61.952$ \(\Q\) None \(-2\) \(3\) \(0\) \(-7\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.j 1050.a 1.a $1$ $61.952$ \(\Q\) None \(-2\) \(3\) \(0\) \(7\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.k 1050.a 1.a $1$ $61.952$ \(\Q\) None \(-2\) \(3\) \(0\) \(7\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.l 1050.a 1.a $1$ $61.952$ \(\Q\) None \(-2\) \(3\) \(0\) \(7\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.m 1050.a 1.a $1$ $61.952$ \(\Q\) None \(2\) \(-3\) \(0\) \(-7\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.n 1050.a 1.a $1$ $61.952$ \(\Q\) None \(2\) \(-3\) \(0\) \(-7\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.o 1050.a 1.a $1$ $61.952$ \(\Q\) None \(2\) \(-3\) \(0\) \(7\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.p 1050.a 1.a $1$ $61.952$ \(\Q\) None \(2\) \(-3\) \(0\) \(7\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.q 1050.a 1.a $1$ $61.952$ \(\Q\) None \(2\) \(-3\) \(0\) \(7\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.r 1050.a 1.a $1$ $61.952$ \(\Q\) None \(2\) \(3\) \(0\) \(-7\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.s 1050.a 1.a $1$ $61.952$ \(\Q\) None \(2\) \(3\) \(0\) \(-7\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.t 1050.a 1.a $1$ $61.952$ \(\Q\) None \(2\) \(3\) \(0\) \(-7\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.u 1050.a 1.a $1$ $61.952$ \(\Q\) None \(2\) \(3\) \(0\) \(-7\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.v 1050.a 1.a $1$ $61.952$ \(\Q\) None \(2\) \(3\) \(0\) \(-7\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.w 1050.a 1.a $1$ $61.952$ \(\Q\) None \(2\) \(3\) \(0\) \(7\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.x 1050.a 1.a $1$ $61.952$ \(\Q\) None \(2\) \(3\) \(0\) \(7\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.y 1050.a 1.a $2$ $61.952$ \(\Q(\sqrt{8761}) \) None \(-4\) \(-6\) \(0\) \(-14\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.z 1050.a 1.a $2$ $61.952$ \(\Q(\sqrt{106}) \) None \(-4\) \(-6\) \(0\) \(-14\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.ba 1050.a 1.a $2$ $61.952$ \(\Q(\sqrt{21}) \) None \(-4\) \(-6\) \(0\) \(14\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.bb 1050.a 1.a $2$ $61.952$ \(\Q(\sqrt{1129}) \) None \(-4\) \(6\) \(0\) \(-14\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.bc 1050.a 1.a $2$ $61.952$ \(\Q(\sqrt{6}) \) None \(-4\) \(6\) \(0\) \(-14\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.bd 1050.a 1.a $2$ $61.952$ \(\Q(\sqrt{6001}) \) None \(-4\) \(6\) \(0\) \(14\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.be 1050.a 1.a $2$ $61.952$ \(\Q(\sqrt{6001}) \) None \(4\) \(-6\) \(0\) \(-14\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.bf 1050.a 1.a $2$ $61.952$ \(\Q(\sqrt{1129}) \) None \(4\) \(-6\) \(0\) \(14\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.bg 1050.a 1.a $2$ $61.952$ \(\Q(\sqrt{6}) \) None \(4\) \(-6\) \(0\) \(14\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.bh 1050.a 1.a $2$ $61.952$ \(\Q(\sqrt{21}) \) None \(4\) \(6\) \(0\) \(-14\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.bi 1050.a 1.a $2$ $61.952$ \(\Q(\sqrt{8761}) \) None \(4\) \(6\) \(0\) \(14\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.bj 1050.a 1.a $3$ $61.952$ 3.3.104916.1 None \(-6\) \(-9\) \(0\) \(-21\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.bk 1050.a 1.a $3$ $61.952$ 3.3.62004.1 None \(-6\) \(9\) \(0\) \(21\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.bl 1050.a 1.a $3$ $61.952$ 3.3.62004.1 None \(6\) \(-9\) \(0\) \(-21\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.bm 1050.a 1.a $3$ $61.952$ 3.3.104916.1 None \(6\) \(9\) \(0\) \(21\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1050))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1050)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 2}\)