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Properties

Label 1950.2.a
Level $1950$
Weight $2$
Character orbit 1950.a
Rep. character $\chi_{1950}(1,\cdot)$
Character field $\Q$
Dimension $38$
Newform subspaces $33$
Sturm bound $840$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 33 \)
Sturm bound: \(840\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\), \(23\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1950))\).

Total New Old
Modular forms 444 38 406
Cusp forms 397 38 359
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.

\(2\)\(3\)\(5\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(+\)\(-\)$-$\(4\)
\(+\)\(+\)\(-\)\(+\)$-$\(3\)
\(+\)\(+\)\(-\)\(-\)$+$\(1\)
\(+\)\(-\)\(+\)\(+\)$-$\(2\)
\(+\)\(-\)\(+\)\(-\)$+$\(2\)
\(+\)\(-\)\(-\)\(+\)$+$\(3\)
\(+\)\(-\)\(-\)\(-\)$-$\(3\)
\(-\)\(+\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(+\)\(-\)$+$\(2\)
\(-\)\(+\)\(-\)\(+\)$+$\(3\)
\(-\)\(+\)\(-\)\(-\)$-$\(3\)
\(-\)\(-\)\(+\)\(+\)$+$\(1\)
\(-\)\(-\)\(+\)\(-\)$-$\(4\)
\(-\)\(-\)\(-\)\(+\)$-$\(3\)
\(-\)\(-\)\(-\)\(-\)$+$\(1\)
Plus space\(+\)\(14\)
Minus space\(-\)\(24\)

Trace form

\( 38 q + 38 q^{4} - 2 q^{6} - 12 q^{7} + 38 q^{9} + O(q^{10}) \) \( 38 q + 38 q^{4} - 2 q^{6} - 12 q^{7} + 38 q^{9} - 8 q^{11} + 2 q^{13} + 4 q^{14} + 38 q^{16} + 4 q^{17} + 4 q^{19} - 4 q^{21} - 8 q^{22} - 8 q^{23} - 2 q^{24} - 12 q^{28} - 12 q^{29} - 12 q^{31} + 8 q^{33} + 38 q^{36} + 20 q^{37} - 4 q^{38} + 4 q^{42} - 16 q^{43} - 8 q^{44} - 8 q^{46} + 8 q^{47} + 30 q^{49} + 2 q^{52} + 28 q^{53} - 2 q^{54} + 4 q^{56} - 20 q^{57} + 16 q^{58} - 4 q^{61} + 4 q^{62} - 12 q^{63} + 38 q^{64} + 8 q^{66} + 20 q^{67} + 4 q^{68} + 8 q^{69} - 32 q^{71} + 20 q^{73} + 8 q^{74} + 4 q^{76} + 48 q^{77} + 2 q^{78} - 16 q^{79} + 38 q^{81} + 20 q^{82} - 4 q^{84} - 24 q^{86} + 16 q^{87} - 8 q^{88} + 24 q^{89} - 12 q^{91} - 8 q^{92} - 12 q^{93} + 8 q^{94} - 2 q^{96} + 20 q^{97} + 16 q^{98} - 8 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1950))\) 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 13
1950.2.a.a 1950.a 1.a $1$ $15.571$ \(\Q\) None \(-1\) \(-1\) \(0\) \(-3\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-3q^{7}-q^{8}+\cdots\)
1950.2.a.b 1950.a 1.a $1$ $15.571$ \(\Q\) None \(-1\) \(-1\) \(0\) \(-2\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-2q^{7}-q^{8}+\cdots\)
1950.2.a.c 1950.a 1.a $1$ $15.571$ \(\Q\) None \(-1\) \(-1\) \(0\) \(-2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-2q^{7}-q^{8}+\cdots\)
1950.2.a.d 1950.a 1.a $1$ $15.571$ \(\Q\) None \(-1\) \(-1\) \(0\) \(4\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+4q^{7}-q^{8}+\cdots\)
1950.2.a.e 1950.a 1.a $1$ $15.571$ \(\Q\) None \(-1\) \(-1\) \(0\) \(4\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+4q^{7}-q^{8}+\cdots\)
1950.2.a.f 1950.a 1.a $1$ $15.571$ \(\Q\) None \(-1\) \(1\) \(0\) \(-4\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-4q^{7}-q^{8}+\cdots\)
1950.2.a.g 1950.a 1.a $1$ $15.571$ \(\Q\) None \(-1\) \(1\) \(0\) \(-4\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-4q^{7}-q^{8}+\cdots\)
1950.2.a.h 1950.a 1.a $1$ $15.571$ \(\Q\) None \(-1\) \(1\) \(0\) \(-2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-2q^{7}-q^{8}+\cdots\)
1950.2.a.i 1950.a 1.a $1$ $15.571$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
1950.2.a.j 1950.a 1.a $1$ $15.571$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
1950.2.a.k 1950.a 1.a $1$ $15.571$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
1950.2.a.l 1950.a 1.a $1$ $15.571$ \(\Q\) None \(-1\) \(1\) \(0\) \(1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+q^{7}-q^{8}+\cdots\)
1950.2.a.m 1950.a 1.a $1$ $15.571$ \(\Q\) None \(-1\) \(1\) \(0\) \(1\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+q^{7}-q^{8}+\cdots\)
1950.2.a.n 1950.a 1.a $1$ $15.571$ \(\Q\) None \(1\) \(-1\) \(0\) \(-4\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-4q^{7}+q^{8}+\cdots\)
1950.2.a.o 1950.a 1.a $1$ $15.571$ \(\Q\) None \(1\) \(-1\) \(0\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
1950.2.a.p 1950.a 1.a $1$ $15.571$ \(\Q\) None \(1\) \(-1\) \(0\) \(-1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-q^{7}+q^{8}+\cdots\)
1950.2.a.q 1950.a 1.a $1$ $15.571$ \(\Q\) None \(1\) \(-1\) \(0\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-q^{7}+q^{8}+\cdots\)
1950.2.a.r 1950.a 1.a $1$ $15.571$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
1950.2.a.s 1950.a 1.a $1$ $15.571$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
1950.2.a.t 1950.a 1.a $1$ $15.571$ \(\Q\) None \(1\) \(-1\) \(0\) \(4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+4q^{7}+q^{8}+\cdots\)
1950.2.a.u 1950.a 1.a $1$ $15.571$ \(\Q\) None \(1\) \(-1\) \(0\) \(4\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+4q^{7}+q^{8}+\cdots\)
1950.2.a.v 1950.a 1.a $1$ $15.571$ \(\Q\) None \(1\) \(1\) \(0\) \(-4\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
1950.2.a.w 1950.a 1.a $1$ $15.571$ \(\Q\) None \(1\) \(1\) \(0\) \(-4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
1950.2.a.x 1950.a 1.a $1$ $15.571$ \(\Q\) None \(1\) \(1\) \(0\) \(-4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
1950.2.a.y 1950.a 1.a $1$ $15.571$ \(\Q\) None \(1\) \(1\) \(0\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
1950.2.a.z 1950.a 1.a $1$ $15.571$ \(\Q\) None \(1\) \(1\) \(0\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots\)
1950.2.a.ba 1950.a 1.a $1$ $15.571$ \(\Q\) None \(1\) \(1\) \(0\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots\)
1950.2.a.bb 1950.a 1.a $1$ $15.571$ \(\Q\) None \(1\) \(1\) \(0\) \(3\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+3q^{7}+q^{8}+\cdots\)
1950.2.a.bc 1950.a 1.a $2$ $15.571$ \(\Q(\sqrt{41}) \) None \(-2\) \(-2\) \(0\) \(-3\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+(-1-\beta )q^{7}+\cdots\)
1950.2.a.bd 1950.a 1.a $2$ $15.571$ \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(0\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+\beta q^{7}-q^{8}+\cdots\)
1950.2.a.be 1950.a 1.a $2$ $15.571$ \(\Q(\sqrt{5}) \) None \(-2\) \(2\) \(0\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+(1+\beta )q^{7}+\cdots\)
1950.2.a.bf 1950.a 1.a $2$ $15.571$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(0\) \(-2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+(-1-\beta )q^{7}+\cdots\)
1950.2.a.bg 1950.a 1.a $2$ $15.571$ \(\Q(\sqrt{41}) \) None \(2\) \(2\) \(0\) \(3\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+(1+\beta )q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1950))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1950)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(325))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(390))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(650))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(975))\)\(^{\oplus 2}\)