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Properties

Label 9898.2.a
Level $9898$
Weight $2$
Character orbit 9898.a
Rep. character $\chi_{9898}(1,\cdot)$
Character field $\Q$
Dimension $343$
Newform subspaces $40$
Sturm bound $2856$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 9898 = 2 \cdot 7^{2} \cdot 101 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9898.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 40 \)
Sturm bound: \(2856\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(9898))\).

Total New Old
Modular forms 1444 343 1101
Cusp forms 1413 343 1070
Eisenstein series 31 0 31

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

\(2\)\(7\)\(101\)FrickeDim
\(+\)\(+\)\(+\)$+$\(45\)
\(+\)\(+\)\(-\)$-$\(41\)
\(+\)\(-\)\(+\)$-$\(40\)
\(+\)\(-\)\(-\)$+$\(46\)
\(-\)\(+\)\(+\)$-$\(46\)
\(-\)\(+\)\(-\)$+$\(36\)
\(-\)\(-\)\(+\)$+$\(37\)
\(-\)\(-\)\(-\)$-$\(52\)
Plus space\(+\)\(164\)
Minus space\(-\)\(179\)

Trace form

\( 343 q - q^{2} - 4 q^{3} + 343 q^{4} - 4 q^{5} - 2 q^{6} - q^{8} + 349 q^{9} + O(q^{10}) \) \( 343 q - q^{2} - 4 q^{3} + 343 q^{4} - 4 q^{5} - 2 q^{6} - q^{8} + 349 q^{9} + 2 q^{10} + 8 q^{11} - 4 q^{12} + 4 q^{15} + 343 q^{16} + 6 q^{17} - 5 q^{18} + 2 q^{19} - 4 q^{20} - 6 q^{22} + 12 q^{23} - 2 q^{24} + 339 q^{25} + 10 q^{26} + 20 q^{27} + 6 q^{29} + 4 q^{31} - q^{32} - 16 q^{33} - 2 q^{34} + 349 q^{36} - 44 q^{37} - 8 q^{39} + 2 q^{40} + 30 q^{41} - 34 q^{43} + 8 q^{44} - 4 q^{45} + 8 q^{46} + 12 q^{47} - 4 q^{48} + 9 q^{50} + 44 q^{51} + 2 q^{53} - 32 q^{54} - 8 q^{55} - 36 q^{57} + 4 q^{59} + 4 q^{60} + 38 q^{61} + 12 q^{62} + 343 q^{64} + 48 q^{65} + 16 q^{66} - 8 q^{67} + 6 q^{68} - 20 q^{69} + 16 q^{71} - 5 q^{72} + 6 q^{73} + 14 q^{74} - 24 q^{75} + 2 q^{76} - 4 q^{78} - 4 q^{80} + 351 q^{81} + 2 q^{82} - 28 q^{83} + 20 q^{85} - 12 q^{86} + 4 q^{87} - 6 q^{88} - 38 q^{89} + 22 q^{90} + 12 q^{92} - 60 q^{93} + 36 q^{94} + 20 q^{95} - 2 q^{96} + 6 q^{97} - 80 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(9898))\) 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 7 101
9898.2.a.a 9898.a 1.a $1$ $79.036$ \(\Q\) None \(-1\) \(-1\) \(-2\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-2q^{5}+q^{6}-q^{8}+\cdots\)
9898.2.a.b 9898.a 1.a $1$ $79.036$ \(\Q\) None \(-1\) \(0\) \(-2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-2q^{5}-q^{8}-3q^{9}+2q^{10}+\cdots\)
9898.2.a.c 9898.a 1.a $1$ $79.036$ \(\Q\) None \(1\) \(-3\) \(-3\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}+q^{4}-3q^{5}-3q^{6}+q^{8}+\cdots\)
9898.2.a.d 9898.a 1.a $1$ $79.036$ \(\Q\) None \(1\) \(-2\) \(2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}+2q^{5}-2q^{6}+q^{8}+\cdots\)
9898.2.a.e 9898.a 1.a $1$ $79.036$ \(\Q\) None \(1\) \(-1\) \(-3\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-3q^{5}-q^{6}+q^{8}+\cdots\)
9898.2.a.f 9898.a 1.a $1$ $79.036$ \(\Q\) None \(1\) \(-1\) \(4\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+4q^{5}-q^{6}+q^{8}+\cdots\)
9898.2.a.g 9898.a 1.a $1$ $79.036$ \(\Q\) None \(1\) \(1\) \(-4\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-4q^{5}+q^{6}+q^{8}+\cdots\)
9898.2.a.h 9898.a 1.a $1$ $79.036$ \(\Q\) None \(1\) \(1\) \(3\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}+q^{8}+\cdots\)
9898.2.a.i 9898.a 1.a $1$ $79.036$ \(\Q\) None \(1\) \(2\) \(-2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}-2q^{5}+2q^{6}+q^{8}+\cdots\)
9898.2.a.j 9898.a 1.a $1$ $79.036$ \(\Q\) None \(1\) \(3\) \(3\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}+q^{4}+3q^{5}+3q^{6}+q^{8}+\cdots\)
9898.2.a.k 9898.a 1.a $2$ $79.036$ \(\Q(\sqrt{2}) \) None \(-2\) \(-6\) \(4\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}+q^{4}+(2+\beta )q^{5}+3q^{6}+\cdots\)
9898.2.a.l 9898.a 1.a $2$ $79.036$ \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta )q^{3}+q^{4}+q^{5}+(1+\cdots)q^{6}+\cdots\)
9898.2.a.m 9898.a 1.a $2$ $79.036$ \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(-2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta )q^{3}+q^{4}-q^{5}+(-1+\cdots)q^{6}+\cdots\)
9898.2.a.n 9898.a 1.a $2$ $79.036$ \(\Q(\sqrt{2}) \) None \(-2\) \(6\) \(-4\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}+(-2-\beta )q^{5}-3q^{6}+\cdots\)
9898.2.a.o 9898.a 1.a $3$ $79.036$ \(\Q(\zeta_{18})^+\) None \(-3\) \(3\) \(3\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta _{1})q^{3}+q^{4}+(1-\beta _{1}-\beta _{2})q^{5}+\cdots\)
9898.2.a.p 9898.a 1.a $4$ $79.036$ 4.4.4525.1 None \(-4\) \(-1\) \(6\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+(2+\beta _{3})q^{5}+\beta _{1}q^{6}+\cdots\)
9898.2.a.q 9898.a 1.a $4$ $79.036$ 4.4.10273.1 None \(4\) \(1\) \(-3\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{2}q^{3}+q^{4}+(-1+\beta _{3})q^{5}+\cdots\)
9898.2.a.r 9898.a 1.a $4$ $79.036$ \(\Q(\zeta_{15})^+\) None \(4\) \(3\) \(2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta _{2}+\beta _{3})q^{3}+q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
9898.2.a.s 9898.a 1.a $4$ $79.036$ 4.4.1957.1 None \(4\) \(3\) \(8\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta _{2})q^{3}+q^{4}+(2-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
9898.2.a.t 9898.a 1.a $5$ $79.036$ 5.5.301909.1 None \(-5\) \(0\) \(6\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{3}q^{3}+q^{4}+(1-\beta _{4})q^{5}-\beta _{3}q^{6}+\cdots\)
9898.2.a.u 9898.a 1.a $7$ $79.036$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(-3\) \(-11\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{3}q^{3}+q^{4}+(-2+\beta _{6})q^{5}+\cdots\)
9898.2.a.v 9898.a 1.a $7$ $79.036$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(1\) \(-3\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{4}q^{3}+q^{4}-\beta _{5}q^{5}-\beta _{4}q^{6}+\cdots\)
9898.2.a.w 9898.a 1.a $8$ $79.036$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(-6\) \(-7\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1+\beta _{3}+\cdots)q^{5}+\cdots\)
9898.2.a.x 9898.a 1.a $9$ $79.036$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(-4\) \(-1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{3}q^{5}-\beta _{1}q^{6}+\cdots\)
9898.2.a.y 9898.a 1.a $11$ $79.036$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(11\) \(-2\) \(-7\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+(-1+\beta _{3})q^{5}+\cdots\)
9898.2.a.z 9898.a 1.a $11$ $79.036$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(11\) \(2\) \(7\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+(1-\beta _{3})q^{5}+\beta _{1}q^{6}+\cdots\)
9898.2.a.ba 9898.a 1.a $12$ $79.036$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-12\) \(-3\) \(-5\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{5}q^{5}+\beta _{1}q^{6}+\cdots\)
9898.2.a.bb 9898.a 1.a $12$ $79.036$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-12\) \(3\) \(5\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{5}q^{5}-\beta _{1}q^{6}+\cdots\)
9898.2.a.bc 9898.a 1.a $12$ $79.036$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(12\) \(0\) \(-1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{5}q^{5}+\beta _{1}q^{6}+\cdots\)
9898.2.a.bd 9898.a 1.a $12$ $79.036$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(12\) \(0\) \(1\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{4}q^{5}+\beta _{1}q^{6}+\cdots\)
9898.2.a.be 9898.a 1.a $13$ $79.036$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(-13\) \(-2\) \(1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{2}q^{5}+\beta _{1}q^{6}+\cdots\)
9898.2.a.bf 9898.a 1.a $13$ $79.036$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(-13\) \(2\) \(-1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{2}q^{5}-\beta _{1}q^{6}+\cdots\)
9898.2.a.bg 9898.a 1.a $19$ $79.036$ \(\mathbb{Q}[x]/(x^{19} - \cdots)\) None \(-19\) \(0\) \(-1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{8}q^{5}-\beta _{1}q^{6}+\cdots\)
9898.2.a.bh 9898.a 1.a $19$ $79.036$ \(\mathbb{Q}[x]/(x^{19} - \cdots)\) None \(-19\) \(0\) \(1\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{8}q^{5}+\beta _{1}q^{6}+\cdots\)
9898.2.a.bi 9898.a 1.a $20$ $79.036$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(20\) \(-2\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{7}q^{5}-\beta _{1}q^{6}+\cdots\)
9898.2.a.bj 9898.a 1.a $20$ $79.036$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(20\) \(2\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{7}q^{5}+\beta _{1}q^{6}+\cdots\)
9898.2.a.bk 9898.a 1.a $24$ $79.036$ None \(-24\) \(0\) \(-14\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$
9898.2.a.bl 9898.a 1.a $24$ $79.036$ None \(-24\) \(0\) \(14\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$
9898.2.a.bm 9898.a 1.a $24$ $79.036$ None \(24\) \(-6\) \(-18\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$
9898.2.a.bn 9898.a 1.a $24$ $79.036$ None \(24\) \(6\) \(18\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(9898))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(9898)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(101))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(202))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(707))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1414))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4949))\)\(^{\oplus 2}\)