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Properties

Label 3038.2.a
Level $3038$
Weight $2$
Character orbit 3038.a
Rep. character $\chi_{3038}(1,\cdot)$
Character field $\Q$
Dimension $102$
Newform subspaces $41$
Sturm bound $896$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 3038 = 2 \cdot 7^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3038.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 41 \)
Sturm bound: \(896\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3038))\).

Total New Old
Modular forms 464 102 362
Cusp forms 433 102 331
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.

\(2\)\(7\)\(31\)FrickeDim
\(+\)\(+\)\(+\)$+$\(11\)
\(+\)\(+\)\(-\)$-$\(15\)
\(+\)\(-\)\(+\)$-$\(16\)
\(+\)\(-\)\(-\)$+$\(10\)
\(-\)\(+\)\(+\)$-$\(13\)
\(-\)\(+\)\(-\)$+$\(9\)
\(-\)\(-\)\(+\)$+$\(11\)
\(-\)\(-\)\(-\)$-$\(17\)
Plus space\(+\)\(41\)
Minus space\(-\)\(61\)

Trace form

\( 102 q - 2 q^{2} - 2 q^{3} + 102 q^{4} + 4 q^{5} + 2 q^{6} - 2 q^{8} + 110 q^{9} + O(q^{10}) \) \( 102 q - 2 q^{2} - 2 q^{3} + 102 q^{4} + 4 q^{5} + 2 q^{6} - 2 q^{8} + 110 q^{9} - 4 q^{10} + 14 q^{11} - 2 q^{12} - 14 q^{13} + 20 q^{15} + 102 q^{16} + 8 q^{17} - 2 q^{18} - 12 q^{19} + 4 q^{20} - 6 q^{22} + 8 q^{23} + 2 q^{24} + 90 q^{25} - 2 q^{26} - 8 q^{27} - 10 q^{29} + 12 q^{30} - 2 q^{32} - 8 q^{34} + 110 q^{36} - 2 q^{37} + 4 q^{38} + 36 q^{39} - 4 q^{40} - 4 q^{41} + 2 q^{43} + 14 q^{44} + 60 q^{45} + 16 q^{46} + 4 q^{47} - 2 q^{48} - 22 q^{50} + 4 q^{51} - 14 q^{52} - 22 q^{53} + 8 q^{54} - 4 q^{55} + 24 q^{57} - 30 q^{58} - 8 q^{59} + 20 q^{60} - 6 q^{61} + 4 q^{62} + 102 q^{64} - 36 q^{65} + 16 q^{66} - 16 q^{67} + 8 q^{68} + 24 q^{69} + 32 q^{71} - 2 q^{72} + 12 q^{73} - 14 q^{74} + 50 q^{75} - 12 q^{76} + 28 q^{78} + 36 q^{79} + 4 q^{80} + 166 q^{81} - 4 q^{82} + 2 q^{83} - 16 q^{85} + 6 q^{86} + 60 q^{87} - 6 q^{88} + 44 q^{89} + 4 q^{90} + 8 q^{92} - 2 q^{93} + 44 q^{94} + 64 q^{95} + 2 q^{96} - 44 q^{97} + 2 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3038))\) 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 31
3038.2.a.a 3038.a 1.a $1$ $24.259$ \(\Q\) None \(-1\) \(-3\) \(3\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}+q^{4}+3q^{5}+3q^{6}-q^{8}+\cdots\)
3038.2.a.b 3038.a 1.a $1$ $24.259$ \(\Q\) None \(-1\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{8}-3q^{9}-2q^{11}+2q^{13}+\cdots\)
3038.2.a.c 3038.a 1.a $1$ $24.259$ \(\Q\) None \(-1\) \(0\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{8}-3q^{9}-4q^{13}+q^{16}+\cdots\)
3038.2.a.d 3038.a 1.a $1$ $24.259$ \(\Q\) None \(-1\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{8}-3q^{9}+4q^{13}+q^{16}+\cdots\)
3038.2.a.e 3038.a 1.a $1$ $24.259$ \(\Q\) None \(-1\) \(3\) \(-3\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}-3q^{5}-3q^{6}-q^{8}+\cdots\)
3038.2.a.f 3038.a 1.a $1$ $24.259$ \(\Q\) None \(1\) \(-3\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}+q^{4}-3q^{6}+q^{8}+6q^{9}+\cdots\)
3038.2.a.g 3038.a 1.a $1$ $24.259$ \(\Q\) None \(1\) \(-2\) \(-2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-2q^{5}-2q^{6}+q^{8}+\cdots\)
3038.2.a.h 3038.a 1.a $1$ $24.259$ \(\Q\) None \(1\) \(-1\) \(-4\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-4q^{5}-q^{6}+q^{8}+\cdots\)
3038.2.a.i 3038.a 1.a $1$ $24.259$ \(\Q\) None \(1\) \(-1\) \(-3\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-3q^{5}-q^{6}+q^{8}+\cdots\)
3038.2.a.j 3038.a 1.a $1$ $24.259$ \(\Q\) None \(1\) \(0\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2q^{5}+q^{8}-3q^{9}+2q^{10}+\cdots\)
3038.2.a.k 3038.a 1.a $1$ $24.259$ \(\Q\) None \(1\) \(1\) \(4\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+4q^{5}+q^{6}+q^{8}+\cdots\)
3038.2.a.l 3038.a 1.a $1$ $24.259$ \(\Q\) None \(1\) \(2\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{5}+2q^{6}+q^{8}+\cdots\)
3038.2.a.m 3038.a 1.a $1$ $24.259$ \(\Q\) None \(1\) \(3\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}+q^{4}+3q^{6}+q^{8}+6q^{9}+\cdots\)
3038.2.a.n 3038.a 1.a $1$ $24.259$ \(\Q\) None \(1\) \(3\) \(3\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}+q^{4}+3q^{5}+3q^{6}+q^{8}+\cdots\)
3038.2.a.o 3038.a 1.a $2$ $24.259$ \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta )q^{3}+q^{4}-2\beta q^{5}+\cdots\)
3038.2.a.p 3038.a 1.a $2$ $24.259$ \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(2\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta )q^{3}+q^{4}+(1+2\beta )q^{5}+\cdots\)
3038.2.a.q 3038.a 1.a $2$ $24.259$ \(\Q(\sqrt{17}) \) None \(-2\) \(-1\) \(3\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta q^{3}+q^{4}+(2-\beta )q^{5}+\beta q^{6}+\cdots\)
3038.2.a.r 3038.a 1.a $2$ $24.259$ \(\Q(\sqrt{17}) \) None \(-2\) \(1\) \(-3\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}+(-2+\beta )q^{5}-\beta q^{6}+\cdots\)
3038.2.a.s 3038.a 1.a $2$ $24.259$ \(\Q(\sqrt{2}) \) None \(2\) \(-4\) \(-4\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}+(-2+\beta )q^{5}-2q^{6}+\cdots\)
3038.2.a.t 3038.a 1.a $2$ $24.259$ \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(-2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-1+2\beta )q^{5}+\cdots\)
3038.2.a.u 3038.a 1.a $2$ $24.259$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(-2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
3038.2.a.v 3038.a 1.a $2$ $24.259$ \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta )q^{3}+q^{4}+2\beta q^{5}+\cdots\)
3038.2.a.w 3038.a 1.a $2$ $24.259$ \(\Q(\sqrt{17}) \) None \(2\) \(-1\) \(-3\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta q^{3}+q^{4}+(-2+\beta )q^{5}-\beta q^{6}+\cdots\)
3038.2.a.x 3038.a 1.a $2$ $24.259$ \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}+2\beta q^{5}+(1+\cdots)q^{6}+\cdots\)
3038.2.a.y 3038.a 1.a $2$ $24.259$ \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+2\beta )q^{5}+\cdots\)
3038.2.a.z 3038.a 1.a $2$ $24.259$ \(\Q(\sqrt{5}) \) None \(2\) \(2\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1-\beta )q^{5}+\cdots\)
3038.2.a.ba 3038.a 1.a $2$ $24.259$ \(\Q(\sqrt{2}) \) None \(2\) \(4\) \(4\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+(2-\beta )q^{5}+2q^{6}+\cdots\)
3038.2.a.bb 3038.a 1.a $3$ $24.259$ 3.3.257.1 None \(-3\) \(-1\) \(1\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{5}+\beta _{1}q^{6}+\cdots\)
3038.2.a.bc 3038.a 1.a $3$ $24.259$ 3.3.257.1 None \(-3\) \(1\) \(-1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{5}-\beta _{1}q^{6}+\cdots\)
3038.2.a.bd 3038.a 1.a $3$ $24.259$ 3.3.568.1 None \(-3\) \(2\) \(2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}+(1+\beta _{2})q^{5}+\cdots\)
3038.2.a.be 3038.a 1.a $3$ $24.259$ 3.3.993.1 None \(3\) \(-1\) \(-5\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+(-2+\beta _{1})q^{5}+\cdots\)
3038.2.a.bf 3038.a 1.a $3$ $24.259$ \(\Q(\zeta_{14})^+\) None \(3\) \(-1\) \(1\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{5}-\beta _{1}q^{6}+\cdots\)
3038.2.a.bg 3038.a 1.a $3$ $24.259$ 3.3.568.1 None \(3\) \(-1\) \(1\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{2}q^{3}+q^{4}-\beta _{2}q^{5}+\beta _{2}q^{6}+\cdots\)
3038.2.a.bh 3038.a 1.a $3$ $24.259$ \(\Q(\zeta_{14})^+\) None \(3\) \(1\) \(-1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{5}+\beta _{1}q^{6}+\cdots\)
3038.2.a.bi 3038.a 1.a $3$ $24.259$ 3.3.993.1 None \(3\) \(1\) \(5\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+(2-\beta _{1})q^{5}+\beta _{1}q^{6}+\cdots\)
3038.2.a.bj 3038.a 1.a $4$ $24.259$ 4.4.14336.1 None \(4\) \(-4\) \(-4\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1-\beta _{2}+\cdots)q^{5}+\cdots\)
3038.2.a.bk 3038.a 1.a $4$ $24.259$ 4.4.14336.1 None \(4\) \(4\) \(4\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta _{1})q^{3}+q^{4}+(1+\beta _{2})q^{5}+\cdots\)
3038.2.a.bl 3038.a 1.a $7$ $24.259$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(-1\) \(1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+(-\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots\)
3038.2.a.bm 3038.a 1.a $7$ $24.259$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(1\) \(-1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+(\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
3038.2.a.bn 3038.a 1.a $8$ $24.259$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(-4\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1-\beta _{3})q^{3}+q^{4}+\beta _{1}q^{5}+\cdots\)
3038.2.a.bo 3038.a 1.a $8$ $24.259$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(4\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta _{3})q^{3}+q^{4}-\beta _{1}q^{5}+(-1+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3038))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3038)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(217))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(434))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1519))\)\(^{\oplus 2}\)