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Properties

Label 1014.6.a
Level $1014$
Weight $6$
Character orbit 1014.a
Rep. character $\chi_{1014}(1,\cdot)$
Character field $\Q$
Dimension $129$
Newform subspaces $33$
Sturm bound $1092$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1014.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 33 \)
Sturm bound: \(1092\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(1014))\).

Total New Old
Modular forms 938 129 809
Cusp forms 882 129 753
Eisenstein series 56 0 56

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

\(2\)\(3\)\(13\)FrickeDim
\(+\)\(+\)\(+\)$+$\(14\)
\(+\)\(+\)\(-\)$-$\(18\)
\(+\)\(-\)\(+\)$-$\(17\)
\(+\)\(-\)\(-\)$+$\(16\)
\(-\)\(+\)\(+\)$-$\(17\)
\(-\)\(+\)\(-\)$+$\(15\)
\(-\)\(-\)\(+\)$+$\(13\)
\(-\)\(-\)\(-\)$-$\(19\)
Plus space\(+\)\(58\)
Minus space\(-\)\(71\)

Trace form

\( 129 q - 4 q^{2} + 9 q^{3} + 2064 q^{4} - 22 q^{5} - 36 q^{6} + 372 q^{7} - 64 q^{8} + 10449 q^{9} + O(q^{10}) \) \( 129 q - 4 q^{2} + 9 q^{3} + 2064 q^{4} - 22 q^{5} - 36 q^{6} + 372 q^{7} - 64 q^{8} + 10449 q^{9} + 760 q^{10} + 692 q^{11} + 144 q^{12} - 704 q^{14} - 198 q^{15} + 33024 q^{16} - 3026 q^{17} - 324 q^{18} - 1832 q^{19} - 352 q^{20} - 180 q^{21} - 3072 q^{22} + 6520 q^{23} - 576 q^{24} + 72319 q^{25} + 729 q^{27} + 5952 q^{28} - 9486 q^{29} + 5976 q^{30} - 13284 q^{31} - 1024 q^{32} - 2412 q^{33} - 13704 q^{34} - 3216 q^{35} + 167184 q^{36} + 32038 q^{37} - 9872 q^{38} + 12160 q^{40} - 298 q^{41} + 720 q^{42} + 7316 q^{43} + 11072 q^{44} - 1782 q^{45} + 6592 q^{46} - 88312 q^{47} + 2304 q^{48} + 364213 q^{49} + 37444 q^{50} + 22122 q^{51} + 86738 q^{53} - 2916 q^{54} - 48496 q^{55} - 11264 q^{56} - 27936 q^{57} - 28280 q^{58} + 99196 q^{59} - 3168 q^{60} - 12430 q^{61} + 106848 q^{62} + 30132 q^{63} + 528384 q^{64} + 9648 q^{66} + 107184 q^{67} - 48416 q^{68} + 16056 q^{69} + 14752 q^{70} + 98680 q^{71} - 5184 q^{72} - 1702 q^{73} - 37624 q^{74} + 48807 q^{75} - 29312 q^{76} - 269640 q^{77} - 120476 q^{79} - 5632 q^{80} + 846369 q^{81} + 151640 q^{82} - 40572 q^{83} - 2880 q^{84} - 69132 q^{85} - 36976 q^{86} - 1422 q^{87} - 49152 q^{88} + 96782 q^{89} + 61560 q^{90} + 104320 q^{92} + 138420 q^{93} + 105536 q^{94} - 91136 q^{95} - 9216 q^{96} + 11618 q^{97} - 33060 q^{98} + 56052 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(1014))\) 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 13
1014.6.a.a 1014.a 1.a $1$ $162.629$ \(\Q\) None \(-4\) \(-9\) \(-56\) \(68\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}-56q^{5}+6^{2}q^{6}+\cdots\)
1014.6.a.b 1014.a 1.a $1$ $162.629$ \(\Q\) None \(-4\) \(-9\) \(-4\) \(146\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}-4q^{5}+6^{2}q^{6}+\cdots\)
1014.6.a.c 1014.a 1.a $1$ $162.629$ \(\Q\) None \(-4\) \(-9\) \(66\) \(-176\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+66q^{5}+6^{2}q^{6}+\cdots\)
1014.6.a.d 1014.a 1.a $1$ $162.629$ \(\Q\) None \(-4\) \(9\) \(88\) \(176\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+88q^{5}-6^{2}q^{6}+\cdots\)
1014.6.a.e 1014.a 1.a $1$ $162.629$ \(\Q\) None \(4\) \(-9\) \(-4\) \(8\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-4q^{5}-6^{2}q^{6}+\cdots\)
1014.6.a.f 1014.a 1.a $1$ $162.629$ \(\Q\) None \(4\) \(9\) \(-76\) \(-100\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}-76q^{5}+6^{2}q^{6}+\cdots\)
1014.6.a.g 1014.a 1.a $1$ $162.629$ \(\Q\) None \(4\) \(9\) \(-24\) \(238\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}-24q^{5}+6^{2}q^{6}+\cdots\)
1014.6.a.h 1014.a 1.a $2$ $162.629$ \(\Q(\sqrt{313}) \) None \(-8\) \(-18\) \(-3\) \(113\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(1-5\beta )q^{5}+\cdots\)
1014.6.a.i 1014.a 1.a $2$ $162.629$ \(\Q(\sqrt{3241}) \) None \(-8\) \(18\) \(-10\) \(-138\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-5-\beta )q^{5}+\cdots\)
1014.6.a.j 1014.a 1.a $2$ $162.629$ \(\Q(\sqrt{649}) \) None \(-8\) \(18\) \(25\) \(39\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(13-\beta )q^{5}+\cdots\)
1014.6.a.k 1014.a 1.a $2$ $162.629$ \(\Q(\sqrt{433}) \) None \(8\) \(-18\) \(-2\) \(150\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(-1-5\beta )q^{5}+\cdots\)
1014.6.a.l 1014.a 1.a $2$ $162.629$ \(\Q(\sqrt{313}) \) None \(8\) \(-18\) \(3\) \(-113\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(-1+5\beta )q^{5}+\cdots\)
1014.6.a.m 1014.a 1.a $2$ $162.629$ \(\Q(\sqrt{649}) \) None \(8\) \(18\) \(-25\) \(-39\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(-12-\beta )q^{5}+\cdots\)
1014.6.a.n 1014.a 1.a $3$ $162.629$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-12\) \(-27\) \(-36\) \(-79\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(-12-\beta _{2})q^{5}+\cdots\)
1014.6.a.o 1014.a 1.a $3$ $162.629$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-12\) \(-27\) \(40\) \(170\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(13+\beta _{1})q^{5}+\cdots\)
1014.6.a.p 1014.a 1.a $3$ $162.629$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-12\) \(27\) \(36\) \(191\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(11-2\beta _{1}+\cdots)q^{5}+\cdots\)
1014.6.a.q 1014.a 1.a $3$ $162.629$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(12\) \(-27\) \(-40\) \(-170\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(-13-\beta _{1}+\cdots)q^{5}+\cdots\)
1014.6.a.r 1014.a 1.a $3$ $162.629$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(12\) \(-27\) \(36\) \(79\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(12+\beta _{2})q^{5}+\cdots\)
1014.6.a.s 1014.a 1.a $3$ $162.629$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(12\) \(27\) \(-36\) \(-191\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(-11+2\beta _{1}+\cdots)q^{5}+\cdots\)
1014.6.a.t 1014.a 1.a $4$ $162.629$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-16\) \(36\) \(-10\) \(72\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-2-\beta _{1}+\cdots)q^{5}+\cdots\)
1014.6.a.u 1014.a 1.a $4$ $162.629$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(16\) \(36\) \(10\) \(-72\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(2+\beta _{1})q^{5}+\cdots\)
1014.6.a.v 1014.a 1.a $6$ $162.629$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-24\) \(-54\) \(-40\) \(10\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(-7-9\beta _{1}+\cdots)q^{5}+\cdots\)
1014.6.a.w 1014.a 1.a $6$ $162.629$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-24\) \(-54\) \(30\) \(26\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(4+\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
1014.6.a.x 1014.a 1.a $6$ $162.629$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-24\) \(54\) \(-140\) \(-186\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-23+\beta _{3}+\cdots)q^{5}+\cdots\)
1014.6.a.y 1014.a 1.a $6$ $162.629$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-24\) \(54\) \(-50\) \(202\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-8-\beta _{1}+\cdots)q^{5}+\cdots\)
1014.6.a.z 1014.a 1.a $6$ $162.629$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(24\) \(-54\) \(-30\) \(-26\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(-4-\beta _{1}+\cdots)q^{5}+\cdots\)
1014.6.a.ba 1014.a 1.a $6$ $162.629$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(24\) \(-54\) \(40\) \(-10\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(7+9\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)
1014.6.a.bb 1014.a 1.a $6$ $162.629$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(24\) \(54\) \(50\) \(-202\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(10+\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots\)
1014.6.a.bc 1014.a 1.a $6$ $162.629$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(24\) \(54\) \(140\) \(186\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(23-\beta _{3})q^{5}+\cdots\)
1014.6.a.bd 1014.a 1.a $9$ $162.629$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-36\) \(-81\) \(-3\) \(-131\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
1014.6.a.be 1014.a 1.a $9$ $162.629$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-36\) \(81\) \(-39\) \(-229\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-5+\beta _{2}+\cdots)q^{5}+\cdots\)
1014.6.a.bf 1014.a 1.a $9$ $162.629$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(36\) \(-81\) \(3\) \(131\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots\)
1014.6.a.bg 1014.a 1.a $9$ $162.629$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(36\) \(81\) \(39\) \(229\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(5-\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(1014))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(1014)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(507))\)\(^{\oplus 2}\)