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Properties

Label 294.6.a
Level $294$
Weight $6$
Character orbit 294.a
Rep. character $\chi_{294}(1,\cdot)$
Character field $\Q$
Dimension $33$
Newform subspaces $23$
Sturm bound $336$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 294.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 23 \)
Sturm bound: \(336\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(294))\).

Total New Old
Modular forms 296 33 263
Cusp forms 264 33 231
Eisenstein series 32 0 32

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

\(2\)\(3\)\(7\)FrickeDim
\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(-\)$-$\(5\)
\(+\)\(-\)\(+\)$-$\(4\)
\(+\)\(-\)\(-\)$+$\(4\)
\(-\)\(+\)\(+\)$-$\(4\)
\(-\)\(+\)\(-\)$+$\(4\)
\(-\)\(-\)\(+\)$+$\(3\)
\(-\)\(-\)\(-\)$-$\(6\)
Plus space\(+\)\(14\)
Minus space\(-\)\(19\)

Trace form

\( 33 q + 4 q^{2} + 9 q^{3} + 528 q^{4} + 22 q^{5} + 36 q^{6} + 64 q^{8} + 2673 q^{9} + O(q^{10}) \) \( 33 q + 4 q^{2} + 9 q^{3} + 528 q^{4} + 22 q^{5} + 36 q^{6} + 64 q^{8} + 2673 q^{9} - 360 q^{10} - 652 q^{11} + 144 q^{12} + 694 q^{13} + 198 q^{15} + 8448 q^{16} - 1054 q^{17} + 324 q^{18} + 3124 q^{19} + 352 q^{20} - 1696 q^{22} + 5424 q^{23} + 576 q^{24} + 24427 q^{25} - 4392 q^{26} + 729 q^{27} + 1262 q^{29} - 5976 q^{30} + 616 q^{31} + 1024 q^{32} + 180 q^{33} + 13704 q^{34} + 42768 q^{36} - 31658 q^{37} + 2192 q^{38} - 10134 q^{39} - 5760 q^{40} - 20742 q^{41} + 3684 q^{43} - 10432 q^{44} + 1782 q^{45} + 14080 q^{46} + 13776 q^{47} + 2304 q^{48} - 89124 q^{50} + 11718 q^{51} + 11104 q^{52} - 63538 q^{53} + 2916 q^{54} - 69672 q^{55} + 80100 q^{57} + 2440 q^{58} - 80900 q^{59} + 3168 q^{60} + 86566 q^{61} + 106848 q^{62} + 135168 q^{64} + 146500 q^{65} + 37008 q^{66} + 123956 q^{67} - 16864 q^{68} - 88920 q^{69} + 161888 q^{71} + 5184 q^{72} + 105274 q^{73} + 349528 q^{74} + 48807 q^{75} + 49984 q^{76} - 48024 q^{78} + 212748 q^{79} + 5632 q^{80} + 216513 q^{81} + 13800 q^{82} + 62612 q^{83} + 158524 q^{85} - 172368 q^{86} + 188982 q^{87} - 27136 q^{88} - 121398 q^{89} - 29160 q^{90} + 86784 q^{92} + 235656 q^{93} + 129024 q^{94} + 302848 q^{95} + 9216 q^{96} - 163646 q^{97} - 52812 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(294))\) 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 7
294.6.a.a 294.a 1.a $1$ $47.153$ \(\Q\) None \(-4\) \(-9\) \(-26\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}-26q^{5}+6^{2}q^{6}+\cdots\)
294.6.a.b 294.a 1.a $1$ $47.153$ \(\Q\) None \(-4\) \(-9\) \(-26\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}-26q^{5}+6^{2}q^{6}+\cdots\)
294.6.a.c 294.a 1.a $1$ $47.153$ \(\Q\) None \(-4\) \(-9\) \(72\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+72q^{5}+6^{2}q^{6}+\cdots\)
294.6.a.d 294.a 1.a $1$ $47.153$ \(\Q\) None \(-4\) \(-9\) \(86\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+86q^{5}+6^{2}q^{6}+\cdots\)
294.6.a.e 294.a 1.a $1$ $47.153$ \(\Q\) None \(-4\) \(9\) \(-86\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-86q^{5}-6^{2}q^{6}+\cdots\)
294.6.a.f 294.a 1.a $1$ $47.153$ \(\Q\) None \(-4\) \(9\) \(-44\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-44q^{5}-6^{2}q^{6}+\cdots\)
294.6.a.g 294.a 1.a $1$ $47.153$ \(\Q\) None \(-4\) \(9\) \(26\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+26q^{5}-6^{2}q^{6}+\cdots\)
294.6.a.h 294.a 1.a $1$ $47.153$ \(\Q\) None \(-4\) \(9\) \(54\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+54q^{5}-6^{2}q^{6}+\cdots\)
294.6.a.i 294.a 1.a $1$ $47.153$ \(\Q\) None \(4\) \(-9\) \(-24\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-24q^{5}-6^{2}q^{6}+\cdots\)
294.6.a.j 294.a 1.a $1$ $47.153$ \(\Q\) None \(4\) \(-9\) \(6\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+6q^{5}-6^{2}q^{6}+\cdots\)
294.6.a.k 294.a 1.a $1$ $47.153$ \(\Q\) None \(4\) \(9\) \(-76\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}-76q^{5}+6^{2}q^{6}+\cdots\)
294.6.a.l 294.a 1.a $1$ $47.153$ \(\Q\) None \(4\) \(9\) \(-6\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}-6q^{5}+6^{2}q^{6}+\cdots\)
294.6.a.m 294.a 1.a $1$ $47.153$ \(\Q\) None \(4\) \(9\) \(66\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+66q^{5}+6^{2}q^{6}+\cdots\)
294.6.a.n 294.a 1.a $2$ $47.153$ \(\Q(\sqrt{2}) \) None \(-8\) \(-18\) \(-108\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(-54+5\beta )q^{5}+\cdots\)
294.6.a.o 294.a 1.a $2$ $47.153$ \(\Q(\sqrt{505}) \) None \(-8\) \(-18\) \(-17\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(-9-\beta )q^{5}+\cdots\)
294.6.a.p 294.a 1.a $2$ $47.153$ \(\Q(\sqrt{505}) \) None \(-8\) \(18\) \(17\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(9+\beta )q^{5}+\cdots\)
294.6.a.q 294.a 1.a $2$ $47.153$ \(\Q(\sqrt{2}) \) None \(-8\) \(18\) \(108\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(54+5\beta )q^{5}+\cdots\)
294.6.a.r 294.a 1.a $2$ $47.153$ \(\Q(\sqrt{9601}) \) None \(8\) \(-18\) \(-53\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(-26-\beta )q^{5}+\cdots\)
294.6.a.s 294.a 1.a $2$ $47.153$ \(\Q(\sqrt{4705}) \) None \(8\) \(-18\) \(-18\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(-9-\beta )q^{5}+\cdots\)
294.6.a.t 294.a 1.a $2$ $47.153$ \(\Q(\sqrt{2}) \) None \(8\) \(-18\) \(108\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(54+5\beta )q^{5}+\cdots\)
294.6.a.u 294.a 1.a $2$ $47.153$ \(\Q(\sqrt{2}) \) None \(8\) \(18\) \(-108\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(-54+5\beta )q^{5}+\cdots\)
294.6.a.v 294.a 1.a $2$ $47.153$ \(\Q(\sqrt{4705}) \) None \(8\) \(18\) \(18\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(9-\beta )q^{5}+\cdots\)
294.6.a.w 294.a 1.a $2$ $47.153$ \(\Q(\sqrt{9601}) \) None \(8\) \(18\) \(53\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(3^{3}-\beta )q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(294))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(294)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)