Properties

Label 798.4.a
Level $798$
Weight $4$
Character orbit 798.a
Rep. character $\chi_{798}(1,\cdot)$
Character field $\Q$
Dimension $56$
Newform subspaces $18$
Sturm bound $640$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 798.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(640\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(798))\).

Total New Old
Modular forms 488 56 432
Cusp forms 472 56 416
Eisenstein series 16 0 16

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

\(2\)\(3\)\(7\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(5\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(5\)
Plus space\(+\)\(32\)
Minus space\(-\)\(24\)

Trace form

\( 56 q - 8 q^{2} + 224 q^{4} - 40 q^{5} - 32 q^{8} + 504 q^{9} + O(q^{10}) \) \( 56 q - 8 q^{2} + 224 q^{4} - 40 q^{5} - 32 q^{8} + 504 q^{9} - 80 q^{10} + 160 q^{11} - 40 q^{13} + 896 q^{16} - 264 q^{17} - 72 q^{18} - 160 q^{20} - 416 q^{23} + 1400 q^{25} + 144 q^{26} - 24 q^{29} + 240 q^{30} + 112 q^{31} - 128 q^{32} + 384 q^{33} - 16 q^{34} + 2016 q^{36} - 104 q^{37} + 48 q^{39} - 320 q^{40} - 1336 q^{41} + 168 q^{42} + 1104 q^{43} + 640 q^{44} - 360 q^{45} - 48 q^{46} + 848 q^{47} + 2744 q^{49} - 1400 q^{50} - 360 q^{51} - 160 q^{52} - 2216 q^{53} - 720 q^{55} + 228 q^{57} + 384 q^{58} + 2656 q^{59} - 1072 q^{61} + 736 q^{62} + 3584 q^{64} + 240 q^{65} - 1008 q^{66} - 1240 q^{67} - 1056 q^{68} + 960 q^{69} - 96 q^{71} - 288 q^{72} + 1024 q^{73} - 2384 q^{74} + 2112 q^{75} - 528 q^{78} - 768 q^{79} - 640 q^{80} + 4536 q^{81} - 800 q^{82} - 4944 q^{83} + 1152 q^{85} + 2624 q^{86} - 2112 q^{87} - 2072 q^{89} - 720 q^{90} - 840 q^{91} - 1664 q^{92} + 720 q^{93} - 96 q^{94} - 376 q^{97} - 392 q^{98} + 1440 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(798))\) 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 19
798.4.a.a 798.a 1.a $1$ $47.084$ \(\Q\) None \(-2\) \(-3\) \(-12\) \(7\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}-12q^{5}+6q^{6}+\cdots\)
798.4.a.b 798.a 1.a $1$ $47.084$ \(\Q\) None \(2\) \(-3\) \(-10\) \(7\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-10q^{5}-6q^{6}+\cdots\)
798.4.a.c 798.a 1.a $1$ $47.084$ \(\Q\) None \(2\) \(-3\) \(0\) \(7\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
798.4.a.d 798.a 1.a $2$ $47.084$ \(\Q(\sqrt{2}) \) None \(4\) \(6\) \(-20\) \(14\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(-10+3\beta )q^{5}+\cdots\)
798.4.a.e 798.a 1.a $3$ $47.084$ 3.3.22397.1 None \(-6\) \(-9\) \(0\) \(-21\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+\beta _{2}q^{5}+6q^{6}+\cdots\)
798.4.a.f 798.a 1.a $3$ $47.084$ 3.3.93944.1 None \(-6\) \(-9\) \(10\) \(21\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(3-\beta _{2})q^{5}+\cdots\)
798.4.a.g 798.a 1.a $3$ $47.084$ 3.3.42440.1 None \(-6\) \(9\) \(-20\) \(21\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(-7-\beta _{1})q^{5}+\cdots\)
798.4.a.h 798.a 1.a $3$ $47.084$ 3.3.3221.1 None \(6\) \(-9\) \(0\) \(-21\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+\beta _{2}q^{5}-6q^{6}+\cdots\)
798.4.a.i 798.a 1.a $3$ $47.084$ 3.3.12092.1 None \(6\) \(9\) \(-20\) \(-21\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(-7+\beta _{1})q^{5}+\cdots\)
798.4.a.j 798.a 1.a $3$ $47.084$ 3.3.57553.1 None \(6\) \(9\) \(10\) \(-21\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(3-\beta _{1})q^{5}+\cdots\)
798.4.a.k 798.a 1.a $4$ $47.084$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(-12\) \(0\) \(-28\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+\beta _{1}q^{5}+6q^{6}+\cdots\)
798.4.a.l 798.a 1.a $4$ $47.084$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(-12\) \(12\) \(28\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(3-\beta _{1})q^{5}+\cdots\)
798.4.a.m 798.a 1.a $4$ $47.084$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(12\) \(-10\) \(-28\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(-2-\beta _{1})q^{5}+\cdots\)
798.4.a.n 798.a 1.a $4$ $47.084$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(12\) \(10\) \(-28\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(3+\beta _{2})q^{5}+\cdots\)
798.4.a.o 798.a 1.a $4$ $47.084$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(12\) \(10\) \(28\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(2+\beta _{1})q^{5}+\cdots\)
798.4.a.p 798.a 1.a $4$ $47.084$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(-12\) \(-10\) \(-28\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(-2+\beta _{1})q^{5}+\cdots\)
798.4.a.q 798.a 1.a $4$ $47.084$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(-12\) \(-10\) \(28\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(-2-\beta _{1})q^{5}+\cdots\)
798.4.a.r 798.a 1.a $5$ $47.084$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(10\) \(15\) \(20\) \(35\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(4+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(798))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(798)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(133))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(266))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(399))\)\(^{\oplus 2}\)