Properties

Label 784.4.a
Level $784$
Weight $4$
Character orbit 784.a
Rep. character $\chi_{784}(1,\cdot)$
Character field $\Q$
Dimension $59$
Newform subspaces $34$
Sturm bound $448$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 34 \)
Sturm bound: \(448\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(784))\).

Total New Old
Modular forms 360 64 296
Cusp forms 312 59 253
Eisenstein series 48 5 43

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

\(2\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(16\)
\(+\)\(-\)\(-\)\(15\)
\(-\)\(+\)\(-\)\(13\)
\(-\)\(-\)\(+\)\(15\)
Plus space\(+\)\(31\)
Minus space\(-\)\(28\)

Trace form

\( 59 q - 4 q^{3} + 477 q^{9} + O(q^{10}) \) \( 59 q - 4 q^{3} + 477 q^{9} + 22 q^{11} + 24 q^{13} - 58 q^{15} + 52 q^{17} + 20 q^{19} - 194 q^{23} + 1299 q^{25} - 424 q^{27} - 62 q^{29} + 104 q^{31} + 208 q^{33} - 168 q^{37} - 100 q^{39} + 100 q^{41} - 116 q^{43} - 112 q^{45} - 264 q^{47} + 1318 q^{51} + 528 q^{53} + 1168 q^{55} + 554 q^{57} + 292 q^{59} + 752 q^{61} + 392 q^{65} + 662 q^{67} + 304 q^{69} + 400 q^{71} - 612 q^{73} - 2236 q^{75} - 810 q^{79} + 2227 q^{81} - 876 q^{83} + 1442 q^{85} - 776 q^{87} - 564 q^{89} + 1334 q^{93} - 1478 q^{95} - 428 q^{97} - 384 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\) 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
784.4.a.a 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(-10\) \(8\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-10q^{3}+8q^{5}+73q^{9}+40q^{11}+\cdots\)
784.4.a.b 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(-7\) \(7\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-7q^{3}+7q^{5}+22q^{9}+5q^{11}-14q^{13}+\cdots\)
784.4.a.c 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(-5\) \(9\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-5q^{3}+9q^{5}-2q^{9}+57q^{11}+70q^{13}+\cdots\)
784.4.a.d 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(-4\) \(-12\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{3}-12q^{5}-11q^{9}-12q^{11}+\cdots\)
784.4.a.e 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(-4\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{3}+2q^{5}-11q^{9}+44q^{11}-22q^{13}+\cdots\)
784.4.a.f 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(-4\) \(20\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{3}+20q^{5}-11q^{9}-44q^{11}+\cdots\)
784.4.a.g 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(-2\) \(-16\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-2^{4}q^{5}-23q^{9}+8q^{11}-28q^{13}+\cdots\)
784.4.a.h 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(-2\) \(12\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+12q^{5}-23q^{9}-48q^{11}+\cdots\)
784.4.a.i 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(-2\) \(16\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+2^{4}q^{5}-23q^{9}-24q^{11}+\cdots\)
784.4.a.j 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(-1\) \(-7\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-7q^{5}-26q^{9}-35q^{11}-66q^{13}+\cdots\)
784.4.a.k 784.a 1.a $1$ $46.257$ \(\Q\) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-3^{3}q^{9}+68q^{11}+40q^{23}-5^{3}q^{25}+\cdots\)
784.4.a.l 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(1\) \(7\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+7q^{5}-26q^{9}-35q^{11}+66q^{13}+\cdots\)
784.4.a.m 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(4\) \(-20\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{3}-20q^{5}-11q^{9}-44q^{11}+\cdots\)
784.4.a.n 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(4\) \(-6\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{3}-6q^{5}-11q^{9}+12q^{11}+82q^{13}+\cdots\)
784.4.a.o 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(4\) \(12\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{3}+12q^{5}-11q^{9}-12q^{11}+\cdots\)
784.4.a.p 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(5\) \(-9\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+5q^{3}-9q^{5}-2q^{9}+57q^{11}-70q^{13}+\cdots\)
784.4.a.q 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(6\) \(-8\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+6q^{3}-8q^{5}+9q^{9}-56q^{11}+28q^{13}+\cdots\)
784.4.a.r 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(7\) \(-7\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+7q^{3}-7q^{5}+22q^{9}+5q^{11}+14q^{13}+\cdots\)
784.4.a.s 784.a 1.a $1$ $46.257$ \(\Q\) None \(0\) \(8\) \(14\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{3}+14q^{5}+37q^{9}+28q^{11}+\cdots\)
784.4.a.t 784.a 1.a $2$ $46.257$ \(\Q(\sqrt{57}) \) None \(0\) \(-2\) \(-22\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+(-11-\beta )q^{5}+(31+\cdots)q^{9}+\cdots\)
784.4.a.u 784.a 1.a $2$ $46.257$ \(\Q(\sqrt{37}) \) None \(0\) \(0\) \(-14\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(-7-2\beta )q^{5}+10q^{9}+(-2^{4}+\cdots)q^{11}+\cdots\)
784.4.a.v 784.a 1.a $2$ $46.257$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+10\beta q^{5}-5^{2}q^{9}-54q^{11}+\cdots\)
784.4.a.w 784.a 1.a $2$ $46.257$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+2\beta q^{5}-5^{2}q^{9}+26q^{11}+\cdots\)
784.4.a.x 784.a 1.a $2$ $46.257$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-\beta q^{5}+5q^{9}+4q^{11}+\beta q^{13}+\cdots\)
784.4.a.y 784.a 1.a $2$ $46.257$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+5\beta q^{3}-14\beta q^{5}+23q^{9}+14q^{11}+\cdots\)
784.4.a.z 784.a 1.a $2$ $46.257$ \(\Q(\sqrt{22}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+\beta q^{5}+61q^{9}-20q^{11}+7\beta q^{13}+\cdots\)
784.4.a.ba 784.a 1.a $2$ $46.257$ \(\Q(\sqrt{37}) \) None \(0\) \(0\) \(14\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(7-2\beta )q^{5}+10q^{9}+(-2^{4}+\cdots)q^{11}+\cdots\)
784.4.a.bb 784.a 1.a $3$ $46.257$ 3.3.1929.1 None \(0\) \(-7\) \(3\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots\)
784.4.a.bc 784.a 1.a $3$ $46.257$ 3.3.1929.1 None \(0\) \(-1\) \(13\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(4+\beta _{1})q^{5}+(15+3\beta _{1}-2\beta _{2})q^{9}+\cdots\)
784.4.a.bd 784.a 1.a $3$ $46.257$ 3.3.1929.1 None \(0\) \(1\) \(-13\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+(-4-\beta _{1})q^{5}+(15+3\beta _{1}+\cdots)q^{9}+\cdots\)
784.4.a.be 784.a 1.a $3$ $46.257$ 3.3.1929.1 None \(0\) \(7\) \(-3\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+\cdots\)
784.4.a.bf 784.a 1.a $4$ $46.257$ \(\Q(\sqrt{2}, \sqrt{65})\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+\beta _{3}q^{5}+(10-3\beta _{1})q^{9}+(-5^{2}+\cdots)q^{11}+\cdots\)
784.4.a.bg 784.a 1.a $4$ $46.257$ \(\Q(\sqrt{2}, \sqrt{65})\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(2\beta _{1}+\beta _{3})q^{3}+(3\beta _{1}+\beta _{3})q^{5}+(10+\cdots)q^{9}+\cdots\)
784.4.a.bh 784.a 1.a $4$ $46.257$ \(\Q(\sqrt{2}, \sqrt{113})\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(2\beta _{1}+\beta _{3})q^{3}+(9\beta _{1}-\beta _{3})q^{5}+(34+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(784))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(784)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 2}\)