Properties

Label 5776.2.a
Level $5776$
Weight $2$
Character orbit 5776.a
Rep. character $\chi_{5776}(1,\cdot)$
Character field $\Q$
Dimension $162$
Newform subspaces $57$
Sturm bound $1520$
Trace bound $15$

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Defining parameters

Level: \( N \) \(=\) \( 5776 = 2^{4} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5776.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 57 \)
Sturm bound: \(1520\)
Trace bound: \(15\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5776))\).

Total New Old
Modular forms 820 179 641
Cusp forms 701 162 539
Eisenstein series 119 17 102

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

\(2\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(40\)
\(+\)\(-\)\(-\)\(45\)
\(-\)\(+\)\(-\)\(41\)
\(-\)\(-\)\(+\)\(36\)
Plus space\(+\)\(76\)
Minus space\(-\)\(86\)

Trace form

\( 162 q + 2 q^{5} + 2 q^{7} + 144 q^{9} + O(q^{10}) \) \( 162 q + 2 q^{5} + 2 q^{7} + 144 q^{9} - 2 q^{11} + 2 q^{13} - 12 q^{15} - 2 q^{17} + 8 q^{21} + 10 q^{23} + 132 q^{25} + 2 q^{29} + 8 q^{33} - 12 q^{35} + 2 q^{37} + 14 q^{39} - 2 q^{41} + 14 q^{43} + 18 q^{45} - 14 q^{47} + 116 q^{49} - 4 q^{51} - 6 q^{53} + 24 q^{55} + 16 q^{59} + 10 q^{61} + 30 q^{63} - 12 q^{65} + 20 q^{67} + 24 q^{69} - 4 q^{71} - 2 q^{73} + 4 q^{75} - 10 q^{77} + 8 q^{79} + 82 q^{81} + 2 q^{83} + 12 q^{85} - 12 q^{87} - 18 q^{89} + 40 q^{91} - 8 q^{93} - 10 q^{97} - 58 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5776))\) 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 19
5776.2.a.a 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(-3\) \(2\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+2q^{5}+3q^{7}+6q^{9}+2q^{11}+\cdots\)
5776.2.a.b 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(-2\) \(-1\) \(3\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-q^{5}+3q^{7}+q^{9}+3q^{11}+\cdots\)
5776.2.a.c 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(-2\) \(3\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+3q^{5}+q^{7}+q^{9}-3q^{11}+\cdots\)
5776.2.a.d 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(-1\) \(-4\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{5}-3q^{7}-2q^{9}-2q^{11}+\cdots\)
5776.2.a.e 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(-1\) \(-4\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{5}-2q^{9}-3q^{11}+2q^{13}+\cdots\)
5776.2.a.f 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{9}+4q^{11}+q^{13}+\cdots\)
5776.2.a.g 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(-1\) \(0\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+4q^{7}-2q^{9}-3q^{11}+2q^{13}+\cdots\)
5776.2.a.h 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(-1\) \(3\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{5}-2q^{9}+4q^{11}-5q^{13}+\cdots\)
5776.2.a.i 5776.a 1.a $1$ $46.122$ \(\Q\) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(-1\) \(-3\) $-$ $+$ $N(\mathrm{U}(1))$ \(q-q^{5}-3q^{7}-3q^{9}+5q^{11}-7q^{17}+\cdots\)
5776.2.a.j 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(1\) \(-4\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{5}-2q^{9}-3q^{11}-2q^{13}+\cdots\)
5776.2.a.k 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(1\) \(-1\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{9}+4q^{11}-q^{13}+\cdots\)
5776.2.a.l 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(1\) \(0\) \(-3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{7}-2q^{9}-2q^{11}-q^{13}+\cdots\)
5776.2.a.m 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}-2q^{9}+6q^{11}-5q^{13}+\cdots\)
5776.2.a.n 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(1\) \(0\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{7}-2q^{9}-3q^{11}-2q^{13}+\cdots\)
5776.2.a.o 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(1\) \(3\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{5}-2q^{9}+4q^{11}+5q^{13}+\cdots\)
5776.2.a.p 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(2\) \(-1\) \(3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{5}+3q^{7}+q^{9}-5q^{11}+\cdots\)
5776.2.a.q 5776.a 1.a $1$ $46.122$ \(\Q\) None \(0\) \(3\) \(2\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+2q^{5}+3q^{7}+6q^{9}+2q^{11}+\cdots\)
5776.2.a.r 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{5}) \) None \(0\) \(-4\) \(1\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+\beta q^{5}+(2-2\beta )q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
5776.2.a.s 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{5}) \) None \(0\) \(-3\) \(2\) \(-6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+(2-2\beta )q^{5}-3q^{7}+\cdots\)
5776.2.a.t 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-5\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2\beta q^{3}+(-3+\beta )q^{5}+(2-2\beta )q^{7}+\cdots\)
5776.2.a.u 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(5\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2\beta q^{3}+(2+\beta )q^{5}+2q^{7}+(1+4\beta )q^{9}+\cdots\)
5776.2.a.v 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(-3\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(-1-\beta )q^{5}-q^{7}+(1+\beta )q^{9}+\cdots\)
5776.2.a.w 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{5}) \) None \(0\) \(-1\) \(-2\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}-2\beta q^{5}+(-1-2\beta )q^{7}+(-2+\cdots)q^{9}+\cdots\)
5776.2.a.x 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{5}) \) None \(0\) \(-1\) \(-2\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}-2\beta q^{5}+(-3+2\beta )q^{7}+(-2+\cdots)q^{9}+\cdots\)
5776.2.a.y 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{57}) \) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(1\) \(3\) $-$ $+$ $N(\mathrm{U}(1))$ \(q+\beta q^{5}+(2-\beta )q^{7}-3q^{9}+(-2-\beta )q^{11}+\cdots\)
5776.2.a.z 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(2\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(1-\beta )q^{5}+(-1-\beta )q^{7}+4q^{9}+\cdots\)
5776.2.a.ba 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(2\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(1+\beta )q^{5}+(-1+\beta )q^{7}+4q^{9}+\cdots\)
5776.2.a.bb 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(-3\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(-1-\beta )q^{5}-q^{7}+(1+\beta )q^{9}+\cdots\)
5776.2.a.bc 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{5}) \) None \(0\) \(1\) \(-2\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-2\beta q^{5}+(-1-2\beta )q^{7}+(-2+\cdots)q^{9}+\cdots\)
5776.2.a.bd 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{5}) \) None \(0\) \(1\) \(-2\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-2\beta q^{5}+(-3+2\beta )q^{7}+(-2+\cdots)q^{9}+\cdots\)
5776.2.a.be 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(-5\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2\beta q^{3}+(-3+\beta )q^{5}+(2-2\beta )q^{7}+\cdots\)
5776.2.a.bf 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(5\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2\beta q^{3}+(2+\beta )q^{5}+2q^{7}+(1+4\beta )q^{9}+\cdots\)
5776.2.a.bg 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{5}) \) None \(0\) \(3\) \(2\) \(-6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(2-2\beta )q^{5}-3q^{7}+(-1+\cdots)q^{9}+\cdots\)
5776.2.a.bh 5776.a 1.a $2$ $46.122$ \(\Q(\sqrt{5}) \) None \(0\) \(4\) \(1\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+(1-\beta )q^{5}+2\beta q^{7}+q^{9}+(-4+\cdots)q^{11}+\cdots\)
5776.2.a.bi 5776.a 1.a $3$ $46.122$ \(\Q(\zeta_{18})^+\) None \(0\) \(-3\) \(-3\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{3}+(-1+\beta _{1})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
5776.2.a.bj 5776.a 1.a $3$ $46.122$ \(\Q(\zeta_{18})^+\) None \(0\) \(-3\) \(-3\) \(6\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-2\beta _{1}+\beta _{2})q^{3}+(-1-\beta _{2})q^{5}+\cdots\)
5776.2.a.bk 5776.a 1.a $3$ $46.122$ 3.3.316.1 None \(0\) \(-1\) \(1\) \(2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}+\beta _{2})q^{3}+\beta _{1}q^{5}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
5776.2.a.bl 5776.a 1.a $3$ $46.122$ \(\Q(\zeta_{18})^+\) None \(0\) \(0\) \(0\) \(3\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(\beta _{1}-\beta _{2})q^{5}+(1+2\beta _{1})q^{7}+\cdots\)
5776.2.a.bm 5776.a 1.a $3$ $46.122$ \(\Q(\zeta_{18})^+\) None \(0\) \(0\) \(0\) \(3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(\beta _{1}-\beta _{2})q^{5}+(1+2\beta _{1})q^{7}+\cdots\)
5776.2.a.bn 5776.a 1.a $3$ $46.122$ \(\Q(\zeta_{18})^+\) None \(0\) \(0\) \(6\) \(-6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+2q^{5}+(-2-2\beta _{2})q^{7}+(-1+\cdots)q^{9}+\cdots\)
5776.2.a.bo 5776.a 1.a $3$ $46.122$ \(\Q(\zeta_{18})^+\) None \(0\) \(0\) \(6\) \(-6\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+2q^{5}+(-2-2\beta _{2})q^{7}+(-1+\cdots)q^{9}+\cdots\)
5776.2.a.bp 5776.a 1.a $3$ $46.122$ 3.3.961.1 None \(0\) \(1\) \(1\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-\beta _{2}q^{5}+(-2+\beta _{1}-\beta _{2})q^{7}+\cdots\)
5776.2.a.bq 5776.a 1.a $3$ $46.122$ 3.3.316.1 None \(0\) \(1\) \(1\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(\beta _{1}-\beta _{2})q^{3}+\beta _{1}q^{5}+(1-\beta _{1}-\beta _{2})q^{7}+\cdots\)
5776.2.a.br 5776.a 1.a $3$ $46.122$ \(\Q(\zeta_{18})^+\) None \(0\) \(3\) \(-3\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{3}+(-1+\beta _{1})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
5776.2.a.bs 5776.a 1.a $3$ $46.122$ \(\Q(\zeta_{18})^+\) None \(0\) \(3\) \(-3\) \(6\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+2\beta _{1}-\beta _{2})q^{3}+(-1-\beta _{2})q^{5}+\cdots\)
5776.2.a.bt 5776.a 1.a $4$ $46.122$ \(\Q(\zeta_{20})^+\) None \(0\) \(-2\) \(-2\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2}-\beta _{3})q^{3}+(-1-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
5776.2.a.bu 5776.a 1.a $4$ $46.122$ \(\Q(\zeta_{20})^+\) None \(0\) \(0\) \(-4\) \(8\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-2-2\beta _{2})q^{5}+(1-2\beta _{2}+\cdots)q^{7}+\cdots\)
5776.2.a.bv 5776.a 1.a $4$ $46.122$ \(\Q(\zeta_{20})^+\) None \(0\) \(2\) \(-2\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2}+\beta _{3})q^{3}+(-1-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
5776.2.a.bw 5776.a 1.a $6$ $46.122$ 6.6.20319417.1 None \(0\) \(-3\) \(-3\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}-\beta _{2}-\beta _{3})q^{3}-\beta _{4}q^{5}+(1+\cdots)q^{7}+\cdots\)
5776.2.a.bx 5776.a 1.a $6$ $46.122$ 6.6.3022625.1 None \(0\) \(-3\) \(2\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{3}+(-\beta _{1}-\beta _{5})q^{5}+(-\beta _{2}+\cdots)q^{7}+\cdots\)
5776.2.a.by 5776.a 1.a $6$ $46.122$ 6.6.20319417.1 None \(0\) \(3\) \(-3\) \(3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(\beta _{1}+\beta _{2}+\beta _{3})q^{3}-\beta _{4}q^{5}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
5776.2.a.bz 5776.a 1.a $6$ $46.122$ 6.6.3022625.1 None \(0\) \(3\) \(2\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{3}+(-\beta _{1}-\beta _{5})q^{5}+(-\beta _{2}+\cdots)q^{7}+\cdots\)
5776.2.a.ca 5776.a 1.a $8$ $46.122$ 8.8.\(\cdots\).1 None \(0\) \(-6\) \(-2\) \(2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-1+\beta _{3}+\beta _{4}-\beta _{5}+\cdots)q^{5}+\cdots\)
5776.2.a.cb 5776.a 1.a $8$ $46.122$ 8.8.\(\cdots\).1 None \(0\) \(0\) \(14\) \(-8\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(\beta _{3}+\beta _{5})q^{3}+(1+\beta _{1}+\beta _{4})q^{5}+(-2+\cdots)q^{7}+\cdots\)
5776.2.a.cc 5776.a 1.a $8$ $46.122$ 8.8.\(\cdots\).1 None \(0\) \(6\) \(-2\) \(2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(-1+\beta _{3}+\beta _{4}-\beta _{5}+\cdots)q^{5}+\cdots\)
5776.2.a.cd 5776.a 1.a $9$ $46.122$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-3\) \(3\) \(-9\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(\beta _{4}+\beta _{5})q^{5}+(-1-\beta _{6}+\cdots)q^{7}+\cdots\)
5776.2.a.ce 5776.a 1.a $9$ $46.122$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(3\) \(3\) \(-9\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(\beta _{4}+\beta _{5})q^{5}+(-1-\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5776))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5776)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(304))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(361))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(722))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1444))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2888))\)\(^{\oplus 2}\)