Properties

Label 2028.4.a
Level $2028$
Weight $4$
Character orbit 2028.a
Rep. character $\chi_{2028}(1,\cdot)$
Character field $\Q$
Dimension $77$
Newform subspaces $19$
Sturm bound $1456$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2028.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 19 \)
Sturm bound: \(1456\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2028))\).

Total New Old
Modular forms 1134 77 1057
Cusp forms 1050 77 973
Eisenstein series 84 0 84

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
\(-\)\(+\)\(+\)\(-\)\(20\)
\(-\)\(+\)\(-\)\(+\)\(19\)
\(-\)\(-\)\(+\)\(+\)\(21\)
\(-\)\(-\)\(-\)\(-\)\(17\)
Plus space\(+\)\(40\)
Minus space\(-\)\(37\)

Trace form

\( 77 q - 3 q^{3} + 2 q^{5} + 12 q^{7} + 693 q^{9} + O(q^{10}) \) \( 77 q - 3 q^{3} + 2 q^{5} + 12 q^{7} + 693 q^{9} + 20 q^{11} - 30 q^{15} + 46 q^{17} + 136 q^{19} + 60 q^{21} - 8 q^{23} + 2131 q^{25} - 27 q^{27} + 90 q^{29} - 556 q^{31} - 84 q^{33} + 672 q^{35} + 454 q^{37} + 14 q^{41} - 620 q^{43} + 18 q^{45} + 584 q^{47} + 3617 q^{49} + 66 q^{51} - 742 q^{53} - 576 q^{55} - 552 q^{57} + 1324 q^{59} + 626 q^{61} + 108 q^{63} - 112 q^{67} - 1056 q^{69} + 2392 q^{71} - 14 q^{73} - 1077 q^{75} + 1560 q^{77} - 1300 q^{79} + 6237 q^{81} - 3324 q^{83} + 1220 q^{85} + 282 q^{87} + 2390 q^{89} + 372 q^{93} - 4112 q^{95} + 1514 q^{97} + 180 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2028))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 13
2028.4.a.a 2028.a 1.a $1$ $119.656$ \(\Q\) None 156.4.a.a \(0\) \(-3\) \(6\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+6q^{5}+4q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
2028.4.a.b 2028.a 1.a $1$ $119.656$ \(\Q\) None 156.4.a.b \(0\) \(3\) \(2\) \(32\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+2q^{5}+2^{5}q^{7}+9q^{9}+68q^{11}+\cdots\)
2028.4.a.c 2028.a 1.a $1$ $119.656$ \(\Q\) None 12.4.a.a \(0\) \(3\) \(18\) \(-8\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+18q^{5}-8q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
2028.4.a.d 2028.a 1.a $2$ $119.656$ \(\Q(\sqrt{22}) \) None 156.4.a.c \(0\) \(-6\) \(0\) \(-8\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+\beta q^{5}+(-4-3\beta )q^{7}+9q^{9}+\cdots\)
2028.4.a.e 2028.a 1.a $2$ $119.656$ \(\Q(\sqrt{10}) \) None 156.4.a.d \(0\) \(6\) \(-24\) \(-8\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-12+\beta )q^{5}+(-4-3\beta )q^{7}+\cdots\)
2028.4.a.f 2028.a 1.a $2$ $119.656$ \(\Q(\sqrt{3}) \) None 156.4.q.b \(0\) \(6\) \(-24\) \(-6\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-12+3\beta )q^{5}+(-3+\beta )q^{7}+\cdots\)
2028.4.a.g 2028.a 1.a $2$ $119.656$ \(\Q(\sqrt{3}) \) None 156.4.b.a \(0\) \(6\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+2\beta q^{5}-3\beta q^{7}+9q^{9}-5\beta q^{11}+\cdots\)
2028.4.a.h 2028.a 1.a $2$ $119.656$ \(\Q(\sqrt{3}) \) None 156.4.q.a \(0\) \(6\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+2\beta q^{5}-11\beta q^{7}+9q^{9}+22\beta q^{11}+\cdots\)
2028.4.a.i 2028.a 1.a $2$ $119.656$ \(\Q(\sqrt{3}) \) None 156.4.q.b \(0\) \(6\) \(24\) \(6\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(12+3\beta )q^{5}+(3+\beta )q^{7}+9q^{9}+\cdots\)
2028.4.a.j 2028.a 1.a $4$ $119.656$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 156.4.i.b \(0\) \(-12\) \(-3\) \(15\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-1+\beta _{2})q^{5}+(4-\beta _{1})q^{7}+\cdots\)
2028.4.a.k 2028.a 1.a $4$ $119.656$ 4.4.47664588.1 None 156.4.b.b \(0\) \(-12\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+\beta _{1}q^{5}-\beta _{2}q^{7}+9q^{9}+(3\beta _{1}+\cdots)q^{11}+\cdots\)
2028.4.a.l 2028.a 1.a $4$ $119.656$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 156.4.i.b \(0\) \(-12\) \(3\) \(-15\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(1-\beta _{2})q^{5}+(-4+\beta _{1})q^{7}+\cdots\)
2028.4.a.m 2028.a 1.a $4$ $119.656$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 156.4.i.a \(0\) \(12\) \(-7\) \(-11\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-2+\beta _{2})q^{5}+(-3+\beta _{1}+\cdots)q^{7}+\cdots\)
2028.4.a.n 2028.a 1.a $4$ $119.656$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 156.4.i.a \(0\) \(12\) \(7\) \(11\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(2-\beta _{2})q^{5}+(3-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
2028.4.a.o 2028.a 1.a $6$ $119.656$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 156.4.q.c \(0\) \(-18\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(\beta _{1}-\beta _{2})q^{5}+(\beta _{2}+\beta _{4})q^{7}+\cdots\)
2028.4.a.p 2028.a 1.a $9$ $119.656$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 2028.4.a.p \(0\) \(-27\) \(-3\) \(-47\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-\beta _{1}q^{5}+(-6-\beta _{2}+\beta _{7})q^{7}+\cdots\)
2028.4.a.q 2028.a 1.a $9$ $119.656$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 2028.4.a.p \(0\) \(-27\) \(3\) \(47\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+\beta _{1}q^{5}+(6+\beta _{2}-\beta _{7})q^{7}+\cdots\)
2028.4.a.r 2028.a 1.a $9$ $119.656$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 2028.4.a.r \(0\) \(27\) \(-11\) \(-37\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-1-\beta _{4})q^{5}+(-4-\beta _{5}+\cdots)q^{7}+\cdots\)
2028.4.a.s 2028.a 1.a $9$ $119.656$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 2028.4.a.r \(0\) \(27\) \(11\) \(37\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(1+\beta _{4})q^{5}+(4+\beta _{5})q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2028))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2028)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(507))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(676))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1014))\)\(^{\oplus 2}\)