Properties

Label 1428.2.a
Level $1428$
Weight $2$
Character orbit 1428.a
Rep. character $\chi_{1428}(1,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $10$
Sturm bound $576$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1428.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(576\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1428))\).

Total New Old
Modular forms 300 16 284
Cusp forms 277 16 261
Eisenstein series 23 0 23

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

\(2\)\(3\)\(7\)\(17\)FrickeDim.
\(-\)\(+\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(2\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(6\)
Minus space\(-\)\(10\)

Trace form

\( 16 q + 16 q^{9} + O(q^{10}) \) \( 16 q + 16 q^{9} + 4 q^{13} + 4 q^{15} + 4 q^{19} - 8 q^{23} + 28 q^{25} + 8 q^{29} - 16 q^{31} + 4 q^{33} + 8 q^{35} - 8 q^{37} - 8 q^{41} + 20 q^{43} - 40 q^{47} + 16 q^{49} - 4 q^{51} + 24 q^{53} - 12 q^{55} - 8 q^{57} + 24 q^{59} + 8 q^{61} + 48 q^{65} - 16 q^{67} + 12 q^{69} - 24 q^{71} + 16 q^{79} + 16 q^{81} + 16 q^{83} - 4 q^{85} - 16 q^{87} - 16 q^{89} - 8 q^{91} - 16 q^{93} + 8 q^{95} + 24 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1428))\) 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 17
1428.2.a.a 1428.a 1.a $1$ $11.403$ \(\Q\) None \(0\) \(-1\) \(-3\) \(-1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{5}-q^{7}+q^{9}-5q^{11}+q^{13}+\cdots\)
1428.2.a.b 1428.a 1.a $1$ $11.403$ \(\Q\) None \(0\) \(-1\) \(1\) \(1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{7}+q^{9}-q^{11}-7q^{13}+\cdots\)
1428.2.a.c 1428.a 1.a $1$ $11.403$ \(\Q\) None \(0\) \(-1\) \(2\) \(-1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}-q^{7}+q^{9}+6q^{13}-2q^{15}+\cdots\)
1428.2.a.d 1428.a 1.a $1$ $11.403$ \(\Q\) None \(0\) \(1\) \(-3\) \(1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{5}+q^{7}+q^{9}-3q^{11}-q^{13}+\cdots\)
1428.2.a.e 1428.a 1.a $1$ $11.403$ \(\Q\) None \(0\) \(1\) \(2\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+q^{7}+q^{9}-2q^{13}+2q^{15}+\cdots\)
1428.2.a.f 1428.a 1.a $2$ $11.403$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(-4\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(-2+\beta )q^{5}-q^{7}+q^{9}+(3+\cdots)q^{11}+\cdots\)
1428.2.a.g 1428.a 1.a $2$ $11.403$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(-2\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1+\beta )q^{5}-q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
1428.2.a.h 1428.a 1.a $2$ $11.403$ \(\Q(\sqrt{10}) \) None \(0\) \(2\) \(2\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(1+\beta )q^{5}+q^{7}+q^{9}+3q^{11}+\cdots\)
1428.2.a.i 1428.a 1.a $2$ $11.403$ \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(3\) \(-2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(1+\beta )q^{5}-q^{7}+q^{9}+(1-\beta )q^{11}+\cdots\)
1428.2.a.j 1428.a 1.a $3$ $11.403$ 3.3.4764.1 None \(0\) \(-3\) \(2\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(1-\beta _{1})q^{5}+q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1428))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1428)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(119))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(204))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(238))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(357))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(476))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(714))\)\(^{\oplus 2}\)