Properties

Label 3528.2.a
Level $3528$
Weight $2$
Character orbit 3528.a
Rep. character $\chi_{3528}(1,\cdot)$
Character field $\Q$
Dimension $51$
Newform subspaces $39$
Sturm bound $1344$
Trace bound $23$

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Defining parameters

Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 39 \)
Sturm bound: \(1344\)
Trace bound: \(23\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3528))\).

Total New Old
Modular forms 736 51 685
Cusp forms 609 51 558
Eisenstein series 127 0 127

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

\(2\)\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(6\)
\(+\)\(-\)\(+\)\(-\)\(9\)
\(+\)\(-\)\(-\)\(+\)\(7\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(8\)
Plus space\(+\)\(24\)
Minus space\(-\)\(27\)

Trace form

\( 51 q + O(q^{10}) \) \( 51 q + 4 q^{13} - 2 q^{17} - 2 q^{19} - 12 q^{23} + 53 q^{25} + 10 q^{29} - 4 q^{31} - 6 q^{37} - 18 q^{41} - 4 q^{43} - 12 q^{47} - 30 q^{53} - 16 q^{55} + 26 q^{59} + 8 q^{61} + 8 q^{65} - 24 q^{67} + 8 q^{71} + 14 q^{73} + 36 q^{79} + 26 q^{83} - 28 q^{85} + 2 q^{89} + 20 q^{95} + 26 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3528))\) 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
3528.2.a.a 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(-4\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{5}-3q^{13}+4q^{17}+7q^{19}-4q^{23}+\cdots\)
3528.2.a.b 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(-4\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{5}-2q^{17}+2q^{19}-8q^{23}+11q^{25}+\cdots\)
3528.2.a.c 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(-4\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{5}+3q^{13}+4q^{17}-7q^{19}+4q^{23}+\cdots\)
3528.2.a.d 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(-2\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{5}-4q^{11}+2q^{13}+2q^{17}+4q^{19}+\cdots\)
3528.2.a.e 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(-2\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{5}+2q^{11}-2q^{13}-6q^{17}+4q^{19}+\cdots\)
3528.2.a.f 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(-2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{5}+6q^{11}-3q^{13}-4q^{17}-5q^{19}+\cdots\)
3528.2.a.g 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(-2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{5}+6q^{11}+6q^{13}+2q^{17}-4q^{19}+\cdots\)
3528.2.a.h 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-5q^{11}-2q^{13}+6q^{17}-2q^{19}+\cdots\)
3528.2.a.i 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-3q^{11}-4q^{13}+4q^{19}-8q^{23}+\cdots\)
3528.2.a.j 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-3q^{11}+6q^{13}-5q^{17}-q^{19}+\cdots\)
3528.2.a.k 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+q^{11}-2q^{13}+3q^{17}-5q^{19}+\cdots\)
3528.2.a.l 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+5q^{11}+2q^{13}+6q^{17}+2q^{19}+\cdots\)
3528.2.a.m 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{13}+4q^{17}+4q^{19}-4q^{23}+\cdots\)
3528.2.a.n 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{13}-4q^{17}-4q^{19}-4q^{23}+\cdots\)
3528.2.a.o 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-5q^{11}+2q^{13}-6q^{17}+2q^{19}+\cdots\)
3528.2.a.p 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-3q^{11}-6q^{13}+5q^{17}+q^{19}+\cdots\)
3528.2.a.q 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-3q^{11}+4q^{13}-4q^{19}-8q^{23}+\cdots\)
3528.2.a.r 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+q^{11}+2q^{13}-3q^{17}+5q^{19}+\cdots\)
3528.2.a.s 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+5q^{11}-2q^{13}-6q^{17}-2q^{19}+\cdots\)
3528.2.a.t 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(2\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}-6q^{11}+6q^{13}-2q^{17}-4q^{19}+\cdots\)
3528.2.a.u 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}-2q^{11}-2q^{13}+6q^{17}+4q^{19}+\cdots\)
3528.2.a.v 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}-6q^{13}-2q^{17}-4q^{19}+4q^{23}+\cdots\)
3528.2.a.w 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}+2q^{13}+6q^{17}+4q^{19}+4q^{23}+\cdots\)
3528.2.a.x 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(2\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}+4q^{11}-2q^{13}-6q^{17}-8q^{19}+\cdots\)
3528.2.a.y 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}+6q^{11}+3q^{13}+4q^{17}+5q^{19}+\cdots\)
3528.2.a.z 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(4\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{5}-3q^{13}-4q^{17}+7q^{19}+4q^{23}+\cdots\)
3528.2.a.ba 3528.a 1.a $1$ $28.171$ \(\Q\) None \(0\) \(0\) \(4\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{5}+3q^{13}-4q^{17}-7q^{19}-4q^{23}+\cdots\)
3528.2.a.bb 3528.a 1.a $2$ $28.171$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-4\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{5}+(-2+2\beta )q^{11}-\beta q^{13}+\cdots\)
3528.2.a.bc 3528.a 1.a $2$ $28.171$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-4\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{5}+(2+2\beta )q^{11}+3\beta q^{13}+\cdots\)
3528.2.a.bd 3528.a 1.a $2$ $28.171$ \(\Q(\sqrt{57}) \) None \(0\) \(0\) \(-1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{5}+\beta q^{11}+(-3+\beta )q^{13}-4q^{17}+\cdots\)
3528.2.a.be 3528.a 1.a $2$ $28.171$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2\beta q^{5}-6q^{11}+4\beta q^{13}+\beta q^{17}+\cdots\)
3528.2.a.bf 3528.a 1.a $2$ $28.171$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}-2q^{11}-2\beta q^{13}+\beta q^{17}+\cdots\)
3528.2.a.bg 3528.a 1.a $2$ $28.171$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{5}-\beta q^{13}-\beta q^{17}-4q^{23}-3q^{25}+\cdots\)
3528.2.a.bh 3528.a 1.a $2$ $28.171$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+\beta q^{13}-\beta q^{17}+4q^{23}-3q^{25}+\cdots\)
3528.2.a.bi 3528.a 1.a $2$ $28.171$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+2q^{11}+2\beta q^{13}+\beta q^{17}+\cdots\)
3528.2.a.bj 3528.a 1.a $2$ $28.171$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+4q^{11}+\beta q^{13}-2\beta q^{17}+\cdots\)
3528.2.a.bk 3528.a 1.a $2$ $28.171$ \(\Q(\sqrt{57}) \) None \(0\) \(0\) \(1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+\beta q^{11}+(3-\beta )q^{13}+4q^{17}+\cdots\)
3528.2.a.bl 3528.a 1.a $2$ $28.171$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(4\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{5}+(-2-2\beta )q^{11}-\beta q^{13}+\cdots\)
3528.2.a.bm 3528.a 1.a $2$ $28.171$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(4\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{5}+(2-2\beta )q^{11}+3\beta q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3528))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3528)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(441))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(504))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(588))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(882))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1176))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1764))\)\(^{\oplus 2}\)