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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3528))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
3528.2.a.a \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(-4\) \(0\) \(+\) \(+\) \(+\) \(q-4q^{5}-3q^{13}+4q^{17}+7q^{19}-4q^{23}+\cdots\)
3528.2.a.b \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(-4\) \(0\) \(-\) \(-\) \(-\) \(q-4q^{5}-2q^{17}+2q^{19}-8q^{23}+11q^{25}+\cdots\)
3528.2.a.c \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(-4\) \(0\) \(-\) \(+\) \(-\) \(q-4q^{5}+3q^{13}+4q^{17}-7q^{19}+4q^{23}+\cdots\)
3528.2.a.d \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(-2\) \(0\) \(+\) \(-\) \(-\) \(q-2q^{5}-4q^{11}+2q^{13}+2q^{17}+4q^{19}+\cdots\)
3528.2.a.e \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(-2\) \(0\) \(-\) \(+\) \(-\) \(q-2q^{5}+2q^{11}-2q^{13}-6q^{17}+4q^{19}+\cdots\)
3528.2.a.f \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(-2\) \(0\) \(-\) \(-\) \(+\) \(q-2q^{5}+6q^{11}-3q^{13}-4q^{17}-5q^{19}+\cdots\)
3528.2.a.g \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(-2\) \(0\) \(+\) \(+\) \(-\) \(q-2q^{5}+6q^{11}+6q^{13}+2q^{17}-4q^{19}+\cdots\)
3528.2.a.h \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(-1\) \(0\) \(+\) \(+\) \(-\) \(q-q^{5}-5q^{11}-2q^{13}+6q^{17}-2q^{19}+\cdots\)
3528.2.a.i \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(-1\) \(0\) \(-\) \(-\) \(-\) \(q-q^{5}-3q^{11}-4q^{13}+4q^{19}-8q^{23}+\cdots\)
3528.2.a.j \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(-1\) \(0\) \(-\) \(-\) \(-\) \(q-q^{5}-3q^{11}+6q^{13}-5q^{17}-q^{19}+\cdots\)
3528.2.a.k \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(-1\) \(0\) \(+\) \(-\) \(-\) \(q-q^{5}+q^{11}-2q^{13}+3q^{17}-5q^{19}+\cdots\)
3528.2.a.l \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(-1\) \(0\) \(-\) \(+\) \(+\) \(q-q^{5}+5q^{11}+2q^{13}+6q^{17}+2q^{19}+\cdots\)
3528.2.a.m \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q-4q^{13}+4q^{17}+4q^{19}-4q^{23}+\cdots\)
3528.2.a.n \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q+4q^{13}-4q^{17}-4q^{19}-4q^{23}+\cdots\)
3528.2.a.o \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(1\) \(0\) \(+\) \(+\) \(+\) \(q+q^{5}-5q^{11}+2q^{13}-6q^{17}+2q^{19}+\cdots\)
3528.2.a.p \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(1\) \(0\) \(-\) \(-\) \(+\) \(q+q^{5}-3q^{11}-6q^{13}+5q^{17}+q^{19}+\cdots\)
3528.2.a.q \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(1\) \(0\) \(-\) \(-\) \(+\) \(q+q^{5}-3q^{11}+4q^{13}-4q^{19}-8q^{23}+\cdots\)
3528.2.a.r \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(1\) \(0\) \(+\) \(-\) \(+\) \(q+q^{5}+q^{11}+2q^{13}-3q^{17}+5q^{19}+\cdots\)
3528.2.a.s \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(1\) \(0\) \(-\) \(+\) \(-\) \(q+q^{5}+5q^{11}-2q^{13}-6q^{17}-2q^{19}+\cdots\)
3528.2.a.t \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(2\) \(0\) \(-\) \(+\) \(-\) \(q+2q^{5}-6q^{11}+6q^{13}-2q^{17}-4q^{19}+\cdots\)
3528.2.a.u \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(2\) \(0\) \(+\) \(+\) \(-\) \(q+2q^{5}-2q^{11}-2q^{13}+6q^{17}+4q^{19}+\cdots\)
3528.2.a.v \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q+2q^{5}-6q^{13}-2q^{17}-4q^{19}+4q^{23}+\cdots\)
3528.2.a.w \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q+2q^{5}+2q^{13}+6q^{17}+4q^{19}+4q^{23}+\cdots\)
3528.2.a.x \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(2\) \(0\) \(+\) \(-\) \(-\) \(q+2q^{5}+4q^{11}-2q^{13}-6q^{17}-8q^{19}+\cdots\)
3528.2.a.y \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q+2q^{5}+6q^{11}+3q^{13}+4q^{17}+5q^{19}+\cdots\)
3528.2.a.z \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(4\) \(0\) \(-\) \(+\) \(+\) \(q+4q^{5}-3q^{13}-4q^{17}+7q^{19}+4q^{23}+\cdots\)
3528.2.a.ba \(1\) \(28.171\) \(\Q\) None \(0\) \(0\) \(4\) \(0\) \(+\) \(+\) \(-\) \(q+4q^{5}+3q^{13}-4q^{17}-7q^{19}-4q^{23}+\cdots\)
3528.2.a.bb \(2\) \(28.171\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-4\) \(0\) \(-\) \(-\) \(+\) \(q+(-2+\beta )q^{5}+(-2+2\beta )q^{11}-\beta q^{13}+\cdots\)
3528.2.a.bc \(2\) \(28.171\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-4\) \(0\) \(+\) \(-\) \(+\) \(q+(-2+\beta )q^{5}+(2+2\beta )q^{11}+3\beta q^{13}+\cdots\)
3528.2.a.bd \(2\) \(28.171\) \(\Q(\sqrt{57}) \) None \(0\) \(0\) \(-1\) \(0\) \(+\) \(-\) \(-\) \(q-\beta q^{5}+\beta q^{11}+(-3+\beta )q^{13}-4q^{17}+\cdots\)
3528.2.a.be \(2\) \(28.171\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+2\beta q^{5}-6q^{11}+4\beta q^{13}+\beta q^{17}+\cdots\)
3528.2.a.bf \(2\) \(28.171\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+\beta q^{5}-2q^{11}-2\beta q^{13}+\beta q^{17}+\cdots\)
3528.2.a.bg \(2\) \(28.171\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q+\beta q^{5}-\beta q^{13}-\beta q^{17}-4q^{23}-3q^{25}+\cdots\)
3528.2.a.bh \(2\) \(28.171\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+\beta q^{5}+\beta q^{13}-\beta q^{17}+4q^{23}-3q^{25}+\cdots\)
3528.2.a.bi \(2\) \(28.171\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q+\beta q^{5}+2q^{11}+2\beta q^{13}+\beta q^{17}+\cdots\)
3528.2.a.bj \(2\) \(28.171\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+\beta q^{5}+4q^{11}+\beta q^{13}-2\beta q^{17}+\cdots\)
3528.2.a.bk \(2\) \(28.171\) \(\Q(\sqrt{57}) \) None \(0\) \(0\) \(1\) \(0\) \(+\) \(-\) \(+\) \(q+\beta q^{5}+\beta q^{11}+(3-\beta )q^{13}+4q^{17}+\cdots\)
3528.2.a.bl \(2\) \(28.171\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(4\) \(0\) \(-\) \(-\) \(+\) \(q+(2+\beta )q^{5}+(-2-2\beta )q^{11}-\beta q^{13}+\cdots\)
3528.2.a.bm \(2\) \(28.171\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(4\) \(0\) \(+\) \(-\) \(+\) \(q+(2+\beta )q^{5}+(2-2\beta )q^{11}+3\beta q^{13}+\cdots\)

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

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