Properties

Label 1638.2.a
Level $1638$
Weight $2$
Character orbit 1638.a
Rep. character $\chi_{1638}(1,\cdot)$
Character field $\Q$
Dimension $30$
Newform subspaces $25$
Sturm bound $672$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 25 \)
Sturm bound: \(672\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1638))\).

Total New Old
Modular forms 352 30 322
Cusp forms 321 30 291
Eisenstein series 31 0 31

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

\(2\)\(3\)\(7\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(2\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(12\)
Minus space\(-\)\(18\)

Trace form

\( 30 q - 2 q^{2} + 30 q^{4} - 8 q^{5} - 2 q^{8} + O(q^{10}) \) \( 30 q - 2 q^{2} + 30 q^{4} - 8 q^{5} - 2 q^{8} - 8 q^{11} + 30 q^{16} + 4 q^{17} + 20 q^{19} - 8 q^{20} + 4 q^{22} + 4 q^{23} + 50 q^{25} + 12 q^{29} + 8 q^{31} - 2 q^{32} + 12 q^{34} + 4 q^{35} + 12 q^{37} + 4 q^{38} + 12 q^{41} + 8 q^{43} - 8 q^{44} + 8 q^{46} + 24 q^{47} + 30 q^{49} + 2 q^{50} - 4 q^{53} + 8 q^{55} - 12 q^{58} - 20 q^{59} - 56 q^{61} + 32 q^{62} + 30 q^{64} - 4 q^{65} + 4 q^{68} - 4 q^{70} - 16 q^{71} - 52 q^{73} + 20 q^{76} - 44 q^{79} - 8 q^{80} + 4 q^{82} + 12 q^{83} - 48 q^{85} - 16 q^{86} + 4 q^{88} + 4 q^{89} + 2 q^{91} + 4 q^{92} + 8 q^{95} + 4 q^{97} - 2 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1638))\) 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 7 13
1638.2.a.a 1638.a 1.a $1$ $13.079$ \(\Q\) None 546.2.a.e \(-1\) \(0\) \(-3\) \(-1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-3q^{5}-q^{7}-q^{8}+3q^{10}+\cdots\)
1638.2.a.b 1638.a 1.a $1$ $13.079$ \(\Q\) None 1638.2.a.b \(-1\) \(0\) \(-3\) \(1\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-3q^{5}+q^{7}-q^{8}+3q^{10}+\cdots\)
1638.2.a.c 1638.a 1.a $1$ $13.079$ \(\Q\) None 182.2.a.c \(-1\) \(0\) \(-2\) \(-1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-2q^{5}-q^{7}-q^{8}+2q^{10}+\cdots\)
1638.2.a.d 1638.a 1.a $1$ $13.079$ \(\Q\) None 546.2.a.g \(-1\) \(0\) \(-2\) \(1\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-2q^{5}+q^{7}-q^{8}+2q^{10}+\cdots\)
1638.2.a.e 1638.a 1.a $1$ $13.079$ \(\Q\) None 1638.2.a.e \(-1\) \(0\) \(0\) \(1\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{7}-q^{8}+q^{13}-q^{14}+\cdots\)
1638.2.a.f 1638.a 1.a $1$ $13.079$ \(\Q\) None 182.2.a.d \(-1\) \(0\) \(0\) \(1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{7}-q^{8}+3q^{11}+q^{13}+\cdots\)
1638.2.a.g 1638.a 1.a $1$ $13.079$ \(\Q\) None 1638.2.a.g \(-1\) \(0\) \(1\) \(-1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-q^{7}-q^{8}-q^{10}+\cdots\)
1638.2.a.h 1638.a 1.a $1$ $13.079$ \(\Q\) None 546.2.a.f \(-1\) \(0\) \(1\) \(1\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+q^{7}-q^{8}-q^{10}+\cdots\)
1638.2.a.i 1638.a 1.a $1$ $13.079$ \(\Q\) None 1638.2.a.i \(-1\) \(0\) \(3\) \(1\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+3q^{5}+q^{7}-q^{8}-3q^{10}+\cdots\)
1638.2.a.j 1638.a 1.a $1$ $13.079$ \(\Q\) None 182.2.a.e \(-1\) \(0\) \(4\) \(-1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+4q^{5}-q^{7}-q^{8}-4q^{10}+\cdots\)
1638.2.a.k 1638.a 1.a $1$ $13.079$ \(\Q\) None 182.2.a.a \(1\) \(0\) \(-4\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-4q^{5}-q^{7}+q^{8}-4q^{10}+\cdots\)
1638.2.a.l 1638.a 1.a $1$ $13.079$ \(\Q\) None 546.2.a.d \(1\) \(0\) \(-3\) \(1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-3q^{5}+q^{7}+q^{8}-3q^{10}+\cdots\)
1638.2.a.m 1638.a 1.a $1$ $13.079$ \(\Q\) None 1638.2.a.i \(1\) \(0\) \(-3\) \(1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-3q^{5}+q^{7}+q^{8}-3q^{10}+\cdots\)
1638.2.a.n 1638.a 1.a $1$ $13.079$ \(\Q\) None 1638.2.a.g \(1\) \(0\) \(-1\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-q^{7}+q^{8}-q^{10}+\cdots\)
1638.2.a.o 1638.a 1.a $1$ $13.079$ \(\Q\) None 546.2.a.c \(1\) \(0\) \(-1\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-q^{7}+q^{8}-q^{10}+\cdots\)
1638.2.a.p 1638.a 1.a $1$ $13.079$ \(\Q\) None 1638.2.a.e \(1\) \(0\) \(0\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{7}+q^{8}+q^{13}+q^{14}+\cdots\)
1638.2.a.q 1638.a 1.a $1$ $13.079$ \(\Q\) None 182.2.a.b \(1\) \(0\) \(0\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{7}+q^{8}+5q^{11}-q^{13}+\cdots\)
1638.2.a.r 1638.a 1.a $1$ $13.079$ \(\Q\) None 546.2.a.a \(1\) \(0\) \(1\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}+q^{10}+\cdots\)
1638.2.a.s 1638.a 1.a $1$ $13.079$ \(\Q\) None 546.2.a.b \(1\) \(0\) \(2\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2q^{5}-q^{7}+q^{8}+2q^{10}+\cdots\)
1638.2.a.t 1638.a 1.a $1$ $13.079$ \(\Q\) None 1638.2.a.b \(1\) \(0\) \(3\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+3q^{5}+q^{7}+q^{8}+3q^{10}+\cdots\)
1638.2.a.u 1638.a 1.a $2$ $13.079$ \(\Q(\sqrt{17}) \) None 546.2.a.j \(-2\) \(0\) \(-3\) \(-2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-1-\beta )q^{5}-q^{7}-q^{8}+\cdots\)
1638.2.a.v 1638.a 1.a $2$ $13.079$ \(\Q(\sqrt{33}) \) None 1638.2.a.v \(-2\) \(0\) \(-1\) \(-2\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-\beta q^{5}-q^{7}-q^{8}+\beta q^{10}+\cdots\)
1638.2.a.w 1638.a 1.a $2$ $13.079$ \(\Q(\sqrt{41}) \) None 546.2.a.i \(-2\) \(0\) \(1\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+\beta q^{5}+q^{7}-q^{8}-\beta q^{10}+\cdots\)
1638.2.a.x 1638.a 1.a $2$ $13.079$ \(\Q(\sqrt{33}) \) None 1638.2.a.v \(2\) \(0\) \(1\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+\beta q^{5}-q^{7}+q^{8}+\beta q^{10}+\cdots\)
1638.2.a.y 1638.a 1.a $2$ $13.079$ \(\Q(\sqrt{57}) \) None 546.2.a.h \(2\) \(0\) \(1\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+\beta q^{5}+q^{7}+q^{8}+\beta q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1638))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1638)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(182))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(234))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(273))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(546))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(819))\)\(^{\oplus 2}\)