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Properties

Label 3380.2.a
Level $3380$
Weight $2$
Character orbit 3380.a
Rep. character $\chi_{3380}(1,\cdot)$
Character field $\Q$
Dimension $51$
Newform subspaces $19$
Sturm bound $1092$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 19 \)
Sturm bound: \(1092\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3380))\).

Total New Old
Modular forms 588 51 537
Cusp forms 505 51 454
Eisenstein series 83 0 83

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

\(2\)\(5\)\(13\)FrickeDim
\(-\)\(+\)\(+\)$-$\(16\)
\(-\)\(+\)\(-\)$+$\(10\)
\(-\)\(-\)\(+\)$+$\(9\)
\(-\)\(-\)\(-\)$-$\(16\)
Plus space\(+\)\(19\)
Minus space\(-\)\(32\)

Trace form

\( 51 q - 2 q^{3} - q^{5} - 2 q^{7} + 43 q^{9} + O(q^{10}) \) \( 51 q - 2 q^{3} - q^{5} - 2 q^{7} + 43 q^{9} - 4 q^{11} - 2 q^{15} + 6 q^{17} - 4 q^{19} + 8 q^{21} + 10 q^{23} + 51 q^{25} - 8 q^{27} - 2 q^{29} + 16 q^{31} + 4 q^{33} + 6 q^{35} - 10 q^{37} - 2 q^{41} + 10 q^{43} - 9 q^{45} + 22 q^{47} + 39 q^{49} + 16 q^{51} + 26 q^{53} + 12 q^{57} + 12 q^{59} - 18 q^{61} + 46 q^{63} - 10 q^{67} + 16 q^{69} + 20 q^{71} - 26 q^{73} - 2 q^{75} + 36 q^{77} + 4 q^{79} + 63 q^{81} - 6 q^{83} - 6 q^{85} + 40 q^{87} - 2 q^{89} + 12 q^{93} - 12 q^{95} - 30 q^{97} - 40 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3380))\) 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 5 13
3380.2.a.a 3380.a 1.a $1$ $26.989$ \(\Q\) None \(0\) \(-3\) \(-1\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-q^{5}-3q^{7}+6q^{9}-3q^{11}+\cdots\)
3380.2.a.b 3380.a 1.a $1$ $26.989$ \(\Q\) None \(0\) \(-3\) \(1\) \(3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+q^{5}+3q^{7}+6q^{9}+3q^{11}+\cdots\)
3380.2.a.c 3380.a 1.a $1$ $26.989$ \(\Q\) None \(0\) \(-2\) \(1\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}-2q^{7}+q^{9}-2q^{15}+\cdots\)
3380.2.a.d 3380.a 1.a $1$ $26.989$ \(\Q\) None \(0\) \(-1\) \(-1\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+5q^{7}-2q^{9}-5q^{11}+\cdots\)
3380.2.a.e 3380.a 1.a $1$ $26.989$ \(\Q\) None \(0\) \(-1\) \(1\) \(-5\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-5q^{7}-2q^{9}+5q^{11}+\cdots\)
3380.2.a.f 3380.a 1.a $1$ $26.989$ \(\Q\) None \(0\) \(1\) \(-1\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-q^{7}-2q^{9}+3q^{11}-q^{15}+\cdots\)
3380.2.a.g 3380.a 1.a $1$ $26.989$ \(\Q\) None \(0\) \(1\) \(-1\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+q^{7}-2q^{9}-3q^{11}-q^{15}+\cdots\)
3380.2.a.h 3380.a 1.a $1$ $26.989$ \(\Q\) None \(0\) \(1\) \(1\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-q^{7}-2q^{9}+3q^{11}+q^{15}+\cdots\)
3380.2.a.i 3380.a 1.a $1$ $26.989$ \(\Q\) None \(0\) \(1\) \(1\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+q^{7}-2q^{9}-3q^{11}+q^{15}+\cdots\)
3380.2.a.j 3380.a 1.a $1$ $26.989$ \(\Q\) None \(0\) \(2\) \(1\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots\)
3380.2.a.k 3380.a 1.a $3$ $26.989$ \(\Q(\zeta_{14})^+\) None \(0\) \(-1\) \(-3\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-q^{5}-\beta _{1}q^{7}+(-1+\beta _{2})q^{9}+\cdots\)
3380.2.a.l 3380.a 1.a $3$ $26.989$ \(\Q(\zeta_{14})^+\) None \(0\) \(-1\) \(3\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+q^{5}+\beta _{1}q^{7}+(-1+\beta _{2})q^{9}+\cdots\)
3380.2.a.m 3380.a 1.a $3$ $26.989$ 3.3.756.1 None \(0\) \(0\) \(-3\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-q^{5}-\beta _{2}q^{7}+(1+\beta _{2})q^{9}+\cdots\)
3380.2.a.n 3380.a 1.a $3$ $26.989$ 3.3.756.1 None \(0\) \(0\) \(3\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+q^{5}+\beta _{2}q^{7}+(1+\beta _{2})q^{9}+\cdots\)
3380.2.a.o 3380.a 1.a $3$ $26.989$ 3.3.564.1 None \(0\) \(2\) \(-3\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{3}-q^{5}+(1+\beta _{1})q^{7}+(4+\cdots)q^{9}+\cdots\)
3380.2.a.p 3380.a 1.a $4$ $26.989$ 4.4.4752.1 None \(0\) \(2\) \(-4\) \(-6\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{3}-q^{5}+(-2-\beta _{2}+\beta _{3})q^{7}+\cdots\)
3380.2.a.q 3380.a 1.a $4$ $26.989$ 4.4.4752.1 None \(0\) \(2\) \(4\) \(6\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{3}+q^{5}+(2+\beta _{2}-\beta _{3})q^{7}+\cdots\)
3380.2.a.r 3380.a 1.a $9$ $26.989$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-1\) \(-9\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-q^{5}+(-1+\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots\)
3380.2.a.s 3380.a 1.a $9$ $26.989$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-1\) \(9\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+q^{5}+(1-\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3380))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3380)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(676))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(845))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1690))\)\(^{\oplus 2}\)