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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3380))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 13
3380.2.a.a \(1\) \(26.989\) \(\Q\) None \(0\) \(-3\) \(-1\) \(-3\) \(-\) \(+\) \(+\) \(q-3q^{3}-q^{5}-3q^{7}+6q^{9}-3q^{11}+\cdots\)
3380.2.a.b \(1\) \(26.989\) \(\Q\) None \(0\) \(-3\) \(1\) \(3\) \(-\) \(-\) \(+\) \(q-3q^{3}+q^{5}+3q^{7}+6q^{9}+3q^{11}+\cdots\)
3380.2.a.c \(1\) \(26.989\) \(\Q\) None \(0\) \(-2\) \(1\) \(-2\) \(-\) \(-\) \(+\) \(q-2q^{3}+q^{5}-2q^{7}+q^{9}-2q^{15}+\cdots\)
3380.2.a.d \(1\) \(26.989\) \(\Q\) None \(0\) \(-1\) \(-1\) \(5\) \(-\) \(+\) \(+\) \(q-q^{3}-q^{5}+5q^{7}-2q^{9}-5q^{11}+\cdots\)
3380.2.a.e \(1\) \(26.989\) \(\Q\) None \(0\) \(-1\) \(1\) \(-5\) \(-\) \(-\) \(+\) \(q-q^{3}+q^{5}-5q^{7}-2q^{9}+5q^{11}+\cdots\)
3380.2.a.f \(1\) \(26.989\) \(\Q\) None \(0\) \(1\) \(-1\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{5}-q^{7}-2q^{9}+3q^{11}-q^{15}+\cdots\)
3380.2.a.g \(1\) \(26.989\) \(\Q\) None \(0\) \(1\) \(-1\) \(1\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{5}+q^{7}-2q^{9}-3q^{11}-q^{15}+\cdots\)
3380.2.a.h \(1\) \(26.989\) \(\Q\) None \(0\) \(1\) \(1\) \(-1\) \(-\) \(-\) \(+\) \(q+q^{3}+q^{5}-q^{7}-2q^{9}+3q^{11}+q^{15}+\cdots\)
3380.2.a.i \(1\) \(26.989\) \(\Q\) None \(0\) \(1\) \(1\) \(1\) \(-\) \(-\) \(+\) \(q+q^{3}+q^{5}+q^{7}-2q^{9}-3q^{11}+q^{15}+\cdots\)
3380.2.a.j \(1\) \(26.989\) \(\Q\) None \(0\) \(2\) \(1\) \(-2\) \(-\) \(-\) \(+\) \(q+2q^{3}+q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots\)
3380.2.a.k \(3\) \(26.989\) \(\Q(\zeta_{14})^+\) None \(0\) \(-1\) \(-3\) \(-1\) \(-\) \(+\) \(-\) \(q-\beta _{1}q^{3}-q^{5}-\beta _{1}q^{7}+(-1+\beta _{2})q^{9}+\cdots\)
3380.2.a.l \(3\) \(26.989\) \(\Q(\zeta_{14})^+\) None \(0\) \(-1\) \(3\) \(1\) \(-\) \(-\) \(+\) \(q-\beta _{1}q^{3}+q^{5}+\beta _{1}q^{7}+(-1+\beta _{2})q^{9}+\cdots\)
3380.2.a.m \(3\) \(26.989\) 3.3.756.1 None \(0\) \(0\) \(-3\) \(0\) \(-\) \(+\) \(-\) \(q+\beta _{1}q^{3}-q^{5}-\beta _{2}q^{7}+(1+\beta _{2})q^{9}+\cdots\)
3380.2.a.n \(3\) \(26.989\) 3.3.756.1 None \(0\) \(0\) \(3\) \(0\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{3}+q^{5}+\beta _{2}q^{7}+(1+\beta _{2})q^{9}+\cdots\)
3380.2.a.o \(3\) \(26.989\) 3.3.564.1 None \(0\) \(2\) \(-3\) \(2\) \(-\) \(+\) \(+\) \(q+(1+\beta _{2})q^{3}-q^{5}+(1+\beta _{1})q^{7}+(4+\cdots)q^{9}+\cdots\)
3380.2.a.p \(4\) \(26.989\) 4.4.4752.1 None \(0\) \(2\) \(-4\) \(-6\) \(-\) \(+\) \(-\) \(q+(1+\beta _{2})q^{3}-q^{5}+(-2-\beta _{2}+\beta _{3})q^{7}+\cdots\)
3380.2.a.q \(4\) \(26.989\) 4.4.4752.1 None \(0\) \(2\) \(4\) \(6\) \(-\) \(-\) \(-\) \(q+(1+\beta _{2})q^{3}+q^{5}+(2+\beta _{2}-\beta _{3})q^{7}+\cdots\)
3380.2.a.r \(9\) \(26.989\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-1\) \(-9\) \(-1\) \(-\) \(+\) \(+\) \(q-\beta _{1}q^{3}-q^{5}+(-1+\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots\)
3380.2.a.s \(9\) \(26.989\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-1\) \(9\) \(1\) \(-\) \(-\) \(-\) \(q-\beta _{1}q^{3}+q^{5}+(1-\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)

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

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