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Properties

Label 3420.2.a
Level $3420$
Weight $2$
Character orbit 3420.a
Rep. character $\chi_{3420}(1,\cdot)$
Character field $\Q$
Dimension $30$
Newform subspaces $15$
Sturm bound $1440$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(1440\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3420))\).

Total New Old
Modular forms 744 30 714
Cusp forms 697 30 667
Eisenstein series 47 0 47

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

\(2\)\(3\)\(5\)\(19\)FrickeDim
\(-\)\(+\)\(+\)\(+\)$-$\(3\)
\(-\)\(+\)\(+\)\(-\)$+$\(3\)
\(-\)\(+\)\(-\)\(+\)$+$\(3\)
\(-\)\(+\)\(-\)\(-\)$-$\(3\)
\(-\)\(-\)\(+\)\(+\)$+$\(5\)
\(-\)\(-\)\(+\)\(-\)$-$\(5\)
\(-\)\(-\)\(-\)\(+\)$-$\(4\)
\(-\)\(-\)\(-\)\(-\)$+$\(4\)
Plus space\(+\)\(15\)
Minus space\(-\)\(15\)

Trace form

\( 30 q - 2 q^{5} + O(q^{10}) \) \( 30 q - 2 q^{5} - 4 q^{13} - 12 q^{17} + 8 q^{23} + 30 q^{25} - 4 q^{29} - 8 q^{31} - 8 q^{35} + 8 q^{37} - 4 q^{41} - 8 q^{43} - 16 q^{47} + 38 q^{49} - 32 q^{53} + 12 q^{61} - 12 q^{67} - 64 q^{71} + 4 q^{73} - 24 q^{77} - 12 q^{85} + 28 q^{89} - 16 q^{91} - 16 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3420))\) 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 5 19
3420.2.a.a 3420.a 1.a $1$ $27.309$ \(\Q\) None \(0\) \(0\) \(1\) \(-4\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-4q^{7}-2q^{11}+6q^{13}+2q^{17}+\cdots\)
3420.2.a.b 3420.a 1.a $1$ $27.309$ \(\Q\) None \(0\) \(0\) \(1\) \(-2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}-4q^{11}+2q^{17}+q^{19}+\cdots\)
3420.2.a.c 3420.a 1.a $1$ $27.309$ \(\Q\) None \(0\) \(0\) \(1\) \(-2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}-4q^{13}+2q^{17}-q^{19}+\cdots\)
3420.2.a.d 3420.a 1.a $1$ $27.309$ \(\Q\) None \(0\) \(0\) \(1\) \(-2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}+4q^{11}-4q^{13}-6q^{17}+\cdots\)
3420.2.a.e 3420.a 1.a $1$ $27.309$ \(\Q\) None \(0\) \(0\) \(1\) \(2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+2q^{7}-4q^{13}-6q^{17}+q^{19}+\cdots\)
3420.2.a.f 3420.a 1.a $1$ $27.309$ \(\Q\) None \(0\) \(0\) \(1\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+2q^{7}+6q^{13}-2q^{17}-q^{19}+\cdots\)
3420.2.a.g 3420.a 1.a $2$ $27.309$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-2\) \(-4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(-2+2\beta )q^{7}+2q^{11}+(-2+\cdots)q^{13}+\cdots\)
3420.2.a.h 3420.a 1.a $2$ $27.309$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-2\) \(4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+2q^{7}-2\beta q^{11}+(-1+\beta )q^{13}+\cdots\)
3420.2.a.i 3420.a 1.a $2$ $27.309$ \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(2\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+(1+\beta )q^{7}+(1-\beta )q^{11}+(1-\beta )q^{13}+\cdots\)
3420.2.a.j 3420.a 1.a $3$ $27.309$ 3.3.404.1 None \(0\) \(0\) \(-3\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(-1+\beta _{2})q^{7}+(1-\beta _{1}+2\beta _{2})q^{11}+\cdots\)
3420.2.a.k 3420.a 1.a $3$ $27.309$ 3.3.1524.1 None \(0\) \(0\) \(-3\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-\beta _{1}q^{7}+(1+\beta _{2})q^{11}+(2-\beta _{1}+\cdots)q^{13}+\cdots\)
3420.2.a.l 3420.a 1.a $3$ $27.309$ 3.3.564.1 None \(0\) \(0\) \(-3\) \(2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+(\beta _{1}-\beta _{2})q^{7}+(-1-\beta _{1})q^{11}+\cdots\)
3420.2.a.m 3420.a 1.a $3$ $27.309$ 3.3.564.1 None \(0\) \(0\) \(-3\) \(4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+(1-\beta _{2})q^{7}+(-2-\beta _{1}+\beta _{2})q^{11}+\cdots\)
3420.2.a.n 3420.a 1.a $3$ $27.309$ 3.3.404.1 None \(0\) \(0\) \(3\) \(-2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+(-1+\beta _{2})q^{7}+(-1+\beta _{1}-2\beta _{2})q^{11}+\cdots\)
3420.2.a.o 3420.a 1.a $3$ $27.309$ 3.3.564.1 None \(0\) \(0\) \(3\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(\beta _{1}-\beta _{2})q^{7}+(1+\beta _{1})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3420))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3420)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(285))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(342))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(380))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(570))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(684))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(855))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1140))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1710))\)\(^{\oplus 2}\)