Properties

Label 605.2.a
Level $605$
Weight $2$
Character orbit 605.a
Rep. character $\chi_{605}(1,\cdot)$
Character field $\Q$
Dimension $37$
Newform subspaces $13$
Sturm bound $132$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(132\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(605))\).

Total New Old
Modular forms 78 37 41
Cusp forms 55 37 18
Eisenstein series 23 0 23

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

\(5\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(6\)
\(+\)\(-\)\(-\)\(12\)
\(-\)\(+\)\(-\)\(12\)
\(-\)\(-\)\(+\)\(7\)
Plus space\(+\)\(13\)
Minus space\(-\)\(24\)

Trace form

\( 37 q - 3 q^{2} + 39 q^{4} + q^{5} + 8 q^{6} + 4 q^{7} - 3 q^{8} + 33 q^{9} + O(q^{10}) \) \( 37 q - 3 q^{2} + 39 q^{4} + q^{5} + 8 q^{6} + 4 q^{7} - 3 q^{8} + 33 q^{9} + q^{10} + 16 q^{12} + 6 q^{13} + 31 q^{16} - 14 q^{17} - 7 q^{18} + 4 q^{19} - q^{20} - 8 q^{23} + 8 q^{24} + 37 q^{25} - 10 q^{26} + 4 q^{28} - 10 q^{29} - 8 q^{30} + 4 q^{31} + q^{32} - 34 q^{34} - 4 q^{35} + 7 q^{36} - 2 q^{37} - 20 q^{38} + 16 q^{39} + 9 q^{40} - 14 q^{41} - 44 q^{42} + 8 q^{43} + 13 q^{45} + 4 q^{46} + 4 q^{47} - 20 q^{48} + 41 q^{49} - 3 q^{50} + 16 q^{51} - 6 q^{52} - 14 q^{53} + 16 q^{54} - 20 q^{56} - 6 q^{58} - 8 q^{59} - 16 q^{60} + 6 q^{61} + 8 q^{62} + 20 q^{63} + 39 q^{64} - 10 q^{65} + 4 q^{67} - 18 q^{68} - 24 q^{69} - 8 q^{70} - 20 q^{71} - 39 q^{72} - 6 q^{73} + 22 q^{74} - 4 q^{76} - 52 q^{78} - 16 q^{79} - q^{80} + 13 q^{81} - 22 q^{82} + 16 q^{83} - 32 q^{84} + 2 q^{85} - 16 q^{86} - 32 q^{87} - 22 q^{89} + 13 q^{90} + 48 q^{91} - 12 q^{92} + 56 q^{93} + 4 q^{94} + 4 q^{95} + 8 q^{96} + 46 q^{97} + 13 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(605))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 11
605.2.a.a 605.a 1.a $1$ $4.831$ \(\Q\) None \(-1\) \(-3\) \(1\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}-q^{4}+q^{5}+3q^{6}-3q^{7}+\cdots\)
605.2.a.b 605.a 1.a $1$ $4.831$ \(\Q\) None \(-1\) \(0\) \(1\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+q^{5}+3q^{8}-3q^{9}-q^{10}+\cdots\)
605.2.a.c 605.a 1.a $1$ $4.831$ \(\Q\) None \(1\) \(-3\) \(1\) \(3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}-q^{4}+q^{5}-3q^{6}+3q^{7}+\cdots\)
605.2.a.d 605.a 1.a $2$ $4.831$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(-2\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+2\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
605.2.a.e 605.a 1.a $2$ $4.831$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(-2\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{3}+q^{4}-q^{5}-\beta q^{6}-\beta q^{7}+\cdots\)
605.2.a.f 605.a 1.a $2$ $4.831$ \(\Q(\sqrt{3}) \) None \(0\) \(4\) \(2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+2q^{3}+q^{4}+q^{5}+2\beta q^{6}+\cdots\)
605.2.a.g 605.a 1.a $3$ $4.831$ 3.3.404.1 None \(-1\) \(1\) \(-3\) \(1\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+\beta _{1}q^{3}+(3+\beta _{1}-\beta _{2})q^{4}+\cdots\)
605.2.a.h 605.a 1.a $3$ $4.831$ 3.3.404.1 None \(1\) \(1\) \(-3\) \(-1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+\beta _{1}q^{3}+(3+\beta _{1}-\beta _{2})q^{4}+\cdots\)
605.2.a.i 605.a 1.a $4$ $4.831$ 4.4.2525.1 None \(-3\) \(-2\) \(4\) \(-11\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{3})q^{2}-\beta _{1}q^{3}+(1+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
605.2.a.j 605.a 1.a $4$ $4.831$ 4.4.725.1 None \(-1\) \(0\) \(-4\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-1+\beta _{2}+\beta _{3})q^{3}+(-1+\cdots)q^{4}+\cdots\)
605.2.a.k 605.a 1.a $4$ $4.831$ 4.4.725.1 None \(1\) \(0\) \(-4\) \(3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1+\beta _{2}+\beta _{3})q^{3}+(-1+\cdots)q^{4}+\cdots\)
605.2.a.l 605.a 1.a $4$ $4.831$ 4.4.2525.1 None \(3\) \(-2\) \(4\) \(11\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2}-\beta _{3})q^{2}+(-1+\beta _{1})q^{3}+\cdots\)
605.2.a.m 605.a 1.a $6$ $4.831$ 6.6.27433728.1 None \(0\) \(6\) \(6\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(605))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(605)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 2}\)