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Properties

Label 605.4.a
Level $605$
Weight $4$
Character orbit 605.a
Rep. character $\chi_{605}(1,\cdot)$
Character field $\Q$
Dimension $109$
Newform subspaces $20$
Sturm bound $264$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 20 \)
Sturm bound: \(264\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(605))\).

Total New Old
Modular forms 210 109 101
Cusp forms 186 109 77
Eisenstein series 24 0 24

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
\(+\)\(+\)$+$\(30\)
\(+\)\(-\)$-$\(25\)
\(-\)\(+\)$-$\(24\)
\(-\)\(-\)$+$\(30\)
Plus space\(+\)\(60\)
Minus space\(-\)\(49\)

Trace form

\( 109 q + 4 q^{2} - 2 q^{3} + 420 q^{4} - 5 q^{5} - 48 q^{6} + 34 q^{7} + 12 q^{8} + 1041 q^{9} + O(q^{10}) \) \( 109 q + 4 q^{2} - 2 q^{3} + 420 q^{4} - 5 q^{5} - 48 q^{6} + 34 q^{7} + 12 q^{8} + 1041 q^{9} + 40 q^{10} - 32 q^{12} + 30 q^{13} + 12 q^{14} - 50 q^{15} + 1860 q^{16} - 90 q^{17} + 420 q^{18} + 212 q^{19} - 80 q^{20} + 220 q^{21} - 190 q^{23} + 248 q^{24} + 2725 q^{25} + 8 q^{26} - 164 q^{27} - 76 q^{28} + 102 q^{29} + 240 q^{30} + 108 q^{31} + 680 q^{32} - 76 q^{34} - 10 q^{35} + 5068 q^{36} + 714 q^{37} - 108 q^{38} - 132 q^{39} + 240 q^{40} - 194 q^{41} + 668 q^{42} - 754 q^{43} + 115 q^{45} - 136 q^{46} - 482 q^{47} + 1644 q^{48} + 4845 q^{49} + 100 q^{50} - 1764 q^{51} - 796 q^{52} - 1270 q^{53} + 104 q^{54} + 16 q^{56} - 1864 q^{57} - 2896 q^{58} + 1732 q^{59} + 260 q^{60} + 418 q^{61} - 1664 q^{62} + 2498 q^{63} + 6892 q^{64} - 350 q^{65} - 126 q^{67} + 2364 q^{68} + 1716 q^{69} - 40 q^{70} - 1164 q^{71} + 444 q^{72} + 2334 q^{73} - 3208 q^{74} - 50 q^{75} + 4352 q^{76} + 220 q^{78} + 664 q^{79} - 2320 q^{80} + 7153 q^{81} + 2616 q^{82} - 114 q^{83} + 2000 q^{84} + 930 q^{85} - 80 q^{86} + 4916 q^{87} + 210 q^{89} + 280 q^{90} - 1708 q^{91} - 4652 q^{92} - 6136 q^{93} - 2224 q^{94} + 1020 q^{95} - 3976 q^{96} - 2722 q^{97} - 6148 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(605))\) 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
605.4.a.a 605.a 1.a $1$ $35.696$ \(\Q\) None \(-1\) \(-5\) \(5\) \(-29\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-5q^{3}-7q^{4}+5q^{5}+5q^{6}+\cdots\)
605.4.a.b 605.a 1.a $1$ $35.696$ \(\Q\) None \(-1\) \(-3\) \(-5\) \(9\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}-7q^{4}-5q^{5}+3q^{6}+\cdots\)
605.4.a.c 605.a 1.a $1$ $35.696$ \(\Q\) None \(1\) \(-5\) \(5\) \(29\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-5q^{3}-7q^{4}+5q^{5}-5q^{6}+\cdots\)
605.4.a.d 605.a 1.a $1$ $35.696$ \(\Q\) None \(4\) \(2\) \(-5\) \(-6\) $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2q^{3}+8q^{4}-5q^{5}+8q^{6}+\cdots\)
605.4.a.e 605.a 1.a $2$ $35.696$ \(\Q(\sqrt{5}) \) None \(0\) \(4\) \(-10\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+2q^{3}-3q^{4}-5q^{5}-2\beta q^{6}+\cdots\)
605.4.a.f 605.a 1.a $2$ $35.696$ \(\Q(\sqrt{15}) \) None \(0\) \(4\) \(10\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+2q^{3}+7q^{4}+5q^{5}+2\beta q^{6}+\cdots\)
605.4.a.g 605.a 1.a $2$ $35.696$ \(\Q(\sqrt{17}) \) None \(7\) \(-3\) \(10\) \(25\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(4-\beta )q^{2}+(-2+\beta )q^{3}+(12-7\beta )q^{4}+\cdots\)
605.4.a.h 605.a 1.a $3$ $35.696$ 3.3.568.1 None \(-5\) \(-3\) \(-15\) \(15\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1}+\beta _{2})q^{2}+(-2-3\beta _{2})q^{3}+\cdots\)
605.4.a.i 605.a 1.a $4$ $35.696$ 4.4.883060.1 None \(-3\) \(0\) \(-20\) \(-6\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(\beta _{1}+\beta _{2}+\beta _{3})q^{3}+\cdots\)
605.4.a.j 605.a 1.a $4$ $35.696$ 4.4.1539480.1 None \(-1\) \(9\) \(20\) \(-9\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2-\beta _{2})q^{3}+(5+\beta _{1}+\beta _{3})q^{4}+\cdots\)
605.4.a.k 605.a 1.a $4$ $35.696$ 4.4.1236492.1 None \(0\) \(2\) \(-20\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{3})q^{3}+(9+\beta _{3})q^{4}+\cdots\)
605.4.a.l 605.a 1.a $4$ $35.696$ 4.4.883060.1 None \(3\) \(0\) \(-20\) \(6\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(\beta _{1}+\beta _{2}+\beta _{3})q^{3}+(2+\cdots)q^{4}+\cdots\)
605.4.a.m 605.a 1.a $6$ $35.696$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-3\) \(12\) \(30\) \(10\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(3-\beta _{1}+\beta _{4})q^{3}+\cdots\)
605.4.a.n 605.a 1.a $6$ $35.696$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(3\) \(12\) \(30\) \(-10\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(3-\beta _{1}+\beta _{4})q^{3}+(5+\cdots)q^{4}+\cdots\)
605.4.a.o 605.a 1.a $8$ $35.696$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-24\) \(40\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(\beta _{1}+\beta _{3})q^{2}+(-3-\beta _{4})q^{3}+(2+\beta _{4}+\cdots)q^{4}+\cdots\)
605.4.a.p 605.a 1.a $12$ $35.696$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-7\) \(-3\) \(60\) \(-97\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-\beta _{6}q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
605.4.a.q 605.a 1.a $12$ $35.696$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-1\) \(1\) \(-60\) \(-41\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+\beta _{4}q^{3}+(3+\beta _{2})q^{4}-5q^{5}+\cdots\)
605.4.a.r 605.a 1.a $12$ $35.696$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(-60\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(6+\beta _{2}-\beta _{3})q^{4}+\cdots\)
605.4.a.s 605.a 1.a $12$ $35.696$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(1\) \(1\) \(-60\) \(41\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(3+\beta _{2})q^{4}-5q^{5}+\cdots\)
605.4.a.t 605.a 1.a $12$ $35.696$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(7\) \(-3\) \(60\) \(97\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-\beta _{6}q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(605))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(605)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 2}\)