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Properties

Label 361.4.a
Level $361$
Weight $4$
Character orbit 361.a
Rep. character $\chi_{361}(1,\cdot)$
Character field $\Q$
Dimension $77$
Newform subspaces $15$
Sturm bound $126$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 361.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(126\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(361))\).

Total New Old
Modular forms 105 94 11
Cusp forms 85 77 8
Eisenstein series 20 17 3

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

\(19\)Dim
\(+\)\(41\)
\(-\)\(36\)

Trace form

\( 77 q + 4 q^{3} + 266 q^{4} - 6 q^{5} + 50 q^{6} + 26 q^{7} - 48 q^{8} + 521 q^{9} + O(q^{10}) \) \( 77 q + 4 q^{3} + 266 q^{4} - 6 q^{5} + 50 q^{6} + 26 q^{7} - 48 q^{8} + 521 q^{9} + 52 q^{10} + 52 q^{11} + 120 q^{12} - 76 q^{13} - 4 q^{14} - 200 q^{15} + 902 q^{16} + 96 q^{17} - 144 q^{18} - 170 q^{20} + 80 q^{21} - 280 q^{22} - 110 q^{23} + 410 q^{24} + 1199 q^{25} - 402 q^{26} + 232 q^{27} + 514 q^{28} - 128 q^{29} + 516 q^{30} + 84 q^{31} - 624 q^{32} - 140 q^{33} - 608 q^{34} + 772 q^{35} + 328 q^{36} + 540 q^{37} + 198 q^{39} + 324 q^{40} - 1196 q^{41} - 1850 q^{42} + 556 q^{43} + 464 q^{44} - 434 q^{45} + 788 q^{46} + 158 q^{47} + 1056 q^{48} + 1523 q^{49} + 1696 q^{50} - 652 q^{51} - 708 q^{52} - 1252 q^{53} + 1978 q^{54} - 124 q^{55} + 684 q^{56} - 1628 q^{58} - 460 q^{59} - 1420 q^{60} - 646 q^{61} + 1238 q^{62} + 1638 q^{63} + 548 q^{64} + 372 q^{65} + 1302 q^{66} + 1168 q^{67} + 1876 q^{68} + 108 q^{69} - 2436 q^{70} - 600 q^{71} - 2784 q^{72} - 776 q^{73} + 290 q^{74} - 1936 q^{75} - 30 q^{77} + 1652 q^{78} - 348 q^{79} + 996 q^{80} + 1581 q^{81} - 86 q^{82} + 928 q^{83} - 852 q^{84} + 56 q^{85} - 3180 q^{86} + 832 q^{87} + 792 q^{88} + 340 q^{89} + 1960 q^{90} + 336 q^{91} + 1236 q^{92} - 584 q^{93} - 408 q^{94} - 584 q^{96} + 1692 q^{97} + 1784 q^{98} - 1056 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(361))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 19
361.4.a.a 361.a 1.a $1$ $21.300$ \(\Q\) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(14\) \(-36\) $+$ $N(\mathrm{U}(1))$ \(q-8q^{4}+14q^{5}-6^{2}q^{7}-3^{3}q^{9}+40q^{11}+\cdots\)
361.4.a.b 361.a 1.a $1$ $21.300$ \(\Q\) None \(3\) \(5\) \(-12\) \(11\) $-$ $\mathrm{SU}(2)$ \(q+3q^{2}+5q^{3}+q^{4}-12q^{5}+15q^{6}+\cdots\)
361.4.a.c 361.a 1.a $2$ $21.300$ \(\Q(\sqrt{5}) \) None \(-4\) \(0\) \(-19\) \(-18\) $-$ $\mathrm{SU}(2)$ \(q+(-1-2\beta )q^{2}+(2-4\beta )q^{3}+(-3+\cdots)q^{4}+\cdots\)
361.4.a.d 361.a 1.a $2$ $21.300$ \(\Q(\sqrt{55}) \) None \(-2\) \(0\) \(-14\) \(-14\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}-7q^{4}+(-7+\beta )q^{5}+\cdots\)
361.4.a.e 361.a 1.a $2$ $21.300$ \(\Q(\sqrt{73}) \) None \(-1\) \(-2\) \(19\) \(20\) $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+(10+\beta )q^{4}+(9+\beta )q^{5}+\cdots\)
361.4.a.f 361.a 1.a $2$ $21.300$ \(\Q(\sqrt{73}) \) None \(1\) \(2\) \(19\) \(20\) $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+(10+\beta )q^{4}+(9+\beta )q^{5}+\cdots\)
361.4.a.g 361.a 1.a $2$ $21.300$ \(\Q(\sqrt{55}) \) None \(2\) \(0\) \(-14\) \(-14\) $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}-7q^{4}+(-7-\beta )q^{5}+\cdots\)
361.4.a.h 361.a 1.a $2$ $21.300$ \(\Q(\sqrt{5}) \) None \(4\) \(0\) \(-19\) \(-18\) $-$ $\mathrm{SU}(2)$ \(q+(1+2\beta )q^{2}+(-2+4\beta )q^{3}+(-3+\cdots)q^{4}+\cdots\)
361.4.a.i 361.a 1.a $3$ $21.300$ 3.3.3144.1 None \(-3\) \(-1\) \(14\) \(-35\) $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1}+\beta _{2})q^{2}+(\beta _{1}-2\beta _{2})q^{3}+\cdots\)
361.4.a.j 361.a 1.a $4$ $21.300$ 4.4.205768.1 None \(0\) \(0\) \(-30\) \(40\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(6+\beta _{2})q^{4}+(-5+\cdots)q^{5}+\cdots\)
361.4.a.k 361.a 1.a $6$ $21.300$ 6.6.3365875125.1 None \(-1\) \(-2\) \(-17\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+(-\beta _{2}+\beta _{3})q^{2}+(1-2\beta _{2}-\beta _{3}-\beta _{5})q^{3}+\cdots\)
361.4.a.l 361.a 1.a $6$ $21.300$ 6.6.3365875125.1 None \(1\) \(2\) \(-17\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+(\beta _{2}-\beta _{3})q^{2}+(-1+2\beta _{2}+\beta _{3}+\beta _{5})q^{3}+\cdots\)
361.4.a.m 361.a 1.a $12$ $21.300$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-6\) \(-18\) \(12\) \(-3\) $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-2+\beta _{4})q^{3}+(4+\cdots)q^{4}+\cdots\)
361.4.a.n 361.a 1.a $12$ $21.300$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(6\) \(18\) \(12\) \(-3\) $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(2-\beta _{4})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
361.4.a.o 361.a 1.a $20$ $21.300$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(0\) \(46\) \(76\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-\beta _{5}q^{3}+(5+\beta _{2})q^{4}+(2-\beta _{13}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(361))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(361)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)