Properties

Label 363.4.a
Level $363$
Weight $4$
Character orbit 363.a
Rep. character $\chi_{363}(1,\cdot)$
Character field $\Q$
Dimension $54$
Newform subspaces $22$
Sturm bound $176$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 22 \)
Sturm bound: \(176\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(363))\).

Total New Old
Modular forms 144 54 90
Cusp forms 120 54 66
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.

\(3\)\(11\)FrickeDim
\(+\)\(+\)\(+\)\(14\)
\(+\)\(-\)\(-\)\(13\)
\(-\)\(+\)\(-\)\(10\)
\(-\)\(-\)\(+\)\(17\)
Plus space\(+\)\(31\)
Minus space\(-\)\(23\)

Trace form

\( 54 q + 4 q^{2} + 196 q^{4} + 16 q^{5} + 12 q^{6} + 32 q^{7} - 36 q^{8} + 486 q^{9} + O(q^{10}) \) \( 54 q + 4 q^{2} + 196 q^{4} + 16 q^{5} + 12 q^{6} + 32 q^{7} - 36 q^{8} + 486 q^{9} + 88 q^{10} + 24 q^{12} + 116 q^{13} + 132 q^{14} - 60 q^{15} + 804 q^{16} - 152 q^{17} + 36 q^{18} - 8 q^{19} + 556 q^{20} + 84 q^{21} - 168 q^{23} + 324 q^{24} + 1354 q^{25} - 572 q^{26} + 192 q^{28} + 120 q^{29} - 336 q^{30} - 180 q^{31} - 364 q^{32} - 48 q^{34} - 920 q^{35} + 1764 q^{36} + 352 q^{37} + 592 q^{38} + 336 q^{39} + 168 q^{40} + 760 q^{41} - 828 q^{42} - 200 q^{43} + 144 q^{45} + 464 q^{46} + 616 q^{47} + 816 q^{48} + 2266 q^{49} - 148 q^{50} + 624 q^{51} - 264 q^{52} + 808 q^{53} + 108 q^{54} - 300 q^{56} - 84 q^{57} - 668 q^{58} + 688 q^{59} + 600 q^{60} + 1180 q^{61} - 3032 q^{62} + 288 q^{63} + 3688 q^{64} + 880 q^{65} + 1792 q^{67} - 1000 q^{68} - 1320 q^{69} + 1740 q^{70} - 1144 q^{71} - 324 q^{72} - 724 q^{73} - 2256 q^{74} - 528 q^{75} - 2048 q^{76} - 456 q^{78} + 144 q^{79} + 6992 q^{80} + 4374 q^{81} + 3324 q^{82} - 264 q^{83} + 2880 q^{84} + 1840 q^{85} - 796 q^{86} - 1368 q^{87} + 880 q^{89} + 792 q^{90} - 5840 q^{91} - 2496 q^{92} - 1392 q^{93} - 5104 q^{94} + 168 q^{95} + 3084 q^{96} - 348 q^{97} + 8580 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(363))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 11
363.4.a.a 363.a 1.a $1$ $21.418$ \(\Q\) None 363.4.a.a \(-4\) \(-3\) \(-13\) \(-26\) $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-3q^{3}+8q^{4}-13q^{5}+12q^{6}+\cdots\)
363.4.a.b 363.a 1.a $1$ $21.418$ \(\Q\) None 363.4.a.b \(-3\) \(3\) \(-12\) \(-12\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}+3q^{3}+q^{4}-12q^{5}-9q^{6}+\cdots\)
363.4.a.c 363.a 1.a $1$ $21.418$ \(\Q\) None 363.4.a.c \(-1\) \(-3\) \(7\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}-7q^{4}+7q^{5}+3q^{6}+\cdots\)
363.4.a.d 363.a 1.a $1$ $21.418$ \(\Q\) None 33.4.a.b \(1\) \(-3\) \(-4\) \(26\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}-7q^{4}-4q^{5}-3q^{6}+\cdots\)
363.4.a.e 363.a 1.a $1$ $21.418$ \(\Q\) None 363.4.a.c \(1\) \(-3\) \(7\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}-7q^{4}+7q^{5}-3q^{6}+\cdots\)
363.4.a.f 363.a 1.a $1$ $21.418$ \(\Q\) None 363.4.a.b \(3\) \(3\) \(-12\) \(12\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}+3q^{3}+q^{4}-12q^{5}+9q^{6}+\cdots\)
363.4.a.g 363.a 1.a $1$ $21.418$ \(\Q\) None 363.4.a.a \(4\) \(-3\) \(-13\) \(26\) $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-3q^{3}+8q^{4}-13q^{5}-12q^{6}+\cdots\)
363.4.a.h 363.a 1.a $1$ $21.418$ \(\Q\) None 33.4.a.a \(5\) \(3\) \(-14\) \(32\) $-$ $-$ $\mathrm{SU}(2)$ \(q+5q^{2}+3q^{3}+17q^{4}-14q^{5}+15q^{6}+\cdots\)
363.4.a.i 363.a 1.a $2$ $21.418$ \(\Q(\sqrt{97}) \) None 33.4.a.c \(-1\) \(-6\) \(-14\) \(-24\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-3q^{3}+(2^{4}+\beta )q^{4}+(-6-2\beta )q^{5}+\cdots\)
363.4.a.j 363.a 1.a $2$ $21.418$ \(\Q(\sqrt{33}) \) None 33.4.a.d \(-1\) \(6\) \(16\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+3q^{3}+\beta q^{4}+(10-4\beta )q^{5}+\cdots\)
363.4.a.k 363.a 1.a $2$ $21.418$ \(\Q(\sqrt{5}) \) None 363.4.a.k \(0\) \(-6\) \(-4\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-3q^{3}-3q^{4}-2q^{5}+3\beta q^{6}+\cdots\)
363.4.a.l 363.a 1.a $2$ $21.418$ \(\Q(\sqrt{3}) \) None 363.4.a.l \(0\) \(-6\) \(18\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+3\beta q^{2}-3q^{3}+19q^{4}+9q^{5}-9\beta q^{6}+\cdots\)
363.4.a.m 363.a 1.a $2$ $21.418$ \(\Q(\sqrt{3}) \) None 363.4.a.m \(0\) \(6\) \(-6\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+3q^{3}-5q^{4}-3q^{5}+3\beta q^{6}+\cdots\)
363.4.a.n 363.a 1.a $2$ $21.418$ \(\Q(\sqrt{5}) \) None 33.4.e.a \(0\) \(6\) \(-13\) \(-44\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(2-4\beta )q^{2}+3q^{3}+12q^{4}+(-12+\cdots)q^{5}+\cdots\)
363.4.a.o 363.a 1.a $2$ $21.418$ \(\Q(\sqrt{5}) \) None 33.4.e.a \(0\) \(6\) \(-13\) \(44\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(2-4\beta )q^{2}+3q^{3}+12q^{4}+(-1+\cdots)q^{5}+\cdots\)
363.4.a.p 363.a 1.a $4$ $21.418$ 4.4.5225.1 None 33.4.e.b \(-3\) \(12\) \(12\) \(-11\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1}-\beta _{2})q^{2}+3q^{3}+(-2+\cdots)q^{4}+\cdots\)
363.4.a.q 363.a 1.a $4$ $21.418$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 363.4.a.q \(-1\) \(12\) \(14\) \(20\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(6+\beta _{1}+\beta _{3})q^{4}+\cdots\)
363.4.a.r 363.a 1.a $4$ $21.418$ \(\Q(\sqrt{3}, \sqrt{5})\) None 363.4.a.r \(0\) \(-12\) \(16\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}+\beta _{3})q^{2}-3q^{3}+2\beta _{2}q^{4}+\cdots\)
363.4.a.s 363.a 1.a $4$ $21.418$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 363.4.a.q \(1\) \(12\) \(14\) \(-20\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(6+\beta _{1}+\beta _{3})q^{4}+\cdots\)
363.4.a.t 363.a 1.a $4$ $21.418$ 4.4.5225.1 None 33.4.e.b \(3\) \(12\) \(12\) \(11\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1}+\beta _{2})q^{2}+3q^{3}+(-2+3\beta _{1}+\cdots)q^{4}+\cdots\)
363.4.a.u 363.a 1.a $6$ $21.418$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 33.4.e.c \(-5\) \(-18\) \(9\) \(-1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-3q^{3}+(3-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
363.4.a.v 363.a 1.a $6$ $21.418$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 33.4.e.c \(5\) \(-18\) \(9\) \(1\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-3q^{3}+(3-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(363))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(363)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 2}\)