Properties

Label 1386.4.a
Level $1386$
Weight $4$
Character orbit 1386.a
Rep. character $\chi_{1386}(1,\cdot)$
Character field $\Q$
Dimension $74$
Newform subspaces $38$
Sturm bound $1152$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 38 \)
Sturm bound: \(1152\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1386))\).

Total New Old
Modular forms 880 74 806
Cusp forms 848 74 774
Eisenstein series 32 0 32

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

\(2\)\(3\)\(7\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(5\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(6\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(6\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(5\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(5\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(8\)
Plus space\(+\)\(43\)
Minus space\(-\)\(31\)

Trace form

\( 74 q + 4 q^{2} + 296 q^{4} + 4 q^{5} + 16 q^{8} + O(q^{10}) \) \( 74 q + 4 q^{2} + 296 q^{4} + 4 q^{5} + 16 q^{8} + 40 q^{10} + 22 q^{11} + 52 q^{13} + 56 q^{14} + 1184 q^{16} - 60 q^{17} - 80 q^{19} + 16 q^{20} - 44 q^{22} - 572 q^{23} + 1434 q^{25} - 112 q^{26} + 140 q^{29} - 52 q^{31} + 64 q^{32} + 152 q^{34} + 112 q^{35} + 496 q^{37} - 168 q^{38} + 160 q^{40} + 852 q^{41} + 344 q^{43} + 88 q^{44} + 160 q^{46} + 1760 q^{47} + 3626 q^{49} - 52 q^{50} + 208 q^{52} + 2220 q^{53} + 792 q^{55} + 224 q^{56} + 296 q^{58} - 1856 q^{59} + 236 q^{61} + 560 q^{62} + 4736 q^{64} + 304 q^{65} + 1500 q^{67} - 240 q^{68} - 280 q^{70} + 652 q^{71} + 5148 q^{73} + 728 q^{74} - 320 q^{76} + 308 q^{77} + 4264 q^{79} + 64 q^{80} - 488 q^{82} + 1072 q^{83} - 1648 q^{85} + 3504 q^{86} - 176 q^{88} + 768 q^{89} - 1204 q^{91} - 2288 q^{92} + 2320 q^{94} - 6032 q^{95} + 1480 q^{97} + 196 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1386))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 7 11
1386.4.a.a $1$ $81.777$ \(\Q\) None \(-2\) \(0\) \(-18\) \(7\) $+$ $-$ $-$ $-$ \(q-2q^{2}+4q^{4}-18q^{5}+7q^{7}-8q^{8}+\cdots\)
1386.4.a.b $1$ $81.777$ \(\Q\) None \(-2\) \(0\) \(-3\) \(7\) $+$ $-$ $-$ $-$ \(q-2q^{2}+4q^{4}-3q^{5}+7q^{7}-8q^{8}+\cdots\)
1386.4.a.c $1$ $81.777$ \(\Q\) None \(-2\) \(0\) \(-3\) \(7\) $+$ $-$ $-$ $-$ \(q-2q^{2}+4q^{4}-3q^{5}+7q^{7}-8q^{8}+\cdots\)
1386.4.a.d $1$ $81.777$ \(\Q\) None \(-2\) \(0\) \(-1\) \(-7\) $+$ $-$ $+$ $-$ \(q-2q^{2}+4q^{4}-q^{5}-7q^{7}-8q^{8}+\cdots\)
1386.4.a.e $1$ $81.777$ \(\Q\) None \(-2\) \(0\) \(4\) \(-7\) $+$ $-$ $+$ $-$ \(q-2q^{2}+4q^{4}+4q^{5}-7q^{7}-8q^{8}+\cdots\)
1386.4.a.f $1$ $81.777$ \(\Q\) None \(-2\) \(0\) \(13\) \(-7\) $+$ $-$ $+$ $+$ \(q-2q^{2}+4q^{4}+13q^{5}-7q^{7}-8q^{8}+\cdots\)
1386.4.a.g $1$ $81.777$ \(\Q\) None \(-2\) \(0\) \(14\) \(7\) $+$ $-$ $-$ $-$ \(q-2q^{2}+4q^{4}+14q^{5}+7q^{7}-8q^{8}+\cdots\)
1386.4.a.h $1$ $81.777$ \(\Q\) None \(-2\) \(0\) \(17\) \(7\) $+$ $-$ $-$ $-$ \(q-2q^{2}+4q^{4}+17q^{5}+7q^{7}-8q^{8}+\cdots\)
1386.4.a.i $1$ $81.777$ \(\Q\) None \(2\) \(0\) \(-11\) \(-7\) $-$ $-$ $+$ $+$ \(q+2q^{2}+4q^{4}-11q^{5}-7q^{7}+8q^{8}+\cdots\)
1386.4.a.j $1$ $81.777$ \(\Q\) None \(2\) \(0\) \(-2\) \(-7\) $-$ $-$ $+$ $+$ \(q+2q^{2}+4q^{4}-2q^{5}-7q^{7}+8q^{8}+\cdots\)
1386.4.a.k $1$ $81.777$ \(\Q\) None \(2\) \(0\) \(1\) \(-7\) $-$ $-$ $+$ $-$ \(q+2q^{2}+4q^{4}+q^{5}-7q^{7}+8q^{8}+\cdots\)
1386.4.a.l $1$ $81.777$ \(\Q\) None \(2\) \(0\) \(7\) \(7\) $-$ $-$ $-$ $+$ \(q+2q^{2}+4q^{4}+7q^{5}+7q^{7}+8q^{8}+\cdots\)
1386.4.a.m $1$ $81.777$ \(\Q\) None \(2\) \(0\) \(14\) \(-7\) $-$ $-$ $+$ $+$ \(q+2q^{2}+4q^{4}+14q^{5}-7q^{7}+8q^{8}+\cdots\)
1386.4.a.n $1$ $81.777$ \(\Q\) None \(2\) \(0\) \(21\) \(7\) $-$ $-$ $-$ $-$ \(q+2q^{2}+4q^{4}+21q^{5}+7q^{7}+8q^{8}+\cdots\)
1386.4.a.o $2$ $81.777$ \(\Q(\sqrt{193}) \) None \(-4\) \(0\) \(-19\) \(14\) $+$ $+$ $-$ $-$ \(q-2q^{2}+4q^{4}+(-9-\beta )q^{5}+7q^{7}+\cdots\)
1386.4.a.p $2$ $81.777$ \(\Q(\sqrt{2}) \) None \(-4\) \(0\) \(2\) \(14\) $+$ $+$ $-$ $+$ \(q-2q^{2}+4q^{4}+(1+2\beta )q^{5}+7q^{7}+\cdots\)
1386.4.a.q $2$ $81.777$ \(\Q(\sqrt{817}) \) None \(-4\) \(0\) \(3\) \(-14\) $+$ $-$ $+$ $+$ \(q-2q^{2}+4q^{4}+(1+\beta )q^{5}-7q^{7}-8q^{8}+\cdots\)
1386.4.a.r $2$ $81.777$ \(\Q(\sqrt{793}) \) None \(-4\) \(0\) \(3\) \(-14\) $+$ $-$ $+$ $-$ \(q-2q^{2}+4q^{4}+(1+\beta )q^{5}-7q^{7}-8q^{8}+\cdots\)
1386.4.a.s $2$ $81.777$ \(\Q(\sqrt{57}) \) None \(-4\) \(0\) \(17\) \(-14\) $+$ $-$ $+$ $-$ \(q-2q^{2}+4q^{4}+(10-3\beta )q^{5}-7q^{7}+\cdots\)
1386.4.a.t $2$ $81.777$ \(\Q(\sqrt{89}) \) None \(-4\) \(0\) \(17\) \(14\) $+$ $-$ $-$ $+$ \(q-2q^{2}+4q^{4}+(9-\beta )q^{5}+7q^{7}-8q^{8}+\cdots\)
1386.4.a.u $2$ $81.777$ \(\Q(\sqrt{37}) \) None \(4\) \(0\) \(-26\) \(14\) $-$ $-$ $-$ $+$ \(q+2q^{2}+4q^{4}+(-13-\beta )q^{5}+7q^{7}+\cdots\)
1386.4.a.v $2$ $81.777$ \(\Q(\sqrt{113}) \) None \(4\) \(0\) \(-14\) \(14\) $-$ $-$ $-$ $-$ \(q+2q^{2}+4q^{4}+(-7-\beta )q^{5}+7q^{7}+\cdots\)
1386.4.a.w $2$ $81.777$ \(\Q(\sqrt{217}) \) None \(4\) \(0\) \(-7\) \(-14\) $-$ $-$ $+$ $-$ \(q+2q^{2}+4q^{4}+(-3-\beta )q^{5}-7q^{7}+\cdots\)
1386.4.a.x $2$ $81.777$ \(\Q(\sqrt{697}) \) None \(4\) \(0\) \(-7\) \(-14\) $-$ $-$ $+$ $-$ \(q+2q^{2}+4q^{4}+(-3-\beta )q^{5}-7q^{7}+\cdots\)
1386.4.a.y $2$ $81.777$ \(\Q(\sqrt{177}) \) None \(4\) \(0\) \(-3\) \(14\) $-$ $-$ $-$ $+$ \(q+2q^{2}+4q^{4}+(-1-\beta )q^{5}+7q^{7}+\cdots\)
1386.4.a.z $2$ $81.777$ \(\Q(\sqrt{2}) \) None \(4\) \(0\) \(-2\) \(14\) $-$ $+$ $-$ $-$ \(q+2q^{2}+4q^{4}+(-1+2\beta )q^{5}+7q^{7}+\cdots\)
1386.4.a.ba $2$ $81.777$ \(\Q(\sqrt{137}) \) None \(4\) \(0\) \(7\) \(14\) $-$ $-$ $-$ $-$ \(q+2q^{2}+4q^{4}+(4-\beta )q^{5}+7q^{7}+8q^{8}+\cdots\)
1386.4.a.bb $2$ $81.777$ \(\Q(\sqrt{193}) \) None \(4\) \(0\) \(19\) \(14\) $-$ $+$ $-$ $+$ \(q+2q^{2}+4q^{4}+(10-\beta )q^{5}+7q^{7}+\cdots\)
1386.4.a.bc $3$ $81.777$ 3.3.7636.1 None \(-6\) \(0\) \(-26\) \(-21\) $+$ $-$ $+$ $+$ \(q-2q^{2}+4q^{4}+(-9+\beta _{2})q^{5}-7q^{7}+\cdots\)
1386.4.a.bd $3$ $81.777$ 3.3.843032.1 None \(-6\) \(0\) \(-13\) \(21\) $+$ $-$ $-$ $+$ \(q-2q^{2}+4q^{4}+(-4-\beta _{1})q^{5}+7q^{7}+\cdots\)
1386.4.a.be $3$ $81.777$ 3.3.21324.1 None \(-6\) \(0\) \(-8\) \(-21\) $+$ $+$ $+$ $-$ \(q-2q^{2}+4q^{4}+(-3+\beta _{2})q^{5}-7q^{7}+\cdots\)
1386.4.a.bf $3$ $81.777$ 3.3.195128.1 None \(-6\) \(0\) \(11\) \(21\) $+$ $+$ $-$ $-$ \(q-2q^{2}+4q^{4}+(4+\beta _{2})q^{5}+7q^{7}+\cdots\)
1386.4.a.bg $3$ $81.777$ 3.3.195128.1 None \(6\) \(0\) \(-11\) \(21\) $-$ $+$ $-$ $+$ \(q+2q^{2}+4q^{4}+(-4-\beta _{2})q^{5}+7q^{7}+\cdots\)
1386.4.a.bh $3$ $81.777$ 3.3.1028796.1 None \(6\) \(0\) \(-7\) \(-21\) $-$ $-$ $+$ $+$ \(q+2q^{2}+4q^{4}+(-2-\beta _{1})q^{5}-7q^{7}+\cdots\)
1386.4.a.bi $3$ $81.777$ 3.3.768425.1 None \(6\) \(0\) \(7\) \(21\) $-$ $-$ $-$ $-$ \(q+2q^{2}+4q^{4}+(2-\beta _{1})q^{5}+7q^{7}+\cdots\)
1386.4.a.bj $3$ $81.777$ 3.3.21324.1 None \(6\) \(0\) \(8\) \(-21\) $-$ $+$ $+$ $+$ \(q+2q^{2}+4q^{4}+(3-\beta _{2})q^{5}-7q^{7}+\cdots\)
1386.4.a.bk $4$ $81.777$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(0\) \(-18\) \(-28\) $+$ $+$ $+$ $+$ \(q-2q^{2}+4q^{4}+(-4+\beta _{2})q^{5}-7q^{7}+\cdots\)
1386.4.a.bl $4$ $81.777$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(0\) \(18\) \(-28\) $-$ $+$ $+$ $-$ \(q+2q^{2}+4q^{4}+(4-\beta _{2})q^{5}-7q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1386))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1386)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(231))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(462))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(693))\)\(^{\oplus 2}\)