Properties

Label 1008.6.a
Level $1008$
Weight $6$
Character orbit 1008.a
Rep. character $\chi_{1008}(1,\cdot)$
Character field $\Q$
Dimension $75$
Newform subspaces $51$
Sturm bound $1152$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1008.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 51 \)
Sturm bound: \(1152\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(1008))\).

Total New Old
Modular forms 984 75 909
Cusp forms 936 75 861
Eisenstein series 48 0 48

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

\(2\)\(3\)\(7\)FrickeDim
\(+\)\(+\)\(+\)$+$\(8\)
\(+\)\(+\)\(-\)$-$\(8\)
\(+\)\(-\)\(+\)$-$\(11\)
\(+\)\(-\)\(-\)$+$\(11\)
\(-\)\(+\)\(+\)$-$\(8\)
\(-\)\(+\)\(-\)$+$\(6\)
\(-\)\(-\)\(+\)$+$\(11\)
\(-\)\(-\)\(-\)$-$\(12\)
Plus space\(+\)\(36\)
Minus space\(-\)\(39\)

Trace form

\( 75 q + 38 q^{5} - 49 q^{7} + O(q^{10}) \) \( 75 q + 38 q^{5} - 49 q^{7} - 424 q^{11} - 122 q^{13} - 802 q^{17} - 2360 q^{19} + 2756 q^{23} + 45317 q^{25} - 2246 q^{29} - 7160 q^{31} - 7350 q^{35} - 7206 q^{37} + 4566 q^{41} + 3732 q^{43} - 29184 q^{47} + 180075 q^{49} - 32486 q^{53} - 70408 q^{55} - 60264 q^{59} - 38922 q^{61} + 31460 q^{65} + 126572 q^{67} + 119884 q^{71} - 66218 q^{73} - 3724 q^{77} - 49720 q^{79} - 122632 q^{83} + 131684 q^{85} - 37186 q^{89} + 49686 q^{91} + 303432 q^{95} + 154662 q^{97} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(1008))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 7
1008.6.a.a 1008.a 1.a $1$ $161.667$ \(\Q\) None 21.6.a.c \(0\) \(0\) \(-94\) \(49\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-94q^{5}+7^{2}q^{7}+52q^{11}-770q^{13}+\cdots\)
1008.6.a.b 1008.a 1.a $1$ $161.667$ \(\Q\) None 14.6.a.a \(0\) \(0\) \(-84\) \(-49\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-84q^{5}-7^{2}q^{7}-336q^{11}+584q^{13}+\cdots\)
1008.6.a.c 1008.a 1.a $1$ $161.667$ \(\Q\) None 21.6.a.a \(0\) \(0\) \(-78\) \(-49\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-78q^{5}-7^{2}q^{7}+444q^{11}-442q^{13}+\cdots\)
1008.6.a.d 1008.a 1.a $1$ $161.667$ \(\Q\) None 42.6.a.e \(0\) \(0\) \(-76\) \(49\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-76q^{5}+7^{2}q^{7}+650q^{11}+762q^{13}+\cdots\)
1008.6.a.e 1008.a 1.a $1$ $161.667$ \(\Q\) None 168.6.a.c \(0\) \(0\) \(-74\) \(-49\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-74q^{5}-7^{2}q^{7}+6^{3}q^{11}-186q^{13}+\cdots\)
1008.6.a.f 1008.a 1.a $1$ $161.667$ \(\Q\) None 126.6.a.e \(0\) \(0\) \(-54\) \(-49\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-54q^{5}-7^{2}q^{7}+594q^{11}+26q^{13}+\cdots\)
1008.6.a.g 1008.a 1.a $1$ $161.667$ \(\Q\) None 42.6.a.b \(0\) \(0\) \(-44\) \(49\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-44q^{5}+7^{2}q^{7}-470q^{11}-1158q^{13}+\cdots\)
1008.6.a.h 1008.a 1.a $1$ $161.667$ \(\Q\) None 56.6.a.b \(0\) \(0\) \(-32\) \(-49\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{5}q^{5}-7^{2}q^{7}-624q^{11}-708q^{13}+\cdots\)
1008.6.a.i 1008.a 1.a $1$ $161.667$ \(\Q\) None 126.6.a.d \(0\) \(0\) \(-26\) \(49\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-26q^{5}+7^{2}q^{7}-470q^{11}+642q^{13}+\cdots\)
1008.6.a.j 1008.a 1.a $1$ $161.667$ \(\Q\) None 42.6.a.d \(0\) \(0\) \(-26\) \(49\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-26q^{5}+7^{2}q^{7}+664q^{11}+318q^{13}+\cdots\)
1008.6.a.k 1008.a 1.a $1$ $161.667$ \(\Q\) None 42.6.a.f \(0\) \(0\) \(-24\) \(-49\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-24q^{5}-7^{2}q^{7}+66q^{11}+98q^{13}+\cdots\)
1008.6.a.l 1008.a 1.a $1$ $161.667$ \(\Q\) None 28.6.a.b \(0\) \(0\) \(-16\) \(49\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{5}+7^{2}q^{7}+8q^{11}+684q^{13}+\cdots\)
1008.6.a.m 1008.a 1.a $1$ $161.667$ \(\Q\) None 168.6.a.f \(0\) \(0\) \(-14\) \(49\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-14q^{5}+7^{2}q^{7}-700q^{11}+158q^{13}+\cdots\)
1008.6.a.n 1008.a 1.a $1$ $161.667$ \(\Q\) None 14.6.a.b \(0\) \(0\) \(-10\) \(49\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-10q^{5}+7^{2}q^{7}-340q^{11}-294q^{13}+\cdots\)
1008.6.a.o 1008.a 1.a $1$ $161.667$ \(\Q\) None 84.6.a.a \(0\) \(0\) \(-6\) \(-49\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-6q^{5}-7^{2}q^{7}-108q^{11}-346q^{13}+\cdots\)
1008.6.a.p 1008.a 1.a $1$ $161.667$ \(\Q\) None 56.6.a.a \(0\) \(0\) \(-4\) \(-49\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{5}-7^{2}q^{7}-240q^{11}-744q^{13}+\cdots\)
1008.6.a.q 1008.a 1.a $1$ $161.667$ \(\Q\) None 168.6.a.b \(0\) \(0\) \(-4\) \(49\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{5}+7^{2}q^{7}+370q^{11}+122q^{13}+\cdots\)
1008.6.a.r 1008.a 1.a $1$ $161.667$ \(\Q\) None 126.6.a.d \(0\) \(0\) \(26\) \(49\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+26q^{5}+7^{2}q^{7}+470q^{11}+642q^{13}+\cdots\)
1008.6.a.s 1008.a 1.a $1$ $161.667$ \(\Q\) None 168.6.a.a \(0\) \(0\) \(34\) \(-49\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+34q^{5}-7^{2}q^{7}-756q^{11}+678q^{13}+\cdots\)
1008.6.a.t 1008.a 1.a $1$ $161.667$ \(\Q\) None 21.6.a.b \(0\) \(0\) \(34\) \(49\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+34q^{5}+7^{2}q^{7}-340q^{11}+454q^{13}+\cdots\)
1008.6.a.u 1008.a 1.a $1$ $161.667$ \(\Q\) None 84.6.a.b \(0\) \(0\) \(34\) \(49\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+34q^{5}+7^{2}q^{7}-332q^{11}-1026q^{13}+\cdots\)
1008.6.a.v 1008.a 1.a $1$ $161.667$ \(\Q\) None 168.6.a.e \(0\) \(0\) \(38\) \(49\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+38q^{5}+7^{2}q^{7}+600q^{11}-674q^{13}+\cdots\)
1008.6.a.w 1008.a 1.a $1$ $161.667$ \(\Q\) None 126.6.a.e \(0\) \(0\) \(54\) \(-49\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+54q^{5}-7^{2}q^{7}-594q^{11}+26q^{13}+\cdots\)
1008.6.a.x 1008.a 1.a $1$ $161.667$ \(\Q\) None 42.6.a.a \(0\) \(0\) \(54\) \(-49\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+54q^{5}-7^{2}q^{7}+6^{3}q^{11}+998q^{13}+\cdots\)
1008.6.a.y 1008.a 1.a $1$ $161.667$ \(\Q\) None 7.6.a.a \(0\) \(0\) \(56\) \(49\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+56q^{5}+7^{2}q^{7}+232q^{11}-140q^{13}+\cdots\)
1008.6.a.z 1008.a 1.a $1$ $161.667$ \(\Q\) None 168.6.a.d \(0\) \(0\) \(64\) \(-49\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{6}q^{5}-7^{2}q^{7}-54q^{11}+738q^{13}+\cdots\)
1008.6.a.ba 1008.a 1.a $1$ $161.667$ \(\Q\) None 42.6.a.c \(0\) \(0\) \(72\) \(-49\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+72q^{5}-7^{2}q^{7}-414q^{11}-1054q^{13}+\cdots\)
1008.6.a.bb 1008.a 1.a $1$ $161.667$ \(\Q\) None 28.6.a.a \(0\) \(0\) \(96\) \(-49\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+96q^{5}-7^{2}q^{7}-720q^{11}+572q^{13}+\cdots\)
1008.6.a.bc 1008.a 1.a $1$ $161.667$ \(\Q\) None 21.6.a.d \(0\) \(0\) \(106\) \(49\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+106q^{5}+7^{2}q^{7}+92q^{11}+670q^{13}+\cdots\)
1008.6.a.bd 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{345}) \) None 56.6.a.e \(0\) \(0\) \(-82\) \(98\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-41-\beta )q^{5}+7^{2}q^{7}+(170+2\beta )q^{11}+\cdots\)
1008.6.a.be 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{429}) \) None 504.6.a.j \(0\) \(0\) \(-80\) \(-98\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-40-\beta )q^{5}-7^{2}q^{7}+(6^{3}+3\beta )q^{11}+\cdots\)
1008.6.a.bf 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{505}) \) None 84.6.a.d \(0\) \(0\) \(-78\) \(-98\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-39-\beta )q^{5}-7^{2}q^{7}+(87-5\beta )q^{11}+\cdots\)
1008.6.a.bg 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{106}) \) None 504.6.a.k \(0\) \(0\) \(-76\) \(-98\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-38+\beta )q^{5}-7^{2}q^{7}+(282+5\beta )q^{11}+\cdots\)
1008.6.a.bh 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{37}) \) None 504.6.a.l \(0\) \(0\) \(-48\) \(98\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-24-5\beta )q^{5}+7^{2}q^{7}+(184+37\beta )q^{11}+\cdots\)
1008.6.a.bi 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{193}) \) None 56.6.a.d \(0\) \(0\) \(-42\) \(98\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-21-5\beta )q^{5}+7^{2}q^{7}+(-358+\cdots)q^{11}+\cdots\)
1008.6.a.bj 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{114}) \) None 504.6.a.n \(0\) \(0\) \(-28\) \(98\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-14+\beta )q^{5}+7^{2}q^{7}+(-298+7\beta )q^{11}+\cdots\)
1008.6.a.bk 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{91}) \) None 252.6.a.g \(0\) \(0\) \(0\) \(-98\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{5}-7^{2}q^{7}+9\beta q^{11}-670q^{13}+\cdots\)
1008.6.a.bl 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{7}) \) None 63.6.a.g \(0\) \(0\) \(0\) \(98\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+7\beta q^{5}+7^{2}q^{7}-43\beta q^{11}-518q^{13}+\cdots\)
1008.6.a.bm 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{1099}) \) None 252.6.a.f \(0\) \(0\) \(0\) \(98\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+7^{2}q^{7}-5\beta q^{11}-54q^{13}+\cdots\)
1008.6.a.bn 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{1129}) \) None 168.6.a.j \(0\) \(0\) \(0\) \(98\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{5}+7^{2}q^{7}+(50-\beta )q^{11}+(270+\cdots)q^{13}+\cdots\)
1008.6.a.bo 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{5569}) \) None 84.6.a.c \(0\) \(0\) \(6\) \(98\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{5}+7^{2}q^{7}+(-45+7\beta )q^{11}+\cdots\)
1008.6.a.bp 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{4281}) \) None 168.6.a.i \(0\) \(0\) \(10\) \(-98\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(5-\beta )q^{5}-7^{2}q^{7}+(3^{3}+3\beta )q^{11}+\cdots\)
1008.6.a.bq 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{57}) \) None 7.6.a.b \(0\) \(0\) \(18\) \(-98\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(9+5\beta )q^{5}-7^{2}q^{7}+(198+62\beta )q^{11}+\cdots\)
1008.6.a.br 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{114}) \) None 504.6.a.n \(0\) \(0\) \(28\) \(98\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(14+\beta )q^{5}+7^{2}q^{7}+(298+7\beta )q^{11}+\cdots\)
1008.6.a.bs 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{37}) \) None 504.6.a.l \(0\) \(0\) \(48\) \(98\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(24+5\beta )q^{5}+7^{2}q^{7}+(-184-37\beta )q^{11}+\cdots\)
1008.6.a.bt 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{177}) \) None 56.6.a.c \(0\) \(0\) \(62\) \(-98\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(31-5\beta )q^{5}-7^{2}q^{7}+(486+6\beta )q^{11}+\cdots\)
1008.6.a.bu 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{249}) \) None 168.6.a.h \(0\) \(0\) \(64\) \(-98\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(2^{5}-\beta )q^{5}-7^{2}q^{7}+(270-15\beta )q^{11}+\cdots\)
1008.6.a.bv 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{106}) \) None 504.6.a.k \(0\) \(0\) \(76\) \(-98\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(38+\beta )q^{5}-7^{2}q^{7}+(-282+5\beta )q^{11}+\cdots\)
1008.6.a.bw 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{193}) \) None 168.6.a.g \(0\) \(0\) \(78\) \(98\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(39-5\beta )q^{5}+7^{2}q^{7}+(-185+37\beta )q^{11}+\cdots\)
1008.6.a.bx 1008.a 1.a $2$ $161.667$ \(\Q(\sqrt{429}) \) None 504.6.a.j \(0\) \(0\) \(80\) \(-98\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(40-\beta )q^{5}-7^{2}q^{7}+(-6^{3}+3\beta )q^{11}+\cdots\)
1008.6.a.by 1008.a 1.a $4$ $161.667$ 4.4.358541904.1 None 63.6.a.h \(0\) \(0\) \(0\) \(-196\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{5}-7^{2}q^{7}+(3\beta _{1}-2\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(1008))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(1008)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 20}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 18}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 16}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 15}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(504))\)\(^{\oplus 2}\)