Properties

Label 576.8.a
Level $576$
Weight $8$
Character orbit 576.a
Rep. character $\chi_{576}(1,\cdot)$
Character field $\Q$
Dimension $69$
Newform subspaces $46$
Sturm bound $768$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 576.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 46 \)
Sturm bound: \(768\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(576))\).

Total New Old
Modular forms 696 71 625
Cusp forms 648 69 579
Eisenstein series 48 2 46

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

\(2\)\(3\)FrickeDim.
\(+\)\(+\)\(+\)\(15\)
\(+\)\(-\)\(-\)\(20\)
\(-\)\(+\)\(-\)\(13\)
\(-\)\(-\)\(+\)\(21\)
Plus space\(+\)\(36\)
Minus space\(-\)\(33\)

Trace form

\( 69q - 2q^{5} + O(q^{10}) \) \( 69q - 2q^{5} + 7066q^{13} + 2910q^{17} + 1031747q^{25} - 51690q^{29} - 218446q^{37} - 1042490q^{41} + 7411885q^{49} + 2958174q^{53} + 1972458q^{61} - 177060q^{65} - 1342654q^{73} + 83680q^{77} - 17993692q^{85} - 10004970q^{89} + 13021994q^{97} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(576))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3
576.8.a.a \(1\) \(179.934\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-556\) \(0\) \(-\) \(+\) \(q-556q^{5}-8898q^{13}+5816q^{17}+\cdots\)
576.8.a.b \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(-530\) \(-120\) \(-\) \(-\) \(q-530q^{5}-120q^{7}+7196q^{11}+9626q^{13}+\cdots\)
576.8.a.c \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(-530\) \(120\) \(+\) \(-\) \(q-530q^{5}+120q^{7}-7196q^{11}+9626q^{13}+\cdots\)
576.8.a.d \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(-378\) \(-832\) \(+\) \(-\) \(q-378q^{5}-832q^{7}-2484q^{11}-14870q^{13}+\cdots\)
576.8.a.e \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(-378\) \(832\) \(-\) \(-\) \(q-378q^{5}+832q^{7}+2484q^{11}-14870q^{13}+\cdots\)
576.8.a.f \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(-210\) \(-1016\) \(-\) \(-\) \(q-210q^{5}-1016q^{7}-1092q^{11}+\cdots\)
576.8.a.g \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(-210\) \(1016\) \(+\) \(-\) \(q-210q^{5}+1016q^{7}+1092q^{11}+\cdots\)
576.8.a.h \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(-114\) \(-1576\) \(+\) \(-\) \(q-114q^{5}-1576q^{7}+7332q^{11}+\cdots\)
576.8.a.i \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(-114\) \(1576\) \(-\) \(-\) \(q-114q^{5}+1576q^{7}-7332q^{11}+\cdots\)
576.8.a.j \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(-82\) \(-456\) \(+\) \(-\) \(q-82q^{5}-456q^{7}-2524q^{11}+10778q^{13}+\cdots\)
576.8.a.k \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(-82\) \(456\) \(-\) \(-\) \(q-82q^{5}+456q^{7}+2524q^{11}+10778q^{13}+\cdots\)
576.8.a.l \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(-70\) \(-92\) \(+\) \(-\) \(q-70q^{5}-92q^{7}-3124q^{11}-1174q^{13}+\cdots\)
576.8.a.m \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(-70\) \(92\) \(+\) \(-\) \(q-70q^{5}+92q^{7}+3124q^{11}-1174q^{13}+\cdots\)
576.8.a.n \(1\) \(179.934\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-58\) \(0\) \(-\) \(-\) \(q-58q^{5}+8898q^{13}+40094q^{17}+\cdots\)
576.8.a.o \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(-26\) \(-1056\) \(-\) \(-\) \(q-26q^{5}-1056q^{7}-6412q^{11}-5206q^{13}+\cdots\)
576.8.a.p \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(-26\) \(1056\) \(+\) \(-\) \(q-26q^{5}+1056q^{7}+6412q^{11}-5206q^{13}+\cdots\)
576.8.a.q \(1\) \(179.934\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-508\) \(+\) \(+\) \(q-508q^{7}+14614q^{13}+57448q^{19}+\cdots\)
576.8.a.r \(1\) \(179.934\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(508\) \(-\) \(+\) \(q+508q^{7}+14614q^{13}-57448q^{19}+\cdots\)
576.8.a.s \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(110\) \(-504\) \(-\) \(-\) \(q+110q^{5}-504q^{7}-3812q^{11}-9574q^{13}+\cdots\)
576.8.a.t \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(110\) \(504\) \(+\) \(-\) \(q+110q^{5}+504q^{7}+3812q^{11}-9574q^{13}+\cdots\)
576.8.a.u \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(270\) \(-1112\) \(-\) \(-\) \(q+270q^{5}-1112q^{7}+5724q^{11}+\cdots\)
576.8.a.v \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(270\) \(1112\) \(+\) \(-\) \(q+270q^{5}+1112q^{7}-5724q^{11}+\cdots\)
576.8.a.w \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(390\) \(-64\) \(+\) \(-\) \(q+390q^{5}-2^{6}q^{7}-948q^{11}+5098q^{13}+\cdots\)
576.8.a.x \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(390\) \(64\) \(-\) \(-\) \(q+390q^{5}+2^{6}q^{7}+948q^{11}+5098q^{13}+\cdots\)
576.8.a.y \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(430\) \(-1224\) \(+\) \(-\) \(q+430q^{5}-1224q^{7}-3164q^{11}+\cdots\)
576.8.a.z \(1\) \(179.934\) \(\Q\) None \(0\) \(0\) \(430\) \(1224\) \(-\) \(-\) \(q+430q^{5}+1224q^{7}+3164q^{11}+\cdots\)
576.8.a.ba \(1\) \(179.934\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(556\) \(0\) \(-\) \(+\) \(q+556q^{5}-8898q^{13}-5816q^{17}+\cdots\)
576.8.a.bb \(2\) \(179.934\) \(\Q(\sqrt{435}) \) None \(0\) \(0\) \(-280\) \(0\) \(-\) \(+\) \(q-140q^{5}+\beta q^{7}+4\beta q^{11}+2238q^{13}+\cdots\)
576.8.a.bc \(2\) \(179.934\) \(\Q(\sqrt{46}) \) None \(0\) \(0\) \(-224\) \(-840\) \(-\) \(+\) \(q+(-112+\beta )q^{5}+(-420+4\beta )q^{7}+\cdots\)
576.8.a.bd \(2\) \(179.934\) \(\Q(\sqrt{46}) \) None \(0\) \(0\) \(-224\) \(840\) \(+\) \(+\) \(q+(-112+\beta )q^{5}+(420-4\beta )q^{7}+(320+\cdots)q^{11}+\cdots\)
576.8.a.be \(2\) \(179.934\) \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(-180\) \(-1248\) \(+\) \(-\) \(q+(-90+8\beta )q^{5}+(-624+14\beta )q^{7}+\cdots\)
576.8.a.bf \(2\) \(179.934\) \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(-180\) \(1248\) \(+\) \(-\) \(q+(-90+8\beta )q^{5}+(624-14\beta )q^{7}+\cdots\)
576.8.a.bg \(2\) \(179.934\) \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(-28\) \(-936\) \(-\) \(-\) \(q+(-14+3\beta )q^{5}+(-468-7\beta )q^{7}+\cdots\)
576.8.a.bh \(2\) \(179.934\) \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(-28\) \(936\) \(-\) \(-\) \(q+(-14+3\beta )q^{5}+(468+7\beta )q^{7}+(-1692+\cdots)q^{11}+\cdots\)
576.8.a.bi \(2\) \(179.934\) \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(0\) \(-520\) \(-\) \(+\) \(q+\beta q^{5}-260q^{7}-20\beta q^{11}-6890q^{13}+\cdots\)
576.8.a.bj \(2\) \(179.934\) \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(0\) \(520\) \(+\) \(+\) \(q+\beta q^{5}+260q^{7}+20\beta q^{11}-6890q^{13}+\cdots\)
576.8.a.bk \(2\) \(179.934\) \(\Q(\sqrt{15}) \) None \(0\) \(0\) \(140\) \(0\) \(-\) \(-\) \(q+70q^{5}+2\beta q^{7}-13\beta q^{11}-13758q^{13}+\cdots\)
576.8.a.bl \(2\) \(179.934\) \(\Q(\sqrt{235}) \) None \(0\) \(0\) \(180\) \(-1032\) \(+\) \(-\) \(q+(90+\beta )q^{5}+(-516+5\beta )q^{7}+(-1420+\cdots)q^{11}+\cdots\)
576.8.a.bm \(2\) \(179.934\) \(\Q(\sqrt{235}) \) None \(0\) \(0\) \(180\) \(1032\) \(+\) \(-\) \(q+(90+\beta )q^{5}+(516-5\beta )q^{7}+(1420+\cdots)q^{11}+\cdots\)
576.8.a.bn \(2\) \(179.934\) \(\Q(\sqrt{46}) \) None \(0\) \(0\) \(196\) \(-504\) \(-\) \(-\) \(q+(98+\beta )q^{5}+(-252-3\beta )q^{7}+(-828+\cdots)q^{11}+\cdots\)
576.8.a.bo \(2\) \(179.934\) \(\Q(\sqrt{46}) \) None \(0\) \(0\) \(196\) \(504\) \(-\) \(-\) \(q+(98+\beta )q^{5}+(252+3\beta )q^{7}+(828+\cdots)q^{11}+\cdots\)
576.8.a.bp \(2\) \(179.934\) \(\Q(\sqrt{46}) \) None \(0\) \(0\) \(224\) \(-840\) \(-\) \(+\) \(q+(112+\beta )q^{5}+(-420-4\beta )q^{7}+(320+\cdots)q^{11}+\cdots\)
576.8.a.bq \(2\) \(179.934\) \(\Q(\sqrt{46}) \) None \(0\) \(0\) \(224\) \(840\) \(+\) \(+\) \(q+(112+\beta )q^{5}+(420+4\beta )q^{7}+(-320+\cdots)q^{11}+\cdots\)
576.8.a.br \(2\) \(179.934\) \(\Q(\sqrt{435}) \) None \(0\) \(0\) \(280\) \(0\) \(-\) \(+\) \(q+140q^{5}+\beta q^{7}-4\beta q^{11}+2238q^{13}+\cdots\)
576.8.a.bs \(4\) \(179.934\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+\beta _{2}q^{5}+\beta _{3}q^{7}+(-160-\beta _{1})q^{11}+\cdots\)
576.8.a.bt \(4\) \(179.934\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+\beta _{2}q^{5}-\beta _{3}q^{7}+(160+\beta _{1})q^{11}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(576))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(576)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 18}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 14}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 7}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 2}\)