Properties

Label 1008.4.a
Level $1008$
Weight $4$
Character orbit 1008.a
Rep. character $\chi_{1008}(1,\cdot)$
Character field $\Q$
Dimension $45$
Newform subspaces $34$
Sturm bound $768$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1008.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 34 \)
Sturm bound: \(768\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1008))\).

Total New Old
Modular forms 600 45 555
Cusp forms 552 45 507
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.
\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(7\)
\(+\)\(-\)\(-\)\(+\)\(7\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(6\)
Plus space\(+\)\(24\)
Minus space\(-\)\(21\)

Trace form

\( 45q + 2q^{5} + 7q^{7} + O(q^{10}) \) \( 45q + 2q^{5} + 7q^{7} + 56q^{11} - 46q^{13} + 50q^{17} - 24q^{19} - 220q^{23} + 1147q^{25} - 242q^{29} + 264q^{31} + 210q^{35} + 62q^{37} + 138q^{41} - 892q^{43} - 1536q^{47} + 2205q^{49} - 98q^{53} + 1080q^{55} + 1080q^{59} + 834q^{61} - 580q^{65} - 228q^{67} - 2900q^{71} + 458q^{73} + 476q^{77} + 3016q^{79} + 1112q^{83} - 1588q^{85} + 1730q^{89} - 546q^{91} - 1848q^{95} + 906q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1008))\) 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
1008.4.a.a \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(-22\) \(7\) \(-\) \(+\) \(-\) \(q-22q^{5}+7q^{7}+26q^{11}-54q^{13}+\cdots\)
1008.4.a.b \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(-18\) \(-7\) \(-\) \(-\) \(+\) \(q-18q^{5}-7q^{7}-72q^{11}-34q^{13}+\cdots\)
1008.4.a.c \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(-16\) \(7\) \(-\) \(-\) \(-\) \(q-2^{4}q^{5}+7q^{7}-8q^{11}+28q^{13}+\cdots\)
1008.4.a.d \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(-14\) \(7\) \(-\) \(-\) \(-\) \(q-14q^{5}+7q^{7}+4q^{11}+54q^{13}+\cdots\)
1008.4.a.e \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(-8\) \(7\) \(+\) \(-\) \(-\) \(q-8q^{5}+7q^{7}+56q^{11}-28q^{13}+\cdots\)
1008.4.a.f \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(-6\) \(-7\) \(-\) \(-\) \(+\) \(q-6q^{5}-7q^{7}-12q^{11}-82q^{13}+\cdots\)
1008.4.a.g \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(-6\) \(-7\) \(-\) \(+\) \(+\) \(q-6q^{5}-7q^{7}+30q^{11}+2q^{13}-66q^{17}+\cdots\)
1008.4.a.h \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(-6\) \(-7\) \(-\) \(-\) \(+\) \(q-6q^{5}-7q^{7}+6^{2}q^{11}+62q^{13}+\cdots\)
1008.4.a.i \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(-4\) \(7\) \(+\) \(-\) \(-\) \(q-4q^{5}+7q^{7}-26q^{11}+2q^{13}+6^{2}q^{17}+\cdots\)
1008.4.a.j \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(-2\) \(7\) \(-\) \(-\) \(-\) \(q-2q^{5}+7q^{7}-8q^{11}-42q^{13}+2q^{17}+\cdots\)
1008.4.a.k \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(2\) \(-7\) \(+\) \(-\) \(+\) \(q+2q^{5}-7q^{7}+12q^{11}-66q^{13}+\cdots\)
1008.4.a.l \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(2\) \(7\) \(+\) \(-\) \(-\) \(q+2q^{5}+7q^{7}+52q^{11}+86q^{13}+\cdots\)
1008.4.a.m \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(4\) \(7\) \(-\) \(-\) \(-\) \(q+4q^{5}+7q^{7}+62q^{11}-62q^{13}+\cdots\)
1008.4.a.n \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(6\) \(-7\) \(-\) \(+\) \(+\) \(q+6q^{5}-7q^{7}-30q^{11}+2q^{13}+66q^{17}+\cdots\)
1008.4.a.o \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(8\) \(7\) \(-\) \(-\) \(-\) \(q+8q^{5}+7q^{7}-40q^{11}-12q^{13}+\cdots\)
1008.4.a.p \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(10\) \(-7\) \(+\) \(-\) \(+\) \(q+10q^{5}-7q^{7}-12q^{11}+30q^{13}+\cdots\)
1008.4.a.q \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(10\) \(7\) \(+\) \(-\) \(-\) \(q+10q^{5}+7q^{7}-52q^{11}-10q^{13}+\cdots\)
1008.4.a.r \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(12\) \(-7\) \(-\) \(-\) \(+\) \(q+12q^{5}-7q^{7}+48q^{11}+56q^{13}+\cdots\)
1008.4.a.s \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(14\) \(7\) \(-\) \(-\) \(-\) \(q+14q^{5}+7q^{7}-28q^{11}+18q^{13}+\cdots\)
1008.4.a.t \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(16\) \(-7\) \(+\) \(-\) \(+\) \(q+2^{4}q^{5}-7q^{7}-18q^{11}-54q^{13}+\cdots\)
1008.4.a.u \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(16\) \(7\) \(+\) \(-\) \(-\) \(q+2^{4}q^{5}+7q^{7}+24q^{11}-68q^{13}+\cdots\)
1008.4.a.v \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(18\) \(-7\) \(-\) \(-\) \(+\) \(q+18q^{5}-7q^{7}-6^{2}q^{11}-34q^{13}+\cdots\)
1008.4.a.w \(1\) \(59.474\) \(\Q\) None \(0\) \(0\) \(22\) \(7\) \(-\) \(+\) \(-\) \(q+22q^{5}+7q^{7}-26q^{11}-54q^{13}+\cdots\)
1008.4.a.x \(2\) \(59.474\) \(\Q(\sqrt{57}) \) None \(0\) \(0\) \(-22\) \(-14\) \(+\) \(-\) \(+\) \(q+(-11-\beta )q^{5}-7q^{7}+(18+6\beta )q^{11}+\cdots\)
1008.4.a.y \(2\) \(59.474\) \(\Q(\sqrt{177}) \) None \(0\) \(0\) \(-14\) \(-14\) \(+\) \(-\) \(+\) \(q+(-7-\beta )q^{5}-7q^{7}+(9-3\beta )q^{11}+\cdots\)
1008.4.a.z \(2\) \(59.474\) \(\Q(\sqrt{22}) \) None \(0\) \(0\) \(-12\) \(14\) \(+\) \(+\) \(-\) \(q+(-6+\beta )q^{5}+7q^{7}+(2+\beta )q^{11}+\cdots\)
1008.4.a.ba \(2\) \(59.474\) \(\Q(\sqrt{57}) \) None \(0\) \(0\) \(-6\) \(-14\) \(-\) \(-\) \(+\) \(q+(-3-\beta )q^{5}-7q^{7}+(-3+5\beta )q^{11}+\cdots\)
1008.4.a.bb \(2\) \(59.474\) \(\Q(\sqrt{30}) \) None \(0\) \(0\) \(-4\) \(-14\) \(+\) \(+\) \(+\) \(q+(-2+\beta )q^{5}-7q^{7}+(18+3\beta )q^{11}+\cdots\)
1008.4.a.bc \(2\) \(59.474\) \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(-14\) \(-\) \(+\) \(+\) \(q+\beta q^{5}-7q^{7}-\beta q^{11}+26q^{13}-5\beta q^{17}+\cdots\)
1008.4.a.bd \(2\) \(59.474\) \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(14\) \(-\) \(+\) \(-\) \(q+\beta q^{5}+7q^{7}+13\beta q^{11}-30q^{13}+\cdots\)
1008.4.a.be \(2\) \(59.474\) \(\Q(\sqrt{19}) \) None \(0\) \(0\) \(0\) \(14\) \(-\) \(+\) \(-\) \(q+\beta q^{5}+7q^{7}+5\beta q^{11}+82q^{13}+\cdots\)
1008.4.a.bf \(2\) \(59.474\) \(\Q(\sqrt{30}) \) None \(0\) \(0\) \(4\) \(-14\) \(+\) \(+\) \(+\) \(q+(2+\beta )q^{5}-7q^{7}+(-18+3\beta )q^{11}+\cdots\)
1008.4.a.bg \(2\) \(59.474\) \(\Q(\sqrt{337}) \) None \(0\) \(0\) \(6\) \(14\) \(+\) \(-\) \(-\) \(q+(3+\beta )q^{5}+7q^{7}+(13+\beta )q^{11}+(48+\cdots)q^{13}+\cdots\)
1008.4.a.bh \(2\) \(59.474\) \(\Q(\sqrt{22}) \) None \(0\) \(0\) \(12\) \(14\) \(+\) \(+\) \(-\) \(q+(6+\beta )q^{5}+7q^{7}+(-2+\beta )q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1008))\) into lower level spaces

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