Properties

Label 6048.2.a
Level 6048
Weight 2
Character orbit a
Rep. character \(\chi_{6048}(1,\cdot)\)
Character field \(\Q\)
Dimension 96
Newforms 48
Sturm bound 2304
Trace bound 13

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Defining parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.a (trivial)
Character field: \(\Q\)
Newforms: \( 48 \)
Sturm bound: \(2304\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6048))\).

Total New Old
Modular forms 1200 96 1104
Cusp forms 1105 96 1009
Eisenstein series 95 0 95

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.
\(+\)\(+\)\(+\)\(+\)\(11\)
\(+\)\(+\)\(-\)\(-\)\(13\)
\(+\)\(-\)\(+\)\(-\)\(13\)
\(+\)\(-\)\(-\)\(+\)\(11\)
\(-\)\(+\)\(+\)\(-\)\(13\)
\(-\)\(+\)\(-\)\(+\)\(11\)
\(-\)\(-\)\(+\)\(+\)\(11\)
\(-\)\(-\)\(-\)\(-\)\(13\)
Plus space\(+\)\(44\)
Minus space\(-\)\(52\)

Trace form

\( 96q + O(q^{10}) \) \( 96q + 96q^{25} - 64q^{37} + 96q^{49} - 64q^{61} - 48q^{85} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6048))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
6048.2.a.a \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(-3\) \(-1\) \(-\) \(-\) \(+\) \(q-3q^{5}-q^{7}-2q^{11}-4q^{13}+5q^{17}+\cdots\)
6048.2.a.b \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(-3\) \(-1\) \(+\) \(+\) \(+\) \(q-3q^{5}-q^{7}-q^{11}+2q^{13}-2q^{17}+\cdots\)
6048.2.a.c \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(-3\) \(1\) \(+\) \(-\) \(-\) \(q-3q^{5}+q^{7}+q^{11}+2q^{13}-2q^{17}+\cdots\)
6048.2.a.d \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(-3\) \(1\) \(-\) \(+\) \(-\) \(q-3q^{5}+q^{7}+2q^{11}-4q^{13}+5q^{17}+\cdots\)
6048.2.a.e \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(-1\) \(-1\) \(-\) \(+\) \(+\) \(q-q^{5}-q^{7}-3q^{11}+2q^{13}+2q^{17}+\cdots\)
6048.2.a.f \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(-1\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{5}-q^{7}-2q^{11}-q^{17}+4q^{19}+\cdots\)
6048.2.a.g \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(-1\) \(-1\) \(-\) \(+\) \(+\) \(q-q^{5}-q^{7}+2q^{11}+4q^{13}+3q^{17}+\cdots\)
6048.2.a.h \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(-1\) \(1\) \(+\) \(-\) \(-\) \(q-q^{5}+q^{7}-2q^{11}+4q^{13}+3q^{17}+\cdots\)
6048.2.a.i \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(-1\) \(1\) \(+\) \(-\) \(-\) \(q-q^{5}+q^{7}+2q^{11}-q^{17}-4q^{19}+\cdots\)
6048.2.a.j \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(-1\) \(1\) \(-\) \(-\) \(-\) \(q-q^{5}+q^{7}+3q^{11}+2q^{13}+2q^{17}+\cdots\)
6048.2.a.k \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q-q^{7}-4q^{11}+q^{13}+5q^{17}+2q^{19}+\cdots\)
6048.2.a.l \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{7}+4q^{11}+q^{13}-5q^{17}+2q^{19}+\cdots\)
6048.2.a.m \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+q^{7}-4q^{11}+q^{13}-5q^{17}-2q^{19}+\cdots\)
6048.2.a.n \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q+q^{7}+4q^{11}+q^{13}+5q^{17}-2q^{19}+\cdots\)
6048.2.a.o \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(1\) \(-1\) \(+\) \(-\) \(+\) \(q+q^{5}-q^{7}-2q^{11}+4q^{13}-3q^{17}+\cdots\)
6048.2.a.p \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(1\) \(-1\) \(-\) \(-\) \(+\) \(q+q^{5}-q^{7}+2q^{11}+q^{17}+4q^{19}+\cdots\)
6048.2.a.q \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(1\) \(-1\) \(+\) \(+\) \(+\) \(q+q^{5}-q^{7}+3q^{11}+2q^{13}-2q^{17}+\cdots\)
6048.2.a.r \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(1\) \(1\) \(+\) \(-\) \(-\) \(q+q^{5}+q^{7}-3q^{11}+2q^{13}-2q^{17}+\cdots\)
6048.2.a.s \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(1\) \(1\) \(-\) \(+\) \(-\) \(q+q^{5}+q^{7}-2q^{11}+q^{17}-4q^{19}+\cdots\)
6048.2.a.t \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(1\) \(1\) \(-\) \(+\) \(-\) \(q+q^{5}+q^{7}+2q^{11}+4q^{13}-3q^{17}+\cdots\)
6048.2.a.u \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(3\) \(-1\) \(-\) \(+\) \(+\) \(q+3q^{5}-q^{7}+q^{11}+2q^{13}+2q^{17}+\cdots\)
6048.2.a.v \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(3\) \(-1\) \(+\) \(+\) \(+\) \(q+3q^{5}-q^{7}+2q^{11}-4q^{13}-5q^{17}+\cdots\)
6048.2.a.w \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(3\) \(1\) \(+\) \(-\) \(-\) \(q+3q^{5}+q^{7}-2q^{11}-4q^{13}-5q^{17}+\cdots\)
6048.2.a.x \(1\) \(48.294\) \(\Q\) None \(0\) \(0\) \(3\) \(1\) \(-\) \(-\) \(-\) \(q+3q^{5}+q^{7}-q^{11}+2q^{13}+2q^{17}+\cdots\)
6048.2.a.y \(2\) \(48.294\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-2\) \(-2\) \(+\) \(+\) \(+\) \(q+(-1+\beta )q^{5}-q^{7}+(-2-\beta )q^{11}+\cdots\)
6048.2.a.z \(2\) \(48.294\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-2\) \(-2\) \(+\) \(-\) \(+\) \(q+(-1+\beta )q^{5}-q^{7}+(1-\beta )q^{11}-2\beta q^{13}+\cdots\)
6048.2.a.ba \(2\) \(48.294\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-2\) \(2\) \(-\) \(+\) \(-\) \(q+(-1+\beta )q^{5}+q^{7}+(-1+\beta )q^{11}+\cdots\)
6048.2.a.bb \(2\) \(48.294\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-2\) \(2\) \(-\) \(-\) \(-\) \(q+(-1+\beta )q^{5}+q^{7}+(2+\beta )q^{11}+\beta q^{13}+\cdots\)
6048.2.a.bc \(2\) \(48.294\) \(\Q(\sqrt{17}) \) None \(0\) \(0\) \(-1\) \(-2\) \(-\) \(+\) \(+\) \(q-\beta q^{5}-q^{7}-2\beta q^{11}+(-5+\beta )q^{13}+\cdots\)
6048.2.a.bd \(2\) \(48.294\) \(\Q(\sqrt{17}) \) None \(0\) \(0\) \(-1\) \(2\) \(+\) \(-\) \(-\) \(q-\beta q^{5}+q^{7}+2\beta q^{11}+(-5+\beta )q^{13}+\cdots\)
6048.2.a.be \(2\) \(48.294\) \(\Q(\sqrt{17}) \) None \(0\) \(0\) \(1\) \(-2\) \(+\) \(-\) \(+\) \(q+\beta q^{5}-q^{7}+2\beta q^{11}+(-5+\beta )q^{13}+\cdots\)
6048.2.a.bf \(2\) \(48.294\) \(\Q(\sqrt{17}) \) None \(0\) \(0\) \(1\) \(2\) \(-\) \(+\) \(-\) \(q+\beta q^{5}+q^{7}-2\beta q^{11}+(-5+\beta )q^{13}+\cdots\)
6048.2.a.bg \(2\) \(48.294\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(2\) \(-2\) \(-\) \(-\) \(+\) \(q+(1+\beta )q^{5}-q^{7}+(-1-\beta )q^{11}+2\beta q^{13}+\cdots\)
6048.2.a.bh \(2\) \(48.294\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(2\) \(-2\) \(-\) \(-\) \(+\) \(q+(1+\beta )q^{5}-q^{7}+(2-\beta )q^{11}-\beta q^{13}+\cdots\)
6048.2.a.bi \(2\) \(48.294\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(2\) \(2\) \(+\) \(+\) \(-\) \(q+(1+\beta )q^{5}+q^{7}+(-2+\beta )q^{11}-\beta q^{13}+\cdots\)
6048.2.a.bj \(2\) \(48.294\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(2\) \(2\) \(+\) \(+\) \(-\) \(q+(1+\beta )q^{5}+q^{7}+(1+\beta )q^{11}+2\beta q^{13}+\cdots\)
6048.2.a.bk \(4\) \(48.294\) 4.4.39528.1 None \(0\) \(0\) \(-2\) \(-4\) \(+\) \(-\) \(+\) \(q+(-1-\beta _{2})q^{5}-q^{7}-\beta _{1}q^{11}+(2+\cdots)q^{13}+\cdots\)
6048.2.a.bl \(4\) \(48.294\) 4.4.25808.1 None \(0\) \(0\) \(-2\) \(-4\) \(-\) \(+\) \(+\) \(q+(-1-\beta _{2})q^{5}-q^{7}+(1+\beta _{1})q^{11}+\cdots\)
6048.2.a.bm \(4\) \(48.294\) 4.4.22896.1 None \(0\) \(0\) \(-2\) \(-4\) \(-\) \(-\) \(+\) \(q+\beta _{2}q^{5}-q^{7}-\beta _{1}q^{11}+(\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)
6048.2.a.bn \(4\) \(48.294\) 4.4.25808.1 None \(0\) \(0\) \(-2\) \(4\) \(+\) \(-\) \(-\) \(q+(-1-\beta _{2})q^{5}+q^{7}+(-1-\beta _{1})q^{11}+\cdots\)
6048.2.a.bo \(4\) \(48.294\) 4.4.22896.1 None \(0\) \(0\) \(-2\) \(4\) \(-\) \(+\) \(-\) \(q+\beta _{2}q^{5}+q^{7}+\beta _{1}q^{11}+(\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)
6048.2.a.bp \(4\) \(48.294\) 4.4.39528.1 None \(0\) \(0\) \(-2\) \(4\) \(+\) \(+\) \(-\) \(q+(-1-\beta _{2})q^{5}+q^{7}+\beta _{1}q^{11}+(2+\cdots)q^{13}+\cdots\)
6048.2.a.bq \(4\) \(48.294\) 4.4.25808.1 None \(0\) \(0\) \(2\) \(-4\) \(+\) \(+\) \(+\) \(q+(1+\beta _{2})q^{5}-q^{7}+(-1-\beta _{1})q^{11}+\cdots\)
6048.2.a.br \(4\) \(48.294\) 4.4.22896.1 None \(0\) \(0\) \(2\) \(-4\) \(+\) \(-\) \(+\) \(q-\beta _{2}q^{5}-q^{7}+\beta _{1}q^{11}+(\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)
6048.2.a.bs \(4\) \(48.294\) 4.4.39528.1 None \(0\) \(0\) \(2\) \(-4\) \(-\) \(+\) \(+\) \(q+(1+\beta _{2})q^{5}-q^{7}+\beta _{1}q^{11}+(2+\beta _{1}+\cdots)q^{13}+\cdots\)
6048.2.a.bt \(4\) \(48.294\) 4.4.39528.1 None \(0\) \(0\) \(2\) \(4\) \(-\) \(-\) \(-\) \(q+(1+\beta _{2})q^{5}+q^{7}-\beta _{1}q^{11}+(2+\beta _{1}+\cdots)q^{13}+\cdots\)
6048.2.a.bu \(4\) \(48.294\) 4.4.25808.1 None \(0\) \(0\) \(2\) \(4\) \(-\) \(-\) \(-\) \(q+(1+\beta _{2})q^{5}+q^{7}+(1+\beta _{1})q^{11}+(-1+\cdots)q^{13}+\cdots\)
6048.2.a.bv \(4\) \(48.294\) 4.4.22896.1 None \(0\) \(0\) \(2\) \(4\) \(+\) \(+\) \(-\) \(q-\beta _{2}q^{5}+q^{7}-\beta _{1}q^{11}+(\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6048))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6048)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(189))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(378))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(432))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(504))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(672))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(756))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(864))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1008))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1512))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2016))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3024))\)\(^{\oplus 2}\)