Properties

Label 448.4.a
Level $448$
Weight $4$
Character orbit 448.a
Rep. character $\chi_{448}(1,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $24$
Sturm bound $256$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(256\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(448))\).

Total New Old
Modular forms 204 36 168
Cusp forms 180 36 144
Eisenstein series 24 0 24

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

\(2\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(9\)
\(+\)\(-\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(9\)
\(-\)\(-\)\(+\)\(10\)
Plus space\(+\)\(19\)
Minus space\(-\)\(17\)

Trace form

\( 36q + 324q^{9} + O(q^{10}) \) \( 36q + 324q^{9} + 104q^{17} + 812q^{25} + 200q^{29} + 464q^{33} + 8q^{37} - 392q^{41} - 1968q^{45} + 1764q^{49} - 1192q^{53} - 688q^{57} + 1824q^{61} - 1536q^{65} + 2544q^{69} - 296q^{73} - 952q^{77} + 1076q^{81} - 480q^{85} + 88q^{89} + 9216q^{93} - 1816q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(448))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7
448.4.a.a \(1\) \(26.433\) \(\Q\) None \(0\) \(-10\) \(8\) \(7\) \(-\) \(-\) \(q-10q^{3}+8q^{5}+7q^{7}+73q^{9}-40q^{11}+\cdots\)
448.4.a.b \(1\) \(26.433\) \(\Q\) None \(0\) \(-8\) \(14\) \(-7\) \(+\) \(+\) \(q-8q^{3}+14q^{5}-7q^{7}+37q^{9}+28q^{11}+\cdots\)
448.4.a.c \(1\) \(26.433\) \(\Q\) None \(0\) \(-6\) \(-8\) \(-7\) \(+\) \(+\) \(q-6q^{3}-8q^{5}-7q^{7}+9q^{9}-56q^{11}+\cdots\)
448.4.a.d \(1\) \(26.433\) \(\Q\) None \(0\) \(-4\) \(-6\) \(7\) \(+\) \(-\) \(q-4q^{3}-6q^{5}+7q^{7}-11q^{9}+12q^{11}+\cdots\)
448.4.a.e \(1\) \(26.433\) \(\Q\) None \(0\) \(-2\) \(-16\) \(7\) \(-\) \(-\) \(q-2q^{3}-2^{4}q^{5}+7q^{7}-23q^{9}-8q^{11}+\cdots\)
448.4.a.f \(1\) \(26.433\) \(\Q\) None \(0\) \(-2\) \(0\) \(-7\) \(+\) \(+\) \(q-2q^{3}-7q^{7}-23q^{9}+20q^{11}+20q^{13}+\cdots\)
448.4.a.g \(1\) \(26.433\) \(\Q\) None \(0\) \(-2\) \(12\) \(-7\) \(-\) \(+\) \(q-2q^{3}+12q^{5}-7q^{7}-23q^{9}+48q^{11}+\cdots\)
448.4.a.h \(1\) \(26.433\) \(\Q\) None \(0\) \(-2\) \(16\) \(7\) \(-\) \(-\) \(q-2q^{3}+2^{4}q^{5}+7q^{7}-23q^{9}+24q^{11}+\cdots\)
448.4.a.i \(1\) \(26.433\) \(\Q\) None \(0\) \(2\) \(-16\) \(-7\) \(+\) \(+\) \(q+2q^{3}-2^{4}q^{5}-7q^{7}-23q^{9}+8q^{11}+\cdots\)
448.4.a.j \(1\) \(26.433\) \(\Q\) None \(0\) \(2\) \(0\) \(7\) \(+\) \(-\) \(q+2q^{3}+7q^{7}-23q^{9}-20q^{11}+20q^{13}+\cdots\)
448.4.a.k \(1\) \(26.433\) \(\Q\) None \(0\) \(2\) \(12\) \(7\) \(+\) \(-\) \(q+2q^{3}+12q^{5}+7q^{7}-23q^{9}-48q^{11}+\cdots\)
448.4.a.l \(1\) \(26.433\) \(\Q\) None \(0\) \(2\) \(16\) \(-7\) \(+\) \(+\) \(q+2q^{3}+2^{4}q^{5}-7q^{7}-23q^{9}-24q^{11}+\cdots\)
448.4.a.m \(1\) \(26.433\) \(\Q\) None \(0\) \(4\) \(-6\) \(-7\) \(-\) \(+\) \(q+4q^{3}-6q^{5}-7q^{7}-11q^{9}-12q^{11}+\cdots\)
448.4.a.n \(1\) \(26.433\) \(\Q\) None \(0\) \(6\) \(-8\) \(7\) \(-\) \(-\) \(q+6q^{3}-8q^{5}+7q^{7}+9q^{9}+56q^{11}+\cdots\)
448.4.a.o \(1\) \(26.433\) \(\Q\) None \(0\) \(8\) \(14\) \(7\) \(-\) \(-\) \(q+8q^{3}+14q^{5}+7q^{7}+37q^{9}-28q^{11}+\cdots\)
448.4.a.p \(1\) \(26.433\) \(\Q\) None \(0\) \(10\) \(8\) \(-7\) \(+\) \(+\) \(q+10q^{3}+8q^{5}-7q^{7}+73q^{9}+40q^{11}+\cdots\)
448.4.a.q \(2\) \(26.433\) \(\Q(\sqrt{37}) \) None \(0\) \(-6\) \(6\) \(-14\) \(-\) \(+\) \(q+(-3-\beta )q^{3}+(3-\beta )q^{5}-7q^{7}+(19+\cdots)q^{9}+\cdots\)
448.4.a.r \(2\) \(26.433\) \(\Q(\sqrt{57}) \) None \(0\) \(-2\) \(-22\) \(-14\) \(-\) \(+\) \(q+(-1-\beta )q^{3}+(-11-\beta )q^{5}-7q^{7}+\cdots\)
448.4.a.s \(2\) \(26.433\) \(\Q(\sqrt{57}) \) None \(0\) \(2\) \(-22\) \(14\) \(+\) \(-\) \(q+(1+\beta )q^{3}+(-11-\beta )q^{5}+7q^{7}+\cdots\)
448.4.a.t \(2\) \(26.433\) \(\Q(\sqrt{37}) \) None \(0\) \(6\) \(6\) \(14\) \(-\) \(-\) \(q+(3+\beta )q^{3}+(3-\beta )q^{5}+7q^{7}+(19+\cdots)q^{9}+\cdots\)
448.4.a.u \(3\) \(26.433\) 3.3.2981.1 None \(0\) \(-8\) \(-10\) \(21\) \(+\) \(-\) \(q+(-3-\beta _{1})q^{3}+(-3-\beta _{2})q^{5}+7q^{7}+\cdots\)
448.4.a.v \(3\) \(26.433\) 3.3.621.1 None \(0\) \(0\) \(6\) \(-21\) \(-\) \(+\) \(q-\beta _{1}q^{3}+(2+2\beta _{1}+\beta _{2})q^{5}-7q^{7}+\cdots\)
448.4.a.w \(3\) \(26.433\) 3.3.621.1 None \(0\) \(0\) \(6\) \(21\) \(-\) \(-\) \(q+\beta _{1}q^{3}+(2+2\beta _{1}+\beta _{2})q^{5}+7q^{7}+\cdots\)
448.4.a.x \(3\) \(26.433\) 3.3.2981.1 None \(0\) \(8\) \(-10\) \(-21\) \(+\) \(+\) \(q+(3+\beta _{1})q^{3}+(-3-\beta _{2})q^{5}-7q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(448))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(448)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 2}\)