Properties

Label 448.6.a
Level $448$
Weight $6$
Character orbit 448.a
Rep. character $\chi_{448}(1,\cdot)$
Character field $\Q$
Dimension $60$
Newform subspaces $32$
Sturm bound $384$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 448.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 32 \)
Sturm bound: \(384\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(448))\).

Total New Old
Modular forms 332 60 272
Cusp forms 308 60 248
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.
\(+\)\(+\)\(+\)\(15\)
\(+\)\(-\)\(-\)\(16\)
\(-\)\(+\)\(-\)\(15\)
\(-\)\(-\)\(+\)\(14\)
Plus space\(+\)\(29\)
Minus space\(-\)\(31\)

Trace form

\( 60 q + 4860 q^{9} + O(q^{10}) \) \( 60 q + 4860 q^{9} - 808 q^{17} + 43732 q^{25} - 4072 q^{29} - 11344 q^{33} + 10648 q^{37} + 28168 q^{41} + 72048 q^{45} + 144060 q^{49} - 52984 q^{53} + 61616 q^{57} - 96160 q^{61} - 7680 q^{65} - 240432 q^{69} - 10072 q^{73} + 7448 q^{77} + 436076 q^{81} + 264800 q^{85} - 70232 q^{89} - 728064 q^{93} - 160808 q^{97} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 7
448.6.a.a 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(-30\) \(-32\) \(49\) $+$ $-$ $\mathrm{SU}(2)$ \(q-30q^{3}-2^{5}q^{5}+7^{2}q^{7}+657q^{9}+\cdots\)
448.6.a.b 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(-26\) \(-16\) \(-49\) $+$ $+$ $\mathrm{SU}(2)$ \(q-26q^{3}-2^{4}q^{5}-7^{2}q^{7}+433q^{9}+\cdots\)
448.6.a.c 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(-14\) \(56\) \(49\) $-$ $-$ $\mathrm{SU}(2)$ \(q-14q^{3}+56q^{5}+7^{2}q^{7}-47q^{9}+\cdots\)
448.6.a.d 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(-14\) \(64\) \(-49\) $-$ $+$ $\mathrm{SU}(2)$ \(q-14q^{3}+2^{6}q^{5}-7^{2}q^{7}-47q^{9}+\cdots\)
448.6.a.e 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(-10\) \(-84\) \(49\) $+$ $-$ $\mathrm{SU}(2)$ \(q-10q^{3}-84q^{5}+7^{2}q^{7}-143q^{9}+\cdots\)
448.6.a.f 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(-8\) \(-10\) \(-49\) $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{3}-10q^{5}-7^{2}q^{7}-179q^{9}+\cdots\)
448.6.a.g 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(-6\) \(-4\) \(-49\) $-$ $+$ $\mathrm{SU}(2)$ \(q-6q^{3}-4q^{5}-7^{2}q^{7}-207q^{9}-240q^{11}+\cdots\)
448.6.a.h 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(-2\) \(96\) \(-49\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+96q^{5}-7^{2}q^{7}-239q^{9}+\cdots\)
448.6.a.i 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(2\) \(96\) \(49\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+96q^{5}+7^{2}q^{7}-239q^{9}+\cdots\)
448.6.a.j 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(6\) \(-4\) \(49\) $+$ $-$ $\mathrm{SU}(2)$ \(q+6q^{3}-4q^{5}+7^{2}q^{7}-207q^{9}+240q^{11}+\cdots\)
448.6.a.k 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(8\) \(-10\) \(49\) $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{3}-10q^{5}+7^{2}q^{7}-179q^{9}+\cdots\)
448.6.a.l 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(10\) \(-84\) \(-49\) $-$ $+$ $\mathrm{SU}(2)$ \(q+10q^{3}-84q^{5}-7^{2}q^{7}-143q^{9}+\cdots\)
448.6.a.m 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(14\) \(56\) \(-49\) $+$ $+$ $\mathrm{SU}(2)$ \(q+14q^{3}+56q^{5}-7^{2}q^{7}-47q^{9}+\cdots\)
448.6.a.n 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(14\) \(64\) \(49\) $-$ $-$ $\mathrm{SU}(2)$ \(q+14q^{3}+2^{6}q^{5}+7^{2}q^{7}-47q^{9}+\cdots\)
448.6.a.o 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(26\) \(-16\) \(49\) $-$ $-$ $\mathrm{SU}(2)$ \(q+26q^{3}-2^{4}q^{5}+7^{2}q^{7}+433q^{9}+\cdots\)
448.6.a.p 448.a 1.a $1$ $71.852$ \(\Q\) None \(0\) \(30\) \(-32\) \(-49\) $-$ $+$ $\mathrm{SU}(2)$ \(q+30q^{3}-2^{5}q^{5}-7^{2}q^{7}+657q^{9}+\cdots\)
448.6.a.q 448.a 1.a $2$ $71.852$ \(\Q(\sqrt{177}) \) None \(0\) \(-26\) \(62\) \(-98\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-13-\beta )q^{3}+(31+5\beta )q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.r 448.a 1.a $2$ $71.852$ \(\Q(\sqrt{193}) \) None \(0\) \(-14\) \(-42\) \(98\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-7-\beta )q^{3}+(-21-5\beta )q^{5}+7^{2}q^{7}+\cdots\)
448.6.a.s 448.a 1.a $2$ $71.852$ \(\Q(\sqrt{61}) \) None \(0\) \(-14\) \(-34\) \(98\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-7-\beta )q^{3}+(-17+7\beta )q^{5}+7^{2}q^{7}+\cdots\)
448.6.a.t 448.a 1.a $2$ $71.852$ \(\Q(\sqrt{345}) \) None \(0\) \(-6\) \(-82\) \(98\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta )q^{3}+(-41+3\beta )q^{5}+7^{2}q^{7}+\cdots\)
448.6.a.u 448.a 1.a $2$ $71.852$ \(\Q(\sqrt{57}) \) None \(0\) \(-6\) \(18\) \(-98\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-3-3\beta )q^{3}+(9-5\beta )q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.v 448.a 1.a $2$ $71.852$ \(\Q(\sqrt{345}) \) None \(0\) \(6\) \(-82\) \(-98\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{3}+(-41+3\beta )q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.w 448.a 1.a $2$ $71.852$ \(\Q(\sqrt{57}) \) None \(0\) \(6\) \(18\) \(98\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(3+3\beta )q^{3}+(9-5\beta )q^{5}+7^{2}q^{7}+\cdots\)
448.6.a.x 448.a 1.a $2$ $71.852$ \(\Q(\sqrt{193}) \) None \(0\) \(14\) \(-42\) \(-98\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(7-\beta )q^{3}+(-21+5\beta )q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.y 448.a 1.a $2$ $71.852$ \(\Q(\sqrt{61}) \) None \(0\) \(14\) \(-34\) \(-98\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(7-\beta )q^{3}+(-17-7\beta )q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.z 448.a 1.a $2$ $71.852$ \(\Q(\sqrt{177}) \) None \(0\) \(26\) \(62\) \(98\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(13-\beta )q^{3}+(31-5\beta )q^{5}+7^{2}q^{7}+\cdots\)
448.6.a.ba 448.a 1.a $3$ $71.852$ 3.3.367637.1 None \(0\) \(-8\) \(14\) \(-147\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{1})q^{3}+(5+\beta _{2})q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.bb 448.a 1.a $3$ $71.852$ 3.3.367637.1 None \(0\) \(8\) \(14\) \(147\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{3}+(5+\beta _{2})q^{5}+7^{2}q^{7}+\cdots\)
448.6.a.bc 448.a 1.a $4$ $71.852$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-18\) \(30\) \(196\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-4-\beta _{1})q^{3}+(7+\beta _{1}+\beta _{2})q^{5}+\cdots\)
448.6.a.bd 448.a 1.a $4$ $71.852$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(18\) \(30\) \(-196\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(4+\beta _{1})q^{3}+(7+\beta _{1}+\beta _{2})q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.be 448.a 1.a $5$ $71.852$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-10\) \(-36\) \(-245\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{3}+(-7+\beta _{3})q^{5}-7^{2}q^{7}+\cdots\)
448.6.a.bf 448.a 1.a $5$ $71.852$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(10\) \(-36\) \(245\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{3}+(-7+\beta _{3})q^{5}+7^{2}q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(448))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(448)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 7}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 2}\)