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Properties

Label 1050.6.a
Level $1050$
Weight $6$
Character orbit 1050.a
Rep. character $\chi_{1050}(1,\cdot)$
Character field $\Q$
Dimension $94$
Newform subspaces $45$
Sturm bound $1440$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1050.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 45 \)
Sturm bound: \(1440\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(1050))\).

Total New Old
Modular forms 1224 94 1130
Cusp forms 1176 94 1082
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.

\(2\)\(3\)\(5\)\(7\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(+\)\(5\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(6\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(6\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(6\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(6\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(6\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(6\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(7\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(5\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(7\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(7\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(5\)
Plus space\(+\)\(42\)
Minus space\(-\)\(52\)

Trace form

\( 94 q + 8 q^{2} + 1504 q^{4} + 128 q^{8} + 7614 q^{9} + O(q^{10}) \) \( 94 q + 8 q^{2} + 1504 q^{4} + 128 q^{8} + 7614 q^{9} + 712 q^{11} - 1580 q^{13} + 24064 q^{16} - 3420 q^{17} + 648 q^{18} + 2944 q^{19} - 882 q^{21} + 992 q^{22} - 9712 q^{23} + 3120 q^{26} - 15628 q^{29} - 6704 q^{31} + 2048 q^{32} + 5184 q^{33} - 36752 q^{34} + 121824 q^{36} - 16828 q^{37} - 6016 q^{38} + 11160 q^{39} - 37876 q^{41} + 3528 q^{42} + 11640 q^{43} + 11392 q^{44} - 1024 q^{46} - 47248 q^{47} + 225694 q^{49} - 42696 q^{51} - 25280 q^{52} - 81300 q^{53} - 13104 q^{57} + 432 q^{58} - 35888 q^{59} - 136692 q^{61} + 34368 q^{62} + 385024 q^{64} + 15552 q^{66} + 134728 q^{67} - 54720 q^{68} - 46656 q^{69} + 89024 q^{71} + 10368 q^{72} - 16684 q^{73} - 29680 q^{74} + 47104 q^{76} + 47824 q^{77} - 3168 q^{78} + 42752 q^{79} + 616734 q^{81} + 13264 q^{82} + 37840 q^{83} - 14112 q^{84} - 44896 q^{86} + 9648 q^{87} + 15872 q^{88} + 388428 q^{89} - 60368 q^{91} - 155392 q^{92} + 66888 q^{93} - 84544 q^{94} - 33484 q^{97} + 19208 q^{98} + 57672 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(1050))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 7
1050.6.a.a 1050.a 1.a $1$ $168.403$ \(\Q\) None 42.6.a.f \(-4\) \(-9\) \(0\) \(-49\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.b 1050.a 1.a $1$ $168.403$ \(\Q\) None 210.6.a.j \(-4\) \(-9\) \(0\) \(49\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.c 1050.a 1.a $1$ $168.403$ \(\Q\) None 210.6.a.h \(-4\) \(9\) \(0\) \(-49\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.d 1050.a 1.a $1$ $168.403$ \(\Q\) None 210.6.a.g \(-4\) \(9\) \(0\) \(49\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.e 1050.a 1.a $1$ $168.403$ \(\Q\) None 210.6.a.i \(-4\) \(9\) \(0\) \(49\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.f 1050.a 1.a $1$ $168.403$ \(\Q\) None 42.6.a.e \(-4\) \(9\) \(0\) \(49\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.g 1050.a 1.a $1$ $168.403$ \(\Q\) None 42.6.a.c \(4\) \(-9\) \(0\) \(-49\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.h 1050.a 1.a $1$ $168.403$ \(\Q\) None 210.6.a.e \(4\) \(-9\) \(0\) \(-49\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.i 1050.a 1.a $1$ $168.403$ \(\Q\) None 210.6.a.d \(4\) \(-9\) \(0\) \(49\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.j 1050.a 1.a $1$ $168.403$ \(\Q\) None 210.6.a.f \(4\) \(-9\) \(0\) \(49\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.k 1050.a 1.a $1$ $168.403$ \(\Q\) None 42.6.a.d \(4\) \(-9\) \(0\) \(49\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.l 1050.a 1.a $1$ $168.403$ \(\Q\) None 210.6.a.c \(4\) \(9\) \(0\) \(-49\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.m 1050.a 1.a $1$ $168.403$ \(\Q\) None 210.6.a.a \(4\) \(9\) \(0\) \(-49\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.n 1050.a 1.a $1$ $168.403$ \(\Q\) None 42.6.a.a \(4\) \(9\) \(0\) \(-49\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.o 1050.a 1.a $1$ $168.403$ \(\Q\) None 42.6.a.b \(4\) \(9\) \(0\) \(49\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.p 1050.a 1.a $1$ $168.403$ \(\Q\) None 210.6.a.b \(4\) \(9\) \(0\) \(49\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.q 1050.a 1.a $2$ $168.403$ \(\Q(\sqrt{499}) \) None 1050.6.a.q \(-8\) \(-18\) \(0\) \(-98\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.r 1050.a 1.a $2$ $168.403$ \(\Q(\sqrt{27169}) \) None 210.6.a.n \(-8\) \(-18\) \(0\) \(-98\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.s 1050.a 1.a $2$ $168.403$ \(\Q(\sqrt{46}) \) None 1050.6.a.s \(-8\) \(-18\) \(0\) \(98\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.t 1050.a 1.a $2$ $168.403$ \(\Q(\sqrt{62689}) \) None 210.6.a.o \(-8\) \(-18\) \(0\) \(98\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.u 1050.a 1.a $2$ $168.403$ \(\Q(\sqrt{176089}) \) None 210.6.a.m \(-8\) \(18\) \(0\) \(-98\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.v 1050.a 1.a $2$ $168.403$ \(\Q(\sqrt{2059}) \) None 1050.6.a.v \(-8\) \(18\) \(0\) \(-98\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.w 1050.a 1.a $2$ $168.403$ \(\Q(\sqrt{274}) \) None 1050.6.a.w \(-8\) \(18\) \(0\) \(98\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.x 1050.a 1.a $2$ $168.403$ \(\Q(\sqrt{274}) \) None 1050.6.a.w \(8\) \(-18\) \(0\) \(-98\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.y 1050.a 1.a $2$ $168.403$ \(\Q(\sqrt{1066}) \) None 210.6.a.l \(8\) \(-18\) \(0\) \(-98\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.z 1050.a 1.a $2$ $168.403$ \(\Q(\sqrt{2059}) \) None 1050.6.a.v \(8\) \(-18\) \(0\) \(98\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.ba 1050.a 1.a $2$ $168.403$ \(\Q(\sqrt{46}) \) None 1050.6.a.s \(8\) \(18\) \(0\) \(-98\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.bb 1050.a 1.a $2$ $168.403$ \(\Q(\sqrt{499}) \) None 1050.6.a.q \(8\) \(18\) \(0\) \(98\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.bc 1050.a 1.a $2$ $168.403$ \(\Q(\sqrt{116209}) \) None 210.6.a.k \(8\) \(18\) \(0\) \(98\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.bd 1050.a 1.a $3$ $168.403$ 3.3.74004.1 None 210.6.g.a \(-12\) \(-27\) \(0\) \(-147\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.be 1050.a 1.a $3$ $168.403$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 1050.6.a.be \(-12\) \(-27\) \(0\) \(-147\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.bf 1050.a 1.a $3$ $168.403$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 1050.6.a.bf \(-12\) \(-27\) \(0\) \(147\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.bg 1050.a 1.a $3$ $168.403$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 1050.6.a.bg \(-12\) \(27\) \(0\) \(-147\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.bh 1050.a 1.a $3$ $168.403$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 1050.6.a.bh \(-12\) \(27\) \(0\) \(147\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.bi 1050.a 1.a $3$ $168.403$ 3.3.9428.1 None 210.6.g.b \(-12\) \(27\) \(0\) \(147\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.bj 1050.a 1.a $3$ $168.403$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 1050.6.a.bh \(12\) \(-27\) \(0\) \(-147\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.bk 1050.a 1.a $3$ $168.403$ 3.3.9428.1 None 210.6.g.b \(12\) \(-27\) \(0\) \(-147\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.bl 1050.a 1.a $3$ $168.403$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 1050.6.a.bg \(12\) \(-27\) \(0\) \(147\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.bm 1050.a 1.a $3$ $168.403$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 1050.6.a.bf \(12\) \(27\) \(0\) \(-147\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.bn 1050.a 1.a $3$ $168.403$ 3.3.74004.1 None 210.6.g.a \(12\) \(27\) \(0\) \(147\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.bo 1050.a 1.a $3$ $168.403$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 1050.6.a.be \(12\) \(27\) \(0\) \(147\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.bp 1050.a 1.a $4$ $168.403$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 210.6.g.c \(-16\) \(-36\) \(0\) \(196\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.bq 1050.a 1.a $4$ $168.403$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 210.6.g.d \(-16\) \(36\) \(0\) \(-196\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-6^{2}q^{6}-7^{2}q^{7}+\cdots\)
1050.6.a.br 1050.a 1.a $4$ $168.403$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 210.6.g.d \(16\) \(-36\) \(0\) \(196\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-6^{2}q^{6}+7^{2}q^{7}+\cdots\)
1050.6.a.bs 1050.a 1.a $4$ $168.403$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 210.6.g.c \(16\) \(36\) \(0\) \(-196\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+6^{2}q^{6}-7^{2}q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(1050))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(1050)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 16}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 2}\)