Properties

Label 6030.2.a
Level $6030$
Weight $2$
Character orbit 6030.a
Rep. character $\chi_{6030}(1,\cdot)$
Character field $\Q$
Dimension $110$
Newform subspaces $51$
Sturm bound $2448$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6030.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 51 \)
Sturm bound: \(2448\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\), \(23\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6030))\).

Total New Old
Modular forms 1240 110 1130
Cusp forms 1209 110 1099
Eisenstein series 31 0 31

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

\(2\)\(3\)\(5\)\(67\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(4\)
\(+\)\(+\)\(+\)\(-\)$-$\(8\)
\(+\)\(+\)\(-\)\(+\)$-$\(6\)
\(+\)\(+\)\(-\)\(-\)$+$\(4\)
\(+\)\(-\)\(+\)\(+\)$-$\(9\)
\(+\)\(-\)\(+\)\(-\)$+$\(7\)
\(+\)\(-\)\(-\)\(+\)$+$\(8\)
\(+\)\(-\)\(-\)\(-\)$-$\(9\)
\(-\)\(+\)\(+\)\(+\)$-$\(6\)
\(-\)\(+\)\(+\)\(-\)$+$\(4\)
\(-\)\(+\)\(-\)\(+\)$+$\(4\)
\(-\)\(+\)\(-\)\(-\)$-$\(8\)
\(-\)\(-\)\(+\)\(+\)$+$\(8\)
\(-\)\(-\)\(+\)\(-\)$-$\(9\)
\(-\)\(-\)\(-\)\(+\)$-$\(9\)
\(-\)\(-\)\(-\)\(-\)$+$\(7\)
Plus space\(+\)\(46\)
Minus space\(-\)\(64\)

Trace form

\( 110 q + 110 q^{4} + O(q^{10}) \) \( 110 q + 110 q^{4} + 2 q^{10} - 4 q^{11} + 4 q^{14} + 110 q^{16} - 4 q^{17} + 8 q^{19} + 4 q^{22} - 12 q^{23} + 110 q^{25} - 4 q^{26} - 28 q^{29} - 8 q^{31} - 16 q^{34} - 24 q^{37} + 16 q^{38} + 2 q^{40} - 20 q^{41} - 8 q^{43} - 4 q^{44} + 12 q^{46} - 4 q^{47} + 98 q^{49} + 4 q^{56} - 8 q^{58} - 48 q^{59} - 8 q^{61} - 8 q^{62} + 110 q^{64} - 12 q^{65} + 2 q^{67} - 4 q^{68} + 4 q^{71} + 28 q^{73} + 4 q^{74} + 8 q^{76} - 32 q^{77} + 48 q^{79} + 8 q^{82} + 8 q^{83} + 24 q^{85} + 4 q^{88} + 56 q^{89} + 72 q^{91} - 12 q^{92} - 12 q^{94} - 16 q^{95} + 8 q^{97} - 16 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6030))\) 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 3 5 67
6030.2.a.a 6030.a 1.a $1$ $48.150$ \(\Q\) None \(-1\) \(0\) \(-1\) \(-1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}+q^{10}+\cdots\)
6030.2.a.b 6030.a 1.a $1$ $48.150$ \(\Q\) None \(-1\) \(0\) \(-1\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-q^{8}+q^{10}-2q^{11}+\cdots\)
6030.2.a.c 6030.a 1.a $1$ $48.150$ \(\Q\) None \(-1\) \(0\) \(-1\) \(4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+4q^{7}-q^{8}+q^{10}+\cdots\)
6030.2.a.d 6030.a 1.a $1$ $48.150$ \(\Q\) None \(-1\) \(0\) \(1\) \(-5\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-5q^{7}-q^{8}-q^{10}+\cdots\)
6030.2.a.e 6030.a 1.a $1$ $48.150$ \(\Q\) None \(-1\) \(0\) \(1\) \(-4\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-q^{10}+\cdots\)
6030.2.a.f 6030.a 1.a $1$ $48.150$ \(\Q\) None \(-1\) \(0\) \(1\) \(-2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-2q^{7}-q^{8}-q^{10}+\cdots\)
6030.2.a.g 6030.a 1.a $1$ $48.150$ \(\Q\) None \(-1\) \(0\) \(1\) \(-2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-2q^{7}-q^{8}-q^{10}+\cdots\)
6030.2.a.h 6030.a 1.a $1$ $48.150$ \(\Q\) None \(-1\) \(0\) \(1\) \(-2\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-2q^{7}-q^{8}-q^{10}+\cdots\)
6030.2.a.i 6030.a 1.a $1$ $48.150$ \(\Q\) None \(-1\) \(0\) \(1\) \(-2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-2q^{7}-q^{8}-q^{10}+\cdots\)
6030.2.a.j 6030.a 1.a $1$ $48.150$ \(\Q\) None \(-1\) \(0\) \(1\) \(1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+q^{7}-q^{8}-q^{10}+\cdots\)
6030.2.a.k 6030.a 1.a $1$ $48.150$ \(\Q\) None \(-1\) \(0\) \(1\) \(2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
6030.2.a.l 6030.a 1.a $1$ $48.150$ \(\Q\) None \(-1\) \(0\) \(1\) \(2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
6030.2.a.m 6030.a 1.a $1$ $48.150$ \(\Q\) None \(-1\) \(0\) \(1\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
6030.2.a.n 6030.a 1.a $1$ $48.150$ \(\Q\) None \(1\) \(0\) \(-1\) \(-3\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-3q^{7}+q^{8}-q^{10}+\cdots\)
6030.2.a.o 6030.a 1.a $1$ $48.150$ \(\Q\) None \(1\) \(0\) \(-1\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-2q^{7}+q^{8}-q^{10}+\cdots\)
6030.2.a.p 6030.a 1.a $1$ $48.150$ \(\Q\) None \(1\) \(0\) \(-1\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-2q^{7}+q^{8}-q^{10}+\cdots\)
6030.2.a.q 6030.a 1.a $1$ $48.150$ \(\Q\) None \(1\) \(0\) \(-1\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-2q^{7}+q^{8}-q^{10}+\cdots\)
6030.2.a.r 6030.a 1.a $1$ $48.150$ \(\Q\) None \(1\) \(0\) \(-1\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-2q^{7}+q^{8}-q^{10}+\cdots\)
6030.2.a.s 6030.a 1.a $1$ $48.150$ \(\Q\) None \(1\) \(0\) \(-1\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-q^{7}+q^{8}-q^{10}+\cdots\)
6030.2.a.t 6030.a 1.a $1$ $48.150$ \(\Q\) None \(1\) \(0\) \(-1\) \(1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+q^{7}+q^{8}-q^{10}+\cdots\)
6030.2.a.u 6030.a 1.a $1$ $48.150$ \(\Q\) None \(1\) \(0\) \(-1\) \(2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots\)
6030.2.a.v 6030.a 1.a $1$ $48.150$ \(\Q\) None \(1\) \(0\) \(-1\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots\)
6030.2.a.w 6030.a 1.a $1$ $48.150$ \(\Q\) None \(1\) \(0\) \(-1\) \(2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots\)
6030.2.a.x 6030.a 1.a $1$ $48.150$ \(\Q\) None \(1\) \(0\) \(1\) \(0\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+q^{8}+q^{10}-4q^{11}+\cdots\)
6030.2.a.y 6030.a 1.a $1$ $48.150$ \(\Q\) None \(1\) \(0\) \(1\) \(4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+4q^{7}+q^{8}+q^{10}+\cdots\)
6030.2.a.z 6030.a 1.a $2$ $48.150$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(-2\) \(-6\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+(-3+\beta )q^{7}-q^{8}+\cdots\)
6030.2.a.ba 6030.a 1.a $2$ $48.150$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(-2\) \(-4\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+(-2+2\beta )q^{7}-q^{8}+\cdots\)
6030.2.a.bb 6030.a 1.a $2$ $48.150$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(-2\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+\beta q^{7}-q^{8}+q^{10}+\cdots\)
6030.2.a.bc 6030.a 1.a $2$ $48.150$ \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(2\) \(4\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+(2+\beta )q^{7}-q^{8}+\cdots\)
6030.2.a.bd 6030.a 1.a $2$ $48.150$ \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(-2\) \(4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+(2+\beta )q^{7}+q^{8}+\cdots\)
6030.2.a.be 6030.a 1.a $2$ $48.150$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(2\) \(-4\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+(-2+2\beta )q^{7}+q^{8}+\cdots\)
6030.2.a.bf 6030.a 1.a $2$ $48.150$ \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(2\) \(-2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+(-1+\beta )q^{7}+q^{8}+\cdots\)
6030.2.a.bg 6030.a 1.a $2$ $48.150$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(2\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+\beta q^{7}+q^{8}+q^{10}+\cdots\)
6030.2.a.bh 6030.a 1.a $2$ $48.150$ \(\Q(\sqrt{17}) \) None \(2\) \(0\) \(2\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+\beta q^{7}+q^{8}+q^{10}+\cdots\)
6030.2.a.bi 6030.a 1.a $2$ $48.150$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(2\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+(1+\beta )q^{7}+q^{8}+\cdots\)
6030.2.a.bj 6030.a 1.a $3$ $48.150$ 3.3.404.1 None \(-3\) \(0\) \(-3\) \(1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+\beta _{1}q^{7}-q^{8}+q^{10}+\cdots\)
6030.2.a.bk 6030.a 1.a $3$ $48.150$ 3.3.316.1 None \(-3\) \(0\) \(-3\) \(2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+(1-\beta _{1})q^{7}-q^{8}+\cdots\)
6030.2.a.bl 6030.a 1.a $3$ $48.150$ 3.3.568.1 None \(-3\) \(0\) \(3\) \(-2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+(-1-\beta _{2})q^{7}-q^{8}+\cdots\)
6030.2.a.bm 6030.a 1.a $3$ $48.150$ 3.3.568.1 None \(-3\) \(0\) \(3\) \(-1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-\beta _{1}q^{7}-q^{8}-q^{10}+\cdots\)
6030.2.a.bn 6030.a 1.a $3$ $48.150$ 3.3.316.1 None \(-3\) \(0\) \(3\) \(2\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+(\beta _{1}-\beta _{2})q^{7}-q^{8}+\cdots\)
6030.2.a.bo 6030.a 1.a $3$ $48.150$ 3.3.756.1 None \(-3\) \(0\) \(3\) \(3\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+(1-\beta _{1})q^{7}-q^{8}+\cdots\)
6030.2.a.bp 6030.a 1.a $3$ $48.150$ 3.3.3132.1 None \(-3\) \(0\) \(3\) \(3\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+(1+\beta _{1})q^{7}-q^{8}+\cdots\)
6030.2.a.bq 6030.a 1.a $3$ $48.150$ 3.3.568.1 None \(3\) \(0\) \(-3\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+(-1-\beta _{2})q^{7}+q^{8}+\cdots\)
6030.2.a.br 6030.a 1.a $3$ $48.150$ 3.3.148.1 None \(3\) \(0\) \(-3\) \(1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+\beta _{1}q^{7}+q^{8}-q^{10}+\cdots\)
6030.2.a.bs 6030.a 1.a $4$ $48.150$ 4.4.11324.1 None \(4\) \(0\) \(-4\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+(-\beta _{2}+\beta _{3})q^{7}+\cdots\)
6030.2.a.bt 6030.a 1.a $4$ $48.150$ 4.4.15188.1 None \(4\) \(0\) \(4\) \(-5\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+(-1-\beta _{1}-\beta _{3})q^{7}+\cdots\)
6030.2.a.bu 6030.a 1.a $4$ $48.150$ 4.4.70292.1 None \(4\) \(0\) \(4\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+\beta _{1}q^{7}+q^{8}+q^{10}+\cdots\)
6030.2.a.bv 6030.a 1.a $5$ $48.150$ 5.5.6517908.1 None \(-5\) \(0\) \(-5\) \(-1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+\beta _{2}q^{7}-q^{8}+q^{10}+\cdots\)
6030.2.a.bw 6030.a 1.a $5$ $48.150$ 5.5.31460256.1 None \(5\) \(0\) \(-5\) \(5\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+(1-\beta _{2})q^{7}+q^{8}+\cdots\)
6030.2.a.bx 6030.a 1.a $8$ $48.150$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(0\) \(-8\) \(4\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+(1-\beta _{6})q^{7}-q^{8}+\cdots\)
6030.2.a.by 6030.a 1.a $8$ $48.150$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(0\) \(8\) \(4\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+(1-\beta _{6})q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6030))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6030)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(201))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(335))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(402))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(603))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(670))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1005))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1206))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2010))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3015))\)\(^{\oplus 2}\)