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Properties

Label 7230.2.a
Level $7230$
Weight $2$
Character orbit 7230.a
Rep. character $\chi_{7230}(1,\cdot)$
Character field $\Q$
Dimension $161$
Newform subspaces $45$
Sturm bound $2904$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 7230 = 2 \cdot 3 \cdot 5 \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7230.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 45 \)
Sturm bound: \(2904\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(7230))\).

Total New Old
Modular forms 1460 161 1299
Cusp forms 1445 161 1284
Eisenstein series 15 0 15

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

\(2\)\(3\)\(5\)\(241\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(+\)\(9\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(11\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(10\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(10\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(13\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(8\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(12\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(10\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(10\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(9\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(11\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(8\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(12\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(13\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(8\)
Plus space\(+\)\(69\)
Minus space\(-\)\(92\)

Trace form

\( 161 q + q^{2} + q^{3} + 161 q^{4} + q^{5} + q^{6} + 8 q^{7} + q^{8} + 161 q^{9} + O(q^{10}) \) \( 161 q + q^{2} + q^{3} + 161 q^{4} + q^{5} + q^{6} + 8 q^{7} + q^{8} + 161 q^{9} + q^{10} - 4 q^{11} + q^{12} - 10 q^{13} - 8 q^{14} + q^{15} + 161 q^{16} - 14 q^{17} + q^{18} + 20 q^{19} + q^{20} + 8 q^{21} - 12 q^{22} - 8 q^{23} + q^{24} + 161 q^{25} - 2 q^{26} + q^{27} + 8 q^{28} - 2 q^{29} + q^{30} - 16 q^{31} + q^{32} + 4 q^{33} + 2 q^{34} - 8 q^{35} + 161 q^{36} + 14 q^{37} - 28 q^{38} + 14 q^{39} + q^{40} - 22 q^{41} - 16 q^{42} + 4 q^{43} - 4 q^{44} + q^{45} - 8 q^{46} + q^{48} + 153 q^{49} + q^{50} - 14 q^{51} - 10 q^{52} - 10 q^{53} + q^{54} - 4 q^{55} - 8 q^{56} + 12 q^{57} - 34 q^{58} + 28 q^{59} + q^{60} - 2 q^{61} + 8 q^{63} + 161 q^{64} + 14 q^{65} - 4 q^{66} + 20 q^{67} - 14 q^{68} - 24 q^{69} - 40 q^{71} + q^{72} - 38 q^{73} - 10 q^{74} + q^{75} + 20 q^{76} - 64 q^{77} - 10 q^{78} - 48 q^{79} + q^{80} + 161 q^{81} - 38 q^{82} - 44 q^{83} + 8 q^{84} - 6 q^{85} - 20 q^{86} - 2 q^{87} - 12 q^{88} - 54 q^{89} + q^{90} - 8 q^{92} + 8 q^{93} - 16 q^{94} - 28 q^{95} + q^{96} - 46 q^{97} + 25 q^{98} - 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(7230))\) 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 5 241
7230.2.a.a 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.a \(-1\) \(-1\) \(-1\) \(-2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
7230.2.a.b 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.b \(-1\) \(-1\) \(-1\) \(-2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
7230.2.a.c 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.c \(-1\) \(-1\) \(-1\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
7230.2.a.d 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.d \(-1\) \(-1\) \(1\) \(-4\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-4q^{7}+\cdots\)
7230.2.a.e 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.e \(-1\) \(-1\) \(1\) \(-2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
7230.2.a.f 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.f \(-1\) \(-1\) \(1\) \(-2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
7230.2.a.g 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.g \(-1\) \(-1\) \(1\) \(4\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+4q^{7}+\cdots\)
7230.2.a.h 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.h \(-1\) \(1\) \(-1\) \(-2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-2q^{7}+\cdots\)
7230.2.a.i 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.i \(-1\) \(1\) \(-1\) \(2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
7230.2.a.j 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.j \(-1\) \(1\) \(1\) \(-2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-2q^{7}+\cdots\)
7230.2.a.k 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.k \(-1\) \(1\) \(1\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+2q^{7}+\cdots\)
7230.2.a.l 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.l \(-1\) \(1\) \(1\) \(5\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+5q^{7}+\cdots\)
7230.2.a.m 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.m \(1\) \(-1\) \(1\) \(-4\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-4q^{7}+\cdots\)
7230.2.a.n 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.n \(1\) \(-1\) \(1\) \(-3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-3q^{7}+\cdots\)
7230.2.a.o 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.o \(1\) \(-1\) \(1\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
7230.2.a.p 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.p \(1\) \(-1\) \(1\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
7230.2.a.q 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.q \(1\) \(-1\) \(1\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
7230.2.a.r 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.r \(1\) \(-1\) \(1\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+2q^{7}+\cdots\)
7230.2.a.s 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.s \(1\) \(-1\) \(1\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+3q^{7}+\cdots\)
7230.2.a.t 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.t \(1\) \(1\) \(1\) \(-4\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-4q^{7}+\cdots\)
7230.2.a.u 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.u \(1\) \(1\) \(1\) \(-2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
7230.2.a.v 7230.a 1.a $1$ $57.732$ \(\Q\) None 7230.2.a.v \(1\) \(1\) \(1\) \(-2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
7230.2.a.w 7230.a 1.a $2$ $57.732$ \(\Q(\sqrt{17}) \) None 7230.2.a.w \(-2\) \(2\) \(2\) \(-3\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
7230.2.a.x 7230.a 1.a $2$ $57.732$ \(\Q(\sqrt{3}) \) None 7230.2.a.x \(-2\) \(2\) \(2\) \(8\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+4q^{7}+\cdots\)
7230.2.a.y 7230.a 1.a $2$ $57.732$ \(\Q(\sqrt{2}) \) None 7230.2.a.y \(2\) \(-2\) \(-2\) \(0\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+2\beta q^{7}+\cdots\)
7230.2.a.z 7230.a 1.a $2$ $57.732$ \(\Q(\sqrt{5}) \) None 7230.2.a.z \(2\) \(2\) \(-2\) \(0\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
7230.2.a.ba 7230.a 1.a $3$ $57.732$ 3.3.148.1 None 7230.2.a.ba \(-3\) \(-3\) \(-3\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+(\beta _{1}-\beta _{2})q^{7}+\cdots\)
7230.2.a.bb 7230.a 1.a $4$ $57.732$ \(\Q(\sqrt{2}, \sqrt{5})\) None 7230.2.a.bb \(4\) \(-4\) \(-4\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+\beta _{2}q^{7}+\cdots\)
7230.2.a.bc 7230.a 1.a $5$ $57.732$ 5.5.1033709.1 None 7230.2.a.bc \(-5\) \(-5\) \(-5\) \(3\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+(1+\beta _{3}+\cdots)q^{7}+\cdots\)
7230.2.a.bd 7230.a 1.a $5$ $57.732$ 5.5.240133.1 None 7230.2.a.bd \(-5\) \(5\) \(-5\) \(-1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
7230.2.a.be 7230.a 1.a $5$ $57.732$ 5.5.4462989.1 None 7230.2.a.be \(-5\) \(5\) \(5\) \(3\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+(1+\beta _{4})q^{7}+\cdots\)
7230.2.a.bf 7230.a 1.a $5$ $57.732$ 5.5.216637.1 None 7230.2.a.bf \(5\) \(5\) \(5\) \(-7\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+(-2+\cdots)q^{7}+\cdots\)
7230.2.a.bg 7230.a 1.a $6$ $57.732$ 6.6.45040428.1 None 7230.2.a.bg \(-6\) \(-6\) \(6\) \(1\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-\beta _{2}q^{7}+\cdots\)
7230.2.a.bh 7230.a 1.a $6$ $57.732$ 6.6.354280684.1 None 7230.2.a.bh \(6\) \(-6\) \(-6\) \(1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+\beta _{3}q^{7}+\cdots\)
7230.2.a.bi 7230.a 1.a $6$ $57.732$ 6.6.7783241.1 None 7230.2.a.bi \(6\) \(-6\) \(6\) \(7\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+(2+\beta _{1}+\cdots)q^{7}+\cdots\)
7230.2.a.bj 7230.a 1.a $6$ $57.732$ 6.6.1767625.1 None 7230.2.a.bj \(6\) \(6\) \(-6\) \(-9\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+(-1+\cdots)q^{7}+\cdots\)
7230.2.a.bk 7230.a 1.a $6$ $57.732$ 6.6.624373996.1 None 7230.2.a.bk \(6\) \(6\) \(-6\) \(3\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+\beta _{1}q^{7}+\cdots\)
7230.2.a.bl 7230.a 1.a $6$ $57.732$ 6.6.19680064.1 None 7230.2.a.bl \(6\) \(6\) \(-6\) \(8\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
7230.2.a.bm 7230.a 1.a $7$ $57.732$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 7230.2.a.bm \(7\) \(-7\) \(7\) \(-3\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+(-\beta _{5}+\cdots)q^{7}+\cdots\)
7230.2.a.bn 7230.a 1.a $8$ $57.732$ 8.8.5817206249.1 None 7230.2.a.bn \(-8\) \(8\) \(8\) \(-11\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
7230.2.a.bo 7230.a 1.a $8$ $57.732$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 7230.2.a.bo \(8\) \(-8\) \(-8\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+(\beta _{1}-\beta _{5}+\cdots)q^{7}+\cdots\)
7230.2.a.bp 7230.a 1.a $9$ $57.732$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 7230.2.a.bp \(-9\) \(-9\) \(-9\) \(-1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+(\beta _{4}+\beta _{7}+\cdots)q^{7}+\cdots\)
7230.2.a.bq 7230.a 1.a $10$ $57.732$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 7230.2.a.bq \(-10\) \(-10\) \(10\) \(3\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+\beta _{5}q^{7}+\cdots\)
7230.2.a.br 7230.a 1.a $13$ $57.732$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None 7230.2.a.br \(-13\) \(13\) \(-13\) \(9\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+(1-\beta _{8}+\cdots)q^{7}+\cdots\)
7230.2.a.bs 7230.a 1.a $13$ $57.732$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None 7230.2.a.bs \(13\) \(13\) \(13\) \(11\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+(1+\beta _{4}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(7230))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(7230)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(241))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(482))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(723))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1205))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1446))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2410))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3615))\)\(^{\oplus 2}\)