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Properties

Label 7569.2.a
Level $7569$
Weight $2$
Character orbit 7569.a
Rep. character $\chi_{7569}(1,\cdot)$
Character field $\Q$
Dimension $325$
Newform subspaces $49$
Sturm bound $1740$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 7569 = 3^{2} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7569.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 49 \)
Sturm bound: \(1740\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(2\), \(5\), \(7\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(7569))\).

Total New Old
Modular forms 930 352 578
Cusp forms 811 325 486
Eisenstein series 119 27 92

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

\(3\)\(29\)FrickeDim
\(+\)\(+\)\(+\)\(61\)
\(+\)\(-\)\(-\)\(75\)
\(-\)\(+\)\(-\)\(98\)
\(-\)\(-\)\(+\)\(91\)
Plus space\(+\)\(152\)
Minus space\(-\)\(173\)

Trace form

\( 325 q + q^{2} + 317 q^{4} - 3 q^{8} + O(q^{10}) \) \( 325 q + q^{2} + 317 q^{4} - 3 q^{8} + 4 q^{10} - 2 q^{11} + 4 q^{13} - 4 q^{14} + 301 q^{16} + 6 q^{17} - 24 q^{20} + 273 q^{25} + 8 q^{26} - 10 q^{28} - 6 q^{31} + 5 q^{32} - 28 q^{34} - 4 q^{35} + 10 q^{37} + 2 q^{38} + 16 q^{40} + 10 q^{41} + 2 q^{43} - 14 q^{44} + 12 q^{46} - 14 q^{47} + 257 q^{49} + 53 q^{50} + 22 q^{52} + 24 q^{53} - 14 q^{55} + 32 q^{56} + 14 q^{59} + 14 q^{61} + 12 q^{62} + 239 q^{64} - 4 q^{65} - 18 q^{67} + 38 q^{68} + 8 q^{70} - 36 q^{71} - 10 q^{73} - 8 q^{74} - 24 q^{76} - 8 q^{77} + 2 q^{79} - 22 q^{80} + 28 q^{82} - 14 q^{83} + 24 q^{85} + 36 q^{86} - 50 q^{88} - 6 q^{89} + 20 q^{91} + 50 q^{92} + 22 q^{94} - 44 q^{95} - 34 q^{97} - 11 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(7569))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 29
7569.2.a.a 7569.a 1.a $1$ $60.439$ \(\Q\) \(\Q(\sqrt{-3}) \) 7569.2.a.a \(0\) \(0\) \(0\) \(-1\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-2q^{4}-q^{7}+2q^{13}+4q^{16}-q^{19}+\cdots\)
7569.2.a.b 7569.a 1.a $1$ $60.439$ \(\Q\) \(\Q(\sqrt{-3}) \) 7569.2.a.a \(0\) \(0\) \(0\) \(-1\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-2q^{4}-q^{7}+2q^{13}+4q^{16}+q^{19}+\cdots\)
7569.2.a.c 7569.a 1.a $2$ $60.439$ \(\Q(\sqrt{2}) \) None 29.2.a.a \(-2\) \(0\) \(2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+q^{5}+2\beta q^{7}+\cdots\)
7569.2.a.d 7569.a 1.a $2$ $60.439$ \(\Q(\sqrt{5}) \) None 841.2.a.a \(-1\) \(0\) \(-1\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}+(-2+3\beta )q^{5}+\cdots\)
7569.2.a.e 7569.a 1.a $2$ $60.439$ \(\Q(\sqrt{5}) \) None 2523.2.a.b \(-1\) \(0\) \(1\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}+(1-\beta )q^{5}+(2+\cdots)q^{7}+\cdots\)
7569.2.a.f 7569.a 1.a $2$ $60.439$ \(\Q(\sqrt{5}) \) None 87.2.c.a \(-1\) \(0\) \(2\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}+2\beta q^{5}+(-1+\cdots)q^{7}+\cdots\)
7569.2.a.g 7569.a 1.a $2$ $60.439$ \(\Q(\sqrt{5}) \) None 2523.2.a.a \(-1\) \(0\) \(3\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}+(1+\beta )q^{5}+(-1+\cdots)q^{7}+\cdots\)
7569.2.a.h 7569.a 1.a $2$ $60.439$ \(\Q(\sqrt{5}) \) None 261.2.a.a \(-1\) \(0\) \(4\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}+2q^{5}+(-1+\cdots)q^{7}+\cdots\)
7569.2.a.i 7569.a 1.a $2$ $60.439$ \(\Q(\sqrt{5}) \) None 29.2.b.a \(0\) \(0\) \(-6\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+3q^{4}-3q^{5}+2q^{7}-\beta q^{8}+\cdots\)
7569.2.a.j 7569.a 1.a $2$ $60.439$ \(\Q(\sqrt{5}) \) None 261.2.a.a \(1\) \(0\) \(-4\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}-2q^{5}+(-1+\cdots)q^{7}+\cdots\)
7569.2.a.k 7569.a 1.a $2$ $60.439$ \(\Q(\sqrt{5}) \) None 87.2.a.a \(1\) \(0\) \(-2\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}+(-2+2\beta )q^{5}+\cdots\)
7569.2.a.l 7569.a 1.a $2$ $60.439$ \(\Q(\sqrt{5}) \) None 841.2.a.a \(1\) \(0\) \(-1\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}+(-2+3\beta )q^{5}+\cdots\)
7569.2.a.m 7569.a 1.a $2$ $60.439$ \(\Q(\sqrt{5}) \) None 2523.2.a.b \(1\) \(0\) \(1\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}+(1-\beta )q^{5}+(2+\cdots)q^{7}+\cdots\)
7569.2.a.n 7569.a 1.a $2$ $60.439$ \(\Q(\sqrt{5}) \) None 87.2.c.a \(1\) \(0\) \(2\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}+2\beta q^{5}+(-1+\cdots)q^{7}+\cdots\)
7569.2.a.o 7569.a 1.a $2$ $60.439$ \(\Q(\sqrt{5}) \) None 2523.2.a.a \(1\) \(0\) \(3\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}+(1+\beta )q^{5}+(-1+\cdots)q^{7}+\cdots\)
7569.2.a.p 7569.a 1.a $3$ $60.439$ \(\Q(\zeta_{14})^+\) None 29.2.d.a \(-1\) \(0\) \(3\) \(-3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+\beta _{2}q^{4}+(2-2\beta _{1}+\beta _{2})q^{5}+\cdots\)
7569.2.a.q 7569.a 1.a $3$ $60.439$ 3.3.733.1 None 2523.2.a.i \(-1\) \(0\) \(-3\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
7569.2.a.r 7569.a 1.a $3$ $60.439$ \(\Q(\zeta_{14})^+\) None 29.2.d.a \(1\) \(0\) \(3\) \(-3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(2-2\beta _{1}+\beta _{2})q^{5}+\cdots\)
7569.2.a.s 7569.a 1.a $3$ $60.439$ 3.3.733.1 None 2523.2.a.i \(1\) \(0\) \(-3\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
7569.2.a.t 7569.a 1.a $3$ $60.439$ 3.3.229.1 None 87.2.a.b \(2\) \(0\) \(0\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{2}+(2+\beta _{1})q^{4}+2\beta _{1}q^{5}+\cdots\)
7569.2.a.u 7569.a 1.a $4$ $60.439$ \(\Q(\zeta_{15})^+\) \(\Q(\sqrt{-3}) \) 7569.2.a.u \(0\) \(0\) \(0\) \(1\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-2q^{4}+(-1+3\beta _{1}-\beta _{3})q^{7}+(4\beta _{2}+\cdots)q^{13}+\cdots\)
7569.2.a.v 7569.a 1.a $4$ $60.439$ \(\Q(\zeta_{15})^+\) \(\Q(\sqrt{-3}) \) 7569.2.a.u \(0\) \(0\) \(0\) \(1\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-2q^{4}+(-1+3\beta _{1}-\beta _{3})q^{7}+(4\beta _{2}+\cdots)q^{13}+\cdots\)
7569.2.a.w 7569.a 1.a $4$ $60.439$ 4.4.7600.1 None 7569.2.a.w \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+\beta _{3}q^{5}+(-2+\cdots)q^{7}+\cdots\)
7569.2.a.x 7569.a 1.a $4$ $60.439$ 4.4.7600.1 None 7569.2.a.w \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}-\beta _{3}q^{5}+(-2+\cdots)q^{7}+\cdots\)
7569.2.a.y 7569.a 1.a $6$ $60.439$ 6.6.11973625.1 None 841.2.a.g \(-2\) \(0\) \(2\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2-\beta _{3}+\beta _{4})q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots\)
7569.2.a.z 7569.a 1.a $6$ $60.439$ 6.6.8902000.1 None 2523.2.a.k \(-1\) \(0\) \(-5\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2+\beta _{2}-\beta _{3}+\beta _{4})q^{4}+(-1+\cdots)q^{5}+\cdots\)
7569.2.a.ba 7569.a 1.a $6$ $60.439$ 6.6.160016229.2 \(\Q(\sqrt{-87}) \) 261.2.c.c \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots\)
7569.2.a.bb 7569.a 1.a $6$ $60.439$ 6.6.8902000.1 None 2523.2.a.k \(1\) \(0\) \(-5\) \(3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2}-\beta _{3}+\beta _{4})q^{4}+(-1+\cdots)q^{5}+\cdots\)
7569.2.a.bc 7569.a 1.a $6$ $60.439$ 6.6.11973625.1 None 841.2.a.g \(2\) \(0\) \(2\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2-\beta _{3}+\beta _{4})q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots\)
7569.2.a.bd 7569.a 1.a $8$ $60.439$ 8.8.2841328125.1 None 841.2.a.i \(-4\) \(0\) \(1\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(\beta _{3}-\beta _{6})q^{5}+\cdots\)
7569.2.a.be 7569.a 1.a $8$ $60.439$ 8.8.5878828125.1 None 2523.2.a.m \(-1\) \(0\) \(9\) \(-12\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(\beta _{1}-\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
7569.2.a.bf 7569.a 1.a $8$ $60.439$ 8.8.\(\cdots\).2 None 7569.2.a.bf \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}-\beta _{6}+\cdots)q^{5}+\cdots\)
7569.2.a.bg 7569.a 1.a $8$ $60.439$ 8.8.\(\cdots\).2 None 7569.2.a.bf \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(\beta _{1}+\beta _{6}-\beta _{7})q^{5}+\cdots\)
7569.2.a.bh 7569.a 1.a $8$ $60.439$ 8.8.5878828125.1 None 2523.2.a.m \(1\) \(0\) \(9\) \(-12\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}+\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
7569.2.a.bi 7569.a 1.a $8$ $60.439$ 8.8.2841328125.1 None 841.2.a.i \(4\) \(0\) \(1\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(\beta _{3}-\beta _{6})q^{5}+\cdots\)
7569.2.a.bj 7569.a 1.a $9$ $60.439$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 87.2.g.a \(-5\) \(0\) \(4\) \(5\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(1-\beta _{1}+\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
7569.2.a.bk 7569.a 1.a $9$ $60.439$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 87.2.g.b \(-1\) \(0\) \(0\) \(5\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1-\beta _{4}+\beta _{5}+\cdots)q^{5}+\cdots\)
7569.2.a.bl 7569.a 1.a $9$ $60.439$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 87.2.g.b \(1\) \(0\) \(0\) \(5\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1-\beta _{4}+\beta _{5}+\cdots)q^{5}+\cdots\)
7569.2.a.bm 7569.a 1.a $9$ $60.439$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 87.2.g.a \(5\) \(0\) \(4\) \(5\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(1-\beta _{1}+\beta _{3}+\beta _{4})q^{4}+\cdots\)
7569.2.a.bn 7569.a 1.a $12$ $60.439$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 87.2.i.a \(-6\) \(0\) \(-2\) \(-10\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(1+\beta _{2})q^{4}+(\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)
7569.2.a.bo 7569.a 1.a $12$ $60.439$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 2523.2.a.t \(-1\) \(0\) \(-7\) \(9\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{3}-\beta _{6}+\beta _{9})q^{4}+(-1+\cdots)q^{5}+\cdots\)
7569.2.a.bp 7569.a 1.a $12$ $60.439$ 12.12.\(\cdots\).1 None 29.2.e.a \(0\) \(0\) \(-8\) \(10\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{11}q^{2}+(1-\beta _{1}-\beta _{2}+\beta _{7}-\beta _{10}+\cdots)q^{4}+\cdots\)
7569.2.a.bq 7569.a 1.a $12$ $60.439$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 261.2.k.d \(0\) \(0\) \(0\) \(-14\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{11}q^{5}+(-1+\cdots)q^{7}+\cdots\)
7569.2.a.br 7569.a 1.a $12$ $60.439$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 261.2.k.d \(0\) \(0\) \(0\) \(-14\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{11}q^{5}+(-1+\cdots)q^{7}+\cdots\)
7569.2.a.bs 7569.a 1.a $12$ $60.439$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 2523.2.a.t \(1\) \(0\) \(-7\) \(9\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{3}-\beta _{6}+\beta _{9})q^{4}+(-1+\cdots)q^{5}+\cdots\)
7569.2.a.bt 7569.a 1.a $12$ $60.439$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 87.2.i.a \(6\) \(0\) \(-2\) \(-10\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(1+\beta _{2})q^{4}+(\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)
7569.2.a.bu 7569.a 1.a $16$ $60.439$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 7569.2.a.bu \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{1}-\beta _{10}+\cdots)q^{5}+\cdots\)
7569.2.a.bv 7569.a 1.a $16$ $60.439$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 7569.2.a.bu \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{1}+\beta _{10})q^{5}+\cdots\)
7569.2.a.bw 7569.a 1.a $36$ $60.439$ None 261.2.o.c \(0\) \(0\) \(0\) \(28\) $+$ $-$ $\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(7569))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(7569)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(261))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(841))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2523))\)\(^{\oplus 2}\)