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Properties

Label 690.4.a
Level $690$
Weight $4$
Character orbit 690.a
Rep. character $\chi_{690}(1,\cdot)$
Character field $\Q$
Dimension $44$
Newform subspaces $20$
Sturm bound $576$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 690.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 20 \)
Sturm bound: \(576\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(690))\).

Total New Old
Modular forms 440 44 396
Cusp forms 424 44 380
Eisenstein series 16 0 16

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

\(2\)\(3\)\(5\)\(23\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(+\)\(-\)\(-\)$+$\(4\)
\(+\)\(-\)\(+\)\(+\)$-$\(3\)
\(+\)\(-\)\(+\)\(-\)$+$\(2\)
\(+\)\(-\)\(-\)\(+\)$+$\(3\)
\(+\)\(-\)\(-\)\(-\)$-$\(2\)
\(-\)\(+\)\(+\)\(+\)$-$\(3\)
\(-\)\(+\)\(+\)\(-\)$+$\(3\)
\(-\)\(+\)\(-\)\(+\)$+$\(4\)
\(-\)\(+\)\(-\)\(-\)$-$\(2\)
\(-\)\(-\)\(+\)\(+\)$+$\(3\)
\(-\)\(-\)\(+\)\(-\)$-$\(2\)
\(-\)\(-\)\(-\)\(+\)$-$\(1\)
\(-\)\(-\)\(-\)\(-\)$+$\(4\)
Plus space\(+\)\(26\)
Minus space\(-\)\(18\)

Trace form

\( 44 q - 12 q^{3} + 176 q^{4} - 56 q^{7} + 396 q^{9} + O(q^{10}) \) \( 44 q - 12 q^{3} + 176 q^{4} - 56 q^{7} + 396 q^{9} - 48 q^{12} + 160 q^{14} + 704 q^{16} + 16 q^{17} - 152 q^{19} - 168 q^{21} - 224 q^{22} + 1100 q^{25} - 108 q^{27} - 224 q^{28} - 336 q^{29} + 408 q^{31} + 384 q^{33} + 80 q^{35} + 1584 q^{36} + 472 q^{37} - 992 q^{38} + 168 q^{39} + 576 q^{41} + 392 q^{43} + 1824 q^{47} - 192 q^{48} + 1268 q^{49} - 480 q^{51} + 1744 q^{53} + 640 q^{56} - 840 q^{57} + 464 q^{58} + 704 q^{59} - 1368 q^{61} - 1408 q^{62} - 504 q^{63} + 2816 q^{64} - 1352 q^{67} + 64 q^{68} + 560 q^{70} - 512 q^{71} - 952 q^{73} + 3328 q^{74} - 300 q^{75} - 608 q^{76} + 960 q^{77} + 624 q^{78} + 3176 q^{79} + 3564 q^{81} + 3840 q^{82} + 3664 q^{83} - 672 q^{84} - 440 q^{85} + 3904 q^{86} - 1392 q^{87} - 896 q^{88} + 1328 q^{89} + 3184 q^{91} + 576 q^{93} - 1632 q^{94} - 80 q^{95} + 1544 q^{97} + 1024 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(690))\) 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 23
690.4.a.a 690.a 1.a $1$ $40.711$ \(\Q\) None \(-2\) \(-3\) \(5\) \(-19\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
690.4.a.b 690.a 1.a $1$ $40.711$ \(\Q\) None \(-2\) \(-3\) \(5\) \(-18\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
690.4.a.c 690.a 1.a $1$ $40.711$ \(\Q\) None \(-2\) \(-3\) \(5\) \(16\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
690.4.a.d 690.a 1.a $1$ $40.711$ \(\Q\) None \(-2\) \(3\) \(-5\) \(-16\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
690.4.a.e 690.a 1.a $1$ $40.711$ \(\Q\) None \(-2\) \(3\) \(-5\) \(-5\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
690.4.a.f 690.a 1.a $1$ $40.711$ \(\Q\) None \(2\) \(3\) \(-5\) \(-20\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
690.4.a.g 690.a 1.a $1$ $40.711$ \(\Q\) None \(2\) \(3\) \(-5\) \(-5\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
690.4.a.h 690.a 1.a $1$ $40.711$ \(\Q\) None \(2\) \(3\) \(5\) \(-7\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
690.4.a.i 690.a 1.a $2$ $40.711$ \(\Q(\sqrt{6}) \) None \(-4\) \(6\) \(10\) \(-26\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
690.4.a.j 690.a 1.a $2$ $40.711$ \(\Q(\sqrt{14}) \) None \(4\) \(-6\) \(10\) \(-2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
690.4.a.k 690.a 1.a $3$ $40.711$ 3.3.471057.3 None \(-6\) \(-9\) \(-15\) \(-35\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
690.4.a.l 690.a 1.a $3$ $40.711$ 3.3.617756.1 None \(-6\) \(-9\) \(-15\) \(30\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
690.4.a.m 690.a 1.a $3$ $40.711$ 3.3.931848.1 None \(-6\) \(-9\) \(15\) \(6\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
690.4.a.n 690.a 1.a $3$ $40.711$ 3.3.207308.1 None \(-6\) \(9\) \(-15\) \(2\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
690.4.a.o 690.a 1.a $3$ $40.711$ 3.3.460593.1 None \(-6\) \(9\) \(15\) \(-3\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
690.4.a.p 690.a 1.a $3$ $40.711$ 3.3.162793.1 None \(6\) \(-9\) \(-15\) \(3\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
690.4.a.q 690.a 1.a $3$ $40.711$ 3.3.396732.1 None \(6\) \(-9\) \(-15\) \(12\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
690.4.a.r 690.a 1.a $3$ $40.711$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(6\) \(9\) \(-15\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
690.4.a.s 690.a 1.a $4$ $40.711$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(-12\) \(20\) \(7\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
690.4.a.t 690.a 1.a $4$ $40.711$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(12\) \(20\) \(26\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(690))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(690)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(230))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 2}\)