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Properties

Label 1690.4.a
Level $1690$
Weight $4$
Character orbit 1690.a
Rep. character $\chi_{1690}(1,\cdot)$
Character field $\Q$
Dimension $155$
Newform subspaces $40$
Sturm bound $1092$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1690.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 40 \)
Sturm bound: \(1092\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1690))\).

Total New Old
Modular forms 848 155 693
Cusp forms 792 155 637
Eisenstein series 56 0 56

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

\(2\)\(5\)\(13\)FrickeDim
\(+\)\(+\)\(+\)$+$\(17\)
\(+\)\(+\)\(-\)$-$\(21\)
\(+\)\(-\)\(+\)$-$\(18\)
\(+\)\(-\)\(-\)$+$\(21\)
\(-\)\(+\)\(+\)$-$\(18\)
\(-\)\(+\)\(-\)$+$\(21\)
\(-\)\(-\)\(+\)$+$\(24\)
\(-\)\(-\)\(-\)$-$\(15\)
Plus space\(+\)\(83\)
Minus space\(-\)\(72\)

Trace form

\( 155 q + 2 q^{2} + 4 q^{3} + 620 q^{4} + 5 q^{5} - 16 q^{6} + 4 q^{7} + 8 q^{8} + 1359 q^{9} + O(q^{10}) \) \( 155 q + 2 q^{2} + 4 q^{3} + 620 q^{4} + 5 q^{5} - 16 q^{6} + 4 q^{7} + 8 q^{8} + 1359 q^{9} - 10 q^{10} - 112 q^{11} + 16 q^{12} - 80 q^{14} + 40 q^{15} + 2480 q^{16} - 14 q^{17} - 150 q^{18} + 360 q^{19} + 20 q^{20} + 344 q^{21} + 48 q^{22} + 188 q^{23} - 64 q^{24} + 3875 q^{25} - 272 q^{27} + 16 q^{28} + 126 q^{29} - 200 q^{30} + 504 q^{31} + 32 q^{32} + 208 q^{33} - 12 q^{34} + 100 q^{35} + 5436 q^{36} + 358 q^{37} + 48 q^{38} - 40 q^{40} - 458 q^{41} + 32 q^{42} - 540 q^{43} - 448 q^{44} - 135 q^{45} + 320 q^{46} + 524 q^{47} + 64 q^{48} + 8511 q^{49} + 50 q^{50} - 424 q^{51} + 1578 q^{53} - 1120 q^{54} - 60 q^{55} - 320 q^{56} + 2192 q^{57} + 604 q^{58} + 896 q^{59} + 160 q^{60} - 1338 q^{61} - 240 q^{62} - 428 q^{63} + 9920 q^{64} - 80 q^{66} + 824 q^{67} - 56 q^{68} + 360 q^{69} - 40 q^{70} - 960 q^{71} - 600 q^{72} - 1358 q^{73} + 1732 q^{74} + 100 q^{75} + 1440 q^{76} - 1568 q^{77} + 2664 q^{79} + 80 q^{80} + 13907 q^{81} - 2588 q^{82} + 3048 q^{83} + 1376 q^{84} - 1110 q^{85} - 1296 q^{86} - 504 q^{87} + 192 q^{88} - 2474 q^{89} - 370 q^{90} + 752 q^{92} - 4584 q^{93} + 688 q^{94} + 460 q^{95} - 256 q^{96} - 2638 q^{97} + 3890 q^{98} + 1064 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1690))\) 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 5 13
1690.4.a.a 1690.a 1.a $1$ $99.713$ \(\Q\) None \(-2\) \(-8\) \(-5\) \(4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-8q^{3}+4q^{4}-5q^{5}+2^{4}q^{6}+\cdots\)
1690.4.a.b 1690.a 1.a $1$ $99.713$ \(\Q\) None \(-2\) \(-5\) \(5\) \(7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-5q^{3}+4q^{4}+5q^{5}+10q^{6}+\cdots\)
1690.4.a.c 1690.a 1.a $1$ $99.713$ \(\Q\) None \(-2\) \(-4\) \(5\) \(8\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-4q^{3}+4q^{4}+5q^{5}+8q^{6}+\cdots\)
1690.4.a.d 1690.a 1.a $1$ $99.713$ \(\Q\) None \(-2\) \(-2\) \(5\) \(-5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-2q^{3}+4q^{4}+5q^{5}+4q^{6}+\cdots\)
1690.4.a.e 1690.a 1.a $1$ $99.713$ \(\Q\) None \(-2\) \(2\) \(5\) \(-24\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{3}+4q^{4}+5q^{5}-4q^{6}+\cdots\)
1690.4.a.f 1690.a 1.a $1$ $99.713$ \(\Q\) None \(-2\) \(4\) \(5\) \(23\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{3}+4q^{4}+5q^{5}-8q^{6}+\cdots\)
1690.4.a.g 1690.a 1.a $1$ $99.713$ \(\Q\) None \(-2\) \(5\) \(5\) \(-19\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+5q^{3}+4q^{4}+5q^{5}-10q^{6}+\cdots\)
1690.4.a.h 1690.a 1.a $1$ $99.713$ \(\Q\) None \(2\) \(-5\) \(-5\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-5q^{3}+4q^{4}-5q^{5}-10q^{6}+\cdots\)
1690.4.a.i 1690.a 1.a $1$ $99.713$ \(\Q\) None \(2\) \(-2\) \(-5\) \(-8\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-2q^{3}+4q^{4}-5q^{5}-4q^{6}+\cdots\)
1690.4.a.j 1690.a 1.a $1$ $99.713$ \(\Q\) None \(2\) \(-2\) \(-5\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-2q^{3}+4q^{4}-5q^{5}-4q^{6}+\cdots\)
1690.4.a.k 1690.a 1.a $1$ $99.713$ \(\Q\) None \(2\) \(2\) \(-5\) \(24\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{3}+4q^{4}-5q^{5}+4q^{6}+\cdots\)
1690.4.a.l 1690.a 1.a $1$ $99.713$ \(\Q\) None \(2\) \(4\) \(-5\) \(-23\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{3}+4q^{4}-5q^{5}+8q^{6}+\cdots\)
1690.4.a.m 1690.a 1.a $1$ $99.713$ \(\Q\) None \(2\) \(5\) \(-5\) \(19\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+5q^{3}+4q^{4}-5q^{5}+10q^{6}+\cdots\)
1690.4.a.n 1690.a 1.a $2$ $99.713$ \(\Q(\sqrt{3}) \) None \(-4\) \(-2\) \(10\) \(10\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1+\beta )q^{3}+4q^{4}+5q^{5}+\cdots\)
1690.4.a.o 1690.a 1.a $2$ $99.713$ \(\Q(\sqrt{51}) \) None \(-4\) \(2\) \(10\) \(-16\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1+\beta )q^{3}+4q^{4}+5q^{5}+(-2+\cdots)q^{6}+\cdots\)
1690.4.a.p 1690.a 1.a $2$ $99.713$ \(\Q(\sqrt{10}) \) None \(-4\) \(12\) \(-10\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(6+\beta )q^{3}+4q^{4}-5q^{5}+(-12+\cdots)q^{6}+\cdots\)
1690.4.a.q 1690.a 1.a $2$ $99.713$ \(\Q(\sqrt{30}) \) None \(4\) \(-8\) \(-10\) \(20\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-4+\beta )q^{3}+4q^{4}-5q^{5}+\cdots\)
1690.4.a.r 1690.a 1.a $2$ $99.713$ \(\Q(\sqrt{3}) \) None \(4\) \(-2\) \(-10\) \(-10\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1+\beta )q^{3}+4q^{4}-5q^{5}+\cdots\)
1690.4.a.s 1690.a 1.a $2$ $99.713$ \(\Q(\sqrt{19}) \) None \(4\) \(2\) \(10\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1+\beta )q^{3}+4q^{4}+5q^{5}+(2+\cdots)q^{6}+\cdots\)
1690.4.a.t 1690.a 1.a $2$ $99.713$ \(\Q(\sqrt{5}) \) None \(4\) \(2\) \(10\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1+3\beta )q^{3}+4q^{4}+5q^{5}+\cdots\)
1690.4.a.u 1690.a 1.a $3$ $99.713$ 3.3.18708.1 None \(-6\) \(0\) \(-15\) \(14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta _{1}q^{3}+4q^{4}-5q^{5}-2\beta _{1}q^{6}+\cdots\)
1690.4.a.v 1690.a 1.a $3$ $99.713$ 3.3.134228.1 None \(-6\) \(4\) \(15\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots\)
1690.4.a.w 1690.a 1.a $3$ $99.713$ 3.3.18708.1 None \(6\) \(0\) \(15\) \(-14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta _{1}q^{3}+4q^{4}+5q^{5}+2\beta _{1}q^{6}+\cdots\)
1690.4.a.x 1690.a 1.a $3$ $99.713$ 3.3.134228.1 None \(6\) \(4\) \(-15\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1+\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots\)
1690.4.a.y 1690.a 1.a $4$ $99.713$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(4\) \(-20\) \(6\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1-\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots\)
1690.4.a.z 1690.a 1.a $4$ $99.713$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(8\) \(-20\) \(-44\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(2+\beta _{2})q^{3}+4q^{4}-5q^{5}+\cdots\)
1690.4.a.ba 1690.a 1.a $4$ $99.713$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(4\) \(20\) \(-6\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots\)
1690.4.a.bb 1690.a 1.a $4$ $99.713$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(8\) \(20\) \(44\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(2+\beta _{2})q^{3}+4q^{4}+5q^{5}+\cdots\)
1690.4.a.bc 1690.a 1.a $6$ $99.713$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(-12\) \(-30\) \(-8\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-2+\beta _{1}-\beta _{3})q^{3}+4q^{4}+\cdots\)
1690.4.a.bd 1690.a 1.a $6$ $99.713$ 6.6.1104654481.1 None \(-12\) \(-6\) \(-30\) \(44\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-\beta _{1}-\beta _{2}+\beta _{3})q^{3}+4q^{4}+\cdots\)
1690.4.a.be 1690.a 1.a $6$ $99.713$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(12\) \(-12\) \(30\) \(8\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-2+\beta _{1}-\beta _{3})q^{3}+4q^{4}+\cdots\)
1690.4.a.bf 1690.a 1.a $6$ $99.713$ 6.6.1104654481.1 None \(12\) \(-6\) \(30\) \(-44\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-\beta _{1}-\beta _{2}+\beta _{3})q^{3}+4q^{4}+\cdots\)
1690.4.a.bg 1690.a 1.a $8$ $99.713$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-16\) \(0\) \(40\) \(-10\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-\beta _{1}q^{3}+4q^{4}+5q^{5}+2\beta _{1}q^{6}+\cdots\)
1690.4.a.bh 1690.a 1.a $8$ $99.713$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(16\) \(0\) \(-40\) \(10\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-\beta _{1}q^{3}+4q^{4}-5q^{5}-2\beta _{1}q^{6}+\cdots\)
1690.4.a.bi 1690.a 1.a $9$ $99.713$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-18\) \(1\) \(45\) \(-3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-\beta _{2}q^{3}+4q^{4}+5q^{5}+2\beta _{2}q^{6}+\cdots\)
1690.4.a.bj 1690.a 1.a $9$ $99.713$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-18\) \(5\) \(45\) \(48\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1+\beta _{3})q^{3}+4q^{4}+5q^{5}+\cdots\)
1690.4.a.bk 1690.a 1.a $9$ $99.713$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(18\) \(1\) \(-45\) \(3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-\beta _{2}q^{3}+4q^{4}-5q^{5}-2\beta _{2}q^{6}+\cdots\)
1690.4.a.bl 1690.a 1.a $9$ $99.713$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(18\) \(5\) \(-45\) \(-48\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1+\beta _{3})q^{3}+4q^{4}-5q^{5}+\cdots\)
1690.4.a.bm 1690.a 1.a $12$ $99.713$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-24\) \(-2\) \(-60\) \(-7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta _{1}q^{3}+4q^{4}-5q^{5}-2\beta _{1}q^{6}+\cdots\)
1690.4.a.bn 1690.a 1.a $12$ $99.713$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(24\) \(-2\) \(60\) \(7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta _{1}q^{3}+4q^{4}+5q^{5}+2\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1690))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1690)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(845))\)\(^{\oplus 2}\)