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Properties

Label 1210.4.a
Level $1210$
Weight $4$
Character orbit 1210.a
Rep. character $\chi_{1210}(1,\cdot)$
Character field $\Q$
Dimension $109$
Newform subspaces $36$
Sturm bound $792$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1210 = 2 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1210.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 36 \)
Sturm bound: \(792\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1210))\).

Total New Old
Modular forms 618 109 509
Cusp forms 570 109 461
Eisenstein series 48 0 48

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

\(2\)\(5\)\(11\)FrickeDim
\(+\)\(+\)\(+\)$+$\(12\)
\(+\)\(+\)\(-\)$-$\(15\)
\(+\)\(-\)\(+\)$-$\(13\)
\(+\)\(-\)\(-\)$+$\(15\)
\(-\)\(+\)\(+\)$-$\(12\)
\(-\)\(+\)\(-\)$+$\(15\)
\(-\)\(-\)\(+\)$+$\(17\)
\(-\)\(-\)\(-\)$-$\(10\)
Plus space\(+\)\(59\)
Minus space\(-\)\(50\)

Trace form

\( 109 q - 2 q^{2} - 4 q^{3} + 436 q^{4} + 5 q^{5} + 24 q^{6} - 4 q^{7} - 8 q^{8} + 925 q^{9} + O(q^{10}) \) \( 109 q - 2 q^{2} - 4 q^{3} + 436 q^{4} + 5 q^{5} + 24 q^{6} - 4 q^{7} - 8 q^{8} + 925 q^{9} - 10 q^{10} - 16 q^{12} - 142 q^{13} - 80 q^{14} + 40 q^{15} + 1744 q^{16} + 62 q^{17} - 74 q^{18} - 212 q^{19} + 20 q^{20} - 304 q^{21} - 148 q^{23} + 96 q^{24} + 2725 q^{25} + 116 q^{26} - 352 q^{27} - 16 q^{28} + 126 q^{29} - 200 q^{30} - 104 q^{31} - 32 q^{32} - 132 q^{34} + 100 q^{35} + 3700 q^{36} - 562 q^{37} + 928 q^{38} + 1152 q^{39} - 40 q^{40} + 994 q^{41} + 896 q^{42} + 640 q^{43} + 185 q^{45} + 800 q^{46} + 52 q^{47} - 64 q^{48} + 4505 q^{49} - 50 q^{50} + 2400 q^{51} - 568 q^{52} + 1766 q^{53} + 768 q^{54} - 320 q^{56} + 368 q^{57} + 316 q^{58} - 1244 q^{59} + 160 q^{60} + 750 q^{61} + 384 q^{62} - 1012 q^{63} + 6976 q^{64} + 610 q^{65} + 2012 q^{67} + 248 q^{68} + 48 q^{69} - 40 q^{70} - 784 q^{71} - 296 q^{72} - 1194 q^{73} - 1068 q^{74} - 100 q^{75} - 848 q^{76} - 1888 q^{78} - 1584 q^{79} + 80 q^{80} + 8693 q^{81} + 2588 q^{82} + 2888 q^{83} - 1216 q^{84} - 1450 q^{85} - 600 q^{86} - 440 q^{87} - 2366 q^{89} - 370 q^{90} + 9504 q^{91} - 592 q^{92} + 10176 q^{93} + 704 q^{94} - 500 q^{95} + 384 q^{96} + 7038 q^{97} + 2702 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1210))\) 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 11
1210.4.a.a 1210.a 1.a $1$ $71.392$ \(\Q\) None \(-2\) \(-8\) \(-5\) \(-26\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-8q^{3}+4q^{4}-5q^{5}+2^{4}q^{6}+\cdots\)
1210.4.a.b 1210.a 1.a $1$ $71.392$ \(\Q\) None \(-2\) \(-8\) \(5\) \(4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-8q^{3}+4q^{4}+5q^{5}+2^{4}q^{6}+\cdots\)
1210.4.a.c 1210.a 1.a $1$ $71.392$ \(\Q\) None \(-2\) \(-5\) \(-5\) \(11\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-5q^{3}+4q^{4}-5q^{5}+10q^{6}+\cdots\)
1210.4.a.d 1210.a 1.a $1$ $71.392$ \(\Q\) None \(-2\) \(-4\) \(-5\) \(22\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-4q^{3}+4q^{4}-5q^{5}+8q^{6}+\cdots\)
1210.4.a.e 1210.a 1.a $1$ $71.392$ \(\Q\) None \(-2\) \(1\) \(5\) \(-23\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+4q^{4}+5q^{5}-2q^{6}+\cdots\)
1210.4.a.f 1210.a 1.a $1$ $71.392$ \(\Q\) None \(-2\) \(2\) \(-5\) \(-24\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{3}+4q^{4}-5q^{5}-4q^{6}+\cdots\)
1210.4.a.g 1210.a 1.a $1$ $71.392$ \(\Q\) None \(-2\) \(7\) \(-5\) \(-11\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+7q^{3}+4q^{4}-5q^{5}-14q^{6}+\cdots\)
1210.4.a.h 1210.a 1.a $1$ $71.392$ \(\Q\) None \(-2\) \(8\) \(5\) \(12\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+8q^{3}+4q^{4}+5q^{5}-2^{4}q^{6}+\cdots\)
1210.4.a.i 1210.a 1.a $1$ $71.392$ \(\Q\) None \(2\) \(-7\) \(5\) \(35\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-7q^{3}+4q^{4}+5q^{5}-14q^{6}+\cdots\)
1210.4.a.j 1210.a 1.a $1$ $71.392$ \(\Q\) None \(2\) \(-5\) \(-5\) \(-11\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-5q^{3}+4q^{4}-5q^{5}-10q^{6}+\cdots\)
1210.4.a.k 1210.a 1.a $1$ $71.392$ \(\Q\) None \(2\) \(2\) \(-5\) \(24\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{3}+4q^{4}-5q^{5}+4q^{6}+\cdots\)
1210.4.a.l 1210.a 1.a $1$ $71.392$ \(\Q\) None \(2\) \(4\) \(-5\) \(30\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{3}+4q^{4}-5q^{5}+8q^{6}+\cdots\)
1210.4.a.m 1210.a 1.a $1$ $71.392$ \(\Q\) None \(2\) \(4\) \(5\) \(-20\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{3}+4q^{4}+5q^{5}+8q^{6}+\cdots\)
1210.4.a.n 1210.a 1.a $2$ $71.392$ \(\Q(\sqrt{22}) \) None \(-4\) \(-6\) \(10\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-3+\beta )q^{3}+4q^{4}+5q^{5}+\cdots\)
1210.4.a.o 1210.a 1.a $2$ $71.392$ \(\Q(\sqrt{31}) \) None \(-4\) \(0\) \(10\) \(36\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta q^{3}+4q^{4}+5q^{5}-2\beta q^{6}+\cdots\)
1210.4.a.p 1210.a 1.a $2$ $71.392$ \(\Q(\sqrt{22}) \) None \(4\) \(-6\) \(10\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-3+\beta )q^{3}+4q^{4}+5q^{5}+\cdots\)
1210.4.a.q 1210.a 1.a $2$ $71.392$ \(\Q(\sqrt{31}) \) None \(4\) \(0\) \(10\) \(-36\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta q^{3}+4q^{4}+5q^{5}+2\beta q^{6}+\cdots\)
1210.4.a.r 1210.a 1.a $2$ $71.392$ \(\Q(\sqrt{177}) \) None \(4\) \(7\) \(-10\) \(-27\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(4-\beta )q^{3}+4q^{4}-5q^{5}+(8+\cdots)q^{6}+\cdots\)
1210.4.a.s 1210.a 1.a $3$ $71.392$ 3.3.201876.1 None \(-6\) \(-7\) \(-15\) \(9\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-2-\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots\)
1210.4.a.t 1210.a 1.a $3$ $71.392$ 3.3.300520.1 None \(-6\) \(-3\) \(15\) \(-9\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1-\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots\)
1210.4.a.u 1210.a 1.a $3$ $71.392$ 3.3.515892.1 None \(-6\) \(1\) \(-15\) \(-21\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta _{1}q^{3}+4q^{4}-5q^{5}-2\beta _{1}q^{6}+\cdots\)
1210.4.a.v 1210.a 1.a $3$ $71.392$ 3.3.201876.1 None \(6\) \(-7\) \(-15\) \(-9\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-2-\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots\)
1210.4.a.w 1210.a 1.a $3$ $71.392$ 3.3.300520.1 None \(6\) \(-3\) \(15\) \(9\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1-\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots\)
1210.4.a.x 1210.a 1.a $3$ $71.392$ 3.3.515892.1 None \(6\) \(1\) \(-15\) \(21\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta _{1}q^{3}+4q^{4}-5q^{5}+2\beta _{1}q^{6}+\cdots\)
1210.4.a.y 1210.a 1.a $4$ $71.392$ 4.4.52525.1 None \(-8\) \(-6\) \(20\) \(25\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-2-\beta _{1}-2\beta _{2})q^{3}+4q^{4}+\cdots\)
1210.4.a.z 1210.a 1.a $4$ $71.392$ 4.4.518544.2 None \(-8\) \(6\) \(-20\) \(34\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(2+\beta _{1}+\beta _{2})q^{3}+4q^{4}-5q^{5}+\cdots\)
1210.4.a.ba 1210.a 1.a $4$ $71.392$ 4.4.52525.1 None \(8\) \(-6\) \(20\) \(-25\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-2-\beta _{1}-2\beta _{2})q^{3}+4q^{4}+\cdots\)
1210.4.a.bb 1210.a 1.a $4$ $71.392$ 4.4.518544.2 None \(8\) \(6\) \(-20\) \(-34\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(2+\beta _{1}+\beta _{2})q^{3}+4q^{4}-5q^{5}+\cdots\)
1210.4.a.bc 1210.a 1.a $6$ $71.392$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(-5\) \(-30\) \(25\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1+\beta _{2})q^{3}+4q^{4}-5q^{5}+\cdots\)
1210.4.a.bd 1210.a 1.a $6$ $71.392$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(2\) \(-30\) \(-16\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta _{1}q^{3}+4q^{4}-5q^{5}-2\beta _{1}q^{6}+\cdots\)
1210.4.a.be 1210.a 1.a $6$ $71.392$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(6\) \(30\) \(-12\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1+\beta _{1}+\beta _{2})q^{3}+4q^{4}+5q^{5}+\cdots\)
1210.4.a.bf 1210.a 1.a $6$ $71.392$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(12\) \(-5\) \(-30\) \(-25\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1+\beta _{2})q^{3}+4q^{4}-5q^{5}+\cdots\)
1210.4.a.bg 1210.a 1.a $6$ $71.392$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(12\) \(2\) \(-30\) \(16\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta _{1}q^{3}+4q^{4}-5q^{5}+2\beta _{1}q^{6}+\cdots\)
1210.4.a.bh 1210.a 1.a $6$ $71.392$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(12\) \(6\) \(30\) \(12\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1+\beta _{1}+\beta _{2})q^{3}+4q^{4}+5q^{5}+\cdots\)
1210.4.a.bi 1210.a 1.a $8$ $71.392$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-16\) \(11\) \(40\) \(-16\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots\)
1210.4.a.bj 1210.a 1.a $8$ $71.392$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(16\) \(11\) \(40\) \(16\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1210))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1210)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(605))\)\(^{\oplus 2}\)