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Properties

Label 2070.4.a
Level $2070$
Weight $4$
Character orbit 2070.a
Rep. character $\chi_{2070}(1,\cdot)$
Character field $\Q$
Dimension $110$
Newform subspaces $42$
Sturm bound $1728$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 42 \)
Sturm bound: \(1728\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2070))\).

Total New Old
Modular forms 1312 110 1202
Cusp forms 1280 110 1170
Eisenstein series 32 0 32

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.
\(+\)\(+\)\(+\)\(+\)\(+\)\(5\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(6\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(6\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(5\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(7\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(9\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(9\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(7\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(6\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(10\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(7\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(10\)
Plus space\(+\)\(60\)
Minus space\(-\)\(50\)

Trace form

\( 110 q + 4 q^{2} + 440 q^{4} - 104 q^{7} + 16 q^{8} + O(q^{10}) \) \( 110 q + 4 q^{2} + 440 q^{4} - 104 q^{7} + 16 q^{8} - 84 q^{11} - 60 q^{13} - 80 q^{14} + 1760 q^{16} - 84 q^{17} - 92 q^{19} + 40 q^{22} + 2750 q^{25} - 416 q^{28} - 248 q^{29} + 24 q^{31} + 64 q^{32} + 360 q^{34} + 100 q^{35} + 1144 q^{37} + 72 q^{38} - 268 q^{41} + 428 q^{43} - 336 q^{44} - 912 q^{47} + 6206 q^{49} + 100 q^{50} - 240 q^{52} - 872 q^{53} - 320 q^{56} - 232 q^{58} - 260 q^{59} - 3408 q^{61} + 1696 q^{62} + 7040 q^{64} - 1280 q^{65} - 1220 q^{67} - 336 q^{68} - 280 q^{70} + 792 q^{71} - 4300 q^{73} - 1664 q^{74} - 368 q^{76} + 752 q^{77} - 3088 q^{79} + 1752 q^{82} - 428 q^{83} - 1220 q^{85} - 584 q^{86} + 160 q^{88} + 1596 q^{89} + 1816 q^{91} + 3696 q^{94} + 40 q^{95} + 5612 q^{97} - 220 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2070))\) 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
2070.4.a.a 2070.a 1.a $1$ $122.134$ \(\Q\) None \(-2\) \(0\) \(-5\) \(-32\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}-2^{5}q^{7}-8q^{8}+\cdots\)
2070.4.a.b 2070.a 1.a $1$ $122.134$ \(\Q\) None \(-2\) \(0\) \(-5\) \(-7\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}-7q^{7}-8q^{8}+\cdots\)
2070.4.a.c 2070.a 1.a $1$ $122.134$ \(\Q\) None \(-2\) \(0\) \(-5\) \(2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}+2q^{7}-8q^{8}+\cdots\)
2070.4.a.d 2070.a 1.a $1$ $122.134$ \(\Q\) None \(-2\) \(0\) \(5\) \(-20\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}-20q^{7}-8q^{8}+\cdots\)
2070.4.a.e 2070.a 1.a $1$ $122.134$ \(\Q\) None \(-2\) \(0\) \(5\) \(-18\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}-18q^{7}-8q^{8}+\cdots\)
2070.4.a.f 2070.a 1.a $1$ $122.134$ \(\Q\) None \(-2\) \(0\) \(5\) \(-5\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}-5q^{7}-8q^{8}+\cdots\)
2070.4.a.g 2070.a 1.a $1$ $122.134$ \(\Q\) None \(2\) \(0\) \(-5\) \(-19\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}-19q^{7}+8q^{8}+\cdots\)
2070.4.a.h 2070.a 1.a $1$ $122.134$ \(\Q\) None \(2\) \(0\) \(-5\) \(-18\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}-18q^{7}+8q^{8}+\cdots\)
2070.4.a.i 2070.a 1.a $1$ $122.134$ \(\Q\) None \(2\) \(0\) \(-5\) \(16\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}+2^{4}q^{7}+8q^{8}+\cdots\)
2070.4.a.j 2070.a 1.a $1$ $122.134$ \(\Q\) None \(2\) \(0\) \(-5\) \(20\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}+20q^{7}+8q^{8}+\cdots\)
2070.4.a.k 2070.a 1.a $1$ $122.134$ \(\Q\) None \(2\) \(0\) \(5\) \(-16\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}-2^{4}q^{7}+8q^{8}+\cdots\)
2070.4.a.l 2070.a 1.a $1$ $122.134$ \(\Q\) None \(2\) \(0\) \(5\) \(-5\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}-5q^{7}+8q^{8}+\cdots\)
2070.4.a.m 2070.a 1.a $1$ $122.134$ \(\Q\) None \(2\) \(0\) \(5\) \(2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+2q^{7}+8q^{8}+\cdots\)
2070.4.a.n 2070.a 1.a $1$ $122.134$ \(\Q\) None \(2\) \(0\) \(5\) \(3\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+3q^{7}+8q^{8}+\cdots\)
2070.4.a.o 2070.a 1.a $1$ $122.134$ \(\Q\) None \(2\) \(0\) \(5\) \(12\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+12q^{7}+8q^{8}+\cdots\)
2070.4.a.p 2070.a 1.a $2$ $122.134$ \(\Q(\sqrt{41}) \) None \(-4\) \(0\) \(-10\) \(-26\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}+(-13-\beta )q^{7}+\cdots\)
2070.4.a.q 2070.a 1.a $2$ $122.134$ \(\Q(\sqrt{14}) \) None \(-4\) \(0\) \(-10\) \(-2\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}+(-1+4\beta )q^{7}+\cdots\)
2070.4.a.r 2070.a 1.a $2$ $122.134$ \(\Q(\sqrt{6}) \) None \(4\) \(0\) \(-10\) \(-26\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}+(-13+2\beta )q^{7}+\cdots\)
2070.4.a.s 2070.a 1.a $2$ $122.134$ \(\Q(\sqrt{73}) \) None \(4\) \(0\) \(-10\) \(-17\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}+(-8-\beta )q^{7}+\cdots\)
2070.4.a.t 2070.a 1.a $2$ $122.134$ \(\Q(\sqrt{41}) \) None \(4\) \(0\) \(10\) \(-26\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+(-13-\beta )q^{7}+\cdots\)
2070.4.a.u 2070.a 1.a $3$ $122.134$ 3.3.19544.1 None \(-6\) \(0\) \(-15\) \(-16\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}+(-6+3\beta _{1}+\cdots)q^{7}+\cdots\)
2070.4.a.v 2070.a 1.a $3$ $122.134$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-6\) \(0\) \(15\) \(-2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}+(-1+\beta _{1}-\beta _{2})q^{7}+\cdots\)
2070.4.a.w 2070.a 1.a $3$ $122.134$ 3.3.162793.1 None \(-6\) \(0\) \(15\) \(3\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}+(1-\beta _{1}+\beta _{2})q^{7}+\cdots\)
2070.4.a.x 2070.a 1.a $3$ $122.134$ 3.3.396732.1 None \(-6\) \(0\) \(15\) \(12\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}+(4-\beta _{1}+\beta _{2})q^{7}+\cdots\)
2070.4.a.y 2070.a 1.a $3$ $122.134$ 3.3.460593.1 None \(6\) \(0\) \(-15\) \(-3\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}+(-1+\beta _{2})q^{7}+\cdots\)
2070.4.a.z 2070.a 1.a $3$ $122.134$ 3.3.931848.1 None \(6\) \(0\) \(-15\) \(6\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}+(1+2\beta _{1}-\beta _{2})q^{7}+\cdots\)
2070.4.a.ba 2070.a 1.a $3$ $122.134$ 3.3.318165.1 None \(6\) \(0\) \(-15\) \(7\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}+(1+3\beta _{1}-\beta _{2})q^{7}+\cdots\)
2070.4.a.bb 2070.a 1.a $3$ $122.134$ 3.3.471057.3 None \(6\) \(0\) \(15\) \(-35\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+(-12-\beta _{1}+\cdots)q^{7}+\cdots\)
2070.4.a.bc 2070.a 1.a $3$ $122.134$ 3.3.19544.1 None \(6\) \(0\) \(15\) \(-16\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+(-6+3\beta _{1}+\cdots)q^{7}+\cdots\)
2070.4.a.bd 2070.a 1.a $3$ $122.134$ 3.3.207308.1 None \(6\) \(0\) \(15\) \(2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+(1-2\beta _{1}+\beta _{2})q^{7}+\cdots\)
2070.4.a.be 2070.a 1.a $3$ $122.134$ 3.3.617756.1 None \(6\) \(0\) \(15\) \(30\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+(10+\beta _{2})q^{7}+\cdots\)
2070.4.a.bf 2070.a 1.a $4$ $122.134$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(0\) \(-20\) \(7\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}+(2-\beta _{1})q^{7}+\cdots\)
2070.4.a.bg 2070.a 1.a $4$ $122.134$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(0\) \(-20\) \(8\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}+(1+\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)
2070.4.a.bh 2070.a 1.a $4$ $122.134$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(0\) \(-20\) \(26\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}+(6-\beta _{1})q^{7}+\cdots\)
2070.4.a.bi 2070.a 1.a $4$ $122.134$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(0\) \(20\) \(26\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}+(6-\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
2070.4.a.bj 2070.a 1.a $4$ $122.134$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(0\) \(20\) \(-1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+(2\beta _{1}+\beta _{2})q^{7}+\cdots\)
2070.4.a.bk 2070.a 1.a $5$ $122.134$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-10\) \(0\) \(-25\) \(12\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}+(2-\beta _{2})q^{7}+\cdots\)
2070.4.a.bl 2070.a 1.a $5$ $122.134$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-10\) \(0\) \(25\) \(0\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}+(-\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
2070.4.a.bm 2070.a 1.a $5$ $122.134$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(10\) \(0\) \(-25\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}+(-\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
2070.4.a.bn 2070.a 1.a $5$ $122.134$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(10\) \(0\) \(25\) \(12\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+(2-\beta _{2})q^{7}+\cdots\)
2070.4.a.bo 2070.a 1.a $6$ $122.134$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(0\) \(30\) \(0\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}-\beta _{1}q^{7}-8q^{8}+\cdots\)
2070.4.a.bp 2070.a 1.a $6$ $122.134$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(12\) \(0\) \(-30\) \(0\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}-\beta _{1}q^{7}+8q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2070))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2070)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(207))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(230))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(414))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(690))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1035))\)\(^{\oplus 2}\)