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Properties

Label 1470.4.a
Level $1470$
Weight $4$
Character orbit 1470.a
Rep. character $\chi_{1470}(1,\cdot)$
Character field $\Q$
Dimension $82$
Newform subspaces $51$
Sturm bound $1344$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1470.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 51 \)
Sturm bound: \(1344\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1470))\).

Total New Old
Modular forms 1040 82 958
Cusp forms 976 82 894
Eisenstein series 64 0 64

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

\(2\)\(3\)\(5\)\(7\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(6\)
\(+\)\(+\)\(+\)\(-\)$-$\(5\)
\(+\)\(+\)\(-\)\(+\)$-$\(5\)
\(+\)\(+\)\(-\)\(-\)$+$\(5\)
\(+\)\(-\)\(+\)\(+\)$-$\(3\)
\(+\)\(-\)\(+\)\(-\)$+$\(7\)
\(+\)\(-\)\(-\)\(+\)$+$\(6\)
\(+\)\(-\)\(-\)\(-\)$-$\(4\)
\(-\)\(+\)\(+\)\(+\)$-$\(5\)
\(-\)\(+\)\(+\)\(-\)$+$\(5\)
\(-\)\(+\)\(-\)\(+\)$+$\(6\)
\(-\)\(+\)\(-\)\(-\)$-$\(5\)
\(-\)\(-\)\(+\)\(+\)$+$\(6\)
\(-\)\(-\)\(+\)\(-\)$-$\(4\)
\(-\)\(-\)\(-\)\(+\)$-$\(3\)
\(-\)\(-\)\(-\)\(-\)$+$\(7\)
Plus space\(+\)\(48\)
Minus space\(-\)\(34\)

Trace form

\( 82 q - 6 q^{3} + 328 q^{4} + 738 q^{9} + O(q^{10}) \) \( 82 q - 6 q^{3} + 328 q^{4} + 738 q^{9} + 20 q^{10} - 52 q^{11} - 24 q^{12} - 32 q^{13} + 1312 q^{16} - 56 q^{17} - 224 q^{19} - 200 q^{22} + 72 q^{23} + 2050 q^{25} - 232 q^{26} - 54 q^{27} + 104 q^{29} - 60 q^{30} + 144 q^{31} - 60 q^{33} + 168 q^{34} + 2952 q^{36} - 1600 q^{37} - 560 q^{38} - 1608 q^{39} + 80 q^{40} + 204 q^{41} - 1120 q^{43} - 208 q^{44} - 160 q^{46} + 1328 q^{47} - 96 q^{48} + 696 q^{51} - 128 q^{52} + 1824 q^{53} + 420 q^{55} + 696 q^{57} - 360 q^{58} - 868 q^{59} - 84 q^{61} - 1136 q^{62} + 5248 q^{64} - 460 q^{65} + 72 q^{66} + 2056 q^{67} - 224 q^{68} - 1176 q^{69} + 2216 q^{71} + 3924 q^{73} + 776 q^{74} - 150 q^{75} - 896 q^{76} - 840 q^{78} + 5856 q^{79} + 6642 q^{81} + 96 q^{82} + 5792 q^{83} - 1460 q^{85} + 2880 q^{86} + 648 q^{87} - 800 q^{88} + 1652 q^{89} + 180 q^{90} + 288 q^{92} + 7344 q^{93} - 1904 q^{94} + 1080 q^{95} - 36 q^{97} - 468 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1470))\) 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 7
1470.4.a.a 1470.a 1.a $1$ $86.733$ \(\Q\) None \(-2\) \(-3\) \(-5\) \(0\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
1470.4.a.b 1470.a 1.a $1$ $86.733$ \(\Q\) None \(-2\) \(-3\) \(-5\) \(0\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
1470.4.a.c 1470.a 1.a $1$ $86.733$ \(\Q\) None \(-2\) \(-3\) \(-5\) \(0\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
1470.4.a.d 1470.a 1.a $1$ $86.733$ \(\Q\) None \(-2\) \(-3\) \(-5\) \(0\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
1470.4.a.e 1470.a 1.a $1$ $86.733$ \(\Q\) None \(-2\) \(-3\) \(-5\) \(0\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
1470.4.a.f 1470.a 1.a $1$ $86.733$ \(\Q\) None \(-2\) \(-3\) \(5\) \(0\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
1470.4.a.g 1470.a 1.a $1$ $86.733$ \(\Q\) None \(-2\) \(-3\) \(5\) \(0\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
1470.4.a.h 1470.a 1.a $1$ $86.733$ \(\Q\) None \(-2\) \(-3\) \(5\) \(0\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
1470.4.a.i 1470.a 1.a $1$ $86.733$ \(\Q\) None \(-2\) \(3\) \(-5\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
1470.4.a.j 1470.a 1.a $1$ $86.733$ \(\Q\) None \(-2\) \(3\) \(-5\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
1470.4.a.k 1470.a 1.a $1$ $86.733$ \(\Q\) None \(-2\) \(3\) \(-5\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
1470.4.a.l 1470.a 1.a $1$ $86.733$ \(\Q\) None \(-2\) \(3\) \(5\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
1470.4.a.m 1470.a 1.a $1$ $86.733$ \(\Q\) None \(-2\) \(3\) \(5\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
1470.4.a.n 1470.a 1.a $1$ $86.733$ \(\Q\) None \(-2\) \(3\) \(5\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
1470.4.a.o 1470.a 1.a $1$ $86.733$ \(\Q\) None \(-2\) \(3\) \(5\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
1470.4.a.p 1470.a 1.a $1$ $86.733$ \(\Q\) None \(2\) \(-3\) \(-5\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
1470.4.a.q 1470.a 1.a $1$ $86.733$ \(\Q\) None \(2\) \(-3\) \(-5\) \(0\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
1470.4.a.r 1470.a 1.a $1$ $86.733$ \(\Q\) None \(2\) \(-3\) \(5\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
1470.4.a.s 1470.a 1.a $1$ $86.733$ \(\Q\) None \(2\) \(-3\) \(5\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
1470.4.a.t 1470.a 1.a $1$ $86.733$ \(\Q\) None \(2\) \(-3\) \(5\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
1470.4.a.u 1470.a 1.a $1$ $86.733$ \(\Q\) None \(2\) \(-3\) \(5\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
1470.4.a.v 1470.a 1.a $1$ $86.733$ \(\Q\) None \(2\) \(-3\) \(5\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
1470.4.a.w 1470.a 1.a $1$ $86.733$ \(\Q\) None \(2\) \(3\) \(-5\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
1470.4.a.x 1470.a 1.a $1$ $86.733$ \(\Q\) None \(2\) \(3\) \(-5\) \(0\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
1470.4.a.y 1470.a 1.a $1$ $86.733$ \(\Q\) None \(2\) \(3\) \(-5\) \(0\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
1470.4.a.z 1470.a 1.a $1$ $86.733$ \(\Q\) None \(2\) \(3\) \(-5\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
1470.4.a.ba 1470.a 1.a $1$ $86.733$ \(\Q\) None \(2\) \(3\) \(5\) \(0\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
1470.4.a.bb 1470.a 1.a $1$ $86.733$ \(\Q\) None \(2\) \(3\) \(5\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
1470.4.a.bc 1470.a 1.a $1$ $86.733$ \(\Q\) None \(2\) \(3\) \(5\) \(0\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
1470.4.a.bd 1470.a 1.a $1$ $86.733$ \(\Q\) None \(2\) \(3\) \(5\) \(0\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
1470.4.a.be 1470.a 1.a $2$ $86.733$ \(\Q(\sqrt{15}) \) None \(-4\) \(-6\) \(-10\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
1470.4.a.bf 1470.a 1.a $2$ $86.733$ \(\Q(\sqrt{505}) \) None \(-4\) \(-6\) \(10\) \(0\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
1470.4.a.bg 1470.a 1.a $2$ $86.733$ \(\Q(\sqrt{2}) \) None \(-4\) \(-6\) \(10\) \(0\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
1470.4.a.bh 1470.a 1.a $2$ $86.733$ \(\Q(\sqrt{505}) \) None \(-4\) \(6\) \(-10\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
1470.4.a.bi 1470.a 1.a $2$ $86.733$ \(\Q(\sqrt{2}) \) None \(-4\) \(6\) \(-10\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
1470.4.a.bj 1470.a 1.a $2$ $86.733$ \(\Q(\sqrt{15}) \) None \(-4\) \(6\) \(10\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
1470.4.a.bk 1470.a 1.a $2$ $86.733$ \(\Q(\sqrt{2}) \) None \(4\) \(-6\) \(-10\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
1470.4.a.bl 1470.a 1.a $2$ $86.733$ \(\Q(\sqrt{3441}) \) None \(4\) \(-6\) \(-10\) \(0\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
1470.4.a.bm 1470.a 1.a $2$ $86.733$ \(\Q(\sqrt{106}) \) None \(4\) \(-6\) \(-10\) \(0\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
1470.4.a.bn 1470.a 1.a $2$ $86.733$ \(\Q(\sqrt{295}) \) None \(4\) \(-6\) \(-10\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
1470.4.a.bo 1470.a 1.a $2$ $86.733$ \(\Q(\sqrt{46}) \) None \(4\) \(-6\) \(10\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
1470.4.a.bp 1470.a 1.a $2$ $86.733$ \(\Q(\sqrt{46}) \) None \(4\) \(6\) \(-10\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
1470.4.a.bq 1470.a 1.a $2$ $86.733$ \(\Q(\sqrt{2}) \) None \(4\) \(6\) \(10\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
1470.4.a.br 1470.a 1.a $2$ $86.733$ \(\Q(\sqrt{3441}) \) None \(4\) \(6\) \(10\) \(0\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
1470.4.a.bs 1470.a 1.a $2$ $86.733$ \(\Q(\sqrt{295}) \) None \(4\) \(6\) \(10\) \(0\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
1470.4.a.bt 1470.a 1.a $3$ $86.733$ 3.3.792520.1 None \(-6\) \(-9\) \(15\) \(0\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
1470.4.a.bu 1470.a 1.a $3$ $86.733$ 3.3.792520.1 None \(-6\) \(9\) \(-15\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
1470.4.a.bv 1470.a 1.a $4$ $86.733$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(-12\) \(-20\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
1470.4.a.bw 1470.a 1.a $4$ $86.733$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(12\) \(20\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
1470.4.a.bx 1470.a 1.a $4$ $86.733$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(-12\) \(20\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
1470.4.a.by 1470.a 1.a $4$ $86.733$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(12\) \(-20\) \(0\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1470))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1470)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(490))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(735))\)\(^{\oplus 2}\)