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Properties

Label 735.4.a
Level $735$
Weight $4$
Character orbit 735.a
Rep. character $\chi_{735}(1,\cdot)$
Character field $\Q$
Dimension $82$
Newform subspaces $29$
Sturm bound $448$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 735.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 29 \)
Sturm bound: \(448\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(735))\).

Total New Old
Modular forms 352 82 270
Cusp forms 320 82 238
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.

\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(11\)
\(+\)\(+\)\(-\)\(-\)\(10\)
\(+\)\(-\)\(+\)\(-\)\(7\)
\(+\)\(-\)\(-\)\(+\)\(13\)
\(-\)\(+\)\(+\)\(-\)\(9\)
\(-\)\(+\)\(-\)\(+\)\(11\)
\(-\)\(-\)\(+\)\(+\)\(13\)
\(-\)\(-\)\(-\)\(-\)\(8\)
Plus space\(+\)\(48\)
Minus space\(-\)\(34\)

Trace form

\( 82 q - 4 q^{2} + 354 q^{4} - 6 q^{6} + 36 q^{8} + 738 q^{9} + O(q^{10}) \) \( 82 q - 4 q^{2} + 354 q^{4} - 6 q^{6} + 36 q^{8} + 738 q^{9} - 50 q^{10} + 52 q^{11} + 24 q^{12} - 64 q^{13} + 30 q^{15} + 1338 q^{16} + 24 q^{17} - 36 q^{18} + 248 q^{19} + 40 q^{20} + 196 q^{22} + 368 q^{23} - 342 q^{24} + 2050 q^{25} - 84 q^{26} - 312 q^{29} + 60 q^{30} - 200 q^{31} + 1124 q^{32} - 36 q^{33} + 780 q^{34} + 3186 q^{36} + 1152 q^{37} + 1040 q^{38} + 1020 q^{39} - 450 q^{40} - 372 q^{41} + 1512 q^{43} + 1468 q^{44} + 1480 q^{46} - 504 q^{47} + 912 q^{48} - 100 q^{50} - 780 q^{51} + 296 q^{52} - 1184 q^{53} - 54 q^{54} - 740 q^{55} - 528 q^{57} + 244 q^{58} + 180 q^{59} - 90 q^{60} + 988 q^{61} + 384 q^{62} + 7234 q^{64} + 580 q^{65} - 996 q^{66} - 400 q^{67} + 2936 q^{68} + 1152 q^{69} + 152 q^{71} + 324 q^{72} - 2124 q^{73} + 140 q^{74} + 3936 q^{76} + 3984 q^{78} - 4768 q^{79} + 1200 q^{80} + 6642 q^{81} + 1984 q^{82} - 3368 q^{83} + 780 q^{85} + 12208 q^{86} - 1164 q^{87} + 9148 q^{88} + 2036 q^{89} - 450 q^{90} + 8872 q^{92} - 2304 q^{93} + 288 q^{94} - 520 q^{95} - 2262 q^{96} + 3196 q^{97} + 468 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(735))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 7
735.4.a.a 735.a 1.a $1$ $43.366$ \(\Q\) None \(-4\) \(-3\) \(-5\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-3q^{3}+8q^{4}-5q^{5}+12q^{6}+\cdots\)
735.4.a.b 735.a 1.a $1$ $43.366$ \(\Q\) None \(-4\) \(3\) \(5\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+3q^{3}+8q^{4}+5q^{5}-12q^{6}+\cdots\)
735.4.a.c 735.a 1.a $1$ $43.366$ \(\Q\) None \(0\) \(3\) \(-5\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-8q^{4}-5q^{5}+9q^{9}+42q^{11}+\cdots\)
735.4.a.d 735.a 1.a $1$ $43.366$ \(\Q\) None \(1\) \(-3\) \(-5\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}-7q^{4}-5q^{5}-3q^{6}+\cdots\)
735.4.a.e 735.a 1.a $1$ $43.366$ \(\Q\) None \(1\) \(-3\) \(-5\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}-7q^{4}-5q^{5}-3q^{6}+\cdots\)
735.4.a.f 735.a 1.a $1$ $43.366$ \(\Q\) None \(1\) \(3\) \(5\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}-7q^{4}+5q^{5}+3q^{6}+\cdots\)
735.4.a.g 735.a 1.a $1$ $43.366$ \(\Q\) None \(3\) \(-3\) \(-5\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}-3q^{3}+q^{4}-5q^{5}-9q^{6}+\cdots\)
735.4.a.h 735.a 1.a $1$ $43.366$ \(\Q\) None \(3\) \(3\) \(5\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}+3q^{3}+q^{4}+5q^{5}+9q^{6}+\cdots\)
735.4.a.i 735.a 1.a $1$ $43.366$ \(\Q\) None \(3\) \(3\) \(5\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}+3q^{3}+q^{4}+5q^{5}+9q^{6}+\cdots\)
735.4.a.j 735.a 1.a $1$ $43.366$ \(\Q\) None \(5\) \(3\) \(-5\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+5q^{2}+3q^{3}+17q^{4}-5q^{5}+15q^{6}+\cdots\)
735.4.a.k 735.a 1.a $2$ $43.366$ \(\Q(\sqrt{17}) \) None \(-7\) \(6\) \(10\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta )q^{2}+3q^{3}+(5+7\beta )q^{4}+\cdots\)
735.4.a.l 735.a 1.a $2$ $43.366$ \(\Q(\sqrt{2}) \) None \(-4\) \(-6\) \(-10\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{2}-3q^{3}+(-2-4\beta )q^{4}+\cdots\)
735.4.a.m 735.a 1.a $2$ $43.366$ \(\Q(\sqrt{5}) \) None \(-4\) \(-6\) \(10\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta )q^{2}-3q^{3}+(1+4\beta )q^{4}+\cdots\)
735.4.a.n 735.a 1.a $2$ $43.366$ \(\Q(\sqrt{2}) \) None \(-4\) \(6\) \(10\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{2}+3q^{3}+(-2-4\beta )q^{4}+\cdots\)
735.4.a.o 735.a 1.a $2$ $43.366$ \(\Q(\sqrt{2}) \) None \(-2\) \(6\) \(-10\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+3q^{3}+(1-2\beta )q^{4}+\cdots\)
735.4.a.p 735.a 1.a $2$ $43.366$ \(\Q(\sqrt{65}) \) None \(1\) \(-6\) \(-10\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-3q^{3}+(8+\beta )q^{4}-5q^{5}-3\beta q^{6}+\cdots\)
735.4.a.q 735.a 1.a $2$ $43.366$ \(\Q(\sqrt{41}) \) None \(3\) \(-6\) \(10\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-3q^{3}+(3+3\beta )q^{4}+5q^{5}+\cdots\)
735.4.a.r 735.a 1.a $3$ $43.366$ 3.3.4892.1 None \(3\) \(-9\) \(15\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-3q^{3}+(-\beta _{1}+\beta _{2})q^{4}+\cdots\)
735.4.a.s 735.a 1.a $3$ $43.366$ 3.3.4892.1 None \(3\) \(9\) \(-15\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3q^{3}+(-\beta _{1}+\beta _{2})q^{4}+\cdots\)
735.4.a.t 735.a 1.a $4$ $43.366$ 4.4.51264.1 None \(-6\) \(-12\) \(20\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1}+\beta _{2})q^{2}-3q^{3}+(6-\beta _{1}+\cdots)q^{4}+\cdots\)
735.4.a.u 735.a 1.a $4$ $43.366$ 4.4.51264.1 None \(-6\) \(12\) \(-20\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1}+\beta _{2})q^{2}+3q^{3}+(6-\beta _{1}+\cdots)q^{4}+\cdots\)
735.4.a.v 735.a 1.a $4$ $43.366$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(1\) \(-12\) \(20\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(10+\beta _{2})q^{4}+5q^{5}+\cdots\)
735.4.a.w 735.a 1.a $4$ $43.366$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(1\) \(12\) \(-20\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(10+\beta _{2})q^{4}-5q^{5}+\cdots\)
735.4.a.x 735.a 1.a $5$ $43.366$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-1\) \(-15\) \(25\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
735.4.a.y 735.a 1.a $5$ $43.366$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-1\) \(15\) \(-25\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
735.4.a.z 735.a 1.a $5$ $43.366$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(3\) \(-15\) \(-25\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-3q^{3}+(5+\beta _{2})q^{4}-5q^{5}+\cdots\)
735.4.a.ba 735.a 1.a $5$ $43.366$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(3\) \(15\) \(25\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3q^{3}+(5+\beta _{2})q^{4}+5q^{5}+\cdots\)
735.4.a.bb 735.a 1.a $8$ $43.366$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(2\) \(-24\) \(-40\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(6+\beta _{1}+\beta _{2})q^{4}+\cdots\)
735.4.a.bc 735.a 1.a $8$ $43.366$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(2\) \(24\) \(40\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(6+\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(735))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(735)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 2}\)