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Properties

Label 1155.4.a
Level $1155$
Weight $4$
Character orbit 1155.a
Rep. character $\chi_{1155}(1,\cdot)$
Character field $\Q$
Dimension $120$
Newform subspaces $23$
Sturm bound $768$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1155.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 23 \)
Sturm bound: \(768\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(2\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1155))\).

Total New Old
Modular forms 584 120 464
Cusp forms 568 120 448
Eisenstein series 16 0 16

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

\(3\)\(5\)\(7\)\(11\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(8\)
\(+\)\(+\)\(+\)\(-\)$-$\(8\)
\(+\)\(+\)\(-\)\(+\)$-$\(7\)
\(+\)\(+\)\(-\)\(-\)$+$\(9\)
\(+\)\(-\)\(+\)\(+\)$-$\(7\)
\(+\)\(-\)\(+\)\(-\)$+$\(7\)
\(+\)\(-\)\(-\)\(+\)$+$\(8\)
\(+\)\(-\)\(-\)\(-\)$-$\(6\)
\(-\)\(+\)\(+\)\(+\)$-$\(6\)
\(-\)\(+\)\(+\)\(-\)$+$\(8\)
\(-\)\(+\)\(-\)\(+\)$+$\(9\)
\(-\)\(+\)\(-\)\(-\)$-$\(5\)
\(-\)\(-\)\(+\)\(+\)$+$\(9\)
\(-\)\(-\)\(+\)\(-\)$-$\(7\)
\(-\)\(-\)\(-\)\(+\)$-$\(6\)
\(-\)\(-\)\(-\)\(-\)$+$\(10\)
Plus space\(+\)\(68\)
Minus space\(-\)\(52\)

Trace form

\( 120 q + 504 q^{4} - 24 q^{6} + 1080 q^{9} + O(q^{10}) \) \( 120 q + 504 q^{4} - 24 q^{6} + 1080 q^{9} - 120 q^{10} + 120 q^{15} + 1848 q^{16} + 256 q^{17} + 208 q^{19} + 176 q^{22} + 272 q^{23} - 648 q^{24} + 3000 q^{25} + 160 q^{26} - 320 q^{29} - 608 q^{31} - 432 q^{34} + 4536 q^{36} - 112 q^{37} + 672 q^{38} - 840 q^{40} + 1440 q^{41} + 336 q^{42} + 1168 q^{43} - 208 q^{46} - 1680 q^{47} + 5880 q^{49} - 1248 q^{51} - 288 q^{53} - 216 q^{54} - 544 q^{58} + 928 q^{59} - 360 q^{60} + 2336 q^{61} + 6112 q^{62} + 11928 q^{64} + 640 q^{65} + 3664 q^{67} + 8224 q^{68} + 2016 q^{69} + 560 q^{70} + 2144 q^{71} - 3936 q^{73} + 8320 q^{74} + 4784 q^{76} + 192 q^{78} + 704 q^{79} + 3520 q^{80} + 9720 q^{81} + 2208 q^{82} - 704 q^{83} - 480 q^{85} - 3520 q^{86} + 1728 q^{87} + 2112 q^{88} + 368 q^{89} - 1080 q^{90} - 1680 q^{91} + 352 q^{92} + 1488 q^{93} + 1680 q^{94} - 80 q^{95} - 4488 q^{96} + 3088 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1155))\) 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 11
1155.4.a.a 1155.a 1.a $1$ $68.147$ \(\Q\) None \(-3\) \(-3\) \(-5\) \(7\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}-3q^{3}+q^{4}-5q^{5}+9q^{6}+\cdots\)
1155.4.a.b 1155.a 1.a $1$ $68.147$ \(\Q\) None \(-3\) \(-3\) \(-5\) \(7\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}-3q^{3}+q^{4}-5q^{5}+9q^{6}+\cdots\)
1155.4.a.c 1155.a 1.a $1$ $68.147$ \(\Q\) None \(0\) \(-3\) \(-5\) \(7\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-8q^{4}-5q^{5}+7q^{7}+9q^{9}+\cdots\)
1155.4.a.d 1155.a 1.a $1$ $68.147$ \(\Q\) None \(1\) \(3\) \(5\) \(-7\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}-7q^{4}+5q^{5}+3q^{6}+\cdots\)
1155.4.a.e 1155.a 1.a $1$ $68.147$ \(\Q\) None \(3\) \(-3\) \(-5\) \(7\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}-3q^{3}+q^{4}-5q^{5}-9q^{6}+\cdots\)
1155.4.a.f 1155.a 1.a $1$ $68.147$ \(\Q\) None \(3\) \(3\) \(-5\) \(7\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}+3q^{3}+q^{4}-5q^{5}+9q^{6}+\cdots\)
1155.4.a.g 1155.a 1.a $3$ $68.147$ 3.3.39484.1 None \(1\) \(-9\) \(-15\) \(21\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(9-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1155.4.a.h 1155.a 1.a $4$ $68.147$ 4.4.978700.1 None \(-4\) \(12\) \(-20\) \(28\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+3q^{3}+(3-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1155.4.a.i 1155.a 1.a $4$ $68.147$ 4.4.40709.1 None \(-1\) \(-12\) \(-20\) \(-28\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}-3q^{3}+(3-\beta _{1}+\beta _{3})q^{4}+\cdots\)
1155.4.a.j 1155.a 1.a $4$ $68.147$ 4.4.7885848.1 None \(5\) \(-12\) \(-20\) \(-28\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}-3q^{3}+(5+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1155.4.a.k 1155.a 1.a $6$ $68.147$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-7\) \(18\) \(30\) \(-42\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+3q^{3}+(4+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1155.4.a.l 1155.a 1.a $6$ $68.147$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-5\) \(-18\) \(30\) \(42\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-3q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1155.4.a.m 1155.a 1.a $6$ $68.147$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-4\) \(18\) \(30\) \(42\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+3q^{3}+(2-\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)
1155.4.a.n 1155.a 1.a $6$ $68.147$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(1\) \(18\) \(-30\) \(-42\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{2})q^{4}-5q^{5}+\cdots\)
1155.4.a.o 1155.a 1.a $7$ $68.147$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-3\) \(-21\) \(35\) \(-49\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(5+\beta _{2})q^{4}+5q^{5}+\cdots\)
1155.4.a.p 1155.a 1.a $7$ $68.147$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(3\) \(-21\) \(35\) \(-49\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1155.4.a.q 1155.a 1.a $8$ $68.147$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(1\) \(-24\) \(40\) \(56\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(5+\beta _{2})q^{4}+5q^{5}+\cdots\)
1155.4.a.r 1155.a 1.a $8$ $68.147$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(2\) \(-24\) \(-40\) \(-56\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(4+\beta _{2})q^{4}-5q^{5}+\cdots\)
1155.4.a.s 1155.a 1.a $8$ $68.147$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(3\) \(24\) \(-40\) \(-56\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(5+\beta _{2})q^{4}-5q^{5}+\cdots\)
1155.4.a.t 1155.a 1.a $9$ $68.147$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-4\) \(27\) \(45\) \(-63\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1155.4.a.u 1155.a 1.a $9$ $68.147$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(1\) \(27\) \(-45\) \(63\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(6+\beta _{2})q^{4}-5q^{5}+\cdots\)
1155.4.a.v 1155.a 1.a $9$ $68.147$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(4\) \(-27\) \(-45\) \(63\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1155.4.a.w 1155.a 1.a $10$ $68.147$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(6\) \(30\) \(50\) \(70\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3q^{3}+(6-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1155))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1155)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(231))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(385))\)\(^{\oplus 2}\)