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Properties

Label 2175.4.a
Level $2175$
Weight $4$
Character orbit 2175.a
Rep. character $\chi_{2175}(1,\cdot)$
Character field $\Q$
Dimension $266$
Newform subspaces $28$
Sturm bound $1200$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2175.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 28 \)
Sturm bound: \(1200\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2175))\).

Total New Old
Modular forms 912 266 646
Cusp forms 888 266 622
Eisenstein series 24 0 24

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

\(3\)\(5\)\(29\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(35\)
\(+\)\(+\)\(-\)\(-\)\(26\)
\(+\)\(-\)\(+\)\(-\)\(33\)
\(+\)\(-\)\(-\)\(+\)\(39\)
\(-\)\(+\)\(+\)\(-\)\(28\)
\(-\)\(+\)\(-\)\(+\)\(37\)
\(-\)\(-\)\(+\)\(+\)\(37\)
\(-\)\(-\)\(-\)\(-\)\(31\)
Plus space\(+\)\(148\)
Minus space\(-\)\(118\)

Trace form

\( 266 q + 1052 q^{4} + 12 q^{6} - 24 q^{7} - 84 q^{8} + 2394 q^{9} + O(q^{10}) \) \( 266 q + 1052 q^{4} + 12 q^{6} - 24 q^{7} - 84 q^{8} + 2394 q^{9} + 40 q^{11} - 48 q^{12} + 44 q^{13} - 60 q^{14} + 4252 q^{16} + 96 q^{17} - 24 q^{19} - 120 q^{21} - 62 q^{22} - 216 q^{23} + 414 q^{24} + 652 q^{26} + 414 q^{28} - 484 q^{31} + 288 q^{32} + 1378 q^{34} + 9468 q^{36} + 56 q^{37} + 1528 q^{38} - 816 q^{39} - 848 q^{41} - 894 q^{42} - 176 q^{43} + 1996 q^{44} + 1012 q^{46} + 792 q^{47} - 480 q^{48} + 12198 q^{49} + 408 q^{51} + 154 q^{52} - 1160 q^{53} + 108 q^{54} + 1392 q^{56} + 780 q^{57} + 116 q^{58} - 1012 q^{59} - 3376 q^{61} + 944 q^{62} - 216 q^{63} + 19842 q^{64} + 744 q^{66} - 1324 q^{67} - 380 q^{68} - 984 q^{69} - 1944 q^{71} - 756 q^{72} - 1132 q^{73} - 1768 q^{74} + 4488 q^{76} - 944 q^{77} - 1554 q^{78} + 924 q^{79} + 21546 q^{81} + 4948 q^{82} - 980 q^{83} + 936 q^{84} - 180 q^{86} + 522 q^{87} - 3304 q^{88} + 1768 q^{89} + 3892 q^{91} + 2876 q^{92} - 2796 q^{93} + 7038 q^{94} + 10764 q^{96} - 92 q^{97} + 2836 q^{98} + 360 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2175))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 29
2175.4.a.a 2175.a 1.a $1$ $128.329$ \(\Q\) None 435.4.a.c \(-5\) \(3\) \(0\) \(-16\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-5q^{2}+3q^{3}+17q^{4}-15q^{6}-2^{4}q^{7}+\cdots\)
2175.4.a.b 2175.a 1.a $1$ $128.329$ \(\Q\) None 435.4.a.b \(1\) \(-3\) \(0\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}-7q^{4}-3q^{6}-4q^{7}+\cdots\)
2175.4.a.c 2175.a 1.a $1$ $128.329$ \(\Q\) None 435.4.a.a \(2\) \(3\) \(0\) \(-29\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}-4q^{4}+6q^{6}-29q^{7}+\cdots\)
2175.4.a.d 2175.a 1.a $2$ $128.329$ \(\Q(\sqrt{34}) \) None 435.4.a.e \(0\) \(-6\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-8q^{4}+\beta q^{7}+9q^{9}+(-26+\cdots)q^{11}+\cdots\)
2175.4.a.e 2175.a 1.a $2$ $128.329$ \(\Q(\sqrt{41}) \) None 435.4.a.d \(1\) \(-6\) \(0\) \(13\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-3q^{3}+(2+\beta )q^{4}-3\beta q^{6}+\cdots\)
2175.4.a.f 2175.a 1.a $2$ $128.329$ \(\Q(\sqrt{41}) \) None 87.4.a.b \(1\) \(6\) \(0\) \(24\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+3q^{3}+(2+\beta )q^{4}+3\beta q^{6}+\cdots\)
2175.4.a.g 2175.a 1.a $2$ $128.329$ \(\Q(\sqrt{17}) \) None 87.4.a.a \(5\) \(-6\) \(0\) \(24\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta )q^{2}-3q^{3}+(5-5\beta )q^{4}+(-9+\cdots)q^{6}+\cdots\)
2175.4.a.h 2175.a 1.a $5$ $128.329$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 87.4.a.d \(-3\) \(-15\) \(0\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-3q^{3}+(6+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
2175.4.a.i 2175.a 1.a $5$ $128.329$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 87.4.a.c \(-3\) \(15\) \(0\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+3q^{3}+(6-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
2175.4.a.j 2175.a 1.a $5$ $128.329$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 435.4.a.f \(-2\) \(15\) \(0\) \(29\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(2+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
2175.4.a.k 2175.a 1.a $6$ $128.329$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 435.4.a.h \(-1\) \(-18\) \(0\) \(-47\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(8+\beta _{1}+\beta _{2})q^{4}+\cdots\)
2175.4.a.l 2175.a 1.a $6$ $128.329$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 435.4.a.g \(1\) \(18\) \(0\) \(-23\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(2+\beta _{1}+\beta _{2}+\beta _{4}+\cdots)q^{4}+\cdots\)
2175.4.a.m 2175.a 1.a $7$ $128.329$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 435.4.a.j \(1\) \(-21\) \(0\) \(37\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\cdots\)
2175.4.a.n 2175.a 1.a $7$ $128.329$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 435.4.a.i \(2\) \(21\) \(0\) \(50\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(3+\beta _{1}+\beta _{2})q^{4}+\cdots\)
2175.4.a.o 2175.a 1.a $8$ $128.329$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 435.4.a.k \(4\) \(24\) \(0\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3q^{3}+(5+\beta _{2})q^{4}+(3+\cdots)q^{6}+\cdots\)
2175.4.a.p 2175.a 1.a $10$ $128.329$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 435.4.a.l \(-4\) \(-30\) \(0\) \(-75\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(7+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
2175.4.a.q 2175.a 1.a $10$ $128.329$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 2175.4.a.q \(-1\) \(30\) \(0\) \(-12\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(2+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
2175.4.a.r 2175.a 1.a $10$ $128.329$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 2175.4.a.q \(1\) \(-30\) \(0\) \(12\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(2+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
2175.4.a.s 2175.a 1.a $12$ $128.329$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 2175.4.a.s \(-1\) \(36\) \(0\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(3+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
2175.4.a.t 2175.a 1.a $12$ $128.329$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 2175.4.a.s \(1\) \(-36\) \(0\) \(14\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(3+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
2175.4.a.u 2175.a 1.a $16$ $128.329$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 2175.4.a.u \(-3\) \(-48\) \(0\) \(12\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
2175.4.a.v 2175.a 1.a $16$ $128.329$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 2175.4.a.u \(3\) \(48\) \(0\) \(-12\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
2175.4.a.w 2175.a 1.a $18$ $128.329$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None 2175.4.a.w \(-3\) \(-54\) \(0\) \(14\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(5+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
2175.4.a.x 2175.a 1.a $18$ $128.329$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None 2175.4.a.w \(3\) \(54\) \(0\) \(-14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(5+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
2175.4.a.y 2175.a 1.a $21$ $128.329$ None 435.4.c.a \(-7\) \(63\) \(0\) \(-34\) $-$ $-$ $-$ $\mathrm{SU}(2)$
2175.4.a.z 2175.a 1.a $21$ $128.329$ None 435.4.c.b \(-5\) \(-63\) \(0\) \(-22\) $+$ $-$ $+$ $\mathrm{SU}(2)$
2175.4.a.ba 2175.a 1.a $21$ $128.329$ None 435.4.c.b \(5\) \(63\) \(0\) \(22\) $-$ $-$ $+$ $\mathrm{SU}(2)$
2175.4.a.bb 2175.a 1.a $21$ $128.329$ None 435.4.c.a \(7\) \(-63\) \(0\) \(34\) $+$ $-$ $-$ $\mathrm{SU}(2)$

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2175))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2175)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(435))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(725))\)\(^{\oplus 2}\)