Properties

Label 2175.2.a
Level $2175$
Weight $2$
Character orbit 2175.a
Rep. character $\chi_{2175}(1,\cdot)$
Character field $\Q$
Dimension $88$
Newform subspaces $30$
Sturm bound $600$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 30 \)
Sturm bound: \(600\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2175))\).

Total New Old
Modular forms 312 88 224
Cusp forms 289 88 201
Eisenstein series 23 0 23

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
\(+\)\(+\)\(+\)\(+\)\(7\)
\(+\)\(+\)\(-\)\(-\)\(16\)
\(+\)\(-\)\(+\)\(-\)\(14\)
\(+\)\(-\)\(-\)\(+\)\(8\)
\(-\)\(+\)\(+\)\(-\)\(14\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(9\)
\(-\)\(-\)\(-\)\(-\)\(15\)
Plus space\(+\)\(29\)
Minus space\(-\)\(59\)

Trace form

\( 88 q - 2 q^{3} + 90 q^{4} + 4 q^{6} + 12 q^{8} + 88 q^{9} + O(q^{10}) \) \( 88 q - 2 q^{3} + 90 q^{4} + 4 q^{6} + 12 q^{8} + 88 q^{9} + 8 q^{11} + 2 q^{12} - 12 q^{13} + 36 q^{14} + 118 q^{16} + 12 q^{19} - 12 q^{21} + 30 q^{22} + 12 q^{23} + 6 q^{24} + 44 q^{26} - 2 q^{27} + 10 q^{28} + 4 q^{31} + 24 q^{32} - 2 q^{34} + 90 q^{36} - 8 q^{37} - 16 q^{38} - 8 q^{39} + 8 q^{41} + 14 q^{42} + 20 q^{44} + 44 q^{46} + 24 q^{47} + 2 q^{48} + 96 q^{49} + 16 q^{51} - 34 q^{52} - 4 q^{53} + 4 q^{54} + 96 q^{56} - 12 q^{57} - 2 q^{58} + 40 q^{59} + 28 q^{61} + 4 q^{62} + 132 q^{64} + 16 q^{66} + 40 q^{67} + 20 q^{68} + 32 q^{69} + 84 q^{71} + 12 q^{72} + 4 q^{74} + 56 q^{76} + 56 q^{77} + 10 q^{78} - 8 q^{79} + 88 q^{81} + 28 q^{82} + 56 q^{83} - 64 q^{84} + 84 q^{86} - 6 q^{87} + 76 q^{88} + 32 q^{89} + 4 q^{91} + 4 q^{92} + 4 q^{93} - 74 q^{94} - 24 q^{96} - 16 q^{97} + 44 q^{98} + 8 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2175))\) 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
2175.2.a.a 2175.a 1.a $1$ $17.367$ \(\Q\) None 435.2.c.a \(-2\) \(1\) \(0\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+2q^{4}-2q^{6}-2q^{7}+\cdots\)
2175.2.a.b 2175.a 1.a $1$ $17.367$ \(\Q\) None 435.2.a.d \(-1\) \(-1\) \(0\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}+q^{6}-4q^{7}+3q^{8}+\cdots\)
2175.2.a.c 2175.a 1.a $1$ $17.367$ \(\Q\) None 435.2.c.b \(-1\) \(-1\) \(0\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}+q^{6}+2q^{7}+3q^{8}+\cdots\)
2175.2.a.d 2175.a 1.a $1$ $17.367$ \(\Q\) None 435.2.a.c \(0\) \(-1\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}-2q^{7}+q^{9}+3q^{11}+\cdots\)
2175.2.a.e 2175.a 1.a $1$ $17.367$ \(\Q\) None 2175.2.a.e \(0\) \(-1\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}+q^{7}+q^{9}-2q^{11}+2q^{12}+\cdots\)
2175.2.a.f 2175.a 1.a $1$ $17.367$ \(\Q\) None 2175.2.a.e \(0\) \(1\) \(0\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-q^{7}+q^{9}-2q^{11}-2q^{12}+\cdots\)
2175.2.a.g 2175.a 1.a $1$ $17.367$ \(\Q\) None 435.2.a.b \(0\) \(1\) \(0\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}+2q^{7}+q^{9}+q^{11}-2q^{12}+\cdots\)
2175.2.a.h 2175.a 1.a $1$ $17.367$ \(\Q\) None 435.2.a.a \(1\) \(-1\) \(0\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-q^{6}+4q^{7}-3q^{8}+\cdots\)
2175.2.a.i 2175.a 1.a $1$ $17.367$ \(\Q\) None 435.2.c.b \(1\) \(1\) \(0\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}+q^{6}-2q^{7}-3q^{8}+\cdots\)
2175.2.a.j 2175.a 1.a $1$ $17.367$ \(\Q\) None 435.2.c.a \(2\) \(-1\) \(0\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+2q^{4}-2q^{6}+2q^{7}+\cdots\)
2175.2.a.k 2175.a 1.a $2$ $17.367$ \(\Q(\sqrt{2}) \) None 435.2.c.c \(-2\) \(2\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}-q^{6}-\beta q^{7}+3q^{8}+\cdots\)
2175.2.a.l 2175.a 1.a $2$ $17.367$ \(\Q(\sqrt{5}) \) None 87.2.a.a \(-1\) \(-2\) \(0\) \(4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
2175.2.a.m 2175.a 1.a $2$ $17.367$ \(\Q(\sqrt{17}) \) None 435.2.a.h \(-1\) \(-2\) \(0\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+(2+\beta )q^{4}+\beta q^{6}+(-2+\cdots)q^{7}+\cdots\)
2175.2.a.n 2175.a 1.a $2$ $17.367$ \(\Q(\sqrt{2}) \) None 2175.2.a.n \(0\) \(-2\) \(0\) \(2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{3}-\beta q^{6}+q^{7}-2\beta q^{8}+\cdots\)
2175.2.a.o 2175.a 1.a $2$ $17.367$ \(\Q(\sqrt{5}) \) None 435.2.a.g \(0\) \(-2\) \(0\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+3q^{4}+\beta q^{6}-2q^{7}+\cdots\)
2175.2.a.p 2175.a 1.a $2$ $17.367$ \(\Q(\sqrt{2}) \) None 2175.2.a.n \(0\) \(2\) \(0\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+\beta q^{6}-q^{7}-2\beta q^{8}+\cdots\)
2175.2.a.q 2175.a 1.a $2$ $17.367$ \(\Q(\sqrt{5}) \) None 435.2.a.e \(1\) \(-2\) \(0\) \(6\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
2175.2.a.r 2175.a 1.a $2$ $17.367$ \(\Q(\sqrt{21}) \) None 435.2.a.f \(1\) \(-2\) \(0\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{3}+(3+\beta )q^{4}-\beta q^{6}-q^{7}+\cdots\)
2175.2.a.s 2175.a 1.a $2$ $17.367$ \(\Q(\sqrt{2}) \) None 435.2.c.c \(2\) \(-2\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-q^{6}+\beta q^{7}-3q^{8}+\cdots\)
2175.2.a.t 2175.a 1.a $3$ $17.367$ 3.3.229.1 None 87.2.a.b \(-2\) \(3\) \(0\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}+q^{3}+(2+\beta _{1})q^{4}+\cdots\)
2175.2.a.u 2175.a 1.a $3$ $17.367$ 3.3.469.1 None 435.2.a.i \(-1\) \(3\) \(0\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
2175.2.a.v 2175.a 1.a $4$ $17.367$ 4.4.2225.1 None 435.2.a.j \(3\) \(4\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
2175.2.a.w 2175.a 1.a $5$ $17.367$ 5.5.246832.1 None 435.2.c.e \(-3\) \(-5\) \(0\) \(-8\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-q^{3}+(1-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
2175.2.a.x 2175.a 1.a $5$ $17.367$ 5.5.331312.1 None 435.2.c.d \(-2\) \(5\) \(0\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}+q^{3}+(1+\beta _{1}-\beta _{3}-\beta _{4})q^{4}+\cdots\)
2175.2.a.y 2175.a 1.a $5$ $17.367$ 5.5.331312.1 None 435.2.c.d \(2\) \(-5\) \(0\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{3}q^{2}-q^{3}+(1+\beta _{1}-\beta _{3}-\beta _{4})q^{4}+\cdots\)
2175.2.a.z 2175.a 1.a $5$ $17.367$ 5.5.246832.1 None 435.2.c.e \(3\) \(5\) \(0\) \(8\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+q^{3}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
2175.2.a.ba 2175.a 1.a $7$ $17.367$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 2175.2.a.ba \(-2\) \(-7\) \(0\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
2175.2.a.bb 2175.a 1.a $7$ $17.367$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 2175.2.a.ba \(2\) \(7\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
2175.2.a.bc 2175.a 1.a $8$ $17.367$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 2175.2.a.bc \(-2\) \(-8\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
2175.2.a.bd 2175.a 1.a $8$ $17.367$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 2175.2.a.bc \(2\) \(8\) \(0\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2175))\) into lower level spaces

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