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Properties

Label 3675.2.a
Level $3675$
Weight $2$
Character orbit 3675.a
Rep. character $\chi_{3675}(1,\cdot)$
Character field $\Q$
Dimension $129$
Newform subspaces $54$
Sturm bound $1120$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 54 \)
Sturm bound: \(1120\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(2\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3675))\).

Total New Old
Modular forms 608 129 479
Cusp forms 513 129 384
Eisenstein series 95 0 95

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
\(+\)\(+\)\(+\)$+$\(14\)
\(+\)\(+\)\(-\)$-$\(18\)
\(+\)\(-\)\(+\)$-$\(16\)
\(+\)\(-\)\(-\)$+$\(17\)
\(-\)\(+\)\(+\)$-$\(18\)
\(-\)\(+\)\(-\)$+$\(12\)
\(-\)\(-\)\(+\)$+$\(13\)
\(-\)\(-\)\(-\)$-$\(21\)
Plus space\(+\)\(56\)
Minus space\(-\)\(73\)

Trace form

\( 129 q - 3 q^{2} - q^{3} + 123 q^{4} - q^{6} - 3 q^{8} + 129 q^{9} + O(q^{10}) \) \( 129 q - 3 q^{2} - q^{3} + 123 q^{4} - q^{6} - 3 q^{8} + 129 q^{9} + 4 q^{11} - 7 q^{12} - 10 q^{13} + 115 q^{16} - 6 q^{17} - 3 q^{18} - 2 q^{19} - 20 q^{22} + 16 q^{23} - 9 q^{24} + 10 q^{26} - q^{27} + 10 q^{29} + 6 q^{31} - 19 q^{32} + 4 q^{33} + 26 q^{34} + 123 q^{36} + 8 q^{37} + 4 q^{38} - 2 q^{39} + 38 q^{41} + 14 q^{43} - 4 q^{44} - 24 q^{46} + 24 q^{47} + q^{48} - 6 q^{51} + 10 q^{52} - 54 q^{53} - q^{54} + 6 q^{57} - 34 q^{58} + 32 q^{59} - 4 q^{61} + 24 q^{62} + 107 q^{64} + 20 q^{66} + 26 q^{67} - 10 q^{68} - 4 q^{69} + 28 q^{71} - 3 q^{72} - 14 q^{73} - 10 q^{74} - 16 q^{76} + 22 q^{78} - 10 q^{79} + 129 q^{81} - 18 q^{82} - 20 q^{83} + 12 q^{86} + 2 q^{87} + 76 q^{88} - 2 q^{89} + 104 q^{92} + 18 q^{93} - 16 q^{94} + 31 q^{96} + 10 q^{97} + 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\) 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
3675.2.a.a 3675.a 1.a $1$ $29.345$ \(\Q\) None \(-2\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+q^{9}-2q^{11}+\cdots\)
3675.2.a.b 3675.a 1.a $1$ $29.345$ \(\Q\) None \(-2\) \(-1\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+q^{9}+2q^{11}+\cdots\)
3675.2.a.c 3675.a 1.a $1$ $29.345$ \(\Q\) None \(-2\) \(1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+2q^{4}-2q^{6}+q^{9}-2q^{11}+\cdots\)
3675.2.a.d 3675.a 1.a $1$ $29.345$ \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}+q^{6}+3q^{8}+q^{9}+\cdots\)
3675.2.a.e 3675.a 1.a $1$ $29.345$ \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}+q^{6}+3q^{8}+q^{9}+\cdots\)
3675.2.a.f 3675.a 1.a $1$ $29.345$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}-q^{6}+3q^{8}+q^{9}+\cdots\)
3675.2.a.g 3675.a 1.a $1$ $29.345$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}-q^{6}+3q^{8}+q^{9}+\cdots\)
3675.2.a.h 3675.a 1.a $1$ $29.345$ \(\Q\) None \(0\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}+q^{9}+2q^{12}+q^{13}+\cdots\)
3675.2.a.i 3675.a 1.a $1$ $29.345$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}+q^{9}-2q^{12}-q^{13}+\cdots\)
3675.2.a.j 3675.a 1.a $1$ $29.345$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-q^{6}-3q^{8}+q^{9}+\cdots\)
3675.2.a.k 3675.a 1.a $1$ $29.345$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-q^{6}-3q^{8}+q^{9}+\cdots\)
3675.2.a.l 3675.a 1.a $1$ $29.345$ \(\Q\) None \(1\) \(1\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}+q^{9}+\cdots\)
3675.2.a.m 3675.a 1.a $1$ $29.345$ \(\Q\) None \(1\) \(1\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}+q^{9}+\cdots\)
3675.2.a.n 3675.a 1.a $1$ $29.345$ \(\Q\) None \(1\) \(1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}+q^{9}+\cdots\)
3675.2.a.o 3675.a 1.a $1$ $29.345$ \(\Q\) None \(2\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+2q^{4}-2q^{6}+q^{9}-6q^{11}+\cdots\)
3675.2.a.p 3675.a 1.a $1$ $29.345$ \(\Q\) None \(2\) \(1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+q^{3}+2q^{4}+2q^{6}+q^{9}-6q^{11}+\cdots\)
3675.2.a.q 3675.a 1.a $1$ $29.345$ \(\Q\) None \(2\) \(1\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+q^{3}+2q^{4}+2q^{6}+q^{9}+2q^{11}+\cdots\)
3675.2.a.r 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{5}) \) None \(-3\) \(2\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+q^{3}+3\beta q^{4}+(-1+\cdots)q^{6}+\cdots\)
3675.2.a.s 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}+(1+\cdots)q^{6}+\cdots\)
3675.2.a.t 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}+(-1+\cdots)q^{6}+\cdots\)
3675.2.a.u 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{5}) \) None \(-1\) \(-2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
3675.2.a.v 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{5}) \) None \(-1\) \(2\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
3675.2.a.w 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{13}) \) None \(-1\) \(2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+q^{3}+(1+\beta )q^{4}-\beta q^{6}-3q^{8}+\cdots\)
3675.2.a.x 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{3}-\beta q^{6}-2\beta q^{8}+q^{9}+\cdots\)
3675.2.a.y 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+3q^{4}+\beta q^{6}-\beta q^{8}+\cdots\)
3675.2.a.z 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+\beta q^{6}-2\beta q^{8}+q^{9}+\cdots\)
3675.2.a.ba 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{5}) \) None \(1\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
3675.2.a.bb 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{13}) \) None \(1\) \(-2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{3}+(1+\beta )q^{4}-\beta q^{6}+3q^{8}+\cdots\)
3675.2.a.bc 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
3675.2.a.bd 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-q^{3}+(1+2\beta )q^{4}+(-1+\cdots)q^{6}+\cdots\)
3675.2.a.be 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-q^{3}+(2+2\beta )q^{4}+(-1+\cdots)q^{6}+\cdots\)
3675.2.a.bf 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}+(1+\beta )q^{6}+\cdots\)
3675.2.a.bg 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+q^{3}+(2+2\beta )q^{4}+(1+\beta )q^{6}+\cdots\)
3675.2.a.bh 3675.a 1.a $2$ $29.345$ \(\Q(\sqrt{5}) \) None \(3\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-q^{3}+3\beta q^{4}+(-1-\beta )q^{6}+\cdots\)
3675.2.a.bi 3675.a 1.a $3$ $29.345$ 3.3.148.1 None \(-1\) \(-3\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
3675.2.a.bj 3675.a 1.a $3$ $29.345$ 3.3.148.1 None \(1\) \(3\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
3675.2.a.bk 3675.a 1.a $4$ $29.345$ 4.4.4352.1 None \(-4\) \(-4\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1}+\beta _{3})q^{2}-q^{3}+(2+\beta _{1}+\cdots)q^{4}+\cdots\)
3675.2.a.bl 3675.a 1.a $4$ $29.345$ 4.4.4352.1 None \(-4\) \(4\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1}+\beta _{3})q^{2}+q^{3}+(2+\beta _{1}+\cdots)q^{4}+\cdots\)
3675.2.a.bm 3675.a 1.a $4$ $29.345$ 4.4.2624.1 None \(-2\) \(-4\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-q^{3}+\beta _{2}q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
3675.2.a.bn 3675.a 1.a $4$ $29.345$ 4.4.11344.1 None \(-2\) \(-4\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
3675.2.a.bo 3675.a 1.a $4$ $29.345$ 4.4.2624.1 None \(-2\) \(4\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+q^{3}+\beta _{2}q^{4}+(-1+\cdots)q^{6}+\cdots\)
3675.2.a.bp 3675.a 1.a $4$ $29.345$ 4.4.11344.1 None \(-2\) \(4\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
3675.2.a.bq 3675.a 1.a $4$ $29.345$ 4.4.88404.1 None \(-1\) \(-4\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
3675.2.a.br 3675.a 1.a $4$ $29.345$ 4.4.88404.1 None \(-1\) \(4\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
3675.2.a.bs 3675.a 1.a $4$ $29.345$ \(\Q(\zeta_{24})^+\) None \(0\) \(-4\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}-\beta _{1}q^{6}+\beta _{3}q^{8}+\cdots\)
3675.2.a.bt 3675.a 1.a $4$ $29.345$ 4.4.4400.1 None \(0\) \(-4\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}-q^{3}+(2-\beta _{3})q^{4}+\beta _{2}q^{6}+\cdots\)
3675.2.a.bu 3675.a 1.a $4$ $29.345$ \(\Q(\zeta_{24})^+\) None \(0\) \(4\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+\beta _{1}q^{6}+\beta _{3}q^{8}+\cdots\)
3675.2.a.bv 3675.a 1.a $4$ $29.345$ 4.4.4400.1 None \(0\) \(4\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+q^{3}+(2-\beta _{3})q^{4}-\beta _{2}q^{6}+\cdots\)
3675.2.a.bw 3675.a 1.a $4$ $29.345$ 4.4.88404.1 None \(1\) \(-4\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
3675.2.a.bx 3675.a 1.a $4$ $29.345$ 4.4.88404.1 None \(1\) \(4\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
3675.2.a.by 3675.a 1.a $4$ $29.345$ 4.4.2624.1 None \(2\) \(-4\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-q^{3}+\beta _{2}q^{4}+(-1+\beta _{1}+\cdots)q^{6}+\cdots\)
3675.2.a.bz 3675.a 1.a $4$ $29.345$ 4.4.11344.1 None \(2\) \(-4\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
3675.2.a.ca 3675.a 1.a $4$ $29.345$ 4.4.2624.1 None \(2\) \(4\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+q^{3}+\beta _{2}q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
3675.2.a.cb 3675.a 1.a $4$ $29.345$ 4.4.11344.1 None \(2\) \(4\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3675))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3675)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(735))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\)\(^{\oplus 2}\)