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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 7
3675.2.a.a \(1\) \(29.345\) \(\Q\) None \(-2\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+q^{9}-2q^{11}+\cdots\)
3675.2.a.b \(1\) \(29.345\) \(\Q\) None \(-2\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+q^{9}+2q^{11}+\cdots\)
3675.2.a.c \(1\) \(29.345\) \(\Q\) None \(-2\) \(1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-2q^{2}+q^{3}+2q^{4}-2q^{6}+q^{9}-2q^{11}+\cdots\)
3675.2.a.d \(1\) \(29.345\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}-q^{4}+q^{6}+3q^{8}+q^{9}+\cdots\)
3675.2.a.e \(1\) \(29.345\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}-q^{4}+q^{6}+3q^{8}+q^{9}+\cdots\)
3675.2.a.f \(1\) \(29.345\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}-q^{4}-q^{6}+3q^{8}+q^{9}+\cdots\)
3675.2.a.g \(1\) \(29.345\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}-q^{4}-q^{6}+3q^{8}+q^{9}+\cdots\)
3675.2.a.h \(1\) \(29.345\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-q^{3}-2q^{4}+q^{9}+2q^{12}+q^{13}+\cdots\)
3675.2.a.i \(1\) \(29.345\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+q^{3}-2q^{4}+q^{9}-2q^{12}-q^{13}+\cdots\)
3675.2.a.j \(1\) \(29.345\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}-q^{4}-q^{6}-3q^{8}+q^{9}+\cdots\)
3675.2.a.k \(1\) \(29.345\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}-q^{4}-q^{6}-3q^{8}+q^{9}+\cdots\)
3675.2.a.l \(1\) \(29.345\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}+q^{9}+\cdots\)
3675.2.a.m \(1\) \(29.345\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}+q^{9}+\cdots\)
3675.2.a.n \(1\) \(29.345\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}+q^{9}+\cdots\)
3675.2.a.o \(1\) \(29.345\) \(\Q\) None \(2\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q+2q^{2}-q^{3}+2q^{4}-2q^{6}+q^{9}-6q^{11}+\cdots\)
3675.2.a.p \(1\) \(29.345\) \(\Q\) None \(2\) \(1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+2q^{2}+q^{3}+2q^{4}+2q^{6}+q^{9}-6q^{11}+\cdots\)
3675.2.a.q \(1\) \(29.345\) \(\Q\) None \(2\) \(1\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+2q^{2}+q^{3}+2q^{4}+2q^{6}+q^{9}+2q^{11}+\cdots\)
3675.2.a.r \(2\) \(29.345\) \(\Q(\sqrt{5}) \) None \(-3\) \(2\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+(-1-\beta )q^{2}+q^{3}+3\beta q^{4}+(-1+\cdots)q^{6}+\cdots\)
3675.2.a.s \(2\) \(29.345\) \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}+(1+\cdots)q^{6}+\cdots\)
3675.2.a.t \(2\) \(29.345\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}+(-1+\cdots)q^{6}+\cdots\)
3675.2.a.u \(2\) \(29.345\) \(\Q(\sqrt{5}) \) None \(-1\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
3675.2.a.v \(2\) \(29.345\) \(\Q(\sqrt{5}) \) None \(-1\) \(2\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q-\beta q^{2}+q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
3675.2.a.w \(2\) \(29.345\) \(\Q(\sqrt{13}) \) None \(-1\) \(2\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-\beta q^{2}+q^{3}+(1+\beta )q^{4}-\beta q^{6}-3q^{8}+\cdots\)
3675.2.a.x \(2\) \(29.345\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q+\beta q^{2}-q^{3}-\beta q^{6}-2\beta q^{8}+q^{9}+\cdots\)
3675.2.a.y \(2\) \(29.345\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-\beta q^{2}-q^{3}+3q^{4}+\beta q^{6}-\beta q^{8}+\cdots\)
3675.2.a.z \(2\) \(29.345\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+\beta q^{2}+q^{3}+\beta q^{6}-2\beta q^{8}+q^{9}+\cdots\)
3675.2.a.ba \(2\) \(29.345\) \(\Q(\sqrt{5}) \) None \(1\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
3675.2.a.bb \(2\) \(29.345\) \(\Q(\sqrt{13}) \) None \(1\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q+\beta q^{2}-q^{3}+(1+\beta )q^{4}-\beta q^{6}+3q^{8}+\cdots\)
3675.2.a.bc \(2\) \(29.345\) \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
3675.2.a.bd \(2\) \(29.345\) \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q+(1+\beta )q^{2}-q^{3}+(1+2\beta )q^{4}+(-1+\cdots)q^{6}+\cdots\)
3675.2.a.be \(2\) \(29.345\) \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q+(1+\beta )q^{2}-q^{3}+(2+2\beta )q^{4}+(-1+\cdots)q^{6}+\cdots\)
3675.2.a.bf \(2\) \(29.345\) \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}+(1+\beta )q^{6}+\cdots\)
3675.2.a.bg \(2\) \(29.345\) \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+(1+\beta )q^{2}+q^{3}+(2+2\beta )q^{4}+(1+\beta )q^{6}+\cdots\)
3675.2.a.bh \(2\) \(29.345\) \(\Q(\sqrt{5}) \) None \(3\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q+(1+\beta )q^{2}-q^{3}+3\beta q^{4}+(-1-\beta )q^{6}+\cdots\)
3675.2.a.bi \(3\) \(29.345\) 3.3.148.1 None \(-1\) \(-3\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
3675.2.a.bj \(3\) \(29.345\) 3.3.148.1 None \(1\) \(3\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
3675.2.a.bk \(4\) \(29.345\) 4.4.4352.1 None \(-4\) \(-4\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q+(-1-\beta _{1}+\beta _{3})q^{2}-q^{3}+(2+\beta _{1}+\cdots)q^{4}+\cdots\)
3675.2.a.bl \(4\) \(29.345\) 4.4.4352.1 None \(-4\) \(4\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+(-1-\beta _{1}+\beta _{3})q^{2}+q^{3}+(2+\beta _{1}+\cdots)q^{4}+\cdots\)
3675.2.a.bm \(4\) \(29.345\) 4.4.2624.1 None \(-2\) \(-4\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+(-1+\beta _{1})q^{2}-q^{3}+\beta _{2}q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
3675.2.a.bn \(4\) \(29.345\) 4.4.11344.1 None \(-2\) \(-4\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q+(-1+\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
3675.2.a.bo \(4\) \(29.345\) 4.4.2624.1 None \(-2\) \(4\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+(-1+\beta _{1})q^{2}+q^{3}+\beta _{2}q^{4}+(-1+\cdots)q^{6}+\cdots\)
3675.2.a.bp \(4\) \(29.345\) 4.4.11344.1 None \(-2\) \(4\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+(-1+\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
3675.2.a.bq \(4\) \(29.345\) 4.4.88404.1 None \(-1\) \(-4\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
3675.2.a.br \(4\) \(29.345\) 4.4.88404.1 None \(-1\) \(4\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
3675.2.a.bs \(4\) \(29.345\) \(\Q(\zeta_{24})^+\) None \(0\) \(-4\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}-\beta _{1}q^{6}+\beta _{3}q^{8}+\cdots\)
3675.2.a.bt \(4\) \(29.345\) 4.4.4400.1 None \(0\) \(-4\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q-\beta _{2}q^{2}-q^{3}+(2-\beta _{3})q^{4}+\beta _{2}q^{6}+\cdots\)
3675.2.a.bu \(4\) \(29.345\) \(\Q(\zeta_{24})^+\) None \(0\) \(4\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+\beta _{1}q^{6}+\beta _{3}q^{8}+\cdots\)
3675.2.a.bv \(4\) \(29.345\) 4.4.4400.1 None \(0\) \(4\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q-\beta _{2}q^{2}+q^{3}+(2-\beta _{3})q^{4}-\beta _{2}q^{6}+\cdots\)
3675.2.a.bw \(4\) \(29.345\) 4.4.88404.1 None \(1\) \(-4\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
3675.2.a.bx \(4\) \(29.345\) 4.4.88404.1 None \(1\) \(4\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
3675.2.a.by \(4\) \(29.345\) 4.4.2624.1 None \(2\) \(-4\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q+(1-\beta _{1})q^{2}-q^{3}+\beta _{2}q^{4}+(-1+\beta _{1}+\cdots)q^{6}+\cdots\)
3675.2.a.bz \(4\) \(29.345\) 4.4.11344.1 None \(2\) \(-4\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+(1-\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
3675.2.a.ca \(4\) \(29.345\) 4.4.2624.1 None \(2\) \(4\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+(1-\beta _{1})q^{2}+q^{3}+\beta _{2}q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
3675.2.a.cb \(4\) \(29.345\) 4.4.11344.1 None \(2\) \(4\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+(1-\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)

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

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