Properties

Label 3450.2.a
Level $3450$
Weight $2$
Character orbit 3450.a
Rep. character $\chi_{3450}(1,\cdot)$
Character field $\Q$
Dimension $70$
Newform subspaces $46$
Sturm bound $1440$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 46 \)
Sturm bound: \(1440\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3450))\).

Total New Old
Modular forms 744 70 674
Cusp forms 697 70 627
Eisenstein series 47 0 47

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

\(2\)\(3\)\(5\)\(23\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(5\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(5\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(6\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(6\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(6\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(7\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(25\)
Minus space\(-\)\(45\)

Trace form

\( 70q - 2q^{2} + 2q^{3} + 70q^{4} - 2q^{6} - 2q^{8} + 70q^{9} + O(q^{10}) \) \( 70q - 2q^{2} + 2q^{3} + 70q^{4} - 2q^{6} - 2q^{8} + 70q^{9} + 2q^{12} + 4q^{13} + 8q^{14} + 70q^{16} + 4q^{17} - 2q^{18} + 20q^{19} + 20q^{21} + 4q^{22} - 2q^{24} + 28q^{26} + 2q^{27} + 28q^{29} + 16q^{31} - 2q^{32} + 4q^{33} + 28q^{34} + 70q^{36} - 8q^{37} + 8q^{38} + 28q^{39} + 4q^{41} - 4q^{42} + 28q^{43} - 4q^{46} - 8q^{47} + 2q^{48} + 102q^{49} + 8q^{51} + 4q^{52} - 12q^{53} - 2q^{54} + 8q^{56} - 4q^{57} - 20q^{58} - 8q^{59} + 56q^{61} + 8q^{62} + 70q^{64} - 12q^{67} + 4q^{68} - 4q^{69} - 56q^{71} - 2q^{72} - 20q^{73} + 20q^{74} + 20q^{76} + 16q^{77} - 4q^{78} + 40q^{79} + 70q^{81} + 20q^{82} + 8q^{83} + 20q^{84} - 16q^{86} + 12q^{87} + 4q^{88} + 44q^{89} + 48q^{91} + 32q^{93} - 8q^{94} - 2q^{96} + 36q^{97} - 18q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3450))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5 23
3450.2.a.a \(1\) \(27.548\) \(\Q\) None \(-1\) \(-1\) \(0\) \(-5\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-5q^{7}-q^{8}+\cdots\)
3450.2.a.b \(1\) \(27.548\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
3450.2.a.c \(1\) \(27.548\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
3450.2.a.d \(1\) \(27.548\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
3450.2.a.e \(1\) \(27.548\) \(\Q\) None \(-1\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+q^{7}-q^{8}+\cdots\)
3450.2.a.f \(1\) \(27.548\) \(\Q\) None \(-1\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+q^{7}-q^{8}+\cdots\)
3450.2.a.g \(1\) \(27.548\) \(\Q\) None \(-1\) \(-1\) \(0\) \(3\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+3q^{7}-q^{8}+\cdots\)
3450.2.a.h \(1\) \(27.548\) \(\Q\) None \(-1\) \(1\) \(0\) \(-3\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-3q^{7}-q^{8}+\cdots\)
3450.2.a.i \(1\) \(27.548\) \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
3450.2.a.j \(1\) \(27.548\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
3450.2.a.k \(1\) \(27.548\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
3450.2.a.l \(1\) \(27.548\) \(\Q\) None \(-1\) \(1\) \(0\) \(2\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
3450.2.a.m \(1\) \(27.548\) \(\Q\) None \(-1\) \(1\) \(0\) \(5\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}+5q^{7}-q^{8}+\cdots\)
3450.2.a.n \(1\) \(27.548\) \(\Q\) None \(1\) \(-1\) \(0\) \(-5\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}-5q^{7}+q^{8}+\cdots\)
3450.2.a.o \(1\) \(27.548\) \(\Q\) None \(1\) \(-1\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
3450.2.a.p \(1\) \(27.548\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
3450.2.a.q \(1\) \(27.548\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
3450.2.a.r \(1\) \(27.548\) \(\Q\) None \(1\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots\)
3450.2.a.s \(1\) \(27.548\) \(\Q\) None \(1\) \(-1\) \(0\) \(3\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+3q^{7}+q^{8}+\cdots\)
3450.2.a.t \(1\) \(27.548\) \(\Q\) None \(1\) \(1\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
3450.2.a.u \(1\) \(27.548\) \(\Q\) None \(1\) \(1\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
3450.2.a.v \(1\) \(27.548\) \(\Q\) None \(1\) \(1\) \(0\) \(-3\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}-3q^{7}+q^{8}+\cdots\)
3450.2.a.w \(1\) \(27.548\) \(\Q\) None \(1\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}-q^{7}+q^{8}+\cdots\)
3450.2.a.x \(1\) \(27.548\) \(\Q\) None \(1\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}-q^{7}+q^{8}+\cdots\)
3450.2.a.y \(1\) \(27.548\) \(\Q\) None \(1\) \(1\) \(0\) \(2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots\)
3450.2.a.z \(1\) \(27.548\) \(\Q\) None \(1\) \(1\) \(0\) \(2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots\)
3450.2.a.ba \(1\) \(27.548\) \(\Q\) None \(1\) \(1\) \(0\) \(2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots\)
3450.2.a.bb \(1\) \(27.548\) \(\Q\) None \(1\) \(1\) \(0\) \(5\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+5q^{7}+q^{8}+\cdots\)
3450.2.a.bc \(2\) \(27.548\) \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(0\) \(-8\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-4q^{7}-q^{8}+\cdots\)
3450.2.a.bd \(2\) \(27.548\) \(\Q(\sqrt{57}) \) None \(-2\) \(-2\) \(0\) \(-6\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-3q^{7}-q^{8}+\cdots\)
3450.2.a.be \(2\) \(27.548\) \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-2\beta q^{7}-q^{8}+\cdots\)
3450.2.a.bf \(2\) \(27.548\) \(\Q(\sqrt{6}) \) None \(-2\) \(-2\) \(0\) \(2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+(1+\beta )q^{7}+\cdots\)
3450.2.a.bg \(2\) \(27.548\) \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(4\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots\)
3450.2.a.bh \(2\) \(27.548\) \(\Q(\sqrt{73}) \) None \(-2\) \(2\) \(0\) \(-6\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-3q^{7}-q^{8}+\cdots\)
3450.2.a.bi \(2\) \(27.548\) \(\Q(\sqrt{17}) \) None \(-2\) \(2\) \(0\) \(2\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}+(1+\beta )q^{7}+\cdots\)
3450.2.a.bj \(2\) \(27.548\) \(\Q(\sqrt{73}) \) None \(2\) \(-2\) \(0\) \(6\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+3q^{7}+q^{8}+\cdots\)
3450.2.a.bk \(2\) \(27.548\) \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(-4\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots\)
3450.2.a.bl \(2\) \(27.548\) \(\Q(\sqrt{6}) \) None \(2\) \(2\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+(-1+\beta )q^{7}+\cdots\)
3450.2.a.bm \(2\) \(27.548\) \(\Q(\sqrt{57}) \) None \(2\) \(2\) \(0\) \(6\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+3q^{7}+q^{8}+\cdots\)
3450.2.a.bn \(2\) \(27.548\) \(\Q(\sqrt{5}) \) None \(2\) \(2\) \(0\) \(8\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+4q^{7}+q^{8}+\cdots\)
3450.2.a.bo \(3\) \(27.548\) 3.3.148.1 None \(-3\) \(3\) \(0\) \(-2\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
3450.2.a.bp \(3\) \(27.548\) 3.3.3368.1 None \(-3\) \(3\) \(0\) \(1\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}+\beta _{1}q^{7}-q^{8}+\cdots\)
3450.2.a.bq \(3\) \(27.548\) 3.3.568.1 None \(-3\) \(3\) \(0\) \(6\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
3450.2.a.br \(3\) \(27.548\) 3.3.568.1 None \(3\) \(-3\) \(0\) \(-6\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
3450.2.a.bs \(3\) \(27.548\) 3.3.3368.1 None \(3\) \(-3\) \(0\) \(-1\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}-\beta _{1}q^{7}+q^{8}+\cdots\)
3450.2.a.bt \(3\) \(27.548\) 3.3.148.1 None \(3\) \(-3\) \(0\) \(2\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+(1+\beta _{1})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3450))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3450)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(230))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(575))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(690))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1150))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1725))\)\(^{\oplus 2}\)