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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

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

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 23
3450.2.a.a 3450.a 1.a $1$ $27.548$ \(\Q\) None \(-1\) \(-1\) \(0\) \(-5\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-5q^{7}-q^{8}+\cdots\)
3450.2.a.b 3450.a 1.a $1$ $27.548$ \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
3450.2.a.c 3450.a 1.a $1$ $27.548$ \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
3450.2.a.d 3450.a 1.a $1$ $27.548$ \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
3450.2.a.e 3450.a 1.a $1$ $27.548$ \(\Q\) None \(-1\) \(-1\) \(0\) \(1\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+q^{7}-q^{8}+\cdots\)
3450.2.a.f 3450.a 1.a $1$ $27.548$ \(\Q\) None \(-1\) \(-1\) \(0\) \(1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+q^{7}-q^{8}+\cdots\)
3450.2.a.g 3450.a 1.a $1$ $27.548$ \(\Q\) None \(-1\) \(-1\) \(0\) \(3\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+3q^{7}-q^{8}+\cdots\)
3450.2.a.h 3450.a 1.a $1$ $27.548$ \(\Q\) None \(-1\) \(1\) \(0\) \(-3\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-3q^{7}-q^{8}+\cdots\)
3450.2.a.i 3450.a 1.a $1$ $27.548$ \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
3450.2.a.j 3450.a 1.a $1$ $27.548$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
3450.2.a.k 3450.a 1.a $1$ $27.548$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
3450.2.a.l 3450.a 1.a $1$ $27.548$ \(\Q\) None \(-1\) \(1\) \(0\) \(2\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
3450.2.a.m 3450.a 1.a $1$ $27.548$ \(\Q\) None \(-1\) \(1\) \(0\) \(5\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+5q^{7}-q^{8}+\cdots\)
3450.2.a.n 3450.a 1.a $1$ $27.548$ \(\Q\) None \(1\) \(-1\) \(0\) \(-5\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-5q^{7}+q^{8}+\cdots\)
3450.2.a.o 3450.a 1.a $1$ $27.548$ \(\Q\) None \(1\) \(-1\) \(0\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
3450.2.a.p 3450.a 1.a $1$ $27.548$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
3450.2.a.q 3450.a 1.a $1$ $27.548$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
3450.2.a.r 3450.a 1.a $1$ $27.548$ \(\Q\) None \(1\) \(-1\) \(0\) \(1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots\)
3450.2.a.s 3450.a 1.a $1$ $27.548$ \(\Q\) None \(1\) \(-1\) \(0\) \(3\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+3q^{7}+q^{8}+\cdots\)
3450.2.a.t 3450.a 1.a $1$ $27.548$ \(\Q\) None \(1\) \(1\) \(0\) \(-4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
3450.2.a.u 3450.a 1.a $1$ $27.548$ \(\Q\) None \(1\) \(1\) \(0\) \(-4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
3450.2.a.v 3450.a 1.a $1$ $27.548$ \(\Q\) None \(1\) \(1\) \(0\) \(-3\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-3q^{7}+q^{8}+\cdots\)
3450.2.a.w 3450.a 1.a $1$ $27.548$ \(\Q\) None \(1\) \(1\) \(0\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-q^{7}+q^{8}+\cdots\)
3450.2.a.x 3450.a 1.a $1$ $27.548$ \(\Q\) None \(1\) \(1\) \(0\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-q^{7}+q^{8}+\cdots\)
3450.2.a.y 3450.a 1.a $1$ $27.548$ \(\Q\) None \(1\) \(1\) \(0\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots\)
3450.2.a.z 3450.a 1.a $1$ $27.548$ \(\Q\) None \(1\) \(1\) \(0\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots\)
3450.2.a.ba 3450.a 1.a $1$ $27.548$ \(\Q\) None \(1\) \(1\) \(0\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots\)
3450.2.a.bb 3450.a 1.a $1$ $27.548$ \(\Q\) None \(1\) \(1\) \(0\) \(5\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+5q^{7}+q^{8}+\cdots\)
3450.2.a.bc 3450.a 1.a $2$ $27.548$ \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(0\) \(-8\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-4q^{7}-q^{8}+\cdots\)
3450.2.a.bd 3450.a 1.a $2$ $27.548$ \(\Q(\sqrt{57}) \) None \(-2\) \(-2\) \(0\) \(-6\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-3q^{7}-q^{8}+\cdots\)
3450.2.a.be 3450.a 1.a $2$ $27.548$ \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(0\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-2\beta q^{7}-q^{8}+\cdots\)
3450.2.a.bf 3450.a 1.a $2$ $27.548$ \(\Q(\sqrt{6}) \) None \(-2\) \(-2\) \(0\) \(2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+(1+\beta )q^{7}+\cdots\)
3450.2.a.bg 3450.a 1.a $2$ $27.548$ \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(4\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots\)
3450.2.a.bh 3450.a 1.a $2$ $27.548$ \(\Q(\sqrt{73}) \) None \(-2\) \(2\) \(0\) \(-6\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-3q^{7}-q^{8}+\cdots\)
3450.2.a.bi 3450.a 1.a $2$ $27.548$ \(\Q(\sqrt{17}) \) None \(-2\) \(2\) \(0\) \(2\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+(1+\beta )q^{7}+\cdots\)
3450.2.a.bj 3450.a 1.a $2$ $27.548$ \(\Q(\sqrt{73}) \) None \(2\) \(-2\) \(0\) \(6\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+3q^{7}+q^{8}+\cdots\)
3450.2.a.bk 3450.a 1.a $2$ $27.548$ \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(-4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots\)
3450.2.a.bl 3450.a 1.a $2$ $27.548$ \(\Q(\sqrt{6}) \) None \(2\) \(2\) \(0\) \(-2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+(-1+\beta )q^{7}+\cdots\)
3450.2.a.bm 3450.a 1.a $2$ $27.548$ \(\Q(\sqrt{57}) \) None \(2\) \(2\) \(0\) \(6\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+3q^{7}+q^{8}+\cdots\)
3450.2.a.bn 3450.a 1.a $2$ $27.548$ \(\Q(\sqrt{5}) \) None \(2\) \(2\) \(0\) \(8\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+4q^{7}+q^{8}+\cdots\)
3450.2.a.bo 3450.a 1.a $3$ $27.548$ 3.3.148.1 None \(-3\) \(3\) \(0\) \(-2\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
3450.2.a.bp 3450.a 1.a $3$ $27.548$ 3.3.3368.1 None \(-3\) \(3\) \(0\) \(1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+\beta _{1}q^{7}-q^{8}+\cdots\)
3450.2.a.bq 3450.a 1.a $3$ $27.548$ 3.3.568.1 None \(-3\) \(3\) \(0\) \(6\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
3450.2.a.br 3450.a 1.a $3$ $27.548$ 3.3.568.1 None \(3\) \(-3\) \(0\) \(-6\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
3450.2.a.bs 3450.a 1.a $3$ $27.548$ 3.3.3368.1 None \(3\) \(-3\) \(0\) \(-1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-\beta _{1}q^{7}+q^{8}+\cdots\)
3450.2.a.bt 3450.a 1.a $3$ $27.548$ 3.3.148.1 None \(3\) \(-3\) \(0\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+(1+\beta _{1})q^{7}+\cdots\)

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

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