Properties

Label 3549.2.a
Level $3549$
Weight $2$
Character orbit 3549.a
Rep. character $\chi_{3549}(1,\cdot)$
Character field $\Q$
Dimension $156$
Newform subspaces $34$
Sturm bound $970$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 34 \)
Sturm bound: \(970\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3549))\).

Total New Old
Modular forms 512 156 356
Cusp forms 457 156 301
Eisenstein series 55 0 55

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

\(3\)\(7\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(18\)
\(+\)\(+\)\(-\)\(-\)\(21\)
\(+\)\(-\)\(+\)\(-\)\(18\)
\(+\)\(-\)\(-\)\(+\)\(21\)
\(-\)\(+\)\(+\)\(-\)\(25\)
\(-\)\(+\)\(-\)\(+\)\(15\)
\(-\)\(-\)\(+\)\(+\)\(11\)
\(-\)\(-\)\(-\)\(-\)\(27\)
Plus space\(+\)\(65\)
Minus space\(-\)\(91\)

Trace form

\( 156q + 152q^{4} + 8q^{5} - 4q^{6} - 2q^{7} + 156q^{9} + O(q^{10}) \) \( 156q + 152q^{4} + 8q^{5} - 4q^{6} - 2q^{7} + 156q^{9} + 8q^{10} + 8q^{12} - 2q^{14} + 152q^{16} + 16q^{17} - 8q^{19} + 40q^{20} - 2q^{21} + 24q^{22} + 8q^{23} - 12q^{24} + 180q^{25} - 6q^{28} + 8q^{29} + 8q^{30} + 8q^{31} + 40q^{32} + 32q^{34} + 4q^{35} + 152q^{36} - 16q^{37} - 8q^{38} + 16q^{40} + 16q^{41} + 2q^{42} + 24q^{43} - 16q^{44} + 8q^{45} - 16q^{46} + 32q^{48} + 156q^{49} - 24q^{50} + 8q^{53} - 4q^{54} + 32q^{55} - 18q^{56} - 16q^{57} + 24q^{59} + 16q^{60} - 16q^{61} + 32q^{62} - 2q^{63} + 168q^{64} + 24q^{66} + 32q^{68} - 16q^{69} + 4q^{70} - 24q^{71} + 8q^{73} + 56q^{74} + 64q^{76} + 32q^{79} + 48q^{80} + 156q^{81} + 16q^{83} - 6q^{84} - 40q^{85} - 48q^{86} + 16q^{87} + 120q^{88} + 40q^{89} + 8q^{90} + 32q^{92} - 32q^{93} - 32q^{94} + 32q^{95} - 28q^{96} + 8q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3549))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7 13
3549.2.a.a \(1\) \(28.339\) \(\Q\) None \(-2\) \(1\) \(-3\) \(-1\) \(-\) \(+\) \(-\) \(q-2q^{2}+q^{3}+2q^{4}-3q^{5}-2q^{6}+\cdots\)
3549.2.a.b \(1\) \(28.339\) \(\Q\) None \(-2\) \(1\) \(-1\) \(1\) \(-\) \(-\) \(+\) \(q-2q^{2}+q^{3}+2q^{4}-q^{5}-2q^{6}+q^{7}+\cdots\)
3549.2.a.c \(1\) \(28.339\) \(\Q\) None \(1\) \(1\) \(2\) \(1\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}-q^{4}+2q^{5}+q^{6}+q^{7}+\cdots\)
3549.2.a.d \(1\) \(28.339\) \(\Q\) None \(2\) \(-1\) \(1\) \(-1\) \(+\) \(+\) \(+\) \(q+2q^{2}-q^{3}+2q^{4}+q^{5}-2q^{6}-q^{7}+\cdots\)
3549.2.a.e \(1\) \(28.339\) \(\Q\) None \(2\) \(1\) \(3\) \(1\) \(-\) \(-\) \(-\) \(q+2q^{2}+q^{3}+2q^{4}+3q^{5}+2q^{6}+\cdots\)
3549.2.a.f \(2\) \(28.339\) \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}+(1+\cdots)q^{6}+\cdots\)
3549.2.a.g \(3\) \(28.339\) \(\Q(\zeta_{14})^+\) None \(-2\) \(-3\) \(0\) \(3\) \(+\) \(-\) \(-\) \(q+(-1-\beta _{2})q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
3549.2.a.h \(3\) \(28.339\) 3.3.169.1 None \(-2\) \(-3\) \(0\) \(-3\) \(+\) \(+\) \(+\) \(q+(-1+\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
3549.2.a.i \(3\) \(28.339\) \(\Q(\zeta_{14})^+\) None \(-2\) \(3\) \(0\) \(3\) \(-\) \(-\) \(+\) \(q+(-1-\beta _{2})q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
3549.2.a.j \(3\) \(28.339\) \(\Q(\zeta_{14})^+\) None \(-1\) \(3\) \(3\) \(-3\) \(-\) \(+\) \(-\) \(q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)
3549.2.a.k \(3\) \(28.339\) 3.3.148.1 None \(-1\) \(3\) \(-2\) \(3\) \(-\) \(-\) \(-\) \(q-\beta _{1}q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
3549.2.a.l \(3\) \(28.339\) 3.3.473.1 None \(0\) \(-3\) \(-2\) \(-3\) \(+\) \(+\) \(+\) \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
3549.2.a.m \(3\) \(28.339\) 3.3.473.1 None \(0\) \(-3\) \(2\) \(3\) \(+\) \(-\) \(+\) \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
3549.2.a.n \(3\) \(28.339\) \(\Q(\zeta_{18})^+\) None \(0\) \(3\) \(-6\) \(3\) \(-\) \(-\) \(+\) \(q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(-2-\beta _{1}+\cdots)q^{5}+\cdots\)
3549.2.a.o \(3\) \(28.339\) \(\Q(\zeta_{18})^+\) None \(0\) \(3\) \(6\) \(-3\) \(-\) \(+\) \(+\) \(q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(2+\beta _{1}-\beta _{2})q^{5}+\cdots\)
3549.2.a.p \(3\) \(28.339\) \(\Q(\zeta_{14})^+\) None \(1\) \(3\) \(-3\) \(3\) \(-\) \(-\) \(+\) \(q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
3549.2.a.q \(3\) \(28.339\) 3.3.148.1 None \(1\) \(3\) \(2\) \(-3\) \(-\) \(+\) \(-\) \(q+\beta _{1}q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots\)
3549.2.a.r \(3\) \(28.339\) \(\Q(\zeta_{14})^+\) None \(2\) \(-3\) \(0\) \(-3\) \(+\) \(+\) \(+\) \(q+(1-\beta _{1})q^{2}-q^{3}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
3549.2.a.s \(3\) \(28.339\) 3.3.169.1 None \(2\) \(-3\) \(0\) \(3\) \(+\) \(-\) \(+\) \(q+(1-\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
3549.2.a.t \(3\) \(28.339\) 3.3.316.1 None \(2\) \(-3\) \(3\) \(3\) \(+\) \(-\) \(+\) \(q+(1-\beta _{1})q^{2}-q^{3}+(2-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
3549.2.a.u \(3\) \(28.339\) \(\Q(\zeta_{14})^+\) None \(2\) \(3\) \(0\) \(-3\) \(-\) \(+\) \(+\) \(q+(1-\beta _{1})q^{2}+q^{3}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
3549.2.a.v \(4\) \(28.339\) 4.4.64436.1 None \(-1\) \(-4\) \(-3\) \(4\) \(+\) \(-\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(-1-\beta _{2}+\cdots)q^{5}+\cdots\)
3549.2.a.w \(4\) \(28.339\) 4.4.17428.1 None \(-1\) \(4\) \(3\) \(-4\) \(-\) \(+\) \(+\) \(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+(1+\beta _{2}+\cdots)q^{5}+\cdots\)
3549.2.a.x \(4\) \(28.339\) 4.4.64436.1 None \(1\) \(-4\) \(3\) \(-4\) \(+\) \(+\) \(-\) \(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(1+\beta _{2}+\cdots)q^{5}+\cdots\)
3549.2.a.y \(6\) \(28.339\) 6.6.121819537.1 None \(-2\) \(-6\) \(3\) \(-6\) \(+\) \(+\) \(+\) \(q+\beta _{2}q^{2}-q^{3}+(-1-\beta _{2}+\beta _{4})q^{4}+\cdots\)
3549.2.a.z \(6\) \(28.339\) 6.6.121819537.1 None \(2\) \(-6\) \(-3\) \(6\) \(+\) \(-\) \(-\) \(q-\beta _{2}q^{2}-q^{3}+(-1-\beta _{2}+\beta _{4})q^{4}+\cdots\)
3549.2.a.ba \(8\) \(28.339\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-2\) \(-8\) \(-6\) \(8\) \(+\) \(-\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{4}+\beta _{5}+\cdots)q^{4}+\cdots\)
3549.2.a.bb \(8\) \(28.339\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-2\) \(8\) \(-2\) \(-8\) \(-\) \(+\) \(-\) \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\beta _{3}q^{5}+\cdots\)
3549.2.a.bc \(8\) \(28.339\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(2\) \(-8\) \(6\) \(-8\) \(+\) \(+\) \(-\) \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{4}+\beta _{5}+\cdots)q^{4}+\cdots\)
3549.2.a.bd \(8\) \(28.339\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(2\) \(8\) \(2\) \(8\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-\beta _{3}q^{5}+\cdots\)
3549.2.a.be \(9\) \(28.339\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-1\) \(-9\) \(9\) \(9\) \(+\) \(-\) \(+\) \(q-\beta _{1}q^{2}-q^{3}+(\beta _{1}-\beta _{2}+\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
3549.2.a.bf \(9\) \(28.339\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(1\) \(-9\) \(-9\) \(-9\) \(+\) \(+\) \(-\) \(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}-\beta _{2}+\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
3549.2.a.bg \(15\) \(28.339\) \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(-2\) \(15\) \(9\) \(15\) \(-\) \(-\) \(-\) \(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+(1-\beta _{13}+\cdots)q^{5}+\cdots\)
3549.2.a.bh \(15\) \(28.339\) \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(2\) \(15\) \(-9\) \(-15\) \(-\) \(+\) \(+\) \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+(-1+\beta _{13}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3549))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3549)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(273))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(507))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1183))\)\(^{\oplus 2}\)