Properties

Label 4598.2.a
Level $4598$
Weight $2$
Character orbit 4598.a
Rep. character $\chi_{4598}(1,\cdot)$
Character field $\Q$
Dimension $163$
Newform subspaces $56$
Sturm bound $1320$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 56 \)
Sturm bound: \(1320\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4598))\).

Total New Old
Modular forms 684 163 521
Cusp forms 637 163 474
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.

\(2\)\(11\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(16\)
\(+\)\(+\)\(-\)\(-\)\(23\)
\(+\)\(-\)\(+\)\(-\)\(24\)
\(+\)\(-\)\(-\)\(+\)\(19\)
\(-\)\(+\)\(+\)\(-\)\(25\)
\(-\)\(+\)\(-\)\(+\)\(14\)
\(-\)\(-\)\(+\)\(+\)\(16\)
\(-\)\(-\)\(-\)\(-\)\(26\)
Plus space\(+\)\(65\)
Minus space\(-\)\(98\)

Trace form

\( 163 q - q^{2} + 4 q^{3} + 163 q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{7} - q^{8} + 169 q^{9} + O(q^{10}) \) \( 163 q - q^{2} + 4 q^{3} + 163 q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{7} - q^{8} + 169 q^{9} - 2 q^{10} + 4 q^{12} - 2 q^{13} - 4 q^{14} + 20 q^{15} + 163 q^{16} + 8 q^{17} + 3 q^{18} + q^{19} + 2 q^{20} + 20 q^{21} + 22 q^{23} - 2 q^{24} + 173 q^{25} - 4 q^{26} + 16 q^{27} + 6 q^{28} - 10 q^{29} - 12 q^{30} + 28 q^{31} - q^{32} - 2 q^{34} + 20 q^{35} + 169 q^{36} + 26 q^{37} - 3 q^{38} + 38 q^{39} - 2 q^{40} + 6 q^{41} - 10 q^{42} + 8 q^{43} + 30 q^{45} - 12 q^{46} + 20 q^{47} + 4 q^{48} + 177 q^{49} - 15 q^{50} + 32 q^{51} - 2 q^{52} + 30 q^{53} - 14 q^{54} - 4 q^{56} + 2 q^{57} - 4 q^{58} - 4 q^{59} + 20 q^{60} + 2 q^{61} + 4 q^{62} + 40 q^{63} + 163 q^{64} - 16 q^{65} + 20 q^{67} + 8 q^{68} + 4 q^{69} + 36 q^{70} + 36 q^{71} + 3 q^{72} + 12 q^{73} - 6 q^{74} + 28 q^{75} + q^{76} + 44 q^{78} + 36 q^{79} + 2 q^{80} + 219 q^{81} + 6 q^{82} - 8 q^{83} + 20 q^{84} - 40 q^{85} + 24 q^{86} + 14 q^{87} + 66 q^{89} + 2 q^{90} + 40 q^{91} + 22 q^{92} + 12 q^{93} - 8 q^{94} + 2 q^{95} - 2 q^{96} - 14 q^{97} + 15 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 11 19
4598.2.a.a \(1\) \(36.715\) \(\Q\) None \(-1\) \(-1\) \(-4\) \(-3\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}-4q^{5}+q^{6}-3q^{7}+\cdots\)
4598.2.a.b \(1\) \(36.715\) \(\Q\) None \(-1\) \(-1\) \(-2\) \(3\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}-2q^{5}+q^{6}+3q^{7}+\cdots\)
4598.2.a.c \(1\) \(36.715\) \(\Q\) None \(-1\) \(-1\) \(2\) \(3\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}+3q^{7}+\cdots\)
4598.2.a.d \(1\) \(36.715\) \(\Q\) None \(-1\) \(0\) \(-1\) \(4\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}-q^{5}+4q^{7}-q^{8}-3q^{9}+\cdots\)
4598.2.a.e \(1\) \(36.715\) \(\Q\) None \(-1\) \(0\) \(0\) \(-3\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}-3q^{7}-q^{8}-3q^{9}-q^{13}+\cdots\)
4598.2.a.f \(1\) \(36.715\) \(\Q\) None \(-1\) \(0\) \(2\) \(-2\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}+2q^{5}-2q^{7}-q^{8}-3q^{9}+\cdots\)
4598.2.a.g \(1\) \(36.715\) \(\Q\) None \(-1\) \(0\) \(4\) \(-1\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}+4q^{5}-q^{7}-q^{8}-3q^{9}+\cdots\)
4598.2.a.h \(1\) \(36.715\) \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
4598.2.a.i \(1\) \(36.715\) \(\Q\) None \(-1\) \(2\) \(-2\) \(-4\) \(+\) \(+\) \(-\) \(q-q^{2}+2q^{3}+q^{4}-2q^{5}-2q^{6}-4q^{7}+\cdots\)
4598.2.a.j \(1\) \(36.715\) \(\Q\) None \(-1\) \(3\) \(-2\) \(-1\) \(+\) \(-\) \(+\) \(q-q^{2}+3q^{3}+q^{4}-2q^{5}-3q^{6}-q^{7}+\cdots\)
4598.2.a.k \(1\) \(36.715\) \(\Q\) None \(-1\) \(3\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q-q^{2}+3q^{3}+q^{4}-3q^{6}-q^{8}+6q^{9}+\cdots\)
4598.2.a.l \(1\) \(36.715\) \(\Q\) None \(1\) \(-1\) \(2\) \(-3\) \(-\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}-3q^{7}+\cdots\)
4598.2.a.m \(1\) \(36.715\) \(\Q\) None \(1\) \(0\) \(-1\) \(-4\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}-q^{5}-4q^{7}+q^{8}-3q^{9}+\cdots\)
4598.2.a.n \(1\) \(36.715\) \(\Q\) None \(1\) \(0\) \(0\) \(3\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+3q^{7}+q^{8}-3q^{9}+q^{13}+\cdots\)
4598.2.a.o \(1\) \(36.715\) \(\Q\) None \(1\) \(0\) \(4\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+4q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
4598.2.a.p \(1\) \(36.715\) \(\Q\) None \(1\) \(1\) \(0\) \(1\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{7}+q^{8}+\cdots\)
4598.2.a.q \(1\) \(36.715\) \(\Q\) None \(1\) \(1\) \(0\) \(1\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{7}+q^{8}+\cdots\)
4598.2.a.r \(1\) \(36.715\) \(\Q\) None \(1\) \(2\) \(-2\) \(4\) \(-\) \(+\) \(+\) \(q+q^{2}+2q^{3}+q^{4}-2q^{5}+2q^{6}+4q^{7}+\cdots\)
4598.2.a.s \(1\) \(36.715\) \(\Q\) None \(1\) \(3\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+q^{2}+3q^{3}+q^{4}+3q^{6}+q^{8}+6q^{9}+\cdots\)
4598.2.a.t \(2\) \(36.715\) \(\Q(\sqrt{5}) \) None \(-2\) \(-4\) \(1\) \(3\) \(+\) \(-\) \(+\) \(q-q^{2}-2q^{3}+q^{4}+(1-\beta )q^{5}+2q^{6}+\cdots\)
4598.2.a.u \(2\) \(36.715\) \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(-3\) \(-1\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}+q^{6}+\cdots\)
4598.2.a.v \(2\) \(36.715\) \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(1\) \(1\) \(+\) \(-\) \(+\) \(q-q^{2}-2\beta q^{3}+q^{4}+(-1+3\beta )q^{5}+\cdots\)
4598.2.a.w \(2\) \(36.715\) \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(2\) \(4\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+(1+\beta )q^{5}+q^{6}+\cdots\)
4598.2.a.x \(2\) \(36.715\) \(\Q(\sqrt{5}) \) None \(-2\) \(-1\) \(-1\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{2}-\beta q^{3}+q^{4}-\beta q^{5}+\beta q^{6}+(1+\cdots)q^{7}+\cdots\)
4598.2.a.y \(2\) \(36.715\) \(\Q(\sqrt{21}) \) None \(-2\) \(-1\) \(3\) \(5\) \(+\) \(-\) \(+\) \(q-q^{2}-\beta q^{3}+q^{4}+(2-\beta )q^{5}+\beta q^{6}+\cdots\)
4598.2.a.z \(2\) \(36.715\) \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(-1\) \(1\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}+(1-3\beta )q^{5}+(1-\beta )q^{7}+\cdots\)
4598.2.a.ba \(2\) \(36.715\) \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(-1\) \(1\) \(+\) \(-\) \(+\) \(q-q^{2}+(1-2\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
4598.2.a.bb \(2\) \(36.715\) \(\Q(\sqrt{5}) \) None \(2\) \(-4\) \(1\) \(-3\) \(-\) \(+\) \(-\) \(q+q^{2}-2q^{3}+q^{4}+\beta q^{5}-2q^{6}-3\beta q^{7}+\cdots\)
4598.2.a.bc \(2\) \(36.715\) \(\Q(\sqrt{13}) \) None \(2\) \(-3\) \(-1\) \(1\) \(-\) \(-\) \(+\) \(q+q^{2}+(-1-\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
4598.2.a.bd \(2\) \(36.715\) \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(-3\) \(1\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}-q^{6}+\cdots\)
4598.2.a.be \(2\) \(36.715\) \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(1\) \(-1\) \(-\) \(+\) \(-\) \(q+q^{2}-2\beta q^{3}+q^{4}+(-1+3\beta )q^{5}+\cdots\)
4598.2.a.bf \(2\) \(36.715\) \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(2\) \(-4\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+(1+\beta )q^{5}-q^{6}+\cdots\)
4598.2.a.bg \(2\) \(36.715\) \(\Q(\sqrt{5}) \) None \(2\) \(-1\) \(-1\) \(1\) \(-\) \(+\) \(-\) \(q+q^{2}-\beta q^{3}+q^{4}-\beta q^{5}-\beta q^{6}+(-1+\cdots)q^{7}+\cdots\)
4598.2.a.bh \(2\) \(36.715\) \(\Q(\sqrt{5}) \) None \(2\) \(0\) \(-1\) \(-1\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}+(1-3\beta )q^{5}+(-1+\beta )q^{7}+\cdots\)
4598.2.a.bi \(2\) \(36.715\) \(\Q(\sqrt{5}) \) None \(2\) \(0\) \(-1\) \(-1\) \(-\) \(+\) \(-\) \(q+q^{2}+(1-2\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
4598.2.a.bj \(2\) \(36.715\) \(\Q(\sqrt{17}) \) None \(2\) \(1\) \(4\) \(-3\) \(-\) \(-\) \(-\) \(q+q^{2}+\beta q^{3}+q^{4}+2q^{5}+\beta q^{6}+(-2+\cdots)q^{7}+\cdots\)
4598.2.a.bk \(3\) \(36.715\) 3.3.568.1 None \(-3\) \(-2\) \(-1\) \(2\) \(+\) \(-\) \(-\) \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}-\beta _{1}q^{5}+\cdots\)
4598.2.a.bl \(3\) \(36.715\) 3.3.404.1 None \(-3\) \(1\) \(4\) \(3\) \(+\) \(-\) \(+\) \(q-q^{2}+\beta _{2}q^{3}+q^{4}+(1+2\beta _{1}-\beta _{2})q^{5}+\cdots\)
4598.2.a.bm \(3\) \(36.715\) 3.3.469.1 None \(-3\) \(1\) \(5\) \(-1\) \(+\) \(-\) \(-\) \(q-q^{2}+\beta _{1}q^{3}+q^{4}+(2-\beta _{1})q^{5}-\beta _{1}q^{6}+\cdots\)
4598.2.a.bn \(3\) \(36.715\) 3.3.568.1 None \(3\) \(-2\) \(-1\) \(-2\) \(-\) \(-\) \(+\) \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}-\beta _{1}q^{5}+\cdots\)
4598.2.a.bo \(3\) \(36.715\) 3.3.621.1 None \(3\) \(0\) \(-3\) \(6\) \(-\) \(-\) \(-\) \(q+q^{2}+\beta _{1}q^{3}+q^{4}+(-1-\beta _{1}-\beta _{2})q^{5}+\cdots\)
4598.2.a.bp \(3\) \(36.715\) 3.3.404.1 None \(3\) \(1\) \(4\) \(-3\) \(-\) \(-\) \(-\) \(q+q^{2}+\beta _{2}q^{3}+q^{4}+(1+2\beta _{1}-\beta _{2})q^{5}+\cdots\)
4598.2.a.bq \(4\) \(36.715\) 4.4.7488.1 None \(-4\) \(-2\) \(-2\) \(4\) \(+\) \(+\) \(+\) \(q-q^{2}+(-1-\beta _{3})q^{3}+q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
4598.2.a.br \(4\) \(36.715\) 4.4.33452.1 None \(-4\) \(1\) \(3\) \(1\) \(+\) \(+\) \(-\) \(q-q^{2}+\beta _{1}q^{3}+q^{4}+(1+\beta _{2}+\beta _{3})q^{5}+\cdots\)
4598.2.a.bs \(4\) \(36.715\) 4.4.258228.1 None \(-4\) \(2\) \(-2\) \(-3\) \(+\) \(-\) \(+\) \(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}-\beta _{1}q^{5}+(-1+\cdots)q^{6}+\cdots\)
4598.2.a.bt \(4\) \(36.715\) 4.4.7488.1 None \(4\) \(-2\) \(-2\) \(-4\) \(-\) \(+\) \(-\) \(q+q^{2}+(-1-\beta _{3})q^{3}+q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
4598.2.a.bu \(4\) \(36.715\) 4.4.33452.1 None \(4\) \(1\) \(3\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{2}+\beta _{1}q^{3}+q^{4}+(1+\beta _{2}+\beta _{3})q^{5}+\cdots\)
4598.2.a.bv \(4\) \(36.715\) 4.4.258228.1 None \(4\) \(2\) \(-2\) \(3\) \(-\) \(-\) \(-\) \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}-\beta _{1}q^{5}+(1+\cdots)q^{6}+\cdots\)
4598.2.a.bw \(8\) \(36.715\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(-2\) \(-2\) \(14\) \(+\) \(+\) \(-\) \(q-q^{2}-\beta _{1}q^{3}+q^{4}+(-\beta _{1}-\beta _{3}-\beta _{6}+\cdots)q^{5}+\cdots\)
4598.2.a.bx \(8\) \(36.715\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(0\) \(2\) \(-8\) \(+\) \(-\) \(-\) \(q-q^{2}+(\beta _{2}+\beta _{4}-\beta _{6})q^{3}+q^{4}-\beta _{3}q^{5}+\cdots\)
4598.2.a.by \(8\) \(36.715\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(8\) \(0\) \(-4\) \(+\) \(+\) \(-\) \(q-q^{2}+(1+\beta _{6})q^{3}+q^{4}+(-\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
4598.2.a.bz \(8\) \(36.715\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(-2\) \(-2\) \(-14\) \(-\) \(-\) \(+\) \(q+q^{2}-\beta _{1}q^{3}+q^{4}+(-\beta _{1}-\beta _{3}-\beta _{6}+\cdots)q^{5}+\cdots\)
4598.2.a.ca \(8\) \(36.715\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(0\) \(2\) \(8\) \(-\) \(+\) \(+\) \(q+q^{2}+(\beta _{2}+\beta _{4}-\beta _{6})q^{3}+q^{4}-\beta _{3}q^{5}+\cdots\)
4598.2.a.cb \(8\) \(36.715\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(8\) \(0\) \(4\) \(-\) \(+\) \(+\) \(q+q^{2}+(1+\beta _{6})q^{3}+q^{4}+(-\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
4598.2.a.cc \(10\) \(36.715\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-10\) \(2\) \(-3\) \(-11\) \(+\) \(+\) \(+\) \(q-q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{3}q^{5}-\beta _{1}q^{6}+\cdots\)
4598.2.a.cd \(10\) \(36.715\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(10\) \(2\) \(-3\) \(11\) \(-\) \(-\) \(-\) \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{3}q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4598))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4598)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(209))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(418))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2299))\)\(^{\oplus 2}\)