Properties

Label 8619.2.a
Level $8619$
Weight $2$
Character orbit 8619.a
Rep. character $\chi_{8619}(1,\cdot)$
Character field $\Q$
Dimension $414$
Newform subspaces $50$
Sturm bound $2184$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8619.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 50 \)
Sturm bound: \(2184\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8619))\).

Total New Old
Modular forms 1120 414 706
Cusp forms 1065 414 651
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.

\(3\)\(13\)\(17\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(48\)
\(+\)\(+\)\(-\)\(-\)\(58\)
\(+\)\(-\)\(+\)\(-\)\(56\)
\(+\)\(-\)\(-\)\(+\)\(44\)
\(-\)\(+\)\(+\)\(-\)\(55\)
\(-\)\(+\)\(-\)\(+\)\(45\)
\(-\)\(-\)\(+\)\(+\)\(48\)
\(-\)\(-\)\(-\)\(-\)\(60\)
Plus space\(+\)\(185\)
Minus space\(-\)\(229\)

Trace form

\( 414q - 4q^{2} + 2q^{3} + 412q^{4} - 2q^{6} - 4q^{7} - 24q^{8} + 414q^{9} + O(q^{10}) \) \( 414q - 4q^{2} + 2q^{3} + 412q^{4} - 2q^{6} - 4q^{7} - 24q^{8} + 414q^{9} - 4q^{10} + 14q^{12} - 8q^{14} - 2q^{15} + 416q^{16} - 4q^{18} + 18q^{19} + 8q^{20} + 12q^{21} + 20q^{22} + 8q^{23} - 6q^{24} + 416q^{25} + 2q^{27} - 24q^{28} - 8q^{29} + 8q^{30} - 32q^{32} - 10q^{33} - 4q^{34} - 4q^{35} + 412q^{36} + 4q^{37} - 20q^{38} + 40q^{40} + 24q^{41} + 16q^{42} + 14q^{43} - 24q^{44} - 20q^{46} - 28q^{47} + 30q^{48} + 446q^{49} - 8q^{50} + 4q^{51} - 12q^{53} - 2q^{54} + 26q^{55} + 16q^{57} + 8q^{58} + 8q^{59} + 40q^{60} + 20q^{61} + 48q^{62} - 4q^{63} + 392q^{64} - 12q^{66} - 32q^{67} - 2q^{69} + 64q^{70} - 24q^{72} - 16q^{74} + 30q^{75} + 108q^{76} + 12q^{77} + 52q^{79} + 72q^{80} + 414q^{81} + 12q^{82} + 60q^{83} + 24q^{84} - 2q^{85} + 4q^{86} + 8q^{87} + 56q^{88} + 28q^{89} - 4q^{90} + 96q^{92} + 12q^{93} + 64q^{94} + 40q^{95} + 26q^{96} + 16q^{97} - 4q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 13 17
8619.2.a.a \(1\) \(68.823\) \(\Q\) None \(-2\) \(-1\) \(-4\) \(3\) \(+\) \(+\) \(-\) \(q-2q^{2}-q^{3}+2q^{4}-4q^{5}+2q^{6}+\cdots\)
8619.2.a.b \(1\) \(68.823\) \(\Q\) None \(-2\) \(-1\) \(2\) \(-4\) \(+\) \(+\) \(-\) \(q-2q^{2}-q^{3}+2q^{4}+2q^{5}+2q^{6}+\cdots\)
8619.2.a.c \(1\) \(68.823\) \(\Q\) None \(-2\) \(1\) \(3\) \(2\) \(-\) \(+\) \(-\) \(q-2q^{2}+q^{3}+2q^{4}+3q^{5}-2q^{6}+\cdots\)
8619.2.a.d \(1\) \(68.823\) \(\Q\) None \(-1\) \(-1\) \(1\) \(-2\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}-q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
8619.2.a.e \(1\) \(68.823\) \(\Q\) None \(-1\) \(-1\) \(4\) \(-2\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}-q^{4}+4q^{5}+q^{6}-2q^{7}+\cdots\)
8619.2.a.f \(1\) \(68.823\) \(\Q\) None \(-1\) \(1\) \(0\) \(-4\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}-q^{4}-q^{6}-4q^{7}+3q^{8}+\cdots\)
8619.2.a.g \(1\) \(68.823\) \(\Q\) None \(0\) \(1\) \(-3\) \(4\) \(-\) \(+\) \(+\) \(q+q^{3}-2q^{4}-3q^{5}+4q^{7}+q^{9}+3q^{11}+\cdots\)
8619.2.a.h \(1\) \(68.823\) \(\Q\) None \(1\) \(-1\) \(-1\) \(2\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}-q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
8619.2.a.i \(1\) \(68.823\) \(\Q\) None \(1\) \(-1\) \(2\) \(0\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}-q^{4}+2q^{5}-q^{6}-3q^{8}+\cdots\)
8619.2.a.j \(1\) \(68.823\) \(\Q\) None \(1\) \(1\) \(0\) \(2\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}-q^{4}+q^{6}+2q^{7}-3q^{8}+\cdots\)
8619.2.a.k \(1\) \(68.823\) \(\Q\) None \(1\) \(1\) \(0\) \(4\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}-q^{4}+q^{6}+4q^{7}-3q^{8}+\cdots\)
8619.2.a.l \(1\) \(68.823\) \(\Q\) None \(2\) \(-1\) \(-2\) \(4\) \(+\) \(+\) \(-\) \(q+2q^{2}-q^{3}+2q^{4}-2q^{5}-2q^{6}+\cdots\)
8619.2.a.m \(1\) \(68.823\) \(\Q\) None \(2\) \(-1\) \(4\) \(-3\) \(+\) \(+\) \(-\) \(q+2q^{2}-q^{3}+2q^{4}+4q^{5}-2q^{6}+\cdots\)
8619.2.a.n \(1\) \(68.823\) \(\Q\) None \(2\) \(1\) \(-3\) \(-2\) \(-\) \(+\) \(-\) \(q+2q^{2}+q^{3}+2q^{4}-3q^{5}+2q^{6}+\cdots\)
8619.2.a.o \(2\) \(68.823\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q-q^{3}-2q^{4}-\beta q^{5}+q^{9}-3\beta q^{11}+\cdots\)
8619.2.a.p \(2\) \(68.823\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q-q^{3}-2q^{4}-\beta q^{5}-2\beta q^{7}+q^{9}+\cdots\)
8619.2.a.q \(2\) \(68.823\) \(\Q(\sqrt{17}) \) None \(1\) \(-2\) \(-3\) \(0\) \(+\) \(+\) \(-\) \(q+\beta q^{2}-q^{3}+(2+\beta )q^{4}+(-1-\beta )q^{5}+\cdots\)
8619.2.a.r \(3\) \(68.823\) \(\Q(\zeta_{14})^+\) None \(-2\) \(3\) \(-1\) \(5\) \(-\) \(+\) \(+\) \(q+(-1-\beta _{2})q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
8619.2.a.s \(3\) \(68.823\) 3.3.148.1 None \(1\) \(-3\) \(0\) \(4\) \(+\) \(+\) \(+\) \(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
8619.2.a.t \(3\) \(68.823\) \(\Q(\zeta_{14})^+\) None \(2\) \(3\) \(1\) \(-5\) \(-\) \(-\) \(+\) \(q+(1-\beta _{1})q^{2}+q^{3}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
8619.2.a.u \(3\) \(68.823\) 3.3.148.1 None \(3\) \(3\) \(6\) \(2\) \(-\) \(+\) \(+\) \(q+(1+\beta _{2})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
8619.2.a.v \(4\) \(68.823\) \(\Q(\zeta_{24})^+\) None \(0\) \(4\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q-\beta _{2}q^{2}+q^{3}+(-\beta _{1}-\beta _{2})q^{5}-\beta _{2}q^{6}+\cdots\)
8619.2.a.w \(5\) \(68.823\) 5.5.153424.1 None \(-3\) \(5\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q+(-1-\beta _{3})q^{2}+q^{3}+(2+\beta _{3}-\beta _{4})q^{4}+\cdots\)
8619.2.a.x \(5\) \(68.823\) 5.5.1004368.1 None \(-1\) \(-5\) \(-2\) \(-8\) \(+\) \(+\) \(+\) \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{4}q^{5}+\cdots\)
8619.2.a.y \(6\) \(68.823\) 6.6.15187408.1 None \(-4\) \(6\) \(-4\) \(-2\) \(-\) \(+\) \(-\) \(q+(-1+\beta _{1})q^{2}+q^{3}+(1-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
8619.2.a.z \(6\) \(68.823\) 6.6.83831632.1 None \(-2\) \(-6\) \(0\) \(-2\) \(+\) \(+\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
8619.2.a.ba \(6\) \(68.823\) 6.6.485125.1 None \(-2\) \(6\) \(4\) \(5\) \(-\) \(+\) \(-\) \(q-\beta _{1}q^{2}+q^{3}+(\beta _{3}+\beta _{4})q^{4}+(2-\beta _{1}+\cdots)q^{5}+\cdots\)
8619.2.a.bb \(6\) \(68.823\) 6.6.31337472.1 None \(0\) \(-6\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q+\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}+(\beta _{1}-\beta _{5})q^{5}+\cdots\)
8619.2.a.bc \(6\) \(68.823\) 6.6.485125.1 None \(2\) \(6\) \(-4\) \(-5\) \(-\) \(+\) \(-\) \(q+\beta _{1}q^{2}+q^{3}+(\beta _{3}+\beta _{4})q^{4}+(-2+\cdots)q^{5}+\cdots\)
8619.2.a.bd \(8\) \(68.823\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-8\) \(-1\) \(-2\) \(+\) \(+\) \(+\) \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(\beta _{1}+\beta _{7})q^{5}+\cdots\)
8619.2.a.be \(8\) \(68.823\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-8\) \(1\) \(2\) \(+\) \(+\) \(+\) \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
8619.2.a.bf \(8\) \(68.823\) 8.8.\(\cdots\).1 None \(0\) \(-8\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+(\beta _{1}+\beta _{7})q^{2}-q^{3}+(2+\beta _{5})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
8619.2.a.bg \(8\) \(68.823\) 8.8.\(\cdots\).1 None \(0\) \(8\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
8619.2.a.bh \(9\) \(68.823\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-1\) \(-9\) \(-5\) \(3\) \(+\) \(+\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2}+\beta _{3})q^{4}+(-1+\cdots)q^{5}+\cdots\)
8619.2.a.bi \(9\) \(68.823\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(1\) \(-9\) \(5\) \(-3\) \(+\) \(+\) \(-\) \(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2}+\beta _{3})q^{4}+(1+\cdots)q^{5}+\cdots\)
8619.2.a.bj \(10\) \(68.823\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(10\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+\beta _{8}q^{5}+\cdots\)
8619.2.a.bk \(11\) \(68.823\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(0\) \(11\) \(0\) \(-3\) \(-\) \(+\) \(+\) \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots\)
8619.2.a.bl \(11\) \(68.823\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(0\) \(11\) \(0\) \(3\) \(-\) \(+\) \(+\) \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{4}q^{5}+\cdots\)
8619.2.a.bm \(12\) \(68.823\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-12\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q+\beta _{7}q^{2}-q^{3}+(1-\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
8619.2.a.bn \(16\) \(68.823\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(16\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{14}q^{5}+\cdots\)
8619.2.a.bo \(20\) \(68.823\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(20\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{15}q^{5}+\cdots\)
8619.2.a.bp \(21\) \(68.823\) None \(-9\) \(21\) \(-26\) \(5\) \(-\) \(-\) \(+\)
8619.2.a.bq \(21\) \(68.823\) None \(9\) \(21\) \(26\) \(-5\) \(-\) \(+\) \(+\)
8619.2.a.br \(22\) \(68.823\) None \(0\) \(-22\) \(0\) \(0\) \(+\) \(-\) \(+\)
8619.2.a.bs \(24\) \(68.823\) None \(-11\) \(24\) \(-23\) \(-10\) \(-\) \(+\) \(-\)
8619.2.a.bt \(24\) \(68.823\) None \(-7\) \(-24\) \(13\) \(12\) \(+\) \(-\) \(+\)
8619.2.a.bu \(24\) \(68.823\) None \(-1\) \(-24\) \(-11\) \(-2\) \(+\) \(-\) \(-\)
8619.2.a.bv \(24\) \(68.823\) None \(1\) \(-24\) \(11\) \(2\) \(+\) \(+\) \(-\)
8619.2.a.bw \(24\) \(68.823\) None \(7\) \(-24\) \(-13\) \(-12\) \(+\) \(+\) \(+\)
8619.2.a.bx \(24\) \(68.823\) None \(11\) \(24\) \(23\) \(10\) \(-\) \(-\) \(-\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8619))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8619)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(221))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(507))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(663))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2873))\)\(^{\oplus 2}\)