Properties

Label 3211.2.a
Level $3211$
Weight $2$
Character orbit 3211.a
Rep. character $\chi_{3211}(1,\cdot)$
Character field $\Q$
Dimension $232$
Newform subspaces $18$
Sturm bound $606$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 3211 = 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3211.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(606\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3211))\).

Total New Old
Modular forms 316 232 84
Cusp forms 289 232 57
Eisenstein series 27 0 27

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

\(13\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(52\)
\(+\)\(-\)\(-\)\(66\)
\(-\)\(+\)\(-\)\(63\)
\(-\)\(-\)\(+\)\(51\)
Plus space\(+\)\(103\)
Minus space\(-\)\(129\)

Trace form

\( 232q + q^{2} + 2q^{3} + 233q^{4} + q^{5} + 7q^{7} - 3q^{8} + 232q^{9} + O(q^{10}) \) \( 232q + q^{2} + 2q^{3} + 233q^{4} + q^{5} + 7q^{7} - 3q^{8} + 232q^{9} - 10q^{10} + 3q^{11} + 4q^{12} + 8q^{14} + 10q^{15} + 227q^{16} - 13q^{17} + q^{18} + 2q^{19} + 10q^{21} - 4q^{22} + 8q^{23} - 4q^{24} + 237q^{25} - 4q^{27} + 34q^{28} + 12q^{29} + 8q^{30} + 8q^{31} + 25q^{32} + 6q^{33} - 22q^{34} - 27q^{35} + 245q^{36} - 4q^{37} - 3q^{38} - 34q^{40} - 4q^{41} - 28q^{42} - 13q^{43} - 2q^{44} + 21q^{45} + 5q^{47} - 4q^{48} + 257q^{49} - 25q^{50} - 18q^{51} + 2q^{53} + 4q^{54} - 11q^{55} + 36q^{56} + 2q^{57} + 10q^{58} - 26q^{59} + 56q^{60} - 15q^{61} - 28q^{62} + 11q^{63} + 199q^{64} - 20q^{66} + 12q^{67} - 60q^{68} + 4q^{69} - 4q^{70} - 38q^{71} + 5q^{72} + 15q^{73} + 42q^{74} - 60q^{75} + 5q^{76} - 19q^{77} + 8q^{79} - 6q^{80} + 244q^{81} - 54q^{82} - 4q^{83} + 64q^{84} - 37q^{85} - 8q^{86} - 28q^{87} + 36q^{88} + 14q^{89} - 66q^{90} + 44q^{92} - 40q^{93} - 32q^{94} - 7q^{95} + 4q^{96} - 10q^{97} - 79q^{98} - 17q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3211))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 13 19
3211.2.a.a \(1\) \(25.640\) \(\Q\) None \(0\) \(-2\) \(-3\) \(1\) \(+\) \(+\) \(q-2q^{3}-2q^{4}-3q^{5}+q^{7}+q^{9}-3q^{11}+\cdots\)
3211.2.a.b \(2\) \(25.640\) \(\Q(\sqrt{5}) \) None \(-1\) \(-2\) \(-2\) \(4\) \(+\) \(-\) \(q-\beta q^{2}+(-2+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
3211.2.a.c \(3\) \(25.640\) \(\Q(\zeta_{18})^+\) None \(3\) \(-3\) \(3\) \(3\) \(+\) \(+\) \(q+(1-\beta _{1})q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+\cdots\)
3211.2.a.d \(4\) \(25.640\) 4.4.6809.1 None \(3\) \(-1\) \(8\) \(2\) \(+\) \(-\) \(q+(1+\beta _{3})q^{2}+(\beta _{1}+\beta _{3})q^{3}+(1-\beta _{2}+\cdots)q^{4}+\cdots\)
3211.2.a.e \(5\) \(25.640\) 5.5.288565.1 None \(-4\) \(3\) \(-3\) \(1\) \(+\) \(+\) \(q+(-1+\beta _{1})q^{2}+(1-\beta _{2})q^{3}+(1-\beta _{1}+\cdots)q^{4}+\cdots\)
3211.2.a.f \(5\) \(25.640\) 5.5.2655049.1 None \(0\) \(3\) \(-2\) \(-4\) \(+\) \(-\) \(q-\beta _{1}q^{2}+(1+\beta _{3})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
3211.2.a.g \(10\) \(25.640\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-3\) \(0\) \(-8\) \(-2\) \(-\) \(-\) \(q-\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
3211.2.a.h \(10\) \(25.640\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(3\) \(0\) \(8\) \(2\) \(-\) \(+\) \(q+\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{4}+\cdots)q^{5}+\cdots\)
3211.2.a.i \(11\) \(25.640\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(-4\) \(0\) \(-6\) \(-2\) \(+\) \(+\) \(q-\beta _{1}q^{2}-\beta _{5}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
3211.2.a.j \(11\) \(25.640\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(-2\) \(0\) \(-10\) \(-2\) \(+\) \(+\) \(q-\beta _{1}q^{2}-\beta _{5}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
3211.2.a.k \(11\) \(25.640\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(2\) \(0\) \(10\) \(2\) \(+\) \(-\) \(q+\beta _{1}q^{2}-\beta _{5}q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{7}+\cdots)q^{5}+\cdots\)
3211.2.a.l \(11\) \(25.640\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(4\) \(0\) \(6\) \(2\) \(+\) \(-\) \(q+\beta _{1}q^{2}-\beta _{5}q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{8}+\cdots)q^{5}+\cdots\)
3211.2.a.m \(20\) \(25.640\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-6\) \(0\) \(-16\) \(-4\) \(-\) \(-\) \(q-\beta _{1}q^{2}-\beta _{11}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
3211.2.a.n \(20\) \(25.640\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(6\) \(0\) \(16\) \(4\) \(-\) \(+\) \(q+\beta _{1}q^{2}-\beta _{11}q^{3}+(1+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
3211.2.a.o \(21\) \(25.640\) None \(-1\) \(-5\) \(4\) \(1\) \(+\) \(+\)
3211.2.a.p \(21\) \(25.640\) None \(1\) \(-5\) \(-4\) \(-1\) \(-\) \(-\)
3211.2.a.q \(33\) \(25.640\) None \(-1\) \(7\) \(5\) \(6\) \(+\) \(-\)
3211.2.a.r \(33\) \(25.640\) None \(1\) \(7\) \(-5\) \(-6\) \(-\) \(+\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3211))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3211)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(247))\)\(^{\oplus 2}\)