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Properties

Label 3481.2.a
Level $3481$
Weight $2$
Character orbit 3481.a
Rep. character $\chi_{3481}(1,\cdot)$
Character field $\Q$
Dimension $256$
Newform subspaces $17$
Sturm bound $590$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 3481 = 59^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3481.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(590\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3481))\).

Total New Old
Modular forms 325 313 12
Cusp forms 266 256 10
Eisenstein series 59 57 2

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

\(59\)Dim
\(+\)\(121\)
\(-\)\(135\)

Trace form

\( 256 q + 2 q^{3} + 224 q^{4} - 2 q^{5} + 4 q^{6} - 2 q^{7} + 6 q^{8} + 198 q^{9} + O(q^{10}) \) \( 256 q + 2 q^{3} + 224 q^{4} - 2 q^{5} + 4 q^{6} - 2 q^{7} + 6 q^{8} + 198 q^{9} + 8 q^{10} + 2 q^{11} + 22 q^{12} - 8 q^{13} + 18 q^{14} + 12 q^{15} + 164 q^{16} + 4 q^{17} + 2 q^{18} - 6 q^{19} - 6 q^{20} - 12 q^{21} - 8 q^{22} + 8 q^{23} - 6 q^{24} + 138 q^{25} - 8 q^{26} + 14 q^{27} + 2 q^{28} - 14 q^{29} - 14 q^{30} + 2 q^{32} + 14 q^{33} + 2 q^{34} + 12 q^{35} + 74 q^{36} - 18 q^{37} + 18 q^{39} + 18 q^{40} + 10 q^{41} - 34 q^{42} + 4 q^{43} - 12 q^{44} + 2 q^{45} - 16 q^{46} + 20 q^{47} + 54 q^{48} + 86 q^{49} - 8 q^{50} + 12 q^{51} - 28 q^{52} + 10 q^{53} + 26 q^{54} + 20 q^{55} + 38 q^{56} + 6 q^{57} + 38 q^{58} - 18 q^{60} - 22 q^{61} - 48 q^{62} + 24 q^{63} + 40 q^{64} + 16 q^{65} - 28 q^{66} + 26 q^{68} + 4 q^{69} + 24 q^{70} - 28 q^{72} + 8 q^{73} + 4 q^{75} - 6 q^{76} - 2 q^{77} - 20 q^{78} - 10 q^{79} - 42 q^{80} + 32 q^{81} + 48 q^{82} - 6 q^{83} - 16 q^{84} - 40 q^{85} + 16 q^{86} - 8 q^{87} + 24 q^{88} - 10 q^{89} + 8 q^{90} - 6 q^{91} - 4 q^{92} - 6 q^{93} + 44 q^{94} - 14 q^{95} - 42 q^{96} + 22 q^{97} - 24 q^{98} + 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3481))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 59
3481.2.a.a 3481.a 1.a $2$ $27.796$ \(\Q(\sqrt{5}) \) None 3481.2.a.a \(0\) \(-2\) \(-2\) \(-6\) $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+3q^{4}-q^{5}+\beta q^{6}-3q^{7}+\cdots\)
3481.2.a.b 3481.a 1.a $2$ $27.796$ \(\Q(\sqrt{3}) \) None 3481.2.a.b \(0\) \(4\) \(-6\) \(8\) $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+2q^{3}+q^{4}-3q^{5}+2\beta q^{6}+\cdots\)
3481.2.a.c 3481.a 1.a $3$ $27.796$ 3.3.1593.1 \(\Q(\sqrt{-59}) \) 3481.2.a.c \(0\) \(0\) \(0\) \(0\) $+$ $N(\mathrm{U}(1))$ \(q-\beta _{1}q^{3}-2q^{4}+\beta _{2}q^{5}+(\beta _{1}-\beta _{2})q^{7}+\cdots\)
3481.2.a.d 3481.a 1.a $5$ $27.796$ 5.5.138136.1 None 59.2.a.a \(0\) \(-2\) \(2\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}+\beta _{3})q^{2}+(-1-\beta _{2}+\beta _{4})q^{3}+\cdots\)
3481.2.a.e 3481.a 1.a $5$ $27.796$ 5.5.245992.1 None 3481.2.a.e \(0\) \(1\) \(-1\) \(5\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-1+\beta _{2}-\beta _{3}-\beta _{4})q^{3}+\cdots\)
3481.2.a.f 3481.a 1.a $5$ $27.796$ 5.5.245992.1 None 3481.2.a.e \(0\) \(1\) \(-1\) \(5\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1+\beta _{2}-\beta _{3}-\beta _{4})q^{3}+\cdots\)
3481.2.a.g 3481.a 1.a $6$ $27.796$ 6.6.303369408.1 None 3481.2.a.g \(0\) \(-6\) \(2\) \(-10\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{2}q^{5}+\cdots\)
3481.2.a.h 3481.a 1.a $8$ $27.796$ \(\Q(\zeta_{60})^+\) None 3481.2.a.h \(0\) \(-8\) \(-18\) \(-6\) $+$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}+\beta _{6})q^{2}+(-1+\beta _{2}-\beta _{3}+\cdots)q^{3}+\cdots\)
3481.2.a.i 3481.a 1.a $8$ $27.796$ \(\Q(\zeta_{60})^+\) None 3481.2.a.i \(0\) \(-4\) \(-4\) \(2\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-\beta _{2}-\beta _{3}+\beta _{4})q^{3}+\beta _{2}q^{4}+\cdots\)
3481.2.a.j 3481.a 1.a $12$ $27.796$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 3481.2.a.j \(-1\) \(4\) \(1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-\beta _{7}q^{3}+(1+\beta _{2})q^{4}-\beta _{11}q^{5}+\cdots\)
3481.2.a.k 3481.a 1.a $12$ $27.796$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 3481.2.a.j \(1\) \(4\) \(1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-\beta _{7}q^{3}+(1+\beta _{2})q^{4}-\beta _{11}q^{5}+\cdots\)
3481.2.a.l 3481.a 1.a $16$ $27.796$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 3481.2.a.l \(0\) \(4\) \(6\) \(-14\) $+$ $\mathrm{SU}(2)$ \(q+(\beta _{1}-\beta _{10}-\beta _{12})q^{2}+(1+\beta _{6}-\beta _{13}+\cdots)q^{3}+\cdots\)
3481.2.a.m 3481.a 1.a $20$ $27.796$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None 3481.2.a.m \(-5\) \(4\) \(11\) \(5\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-\beta _{8}q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{15}+\cdots)q^{5}+\cdots\)
3481.2.a.n 3481.a 1.a $20$ $27.796$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None 3481.2.a.n \(0\) \(-8\) \(-8\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{17}q^{3}+(1+\beta _{13}-\beta _{14}+\cdots)q^{4}+\cdots\)
3481.2.a.o 3481.a 1.a $20$ $27.796$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None 3481.2.a.m \(5\) \(4\) \(11\) \(5\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-\beta _{8}q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{15}+\cdots)q^{5}+\cdots\)
3481.2.a.p 3481.a 1.a $56$ $27.796$ None 59.2.c.a \(-13\) \(3\) \(2\) \(3\) $+$ $\mathrm{SU}(2)$
3481.2.a.q 3481.a 1.a $56$ $27.796$ None 59.2.c.a \(13\) \(3\) \(2\) \(3\) $-$ $\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3481))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3481)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 2}\)