Properties

Label 3969.2.a
Level $3969$
Weight $2$
Character orbit 3969.a
Rep. character $\chi_{3969}(1,\cdot)$
Character field $\Q$
Dimension $154$
Newform subspaces $36$
Sturm bound $1008$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 36 \)
Sturm bound: \(1008\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\), \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3969))\).

Total New Old
Modular forms 552 174 378
Cusp forms 457 154 303
Eisenstein series 95 20 75

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

\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(34\)
\(+\)\(-\)\(-\)\(42\)
\(-\)\(+\)\(-\)\(42\)
\(-\)\(-\)\(+\)\(36\)
Plus space\(+\)\(70\)
Minus space\(-\)\(84\)

Trace form

\( 154 q + 142 q^{4} + O(q^{10}) \) \( 154 q + 142 q^{4} - 6 q^{10} - 10 q^{13} + 130 q^{16} + 8 q^{19} + 36 q^{22} + 112 q^{25} - 4 q^{31} + 6 q^{34} - 10 q^{37} - 18 q^{40} + 20 q^{43} + 36 q^{46} + 2 q^{52} + 18 q^{58} - 22 q^{61} + 106 q^{64} + 44 q^{67} + 38 q^{73} + 56 q^{76} - 4 q^{79} + 48 q^{82} - 30 q^{85} + 96 q^{88} + 60 q^{94} - 4 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 7
3969.2.a.a 3969.a 1.a $1$ $31.693$ \(\Q\) None \(-1\) \(0\) \(-1\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-q^{5}+3q^{8}+q^{10}-5q^{11}+\cdots\)
3969.2.a.b 3969.a 1.a $1$ $31.693$ \(\Q\) None \(-1\) \(0\) \(-1\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-q^{5}+3q^{8}+q^{10}-2q^{11}+\cdots\)
3969.2.a.c 3969.a 1.a $1$ $31.693$ \(\Q\) None \(-1\) \(0\) \(1\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+q^{5}+3q^{8}-q^{10}-5q^{11}+\cdots\)
3969.2.a.d 3969.a 1.a $1$ $31.693$ \(\Q\) None \(1\) \(0\) \(-1\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-q^{5}-3q^{8}-q^{10}+5q^{11}+\cdots\)
3969.2.a.e 3969.a 1.a $1$ $31.693$ \(\Q\) None \(1\) \(0\) \(1\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+q^{5}-3q^{8}+q^{10}+2q^{11}+\cdots\)
3969.2.a.f 3969.a 1.a $1$ $31.693$ \(\Q\) None \(1\) \(0\) \(1\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+q^{5}-3q^{8}+q^{10}+5q^{11}+\cdots\)
3969.2.a.g 3969.a 1.a $2$ $31.693$ \(\Q(\sqrt{15}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+\beta q^{5}+3q^{8}-\beta q^{10}+\cdots\)
3969.2.a.h 3969.a 1.a $2$ $31.693$ \(\Q(\sqrt{21}) \) \(\Q(\sqrt{-7}) \) \(-1\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-\beta q^{2}+(3+\beta )q^{4}+(-5-2\beta )q^{8}+\cdots\)
3969.2.a.i 3969.a 1.a $2$ $31.693$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}+\beta q^{5}-\beta q^{8}+3q^{10}+\cdots\)
3969.2.a.j 3969.a 1.a $2$ $31.693$ \(\Q(\sqrt{21}) \) \(\Q(\sqrt{-7}) \) \(1\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+\beta q^{2}+(3+\beta )q^{4}+(5+2\beta )q^{8}+(1+\cdots)q^{11}+\cdots\)
3969.2.a.k 3969.a 1.a $2$ $31.693$ \(\Q(\sqrt{15}) \) None \(2\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+\beta q^{5}-3q^{8}+\beta q^{10}+\cdots\)
3969.2.a.l 3969.a 1.a $3$ $31.693$ \(\Q(\zeta_{18})^+\) None \(-3\) \(0\) \(3\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
3969.2.a.m 3969.a 1.a $3$ $31.693$ 3.3.321.1 None \(-1\) \(0\) \(5\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(2+\beta _{2})q^{5}+\cdots\)
3969.2.a.n 3969.a 1.a $3$ $31.693$ 3.3.621.1 None \(0\) \(0\) \(-3\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
3969.2.a.o 3969.a 1.a $3$ $31.693$ 3.3.621.1 None \(0\) \(0\) \(3\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+(1+\beta _{2})q^{5}+\cdots\)
3969.2.a.p 3969.a 1.a $3$ $31.693$ 3.3.321.1 None \(1\) \(0\) \(-5\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(-2-\beta _{2})q^{5}+\cdots\)
3969.2.a.q 3969.a 1.a $3$ $31.693$ \(\Q(\zeta_{18})^+\) None \(3\) \(0\) \(-3\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
3969.2.a.r 3969.a 1.a $4$ $31.693$ \(\Q(\sqrt{2}, \sqrt{5})\) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{2})q^{2}-\beta _{2}q^{4}-\beta _{1}q^{5}+(1+\cdots)q^{8}+\cdots\)
3969.2.a.s 3969.a 1.a $4$ $31.693$ 4.4.14013.1 None \(-1\) \(0\) \(-2\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
3969.2.a.t 3969.a 1.a $4$ $31.693$ 4.4.14013.1 None \(-1\) \(0\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
3969.2.a.u 3969.a 1.a $4$ $31.693$ \(\Q(\sqrt{3}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}-\beta _{2}q^{4}+2\beta _{1}q^{5}+(\beta _{1}+\beta _{3})q^{8}+\cdots\)
3969.2.a.v 3969.a 1.a $4$ $31.693$ \(\Q(\sqrt{3}, \sqrt{7})\) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q+\beta _{3}q^{2}-\beta _{2}q^{4}+(\beta _{1}+\beta _{3})q^{8}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
3969.2.a.w 3969.a 1.a $4$ $31.693$ 4.4.14013.1 None \(1\) \(0\) \(-2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
3969.2.a.x 3969.a 1.a $4$ $31.693$ 4.4.14013.1 None \(1\) \(0\) \(2\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
3969.2.a.y 3969.a 1.a $4$ $31.693$ \(\Q(\sqrt{2}, \sqrt{5})\) None \(2\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{2})q^{2}-\beta _{2}q^{4}-\beta _{1}q^{5}+(-1+\cdots)q^{8}+\cdots\)
3969.2.a.z 3969.a 1.a $5$ $31.693$ 5.5.574857.1 None \(-2\) \(0\) \(-4\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{4})q^{5}+\cdots\)
3969.2.a.ba 3969.a 1.a $5$ $31.693$ 5.5.574857.1 None \(-2\) \(0\) \(4\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{4})q^{5}+\cdots\)
3969.2.a.bb 3969.a 1.a $5$ $31.693$ 5.5.574857.1 None \(2\) \(0\) \(-4\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{4})q^{5}+\cdots\)
3969.2.a.bc 3969.a 1.a $5$ $31.693$ 5.5.574857.1 None \(2\) \(0\) \(4\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{4})q^{5}+\cdots\)
3969.2.a.bd 3969.a 1.a $6$ $31.693$ 6.6.59351616.1 None \(-2\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(1+\beta _{2}+\beta _{4})q^{4}-\beta _{1}q^{5}+\cdots\)
3969.2.a.be 3969.a 1.a $6$ $31.693$ 6.6.59351616.1 None \(2\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+(1+\beta _{2}+\beta _{4})q^{4}+\beta _{1}q^{5}+\cdots\)
3969.2.a.bf 3969.a 1.a $8$ $31.693$ 8.8.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
3969.2.a.bg 3969.a 1.a $8$ $31.693$ 8.8.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(\beta _{1}-\beta _{3})q^{5}+\cdots\)
3969.2.a.bh 3969.a 1.a $12$ $31.693$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-4\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(1+\beta _{8})q^{4}-\beta _{10}q^{5}+(-1+\cdots)q^{8}+\cdots\)
3969.2.a.bi 3969.a 1.a $12$ $31.693$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(4\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+(1+\beta _{8})q^{4}-\beta _{10}q^{5}+(1+\cdots)q^{8}+\cdots\)
3969.2.a.bj 3969.a 1.a $16$ $31.693$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{5}q^{2}+(2-\beta _{12})q^{4}+(\beta _{3}-\beta _{9})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3969))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3969)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(189))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(441))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(567))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1323))\)\(^{\oplus 2}\)