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Properties

Label 6422.2.a
Level $6422$
Weight $2$
Character orbit 6422.a
Rep. character $\chi_{6422}(1,\cdot)$
Character field $\Q$
Dimension $233$
Newform subspaces $44$
Sturm bound $1820$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 44 \)
Sturm bound: \(1820\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6422))\).

Total New Old
Modular forms 938 233 705
Cusp forms 883 233 650
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.

\(2\)\(13\)\(19\)FrickeDim
\(+\)\(+\)\(+\)$+$\(26\)
\(+\)\(+\)\(-\)$-$\(30\)
\(+\)\(-\)\(+\)$-$\(33\)
\(+\)\(-\)\(-\)$+$\(27\)
\(-\)\(+\)\(+\)$-$\(34\)
\(-\)\(+\)\(-\)$+$\(23\)
\(-\)\(-\)\(+\)$+$\(24\)
\(-\)\(-\)\(-\)$-$\(36\)
Plus space\(+\)\(100\)
Minus space\(-\)\(133\)

Trace form

\( 233 q + q^{2} - 4 q^{3} + 233 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + q^{8} + 235 q^{9} + O(q^{10}) \) \( 233 q + q^{2} - 4 q^{3} + 233 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + q^{8} + 235 q^{9} + 2 q^{10} + 8 q^{11} - 4 q^{12} - 4 q^{14} - 4 q^{15} + 233 q^{16} + 8 q^{17} + 13 q^{18} - q^{19} - 6 q^{20} - 4 q^{21} - 4 q^{22} - 14 q^{23} - 6 q^{24} + 223 q^{25} - 16 q^{27} - 6 q^{28} + 2 q^{29} + 12 q^{30} - 20 q^{31} + q^{32} + 18 q^{34} + 36 q^{35} + 235 q^{36} - 2 q^{37} + 3 q^{38} + 2 q^{40} - 6 q^{41} + 26 q^{42} - 12 q^{43} + 8 q^{44} - 10 q^{45} + 4 q^{46} + 4 q^{47} - 4 q^{48} + 211 q^{49} + 15 q^{50} + 32 q^{51} + 26 q^{53} - 18 q^{54} - 8 q^{55} - 4 q^{56} - 2 q^{57} - 12 q^{59} - 4 q^{60} + 10 q^{61} + 28 q^{62} + 32 q^{63} + 233 q^{64} + 36 q^{66} - 4 q^{67} + 8 q^{68} + 12 q^{69} - 4 q^{70} + 12 q^{71} + 13 q^{72} - 36 q^{73} + 18 q^{74} + 52 q^{75} - q^{76} + 20 q^{77} + 20 q^{79} - 6 q^{80} + 241 q^{81} + 26 q^{82} + 48 q^{83} - 4 q^{84} + 40 q^{85} + 24 q^{86} + 26 q^{87} - 4 q^{88} + 6 q^{89} + 38 q^{90} - 14 q^{92} + 20 q^{93} - 24 q^{94} - 2 q^{95} - 6 q^{96} + 38 q^{97} + 33 q^{98} + 52 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 13 19
6422.2.a.a 6422.a 1.a $1$ $51.280$ \(\Q\) None \(-1\) \(-1\) \(1\) \(3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+3q^{7}+\cdots\)
6422.2.a.b 6422.a 1.a $1$ $51.280$ \(\Q\) None \(-1\) \(-1\) \(4\) \(-3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+4q^{5}+q^{6}-3q^{7}+\cdots\)
6422.2.a.c 6422.a 1.a $1$ $51.280$ \(\Q\) None \(-1\) \(1\) \(3\) \(-1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+3q^{5}-q^{6}-q^{7}+\cdots\)
6422.2.a.d 6422.a 1.a $1$ $51.280$ \(\Q\) None \(-1\) \(2\) \(-3\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-3q^{5}-2q^{6}-q^{8}+\cdots\)
6422.2.a.e 6422.a 1.a $1$ $51.280$ \(\Q\) None \(1\) \(-1\) \(-1\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
6422.2.a.f 6422.a 1.a $1$ $51.280$ \(\Q\) None \(1\) \(0\) \(-2\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-2q^{5}-4q^{7}+q^{8}-3q^{9}+\cdots\)
6422.2.a.g 6422.a 1.a $1$ $51.280$ \(\Q\) None \(1\) \(1\) \(-3\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-3q^{5}+q^{6}+q^{7}+\cdots\)
6422.2.a.h 6422.a 1.a $1$ $51.280$ \(\Q\) None \(1\) \(1\) \(0\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{7}+q^{8}+\cdots\)
6422.2.a.i 6422.a 1.a $1$ $51.280$ \(\Q\) None \(1\) \(2\) \(3\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+3q^{5}+2q^{6}+q^{8}+\cdots\)
6422.2.a.j 6422.a 1.a $1$ $51.280$ \(\Q\) None \(1\) \(3\) \(3\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}+q^{4}+3q^{5}+3q^{6}-3q^{7}+\cdots\)
6422.2.a.k 6422.a 1.a $3$ $51.280$ \(\Q(\zeta_{14})^+\) None \(-3\) \(-3\) \(-5\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+q^{4}+(-2+\cdots)q^{5}+\cdots\)
6422.2.a.l 6422.a 1.a $3$ $51.280$ \(\Q(\zeta_{14})^+\) None \(-3\) \(-3\) \(3\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+q^{4}+(2\beta _{1}+\cdots)q^{5}+\cdots\)
6422.2.a.m 6422.a 1.a $3$ $51.280$ 3.3.169.1 None \(-3\) \(-2\) \(-2\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
6422.2.a.n 6422.a 1.a $3$ $51.280$ \(\Q(\zeta_{14})^+\) None \(-3\) \(-1\) \(7\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+(3-\beta _{1}+\beta _{2})q^{5}+\cdots\)
6422.2.a.o 6422.a 1.a $3$ $51.280$ 3.3.1129.1 None \(-3\) \(0\) \(0\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots\)
6422.2.a.p 6422.a 1.a $3$ $51.280$ 3.3.1129.1 None \(-3\) \(0\) \(2\) \(3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+(1+\beta _{1}+\beta _{2})q^{5}+\cdots\)
6422.2.a.q 6422.a 1.a $3$ $51.280$ \(\Q(\zeta_{14})^+\) None \(-3\) \(5\) \(-5\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(2-\beta _{1})q^{3}+q^{4}+(-2-\beta _{2})q^{5}+\cdots\)
6422.2.a.r 6422.a 1.a $3$ $51.280$ \(\Q(\zeta_{14})^+\) None \(3\) \(-3\) \(-3\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+q^{4}+(-2\beta _{1}+\cdots)q^{5}+\cdots\)
6422.2.a.s 6422.a 1.a $3$ $51.280$ \(\Q(\zeta_{18})^+\) None \(3\) \(-3\) \(-3\) \(6\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-2\beta _{1}+\beta _{2})q^{3}+q^{4}+\cdots\)
6422.2.a.t 6422.a 1.a $3$ $51.280$ \(\Q(\zeta_{14})^+\) None \(3\) \(-3\) \(5\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+q^{4}+(2+\cdots)q^{5}+\cdots\)
6422.2.a.u 6422.a 1.a $3$ $51.280$ 3.3.169.1 None \(3\) \(-2\) \(2\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots\)
6422.2.a.v 6422.a 1.a $3$ $51.280$ \(\Q(\zeta_{14})^+\) None \(3\) \(-1\) \(-7\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+(-3+\beta _{1}-\beta _{2})q^{5}+\cdots\)
6422.2.a.w 6422.a 1.a $3$ $51.280$ 3.3.361.1 None \(3\) \(-1\) \(-1\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{2}q^{5}-\beta _{1}q^{6}+\cdots\)
6422.2.a.x 6422.a 1.a $3$ $51.280$ 3.3.1129.1 None \(3\) \(0\) \(-2\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+(-1-\beta _{1}-\beta _{2})q^{5}+\cdots\)
6422.2.a.y 6422.a 1.a $3$ $51.280$ 3.3.1129.1 None \(3\) \(0\) \(0\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
6422.2.a.z 6422.a 1.a $4$ $51.280$ 4.4.10273.1 None \(-4\) \(-2\) \(-1\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+(-\beta _{1}+\beta _{3})q^{5}+\cdots\)
6422.2.a.ba 6422.a 1.a $4$ $51.280$ 4.4.16609.1 None \(-4\) \(2\) \(-2\) \(-7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{2}q^{3}+q^{4}+(-1+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
6422.2.a.bb 6422.a 1.a $4$ $51.280$ 4.4.10273.1 None \(4\) \(-2\) \(1\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+(\beta _{1}-\beta _{3})q^{5}+\cdots\)
6422.2.a.bc 6422.a 1.a $6$ $51.280$ 6.6.316645497.1 None \(-6\) \(2\) \(-2\) \(-1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{2}q^{5}-\beta _{1}q^{6}+\cdots\)
6422.2.a.bd 6422.a 1.a $6$ $51.280$ 6.6.316645497.1 None \(6\) \(2\) \(2\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{2}q^{5}+\beta _{1}q^{6}+\cdots\)
6422.2.a.be 6422.a 1.a $7$ $51.280$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(2\) \(-2\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{2}q^{5}-\beta _{1}q^{6}+\cdots\)
6422.2.a.bf 6422.a 1.a $7$ $51.280$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(7\) \(2\) \(2\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{2}q^{5}+\beta _{1}q^{6}+\cdots\)
6422.2.a.bg 6422.a 1.a $8$ $51.280$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(-4\) \(2\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{6}q^{5}+\beta _{1}q^{6}+\cdots\)
6422.2.a.bh 6422.a 1.a $8$ $51.280$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(-4\) \(-2\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{6}q^{5}-\beta _{1}q^{6}+\cdots\)
6422.2.a.bi 6422.a 1.a $9$ $51.280$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-9\) \(-5\) \(-1\) \(13\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{7})q^{3}+q^{4}+\beta _{4}q^{5}+\cdots\)
6422.2.a.bj 6422.a 1.a $9$ $51.280$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-9\) \(1\) \(-1\) \(13\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{6}q^{3}+q^{4}+(-1+\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)
6422.2.a.bk 6422.a 1.a $9$ $51.280$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(-5\) \(1\) \(-13\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{7})q^{3}+q^{4}-\beta _{4}q^{5}+\cdots\)
6422.2.a.bl 6422.a 1.a $9$ $51.280$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(1\) \(1\) \(-13\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{6}q^{3}+q^{4}+(1-\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots\)
6422.2.a.bm 6422.a 1.a $14$ $51.280$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(-14\) \(4\) \(2\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{4}q^{5}-\beta _{1}q^{6}+\cdots\)
6422.2.a.bn 6422.a 1.a $14$ $51.280$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(14\) \(4\) \(-2\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{4}q^{5}+\beta _{1}q^{6}+\cdots\)
6422.2.a.bo 6422.a 1.a $15$ $51.280$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(-15\) \(0\) \(-1\) \(-18\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{10}q^{5}-\beta _{1}q^{6}+\cdots\)
6422.2.a.bp 6422.a 1.a $15$ $51.280$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(-15\) \(4\) \(-3\) \(-10\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{13}q^{5}-\beta _{1}q^{6}+\cdots\)
6422.2.a.bq 6422.a 1.a $15$ $51.280$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(15\) \(0\) \(1\) \(18\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{10}q^{5}+\beta _{1}q^{6}+\cdots\)
6422.2.a.br 6422.a 1.a $15$ $51.280$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(15\) \(4\) \(3\) \(10\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{13}q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6422))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6422)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(247))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(494))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3211))\)\(^{\oplus 2}\)