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Properties

Label 1400.2.q
Level $1400$
Weight $2$
Character orbit 1400.q
Rep. character $\chi_{1400}(401,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $76$
Newform subspaces $15$
Sturm bound $480$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.q (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 15 \)
Sturm bound: \(480\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1400, [\chi])\).

Total New Old
Modular forms 528 76 452
Cusp forms 432 76 356
Eisenstein series 96 0 96

Trace form

\( 76 q - 2 q^{3} - 4 q^{7} - 36 q^{9} + O(q^{10}) \) \( 76 q - 2 q^{3} - 4 q^{7} - 36 q^{9} + 2 q^{11} - 8 q^{13} + 2 q^{17} + 10 q^{19} + 18 q^{21} - 10 q^{23} + 4 q^{27} + 16 q^{29} + 10 q^{31} - 6 q^{33} - 6 q^{37} + 4 q^{39} + 24 q^{41} - 32 q^{43} - 6 q^{47} - 20 q^{49} + 22 q^{51} - 6 q^{53} - 36 q^{57} + 18 q^{59} + 2 q^{61} + 40 q^{63} + 14 q^{67} - 28 q^{69} + 26 q^{73} + 34 q^{77} + 30 q^{79} - 22 q^{81} + 72 q^{83} + 24 q^{87} - 10 q^{89} + 16 q^{91} + 46 q^{93} + 8 q^{97} - 112 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(1400, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1400.2.q.a 1400.q 7.c $2$ $11.179$ \(\Q(\sqrt{-3}) \) None 280.2.q.c \(0\) \(-2\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{3}+(-3+2\zeta_{6})q^{7}+\cdots\)
1400.2.q.b 1400.q 7.c $2$ $11.179$ \(\Q(\sqrt{-3}) \) None 1400.2.q.b \(0\) \(-2\) \(0\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{3}+(2-3\zeta_{6})q^{7}-\zeta_{6}q^{9}+\cdots\)
1400.2.q.c 1400.q 7.c $2$ $11.179$ \(\Q(\sqrt{-3}) \) None 280.2.q.b \(0\) \(-1\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+(1-3\zeta_{6})q^{7}+2\zeta_{6}q^{9}+\cdots\)
1400.2.q.d 1400.q 7.c $2$ $11.179$ \(\Q(\sqrt{-3}) \) None 56.2.i.b \(0\) \(-1\) \(0\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+(1+2\zeta_{6})q^{7}+2\zeta_{6}q^{9}+\cdots\)
1400.2.q.e 1400.q 7.c $2$ $11.179$ \(\Q(\sqrt{-3}) \) None 280.2.q.a \(0\) \(1\) \(0\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+(3-\zeta_{6})q^{7}+2\zeta_{6}q^{9}+\cdots\)
1400.2.q.f 1400.q 7.c $2$ $11.179$ \(\Q(\sqrt{-3}) \) None 1400.2.q.b \(0\) \(2\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{3}+(-2+3\zeta_{6})q^{7}-\zeta_{6}q^{9}+\cdots\)
1400.2.q.g 1400.q 7.c $2$ $11.179$ \(\Q(\sqrt{-3}) \) None 56.2.i.a \(0\) \(3\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{3}+(-3+2\zeta_{6})q^{7}-6\zeta_{6}q^{9}+\cdots\)
1400.2.q.h 1400.q 7.c $4$ $11.179$ \(\Q(\sqrt{2}, \sqrt{-3})\) None 280.2.q.d \(0\) \(-2\) \(0\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1400.2.q.i 1400.q 7.c $6$ $11.179$ \(\Q(\zeta_{18})\) None 1400.2.q.i \(0\) \(-3\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{18}+\zeta_{18}^{3}-\zeta_{18}^{4}-\zeta_{18}^{5})q^{3}+\cdots\)
1400.2.q.j 1400.q 7.c $6$ $11.179$ 6.0.11337408.1 None 280.2.q.e \(0\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{3}+\beta _{4}-\beta _{5})q^{3}+(-1-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
1400.2.q.k 1400.q 7.c $6$ $11.179$ \(\Q(\zeta_{18})\) None 1400.2.q.i \(0\) \(3\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\zeta_{18}^{2}+\zeta_{18}^{3}-\zeta_{18}^{4})q^{3}+(2\zeta_{18}+\cdots)q^{7}+\cdots\)
1400.2.q.l 1400.q 7.c $8$ $11.179$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 1400.2.q.l \(0\) \(-3\) \(0\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{5})q^{3}+(-\beta _{1}-\beta _{3}+\beta _{4})q^{7}+\cdots\)
1400.2.q.m 1400.q 7.c $8$ $11.179$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 1400.2.q.l \(0\) \(3\) \(0\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}-\beta _{5})q^{3}+(\beta _{1}+\beta _{3}-\beta _{4})q^{7}+\cdots\)
1400.2.q.n 1400.q 7.c $12$ $11.179$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 280.2.bg.a \(0\) \(-1\) \(0\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{3})q^{3}-\beta _{4}q^{7}+(\beta _{6}-\beta _{10}+\cdots)q^{9}+\cdots\)
1400.2.q.o 1400.q 7.c $12$ $11.179$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 280.2.bg.a \(0\) \(1\) \(0\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}-\beta _{3})q^{3}+\beta _{4}q^{7}+(\beta _{6}-\beta _{10}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1400, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1400, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(350, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(700, [\chi])\)\(^{\oplus 2}\)