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Properties

Label 1700.2.o
Level $1700$
Weight $2$
Character orbit 1700.o
Rep. character $\chi_{1700}(701,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $56$
Newform subspaces $7$
Sturm bound $540$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 1700 = 2^{2} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1700.o (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 17 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 7 \)
Sturm bound: \(540\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1700, [\chi])\).

Total New Old
Modular forms 576 56 520
Cusp forms 504 56 448
Eisenstein series 72 0 72

Trace form

\( 56 q - 2 q^{3} - 2 q^{7} + O(q^{10}) \) \( 56 q - 2 q^{3} - 2 q^{7} + 2 q^{11} - 12 q^{13} + 4 q^{17} + 12 q^{21} - 2 q^{23} - 8 q^{27} - 14 q^{31} + 28 q^{33} + 8 q^{37} + 4 q^{41} + 24 q^{47} - 18 q^{51} + 20 q^{57} - 24 q^{61} - 38 q^{63} - 40 q^{67} - 20 q^{69} + 2 q^{71} + 28 q^{73} + 14 q^{79} - 20 q^{81} + 76 q^{89} - 8 q^{91} + 60 q^{97} + 6 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(1700, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1700.2.o.a 1700.o 17.c $2$ $13.575$ \(\Q(\sqrt{-1}) \) None 340.2.m.a \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2-2i)q^{3}+5iq^{9}+(-2+2i)q^{11}+\cdots\)
1700.2.o.b 1700.o 17.c $2$ $13.575$ \(\Q(\sqrt{-1}) \) None 340.2.m.a \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2+2i)q^{3}+5iq^{9}+(-2+2i)q^{11}+\cdots\)
1700.2.o.c 1700.o 17.c $4$ $13.575$ \(\Q(i, \sqrt{13})\) None 68.2.e.a \(0\) \(2\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\beta _{1}+\beta _{3})q^{3}+(-1+\beta _{2})q^{7}+\cdots\)
1700.2.o.d 1700.o 17.c $12$ $13.575$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 340.2.o.a \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\beta _{5}-\beta _{8}+\beta _{9})q^{3}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
1700.2.o.e 1700.o 17.c $12$ $13.575$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 1700.2.o.e \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{3}q^{3}+\beta _{11}q^{7}+(-\beta _{9}-\beta _{10}+\beta _{11})q^{9}+\cdots\)
1700.2.o.f 1700.o 17.c $12$ $13.575$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 340.2.m.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{4}q^{3}+(-\beta _{1}-\beta _{5}+\beta _{6})q^{7}+(-\beta _{3}+\cdots)q^{9}+\cdots\)
1700.2.o.g 1700.o 17.c $12$ $13.575$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 1700.2.o.e \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{3}-\beta _{9}q^{7}+(\beta _{9}+\beta _{10}-\beta _{11})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1700, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1700, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(34, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(68, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(170, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(340, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(425, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(850, [\chi])\)\(^{\oplus 2}\)