Properties

Label 100.7.b
Level $100$
Weight $7$
Character orbit 100.b
Rep. character $\chi_{100}(51,\cdot)$
Character field $\Q$
Dimension $54$
Newform subspaces $8$
Sturm bound $105$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 100.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(105\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(100, [\chi])\).

Total New Old
Modular forms 96 60 36
Cusp forms 84 54 30
Eisenstein series 12 6 6

Trace form

\( 54 q + 6 q^{2} + 22 q^{4} + 26 q^{6} + 264 q^{8} - 12122 q^{9} + O(q^{10}) \) \( 54 q + 6 q^{2} + 22 q^{4} + 26 q^{6} + 264 q^{8} - 12122 q^{9} - 1480 q^{12} + 2108 q^{13} - 3324 q^{14} - 11406 q^{16} + 2692 q^{17} - 14866 q^{18} + 232 q^{21} + 11280 q^{22} + 19726 q^{24} + 22292 q^{26} - 120 q^{28} + 15404 q^{29} - 1664 q^{32} - 53360 q^{33} + 15838 q^{34} + 65120 q^{36} - 66628 q^{37} + 41840 q^{38} - 6532 q^{41} + 99960 q^{42} - 37350 q^{44} - 389284 q^{46} - 82560 q^{48} - 890002 q^{49} - 384088 q^{52} + 63548 q^{53} - 590018 q^{54} - 186924 q^{56} + 746880 q^{57} - 452692 q^{58} - 351132 q^{61} + 656080 q^{62} + 1377682 q^{64} + 1317570 q^{66} + 1144488 q^{68} + 1077032 q^{69} - 2480184 q^{72} - 162172 q^{73} - 654952 q^{74} - 612390 q^{76} - 670800 q^{77} - 1176880 q^{78} + 1867406 q^{81} + 3052052 q^{82} + 4981772 q^{84} - 959184 q^{86} + 993120 q^{88} - 1943596 q^{89} - 4744680 q^{92} + 1050160 q^{93} - 3790784 q^{94} + 1823726 q^{96} + 569492 q^{97} - 1136866 q^{98} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(100, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
100.7.b.a 100.b 4.b $1$ $23.005$ \(\Q\) \(\Q(\sqrt{-1}) \) \(-8\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-8q^{2}+2^{6}q^{4}-2^{9}q^{8}+3^{6}q^{9}+1656q^{13}+\cdots\)
100.7.b.b 100.b 4.b $1$ $23.005$ \(\Q\) \(\Q(\sqrt{-1}) \) \(8\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+8q^{2}+2^{6}q^{4}+2^{9}q^{8}+3^{6}q^{9}-1656q^{13}+\cdots\)
100.7.b.c 100.b 4.b $2$ $23.005$ \(\Q(\sqrt{-15}) \) None \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-2+\beta )q^{2}-4\beta q^{3}+(-56-4\beta )q^{4}+\cdots\)
100.7.b.d 100.b 4.b $2$ $23.005$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-5}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2iq^{2}+11iq^{3}-2^{6}q^{4}+352q^{6}+\cdots\)
100.7.b.e 100.b 4.b $12$ $23.005$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-5\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(1+\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
100.7.b.f 100.b 4.b $12$ $23.005$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(\beta _{1}-\beta _{2})q^{3}+(-6+\beta _{5}+\cdots)q^{4}+\cdots\)
100.7.b.g 100.b 4.b $12$ $23.005$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(5\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(1+\beta _{1}-\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
100.7.b.h 100.b 4.b $12$ $23.005$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(10\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1-\beta _{1})q^{2}+(-\beta _{1}-\beta _{7})q^{3}+(13+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(100, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(100, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 2}\)