Properties

Label 400.5.p
Level $400$
Weight $5$
Character orbit 400.p
Rep. character $\chi_{400}(193,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $70$
Newform subspaces $17$
Sturm bound $300$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 400.p (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 17 \)
Sturm bound: \(300\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(400, [\chi])\).

Total New Old
Modular forms 516 74 442
Cusp forms 444 70 374
Eisenstein series 72 4 68

Trace form

\( 70 q - 2 q^{3} - 2 q^{7} + 4 q^{11} + 122 q^{13} + 122 q^{17} + 1212 q^{21} - 1442 q^{23} + 1120 q^{27} + 260 q^{31} - 1276 q^{33} - 1398 q^{37} + 2492 q^{41} + 4478 q^{43} - 4322 q^{47} + 6532 q^{51} - 2518 q^{53}+ \cdots + 7722 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{5}^{\mathrm{new}}(400, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
400.5.p.a 400.p 5.c $2$ $41.348$ \(\Q(\sqrt{-1}) \) None 5.5.c.a \(0\) \(-12\) \(0\) \(-52\) $\mathrm{SU}(2)[C_{4}]$ \(q+(6 i-6)q^{3}+(-26 i-26)q^{7}+\cdots\)
400.5.p.b 400.p 5.c $2$ $41.348$ \(\Q(\sqrt{-1}) \) None 10.5.c.b \(0\) \(2\) \(0\) \(-38\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-i+1)q^{3}+(-19 i-19)q^{7}+\cdots\)
400.5.p.c 400.p 5.c $2$ $41.348$ \(\Q(\sqrt{-1}) \) None 10.5.c.a \(0\) \(18\) \(0\) \(58\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-9 i+9)q^{3}+(29 i+29)q^{7}+\cdots\)
400.5.p.d 400.p 5.c $2$ $41.348$ \(\Q(\sqrt{-1}) \) None 40.5.l.a \(0\) \(20\) \(0\) \(-84\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-10 i+10)q^{3}+(-42 i-42)q^{7}+\cdots\)
400.5.p.e 400.p 5.c $4$ $41.348$ \(\Q(i, \sqrt{6})\) None 50.5.c.c \(0\) \(-24\) \(0\) \(-144\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-6+\beta _{1}-6\beta _{2})q^{3}+(-6^{2}+6^{2}\beta _{2}+\cdots)q^{7}+\cdots\)
400.5.p.f 400.p 5.c $4$ $41.348$ \(\Q(i, \sqrt{61})\) None 200.5.l.b \(0\) \(-16\) \(0\) \(48\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-4+4\beta _{1}-\beta _{3})q^{3}+(12+12\beta _{1}+\cdots)q^{7}+\cdots\)
400.5.p.g 400.p 5.c $4$ $41.348$ \(\Q(i, \sqrt{29})\) None 40.5.l.b \(0\) \(-12\) \(0\) \(44\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-3+3\beta _{1})q^{3}+(11+11\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
400.5.p.h 400.p 5.c $4$ $41.348$ \(\Q(i, \sqrt{241})\) None 20.5.f.a \(0\) \(-10\) \(0\) \(110\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2+2\beta _{1}+\beta _{3})q^{3}+(26+29\beta _{1}+\cdots)q^{7}+\cdots\)
400.5.p.i 400.p 5.c $4$ $41.348$ \(\Q(i, \sqrt{69})\) None 100.5.f.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{3}q^{3}+7\beta _{2}q^{7}-57\beta _{1}q^{9}-180q^{11}+\cdots\)
400.5.p.j 400.p 5.c $4$ $41.348$ \(\Q(i, \sqrt{6})\) None 25.5.c.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+7\beta _{1}q^{3}+28\beta _{3}q^{7}+66\beta _{2}q^{9}-117q^{11}+\cdots\)
400.5.p.k 400.p 5.c $4$ $41.348$ \(\Q(i, \sqrt{6})\) None 100.5.f.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{3}-12\beta _{3}q^{7}-78\beta _{2}q^{9}-45q^{11}+\cdots\)
400.5.p.l 400.p 5.c $4$ $41.348$ \(\Q(i, \sqrt{21})\) None 25.5.c.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{3}q^{3}-9\beta _{2}q^{7}+39\beta _{1}q^{9}+108q^{11}+\cdots\)
400.5.p.m 400.p 5.c $4$ $41.348$ \(\Q(i, \sqrt{61})\) None 200.5.l.b \(0\) \(16\) \(0\) \(-48\) $\mathrm{SU}(2)[C_{4}]$ \(q+(4+4\beta _{1}-\beta _{2})q^{3}+(-12+12\beta _{1}+\cdots)q^{7}+\cdots\)
400.5.p.n 400.p 5.c $4$ $41.348$ \(\Q(i, \sqrt{6})\) None 50.5.c.c \(0\) \(24\) \(0\) \(144\) $\mathrm{SU}(2)[C_{4}]$ \(q+(6+\beta _{1}+6\beta _{2})q^{3}+(6^{2}-6^{2}\beta _{2}+4\beta _{3})q^{7}+\cdots\)
400.5.p.o 400.p 5.c $6$ $41.348$ 6.0.313431616.3 None 40.5.l.c \(0\) \(-8\) \(0\) \(-40\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\beta _{1}-\beta _{3})q^{3}+(-6+6\beta _{1}+\cdots)q^{7}+\cdots\)
400.5.p.p 400.p 5.c $8$ $41.348$ 8.0.\(\cdots\).18 None 200.5.l.f \(0\) \(-8\) \(0\) \(-48\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\beta _{1}+\beta _{3})q^{3}+(-6+6\beta _{1}+\cdots)q^{7}+\cdots\)
400.5.p.q 400.p 5.c $8$ $41.348$ 8.0.\(\cdots\).18 None 200.5.l.f \(0\) \(8\) \(0\) \(48\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\beta _{1}-\beta _{2})q^{3}+(6+6\beta _{1}-2\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(400, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(400, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)