Properties

Label 650.6.a
Level $650$
Weight $6$
Character orbit 650.a
Rep. character $\chi_{650}(1,\cdot)$
Character field $\Q$
Dimension $95$
Newform subspaces $23$
Sturm bound $630$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 650.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 23 \)
Sturm bound: \(630\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(650))\).

Total New Old
Modular forms 536 95 441
Cusp forms 512 95 417
Eisenstein series 24 0 24

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(5\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(12\)
\(+\)\(+\)\(-\)\(-\)\(12\)
\(+\)\(-\)\(+\)\(-\)\(13\)
\(+\)\(-\)\(-\)\(+\)\(11\)
\(-\)\(+\)\(+\)\(-\)\(12\)
\(-\)\(+\)\(-\)\(+\)\(9\)
\(-\)\(-\)\(+\)\(+\)\(11\)
\(-\)\(-\)\(-\)\(-\)\(15\)
Plus space\(+\)\(43\)
Minus space\(-\)\(52\)

Trace form

\( 95 q - 4 q^{2} + 26 q^{3} + 1520 q^{4} - 16 q^{6} - 312 q^{7} - 64 q^{8} + 7461 q^{9} + O(q^{10}) \) \( 95 q - 4 q^{2} + 26 q^{3} + 1520 q^{4} - 16 q^{6} - 312 q^{7} - 64 q^{8} + 7461 q^{9} - 532 q^{11} + 416 q^{12} - 169 q^{13} - 168 q^{14} + 24320 q^{16} - 216 q^{17} - 2900 q^{18} + 112 q^{19} - 6052 q^{21} + 6400 q^{22} + 8988 q^{23} - 256 q^{24} + 2028 q^{26} + 134 q^{27} - 4992 q^{28} + 4570 q^{29} - 7536 q^{31} - 1024 q^{32} + 16236 q^{33} + 6168 q^{34} + 119376 q^{36} - 24178 q^{37} - 16624 q^{38} + 34150 q^{41} + 36424 q^{42} + 34070 q^{43} - 8512 q^{44} + 20176 q^{46} + 38672 q^{47} + 6656 q^{48} + 214137 q^{49} - 78026 q^{51} - 2704 q^{52} - 13618 q^{53} + 25184 q^{54} - 2688 q^{56} - 75864 q^{57} + 17464 q^{58} + 22644 q^{59} + 22682 q^{61} - 42016 q^{62} - 63984 q^{63} + 389120 q^{64} + 24656 q^{66} - 75064 q^{67} - 3456 q^{68} - 177060 q^{69} - 27776 q^{71} - 46400 q^{72} - 40618 q^{73} - 10608 q^{74} + 1792 q^{76} + 98064 q^{77} - 12168 q^{78} - 208180 q^{79} + 779583 q^{81} - 37288 q^{82} - 30048 q^{83} - 96832 q^{84} + 453504 q^{86} + 564960 q^{87} + 102400 q^{88} + 608798 q^{89} - 57798 q^{91} + 143808 q^{92} + 492224 q^{93} - 138472 q^{94} - 4096 q^{96} + 221234 q^{97} - 69028 q^{98} + 1311924 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(650))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5 13
650.6.a.a 650.a 1.a $1$ $104.249$ \(\Q\) None 26.6.a.a \(4\) \(0\) \(0\) \(170\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+170q^{7}+2^{6}q^{8}+\cdots\)
650.6.a.b 650.a 1.a $2$ $104.249$ \(\Q(\sqrt{849}) \) None 26.6.a.c \(-8\) \(-9\) \(0\) \(-155\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+(-4-\beta )q^{3}+2^{4}q^{4}+(2^{4}+\cdots)q^{6}+\cdots\)
650.6.a.c 650.a 1.a $2$ $104.249$ \(\Q(\sqrt{14}) \) None 130.6.a.e \(-8\) \(16\) \(0\) \(252\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+(8+3\beta )q^{3}+2^{4}q^{4}+(-2^{5}+\cdots)q^{6}+\cdots\)
650.6.a.d 650.a 1.a $2$ $104.249$ \(\Q(\sqrt{235}) \) None 130.6.a.d \(-8\) \(26\) \(0\) \(-160\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+(13+\beta )q^{3}+2^{4}q^{4}+(-52+\cdots)q^{6}+\cdots\)
650.6.a.e 650.a 1.a $2$ $104.249$ \(\Q(\sqrt{2785}) \) None 26.6.a.b \(8\) \(-9\) \(0\) \(-327\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+(-4-\beta )q^{3}+2^{4}q^{4}+(-2^{4}+\cdots)q^{6}+\cdots\)
650.6.a.f 650.a 1.a $2$ $104.249$ \(\Q(\sqrt{145}) \) None 130.6.a.c \(8\) \(-6\) \(0\) \(-130\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+(-3-\beta )q^{3}+2^{4}q^{4}+(-12+\cdots)q^{6}+\cdots\)
650.6.a.g 650.a 1.a $2$ $104.249$ \(\Q(\sqrt{19}) \) None 130.6.a.b \(8\) \(-6\) \(0\) \(208\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+(-3+3\beta )q^{3}+2^{4}q^{4}+(-12+\cdots)q^{6}+\cdots\)
650.6.a.h 650.a 1.a $2$ $104.249$ \(\Q(\sqrt{10}) \) None 130.6.a.a \(8\) \(4\) \(0\) \(-300\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+(2+\beta )q^{3}+2^{4}q^{4}+(8+4\beta )q^{6}+\cdots\)
650.6.a.i 650.a 1.a $3$ $104.249$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 130.6.a.g \(-12\) \(8\) \(0\) \(-42\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+(3+\beta _{1})q^{3}+2^{4}q^{4}+(-12+\cdots)q^{6}+\cdots\)
650.6.a.j 650.a 1.a $3$ $104.249$ 3.3.1458804.1 None 130.6.a.f \(12\) \(22\) \(0\) \(234\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+(7+\beta _{1})q^{3}+2^{4}q^{4}+(28+4\beta _{1}+\cdots)q^{6}+\cdots\)
650.6.a.k 650.a 1.a $4$ $104.249$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 130.6.a.h \(-16\) \(-20\) \(0\) \(-62\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+(-5+\beta _{1})q^{3}+2^{4}q^{4}+(20+\cdots)q^{6}+\cdots\)
650.6.a.l 650.a 1.a $4$ $104.249$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 650.6.a.l \(-16\) \(20\) \(0\) \(178\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+(5-\beta _{1})q^{3}+2^{4}q^{4}+(-20+\cdots)q^{6}+\cdots\)
650.6.a.m 650.a 1.a $4$ $104.249$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 650.6.a.l \(16\) \(-20\) \(0\) \(-178\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+(-5+\beta _{1})q^{3}+2^{4}q^{4}+(-20+\cdots)q^{6}+\cdots\)
650.6.a.n 650.a 1.a $5$ $104.249$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 650.6.a.n \(-20\) \(2\) \(0\) \(-157\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}-4\beta _{1}q^{6}+\cdots\)
650.6.a.o 650.a 1.a $5$ $104.249$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 650.6.a.o \(-20\) \(9\) \(0\) \(-42\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+(2-\beta _{1})q^{3}+2^{4}q^{4}+(-8+\cdots)q^{6}+\cdots\)
650.6.a.p 650.a 1.a $5$ $104.249$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 650.6.a.o \(20\) \(-9\) \(0\) \(42\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+(-2+\beta _{1})q^{3}+2^{4}q^{4}+(-8+\cdots)q^{6}+\cdots\)
650.6.a.q 650.a 1.a $5$ $104.249$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 650.6.a.n \(20\) \(-2\) \(0\) \(157\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-\beta _{1}q^{3}+2^{4}q^{4}-4\beta _{1}q^{6}+\cdots\)
650.6.a.r 650.a 1.a $6$ $104.249$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 130.6.b.a \(-24\) \(-14\) \(0\) \(154\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+(-2+\beta _{1})q^{3}+2^{4}q^{4}+(8+\cdots)q^{6}+\cdots\)
650.6.a.s 650.a 1.a $6$ $104.249$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 650.6.a.s \(-24\) \(-9\) \(0\) \(63\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+(-1-\beta _{1})q^{3}+2^{4}q^{4}+(4+\cdots)q^{6}+\cdots\)
650.6.a.t 650.a 1.a $6$ $104.249$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 650.6.a.s \(24\) \(9\) \(0\) \(-63\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+(1+\beta _{1})q^{3}+2^{4}q^{4}+(4+4\beta _{1}+\cdots)q^{6}+\cdots\)
650.6.a.u 650.a 1.a $6$ $104.249$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 130.6.b.a \(24\) \(14\) \(0\) \(-154\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+(2-\beta _{1})q^{3}+2^{4}q^{4}+(8-4\beta _{1}+\cdots)q^{6}+\cdots\)
650.6.a.v 650.a 1.a $9$ $104.249$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 130.6.b.b \(-36\) \(-14\) \(0\) \(-164\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+(-2+\beta _{1})q^{3}+2^{4}q^{4}+(8+\cdots)q^{6}+\cdots\)
650.6.a.w 650.a 1.a $9$ $104.249$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 130.6.b.b \(36\) \(14\) \(0\) \(164\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+(2-\beta _{1})q^{3}+2^{4}q^{4}+(8-4\beta _{1}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(650))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(650)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(325))\)\(^{\oplus 2}\)