Properties

Label 208.8.a
Level $208$
Weight $8$
Character orbit 208.a
Rep. character $\chi_{208}(1,\cdot)$
Character field $\Q$
Dimension $42$
Newform subspaces $15$
Sturm bound $224$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 208.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(224\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(208))\).

Total New Old
Modular forms 202 42 160
Cusp forms 190 42 148
Eisenstein series 12 0 12

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

\(2\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(11\)
\(+\)\(-\)\(-\)\(10\)
\(-\)\(+\)\(-\)\(10\)
\(-\)\(-\)\(+\)\(11\)
Plus space\(+\)\(22\)
Minus space\(-\)\(20\)

Trace form

\( 42 q + 686 q^{7} + 28382 q^{9} + O(q^{10}) \) \( 42 q + 686 q^{7} + 28382 q^{9} - 9190 q^{11} + 50952 q^{15} + 23972 q^{17} - 101738 q^{19} + 2344 q^{21} + 230984 q^{23} + 676738 q^{25} - 366756 q^{27} - 179712 q^{29} + 446770 q^{31} - 176184 q^{33} - 733824 q^{35} + 415576 q^{37} + 237276 q^{39} + 300604 q^{41} - 559516 q^{43} + 1009000 q^{45} - 476650 q^{47} + 5957650 q^{49} + 379444 q^{51} - 565268 q^{53} - 4278044 q^{55} - 1209600 q^{57} - 3696822 q^{59} - 3192412 q^{61} - 459002 q^{63} + 1819198 q^{67} + 8418904 q^{69} + 1893642 q^{71} + 8118100 q^{73} + 23826716 q^{75} - 6960692 q^{77} - 10823644 q^{79} + 11008650 q^{81} - 2804594 q^{83} + 14693376 q^{85} - 21803704 q^{87} - 4158212 q^{89} - 4521426 q^{91} - 8125488 q^{93} + 17624484 q^{95} - 16149148 q^{97} + 15150762 q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(208))\) 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 13
208.8.a.a 208.a 1.a $1$ $64.976$ \(\Q\) None 26.8.a.c \(0\) \(27\) \(-245\) \(587\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3^{3}q^{3}-245q^{5}+587q^{7}-1458q^{9}+\cdots\)
208.8.a.b 208.a 1.a $1$ $64.976$ \(\Q\) None 52.8.a.a \(0\) \(28\) \(-418\) \(-124\) $-$ $-$ $\mathrm{SU}(2)$ \(q+28q^{3}-418q^{5}-124q^{7}-1403q^{9}+\cdots\)
208.8.a.c 208.a 1.a $1$ $64.976$ \(\Q\) None 26.8.a.a \(0\) \(39\) \(385\) \(293\) $-$ $-$ $\mathrm{SU}(2)$ \(q+39q^{3}+385q^{5}+293q^{7}-666q^{9}+\cdots\)
208.8.a.d 208.a 1.a $1$ $64.976$ \(\Q\) None 13.8.a.a \(0\) \(73\) \(-295\) \(-1373\) $-$ $-$ $\mathrm{SU}(2)$ \(q+73q^{3}-295q^{5}-1373q^{7}+3142q^{9}+\cdots\)
208.8.a.e 208.a 1.a $1$ $64.976$ \(\Q\) None 26.8.a.b \(0\) \(87\) \(321\) \(181\) $-$ $-$ $\mathrm{SU}(2)$ \(q+87q^{3}+321q^{5}+181q^{7}+5382q^{9}+\cdots\)
208.8.a.f 208.a 1.a $2$ $64.976$ \(\Q(\sqrt{2305}) \) None 26.8.a.e \(0\) \(-87\) \(215\) \(-705\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-43-\beta )q^{3}+(105+5\beta )q^{5}+(-377+\cdots)q^{7}+\cdots\)
208.8.a.g 208.a 1.a $2$ $64.976$ \(\Q(\sqrt{337}) \) None 13.8.a.b \(0\) \(-45\) \(-353\) \(2009\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-21-3\beta )q^{3}+(-171-11\beta )q^{5}+\cdots\)
208.8.a.h 208.a 1.a $2$ $64.976$ \(\Q(\sqrt{105}) \) None 26.8.a.d \(0\) \(12\) \(-146\) \(1780\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(6+7\beta )q^{3}+(-73-6^{2}\beta )q^{5}+(890+\cdots)q^{7}+\cdots\)
208.8.a.i 208.a 1.a $3$ $64.976$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 52.8.a.c \(0\) \(-28\) \(520\) \(736\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-9+\beta _{1})q^{3}+(175+5\beta _{2})q^{5}+(245+\cdots)q^{7}+\cdots\)
208.8.a.j 208.a 1.a $3$ $64.976$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 52.8.a.b \(0\) \(0\) \(8\) \(342\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(3+2\beta _{1}-\beta _{2})q^{5}+(116+\cdots)q^{7}+\cdots\)
208.8.a.k 208.a 1.a $4$ $64.976$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 13.8.a.c \(0\) \(-80\) \(258\) \(-1692\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-20-\beta _{2})q^{3}+(63+\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots\)
208.8.a.l 208.a 1.a $4$ $64.976$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 104.8.a.a \(0\) \(71\) \(-263\) \(-231\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(18+\beta _{1})q^{3}+(-66+\beta _{1}-\beta _{3})q^{5}+\cdots\)
208.8.a.m 208.a 1.a $5$ $64.976$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 104.8.a.b \(0\) \(-57\) \(13\) \(1803\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-11-\beta _{1})q^{3}+(3-2\beta _{1}-\beta _{3})q^{5}+\cdots\)
208.8.a.n 208.a 1.a $6$ $64.976$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 104.8.a.d \(0\) \(-84\) \(-112\) \(-1472\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-14-\beta _{2})q^{3}+(-19-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
208.8.a.o 208.a 1.a $6$ $64.976$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 104.8.a.c \(0\) \(44\) \(112\) \(-1448\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(7+\beta _{1})q^{3}+(18+2\beta _{1}+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(208))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(208)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 2}\)