Properties

Label 208.10.a
Level $208$
Weight $10$
Character orbit 208.a
Rep. character $\chi_{208}(1,\cdot)$
Character field $\Q$
Dimension $54$
Newform subspaces $13$
Sturm bound $280$
Trace bound $3$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 208.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(280\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(208))\).

Total New Old
Modular forms 258 54 204
Cusp forms 246 54 192
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
\(+\)\(+\)\(+\)\(13\)
\(+\)\(-\)\(-\)\(14\)
\(-\)\(+\)\(-\)\(14\)
\(-\)\(-\)\(+\)\(13\)
Plus space\(+\)\(26\)
Minus space\(-\)\(28\)

Trace form

\( 54 q - 4802 q^{7} + 382034 q^{9} + O(q^{10}) \) \( 54 q - 4802 q^{7} + 382034 q^{9} + 21986 q^{11} - 81528 q^{15} - 242212 q^{17} + 1262814 q^{19} + 634168 q^{21} - 1020280 q^{23} + 21856302 q^{25} - 527028 q^{27} - 4511232 q^{29} - 11885838 q^{31} + 7018680 q^{33} + 41639040 q^{35} - 1214776 q^{37} - 9253764 q^{39} - 4071260 q^{41} + 45987380 q^{43} - 13805000 q^{45} - 224367370 q^{47} + 269172846 q^{49} + 207371428 q^{51} + 64563556 q^{53} - 144893436 q^{55} + 89709312 q^{57} + 218112498 q^{59} + 145215404 q^{61} - 559687034 q^{63} + 218286534 q^{67} - 72690488 q^{69} - 614785734 q^{71} + 111758796 q^{73} - 702852148 q^{75} - 321924380 q^{77} + 994007268 q^{79} + 3011851446 q^{81} - 1915074314 q^{83} - 40098816 q^{85} + 403827080 q^{87} - 842440028 q^{89} + 411449766 q^{91} + 1205123184 q^{93} + 4493399076 q^{95} + 335423196 q^{97} - 3586752702 q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(208))\) 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
208.10.a.a 208.a 1.a $1$ $107.127$ \(\Q\) None 26.10.a.b \(0\) \(-192\) \(-1310\) \(5810\) $-$ $+$ $\mathrm{SU}(2)$ \(q-192q^{3}-1310q^{5}+5810q^{7}+17181q^{9}+\cdots\)
208.10.a.b 208.a 1.a $1$ $107.127$ \(\Q\) None 26.10.a.c \(0\) \(-75\) \(-1979\) \(10115\) $-$ $-$ $\mathrm{SU}(2)$ \(q-75q^{3}-1979q^{5}+10115q^{7}-14058q^{9}+\cdots\)
208.10.a.c 208.a 1.a $1$ $107.127$ \(\Q\) None 26.10.a.a \(0\) \(273\) \(1015\) \(-3955\) $-$ $+$ $\mathrm{SU}(2)$ \(q+273q^{3}+1015q^{5}-3955q^{7}+54846q^{9}+\cdots\)
208.10.a.d 208.a 1.a $3$ $107.127$ 3.3.2119705.1 None 26.10.a.e \(0\) \(-156\) \(-1272\) \(-17058\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-52+\beta _{1}+\beta _{2})q^{3}+(-424-18\beta _{1}+\cdots)q^{5}+\cdots\)
208.10.a.e 208.a 1.a $3$ $107.127$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 26.10.a.d \(0\) \(0\) \(248\) \(2956\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(79-11\beta _{1}-9\beta _{2})q^{5}+(972+\cdots)q^{7}+\cdots\)
208.10.a.f 208.a 1.a $4$ $107.127$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 52.10.a.a \(0\) \(147\) \(-1947\) \(-10251\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(37+\beta _{1})q^{3}+(-486+2\beta _{1}+\beta _{3})q^{5}+\cdots\)
208.10.a.g 208.a 1.a $4$ $107.127$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 13.10.a.a \(0\) \(163\) \(471\) \(11241\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(41+\beta _{3})q^{3}+(118-2\beta _{1}+7\beta _{2}+\cdots)q^{5}+\cdots\)
208.10.a.h 208.a 1.a $5$ $107.127$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 13.10.a.b \(0\) \(-161\) \(1803\) \(-10099\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2^{5}+\beta _{3})q^{3}+(19^{2}+3\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
208.10.a.i 208.a 1.a $5$ $107.127$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 52.10.a.b \(0\) \(147\) \(1721\) \(-3317\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(29-\beta _{1})q^{3}+(345+\beta _{1}+\beta _{2})q^{5}+\cdots\)
208.10.a.j 208.a 1.a $6$ $107.127$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 104.10.a.b \(0\) \(-60\) \(176\) \(-3416\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-10+\beta _{1})q^{3}+(29-2\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
208.10.a.k 208.a 1.a $6$ $107.127$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 104.10.a.a \(0\) \(68\) \(-176\) \(9664\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(11+\beta _{1})q^{3}+(-29+\beta _{2})q^{5}+(1610+\cdots)q^{7}+\cdots\)
208.10.a.l 208.a 1.a $7$ $107.127$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 104.10.a.c \(0\) \(-13\) \(-801\) \(1091\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{3}+(-114+\beta _{2})q^{5}+\cdots\)
208.10.a.m 208.a 1.a $8$ $107.127$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 104.10.a.d \(0\) \(-141\) \(2051\) \(2417\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-18+\beta _{1})q^{3}+(2^{8}+2\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(208))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(208)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 5}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 2}\)