Properties

Label 207.8.a
Level $207$
Weight $8$
Character orbit 207.a
Rep. character $\chi_{207}(1,\cdot)$
Character field $\Q$
Dimension $63$
Newform subspaces $8$
Sturm bound $192$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 207.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(192\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(207))\).

Total New Old
Modular forms 172 63 109
Cusp forms 164 63 101
Eisenstein series 8 0 8

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

\(3\)\(23\)FrickeDim.
\(+\)\(+\)\(+\)\(12\)
\(+\)\(-\)\(-\)\(12\)
\(-\)\(+\)\(-\)\(18\)
\(-\)\(-\)\(+\)\(21\)
Plus space\(+\)\(33\)
Minus space\(-\)\(30\)

Trace form

\( 63 q + 3840 q^{4} + 388 q^{5} - 622 q^{7} - 2301 q^{8} + O(q^{10}) \) \( 63 q + 3840 q^{4} + 388 q^{5} - 622 q^{7} - 2301 q^{8} + 13602 q^{10} - 6270 q^{11} - 27140 q^{13} - 47284 q^{14} + 253568 q^{16} + 54540 q^{17} - 3232 q^{19} - 66608 q^{20} + 328052 q^{22} + 36501 q^{23} + 924093 q^{25} - 44309 q^{26} - 655166 q^{28} + 323992 q^{29} - 10520 q^{31} - 234068 q^{32} - 550534 q^{34} + 419600 q^{35} + 790772 q^{37} - 729792 q^{38} + 2536678 q^{40} - 869988 q^{41} - 2354396 q^{43} + 136610 q^{44} - 194672 q^{46} + 1800416 q^{47} + 9381947 q^{49} + 1127120 q^{50} - 3631709 q^{52} + 919298 q^{53} - 3700700 q^{55} - 8557654 q^{56} + 7994619 q^{58} + 3244464 q^{59} + 1035198 q^{61} + 3973947 q^{62} + 27264643 q^{64} - 353326 q^{65} - 9767758 q^{67} + 2096216 q^{68} - 18875156 q^{70} + 11648844 q^{71} + 9874056 q^{73} - 8502630 q^{74} + 4135256 q^{76} + 11787232 q^{77} + 6419924 q^{79} + 6789622 q^{80} - 11513635 q^{82} - 3900694 q^{83} - 1484804 q^{85} + 33483674 q^{86} + 24462596 q^{88} + 15922358 q^{89} - 8363866 q^{91} + 4672128 q^{92} - 29511367 q^{94} - 7015012 q^{95} + 75401160 q^{97} + 15767316 q^{98} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(207))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 23
207.8.a.a 207.a 1.a $5$ $64.664$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(266\) \(-496\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(54+2\beta _{1}+\beta _{3}+\beta _{4})q^{4}+\cdots\)
207.8.a.b 207.a 1.a $5$ $64.664$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(16\) \(0\) \(56\) \(-1156\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta _{1})q^{2}+(51-7\beta _{1}-3\beta _{2}+\beta _{4})q^{4}+\cdots\)
207.8.a.c 207.a 1.a $6$ $64.664$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(8\) \(0\) \(372\) \(-1104\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+(29+3\beta _{1}+\beta _{2})q^{4}+\cdots\)
207.8.a.d 207.a 1.a $7$ $64.664$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(0\) \(516\) \(1018\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(93+\beta _{1}+\beta _{2})q^{4}+(74+10\beta _{1}+\cdots)q^{5}+\cdots\)
207.8.a.e 207.a 1.a $8$ $64.664$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-24\) \(0\) \(-378\) \(126\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{1})q^{2}+(70-4\beta _{1}+\beta _{2})q^{4}+\cdots\)
207.8.a.f 207.a 1.a $8$ $64.664$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(-444\) \(1446\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(80+2\beta _{1}+\beta _{2})q^{4}+(-55+\cdots)q^{5}+\cdots\)
207.8.a.g 207.a 1.a $12$ $64.664$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-16\) \(0\) \(-500\) \(-228\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+(53+2\beta _{1}+\beta _{2})q^{4}+\cdots\)
207.8.a.h 207.a 1.a $12$ $64.664$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(16\) \(0\) \(500\) \(-228\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+(53+2\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(207))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(207)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 2}\)