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Properties

Label 26.8.a
Level $26$
Weight $8$
Character orbit 26.a
Rep. character $\chi_{26}(1,\cdot)$
Character field $\Q$
Dimension $7$
Newform subspaces $5$
Sturm bound $28$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 26.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(28\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(26))\).

Total New Old
Modular forms 27 7 20
Cusp forms 23 7 16
Eisenstein series 4 0 4

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
\(+\)\(+\)$+$\(2\)
\(+\)\(-\)$-$\(1\)
\(-\)\(+\)$-$\(1\)
\(-\)\(-\)$+$\(3\)
Plus space\(+\)\(5\)
Minus space\(-\)\(2\)

Trace form

\( 7 q + 8 q^{2} - 78 q^{3} + 448 q^{4} + 530 q^{5} + 192 q^{6} - 2136 q^{7} + 512 q^{8} + 9809 q^{9} + O(q^{10}) \) \( 7 q + 8 q^{2} - 78 q^{3} + 448 q^{4} + 530 q^{5} + 192 q^{6} - 2136 q^{7} + 512 q^{8} + 9809 q^{9} + 416 q^{10} + 10024 q^{11} - 4992 q^{12} + 2197 q^{13} + 16080 q^{14} + 32584 q^{15} + 28672 q^{16} - 14208 q^{17} - 6680 q^{18} - 11052 q^{19} + 33920 q^{20} + 67588 q^{21} - 7840 q^{22} - 241564 q^{23} + 12288 q^{24} + 99159 q^{25} + 52728 q^{26} - 266958 q^{27} - 136704 q^{28} + 81806 q^{29} - 359824 q^{30} - 273328 q^{31} + 32768 q^{32} - 487148 q^{33} + 448400 q^{34} + 100394 q^{35} + 627776 q^{36} - 504654 q^{37} - 521728 q^{38} + 26624 q^{40} + 367002 q^{41} - 448048 q^{42} + 1091430 q^{43} + 641536 q^{44} + 1318118 q^{45} + 1580064 q^{46} + 778576 q^{47} - 319488 q^{48} - 548667 q^{49} - 2353416 q^{50} + 4982374 q^{51} + 140608 q^{52} - 1353966 q^{53} - 1785312 q^{54} + 222920 q^{55} + 1029120 q^{56} + 4422688 q^{57} + 1032528 q^{58} - 2585068 q^{59} + 2085376 q^{60} - 9677322 q^{61} + 414912 q^{62} - 10443408 q^{63} + 1835008 q^{64} + 2882464 q^{65} - 8338048 q^{66} + 2686808 q^{67} - 909312 q^{68} - 2837580 q^{69} - 2736928 q^{70} + 4987264 q^{71} - 427520 q^{72} + 15472994 q^{73} - 2536096 q^{74} - 16197820 q^{75} - 707328 q^{76} + 4200232 q^{77} + 949104 q^{78} - 9290380 q^{79} + 2170880 q^{80} + 4893767 q^{81} + 4507856 q^{82} + 11343624 q^{83} + 4325632 q^{84} + 19368848 q^{85} + 167808 q^{86} - 20133240 q^{87} - 501760 q^{88} - 24609366 q^{89} + 31802128 q^{90} + 5707806 q^{91} - 15460096 q^{92} - 363312 q^{93} + 6655568 q^{94} - 23761284 q^{95} + 786432 q^{96} + 25865262 q^{97} + 5970504 q^{98} + 63309740 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(26))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 13
26.8.a.a 26.a 1.a $1$ $8.122$ \(\Q\) None \(-8\) \(-39\) \(385\) \(-293\) $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}-39q^{3}+2^{6}q^{4}+385q^{5}+\cdots\)
26.8.a.b 26.a 1.a $1$ $8.122$ \(\Q\) None \(8\) \(-87\) \(321\) \(-181\) $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}-87q^{3}+2^{6}q^{4}+321q^{5}+\cdots\)
26.8.a.c 26.a 1.a $1$ $8.122$ \(\Q\) None \(8\) \(-27\) \(-245\) \(-587\) $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}-245q^{5}+\cdots\)
26.8.a.d 26.a 1.a $2$ $8.122$ \(\Q(\sqrt{105}) \) None \(-16\) \(-12\) \(-146\) \(-1780\) $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+(-6-7\beta )q^{3}+2^{6}q^{4}+(-73+\cdots)q^{5}+\cdots\)
26.8.a.e 26.a 1.a $2$ $8.122$ \(\Q(\sqrt{2305}) \) None \(16\) \(87\) \(215\) \(705\) $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+(44-\beta )q^{3}+2^{6}q^{4}+(110+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(26))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(26)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 2}\)