Properties

Label 7.6.a
Level $7$
Weight $6$
Character orbit 7.a
Rep. character $\chi_{7}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $2$
Sturm bound $4$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 7.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(4\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(7))\).

Total New Old
Modular forms 5 3 2
Cusp forms 3 3 0
Eisenstein series 2 0 2

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

\(7\)Dim.
\(+\)\(1\)
\(-\)\(2\)

Trace form

\( 3 q - q^{2} - 20 q^{3} + 73 q^{4} - 74 q^{5} - 58 q^{6} + 49 q^{7} - 369 q^{8} + 511 q^{9} + O(q^{10}) \) \( 3 q - q^{2} - 20 q^{3} + 73 q^{4} - 74 q^{5} - 58 q^{6} + 49 q^{7} - 369 q^{8} + 511 q^{9} + 764 q^{10} + 628 q^{11} - 2506 q^{12} - 490 q^{13} + 931 q^{14} - 872 q^{15} + 1537 q^{16} + 78 q^{17} + 4007 q^{18} - 3364 q^{19} - 1288 q^{20} + 392 q^{21} - 4072 q^{22} + 3912 q^{23} + 3186 q^{24} - 3227 q^{25} + 3416 q^{26} - 2312 q^{27} - 3087 q^{28} + 10114 q^{29} - 14608 q^{30} - 7664 q^{31} - 8849 q^{32} + 16768 q^{33} + 23154 q^{34} + 1862 q^{35} + 7433 q^{36} - 4166 q^{37} - 14230 q^{38} - 18536 q^{39} + 23376 q^{40} - 24010 q^{41} - 16562 q^{42} + 7860 q^{43} - 15040 q^{44} + 7870 q^{45} + 7344 q^{46} + 21024 q^{47} + 21278 q^{48} + 7203 q^{49} - 19811 q^{50} + 31704 q^{51} + 21924 q^{52} + 11730 q^{53} - 78508 q^{54} - 51896 q^{55} + 17199 q^{56} + 14248 q^{57} + 3134 q^{58} - 46668 q^{59} + 55328 q^{60} - 39106 q^{61} + 91740 q^{62} + 29645 q^{63} - 89151 q^{64} + 46900 q^{65} + 99296 q^{66} - 23620 q^{67} - 132090 q^{68} - 128928 q^{69} - 17444 q^{70} + 38856 q^{71} + 25695 q^{72} + 85534 q^{73} + 147414 q^{74} + 40340 q^{75} - 19446 q^{76} + 8036 q^{77} - 92680 q^{78} + 83040 q^{79} - 150016 q^{80} - 106493 q^{81} + 121282 q^{82} + 97020 q^{83} - 29498 q^{84} + 58572 q^{85} - 258848 q^{86} - 111032 q^{87} - 124176 q^{88} + 33694 q^{89} + 67532 q^{90} - 10290 q^{91} + 274944 q^{92} + 14736 q^{93} + 66228 q^{94} + 29752 q^{95} + 293986 q^{96} - 37730 q^{97} - 2401 q^{98} - 27644 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(7))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 7
7.6.a.a 7.a 1.a $1$ $1.123$ \(\Q\) None \(-10\) \(-14\) \(-56\) \(-49\) $+$ $\mathrm{SU}(2)$ \(q-10q^{2}-14q^{3}+68q^{4}-56q^{5}+\cdots\)
7.6.a.b 7.a 1.a $2$ $1.123$ \(\Q(\sqrt{57}) \) None \(9\) \(-6\) \(-18\) \(98\) $-$ $\mathrm{SU}(2)$ \(q+(5-\beta )q^{2}+(-6+6\beta )q^{3}+(7-9\beta )q^{4}+\cdots\)