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Properties

Label 7.8.a
Level $7$
Weight $8$
Character orbit 7.a
Rep. character $\chi_{7}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $2$
Sturm bound $5$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 7.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(5\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(7))\).

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
\(+\)\(2\)
\(-\)\(1\)

Trace form

\( 3 q - 9 q^{2} + 52 q^{3} + 89 q^{4} + 246 q^{5} - 754 q^{6} - 343 q^{7} + 135 q^{8} + 1351 q^{9} + O(q^{10}) \) \( 3 q - 9 q^{2} + 52 q^{3} + 89 q^{4} + 246 q^{5} - 754 q^{6} - 343 q^{7} + 135 q^{8} + 1351 q^{9} - 4316 q^{10} - 2724 q^{11} + 14966 q^{12} - 2618 q^{13} - 1029 q^{14} + 27688 q^{15} - 31807 q^{16} - 15474 q^{17} - 81433 q^{18} + 88180 q^{19} + 50568 q^{20} - 46648 q^{21} + 79312 q^{22} - 97848 q^{23} - 82590 q^{24} - 129619 q^{25} + 408576 q^{26} + 150040 q^{27} - 93639 q^{28} + 62250 q^{29} - 403408 q^{30} - 241504 q^{31} - 55689 q^{32} + 267184 q^{33} - 53982 q^{34} - 142002 q^{35} + 443393 q^{36} + 231026 q^{37} - 10350 q^{38} - 427448 q^{39} - 163680 q^{40} + 224406 q^{41} + 431494 q^{42} - 652548 q^{43} + 619128 q^{44} + 1141342 q^{45} + 536736 q^{46} - 1608144 q^{47} - 1700578 q^{48} + 352947 q^{49} - 913011 q^{50} + 797736 q^{51} - 441084 q^{52} + 2799642 q^{53} - 3415420 q^{54} + 435272 q^{55} + 859215 q^{56} - 1525640 q^{57} + 3776230 q^{58} + 2139420 q^{59} + 2899904 q^{60} - 3853394 q^{61} - 1398948 q^{62} - 753571 q^{63} - 762831 q^{64} - 2963772 q^{65} - 325168 q^{66} + 2080756 q^{67} + 1572438 q^{68} + 3565872 q^{69} + 1826132 q^{70} - 5995704 q^{71} + 1073775 q^{72} + 2480222 q^{73} - 11488242 q^{74} + 3087548 q^{75} - 3237990 q^{76} - 2885316 q^{77} + 16500344 q^{78} - 8077680 q^{79} - 5520624 q^{80} - 264797 q^{81} + 12216722 q^{82} + 12355812 q^{83} - 2482634 q^{84} + 2330604 q^{85} + 4187256 q^{86} - 8891240 q^{87} - 10690440 q^{88} + 15330750 q^{89} - 19124332 q^{90} - 2636298 q^{91} + 7864176 q^{92} - 22560816 q^{93} - 11211372 q^{94} - 2948280 q^{95} + 26043346 q^{96} + 20639486 q^{97} - 1058841 q^{98} - 4554068 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(7))\) 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
7.8.a.a 7.a 1.a $1$ $2.187$ \(\Q\) None \(-6\) \(-42\) \(-84\) \(343\) $-$ $\mathrm{SU}(2)$ \(q-6q^{2}-42q^{3}-92q^{4}-84q^{5}+252q^{6}+\cdots\)
7.8.a.b 7.a 1.a $2$ $2.187$ \(\Q(\sqrt{865}) \) None \(-3\) \(94\) \(330\) \(-686\) $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(46+2\beta )q^{3}+(89+3\beta )q^{4}+\cdots\)