Properties

Label 7.7.b
Level 7
Weight 7
Character orbit b
Rep. character \(\chi_{7}(6,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 4
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 7.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
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 \(M_{7}(7, [\chi])\).

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

Trace form

\( 3q - 7q^{2} + 17q^{4} - 77q^{7} + 601q^{8} - 1893q^{9} + O(q^{10}) \) \( 3q - 7q^{2} + 17q^{4} - 77q^{7} + 601q^{8} - 1893q^{9} + 3710q^{11} - 5215q^{14} + 4080q^{15} - 13087q^{16} + 27537q^{18} - 28560q^{21} + 3674q^{22} - 13258q^{23} + 42795q^{25} - 5831q^{28} + 1070q^{29} - 32640q^{30} - 16983q^{32} + 28560q^{35} + 12393q^{36} + 107198q^{37} - 199920q^{39} + 228480q^{42} - 63778q^{43} + 33354q^{44} - 280414q^{46} - 46893q^{49} - 76735q^{50} + 538560q^{51} - 102466q^{53} + 281281q^{56} - 281520q^{57} - 369334q^{58} - 598773q^{63} + 684721q^{64} + 199920q^{65} + 936558q^{67} - 228480q^{70} - 611290q^{71} - 1650831q^{72} + 862714q^{74} - 440482q^{77} + 1599360q^{78} - 140490q^{79} + 994563q^{81} - 538560q^{85} - 1642214q^{86} + 65050q^{88} - 1399440q^{91} - 386478q^{92} + 2480640q^{93} + 281520q^{95} + 2375177q^{98} - 861330q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(7, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
7.7.b.a \(1\) \(1.610\) \(\Q\) \(\Q(\sqrt{-7}) \) \(9\) \(0\) \(0\) \(-343\) \(q+9q^{2}+17q^{4}-7^{3}q^{7}-423q^{8}+\cdots\)
7.7.b.b \(2\) \(1.610\) \(\Q(\sqrt{-510}) \) None \(-16\) \(0\) \(0\) \(266\) \(q-8q^{2}+\beta q^{3}-\beta q^{5}-8\beta q^{6}+(133+\cdots)q^{7}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 9 T + 64 T^{2} \))(\( ( 1 + 8 T + 64 T^{2} )^{2} \))
$3$ (\( ( 1 - 27 T )( 1 + 27 T ) \))(\( 1 + 582 T^{2} + 531441 T^{4} \))
$5$ (\( ( 1 - 125 T )( 1 + 125 T ) \))(\( 1 - 29210 T^{2} + 244140625 T^{4} \))
$7$ (\( 1 + 343 T \))(\( 1 - 266 T + 117649 T^{2} \))
$11$ (\( 1 - 1962 T + 1771561 T^{2} \))(\( ( 1 - 874 T + 1771561 T^{2} )^{2} \))
$13$ (\( ( 1 - 2197 T )( 1 + 2197 T ) \))(\( 1 - 4755578 T^{2} + 23298085122481 T^{4} \))
$17$ (\( ( 1 - 4913 T )( 1 + 4913 T ) \))(\( 1 - 12730178 T^{2} + 582622237229761 T^{4} \))
$19$ (\( ( 1 - 6859 T )( 1 + 6859 T ) \))(\( 1 - 84379322 T^{2} + 2213314919066161 T^{4} \))
$23$ (\( 1 + 22734 T + 148035889 T^{2} \))(\( ( 1 - 4738 T + 148035889 T^{2} )^{2} \))
$29$ (\( 1 + 21222 T + 594823321 T^{2} \))(\( ( 1 - 11146 T + 594823321 T^{2} )^{2} \))
$31$ (\( ( 1 - 29791 T )( 1 + 29791 T ) \))(\( 1 - 1020892802 T^{2} + 787662783788549761 T^{4} \))
$37$ (\( 1 - 101194 T + 2565726409 T^{2} \))(\( ( 1 - 3002 T + 2565726409 T^{2} )^{2} \))
$41$ (\( ( 1 - 68921 T )( 1 + 68921 T ) \))(\( 1 - 6189133442 T^{2} + 22563490300366186081 T^{4} \))
$43$ (\( 1 + 126614 T + 6321363049 T^{2} \))(\( ( 1 - 31418 T + 6321363049 T^{2} )^{2} \))
$47$ (\( ( 1 - 103823 T )( 1 + 103823 T ) \))(\( 1 - 16309886018 T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \))
$53$ (\( 1 - 50346 T + 22164361129 T^{2} \))(\( ( 1 + 76406 T + 22164361129 T^{2} )^{2} \))
$59$ (\( ( 1 - 205379 T )( 1 + 205379 T ) \))(\( 1 - 71539567322 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \))
$61$ (\( ( 1 - 226981 T )( 1 + 226981 T ) \))(\( 1 - 27356175482 T^{2} + \)\(26\!\cdots\!21\)\( T^{4} \))
$67$ (\( 1 + 53926 T + 90458382169 T^{2} \))(\( ( 1 - 495242 T + 90458382169 T^{2} )^{2} \))
$71$ (\( 1 + 242478 T + 128100283921 T^{2} \))(\( ( 1 + 184406 T + 128100283921 T^{2} )^{2} \))
$73$ (\( ( 1 - 389017 T )( 1 + 389017 T ) \))(\( 1 - 298950552578 T^{2} + \)\(22\!\cdots\!21\)\( T^{4} \))
$79$ (\( 1 - 929378 T + 243087455521 T^{2} \))(\( ( 1 + 534934 T + 243087455521 T^{2} )^{2} \))
$83$ (\( ( 1 - 571787 T )( 1 + 571787 T ) \))(\( 1 - 142873131578 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \))
$89$ (\( ( 1 - 704969 T )( 1 + 704969 T ) \))(\( 1 - 597656180162 T^{2} + \)\(24\!\cdots\!21\)\( T^{4} \))
$97$ (\( ( 1 - 912673 T )( 1 + 912673 T ) \))(\( 1 - 1002631840898 T^{2} + \)\(69\!\cdots\!41\)\( T^{4} \))
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