Properties

Label 7.4
Level 7
Weight 4
Dimension 3
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 16
Trace bound 1

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Defining parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(7))\).

Total New Old
Modular forms 9 7 2
Cusp forms 3 3 0
Eisenstein series 6 4 2

Trace form

\( 3q - 3q^{2} - 9q^{3} - 3q^{4} + 9q^{5} + 30q^{6} + 21q^{7} - 33q^{8} - 45q^{9} + O(q^{10}) \) \( 3q - 3q^{2} - 9q^{3} - 3q^{4} + 9q^{5} + 30q^{6} + 21q^{7} - 33q^{8} - 45q^{9} - 30q^{10} - 3q^{11} + 42q^{12} + 21q^{14} + 66q^{15} + 57q^{16} + 75q^{17} - 21q^{18} - 159q^{19} - 168q^{20} - 231q^{21} - 12q^{22} + 207q^{23} + 138q^{24} + 207q^{25} + 30q^{27} + 189q^{28} + 6q^{29} - 66q^{30} - 135q^{31} - 321q^{32} + 51q^{33} - 138q^{34} - 63q^{35} - 15q^{36} - 465q^{37} + 12q^{38} + 42q^{39} + 408q^{40} + 882q^{41} + 378q^{42} - 120q^{43} + 36q^{44} - 522q^{45} + 270q^{46} - 201q^{47} - 306q^{48} + 147q^{49} - 435q^{50} + 39q^{51} - 252q^{52} - 465q^{53} - 30q^{54} - 198q^{55} - 777q^{56} + 906q^{57} - 6q^{58} + 915q^{59} + 420q^{60} - 75q^{61} + 576q^{62} + 315q^{63} + 729q^{64} + 546q^{65} + 54q^{66} - 171q^{67} - 462q^{68} - 2322q^{69} - 378q^{70} - 1632q^{71} + 183q^{72} + 411q^{73} - 192q^{74} + 270q^{75} + 378q^{76} + 231q^{77} - 336q^{78} + 543q^{79} + 768q^{80} + 1260q^{81} - 882q^{82} + 882q^{83} - 294q^{84} + 570q^{85} + 120q^{86} - 186q^{87} - 240q^{88} + 1059q^{89} + 984q^{90} - 588q^{91} + 936q^{92} - 1053q^{93} - 1374q^{94} - 2103q^{95} - 798q^{96} - 1470q^{97} + 1029q^{98} - 36q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(7))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
7.4.a \(\chi_{7}(1, \cdot)\) 7.4.a.a 1 1
7.4.c \(\chi_{7}(2, \cdot)\) 7.4.c.a 2 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 8 T^{2} \))(\( 1 + 2 T - 4 T^{2} + 16 T^{3} + 64 T^{4} \))
$3$ (\( 1 + 2 T + 27 T^{2} \))(\( 1 + 7 T + 22 T^{2} + 189 T^{3} + 729 T^{4} \))
$5$ (\( 1 - 16 T + 125 T^{2} \))(\( 1 + 7 T - 76 T^{2} + 875 T^{3} + 15625 T^{4} \))
$7$ (\( 1 + 7 T \))(\( 1 - 28 T + 343 T^{2} \))
$11$ (\( 1 + 8 T + 1331 T^{2} \))(\( 1 - 5 T - 1306 T^{2} - 6655 T^{3} + 1771561 T^{4} \))
$13$ (\( 1 - 28 T + 2197 T^{2} \))(\( ( 1 + 14 T + 2197 T^{2} )^{2} \))
$17$ (\( 1 - 54 T + 4913 T^{2} \))(\( 1 - 21 T - 4472 T^{2} - 103173 T^{3} + 24137569 T^{4} \))
$19$ (\( 1 + 110 T + 6859 T^{2} \))(\( 1 + 49 T - 4458 T^{2} + 336091 T^{3} + 47045881 T^{4} \))
$23$ (\( 1 - 48 T + 12167 T^{2} \))(\( 1 - 159 T + 13114 T^{2} - 1934553 T^{3} + 148035889 T^{4} \))
$29$ (\( 1 + 110 T + 24389 T^{2} \))(\( ( 1 - 58 T + 24389 T^{2} )^{2} \))
$31$ (\( 1 - 12 T + 29791 T^{2} \))(\( 1 + 147 T - 8182 T^{2} + 4379277 T^{3} + 887503681 T^{4} \))
$37$ (\( 1 + 246 T + 50653 T^{2} \))(\( 1 + 219 T - 2692 T^{2} + 11093007 T^{3} + 2565726409 T^{4} \))
$41$ (\( 1 - 182 T + 68921 T^{2} \))(\( ( 1 - 350 T + 68921 T^{2} )^{2} \))
$43$ (\( 1 - 128 T + 79507 T^{2} \))(\( ( 1 + 124 T + 79507 T^{2} )^{2} \))
$47$ (\( 1 - 324 T + 103823 T^{2} \))(\( 1 + 525 T + 171802 T^{2} + 54507075 T^{3} + 10779215329 T^{4} \))
$53$ (\( 1 + 162 T + 148877 T^{2} \))(\( 1 + 303 T - 57068 T^{2} + 45109731 T^{3} + 22164361129 T^{4} \))
$59$ (\( 1 - 810 T + 205379 T^{2} \))(\( 1 - 105 T - 194354 T^{2} - 21564795 T^{3} + 42180533641 T^{4} \))
$61$ (\( 1 + 488 T + 226981 T^{2} \))(\( 1 - 413 T - 56412 T^{2} - 93743153 T^{3} + 51520374361 T^{4} \))
$67$ (\( 1 - 244 T + 300763 T^{2} \))(\( 1 + 415 T - 128538 T^{2} + 124816645 T^{3} + 90458382169 T^{4} \))
$71$ (\( 1 + 768 T + 357911 T^{2} \))(\( ( 1 + 432 T + 357911 T^{2} )^{2} \))
$73$ (\( 1 + 702 T + 389017 T^{2} \))(\( 1 - 1113 T + 849752 T^{2} - 432975921 T^{3} + 151334226289 T^{4} \))
$79$ (\( 1 - 440 T + 493039 T^{2} \))(\( 1 - 103 T - 482430 T^{2} - 50783017 T^{3} + 243087455521 T^{4} \))
$83$ (\( 1 + 1302 T + 571787 T^{2} \))(\( ( 1 - 1092 T + 571787 T^{2} )^{2} \))
$89$ (\( 1 - 730 T + 704969 T^{2} \))(\( 1 - 329 T - 596728 T^{2} - 231934801 T^{3} + 496981290961 T^{4} \))
$97$ (\( 1 - 294 T + 912673 T^{2} \))(\( ( 1 + 882 T + 912673 T^{2} )^{2} \))
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