Properties

Label 13.4.a
Level 13
Weight 4
Character orbit a
Rep. character \(\chi_{13}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 4
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 13.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 \(M_{4}(\Gamma_0(13))\).

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.

\(13\)Dim.
\(+\)\(2\)
\(-\)\(1\)

Trace form

\( 3q - 4q^{2} - 2q^{3} + 10q^{4} - 10q^{5} + 12q^{6} - 22q^{7} - 48q^{8} + 57q^{9} + O(q^{10}) \) \( 3q - 4q^{2} - 2q^{3} + 10q^{4} - 10q^{5} + 12q^{6} - 22q^{7} - 48q^{8} + 57q^{9} + 42q^{10} + 54q^{11} - 162q^{12} - 13q^{13} + 154q^{14} + 16q^{15} + 50q^{16} + 96q^{17} - 220q^{18} - 210q^{19} - 100q^{20} - 212q^{21} + 272q^{22} + 100q^{23} + 588q^{24} - 313q^{25} - 78q^{26} + 370q^{27} - 96q^{28} - 126q^{29} - 202q^{30} + 110q^{31} - 208q^{32} + 76q^{33} - 520q^{34} + 198q^{35} + 124q^{36} + 78q^{37} + 316q^{38} - 156q^{39} + 226q^{40} + 106q^{41} - 258q^{42} + 86q^{43} - 620q^{44} - 334q^{45} + 476q^{46} + 330q^{47} - 338q^{48} + 209q^{49} + 236q^{50} - 58q^{51} + 312q^{52} - 550q^{53} - 84q^{54} + 164q^{55} - 430q^{56} + 1488q^{57} + 1204q^{58} - 662q^{59} + 1008q^{60} - 1114q^{61} - 1312q^{62} - 1846q^{63} + 482q^{64} - 52q^{65} - 1728q^{66} + 546q^{67} + 1098q^{68} + 1468q^{69} - 580q^{70} - 122q^{71} + 360q^{72} + 554q^{73} + 802q^{74} + 16q^{75} - 2120q^{76} + 1100q^{77} + 754q^{78} + 296q^{79} - 692q^{80} - 717q^{81} - 1608q^{82} + 1650q^{83} + 2956q^{84} - 712q^{85} + 2364q^{86} - 1984q^{87} - 72q^{88} - 1910q^{89} + 1020q^{90} - 52q^{91} - 2420q^{92} - 720q^{93} - 286q^{94} + 736q^{95} - 1268q^{96} - 858q^{97} + 220q^{98} - 702q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(13))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 13
13.4.a.a \(1\) \(0.767\) \(\Q\) None \(-5\) \(-7\) \(-7\) \(-13\) \(-\) \(q-5q^{2}-7q^{3}+17q^{4}-7q^{5}+35q^{6}+\cdots\)
13.4.a.b \(2\) \(0.767\) \(\Q(\sqrt{17}) \) None \(1\) \(5\) \(-3\) \(-9\) \(+\) \(q+\beta q^{2}+(4-3\beta )q^{3}+(-4+\beta )q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 5 T + 8 T^{2} \))(\( 1 - T + 12 T^{2} - 8 T^{3} + 64 T^{4} \))
$3$ (\( 1 + 7 T + 27 T^{2} \))(\( 1 - 5 T + 22 T^{2} - 135 T^{3} + 729 T^{4} \))
$5$ (\( 1 + 7 T + 125 T^{2} \))(\( 1 + 3 T + 248 T^{2} + 375 T^{3} + 15625 T^{4} \))
$7$ (\( 1 + 13 T + 343 T^{2} \))(\( 1 + 9 T + 192 T^{2} + 3087 T^{3} + 117649 T^{4} \))
$11$ (\( 1 + 26 T + 1331 T^{2} \))(\( 1 - 80 T + 3650 T^{2} - 106480 T^{3} + 1771561 T^{4} \))
$13$ (\( 1 - 13 T \))(\( ( 1 + 13 T )^{2} \))
$17$ (\( 1 - 77 T + 4913 T^{2} \))(\( 1 - 19 T + 8688 T^{2} - 93347 T^{3} + 24137569 T^{4} \))
$19$ (\( 1 + 126 T + 6859 T^{2} \))(\( 1 + 84 T + 11130 T^{2} + 576156 T^{3} + 47045881 T^{4} \))
$23$ (\( 1 + 96 T + 12167 T^{2} \))(\( 1 - 196 T + 33326 T^{2} - 2384732 T^{3} + 148035889 T^{4} \))
$29$ (\( 1 + 82 T + 24389 T^{2} \))(\( 1 + 44 T + 10094 T^{2} + 1073116 T^{3} + 594823321 T^{4} \))
$31$ (\( 1 - 196 T + 29791 T^{2} \))(\( 1 + 86 T + 56518 T^{2} + 2562026 T^{3} + 887503681 T^{4} \))
$37$ (\( 1 + 131 T + 50653 T^{2} \))(\( 1 - 209 T + 112120 T^{2} - 10586477 T^{3} + 2565726409 T^{4} \))
$41$ (\( 1 - 336 T + 68921 T^{2} \))(\( 1 + 230 T + 149010 T^{2} + 15851830 T^{3} + 4750104241 T^{4} \))
$43$ (\( 1 + 201 T + 79507 T^{2} \))(\( 1 - 287 T + 92698 T^{2} - 22818509 T^{3} + 6321363049 T^{4} \))
$47$ (\( 1 + 105 T + 103823 T^{2} \))(\( 1 - 435 T + 192728 T^{2} - 45163005 T^{3} + 10779215329 T^{4} \))
$53$ (\( 1 + 432 T + 148877 T^{2} \))(\( 1 + 118 T + 297410 T^{2} + 17567486 T^{3} + 22164361129 T^{4} \))
$59$ (\( 1 + 294 T + 205379 T^{2} \))(\( 1 + 368 T + 379266 T^{2} + 75579472 T^{3} + 42180533641 T^{4} \))
$61$ (\( 1 + 56 T + 226981 T^{2} \))(\( 1 + 1058 T + 580378 T^{2} + 240145898 T^{3} + 51520374361 T^{4} \))
$67$ (\( 1 - 478 T + 300763 T^{2} \))(\( 1 - 68 T + 373930 T^{2} - 20451884 T^{3} + 90458382169 T^{4} \))
$71$ (\( 1 - 9 T + 357911 T^{2} \))(\( 1 + 131 T + 493328 T^{2} + 46886341 T^{3} + 128100283921 T^{4} \))
$73$ (\( 1 - 98 T + 389017 T^{2} \))(\( 1 - 456 T + 542718 T^{2} - 177391752 T^{3} + 151334226289 T^{4} \))
$79$ (\( 1 - 1304 T + 493039 T^{2} \))(\( 1 + 1008 T + 1233294 T^{2} + 496983312 T^{3} + 243087455521 T^{4} \))
$83$ (\( 1 + 308 T + 571787 T^{2} \))(\( 1 - 1958 T + 1961238 T^{2} - 1119558946 T^{3} + 326940373369 T^{4} \))
$89$ (\( 1 + 1190 T + 704969 T^{2} \))(\( 1 + 720 T + 899726 T^{2} + 507577680 T^{3} + 496981290961 T^{4} \))
$97$ (\( 1 - 70 T + 912673 T^{2} \))(\( 1 + 928 T + 943870 T^{2} + 846960544 T^{3} + 832972004929 T^{4} \))
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