Properties

Label 63.4.a
Level 63
Weight 4
Character orbit a
Rep. character \(\chi_{63}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newform subspaces 5
Sturm bound 32
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(32\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(63))\).

Total New Old
Modular forms 28 7 21
Cusp forms 20 7 13
Eisenstein series 8 0 8

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

\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(5\)
Minus space\(-\)\(2\)

Trace form

\( 7q + 3q^{2} + 41q^{4} - 7q^{7} + 51q^{8} + O(q^{10}) \) \( 7q + 3q^{2} + 41q^{4} - 7q^{7} + 51q^{8} + 32q^{10} - 12q^{11} + 112q^{13} + 63q^{14} - 19q^{16} - 174q^{17} - 110q^{19} - 60q^{20} - 292q^{22} - 12q^{23} + 5q^{25} - 144q^{26} - 35q^{28} + 270q^{29} + 156q^{31} + 651q^{32} - 462q^{34} + 168q^{35} + 30q^{37} - 1470q^{38} - 456q^{40} + 834q^{41} + 632q^{43} - 1320q^{44} - 396q^{46} - 876q^{47} + 343q^{49} + 1329q^{50} + 188q^{52} + 1506q^{53} + 64q^{55} + 567q^{56} + 1586q^{58} - 1602q^{59} + 1684q^{61} + 1836q^{62} - 551q^{64} - 672q^{65} + 1300q^{67} - 114q^{68} - 392q^{70} + 1212q^{71} - 522q^{73} + 534q^{74} - 502q^{76} + 672q^{77} - 1840q^{79} - 4140q^{80} - 134q^{82} - 906q^{83} - 1728q^{85} - 1752q^{86} - 3180q^{88} - 1650q^{89} - 1036q^{91} + 3528q^{92} + 2436q^{94} + 1104q^{95} + 1146q^{97} + 147q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(63))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7
63.4.a.a \(1\) \(3.717\) \(\Q\) None \(-4\) \(0\) \(4\) \(-7\) \(-\) \(+\) \(q-4q^{2}+8q^{4}+4q^{5}-7q^{7}-2^{4}q^{10}+\cdots\)
63.4.a.b \(1\) \(3.717\) \(\Q\) None \(1\) \(0\) \(-16\) \(-7\) \(-\) \(+\) \(q+q^{2}-7q^{4}-2^{4}q^{5}-7q^{7}-15q^{8}+\cdots\)
63.4.a.c \(1\) \(3.717\) \(\Q\) None \(3\) \(0\) \(18\) \(7\) \(-\) \(-\) \(q+3q^{2}+q^{4}+18q^{5}+7q^{7}-21q^{8}+\cdots\)
63.4.a.d \(2\) \(3.717\) \(\Q(\sqrt{19}) \) None \(0\) \(0\) \(0\) \(-14\) \(+\) \(+\) \(q+\beta q^{2}+11q^{4}+2\beta q^{5}-7q^{7}+3\beta q^{8}+\cdots\)
63.4.a.e \(2\) \(3.717\) \(\Q(\sqrt{57}) \) None \(3\) \(0\) \(-6\) \(14\) \(-\) \(-\) \(q+(1+\beta )q^{2}+(7+3\beta )q^{4}+(-2-2\beta )q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(63))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(63)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T + 8 T^{2} \))(\( 1 - T + 8 T^{2} \))(\( 1 - 3 T + 8 T^{2} \))(\( 1 - 3 T^{2} + 64 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 24 T^{3} + 64 T^{4} \))
$3$ 1
$5$ (\( 1 - 4 T + 125 T^{2} \))(\( 1 + 16 T + 125 T^{2} \))(\( 1 - 18 T + 125 T^{2} \))(\( 1 + 174 T^{2} + 15625 T^{4} \))(\( 1 + 6 T + 202 T^{2} + 750 T^{3} + 15625 T^{4} \))
$7$ (\( 1 + 7 T \))(\( 1 + 7 T \))(\( 1 - 7 T \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 - 7 T )^{2} \))
$11$ (\( 1 + 62 T + 1331 T^{2} \))(\( 1 - 8 T + 1331 T^{2} \))(\( 1 - 36 T + 1331 T^{2} \))(\( 1 + 762 T^{2} + 1771561 T^{4} \))(\( 1 - 6 T + 1246 T^{2} - 7986 T^{3} + 1771561 T^{4} \))
$13$ (\( 1 + 62 T + 2197 T^{2} \))(\( 1 - 28 T + 2197 T^{2} \))(\( 1 + 34 T + 2197 T^{2} \))(\( ( 1 - 82 T + 2197 T^{2} )^{2} \))(\( 1 - 16 T + 2406 T^{2} - 35152 T^{3} + 4826809 T^{4} \))
$17$ (\( 1 + 84 T + 4913 T^{2} \))(\( 1 + 54 T + 4913 T^{2} \))(\( 1 + 42 T + 4913 T^{2} \))(\( 1 + 3670 T^{2} + 24137569 T^{4} \))(\( 1 - 6 T + 9778 T^{2} - 29478 T^{3} + 24137569 T^{4} \))
$19$ (\( 1 - 100 T + 6859 T^{2} \))(\( 1 + 110 T + 6859 T^{2} \))(\( 1 + 124 T + 6859 T^{2} \))(\( ( 1 + 20 T + 6859 T^{2} )^{2} \))(\( 1 - 64 T + 6534 T^{2} - 438976 T^{3} + 47045881 T^{4} \))
$23$ (\( 1 - 42 T + 12167 T^{2} \))(\( 1 + 48 T + 12167 T^{2} \))(\( 1 + 12167 T^{2} \))(\( 1 + 7234 T^{2} + 148035889 T^{4} \))(\( 1 + 6 T + 7870 T^{2} + 73002 T^{3} + 148035889 T^{4} \))
$29$ (\( 1 - 10 T + 24389 T^{2} \))(\( 1 - 110 T + 24389 T^{2} \))(\( 1 + 102 T + 24389 T^{2} \))(\( 1 - 10806 T^{2} + 594823321 T^{4} \))(\( 1 - 252 T + 56446 T^{2} - 6146028 T^{3} + 594823321 T^{4} \))
$31$ (\( 1 + 48 T + 29791 T^{2} \))(\( 1 - 12 T + 29791 T^{2} \))(\( 1 + 160 T + 29791 T^{2} \))(\( ( 1 - 156 T + 29791 T^{2} )^{2} \))(\( 1 - 40 T - 13890 T^{2} - 1191640 T^{3} + 887503681 T^{4} \))
$37$ (\( 1 + 246 T + 50653 T^{2} \))(\( 1 + 246 T + 50653 T^{2} \))(\( 1 - 398 T + 50653 T^{2} \))(\( ( 1 - 186 T + 50653 T^{2} )^{2} \))(\( 1 + 248 T + 98214 T^{2} + 12561944 T^{3} + 2565726409 T^{4} \))
$41$ (\( 1 - 248 T + 68921 T^{2} \))(\( 1 + 182 T + 68921 T^{2} \))(\( 1 - 318 T + 68921 T^{2} \))(\( 1 + 110406 T^{2} + 4750104241 T^{4} \))(\( 1 - 450 T + 175642 T^{2} - 31014450 T^{3} + 4750104241 T^{4} \))
$43$ (\( 1 - 68 T + 79507 T^{2} \))(\( 1 - 128 T + 79507 T^{2} \))(\( 1 + 268 T + 79507 T^{2} \))(\( ( 1 - 164 T + 79507 T^{2} )^{2} \))(\( 1 - 376 T + 161526 T^{2} - 29894632 T^{3} + 6321363049 T^{4} \))
$47$ (\( 1 + 324 T + 103823 T^{2} \))(\( 1 + 324 T + 103823 T^{2} \))(\( 1 + 240 T + 103823 T^{2} \))(\( 1 - 13970 T^{2} + 10779215329 T^{4} \))(\( 1 - 12 T + 141790 T^{2} - 1245876 T^{3} + 10779215329 T^{4} \))
$53$ (\( 1 + 258 T + 148877 T^{2} \))(\( 1 - 162 T + 148877 T^{2} \))(\( 1 - 498 T + 148877 T^{2} \))(\( 1 + 273130 T^{2} + 22164361129 T^{4} \))(\( 1 - 1104 T + 602230 T^{2} - 164360208 T^{3} + 22164361129 T^{4} \))
$59$ (\( 1 + 120 T + 205379 T^{2} \))(\( 1 + 810 T + 205379 T^{2} \))(\( 1 - 132 T + 205379 T^{2} \))(\( 1 + 386134 T^{2} + 42180533641 T^{4} \))(\( 1 + 804 T + 380614 T^{2} + 165124716 T^{3} + 42180533641 T^{4} \))
$61$ (\( 1 - 622 T + 226981 T^{2} \))(\( 1 + 488 T + 226981 T^{2} \))(\( 1 - 398 T + 226981 T^{2} \))(\( ( 1 - 790 T + 226981 T^{2} )^{2} \))(\( 1 + 428 T + 425886 T^{2} + 97147868 T^{3} + 51520374361 T^{4} \))
$67$ (\( 1 - 904 T + 300763 T^{2} \))(\( 1 - 244 T + 300763 T^{2} \))(\( 1 - 92 T + 300763 T^{2} \))(\( ( 1 + 44 T + 300763 T^{2} )^{2} \))(\( 1 - 148 T + 440790 T^{2} - 44512924 T^{3} + 90458382169 T^{4} \))
$71$ (\( 1 - 678 T + 357911 T^{2} \))(\( 1 - 768 T + 357911 T^{2} \))(\( 1 - 720 T + 357911 T^{2} \))(\( 1 + 518146 T^{2} + 128100283921 T^{4} \))(\( 1 + 954 T + 930526 T^{2} + 341447094 T^{3} + 128100283921 T^{4} \))
$73$ (\( 1 + 642 T + 389017 T^{2} \))(\( 1 + 702 T + 389017 T^{2} \))(\( 1 + 502 T + 389017 T^{2} \))(\( ( 1 - 126 T + 389017 T^{2} )^{2} \))(\( 1 - 1072 T + 1063278 T^{2} - 417026224 T^{3} + 151334226289 T^{4} \))
$79$ (\( 1 - 740 T + 493039 T^{2} \))(\( 1 - 440 T + 493039 T^{2} \))(\( 1 + 1024 T + 493039 T^{2} \))(\( ( 1 + 712 T + 493039 T^{2} )^{2} \))(\( 1 + 572 T + 901662 T^{2} + 282018308 T^{3} + 243087455521 T^{4} \))
$83$ (\( 1 + 468 T + 571787 T^{2} \))(\( 1 - 1302 T + 571787 T^{2} \))(\( 1 - 204 T + 571787 T^{2} \))(\( 1 - 1001450 T^{2} + 326940373369 T^{4} \))(\( 1 + 1944 T + 1957030 T^{2} + 1111553928 T^{3} + 326940373369 T^{4} \))
$89$ (\( 1 + 200 T + 704969 T^{2} \))(\( 1 + 730 T + 704969 T^{2} \))(\( 1 + 354 T + 704969 T^{2} \))(\( 1 - 709626 T^{2} + 496981290961 T^{4} \))(\( 1 + 366 T + 1156090 T^{2} + 258018654 T^{3} + 496981290961 T^{4} \))
$97$ (\( 1 + 1266 T + 912673 T^{2} \))(\( 1 - 294 T + 912673 T^{2} \))(\( 1 + 286 T + 912673 T^{2} \))(\( ( 1 - 798 T + 912673 T^{2} )^{2} \))(\( 1 - 808 T + 903054 T^{2} - 737439784 T^{3} + 832972004929 T^{4} \))
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