Properties

Label 163.2.a
Level 163
Weight 2
Character orbit a
Rep. character \(\chi_{163}(1,\cdot)\)
Character field \(\Q\)
Dimension 13
Newform subspaces 3
Sturm bound 27
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 163 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 163.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(27\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(163))\).

Total New Old
Modular forms 14 14 0
Cusp forms 13 13 0
Eisenstein series 1 1 0

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

\(163\)Dim.
\(+\)\(6\)
\(-\)\(7\)

Trace form

\( 13q - 2q^{2} - 4q^{3} + 12q^{4} - 2q^{5} - 6q^{6} - 4q^{7} - 12q^{8} + 7q^{9} + O(q^{10}) \) \( 13q - 2q^{2} - 4q^{3} + 12q^{4} - 2q^{5} - 6q^{6} - 4q^{7} - 12q^{8} + 7q^{9} + 6q^{10} - 2q^{11} - 10q^{12} + 2q^{14} + 14q^{16} - 8q^{17} - 2q^{18} - 4q^{19} - 10q^{20} - 18q^{21} - 2q^{22} + 10q^{24} + 25q^{25} + 2q^{26} - 22q^{27} - 16q^{28} - 10q^{29} + 30q^{30} - 10q^{31} - 26q^{32} - 12q^{33} + 24q^{34} + 2q^{35} + 18q^{36} - 6q^{37} + 6q^{38} + 10q^{39} + 24q^{40} + 11q^{41} + 40q^{42} - 7q^{43} + 12q^{44} - 14q^{45} - 36q^{46} + 9q^{47} - 16q^{48} + q^{49} + 14q^{50} + 28q^{51} - 14q^{52} - 7q^{53} - 24q^{54} + 8q^{55} + 20q^{57} + 6q^{58} + 6q^{59} - 54q^{60} - 3q^{61} + 26q^{62} + 16q^{63} - 10q^{64} + 44q^{65} - 18q^{66} - 22q^{67} - 24q^{68} + 4q^{69} - 38q^{70} + 23q^{71} - 40q^{72} - 14q^{73} - 12q^{74} + 10q^{75} - 26q^{76} - 10q^{77} - 2q^{78} - 22q^{79} - 36q^{80} + 13q^{81} + 11q^{83} - 64q^{84} + 40q^{85} + 48q^{86} + 50q^{87} - 22q^{88} + 4q^{89} + 14q^{90} - 28q^{91} + 42q^{92} - 30q^{93} + 56q^{94} - 30q^{95} + 84q^{96} + 33q^{97} - 14q^{98} + 18q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(163))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 163
163.2.a.a \(1\) \(1.302\) \(\Q\) None \(0\) \(0\) \(-4\) \(2\) \(+\) \(q-2q^{4}-4q^{5}+2q^{7}-3q^{9}-6q^{11}+\cdots\)
163.2.a.b \(5\) \(1.302\) 5.5.65657.1 None \(-5\) \(-5\) \(-9\) \(-6\) \(+\) \(q+(-1+\beta _{2}+\beta _{3})q^{2}+(-1+\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots\)
163.2.a.c \(7\) \(1.302\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(3\) \(1\) \(11\) \(0\) \(-\) \(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(1+\beta _{2})q^{4}+(2-\beta _{6})q^{5}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} \))(\( 1 + 5 T + 13 T^{2} + 25 T^{3} + 42 T^{4} + 63 T^{5} + 84 T^{6} + 100 T^{7} + 104 T^{8} + 80 T^{9} + 32 T^{10} \))(\( 1 - 3 T + 9 T^{2} - 17 T^{3} + 34 T^{4} - 51 T^{5} + 84 T^{6} - 110 T^{7} + 168 T^{8} - 204 T^{9} + 272 T^{10} - 272 T^{11} + 288 T^{12} - 192 T^{13} + 128 T^{14} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 + 5 T + 16 T^{2} + 37 T^{3} + 71 T^{4} + 123 T^{5} + 213 T^{6} + 333 T^{7} + 432 T^{8} + 405 T^{9} + 243 T^{10} \))(\( 1 - T + 10 T^{2} - 5 T^{3} + 50 T^{4} - 18 T^{5} + 205 T^{6} - 74 T^{7} + 615 T^{8} - 162 T^{9} + 1350 T^{10} - 405 T^{11} + 2430 T^{12} - 729 T^{13} + 2187 T^{14} \))
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 + 9 T + 48 T^{2} + 192 T^{3} + 594 T^{4} + 1469 T^{5} + 2970 T^{6} + 4800 T^{7} + 6000 T^{8} + 5625 T^{9} + 3125 T^{10} \))(\( 1 - 11 T + 76 T^{2} - 374 T^{3} + 1477 T^{4} - 4806 T^{5} + 13394 T^{6} - 32086 T^{7} + 66970 T^{8} - 120150 T^{9} + 184625 T^{10} - 233750 T^{11} + 237500 T^{12} - 171875 T^{13} + 78125 T^{14} \))
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 + 6 T + 30 T^{2} + 120 T^{3} + 403 T^{4} + 1091 T^{5} + 2821 T^{6} + 5880 T^{7} + 10290 T^{8} + 14406 T^{9} + 16807 T^{10} \))(\( 1 + 28 T^{2} - 18 T^{3} + 398 T^{4} - 389 T^{5} + 3763 T^{6} - 3840 T^{7} + 26341 T^{8} - 19061 T^{9} + 136514 T^{10} - 43218 T^{11} + 470596 T^{12} + 823543 T^{14} \))
$11$ (\( 1 + 6 T + 11 T^{2} \))(\( 1 - 2 T + 29 T^{2} - 31 T^{3} + 320 T^{4} - 195 T^{5} + 3520 T^{6} - 3751 T^{7} + 38599 T^{8} - 29282 T^{9} + 161051 T^{10} \))(\( 1 - 2 T + 57 T^{2} - 65 T^{3} + 1407 T^{4} - 731 T^{5} + 21283 T^{6} - 5664 T^{7} + 234113 T^{8} - 88451 T^{9} + 1872717 T^{10} - 951665 T^{11} + 9179907 T^{12} - 3543122 T^{13} + 19487171 T^{14} \))
$13$ (\( 1 - 4 T + 13 T^{2} \))(\( 1 + 14 T + 138 T^{2} + 901 T^{3} + 4716 T^{4} + 18755 T^{5} + 61308 T^{6} + 152269 T^{7} + 303186 T^{8} + 399854 T^{9} + 371293 T^{10} \))(\( 1 - 10 T + 102 T^{2} - 631 T^{3} + 3771 T^{4} - 17291 T^{5} + 76660 T^{6} - 280562 T^{7} + 996580 T^{8} - 2922179 T^{9} + 8284887 T^{10} - 18021991 T^{11} + 37871886 T^{12} - 48268090 T^{13} + 62748517 T^{14} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 + 21 T + 254 T^{2} + 2079 T^{3} + 12705 T^{4} + 59379 T^{5} + 215985 T^{6} + 600831 T^{7} + 1247902 T^{8} + 1753941 T^{9} + 1419857 T^{10} \))(\( 1 - 13 T + 166 T^{2} - 1329 T^{3} + 9840 T^{4} - 56274 T^{5} + 296357 T^{6} - 1272982 T^{7} + 5038069 T^{8} - 16263186 T^{9} + 48343920 T^{10} - 110999409 T^{11} + 235696262 T^{12} - 313788397 T^{13} + 410338673 T^{14} \))
$19$ (\( 1 + 6 T + 19 T^{2} \))(\( 1 - 7 T + 51 T^{2} - 185 T^{3} + 989 T^{4} - 2987 T^{5} + 18791 T^{6} - 66785 T^{7} + 349809 T^{8} - 912247 T^{9} + 2476099 T^{10} \))(\( 1 + 5 T + 91 T^{2} + 469 T^{3} + 3938 T^{4} + 20122 T^{5} + 107766 T^{6} + 493646 T^{7} + 2047554 T^{8} + 7264042 T^{9} + 27010742 T^{10} + 61120549 T^{11} + 225325009 T^{12} + 235229405 T^{13} + 893871739 T^{14} \))
$23$ (\( 1 - 6 T + 23 T^{2} \))(\( 1 + 8 T + 60 T^{2} + 182 T^{3} + 261 T^{4} - 905 T^{5} + 6003 T^{6} + 96278 T^{7} + 730020 T^{8} + 2238728 T^{9} + 6436343 T^{10} \))(\( 1 - 2 T + 118 T^{2} - 250 T^{3} + 6608 T^{4} - 13499 T^{5} + 227743 T^{6} - 405686 T^{7} + 5238089 T^{8} - 7140971 T^{9} + 80399536 T^{10} - 69960250 T^{11} + 759488474 T^{12} - 296071778 T^{13} + 3404825447 T^{14} \))
$29$ (\( 1 + 4 T + 29 T^{2} \))(\( 1 + 13 T + 191 T^{2} + 1506 T^{3} + 12237 T^{4} + 65439 T^{5} + 354873 T^{6} + 1266546 T^{7} + 4658299 T^{8} + 9194653 T^{9} + 20511149 T^{10} \))(\( 1 - 7 T + 127 T^{2} - 646 T^{3} + 8098 T^{4} - 34888 T^{5} + 334936 T^{6} - 1194456 T^{7} + 9713144 T^{8} - 29340808 T^{9} + 197502122 T^{10} - 456903526 T^{11} + 2604915923 T^{12} - 4163763247 T^{13} + 17249876309 T^{14} \))
$31$ (\( 1 + 6 T + 31 T^{2} \))(\( 1 - 7 T + 81 T^{2} - 635 T^{3} + 4061 T^{4} - 24611 T^{5} + 125891 T^{6} - 610235 T^{7} + 2413071 T^{8} - 6464647 T^{9} + 28629151 T^{10} \))(\( 1 + 11 T + 141 T^{2} + 989 T^{3} + 7342 T^{4} + 36732 T^{5} + 225056 T^{6} + 1015190 T^{7} + 6976736 T^{8} + 35299452 T^{9} + 218725522 T^{10} + 913362269 T^{11} + 4036710291 T^{12} + 9762540491 T^{13} + 27512614111 T^{14} \))
$37$ (\( 1 + 8 T + 37 T^{2} \))(\( 1 + T + 162 T^{2} + 98 T^{3} + 11176 T^{4} + 4621 T^{5} + 413512 T^{6} + 134162 T^{7} + 8205786 T^{8} + 1874161 T^{9} + 69343957 T^{10} \))(\( 1 - 3 T + 94 T^{2} - 406 T^{3} + 4983 T^{4} - 20842 T^{5} + 252248 T^{6} - 733312 T^{7} + 9333176 T^{8} - 28532698 T^{9} + 252403899 T^{10} - 760909366 T^{11} + 6518331958 T^{12} - 7697179227 T^{13} + 94931877133 T^{14} \))
$41$ (\( 1 - 3 T + 41 T^{2} \))(\( 1 + 9 T + 142 T^{2} + 811 T^{3} + 7557 T^{4} + 35461 T^{5} + 309837 T^{6} + 1363291 T^{7} + 9786782 T^{8} + 25431849 T^{9} + 115856201 T^{10} \))(\( 1 - 17 T + 283 T^{2} - 2852 T^{3} + 28544 T^{4} - 214023 T^{5} + 1647302 T^{6} - 10274539 T^{7} + 67539382 T^{8} - 359772663 T^{9} + 1967281024 T^{10} - 8059070372 T^{11} + 32787304883 T^{12} - 80751772097 T^{13} + 194754273881 T^{14} \))
$43$ (\( 1 - 7 T + 43 T^{2} \))(\( 1 + 4 T + 166 T^{2} + 672 T^{3} + 12749 T^{4} + 42159 T^{5} + 548207 T^{6} + 1242528 T^{7} + 13198162 T^{8} + 13675204 T^{9} + 147008443 T^{10} \))(\( 1 + 10 T + 141 T^{2} + 1062 T^{3} + 9368 T^{4} + 61649 T^{5} + 460654 T^{6} + 2932885 T^{7} + 19808122 T^{8} + 113989001 T^{9} + 744821576 T^{10} + 3630766662 T^{11} + 20728190463 T^{12} + 63213630490 T^{13} + 271818611107 T^{14} \))
$47$ (\( 1 - T + 47 T^{2} \))(\( 1 + 3 T + 210 T^{2} + 535 T^{3} + 18697 T^{4} + 37089 T^{5} + 878759 T^{6} + 1181815 T^{7} + 21802830 T^{8} + 14639043 T^{9} + 229345007 T^{10} \))(\( 1 - 11 T + 43 T^{2} + 314 T^{3} - 2804 T^{4} + 5383 T^{5} + 131912 T^{6} - 1116863 T^{7} + 6199864 T^{8} + 11891047 T^{9} - 291119692 T^{10} + 1532219834 T^{11} + 9861835301 T^{12} - 118571368619 T^{13} + 506623120463 T^{14} \))
$53$ (\( 1 + 9 T + 53 T^{2} \))(\( 1 + 16 T + 324 T^{2} + 3253 T^{3} + 36558 T^{4} + 254243 T^{5} + 1937574 T^{6} + 9137677 T^{7} + 48236148 T^{8} + 126247696 T^{9} + 418195493 T^{10} \))(\( 1 - 18 T + 375 T^{2} - 4005 T^{3} + 48141 T^{4} - 371978 T^{5} + 3458860 T^{6} - 22383137 T^{7} + 183319580 T^{8} - 1044886202 T^{9} + 7167087657 T^{10} - 31601376405 T^{11} + 156823309875 T^{12} - 398958500322 T^{13} + 1174711139837 T^{14} \))
$59$ (\( 1 + 2 T + 59 T^{2} \))(\( 1 + 3 T + 110 T^{2} + 828 T^{3} + 7192 T^{4} + 75139 T^{5} + 424328 T^{6} + 2882268 T^{7} + 22591690 T^{8} + 36352083 T^{9} + 714924299 T^{10} \))(\( 1 - 11 T + 252 T^{2} - 1934 T^{3} + 28087 T^{4} - 191672 T^{5} + 2291334 T^{6} - 13940160 T^{7} + 135188706 T^{8} - 667210232 T^{9} + 5768479973 T^{10} - 23434976174 T^{11} + 180160923348 T^{12} - 463985870051 T^{13} + 2488651484819 T^{14} \))
$61$ (\( 1 - 3 T + 61 T^{2} \))(\( 1 + 10 T + 280 T^{2} + 1878 T^{3} + 30705 T^{4} + 152779 T^{5} + 1873005 T^{6} + 6988038 T^{7} + 63554680 T^{8} + 138458410 T^{9} + 844596301 T^{10} \))(\( 1 - 4 T + 187 T^{2} - 510 T^{3} + 16426 T^{4} - 42911 T^{5} + 1151882 T^{6} - 3243451 T^{7} + 70264802 T^{8} - 159671831 T^{9} + 3728389906 T^{10} - 7061378910 T^{11} + 157939508287 T^{12} - 206081497444 T^{13} + 3142742836021 T^{14} \))
$67$ (\( 1 + 2 T + 67 T^{2} \))(\( 1 + 2 T + 248 T^{2} + 653 T^{3} + 27862 T^{4} + 69141 T^{5} + 1866754 T^{6} + 2931317 T^{7} + 74589224 T^{8} + 40302242 T^{9} + 1350125107 T^{10} \))(\( 1 + 18 T + 402 T^{2} + 4949 T^{3} + 70617 T^{4} + 686165 T^{5} + 7367052 T^{6} + 57501620 T^{7} + 493592484 T^{8} + 3080194685 T^{9} + 21238980771 T^{10} + 99727897829 T^{11} + 542750293014 T^{12} + 1628250879042 T^{13} + 6060711605323 T^{14} \))
$71$ (\( 1 + 5 T + 71 T^{2} \))(\( 1 - 31 T + 561 T^{2} - 7366 T^{3} + 81729 T^{4} - 751573 T^{5} + 5802759 T^{6} - 37132006 T^{7} + 200788071 T^{8} - 787762111 T^{9} + 1804229351 T^{10} \))(\( 1 + 3 T + 178 T^{2} + 793 T^{3} + 13251 T^{4} + 107538 T^{5} + 826217 T^{6} + 9409813 T^{7} + 58661407 T^{8} + 542099058 T^{9} + 4742678661 T^{10} + 20151463033 T^{11} + 321152824478 T^{12} + 384300851763 T^{13} + 9095120158391 T^{14} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 + 14 T + 271 T^{2} + 2103 T^{3} + 27252 T^{4} + 168363 T^{5} + 1989396 T^{6} + 11206887 T^{7} + 105423607 T^{8} + 397575374 T^{9} + 2073071593 T^{10} \))(\( 1 - 2 T + 363 T^{2} - 1347 T^{3} + 59673 T^{4} - 291847 T^{5} + 6116143 T^{6} - 29811958 T^{7} + 446478439 T^{8} - 1555252663 T^{9} + 23213811441 T^{10} - 38252430627 T^{11} + 752524988259 T^{12} - 302668452578 T^{13} + 11047398519097 T^{14} \))
$79$ (\( 1 + 8 T + 79 T^{2} \))(\( 1 + 14 T + 302 T^{2} + 2853 T^{3} + 38342 T^{4} + 276945 T^{5} + 3029018 T^{6} + 17805573 T^{7} + 148897778 T^{8} + 545301134 T^{9} + 3077056399 T^{10} \))(\( 1 + 200 T^{2} - 121 T^{3} + 21111 T^{4} - 85025 T^{5} + 1701860 T^{6} - 10725940 T^{7} + 134446940 T^{8} - 530641025 T^{9} + 10408546329 T^{10} - 4712959801 T^{11} + 615411279800 T^{12} + 19203908986159 T^{14} \))
$83$ (\( 1 - 5 T + 83 T^{2} \))(\( 1 + 12 T + 384 T^{2} + 3707 T^{3} + 61198 T^{4} + 450375 T^{5} + 5079434 T^{6} + 25537523 T^{7} + 219566208 T^{8} + 569499852 T^{9} + 3939040643 T^{10} \))(\( 1 - 18 T + 511 T^{2} - 6019 T^{3} + 100973 T^{4} - 894832 T^{5} + 11680108 T^{6} - 86133917 T^{7} + 969448964 T^{8} - 6164497648 T^{9} + 57735048751 T^{10} - 285651634099 T^{11} + 2012849768573 T^{12} - 5884926720642 T^{13} + 27136050989627 T^{14} \))
$89$ (\( 1 + 14 T + 89 T^{2} \))(\( 1 + 150 T^{2} + 1530 T^{3} + 8043 T^{4} + 235795 T^{5} + 715827 T^{6} + 12119130 T^{7} + 105745350 T^{8} + 5584059449 T^{10} \))(\( 1 - 18 T + 618 T^{2} - 8390 T^{3} + 159538 T^{4} - 1707859 T^{5} + 23082447 T^{6} - 196461066 T^{7} + 2054337783 T^{8} - 13527951139 T^{9} + 112469344322 T^{10} - 526407401990 T^{11} + 3450948739482 T^{12} - 8945663237298 T^{13} + 44231334895529 T^{14} \))
$97$ (\( 1 + 11 T + 97 T^{2} \))(\( 1 - 23 T + 414 T^{2} - 3296 T^{3} + 24642 T^{4} - 80747 T^{5} + 2390274 T^{6} - 31012064 T^{7} + 377846622 T^{8} - 2036173463 T^{9} + 8587340257 T^{10} \))(\( 1 - 21 T + 439 T^{2} - 4879 T^{3} + 68083 T^{4} - 712034 T^{5} + 9819956 T^{6} - 93024973 T^{7} + 952535732 T^{8} - 6699527906 T^{9} + 62137515859 T^{10} - 431934361999 T^{11} + 3769842372823 T^{12} - 17492412103509 T^{13} + 80798284478113 T^{14} \))
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