Properties

Label 167.2
Level 167
Weight 2
Dimension 1080
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 4648
Trace bound 1

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Defining parameters

Level: \( N \) = \( 167 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(4648\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(167))\).

Total New Old
Modular forms 1245 1245 0
Cusp forms 1080 1080 0
Eisenstein series 165 165 0

Trace form

\( 1080q - 80q^{2} - 79q^{3} - 76q^{4} - 77q^{5} - 71q^{6} - 75q^{7} - 68q^{8} - 70q^{9} + O(q^{10}) \) \( 1080q - 80q^{2} - 79q^{3} - 76q^{4} - 77q^{5} - 71q^{6} - 75q^{7} - 68q^{8} - 70q^{9} - 65q^{10} - 71q^{11} - 55q^{12} - 69q^{13} - 59q^{14} - 59q^{15} - 52q^{16} - 65q^{17} - 44q^{18} - 63q^{19} - 41q^{20} - 51q^{21} - 47q^{22} - 59q^{23} - 23q^{24} - 52q^{25} - 41q^{26} - 43q^{27} - 27q^{28} - 53q^{29} - 11q^{30} - 51q^{31} - 20q^{32} - 35q^{33} - 29q^{34} - 35q^{35} + 8q^{36} - 45q^{37} - 23q^{38} - 27q^{39} + 7q^{40} - 41q^{41} + 13q^{42} - 39q^{43} + q^{44} - 5q^{45} - 11q^{46} - 35q^{47} + 41q^{48} - 26q^{49} + 10q^{50} - 11q^{51} + 15q^{52} - 29q^{53} + 37q^{54} - 11q^{55} + 37q^{56} - 3q^{57} + 7q^{58} - 23q^{59} + 85q^{60} - 21q^{61} + 13q^{62} + 21q^{63} + 44q^{64} + q^{65} + 61q^{66} - 15q^{67} + 43q^{68} + 13q^{69} + 61q^{70} - 11q^{71} + 112q^{72} - 9q^{73} + 31q^{74} + 41q^{75} + 57q^{76} + 13q^{77} + 85q^{78} - 3q^{79} + 103q^{80} + 38q^{81} + 43q^{82} + q^{83} + 141q^{84} + 25q^{85} + 49q^{86} + 37q^{87} + 97q^{88} + 7q^{89} + 151q^{90} + 29q^{91} + 85q^{92} + 45q^{93} + 61q^{94} + 37q^{95} + 169q^{96} + 15q^{97} + 88q^{98} + 73q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(167))\)

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
167.2.a \(\chi_{167}(1, \cdot)\) 167.2.a.a 2 1
167.2.a.b 12
167.2.c \(\chi_{167}(2, \cdot)\) 167.2.c.a 1066 82

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 - 2 T + 7 T^{2} - 11 T^{3} + 27 T^{4} - 35 T^{5} + 71 T^{6} - 87 T^{7} + 175 T^{8} - 223 T^{9} + 447 T^{10} - 558 T^{11} + 989 T^{12} - 1116 T^{13} + 1788 T^{14} - 1784 T^{15} + 2800 T^{16} - 2784 T^{17} + 4544 T^{18} - 4480 T^{19} + 6912 T^{20} - 5632 T^{21} + 7168 T^{22} - 4096 T^{23} + 4096 T^{24} \))
$3$ (\( 1 + T + 5 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - 3 T + 14 T^{2} - 28 T^{3} + 79 T^{4} - 120 T^{5} + 267 T^{6} - 376 T^{7} + 859 T^{8} - 1572 T^{9} + 3591 T^{10} - 6916 T^{11} + 12959 T^{12} - 20748 T^{13} + 32319 T^{14} - 42444 T^{15} + 69579 T^{16} - 91368 T^{17} + 194643 T^{18} - 262440 T^{19} + 518319 T^{20} - 551124 T^{21} + 826686 T^{22} - 531441 T^{23} + 531441 T^{24} \))
$5$ (\( ( 1 + T + 5 T^{2} )^{2} \))(\( 1 - 4 T + 19 T^{2} - 68 T^{3} + 248 T^{4} - 796 T^{5} + 2479 T^{6} - 6892 T^{7} + 19111 T^{8} - 48832 T^{9} + 123526 T^{10} - 288688 T^{11} + 667744 T^{12} - 1443440 T^{13} + 3088150 T^{14} - 6104000 T^{15} + 11944375 T^{16} - 21537500 T^{17} + 38734375 T^{18} - 62187500 T^{19} + 96875000 T^{20} - 132812500 T^{21} + 185546875 T^{22} - 195312500 T^{23} + 244140625 T^{24} \))
$7$ (\( 1 + 5 T + 19 T^{2} + 35 T^{3} + 49 T^{4} \))(\( 1 - 11 T + 88 T^{2} - 512 T^{3} + 2549 T^{4} - 10848 T^{5} + 41869 T^{6} - 146188 T^{7} + 478105 T^{8} - 1460284 T^{9} + 4272913 T^{10} - 11903480 T^{11} + 32169505 T^{12} - 83324360 T^{13} + 209372737 T^{14} - 500877412 T^{15} + 1147930105 T^{16} - 2456981716 T^{17} + 4925845981 T^{18} - 8933794464 T^{19} + 14694477749 T^{20} - 20661046784 T^{21} + 24857821912 T^{22} - 21750594173 T^{23} + 13841287201 T^{24} \))
$11$ (\( ( 1 + 11 T^{2} )^{2} \))(\( 1 + 55 T^{2} - 12 T^{3} + 1596 T^{4} - 688 T^{5} + 31920 T^{6} - 20160 T^{7} + 495320 T^{8} - 392868 T^{9} + 6450124 T^{10} - 5620672 T^{11} + 74530936 T^{12} - 61827392 T^{13} + 780465004 T^{14} - 522907308 T^{15} + 7251980120 T^{16} - 3246788160 T^{17} + 56548227120 T^{18} - 13407173648 T^{19} + 342116774076 T^{20} - 28295372292 T^{21} + 1426558353055 T^{22} + 3138428376721 T^{24} \))
$13$ (\( 1 + 5 T + 31 T^{2} + 65 T^{3} + 169 T^{4} \))(\( 1 - 9 T + 109 T^{2} - 645 T^{3} + 4648 T^{4} - 20861 T^{5} + 117121 T^{6} - 441833 T^{7} + 2219655 T^{8} - 7677394 T^{9} + 36384794 T^{10} - 117891114 T^{11} + 516138240 T^{12} - 1532584482 T^{13} + 6149030186 T^{14} - 16867234618 T^{15} + 63395566455 T^{16} - 164049500069 T^{17} + 565320696889 T^{18} - 1308996813137 T^{19} + 3791516391208 T^{20} - 6839902095585 T^{21} + 15026575611541 T^{22} - 16129443546333 T^{23} + 23298085122481 T^{24} \))
$17$ (\( 1 + 5 T + 39 T^{2} + 85 T^{3} + 289 T^{4} \))(\( 1 - 3 T + 89 T^{2} - 271 T^{3} + 3764 T^{4} - 10083 T^{5} + 102997 T^{6} - 196615 T^{7} + 2062567 T^{8} - 2024678 T^{9} + 33284978 T^{10} - 9244702 T^{11} + 527249480 T^{12} - 157159934 T^{13} + 9619358642 T^{14} - 9947243014 T^{15} + 172267658407 T^{16} - 279165184055 T^{17} + 2486097194293 T^{18} - 4137444839859 T^{19} + 26256751007924 T^{20} - 32137314530687 T^{21} + 179423457139961 T^{22} - 102815688922899 T^{23} + 582622237229761 T^{24} \))
$19$ (\( 1 + 18 T^{2} + 361 T^{4} \))(\( 1 + 123 T^{2} - 44 T^{3} + 7924 T^{4} - 5052 T^{5} + 346656 T^{6} - 289044 T^{7} + 11384560 T^{8} - 10701760 T^{9} + 295368344 T^{10} - 279137696 T^{11} + 6209526260 T^{12} - 5303616224 T^{13} + 106627972184 T^{14} - 73403371840 T^{15} + 1483647243760 T^{16} - 715701559356 T^{17} + 16308736923936 T^{18} - 4515840025428 T^{19} + 134577753536884 T^{20} - 14198258702276 T^{21} + 754121149709523 T^{22} + 2213314919066161 T^{24} \))
$23$ (\( 1 - T + 45 T^{2} - 23 T^{3} + 529 T^{4} \))(\( 1 - T + 111 T^{2} + 17 T^{3} + 6044 T^{4} + 5251 T^{5} + 232339 T^{6} + 271053 T^{7} + 7190087 T^{8} + 9335894 T^{9} + 189133806 T^{10} + 264460122 T^{11} + 4496011768 T^{12} + 6082582806 T^{13} + 100051783374 T^{14} + 113589822298 T^{15} + 2012081136167 T^{16} + 1744590079179 T^{17} + 34394510414371 T^{18} + 17878738422197 T^{19} + 473311595038364 T^{20} + 30619595244871 T^{21} + 4598342744715039 T^{22} - 952809757913927 T^{23} + 21914624432020321 T^{24} \))
$29$ (\( 1 - 8 T + 69 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 212 T^{2} + 1082 T^{3} + 21101 T^{4} + 89188 T^{5} + 1306403 T^{6} + 4417266 T^{7} + 57247721 T^{8} + 150616432 T^{9} + 1974753899 T^{10} + 4219750520 T^{11} + 59628660633 T^{12} + 122372765080 T^{13} + 1660768029059 T^{14} + 3673384160048 T^{15} + 40490225356601 T^{16} + 90603201098634 T^{17} + 777078971024363 T^{18} + 1538481968247092 T^{19} + 10555699559890061 T^{20} + 15696731945890258 T^{21} + 89189933459642612 T^{22} + 73203058594234974 T^{23} + 353814783205469041 T^{24} \))
$31$ (\( 1 - 6 T + 66 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 - 2 T + 215 T^{2} - 552 T^{3} + 24140 T^{4} - 66360 T^{5} + 1836432 T^{6} - 5016820 T^{7} + 103696828 T^{8} - 270132408 T^{9} + 4543215564 T^{10} - 10915998044 T^{11} + 157730757196 T^{12} - 338395939364 T^{13} + 4366030157004 T^{14} - 8047514566728 T^{15} + 95766198291388 T^{16} - 143627297319820 T^{17} + 1629840159906192 T^{18} - 1825737072405960 T^{19} + 20588789643825740 T^{20} - 14594671432690392 T^{21} + 176220081700872215 T^{22} - 50816953792809662 T^{23} + 787662783788549761 T^{24} \))
$37$ (\( 1 + 12 T + 105 T^{2} + 444 T^{3} + 1369 T^{4} \))(\( 1 - 34 T + 721 T^{2} - 10994 T^{3} + 136136 T^{4} - 1426586 T^{5} + 13262837 T^{6} - 111888490 T^{7} + 878750471 T^{8} - 6481238924 T^{9} + 45307548946 T^{10} - 299446846172 T^{11} + 1874841412864 T^{12} - 11079533308364 T^{13} + 62026034507074 T^{14} - 328294195217372 T^{15} + 1646919861479831 T^{16} - 7758790639354930 T^{17} + 34028811149162333 T^{18} - 135428486871657938 T^{19} + 478174902938989256 T^{20} - 1428799367307076538 T^{21} + 3466989332513269129 T^{22} - 6049199140501654042 T^{23} + 6582952005840035281 T^{24} \))
$41$ (\( 1 - 2 T + 3 T^{2} - 82 T^{3} + 1681 T^{4} \))(\( 1 + 14 T + 339 T^{2} + 3588 T^{3} + 53932 T^{4} + 480564 T^{5} + 5599015 T^{6} + 43529766 T^{7} + 424099143 T^{8} + 2922923404 T^{9} + 24710309430 T^{10} + 152033575416 T^{11} + 1136390796760 T^{12} + 6233376592056 T^{13} + 41538030151830 T^{14} + 201450803927084 T^{15} + 1198402818422823 T^{16} + 5043193319178966 T^{17} + 26595904896922615 T^{18} + 93591892873348884 T^{19} + 430642987456953772 T^{20} + 1174646380605532068 T^{21} + 4550281506141663939 T^{22} + 7704606444027478174 T^{23} + 22563490300366186081 T^{24} \))
$43$ (\( 1 + 6 T + 15 T^{2} + 258 T^{3} + 1849 T^{4} \))(\( 1 - 6 T + 255 T^{2} - 1748 T^{3} + 32440 T^{4} - 226068 T^{5} + 2746475 T^{6} - 18002290 T^{7} + 172552423 T^{8} - 1047703492 T^{9} + 8712770270 T^{10} - 50501656128 T^{11} + 388449379296 T^{12} - 2171571213504 T^{13} + 16109912229230 T^{14} - 83299761538444 T^{15} + 589922396304823 T^{16} - 2646488623334470 T^{17} + 17361465580002275 T^{18} - 61449489775737276 T^{19} + 379165217005376440 T^{20} - 878531885665601564 T^{21} + 5510927989887483495 T^{22} - 5575762436827336242 T^{23} + 39959630797262576401 T^{24} \))
$47$ (\( ( 1 - 7 T + 47 T^{2} )^{2} \))(\( 1 - 2 T + 336 T^{2} - 344 T^{3} + 54457 T^{4} - 3732 T^{5} + 5722257 T^{6} + 4378892 T^{7} + 444390369 T^{8} + 603623948 T^{9} + 27487693525 T^{10} + 44089368444 T^{11} + 1409831184861 T^{12} + 2072200316868 T^{13} + 60720314996725 T^{14} + 62670049153204 T^{15} + 2168483240192289 T^{16} + 1004277016392244 T^{17} + 61681440370877553 T^{18} - 1890717485567916 T^{19} + 1296691237739518777 T^{20} - 384980882747351848 T^{21} + 17673308431238896464 T^{22} - 4944318430168024606 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( 1 + 10 T + 126 T^{2} + 530 T^{3} + 2809 T^{4} \))(\( 1 - 10 T + 416 T^{2} - 3966 T^{3} + 86330 T^{4} - 761902 T^{5} + 11674400 T^{6} - 94151882 T^{7} + 1144126111 T^{8} - 8383469412 T^{9} + 85931558464 T^{10} - 568900866028 T^{11} + 5098746497836 T^{12} - 30151745899484 T^{13} + 241381747725376 T^{14} - 1248105775650324 T^{15} + 9027705340449391 T^{16} - 39373892709867826 T^{17} + 258755617564397600 T^{18} - 895014766864089974 T^{19} + 5374879073212795130 T^{20} - 13086862405087259478 T^{21} + 72753187672053428384 T^{22} - 92690359293721915970 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( 1 + 2 T + 114 T^{2} + 118 T^{3} + 3481 T^{4} \))(\( 1 + 28 T + 668 T^{2} + 10196 T^{3} + 145714 T^{4} + 1636716 T^{5} + 18251276 T^{6} + 172162052 T^{7} + 1672497151 T^{8} + 14180893624 T^{9} + 126122113016 T^{10} + 986576008488 T^{11} + 8091055420924 T^{12} + 58207984500792 T^{13} + 439031075408696 T^{14} + 2912457751603496 T^{15} + 20266251750138511 T^{16} + 123082834340501548 T^{17} + 769848561309175916 T^{18} + 4073215703627014404 T^{19} + 21395250385076030194 T^{20} + 88327905367005758044 T^{21} + \)\(34\!\cdots\!68\)\( T^{22} + \)\(84\!\cdots\!52\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( 1 + 117 T^{2} + 3721 T^{4} \))(\( 1 - 2 T + 436 T^{2} - 1038 T^{3} + 96193 T^{4} - 245604 T^{5} + 14171095 T^{6} - 36275338 T^{7} + 1548507881 T^{8} - 3796228844 T^{9} + 131946978859 T^{10} - 299278550144 T^{11} + 8976427949089 T^{12} - 18255991558784 T^{13} + 490974708334339 T^{14} - 861671819239964 T^{15} + 21440393907572921 T^{16} - 30638016292324738 T^{17} + 730100119505295295 T^{18} - 771870211498101684 T^{19} + 18440901559147451233 T^{20} - 12138523644361838358 T^{21} + \)\(31\!\cdots\!36\)\( T^{22} - 87027835222871677322 T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))
$67$ (\( 1 + 4 T + 133 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 28 T + 849 T^{2} - 15350 T^{3} + 282852 T^{4} - 3942978 T^{5} + 55516261 T^{6} - 642715788 T^{7} + 7500440119 T^{8} - 74700374352 T^{9} + 749275246162 T^{10} - 6523249807428 T^{11} + 57171174721064 T^{12} - 437057737097676 T^{13} + 3363496580021218 T^{14} - 22467108691230576 T^{15} + 151142276391223399 T^{16} - 867746722044089316 T^{17} + 5021911154131950109 T^{18} - 23897252524133271894 T^{19} + \)\(11\!\cdots\!32\)\( T^{20} - \)\(41\!\cdots\!50\)\( T^{21} + \)\(15\!\cdots\!01\)\( T^{22} - \)\(34\!\cdots\!24\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))
$71$ (\( 1 + 11 T + 141 T^{2} + 781 T^{3} + 5041 T^{4} \))(\( 1 + 9 T + 567 T^{2} + 5311 T^{3} + 156896 T^{4} + 1466349 T^{5} + 28329979 T^{6} + 252668907 T^{7} + 3739585015 T^{8} + 30670653834 T^{9} + 379907284894 T^{10} + 2802006636918 T^{11} + 30339404350608 T^{12} + 198942471221178 T^{13} + 1915112623150654 T^{14} + 10977364384380774 T^{15} + 95029141473560215 T^{16} + 455872658094489357 T^{17} + 3629078353375967659 T^{18} + 13336620349136484459 T^{19} + \)\(10\!\cdots\!56\)\( T^{20} + \)\(24\!\cdots\!41\)\( T^{21} + \)\(18\!\cdots\!67\)\( T^{22} + \)\(20\!\cdots\!39\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))
$73$ (\( 1 + 19 T + 225 T^{2} + 1387 T^{3} + 5329 T^{4} \))(\( 1 - 61 T + 2263 T^{2} - 61453 T^{3} + 1350684 T^{4} - 25082241 T^{5} + 405789051 T^{6} - 5819222145 T^{7} + 74966261383 T^{8} - 874663299866 T^{9} + 9301701526878 T^{10} - 90483767786106 T^{11} + 807146180879736 T^{12} - 6605315048385738 T^{13} + 49568767436732862 T^{14} - 340258892923971722 T^{15} + 2128909957623427303 T^{16} - 12063664122156026985 T^{17} + 61409772069632561739 T^{18} - \)\(27\!\cdots\!77\)\( T^{19} + \)\(10\!\cdots\!04\)\( T^{20} - \)\(36\!\cdots\!89\)\( T^{21} + \)\(97\!\cdots\!87\)\( T^{22} - \)\(19\!\cdots\!97\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))
$79$ (\( 1 - 2 T + 139 T^{2} - 158 T^{3} + 6241 T^{4} \))(\( 1 + 427 T^{2} - 1180 T^{3} + 95292 T^{4} - 474180 T^{5} + 15149935 T^{6} - 94423472 T^{7} + 1950616551 T^{8} - 12357400976 T^{9} + 208925853190 T^{10} - 1214936220104 T^{11} + 18336406212664 T^{12} - 95979961388216 T^{13} + 1303906249758790 T^{14} - 6092680619806064 T^{15} + 75976672661390631 T^{16} - 290546348733397328 T^{17} + 3682759150458541135 T^{18} - 9106109563056874620 T^{19} + \)\(14\!\cdots\!12\)\( T^{20} - \)\(14\!\cdots\!20\)\( T^{21} + \)\(40\!\cdots\!27\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \))
$83$ (\( 1 + 2 T + 87 T^{2} + 166 T^{3} + 6889 T^{4} \))(\( 1 + 16 T + 651 T^{2} + 9008 T^{3} + 209704 T^{4} + 2612920 T^{5} + 44543431 T^{6} + 502710712 T^{7} + 6909206119 T^{8} + 70557447080 T^{9} + 822114410918 T^{10} + 7547188829640 T^{11} + 76761076758592 T^{12} + 626416672860120 T^{13} + 5663546176814102 T^{14} + 40343830993531960 T^{15} + 327899321850666199 T^{16} + 1980197926239467816 T^{17} + 14563045962276289039 T^{18} + 70904330351816180840 T^{19} + \)\(47\!\cdots\!64\)\( T^{20} + \)\(16\!\cdots\!24\)\( T^{21} + \)\(10\!\cdots\!99\)\( T^{22} + \)\(20\!\cdots\!72\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))
$89$ (\( 1 + 2 T + 54 T^{2} + 178 T^{3} + 7921 T^{4} \))(\( 1 + 12 T + 697 T^{2} + 6680 T^{3} + 227732 T^{4} + 1834756 T^{5} + 48216552 T^{6} + 338563596 T^{7} + 7544000400 T^{8} + 47120413712 T^{9} + 925586563900 T^{10} + 5192252357770 T^{11} + 91437739905100 T^{12} + 462110459841530 T^{13} + 7331571172651900 T^{14} + 33218430934134928 T^{15} + 473327491200896400 T^{16} + 1890559247331218604 T^{17} + 23962724258648186472 T^{18} + 81153707087581205924 T^{19} + \)\(89\!\cdots\!92\)\( T^{20} + \)\(23\!\cdots\!20\)\( T^{21} + \)\(21\!\cdots\!97\)\( T^{22} + \)\(33\!\cdots\!68\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))
$97$ (\( 1 + 21 T + 293 T^{2} + 2037 T^{3} + 9409 T^{4} \))(\( 1 - 73 T + 3010 T^{2} - 87102 T^{3} + 1967135 T^{4} - 36575228 T^{5} + 582031485 T^{6} - 8144116778 T^{7} + 102571284139 T^{8} - 1186076744056 T^{9} + 12844235600713 T^{10} - 132620387939348 T^{11} + 1324602883633461 T^{12} - 12864177630116756 T^{13} + 120851412767108617 T^{14} - 1082500220227821688 T^{15} + 9080562036072374059 T^{16} - 69936301865428531946 T^{17} + \)\(48\!\cdots\!65\)\( T^{18} - \)\(29\!\cdots\!64\)\( T^{19} + \)\(15\!\cdots\!35\)\( T^{20} - \)\(66\!\cdots\!34\)\( T^{21} + \)\(22\!\cdots\!90\)\( T^{22} - \)\(52\!\cdots\!69\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))
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