Properties

Label 668.2.a
Level 668
Weight 2
Character orbit a
Rep. character \(\chi_{668}(1,\cdot)\)
Character field \(\Q\)
Dimension 14
Newform subspaces 3
Sturm bound 168
Trace bound 1

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Defining parameters

Level: \( N \) = \( 668 = 2^{2} \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 668.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(168\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(668))\).

Total New Old
Modular forms 87 14 73
Cusp forms 82 14 68
Eisenstein series 5 0 5

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

\(2\)\(167\)FrickeDim.
\(-\)\(+\)\(-\)\(7\)
\(-\)\(-\)\(+\)\(7\)
Plus space\(+\)\(7\)
Minus space\(-\)\(7\)

Trace form

\( 14q + 2q^{5} + 14q^{9} + O(q^{10}) \) \( 14q + 2q^{5} + 14q^{9} - 2q^{11} - 4q^{13} - 14q^{15} - 2q^{17} - 2q^{19} - 14q^{23} + 6q^{25} - 2q^{29} + 2q^{31} - 16q^{33} + 2q^{35} - 10q^{37} - 6q^{39} - 4q^{41} - 4q^{43} + 2q^{45} - 2q^{47} + 30q^{49} + 2q^{51} + 6q^{55} + 4q^{57} - 6q^{59} - 2q^{61} - 14q^{63} - 22q^{65} - 12q^{67} + 14q^{69} + 8q^{71} + 18q^{73} + 26q^{75} + 2q^{79} + 6q^{81} + 22q^{83} - 34q^{85} - 2q^{87} - 14q^{89} + 6q^{91} - 32q^{93} + 8q^{95} - 44q^{97} - 2q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(668))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 167
668.2.a.a \(2\) \(5.334\) \(\Q(\sqrt{13}) \) None \(0\) \(1\) \(-6\) \(3\) \(-\) \(+\) \(q+\beta q^{3}-3q^{5}+(1+\beta )q^{7}+\beta q^{9}+(4+\cdots)q^{13}+\cdots\)
668.2.a.b \(5\) \(5.334\) 5.5.826865.1 None \(0\) \(3\) \(10\) \(9\) \(-\) \(+\) \(q+(1-\beta _{2})q^{3}+2q^{5}+(2-\beta _{3})q^{7}+(2+\cdots)q^{9}+\cdots\)
668.2.a.c \(7\) \(5.334\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-4\) \(-2\) \(-12\) \(-\) \(-\) \(q+(-1-\beta _{3})q^{3}+(\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(668))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(668)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(167))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(334))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))
$3$ (\( 1 - T + 3 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 - 3 T + 9 T^{2} - 19 T^{3} + 45 T^{4} - 83 T^{5} + 135 T^{6} - 171 T^{7} + 243 T^{8} - 243 T^{9} + 243 T^{10} \))(\( 1 + 4 T + 15 T^{2} + 40 T^{3} + 93 T^{4} + 185 T^{5} + 355 T^{6} + 599 T^{7} + 1065 T^{8} + 1665 T^{9} + 2511 T^{10} + 3240 T^{11} + 3645 T^{12} + 2916 T^{13} + 2187 T^{14} \))
$5$ (\( ( 1 + 3 T + 5 T^{2} )^{2} \))(\( ( 1 - 2 T + 5 T^{2} )^{5} \))(\( 1 + 2 T + 18 T^{2} + 18 T^{3} + 128 T^{4} - 2 T^{5} + 545 T^{6} - 452 T^{7} + 2725 T^{8} - 50 T^{9} + 16000 T^{10} + 11250 T^{11} + 56250 T^{12} + 31250 T^{13} + 78125 T^{14} \))
$7$ (\( 1 - 3 T + 13 T^{2} - 21 T^{3} + 49 T^{4} \))(\( 1 - 9 T + 51 T^{2} - 223 T^{3} + 787 T^{4} - 2265 T^{5} + 5509 T^{6} - 10927 T^{7} + 17493 T^{8} - 21609 T^{9} + 16807 T^{10} \))(\( 1 + 12 T + 87 T^{2} + 466 T^{3} + 2053 T^{4} + 7675 T^{5} + 24829 T^{6} + 70131 T^{7} + 173803 T^{8} + 376075 T^{9} + 704179 T^{10} + 1118866 T^{11} + 1462209 T^{12} + 1411788 T^{13} + 823543 T^{14} \))
$11$ (\( ( 1 + 11 T^{2} )^{2} \))(\( 1 - 5 T + 43 T^{2} - 191 T^{3} + 849 T^{4} - 3017 T^{5} + 9339 T^{6} - 23111 T^{7} + 57233 T^{8} - 73205 T^{9} + 161051 T^{10} \))(\( 1 + 7 T + 49 T^{2} + 211 T^{3} + 1044 T^{4} + 3820 T^{5} + 15876 T^{6} + 49868 T^{7} + 174636 T^{8} + 462220 T^{9} + 1389564 T^{10} + 3089251 T^{11} + 7891499 T^{12} + 12400927 T^{13} + 19487171 T^{14} \))
$13$ (\( 1 - 9 T + 43 T^{2} - 117 T^{3} + 169 T^{4} \))(\( 1 + 4 T + 45 T^{2} + 160 T^{3} + 1006 T^{4} + 2840 T^{5} + 13078 T^{6} + 27040 T^{7} + 98865 T^{8} + 114244 T^{9} + 371293 T^{10} \))(\( 1 + 9 T + 68 T^{2} + 356 T^{3} + 1658 T^{4} + 6631 T^{5} + 26213 T^{6} + 92808 T^{7} + 340769 T^{8} + 1120639 T^{9} + 3642626 T^{10} + 10167716 T^{11} + 25247924 T^{12} + 43441281 T^{13} + 62748517 T^{14} \))
$17$ (\( 1 - T + 31 T^{2} - 17 T^{3} + 289 T^{4} \))(\( 1 + 2 T + 25 T^{2} + 96 T^{3} + 662 T^{4} + 1308 T^{5} + 11254 T^{6} + 27744 T^{7} + 122825 T^{8} + 167042 T^{9} + 1419857 T^{10} \))(\( 1 + T + 66 T^{2} + 26 T^{3} + 2212 T^{4} + 199 T^{5} + 51705 T^{6} + 1532 T^{7} + 878985 T^{8} + 57511 T^{9} + 10867556 T^{10} + 2171546 T^{11} + 93710562 T^{12} + 24137569 T^{13} + 410338673 T^{14} \))
$19$ (\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( 1 - 5 T + 53 T^{2} - 279 T^{3} + 1821 T^{4} - 6547 T^{5} + 34599 T^{6} - 100719 T^{7} + 363527 T^{8} - 651605 T^{9} + 2476099 T^{10} \))(\( 1 + 11 T + 123 T^{2} + 943 T^{3} + 6780 T^{4} + 38558 T^{5} + 207830 T^{6} + 937252 T^{7} + 3948770 T^{8} + 13919438 T^{9} + 46504020 T^{10} + 122892703 T^{11} + 304560177 T^{12} + 517504691 T^{13} + 893871739 T^{14} \))
$23$ (\( 1 + T + 43 T^{2} + 23 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 63 T^{2} - 192 T^{3} + 1414 T^{4} - 2452 T^{5} + 32522 T^{6} - 101568 T^{7} + 766521 T^{8} - 1679046 T^{9} + 6436343 T^{10} \))(\( 1 + 19 T + 232 T^{2} + 2074 T^{3} + 15558 T^{4} + 100741 T^{5} + 582135 T^{6} + 2958428 T^{7} + 13389105 T^{8} + 53291989 T^{9} + 189294186 T^{10} + 580390234 T^{11} + 1493231576 T^{12} + 2812681891 T^{13} + 3404825447 T^{14} \))
$29$ (\( 1 - 8 T + 61 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 + 5 T + 117 T^{2} + 541 T^{3} + 6005 T^{4} + 22981 T^{5} + 174145 T^{6} + 454981 T^{7} + 2853513 T^{8} + 3536405 T^{9} + 20511149 T^{10} \))(\( 1 + 5 T + 162 T^{2} + 702 T^{3} + 12203 T^{4} + 45007 T^{5} + 550271 T^{6} + 1671295 T^{7} + 15957859 T^{8} + 37850887 T^{9} + 297618967 T^{10} + 496511262 T^{11} + 3322806138 T^{12} + 2974116605 T^{13} + 17249876309 T^{14} \))
$31$ (\( 1 - 6 T + 58 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 - 9 T + 119 T^{2} - 909 T^{3} + 6257 T^{4} - 39425 T^{5} + 193967 T^{6} - 873549 T^{7} + 3545129 T^{8} - 8311689 T^{9} + 28629151 T^{10} \))(\( 1 + 13 T + 95 T^{2} + 305 T^{3} + 648 T^{4} - 560 T^{5} + 20132 T^{6} + 105568 T^{7} + 624092 T^{8} - 538160 T^{9} + 19304568 T^{10} + 281673905 T^{11} + 2719769345 T^{12} + 11537547853 T^{13} + 27512614111 T^{14} \))
$37$ (\( 1 - 8 T + 77 T^{2} - 296 T^{3} + 1369 T^{4} \))(\( 1 - 8 T + 141 T^{2} - 1104 T^{3} + 8998 T^{4} - 60080 T^{5} + 332926 T^{6} - 1511376 T^{7} + 7142073 T^{8} - 14993288 T^{9} + 69343957 T^{10} \))(\( 1 + 26 T + 404 T^{2} + 4432 T^{3} + 39662 T^{4} + 308446 T^{5} + 2171097 T^{6} + 13871632 T^{7} + 80330589 T^{8} + 422262574 T^{9} + 2008999286 T^{10} + 8306281552 T^{11} + 28014958628 T^{12} + 66708886634 T^{13} + 94931877133 T^{14} \))
$41$ (\( 1 - 2 T + 31 T^{2} - 82 T^{3} + 1681 T^{4} \))(\( 1 + 4 T + 109 T^{2} + 424 T^{3} + 6474 T^{4} + 20392 T^{5} + 265434 T^{6} + 712744 T^{7} + 7512389 T^{8} + 11303044 T^{9} + 115856201 T^{10} \))(\( 1 + 2 T + 146 T^{2} + 266 T^{3} + 11268 T^{4} + 21566 T^{5} + 619985 T^{6} + 1112908 T^{7} + 25419385 T^{8} + 36252446 T^{9} + 776601828 T^{10} + 751652426 T^{11} + 16915005346 T^{12} + 9500208482 T^{13} + 194754273881 T^{14} \))
$43$ (\( 1 - 12 T + 109 T^{2} - 516 T^{3} + 1849 T^{4} \))(\( 1 - 8 T + 135 T^{2} - 992 T^{3} + 9706 T^{4} - 56752 T^{5} + 417358 T^{6} - 1834208 T^{7} + 10733445 T^{8} - 27350408 T^{9} + 147008443 T^{10} \))(\( 1 + 24 T + 514 T^{2} + 7062 T^{3} + 86260 T^{4} + 816360 T^{5} + 6931999 T^{6} + 47907988 T^{7} + 298075957 T^{8} + 1509449640 T^{9} + 6858273820 T^{10} + 24143572662 T^{11} + 75562339702 T^{12} + 151712713176 T^{13} + 271818611107 T^{14} \))
$47$ (\( 1 + 4 T + 85 T^{2} + 188 T^{3} + 2209 T^{4} \))(\( 1 - 13 T + 261 T^{2} - 2283 T^{3} + 25333 T^{4} - 157087 T^{5} + 1190651 T^{6} - 5043147 T^{7} + 27097803 T^{8} - 63435853 T^{9} + 229345007 T^{10} \))(\( 1 + 11 T + 154 T^{2} + 1124 T^{3} + 10727 T^{4} + 58321 T^{5} + 473997 T^{6} + 2317571 T^{7} + 22277859 T^{8} + 128831089 T^{9} + 1113709321 T^{10} + 5484761444 T^{11} + 35319131078 T^{12} + 118571368619 T^{13} + 506623120463 T^{14} \))
$53$ (\( 1 + 2 T + 94 T^{2} + 106 T^{3} + 2809 T^{4} \))(\( 1 + 2 T + 141 T^{2} + 216 T^{3} + 11910 T^{4} + 17068 T^{5} + 631230 T^{6} + 606744 T^{7} + 20991657 T^{8} + 15780962 T^{9} + 418195493 T^{10} \))(\( 1 - 4 T + 207 T^{2} - 920 T^{3} + 21673 T^{4} - 86044 T^{5} + 1568415 T^{6} - 5154512 T^{7} + 83125995 T^{8} - 241697596 T^{9} + 3226611221 T^{10} - 7259242520 T^{11} + 86566467051 T^{12} - 88657444516 T^{13} + 1174711139837 T^{14} \))
$59$ (\( 1 + 6 T + 10 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 4 T + 59 T^{2} - 120 T^{3} + 4414 T^{4} - 36008 T^{5} + 260426 T^{6} - 417720 T^{7} + 12117361 T^{8} - 48469444 T^{9} + 714924299 T^{10} \))(\( 1 + 4 T + 321 T^{2} + 944 T^{3} + 46921 T^{4} + 103612 T^{5} + 4159569 T^{6} + 7292384 T^{7} + 245414571 T^{8} + 360673372 T^{9} + 9636588059 T^{10} + 11438788784 T^{11} + 229490699979 T^{12} + 168722134564 T^{13} + 2488651484819 T^{14} \))
$61$ (\( 1 + 8 T + 21 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 - 11 T + 229 T^{2} - 2491 T^{3} + 24489 T^{4} - 221163 T^{5} + 1493829 T^{6} - 9269011 T^{7} + 51978649 T^{8} - 152304251 T^{9} + 844596301 T^{10} \))(\( 1 + 5 T + 210 T^{2} + 398 T^{3} + 22363 T^{4} + 26651 T^{5} + 1945003 T^{6} + 2595919 T^{7} + 118645183 T^{8} + 99168371 T^{9} + 5075976103 T^{10} + 5510644718 T^{11} + 177365223210 T^{12} + 257601871805 T^{13} + 3142742836021 T^{14} \))
$67$ (\( 1 - 2 T + 83 T^{2} - 134 T^{3} + 4489 T^{4} \))(\( 1 - 28 T + 491 T^{2} - 5616 T^{3} + 53302 T^{4} - 436232 T^{5} + 3571234 T^{6} - 25210224 T^{7} + 147674633 T^{8} - 564231388 T^{9} + 1350125107 T^{10} \))(\( 1 + 42 T + 1060 T^{2} + 19258 T^{3} + 278042 T^{4} + 3324230 T^{5} + 33867347 T^{6} + 297649804 T^{7} + 2269112249 T^{8} + 14922468470 T^{9} + 83624746046 T^{10} + 388070288218 T^{11} + 1431132613420 T^{12} + 3799252051098 T^{13} + 6060711605323 T^{14} \))
$71$ (\( 1 + 3 T + 115 T^{2} + 213 T^{3} + 5041 T^{4} \))(\( 1 - 2 T + 175 T^{2} - 320 T^{3} + 17542 T^{4} - 36060 T^{5} + 1245482 T^{6} - 1613120 T^{7} + 62634425 T^{8} - 50823362 T^{9} + 1804229351 T^{10} \))(\( 1 - 9 T + 320 T^{2} - 2246 T^{3} + 49410 T^{4} - 293863 T^{5} + 5051659 T^{6} - 25462260 T^{7} + 358667789 T^{8} - 1481363383 T^{9} + 17684382510 T^{10} - 57074635526 T^{11} + 577353392320 T^{12} - 1152902555289 T^{13} + 9095120158391 T^{14} \))
$73$ (\( 1 + 17 T + 189 T^{2} + 1241 T^{3} + 5329 T^{4} \))(\( 1 - 8 T + 241 T^{2} - 616 T^{3} + 19206 T^{4} + 4320 T^{5} + 1402038 T^{6} - 3282664 T^{7} + 93753097 T^{8} - 227185928 T^{9} + 2073071593 T^{10} \))(\( 1 - 27 T + 614 T^{2} - 10216 T^{3} + 143660 T^{4} - 1707773 T^{5} + 17858693 T^{6} - 161388576 T^{7} + 1303684589 T^{8} - 9100722317 T^{9} + 55886182220 T^{10} - 290116430056 T^{11} + 1272865958102 T^{12} - 4086024109803 T^{13} + 11047398519097 T^{14} \))
$79$ (\( 1 - 20 T + 245 T^{2} - 1580 T^{3} + 6241 T^{4} \))(\( 1 + 10 T + 139 T^{2} + 1552 T^{3} + 15962 T^{4} + 195500 T^{5} + 1260998 T^{6} + 9686032 T^{7} + 68532421 T^{8} + 389500810 T^{9} + 3077056399 T^{10} \))(\( 1 + 8 T + 314 T^{2} + 2854 T^{3} + 52244 T^{4} + 477640 T^{5} + 5822187 T^{6} + 47351668 T^{7} + 459952773 T^{8} + 2980951240 T^{9} + 25758329516 T^{10} + 111163531174 T^{11} + 966195709286 T^{12} + 1944699644168 T^{13} + 19203908986159 T^{14} \))
$83$ (\( 1 - 4 T + 157 T^{2} - 332 T^{3} + 6889 T^{4} \))(\( 1 - 2 T + 179 T^{2} - 656 T^{3} + 20622 T^{4} - 69980 T^{5} + 1711626 T^{6} - 4519184 T^{7} + 102349873 T^{8} - 94916642 T^{9} + 3939040643 T^{10} \))(\( 1 - 16 T + 238 T^{2} - 1826 T^{3} + 19504 T^{4} - 186096 T^{5} + 2712391 T^{6} - 24177788 T^{7} + 225128453 T^{8} - 1282015344 T^{9} + 11152133648 T^{10} - 86658894146 T^{11} + 937491673034 T^{12} - 5231045973904 T^{13} + 27136050989627 T^{14} \))
$89$ (\( 1 + 6 T + 70 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 17 T + 485 T^{2} + 5689 T^{3} + 89097 T^{4} + 743149 T^{5} + 7929633 T^{6} + 45062569 T^{7} + 341909965 T^{8} + 1066618097 T^{9} + 5584059449 T^{10} \))(\( 1 - 9 T + 261 T^{2} - 1817 T^{3} + 33900 T^{4} - 261500 T^{5} + 4225720 T^{6} - 30821292 T^{7} + 376089080 T^{8} - 2071341500 T^{9} + 23898449100 T^{10} - 114002651897 T^{11} + 1457439516189 T^{12} - 4472831618649 T^{13} + 44231334895529 T^{14} \))
$97$ (\( 1 + 9 T + 133 T^{2} + 873 T^{3} + 9409 T^{4} \))(\( 1 + 27 T + 521 T^{2} + 8059 T^{3} + 104425 T^{4} + 1093259 T^{5} + 10129225 T^{6} + 75827131 T^{7} + 475502633 T^{8} + 2390290587 T^{9} + 8587340257 T^{10} \))(\( 1 + 8 T + 417 T^{2} + 3412 T^{3} + 94457 T^{4} + 671935 T^{5} + 13479577 T^{6} + 82425871 T^{7} + 1307518969 T^{8} + 6322236415 T^{9} + 86208353561 T^{10} + 302061906772 T^{11} + 3580920887169 T^{12} + 6663776039432 T^{13} + 80798284478113 T^{14} \))
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