Properties

Label 38.9.b.a
Level 38
Weight 9
Character orbit 38.b
Analytic conductor 15.480
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 38.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.4803871823\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 46118 x^{10} + 738386961 x^{8} + 5214446299656 x^{6} + 17370647163698184 x^{4} + 24830681474333400768 x^{2} + 9218779084612644462864\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{21} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + ( \beta_{6} - \beta_{7} ) q^{3} -128 q^{4} + ( 47 - \beta_{2} ) q^{5} + ( 150 - 2 \beta_{1} ) q^{6} + ( -452 + \beta_{1} + \beta_{4} ) q^{7} -128 \beta_{7} q^{8} + ( -1298 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{7} q^{2} + ( \beta_{6} - \beta_{7} ) q^{3} -128 q^{4} + ( 47 - \beta_{2} ) q^{5} + ( 150 - 2 \beta_{1} ) q^{6} + ( -452 + \beta_{1} + \beta_{4} ) q^{7} -128 \beta_{7} q^{8} + ( -1298 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{9} + ( 47 \beta_{7} + 2 \beta_{10} ) q^{10} + ( -1046 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{11} + ( -128 \beta_{6} + 128 \beta_{7} ) q^{12} + ( -57 \beta_{6} + 269 \beta_{7} - \beta_{8} + 3 \beta_{9} - 4 \beta_{10} ) q^{13} + ( 82 \beta_{6} - 439 \beta_{7} - 2 \beta_{8} + \beta_{11} ) q^{14} + ( 15 \beta_{6} - 48 \beta_{7} - 3 \beta_{8} + 4 \beta_{9} - 7 \beta_{10} ) q^{15} + 16384 q^{16} + ( 22569 - 16 \beta_{1} + 4 \beta_{2} - 9 \beta_{3} - 28 \beta_{4} + \beta_{5} ) q^{17} + ( -82 \beta_{6} - 1311 \beta_{7} - 4 \beta_{8} - 6 \beta_{9} + 2 \beta_{10} ) q^{18} + ( 3430 + 78 \beta_{1} - 11 \beta_{2} + 11 \beta_{3} - 31 \beta_{4} - 169 \beta_{6} - 683 \beta_{7} - 3 \beta_{8} + 5 \beta_{9} + \beta_{10} + 6 \beta_{11} ) q^{19} + ( -6016 + 128 \beta_{2} ) q^{20} + ( -1194 \beta_{6} + 5261 \beta_{7} - 4 \beta_{8} + 7 \beta_{9} + 7 \beta_{10} - 12 \beta_{11} ) q^{21} + ( -176 \beta_{6} - 1079 \beta_{7} - 6 \beta_{8} + 10 \beta_{9} - 14 \beta_{10} - 15 \beta_{11} ) q^{22} + ( -68761 + 201 \beta_{1} + 59 \beta_{2} - 11 \beta_{3} + 10 \beta_{4} + 41 \beta_{5} ) q^{23} + ( -19200 + 256 \beta_{1} ) q^{24} + ( 72321 - 216 \beta_{1} - 289 \beta_{2} - 5 \beta_{3} - 74 \beta_{4} + 78 \beta_{5} ) q^{25} + ( -35912 + 110 \beta_{1} - 220 \beta_{2} + 18 \beta_{3} - 28 \beta_{4} + 40 \beta_{5} ) q^{26} + ( -230 \beta_{6} - 11085 \beta_{7} - 41 \beta_{8} + 24 \beta_{9} - 47 \beta_{10} - 30 \beta_{11} ) q^{27} + ( 57856 - 128 \beta_{1} - 128 \beta_{4} ) q^{28} + ( -1224 \beta_{6} + 3357 \beta_{7} - 30 \beta_{8} + 63 \beta_{9} - 9 \beta_{10} - 36 \beta_{11} ) q^{29} + ( 6086 - 12 \beta_{1} - 380 \beta_{2} + 34 \beta_{3} - 124 \beta_{4} + 40 \beta_{5} ) q^{30} + ( -1553 \beta_{6} - 21294 \beta_{7} - 59 \beta_{8} - 2 \beta_{9} - 57 \beta_{10} ) q^{31} + 16384 \beta_{7} q^{32} + ( -159 \beta_{6} - 9342 \beta_{7} - 39 \beta_{8} - 104 \beta_{9} + 149 \beta_{10} + 90 \beta_{11} ) q^{33} + ( -1698 \beta_{6} + 22279 \beta_{7} + 20 \beta_{8} + 66 \beta_{9} - 44 \beta_{10} - 84 \beta_{11} ) q^{34} + ( -100278 + 790 \beta_{1} + 1211 \beta_{2} - 126 \beta_{3} + 433 \beta_{4} - 49 \beta_{5} ) q^{35} + ( 166144 + 256 \beta_{1} + 128 \beta_{2} - 256 \beta_{4} - 128 \beta_{5} ) q^{36} + ( 123 \beta_{6} + 1294 \beta_{7} - 5 \beta_{8} + 132 \beta_{9} - 119 \beta_{10} + 120 \beta_{11} ) q^{37} + ( 83044 + 398 \beta_{1} - 4 \beta_{2} - 34 \beta_{3} - 260 \beta_{4} + 152 \beta_{5} + 4654 \beta_{6} + 4303 \beta_{7} + 106 \beta_{8} - 88 \beta_{9} + 66 \beta_{10} + 35 \beta_{11} ) q^{38} + ( 481905 - 2739 \beta_{1} + 2583 \beta_{2} - 189 \beta_{3} + 72 \beta_{4} - 201 \beta_{5} ) q^{39} + ( -6016 \beta_{7} - 256 \beta_{10} ) q^{40} + ( -8933 \beta_{6} + 30906 \beta_{7} + 217 \beta_{8} - 14 \beta_{9} + 315 \beta_{10} + 126 \beta_{11} ) q^{41} + ( -699858 + 2306 \beta_{1} + 840 \beta_{2} + 196 \beta_{3} + 136 \beta_{4} - 112 \beta_{5} ) q^{42} + ( 634410 - 1102 \beta_{1} - 3403 \beta_{2} - 32 \beta_{3} + 723 \beta_{4} + 63 \beta_{5} ) q^{43} + ( 133888 + 256 \beta_{1} - 384 \beta_{2} + 256 \beta_{3} + 128 \beta_{4} - 128 \beta_{5} ) q^{44} + ( 187283 - 1546 \beta_{1} - 2273 \beta_{2} - 162 \beta_{3} + 88 \beta_{4} - 352 \beta_{5} ) q^{45} + ( 13234 \beta_{6} - 66402 \beta_{7} - 64 \beta_{8} - 158 \beta_{9} - 162 \beta_{10} - 138 \beta_{11} ) q^{46} + ( -1688259 - 2650 \beta_{1} + 1933 \beta_{2} + 169 \beta_{3} + 1593 \beta_{4} + 45 \beta_{5} ) q^{47} + ( 16384 \beta_{6} - 16384 \beta_{7} ) q^{48} + ( -1622284 - 2564 \beta_{1} - 3416 \beta_{2} - 175 \beta_{3} + 12 \beta_{4} + 21 \beta_{5} ) q^{49} + ( -14476 \beta_{6} + 70152 \beta_{7} + 128 \beta_{8} - 428 \beta_{9} + 558 \beta_{10} - 260 \beta_{11} ) q^{50} + ( 57903 \beta_{6} - 118647 \beta_{7} - 608 \beta_{9} + 626 \beta_{10} + 708 \beta_{11} ) q^{51} + ( 7296 \beta_{6} - 34432 \beta_{7} + 128 \beta_{8} - 384 \beta_{9} + 512 \beta_{10} ) q^{52} + ( 21848 \beta_{6} + 60699 \beta_{7} + 716 \beta_{8} + 383 \beta_{9} + 417 \beta_{10} - 360 \beta_{11} ) q^{53} + ( 1411376 + 650 \beta_{1} - 1604 \beta_{2} + 702 \beta_{3} - 1220 \beta_{4} - 424 \beta_{5} ) q^{54} + ( -1238469 + 4378 \beta_{1} - 1719 \beta_{2} + 135 \beta_{3} + 2119 \beta_{4} - 213 \beta_{5} ) q^{55} + ( -10496 \beta_{6} + 56192 \beta_{7} + 256 \beta_{8} - 128 \beta_{11} ) q^{56} + ( 1201239 - 2196 \beta_{1} + 4497 \beta_{2} + 234 \beta_{3} + 1206 \beta_{4} - 513 \beta_{5} + 18495 \beta_{6} + 277822 \beta_{7} + 463 \beta_{8} + 704 \beta_{9} - 1459 \beta_{10} - 90 \beta_{11} ) q^{57} + ( -460542 + 2202 \beta_{1} + 1152 \beta_{2} + 864 \beta_{3} - 192 \beta_{4} + 192 \beta_{5} ) q^{58} + ( -109858 \beta_{6} + 13197 \beta_{7} + 521 \beta_{8} - 976 \beta_{9} - 939 \beta_{10} - 18 \beta_{11} ) q^{59} + ( -1920 \beta_{6} + 6144 \beta_{7} + 384 \beta_{8} - 512 \beta_{9} + 896 \beta_{10} ) q^{60} + ( -3451227 + 1778 \beta_{1} + 7603 \beta_{2} - 688 \beta_{3} - 848 \beta_{4} - 108 \beta_{5} ) q^{61} + ( 2688514 + 3944 \beta_{1} - 2956 \beta_{2} + 346 \beta_{3} - 3084 \beta_{4} - 504 \beta_{5} ) q^{62} + ( 7018301 + 1310 \beta_{1} + 2443 \beta_{2} - 693 \beta_{3} - 587 \beta_{4} - 1057 \beta_{5} ) q^{63} -2097152 q^{64} + ( 134500 \beta_{6} - 745764 \beta_{7} - 1046 \beta_{8} - 86 \beta_{9} - 3396 \beta_{10} + 360 \beta_{11} ) q^{65} + ( 1193042 + 2208 \beta_{1} + 7012 \beta_{2} - 1262 \beta_{3} - 5020 \beta_{4} - 536 \beta_{5} ) q^{66} + ( 21781 \beta_{6} + 416315 \beta_{7} - 870 \beta_{8} + 514 \beta_{9} + 5000 \beta_{10} + 900 \beta_{11} ) q^{67} + ( -2888832 + 2048 \beta_{1} - 512 \beta_{2} + 1152 \beta_{3} + 3584 \beta_{4} - 128 \beta_{5} ) q^{68} + ( -175941 \beta_{6} + 845073 \beta_{7} - 537 \beta_{8} + 595 \beta_{9} - 1030 \beta_{10} + 96 \beta_{11} ) q^{69} + ( 55344 \beta_{6} - 91849 \beta_{7} - 1370 \beta_{8} + 1302 \beta_{9} - 2926 \beta_{10} - 225 \beta_{11} ) q^{70} + ( 55658 \beta_{6} - 570954 \beta_{7} - 1282 \beta_{8} - 58 \beta_{9} + 5160 \beta_{10} - 516 \beta_{11} ) q^{71} + ( 10496 \beta_{6} + 167808 \beta_{7} + 512 \beta_{8} + 768 \beta_{9} - 256 \beta_{10} ) q^{72} + ( 7341843 - 40438 \beta_{1} - 6790 \beta_{2} - 1247 \beta_{3} - 2352 \beta_{4} + 3579 \beta_{5} ) q^{73} + ( -175162 - 8 \beta_{1} - 9380 \beta_{2} - 882 \beta_{3} - 2084 \beta_{4} + 3992 \beta_{5} ) q^{74} + ( -163310 \beta_{6} - 980663 \beta_{7} - 2641 \beta_{8} + 1366 \beta_{9} - 6443 \beta_{10} + 630 \beta_{11} ) q^{75} + ( -439040 - 9984 \beta_{1} + 1408 \beta_{2} - 1408 \beta_{3} + 3968 \beta_{4} + 21632 \beta_{6} + 87424 \beta_{7} + 384 \beta_{8} - 640 \beta_{9} - 128 \beta_{10} - 768 \beta_{11} ) q^{76} + ( -6569933 + 37098 \beta_{1} - 21749 \beta_{2} - 756 \beta_{3} - 576 \beta_{4} + 2604 \beta_{5} ) q^{77} + ( -179790 \beta_{6} + 449592 \beta_{7} - 900 \beta_{8} + 2718 \beta_{9} - 5922 \beta_{10} - 660 \beta_{11} ) q^{78} + ( -260753 \beta_{6} - 578942 \beta_{7} + 611 \beta_{8} + 2260 \beta_{9} + 4735 \beta_{10} - 2484 \beta_{11} ) q^{79} + ( 770048 - 16384 \beta_{2} ) q^{80} + ( -8431877 + 33496 \beta_{1} + 23570 \beta_{2} - 1044 \beta_{3} - 448 \beta_{4} + 2794 \beta_{5} ) q^{81} + ( -4144934 + 15920 \beta_{1} + 14420 \beta_{2} - 2870 \beta_{3} + 8148 \beta_{4} + 3528 \beta_{5} ) q^{82} + ( -4656328 - 32862 \beta_{1} + 8162 \beta_{2} + 3148 \beta_{3} + 3154 \beta_{4} + 3584 \beta_{5} ) q^{83} + ( 152832 \beta_{6} - 673408 \beta_{7} + 512 \beta_{8} - 896 \beta_{9} - 896 \beta_{10} + 1536 \beta_{11} ) q^{84} + ( 2134697 + 9284 \beta_{1} - 37895 \beta_{2} + 3146 \beta_{3} + 1276 \beta_{4} + 2046 \beta_{5} ) q^{85} + ( -57524 \beta_{6} + 623825 \beta_{7} - 1574 \beta_{8} - 122 \beta_{9} + 6678 \beta_{10} + 405 \beta_{11} ) q^{86} + ( 9922773 - 29133 \beta_{1} + 36879 \beta_{2} - 3999 \beta_{3} - 16440 \beta_{4} - 2649 \beta_{5} ) q^{87} + ( 22528 \beta_{6} + 138112 \beta_{7} + 768 \beta_{8} - 1280 \beta_{9} + 1792 \beta_{10} + 1920 \beta_{11} ) q^{88} + ( 304979 \beta_{6} + 1283328 \beta_{7} - 1915 \beta_{8} + 2228 \beta_{9} - 6495 \beta_{10} - 5454 \beta_{11} ) q^{89} + ( -104120 \beta_{6} + 167559 \beta_{7} - 824 \beta_{8} + 3408 \beta_{9} + 3898 \beta_{10} - 180 \beta_{11} ) q^{90} + ( 119836 \beta_{6} + 379691 \beta_{7} + 1515 \beta_{8} - 6482 \beta_{9} + 14399 \beta_{10} + 1542 \beta_{11} ) q^{91} + ( 8801408 - 25728 \beta_{1} - 7552 \beta_{2} + 1408 \beta_{3} - 1280 \beta_{4} - 5248 \beta_{5} ) q^{92} + ( 8736990 + 109728 \beta_{1} + 33342 \beta_{2} + 3348 \beta_{3} - 10032 \beta_{4} - 7722 \beta_{5} ) q^{93} + ( -137096 \beta_{6} - 1712815 \beta_{7} - 2510 \beta_{8} - 1622 \beta_{9} - 3190 \beta_{10} + 2517 \beta_{11} ) q^{94} + ( 6814487 - 66978 \beta_{1} - 26821 \beta_{2} + 2273 \beta_{3} - 23143 \beta_{4} - 1805 \beta_{5} + 271108 \beta_{6} - 121212 \beta_{7} - 2228 \beta_{8} - 1100 \beta_{9} - 13938 \beta_{10} - 1320 \beta_{11} ) q^{95} + ( 2457600 - 32768 \beta_{1} ) q^{96} + ( -294998 \beta_{6} + 1610242 \beta_{7} + 3200 \beta_{8} - 44 \beta_{9} - 22376 \beta_{10} - 1260 \beta_{11} ) q^{97} + ( -167170 \beta_{6} - 1651584 \beta_{7} - 724 \beta_{8} + 1274 \beta_{9} + 6132 \beta_{10} - 1080 \beta_{11} ) q^{98} + ( -7164160 + 138812 \beta_{1} - 26159 \beta_{2} + 3696 \beta_{3} + 2227 \beta_{4} + 3179 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 1536q^{4} + 558q^{5} + 1792q^{6} - 5422q^{7} - 15592q^{9} + O(q^{10}) \) \( 12q - 1536q^{4} + 558q^{5} + 1792q^{6} - 5422q^{7} - 15592q^{9} - 12546q^{11} + 196608q^{16} + 270810q^{17} + 41512q^{19} - 71424q^{20} - 823956q^{23} - 229376q^{24} + 865538q^{25} - 431616q^{26} + 694016q^{28} + 71168q^{30} - 1194378q^{35} + 1995776q^{36} + 998784q^{38} + 5786100q^{39} - 8383744q^{42} + 7586646q^{43} + 1605888q^{44} + 2226046q^{45} - 20260530q^{47} - 19498842q^{49} + 16933888q^{54} - 14858554q^{55} + 14430564q^{57} - 5506560q^{58} - 41363266q^{61} + 32266752q^{62} + 84235798q^{63} - 25165824q^{64} + 14371328q^{66} - 34663680q^{68} + 87906498q^{73} - 2149632q^{74} - 5313536q^{76} - 78817962q^{77} + 9142272q^{80} - 100904812q^{81} - 49609728q^{82} - 55944960q^{83} + 25440254q^{85} + 119189604q^{87} + 105466368q^{92} + 105500856q^{93} + 81396774q^{95} + 29360128q^{96} - 85554938q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 46118 x^{10} + 738386961 x^{8} + 5214446299656 x^{6} + 17370647163698184 x^{4} + 24830681474333400768 x^{2} + 9218779084612644462864\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-627859595 \nu^{10} - 26324430503956 \nu^{8} - 353286216457517619 \nu^{6} - 1793134315636569559098 \nu^{4} - 3384671207395419974567700 \nu^{2} - 1373838894169787985475717944\)\()/ \)\(30\!\cdots\!08\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-24121866715514597 \nu^{10} - 1017763065922454486260 \nu^{8} - 13814942804320117626042957 \nu^{6} - 71508680718346538969766820686 \nu^{4} - 137813144295779199390399764575500 \nu^{2} - 55983272825769961557419137351953576\)\()/ \)\(50\!\cdots\!60\)\( \)
\(\beta_{3}\)\(=\)\((\)\(35029922688763993 \nu^{10} + 1456040578930597931300 \nu^{8} + 19223880793457955507632673 \nu^{6} + 94663934306522838531873005094 \nu^{4} + 172623436452128678415220175093820 \nu^{2} + 70177834995932192859188014024824264\)\()/ \)\(21\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-618986366251557679 \nu^{10} - 26065639628469526607900 \nu^{8} - 352601062796314536167993319 \nu^{6} - 1815096393742013789389000746282 \nu^{4} - 3483662457189746493031161575476260 \nu^{2} - 1425882270081404839137844752911455992\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(86539462063981597 \nu^{10} + 3649971332118746402900 \nu^{8} + 49517157755370789665038917 \nu^{6} + 256209719211152744199099271326 \nu^{4} + 495107109128898936934582066759980 \nu^{2} + 205627696128311115390625622010929256\)\()/ \)\(28\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(13946998708921 \nu^{11} + 584451732924588668 \nu^{9} + 7834806036614273926833 \nu^{7} + 39664871910263669791484574 \nu^{5} + 74464449231549988985860204380 \nu^{3} + 41025489550497873181827514943784 \nu\)\()/ \)\(11\!\cdots\!56\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-13946998708921 \nu^{11} - 584451732924588668 \nu^{9} - 7834806036614273926833 \nu^{7} - 39664871910263669791484574 \nu^{5} - 74464449231549988985860204380 \nu^{3} - 29571354492593489110032440914728 \nu\)\()/ \)\(14\!\cdots\!32\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-8367191740264433969 \nu^{11} - 569265796886210505271396 \nu^{9} - 13043198936030284433391396729 \nu^{7} - 115344541421528164908248156011542 \nu^{5} - 324807244519316701149246211086354012 \nu^{3} - 165069551945580035325154407503925815304 \nu\)\()/ \)\(13\!\cdots\!88\)\( \)
\(\beta_{9}\)\(=\)\((\)\(19647103015700413779857 \nu^{11} + 826158121195012989052943380 \nu^{9} + 11140647613003328885517461706297 \nu^{7} + 56906307870372630615163382493453126 \nu^{5} + 107053061910302206543189523066577769500 \nu^{3} + 39852758935436971475106252934886352276936 \nu\)\()/ \)\(59\!\cdots\!60\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-42657931362165516833629 \nu^{11} - 1796051477157354058847378900 \nu^{9} - 24289830659965640898513821643909 \nu^{7} - 125006671730092999818511359218758302 \nu^{5} - 240201662798490922205337951714577332780 \nu^{3} - 99733967972159372895159102619708304862312 \nu\)\()/ \)\(59\!\cdots\!60\)\( \)
\(\beta_{11}\)\(=\)\((\)\(64801682891250922204057 \nu^{11} + 2734016466856695724028659400 \nu^{9} + 37110870611229586637650549596777 \nu^{7} + 192145367348700068223448117559922906 \nu^{5} + 370777562231676662913796090387390289580 \nu^{3} + 150825559923347746278629395173856721714136 \nu\)\()/ \)\(37\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + 8 \beta_{6}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{5} + 4 \beta_{4} - 2 \beta_{2} - 13 \beta_{1} - 15367\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-120 \beta_{11} - 161 \beta_{10} + 15 \beta_{9} - 218 \beta_{8} - 96909 \beta_{7} - 56459 \beta_{6}\)\()/4\)
\(\nu^{4}\)\(=\)\(-19769 \beta_{5} - 47248 \beta_{4} + 2115 \beta_{3} + 37007 \beta_{2} + 197287 \beta_{1} + 108259783\)
\(\nu^{5}\)\(=\)\((\)\(2409450 \beta_{11} + 2214367 \beta_{10} + 476619 \beta_{9} + 5477746 \beta_{8} + 2937881826 \beta_{7} + 1006000693 \beta_{6}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(781327466 \beta_{5} + 1948624906 \beta_{4} - 159513309 \beta_{3} - 1389463040 \beta_{2} - 9545300437 \beta_{1} - 3853054598260\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-44028806610 \beta_{11} - 24374350019 \beta_{10} - 26067322407 \beta_{9} - 118950691874 \beta_{8} - 71468966199420 \beta_{7} - 19541503881929 \beta_{6}\)\()/4\)
\(\nu^{8}\)\(=\)\(-7899466534745 \beta_{5} - 19941863755858 \beta_{4} + 2150472839598 \beta_{3} + 12265474994369 \beta_{2} + 107834804471680 \beta_{1} + 37393690537545847\)
\(\nu^{9}\)\(=\)\((\)\(824936798643090 \beta_{11} + 209168343586807 \beta_{10} + 775901785287075 \beta_{9} + 2522233684836994 \beta_{8} + 1620840870923721192 \beta_{7} + 394084193457700285 \beta_{6}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(324902725272216098 \beta_{5} + 824068680906391942 \beta_{4} - 102652011517318191 \beta_{3} - 447286066701150692 \beta_{2} - 4728352058833047115 \beta_{1} - 1507483904910684245284\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-16047421964734657890 \beta_{11} - 510812645400404687 \beta_{10} - 19306427997933171075 \beta_{9} - 53288102208612841034 \beta_{8} - 35612874290211637582152 \beta_{7} - 8104703045671676003357 \beta_{6}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
145.414i
64.6816i
23.4825i
61.5968i
63.2768i
111.533i
111.533i
63.2768i
61.5968i
23.4825i
64.6816i
145.414i
11.3137i 132.686i −128.000 −12.7536 −1501.17 −1217.09 1448.15i −11044.5 144.290i
37.2 11.3137i 51.9537i −128.000 629.221 −587.789 2565.52 1448.15i 3861.81 7118.82i
37.3 11.3137i 10.7546i −128.000 −919.278 −121.674 −343.629 1448.15i 6445.34 10400.4i
37.4 11.3137i 74.3247i −128.000 −154.845 840.888 1585.07 1448.15i 1036.83 1751.87i
37.5 11.3137i 76.0047i −128.000 1155.75 859.895 −2869.50 1448.15i 784.279 13075.8i
37.6 11.3137i 124.261i −128.000 −419.091 1405.85 −2431.38 1448.15i −8879.72 4741.48i
37.7 11.3137i 124.261i −128.000 −419.091 1405.85 −2431.38 1448.15i −8879.72 4741.48i
37.8 11.3137i 76.0047i −128.000 1155.75 859.895 −2869.50 1448.15i 784.279 13075.8i
37.9 11.3137i 74.3247i −128.000 −154.845 840.888 1585.07 1448.15i 1036.83 1751.87i
37.10 11.3137i 10.7546i −128.000 −919.278 −121.674 −343.629 1448.15i 6445.34 10400.4i
37.11 11.3137i 51.9537i −128.000 629.221 −587.789 2565.52 1448.15i 3861.81 7118.82i
37.12 11.3137i 132.686i −128.000 −12.7536 −1501.17 −1217.09 1448.15i −11044.5 144.290i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.9.b.a 12
3.b odd 2 1 342.9.d.a 12
4.b odd 2 1 304.9.e.e 12
19.b odd 2 1 inner 38.9.b.a 12
57.d even 2 1 342.9.d.a 12
76.d even 2 1 304.9.e.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.9.b.a 12 1.a even 1 1 trivial
38.9.b.a 12 19.b odd 2 1 inner
304.9.e.e 12 4.b odd 2 1
304.9.e.e 12 76.d even 2 1
342.9.d.a 12 3.b odd 2 1
342.9.d.a 12 57.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 128 T^{2} )^{6} \)
$3$ \( 1 - 31570 T^{2} + 549133431 T^{4} - 6843854959362 T^{6} + 68397361434579483 T^{8} - \)\(57\!\cdots\!56\)\( T^{10} + \)\(40\!\cdots\!66\)\( T^{12} - \)\(24\!\cdots\!76\)\( T^{14} + \)\(12\!\cdots\!03\)\( T^{16} - \)\(54\!\cdots\!82\)\( T^{18} + \)\(18\!\cdots\!11\)\( T^{20} - \)\(46\!\cdots\!70\)\( T^{22} + \)\(63\!\cdots\!21\)\( T^{24} \)
$5$ \( ( 1 - 279 T + 994411 T^{2} - 474127296 T^{3} + 505992934361 T^{4} - 295237796547405 T^{5} + 211599402340940350 T^{6} - \)\(11\!\cdots\!25\)\( T^{7} + \)\(77\!\cdots\!25\)\( T^{8} - \)\(28\!\cdots\!00\)\( T^{9} + \)\(23\!\cdots\!75\)\( T^{10} - \)\(25\!\cdots\!75\)\( T^{11} + \)\(35\!\cdots\!25\)\( T^{12} )^{2} \)
$7$ \( ( 1 + 2711 T + 25843874 T^{2} + 54118130997 T^{3} + 309079144016020 T^{4} + 526652853450386131 T^{5} + \)\(22\!\cdots\!64\)\( T^{6} + \)\(30\!\cdots\!31\)\( T^{7} + \)\(10\!\cdots\!20\)\( T^{8} + \)\(10\!\cdots\!97\)\( T^{9} + \)\(28\!\cdots\!74\)\( T^{10} + \)\(17\!\cdots\!11\)\( T^{11} + \)\(36\!\cdots\!01\)\( T^{12} )^{2} \)
$11$ \( ( 1 + 6273 T + 702136351 T^{2} + 6228795449472 T^{3} + 237818010620983361 T^{4} + \)\(25\!\cdots\!51\)\( T^{5} + \)\(56\!\cdots\!42\)\( T^{6} + \)\(55\!\cdots\!31\)\( T^{7} + \)\(10\!\cdots\!21\)\( T^{8} + \)\(61\!\cdots\!52\)\( T^{9} + \)\(14\!\cdots\!71\)\( T^{10} + \)\(28\!\cdots\!73\)\( T^{11} + \)\(97\!\cdots\!81\)\( T^{12} )^{2} \)
$13$ \( 1 - 4839667362 T^{2} + 10528682202260514135 T^{4} - \)\(14\!\cdots\!30\)\( T^{6} + \)\(14\!\cdots\!71\)\( T^{8} - \)\(12\!\cdots\!80\)\( T^{10} + \)\(10\!\cdots\!94\)\( T^{12} - \)\(86\!\cdots\!80\)\( T^{14} + \)\(63\!\cdots\!51\)\( T^{16} - \)\(41\!\cdots\!30\)\( T^{18} + \)\(20\!\cdots\!35\)\( T^{20} - \)\(63\!\cdots\!62\)\( T^{22} + \)\(86\!\cdots\!41\)\( T^{24} \)
$17$ \( ( 1 - 135405 T + 26682056806 T^{2} - 2582482136995467 T^{3} + \)\(23\!\cdots\!08\)\( T^{4} - \)\(20\!\cdots\!01\)\( T^{5} + \)\(13\!\cdots\!96\)\( T^{6} - \)\(14\!\cdots\!41\)\( T^{7} + \)\(11\!\cdots\!48\)\( T^{8} - \)\(87\!\cdots\!07\)\( T^{9} + \)\(63\!\cdots\!66\)\( T^{10} - \)\(22\!\cdots\!05\)\( T^{11} + \)\(11\!\cdots\!41\)\( T^{12} )^{2} \)
$19$ \( 1 - 41512 T - 18283117042 T^{2} + 4457210163533304 T^{3} - \)\(38\!\cdots\!73\)\( T^{4} - \)\(64\!\cdots\!28\)\( T^{5} + \)\(14\!\cdots\!60\)\( T^{6} - \)\(10\!\cdots\!48\)\( T^{7} - \)\(11\!\cdots\!13\)\( T^{8} + \)\(21\!\cdots\!84\)\( T^{9} - \)\(15\!\cdots\!62\)\( T^{10} - \)\(58\!\cdots\!12\)\( T^{11} + \)\(23\!\cdots\!41\)\( T^{12} \)
$23$ \( ( 1 + 411978 T + 378210302887 T^{2} + 107714123506523250 T^{3} + \)\(59\!\cdots\!23\)\( T^{4} + \)\(13\!\cdots\!00\)\( T^{5} + \)\(56\!\cdots\!34\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} + \)\(36\!\cdots\!03\)\( T^{8} + \)\(51\!\cdots\!50\)\( T^{9} + \)\(14\!\cdots\!27\)\( T^{10} + \)\(12\!\cdots\!78\)\( T^{11} + \)\(23\!\cdots\!81\)\( T^{12} )^{2} \)
$29$ \( 1 - 3871047956610 T^{2} + \)\(74\!\cdots\!87\)\( T^{4} - \)\(92\!\cdots\!78\)\( T^{6} + \)\(84\!\cdots\!03\)\( T^{8} - \)\(59\!\cdots\!28\)\( T^{10} + \)\(33\!\cdots\!90\)\( T^{12} - \)\(14\!\cdots\!88\)\( T^{14} + \)\(52\!\cdots\!23\)\( T^{16} - \)\(14\!\cdots\!58\)\( T^{18} + \)\(29\!\cdots\!47\)\( T^{20} - \)\(37\!\cdots\!10\)\( T^{22} + \)\(24\!\cdots\!21\)\( T^{24} \)
$31$ \( 1 - 7689804955764 T^{2} + \)\(28\!\cdots\!10\)\( T^{4} - \)\(65\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!91\)\( T^{8} - \)\(13\!\cdots\!40\)\( T^{10} + \)\(13\!\cdots\!04\)\( T^{12} - \)\(98\!\cdots\!40\)\( T^{14} + \)\(57\!\cdots\!51\)\( T^{16} - \)\(25\!\cdots\!20\)\( T^{18} + \)\(78\!\cdots\!10\)\( T^{20} - \)\(15\!\cdots\!64\)\( T^{22} + \)\(14\!\cdots\!81\)\( T^{24} \)
$37$ \( 1 - 20635084132020 T^{2} + \)\(23\!\cdots\!02\)\( T^{4} - \)\(18\!\cdots\!04\)\( T^{6} + \)\(11\!\cdots\!43\)\( T^{8} - \)\(54\!\cdots\!72\)\( T^{10} + \)\(21\!\cdots\!44\)\( T^{12} - \)\(67\!\cdots\!52\)\( T^{14} + \)\(17\!\cdots\!83\)\( T^{16} - \)\(35\!\cdots\!84\)\( T^{18} + \)\(54\!\cdots\!22\)\( T^{20} - \)\(58\!\cdots\!20\)\( T^{22} + \)\(35\!\cdots\!41\)\( T^{24} \)
$41$ \( 1 - 53751573622644 T^{2} + \)\(15\!\cdots\!42\)\( T^{4} - \)\(29\!\cdots\!08\)\( T^{6} + \)\(43\!\cdots\!67\)\( T^{8} - \)\(48\!\cdots\!64\)\( T^{10} + \)\(43\!\cdots\!12\)\( T^{12} - \)\(31\!\cdots\!24\)\( T^{14} + \)\(17\!\cdots\!27\)\( T^{16} - \)\(77\!\cdots\!68\)\( T^{18} + \)\(25\!\cdots\!62\)\( T^{20} - \)\(56\!\cdots\!44\)\( T^{22} + \)\(67\!\cdots\!41\)\( T^{24} \)
$43$ \( ( 1 - 3793323 T + 55073885573535 T^{2} - \)\(18\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!21\)\( T^{4} - \)\(38\!\cdots\!25\)\( T^{5} + \)\(21\!\cdots\!74\)\( T^{6} - \)\(45\!\cdots\!25\)\( T^{7} + \)\(19\!\cdots\!21\)\( T^{8} - \)\(29\!\cdots\!80\)\( T^{9} + \)\(10\!\cdots\!35\)\( T^{10} - \)\(82\!\cdots\!23\)\( T^{11} + \)\(25\!\cdots\!01\)\( T^{12} )^{2} \)
$47$ \( ( 1 + 10130265 T + 141280852430215 T^{2} + \)\(94\!\cdots\!68\)\( T^{3} + \)\(77\!\cdots\!65\)\( T^{4} + \)\(39\!\cdots\!79\)\( T^{5} + \)\(23\!\cdots\!02\)\( T^{6} + \)\(93\!\cdots\!19\)\( T^{7} + \)\(43\!\cdots\!65\)\( T^{8} + \)\(12\!\cdots\!08\)\( T^{9} + \)\(45\!\cdots\!15\)\( T^{10} + \)\(77\!\cdots\!65\)\( T^{11} + \)\(18\!\cdots\!61\)\( T^{12} )^{2} \)
$53$ \( 1 - 225146627585250 T^{2} + \)\(33\!\cdots\!23\)\( T^{4} - \)\(33\!\cdots\!30\)\( T^{6} + \)\(28\!\cdots\!83\)\( T^{8} - \)\(20\!\cdots\!40\)\( T^{10} + \)\(13\!\cdots\!22\)\( T^{12} - \)\(79\!\cdots\!40\)\( T^{14} + \)\(42\!\cdots\!03\)\( T^{16} - \)\(19\!\cdots\!30\)\( T^{18} + \)\(74\!\cdots\!63\)\( T^{20} - \)\(19\!\cdots\!50\)\( T^{22} + \)\(33\!\cdots\!21\)\( T^{24} \)
$59$ \( 1 - 660189424136946 T^{2} + \)\(25\!\cdots\!27\)\( T^{4} - \)\(74\!\cdots\!78\)\( T^{6} + \)\(16\!\cdots\!83\)\( T^{8} - \)\(31\!\cdots\!28\)\( T^{10} + \)\(50\!\cdots\!62\)\( T^{12} - \)\(68\!\cdots\!48\)\( T^{14} + \)\(78\!\cdots\!23\)\( T^{16} - \)\(74\!\cdots\!38\)\( T^{18} + \)\(56\!\cdots\!47\)\( T^{20} - \)\(30\!\cdots\!46\)\( T^{22} + \)\(10\!\cdots\!41\)\( T^{24} \)
$61$ \( ( 1 + 20681633 T + 1161765037087475 T^{2} + \)\(18\!\cdots\!12\)\( T^{3} + \)\(55\!\cdots\!97\)\( T^{4} + \)\(67\!\cdots\!71\)\( T^{5} + \)\(14\!\cdots\!46\)\( T^{6} + \)\(12\!\cdots\!51\)\( T^{7} + \)\(20\!\cdots\!17\)\( T^{8} + \)\(12\!\cdots\!92\)\( T^{9} + \)\(15\!\cdots\!75\)\( T^{10} + \)\(53\!\cdots\!33\)\( T^{11} + \)\(49\!\cdots\!81\)\( T^{12} )^{2} \)
$67$ \( 1 - 1755708948029538 T^{2} + \)\(16\!\cdots\!27\)\( T^{4} - \)\(11\!\cdots\!50\)\( T^{6} + \)\(66\!\cdots\!59\)\( T^{8} - \)\(34\!\cdots\!60\)\( T^{10} + \)\(15\!\cdots\!46\)\( T^{12} - \)\(56\!\cdots\!60\)\( T^{14} + \)\(18\!\cdots\!99\)\( T^{16} - \)\(51\!\cdots\!50\)\( T^{18} + \)\(12\!\cdots\!67\)\( T^{20} - \)\(21\!\cdots\!38\)\( T^{22} + \)\(20\!\cdots\!81\)\( T^{24} \)
$71$ \( 1 - 3698562871481892 T^{2} + \)\(77\!\cdots\!38\)\( T^{4} - \)\(11\!\cdots\!36\)\( T^{6} + \)\(12\!\cdots\!67\)\( T^{8} - \)\(10\!\cdots\!68\)\( T^{10} + \)\(78\!\cdots\!60\)\( T^{12} - \)\(45\!\cdots\!28\)\( T^{14} + \)\(21\!\cdots\!47\)\( T^{16} - \)\(81\!\cdots\!96\)\( T^{18} + \)\(23\!\cdots\!78\)\( T^{20} - \)\(46\!\cdots\!92\)\( T^{22} + \)\(52\!\cdots\!21\)\( T^{24} \)
$73$ \( ( 1 - 43953249 T + 3270766908915330 T^{2} - \)\(96\!\cdots\!35\)\( T^{3} + \)\(43\!\cdots\!36\)\( T^{4} - \)\(10\!\cdots\!85\)\( T^{5} + \)\(38\!\cdots\!04\)\( T^{6} - \)\(80\!\cdots\!85\)\( T^{7} + \)\(28\!\cdots\!96\)\( T^{8} - \)\(50\!\cdots\!35\)\( T^{9} + \)\(13\!\cdots\!30\)\( T^{10} - \)\(14\!\cdots\!49\)\( T^{11} + \)\(27\!\cdots\!81\)\( T^{12} )^{2} \)
$79$ \( 1 - 5922531389848308 T^{2} + \)\(24\!\cdots\!02\)\( T^{4} - \)\(74\!\cdots\!32\)\( T^{6} + \)\(17\!\cdots\!43\)\( T^{8} - \)\(35\!\cdots\!12\)\( T^{10} + \)\(59\!\cdots\!72\)\( T^{12} - \)\(82\!\cdots\!52\)\( T^{14} + \)\(95\!\cdots\!63\)\( T^{16} - \)\(90\!\cdots\!52\)\( T^{18} + \)\(68\!\cdots\!62\)\( T^{20} - \)\(38\!\cdots\!08\)\( T^{22} + \)\(14\!\cdots\!21\)\( T^{24} \)
$83$ \( ( 1 + 27972480 T + 9462519986864926 T^{2} + \)\(22\!\cdots\!08\)\( T^{3} + \)\(44\!\cdots\!63\)\( T^{4} + \)\(90\!\cdots\!04\)\( T^{5} + \)\(12\!\cdots\!16\)\( T^{6} + \)\(20\!\cdots\!64\)\( T^{7} + \)\(22\!\cdots\!03\)\( T^{8} + \)\(26\!\cdots\!68\)\( T^{9} + \)\(24\!\cdots\!86\)\( T^{10} + \)\(16\!\cdots\!80\)\( T^{11} + \)\(13\!\cdots\!41\)\( T^{12} )^{2} \)
$89$ \( 1 - 22501843998186012 T^{2} + \)\(26\!\cdots\!98\)\( T^{4} - \)\(20\!\cdots\!16\)\( T^{6} + \)\(12\!\cdots\!87\)\( T^{8} - \)\(62\!\cdots\!68\)\( T^{10} + \)\(26\!\cdots\!60\)\( T^{12} - \)\(96\!\cdots\!48\)\( T^{14} + \)\(30\!\cdots\!27\)\( T^{16} - \)\(77\!\cdots\!96\)\( T^{18} + \)\(15\!\cdots\!18\)\( T^{20} - \)\(20\!\cdots\!12\)\( T^{22} + \)\(13\!\cdots\!61\)\( T^{24} \)
$97$ \( 1 - 38837432167990116 T^{2} + \)\(83\!\cdots\!22\)\( T^{4} - \)\(13\!\cdots\!96\)\( T^{6} + \)\(16\!\cdots\!15\)\( T^{8} - \)\(16\!\cdots\!28\)\( T^{10} + \)\(14\!\cdots\!48\)\( T^{12} - \)\(10\!\cdots\!88\)\( T^{14} + \)\(61\!\cdots\!15\)\( T^{16} - \)\(30\!\cdots\!56\)\( T^{18} + \)\(11\!\cdots\!82\)\( T^{20} - \)\(33\!\cdots\!16\)\( T^{22} + \)\(53\!\cdots\!21\)\( T^{24} \)
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