Properties

Label 1331.1.d.a
Level $1331$
Weight $1$
Character orbit 1331.d
Analytic conductor $0.664$
Analytic rank $0$
Dimension $20$
Projective image $D_{11}$
CM discriminant -11
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1331 = 11^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1331.d (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.664255531815\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{10})\)
Coefficient field: 20.0.1402274470934209014892578125.1
Defining polynomial: \(x^{20} - x^{19} + 5 x^{18} - 6 x^{17} + 20 x^{16} - 11 x^{15} + 59 x^{14} - 30 x^{13} + 179 x^{12} - 109 x^{11} + 260 x^{10} - 128 x^{9} + 334 x^{8} + 82 x^{7} + 199 x^{6} + 22 x^{5} + 146 x^{4} - 41 x^{3} + 12 x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{11}\)
Projective field: Galois closure of 11.1.4177248169415651.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{13} q^{3} + \beta_{9} q^{4} + ( \beta_{8} + \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{5} + ( \beta_{17} + \beta_{19} ) q^{9} +O(q^{10})\) \( q -\beta_{13} q^{3} + \beta_{9} q^{4} + ( \beta_{8} + \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{5} + ( \beta_{17} + \beta_{19} ) q^{9} + \beta_{3} q^{12} + ( -\beta_{5} - \beta_{7} ) q^{15} -\beta_{11} q^{16} + ( \beta_{1} + \beta_{3} - \beta_{8} - \beta_{13} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{20} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{23} + ( -1 + \beta_{5} - \beta_{6} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{16} - \beta_{17} ) q^{25} + ( -\beta_{8} - \beta_{12} ) q^{27} + \beta_{18} q^{31} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} - \beta_{10} + \beta_{11} - \beta_{15} - \beta_{17} - \beta_{19} ) q^{36} + ( -\beta_{1} + \beta_{5} + \beta_{7} - \beta_{9} + \beta_{10} ) q^{37} + ( -1 - \beta_{3} - \beta_{6} ) q^{45} + ( \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} + \beta_{9} - \beta_{10} - \beta_{15} - \beta_{19} ) q^{47} + \beta_{1} q^{48} -\beta_{11} q^{49} + \beta_{16} q^{53} -\beta_{5} q^{59} + ( -\beta_{12} - \beta_{14} ) q^{60} + \beta_{17} q^{64} + \beta_{4} q^{67} + ( 1 - \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} + \beta_{16} + \beta_{17} + \beta_{19} ) q^{69} + \beta_{15} q^{71} + ( -\beta_{16} - \beta_{17} - \beta_{18} ) q^{75} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{80} + ( -\beta_{1} + \beta_{5} + \beta_{7} ) q^{81} + \beta_{6} q^{89} -\beta_{7} q^{92} + ( \beta_{8} + \beta_{11} + \beta_{12} - \beta_{15} ) q^{93} + \beta_{18} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + q^{3} - 5q^{4} + q^{5} - 4q^{9} + O(q^{10}) \) \( 20q + q^{3} - 5q^{4} + q^{5} - 4q^{9} - 4q^{12} + 2q^{15} - 5q^{16} + q^{20} - 4q^{23} - 4q^{25} + 2q^{27} + q^{31} - 4q^{36} + q^{37} - 12q^{45} + q^{47} + q^{48} - 5q^{49} + q^{53} + q^{59} + 2q^{60} - 5q^{64} - 4q^{67} + 2q^{69} + q^{71} + 3q^{75} + q^{80} - 3q^{81} - 4q^{89} + q^{92} + 2q^{93} + q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - x^{19} + 5 x^{18} - 6 x^{17} + 20 x^{16} - 11 x^{15} + 59 x^{14} - 30 x^{13} + 179 x^{12} - 109 x^{11} + 260 x^{10} - 128 x^{9} + 334 x^{8} + 82 x^{7} + 199 x^{6} + 22 x^{5} + 146 x^{4} - 41 x^{3} + 12 x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-126239 \nu^{19} + 631195 \nu^{18} - 757434 \nu^{17} + 2524780 \nu^{16} - 3335154 \nu^{15} + 7448101 \nu^{14} - 3787170 \nu^{13} + 22596781 \nu^{12} - 13760051 \nu^{11} + 73415636 \nu^{10} - 16158592 \nu^{9} + 42163826 \nu^{8} + 10351598 \nu^{7} + 25121561 \nu^{6} + 138931963 \nu^{5} + 18430894 \nu^{4} - 5175799 \nu^{3} + 1514868 \nu^{2} - 378717 \nu - 189837059\)\()/99123971\)
\(\beta_{3}\)\(=\)\((\)\(-171700 \nu^{19} + 858500 \nu^{18} - 1030200 \nu^{17} + 3434000 \nu^{16} - 4744285 \nu^{15} + 10130300 \nu^{14} - 5151000 \nu^{13} + 30734300 \nu^{12} - 18715300 \nu^{11} + 96284406 \nu^{10} - 21977600 \nu^{9} + 57347800 \nu^{8} + 14079400 \nu^{7} + 34168300 \nu^{6} + 261874495 \nu^{5} + 25068200 \nu^{4} - 7039700 \nu^{3} + 2060400 \nu^{2} - 515100 \nu - 27559952\)\()/99123971\)
\(\beta_{4}\)\(=\)\((\)\(387795 \nu^{19} - 1938975 \nu^{18} + 2326770 \nu^{17} - 7755900 \nu^{16} + 10354441 \nu^{15} - 22879905 \nu^{14} + 11633850 \nu^{13} - 69415305 \nu^{12} + 42269655 \nu^{11} - 222157770 \nu^{10} + 49637760 \nu^{9} - 129523530 \nu^{8} - 31799190 \nu^{7} - 77171205 \nu^{6} - 451938463 \nu^{5} - 56618070 \nu^{4} + 15899595 \nu^{3} - 4653540 \nu^{2} + 1163385 \nu + 165561080\)\()/99123971\)
\(\beta_{5}\)\(=\)\((\)\(-504956 \nu^{19} + 126239 \nu^{18} - 1767346 \nu^{17} + 810374 \nu^{16} - 6059472 \nu^{15} - 3660931 \nu^{14} - 18809611 \nu^{13} - 8836730 \nu^{12} - 59655585 \nu^{11} - 16663548 \nu^{10} - 26005234 \nu^{9} - 52515424 \nu^{8} - 35473159 \nu^{7} - 164053524 \nu^{6} - 21208152 \nu^{5} - 13255095 \nu^{4} + 3660931 \nu^{3} - 1136151 \nu^{2} + 289339747 \nu - 126239\)\()/99123971\)
\(\beta_{6}\)\(=\)\((\)\(451714 \nu^{19} - 2258570 \nu^{18} + 2710284 \nu^{17} - 9034280 \nu^{16} + 12429463 \nu^{15} - 26651126 \nu^{14} + 13551420 \nu^{13} - 80856806 \nu^{12} + 49236826 \nu^{11} - 253825447 \nu^{10} + 57819392 \nu^{9} - 150872476 \nu^{8} - 37040548 \nu^{7} - 89891086 \nu^{6} - 642688246 \nu^{5} - 65950244 \nu^{4} + 18520274 \nu^{3} - 5420568 \nu^{2} + 1355142 \nu + 80721890\)\()/99123971\)
\(\beta_{7}\)\(=\)\((\)\(2056136 \nu^{19} - 514034 \nu^{18} + 7196476 \nu^{17} - 3408915 \nu^{16} + 24673632 \nu^{15} + 14906986 \nu^{14} + 76591066 \nu^{13} + 35982380 \nu^{12} + 239543700 \nu^{11} + 67852488 \nu^{10} + 105891004 \nu^{9} + 213838144 \nu^{8} + 144443554 \nu^{7} + 693163192 \nu^{6} + 86357712 \nu^{5} + 53973570 \nu^{4} - 14906986 \nu^{3} + 4626306 \nu^{2} - 456064212 \nu + 514034\)\()/99123971\)
\(\beta_{8}\)\(=\)\((\)\(-2518527 \nu^{19} + 5037054 \nu^{18} - 14154492 \nu^{17} + 26864288 \nu^{16} - 61284157 \nu^{15} + 73037283 \nu^{14} - 159506710 \nu^{13} + 216611708 \nu^{12} - 476841112 \nu^{11} + 700150506 \nu^{10} - 779064352 \nu^{9} + 885681995 \nu^{8} - 986330048 \nu^{7} + 527211652 \nu^{6} - 14271653 \nu^{5} + 514619017 \nu^{4} - 145235057 \nu^{3} + 388221900 \nu^{2} - 10913617 \nu + 3358036\)\()/99123971\)
\(\beta_{9}\)\(=\)\((\)\(3358036 \nu^{19} - 839509 \nu^{18} + 11753126 \nu^{17} - 5993724 \nu^{16} + 40296432 \nu^{15} + 24345761 \nu^{14} + 125086841 \nu^{13} + 58765630 \nu^{12} + 384476736 \nu^{11} + 110815188 \nu^{10} + 172938854 \nu^{9} + 349235744 \nu^{8} + 235902029 \nu^{7} + 1261689000 \nu^{6} + 141037512 \nu^{5} + 88148445 \nu^{4} - 24345761 \nu^{3} + 7555581 \nu^{2} - 347925468 \nu + 839509\)\()/99123971\)
\(\beta_{10}\)\(=\)\((\)\(-4222416 \nu^{19} + 1055604 \nu^{18} - 14778456 \nu^{17} + 7281980 \nu^{16} - 50668992 \nu^{15} - 30612516 \nu^{14} - 157284996 \nu^{13} - 73892280 \nu^{12} - 486795745 \nu^{11} - 139339728 \nu^{10} - 217454424 \nu^{9} - 439131264 \nu^{8} - 296624724 \nu^{7} - 1494755873 \nu^{6} - 177341472 \nu^{5} - 110838420 \nu^{4} + 30612516 \nu^{3} - 9500436 \nu^{2} + 585698852 \nu - 1055604\)\()/99123971\)
\(\beta_{11}\)\(=\)\((\)\(6064056 \nu^{19} - 12128112 \nu^{18} + 34426370 \nu^{17} - 64683264 \nu^{16} + 147558696 \nu^{15} - 175857624 \nu^{14} + 384056880 \nu^{13} - 516359966 \nu^{12} + 1148127936 \nu^{11} - 1685807568 \nu^{10} + 1875814656 \nu^{9} - 2132526360 \nu^{8} + 2257494941 \nu^{7} - 1269409056 \nu^{6} + 34362984 \nu^{5} - 1239088776 \nu^{4} + 349693896 \nu^{3} - 1319988963 \nu^{2} + 26277576 \nu - 8085408\)\()/99123971\)
\(\beta_{12}\)\(=\)\((\)\(7040481 \nu^{19} - 14080962 \nu^{18} + 39652961 \nu^{17} - 75098464 \nu^{16} + 171318371 \nu^{15} - 204173949 \nu^{14} + 445897130 \nu^{13} - 604467030 \nu^{12} + 1332997736 \nu^{11} - 1957253718 \nu^{10} + 2177855456 \nu^{9} - 2475902485 \nu^{8} + 2719264949 \nu^{7} - 1473807356 \nu^{6} + 39896059 \nu^{5} - 1438604951 \nu^{4} + 406001071 \nu^{3} - 1128349048 \nu^{2} + 30508751 \nu - 9387308\)\()/99123971\)
\(\beta_{13}\)\(=\)\((\)\(8410883 \nu^{19} - 8284644 \nu^{18} + 41423220 \nu^{17} - 49707864 \nu^{16} + 165692880 \nu^{15} - 89184559 \nu^{14} + 488793996 \nu^{13} - 248539320 \nu^{12} + 1482951276 \nu^{11} - 903026196 \nu^{10} + 2113413944 \nu^{9} - 1060434432 \nu^{8} + 2767071096 \nu^{7} + 679340808 \nu^{6} + 1648644156 \nu^{5} + 46107463 \nu^{4} + 1209558024 \nu^{3} - 339670404 \nu^{2} + 99415728 \nu - 24853932\)\()/99123971\)
\(\beta_{14}\)\(=\)\((\)\(-10207293 \nu^{19} + 20414586 \nu^{18} - 57705477 \nu^{17} + 108877792 \nu^{16} - 248377463 \nu^{15} + 296011497 \nu^{14} - 646461890 \nu^{13} + 873483749 \nu^{12} - 1932580808 \nu^{11} + 2837627454 \nu^{10} - 3157455968 \nu^{9} + 3589564705 \nu^{8} - 3867782710 \nu^{7} + 2136726668 \nu^{6} - 57841327 \nu^{5} + 2085690203 \nu^{4} - 588620563 \nu^{3} + 1764716892 \nu^{2} - 44231603 \nu + 13609724\)\()/99123971\)
\(\beta_{15}\)\(=\)\((\)\(12128112 \nu^{19} - 24256224 \nu^{18} + 68852740 \nu^{17} - 129366528 \nu^{16} + 295117392 \nu^{15} - 351715248 \nu^{14} + 768113760 \nu^{13} - 1032719932 \nu^{12} + 2296255872 \nu^{11} - 3371615136 \nu^{10} + 3751629312 \nu^{9} - 4265052720 \nu^{8} + 4514989882 \nu^{7} - 2538818112 \nu^{6} + 68725968 \nu^{5} - 2478177552 \nu^{4} + 699387792 \nu^{3} - 2540853955 \nu^{2} + 52555152 \nu - 16170816\)\()/99123971\)
\(\beta_{16}\)\(=\)\((\)\(-18192168 \nu^{19} + 12318270 \nu^{18} - 84896784 \nu^{17} + 78832728 \nu^{16} - 327459024 \nu^{15} + 78832728 \nu^{14} - 1003314858 \nu^{13} + 187985736 \nu^{12} - 3074476392 \nu^{11} + 897480288 \nu^{10} - 4068981576 \nu^{9} + 696300711 \nu^{8} - 5299984944 \nu^{7} - 3517152480 \nu^{6} - 4117494024 \nu^{5} - 1606974840 \nu^{4} - 3114964030 \nu^{3} - 139473288 \nu^{2} + 30320280 \nu - 18192168\)\()/99123971\)
\(\beta_{17}\)\(=\)\((\)\(-24853932 \nu^{19} + 16443049 \nu^{18} - 115985016 \nu^{17} + 107700372 \nu^{16} - 447370776 \nu^{15} + 107700372 \nu^{14} - 1377197429 \nu^{13} + 256823964 \nu^{12} - 4200314508 \nu^{11} + 1226127312 \nu^{10} - 5558996124 \nu^{9} + 1067889352 \nu^{8} - 7240778856 \nu^{7} - 4805093520 \nu^{6} - 5625273276 \nu^{5} - 2195430660 \nu^{4} - 3674781535 \nu^{3} - 190546812 \nu^{2} + 41423220 \nu - 24853932\)\()/99123971\)
\(\beta_{18}\)\(=\)\((\)\(29941563 \nu^{19} - 20204673 \nu^{18} + 139727294 \nu^{17} - 129746773 \nu^{16} + 538948134 \nu^{15} - 129746773 \nu^{14} + 1652821401 \nu^{13} - 309396151 \nu^{12} + 5060124147 \nu^{11} - 1477117108 \nu^{10} + 6696929591 \nu^{9} - 1164345531 \nu^{8} + 8722975354 \nu^{7} + 5788702180 \nu^{6} + 6776773759 \nu^{5} + 2644838065 \nu^{4} + 4878693571 \nu^{3} + 229551983 \nu^{2} - 49902605 \nu + 29941563\)\()/99123971\)
\(\beta_{19}\)\(=\)\((\)\(47189337 \nu^{19} - 31324241 \nu^{18} + 220216906 \nu^{17} - 204487127 \nu^{16} + 849408066 \nu^{15} - 204487127 \nu^{14} + 2613338960 \nu^{13} - 487623149 \nu^{12} + 7974997953 \nu^{11} - 2328007292 \nu^{10} + 10554681709 \nu^{9} - 1990636674 \nu^{8} + 13747826846 \nu^{7} + 9123271820 \nu^{6} + 10680519941 \nu^{5} + 4168391435 \nu^{4} + 7064600777 \nu^{3} + 361784917 \nu^{2} - 78648895 \nu + 47189337\)\()/99123971\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{15} - 2 \beta_{11}\)
\(\nu^{3}\)\(=\)\(\beta_{16} + 3 \beta_{13} + 3 \beta_{8} - 3 \beta_{3} - 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-3 \beta_{19} + \beta_{18} - 5 \beta_{17} + \beta_{16} - 3 \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + 5 \beta_{11} - 3 \beta_{10} - 2 \beta_{9} - 4 \beta_{7} - 4 \beta_{5} - \beta_{4} - \beta_{3} - 4 \beta_{2} + 3 \beta_{1} - 6\)
\(\nu^{5}\)\(=\)\(4 \beta_{6} - \beta_{4} + 9 \beta_{3} - \beta_{2} - 1\)
\(\nu^{6}\)\(=\)\(9 \beta_{10} + 5 \beta_{9} + 14 \beta_{7} + 15 \beta_{5} - 15 \beta_{1}\)
\(\nu^{7}\)\(=\)\(-\beta_{15} - 6 \beta_{14} - 20 \beta_{12} + \beta_{11} - 34 \beta_{8}\)
\(\nu^{8}\)\(=\)\(28 \beta_{19} - 20 \beta_{18} + 42 \beta_{17} - 27 \beta_{16} - 28 \beta_{13} - 28 \beta_{8} + 28 \beta_{3} + 28 \beta_{1}\)
\(\nu^{9}\)\(=\)\(8 \beta_{19} - 27 \beta_{18} + 9 \beta_{17} - 75 \beta_{16} + 8 \beta_{15} + 27 \beta_{14} - 117 \beta_{13} + 75 \beta_{12} - 9 \beta_{11} + 8 \beta_{10} + \beta_{9} + 35 \beta_{7} - 48 \beta_{6} + 83 \beta_{5} + 27 \beta_{4} + 27 \beta_{3} + 35 \beta_{2} - 8 \beta_{1} + 36\)
\(\nu^{10}\)\(=\)\(-35 \beta_{6} + 75 \beta_{4} - 44 \beta_{3} + 165 \beta_{2} + 207\)
\(\nu^{11}\)\(=\)\(-44 \beta_{10} - 9 \beta_{9} - 154 \beta_{7} - 319 \beta_{5} + 451 \beta_{1}\)
\(\nu^{12}\)\(=\)\(297 \beta_{15} + 275 \beta_{14} + 429 \beta_{12} - 429 \beta_{11} + 482 \beta_{8}\)
\(\nu^{13}\)\(=\)\(-207 \beta_{19} + 429 \beta_{18} - 260 \beta_{17} + 1001 \beta_{16} + 1430 \beta_{13} + 1430 \beta_{8} - 1430 \beta_{3} - 1430 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-1001 \beta_{19} + 1001 \beta_{18} - 1430 \beta_{17} + 1637 \beta_{16} - 1001 \beta_{15} - 1001 \beta_{14} + 1897 \beta_{13} - 1637 \beta_{12} + 1430 \beta_{11} - 1001 \beta_{10} - 429 \beta_{9} - 2002 \beta_{7} + 636 \beta_{6} - 2638 \beta_{5} - 1001 \beta_{4} - 1001 \beta_{3} - 2002 \beta_{2} + 1001 \beta_{1} - 2431\)
\(\nu^{15}\)\(=\)\(2002 \beta_{6} - 1637 \beta_{4} + 3432 \beta_{3} - 2533 \beta_{2} - 2793\)
\(\nu^{16}\)\(=\)\(3432 \beta_{10} + 1430 \beta_{9} + 7071 \beta_{7} + 9604 \beta_{5} - 10760 \beta_{1}\)
\(\nu^{17}\)\(=\)\(-3689 \beta_{15} - 6172 \beta_{14} - 13243 \beta_{12} + 4845 \beta_{11} - 18105 \beta_{8}\)
\(\nu^{18}\)\(=\)\(11933 \beta_{19} - 13243 \beta_{18} + 16795 \beta_{17} - 23104 \beta_{16} - 27949 \beta_{13} - 27949 \beta_{8} + 27949 \beta_{3} + 27949 \beta_{1}\)
\(\nu^{19}\)\(=\)\(14706 \beta_{19} - 23104 \beta_{18} + 19551 \beta_{17} - 48280 \beta_{16} + 14706 \beta_{15} + 23104 \beta_{14} - 65075 \beta_{13} + 48280 \beta_{12} - 19551 \beta_{11} + 14706 \beta_{10} + 4845 \beta_{9} + 37810 \beta_{7} - 25176 \beta_{6} + 62986 \beta_{5} + 23104 \beta_{4} + 23104 \beta_{3} + 37810 \beta_{2} - 14706 \beta_{1} + 42655\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1331\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.592999 1.82506i
−0.404726 1.24562i
−0.0879554 0.270699i
0.256741 + 0.790166i
0.519923 + 1.60016i
−1.36118 0.988953i
−0.672156 0.488350i
0.230270 + 0.167301i
1.05959 + 0.769835i
1.55249 + 1.12795i
−1.36118 + 0.988953i
−0.672156 + 0.488350i
0.230270 0.167301i
1.05959 0.769835i
1.55249 1.12795i
−0.592999 + 1.82506i
−0.404726 + 1.24562i
−0.0879554 + 0.270699i
0.256741 0.790166i
0.519923 1.60016i
0 −0.592999 + 1.82506i 0.309017 + 0.951057i −0.672156 + 0.488350i 0 0 0 −2.17019 1.57674i 0
161.2 0 −0.404726 + 1.24562i 0.309017 + 0.951057i 1.55249 1.12795i 0 0 0 −0.578747 0.420484i 0
161.3 0 −0.0879554 + 0.270699i 0.309017 + 0.951057i −1.36118 + 0.988953i 0 0 0 0.743475 + 0.540166i 0
161.4 0 0.256741 0.790166i 0.309017 + 0.951057i 0.230270 0.167301i 0 0 0 0.250570 + 0.182050i 0
161.5 0 0.519923 1.60016i 0.309017 + 0.951057i 1.05959 0.769835i 0 0 0 −1.48117 1.07613i 0
596.1 0 −1.36118 + 0.988953i −0.809017 0.587785i −0.404726 1.24562i 0 0 0 0.565758 1.74122i 0
596.2 0 −0.672156 + 0.488350i −0.809017 0.587785i −0.0879554 0.270699i 0 0 0 −0.0957092 + 0.294563i 0
596.3 0 0.230270 0.167301i −0.809017 0.587785i 0.519923 + 1.60016i 0 0 0 −0.283982 + 0.874008i 0
596.4 0 1.05959 0.769835i −0.809017 0.587785i −0.592999 1.82506i 0 0 0 0.221062 0.680358i 0
596.5 0 1.55249 1.12795i −0.809017 0.587785i 0.256741 + 0.790166i 0 0 0 0.828940 2.55122i 0
699.1 0 −1.36118 0.988953i −0.809017 + 0.587785i −0.404726 + 1.24562i 0 0 0 0.565758 + 1.74122i 0
699.2 0 −0.672156 0.488350i −0.809017 + 0.587785i −0.0879554 + 0.270699i 0 0 0 −0.0957092 0.294563i 0
699.3 0 0.230270 + 0.167301i −0.809017 + 0.587785i 0.519923 1.60016i 0 0 0 −0.283982 0.874008i 0
699.4 0 1.05959 + 0.769835i −0.809017 + 0.587785i −0.592999 + 1.82506i 0 0 0 0.221062 + 0.680358i 0
699.5 0 1.55249 + 1.12795i −0.809017 + 0.587785i 0.256741 0.790166i 0 0 0 0.828940 + 2.55122i 0
1207.1 0 −0.592999 1.82506i 0.309017 0.951057i −0.672156 0.488350i 0 0 0 −2.17019 + 1.57674i 0
1207.2 0 −0.404726 1.24562i 0.309017 0.951057i 1.55249 + 1.12795i 0 0 0 −0.578747 + 0.420484i 0
1207.3 0 −0.0879554 0.270699i 0.309017 0.951057i −1.36118 0.988953i 0 0 0 0.743475 0.540166i 0
1207.4 0 0.256741 + 0.790166i 0.309017 0.951057i 0.230270 + 0.167301i 0 0 0 0.250570 0.182050i 0
1207.5 0 0.519923 + 1.60016i 0.309017 0.951057i 1.05959 + 0.769835i 0 0 0 −1.48117 + 1.07613i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1207.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
11.c even 5 3 inner
11.d odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1331.1.d.a 20
11.b odd 2 1 CM 1331.1.d.a 20
11.c even 5 1 1331.1.b.a 5
11.c even 5 3 inner 1331.1.d.a 20
11.d odd 10 1 1331.1.b.a 5
11.d odd 10 3 inner 1331.1.d.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1331.1.b.a 5 11.c even 5 1
1331.1.b.a 5 11.d odd 10 1
1331.1.d.a 20 1.a even 1 1 trivial
1331.1.d.a 20 11.b odd 2 1 CM
1331.1.d.a 20 11.c even 5 3 inner
1331.1.d.a 20 11.d odd 10 3 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1331, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( 1 - 3 T + 12 T^{2} - 41 T^{3} + 146 T^{4} + 22 T^{5} + 199 T^{6} + 82 T^{7} + 334 T^{8} - 128 T^{9} + 260 T^{10} - 109 T^{11} + 179 T^{12} - 30 T^{13} + 59 T^{14} - 11 T^{15} + 20 T^{16} - 6 T^{17} + 5 T^{18} - T^{19} + T^{20} \)
$5$ \( 1 - 3 T + 12 T^{2} - 41 T^{3} + 146 T^{4} + 22 T^{5} + 199 T^{6} + 82 T^{7} + 334 T^{8} - 128 T^{9} + 260 T^{10} - 109 T^{11} + 179 T^{12} - 30 T^{13} + 59 T^{14} - 11 T^{15} + 20 T^{16} - 6 T^{17} + 5 T^{18} - T^{19} + T^{20} \)
$7$ \( T^{20} \)
$11$ \( T^{20} \)
$13$ \( T^{20} \)
$17$ \( T^{20} \)
$19$ \( T^{20} \)
$23$ \( ( 1 + 3 T - 3 T^{2} - 4 T^{3} + T^{4} + T^{5} )^{4} \)
$29$ \( T^{20} \)
$31$ \( 1 - 3 T + 12 T^{2} - 41 T^{3} + 146 T^{4} + 22 T^{5} + 199 T^{6} + 82 T^{7} + 334 T^{8} - 128 T^{9} + 260 T^{10} - 109 T^{11} + 179 T^{12} - 30 T^{13} + 59 T^{14} - 11 T^{15} + 20 T^{16} - 6 T^{17} + 5 T^{18} - T^{19} + T^{20} \)
$37$ \( 1 - 3 T + 12 T^{2} - 41 T^{3} + 146 T^{4} + 22 T^{5} + 199 T^{6} + 82 T^{7} + 334 T^{8} - 128 T^{9} + 260 T^{10} - 109 T^{11} + 179 T^{12} - 30 T^{13} + 59 T^{14} - 11 T^{15} + 20 T^{16} - 6 T^{17} + 5 T^{18} - T^{19} + T^{20} \)
$41$ \( T^{20} \)
$43$ \( T^{20} \)
$47$ \( 1 - 3 T + 12 T^{2} - 41 T^{3} + 146 T^{4} + 22 T^{5} + 199 T^{6} + 82 T^{7} + 334 T^{8} - 128 T^{9} + 260 T^{10} - 109 T^{11} + 179 T^{12} - 30 T^{13} + 59 T^{14} - 11 T^{15} + 20 T^{16} - 6 T^{17} + 5 T^{18} - T^{19} + T^{20} \)
$53$ \( 1 - 3 T + 12 T^{2} - 41 T^{3} + 146 T^{4} + 22 T^{5} + 199 T^{6} + 82 T^{7} + 334 T^{8} - 128 T^{9} + 260 T^{10} - 109 T^{11} + 179 T^{12} - 30 T^{13} + 59 T^{14} - 11 T^{15} + 20 T^{16} - 6 T^{17} + 5 T^{18} - T^{19} + T^{20} \)
$59$ \( 1 - 3 T + 12 T^{2} - 41 T^{3} + 146 T^{4} + 22 T^{5} + 199 T^{6} + 82 T^{7} + 334 T^{8} - 128 T^{9} + 260 T^{10} - 109 T^{11} + 179 T^{12} - 30 T^{13} + 59 T^{14} - 11 T^{15} + 20 T^{16} - 6 T^{17} + 5 T^{18} - T^{19} + T^{20} \)
$61$ \( T^{20} \)
$67$ \( ( 1 + 3 T - 3 T^{2} - 4 T^{3} + T^{4} + T^{5} )^{4} \)
$71$ \( 1 - 3 T + 12 T^{2} - 41 T^{3} + 146 T^{4} + 22 T^{5} + 199 T^{6} + 82 T^{7} + 334 T^{8} - 128 T^{9} + 260 T^{10} - 109 T^{11} + 179 T^{12} - 30 T^{13} + 59 T^{14} - 11 T^{15} + 20 T^{16} - 6 T^{17} + 5 T^{18} - T^{19} + T^{20} \)
$73$ \( T^{20} \)
$79$ \( T^{20} \)
$83$ \( T^{20} \)
$89$ \( ( 1 + 3 T - 3 T^{2} - 4 T^{3} + T^{4} + T^{5} )^{4} \)
$97$ \( 1 - 3 T + 12 T^{2} - 41 T^{3} + 146 T^{4} + 22 T^{5} + 199 T^{6} + 82 T^{7} + 334 T^{8} - 128 T^{9} + 260 T^{10} - 109 T^{11} + 179 T^{12} - 30 T^{13} + 59 T^{14} - 11 T^{15} + 20 T^{16} - 6 T^{17} + 5 T^{18} - T^{19} + T^{20} \)
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