Properties

Label 8046.2.a.r
Level 8046
Weight 2
Character orbit 8046.a
Self dual yes
Analytic conductor 64.248
Analytic rank 0
Dimension 14
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8046.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.2476334663\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta_{1} q^{5} -\beta_{11} q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + \beta_{1} q^{5} -\beta_{11} q^{7} + q^{8} + \beta_{1} q^{10} + ( \beta_{9} - \beta_{12} ) q^{11} + ( \beta_{4} - \beta_{11} ) q^{13} -\beta_{11} q^{14} + q^{16} + ( 1 - \beta_{3} ) q^{17} + ( 1 + \beta_{5} + \beta_{7} - \beta_{9} - \beta_{11} + \beta_{12} ) q^{19} + \beta_{1} q^{20} + ( \beta_{9} - \beta_{12} ) q^{22} + ( 2 + \beta_{2} - \beta_{12} ) q^{23} + ( 2 + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} - \beta_{12} ) q^{25} + ( \beta_{4} - \beta_{11} ) q^{26} -\beta_{11} q^{28} + ( 1 + \beta_{1} - \beta_{4} - \beta_{7} - \beta_{8} ) q^{29} + ( 1 - \beta_{7} + \beta_{9} - \beta_{10} ) q^{31} + q^{32} + ( 1 - \beta_{3} ) q^{34} + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{11} + \beta_{13} ) q^{35} + ( \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{37} + ( 1 + \beta_{5} + \beta_{7} - \beta_{9} - \beta_{11} + \beta_{12} ) q^{38} + \beta_{1} q^{40} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{10} + \beta_{12} - \beta_{13} ) q^{41} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{43} + ( \beta_{9} - \beta_{12} ) q^{44} + ( 2 + \beta_{2} - \beta_{12} ) q^{46} + ( 2 - \beta_{9} - \beta_{13} ) q^{47} + ( 3 - 2 \beta_{1} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{49} + ( 2 + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} - \beta_{12} ) q^{50} + ( \beta_{4} - \beta_{11} ) q^{52} + ( 1 - \beta_{2} - \beta_{5} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{53} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{55} -\beta_{11} q^{56} + ( 1 + \beta_{1} - \beta_{4} - \beta_{7} - \beta_{8} ) q^{58} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} + 2 \beta_{9} + \beta_{11} - \beta_{12} ) q^{59} + ( 2 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{10} + \beta_{12} - \beta_{13} ) q^{61} + ( 1 - \beta_{7} + \beta_{9} - \beta_{10} ) q^{62} + q^{64} + ( -3 + \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{65} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{7} - \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{67} + ( 1 - \beta_{3} ) q^{68} + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{11} + \beta_{13} ) q^{70} + ( 3 - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{12} ) q^{71} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} ) q^{73} + ( \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{74} + ( 1 + \beta_{5} + \beta_{7} - \beta_{9} - \beta_{11} + \beta_{12} ) q^{76} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{77} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{79} + \beta_{1} q^{80} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{10} + \beta_{12} - \beta_{13} ) q^{82} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{8} - \beta_{9} + 2 \beta_{12} - 2 \beta_{13} ) q^{83} + ( -3 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} - \beta_{12} + 2 \beta_{13} ) q^{85} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{86} + ( \beta_{9} - \beta_{12} ) q^{88} + ( \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} + 2 \beta_{12} - \beta_{13} ) q^{89} + ( 7 - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} + \beta_{13} ) q^{91} + ( 2 + \beta_{2} - \beta_{12} ) q^{92} + ( 2 - \beta_{9} - \beta_{13} ) q^{94} + ( 2 \beta_{1} - 2 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{11} - \beta_{13} ) q^{95} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{97} + ( 3 - 2 \beta_{1} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q + 14q^{2} + 14q^{4} + 2q^{5} + 4q^{7} + 14q^{8} + O(q^{10}) \) \( 14q + 14q^{2} + 14q^{4} + 2q^{5} + 4q^{7} + 14q^{8} + 2q^{10} + 2q^{11} + 4q^{13} + 4q^{14} + 14q^{16} + 9q^{17} + 14q^{19} + 2q^{20} + 2q^{22} + 30q^{23} + 18q^{25} + 4q^{26} + 4q^{28} + 6q^{29} + 11q^{31} + 14q^{32} + 9q^{34} - 18q^{35} + 13q^{37} + 14q^{38} + 2q^{40} - 2q^{41} + 12q^{43} + 2q^{44} + 30q^{46} + 21q^{47} + 32q^{49} + 18q^{50} + 4q^{52} + 22q^{53} - 7q^{55} + 4q^{56} + 6q^{58} + 14q^{59} + 31q^{61} + 11q^{62} + 14q^{64} + 24q^{67} + 9q^{68} - 18q^{70} + 28q^{71} + 24q^{73} + 13q^{74} + 14q^{76} + 16q^{77} + 65q^{79} + 2q^{80} - 2q^{82} - 15q^{83} - 19q^{85} + 12q^{86} + 2q^{88} - 11q^{89} + 68q^{91} + 30q^{92} + 21q^{94} + 8q^{95} + 23q^{97} + 32q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 2 x^{13} - 42 x^{12} + 70 x^{11} + 680 x^{10} - 882 x^{9} - 5559 x^{8} + 5066 x^{7} + 24455 x^{6} - 12990 x^{5} - 55580 x^{4} + 9808 x^{3} + 53551 x^{2} + 6282 x - 7083\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-6885747582 \nu^{13} + 33914318185 \nu^{12} + 202430140259 \nu^{11} - 1106373672733 \nu^{10} - 1870674412587 \nu^{9} + 12477302034149 \nu^{8} + 6907753319984 \nu^{7} - 63433606982404 \nu^{6} - 11816541353639 \nu^{5} + 149813081971647 \nu^{4} + 23331243759189 \nu^{3} - 143402427852725 \nu^{2} - 36206262335215 \nu + 19597736159910\)\()/ 837106466514 \)
\(\beta_{3}\)\(=\)\((\)\(-11004140892 \nu^{13} + 49436090470 \nu^{12} + 340112131925 \nu^{11} - 1615645352599 \nu^{10} - 3518878903680 \nu^{9} + 18377007078962 \nu^{8} + 16571607563132 \nu^{7} - 95500280471449 \nu^{6} - 40945399675328 \nu^{5} + 234011215846473 \nu^{4} + 63297600417105 \nu^{3} - 232233141936539 \nu^{2} - 56298435608683 \nu + 34191931183770\)\()/ 837106466514 \)
\(\beta_{4}\)\(=\)\((\)\(-19118330939 \nu^{13} + 86554061461 \nu^{12} + 595251271041 \nu^{11} - 2872804185614 \nu^{10} - 6118651043878 \nu^{9} + 33227082074766 \nu^{8} + 26870136857865 \nu^{7} - 173410067107969 \nu^{6} - 54385976946928 \nu^{5} + 418064433687984 \nu^{4} + 69177996434839 \nu^{3} - 401075285306924 \nu^{2} - 68967721612466 \nu + 53337950263866\)\()/ 837106466514 \)
\(\beta_{5}\)\(=\)\((\)\(-23639242991 \nu^{13} + 109271241761 \nu^{12} + 703785403742 \nu^{11} - 3523804015461 \nu^{10} - 6629286791539 \nu^{9} + 38941941028796 \nu^{8} + 24638483686895 \nu^{7} - 191223255135815 \nu^{6} - 34950317548215 \nu^{5} + 426625795784475 \nu^{4} + 35993911720081 \nu^{3} - 377615249540662 \nu^{2} - 56296105207467 \nu + 49744016665776\)\()/ 837106466514 \)
\(\beta_{6}\)\(=\)\((\)\(-25609954388 \nu^{13} + 109910586859 \nu^{12} + 842563500174 \nu^{11} - 3739110618812 \nu^{10} - 9624887605363 \nu^{9} + 45107127742866 \nu^{8} + 50850635857116 \nu^{7} - 250882871913973 \nu^{6} - 135245303862964 \nu^{5} + 659898648295785 \nu^{4} + 189622681806508 \nu^{3} - 696858166252685 \nu^{2} - 135681262744841 \nu + 105376014149448\)\()/ 837106466514 \)
\(\beta_{7}\)\(=\)\((\)\(31764401845 \nu^{13} - 134861372299 \nu^{12} - 1026991737247 \nu^{11} + 4499074114704 \nu^{10} + 11379461904392 \nu^{9} - 52596458855725 \nu^{8} - 57437914514824 \nu^{7} + 279816457466485 \nu^{6} + 143968104585861 \nu^{5} - 694429136478564 \nu^{4} - 192408481665842 \nu^{3} + 688027259578091 \nu^{2} + 132289761077346 \nu - 101151393317133\)\()/ 837106466514 \)
\(\beta_{8}\)\(=\)\((\)\(37435189105 \nu^{13} - 162933719250 \nu^{12} - 1191384212828 \nu^{11} + 5420978081009 \nu^{10} + 12830895743357 \nu^{9} - 63127448399537 \nu^{8} - 61998638358803 \nu^{7} + 334462797242739 \nu^{6} + 148254476159677 \nu^{5} - 827159607477825 \nu^{4} - 203381316447602 \nu^{3} + 817249872781809 \nu^{2} + 162909865954943 \nu - 113370111470034\)\()/ 837106466514 \)
\(\beta_{9}\)\(=\)\((\)\(37752739446 \nu^{13} - 161274827528 \nu^{12} - 1202197738381 \nu^{11} + 5328164864492 \nu^{10} + 12968704565682 \nu^{9} - 61372297713001 \nu^{8} - 62896893877735 \nu^{7} + 320445662317190 \nu^{6} + 150472617351805 \nu^{5} - 778793817550197 \nu^{4} - 201531065820906 \nu^{3} + 758481182157460 \nu^{2} + 153671579418602 \nu - 109196500860438\)\()/ 837106466514 \)
\(\beta_{10}\)\(=\)\((\)\(-38350226953 \nu^{13} + 161631948769 \nu^{12} + 1243644226300 \nu^{11} - 5401712538693 \nu^{10} - 13841584361987 \nu^{9} + 63333380868157 \nu^{8} + 70296111977863 \nu^{7} - 338261095735141 \nu^{6} - 178318841583876 \nu^{5} + 844077466448136 \nu^{4} + 244176912750701 \nu^{3} - 842622003635849 \nu^{2} - 170600783335470 \nu + 123665771390928\)\()/ 837106466514 \)
\(\beta_{11}\)\(=\)\((\)\(-38926641911 \nu^{13} + 173061237061 \nu^{12} + 1207022732481 \nu^{11} - 5680374898316 \nu^{10} - 12367234402564 \nu^{9} + 64701425405268 \nu^{8} + 54689127249561 \nu^{7} - 332421010230685 \nu^{6} - 113117298126910 \nu^{5} + 790930436132310 \nu^{4} + 141244421821633 \nu^{3} - 754127045359874 \nu^{2} - 128265388501436 \nu + 106272852513750\)\()/ 837106466514 \)
\(\beta_{12}\)\(=\)\((\)\(-48334159531 \nu^{13} + 220483599101 \nu^{12} + 1494088890444 \nu^{11} - 7282180176589 \nu^{10} - 15243485778272 \nu^{9} + 83843136303042 \nu^{8} + 67191645924951 \nu^{7} - 438307828557386 \nu^{6} - 140131255986980 \nu^{5} + 1069606585109949 \nu^{4} + 184820997223490 \nu^{3} - 1049938391405821 \nu^{2} - 184286450571781 \nu + 150684116159928\)\()/ 837106466514 \)
\(\beta_{13}\)\(=\)\((\)\(-102355805795 \nu^{13} + 456007662555 \nu^{12} + 3186087569260 \nu^{11} - 15003756789322 \nu^{10} - 32990628819313 \nu^{9} + 171744662126800 \nu^{8} + 149960354490979 \nu^{7} - 890755378335540 \nu^{6} - 330103734664163 \nu^{5} + 2151287002908579 \nu^{4} + 441253708065913 \nu^{3} - 2084788718214012 \nu^{2} - 392875051320628 \nu + 295489137970695\)\()/ 837106466514 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{12} - \beta_{10} - \beta_{8} + \beta_{6} + \beta_{5} + 7\)
\(\nu^{3}\)\(=\)\(-\beta_{13} + 2 \beta_{11} - 2 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + 9 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-6 \beta_{13} - 9 \beta_{12} - 2 \beta_{11} - 15 \beta_{10} - 4 \beta_{9} - 18 \beta_{8} - 2 \beta_{7} + 15 \beta_{6} + 21 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 84\)
\(\nu^{5}\)\(=\)\(-24 \beta_{13} - 3 \beta_{12} + 51 \beta_{11} - 45 \beta_{10} - 14 \beta_{9} - 29 \beta_{8} - 26 \beta_{7} + 3 \beta_{6} - 17 \beta_{5} - 9 \beta_{4} + 20 \beta_{3} + 26 \beta_{2} + 113 \beta_{1} + 70\)
\(\nu^{6}\)\(=\)\(-147 \beta_{13} - 92 \beta_{12} - 37 \beta_{11} - 251 \beta_{10} - 91 \beta_{9} - 330 \beta_{8} - 61 \beta_{7} + 229 \beta_{6} + 372 \beta_{5} - 132 \beta_{4} + 44 \beta_{3} + 31 \beta_{2} + 59 \beta_{1} + 1249\)
\(\nu^{7}\)\(=\)\(-495 \beta_{13} - 61 \beta_{12} + 968 \beta_{11} - 869 \beta_{10} - 225 \beta_{9} - 659 \beta_{8} - 525 \beta_{7} + 97 \beta_{6} - 208 \beta_{5} - 291 \beta_{4} + 341 \beta_{3} + 532 \beta_{2} + 1698 \beta_{1} + 1468\)
\(\nu^{8}\)\(=\)\(-2920 \beta_{13} - 1131 \beta_{12} - 437 \beta_{11} - 4441 \beta_{10} - 1716 \beta_{9} - 6010 \beta_{8} - 1374 \beta_{7} + 3705 \beta_{6} + 6387 \beta_{5} - 2754 \beta_{4} + 804 \beta_{3} + 783 \beta_{2} + 1410 \beta_{1} + 20427\)
\(\nu^{9}\)\(=\)\(-9734 \beta_{13} - 1052 \beta_{12} + 17068 \beta_{11} - 16349 \beta_{10} - 4100 \beta_{9} - 13751 \beta_{8} - 9918 \beta_{7} + 2531 \beta_{6} - 1703 \beta_{5} - 6883 \beta_{4} + 5735 \beta_{3} + 10034 \beta_{2} + 27763 \beta_{1} + 30197\)
\(\nu^{10}\)\(=\)\(-54851 \beta_{13} - 16214 \beta_{12} - 2601 \beta_{11} - 80429 \beta_{10} - 31101 \beta_{9} - 108802 \beta_{8} - 28167 \beta_{7} + 62247 \beta_{6} + 109206 \beta_{5} - 53104 \beta_{4} + 14470 \beta_{3} + 17745 \beta_{2} + 31378 \beta_{1} + 348706\)
\(\nu^{11}\)\(=\)\(-187187 \beta_{13} - 18966 \beta_{12} + 295104 \beta_{11} - 305918 \beta_{10} - 78259 \beta_{9} - 275524 \beta_{8} - 183285 \beta_{7} + 60180 \beta_{6} + 5121 \beta_{5} - 145199 \beta_{4} + 98091 \beta_{3} + 183650 \beta_{2} + 472087 \beta_{1} + 613563\)
\(\nu^{12}\)\(=\)\(-1012139 \beta_{13} - 256618 \beta_{12} + 47933 \beta_{11} - 1470566 \beta_{10} - 559263 \beta_{9} - 1965398 \beta_{8} - 555341 \beta_{7} + 1068724 \beta_{6} + 1872765 \beta_{5} - 990846 \beta_{4} + 264711 \beta_{3} + 376610 \beta_{2} + 671115 \beta_{1} + 6080293\)
\(\nu^{13}\)\(=\)\(-3556677 \beta_{13} - 363417 \beta_{12} + 5088500 \beta_{11} - 5713758 \beta_{10} - 1507447 \beta_{9} - 5398364 \beta_{8} - 3360057 \beta_{7} + 1345486 \beta_{6} + 690113 \beta_{5} - 2905992 \beta_{4} + 1710688 \beta_{3} + 3324128 \beta_{2} + 8189616 \beta_{1} + 12308542\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.97637
−2.45968
−2.42130
−2.39851
−1.53522
−1.46245
−0.541630
0.316300
1.69938
2.04034
2.34343
2.95487
3.13855
4.30228
1.00000 0 1.00000 −3.97637 0 3.97870 1.00000 0 −3.97637
1.2 1.00000 0 1.00000 −2.45968 0 −3.97455 1.00000 0 −2.45968
1.3 1.00000 0 1.00000 −2.42130 0 −4.62365 1.00000 0 −2.42130
1.4 1.00000 0 1.00000 −2.39851 0 2.84704 1.00000 0 −2.39851
1.5 1.00000 0 1.00000 −1.53522 0 4.84041 1.00000 0 −1.53522
1.6 1.00000 0 1.00000 −1.46245 0 −0.313566 1.00000 0 −1.46245
1.7 1.00000 0 1.00000 −0.541630 0 3.83971 1.00000 0 −0.541630
1.8 1.00000 0 1.00000 0.316300 0 −2.35786 1.00000 0 0.316300
1.9 1.00000 0 1.00000 1.69938 0 −0.423602 1.00000 0 1.69938
1.10 1.00000 0 1.00000 2.04034 0 2.45177 1.00000 0 2.04034
1.11 1.00000 0 1.00000 2.34343 0 0.305937 1.00000 0 2.34343
1.12 1.00000 0 1.00000 2.95487 0 2.46629 1.00000 0 2.95487
1.13 1.00000 0 1.00000 3.13855 0 −2.45592 1.00000 0 3.13855
1.14 1.00000 0 1.00000 4.30228 0 −2.58072 1.00000 0 4.30228
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8046.2.a.r yes 14
3.b odd 2 1 8046.2.a.q 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8046.2.a.q 14 3.b odd 2 1
8046.2.a.r yes 14 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(149\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

\(T_{5}^{14} - \cdots\)
\(T_{11}^{14} - \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{14} \)
$3$ \( \)
$5$ \( 1 - 2 T + 28 T^{2} - 60 T^{3} + 435 T^{4} - 932 T^{5} + 4641 T^{6} - 9874 T^{7} + 37720 T^{8} - 79480 T^{9} + 249270 T^{10} - 522542 T^{11} + 1421451 T^{12} - 2950348 T^{13} + 7372677 T^{14} - 14751740 T^{15} + 35536275 T^{16} - 65317750 T^{17} + 155793750 T^{18} - 248375000 T^{19} + 589375000 T^{20} - 771406250 T^{21} + 1812890625 T^{22} - 1820312500 T^{23} + 4248046875 T^{24} - 2929687500 T^{25} + 6835937500 T^{26} - 2441406250 T^{27} + 6103515625 T^{28} \)
$7$ \( 1 - 4 T + 41 T^{2} - 141 T^{3} + 886 T^{4} - 2702 T^{5} + 13194 T^{6} - 36749 T^{7} + 152933 T^{8} - 396432 T^{9} + 1481221 T^{10} - 3629234 T^{11} + 12465202 T^{12} - 28914434 T^{13} + 92675285 T^{14} - 202401038 T^{15} + 610794898 T^{16} - 1244827262 T^{17} + 3556411621 T^{18} - 6662832624 T^{19} + 17992414517 T^{20} - 30264381707 T^{21} + 76060784394 T^{22} - 109035446114 T^{23} + 250273070614 T^{24} - 278803070763 T^{25} + 567492775241 T^{26} - 387556041628 T^{27} + 678223072849 T^{28} \)
$11$ \( 1 - 2 T + 61 T^{2} - 136 T^{3} + 2029 T^{4} - 4541 T^{5} + 47979 T^{6} - 102861 T^{7} + 894464 T^{8} - 1797910 T^{9} + 13870329 T^{10} - 26073088 T^{11} + 184810378 T^{12} - 326416620 T^{13} + 2161407870 T^{14} - 3590582820 T^{15} + 22362055738 T^{16} - 34703280128 T^{17} + 203075486889 T^{18} - 289555203410 T^{19} + 1584597538304 T^{20} - 2004469896231 T^{21} + 10284724751499 T^{22} - 10707440464831 T^{23} + 52627034515429 T^{24} - 38802387203096 T^{25} + 191444130979981 T^{26} - 69045424287862 T^{27} + 379749833583241 T^{28} \)
$13$ \( 1 - 4 T + 55 T^{2} - 164 T^{3} + 1841 T^{4} - 5834 T^{5} + 51222 T^{6} - 152916 T^{7} + 1105419 T^{8} - 3168570 T^{9} + 20685432 T^{10} - 57342178 T^{11} + 333039468 T^{12} - 858558930 T^{13} + 4596254473 T^{14} - 11161266090 T^{15} + 56283670092 T^{16} - 125980765066 T^{17} + 590796623352 T^{18} - 1176467861010 T^{19} + 5335646377971 T^{20} - 9595252225572 T^{21} + 41783358991062 T^{22} - 61866649342082 T^{23} + 253797483494009 T^{24} - 293914304622068 T^{25} + 1281394681736455 T^{26} - 1211500426369012 T^{27} + 3937376385699289 T^{28} \)
$17$ \( 1 - 9 T + 116 T^{2} - 734 T^{3} + 5894 T^{4} - 33374 T^{5} + 217597 T^{6} - 1158998 T^{7} + 6467809 T^{8} - 31916274 T^{9} + 160080026 T^{10} - 734570171 T^{11} + 3403169256 T^{12} - 14440220552 T^{13} + 62194715466 T^{14} - 245483749384 T^{15} + 983515914984 T^{16} - 3608943250123 T^{17} + 13370043851546 T^{18} - 45316545052818 T^{19} + 156117186016321 T^{20} - 475581701329654 T^{21} + 1517903891889277 T^{22} - 3957751790210878 T^{23} + 11882268049246406 T^{24} - 25155571889802622 T^{25} + 67584179518652276 T^{26} - 89141202296153433 T^{27} + 168377826559400929 T^{28} \)
$19$ \( 1 - 14 T + 169 T^{2} - 1403 T^{3} + 11488 T^{4} - 78175 T^{5} + 522320 T^{6} - 3035681 T^{7} + 17497829 T^{8} - 90997285 T^{9} + 473230201 T^{10} - 2251728068 T^{11} + 10805853754 T^{12} - 48136741502 T^{13} + 217617850140 T^{14} - 914598088538 T^{15} + 3900913205194 T^{16} - 15444602818412 T^{17} + 61671833024521 T^{18} - 225318286391215 T^{19} + 823200780892349 T^{20} - 2713509454519259 T^{21} + 8870854647575120 T^{22} - 25226110773873325 T^{23} + 70433689169617888 T^{24} - 163435833234201257 T^{25} + 374050221322181209 T^{26} - 588741768471598826 T^{27} + 799006685782884121 T^{28} \)
$23$ \( 1 - 30 T + 584 T^{2} - 8287 T^{3} + 96925 T^{4} - 963908 T^{5} + 8497143 T^{6} - 67352281 T^{7} + 489924256 T^{8} - 3292775643 T^{9} + 20674520004 T^{10} - 121569332525 T^{11} + 673577436637 T^{12} - 3516529515759 T^{13} + 17358630554967 T^{14} - 80880178862457 T^{15} + 356322463980973 T^{16} - 1479134068831675 T^{17} + 5785578352439364 T^{18} - 21193433460393549 T^{19} + 72526372779623584 T^{20} - 229322760262294607 T^{21} + 665419640403552183 T^{22} - 1736145459605477404 T^{23} + 4015264599382929325 T^{24} - 7895934463832713049 T^{25} + 12798140668299867464 T^{26} - 15121090858094021490 T^{27} + 11592836324538749809 T^{28} \)
$29$ \( 1 - 6 T + 181 T^{2} - 1020 T^{3} + 14668 T^{4} - 73882 T^{5} + 710119 T^{6} - 2807471 T^{7} + 22480794 T^{8} - 45950648 T^{9} + 431717642 T^{10} + 836466161 T^{11} + 2151226620 T^{12} + 73261063515 T^{13} - 88428928137 T^{14} + 2124570841935 T^{15} + 1809181587420 T^{16} + 20400573200629 T^{17} + 305345685551402 T^{18} - 942500587774552 T^{19} + 13372100545796874 T^{20} - 48428527491104539 T^{21} + 355234482525452359 T^{22} - 1071816958989153458 T^{23} + 6170933698047348268 T^{24} - 12444519961019945580 T^{25} + 64040475760189896421 T^{26} - 61563772277751613134 T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \)
$31$ \( 1 - 11 T + 293 T^{2} - 2401 T^{3} + 37974 T^{4} - 249434 T^{5} + 3040390 T^{6} - 16528313 T^{7} + 172961258 T^{8} - 792069810 T^{9} + 7626311273 T^{10} - 30088144642 T^{11} + 280194802802 T^{12} - 994265734303 T^{13} + 9112906214803 T^{14} - 30822237763393 T^{15} + 269267205492722 T^{16} - 896355917029822 T^{17} + 7043058613152233 T^{18} - 22676286193031310 T^{19} + 153503753145390698 T^{20} - 454737097474824743 T^{21} + 2593121381325241990 T^{22} - 6594940714024810214 T^{23} + 31124564569808937174 T^{24} - 61005753028267999231 T^{25} + \)\(23\!\cdots\!73\)\( T^{26} - \)\(26\!\cdots\!01\)\( T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \)
$37$ \( 1 - 13 T + 281 T^{2} - 3111 T^{3} + 40264 T^{4} - 367589 T^{5} + 3638571 T^{6} - 28152253 T^{7} + 231032713 T^{8} - 1552251607 T^{9} + 11156663741 T^{10} - 67286788305 T^{11} + 447921138414 T^{12} - 2557243358182 T^{13} + 16700844235102 T^{14} - 94618004252734 T^{15} + 613204038488766 T^{16} - 3408277688013165 T^{17} + 20909384073496301 T^{18} - 107639268688988899 T^{19} + 592766733087017617 T^{20} - 2672546222813130649 T^{21} + 12780405879132786891 T^{22} - 47772505969532559353 T^{23} + \)\(19\!\cdots\!36\)\( T^{24} - \)\(55\!\cdots\!43\)\( T^{25} + \)\(18\!\cdots\!61\)\( T^{26} - \)\(31\!\cdots\!61\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \)
$41$ \( 1 + 2 T + 310 T^{2} + 1067 T^{3} + 48772 T^{4} + 213511 T^{5} + 5293181 T^{6} + 25200647 T^{7} + 441132414 T^{8} + 2099126214 T^{9} + 29405656103 T^{10} + 134309132098 T^{11} + 1599620490390 T^{12} + 6841742496901 T^{13} + 71944474959847 T^{14} + 280511442372941 T^{15} + 2688962044345590 T^{16} + 9256719693326258 T^{17} + 83093356195269383 T^{18} + 243196788573553014 T^{19} + 2095424950583967774 T^{20} + 4907933707816401007 T^{21} + 42265654509203923901 T^{22} + 69899644194389007071 T^{23} + \)\(65\!\cdots\!72\)\( T^{24} + \)\(58\!\cdots\!47\)\( T^{25} + \)\(69\!\cdots\!10\)\( T^{26} + \)\(18\!\cdots\!42\)\( T^{27} + \)\(37\!\cdots\!61\)\( T^{28} \)
$43$ \( 1 - 12 T + 313 T^{2} - 2941 T^{3} + 46354 T^{4} - 364249 T^{5} + 4500456 T^{6} - 30371096 T^{7} + 323732139 T^{8} - 1907284425 T^{9} + 18605092085 T^{10} - 97945613665 T^{11} + 915170309538 T^{12} - 4461374153680 T^{13} + 40823650298033 T^{14} - 191839088608240 T^{15} + 1692149902335762 T^{16} - 7787361905663155 T^{17} + 63607107425290085 T^{18} - 280386913677400275 T^{19} + 2046428381248331811 T^{20} - 8255429132517363272 T^{21} + 52602231068531086056 T^{22} - \)\(18\!\cdots\!07\)\( T^{23} + \)\(10\!\cdots\!46\)\( T^{24} - \)\(27\!\cdots\!87\)\( T^{25} + \)\(12\!\cdots\!13\)\( T^{26} - \)\(20\!\cdots\!16\)\( T^{27} + \)\(73\!\cdots\!49\)\( T^{28} \)
$47$ \( 1 - 21 T + 642 T^{2} - 9579 T^{3} + 173282 T^{4} - 2058501 T^{5} + 28081277 T^{6} - 281260080 T^{7} + 3164831869 T^{8} - 27593886580 T^{9} + 267463992132 T^{10} - 2066070951610 T^{11} + 17645268146416 T^{12} - 121770185396311 T^{13} + 926763873038362 T^{14} - 5723198713626617 T^{15} + 38978397335432944 T^{16} - 214505684409005030 T^{17} + 1305138960590669892 T^{18} - 6328520110847306060 T^{19} + 34114404196032519901 T^{20} - \)\(14\!\cdots\!40\)\( T^{21} + \)\(66\!\cdots\!97\)\( T^{22} - \)\(23\!\cdots\!67\)\( T^{23} + \)\(91\!\cdots\!18\)\( T^{24} - \)\(23\!\cdots\!37\)\( T^{25} + \)\(74\!\cdots\!22\)\( T^{26} - \)\(11\!\cdots\!67\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \)
$53$ \( 1 - 22 T + 600 T^{2} - 8891 T^{3} + 149025 T^{4} - 1767428 T^{5} + 23162405 T^{6} - 236054523 T^{7} + 2622232050 T^{8} - 23676784217 T^{9} + 231170051836 T^{10} - 1878471829095 T^{11} + 16435898992365 T^{12} - 121114546059821 T^{13} + 959063276750599 T^{14} - 6419070941170513 T^{15} + 46168440269553285 T^{16} - 279661250500176315 T^{17} + 1824042901780973116 T^{18} - 9901524448282933981 T^{19} + 58120098120237984450 T^{20} - \)\(27\!\cdots\!51\)\( T^{21} + \)\(14\!\cdots\!05\)\( T^{22} - \)\(58\!\cdots\!24\)\( T^{23} + \)\(26\!\cdots\!25\)\( T^{24} - \)\(82\!\cdots\!27\)\( T^{25} + \)\(29\!\cdots\!00\)\( T^{26} - \)\(57\!\cdots\!06\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \)
$59$ \( 1 - 14 T + 581 T^{2} - 7377 T^{3} + 164702 T^{4} - 1883247 T^{5} + 30220988 T^{6} - 310284265 T^{7} + 4012968527 T^{8} - 37055439705 T^{9} + 408802361169 T^{10} - 3406965871622 T^{11} + 33055223215074 T^{12} - 249122296374202 T^{13} + 2161859424644700 T^{14} - 14698215486077918 T^{15} + 115065232011672594 T^{16} - 699719243747854738 T^{17} + 4953605787937155009 T^{18} - 26491834255233891795 T^{19} + \)\(16\!\cdots\!07\)\( T^{20} - \)\(77\!\cdots\!35\)\( T^{21} + \)\(44\!\cdots\!48\)\( T^{22} - \)\(16\!\cdots\!33\)\( T^{23} + \)\(84\!\cdots\!02\)\( T^{24} - \)\(22\!\cdots\!43\)\( T^{25} + \)\(10\!\cdots\!61\)\( T^{26} - \)\(14\!\cdots\!06\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \)
$61$ \( 1 - 31 T + 799 T^{2} - 14275 T^{3} + 232164 T^{4} - 3132074 T^{5} + 39576049 T^{6} - 440162687 T^{7} + 4672193963 T^{8} - 45028609730 T^{9} + 421062079995 T^{10} - 3650714835224 T^{11} + 31205726643312 T^{12} - 251150250443459 T^{13} + 2015313166590698 T^{14} - 15320165277050999 T^{15} + 116116508839763952 T^{16} - 828642904013978744 T^{17} + 5829958610740050795 T^{18} - 38030997217130608730 T^{19} + \)\(24\!\cdots\!43\)\( T^{20} - \)\(13\!\cdots\!27\)\( T^{21} + \)\(75\!\cdots\!69\)\( T^{22} - \)\(36\!\cdots\!34\)\( T^{23} + \)\(16\!\cdots\!64\)\( T^{24} - \)\(62\!\cdots\!75\)\( T^{25} + \)\(21\!\cdots\!79\)\( T^{26} - \)\(50\!\cdots\!11\)\( T^{27} + \)\(98\!\cdots\!41\)\( T^{28} \)
$67$ \( 1 - 24 T + 645 T^{2} - 10579 T^{3} + 173398 T^{4} - 2229369 T^{5} + 27927266 T^{6} - 300633461 T^{7} + 3134279069 T^{8} - 29546445593 T^{9} + 270345477209 T^{10} - 2329683918478 T^{11} + 19689782826956 T^{12} - 162614886738420 T^{13} + 1334836555870912 T^{14} - 10895197411474140 T^{15} + 88387435110205484 T^{16} - 700682724373198714 T^{17} + 5447764423041301289 T^{18} - 39891398017718803451 T^{19} + \)\(28\!\cdots\!61\)\( T^{20} - \)\(18\!\cdots\!03\)\( T^{21} + \)\(11\!\cdots\!06\)\( T^{22} - \)\(60\!\cdots\!43\)\( T^{23} + \)\(31\!\cdots\!02\)\( T^{24} - \)\(12\!\cdots\!57\)\( T^{25} + \)\(52\!\cdots\!45\)\( T^{26} - \)\(13\!\cdots\!88\)\( T^{27} + \)\(36\!\cdots\!29\)\( T^{28} \)
$71$ \( 1 - 28 T + 793 T^{2} - 15570 T^{3} + 289339 T^{4} - 4536466 T^{5} + 66735100 T^{6} - 884178035 T^{7} + 11020890785 T^{8} - 127178786894 T^{9} + 1385246841317 T^{10} - 14162754693270 T^{11} + 137026427434000 T^{12} - 1252415837062688 T^{13} + 10846731641608785 T^{14} - 88921524431450848 T^{15} + 690750220694794000 T^{16} - 5069005695022958970 T^{17} + 35201450837805223877 T^{18} - \)\(22\!\cdots\!94\)\( T^{19} + \)\(14\!\cdots\!85\)\( T^{20} - \)\(80\!\cdots\!85\)\( T^{21} + \)\(43\!\cdots\!00\)\( T^{22} - \)\(20\!\cdots\!46\)\( T^{23} + \)\(94\!\cdots\!39\)\( T^{24} - \)\(35\!\cdots\!70\)\( T^{25} + \)\(13\!\cdots\!13\)\( T^{26} - \)\(32\!\cdots\!08\)\( T^{27} + \)\(82\!\cdots\!81\)\( T^{28} \)
$73$ \( 1 - 24 T + 559 T^{2} - 10259 T^{3} + 166212 T^{4} - 2371579 T^{5} + 31144574 T^{6} - 379413212 T^{7} + 4292139247 T^{8} - 45848681295 T^{9} + 466387972197 T^{10} - 4520665874695 T^{11} + 42080295557530 T^{12} - 377765144648286 T^{13} + 3289457070452829 T^{14} - 27576855559324878 T^{15} + 224245895026077370 T^{16} - 1758615876576224815 T^{17} + 13244598033951705477 T^{18} - 95047598769174952935 T^{19} + \)\(64\!\cdots\!83\)\( T^{20} - \)\(41\!\cdots\!64\)\( T^{21} + \)\(25\!\cdots\!94\)\( T^{22} - \)\(13\!\cdots\!27\)\( T^{23} + \)\(71\!\cdots\!88\)\( T^{24} - \)\(32\!\cdots\!43\)\( T^{25} + \)\(12\!\cdots\!39\)\( T^{26} - \)\(40\!\cdots\!92\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \)
$79$ \( 1 - 65 T + 2695 T^{2} - 81273 T^{3} + 1994435 T^{4} - 41257851 T^{5} + 745630895 T^{6} - 11965350812 T^{7} + 173460480090 T^{8} - 2292541231594 T^{9} + 27902035275023 T^{10} - 314417424571610 T^{11} + 3300424334374354 T^{12} - 32356362354243161 T^{13} + 297156786296087190 T^{14} - 2556152625985209719 T^{15} + 20597948270830343314 T^{16} - \)\(15\!\cdots\!90\)\( T^{17} + \)\(10\!\cdots\!63\)\( T^{18} - \)\(70\!\cdots\!06\)\( T^{19} + \)\(42\!\cdots\!90\)\( T^{20} - \)\(22\!\cdots\!08\)\( T^{21} + \)\(11\!\cdots\!95\)\( T^{22} - \)\(49\!\cdots\!69\)\( T^{23} + \)\(18\!\cdots\!35\)\( T^{24} - \)\(60\!\cdots\!67\)\( T^{25} + \)\(15\!\cdots\!95\)\( T^{26} - \)\(30\!\cdots\!35\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \)
$83$ \( 1 + 15 T + 316 T^{2} + 3384 T^{3} + 55867 T^{4} + 540953 T^{5} + 8059014 T^{6} + 79440974 T^{7} + 1044683618 T^{8} + 9433281291 T^{9} + 111583188986 T^{10} + 972398968994 T^{11} + 10784971542554 T^{12} + 91456205304469 T^{13} + 961321029078984 T^{14} + 7590865040270927 T^{15} + 74297668956654506 T^{16} + 556005089284172278 T^{17} + 5295550801101252506 T^{18} + 37158078402100510113 T^{19} + \)\(34\!\cdots\!42\)\( T^{20} + \)\(21\!\cdots\!98\)\( T^{21} + \)\(18\!\cdots\!74\)\( T^{22} + \)\(10\!\cdots\!59\)\( T^{23} + \)\(86\!\cdots\!83\)\( T^{24} + \)\(43\!\cdots\!28\)\( T^{25} + \)\(33\!\cdots\!76\)\( T^{26} + \)\(13\!\cdots\!45\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \)
$89$ \( 1 + 11 T + 637 T^{2} + 4666 T^{3} + 194340 T^{4} + 988871 T^{5} + 40302057 T^{6} + 140952095 T^{7} + 6481748951 T^{8} + 15564145255 T^{9} + 861119774688 T^{10} + 1481669315671 T^{11} + 96912973093927 T^{12} + 132503710750456 T^{13} + 9321434523205205 T^{14} + 11792830256790584 T^{15} + 767647659876995767 T^{16} + 1044530935799269199 T^{17} + 54028584433340195808 T^{18} + 86911112376791264495 T^{19} + \)\(32\!\cdots\!11\)\( T^{20} + \)\(62\!\cdots\!55\)\( T^{21} + \)\(15\!\cdots\!17\)\( T^{22} + \)\(34\!\cdots\!39\)\( T^{23} + \)\(60\!\cdots\!40\)\( T^{24} + \)\(12\!\cdots\!74\)\( T^{25} + \)\(15\!\cdots\!77\)\( T^{26} + \)\(24\!\cdots\!59\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \)
$97$ \( 1 - 23 T + 725 T^{2} - 11408 T^{3} + 224098 T^{4} - 2793230 T^{5} + 41901368 T^{6} - 415274078 T^{7} + 5095591937 T^{8} - 38237772108 T^{9} + 402889459893 T^{10} - 1768490206825 T^{11} + 18928181782076 T^{12} + 18318177648168 T^{13} + 831994608674812 T^{14} + 1776863231872296 T^{15} + 178095262387553084 T^{16} - 1614053262533593225 T^{17} + 35667514206805626933 T^{18} - \)\(32\!\cdots\!56\)\( T^{19} + \)\(42\!\cdots\!73\)\( T^{20} - \)\(33\!\cdots\!14\)\( T^{21} + \)\(32\!\cdots\!48\)\( T^{22} - \)\(21\!\cdots\!10\)\( T^{23} + \)\(16\!\cdots\!02\)\( T^{24} - \)\(81\!\cdots\!24\)\( T^{25} + \)\(50\!\cdots\!25\)\( T^{26} - \)\(15\!\cdots\!71\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \)
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