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: \( 1 \)
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.

Atkin-Lehner signs

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

Hecke kernels

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

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