# Properties

 Label 1148.2.i.d Level $1148$ Weight $2$ Character orbit 1148.i Analytic conductor $9.167$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1148 = 2^{2} \cdot 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1148.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.16682615204$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 14 x^{14} - 8 x^{13} + 136 x^{12} - 87 x^{11} + 706 x^{10} - 568 x^{9} + 2685 x^{8} - 2100 x^{7} + 5529 x^{6} - 4919 x^{5} + 8145 x^{4} - 5182 x^{3} + 2775 x^{2} - 660 x + 121$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + \beta_{14} q^{5} + ( -\beta_{5} + \beta_{8} ) q^{7} + ( \beta_{2} + \beta_{10} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + \beta_{14} q^{5} + ( -\beta_{5} + \beta_{8} ) q^{7} + ( \beta_{2} + \beta_{10} ) q^{9} + ( -\beta_{1} + \beta_{7} + \beta_{8} + \beta_{11} ) q^{11} + ( -1 + \beta_{4} + \beta_{13} ) q^{13} + ( -\beta_{4} + \beta_{8} ) q^{15} + ( \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{17} + ( \beta_{2} + \beta_{10} - \beta_{15} ) q^{19} + ( 1 - \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{21} + ( -\beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{10} ) q^{23} + ( \beta_{1} + \beta_{10} - \beta_{13} + \beta_{14} ) q^{25} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{8} - \beta_{9} + \beta_{12} + \beta_{13} ) q^{27} + ( \beta_{2} + \beta_{8} + \beta_{9} - \beta_{13} ) q^{29} + ( \beta_{1} - \beta_{7} + 2 \beta_{8} + \beta_{10} + 2 \beta_{11} + \beta_{13} - \beta_{14} ) q^{31} + ( -3 - \beta_{3} + \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{11} - \beta_{14} + \beta_{15} ) q^{33} + ( -\beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{35} + ( 4 + \beta_{5} - \beta_{6} - 4 \beta_{7} - \beta_{11} + \beta_{14} - \beta_{15} ) q^{37} + ( \beta_{1} - \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{9} - \beta_{10} - \beta_{13} + \beta_{14} ) q^{39} + q^{41} + ( -1 + \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} ) q^{43} + ( -\beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{8} + \beta_{9} + 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{45} + ( -3 - \beta_{3} + \beta_{5} + 3 \beta_{7} - 2 \beta_{14} + \beta_{15} ) q^{47} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{7} + 2 \beta_{10} - 2 \beta_{12} - \beta_{15} ) q^{49} + ( 2 - 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{10} - \beta_{11} + \beta_{14} ) q^{51} + ( \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{9} - 2 \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{53} + ( -\beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{8} + \beta_{9} + \beta_{12} + \beta_{13} ) q^{55} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{8} - 2 \beta_{9} ) q^{57} + ( 3 \beta_{1} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{59} + ( -\beta_{2} + 3 \beta_{5} - \beta_{10} + 2 \beta_{11} - \beta_{14} ) q^{61} + ( -2 \beta_{2} - \beta_{4} + \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{10} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{63} + ( 3 - \beta_{3} + 2 \beta_{5} - 3 \beta_{7} + \beta_{11} - \beta_{14} ) q^{65} + ( -\beta_{1} - \beta_{8} - \beta_{11} + \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{67} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{9} - \beta_{12} ) q^{69} + ( 1 - \beta_{1} + \beta_{3} + \beta_{8} - 2 \beta_{9} - 2 \beta_{12} + \beta_{13} ) q^{71} + ( 2 \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} - 2 \beta_{15} ) q^{73} + ( 2 - 2 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{11} + \beta_{14} - \beta_{15} ) q^{75} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} + 4 \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{13} - \beta_{14} ) q^{77} + ( -2 - 3 \beta_{3} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{11} + 2 \beta_{14} + \beta_{15} ) q^{79} + ( \beta_{1} + 2 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + \beta_{10} + 3 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{81} + ( -6 - \beta_{2} + 2 \beta_{4} - \beta_{8} - \beta_{9} - 3 \beta_{12} ) q^{83} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{8} + 3 \beta_{9} - 2 \beta_{13} ) q^{85} + ( 3 \beta_{1} - \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{87} + ( 1 - \beta_{2} + 3 \beta_{3} + 5 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} - 3 \beta_{14} - \beta_{15} ) q^{89} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{12} - 2 \beta_{14} ) q^{91} + ( 2 - 2 \beta_{2} - \beta_{3} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 2 \beta_{10} + 2 \beta_{11} - \beta_{14} + \beta_{15} ) q^{93} + ( -2 \beta_{1} - \beta_{4} + \beta_{6} + 3 \beta_{7} + \beta_{8} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{95} + ( -3 - 2 \beta_{2} - 2 \beta_{8} + \beta_{9} - 2 \beta_{12} - \beta_{13} ) q^{97} + ( -1 + 2 \beta_{1} - 2 \beta_{3} - \beta_{9} - \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 4q^{9} + O(q^{10})$$ $$16q - 4q^{9} + 8q^{11} - 14q^{13} - 2q^{15} + q^{17} - 4q^{19} + 13q^{21} + 3q^{23} + 4q^{25} - 24q^{27} - 8q^{29} - 4q^{31} - 23q^{33} + 12q^{35} + 31q^{37} - 5q^{39} + 16q^{41} - 16q^{43} - q^{45} - 24q^{47} + 16q^{49} + 23q^{51} + q^{53} + 4q^{55} - 30q^{57} - 4q^{59} + 4q^{61} + 23q^{63} + 24q^{65} - 42q^{69} + 16q^{71} - 11q^{73} + 15q^{75} + 25q^{77} - 14q^{79} + 28q^{81} - 84q^{83} - 40q^{85} - 25q^{87} + 11q^{89} + 7q^{91} + 27q^{93} + 15q^{95} - 32q^{97} - 16q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 14 x^{14} - 8 x^{13} + 136 x^{12} - 87 x^{11} + 706 x^{10} - 568 x^{9} + 2685 x^{8} - 2100 x^{7} + 5529 x^{6} - 4919 x^{5} + 8145 x^{4} - 5182 x^{3} + 2775 x^{2} - 660 x + 121$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$780476607889860 \nu^{15} - 90657693823740 \nu^{14} + 10806309789157715 \nu^{13} - 6485068468898700 \nu^{12} + 105335780848540510 \nu^{11} - 66921393935742251 \nu^{10} + 537049880243353760 \nu^{9} - 378221909206358939 \nu^{8} + 2015604178438858179 \nu^{7} - 1299260518357132716 \nu^{6} + 3883000495870123846 \nu^{5} - 1869054719901640995 \nu^{4} + 5322936685577548599 \nu^{3} + 12994566126392139 \nu^{2} - 25305199101655440 \nu + 8034654060157036513$$$$)/ 2715054164464095114$$ $$\beta_{3}$$ $$=$$ $$($$$$913292836654949 \nu^{15} + 780476607889860 \nu^{14} + 12695442019345546 \nu^{13} + 3499967095918123 \nu^{12} + 117722757316174364 \nu^{11} + 25879304059559947 \nu^{10} + 577863348742651743 \nu^{9} + 18299549023342728 \nu^{8} + 2073969357212179126 \nu^{7} + 97689221463465279 \nu^{6} + 3750335575508080305 \nu^{5} - 609486967635570285 \nu^{4} + 5569715434652918610 \nu^{3} + 590253206031602881 \nu^{2} + 2547382187843875614 \nu - 628078471293921780$$$$)/ 2715054164464095114$$ $$\beta_{4}$$ $$=$$ $$($$$$4572267508647047 \nu^{15} + 3812484629844360 \nu^{14} + 56677408200185342 \nu^{13} + 9305383784116747 \nu^{12} + 523330485571362028 \nu^{11} + 154194179133320250 \nu^{10} + 2485803977795964337 \nu^{9} + 252846882325812290 \nu^{8} + 9150246496112283838 \nu^{7} + 2207270426229063267 \nu^{6} + 16442760501465139364 \nu^{5} - 2134144151886343350 \nu^{4} + 19906625080003062690 \nu^{3} + 2847832657552212838 \nu^{2} - 799584084296194560 \nu - 4049415427203613834$$$$)/ 4072581246696142671$$ $$\beta_{5}$$ $$=$$ $$($$$$155044131612905963 \nu^{15} - 178204148988571706 \nu^{14} + 1927451091102151942 \nu^{13} - 4085450739326494695 \nu^{12} + 18934153970045508190 \nu^{11} - 39407105956774649875 \nu^{10} + 95619591623053842749 \nu^{9} - 219330848018062886340 \nu^{8} + 376589330624672017654 \nu^{7} - 767752255836227766523 \nu^{6} + 782728947339221832921 \nu^{5} - 1540678187372657723355 \nu^{4} + 1277346186935558250330 \nu^{3} - 1629897809507746106657 \nu^{2} + 253720565374344881942 \nu - 54137605536294027660$$$$)/ 89596787427315138762$$ $$\beta_{6}$$ $$=$$ $$($$$$80352162859543897 \nu^{15} + 62656408743567419 \nu^{14} + 1087951016207688158 \nu^{13} + 84120867721335663 \nu^{12} + 9819613952551524221 \nu^{11} + 305130039696101887 \nu^{10} + 48588630007357744024 \nu^{9} - 8390538382601918157 \nu^{8} + 174195568380603844088 \nu^{7} - 22914601517441937377 \nu^{6} + 351933409938123713853 \nu^{5} - 105791547455390704452 \nu^{4} + 473509252635343784655 \nu^{3} - 99278396820069455332 \nu^{2} + 203832942589769042053 \nu - 49932150664970082960$$$$)/ 44798393713657569381$$ $$\beta_{7}$$ $$=$$ $$($$$$57098042844901980 \nu^{15} + 10046221203204439 \nu^{14} + 807957842515416180 \nu^{13} - 317134480546414834 \nu^{12} + 7803833464961768633 \nu^{11} - 3672579397028554256 \nu^{10} + 40595890593155957297 \nu^{9} - 26075191499735155467 \nu^{8} + 153509540077818586308 \nu^{7} - 97092227044960187614 \nu^{6} + 316769660325561165489 \nu^{5} - 239611581423483956265 \nu^{4} + 458359202327735353965 \nu^{3} - 234615188241099955650 \nu^{2} + 164939854160950626191 \nu - 9663504211352675046$$$$)/ 29865595809105046254$$ $$\beta_{8}$$ $$=$$ $$($$$$-13768306337420915 \nu^{15} - 10654557861806460 \nu^{14} - 196190324589487490 \nu^{13} - 42618731780100016 \nu^{12} - 1828799575469276860 \nu^{11} - 234208043010112029 \nu^{10} - 9114112815581019613 \nu^{9} + 450883738373513644 \nu^{8} - 32516534755885227334 \nu^{7} + 1804220478619772214 \nu^{6} - 59243082582111251291 \nu^{5} + 11934414817022812395 \nu^{4} - 73536347235924406872 \nu^{3} - 8209698241432178119 \nu^{2} + 2372379908735092740 \nu - 11764752278995496543$$$$)/ 4072581246696142671$$ $$\beta_{9}$$ $$=$$ $$($$$$-28084790708038187 \nu^{15} - 32072467399588440 \nu^{14} - 410086386678702503 \nu^{13} - 224733207884142661 \nu^{12} - 3756035289132322702 \nu^{11} - 1644350388232390080 \nu^{10} - 18638450843813729299 \nu^{9} - 4170061753174424855 \nu^{8} - 64718726468327036347 \nu^{7} - 11481250003078998417 \nu^{6} - 114865264453888455221 \nu^{5} + 7415997838355686800 \nu^{4} - 131655043690298281791 \nu^{3} - 23514324919696431172 \nu^{2} + 6483231491927948340 \nu + 132418327154742073$$$$)/ 8145162493392285342$$ $$\beta_{10}$$ $$=$$ $$($$$$-179879371221494400 \nu^{15} - 29141428977552177 \nu^{14} - 2542742935226983405 \nu^{13} + 1022739194797130202 \nu^{12} - 24570193984219251509 \nu^{11} + 11753873524378827529 \nu^{10} - 127695220462144763251 \nu^{9} + 82386015500475414730 \nu^{8} - 482700266196283198893 \nu^{7} + 305568546836809022718 \nu^{6} - 993021986431254858773 \nu^{5} + 739394346189369919740 \nu^{4} - 1433629910524559096484 \nu^{3} + 733568220305014599675 \nu^{2} - 494541205292733668733 \nu + 30206105399645762257$$$$)/ 29865595809105046254$$ $$\beta_{11}$$ $$=$$ $$($$$$-620256025308244210 \nu^{15} + 50103432857831551 \nu^{14} - 8638680375870167960 \nu^{13} + 5809893155253891246 \nu^{12} - 83982043314345290699 \nu^{11} + 61873258476416051714 \nu^{10} - 437022940638726066505 \nu^{9} + 392965187406109033461 \nu^{8} - 1667930633774952024110 \nu^{7} + 1443437174172852152354 \nu^{6} - 3461836791764849857929 \nu^{5} + 3308104027417853453307 \nu^{4} - 5105599504454018633775 \nu^{3} + 3605627230677670548928 \nu^{2} - 1686471009701311889821 \nu + 399293950415621241420$$$$)/ 89596787427315138762$$ $$\beta_{12}$$ $$=$$ $$($$$$-86629913538641744 \nu^{15} - 77242804403670180 \nu^{14} - 1236835964953177307 \nu^{13} - 381284216155455616 \nu^{12} - 11460782503377672238 \nu^{11} - 2663333790829272573 \nu^{10} - 57242336149328628010 \nu^{9} - 2411817133067431037 \nu^{8} - 201653889660631616011 \nu^{7} - 8394903393088675650 \nu^{6} - 364267988458938672326 \nu^{5} + 57221674384605398715 \nu^{4} - 409157911452156282963 \nu^{3} - 58286360548016280931 \nu^{2} + 16522759326121596240 \nu - 47318936372584568483$$$$)/ 8145162493392285342$$ $$\beta_{13}$$ $$=$$ $$($$$$97439698231600042 \nu^{15} + 78458574927513780 \nu^{14} + 1377142928376159829 \nu^{13} + 313743183676980932 \nu^{12} + 12812180382380897186 \nu^{11} + 2076353231317732509 \nu^{10} + 63888911527086652802 \nu^{9} - 1250853133884803369 \nu^{8} + 227030960526941146685 \nu^{7} - 1331667690206489862 \nu^{6} + 412494921314283552724 \nu^{5} - 77209646535123967995 \nu^{4} + 484007321203375662927 \nu^{3} + 60006246503279072795 \nu^{2} - 17223451703662159560 \nu + 45600882501242652745$$$$)/ 8145162493392285342$$ $$\beta_{14}$$ $$=$$ $$($$$$-701394259253807519 \nu^{15} + 201746312306747582 \nu^{14} - 9660382162145146246 \nu^{13} + 8392699134716861304 \nu^{12} - 94818453987311438191 \nu^{11} + 86894731226528281921 \nu^{10} - 490911922589610353345 \nu^{9} + 525991023983309494242 \nu^{8} - 1885421991512997903409 \nu^{7} + 1931262538204968412792 \nu^{6} - 3873575309990599570968 \nu^{5} + 4282702635188819483166 \nu^{4} - 5867610390280262955450 \nu^{3} + 4662497746928432728640 \nu^{2} - 1824736229540487564698 \nu + 428028607184168802240$$$$)/ 44798393713657569381$$ $$\beta_{15}$$ $$=$$ $$($$$$-64903022546980538 \nu^{15} + 15750424683004205 \nu^{14} - 888366754218911572 \nu^{13} + 746043927916644528 \nu^{12} - 8675846214483398614 \nu^{11} + 7706799959470290022 \nu^{10} - 44613237399977247962 \nu^{9} + 47019028360987770429 \nu^{8} - 170635286186528126926 \nu^{7} + 171251781949256722432 \nu^{6} - 347839987638179750526 \nu^{5} + 380469898316401499424 \nu^{4} - 528201088172267201040 \nu^{3} + 400975883068341362333 \nu^{2} - 165455963769864967928 \nu + 38855675475934455600$$$$)/ 4072581246696142671$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{10} + 3 \beta_{7} + \beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{13} - \beta_{12} + \beta_{9} - \beta_{8} + 5 \beta_{3} - \beta_{2} - 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + 3 \beta_{11} - 8 \beta_{10} - 2 \beta_{9} + 3 \beta_{8} - 15 \beta_{7} + 2 \beta_{5} + \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-10 \beta_{15} + 10 \beta_{14} - 12 \beta_{11} + 13 \beta_{10} + 9 \beta_{7} - \beta_{6} - 10 \beta_{5} - 28 \beta_{3} + 13 \beta_{2} - 9$$ $$\nu^{6}$$ $$=$$ $$-16 \beta_{13} - 15 \beta_{12} + 23 \beta_{9} - 36 \beta_{8} + 2 \beta_{4} + 15 \beta_{3} - 62 \beta_{2} - 15 \beta_{1} + 87$$ $$\nu^{7}$$ $$=$$ $$83 \beta_{15} - 85 \beta_{14} + 85 \beta_{13} + 83 \beta_{12} + 116 \beta_{11} - 126 \beta_{10} - 86 \beta_{9} + 116 \beta_{8} - 84 \beta_{7} + 14 \beta_{6} + 86 \beta_{5} - 14 \beta_{4} + 173 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-159 \beta_{15} + 173 \beta_{14} - 341 \beta_{11} + 484 \beta_{10} + 559 \beta_{7} - 32 \beta_{6} - 211 \beta_{5} - 167 \beta_{3} + 484 \beta_{2} - 559$$ $$\nu^{9}$$ $$=$$ $$-698 \beta_{13} - 666 \beta_{12} + 709 \beta_{9} - 1030 \beta_{8} + 144 \beta_{4} + 1157 \beta_{3} - 1119 \beta_{2} - 1157 \beta_{1} + 774$$ $$\nu^{10}$$ $$=$$ $$1483 \beta_{15} - 1627 \beta_{14} + 1627 \beta_{13} + 1483 \beta_{12} + 2991 \beta_{11} - 3828 \beta_{10} - 1818 \beta_{9} + 2991 \beta_{8} - 3881 \beta_{7} + 353 \beta_{6} + 1818 \beta_{5} - 353 \beta_{4} + 1637 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$-5336 \beta_{15} + 5689 \beta_{14} - 8799 \beta_{11} + 9586 \beta_{10} + 6921 \beta_{7} - 1317 \beta_{6} - 5778 \beta_{5} - 8236 \beta_{3} + 9586 \beta_{2} - 6921$$ $$\nu^{12}$$ $$=$$ $$-14366 \beta_{13} - 13049 \beta_{12} + 15309 \beta_{9} - 25391 \beta_{8} + 3374 \beta_{4} + 14979 \beta_{3} - 30640 \beta_{2} - 14979 \beta_{1} + 28516$$ $$\nu^{13}$$ $$=$$ $$42982 \beta_{15} - 46356 \beta_{14} + 46356 \beta_{13} + 42982 \beta_{12} + 73771 \beta_{11} - 80685 \beta_{10} - 47023 \beta_{9} + 73771 \beta_{8} - 60268 \beta_{7} + 11399 \beta_{6} + 47023 \beta_{5} - 11399 \beta_{4} + 61438 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-111474 \beta_{15} + 122873 \beta_{14} - 212211 \beta_{11} + 247477 \beta_{10} + 217976 \beta_{7} - 30122 \beta_{6} - 127617 \beta_{5} - 131633 \beta_{3} + 247477 \beta_{2} - 217976$$ $$\nu^{15}$$ $$=$$ $$-378336 \beta_{13} - 348214 \beta_{12} + 383301 \beta_{9} - 612683 \beta_{8} + 95993 \beta_{4} + 473839 \beta_{3} - 672673 \beta_{2} - 473839 \beta_{1} + 514759$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times$$.

 $$n$$ $$493$$ $$575$$ $$785$$ $$\chi(n)$$ $$-1 + \beta_{7}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1
 1.14440 − 1.98216i 1.12452 − 1.94772i 0.666692 − 1.15474i 0.251812 − 0.436150i 0.154058 − 0.266837i −0.856454 + 1.48342i −1.05150 + 1.82125i −1.43353 + 2.48295i 1.14440 + 1.98216i 1.12452 + 1.94772i 0.666692 + 1.15474i 0.251812 + 0.436150i 0.154058 + 0.266837i −0.856454 − 1.48342i −1.05150 − 1.82125i −1.43353 − 2.48295i
0 −1.14440 + 1.98216i 0 −0.175017 0.303139i 0 −0.303923 + 2.62824i 0 −1.11932 1.93871i 0
165.2 0 −1.12452 + 1.94772i 0 0.735486 + 1.27390i 0 0.221110 2.63650i 0 −1.02909 1.78243i 0
165.3 0 −0.666692 + 1.15474i 0 −0.883435 1.53015i 0 2.62366 0.341202i 0 0.611043 + 1.05836i 0
165.4 0 −0.251812 + 0.436150i 0 −1.11864 1.93755i 0 −1.70106 2.02642i 0 1.37318 + 2.37842i 0
165.5 0 −0.154058 + 0.266837i 0 1.65062 + 2.85896i 0 −2.57062 0.626020i 0 1.45253 + 2.51586i 0
165.6 0 0.856454 1.48342i 0 −1.07560 1.86299i 0 1.55192 + 2.14279i 0 0.0329726 + 0.0571103i 0
165.7 0 1.05150 1.82125i 0 1.46762 + 2.54199i 0 2.64377 0.102404i 0 −0.711302 1.23201i 0
165.8 0 1.43353 2.48295i 0 −0.601031 1.04102i 0 −2.46485 + 0.961519i 0 −2.61002 4.52069i 0
821.1 0 −1.14440 1.98216i 0 −0.175017 + 0.303139i 0 −0.303923 2.62824i 0 −1.11932 + 1.93871i 0
821.2 0 −1.12452 1.94772i 0 0.735486 1.27390i 0 0.221110 + 2.63650i 0 −1.02909 + 1.78243i 0
821.3 0 −0.666692 1.15474i 0 −0.883435 + 1.53015i 0 2.62366 + 0.341202i 0 0.611043 1.05836i 0
821.4 0 −0.251812 0.436150i 0 −1.11864 + 1.93755i 0 −1.70106 + 2.02642i 0 1.37318 2.37842i 0
821.5 0 −0.154058 0.266837i 0 1.65062 2.85896i 0 −2.57062 + 0.626020i 0 1.45253 2.51586i 0
821.6 0 0.856454 + 1.48342i 0 −1.07560 + 1.86299i 0 1.55192 2.14279i 0 0.0329726 0.0571103i 0
821.7 0 1.05150 + 1.82125i 0 1.46762 2.54199i 0 2.64377 + 0.102404i 0 −0.711302 + 1.23201i 0
821.8 0 1.43353 + 2.48295i 0 −0.601031 + 1.04102i 0 −2.46485 0.961519i 0 −2.61002 + 4.52069i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 821.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.i.d 16
7.c even 3 1 inner 1148.2.i.d 16
7.c even 3 1 8036.2.a.m 8
7.d odd 6 1 8036.2.a.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.i.d 16 1.a even 1 1 trivial
1148.2.i.d 16 7.c even 3 1 inner
8036.2.a.m 8 7.c even 3 1
8036.2.a.n 8 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1148, [\chi])$$:

 $$T_{3}^{16} + \cdots$$ $$T_{11}^{16} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$121 + 660 T + 2775 T^{2} + 5182 T^{3} + 8145 T^{4} + 4919 T^{5} + 5529 T^{6} + 2100 T^{7} + 2685 T^{8} + 568 T^{9} + 706 T^{10} + 87 T^{11} + 136 T^{12} + 8 T^{13} + 14 T^{14} + T^{16}$$
$5$ $$2601 + 9996 T + 32551 T^{2} + 34474 T^{3} + 41155 T^{4} + 25573 T^{5} + 24443 T^{6} + 12234 T^{7} + 8989 T^{8} + 2840 T^{9} + 1678 T^{10} + 333 T^{11} + 226 T^{12} + 24 T^{13} + 18 T^{14} + T^{16}$$
$7$ $$5764801 - 941192 T^{2} - 151263 T^{3} - 76832 T^{4} - 9261 T^{5} + 17983 T^{6} + 2709 T^{7} - 20 T^{8} + 387 T^{9} + 367 T^{10} - 27 T^{11} - 32 T^{12} - 9 T^{13} - 8 T^{14} + T^{16}$$
$11$ $$9 + 231 T + 6724 T^{2} - 21329 T^{3} + 57974 T^{4} - 61266 T^{5} + 65663 T^{6} - 29699 T^{7} + 32607 T^{8} - 14203 T^{9} + 8265 T^{10} - 2156 T^{11} + 884 T^{12} - 196 T^{13} + 63 T^{14} - 8 T^{15} + T^{16}$$
$13$ $$( -634 + 99 T + 1029 T^{2} + 222 T^{3} - 433 T^{4} - 223 T^{5} - 17 T^{6} + 7 T^{7} + T^{8} )^{2}$$
$17$ $$10673289 + 3293136 T + 10160397 T^{2} + 1177416 T^{3} + 6278742 T^{4} + 561834 T^{5} + 1857249 T^{6} - 134955 T^{7} + 335287 T^{8} - 17599 T^{9} + 29739 T^{10} - 2124 T^{11} + 1794 T^{12} - 67 T^{13} + 50 T^{14} - T^{15} + T^{16}$$
$19$ $$314721 - 360162 T + 969237 T^{2} - 366684 T^{3} + 1364850 T^{4} - 506256 T^{5} + 1034320 T^{6} + 104065 T^{7} + 187057 T^{8} + 15413 T^{9} + 19989 T^{10} + 2081 T^{11} + 1275 T^{12} + 98 T^{13} + 50 T^{14} + 4 T^{15} + T^{16}$$
$23$ $$81 - 2097 T + 51949 T^{2} - 66700 T^{3} + 152319 T^{4} - 193626 T^{5} + 343514 T^{6} - 352816 T^{7} + 284930 T^{8} - 152353 T^{9} + 62823 T^{10} - 17426 T^{11} + 3720 T^{12} - 459 T^{13} + 56 T^{14} - 3 T^{15} + T^{16}$$
$29$ $$( 486 - 8261 T - 3266 T^{2} + 5347 T^{3} + 2275 T^{4} - 296 T^{5} - 94 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$31$ $$3567553441 + 8243079832 T + 18299535835 T^{2} + 4027790958 T^{3} + 3014321107 T^{4} + 706133367 T^{5} + 351532899 T^{6} + 65544246 T^{7} + 18985199 T^{8} + 2333400 T^{9} + 562830 T^{10} + 53467 T^{11} + 11398 T^{12} + 636 T^{13} + 128 T^{14} + 4 T^{15} + T^{16}$$
$37$ $$1547536 + 10708352 T + 61230972 T^{2} + 104132216 T^{3} + 161598129 T^{4} - 27845615 T^{5} + 41843321 T^{6} - 24752374 T^{7} + 17581140 T^{8} - 6987741 T^{9} + 2112214 T^{10} - 435792 T^{11} + 68432 T^{12} - 7527 T^{13} + 614 T^{14} - 31 T^{15} + T^{16}$$
$41$ $$( -1 + T )^{16}$$
$43$ $$( -20588 - 23301 T - 208 T^{2} + 5381 T^{3} + 731 T^{4} - 372 T^{5} - 56 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$47$ $$539770865481 + 372555193881 T + 218659637083 T^{2} + 84755286508 T^{3} + 31845125964 T^{4} + 9803063762 T^{5} + 2819073163 T^{6} + 658812322 T^{7} + 135873802 T^{8} + 22799563 T^{9} + 3440435 T^{10} + 436785 T^{11} + 51972 T^{12} + 5036 T^{13} + 431 T^{14} + 24 T^{15} + T^{16}$$
$53$ $$5363072289 - 771949053 T + 2586021916 T^{2} + 29760381 T^{3} + 869883060 T^{4} - 599025 T^{5} + 118844725 T^{6} + 847593 T^{7} + 11703331 T^{8} + 91810 T^{9} + 536275 T^{10} + 30753 T^{11} + 16556 T^{12} + 428 T^{13} + 145 T^{14} - T^{15} + T^{16}$$
$59$ $$126765081 + 108232767 T + 238484035 T^{2} + 201859492 T^{3} + 337769768 T^{4} + 246125510 T^{5} + 172975087 T^{6} + 55306880 T^{7} + 17361620 T^{8} + 2257871 T^{9} + 670149 T^{10} + 70981 T^{11} + 16870 T^{12} + 664 T^{13} + 147 T^{14} + 4 T^{15} + T^{16}$$
$61$ $$5224976656 - 11001046528 T + 30374540964 T^{2} + 15241916592 T^{3} + 9448805561 T^{4} + 1937877994 T^{5} + 637573806 T^{6} + 70799694 T^{7} + 26098651 T^{8} + 1790562 T^{9} + 691305 T^{10} + 13487 T^{11} + 12583 T^{12} - 2 T^{13} + 149 T^{14} - 4 T^{15} + T^{16}$$
$67$ $$4942280319376 - 288565941448 T + 706028114824 T^{2} + 10747306026 T^{3} + 68514758863 T^{4} + 1114716791 T^{5} + 3130839530 T^{6} + 68163013 T^{7} + 101230149 T^{8} + 1600138 T^{9} + 1892826 T^{10} + 19961 T^{11} + 25619 T^{12} + 136 T^{13} + 196 T^{14} + T^{16}$$
$71$ $$( 896976 + 140655 T - 349614 T^{2} - 64500 T^{3} + 16094 T^{4} + 1439 T^{5} - 227 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$73$ $$7458361 + 9943571 T + 26327447 T^{2} + 39504600 T^{3} + 78279719 T^{4} + 92025927 T^{5} + 83259850 T^{6} + 50994939 T^{7} + 23776548 T^{8} + 7744715 T^{9} + 1902842 T^{10} + 303847 T^{11} + 35871 T^{12} + 1952 T^{13} + 231 T^{14} + 11 T^{15} + T^{16}$$
$79$ $$215074265121 - 347719186341 T + 496210820931 T^{2} - 214915632300 T^{3} + 106155546246 T^{4} + 20619515976 T^{5} + 11129853925 T^{6} + 1229703358 T^{7} + 302533862 T^{8} + 27433561 T^{9} + 5374135 T^{10} + 391471 T^{11} + 53640 T^{12} + 2898 T^{13} + 345 T^{14} + 14 T^{15} + T^{16}$$
$83$ $$( -3922452 + 2151213 T + 518556 T^{2} - 132669 T^{3} - 32249 T^{4} + 522 T^{5} + 536 T^{6} + 42 T^{7} + T^{8} )^{2}$$
$89$ $$5289507441 - 42712505307 T + 252843208488 T^{2} - 683947059039 T^{3} + 1360483326198 T^{4} - 487825293588 T^{5} + 138060258951 T^{6} - 20286059469 T^{7} + 2764574302 T^{8} - 210335050 T^{9} + 24009068 T^{10} - 1367872 T^{11} + 153235 T^{12} - 4244 T^{13} + 485 T^{14} - 11 T^{15} + T^{16}$$
$97$ $$( -69682 + 103607 T + 43765 T^{2} - 27642 T^{3} - 16692 T^{4} - 2761 T^{5} - 83 T^{6} + 16 T^{7} + T^{8} )^{2}$$