# Properties

 Label 1764.2.b.n Level $1764$ Weight $2$ Character orbit 1764.b Analytic conductor $14.086$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1764.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.0856109166$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 4 x^{14} + 54 x^{12} - 112 x^{11} - 104 x^{10} + 1312 x^{9} - 3159 x^{8} + 2544 x^{7} + 4132 x^{6} - 16824 x^{5} + 27780 x^{4} - 26200 x^{3} + 14608 x^{2} - 4784 x + 782$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}\cdot 7^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 4 x^{14} + 54 x^{12} - 112 x^{11} - 104 x^{10} + 1312 x^{9} - 3159 x^{8} + 2544 x^{7} + 4132 x^{6} - 16824 x^{5} + 27780 x^{4} - 26200 x^{3} + 14608 x^{2} - 4784 x + 782$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$68460609096708 \nu^{15} - 4470673662808 \nu^{14} + 221129789385746 \nu^{13} - 100263474949006 \nu^{12} + 3386328542280566 \nu^{11} - 8346277362814708 \nu^{10} - 9975218138394550 \nu^{9} + 91208721367956313 \nu^{8} - 213323163203640222 \nu^{7} + 132548534496325226 \nu^{6} + 345859102636619816 \nu^{5} - 1157428177735265097 \nu^{4} + 1741603948057338728 \nu^{3} - 1414776392578940182 \nu^{2} + 581795728509992600 \nu - 115526876966468324$$$$)/ 14997061542603511$$ $$\beta_{2}$$ $$=$$ $$($$$$-327417918828952 \nu^{15} - 912489737968762 \nu^{14} - 2464634453668515 \nu^{13} - 4941371389549504 \nu^{12} - 23754738088595688 \nu^{11} - 19398225654807318 \nu^{10} + 66290267686202447 \nu^{9} - 283670421349742603 \nu^{8} + 13792080161554432 \nu^{7} + 733013280422225699 \nu^{6} - 1490194499653969402 \nu^{5} + 1222036940832765734 \nu^{4} + 954240083858638532 \nu^{3} - 3193825681085770178 \nu^{2} + 2032044391115129084 \nu - 472453024841471228$$$$)/ 66415558260101263$$ $$\beta_{3}$$ $$=$$ $$($$$$1555384052304487487 \nu^{15} + 1339604670703050332 \nu^{14} + 7531743358669999721 \nu^{13} + 6652426924136514709 \nu^{12} + 90531377909982386585 \nu^{11} - 95362399007409087852 \nu^{10} - 234348639348792239443 \nu^{9} + 1831404745105328437098 \nu^{8} - 3359157263983789674166 \nu^{7} + 1242927061655737977845 \nu^{6} + 7200798187189625037522 \nu^{5} - 19889480124946128542187 \nu^{4} + 26793508912799306677644 \nu^{3} - 19451040555700317738042 \nu^{2} + 8336186753002420522326 \nu - 1867599546248460713688$$$$)/$$$$13\!\cdots\!21$$ $$\beta_{4}$$ $$=$$ $$($$$$-318349714014117155 \nu^{15} - 169063652800363986 \nu^{14} - 1345989061870001853 \nu^{13} - 672215985049491150 \nu^{12} - 17423386714752992097 \nu^{11} + 26627463087863738529 \nu^{10} + 48407068192275527973 \nu^{9} - 391414509363635876612 \nu^{8} + 794214729102405745630 \nu^{7} - 373255694762441913900 \nu^{6} - 1523787615264866790274 \nu^{5} + 4520871847151325019132 \nu^{4} - 6370293219085152249418 \nu^{3} + 4835892465797678085258 \nu^{2} - 2036176636502016272590 \nu + 487073062122041378648$$$$)/ 18662771871088454903$$ $$\beta_{5}$$ $$=$$ $$($$$$473856923359748 \nu^{15} + 383840295516265 \nu^{14} + 2174756443991264 \nu^{13} + 1749715980623946 \nu^{12} + 26886219324590582 \nu^{11} - 31321770680120666 \nu^{10} - 76289123157682859 \nu^{9} + 563049199499742674 \nu^{8} - 1035407239122975332 \nu^{7} + 327835676897206279 \nu^{6} + 2307827064302842291 \nu^{5} - 6132682691001785589 \nu^{4} + 8024738563480367422 \nu^{3} - 5446994131157254103 \nu^{2} + 1919144838354128326 \nu - 358146927872032900$$$$)/ 14997061542603511$$ $$\beta_{6}$$ $$=$$ $$($$$$5500492503984100227 \nu^{15} + 4995786690360147679 \nu^{14} + 26478219286468497838 \nu^{13} + 23828334411265339223 \nu^{12} + 318208926056455796405 \nu^{11} - 328132761419129144004 \nu^{10} - 874281158143623647920 \nu^{9} + 6416571148191675290724 \nu^{8} - 11527558565646261639380 \nu^{7} + 3479438641302538497722 \nu^{6} + 25844146842968921774278 \nu^{5} - 68798329378779488859130 \nu^{4} + 90036644526399417686818 \nu^{3} - 62326650020669503237070 \nu^{2} + 24339909928772330731036 \nu - 5087914532726904122076$$$$)/$$$$13\!\cdots\!21$$ $$\beta_{7}$$ $$=$$ $$($$$$5952725304690144831 \nu^{15} + 4913373504941792009 \nu^{14} + 27689331148288188771 \nu^{13} + 22550928196542473072 \nu^{12} + 339010718516081477989 \nu^{11} - 388470965209901181124 \nu^{10} - 950764010116474908521 \nu^{9} + 7026897442921298831606 \nu^{8} - 12970938757912323132088 \nu^{7} + 4252322434555389872600 \nu^{6} + 28339859091945305749350 \nu^{5} - 76671489479188666427654 \nu^{4} + 101240151222016002112922 \nu^{3} - 70636213458116768776102 \nu^{2} + 26778488614860426571854 \nu - 5496866961938737811274$$$$)/$$$$13\!\cdots\!21$$ $$\beta_{8}$$ $$=$$ $$($$$$25902327859007496 \nu^{15} + 21577079064058868 \nu^{14} + 121685367454370832 \nu^{13} + 101698244496858797 \nu^{12} + 1483369668724706806 \nu^{11} - 1665246494940919959 \nu^{10} - 4078856972771178701 \nu^{9} + 30589633647423284589 \nu^{8} - 56420217111824698700 \nu^{7} + 19005489443147365107 \nu^{6} + 123088666360044540971 \nu^{5} - 334207906277498618299 \nu^{4} + 442421329376818071598 \nu^{3} - 309216748734375465357 \nu^{2} + 117379544690231687414 \nu - 21432466723742801850$$$$)/ 464908907820708841$$ $$\beta_{9}$$ $$=$$ $$($$$$7311390845285256291 \nu^{15} + 5144247759478178881 \nu^{14} + 33052494395288935583 \nu^{13} + 23464216531976602108 \nu^{12} + 412304745350307442286 \nu^{11} - 527646370388946126359 \nu^{10} - 1119851439923064088482 \nu^{9} + 8796643036979951155484 \nu^{8} - 16933879259688981418781 \nu^{7} + 6901825506220311144657 \nu^{6} + 34736847577918930053593 \nu^{5} - 98539036507797617339480 \nu^{4} + 134668567438981170920014 \nu^{3} - 98761332546754697649693 \nu^{2} + 39340148768838995107316 \nu - 8239028835957057526486$$$$)/$$$$13\!\cdots\!21$$ $$\beta_{10}$$ $$=$$ $$($$$$26598978070628094 \nu^{15} + 20754861575511394 \nu^{14} + 121945297597469092 \nu^{13} + 94685759422792651 \nu^{12} + 1507636575022232512 \nu^{11} - 1803486805708866118 \nu^{10} - 4207419896042615164 \nu^{9} + 31667827257227959771 \nu^{8} - 59195530313521122086 \nu^{7} + 20748972299418636178 \nu^{6} + 127442752230905249876 \nu^{5} - 348322567732844885217 \nu^{4} + 464281601916429123172 \nu^{3} - 327276480336990007534 \nu^{2} + 123913467597404059828 \nu - 24957690805039740937$$$$)/ 464908907820708841$$ $$\beta_{11}$$ $$=$$ $$($$$$8448544333599963272 \nu^{15} + 6459958212915796841 \nu^{14} + 38949945226253425308 \nu^{13} + 30128663474405533398 \nu^{12} + 480593764804618359742 \nu^{11} - 577012814267616188201 \nu^{10} - 1305900993760500402942 \nu^{9} + 10081844323059287945776 \nu^{8} - 19013659824524795362304 \nu^{7} + 7168427247637577844956 \nu^{6} + 40100295800739031719712 \nu^{5} - 111495933370675259317678 \nu^{4} + 150268019875044278229648 \nu^{3} - 108297352271035265759942 \nu^{2} + 42615858527687211502860 \nu - 8762536382973257608560$$$$)/$$$$13\!\cdots\!21$$ $$\beta_{12}$$ $$=$$ $$($$$$-1244394735841324 \nu^{15} - 1015254487117297 \nu^{14} - 5815471027979606 \nu^{13} - 4747356334044193 \nu^{12} - 71097289196119918 \nu^{11} + 81359376829844749 \nu^{10} + 195369841406174707 \nu^{9} - 1472274308570746587 \nu^{8} + 2732180783018590880 \nu^{7} - 950033829183946954 \nu^{6} - 5888232645967983481 \nu^{5} + 16126119659137989335 \nu^{4} - 21484874797011548774 \nu^{3} + 15258438472298117941 \nu^{2} - 5959302325712781184 \nu + 1231061099793606132$$$$)/ 14997061542603511$$ $$\beta_{13}$$ $$=$$ $$($$$$-38837071408467006 \nu^{15} - 30709761853974942 \nu^{14} - 180300822619285700 \nu^{13} - 143011704542588177 \nu^{12} - 2212341935495582864 \nu^{11} + 2600106172853103294 \nu^{10} + 6064169002183889708 \nu^{9} - 46102238907939724050 \nu^{8} + 86397532556884843782 \nu^{7} - 31309069072667929038 \nu^{6} - 183745007679448037388 \nu^{5} + 508101002775460820933 \nu^{4} - 681018271981825579836 \nu^{3} + 488427055994476300206 \nu^{2} - 192955113465912973604 \nu + 40106176486555581423$$$$)/ 464908907820708841$$ $$\beta_{14}$$ $$=$$ $$($$$$15998972551103279917 \nu^{15} + 13146545739463821814 \nu^{14} + 74380372178245371088 \nu^{13} + 60446949971898835049 \nu^{12} + 911342693405039975939 \nu^{11} - 1046558893404974928844 \nu^{10} - 2549704930836326331342 \nu^{9} + 18900925418135661838222 \nu^{8} - 34925501838727448675442 \nu^{7} + 11538713375264870681853 \nu^{6} + 76153154961304809095912 \nu^{5} - 206238287624443883630230 \nu^{4} + 272728671644159240844356 \nu^{3} - 190857346846836674542978 \nu^{2} + 73063629911885826958694 \nu - 15209681969564397709940$$$$)/$$$$13\!\cdots\!21$$ $$\beta_{15}$$ $$=$$ $$($$$$2407697249379471303 \nu^{15} + 1962337849022174960 \nu^{14} + 11269452242416570801 \nu^{13} + 9250267627009115113 \nu^{12} + 137786151121143971234 \nu^{11} - 157048626672931238306 \nu^{10} - 375855880039528846246 \nu^{9} + 2851912517811313200749 \nu^{8} - 5288343572174182331263 \nu^{7} + 1853689065595543927095 \nu^{6} + 11414646474950630716223 \nu^{5} - 31237695433043179805246 \nu^{4} + 41586859446763682767190 \nu^{3} - 29458323721466198065049 \nu^{2} + 11345194533084915125680 \nu - 2256954365421484403296$$$$)/ 18662771871088454903$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{15} + 3 \beta_{14} - 7 \beta_{11} - \beta_{9} - 7 \beta_{7} - \beta_{6} + 7 \beta_{3} + 7 \beta_{1}$$$$)/14$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{15} - 2 \beta_{14} + 14 \beta_{13} - 18 \beta_{12} + 7 \beta_{11} - 14 \beta_{10} - 4 \beta_{9} + 6 \beta_{8} + 7 \beta_{7} - 4 \beta_{6} + 2 \beta_{5} - 7 \beta_{4} - 4 \beta_{2} + 14 \beta_{1} - 14$$$$)/14$$ $$\nu^{3}$$ $$=$$ $$($$$$14 \beta_{15} - 7 \beta_{14} - 21 \beta_{13} + 39 \beta_{12} + 7 \beta_{11} + 42 \beta_{10} - 28 \beta_{9} - 6 \beta_{8} - 28 \beta_{7} + 35 \beta_{6} - 9 \beta_{5} - 28 \beta_{4} - 42 \beta_{3} + 18 \beta_{2} - 49 \beta_{1} + 21$$$$)/14$$ $$\nu^{4}$$ $$=$$ $$($$$$-26 \beta_{15} + 16 \beta_{14} - 35 \beta_{13} + 68 \beta_{12} - 7 \beta_{11} + 49 \beta_{10} + 46 \beta_{9} + 24 \beta_{8} + 21 \beta_{7} - 38 \beta_{6} - 48 \beta_{5} + 49 \beta_{4} + 56 \beta_{3} - 30 \beta_{2} + 7 \beta_{1} - 56$$$$)/7$$ $$\nu^{5}$$ $$=$$ $$($$$$-161 \beta_{15} - 196 \beta_{14} + 315 \beta_{13} - 590 \beta_{12} + 350 \beta_{11} - 490 \beta_{10} - 105 \beta_{9} - 130 \beta_{8} + 637 \beta_{7} + 400 \beta_{5} + 161 \beta_{4} + 196 \beta_{3} + 180 \beta_{2} + 63 \beta_{1} + 245$$$$)/14$$ $$\nu^{6}$$ $$=$$ $$($$$$422 \beta_{15} + 317 \beta_{14} - 420 \beta_{13} + 610 \beta_{12} - 672 \beta_{11} + 560 \beta_{10} + 60 \beta_{9} - 110 \beta_{8} - 1442 \beta_{7} + 270 \beta_{6} - 130 \beta_{5} - 546 \beta_{4} - 763 \beta_{3} + 141 \beta_{2} - 1015 \beta_{1} + 1064$$$$)/7$$ $$\nu^{7}$$ $$=$$ $$($$$$-1152 \beta_{15} + 416 \beta_{14} + 1617 \beta_{13} - 1638 \beta_{12} - 595 \beta_{11} - 2254 \beta_{10} + 1854 \beta_{9} + 1869 \beta_{8} + 2814 \beta_{7} - 2164 \beta_{6} - 1239 \beta_{5} + 3388 \beta_{4} + 4914 \beta_{3} - 2716 \beta_{2} + 9709 \beta_{1} - 11907$$$$)/14$$ $$\nu^{8}$$ $$=$$ $$($$$$-680 \beta_{15} - 4208 \beta_{14} + 3115 \beta_{13} - 6244 \beta_{12} + 7336 \beta_{11} - 4263 \beta_{10} - 5280 \beta_{9} - 2072 \beta_{8} + 6188 \beta_{7} + 3148 \beta_{6} + 4564 \beta_{5} - 4256 \beta_{4} - 5908 \beta_{3} + 2884 \beta_{2} - 4634 \beta_{1} + 9772$$$$)/7$$ $$\nu^{9}$$ $$=$$ $$($$$$20667 \beta_{15} + 24972 \beta_{14} - 52143 \beta_{13} + 87744 \beta_{12} - 47943 \beta_{11} + 74214 \beta_{10} + 14209 \beta_{9} + 1440 \beta_{8} - 78106 \beta_{7} + 4150 \beta_{6} - 37992 \beta_{5} - 9716 \beta_{4} - 13580 \beta_{3} - 1968 \beta_{2} - 54719 \beta_{1} + 24885$$$$)/14$$ $$\nu^{10}$$ $$=$$ $$($$$$-49773 \beta_{15} - 15858 \beta_{14} + 61257 \beta_{13} - 91989 \beta_{12} + 42483 \beta_{11} - 86562 \beta_{10} + 26937 \beta_{9} + 20058 \beta_{8} + 139713 \beta_{7} - 54333 \beta_{6} + 18145 \beta_{5} + 80458 \beta_{4} + 113596 \beta_{3} - 28093 \beta_{2} + 143997 \beta_{1} - 143297$$$$)/7$$ $$\nu^{11}$$ $$=$$ $$($$$$168752 \beta_{15} - 125850 \beta_{14} + 45353 \beta_{13} - 193248 \beta_{12} + 158557 \beta_{11} - 65296 \beta_{10} - 346956 \beta_{9} - 242275 \beta_{8} - 273000 \beta_{7} + 365910 \beta_{6} + 323389 \beta_{5} - 495488 \beta_{4} - 702338 \beta_{3} + 343596 \beta_{2} - 829759 \beta_{1} + 1215907$$$$)/14$$ $$\nu^{12}$$ $$=$$ $$($$$$248970 \beta_{15} + 554352 \beta_{14} - 683242 \beta_{13} + 1268366 \beta_{12} - 1002050 \beta_{11} + 964677 \beta_{10} + 536090 \beta_{9} + 285378 \beta_{8} - 1250830 \beta_{7} - 201262 \beta_{6} - 809932 \beta_{5} + 165634 \beta_{4} + 232960 \beta_{3} - 403542 \beta_{2} + 216783 \beta_{1} - 989653$$$$)/7$$ $$\nu^{13}$$ $$=$$ $$($$$$-3723771 \beta_{15} - 3352582 \beta_{14} + 6092905 \beta_{13} - 9815208 \beta_{12} + 6721715 \beta_{11} - 8620248 \beta_{10} - 1012029 \beta_{9} + 494416 \beta_{8} + 13076630 \beta_{7} - 1914294 \beta_{6} + 3571360 \beta_{5} + 3561852 \beta_{4} + 5037298 \beta_{3} - 702468 \beta_{2} + 9279081 \beta_{1} - 7018557$$$$)/14$$ $$\nu^{14}$$ $$=$$ $$($$$$5365394 \beta_{15} + 517824 \beta_{14} - 6184157 \beta_{13} + 8209161 \beta_{12} - 2486463 \beta_{11} + 8747046 \beta_{10} - 4630516 \beta_{9} - 3998767 \beta_{8} - 13988520 \beta_{7} + 7065441 \beta_{6} + 595888 \beta_{5} - 10480421 \beta_{4} - 14816676 \beta_{3} + 5651431 \beta_{2} - 22382745 \beta_{1} + 25481323$$$$)/7$$ $$\nu^{15}$$ $$=$$ $$($$$$-7914484 \beta_{15} + 31441000 \beta_{14} - 20980519 \beta_{13} + 49492304 \beta_{12} - 50219043 \beta_{11} + 29690458 \beta_{10} + 52372088 \beta_{9} + 29316639 \beta_{8} - 15267476 \beta_{7} - 42652016 \beta_{6} - 49829581 \beta_{5} + 55745130 \beta_{4} + 78857142 \beta_{3} - 41465038 \beta_{2} + 83849479 \beta_{1} - 139549431$$$$)/14$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 0.271975 + 0.405613i 0.271975 + 1.48800i 0.271975 − 1.48800i 0.271975 − 0.405613i 0.812979 + 2.57288i 0.812979 − 0.0402481i 0.812979 + 0.0402481i 0.812979 − 2.57288i −2.22719 − 0.0402481i −2.22719 + 2.57288i −2.22719 − 2.57288i −2.22719 + 0.0402481i 1.14224 + 1.48800i 1.14224 + 0.405613i 1.14224 − 0.405613i 1.14224 − 1.48800i
−1.05050 0.946809i 0 0.207107 + 1.98925i 1.60804i 0 0 1.66587 2.28580i 0 −1.52250 + 1.68925i
1567.2 −1.05050 0.946809i 0 0.207107 + 1.98925i 1.60804i 0 0 1.66587 2.28580i 0 1.52250 1.68925i
1567.3 −1.05050 + 0.946809i 0 0.207107 1.98925i 1.60804i 0 0 1.66587 + 2.28580i 0 1.52250 + 1.68925i
1567.4 −1.05050 + 0.946809i 0 0.207107 1.98925i 1.60804i 0 0 1.66587 + 2.28580i 0 −1.52250 1.68925i
1567.5 −0.629640 1.26631i 0 −1.20711 + 1.59465i 2.32685i 0 0 2.77937 + 0.524525i 0 −2.94652 + 1.46508i
1567.6 −0.629640 1.26631i 0 −1.20711 + 1.59465i 2.32685i 0 0 2.77937 + 0.524525i 0 2.94652 1.46508i
1567.7 −0.629640 + 1.26631i 0 −1.20711 1.59465i 2.32685i 0 0 2.77937 0.524525i 0 2.94652 + 1.46508i
1567.8 −0.629640 + 1.26631i 0 −1.20711 1.59465i 2.32685i 0 0 2.77937 0.524525i 0 −2.94652 1.46508i
1567.9 0.629640 1.26631i 0 −1.20711 1.59465i 2.32685i 0 0 −2.77937 + 0.524525i 0 −2.94652 1.46508i
1567.10 0.629640 1.26631i 0 −1.20711 1.59465i 2.32685i 0 0 −2.77937 + 0.524525i 0 2.94652 + 1.46508i
1567.11 0.629640 + 1.26631i 0 −1.20711 + 1.59465i 2.32685i 0 0 −2.77937 0.524525i 0 2.94652 1.46508i
1567.12 0.629640 + 1.26631i 0 −1.20711 + 1.59465i 2.32685i 0 0 −2.77937 0.524525i 0 −2.94652 + 1.46508i
1567.13 1.05050 0.946809i 0 0.207107 1.98925i 1.60804i 0 0 −1.66587 2.28580i 0 −1.52250 1.68925i
1567.14 1.05050 0.946809i 0 0.207107 1.98925i 1.60804i 0 0 −1.66587 2.28580i 0 1.52250 + 1.68925i
1567.15 1.05050 + 0.946809i 0 0.207107 + 1.98925i 1.60804i 0 0 −1.66587 + 2.28580i 0 1.52250 1.68925i
1567.16 1.05050 + 0.946809i 0 0.207107 + 1.98925i 1.60804i 0 0 −1.66587 + 2.28580i 0 −1.52250 + 1.68925i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1567.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.b.n 16
3.b odd 2 1 inner 1764.2.b.n 16
4.b odd 2 1 inner 1764.2.b.n 16
7.b odd 2 1 inner 1764.2.b.n 16
12.b even 2 1 inner 1764.2.b.n 16
21.c even 2 1 inner 1764.2.b.n 16
28.d even 2 1 inner 1764.2.b.n 16
84.h odd 2 1 inner 1764.2.b.n 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.b.n 16 1.a even 1 1 trivial
1764.2.b.n 16 3.b odd 2 1 inner
1764.2.b.n 16 4.b odd 2 1 inner
1764.2.b.n 16 7.b odd 2 1 inner
1764.2.b.n 16 12.b even 2 1 inner
1764.2.b.n 16 21.c even 2 1 inner
1764.2.b.n 16 28.d even 2 1 inner
1764.2.b.n 16 84.h odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1764, [\chi])$$:

 $$T_{5}^{4} + 8 T_{5}^{2} + 14$$ $$T_{11}^{4} + 20 T_{11}^{2} + 92$$ $$T_{19}^{4} - 88 T_{19}^{2} + 1288$$ $$T_{29}^{4} - 84 T_{29}^{2} + 1372$$ $$T_{53}^{4} - 80 T_{53}^{2} + 448$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 + 8 T^{2} + 7 T^{4} + 2 T^{6} + T^{8} )^{2}$$
$3$ $$T^{16}$$
$5$ $$( 14 + 8 T^{2} + T^{4} )^{4}$$
$7$ $$T^{16}$$
$11$ $$( 92 + 20 T^{2} + T^{4} )^{4}$$
$13$ $$( 98 + 20 T^{2} + T^{4} )^{4}$$
$17$ $$( 14 + 8 T^{2} + T^{4} )^{4}$$
$19$ $$( 1288 - 88 T^{2} + T^{4} )^{4}$$
$23$ $$( 92 + 44 T^{2} + T^{4} )^{4}$$
$29$ $$( 1372 - 84 T^{2} + T^{4} )^{4}$$
$31$ $$( 1288 - 120 T^{2} + T^{4} )^{4}$$
$37$ $$( -2 + 8 T + T^{2} )^{8}$$
$41$ $$( 1134 + 144 T^{2} + T^{4} )^{4}$$
$43$ $$( 2576 + 104 T^{2} + T^{4} )^{4}$$
$47$ $$( 9016 - 192 T^{2} + T^{4} )^{4}$$
$53$ $$( 448 - 80 T^{2} + T^{4} )^{4}$$
$59$ $$( 9016 - 192 T^{2} + T^{4} )^{4}$$
$61$ $$( 98 + 20 T^{2} + T^{4} )^{4}$$
$67$ $$( 10304 + 208 T^{2} + T^{4} )^{4}$$
$71$ $$( 7452 + 180 T^{2} + T^{4} )^{4}$$
$73$ $$( 4802 + 164 T^{2} + T^{4} )^{4}$$
$79$ $$( 2576 + 248 T^{2} + T^{4} )^{4}$$
$83$ $$( 36064 - 384 T^{2} + T^{4} )^{4}$$
$89$ $$( 14 + 184 T^{2} + T^{4} )^{4}$$
$97$ $$( 4802 + 196 T^{2} + T^{4} )^{4}$$