# Properties

 Label 84.9.m.a Level $84$ Weight $9$ Character orbit 84.m Analytic conductor $34.220$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 84.m (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$34.2198032451$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 38255 x^{8} + 1483053595 x^{6} - 139470625170 x^{5} + 5194605060018 x^{4} - 71600654137860 x^{3} + 119846615988780 x^{2} + \cdots + 15\!\cdots\!00$$ x^10 - 38255*x^8 + 1483053595*x^6 - 139470625170*x^5 + 5194605060018*x^4 - 71600654137860*x^3 + 119846615988780*x^2 + 4263507454127400*x + 15758720495290800 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{10}\cdot 3^{8}\cdot 7^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (27 \beta_1 - 54) q^{3} + (\beta_{5} + 93 \beta_1 + 92) q^{5} + (\beta_{9} - \beta_{4} + \beta_{3} - 35 \beta_1 + 139) q^{7} + ( - 2187 \beta_1 + 2187) q^{9}+O(q^{10})$$ q + (27*b1 - 54) * q^3 + (b5 + 93*b1 + 92) * q^5 + (b9 - b4 + b3 - 35*b1 + 139) * q^7 + (-2187*b1 + 2187) * q^9 $$q + (27 \beta_1 - 54) q^{3} + (\beta_{5} + 93 \beta_1 + 92) q^{5} + (\beta_{9} - \beta_{4} + \beta_{3} - 35 \beta_1 + 139) q^{7} + ( - 2187 \beta_1 + 2187) q^{9} + (2 \beta_{9} + \beta_{8} + \beta_{7} + 4 \beta_{6} + 10 \beta_{5} - 3 \beta_{3} - 5 \beta_{2} + \cdots - 6) q^{11}+ \cdots + (4374 \beta_{9} + 2187 \beta_{8} + 2187 \beta_{6} + 10935 \beta_{5} + \cdots - 395847) q^{99}+O(q^{100})$$ q + (27*b1 - 54) * q^3 + (b5 + 93*b1 + 92) * q^5 + (b9 - b4 + b3 - 35*b1 + 139) * q^7 + (-2187*b1 + 2187) * q^9 + (2*b9 + b8 + b7 + 4*b6 + 10*b5 - 3*b3 - 5*b2 - 170*b1 - 6) * q^11 + (14*b9 - b8 - 2*b7 + 5*b6 + 9*b5 - 6*b4 - 2*b3 - 9*b2 + 6087*b1 - 3044) * q^13 + (-27*b5 - 27*b2 - 7479) * q^15 + (18*b9 + b8 - b7 - 5*b6 + 6*b4 - 11*b3 + 41*b2 + 894*b1 - 1823) * q^17 + (12*b9 - 4*b8 - 2*b7 + 4*b6 - 72*b5 - 22*b4 - 7*b3 - 1984*b1 - 1913) * q^19 + (-27*b9 - 27*b6 + 54*b4 - 27*b3 + 4698*b1 - 6561) * q^21 + (-27*b9 + 16*b7 - 24*b6 - 149*b5 - 4*b4 - 2*b3 + 298*b2 - 62712*b1 + 62565) * q^23 + (-71*b9 + 6*b8 + 6*b7 + 99*b6 + 414*b5 + 113*b4 - 78*b3 - 207*b2 - 4121*b1 - 228) * q^25 + (118098*b1 - 59049) * q^27 + (-92*b9 - 22*b8 - 46*b6 + 168*b5 + 161*b4 + 208*b3 + 168*b2 - 46*b1 - 29031) * q^29 + (-216*b9 + 39*b8 - 39*b7 + 156*b6 + 18*b4 + 273*b3 - 396*b2 - 16058*b1 + 32395) * q^31 + (-108*b9 - 54*b8 - 27*b7 - 135*b6 - 405*b5 + 54*b4 + 189*b3 + 4563*b1 + 5049) * q^33 + (-852*b9 + 35*b8 - 21*b7 - 148*b6 - 1043*b5 + 433*b4 + 604*b3 + 266*b2 - 44997*b1 + 143800) * q^35 + (-1393*b9 + 18*b7 + 754*b6 + 585*b5 + 386*b4 + 193*b3 - 1170*b2 - 381634*b1 + 382026) * q^37 + (-594*b9 + 81*b8 + 81*b7 - 216*b6 - 486*b5 + 540*b4 + 189*b3 + 243*b2 - 246537*b1 + 270) * q^39 + (-2230*b9 + 13*b8 + 26*b7 + 928*b6 + 2338*b5 + 1433*b4 - 305*b3 - 2338*b2 + 887650*b1 - 443984) * q^41 + (-798*b9 - 33*b8 - 399*b6 + 828*b5 + 779*b4 + 1962*b3 + 828*b2 - 399*b1 - 86376) * q^43 + (2187*b2 - 203391*b1 + 404595) * q^45 + (-1776*b9 - 30*b8 - 15*b7 - 1065*b6 - 1343*b5 + 2472*b4 + 1590*b3 - 20622*b1 - 18739) * q^47 + (-1241*b9 - 259*b8 - 119*b7 + 290*b6 - 2709*b5 + 1168*b4 + 264*b3 + 8253*b2 - 363626*b1 + 1161929) * q^49 + (-1134*b9 + 81*b7 + 567*b6 + 1107*b5 + 324*b4 + 162*b3 - 2214*b2 - 72252*b1 + 73197) * q^51 + (-4269*b9 - 105*b8 - 105*b7 + 975*b6 - 5014*b5 + 4617*b4 - 801*b3 + 2507*b2 - 2133843*b1 + 2333) * q^53 + (-4666*b9 + 74*b8 + 148*b7 + 788*b6 + 6480*b5 + 2645*b4 - 238*b3 - 6480*b2 + 3079054*b1 - 1539683) * q^55 + (-54*b9 + 162*b8 - 27*b6 + 1944*b5 + 918*b4 + 297*b3 + 1944*b2 - 27*b1 + 156870) * q^57 + (-4406*b9 - 120*b8 + 120*b7 + 2199*b6 + 232*b4 + 4518*b3 - 10160*b2 - 1223884*b1 + 2455609) * q^59 + (-3693*b9 + 886*b8 + 443*b7 - 327*b6 - 9756*b5 + 7502*b4 + 712*b3 - 931566*b1 - 921868) * q^61 + (2187*b6 - 2187*b4 - 303993*b1 + 227448) * q^63 + (-1365*b9 - 890*b7 + 7530*b6 + 2127*b5 + 3320*b4 + 1660*b3 - 4254*b2 + 2994468*b1 - 2994001) * q^65 + (-6582*b9 - 348*b8 - 348*b7 - 4897*b6 - 8820*b5 + 4484*b4 + 3848*b3 + 4410*b2 - 4147853*b1 + 5459) * q^67 + (1566*b9 - 432*b8 - 864*b7 + 1350*b6 + 12069*b5 - 621*b4 - 594*b3 - 12069*b2 + 3386502*b1 - 1693332) * q^69 + (-2104*b9 - 95*b8 - 1052*b6 + 27373*b5 - 242*b4 + 5165*b3 + 27373*b2 - 1052*b1 + 11281684) * q^71 + (-3284*b9 - 871*b8 + 871*b7 + 2262*b6 + 2982*b4 + 5395*b3 - 18513*b2 - 2890169*b1 + 5795718) * q^73 + (783*b9 - 324*b8 - 162*b7 - 3240*b6 - 16767*b5 - 4968*b4 + 4779*b3 + 110700*b1 + 129168) * q^75 + (-11421*b9 + 2009*b8 + 2156*b7 + 6458*b6 - 5194*b5 + 19271*b4 + 1221*b3 + 21854*b2 - 6675651*b1 - 6548339) * q^77 + (-9692*b9 - 297*b7 - 8083*b6 - 2466*b5 - 4190*b4 - 2095*b3 + 4932*b2 + 8441359*b1 - 8441730) * q^79 - 4782969*b1 * q^81 + (-27720*b9 + 825*b8 + 1650*b7 - 3081*b6 + 32788*b5 + 13383*b4 + 1302*b3 - 32788*b2 + 8232315*b1 - 4115919) * q^83 + (2664*b9 + 567*b8 + 1332*b6 + 16596*b5 - 1223*b4 - 6093*b3 + 16596*b2 + 1332*b1 + 14245912) * q^85 + (621*b9 + 594*b8 - 594*b7 - 3132*b6 - 6831*b4 - 6858*b3 - 13608*b2 - 773523*b1 + 1564380) * q^87 + (-2338*b9 - 3328*b8 - 1664*b7 + 440*b6 - 78888*b5 + 3452*b4 - 1492*b3 + 4447018*b1 + 4526518) * q^89 + (15039*b9 - 2394*b8 - 3297*b7 - 18619*b6 - 81774*b5 - 24042*b4 + 3531*b3 + 65772*b2 - 10971302*b1 - 19117109) * q^91 + (11178*b9 + 3159*b7 - 15795*b6 - 10692*b5 - 6318*b4 - 3159*b3 + 21384*b2 + 1297539*b1 - 1305072) * q^93 + (11690*b9 - 2075*b8 - 2075*b7 + 7225*b6 - 29062*b5 - 9630*b4 - 6195*b3 + 14531*b2 - 28151474*b1 + 13501) * q^95 + (79550*b9 - 861*b8 - 1722*b7 + 2877*b6 + 114975*b5 - 39390*b4 - 1246*b3 - 114975*b2 + 45846697*b1 - 22923541) * q^97 + (4374*b9 + 2187*b8 + 2187*b6 + 10935*b5 - 4374*b4 - 8748*b3 + 10935*b2 + 2187*b1 - 395847) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9}+O(q^{10})$$ 10 * q - 405 * q^3 + 1389 * q^5 + 1217 * q^7 + 10935 * q^9 $$10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9} - 879 q^{11} - 75006 q^{15} - 13674 q^{17} - 29268 q^{19} - 42363 q^{21} + 312732 q^{23} - 22052 q^{25} - 289794 q^{29} + 242787 q^{31} + 71199 q^{33} + 1209372 q^{35} + 1913308 q^{37} - 1232334 q^{39} - 861848 q^{43} + 3037743 q^{45} - 305448 q^{47} + 9821659 q^{49} + 369198 q^{51} - 10663233 q^{53} + 1580472 q^{57} + 18410871 q^{59} - 13937808 q^{61} + 769824 q^{63} - 14966808 q^{65} - 20722822 q^{67} + 113032584 q^{71} + 43436322 q^{73} + 1786212 q^{75} - 98823405 q^{77} - 42189637 q^{79} - 23914845 q^{81} + 142602108 q^{85} + 11736657 q^{87} + 67171914 q^{89} - 246091266 q^{91} - 6555249 q^{93} - 140649894 q^{95} - 3844746 q^{99}+O(q^{100})$$ 10 * q - 405 * q^3 + 1389 * q^5 + 1217 * q^7 + 10935 * q^9 - 879 * q^11 - 75006 * q^15 - 13674 * q^17 - 29268 * q^19 - 42363 * q^21 + 312732 * q^23 - 22052 * q^25 - 289794 * q^29 + 242787 * q^31 + 71199 * q^33 + 1209372 * q^35 + 1913308 * q^37 - 1232334 * q^39 - 861848 * q^43 + 3037743 * q^45 - 305448 * q^47 + 9821659 * q^49 + 369198 * q^51 - 10663233 * q^53 + 1580472 * q^57 + 18410871 * q^59 - 13937808 * q^61 + 769824 * q^63 - 14966808 * q^65 - 20722822 * q^67 + 113032584 * q^71 + 43436322 * q^73 + 1786212 * q^75 - 98823405 * q^77 - 42189637 * q^79 - 23914845 * q^81 + 142602108 * q^85 + 11736657 * q^87 + 67171914 * q^89 - 246091266 * q^91 - 6555249 * q^93 - 140649894 * q^95 - 3844746 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 38255 x^{8} + 1483053595 x^{6} - 139470625170 x^{5} + 5194605060018 x^{4} - 71600654137860 x^{3} + 119846615988780 x^{2} + \cdots + 15\!\cdots\!00$$ :

 $$\beta_{1}$$ $$=$$ $$( 27\!\cdots\!39 \nu^{9} + \cdots + 25\!\cdots\!80 ) / 19\!\cdots\!80$$ (272153962454078370610521305639*v^9 - 7804201098746205229207193944284*v^8 - 10508963126474563853994149197651605*v^7 + 295816416513335686392206334903581040*v^6 + 407247873947801469805329113847737444505*v^5 - 49432323370232420671816106511466809823710*v^4 + 2361167606711182835213093021122334838884122*v^3 - 50311808783007248050547388838766467752232152*v^2 + 310007330258636047056371023027532001368945580*v + 2560106014199641516715702430770984676831048480) / 1918217540567141519798952270265420544310290280 $$\beta_{2}$$ $$=$$ $$( 53\!\cdots\!52 \nu^{9} + \cdots - 10\!\cdots\!10 ) / 57\!\cdots\!30$$ (532630612913952831343276059927313052*v^9 + 7870412045695935977777751606167139423*v^8 - 19994495655697126596428117946280492544005*v^7 - 294398589639692400218010034596205124844645*v^6 + 775352444641497378675785892510643463540083910*v^5 - 62876999260667221030581873809544404857848115080*v^4 + 2233381193858871090009298488823372903064410792956*v^3 - 40242428444367127486561034996929984578050403227196*v^2 + 586448131599276055459085751241829208346571679040140*v - 1078735705906769675415332633077119427873704893649810) / 5744581979613447066417912311377368175073241816030 $$\beta_{3}$$ $$=$$ $$( - 20\!\cdots\!70 \nu^{9} + \cdots - 25\!\cdots\!70 ) / 19\!\cdots\!10$$ (-204614186603676854088703130386730570*v^9 - 3246451082198705599763328915997444869*v^8 + 7627898151160140166317081586356955394110*v^7 + 118019207271863297887038018968875916038890*v^6 - 296034674386757575598878888639092410056688255*v^5 + 23946551205117981449732536802869245497095310730*v^4 - 898604579864163378810330486014199328406070050970*v^3 + 16944013800571923749735267866200265950623106507168*v^2 - 263577652563944625609641101301157681568421512160460*v - 2598814389843358830802168363680544682375898814149870) / 1914860659871149022139304103792456058357747272010 $$\beta_{4}$$ $$=$$ $$( 42\!\cdots\!33 \nu^{9} + \cdots + 33\!\cdots\!20 ) / 24\!\cdots\!65$$ (4204770312641237494384848933*v^9 + 48826521187590904857733922530*v^8 - 159590479431792254941050535412205*v^7 - 1820481341930148198652470129183830*v^6 + 6188375721497063895916391112390776055*v^5 - 515871636771755277492400915734308895140*v^4 + 16872517881168933617029012438624753922319*v^3 - 152333793468636799121418589411954938505920*v^2 - 1819449872910862947016101531720000456316920*v + 33464348625032329313820696140810334992737920) / 24018431490963621383815999754761571666265 $$\beta_{5}$$ $$=$$ $$( - 16\!\cdots\!17 \nu^{9} + \cdots - 42\!\cdots\!50 ) / 57\!\cdots\!30$$ (-1618439339072742924425118221464244117*v^9 - 2272436844388429587685896996872381793*v^8 + 62081366160749117215994564164255013429395*v^7 + 95583868167182138326520184262962142427190*v^6 - 2406385278346685811283500413143991438396776550*v^5 + 222027676673540012619104989929351376380509513790*v^4 - 7851473574291654631049349409823274193323312329526*v^3 + 93300647143174493333824794251437218022775609173896*v^2 + 34832055152195878846278109115358683323428396428460*v - 4294548229985885807395075384212369224834365652847450) / 5744581979613447066417912311377368175073241816030 $$\beta_{6}$$ $$=$$ $$( - 21\!\cdots\!39 \nu^{9} + \cdots - 14\!\cdots\!80 ) / 76\!\cdots\!40$$ (-2179483260990883745836929408241933939*v^9 - 75268334323461281196264785746980750240*v^8 + 81995458216232489810648481079270894506125*v^7 + 2848568278746643391370152984756331778875660*v^6 - 3180336698686086901112384376876310259729577085*v^5 + 193500313102657780166256211917881434787130996510*v^4 - 2838752209987202470948260202434074813749717371762*v^3 - 86572196292660480900155655360631781758804887647640*v^2 + 1020101344318079450977636648620781996729425256488420*v - 1485761827155484536329503772546514717139114517722680) / 7659442639484596088557216415169824233430989088040 $$\beta_{7}$$ $$=$$ $$( 31\!\cdots\!97 \nu^{9} + \cdots - 16\!\cdots\!40 ) / 10\!\cdots\!20$$ (318443371724602025613880767677021597*v^9 + 36578504403459235578355632876886077432*v^8 - 11582160003070321206854790597848640629635*v^7 - 1399932730182171774073007736150029718527940*v^6 + 448478185031561734110814343907563332462607675*v^5 + 9805364357520911139214322426785574771683111590*v^4 - 2525897975933825053216766986534381152420318526794*v^3 + 84864340784719869974155988031904158946959730446696*v^2 - 464422025971117234463332559981837535006056401378860*v - 1631428644599401802435045072699269149239259600518040) / 1094206091354942298365316630738546319061569869720 $$\beta_{8}$$ $$=$$ $$( 68\!\cdots\!98 \nu^{9} + \cdots - 49\!\cdots\!50 ) / 19\!\cdots\!10$$ (684070894048704959508734332969630298*v^9 - 27004682297995660733339267495566312551*v^8 - 27543685197067700292026033244281398806220*v^7 + 991271384753550460027738691536799868051660*v^6 + 1066738181227501224504906927562310518687879685*v^5 - 133799452544367282982944886967264428178391771690*v^4 + 5303505008538672248969271906736760067256100165424*v^3 - 61307193571096435941381695732933183527777990379088*v^2 - 241821570560664249202218062952866912341391278432260*v - 4966690042660575942850729595967244584029629695676950) / 1914860659871149022139304103792456058357747272010 $$\beta_{9}$$ $$=$$ $$( - 21\!\cdots\!87 \nu^{9} + \cdots - 30\!\cdots\!00 ) / 58\!\cdots\!70$$ (-21285468882417707370951983948134087*v^9 + 166802970724034577546937084981332038*v^8 + 814169950798001130867605097419409420375*v^7 - 6432943521603493526685349644701775026650*v^6 - 31568804114576470524035623949368973401800165*v^5 + 3217746710447923589952203054212526480124886640*v^4 - 133802328271531062019350797042258591132398654316*v^3 + 2339759522782711753070849660808964247862078994824*v^2 - 15346834890558830124294893024572179403018021941240*v - 30733528526416724760261931925663268401776539663600) / 58026080602156030973918306175528971465386280970
 $$\nu$$ $$=$$ $$( -7\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - 7\beta_{4} - \beta_{3} - 18\beta_{2} + 10\beta _1 - 2 ) / 252$$ (-7*b9 - b8 + b7 - b6 - 7*b4 - b3 - 18*b2 + 10*b1 - 2) / 252 $$\nu^{2}$$ $$=$$ $$( - 133 \beta_{9} - 25 \beta_{7} + 641 \beta_{6} - 612 \beta_{5} + 308 \beta_{4} + 154 \beta_{3} + 1224 \beta_{2} - 1286234 \beta _1 + 1285468 ) / 84$$ (-133*b9 - 25*b7 + 641*b6 - 612*b5 + 308*b4 + 154*b3 + 1224*b2 - 1286234*b1 + 1285468) / 84 $$\nu^{3}$$ $$=$$ $$( - 98350 \beta_{9} + 9962 \beta_{8} + 19924 \beta_{7} - 44872 \beta_{6} + 204264 \beta_{5} + 40859 \beta_{4} + 18278 \beta_{3} - 204264 \beta_{2} + 92222164 \beta _1 - 46106924 ) / 63$$ (-98350*b9 + 9962*b8 + 19924*b7 - 44872*b6 + 204264*b5 + 40859*b4 + 18278*b3 - 204264*b2 + 92222164*b1 - 46106924) / 63 $$\nu^{4}$$ $$=$$ $$( 7241731 \beta_{9} - 1087531 \beta_{8} - 1087531 \beta_{7} + 15842271 \beta_{6} - 31426884 \beta_{5} - 1339835 \beta_{4} - 12891323 \beta_{3} + 15713442 \beta_{2} + \cdots + 12762494 ) / 42$$ (7241731*b9 - 1087531*b8 - 1087531*b7 + 15842271*b6 - 31426884*b5 - 1339835*b4 - 12891323*b3 + 15713442*b2 - 25264706626*b1 + 12762494) / 42 $$\nu^{5}$$ $$=$$ $$( - 2674551586 \beta_{9} + 817640630 \beta_{8} + 408820315 \beta_{7} - 1888573513 \beta_{6} + 9578395467 \beta_{5} + 3869349974 \beta_{4} + \cdots + 2924353887244 ) / 63$$ (-2674551586*b9 + 817640630*b8 + 408820315*b7 - 1888573513*b6 + 9578395467*b5 + 3869349974*b4 + 3037270427*b3 + 2933192406112*b1 + 2924353887244) / 63 $$\nu^{6}$$ $$=$$ $$( 17306620562 \beta_{9} - 4718798746 \beta_{8} + 8653310281 \beta_{6} - 55189236609 \beta_{5} - 28925053171 \beta_{4} - 47985350151 \beta_{3} + \cdots - 77852302903406 ) / 3$$ (17306620562*b9 - 4718798746*b8 + 8653310281*b6 - 55189236609*b5 - 28925053171*b4 - 47985350151*b3 - 55189236609*b2 + 8653310281*b1 - 77852302903406) / 3 $$\nu^{7}$$ $$=$$ $$( 38202992333786 \beta_{9} + 17818961417759 \beta_{8} - 17818961417759 \beta_{7} + 62423963401070 \beta_{6} + 127412996300408 \beta_{4} + \cdots + 31\!\cdots\!16 ) / 63$$ (38202992333786*b9 + 17818961417759*b8 - 17818961417759*b7 + 62423963401070*b6 + 127412996300408*b4 + 107028965384381*b3 + 456393445840359*b2 - 159209696525729543*b1 + 317918394603635416) / 63 $$\nu^{8}$$ $$=$$ $$( - 51\!\cdots\!84 \beta_{9} + \cdots - 24\!\cdots\!06 ) / 21$$ (-5175651354674584*b9 + 1771166158318955*b7 - 12355733903245207*b6 + 18814139744811489*b5 - 5292283872463126*b4 - 2646141936231563*b3 - 37628279489622978*b2 + 24692408294605392058*b1 - 24670948012924349006) / 21 $$\nu^{9}$$ $$=$$ $$( 74\!\cdots\!70 \beta_{9} + \cdots + 81\!\cdots\!52 ) / 63$$ (7446890317225525670*b9 - 810177308705867941*b8 - 1620354617411735882*b7 + 8770729461200014061*b6 - 21874016157841757997*b5 - 1339986877350003442*b4 - 3193635589968627334*b3 + 21874016157841757997*b2 - 16274286664182963265697*b1 + 8135951602950850253152) / 63

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$43$$ $$73$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 - \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 −3.70642 + 2.13991i 38.0902 − 21.9914i 139.521 − 80.5522i −190.225 + 109.826i 16.3207 − 9.42278i −3.70642 − 2.13991i 38.0902 + 21.9914i 139.521 + 80.5522i −190.225 − 109.826i 16.3207 + 9.42278i
0 −40.5000 + 23.3827i 0 −489.814 282.795i 0 −1456.69 1908.63i 0 1093.50 1894.00i 0
61.2 0 −40.5000 + 23.3827i 0 −374.307 216.106i 0 83.7630 + 2399.54i 0 1093.50 1894.00i 0
61.3 0 −40.5000 + 23.3827i 0 111.592 + 64.4274i 0 2392.99 + 195.963i 0 1093.50 1894.00i 0
61.4 0 −40.5000 + 23.3827i 0 656.878 + 379.249i 0 1906.96 1458.87i 0 1093.50 1894.00i 0
61.5 0 −40.5000 + 23.3827i 0 790.151 + 456.194i 0 −2318.52 + 623.902i 0 1093.50 1894.00i 0
73.1 0 −40.5000 23.3827i 0 −489.814 + 282.795i 0 −1456.69 + 1908.63i 0 1093.50 + 1894.00i 0
73.2 0 −40.5000 23.3827i 0 −374.307 + 216.106i 0 83.7630 2399.54i 0 1093.50 + 1894.00i 0
73.3 0 −40.5000 23.3827i 0 111.592 64.4274i 0 2392.99 195.963i 0 1093.50 + 1894.00i 0
73.4 0 −40.5000 23.3827i 0 656.878 379.249i 0 1906.96 + 1458.87i 0 1093.50 + 1894.00i 0
73.5 0 −40.5000 23.3827i 0 790.151 456.194i 0 −2318.52 623.902i 0 1093.50 + 1894.00i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.5 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.9.m.a 10
3.b odd 2 1 252.9.z.b 10
7.d odd 6 1 inner 84.9.m.a 10
21.g even 6 1 252.9.z.b 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.9.m.a 10 1.a even 1 1 trivial
84.9.m.a 10 7.d odd 6 1 inner
252.9.z.b 10 3.b odd 2 1
252.9.z.b 10 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} - 1389 T_{5}^{9} - 876 T_{5}^{8} + 894492387 T_{5}^{7} - 19821737736 T_{5}^{6} - 381886886416725 T_{5}^{5} + \cdots + 47\!\cdots\!00$$ acting on $$S_{9}^{\mathrm{new}}(84, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$(T^{2} + 81 T + 2187)^{5}$$
$5$ $$T^{10} - 1389 T^{9} + \cdots + 47\!\cdots\!00$$
$7$ $$T^{10} - 1217 T^{9} + \cdots + 63\!\cdots\!01$$
$11$ $$T^{10} + 879 T^{9} + \cdots + 15\!\cdots\!00$$
$13$ $$T^{10} + 4745027382 T^{8} + \cdots + 67\!\cdots\!12$$
$17$ $$T^{10} + 13674 T^{9} + \cdots + 77\!\cdots\!00$$
$19$ $$T^{10} + 29268 T^{9} + \cdots + 22\!\cdots\!28$$
$23$ $$T^{10} - 312732 T^{9} + \cdots + 38\!\cdots\!56$$
$29$ $$(T^{5} + 144897 T^{4} + \cdots + 37\!\cdots\!32)^{2}$$
$31$ $$T^{10} - 242787 T^{9} + \cdots + 46\!\cdots\!43$$
$37$ $$T^{10} - 1913308 T^{9} + \cdots + 82\!\cdots\!24$$
$41$ $$T^{10} + 44132018588988 T^{8} + \cdots + 31\!\cdots\!92$$
$43$ $$(T^{5} + 430924 T^{4} + \cdots + 41\!\cdots\!00)^{2}$$
$47$ $$T^{10} + 305448 T^{9} + \cdots + 51\!\cdots\!00$$
$53$ $$T^{10} + 10663233 T^{9} + \cdots + 16\!\cdots\!96$$
$59$ $$T^{10} - 18410871 T^{9} + \cdots + 20\!\cdots\!92$$
$61$ $$T^{10} + 13937808 T^{9} + \cdots + 15\!\cdots\!00$$
$67$ $$T^{10} + 20722822 T^{9} + \cdots + 62\!\cdots\!04$$
$71$ $$(T^{5} - 56516292 T^{4} + \cdots - 22\!\cdots\!80)^{2}$$
$73$ $$T^{10} - 43436322 T^{9} + \cdots + 77\!\cdots\!00$$
$79$ $$T^{10} + 42189637 T^{9} + \cdots + 95\!\cdots\!25$$
$83$ $$T^{10} + \cdots + 34\!\cdots\!72$$
$89$ $$T^{10} - 67171914 T^{9} + \cdots + 26\!\cdots\!92$$
$97$ $$T^{10} + \cdots + 12\!\cdots\!00$$