# Properties

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

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$1148 = 2^{2} \cdot 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1148.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.16682615204$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 3 x^{15} + 12 x^{14} - 19 x^{13} + 49 x^{12} - 91 x^{11} + 269 x^{10} - 367 x^{9} + 1058 x^{8} - 1675 x^{7} + 2755 x^{6} - 2800 x^{5} + 4032 x^{4} - 2186 x^{3} + 2189 x^{2} + 76 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} ) q^{3} + ( -1 + \beta_{9} - \beta_{10} ) q^{5} + \beta_{9} q^{7} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} ) q^{3} + ( -1 + \beta_{9} - \beta_{10} ) q^{5} + \beta_{9} q^{7} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{9} + ( -2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{13} + \beta_{15} ) q^{11} + ( -2 - \beta_{1} + \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{14} - \beta_{15} ) q^{13} + ( -1 + \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{14} - \beta_{15} ) q^{15} + ( -2 \beta_{4} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{17} + ( 2 + \beta_{1} + \beta_{6} - 2 \beta_{8} + 2 \beta_{12} + \beta_{14} ) q^{19} + \beta_{5} q^{21} + ( 3 \beta_{8} + 3 \beta_{9} - \beta_{15} ) q^{23} + ( -\beta_{1} + \beta_{5} - \beta_{6} + 2 \beta_{10} + 2 \beta_{13} - \beta_{14} ) q^{25} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{14} - \beta_{15} ) q^{27} + ( -1 + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} - 3 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{29} + ( \beta_{1} + \beta_{4} - \beta_{5} + 3 \beta_{7} + \beta_{8} - 3 \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} ) q^{31} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{11} - 2 \beta_{13} - 2 \beta_{15} ) q^{33} + ( -\beta_{8} - \beta_{9} + \beta_{13} ) q^{35} + ( -1 - \beta_{1} + 6 \beta_{2} + 3 \beta_{3} + \beta_{4} + 3 \beta_{6} + 2 \beta_{7} + \beta_{9} + 3 \beta_{11} - \beta_{12} - 3 \beta_{14} - 3 \beta_{15} ) q^{37} + ( -\beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{14} - \beta_{15} ) q^{39} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} + \beta_{10} - 2 \beta_{11} + 3 \beta_{12} + 2 \beta_{14} ) q^{41} + ( 1 + \beta_{1} - 4 \beta_{2} + 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{43} + ( -1 + \beta_{1} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{15} ) q^{45} + ( -4 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 4 \beta_{7} + 5 \beta_{8} + 5 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 3 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{47} -\beta_{8} q^{49} + ( \beta_{1} + 3 \beta_{2} + \beta_{4} - \beta_{5} - 2 \beta_{8} - 2 \beta_{11} - \beta_{13} ) q^{51} + ( -3 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{7} + 3 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{53} + ( 1 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{12} - \beta_{13} ) q^{55} + ( -3 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} + 3 \beta_{8} + 2 \beta_{9} + \beta_{11} - 3 \beta_{12} - 3 \beta_{14} ) q^{57} + ( 3 + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - \beta_{10} + 3 \beta_{15} ) q^{59} + ( -3 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} + 3 \beta_{8} + \beta_{9} + 4 \beta_{10} + 2 \beta_{11} + \beta_{12} + 4 \beta_{13} - 2 \beta_{14} ) q^{61} + ( -\beta_{1} + \beta_{5} - \beta_{6} - \beta_{14} ) q^{63} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{65} + ( -3 + 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{6} - \beta_{7} + 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{12} - 3 \beta_{14} ) q^{67} + ( 2 - \beta_{1} - 3 \beta_{2} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{69} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + 3 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{71} + ( 6 - 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{73} + ( 1 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{75} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{12} - \beta_{14} ) q^{77} + ( 4 - 4 \beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{79} + ( -4 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{81} + ( -3 - 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{12} + \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{83} + ( 3 + \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} - 4 \beta_{7} + 4 \beta_{8} - 2 \beta_{12} - 2 \beta_{13} ) q^{85} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} + \beta_{14} - 2 \beta_{15} ) q^{87} + ( 4 + 2 \beta_{1} + 2 \beta_{6} - 4 \beta_{8} - \beta_{9} + 3 \beta_{10} + 3 \beta_{12} + 3 \beta_{13} + 4 \beta_{14} ) q^{89} + ( -2 + \beta_{2} - \beta_{14} - \beta_{15} ) q^{91} + ( \beta_{1} - \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{8} + 3 \beta_{10} + 3 \beta_{12} + 3 \beta_{13} + \beta_{15} ) q^{93} + ( 5 - \beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{6} + 5 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{95} + ( -3 - 4 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{6} - 5 \beta_{7} + 3 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} - \beta_{12} - 4 \beta_{14} - 4 \beta_{15} ) q^{97} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} + 3 \beta_{8} - \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 2q^{3} - 13q^{5} + 4q^{7} + 2q^{9} + O(q^{10})$$ $$16q - 2q^{3} - 13q^{5} + 4q^{7} + 2q^{9} - q^{11} - 6q^{13} - q^{17} + 15q^{19} + 2q^{21} + 27q^{23} - 3q^{25} + 28q^{27} - q^{29} - 14q^{31} - 13q^{33} - 12q^{35} - 16q^{37} + 10q^{39} + 26q^{41} + 5q^{43} - 9q^{45} - 14q^{47} - 4q^{49} + 4q^{51} - 20q^{53} + 10q^{55} - 13q^{57} - 47q^{61} + 3q^{63} - 29q^{65} - 27q^{67} + 15q^{69} - 11q^{71} + 70q^{73} + 14q^{75} + q^{77} + 30q^{79} - 72q^{81} - 78q^{83} + 72q^{85} + 21q^{87} + 17q^{89} - 24q^{91} - 7q^{93} + 27q^{95} - 17q^{97} + 13q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{15} + 12 x^{14} - 19 x^{13} + 49 x^{12} - 91 x^{11} + 269 x^{10} - 367 x^{9} + 1058 x^{8} - 1675 x^{7} + 2755 x^{6} - 2800 x^{5} + 4032 x^{4} - 2186 x^{3} + 2189 x^{2} + 76 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-48178164327148691955 \nu^{15} - 477186355399061442124 \nu^{14} + 765214320927420205186 \nu^{13} - 5057398534594904646535 \nu^{12} + 3121464387555847093535 \nu^{11} - 16673753187579169611905 \nu^{10} + 17762544006117153746885 \nu^{9} - 104087965543022390202133 \nu^{8} + 32152675260990418049368 \nu^{7} - 382744590680244763843441 \nu^{6} + 315107768605336503658471 \nu^{5} - 769812163220469704027673 \nu^{4} - 52818197762278676124986 \nu^{3} - 770685121920395044641092 \nu^{2} - 1122536262414136465777839 \nu - 37604091962631649632244$$$$)/$$$$10\!\cdots\!27$$ $$\beta_{3}$$ $$=$$ $$($$$$448273133278566247238 \nu^{15} - 1512541918523234823556 \nu^{14} + 5989726839536483627452 \nu^{13} - 9077548044609052323300 \nu^{12} + 22543724273443078345130 \nu^{11} - 35372137859957990765351 \nu^{10} + 123977762482704451640412 \nu^{9} - 160883655146463447542560 \nu^{8} + 457238656508711204264194 \nu^{7} - 620856404530228358037790 \nu^{6} + 1401576743317291664686349 \nu^{5} - 574205169774850952128706 \nu^{4} + 752956752555178705056390 \nu^{3} + 405412769530229125089080 \nu^{2} + 13236645784802464419326 \nu + 3311830514591337987223063$$$$)/$$$$10\!\cdots\!27$$ $$\beta_{4}$$ $$=$$ $$($$$$-691677515754976523241 \nu^{15} + 2922835546494050406323 \nu^{14} - 10777299818076872659333 \nu^{13} + 19482741001314285643405 \nu^{12} - 39637639408281057071680 \nu^{11} + 66396395971994103695427 \nu^{10} - 216182194518223629105857 \nu^{9} + 360824293776248463431482 \nu^{8} - 812586653437493849092845 \nu^{7} + 1303478668639530034900799 \nu^{6} - 2436903629033132203150689 \nu^{5} + 1512433091691958880061932 \nu^{4} - 1254345565286042317243552 \nu^{3} - 85147439206521854413139 \nu^{2} - 1984477621821553369727 \nu - 2543394590033437790203327$$$$)/$$$$10\!\cdots\!27$$ $$\beta_{5}$$ $$=$$ $$($$$$770766983612407585392 \nu^{15} - 3427439564944663107690 \nu^{14} + 11985047556174877420365 \nu^{13} - 25433869386472644534793 \nu^{12} + 50090496469881220529233 \nu^{11} - 105940512854774931832572 \nu^{10} + 272679070929387063463333 \nu^{9} - 499814358394475775588689 \nu^{8} + 1018113992051629604716702 \nu^{7} - 2091764410921831565098331 \nu^{6} + 3256626744011737401759081 \nu^{5} - 3639515347237117350511823 \nu^{4} + 4073557016193011560685656 \nu^{3} - 3167035634109794878534042 \nu^{2} + 1663380747799736472784740 \nu + 21571003979255052244985$$$$)/$$$$10\!\cdots\!27$$ $$\beta_{6}$$ $$=$$ $$($$$$992392863041497548098 \nu^{15} - 2781063583875571350656 \nu^{14} + 11202890052521703381407 \nu^{13} - 16781117246854820564288 \nu^{12} + 45483273950231995670378 \nu^{11} - 84617092017357923338096 \nu^{10} + 252708116291651418225654 \nu^{9} - 317967910824103671981754 \nu^{8} + 992165676210146805271206 \nu^{7} - 1536002027037989593211596 \nu^{6} + 2444653255425595274915934 \nu^{5} - 2436934175046581544193018 \nu^{4} + 4249421050539653617008974 \nu^{3} - 1784689021017620979893686 \nu^{2} + 1710250874261042241737390 \nu + 36412528111611310310792$$$$)/$$$$10\!\cdots\!27$$ $$\beta_{7}$$ $$=$$ $$($$$$21571003979255052244985 \nu^{15} - 65483778921377564320347 \nu^{14} + 262279487316005290047510 \nu^{13} - 421834123162020870075080 \nu^{12} + 1082413064369970204539058 \nu^{11} - 2013051858582090974822868 \nu^{10} + 5908540583274383985733537 \nu^{9} - 8189237531315991237372828 \nu^{8} + 23321936568446321050782819 \nu^{7} - 37149545657303842115066577 \nu^{6} + 61519880373769500500032006 \nu^{5} - 63655437885925883687717081 \nu^{4} + 90613803391593488002291343 \nu^{3} - 51227771714844555768222866 \nu^{2} + 50385963344699104242806207 \nu - 23984445376352502165880$$$$)/$$$$10\!\cdots\!27$$ $$\beta_{8}$$ $$=$$ $$($$$$22762567830275391566437 \nu^{15} - 68114255775724233428560 \nu^{14} + 273320003588974729112154 \nu^{13} - 432505731957958193596075 \nu^{12} + 1118961177288517106079383 \nu^{11} - 2072879430687154010311087 \nu^{10} + 6127780413993918170254124 \nu^{9} - 8356070804342687196373173 \nu^{8} + 24160555246458713990695148 \nu^{7} - 38121096756291773255178103 \nu^{6} + 62883892165612300973577229 \nu^{5} - 64232055644751902009208276 \nu^{4} + 92211442500640444898165116 \nu^{3} - 49635927440397642584032950 \nu^{2} + 50438922471644610992929857 \nu + 1750661060789895643673550$$$$)/$$$$10\!\cdots\!27$$ $$\beta_{9}$$ $$=$$ $$($$$$36412528111611310310792 \nu^{15} - 110229977197875428480474 \nu^{14} + 439731400923211295080160 \nu^{13} - 703040924173136599286455 \nu^{12} + 1800994994715809025793096 \nu^{11} - 3359023332106861233952450 \nu^{10} + 9879587154040800396941144 \nu^{9} - 13616105933253002302286318 \nu^{8} + 38842422652908869980799690 \nu^{7} - 61983150263159091575847806 \nu^{6} + 101852516974527149499443556 \nu^{5} - 104399731967937264145133534 \nu^{4} + 149252247521063384717306362 \nu^{3} - 83847207502521977956400286 \nu^{2} + 81491713057334779250217374 \nu + 1057101262221417341882802$$$$)/$$$$10\!\cdots\!27$$ $$\beta_{10}$$ $$=$$ $$($$$$50464608876899524776768 \nu^{15} - 153022373941326399471359 \nu^{14} + 610650172361290276693352 \nu^{13} - 978187190001872863268032 \nu^{12} + 2503939676188689684200631 \nu^{11} - 4670880045790741210065862 \nu^{10} + 13732496957722901213192919 \nu^{9} - 18976883480570911518375660 \nu^{8} + 54029466815154570194034355 \nu^{7} - 86202331996552868233088880 \nu^{6} + 141961402155353446997491127 \nu^{5} - 146057070409967860156519639 \nu^{4} + 208400026900270914288618007 \nu^{3} - 117355378752221590593001179 \nu^{2} + 116937654016600649442467379 \nu - 54957208433357394522853$$$$)/$$$$10\!\cdots\!27$$ $$\beta_{11}$$ $$=$$ $$($$$$-68483818496075095992949 \nu^{15} + 205048591631148967481449 \nu^{14} - 822034357917857820186036 \nu^{13} + 1300661172212675964974649 \nu^{12} - 3362574190384969671776971 \nu^{11} + 6232883855927973453145969 \nu^{10} - 18429581511893880438932584 \nu^{9} + 25125998385915819189735997 \nu^{8} - 72607921757932660771937998 \nu^{7} + 114652679351129050235628365 \nu^{6} - 188993442338306514511213069 \nu^{5} + 192448073907499370524546990 \nu^{4} - 277019009279512427354743890 \nu^{3} + 149369879424129723643147982 \nu^{2} - 151277758085454290475444915 \nu - 5250990789506645433472552$$$$)/$$$$10\!\cdots\!27$$ $$\beta_{12}$$ $$=$$ $$($$$$-84624843314501624262178 \nu^{15} + 258147185960475164397760 \nu^{14} - 1027296735343530437657844 \nu^{13} + 1654095951480456458094796 \nu^{12} - 4210030391385674740778786 \nu^{11} + 7867996816633917651798412 \nu^{10} - 23074579980017432740505183 \nu^{9} + 32036182819591818833878905 \nu^{8} - 90718548900363214519146938 \nu^{7} + 145452711717574087206756763 \nu^{6} - 238729466691906098870027347 \nu^{5} + 245242997522638395843821700 \nu^{4} - 348329692280883837766444321 \nu^{3} + 195721044453551078896236510 \nu^{2} - 189002707049178903129762867 \nu - 2451708634506897180581808$$$$)/$$$$10\!\cdots\!27$$ $$\beta_{13}$$ $$=$$ $$($$$$86026924992664105959758 \nu^{15} - 258393934287281765223435 \nu^{14} + 1033006095887048661027371 \nu^{13} - 1639972138892209578246211 \nu^{12} + 4225793708040688514193771 \nu^{11} - 7863011305649698757644382 \nu^{10} + 23189923761796556345751458 \nu^{9} - 31693714408485555403947779 \nu^{8} + 91220562904582059902781766 \nu^{7} - 144820353571319950293037047 \nu^{6} + 237858036170203319774612132 \nu^{5} - 242752693710312824692138237 \nu^{4} + 349845626359338300061464862 \nu^{3} - 191191559623801835806192103 \nu^{2} + 189155600328018408873803149 \nu + 4025578405843773216120461$$$$)/$$$$10\!\cdots\!27$$ $$\beta_{14}$$ $$=$$ $$($$$$109015958455404840969670 \nu^{15} - 331336307574695377198456 \nu^{14} + 1319976360273287059279438 \nu^{13} - 2117775524659027621829870 \nu^{12} + 5407592206667076302238143 \nu^{11} - 10098393417158000710351826 \nu^{10} + 29658732416760136836061111 \nu^{9} - 41030164247329379010465889 \nu^{8} + 116553216274568092741844566 \nu^{7} - 186505213173361116699430153 \nu^{6} + 306369524412167590625173815 \nu^{5} - 314401777076002328241719407 \nu^{4} + 447580878528843512095595768 \nu^{3} - 251827597079016311186391687 \nu^{2} + 243331897003901235400249945 \nu + 3156462262531895767582599$$$$)/$$$$10\!\cdots\!27$$ $$\beta_{15}$$ $$=$$ $$($$$$-111697137273787508545218 \nu^{15} + 335845518403840306370400 \nu^{14} - 1341526454890705302022067 \nu^{13} + 2128814684634348519577325 \nu^{12} - 5476828605557021940438363 \nu^{11} + 10191209960298123385329122 \nu^{10} - 30079538078872281552635368 \nu^{9} + 41146175095449392404488023 \nu^{8} - 118242394279392739902569326 \nu^{7} + 187730940382281513643873875 \nu^{6} - 308168218505987473807552114 \nu^{5} + 313963224642022063223489075 \nu^{4} - 451067520882397952942673275 \nu^{3} + 246789917863553018048269458 \nu^{2} - 244597009266065570076778265 \nu - 5205353413253994311232723$$$$)/$$$$10\!\cdots\!27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{14} - 3 \beta_{9} + \beta_{6} - \beta_{5} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} + \beta_{14} - \beta_{13} - \beta_{11} + \beta_{9} + \beta_{8} - 2 \beta_{7} + 5 \beta_{6} - \beta_{5} + \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{15} - \beta_{13} - 7 \beta_{11} + \beta_{9} - 13 \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-9 \beta_{15} - 9 \beta_{14} - 11 \beta_{8} + 11 \beta_{7} - 30 \beta_{6} + 30 \beta_{5} + 7 \beta_{4} - \beta_{3} - 19 \beta_{2} - 28 \beta_{1} + 20$$ $$\nu^{6}$$ $$=$$ $$-46 \beta_{15} - 34 \beta_{14} - 9 \beta_{12} + 46 \beta_{11} - 3 \beta_{10} - 18 \beta_{9} + 82 \beta_{7} - 13 \beta_{6} + 9 \beta_{4} + 34 \beta_{3} + 21 \beta_{2} + 10 \beta_{1} + 18$$ $$\nu^{7}$$ $$=$$ $$16 \beta_{14} - 3 \beta_{13} - 43 \beta_{12} + 68 \beta_{11} - 3 \beta_{10} - 67 \beta_{9} + 94 \beta_{8} + 111 \beta_{6} - 200 \beta_{5} + 68 \beta_{3} + 68 \beta_{2} + 111 \beta_{1} - 94$$ $$\nu^{8}$$ $$=$$ $$311 \beta_{15} + 114 \beta_{14} + 24 \beta_{13} - 114 \beta_{11} + 65 \beta_{10} + 348 \beta_{9} + 348 \beta_{8} - 538 \beta_{7} + 313 \beta_{6} - 233 \beta_{5} - 65 \beta_{4} + 119 \beta_{2} + 114 \beta_{1} - 538$$ $$\nu^{9}$$ $$=$$ $$495 \beta_{15} + 221 \beta_{13} + 270 \beta_{12} - 665 \beta_{11} + 270 \beta_{10} + 741 \beta_{9} - 486 \beta_{8} - 741 \beta_{7} - 495 \beta_{6} + 758 \beta_{5} - 221 \beta_{4} - 495 \beta_{3} - 15 \beta_{2} - 758 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-977 \beta_{15} - 977 \beta_{14} + 446 \beta_{13} + 446 \beta_{12} - 1679 \beta_{8} + 1679 \beta_{7} - 3746 \beta_{6} + 3746 \beta_{5} - 51 \beta_{4} - 1196 \beta_{3} - 1803 \beta_{2} - 2780 \beta_{1} + 3735$$ $$\nu^{11}$$ $$=$$ $$-5118 \beta_{15} - 1533 \beta_{14} - 531 \beta_{12} + 5118 \beta_{11} - 1778 \beta_{10} - 5653 \beta_{9} + 9201 \beta_{7} - 2261 \beta_{6} + 531 \beta_{4} + 1533 \beta_{3} - 728 \beta_{2} + 2806 \beta_{1} + 5653$$ $$\nu^{12}$$ $$=$$ $$7638 \beta_{14} - 3054 \beta_{13} - 3340 \beta_{12} + 7910 \beta_{11} - 3054 \beta_{10} - 13005 \beta_{9} + 13708 \beta_{8} + 23008 \beta_{6} - 27558 \beta_{5} + 7910 \beta_{3} + 7910 \beta_{2} + 23008 \beta_{1} - 13708$$ $$\nu^{13}$$ $$=$$ $$38808 \beta_{15} + 26062 \beta_{14} - 7352 \beta_{13} - 26062 \beta_{11} + 4856 \beta_{10} + 26144 \beta_{9} + 26144 \beta_{8} - 68642 \beta_{7} + 61929 \beta_{6} - 44340 \beta_{5} - 4856 \beta_{4} + 18278 \beta_{2} + 26062 \beta_{1} - 68642$$ $$\nu^{14}$$ $$=$$ $$61942 \beta_{15} + 5394 \beta_{13} + 26600 \beta_{12} - 112945 \beta_{11} + 26600 \beta_{10} + 107496 \beta_{9} - 86719 \beta_{8} - 107496 \beta_{7} - 61942 \beta_{6} + 118294 \beta_{5} - 5394 \beta_{4} - 61942 \beta_{3} - 27908 \beta_{2} - 118294 \beta_{1}$$ $$\nu^{15}$$ $$=$$ $$-190503 \beta_{15} - 190503 \beta_{14} + 40736 \beta_{13} + 40736 \beta_{12} - 317343 \beta_{8} + 317343 \beta_{7} - 551227 \beta_{6} + 551227 \beta_{5} + 45609 \beta_{4} - 101370 \beta_{3} - 217886 \beta_{2} - 408389 \beta_{1} + 511184$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
57.1
 1.75205 + 1.27294i 1.25138 + 0.909177i −0.0169945 − 0.0123472i −1.67741 − 1.21871i 1.75205 − 1.27294i 1.25138 − 0.909177i −0.0169945 + 0.0123472i −1.67741 + 1.21871i −0.567301 + 1.74597i −0.406200 + 1.25015i 0.321891 − 0.990677i 0.842594 − 2.59324i −0.567301 − 1.74597i −0.406200 − 1.25015i 0.321891 + 0.990677i 0.842594 + 2.59324i
0 −2.16565 0 −0.636054 0.462120i 0 −0.309017 0.951057i 0 1.69003 0
57.2 0 −1.54678 0 −2.70780 1.96733i 0 −0.309017 0.951057i 0 −0.607457 0
57.3 0 0.0210063 0 0.567750 + 0.412494i 0 −0.309017 0.951057i 0 −2.99956 0
57.4 0 2.07339 0 −2.15094 1.56275i 0 −0.309017 0.951057i 0 1.29895 0
141.1 0 −2.16565 0 −0.636054 + 0.462120i 0 −0.309017 + 0.951057i 0 1.69003 0
141.2 0 −1.54678 0 −2.70780 + 1.96733i 0 −0.309017 + 0.951057i 0 −0.607457 0
141.3 0 0.0210063 0 0.567750 0.412494i 0 −0.309017 + 0.951057i 0 −2.99956 0
141.4 0 2.07339 0 −2.15094 + 1.56275i 0 −0.309017 + 0.951057i 0 1.29895 0
365.1 0 −1.83583 0 −0.827871 + 2.54793i 0 0.809017 + 0.587785i 0 0.370254 0
365.2 0 −1.31449 0 0.558990 1.72039i 0 0.809017 + 0.587785i 0 −1.27211 0
365.3 0 1.04166 0 −0.980821 + 3.01866i 0 0.809017 + 0.587785i 0 −1.91494 0
365.4 0 2.72669 0 −0.323246 + 0.994850i 0 0.809017 + 0.587785i 0 4.43484 0
953.1 0 −1.83583 0 −0.827871 2.54793i 0 0.809017 0.587785i 0 0.370254 0
953.2 0 −1.31449 0 0.558990 + 1.72039i 0 0.809017 0.587785i 0 −1.27211 0
953.3 0 1.04166 0 −0.980821 3.01866i 0 0.809017 0.587785i 0 −1.91494 0
953.4 0 2.72669 0 −0.323246 0.994850i 0 0.809017 0.587785i 0 4.43484 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 953.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.n.c 16
41.d even 5 1 inner 1148.2.n.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.n.c 16 1.a even 1 1 trivial
1148.2.n.c 16 41.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1148, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 1 - 47 T - 30 T^{2} + 57 T^{3} + 40 T^{4} - 15 T^{5} - 12 T^{6} + T^{7} + T^{8} )^{2}$$
$5$ $$6241 + 9480 T + 7081 T^{2} + 1784 T^{3} + 14858 T^{4} + 31218 T^{5} + 50911 T^{6} + 48003 T^{7} + 36864 T^{8} + 23033 T^{9} + 12137 T^{10} + 5195 T^{11} + 1797 T^{12} + 479 T^{13} + 96 T^{14} + 13 T^{15} + T^{16}$$
$7$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$
$11$ $$1739761 - 349535 T + 569383 T^{2} + 348428 T^{3} + 671651 T^{4} - 508232 T^{5} + 713193 T^{6} + 11648 T^{7} + 153830 T^{8} - 12506 T^{9} + 6027 T^{10} + 203 T^{11} + 863 T^{12} + 123 T^{13} + 42 T^{14} + T^{15} + T^{16}$$
$13$ $$1 - 16 T + 290 T^{2} - 2010 T^{3} + 11947 T^{4} - 35512 T^{5} + 60430 T^{6} - 50060 T^{7} + 32769 T^{8} + 5804 T^{9} - 1562 T^{10} - 906 T^{11} + 743 T^{12} + 74 T^{13} + 22 T^{14} + 6 T^{15} + T^{16}$$
$17$ $$151321 + 274634 T + 750751 T^{2} + 1519552 T^{3} + 3115487 T^{4} + 2205890 T^{5} + 5209338 T^{6} + 2997987 T^{7} + 1828557 T^{8} + 477197 T^{9} + 78955 T^{10} + 10600 T^{11} + 2265 T^{12} + 124 T^{13} - 9 T^{14} + T^{15} + T^{16}$$
$19$ $$524176 - 1587008 T + 2366240 T^{2} - 2064318 T^{3} + 2066883 T^{4} - 1562971 T^{5} + 917056 T^{6} - 324816 T^{7} + 278771 T^{8} - 13438 T^{9} + 22293 T^{10} + 664 T^{11} + 2275 T^{12} - 638 T^{13} + 136 T^{14} - 15 T^{15} + T^{16}$$
$23$ $$57121 + 1271958 T + 9348023 T^{2} - 54259566 T^{3} + 118841907 T^{4} - 128715180 T^{5} + 91509862 T^{6} - 46878927 T^{7} + 18474605 T^{8} - 5675307 T^{9} + 1394017 T^{10} - 270570 T^{11} + 41727 T^{12} - 4956 T^{13} + 443 T^{14} - 27 T^{15} + T^{16}$$
$29$ $$9339136 + 157506240 T + 1006059664 T^{2} - 271856308 T^{3} + 442772369 T^{4} - 70048944 T^{5} + 61250722 T^{6} - 12040767 T^{7} + 2377467 T^{8} + 557144 T^{9} + 276911 T^{10} + 1034 T^{11} + 3753 T^{12} - 505 T^{13} + 93 T^{14} + T^{15} + T^{16}$$
$31$ $$516961 + 4170919 T + 79191875 T^{2} + 45239388 T^{3} + 69518717 T^{4} - 11153056 T^{5} - 5159915 T^{6} + 2301331 T^{7} + 1509881 T^{8} - 15339 T^{9} - 43874 T^{10} + 6162 T^{11} + 6741 T^{12} + 1276 T^{13} + 209 T^{14} + 14 T^{15} + T^{16}$$
$37$ $$121773779521 - 119581606519 T + 81039594968 T^{2} - 50980225421 T^{3} + 32900585575 T^{4} - 7770133515 T^{5} + 162205369 T^{6} + 288667673 T^{7} + 97195946 T^{8} + 16408253 T^{9} + 3686049 T^{10} + 490112 T^{11} + 54578 T^{12} + 4326 T^{13} + 359 T^{14} + 16 T^{15} + T^{16}$$
$41$ $$7984925229121 - 5063611120906 T + 2327551078090 T^{2} - 790139290820 T^{3} + 226385842515 T^{4} - 54627335968 T^{5} + 11603939638 T^{6} - 2157394580 T^{7} + 358613765 T^{8} - 52619380 T^{9} + 6902998 T^{10} - 792608 T^{11} + 80115 T^{12} - 6820 T^{13} + 490 T^{14} - 26 T^{15} + T^{16}$$
$43$ $$2781276656656 - 3466724609816 T + 2076827097692 T^{2} - 714595460598 T^{3} + 195626855223 T^{4} - 43558888903 T^{5} + 8435954512 T^{6} - 1290473024 T^{7} + 175302885 T^{8} - 17632668 T^{9} + 1829033 T^{10} - 140948 T^{11} + 13781 T^{12} - 598 T^{13} + 138 T^{14} - 5 T^{15} + T^{16}$$
$47$ $$59136998761 + 11050631002 T + 5432482324 T^{2} + 2002009434 T^{3} + 2921727461 T^{4} + 636547846 T^{5} + 689645860 T^{6} + 202664320 T^{7} + 95311882 T^{8} + 20930230 T^{9} + 3230380 T^{10} + 395732 T^{11} + 49476 T^{12} + 4162 T^{13} + 266 T^{14} + 14 T^{15} + T^{16}$$
$53$ $$19447975936 - 45460424704 T + 190645628928 T^{2} - 124499712000 T^{3} + 142888692480 T^{4} - 3519795712 T^{5} + 1155595520 T^{6} - 238612544 T^{7} + 76411136 T^{8} + 17383680 T^{9} + 2223200 T^{10} + 28224 T^{11} + 848 T^{12} + 776 T^{13} + 232 T^{14} + 20 T^{15} + T^{16}$$
$59$ $$2792638712161 + 2996366500570 T + 1069912512052 T^{2} - 484314722170 T^{3} + 451168035591 T^{4} - 113607826590 T^{5} + 21019734996 T^{6} - 1916517830 T^{7} + 488225524 T^{8} - 17120540 T^{9} + 3391188 T^{10} - 68620 T^{11} + 26774 T^{12} - 890 T^{13} + 204 T^{14} + T^{16}$$
$61$ $$594007496208361 + 241572665917896 T + 74863737386043 T^{2} + 14203770537734 T^{3} + 2316218899697 T^{4} + 356470790644 T^{5} + 70219842528 T^{6} + 14628346301 T^{7} + 2696744539 T^{8} + 397027505 T^{9} + 46953699 T^{10} + 4459668 T^{11} + 349803 T^{12} + 22840 T^{13} + 1225 T^{14} + 47 T^{15} + T^{16}$$
$67$ $$651140560711936 - 72178063622656 T + 12471227016704 T^{2} + 260811782592 T^{3} + 1069351381504 T^{4} + 127806806656 T^{5} + 38827949536 T^{6} + 3504332224 T^{7} + 708571440 T^{8} + 38647664 T^{9} + 5906712 T^{10} + 335332 T^{11} + 63365 T^{12} + 6191 T^{13} + 563 T^{14} + 27 T^{15} + T^{16}$$
$71$ $$53983463854336 + 11525455249664 T + 7935784473216 T^{2} + 2928774996416 T^{3} + 769980999680 T^{4} + 163978679200 T^{5} + 34766243040 T^{6} + 5280900928 T^{7} + 544416672 T^{8} + 38087488 T^{9} + 3553528 T^{10} + 338850 T^{11} + 32797 T^{12} + 2398 T^{13} + 246 T^{14} + 11 T^{15} + T^{16}$$
$73$ $$( -108821 - 690213 T + 246281 T^{2} + 80118 T^{3} - 34250 T^{4} + 2034 T^{5} + 289 T^{6} - 35 T^{7} + T^{8} )^{2}$$
$79$ $$( -2357429 + 1080511 T + 126154 T^{2} - 100125 T^{3} + 2668 T^{4} + 2349 T^{5} - 130 T^{6} - 15 T^{7} + T^{8} )^{2}$$
$83$ $$( -176521 + 250689 T + 21300 T^{2} - 43821 T^{3} - 5862 T^{4} + 1949 T^{5} + 502 T^{6} + 39 T^{7} + T^{8} )^{2}$$
$89$ $$431729157721 + 10120451043905 T + 89532194626822 T^{2} - 41572104673511 T^{3} + 13234730149561 T^{4} - 2650008287309 T^{5} + 406635291087 T^{6} - 44189428376 T^{7} + 4185748440 T^{8} - 273617692 T^{9} + 20854943 T^{10} - 973496 T^{11} + 77333 T^{12} - 3384 T^{13} + 293 T^{14} - 17 T^{15} + T^{16}$$
$97$ $$19920234577681 + 37563982625658 T + 28104159617263 T^{2} + 573488189498 T^{3} + 1939924406746 T^{4} - 660882247870 T^{5} + 132764189907 T^{6} - 12608971685 T^{7} + 1606515872 T^{8} - 121687425 T^{9} + 11077927 T^{10} - 260045 T^{11} + 63381 T^{12} + 2367 T^{13} + 368 T^{14} + 17 T^{15} + T^{16}$$
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