# Properties

 Label 108.5.d.a Level 108 Weight 5 Character orbit 108.d Analytic conductor 11.164 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 108.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1639560131$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{32}\cdot 3^{26}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{2} ) q^{4} + ( -\beta_{1} - \beta_{7} ) q^{5} + \beta_{4} q^{7} + ( -\beta_{1} + \beta_{11} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{2} ) q^{4} + ( -\beta_{1} - \beta_{7} ) q^{5} + \beta_{4} q^{7} + ( -\beta_{1} + \beta_{11} ) q^{8} + ( -12 + \beta_{2} + \beta_{12} ) q^{10} + ( -3 \beta_{1} - \beta_{3} - \beta_{11} ) q^{11} + ( -22 + \beta_{10} + \beta_{12} ) q^{13} + ( \beta_{1} - \beta_{6} + \beta_{7} - \beta_{15} ) q^{14} + ( -13 + \beta_{2} + 3 \beta_{4} + \beta_{10} - \beta_{13} ) q^{16} + ( -10 \beta_{1} - \beta_{7} + \beta_{9} + \beta_{11} - \beta_{14} - \beta_{15} ) q^{17} + ( -1 - 8 \beta_{2} + \beta_{5} + \beta_{8} + \beta_{13} ) q^{19} + ( -15 \beta_{1} - 3 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - \beta_{11} - \beta_{14} ) q^{20} + ( 48 + \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{8} - 4 \beta_{10} ) q^{22} + ( -\beta_{3} - 3 \beta_{6} - 2 \beta_{7} + 5 \beta_{11} + 2 \beta_{14} - 2 \beta_{15} ) q^{23} + ( 98 - 8 \beta_{2} - 2 \beta_{8} - \beta_{10} - 5 \beta_{12} + 2 \beta_{13} ) q^{25} + ( -24 \beta_{1} - 4 \beta_{3} - 4 \beta_{6} - 16 \beta_{7} - 2 \beta_{9} ) q^{26} + ( 23 + \beta_{2} - 10 \beta_{4} + 2 \beta_{5} - 2 \beta_{10} - 2 \beta_{13} ) q^{28} + ( 8 \beta_{1} - 2 \beta_{3} + 2 \beta_{6} - 5 \beta_{7} - 3 \beta_{9} + 9 \beta_{11} - 3 \beta_{14} + \beta_{15} ) q^{29} + ( -3 + 16 \beta_{2} + 5 \beta_{4} - \beta_{5} + 5 \beta_{8} + 6 \beta_{10} - 6 \beta_{12} + \beta_{13} ) q^{31} + ( -17 \beta_{1} + 2 \beta_{3} + 4 \beta_{6} - 24 \beta_{7} - 2 \beta_{9} + \beta_{11} + 2 \beta_{14} - 4 \beta_{15} ) q^{32} + ( -154 + 12 \beta_{2} - 20 \beta_{4} - 2 \beta_{5} + 4 \beta_{12} - 4 \beta_{13} ) q^{34} + ( 51 \beta_{1} - 4 \beta_{3} + \beta_{6} + 8 \beta_{9} - 4 \beta_{11} ) q^{35} + ( 200 + 48 \beta_{2} + 8 \beta_{5} - 4 \beta_{8} + 3 \beta_{10} - 5 \beta_{12} + 4 \beta_{13} ) q^{37} + ( -2 \beta_{1} - 10 \beta_{6} + 6 \beta_{7} + 14 \beta_{9} + 8 \beta_{11} - 4 \beta_{14} + 2 \beta_{15} ) q^{38} + ( -171 + 9 \beta_{2} - 11 \beta_{4} - 10 \beta_{5} - 4 \beta_{8} - 9 \beta_{10} + 4 \beta_{12} - 3 \beta_{13} ) q^{40} + ( 4 \beta_{1} + 4 \beta_{3} - 4 \beta_{6} - 7 \beta_{7} + \beta_{9} - 15 \beta_{11} + 3 \beta_{14} - 5 \beta_{15} ) q^{41} + ( 6 \beta_{4} + 8 \beta_{8} + 8 \beta_{13} ) q^{43} + ( 59 \beta_{1} + \beta_{3} + 22 \beta_{6} + 62 \beta_{7} - 16 \beta_{9} - 3 \beta_{11} - \beta_{14} ) q^{44} + ( 6 + 2 \beta_{2} + 22 \beta_{4} + 10 \beta_{5} - 2 \beta_{8} - 4 \beta_{10} - 8 \beta_{13} ) q^{46} + ( -194 \beta_{1} + 2 \beta_{6} + 4 \beta_{7} - 24 \beta_{9} - 12 \beta_{11} - 4 \beta_{14} + 4 \beta_{15} ) q^{47} + ( -76 - 88 \beta_{2} - 8 \beta_{5} - 6 \beta_{8} + 9 \beta_{10} - 3 \beta_{12} + 6 \beta_{13} ) q^{49} + ( 132 \beta_{1} + 4 \beta_{3} - 36 \beta_{6} + 64 \beta_{7} - 10 \beta_{9} + 8 \beta_{14} ) q^{50} + ( 160 + 16 \beta_{2} + 12 \beta_{4} - 8 \beta_{5} - 12 \beta_{10} + 16 \beta_{12} - 4 \beta_{13} ) q^{52} + ( -233 \beta_{1} - 2 \beta_{3} + 2 \beta_{6} + 2 \beta_{7} + 29 \beta_{9} + \beta_{11} + 5 \beta_{14} + 9 \beta_{15} ) q^{53} + ( -32 - 176 \beta_{2} + 41 \beta_{4} + 24 \beta_{5} + 12 \beta_{8} + 12 \beta_{10} - 12 \beta_{12} + 4 \beta_{13} ) q^{55} + ( -19 \beta_{1} + 8 \beta_{3} + 48 \beta_{6} - 8 \beta_{7} + 44 \beta_{9} + 3 \beta_{11} + 8 \beta_{15} ) q^{56} + ( 158 - 14 \beta_{2} + 4 \beta_{4} - 18 \beta_{5} + 8 \beta_{10} + 10 \beta_{12} - 12 \beta_{13} ) q^{58} + ( 104 \beta_{1} + 18 \beta_{3} + 14 \beta_{6} + 4 \beta_{7} + 8 \beta_{9} + 6 \beta_{11} - 4 \beta_{14} + 4 \beta_{15} ) q^{59} + ( -180 - 16 \beta_{2} - 4 \beta_{8} + 18 \beta_{10} + 10 \beta_{12} + 4 \beta_{13} ) q^{61} + ( 31 \beta_{1} + 24 \beta_{3} - 51 \beta_{6} - 89 \beta_{7} - 54 \beta_{9} - 8 \beta_{11} - 4 \beta_{14} + \beta_{15} ) q^{62} + ( 59 + 29 \beta_{2} - 25 \beta_{4} + 20 \beta_{5} - 8 \beta_{8} - 3 \beta_{10} + 24 \beta_{12} - 5 \beta_{13} ) q^{64} + ( 714 \beta_{1} + 4 \beta_{3} - 4 \beta_{6} + 55 \beta_{7} - 79 \beta_{9} - 23 \beta_{11} + 11 \beta_{14} + 3 \beta_{15} ) q^{65} + ( 32 + 176 \beta_{2} + 30 \beta_{4} - 24 \beta_{5} + 4 \beta_{8} - 12 \beta_{10} + 12 \beta_{12} + 12 \beta_{13} ) q^{67} + ( -204 \beta_{1} + 80 \beta_{6} - 80 \beta_{7} - 16 \beta_{9} - 4 \beta_{11} + 16 \beta_{15} ) q^{68} + ( -886 - 59 \beta_{2} - 5 \beta_{4} - 8 \beta_{5} - 5 \beta_{8} - 16 \beta_{10} ) q^{70} + ( 674 \beta_{1} + 29 \beta_{3} - 11 \beta_{6} + 2 \beta_{7} + 72 \beta_{9} + 23 \beta_{11} - 2 \beta_{14} + 2 \beta_{15} ) q^{71} + ( 543 + 248 \beta_{2} + 32 \beta_{5} - 2 \beta_{8} + 5 \beta_{10} + \beta_{12} + 2 \beta_{13} ) q^{73} + ( 202 \beta_{1} - 12 \beta_{3} - 92 \beta_{6} + 48 \beta_{7} + 98 \beta_{9} - 64 \beta_{11} + 16 \beta_{14} ) q^{74} + ( -2070 - 2 \beta_{2} + 8 \beta_{4} - 20 \beta_{5} + 16 \beta_{8} + 16 \beta_{10} - 8 \beta_{13} ) q^{76} + ( -153 \beta_{1} - 8 \beta_{3} + 8 \beta_{6} + 89 \beta_{7} + 26 \beta_{9} + 26 \beta_{11} - 2 \beta_{14} + 14 \beta_{15} ) q^{77} + ( -19 + 48 \beta_{2} + 22 \beta_{4} - \beta_{5} + 5 \beta_{8} + 30 \beta_{10} - 30 \beta_{12} - 15 \beta_{13} ) q^{79} + ( -149 \beta_{1} - 6 \beta_{3} + 100 \beta_{6} + 96 \beta_{7} - 110 \beta_{9} - 11 \beta_{11} + 2 \beta_{14} + 4 \beta_{15} ) q^{80} + ( 14 + 14 \beta_{2} - 68 \beta_{4} + 30 \beta_{5} - 16 \beta_{10} + 6 \beta_{12} + 12 \beta_{13} ) q^{82} + ( -1179 \beta_{1} + 17 \beta_{3} + 14 \beta_{6} - 8 \beta_{7} - 152 \beta_{9} + 41 \beta_{11} + 8 \beta_{14} - 8 \beta_{15} ) q^{83} + ( 510 - 272 \beta_{2} - 32 \beta_{5} - 4 \beta_{8} - 39 \beta_{10} - 47 \beta_{12} + 4 \beta_{13} ) q^{85} + ( 54 \beta_{1} - 86 \beta_{6} + 54 \beta_{7} - 16 \beta_{9} - 32 \beta_{14} + 10 \beta_{15} ) q^{86} + ( 2975 - 57 \beta_{2} - 45 \beta_{4} - 2 \beta_{5} - 20 \beta_{8} + \beta_{10} - 60 \beta_{12} + 3 \beta_{13} ) q^{88} + ( -1126 \beta_{1} + 8 \beta_{3} - 8 \beta_{6} + 30 \beta_{7} + 148 \beta_{9} - 20 \beta_{11} - 4 \beta_{14} - 20 \beta_{15} ) q^{89} + ( -3 - 184 \beta_{2} - 138 \beta_{4} + 19 \beta_{5} - 13 \beta_{8} - 24 \beta_{10} + 24 \beta_{12} + 3 \beta_{13} ) q^{91} + ( -106 \beta_{1} + 22 \beta_{3} + 100 \beta_{6} - 12 \beta_{7} + 192 \beta_{9} + 10 \beta_{11} + 10 \beta_{14} - 32 \beta_{15} ) q^{92} + ( 3308 + 186 \beta_{2} - 42 \beta_{4} - 24 \beta_{5} + 6 \beta_{8} + 16 \beta_{13} ) q^{94} + ( 240 \beta_{1} - 59 \beta_{3} - 33 \beta_{6} + 10 \beta_{7} + 48 \beta_{9} - 89 \beta_{11} - 10 \beta_{14} + 10 \beta_{15} ) q^{95} + ( -417 - 16 \beta_{2} - 8 \beta_{5} + 12 \beta_{8} - 24 \beta_{10} - 12 \beta_{13} ) q^{97} + ( 58 \beta_{1} - 36 \beta_{3} - 156 \beta_{6} - 182 \beta_{9} + 64 \beta_{11} + 24 \beta_{14} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 14q^{4} + O(q^{10})$$ $$16q - 14q^{4} - 202q^{10} - 352q^{13} - 206q^{16} + 738q^{22} + 1632q^{25} + 342q^{28} - 2536q^{34} + 3200q^{37} - 2854q^{40} + 36q^{46} - 896q^{49} + 2288q^{52} + 2492q^{58} - 2752q^{61} + 682q^{64} - 14166q^{70} + 8240q^{73} - 33084q^{76} + 68q^{82} + 8800q^{85} + 48294q^{88} + 52596q^{94} - 6928q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 6 x^{14} - 22 x^{13} + 19 x^{12} + 18 x^{11} + 1423 x^{10} + 660 x^{9} - 7353 x^{8} - 22934 x^{7} - 36353 x^{6} - 16248 x^{5} + 360646 x^{4} + 1077384 x^{3} + 2005641 x^{2} + 2990790 x + 2924100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$761552187088980301453 \nu^{15} - 1491959421666769906121 \nu^{14} + 3344302464758942156688 \nu^{13} - 15901792535677858594141 \nu^{12} + 3014388319994478907042 \nu^{11} + 101321885624845296988149 \nu^{10} + 807839980494143961894094 \nu^{9} + 1321948851093850161787065 \nu^{8} - 9714576113455613575817754 \nu^{7} - 15844891720438754897824757 \nu^{6} - 36203030832393339219546224 \nu^{5} + 98184628842960984477353481 \nu^{4} + 280667703098769389276470423 \nu^{3} + 472903522110138858905363022 \nu^{2} + 1106227162921563762411701628 \nu + 1040662983654216954539591760$$$$)/$$$$40\!\cdots\!40$$ $$\beta_{2}$$ $$=$$ $$($$$$5067345804644806447 \nu^{15} - 33589566366428217980 \nu^{14} + 101379323293995621549 \nu^{13} - 231129995514191060281 \nu^{12} + 409467356887273180567 \nu^{11} + 139142383633946850999 \nu^{10} + 3419939050357931394025 \nu^{9} - 18916799345583144301377 \nu^{8} - 36352034651388603114855 \nu^{7} + 196438478471418483364897 \nu^{6} - 118508651500636990206083 \nu^{5} - 163865873668959862391787 \nu^{4} + 651707629697127498278140 \nu^{3} - 424761682449882564783639 \nu^{2} - 388979327283261444715410 \nu + 6569736067775022440964228$$$$)/$$$$53\!\cdots\!84$$ $$\beta_{3}$$ $$=$$ $$($$$$-644725549798304743577 \nu^{15} + 7487651363333042856889 \nu^{14} - 47840655887018324414052 \nu^{13} + 166353924107243206678049 \nu^{12} - 477600922375981789902398 \nu^{11} + 1313938545703610318822319 \nu^{10} - 3214761893198220545262866 \nu^{9} + 11193185082007915782542955 \nu^{8} - 39631352782981662572616234 \nu^{7} + 28425015334904667368686033 \nu^{6} + 101078044756304262675923836 \nu^{5} + 266287512393214094616114531 \nu^{4} - 260234319729302891558887127 \nu^{3} + 396603607397221369513765422 \nu^{2} - 7148782001037310240432059252 \nu - 5594889130775964060282591840$$$$)/$$$$67\!\cdots\!40$$ $$\beta_{4}$$ $$=$$ $$($$$$-24996928099783629637 \nu^{15} + 169014394422191295347 \nu^{14} - 514229095182565523766 \nu^{13} + 943429786549289011633 \nu^{12} - 996465397833856831648 \nu^{11} - 3883078592278380293757 \nu^{10} - 5711790625148073147112 \nu^{9} + 78735617182631193745383 \nu^{8} + 181842088526446427093376 \nu^{7} - 1253151104355899035632115 \nu^{6} + 1287608868078035067083210 \nu^{5} + 2100077492493794813298051 \nu^{4} - 4357719988054330165893097 \nu^{3} + 4026951771531907148789868 \nu^{2} + 2061237960871966378789920 \nu - 91237402007135267828489784$$$$)/$$$$21\!\cdots\!36$$ $$\beta_{5}$$ $$=$$ $$($$$$3727952873685311603 \nu^{15} + 5521182694705034291 \nu^{14} - 23399524664310108990 \nu^{13} - 365128169563651976183 \nu^{12} + 1674764379870479392520 \nu^{11} - 5234880775860745311093 \nu^{10} + 24484130999981440613888 \nu^{9} - 23825558874558717995985 \nu^{8} + 4219283013424309199448 \nu^{7} - 784453953085542869398219 \nu^{6} + 1151472672597677905818434 \nu^{5} + 2115612308434572649960539 \nu^{4} + 2772813326813228070194327 \nu^{3} + 3164089909644561264567876 \nu^{2} + 234218914467245497603440 \nu - 81861404862764062906547832$$$$)/$$$$26\!\cdots\!92$$ $$\beta_{6}$$ $$=$$ $$($$$$11508190057396542531277 \nu^{15} - 33265779773123228043929 \nu^{14} + 107802224324647645098432 \nu^{13} - 372151105032278945541469 \nu^{12} + 609008125043123316506578 \nu^{11} - 409260385034556574352619 \nu^{10} + 16582961987520332900382046 \nu^{9} - 5850862148936619852277095 \nu^{8} - 62869813784790621932350026 \nu^{7} - 218823911866770907233277973 \nu^{6} - 253981735005402302914630976 \nu^{5} - 76174259696263557320369271 \nu^{4} + 4091644947625059543128831527 \nu^{3} + 8010683204712207413494147038 \nu^{2} + 22780812172987197387062926812 \nu + 22688894231542180842960353040$$$$)/$$$$40\!\cdots\!40$$ $$\beta_{7}$$ $$=$$ $$($$$$-14598569399574588951649 \nu^{15} + 50390477692181007219533 \nu^{14} - 238900329205779616625784 \nu^{13} + 881460226366638816943753 \nu^{12} - 1878213100594450085425186 \nu^{11} + 3650722482873078021850383 \nu^{10} - 26783582609697969420831262 \nu^{9} + 20294517900294482343208155 \nu^{8} - 13097904209502776702329158 \nu^{7} + 370017574284926921473527281 \nu^{6} + 1014104166900657156964503272 \nu^{5} - 264730445852857656591203013 \nu^{4} - 5847272761554321786848897659 \nu^{3} - 12797505364056679985760036126 \nu^{2} - 31864767299043237143858602044 \nu - 33639309648809410319714418480$$$$)/$$$$40\!\cdots\!40$$ $$\beta_{8}$$ $$=$$ $$($$$$-161479646537937236885 \nu^{15} + 784534032521563760731 \nu^{14} - 2248110713933871056466 \nu^{13} + 11560334720004254498453 \nu^{12} - 40536240086606735313188 \nu^{11} + 89415261518206643357271 \nu^{10} - 415631336851833307748732 \nu^{9} + 920710549879708401978363 \nu^{8} + 880275269198360219532228 \nu^{7} + 4441432971420919107670873 \nu^{6} - 17357939333393815909822994 \nu^{5} - 33307592455613217050577393 \nu^{4} + 38906234025999215003629339 \nu^{3} - 45092591986332361797813168 \nu^{2} + 6057724171262482583821080 \nu + 548017354969978352829867000$$$$)/$$$$42\!\cdots\!72$$ $$\beta_{9}$$ $$=$$ $$($$$$584289809742710006119 \nu^{15} - 1170945684771979401743 \nu^{14} + 1943426810952480585054 \nu^{13} - 7990413294382484438173 \nu^{12} - 4428587027386692977864 \nu^{11} + 68590921319627110618497 \nu^{10} + 715044399161095193008132 \nu^{9} + 527588145155386360056645 \nu^{8} - 6831302071953670043743872 \nu^{7} - 11332505235758605359584681 \nu^{6} - 13462638542996834045929802 \nu^{5} + 24139119760298763058317153 \nu^{4} + 226647859384908572503300099 \nu^{3} + 579028317415366310872204716 \nu^{2} + 605000690633487855944154384 \nu + 828929021894564933067290280$$$$)/$$$$12\!\cdots\!20$$ $$\beta_{10}$$ $$=$$ $$($$$$-394486909086153672407 \nu^{15} + 2064696964331456007289 \nu^{14} - 6159998151995533868094 \nu^{13} + 18046614324158982361583 \nu^{12} - 43694569816143556601492 \nu^{11} + 46871951597620133027109 \nu^{10} - 547043257159049002608092 \nu^{9} + 1449213238584959153226417 \nu^{8} + 2253174582549040670002932 \nu^{7} - 3539467101970109354503253 \nu^{6} - 4753143307444231156849406 \nu^{5} - 12138542121388862651170947 \nu^{4} - 44463333236017482325992479 \nu^{3} - 6795234561761300192330952 \nu^{2} + 22099153019210681187696120 \nu - 53580587133261850172436600$$$$)/$$$$42\!\cdots\!72$$ $$\beta_{11}$$ $$=$$ $$($$$$61315938126590302212769 \nu^{15} - 143996280454582399729733 \nu^{14} + 397746596742611857807224 \nu^{13} - 1550227781954882422325713 \nu^{12} + 1447581124060920506492626 \nu^{11} + 3197027792721695201172537 \nu^{10} + 79199343123106661106720382 \nu^{9} + 35716304270241057169140525 \nu^{8} - 564333154007148374960933802 \nu^{7} - 1257651471188281952464360121 \nu^{6} - 2251132999906952287759133192 \nu^{5} + 3768349639700568267832012653 \nu^{4} + 23875856856221837813107707499 \nu^{3} + 53191805132830946580547885566 \nu^{2} + 86241882498005925832425377724 \nu + 97766553013483949833702060080$$$$)/$$$$40\!\cdots\!40$$ $$\beta_{12}$$ $$=$$ $$($$$$-894889092286227197971 \nu^{15} + 4291676808621833692445 \nu^{14} - 12739561422185392290246 \nu^{13} + 43366373334147120638971 \nu^{12} - 107532359718783135563668 \nu^{11} + 147418431461165146725129 \nu^{10} - 1435146542395456023496252 \nu^{9} + 3270676777102305558718533 \nu^{8} + 4700288397313157242069620 \nu^{7} + 5092207886362586521900295 \nu^{6} - 20737272401540919401667718 \nu^{5} - 45264490308317476986125727 \nu^{4} - 198169407711562161640128523 \nu^{3} - 43802943362796704738322744 \nu^{2} + 44302036482213263534699640 \nu + 415499581570454302207402728$$$$)/$$$$42\!\cdots\!72$$ $$\beta_{13}$$ $$=$$ $$($$$$980764247836442600635 \nu^{15} - 5493864824105826883925 \nu^{14} + 16491312447116624015646 \nu^{13} - 43628336410385269262347 \nu^{12} + 90289225732799390198668 \nu^{11} - 47003882149805572742889 \nu^{10} + 1091580927180863779673140 \nu^{9} - 3454526087741301798051333 \nu^{8} - 5969193838474978257582060 \nu^{7} + 18159280543169214593877145 \nu^{6} - 4231589552251201591927202 \nu^{5} + 1154891639816981277564207 \nu^{4} + 198931620139748606775206971 \nu^{3} - 28057187339545187346248736 \nu^{2} - 61341752375838554686440840 \nu + 1217605006446683899455174456$$$$)/$$$$42\!\cdots\!72$$ $$\beta_{14}$$ $$=$$ $$($$$$16251623242025946916373 \nu^{15} - 19769340192498340593781 \nu^{14} - 51333484927903529912652 \nu^{13} + 161186637766282020676579 \nu^{12} - 1227568479475740679936858 \nu^{11} + 5205919776027649302720909 \nu^{10} + 13565929650916564850773994 \nu^{9} + 38829334914279143161365825 \nu^{8} - 300227923013663114744253534 \nu^{7} - 253729211901876050017612237 \nu^{6} + 26804223163248822851532116 \nu^{5} + 981693882130490498102835561 \nu^{4} + 5698421299164518318380435643 \nu^{3} + 14179258105370254370364795882 \nu^{2} + 9416701999875129527462265828 \nu + 14906580901344096064557233760$$$$)/$$$$67\!\cdots\!40$$ $$\beta_{15}$$ $$=$$ $$($$$$-1525510080235744328053 \nu^{15} + 3665537967228523358711 \nu^{14} - 10053956947939048350813 \nu^{13} + 37923557350211977969426 \nu^{12} - 37391847682323120690397 \nu^{11} - 56610234698652172353624 \nu^{10} - 2033098552178556859049689 \nu^{9} - 537425576506390953371400 \nu^{8} + 13279890991562223538810509 \nu^{7} + 30749460728687223529713032 \nu^{6} + 48033386791453654428272279 \nu^{5} - 59992373318898496772090226 \nu^{4} - 586377101747628177724497688 \nu^{3} - 1341306594475702335606504537 \nu^{2} - 2202476208171216859525781718 \nu - 2528107798275748681649121060$$$$)/$$$$56\!\cdots\!20$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{12} + 3 \beta_{11} - \beta_{10} - 3 \beta_{9} + 6 \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 8 \beta_{2} - 26 \beta_{1} + 29$$$$)/216$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{15} + \beta_{12} + 12 \beta_{11} + 2 \beta_{10} - 3 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} - 3 \beta_{6} + \beta_{5} - 13 \beta_{4} + 2 \beta_{2} - 306 \beta_{1} - 109$$$$)/216$$ $$\nu^{3}$$ $$=$$ $$($$$$11 \beta_{13} + 11 \beta_{12} + 7 \beta_{10} - \beta_{8} + 23 \beta_{5} - 90 \beta_{4} - 104 \beta_{2} + 499$$$$)/216$$ $$\nu^{4}$$ $$=$$ $$($$$$-78 \beta_{15} - 24 \beta_{14} + 4 \beta_{13} + 21 \beta_{12} - 3 \beta_{11} - 39 \beta_{10} - 153 \beta_{9} - 2 \beta_{8} + 156 \beta_{7} - 73 \beta_{6} + 27 \beta_{5} - 164 \beta_{4} + 27 \beta_{3} - 252 \beta_{2} - 2 \beta_{1} + 1239$$$$)/216$$ $$\nu^{5}$$ $$=$$ $$($$$$-270 \beta_{15} + 45 \beta_{14} + 53 \beta_{13} - 22 \beta_{12} - 66 \beta_{11} + 145 \beta_{10} - 1140 \beta_{9} - 44 \beta_{8} + 30 \beta_{7} - 346 \beta_{6} - 64 \beta_{5} + 515 \beta_{4} + 75 \beta_{3} + 158 \beta_{2} - 5018 \beta_{1} - 4786$$$$)/216$$ $$\nu^{6}$$ $$=$$ $$($$$$629 \beta_{13} + 101 \beta_{12} + 685 \beta_{10} - 355 \beta_{8} - 427 \beta_{5} + 2118 \beta_{4} - 4496 \beta_{2} - 120359$$$$)/216$$ $$\nu^{7}$$ $$=$$ $$($$$$1011 \beta_{15} - 1761 \beta_{14} + 885 \beta_{13} - 50 \beta_{12} - 813 \beta_{11} + 884 \beta_{10} + 10656 \beta_{9} - 659 \beta_{8} + 3585 \beta_{7} + 12286 \beta_{6} - 1022 \beta_{5} - 504 \beta_{4} + 2844 \beta_{3} - 15560 \beta_{2} + 72158 \beta_{1} - 294398$$$$)/216$$ $$\nu^{8}$$ $$=$$ $$($$$$-2823 \beta_{15} - 4122 \beta_{14} - 1754 \beta_{13} + 615 \beta_{12} - 14604 \beta_{11} - 3120 \beta_{10} + 20463 \beta_{9} + 2971 \beta_{8} + 57 \beta_{7} + 35363 \beta_{6} + 2475 \beta_{5} + 18895 \beta_{4} + 12510 \beta_{3} + 53202 \beta_{2} + 344254 \beta_{1} + 935853$$$$)/216$$ $$\nu^{9}$$ $$=$$ $$($$$$-7989 \beta_{13} - 3365 \beta_{12} - 18649 \beta_{10} + 20047 \beta_{8} - 7217 \beta_{5} + 224214 \beta_{4} + 304504 \beta_{2} + 3424123$$$$)/216$$ $$\nu^{10}$$ $$=$$ $$($$$$179142 \beta_{15} + 27996 \beta_{14} - 8616 \beta_{13} - 26495 \beta_{12} + 256005 \beta_{11} + 4853 \beta_{10} + 86295 \beta_{9} + 22706 \beta_{8} - 55872 \beta_{7} + 82755 \beta_{6} - 56357 \beta_{5} + 469060 \beta_{4} - 67929 \beta_{3} + 426660 \beta_{2} - 3165234 \beta_{1} - 206529$$$$)/216$$ $$\nu^{11}$$ $$=$$ $$($$$$734322 \beta_{15} + 29925 \beta_{14} - 23563 \beta_{13} + 124586 \beta_{12} + 569910 \beta_{11} - 192887 \beta_{10} + 1379976 \beta_{9} + 3164 \beta_{8} - 390354 \beta_{7} + 1023406 \beta_{6} + 269660 \beta_{5} - 1592181 \beta_{4} - 100341 \beta_{3} - 922034 \beta_{2} - 1057402 \beta_{1} + 17935234$$$$)/216$$ $$\nu^{12}$$ $$=$$ $$($$$$-761411 \beta_{13} + 660669 \beta_{12} - 2164587 \beta_{10} + 613557 \beta_{8} + 1843941 \beta_{5} - 8581530 \beta_{4} + 1089232 \beta_{2} + 232789369$$$$)/216$$ $$\nu^{13}$$ $$=$$ $$($$$$-5858433 \beta_{15} + 3139371 \beta_{14} - 2150335 \beta_{13} + 526994 \beta_{12} + 1480539 \beta_{11} - 3981368 \beta_{10} - 34959216 \beta_{9} + 1816953 \beta_{8} - 3941859 \beta_{7} - 26151518 \beta_{6} + 2723330 \beta_{5} - 6857312 \beta_{4} - 3254592 \beta_{3} + 19713048 \beta_{2} - 204851986 \beta_{1} + 579594810$$$$)/216$$ $$\nu^{14}$$ $$=$$ $$($$$$-6197223 \beta_{15} + 14141502 \beta_{14} + 8016662 \beta_{13} - 183529 \beta_{12} + 21641868 \beta_{11} + 11620120 \beta_{10} - 106548813 \beta_{9} - 7742349 \beta_{8} - 21773463 \beta_{7} - 97697109 \beta_{6} - 6842257 \beta_{5} - 14271785 \beta_{4} - 21812322 \beta_{3} - 121025982 \beta_{2} - 894364770 \beta_{1} - 2350843623$$$$)/216$$ $$\nu^{15}$$ $$=$$ $$($$$$44334675 \beta_{13} + 8238067 \beta_{12} + 55447631 \beta_{10} - 55186649 \beta_{8} - 16065713 \beta_{5} - 382179978 \beta_{4} - 995971112 \beta_{2} - 14690631509$$$$)/216$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$55$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 −1.64756 − 0.974628i −1.64756 + 0.974628i −1.81633 + 0.757118i −1.81633 − 0.757118i 2.91300 + 0.609585i 2.91300 − 0.609585i 3.07345 + 2.03963i 3.07345 − 2.03963i 0.229644 + 3.68150i 0.229644 − 3.68150i −1.98442 − 2.21794i −1.98442 + 2.21794i 0.252484 + 1.95155i 0.252484 − 1.95155i −0.0202748 − 1.91414i −0.0202748 + 1.91414i
−3.81409 1.20528i 0 13.0946 + 9.19411i −13.3855 0 25.4062i −38.8625 50.8499i 0 51.0536 + 16.1333i
55.2 −3.81409 + 1.20528i 0 13.0946 9.19411i −13.3855 0 25.4062i −38.8625 + 50.8499i 0 51.0536 16.1333i
55.3 −3.36416 2.16389i 0 6.63513 + 14.5594i 43.0579 0 14.1139i 9.18328 63.3377i 0 −144.853 93.1726i
55.4 −3.36416 + 2.16389i 0 6.63513 14.5594i 43.0579 0 14.1139i 9.18328 + 63.3377i 0 −144.853 + 93.1726i
55.5 −1.59266 3.66925i 0 −10.9268 + 11.6878i −29.4580 0 32.9098i 60.2882 + 21.4787i 0 46.9167 + 108.089i
55.6 −1.59266 + 3.66925i 0 −10.9268 11.6878i −29.4580 0 32.9098i 60.2882 21.4787i 0 46.9167 108.089i
55.7 −1.35962 3.76184i 0 −12.3029 + 10.2293i 2.66014 0 88.8835i 55.2083 + 32.3735i 0 −3.61678 10.0070i
55.8 −1.35962 + 3.76184i 0 −12.3029 10.2293i 2.66014 0 88.8835i 55.2083 32.3735i 0 −3.61678 + 10.0070i
55.9 1.35962 3.76184i 0 −12.3029 10.2293i −2.66014 0 88.8835i −55.2083 + 32.3735i 0 −3.61678 + 10.0070i
55.10 1.35962 + 3.76184i 0 −12.3029 + 10.2293i −2.66014 0 88.8835i −55.2083 32.3735i 0 −3.61678 10.0070i
55.11 1.59266 3.66925i 0 −10.9268 11.6878i 29.4580 0 32.9098i −60.2882 + 21.4787i 0 46.9167 108.089i
55.12 1.59266 + 3.66925i 0 −10.9268 + 11.6878i 29.4580 0 32.9098i −60.2882 21.4787i 0 46.9167 + 108.089i
55.13 3.36416 2.16389i 0 6.63513 14.5594i −43.0579 0 14.1139i −9.18328 63.3377i 0 −144.853 + 93.1726i
55.14 3.36416 + 2.16389i 0 6.63513 + 14.5594i −43.0579 0 14.1139i −9.18328 + 63.3377i 0 −144.853 93.1726i
55.15 3.81409 1.20528i 0 13.0946 9.19411i 13.3855 0 25.4062i 38.8625 50.8499i 0 51.0536 16.1333i
55.16 3.81409 + 1.20528i 0 13.0946 + 9.19411i 13.3855 0 25.4062i 38.8625 + 50.8499i 0 51.0536 + 16.1333i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 55.16 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
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.5.d.a 16
3.b odd 2 1 inner 108.5.d.a 16
4.b odd 2 1 inner 108.5.d.a 16
12.b even 2 1 inner 108.5.d.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.5.d.a 16 1.a even 1 1 trivial
108.5.d.a 16 3.b odd 2 1 inner
108.5.d.a 16 4.b odd 2 1 inner
108.5.d.a 16 12.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 2908 T_{5}^{6} + 2117022 T_{5}^{4} - 303093292 T_{5}^{2} + 2039817193$$ acting on $$S_{5}^{\mathrm{new}}(108, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 7 T^{2} + 76 T^{4} + 304 T^{6} + 94720 T^{8} + 77824 T^{10} + 4980736 T^{12} + 117440512 T^{14} + 4294967296 T^{16}$$
$3$ 1
$5$ $$( 1 + 2092 T^{2} + 2149522 T^{4} + 1622274208 T^{6} + 1066877108443 T^{8} + 633700862500000 T^{10} + 327991027832031250 T^{12} +$$$$12\!\cdots\!00$$$$T^{14} +$$$$23\!\cdots\!25$$$$T^{16} )^{2}$$
$7$ $$( 1 - 9380 T^{2} + 36105250 T^{4} - 73168453568 T^{6} + 129372615567787 T^{8} - 421801574297259968 T^{10} +$$$$11\!\cdots\!50$$$$T^{12} -$$$$17\!\cdots\!80$$$$T^{14} +$$$$11\!\cdots\!01$$$$T^{16} )^{2}$$
$11$ $$( 1 - 48044 T^{2} + 1167571642 T^{4} - 20151290866448 T^{6} + 304446182867375107 T^{8} -$$$$43\!\cdots\!88$$$$T^{10} +$$$$53\!\cdots\!62$$$$T^{12} -$$$$47\!\cdots\!04$$$$T^{14} +$$$$21\!\cdots\!21$$$$T^{16} )^{2}$$
$13$ $$( 1 + 88 T + 63508 T^{2} + 5077384 T^{3} + 2650180198 T^{4} + 145015164424 T^{5} + 51805426629268 T^{6} + 2050231490778328 T^{7} + 665416609183179841 T^{8} )^{4}$$
$17$ $$( 1 + 239464 T^{2} + 41438060380 T^{4} + 5059782297489112 T^{6} +$$$$47\!\cdots\!02$$$$T^{8} +$$$$35\!\cdots\!92$$$$T^{10} +$$$$20\!\cdots\!80$$$$T^{12} +$$$$81\!\cdots\!44$$$$T^{14} +$$$$23\!\cdots\!61$$$$T^{16} )^{2}$$
$19$ $$( 1 - 680120 T^{2} + 233557156252 T^{4} - 51394020863660552 T^{6} +$$$$79\!\cdots\!98$$$$T^{8} -$$$$87\!\cdots\!32$$$$T^{10} +$$$$67\!\cdots\!12$$$$T^{12} -$$$$33\!\cdots\!20$$$$T^{14} +$$$$83\!\cdots\!61$$$$T^{16} )^{2}$$
$23$ $$( 1 - 675608 T^{2} + 314533671388 T^{4} - 115630955446337192 T^{6} +$$$$35\!\cdots\!18$$$$T^{8} -$$$$90\!\cdots\!52$$$$T^{10} +$$$$19\!\cdots\!68$$$$T^{12} -$$$$32\!\cdots\!28$$$$T^{14} +$$$$37\!\cdots\!21$$$$T^{16} )^{2}$$
$29$ $$( 1 + 2998744 T^{2} + 4863922457692 T^{4} + 5416371986440228072 T^{6} +$$$$44\!\cdots\!14$$$$T^{8} +$$$$27\!\cdots\!92$$$$T^{10} +$$$$12\!\cdots\!32$$$$T^{12} +$$$$37\!\cdots\!64$$$$T^{14} +$$$$62\!\cdots\!41$$$$T^{16} )^{2}$$
$31$ $$( 1 - 1328756 T^{2} + 2683287954034 T^{4} - 2424947225563311392 T^{6} +$$$$29\!\cdots\!59$$$$T^{8} -$$$$20\!\cdots\!72$$$$T^{10} +$$$$19\!\cdots\!54$$$$T^{12} -$$$$82\!\cdots\!76$$$$T^{14} +$$$$52\!\cdots\!61$$$$T^{16} )^{2}$$
$37$ $$( 1 - 800 T + 1852732 T^{2} - 3784975328 T^{3} + 1716048721222 T^{4} - 7093653145699808 T^{5} + 6507683083621962172 T^{6} -$$$$52\!\cdots\!00$$$$T^{7} +$$$$12\!\cdots\!41$$$$T^{8} )^{4}$$
$41$ $$( 1 + 14045080 T^{2} + 100482710328412 T^{4} +$$$$46\!\cdots\!84$$$$T^{6} +$$$$15\!\cdots\!54$$$$T^{8} +$$$$37\!\cdots\!64$$$$T^{10} +$$$$64\!\cdots\!92$$$$T^{12} +$$$$71\!\cdots\!80$$$$T^{14} +$$$$40\!\cdots\!81$$$$T^{16} )^{2}$$
$43$ $$( 1 - 11990648 T^{2} + 69115070774812 T^{4} -$$$$29\!\cdots\!24$$$$T^{6} +$$$$10\!\cdots\!18$$$$T^{8} -$$$$34\!\cdots\!24$$$$T^{10} +$$$$94\!\cdots\!12$$$$T^{12} -$$$$19\!\cdots\!48$$$$T^{14} +$$$$18\!\cdots\!01$$$$T^{16} )^{2}$$
$47$ $$( 1 - 27701336 T^{2} + 374201000701276 T^{4} -$$$$31\!\cdots\!48$$$$T^{6} +$$$$18\!\cdots\!14$$$$T^{8} -$$$$75\!\cdots\!28$$$$T^{10} +$$$$21\!\cdots\!96$$$$T^{12} -$$$$37\!\cdots\!16$$$$T^{14} +$$$$32\!\cdots\!41$$$$T^{16} )^{2}$$
$53$ $$( 1 + 35837884 T^{2} + 582188420839618 T^{4} +$$$$60\!\cdots\!16$$$$T^{6} +$$$$50\!\cdots\!79$$$$T^{8} +$$$$37\!\cdots\!76$$$$T^{10} +$$$$22\!\cdots\!78$$$$T^{12} +$$$$86\!\cdots\!04$$$$T^{14} +$$$$15\!\cdots\!41$$$$T^{16} )^{2}$$
$59$ $$( 1 - 70312136 T^{2} + 2419955263154716 T^{4} -$$$$51\!\cdots\!68$$$$T^{6} +$$$$75\!\cdots\!74$$$$T^{8} -$$$$76\!\cdots\!28$$$$T^{10} +$$$$52\!\cdots\!56$$$$T^{12} -$$$$22\!\cdots\!96$$$$T^{14} +$$$$46\!\cdots\!81$$$$T^{16} )^{2}$$
$61$ $$( 1 + 688 T + 35342980 T^{2} + 28994874640 T^{3} + 632810106882118 T^{4} + 401458424080372240 T^{5} +$$$$67\!\cdots\!80$$$$T^{6} +$$$$18\!\cdots\!48$$$$T^{7} +$$$$36\!\cdots\!61$$$$T^{8} )^{4}$$
$67$ $$( 1 - 29579000 T^{2} + 1274223737389084 T^{4} -$$$$30\!\cdots\!28$$$$T^{6} +$$$$73\!\cdots\!58$$$$T^{8} -$$$$12\!\cdots\!48$$$$T^{10} +$$$$21\!\cdots\!04$$$$T^{12} -$$$$19\!\cdots\!00$$$$T^{14} +$$$$27\!\cdots\!61$$$$T^{16} )^{2}$$
$71$ $$( 1 - 86649128 T^{2} + 3567527319446236 T^{4} -$$$$11\!\cdots\!36$$$$T^{6} +$$$$32\!\cdots\!90$$$$T^{8} -$$$$74\!\cdots\!96$$$$T^{10} +$$$$14\!\cdots\!56$$$$T^{12} -$$$$23\!\cdots\!68$$$$T^{14} +$$$$17\!\cdots\!41$$$$T^{16} )^{2}$$
$73$ $$( 1 - 2060 T + 55108594 T^{2} - 101224244576 T^{3} + 2192760100160827 T^{4} - 2874590492512190816 T^{5} +$$$$44\!\cdots\!14$$$$T^{6} -$$$$47\!\cdots\!60$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8} )^{4}$$
$79$ $$( 1 - 162885800 T^{2} + 12409200314583772 T^{4} -$$$$63\!\cdots\!36$$$$T^{6} +$$$$26\!\cdots\!06$$$$T^{8} -$$$$96\!\cdots\!96$$$$T^{10} +$$$$28\!\cdots\!12$$$$T^{12} -$$$$56\!\cdots\!00$$$$T^{14} +$$$$52\!\cdots\!41$$$$T^{16} )^{2}$$
$83$ $$( 1 - 140155436 T^{2} + 8287556309932090 T^{4} -$$$$42\!\cdots\!92$$$$T^{6} +$$$$22\!\cdots\!03$$$$T^{8} -$$$$95\!\cdots\!72$$$$T^{10} +$$$$42\!\cdots\!90$$$$T^{12} -$$$$16\!\cdots\!56$$$$T^{14} +$$$$25\!\cdots\!61$$$$T^{16} )^{2}$$
$89$ $$( 1 + 257253400 T^{2} + 38619499284608860 T^{4} +$$$$38\!\cdots\!44$$$$T^{6} +$$$$28\!\cdots\!58$$$$T^{8} +$$$$15\!\cdots\!64$$$$T^{10} +$$$$59\!\cdots\!60$$$$T^{12} +$$$$15\!\cdots\!00$$$$T^{14} +$$$$24\!\cdots\!21$$$$T^{16} )^{2}$$
$97$ $$( 1 + 1732 T + 297359818 T^{2} + 428249450896 T^{3} + 37002869299122643 T^{4} + 37912615976467685776 T^{5} +$$$$23\!\cdots\!98$$$$T^{6} +$$$$12\!\cdots\!12$$$$T^{7} +$$$$61\!\cdots\!21$$$$T^{8} )^{4}$$