# Properties

 Label 354.8.a.e Level 354 Weight 8 Character orbit 354.a Self dual Yes Analytic conductor 110.584 Analytic rank 1 Dimension 9 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$354 = 2 \cdot 3 \cdot 59$$ Weight: $$k$$ = $$8$$ Character orbit: $$[\chi]$$ = 354.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$110.584299021$$ Analytic rank: $$1$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 3^{5}\cdot 5\cdot 7$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -8 q^{2} -27 q^{3} + 64 q^{4} + ( -26 + \beta_{3} ) q^{5} + 216 q^{6} + ( -37 + \beta_{2} - \beta_{3} ) q^{7} -512 q^{8} + 729 q^{9} +O(q^{10})$$ $$q -8 q^{2} -27 q^{3} + 64 q^{4} + ( -26 + \beta_{3} ) q^{5} + 216 q^{6} + ( -37 + \beta_{2} - \beta_{3} ) q^{7} -512 q^{8} + 729 q^{9} + ( 208 - 8 \beta_{3} ) q^{10} + ( -608 + 2 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{7} ) q^{11} -1728 q^{12} + ( 346 - \beta_{1} + 4 \beta_{2} + 9 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} - 2 \beta_{8} ) q^{13} + ( 296 - 8 \beta_{2} + 8 \beta_{3} ) q^{14} + ( 702 - 27 \beta_{3} ) q^{15} + 4096 q^{16} + ( -416 - 3 \beta_{1} - 10 \beta_{2} + 15 \beta_{3} + \beta_{4} - 2 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} + 7 \beta_{8} ) q^{17} -5832 q^{18} + ( 62 - 14 \beta_{1} + 4 \beta_{2} + 33 \beta_{3} - \beta_{4} + 6 \beta_{5} + 14 \beta_{6} + \beta_{7} + \beta_{8} ) q^{19} + ( -1664 + 64 \beta_{3} ) q^{20} + ( 999 - 27 \beta_{2} + 27 \beta_{3} ) q^{21} + ( 4864 - 16 \beta_{1} - 8 \beta_{3} + 8 \beta_{4} + 16 \beta_{7} ) q^{22} + ( -10265 + 4 \beta_{1} - 37 \beta_{2} + 40 \beta_{3} - 41 \beta_{4} - 8 \beta_{5} - 8 \beta_{6} + 17 \beta_{7} + 4 \beta_{8} ) q^{23} + 13824 q^{24} + ( 8387 + 13 \beta_{1} - 7 \beta_{2} - 181 \beta_{3} + 55 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - 25 \beta_{7} - 5 \beta_{8} ) q^{25} + ( -2768 + 8 \beta_{1} - 32 \beta_{2} - 72 \beta_{3} - 16 \beta_{4} - 16 \beta_{7} + 16 \beta_{8} ) q^{26} -19683 q^{27} + ( -2368 + 64 \beta_{2} - 64 \beta_{3} ) q^{28} + ( -4521 - 36 \beta_{1} - 55 \beta_{2} - 91 \beta_{3} + 15 \beta_{4} - 8 \beta_{5} - 14 \beta_{6} - 17 \beta_{7} - 40 \beta_{8} ) q^{29} + ( -5616 + 216 \beta_{3} ) q^{30} + ( -35981 + 107 \beta_{1} - 14 \beta_{2} - 223 \beta_{3} + 8 \beta_{4} + 37 \beta_{5} - 29 \beta_{6} + 47 \beta_{7} - 2 \beta_{8} ) q^{31} -32768 q^{32} + ( 16416 - 54 \beta_{1} - 27 \beta_{3} + 27 \beta_{4} + 54 \beta_{7} ) q^{33} + ( 3328 + 24 \beta_{1} + 80 \beta_{2} - 120 \beta_{3} - 8 \beta_{4} + 16 \beta_{5} + 48 \beta_{6} - 32 \beta_{7} - 56 \beta_{8} ) q^{34} + ( -46075 + 109 \beta_{1} - 111 \beta_{2} - 237 \beta_{3} - 20 \beta_{4} + 46 \beta_{5} + 22 \beta_{6} - 120 \beta_{7} ) q^{35} + 46656 q^{36} + ( 37877 - 189 \beta_{1} + 32 \beta_{2} - 458 \beta_{3} - 15 \beta_{4} - 71 \beta_{5} - 7 \beta_{6} - 40 \beta_{7} - 10 \beta_{8} ) q^{37} + ( -496 + 112 \beta_{1} - 32 \beta_{2} - 264 \beta_{3} + 8 \beta_{4} - 48 \beta_{5} - 112 \beta_{6} - 8 \beta_{7} - 8 \beta_{8} ) q^{38} + ( -9342 + 27 \beta_{1} - 108 \beta_{2} - 243 \beta_{3} - 54 \beta_{4} - 54 \beta_{7} + 54 \beta_{8} ) q^{39} + ( 13312 - 512 \beta_{3} ) q^{40} + ( -11555 + 42 \beta_{1} - 24 \beta_{2} - 337 \beta_{3} + 66 \beta_{4} - 93 \beta_{5} + 36 \beta_{6} + 113 \beta_{7} + 119 \beta_{8} ) q^{41} + ( -7992 + 216 \beta_{2} - 216 \beta_{3} ) q^{42} + ( 111169 - 28 \beta_{1} - 161 \beta_{2} - 394 \beta_{3} - 293 \beta_{4} - 58 \beta_{5} + 158 \beta_{6} + 106 \beta_{7} - 67 \beta_{8} ) q^{43} + ( -38912 + 128 \beta_{1} + 64 \beta_{3} - 64 \beta_{4} - 128 \beta_{7} ) q^{44} + ( -18954 + 729 \beta_{3} ) q^{45} + ( 82120 - 32 \beta_{1} + 296 \beta_{2} - 320 \beta_{3} + 328 \beta_{4} + 64 \beta_{5} + 64 \beta_{6} - 136 \beta_{7} - 32 \beta_{8} ) q^{46} + ( -40638 + 111 \beta_{1} - 121 \beta_{2} + 54 \beta_{3} + 32 \beta_{4} + 181 \beta_{5} + 54 \beta_{6} + 30 \beta_{7} + 116 \beta_{8} ) q^{47} -110592 q^{48} + ( 256966 - 568 \beta_{1} + 15 \beta_{2} - 770 \beta_{3} - 140 \beta_{4} + 100 \beta_{5} - 313 \beta_{6} + 156 \beta_{7} + 5 \beta_{8} ) q^{49} + ( -67096 - 104 \beta_{1} + 56 \beta_{2} + 1448 \beta_{3} - 440 \beta_{4} + 24 \beta_{5} - 32 \beta_{6} + 200 \beta_{7} + 40 \beta_{8} ) q^{50} + ( 11232 + 81 \beta_{1} + 270 \beta_{2} - 405 \beta_{3} - 27 \beta_{4} + 54 \beta_{5} + 162 \beta_{6} - 108 \beta_{7} - 189 \beta_{8} ) q^{51} + ( 22144 - 64 \beta_{1} + 256 \beta_{2} + 576 \beta_{3} + 128 \beta_{4} + 128 \beta_{7} - 128 \beta_{8} ) q^{52} + ( 224690 - 278 \beta_{1} + 288 \beta_{2} - 577 \beta_{3} + 834 \beta_{4} + 80 \beta_{5} + 132 \beta_{6} - 318 \beta_{7} + 4 \beta_{8} ) q^{53} + 157464 q^{54} + ( 125080 + 115 \beta_{1} + 440 \beta_{2} - 1195 \beta_{3} - 185 \beta_{4} + 10 \beta_{5} - 120 \beta_{6} + 295 \beta_{7} + 330 \beta_{8} ) q^{55} + ( 18944 - 512 \beta_{2} + 512 \beta_{3} ) q^{56} + ( -1674 + 378 \beta_{1} - 108 \beta_{2} - 891 \beta_{3} + 27 \beta_{4} - 162 \beta_{5} - 378 \beta_{6} - 27 \beta_{7} - 27 \beta_{8} ) q^{57} + ( 36168 + 288 \beta_{1} + 440 \beta_{2} + 728 \beta_{3} - 120 \beta_{4} + 64 \beta_{5} + 112 \beta_{6} + 136 \beta_{7} + 320 \beta_{8} ) q^{58} + 205379 q^{59} + ( 44928 - 1728 \beta_{3} ) q^{60} + ( 566576 + 377 \beta_{1} + 210 \beta_{2} - 410 \beta_{3} - 34 \beta_{4} - 560 \beta_{5} + 293 \beta_{6} + 455 \beta_{7} - 4 \beta_{8} ) q^{61} + ( 287848 - 856 \beta_{1} + 112 \beta_{2} + 1784 \beta_{3} - 64 \beta_{4} - 296 \beta_{5} + 232 \beta_{6} - 376 \beta_{7} + 16 \beta_{8} ) q^{62} + ( -26973 + 729 \beta_{2} - 729 \beta_{3} ) q^{63} + 262144 q^{64} + ( 834126 - 40 \beta_{1} - 1140 \beta_{2} + 739 \beta_{3} - 10 \beta_{4} + 180 \beta_{5} + 85 \beta_{6} - 105 \beta_{7} - 555 \beta_{8} ) q^{65} + ( -131328 + 432 \beta_{1} + 216 \beta_{3} - 216 \beta_{4} - 432 \beta_{7} ) q^{66} + ( 1215582 + 203 \beta_{1} - 226 \beta_{2} - 3575 \beta_{3} + 561 \beta_{4} + 158 \beta_{5} - 984 \beta_{6} - 537 \beta_{7} - 74 \beta_{8} ) q^{67} + ( -26624 - 192 \beta_{1} - 640 \beta_{2} + 960 \beta_{3} + 64 \beta_{4} - 128 \beta_{5} - 384 \beta_{6} + 256 \beta_{7} + 448 \beta_{8} ) q^{68} + ( 277155 - 108 \beta_{1} + 999 \beta_{2} - 1080 \beta_{3} + 1107 \beta_{4} + 216 \beta_{5} + 216 \beta_{6} - 459 \beta_{7} - 108 \beta_{8} ) q^{69} + ( 368600 - 872 \beta_{1} + 888 \beta_{2} + 1896 \beta_{3} + 160 \beta_{4} - 368 \beta_{5} - 176 \beta_{6} + 960 \beta_{7} ) q^{70} + ( 397941 - 1143 \beta_{1} + 344 \beta_{2} + 6206 \beta_{3} - 860 \beta_{4} - 363 \beta_{5} + 264 \beta_{6} + 476 \beta_{7} - 524 \beta_{8} ) q^{71} -373248 q^{72} + ( 1439021 + 2004 \beta_{1} - 1966 \beta_{2} - 1679 \beta_{3} + 387 \beta_{4} + 719 \beta_{5} - 1071 \beta_{6} + 370 \beta_{7} + 240 \beta_{8} ) q^{73} + ( -303016 + 1512 \beta_{1} - 256 \beta_{2} + 3664 \beta_{3} + 120 \beta_{4} + 568 \beta_{5} + 56 \beta_{6} + 320 \beta_{7} + 80 \beta_{8} ) q^{74} + ( -226449 - 351 \beta_{1} + 189 \beta_{2} + 4887 \beta_{3} - 1485 \beta_{4} + 81 \beta_{5} - 108 \beta_{6} + 675 \beta_{7} + 135 \beta_{8} ) q^{75} + ( 3968 - 896 \beta_{1} + 256 \beta_{2} + 2112 \beta_{3} - 64 \beta_{4} + 384 \beta_{5} + 896 \beta_{6} + 64 \beta_{7} + 64 \beta_{8} ) q^{76} + ( 787223 + 167 \beta_{1} - 1124 \beta_{2} + 9977 \beta_{3} - 965 \beta_{4} + 309 \beta_{5} + 1317 \beta_{6} - 1242 \beta_{7} - 1132 \beta_{8} ) q^{77} + ( 74736 - 216 \beta_{1} + 864 \beta_{2} + 1944 \beta_{3} + 432 \beta_{4} + 432 \beta_{7} - 432 \beta_{8} ) q^{78} + ( 831143 - 1276 \beta_{1} + 788 \beta_{2} + 2594 \beta_{3} + 217 \beta_{4} - 1107 \beta_{5} + 1610 \beta_{6} - 1509 \beta_{7} - 692 \beta_{8} ) q^{79} + ( -106496 + 4096 \beta_{3} ) q^{80} + 531441 q^{81} + ( 92440 - 336 \beta_{1} + 192 \beta_{2} + 2696 \beta_{3} - 528 \beta_{4} + 744 \beta_{5} - 288 \beta_{6} - 904 \beta_{7} - 952 \beta_{8} ) q^{82} + ( 303351 + 1424 \beta_{1} - 650 \beta_{2} + 3687 \beta_{3} - 1584 \beta_{4} - 59 \beta_{5} + 892 \beta_{6} + 1763 \beta_{7} + 239 \beta_{8} ) q^{83} + ( 63936 - 1728 \beta_{2} + 1728 \beta_{3} ) q^{84} + ( 1313738 + 2297 \beta_{1} - 518 \beta_{2} - 3429 \beta_{3} + 2500 \beta_{4} - 632 \beta_{5} + 281 \beta_{6} - 935 \beta_{7} + 900 \beta_{8} ) q^{85} + ( -889352 + 224 \beta_{1} + 1288 \beta_{2} + 3152 \beta_{3} + 2344 \beta_{4} + 464 \beta_{5} - 1264 \beta_{6} - 848 \beta_{7} + 536 \beta_{8} ) q^{86} + ( 122067 + 972 \beta_{1} + 1485 \beta_{2} + 2457 \beta_{3} - 405 \beta_{4} + 216 \beta_{5} + 378 \beta_{6} + 459 \beta_{7} + 1080 \beta_{8} ) q^{87} + ( 311296 - 1024 \beta_{1} - 512 \beta_{3} + 512 \beta_{4} + 1024 \beta_{7} ) q^{88} + ( -1110551 + 983 \beta_{1} - 828 \beta_{2} + 9685 \beta_{3} - 2198 \beta_{4} - 775 \beta_{5} - 2540 \beta_{6} + 2852 \beta_{7} + 1773 \beta_{8} ) q^{89} + ( 151632 - 5832 \beta_{3} ) q^{90} + ( 2164461 - 863 \beta_{1} - 2481 \beta_{2} - 17367 \beta_{3} - 2089 \beta_{4} + 1562 \beta_{5} - 2733 \beta_{6} - 31 \beta_{7} + 2559 \beta_{8} ) q^{91} + ( -656960 + 256 \beta_{1} - 2368 \beta_{2} + 2560 \beta_{3} - 2624 \beta_{4} - 512 \beta_{5} - 512 \beta_{6} + 1088 \beta_{7} + 256 \beta_{8} ) q^{92} + ( 971487 - 2889 \beta_{1} + 378 \beta_{2} + 6021 \beta_{3} - 216 \beta_{4} - 999 \beta_{5} + 783 \beta_{6} - 1269 \beta_{7} + 54 \beta_{8} ) q^{93} + ( 325104 - 888 \beta_{1} + 968 \beta_{2} - 432 \beta_{3} - 256 \beta_{4} - 1448 \beta_{5} - 432 \beta_{6} - 240 \beta_{7} - 928 \beta_{8} ) q^{94} + ( 2682691 - 3201 \beta_{1} + 2204 \beta_{2} - 15848 \beta_{3} + 6420 \beta_{4} - 9 \beta_{5} + 1637 \beta_{6} - 7295 \beta_{7} - 2840 \beta_{8} ) q^{95} + 884736 q^{96} + ( 4242613 - 1476 \beta_{1} - 3475 \beta_{2} - 6046 \beta_{3} - 714 \beta_{4} + 1636 \beta_{5} + 3625 \beta_{6} - 4689 \beta_{7} - 1714 \beta_{8} ) q^{97} + ( -2055728 + 4544 \beta_{1} - 120 \beta_{2} + 6160 \beta_{3} + 1120 \beta_{4} - 800 \beta_{5} + 2504 \beta_{6} - 1248 \beta_{7} - 40 \beta_{8} ) q^{98} + ( -443232 + 1458 \beta_{1} + 729 \beta_{3} - 729 \beta_{4} - 1458 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9q - 72q^{2} - 243q^{3} + 576q^{4} - 230q^{5} + 1944q^{6} - 340q^{7} - 4608q^{8} + 6561q^{9} + O(q^{10})$$ $$9q - 72q^{2} - 243q^{3} + 576q^{4} - 230q^{5} + 1944q^{6} - 340q^{7} - 4608q^{8} + 6561q^{9} + 1840q^{10} - 5472q^{11} - 15552q^{12} + 3144q^{13} + 2720q^{14} + 6210q^{15} + 36864q^{16} - 3662q^{17} - 52488q^{18} + 692q^{19} - 14720q^{20} + 9180q^{21} + 43776q^{22} - 92046q^{23} + 124416q^{24} + 74731q^{25} - 25152q^{26} - 177147q^{27} - 21760q^{28} - 41060q^{29} - 49680q^{30} - 324504q^{31} - 294912q^{32} + 147744q^{33} + 29296q^{34} - 415602q^{35} + 419904q^{36} + 338612q^{37} - 5536q^{38} - 84888q^{39} + 117760q^{40} - 104312q^{41} - 73440q^{42} + 1000602q^{43} - 350208q^{44} - 167670q^{45} + 736368q^{46} - 365148q^{47} - 995328q^{48} + 2307505q^{49} - 597848q^{50} + 98874q^{51} + 201216q^{52} + 2017498q^{53} + 1417176q^{54} + 1120520q^{55} + 174080q^{56} - 18684q^{57} + 328480q^{58} + 1848411q^{59} + 397440q^{60} + 5102340q^{61} + 2596032q^{62} - 247860q^{63} + 2359296q^{64} + 7512810q^{65} - 1181952q^{66} + 10920464q^{67} - 234368q^{68} + 2485242q^{69} + 3324816q^{70} + 3607024q^{71} - 3359232q^{72} + 12949418q^{73} - 2708896q^{74} - 2017737q^{75} + 44288q^{76} + 7127994q^{77} + 679104q^{78} + 7489472q^{79} - 942080q^{80} + 4782969q^{81} + 834496q^{82} + 2760502q^{83} + 587520q^{84} + 11815354q^{85} - 8004816q^{86} + 1108620q^{87} + 2801664q^{88} - 9948196q^{89} + 1341360q^{90} + 19400656q^{91} - 5890944q^{92} + 8761608q^{93} + 2921184q^{94} + 24045208q^{95} + 7962624q^{96} + 38157642q^{97} - 18460040q^{98} - 3989088q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 4 x^{8} - 260588 x^{7} - 2627755 x^{6} + 16696953355 x^{5} + 808091684078 x^{4} - 191825159847522 x^{3} - 12153827093891319 x^{2} + 237398587398357570 x + 15300107670464479800$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3668939840838464900208001479756826739 \nu^{8} - 808927757691660685532640411115107139705 \nu^{7} + 1024232305562256697349930174921364047203213 \nu^{6} + 212610580222705259305153124089274632924641212 \nu^{5} - 75920706305805909138164910976987871519071561109 \nu^{4} - 14198952877994518486481308166268388028461601892661 \nu^{3} + 1077202071723610663643339322640899163240208989312515 \nu^{2} + 110074072158696633046291681610817452915364371270148690 \nu - 2027834810736579130218015207089154601444816851262259400$$$$)/$$$$25\!\cdots\!00$$ $$\beta_{2}$$ $$=$$ $$($$$$-118684004847550059995542261743618877787 \nu^{8} + 13595699219220410996763709815180968169899 \nu^{7} + 30481371877807375743089022476056177451694229 \nu^{6} - 3024192307844304357116007988924504830312840232 \nu^{5} - 1922596629885236661623187863842062927199207654049 \nu^{4} + 103699244460244789558284990033847253771063309891891 \nu^{3} + 28085418254449554328902761310538957319792572619315871 \nu^{2} - 382066949961677418104366215782574455171318015023080230 \nu - 82923706136157192792692017368131668590520608308631352200$$$$)/$$$$50\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$-30918435317443156228807597915841337847 \nu^{8} + 3711933721783212859865161893471312142279 \nu^{7} + 7719432430908895424475638453797638120292889 \nu^{6} - 825141565071695543897330770845570883887518312 \nu^{5} - 444607696400972519555722297814460838206382362229 \nu^{4} + 28796906905906049775833620338336523148969058584911 \nu^{3} + 4091864013234158861313598244750788959450083475778731 \nu^{2} - 154186562029489947330374028704137958411651456534163870 \nu - 5866505261182097230307101835202254223855567679656393000$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$-318346690634034368473388623817452193383 \nu^{8} + 29524247885641074757696978878953649672391 \nu^{7} + 79954026153943303988433609082440025706807561 \nu^{6} - 6306084734750921318874707963766803202932692088 \nu^{5} - 4663310076628940346921864556576657455273674847541 \nu^{4} + 167544239394045192027393523996789645313102088060319 \nu^{3} + 40449564277301131082425014715196369287298989265108139 \nu^{2} - 356910380139749914086961920172649359385781254132019870 \nu + 18341088375876563304019673790659987067685186088026138200$$$$)/$$$$50\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$994989197480974901132417176878043862411 \nu^{8} - 80487732975776562781914311641126142686907 \nu^{7} - 252880741903334146927288918493442503118261637 \nu^{6} + 16887482994280777627315395389364439379555021416 \nu^{5} + 15251630441157334077543879746241556671884163610577 \nu^{4} - 372984438916463045266074516946704333801386818326083 \nu^{3} - 156688934826991019487174603181202478108491480983533903 \nu^{2} - 519087784007291069767282093447308206760109342682999610 \nu + 170536040550175901532968458378464126389486546422809054600$$$$)/$$$$50\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$343883527816357216173114211852959877687 \nu^{8} - 32199428189865214136469283558127322037799 \nu^{7} - 86828181697893048469423336655249951235850329 \nu^{6} + 6850882490986822323610434493952680305370034632 \nu^{5} + 5157702614710903890188443308645317525834042363349 \nu^{4} - 176376640121537010890823039863823065662417633842191 \nu^{3} - 52291904204423494126970708285245358169326839638416971 \nu^{2} - 73023582675783362477543204436452132177915836802025570 \nu + 77006632551331802382072715087977635057390902132337836200$$$$)/$$$$16\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$206267896829254349511224388063445300199 \nu^{8} - 17291292699056025928059149170821621093803 \nu^{7} - 52240207679110604791748286552129880570658233 \nu^{6} + 3612837746963555301204000567795550672694760724 \nu^{5} + 3121813203000535098371355426778026920870559330113 \nu^{4} - 79542398652486879592363498787696402203982378745887 \nu^{3} - 31193007741511147854424399553870945417110675241829687 \nu^{2} - 58998475006516095505055677529803116478719268419862890 \nu + 38391130682115345448181474326295742567362258757557483400$$$$)/$$$$63\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$-33420522753055636948940132034854277909 \nu^{8} + 2564671763761851894228729142951429296133 \nu^{7} + 8534473203765996872973613103837858546008403 \nu^{6} - 531815687794417709774989855680608938127892944 \nu^{5} - 522183859029337187069718019940855438842372426543 \nu^{4} + 10719879155884022086517718381188220339174378456997 \nu^{3} + 5785830220549247645774640772969558173804580724359817 \nu^{2} + 3082733564840701057422315887334955185869278943556870 \nu - 9484897774275819470615195419733617188435922954531066200$$$$)/$$$$84\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$4 \beta_{8} + 9 \beta_{7} - 10 \beta_{6} + \beta_{5} + 2 \beta_{4} - 7 \beta_{3} - 16 \beta_{2} - 2 \beta_{1} + 39$$$$)/90$$ $$\nu^{2}$$ $$=$$ $$($$$$136 \beta_{8} + 441 \beta_{7} - 985 \beta_{6} - 281 \beta_{5} - 4627 \beta_{4} + 6002 \beta_{3} - 2959 \beta_{2} + 757 \beta_{1} + 5208246$$$$)/90$$ $$\nu^{3}$$ $$=$$ $$($$$$660296 \beta_{8} + 1069071 \beta_{7} - 1085315 \beta_{6} + 487904 \beta_{5} + 446743 \beta_{4} - 394313 \beta_{3} - 2159819 \beta_{2} - 259723 \beta_{1} + 109786731$$$$)/90$$ $$\nu^{4}$$ $$=$$ $$($$$$6903992 \beta_{8} - 2941023 \beta_{7} - 69045830 \beta_{6} - 99414577 \beta_{5} - 795610454 \beta_{4} + 789215059 \beta_{3} - 502769048 \beta_{2} + 28609274 \beta_{1} + 690305346777$$$$)/90$$ $$\nu^{5}$$ $$=$$ $$($$$$34485758882 \beta_{8} + 46859996827 \beta_{7} - 51074404415 \beta_{6} + 33706022693 \beta_{5} + 30061759051 \beta_{4} - 44405556036 \beta_{3} - 103765445963 \beta_{2} - 18670834021 \beta_{1} + 1361822594772$$$$)/30$$ $$\nu^{6}$$ $$=$$ $$($$$$-33034404038 \beta_{8} - 593000696095 \beta_{7} - 93645172897 \beta_{6} - 1365304900840 \beta_{5} - 8907045647747 \beta_{4} + 7093259705175 \beta_{3} - 5107933147123 \beta_{2} - 302146439705 \beta_{1} + 6980526052958887$$$$)/6$$ $$\nu^{7}$$ $$=$$ $$($$$$16707747648301018 \beta_{8} + 20283845213355273 \beta_{7} - 23115022462413850 \beta_{6} + 18291518699334007 \beta_{5} + 17308064143451654 \beta_{4} - 30756049592374279 \beta_{3} - 47277178614962902 \beta_{2} - 11211538232174354 \beta_{1} - 1886703060504560037$$$$)/90$$ $$\nu^{8}$$ $$=$$ $$($$$$-365131293680344988 \beta_{8} - 2485900825788308103 \beta_{7} + 864923096880567395 \beta_{6} - 3773926035695805917 \beta_{5} - 22451697906264216799 \beta_{4} + 15511088503777472174 \beta_{3} - 11591250765241236943 \beta_{2} - 1525867541454614591 \beta_{1} + 16654625647993278684522$$$$)/90$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 37.3239 −93.7830 115.905 401.344 −415.350 −237.937 315.149 −36.9079 −81.7436
−8.00000 −27.0000 64.0000 −551.934 216.000 −627.747 −512.000 729.000 4415.47
1.2 −8.00000 −27.0000 64.0000 −297.331 216.000 367.966 −512.000 729.000 2378.65
1.3 −8.00000 −27.0000 64.0000 −290.312 216.000 1712.32 −512.000 729.000 2322.49
1.4 −8.00000 −27.0000 64.0000 −140.027 216.000 −1605.75 −512.000 729.000 1120.22
1.5 −8.00000 −27.0000 64.0000 33.0058 216.000 651.589 −512.000 729.000 −264.046
1.6 −8.00000 −27.0000 64.0000 137.747 216.000 1182.79 −512.000 729.000 −1101.98
1.7 −8.00000 −27.0000 64.0000 253.154 216.000 −755.716 −512.000 729.000 −2025.23
1.8 −8.00000 −27.0000 64.0000 290.127 216.000 −1124.72 −512.000 729.000 −2321.02
1.9 −8.00000 −27.0000 64.0000 335.569 216.000 −140.718 −512.000 729.000 −2684.55
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$59$$ $$-1$$

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{5}^{9} + \cdots$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(354))$$.