# Properties

 Label 405.4.e.w Level $405$ Weight $4$ Character orbit 405.e Analytic conductor $23.896$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 405.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.8957735523$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} + 2 x^{10} + 32 x^{9} + 583 x^{8} - 624 x^{7} + 594 x^{6} + 9450 x^{5} + 90513 x^{4} - 20304 x^{3} + 10368 x^{2} + 124416 x + 746496$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{9}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{2} - \beta_{3} ) q^{2} + ( -6 + 6 \beta_{2} + \beta_{5} + \beta_{9} ) q^{4} + ( -5 + 5 \beta_{2} ) q^{5} + ( \beta_{1} - 7 \beta_{2} - 2 \beta_{3} + \beta_{10} ) q^{7} + ( 12 + 4 \beta_{3} - 2 \beta_{6} + \beta_{8} - 4 \beta_{9} - \beta_{10} ) q^{8} +O(q^{10})$$ $$q + ( -\beta_{2} - \beta_{3} ) q^{2} + ( -6 + 6 \beta_{2} + \beta_{5} + \beta_{9} ) q^{4} + ( -5 + 5 \beta_{2} ) q^{5} + ( \beta_{1} - 7 \beta_{2} - 2 \beta_{3} + \beta_{10} ) q^{7} + ( 12 + 4 \beta_{3} - 2 \beta_{6} + \beta_{8} - 4 \beta_{9} - \beta_{10} ) q^{8} + ( 5 + 5 \beta_{3} - 5 \beta_{9} ) q^{10} + ( 2 \beta_{1} - 14 \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{11} ) q^{11} + ( -4 + 4 \beta_{2} - \beta_{4} - \beta_{5} - 3 \beta_{7} + 2 \beta_{8} + \beta_{11} ) q^{13} + ( -32 + 32 \beta_{2} + 3 \beta_{4} + 4 \beta_{5} + \beta_{7} - \beta_{8} + 7 \beta_{9} - 3 \beta_{11} ) q^{14} + ( -14 \beta_{2} - 16 \beta_{3} + 3 \beta_{10} + 2 \beta_{11} ) q^{16} + ( 19 + 2 \beta_{1} - 6 \beta_{3} + 3 \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{8} + 6 \beta_{9} + \beta_{10} ) q^{17} + ( -7 - 4 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} ) q^{19} + ( -30 \beta_{2} - 5 \beta_{3} - 5 \beta_{5} + 5 \beta_{6} ) q^{20} + ( 5 - 5 \beta_{2} + 2 \beta_{4} - \beta_{5} - 10 \beta_{7} + 2 \beta_{8} + 20 \beta_{9} - 2 \beta_{11} ) q^{22} + ( -34 + 34 \beta_{2} + 6 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + 3 \beta_{8} - 6 \beta_{9} - 6 \beta_{11} ) q^{23} -25 \beta_{2} q^{25} + ( 2 + 15 \beta_{1} - 2 \beta_{3} + 7 \beta_{4} + 5 \beta_{6} + 15 \beta_{7} - 6 \beta_{8} + 2 \beta_{9} + 6 \beta_{10} ) q^{26} + ( 70 - 29 \beta_{1} + 40 \beta_{3} - 3 \beta_{4} - 13 \beta_{6} - 29 \beta_{7} + 6 \beta_{8} - 40 \beta_{9} - 6 \beta_{10} ) q^{28} + ( -8 \beta_{1} - 52 \beta_{2} - 12 \beta_{3} - \beta_{5} + \beta_{6} + 4 \beta_{10} - 3 \beta_{11} ) q^{29} + ( 13 - 13 \beta_{2} - 4 \beta_{4} - 10 \beta_{5} - 6 \beta_{7} - 3 \beta_{8} + 16 \beta_{9} + 4 \beta_{11} ) q^{31} + ( -114 + 114 \beta_{2} + 6 \beta_{4} + 6 \beta_{5} + 24 \beta_{7} - \beta_{8} - 18 \beta_{9} - 6 \beta_{11} ) q^{32} + ( -38 \beta_{1} + 60 \beta_{2} - 25 \beta_{3} + 7 \beta_{5} - 7 \beta_{6} - 9 \beta_{10} - 4 \beta_{11} ) q^{34} + ( 35 - 5 \beta_{1} + 10 \beta_{3} - 5 \beta_{7} + 5 \beta_{8} - 10 \beta_{9} - 5 \beta_{10} ) q^{35} + ( -29 - 28 \beta_{1} + 10 \beta_{3} - 3 \beta_{4} + 5 \beta_{6} - 28 \beta_{7} - 5 \beta_{8} - 10 \beta_{9} + 5 \beta_{10} ) q^{37} + ( -20 \beta_{1} + 9 \beta_{2} + 31 \beta_{3} + 14 \beta_{5} - 14 \beta_{6} - 14 \beta_{10} - 4 \beta_{11} ) q^{38} + ( -60 + 60 \beta_{2} + 10 \beta_{5} - 5 \beta_{8} + 20 \beta_{9} ) q^{40} + ( -62 + 62 \beta_{2} + 10 \beta_{4} - 26 \beta_{5} + 7 \beta_{7} - 2 \beta_{8} + 16 \beta_{9} - 10 \beta_{11} ) q^{41} + ( -38 \beta_{1} - 99 \beta_{2} - 42 \beta_{3} - 25 \beta_{5} + 25 \beta_{6} + \beta_{10} + \beta_{11} ) q^{43} + ( 120 + 2 \beta_{1} + 5 \beta_{3} + 6 \beta_{4} - 7 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 5 \beta_{9} - 3 \beta_{10} ) q^{44} + ( -28 - 74 \beta_{1} + 22 \beta_{3} + 4 \beta_{4} + 14 \beta_{6} - 74 \beta_{7} + 13 \beta_{8} - 22 \beta_{9} - 13 \beta_{10} ) q^{46} + ( -14 \beta_{1} - 23 \beta_{2} + 44 \beta_{3} + 29 \beta_{5} - 29 \beta_{6} + 6 \beta_{10} - 9 \beta_{11} ) q^{47} + ( -15 + 15 \beta_{2} + 12 \beta_{4} + 16 \beta_{5} + 12 \beta_{7} - 15 \beta_{8} + 26 \beta_{9} - 12 \beta_{11} ) q^{49} + ( -25 + 25 \beta_{2} + 25 \beta_{9} ) q^{50} + ( -75 \beta_{1} + 82 \beta_{2} + 28 \beta_{3} + 11 \beta_{5} - 11 \beta_{6} - 16 \beta_{10} - 19 \beta_{11} ) q^{52} + ( 127 - 58 \beta_{1} - 20 \beta_{3} - 5 \beta_{4} + 27 \beta_{6} - 58 \beta_{7} + 24 \beta_{8} + 20 \beta_{9} - 24 \beta_{10} ) q^{53} + ( 70 - 10 \beta_{1} - 10 \beta_{3} + 5 \beta_{4} - 5 \beta_{6} - 10 \beta_{7} + 10 \beta_{9} ) q^{55} + ( 57 \beta_{1} - 410 \beta_{2} - 92 \beta_{3} - 33 \beta_{5} + 33 \beta_{6} + 20 \beta_{10} + 17 \beta_{11} ) q^{56} + ( -169 + 169 \beta_{2} + 21 \beta_{5} - 44 \beta_{7} + 4 \beta_{8} + 58 \beta_{9} ) q^{58} + ( -294 + 294 \beta_{2} + 6 \beta_{5} - 15 \beta_{7} - 8 \beta_{8} - 30 \beta_{9} ) q^{59} + ( -30 \beta_{1} + 84 \beta_{2} + 72 \beta_{3} + 38 \beta_{5} - 38 \beta_{6} - 2 \beta_{10} + 22 \beta_{11} ) q^{61} + ( 155 + 54 \beta_{1} - 73 \beta_{3} - 12 \beta_{6} + 54 \beta_{7} - 19 \beta_{8} + 73 \beta_{9} + 19 \beta_{10} ) q^{62} + ( -158 - 96 \beta_{1} + 22 \beta_{3} - 10 \beta_{4} + 10 \beta_{6} - 96 \beta_{7} + \beta_{8} - 22 \beta_{9} - \beta_{10} ) q^{64} + ( -15 \beta_{1} - 20 \beta_{2} + 5 \beta_{5} - 5 \beta_{6} - 10 \beta_{10} - 5 \beta_{11} ) q^{65} + ( 36 - 36 \beta_{2} + 10 \beta_{4} - 30 \beta_{5} + 8 \beta_{7} + 12 \beta_{8} - 164 \beta_{9} - 10 \beta_{11} ) q^{67} + ( -28 + 28 \beta_{2} - 32 \beta_{4} - 8 \beta_{5} - 70 \beta_{7} + 20 \beta_{8} - 54 \beta_{9} + 32 \beta_{11} ) q^{68} + ( 5 \beta_{1} - 160 \beta_{2} - 35 \beta_{3} - 20 \beta_{5} + 20 \beta_{6} + 5 \beta_{10} + 15 \beta_{11} ) q^{70} + ( 268 + 52 \beta_{1} + 82 \beta_{3} - 9 \beta_{4} + 9 \beta_{6} + 52 \beta_{7} - 40 \beta_{8} - 82 \beta_{9} + 40 \beta_{10} ) q^{71} + ( 101 - 50 \beta_{1} - 138 \beta_{3} + 7 \beta_{4} - 33 \beta_{6} - 50 \beta_{7} - 23 \beta_{8} + 138 \beta_{9} + 23 \beta_{10} ) q^{73} + ( 64 \beta_{1} - 200 \beta_{2} + 59 \beta_{3} + 5 \beta_{5} - 5 \beta_{6} - \beta_{10} + 18 \beta_{11} ) q^{74} + ( 374 - 374 \beta_{2} - 32 \beta_{4} - 41 \beta_{5} - 100 \beta_{7} + 8 \beta_{8} - 77 \beta_{9} + 32 \beta_{11} ) q^{76} + ( -123 + 123 \beta_{2} - 9 \beta_{4} + 19 \beta_{5} + 40 \beta_{7} - 5 \beta_{8} - 46 \beta_{9} + 9 \beta_{11} ) q^{77} + ( -10 \beta_{1} + 172 \beta_{2} + 144 \beta_{3} - 12 \beta_{5} + 12 \beta_{6} + 32 \beta_{10} - 12 \beta_{11} ) q^{79} + ( 70 + 80 \beta_{3} - 10 \beta_{4} + 15 \beta_{8} - 80 \beta_{9} - 15 \beta_{10} ) q^{80} + ( 257 - 127 \beta_{1} - 94 \beta_{3} - 11 \beta_{4} + 6 \beta_{6} - 127 \beta_{7} + 6 \beta_{8} + 94 \beta_{9} - 6 \beta_{10} ) q^{82} + ( -34 \beta_{1} - 369 \beta_{2} - 6 \beta_{3} - 45 \beta_{5} + 45 \beta_{6} - 27 \beta_{10} - 19 \beta_{11} ) q^{83} + ( -95 + 95 \beta_{2} - 15 \beta_{4} + 5 \beta_{5} - 10 \beta_{7} + 5 \beta_{8} - 30 \beta_{9} + 15 \beta_{11} ) q^{85} + ( -552 + 552 \beta_{2} - 36 \beta_{4} + 69 \beta_{5} - 26 \beta_{7} - 29 \beta_{8} + 249 \beta_{9} + 36 \beta_{11} ) q^{86} + ( 6 \beta_{1} - 214 \beta_{2} - 2 \beta_{3} - 26 \beta_{5} + 26 \beta_{6} + 8 \beta_{10} - 12 \beta_{11} ) q^{88} + ( 474 - 76 \beta_{1} - 96 \beta_{3} - 21 \beta_{4} - 33 \beta_{6} - 76 \beta_{7} - 18 \beta_{8} + 96 \beta_{9} + 18 \beta_{10} ) q^{89} + ( -538 + 44 \beta_{1} - 106 \beta_{3} + 18 \beta_{4} + 34 \beta_{6} + 44 \beta_{7} - 21 \beta_{8} + 106 \beta_{9} + 21 \beta_{10} ) q^{91} + ( 10 \beta_{1} - 142 \beta_{2} + 64 \beta_{3} - 50 \beta_{5} + 50 \beta_{6} + 11 \beta_{10} + 52 \beta_{11} ) q^{92} + ( 644 - 644 \beta_{2} - 2 \beta_{4} - 61 \beta_{5} - 122 \beta_{7} + 50 \beta_{8} - 151 \beta_{9} + 2 \beta_{11} ) q^{94} + ( 35 - 35 \beta_{2} - 10 \beta_{4} - 20 \beta_{5} + 20 \beta_{7} + 20 \beta_{8} - 10 \beta_{9} + 10 \beta_{11} ) q^{95} + ( -72 \beta_{1} + 125 \beta_{2} + 30 \beta_{3} + 13 \beta_{5} - 13 \beta_{6} + 39 \beta_{10} - 41 \beta_{11} ) q^{97} + ( 345 - 156 \beta_{1} + 111 \beta_{3} - 42 \beta_{4} - 72 \beta_{6} - 156 \beta_{7} + 67 \beta_{8} - 111 \beta_{9} - 67 \beta_{10} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 4 q^{2} - 34 q^{4} - 30 q^{5} - 40 q^{7} + 132 q^{8} + O(q^{10})$$ $$12 q - 4 q^{2} - 34 q^{4} - 30 q^{5} - 40 q^{7} + 132 q^{8} + 40 q^{10} - 88 q^{11} - 20 q^{13} - 180 q^{14} - 58 q^{16} + 248 q^{17} - 92 q^{19} - 170 q^{20} + 74 q^{22} - 210 q^{23} - 150 q^{25} + 8 q^{26} + 704 q^{28} - 296 q^{29} + 104 q^{31} - 722 q^{32} + 428 q^{34} + 400 q^{35} - 408 q^{37} + 20 q^{38} - 330 q^{40} - 344 q^{41} - 512 q^{43} + 1432 q^{44} - 372 q^{46} - 238 q^{47} - 68 q^{49} - 100 q^{50} + 468 q^{52} + 1700 q^{53} + 880 q^{55} - 2316 q^{56} - 890 q^{58} - 1840 q^{59} + 364 q^{61} + 2076 q^{62} - 1980 q^{64} - 100 q^{65} - 88 q^{67} - 236 q^{68} - 900 q^{70} + 2728 q^{71} + 1672 q^{73} - 1316 q^{74} + 2106 q^{76} - 840 q^{77} + 680 q^{79} + 580 q^{80} + 3484 q^{82} - 2148 q^{83} - 620 q^{85} - 2872 q^{86} - 1296 q^{88} + 6000 q^{89} - 6116 q^{91} - 1002 q^{92} + 3662 q^{94} + 230 q^{95} + 612 q^{97} + 3964 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 2 x^{10} + 32 x^{9} + 583 x^{8} - 624 x^{7} + 594 x^{6} + 9450 x^{5} + 90513 x^{4} - 20304 x^{3} + 10368 x^{2} + 124416 x + 746496$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$39168669767185 \nu^{11} + 622492575406102 \nu^{10} - 116378803247278 \nu^{9} - 4437769721904304 \nu^{8} + 76487510388819703 \nu^{7} + 415285293653958168 \nu^{6} - 310931437403832750 \nu^{5} - 443929146343266438 \nu^{4} + 17810423563382896401 \nu^{3} + 83726076442829728680 \nu^{2} - 66823261574041484640 \nu + 5737362693292717056$$$$)/ 53110961918205557760$$ $$\beta_{2}$$ $$=$$ $$($$$$-21833561 \nu^{11} - 6933902 \nu^{10} + 191663726 \nu^{9} - 824768992 \nu^{8} - 19024412783 \nu^{7} + 8759970096 \nu^{6} + 9602073054 \nu^{5} - 242401187034 \nu^{4} - 3561497041833 \nu^{3} + 692957538000 \nu^{2} - 10161675967968 \nu + 11197076020992$$$$)/ 28100111293440$$ $$\beta_{3}$$ $$=$$ $$($$$$8315698756077 \nu^{11} + 101245330415030 \nu^{10} - 394560515037286 \nu^{9} + 1057574546275072 \nu^{8} + 7683197260871035 \nu^{7} + 48297387784200512 \nu^{6} - 116794756095706998 \nu^{5} + 265856737439346114 \nu^{4} + 1425180458645681949 \nu^{3} + 5529771972303382080 \nu^{2} - 4139103064502083104 \nu + 1560323092464854784$$$$)/ 4425913493183796480$$ $$\beta_{4}$$ $$=$$ $$($$$$4200645557245 \nu^{11} + 6160897283854 \nu^{10} - 314135929010806 \nu^{9} + 3132646358261552 \nu^{8} - 3889069452974549 \nu^{7} + 1955376027463176 \nu^{6} - 57814071395165910 \nu^{5} + 1126387696678531794 \nu^{4} - 1077777736999692003 \nu^{3} + 79352387692368120 \nu^{2} + 2581814601408702240 \nu + 62107963971000458112$$$$)/ 2212956746591898240$$ $$\beta_{5}$$ $$=$$ $$($$$$-335658953887093 \nu^{11} + 1171025533283690 \nu^{10} - 8666619525148106 \nu^{9} + 50196423604163872 \nu^{8} - 313704453000911155 \nu^{7} + 295373042049020592 \nu^{6} - 1436493017218079898 \nu^{5} + 11741310942738453774 \nu^{4} - 64803099267637148421 \nu^{3} + 5271451684560213840 \nu^{2} - 192953421149110468704 \nu - 174266393008948736256$$$$)/$$$$15\!\cdots\!80$$ $$\beta_{6}$$ $$=$$ $$($$$$11005996995395 \nu^{11} - 8278688448046 \nu^{10} - 261584063656106 \nu^{9} + 2809757609441872 \nu^{8} + 952726608368501 \nu^{7} - 2440351842639624 \nu^{6} - 41491125321905610 \nu^{5} + 700433575797000654 \nu^{4} - 39409425332857053 \nu^{3} - 35284815360850680 \nu^{2} - 206965861006966560 \nu - 4842346192147027008$$$$)/ 3319435119887847360$$ $$\beta_{7}$$ $$=$$ $$($$$$238036479862585 \nu^{11} - 1222449555829658 \nu^{10} + 2942151887057762 \nu^{9} - 677665458581104 \nu^{8} + 148598736796672783 \nu^{7} - 597691959138682632 \nu^{6} + 614420351455033890 \nu^{5} + 662850739681255242 \nu^{4} + 22923435786759202041 \nu^{3} - 88414990146924296760 \nu^{2} - 44666427965491905120 \nu + 17268670504516340736$$$$)/ 53110961918205557760$$ $$\beta_{8}$$ $$=$$ $$($$$$-1081657721587607 \nu^{11} - 357550815547730 \nu^{10} + 17604901276198706 \nu^{9} - 64482188744355232 \nu^{8} - 904614383398756385 \nu^{7} + 2237153490596534928 \nu^{6} + 3109902766563363138 \nu^{5} - 17333693087164935174 \nu^{4} - 151757889547035749319 \nu^{3} + 567820946400626294640 \nu^{2} + 330054951944179314144 \nu - 225525340579288345344$$$$)/$$$$15\!\cdots\!80$$ $$\beta_{9}$$ $$=$$ $$($$$$-96943826623309 \nu^{11} + 480126841023914 \nu^{10} - 441998532012314 \nu^{9} - 4147237826236352 \nu^{8} - 37432400654764699 \nu^{7} + 198005263782509472 \nu^{6} - 64410209363731914 \nu^{5} - 1084777218670144194 \nu^{4} - 3488004203931395901 \nu^{3} + 17774459217805263840 \nu^{2} + 11311092917146506528 \nu - 7383692416832315136$$$$)/ 13277740479551389440$$ $$\beta_{10}$$ $$=$$ $$($$$$1828239864155833 \nu^{11} - 5135036083007858 \nu^{10} + 11308370778733778 \nu^{9} + 67186055714148704 \nu^{8} + 710053265142888463 \nu^{7} + 792571975775440656 \nu^{6} + 450734978859702498 \nu^{5} + 18141585956481889818 \nu^{4} + 109507911984962575497 \nu^{3} + 533504254636404411120 \nu^{2} - 419473500997898113056 \nu + 227037510271026600192$$$$)/$$$$15\!\cdots\!80$$ $$\beta_{11}$$ $$=$$ $$($$$$-2442175499886815 \nu^{11} + 7080096183948958 \nu^{10} - 25311276149957662 \nu^{9} - 22787594734230496 \nu^{8} - 1298135621615504153 \nu^{7} + 1752882046900662192 \nu^{6} - 8532089400267944910 \nu^{5} - 4244172237278355222 \nu^{4} - 174847215049808235471 \nu^{3} - 4245140807314092720 \nu^{2} - 497882512957950381600 \nu + 549583910441667857664$$$$)/ 53110961918205557760$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - 2 \beta_{5} + \beta_{3} - 2 \beta_{2} + 2$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{10} - 5 \beta_{9} - \beta_{8} - 8 \beta_{7} + \beta_{6} - 2 \beta_{5} - 5 \beta_{3} + 8 \beta_{1}$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{11} + 8 \beta_{10} + 12 \beta_{9} - 9 \beta_{8} - 8 \beta_{7} + 7 \beta_{6} - 17 \beta_{5} - 14 \beta_{3} + 17 \beta_{2} - 4 \beta_{1} - 35$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$35 \beta_{10} + 35 \beta_{9} - 35 \beta_{8} - 24 \beta_{7} - 81 \beta_{6} + 40 \beta_{4} - 35 \beta_{3} - 24 \beta_{1} - 1298$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$-26 \beta_{11} + 365 \beta_{10} + 501 \beta_{9} - 279 \beta_{8} - 184 \beta_{7} - 451 \beta_{6} + 644 \beta_{5} + 48 \beta_{4} - 377 \beta_{3} - 1308 \beta_{2} - 464 \beta_{1} - 904$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$($$$$16 \beta_{11} + 325 \beta_{10} + 758 \beta_{9} + 325 \beta_{8} + 1288 \beta_{7} - 488 \beta_{6} + 976 \beta_{5} - 8 \beta_{4} + 758 \beta_{3} - 2818 \beta_{2} - 1288 \beta_{1} + 1409$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-144 \beta_{11} - 5057 \beta_{10} - 5627 \beta_{9} + 7763 \beta_{8} + 11704 \beta_{7} - 2351 \beta_{6} + 12820 \beta_{5} - 856 \beta_{4} + 8489 \beta_{3} - 36254 \beta_{2} + 3896 \beta_{1} + 58398$$$$)/6$$ $$\nu^{8}$$ $$=$$ $$($$$$-24965 \beta_{10} - 17315 \beta_{9} + 24965 \beta_{8} + 32472 \beta_{7} + 46041 \beta_{6} - 13252 \beta_{4} + 17315 \beta_{3} + 32472 \beta_{1} + 480200$$$$)/6$$ $$\nu^{9}$$ $$=$$ $$($$$$-2095 \beta_{11} - 84984 \beta_{10} - 71512 \beta_{9} + 47317 \beta_{8} + 42152 \beta_{7} + 122651 \beta_{6} - 132301 \beta_{5} - 10160 \beta_{4} + 41934 \beta_{3} + 464143 \beta_{2} + 142012 \beta_{1} + 260571$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$-61008 \beta_{11} - 360417 \beta_{10} - 395679 \beta_{9} - 360417 \beta_{8} - 1058408 \beta_{7} + 612127 \beta_{6} - 1224254 \beta_{5} + 30504 \beta_{4} - 395679 \beta_{3} + 5103652 \beta_{2} + 1058408 \beta_{1} - 2551826$$$$)/6$$ $$\nu^{11}$$ $$=$$ $$($$$$-216770 \beta_{11} + 1828317 \beta_{10} + 1259071 \beta_{9} - 3796381 \beta_{8} - 6773512 \beta_{7} - 139747 \beta_{6} - 5624698 \beta_{5} + 629000 \beta_{4} - 2414697 \beta_{3} + 22943700 \beta_{2} - 1876496 \beta_{1} - 34930520$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$$.

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
136.1
 2.93142 + 2.93142i −1.16241 + 1.16241i −2.82176 − 2.82176i 3.41462 − 3.41462i 1.25636 + 1.25636i −2.61824 + 2.61824i 2.93142 − 2.93142i −1.16241 − 1.16241i −2.82176 + 2.82176i 3.41462 + 3.41462i 1.25636 − 1.25636i −2.61824 − 2.61824i
−2.61668 + 4.53223i 0 −9.69405 16.7906i −2.50000 4.33013i 0 −16.5090 + 28.5945i 59.5981 0 26.1668
136.2 −2.26722 + 3.92694i 0 −6.28058 10.8783i −2.50000 4.33013i 0 1.31809 2.28300i 20.6823 0 22.6722
136.3 −1.03663 + 1.79550i 0 1.85079 + 3.20567i −2.50000 4.33013i 0 2.33056 4.03665i −24.2604 0 10.3663
136.4 0.0745751 0.129168i 0 3.98888 + 6.90894i −2.50000 4.33013i 0 −10.0712 + 17.4439i 2.38308 0 −0.745751
136.5 1.78729 3.09567i 0 −2.38879 4.13751i −2.50000 4.33013i 0 −7.07987 + 12.2627i 11.5188 0 −17.8729
136.6 2.05867 3.56572i 0 −4.47625 7.75309i −2.50000 4.33013i 0 10.0115 17.3404i −3.92177 0 −20.5867
271.1 −2.61668 4.53223i 0 −9.69405 + 16.7906i −2.50000 + 4.33013i 0 −16.5090 28.5945i 59.5981 0 26.1668
271.2 −2.26722 3.92694i 0 −6.28058 + 10.8783i −2.50000 + 4.33013i 0 1.31809 + 2.28300i 20.6823 0 22.6722
271.3 −1.03663 1.79550i 0 1.85079 3.20567i −2.50000 + 4.33013i 0 2.33056 + 4.03665i −24.2604 0 10.3663
271.4 0.0745751 + 0.129168i 0 3.98888 6.90894i −2.50000 + 4.33013i 0 −10.0712 17.4439i 2.38308 0 −0.745751
271.5 1.78729 + 3.09567i 0 −2.38879 + 4.13751i −2.50000 + 4.33013i 0 −7.07987 12.2627i 11.5188 0 −17.8729
271.6 2.05867 + 3.56572i 0 −4.47625 + 7.75309i −2.50000 + 4.33013i 0 10.0115 + 17.3404i −3.92177 0 −20.5867
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 271.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.4.e.w 12
3.b odd 2 1 405.4.e.x 12
9.c even 3 1 405.4.a.l yes 6
9.c even 3 1 inner 405.4.e.w 12
9.d odd 6 1 405.4.a.k 6
9.d odd 6 1 405.4.e.x 12
45.h odd 6 1 2025.4.a.z 6
45.j even 6 1 2025.4.a.y 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.k 6 9.d odd 6 1
405.4.a.l yes 6 9.c even 3 1
405.4.e.w 12 1.a even 1 1 trivial
405.4.e.w 12 9.c even 3 1 inner
405.4.e.x 12 3.b odd 2 1
405.4.e.x 12 9.d odd 6 1
2025.4.a.y 6 45.j even 6 1
2025.4.a.z 6 45.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{12} + \cdots$$ $$11\!\cdots\!96$$$$T_{7}^{2} -$$$$26\!\cdots\!44$$$$T_{7} +$$$$53\!\cdots\!76$$">$$T_{7}^{12} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$11664 - 73872 T + 498744 T^{2} + 171864 T^{3} + 153472 T^{4} + 13252 T^{5} + 18586 T^{6} + 2026 T^{7} + 1243 T^{8} + 88 T^{9} + 49 T^{10} + 4 T^{11} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$( 25 + 5 T + T^{2} )^{6}$$
$7$ $$5368136821776 - 2647910494944 T + 1191988160496 T^{2} - 143876813760 T^{3} + 23417174988 T^{4} + 1439479296 T^{5} + 293906532 T^{6} + 10054356 T^{7} + 874429 T^{8} + 27280 T^{9} + 1863 T^{10} + 40 T^{11} + T^{12}$$
$11$ $$1433067563664 - 511059770496 T + 383079496716 T^{2} + 82440186528 T^{3} + 24396229897 T^{4} + 2159721016 T^{5} + 315051034 T^{6} + 22237312 T^{7} + 2869843 T^{8} + 142624 T^{9} + 6226 T^{10} + 88 T^{11} + T^{12}$$
$13$ $$34519309364137984 - 2104552442798080 T + 510205862948864 T^{2} - 11449770135552 T^{3} + 4444220055232 T^{4} - 93469769216 T^{5} + 17427630400 T^{6} + 351507808 T^{7} + 20235313 T^{8} + 96564 T^{9} + 4919 T^{10} + 20 T^{11} + T^{12}$$
$17$ $$( 105966288 - 34954656 T - 3790268 T^{2} + 762824 T^{3} - 5004 T^{4} - 124 T^{5} + T^{6} )^{2}$$
$19$ $$( -36821611175 + 1527158350 T + 46339391 T^{2} - 872156 T^{3} - 18169 T^{4} + 46 T^{5} + T^{6} )^{2}$$
$23$ $$11\!\cdots\!24$$$$+$$$$99\!\cdots\!20$$$$T +$$$$84\!\cdots\!04$$$$T^{2} + 7572918651097847040 T^{3} + 185560597052855856 T^{4} + 893549898701280 T^{5} + 26052899818824 T^{6} + 109962941040 T^{7} + 1644156369 T^{8} + 5203530 T^{9} + 69879 T^{10} + 210 T^{11} + T^{12}$$
$29$ $$40\!\cdots\!44$$$$+$$$$57\!\cdots\!12$$$$T +$$$$21\!\cdots\!80$$$$T^{2} + 2288402682504676704 T^{3} + 65553875537486305 T^{4} + 670359152223368 T^{5} + 11502115930846 T^{6} + 96355391264 T^{7} + 1288780675 T^{8} + 9697088 T^{9} + 76990 T^{10} + 296 T^{11} + T^{12}$$
$31$ $$48\!\cdots\!56$$$$- 87408210595735306656 T + 37161778657121314200 T^{2} + 830178716262723828 T^{3} + 253833420007126305 T^{4} + 855647498603340 T^{5} + 31584379870242 T^{6} + 26886333900 T^{7} + 3029716915 T^{8} + 3302872 T^{9} + 69090 T^{10} - 104 T^{11} + T^{12}$$
$37$ $$( 12008297128192 + 211123326144 T - 305431068 T^{2} - 20257048 T^{3} - 74400 T^{4} + 204 T^{5} + T^{6} )^{2}$$
$41$ $$74\!\cdots\!09$$$$-$$$$10\!\cdots\!52$$$$T +$$$$23\!\cdots\!69$$$$T^{2} +$$$$43\!\cdots\!64$$$$T^{3} +$$$$14\!\cdots\!62$$$$T^{4} - 71785826905652008 T^{5} + 3638232970571101 T^{6} + 3061460784536 T^{7} + 40567964278 T^{8} + 32702408 T^{9} + 303529 T^{10} + 344 T^{11} + T^{12}$$
$43$ $$32\!\cdots\!00$$$$-$$$$63\!\cdots\!00$$$$T +$$$$13\!\cdots\!00$$$$T^{2} -$$$$34\!\cdots\!80$$$$T^{3} +$$$$35\!\cdots\!04$$$$T^{4} + 389605574404146208 T^{5} + 7260140010772000 T^{6} + 10580589884944 T^{7} + 67412716924 T^{8} + 90657648 T^{9} + 434696 T^{10} + 512 T^{11} + T^{12}$$
$47$ $$24\!\cdots\!00$$$$-$$$$75\!\cdots\!00$$$$T +$$$$34\!\cdots\!00$$$$T^{2} +$$$$27\!\cdots\!20$$$$T^{3} +$$$$65\!\cdots\!44$$$$T^{4} - 737568768619252472 T^{5} + 17059335428457412 T^{6} + 27410378189476 T^{7} + 180287525485 T^{8} + 71280238 T^{9} + 484567 T^{10} + 238 T^{11} + T^{12}$$
$53$ $$( 15741889277692500 - 87258267127200 T - 7800157964 T^{2} + 601620140 T^{3} - 550155 T^{4} - 850 T^{5} + T^{6} )^{2}$$
$59$ $$28\!\cdots\!89$$$$-$$$$42\!\cdots\!00$$$$T +$$$$17\!\cdots\!13$$$$T^{2} +$$$$30\!\cdots\!80$$$$T^{3} +$$$$32\!\cdots\!22$$$$T^{4} + 20963259757696045360 T^{5} + 98093299140313117 T^{6} + 327172692495280 T^{7} + 829401317398 T^{8} + 1527016720 T^{9} + 2086153 T^{10} + 1840 T^{11} + T^{12}$$
$61$ $$41\!\cdots\!96$$$$+$$$$14\!\cdots\!44$$$$T +$$$$10\!\cdots\!44$$$$T^{2} -$$$$79\!\cdots\!96$$$$T^{3} +$$$$90\!\cdots\!08$$$$T^{4} - 39374994072408743936 T^{5} + 338087064255877120 T^{6} - 175701867321344 T^{7} + 845842550848 T^{8} - 268012704 T^{9} + 1146584 T^{10} - 364 T^{11} + T^{12}$$
$67$ $$38\!\cdots\!00$$$$+$$$$33\!\cdots\!00$$$$T +$$$$22\!\cdots\!00$$$$T^{2} +$$$$77\!\cdots\!80$$$$T^{3} +$$$$22\!\cdots\!24$$$$T^{4} +$$$$34\!\cdots\!52$$$$T^{5} + 610352230312202880 T^{6} + 484487372587776 T^{7} + 1368463483504 T^{8} + 463002112 T^{9} + 1321416 T^{10} + 88 T^{11} + T^{12}$$
$71$ $$( -11833359413893884 - 215551314974244 T - 225597721799 T^{2} + 1811192200 T^{3} - 995994 T^{4} - 1364 T^{5} + T^{6} )^{2}$$
$73$ $$( -10768933998500336 - 63694541190656 T + 254161845668 T^{2} + 781045192 T^{3} - 1130908 T^{4} - 836 T^{5} + T^{6} )^{2}$$
$79$ $$25\!\cdots\!84$$$$+$$$$39\!\cdots\!40$$$$T +$$$$67\!\cdots\!56$$$$T^{2} -$$$$67\!\cdots\!24$$$$T^{3} +$$$$41\!\cdots\!72$$$$T^{4} -$$$$34\!\cdots\!68$$$$T^{5} + 1662445816879656000 T^{6} - 1458845818702464 T^{7} + 2977345384288 T^{8} - 1073626208 T^{9} + 2090196 T^{10} - 680 T^{11} + T^{12}$$
$83$ $$17\!\cdots\!24$$$$-$$$$12\!\cdots\!96$$$$T +$$$$28\!\cdots\!12$$$$T^{2} +$$$$21\!\cdots\!64$$$$T^{3} +$$$$18\!\cdots\!68$$$$T^{4} +$$$$44\!\cdots\!60$$$$T^{5} + 1098572062595648400 T^{6} + 1409651438515200 T^{7} + 2716237295532 T^{8} + 3104000352 T^{9} + 3958812 T^{10} + 2148 T^{11} + T^{12}$$
$89$ $$( 167521946305912848 - 24233669446680 T - 1254009337479 T^{2} + 1004808240 T^{3} + 2101338 T^{4} - 3000 T^{5} + T^{6} )^{2}$$
$97$ $$96\!\cdots\!76$$$$+$$$$95\!\cdots\!48$$$$T +$$$$40\!\cdots\!16$$$$T^{2} -$$$$75\!\cdots\!48$$$$T^{3} +$$$$85\!\cdots\!28$$$$T^{4} -$$$$66\!\cdots\!92$$$$T^{5} + 9909273498792694080 T^{6} - 1976707801805472 T^{7} + 7798605716988 T^{8} - 1152080528 T^{9} + 3542256 T^{10} - 612 T^{11} + T^{12}$$