# 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: $$x^{12} - 2 x^{11} + 2 x^{10} + 32 x^{9} + 583 x^{8} - 624 x^{7} + 594 x^{6} + 9450 x^{5} + 90513 x^{4} - 20304 x^{3} + 10368 x^{2} + 124416 x + 746496$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - \beta_{2}) q^{2} + (\beta_{9} + \beta_{5} + 6 \beta_{2} - 6) q^{4} + (5 \beta_{2} - 5) q^{5} + (\beta_{10} - 2 \beta_{3} - 7 \beta_{2} + \beta_1) q^{7} + ( - \beta_{10} - 4 \beta_{9} + \beta_{8} - 2 \beta_{6} + 4 \beta_{3} + 12) q^{8}+O(q^{10})$$ q + (-b3 - b2) * q^2 + (b9 + b5 + 6*b2 - 6) * q^4 + (5*b2 - 5) * q^5 + (b10 - 2*b3 - 7*b2 + b1) * q^7 + (-b10 - 4*b9 + b8 - 2*b6 + 4*b3 + 12) * q^8 $$q + ( - \beta_{3} - \beta_{2}) q^{2} + (\beta_{9} + \beta_{5} + 6 \beta_{2} - 6) q^{4} + (5 \beta_{2} - 5) q^{5} + (\beta_{10} - 2 \beta_{3} - 7 \beta_{2} + \beta_1) q^{7} + ( - \beta_{10} - 4 \beta_{9} + \beta_{8} - 2 \beta_{6} + 4 \beta_{3} + 12) q^{8} + ( - 5 \beta_{9} + 5 \beta_{3} + 5) q^{10} + ( - \beta_{11} + \beta_{6} - \beta_{5} + 2 \beta_{3} - 14 \beta_{2} + 2 \beta_1) q^{11} + (\beta_{11} + 2 \beta_{8} - 3 \beta_{7} - \beta_{5} - \beta_{4} + 4 \beta_{2} - 4) q^{13} + ( - 3 \beta_{11} + 7 \beta_{9} - \beta_{8} + \beta_{7} + 4 \beta_{5} + 3 \beta_{4} + \cdots - 32) q^{14}+ \cdots + ( - 67 \beta_{10} - 111 \beta_{9} + 67 \beta_{8} - 156 \beta_{7} + \cdots + 345) q^{98}+O(q^{100})$$ q + (-b3 - b2) * q^2 + (b9 + b5 + 6*b2 - 6) * q^4 + (5*b2 - 5) * q^5 + (b10 - 2*b3 - 7*b2 + b1) * q^7 + (-b10 - 4*b9 + b8 - 2*b6 + 4*b3 + 12) * q^8 + (-5*b9 + 5*b3 + 5) * q^10 + (-b11 + b6 - b5 + 2*b3 - 14*b2 + 2*b1) * q^11 + (b11 + 2*b8 - 3*b7 - b5 - b4 + 4*b2 - 4) * q^13 + (-3*b11 + 7*b9 - b8 + b7 + 4*b5 + 3*b4 + 32*b2 - 32) * q^14 + (2*b11 + 3*b10 - 16*b3 - 14*b2) * q^16 + (b10 + 6*b9 - b8 + 2*b7 - b6 + 3*b4 - 6*b3 + 2*b1 + 19) * q^17 + (4*b10 + 2*b9 - 4*b8 - 4*b7 + 4*b6 + 2*b4 - 2*b3 - 4*b1 - 7) * q^19 + (5*b6 - 5*b5 - 5*b3 - 30*b2) * q^20 + (-2*b11 + 20*b9 + 2*b8 - 10*b7 - b5 + 2*b4 - 5*b2 + 5) * q^22 + (-6*b11 - 6*b9 + 3*b8 + 2*b7 - 2*b5 + 6*b4 + 34*b2 - 34) * q^23 - 25*b2 * q^25 + (6*b10 + 2*b9 - 6*b8 + 15*b7 + 5*b6 + 7*b4 - 2*b3 + 15*b1 + 2) * q^26 + (-6*b10 - 40*b9 + 6*b8 - 29*b7 - 13*b6 - 3*b4 + 40*b3 - 29*b1 + 70) * q^28 + (-3*b11 + 4*b10 + b6 - b5 - 12*b3 - 52*b2 - 8*b1) * q^29 + (4*b11 + 16*b9 - 3*b8 - 6*b7 - 10*b5 - 4*b4 - 13*b2 + 13) * q^31 + (-6*b11 - 18*b9 - b8 + 24*b7 + 6*b5 + 6*b4 + 114*b2 - 114) * q^32 + (-4*b11 - 9*b10 - 7*b6 + 7*b5 - 25*b3 + 60*b2 - 38*b1) * q^34 + (-5*b10 - 10*b9 + 5*b8 - 5*b7 + 10*b3 - 5*b1 + 35) * q^35 + (5*b10 - 10*b9 - 5*b8 - 28*b7 + 5*b6 - 3*b4 + 10*b3 - 28*b1 - 29) * q^37 + (-4*b11 - 14*b10 - 14*b6 + 14*b5 + 31*b3 + 9*b2 - 20*b1) * q^38 + (20*b9 - 5*b8 + 10*b5 + 60*b2 - 60) * q^40 + (-10*b11 + 16*b9 - 2*b8 + 7*b7 - 26*b5 + 10*b4 + 62*b2 - 62) * q^41 + (b11 + b10 + 25*b6 - 25*b5 - 42*b3 - 99*b2 - 38*b1) * q^43 + (-3*b10 - 5*b9 + 3*b8 + 2*b7 - 7*b6 + 6*b4 + 5*b3 + 2*b1 + 120) * q^44 + (-13*b10 - 22*b9 + 13*b8 - 74*b7 + 14*b6 + 4*b4 + 22*b3 - 74*b1 - 28) * q^46 + (-9*b11 + 6*b10 - 29*b6 + 29*b5 + 44*b3 - 23*b2 - 14*b1) * q^47 + (-12*b11 + 26*b9 - 15*b8 + 12*b7 + 16*b5 + 12*b4 + 15*b2 - 15) * q^49 + (25*b9 + 25*b2 - 25) * q^50 + (-19*b11 - 16*b10 - 11*b6 + 11*b5 + 28*b3 + 82*b2 - 75*b1) * q^52 + (-24*b10 + 20*b9 + 24*b8 - 58*b7 + 27*b6 - 5*b4 - 20*b3 - 58*b1 + 127) * q^53 + (10*b9 - 10*b7 - 5*b6 + 5*b4 - 10*b3 - 10*b1 + 70) * q^55 + (17*b11 + 20*b10 + 33*b6 - 33*b5 - 92*b3 - 410*b2 + 57*b1) * q^56 + (58*b9 + 4*b8 - 44*b7 + 21*b5 + 169*b2 - 169) * q^58 + (-30*b9 - 8*b8 - 15*b7 + 6*b5 + 294*b2 - 294) * q^59 + (22*b11 - 2*b10 - 38*b6 + 38*b5 + 72*b3 + 84*b2 - 30*b1) * q^61 + (19*b10 + 73*b9 - 19*b8 + 54*b7 - 12*b6 - 73*b3 + 54*b1 + 155) * q^62 + (-b10 - 22*b9 + b8 - 96*b7 + 10*b6 - 10*b4 + 22*b3 - 96*b1 - 158) * q^64 + (-5*b11 - 10*b10 - 5*b6 + 5*b5 - 20*b2 - 15*b1) * q^65 + (-10*b11 - 164*b9 + 12*b8 + 8*b7 - 30*b5 + 10*b4 - 36*b2 + 36) * q^67 + (32*b11 - 54*b9 + 20*b8 - 70*b7 - 8*b5 - 32*b4 + 28*b2 - 28) * q^68 + (15*b11 + 5*b10 + 20*b6 - 20*b5 - 35*b3 - 160*b2 + 5*b1) * q^70 + (40*b10 - 82*b9 - 40*b8 + 52*b7 + 9*b6 - 9*b4 + 82*b3 + 52*b1 + 268) * q^71 + (23*b10 + 138*b9 - 23*b8 - 50*b7 - 33*b6 + 7*b4 - 138*b3 - 50*b1 + 101) * q^73 + (18*b11 - b10 - 5*b6 + 5*b5 + 59*b3 - 200*b2 + 64*b1) * q^74 + (32*b11 - 77*b9 + 8*b8 - 100*b7 - 41*b5 - 32*b4 - 374*b2 + 374) * q^76 + (9*b11 - 46*b9 - 5*b8 + 40*b7 + 19*b5 - 9*b4 + 123*b2 - 123) * q^77 + (-12*b11 + 32*b10 + 12*b6 - 12*b5 + 144*b3 + 172*b2 - 10*b1) * q^79 + (-15*b10 - 80*b9 + 15*b8 - 10*b4 + 80*b3 + 70) * q^80 + (-6*b10 + 94*b9 + 6*b8 - 127*b7 + 6*b6 - 11*b4 - 94*b3 - 127*b1 + 257) * q^82 + (-19*b11 - 27*b10 + 45*b6 - 45*b5 - 6*b3 - 369*b2 - 34*b1) * q^83 + (15*b11 - 30*b9 + 5*b8 - 10*b7 + 5*b5 - 15*b4 + 95*b2 - 95) * q^85 + (36*b11 + 249*b9 - 29*b8 - 26*b7 + 69*b5 - 36*b4 + 552*b2 - 552) * q^86 + (-12*b11 + 8*b10 + 26*b6 - 26*b5 - 2*b3 - 214*b2 + 6*b1) * q^88 + (18*b10 + 96*b9 - 18*b8 - 76*b7 - 33*b6 - 21*b4 - 96*b3 - 76*b1 + 474) * q^89 + (21*b10 + 106*b9 - 21*b8 + 44*b7 + 34*b6 + 18*b4 - 106*b3 + 44*b1 - 538) * q^91 + (52*b11 + 11*b10 + 50*b6 - 50*b5 + 64*b3 - 142*b2 + 10*b1) * q^92 + (2*b11 - 151*b9 + 50*b8 - 122*b7 - 61*b5 - 2*b4 - 644*b2 + 644) * q^94 + (10*b11 - 10*b9 + 20*b8 + 20*b7 - 20*b5 - 10*b4 - 35*b2 + 35) * q^95 + (-41*b11 + 39*b10 - 13*b6 + 13*b5 + 30*b3 + 125*b2 - 72*b1) * q^97 + (-67*b10 - 111*b9 + 67*b8 - 156*b7 - 72*b6 - 42*b4 + 111*b3 - 156*b1 + 345) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 4 q^{2} - 34 q^{4} - 30 q^{5} - 40 q^{7} + 132 q^{8}+O(q^{10})$$ 12 * q - 4 * q^2 - 34 * q^4 - 30 * q^5 - 40 * q^7 + 132 * q^8 $$12 q - 4 q^{2} - 34 q^{4} - 30 q^{5} - 40 q^{7} + 132 q^{8} + 40 q^{10} - 88 q^{11} - 20 q^{13} - 180 q^{14} - 58 q^{16} + 248 q^{17} - 92 q^{19} - 170 q^{20} + 74 q^{22} - 210 q^{23} - 150 q^{25} + 8 q^{26} + 704 q^{28} - 296 q^{29} + 104 q^{31} - 722 q^{32} + 428 q^{34} + 400 q^{35} - 408 q^{37} + 20 q^{38} - 330 q^{40} - 344 q^{41} - 512 q^{43} + 1432 q^{44} - 372 q^{46} - 238 q^{47} - 68 q^{49} - 100 q^{50} + 468 q^{52} + 1700 q^{53} + 880 q^{55} - 2316 q^{56} - 890 q^{58} - 1840 q^{59} + 364 q^{61} + 2076 q^{62} - 1980 q^{64} - 100 q^{65} - 88 q^{67} - 236 q^{68} - 900 q^{70} + 2728 q^{71} + 1672 q^{73} - 1316 q^{74} + 2106 q^{76} - 840 q^{77} + 680 q^{79} + 580 q^{80} + 3484 q^{82} - 2148 q^{83} - 620 q^{85} - 2872 q^{86} - 1296 q^{88} + 6000 q^{89} - 6116 q^{91} - 1002 q^{92} + 3662 q^{94} + 230 q^{95} + 612 q^{97} + 3964 q^{98}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$( 39168669767185 \nu^{11} + 622492575406102 \nu^{10} - 116378803247278 \nu^{9} + \cdots + 57\!\cdots\!56 ) / 53\!\cdots\!60$$ (39168669767185*v^11 + 622492575406102*v^10 - 116378803247278*v^9 - 4437769721904304*v^8 + 76487510388819703*v^7 + 415285293653958168*v^6 - 310931437403832750*v^5 - 443929146343266438*v^4 + 17810423563382896401*v^3 + 83726076442829728680*v^2 - 66823261574041484640*v + 5737362693292717056) / 53110961918205557760 $$\beta_{2}$$ $$=$$ $$( - 21833561 \nu^{11} - 6933902 \nu^{10} + 191663726 \nu^{9} - 824768992 \nu^{8} - 19024412783 \nu^{7} + 8759970096 \nu^{6} + \cdots + 11197076020992 ) / 28100111293440$$ (-21833561*v^11 - 6933902*v^10 + 191663726*v^9 - 824768992*v^8 - 19024412783*v^7 + 8759970096*v^6 + 9602073054*v^5 - 242401187034*v^4 - 3561497041833*v^3 + 692957538000*v^2 - 10161675967968*v + 11197076020992) / 28100111293440 $$\beta_{3}$$ $$=$$ $$( 8315698756077 \nu^{11} + 101245330415030 \nu^{10} - 394560515037286 \nu^{9} + \cdots + 15\!\cdots\!84 ) / 44\!\cdots\!80$$ (8315698756077*v^11 + 101245330415030*v^10 - 394560515037286*v^9 + 1057574546275072*v^8 + 7683197260871035*v^7 + 48297387784200512*v^6 - 116794756095706998*v^5 + 265856737439346114*v^4 + 1425180458645681949*v^3 + 5529771972303382080*v^2 - 4139103064502083104*v + 1560323092464854784) / 4425913493183796480 $$\beta_{4}$$ $$=$$ $$( 4200645557245 \nu^{11} + 6160897283854 \nu^{10} - 314135929010806 \nu^{9} + \cdots + 62\!\cdots\!12 ) / 22\!\cdots\!40$$ (4200645557245*v^11 + 6160897283854*v^10 - 314135929010806*v^9 + 3132646358261552*v^8 - 3889069452974549*v^7 + 1955376027463176*v^6 - 57814071395165910*v^5 + 1126387696678531794*v^4 - 1077777736999692003*v^3 + 79352387692368120*v^2 + 2581814601408702240*v + 62107963971000458112) / 2212956746591898240 $$\beta_{5}$$ $$=$$ $$( - 335658953887093 \nu^{11} + \cdots - 17\!\cdots\!56 ) / 15\!\cdots\!80$$ (-335658953887093*v^11 + 1171025533283690*v^10 - 8666619525148106*v^9 + 50196423604163872*v^8 - 313704453000911155*v^7 + 295373042049020592*v^6 - 1436493017218079898*v^5 + 11741310942738453774*v^4 - 64803099267637148421*v^3 + 5271451684560213840*v^2 - 192953421149110468704*v - 174266393008948736256) / 159332885754616673280 $$\beta_{6}$$ $$=$$ $$( 11005996995395 \nu^{11} - 8278688448046 \nu^{10} - 261584063656106 \nu^{9} + \cdots - 48\!\cdots\!08 ) / 33\!\cdots\!60$$ (11005996995395*v^11 - 8278688448046*v^10 - 261584063656106*v^9 + 2809757609441872*v^8 + 952726608368501*v^7 - 2440351842639624*v^6 - 41491125321905610*v^5 + 700433575797000654*v^4 - 39409425332857053*v^3 - 35284815360850680*v^2 - 206965861006966560*v - 4842346192147027008) / 3319435119887847360 $$\beta_{7}$$ $$=$$ $$( 238036479862585 \nu^{11} + \cdots + 17\!\cdots\!36 ) / 53\!\cdots\!60$$ (238036479862585*v^11 - 1222449555829658*v^10 + 2942151887057762*v^9 - 677665458581104*v^8 + 148598736796672783*v^7 - 597691959138682632*v^6 + 614420351455033890*v^5 + 662850739681255242*v^4 + 22923435786759202041*v^3 - 88414990146924296760*v^2 - 44666427965491905120*v + 17268670504516340736) / 53110961918205557760 $$\beta_{8}$$ $$=$$ $$( - 10\!\cdots\!07 \nu^{11} - 357550815547730 \nu^{10} + \cdots - 22\!\cdots\!44 ) / 15\!\cdots\!80$$ (-1081657721587607*v^11 - 357550815547730*v^10 + 17604901276198706*v^9 - 64482188744355232*v^8 - 904614383398756385*v^7 + 2237153490596534928*v^6 + 3109902766563363138*v^5 - 17333693087164935174*v^4 - 151757889547035749319*v^3 + 567820946400626294640*v^2 + 330054951944179314144*v - 225525340579288345344) / 159332885754616673280 $$\beta_{9}$$ $$=$$ $$( - 96943826623309 \nu^{11} + 480126841023914 \nu^{10} - 441998532012314 \nu^{9} + \cdots - 73\!\cdots\!36 ) / 13\!\cdots\!40$$ (-96943826623309*v^11 + 480126841023914*v^10 - 441998532012314*v^9 - 4147237826236352*v^8 - 37432400654764699*v^7 + 198005263782509472*v^6 - 64410209363731914*v^5 - 1084777218670144194*v^4 - 3488004203931395901*v^3 + 17774459217805263840*v^2 + 11311092917146506528*v - 7383692416832315136) / 13277740479551389440 $$\beta_{10}$$ $$=$$ $$( 18\!\cdots\!33 \nu^{11} + \cdots + 22\!\cdots\!92 ) / 15\!\cdots\!80$$ (1828239864155833*v^11 - 5135036083007858*v^10 + 11308370778733778*v^9 + 67186055714148704*v^8 + 710053265142888463*v^7 + 792571975775440656*v^6 + 450734978859702498*v^5 + 18141585956481889818*v^4 + 109507911984962575497*v^3 + 533504254636404411120*v^2 - 419473500997898113056*v + 227037510271026600192) / 159332885754616673280 $$\beta_{11}$$ $$=$$ $$( - 24\!\cdots\!15 \nu^{11} + \cdots + 54\!\cdots\!64 ) / 53\!\cdots\!60$$ (-2442175499886815*v^11 + 7080096183948958*v^10 - 25311276149957662*v^9 - 22787594734230496*v^8 - 1298135621615504153*v^7 + 1752882046900662192*v^6 - 8532089400267944910*v^5 - 4244172237278355222*v^4 - 174847215049808235471*v^3 - 4245140807314092720*v^2 - 497882512957950381600*v + 549583910441667857664) / 53110961918205557760
 $$\nu$$ $$=$$ $$( -\beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - 2\beta_{5} + \beta_{3} - 2\beta_{2} + 2 ) / 6$$ (-b10 - b9 + b8 + b6 - 2*b5 + b3 - 2*b2 + 2) / 6 $$\nu^{2}$$ $$=$$ $$( -\beta_{10} - 5\beta_{9} - \beta_{8} - 8\beta_{7} + \beta_{6} - 2\beta_{5} - 5\beta_{3} + 8\beta_1 ) / 6$$ (-b10 - 5*b9 - b8 - 8*b7 + b6 - 2*b5 - 5*b3 + 8*b1) / 6 $$\nu^{3}$$ $$=$$ $$( \beta_{11} + 8 \beta_{10} + 12 \beta_{9} - 9 \beta_{8} - 8 \beta_{7} + 7 \beta_{6} - 17 \beta_{5} - 14 \beta_{3} + 17 \beta_{2} - 4 \beta _1 - 35 ) / 3$$ (b11 + 8*b10 + 12*b9 - 9*b8 - 8*b7 + 7*b6 - 17*b5 - 14*b3 + 17*b2 - 4*b1 - 35) / 3 $$\nu^{4}$$ $$=$$ $$( 35 \beta_{10} + 35 \beta_{9} - 35 \beta_{8} - 24 \beta_{7} - 81 \beta_{6} + 40 \beta_{4} - 35 \beta_{3} - 24 \beta _1 - 1298 ) / 6$$ (35*b10 + 35*b9 - 35*b8 - 24*b7 - 81*b6 + 40*b4 - 35*b3 - 24*b1 - 1298) / 6 $$\nu^{5}$$ $$=$$ $$( - 26 \beta_{11} + 365 \beta_{10} + 501 \beta_{9} - 279 \beta_{8} - 184 \beta_{7} - 451 \beta_{6} + 644 \beta_{5} + 48 \beta_{4} - 377 \beta_{3} - 1308 \beta_{2} - 464 \beta _1 - 904 ) / 6$$ (-26*b11 + 365*b10 + 501*b9 - 279*b8 - 184*b7 - 451*b6 + 644*b5 + 48*b4 - 377*b3 - 1308*b2 - 464*b1 - 904) / 6 $$\nu^{6}$$ $$=$$ $$( 16 \beta_{11} + 325 \beta_{10} + 758 \beta_{9} + 325 \beta_{8} + 1288 \beta_{7} - 488 \beta_{6} + 976 \beta_{5} - 8 \beta_{4} + 758 \beta_{3} - 2818 \beta_{2} - 1288 \beta _1 + 1409 ) / 3$$ (16*b11 + 325*b10 + 758*b9 + 325*b8 + 1288*b7 - 488*b6 + 976*b5 - 8*b4 + 758*b3 - 2818*b2 - 1288*b1 + 1409) / 3 $$\nu^{7}$$ $$=$$ $$( - 144 \beta_{11} - 5057 \beta_{10} - 5627 \beta_{9} + 7763 \beta_{8} + 11704 \beta_{7} - 2351 \beta_{6} + 12820 \beta_{5} - 856 \beta_{4} + 8489 \beta_{3} - 36254 \beta_{2} + 3896 \beta _1 + 58398 ) / 6$$ (-144*b11 - 5057*b10 - 5627*b9 + 7763*b8 + 11704*b7 - 2351*b6 + 12820*b5 - 856*b4 + 8489*b3 - 36254*b2 + 3896*b1 + 58398) / 6 $$\nu^{8}$$ $$=$$ $$( - 24965 \beta_{10} - 17315 \beta_{9} + 24965 \beta_{8} + 32472 \beta_{7} + 46041 \beta_{6} - 13252 \beta_{4} + 17315 \beta_{3} + 32472 \beta _1 + 480200 ) / 6$$ (-24965*b10 - 17315*b9 + 24965*b8 + 32472*b7 + 46041*b6 - 13252*b4 + 17315*b3 + 32472*b1 + 480200) / 6 $$\nu^{9}$$ $$=$$ $$( - 2095 \beta_{11} - 84984 \beta_{10} - 71512 \beta_{9} + 47317 \beta_{8} + 42152 \beta_{7} + 122651 \beta_{6} - 132301 \beta_{5} - 10160 \beta_{4} + 41934 \beta_{3} + 464143 \beta_{2} + \cdots + 260571 ) / 3$$ (-2095*b11 - 84984*b10 - 71512*b9 + 47317*b8 + 42152*b7 + 122651*b6 - 132301*b5 - 10160*b4 + 41934*b3 + 464143*b2 + 142012*b1 + 260571) / 3 $$\nu^{10}$$ $$=$$ $$( - 61008 \beta_{11} - 360417 \beta_{10} - 395679 \beta_{9} - 360417 \beta_{8} - 1058408 \beta_{7} + 612127 \beta_{6} - 1224254 \beta_{5} + 30504 \beta_{4} - 395679 \beta_{3} + \cdots - 2551826 ) / 6$$ (-61008*b11 - 360417*b10 - 395679*b9 - 360417*b8 - 1058408*b7 + 612127*b6 - 1224254*b5 + 30504*b4 - 395679*b3 + 5103652*b2 + 1058408*b1 - 2551826) / 6 $$\nu^{11}$$ $$=$$ $$( - 216770 \beta_{11} + 1828317 \beta_{10} + 1259071 \beta_{9} - 3796381 \beta_{8} - 6773512 \beta_{7} - 139747 \beta_{6} - 5624698 \beta_{5} + 629000 \beta_{4} - 2414697 \beta_{3} + \cdots - 34930520 ) / 6$$ (-216770*b11 + 1828317*b10 + 1259071*b9 - 3796381*b8 - 6773512*b7 - 139747*b6 - 5624698*b5 + 629000*b4 - 2414697*b3 + 22943700*b2 - 1876496*b1 - 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 $$S_{4}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{12} + 4 T_{2}^{11} + 49 T_{2}^{10} + 88 T_{2}^{9} + 1243 T_{2}^{8} + 2026 T_{2}^{7} + 18586 T_{2}^{6} + 13252 T_{2}^{5} + 153472 T_{2}^{4} + 171864 T_{2}^{3} + 498744 T_{2}^{2} - 73872 T_{2} + 11664$$ T2^12 + 4*T2^11 + 49*T2^10 + 88*T2^9 + 1243*T2^8 + 2026*T2^7 + 18586*T2^6 + 13252*T2^5 + 153472*T2^4 + 171864*T2^3 + 498744*T2^2 - 73872*T2 + 11664 $$T_{7}^{12} + 40 T_{7}^{11} + 1863 T_{7}^{10} + 27280 T_{7}^{9} + 874429 T_{7}^{8} + 10054356 T_{7}^{7} + 293906532 T_{7}^{6} + 1439479296 T_{7}^{5} + 23417174988 T_{7}^{4} + \cdots + 5368136821776$$ T7^12 + 40*T7^11 + 1863*T7^10 + 27280*T7^9 + 874429*T7^8 + 10054356*T7^7 + 293906532*T7^6 + 1439479296*T7^5 + 23417174988*T7^4 - 143876813760*T7^3 + 1191988160496*T7^2 - 2647910494944*T7 + 5368136821776

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 4 T^{11} + 49 T^{10} + \cdots + 11664$$
$3$ $$T^{12}$$
$5$ $$(T^{2} + 5 T + 25)^{6}$$
$7$ $$T^{12} + 40 T^{11} + \cdots + 5368136821776$$
$11$ $$T^{12} + 88 T^{11} + \cdots + 1433067563664$$
$13$ $$T^{12} + 20 T^{11} + \cdots + 34\!\cdots\!84$$
$17$ $$(T^{6} - 124 T^{5} - 5004 T^{4} + \cdots + 105966288)^{2}$$
$19$ $$(T^{6} + 46 T^{5} - 18169 T^{4} + \cdots - 36821611175)^{2}$$
$23$ $$T^{12} + 210 T^{11} + \cdots + 11\!\cdots\!24$$
$29$ $$T^{12} + 296 T^{11} + \cdots + 40\!\cdots\!44$$
$31$ $$T^{12} - 104 T^{11} + \cdots + 48\!\cdots\!56$$
$37$ $$(T^{6} + 204 T^{5} + \cdots + 12008297128192)^{2}$$
$41$ $$T^{12} + 344 T^{11} + \cdots + 74\!\cdots\!09$$
$43$ $$T^{12} + 512 T^{11} + \cdots + 32\!\cdots\!00$$
$47$ $$T^{12} + 238 T^{11} + \cdots + 24\!\cdots\!00$$
$53$ $$(T^{6} - 850 T^{5} + \cdots + 15\!\cdots\!00)^{2}$$
$59$ $$T^{12} + 1840 T^{11} + \cdots + 28\!\cdots\!89$$
$61$ $$T^{12} - 364 T^{11} + \cdots + 41\!\cdots\!96$$
$67$ $$T^{12} + 88 T^{11} + \cdots + 38\!\cdots\!00$$
$71$ $$(T^{6} - 1364 T^{5} + \cdots - 11\!\cdots\!84)^{2}$$
$73$ $$(T^{6} - 836 T^{5} + \cdots - 10\!\cdots\!36)^{2}$$
$79$ $$T^{12} - 680 T^{11} + \cdots + 25\!\cdots\!84$$
$83$ $$T^{12} + 2148 T^{11} + \cdots + 17\!\cdots\!24$$
$89$ $$(T^{6} - 3000 T^{5} + \cdots + 16\!\cdots\!48)^{2}$$
$97$ $$T^{12} - 612 T^{11} + \cdots + 96\!\cdots\!76$$