# Properties

 Label 54.7.d.a Level $54$ Weight $7$ Character orbit 54.d Analytic conductor $12.423$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,7,Mod(17,54)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

N = Newforms(chi, 7, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("54.17");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$54 = 2 \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 54.d (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$12.4229205155$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600$$ x^12 + 370*x^10 + 51793*x^8 + 3491832*x^6 + 117603792*x^4 + 1832032512*x^2 + 10453017600 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}\cdot 3^{18}$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} - 32 \beta_{2} q^{4} + ( - \beta_{7} - 24 \beta_{2} - 48) q^{5} + (\beta_{10} + \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} - 16 \beta_{4} - 8 \beta_{3} + 40 \beta_{2} + \cdots + 40) q^{7}+ \cdots + (32 \beta_{4} + 32 \beta_{3}) q^{8}+O(q^{10})$$ q + b3 * q^2 - 32*b2 * q^4 + (-b7 - 24*b2 - 48) * q^5 + (b10 + b8 + 2*b7 - b6 + b5 - 16*b4 - 8*b3 + 40*b2 + 40) * q^7 + (32*b4 + 32*b3) * q^8 $$q + \beta_{3} q^{2} - 32 \beta_{2} q^{4} + ( - \beta_{7} - 24 \beta_{2} - 48) q^{5} + (\beta_{10} + \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} - 16 \beta_{4} - 8 \beta_{3} + 40 \beta_{2} + \cdots + 40) q^{7}+ \cdots + (254 \beta_{11} - 1132 \beta_{10} - 1132 \beta_{9} - 390 \beta_{8} + \cdots + 302352) q^{98}+O(q^{100})$$ q + b3 * q^2 - 32*b2 * q^4 + (-b7 - 24*b2 - 48) * q^5 + (b10 + b8 + 2*b7 - b6 + b5 - 16*b4 - 8*b3 + 40*b2 + 40) * q^7 + (32*b4 + 32*b3) * q^8 + (2*b10 - 2*b9 - 2*b8 + 2*b7 + 3*b6 + 3*b5 + 24*b4 - 24*b3 - b1) * q^10 + (-b11 + 6*b10 - 2*b9 + 6*b6 + 4*b5 - 19*b3 + 21*b2 - 2*b1 - 21) * q^11 + (-8*b11 - 3*b10 - 2*b9 + 2*b8 + b7 - b6 - 2*b5 + 50*b4 + 100*b3 - 280*b2 - 8*b1) * q^13 + (b11 + 5*b10 + 3*b9 - 2*b7 - 8*b6 - 3*b5 - 36*b4 + 264*b2 - b1 + 528) * q^14 + (-1024*b2 - 1024) * q^16 + (20*b11 + 5*b10 + 5*b9 + 37*b8 + 37*b7 - 11*b6 + 11*b5 + 154*b4 + 154*b3 - 2586*b2 + 10*b1 - 1293) * q^17 + (-8*b10 + 8*b9 - b8 + b7 - 27*b6 - 27*b5 + 183*b4 - 183*b3 - 35*b1 - 235) * q^19 + (32*b8 + 768*b2 - 768) * q^20 + (-10*b11 + 18*b10 + 2*b9 + 40*b8 + 20*b7 + 16*b6 + 2*b5 - 23*b4 - 46*b3 + 600*b2 - 10*b1) * q^22 + (40*b11 - b10 + 3*b9 + 9*b7 - 2*b6 - 3*b5 + 382*b4 + 4236*b2 - 40*b1 + 8472) * q^23 + (40*b11 + 30*b9 + 87*b8 + 174*b7 - 30*b5 + 1100*b4 + 550*b3 + 1349*b2 + 1349) * q^25 + (34*b11 + 28*b10 + 28*b9 + 6*b8 + 6*b7 + 23*b6 - 23*b5 + 294*b4 + 294*b3 - 3360*b2 + 17*b1 - 1680) * q^26 + (32*b10 - 32*b9 - 32*b8 + 32*b7 - 256*b4 + 256*b3 + 1280) * q^28 + (-30*b11 - 90*b10 + 39*b9 + 95*b8 - 90*b6 - 51*b5 + 804*b3 + 5394*b2 - 60*b1 - 5394) * q^29 + (-14*b11 - 63*b10 - 39*b9 + 30*b8 + 15*b7 - 24*b6 - 39*b5 - 2036*b4 - 4072*b3 - 3580*b2 - 14*b1) * q^31 + 1024*b4 * q^32 + (-63*b11 + 71*b10 + 21*b9 - 94*b8 - 188*b7 - 71*b6 + 50*b5 + 2590*b4 + 1295*b3 - 4560*b2 - 4560) * q^34 + (-56*b11 - 23*b10 - 23*b9 - 69*b8 - 69*b7 - 52*b6 + 52*b5 - 826*b4 - 826*b3 - 25788*b2 - 28*b1 - 12894) * q^35 + (-109*b10 + 109*b9 + 10*b8 - 10*b7 + 3*b6 + 3*b5 - 3836*b4 + 3836*b3 - 26*b1 - 2128) * q^37 + (87*b11 + 189*b10 - 129*b9 - 38*b8 + 189*b6 + 60*b5 - 187*b3 + 5424*b2 + 174*b1 - 5424) * q^38 + (-32*b11 + 96*b10 + 32*b9 - 128*b8 - 64*b7 + 64*b6 + 32*b5 - 768*b4 - 1536*b3 - 32*b1) * q^40 + (-100*b11 + 61*b10 - 156*b9 + 162*b7 + 95*b6 + 156*b5 - 2878*b4 + 22809*b2 + 100*b1 + 45618) * q^41 + (27*b11 - 150*b10 - 195*b9 - 180*b8 - 360*b7 + 150*b6 + 45*b5 + 11274*b4 + 5637*b3 + 11905*b2 + 11905) * q^43 + (-64*b11 + 128*b10 + 128*b9 + 64*b6 - 64*b5 - 608*b4 - 608*b3 + 1344*b2 - 32*b1 + 672) * q^44 + (298*b10 - 298*b9 + 170*b8 - 170*b7 + 369*b6 + 369*b5 - 4318*b4 + 4318*b3 + 257*b1 - 11256) * q^46 + (-32*b11 + 3*b10 + 170*b9 - 751*b8 + 3*b6 + 173*b5 - 3776*b3 + 19314*b2 - 64*b1 - 19314) * q^47 + (284*b11 - 147*b10 + 134*b9 - 266*b8 - 133*b7 - 281*b6 + 134*b5 - 9344*b4 - 18688*b3 + 22659*b2 + 284*b1) * q^49 + (-227*b11 - 67*b10 + 255*b9 - 562*b7 - 188*b6 - 255*b5 - 1489*b4 - 17280*b2 + 227*b1 - 34560) * q^50 + (-256*b11 - 64*b10 - 96*b9 + 32*b8 + 64*b7 + 64*b6 + 32*b5 + 3200*b4 + 1600*b3 - 8960*b2 - 8960) * q^52 + (-432*b11 - 729*b10 - 729*b9 - 410*b8 - 410*b7 + 81*b6 - 81*b5 - 1182*b4 - 1182*b3 - 22596*b2 - 216*b1 - 11298) * q^53 + (-426*b10 + 426*b9 + 3*b8 - 3*b7 - 519*b6 - 519*b5 - 13552*b4 + 13552*b3 + 418*b1 + 48366) * q^55 + (-32*b11 - 96*b10 - 160*b9 + 64*b8 - 96*b6 - 256*b5 + 1152*b3 - 8448*b2 - 64*b1 + 8448) * q^56 + (343*b11 + 483*b10 + 209*b9 - 548*b8 - 274*b7 + 274*b6 + 209*b5 - 5256*b4 - 10512*b3 - 26544*b2 + 343*b1) * q^58 + (5*b11 - 629*b10 + 240*b9 + 8*b7 + 389*b6 - 240*b5 - 2959*b4 - 20541*b2 - 5*b1 - 41082) * q^59 + (154*b11 - 24*b10 + 363*b9 - 189*b8 - 378*b7 + 24*b6 - 387*b5 + 20360*b4 + 10180*b3 + 22624*b2 + 22624) * q^61 + (122*b11 - 40*b10 - 40*b9 - 442*b8 - 442*b7 + 151*b6 - 151*b5 + 3578*b4 + 3578*b3 + 129456*b2 + 61*b1 + 64728) * q^62 - 32768 * q^64 + (-64*b11 - 480*b10 + 151*b9 + 341*b8 - 480*b6 - 329*b5 + 452*b3 - 41880*b2 - 128*b1 + 41880) * q^65 + (3*b11 - 396*b10 - 528*b9 + 1896*b8 + 948*b7 + 132*b6 - 528*b5 - 15033*b4 - 30066*b3 + 48323*b2 + 3*b1) * q^67 + (320*b11 + 352*b10 - 192*b9 + 1184*b7 - 160*b6 + 192*b5 + 4928*b4 - 41376*b2 - 320*b1 - 82752) * q^68 + (491*b11 + 149*b10 + 27*b9 + 158*b8 + 316*b7 - 149*b6 + 122*b5 + 26108*b4 + 13054*b3 + 25992*b2 + 25992) * q^70 + (848*b11 + 1391*b10 + 1391*b9 + 1240*b8 + 1240*b7 - 425*b6 + 425*b5 - 13346*b4 - 13346*b3 + 108672*b2 + 424*b1 + 54336) * q^71 + (399*b10 - 399*b9 - 327*b8 + 327*b7 + 699*b6 + 699*b5 - 17810*b4 + 17810*b3 - 1186*b1 - 81475) * q^73 + (260*b11 - 12*b10 + 604*b9 + 504*b8 - 12*b6 + 592*b5 - 1634*b3 - 122064*b2 + 520*b1 + 122064) * q^74 + (-1120*b11 - 864*b10 - 608*b9 - 64*b8 - 32*b7 - 256*b6 - 608*b5 - 5856*b4 - 11712*b3 + 7520*b2 - 1120*b1) * q^76 + (494*b11 + 1420*b10 - 858*b9 + 119*b7 - 562*b6 + 858*b5 + 18986*b4 + 8844*b2 - 494*b1 + 17688) * q^77 + (-158*b11 + 539*b10 + 153*b9 + 539*b8 + 1078*b7 - 539*b6 + 386*b5 + 83808*b4 + 41904*b3 - 127466*b2 - 127466) * q^79 + (1024*b8 + 1024*b7 + 49152*b2 + 24576) * q^80 + (-880*b10 + 880*b9 + 304*b8 - 304*b7 - 1296*b6 - 1296*b5 - 22487*b4 + 22487*b3 - 872*b1 + 89424) * q^82 + (166*b11 + 921*b10 - 1432*b9 + 2621*b8 + 921*b6 - 511*b5 + 29812*b3 + 22050*b2 + 332*b1 - 22050) * q^83 + (-2114*b11 + 1407*b10 + 804*b9 - 3024*b8 - 1512*b7 + 603*b6 + 804*b5 - 43856*b4 - 87712*b3 - 269928*b2 - 2114*b1) * q^85 + (216*b11 - 276*b10 - 1152*b9 - 408*b7 + 1428*b6 + 1152*b5 - 12169*b4 - 181368*b2 - 216*b1 - 362736) * q^86 + (-320*b11 + 64*b10 + 576*b9 + 640*b8 + 1280*b7 - 64*b6 - 512*b5 - 1472*b4 - 736*b3 + 19200*b2 + 19200) * q^88 + (536*b11 + 953*b10 + 953*b9 - 206*b8 - 206*b7 + 31*b6 - 31*b5 + 61774*b4 + 61774*b3 - 280668*b2 + 268*b1 - 140334) * q^89 + (2586*b10 - 2586*b9 - 921*b8 + 921*b7 + 1065*b6 + 1065*b5 - 25740*b4 + 25740*b3 - 180*b1 + 29632) * q^91 + (-1280*b11 - 96*b10 + 32*b9 - 288*b8 - 96*b6 - 64*b5 - 12224*b3 - 135552*b2 - 2560*b1 + 135552) * q^92 + (1277*b11 - 1587*b10 + 339*b9 + 4644*b8 + 2322*b7 - 1926*b6 + 339*b5 - 18910*b4 - 37820*b3 + 122808*b2 + 1277*b1) * q^94 + (176*b11 - 494*b10 + 3210*b9 - 3004*b7 - 2716*b6 - 3210*b5 + 8252*b4 + 116070*b2 - 176*b1 + 232140) * q^95 + (-430*b11 - 28*b10 - 2319*b9 + 2912*b8 + 5824*b7 + 28*b6 + 2291*b5 + 60128*b4 + 30064*b3 - 6479*b2 - 6479) * q^97 + (254*b11 - 1132*b10 - 1132*b9 - 390*b8 - 390*b7 - 1799*b6 + 1799*b5 - 22397*b4 - 22397*b3 + 604704*b2 + 127*b1 + 302352) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 192 q^{4} - 432 q^{5} + 240 q^{7}+O(q^{10})$$ 12 * q + 192 * q^4 - 432 * q^5 + 240 * q^7 $$12 q + 192 q^{4} - 432 q^{5} + 240 q^{7} - 378 q^{11} + 1680 q^{13} + 4752 q^{14} - 6144 q^{16} - 2820 q^{19} - 13824 q^{20} - 3600 q^{22} + 76248 q^{23} + 8094 q^{25} + 15360 q^{28} - 97092 q^{29} + 21480 q^{31} - 27360 q^{34} - 25536 q^{37} - 97632 q^{38} + 410562 q^{41} + 71430 q^{43} - 135072 q^{46} - 347652 q^{47} - 135954 q^{49} - 311040 q^{50} - 53760 q^{52} + 580392 q^{55} + 152064 q^{56} + 159264 q^{58} - 369738 q^{59} + 135744 q^{61} - 393216 q^{64} + 753840 q^{65} - 289938 q^{67} - 744768 q^{68} + 155952 q^{70} - 977700 q^{73} + 2197152 q^{74} - 45120 q^{76} + 159192 q^{77} - 764796 q^{79} + 1073088 q^{82} - 396900 q^{83} + 1619568 q^{85} - 3264624 q^{86} + 115200 q^{88} + 355584 q^{91} + 2439936 q^{92} - 736848 q^{94} + 2089260 q^{95} - 38874 q^{97}+O(q^{100})$$ 12 * q + 192 * q^4 - 432 * q^5 + 240 * q^7 - 378 * q^11 + 1680 * q^13 + 4752 * q^14 - 6144 * q^16 - 2820 * q^19 - 13824 * q^20 - 3600 * q^22 + 76248 * q^23 + 8094 * q^25 + 15360 * q^28 - 97092 * q^29 + 21480 * q^31 - 27360 * q^34 - 25536 * q^37 - 97632 * q^38 + 410562 * q^41 + 71430 * q^43 - 135072 * q^46 - 347652 * q^47 - 135954 * q^49 - 311040 * q^50 - 53760 * q^52 + 580392 * q^55 + 152064 * q^56 + 159264 * q^58 - 369738 * q^59 + 135744 * q^61 - 393216 * q^64 + 753840 * q^65 - 289938 * q^67 - 744768 * q^68 + 155952 * q^70 - 977700 * q^73 + 2197152 * q^74 - 45120 * q^76 + 159192 * q^77 - 764796 * q^79 + 1073088 * q^82 - 396900 * q^83 + 1619568 * q^85 - 3264624 * q^86 + 115200 * q^88 + 355584 * q^91 + 2439936 * q^92 - 736848 * q^94 + 2089260 * q^95 - 38874 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600$$ :

 $$\beta_{1}$$ $$=$$ $$( 7 \nu^{10} - 806378 \nu^{8} - 198775097 \nu^{6} - 12957175728 \nu^{4} - 57552855264 \nu^{2} + 7369432971840 ) / 18521148960$$ (7*v^10 - 806378*v^8 - 198775097*v^6 - 12957175728*v^4 - 57552855264*v^2 + 7369432971840) / 18521148960 $$\beta_{2}$$ $$=$$ $$( - 119 \nu^{11} - 59366 \nu^{9} - 10447223 \nu^{7} - 794976432 \nu^{5} - 25420007664 \nu^{3} - 246402608256 \nu - 25705589760 ) / 51411179520$$ (-119*v^11 - 59366*v^9 - 10447223*v^7 - 794976432*v^5 - 25420007664*v^3 - 246402608256*v - 25705589760) / 51411179520 $$\beta_{3}$$ $$=$$ $$( 79977 \nu^{11} - 234158 \nu^{10} + 22976562 \nu^{9} - 160687484 \nu^{8} + 2207867961 \nu^{7} - 34293527246 \nu^{6} + \cdots - 10\!\cdots\!40 ) / 10520012609280$$ (79977*v^11 - 234158*v^10 + 22976562*v^9 - 160687484*v^8 + 2207867961*v^7 - 34293527246*v^6 + 86986802664*v^5 - 2987464567824*v^4 + 1232368368624*v^3 - 104865602354208*v^2 - 2418745339008*v - 1060766629021440) / 10520012609280 $$\beta_{4}$$ $$=$$ $$( 79977 \nu^{11} + 234158 \nu^{10} + 22976562 \nu^{9} + 160687484 \nu^{8} + 2207867961 \nu^{7} + 34293527246 \nu^{6} + \cdots + 10\!\cdots\!40 ) / 10520012609280$$ (79977*v^11 + 234158*v^10 + 22976562*v^9 + 160687484*v^8 + 2207867961*v^7 + 34293527246*v^6 + 86986802664*v^5 + 2987464567824*v^4 + 1232368368624*v^3 + 104865602354208*v^2 - 2418745339008*v + 1060766629021440) / 10520012609280 $$\beta_{5}$$ $$=$$ $$( - 1650933 \nu^{11} + 524294672 \nu^{10} + 355215438 \nu^{9} + 168117019424 \nu^{8} + 197920941003 \nu^{7} + \cdots + 14\!\cdots\!80 ) / 42080050437120$$ (-1650933*v^11 + 524294672*v^10 + 355215438*v^9 + 168117019424*v^8 + 197920941003*v^7 + 19054970083856*v^6 + 21471003767952*v^5 + 944832586817664*v^4 + 788574211686576*v^3 + 19949535057095424*v^2 + 6905882160733824*v + 142852860276986880) / 42080050437120 $$\beta_{6}$$ $$=$$ $$( 1650933 \nu^{11} + 524294672 \nu^{10} - 355215438 \nu^{9} + 168117019424 \nu^{8} - 197920941003 \nu^{7} + 19054970083856 \nu^{6} + \cdots + 14\!\cdots\!80 ) / 42080050437120$$ (1650933*v^11 + 524294672*v^10 - 355215438*v^9 + 168117019424*v^8 - 197920941003*v^7 + 19054970083856*v^6 - 21471003767952*v^5 + 944832586817664*v^4 - 788574211686576*v^3 + 19949535057095424*v^2 - 6905882160733824*v + 142852860276986880) / 42080050437120 $$\beta_{7}$$ $$=$$ $$( - 2804005 \nu^{11} + 2835456 \nu^{10} - 937140106 \nu^{9} - 243106272 \nu^{8} - 112322159605 \nu^{7} - 213790915008 \nu^{6} + \cdots - 78\!\cdots\!80 ) / 4675561159680$$ (-2804005*v^11 + 2835456*v^10 - 937140106*v^9 - 243106272*v^8 - 112322159605*v^7 - 213790915008*v^6 - 5974025513640*v^5 - 22788957292512*v^4 - 137415533407920*v^3 - 805353881336064*v^2 - 1027276831016064*v - 7825356164090880) / 4675561159680 $$\beta_{8}$$ $$=$$ $$( - 2804005 \nu^{11} - 2835456 \nu^{10} - 937140106 \nu^{9} + 243106272 \nu^{8} - 112322159605 \nu^{7} + 213790915008 \nu^{6} + \cdots + 78\!\cdots\!80 ) / 4675561159680$$ (-2804005*v^11 - 2835456*v^10 - 937140106*v^9 + 243106272*v^8 - 112322159605*v^7 + 213790915008*v^6 - 5974025513640*v^5 + 22788957292512*v^4 - 137415533407920*v^3 + 805353881336064*v^2 - 1027276831016064*v + 7825356164090880) / 4675561159680 $$\beta_{9}$$ $$=$$ $$( - 24333735 \nu^{11} + 53593072 \nu^{10} - 8404314654 \nu^{9} + 14407397248 \nu^{8} - 1054979118327 \nu^{7} + \cdots - 17\!\cdots\!00 ) / 21040025218560$$ (-24333735*v^11 + 53593072*v^10 - 8404314654*v^9 + 14407397248*v^8 - 1054979118327*v^7 + 1125188911792*v^6 - 59515405647048*v^5 + 15049212117408*v^4 - 1456094103743952*v^3 - 1050505429201920*v^2 - 11427200973304704*v - 17312246722252800) / 21040025218560 $$\beta_{10}$$ $$=$$ $$( - 24333735 \nu^{11} - 53593072 \nu^{10} - 8404314654 \nu^{9} - 14407397248 \nu^{8} - 1054979118327 \nu^{7} + \cdots + 17\!\cdots\!00 ) / 21040025218560$$ (-24333735*v^11 - 53593072*v^10 - 8404314654*v^9 - 14407397248*v^8 - 1054979118327*v^7 - 1125188911792*v^6 - 59515405647048*v^5 - 15049212117408*v^4 - 1456094103743952*v^3 + 1050505429201920*v^2 - 11427200973304704*v + 17312246722252800) / 21040025218560 $$\beta_{11}$$ $$=$$ $$( 169346043 \nu^{11} - 7952 \nu^{10} + 57570639966 \nu^{9} + 916045408 \nu^{8} + 7121624056155 \nu^{7} + 225808510192 \nu^{6} + \cdots - 83\!\cdots\!40 ) / 42080050437120$$ (169346043*v^11 - 7952*v^10 + 57570639966*v^9 + 916045408*v^8 + 7121624056155*v^7 + 225808510192*v^6 + 400258044077760*v^5 + 14719351627008*v^4 + 10027586431961712*v^3 + 65380043579904*v^2 + 84660101328680064*v - 8371675856010240) / 42080050437120
 $$\nu$$ $$=$$ $$( -\beta_{10} - \beta_{9} + 2\beta_{8} + 2\beta_{7} + \beta_{6} - \beta_{5} - 5\beta_{4} - 5\beta_{3} - 36\beta_{2} - 18 ) / 54$$ (-b10 - b9 + 2*b8 + 2*b7 + b6 - b5 - 5*b4 - 5*b3 - 36*b2 - 18) / 54 $$\nu^{2}$$ $$=$$ $$( - 5 \beta_{10} + 5 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{5} + 42 \beta_{4} - 42 \beta_{3} + 8 \beta _1 - 3330 ) / 54$$ (-5*b10 + 5*b9 + 2*b8 - 2*b7 - b6 - b5 + 42*b4 - 42*b3 + 8*b1 - 3330) / 54 $$\nu^{3}$$ $$=$$ $$( 32 \beta_{11} + 95 \beta_{10} + 95 \beta_{9} - 130 \beta_{8} - 130 \beta_{7} - 107 \beta_{6} + 107 \beta_{5} - 1994 \beta_{4} - 1994 \beta_{3} + 11340 \beta_{2} + 16 \beta _1 + 5670 ) / 54$$ (32*b11 + 95*b10 + 95*b9 - 130*b8 - 130*b7 - 107*b6 + 107*b5 - 1994*b4 - 1994*b3 + 11340*b2 + 16*b1 + 5670) / 54 $$\nu^{4}$$ $$=$$ $$( 977 \beta_{10} - 977 \beta_{9} - 266 \beta_{8} + 266 \beta_{7} + 205 \beta_{6} + 205 \beta_{5} - 10186 \beta_{4} + 10186 \beta_{3} - 892 \beta _1 + 299826 ) / 54$$ (977*b10 - 977*b9 - 266*b8 + 266*b7 + 205*b6 + 205*b5 - 10186*b4 + 10186*b3 - 892*b1 + 299826) / 54 $$\nu^{5}$$ $$=$$ $$( - 5000 \beta_{11} - 8731 \beta_{10} - 8731 \beta_{9} + 9266 \beta_{8} + 9266 \beta_{7} + 14143 \beta_{6} - 14143 \beta_{5} + 352906 \beta_{4} + 352906 \beta_{3} - 1972044 \beta_{2} + \cdots - 986022 ) / 54$$ (-5000*b11 - 8731*b10 - 8731*b9 + 9266*b8 + 9266*b7 + 14143*b6 - 14143*b5 + 352906*b4 + 352906*b3 - 1972044*b2 - 2500*b1 - 986022) / 54 $$\nu^{6}$$ $$=$$ $$( - 151881 \beta_{10} + 151881 \beta_{9} + 29178 \beta_{8} - 29178 \beta_{7} - 32913 \beta_{6} - 32913 \beta_{5} + 1836802 \beta_{4} - 1836802 \beta_{3} + 95660 \beta _1 - 32744394 ) / 54$$ (-151881*b10 + 151881*b9 + 29178*b8 - 29178*b7 - 32913*b6 - 32913*b5 + 1836802*b4 - 1836802*b3 + 95660*b1 - 32744394) / 54 $$\nu^{7}$$ $$=$$ $$( 731688 \beta_{11} + 902903 \beta_{10} + 902903 \beta_{9} - 643402 \beta_{8} - 643402 \beta_{7} - 1956479 \beta_{6} + 1956479 \beta_{5} - 51231530 \beta_{4} - 51231530 \beta_{3} + \cdots + 150200550 ) / 54$$ (731688*b11 + 902903*b10 + 902903*b9 - 643402*b8 - 643402*b7 - 1956479*b6 + 1956479*b5 - 51231530*b4 - 51231530*b3 + 300401100*b2 + 365844*b1 + 150200550) / 54 $$\nu^{8}$$ $$=$$ $$( 22070137 \beta_{10} - 22070137 \beta_{9} - 3058234 \beta_{8} + 3058234 \beta_{7} + 4884461 \beta_{6} + 4884461 \beta_{5} - 291725610 \beta_{4} + 291725610 \beta_{3} + \cdots + 3980601810 ) / 54$$ (22070137*b10 - 22070137*b9 - 3058234*b8 + 3058234*b7 + 4884461*b6 + 4884461*b5 - 291725610*b4 + 291725610*b3 - 11046940*b1 + 3980601810) / 54 $$\nu^{9}$$ $$=$$ $$( - 105940840 \beta_{11} - 105108331 \beta_{10} - 105108331 \beta_{9} + 40008722 \beta_{8} + 40008722 \beta_{7} + 272985031 \beta_{6} - 272985031 \beta_{5} + \cdots - 21849040278 ) / 54$$ (-105940840*b11 - 105108331*b10 - 105108331*b9 + 40008722*b8 + 40008722*b7 + 272985031*b6 - 272985031*b5 + 7145263930*b4 + 7145263930*b3 - 43698080556*b2 - 52970420*b1 - 21849040278) / 54 $$\nu^{10}$$ $$=$$ $$( - 3127309105 \beta_{10} + 3127309105 \beta_{9} + 323447242 \beta_{8} - 323447242 \beta_{7} - 699243761 \beta_{6} - 699243761 \beta_{5} + 43544247842 \beta_{4} + \cdots - 514655128170 ) / 54$$ (-3127309105*b10 + 3127309105*b9 + 323447242*b8 - 323447242*b7 - 699243761*b6 - 699243761*b5 + 43544247842*b4 - 43544247842*b3 + 1368647468*b1 - 514655128170) / 54 $$\nu^{11}$$ $$=$$ $$( 15181664872 \beta_{11} + 13272884855 \beta_{10} + 13272884855 \beta_{9} - 1746634090 \beta_{8} - 1746634090 \beta_{7} - 38118238727 \beta_{6} + \cdots + 3115054097910 ) / 54$$ (15181664872*b11 + 13272884855*b10 + 13272884855*b9 - 1746634090*b8 - 1746634090*b7 - 38118238727*b6 + 38118238727*b5 - 988159363994*b4 - 988159363994*b3 + 6230108195820*b2 + 7590832436*b1 + 3115054097910) / 54

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/54\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$\chi(n)$$ $$-\beta_{2}$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 8.88570i − 3.87527i − 8.15670i 4.28281i − 11.8022i 7.20150i − 8.88570i 3.87527i 8.15670i − 4.28281i 11.8022i − 7.20150i
−4.89898 + 2.82843i 0 16.0000 27.7128i −202.253 116.771i 0 95.5752 + 165.541i 181.019i 0 1321.11
17.2 −4.89898 + 2.82843i 0 16.0000 27.7128i −1.59771 0.922438i 0 6.34411 + 10.9883i 181.019i 0 10.4362
17.3 −4.89898 + 2.82843i 0 16.0000 27.7128i 95.8504 + 55.3393i 0 −163.169 282.617i 181.019i 0 −626.092
17.4 4.89898 2.82843i 0 16.0000 27.7128i −156.951 90.6160i 0 104.306 + 180.663i 181.019i 0 −1025.20
17.5 4.89898 2.82843i 0 16.0000 27.7128i 9.39126 + 5.42205i 0 322.041 + 557.792i 181.019i 0 61.3435
17.6 4.89898 2.82843i 0 16.0000 27.7128i 39.5602 + 22.8401i 0 −245.097 424.521i 181.019i 0 258.406
35.1 −4.89898 2.82843i 0 16.0000 + 27.7128i −202.253 + 116.771i 0 95.5752 165.541i 181.019i 0 1321.11
35.2 −4.89898 2.82843i 0 16.0000 + 27.7128i −1.59771 + 0.922438i 0 6.34411 10.9883i 181.019i 0 10.4362
35.3 −4.89898 2.82843i 0 16.0000 + 27.7128i 95.8504 55.3393i 0 −163.169 + 282.617i 181.019i 0 −626.092
35.4 4.89898 + 2.82843i 0 16.0000 + 27.7128i −156.951 + 90.6160i 0 104.306 180.663i 181.019i 0 −1025.20
35.5 4.89898 + 2.82843i 0 16.0000 + 27.7128i 9.39126 5.42205i 0 322.041 557.792i 181.019i 0 61.3435
35.6 4.89898 + 2.82843i 0 16.0000 + 27.7128i 39.5602 22.8401i 0 −245.097 + 424.521i 181.019i 0 258.406
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 35.6 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 54.7.d.a 12
3.b odd 2 1 18.7.d.a 12
4.b odd 2 1 432.7.q.b 12
9.c even 3 1 18.7.d.a 12
9.c even 3 1 162.7.b.c 12
9.d odd 6 1 inner 54.7.d.a 12
9.d odd 6 1 162.7.b.c 12
12.b even 2 1 144.7.q.c 12
36.f odd 6 1 144.7.q.c 12
36.h even 6 1 432.7.q.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.7.d.a 12 3.b odd 2 1
18.7.d.a 12 9.c even 3 1
54.7.d.a 12 1.a even 1 1 trivial
54.7.d.a 12 9.d odd 6 1 inner
144.7.q.c 12 12.b even 2 1
144.7.q.c 12 36.f odd 6 1
162.7.b.c 12 9.c even 3 1
162.7.b.c 12 9.d odd 6 1
432.7.q.b 12 4.b odd 2 1
432.7.q.b 12 36.h even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(54, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 32 T^{2} + 1024)^{3}$$
$3$ $$T^{12}$$
$5$ $$T^{12} + 432 T^{11} + \cdots + 18\!\cdots\!00$$
$7$ $$T^{12} - 240 T^{11} + \cdots + 27\!\cdots\!00$$
$11$ $$T^{12} + 378 T^{11} + \cdots + 83\!\cdots\!09$$
$13$ $$T^{12} - 1680 T^{11} + \cdots + 98\!\cdots\!16$$
$17$ $$T^{12} + 215347950 T^{10} + \cdots + 64\!\cdots\!00$$
$19$ $$(T^{6} + 1410 T^{5} + \cdots - 71\!\cdots\!00)^{2}$$
$23$ $$T^{12} - 76248 T^{11} + \cdots + 18\!\cdots\!84$$
$29$ $$T^{12} + 97092 T^{11} + \cdots + 50\!\cdots\!00$$
$31$ $$T^{12} - 21480 T^{11} + \cdots + 22\!\cdots\!00$$
$37$ $$(T^{6} + 12768 T^{5} + \cdots + 54\!\cdots\!48)^{2}$$
$41$ $$T^{12} - 410562 T^{11} + \cdots + 10\!\cdots\!25$$
$43$ $$T^{12} - 71430 T^{11} + \cdots + 19\!\cdots\!25$$
$47$ $$T^{12} + 347652 T^{11} + \cdots + 11\!\cdots\!84$$
$53$ $$T^{12} + 154278905256 T^{10} + \cdots + 27\!\cdots\!00$$
$59$ $$T^{12} + 369738 T^{11} + \cdots + 10\!\cdots\!25$$
$61$ $$T^{12} - 135744 T^{11} + \cdots + 10\!\cdots\!04$$
$67$ $$T^{12} + 289938 T^{11} + \cdots + 50\!\cdots\!25$$
$71$ $$T^{12} + 745338015792 T^{10} + \cdots + 58\!\cdots\!76$$
$73$ $$(T^{6} + 488850 T^{5} + \cdots - 49\!\cdots\!60)^{2}$$
$79$ $$T^{12} + 764796 T^{11} + \cdots + 34\!\cdots\!00$$
$83$ $$T^{12} + 396900 T^{11} + \cdots + 42\!\cdots\!36$$
$89$ $$T^{12} + 1318308583464 T^{10} + \cdots + 74\!\cdots\!00$$
$97$ $$T^{12} + 38874 T^{11} + \cdots + 16\!\cdots\!25$$