# Properties

 Label 147.6.e.p Level $147$ Weight $6$ Character orbit 147.e Analytic conductor $23.576$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 147.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.5764215125$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} + 63 x^{10} - 126 x^{9} + 2784 x^{8} - 5290 x^{7} + 62015 x^{6} - 99530 x^{5} + 973971 x^{4} - 1176024 x^{3} + 5644794 x^{2} + 4339328 x + 5466244$$ x^12 - 2*x^11 + 63*x^10 - 126*x^9 + 2784*x^8 - 5290*x^7 + 62015*x^6 - 99530*x^5 + 973971*x^4 - 1176024*x^3 + 5644794*x^2 + 4339328*x + 5466244 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{8}\cdot 7^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + ( - 9 \beta_{2} - 9) q^{3} + (\beta_{7} + \beta_{5} + \beta_{4} - 25 \beta_{2} - 25) q^{4} + (\beta_{9} + \beta_{8} + 2 \beta_{4} + 2 \beta_{3} + 16 \beta_{2} - 2 \beta_1) q^{5} + (9 \beta_{6} + 9 \beta_1) q^{6} + ( - 2 \beta_{11} - 3 \beta_{8} + 2 \beta_{7} + 25 \beta_{6} - 9 \beta_{3} + \cdots - 28) q^{8}+ \cdots + 81 \beta_{2} q^{9}+O(q^{10})$$ q - b1 * q^2 + (-9*b2 - 9) * q^3 + (b7 + b5 + b4 - 25*b2 - 25) * q^4 + (b9 + b8 + 2*b4 + 2*b3 + 16*b2 - 2*b1) * q^5 + (9*b6 + 9*b1) * q^6 + (-2*b11 - 3*b8 + 2*b7 + 25*b6 - 9*b3 + 25*b1 - 28) * q^8 + 81*b2 * q^9 $$q - \beta_1 q^{2} + ( - 9 \beta_{2} - 9) q^{3} + (\beta_{7} + \beta_{5} + \beta_{4} - 25 \beta_{2} - 25) q^{4} + (\beta_{9} + \beta_{8} + 2 \beta_{4} + 2 \beta_{3} + 16 \beta_{2} - 2 \beta_1) q^{5} + (9 \beta_{6} + 9 \beta_1) q^{6} + ( - 2 \beta_{11} - 3 \beta_{8} + 2 \beta_{7} + 25 \beta_{6} - 9 \beta_{3} + \cdots - 28) q^{8}+ \cdots + (567 \beta_{11} - 243 \beta_{8} + 648 \beta_{7} - 972 \beta_{6} + \cdots + 8667) q^{99}+O(q^{100})$$ q - b1 * q^2 + (-9*b2 - 9) * q^3 + (b7 + b5 + b4 - 25*b2 - 25) * q^4 + (b9 + b8 + 2*b4 + 2*b3 + 16*b2 - 2*b1) * q^5 + (9*b6 + 9*b1) * q^6 + (-2*b11 - 3*b8 + 2*b7 + 25*b6 - 9*b3 + 25*b1 - 28) * q^8 + 81*b2 * q^9 + (3*b11 + 3*b10 + 3*b9 + 9*b7 - 15*b6 + 9*b5 + 19*b4 - 138*b2 - 138) * q^10 + (-7*b11 - 7*b10 - 3*b9 - 8*b7 + 12*b6 - 8*b5 - 6*b4 - 107*b2 - 107) * q^11 + (-9*b5 - 9*b4 - 9*b3 + 225*b2) * q^12 + (3*b11 + 2*b8 - 8*b7 + 34*b6 - 12*b3 + 34*b1 + 215) * q^13 + (-9*b8 + 18*b6 - 18*b3 + 18*b1 + 144) * q^15 + (-12*b10 - 18*b9 - 18*b8 - 33*b5 - 27*b4 - 27*b3 + 801*b2 + 126*b1) * q^16 + (-17*b11 - 17*b10 - 7*b9 - 8*b7 + 32*b6 - 8*b5 + 31*b4 - 521*b2 - 521) * q^17 - 81*b6 * q^18 + (3*b10 - 12*b9 - 12*b8 + 24*b5 - 55*b4 - 55*b3 + 231*b2 - 174*b1) * q^19 + (-14*b11 - 5*b8 + 7*b7 + 335*b6 - 100*b3 + 335*b1 - 41) * q^20 + (36*b11 + 18*b8 + 32*b7 - 30*b6 + 238*b3 - 30*b1 - 664) * q^22 + (-51*b10 - 45*b9 - 45*b8 + 24*b5 - 759*b2 + 16*b1) * q^23 + (18*b11 + 18*b10 - 27*b9 - 18*b7 - 225*b6 - 18*b5 - 81*b4 + 252*b2 + 252) * q^24 + (-18*b11 - 18*b10 - 36*b9 + 16*b7 + 228*b6 + 16*b5 - 290*b4 - 883*b2 - 883) * q^25 + (-25*b10 + 9*b9 + 9*b8 - 7*b5 + 45*b4 + 45*b3 + 2206*b2 - 443*b1) * q^26 + 729 * q^27 + (63*b11 - 54*b8 + 24*b7 - 86*b6 + 201*b3 - 86*b1 - 837) * q^29 + (-27*b10 - 27*b9 - 27*b8 - 81*b5 - 171*b4 - 171*b3 + 1242*b2 - 135*b1) * q^30 + (63*b11 + 63*b10 + 88*b9 - 56*b7 + 94*b6 - 56*b5 - 215*b4 - 673*b2 - 673) * q^31 + (26*b11 + 26*b10 - 3*b9 - 128*b7 - 781*b6 - 128*b5 - 687*b4 + 6490*b2 + 6490) * q^32 + (63*b10 + 27*b9 + 27*b8 + 72*b5 + 54*b4 + 54*b3 + 963*b2 + 108*b1) * q^33 + (99*b11 - 25*b8 + 139*b7 + 655*b6 + 501*b3 + 655*b1 - 2908) * q^34 + (-81*b7 + 81*b3 + 2025) * q^36 + (-42*b10 + 54*b9 + 54*b8 - 96*b5 - 300*b4 - 300*b3 + 3758*b2 - 24*b1) * q^37 + (-148*b11 - 148*b10 - 22*b9 + 62*b7 + 318*b6 + 62*b5 + 292*b4 - 8946*b2 - 8946) * q^38 + (-27*b11 - 27*b10 + 18*b9 + 72*b7 - 306*b6 + 72*b5 - 108*b4 - 1935*b2 - 1935) * q^39 + (-30*b10 - 37*b9 - 37*b8 - 277*b5 + 446*b4 + 446*b3 + 17155*b2 + 1195*b1) * q^40 + (-151*b11 + 65*b8 - 136*b7 - 656*b6 - 23*b3 - 656*b1 + 4961) * q^41 + (-150*b11 - 90*b8 + 32*b7 + 120*b6 + 208*b3 + 120*b1 - 1218) * q^43 + (236*b10 + 132*b9 + 132*b8 + 322*b5 - 990*b4 - 990*b3 - 10718*b2 + 80*b1) * q^44 + (-81*b9 - 162*b6 - 162*b4 - 1296*b2 - 1296) * q^45 + (36*b11 + 36*b10 + 54*b9 + 140*b7 + 2394*b6 + 140*b5 - 1498*b4 - 876*b2 - 876) * q^46 + (-47*b10 + 72*b9 + 72*b8 + 184*b5 - 381*b4 - 381*b3 + 8645*b2 + 206*b1) * q^47 + (-108*b11 + 162*b8 - 297*b7 - 1134*b6 + 243*b3 - 1134*b1 + 7209) * q^48 + (-320*b11 + 276*b8 - 100*b7 + 909*b6 + 312*b3 + 909*b1 - 7180) * q^50 + (153*b10 + 63*b9 + 63*b8 + 72*b5 - 279*b4 - 279*b3 + 4689*b2 + 288*b1) * q^51 + (246*b11 + 246*b10 + b9 + 445*b7 - 881*b6 + 445*b5 + 458*b4 - 19507*b2 - 19507) * q^52 + (-58*b11 - 58*b10 + 132*b9 + 208*b7 + 996*b6 + 208*b5 - 1614*b4 - 13832*b2 - 13832) * q^53 - 729*b1 * q^54 + (417*b11 - 536*b8 + 248*b7 - 2686*b6 - 147*b3 - 2686*b1 + 2977) * q^55 + (27*b11 + 108*b8 + 216*b7 + 1566*b6 + 495*b3 + 1566*b1 + 2079) * q^57 + (408*b10 + 162*b9 + 162*b8 + 434*b5 - 2452*b4 - 2452*b3 - 11682*b2 - 222*b1) * q^58 + (-115*b11 - 115*b10 - 100*b9 - 280*b7 + 1794*b6 - 280*b5 - 941*b4 - 2115*b2 - 2115) * q^59 + (126*b11 + 126*b10 - 45*b9 - 63*b7 - 3015*b6 - 63*b5 - 900*b4 + 369*b2 + 369) * q^60 + (-225*b10 - 328*b9 - 328*b8 + 200*b5 - 74*b4 - 74*b3 + 8803*b2 + 1186*b1) * q^61 + (-148*b11 + 302*b8 - 178*b7 - 2918*b6 - 2492*b3 - 2918*b1 + 2894) * q^62 + (396*b8 - 737*b7 - 8580*b6 + 125*b3 - 8580*b1 + 36125) * q^64 + (-205*b10 + 132*b9 + 132*b8 - 392*b5 - 3*b4 - 3*b3 - 467*b2 - 2942*b1) * q^65 + (-324*b11 - 324*b10 + 162*b9 - 288*b7 + 270*b6 - 288*b5 + 2142*b4 + 5976*b2 + 5976) * q^66 + (354*b11 + 354*b10 + 414*b9 - 304*b7 + 2496*b6 - 304*b5 - 148*b4 - 1794*b2 - 1794) * q^67 + (646*b10 + 73*b9 + 73*b8 + 19*b5 - 3712*b4 - 3712*b3 + 7407*b2 + 2901*b1) * q^68 + (-459*b11 + 405*b8 + 216*b7 - 144*b6 - 144*b1 - 6831) * q^69 + (549*b11 - 243*b8 + 120*b7 + 3200*b6 + 2364*b3 + 3200*b1 - 19359) * q^71 + (-162*b10 + 243*b9 + 243*b8 + 162*b5 + 729*b4 + 729*b3 - 2268*b2 - 2025*b1) * q^72 + (138*b11 + 138*b10 + 266*b9 + 32*b7 + 6584*b6 + 32*b5 - 787*b4 - 12854*b2 - 12854) * q^73 + (168*b11 + 168*b10 - 396*b9 + 192*b7 - 7124*b6 + 192*b5 - 324*b4 + 7008*b2 + 7008) * q^74 + (162*b10 + 324*b9 + 324*b8 - 144*b5 + 2610*b4 + 2610*b3 + 7947*b2 + 2052*b1) * q^75 + (432*b11 - 926*b8 + 830*b7 + 8234*b6 + 528*b3 + 8234*b1 - 35078) * q^76 + (-225*b11 - 81*b8 - 63*b7 + 3987*b6 - 405*b3 + 3987*b1 + 19854) * q^78 + (-312*b10 - 468*b9 - 468*b8 + 272*b5 + 3992*b4 + 3992*b3 + 19592*b2 + 3240*b1) * q^79 + (882*b11 + 882*b10 + 45*b9 - 777*b7 - 12525*b6 - 777*b5 - 2424*b4 + 55959*b2 + 55959) * q^80 + (-6561*b2 - 6561) * q^81 + (-819*b10 + 163*b9 + 163*b8 + 37*b5 + 5841*b4 + 5841*b3 - 34012*b2 - 6397*b1) * q^82 + (142*b11 + 764*b8 + 16*b7 - 828*b6 + 622*b3 - 828*b1 + 17178) * q^83 + (888*b11 - 1134*b8 + 976*b7 - 1428*b6 - 514*b3 - 1428*b1 - 3110) * q^85 + (-236*b10 - 96*b9 - 96*b8 - 1444*b5 + 192*b4 + 192*b3 + 1616*b2 + 3172*b1) * q^86 + (-567*b11 - 567*b10 - 486*b9 - 216*b7 + 774*b6 - 216*b5 + 1809*b4 + 7533*b2 + 7533) * q^87 + (-1380*b11 - 1380*b10 - 1026*b9 - 716*b7 + 12162*b6 - 716*b5 + 3034*b4 + 11784*b2 + 11784) * q^88 + (-355*b10 + 169*b9 + 169*b8 - 1000*b5 - 6481*b4 - 6481*b3 - 7483*b2 - 4588*b1) * q^89 + (-243*b11 + 243*b8 - 729*b7 + 1215*b6 + 1539*b3 + 1215*b1 + 11178) * q^90 + (-340*b11 - 240*b8 + 286*b7 + 180*b6 - 6930*b3 + 180*b1 - 75050) * q^92 + (-567*b10 - 792*b9 - 792*b8 + 504*b5 + 1935*b4 + 1935*b3 + 6057*b2 + 846*b1) * q^93 + (-720*b11 - 720*b10 + 106*b9 + 286*b7 - 4154*b6 + 286*b5 + 2584*b4 + 21314*b2 + 21314) * q^94 + (338*b11 + 338*b10 + 366*b9 + 1264*b7 - 1856*b6 + 1264*b5 + 9324*b4 + 20578*b2 + 20578) * q^95 + (-234*b10 + 27*b9 + 27*b8 + 1152*b5 + 6183*b4 + 6183*b3 - 58410*b2 - 7029*b1) * q^96 + (432*b11 - 22*b8 - 32*b7 + 6220*b6 + 6723*b3 + 6220*b1 + 9896) * q^97 + (567*b11 - 243*b8 + 648*b7 - 972*b6 - 486*b3 - 972*b1 + 8667) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 2 q^{2} - 54 q^{3} - 150 q^{4} - 100 q^{5} + 36 q^{6} - 228 q^{8} - 486 q^{9}+O(q^{10})$$ 12 * q - 2 * q^2 - 54 * q^3 - 150 * q^4 - 100 * q^5 + 36 * q^6 - 228 * q^8 - 486 * q^9 $$12 q - 2 q^{2} - 54 q^{3} - 150 q^{4} - 100 q^{5} + 36 q^{6} - 228 q^{8} - 486 q^{9} - 864 q^{10} - 604 q^{11} - 1350 q^{12} + 2704 q^{13} + 1800 q^{15} - 4578 q^{16} - 3028 q^{17} - 162 q^{18} - 1728 q^{19} + 904 q^{20} - 8232 q^{22} + 4484 q^{23} + 1026 q^{24} - 4806 q^{25} - 14172 q^{26} + 8748 q^{27} - 10640 q^{29} - 7776 q^{30} - 3976 q^{31} + 37326 q^{32} - 5436 q^{33} - 32672 q^{34} + 24300 q^{36} - 22680 q^{37} - 52744 q^{38} - 12168 q^{39} - 100600 q^{40} + 57512 q^{41} - 13536 q^{43} + 64940 q^{44} - 8100 q^{45} - 540 q^{46} - 51552 q^{47} + 82404 q^{48} - 81244 q^{50} - 27252 q^{51} - 119296 q^{52} - 80884 q^{53} - 1458 q^{54} + 23312 q^{55} + 31104 q^{57} + 70464 q^{58} - 8872 q^{59} - 4068 q^{60} - 50896 q^{61} + 23648 q^{62} + 399180 q^{64} - 3492 q^{65} + 37044 q^{66} - 6480 q^{67} - 37348 q^{68} - 80712 q^{69} - 221704 q^{71} + 9234 q^{72} - 64232 q^{73} + 27464 q^{74} - 43254 q^{75} - 389728 q^{76} + 255096 q^{78} - 111696 q^{79} + 308940 q^{80} - 39366 q^{81} + 189640 q^{82} + 202256 q^{83} - 46584 q^{85} - 3824 q^{86} + 47880 q^{87} + 97788 q^{88} + 35012 q^{89} + 139968 q^{90} - 898520 q^{92} - 35784 q^{93} + 121016 q^{94} + 119080 q^{95} + 335934 q^{96} + 141904 q^{97} + 97848 q^{99}+O(q^{100})$$ 12 * q - 2 * q^2 - 54 * q^3 - 150 * q^4 - 100 * q^5 + 36 * q^6 - 228 * q^8 - 486 * q^9 - 864 * q^10 - 604 * q^11 - 1350 * q^12 + 2704 * q^13 + 1800 * q^15 - 4578 * q^16 - 3028 * q^17 - 162 * q^18 - 1728 * q^19 + 904 * q^20 - 8232 * q^22 + 4484 * q^23 + 1026 * q^24 - 4806 * q^25 - 14172 * q^26 + 8748 * q^27 - 10640 * q^29 - 7776 * q^30 - 3976 * q^31 + 37326 * q^32 - 5436 * q^33 - 32672 * q^34 + 24300 * q^36 - 22680 * q^37 - 52744 * q^38 - 12168 * q^39 - 100600 * q^40 + 57512 * q^41 - 13536 * q^43 + 64940 * q^44 - 8100 * q^45 - 540 * q^46 - 51552 * q^47 + 82404 * q^48 - 81244 * q^50 - 27252 * q^51 - 119296 * q^52 - 80884 * q^53 - 1458 * q^54 + 23312 * q^55 + 31104 * q^57 + 70464 * q^58 - 8872 * q^59 - 4068 * q^60 - 50896 * q^61 + 23648 * q^62 + 399180 * q^64 - 3492 * q^65 + 37044 * q^66 - 6480 * q^67 - 37348 * q^68 - 80712 * q^69 - 221704 * q^71 + 9234 * q^72 - 64232 * q^73 + 27464 * q^74 - 43254 * q^75 - 389728 * q^76 + 255096 * q^78 - 111696 * q^79 + 308940 * q^80 - 39366 * q^81 + 189640 * q^82 + 202256 * q^83 - 46584 * q^85 - 3824 * q^86 + 47880 * q^87 + 97788 * q^88 + 35012 * q^89 + 139968 * q^90 - 898520 * q^92 - 35784 * q^93 + 121016 * q^94 + 119080 * q^95 + 335934 * q^96 + 141904 * q^97 + 97848 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 63 x^{10} - 126 x^{9} + 2784 x^{8} - 5290 x^{7} + 62015 x^{6} - 99530 x^{5} + 973971 x^{4} - 1176024 x^{3} + 5644794 x^{2} + 4339328 x + 5466244$$ :

 $$\beta_{1}$$ $$=$$ $$( - 22\!\cdots\!63 \nu^{11} + \cdots - 53\!\cdots\!20 ) / 82\!\cdots\!02$$ (-22940144244931089763*v^11 - 3038545450186586446973*v^10 - 26131237974406419382629*v^9 - 177148912886630931899425*v^8 - 1171977996990008298747354*v^7 - 7006947392310859952472388*v^6 - 40788607526986404577984255*v^5 - 131464912894225437940986791*v^4 - 544742107360979765026223277*v^3 - 1593880650876785305074543367*v^2 - 6192628378330469523706273542*v - 5312192750520319842951993220) / 823792125413448096897931002 $$\beta_{2}$$ $$=$$ $$( - 74\!\cdots\!16 \nu^{11} + \cdots - 25\!\cdots\!72 ) / 42\!\cdots\!22$$ (-74017405486250716*v^11 + 208838138667100562*v^10 - 4565510549861301170*v^9 + 12560312084710410247*v^8 - 201467508660928760044*v^7 + 535361150914196233104*v^6 - 4385399005576085002006*v^5 + 10066587304087074527078*v^4 - 68256831693782948072498*v^3 + 144332409961229777524483*v^2 - 345446156158343882363740*v - 256440602205707239215672) / 424416344880704841266322 $$\beta_{3}$$ $$=$$ $$( - 1293273694 \nu^{11} - 2504891733 \nu^{10} - 58398312001 \nu^{9} - 116805928632 \nu^{8} - 3055159408682 \nu^{7} + \cdots - 19\!\cdots\!72 ) / 18\!\cdots\!74$$ (-1293273694*v^11 - 2504891733*v^10 - 58398312001*v^9 - 116805928632*v^8 - 3055159408682*v^7 - 4688491956128*v^6 - 57351081264574*v^5 - 88679977716417*v^4 - 1145670345942085*v^3 - 1591256802005214*v^2 - 1419056815456112*v - 19097668648781072) / 1879904806270074 $$\beta_{4}$$ $$=$$ $$( - 63\!\cdots\!97 \nu^{11} + \cdots + 13\!\cdots\!02 ) / 58\!\cdots\!43$$ (-63425667576406489997*v^11 + 175014220380253722675*v^10 - 3827831175876896702282*v^9 + 9978913591470021635823*v^8 - 154670545214410850870134*v^7 + 424369668155373148326452*v^6 - 3550550855853717430864559*v^5 + 8006942818707963465784836*v^4 - 54594672875142946730522540*v^3 + 113721738065652181131926793*v^2 - 544912962373405029889955293*v + 132903605090650564529783102) / 58842294672389149778423643 $$\beta_{5}$$ $$=$$ $$( 10\!\cdots\!82 \nu^{11} + \cdots - 23\!\cdots\!16 ) / 91\!\cdots\!78$$ (107705297845866006982*v^11 - 2457275938909575867378*v^10 - 1140994831888406684677*v^9 - 166289753272709259584550*v^8 - 61525592714674851748304*v^7 - 5835522746262072258185282*v^6 - 7893798954478637000317922*v^5 - 109555993754531810357411274*v^4 - 108375708398801026426535425*v^3 - 845459525156435556542761116*v^2 - 2584877641169445214629409886*v - 2342735488567530359831294816) / 91532458379272010766436778 $$\beta_{6}$$ $$=$$ $$( - 13\!\cdots\!34 \nu^{11} + \cdots - 12\!\cdots\!62 ) / 82\!\cdots\!02$$ (-1359019473038182889734*v^11 - 752737776898658436101*v^10 - 59325880598205734683578*v^9 + 2919913422284846941259*v^8 - 2083265915013574037130072*v^7 - 155481405079037576662408*v^6 - 20314656155465719692311752*v^5 + 18184256756799931517438425*v^4 - 122160111944138110022270958*v^3 - 241596680310926293314864925*v^2 + 4567357062992053145021959146*v - 120597259734034536956553862) / 823792125413448096897931002 $$\beta_{7}$$ $$=$$ $$( 24\!\cdots\!52 \nu^{11} + \cdots + 13\!\cdots\!64 ) / 54\!\cdots\!34$$ (2464787564900969652*v^11 + 20960090940092719995*v^10 + 147843268240512381655*v^9 + 940176305087010416676*v^8 + 5685852490593409781154*v^7 + 33155097643713254610806*v^6 + 106679246496489729800928*v^5 + 437944638349477269246807*v^4 + 1171343352483340064080327*v^3 + 4994584744207813573132530*v^2 + 4285966128820018824518568*v + 13616207669803588250712164) / 548098553169293477643334 $$\beta_{8}$$ $$=$$ $$( - 26\!\cdots\!60 \nu^{11} + \cdots - 39\!\cdots\!52 ) / 49\!\cdots\!06$$ (-26899152544270397760*v^11 + 33112687019152545439*v^10 - 1638789529801478055059*v^9 + 1348154702247278956700*v^8 - 64265428797222686024158*v^7 + 33903802215030980761240*v^6 - 1206671770264567040390556*v^5 - 428665862085260835685061*v^4 - 12570471057663436642215767*v^3 - 22768554798280861887279958*v^2 - 21189715433454272363164072*v - 396438298318478959057791752) / 4932886978523641298790006 $$\beta_{9}$$ $$=$$ $$( - 31\!\cdots\!85 \nu^{11} + \cdots + 11\!\cdots\!66 ) / 41\!\cdots\!01$$ (-3106417486473775610985*v^11 + 12342303705178089144622*v^10 - 225217127457056777953586*v^9 + 839481933538358644807787*v^8 - 10617174392790390571238974*v^7 + 36165437873058260519733412*v^6 - 266634921403029927539684691*v^5 + 769156302163975883732654245*v^4 - 4416213322544644944647051372*v^3 + 10024044949441703217080053661*v^2 - 27073204355928645599383246603*v + 11445832576094451029609406166) / 411896062706724048448965501 $$\beta_{10}$$ $$=$$ $$( - 86\!\cdots\!74 \nu^{11} + \cdots + 21\!\cdots\!48 ) / 82\!\cdots\!02$$ (-8633853285626504553874*v^11 + 57990014951032465590055*v^10 - 313892105838838737820431*v^9 + 3547878248595484307793965*v^8 - 13657167348010097631902730*v^7 + 140595376219660688131526312*v^6 - 135242794874950703001540196*v^5 + 2640684563630434993444170109*v^4 - 3105134903031672206984220591*v^3 + 29260859512314453916039824863*v^2 + 19691445421164430756419627582*v + 21987494373188120483867077148) / 823792125413448096897931002 $$\beta_{11}$$ $$=$$ $$( - 66\!\cdots\!56 \nu^{11} + \cdots - 38\!\cdots\!66 ) / 49\!\cdots\!06$$ (-66189921555356533556*v^11 - 300073551311318479339*v^10 - 4062290121503241597705*v^9 - 13597585346057218772708*v^8 - 154910563217250766009794*v^7 - 531133525638151136982128*v^6 - 2907379468371290522165408*v^5 - 7394332013666855949510535*v^4 - 28732598538727146686894997*v^3 - 102273060300388923078867602*v^2 - 89420227193411237320762152*v - 38056249342653968586186866) / 4932886978523641298790006
 $$\nu$$ $$=$$ $$( -\beta_{11} - \beta_{10} + \beta_{9} + 12\beta_{6} - 4\beta_{4} + 5\beta_{2} + 5 ) / 28$$ (-b11 - b10 + b9 + 12*b6 - 4*b4 + 5*b2 + 5) / 28 $$\nu^{2}$$ $$=$$ $$( -3\beta_{10} - 5\beta_{9} - 5\beta_{8} - 4\beta_{5} - 4\beta_{4} - 4\beta_{3} + 567\beta_{2} - 4\beta_1 ) / 28$$ (-3*b10 - 5*b9 - 5*b8 - 4*b5 - 4*b4 - 4*b3 + 567*b2 - 4*b1) / 28 $$\nu^{3}$$ $$=$$ $$( 29\beta_{11} + 28\beta_{8} + 4\beta_{7} - 302\beta_{6} - 25\beta_{3} - 302\beta _1 + 129 ) / 28$$ (29*b11 + 28*b8 + 4*b7 - 302*b6 - 25*b3 - 302*b1 + 129) / 28 $$\nu^{4}$$ $$=$$ $$( 99 \beta_{11} + 99 \beta_{10} + 155 \beta_{9} + 176 \beta_{7} + 80 \beta_{6} + 176 \beta_{5} - 74 \beta_{4} - 14843 \beta_{2} - 14843 ) / 28$$ (99*b11 + 99*b10 + 155*b9 + 176*b7 + 80*b6 + 176*b5 - 74*b4 - 14843*b2 - 14843) / 28 $$\nu^{5}$$ $$=$$ $$( 862 \beta_{10} - 737 \beta_{9} - 737 \beta_{8} + 192 \beta_{5} - 643 \beta_{4} - 643 \beta_{3} + 9528 \beta_{2} + 8710 \beta_1 ) / 28$$ (862*b10 - 737*b9 - 737*b8 + 192*b5 - 643*b4 - 643*b3 + 9528*b2 + 8710*b1) / 28 $$\nu^{6}$$ $$=$$ $$( -1530\beta_{11} + 2263\beta_{8} - 3258\beta_{7} - 2818\beta_{6} - 2521\beta_{3} - 2818\beta _1 + 206808 ) / 14$$ (-1530*b11 + 2263*b8 - 3258*b7 - 2818*b6 - 2521*b3 - 2818*b1 + 206808) / 14 $$\nu^{7}$$ $$=$$ $$( - 25517 \beta_{11} - 25517 \beta_{10} + 20051 \beta_{9} - 6548 \beta_{7} + 263676 \beta_{6} - 6548 \beta_{5} + 45742 \beta_{4} - 360335 \beta_{2} - 360335 ) / 28$$ (-25517*b11 - 25517*b10 + 20051*b9 - 6548*b7 + 263676*b6 - 6548*b5 + 45742*b4 - 360335*b2 - 360335) / 28 $$\nu^{8}$$ $$=$$ $$( - 94089 \beta_{10} - 130713 \beta_{9} - 130713 \beta_{8} - 225276 \beta_{5} + 185130 \beta_{4} + 185130 \beta_{3} + 11964049 \beta_{2} + 222648 \beta_1 ) / 28$$ (-94089*b10 - 130713*b9 - 130713*b8 - 225276*b5 + 185130*b4 + 185130*b3 + 11964049*b2 + 222648*b1) / 28 $$\nu^{9}$$ $$=$$ $$( 760765 \beta_{11} + 558022 \beta_{8} + 197268 \beta_{7} - 8160666 \beta_{6} + 1927745 \beta_{3} - 8160666 \beta _1 + 11579965 ) / 28$$ (760765*b11 + 558022*b8 + 197268*b7 - 8160666*b6 + 1927745*b3 - 8160666*b1 + 11579965) / 28 $$\nu^{10}$$ $$=$$ $$( 2911581 \beta_{11} + 2911581 \beta_{10} + 3777797 \beta_{9} + 7523248 \beta_{7} + 7701176 \beta_{6} + 7523248 \beta_{5} - 5819534 \beta_{4} - 354486141 \beta_{2} + \cdots - 354486141 ) / 28$$ (2911581*b11 + 2911581*b10 + 3777797*b9 + 7523248*b7 + 7701176*b6 + 7523248*b5 - 5819534*b4 - 354486141*b2 - 354486141) / 28 $$\nu^{11}$$ $$=$$ $$( 22912658 \beta_{10} - 15769549 \beta_{9} - 15769549 \beta_{8} + 5582560 \beta_{5} - 70608227 \beta_{4} - 70608227 \beta_{3} + 349389564 \beta_{2} + 255024446 \beta_1 ) / 28$$ (22912658*b10 - 15769549*b9 - 15769549*b8 + 5582560*b5 - 70608227*b4 - 70608227*b3 + 349389564*b2 + 255024446*b1) / 28

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$$.

 $$n$$ $$50$$ $$52$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 2.79840 − 4.84697i 1.87676 − 3.25065i −2.55045 + 4.41750i 2.13607 − 3.69977i −0.455061 + 0.788188i −2.80572 + 4.85966i 2.79840 + 4.84697i 1.87676 + 3.25065i −2.55045 − 4.41750i 2.13607 + 3.69977i −0.455061 − 0.788188i −2.80572 − 4.85966i
−5.00445 8.66795i −4.50000 + 7.79423i −34.0890 + 59.0438i −35.1756 60.9260i 90.0800 0 362.101 −40.5000 70.1481i −352.069 + 609.801i
67.2 −4.10431 7.10888i −4.50000 + 7.79423i −17.6908 + 30.6414i −14.6130 25.3104i 73.8777 0 27.7583 −40.5000 70.1481i −119.952 + 207.764i
67.3 −1.69017 2.92745i −4.50000 + 7.79423i 10.2867 17.8171i 27.2626 + 47.2203i 30.4230 0 −177.715 −40.5000 70.1481i 92.1567 159.620i
67.4 1.54581 + 2.67743i −4.50000 + 7.79423i 11.2209 19.4352i −6.89630 11.9448i −27.8246 0 168.314 −40.5000 70.1481i 21.3208 36.9287i
67.5 2.65908 + 4.60566i −4.50000 + 7.79423i 1.85862 3.21922i −51.7353 89.6081i −47.8634 0 189.950 −40.5000 70.1481i 275.136 476.550i
67.6 5.59404 + 9.68915i −4.50000 + 7.79423i −46.5865 + 80.6901i 31.1575 + 53.9664i −100.693 0 −684.407 −40.5000 70.1481i −348.592 + 603.780i
79.1 −5.00445 + 8.66795i −4.50000 7.79423i −34.0890 59.0438i −35.1756 + 60.9260i 90.0800 0 362.101 −40.5000 + 70.1481i −352.069 609.801i
79.2 −4.10431 + 7.10888i −4.50000 7.79423i −17.6908 30.6414i −14.6130 + 25.3104i 73.8777 0 27.7583 −40.5000 + 70.1481i −119.952 207.764i
79.3 −1.69017 + 2.92745i −4.50000 7.79423i 10.2867 + 17.8171i 27.2626 47.2203i 30.4230 0 −177.715 −40.5000 + 70.1481i 92.1567 + 159.620i
79.4 1.54581 2.67743i −4.50000 7.79423i 11.2209 + 19.4352i −6.89630 + 11.9448i −27.8246 0 168.314 −40.5000 + 70.1481i 21.3208 + 36.9287i
79.5 2.65908 4.60566i −4.50000 7.79423i 1.85862 + 3.21922i −51.7353 + 89.6081i −47.8634 0 189.950 −40.5000 + 70.1481i 275.136 + 476.550i
79.6 5.59404 9.68915i −4.50000 7.79423i −46.5865 80.6901i 31.1575 53.9664i −100.693 0 −684.407 −40.5000 + 70.1481i −348.592 603.780i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 79.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.6.e.p 12
7.b odd 2 1 147.6.e.q 12
7.c even 3 1 147.6.a.o yes 6
7.c even 3 1 inner 147.6.e.p 12
7.d odd 6 1 147.6.a.n 6
7.d odd 6 1 147.6.e.q 12
21.g even 6 1 441.6.a.bb 6
21.h odd 6 1 441.6.a.ba 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.a.n 6 7.d odd 6 1
147.6.a.o yes 6 7.c even 3 1
147.6.e.p 12 1.a even 1 1 trivial
147.6.e.p 12 7.c even 3 1 inner
147.6.e.q 12 7.b odd 2 1
147.6.e.q 12 7.d odd 6 1
441.6.a.ba 6 21.h odd 6 1
441.6.a.bb 6 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}^{12} + 2 T_{2}^{11} + 173 T_{2}^{10} + 334 T_{2}^{9} + 22761 T_{2}^{8} + 35152 T_{2}^{7} + 1095976 T_{2}^{6} - 838240 T_{2}^{5} + 34682928 T_{2}^{4} - 6786304 T_{2}^{3} + 348755072 T_{2}^{2} + \cdots + 2609983744$$ T2^12 + 2*T2^11 + 173*T2^10 + 334*T2^9 + 22761*T2^8 + 35152*T2^7 + 1095976*T2^6 - 838240*T2^5 + 34682928*T2^4 - 6786304*T2^3 + 348755072*T2^2 - 217430528*T2 + 2609983744 $$T_{5}^{12} + 100 T_{5}^{11} + 16778 T_{5}^{10} + 624824 T_{5}^{9} + 101403192 T_{5}^{8} + 3439946000 T_{5}^{7} + 413764540744 T_{5}^{6} + 7702343637280 T_{5}^{5} + \cdots + 99\!\cdots\!44$$ T5^12 + 100*T5^11 + 16778*T5^10 + 624824*T5^9 + 101403192*T5^8 + 3439946000*T5^7 + 413764540744*T5^6 + 7702343637280*T5^5 + 785817679635168*T5^4 + 22262009855728576*T5^3 + 823684539529925792*T5^2 + 9563304139310807168*T5 + 99400148502700385344

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 2 T^{11} + \cdots + 2609983744$$
$3$ $$(T^{2} + 9 T + 81)^{6}$$
$5$ $$T^{12} + 100 T^{11} + \cdots + 99\!\cdots\!44$$
$7$ $$T^{12}$$
$11$ $$T^{12} + 604 T^{11} + \cdots + 30\!\cdots\!16$$
$13$ $$(T^{6} - 1352 T^{5} + \cdots + 88\!\cdots\!04)^{2}$$
$17$ $$T^{12} + 3028 T^{11} + \cdots + 60\!\cdots\!44$$
$19$ $$T^{12} + 1728 T^{11} + \cdots + 13\!\cdots\!24$$
$23$ $$T^{12} - 4484 T^{11} + \cdots + 11\!\cdots\!04$$
$29$ $$(T^{6} + 5320 T^{5} + \cdots - 45\!\cdots\!84)^{2}$$
$31$ $$T^{12} + 3976 T^{11} + \cdots + 10\!\cdots\!16$$
$37$ $$T^{12} + 22680 T^{11} + \cdots + 63\!\cdots\!16$$
$41$ $$(T^{6} - 28756 T^{5} + \cdots + 18\!\cdots\!56)^{2}$$
$43$ $$(T^{6} + 6768 T^{5} + \cdots - 28\!\cdots\!68)^{2}$$
$47$ $$T^{12} + 51552 T^{11} + \cdots + 16\!\cdots\!84$$
$53$ $$T^{12} + 80884 T^{11} + \cdots + 87\!\cdots\!44$$
$59$ $$T^{12} + 8872 T^{11} + \cdots + 11\!\cdots\!04$$
$61$ $$T^{12} + 50896 T^{11} + \cdots + 34\!\cdots\!96$$
$67$ $$T^{12} + 6480 T^{11} + \cdots + 62\!\cdots\!24$$
$71$ $$(T^{6} + 110852 T^{5} + \cdots - 29\!\cdots\!48)^{2}$$
$73$ $$T^{12} + 64232 T^{11} + \cdots + 10\!\cdots\!56$$
$79$ $$T^{12} + 111696 T^{11} + \cdots + 10\!\cdots\!24$$
$83$ $$(T^{6} - 101128 T^{5} + \cdots - 19\!\cdots\!68)^{2}$$
$89$ $$T^{12} - 35012 T^{11} + \cdots + 23\!\cdots\!84$$
$97$ $$(T^{6} - 70952 T^{5} + \cdots - 13\!\cdots\!88)^{2}$$