# Properties

 Label 147.4.e.n Level $147$ Weight $4$ Character orbit 147.e Analytic conductor $8.673$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,4,Mod(67,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("147.67");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 147.e (of order $$3$$, 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: $$8.67328077084$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.9924270768.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + 25x^{4} + 12x^{3} + 582x^{2} - 144x + 36$$ x^6 - x^5 + 25*x^4 + 12*x^3 + 582*x^2 - 144*x + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (3 \beta_{4} - 3) q^{3} + (\beta_{5} + 8 \beta_{4} + \beta_{2} + \beta_1 - 8) q^{4} + (\beta_{5} + 4 \beta_{4} - \beta_{3} - \beta_1) q^{5} - 3 \beta_{2} q^{6} + ( - \beta_{3} - 9 \beta_{2} + 10) q^{8} - 9 \beta_{4} q^{9}+O(q^{10})$$ q - b1 * q^2 + (3*b4 - 3) * q^3 + (b5 + 8*b4 + b2 + b1 - 8) * q^4 + (b5 + 4*b4 - b3 - b1) * q^5 - 3*b2 * q^6 + (-b3 - 9*b2 + 10) * q^8 - 9*b4 * q^9 $$q - \beta_1 q^{2} + (3 \beta_{4} - 3) q^{3} + (\beta_{5} + 8 \beta_{4} + \beta_{2} + \beta_1 - 8) q^{4} + (\beta_{5} + 4 \beta_{4} - \beta_{3} - \beta_1) q^{5} - 3 \beta_{2} q^{6} + ( - \beta_{3} - 9 \beta_{2} + 10) q^{8} - 9 \beta_{4} q^{9} + (\beta_{5} + 22 \beta_{4} - 11 \beta_{2} - 11 \beta_1 - 22) q^{10} + ( - 3 \beta_{5} + 12 \beta_{4} - \beta_{2} - \beta_1 - 12) q^{11} + ( - 3 \beta_{5} - 24 \beta_{4} + 3 \beta_{3} - 3 \beta_1) q^{12} + ( - \beta_{3} + 5 \beta_{2} - 19) q^{13} + (3 \beta_{3} - 3 \beta_{2} - 12) q^{15} + ( - \beta_{5} - 74 \beta_{4} + \beta_{3} - 19 \beta_1) q^{16} + ( - 4 \beta_{5} - 16 \beta_{4} + 16) q^{17} + (9 \beta_{2} + 9 \beta_1) q^{18} + ( - \beta_{5} - 65 \beta_{4} + \beta_{3} - 7 \beta_1) q^{19} + (3 \beta_{3} - 11 \beta_{2} - 150) q^{20} + (\beta_{3} + 13 \beta_{2} + 2) q^{22} + (4 \beta_{5} - 80 \beta_{4} - 4 \beta_{3} + 24 \beta_1) q^{23} + (3 \beta_{5} + 30 \beta_{4} + 27 \beta_{2} + 27 \beta_1 - 30) q^{24} + (\beta_{5} + 53 \beta_{4} - 29 \beta_{2} - 29 \beta_1 - 53) q^{25} + (5 \beta_{5} + 86 \beta_{4} - 5 \beta_{3} + 16 \beta_1) q^{26} + 27 q^{27} + ( - 5 \beta_{3} + 25 \beta_{2} + 26) q^{29} + ( - 3 \beta_{5} - 66 \beta_{4} + 3 \beta_{3} + 33 \beta_1) q^{30} + (2 \beta_{5} + 39 \beta_{4} - 22 \beta_{2} - 22 \beta_1 - 39) q^{31} + (11 \beta_{5} + 218 \beta_{4} + 29 \beta_{2} + 29 \beta_1 - 218) q^{32} + (9 \beta_{5} - 36 \beta_{4} - 9 \beta_{3} + 3 \beta_1) q^{33} + (48 \beta_{2} + 24) q^{34} + ( - 9 \beta_{3} - 9 \beta_{2} + 72) q^{36} + ( - \beta_{5} - 81 \beta_{4} + \beta_{3} - 19 \beta_1) q^{37} + (7 \beta_{5} + 106 \beta_{4} + 80 \beta_{2} + 80 \beta_1 - 106) q^{38} + (3 \beta_{5} - 57 \beta_{4} - 15 \beta_{2} - 15 \beta_1 + 57) q^{39} + ( - 3 \beta_{5} - 18 \beta_{4} + 3 \beta_{3} + 75 \beta_1) q^{40} + ( - 14 \beta_{3} - 2 \beta_{2} - 82) q^{41} + (3 \beta_{3} + 69 \beta_{2} + 143) q^{43} + ( - 11 \beta_{5} + 298 \beta_{4} + 11 \beta_{3} + 11 \beta_1) q^{44} + ( - 9 \beta_{5} - 36 \beta_{4} + 9 \beta_{2} + 9 \beta_1 + 36) q^{45} + ( - 24 \beta_{5} - 360 \beta_{4} + 24 \beta_{2} + 24 \beta_1 + 360) q^{46} + ( - 28 \beta_{5} - 46 \beta_{4} + 28 \beta_{3} - 72 \beta_1) q^{47} + ( - 3 \beta_{3} - 57 \beta_{2} + 222) q^{48} + (29 \beta_{3} - 32 \beta_{2} - 470) q^{50} + (12 \beta_{5} + 48 \beta_{4} - 12 \beta_{3}) q^{51} + ( - 24 \beta_{5} - 74 \beta_{4} - 102 \beta_{2} - 102 \beta_1 + 74) q^{52} + ( - 11 \beta_{5} + 154 \beta_{4} - 69 \beta_{2} - 69 \beta_1 - 154) q^{53} - 27 \beta_1 q^{54} + (25 \beta_{3} + 19 \beta_{2} + 350) q^{55} + ( - 3 \beta_{3} - 21 \beta_{2} + 195) q^{57} + (25 \beta_{5} + 430 \beta_{4} - 25 \beta_{3} - 41 \beta_1) q^{58} + (29 \beta_{5} - 358 \beta_{4} - 69 \beta_{2} - 69 \beta_1 + 358) q^{59} + ( - 9 \beta_{5} - 450 \beta_{4} + 33 \beta_{2} + 33 \beta_1 + 450) q^{60} + (20 \beta_{5} + 10 \beta_{4} - 20 \beta_{3} - 100 \beta_1) q^{61} + (22 \beta_{3} - 33 \beta_{2} - 364) q^{62} + ( - 21 \beta_{3} - 183 \beta_{2} - 194) q^{64} + ( - 18 \beta_{5} + 174 \beta_{4} + 18 \beta_{3} - 50 \beta_1) q^{65} + ( - 3 \beta_{5} + 6 \beta_{4} - 39 \beta_{2} - 39 \beta_1 - 6) q^{66} + (47 \beta_{5} - 215 \beta_{4} + 17 \beta_{2} + 17 \beta_1 + 215) q^{67} + (16 \beta_{5} + 640 \beta_{4} - 16 \beta_{3} + 24 \beta_1) q^{68} + (12 \beta_{3} + 72 \beta_{2} + 240) q^{69} + ( - 12 \beta_{3} - 120 \beta_{2} + 66) q^{71} + ( - 9 \beta_{5} - 90 \beta_{4} + 9 \beta_{3} - 81 \beta_1) q^{72} + (23 \beta_{5} - 363 \beta_{4} + 101 \beta_{2} + 101 \beta_1 + 363) q^{73} + (19 \beta_{5} + 298 \beta_{4} + 108 \beta_{2} + 108 \beta_1 - 298) q^{74} + ( - 3 \beta_{5} - 159 \beta_{4} + 3 \beta_{3} + 87 \beta_1) q^{75} + ( - 72 \beta_{3} - 186 \beta_{2} + 718) q^{76} + (15 \beta_{3} + 48 \beta_{2} - 258) q^{78} + ( - 48 \beta_{5} - 299 \beta_{4} + 48 \beta_{3} + 36 \beta_1) q^{79} + ( - 51 \beta_{5} - 18 \beta_{4} - 121 \beta_{2} - 121 \beta_1 + 18) q^{80} + (81 \beta_{4} - 81) q^{81} + ( - 2 \beta_{5} + 52 \beta_{4} + 2 \beta_{3} - 32 \beta_1) q^{82} + ( - 27 \beta_{3} + 51 \beta_{2} - 156) q^{83} + (72 \beta_{2} + 624) q^{85} + (69 \beta_{5} + 1086 \beta_{4} - 69 \beta_{3} - 50 \beta_1) q^{86} + (15 \beta_{5} + 78 \beta_{4} - 75 \beta_{2} - 75 \beta_1 - 78) q^{87} + ( - 3 \beta_{5} - 258 \beta_{4} - 117 \beta_{2} - 117 \beta_1 + 258) q^{88} + (22 \beta_{5} + 532 \beta_{4} - 22 \beta_{3} + 170 \beta_1) q^{89} + ( - 9 \beta_{3} + 99 \beta_{2} + 198) q^{90} + ( - 56 \beta_{3} + 336 \beta_{2} - 112) q^{92} + ( - 6 \beta_{5} - 117 \beta_{4} + 6 \beta_{3} + 66 \beta_1) q^{93} + (72 \beta_{5} + 984 \beta_{4} + 342 \beta_{2} + 342 \beta_1 - 984) q^{94} + ( - 54 \beta_{5} - 246 \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 246) q^{95} + ( - 33 \beta_{5} - 654 \beta_{4} + 33 \beta_{3} - 87 \beta_1) q^{96} + (49 \beta_{3} - 53 \beta_{2} - 24) q^{97} + (27 \beta_{3} + 9 \beta_{2} + 108) q^{99}+O(q^{100})$$ q - b1 * q^2 + (3*b4 - 3) * q^3 + (b5 + 8*b4 + b2 + b1 - 8) * q^4 + (b5 + 4*b4 - b3 - b1) * q^5 - 3*b2 * q^6 + (-b3 - 9*b2 + 10) * q^8 - 9*b4 * q^9 + (b5 + 22*b4 - 11*b2 - 11*b1 - 22) * q^10 + (-3*b5 + 12*b4 - b2 - b1 - 12) * q^11 + (-3*b5 - 24*b4 + 3*b3 - 3*b1) * q^12 + (-b3 + 5*b2 - 19) * q^13 + (3*b3 - 3*b2 - 12) * q^15 + (-b5 - 74*b4 + b3 - 19*b1) * q^16 + (-4*b5 - 16*b4 + 16) * q^17 + (9*b2 + 9*b1) * q^18 + (-b5 - 65*b4 + b3 - 7*b1) * q^19 + (3*b3 - 11*b2 - 150) * q^20 + (b3 + 13*b2 + 2) * q^22 + (4*b5 - 80*b4 - 4*b3 + 24*b1) * q^23 + (3*b5 + 30*b4 + 27*b2 + 27*b1 - 30) * q^24 + (b5 + 53*b4 - 29*b2 - 29*b1 - 53) * q^25 + (5*b5 + 86*b4 - 5*b3 + 16*b1) * q^26 + 27 * q^27 + (-5*b3 + 25*b2 + 26) * q^29 + (-3*b5 - 66*b4 + 3*b3 + 33*b1) * q^30 + (2*b5 + 39*b4 - 22*b2 - 22*b1 - 39) * q^31 + (11*b5 + 218*b4 + 29*b2 + 29*b1 - 218) * q^32 + (9*b5 - 36*b4 - 9*b3 + 3*b1) * q^33 + (48*b2 + 24) * q^34 + (-9*b3 - 9*b2 + 72) * q^36 + (-b5 - 81*b4 + b3 - 19*b1) * q^37 + (7*b5 + 106*b4 + 80*b2 + 80*b1 - 106) * q^38 + (3*b5 - 57*b4 - 15*b2 - 15*b1 + 57) * q^39 + (-3*b5 - 18*b4 + 3*b3 + 75*b1) * q^40 + (-14*b3 - 2*b2 - 82) * q^41 + (3*b3 + 69*b2 + 143) * q^43 + (-11*b5 + 298*b4 + 11*b3 + 11*b1) * q^44 + (-9*b5 - 36*b4 + 9*b2 + 9*b1 + 36) * q^45 + (-24*b5 - 360*b4 + 24*b2 + 24*b1 + 360) * q^46 + (-28*b5 - 46*b4 + 28*b3 - 72*b1) * q^47 + (-3*b3 - 57*b2 + 222) * q^48 + (29*b3 - 32*b2 - 470) * q^50 + (12*b5 + 48*b4 - 12*b3) * q^51 + (-24*b5 - 74*b4 - 102*b2 - 102*b1 + 74) * q^52 + (-11*b5 + 154*b4 - 69*b2 - 69*b1 - 154) * q^53 - 27*b1 * q^54 + (25*b3 + 19*b2 + 350) * q^55 + (-3*b3 - 21*b2 + 195) * q^57 + (25*b5 + 430*b4 - 25*b3 - 41*b1) * q^58 + (29*b5 - 358*b4 - 69*b2 - 69*b1 + 358) * q^59 + (-9*b5 - 450*b4 + 33*b2 + 33*b1 + 450) * q^60 + (20*b5 + 10*b4 - 20*b3 - 100*b1) * q^61 + (22*b3 - 33*b2 - 364) * q^62 + (-21*b3 - 183*b2 - 194) * q^64 + (-18*b5 + 174*b4 + 18*b3 - 50*b1) * q^65 + (-3*b5 + 6*b4 - 39*b2 - 39*b1 - 6) * q^66 + (47*b5 - 215*b4 + 17*b2 + 17*b1 + 215) * q^67 + (16*b5 + 640*b4 - 16*b3 + 24*b1) * q^68 + (12*b3 + 72*b2 + 240) * q^69 + (-12*b3 - 120*b2 + 66) * q^71 + (-9*b5 - 90*b4 + 9*b3 - 81*b1) * q^72 + (23*b5 - 363*b4 + 101*b2 + 101*b1 + 363) * q^73 + (19*b5 + 298*b4 + 108*b2 + 108*b1 - 298) * q^74 + (-3*b5 - 159*b4 + 3*b3 + 87*b1) * q^75 + (-72*b3 - 186*b2 + 718) * q^76 + (15*b3 + 48*b2 - 258) * q^78 + (-48*b5 - 299*b4 + 48*b3 + 36*b1) * q^79 + (-51*b5 - 18*b4 - 121*b2 - 121*b1 + 18) * q^80 + (81*b4 - 81) * q^81 + (-2*b5 + 52*b4 + 2*b3 - 32*b1) * q^82 + (-27*b3 + 51*b2 - 156) * q^83 + (72*b2 + 624) * q^85 + (69*b5 + 1086*b4 - 69*b3 - 50*b1) * q^86 + (15*b5 + 78*b4 - 75*b2 - 75*b1 - 78) * q^87 + (-3*b5 - 258*b4 - 117*b2 - 117*b1 + 258) * q^88 + (22*b5 + 532*b4 - 22*b3 + 170*b1) * q^89 + (-9*b3 + 99*b2 + 198) * q^90 + (-56*b3 + 336*b2 - 112) * q^92 + (-6*b5 - 117*b4 + 6*b3 + 66*b1) * q^93 + (72*b5 + 984*b4 + 342*b2 + 342*b1 - 984) * q^94 + (-54*b5 - 246*b4 + 2*b2 + 2*b1 + 246) * q^95 + (-33*b5 - 654*b4 + 33*b3 - 87*b1) * q^96 + (49*b3 - 53*b2 - 24) * q^97 + (27*b3 + 9*b2 + 108) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - q^{2} - 9 q^{3} - 25 q^{4} + 11 q^{5} + 6 q^{6} + 78 q^{8} - 27 q^{9}+O(q^{10})$$ 6 * q - q^2 - 9 * q^3 - 25 * q^4 + 11 * q^5 + 6 * q^6 + 78 * q^8 - 27 * q^9 $$6 q - q^{2} - 9 q^{3} - 25 q^{4} + 11 q^{5} + 6 q^{6} + 78 q^{8} - 27 q^{9} - 55 q^{10} - 35 q^{11} - 75 q^{12} - 124 q^{13} - 66 q^{15} - 241 q^{16} + 48 q^{17} - 9 q^{18} - 202 q^{19} - 878 q^{20} - 14 q^{22} - 216 q^{23} - 117 q^{24} - 130 q^{25} + 274 q^{26} + 162 q^{27} + 106 q^{29} - 165 q^{30} - 95 q^{31} - 683 q^{32} - 105 q^{33} + 48 q^{34} + 450 q^{36} - 262 q^{37} - 398 q^{38} + 186 q^{39} + 21 q^{40} - 488 q^{41} + 720 q^{43} + 905 q^{44} + 99 q^{45} + 1056 q^{46} - 210 q^{47} + 1446 q^{48} - 2756 q^{50} + 144 q^{51} + 324 q^{52} - 393 q^{53} - 27 q^{54} + 2062 q^{55} + 1212 q^{57} + 1249 q^{58} + 1143 q^{59} + 1317 q^{60} - 70 q^{61} - 2118 q^{62} - 798 q^{64} + 472 q^{65} + 21 q^{66} + 628 q^{67} + 1944 q^{68} + 1296 q^{69} + 636 q^{71} - 351 q^{72} + 988 q^{73} - 1002 q^{74} - 390 q^{75} + 4680 q^{76} - 1644 q^{78} - 861 q^{79} + 175 q^{80} - 243 q^{81} + 124 q^{82} - 1038 q^{83} + 3600 q^{85} + 3208 q^{86} - 159 q^{87} + 891 q^{88} + 1766 q^{89} + 990 q^{90} - 1344 q^{92} - 285 q^{93} - 3294 q^{94} + 736 q^{95} - 2049 q^{96} - 38 q^{97} + 630 q^{99}+O(q^{100})$$ 6 * q - q^2 - 9 * q^3 - 25 * q^4 + 11 * q^5 + 6 * q^6 + 78 * q^8 - 27 * q^9 - 55 * q^10 - 35 * q^11 - 75 * q^12 - 124 * q^13 - 66 * q^15 - 241 * q^16 + 48 * q^17 - 9 * q^18 - 202 * q^19 - 878 * q^20 - 14 * q^22 - 216 * q^23 - 117 * q^24 - 130 * q^25 + 274 * q^26 + 162 * q^27 + 106 * q^29 - 165 * q^30 - 95 * q^31 - 683 * q^32 - 105 * q^33 + 48 * q^34 + 450 * q^36 - 262 * q^37 - 398 * q^38 + 186 * q^39 + 21 * q^40 - 488 * q^41 + 720 * q^43 + 905 * q^44 + 99 * q^45 + 1056 * q^46 - 210 * q^47 + 1446 * q^48 - 2756 * q^50 + 144 * q^51 + 324 * q^52 - 393 * q^53 - 27 * q^54 + 2062 * q^55 + 1212 * q^57 + 1249 * q^58 + 1143 * q^59 + 1317 * q^60 - 70 * q^61 - 2118 * q^62 - 798 * q^64 + 472 * q^65 + 21 * q^66 + 628 * q^67 + 1944 * q^68 + 1296 * q^69 + 636 * q^71 - 351 * q^72 + 988 * q^73 - 1002 * q^74 - 390 * q^75 + 4680 * q^76 - 1644 * q^78 - 861 * q^79 + 175 * q^80 - 243 * q^81 + 124 * q^82 - 1038 * q^83 + 3600 * q^85 + 3208 * q^86 - 159 * q^87 + 891 * q^88 + 1766 * q^89 + 990 * q^90 - 1344 * q^92 - 285 * q^93 - 3294 * q^94 + 736 * q^95 - 2049 * q^96 - 38 * q^97 + 630 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 25x^{4} + 12x^{3} + 582x^{2} - 144x + 36$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} - 25\nu^{4} + 625\nu^{3} - 582\nu^{2} + 144\nu - 3600 ) / 14406$$ (v^5 - 25*v^4 + 625*v^3 - 582*v^2 + 144*v - 3600) / 14406 $$\beta_{3}$$ $$=$$ $$( -25\nu^{5} + 625\nu^{4} - 1219\nu^{3} + 14550\nu^{2} - 3600\nu + 234060 ) / 14406$$ (-25*v^5 + 625*v^4 - 1219*v^3 + 14550*v^2 - 3600*v + 234060) / 14406 $$\beta_{4}$$ $$=$$ $$( 100\nu^{5} - 99\nu^{4} + 2475\nu^{3} + 1825\nu^{2} + 57618\nu + 150 ) / 14406$$ (100*v^5 - 99*v^4 + 2475*v^3 + 1825*v^2 + 57618*v + 150) / 14406 $$\beta_{5}$$ $$=$$ $$( -1601\nu^{5} + 1609\nu^{4} - 40225\nu^{3} - 14212\nu^{2} - 936438\nu + 231696 ) / 14406$$ (-1601*v^5 + 1609*v^4 - 40225*v^3 - 14212*v^2 - 936438*v + 231696) / 14406
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 16\beta_{4} + \beta_{2} + \beta _1 - 16$$ b5 + 16*b4 + b2 + b1 - 16 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 25\beta_{2} - 10$$ b3 + 25*b2 - 10 $$\nu^{4}$$ $$=$$ $$-25\beta_{5} - 394\beta_{4} + 25\beta_{3} - 43\beta_1$$ -25*b5 - 394*b4 + 25*b3 - 43*b1 $$\nu^{5}$$ $$=$$ $$-43\beta_{5} - 538\beta_{4} - 637\beta_{2} - 637\beta _1 + 538$$ -43*b5 - 538*b4 - 637*b2 - 637*b1 + 538

## Character values

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

 $$n$$ $$50$$ $$52$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 2.65415 + 4.59712i 0.124036 + 0.214837i −2.27818 − 3.94593i 2.65415 − 4.59712i 0.124036 − 0.214837i −2.27818 + 3.94593i
−2.65415 4.59712i −1.50000 + 2.59808i −10.0890 + 17.4746i 2.78070 + 4.81631i 15.9249 0 64.6443 −4.50000 7.79423i 14.7608 25.5664i
67.2 −0.124036 0.214837i −1.50000 + 2.59808i 3.96923 6.87491i −6.21730 10.7687i 0.744216 0 −3.95388 −4.50000 7.79423i −1.54234 + 2.67141i
67.3 2.27818 + 3.94593i −1.50000 + 2.59808i −6.38024 + 11.0509i 8.93660 + 15.4786i −13.6691 0 −21.6905 −4.50000 7.79423i −40.7184 + 70.5264i
79.1 −2.65415 + 4.59712i −1.50000 2.59808i −10.0890 17.4746i 2.78070 4.81631i 15.9249 0 64.6443 −4.50000 + 7.79423i 14.7608 + 25.5664i
79.2 −0.124036 + 0.214837i −1.50000 2.59808i 3.96923 + 6.87491i −6.21730 + 10.7687i 0.744216 0 −3.95388 −4.50000 + 7.79423i −1.54234 2.67141i
79.3 2.27818 3.94593i −1.50000 2.59808i −6.38024 11.0509i 8.93660 15.4786i −13.6691 0 −21.6905 −4.50000 + 7.79423i −40.7184 70.5264i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 79.3 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 147.4.e.n 6
3.b odd 2 1 441.4.e.w 6
7.b odd 2 1 21.4.e.b 6
7.c even 3 1 147.4.a.m 3
7.c even 3 1 inner 147.4.e.n 6
7.d odd 6 1 21.4.e.b 6
7.d odd 6 1 147.4.a.l 3
21.c even 2 1 63.4.e.c 6
21.g even 6 1 63.4.e.c 6
21.g even 6 1 441.4.a.s 3
21.h odd 6 1 441.4.a.t 3
21.h odd 6 1 441.4.e.w 6
28.d even 2 1 336.4.q.k 6
28.f even 6 1 336.4.q.k 6
28.f even 6 1 2352.4.a.ci 3
28.g odd 6 1 2352.4.a.cg 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.b 6 7.b odd 2 1
21.4.e.b 6 7.d odd 6 1
63.4.e.c 6 21.c even 2 1
63.4.e.c 6 21.g even 6 1
147.4.a.l 3 7.d odd 6 1
147.4.a.m 3 7.c even 3 1
147.4.e.n 6 1.a even 1 1 trivial
147.4.e.n 6 7.c even 3 1 inner
336.4.q.k 6 28.d even 2 1
336.4.q.k 6 28.f even 6 1
441.4.a.s 3 21.g even 6 1
441.4.a.t 3 21.h odd 6 1
441.4.e.w 6 3.b odd 2 1
441.4.e.w 6 21.h odd 6 1
2352.4.a.cg 3 28.g odd 6 1
2352.4.a.ci 3 28.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}^{6} + T_{2}^{5} + 25T_{2}^{4} - 12T_{2}^{3} + 582T_{2}^{2} + 144T_{2} + 36$$ T2^6 + T2^5 + 25*T2^4 - 12*T2^3 + 582*T2^2 + 144*T2 + 36 $$T_{5}^{6} - 11T_{5}^{5} + 313T_{5}^{4} - 360T_{5}^{3} + 50460T_{5}^{2} - 237312T_{5} + 1527696$$ T5^6 - 11*T5^5 + 313*T5^4 - 360*T5^3 + 50460*T5^2 - 237312*T5 + 1527696

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{5} + 25 T^{4} - 12 T^{3} + \cdots + 36$$
$3$ $$(T^{2} + 3 T + 9)^{3}$$
$5$ $$T^{6} - 11 T^{5} + 313 T^{4} + \cdots + 1527696$$
$7$ $$T^{6}$$
$11$ $$T^{6} + 35 T^{5} + 2593 T^{4} + \cdots + 91470096$$
$13$ $$(T^{3} + 62 T^{2} + 425 T - 18452)^{2}$$
$17$ $$T^{6} - 48 T^{5} + \cdots + 12745506816$$
$19$ $$T^{6} + 202 T^{5} + \cdots + 54664310416$$
$23$ $$T^{6} + 216 T^{5} + \cdots + 2498119335936$$
$29$ $$(T^{3} - 53 T^{2} - 20472 T - 824976)^{2}$$
$31$ $$T^{6} + 95 T^{5} + \cdots + 139783329$$
$37$ $$T^{6} + 262 T^{5} + \cdots + 2415919104$$
$41$ $$(T^{3} + 244 T^{2} - 18780 T + 300384)^{2}$$
$43$ $$(T^{3} - 360 T^{2} - 72363 T + 18269746)^{2}$$
$47$ $$T^{6} + 210 T^{5} + \cdots + 26205471480384$$
$53$ $$T^{6} + 393 T^{5} + \cdots + 11\!\cdots\!64$$
$59$ $$T^{6} - 1143 T^{5} + \cdots + 10\!\cdots\!36$$
$61$ $$T^{6} + 70 T^{5} + \cdots + 71\!\cdots\!00$$
$67$ $$T^{6} + \cdots + 783608160972004$$
$71$ $$(T^{3} - 318 T^{2} - 330804 T - 28535976)^{2}$$
$73$ $$T^{6} - 988 T^{5} + \cdots + 20\!\cdots\!24$$
$79$ $$T^{6} + 861 T^{5} + \cdots + 37\!\cdots\!69$$
$83$ $$(T^{3} + 519 T^{2} - 131616 T - 47916036)^{2}$$
$89$ $$T^{6} + \cdots + 169118164647936$$
$97$ $$(T^{3} + 19 T^{2} - 569600 T - 44776452)^{2}$$