# Properties

 Label 1421.4.a.e Level $1421$ Weight $4$ Character orbit 1421.a Self dual yes Analytic conductor $83.842$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1421 = 7^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1421.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$83.8417141182$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.13458092.1 Defining polynomial: $$x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8$$ x^5 - x^4 - 14*x^3 + 18*x^2 + 20*x - 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 29) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{4} + \beta_{3} + \beta_1 - 1) q^{3} + (\beta_{4} + 2 \beta_{3} + 2 \beta_1 + 6) q^{4} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 3) q^{5} + ( - \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 6) q^{6} + ( - 4 \beta_{4} - 4 \beta_{3} - 8 \beta_{2} - 9 \beta_1 - 16) q^{8} + ( - 6 \beta_{4} - \beta_{3} + 5 \beta_{2} + \beta_1 + 2) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b4 + b3 + b1 - 1) * q^3 + (b4 + 2*b3 + 2*b1 + 6) * q^4 + (-2*b4 + b3 + b2 - b1 - 3) * q^5 + (-b4 - 2*b3 - 4*b2 - 4*b1 - 6) * q^6 + (-4*b4 - 4*b3 - 8*b2 - 9*b1 - 16) * q^8 + (-6*b4 - b3 + 5*b2 + b1 + 2) * q^9 $$q - \beta_1 q^{2} + (\beta_{4} + \beta_{3} + \beta_1 - 1) q^{3} + (\beta_{4} + 2 \beta_{3} + 2 \beta_1 + 6) q^{4} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 3) q^{5} + ( - \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 6) q^{6} + ( - 4 \beta_{4} - 4 \beta_{3} - 8 \beta_{2} - 9 \beta_1 - 16) q^{8} + ( - 6 \beta_{4} - \beta_{3} + 5 \beta_{2} + \beta_1 + 2) q^{9} + ( - 6 \beta_{4} - 4 \beta_{2} - \beta_1 + 12) q^{10} + (9 \beta_{4} + 5 \beta_{3} + 10 \beta_{2} + 7 \beta_1 + 3) q^{11} + (2 \beta_{4} + 8 \beta_{3} + 8 \beta_{2} + 13 \beta_1 + 44) q^{12} + (6 \beta_{4} - 5 \beta_{3} + 9 \beta_{2} + 3 \beta_1 - 5) q^{13} + (11 \beta_{4} - \beta_{3} - 14 \beta_{2} + 3 \beta_1 - 5) q^{15} + (9 \beta_{4} + 18 \beta_{3} + 16 \beta_{2} + 30 \beta_1 + 30) q^{16} + ( - 4 \beta_{3} - 10 \beta_{2} - 10) q^{17} + ( - 16 \beta_{4} - 12 \beta_{3} + 4 \beta_{2} - 6 \beta_1 - 32) q^{18} + ( - 4 \beta_{4} + 2 \beta_{3} - 8 \beta_1 - 44) q^{19} + (9 \beta_{4} + 2 \beta_{3} - 8 \beta_{2} - 8 \beta_1 + 6) q^{20} + ( - 9 \beta_{4} - 34 \beta_{3} - 20 \beta_{2} - 28 \beta_1 - 22) q^{22} + (24 \beta_{4} + 14 \beta_{3} - 2 \beta_{2} + 46) q^{23} + ( - 25 \beta_{4} - 26 \beta_{3} - 68 \beta_1 - 78) q^{24} + ( - 2 \beta_{4} - 21 \beta_{3} + 11 \beta_{2} - 17 \beta_1 + 32) q^{25} + (10 \beta_{4} - 24 \beta_{3} + 20 \beta_{2} + 25 \beta_1 - 20) q^{26} + (27 \beta_{4} - 11 \beta_{3} - 32 \beta_{2} + 11 \beta_1 - 51) q^{27} - 29 q^{29} + (35 \beta_{4} + 22 \beta_{3} + 4 \beta_{2} + 14 \beta_1 - 30) q^{30} + ( - 5 \beta_{4} - 7 \beta_{3} + 14 \beta_{2} + 19 \beta_1 - 93) q^{31} + ( - 32 \beta_{4} - 60 \beta_{3} - 8 \beta_{2} - 81 \beta_1 - 152) q^{32} + ( - 14 \beta_{4} - \beta_{3} - 9 \beta_{2} + 39 \beta_1 + 113) q^{33} + (18 \beta_{4} + 20 \beta_{3} + 16 \beta_{2} + 26 \beta_1 - 36) q^{34} + (42 \beta_{4} + 12 \beta_{3} + 8 \beta_{2} + 68 \beta_1 - 36) q^{36} + ( - 28 \beta_{4} - 22 \beta_{3} + 30 \beta_{2} + 22 \beta_1 + 48) q^{37} + ( - 4 \beta_{4} + 16 \beta_{3} - 8 \beta_{2} + 48 \beta_1 + 104) q^{38} + ( - 11 \beta_{4} - 11 \beta_{3} - 16 \beta_{2} + 11 \beta_1 - 75) q^{39} + (78 \beta_{4} + 32 \beta_{3} + 24 \beta_{2} + 19 \beta_1 + 44) q^{40} + ( - 44 \beta_{4} - 22 \beta_{3} - 16 \beta_{2} - 18 \beta_1 + 216) q^{41} + ( - 11 \beta_{4} + 25 \beta_{3} + 22 \beta_{2} - 29 \beta_1 - 49) q^{43} + (26 \beta_{4} + 56 \beta_{3} + 56 \beta_{2} + 149 \beta_1 + 156) q^{44} + ( - 56 \beta_{4} - 48 \beta_{3} + 78 \beta_{2} - 42 \beta_1 + 238) q^{45} + (22 \beta_{4} + 4 \beta_{3} - 56 \beta_{2} - 78 \beta_1 + 148) q^{46} + (7 \beta_{4} + 41 \beta_{3} + 16 \beta_{2} - 75 \beta_1 - 45) q^{47} + (54 \beta_{4} + 72 \beta_{3} + 40 \beta_{2} + 189 \beta_1 + 396) q^{48} + (44 \beta_{4} + 12 \beta_{3} + 84 \beta_{2} + 84 \beta_1 + 168) q^{50} + ( - 40 \beta_{4} - 10 \beta_{3} + 54 \beta_{2} - 44 \beta_1 + 6) q^{51} + ( - 25 \beta_{4} - 50 \beta_{3} + 24 \beta_{2} + 52 \beta_1 - 326) q^{52} + (10 \beta_{4} + 23 \beta_{3} + 17 \beta_{2} + 75 \beta_1 - 109) q^{53} + (97 \beta_{4} + 42 \beta_{3} + 44 \beta_{2} + 100 \beta_1 - 154) q^{54} + ( - 101 \beta_{4} - 41 \beta_{3} - 112 \beta_{2} + 13 \beta_1 - 113) q^{55} + ( - 28 \beta_{4} - 52 \beta_{3} - 42 \beta_{2} - 62 \beta_1 + 4) q^{57} + 29 \beta_1 q^{58} + (16 \beta_{4} - 30 \beta_{3} + 6 \beta_{2} - 52 \beta_1 - 90) q^{59} + ( - 80 \beta_{4} - 28 \beta_{3} + 24 \beta_{2} - 75 \beta_1 + 80) q^{60} + ( - 12 \beta_{4} - 60 \beta_{3} - 66 \beta_{2} + 20 \beta_1 - 114) q^{61} + ( - 29 \beta_{4} - 66 \beta_{3} + 28 \beta_{2} + 78 \beta_1 - 286) q^{62} + (73 \beta_{4} + 34 \beta_{3} + 112 \beta_{2} + 282 \beta_1 + 510) q^{64} + ( - 146 \beta_{4} - 15 \beta_{3} - 49 \beta_{2} + 51 \beta_1 - 373) q^{65} + ( - 56 \beta_{4} - 60 \beta_{3} + 4 \beta_{2} - 201 \beta_1 - 624) q^{66} + ( - 44 \beta_{3} - 140 \beta_{2} - 88 \beta_1 + 280) q^{67} + ( - 46 \beta_{4} - 52 \beta_{3} - 78 \beta_1 - 100) q^{68} + ( - 104 \beta_{4} - 2 \beta_{3} + 44 \beta_{2} - 18 \beta_1 + 334) q^{69} + (6 \beta_{4} - 122 \beta_{3} - 96 \beta_{2} - 58 \beta_1 - 122) q^{71} + (112 \beta_{4} - 56 \beta_{3} - 80 \beta_{2} - 58 \beta_1 - 464) q^{72} + (32 \beta_{4} + 38 \beta_{3} + 66 \beta_{2} + 106 \beta_1 + 384) q^{73} + ( - 64 \beta_{4} - 104 \beta_{3} + 88 \beta_{2} - 32 \beta_1 - 448) q^{74} + (60 \beta_{4} + 2 \beta_{3} - 106 \beta_{2} - 18 \beta_1 - 548) q^{75} + ( - 48 \beta_{4} - 96 \beta_{3} - 64 \beta_{2} - 204 \beta_1 - 288) q^{76} + (5 \beta_{4} + 10 \beta_{3} + 44 \beta_{2} + 86 \beta_1 - 274) q^{78} + ( - 177 \beta_{4} - 63 \beta_{3} - 140 \beta_{2} - 139 \beta_1 + 27) q^{79} + ( - 23 \beta_{4} - 102 \beta_{3} - 64 \beta_{2} - 68 \beta_1 + 174) q^{80} + ( - 136 \beta_{4} - 56 \beta_{3} + 134 \beta_{2} - 222 \beta_1 + 241) q^{81} + ( - 10 \beta_{4} + 68 \beta_{3} + 88 \beta_{2} - 136 \beta_1 - 44) q^{82} + ( - 60 \beta_{4} - 14 \beta_{3} + 42 \beta_{2} + 96 \beta_1 - 146) q^{83} + (136 \beta_{4} + 66 \beta_{3} + 20 \beta_{2} + 50 \beta_1 - 266) q^{85} + ( - 65 \beta_{4} + 14 \beta_{3} - 100 \beta_{2} - 4 \beta_1 + 506) q^{86} + ( - 29 \beta_{4} - 29 \beta_{3} - 29 \beta_1 + 29) q^{87} + ( - 193 \beta_{4} - 138 \beta_{3} - 64 \beta_{2} - 428 \beta_1 - 1470) q^{88} + ( - 152 \beta_{4} - 10 \beta_{3} - 100 \beta_{2} - 66 \beta_1 - 196) q^{89} + ( - 52 \beta_{4} - 72 \beta_{3} + 192 \beta_{2} - 18 \beta_1 + 328) q^{90} + ( - 22 \beta_{4} + 156 \beta_{3} + 14 \beta_1 + 716) q^{92} + ( - 2 \beta_{4} - 45 \beta_{3} + 3 \beta_{2} + 33 \beta_1 - 19) q^{93} + ( - 9 \beta_{4} + 118 \beta_{3} - 164 \beta_{2} + 38 \beta_1 + 1274) q^{94} + (64 \beta_{4} - 68 \beta_{3} - 38 \beta_{2} + 2 \beta_1 + 476) q^{95} + ( - 65 \beta_{4} - 250 \beta_{3} - 288 \beta_{2} - 464 \beta_1 - 1438) q^{96} + (32 \beta_{4} - 78 \beta_{3} - 48 \beta_{2} - 34 \beta_1 - 296) q^{97} + ( - 34 \beta_{4} + 84 \beta_{3} - 96 \beta_{2} + 94 \beta_1 - 6) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b4 + b3 + b1 - 1) * q^3 + (b4 + 2*b3 + 2*b1 + 6) * q^4 + (-2*b4 + b3 + b2 - b1 - 3) * q^5 + (-b4 - 2*b3 - 4*b2 - 4*b1 - 6) * q^6 + (-4*b4 - 4*b3 - 8*b2 - 9*b1 - 16) * q^8 + (-6*b4 - b3 + 5*b2 + b1 + 2) * q^9 + (-6*b4 - 4*b2 - b1 + 12) * q^10 + (9*b4 + 5*b3 + 10*b2 + 7*b1 + 3) * q^11 + (2*b4 + 8*b3 + 8*b2 + 13*b1 + 44) * q^12 + (6*b4 - 5*b3 + 9*b2 + 3*b1 - 5) * q^13 + (11*b4 - b3 - 14*b2 + 3*b1 - 5) * q^15 + (9*b4 + 18*b3 + 16*b2 + 30*b1 + 30) * q^16 + (-4*b3 - 10*b2 - 10) * q^17 + (-16*b4 - 12*b3 + 4*b2 - 6*b1 - 32) * q^18 + (-4*b4 + 2*b3 - 8*b1 - 44) * q^19 + (9*b4 + 2*b3 - 8*b2 - 8*b1 + 6) * q^20 + (-9*b4 - 34*b3 - 20*b2 - 28*b1 - 22) * q^22 + (24*b4 + 14*b3 - 2*b2 + 46) * q^23 + (-25*b4 - 26*b3 - 68*b1 - 78) * q^24 + (-2*b4 - 21*b3 + 11*b2 - 17*b1 + 32) * q^25 + (10*b4 - 24*b3 + 20*b2 + 25*b1 - 20) * q^26 + (27*b4 - 11*b3 - 32*b2 + 11*b1 - 51) * q^27 - 29 * q^29 + (35*b4 + 22*b3 + 4*b2 + 14*b1 - 30) * q^30 + (-5*b4 - 7*b3 + 14*b2 + 19*b1 - 93) * q^31 + (-32*b4 - 60*b3 - 8*b2 - 81*b1 - 152) * q^32 + (-14*b4 - b3 - 9*b2 + 39*b1 + 113) * q^33 + (18*b4 + 20*b3 + 16*b2 + 26*b1 - 36) * q^34 + (42*b4 + 12*b3 + 8*b2 + 68*b1 - 36) * q^36 + (-28*b4 - 22*b3 + 30*b2 + 22*b1 + 48) * q^37 + (-4*b4 + 16*b3 - 8*b2 + 48*b1 + 104) * q^38 + (-11*b4 - 11*b3 - 16*b2 + 11*b1 - 75) * q^39 + (78*b4 + 32*b3 + 24*b2 + 19*b1 + 44) * q^40 + (-44*b4 - 22*b3 - 16*b2 - 18*b1 + 216) * q^41 + (-11*b4 + 25*b3 + 22*b2 - 29*b1 - 49) * q^43 + (26*b4 + 56*b3 + 56*b2 + 149*b1 + 156) * q^44 + (-56*b4 - 48*b3 + 78*b2 - 42*b1 + 238) * q^45 + (22*b4 + 4*b3 - 56*b2 - 78*b1 + 148) * q^46 + (7*b4 + 41*b3 + 16*b2 - 75*b1 - 45) * q^47 + (54*b4 + 72*b3 + 40*b2 + 189*b1 + 396) * q^48 + (44*b4 + 12*b3 + 84*b2 + 84*b1 + 168) * q^50 + (-40*b4 - 10*b3 + 54*b2 - 44*b1 + 6) * q^51 + (-25*b4 - 50*b3 + 24*b2 + 52*b1 - 326) * q^52 + (10*b4 + 23*b3 + 17*b2 + 75*b1 - 109) * q^53 + (97*b4 + 42*b3 + 44*b2 + 100*b1 - 154) * q^54 + (-101*b4 - 41*b3 - 112*b2 + 13*b1 - 113) * q^55 + (-28*b4 - 52*b3 - 42*b2 - 62*b1 + 4) * q^57 + 29*b1 * q^58 + (16*b4 - 30*b3 + 6*b2 - 52*b1 - 90) * q^59 + (-80*b4 - 28*b3 + 24*b2 - 75*b1 + 80) * q^60 + (-12*b4 - 60*b3 - 66*b2 + 20*b1 - 114) * q^61 + (-29*b4 - 66*b3 + 28*b2 + 78*b1 - 286) * q^62 + (73*b4 + 34*b3 + 112*b2 + 282*b1 + 510) * q^64 + (-146*b4 - 15*b3 - 49*b2 + 51*b1 - 373) * q^65 + (-56*b4 - 60*b3 + 4*b2 - 201*b1 - 624) * q^66 + (-44*b3 - 140*b2 - 88*b1 + 280) * q^67 + (-46*b4 - 52*b3 - 78*b1 - 100) * q^68 + (-104*b4 - 2*b3 + 44*b2 - 18*b1 + 334) * q^69 + (6*b4 - 122*b3 - 96*b2 - 58*b1 - 122) * q^71 + (112*b4 - 56*b3 - 80*b2 - 58*b1 - 464) * q^72 + (32*b4 + 38*b3 + 66*b2 + 106*b1 + 384) * q^73 + (-64*b4 - 104*b3 + 88*b2 - 32*b1 - 448) * q^74 + (60*b4 + 2*b3 - 106*b2 - 18*b1 - 548) * q^75 + (-48*b4 - 96*b3 - 64*b2 - 204*b1 - 288) * q^76 + (5*b4 + 10*b3 + 44*b2 + 86*b1 - 274) * q^78 + (-177*b4 - 63*b3 - 140*b2 - 139*b1 + 27) * q^79 + (-23*b4 - 102*b3 - 64*b2 - 68*b1 + 174) * q^80 + (-136*b4 - 56*b3 + 134*b2 - 222*b1 + 241) * q^81 + (-10*b4 + 68*b3 + 88*b2 - 136*b1 - 44) * q^82 + (-60*b4 - 14*b3 + 42*b2 + 96*b1 - 146) * q^83 + (136*b4 + 66*b3 + 20*b2 + 50*b1 - 266) * q^85 + (-65*b4 + 14*b3 - 100*b2 - 4*b1 + 506) * q^86 + (-29*b4 - 29*b3 - 29*b1 + 29) * q^87 + (-193*b4 - 138*b3 - 64*b2 - 428*b1 - 1470) * q^88 + (-152*b4 - 10*b3 - 100*b2 - 66*b1 - 196) * q^89 + (-52*b4 - 72*b3 + 192*b2 - 18*b1 + 328) * q^90 + (-22*b4 + 156*b3 + 14*b1 + 716) * q^92 + (-2*b4 - 45*b3 + 3*b2 + 33*b1 - 19) * q^93 + (-9*b4 + 118*b3 - 164*b2 + 38*b1 + 1274) * q^94 + (64*b4 - 68*b3 - 38*b2 + 2*b1 + 476) * q^95 + (-65*b4 - 250*b3 - 288*b2 - 464*b1 - 1438) * q^96 + (32*b4 - 78*b3 - 48*b2 - 34*b1 - 296) * q^97 + (-34*b4 + 84*b3 - 96*b2 + 94*b1 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 8 q^{3} + 26 q^{4} - 10 q^{5} - 34 q^{6} - 84 q^{8} + 33 q^{9}+O(q^{10})$$ 5 * q - 8 * q^3 + 26 * q^4 - 10 * q^5 - 34 * q^6 - 84 * q^8 + 33 * q^9 $$5 q - 8 q^{3} + 26 q^{4} - 10 q^{5} - 34 q^{6} - 84 q^{8} + 33 q^{9} + 64 q^{10} + 12 q^{11} + 224 q^{12} - 14 q^{13} - 74 q^{15} + 146 q^{16} - 66 q^{17} - 108 q^{18} - 214 q^{19} - 6 q^{20} - 98 q^{22} + 164 q^{23} - 314 q^{24} + 207 q^{25} - 56 q^{26} - 362 q^{27} - 145 q^{29} - 234 q^{30} - 420 q^{31} - 652 q^{32} + 576 q^{33} - 204 q^{34} - 260 q^{36} + 378 q^{37} + 496 q^{38} - 374 q^{39} + 80 q^{40} + 1158 q^{41} - 204 q^{43} + 784 q^{44} + 1506 q^{45} + 580 q^{46} - 248 q^{47} + 1880 q^{48} + 908 q^{50} + 228 q^{51} - 1482 q^{52} - 554 q^{53} - 918 q^{54} - 546 q^{55} + 44 q^{57} - 440 q^{59} + 636 q^{60} - 618 q^{61} - 1250 q^{62} + 2594 q^{64} - 1656 q^{65} - 2940 q^{66} + 1164 q^{67} - 356 q^{68} + 1968 q^{69} - 692 q^{71} - 2648 q^{72} + 1950 q^{73} - 1832 q^{74} - 3074 q^{75} - 1376 q^{76} - 1302 q^{78} + 272 q^{79} + 890 q^{80} + 1801 q^{81} - 92 q^{82} - 512 q^{83} - 1628 q^{85} + 2446 q^{86} + 232 q^{87} - 6954 q^{88} - 866 q^{89} + 2200 q^{90} + 3468 q^{92} - 40 q^{93} + 5942 q^{94} + 2244 q^{95} - 7386 q^{96} - 1562 q^{97} - 238 q^{99}+O(q^{100})$$ 5 * q - 8 * q^3 + 26 * q^4 - 10 * q^5 - 34 * q^6 - 84 * q^8 + 33 * q^9 + 64 * q^10 + 12 * q^11 + 224 * q^12 - 14 * q^13 - 74 * q^15 + 146 * q^16 - 66 * q^17 - 108 * q^18 - 214 * q^19 - 6 * q^20 - 98 * q^22 + 164 * q^23 - 314 * q^24 + 207 * q^25 - 56 * q^26 - 362 * q^27 - 145 * q^29 - 234 * q^30 - 420 * q^31 - 652 * q^32 + 576 * q^33 - 204 * q^34 - 260 * q^36 + 378 * q^37 + 496 * q^38 - 374 * q^39 + 80 * q^40 + 1158 * q^41 - 204 * q^43 + 784 * q^44 + 1506 * q^45 + 580 * q^46 - 248 * q^47 + 1880 * q^48 + 908 * q^50 + 228 * q^51 - 1482 * q^52 - 554 * q^53 - 918 * q^54 - 546 * q^55 + 44 * q^57 - 440 * q^59 + 636 * q^60 - 618 * q^61 - 1250 * q^62 + 2594 * q^64 - 1656 * q^65 - 2940 * q^66 + 1164 * q^67 - 356 * q^68 + 1968 * q^69 - 692 * q^71 - 2648 * q^72 + 1950 * q^73 - 1832 * q^74 - 3074 * q^75 - 1376 * q^76 - 1302 * q^78 + 272 * q^79 + 890 * q^80 + 1801 * q^81 - 92 * q^82 - 512 * q^83 - 1628 * q^85 + 2446 * q^86 + 232 * q^87 - 6954 * q^88 - 866 * q^89 + 2200 * q^90 + 3468 * q^92 - 40 * q^93 + 5942 * q^94 + 2244 * q^95 - 7386 * q^96 - 1562 * q^97 - 238 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + \nu^{2} - 8\nu - 2 ) / 2$$ (v^3 + v^2 - 8*v - 2) / 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} - \nu^{2} + 12\nu + 2 ) / 2$$ (-v^3 - v^2 + 12*v + 2) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 6$$ v^2 - 6 $$\beta_{4}$$ $$=$$ $$( \nu^{4} - 13\nu^{2} + 6\nu + 14 ) / 2$$ (v^4 - 13*v^2 + 6*v + 14) / 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 6$$ b3 + 6 $$\nu^{3}$$ $$=$$ $$-\beta_{3} + 4\beta_{2} + 6\beta _1 - 4$$ -b3 + 4*b2 + 6*b1 - 4 $$\nu^{4}$$ $$=$$ $$2\beta_{4} + 13\beta_{3} - 3\beta_{2} - 3\beta _1 + 64$$ 2*b4 + 13*b3 - 3*b2 - 3*b1 + 64

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.03898 −0.957567 2.27399 0.328194 −3.68360
−5.49488 6.46343 22.1937 −2.14270 −35.5158 0 −77.9928 14.7760 11.7739
1.2 −2.84972 −4.64574 0.120922 −12.8729 13.2391 0 22.4532 −5.41713 36.6841
1.3 1.63099 −9.87991 −5.33986 16.8209 −16.1141 0 −21.7572 70.6126 27.4348
1.4 2.24125 −1.84328 −2.97681 −18.3339 −4.13124 0 −24.6017 −23.6023 −41.0908
1.5 4.47236 1.90549 12.0020 6.52855 8.52204 0 17.8986 −23.3691 29.1981
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$29$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1421.4.a.e 5
7.b odd 2 1 29.4.a.b 5
21.c even 2 1 261.4.a.f 5
28.d even 2 1 464.4.a.l 5
35.c odd 2 1 725.4.a.c 5
56.e even 2 1 1856.4.a.bb 5
56.h odd 2 1 1856.4.a.y 5
203.c odd 2 1 841.4.a.b 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.4.a.b 5 7.b odd 2 1
261.4.a.f 5 21.c even 2 1
464.4.a.l 5 28.d even 2 1
725.4.a.c 5 35.c odd 2 1
841.4.a.b 5 203.c odd 2 1
1421.4.a.e 5 1.a even 1 1 trivial
1856.4.a.y 5 56.h odd 2 1
1856.4.a.bb 5 56.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1421))$$:

 $$T_{2}^{5} - 33T_{2}^{3} + 28T_{2}^{2} + 192T_{2} - 256$$ T2^5 - 33*T2^3 + 28*T2^2 + 192*T2 - 256 $$T_{3}^{5} + 8T_{3}^{4} - 52T_{3}^{3} - 322T_{3}^{2} + 187T_{3} + 1042$$ T3^5 + 8*T3^4 - 52*T3^3 - 322*T3^2 + 187*T3 + 1042

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - 33 T^{3} + 28 T^{2} + \cdots - 256$$
$3$ $$T^{5} + 8 T^{4} - 52 T^{3} + \cdots + 1042$$
$5$ $$T^{5} + 10 T^{4} - 366 T^{3} + \cdots + 55534$$
$7$ $$T^{5}$$
$11$ $$T^{5} - 12 T^{4} - 4892 T^{3} + \cdots + 30997958$$
$13$ $$T^{5} + 14 T^{4} - 7558 T^{3} + \cdots + 13078418$$
$17$ $$T^{5} + 66 T^{4} - 2444 T^{3} + \cdots - 19935872$$
$19$ $$T^{5} + 214 T^{4} + \cdots - 19441152$$
$23$ $$T^{5} - 164 T^{4} + \cdots - 7938109184$$
$29$ $$(T + 29)^{5}$$
$31$ $$T^{5} + 420 T^{4} + 45552 T^{3} + \cdots + 2094346$$
$37$ $$T^{5} - 378 T^{4} + \cdots + 23564115968$$
$41$ $$T^{5} - 1158 T^{4} + \cdots - 59613728000$$
$43$ $$T^{5} + 204 T^{4} + \cdots + 198643410886$$
$47$ $$T^{5} + 248 T^{4} + \cdots + 203435244846$$
$53$ $$T^{5} + 554 T^{4} + \cdots - 786854101018$$
$59$ $$T^{5} + 440 T^{4} + \cdots - 109032704000$$
$61$ $$T^{5} + 618 T^{4} + \cdots - 2140697762176$$
$67$ $$T^{5} - 1164 T^{4} + \cdots - 39308070146048$$
$71$ $$T^{5} + 692 T^{4} + \cdots + 98341318953856$$
$73$ $$T^{5} - 1950 T^{4} + \cdots - 7201878016$$
$79$ $$T^{5} + \cdots - 240961986300538$$
$83$ $$T^{5} + 512 T^{4} + \cdots - 6057622580224$$
$89$ $$T^{5} + 866 T^{4} + \cdots - 21549994365568$$
$97$ $$T^{5} + 1562 T^{4} + \cdots + 20480102175488$$