# Properties

 Label 114.4.e.e Level $114$ Weight $4$ Character orbit 114.e Analytic conductor $6.726$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 114.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.72621774065$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.6967728.1 Defining polynomial: $$x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1$$ x^6 - x^5 + 8*x^4 + 5*x^3 + 50*x^2 - 7*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \beta_{3} + 2) q^{2} + (3 \beta_{3} + 3) q^{3} + 4 \beta_{3} q^{4} + ( - \beta_{5} - \beta_{3} + \beta_1 - 1) q^{5} + 6 \beta_{3} q^{6} + ( - 2 \beta_{4} + 2 \beta_{2} + \beta_1 - 6) q^{7} - 8 q^{8} + 9 \beta_{3} q^{9}+O(q^{10})$$ q + (2*b3 + 2) * q^2 + (3*b3 + 3) * q^3 + 4*b3 * q^4 + (-b5 - b3 + b1 - 1) * q^5 + 6*b3 * q^6 + (-2*b4 + 2*b2 + b1 - 6) * q^7 - 8 * q^8 + 9*b3 * q^9 $$q + (2 \beta_{3} + 2) q^{2} + (3 \beta_{3} + 3) q^{3} + 4 \beta_{3} q^{4} + ( - \beta_{5} - \beta_{3} + \beta_1 - 1) q^{5} + 6 \beta_{3} q^{6} + ( - 2 \beta_{4} + 2 \beta_{2} + \beta_1 - 6) q^{7} - 8 q^{8} + 9 \beta_{3} q^{9} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{10} + (\beta_{4} - \beta_{2} + 2 \beta_1 - 18) q^{11} - 12 q^{12} + (4 \beta_{4} + 23 \beta_{3} + 2 \beta_{2} + 2) q^{13} + ( - 2 \beta_{5} + 4 \beta_{4} - 16 \beta_{3} + 8 \beta_{2} + 2 \beta_1 - 8) q^{14} + ( - 3 \beta_{5} - 3 \beta_{3}) q^{15} + ( - 16 \beta_{3} - 16) q^{16} + (3 \beta_{5} - \beta_{4} + 18 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 16) q^{17} - 18 q^{18} + ( - 3 \beta_{5} + 3 \beta_{4} + 58 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 48) q^{19} + ( - 4 \beta_1 + 4) q^{20} + ( - 3 \beta_{5} + 6 \beta_{4} - 24 \beta_{3} + 12 \beta_{2} + 3 \beta_1 - 12) q^{21} + ( - 4 \beta_{5} - 2 \beta_{4} - 34 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 38) q^{22} + ( - 5 \beta_{5} - 20 \beta_{4} - 71 \beta_{3} - 10 \beta_{2} - 10) q^{23} + ( - 24 \beta_{3} - 24) q^{24} + (11 \beta_{5} - 14 \beta_{4} + 87 \beta_{3} - 7 \beta_{2} - 7) q^{25} + (4 \beta_{4} - 4 \beta_{2} - 50) q^{26} - 27 q^{27} + ( - 4 \beta_{5} + 16 \beta_{4} - 32 \beta_{3} + 8 \beta_{2} + 8) q^{28} + ( - \beta_{5} + 26 \beta_{4} - 16 \beta_{3} + 13 \beta_{2} + 13) q^{29} + ( - 6 \beta_1 + 6) q^{30} + ( - 3 \beta_{4} + 3 \beta_{2} + 8 \beta_1 + 33) q^{31} - 32 \beta_{3} q^{32} + ( - 6 \beta_{5} - 3 \beta_{4} - 51 \beta_{3} - 6 \beta_{2} + 6 \beta_1 - 57) q^{33} + (6 \beta_{5} - 4 \beta_{4} + 36 \beta_{3} - 2 \beta_{2} - 2) q^{34} + (18 \beta_{5} + 11 \beta_{4} + 93 \beta_{3} + 22 \beta_{2} - 18 \beta_1 + 115) q^{35} + ( - 36 \beta_{3} - 36) q^{36} + ( - 13 \beta_{4} + 13 \beta_{2} + 23 \beta_1 + 94) q^{37} + (4 \beta_{5} - 4 \beta_{4} + 100 \beta_{3} - 10 \beta_{2} - 10 \beta_1 - 26) q^{38} + (6 \beta_{4} - 6 \beta_{2} - 75) q^{39} + (8 \beta_{5} + 8 \beta_{3} - 8 \beta_1 + 8) q^{40} + (4 \beta_{5} - 26 \beta_{4} + 22 \beta_{3} - 52 \beta_{2} - 4 \beta_1 - 30) q^{41} + ( - 6 \beta_{5} + 24 \beta_{4} - 48 \beta_{3} + 12 \beta_{2} + 12) q^{42} + ( - 7 \beta_{5} - 18 \beta_{4} + 126 \beta_{3} - 36 \beta_{2} + 7 \beta_1 + 90) q^{43} + ( - 8 \beta_{5} - 8 \beta_{4} - 68 \beta_{3} - 4 \beta_{2} - 4) q^{44} + ( - 9 \beta_1 + 9) q^{45} + ( - 20 \beta_{4} + 20 \beta_{2} - 10 \beta_1 + 162) q^{46} + ( - 14 \beta_{5} + 36 \beta_{4} - 278 \beta_{3} + 18 \beta_{2} + 18) q^{47} - 48 \beta_{3} q^{48} + ( - 13 \beta_{4} + 13 \beta_{2} + 11 \beta_1 + 377) q^{49} + ( - 14 \beta_{4} + 14 \beta_{2} + 22 \beta_1 - 160) q^{50} + (9 \beta_{5} - 6 \beta_{4} + 54 \beta_{3} - 3 \beta_{2} - 3) q^{51} + ( - 8 \beta_{4} - 92 \beta_{3} - 16 \beta_{2} - 108) q^{52} + (28 \beta_{5} - 26 \beta_{4} - 17 \beta_{3} - 13 \beta_{2} - 13) q^{53} + ( - 54 \beta_{3} - 54) q^{54} + (37 \beta_{5} - 23 \beta_{4} + 502 \beta_{3} - 46 \beta_{2} - 37 \beta_1 + 456) q^{55} + (16 \beta_{4} - 16 \beta_{2} - 8 \beta_1 + 48) q^{56} + (6 \beta_{5} - 6 \beta_{4} + 150 \beta_{3} - 15 \beta_{2} - 15 \beta_1 - 39) q^{57} + (26 \beta_{4} - 26 \beta_{2} - 2 \beta_1 + 6) q^{58} + ( - 9 \beta_{5} + 24 \beta_{4} - 339 \beta_{3} + 48 \beta_{2} + 9 \beta_1 - 291) q^{59} + (12 \beta_{5} + 12 \beta_{3} - 12 \beta_1 + 12) q^{60} + (30 \beta_{5} - 4 \beta_{4} - 121 \beta_{3} - 2 \beta_{2} - 2) q^{61} + ( - 16 \beta_{5} + 6 \beta_{4} + 60 \beta_{3} + 12 \beta_{2} + 16 \beta_1 + 72) q^{62} + ( - 9 \beta_{5} + 36 \beta_{4} - 72 \beta_{3} + 18 \beta_{2} + 18) q^{63} + 64 q^{64} + (18 \beta_{4} - 18 \beta_{2} - 23 \beta_1 + 131) q^{65} + ( - 12 \beta_{5} - 12 \beta_{4} - 102 \beta_{3} - 6 \beta_{2} - 6) q^{66} + (26 \beta_{5} - 10 \beta_{4} + 34 \beta_{3} - 5 \beta_{2} - 5) q^{67} + ( - 4 \beta_{4} + 4 \beta_{2} + 12 \beta_1 - 68) q^{68} + ( - 30 \beta_{4} + 30 \beta_{2} - 15 \beta_1 + 243) q^{69} + (36 \beta_{5} + 44 \beta_{4} + 186 \beta_{3} + 22 \beta_{2} + 22) q^{70} + ( - 31 \beta_{5} - \beta_{4} - 334 \beta_{3} - 2 \beta_{2} + 31 \beta_1 - 336) q^{71} - 72 \beta_{3} q^{72} + ( - 14 \beta_{5} - 40 \beta_{4} + 5 \beta_{3} - 80 \beta_{2} + 14 \beta_1 - 75) q^{73} + ( - 46 \beta_{5} + 26 \beta_{4} + 162 \beta_{3} + 52 \beta_{2} + \cdots + 214) q^{74}+ \cdots + ( - 18 \beta_{5} - 18 \beta_{4} - 153 \beta_{3} - 9 \beta_{2} - 9) q^{99}+O(q^{100})$$ q + (2*b3 + 2) * q^2 + (3*b3 + 3) * q^3 + 4*b3 * q^4 + (-b5 - b3 + b1 - 1) * q^5 + 6*b3 * q^6 + (-2*b4 + 2*b2 + b1 - 6) * q^7 - 8 * q^8 + 9*b3 * q^9 + (-2*b5 - 2*b3) * q^10 + (b4 - b2 + 2*b1 - 18) * q^11 - 12 * q^12 + (4*b4 + 23*b3 + 2*b2 + 2) * q^13 + (-2*b5 + 4*b4 - 16*b3 + 8*b2 + 2*b1 - 8) * q^14 + (-3*b5 - 3*b3) * q^15 + (-16*b3 - 16) * q^16 + (3*b5 - b4 + 18*b3 - 2*b2 - 3*b1 + 16) * q^17 - 18 * q^18 + (-3*b5 + 3*b4 + 58*b3 - 2*b2 - 2*b1 + 48) * q^19 + (-4*b1 + 4) * q^20 + (-3*b5 + 6*b4 - 24*b3 + 12*b2 + 3*b1 - 12) * q^21 + (-4*b5 - 2*b4 - 34*b3 - 4*b2 + 4*b1 - 38) * q^22 + (-5*b5 - 20*b4 - 71*b3 - 10*b2 - 10) * q^23 + (-24*b3 - 24) * q^24 + (11*b5 - 14*b4 + 87*b3 - 7*b2 - 7) * q^25 + (4*b4 - 4*b2 - 50) * q^26 - 27 * q^27 + (-4*b5 + 16*b4 - 32*b3 + 8*b2 + 8) * q^28 + (-b5 + 26*b4 - 16*b3 + 13*b2 + 13) * q^29 + (-6*b1 + 6) * q^30 + (-3*b4 + 3*b2 + 8*b1 + 33) * q^31 - 32*b3 * q^32 + (-6*b5 - 3*b4 - 51*b3 - 6*b2 + 6*b1 - 57) * q^33 + (6*b5 - 4*b4 + 36*b3 - 2*b2 - 2) * q^34 + (18*b5 + 11*b4 + 93*b3 + 22*b2 - 18*b1 + 115) * q^35 + (-36*b3 - 36) * q^36 + (-13*b4 + 13*b2 + 23*b1 + 94) * q^37 + (4*b5 - 4*b4 + 100*b3 - 10*b2 - 10*b1 - 26) * q^38 + (6*b4 - 6*b2 - 75) * q^39 + (8*b5 + 8*b3 - 8*b1 + 8) * q^40 + (4*b5 - 26*b4 + 22*b3 - 52*b2 - 4*b1 - 30) * q^41 + (-6*b5 + 24*b4 - 48*b3 + 12*b2 + 12) * q^42 + (-7*b5 - 18*b4 + 126*b3 - 36*b2 + 7*b1 + 90) * q^43 + (-8*b5 - 8*b4 - 68*b3 - 4*b2 - 4) * q^44 + (-9*b1 + 9) * q^45 + (-20*b4 + 20*b2 - 10*b1 + 162) * q^46 + (-14*b5 + 36*b4 - 278*b3 + 18*b2 + 18) * q^47 - 48*b3 * q^48 + (-13*b4 + 13*b2 + 11*b1 + 377) * q^49 + (-14*b4 + 14*b2 + 22*b1 - 160) * q^50 + (9*b5 - 6*b4 + 54*b3 - 3*b2 - 3) * q^51 + (-8*b4 - 92*b3 - 16*b2 - 108) * q^52 + (28*b5 - 26*b4 - 17*b3 - 13*b2 - 13) * q^53 + (-54*b3 - 54) * q^54 + (37*b5 - 23*b4 + 502*b3 - 46*b2 - 37*b1 + 456) * q^55 + (16*b4 - 16*b2 - 8*b1 + 48) * q^56 + (6*b5 - 6*b4 + 150*b3 - 15*b2 - 15*b1 - 39) * q^57 + (26*b4 - 26*b2 - 2*b1 + 6) * q^58 + (-9*b5 + 24*b4 - 339*b3 + 48*b2 + 9*b1 - 291) * q^59 + (12*b5 + 12*b3 - 12*b1 + 12) * q^60 + (30*b5 - 4*b4 - 121*b3 - 2*b2 - 2) * q^61 + (-16*b5 + 6*b4 + 60*b3 + 12*b2 + 16*b1 + 72) * q^62 + (-9*b5 + 36*b4 - 72*b3 + 18*b2 + 18) * q^63 + 64 * q^64 + (18*b4 - 18*b2 - 23*b1 + 131) * q^65 + (-12*b5 - 12*b4 - 102*b3 - 6*b2 - 6) * q^66 + (26*b5 - 10*b4 + 34*b3 - 5*b2 - 5) * q^67 + (-4*b4 + 4*b2 + 12*b1 - 68) * q^68 + (-30*b4 + 30*b2 - 15*b1 + 243) * q^69 + (36*b5 + 44*b4 + 186*b3 + 22*b2 + 22) * q^70 + (-31*b5 - b4 - 334*b3 - 2*b2 + 31*b1 - 336) * q^71 - 72*b3 * q^72 + (-14*b5 - 40*b4 + 5*b3 - 80*b2 + 14*b1 - 75) * q^73 + (-46*b5 + 26*b4 + 162*b3 + 52*b2 + 46*b1 + 214) * q^74 + (-21*b4 + 21*b2 + 33*b1 - 240) * q^75 + (20*b5 - 20*b4 - 32*b3 - 12*b2 - 12*b1 - 244) * q^76 + (16*b4 - 16*b2 - 69*b1 + 11) * q^77 + (-12*b4 - 138*b3 - 24*b2 - 162) * q^78 + (-18*b5 + 43*b4 + 58*b3 + 86*b2 + 18*b1 + 144) * q^79 + (16*b5 + 16*b3) * q^80 + (-81*b3 - 81) * q^81 + (8*b5 - 104*b4 + 44*b3 - 52*b2 - 52) * q^82 + (-17*b4 + 17*b2 - 21*b1 - 709) * q^83 + (24*b4 - 24*b2 - 12*b1 + 72) * q^84 + (-48*b5 + 24*b4 - 588*b3 + 12*b2 + 12) * q^85 + (-14*b5 - 72*b4 + 252*b3 - 36*b2 - 36) * q^86 + (39*b4 - 39*b2 - 3*b1 + 9) * q^87 + (-8*b4 + 8*b2 - 16*b1 + 144) * q^88 + (24*b5 - 2*b4 + 381*b3 - b2 - 1) * q^89 + (18*b5 + 18*b3 - 18*b1 + 18) * q^90 + (-59*b5 + 124*b4 + 374*b3 + 62*b2 + 62) * q^91 + (20*b5 + 40*b4 + 284*b3 + 80*b2 - 20*b1 + 364) * q^92 + (-24*b5 + 9*b4 + 90*b3 + 18*b2 + 24*b1 + 108) * q^93 + (36*b4 - 36*b2 - 28*b1 + 520) * q^94 + (-70*b5 - 25*b4 - 325*b3 + 4*b2 + 42*b1 - 875) * q^95 + 96 * q^96 + (-7*b5 - 7*b4 - 456*b3 - 14*b2 + 7*b1 - 470) * q^97 + (-22*b5 + 26*b4 + 728*b3 + 52*b2 + 22*b1 + 780) * q^98 + (-18*b5 - 18*b4 - 153*b3 - 9*b2 - 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{2} + 9 q^{3} - 12 q^{4} - 2 q^{5} - 18 q^{6} - 34 q^{7} - 48 q^{8} - 27 q^{9}+O(q^{10})$$ 6 * q + 6 * q^2 + 9 * q^3 - 12 * q^4 - 2 * q^5 - 18 * q^6 - 34 * q^7 - 48 * q^8 - 27 * q^9 $$6 q + 6 q^{2} + 9 q^{3} - 12 q^{4} - 2 q^{5} - 18 q^{6} - 34 q^{7} - 48 q^{8} - 27 q^{9} + 4 q^{10} - 104 q^{11} - 72 q^{12} - 75 q^{13} - 34 q^{14} + 6 q^{15} - 48 q^{16} + 48 q^{17} - 108 q^{18} + 104 q^{19} + 16 q^{20} - 51 q^{21} - 104 q^{22} + 238 q^{23} - 72 q^{24} - 229 q^{25} - 300 q^{26} - 162 q^{27} + 68 q^{28} + 8 q^{29} + 24 q^{30} + 214 q^{31} + 96 q^{32} - 156 q^{33} - 96 q^{34} + 294 q^{35} - 108 q^{36} + 610 q^{37} - 430 q^{38} - 450 q^{39} + 16 q^{40} - 16 q^{41} + 102 q^{42} + 331 q^{43} + 208 q^{44} + 36 q^{45} + 952 q^{46} + 766 q^{47} + 144 q^{48} + 2284 q^{49} - 916 q^{50} - 144 q^{51} - 300 q^{52} + 118 q^{53} - 162 q^{54} + 1400 q^{55} + 272 q^{56} - 645 q^{57} + 32 q^{58} - 936 q^{59} + 24 q^{60} + 399 q^{61} + 214 q^{62} + 153 q^{63} + 384 q^{64} + 740 q^{65} + 312 q^{66} - 61 q^{67} - 384 q^{68} + 1428 q^{69} - 588 q^{70} - 974 q^{71} + 216 q^{72} - 91 q^{73} + 610 q^{74} - 1374 q^{75} - 1276 q^{76} - 72 q^{77} - 450 q^{78} + 321 q^{79} - 32 q^{80} - 243 q^{81} + 32 q^{82} - 4296 q^{83} + 408 q^{84} + 1680 q^{85} - 662 q^{86} + 48 q^{87} + 832 q^{88} - 1116 q^{89} + 36 q^{90} - 1367 q^{91} + 952 q^{92} + 321 q^{93} + 3064 q^{94} - 4198 q^{95} + 576 q^{96} - 1382 q^{97} + 2284 q^{98} + 468 q^{99}+O(q^{100})$$ 6 * q + 6 * q^2 + 9 * q^3 - 12 * q^4 - 2 * q^5 - 18 * q^6 - 34 * q^7 - 48 * q^8 - 27 * q^9 + 4 * q^10 - 104 * q^11 - 72 * q^12 - 75 * q^13 - 34 * q^14 + 6 * q^15 - 48 * q^16 + 48 * q^17 - 108 * q^18 + 104 * q^19 + 16 * q^20 - 51 * q^21 - 104 * q^22 + 238 * q^23 - 72 * q^24 - 229 * q^25 - 300 * q^26 - 162 * q^27 + 68 * q^28 + 8 * q^29 + 24 * q^30 + 214 * q^31 + 96 * q^32 - 156 * q^33 - 96 * q^34 + 294 * q^35 - 108 * q^36 + 610 * q^37 - 430 * q^38 - 450 * q^39 + 16 * q^40 - 16 * q^41 + 102 * q^42 + 331 * q^43 + 208 * q^44 + 36 * q^45 + 952 * q^46 + 766 * q^47 + 144 * q^48 + 2284 * q^49 - 916 * q^50 - 144 * q^51 - 300 * q^52 + 118 * q^53 - 162 * q^54 + 1400 * q^55 + 272 * q^56 - 645 * q^57 + 32 * q^58 - 936 * q^59 + 24 * q^60 + 399 * q^61 + 214 * q^62 + 153 * q^63 + 384 * q^64 + 740 * q^65 + 312 * q^66 - 61 * q^67 - 384 * q^68 + 1428 * q^69 - 588 * q^70 - 974 * q^71 + 216 * q^72 - 91 * q^73 + 610 * q^74 - 1374 * q^75 - 1276 * q^76 - 72 * q^77 - 450 * q^78 + 321 * q^79 - 32 * q^80 - 243 * q^81 + 32 * q^82 - 4296 * q^83 + 408 * q^84 + 1680 * q^85 - 662 * q^86 + 48 * q^87 + 832 * q^88 - 1116 * q^89 + 36 * q^90 - 1367 * q^91 + 952 * q^92 + 321 * q^93 + 3064 * q^94 - 4198 * q^95 + 576 * q^96 - 1382 * q^97 + 2284 * q^98 + 468 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 10\nu^{5} - 80\nu^{4} + 116\nu^{3} - 500\nu^{2} + 70\nu - 2525 ) / 131$$ (10*v^5 - 80*v^4 + 116*v^3 - 500*v^2 + 70*v - 2525) / 131 $$\beta_{2}$$ $$=$$ $$( -52\nu^{5} + 23\nu^{4} - 184\nu^{3} - 544\nu^{2} - 1936\nu + 161 ) / 393$$ (-52*v^5 + 23*v^4 - 184*v^3 - 544*v^2 - 1936*v + 161) / 393 $$\beta_{3}$$ $$=$$ $$( -56\nu^{5} + 55\nu^{4} - 440\nu^{3} - 344\nu^{2} - 2750\nu - 8 ) / 393$$ (-56*v^5 + 55*v^4 - 440*v^3 - 344*v^2 - 2750*v - 8) / 393 $$\beta_{4}$$ $$=$$ $$( -58\nu^{5} + 71\nu^{4} - 568\nu^{3} - 244\nu^{2} - 1978\nu - 289 ) / 393$$ (-58*v^5 + 71*v^4 - 568*v^3 - 244*v^2 - 1978*v - 289) / 393 $$\beta_{5}$$ $$=$$ $$( -1036\nu^{5} + 821\nu^{4} - 8140\nu^{3} - 6364\nu^{2} - 52840\nu - 148 ) / 393$$ (-1036*v^5 + 821*v^4 - 8140*v^3 - 6364*v^2 - 52840*v - 148) / 393
 $$\nu$$ $$=$$ $$( 2\beta_{4} - 3\beta_{3} + \beta_{2} + 1 ) / 6$$ (2*b4 - 3*b3 + b2 + 1) / 6 $$\nu^{2}$$ $$=$$ $$( 3\beta_{5} + \beta_{4} - 60\beta_{3} + 2\beta_{2} - 3\beta _1 - 58 ) / 12$$ (3*b5 + b4 - 60*b3 + 2*b2 - 3*b1 - 58) / 12 $$\nu^{3}$$ $$=$$ $$( -5\beta_{4} + 5\beta_{2} - \beta _1 - 25 ) / 4$$ (-5*b4 + 5*b2 - b1 - 25) / 4 $$\nu^{4}$$ $$=$$ $$( -6\beta_{5} - 10\beta_{4} + 126\beta_{3} - 5\beta_{2} - 5 ) / 3$$ (-6*b5 - 10*b4 + 126*b3 - 5*b2 - 5) / 3 $$\nu^{5}$$ $$=$$ $$( -21\beta_{5} - 62\beta_{4} + 537\beta_{3} - 124\beta_{2} + 21\beta _1 + 413 ) / 6$$ (-21*b5 - 62*b4 + 537*b3 - 124*b2 + 21*b1 + 413) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/114\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$97$$ $$\chi(n)$$ $$1$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.0702177 − 0.121621i −1.13654 + 1.96854i 1.56632 − 2.71294i 0.0702177 + 0.121621i −1.13654 − 1.96854i 1.56632 + 2.71294i
1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −10.1010 17.4954i −3.00000 + 5.19615i −22.8872 −8.00000 −4.50000 + 7.79423i 20.2020 34.9909i
7.2 1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 2.60679 + 4.51510i −3.00000 + 5.19615i 31.4905 −8.00000 −4.50000 + 7.79423i −5.21359 + 9.03020i
7.3 1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 6.49420 + 11.2483i −3.00000 + 5.19615i −25.6033 −8.00000 −4.50000 + 7.79423i −12.9884 + 22.4966i
49.1 1.00000 1.73205i 1.50000 2.59808i −2.00000 3.46410i −10.1010 + 17.4954i −3.00000 5.19615i −22.8872 −8.00000 −4.50000 7.79423i 20.2020 + 34.9909i
49.2 1.00000 1.73205i 1.50000 2.59808i −2.00000 3.46410i 2.60679 4.51510i −3.00000 5.19615i 31.4905 −8.00000 −4.50000 7.79423i −5.21359 9.03020i
49.3 1.00000 1.73205i 1.50000 2.59808i −2.00000 3.46410i 6.49420 11.2483i −3.00000 5.19615i −25.6033 −8.00000 −4.50000 7.79423i −12.9884 22.4966i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.4.e.e 6
3.b odd 2 1 342.4.g.g 6
19.c even 3 1 inner 114.4.e.e 6
19.c even 3 1 2166.4.a.s 3
19.d odd 6 1 2166.4.a.w 3
57.h odd 6 1 342.4.g.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.e.e 6 1.a even 1 1 trivial
114.4.e.e 6 19.c even 3 1 inner
342.4.g.g 6 3.b odd 2 1
342.4.g.g 6 57.h odd 6 1
2166.4.a.s 3 19.c even 3 1
2166.4.a.w 3 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 2T_{5}^{5} + 304T_{5}^{4} - 3336T_{5}^{3} + 87264T_{5}^{2} - 410400T_{5} + 1871424$$ acting on $$S_{4}^{\mathrm{new}}(114, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 4)^{3}$$
$3$ $$(T^{2} - 3 T + 9)^{3}$$
$5$ $$T^{6} + 2 T^{5} + 304 T^{4} + \cdots + 1871424$$
$7$ $$(T^{3} + 17 T^{2} - 941 T - 18453)^{2}$$
$11$ $$(T^{3} + 52 T^{2} - 888 T - 32688)^{2}$$
$13$ $$T^{6} + 75 T^{5} + \cdots + 174424849$$
$17$ $$T^{6} - 48 T^{5} + \cdots + 107495424$$
$19$ $$T^{6} - 104 T^{5} + \cdots + 322687697779$$
$23$ $$T^{6} - 238 T^{5} + \cdots + 37266922552896$$
$29$ $$T^{6} - 8 T^{5} + \cdots + 93900570624$$
$31$ $$(T^{3} - 107 T^{2} - 14005 T + 1435247)^{2}$$
$37$ $$(T^{3} - 305 T^{2} - 125173 T + 28900349)^{2}$$
$41$ $$T^{6} + 16 T^{5} + \cdots + 49106682003456$$
$43$ $$T^{6} - 331 T^{5} + \cdots + 9744392803201$$
$47$ $$T^{6} + \cdots + 407898773822016$$
$53$ $$T^{6} - 118 T^{5} + \cdots + 16\!\cdots\!76$$
$59$ $$T^{6} + 936 T^{5} + \cdots + 10\!\cdots\!44$$
$61$ $$T^{6} - 399 T^{5} + \cdots + 61\!\cdots\!69$$
$67$ $$T^{6} + \cdots + 762088088919249$$
$71$ $$T^{6} + \cdots + 288657109767744$$
$73$ $$T^{6} + 91 T^{5} + \cdots + 27\!\cdots\!21$$
$79$ $$T^{6} - 321 T^{5} + \cdots + 40\!\cdots\!29$$
$83$ $$(T^{3} + 2148 T^{2} + 1271664 T + 211415616)^{2}$$
$89$ $$T^{6} + \cdots + 121838150433024$$
$97$ $$T^{6} + 1382 T^{5} + \cdots + 62\!\cdots\!44$$