# Properties

 Label 38.4.e.a Level $38$ Weight $4$ Character orbit 38.e Analytic conductor $2.242$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 38.e (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.24207258022$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 6 x^{11} - 135 x^{10} + 730 x^{9} + 7953 x^{8} - 36258 x^{7} - 262940 x^{6} + 918855 x^{5} + 5157591 x^{4} - 11890401 x^{3} - 56759508 x^{2} + \cdots + 272110107$$ x^12 - 6*x^11 - 135*x^10 + 730*x^9 + 7953*x^8 - 36258*x^7 - 262940*x^6 + 918855*x^5 + 5157591*x^4 - 11890401*x^3 - 56759508*x^2 + 62864118*x + 272110107 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - 2 \beta_{3} q^{2} + ( - 2 \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{4} + 2 \beta_{2} - 1) q^{3} - 4 \beta_{6} q^{4} + (\beta_{10} + \beta_{9} + 3 \beta_{8} + \beta_{6} + \beta_{5} - 3 \beta_{3} - 2 \beta_1 + 1) q^{5} + (2 \beta_{8} + 4 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2}) q^{6} + (\beta_{9} - 7 \beta_{8} - 4 \beta_{7} + 2 \beta_{6} + 2 \beta_{3} + 6 \beta_{2} - \beta_1) q^{7} + ( - 8 \beta_{7} - 8) q^{8} + (3 \beta_{11} + 3 \beta_{10} + 3 \beta_{7} + 3 \beta_{4} - 3 \beta_{3} - 7 \beta_{2} - 3 \beta_1) q^{9}+O(q^{10})$$ q - 2*b3 * q^2 + (-2*b8 - b7 + 2*b6 - b4 + 2*b2 - 1) * q^3 - 4*b6 * q^4 + (b10 + b9 + 3*b8 + b6 + b5 - 3*b3 - 2*b1 + 1) * q^5 + (2*b8 + 4*b7 - 2*b6 + 2*b5 - 2*b2) * q^6 + (b9 - 7*b8 - 4*b7 + 2*b6 + 2*b3 + 6*b2 - b1) * q^7 + (-8*b7 - 8) * q^8 + (3*b11 + 3*b10 + 3*b7 + 3*b4 - 3*b3 - 7*b2 - 3*b1) * q^9 $$q - 2 \beta_{3} q^{2} + ( - 2 \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{4} + 2 \beta_{2} - 1) q^{3} - 4 \beta_{6} q^{4} + (\beta_{10} + \beta_{9} + 3 \beta_{8} + \beta_{6} + \beta_{5} - 3 \beta_{3} - 2 \beta_1 + 1) q^{5} + (2 \beta_{8} + 4 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2}) q^{6} + (\beta_{9} - 7 \beta_{8} - 4 \beta_{7} + 2 \beta_{6} + 2 \beta_{3} + 6 \beta_{2} - \beta_1) q^{7} + ( - 8 \beta_{7} - 8) q^{8} + (3 \beta_{11} + 3 \beta_{10} + 3 \beta_{7} + 3 \beta_{4} - 3 \beta_{3} - 7 \beta_{2} - 3 \beta_1) q^{9} + ( - 4 \beta_{11} - 2 \beta_{9} + 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + 12 \beta_{2} + \cdots + 2) q^{10}+ \cdots + ( - 55 \beta_{11} - 138 \beta_{10} + 55 \beta_{9} + 183 \beta_{8} + \cdots + 443) q^{99}+O(q^{100})$$ q - 2*b3 * q^2 + (-2*b8 - b7 + 2*b6 - b4 + 2*b2 - 1) * q^3 - 4*b6 * q^4 + (b10 + b9 + 3*b8 + b6 + b5 - 3*b3 - 2*b1 + 1) * q^5 + (2*b8 + 4*b7 - 2*b6 + 2*b5 - 2*b2) * q^6 + (b9 - 7*b8 - 4*b7 + 2*b6 + 2*b3 + 6*b2 - b1) * q^7 + (-8*b7 - 8) * q^8 + (3*b11 + 3*b10 + 3*b7 + 3*b4 - 3*b3 - 7*b2 - 3*b1) * q^9 + (-4*b11 - 2*b9 + 2*b7 - 2*b6 - 2*b4 + 12*b2 + 4*b1 + 2) * q^10 + (5*b11 - 2*b10 - 13*b8 - 4*b7 + 16*b6 - 3*b5 + 2*b4 - 3*b3 - 4*b2 - 6) * q^11 + (-4*b9 - 4*b8 - 4*b7 + 4*b6 + 4*b3 + 8*b2 + 4*b1) * q^12 + (-b11 - 2*b9 + 10*b8 + 7*b7 - 7*b6 - 5*b5 - 2*b4 - 7*b3 - 5*b2 + b1 + 9) * q^13 + (-2*b11 + 6*b8 + 4*b7 - 10*b6 - 2*b4 - 6*b3 - 12*b2 - 8) * q^14 + (-3*b11 - 9*b10 + 3*b9 + 19*b8 + 34*b7 + 3*b6 + 9*b5 + 3*b4 + 28*b3 + 3*b1 + 44) * q^15 + 16*b8 * q^16 + (-5*b11 - b10 + 7*b9 - 33*b7 - 21*b6 - 6*b5 + 22*b3 - 33*b2 - 6*b1 - 6) * q^17 + (-6*b11 - 6*b10 - 6*b8 - 12*b6 + 6*b3 + 6*b1 + 8) * q^18 + (b11 + 4*b10 + 2*b9 + 10*b7 + 9*b6 - 13*b5 - 7*b4 + 18*b2 + 3*b1 - 29) * q^19 + (8*b11 + 8*b10 - 4*b7 - 4*b5 + 4*b4 - 20) * q^20 + (2*b11 + 5*b10 - 5*b9 - 5*b7 + 18*b6 + 10*b3 - 5*b2 - 23) * q^21 + (-10*b10 + 6*b9 + 2*b8 + 32*b7 - 26*b6 + 6*b5 - 40*b2 - 10*b1 + 30) * q^22 + (4*b11 - b10 - 4*b9 + 7*b8 - 12*b7 + 38*b6 + b5 + 13*b4 - 17*b3 + 13*b1 - 23) * q^23 + (8*b11 + 16*b8 + 8*b7 + 8*b4 - 16*b3 - 16*b2 - 8) * q^24 + (-9*b11 - 18*b9 - 57*b8 - 39*b7 + 25*b6 - 18*b4 + 43*b2 + 9*b1 + 18) * q^25 + (2*b11 + 2*b10 + 10*b9 - 20*b8 - 14*b7 + 6*b6 + 2*b5 + 4*b4 + 6*b3 + 10*b2 - 10*b1 - 2) * q^26 + (-9*b11 - 9*b10 - 8*b9 + 49*b8 + 40*b7 - 49*b6 + 18*b5 + 9*b4 + 31) * q^27 + (4*b10 - 8*b8 - 20*b7 + 28*b3 + 16*b2 + 8) * q^28 + (b11 + 26*b10 - 3*b9 - 36*b8 + 26*b7 - 3*b6 + 4*b4 + 13*b3 + 91*b2 - b1 + 39) * q^29 + (6*b11 + 6*b10 - 18*b9 - 56*b8 + 6*b7 + 94*b6 - 12*b5 - 6*b4 - 38*b3 + 38*b2 + 12) * q^30 + (13*b11 - 25*b10 + 9*b9 + 6*b8 - 4*b7 - 34*b6 + 13*b5 - 12*b4 - 72*b3 - 15*b2 - 9*b1 - 13) * q^31 + (32*b6 + 32*b2) * q^32 + (3*b11 + 21*b10 + 21*b9 + 43*b8 - 51*b7 - 138*b6 + 4*b5 + 3*b4 - 43*b3 - 111*b2 - 25*b1 - 141) * q^33 + (-12*b11 + 10*b10 + 12*b9 + 40*b8 - 42*b7 + 36*b6 - 10*b5 - 14*b4 - 20*b3 - 14*b1 + 34) * q^34 + (15*b9 + 15*b8 - 23*b7 + 23*b6 + 15*b5 + 15*b4 + 19*b2 - 34) * q^35 + (12*b11 + 12*b10 - 24*b7 - 12*b5 - 16*b3 - 24*b2 - 12*b1 - 12) * q^36 + (-19*b11 - 19*b10 - 68*b8 + 27*b7 - 33*b6 + 27*b5 - 27*b4 + 11*b3 - b2 + 21*b1 + 96) * q^37 + (6*b11 - 2*b10 + 26*b9 - 50*b8 + 18*b7 + 26*b6 + 16*b5 - 4*b4 + 96*b3 - 18*b1 - 20) * q^38 + (6*b11 + 6*b10 + 12*b8 + 4*b7 + 26*b6 + 4*b5 - 4*b4 - b3 + 17*b2 - 8*b1 - 98) * q^39 + (-16*b10 + 8*b9 + 8*b6 + 8*b5 + 24*b3 + 8*b1 - 16) * q^40 + (7*b10 - 21*b9 - 50*b8 + 117*b7 - 135*b6 - 21*b5 - 3*b4 - 72*b2 + 7*b1 + 82) * q^41 + (-4*b10 + 20*b8 + 36*b7 + 14*b6 + 4*b5 + 10*b4 + 32*b3 + 10*b1 + 42) * q^42 + (-7*b11 + 11*b10 + 11*b9 + 337*b8 + 50*b7 + 18*b6 - 15*b5 - 7*b4 - 337*b3 - 36*b2 + 4*b1 + 25) * q^43 + (-20*b11 - 12*b9 - 64*b8 - 52*b7 - 4*b6 - 12*b4 + 12*b3 + 8*b2 - 8*b1 + 28) * q^44 + (-28*b11 - b10 - 27*b9 - 255*b8 - 202*b7 + 87*b6 - 28*b5 - 29*b4 + 114*b3 + 282*b2 + 27*b1 + 28) * q^45 + (26*b11 - 8*b10 - 2*b9 + 34*b8 + 76*b7 - 20*b6 - 18*b5 + 8*b4 - 14*b3 - 20*b2 + 68) * q^46 + (5*b11 - 31*b10 - 29*b9 + 58*b8 - 155*b7 - 29*b6 + 34*b4 + 126*b3 - 178*b2 - 5*b1 - 29) * q^47 + (-16*b10 - 16*b8 + 16*b3 + 16*b2 + 16) * q^48 + (-10*b11 - 3*b10 + 7*b9 - 53*b8 + 237*b7 + 64*b6 + 13*b5 + 3*b4 - 11*b3 + 18*b2 + 234) * q^49 + (18*b11 + 18*b10 + 114*b8 + 50*b7 - 114*b6 + 18*b5 + 36*b4 - 114*b3 - 114*b2 - 18) * q^50 + (25*b11 - 3*b9 + 9*b8 - 5*b7 + 216*b6 - 17*b5 - 3*b4 + 165*b3 + 219*b2 + 28*b1 - 215) * q^51 + (-20*b11 - 4*b10 - 4*b9 + 12*b8 + 12*b7 - 32*b6 - 4*b5 - 20*b4 - 12*b3 - 20*b2 + 8*b1 - 12) * q^52 + (33*b11 + 19*b10 - 33*b9 - 43*b8 + 106*b7 + 96*b6 - 19*b5 + 43*b4 + 92*b3 + 43*b1 + 6) * q^53 + (18*b10 - 36*b9 - 28*b8 - 98*b7 + 46*b6 - 36*b5 + 16*b4 + 82*b2 + 18*b1 - 80) * q^54 + (-48*b11 - 49*b10 + 41*b9 - 199*b7 - 273*b6 + 8*b5 + 180*b3 - 199*b2 + 8*b1 + 66) * q^55 + (40*b8 + 40*b6 - 56*b3 - 8*b2 + 8*b1 - 32) * q^56 + (13*b11 + 21*b10 - 7*b9 - 102*b8 + 153*b7 - 64*b6 + 3*b5 + 42*b4 + 132*b3 - 266*b2 - 39*b1 - 132) * q^57 + (-2*b11 - 2*b10 - 46*b8 - 6*b7 - 60*b6 - 6*b5 + 6*b4 - 26*b3 - 34*b2 + 52*b1 - 190) * q^58 + (23*b11 - 24*b10 + 36*b9 - 15*b7 + 382*b6 - 12*b5 + 48*b3 - 15*b2 - 12*b1 - 385) * q^59 + (-12*b10 + 24*b9 + 188*b7 - 200*b6 + 24*b5 + 36*b4 - 148*b2 - 12*b1 + 100) * q^60 + (9*b11 + 61*b10 - 9*b9 + 306*b8 - 133*b7 + 97*b6 - 61*b5 - 46*b4 - 81*b3 - 46*b1 + 271) * q^61 + (-18*b11 - 26*b10 - 26*b9 + 16*b8 - 68*b7 - 82*b6 + 50*b5 - 18*b4 - 16*b3 + 30*b2 - 24*b1 - 64) * q^62 + (15*b11 + 45*b9 - 20*b8 - 36*b7 + 8*b6 + 29*b5 + 45*b4 - 20*b3 - 37*b2 - 30*b1 - 10) * q^63 + 64*b7 * q^64 + (-7*b11 + 14*b10 + 173*b8 + 86*b7 - 247*b6 - 7*b5 - 14*b4 + 74*b3 - 53*b2 + 100) * q^65 + (-50*b11 - 6*b10 - 8*b9 + 68*b8 - 276*b7 - 8*b6 - 42*b4 + 216*b3 + 172*b2 + 50*b1 - 60) * q^66 + (24*b11 - 17*b10 + 19*b9 - 317*b8 + 115*b7 + 19*b6 + 5*b4 + 183*b3 + 315*b2 - 24*b1 + 298) * q^67 + (-28*b11 + 24*b10 + 20*b9 + 40*b8 + 72*b7 + 40*b6 + 4*b5 - 24*b4 - 80*b3 + 132*b2 + 96) * q^68 + (-7*b11 - 12*b10 + 35*b9 - 103*b8 + 28*b7 - 206*b6 - 7*b5 - 19*b4 - 211*b3 + 68*b2 - 35*b1 + 7) * q^69 + (-30*b9 + 46*b8 + 46*b7 + 30*b6 - 30*b5 - 30*b4 - 8*b3 + 60*b2 + 30*b1 + 8) * q^70 + (-35*b11 - 68*b10 - 68*b9 + 254*b8 + 48*b7 + 110*b6 + 21*b5 - 35*b4 - 254*b3 + 165*b2 + 47*b1 + 145) * q^71 + (-24*b11 - 24*b10 + 24*b9 + 24*b8 - 32*b6 + 24*b5 + 24) * q^72 + (-18*b10 + 29*b9 - 473*b8 - 101*b7 + 140*b6 + 29*b5 - 10*b4 - 81*b2 - 18*b1 + 73) * q^73 + (42*b11 + 38*b10 - 54*b9 - 66*b7 - 56*b6 + 16*b5 - 196*b3 - 66*b2 + 16*b1 - 26) * q^74 + (111*b11 + 111*b10 + 228*b8 - 27*b7 + 285*b6 - 27*b5 + 27*b4 - 583*b3 - 368*b2 - 70*b1 - 133) * q^75 + (-36*b11 - 12*b10 - 32*b9 - 56*b8 + 52*b7 + 128*b6 + 20*b5 - 52*b4 + 68*b3 - 12*b2 + 28*b1 + 40) * q^76 + (-11*b11 - 11*b10 - 98*b8 + 10*b7 - 89*b6 + 10*b5 - 10*b4 - 93*b3 - 222*b2 - 40*b1 - 93) * q^77 + (-16*b11 - 12*b10 - 8*b9 + 52*b7 + 26*b6 + 20*b5 + 200*b3 + 52*b2 + 20*b1 + 6) * q^78 + (18*b10 + 17*b9 - 203*b8 + 334*b7 - 286*b6 + 17*b5 - 31*b4 - 90*b2 + 18*b1 + 139) * q^79 + (16*b11 - 16*b9 + 16*b7 + 80*b6 - 16*b4 - 16*b1 + 16) * q^80 + (21*b11 + 21*b10 + 21*b9 + 236*b8 - 417*b7 + 96*b6 - 27*b5 + 21*b4 - 236*b3 + 471*b2 + 6*b1 + 75) * q^81 + (14*b11 + 42*b9 - 234*b8 - 270*b7 - 122*b6 + 6*b5 + 42*b4 + 98*b3 - 164*b2 - 28*b1 - 126) * q^82 + (-17*b11 + 30*b10 - 15*b9 - 390*b8 - 162*b7 - 12*b6 - 17*b5 + 13*b4 + 35*b3 + 405*b2 + 15*b1 + 17) * q^83 + (20*b11 - 8*b9 - 64*b8 + 28*b7 + 104*b6 - 20*b5 - 40*b3 + 20*b2 + 28) * q^84 + (50*b11 + 94*b9 + 385*b8 - 384*b7 + 94*b6 - 44*b4 - 95*b3 - 69*b2 - 50*b1 - 479) * q^85 + (8*b11 + 14*b10 + 30*b9 - 152*b8 + 36*b7 + 30*b6 - 22*b4 + 86*b3 + 702*b2 - 8*b1 + 122) * q^86 + (-28*b11 - 14*b10 - 34*b9 - 109*b8 + 513*b7 + 132*b6 + 42*b5 + 14*b4 - 23*b3 + 37*b2 + 499) * q^87 + (-16*b11 + 40*b10 + 128*b8 - 8*b7 - 160*b6 - 16*b5 + 24*b4 - 104*b3 - 128*b2 + 16) * q^88 + (-67*b11 + 24*b9 - 57*b8 + 47*b7 + 240*b6 + 128*b5 + 24*b4 + 606*b3 + 216*b2 - 91*b1 - 472) * q^89 + (54*b11 + 56*b10 + 56*b9 + 402*b8 + 174*b7 - 280*b6 + 2*b5 + 54*b4 - 402*b3 - 564*b2 - 58*b1 - 334) * q^90 + (23*b11 + 41*b10 - 23*b9 + 36*b8 + 66*b7 - 225*b6 - 41*b5 - 32*b4 + 84*b3 - 32*b1 + 152) * q^91 + (-52*b10 + 36*b9 - 184*b8 - 40*b7 + 72*b6 + 36*b5 + 4*b4 - 4*b2 - 52*b1 - 52) * q^92 + (-3*b11 + 12*b10 - 65*b9 - 551*b7 - 385*b6 + 53*b5 + 209*b3 - 551*b2 + 53*b1 - 219) * q^93 + (-10*b11 - 10*b10 + 368*b8 - 58*b7 + 242*b6 - 58*b5 + 58*b4 - 252*b3 - 72*b2 - 62*b1 + 288) * q^94 + (19*b11 - 57*b10 - 19*b9 + 95*b8 + 209*b7 - 456*b6 + 38*b4 + 665*b3 - 247*b2 + 95*b1 + 57) * q^95 + (-32*b3 - 64*b2 - 32*b1 - 32) * q^96 + (-125*b11 + 68*b10 - 72*b9 + 173*b7 + 642*b6 + 4*b5 - 306*b3 + 173*b2 + 4*b1 - 473) * q^97 + (20*b10 - 26*b9 - 462*b8 + 128*b7 - 140*b6 - 26*b5 - 14*b4 - 78*b2 + 20*b1 + 112) * q^98 + (-55*b11 - 138*b10 + 55*b9 + 183*b8 + 396*b7 - 52*b6 + 138*b5 + 53*b4 + 313*b3 + 53*b1 + 443) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 9 q^{3} - 18 q^{6} + 21 q^{7} - 48 q^{8} - 27 q^{9}+O(q^{10})$$ 12 * q - 9 * q^3 - 18 * q^6 + 21 * q^7 - 48 * q^8 - 27 * q^9 $$12 q - 9 q^{3} - 18 q^{6} + 21 q^{7} - 48 q^{8} - 27 q^{9} - 9 q^{11} + 36 q^{12} + 39 q^{13} - 138 q^{14} + 423 q^{15} + 69 q^{17} + 132 q^{18} - 462 q^{19} - 216 q^{20} - 279 q^{21} + 204 q^{22} - 66 q^{23} - 72 q^{24} + 342 q^{25} + 48 q^{26} + 189 q^{27} + 192 q^{28} + 159 q^{29} + 72 q^{31} - 1560 q^{33} + 408 q^{34} - 135 q^{35} - 108 q^{36} + 1116 q^{37} - 294 q^{38} - 1248 q^{39} + 147 q^{41} + 414 q^{42} - 117 q^{43} + 408 q^{44} + 1296 q^{45} + 528 q^{46} + 783 q^{47} + 288 q^{48} + 1413 q^{49} - 354 q^{50} - 2301 q^{51} - 348 q^{52} - 249 q^{53} - 540 q^{54} + 2187 q^{55} - 336 q^{56} - 2670 q^{57} - 1932 q^{58} - 4248 q^{59} + 324 q^{60} + 3114 q^{61} - 438 q^{62} + 363 q^{63} - 384 q^{64} + 495 q^{65} + 822 q^{66} + 3060 q^{67} + 408 q^{68} - 237 q^{69} - 270 q^{70} + 1686 q^{71} + 432 q^{72} + 1626 q^{73} + 90 q^{74} - 1854 q^{75} - 1416 q^{77} - 108 q^{78} - 327 q^{79} + 3483 q^{81} + 294 q^{82} + 927 q^{83} + 204 q^{84} - 3294 q^{85} + 1188 q^{86} + 2892 q^{87} - 72 q^{88} - 6366 q^{89} - 5076 q^{90} + 840 q^{91} - 156 q^{92} + 870 q^{93} + 3432 q^{94} + 513 q^{95} - 576 q^{96} - 8052 q^{97} + 378 q^{98} + 4494 q^{99}+O(q^{100})$$ 12 * q - 9 * q^3 - 18 * q^6 + 21 * q^7 - 48 * q^8 - 27 * q^9 - 9 * q^11 + 36 * q^12 + 39 * q^13 - 138 * q^14 + 423 * q^15 + 69 * q^17 + 132 * q^18 - 462 * q^19 - 216 * q^20 - 279 * q^21 + 204 * q^22 - 66 * q^23 - 72 * q^24 + 342 * q^25 + 48 * q^26 + 189 * q^27 + 192 * q^28 + 159 * q^29 + 72 * q^31 - 1560 * q^33 + 408 * q^34 - 135 * q^35 - 108 * q^36 + 1116 * q^37 - 294 * q^38 - 1248 * q^39 + 147 * q^41 + 414 * q^42 - 117 * q^43 + 408 * q^44 + 1296 * q^45 + 528 * q^46 + 783 * q^47 + 288 * q^48 + 1413 * q^49 - 354 * q^50 - 2301 * q^51 - 348 * q^52 - 249 * q^53 - 540 * q^54 + 2187 * q^55 - 336 * q^56 - 2670 * q^57 - 1932 * q^58 - 4248 * q^59 + 324 * q^60 + 3114 * q^61 - 438 * q^62 + 363 * q^63 - 384 * q^64 + 495 * q^65 + 822 * q^66 + 3060 * q^67 + 408 * q^68 - 237 * q^69 - 270 * q^70 + 1686 * q^71 + 432 * q^72 + 1626 * q^73 + 90 * q^74 - 1854 * q^75 - 1416 * q^77 - 108 * q^78 - 327 * q^79 + 3483 * q^81 + 294 * q^82 + 927 * q^83 + 204 * q^84 - 3294 * q^85 + 1188 * q^86 + 2892 * q^87 - 72 * q^88 - 6366 * q^89 - 5076 * q^90 + 840 * q^91 - 156 * q^92 + 870 * q^93 + 3432 * q^94 + 513 * q^95 - 576 * q^96 - 8052 * q^97 + 378 * q^98 + 4494 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} - 135 x^{10} + 730 x^{9} + 7953 x^{8} - 36258 x^{7} - 262940 x^{6} + 918855 x^{5} + 5157591 x^{4} - 11890401 x^{3} - 56759508 x^{2} + \cdots + 272110107$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 177221217078 \nu^{11} + 3257235752468 \nu^{10} + 7448859189417 \nu^{9} - 348800270203041 \nu^{8} + \cdots - 21\!\cdots\!68 ) / 20\!\cdots\!27$$ (-177221217078*v^11 + 3257235752468*v^10 + 7448859189417*v^9 - 348800270203041*v^8 + 246725060413890*v^7 + 15490888685337464*v^6 - 19811196354376926*v^5 - 357754300688799514*v^4 + 400473915693733842*v^3 + 4306750761297997378*v^2 - 2692306104561693094*v - 21338093845933036168) / 207201391860211327 $$\beta_{3}$$ $$=$$ $$( 177221217078 \nu^{11} + 1307802364610 \nu^{10} - 30274049774807 \nu^{9} - 164426429455098 \nu^{8} + \cdots - 19\!\cdots\!62 ) / 20\!\cdots\!27$$ (177221217078*v^11 + 1307802364610*v^10 - 30274049774807*v^9 - 164426429455098*v^8 + 1943132881731006*v^7 + 8679404020188818*v^6 - 60460050360167694*v^5 - 238663501216099822*v^4 + 928747583953979022*v^3 + 3391717877058441844*v^2 - 5684523648618559763*v - 19685341428196901362) / 207201391860211327 $$\beta_{4}$$ $$=$$ $$( - 488144071095 \nu^{11} + 17846740004059 \nu^{10} - 13839005362779 \nu^{9} + \cdots - 20\!\cdots\!46 ) / 20\!\cdots\!27$$ (-488144071095*v^11 + 17846740004059*v^10 - 13839005362779*v^9 - 2106984070175911*v^8 + 4541779588170822*v^7 + 102961725378166441*v^6 - 217798892218728099*v^5 - 2625515013443047084*v^4 + 4208642241669180665*v^3 + 35298117216643037198*v^2 - 30071865827575778859*v - 200899777020391137346) / 207201391860211327 $$\beta_{5}$$ $$=$$ $$( 488144071095 \nu^{11} + 12477155222014 \nu^{10} - 137780470767586 \nu^{9} + \cdots - 19\!\cdots\!88 ) / 20\!\cdots\!27$$ (488144071095*v^11 + 12477155222014*v^10 - 137780470767586*v^9 - 1508975589988942*v^8 + 10831775909270780*v^7 + 78118444919969751*v^6 - 379885864716473617*v^5 - 2156767180008845078*v^4 + 6370302460492549118*v^3 + 31574050688047386454*v^2 - 42191996299443789347*v - 194202797254829741988) / 207201391860211327 $$\beta_{6}$$ $$=$$ $$( - 545356750193 \nu^{11} + 639178329540 \nu^{10} + 77274249510048 \nu^{9} - 43408610678418 \nu^{8} + \cdots + 19\!\cdots\!01 ) / 20\!\cdots\!27$$ (-545356750193*v^11 + 639178329540*v^10 + 77274249510048*v^9 - 43408610678418*v^8 - 4452252157863755*v^7 - 199135495484657*v^6 + 130514673620858285*v^5 + 80929955889112716*v^4 - 1946446810652341253*v^3 - 2244991753408961063*v^2 + 11804378059999411394*v + 19269854158760561501) / 207201391860211327 $$\beta_{7}$$ $$=$$ $$( 1581666 \nu^{11} - 8699163 \nu^{10} - 191254825 \nu^{9} + 925890435 \nu^{8} + 9722082222 \nu^{7} - 38409003948 \nu^{6} + \cdots + 11710200330886 ) / 486606417103$$ (1581666*v^11 - 8699163*v^10 - 191254825*v^9 + 925890435*v^8 + 9722082222*v^7 - 38409003948*v^6 - 260049871314*v^5 + 748351094985*v^4 + 3667777839865*v^3 - 6269852665791*v^2 - 21765274073007*v + 11710200330886) / 486606417103 $$\beta_{8}$$ $$=$$ $$( 722577967271 \nu^{11} - 4051943557973 \nu^{10} - 83945461319640 \nu^{9} + 426412366504371 \nu^{8} + \cdots + 74\!\cdots\!83 ) / 20\!\cdots\!27$$ (722577967271*v^11 - 4051943557973*v^10 - 83945461319640*v^9 + 426412366504371*v^8 + 4064047270679267*v^7 - 17527628089879562*v^6 - 103497407053708736*v^5 + 342676177232219537*v^4 + 1402092497979595547*v^3 - 2996806964224757833*v^2 - 8103858975645019699*v + 7404279427818802783) / 207201391860211327 $$\beta_{9}$$ $$=$$ $$( 882100559090 \nu^{11} + 8843624294225 \nu^{10} - 174667730262753 \nu^{9} + \cdots - 20\!\cdots\!59 ) / 20\!\cdots\!27$$ (882100559090*v^11 + 8843624294225*v^10 - 174667730262753*v^9 - 1173088304769666*v^8 + 12516666641744675*v^7 + 66554394063895991*v^6 - 430697507264496290*v^5 - 1992730817819811185*v^4 + 7284724960989936728*v^3 + 31203967995735408752*v^2 - 48949073051985878985*v - 202532641899861005459) / 207201391860211327 $$\beta_{10}$$ $$=$$ $$( - 1648551699807 \nu^{11} + 11116259193530 \nu^{10} + 165757370191268 \nu^{9} + \cdots - 21\!\cdots\!01 ) / 20\!\cdots\!27$$ (-1648551699807*v^11 + 11116259193530*v^10 + 165757370191268*v^9 - 1065326603875755*v^8 - 6944287407575079*v^7 + 40202711062797866*v^6 + 151876342577052782*v^5 - 733168791455953581*v^4 - 1726174510506912575*v^3 + 6362770351786971041*v^2 + 7777903016411569028*v - 21134063485749103201) / 207201391860211327 $$\beta_{11}$$ $$=$$ $$( - 1648551699807 \nu^{11} + 7017809504347 \nu^{10} + 186249618637183 \nu^{9} - 654710361086352 \nu^{8} + \cdots + 92\!\cdots\!83 ) / 20\!\cdots\!27$$ (-1648551699807*v^11 + 7017809504347*v^10 + 186249618637183*v^9 - 654710361086352*v^8 - 8709705869408181*v^7 + 22740043057351849*v^6 + 210529378653279533*v^5 - 334306486848652462*v^4 - 2623754819217589847*v^3 + 1262244190577485046*v^2 + 13337295222073937409*v + 9268488754807344483) / 207201391860211327
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{11} + \beta_{10} - 3\beta_{8} - 3\beta_{6} - \beta_{3} - 4\beta_{2} + \beta _1 + 26$$ -b11 + b10 - 3*b8 - 3*b6 - b3 - 4*b2 + b1 + 26 $$\nu^{3}$$ $$=$$ $$3 \beta_{10} - 15 \beta_{8} + 10 \beta_{7} + 6 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 28 \beta_{3} - 25 \beta_{2} + 25 \beta _1 + 32$$ 3*b10 - 15*b8 + 10*b7 + 6*b6 + 3*b5 - 3*b4 + 28*b3 - 25*b2 + 25*b1 + 32 $$\nu^{4}$$ $$=$$ $$- 53 \beta_{11} + 59 \beta_{10} + 10 \beta_{9} - 217 \beta_{8} + 15 \beta_{7} - 165 \beta_{6} - 3 \beta_{5} - 15 \beta_{4} + 37 \beta_{3} - 256 \beta_{2} + 44 \beta _1 + 698$$ -53*b11 + 59*b10 + 10*b9 - 217*b8 + 15*b7 - 165*b6 - 3*b5 - 15*b4 + 37*b3 - 256*b2 + 44*b1 + 698 $$\nu^{5}$$ $$=$$ $$- 81 \beta_{11} + 194 \beta_{10} + 25 \beta_{9} - 1321 \beta_{8} + 892 \beta_{7} + 381 \beta_{6} + 162 \beta_{5} - 207 \beta_{4} + 1665 \beta_{3} - 1414 \beta_{2} + 615 \beta _1 + 1854$$ -81*b11 + 194*b10 + 25*b9 - 1321*b8 + 892*b7 + 381*b6 + 162*b5 - 207*b4 + 1665*b3 - 1414*b2 + 615*b1 + 1854 $$\nu^{6}$$ $$=$$ $$- 2280 \beta_{11} + 2604 \beta_{10} + 917 \beta_{9} - 11518 \beta_{8} + 2205 \beta_{7} - 5675 \beta_{6} - 219 \beta_{5} - 1296 \beta_{4} + 5259 \beta_{3} - 11343 \beta_{2} + 1302 \beta _1 + 19896$$ -2280*b11 + 2604*b10 + 917*b9 - 11518*b8 + 2205*b7 - 5675*b6 - 219*b5 - 1296*b4 + 5259*b3 - 11343*b2 + 1302*b1 + 19896 $$\nu^{7}$$ $$=$$ $$- 6561 \beta_{11} + 9574 \beta_{10} + 3122 \beta_{9} - 76494 \beta_{8} + 52916 \beta_{7} + 19598 \beta_{6} + 5456 \beta_{5} - 10601 \beta_{4} + 78989 \beta_{3} - 60371 \beta_{2} + 14109 \beta _1 + 81501$$ -6561*b11 + 9574*b10 + 3122*b9 - 76494*b8 + 52916*b7 + 19598*b6 + 5456*b5 - 10601*b4 + 78989*b3 - 60371*b2 + 14109*b1 + 81501 $$\nu^{8}$$ $$=$$ $$- 93098 \beta_{11} + 103652 \beta_{10} + 56038 \beta_{9} - 531544 \beta_{8} + 177506 \beta_{7} - 126516 \beta_{6} - 14142 \beta_{5} - 73372 \beta_{4} + 339489 \beta_{3} - 419971 \beta_{2} + \cdots + 609442$$ -93098*b11 + 103652*b10 + 56038*b9 - 531544*b8 + 177506*b7 - 126516*b6 - 14142*b5 - 73372*b4 + 339489*b3 - 419971*b2 + 26559*b1 + 609442 $$\nu^{9}$$ $$=$$ $$- 366048 \beta_{11} + 423666 \beta_{10} + 233544 \beta_{9} - 3638850 \beta_{8} + 2627538 \beta_{7} + 1015026 \beta_{6} + 112374 \beta_{5} - 475506 \beta_{4} + 3478249 \beta_{3} + \cdots + 3220296$$ -366048*b11 + 423666*b10 + 233544*b9 - 3638850*b8 + 2627538*b7 + 1015026*b6 + 112374*b5 - 475506*b4 + 3478249*b3 - 2250730*b2 + 261745*b1 + 3220296 $$\nu^{10}$$ $$=$$ $$- 3739994 \beta_{11} + 3951167 \beta_{10} + 2861082 \beta_{9} - 22427599 \beta_{8} + 10768116 \beta_{7} - 48010 \beta_{6} - 902652 \beta_{5} - 3405306 \beta_{4} + \cdots + 19795826$$ -3739994*b11 + 3951167*b10 + 2861082*b9 - 22427599*b8 + 10768116*b7 - 48010*b6 - 902652*b5 - 3405306*b4 + 17512846*b3 - 13691236*b2 + 64789*b1 + 19795826 $$\nu^{11}$$ $$=$$ $$- 17577635 \beta_{11} + 17659182 \beta_{10} + 13629198 \beta_{9} - 153343365 \beta_{8} + 118000897 \beta_{7} + 53163670 \beta_{6} - 854642 \beta_{5} - 19566517 \beta_{4} + \cdots + 119730930$$ -17577635*b11 + 17659182*b10 + 13629198*b9 - 153343365*b8 + 118000897*b7 + 53163670*b6 - 854642*b5 - 19566517*b4 + 147712096*b3 - 75222735*b2 + 1401338*b1 + 119730930

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$\chi(n)$$ $$-\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 5.05412 − 0.342020i −4.05412 − 0.342020i 5.30460 − 0.984808i −4.30460 − 0.984808i 5.30460 + 0.984808i −4.30460 + 0.984808i 5.05412 + 0.342020i −4.05412 + 0.342020i −5.28151 + 0.642788i 6.28151 + 0.642788i −5.28151 − 0.642788i 6.28151 − 0.642788i
−1.87939 + 0.684040i −1.54081 + 8.73839i 3.06418 2.57115i −15.3487 12.8791i −3.08163 17.4768i 4.71270 + 8.16264i −4.00000 + 6.92820i −48.6136 17.6939i 37.6558 + 13.7056i
5.2 −1.87939 + 0.684040i 0.0408138 0.231467i 3.06418 2.57115i 8.45426 + 7.09396i 0.0816277 + 0.462933i 9.26682 + 16.0506i −4.00000 + 6.92820i 25.3198 + 9.21565i −20.7414 7.54924i
9.1 0.347296 + 1.96962i −4.43054 + 3.71766i −3.75877 + 1.36808i −5.82452 2.11995i −8.86108 7.43533i −5.61124 + 9.71895i −4.00000 6.92820i 1.12015 6.35271i 2.15265 12.2083i
9.2 0.347296 + 1.96962i 2.93054 2.45902i −3.75877 + 1.36808i 14.2818 + 5.19813i 5.86108 + 4.91803i −0.806634 + 1.39713i −4.00000 6.92820i −2.14719 + 12.1773i −5.27832 + 29.9349i
17.1 0.347296 1.96962i −4.43054 3.71766i −3.75877 1.36808i −5.82452 + 2.11995i −8.86108 + 7.43533i −5.61124 9.71895i −4.00000 + 6.92820i 1.12015 + 6.35271i 2.15265 + 12.2083i
17.2 0.347296 1.96962i 2.93054 + 2.45902i −3.75877 1.36808i 14.2818 5.19813i 5.86108 4.91803i −0.806634 1.39713i −4.00000 + 6.92820i −2.14719 12.1773i −5.27832 29.9349i
23.1 −1.87939 0.684040i −1.54081 8.73839i 3.06418 + 2.57115i −15.3487 + 12.8791i −3.08163 + 17.4768i 4.71270 8.16264i −4.00000 6.92820i −48.6136 + 17.6939i 37.6558 13.7056i
23.2 −1.87939 0.684040i 0.0408138 + 0.231467i 3.06418 + 2.57115i 8.45426 7.09396i 0.0816277 0.462933i 9.26682 16.0506i −4.00000 6.92820i 25.3198 9.21565i −20.7414 + 7.54924i
25.1 1.53209 1.28558i −6.18284 2.25037i 0.694593 3.93923i −1.97089 11.1775i −12.3657 + 4.50074i 4.35993 7.55162i −4.00000 6.92820i 12.4801 + 10.4721i −17.3890 14.5911i
25.2 1.53209 1.28558i 4.68284 + 1.70441i 0.694593 3.93923i 0.408053 + 2.31419i 9.36568 3.40883i −1.42158 + 2.46225i −4.00000 6.92820i −1.65925 1.39227i 3.60023 + 3.02095i
35.1 1.53209 + 1.28558i −6.18284 + 2.25037i 0.694593 + 3.93923i −1.97089 + 11.1775i −12.3657 4.50074i 4.35993 + 7.55162i −4.00000 + 6.92820i 12.4801 10.4721i −17.3890 + 14.5911i
35.2 1.53209 + 1.28558i 4.68284 1.70441i 0.694593 + 3.93923i 0.408053 2.31419i 9.36568 + 3.40883i −1.42158 2.46225i −4.00000 + 6.92820i −1.65925 + 1.39227i 3.60023 3.02095i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 35.2 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.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.4.e.a 12
19.e even 9 1 inner 38.4.e.a 12
19.e even 9 1 722.4.a.p 6
19.f odd 18 1 722.4.a.o 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.e.a 12 1.a even 1 1 trivial
38.4.e.a 12 19.e even 9 1 inner
722.4.a.o 6 19.f odd 18 1
722.4.a.p 6 19.e even 9 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 9 T_{3}^{11} + 54 T_{3}^{10} + 18 T_{3}^{9} - 3078 T_{3}^{8} - 12789 T_{3}^{7} + 104059 T_{3}^{6} + 455373 T_{3}^{5} - 1524069 T_{3}^{4} - 7640667 T_{3}^{3} + 41991777 T_{3}^{2} + \cdots + 2289169$$ acting on $$S_{4}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{6} + 8 T^{3} + 64)^{2}$$
$3$ $$T^{12} + 9 T^{11} + 54 T^{10} + \cdots + 2289169$$
$5$ $$T^{12} - 171 T^{10} + \cdots + 308668025241$$
$7$ $$T^{12} - 21 T^{11} + \cdots + 6147971281$$
$11$ $$T^{12} + 9 T^{11} + \cdots + 55\!\cdots\!09$$
$13$ $$T^{12} - 39 T^{11} + \cdots + 25\!\cdots\!61$$
$17$ $$T^{12} - 69 T^{11} + \cdots + 41\!\cdots\!89$$
$19$ $$T^{12} + 462 T^{11} + \cdots + 10\!\cdots\!41$$
$23$ $$T^{12} + 66 T^{11} + \cdots + 16\!\cdots\!24$$
$29$ $$T^{12} - 159 T^{11} + \cdots + 42\!\cdots\!49$$
$31$ $$T^{12} - 72 T^{11} + \cdots + 12\!\cdots\!09$$
$37$ $$(T^{6} - 558 T^{5} + \cdots - 20292669795592)^{2}$$
$41$ $$T^{12} - 147 T^{11} + \cdots + 38\!\cdots\!69$$
$43$ $$T^{12} + 117 T^{11} + \cdots + 50\!\cdots\!21$$
$47$ $$T^{12} - 783 T^{11} + \cdots + 11\!\cdots\!81$$
$53$ $$T^{12} + 249 T^{11} + \cdots + 10\!\cdots\!69$$
$59$ $$T^{12} + 4248 T^{11} + \cdots + 36\!\cdots\!09$$
$61$ $$T^{12} - 3114 T^{11} + \cdots + 28\!\cdots\!49$$
$67$ $$T^{12} - 3060 T^{11} + \cdots + 46\!\cdots\!24$$
$71$ $$T^{12} - 1686 T^{11} + \cdots + 82\!\cdots\!04$$
$73$ $$T^{12} - 1626 T^{11} + \cdots + 20\!\cdots\!64$$
$79$ $$T^{12} + 327 T^{11} + \cdots + 63\!\cdots\!89$$
$83$ $$T^{12} - 927 T^{11} + \cdots + 42\!\cdots\!49$$
$89$ $$T^{12} + 6366 T^{11} + \cdots + 80\!\cdots\!69$$
$97$ $$T^{12} + 8052 T^{11} + \cdots + 11\!\cdots\!89$$