# Properties

 Label 38.4.c.c Level $38$ Weight $4$ Character orbit 38.c Analytic conductor $2.242$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 38.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.24207258022$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - x^{5} + 64x^{4} + 33x^{3} + 3984x^{2} - 945x + 225$$ x^6 - x^5 + 64*x^4 + 33*x^3 + 3984*x^2 - 945*x + 225 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{4} + 2) q^{2} + (2 \beta_{4} - \beta_{2} - \beta_1 - 2) q^{3} - 4 \beta_{4} q^{4} + ( - \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 - 1) q^{5} + (4 \beta_{4} - 2 \beta_1) q^{6} + (\beta_{2} + 9) q^{7} - 8 q^{8} + (2 \beta_{5} - 20 \beta_{4} - 2 \beta_{3} + 4 \beta_1) q^{9}+O(q^{10})$$ q + (-2*b4 + 2) * q^2 + (2*b4 - b2 - b1 - 2) * q^3 - 4*b4 * q^4 + (-b5 + b4 - b2 - b1 - 1) * q^5 + (4*b4 - 2*b1) * q^6 + (b2 + 9) * q^7 - 8 * q^8 + (2*b5 - 20*b4 - 2*b3 + 4*b1) * q^9 $$q + ( - 2 \beta_{4} + 2) q^{2} + (2 \beta_{4} - \beta_{2} - \beta_1 - 2) q^{3} - 4 \beta_{4} q^{4} + ( - \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 - 1) q^{5} + (4 \beta_{4} - 2 \beta_1) q^{6} + (\beta_{2} + 9) q^{7} - 8 q^{8} + (2 \beta_{5} - 20 \beta_{4} - 2 \beta_{3} + 4 \beta_1) q^{9} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{10} + (3 \beta_{3} - \beta_{2} + 2) q^{11} + (4 \beta_{2} + 8) q^{12} + (\beta_{5} + 44 \beta_{4} - \beta_{3} - 4 \beta_1) q^{13} + ( - 18 \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 18) q^{14} + (3 \beta_{5} - 31 \beta_{4} - 3 \beta_{3} + 13 \beta_1) q^{15} + (16 \beta_{4} - 16) q^{16} + ( - 3 \beta_{5} + 18 \beta_{4} - 18) q^{17} + ( - 4 \beta_{3} - 8 \beta_{2} - 40) q^{18} + (3 \beta_{5} + 26 \beta_{4} - 2 \beta_{3} + 8 \beta_{2} + 13 \beta_1 - 6) q^{19} + (4 \beta_{3} + 4 \beta_{2} + 4) q^{20} + ( - 2 \beta_{5} + 61 \beta_{4} - 11 \beta_{2} - 11 \beta_1 - 61) q^{21} + (6 \beta_{5} - 4 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 4) q^{22} + (\beta_{5} + 17 \beta_{4} - \beta_{3} - 5 \beta_1) q^{23} + ( - 16 \beta_{4} + 8 \beta_{2} + 8 \beta_1 + 16) q^{24} + ( - 9 \beta_{5} - 113 \beta_{4} + 9 \beta_{3} + 10 \beta_1) q^{25} + ( - 2 \beta_{3} + 8 \beta_{2} + 88) q^{26} + (10 \beta_{3} + 21 \beta_{2} + 130) q^{27} + ( - 36 \beta_{4} + 4 \beta_1) q^{28} + (5 \beta_{5} - 35 \beta_{4} - 5 \beta_{3} - 25 \beta_1) q^{29} + ( - 6 \beta_{3} - 26 \beta_{2} - 62) q^{30} + ( - 6 \beta_{3} + 23 \beta_{2} - 11) q^{31} + 32 \beta_{4} q^{32} + ( - \beta_{5} - 81 \beta_{4} - 30 \beta_{2} - 30 \beta_1 + 81) q^{33} + ( - 6 \beta_{5} + 36 \beta_{4} + 6 \beta_{3}) q^{34} + ( - 10 \beta_{5} + 38 \beta_{4} - 20 \beta_{2} - 20 \beta_1 - 38) q^{35} + ( - 8 \beta_{5} + 80 \beta_{4} - 16 \beta_{2} - 16 \beta_1 - 80) q^{36} + (6 \beta_{3} + 5 \beta_{2} - 59) q^{37} + (2 \beta_{5} + 12 \beta_{4} - 6 \beta_{3} - 10 \beta_{2} + 16 \beta_1 + 40) q^{38} + ( - 7 \beta_{3} - 42 \beta_{2} - 274) q^{39} + (8 \beta_{5} - 8 \beta_{4} + 8 \beta_{2} + 8 \beta_1 + 8) q^{40} + ( - 4 \beta_{5} - 147 \beta_{4} - 30 \beta_{2} - 30 \beta_1 + 147) q^{41} + ( - 4 \beta_{5} + 122 \beta_{4} + 4 \beta_{3} - 22 \beta_1) q^{42} + (7 \beta_{5} + 8 \beta_{4} + 42 \beta_{2} + 42 \beta_1 - 8) q^{43} + (12 \beta_{5} - 8 \beta_{4} - 12 \beta_{3} - 4 \beta_1) q^{44} + (2 \beta_{3} + 60 \beta_{2} + 552) q^{45} + ( - 2 \beta_{3} + 10 \beta_{2} + 34) q^{46} + ( - 5 \beta_{5} - 85 \beta_{4} + 5 \beta_{3} + 19 \beta_1) q^{47} + ( - 32 \beta_{4} + 16 \beta_1) q^{48} + (2 \beta_{3} + 18 \beta_{2} - 219) q^{49} + (18 \beta_{3} - 20 \beta_{2} - 226) q^{50} + (3 \beta_{5} + 6 \beta_{4} - 3 \beta_{3} + 48 \beta_1) q^{51} + ( - 4 \beta_{5} - 176 \beta_{4} + 16 \beta_{2} + 16 \beta_1 + 176) q^{52} + (5 \beta_{5} - 5 \beta_{3} + 24 \beta_1) q^{53} + (20 \beta_{5} - 260 \beta_{4} + 42 \beta_{2} + 42 \beta_1 + 260) q^{54} + (32 \beta_{5} + 597 \beta_{4} + 15 \beta_{2} + 15 \beta_1 - 597) q^{55} + ( - 8 \beta_{2} - 72) q^{56} + ( - 17 \beta_{5} + 318 \beta_{4} + 29 \beta_{3} + 10 \beta_{2} + \cdots + 147) q^{57}+ \cdots + ( - 20 \beta_{5} - 1060 \beta_{4} + 20 \beta_{3} + 16 \beta_1) q^{99}+O(q^{100})$$ q + (-2*b4 + 2) * q^2 + (2*b4 - b2 - b1 - 2) * q^3 - 4*b4 * q^4 + (-b5 + b4 - b2 - b1 - 1) * q^5 + (4*b4 - 2*b1) * q^6 + (b2 + 9) * q^7 - 8 * q^8 + (2*b5 - 20*b4 - 2*b3 + 4*b1) * q^9 + (-2*b5 + 2*b4 + 2*b3 - 2*b1) * q^10 + (3*b3 - b2 + 2) * q^11 + (4*b2 + 8) * q^12 + (b5 + 44*b4 - b3 - 4*b1) * q^13 + (-18*b4 + 2*b2 + 2*b1 + 18) * q^14 + (3*b5 - 31*b4 - 3*b3 + 13*b1) * q^15 + (16*b4 - 16) * q^16 + (-3*b5 + 18*b4 - 18) * q^17 + (-4*b3 - 8*b2 - 40) * q^18 + (3*b5 + 26*b4 - 2*b3 + 8*b2 + 13*b1 - 6) * q^19 + (4*b3 + 4*b2 + 4) * q^20 + (-2*b5 + 61*b4 - 11*b2 - 11*b1 - 61) * q^21 + (6*b5 - 4*b4 - 2*b2 - 2*b1 + 4) * q^22 + (b5 + 17*b4 - b3 - 5*b1) * q^23 + (-16*b4 + 8*b2 + 8*b1 + 16) * q^24 + (-9*b5 - 113*b4 + 9*b3 + 10*b1) * q^25 + (-2*b3 + 8*b2 + 88) * q^26 + (10*b3 + 21*b2 + 130) * q^27 + (-36*b4 + 4*b1) * q^28 + (5*b5 - 35*b4 - 5*b3 - 25*b1) * q^29 + (-6*b3 - 26*b2 - 62) * q^30 + (-6*b3 + 23*b2 - 11) * q^31 + 32*b4 * q^32 + (-b5 - 81*b4 - 30*b2 - 30*b1 + 81) * q^33 + (-6*b5 + 36*b4 + 6*b3) * q^34 + (-10*b5 + 38*b4 - 20*b2 - 20*b1 - 38) * q^35 + (-8*b5 + 80*b4 - 16*b2 - 16*b1 - 80) * q^36 + (6*b3 + 5*b2 - 59) * q^37 + (2*b5 + 12*b4 - 6*b3 - 10*b2 + 16*b1 + 40) * q^38 + (-7*b3 - 42*b2 - 274) * q^39 + (8*b5 - 8*b4 + 8*b2 + 8*b1 + 8) * q^40 + (-4*b5 - 147*b4 - 30*b2 - 30*b1 + 147) * q^41 + (-4*b5 + 122*b4 + 4*b3 - 22*b1) * q^42 + (7*b5 + 8*b4 + 42*b2 + 42*b1 - 8) * q^43 + (12*b5 - 8*b4 - 12*b3 - 4*b1) * q^44 + (2*b3 + 60*b2 + 552) * q^45 + (-2*b3 + 10*b2 + 34) * q^46 + (-5*b5 - 85*b4 + 5*b3 + 19*b1) * q^47 + (-32*b4 + 16*b1) * q^48 + (2*b3 + 18*b2 - 219) * q^49 + (18*b3 - 20*b2 - 226) * q^50 + (3*b5 + 6*b4 - 3*b3 + 48*b1) * q^51 + (-4*b5 - 176*b4 + 16*b2 + 16*b1 + 176) * q^52 + (5*b5 - 5*b3 + 24*b1) * q^53 + (20*b5 - 260*b4 + 42*b2 + 42*b1 + 260) * q^54 + (32*b5 + 597*b4 + 15*b2 + 15*b1 - 597) * q^55 + (-8*b2 - 72) * q^56 + (-17*b5 + 318*b4 + 29*b3 + 10*b2 - 20*b1 + 147) * q^57 + (-10*b3 + 50*b2 - 70) * q^58 + (-8*b5 + 358*b4 - b2 - b1 - 358) * q^59 + (-12*b5 + 124*b4 - 52*b2 - 52*b1 - 124) * q^60 + (b5 - 337*b4 - b3 + 29*b1) * q^61 + (-12*b5 + 22*b4 + 46*b2 + 46*b1 - 22) * q^62 + (24*b5 - 324*b4 - 24*b3 + 76*b1) * q^63 + 64 * q^64 + (-59*b3 - 90*b2 + 48) * q^65 + (-2*b5 - 162*b4 + 2*b3 - 60*b1) * q^66 + (-16*b5 + 320*b4 + 16*b3 - 67*b1) * q^67 + (12*b3 + 72) * q^68 + (-9*b3 - 17*b2 - 263) * q^69 + (-20*b5 + 76*b4 + 20*b3 - 40*b1) * q^70 + (17*b5 - 710*b4 - 22*b2 - 22*b1 + 710) * q^71 + (-16*b5 + 160*b4 + 16*b3 - 32*b1) * q^72 + (10*b5 - 221*b4 - 14*b2 - 14*b1 + 221) * q^73 + (12*b5 + 118*b4 + 10*b2 + 10*b1 - 118) * q^74 + (11*b3 + 43*b2 + 782) * q^75 + (-8*b5 - 80*b4 - 4*b3 - 52*b2 - 20*b1 + 104) * q^76 + (22*b3 + 23*b2 - 67) * q^77 + (-14*b5 + 548*b4 - 84*b2 - 84*b1 - 548) * q^78 + (-29*b5 - 570*b4 + 68*b2 + 68*b1 + 570) * q^79 + (16*b5 - 16*b4 - 16*b3 + 16*b1) * q^80 + (2*b5 + 483*b4 - 164*b2 - 164*b1 - 483) * q^81 + (-8*b5 - 294*b4 + 8*b3 - 60*b1) * q^82 + (21*b3 - 105*b2 + 168) * q^83 + (8*b3 + 44*b2 + 244) * q^84 + (-15*b5 - 642*b4 + 15*b3 + 12*b1) * q^85 + (14*b5 + 16*b4 - 14*b3 + 84*b1) * q^86 + (-45*b3 + 35*b2 - 1075) * q^87 + (-24*b3 + 8*b2 - 16) * q^88 + (-5*b5 + 262*b4 + 5*b3 - 88*b1) * q^89 + (4*b5 - 1104*b4 + 120*b2 + 120*b1 + 1104) * q^90 + (582*b4 - 70*b1) * q^91 + (-4*b5 - 68*b4 + 20*b2 + 20*b1 + 68) * q^92 + (-40*b5 + 1051*b4 + 25*b2 + 25*b1 - 1051) * q^93 + (10*b3 - 38*b2 - 170) * q^94 + (9*b5 + 434*b4 - 46*b3 + 31*b2 - 80*b1 + 541) * q^95 + (-32*b2 - 64) * q^96 + (42*b5 + 247*b4 + 202*b2 + 202*b1 - 247) * q^97 + (4*b5 + 438*b4 + 36*b2 + 36*b1 - 438) * q^98 + (-20*b5 - 1060*b4 + 20*b3 + 16*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{2} - 5 q^{3} - 12 q^{4} - q^{5} + 10 q^{6} + 52 q^{7} - 48 q^{8} - 54 q^{9}+O(q^{10})$$ 6 * q + 6 * q^2 - 5 * q^3 - 12 * q^4 - q^5 + 10 * q^6 + 52 * q^7 - 48 * q^8 - 54 * q^9 $$6 q + 6 q^{2} - 5 q^{3} - 12 q^{4} - q^{5} + 10 q^{6} + 52 q^{7} - 48 q^{8} - 54 q^{9} + 2 q^{10} + 8 q^{11} + 40 q^{12} + 129 q^{13} + 52 q^{14} - 77 q^{15} - 48 q^{16} - 51 q^{17} - 216 q^{18} + 40 q^{19} + 8 q^{20} - 170 q^{21} + 8 q^{22} + 47 q^{23} + 40 q^{24} - 338 q^{25} + 516 q^{26} + 718 q^{27} - 104 q^{28} - 125 q^{29} - 308 q^{30} - 100 q^{31} + 96 q^{32} + 274 q^{33} + 102 q^{34} - 84 q^{35} - 216 q^{36} - 376 q^{37} + 322 q^{38} - 1546 q^{39} + 8 q^{40} + 475 q^{41} + 340 q^{42} - 73 q^{43} - 16 q^{44} + 3188 q^{45} + 188 q^{46} - 241 q^{47} - 80 q^{48} - 1354 q^{49} - 1352 q^{50} + 69 q^{51} + 516 q^{52} + 29 q^{53} + 718 q^{54} - 1838 q^{55} - 416 q^{56} + 1755 q^{57} - 500 q^{58} - 1065 q^{59} - 308 q^{60} - 981 q^{61} - 100 q^{62} - 872 q^{63} + 384 q^{64} + 586 q^{65} - 548 q^{66} + 877 q^{67} + 408 q^{68} - 1526 q^{69} + 168 q^{70} + 2135 q^{71} + 432 q^{72} + 667 q^{73} - 376 q^{74} + 4584 q^{75} + 484 q^{76} - 492 q^{77} - 1546 q^{78} + 1671 q^{79} - 16 q^{80} - 1287 q^{81} - 950 q^{82} + 1176 q^{83} + 1360 q^{84} - 1929 q^{85} + 146 q^{86} - 6430 q^{87} - 64 q^{88} + 693 q^{89} + 3188 q^{90} + 1676 q^{91} + 188 q^{92} - 3138 q^{93} - 964 q^{94} + 4489 q^{95} - 320 q^{96} - 985 q^{97} - 1354 q^{98} - 3184 q^{99}+O(q^{100})$$ 6 * q + 6 * q^2 - 5 * q^3 - 12 * q^4 - q^5 + 10 * q^6 + 52 * q^7 - 48 * q^8 - 54 * q^9 + 2 * q^10 + 8 * q^11 + 40 * q^12 + 129 * q^13 + 52 * q^14 - 77 * q^15 - 48 * q^16 - 51 * q^17 - 216 * q^18 + 40 * q^19 + 8 * q^20 - 170 * q^21 + 8 * q^22 + 47 * q^23 + 40 * q^24 - 338 * q^25 + 516 * q^26 + 718 * q^27 - 104 * q^28 - 125 * q^29 - 308 * q^30 - 100 * q^31 + 96 * q^32 + 274 * q^33 + 102 * q^34 - 84 * q^35 - 216 * q^36 - 376 * q^37 + 322 * q^38 - 1546 * q^39 + 8 * q^40 + 475 * q^41 + 340 * q^42 - 73 * q^43 - 16 * q^44 + 3188 * q^45 + 188 * q^46 - 241 * q^47 - 80 * q^48 - 1354 * q^49 - 1352 * q^50 + 69 * q^51 + 516 * q^52 + 29 * q^53 + 718 * q^54 - 1838 * q^55 - 416 * q^56 + 1755 * q^57 - 500 * q^58 - 1065 * q^59 - 308 * q^60 - 981 * q^61 - 100 * q^62 - 872 * q^63 + 384 * q^64 + 586 * q^65 - 548 * q^66 + 877 * q^67 + 408 * q^68 - 1526 * q^69 + 168 * q^70 + 2135 * q^71 + 432 * q^72 + 667 * q^73 - 376 * q^74 + 4584 * q^75 + 484 * q^76 - 492 * q^77 - 1546 * q^78 + 1671 * q^79 - 16 * q^80 - 1287 * q^81 - 950 * q^82 + 1176 * q^83 + 1360 * q^84 - 1929 * q^85 + 146 * q^86 - 6430 * q^87 - 64 * q^88 + 693 * q^89 + 3188 * q^90 + 1676 * q^91 + 188 * q^92 - 3138 * q^93 - 964 * q^94 + 4489 * q^95 - 320 * q^96 - 985 * q^97 - 1354 * q^98 - 3184 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 64x^{4} + 33x^{3} + 3984x^{2} - 945x + 225$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} - 64\nu^{4} + 4096\nu^{3} - 3984\nu^{2} + 945\nu - 60480 ) / 254031$$ (v^5 - 64*v^4 + 4096*v^3 - 3984*v^2 + 945*v - 60480) / 254031 $$\beta_{3}$$ $$=$$ $$( 21\nu^{5} - 1344\nu^{4} + 1339\nu^{3} - 83664\nu^{2} + 19845\nu - 3641036 ) / 169354$$ (21*v^5 - 1344*v^4 + 1339*v^3 - 83664*v^2 + 19845*v - 3641036) / 169354 $$\beta_{4}$$ $$=$$ $$( 1344\nu^{5} - 1339\nu^{4} + 85696\nu^{3} + 64832\nu^{2} + 5334576\nu + 4800 ) / 1270155$$ (1344*v^5 - 1339*v^4 + 85696*v^3 + 64832*v^2 + 5334576*v + 4800) / 1270155 $$\beta_{5}$$ $$=$$ $$( 57792\nu^{5} - 57577\nu^{4} + 3684928\nu^{3} + 1517621\nu^{2} + 229386768\nu - 54410265 ) / 2540310$$ (57792*v^5 - 57577*v^4 + 3684928*v^3 + 1517621*v^2 + 229386768*v - 54410265) / 2540310
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-2\beta_{5} + 43\beta_{4} - 43$$ -2*b5 + 43*b4 - 43 $$\nu^{3}$$ $$=$$ $$-2\beta_{3} + 63\beta_{2} - 28$$ -2*b3 + 63*b2 - 28 $$\nu^{4}$$ $$=$$ $$128\beta_{5} - 2737\beta_{4} - 128\beta_{3} - 48\beta_1$$ 128*b5 - 2737*b4 - 128*b3 - 48*b1 $$\nu^{5}$$ $$=$$ $$224\beta_{5} - 3856\beta_{4} - 4017\beta_{2} - 4017\beta _1 + 3856$$ 224*b5 - 3856*b4 - 4017*b2 - 4017*b1 + 3856

## Character values

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

 $$n$$ $$21$$ $$\chi(n)$$ $$-\beta_{4}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −3.78825 + 6.56144i 0.118706 − 0.205606i 4.16954 − 7.22186i −3.78825 − 6.56144i 0.118706 + 0.205606i 4.16954 + 7.22186i
1.00000 + 1.73205i −4.78825 8.29349i −2.00000 + 3.46410i −7.88908 13.6643i 9.57650 16.5870i 16.5765 −8.00000 −32.3546 + 56.0399i 15.7782 27.3286i
7.2 1.00000 + 1.73205i −0.881294 1.52645i −2.00000 + 3.46410i 10.3546 + 17.9347i 1.76259 3.05289i 8.76259 −8.00000 11.9466 20.6922i −20.7092 + 35.8694i
7.3 1.00000 + 1.73205i 3.16954 + 5.48981i −2.00000 + 3.46410i −2.96554 5.13646i −6.33908 + 10.9796i 0.660916 −8.00000 −6.59199 + 11.4177i 5.93108 10.2729i
11.1 1.00000 1.73205i −4.78825 + 8.29349i −2.00000 3.46410i −7.88908 + 13.6643i 9.57650 + 16.5870i 16.5765 −8.00000 −32.3546 56.0399i 15.7782 + 27.3286i
11.2 1.00000 1.73205i −0.881294 + 1.52645i −2.00000 3.46410i 10.3546 17.9347i 1.76259 + 3.05289i 8.76259 −8.00000 11.9466 + 20.6922i −20.7092 35.8694i
11.3 1.00000 1.73205i 3.16954 5.48981i −2.00000 3.46410i −2.96554 + 5.13646i −6.33908 10.9796i 0.660916 −8.00000 −6.59199 11.4177i 5.93108 + 10.2729i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.3 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 38.4.c.c 6
3.b odd 2 1 342.4.g.f 6
4.b odd 2 1 304.4.i.e 6
19.c even 3 1 inner 38.4.c.c 6
19.c even 3 1 722.4.a.j 3
19.d odd 6 1 722.4.a.k 3
57.h odd 6 1 342.4.g.f 6
76.g odd 6 1 304.4.i.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.c.c 6 1.a even 1 1 trivial
38.4.c.c 6 19.c even 3 1 inner
304.4.i.e 6 4.b odd 2 1
304.4.i.e 6 76.g odd 6 1
342.4.g.f 6 3.b odd 2 1
342.4.g.f 6 57.h odd 6 1
722.4.a.j 3 19.c even 3 1
722.4.a.k 3 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 5T_{3}^{5} + 80T_{3}^{4} - 61T_{3}^{3} + 3560T_{3}^{2} + 5885T_{3} + 11449$$ acting on $$S_{4}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 4)^{3}$$
$3$ $$T^{6} + 5 T^{5} + 80 T^{4} + \cdots + 11449$$
$5$ $$T^{6} + T^{5} + 357 T^{4} + \cdots + 3755844$$
$7$ $$(T^{3} - 26 T^{2} + 162 T - 96)^{2}$$
$11$ $$(T^{3} - 4 T^{2} - 3311 T + 49980)^{2}$$
$13$ $$T^{6} - 129 T^{5} + \cdots + 129322384$$
$17$ $$T^{6} + 51 T^{5} + \cdots + 11293737984$$
$19$ $$T^{6} - 40 T^{5} + \cdots + 322687697779$$
$23$ $$T^{6} - 47 T^{5} + \cdots + 4555440036$$
$29$ $$T^{6} + 125 T^{5} + \cdots + 5937750562500$$
$31$ $$(T^{3} + 50 T^{2} - 52150 T + 3809848)^{2}$$
$37$ $$(T^{3} + 188 T^{2} - 658 T - 88004)^{2}$$
$41$ $$T^{6} - 475 T^{5} + \cdots + 81183541856481$$
$43$ $$T^{6} + \cdots + 259289089231936$$
$47$ $$T^{6} + 241 T^{5} + \cdots + 25671752892900$$
$53$ $$T^{6} - 29 T^{5} + \cdots + 10857156800400$$
$59$ $$T^{6} + 1065 T^{5} + \cdots + 12\!\cdots\!21$$
$61$ $$T^{6} + \cdots + 356206057990084$$
$67$ $$T^{6} - 877 T^{5} + \cdots + 964239549849$$
$71$ $$T^{6} - 2135 T^{5} + \cdots + 68\!\cdots\!16$$
$73$ $$T^{6} - 667 T^{5} + \cdots + 713681971209$$
$79$ $$T^{6} - 1671 T^{5} + \cdots + 15\!\cdots\!00$$
$83$ $$(T^{3} - 588 T^{2} - 848043 T - 162474984)^{2}$$
$89$ $$T^{6} - 693 T^{5} + \cdots + 27\!\cdots\!04$$
$97$ $$T^{6} + 985 T^{5} + \cdots + 68\!\cdots\!25$$