# Properties

 Label 1875.4.a.f Level $1875$ Weight $4$ Character orbit 1875.a Self dual yes Analytic conductor $110.629$ Analytic rank $0$ Dimension $14$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1875.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$110.628581261$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Defining polynomial: $$x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + 125779 x^{5} - 718713 x^{4} - 371750 x^{3} + 579848 x^{2} + \cdots + 42064$$ x^14 - 81*x^12 - 7*x^11 + 2512*x^10 + 517*x^9 - 36970*x^8 - 12987*x^7 + 257291*x^6 + 125779*x^5 - 718713*x^4 - 371750*x^3 + 579848*x^2 + 394896*x + 42064 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 5^{3}$$ Twist minimal: no (minimal twist has level 75) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + 3 \beta_1 q^{6} + ( - \beta_{8} - \beta_{5} - \beta_{3} + 2 \beta_1 + 1) q^{7} + ( - \beta_{6} - 3 \beta_{3} - 3 \beta_1 - 3) q^{8} + 9 q^{9}+O(q^{10})$$ q - b1 * q^2 - 3 * q^3 + (b2 + 4) * q^4 + 3*b1 * q^6 + (-b8 - b5 - b3 + 2*b1 + 1) * q^7 + (-b6 - 3*b3 - 3*b1 - 3) * q^8 + 9 * q^9 $$q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + 3 \beta_1 q^{6} + ( - \beta_{8} - \beta_{5} - \beta_{3} + 2 \beta_1 + 1) q^{7} + ( - \beta_{6} - 3 \beta_{3} - 3 \beta_1 - 3) q^{8} + 9 q^{9} + ( - \beta_{13} - 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 4) q^{11} + ( - 3 \beta_{2} - 12) q^{12} + ( - \beta_{13} + 2 \beta_{12} - 2 \beta_{7} - \beta_{6} + \beta_{4} + 5 \beta_{3} - \beta_1 + 15) q^{13} + ( - \beta_{13} + 3 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} - 3 \beta_{5} + \cdots - 26) q^{14}+ \cdots + ( - 9 \beta_{13} - 18 \beta_{9} - 9 \beta_{8} + 9 \beta_{7} - 9 \beta_{6} + \cdots - 36) q^{99}+O(q^{100})$$ q - b1 * q^2 - 3 * q^3 + (b2 + 4) * q^4 + 3*b1 * q^6 + (-b8 - b5 - b3 + 2*b1 + 1) * q^7 + (-b6 - 3*b3 - 3*b1 - 3) * q^8 + 9 * q^9 + (-b13 - 2*b9 - b8 + b7 - b6 - b5 + b4 - b3 - 4) * q^11 + (-3*b2 - 12) * q^12 + (-b13 + 2*b12 - 2*b7 - b6 + b4 + 5*b3 - b1 + 15) * q^13 + (-b13 + 3*b11 - b10 + b9 + b8 - 2*b7 - 3*b5 + b4 - 10*b3 - 2*b2 + 2*b1 - 26) * q^14 + (-b13 - b11 + b10 + 2*b9 - b8 - 2*b6 - b5 - 2*b4 - b2 - 3*b1 + 5) * q^16 + (-b13 + 2*b12 + 2*b11 - 2*b10 + 2*b9 - 2*b7 - 2*b5 - 2*b4 + 7*b3 + b2 + 15) * q^17 - 9*b1 * q^18 + (-2*b13 + b12 + 4*b11 - b10 + b9 - b8 - 2*b7 - 2*b6 - 9*b5 + 2*b3 - 3*b2 - 7*b1 - 14) * q^19 + (3*b8 + 3*b5 + 3*b3 - 6*b1 - 3) * q^21 + (-4*b13 - 2*b12 + 4*b11 - b10 + 3*b9 + 2*b8 - 2*b7 - b6 - 2*b5 - b4 - 15*b3 + 5*b2 + b1 + 15) * q^22 + (-b13 - 3*b12 + b11 - b10 - 4*b9 + 2*b8 + b7 - b6 - 6*b5 + 3*b4 + b3 + 9*b1 + 12) * q^23 + (3*b6 + 9*b3 + 9*b1 + 9) * q^24 + (4*b13 + 6*b12 - 4*b11 - 3*b10 + 7*b9 - 7*b8 - 5*b7 - b6 + 16*b5 - 3*b4 - 6*b3 + 3*b2 - 39*b1 + 5) * q^26 - 27 * q^27 + (b13 + 3*b11 + b10 - 3*b9 - 3*b8 + 6*b7 + 3*b6 - 15*b5 + b4 - 25*b3 + 4*b2 + 40*b1 - 27) * q^28 + (-b13 + 2*b12 - 2*b9 + 2*b8 - 4*b7 - b6 - 7*b5 - 2*b4 + 35*b3 - 4*b1 + 35) * q^29 + (4*b13 + 6*b12 - 4*b11 - 5*b10 - b8 - 6*b7 - 5*b5 - b4 - 8*b3 - 9*b2 - 11*b1 - 71) * q^31 + (-3*b13 - b12 - 3*b11 + 5*b10 - 2*b9 - 3*b8 + 4*b7 - 2*b6 + 15*b5 + 4*b4 + 18*b3 + 2*b2 + 16*b1 + 36) * q^32 + (3*b13 + 6*b9 + 3*b8 - 3*b7 + 3*b6 + 3*b5 - 3*b4 + 3*b3 + 12) * q^33 + (3*b13 - b12 - 6*b11 + 3*b10 - 7*b9 - 4*b8 + b7 - 3*b6 + 10*b5 + 11*b4 - 12*b3 - 6*b2 - 10*b1 - 30) * q^34 + (9*b2 + 36) * q^36 + (-2*b13 - 6*b12 - 4*b11 + 5*b10 - 12*b9 - b8 + 8*b7 - 11*b6 + 13*b5 + 7*b3 - 4*b2 + 6*b1 + 44) * q^37 + (-2*b13 - 2*b12 + 7*b11 + 2*b10 - 10*b8 + 2*b7 - b6 - 11*b5 + 4*b4 - 91*b3 + 19*b2 + 50*b1 + 89) * q^38 + (3*b13 - 6*b12 + 6*b7 + 3*b6 - 3*b4 - 15*b3 + 3*b1 - 45) * q^39 + (-7*b13 - b12 - 6*b11 + 7*b10 - b9 - b8 + 4*b7 - 13*b6 - 17*b5 - b4 + b3 - 17*b2 + 16*b1 - 23) * q^41 + (3*b13 - 9*b11 + 3*b10 - 3*b9 - 3*b8 + 6*b7 + 9*b5 - 3*b4 + 30*b3 + 6*b2 - 6*b1 + 78) * q^42 + (-2*b13 - 2*b12 + 4*b11 - 6*b10 - 8*b9 + 6*b8 + 6*b7 - b6 - 7*b5 + 6*b4 - 8*b3 + 9*b2 - 14*b1 + 52) * q^43 + (2*b13 - 8*b12 + b11 + 12*b10 - 5*b9 - 13*b8 + 19*b7 - 9*b6 - 14*b5 + 5*b4 - 38*b3 + b2 - 14*b1 - 24) * q^44 + (-8*b13 - 5*b12 + 16*b11 + 2*b10 - 2*b9 - 4*b8 + 10*b7 + b6 - 10*b5 - 8*b4 - 91*b3 + 17*b2 - 9*b1 - 68) * q^46 + (-3*b13 - 13*b11 + 3*b10 - 13*b9 + 4*b8 + 4*b7 + 9*b6 - 19*b5 + b4 - 25*b3 + 17*b1 - 23) * q^47 + (3*b13 + 3*b11 - 3*b10 - 6*b9 + 3*b8 + 6*b6 + 3*b5 + 6*b4 + 3*b2 + 9*b1 - 15) * q^48 + (-3*b13 + 3*b12 - 9*b11 - 8*b9 - 12*b8 + 4*b7 - 2*b6 + 57*b5 + 12*b4 + 76*b3 + 10*b2 - 19*b1 + 38) * q^49 + (3*b13 - 6*b12 - 6*b11 + 6*b10 - 6*b9 + 6*b7 + 6*b5 + 6*b4 - 21*b3 - 3*b2 - 45) * q^51 + (18*b13 + 2*b12 - 26*b11 - 8*b10 + 19*b9 + 2*b8 - 16*b7 + 3*b6 + 29*b5 - b4 + 175*b3 + 11*b2 - 53*b1 + 238) * q^52 + (-7*b13 + 10*b12 + 15*b11 + 5*b10 + 3*b9 - 5*b8 - 3*b7 - 13*b6 - 34*b5 - 3*b4 + 61*b3 + 16*b2 + 50*b1 + 102) * q^53 + 27*b1 * q^54 + (4*b13 - 12*b12 - 2*b11 + 2*b10 - 7*b9 + 16*b8 + 6*b7 + 11*b6 - 52*b5 - 3*b4 - 91*b3 - 24*b2 + 49*b1 - 218) * q^56 + (6*b13 - 3*b12 - 12*b11 + 3*b10 - 3*b9 + 3*b8 + 6*b7 + 6*b6 + 27*b5 - 6*b3 + 9*b2 + 21*b1 + 42) * q^57 + (8*b13 + 7*b12 - 2*b10 + 3*b9 - 3*b8 - 7*b7 - 2*b6 + 60*b5 - 3*b4 - 79*b3 + 13*b2 - 85*b1 + 42) * q^58 + (17*b13 - 2*b12 - 4*b11 - 11*b10 + 30*b9 - 14*b8 - 20*b7 + 14*b6 - 19*b4 + 55*b3 + 20*b2 - 39*b1 + 37) * q^59 + (2*b13 + 6*b12 + 35*b11 - 5*b10 + 9*b9 + 5*b8 - 23*b7 + 2*b6 - 29*b5 - 7*b4 - 181*b3 + 28*b2 + 30*b1 - 107) * q^61 + (16*b13 + 20*b12 - 6*b11 - 11*b10 + 19*b9 + 5*b8 - 33*b7 + 18*b6 + 37*b5 - 5*b4 - 15*b3 + 20*b2 + 61*b1 + 112) * q^62 + (-9*b8 - 9*b5 - 9*b3 + 18*b1 + 9) * q^63 + (-b13 + 2*b11 - 12*b10 - 4*b9 + 6*b8 + b7 + 6*b6 + 31*b5 + 8*b4 + 141*b3 - 21*b2 - 44*b1 - 214) * q^64 + (12*b13 + 6*b12 - 12*b11 + 3*b10 - 9*b9 - 6*b8 + 6*b7 + 3*b6 + 6*b5 + 3*b4 + 45*b3 - 15*b2 - 3*b1 - 45) * q^66 + (6*b13 + b12 - 15*b11 - 19*b10 + 2*b9 + 13*b8 + b7 + 18*b6 - 47*b5 - 2*b4 + 104*b3 + 2*b2 - 28*b1 + 201) * q^67 + (4*b13 + 4*b12 - b11 - 7*b10 + 35*b9 - 14*b7 + 17*b6 + 6*b5 - 29*b4 + 29*b3 + 33*b2 - 12*b1 + 89) * q^68 + (3*b13 + 9*b12 - 3*b11 + 3*b10 + 12*b9 - 6*b8 - 3*b7 + 3*b6 + 18*b5 - 9*b4 - 3*b3 - 27*b1 - 36) * q^69 + (-3*b13 + 15*b12 + 2*b11 - 11*b10 - 7*b9 + 4*b8 + 4*b7 + 2*b6 + b5 + 19*b4 - 33*b3 + 40*b2 - 66*b1 + 179) * q^71 + (-9*b6 - 27*b3 - 27*b1 - 27) * q^72 + (6*b13 + 7*b12 + 48*b11 + 6*b10 + b9 + 14*b8 - 10*b7 + 6*b6 - 4*b5 - 25*b4 + 25*b3 + 22*b2 - 36) * q^73 + (-29*b13 - 15*b12 + 9*b10 + 19*b9 + 20*b8 + b7 - 20*b6 + 28*b5 - 29*b4 + 155*b3 + 31*b2 - 99*b1 + 45) * q^74 + (-b13 - 17*b12 + 6*b11 - b10 + 4*b9 + 7*b8 + 8*b7 - 98*b5 + 4*b4 - 194*b3 - 17*b2 - 77*b1 - 506) * q^76 + (-9*b13 + 3*b12 - 19*b11 + 3*b10 - 18*b9 - 17*b8 + 6*b7 - 3*b6 + 23*b5 + 29*b4 + 54*b3 - 6*b2 + 22*b1 + 392) * q^77 + (-12*b13 - 18*b12 + 12*b11 + 9*b10 - 21*b9 + 21*b8 + 15*b7 + 3*b6 - 48*b5 + 9*b4 + 18*b3 - 9*b2 + 117*b1 - 15) * q^78 + (-2*b13 - 21*b12 - 40*b11 - 5*b10 - 15*b9 + 3*b8 + 15*b7 + 3*b6 - 10*b5 + 20*b4 + 122*b3 + 6*b2 - 28*b1 + 7) * q^79 + 81 * q^81 + (-20*b13 - 4*b12 + 5*b11 + 17*b10 + 13*b9 - 15*b8 + 17*b7 - 30*b6 + 73*b5 - 13*b4 - 189*b3 + 28*b2 + 35*b1 - 154) * q^82 + (18*b13 - 12*b12 + 11*b11 + 7*b10 - 7*b9 - 23*b8 + 11*b7 + 3*b6 + 58*b5 + 7*b4 + 60*b3 + 37*b2 - 156*b1 + 422) * q^83 + (-3*b13 - 9*b11 - 3*b10 + 9*b9 + 9*b8 - 18*b7 - 9*b6 + 45*b5 - 3*b4 + 75*b3 - 12*b2 - 120*b1 + 81) * q^84 + (-13*b13 - 20*b12 + 8*b11 + 9*b10 - 12*b9 + 3*b8 + 20*b7 + b6 - 38*b5 - 4*b4 - 131*b3 + 51*b2 - 71*b1 + 240) * q^86 + (3*b13 - 6*b12 + 6*b9 - 6*b8 + 12*b7 + 3*b6 + 21*b5 + 6*b4 - 105*b3 + 12*b1 - 105) * q^87 + (-11*b13 - 12*b12 + 14*b11 + 7*b10 + 17*b9 + 22*b8 + 5*b7 - 4*b6 - 119*b5 - 23*b4 - 45*b3 + 15*b2 + 75*b1 + 210) * q^88 + (16*b13 + 30*b12 + 52*b11 - 12*b10 + 16*b9 - 13*b8 - 24*b7 + 18*b6 + 76*b5 - b4 + 196*b3 + 26*b2 - 91*b1 + 16) * q^89 + (-22*b13 - 9*b12 + 10*b11 + 6*b10 - 47*b9 + b8 + 18*b7 + 23*b6 - 116*b5 + 15*b4 + 217*b3 - 55*b2 + 70*b1 - 16) * q^91 + (-21*b13 - 31*b12 + 10*b11 + 30*b10 - 20*b9 + 8*b8 + 27*b7 - 18*b6 - 157*b5 + 26*b4 - 107*b3 - 21*b2 + 87*b1 + 41) * q^92 + (-12*b13 - 18*b12 + 12*b11 + 15*b10 + 3*b8 + 18*b7 + 15*b5 + 3*b4 + 24*b3 + 27*b2 + 33*b1 + 213) * q^93 + (9*b13 + 6*b12 + 30*b11 - 15*b10 - 14*b9 + 31*b8 - 8*b7 + 13*b6 + 52*b5 + 6*b4 - 275*b3 - 28*b2 - 60*b1 - 176) * q^94 + (9*b13 + 3*b12 + 9*b11 - 15*b10 + 6*b9 + 9*b8 - 12*b7 + 6*b6 - 45*b5 - 12*b4 - 54*b3 - 6*b2 - 48*b1 - 108) * q^96 + (-13*b13 + 16*b12 - 22*b11 - 11*b10 + 4*b9 + 8*b8 + 3*b7 - b6 - 2*b5 + 10*b4 - 356*b3 + 19*b2 - 233*b1 - 151) * q^97 + (-15*b13 + 13*b12 - 28*b11 - 26*b10 + 41*b9 + 7*b8 - 36*b7 - 6*b6 + 102*b5 - 11*b4 + 600*b3 - 26*b2 - 161*b1 + 218) * q^98 + (-9*b13 - 18*b9 - 9*b8 + 9*b7 - 9*b6 - 9*b5 + 9*b4 - 9*b3 - 36) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q - 42 q^{3} + 50 q^{4} + 27 q^{7} - 21 q^{8} + 126 q^{9}+O(q^{10})$$ 14 * q - 42 * q^3 + 50 * q^4 + 27 * q^7 - 21 * q^8 + 126 * q^9 $$14 q - 42 q^{3} + 50 q^{4} + 27 q^{7} - 21 q^{8} + 126 q^{9} - 33 q^{11} - 150 q^{12} + 188 q^{13} - 287 q^{14} + 82 q^{16} + 146 q^{17} - 184 q^{19} - 81 q^{21} + 277 q^{22} + 164 q^{23} + 63 q^{24} + 103 q^{26} - 378 q^{27} - 224 q^{28} + 252 q^{29} - 889 q^{31} + 422 q^{32} + 99 q^{33} - 230 q^{34} + 450 q^{36} + 642 q^{37} + 1833 q^{38} - 564 q^{39} - 164 q^{41} + 861 q^{42} + 696 q^{43} - 6 q^{44} - 419 q^{46} - 92 q^{47} - 246 q^{48} + 81 q^{49} - 438 q^{51} + 1956 q^{52} + 949 q^{53} - 2380 q^{56} + 552 q^{57} + 1041 q^{58} - 81 q^{59} - 496 q^{61} + 1454 q^{62} + 243 q^{63} - 3903 q^{64} - 831 q^{66} + 1926 q^{67} + 685 q^{68} - 492 q^{69} + 2498 q^{71} - 189 q^{72} - 1026 q^{73} - 707 q^{74} - 5704 q^{76} + 5434 q^{77} - 309 q^{78} - 695 q^{79} + 1134 q^{81} - 886 q^{82} + 5315 q^{83} + 672 q^{84} + 3997 q^{86} - 756 q^{87} + 2969 q^{88} - 1424 q^{89} - 1194 q^{91} + 1607 q^{92} + 2667 q^{93} - 629 q^{94} - 1266 q^{96} + 291 q^{97} - 1018 q^{98} - 297 q^{99}+O(q^{100})$$ 14 * q - 42 * q^3 + 50 * q^4 + 27 * q^7 - 21 * q^8 + 126 * q^9 - 33 * q^11 - 150 * q^12 + 188 * q^13 - 287 * q^14 + 82 * q^16 + 146 * q^17 - 184 * q^19 - 81 * q^21 + 277 * q^22 + 164 * q^23 + 63 * q^24 + 103 * q^26 - 378 * q^27 - 224 * q^28 + 252 * q^29 - 889 * q^31 + 422 * q^32 + 99 * q^33 - 230 * q^34 + 450 * q^36 + 642 * q^37 + 1833 * q^38 - 564 * q^39 - 164 * q^41 + 861 * q^42 + 696 * q^43 - 6 * q^44 - 419 * q^46 - 92 * q^47 - 246 * q^48 + 81 * q^49 - 438 * q^51 + 1956 * q^52 + 949 * q^53 - 2380 * q^56 + 552 * q^57 + 1041 * q^58 - 81 * q^59 - 496 * q^61 + 1454 * q^62 + 243 * q^63 - 3903 * q^64 - 831 * q^66 + 1926 * q^67 + 685 * q^68 - 492 * q^69 + 2498 * q^71 - 189 * q^72 - 1026 * q^73 - 707 * q^74 - 5704 * q^76 + 5434 * q^77 - 309 * q^78 - 695 * q^79 + 1134 * q^81 - 886 * q^82 + 5315 * q^83 + 672 * q^84 + 3997 * q^86 - 756 * q^87 + 2969 * q^88 - 1424 * q^89 - 1194 * q^91 + 1607 * q^92 + 2667 * q^93 - 629 * q^94 - 1266 * q^96 + 291 * q^97 - 1018 * q^98 - 297 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + 125779 x^{5} - 718713 x^{4} - 371750 x^{3} + 579848 x^{2} + \cdots + 42064$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 12$$ v^2 - 12 $$\beta_{3}$$ $$=$$ $$( - 154777191 \nu^{13} + 214678293 \nu^{12} + 12768694032 \nu^{11} - 14915595999 \nu^{10} - 404069408515 \nu^{9} + \cdots - 29392108806288 ) / 9573422384000$$ (-154777191*v^13 + 214678293*v^12 + 12768694032*v^11 - 14915595999*v^10 - 404069408515*v^9 + 370272399598*v^8 + 6092827974316*v^7 - 3872467992151*v^6 - 43818200690808*v^5 + 15486244836395*v^4 + 128798752266598*v^3 - 19532525709704*v^2 - 107932745742176*v - 29392108806288) / 9573422384000 $$\beta_{4}$$ $$=$$ $$( - 39842017881 \nu^{13} + 18793858963 \nu^{12} + 3381917381312 \nu^{11} + 617153824991 \nu^{10} - 107039813741365 \nu^{9} + \cdots - 61\!\cdots\!08 ) / 20\!\cdots\!00$$ (-39842017881*v^13 + 18793858963*v^12 + 3381917381312*v^11 + 617153824991*v^10 - 107039813741365*v^9 - 86380604207582*v^8 + 1577495446704356*v^7 + 2255658862839559*v^6 - 11311893366723728*v^5 - 22364379485677555*v^4 + 41656166382191418*v^3 + 80521422793290536*v^2 - 85954039584517216*v - 61602820198807408) / 2039138967792000 $$\beta_{5}$$ $$=$$ $$( - 214678293 \nu^{13} - 231741561 \nu^{12} + 15999036336 \nu^{11} + 15269104723 \nu^{10} - 450292207345 \nu^{9} + \cdots - 6510547762224 ) / 9573422384000$$ (-214678293*v^13 - 231741561*v^12 + 15999036336*v^11 + 15269104723*v^10 - 450292207345*v^9 - 370715223046*v^8 + 5882559371668*v^7 + 3995422441227*v^6 - 34953965143184*v^5 - 17558372991415*v^4 + 77070946463954*v^3 + 18185501095208*v^2 - 31728784810848*v - 6510547762224) / 9573422384000 $$\beta_{6}$$ $$=$$ $$( 464331573 \nu^{13} - 644034879 \nu^{12} - 38306082096 \nu^{11} + 44746787997 \nu^{10} + 1212208225545 \nu^{9} + \cdots + 59456059266864 ) / 9573422384000$$ (464331573*v^13 - 644034879*v^12 - 38306082096*v^11 + 44746787997*v^10 + 1212208225545*v^9 - 1110817198794*v^8 - 18278483922948*v^7 + 11617403976453*v^6 + 131454602072424*v^5 - 46458734509185*v^4 - 376822834415794*v^3 + 58597577129112*v^2 + 141903211930528*v + 59456059266864) / 9573422384000 $$\beta_{7}$$ $$=$$ $$( 52774069869 \nu^{13} - 146787187287 \nu^{12} - 4038749648088 \nu^{11} + 10590731904141 \nu^{10} + 114153083031385 \nu^{9} + \cdots - 24\!\cdots\!08 ) / 679712989264000$$ (52774069869*v^13 - 146787187287*v^12 - 4038749648088*v^11 + 10590731904141*v^10 + 114153083031385*v^9 - 284994761799082*v^8 - 1435508388868644*v^7 + 3540851664674509*v^6 + 7351605029930072*v^5 - 20698866949726305*v^4 - 7399996906970082*v^3 + 52601762509172536*v^2 - 16843498300706016*v - 24134699084790608) / 679712989264000 $$\beta_{8}$$ $$=$$ $$( 198268707541 \nu^{13} - 330612632143 \nu^{12} - 15446080755832 \nu^{11} + 24130842178949 \nu^{10} + 453486628821265 \nu^{9} + \cdots - 30\!\cdots\!12 ) / 20\!\cdots\!00$$ (198268707541*v^13 - 330612632143*v^12 - 15446080755832*v^11 + 24130842178949*v^10 + 453486628821265*v^9 - 626111554905898*v^8 - 6126575299424516*v^7 + 6864964367337301*v^6 + 36372690966037208*v^5 - 30340650774791145*v^4 - 67233637278192898*v^3 + 60972601507328504*v^2 + 9290772958676576*v - 30074258777775312) / 2039138967792000 $$\beta_{9}$$ $$=$$ $$( - 24723383429 \nu^{13} - 19698303673 \nu^{12} + 1994905817228 \nu^{11} + 1654834981079 \nu^{10} - 61648333032485 \nu^{9} + \cdots - 91\!\cdots\!52 ) / 203913896779200$$ (-24723383429*v^13 - 19698303673*v^12 + 1994905817228*v^11 + 1654834981079*v^10 - 61648333032485*v^9 - 54212075098438*v^8 + 907965942281524*v^7 + 849773703313171*v^6 - 6436658577450832*v^5 - 6141712440159795*v^4 + 19640680670547962*v^3 + 15485072554819544*v^2 - 21750605688750304*v - 9141952716611952) / 203913896779200 $$\beta_{10}$$ $$=$$ $$( - 8146109293 \nu^{13} + 6363639039 \nu^{12} + 663666222536 \nu^{11} - 440450615677 \nu^{10} - 20765197179345 \nu^{9} + \cdots - 212680826375024 ) / 53661551784000$$ (-8146109293*v^13 + 6363639039*v^12 + 663666222536*v^11 - 440450615677*v^10 - 20765197179345*v^9 + 10887849198954*v^8 + 310790823668868*v^7 - 117002587471373*v^6 - 2247892398399984*v^5 + 586318929748585*v^4 + 6973604604781954*v^3 - 1764109520445592*v^2 - 7322318299380448*v - 212680826375024) / 53661551784000 $$\beta_{11}$$ $$=$$ $$( - 2089067853 \nu^{13} + 1186234119 \nu^{12} + 166990685056 \nu^{11} - 90007487317 \nu^{10} - 5108559447745 \nu^{9} + \cdots - 228794009350704 ) / 9573422384000$$ (-2089067853*v^13 + 1186234119*v^12 + 166990685056*v^11 - 90007487317*v^10 - 5108559447745*v^9 + 2389171674634*v^8 + 74321331141828*v^7 - 26188788364733*v^6 - 516374760265864*v^5 + 108613804503785*v^4 + 1483963872871634*v^3 - 151211736871832*v^2 - 1216927897475808*v - 228794009350704) / 9573422384000 $$\beta_{12}$$ $$=$$ $$( 268898217 \nu^{13} - 60162011 \nu^{12} - 21662421664 \nu^{11} + 3229526593 \nu^{10} + 662126520605 \nu^{9} - 31650082226 \nu^{8} + \cdots + 16216865165616 ) / 957342238400$$ (268898217*v^13 - 60162011*v^12 - 21662421664*v^11 + 3229526593*v^10 + 662126520605*v^9 - 31650082226*v^8 - 9449742817332*v^7 - 684436038823*v^6 + 61605242598856*v^5 + 10543225997835*v^4 - 145774591408826*v^3 - 13288458871592*v^2 + 70339095459552*v + 16216865165616) / 957342238400 $$\beta_{13}$$ $$=$$ $$( - 629711814493 \nu^{13} + 211929067839 \nu^{12} + 51141259110536 \nu^{11} - 12148429797277 \nu^{10} + \cdots - 10\!\cdots\!24 ) / 20\!\cdots\!00$$ (-629711814493*v^13 + 211929067839*v^12 + 51141259110536*v^11 - 12148429797277*v^10 - 1593816456351345*v^9 + 171646433410554*v^8 + 23644160022864468*v^7 + 951266497072627*v^6 - 166468628891531184*v^5 - 27127244705617415*v^4 + 469758003849131554*v^3 + 70922058848184008*v^2 - 351247214799828448*v - 108688968627954224) / 2039138967792000
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 12$$ b2 + 12 $$\nu^{3}$$ $$=$$ $$\beta_{6} + 3\beta_{3} + 19\beta _1 + 3$$ b6 + 3*b3 + 19*b1 + 3 $$\nu^{4}$$ $$=$$ $$- \beta_{13} - \beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{8} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + 23 \beta_{2} - 3 \beta _1 + 229$$ -b13 - b11 + b10 + 2*b9 - b8 - 2*b6 - b5 - 2*b4 + 23*b2 - 3*b1 + 229 $$\nu^{5}$$ $$=$$ $$3 \beta_{13} + \beta_{12} + 3 \beta_{11} - 5 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - 4 \beta_{7} + 34 \beta_{6} - 15 \beta_{5} - 4 \beta_{4} + 78 \beta_{3} - 2 \beta_{2} + 400 \beta _1 + 60$$ 3*b13 + b12 + 3*b11 - 5*b10 + 2*b9 + 3*b8 - 4*b7 + 34*b6 - 15*b5 - 4*b4 + 78*b3 - 2*b2 + 400*b1 + 60 $$\nu^{6}$$ $$=$$ $$- 41 \beta_{13} - 38 \beta_{11} + 28 \beta_{10} + 76 \beta_{9} - 34 \beta_{8} + \beta_{7} - 74 \beta_{6} - 9 \beta_{5} - 72 \beta_{4} + 141 \beta_{3} + 515 \beta_{2} - 164 \beta _1 + 4850$$ -41*b13 - 38*b11 + 28*b10 + 76*b9 - 34*b8 + b7 - 74*b6 - 9*b5 - 72*b4 + 141*b3 + 515*b2 - 164*b1 + 4850 $$\nu^{7}$$ $$=$$ $$122 \beta_{13} + 49 \beta_{12} + 146 \beta_{11} - 202 \beta_{10} + 106 \beta_{9} + 128 \beta_{8} - 180 \beta_{7} + 931 \beta_{6} - 724 \beta_{5} - 174 \beta_{4} + 1565 \beta_{3} - 104 \beta_{2} + 8783 \beta _1 + 636$$ 122*b13 + 49*b12 + 146*b11 - 202*b10 + 106*b9 + 128*b8 - 180*b7 + 931*b6 - 724*b5 - 174*b4 + 1565*b3 - 104*b2 + 8783*b1 + 636 $$\nu^{8}$$ $$=$$ $$- 1257 \beta_{13} - 9 \beta_{12} - 1152 \beta_{11} + 672 \beta_{10} + 2204 \beta_{9} - 872 \beta_{8} + 23 \beta_{7} - 2125 \beta_{6} + 421 \beta_{5} - 2046 \beta_{4} + 6944 \beta_{3} + 11581 \beta_{2} + \cdots + 107016$$ -1257*b13 - 9*b12 - 1152*b11 + 672*b10 + 2204*b9 - 872*b8 + 23*b7 - 2125*b6 + 421*b5 - 2046*b4 + 6944*b3 + 11581*b2 - 6028*b1 + 107016 $$\nu^{9}$$ $$=$$ $$3623 \beta_{13} + 1534 \beta_{12} + 5048 \beta_{11} - 6120 \beta_{10} + 3642 \beta_{9} + 4046 \beta_{8} - 5813 \beta_{7} + 23756 \beta_{6} - 24641 \beta_{5} - 5368 \beta_{4} + 27815 \beta_{3} - 3748 \beta_{2} + \cdots - 5982$$ 3623*b13 + 1534*b12 + 5048*b11 - 6120*b10 + 3642*b9 + 4046*b8 - 5813*b7 + 23756*b6 - 24641*b5 - 5368*b4 + 27815*b3 - 3748*b2 + 197465*b1 - 5982 $$\nu^{10}$$ $$=$$ $$- 34186 \beta_{13} - 247 \beta_{12} - 32506 \beta_{11} + 15550 \beta_{10} + 58426 \beta_{9} - 20316 \beta_{8} - 212 \beta_{7} - 56078 \beta_{6} + 26260 \beta_{5} - 53402 \beta_{4} + 239954 \beta_{3} + \cdots + 2413393$$ -34186*b13 - 247*b12 - 32506*b11 + 15550*b10 + 58426*b9 - 20316*b8 - 212*b7 - 56078*b6 + 26260*b5 - 53402*b4 + 239954*b3 + 262069*b2 - 188740*b1 + 2413393 $$\nu^{11}$$ $$=$$ $$95312 \beta_{13} + 39355 \beta_{12} + 152581 \beta_{11} - 166261 \beta_{10} + 104498 \beta_{9} + 114853 \beta_{8} - 163757 \beta_{7} + 587178 \beta_{6} - 732762 \beta_{5} - 144636 \beta_{4} + \cdots - 620070$$ 95312*b13 + 39355*b12 + 152581*b11 - 166261*b10 + 104498*b9 + 114853*b8 - 163757*b7 + 587178*b6 - 732762*b5 - 144636*b4 + 438203*b3 - 118030*b2 + 4511346*b1 - 620070 $$\nu^{12}$$ $$=$$ $$- 873549 \beta_{13} + 1297 \beta_{12} - 886792 \beta_{11} + 355354 \beta_{10} + 1487098 \beta_{9} - 455066 \beta_{8} - 37205 \beta_{7} - 1425812 \beta_{6} + 998693 \beta_{5} + \cdots + 55193599$$ -873549*b13 + 1297*b12 - 886792*b11 + 355354*b10 + 1487098*b9 - 455066*b8 - 37205*b7 - 1425812*b6 + 998693*b5 - 1336102*b4 + 7225185*b3 + 5964281*b2 - 5445306*b1 + 55193599 $$\nu^{13}$$ $$=$$ $$2359274 \beta_{13} + 891798 \beta_{12} + 4304259 \beta_{11} - 4273639 \beta_{10} + 2722752 \beta_{9} + 3092949 \beta_{8} - 4288261 \beta_{7} + 14271698 \beta_{6} - 20388722 \beta_{5} + \cdots - 25133913$$ 2359274*b13 + 891798*b12 + 4304259*b11 - 4273639*b10 + 2722752*b9 + 3092949*b8 - 4288261*b7 + 14271698*b6 - 20388722*b5 - 3635478*b4 + 5475971*b3 - 3467652*b2 + 104295128*b1 - 25133913

## 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
 4.81720 4.65416 4.24928 3.90684 1.73740 1.33976 −0.134967 −0.574607 −0.955230 −2.33957 −2.93781 −4.01755 −4.79301 −4.95189
−4.81720 −3.00000 15.2054 0 14.4516 −1.13492 −34.7100 9.00000 0
1.2 −4.65416 −3.00000 13.6612 0 13.9625 26.0445 −26.3483 9.00000 0
1.3 −4.24928 −3.00000 10.0564 0 12.7478 28.2853 −8.73812 9.00000 0
1.4 −3.90684 −3.00000 7.26339 0 11.7205 −22.0918 2.87782 9.00000 0
1.5 −1.73740 −3.00000 −4.98145 0 5.21219 −7.66213 22.5539 9.00000 0
1.6 −1.33976 −3.00000 −6.20504 0 4.01928 12.2101 19.0313 9.00000 0
1.7 0.134967 −3.00000 −7.98178 0 −0.404902 17.3099 −2.15702 9.00000 0
1.8 0.574607 −3.00000 −7.66983 0 −1.72382 2.67744 −9.00399 9.00000 0
1.9 0.955230 −3.00000 −7.08754 0 −2.86569 −12.4836 −14.4121 9.00000 0
1.10 2.33957 −3.00000 −2.52642 0 −7.01870 32.9322 −24.6273 9.00000 0
1.11 2.93781 −3.00000 0.630743 0 −8.81344 −18.9115 −21.6495 9.00000 0
1.12 4.01755 −3.00000 8.14069 0 −12.0526 −1.75849 0.565245 9.00000 0
1.13 4.79301 −3.00000 14.9730 0 −14.3790 −0.140520 33.4216 9.00000 0
1.14 4.95189 −3.00000 16.5212 0 −14.8557 −28.2766 42.1962 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$1$$

## 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 1875.4.a.f 14
5.b even 2 1 1875.4.a.g 14
25.e even 10 2 75.4.g.b 28
75.h odd 10 2 225.4.h.a 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.g.b 28 25.e even 10 2
225.4.h.a 28 75.h odd 10 2
1875.4.a.f 14 1.a even 1 1 trivial
1875.4.a.g 14 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{14} - 81 T_{2}^{12} + 7 T_{2}^{11} + 2512 T_{2}^{10} - 517 T_{2}^{9} - 36970 T_{2}^{8} + 12987 T_{2}^{7} + 257291 T_{2}^{6} - 125779 T_{2}^{5} - 718713 T_{2}^{4} + 371750 T_{2}^{3} + 579848 T_{2}^{2} + \cdots + 42064$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1875))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14} - 81 T^{12} + 7 T^{11} + \cdots + 42064$$
$3$ $$(T + 3)^{14}$$
$5$ $$T^{14}$$
$7$ $$T^{14} - 27 T^{13} + \cdots + 4350562367600$$
$11$ $$T^{14} + 33 T^{13} + \cdots + 57\!\cdots\!96$$
$13$ $$T^{14} - 188 T^{13} + \cdots - 21\!\cdots\!44$$
$17$ $$T^{14} - 146 T^{13} + \cdots + 29\!\cdots\!76$$
$19$ $$T^{14} + 184 T^{13} + \cdots + 37\!\cdots\!00$$
$23$ $$T^{14} - 164 T^{13} + \cdots + 33\!\cdots\!00$$
$29$ $$T^{14} - 252 T^{13} + \cdots + 69\!\cdots\!00$$
$31$ $$T^{14} + 889 T^{13} + \cdots - 98\!\cdots\!00$$
$37$ $$T^{14} - 642 T^{13} + \cdots - 50\!\cdots\!00$$
$41$ $$T^{14} + 164 T^{13} + \cdots + 73\!\cdots\!00$$
$43$ $$T^{14} - 696 T^{13} + \cdots - 55\!\cdots\!16$$
$47$ $$T^{14} + 92 T^{13} + \cdots - 35\!\cdots\!84$$
$53$ $$T^{14} - 949 T^{13} + \cdots - 14\!\cdots\!75$$
$59$ $$T^{14} + 81 T^{13} + \cdots - 44\!\cdots\!00$$
$61$ $$T^{14} + 496 T^{13} + \cdots - 10\!\cdots\!96$$
$67$ $$T^{14} - 1926 T^{13} + \cdots - 22\!\cdots\!44$$
$71$ $$T^{14} - 2498 T^{13} + \cdots + 16\!\cdots\!04$$
$73$ $$T^{14} + 1026 T^{13} + \cdots + 33\!\cdots\!00$$
$79$ $$T^{14} + 695 T^{13} + \cdots + 54\!\cdots\!00$$
$83$ $$T^{14} - 5315 T^{13} + \cdots + 23\!\cdots\!96$$
$89$ $$T^{14} + 1424 T^{13} + \cdots - 11\!\cdots\!00$$
$97$ $$T^{14} - 291 T^{13} + \cdots + 17\!\cdots\!71$$