# Properties

 Label 1815.4.a.bl Level $1815$ Weight $4$ Character orbit 1815.a Self dual yes Analytic conductor $107.088$ Analytic rank $0$ Dimension $12$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,4,Mod(1,1815)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$107.088466660$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{11} - 72 x^{10} + 68 x^{9} + 1852 x^{8} - 1711 x^{7} - 20848 x^{6} + 20766 x^{5} + \cdots + 56080$$ x^12 - x^11 - 72*x^10 + 68*x^9 + 1852*x^8 - 1711*x^7 - 20848*x^6 + 20766*x^5 + 98931*x^4 - 106938*x^3 - 137664*x^2 + 117840*x + 56080 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$11^{3}$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + 5 q^{5} - 3 \beta_1 q^{6} + (\beta_{10} - \beta_1) q^{7} + (\beta_{6} - \beta_{5} + \cdots + 4 \beta_1) q^{8}+ \cdots + 9 q^{9}+O(q^{10})$$ q + b1 * q^2 - 3 * q^3 + (b2 + 4) * q^4 + 5 * q^5 - 3*b1 * q^6 + (b10 - b1) * q^7 + (b6 - b5 + b3 + b2 + 4*b1) * q^8 + 9 * q^9 $$q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + 5 q^{5} - 3 \beta_1 q^{6} + (\beta_{10} - \beta_1) q^{7} + (\beta_{6} - \beta_{5} + \cdots + 4 \beta_1) q^{8}+ \cdots + ( - 20 \beta_{11} + 19 \beta_{10} + \cdots - 95) q^{98}+O(q^{100})$$ q + b1 * q^2 - 3 * q^3 + (b2 + 4) * q^4 + 5 * q^5 - 3*b1 * q^6 + (b10 - b1) * q^7 + (b6 - b5 + b3 + b2 + 4*b1) * q^8 + 9 * q^9 + 5*b1 * q^10 + (-3*b2 - 12) * q^12 + (b11 - b6 + 2*b1 + 3) * q^13 + (b11 - b10 - b9 - 2*b7 + 2*b6 + 2*b5 - b4 + b3 - 2*b2 - b1 - 11) * q^14 - 15 * q^15 + (2*b11 + 3*b8 - 2*b6 + 2*b5 + 2*b4 - b3 + 2*b2 + 7*b1 + 20) * q^16 + (2*b11 + b10 + b9 - b7 + 2*b6 + b5 + b4 + b2 - 9*b1 + 14) * q^17 + 9*b1 * q^18 + (b11 - 3*b10 - 2*b9 + 4*b8 - 2*b7 - b6 + 3*b5 - b3 + b2 + 4*b1 + 15) * q^19 + (5*b2 + 20) * q^20 + (-3*b10 + 3*b1) * q^21 + (3*b11 + 4*b10 + 2*b9 + 3*b6 - 4*b5 + 2*b4 + 2*b2 + 12*b1 + 10) * q^23 + (-3*b6 + 3*b5 - 3*b3 - 3*b2 - 12*b1) * q^24 + 25 * q^25 + (-b11 + 2*b10 - 3*b8 + b7 + 3*b6 - 6*b5 + 2*b3 + 2*b2 + 6*b1 + 21) * q^26 - 27 * q^27 + (-b11 + b10 - b9 + b8 + 4*b7 - 4*b6 + 8*b5 + b4 - 4*b3 - 4*b2 - 25*b1 + 3) * q^28 + (2*b11 - 3*b10 - 3*b9 - b8 - 3*b7 + 2*b6 + 5*b5 - 2*b4 + b3 - 3*b2 - 5*b1 + 34) * q^29 - 15*b1 * q^30 + (-4*b11 - b10 - b9 - b8 + b7 - 6*b6 - b5 - 8*b4 + 4*b3 + 4*b2 + 2*b1 - 50) * q^31 + (-2*b11 + 2*b9 - 3*b8 - 6*b7 + 10*b6 - 10*b5 + 2*b4 + 6*b3 + 5*b2 + 27*b1 + 49) * q^32 + (3*b10 - 3*b9 + 2*b8 - b7 + 2*b6 + 3*b5 + b4 + 4*b3 - 9*b2 + 22*b1 - 119) * q^34 + (5*b10 - 5*b1) * q^35 + (9*b2 + 36) * q^36 + (4*b11 + 6*b10 - 3*b9 + b8 + 5*b7 - 3*b5 + 4*b4 - b3 - 3*b2 + 2*b1 + 5) * q^37 + (-6*b11 + 9*b10 + 8*b9 - 3*b8 - b7 + 9*b6 - 15*b5 + b4 + 10*b3 + 8*b2 + 38*b1 + 40) * q^38 + (-3*b11 + 3*b6 - 6*b1 - 9) * q^39 + (5*b6 - 5*b5 + 5*b3 + 5*b2 + 20*b1) * q^40 + (-4*b11 + 3*b10 + 2*b9 - 4*b8 + 8*b7 - 4*b6 - 4*b5 - 6*b4 - 4*b3 + 16*b2 + b1 + 86) * q^41 + (-3*b11 + 3*b10 + 3*b9 + 6*b7 - 6*b6 - 6*b5 + 3*b4 - 3*b3 + 6*b2 + 3*b1 + 33) * q^42 + (-3*b11 + b10 + 9*b9 - 3*b8 + 9*b7 - 11*b6 - 16*b5 - 2*b4 + 6*b2 - 8*b1 - 4) * q^43 + 45 * q^45 + (3*b11 - 2*b10 - 4*b9 + 3*b8 - 7*b7 - b6 + 6*b5 + 10*b3 + 20*b2 + 23*b1 + 131) * q^46 + (4*b10 + 3*b9 + 3*b7 + 5*b5 - 9*b4 + 8*b3 - b2 + 18*b1 - 49) * q^47 + (-6*b11 - 9*b8 + 6*b6 - 6*b5 - 6*b4 + 3*b3 - 6*b2 - 21*b1 - 60) * q^48 + (b11 + b10 + 4*b9 - 7*b7 + 7*b6 - 3*b5 - 2*b4 - 9*b3 - 11*b1 + 128) * q^49 + 25*b1 * q^50 + (-6*b11 - 3*b10 - 3*b9 + 3*b7 - 6*b6 - 3*b5 - 3*b4 - 3*b2 + 27*b1 - 42) * q^51 + (-b11 - 6*b10 - 6*b9 + 5*b8 + b7 - 5*b6 + 16*b5 + 2*b4 - 7*b3 + 14*b2 + 3*b1 + 78) * q^52 + (12*b11 - b10 - 14*b7 + 12*b6 - 2*b5 + 8*b4 + 9*b3 + 29*b2 - 16*b1 + 58) * q^53 - 27*b1 * q^54 + (-13*b11 - 5*b10 - 5*b9 - 14*b8 + 10*b7 + 10*b6 - 6*b5 + b4 + 2*b3 - 24*b2 - 268) * q^56 + (-3*b11 + 9*b10 + 6*b9 - 12*b8 + 6*b7 + 3*b6 - 9*b5 + 3*b3 - 3*b2 - 12*b1 - 45) * q^57 + (-5*b11 + 17*b10 - 6*b9 - 6*b8 + 12*b7 - 2*b6 + 11*b5 + b4 - 6*b3 - 15*b2 + b1 - 33) * q^58 + (2*b11 - 12*b10 - 22*b9 + 12*b8 - 11*b7 + 2*b6 + 19*b5 + 8*b4 - 3*b3 - 12*b2 - 62*b1 + 83) * q^59 + (-15*b2 - 60) * q^60 + (-7*b11 + 8*b10 - 3*b9 - 4*b8 - 5*b7 + 7*b6 - 7*b5 + 7*b4 + 4*b3 + 41*b2 - 34*b1 + 228) * q^61 + (3*b11 + 7*b10 + 14*b9 - 2*b8 + 8*b7 - 2*b6 - 15*b5 - 7*b4 - 13*b3 + 10*b2 - 35*b1 + 84) * q^62 + (9*b10 - 9*b1) * q^63 + (4*b10 + 10*b9 + 11*b8 + 2*b7 - 24*b6 - 8*b5 - 17*b3 + 25*b2 + 240) * q^64 + (5*b11 - 5*b6 + 10*b1 + 15) * q^65 + (b11 + 3*b10 + 6*b9 + 7*b8 - 11*b7 - 9*b6 + 2*b5 + 9*b4 - 4*b3 - 2*b2 + 65*b1 - 88) * q^67 + (-4*b11 - 11*b10 - 17*b9 + 8*b8 - 3*b7 - 4*b6 + 15*b5 - b4 - 8*b3 + 16*b2 - 107*b1 + 159) * q^68 + (-9*b11 - 12*b10 - 6*b9 - 9*b6 + 12*b5 - 6*b4 - 6*b2 - 36*b1 - 30) * q^69 + (5*b11 - 5*b10 - 5*b9 - 10*b7 + 10*b6 + 10*b5 - 5*b4 + 5*b3 - 10*b2 - 5*b1 - 55) * q^70 + (3*b11 + 9*b9 + 8*b8 + 21*b7 - 11*b6 + 19*b5 - 5*b4 - 15*b3 - 22*b2 - b1 + 58) * q^71 + (9*b6 - 9*b5 + 9*b3 + 9*b2 + 36*b1) * q^72 + (-8*b11 - 2*b10 - b9 - 9*b8 + 5*b7 + 16*b6 - 32*b5 + 16*b4 - 17*b3 + 17*b2 + 78*b1 + 180) * q^73 + (4*b11 - 16*b10 - 25*b9 - 16*b8 - 14*b7 + 26*b6 + 21*b5 + 23*b3 + 15*b2 + 3*b1 - 18) * q^74 - 75 * q^75 + (28*b11 + 3*b10 + 30*b9 + 24*b8 - 3*b7 - 28*b6 - 6*b5 + 13*b4 - 22*b3 + 65*b2 + 37*b1 + 392) * q^76 + (3*b11 - 6*b10 + 9*b8 - 3*b7 - 9*b6 + 18*b5 - 6*b3 - 6*b2 - 18*b1 - 63) * q^78 + (-16*b11 + 9*b10 + 12*b9 - 15*b8 + 19*b7 - 4*b6 + 11*b5 - 17*b4 + 4*b3 + 34*b2 + 15*b1 + 168) * q^79 + (10*b11 + 15*b8 - 10*b6 + 10*b5 + 10*b4 - 5*b3 + 10*b2 + 35*b1 + 100) * q^80 + 81 * q^81 + (-7*b11 - 15*b10 - 5*b9 - 18*b8 + 4*b7 + 2*b6 + 10*b5 - 23*b4 + 17*b3 + 10*b2 + 183*b1 + 15) * q^82 + (13*b11 - 14*b10 + b9 + 4*b8 - 9*b7 + 7*b6 + 13*b5 + 7*b4 + 5*b3 + 10*b2 - 31*b1 - 56) * q^83 + (3*b11 - 3*b10 + 3*b9 - 3*b8 - 12*b7 + 12*b6 - 24*b5 - 3*b4 + 12*b3 + 12*b2 + 75*b1 - 9) * q^84 + (10*b11 + 5*b10 + 5*b9 - 5*b7 + 10*b6 + 5*b5 + 5*b4 + 5*b2 - 45*b1 + 70) * q^85 + (2*b11 - 21*b10 + 25*b9 - b8 + 3*b7 - 21*b6 - 56*b5 - 5*b4 - 5*b3 + 17*b2 + 47*b1 - 124) * q^86 + (-6*b11 + 9*b10 + 9*b9 + 3*b8 + 9*b7 - 6*b6 - 15*b5 + 6*b4 - 3*b3 + 9*b2 + 15*b1 - 102) * q^87 + (3*b11 - b10 + 20*b9 - 11*b8 + 10*b7 - 13*b6 + b5 - 17*b4 - 10*b3 + 31*b2 + 112*b1 - 226) * q^89 + 45*b1 * q^90 + (10*b11 - 24*b10 - 29*b9 + 18*b8 - 29*b7 + 4*b6 + 38*b5 + 29*b4 - 7*b3 - 26*b2 + b1 + 19) * q^91 + (-9*b11 - 10*b10 - 10*b9 + 15*b8 + b7 + 23*b6 - 20*b5 + 6*b4 + 7*b3 + 43*b2 + 223*b1 + 234) * q^92 + (12*b11 + 3*b10 + 3*b9 + 3*b8 - 3*b7 + 18*b6 + 3*b5 + 24*b4 - 12*b3 - 12*b2 - 6*b1 + 150) * q^93 + (11*b11 + 8*b10 + 10*b8 + b7 + 4*b6 + 27*b5 - 6*b4 - 13*b3 + 41*b2 - 70*b1 + 222) * q^94 + (5*b11 - 15*b10 - 10*b9 + 20*b8 - 10*b7 - 5*b6 + 15*b5 - 5*b3 + 5*b2 + 20*b1 + 75) * q^95 + (6*b11 - 6*b9 + 9*b8 + 18*b7 - 30*b6 + 30*b5 - 6*b4 - 18*b3 - 15*b2 - 81*b1 - 147) * q^96 + (15*b11 + 6*b10 - 23*b9 - 16*b8 - 4*b7 - b6 + 7*b5 + b4 + 12*b3 - 14*b2 + 25*b1 + 92) * q^97 + (-20*b11 + 19*b10 + 18*b9 + b7 - 45*b6 + 17*b5 - 23*b4 + 10*b3 - 49*b2 + 71*b1 - 95) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + q^{2} - 36 q^{3} + 49 q^{4} + 60 q^{5} - 3 q^{6} - 5 q^{7} - 3 q^{8} + 108 q^{9}+O(q^{10})$$ 12 * q + q^2 - 36 * q^3 + 49 * q^4 + 60 * q^5 - 3 * q^6 - 5 * q^7 - 3 * q^8 + 108 * q^9 $$12 q + q^{2} - 36 q^{3} + 49 q^{4} + 60 q^{5} - 3 q^{6} - 5 q^{7} - 3 q^{8} + 108 q^{9} + 5 q^{10} - 147 q^{12} + 34 q^{13} - 134 q^{14} - 180 q^{15} + 265 q^{16} + 169 q^{17} + 9 q^{18} + 203 q^{19} + 245 q^{20} + 15 q^{21} + 116 q^{23} + 9 q^{24} + 300 q^{25} + 216 q^{26} - 324 q^{27} + 80 q^{28} + 409 q^{29} - 15 q^{30} - 645 q^{31} + 532 q^{32} - 1442 q^{34} - 25 q^{35} + 441 q^{36} + 40 q^{37} + 404 q^{38} - 102 q^{39} - 15 q^{40} + 1071 q^{41} + 402 q^{42} - 101 q^{43} + 540 q^{45} + 1539 q^{46} - 572 q^{47} - 795 q^{48} + 1563 q^{49} + 25 q^{50} - 507 q^{51} + 1081 q^{52} + 611 q^{53} - 27 q^{54} - 3221 q^{56} - 609 q^{57} - 354 q^{58} + 958 q^{59} - 735 q^{60} + 2620 q^{61} + 1049 q^{62} - 45 q^{63} + 2931 q^{64} + 170 q^{65} - 1027 q^{67} + 1896 q^{68} - 348 q^{69} - 670 q^{70} + 1020 q^{71} - 27 q^{72} + 2244 q^{73} - 283 q^{74} - 900 q^{75} + 4958 q^{76} - 648 q^{78} + 2199 q^{79} + 1325 q^{80} + 972 q^{81} + 362 q^{82} - 602 q^{83} - 240 q^{84} + 845 q^{85} - 1589 q^{86} - 1227 q^{87} - 2413 q^{89} + 45 q^{90} + 302 q^{91} + 3030 q^{92} + 1935 q^{93} + 2885 q^{94} + 1015 q^{95} - 1596 q^{96} + 886 q^{97} - 1236 q^{98}+O(q^{100})$$ 12 * q + q^2 - 36 * q^3 + 49 * q^4 + 60 * q^5 - 3 * q^6 - 5 * q^7 - 3 * q^8 + 108 * q^9 + 5 * q^10 - 147 * q^12 + 34 * q^13 - 134 * q^14 - 180 * q^15 + 265 * q^16 + 169 * q^17 + 9 * q^18 + 203 * q^19 + 245 * q^20 + 15 * q^21 + 116 * q^23 + 9 * q^24 + 300 * q^25 + 216 * q^26 - 324 * q^27 + 80 * q^28 + 409 * q^29 - 15 * q^30 - 645 * q^31 + 532 * q^32 - 1442 * q^34 - 25 * q^35 + 441 * q^36 + 40 * q^37 + 404 * q^38 - 102 * q^39 - 15 * q^40 + 1071 * q^41 + 402 * q^42 - 101 * q^43 + 540 * q^45 + 1539 * q^46 - 572 * q^47 - 795 * q^48 + 1563 * q^49 + 25 * q^50 - 507 * q^51 + 1081 * q^52 + 611 * q^53 - 27 * q^54 - 3221 * q^56 - 609 * q^57 - 354 * q^58 + 958 * q^59 - 735 * q^60 + 2620 * q^61 + 1049 * q^62 - 45 * q^63 + 2931 * q^64 + 170 * q^65 - 1027 * q^67 + 1896 * q^68 - 348 * q^69 - 670 * q^70 + 1020 * q^71 - 27 * q^72 + 2244 * q^73 - 283 * q^74 - 900 * q^75 + 4958 * q^76 - 648 * q^78 + 2199 * q^79 + 1325 * q^80 + 972 * q^81 + 362 * q^82 - 602 * q^83 - 240 * q^84 + 845 * q^85 - 1589 * q^86 - 1227 * q^87 - 2413 * q^89 + 45 * q^90 + 302 * q^91 + 3030 * q^92 + 1935 * q^93 + 2885 * q^94 + 1015 * q^95 - 1596 * q^96 + 886 * q^97 - 1236 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} - 72 x^{10} + 68 x^{9} + 1852 x^{8} - 1711 x^{7} - 20848 x^{6} + 20766 x^{5} + \cdots + 56080$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 12$$ v^2 - 12 $$\beta_{3}$$ $$=$$ $$( 6167 \nu^{11} - 6226 \nu^{10} - 407550 \nu^{9} + 306802 \nu^{8} + 9282906 \nu^{7} + \cdots - 89573168 ) / 1933152$$ (6167*v^11 - 6226*v^10 - 407550*v^9 + 306802*v^8 + 9282906*v^7 - 4177899*v^6 - 88867377*v^5 + 18661711*v^4 + 347802786*v^3 + 2550160*v^2 - 424504448*v - 89573168) / 1933152 $$\beta_{4}$$ $$=$$ $$( - 10859 \nu^{11} - 13370 \nu^{10} + 809770 \nu^{9} + 892386 \nu^{8} - 21568094 \nu^{7} + \cdots + 584253360 ) / 1933152$$ (-10859*v^11 - 13370*v^10 + 809770*v^9 + 892386*v^8 - 21568094*v^7 - 19637877*v^6 + 248584381*v^5 + 155954729*v^4 - 1187670962*v^3 - 401738960*v^2 + 1490807472*v + 584253360) / 1933152 $$\beta_{5}$$ $$=$$ $$( - 13666 \nu^{11} - 10879 \nu^{10} + 980941 \nu^{9} + 776169 \nu^{8} - 24896139 \nu^{7} + \cdots + 565928176 ) / 1933152$$ (-13666*v^11 - 10879*v^10 + 980941*v^9 + 776169*v^8 - 24896139*v^7 - 18185865*v^6 + 270967552*v^5 + 148634765*v^4 - 1219054162*v^3 - 394152976*v^2 + 1470945312*v + 565928176) / 1933152 $$\beta_{6}$$ $$=$$ $$( - 19833 \nu^{11} - 4653 \nu^{10} + 1388491 \nu^{9} + 469367 \nu^{8} - 34179045 \nu^{7} + \cdots + 678699168 ) / 1933152$$ (-19833*v^11 - 4653*v^10 + 1388491*v^9 + 469367*v^8 - 34179045*v^7 - 14007966*v^6 + 359834929*v^5 + 129973054*v^4 - 1564923796*v^3 - 398636288*v^2 + 1856786720*v + 678699168) / 1933152 $$\beta_{7}$$ $$=$$ $$( - 39837 \nu^{11} - 23050 \nu^{10} + 2855554 \nu^{9} + 1750786 \nu^{8} - 72497590 \nu^{7} + \cdots + 1662422576 ) / 3866304$$ (-39837*v^11 - 23050*v^10 + 2855554*v^9 + 1750786*v^8 - 72497590*v^7 - 43512991*v^6 + 791811827*v^5 + 372396203*v^4 - 3567811422*v^3 - 1045373136*v^2 + 4214081824*v + 1662422576) / 3866304 $$\beta_{8}$$ $$=$$ $$( - 1893 \nu^{11} - 890 \nu^{10} + 134058 \nu^{9} + 70090 \nu^{8} - 3346574 \nu^{7} + \cdots + 64425040 ) / 148704$$ (-1893*v^11 - 890*v^10 + 134058*v^9 + 70090*v^8 - 3346574*v^7 - 1768511*v^6 + 35759363*v^5 + 14583923*v^4 - 157496038*v^3 - 37302464*v^2 + 183742720*v + 64425040) / 148704 $$\beta_{9}$$ $$=$$ $$( - 28787 \nu^{11} - 17174 \nu^{10} + 2052466 \nu^{9} + 1301610 \nu^{8} - 51685838 \nu^{7} + \cdots + 1174335600 ) / 1933152$$ (-28787*v^11 - 17174*v^10 + 2052466*v^9 + 1301610*v^8 - 51685838*v^7 - 32260581*v^6 + 558214489*v^5 + 275643989*v^4 - 2493334070*v^3 - 782987792*v^2 + 3010122288*v + 1174335600) / 1933152 $$\beta_{10}$$ $$=$$ $$( 117925 \nu^{11} + 17514 \nu^{10} - 8307546 \nu^{9} - 2099962 \nu^{8} + 206259262 \nu^{7} + \cdots - 4198882416 ) / 7732608$$ (117925*v^11 + 17514*v^10 - 8307546*v^9 - 2099962*v^8 + 206259262*v^7 + 68246639*v^6 - 2194512323*v^5 - 665982659*v^4 + 9625128518*v^3 + 2226362976*v^2 - 11476645504*v - 4198882416) / 7732608 $$\beta_{11}$$ $$=$$ $$( 44689 \nu^{11} + 33838 \nu^{10} - 3220126 \nu^{9} - 2412542 \nu^{8} + 82184834 \nu^{7} + \cdots - 1706830064 ) / 1933152$$ (44689*v^11 + 33838*v^10 - 3220126*v^9 - 2412542*v^8 + 82184834*v^7 + 56212791*v^6 - 901458271*v^5 - 448705507*v^4 + 4086875462*v^3 + 1100797800*v^2 - 4906967360*v - 1706830064) / 1933152
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 12$$ b2 + 12 $$\nu^{3}$$ $$=$$ $$\beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + 20\beta_1$$ b6 - b5 + b3 + b2 + 20*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{11} + 3\beta_{8} - 2\beta_{6} + 2\beta_{5} + 2\beta_{4} - \beta_{3} + 26\beta_{2} + 7\beta _1 + 244$$ 2*b11 + 3*b8 - 2*b6 + 2*b5 + 2*b4 - b3 + 26*b2 + 7*b1 + 244 $$\nu^{5}$$ $$=$$ $$- 2 \beta_{11} + 2 \beta_{9} - 3 \beta_{8} - 6 \beta_{7} + 42 \beta_{6} - 42 \beta_{5} + 2 \beta_{4} + \cdots + 49$$ -2*b11 + 2*b9 - 3*b8 - 6*b7 + 42*b6 - 42*b5 + 2*b4 + 38*b3 + 37*b2 + 475*b1 + 49 $$\nu^{6}$$ $$=$$ $$80 \beta_{11} + 4 \beta_{10} + 10 \beta_{9} + 131 \beta_{8} + 2 \beta_{7} - 104 \beta_{6} + 72 \beta_{5} + \cdots + 5904$$ 80*b11 + 4*b10 + 10*b9 + 131*b8 + 2*b7 - 104*b6 + 72*b5 + 80*b4 - 57*b3 + 681*b2 + 280*b1 + 5904 $$\nu^{7}$$ $$=$$ $$- 110 \beta_{11} - 8 \beta_{10} + 146 \beta_{9} - 149 \beta_{8} - 270 \beta_{7} + 1337 \beta_{6} + \cdots + 1863$$ -110*b11 - 8*b10 + 146*b9 - 149*b8 - 270*b7 + 1337*b6 - 1449*b5 + 42*b4 + 1213*b3 + 1089*b2 + 12311*b1 + 1863 $$\nu^{8}$$ $$=$$ $$2570 \beta_{11} + 244 \beta_{10} + 722 \beta_{9} + 4438 \beta_{8} + 162 \beta_{7} - 4023 \beta_{6} + \cdots + 155160$$ 2570*b11 + 244*b10 + 722*b9 + 4438*b8 + 162*b7 - 4023*b6 + 1991*b5 + 2518*b4 - 2279*b3 + 18466*b2 + 7957*b1 + 155160 $$\nu^{9}$$ $$=$$ $$- 4366 \beta_{11} - 464 \beta_{10} + 6696 \beta_{9} - 5483 \beta_{8} - 9312 \beta_{7} + 39465 \beta_{6} + \cdots + 44680$$ -4366*b11 - 464*b10 + 6696*b9 - 5483*b8 - 9312*b7 + 39465*b6 - 46825*b5 + 234*b4 + 36724*b3 + 29698*b2 + 334403*b1 + 44680 $$\nu^{10}$$ $$=$$ $$77584 \beta_{11} + 9888 \beta_{10} + 33006 \beta_{9} + 139100 \beta_{8} + 7294 \beta_{7} + \cdots + 4255135$$ 77584*b11 + 9888*b10 + 33006*b9 + 139100*b8 + 7294*b7 - 139171*b6 + 51203*b5 + 74380*b4 - 79920*b3 + 512276*b2 + 191918*b1 + 4255135 $$\nu^{11}$$ $$=$$ $$- 153156 \beta_{11} - 18068 \beta_{10} + 255740 \beta_{9} - 181981 \beta_{8} - 294772 \beta_{7} + \cdots + 702235$$ -153156*b11 - 18068*b10 + 255740*b9 - 181981*b8 - 294772*b7 + 1139606*b6 - 1466806*b5 - 20968*b4 + 1089660*b3 + 780938*b2 + 9320152*b1 + 702235

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −5.46545 −3.83557 −3.69805 −3.41710 −1.17661 −0.374476 1.48723 1.51761 2.66682 3.04373 4.90341 5.34844
−5.46545 −3.00000 21.8712 5.00000 16.3964 20.5677 −75.8121 9.00000 −27.3273
1.2 −3.83557 −3.00000 6.71157 5.00000 11.5067 2.37609 4.94185 9.00000 −19.1778
1.3 −3.69805 −3.00000 5.67556 5.00000 11.0941 −34.0461 8.59591 9.00000 −18.4902
1.4 −3.41710 −3.00000 3.67654 5.00000 10.2513 25.9117 14.7737 9.00000 −17.0855
1.5 −1.17661 −3.00000 −6.61558 5.00000 3.52984 −5.87699 17.1969 9.00000 −5.88306
1.6 −0.374476 −3.00000 −7.85977 5.00000 1.12343 −11.4396 5.93911 9.00000 −1.87238
1.7 1.48723 −3.00000 −5.78815 5.00000 −4.46169 −25.2790 −20.5061 9.00000 7.43614
1.8 1.51761 −3.00000 −5.69685 5.00000 −4.55284 25.9264 −20.7865 9.00000 7.58806
1.9 2.66682 −3.00000 −0.888052 5.00000 −8.00047 34.9200 −23.7029 9.00000 13.3341
1.10 3.04373 −3.00000 1.26431 5.00000 −9.13120 −20.6437 −20.5016 9.00000 15.2187
1.11 4.90341 −3.00000 16.0434 5.00000 −14.7102 −0.166503 39.4401 9.00000 24.5170
1.12 5.34844 −3.00000 20.6059 5.00000 −16.0453 −17.2500 67.4218 9.00000 26.7422
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$5$$ $$-1$$
$$11$$ $$-1$$

## 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 1815.4.a.bl 12
11.b odd 2 1 1815.4.a.bi 12
11.d odd 10 2 165.4.m.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.m.c 24 11.d odd 10 2
1815.4.a.bi 12 11.b odd 2 1
1815.4.a.bl 12 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2}^{12} - T_{2}^{11} - 72 T_{2}^{10} + 68 T_{2}^{9} + 1852 T_{2}^{8} - 1711 T_{2}^{7} - 20848 T_{2}^{6} + \cdots + 56080$$ T2^12 - T2^11 - 72*T2^10 + 68*T2^9 + 1852*T2^8 - 1711*T2^7 - 20848*T2^6 + 20766*T2^5 + 98931*T2^4 - 106938*T2^3 - 137664*T2^2 + 117840*T2 + 56080 $$T_{7}^{12} + 5 T_{7}^{11} - 2827 T_{7}^{10} - 19134 T_{7}^{9} + 2818876 T_{7}^{8} + 24860140 T_{7}^{7} + \cdots - 3933260174756$$ T7^12 + 5*T7^11 - 2827*T7^10 - 19134*T7^9 + 2818876*T7^8 + 24860140*T7^7 - 1179475087*T7^6 - 13208675791*T7^5 + 167583328704*T7^4 + 2482016994977*T7^3 + 3771653322849*T7^2 - 23062843751398*T7 - 3933260174756

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - T^{11} + \cdots + 56080$$
$3$ $$(T + 3)^{12}$$
$5$ $$(T - 5)^{12}$$
$7$ $$T^{12} + \cdots - 3933260174756$$
$11$ $$T^{12}$$
$13$ $$T^{12} + \cdots - 19\!\cdots\!99$$
$17$ $$T^{12} + \cdots + 13\!\cdots\!96$$
$19$ $$T^{12} + \cdots + 50\!\cdots\!20$$
$23$ $$T^{12} + \cdots - 33\!\cdots\!51$$
$29$ $$T^{12} + \cdots + 38\!\cdots\!96$$
$31$ $$T^{12} + \cdots + 36\!\cdots\!00$$
$37$ $$T^{12} + \cdots + 61\!\cdots\!29$$
$41$ $$T^{12} + \cdots - 10\!\cdots\!96$$
$43$ $$T^{12} + \cdots - 15\!\cdots\!20$$
$47$ $$T^{12} + \cdots - 10\!\cdots\!75$$
$53$ $$T^{12} + \cdots - 24\!\cdots\!44$$
$59$ $$T^{12} + \cdots - 11\!\cdots\!45$$
$61$ $$T^{12} + \cdots - 14\!\cdots\!20$$
$67$ $$T^{12} + \cdots - 12\!\cdots\!80$$
$71$ $$T^{12} + \cdots + 12\!\cdots\!80$$
$73$ $$T^{12} + \cdots - 13\!\cdots\!44$$
$79$ $$T^{12} + \cdots + 27\!\cdots\!16$$
$83$ $$T^{12} + \cdots + 10\!\cdots\!00$$
$89$ $$T^{12} + \cdots - 66\!\cdots\!04$$
$97$ $$T^{12} + \cdots + 89\!\cdots\!96$$