# Properties

 Label 1815.4.a.bn 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: $$x^{12} - 5 x^{11} - 59 x^{10} + 269 x^{9} + 1318 x^{8} - 5253 x^{7} - 13369 x^{6} + 44853 x^{5} + \cdots + 17600$$ x^12 - 5*x^11 - 59*x^10 + 269*x^9 + 1318*x^8 - 5253*x^7 - 13369*x^6 + 44853*x^5 + 57849*x^4 - 151000*x^3 - 76540*x^2 + 100800*x + 17600 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$5\cdot 11^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + 5 q^{5} + ( - 3 \beta_1 + 3) q^{6} + (\beta_{10} + \beta_1 + 6) q^{7} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 11) q^{8} + 9 q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 + 3 * q^3 + (b2 - b1 + 5) * q^4 + 5 * q^5 + (-3*b1 + 3) * q^6 + (b10 + b1 + 6) * q^7 + (-b3 + b2 - 3*b1 + 11) * q^8 + 9 * q^9 $$q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + 5 q^{5} + ( - 3 \beta_1 + 3) q^{6} + (\beta_{10} + \beta_1 + 6) q^{7} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 11) q^{8} + 9 q^{9} + ( - 5 \beta_1 + 5) q^{10} + (3 \beta_{2} - 3 \beta_1 + 15) q^{12} + ( - \beta_{11} - 2 \beta_{10} + \cdots + 14) q^{13}+ \cdots + ( - 22 \beta_{11} - 43 \beta_{10} + \cdots + 54) q^{98}+O(q^{100})$$ q + (-b1 + 1) * q^2 + 3 * q^3 + (b2 - b1 + 5) * q^4 + 5 * q^5 + (-3*b1 + 3) * q^6 + (b10 + b1 + 6) * q^7 + (-b3 + b2 - 3*b1 + 11) * q^8 + 9 * q^9 + (-5*b1 + 5) * q^10 + (3*b2 - 3*b1 + 15) * q^12 + (-b11 - 2*b10 + b8 + b4 + b2 + 14) * q^13 + (3*b10 + b9 - b8 - 3*b7 - b6 - b5 - 2*b3 - 4*b2 - 6*b1 - 2) * q^14 + 15 * q^15 + (b11 + b10 + b4 - b3 + b2 - 8*b1 + 17) * q^16 + (b11 - 2*b10 + b3 + 3*b2 + 28) * q^17 + (-9*b1 + 9) * q^18 + (b11 + 3*b10 + 2*b9 - 3*b8 - 2*b7 - b6 + b5 - b4 + b3 + 3*b1 + 20) * q^19 + (5*b2 - 5*b1 + 25) * q^20 + (3*b10 + 3*b1 + 18) * q^21 + (-b11 - 2*b10 - b8 + 2*b7 + b4 + 2*b3 + 5*b2 - 4*b1 + 21) * q^23 + (-3*b3 + 3*b2 - 9*b1 + 33) * q^24 + 25 * q^25 + (-2*b10 + b9 + 3*b8 + 3*b7 + b6 + 4*b5 - 3*b4 + 2*b3 - b2 - 24*b1 + 11) * q^26 + 27 * q^27 + (3*b11 + 12*b10 + 5*b9 - b8 - 9*b7 - b6 - b5 - b4 - b3 + 5*b2 + 11*b1 + 44) * q^28 + (-2*b11 - 4*b10 - 6*b9 + 3*b7 - 5*b5 - b4 - 4*b3 + 3*b2 + 7*b1 + 61) * q^29 + (-15*b1 + 15) * q^30 + (-2*b11 + 4*b8 + b7 + 2*b6 + 2*b5 - b4 - b3 - 12*b2 - 7*b1 - 19) * q^31 + (3*b11 + 3*b10 - 4*b8 - 2*b7 - 2*b6 - 5*b5 + 3*b4 + 2*b3 + 4*b2 + 3*b1 + 33) * q^32 + (-b11 - 10*b10 - 4*b9 + 8*b7 + 2*b6 - 2*b5 + 2*b4 - b3 + 3*b2 - 45*b1 + 12) * q^34 + (5*b10 + 5*b1 + 30) * q^35 + (9*b2 - 9*b1 + 45) * q^36 + (-4*b11 - 5*b10 - 4*b9 + b7 - 2*b6 - 7*b5 + 5*b4 - 6*b3 + 5*b2 - 26*b1 + 32) * q^37 + (-b11 + 8*b10 + b9 - 6*b7 + b6 + 10*b5 + 2*b4 - 2*b3 - 5*b2 - 22*b1 - 1) * q^38 + (-3*b11 - 6*b10 + 3*b8 + 3*b4 + 3*b2 + 42) * q^39 + (-5*b3 + 5*b2 - 15*b1 + 55) * q^40 + (-2*b11 - 3*b10 + 4*b9 + 8*b8 + 4*b7 + 6*b6 + 14*b5 + 4*b4 + 4*b2 + 3*b1 + 102) * q^41 + (9*b10 + 3*b9 - 3*b8 - 9*b7 - 3*b6 - 3*b5 - 6*b3 - 12*b2 - 18*b1 - 6) * q^42 + (-5*b11 - 10*b10 - 6*b9 + b8 + 3*b7 - 5*b6 - 10*b5 - 4*b4 + 3*b3 - 3*b2 + 23*b1 + 48) * q^43 + 45 * q^45 + (2*b11 - 8*b10 - b9 + 3*b8 + 3*b7 + 3*b6 + 10*b5 - 5*b4 + 2*b3 + b2 - 55*b1 + 74) * q^46 + (b11 + 11*b10 + 4*b9 + 6*b8 - 4*b7 - 4*b6 - 12*b5 - 6*b4 - b3 - 3*b2 - 13*b1 + 27) * q^47 + (3*b11 + 3*b10 + 3*b4 - 3*b3 + 3*b2 - 24*b1 + 51) * q^48 + (b11 + 2*b10 - 7*b9 + 3*b8 + 9*b7 + 2*b6 + 18*b5 - 6*b4 - 5*b3 + 26*b2 + 8*b1 + 120) * q^49 + (-25*b1 + 25) * q^50 + (3*b11 - 6*b10 + 3*b3 + 9*b2 + 84) * q^51 + (-10*b11 - 14*b10 - 7*b9 - b8 + 9*b7 + 5*b6 + 13*b5 + 3*b4 + 3*b3 + 25*b2 + 5*b1 + 163) * q^52 + (4*b11 + 3*b10 + 2*b8 + 2*b7 + 4*b6 - 3*b5 + 6*b4 + 9*b3 - 7*b2 + 21*b1 + 8) * q^53 + (-27*b1 + 27) * q^54 + (-4*b11 + 25*b10 + 5*b9 + b8 - 5*b7 + 9*b6 - 4*b5 + 6*b4 - 17*b3 - 8*b2 - 47*b1 - 20) * q^56 + (3*b11 + 9*b10 + 6*b9 - 9*b8 - 6*b7 - 3*b6 + 3*b5 - 3*b4 + 3*b3 + 9*b1 + 60) * q^57 + (13*b11 - 2*b10 - 4*b9 - 3*b8 + 9*b7 - 6*b6 - 24*b5 - 9*b4 + 10*b3 + 11*b2 - 68*b1 - 12) * q^58 + (-8*b11 + 5*b10 - b9 - 10*b8 + b7 - 8*b6 - 6*b5 + 5*b4 + b3 - 17*b2 + 39*b1 + 24) * q^59 + (15*b2 - 15*b1 + 75) * q^60 + (8*b11 + b10 - 2*b9 - 5*b8 + 16*b7 + 8*b6 - 22*b5 - 15*b4 + 13*b3 - 17*b1 + 115) * q^61 + (-9*b11 - 2*b10 + 4*b9 + 5*b8 - 3*b7 + 3*b5 + b4 + 11*b3 + 4*b2 + 95*b1 + 42) * q^62 + (9*b10 + 9*b1 + 54) * q^63 + (7*b11 + 9*b10 - 12*b8 - 6*b7 - 20*b5 - 13*b4 - 5*b3 - 30*b2 - 10*b1 - 133) * q^64 + (-5*b11 - 10*b10 + 5*b8 + 5*b4 + 5*b2 + 70) * q^65 + (8*b11 + 14*b10 + 9*b9 - b8 - 18*b7 + 13*b6 + 23*b5 - 13*b4 + 4*b3 + 16*b2 - 45*b1 + 37) * q^67 + (5*b11 - 22*b10 - 12*b9 + 2*b8 + 26*b7 + 6*b6 - 4*b5 - 4*b4 + 11*b3 + 58*b2 - 19*b1 + 312) * q^68 + (-3*b11 - 6*b10 - 3*b8 + 6*b7 + 3*b4 + 6*b3 + 15*b2 - 12*b1 + 63) * q^69 + (15*b10 + 5*b9 - 5*b8 - 15*b7 - 5*b6 - 5*b5 - 10*b3 - 20*b2 - 30*b1 - 10) * q^70 + (-2*b11 - 11*b10 - 12*b9 - 17*b8 + 4*b7 - 14*b6 + 21*b5 + 3*b4 + 10*b3 + 7*b2 - 29*b1 - 49) * q^71 + (-9*b3 + 9*b2 - 27*b1 + 99) * q^72 + (2*b11 - 17*b10 - 4*b9 + 2*b8 + 23*b7 + 9*b6 - 3*b5 + 17*b4 + 9*b2 + 78*b1 + 201) * q^73 + (25*b11 + 19*b10 + 5*b9 + 2*b7 - 3*b6 - 11*b5 - 19*b4 + 6*b3 + 41*b2 - 79*b1 + 401) * q^74 + 75 * q^75 + (-12*b11 + 17*b10 + 3*b9 + 18*b8 - 12*b7 - 23*b6 + 14*b5 + 9*b4 - 18*b3 - 3*b2 + 41*b1 + 101) * q^76 + (-6*b10 + 3*b9 + 9*b8 + 9*b7 + 3*b6 + 12*b5 - 9*b4 + 6*b3 - 3*b2 - 72*b1 + 33) * q^78 + (-7*b11 + 2*b10 + 11*b9 - 8*b8 - 26*b7 - 14*b6 + 11*b5 + 14*b4 + 2*b3 - 47*b2 - 33*b1 + 110) * q^79 + (5*b11 + 5*b10 + 5*b4 - 5*b3 + 5*b2 - 40*b1 + 85) * q^80 + 81 * q^81 + (-20*b11 - 27*b10 + 7*b9 + 11*b8 + b7 - 3*b6 + 45*b5 + 14*b4 - 8*b3 + 8*b2 - 100*b1 + 50) * q^82 + (4*b11 - b10 + 4*b9 + 3*b8 + 6*b7 + 30*b6 + 35*b5 - 15*b4 + 20*b3 + 49*b2 - 33*b1 + 287) * q^83 + (9*b11 + 36*b10 + 15*b9 - 3*b8 - 27*b7 - 3*b6 - 3*b5 - 3*b4 - 3*b3 + 15*b2 + 33*b1 + 132) * q^84 + (5*b11 - 10*b10 + 5*b3 + 15*b2 + 140) * q^85 + (-2*b11 - 25*b10 - 12*b9 + 12*b8 + 24*b7 + 8*b6 - 13*b5 - 21*b4 + 30*b3 - 34*b2 - 55*b1 - 264) * q^86 + (-6*b11 - 12*b10 - 18*b9 + 9*b7 - 15*b5 - 3*b4 - 12*b3 + 9*b2 + 21*b1 + 183) * q^87 + (26*b11 + 17*b10 + 16*b9 - 5*b8 - 11*b7 + 25*b6 - 4*b5 - 18*b4 + 36*b3 - 30*b2 + 32*b1 - 50) * q^89 + (-45*b1 + 45) * q^90 + (-3*b11 + 13*b10 + 18*b9 - 2*b8 - 30*b7 - 13*b6 - 11*b5 + 14*b4 + 4*b3 - 12*b2 + 55*b1 - 401) * q^91 + (-16*b11 - 44*b10 - 17*b9 + 15*b8 + 17*b7 - 5*b6 + 17*b5 + 13*b4 - 3*b3 + 50*b2 + 4*b1 + 474) * q^92 + (-6*b11 + 12*b8 + 3*b7 + 6*b6 + 6*b5 - 3*b4 - 3*b3 - 36*b2 - 21*b1 - 57) * q^93 + (-19*b11 + 29*b10 + 3*b9 + b8 - 25*b7 + 27*b6 - 43*b5 + 8*b4 - 31*b3 - 19*b2 - 78*b1 + 257) * q^94 + (5*b11 + 15*b10 + 10*b9 - 15*b8 - 10*b7 - 5*b6 + 5*b5 - 5*b4 + 5*b3 + 15*b1 + 100) * q^95 + (9*b11 + 9*b10 - 12*b8 - 6*b7 - 6*b6 - 15*b5 + 9*b4 + 6*b3 + 12*b2 + 9*b1 + 99) * q^96 + (22*b11 - 20*b10 + 25*b9 - 9*b8 + 19*b7 - b6 - 19*b5 + 16*b4 + 3*b3 - 4*b2 - 48*b1 + 18) * q^97 + (-22*b11 - 43*b10 - 13*b9 - 12*b8 + 2*b7 - 49*b6 + 19*b5 + 35*b4 - 30*b3 + 47*b2 - 157*b1 + 54) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 7 q^{2} + 36 q^{3} + 49 q^{4} + 60 q^{5} + 21 q^{6} + 77 q^{7} + 111 q^{8} + 108 q^{9}+O(q^{10})$$ 12 * q + 7 * q^2 + 36 * q^3 + 49 * q^4 + 60 * q^5 + 21 * q^6 + 77 * q^7 + 111 * q^8 + 108 * q^9 $$12 q + 7 q^{2} + 36 q^{3} + 49 q^{4} + 60 q^{5} + 21 q^{6} + 77 q^{7} + 111 q^{8} + 108 q^{9} + 35 q^{10} + 147 q^{12} + 172 q^{13} - 30 q^{14} + 180 q^{15} + 161 q^{16} + 317 q^{17} + 63 q^{18} + 237 q^{19} + 245 q^{20} + 231 q^{21} + 210 q^{23} + 333 q^{24} + 300 q^{25} + 8 q^{26} + 324 q^{27} + 542 q^{28} + 759 q^{29} + 105 q^{30} - 193 q^{31} + 410 q^{32} - 78 q^{34} + 385 q^{35} + 441 q^{36} + 286 q^{37} - 168 q^{38} + 516 q^{39} + 555 q^{40} + 1189 q^{41} - 90 q^{42} + 775 q^{43} + 540 q^{45} + 529 q^{46} + 382 q^{47} + 483 q^{48} + 1195 q^{49} + 175 q^{50} + 951 q^{51} + 1741 q^{52} + 275 q^{53} + 189 q^{54} - 419 q^{56} + 711 q^{57} - 418 q^{58} + 646 q^{59} + 735 q^{60} + 1340 q^{61} + 983 q^{62} + 693 q^{63} - 1489 q^{64} + 860 q^{65} - 185 q^{67} + 3322 q^{68} + 630 q^{69} - 150 q^{70} - 932 q^{71} + 999 q^{72} + 2860 q^{73} + 4187 q^{74} + 900 q^{75} + 1594 q^{76} + 24 q^{78} + 1429 q^{79} + 805 q^{80} + 972 q^{81} - 30 q^{82} + 2590 q^{83} + 1626 q^{84} + 1585 q^{85} - 3195 q^{86} + 2277 q^{87} - 473 q^{89} + 315 q^{90} - 4302 q^{91} + 5462 q^{92} - 579 q^{93} + 2875 q^{94} + 1185 q^{95} + 1230 q^{96} + 318 q^{97} - 194 q^{98}+O(q^{100})$$ 12 * q + 7 * q^2 + 36 * q^3 + 49 * q^4 + 60 * q^5 + 21 * q^6 + 77 * q^7 + 111 * q^8 + 108 * q^9 + 35 * q^10 + 147 * q^12 + 172 * q^13 - 30 * q^14 + 180 * q^15 + 161 * q^16 + 317 * q^17 + 63 * q^18 + 237 * q^19 + 245 * q^20 + 231 * q^21 + 210 * q^23 + 333 * q^24 + 300 * q^25 + 8 * q^26 + 324 * q^27 + 542 * q^28 + 759 * q^29 + 105 * q^30 - 193 * q^31 + 410 * q^32 - 78 * q^34 + 385 * q^35 + 441 * q^36 + 286 * q^37 - 168 * q^38 + 516 * q^39 + 555 * q^40 + 1189 * q^41 - 90 * q^42 + 775 * q^43 + 540 * q^45 + 529 * q^46 + 382 * q^47 + 483 * q^48 + 1195 * q^49 + 175 * q^50 + 951 * q^51 + 1741 * q^52 + 275 * q^53 + 189 * q^54 - 419 * q^56 + 711 * q^57 - 418 * q^58 + 646 * q^59 + 735 * q^60 + 1340 * q^61 + 983 * q^62 + 693 * q^63 - 1489 * q^64 + 860 * q^65 - 185 * q^67 + 3322 * q^68 + 630 * q^69 - 150 * q^70 - 932 * q^71 + 999 * q^72 + 2860 * q^73 + 4187 * q^74 + 900 * q^75 + 1594 * q^76 + 24 * q^78 + 1429 * q^79 + 805 * q^80 + 972 * q^81 - 30 * q^82 + 2590 * q^83 + 1626 * q^84 + 1585 * q^85 - 3195 * q^86 + 2277 * q^87 - 473 * q^89 + 315 * q^90 - 4302 * q^91 + 5462 * q^92 - 579 * q^93 + 2875 * q^94 + 1185 * q^95 + 1230 * q^96 + 318 * q^97 - 194 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 5 x^{11} - 59 x^{10} + 269 x^{9} + 1318 x^{8} - 5253 x^{7} - 13369 x^{6} + 44853 x^{5} + \cdots + 17600$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 12$$ v^2 - v - 12 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 17\nu + 14$$ v^3 - 2*v^2 - 17*v + 14 $$\beta_{4}$$ $$=$$ $$( 96262 \nu^{11} - 678861 \nu^{10} - 3890483 \nu^{9} + 30748837 \nu^{8} + 48361337 \nu^{7} + \cdots + 19569545720 ) / 576954640$$ (96262*v^11 - 678861*v^10 - 3890483*v^9 + 30748837*v^8 + 48361337*v^7 - 492252014*v^6 - 69171515*v^5 + 3679847225*v^4 - 2557603615*v^3 - 15355641899*v^2 + 13158086270*v + 19569545720) / 576954640 $$\beta_{5}$$ $$=$$ $$( 111517 \nu^{11} - 717891 \nu^{10} - 5417673 \nu^{9} + 33853987 \nu^{8} + 111044592 \nu^{7} + \cdots + 6277569280 ) / 576954640$$ (111517*v^11 - 717891*v^10 - 5417673*v^9 + 33853987*v^8 + 111044592*v^7 - 588865129*v^6 - 1198561755*v^5 + 4487587975*v^4 + 5977349855*v^3 - 13108478334*v^2 - 7253038920*v + 6277569280) / 576954640 $$\beta_{6}$$ $$=$$ $$( 127626 \nu^{11} - 815837 \nu^{10} - 7380199 \nu^{9} + 46635417 \nu^{8} + 157872765 \nu^{7} + \cdots + 6403216920 ) / 576954640$$ (127626*v^11 - 815837*v^10 - 7380199*v^9 + 46635417*v^8 + 157872765*v^7 - 934962634*v^6 - 1524875283*v^5 + 7642059341*v^4 + 6150472533*v^3 - 21476383063*v^2 - 6096376370*v + 6403216920) / 576954640 $$\beta_{7}$$ $$=$$ $$( 344997 \nu^{11} + 4286939 \nu^{10} - 54344143 \nu^{9} - 171854403 \nu^{8} + 1852765582 \nu^{7} + \cdots + 996822640 ) / 1153909280$$ (344997*v^11 + 4286939*v^10 - 54344143*v^9 - 171854403*v^8 + 1852765582*v^7 + 2298288511*v^6 - 22776905265*v^5 - 10497325415*v^4 + 96597132385*v^3 + 7992440496*v^2 - 63200790100*v + 996822640) / 1153909280 $$\beta_{8}$$ $$=$$ $$( - 578917 \nu^{11} + 467095 \nu^{10} + 48096453 \nu^{9} - 45343183 \nu^{8} - 1361867036 \nu^{7} + \cdots - 22018596800 ) / 1153909280$$ (-578917*v^11 + 467095*v^10 + 48096453*v^9 - 45343183*v^8 - 1361867036*v^7 + 1257981201*v^6 + 16198209623*v^5 - 14189321211*v^4 - 77758255643*v^3 + 56886337010*v^2 + 106024422400*v - 22018596800) / 1153909280 $$\beta_{9}$$ $$=$$ $$( 593265 \nu^{11} - 2943239 \nu^{10} - 30920335 \nu^{9} + 139543951 \nu^{8} + 583355114 \nu^{7} + \cdots + 12399713520 ) / 576954640$$ (593265*v^11 - 2943239*v^10 - 30920335*v^9 + 139543951*v^8 + 583355114*v^7 - 2343590997*v^6 - 4453116903*v^5 + 16990479701*v^4 + 10198258823*v^3 - 46753446776*v^2 + 6155531140*v + 12399713520) / 576954640 $$\beta_{10}$$ $$=$$ $$( 1717069 \nu^{11} - 12271653 \nu^{10} - 60616831 \nu^{9} + 504434173 \nu^{8} + 575213110 \nu^{7} + \cdots - 1046240240 ) / 1153909280$$ (1717069*v^11 - 12271653*v^10 - 60616831*v^9 + 504434173*v^8 + 575213110*v^7 - 6602635481*v^6 + 42148703*v^5 + 28258252009*v^4 - 12588741663*v^3 - 15899848472*v^2 + 4422155660*v - 1046240240) / 1153909280 $$\beta_{11}$$ $$=$$ $$( - 1909593 \nu^{11} + 13629375 \nu^{10} + 68397797 \nu^{9} - 565931847 \nu^{8} + \cdots + 19602612800 ) / 1153909280$$ (-1909593*v^11 + 13629375*v^10 + 68397797*v^9 - 565931847*v^8 - 671935784*v^7 + 7587139509*v^6 + 96194327*v^5 - 34464037179*v^4 + 14242221053*v^3 + 22379037390*v^2 + 10802405880*v + 19602612800) / 1153909280
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 12$$ b2 + b1 + 12 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 19\beta _1 + 10$$ b3 + 2*b2 + 19*b1 + 10 $$\nu^{4}$$ $$=$$ $$\beta_{11} + \beta_{10} + \beta_{4} + 3\beta_{3} + 27\beta_{2} + 42\beta _1 + 232$$ b11 + b10 + b4 + 3*b3 + 27*b2 + 42*b1 + 232 $$\nu^{5}$$ $$=$$ $$2 \beta_{11} + 2 \beta_{10} + 4 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 5 \beta_{5} + 2 \beta_{4} + \cdots + 476$$ 2*b11 + 2*b10 + 4*b8 + 2*b7 + 2*b6 + 5*b5 + 2*b4 + 35*b3 + 89*b2 + 438*b1 + 476 $$\nu^{6}$$ $$=$$ $$44 \beta_{11} + 46 \beta_{10} + 12 \beta_{8} + 6 \beta_{7} + 12 \beta_{6} + 10 \beta_{5} + 24 \beta_{4} + \cdots + 5382$$ 44*b11 + 46*b10 + 12*b8 + 6*b7 + 12*b6 + 10*b5 + 24*b4 + 140*b3 + 740*b2 + 1463*b1 + 5382 $$\nu^{7}$$ $$=$$ $$158 \beta_{11} + 160 \beta_{10} + 12 \beta_{9} + 170 \beta_{8} + 80 \beta_{7} + 84 \beta_{6} + \cdots + 17350$$ 158*b11 + 160*b10 + 12*b9 + 170*b8 + 80*b7 + 84*b6 + 244*b5 + 40*b4 + 1096*b3 + 3105*b2 + 11321*b1 + 17350 $$\nu^{8}$$ $$=$$ $$1686 \beta_{11} + 1752 \beta_{10} + 124 \beta_{9} + 662 \beta_{8} + 284 \beta_{7} + 560 \beta_{6} + \cdots + 139556$$ 1686*b11 + 1752*b10 + 124*b9 + 662*b8 + 284*b7 + 560*b6 + 684*b5 + 280*b4 + 5133*b3 + 20992*b2 + 47291*b1 + 139556 $$\nu^{9}$$ $$=$$ $$7831 \beta_{11} + 7889 \beta_{10} + 1188 \beta_{9} + 5818 \beta_{8} + 2448 \beta_{7} + 2620 \beta_{6} + \cdots + 572862$$ 7831*b11 + 7889*b10 + 1188*b9 + 5818*b8 + 2448*b7 + 2620*b6 + 8792*b5 - 559*b4 + 34045*b3 + 99815*b2 + 312826*b1 + 572862 $$\nu^{10}$$ $$=$$ $$61990 \beta_{11} + 63400 \beta_{10} + 9348 \beta_{9} + 26434 \beta_{8} + 9894 \beta_{7} + 17714 \beta_{6} + \cdots + 3865846$$ 61990*b11 + 63400*b10 + 9348*b9 + 26434*b8 + 9894*b7 + 17714*b6 + 31305*b5 - 6192*b4 + 173757*b3 + 610523*b2 + 1476372*b1 + 3865846 $$\nu^{11}$$ $$=$$ $$323938 \beta_{11} + 324366 \beta_{10} + 68300 \beta_{9} + 188926 \beta_{8} + 69922 \beta_{7} + \cdots + 18092328$$ 323938*b11 + 324366*b10 + 68300*b9 + 188926*b8 + 69922*b7 + 72532*b6 + 289760*b5 - 83864*b4 + 1064022*b3 + 3102834*b2 + 8979181*b1 + 18092328

## 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
 5.58387 5.28037 3.59117 3.07517 2.35682 0.843186 −0.161453 −1.01648 −2.69379 −3.77011 −3.84968 −4.23908
−4.58387 3.00000 13.0119 5.00000 −13.7516 29.4297 −22.9739 9.00000 −22.9194
1.2 −4.28037 3.00000 10.3215 5.00000 −12.8411 9.81276 −9.93705 9.00000 −21.4018
1.3 −2.59117 3.00000 −1.28583 5.00000 −7.77352 −12.5999 24.0612 9.00000 −12.9559
1.4 −2.07517 3.00000 −3.69366 5.00000 −6.22551 −24.2415 24.2664 9.00000 −10.3759
1.5 −1.35682 3.00000 −6.15904 5.00000 −4.07045 36.6735 19.2112 9.00000 −6.78409
1.6 0.156814 3.00000 −7.97541 5.00000 0.470441 3.41884 −2.50516 9.00000 0.784068
1.7 1.16145 3.00000 −6.65103 5.00000 3.48436 4.01638 −17.0165 9.00000 5.80727
1.8 2.01648 3.00000 −3.93383 5.00000 6.04943 17.1804 −24.0643 9.00000 10.0824
1.9 3.69379 3.00000 5.64411 5.00000 11.0814 0.0404379 −8.70216 9.00000 18.4690
1.10 4.77011 3.00000 14.7539 5.00000 14.3103 −30.6111 32.2168 9.00000 23.8505
1.11 4.84968 3.00000 15.5194 5.00000 14.5491 26.0919 36.4669 9.00000 24.2484
1.12 5.23908 3.00000 19.4479 5.00000 15.7172 17.7886 59.9765 9.00000 26.1954
 $$n$$: 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.bn 12
11.b odd 2 1 1815.4.a.bh 12
11.c even 5 2 165.4.m.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.m.a 24 11.c even 5 2
1815.4.a.bh 12 11.b odd 2 1
1815.4.a.bn 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2}^{12} - 7 T_{2}^{11} - 48 T_{2}^{10} + 376 T_{2}^{9} + 754 T_{2}^{8} - 7037 T_{2}^{7} + \cdots - 23536$$ T2^12 - 7*T2^11 - 48*T2^10 + 376*T2^9 + 754*T2^8 - 7037*T2^7 - 4416*T2^6 + 54358*T2^5 + 10333*T2^4 - 166410*T2^3 - 1040*T2^2 + 154272*T2 - 23536 $$T_{7}^{12} - 77 T_{7}^{11} + 309 T_{7}^{10} + 108586 T_{7}^{9} - 2474180 T_{7}^{8} - 25763424 T_{7}^{7} + \cdots - 438449674204$$ T7^12 - 77*T7^11 + 309*T7^10 + 108586*T7^9 - 2474180*T7^8 - 25763424*T7^7 + 1344638405*T7^6 - 9718045329*T7^5 - 104724428240*T7^4 + 1732742973671*T7^3 - 7704067637391*T7^2 + 11151247116998*T7 - 438449674204

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 7 T^{11} + \cdots - 23536$$
$3$ $$(T - 3)^{12}$$
$5$ $$(T - 5)^{12}$$
$7$ $$T^{12} + \cdots - 438449674204$$
$11$ $$T^{12}$$
$13$ $$T^{12} + \cdots - 10\!\cdots\!39$$
$17$ $$T^{12} + \cdots + 23\!\cdots\!00$$
$19$ $$T^{12} + \cdots - 40\!\cdots\!00$$
$23$ $$T^{12} + \cdots + 69\!\cdots\!09$$
$29$ $$T^{12} + \cdots + 65\!\cdots\!00$$
$31$ $$T^{12} + \cdots - 16\!\cdots\!44$$
$37$ $$T^{12} + \cdots + 52\!\cdots\!41$$
$41$ $$T^{12} + \cdots + 95\!\cdots\!76$$
$43$ $$T^{12} + \cdots + 10\!\cdots\!76$$
$47$ $$T^{12} + \cdots + 36\!\cdots\!09$$
$53$ $$T^{12} + \cdots + 20\!\cdots\!16$$
$59$ $$T^{12} + \cdots - 12\!\cdots\!25$$
$61$ $$T^{12} + \cdots + 58\!\cdots\!64$$
$67$ $$T^{12} + \cdots - 13\!\cdots\!44$$
$71$ $$T^{12} + \cdots - 15\!\cdots\!24$$
$73$ $$T^{12} + \cdots + 40\!\cdots\!24$$
$79$ $$T^{12} + \cdots + 53\!\cdots\!00$$
$83$ $$T^{12} + \cdots - 26\!\cdots\!36$$
$89$ $$T^{12} + \cdots - 63\!\cdots\!00$$
$97$ $$T^{12} + \cdots - 48\!\cdots\!00$$