Properties

 Label 3645.2.a.g Level $3645$ Weight $2$ Character orbit 3645.a Self dual yes Analytic conductor $29.105$ Analytic rank $1$ Dimension $15$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3645,2,Mod(1,3645)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3645, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3645.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3645 = 3^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3645.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: $$29.1054715368$$ Analytic rank: $$1$$ Dimension: $$15$$ Coefficient field: $$\mathbb{Q}[x]/(x^{15} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{15} - 3 x^{14} - 15 x^{13} + 47 x^{12} + 84 x^{11} - 279 x^{10} - 219 x^{9} + 783 x^{8} + 279 x^{7} - 1054 x^{6} - 195 x^{5} + 609 x^{4} + 88 x^{3} - 102 x^{2} + 3$$ x^15 - 3*x^14 - 15*x^13 + 47*x^12 + 84*x^11 - 279*x^10 - 219*x^9 + 783*x^8 + 279*x^7 - 1054*x^6 - 195*x^5 + 609*x^4 + 88*x^3 - 102*x^2 + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: no (minimal twist has level 135) 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 $$1,\beta_1,\ldots,\beta_{14}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{14} - 1) q^{7} + ( - \beta_{3} - 1) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b2 + 1) * q^4 + q^5 + (-b14 - 1) * q^7 + (-b3 - 1) * q^8 $$q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{14} - 1) q^{7} + ( - \beta_{3} - 1) q^{8} - \beta_1 q^{10} + (\beta_{11} + \beta_{7} + \beta_{5} - \beta_{4}) q^{11} + (\beta_{14} - \beta_{5} - \beta_{2} - 1) q^{13} + (\beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{9} - \beta_{5} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{14} + (\beta_{11} - \beta_{10} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{16} + ( - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{7} - \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{17} + (\beta_{14} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_1 - 2) q^{19} + (\beta_{2} + 1) q^{20} + (\beta_{12} + \beta_{11} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{22} + ( - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{23} + q^{25} + ( - \beta_{14} - \beta_{13} + \beta_{12} + 2 \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 1) q^{26}+ \cdots + ( - 2 \beta_{14} - \beta_{13} + 4 \beta_{12} + \beta_{11} + 3 \beta_{9} + 2 \beta_{8} + \beta_{6} + \cdots - 1) q^{98}+O(q^{100})$$ q - b1 * q^2 + (b2 + 1) * q^4 + q^5 + (-b14 - 1) * q^7 + (-b3 - 1) * q^8 - b1 * q^10 + (b11 + b7 + b5 - b4) * q^11 + (b14 - b5 - b2 - 1) * q^13 + (b14 + b13 - b12 - b11 - b9 - b5 + b3 - b2 + 2*b1) * q^14 + (b11 - b10 + b4 + b3 - b2 + b1) * q^16 + (-b14 - b13 + b12 - b7 - b5 + 2*b4 + b3 - b2 + b1 - 1) * q^17 + (b14 - b12 - b11 + b10 + b6 + b5 - b4 + b1 - 2) * q^19 + (b2 + 1) * q^20 + (b12 + b11 - b8 + b7 - b6 + b5 - b4 - 2*b3 - b2 - 2) * q^22 + (-b10 + b9 - b8 - b6 - b4 - b3 + b2 - 1) * q^23 + q^25 + (-b14 - b13 + b12 + 2*b9 + b8 - b7 + b5 + b4 + b3 + 2*b2 + 1) * q^26 + (-b13 + b12 - 2*b11 + b10 + b8 - b7 + b4 - b2 - 3) * q^28 + (b14 - b12 - 2*b11 + 2*b10 + b8 - 2*b7 + 2*b6 + b3 + 2*b2 + b1) * q^29 + (b14 + b13 - b11 + b10 - b9 + b8 - b7 - b6 + b5 - 2) * q^31 + (-b14 + 2*b12 + b9 + b5 - 1) * q^32 + (2*b14 + 2*b13 - 2*b12 - 2*b11 + b8 - b7 - b5 - b4 + b3 - b2 + 2*b1 - 2) * q^34 + (-b14 - 1) * q^35 + (-2*b13 + 2*b12 + b11 - 2*b10 + 3*b9 - b6 + b5 + b4 - b3 + b2 - 2*b1 - 1) * q^37 + (-b13 - b12 + b10 - b8 - b4 - b2 + 3*b1 - 2) * q^38 + (-b3 - 1) * q^40 + (2*b13 - b12 - b11 - b9 - b8 + b6 + b4 + 2*b1 + 1) * q^41 + (-b14 + b13 + b12 + b11 - b10 - 2*b9 - b8 - b5 + b4 + b3 - 2*b2 + b1 - 3) * q^43 + (-b13 + b12 + 2*b11 - b10 + b4 + b3 + 2*b1 + 1) * q^44 + (-b14 - 2*b13 + b12 + 2*b10 + b9 + b8 + b7 + 3*b6 + b5 + 2*b3 - b1 - 2) * q^46 + (2*b14 + b11 - 2*b10 + b8 + b7 - 2*b6 - b5 - b3 - 2*b2 + b1 - 2) * q^47 + (b14 - b12 + 2*b11 - b10 - b9 - b8 + b7 - b6 - b5 + b4 - b3 - 2*b2 + 2*b1) * q^49 - b1 * q^50 + (b13 - 2*b12 + b11 - b9 - 2*b8 + b7 + b6 - 2*b4 - b3 - 2*b2 - 3) * q^52 + (-b14 + b13 + 2*b12 - b11 - b9 + b8 + b7 - b5 + b4 + b3 - 1) * q^53 + (b11 + b7 + b5 - b4) * q^55 + (-2*b12 + b11 - b10 - b9 - b8 - b6 - b5 - b4 + b3 - b2 + 3*b1 + 2) * q^56 + (b14 - 3*b12 - b11 + b10 - b8 - 2*b7 - 2*b5 - b2 - 2) * q^58 + (4*b14 + b13 - 3*b12 - 3*b11 + b10 + 2*b8 - 2*b6 - b5 - b4 + 1) * q^59 + (-b14 + b12 + 2*b11 - b10 + b9 - b8 + 2*b7 + b6 + 2*b3 - 3*b1 + 1) * q^61 + (-b14 + 3*b11 - 4*b9 - 2*b8 + 2*b7 - b5 - b4 - b3 - 3*b2 + 4*b1 + 1) * q^62 + (b14 - b12 + b11 - b9 - b8 + b7 - b6 - b5 - 3*b4 - b3 - 2*b2 + b1 - 3) * q^64 + (b14 - b5 - b2 - 1) * q^65 + (b14 - b13 + b10 + b9 + 2*b8 + b4 + b3 + b2 - 2*b1 - 2) * q^67 + (-2*b14 - b13 - b11 + 2*b10 + b7 + 2*b6 + b5 - b4 + b3 + b1 - 3) * q^68 + (b14 + b13 - b12 - b11 - b9 - b5 + b3 - b2 + 2*b1) * q^70 + (-b14 + 2*b11 - 2*b10 - 3*b9 - 2*b8 + b7 - 2*b6 + b4 - 3*b3 - 2*b2 + 3*b1) * q^71 + (-2*b14 + b12 + b11 - 4*b10 - 2*b8 - b6 - b5 + 3*b4 - b3 - b1) * q^73 + (-b13 + b12 + 3*b11 - b10 + 2*b9 - b8 + 2*b7 + 2*b6 + b5 - b4 + b2 + 2) * q^74 + (-b11 + 2*b10 + b9 + b8 - b5 - b4 + 2*b3 - 2*b2 - 4) * q^76 + (b14 + b13 - b12 - 2*b11 + 2*b10 + b9 + b8 + 2*b6 + b3 - b2 - 1) * q^77 + (-b14 + 2*b13 - b11 + b10 - 2*b9 + b7 + 2*b6 + b5 - b4 - b3 + b2 + b1 - 4) * q^79 + (b11 - b10 + b4 + b3 - b2 + b1) * q^80 + (-2*b14 - 2*b13 + b12 + 2*b11 - 2*b10 + b8 - b7 + 2*b4 - b3 + b2 - b1 - 2) * q^82 + (b14 + 2*b13 - b12 + b11 + b10 + b8 + b5 - 2*b4 - b3 + b2 - 3) * q^83 + (-b14 - b13 + b12 - b7 - b5 + 2*b4 + b3 - b2 + b1 - 1) * q^85 + (-b14 + b13 + 2*b12 - b11 - b10 + 2*b8 - b7 - 2*b6 + b5 + b4 - b3 + 3*b1 - 1) * q^86 + (b13 - 2*b11 + 2*b9 + 2*b8 - 2*b7 + b6 + b4 + b2 - 3*b1 - 4) * q^88 + (-b14 - b13 + 3*b12 - 2*b10 + b9 - b8 + 2*b5 + 2*b4 - 2*b3 + 2*b2 + 1) * q^89 + (-b14 - b13 + b12 + b11 - b10 + b9 - b7 + 2*b5 - b4 + 3*b2 - 2*b1 - 3) * q^91 + (5*b14 + 3*b13 - 5*b12 - 2*b11 - b9 - 2*b7 - 3*b6 - 2*b5 - b4 - 4*b2 + 5*b1 + 1) * q^92 + (-4*b14 - b13 + 5*b12 + b11 - b10 + b9 + b7 + 2*b5 + 3*b4 + b2 + b1 - 2) * q^94 + (b14 - b12 - b11 + b10 + b6 + b5 - b4 + b1 - 2) * q^95 + (b14 + b12 - b11 + b10 + 2*b9 + 3*b8 - 3*b7 - b6 - b3 + 3*b2 - 2*b1 - 3) * q^97 + (-2*b14 - b13 + 4*b12 + b11 + 3*b9 + 2*b8 + b6 + 4*b5 + b4 - 2*b3 + 5*b2 - 3*b1 - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$15 q - 3 q^{2} + 9 q^{4} + 15 q^{5} - 12 q^{7} - 9 q^{8}+O(q^{10})$$ 15 * q - 3 * q^2 + 9 * q^4 + 15 * q^5 - 12 * q^7 - 9 * q^8 $$15 q - 3 q^{2} + 9 q^{4} + 15 q^{5} - 12 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} - 3 q^{16} - 12 q^{17} - 24 q^{19} + 9 q^{20} - 18 q^{22} - 18 q^{23} + 15 q^{25} + 9 q^{26} - 30 q^{28} + 3 q^{29} - 24 q^{31} - 6 q^{32} - 18 q^{34} - 12 q^{35} - 24 q^{37} - 18 q^{38} - 9 q^{40} + 9 q^{41} - 42 q^{43} + 12 q^{44} - 30 q^{46} - 21 q^{47} - 3 q^{49} - 3 q^{50} - 36 q^{52} - 18 q^{53} + 30 q^{56} - 30 q^{58} + 6 q^{59} - 15 q^{61} + 36 q^{62} - 27 q^{64} - 12 q^{65} - 45 q^{67} - 36 q^{68} + 12 q^{71} - 21 q^{73} + 21 q^{74} - 48 q^{76} - 9 q^{77} - 48 q^{79} - 3 q^{80} - 24 q^{82} - 33 q^{83} - 12 q^{85} + 15 q^{86} - 54 q^{88} + 9 q^{89} - 51 q^{91} + 33 q^{92} - 30 q^{94} - 24 q^{95} - 30 q^{97} - 15 q^{98}+O(q^{100})$$ 15 * q - 3 * q^2 + 9 * q^4 + 15 * q^5 - 12 * q^7 - 9 * q^8 - 3 * q^10 - 12 * q^13 - 3 * q^16 - 12 * q^17 - 24 * q^19 + 9 * q^20 - 18 * q^22 - 18 * q^23 + 15 * q^25 + 9 * q^26 - 30 * q^28 + 3 * q^29 - 24 * q^31 - 6 * q^32 - 18 * q^34 - 12 * q^35 - 24 * q^37 - 18 * q^38 - 9 * q^40 + 9 * q^41 - 42 * q^43 + 12 * q^44 - 30 * q^46 - 21 * q^47 - 3 * q^49 - 3 * q^50 - 36 * q^52 - 18 * q^53 + 30 * q^56 - 30 * q^58 + 6 * q^59 - 15 * q^61 + 36 * q^62 - 27 * q^64 - 12 * q^65 - 45 * q^67 - 36 * q^68 + 12 * q^71 - 21 * q^73 + 21 * q^74 - 48 * q^76 - 9 * q^77 - 48 * q^79 - 3 * q^80 - 24 * q^82 - 33 * q^83 - 12 * q^85 + 15 * q^86 - 54 * q^88 + 9 * q^89 - 51 * q^91 + 33 * q^92 - 30 * q^94 - 24 * q^95 - 30 * q^97 - 15 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{15} - 3 x^{14} - 15 x^{13} + 47 x^{12} + 84 x^{11} - 279 x^{10} - 219 x^{9} + 783 x^{8} + 279 x^{7} - 1054 x^{6} - 195 x^{5} + 609 x^{4} + 88 x^{3} - 102 x^{2} + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4\nu - 1$$ v^3 - 4*v - 1 $$\beta_{4}$$ $$=$$ $$( \nu^{14} - 2 \nu^{13} - 17 \nu^{12} + 30 \nu^{11} + 114 \nu^{10} - 165 \nu^{9} - 384 \nu^{8} + 399 \nu^{7} + 678 \nu^{6} - 376 \nu^{5} - 571 \nu^{4} + 29 \nu^{3} + 135 \nu^{2} + 69 \nu + 6 ) / 9$$ (v^14 - 2*v^13 - 17*v^12 + 30*v^11 + 114*v^10 - 165*v^9 - 384*v^8 + 399*v^7 + 678*v^6 - 376*v^5 - 571*v^4 + 29*v^3 + 135*v^2 + 69*v + 6) / 9 $$\beta_{5}$$ $$=$$ $$( 2 \nu^{14} - 6 \nu^{13} - 29 \nu^{12} + 93 \nu^{11} + 150 \nu^{10} - 543 \nu^{9} - 315 \nu^{8} + 1482 \nu^{7} + 165 \nu^{6} - 1892 \nu^{5} + 192 \nu^{4} + 977 \nu^{3} - 165 \nu^{2} - 135 \nu + 42 ) / 9$$ (2*v^14 - 6*v^13 - 29*v^12 + 93*v^11 + 150*v^10 - 543*v^9 - 315*v^8 + 1482*v^7 + 165*v^6 - 1892*v^5 + 192*v^4 + 977*v^3 - 165*v^2 - 135*v + 42) / 9 $$\beta_{6}$$ $$=$$ $$( - 4 \nu^{13} + 7 \nu^{12} + 66 \nu^{11} - 99 \nu^{10} - 420 \nu^{9} + 501 \nu^{8} + 1290 \nu^{7} - 1071 \nu^{6} - 1938 \nu^{5} + 826 \nu^{4} + 1235 \nu^{3} - 51 \nu^{2} - 141 \nu + 15 ) / 9$$ (-4*v^13 + 7*v^12 + 66*v^11 - 99*v^10 - 420*v^9 + 501*v^8 + 1290*v^7 - 1071*v^6 - 1938*v^5 + 826*v^4 + 1235*v^3 - 51*v^2 - 141*v + 15) / 9 $$\beta_{7}$$ $$=$$ $$( - 2 \nu^{13} + 6 \nu^{12} + 28 \nu^{11} - 90 \nu^{10} - 136 \nu^{9} + 499 \nu^{8} + 246 \nu^{7} - 1249 \nu^{6} - 29 \nu^{5} + 1361 \nu^{4} - 259 \nu^{3} - 501 \nu^{2} + 102 \nu + 15 ) / 3$$ (-2*v^13 + 6*v^12 + 28*v^11 - 90*v^10 - 136*v^9 + 499*v^8 + 246*v^7 - 1249*v^6 - 29*v^5 + 1361*v^4 - 259*v^3 - 501*v^2 + 102*v + 15) / 3 $$\beta_{8}$$ $$=$$ $$( - 5 \nu^{14} + 14 \nu^{13} + 75 \nu^{12} - 213 \nu^{11} - 420 \nu^{10} + 1209 \nu^{9} + 1089 \nu^{8} - 3153 \nu^{7} - 1329 \nu^{6} + 3722 \nu^{5} + 748 \nu^{4} - 1674 \nu^{3} - 201 \nu^{2} + 201 \nu + 9 ) / 9$$ (-5*v^14 + 14*v^13 + 75*v^12 - 213*v^11 - 420*v^10 + 1209*v^9 + 1089*v^8 - 3153*v^7 - 1329*v^6 + 3722*v^5 + 748*v^4 - 1674*v^3 - 201*v^2 + 201*v + 9) / 9 $$\beta_{9}$$ $$=$$ $$( - 3 \nu^{14} + 7 \nu^{13} + 53 \nu^{12} - 117 \nu^{11} - 372 \nu^{10} + 753 \nu^{9} + 1329 \nu^{8} - 2337 \nu^{7} - 2574 \nu^{6} + 3546 \nu^{5} + 2642 \nu^{4} - 2267 \nu^{3} - 1200 \nu^{2} + \cdots + 57 ) / 9$$ (-3*v^14 + 7*v^13 + 53*v^12 - 117*v^11 - 372*v^10 + 753*v^9 + 1329*v^8 - 2337*v^7 - 2574*v^6 + 3546*v^5 + 2642*v^4 - 2267*v^3 - 1200*v^2 + 303*v + 57) / 9 $$\beta_{10}$$ $$=$$ $$( - 6 \nu^{14} + 22 \nu^{13} + 77 \nu^{12} - 336 \nu^{11} - 309 \nu^{10} + 1917 \nu^{9} + 237 \nu^{8} - 5037 \nu^{7} + 984 \nu^{6} + 6021 \nu^{5} - 1603 \nu^{4} - 2747 \nu^{3} + 357 \nu^{2} + 222 \nu - 15 ) / 9$$ (-6*v^14 + 22*v^13 + 77*v^12 - 336*v^11 - 309*v^10 + 1917*v^9 + 237*v^8 - 5037*v^7 + 984*v^6 + 6021*v^5 - 1603*v^4 - 2747*v^3 + 357*v^2 + 222*v - 15) / 9 $$\beta_{11}$$ $$=$$ $$( - 7 \nu^{14} + 24 \nu^{13} + 94 \nu^{12} - 366 \nu^{11} - 423 \nu^{10} + 2082 \nu^{9} + 621 \nu^{8} - 5436 \nu^{7} + 306 \nu^{6} + 6397 \nu^{5} - 1023 \nu^{4} - 2785 \nu^{3} + 177 \nu^{2} + 180 \nu - 3 ) / 9$$ (-7*v^14 + 24*v^13 + 94*v^12 - 366*v^11 - 423*v^10 + 2082*v^9 + 621*v^8 - 5436*v^7 + 306*v^6 + 6397*v^5 - 1023*v^4 - 2785*v^3 + 177*v^2 + 180*v - 3) / 9 $$\beta_{12}$$ $$=$$ $$( 5 \nu^{14} - 14 \nu^{13} - 81 \nu^{12} + 225 \nu^{11} + 516 \nu^{10} - 1380 \nu^{9} - 1680 \nu^{8} + 4032 \nu^{7} + 3084 \nu^{6} - 5663 \nu^{5} - 3289 \nu^{4} + 3291 \nu^{3} + 1713 \nu^{2} - 372 \nu - 99 ) / 9$$ (5*v^14 - 14*v^13 - 81*v^12 + 225*v^11 + 516*v^10 - 1380*v^9 - 1680*v^8 + 4032*v^7 + 3084*v^6 - 5663*v^5 - 3289*v^4 + 3291*v^3 + 1713*v^2 - 372*v - 99) / 9 $$\beta_{13}$$ $$=$$ $$( - 12 \nu^{14} + 35 \nu^{13} + 178 \nu^{12} - 537 \nu^{11} - 978 \nu^{10} + 3087 \nu^{9} + 2463 \nu^{8} - 8220 \nu^{7} - 2955 \nu^{6} + 10062 \nu^{5} + 2005 \nu^{4} - 4744 \nu^{3} - 1077 \nu^{2} + \cdots + 60 ) / 9$$ (-12*v^14 + 35*v^13 + 178*v^12 - 537*v^11 - 978*v^10 + 3087*v^9 + 2463*v^8 - 8220*v^7 - 2955*v^6 + 10062*v^5 + 2005*v^4 - 4744*v^3 - 1077*v^2 + 363*v + 60) / 9 $$\beta_{14}$$ $$=$$ $$( 3 \nu^{14} - 9 \nu^{13} - 46 \nu^{12} + 142 \nu^{11} + 270 \nu^{10} - 850 \nu^{9} - 782 \nu^{8} + 2403 \nu^{7} + 1253 \nu^{6} - 3221 \nu^{5} - 1248 \nu^{4} + 1740 \nu^{3} + 687 \nu^{2} - 156 \nu - 36 ) / 3$$ (3*v^14 - 9*v^13 - 46*v^12 + 142*v^11 + 270*v^10 - 850*v^9 - 782*v^8 + 2403*v^7 + 1253*v^6 - 3221*v^5 - 1248*v^4 + 1740*v^3 + 687*v^2 - 156*v - 36) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4\beta _1 + 1$$ b3 + 4*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{11} - \beta_{10} + \beta_{4} + \beta_{3} + 5\beta_{2} + \beta _1 + 14$$ b11 - b10 + b4 + b3 + 5*b2 + b1 + 14 $$\nu^{5}$$ $$=$$ $$\beta_{14} - 2\beta_{12} - \beta_{9} - \beta_{5} + 8\beta_{3} + 20\beta _1 + 9$$ b14 - 2*b12 - b9 - b5 + 8*b3 + 20*b1 + 9 $$\nu^{6}$$ $$=$$ $$\beta_{14} - \beta_{12} + 11 \beta_{11} - 10 \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + 7 \beta_{4} + 9 \beta_{3} + 24 \beta_{2} + 11 \beta _1 + 73$$ b14 - b12 + 11*b11 - 10*b10 - b9 - b8 + b7 - b6 - b5 + 7*b4 + 9*b3 + 24*b2 + 11*b1 + 73 $$\nu^{7}$$ $$=$$ $$11 \beta_{14} + \beta_{13} - 22 \beta_{12} + 3 \beta_{11} - 4 \beta_{10} - 12 \beta_{9} - 2 \beta_{8} - \beta_{6} - 13 \beta_{5} + 53 \beta_{3} - \beta_{2} + 111 \beta _1 + 68$$ 11*b14 + b13 - 22*b12 + 3*b11 - 4*b10 - 12*b9 - 2*b8 - b6 - 13*b5 + 53*b3 - b2 + 111*b1 + 68 $$\nu^{8}$$ $$=$$ $$16 \beta_{14} + 2 \beta_{13} - 19 \beta_{12} + 88 \beta_{11} - 77 \beta_{10} - 16 \beta_{9} - 15 \beta_{8} + 12 \beta_{7} - 13 \beta_{6} - 16 \beta_{5} + 38 \beta_{4} + 65 \beta_{3} + 114 \beta_{2} + 96 \beta _1 + 405$$ 16*b14 + 2*b13 - 19*b12 + 88*b11 - 77*b10 - 16*b9 - 15*b8 + 12*b7 - 13*b6 - 16*b5 + 38*b4 + 65*b3 + 114*b2 + 96*b1 + 405 $$\nu^{9}$$ $$=$$ $$95 \beta_{14} + 17 \beta_{13} - 183 \beta_{12} + 47 \beta_{11} - 60 \beta_{10} - 106 \beta_{9} - 31 \beta_{8} + 3 \beta_{7} - 19 \beta_{6} - 119 \beta_{5} - 2 \beta_{4} + 333 \beta_{3} - 18 \beta_{2} + 654 \beta _1 + 484$$ 95*b14 + 17*b13 - 183*b12 + 47*b11 - 60*b10 - 106*b9 - 31*b8 + 3*b7 - 19*b6 - 119*b5 - 2*b4 + 333*b3 - 18*b2 + 654*b1 + 484 $$\nu^{10}$$ $$=$$ $$172 \beta_{14} + 37 \beta_{13} - 219 \beta_{12} + 627 \beta_{11} - 548 \beta_{10} - 167 \beta_{9} - 150 \beta_{8} + 100 \beta_{7} - 124 \beta_{6} - 173 \beta_{5} + 187 \beta_{4} + 445 \beta_{3} + 533 \beta_{2} + 758 \beta _1 + 2345$$ 172*b14 + 37*b13 - 219*b12 + 627*b11 - 548*b10 - 167*b9 - 150*b8 + 100*b7 - 124*b6 - 173*b5 + 187*b4 + 445*b3 + 533*b2 + 758*b1 + 2345 $$\nu^{11}$$ $$=$$ $$757 \beta_{14} + 192 \beta_{13} - 1384 \beta_{12} + 492 \beta_{11} - 614 \beta_{10} - 831 \beta_{9} - 323 \beta_{8} + 49 \beta_{7} - 226 \beta_{6} - 949 \beta_{5} - 39 \beta_{4} + 2058 \beta_{3} - 201 \beta_{2} + 4000 \beta _1 + 3348$$ 757*b14 + 192*b13 - 1384*b12 + 492*b11 - 614*b10 - 831*b9 - 323*b8 + 49*b7 - 226*b6 - 949*b5 - 39*b4 + 2058*b3 - 201*b2 + 4000*b1 + 3348 $$\nu^{12}$$ $$=$$ $$1563 \beta_{14} + 441 \beta_{13} - 2055 \beta_{12} + 4242 \beta_{11} - 3789 \beta_{10} - 1464 \beta_{9} - 1272 \beta_{8} + 723 \beta_{7} - 1053 \beta_{6} - 1572 \beta_{5} + 852 \beta_{4} + 2998 \beta_{3} + \cdots + 14014$$ 1563*b14 + 441*b13 - 2055*b12 + 4242*b11 - 3789*b10 - 1464*b9 - 1272*b8 + 723*b7 - 1053*b6 - 1572*b5 + 852*b4 + 2998*b3 + 2418*b2 + 5668*b1 + 14014 $$\nu^{13}$$ $$=$$ $$5793 \beta_{14} + 1812 \beta_{13} - 10035 \beta_{12} + 4339 \beta_{11} - 5362 \beta_{10} - 6132 \beta_{9} - 2844 \beta_{8} + 519 \beta_{7} - 2193 \beta_{6} - 7077 \beta_{5} - 479 \beta_{4} + 12688 \beta_{3} + \cdots + 22823$$ 5793*b14 + 1812*b13 - 10035*b12 + 4339*b11 - 5362*b10 - 6132*b9 - 2844*b8 + 519*b7 - 2193*b6 - 7077*b5 - 479*b4 + 12688*b3 - 1837*b2 + 25081*b1 + 22823 $$\nu^{14}$$ $$=$$ $$12967 \beta_{14} + 4317 \beta_{13} - 17306 \beta_{12} + 28017 \beta_{11} - 25908 \beta_{10} - 11728 \beta_{9} - 9921 \beta_{8} + 4884 \beta_{7} - 8421 \beta_{6} - 12976 \beta_{5} + 3474 \beta_{4} + \cdots + 85806$$ 12967*b14 + 4317*b13 - 17306*b12 + 28017*b11 - 25908*b10 - 11728*b9 - 9921*b8 + 4884*b7 - 8421*b6 - 12976*b5 + 3474*b4 + 20078*b3 + 10353*b2 + 41039*b1 + 85806

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
 2.60146 2.36967 2.11967 1.63308 1.47820 1.30569 0.300859 0.243507 −0.174249 −0.694378 −0.748857 −1.29880 −1.84992 −1.98404 −2.30187
−2.60146 0 4.76758 1.00000 0 −4.24584 −7.19975 0 −2.60146
1.2 −2.36967 0 3.61532 1.00000 0 −0.197267 −3.82778 0 −2.36967
1.3 −2.11967 0 2.49298 1.00000 0 −0.0632009 −1.04496 0 −2.11967
1.4 −1.63308 0 0.666944 1.00000 0 −3.31953 2.17698 0 −1.63308
1.5 −1.47820 0 0.185067 1.00000 0 3.14304 2.68283 0 −1.47820
1.6 −1.30569 0 −0.295165 1.00000 0 0.875317 2.99678 0 −1.30569
1.7 −0.300859 0 −1.90948 1.00000 0 −4.28937 1.17620 0 −0.300859
1.8 −0.243507 0 −1.94070 1.00000 0 3.96723 0.959589 0 −0.243507
1.9 0.174249 0 −1.96964 1.00000 0 −1.74201 −0.691706 0 0.174249
1.10 0.694378 0 −1.51784 1.00000 0 −0.291597 −2.44271 0 0.694378
1.11 0.748857 0 −1.43921 1.00000 0 −0.530152 −2.57548 0 0.748857
1.12 1.29880 0 −0.313113 1.00000 0 −1.51756 −3.00428 0 1.29880
1.13 1.84992 0 1.42221 1.00000 0 −1.13884 −1.06887 0 1.84992
1.14 1.98404 0 1.93643 1.00000 0 1.64269 −0.126117 0 1.98404
1.15 2.30187 0 3.29862 1.00000 0 −4.29291 2.98925 0 2.30187
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.15 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 3645.2.a.g 15
3.b odd 2 1 3645.2.a.h 15
27.e even 9 2 405.2.k.a 30
27.f odd 18 2 135.2.k.a 30
135.n odd 18 2 675.2.l.d 30
135.q even 36 4 675.2.u.c 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.k.a 30 27.f odd 18 2
405.2.k.a 30 27.e even 9 2
675.2.l.d 30 135.n odd 18 2
675.2.u.c 60 135.q even 36 4
3645.2.a.g 15 1.a even 1 1 trivial
3645.2.a.h 15 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{15} + 3 T_{2}^{14} - 15 T_{2}^{13} - 47 T_{2}^{12} + 84 T_{2}^{11} + 279 T_{2}^{10} - 219 T_{2}^{9} - 783 T_{2}^{8} + 279 T_{2}^{7} + 1054 T_{2}^{6} - 195 T_{2}^{5} - 609 T_{2}^{4} + 88 T_{2}^{3} + 102 T_{2}^{2} - 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3645))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{15} + 3 T^{14} - 15 T^{13} - 47 T^{12} + \cdots - 3$$
$3$ $$T^{15}$$
$5$ $$(T - 1)^{15}$$
$7$ $$T^{15} + 12 T^{14} + 21 T^{13} - 263 T^{12} + \cdots + 27$$
$11$ $$T^{15} - 78 T^{13} - 44 T^{12} + \cdots - 51111$$
$13$ $$T^{15} + 12 T^{14} - 21 T^{13} + \cdots + 75853$$
$17$ $$T^{15} + 12 T^{14} - 51 T^{13} + \cdots + 88719$$
$19$ $$T^{15} + 24 T^{14} + 156 T^{13} + \cdots - 7019$$
$23$ $$T^{15} + 18 T^{14} + \cdots + 208681353$$
$29$ $$T^{15} - 3 T^{14} - 261 T^{13} + \cdots + 70766457$$
$31$ $$T^{15} + 24 T^{14} + \cdots + 2774253969$$
$37$ $$T^{15} + 24 T^{14} + \cdots + 360965557$$
$41$ $$T^{15} - 9 T^{14} + \cdots - 7229668599$$
$43$ $$T^{15} + 42 T^{14} + \cdots - 2166924437$$
$47$ $$T^{15} + 21 T^{14} + \cdots + 1262687781$$
$53$ $$T^{15} + 18 T^{14} + \cdots - 30327689193$$
$59$ $$T^{15} - 6 T^{14} + \cdots + 4279835777463$$
$61$ $$T^{15} + 15 T^{14} + \cdots - 1132793927$$
$67$ $$T^{15} + 45 T^{14} + \cdots - 894784563$$
$71$ $$T^{15} - 12 T^{14} + \cdots - 462163501167$$
$73$ $$T^{15} + 21 T^{14} + \cdots - 72125996039$$
$79$ $$T^{15} + 48 T^{14} + \cdots + 59845453641$$
$83$ $$T^{15} + 33 T^{14} + \cdots - 1117006209153$$
$89$ $$T^{15} - 9 T^{14} + \cdots - 446759170029$$
$97$ $$T^{15} + 30 T^{14} + \cdots + 33134907354553$$