# Properties

 Label 3645.2.a.j Level $3645$ Weight $2$ Character orbit 3645.a Self dual yes Analytic conductor $29.105$ Analytic rank $0$ Dimension $18$ 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: $$0$$ Dimension: $$18$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{18} - 6 x^{17} - 9 x^{16} + 112 x^{15} - 51 x^{14} - 810 x^{13} + 936 x^{12} + 2844 x^{11} - 4536 x^{10} - 4934 x^{9} + 10173 x^{8} + 3672 x^{7} - 11144 x^{6} - 456 x^{5} + \cdots + 9$$ x^18 - 6*x^17 - 9*x^16 + 112*x^15 - 51*x^14 - 810*x^13 + 936*x^12 + 2844*x^11 - 4536*x^10 - 4934*x^9 + 10173*x^8 + 3672*x^7 - 11144*x^6 - 456*x^5 + 5490*x^4 - 354*x^3 - 882*x^2 - 54*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{5}$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{4} + 1) q^{7} + (\beta_{3} + \beta_{2} + 1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 1) * q^4 + q^5 + (-b4 + 1) * q^7 + (b3 + b2 + 1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{4} + 1) q^{7} + (\beta_{3} + \beta_{2} + 1) q^{8} + \beta_1 q^{10} + ( - \beta_{16} + \beta_{15} - \beta_{12} + \beta_{8} - \beta_{7} - \beta_{5} + 1) q^{11} + (\beta_{8} + 1) q^{13} + ( - \beta_{16} + \beta_{13} - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{4} + \beta_1 + 1) q^{14} + ( - \beta_{16} + \beta_{13} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{16} + (\beta_{17} + \beta_{16} - \beta_{15} - \beta_{13} - \beta_{11} + \beta_{9} + \beta_{7} - \beta_{4} - \beta_{3} + 1) q^{17} + ( - \beta_{14} - \beta_{12} + \beta_{6} - \beta_{2} + \beta_1 - 1) q^{19} + (\beta_{2} + 1) q^{20} + (\beta_{17} - \beta_{13} - \beta_{12} - \beta_{11} + \beta_{9} + \beta_{7} - \beta_{5} + 2 \beta_1 - 1) q^{22} + ( - \beta_{17} + \beta_{11} - \beta_{9} - \beta_1 + 2) q^{23} + q^{25} + (2 \beta_{16} - \beta_{15} + \beta_{12} + \beta_{10} - \beta_{8} + 2 \beta_{7} + \beta_{5} + \beta_{2} + 1) q^{26} + ( - \beta_{16} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{3} + \beta_{2} + \cdots + 2) q^{28}+ \cdots + ( - \beta_{16} + \beta_{15} - \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \beta_{11} + 2 \beta_{9} + \cdots + 2) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 1) * q^4 + q^5 + (-b4 + 1) * q^7 + (b3 + b2 + 1) * q^8 + b1 * q^10 + (-b16 + b15 - b12 + b8 - b7 - b5 + 1) * q^11 + (b8 + 1) * q^13 + (-b16 + b13 - b10 + b8 - b7 - b4 + b1 + 1) * q^14 + (-b16 + b13 + b8 - b7 - b6 + b3 + b2 + b1 + 1) * q^16 + (b17 + b16 - b15 - b13 - b11 + b9 + b7 - b4 - b3 + 1) * q^17 + (-b14 - b12 + b6 - b2 + b1 - 1) * q^19 + (b2 + 1) * q^20 + (b17 - b13 - b12 - b11 + b9 + b7 - b5 + 2*b1 - 1) * q^22 + (-b17 + b11 - b9 - b1 + 2) * q^23 + q^25 + (2*b16 - b15 + b12 + b10 - b8 + 2*b7 + b5 + b2 + 1) * q^26 + (-b16 + b14 + b13 + b12 - b10 - b9 - b7 + b3 + b2 + 2) * q^28 + (b16 - b14 + b12 + b10 - b8 + b7 + 2*b5 + 1) * q^29 + (b15 + b14 - b13 - b10 - b7 - b2 - b1) * q^31 + (b17 - b16 - b15 + b13 + b12 + b10 + b7 - b6 + b5 - b4 + 2*b2 + 3) * q^32 + (b15 + b14 - b13 - b10 + b9 - b7 - 2*b5 - b2 + b1 - 1) * q^34 + (-b4 + 1) * q^35 + (-b17 - b16 + 2*b15 - b12 + 2*b11 - b10 - 2*b7 + 2*b6 - 2*b5 + 2*b4 - 1) * q^37 + (-b17 + b15 - b12 + b10 - b9 - b7 - b5 + b4 - b3) * q^38 + (b3 + b2 + 1) * q^40 + (-2*b17 + b11 - b10 - b9 - b8 - b7 + b5 + b3 - b1 + 2) * q^41 + (-b15 - b14 - b11 + 2*b10 - b8 - 2*b6 + b5 + b4 + 2*b1) * q^43 + (b17 + b14 - 2*b13 - b12 - b11 + 2*b9 - b6 - 3*b5 + b4 - b3 + 2*b2 + 2) * q^44 + (2*b16 - 2*b15 - b14 + b13 + b12 + 2*b10 - b9 - b8 + 3*b7 + 2*b6 + 2*b5 - b3 + 2*b1 - 2) * q^46 + (-2*b17 - 2*b16 + 2*b15 - b14 + 2*b13 + b11 - 2*b10 - b9 + b8 - 3*b7 + b6 + b4 - b2 + 1) * q^47 + (b14 - b11 + b10 + 2*b7 - 2*b6 + b5 - b4 + b1 + 1) * q^49 + b1 * q^50 + (-b17 + b16 - b15 - b14 + b13 + b11 + b10 - b9 - b8 + b7 + 2*b5 + b2 + b1) * q^52 + (b17 + b16 + b15 - b12 + b10 + b9 + b6 - b4 - b2 - b1 + 1) * q^53 + (-b16 + b15 - b12 + b8 - b7 - b5 + 1) * q^55 + (-b16 + b14 + b12 + b11 - b10 - 2*b9 - b8 - b7 + 2*b3 + 2*b2 - 2*b1 + 3) * q^56 + (-2*b17 + b16 + b15 - b14 + 2*b13 + b12 + 2*b11 - 2*b7 - b6 + 3*b5 + b4 + b3 - b2 + 2*b1) * q^58 + (2*b17 + 3*b16 - 3*b15 + 2*b14 - 2*b13 + b12 + 2*b10 - 2*b8 + 5*b7 + 2*b6 - b4 - 2*b3 + b2 - b1 + 4) * q^59 + (-b17 - b14 - b13 + b12 - b9 - b8 - 2*b5 + 2*b4 - b3 - 2) * q^61 + (b16 + 2*b15 + b14 + b12 - 3*b10 + 2*b9 - b8 - 2*b7 + b6 + b4 - 2*b2 - b1 - 2) * q^62 + (b17 - b16 - b15 + b12 + 2*b11 + b10 - b9 - b8 + b7 - 2*b4 + b3 + b1 + 1) * q^64 + (b8 + 1) * q^65 + (b17 + 3*b16 - 2*b15 - b14 - 3*b13 + b12 - 3*b11 + b10 + b9 - 2*b8 + 3*b7 - 2*b6 + 2*b5 - b3 + b1 - 1) * q^67 + (-3*b16 + 2*b15 + b14 - b13 - b12 - b11 - 3*b10 + b9 + b8 - 2*b7 - b6 - 2*b5 + b4 - 2*b1 + 2) * q^68 + (-b16 + b13 - b10 + b8 - b7 - b4 + b1 + 1) * q^70 + (b17 + b16 - 2*b15 - b14 + b12 - b11 + b9 + b7 + b6 - b5 + 2*b2 + 3) * q^71 + (b17 + 2*b14 + b13 + b11 - b9 + b7 + b6 - b1 + 1) * q^73 + (b17 + 2*b16 - b15 - 4*b13 - b12 - 3*b11 + b10 - 2*b8 + 3*b7 - b6 + b4 - b3 + 2*b1) * q^74 + (2*b16 - b15 - 2*b13 - b11 - b8 + 4*b7 + b6 + b5 + b4 - b3 - b2 + b1 - 3) * q^76 + (-3*b16 + b15 + b14 + b13 - b12 - b10 + b8 - 2*b7 - b5 - 2*b4 - b2 + 2) * q^77 + (2*b17 + b16 - b15 + b14 + 2*b12 + b10 + 2*b9 - b8 + 2*b7 - b6 - b3) * q^79 + (-b16 + b13 + b8 - b7 - b6 + b3 + b2 + b1 + 1) * q^80 + (-2*b17 + b16 - b14 + 3*b13 + 2*b12 - b9 - b8 - b7 + 2*b6 + 3*b5 + b4 + b2 + 2*b1 - 1) * q^82 + (b15 + b14 + b13 + 2*b12 + b11 - b9 - b8 + b7 + b5 + 2*b4 + b3 + b2 - 3*b1 + 4) * q^83 + (b17 + b16 - b15 - b13 - b11 + b9 + b7 - b4 - b3 + 1) * q^85 + (-b16 - 2*b14 + b11 + 2*b10 + b9 - b8 + b7 + b5 + 2*b4 + b3 + b2 + b1) * q^86 + (2*b17 - b16 + b14 - 5*b13 - b12 - 2*b11 + 3*b9 + b7 - b6 - 4*b5 + b4 + b2 + b1) * q^88 + (b16 - b15 - 2*b14 - b12 - b7 - 2*b6 - b4 - 3*b2 + 3*b1) * q^89 + (b17 + 2*b16 - b15 + 2*b14 + b12 + 2*b11 - b10 + b9 + b8 + b7 + 2*b6 - 2*b4 - b3 - b1) * q^91 + (-2*b17 + 2*b16 - 2*b15 - 2*b14 + 2*b13 + b11 + 2*b10 - 4*b9 + b7 + 3*b5 - b3 + 1) * q^92 + (-2*b17 + 2*b16 + b12 - b11 - b10 - 2*b9 - 2*b8 + b6 + 3*b5 - b4 - 3*b2 + 2*b1 - 2) * q^94 + (-b14 - b12 + b6 - b2 + b1 - 1) * q^95 + (b17 - b15 + b14 - b13 - b12 - b9 - b8 - b7 - 2*b5 - 2*b3) * q^97 + (-b16 + b15 - b14 + 2*b13 - 2*b12 + b11 + 2*b9 + 3*b8 - b7 + b6 - b5 - b3 + 2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q + 6 q^{2} + 18 q^{4} + 18 q^{5} + 12 q^{7} + 18 q^{8}+O(q^{10})$$ 18 * q + 6 * q^2 + 18 * q^4 + 18 * q^5 + 12 * q^7 + 18 * q^8 $$18 q + 6 q^{2} + 18 q^{4} + 18 q^{5} + 12 q^{7} + 18 q^{8} + 6 q^{10} + 24 q^{11} + 12 q^{13} + 12 q^{14} + 18 q^{16} + 12 q^{17} + 18 q^{20} + 6 q^{22} + 24 q^{23} + 18 q^{25} + 12 q^{26} + 24 q^{28} + 24 q^{29} - 6 q^{31} + 42 q^{32} - 12 q^{34} + 12 q^{35} + 6 q^{37} + 12 q^{38} + 18 q^{40} + 24 q^{41} + 24 q^{43} + 48 q^{44} - 30 q^{46} + 24 q^{47} + 12 q^{49} + 6 q^{50} + 6 q^{52} + 24 q^{53} + 24 q^{55} + 36 q^{56} + 12 q^{58} + 48 q^{59} - 24 q^{61} - 30 q^{62} + 12 q^{64} + 12 q^{65} - 6 q^{67} + 36 q^{68} + 12 q^{70} + 48 q^{71} + 6 q^{73} + 36 q^{74} - 42 q^{76} + 24 q^{77} - 6 q^{79} + 18 q^{80} - 12 q^{82} + 60 q^{83} + 12 q^{85} + 36 q^{86} + 24 q^{88} + 24 q^{89} - 42 q^{91} + 6 q^{92} - 36 q^{94} + 6 q^{97} + 42 q^{98}+O(q^{100})$$ 18 * q + 6 * q^2 + 18 * q^4 + 18 * q^5 + 12 * q^7 + 18 * q^8 + 6 * q^10 + 24 * q^11 + 12 * q^13 + 12 * q^14 + 18 * q^16 + 12 * q^17 + 18 * q^20 + 6 * q^22 + 24 * q^23 + 18 * q^25 + 12 * q^26 + 24 * q^28 + 24 * q^29 - 6 * q^31 + 42 * q^32 - 12 * q^34 + 12 * q^35 + 6 * q^37 + 12 * q^38 + 18 * q^40 + 24 * q^41 + 24 * q^43 + 48 * q^44 - 30 * q^46 + 24 * q^47 + 12 * q^49 + 6 * q^50 + 6 * q^52 + 24 * q^53 + 24 * q^55 + 36 * q^56 + 12 * q^58 + 48 * q^59 - 24 * q^61 - 30 * q^62 + 12 * q^64 + 12 * q^65 - 6 * q^67 + 36 * q^68 + 12 * q^70 + 48 * q^71 + 6 * q^73 + 36 * q^74 - 42 * q^76 + 24 * q^77 - 6 * q^79 + 18 * q^80 - 12 * q^82 + 60 * q^83 + 12 * q^85 + 36 * q^86 + 24 * q^88 + 24 * q^89 - 42 * q^91 + 6 * q^92 - 36 * q^94 + 6 * q^97 + 42 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 6 x^{17} - 9 x^{16} + 112 x^{15} - 51 x^{14} - 810 x^{13} + 936 x^{12} + 2844 x^{11} - 4536 x^{10} - 4934 x^{9} + 10173 x^{8} + 3672 x^{7} - 11144 x^{6} - 456 x^{5} + \cdots + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4\nu + 2$$ v^3 - v^2 - 4*v + 2 $$\beta_{4}$$ $$=$$ $$( 260 \nu^{17} - 1019 \nu^{16} - 4489 \nu^{15} + 20229 \nu^{14} + 28191 \nu^{13} - 159909 \nu^{12} - 72663 \nu^{11} + 644193 \nu^{10} + 38247 \nu^{9} - 1407583 \nu^{8} + \cdots + 34029 ) / 7461$$ (260*v^17 - 1019*v^16 - 4489*v^15 + 20229*v^14 + 28191*v^13 - 159909*v^12 - 72663*v^11 + 644193*v^10 + 38247*v^9 - 1407583*v^8 + 114322*v^7 + 1630061*v^6 - 79611*v^5 - 856863*v^4 - 107322*v^3 + 80712*v^2 + 70281*v + 34029) / 7461 $$\beta_{5}$$ $$=$$ $$( - 314 \nu^{17} + 1186 \nu^{16} + 5568 \nu^{15} - 23493 \nu^{14} - 35940 \nu^{13} + 183192 \nu^{12} + 88845 \nu^{11} - 707337 \nu^{10} + 43284 \nu^{9} + 1377325 \nu^{8} + \cdots - 6489 ) / 7461$$ (-314*v^17 + 1186*v^16 + 5568*v^15 - 23493*v^14 - 35940*v^13 + 183192*v^12 + 88845*v^11 - 707337*v^10 + 43284*v^9 + 1377325*v^8 - 662657*v^7 - 1148712*v^6 + 1260999*v^5 + 70770*v^4 - 865035*v^3 + 248256*v^2 + 153702*v - 6489) / 7461 $$\beta_{6}$$ $$=$$ $$( 1273 \nu^{17} - 6255 \nu^{16} - 18175 \nu^{15} + 121896 \nu^{14} + 68688 \nu^{13} - 940854 \nu^{12} + 132228 \nu^{11} + 3669384 \nu^{10} - 1495221 \nu^{9} - 7674794 \nu^{8} + \cdots - 10071 ) / 7461$$ (1273*v^17 - 6255*v^16 - 18175*v^15 + 121896*v^14 + 68688*v^13 - 940854*v^12 + 132228*v^11 + 3669384*v^10 - 1495221*v^9 - 7674794*v^8 + 3598935*v^7 + 8491703*v^6 - 3356310*v^5 - 4631352*v^4 + 980490*v^3 + 993321*v^2 + 56772*v - 10071) / 7461 $$\beta_{7}$$ $$=$$ $$( - 1368 \nu^{17} + 6994 \nu^{16} + 19321 \nu^{15} - 138231 \nu^{14} - 69471 \nu^{13} + 1089723 \nu^{12} - 176988 \nu^{11} - 4390062 \nu^{10} + 1744629 \nu^{9} + \cdots + 33651 ) / 7461$$ (-1368*v^17 + 6994*v^16 + 19321*v^15 - 138231*v^14 - 69471*v^13 + 1089723*v^12 - 176988*v^11 - 4390062*v^10 + 1744629*v^9 + 9656481*v^8 - 4145153*v^7 - 11540648*v^6 + 3868881*v^5 + 7021512*v^4 - 1126749*v^3 - 1731033*v^2 - 83169*v + 33651) / 7461 $$\beta_{8}$$ $$=$$ $$( - 1439 \nu^{17} + 7014 \nu^{16} + 22106 \nu^{15} - 142062 \nu^{14} - 104115 \nu^{13} + 1156950 \nu^{12} + 15615 \nu^{11} - 4867965 \nu^{10} + 1238229 \nu^{9} + \cdots + 37530 ) / 7461$$ (-1439*v^17 + 7014*v^16 + 22106*v^15 - 142062*v^14 - 104115*v^13 + 1156950*v^12 + 15615*v^11 - 4867965*v^10 + 1238229*v^9 + 11336596*v^8 - 3579243*v^7 - 14527561*v^6 + 3643614*v^5 + 9479463*v^4 - 867930*v^3 - 2463246*v^2 - 359163*v + 37530) / 7461 $$\beta_{9}$$ $$=$$ $$( 1534 \nu^{17} - 6924 \nu^{16} - 24910 \nu^{15} + 138501 \nu^{14} + 139716 \nu^{13} - 1106859 \nu^{12} - 261834 \nu^{11} + 4524207 \nu^{10} - 261546 \nu^{9} - 10080209 \nu^{8} + \cdots - 16344 ) / 7461$$ (1534*v^17 - 6924*v^16 - 24910*v^15 + 138501*v^14 + 139716*v^13 - 1106859*v^12 - 261834*v^11 + 4524207*v^10 - 261546*v^9 - 10080209*v^8 + 1420434*v^7 + 12115883*v^6 - 1273752*v^5 - 7305978*v^4 - 62682*v^3 + 1753524*v^2 + 310950*v - 16344) / 7461 $$\beta_{10}$$ $$=$$ $$( 1554 \nu^{17} - 7385 \nu^{16} - 22577 \nu^{15} + 143118 \nu^{14} + 91188 \nu^{13} - 1096968 \nu^{12} + 109644 \nu^{11} + 4242798 \nu^{10} - 1658211 \nu^{9} - 8797869 \nu^{8} + \cdots - 9135 ) / 7461$$ (1554*v^17 - 7385*v^16 - 22577*v^15 + 143118*v^14 + 91188*v^13 - 1096968*v^12 + 109644*v^11 + 4242798*v^10 - 1658211*v^9 - 8797869*v^8 + 4159954*v^7 + 9679471*v^6 - 4015002*v^5 - 5301750*v^4 + 1253103*v^3 + 1167444*v^2 + 56943*v - 9135) / 7461 $$\beta_{11}$$ $$=$$ $$( - 1508 \nu^{17} + 7734 \nu^{16} + 20399 \nu^{15} - 150654 \nu^{14} - 58059 \nu^{13} + 1162245 \nu^{12} - 350022 \nu^{11} - 4529175 \nu^{10} + 2597928 \nu^{9} + 9451750 \nu^{8} + \cdots - 9351 ) / 7461$$ (-1508*v^17 + 7734*v^16 + 20399*v^15 - 150654*v^14 - 58059*v^13 + 1162245*v^12 - 350022*v^11 - 4529175*v^10 + 2597928*v^9 + 9451750*v^8 - 6188850*v^7 - 10367746*v^6 + 6315147*v^5 + 5461734*v^4 - 2383818*v^3 - 1023228*v^2 + 60921*v - 9351) / 7461 $$\beta_{12}$$ $$=$$ $$( 1643 \nu^{17} - 8566 \nu^{16} - 21024 \nu^{15} + 163788 \nu^{14} + 41370 \nu^{13} - 1226670 \nu^{12} + 535884 \nu^{11} + 4542789 \nu^{10} - 3370035 \nu^{9} - 8610109 \nu^{8} + \cdots + 7650 ) / 7461$$ (1643*v^17 - 8566*v^16 - 21024*v^15 + 163788*v^14 + 41370*v^13 - 1226670*v^12 + 535884*v^11 + 4542789*v^10 - 3370035*v^9 - 8610109*v^8 + 7760720*v^7 + 7676733*v^6 - 8030091*v^5 - 2276628*v^4 + 3458052*v^3 - 246852*v^2 - 385857*v + 7650) / 7461 $$\beta_{13}$$ $$=$$ $$( - 1784 \nu^{17} + 8956 \nu^{16} + 24348 \nu^{15} - 172587 \nu^{14} - 75282 \nu^{13} + 1310262 \nu^{12} - 337779 \nu^{11} - 4982064 \nu^{10} + 2642421 \nu^{9} + \cdots + 26955 ) / 7461$$ (-1784*v^17 + 8956*v^16 + 24348*v^15 - 172587*v^14 - 75282*v^13 + 1310262*v^12 - 337779*v^11 - 4982064*v^10 + 2642421*v^9 + 10002577*v^8 - 6071621*v^7 - 10320921*v^6 + 5730195*v^5 + 4979334*v^4 - 1936404*v^3 - 873828*v^2 + 69993*v + 26955) / 7461 $$\beta_{14}$$ $$=$$ $$( - 1844 \nu^{17} + 9510 \nu^{16} + 23981 \nu^{15} - 183951 \nu^{14} - 49074 \nu^{13} + 1402452 \nu^{12} - 629016 \nu^{11} - 5353980 \nu^{10} + 4121586 \nu^{9} + \cdots + 35172 ) / 7461$$ (-1844*v^17 + 9510*v^16 + 23981*v^15 - 183951*v^14 - 49074*v^13 + 1402452*v^12 - 629016*v^11 - 5353980*v^10 + 4121586*v^9 + 10756507*v^8 - 9961143*v^7 - 10931122*v^6 + 10870065*v^5 + 4855866*v^4 - 4821810*v^3 - 600327*v^2 + 458964*v + 35172) / 7461 $$\beta_{15}$$ $$=$$ $$( - 3050 \nu^{17} + 13516 \nu^{16} + 50013 \nu^{15} - 270111 \nu^{14} - 286797 \nu^{13} + 2156217 \nu^{12} + 585423 \nu^{11} - 8803536 \nu^{10} + 291981 \nu^{9} + \cdots + 16047 ) / 7461$$ (-3050*v^17 + 13516*v^16 + 50013*v^15 - 270111*v^14 - 286797*v^13 + 2156217*v^12 + 585423*v^11 - 8803536*v^10 + 291981*v^9 + 19600981*v^8 - 2418785*v^7 - 23552715*v^6 + 2209251*v^5 + 14146125*v^4 + 131976*v^3 - 3303342*v^2 - 519984*v + 16047) / 7461 $$\beta_{16}$$ $$=$$ $$( - 3128 \nu^{17} + 15231 \nu^{16} + 45308 \nu^{15} - 298314 \nu^{14} - 178614 \nu^{13} + 2318343 \nu^{12} - 277404 \nu^{11} - 9129351 \nu^{10} + 3631242 \nu^{9} + \cdots + 11061 ) / 7461$$ (-3128*v^17 + 15231*v^16 + 45308*v^15 - 298314*v^14 - 178614*v^13 + 2318343*v^12 - 277404*v^11 - 9129351*v^10 + 3631242*v^9 + 19357486*v^8 - 9104646*v^7 - 21799537*v^6 + 8861238*v^5 + 12061176*v^4 - 2650614*v^3 - 2554596*v^2 - 285156*v + 11061) / 7461 $$\beta_{17}$$ $$=$$ $$( - 4376 \nu^{17} + 20288 \nu^{16} + 67021 \nu^{15} - 398895 \nu^{14} - 320397 \nu^{13} + 3114258 \nu^{12} + 147978 \nu^{11} - 12325434 \nu^{10} + 3064161 \nu^{9} + \cdots + 20817 ) / 7461$$ (-4376*v^17 + 20288*v^16 + 67021*v^15 - 398895*v^14 - 320397*v^13 + 3114258*v^12 + 147978*v^11 - 12325434*v^10 + 3064161*v^9 + 26247685*v^8 - 8851417*v^7 - 29576411*v^6 + 8827047*v^5 + 16225848*v^4 - 2488125*v^3 - 3362814*v^2 - 400167*v + 20817) / 7461
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta _1 + 1$$ b3 + b2 + 4*b1 + 1 $$\nu^{4}$$ $$=$$ $$-\beta_{16} + \beta_{13} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{3} + 7\beta_{2} + \beta _1 + 15$$ -b16 + b13 + b8 - b7 - b6 + b3 + 7*b2 + b1 + 15 $$\nu^{5}$$ $$=$$ $$\beta_{17} - \beta_{16} - \beta_{15} + \beta_{13} + \beta_{12} + \beta_{10} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 8 \beta_{3} + 10 \beta_{2} + 20 \beta _1 + 11$$ b17 - b16 - b15 + b13 + b12 + b10 + b7 - b6 + b5 - b4 + 8*b3 + 10*b2 + 20*b1 + 11 $$\nu^{6}$$ $$=$$ $$\beta_{17} - 11 \beta_{16} - \beta_{15} + 10 \beta_{13} + \beta_{12} + 2 \beta_{11} + \beta_{10} - \beta_{9} + 9 \beta_{8} - 9 \beta_{7} - 10 \beta_{6} - 2 \beta_{4} + 11 \beta_{3} + 46 \beta_{2} + 11 \beta _1 + 87$$ b17 - 11*b16 - b15 + 10*b13 + b12 + 2*b11 + b10 - b9 + 9*b8 - 9*b7 - 10*b6 - 2*b4 + 11*b3 + 46*b2 + 11*b1 + 87 $$\nu^{7}$$ $$=$$ $$13 \beta_{17} - 15 \beta_{16} - 13 \beta_{15} + 14 \beta_{13} + 12 \beta_{12} + 2 \beta_{11} + 12 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 9 \beta_{7} - 12 \beta_{6} + 10 \beta_{5} - 13 \beta_{4} + 56 \beta_{3} + 81 \beta_{2} + 111 \beta _1 + 96$$ 13*b17 - 15*b16 - 13*b15 + 14*b13 + 12*b12 + 2*b11 + 12*b10 - 2*b9 + 2*b8 + 9*b7 - 12*b6 + 10*b5 - 13*b4 + 56*b3 + 81*b2 + 111*b1 + 96 $$\nu^{8}$$ $$=$$ $$17 \beta_{17} - 95 \beta_{16} - 18 \beta_{15} + \beta_{14} + 81 \beta_{13} + 16 \beta_{12} + 25 \beta_{11} + 15 \beta_{10} - 17 \beta_{9} + 65 \beta_{8} - 62 \beta_{7} - 79 \beta_{6} + \beta_{5} - 29 \beta_{4} + 95 \beta_{3} + 307 \beta_{2} + 94 \beta _1 + 541$$ 17*b17 - 95*b16 - 18*b15 + b14 + 81*b13 + 16*b12 + 25*b11 + 15*b10 - 17*b9 + 65*b8 - 62*b7 - 79*b6 + b5 - 29*b4 + 95*b3 + 307*b2 + 94*b1 + 541 $$\nu^{9}$$ $$=$$ $$120 \beta_{17} - 161 \beta_{16} - 121 \beta_{15} + 2 \beta_{14} + 142 \beta_{13} + 106 \beta_{12} + 35 \beta_{11} + 104 \beta_{10} - 37 \beta_{9} + 34 \beta_{8} + 55 \beta_{7} - 108 \beta_{6} + 72 \beta_{5} - 125 \beta_{4} + 387 \beta_{3} + \cdots + 771$$ 120*b17 - 161*b16 - 121*b15 + 2*b14 + 142*b13 + 106*b12 + 35*b11 + 104*b10 - 37*b9 + 34*b8 + 55*b7 - 108*b6 + 72*b5 - 125*b4 + 387*b3 + 618*b2 + 653*b1 + 771 $$\nu^{10}$$ $$=$$ $$191 \beta_{17} - 760 \beta_{16} - 208 \beta_{15} + 16 \beta_{14} + 621 \beta_{13} + 171 \beta_{12} + 229 \beta_{11} + 155 \beta_{10} - 192 \beta_{9} + 447 \beta_{8} - 394 \beta_{7} - 576 \beta_{6} + 15 \beta_{5} - 301 \beta_{4} + \cdots + 3509$$ 191*b17 - 760*b16 - 208*b15 + 16*b14 + 621*b13 + 171*b12 + 229*b11 + 155*b10 - 192*b9 + 447*b8 - 394*b7 - 576*b6 + 15*b5 - 301*b4 + 755*b3 + 2093*b2 + 739*b1 + 3509 $$\nu^{11}$$ $$=$$ $$981 \beta_{17} - 1490 \beta_{16} - 1004 \beta_{15} + 36 \beta_{14} + 1263 \beta_{13} + 843 \beta_{12} + 398 \beta_{11} + 807 \beta_{10} - 445 \beta_{9} + 384 \beta_{8} + 269 \beta_{7} - 881 \beta_{6} + 460 \beta_{5} - 1078 \beta_{4} + \cdots + 5958$$ 981*b17 - 1490*b16 - 1004*b15 + 36*b14 + 1263*b13 + 843*b12 + 398*b11 + 807*b10 - 445*b9 + 384*b8 + 269*b7 - 881*b6 + 460*b5 - 1078*b4 + 2693*b3 + 4605*b2 + 4005*b1 + 5958 $$\nu^{12}$$ $$=$$ $$1800 \beta_{17} - 5879 \beta_{16} - 1995 \beta_{15} + 174 \beta_{14} + 4676 \beta_{13} + 1557 \beta_{12} + 1886 \beta_{11} + 1377 \beta_{10} - 1831 \beta_{9} + 3057 \beta_{8} - 2446 \beta_{7} - 4062 \beta_{6} + \cdots + 23444$$ 1800*b17 - 5879*b16 - 1995*b15 + 174*b14 + 4676*b13 + 1557*b12 + 1886*b11 + 1377*b10 - 1831*b9 + 3057*b8 - 2446*b7 - 4062*b6 + 152*b5 - 2724*b4 + 5777*b3 + 14519*b2 + 5603*b1 + 23444 $$\nu^{13}$$ $$=$$ $$7596 \beta_{17} - 12704 \beta_{16} - 7920 \beta_{15} + 423 \beta_{14} + 10484 \beta_{13} + 6417 \beta_{12} + 3783 \beta_{11} + 5985 \beta_{10} - 4440 \beta_{9} + 3641 \beta_{8} + 988 \beta_{7} - 6866 \beta_{6} + \cdots + 45099$$ 7596*b17 - 12704*b16 - 7920*b15 + 423*b14 + 10484*b13 + 6417*b12 + 3783*b11 + 5985*b10 - 4440*b9 + 3641*b8 + 988*b7 - 6866*b6 + 2796*b5 - 8805*b4 + 18917*b3 + 33938*b2 + 25403*b1 + 45099 $$\nu^{14}$$ $$=$$ $$15449 \beta_{17} - 44645 \beta_{16} - 17354 \beta_{15} + 1611 \beta_{14} + 34958 \beta_{13} + 13085 \beta_{12} + 14832 \beta_{11} + 11336 \beta_{10} - 15978 \beta_{9} + 21054 \beta_{8} - 15199 \beta_{7} + \cdots + 160117$$ 15449*b17 - 44645*b16 - 17354*b15 + 1611*b14 + 34958*b13 + 13085*b12 + 14832*b11 + 11336*b10 - 15978*b9 + 21054*b8 - 15199*b7 - 28277*b6 + 1334*b5 - 22910*b4 + 43342*b3 + 102035*b2 + 41719*b1 + 160117 $$\nu^{15}$$ $$=$$ $$57224 \beta_{17} - 103030 \beta_{16} - 60878 \beta_{15} + 4113 \beta_{14} + 83564 \beta_{13} + 47900 \beta_{12} + 32761 \beta_{11} + 43544 \beta_{10} - 40013 \beta_{9} + 31437 \beta_{8} + 1059 \beta_{7} + \cdots + 337218$$ 57224*b17 - 103030*b16 - 60878*b15 + 4113*b14 + 83564*b13 + 47900*b12 + 32761*b11 + 43544*b10 - 40013*b9 + 31437*b8 + 1059*b7 - 52175*b6 + 16746*b5 - 69619*b4 + 134017*b3 + 248720*b2 + 165706*b1 + 337218 $$\nu^{16}$$ $$=$$ $$125414 \beta_{17} - 335073 \beta_{16} - 142550 \beta_{15} + 13680 \beta_{14} + 260395 \beta_{13} + 105099 \beta_{12} + 113983 \beta_{11} + 89439 \beta_{10} - 132280 \beta_{9} + 146437 \beta_{8} + \cdots + 1111863$$ 125414*b17 - 335073*b16 - 142550*b15 + 13680*b14 + 260395*b13 + 105099*b12 + 113983*b11 + 89439*b10 - 132280*b9 + 146437*b8 - 95643*b7 - 196290*b6 + 11075*b5 - 184337*b4 + 321631*b3 + 723972*b2 + 307593*b1 + 1111863 $$\nu^{17}$$ $$=$$ $$424612 \beta_{17} - 808918 \beta_{16} - 461019 \beta_{15} + 35975 \beta_{14} + 649323 \beta_{13} + 354392 \beta_{12} + 268769 \beta_{11} + 314484 \beta_{10} - 338935 \beta_{9} + 256675 \beta_{8} + \cdots + 2501963$$ 424612*b17 - 808918*b16 - 461019*b15 + 35975*b14 + 649323*b13 + 354392*b12 + 268769*b11 + 314484*b10 - 338935*b9 + 256675*b8 - 28411*b7 - 390392*b6 + 101189*b5 - 538756*b4 + 956104*b3 + 1817402*b2 + 1106438*b1 + 2501963

## 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
 −2.33155 −2.26753 −1.93622 −1.55859 −1.12960 −0.956682 −0.352457 −0.159695 0.0750551 0.891553 1.04746 1.23218 1.31041 2.03424 2.33441 2.36771 2.69411 2.70519
−2.33155 0 3.43613 1.00000 0 −1.42557 −3.34841 0 −2.33155
1.2 −2.26753 0 3.14168 1.00000 0 0.777155 −2.58880 0 −2.26753
1.3 −1.93622 0 1.74896 1.00000 0 −0.704113 0.486063 0 −1.93622
1.4 −1.55859 0 0.429203 1.00000 0 4.77211 2.44823 0 −1.55859
1.5 −1.12960 0 −0.724011 1.00000 0 1.66257 3.07703 0 −1.12960
1.6 −0.956682 0 −1.08476 1.00000 0 1.77032 2.95113 0 −0.956682
1.7 −0.352457 0 −1.87577 1.00000 0 −0.865944 1.36604 0 −0.352457
1.8 −0.159695 0 −1.97450 1.00000 0 −2.32101 0.634707 0 −0.159695
1.9 0.0750551 0 −1.99437 1.00000 0 −4.31915 −0.299798 0 0.0750551
1.10 0.891553 0 −1.20513 1.00000 0 4.60717 −2.85755 0 0.891553
1.11 1.04746 0 −0.902824 1.00000 0 4.44267 −3.04060 0 1.04746
1.12 1.23218 0 −0.481722 1.00000 0 0.309652 −3.05794 0 1.23218
1.13 1.31041 0 −0.282837 1.00000 0 −2.01323 −2.99144 0 1.31041
1.14 2.03424 0 2.13815 1.00000 0 2.56652 0.281035 0 2.03424
1.15 2.33441 0 3.44947 1.00000 0 1.85122 3.38365 0 2.33441
1.16 2.36771 0 3.60605 1.00000 0 −3.86768 3.80266 0 2.36771
1.17 2.69411 0 5.25825 1.00000 0 2.33612 8.77809 0 2.69411
1.18 2.70519 0 5.31803 1.00000 0 2.42117 8.97588 0 2.70519
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.18 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.j yes 18
3.b odd 2 1 3645.2.a.i 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3645.2.a.i 18 3.b odd 2 1
3645.2.a.j yes 18 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{18} - 6 T_{2}^{17} - 9 T_{2}^{16} + 112 T_{2}^{15} - 51 T_{2}^{14} - 810 T_{2}^{13} + 936 T_{2}^{12} + 2844 T_{2}^{11} - 4536 T_{2}^{10} - 4934 T_{2}^{9} + 10173 T_{2}^{8} + 3672 T_{2}^{7} - 11144 T_{2}^{6} - 456 T_{2}^{5} + \cdots + 9$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3645))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18} - 6 T^{17} - 9 T^{16} + 112 T^{15} + \cdots + 9$$
$3$ $$T^{18}$$
$5$ $$(T - 1)^{18}$$
$7$ $$T^{18} - 12 T^{17} + 3 T^{16} + \cdots - 126144$$
$11$ $$T^{18} - 24 T^{17} + 195 T^{16} + \cdots - 366687$$
$13$ $$T^{18} - 12 T^{17} - 30 T^{16} + \cdots + 541$$
$17$ $$T^{18} - 12 T^{17} - 57 T^{16} + \cdots + 2436813$$
$19$ $$T^{18} - 150 T^{16} + \cdots + 143557201$$
$23$ $$T^{18} - 24 T^{17} + \cdots - 128982699$$
$29$ $$T^{18} - 24 T^{17} + \cdots - 3062298303$$
$31$ $$T^{18} + 6 T^{17} - 261 T^{16} + \cdots + 3405888$$
$37$ $$T^{18} - 6 T^{17} + \cdots - 462226412267$$
$41$ $$T^{18} - 24 T^{17} + \cdots - 32501442396627$$
$43$ $$T^{18} - 24 T^{17} + \cdots + 6246133919317$$
$47$ $$T^{18} - 24 T^{17} + \cdots - 147452956947$$
$53$ $$T^{18} - 24 T^{17} + \cdots - 5114078939583$$
$59$ $$T^{18} - 48 T^{17} + \cdots + 18\!\cdots\!41$$
$61$ $$T^{18} + 24 T^{17} + \cdots - 25\!\cdots\!59$$
$67$ $$T^{18} + 6 T^{17} + \cdots - 72094667182983$$
$71$ $$T^{18} + \cdots - 529074984674403$$
$73$ $$T^{18} - 6 T^{17} - 420 T^{16} + \cdots - 998613683$$
$79$ $$T^{18} + 6 T^{17} + \cdots - 790954750503$$
$83$ $$T^{18} - 60 T^{17} + \cdots + 33896022129$$
$89$ $$T^{18} - 24 T^{17} + \cdots + 15261639728409$$
$97$ $$T^{18} + \cdots - 333200929985747$$