# Properties

 Label 45.2.j.a Level $45$ Weight $2$ Character orbit 45.j Analytic conductor $0.359$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,2,Mod(4,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("45.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 45.j (of order $$6$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$0.359326809096$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} + ( - \beta_{7} + \beta_{5}) q^{3} - \beta_{3} q^{4} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 1) q^{6} + (\beta_{6} - 2 \beta_1) q^{7} + (\beta_{7} - \beta_{6} - \beta_{5}) q^{8} + ( - \beta_{4} - \beta_{3}) q^{9}+O(q^{10})$$ q + b6 * q^2 + (-b7 + b5) * q^3 - b3 * q^4 + (b7 - b6 + b3 + b1) * q^5 + (b3 - b2 - 1) * q^6 + (b6 - 2*b1) * q^7 + (b7 - b6 - b5) * q^8 + (-b4 - b3) * q^9 $$q + \beta_{6} q^{2} + ( - \beta_{7} + \beta_{5}) q^{3} - \beta_{3} q^{4} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 1) q^{6} + (\beta_{6} - 2 \beta_1) q^{7} + (\beta_{7} - \beta_{6} - \beta_{5}) q^{8} + ( - \beta_{4} - \beta_{3}) q^{9} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 1) q^{10} + (\beta_{4} - \beta_{3} + 3 \beta_{2} - 3) q^{11} + (\beta_{7} - 3 \beta_{6} - 2 \beta_{5} + 2 \beta_1) q^{12} + (\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_1) q^{13} - \beta_{3} q^{14} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{15} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 1) q^{16} + ( - 3 \beta_{7} + 2 \beta_{6} + 2 \beta_{5}) q^{17} + (\beta_{7} + 3 \beta_{6} + \beta_{5} - \beta_1) q^{18} + (\beta_{4} - 1) q^{19} + ( - \beta_{6} - 3 \beta_{2} - \beta_1 + 3) q^{20} + (2 \beta_{4} - \beta_{3} - 3 \beta_{2} + 3) q^{21} + (\beta_{7} - \beta_{6} + 4 \beta_{5} + \beta_1) q^{22} + \beta_{5} q^{23} + (\beta_{4} + \beta_{2} + 1) q^{24} + (2 \beta_{6} + \beta_{2} + 2 \beta_1 - 1) q^{25} + ( - \beta_{4} + 3) q^{26} + (3 \beta_{7} - 6 \beta_{6} - 3 \beta_{5} + 3 \beta_1) q^{27} + ( - 3 \beta_{7} + 3 \beta_{6} + 3 \beta_{5}) q^{28} + ( - 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 3) q^{29} + ( - \beta_{7} + 3 \beta_{6} + \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - \beta_1 - 3) q^{30} + ( - \beta_{3} + \beta_{2}) q^{31} + ( - 4 \beta_{7} + 4 \beta_{6} - \beta_{5} - 4 \beta_1) q^{32} + ( - \beta_{7} - 4 \beta_{5} - 2 \beta_1) q^{33} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{34} + (3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - \beta_{4} - 3) q^{35} + (6 \beta_{2} - 3) q^{36} + 3 \beta_{7} q^{37} + ( - 3 \beta_{6} + \beta_1) q^{38} + (\beta_{4} + \beta_{3} + 3 \beta_{2} - 3) q^{39} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{40} + ( - \beta_{3} - 6 \beta_{2}) q^{41} + (\beta_{7} - 3 \beta_{6} - 2 \beta_{5} + 2 \beta_1) q^{42} + ( - 5 \beta_{6} + \beta_1) q^{43} + (3 \beta_{4} - 3) q^{44} + ( - \beta_{7} + 2 \beta_{5} - 6 \beta_{2} - 2 \beta_1 + 3) q^{45} + (\beta_{4} - 2) q^{46} + (2 \beta_{6} + \beta_1) q^{47} + (2 \beta_{7} + 3 \beta_{5} - \beta_1) q^{48} + (3 \beta_{3} - \beta_{2}) q^{49} + (\beta_{5} - 2 \beta_{3} + 6 \beta_{2}) q^{50} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} - 4) q^{51} + (3 \beta_{6} - 3 \beta_1) q^{52} + (2 \beta_{6} + 2 \beta_{5}) q^{53} + (3 \beta_{3} - 6 \beta_{2} + 3) q^{54} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + 3) q^{55} + ( - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 3) q^{56} + ( - \beta_{7} + 3 \beta_{6} - \beta_1) q^{57} + ( - 2 \beta_{7} + 2 \beta_{6} - 5 \beta_{5} - 2 \beta_1) q^{58} + (2 \beta_{3} + 6 \beta_{2}) q^{59} + (3 \beta_{5} + \beta_{4} - 2 \beta_{3} + 3 \beta_1 + 3) q^{60} + (5 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - 2) q^{61} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{5}) q^{62} + ( - 3 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 3 \beta_1) q^{63} + ( - \beta_{4} + 4) q^{64} + ( - 3 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + 3 \beta_1) q^{65} + ( - 5 \beta_{4} + \beta_{3} - 3 \beta_{2} + 9) q^{66} + ( - 5 \beta_{7} + 5 \beta_{6} + \beta_{5} - 5 \beta_1) q^{67} + (5 \beta_{7} - 5 \beta_{6} - 4 \beta_{5} + 5 \beta_1) q^{68} + ( - \beta_{4} - \beta_{2} + 2) q^{69} + ( - \beta_{6} - 3 \beta_{2} - \beta_1 + 3) q^{70} + (3 \beta_{4} + 9) q^{71} + ( - 4 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - 2 \beta_1) q^{72} + 6 \beta_{7} q^{73} + (3 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 3) q^{74} + ( - \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - \beta_1 - 6) q^{75} + (\beta_{3} - 3 \beta_{2}) q^{76} + (3 \beta_{7} - 3 \beta_{6} + 3 \beta_1) q^{77} + ( - \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + \beta_1) q^{78} + (2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 4) q^{79} + ( - \beta_{7} + 4 \beta_{6} + 4 \beta_{5} + \beta_{4} - 6) q^{80} + 9 \beta_{2} q^{81} + (\beta_{7} - 5 \beta_{6} - 5 \beta_{5}) q^{82} + (7 \beta_{6} - 6 \beta_1) q^{83} + ( - 3 \beta_{4} - 3 \beta_{2} - 3) q^{84} + ( - 5 \beta_{7} + 5 \beta_{6} + 4 \beta_{5} + 2 \beta_{3} + 4 \beta_{2} - 5 \beta_1) q^{85} + (5 \beta_{3} - 9 \beta_{2}) q^{86} + (2 \beta_{7} + 5 \beta_{5} + \beta_1) q^{87} + (\beta_{6} + \beta_1) q^{88} + ( - 6 \beta_{4} - 3) q^{89} + ( - 3 \beta_{6} - 6 \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} - 3) q^{90} + ( - 3 \beta_{4} - 3) q^{91} + ( - 2 \beta_{6} + \beta_1) q^{92} + ( - 3 \beta_{6} - 2 \beta_{5} + \beta_1) q^{93} + ( - 2 \beta_{3} + 5 \beta_{2}) q^{94} + ( - 2 \beta_{5} - \beta_{3} + 3 \beta_{2}) q^{95} + (5 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} - 6) q^{96} + ( - 8 \beta_{6} + 10 \beta_1) q^{97} + ( - 3 \beta_{7} - 4 \beta_{6} - 4 \beta_{5}) q^{98} + (6 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 6) q^{99}+O(q^{100})$$ q + b6 * q^2 + (-b7 + b5) * q^3 - b3 * q^4 + (b7 - b6 + b3 + b1) * q^5 + (b3 - b2 - 1) * q^6 + (b6 - 2*b1) * q^7 + (b7 - b6 - b5) * q^8 + (-b4 - b3) * q^9 + (-b7 - b6 - b5 + b4 - 1) * q^10 + (b4 - b3 + 3*b2 - 3) * q^11 + (b7 - 3*b6 - 2*b5 + 2*b1) * q^12 + (b7 - b6 - 2*b5 + b1) * q^13 - b3 * q^14 + (-b7 + 3*b6 + 2*b5 - b4 + b3 + b2 - 2*b1 + 1) * q^15 + (-2*b4 + 2*b3 - b2 + 1) * q^16 + (-3*b7 + 2*b6 + 2*b5) * q^17 + (b7 + 3*b6 + b5 - b1) * q^18 + (b4 - 1) * q^19 + (-b6 - 3*b2 - b1 + 3) * q^20 + (2*b4 - b3 - 3*b2 + 3) * q^21 + (b7 - b6 + 4*b5 + b1) * q^22 + b5 * q^23 + (b4 + b2 + 1) * q^24 + (2*b6 + b2 + 2*b1 - 1) * q^25 + (-b4 + 3) * q^26 + (3*b7 - 6*b6 - 3*b5 + 3*b1) * q^27 + (-3*b7 + 3*b6 + 3*b5) * q^28 + (-2*b4 + 2*b3 - 3*b2 + 3) * q^29 + (-b7 + 3*b6 + b4 - 2*b3 + 3*b2 - b1 - 3) * q^30 + (-b3 + b2) * q^31 + (-4*b7 + 4*b6 - b5 - 4*b1) * q^32 + (-b7 - 4*b5 - 2*b1) * q^33 + (-b4 + b3 + b2 - 1) * q^34 + (3*b7 - 3*b6 - 3*b5 - b4 - 3) * q^35 + (6*b2 - 3) * q^36 + 3*b7 * q^37 + (-3*b6 + b1) * q^38 + (b4 + b3 + 3*b2 - 3) * q^39 + (2*b7 - 2*b6 - b5 - b3 - b2 + 2*b1) * q^40 + (-b3 - 6*b2) * q^41 + (b7 - 3*b6 - 2*b5 + 2*b1) * q^42 + (-5*b6 + b1) * q^43 + (3*b4 - 3) * q^44 + (-b7 + 2*b5 - 6*b2 - 2*b1 + 3) * q^45 + (b4 - 2) * q^46 + (2*b6 + b1) * q^47 + (2*b7 + 3*b5 - b1) * q^48 + (3*b3 - b2) * q^49 + (b5 - 2*b3 + 6*b2) * q^50 + (-2*b4 - b3 - b2 - 4) * q^51 + (3*b6 - 3*b1) * q^52 + (2*b6 + 2*b5) * q^53 + (3*b3 - 6*b2 + 3) * q^54 + (-2*b7 - 2*b6 - 2*b5 - 3*b4 + 3) * q^55 + (-2*b4 + 2*b3 + 3*b2 - 3) * q^56 + (-b7 + 3*b6 - b1) * q^57 + (-2*b7 + 2*b6 - 5*b5 - 2*b1) * q^58 + (2*b3 + 6*b2) * q^59 + (3*b5 + b4 - 2*b3 + 3*b1 + 3) * q^60 + (5*b4 - 5*b3 + 2*b2 - 2) * q^61 + (b7 + 2*b6 + 2*b5) * q^62 + (-3*b7 + 3*b6 + 3*b5 + 3*b1) * q^63 + (-b4 + 4) * q^64 + (-3*b6 + 2*b4 - 2*b3 + 3*b1) * q^65 + (-5*b4 + b3 - 3*b2 + 9) * q^66 + (-5*b7 + 5*b6 + b5 - 5*b1) * q^67 + (5*b7 - 5*b6 - 4*b5 + 5*b1) * q^68 + (-b4 - b2 + 2) * q^69 + (-b6 - 3*b2 - b1 + 3) * q^70 + (3*b4 + 9) * q^71 + (-4*b7 + 3*b6 + 2*b5 - 2*b1) * q^72 + 6*b7 * q^73 + (3*b4 - 3*b3 + 3*b2 - 3) * q^74 + (-b5 - 2*b4 + 4*b3 - b1 - 6) * q^75 + (b3 - 3*b2) * q^76 + (3*b7 - 3*b6 + 3*b1) * q^77 + (-b7 - 3*b6 + 2*b5 + b1) * q^78 + (2*b4 - 2*b3 - 4*b2 + 4) * q^79 + (-b7 + 4*b6 + 4*b5 + b4 - 6) * q^80 + 9*b2 * q^81 + (b7 - 5*b6 - 5*b5) * q^82 + (7*b6 - 6*b1) * q^83 + (-3*b4 - 3*b2 - 3) * q^84 + (-5*b7 + 5*b6 + 4*b5 + 2*b3 + 4*b2 - 5*b1) * q^85 + (5*b3 - 9*b2) * q^86 + (2*b7 + 5*b5 + b1) * q^87 + (b6 + b1) * q^88 + (-6*b4 - 3) * q^89 + (-3*b6 - 6*b5 + b4 + b3 - 3*b2 - 3) * q^90 + (-3*b4 - 3) * q^91 + (-2*b6 + b1) * q^92 + (-3*b6 - 2*b5 + b1) * q^93 + (-2*b3 + 5*b2) * q^94 + (-2*b5 - b3 + 3*b2) * q^95 + (5*b4 - 4*b3 - 3*b2 - 6) * q^96 + (-8*b6 + 10*b1) * q^97 + (-3*b7 - 4*b6 - 4*b5) * q^98 + (6*b4 - 3*b3 + 3*b2 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 12 q^{6}+O(q^{10})$$ 8 * q - 12 * q^6 $$8 q - 12 q^{6} - 8 q^{10} - 12 q^{11} + 12 q^{15} + 4 q^{16} - 8 q^{19} + 12 q^{20} + 12 q^{21} + 12 q^{24} - 4 q^{25} + 24 q^{26} + 12 q^{29} - 12 q^{30} + 4 q^{31} - 4 q^{34} - 24 q^{35} - 12 q^{39} - 4 q^{40} - 24 q^{41} - 24 q^{44} - 16 q^{46} - 4 q^{49} + 24 q^{50} - 36 q^{51} + 24 q^{55} - 12 q^{56} + 24 q^{59} + 24 q^{60} - 8 q^{61} + 32 q^{64} + 60 q^{66} + 12 q^{69} + 12 q^{70} + 72 q^{71} - 12 q^{74} - 48 q^{75} - 12 q^{76} + 16 q^{79} - 48 q^{80} + 36 q^{81} - 36 q^{84} + 16 q^{85} - 36 q^{86} - 24 q^{89} - 36 q^{90} - 24 q^{91} + 20 q^{94} + 12 q^{95} - 60 q^{96} - 36 q^{99}+O(q^{100})$$ 8 * q - 12 * q^6 - 8 * q^10 - 12 * q^11 + 12 * q^15 + 4 * q^16 - 8 * q^19 + 12 * q^20 + 12 * q^21 + 12 * q^24 - 4 * q^25 + 24 * q^26 + 12 * q^29 - 12 * q^30 + 4 * q^31 - 4 * q^34 - 24 * q^35 - 12 * q^39 - 4 * q^40 - 24 * q^41 - 24 * q^44 - 16 * q^46 - 4 * q^49 + 24 * q^50 - 36 * q^51 + 24 * q^55 - 12 * q^56 + 24 * q^59 + 24 * q^60 - 8 * q^61 + 32 * q^64 + 60 * q^66 + 12 * q^69 + 12 * q^70 + 72 * q^71 - 12 * q^74 - 48 * q^75 - 12 * q^76 + 16 * q^79 - 48 * q^80 + 36 * q^81 - 36 * q^84 + 16 * q^85 - 36 * q^86 - 24 * q^89 - 36 * q^90 - 24 * q^91 + 20 * q^94 + 12 * q^95 - 60 * q^96 - 36 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{3} + \zeta_{24}$$ v^3 + v $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{6} + \zeta_{24}^{2}$$ v^6 + v^2 $$\beta_{4}$$ $$=$$ $$-\zeta_{24}^{6} + 2\zeta_{24}^{2}$$ -v^6 + 2*v^2 $$\beta_{5}$$ $$=$$ $$\zeta_{24}^{5} - \zeta_{24}^{3}$$ v^5 - v^3 $$\beta_{6}$$ $$=$$ $$\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}$$ v^7 - v^5 + v $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}$$ -v^5 - v^3 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + \beta_{5} + 2\beta_1 ) / 3$$ (b7 + b5 + 2*b1) / 3 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{4} + \beta_{3} ) / 3$$ (b4 + b3) / 3 $$\zeta_{24}^{3}$$ $$=$$ $$( -\beta_{7} - \beta_{5} + \beta_1 ) / 3$$ (-b7 - b5 + b1) / 3 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{7} + 2\beta_{5} + \beta_1 ) / 3$$ (-b7 + 2*b5 + b1) / 3 $$\zeta_{24}^{6}$$ $$=$$ $$( -\beta_{4} + 2\beta_{3} ) / 3$$ (-b4 + 2*b3) / 3 $$\zeta_{24}^{7}$$ $$=$$ $$( -2\beta_{7} + 3\beta_{6} + \beta_{5} - \beta_1 ) / 3$$ (-2*b7 + 3*b6 + b5 - b1) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$-\beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 − 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 + 0.965926i
−1.67303 + 0.965926i 1.67303 0.448288i 0.866025 1.50000i 0.358719 + 2.20711i −2.36603 + 2.36603i −0.776457 + 0.448288i 0.517638i 2.59808 1.50000i −2.73205 3.34607i
4.2 −0.448288 + 0.258819i 0.448288 + 1.67303i −0.866025 + 1.50000i −0.358719 2.20711i −0.633975 0.633975i 2.89778 1.67303i 1.93185i −2.59808 + 1.50000i 0.732051 + 0.896575i
4.3 0.448288 0.258819i −0.448288 1.67303i −0.866025 + 1.50000i 2.09077 0.792893i −0.633975 0.633975i −2.89778 + 1.67303i 1.93185i −2.59808 + 1.50000i 0.732051 0.896575i
4.4 1.67303 0.965926i −1.67303 + 0.448288i 0.866025 1.50000i −2.09077 + 0.792893i −2.36603 + 2.36603i 0.776457 0.448288i 0.517638i 2.59808 1.50000i −2.73205 + 3.34607i
34.1 −1.67303 0.965926i 1.67303 + 0.448288i 0.866025 + 1.50000i 0.358719 2.20711i −2.36603 2.36603i −0.776457 0.448288i 0.517638i 2.59808 + 1.50000i −2.73205 + 3.34607i
34.2 −0.448288 0.258819i 0.448288 1.67303i −0.866025 1.50000i −0.358719 + 2.20711i −0.633975 + 0.633975i 2.89778 + 1.67303i 1.93185i −2.59808 1.50000i 0.732051 0.896575i
34.3 0.448288 + 0.258819i −0.448288 + 1.67303i −0.866025 1.50000i 2.09077 + 0.792893i −0.633975 + 0.633975i −2.89778 1.67303i 1.93185i −2.59808 1.50000i 0.732051 + 0.896575i
34.4 1.67303 + 0.965926i −1.67303 0.448288i 0.866025 + 1.50000i −2.09077 0.792893i −2.36603 2.36603i 0.776457 + 0.448288i 0.517638i 2.59808 + 1.50000i −2.73205 3.34607i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 34.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.2.j.a 8
3.b odd 2 1 135.2.j.a 8
4.b odd 2 1 720.2.by.d 8
5.b even 2 1 inner 45.2.j.a 8
5.c odd 4 2 225.2.e.d 8
9.c even 3 1 inner 45.2.j.a 8
9.c even 3 1 405.2.b.d 4
9.d odd 6 1 135.2.j.a 8
9.d odd 6 1 405.2.b.c 4
12.b even 2 1 2160.2.by.c 8
15.d odd 2 1 135.2.j.a 8
15.e even 4 2 675.2.e.d 8
20.d odd 2 1 720.2.by.d 8
36.f odd 6 1 720.2.by.d 8
36.h even 6 1 2160.2.by.c 8
45.h odd 6 1 135.2.j.a 8
45.h odd 6 1 405.2.b.c 4
45.j even 6 1 inner 45.2.j.a 8
45.j even 6 1 405.2.b.d 4
45.k odd 12 2 225.2.e.d 8
45.k odd 12 2 2025.2.a.t 4
45.l even 12 2 675.2.e.d 8
45.l even 12 2 2025.2.a.r 4
60.h even 2 1 2160.2.by.c 8
180.n even 6 1 2160.2.by.c 8
180.p odd 6 1 720.2.by.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.j.a 8 1.a even 1 1 trivial
45.2.j.a 8 5.b even 2 1 inner
45.2.j.a 8 9.c even 3 1 inner
45.2.j.a 8 45.j even 6 1 inner
135.2.j.a 8 3.b odd 2 1
135.2.j.a 8 9.d odd 6 1
135.2.j.a 8 15.d odd 2 1
135.2.j.a 8 45.h odd 6 1
225.2.e.d 8 5.c odd 4 2
225.2.e.d 8 45.k odd 12 2
405.2.b.c 4 9.d odd 6 1
405.2.b.c 4 45.h odd 6 1
405.2.b.d 4 9.c even 3 1
405.2.b.d 4 45.j even 6 1
675.2.e.d 8 15.e even 4 2
675.2.e.d 8 45.l even 12 2
720.2.by.d 8 4.b odd 2 1
720.2.by.d 8 20.d odd 2 1
720.2.by.d 8 36.f odd 6 1
720.2.by.d 8 180.p odd 6 1
2025.2.a.r 4 45.l even 12 2
2025.2.a.t 4 45.k odd 12 2
2160.2.by.c 8 12.b even 2 1
2160.2.by.c 8 36.h even 6 1
2160.2.by.c 8 60.h even 2 1
2160.2.by.c 8 180.n even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(45, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 4 T^{6} + 15 T^{4} - 4 T^{2} + \cdots + 1$$
$3$ $$T^{8} - 9T^{4} + 81$$
$5$ $$T^{8} + 2 T^{6} - 21 T^{4} + 50 T^{2} + \cdots + 625$$
$7$ $$T^{8} - 12 T^{6} + 135 T^{4} + \cdots + 81$$
$11$ $$(T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36)^{2}$$
$13$ $$(T^{4} - 6 T^{2} + 36)^{2}$$
$17$ $$(T^{4} + 28 T^{2} + 4)^{2}$$
$19$ $$(T^{2} + 2 T - 2)^{4}$$
$23$ $$T^{8} - 4 T^{6} + 15 T^{4} - 4 T^{2} + \cdots + 1$$
$29$ $$(T^{4} - 6 T^{3} + 39 T^{2} + 18 T + 9)^{2}$$
$31$ $$(T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4)^{2}$$
$37$ $$(T^{2} + 18)^{4}$$
$41$ $$(T^{4} + 12 T^{3} + 111 T^{2} + 396 T + 1089)^{2}$$
$43$ $$T^{8} - 84 T^{6} + 7020 T^{4} + \cdots + 1296$$
$47$ $$T^{8} - 28 T^{6} + 615 T^{4} + \cdots + 28561$$
$53$ $$(T^{4} + 16 T^{2} + 16)^{2}$$
$59$ $$(T^{4} - 12 T^{3} + 120 T^{2} - 288 T + 576)^{2}$$
$61$ $$(T^{4} + 4 T^{3} + 87 T^{2} - 284 T + 5041)^{2}$$
$67$ $$T^{8} - 84 T^{6} + 5535 T^{4} + \cdots + 2313441$$
$71$ $$(T^{2} - 18 T + 54)^{4}$$
$73$ $$(T^{2} + 72)^{4}$$
$79$ $$(T^{4} - 8 T^{3} + 60 T^{2} - 32 T + 16)^{2}$$
$83$ $$T^{8} - 172 T^{6} + \cdots + 47458321$$
$89$ $$(T^{2} + 6 T - 99)^{4}$$
$97$ $$T^{8} - 336 T^{6} + \cdots + 592240896$$