# Properties

 Label 74.2.h.a Level $74$ Weight $2$ Character orbit 74.h Analytic conductor $0.591$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,2,Mod(3,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([13]))

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

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

chi := DirichletCharacter("74.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 74.h (of order $$18$$, degree $$6$$, 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: $$0.590892974957$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{18})$$ Coefficient field: $$\Q(\zeta_{36})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{6} + 1$$ x^12 - x^6 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-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 primitive root of unity $$\zeta_{36}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/74\mathbb{Z}\right)^\times$$.

 $$n$$ $$39$$ $$\chi(n)$$ $$\zeta_{36}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 −0.642788 + 0.766044i 0.642788 − 0.766044i −0.342020 + 0.939693i 0.342020 − 0.939693i −0.642788 − 0.766044i 0.642788 + 0.766044i −0.984808 − 0.173648i 0.984808 + 0.173648i −0.984808 + 0.173648i 0.984808 − 0.173648i −0.342020 − 0.939693i 0.342020 + 0.939693i
−0.642788 + 0.766044i 1.43969 1.20805i −0.173648 0.984808i 0.273629 0.751790i 1.87939i −0.138449 0.0503913i 0.866025 + 0.500000i 0.0923963 0.524005i 0.400019 + 0.692853i
3.2 0.642788 0.766044i 1.43969 1.20805i −0.173648 0.984808i −1.45842 + 4.00698i 1.87939i −3.39364 1.23518i −0.866025 0.500000i 0.0923963 0.524005i 2.13207 + 3.69285i
21.1 −0.342020 + 0.939693i 0.326352 0.118782i −0.766044 0.642788i 2.57176 + 0.453471i 0.347296i −0.361075 + 2.04776i 0.866025 0.500000i −2.20574 + 1.85083i −1.30572 + 2.26157i
21.2 0.342020 0.939693i 0.326352 0.118782i −0.766044 0.642788i 0.839712 + 0.148064i 0.347296i 0.240460 1.36372i −0.866025 + 0.500000i −2.20574 + 1.85083i 0.426333 0.738430i
25.1 −0.642788 0.766044i 1.43969 + 1.20805i −0.173648 + 0.984808i 0.273629 + 0.751790i 1.87939i −0.138449 + 0.0503913i 0.866025 0.500000i 0.0923963 + 0.524005i 0.400019 0.692853i
25.2 0.642788 + 0.766044i 1.43969 + 1.20805i −0.173648 + 0.984808i −1.45842 4.00698i 1.87939i −3.39364 + 1.23518i −0.866025 + 0.500000i 0.0923963 + 0.524005i 2.13207 3.69285i
41.1 −0.984808 0.173648i −0.266044 1.50881i 0.939693 + 0.342020i −1.97937 2.35892i 1.53209i 0.153180 0.128533i −0.866025 0.500000i 0.613341 0.223238i 1.53967 + 2.66679i
41.2 0.984808 + 0.173648i −0.266044 1.50881i 0.939693 + 0.342020i −0.247315 0.294739i 1.53209i −2.50048 + 2.09815i 0.866025 + 0.500000i 0.613341 0.223238i −0.192377 0.333207i
65.1 −0.984808 + 0.173648i −0.266044 + 1.50881i 0.939693 0.342020i −1.97937 + 2.35892i 1.53209i 0.153180 + 0.128533i −0.866025 + 0.500000i 0.613341 + 0.223238i 1.53967 2.66679i
65.2 0.984808 0.173648i −0.266044 + 1.50881i 0.939693 0.342020i −0.247315 + 0.294739i 1.53209i −2.50048 2.09815i 0.866025 0.500000i 0.613341 + 0.223238i −0.192377 + 0.333207i
67.1 −0.342020 0.939693i 0.326352 + 0.118782i −0.766044 + 0.642788i 2.57176 0.453471i 0.347296i −0.361075 2.04776i 0.866025 + 0.500000i −2.20574 1.85083i −1.30572 2.26157i
67.2 0.342020 + 0.939693i 0.326352 + 0.118782i −0.766044 + 0.642788i 0.839712 0.148064i 0.347296i 0.240460 + 1.36372i −0.866025 0.500000i −2.20574 1.85083i 0.426333 + 0.738430i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.2.h.a 12
3.b odd 2 1 666.2.bj.c 12
4.b odd 2 1 592.2.bq.b 12
37.h even 18 1 inner 74.2.h.a 12
37.i odd 36 1 2738.2.a.r 6
37.i odd 36 1 2738.2.a.s 6
111.n odd 18 1 666.2.bj.c 12
148.o odd 18 1 592.2.bq.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.h.a 12 1.a even 1 1 trivial
74.2.h.a 12 37.h even 18 1 inner
592.2.bq.b 12 4.b odd 2 1
592.2.bq.b 12 148.o odd 18 1
666.2.bj.c 12 3.b odd 2 1
666.2.bj.c 12 111.n odd 18 1
2738.2.a.r 6 37.i odd 36 1
2738.2.a.s 6 37.i odd 36 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(74, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - T^{6} + 1$$
$3$ $$(T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + 12 T^{2} + \cdots + 1)^{2}$$
$5$ $$T^{12} + 9 T^{10} - 72 T^{9} + 36 T^{8} + \cdots + 81$$
$7$ $$T^{12} + 12 T^{11} + 66 T^{10} + 218 T^{9} + \cdots + 1$$
$11$ $$T^{12} + 6 T^{11} + 63 T^{10} + \cdots + 408321$$
$13$ $$T^{12} - 6 T^{11} + 30 T^{10} + \cdots + 288369$$
$17$ $$T^{12} + 36 T^{10} + 234 T^{9} + \cdots + 81$$
$19$ $$T^{12} + 18 T^{11} + 135 T^{10} + \cdots + 10439361$$
$23$ $$T^{12} - 63 T^{10} + 3303 T^{8} + \cdots + 431649$$
$29$ $$T^{12} - 18 T^{11} + 126 T^{10} + \cdots + 110889$$
$31$ $$T^{12} + 210 T^{10} + \cdots + 317445489$$
$37$ $$T^{12} - 30 T^{11} + \cdots + 2565726409$$
$41$ $$T^{12} - 24 T^{11} + 216 T^{10} + \cdots + 331776$$
$43$ $$T^{12} + 156 T^{10} + 8910 T^{8} + \cdots + 2277081$$
$47$ $$T^{12} - 6 T^{11} + \cdots + 2027430729$$
$53$ $$T^{12} + 12 T^{11} + \cdots + 45041148441$$
$59$ $$T^{12} + 1404 T^{9} + \cdots + 17477104401$$
$61$ $$T^{12} + 36 T^{11} + 756 T^{10} + \cdots + 331776$$
$67$ $$T^{12} + 30 T^{11} + \cdots + 59697637561$$
$71$ $$T^{12} - 12 T^{11} + \cdots + 4131885551616$$
$73$ $$(T^{6} - 366 T^{4} + 322 T^{3} + \cdots + 94609)^{2}$$
$79$ $$T^{12} - 6 T^{11} + 12 T^{10} + \cdots + 47961$$
$83$ $$T^{12} + 48 T^{11} + 1008 T^{10} + \cdots + 110889$$
$89$ $$T^{12} + 18 T^{11} + \cdots + 687331089$$
$97$ $$T^{12} - 36 T^{11} + \cdots + 27455164416$$