# Properties

 Label 370.2.q.c Level $370$ Weight $2$ Character orbit 370.q Analytic conductor $2.954$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [370,2,Mod(97,370)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(370, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([3, 5]))

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

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

chi := DirichletCharacter("370.97");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.q (of order $$12$$, degree $$4$$, 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: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/370\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$\chi(n)$$ $$-\zeta_{24}^{2}$$ $$-\zeta_{24}^{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 −0.965926 + 0.258819i 0.965926 − 0.258819i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i −0.258819 − 0.965926i 0.258819 + 0.965926i
0.500000 0.866025i 0.241181 0.900100i −0.500000 0.866025i 1.81431 + 1.30701i −0.658919 0.658919i 0.0267682 0.0999004i −1.00000 1.84607 + 1.06583i 2.03906 0.917738i
97.2 0.500000 0.866025i 0.758819 2.83195i −0.500000 0.866025i 0.917738 2.03906i −2.07313 2.07313i −0.490870 + 1.83195i −1.00000 −4.84607 2.79788i −1.30701 1.81431i
103.1 0.500000 + 0.866025i 0.241181 + 0.900100i −0.500000 + 0.866025i 1.81431 1.30701i −0.658919 + 0.658919i 0.0267682 + 0.0999004i −1.00000 1.84607 1.06583i 2.03906 + 0.917738i
103.2 0.500000 + 0.866025i 0.758819 + 2.83195i −0.500000 + 0.866025i 0.917738 + 2.03906i −2.07313 + 2.07313i −0.490870 1.83195i −1.00000 −4.84607 + 2.79788i −1.30701 + 1.81431i
267.1 0.500000 + 0.866025i −0.465926 + 0.124844i −0.500000 + 0.866025i −2.03906 + 0.917738i −0.341081 0.341081i 4.19798 1.12484i −1.00000 −2.39658 + 1.38366i −1.81431 1.30701i
267.2 0.500000 + 0.866025i 1.46593 0.392794i −0.500000 + 0.866025i 1.30701 + 1.81431i 1.07313 + 1.07313i 2.26612 0.607206i −1.00000 −0.603425 + 0.348387i −0.917738 + 2.03906i
273.1 0.500000 0.866025i −0.465926 0.124844i −0.500000 0.866025i −2.03906 0.917738i −0.341081 + 0.341081i 4.19798 + 1.12484i −1.00000 −2.39658 1.38366i −1.81431 + 1.30701i
273.2 0.500000 0.866025i 1.46593 + 0.392794i −0.500000 0.866025i 1.30701 1.81431i 1.07313 1.07313i 2.26612 + 0.607206i −1.00000 −0.603425 0.348387i −0.917738 2.03906i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.p even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.q.c 8
5.c odd 4 1 370.2.r.c yes 8
37.g odd 12 1 370.2.r.c yes 8
185.p even 12 1 inner 370.2.q.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.q.c 8 1.a even 1 1 trivial
370.2.q.c 8 185.p even 12 1 inner
370.2.r.c yes 8 5.c odd 4 1
370.2.r.c yes 8 37.g odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 4T_{3}^{7} + 14T_{3}^{6} - 24T_{3}^{5} + 14T_{3}^{4} - 4T_{3}^{2} + 8T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{4}$$
$3$ $$T^{8} - 4 T^{7} + 14 T^{6} - 24 T^{5} + \cdots + 4$$
$5$ $$T^{8} - 4 T^{7} + 8 T^{6} + 8 T^{5} + \cdots + 625$$
$7$ $$T^{8} - 12 T^{7} + 54 T^{6} - 120 T^{5} + \cdots + 4$$
$11$ $$T^{8} + 48 T^{6} + 584 T^{4} + \cdots + 16$$
$13$ $$T^{8} + 4 T^{7} + 28 T^{6} + 16 T^{5} + \cdots + 64$$
$17$ $$T^{8} - 12 T^{7} + 30 T^{6} + \cdots + 9409$$
$19$ $$T^{8} - 4 T^{7} + 8 T^{6} + \cdots + 10000$$
$23$ $$(T^{4} + 4 T^{3} - 22 T^{2} - 52 T + 142)^{2}$$
$29$ $$T^{8} + 24 T^{7} + 288 T^{6} + \cdots + 81$$
$31$ $$T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 85264$$
$37$ $$T^{8} - 529 T^{4} + 1874161$$
$41$ $$T^{8} - 12 T^{7} - 26 T^{6} + \cdots + 1270129$$
$43$ $$(T^{4} - 8 T^{3} - 58 T^{2} + 404 T - 386)^{2}$$
$47$ $$T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 20394256$$
$53$ $$T^{8} - 16 T^{7} - 40 T^{6} + \cdots + 160000$$
$59$ $$T^{8} + 8 T^{7} + 152 T^{6} + \cdots + 2262016$$
$61$ $$T^{8} - 48 T^{6} - 696 T^{5} + \cdots + 8427409$$
$67$ $$T^{8} + 16 T^{7} + 164 T^{6} + \cdots + 35344$$
$71$ $$T^{8} + 4 T^{7} + 62 T^{6} + \cdots + 21316$$
$73$ $$T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 341056$$
$79$ $$T^{8} + 32 T^{7} + 380 T^{6} + \cdots + 10000$$
$83$ $$T^{8} - 126 T^{6} + \cdots + 240436036$$
$89$ $$T^{8} - 8 T^{7} + 218 T^{6} + \cdots + 332929$$
$97$ $$T^{8} + 836 T^{6} + \cdots + 1120843441$$