# Properties

 Label 70.2.g.a Level $70$ Weight $2$ Character orbit 70.g Analytic conductor $0.559$ 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] = [70,2,Mod(13,70)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(70, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("70.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$70 = 2 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 70.g (of order $$4$$, 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: $$0.558952814149$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{16})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 1$$ x^8 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$\zeta_{16}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/70\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$57$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{16}^{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 −0.382683 + 0.923880i 0.382683 − 0.923880i −0.923880 − 0.382683i 0.923880 + 0.382683i −0.382683 − 0.923880i 0.382683 + 0.923880i −0.923880 + 0.382683i 0.923880 − 0.382683i
−0.707107 + 0.707107i −1.30656 + 1.30656i 1.00000i −0.158513 + 2.23044i 1.84776i −2.47247 + 0.941740i 0.707107 + 0.707107i 0.414214i −1.46508 1.68925i
13.2 −0.707107 + 0.707107i 1.30656 1.30656i 1.00000i 0.158513 2.23044i 1.84776i −0.941740 + 2.47247i 0.707107 + 0.707107i 0.414214i 1.46508 + 1.68925i
13.3 0.707107 0.707107i −0.541196 + 0.541196i 1.00000i 2.23044 + 0.158513i 0.765367i −2.14065 1.55487i −0.707107 0.707107i 2.41421i 1.68925 1.46508i
13.4 0.707107 0.707107i 0.541196 0.541196i 1.00000i −2.23044 0.158513i 0.765367i 1.55487 + 2.14065i −0.707107 0.707107i 2.41421i −1.68925 + 1.46508i
27.1 −0.707107 0.707107i −1.30656 1.30656i 1.00000i −0.158513 2.23044i 1.84776i −2.47247 0.941740i 0.707107 0.707107i 0.414214i −1.46508 + 1.68925i
27.2 −0.707107 0.707107i 1.30656 + 1.30656i 1.00000i 0.158513 + 2.23044i 1.84776i −0.941740 2.47247i 0.707107 0.707107i 0.414214i 1.46508 1.68925i
27.3 0.707107 + 0.707107i −0.541196 0.541196i 1.00000i 2.23044 0.158513i 0.765367i −2.14065 + 1.55487i −0.707107 + 0.707107i 2.41421i 1.68925 + 1.46508i
27.4 0.707107 + 0.707107i 0.541196 + 0.541196i 1.00000i −2.23044 + 0.158513i 0.765367i 1.55487 2.14065i −0.707107 + 0.707107i 2.41421i −1.68925 1.46508i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.4 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.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.2.g.a 8
3.b odd 2 1 630.2.p.a 8
4.b odd 2 1 560.2.bj.c 8
5.b even 2 1 350.2.g.a 8
5.c odd 4 1 inner 70.2.g.a 8
5.c odd 4 1 350.2.g.a 8
7.b odd 2 1 inner 70.2.g.a 8
7.c even 3 2 490.2.l.a 16
7.d odd 6 2 490.2.l.a 16
15.e even 4 1 630.2.p.a 8
20.e even 4 1 560.2.bj.c 8
21.c even 2 1 630.2.p.a 8
28.d even 2 1 560.2.bj.c 8
35.c odd 2 1 350.2.g.a 8
35.f even 4 1 inner 70.2.g.a 8
35.f even 4 1 350.2.g.a 8
35.k even 12 2 490.2.l.a 16
35.l odd 12 2 490.2.l.a 16
105.k odd 4 1 630.2.p.a 8
140.j odd 4 1 560.2.bj.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.g.a 8 1.a even 1 1 trivial
70.2.g.a 8 5.c odd 4 1 inner
70.2.g.a 8 7.b odd 2 1 inner
70.2.g.a 8 35.f even 4 1 inner
350.2.g.a 8 5.b even 2 1
350.2.g.a 8 5.c odd 4 1
350.2.g.a 8 35.c odd 2 1
350.2.g.a 8 35.f even 4 1
490.2.l.a 16 7.c even 3 2
490.2.l.a 16 7.d odd 6 2
490.2.l.a 16 35.k even 12 2
490.2.l.a 16 35.l odd 12 2
560.2.bj.c 8 4.b odd 2 1
560.2.bj.c 8 20.e even 4 1
560.2.bj.c 8 28.d even 2 1
560.2.bj.c 8 140.j odd 4 1
630.2.p.a 8 3.b odd 2 1
630.2.p.a 8 15.e even 4 1
630.2.p.a 8 21.c even 2 1
630.2.p.a 8 105.k odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(70, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 1)^{2}$$
$3$ $$T^{8} + 12T^{4} + 4$$
$5$ $$T^{8} - 48T^{4} + 625$$
$7$ $$T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 2401$$
$11$ $$(T^{2} - 8)^{4}$$
$13$ $$T^{8} + 1548 T^{4} + 334084$$
$17$ $$T^{8} + 768 T^{4} + 16384$$
$19$ $$(T^{4} - 52 T^{2} + 98)^{2}$$
$23$ $$(T^{4} + 4 T^{3} + 8 T^{2} - 8 T + 4)^{2}$$
$29$ $$(T^{4} + 24 T^{2} + 16)^{2}$$
$31$ $$(T^{4} + 16 T^{2} + 32)^{2}$$
$37$ $$(T^{4} - 16 T^{3} + 128 T^{2} - 448 T + 784)^{2}$$
$41$ $$(T^{4} + 16 T^{2} + 32)^{2}$$
$43$ $$(T^{2} - 8 T + 32)^{4}$$
$47$ $$T^{8} + 192T^{4} + 1024$$
$53$ $$(T^{4} + 16 T^{3} + 128 T^{2} - 64 T + 16)^{2}$$
$59$ $$(T^{4} - 100 T^{2} + 1250)^{2}$$
$61$ $$(T^{4} + 148 T^{2} + 4418)^{2}$$
$67$ $$(T^{4} - 8 T^{3} + 32 T^{2} + 1088 T + 18496)^{2}$$
$71$ $$(T^{2} + 4 T + 2)^{4}$$
$73$ $$T^{8} + 3264 T^{4} + \cdots + 2458624$$
$79$ $$(T^{4} + 108 T^{2} + 2116)^{2}$$
$83$ $$T^{8} + 6732 T^{4} + \cdots + 11303044$$
$89$ $$(T^{4} - 416 T^{2} + 36992)^{2}$$
$97$ $$T^{8} + 62208 T^{4} + \cdots + 107495424$$