# Properties

 Label 26.3.f.a Level $26$ Weight $3$ Character orbit 26.f Analytic conductor $0.708$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [26,3,Mod(7,26)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(26, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([11]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("26.7");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 26.f (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: $$0.708448687337$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (7 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8} + ( - 6 \zeta_{12}^{2} + 6) q^{9}+O(q^{10})$$ q + (-z^3 + z^2 + z) * q^2 + (-z^3 - z) * q^3 + 2*z * q^4 + (z^3 + 2*z^2 - 2*z - 1) * q^5 + (-2*z^3 - z^2 + z - 1) * q^6 + (7*z^3 - 3*z^2 - 4*z - 4) * q^7 + (2*z^3 + 2) * q^8 + (-6*z^2 + 6) * q^9 $$q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (7 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8} + ( - 6 \zeta_{12}^{2} + 6) q^{9} + (2 \zeta_{12}^{2} - 4) q^{10} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 4 \zeta_{12} + 1) q^{11} + ( - 4 \zeta_{12}^{2} + 2) q^{12} + ( - 2 \zeta_{12}^{3} + 12 \zeta_{12}^{2} + 8 \zeta_{12} - 9) q^{13} + (7 \zeta_{12}^{3} - 14 \zeta_{12} - 1) q^{14} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12}) q^{15} + 4 \zeta_{12}^{2} q^{16} + (13 \zeta_{12}^{2} - 6 \zeta_{12} + 13) q^{17} + ( - 6 \zeta_{12}^{3} + 6) q^{18} + ( - 17 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 12 \zeta_{12} - 12) q^{19} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{20} + (10 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 10) q^{21} + ( - 2 \zeta_{12}^{3} - 9 \zeta_{12}^{2} + \zeta_{12} + 9) q^{22} + ( - 21 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 21 \zeta_{12} + 4) q^{23} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{24} + 19 \zeta_{12}^{3} q^{25} + (15 \zeta_{12}^{3} + \zeta_{12}^{2} + 5 \zeta_{12} - 4) q^{26} + (15 \zeta_{12}^{3} - 30 \zeta_{12}) q^{27} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 8 \zeta_{12} - 14) q^{28} + ( - 16 \zeta_{12}^{3} - 27 \zeta_{12}^{2} - 16 \zeta_{12}) q^{29} + 6 \zeta_{12} q^{30} + (37 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10 \zeta_{12} - 37) q^{31} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{32} + (9 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 6 \zeta_{12} - 6) q^{33} + ( - 19 \zeta_{12}^{3} + 26 \zeta_{12}^{2} + 26 \zeta_{12} - 19) q^{34} + ( - 2 \zeta_{12}^{3} - 21 \zeta_{12}^{2} + \zeta_{12} + 21) q^{35} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{36} + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 13 \zeta_{12} + 7) q^{37} + (7 \zeta_{12}^{3} - 34 \zeta_{12}^{2} + 17) q^{38} + ( - 15 \zeta_{12}^{3} - 14 \zeta_{12}^{2} + 21 \zeta_{12} + 4) q^{39} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{40} + ( - 26 \zeta_{12}^{3} + 26 \zeta_{12}^{2} + 13 \zeta_{12} - 13) q^{41} + (\zeta_{12}^{3} + 21 \zeta_{12}^{2} + \zeta_{12}) q^{42} + (20 \zeta_{12}^{2} - 39 \zeta_{12} + 20) q^{43} + ( - 10 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 10) q^{44} + (12 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{45} + ( - 4 \zeta_{12}^{3} - 19 \zeta_{12}^{2} + 23 \zeta_{12} + 23) q^{46} + (33 \zeta_{12}^{3} + 33) q^{47} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{48} + ( - 25 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 25 \zeta_{12} + 14) q^{49} + (19 \zeta_{12}^{3} + 19 \zeta_{12}^{2} - 19 \zeta_{12}) q^{50} + ( - 39 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 6) q^{51} + (24 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 18 \zeta_{12} + 4) q^{52} + ( - 22 \zeta_{12}^{3} + 44 \zeta_{12} - 42) q^{53} + ( - 15 \zeta_{12}^{3} + 15 \zeta_{12}^{2} - 15 \zeta_{12} - 30) q^{54} + (9 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 9 \zeta_{12}) q^{55} + ( - 14 \zeta_{12}^{2} - 2 \zeta_{12} - 14) q^{56} + (22 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 7 \zeta_{12} - 22) q^{57} + ( - 32 \zeta_{12}^{3} - 43 \zeta_{12}^{2} - 11 \zeta_{12} + 11) q^{58} + ( - 13 \zeta_{12}^{3} + 23 \zeta_{12}^{2} - 10 \zeta_{12} - 10) q^{59} + (6 \zeta_{12}^{3} + 6) q^{60} + ( - 12 \zeta_{12}^{3} - 39 \zeta_{12}^{2} + 6 \zeta_{12} + 39) q^{61} + (64 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 64 \zeta_{12} - 20) q^{62} + (24 \zeta_{12}^{3} + 24 \zeta_{12}^{2} + 18 \zeta_{12} - 42) q^{63} + 8 \zeta_{12}^{3} q^{64} + ( - 7 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 2 \zeta_{12} - 25) q^{65} + (9 \zeta_{12}^{3} - 18 \zeta_{12} - 3) q^{66} + (23 \zeta_{12}^{3} - 23 \zeta_{12}^{2} + 10 \zeta_{12} + 33) q^{67} + (26 \zeta_{12}^{3} - 12 \zeta_{12}^{2} + 26 \zeta_{12}) q^{68} + ( - 21 \zeta_{12}^{2} - 6 \zeta_{12} - 21) q^{69} + ( - 22 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 22) q^{70} + (25 \zeta_{12}^{3} + 29 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{71} + ( - 12 \zeta_{12}^{2} + 12 \zeta_{12} + 12) q^{72} + ( - 7 \zeta_{12}^{3} - 54 \zeta_{12}^{2} - 54 \zeta_{12} - 7) q^{73} + ( - 14 \zeta_{12}^{3} + 19 \zeta_{12}^{2} + 7 \zeta_{12} - 19) q^{74} + ( - 19 \zeta_{12}^{2} + 38) q^{75} + ( - 10 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 24 \zeta_{12} + 34) q^{76} + (15 \zeta_{12}^{3} + 64 \zeta_{12}^{2} - 32) q^{77} + (2 \zeta_{12}^{3} - 25 \zeta_{12}^{2} + 5 \zeta_{12} + 35) q^{78} + ( - 36 \zeta_{12}^{3} + 72 \zeta_{12} + 48) q^{79} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} - 8) q^{80} - 9 \zeta_{12}^{2} q^{81} + ( - 13 \zeta_{12}^{2} + 39 \zeta_{12} - 13) q^{82} + ( - 25 \zeta_{12}^{3} + 46 \zeta_{12}^{2} - 46 \zeta_{12} + 25) q^{83} + (2 \zeta_{12}^{3} + 22 \zeta_{12}^{2} + 20 \zeta_{12} - 20) q^{84} + ( - 12 \zeta_{12}^{3} + 45 \zeta_{12}^{2} - 33 \zeta_{12} - 33) q^{85} + ( - 59 \zeta_{12}^{3} + 40 \zeta_{12}^{2} + 40 \zeta_{12} - 59) q^{86} + (54 \zeta_{12}^{3} + 48 \zeta_{12}^{2} - 27 \zeta_{12} - 48) q^{87} + ( - 18 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 18 \zeta_{12} + 4) q^{88} + ( - 56 \zeta_{12}^{3} - 56 \zeta_{12}^{2} + 43 \zeta_{12} + 13) q^{89} + (24 \zeta_{12}^{2} - 12) q^{90} + ( - 37 \zeta_{12}^{3} - 25 \zeta_{12}^{2} - 86 \zeta_{12} + 22) q^{91} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} + 42) q^{92} + (17 \zeta_{12}^{3} - 17 \zeta_{12}^{2} + 47 \zeta_{12} + 64) q^{93} + 66 \zeta_{12}^{2} q^{94} + ( - 7 \zeta_{12}^{2} + 51 \zeta_{12} - 7) q^{95} + ( - 4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 4) q^{96} + (69 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 71 \zeta_{12} + 71) q^{97} + ( - 14 \zeta_{12}^{3} - 18 \zeta_{12}^{2} + 32 \zeta_{12} + 32) q^{98} + ( - 24 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12} - 24) q^{99} +O(q^{100})$$ q + (-z^3 + z^2 + z) * q^2 + (-z^3 - z) * q^3 + 2*z * q^4 + (z^3 + 2*z^2 - 2*z - 1) * q^5 + (-2*z^3 - z^2 + z - 1) * q^6 + (7*z^3 - 3*z^2 - 4*z - 4) * q^7 + (2*z^3 + 2) * q^8 + (-6*z^2 + 6) * q^9 + (2*z^2 - 4) * q^10 + (-5*z^3 - 5*z^2 + 4*z + 1) * q^11 + (-4*z^2 + 2) * q^12 + (-2*z^3 + 12*z^2 + 8*z - 9) * q^13 + (7*z^3 - 14*z - 1) * q^14 + (-3*z^3 + 3*z^2 + 3*z) * q^15 + 4*z^2 * q^16 + (13*z^2 - 6*z + 13) * q^17 + (-6*z^3 + 6) * q^18 + (-17*z^3 - 5*z^2 + 12*z - 12) * q^19 + (4*z^3 - 2*z^2 - 2*z - 2) * q^20 + (10*z^3 + z^2 + z + 10) * q^21 + (-2*z^3 - 9*z^2 + z + 9) * q^22 + (-21*z^3 - 2*z^2 + 21*z + 4) * q^23 + (-2*z^3 - 2*z^2 - 2*z + 4) * q^24 + 19*z^3 * q^25 + (15*z^3 + z^2 + 5*z - 4) * q^26 + (15*z^3 - 30*z) * q^27 + (-6*z^3 + 6*z^2 - 8*z - 14) * q^28 + (-16*z^3 - 27*z^2 - 16*z) * q^29 + 6*z * q^30 + (37*z^3 + 10*z^2 - 10*z - 37) * q^31 + (4*z^2 + 4*z - 4) * q^32 + (9*z^3 - 3*z^2 - 6*z - 6) * q^33 + (-19*z^3 + 26*z^2 + 26*z - 19) * q^34 + (-2*z^3 - 21*z^2 + z + 21) * q^35 + (-12*z^3 + 12*z) * q^36 + (6*z^3 + 6*z^2 - 13*z + 7) * q^37 + (7*z^3 - 34*z^2 + 17) * q^38 + (-15*z^3 - 14*z^2 + 21*z + 4) * q^39 + (4*z^3 - 8*z) * q^40 + (-26*z^3 + 26*z^2 + 13*z - 13) * q^41 + (z^3 + 21*z^2 + z) * q^42 + (20*z^2 - 39*z + 20) * q^43 + (-10*z^3 - 2*z^2 + 2*z + 10) * q^44 + (12*z^3 + 6*z^2 - 6*z + 6) * q^45 + (-4*z^3 - 19*z^2 + 23*z + 23) * q^46 + (33*z^3 + 33) * q^47 + (-8*z^3 + 4*z) * q^48 + (-25*z^3 - 7*z^2 + 25*z + 14) * q^49 + (19*z^3 + 19*z^2 - 19*z) * q^50 + (-39*z^3 + 12*z^2 - 6) * q^51 + (24*z^3 + 12*z^2 - 18*z + 4) * q^52 + (-22*z^3 + 44*z - 42) * q^53 + (-15*z^3 + 15*z^2 - 15*z - 30) * q^54 + (9*z^3 + 3*z^2 + 9*z) * q^55 + (-14*z^2 - 2*z - 14) * q^56 + (22*z^3 - 7*z^2 + 7*z - 22) * q^57 + (-32*z^3 - 43*z^2 - 11*z + 11) * q^58 + (-13*z^3 + 23*z^2 - 10*z - 10) * q^59 + (6*z^3 + 6) * q^60 + (-12*z^3 - 39*z^2 + 6*z + 39) * q^61 + (64*z^3 + 10*z^2 - 64*z - 20) * q^62 + (24*z^3 + 24*z^2 + 18*z - 42) * q^63 + 8*z^3 * q^64 + (-7*z^3 - 10*z^2 + 2*z - 25) * q^65 + (9*z^3 - 18*z - 3) * q^66 + (23*z^3 - 23*z^2 + 10*z + 33) * q^67 + (26*z^3 - 12*z^2 + 26*z) * q^68 + (-21*z^2 - 6*z - 21) * q^69 + (-22*z^3 - 2*z^2 + 2*z + 22) * q^70 + (25*z^3 + 29*z^2 + 4*z - 4) * q^71 + (-12*z^2 + 12*z + 12) * q^72 + (-7*z^3 - 54*z^2 - 54*z - 7) * q^73 + (-14*z^3 + 19*z^2 + 7*z - 19) * q^74 + (-19*z^2 + 38) * q^75 + (-10*z^3 - 10*z^2 - 24*z + 34) * q^76 + (15*z^3 + 64*z^2 - 32) * q^77 + (2*z^3 - 25*z^2 + 5*z + 35) * q^78 + (-36*z^3 + 72*z + 48) * q^79 + (-4*z^3 + 4*z^2 - 4*z - 8) * q^80 - 9*z^2 * q^81 + (-13*z^2 + 39*z - 13) * q^82 + (-25*z^3 + 46*z^2 - 46*z + 25) * q^83 + (2*z^3 + 22*z^2 + 20*z - 20) * q^84 + (-12*z^3 + 45*z^2 - 33*z - 33) * q^85 + (-59*z^3 + 40*z^2 + 40*z - 59) * q^86 + (54*z^3 + 48*z^2 - 27*z - 48) * q^87 + (-18*z^3 - 2*z^2 + 18*z + 4) * q^88 + (-56*z^3 - 56*z^2 + 43*z + 13) * q^89 + (24*z^2 - 12) * q^90 + (-37*z^3 - 25*z^2 - 86*z + 22) * q^91 + (-4*z^3 + 8*z + 42) * q^92 + (17*z^3 - 17*z^2 + 47*z + 64) * q^93 + 66*z^2 * q^94 + (-7*z^2 + 51*z - 7) * q^95 + (-4*z^3 - 8*z^2 + 8*z + 4) * q^96 + (69*z^3 - 2*z^2 - 71*z + 71) * q^97 + (-14*z^3 - 18*z^2 + 32*z + 32) * q^98 + (-24*z^3 - 6*z^2 - 6*z - 24) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 6 q^{6} - 22 q^{7} + 8 q^{8} + 12 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 6 * q^6 - 22 * q^7 + 8 * q^8 + 12 * q^9 $$4 q + 2 q^{2} - 6 q^{6} - 22 q^{7} + 8 q^{8} + 12 q^{9} - 12 q^{10} - 6 q^{11} - 12 q^{13} - 4 q^{14} + 6 q^{15} + 8 q^{16} + 78 q^{17} + 24 q^{18} - 58 q^{19} - 12 q^{20} + 42 q^{21} + 18 q^{22} + 12 q^{23} + 12 q^{24} - 14 q^{26} - 44 q^{28} - 54 q^{29} - 128 q^{31} - 8 q^{32} - 30 q^{33} - 24 q^{34} + 42 q^{35} + 40 q^{37} - 12 q^{39} + 42 q^{42} + 120 q^{43} + 36 q^{44} + 36 q^{45} + 54 q^{46} + 132 q^{47} + 42 q^{49} + 38 q^{50} + 40 q^{52} - 168 q^{53} - 90 q^{54} + 6 q^{55} - 84 q^{56} - 102 q^{57} - 42 q^{58} + 6 q^{59} + 24 q^{60} + 78 q^{61} - 60 q^{62} - 120 q^{63} - 120 q^{65} - 12 q^{66} + 86 q^{67} - 24 q^{68} - 126 q^{69} + 84 q^{70} + 42 q^{71} + 24 q^{72} - 136 q^{73} - 38 q^{74} + 114 q^{75} + 116 q^{76} + 90 q^{78} + 192 q^{79} - 24 q^{80} - 18 q^{81} - 78 q^{82} + 192 q^{83} - 36 q^{84} - 42 q^{85} - 156 q^{86} - 96 q^{87} + 12 q^{88} - 60 q^{89} + 38 q^{91} + 168 q^{92} + 222 q^{93} + 132 q^{94} - 42 q^{95} + 280 q^{97} + 92 q^{98} - 108 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 6 * q^6 - 22 * q^7 + 8 * q^8 + 12 * q^9 - 12 * q^10 - 6 * q^11 - 12 * q^13 - 4 * q^14 + 6 * q^15 + 8 * q^16 + 78 * q^17 + 24 * q^18 - 58 * q^19 - 12 * q^20 + 42 * q^21 + 18 * q^22 + 12 * q^23 + 12 * q^24 - 14 * q^26 - 44 * q^28 - 54 * q^29 - 128 * q^31 - 8 * q^32 - 30 * q^33 - 24 * q^34 + 42 * q^35 + 40 * q^37 - 12 * q^39 + 42 * q^42 + 120 * q^43 + 36 * q^44 + 36 * q^45 + 54 * q^46 + 132 * q^47 + 42 * q^49 + 38 * q^50 + 40 * q^52 - 168 * q^53 - 90 * q^54 + 6 * q^55 - 84 * q^56 - 102 * q^57 - 42 * q^58 + 6 * q^59 + 24 * q^60 + 78 * q^61 - 60 * q^62 - 120 * q^63 - 120 * q^65 - 12 * q^66 + 86 * q^67 - 24 * q^68 - 126 * q^69 + 84 * q^70 + 42 * q^71 + 24 * q^72 - 136 * q^73 - 38 * q^74 + 114 * q^75 + 116 * q^76 + 90 * q^78 + 192 * q^79 - 24 * q^80 - 18 * q^81 - 78 * q^82 + 192 * q^83 - 36 * q^84 - 42 * q^85 - 156 * q^86 - 96 * q^87 + 12 * q^88 - 60 * q^89 + 38 * q^91 + 168 * q^92 + 222 * q^93 + 132 * q^94 - 42 * q^95 + 280 * q^97 + 92 * q^98 - 108 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/26\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$\chi(n)$$ $$\zeta_{12}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
1.36603 0.366025i −0.866025 + 1.50000i 1.73205 1.00000i −1.73205 1.73205i −0.633975 + 2.36603i −8.96410 2.40192i 2.00000 2.00000i 3.00000 + 5.19615i −3.00000 1.73205i
11.1 −0.366025 + 1.36603i 0.866025 + 1.50000i −1.73205 1.00000i 1.73205 + 1.73205i −2.36603 + 0.633975i −2.03590 7.59808i 2.00000 2.00000i 3.00000 5.19615i −3.00000 + 1.73205i
15.1 1.36603 + 0.366025i −0.866025 1.50000i 1.73205 + 1.00000i −1.73205 + 1.73205i −0.633975 2.36603i −8.96410 + 2.40192i 2.00000 + 2.00000i 3.00000 5.19615i −3.00000 + 1.73205i
19.1 −0.366025 1.36603i 0.866025 1.50000i −1.73205 + 1.00000i 1.73205 1.73205i −2.36603 0.633975i −2.03590 + 7.59808i 2.00000 + 2.00000i 3.00000 + 5.19615i −3.00000 1.73205i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.3.f.a 4
3.b odd 2 1 234.3.bb.b 4
4.b odd 2 1 208.3.bd.c 4
13.b even 2 1 338.3.f.d 4
13.c even 3 1 338.3.d.d 4
13.c even 3 1 338.3.f.f 4
13.d odd 4 1 338.3.f.c 4
13.d odd 4 1 338.3.f.f 4
13.e even 6 1 338.3.d.e 4
13.e even 6 1 338.3.f.c 4
13.f odd 12 1 inner 26.3.f.a 4
13.f odd 12 1 338.3.d.d 4
13.f odd 12 1 338.3.d.e 4
13.f odd 12 1 338.3.f.d 4
39.k even 12 1 234.3.bb.b 4
52.l even 12 1 208.3.bd.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.f.a 4 1.a even 1 1 trivial
26.3.f.a 4 13.f odd 12 1 inner
208.3.bd.c 4 4.b odd 2 1
208.3.bd.c 4 52.l even 12 1
234.3.bb.b 4 3.b odd 2 1
234.3.bb.b 4 39.k even 12 1
338.3.d.d 4 13.c even 3 1
338.3.d.d 4 13.f odd 12 1
338.3.d.e 4 13.e even 6 1
338.3.d.e 4 13.f odd 12 1
338.3.f.c 4 13.d odd 4 1
338.3.f.c 4 13.e even 6 1
338.3.f.d 4 13.b even 2 1
338.3.f.d 4 13.f odd 12 1
338.3.f.f 4 13.c even 3 1
338.3.f.f 4 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 3T_{3}^{2} + 9$$ acting on $$S_{3}^{\mathrm{new}}(26, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4} + 36$$
$7$ $$T^{4} + 22 T^{3} + 221 T^{2} + \cdots + 5329$$
$11$ $$T^{4} + 6 T^{3} + 45 T^{2} + \cdots + 1521$$
$13$ $$T^{4} + 12 T^{3} + 182 T^{2} + \cdots + 28561$$
$17$ $$T^{4} - 78 T^{3} + 2499 T^{2} + \cdots + 221841$$
$19$ $$T^{4} + 58 T^{3} + 1325 T^{2} + \cdots + 167281$$
$23$ $$T^{4} - 12 T^{3} - 381 T^{2} + \cdots + 184041$$
$29$ $$T^{4} + 54 T^{3} + 2955 T^{2} + \cdots + 1521$$
$31$ $$T^{4} + 128 T^{3} + 8192 T^{2} + \cdots + 3602404$$
$37$ $$T^{4} - 40 T^{3} + 401 T^{2} + \cdots + 11449$$
$41$ $$T^{4} + 1521 T^{2} - 39546 T + 257049$$
$43$ $$T^{4} - 120 T^{3} + 4479 T^{2} + \cdots + 103041$$
$47$ $$(T^{2} - 66 T + 2178)^{2}$$
$53$ $$(T^{2} + 84 T + 312)^{2}$$
$59$ $$T^{4} - 6 T^{3} + 1305 T^{2} + \cdots + 84681$$
$61$ $$T^{4} - 78 T^{3} + 4671 T^{2} + \cdots + 1996569$$
$67$ $$T^{4} - 86 T^{3} + 4985 T^{2} + \cdots + 2399401$$
$71$ $$T^{4} - 42 T^{3} + 3357 T^{2} + \cdots + 154449$$
$73$ $$T^{4} + 136 T^{3} + 9248 T^{2} + \cdots + 4251844$$
$79$ $$(T^{2} - 96 T - 1584)^{2}$$
$83$ $$T^{4} - 192 T^{3} + 18432 T^{2} + \cdots + 2056356$$
$89$ $$T^{4} + 60 T^{3} + 5661 T^{2} + \cdots + 21594609$$
$97$ $$T^{4} - 280 T^{3} + \cdots + 20043529$$