# Properties

 Label 338.3.f.d Level $338$ Weight $3$ Character orbit 338.f Analytic conductor $9.210$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,3,Mod(19,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(12))

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

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

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

chi := DirichletCharacter("338.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 338.f (of order $$12$$, degree $$4$$, not 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: $$9.20983293538$$ 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: no (minimal twist has level 26) 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} + (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} - 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} + (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} + ( - 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} + (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} + ( - 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} + ( - 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 + (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 - 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 + (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 + (-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 + (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 + (-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 + (-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 $$4 q - 2 q^{2} + 6 q^{6} + 22 q^{7} - 8 q^{8} + 12 q^{9} - 12 q^{10} + 6 q^{11} - 4 q^{14} - 6 q^{15} + 8 q^{16} + 78 q^{17} - 24 q^{18} + 58 q^{19} + 12 q^{20} - 42 q^{21} + 18 q^{22} + 12 q^{23} - 12 q^{24} + 44 q^{28} - 54 q^{29} + 128 q^{31} + 8 q^{32} + 30 q^{33} + 24 q^{34} + 42 q^{35} - 40 q^{37} + 42 q^{42} + 120 q^{43} - 36 q^{44} - 36 q^{45} - 54 q^{46} - 132 q^{47} + 42 q^{49} - 38 q^{50} - 168 q^{53} + 90 q^{54} + 6 q^{55} - 84 q^{56} + 102 q^{57} + 42 q^{58} - 6 q^{59} - 24 q^{60} + 78 q^{61} - 60 q^{62} + 120 q^{63} - 12 q^{66} - 86 q^{67} - 24 q^{68} - 126 q^{69} - 84 q^{70} - 42 q^{71} - 24 q^{72} + 136 q^{73} - 38 q^{74} + 114 q^{75} - 116 q^{76} + 192 q^{79} + 24 q^{80} - 18 q^{81} - 78 q^{82} - 192 q^{83} + 36 q^{84} + 42 q^{85} + 156 q^{86} - 96 q^{87} + 12 q^{88} + 60 q^{89} + 168 q^{92} - 222 q^{93} + 132 q^{94} - 42 q^{95} - 280 q^{97} - 92 q^{98} + 108 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 + 6 * q^6 + 22 * q^7 - 8 * q^8 + 12 * q^9 - 12 * q^10 + 6 * q^11 - 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 + 44 * q^28 - 54 * q^29 + 128 * q^31 + 8 * q^32 + 30 * q^33 + 24 * q^34 + 42 * q^35 - 40 * q^37 + 42 * q^42 + 120 * q^43 - 36 * q^44 - 36 * q^45 - 54 * q^46 - 132 * q^47 + 42 * q^49 - 38 * q^50 - 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 - 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 + 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 + 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 $$\left(\mathbb{Z}/338\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$\chi(n)$$ $$\zeta_{12}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i
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
89.1 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
249.1 −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
319.1 −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
 $$n$$: 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 338.3.f.d 4
13.b even 2 1 26.3.f.a 4
13.c even 3 1 338.3.d.e 4
13.c even 3 1 338.3.f.c 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.d 4
13.e even 6 1 338.3.f.f 4
13.f odd 12 1 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 inner 338.3.f.d 4
39.d odd 2 1 234.3.bb.b 4
39.k even 12 1 234.3.bb.b 4
52.b odd 2 1 208.3.bd.c 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 13.b even 2 1
26.3.f.a 4 13.f odd 12 1
208.3.bd.c 4 52.b odd 2 1
208.3.bd.c 4 52.l even 12 1
234.3.bb.b 4 39.d odd 2 1
234.3.bb.b 4 39.k even 12 1
338.3.d.d 4 13.e even 6 1
338.3.d.d 4 13.f odd 12 1
338.3.d.e 4 13.c even 3 1
338.3.d.e 4 13.f odd 12 1
338.3.f.c 4 13.c even 3 1
338.3.f.c 4 13.d odd 4 1
338.3.f.d 4 1.a even 1 1 trivial
338.3.f.d 4 13.f odd 12 1 inner
338.3.f.f 4 13.d odd 4 1
338.3.f.f 4 13.e even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(338, [\chi])$$:

 $$T_{3}^{4} + 3T_{3}^{2} + 9$$ T3^4 + 3*T3^2 + 9 $$T_{5}^{4} + 36$$ T5^4 + 36 $$T_{7}^{4} - 22T_{7}^{3} + 221T_{7}^{2} - 1460T_{7} + 5329$$ T7^4 - 22*T7^3 + 221*T7^2 - 1460*T7 + 5329

## Hecke characteristic polynomials

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