# Properties

 Label 338.3.d.g Level $338$ Weight $3$ Character orbit 338.d Analytic conductor $9.210$ 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] = [338,3,Mod(99,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("338.99");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 338.d (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: $$9.20983293538$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.612074651904.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 74x^{6} + 2067x^{4} - 25778x^{2} + 121801$$ x^8 - 74*x^6 + 2067*x^4 - 25778*x^2 + 121801 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 26) 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + 1) q^{2} + (\beta_{5} + \beta_{3} + \beta_1 - 1) q^{3} - 2 \beta_{5} q^{4} + ( - \beta_{7} + \beta_1) q^{5} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_1 - 1) q^{6} + (\beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{7} + ( - 2 \beta_{5} - 2) q^{8} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + 3 \beta_1 + 9) q^{9}+O(q^{10})$$ q + (-b5 + 1) * q^2 + (b5 + b3 + b1 - 1) * q^3 - 2*b5 * q^4 + (-b7 + b1) * q^5 + (b5 + b4 + b3 + b1 - 1) * q^6 + (b6 + b4 - b3 - b2 - b1 + 3) * q^7 + (-2*b5 - 2) * q^8 + (b7 - b6 + b5 + b3 - b2 + 3*b1 + 9) * q^9 $$q + ( - \beta_{5} + 1) q^{2} + (\beta_{5} + \beta_{3} + \beta_1 - 1) q^{3} - 2 \beta_{5} q^{4} + ( - \beta_{7} + \beta_1) q^{5} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_1 - 1) q^{6} + (\beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{7} + ( - 2 \beta_{5} - 2) q^{8} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + 3 \beta_1 + 9) q^{9} + ( - \beta_{7} - \beta_{6} + 2 \beta_{2} + \beta_1 - 2) q^{10} + (\beta_{6} + 5 \beta_{5} - \beta_{2} + 6) q^{11} + 2 \beta_{4} q^{12} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 5) q^{14} + (2 \beta_{7} - 15 \beta_{5} + \beta_{4} + \beta_{3} - 15 \beta_1 + 15) q^{15} - 4 q^{16} + ( - \beta_{7} - \beta_{6} - 11 \beta_{5} - 3 \beta_{2} - 4 \beta_1 + 3) q^{17} + (2 \beta_{7} - 8 \beta_{5} + \beta_{4} + \beta_{3} + 5 \beta_1 + 8) q^{18} + (\beta_{7} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 10 \beta_1 - 1) q^{19} + ( - 2 \beta_{6} + 4 \beta_{2} - 4) q^{20} + (\beta_{6} - 3 \beta_{5} + 19 \beta_{2} - 22) q^{21} + ( - \beta_{7} + \beta_{6} - \beta_{2} + 11) q^{22} + (\beta_{7} + \beta_{6} - 10 \beta_{5} - \beta_{4} + 3 \beta_{2} + 4 \beta_1 - 3) q^{23} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 2) q^{24} + ( - 22 \beta_{5} - 3 \beta_{4} - 11 \beta_{2} - 11 \beta_1 + 11) q^{25} + (2 \beta_{7} - 2 \beta_{6} + 5 \beta_{5} + 5 \beta_{3} - 16 \beta_{2} + 23 \beta_1 + 3) q^{27} + ( - 2 \beta_{7} - 4 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 4) q^{28} + (\beta_{7} - \beta_{6} - 4 \beta_{5} - 4 \beta_{3} - 3 \beta_1 + 5) q^{29} + (2 \beta_{7} + 2 \beta_{6} - 32 \beta_{5} + 2 \beta_{4} - 18 \beta_{2} - 16 \beta_1 + 18) q^{30} + ( - 5 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \beta_1 + 5) q^{31} + (4 \beta_{5} - 4) q^{32} + ( - \beta_{6} + 20 \beta_{5} - 5 \beta_{4} + 5 \beta_{3} + 17 \beta_{2} + 5 \beta_1 - 7) q^{33} + ( - 2 \beta_{6} - 11 \beta_{5} - 6 \beta_{2} - 5) q^{34} + ( - 3 \beta_{7} + 3 \beta_{6} - 4 \beta_{5} - 4 \beta_{3} - 7 \beta_{2} + 25) q^{35} + (2 \beta_{7} + 2 \beta_{6} - 18 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} + 4 \beta_1 - 2) q^{36} + ( - 6 \beta_{6} + 19 \beta_{5} - 5 \beta_{4} + 5 \beta_{3} + 23 \beta_{2} + \cdots - 14) q^{37}+ \cdots + ( - 2 \beta_{6} + 18 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} - 34 \beta_{2} + \cdots + 44) q^{99}+O(q^{100})$$ q + (-b5 + 1) * q^2 + (b5 + b3 + b1 - 1) * q^3 - 2*b5 * q^4 + (-b7 + b1) * q^5 + (b5 + b4 + b3 + b1 - 1) * q^6 + (b6 + b4 - b3 - b2 - b1 + 3) * q^7 + (-2*b5 - 2) * q^8 + (b7 - b6 + b5 + b3 - b2 + 3*b1 + 9) * q^9 + (-b7 - b6 + 2*b2 + b1 - 2) * q^10 + (b6 + 5*b5 - b2 + 6) * q^11 + 2*b4 * q^12 + (-b7 + b6 - 2*b5 - 2*b3 - b2 - 2*b1 + 5) * q^14 + (2*b7 - 15*b5 + b4 + b3 - 15*b1 + 15) * q^15 - 4 * q^16 + (-b7 - b6 - 11*b5 - 3*b2 - 4*b1 + 3) * q^17 + (2*b7 - 8*b5 + b4 + b3 + 5*b1 + 8) * q^18 + (b7 + b5 - 2*b4 - 2*b3 + 10*b1 - 1) * q^19 + (-2*b6 + 4*b2 - 4) * q^20 + (b6 - 3*b5 + 19*b2 - 22) * q^21 + (-b7 + b6 - b2 + 11) * q^22 + (b7 + b6 - 10*b5 - b4 + 3*b2 + 4*b1 - 3) * q^23 + (-2*b5 + 2*b4 - 2*b3 - 2*b1 + 2) * q^24 + (-22*b5 - 3*b4 - 11*b2 - 11*b1 + 11) * q^25 + (2*b7 - 2*b6 + 5*b5 + 5*b3 - 16*b2 + 23*b1 + 3) * q^27 + (-2*b7 - 4*b5 - 2*b4 - 2*b3 - 2*b1 + 4) * q^28 + (b7 - b6 - 4*b5 - 4*b3 - 3*b1 + 5) * q^29 + (2*b7 + 2*b6 - 32*b5 + 2*b4 - 18*b2 - 16*b1 + 18) * q^30 + (-5*b5 - 3*b4 - 3*b3 + b1 + 5) * q^31 + (4*b5 - 4) * q^32 + (-b6 + 20*b5 - 5*b4 + 5*b3 + 17*b2 + 5*b1 - 7) * q^33 + (-2*b6 - 11*b5 - 6*b2 - 5) * q^34 + (-3*b7 + 3*b6 - 4*b5 - 4*b3 - 7*b2 + 25) * q^35 + (2*b7 + 2*b6 - 18*b5 + 2*b4 + 2*b2 + 4*b1 - 2) * q^36 + (-6*b6 + 19*b5 - 5*b4 + 5*b3 + 23*b2 + 5*b1 - 14) * q^37 + (b7 + b6 + 6*b5 - 4*b4 + 11*b2 + 12*b1 - 11) * q^38 + (2*b7 - 2*b6 + 4*b2 - 2*b1 - 4) * q^40 + (-8*b7 - 10*b5 - 2*b4 - 2*b3 - 25*b1 + 10) * q^41 + (-b7 + b6 + 19*b2 - 20*b1 - 25) * q^42 + (b7 + b6 + 10*b5 - 5*b4 - 7*b2 - 6*b1 + 7) * q^43 + (-2*b7 - 10*b5 + 10) * q^44 + (-7*b7 + 29*b5 + b4 + b3 + 28*b1 - 29) * q^45 + (2*b6 - 9*b5 - b4 + b3 + 6*b2 + b1 - 17) * q^46 + (-2*b6 + 13*b5 - b4 + b3 + 10*b2 + b1 + 1) * q^47 + (-4*b5 - 4*b3 - 4*b1 + 4) * q^48 + (2*b7 + 2*b6 - 25*b5 + 3*b4 - 24*b2 - 22*b1 + 24) * q^49 + (-19*b5 - 3*b4 + 3*b3 - 22*b2 + 3*b1 - 3) * q^50 + (-3*b7 - 3*b6 - 22*b5 + 3*b4 - 13*b2 - 16*b1 + 13) * q^51 + (-5*b7 + 5*b6 - 7*b5 - 7*b3 - 31*b2 + 19*b1 - 7) * q^53 + (4*b7 + 13*b5 + 5*b4 + 5*b3 + 41*b1 - 13) * q^54 + (-5*b7 + 5*b6 - 2*b5 - 2*b3 + 3*b2 - 10*b1 + 45) * q^55 + (-2*b7 - 2*b6 - 4*b5 - 4*b4 + 2*b2 - 2) * q^56 + (7*b7 + 46*b5 + 7*b4 + 7*b3 + 15*b1 - 46) * q^57 + (2*b7 - 5*b5 - 4*b4 - 4*b3 - 2*b1 + 5) * q^58 + (b6 + 30*b5 + 7*b4 - 7*b3 + 15*b2 - 7*b1 + 29) * q^59 + (4*b6 - 34*b5 + 2*b4 - 2*b3 - 36*b2 - 2*b1 + 6) * q^60 + (4*b7 - 4*b6 + 6*b5 + 6*b3 + 8*b2 + 2*b1 - 17) * q^61 + (-4*b5 - 6*b4 + 4*b2 + 4*b1 - 4) * q^62 + (10*b6 - 28*b5 + 14*b4 - 14*b3 + 6*b2 - 14*b1 - 6) * q^63 + 8*b5 * q^64 + (b7 - b6 + 10*b5 + 10*b3 + 17*b2 - 6*b1 + 3) * q^66 + (11*b7 + 8*b5 - b4 - b3 + 19*b1 - 8) * q^67 + (2*b7 - 2*b6 - 6*b2 + 8*b1 - 16) * q^68 + (2*b7 + 2*b6 + 40*b5 + 17*b4 + 12*b2 + 14*b1 - 12) * q^69 + (-6*b7 - 18*b5 - 4*b4 - 4*b3 + 4*b1 + 18) * q^70 + (11*b7 - 25*b5 + 6*b4 + 6*b3 - 10*b1 + 25) * q^71 + (4*b6 - 20*b5 + 2*b4 - 2*b3 + 4*b2 - 2*b1 - 20) * q^72 + (b6 + 30*b5 + 3*b4 - 3*b3 + 24*b2 - 3*b1 + 12) * q^73 + (6*b7 - 6*b6 + 10*b5 + 10*b3 + 23*b2 - 7*b1 - 5) * q^74 + (-14*b7 - 14*b6 + 76*b5 - 3*b4 + 8*b2 - 6*b1 - 8) * q^75 + (2*b6 + 10*b5 - 4*b4 + 4*b3 + 22*b2 + 4*b1 - 20) * q^76 + (7*b7 + 7*b6 + 26*b5 + 11*b4 - 9*b2 - 2*b1 + 9) * q^77 + (-16*b5 - 16*b3 + 24*b2 - 40*b1 + 4) * q^79 + (4*b7 - 4*b1) * q^80 + (12*b7 - 12*b6 + 24*b5 + 24*b3 - 12*b2 + 48*b1 - 27) * q^81 + (-8*b7 - 8*b6 - 16*b5 - 4*b4 - 15*b2 - 23*b1 + 15) * q^82 + (-8*b7 + 9*b5 - 3*b4 - 3*b3 - 43*b1 - 9) * q^83 + (-2*b7 + 6*b5 - 40*b1 - 6) * q^84 + (-11*b6 - 44*b5 + 2*b4 - 2*b3 - 20*b2 - 2*b1 - 20) * q^85 + (2*b6 + 15*b5 - 5*b4 + 5*b3 - 14*b2 + 5*b1 + 19) * q^86 + (-4*b7 + 4*b6 - b5 - b3 - 12*b2 + 7*b1 - 87) * q^87 + (-2*b7 - 2*b6 - 20*b5 + 2*b2 - 2) * q^88 + (b6 + 2*b5 - 6*b4 + 6*b3 - 61*b2 + 6*b1 + 51) * q^89 + (-7*b7 - 7*b6 + 56*b5 + 2*b4 + 34*b2 + 27*b1 - 34) * q^90 + (-2*b7 + 2*b6 + 2*b5 + 2*b3 + 6*b2 - 6*b1 - 28) * q^92 + (-2*b7 + 53*b5 + 3*b4 + 3*b3 - 9*b1 - 53) * q^93 + (2*b7 - 2*b6 + 2*b5 + 2*b3 + 10*b2 - 6*b1 + 12) * q^94 + (-b7 - b6 + 98*b5 - 14*b4 + 67*b2 + 66*b1 - 67) * q^95 + (-4*b5 - 4*b4 - 4*b3 - 4*b1 + 4) * q^96 + (7*b7 - 12*b5 + 10*b4 + 10*b3 + 6*b1 + 12) * q^97 + (4*b6 - 28*b5 + 3*b4 - 3*b3 - 48*b2 - 3*b1 + 26) * q^98 + (-2*b6 + 18*b5 - 4*b4 + 4*b3 - 34*b2 + 4*b1 + 44) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{2} + 6 q^{5} + 10 q^{7} - 16 q^{8} + 84 q^{9}+O(q^{10})$$ 8 * q + 8 * q^2 + 6 * q^5 + 10 * q^7 - 16 * q^8 + 84 * q^9 $$8 q + 8 q^{2} + 6 q^{5} + 10 q^{7} - 16 q^{8} + 84 q^{9} + 42 q^{11} + 20 q^{14} + 60 q^{15} - 32 q^{16} + 84 q^{18} + 22 q^{19} - 12 q^{20} - 102 q^{21} + 84 q^{22} + 72 q^{27} + 20 q^{28} + 12 q^{29} + 32 q^{31} - 32 q^{32} + 54 q^{33} - 60 q^{34} + 156 q^{35} + 32 q^{37} - 24 q^{40} - 12 q^{41} - 204 q^{42} + 84 q^{44} - 102 q^{45} - 108 q^{46} + 60 q^{47} - 88 q^{50} - 132 q^{53} + 72 q^{54} + 324 q^{55} - 294 q^{57} + 12 q^{58} + 234 q^{59} - 120 q^{60} - 72 q^{61} - 156 q^{63} + 108 q^{66} - 14 q^{67} - 120 q^{68} + 156 q^{70} + 162 q^{71} - 168 q^{72} + 166 q^{73} + 64 q^{74} - 44 q^{76} - 96 q^{79} - 24 q^{80} + 24 q^{81} - 240 q^{83} - 204 q^{84} - 234 q^{85} + 132 q^{86} - 720 q^{87} + 210 q^{89} - 216 q^{92} - 444 q^{93} + 120 q^{94} + 146 q^{97} - 16 q^{98} + 252 q^{99}+O(q^{100})$$ 8 * q + 8 * q^2 + 6 * q^5 + 10 * q^7 - 16 * q^8 + 84 * q^9 + 42 * q^11 + 20 * q^14 + 60 * q^15 - 32 * q^16 + 84 * q^18 + 22 * q^19 - 12 * q^20 - 102 * q^21 + 84 * q^22 + 72 * q^27 + 20 * q^28 + 12 * q^29 + 32 * q^31 - 32 * q^32 + 54 * q^33 - 60 * q^34 + 156 * q^35 + 32 * q^37 - 24 * q^40 - 12 * q^41 - 204 * q^42 + 84 * q^44 - 102 * q^45 - 108 * q^46 + 60 * q^47 - 88 * q^50 - 132 * q^53 + 72 * q^54 + 324 * q^55 - 294 * q^57 + 12 * q^58 + 234 * q^59 - 120 * q^60 - 72 * q^61 - 156 * q^63 + 108 * q^66 - 14 * q^67 - 120 * q^68 + 156 * q^70 + 162 * q^71 - 168 * q^72 + 166 * q^73 + 64 * q^74 - 44 * q^76 - 96 * q^79 - 24 * q^80 + 24 * q^81 - 240 * q^83 - 204 * q^84 - 234 * q^85 + 132 * q^86 - 720 * q^87 + 210 * q^89 - 216 * q^92 - 444 * q^93 + 120 * q^94 + 146 * q^97 - 16 * q^98 + 252 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 74x^{6} + 2067x^{4} - 25778x^{2} + 121801$$ :

 $$\beta_{1}$$ $$=$$ $$( 9\nu^{7} - 349\nu^{6} + 32\nu^{5} + 19195\nu^{4} - 8968\nu^{3} - 345859\nu^{2} + 102689\nu + 2068174 ) / 86552$$ (9*v^7 - 349*v^6 + 32*v^5 + 19195*v^4 - 8968*v^3 - 345859*v^2 + 102689*v + 2068174) / 86552 $$\beta_{2}$$ $$=$$ $$( 9\nu^{7} + 349\nu^{6} + 32\nu^{5} - 19195\nu^{4} - 8968\nu^{3} + 345859\nu^{2} + 102689\nu - 1981622 ) / 86552$$ (9*v^7 + 349*v^6 + 32*v^5 - 19195*v^4 - 8968*v^3 + 345859*v^2 + 102689*v - 1981622) / 86552 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - 74\nu^{5} + 1718\nu^{3} - 10073\nu + 1396 ) / 2792$$ (v^7 - 74*v^5 + 1718*v^3 - 10073*v + 1396) / 2792 $$\beta_{4}$$ $$=$$ $$( - 31 \nu^{7} + 1396 \nu^{6} + 2294 \nu^{5} - 76780 \nu^{4} - 53258 \nu^{3} + 1469988 \nu^{2} + 398815 \nu - 9700804 ) / 86552$$ (-31*v^7 + 1396*v^6 + 2294*v^5 - 76780*v^4 - 53258*v^3 + 1469988*v^2 + 398815*v - 9700804) / 86552 $$\beta_{5}$$ $$=$$ $$( -20\nu^{7} + 1131\nu^{5} - 22145\nu^{3} + 148063\nu ) / 21638$$ (-20*v^7 + 1131*v^5 - 22145*v^3 + 148063*v) / 21638 $$\beta_{6}$$ $$=$$ $$( 318 \nu^{7} - 698 \nu^{6} - 16901 \nu^{5} + 49209 \nu^{4} + 292601 \nu^{3} - 1135297 \nu^{2} - 1582807 \nu + 8604595 ) / 86552$$ (318*v^7 - 698*v^6 - 16901*v^5 + 49209*v^4 + 292601*v^3 - 1135297*v^2 - 1582807*v + 8604595) / 86552 $$\beta_{7}$$ $$=$$ $$( - 318 \nu^{7} - 698 \nu^{6} + 16901 \nu^{5} + 49209 \nu^{4} - 292601 \nu^{3} - 1135297 \nu^{2} + 1582807 \nu + 8604595 ) / 86552$$ (-318*v^7 - 698*v^6 + 16901*v^5 + 49209*v^4 - 292601*v^3 - 1135297*v^2 + 1582807*v + 8604595) / 86552
 $$\nu$$ $$=$$ $$( \beta_{5} + 2\beta_{3} + \beta_{2} + \beta _1 - 2 ) / 2$$ (b5 + 2*b3 + b2 + b1 - 2) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{5} + 2\beta_{4} - 5\beta_{2} + 3\beta _1 + 38 ) / 2$$ (-b5 + 2*b4 - 5*b2 + 3*b1 + 38) / 2 $$\nu^{3}$$ $$=$$ $$2\beta_{7} - 2\beta_{6} - 5\beta_{5} + 20\beta_{3} + 14\beta_{2} + 14\beta _1 - 24$$ 2*b7 - 2*b6 - 5*b5 + 20*b3 + 14*b2 + 14*b1 - 24 $$\nu^{4}$$ $$=$$ $$( 8\beta_{7} + 8\beta_{6} - 41\beta_{5} + 82\beta_{4} - 189\beta_{2} + 107\beta _1 + 716 ) / 2$$ (8*b7 + 8*b6 - 41*b5 + 82*b4 - 189*b2 + 107*b1 + 716) / 2 $$\nu^{5}$$ $$=$$ $$( 140\beta_{7} - 140\beta_{6} - 627\beta_{5} + 774\beta_{3} + 827\beta_{2} + 827\beta _1 - 1214 ) / 2$$ (140*b7 - 140*b6 - 627*b5 + 774*b3 + 827*b2 + 827*b1 - 1214) / 2 $$\nu^{6}$$ $$=$$ $$220\beta_{7} + 220\beta_{6} - 632\beta_{5} + 1264\beta_{4} - 2596\beta_{2} + 1332\beta _1 + 6663$$ 220*b7 + 220*b6 - 632*b5 + 1264*b4 - 2596*b2 + 1332*b1 + 6663 $$\nu^{7}$$ $$=$$ $$( 3488\beta_{7} - 3488\beta_{6} - 19145\beta_{5} + 14286\beta_{3} + 23167\beta_{2} + 23167\beta _1 - 30310 ) / 2$$ (3488*b7 - 3488*b6 - 19145*b5 + 14286*b3 + 23167*b2 + 23167*b1 - 30310) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/338\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$\chi(n)$$ $$-\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 −3.90972 + 0.500000i −4.71318 + 0.500000i 3.90972 + 0.500000i 4.71318 + 0.500000i −3.90972 − 0.500000i −4.71318 − 0.500000i 3.90972 − 0.500000i 4.71318 − 0.500000i
1.00000 + 1.00000i −4.77574 2.00000i −5.88981 5.88981i −4.77574 4.77574i 0.251956 0.251956i −2.00000 + 2.00000i 13.8077 11.7796i
99.2 1.00000 + 1.00000i −3.84716 2.00000i 3.77418 + 3.77418i −3.84716 3.84716i 7.25532 7.25532i −2.00000 + 2.00000i 5.80063 7.54837i
99.3 1.00000 + 1.00000i 3.04369 2.00000i 4.79174 + 4.79174i 3.04369 + 3.04369i 3.11407 3.11407i −2.00000 + 2.00000i 0.264067 9.58347i
99.4 1.00000 + 1.00000i 5.57921 2.00000i 0.323893 + 0.323893i 5.57921 + 5.57921i −5.62134 + 5.62134i −2.00000 + 2.00000i 22.1276 0.647786i
239.1 1.00000 1.00000i −4.77574 2.00000i −5.88981 + 5.88981i −4.77574 + 4.77574i 0.251956 + 0.251956i −2.00000 2.00000i 13.8077 11.7796i
239.2 1.00000 1.00000i −3.84716 2.00000i 3.77418 3.77418i −3.84716 + 3.84716i 7.25532 + 7.25532i −2.00000 2.00000i 5.80063 7.54837i
239.3 1.00000 1.00000i 3.04369 2.00000i 4.79174 4.79174i 3.04369 3.04369i 3.11407 + 3.11407i −2.00000 2.00000i 0.264067 9.58347i
239.4 1.00000 1.00000i 5.57921 2.00000i 0.323893 0.323893i 5.57921 5.57921i −5.62134 5.62134i −2.00000 2.00000i 22.1276 0.647786i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 99.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.3.d.g 8
13.b even 2 1 338.3.d.f 8
13.c even 3 1 26.3.f.b 8
13.c even 3 1 338.3.f.h 8
13.d odd 4 1 338.3.d.f 8
13.d odd 4 1 inner 338.3.d.g 8
13.e even 6 1 338.3.f.i 8
13.e even 6 1 338.3.f.j 8
13.f odd 12 1 26.3.f.b 8
13.f odd 12 1 338.3.f.h 8
13.f odd 12 1 338.3.f.i 8
13.f odd 12 1 338.3.f.j 8
39.i odd 6 1 234.3.bb.f 8
39.k even 12 1 234.3.bb.f 8
52.j odd 6 1 208.3.bd.f 8
52.l even 12 1 208.3.bd.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.f.b 8 13.c even 3 1
26.3.f.b 8 13.f odd 12 1
208.3.bd.f 8 52.j odd 6 1
208.3.bd.f 8 52.l even 12 1
234.3.bb.f 8 39.i odd 6 1
234.3.bb.f 8 39.k even 12 1
338.3.d.f 8 13.b even 2 1
338.3.d.f 8 13.d odd 4 1
338.3.d.g 8 1.a even 1 1 trivial
338.3.d.g 8 13.d odd 4 1 inner
338.3.f.h 8 13.c even 3 1
338.3.f.h 8 13.f odd 12 1
338.3.f.i 8 13.e even 6 1
338.3.f.i 8 13.f odd 12 1
338.3.f.j 8 13.e even 6 1
338.3.f.j 8 13.f odd 12 1

## Hecke kernels

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

 $$T_{3}^{4} - 39T_{3}^{2} - 12T_{3} + 312$$ T3^4 - 39*T3^2 - 12*T3 + 312 $$T_{5}^{8} - 6T_{5}^{7} + 18T_{5}^{6} - 90T_{5}^{5} + 4245T_{5}^{4} - 30312T_{5}^{3} + 109512T_{5}^{2} - 64584T_{5} + 19044$$ T5^8 - 6*T5^7 + 18*T5^6 - 90*T5^5 + 4245*T5^4 - 30312*T5^3 + 109512*T5^2 - 64584*T5 + 19044 $$T_{7}^{8} - 10T_{7}^{7} + 50T_{7}^{6} + 146T_{7}^{5} + 5017T_{7}^{4} - 38816T_{7}^{3} + 147968T_{7}^{2} - 69632T_{7} + 16384$$ T7^8 - 10*T7^7 + 50*T7^6 + 146*T7^5 + 5017*T7^4 - 38816*T7^3 + 147968*T7^2 - 69632*T7 + 16384

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 2)^{4}$$
$3$ $$(T^{4} - 39 T^{2} - 12 T + 312)^{2}$$
$5$ $$T^{8} - 6 T^{7} + 18 T^{6} + \cdots + 19044$$
$7$ $$T^{8} - 10 T^{7} + 50 T^{6} + \cdots + 16384$$
$11$ $$T^{8} - 42 T^{7} + 882 T^{6} + \cdots + 389376$$
$13$ $$T^{8}$$
$17$ $$T^{8} + 624 T^{6} + 21630 T^{4} + \cdots + 471969$$
$19$ $$T^{8} - 22 T^{7} + \cdots + 1228362304$$
$23$ $$T^{8} + 1266 T^{6} + \cdots + 2508807744$$
$29$ $$(T^{4} - 6 T^{3} - 930 T^{2} + \cdots + 159549)^{2}$$
$31$ $$T^{8} - 32 T^{7} + \cdots + 8111524096$$
$37$ $$T^{8} - 32 T^{7} + \cdots + 321419829721$$
$41$ $$T^{8} + 12 T^{7} + \cdots + 326485389321$$
$43$ $$T^{8} + 4698 T^{6} + \cdots + 325666531584$$
$47$ $$T^{8} - 60 T^{7} + \cdots + 1853819136$$
$53$ $$(T^{4} + 66 T^{3} - 5319 T^{2} + \cdots - 5234376)^{2}$$
$59$ $$T^{8} - 234 T^{7} + \cdots + 70410089309184$$
$61$ $$(T^{4} + 36 T^{3} - 1710 T^{2} + \cdots + 559869)^{2}$$
$67$ $$T^{8} + 14 T^{7} + \cdots + 2456391674944$$
$71$ $$T^{8} - 162 T^{7} + \cdots + 950999436864$$
$73$ $$T^{8} - 166 T^{7} + \cdots + 3554348548804$$
$79$ $$(T^{4} + 48 T^{3} - 14880 T^{2} + \cdots + 2312448)^{2}$$
$83$ $$T^{8} + \cdots + 154848357540864$$
$89$ $$T^{8} - 210 T^{7} + \cdots + 14950765690884$$
$97$ $$T^{8} - 146 T^{7} + \cdots + 9988090235236$$