# Properties

 Label 735.2.d.d Level $735$ Weight $2$ Character orbit 735.d Analytic conductor $5.869$ 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] = [735,2,Mod(589,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("735.589");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 735.d (of order $$2$$, degree $$1$$, 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: $$5.86900454856$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.2058981376.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1$$ x^8 - 2*x^7 + 2*x^6 + 18*x^4 - 34*x^3 + 32*x^2 - 8*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} - \beta_{4} q^{3} + (\beta_{3} + \beta_1 - 1) q^{4} + ( - \beta_{7} + \beta_{5} - \beta_{2}) q^{5} + (\beta_{3} - 1) q^{6} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4} - 2 \beta_{2}) q^{8} - q^{9}+O(q^{10})$$ q + b6 * q^2 - b4 * q^3 + (b3 + b1 - 1) * q^4 + (-b7 + b5 - b2) * q^5 + (b3 - 1) * q^6 + (-b7 + b5 + 2*b4 - 2*b2) * q^8 - q^9 $$q + \beta_{6} q^{2} - \beta_{4} q^{3} + (\beta_{3} + \beta_1 - 1) q^{4} + ( - \beta_{7} + \beta_{5} - \beta_{2}) q^{5} + (\beta_{3} - 1) q^{6} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4} - 2 \beta_{2}) q^{8} - q^{9} + (\beta_{7} + 2 \beta_{4} - \beta_{2} - 2 \beta_1 - 1) q^{10} + ( - \beta_{7} + \beta_{5} - 2 \beta_{3} + \beta_1 + 2) q^{11} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}) q^{12} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{13} + ( - \beta_{5} - \beta_1 - 1) q^{15} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_1 - 2) q^{16} + ( - \beta_{6} + 2 \beta_{5} - 2 \beta_{2}) q^{17} - \beta_{6} q^{18} + (\beta_{3} - \beta_1 + 2) q^{19} + (\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{20} + ( - \beta_{7} + \beta_{6} - 3 \beta_{4} - \beta_{2}) q^{22} + (2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + 2 \beta_{4}) q^{23} + (\beta_{7} - \beta_{5} - 2 \beta_1) q^{24} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{25} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 3 \beta_1) q^{26} + \beta_{4} q^{27} + ( - 2 \beta_{7} + 2 \beta_{5} + 3 \beta_1 + 4) q^{29} + (\beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{2} - \beta_1 + 1) q^{30} + ( - 2 \beta_1 - 3) q^{31} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{32} + ( - \beta_{7} + 2 \beta_{6} + \beta_{4} - \beta_{2}) q^{33} + (2 \beta_{7} - 2 \beta_{5} - \beta_{3} - 3 \beta_1 - 1) q^{34} + ( - \beta_{3} - \beta_1 + 1) q^{36} + (\beta_{7} - 6 \beta_{4} + \beta_{2}) q^{37} + (\beta_{7} + 3 \beta_{6} + \beta_{5} + 4 \beta_{4}) q^{38} + ( - \beta_{3} - 2 \beta_1) q^{39} + (\beta_{7} + \beta_{6} - \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{40} + (\beta_{7} - \beta_{5} + 2 \beta_{3} - 3 \beta_1 - 2) q^{41} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{2}) q^{43} + ( - \beta_{7} + \beta_{5} - 2) q^{44} + (\beta_{7} - \beta_{5} + \beta_{2}) q^{45} + ( - 5 \beta_{3} + \beta_1 + 7) q^{46} + ( - \beta_{7} - \beta_{5} + 2 \beta_{4}) q^{47} + (2 \beta_{7} + 2 \beta_{2}) q^{48} + ( - \beta_{6} + 2 \beta_{5} + 6 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 2) q^{50} + ( - \beta_{3} - 2 \beta_1 - 1) q^{51} + (3 \beta_{7} + \beta_{6} - 2 \beta_{5} - 9 \beta_{4} + 5 \beta_{2}) q^{52} + (2 \beta_{5} + 6 \beta_{4} - 2 \beta_{2}) q^{53} + ( - \beta_{3} + 1) q^{54} + ( - \beta_{7} + 2 \beta_{6} + \beta_{3} - 3 \beta_{2} + \beta_1) q^{55} + ( - \beta_{6} - \beta_{5} - 4 \beta_{4} + \beta_{2}) q^{57} + ( - 3 \beta_{7} + \beta_{6} + 4 \beta_{5} + 5 \beta_{4} - 7 \beta_{2}) q^{58} + \beta_1 q^{59} + ( - \beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{60} + ( - 2 \beta_{7} + 2 \beta_{5} - 5 \beta_{3} + 3 \beta_1 + 3) q^{61} + (2 \beta_{7} - \beta_{6} + 2 \beta_{4} + 2 \beta_{2}) q^{62} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_1 - 2) q^{64} + ( - \beta_{7} + 4 \beta_{6} - \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{65} + (\beta_{7} - \beta_{5} + \beta_{3} - \beta_1 - 5) q^{66} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - 4 \beta_{2}) q^{67} + (3 \beta_{7} - \beta_{5} - 8 \beta_{4} + 4 \beta_{2}) q^{68} + ( - 2 \beta_{7} + 2 \beta_{5} - 3 \beta_{3} + 5) q^{69} + ( - 2 \beta_{7} + 2 \beta_{5} - 4 \beta_{3} + 5 \beta_1 + 2) q^{71} + (\beta_{7} - \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{72} + (\beta_{7} - 4 \beta_{6} - 2 \beta_{5} + 3 \beta_{2}) q^{73} + ( - \beta_{7} + \beta_{5} + 6 \beta_{3} + 3 \beta_1 - 6) q^{74} + (\beta_{7} - 3 \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{75} + (\beta_{3} + 3 \beta_1 - 3) q^{76} + (2 \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{4} + 3 \beta_{2}) q^{78} + (4 \beta_{7} - 4 \beta_{5} + \beta_{3} - 3 \beta_1 - 2) q^{79} + ( - 4 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 4) q^{80} + q^{81} + (3 \beta_{7} + \beta_{6} + 5 \beta_{4} + 3 \beta_{2}) q^{82} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{4} - 2 \beta_{2}) q^{83} + ( - \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + \beta_{2} - 5) q^{85} + ( - 3 \beta_{7} + 3 \beta_{5} - 2 \beta_{3} + 3 \beta_1 + 10) q^{86} + ( - 2 \beta_{7} + \beta_{5} - \beta_{4} - 3 \beta_{2}) q^{87} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{2}) q^{88} + (\beta_{7} - \beta_{5} + 6 \beta_{3} + 3 \beta_1 - 4) q^{89} + ( - \beta_{7} - 2 \beta_{4} + \beta_{2} + 2 \beta_1 + 1) q^{90} + (3 \beta_{7} - \beta_{5} - 12 \beta_{4} + 4 \beta_{2}) q^{92} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{2}) q^{93} + ( - 2 \beta_{3} - 2 \beta_1 + 4) q^{94} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{95} + 2 \beta_1 q^{96} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{2}) q^{97} + (\beta_{7} - \beta_{5} + 2 \beta_{3} - \beta_1 - 2) q^{99}+O(q^{100})$$ q + b6 * q^2 - b4 * q^3 + (b3 + b1 - 1) * q^4 + (-b7 + b5 - b2) * q^5 + (b3 - 1) * q^6 + (-b7 + b5 + 2*b4 - 2*b2) * q^8 - q^9 + (b7 + 2*b4 - b2 - 2*b1 - 1) * q^10 + (-b7 + b5 - 2*b3 + b1 + 2) * q^11 + (-b6 + b5 + b4 - b2) * q^12 + (-b6 + 2*b5 + b4 - 2*b2) * q^13 + (-b5 - b1 - 1) * q^15 + (2*b7 - 2*b5 - 2*b1 - 2) * q^16 + (-b6 + 2*b5 - 2*b2) * q^17 - b6 * q^18 + (b3 - b1 + 2) * q^19 + (b7 + b6 - b5 - 2*b4 - 2*b3 + 2*b2 - b1) * q^20 + (-b7 + b6 - 3*b4 - b2) * q^22 + (2*b7 - 3*b6 + 2*b5 + 2*b4) * q^23 + (b7 - b5 - 2*b1) * q^24 + (b7 + b6 - 2*b5 - b4 + 3*b3 + b2 - b1 - 2) * q^25 + (2*b7 - 2*b5 - 2*b3 - 3*b1) * q^26 + b4 * q^27 + (-2*b7 + 2*b5 + 3*b1 + 4) * q^29 + (b7 - b5 - b4 + 2*b2 - b1 + 1) * q^30 + (-2*b1 - 3) * q^31 + (-2*b5 - 2*b4 + 2*b2) * q^32 + (-b7 + 2*b6 + b4 - b2) * q^33 + (2*b7 - 2*b5 - b3 - 3*b1 - 1) * q^34 + (-b3 - b1 + 1) * q^36 + (b7 - 6*b4 + b2) * q^37 + (b7 + 3*b6 + b5 + 4*b4) * q^38 + (-b3 - 2*b1) * q^39 + (b7 + b6 - b4 + 3*b3 + b2 + b1 - 5) * q^40 + (b7 - b5 + 2*b3 - 3*b1 - 2) * q^41 + (b7 - 2*b6 - 2*b5 + 3*b2) * q^43 + (-b7 + b5 - 2) * q^44 + (b7 - b5 + b2) * q^45 + (-5*b3 + b1 + 7) * q^46 + (-b7 - b5 + 2*b4) * q^47 + (2*b7 + 2*b2) * q^48 + (-b6 + 2*b5 + 6*b4 + 2*b3 + 2*b1 - 2) * q^50 + (-b3 - 2*b1 - 1) * q^51 + (3*b7 + b6 - 2*b5 - 9*b4 + 5*b2) * q^52 + (2*b5 + 6*b4 - 2*b2) * q^53 + (-b3 + 1) * q^54 + (-b7 + 2*b6 + b3 - 3*b2 + b1) * q^55 + (-b6 - b5 - 4*b4 + b2) * q^57 + (-3*b7 + b6 + 4*b5 + 5*b4 - 7*b2) * q^58 + b1 * q^59 + (-b7 + 2*b6 + b4 + b3 + b2 + 2*b1 - 1) * q^60 + (-2*b7 + 2*b5 - 5*b3 + 3*b1 + 3) * q^61 + (2*b7 - b6 + 2*b4 + 2*b2) * q^62 + (2*b7 - 2*b5 + 2*b3 - 2*b1 - 2) * q^64 + (-b7 + 4*b6 - b5 + 4*b4 + 2*b3 + b2 + b1 - 4) * q^65 + (b7 - b5 + b3 - b1 - 5) * q^66 + (-2*b7 - b6 + 2*b5 - b4 - 4*b2) * q^67 + (3*b7 - b5 - 8*b4 + 4*b2) * q^68 + (-2*b7 + 2*b5 - 3*b3 + 5) * q^69 + (-2*b7 + 2*b5 - 4*b3 + 5*b1 + 2) * q^71 + (b7 - b5 - 2*b4 + 2*b2) * q^72 + (b7 - 4*b6 - 2*b5 + 3*b2) * q^73 + (-b7 + b5 + 6*b3 + 3*b1 - 6) * q^74 + (b7 - 3*b6 - 2*b4 + b3 + b2 + b1 - 1) * q^75 + (b3 + 3*b1 - 3) * q^76 + (2*b7 + 2*b6 - b5 - b4 + 3*b2) * q^78 + (4*b7 - 4*b5 + b3 - 3*b1 - 2) * q^79 + (-4*b6 + 2*b5 + 4*b4 - 2*b3 + 2*b1 - 4) * q^80 + q^81 + (3*b7 + b6 + 5*b4 + 3*b2) * q^82 + (-2*b7 - b6 - 2*b4 - 2*b2) * q^83 + (-b7 + 4*b6 - 2*b5 + 4*b4 + 2*b3 + b2 - 5) * q^85 + (-3*b7 + 3*b5 - 2*b3 + 3*b1 + 10) * q^86 + (-2*b7 + b5 - b4 - 3*b2) * q^87 + (-2*b7 + 2*b5 - 2*b4 - 4*b2) * q^88 + (b7 - b5 + 6*b3 + 3*b1 - 4) * q^89 + (-b7 - 2*b4 + b2 + 2*b1 + 1) * q^90 + (3*b7 - b5 - 12*b4 + 4*b2) * q^92 + (-2*b5 + b4 + 2*b2) * q^93 + (-2*b3 - 2*b1 + 4) * q^94 + (-2*b7 - b6 + 2*b5 + 2*b3 - b2 - b1 + 2) * q^95 + 2*b1 * q^96 + (-b7 + b6 + b4 - b2) * q^97 + (b7 - b5 + 2*b3 - b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4} - 2 q^{5} - 4 q^{6} - 8 q^{9}+O(q^{10})$$ 8 * q - 8 * q^4 - 2 * q^5 - 4 * q^6 - 8 * q^9 $$8 q - 8 q^{4} - 2 q^{5} - 4 q^{6} - 8 q^{9} + 4 q^{10} - 2 q^{15} + 24 q^{19} - 4 q^{20} + 12 q^{24} + 4 q^{25} + 12 q^{26} + 12 q^{29} + 12 q^{30} - 16 q^{31} + 8 q^{34} + 8 q^{36} + 4 q^{39} - 32 q^{40} + 8 q^{41} - 20 q^{44} + 2 q^{45} + 32 q^{46} - 20 q^{50} - 4 q^{51} + 4 q^{54} + 4 q^{55} - 4 q^{59} - 16 q^{60} - 16 q^{61} + 8 q^{64} - 30 q^{65} - 28 q^{66} + 20 q^{69} - 28 q^{71} - 40 q^{74} - 8 q^{75} - 32 q^{76} + 16 q^{79} - 52 q^{80} + 8 q^{81} - 32 q^{85} + 48 q^{86} - 16 q^{89} - 4 q^{90} + 32 q^{94} + 22 q^{95} - 8 q^{96}+O(q^{100})$$ 8 * q - 8 * q^4 - 2 * q^5 - 4 * q^6 - 8 * q^9 + 4 * q^10 - 2 * q^15 + 24 * q^19 - 4 * q^20 + 12 * q^24 + 4 * q^25 + 12 * q^26 + 12 * q^29 + 12 * q^30 - 16 * q^31 + 8 * q^34 + 8 * q^36 + 4 * q^39 - 32 * q^40 + 8 * q^41 - 20 * q^44 + 2 * q^45 + 32 * q^46 - 20 * q^50 - 4 * q^51 + 4 * q^54 + 4 * q^55 - 4 * q^59 - 16 * q^60 - 16 * q^61 + 8 * q^64 - 30 * q^65 - 28 * q^66 + 20 * q^69 - 28 * q^71 - 40 * q^74 - 8 * q^75 - 32 * q^76 + 16 * q^79 - 52 * q^80 + 8 * q^81 - 32 * q^85 + 48 * q^86 - 16 * q^89 - 4 * q^90 + 32 * q^94 + 22 * q^95 - 8 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( -97\nu^{7} + 102\nu^{6} - 30\nu^{5} - 432\nu^{4} - 1769\nu^{3} + 1637\nu^{2} - 408\nu - 4773 ) / 1631$$ (-97*v^7 + 102*v^6 - 30*v^5 - 432*v^4 - 1769*v^3 + 1637*v^2 - 408*v - 4773) / 1631 $$\beta_{2}$$ $$=$$ $$( 145\nu^{7} + 167\nu^{6} - 241\nu^{5} + 444\nu^{4} + 3132\nu^{3} + 2984\nu^{2} - 3930\nu + 4226 ) / 1631$$ (145*v^7 + 167*v^6 - 241*v^5 + 444*v^4 + 3132*v^3 + 2984*v^2 - 3930*v + 4226) / 1631 $$\beta_{3}$$ $$=$$ $$( 485\nu^{7} - 510\nu^{6} + 150\nu^{5} + 529\nu^{4} + 8845\nu^{3} - 8185\nu^{2} + 2040\nu + 4293 ) / 1631$$ (485*v^7 - 510*v^6 + 150*v^5 + 529*v^4 + 8845*v^3 - 8185*v^2 + 2040*v + 4293) / 1631 $$\beta_{4}$$ $$=$$ $$( -552\nu^{7} + 984\nu^{6} - 961\nu^{5} - 138\nu^{4} - 9966\nu^{3} + 16176\nu^{2} - 15353\nu + 2213 ) / 1631$$ (-552*v^7 + 984*v^6 - 961*v^5 - 138*v^4 - 9966*v^3 + 16176*v^2 - 15353*v + 2213) / 1631 $$\beta_{5}$$ $$=$$ $$( -137\nu^{7} + 305\nu^{6} - 309\nu^{5} + 24\nu^{4} - 2400\nu^{3} + 5281\nu^{2} - 4948\nu + 1236 ) / 233$$ (-137*v^7 + 305*v^6 - 309*v^5 + 24*v^4 - 2400*v^3 + 5281*v^2 - 4948*v + 1236) / 233 $$\beta_{6}$$ $$=$$ $$( -1084\nu^{7} + 2216\nu^{6} - 1899\nu^{5} - 271\nu^{4} - 19500\nu^{3} + 38219\nu^{2} - 30067\nu + 4334 ) / 1631$$ (-1084*v^7 + 2216*v^6 - 1899*v^5 - 271*v^4 - 19500*v^3 + 38219*v^2 - 30067*v + 4334) / 1631 $$\beta_{7}$$ $$=$$ $$( -1661\nu^{7} + 2890\nu^{6} - 2481\nu^{5} - 823\nu^{4} - 30006\nu^{3} + 49100\nu^{2} - 37656\nu + 1769 ) / 1631$$ (-1661*v^7 + 2890*v^6 - 2481*v^5 - 823*v^4 - 30006*v^3 + 49100*v^2 - 37656*v + 1769) / 1631
 $$\nu$$ $$=$$ $$( 2\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 1 ) / 2$$ (2*b7 - b6 - b5 - b4 + b3 + b2 - b1 - 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} - 2\beta_{4} - \beta_{2}$$ b5 - 2*b4 - b2 $$\nu^{3}$$ $$=$$ $$( -5\beta_{6} + 5\beta_{5} - 3\beta_{4} - 5\beta_{3} + 3\beta_{2} + 3\beta _1 + 5 ) / 2$$ (-5*b6 + 5*b5 - 3*b4 - 5*b3 + 3*b2 + 3*b1 + 5) / 2 $$\nu^{4}$$ $$=$$ $$-\beta_{3} - 5\beta _1 - 12$$ -b3 - 5*b1 - 12 $$\nu^{5}$$ $$=$$ $$( -34\beta_{7} + 23\beta_{6} + 11\beta_{5} + 13\beta_{4} - 23\beta_{3} - 11\beta_{2} + 11\beta _1 + 21 ) / 2$$ (-34*b7 + 23*b6 + 11*b5 + 13*b4 - 23*b3 - 11*b2 + 11*b1 + 21) / 2 $$\nu^{6}$$ $$=$$ $$-\beta_{7} + 7\beta_{6} - 23\beta_{5} + 35\beta_{4} + 22\beta_{2}$$ -b7 + 7*b6 - 23*b5 + 35*b4 + 22*b2 $$\nu^{7}$$ $$=$$ $$( 103\beta_{6} - 105\beta_{5} + 61\beta_{4} + 103\beta_{3} - 43\beta_{2} - 43\beta _1 - 85 ) / 2$$ (103*b6 - 105*b5 + 61*b4 + 103*b3 - 43*b2 - 43*b1 - 85) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/735\mathbb{Z}\right)^\times$$.

 $$n$$ $$346$$ $$442$$ $$491$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
589.1
 0.769222 − 0.769222i 0.148421 + 0.148421i −1.43917 + 1.43917i 1.52153 + 1.52153i 1.52153 − 1.52153i −1.43917 − 1.43917i 0.148421 − 0.148421i 0.769222 + 0.769222i
2.51658i 1.00000i −4.33317 −1.11922 + 1.93581i −2.51658 0 5.87162i −1.00000 4.87162 + 2.81659i
589.2 1.78165i 1.00000i −1.17429 2.22038 + 0.264435i 1.78165 0 1.47113i −1.00000 0.471131 3.95594i
589.3 1.55241i 1.00000i −0.409975 0.0917505 2.23418i −1.55241 0 2.46837i −1.00000 −3.46837 0.142434i
589.4 0.287336i 1.00000i 1.91744 −2.19291 + 0.437190i 0.287336 0 1.12562i −1.00000 0.125620 + 0.630102i
589.5 0.287336i 1.00000i 1.91744 −2.19291 0.437190i 0.287336 0 1.12562i −1.00000 0.125620 0.630102i
589.6 1.55241i 1.00000i −0.409975 0.0917505 + 2.23418i −1.55241 0 2.46837i −1.00000 −3.46837 + 0.142434i
589.7 1.78165i 1.00000i −1.17429 2.22038 0.264435i 1.78165 0 1.47113i −1.00000 0.471131 + 3.95594i
589.8 2.51658i 1.00000i −4.33317 −1.11922 1.93581i −2.51658 0 5.87162i −1.00000 4.87162 2.81659i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 589.8 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.d.d 8
3.b odd 2 1 2205.2.d.s 8
5.b even 2 1 inner 735.2.d.d 8
5.c odd 4 1 3675.2.a.bp 4
5.c odd 4 1 3675.2.a.bz 4
7.b odd 2 1 735.2.d.e 8
7.c even 3 2 105.2.q.a 16
7.d odd 6 2 735.2.q.g 16
15.d odd 2 1 2205.2.d.s 8
21.c even 2 1 2205.2.d.o 8
21.h odd 6 2 315.2.bf.b 16
28.g odd 6 2 1680.2.di.d 16
35.c odd 2 1 735.2.d.e 8
35.f even 4 1 3675.2.a.bn 4
35.f even 4 1 3675.2.a.cb 4
35.i odd 6 2 735.2.q.g 16
35.j even 6 2 105.2.q.a 16
35.l odd 12 2 525.2.i.h 8
35.l odd 12 2 525.2.i.k 8
105.g even 2 1 2205.2.d.o 8
105.o odd 6 2 315.2.bf.b 16
140.p odd 6 2 1680.2.di.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.q.a 16 7.c even 3 2
105.2.q.a 16 35.j even 6 2
315.2.bf.b 16 21.h odd 6 2
315.2.bf.b 16 105.o odd 6 2
525.2.i.h 8 35.l odd 12 2
525.2.i.k 8 35.l odd 12 2
735.2.d.d 8 1.a even 1 1 trivial
735.2.d.d 8 5.b even 2 1 inner
735.2.d.e 8 7.b odd 2 1
735.2.d.e 8 35.c odd 2 1
735.2.q.g 16 7.d odd 6 2
735.2.q.g 16 35.i odd 6 2
1680.2.di.d 16 28.g odd 6 2
1680.2.di.d 16 140.p odd 6 2
2205.2.d.o 8 21.c even 2 1
2205.2.d.o 8 105.g even 2 1
2205.2.d.s 8 3.b odd 2 1
2205.2.d.s 8 15.d odd 2 1
3675.2.a.bn 4 35.f even 4 1
3675.2.a.bp 4 5.c odd 4 1
3675.2.a.bz 4 5.c odd 4 1
3675.2.a.cb 4 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(735, [\chi])$$:

 $$T_{2}^{8} + 12T_{2}^{6} + 44T_{2}^{4} + 52T_{2}^{2} + 4$$ T2^8 + 12*T2^6 + 44*T2^4 + 52*T2^2 + 4 $$T_{19}^{4} - 12T_{19}^{3} + 38T_{19}^{2} - 40T_{19} + 9$$ T19^4 - 12*T19^3 + 38*T19^2 - 40*T19 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 12 T^{6} + 44 T^{4} + 52 T^{2} + \cdots + 4$$
$3$ $$(T^{2} + 1)^{4}$$
$5$ $$T^{8} + 2 T^{7} - 10 T^{5} - 42 T^{4} + \cdots + 625$$
$7$ $$T^{8}$$
$11$ $$(T^{4} - 18 T^{2} - 14 T + 30)^{2}$$
$13$ $$T^{8} + 60 T^{6} + 1182 T^{4} + \cdots + 16129$$
$17$ $$T^{8} + 60 T^{6} + 1140 T^{4} + \cdots + 17956$$
$19$ $$(T^{4} - 12 T^{3} + 38 T^{2} - 40 T + 9)^{2}$$
$23$ $$T^{8} + 164 T^{6} + 9732 T^{4} + \cdots + 2268036$$
$29$ $$(T^{4} - 6 T^{3} - 38 T^{2} + 190 T - 22)^{2}$$
$31$ $$(T^{4} + 8 T^{3} - 6 T^{2} - 96 T + 61)^{2}$$
$37$ $$T^{8} + 168 T^{6} + 8774 T^{4} + \cdots + 822649$$
$41$ $$(T^{4} - 4 T^{3} - 50 T^{2} + 146 T - 10)^{2}$$
$43$ $$T^{8} + 128 T^{6} + 3390 T^{4} + \cdots + 2401$$
$47$ $$T^{8} + 68 T^{6} + 1296 T^{4} + \cdots + 14400$$
$53$ $$T^{8} + 160 T^{6} + 6272 T^{4} + \cdots + 9216$$
$59$ $$(T^{4} + 2 T^{3} - 6 T^{2} - 6 T + 10)^{2}$$
$61$ $$(T^{4} + 8 T^{3} - 100 T^{2} - 700 T + 500)^{2}$$
$67$ $$T^{8} + 180 T^{6} + 9902 T^{4} + \cdots + 819025$$
$71$ $$(T^{4} + 14 T^{3} - 90 T^{2} - 1334 T - 3202)^{2}$$
$73$ $$T^{8} + 272 T^{6} + 20374 T^{4} + \cdots + 1929321$$
$79$ $$(T^{4} - 8 T^{3} - 154 T^{2} + 684 T + 7081)^{2}$$
$83$ $$T^{8} + 148 T^{6} + 6716 T^{4} + \cdots + 131044$$
$89$ $$(T^{4} + 8 T^{3} - 194 T^{2} - 986 T - 534)^{2}$$
$97$ $$T^{8} + 20 T^{6} + 120 T^{4} + \cdots + 16$$