# Properties

 Label 1014.3.f.j Level $1014$ Weight $3$ Character orbit 1014.f Analytic conductor $27.629$ 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] = [1014,3,Mod(577,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

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

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

chi := DirichletCharacter("1014.577");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1014.f (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: $$27.6294988061$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 82x^{5} + 5053x^{4} - 6736x^{3} + 6728x^{2} + 275384x + 5635876$$ x^8 - 2*x^7 + 2*x^6 + 82*x^5 + 5053*x^4 - 6736*x^3 + 6728*x^2 + 275384*x + 5635876 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 78) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{2} + (\beta_{6} - \beta_{3}) q^{3} + 2 \beta_{2} q^{4} + (\beta_{3} + \beta_{2} + \beta_1) q^{5} + ( - 2 \beta_{3} - \beta_{2} + 1) q^{6} + ( - 2 \beta_{6} - \beta_{5} - \beta_{2} + 1) q^{7} + (2 \beta_{2} - 2) q^{8} + 3 q^{9}+O(q^{10})$$ q + (b2 + 1) * q^2 + (b6 - b3) * q^3 + 2*b2 * q^4 + (b3 + b2 + b1) * q^5 + (-2*b3 - b2 + 1) * q^6 + (-2*b6 - b5 - b2 + 1) * q^7 + (2*b2 - 2) * q^8 + 3 * q^9 $$q + (\beta_{2} + 1) q^{2} + (\beta_{6} - \beta_{3}) q^{3} + 2 \beta_{2} q^{4} + (\beta_{3} + \beta_{2} + \beta_1) q^{5} + ( - 2 \beta_{3} - \beta_{2} + 1) q^{6} + ( - 2 \beta_{6} - \beta_{5} - \beta_{2} + 1) q^{7} + (2 \beta_{2} - 2) q^{8} + 3 q^{9} + (\beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{10} + (4 \beta_{6} + 2 \beta_{5}) q^{11} + ( - 2 \beta_{6} - 2 \beta_{3} - 2 \beta_{2} + 2) q^{12} + ( - 2 \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_1) q^{14} + ( - \beta_{7} - \beta_{6} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{15} - 4 q^{16} + ( - 6 \beta_{6} + \beta_{5} + \beta_{4} - 7 \beta_{3} + \beta_{2} + \beta_1 + 6) q^{17} + (3 \beta_{2} + 3) q^{18} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + 6 \beta_{2} - 2 \beta_1 + 6) q^{19} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_{2} - 2) q^{20} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} - 3) q^{21} + (4 \beta_{6} + 2 \beta_{5} - 4 \beta_{3} - 2 \beta_1 + 4) q^{22} + (2 \beta_{6} + 2 \beta_{4} + 20 \beta_{2} - 2) q^{23} + ( - 4 \beta_{6} - 2 \beta_{2} + 2) q^{24} + (5 \beta_{6} + 2 \beta_{5} + \beta_{4} + 4 \beta_{3} + 30 \beta_{2} + 2 \beta_1 - 5) q^{25} + (3 \beta_{6} - 3 \beta_{3}) q^{27} + (4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{28} + (3 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 5 \beta_{3} + 2 \beta_1 - 10) q^{29} + ( - 3 \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} - 4 \beta_{2} - \beta_1 + 3) q^{30} + ( - \beta_{7} - \beta_{6} - \beta_{4} - 17 \beta_{3} + 3 \beta_{2} - \beta_1 + 21) q^{31} + ( - 4 \beta_{2} - 4) q^{32} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + \cdots + 4) q^{33}+ \cdots + (12 \beta_{6} + 6 \beta_{5}) q^{99}+O(q^{100})$$ q + (b2 + 1) * q^2 + (b6 - b3) * q^3 + 2*b2 * q^4 + (b3 + b2 + b1) * q^5 + (-2*b3 - b2 + 1) * q^6 + (-2*b6 - b5 - b2 + 1) * q^7 + (2*b2 - 2) * q^8 + 3 * q^9 + (b6 + b5 + b3 + 2*b2 + b1 - 1) * q^10 + (4*b6 + 2*b5) * q^11 + (-2*b6 - 2*b3 - 2*b2 + 2) * q^12 + (-2*b6 - b5 + 2*b3 + b1) * q^14 + (-b7 - b6 - b4 - 2*b3 - 2*b2 - b1 + 1) * q^15 - 4 * q^16 + (-6*b6 + b5 + b4 - 7*b3 + b2 + b1 + 6) * q^17 + (3*b2 + 3) * q^18 + (-b7 - b6 - b4 + b3 + 6*b2 - 2*b1 + 6) * q^19 + (2*b6 + 2*b5 + 2*b2 - 2) * q^20 + (-b7 + b6 + b5 + b4 - b3 + 3*b2 - 3) * q^21 + (4*b6 + 2*b5 - 4*b3 - 2*b1 + 4) * q^22 + (2*b6 + 2*b4 + 20*b2 - 2) * q^23 + (-4*b6 - 2*b2 + 2) * q^24 + (5*b6 + 2*b5 + b4 + 4*b3 + 30*b2 + 2*b1 - 5) * q^25 + (3*b6 - 3*b3) * q^27 + (4*b3 + 2*b2 + 2*b1 - 2) * q^28 + (3*b7 - 2*b6 - 2*b5 + 5*b3 + 2*b1 - 10) * q^29 + (-3*b6 - b5 - 2*b4 - b3 - 4*b2 - b1 + 3) * q^30 + (-b7 - b6 - b4 - 17*b3 + 3*b2 - b1 + 21) * q^31 + (-4*b2 - 4) * q^32 + (2*b7 + 2*b6 - 2*b5 - 2*b4 + 2*b3 - 4*b2 + 4) * q^33 + (-b7 - 13*b6 + 2*b5 + b4 - b3 + b2 - 1) * q^34 + (2*b7 - 4*b6 - 2*b5 + 6*b3 + 2*b1 + 50) * q^35 + 6*b2 * q^36 + (17*b6 - 5*b5 + 4*b2 - 4) * q^37 + (-2*b5 - 2*b4 + 2*b3 + 12*b2 - 2*b1) * q^38 + (2*b6 + 2*b5 - 2*b3 - 2*b1 - 2) * q^40 + (3*b7 + 3*b6 + 3*b4 + 10*b3 + 10*b2 + b1 - 3) * q^41 + (-2*b7 + b5 - 2*b3 - b1 - 4) * q^42 + (-20*b6 - 2*b5 - b4 - 19*b3 + 2*b2 - 2*b1 + 20) * q^43 + (-8*b3 - 4*b1 + 8) * q^44 + (3*b3 + 3*b2 + 3*b1) * q^45 + (-2*b7 + 2*b6 + 2*b4 - 2*b3 + 20*b2 - 20) * q^46 + (2*b7 + 6*b6 - 2*b5 - 2*b4 + 2*b3 + 40*b2 - 40) * q^47 + (-4*b6 + 4*b3) * q^48 + (-6*b6 - 2*b5 - 3*b4 - 3*b3 - 9*b2 - 2*b1 + 6) * q^49 + (-b7 + 9*b6 + 4*b5 + b4 - b3 + 30*b2 - 30) * q^50 + (-9*b6 - 2*b5 - b4 - 8*b3 + 13*b2 - 2*b1 + 9) * q^51 + (-3*b7 + 15*b6 + b5 - 18*b3 - b1 + 11) * q^53 + (-6*b3 - 3*b2 + 3) * q^54 + (-4*b7 + 6*b6 + 2*b5 - 10*b3 - 2*b1 - 98) * q^55 + (4*b6 + 2*b5 + 4*b3 + 4*b2 + 2*b1 - 4) * q^56 + (b7 + b6 + b4 - 9*b3 - 8*b2 + 4*b1) * q^57 + (3*b7 + 3*b6 + 3*b4 + 7*b3 - 5*b2 + 4*b1 - 15) * q^58 + (-4*b7 + 12*b6 + 2*b5 + 4*b4 - 4*b3 + 6*b2 - 6) * q^59 + (2*b7 - 4*b6 - 2*b5 - 2*b4 + 2*b3 - 4*b2 + 4) * q^60 + (-4*b7 - 22*b6 - 3*b5 + 18*b3 + 3*b1 + 21) * q^61 + (-18*b6 - b5 - 2*b4 - 16*b3 + 6*b2 - b1 + 18) * q^62 + (-6*b6 - 3*b5 - 3*b2 + 3) * q^63 - 8*b2 * q^64 + (4*b7 + 4*b6 - 2*b5 + 2*b1 + 8) * q^66 + (-b7 - b6 - b4 + 43*b3 + 31*b2 - 3*b1 - 11) * q^67 + (-2*b7 - 14*b6 + 2*b5 + 12*b3 - 2*b1 - 14) * q^68 + (-18*b6 - 2*b5 + 2*b4 - 20*b3 - 20*b2 - 2*b1 + 18) * q^69 + (2*b7 + 2*b6 + 2*b4 + 10*b3 + 56*b2 + 4*b1 + 44) * q^70 + (4*b7 + 4*b6 + 4*b4 + 20*b3 - 8*b2 + 2*b1 - 32) * q^71 + (6*b2 - 6) * q^72 + (b7 + 24*b6 - 6*b5 - b4 + b3 + 15*b2 - 15) * q^73 + (17*b6 - 5*b5 - 17*b3 + 5*b1 + 9) * q^74 + (-29*b6 - 3*b5 - 3*b4 - 26*b3 - 38*b2 - 3*b1 + 29) * q^75 + (2*b7 + 2*b6 - 4*b5 - 2*b4 + 2*b3 + 12*b2 - 12) * q^76 + (8*b6 + 2*b5 + 6*b4 + 2*b3 + 112*b2 + 2*b1 - 8) * q^77 + (5*b7 + 39*b6 - 34*b3 - 12) * q^79 + (-4*b3 - 4*b2 - 4*b1) * q^80 + 9 * q^81 + (13*b6 + b5 + 6*b4 + 7*b3 + 20*b2 + b1 - 13) * q^82 + (2*b7 + 2*b6 + 2*b4 + 30*b3 + 18*b2 - 8*b1 - 14) * q^83 + (-2*b7 - 2*b6 - 2*b4 - 2*b3 - 6*b2 - 2*b1 - 2) * q^84 + (6*b7 + 47*b6 + 3*b5 - 6*b4 + 6*b3 + 49*b2 - 49) * q^85 + (b7 - 39*b6 - 4*b5 - b4 + b3 + 2*b2 - 2) * q^86 + (-b7 - 7*b6 + 5*b5 + 6*b3 - 5*b1 - 5) * q^87 + (-8*b6 - 4*b5 - 8*b3 - 4*b1 + 8) * q^88 + (b7 + 77*b6 - 2*b5 - b4 + b3 + 67*b2 - 67) * q^89 + (3*b6 + 3*b5 + 3*b3 + 6*b2 + 3*b1 - 3) * q^90 + (-4*b7 - 4*b3 - 36) * q^92 + (-22*b3 + 13*b2 + 3*b1 + 35) * q^93 + (4*b7 + 8*b6 - 2*b5 - 4*b3 + 2*b1 - 76) * q^94 + (-46*b6 + 4*b5 - 46*b3 - 96*b2 + 4*b1 + 46) * q^95 + (8*b3 + 4*b2 - 4) * q^96 + (-4*b7 - 4*b6 - 4*b4 + 74*b3 + 36*b2 - 5*b1 - 34) * q^97 + (3*b7 - 9*b6 - 4*b5 - 3*b4 + 3*b3 - 9*b2 + 9) * q^98 + (12*b6 + 6*b5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{2} + 6 q^{5} + 2 q^{7} - 16 q^{8} + 24 q^{9}+O(q^{10})$$ 8 * q + 8 * q^2 + 6 * q^5 + 2 * q^7 - 16 * q^8 + 24 * q^9 $$8 q + 8 q^{2} + 6 q^{5} + 2 q^{7} - 16 q^{8} + 24 q^{9} + 12 q^{11} + 4 q^{14} - 6 q^{15} - 32 q^{16} + 24 q^{18} + 44 q^{19} - 12 q^{20} - 18 q^{21} + 24 q^{22} + 4 q^{28} - 72 q^{29} + 94 q^{31} - 32 q^{32} + 36 q^{33} - 60 q^{34} + 408 q^{35} + 46 q^{37} - 24 q^{40} + 30 q^{41} - 36 q^{42} + 24 q^{44} + 18 q^{45} - 144 q^{46} - 300 q^{47} - 208 q^{50} + 84 q^{53} - 792 q^{55} - 24 q^{57} - 72 q^{58} + 12 q^{59} + 12 q^{60} + 180 q^{61} + 6 q^{63} + 72 q^{66} + 74 q^{67} - 120 q^{68} + 408 q^{70} - 156 q^{71} - 48 q^{72} - 16 q^{73} + 92 q^{74} - 88 q^{76} - 96 q^{79} - 24 q^{80} + 72 q^{81} - 36 q^{84} - 234 q^{85} - 168 q^{86} - 60 q^{87} - 228 q^{89} - 288 q^{92} + 198 q^{93} - 600 q^{94} - 2 q^{97} + 32 q^{98} + 36 q^{99}+O(q^{100})$$ 8 * q + 8 * q^2 + 6 * q^5 + 2 * q^7 - 16 * q^8 + 24 * q^9 + 12 * q^11 + 4 * q^14 - 6 * q^15 - 32 * q^16 + 24 * q^18 + 44 * q^19 - 12 * q^20 - 18 * q^21 + 24 * q^22 + 4 * q^28 - 72 * q^29 + 94 * q^31 - 32 * q^32 + 36 * q^33 - 60 * q^34 + 408 * q^35 + 46 * q^37 - 24 * q^40 + 30 * q^41 - 36 * q^42 + 24 * q^44 + 18 * q^45 - 144 * q^46 - 300 * q^47 - 208 * q^50 + 84 * q^53 - 792 * q^55 - 24 * q^57 - 72 * q^58 + 12 * q^59 + 12 * q^60 + 180 * q^61 + 6 * q^63 + 72 * q^66 + 74 * q^67 - 120 * q^68 + 408 * q^70 - 156 * q^71 - 48 * q^72 - 16 * q^73 + 92 * q^74 - 88 * q^76 - 96 * q^79 - 24 * q^80 + 72 * q^81 - 36 * q^84 - 234 * q^85 - 168 * q^86 - 60 * q^87 - 228 * q^89 - 288 * q^92 + 198 * q^93 - 600 * q^94 - 2 * q^97 + 32 * q^98 + 36 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 2x^{6} + 82x^{5} + 5053x^{4} - 6736x^{3} + 6728x^{2} + 275384x + 5635876$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 4081829 \nu^{7} + 125878448 \nu^{6} - 175318185 \nu^{5} - 167602931 \nu^{4} - 13767350850 \nu^{3} + 914252484693 \nu^{2} + \cdots - 561922481148 ) / 28390156409030$$ (-4081829*v^7 + 125878448*v^6 - 175318185*v^5 - 167602931*v^4 - 13767350850*v^3 + 914252484693*v^2 - 589968133288*v - 561922481148) / 28390156409030 $$\beta_{3}$$ $$=$$ $$( - 207487050 \nu^{7} + 9646046383 \nu^{6} + 263049796262 \nu^{5} + 452388400182 \nu^{4} - 699596394446 \nu^{3} + \cdots + 20\!\cdots\!94 ) / 965265317907020$$ (-207487050*v^7 + 9646046383*v^6 + 263049796262*v^5 + 452388400182*v^4 - 699596394446*v^3 + 25228071565329*v^2 + 744039515950798*v + 2064275292409594) / 965265317907020 $$\beta_{4}$$ $$=$$ $$( 186368592 \nu^{7} + 13656504841 \nu^{6} + 24877044110 \nu^{5} + 30851132578 \nu^{4} + 372305475560 \nu^{3} + \cdots + 132247184234574 ) / 50803437784580$$ (186368592*v^7 + 13656504841*v^6 + 24877044110*v^5 + 30851132578*v^4 + 372305475560*v^3 + 38906811708291*v^2 + 61427229206294*v + 132247184234574) / 50803437784580 $$\beta_{5}$$ $$=$$ $$( 99170 \nu^{7} - 140821 \nu^{6} + 140781 \nu^{5} + 5777701 \nu^{4} + 747057527 \nu^{3} - 473888448 \nu^{2} + 473587124 \nu + 19380524092 ) / 23917570690$$ (99170*v^7 - 140821*v^6 + 140781*v^5 + 5777701*v^4 + 747057527*v^3 - 473888448*v^2 + 473587124*v + 19380524092) / 23917570690 $$\beta_{6}$$ $$=$$ $$( 5690069035 \nu^{7} + 3991178383 \nu^{6} - 3261499808 \nu^{5} - 237335849778 \nu^{4} + 16756634760889 \nu^{3} + \cdots - 378059897144086 ) / 965265317907020$$ (5690069035*v^7 + 3991178383*v^6 - 3261499808*v^5 - 237335849778*v^4 + 16756634760889*v^3 + 6206066365669*v^2 - 10968766639812*v - 378059897144086) / 965265317907020 $$\beta_{7}$$ $$=$$ $$( - 15163829403 \nu^{7} + 4995591495 \nu^{6} + 440871714386 \nu^{5} + 11542895672784 \nu^{4} - 43834774990983 \nu^{3} + \cdots + 30\!\cdots\!86 ) / 965265317907020$$ (-15163829403*v^7 + 4995591495*v^6 + 440871714386*v^5 + 11542895672784*v^4 - 43834774990983*v^3 + 24023479541963*v^2 + 1200974352327728*v + 30969513538197086) / 965265317907020
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{6} - \beta_{4} + 3\beta_{3} + 52\beta_{2} - 2$$ 2*b6 - b4 + 3*b3 + 52*b2 - 2 $$\nu^{3}$$ $$=$$ $$-2\beta_{7} - 44\beta_{6} + 53\beta_{5} + 2\beta_{4} - 2\beta_{3} - 3\beta_{2} + 3$$ -2*b7 - 44*b6 + 53*b5 + 2*b4 - 2*b3 - 3*b2 + 3 $$\nu^{4}$$ $$=$$ $$95\beta_{7} + 251\beta_{6} - 5\beta_{5} - 156\beta_{3} + 5\beta _1 - 2612$$ 95*b7 + 251*b6 - 5*b5 - 156*b3 + 5*b1 - 2612 $$\nu^{5}$$ $$=$$ $$-161\beta_{7} - 161\beta_{6} - 161\beta_{4} + 4239\beta_{3} + 545\beta_{2} - 2863\beta _1 - 3533$$ -161*b7 - 161*b6 - 161*b4 + 4239*b3 + 545*b2 - 2863*b1 - 3533 $$\nu^{6}$$ $$=$$ $$-9054\beta_{6} + 706\beta_{5} + 6941\beta_{4} - 15995\beta_{3} - 149842\beta_{2} + 706\beta _1 + 9054$$ -9054*b6 + 706*b5 + 6941*b4 - 15995*b3 - 149842*b2 + 706*b1 + 9054 $$\nu^{7}$$ $$=$$ $$9760\beta_{7} + 313762\beta_{6} - 156783\beta_{5} - 9760\beta_{4} + 9760\beta_{3} + 57535\beta_{2} - 57535$$ 9760*b7 + 313762*b6 - 156783*b5 - 9760*b4 + 9760*b3 + 57535*b2 - 57535

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 −5.39181 + 5.39181i 5.02578 − 5.02578i −4.04651 + 4.04651i 5.41254 − 5.41254i −5.39181 − 5.39181i 5.02578 + 5.02578i −4.04651 − 4.04651i 5.41254 + 5.41254i
1.00000 1.00000i −1.73205 2.00000i −4.02578 + 4.02578i −1.73205 + 1.73205i −3.65976 3.65976i −2.00000 2.00000i 3.00000 8.05157i
577.2 1.00000 1.00000i −1.73205 2.00000i 6.39181 6.39181i −1.73205 + 1.73205i 6.75784 + 6.75784i −2.00000 2.00000i 3.00000 12.7836i
577.3 1.00000 1.00000i 1.73205 2.00000i −4.41254 + 4.41254i 1.73205 1.73205i −5.77857 5.77857i −2.00000 2.00000i 3.00000 8.82508i
577.4 1.00000 1.00000i 1.73205 2.00000i 5.04651 5.04651i 1.73205 1.73205i 3.68049 + 3.68049i −2.00000 2.00000i 3.00000 10.0930i
775.1 1.00000 + 1.00000i −1.73205 2.00000i −4.02578 4.02578i −1.73205 1.73205i −3.65976 + 3.65976i −2.00000 + 2.00000i 3.00000 8.05157i
775.2 1.00000 + 1.00000i −1.73205 2.00000i 6.39181 + 6.39181i −1.73205 1.73205i 6.75784 6.75784i −2.00000 + 2.00000i 3.00000 12.7836i
775.3 1.00000 + 1.00000i 1.73205 2.00000i −4.41254 4.41254i 1.73205 + 1.73205i −5.77857 + 5.77857i −2.00000 + 2.00000i 3.00000 8.82508i
775.4 1.00000 + 1.00000i 1.73205 2.00000i 5.04651 + 5.04651i 1.73205 + 1.73205i 3.68049 3.68049i −2.00000 + 2.00000i 3.00000 10.0930i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 775.4 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 1014.3.f.j 8
13.b even 2 1 1014.3.f.h 8
13.d odd 4 1 1014.3.f.h 8
13.d odd 4 1 inner 1014.3.f.j 8
13.e even 6 1 78.3.l.c 8
13.f odd 12 1 78.3.l.c 8
39.h odd 6 1 234.3.bb.d 8
39.k even 12 1 234.3.bb.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.3.l.c 8 13.e even 6 1
78.3.l.c 8 13.f odd 12 1
234.3.bb.d 8 39.h odd 6 1
234.3.bb.d 8 39.k even 12 1
1014.3.f.h 8 13.b even 2 1
1014.3.f.h 8 13.d odd 4 1
1014.3.f.j 8 1.a even 1 1 trivial
1014.3.f.j 8 13.d odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1014, [\chi])$$:

 $$T_{5}^{8} - 6T_{5}^{7} + 18T_{5}^{6} + 282T_{5}^{5} + 4065T_{5}^{4} - 11916T_{5}^{3} + 38088T_{5}^{2} + 632592T_{5} + 5253264$$ T5^8 - 6*T5^7 + 18*T5^6 + 282*T5^5 + 4065*T5^4 - 11916*T5^3 + 38088*T5^2 + 632592*T5 + 5253264 $$T_{7}^{8} - 2T_{7}^{7} + 2T_{7}^{6} + 154T_{7}^{5} + 6817T_{7}^{4} - 1672T_{7}^{3} + 1568T_{7}^{2} + 117824T_{7} + 4426816$$ T7^8 - 2*T7^7 + 2*T7^6 + 154*T7^5 + 6817*T7^4 - 1672*T7^3 + 1568*T7^2 + 117824*T7 + 4426816

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 2)^{4}$$
$3$ $$(T^{2} - 3)^{4}$$
$5$ $$T^{8} - 6 T^{7} + 18 T^{6} + \cdots + 5253264$$
$7$ $$T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 4426816$$
$11$ $$T^{8} - 12 T^{7} + \cdots + 973440000$$
$13$ $$T^{8}$$
$17$ $$T^{8} + 1242 T^{6} + \cdots + 4567597056$$
$19$ $$T^{8} - 44 T^{7} + 968 T^{6} + \cdots + 1763584$$
$23$ $$T^{8} + 2760 T^{6} + \cdots + 17831863296$$
$29$ $$(T^{4} + 36 T^{3} - 987 T^{2} + \cdots + 220452)^{2}$$
$31$ $$T^{8} - 94 T^{7} + \cdots + 33544655104$$
$37$ $$T^{8} - 46 T^{7} + \cdots + 10744151716$$
$41$ $$T^{8} - 30 T^{7} + \cdots + 370617958656$$
$43$ $$T^{8} + 7158 T^{6} + \cdots + 1819196698176$$
$47$ $$T^{8} + 300 T^{7} + \cdots + 20494380893184$$
$53$ $$(T^{4} - 42 T^{3} - 2211 T^{2} + \cdots - 109128)^{2}$$
$59$ $$T^{8} - 12 T^{7} + \cdots + 15611728564224$$
$61$ $$(T^{4} - 90 T^{3} - 5682 T^{2} + \cdots - 17180643)^{2}$$
$67$ $$T^{8} - 74 T^{7} + \cdots + 42600893548096$$
$71$ $$T^{8} + 156 T^{7} + \cdots + 1623606027264$$
$73$ $$T^{8} + 16 T^{7} + \cdots + 261324417601$$
$79$ $$(T^{4} + 48 T^{3} - 11667 T^{2} + \cdots - 8209344)^{2}$$
$83$ $$T^{8} - 682176 T^{5} + \cdots + 13865554427904$$
$89$ $$T^{8} + 228 T^{7} + \cdots + 29\!\cdots\!64$$
$97$ $$T^{8} + 2 T^{7} + \cdots + 14\!\cdots\!96$$