# Properties

 Label 555.2.c.b Level $555$ Weight $2$ Character orbit 555.c Analytic conductor $4.432$ 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] = [555,2,Mod(334,555)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(555, 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("555.334");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$555 = 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 555.c (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: $$4.43169731218$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.309760000.3 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1$$ x^8 - 4*x^7 + 8*x^6 - 2*x^5 - x^4 - 2*x^3 + 18*x^2 + 6*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} - \beta_{4}) q^{2} - \beta_{5} q^{3} + ( - \beta_1 + 1) q^{4} + (\beta_{4} + \beta_{2}) q^{5} + \beta_1 q^{6} + ( - \beta_{6} - 2 \beta_{5}) q^{7} + (\beta_{5} - 2 \beta_{4}) q^{8} - q^{9}+O(q^{10})$$ q + (b5 - b4) * q^2 - b5 * q^3 + (-b1 + 1) * q^4 + (b4 + b2) * q^5 + b1 * q^6 + (-b6 - 2*b5) * q^7 + (b5 - 2*b4) * q^8 - q^9 $$q + (\beta_{5} - \beta_{4}) q^{2} - \beta_{5} q^{3} + ( - \beta_1 + 1) q^{4} + (\beta_{4} + \beta_{2}) q^{5} + \beta_1 q^{6} + ( - \beta_{6} - 2 \beta_{5}) q^{7} + (\beta_{5} - 2 \beta_{4}) q^{8} - q^{9} + (\beta_{7} + \beta_{6} + 1) q^{10} + (\beta_{2} - 1) q^{11} - \beta_{4} q^{12} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4}) q^{13} + (\beta_{2} + 2 \beta_1) q^{14} + ( - \beta_{7} - \beta_1 + 1) q^{15} - 3 \beta_1 q^{16} + ( - 2 \beta_{7} + \beta_{5} - \beta_{4}) q^{17} + ( - \beta_{5} + \beta_{4}) q^{18} + ( - \beta_{3} + \beta_{2} + 1) q^{19} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{20} + ( - \beta_{3} - 2) q^{21} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4}) q^{22} + ( - 2 \beta_{6} + \beta_{5} - 2 \beta_{4}) q^{23} + (2 \beta_1 - 1) q^{24} + ( - 2 \beta_{6} + 2 \beta_1 + 1) q^{25} + (\beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{26} + \beta_{5} q^{27} + (\beta_{7} - \beta_{6} - 2 \beta_{4}) q^{28} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{29} + ( - \beta_{5} + \beta_{3} + \beta_{2}) q^{30} + ( - \beta_{2} - 3 \beta_1 + 1) q^{31} + ( - 4 \beta_{5} - \beta_{4}) q^{32} + ( - \beta_{7} + \beta_{5}) q^{33} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{34} + ( - 2 \beta_{7} - \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{35} + (\beta_1 - 1) q^{36} - \beta_{5} q^{37} + (\beta_{6} + \beta_{5} - \beta_{4}) q^{38} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{39} + (\beta_{7} + 2 \beta_{6} - \beta_1 + 3) q^{40} + (\beta_{3} + \beta_{2} - \beta_1 - 6) q^{41} + ( - \beta_{7} - 2 \beta_{5} + 2 \beta_{4}) q^{42} + (2 \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4}) q^{43} + ( - \beta_{3} + \beta_1 - 1) q^{44} + ( - \beta_{4} - \beta_{2}) q^{45} + (2 \beta_{2} - \beta_1 - 2) q^{46} + (3 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4}) q^{47} + (3 \beta_{5} - 3 \beta_{4}) q^{48} + ( - 4 \beta_{3} + 2 \beta_1 - 2) q^{49} + (5 \beta_{5} - 3 \beta_{4} + 2 \beta_{2}) q^{50} + ( - 2 \beta_{2} + \beta_1) q^{51} + (\beta_{7} - \beta_{5}) q^{52} + ( - 2 \beta_{7} + 2 \beta_{6}) q^{53} - \beta_1 q^{54} + ( - \beta_{6} - \beta_{4} - \beta_{2} + \beta_1 + 3) q^{55} + ( - \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{56} + ( - \beta_{7} + \beta_{6} - \beta_{5}) q^{57} + (3 \beta_{7} + \beta_{6} + 6 \beta_{5} - 4 \beta_{4}) q^{58} + (3 \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{59} + (\beta_{6} - \beta_1 + 2) q^{60} + ( - \beta_{2} + 3 \beta_1 - 3) q^{61} + ( - \beta_{7} - \beta_{6} - 5 \beta_{5} + 2 \beta_{4}) q^{62} + (\beta_{6} + 2 \beta_{5}) q^{63} + ( - 2 \beta_1 - 1) q^{64} + (\beta_{7} + \beta_{6} - 5 \beta_{5} + 4 \beta_{4} - \beta_{2} + 1) q^{65} + (\beta_{3} + \beta_{2} - \beta_1) q^{66} + (3 \beta_{7} - 2 \beta_{5}) q^{67} + (2 \beta_{6} - \beta_{5}) q^{68} + ( - 2 \beta_{3} + 2 \beta_1 - 1) q^{69} + ( - \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{70} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 9) q^{71} + ( - \beta_{5} + 2 \beta_{4}) q^{72} + ( - \beta_{7} + 3 \beta_{6} + 6 \beta_{5} - 2 \beta_{4}) q^{73} + \beta_1 q^{74} + ( - 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{75} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{76} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4}) q^{77} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4}) q^{78} + (3 \beta_{3} - 2 \beta_{2} - 7 \beta_1 + 2) q^{79} + (3 \beta_{5} - 3 \beta_{3} - 3 \beta_{2}) q^{80} + q^{81} + (2 \beta_{7} + \beta_{6} - 8 \beta_{5} + 7 \beta_{4}) q^{82} + (2 \beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_{4}) q^{83} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{84} + (\beta_{7} + \beta_{6} - 8 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 1) q^{85} + ( - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 1) q^{86} + ( - \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + 2 \beta_{4}) q^{87} + (\beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4}) q^{88} + (3 \beta_{3} - 4 \beta_1 + 8) q^{89} + ( - \beta_{7} - \beta_{6} - 1) q^{90} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 3) q^{91} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4}) q^{92} + (\beta_{7} + 2 \beta_{5} - 3 \beta_{4}) q^{93} + ( - 3 \beta_{3} - 5 \beta_{2} - 2 \beta_1 - 1) q^{94} + (\beta_{7} - 2 \beta_{6} + \beta_{4} + \beta_{2} - 2 \beta_1 + 5) q^{95} + (\beta_1 - 5) q^{96} + (\beta_{7} + \beta_{6} + \beta_{5} + 5 \beta_{4}) q^{97} + ( - 4 \beta_{7} + 2 \beta_{5}) q^{98} + ( - \beta_{2} + 1) q^{99}+O(q^{100})$$ q + (b5 - b4) * q^2 - b5 * q^3 + (-b1 + 1) * q^4 + (b4 + b2) * q^5 + b1 * q^6 + (-b6 - 2*b5) * q^7 + (b5 - 2*b4) * q^8 - q^9 + (b7 + b6 + 1) * q^10 + (b2 - 1) * q^11 - b4 * q^12 + (-b7 - b6 + b5 - b4) * q^13 + (b2 + 2*b1) * q^14 + (-b7 - b1 + 1) * q^15 - 3*b1 * q^16 + (-2*b7 + b5 - b4) * q^17 + (-b5 + b4) * q^18 + (-b3 + b2 + 1) * q^19 + (b5 + b4 - b3) * q^20 + (-b3 - 2) * q^21 + (b7 + b6 - b5 + b4) * q^22 + (-2*b6 + b5 - 2*b4) * q^23 + (2*b1 - 1) * q^24 + (-2*b6 + 2*b1 + 1) * q^25 + (b3 + 2*b2 - b1 - 1) * q^26 + b5 * q^27 + (b7 - b6 - 2*b4) * q^28 + (2*b3 + b2 + 2*b1 + 2) * q^29 + (-b5 + b3 + b2) * q^30 + (-b2 - 3*b1 + 1) * q^31 + (-4*b5 - b4) * q^32 + (-b7 + b5) * q^33 + (2*b3 + 2*b2 - b1 - 1) * q^34 + (-2*b7 - b5 + 3*b4 + b3 - b2 - 2*b1 + 2) * q^35 + (b1 - 1) * q^36 - b5 * q^37 + (b6 + b5 - b4) * q^38 + (-b3 - b2 + b1) * q^39 + (b7 + 2*b6 - b1 + 3) * q^40 + (b3 + b2 - b1 - 6) * q^41 + (-b7 - 2*b5 + 2*b4) * q^42 + (2*b7 + b6 + 2*b5 - b4) * q^43 + (-b3 + b1 - 1) * q^44 + (-b4 - b2) * q^45 + (2*b2 - b1 - 2) * q^46 + (3*b7 + 2*b6 + 2*b5 - b4) * q^47 + (3*b5 - 3*b4) * q^48 + (-4*b3 + 2*b1 - 2) * q^49 + (5*b5 - 3*b4 + 2*b2) * q^50 + (-2*b2 + b1) * q^51 + (b7 - b5) * q^52 + (-2*b7 + 2*b6) * q^53 - b1 * q^54 + (-b6 - b4 - b2 + b1 + 3) * q^55 + (-b3 + 2*b2 + 4*b1 - 2) * q^56 + (-b7 + b6 - b5) * q^57 + (3*b7 + b6 + 6*b5 - 4*b4) * q^58 + (3*b3 - b2 + 2*b1 + 2) * q^59 + (b6 - b1 + 2) * q^60 + (-b2 + 3*b1 - 3) * q^61 + (-b7 - b6 - 5*b5 + 2*b4) * q^62 + (b6 + 2*b5) * q^63 + (-2*b1 - 1) * q^64 + (b7 + b6 - 5*b5 + 4*b4 - b2 + 1) * q^65 + (b3 + b2 - b1) * q^66 + (3*b7 - 2*b5) * q^67 + (2*b6 - b5) * q^68 + (-2*b3 + 2*b1 - 1) * q^69 + (-b6 - 2*b5 + 2*b3 + 2*b2 + b1 + 3) * q^70 + (b3 - 2*b2 + b1 - 9) * q^71 + (-b5 + 2*b4) * q^72 + (-b7 + 3*b6 + 6*b5 - 2*b4) * q^73 + b1 * q^74 + (-3*b5 + 2*b4 - 2*b3) * q^75 + (-2*b3 + b2 - b1 + 1) * q^76 + (-2*b7 + b6 + b5 + 3*b4) * q^77 + (-2*b7 - b6 + 2*b5 - b4) * q^78 + (3*b3 - 2*b2 - 7*b1 + 2) * q^79 + (3*b5 - 3*b3 - 3*b2) * q^80 + q^81 + (2*b7 + b6 - 8*b5 + 7*b4) * q^82 + (2*b7 - b6 - b5 + 3*b4) * q^83 + (-b3 + b2 + 2*b1 - 2) * q^84 + (b7 + b6 - 8*b5 + 2*b4 - 2*b3 + 1) * q^85 + (-2*b3 - 3*b2 - 2*b1 - 1) * q^86 + (-b7 - 2*b6 - 4*b5 + 2*b4) * q^87 + (b7 + 2*b6 - b5 + 2*b4) * q^88 + (3*b3 - 4*b1 + 8) * q^89 + (-b7 - b6 - 1) * q^90 + (-2*b3 - b2 + b1 - 3) * q^91 + (2*b7 - 2*b6 - 2*b5 - b4) * q^92 + (b7 + 2*b5 - 3*b4) * q^93 + (-3*b3 - 5*b2 - 2*b1 - 1) * q^94 + (b7 - 2*b6 + b4 + b2 - 2*b1 + 5) * q^95 + (b1 - 5) * q^96 + (b7 + b6 + b5 + 5*b4) * q^97 + (-4*b7 + 2*b5) * q^98 + (-b2 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4} + 4 q^{6} - 8 q^{9}+O(q^{10})$$ 8 * q + 4 * q^4 + 4 * q^6 - 8 * q^9 $$8 q + 4 q^{4} + 4 q^{6} - 8 q^{9} + 8 q^{10} - 8 q^{11} + 8 q^{14} + 4 q^{15} - 12 q^{16} + 8 q^{19} - 16 q^{21} + 16 q^{25} - 12 q^{26} + 24 q^{29} - 4 q^{31} - 12 q^{34} + 8 q^{35} - 4 q^{36} + 4 q^{39} + 20 q^{40} - 52 q^{41} - 4 q^{44} - 20 q^{46} - 8 q^{49} + 4 q^{51} - 4 q^{54} + 28 q^{55} + 24 q^{59} + 12 q^{60} - 12 q^{61} - 16 q^{64} + 8 q^{65} - 4 q^{66} + 28 q^{70} - 68 q^{71} + 4 q^{74} + 4 q^{76} - 12 q^{79} + 8 q^{81} - 8 q^{84} + 8 q^{85} - 16 q^{86} + 48 q^{89} - 8 q^{90} - 20 q^{91} - 16 q^{94} + 32 q^{95} - 36 q^{96} + 8 q^{99}+O(q^{100})$$ 8 * q + 4 * q^4 + 4 * q^6 - 8 * q^9 + 8 * q^10 - 8 * q^11 + 8 * q^14 + 4 * q^15 - 12 * q^16 + 8 * q^19 - 16 * q^21 + 16 * q^25 - 12 * q^26 + 24 * q^29 - 4 * q^31 - 12 * q^34 + 8 * q^35 - 4 * q^36 + 4 * q^39 + 20 * q^40 - 52 * q^41 - 4 * q^44 - 20 * q^46 - 8 * q^49 + 4 * q^51 - 4 * q^54 + 28 * q^55 + 24 * q^59 + 12 * q^60 - 12 * q^61 - 16 * q^64 + 8 * q^65 - 4 * q^66 + 28 * q^70 - 68 * q^71 + 4 * q^74 + 4 * q^76 - 12 * q^79 + 8 * q^81 - 8 * q^84 + 8 * q^85 - 16 * q^86 + 48 * q^89 - 8 * q^90 - 20 * q^91 - 16 * q^94 + 32 * q^95 - 36 * q^96 + 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( -8\nu^{7} + 44\nu^{6} - 130\nu^{5} + 211\nu^{4} - 186\nu^{3} + 50\nu^{2} + 26\nu + 403 ) / 245$$ (-8*v^7 + 44*v^6 - 130*v^5 + 211*v^4 - 186*v^3 + 50*v^2 + 26*v + 403) / 245 $$\beta_{2}$$ $$=$$ $$( 11\nu^{7} + 13\nu^{6} - 152\nu^{5} + 451\nu^{4} - 222\nu^{3} - 32\nu^{2} + \nu + 530 ) / 245$$ (11*v^7 + 13*v^6 - 152*v^5 + 451*v^4 - 222*v^3 - 32*v^2 + v + 530) / 245 $$\beta_{3}$$ $$=$$ $$( 13\nu^{7} - 47\nu^{6} + 101\nu^{5} - 55\nu^{4} + 143\nu^{3} - 69\nu^{2} - 30\nu + 319 ) / 245$$ (13*v^7 - 47*v^6 + 101*v^5 - 55*v^4 + 143*v^3 - 69*v^2 - 30*v + 319) / 245 $$\beta_{4}$$ $$=$$ $$( -61\nu^{7} + 262\nu^{6} - 538\nu^{5} + 194\nu^{4} + 162\nu^{3} - 23\nu^{2} - 941\nu - 155 ) / 245$$ (-61*v^7 + 262*v^6 - 538*v^5 + 194*v^4 + 162*v^3 - 23*v^2 - 941*v - 155) / 245 $$\beta_{5}$$ $$=$$ $$( 74\nu^{7} - 309\nu^{6} + 639\nu^{5} - 249\nu^{4} - 19\nu^{3} - 291\nu^{2} + 1401\nu + 229 ) / 245$$ (74*v^7 - 309*v^6 + 639*v^5 - 249*v^4 - 19*v^3 - 291*v^2 + 1401*v + 229) / 245 $$\beta_{6}$$ $$=$$ $$( 87\nu^{7} - 356\nu^{6} + 740\nu^{5} - 304\nu^{4} + 124\nu^{3} - 360\nu^{2} + 1861\nu + 303 ) / 245$$ (87*v^7 - 356*v^6 + 740*v^5 - 304*v^4 + 124*v^3 - 360*v^2 + 1861*v + 303) / 245 $$\beta_{7}$$ $$=$$ $$( 178\nu^{7} - 734\nu^{6} + 1545\nu^{5} - 591\nu^{4} - 149\nu^{3} - 10\nu^{2} + 3219\nu + 527 ) / 245$$ (178*v^7 - 734*v^6 + 1545*v^5 - 591*v^4 - 149*v^3 - 10*v^2 + 3219*v + 527) / 245
 $$\nu$$ $$=$$ $$( \beta_{6} - \beta_{5} - \beta_{3} + 1 ) / 2$$ (b6 - b5 - b3 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{6} - 2\beta_{5} - \beta_{4}$$ b6 - 2*b5 - b4 $$\nu^{3}$$ $$=$$ $$( -\beta_{7} + 4\beta_{6} - 5\beta_{5} - 3\beta_{4} + 4\beta_{3} - \beta_{2} + 3\beta _1 - 8 ) / 2$$ (-b7 + 4*b6 - 5*b5 - 3*b4 + 4*b3 - b2 + 3*b1 - 8) / 2 $$\nu^{4}$$ $$=$$ $$6\beta_{3} - 2\beta_{2} + 7\beta _1 - 15$$ 6*b3 - 2*b2 + 7*b1 - 15 $$\nu^{5}$$ $$=$$ $$( 9\beta_{7} - 21\beta_{6} + 24\beta_{5} + 25\beta_{4} + 21\beta_{3} - 9\beta_{2} + 25\beta _1 - 49 ) / 2$$ (9*b7 - 21*b6 + 24*b5 + 25*b4 + 21*b3 - 9*b2 + 25*b1 - 49) / 2 $$\nu^{6}$$ $$=$$ $$17\beta_{7} - 35\beta_{6} + 39\beta_{5} + 47\beta_{4}$$ 17*b7 - 35*b6 + 39*b5 + 47*b4 $$\nu^{7}$$ $$=$$ $$( 64\beta_{7} - 121\beta_{6} + 127\beta_{5} + 168\beta_{4} - 121\beta_{3} + 64\beta_{2} - 168\beta _1 + 295 ) / 2$$ (64*b7 - 121*b6 + 127*b5 + 168*b4 - 121*b3 + 64*b2 - 168*b1 + 295) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/555\mathbb{Z}\right)^\times$$.

 $$n$$ $$76$$ $$112$$ $$371$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
334.1
 1.16407 + 1.16407i −0.164066 − 0.164066i −0.748606 + 0.748606i 1.74861 − 1.74861i −0.748606 − 0.748606i 1.74861 + 1.74861i 1.16407 − 1.16407i −0.164066 + 0.164066i
1.61803i 1.00000i −0.618034 −2.14896 + 0.618034i 1.61803 0.671869i 2.23607i −1.00000 1.00000 + 3.47709i
334.2 1.61803i 1.00000i −0.618034 2.14896 + 0.618034i 1.61803 3.32813i 2.23607i −1.00000 1.00000 3.47709i
334.3 0.618034i 1.00000i 1.61803 −1.54336 + 1.61803i −0.618034 4.49721i 2.23607i −1.00000 1.00000 + 0.953850i
334.4 0.618034i 1.00000i 1.61803 1.54336 + 1.61803i −0.618034 0.497212i 2.23607i −1.00000 1.00000 0.953850i
334.5 0.618034i 1.00000i 1.61803 −1.54336 1.61803i −0.618034 4.49721i 2.23607i −1.00000 1.00000 0.953850i
334.6 0.618034i 1.00000i 1.61803 1.54336 1.61803i −0.618034 0.497212i 2.23607i −1.00000 1.00000 + 0.953850i
334.7 1.61803i 1.00000i −0.618034 −2.14896 0.618034i 1.61803 0.671869i 2.23607i −1.00000 1.00000 3.47709i
334.8 1.61803i 1.00000i −0.618034 2.14896 0.618034i 1.61803 3.32813i 2.23607i −1.00000 1.00000 + 3.47709i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 334.8 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 555.2.c.b 8
3.b odd 2 1 1665.2.c.d 8
5.b even 2 1 inner 555.2.c.b 8
5.c odd 4 1 2775.2.a.w 4
5.c odd 4 1 2775.2.a.y 4
15.d odd 2 1 1665.2.c.d 8
15.e even 4 1 8325.2.a.bt 4
15.e even 4 1 8325.2.a.bw 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.c.b 8 1.a even 1 1 trivial
555.2.c.b 8 5.b even 2 1 inner
1665.2.c.d 8 3.b odd 2 1
1665.2.c.d 8 15.d odd 2 1
2775.2.a.w 4 5.c odd 4 1
2775.2.a.y 4 5.c odd 4 1
8325.2.a.bt 4 15.e even 4 1
8325.2.a.bw 4 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(555, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 3 T^{2} + 1)^{2}$$
$3$ $$(T^{2} + 1)^{4}$$
$5$ $$T^{8} - 8 T^{6} + 46 T^{4} - 200 T^{2} + \cdots + 625$$
$7$ $$T^{8} + 32 T^{6} + 246 T^{4} + \cdots + 25$$
$11$ $$(T^{4} + 4 T^{3} - T^{2} - 10 T + 5)^{2}$$
$13$ $$T^{8} + 32 T^{6} + 166 T^{4} + \cdots + 25$$
$17$ $$T^{8} + 62 T^{6} + 1171 T^{4} + \cdots + 21025$$
$19$ $$(T^{4} - 4 T^{3} - 11 T^{2} + 30 T - 5)^{2}$$
$23$ $$T^{8} + 84 T^{6} + 1846 T^{4} + \cdots + 1681$$
$29$ $$(T^{4} - 12 T^{3} + 9 T^{2} + 112 T - 49)^{2}$$
$31$ $$(T^{4} + 2 T^{3} - 28 T^{2} - 14 T + 59)^{2}$$
$37$ $$(T^{2} + 1)^{4}$$
$41$ $$(T^{4} + 26 T^{3} + 238 T^{2} + 922 T + 1289)^{2}$$
$43$ $$T^{8} + 78 T^{6} + 691 T^{4} + \cdots + 25$$
$47$ $$T^{8} + 180 T^{6} + 5878 T^{4} + \cdots + 1$$
$53$ $$(T^{4} + 68 T^{2} + 176)^{2}$$
$59$ $$(T^{4} - 12 T^{3} - 41 T^{2} + 812 T - 1949)^{2}$$
$61$ $$(T^{4} + 6 T^{3} - 16 T^{2} - 90 T - 25)^{2}$$
$67$ $$T^{8} + 142 T^{6} + 6351 T^{4} + \cdots + 429025$$
$71$ $$(T^{4} + 34 T^{3} + 391 T^{2} + 1784 T + 2801)^{2}$$
$73$ $$T^{8} + 290 T^{6} + \cdots + 13140625$$
$79$ $$(T^{4} + 6 T^{3} - 221 T^{2} - 2440 T - 6245)^{2}$$
$83$ $$T^{8} + 126 T^{6} + 3391 T^{4} + \cdots + 121$$
$89$ $$(T^{4} - 24 T^{3} + 104 T^{2} + 120 T - 725)^{2}$$
$97$ $$T^{8} + 200 T^{6} + 12238 T^{4} + \cdots + 398161$$