Properties

 Label 322.2.e.b Level $322$ Weight $2$ Character orbit 322.e Analytic conductor $2.571$ 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] = [322,2,Mod(93,322)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(322, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("322.93");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$322 = 2 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 322.e (of order $$3$$, 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: $$2.57118294509$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.1767277521.3 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7$$ x^8 - 2*x^7 + x^6 - 10*x^5 + 38*x^4 - 40*x^3 + 64*x^2 - 38*x + 7 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} + (\beta_{6} - \beta_{5} + \beta_{2} + 1) q^{3} + (\beta_{5} - 1) q^{4} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{6} + \beta_{3}) q^{7} + q^{8} + (\beta_{7} + 2 \beta_{6} - \beta_{5}) q^{9}+O(q^{10})$$ q - b5 * q^2 + (b6 - b5 + b2 + 1) * q^3 + (b5 - 1) * q^4 + (-b7 + b5 - b4 + b3) * q^5 + (-b2 - 1) * q^6 + (-b6 + b3) * q^7 + q^8 + (b7 + 2*b6 - b5) * q^9 $$q - \beta_{5} q^{2} + (\beta_{6} - \beta_{5} + \beta_{2} + 1) q^{3} + (\beta_{5} - 1) q^{4} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{6} + \beta_{3}) q^{7} + q^{8} + (\beta_{7} + 2 \beta_{6} - \beta_{5}) q^{9} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 + 1) q^{10} + (\beta_{7} + \beta_{6} + \beta_{2} + \beta_1) q^{11} + ( - \beta_{6} + \beta_{5}) q^{12} + (\beta_{3} + \beta_{2} - 2) q^{13} + (\beta_{6} + \beta_{4} - \beta_{3} + \beta_{2}) q^{14} + (3 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 1) q^{15} - \beta_{5} q^{16} + ( - 3 \beta_{7} - \beta_{4} - 3 \beta_1) q^{17} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{2} - \beta_1 - 1) q^{18} + (\beta_{7} + 2 \beta_{6} + 3 \beta_{5} + \beta_{4} - \beta_{3}) q^{19} + ( - \beta_{3} - \beta_1 - 1) q^{20} + (\beta_{7} + \beta_{5} + 2 \beta_{4} + \beta_{2} + 2 \beta_1 + 2) q^{21} + ( - \beta_{2} - \beta_1) q^{22} - \beta_{5} q^{23} + (\beta_{6} - \beta_{5} + \beta_{2} + 1) q^{24} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{2} - 2 \beta_1 - 2) q^{25} + (\beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{26} + ( - \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 3) q^{27} + ( - \beta_{4} - \beta_{2}) q^{28} + ( - \beta_{3} - 2 \beta_{2}) q^{29} + (2 \beta_{7} + 3 \beta_{6} + \beta_{5} + 3 \beta_{4} - 3 \beta_{3}) q^{30} + ( - \beta_{7} - \beta_{6} - 4 \beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_1 + 4) q^{31} + (\beta_{5} - 1) q^{32} + (2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{33} + (\beta_{3} + 3 \beta_1) q^{34} + ( - 4 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 + 2) q^{35} + (2 \beta_{2} + \beta_1 + 1) q^{36} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3}) q^{37} + ( - \beta_{7} - 2 \beta_{6} - 3 \beta_{5} - \beta_{4} - 2 \beta_{2} - \beta_1 + 3) q^{38} + (2 \beta_{7} - \beta_{6} + 2 \beta_{4} - \beta_{2} + 2 \beta_1) q^{39} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3}) q^{40} + ( - \beta_{3} + \beta_{2} - 4) q^{41} + (\beta_{7} + \beta_{6} - 3 \beta_{5} - 2 \beta_{3} - \beta_1 + 1) q^{42} + (2 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{43} + ( - \beta_{7} - \beta_{6}) q^{44} + (5 \beta_{7} + 6 \beta_{6} + 5 \beta_{4} + 6 \beta_{2} + 5 \beta_1) q^{45} + (\beta_{5} - 1) q^{46} + ( - 2 \beta_{7} + \beta_{6} + 5 \beta_{5} - \beta_{4} + \beta_{3}) q^{47} + ( - \beta_{2} - 1) q^{48} + ( - 3 \beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{3} - 3 \beta_1 + 2) q^{49} + (\beta_{2} + 2 \beta_1 + 2) q^{50} + ( - 4 \beta_{7} - 6 \beta_{6} - 4 \beta_{5} - 5 \beta_{4} + 5 \beta_{3}) q^{51} + ( - \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{2} + 2) q^{52} + (2 \beta_{7} - 3 \beta_{6} - \beta_{5} + 2 \beta_{4} - 3 \beta_{2} + 2 \beta_1 + 1) q^{53} + ( - 3 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - \beta_{4} + \beta_{3}) q^{54} + (2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 1) q^{55} + ( - \beta_{6} + \beta_{3}) q^{56} + ( - 3 \beta_{3} - \beta_{2} - 4 \beta_1 - 1) q^{57} + ( - 2 \beta_{6} - \beta_{4} + \beta_{3}) q^{58} + (\beta_{7} - \beta_{6} - 4 \beta_{5} - 4 \beta_{4} - \beta_{2} + \beta_1 + 4) q^{59} + ( - 2 \beta_{7} - 3 \beta_{6} - \beta_{5} - 3 \beta_{4} - 3 \beta_{2} - 2 \beta_1 + 1) q^{60} + (3 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3}) q^{61} + (2 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{62} + (5 \beta_{7} + 4 \beta_{6} - \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 5) q^{63} + q^{64} + (2 \beta_{7} - 4 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3}) q^{65} + ( - 2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - \beta_{4} - 3 \beta_{2} - 2 \beta_1 - 2) q^{66} + (\beta_{6} - 4 \beta_{5} - 2 \beta_{4} + \beta_{2} + 4) q^{67} + (3 \beta_{7} + \beta_{4} - \beta_{3}) q^{68} + ( - \beta_{2} - 1) q^{69} + ( - \beta_{7} + \beta_{6} - 3 \beta_{5} - \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 1) q^{70} + ( - 5 \beta_{3} - \beta_{2}) q^{71} + (\beta_{7} + 2 \beta_{6} - \beta_{5}) q^{72} + (\beta_{7} + 3 \beta_{6} + 4 \beta_{4} + 3 \beta_{2} + \beta_1) q^{73} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_{2} + \beta_1) q^{74} + ( - 3 \beta_{7} - 7 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{75} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{76} + (2 \beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 1) q^{77} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{78} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{79} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 + 1) q^{80} + ( - 3 \beta_{7} - 5 \beta_{6} + 2 \beta_{5} - 5 \beta_{4} - 5 \beta_{2} - 3 \beta_1 - 2) q^{81} + (\beta_{6} + 4 \beta_{5} - \beta_{4} + \beta_{3}) q^{82} + ( - 5 \beta_{3} - 4 \beta_{2}) q^{83} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{84} + (\beta_{3} - \beta_{2} - 5 \beta_1 - 12) q^{85} + (\beta_{7} + \beta_{6} + 5 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{86} + ( - 3 \beta_{7} - 2 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 3 \beta_1 - 5) q^{87} + (\beta_{7} + \beta_{6} + \beta_{2} + \beta_1) q^{88} + ( - \beta_{7} - 4 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{89} + ( - 5 \beta_{3} - 6 \beta_{2} - 5 \beta_1) q^{90} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 4) q^{91} + q^{92} + ( - 4 \beta_{7} + \beta_{6} - 4 \beta_{5} - 5 \beta_{4} + 5 \beta_{3}) q^{93} + (2 \beta_{7} - \beta_{6} - 5 \beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_1 + 5) q^{94} + (7 \beta_{6} + \beta_{5} + 7 \beta_{2} - 1) q^{95} + ( - \beta_{6} + \beta_{5}) q^{96} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 9) q^{97} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_1 + 1) q^{98} + ( - 4 \beta_{3} - 6 \beta_{2} - 3 \beta_1 - 8) q^{99}+O(q^{100})$$ q - b5 * q^2 + (b6 - b5 + b2 + 1) * q^3 + (b5 - 1) * q^4 + (-b7 + b5 - b4 + b3) * q^5 + (-b2 - 1) * q^6 + (-b6 + b3) * q^7 + q^8 + (b7 + 2*b6 - b5) * q^9 + (b7 - b5 + b4 + b1 + 1) * q^10 + (b7 + b6 + b2 + b1) * q^11 + (-b6 + b5) * q^12 + (b3 + b2 - 2) * q^13 + (b6 + b4 - b3 + b2) * q^14 + (3*b3 + 3*b2 + 2*b1 - 1) * q^15 - b5 * q^16 + (-3*b7 - b4 - 3*b1) * q^17 + (-b7 - 2*b6 + b5 - 2*b2 - b1 - 1) * q^18 + (b7 + 2*b6 + 3*b5 + b4 - b3) * q^19 + (-b3 - b1 - 1) * q^20 + (b7 + b5 + 2*b4 + b2 + 2*b1 + 2) * q^21 + (-b2 - b1) * q^22 - b5 * q^23 + (b6 - b5 + b2 + 1) * q^24 + (-2*b7 - b6 + 2*b5 - b2 - 2*b1 - 2) * q^25 + (b6 + 2*b5 + b4 - b3) * q^26 + (-b3 - 2*b2 - 3*b1 - 3) * q^27 + (-b4 - b2) * q^28 + (-b3 - 2*b2) * q^29 + (2*b7 + 3*b6 + b5 + 3*b4 - 3*b3) * q^30 + (-b7 - b6 - 4*b5 - 2*b4 - b2 - b1 + 4) * q^31 + (b5 - 1) * q^32 + (2*b7 + 3*b6 - 2*b5 + b4 - b3) * q^33 + (b3 + 3*b1) * q^34 + (-4*b6 + b5 - b4 - b3 - 3*b2 - b1 + 2) * q^35 + (2*b2 + b1 + 1) * q^36 + (-b7 + b6 + b4 - b3) * q^37 + (-b7 - 2*b6 - 3*b5 - b4 - 2*b2 - b1 + 3) * q^38 + (2*b7 - b6 + 2*b4 - b2 + 2*b1) * q^39 + (-b7 + b5 - b4 + b3) * q^40 + (-b3 + b2 - 4) * q^41 + (b7 + b6 - 3*b5 - 2*b3 - b1 + 1) * q^42 + (2*b3 + b2 + b1 - 5) * q^43 + (-b7 - b6) * q^44 + (5*b7 + 6*b6 + 5*b4 + 6*b2 + 5*b1) * q^45 + (b5 - 1) * q^46 + (-2*b7 + b6 + 5*b5 - b4 + b3) * q^47 + (-b2 - 1) * q^48 + (-3*b7 + b5 - 2*b4 + b3 - 3*b1 + 2) * q^49 + (b2 + 2*b1 + 2) * q^50 + (-4*b7 - 6*b6 - 4*b5 - 5*b4 + 5*b3) * q^51 + (-b6 - 2*b5 - b4 - b2 + 2) * q^52 + (2*b7 - 3*b6 - b5 + 2*b4 - 3*b2 + 2*b1 + 1) * q^53 + (-3*b7 - 2*b6 + 3*b5 - b4 + b3) * q^54 + (2*b3 + 3*b2 + 3*b1 + 1) * q^55 + (-b6 + b3) * q^56 + (-3*b3 - b2 - 4*b1 - 1) * q^57 + (-2*b6 - b4 + b3) * q^58 + (b7 - b6 - 4*b5 - 4*b4 - b2 + b1 + 4) * q^59 + (-2*b7 - 3*b6 - b5 - 3*b4 - 3*b2 - 2*b1 + 1) * q^60 + (3*b7 + 4*b6 + 2*b5 - b4 + b3) * q^61 + (2*b3 + b2 + b1 - 4) * q^62 + (5*b7 + 4*b6 - b5 + 3*b4 - 2*b3 + 3*b2 + 2*b1 + 5) * q^63 + q^64 + (2*b7 - 4*b6 - b5 + b4 - b3) * q^65 + (-2*b7 - 3*b6 + 2*b5 - b4 - 3*b2 - 2*b1 - 2) * q^66 + (b6 - 4*b5 - 2*b4 + b2 + 4) * q^67 + (3*b7 + b4 - b3) * q^68 + (-b2 - 1) * q^69 + (-b7 + b6 - 3*b5 - b4 + 2*b3 + 4*b2 + 1) * q^70 + (-5*b3 - b2) * q^71 + (b7 + 2*b6 - b5) * q^72 + (b7 + 3*b6 + 4*b4 + 3*b2 + b1) * q^73 + (b7 - b6 - b4 - b2 + b1) * q^74 + (-3*b7 - 7*b6 + 3*b5 - 2*b4 + 2*b3) * q^75 + (b3 + 2*b2 + b1 - 3) * q^76 + (2*b7 + b6 + 3*b5 + b3 + 3*b2 + 3*b1 - 1) * q^77 + (-2*b3 + b2 - 2*b1) * q^78 + (-b7 - 2*b6 - 2*b5 + b4 - b3) * q^79 + (b7 - b5 + b4 + b1 + 1) * q^80 + (-3*b7 - 5*b6 + 2*b5 - 5*b4 - 5*b2 - 3*b1 - 2) * q^81 + (b6 + 4*b5 - b4 + b3) * q^82 + (-5*b3 - 4*b2) * q^83 + (-2*b7 - b6 + 2*b5 - 2*b4 + 2*b3 - b2 - b1 - 3) * q^84 + (b3 - b2 - 5*b1 - 12) * q^85 + (b7 + b6 + 5*b5 + 2*b4 - 2*b3) * q^86 + (-3*b7 - 2*b6 + 5*b5 - 2*b4 - 2*b2 - 3*b1 - 5) * q^87 + (b7 + b6 + b2 + b1) * q^88 + (-b7 - 4*b6 + b5 + b4 - b3) * q^89 + (-5*b3 - 6*b2 - 5*b1) * q^90 + (-2*b7 + 2*b6 + 2*b5 - b4 - b3 - b1 + 4) * q^91 + q^92 + (-4*b7 + b6 - 4*b5 - 5*b4 + 5*b3) * q^93 + (2*b7 - b6 - 5*b5 + b4 - b2 + 2*b1 + 5) * q^94 + (7*b6 + b5 + 7*b2 - 1) * q^95 + (-b6 + b5) * q^96 + (-b3 + 2*b2 + 2*b1 + 9) * q^97 + (-3*b5 + b4 + b3 + 3*b1 + 1) * q^98 + (-4*b3 - 6*b2 - 3*b1 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{2} + 5 q^{3} - 4 q^{4} + 5 q^{5} - 10 q^{6} + q^{7} + 8 q^{8} - 7 q^{9}+O(q^{10})$$ 8 * q - 4 * q^2 + 5 * q^3 - 4 * q^4 + 5 * q^5 - 10 * q^6 + q^7 + 8 * q^8 - 7 * q^9 $$8 q - 4 q^{2} + 5 q^{3} - 4 q^{4} + 5 q^{5} - 10 q^{6} + q^{7} + 8 q^{8} - 7 q^{9} + 5 q^{10} + 2 q^{11} + 5 q^{12} - 14 q^{13} + q^{14} + 2 q^{15} - 4 q^{16} - 3 q^{17} - 7 q^{18} + 9 q^{19} - 10 q^{20} + 25 q^{21} - 4 q^{22} - 4 q^{23} + 5 q^{24} - 11 q^{25} + 7 q^{26} - 34 q^{27} - 2 q^{28} - 4 q^{29} - q^{30} + 14 q^{31} - 4 q^{32} - 13 q^{33} + 6 q^{34} + 16 q^{35} + 14 q^{36} + 9 q^{38} + q^{39} + 5 q^{40} - 30 q^{41} - 8 q^{42} - 36 q^{43} + 2 q^{44} + 11 q^{45} - 4 q^{46} + 21 q^{47} - 10 q^{48} + 17 q^{49} + 22 q^{50} - 6 q^{51} + 7 q^{52} + 3 q^{53} + 17 q^{54} + 20 q^{55} + q^{56} - 18 q^{57} + 2 q^{58} + 16 q^{59} - q^{60} + q^{61} - 28 q^{62} + 37 q^{63} + 8 q^{64} - 2 q^{65} - 13 q^{66} + 17 q^{67} - 3 q^{68} - 10 q^{69} + 4 q^{70} - 2 q^{71} - 7 q^{72} + 4 q^{73} + 22 q^{75} - 18 q^{76} + 13 q^{77} - 2 q^{78} - 5 q^{79} + 5 q^{80} - 16 q^{81} + 15 q^{82} - 8 q^{83} - 17 q^{84} - 108 q^{85} + 18 q^{86} - 25 q^{87} + 2 q^{88} + 9 q^{89} - 22 q^{90} + 38 q^{91} + 8 q^{92} - 13 q^{93} + 21 q^{94} + 3 q^{95} + 5 q^{96} + 80 q^{97} + 2 q^{98} - 82 q^{99}+O(q^{100})$$ 8 * q - 4 * q^2 + 5 * q^3 - 4 * q^4 + 5 * q^5 - 10 * q^6 + q^7 + 8 * q^8 - 7 * q^9 + 5 * q^10 + 2 * q^11 + 5 * q^12 - 14 * q^13 + q^14 + 2 * q^15 - 4 * q^16 - 3 * q^17 - 7 * q^18 + 9 * q^19 - 10 * q^20 + 25 * q^21 - 4 * q^22 - 4 * q^23 + 5 * q^24 - 11 * q^25 + 7 * q^26 - 34 * q^27 - 2 * q^28 - 4 * q^29 - q^30 + 14 * q^31 - 4 * q^32 - 13 * q^33 + 6 * q^34 + 16 * q^35 + 14 * q^36 + 9 * q^38 + q^39 + 5 * q^40 - 30 * q^41 - 8 * q^42 - 36 * q^43 + 2 * q^44 + 11 * q^45 - 4 * q^46 + 21 * q^47 - 10 * q^48 + 17 * q^49 + 22 * q^50 - 6 * q^51 + 7 * q^52 + 3 * q^53 + 17 * q^54 + 20 * q^55 + q^56 - 18 * q^57 + 2 * q^58 + 16 * q^59 - q^60 + q^61 - 28 * q^62 + 37 * q^63 + 8 * q^64 - 2 * q^65 - 13 * q^66 + 17 * q^67 - 3 * q^68 - 10 * q^69 + 4 * q^70 - 2 * q^71 - 7 * q^72 + 4 * q^73 + 22 * q^75 - 18 * q^76 + 13 * q^77 - 2 * q^78 - 5 * q^79 + 5 * q^80 - 16 * q^81 + 15 * q^82 - 8 * q^83 - 17 * q^84 - 108 * q^85 + 18 * q^86 - 25 * q^87 + 2 * q^88 + 9 * q^89 - 22 * q^90 + 38 * q^91 + 8 * q^92 - 13 * q^93 + 21 * q^94 + 3 * q^95 + 5 * q^96 + 80 * q^97 + 2 * q^98 - 82 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7$$ :

 $$\beta_{1}$$ $$=$$ $$( -14\nu^{7} + 688\nu^{6} - 619\nu^{5} - 193\nu^{4} - 6480\nu^{3} + 17846\nu^{2} - 10595\nu + 23150 ) / 8102$$ (-14*v^7 + 688*v^6 - 619*v^5 - 193*v^4 - 6480*v^3 + 17846*v^2 - 10595*v + 23150) / 8102 $$\beta_{2}$$ $$=$$ $$( 74\nu^{7} - 743\nu^{6} + 957\nu^{5} - 716\nu^{4} + 8788\nu^{3} - 23147\nu^{2} + 13756\nu - 21668 ) / 8102$$ (74*v^7 - 743*v^6 + 957*v^5 - 716*v^4 + 8788*v^3 - 23147*v^2 + 13756*v - 21668) / 8102 $$\beta_{3}$$ $$=$$ $$( -139\nu^{7} + 465\nu^{6} - 648\nu^{5} + 688\nu^{4} - 5887\nu^{3} + 15724\nu^{2} - 9416\nu - 2218 ) / 8102$$ (-139*v^7 + 465*v^6 - 648*v^5 + 688*v^4 - 5887*v^3 + 15724*v^2 - 9416*v - 2218) / 8102 $$\beta_{4}$$ $$=$$ $$( -1269\nu^{7} + 2176\nu^{6} - 262\nu^{5} + 11731\nu^{4} - 43953\nu^{3} + 36565\nu^{2} - 52069\nu + 14432 ) / 8102$$ (-1269*v^7 + 2176*v^6 - 262*v^5 + 11731*v^4 - 43953*v^3 + 36565*v^2 - 52069*v + 14432) / 8102 $$\beta_{5}$$ $$=$$ $$( -962\nu^{7} + 1557\nu^{6} - 288\nu^{5} + 9308\nu^{4} - 33224\nu^{3} + 25443\nu^{2} - 49196\nu + 18369 ) / 4051$$ (-962*v^7 + 1557*v^6 - 288*v^5 + 9308*v^4 - 33224*v^3 + 25443*v^2 - 49196*v + 18369) / 4051 $$\beta_{6}$$ $$=$$ $$( - 4270 \nu^{7} + 7290 \nu^{6} - 2449 \nu^{5} + 42410 \nu^{4} - 149399 \nu^{3} + 128118 \nu^{2} - 241837 \nu + 88979 ) / 8102$$ (-4270*v^7 + 7290*v^6 - 2449*v^5 + 42410*v^4 - 149399*v^3 + 128118*v^2 - 241837*v + 88979) / 8102 $$\beta_{7}$$ $$=$$ $$( 4805\nu^{7} - 8118\nu^{6} + 2087\nu^{5} - 47477\nu^{4} + 167278\nu^{3} - 139939\nu^{2} + 265634\nu - 98045 ) / 8102$$ (4805*v^7 - 8118*v^6 + 2087*v^5 - 47477*v^4 + 167278*v^3 - 139939*v^2 + 265634*v - 98045) / 8102
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} - \beta _1 + 2 ) / 3$$ (b7 + b6 - b5 + 2*b4 - b3 - b2 - b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -5\beta_{7} - 5\beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} - 4\beta_{2} - 4\beta _1 + 2 ) / 3$$ (-5*b7 - 5*b6 - b5 - b4 + 2*b3 - 4*b2 - 4*b1 + 2) / 3 $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{6} - 6\beta_{5} + 2\beta_{4} + \beta_{3} + 3\beta_{2} + \beta _1 + 6$$ -b7 + b6 - 6*b5 + 2*b4 + b3 + 3*b2 + b1 + 6 $$\nu^{4}$$ $$=$$ $$( -7\beta_{7} + 5\beta_{6} - 38\beta_{5} + 16\beta_{4} - 23\beta_{3} - 17\beta_{2} - 17\beta _1 + 1 ) / 3$$ (-7*b7 + 5*b6 - 38*b5 + 16*b4 - 23*b3 - 17*b2 - 17*b1 + 1) / 3 $$\nu^{5}$$ $$=$$ $$( -70\beta_{7} - 70\beta_{6} - 11\beta_{5} - 14\beta_{4} + 4\beta_{3} - 14\beta_{2} - 8\beta _1 - 17 ) / 3$$ (-70*b7 - 70*b6 - 11*b5 - 14*b4 + 4*b3 - 14*b2 - 8*b1 - 17) / 3 $$\nu^{6}$$ $$=$$ $$17\beta_{7} + 37\beta_{6} - 60\beta_{5} + 35\beta_{4} - 16\beta_{3} + 50\beta_{2} + 46\beta _1 + 7$$ 17*b7 + 37*b6 - 60*b5 + 35*b4 - 16*b3 + 50*b2 + 46*b1 + 7 $$\nu^{7}$$ $$=$$ $$( -44\beta_{7} - 38\beta_{6} - 22\beta_{5} - 7\beta_{4} - 301\beta_{3} - 283\beta_{2} - 97\beta _1 - 565 ) / 3$$ (-44*b7 - 38*b6 - 22*b5 - 7*b4 - 301*b3 - 283*b2 - 97*b1 - 565) / 3

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/322\mathbb{Z}\right)^\times$$.

 $$n$$ $$185$$ $$281$$ $$\chi(n)$$ $$-\beta_{5}$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
93.1
 0.373419 + 0.0835272i 2.11692 + 0.978886i 0.0512865 − 1.21608i −1.54162 + 1.88572i 0.373419 − 0.0835272i 2.11692 − 0.978886i 0.0512865 + 1.21608i −1.54162 − 1.88572i
−0.500000 + 0.866025i −0.685837 1.18790i −0.500000 0.866025i 1.57146 2.72186i 1.37167 −1.66774 + 2.05393i 1.00000 0.559256 0.968659i 1.57146 + 2.72186i
93.2 −0.500000 + 0.866025i 0.313577 + 0.543132i −0.500000 0.866025i 0.475705 0.823946i −0.627155 2.62597 + 0.322894i 1.00000 1.30334 2.25745i 0.475705 + 0.823946i
93.3 −0.500000 + 0.866025i 1.26826 + 2.19669i −0.500000 0.866025i −1.34706 + 2.33318i −2.53652 −2.28677 1.33067i 1.00000 −1.71698 + 2.97389i −1.34706 2.33318i
93.4 −0.500000 + 0.866025i 1.60400 + 2.77821i −0.500000 0.866025i 1.79990 3.11751i −3.20800 1.82854 1.91218i 1.00000 −3.64562 + 6.31440i 1.79990 + 3.11751i
277.1 −0.500000 0.866025i −0.685837 + 1.18790i −0.500000 + 0.866025i 1.57146 + 2.72186i 1.37167 −1.66774 2.05393i 1.00000 0.559256 + 0.968659i 1.57146 2.72186i
277.2 −0.500000 0.866025i 0.313577 0.543132i −0.500000 + 0.866025i 0.475705 + 0.823946i −0.627155 2.62597 0.322894i 1.00000 1.30334 + 2.25745i 0.475705 0.823946i
277.3 −0.500000 0.866025i 1.26826 2.19669i −0.500000 + 0.866025i −1.34706 2.33318i −2.53652 −2.28677 + 1.33067i 1.00000 −1.71698 2.97389i −1.34706 + 2.33318i
277.4 −0.500000 0.866025i 1.60400 2.77821i −0.500000 + 0.866025i 1.79990 + 3.11751i −3.20800 1.82854 + 1.91218i 1.00000 −3.64562 6.31440i 1.79990 3.11751i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 277.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.e.b 8
7.c even 3 1 inner 322.2.e.b 8
7.c even 3 1 2254.2.a.w 4
7.d odd 6 1 2254.2.a.ba 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.e.b 8 1.a even 1 1 trivial
322.2.e.b 8 7.c even 3 1 inner
2254.2.a.w 4 7.c even 3 1
2254.2.a.ba 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 5T_{3}^{7} + 22T_{3}^{6} - 37T_{3}^{5} + 71T_{3}^{4} - 37T_{3}^{3} + 142T_{3}^{2} - 77T_{3} + 49$$ acting on $$S_{2}^{\mathrm{new}}(322, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{4}$$
$3$ $$T^{8} - 5 T^{7} + 22 T^{6} - 37 T^{5} + \cdots + 49$$
$5$ $$T^{8} - 5 T^{7} + 28 T^{6} - 59 T^{5} + \cdots + 841$$
$7$ $$T^{8} - T^{7} - 8 T^{6} - 5 T^{5} + \cdots + 2401$$
$11$ $$T^{8} - 2 T^{7} + 16 T^{6} - 6 T^{5} + \cdots + 9$$
$13$ $$(T^{4} + 7 T^{3} + 6 T^{2} - 20 T + 7)^{2}$$
$17$ $$T^{8} + 3 T^{7} + 81 T^{6} + \cdots + 1814409$$
$19$ $$T^{8} - 9 T^{7} + 78 T^{6} + \cdots + 95481$$
$23$ $$(T^{2} + T + 1)^{4}$$
$29$ $$(T^{4} + 2 T^{3} - 27 T^{2} - 56 T - 1)^{2}$$
$31$ $$T^{8} - 14 T^{7} + 154 T^{6} + \cdots + 11449$$
$37$ $$T^{8} + 30 T^{6} + 10 T^{5} + \cdots + 729$$
$41$ $$(T^{4} + 15 T^{3} + 66 T^{2} + 62 T - 63)^{2}$$
$43$ $$(T^{4} + 18 T^{3} + 90 T^{2} + 99 T - 81)^{2}$$
$47$ $$T^{8} - 21 T^{7} + 324 T^{6} + \cdots + 35721$$
$53$ $$T^{8} - 3 T^{7} + 144 T^{6} + \cdots + 13689$$
$59$ $$T^{8} - 16 T^{7} + 334 T^{6} + \cdots + 1320201$$
$61$ $$T^{8} - T^{7} + 199 T^{6} + \cdots + 8346321$$
$67$ $$T^{8} - 17 T^{7} + 229 T^{6} + \cdots + 82369$$
$71$ $$(T^{4} + T^{3} - 216 T^{2} - 196 T + 5243)^{2}$$
$73$ $$T^{8} - 4 T^{7} + 154 T^{6} + \cdots + 19245769$$
$79$ $$T^{8} + 5 T^{7} + 67 T^{6} + \cdots + 145161$$
$83$ $$(T^{4} + 4 T^{3} - 261 T^{2} - 574 T + 17003)^{2}$$
$89$ $$T^{8} - 9 T^{7} + 180 T^{6} + \cdots + 12131289$$
$97$ $$(T^{4} - 40 T^{3} + 519 T^{2} - 2126 T - 637)^{2}$$