# Properties

 Label 95.2.e.c Level $95$ Weight $2$ Character orbit 95.e Analytic conductor $0.759$ 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] = [95,2,Mod(11,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("95.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 95.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: $$0.758578819202$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.4601315889.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9$$ x^8 - x^7 + 6*x^6 - 3*x^5 + 26*x^4 - 14*x^3 + 31*x^2 + 12*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} + ( - \beta_{5} + \beta_1) q^{3} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 1) q^{4} - \beta_{5} q^{5} + ( - \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_1) q^{6} + ( - \beta_{6} + \beta_{2} - 1) q^{7} + (2 \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 3) q^{8} + (\beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 1) q^{9}+O(q^{10})$$ q + b7 * q^2 + (-b5 + b1) * q^3 + (-b7 - b6 + b5 + b4 - 1) * q^4 - b5 * q^5 + (-b7 - b6 + b4 + b3 + b1) * q^6 + (-b6 + b2 - 1) * q^7 + (2*b6 - b4 + b3 - 2*b2 + 3) * q^8 + (b7 + b5 - b4 - 2*b3 - 2*b1 - 1) * q^9 $$q + \beta_{7} q^{2} + ( - \beta_{5} + \beta_1) q^{3} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 1) q^{4} - \beta_{5} q^{5} + ( - \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_1) q^{6} + ( - \beta_{6} + \beta_{2} - 1) q^{7} + (2 \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 3) q^{8} + (\beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 1) q^{9} + ( - \beta_{7} + \beta_{4}) q^{10} + ( - \beta_{6} + 2 \beta_{3} + \beta_{2}) q^{11} + (\beta_{6} - 2 \beta_{4} - \beta_{2} + 1) q^{12} + (\beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_1 - 2) q^{13} + ( - 2 \beta_{7} - \beta_{2} - \beta_1) q^{14} + (\beta_{5} - \beta_{3} - \beta_1 - 1) q^{15} + (4 \beta_{7} - \beta_{5} + 2 \beta_{2} + \beta_1) q^{16} + (\beta_{7} + 2 \beta_1) q^{17} + (\beta_{6} - 2 \beta_{3} - \beta_{2} - 3) q^{18} + (\beta_{4} + 2 \beta_{2} - \beta_1 + 1) q^{19} + (\beta_{6} - \beta_{4} - \beta_{2} + 1) q^{20} + ( - \beta_{7} + \beta_{5} - \beta_1) q^{21} + ( - \beta_{7} + \beta_{2} - 3 \beta_1) q^{22} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_1) q^{23} + (2 \beta_{7} - 6 \beta_{5} + \beta_{2} + 3 \beta_1) q^{24} + (\beta_{5} - 1) q^{25} + ( - \beta_{4} - 2 \beta_{3}) q^{26} + ( - \beta_{6} + 3 \beta_{4} + \beta_{3} + \beta_{2} + 4) q^{27} + (3 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} - 3 \beta_{4} + 4) q^{28} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_1) q^{29} + (\beta_{6} - \beta_{4} - \beta_{3} - \beta_{2}) q^{30} + ( - \beta_{6} - 3 \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{31} + ( - 5 \beta_{7} - 3 \beta_{6} + 6 \beta_{5} + 5 \beta_{4} + \beta_{3} + \beta_1 - 6) q^{32} + ( - 3 \beta_{7} - 6 \beta_{5} + 2 \beta_1) q^{33} + ( - \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 3) q^{34} + (\beta_{5} - \beta_{2}) q^{35} + (2 \beta_{5} - \beta_{2} - \beta_1) q^{36} + (\beta_{6} - 2 \beta_{4} - \beta_{2} - 1) q^{37} + ( - 2 \beta_{7} - \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{38} + (2 \beta_{4} + 3 \beta_{3} + 5) q^{39} + (\beta_{7} - 3 \beta_{5} + 2 \beta_{2} + \beta_1) q^{40} + (2 \beta_{7} + 3 \beta_{5} - 3 \beta_{2} - 2 \beta_1) q^{41} + (2 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_1 + 3) q^{42} + (\beta_{7} + \beta_{5} - \beta_{2} - 4 \beta_1) q^{43} + (\beta_{6} - 3 \beta_{5} + 3) q^{44} + (\beta_{4} + 2 \beta_{3} + 1) q^{45} + ( - \beta_{6} + \beta_{2} + 3) q^{46} + (2 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 3) q^{47} + ( - 5 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 4) q^{48} + (\beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - 3) q^{49} - \beta_{4} q^{50} + (\beta_{7} - \beta_{6} + 6 \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 - 6) q^{51} + (\beta_{7} + \beta_{5} + \beta_{2}) q^{52} + (3 \beta_{7} + 2 \beta_{6} - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{53} + (9 \beta_{5} - 3 \beta_{2} - 2 \beta_1) q^{54} + ( - \beta_{2} + 2 \beta_1) q^{55} + ( - 3 \beta_{6} + 5 \beta_{4} + 3 \beta_{2} - 9) q^{56} + ( - 4 \beta_{7} - 4 \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 + 3) q^{57} + ( - 5 \beta_{6} + 4 \beta_{4} - \beta_{3} + 5 \beta_{2} - 6) q^{58} + 5 \beta_1 q^{59} + (2 \beta_{7} - \beta_{5} + \beta_{2}) q^{60} + ( - 4 \beta_{7} + 2 \beta_{6} - \beta_{5} + 4 \beta_{4} + 1) q^{61} + (\beta_{7} - 9 \beta_{5} + \beta_{2}) q^{62} + ( - 2 \beta_{6} - \beta_{5} + \beta_{3} + \beta_1 + 1) q^{63} + (3 \beta_{6} - 6 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 13) q^{64} + ( - \beta_{6} + \beta_{3} + \beta_{2} + 2) q^{65} + ( - 3 \beta_{7} + \beta_{6} - 9 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 9) q^{66} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 2) q^{67} + (2 \beta_{6} - 5 \beta_{4} + \beta_{3} - 2 \beta_{2} + 3) q^{68} + (\beta_{6} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 3) q^{69} + (2 \beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_1) q^{70} + ( - 3 \beta_{7} - 6 \beta_{5} + \beta_1) q^{71} + (3 \beta_{7} + 4 \beta_{6} + 6 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} - 4 \beta_1 - 6) q^{72} + ( - \beta_{7} + 4 \beta_{5} - \beta_{2} + 3 \beta_1) q^{73} + (2 \beta_{7} - 6 \beta_{5} + 3 \beta_{2} + \beta_1) q^{74} + (\beta_{3} + 1) q^{75} + (2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 5 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 + 8) q^{76} + (2 \beta_{6} + \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 3) q^{77} + (3 \beta_{7} + 6 \beta_{5} + \beta_{2} - 3 \beta_1) q^{78} + ( - 2 \beta_{7} - 5 \beta_{5} + \beta_{2} + \beta_1) q^{79} + ( - 4 \beta_{7} - 2 \beta_{6} + \beta_{5} + 4 \beta_{4} - \beta_{3} - \beta_1 - 1) q^{80} + ( - 2 \beta_{7} - 4 \beta_{5} - 3 \beta_{2} + 2 \beta_1) q^{81} + (4 \beta_{7} + 3 \beta_{6} + 6 \beta_{5} - 4 \beta_{4} + \beta_{3} + \beta_1 - 6) q^{82} + ( - 2 \beta_{6} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{83} + ( - 3 \beta_{6} + 5 \beta_{4} - \beta_{3} + 3 \beta_{2} - 4) q^{84} + ( - \beta_{7} + \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{85} + (\beta_{7} + 4 \beta_{6} + 3 \beta_{5} - \beta_{4} - 3 \beta_{3} - 3 \beta_1 - 3) q^{86} + ( - 2 \beta_{6} + 3 \beta_{4} + \beta_{3} + 2 \beta_{2} - 3) q^{87} + ( - 3 \beta_{6} + 2 \beta_{4} + 5 \beta_{3} + 3 \beta_{2}) q^{88} + ( - 2 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 3 \beta_1 - 3) q^{89} + (3 \beta_{5} + \beta_{2} - 2 \beta_1) q^{90} + (\beta_{6} + \beta_{5} + 2 \beta_{3} + 2 \beta_1 - 1) q^{91} + (\beta_{2} + \beta_1) q^{92} + (3 \beta_{7} + 4 \beta_{5} + 3 \beta_{2} - 5 \beta_1) q^{93} + ( - \beta_{6} + 2 \beta_{4} + 5 \beta_{3} + \beta_{2} - 6) q^{94} + ( - \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{3} + \beta_1) q^{95} + (5 \beta_{6} - 9 \beta_{4} - 5 \beta_{2} + 3) q^{96} + (\beta_{5} - 4 \beta_{2} - 5 \beta_1) q^{97} + ( - \beta_{7} - 3 \beta_{5} + \beta_{2} + 2 \beta_1) q^{98} + (5 \beta_{7} + 12 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} - 5 \beta_1 - 12) q^{99}+O(q^{100})$$ q + b7 * q^2 + (-b5 + b1) * q^3 + (-b7 - b6 + b5 + b4 - 1) * q^4 - b5 * q^5 + (-b7 - b6 + b4 + b3 + b1) * q^6 + (-b6 + b2 - 1) * q^7 + (2*b6 - b4 + b3 - 2*b2 + 3) * q^8 + (b7 + b5 - b4 - 2*b3 - 2*b1 - 1) * q^9 + (-b7 + b4) * q^10 + (-b6 + 2*b3 + b2) * q^11 + (b6 - 2*b4 - b2 + 1) * q^12 + (b6 + 2*b5 - b3 - b1 - 2) * q^13 + (-2*b7 - b2 - b1) * q^14 + (b5 - b3 - b1 - 1) * q^15 + (4*b7 - b5 + 2*b2 + b1) * q^16 + (b7 + 2*b1) * q^17 + (b6 - 2*b3 - b2 - 3) * q^18 + (b4 + 2*b2 - b1 + 1) * q^19 + (b6 - b4 - b2 + 1) * q^20 + (-b7 + b5 - b1) * q^21 + (-b7 + b2 - 3*b1) * q^22 + (-b7 + b6 + b4 + b3 + b1) * q^23 + (2*b7 - 6*b5 + b2 + 3*b1) * q^24 + (b5 - 1) * q^25 + (-b4 - 2*b3) * q^26 + (-b6 + 3*b4 + b3 + b2 + 4) * q^27 + (3*b7 + 2*b6 - 4*b5 - 3*b4 + 4) * q^28 + (2*b7 + 2*b6 - 2*b4 + b3 + b1) * q^29 + (b6 - b4 - b3 - b2) * q^30 + (-b6 - 3*b4 - b3 + b2 - 1) * q^31 + (-5*b7 - 3*b6 + 6*b5 + 5*b4 + b3 + b1 - 6) * q^32 + (-3*b7 - 6*b5 + 2*b1) * q^33 + (-b7 - 3*b6 + 3*b5 + b4 + 2*b3 + 2*b1 - 3) * q^34 + (b5 - b2) * q^35 + (2*b5 - b2 - b1) * q^36 + (b6 - 2*b4 - b2 - 1) * q^37 + (-2*b7 - b6 + 3*b5 + 2*b4 - 3*b3 - b2 - 3*b1) * q^38 + (2*b4 + 3*b3 + 5) * q^39 + (b7 - 3*b5 + 2*b2 + b1) * q^40 + (2*b7 + 3*b5 - 3*b2 - 2*b1) * q^41 + (2*b7 + 2*b6 - 3*b5 - 2*b4 - b3 - b1 + 3) * q^42 + (b7 + b5 - b2 - 4*b1) * q^43 + (b6 - 3*b5 + 3) * q^44 + (b4 + 2*b3 + 1) * q^45 + (-b6 + b2 + 3) * q^46 + (2*b7 - 3*b6 - 3*b5 - 2*b4 + 2*b3 + 2*b1 + 3) * q^47 + (-5*b7 - 4*b6 + 4*b5 + 5*b4 + 2*b3 + 2*b1 - 4) * q^48 + (b6 - b4 - b3 - b2 - 3) * q^49 - b4 * q^50 + (b7 - b6 + 6*b5 - b4 - b3 - b1 - 6) * q^51 + (b7 + b5 + b2) * q^52 + (3*b7 + 2*b6 - 3*b4 - 2*b3 - 2*b1) * q^53 + (9*b5 - 3*b2 - 2*b1) * q^54 + (-b2 + 2*b1) * q^55 + (-3*b6 + 5*b4 + 3*b2 - 9) * q^56 + (-4*b7 - 4*b5 + 3*b4 + b3 - b2 + 3*b1 + 3) * q^57 + (-5*b6 + 4*b4 - b3 + 5*b2 - 6) * q^58 + 5*b1 * q^59 + (2*b7 - b5 + b2) * q^60 + (-4*b7 + 2*b6 - b5 + 4*b4 + 1) * q^61 + (b7 - 9*b5 + b2) * q^62 + (-2*b6 - b5 + b3 + b1 + 1) * q^63 + (3*b6 - 6*b4 + 2*b3 - 3*b2 + 13) * q^64 + (-b6 + b3 + b2 + 2) * q^65 + (-3*b7 + b6 - 9*b5 + 3*b4 + 2*b3 + 2*b1 + 9) * q^66 + (2*b7 + 2*b5 - 2*b4 - 2*b3 - 2*b1 - 2) * q^67 + (2*b6 - 5*b4 + b3 - 2*b2 + 3) * q^68 + (b6 - b4 - 2*b3 - b2 - 3) * q^69 + (2*b7 + b6 - 2*b4 + b3 + b1) * q^70 + (-3*b7 - 6*b5 + b1) * q^71 + (3*b7 + 4*b6 + 6*b5 - 3*b4 - 4*b3 - 4*b1 - 6) * q^72 + (-b7 + 4*b5 - b2 + 3*b1) * q^73 + (2*b7 - 6*b5 + 3*b2 + b1) * q^74 + (b3 + 1) * q^75 + (2*b7 + 3*b6 - 2*b5 - 5*b4 + b3 - 2*b2 + b1 + 8) * q^76 + (2*b6 + b4 - 3*b3 - 2*b2 + 3) * q^77 + (3*b7 + 6*b5 + b2 - 3*b1) * q^78 + (-2*b7 - 5*b5 + b2 + b1) * q^79 + (-4*b7 - 2*b6 + b5 + 4*b4 - b3 - b1 - 1) * q^80 + (-2*b7 - 4*b5 - 3*b2 + 2*b1) * q^81 + (4*b7 + 3*b6 + 6*b5 - 4*b4 + b3 + b1 - 6) * q^82 + (-2*b6 + b4 - 2*b3 + 2*b2) * q^83 + (-3*b6 + 5*b4 - b3 + 3*b2 - 4) * q^84 + (-b7 + b4 - 2*b3 - 2*b1) * q^85 + (b7 + 4*b6 + 3*b5 - b4 - 3*b3 - 3*b1 - 3) * q^86 + (-2*b6 + 3*b4 + b3 + 2*b2 - 3) * q^87 + (-3*b6 + 2*b4 + 5*b3 + 3*b2) * q^88 + (-2*b7 + 3*b6 + 3*b5 + 2*b4 - 3*b3 - 3*b1 - 3) * q^89 + (3*b5 + b2 - 2*b1) * q^90 + (b6 + b5 + 2*b3 + 2*b1 - 1) * q^91 + (b2 + b1) * q^92 + (3*b7 + 4*b5 + 3*b2 - 5*b1) * q^93 + (-b6 + 2*b4 + 5*b3 + b2 - 6) * q^94 + (-b7 - 2*b6 - b5 + b3 + b1) * q^95 + (5*b6 - 9*b4 - 5*b2 + 3) * q^96 + (b5 - 4*b2 - 5*b1) * q^97 + (-b7 - 3*b5 + b2 + 2*b1) * q^98 + (5*b7 + 12*b5 - 5*b4 - 5*b3 - 5*b1 - 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - q^{2} - 3 q^{3} - 5 q^{4} - 4 q^{5} - 2 q^{6} - 8 q^{7} + 24 q^{8} - q^{9}+O(q^{10})$$ 8 * q - q^2 - 3 * q^3 - 5 * q^4 - 4 * q^5 - 2 * q^6 - 8 * q^7 + 24 * q^8 - q^9 $$8 q - q^{2} - 3 q^{3} - 5 q^{4} - 4 q^{5} - 2 q^{6} - 8 q^{7} + 24 q^{8} - q^{9} - q^{10} - 4 q^{11} + 12 q^{12} - 7 q^{13} + q^{14} - 3 q^{15} - 7 q^{16} + q^{17} - 20 q^{18} + 5 q^{19} + 10 q^{20} + 4 q^{21} - 2 q^{22} - 2 q^{23} - 23 q^{24} - 4 q^{25} + 6 q^{26} + 24 q^{27} + 19 q^{28} + q^{29} + 4 q^{30} - 30 q^{32} - 19 q^{33} - 15 q^{34} + 4 q^{35} + 7 q^{36} - 4 q^{37} + 13 q^{38} + 30 q^{39} - 12 q^{40} + 8 q^{41} + 15 q^{42} - q^{43} + 12 q^{44} + 2 q^{45} + 24 q^{46} + 12 q^{47} - 23 q^{48} - 20 q^{49} + 2 q^{50} - 22 q^{51} + 3 q^{52} + 5 q^{53} + 34 q^{54} + 2 q^{55} - 82 q^{56} + 7 q^{57} - 54 q^{58} + 5 q^{59} - 6 q^{60} - 37 q^{62} + 3 q^{63} + 112 q^{64} + 14 q^{65} + 31 q^{66} - 4 q^{67} + 32 q^{68} - 18 q^{69} + q^{70} - 20 q^{71} - 17 q^{72} + 20 q^{73} - 25 q^{74} + 6 q^{75} + 63 q^{76} + 28 q^{77} + 18 q^{78} - 17 q^{79} - 7 q^{80} - 12 q^{81} - 21 q^{82} + 2 q^{83} - 40 q^{84} + q^{85} - 8 q^{86} - 32 q^{87} - 14 q^{88} - 11 q^{89} + 10 q^{90} - 6 q^{91} + q^{92} + 8 q^{93} - 62 q^{94} - 4 q^{95} + 42 q^{96} - q^{97} - 9 q^{98} - 38 q^{99}+O(q^{100})$$ 8 * q - q^2 - 3 * q^3 - 5 * q^4 - 4 * q^5 - 2 * q^6 - 8 * q^7 + 24 * q^8 - q^9 - q^10 - 4 * q^11 + 12 * q^12 - 7 * q^13 + q^14 - 3 * q^15 - 7 * q^16 + q^17 - 20 * q^18 + 5 * q^19 + 10 * q^20 + 4 * q^21 - 2 * q^22 - 2 * q^23 - 23 * q^24 - 4 * q^25 + 6 * q^26 + 24 * q^27 + 19 * q^28 + q^29 + 4 * q^30 - 30 * q^32 - 19 * q^33 - 15 * q^34 + 4 * q^35 + 7 * q^36 - 4 * q^37 + 13 * q^38 + 30 * q^39 - 12 * q^40 + 8 * q^41 + 15 * q^42 - q^43 + 12 * q^44 + 2 * q^45 + 24 * q^46 + 12 * q^47 - 23 * q^48 - 20 * q^49 + 2 * q^50 - 22 * q^51 + 3 * q^52 + 5 * q^53 + 34 * q^54 + 2 * q^55 - 82 * q^56 + 7 * q^57 - 54 * q^58 + 5 * q^59 - 6 * q^60 - 37 * q^62 + 3 * q^63 + 112 * q^64 + 14 * q^65 + 31 * q^66 - 4 * q^67 + 32 * q^68 - 18 * q^69 + q^70 - 20 * q^71 - 17 * q^72 + 20 * q^73 - 25 * q^74 + 6 * q^75 + 63 * q^76 + 28 * q^77 + 18 * q^78 - 17 * q^79 - 7 * q^80 - 12 * q^81 - 21 * q^82 + 2 * q^83 - 40 * q^84 + q^85 - 8 * q^86 - 32 * q^87 - 14 * q^88 - 11 * q^89 + 10 * q^90 - 6 * q^91 + q^92 + 8 * q^93 - 62 * q^94 - 4 * q^95 + 42 * q^96 - q^97 - 9 * q^98 - 38 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -26\nu^{7} - 189\nu^{6} + 729\nu^{5} - 911\nu^{4} + 3051\nu^{3} - 3618\nu^{2} + 14317\nu - 1215 ) / 4243$$ (-26*v^7 - 189*v^6 + 729*v^5 - 911*v^4 + 3051*v^3 - 3618*v^2 + 14317*v - 1215) / 4243 $$\beta_{3}$$ $$=$$ $$( 115\nu^{7} + 20\nu^{6} + 529\nu^{5} + 276\nu^{4} + 3314\nu^{3} + 989\nu^{2} + 483\nu + 1947 ) / 4243$$ (115*v^7 + 20*v^6 + 529*v^5 + 276*v^4 + 3314*v^3 + 989*v^2 + 483*v + 1947) / 4243 $$\beta_{4}$$ $$=$$ $$( -135\nu^{7} + 161\nu^{6} - 621\nu^{5} - 324\nu^{4} - 2599\nu^{3} - 1161\nu^{2} - 567\nu - 11694 ) / 4243$$ (-135*v^7 + 161*v^6 - 621*v^5 - 324*v^4 - 2599*v^3 - 1161*v^2 - 567*v - 11694) / 4243 $$\beta_{5}$$ $$=$$ $$( -649\nu^{7} + 994\nu^{6} - 3834\nu^{5} + 3534\nu^{4} - 16046\nu^{3} + 19028\nu^{2} - 17152\nu + 6390 ) / 12729$$ (-649*v^7 + 994*v^6 - 3834*v^5 + 3534*v^4 - 16046*v^3 + 19028*v^2 - 17152*v + 6390) / 12729 $$\beta_{6}$$ $$=$$ $$( 434\nu^{7} - 109\nu^{6} + 2845\nu^{5} + 193\nu^{4} + 12064\nu^{3} + 338\nu^{2} + 16249\nu + 6573 ) / 4243$$ (434*v^7 - 109*v^6 + 2845*v^5 + 193*v^4 + 12064*v^3 + 338*v^2 + 16249*v + 6573) / 4243 $$\beta_{7}$$ $$=$$ $$( 514\nu^{7} - 833\nu^{6} + 3213\nu^{5} - 3858\nu^{4} + 13447\nu^{3} - 15946\nu^{2} + 16585\nu - 5355 ) / 4243$$ (514*v^7 - 833*v^6 + 3213*v^5 - 3858*v^4 + 13447*v^3 - 15946*v^2 + 16585*v - 5355) / 4243
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} + 3\beta_{5} - \beta_{4} - 3$$ b7 + 3*b5 - b4 - 3 $$\nu^{3}$$ $$=$$ $$-\beta_{6} + 4\beta_{3} + \beta_{2}$$ -b6 + 4*b3 + b2 $$\nu^{4}$$ $$=$$ $$-5\beta_{7} - 12\beta_{5} + \beta_{2}$$ -5*b7 - 12*b5 + b2 $$\nu^{5}$$ $$=$$ $$-\beta_{7} + 6\beta_{6} + \beta_{4} - 17\beta_{3} - 17\beta_1$$ -b7 + 6*b6 + b4 - 17*b3 - 17*b1 $$\nu^{6}$$ $$=$$ $$7\beta_{6} + 23\beta_{4} - \beta_{3} - 7\beta_{2} + 51$$ 7*b6 + 23*b4 - b3 - 7*b2 + 51 $$\nu^{7}$$ $$=$$ $$8\beta_{7} + 3\beta_{5} - 30\beta_{2} + 74\beta_1$$ 8*b7 + 3*b5 - 30*b2 + 74*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/95\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$-1 + \beta_{5}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −0.245959 + 0.426014i 0.689667 − 1.19454i −1.02359 + 1.77290i 1.07988 − 1.87040i −0.245959 − 0.426014i 0.689667 + 1.19454i −1.02359 − 1.77290i 1.07988 + 1.87040i
−1.37901 + 2.38851i −0.745959 + 1.29204i −2.80333 4.85550i −0.500000 + 0.866025i −2.05737 3.56347i −2.84864 9.94721 0.387090 + 0.670459i −1.37901 2.38851i
11.2 −0.548719 + 0.950409i 0.189667 0.328513i 0.397815 + 0.689035i −0.500000 + 0.866025i 0.208148 + 0.360522i 1.89307 −3.06803 1.42805 + 2.47346i −0.548719 0.950409i
11.3 0.595455 1.03136i −1.52359 + 2.63893i 0.290867 + 0.503797i −0.500000 + 0.866025i 1.81445 + 3.14272i −0.609175 3.07461 −3.14263 5.44319i 0.595455 + 1.03136i
11.4 0.832272 1.44154i 0.579878 1.00438i −0.385355 0.667454i −0.500000 + 0.866025i −0.965233 1.67183i −2.43525 2.04621 0.827483 + 1.43324i 0.832272 + 1.44154i
26.1 −1.37901 2.38851i −0.745959 1.29204i −2.80333 + 4.85550i −0.500000 0.866025i −2.05737 + 3.56347i −2.84864 9.94721 0.387090 0.670459i −1.37901 + 2.38851i
26.2 −0.548719 0.950409i 0.189667 + 0.328513i 0.397815 0.689035i −0.500000 0.866025i 0.208148 0.360522i 1.89307 −3.06803 1.42805 2.47346i −0.548719 + 0.950409i
26.3 0.595455 + 1.03136i −1.52359 2.63893i 0.290867 0.503797i −0.500000 0.866025i 1.81445 3.14272i −0.609175 3.07461 −3.14263 + 5.44319i 0.595455 1.03136i
26.4 0.832272 + 1.44154i 0.579878 + 1.00438i −0.385355 + 0.667454i −0.500000 0.866025i −0.965233 + 1.67183i −2.43525 2.04621 0.827483 1.43324i 0.832272 1.44154i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.2.e.c 8
3.b odd 2 1 855.2.k.h 8
4.b odd 2 1 1520.2.q.o 8
5.b even 2 1 475.2.e.e 8
5.c odd 4 2 475.2.j.c 16
19.c even 3 1 inner 95.2.e.c 8
19.c even 3 1 1805.2.a.o 4
19.d odd 6 1 1805.2.a.i 4
57.h odd 6 1 855.2.k.h 8
76.g odd 6 1 1520.2.q.o 8
95.h odd 6 1 9025.2.a.bp 4
95.i even 6 1 475.2.e.e 8
95.i even 6 1 9025.2.a.bg 4
95.m odd 12 2 475.2.j.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.c 8 1.a even 1 1 trivial
95.2.e.c 8 19.c even 3 1 inner
475.2.e.e 8 5.b even 2 1
475.2.e.e 8 95.i even 6 1
475.2.j.c 16 5.c odd 4 2
475.2.j.c 16 95.m odd 12 2
855.2.k.h 8 3.b odd 2 1
855.2.k.h 8 57.h odd 6 1
1520.2.q.o 8 4.b odd 2 1
1520.2.q.o 8 76.g odd 6 1
1805.2.a.i 4 19.d odd 6 1
1805.2.a.o 4 19.c even 3 1
9025.2.a.bg 4 95.i even 6 1
9025.2.a.bp 4 95.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + T_{2}^{7} + 7T_{2}^{6} - 4T_{2}^{5} + 31T_{2}^{4} - 6T_{2}^{3} + 37T_{2}^{2} - 6T_{2} + 36$$ acting on $$S_{2}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + T^{7} + 7 T^{6} - 4 T^{5} + \cdots + 36$$
$3$ $$T^{8} + 3 T^{7} + 11 T^{6} + 4 T^{5} + \cdots + 4$$
$5$ $$(T^{2} + T + 1)^{4}$$
$7$ $$(T^{4} + 4 T^{3} - T^{2} - 15 T - 8)^{2}$$
$11$ $$(T^{4} + 2 T^{3} - 25 T^{2} - 19 T + 3)^{2}$$
$13$ $$T^{8} + 7 T^{7} + 42 T^{6} + 95 T^{5} + \cdots + 256$$
$17$ $$T^{8} - T^{7} + 31 T^{6} + 64 T^{5} + \cdots + 11664$$
$19$ $$T^{8} - 5 T^{7} + 31 T^{6} + \cdots + 130321$$
$23$ $$T^{8} + 2 T^{7} + 21 T^{6} - 48 T^{5} + \cdots + 36$$
$29$ $$T^{8} - T^{7} + 64 T^{6} + 451 T^{5} + \cdots + 19881$$
$31$ $$(T^{4} - 67 T^{2} - 5 T + 1063)^{2}$$
$37$ $$(T^{4} + 2 T^{3} - 31 T^{2} - 123 T - 118)^{2}$$
$41$ $$T^{8} - 8 T^{7} + 151 T^{6} + \cdots + 5008644$$
$43$ $$T^{8} + T^{7} + 99 T^{6} + \cdots + 630436$$
$47$ $$T^{8} - 12 T^{7} + 199 T^{6} + \cdots + 5363856$$
$53$ $$T^{8} - 5 T^{7} + 111 T^{6} + \cdots + 2916$$
$59$ $$T^{8} - 5 T^{7} + 150 T^{6} + \cdots + 3515625$$
$61$ $$T^{8} + 130 T^{6} + 176 T^{5} + \cdots + 9296401$$
$67$ $$T^{8} + 4 T^{7} + 52 T^{6} + \cdots + 4096$$
$71$ $$T^{8} + 20 T^{7} + 309 T^{6} + \cdots + 59049$$
$73$ $$T^{8} - 20 T^{7} + 305 T^{6} + \cdots + 2979076$$
$79$ $$T^{8} + 17 T^{7} + 217 T^{6} + \cdots + 33856$$
$83$ $$(T^{4} - T^{3} - 62 T^{2} + 55 T + 366)^{2}$$
$89$ $$T^{8} + 11 T^{7} + 211 T^{6} + \cdots + 14561856$$
$97$ $$T^{8} + T^{7} + 267 T^{6} + \cdots + 55383364$$