# Properties

 Label 95.2.e.b Level $95$ Weight $2$ Character orbit 95.e Analytic conductor $0.759$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.3518667.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49$$ x^6 - x^5 + 7*x^4 - 8*x^3 + 43*x^2 - 42*x + 49 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 7\nu^{4} - 49\nu^{3} + 43\nu^{2} - 42\nu + 294 ) / 259$$ (-v^5 + 7*v^4 - 49*v^3 + 43*v^2 - 42*v + 294) / 259 $$\beta_{3}$$ $$=$$ $$( -6\nu^{5} + 5\nu^{4} - 35\nu^{3} - \nu^{2} - 215\nu - 49 ) / 259$$ (-6*v^5 + 5*v^4 - 35*v^3 - v^2 - 215*v - 49) / 259 $$\beta_{4}$$ $$=$$ $$( -5\nu^{5} + 35\nu^{4} + 14\nu^{3} + 215\nu^{2} - 210\nu + 952 ) / 259$$ (-5*v^5 + 35*v^4 + 14*v^3 + 215*v^2 - 210*v + 952) / 259 $$\beta_{5}$$ $$=$$ $$( 18\nu^{5} + 22\nu^{4} + 105\nu^{3} + 3\nu^{2} + 608\nu + 147 ) / 259$$ (18*v^5 + 22*v^4 + 105*v^3 + 3*v^2 + 608*v + 147) / 259
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} - 4\beta_{3} + \beta_{2} - 5$$ -b5 + b4 - 4*b3 + b2 - 5 $$\nu^{3}$$ $$=$$ $$\beta_{4} - 5\beta_{2} + 2$$ b4 - 5*b2 + 2 $$\nu^{4}$$ $$=$$ $$7\beta_{5} + 21\beta_{3} + \beta_1$$ 7*b5 + 21*b3 + b1 $$\nu^{5}$$ $$=$$ $$6\beta_{5} - 6\beta_{4} - 25\beta_{3} + 29\beta_{2} - 35\beta _1 - 19$$ 6*b5 - 6*b4 - 25*b3 + 29*b2 - 35*b1 - 19

## Character values

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

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$-1 - \beta_{3}$$ $$1$$

## 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
 1.14257 − 1.97899i 0.610938 − 1.05818i −1.25351 + 2.17114i 1.14257 + 1.97899i 0.610938 + 1.05818i −1.25351 − 2.17114i
−1.14257 + 1.97899i 1.25351 2.17114i −1.61094 2.79023i 0.500000 0.866025i 2.86445 + 4.96137i 3.50702 2.79216 −1.64257 2.84502i 1.14257 + 1.97899i
11.2 −0.610938 + 1.05818i −1.14257 + 1.97899i 0.253509 + 0.439091i 0.500000 0.866025i −1.39608 2.41808i −1.28514 −3.06327 −1.11094 1.92420i 0.610938 + 1.05818i
11.3 1.25351 2.17114i −0.610938 + 1.05818i −2.14257 3.71104i 0.500000 0.866025i 1.53163 + 2.65287i −0.221876 −5.72889 0.753509 + 1.30512i −1.25351 2.17114i
26.1 −1.14257 1.97899i 1.25351 + 2.17114i −1.61094 + 2.79023i 0.500000 + 0.866025i 2.86445 4.96137i 3.50702 2.79216 −1.64257 + 2.84502i 1.14257 1.97899i
26.2 −0.610938 1.05818i −1.14257 1.97899i 0.253509 0.439091i 0.500000 + 0.866025i −1.39608 + 2.41808i −1.28514 −3.06327 −1.11094 + 1.92420i 0.610938 1.05818i
26.3 1.25351 + 2.17114i −0.610938 1.05818i −2.14257 + 3.71104i 0.500000 + 0.866025i 1.53163 2.65287i −0.221876 −5.72889 0.753509 1.30512i −1.25351 + 2.17114i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.3 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.b 6
3.b odd 2 1 855.2.k.g 6
4.b odd 2 1 1520.2.q.j 6
5.b even 2 1 475.2.e.d 6
5.c odd 4 2 475.2.j.b 12
19.c even 3 1 inner 95.2.e.b 6
19.c even 3 1 1805.2.a.h 3
19.d odd 6 1 1805.2.a.g 3
57.h odd 6 1 855.2.k.g 6
76.g odd 6 1 1520.2.q.j 6
95.h odd 6 1 9025.2.a.ba 3
95.i even 6 1 475.2.e.d 6
95.i even 6 1 9025.2.a.z 3
95.m odd 12 2 475.2.j.b 12

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + T_{2}^{5} + 7T_{2}^{4} + 8T_{2}^{3} + 43T_{2}^{2} + 42T_{2} + 49$$ acting on $$S_{2}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{5} + 7 T^{4} + 8 T^{3} + \cdots + 49$$
$3$ $$T^{6} + T^{5} + 7 T^{4} + 8 T^{3} + \cdots + 49$$
$5$ $$(T^{2} - T + 1)^{3}$$
$7$ $$(T^{3} - 2 T^{2} - 5 T - 1)^{2}$$
$11$ $$(T^{3} + 5 T^{2} + 2 T - 1)^{2}$$
$13$ $$(T^{2} - 5 T + 25)^{3}$$
$17$ $$T^{6} + T^{5} + 45 T^{4} - 30 T^{3} + \cdots + 49$$
$19$ $$T^{6} + 133T^{3} + 6859$$
$23$ $$T^{6} + 4 T^{5} + 55 T^{4} + \cdots + 2401$$
$29$ $$T^{6} - 2 T^{5} + 9 T^{4} + 12 T^{3} + \cdots + 1$$
$31$ $$(T^{3} + T^{2} - 6 T - 7)^{2}$$
$37$ $$(T^{3} + 2 T^{2} - 119 T - 227)^{2}$$
$41$ $$T^{6} - 2 T^{5} + 47 T^{4} + \cdots + 1369$$
$43$ $$T^{6} - T^{5} + 45 T^{4} - 198 T^{3} + \cdots + 14641$$
$47$ $$T^{6} + 6 T^{5} + 43 T^{4} + \cdots + 2401$$
$53$ $$T^{6} + 11 T^{5} + 163 T^{4} + \cdots + 96721$$
$59$ $$T^{6} + 6 T^{5} + 43 T^{4} + \cdots + 2401$$
$61$ $$T^{6} - 9 T^{5} + 130 T^{4} + \cdots + 2401$$
$67$ $$T^{6} - 20 T^{5} + 292 T^{4} + \cdots + 7744$$
$71$ $$T^{6} - 29 T^{5} + 605 T^{4} + \cdots + 218089$$
$73$ $$T^{6} - 22 T^{5} + 367 T^{4} + \cdots + 5929$$
$79$ $$T^{6} - 24 T^{5} + 460 T^{4} + \cdots + 61504$$
$83$ $$(T^{3} + 3 T^{2} - 54 T + 77)^{2}$$
$89$ $$T^{6} - 14 T^{5} + 232 T^{4} + \cdots + 3136$$
$97$ $$T^{6} + 7 T^{5} + 115 T^{4} + \cdots + 14641$$