# Properties

 Label 966.2.q.b Level $966$ Weight $2$ Character orbit 966.q Analytic conductor $7.714$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(85,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 0, 8]))

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

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

chi := DirichletCharacter("966.85");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.q (of order $$11$$, degree $$10$$, 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: $$7.71354883526$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\Q(\zeta_{22})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-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 primitive root of unity $$\zeta_{22}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/966\mathbb{Z}\right)^\times$$.

 $$n$$ $$323$$ $$829$$ $$925$$ $$\chi(n)$$ $$1$$ $$1$$ $$\zeta_{22}^{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
85.1
 −0.841254 − 0.540641i 0.142315 + 0.989821i −0.415415 + 0.909632i 0.959493 + 0.281733i −0.415415 − 0.909632i 0.654861 + 0.755750i 0.959493 − 0.281733i 0.142315 − 0.989821i −0.841254 + 0.540641i 0.654861 − 0.755750i
−0.654861 + 0.755750i 0.841254 0.540641i −0.142315 0.989821i 0.544078 + 1.19136i −0.142315 + 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −1.25667 0.368991i
127.1 0.841254 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i 1.61435 0.474017i 0.415415 + 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −0.959493 0.281733i 1.10181 1.27155i
169.1 −0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −0.186393 0.215109i −0.959493 + 0.281733i −0.841254 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i 0.239446 0.153882i
211.1 0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i −0.698939 0.449181i −0.654861 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 0.841254 0.540641i 0.118239 0.822373i
463.1 −0.142315 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i −0.186393 + 0.215109i −0.959493 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i 0.239446 + 0.153882i
547.1 −0.959493 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i −0.273100 + 1.89945i 0.841254 0.540641i −0.415415 0.909632i −0.654861 0.755750i −0.142315 0.989821i 0.797176 1.74557i
673.1 0.415415 0.909632i −0.959493 0.281733i −0.654861 0.755750i −0.698939 + 0.449181i −0.654861 + 0.755750i 0.142315 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.118239 + 0.822373i
715.1 0.841254 + 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i 1.61435 + 0.474017i 0.415415 0.909632i 0.654861 0.755750i −0.142315 + 0.989821i −0.959493 + 0.281733i 1.10181 + 1.27155i
841.1 −0.654861 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i 0.544078 1.19136i −0.142315 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i −1.25667 + 0.368991i
883.1 −0.959493 + 0.281733i −0.654861 0.755750i 0.841254 0.540641i −0.273100 1.89945i 0.841254 + 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.797176 + 1.74557i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 85.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.q.b 10
23.c even 11 1 inner 966.2.q.b 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.q.b 10 1.a even 1 1 trivial
966.2.q.b 10 23.c even 11 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} - 2T_{5}^{9} + 4T_{5}^{8} - 8T_{5}^{7} + 5T_{5}^{6} + T_{5}^{5} - 2T_{5}^{4} + 4T_{5}^{3} + 14T_{5}^{2} + 5T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + T^{9} + \cdots + 1$$
$3$ $$T^{10} + T^{9} + \cdots + 1$$
$5$ $$T^{10} - 2 T^{9} + \cdots + 1$$
$7$ $$T^{10} - T^{9} + \cdots + 1$$
$11$ $$T^{10} - 7 T^{9} + \cdots + 4489$$
$13$ $$T^{10} + 2 T^{9} + \cdots + 1849$$
$17$ $$T^{10} + 6 T^{9} + \cdots + 7921$$
$19$ $$T^{10} - 11 T^{9} + \cdots + 64009$$
$23$ $$T^{10} - 21 T^{9} + \cdots + 6436343$$
$29$ $$T^{10} + 20 T^{9} + \cdots + 62742241$$
$31$ $$T^{10} + \cdots + 122921569$$
$37$ $$T^{10} + \cdots + 108722329$$
$41$ $$T^{10} - 6 T^{9} + \cdots + 1$$
$43$ $$T^{10} + 12 T^{9} + \cdots + 326041$$
$47$ $$(T^{5} + 16 T^{4} + \cdots - 7963)^{2}$$
$53$ $$T^{10} + \cdots + 1256206249$$
$59$ $$T^{10} - 15 T^{9} + \cdots + 1067089$$
$61$ $$T^{10} - 21 T^{9} + \cdots + 73599241$$
$67$ $$T^{10} + 26 T^{9} + \cdots + 591361$$
$71$ $$T^{10} + \cdots + 3335178001$$
$73$ $$T^{10} + 2 T^{9} + \cdots + 4932841$$
$79$ $$T^{10} - 47 T^{9} + \cdots + 214369$$
$83$ $$T^{10} - 5 T^{9} + \cdots + 6285049$$
$89$ $$T^{10} + \cdots + 392951329$$
$97$ $$T^{10} - 2 T^{9} + \cdots + 5139289$$