# Properties

 Label 966.2.l.a Level $966$ Weight $2$ Character orbit 966.l Analytic conductor $7.714$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(47,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5, 0]))

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

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

chi := DirichletCharacter("966.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.l (of order $$6$$, 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: $$7.71354883526$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$323$$ $$829$$ $$925$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{2}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 0.500000i −0.866025 1.50000i 0.500000 + 0.866025i 0.866025 1.50000i 1.73205i 0.500000 + 2.59808i 1.00000i −1.50000 + 2.59808i −1.50000 + 0.866025i
47.2 0.866025 + 0.500000i 0.866025 + 1.50000i 0.500000 + 0.866025i −0.866025 + 1.50000i 1.73205i 0.500000 + 2.59808i 1.00000i −1.50000 + 2.59808i −1.50000 + 0.866025i
185.1 −0.866025 + 0.500000i −0.866025 + 1.50000i 0.500000 0.866025i 0.866025 + 1.50000i 1.73205i 0.500000 2.59808i 1.00000i −1.50000 2.59808i −1.50000 0.866025i
185.2 0.866025 0.500000i 0.866025 1.50000i 0.500000 0.866025i −0.866025 1.50000i 1.73205i 0.500000 2.59808i 1.00000i −1.50000 2.59808i −1.50000 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.l.a 4
3.b odd 2 1 inner 966.2.l.a 4
7.d odd 6 1 inner 966.2.l.a 4
21.g even 6 1 inner 966.2.l.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.l.a 4 1.a even 1 1 trivial
966.2.l.a 4 3.b odd 2 1 inner
966.2.l.a 4 7.d odd 6 1 inner
966.2.l.a 4 21.g even 6 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4} + 3T^{2} + 9$$
$7$ $$(T^{2} - T + 7)^{2}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 3)^{2}$$
$17$ $$T^{4} + 27T^{2} + 729$$
$19$ $$(T^{2} - 6 T + 12)^{2}$$
$23$ $$T^{4} - T^{2} + 1$$
$29$ $$(T^{2} + 36)^{2}$$
$31$ $$(T^{2} - 6 T + 12)^{2}$$
$37$ $$(T^{2} + 4 T + 16)^{2}$$
$41$ $$T^{4}$$
$43$ $$(T - 4)^{4}$$
$47$ $$T^{4} + 27T^{2} + 729$$
$53$ $$T^{4} - 81T^{2} + 6561$$
$59$ $$T^{4} + 108 T^{2} + 11664$$
$61$ $$(T^{2} - 24 T + 192)^{2}$$
$67$ $$(T^{2} + 5 T + 25)^{2}$$
$71$ $$(T^{2} + 225)^{2}$$
$73$ $$(T^{2} - 21 T + 147)^{2}$$
$79$ $$(T^{2} + 8 T + 64)^{2}$$
$83$ $$(T^{2} - 48)^{2}$$
$89$ $$T^{4} + 48T^{2} + 2304$$
$97$ $$(T^{2} + 108)^{2}$$