# Properties

 Label 966.2.l.b 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.

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

## 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 1.50000 0.866025i 0.500000 + 0.866025i −1.73205 + 3.00000i −1.73205 −2.50000 0.866025i 1.00000i 1.50000 2.59808i 3.00000 1.73205i
47.2 0.866025 + 0.500000i 1.50000 0.866025i 0.500000 + 0.866025i 1.73205 3.00000i 1.73205 −2.50000 0.866025i 1.00000i 1.50000 2.59808i 3.00000 1.73205i
185.1 −0.866025 + 0.500000i 1.50000 + 0.866025i 0.500000 0.866025i −1.73205 3.00000i −1.73205 −2.50000 + 0.866025i 1.00000i 1.50000 + 2.59808i 3.00000 + 1.73205i
185.2 0.866025 0.500000i 1.50000 + 0.866025i 0.500000 0.866025i 1.73205 + 3.00000i 1.73205 −2.50000 + 0.866025i 1.00000i 1.50000 + 2.59808i 3.00000 + 1.73205i
 $$n$$: 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.b 4
3.b odd 2 1 inner 966.2.l.b 4
7.d odd 6 1 inner 966.2.l.b 4
21.g even 6 1 inner 966.2.l.b 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 12T_{5}^{2} + 144$$ acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$(T^{2} - 3 T + 3)^{2}$$
$5$ $$T^{4} + 12T^{2} + 144$$
$7$ $$(T^{2} + 5 T + 7)^{2}$$
$11$ $$T^{4} - 9T^{2} + 81$$
$13$ $$(T^{2} + 12)^{2}$$
$17$ $$T^{4} + 3T^{2} + 9$$
$19$ $$(T^{2} - 12 T + 48)^{2}$$
$23$ $$T^{4} - T^{2} + 1$$
$29$ $$(T^{2} + 9)^{2}$$
$31$ $$(T^{2} - 6 T + 12)^{2}$$
$37$ $$(T^{2} - 2 T + 4)^{2}$$
$41$ $$(T^{2} - 12)^{2}$$
$43$ $$(T - 10)^{4}$$
$47$ $$T^{4} + 75T^{2} + 5625$$
$53$ $$T^{4} - 144 T^{2} + 20736$$
$59$ $$T^{4} + 192 T^{2} + 36864$$
$61$ $$(T^{2} + 24 T + 192)^{2}$$
$67$ $$(T^{2} - 4 T + 16)^{2}$$
$71$ $$(T^{2} + 81)^{2}$$
$73$ $$(T^{2} - 15 T + 75)^{2}$$
$79$ $$(T^{2} + 11 T + 121)^{2}$$
$83$ $$(T^{2} - 12)^{2}$$
$89$ $$T^{4} + 48T^{2} + 2304$$
$97$ $$(T^{2} + 12)^{2}$$