# Properties

 Label 966.2.a.k Level $966$ Weight $2$ Character orbit 966.a Self dual yes Analytic conductor $7.714$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(1,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("966.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.a (trivial)

## 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: yes Analytic conductor: $$7.71354883526$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 1.00000 1.00000 3.00000 1.00000 −1.00000 1.00000 1.00000 3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$+1$$
$$23$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.a.k 1
3.b odd 2 1 2898.2.a.a 1
4.b odd 2 1 7728.2.a.j 1
7.b odd 2 1 6762.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.k 1 1.a even 1 1 trivial
2898.2.a.a 1 3.b odd 2 1
6762.2.a.y 1 7.b odd 2 1
7728.2.a.j 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(966))$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{11} - 4$$ T11 - 4 $$T_{13} + 3$$ T13 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T - 3$$
$7$ $$T + 1$$
$11$ $$T - 4$$
$13$ $$T + 3$$
$17$ $$T + 4$$
$19$ $$T$$
$23$ $$T - 1$$
$29$ $$T - 3$$
$31$ $$T + 6$$
$37$ $$T + 9$$
$41$ $$T - 9$$
$43$ $$T + 3$$
$47$ $$T + 7$$
$53$ $$T + 4$$
$59$ $$T - 6$$
$61$ $$T - 10$$
$67$ $$T - 4$$
$71$ $$T + 6$$
$73$ $$T + 8$$
$79$ $$T - 8$$
$83$ $$T - 4$$
$89$ $$T + 14$$
$97$ $$T + 7$$