# Properties

 Label 966.2.a.b Level $966$ Weight $2$ Character orbit 966.a Self dual yes Analytic conductor $7.714$ Analytic rank $1$ 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: $$1$$ 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)

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

## 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 −2.00000 1.00000 −1.00000 −1.00000 1.00000 2.00000
 $$n$$: 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.b 1
3.b odd 2 1 2898.2.a.r 1
4.b odd 2 1 7728.2.a.p 1
7.b odd 2 1 6762.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.b 1 1.a even 1 1 trivial
2898.2.a.r 1 3.b odd 2 1
6762.2.a.r 1 7.b odd 2 1
7728.2.a.p 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(966))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{11}$$ T11 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T + 2$$
$7$ $$T + 1$$
$11$ $$T$$
$13$ $$T - 4$$
$17$ $$T$$
$19$ $$T - 6$$
$23$ $$T + 1$$
$29$ $$T + 6$$
$31$ $$T + 10$$
$37$ $$T + 6$$
$41$ $$T + 2$$
$43$ $$T - 12$$
$47$ $$T - 10$$
$53$ $$T + 10$$
$59$ $$T + 12$$
$61$ $$T + 14$$
$67$ $$T + 12$$
$71$ $$T + 12$$
$73$ $$T - 14$$
$79$ $$T$$
$83$ $$T - 2$$
$89$ $$T + 12$$
$97$ $$T - 12$$