# Properties

 Label 966.2.a.h 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 + q^7 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - q^{12} - q^{13} + q^{14} - q^{15} + q^{16} + 4 q^{17} + q^{18} + 4 q^{19} + q^{20} - q^{21} + q^{23} - q^{24} - 4 q^{25} - q^{26} - q^{27} + q^{28} - q^{29} - q^{30} + 10 q^{31} + q^{32} + 4 q^{34} + q^{35} + q^{36} - 5 q^{37} + 4 q^{38} + q^{39} + q^{40} + 7 q^{41} - q^{42} - 3 q^{43} + q^{45} + q^{46} + 3 q^{47} - q^{48} + q^{49} - 4 q^{50} - 4 q^{51} - q^{52} - q^{54} + q^{56} - 4 q^{57} - q^{58} + 6 q^{59} - q^{60} - 6 q^{61} + 10 q^{62} + q^{63} + q^{64} - q^{65} + 4 q^{67} + 4 q^{68} - q^{69} + q^{70} - 2 q^{71} + q^{72} - 4 q^{73} - 5 q^{74} + 4 q^{75} + 4 q^{76} + q^{78} + 12 q^{79} + q^{80} + q^{81} + 7 q^{82} + 4 q^{83} - q^{84} + 4 q^{85} - 3 q^{86} + q^{87} - 6 q^{89} + q^{90} - q^{91} + q^{92} - 10 q^{93} + 3 q^{94} + 4 q^{95} - q^{96} - q^{97} + q^{98}+O(q^{100})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 + q^7 + q^8 + q^9 + q^10 - q^12 - q^13 + q^14 - q^15 + q^16 + 4 * q^17 + q^18 + 4 * q^19 + q^20 - q^21 + q^23 - q^24 - 4 * q^25 - q^26 - q^27 + q^28 - q^29 - q^30 + 10 * q^31 + q^32 + 4 * q^34 + q^35 + q^36 - 5 * q^37 + 4 * q^38 + q^39 + q^40 + 7 * q^41 - q^42 - 3 * q^43 + q^45 + q^46 + 3 * q^47 - q^48 + q^49 - 4 * q^50 - 4 * q^51 - q^52 - q^54 + q^56 - 4 * q^57 - q^58 + 6 * q^59 - q^60 - 6 * q^61 + 10 * q^62 + q^63 + q^64 - q^65 + 4 * q^67 + 4 * q^68 - q^69 + q^70 - 2 * q^71 + q^72 - 4 * q^73 - 5 * q^74 + 4 * q^75 + 4 * q^76 + q^78 + 12 * q^79 + q^80 + q^81 + 7 * q^82 + 4 * q^83 - q^84 + 4 * q^85 - 3 * q^86 + q^87 - 6 * q^89 + q^90 - q^91 + q^92 - 10 * q^93 + 3 * q^94 + 4 * q^95 - q^96 - 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 1.00000 −1.00000 1.00000 1.00000 1.00000 1.00000
 $$n$$: 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.h 1
3.b odd 2 1 2898.2.a.d 1
4.b odd 2 1 7728.2.a.s 1
7.b odd 2 1 6762.2.a.bi 1

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

 $$T_{5} - 1$$ T5 - 1 $$T_{11}$$ T11 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

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