# Properties

 Label 966.2.a.c 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 $$q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} - q^{12} - 4 q^{13} - q^{14} - 2 q^{15} + q^{16} - 4 q^{17} - q^{18} - 2 q^{19} + 2 q^{20} - q^{21} + 4 q^{22} + q^{23} + q^{24} - q^{25} + 4 q^{26} - q^{27} + q^{28} - 6 q^{29} + 2 q^{30} + 6 q^{31} - q^{32} + 4 q^{33} + 4 q^{34} + 2 q^{35} + q^{36} - 2 q^{37} + 2 q^{38} + 4 q^{39} - 2 q^{40} - 10 q^{41} + q^{42} + 8 q^{43} - 4 q^{44} + 2 q^{45} - q^{46} + 10 q^{47} - q^{48} + q^{49} + q^{50} + 4 q^{51} - 4 q^{52} - 6 q^{53} + q^{54} - 8 q^{55} - q^{56} + 2 q^{57} + 6 q^{58} - 12 q^{59} - 2 q^{60} - 10 q^{61} - 6 q^{62} + q^{63} + q^{64} - 8 q^{65} - 4 q^{66} + 8 q^{67} - 4 q^{68} - q^{69} - 2 q^{70} - 4 q^{71} - q^{72} - 2 q^{73} + 2 q^{74} + q^{75} - 2 q^{76} - 4 q^{77} - 4 q^{78} - 8 q^{79} + 2 q^{80} + q^{81} + 10 q^{82} - 6 q^{83} - q^{84} - 8 q^{85} - 8 q^{86} + 6 q^{87} + 4 q^{88} - 2 q^{90} - 4 q^{91} + q^{92} - 6 q^{93} - 10 q^{94} - 4 q^{95} + q^{96} - 8 q^{97} - q^{98} - 4 q^{99}+O(q^{100})$$ q - q^2 - q^3 + q^4 + 2 * q^5 + q^6 + q^7 - q^8 + q^9 - 2 * q^10 - 4 * q^11 - q^12 - 4 * q^13 - q^14 - 2 * q^15 + q^16 - 4 * q^17 - q^18 - 2 * q^19 + 2 * q^20 - q^21 + 4 * q^22 + q^23 + q^24 - q^25 + 4 * q^26 - q^27 + q^28 - 6 * q^29 + 2 * q^30 + 6 * q^31 - q^32 + 4 * q^33 + 4 * q^34 + 2 * q^35 + q^36 - 2 * q^37 + 2 * q^38 + 4 * q^39 - 2 * q^40 - 10 * q^41 + q^42 + 8 * q^43 - 4 * q^44 + 2 * q^45 - q^46 + 10 * q^47 - q^48 + q^49 + q^50 + 4 * q^51 - 4 * q^52 - 6 * q^53 + q^54 - 8 * q^55 - q^56 + 2 * q^57 + 6 * q^58 - 12 * q^59 - 2 * q^60 - 10 * q^61 - 6 * q^62 + q^63 + q^64 - 8 * q^65 - 4 * q^66 + 8 * q^67 - 4 * q^68 - q^69 - 2 * q^70 - 4 * q^71 - q^72 - 2 * q^73 + 2 * q^74 + q^75 - 2 * q^76 - 4 * q^77 - 4 * q^78 - 8 * q^79 + 2 * q^80 + q^81 + 10 * q^82 - 6 * q^83 - q^84 - 8 * q^85 - 8 * q^86 + 6 * q^87 + 4 * q^88 - 2 * q^90 - 4 * q^91 + q^92 - 6 * q^93 - 10 * q^94 - 4 * q^95 + q^96 - 8 * 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 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.c 1
3.b odd 2 1 2898.2.a.l 1
4.b odd 2 1 7728.2.a.t 1
7.b odd 2 1 6762.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.c 1 1.a even 1 1 trivial
2898.2.a.l 1 3.b odd 2 1
6762.2.a.m 1 7.b odd 2 1
7728.2.a.t 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 $$T_{11} + 4$$ T11 + 4 $$T_{13} + 4$$ T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T - 2$$
$7$ $$T - 1$$
$11$ $$T + 4$$
$13$ $$T + 4$$
$17$ $$T + 4$$
$19$ $$T + 2$$
$23$ $$T - 1$$
$29$ $$T + 6$$
$31$ $$T - 6$$
$37$ $$T + 2$$
$41$ $$T + 10$$
$43$ $$T - 8$$
$47$ $$T - 10$$
$53$ $$T + 6$$
$59$ $$T + 12$$
$61$ $$T + 10$$
$67$ $$T - 8$$
$71$ $$T + 4$$
$73$ $$T + 2$$
$79$ $$T + 8$$
$83$ $$T + 6$$
$89$ $$T$$
$97$ $$T + 8$$