# Properties

 Label 966.2.a.f 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - q^6 + q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} - q^{6} + q^{7} - q^{8} + q^{9} + 6 q^{11} + q^{12} + 2 q^{13} - q^{14} + q^{16} - 6 q^{17} - q^{18} + 2 q^{19} + q^{21} - 6 q^{22} + q^{23} - q^{24} - 5 q^{25} - 2 q^{26} + q^{27} + q^{28} - 6 q^{29} + 8 q^{31} - q^{32} + 6 q^{33} + 6 q^{34} + q^{36} + 8 q^{37} - 2 q^{38} + 2 q^{39} + 6 q^{41} - q^{42} + 2 q^{43} + 6 q^{44} - q^{46} + q^{48} + q^{49} + 5 q^{50} - 6 q^{51} + 2 q^{52} - 12 q^{53} - q^{54} - q^{56} + 2 q^{57} + 6 q^{58} + 8 q^{61} - 8 q^{62} + q^{63} + q^{64} - 6 q^{66} - 10 q^{67} - 6 q^{68} + q^{69} - q^{72} + 14 q^{73} - 8 q^{74} - 5 q^{75} + 2 q^{76} + 6 q^{77} - 2 q^{78} + 8 q^{79} + q^{81} - 6 q^{82} + 6 q^{83} + q^{84} - 2 q^{86} - 6 q^{87} - 6 q^{88} + 6 q^{89} + 2 q^{91} + q^{92} + 8 q^{93} - q^{96} - 10 q^{97} - q^{98} + 6 q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 - q^6 + q^7 - q^8 + q^9 + 6 * q^11 + q^12 + 2 * q^13 - q^14 + q^16 - 6 * q^17 - q^18 + 2 * q^19 + q^21 - 6 * q^22 + q^23 - q^24 - 5 * q^25 - 2 * q^26 + q^27 + q^28 - 6 * q^29 + 8 * q^31 - q^32 + 6 * q^33 + 6 * q^34 + q^36 + 8 * q^37 - 2 * q^38 + 2 * q^39 + 6 * q^41 - q^42 + 2 * q^43 + 6 * q^44 - q^46 + q^48 + q^49 + 5 * q^50 - 6 * q^51 + 2 * q^52 - 12 * q^53 - q^54 - q^56 + 2 * q^57 + 6 * q^58 + 8 * q^61 - 8 * q^62 + q^63 + q^64 - 6 * q^66 - 10 * q^67 - 6 * q^68 + q^69 - q^72 + 14 * q^73 - 8 * q^74 - 5 * q^75 + 2 * q^76 + 6 * q^77 - 2 * q^78 + 8 * q^79 + q^81 - 6 * q^82 + 6 * q^83 + q^84 - 2 * q^86 - 6 * q^87 - 6 * q^88 + 6 * q^89 + 2 * q^91 + q^92 + 8 * q^93 - q^96 - 10 * q^97 - q^98 + 6 * 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 0 −1.00000 1.00000 −1.00000 1.00000 0
 $$n$$: 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.f 1
3.b odd 2 1 2898.2.a.o 1
4.b odd 2 1 7728.2.a.c 1
7.b odd 2 1 6762.2.a.h 1

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

 $$T_{5}$$ T5 $$T_{11} - 6$$ T11 - 6 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T - 6$$
$13$ $$T - 2$$
$17$ $$T + 6$$
$19$ $$T - 2$$
$23$ $$T - 1$$
$29$ $$T + 6$$
$31$ $$T - 8$$
$37$ $$T - 8$$
$41$ $$T - 6$$
$43$ $$T - 2$$
$47$ $$T$$
$53$ $$T + 12$$
$59$ $$T$$
$61$ $$T - 8$$
$67$ $$T + 10$$
$71$ $$T$$
$73$ $$T - 14$$
$79$ $$T - 8$$
$83$ $$T - 6$$
$89$ $$T - 6$$
$97$ $$T + 10$$