# Properties

 Label 966.2.i.f Level $966$ Weight $2$ Character orbit 966.i Analytic conductor $7.714$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(277,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("966.277");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.i (of order $$3$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$7.71354883526$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (\zeta_{6} - 1) q^{4} - q^{6} + ( - 3 \zeta_{6} + 1) q^{7} - q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + z * q^2 + (z - 1) * q^3 + (z - 1) * q^4 - q^6 + (-3*z + 1) * q^7 - q^8 - z * q^9 $$q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (\zeta_{6} - 1) q^{4} - q^{6} + ( - 3 \zeta_{6} + 1) q^{7} - q^{8} - \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{11} - \zeta_{6} q^{12} + 2 q^{13} + ( - 2 \zeta_{6} + 3) q^{14} - \zeta_{6} q^{16} + ( - 3 \zeta_{6} + 3) q^{17} + ( - \zeta_{6} + 1) q^{18} - 2 \zeta_{6} q^{19} + (\zeta_{6} + 2) q^{21} + 3 q^{22} + \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{24} + ( - 5 \zeta_{6} + 5) q^{25} + 2 \zeta_{6} q^{26} + q^{27} + (\zeta_{6} + 2) q^{28} + 3 q^{29} + (2 \zeta_{6} - 2) q^{31} + ( - \zeta_{6} + 1) q^{32} + 3 \zeta_{6} q^{33} + 3 q^{34} + q^{36} - 2 \zeta_{6} q^{37} + ( - 2 \zeta_{6} + 2) q^{38} + (2 \zeta_{6} - 2) q^{39} + 6 q^{41} + (3 \zeta_{6} - 1) q^{42} + 2 q^{43} + 3 \zeta_{6} q^{44} + (\zeta_{6} - 1) q^{46} + 3 \zeta_{6} q^{47} + q^{48} + (3 \zeta_{6} - 8) q^{49} + 5 q^{50} + 3 \zeta_{6} q^{51} + (2 \zeta_{6} - 2) q^{52} + ( - 6 \zeta_{6} + 6) q^{53} + \zeta_{6} q^{54} + (3 \zeta_{6} - 1) q^{56} + 2 q^{57} + 3 \zeta_{6} q^{58} - 2 \zeta_{6} q^{61} - 2 q^{62} + (2 \zeta_{6} - 3) q^{63} + q^{64} + (3 \zeta_{6} - 3) q^{66} + (2 \zeta_{6} - 2) q^{67} + 3 \zeta_{6} q^{68} - q^{69} + 3 q^{71} + \zeta_{6} q^{72} + (11 \zeta_{6} - 11) q^{73} + ( - 2 \zeta_{6} + 2) q^{74} + 5 \zeta_{6} q^{75} + 2 q^{76} + ( - 3 \zeta_{6} - 6) q^{77} - 2 q^{78} - 11 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 6 \zeta_{6} q^{82} + (2 \zeta_{6} - 3) q^{84} + 2 \zeta_{6} q^{86} + (3 \zeta_{6} - 3) q^{87} + (3 \zeta_{6} - 3) q^{88} + 6 \zeta_{6} q^{89} + ( - 6 \zeta_{6} + 2) q^{91} - q^{92} - 2 \zeta_{6} q^{93} + (3 \zeta_{6} - 3) q^{94} + \zeta_{6} q^{96} + 8 q^{97} + ( - 5 \zeta_{6} - 3) q^{98} - 3 q^{99} +O(q^{100})$$ q + z * q^2 + (z - 1) * q^3 + (z - 1) * q^4 - q^6 + (-3*z + 1) * q^7 - q^8 - z * q^9 + (-3*z + 3) * q^11 - z * q^12 + 2 * q^13 + (-2*z + 3) * q^14 - z * q^16 + (-3*z + 3) * q^17 + (-z + 1) * q^18 - 2*z * q^19 + (z + 2) * q^21 + 3 * q^22 + z * q^23 + (-z + 1) * q^24 + (-5*z + 5) * q^25 + 2*z * q^26 + q^27 + (z + 2) * q^28 + 3 * q^29 + (2*z - 2) * q^31 + (-z + 1) * q^32 + 3*z * q^33 + 3 * q^34 + q^36 - 2*z * q^37 + (-2*z + 2) * q^38 + (2*z - 2) * q^39 + 6 * q^41 + (3*z - 1) * q^42 + 2 * q^43 + 3*z * q^44 + (z - 1) * q^46 + 3*z * q^47 + q^48 + (3*z - 8) * q^49 + 5 * q^50 + 3*z * q^51 + (2*z - 2) * q^52 + (-6*z + 6) * q^53 + z * q^54 + (3*z - 1) * q^56 + 2 * q^57 + 3*z * q^58 - 2*z * q^61 - 2 * q^62 + (2*z - 3) * q^63 + q^64 + (3*z - 3) * q^66 + (2*z - 2) * q^67 + 3*z * q^68 - q^69 + 3 * q^71 + z * q^72 + (11*z - 11) * q^73 + (-2*z + 2) * q^74 + 5*z * q^75 + 2 * q^76 + (-3*z - 6) * q^77 - 2 * q^78 - 11*z * q^79 + (z - 1) * q^81 + 6*z * q^82 + (2*z - 3) * q^84 + 2*z * q^86 + (3*z - 3) * q^87 + (3*z - 3) * q^88 + 6*z * q^89 + (-6*z + 2) * q^91 - q^92 - 2*z * q^93 + (3*z - 3) * q^94 + z * q^96 + 8 * q^97 + (-5*z - 3) * q^98 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{3} - q^{4} - 2 q^{6} - q^{7} - 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 - q^3 - q^4 - 2 * q^6 - q^7 - 2 * q^8 - q^9 $$2 q + q^{2} - q^{3} - q^{4} - 2 q^{6} - q^{7} - 2 q^{8} - q^{9} + 3 q^{11} - q^{12} + 4 q^{13} + 4 q^{14} - q^{16} + 3 q^{17} + q^{18} - 2 q^{19} + 5 q^{21} + 6 q^{22} + q^{23} + q^{24} + 5 q^{25} + 2 q^{26} + 2 q^{27} + 5 q^{28} + 6 q^{29} - 2 q^{31} + q^{32} + 3 q^{33} + 6 q^{34} + 2 q^{36} - 2 q^{37} + 2 q^{38} - 2 q^{39} + 12 q^{41} + q^{42} + 4 q^{43} + 3 q^{44} - q^{46} + 3 q^{47} + 2 q^{48} - 13 q^{49} + 10 q^{50} + 3 q^{51} - 2 q^{52} + 6 q^{53} + q^{54} + q^{56} + 4 q^{57} + 3 q^{58} - 2 q^{61} - 4 q^{62} - 4 q^{63} + 2 q^{64} - 3 q^{66} - 2 q^{67} + 3 q^{68} - 2 q^{69} + 6 q^{71} + q^{72} - 11 q^{73} + 2 q^{74} + 5 q^{75} + 4 q^{76} - 15 q^{77} - 4 q^{78} - 11 q^{79} - q^{81} + 6 q^{82} - 4 q^{84} + 2 q^{86} - 3 q^{87} - 3 q^{88} + 6 q^{89} - 2 q^{91} - 2 q^{92} - 2 q^{93} - 3 q^{94} + q^{96} + 16 q^{97} - 11 q^{98} - 6 q^{99}+O(q^{100})$$ 2 * q + q^2 - q^3 - q^4 - 2 * q^6 - q^7 - 2 * q^8 - q^9 + 3 * q^11 - q^12 + 4 * q^13 + 4 * q^14 - q^16 + 3 * q^17 + q^18 - 2 * q^19 + 5 * q^21 + 6 * q^22 + q^23 + q^24 + 5 * q^25 + 2 * q^26 + 2 * q^27 + 5 * q^28 + 6 * q^29 - 2 * q^31 + q^32 + 3 * q^33 + 6 * q^34 + 2 * q^36 - 2 * q^37 + 2 * q^38 - 2 * q^39 + 12 * q^41 + q^42 + 4 * q^43 + 3 * q^44 - q^46 + 3 * q^47 + 2 * q^48 - 13 * q^49 + 10 * q^50 + 3 * q^51 - 2 * q^52 + 6 * q^53 + q^54 + q^56 + 4 * q^57 + 3 * q^58 - 2 * q^61 - 4 * q^62 - 4 * q^63 + 2 * q^64 - 3 * q^66 - 2 * q^67 + 3 * q^68 - 2 * q^69 + 6 * q^71 + q^72 - 11 * q^73 + 2 * q^74 + 5 * q^75 + 4 * q^76 - 15 * q^77 - 4 * q^78 - 11 * q^79 - q^81 + 6 * q^82 - 4 * q^84 + 2 * q^86 - 3 * q^87 - 3 * q^88 + 6 * q^89 - 2 * q^91 - 2 * q^92 - 2 * q^93 - 3 * q^94 + q^96 + 16 * q^97 - 11 * q^98 - 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/966\mathbb{Z}\right)^\times$$.

 $$n$$ $$323$$ $$829$$ $$925$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
277.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 −1.00000 −0.500000 2.59808i −1.00000 −0.500000 0.866025i 0
415.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0 −1.00000 −0.500000 + 2.59808i −1.00000 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.i.f 2
7.c even 3 1 inner 966.2.i.f 2
7.c even 3 1 6762.2.a.o 1
7.d odd 6 1 6762.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.f 2 1.a even 1 1 trivial
966.2.i.f 2 7.c even 3 1 inner
6762.2.a.e 1 7.d odd 6 1
6762.2.a.o 1 7.c even 3 1

## Hecke kernels

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

 $$T_{5}$$ T5 $$T_{11}^{2} - 3T_{11} + 9$$ T11^2 - 3*T11 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T + 7$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$T^{2} - T + 1$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2} + 2T + 4$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T - 2)^{2}$$
$47$ $$T^{2} - 3T + 9$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 2T + 4$$
$67$ $$T^{2} + 2T + 4$$
$71$ $$(T - 3)^{2}$$
$73$ $$T^{2} + 11T + 121$$
$79$ $$T^{2} + 11T + 121$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T - 8)^{2}$$