# Properties

 Label 966.2.f.a Level $966$ Weight $2$ Character orbit 966.f Analytic conductor $7.714$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(461,966)]

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([1, 1, 0]))

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

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

chi := DirichletCharacter("966.461");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.f (of order $$2$$, degree $$1$$, 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + \beta_{2} q^{3} - q^{4} - \beta_{3} q^{5} + \beta_{3} q^{6} + ( - \beta_{2} + 2) q^{7} + \beta_1 q^{8} - 3 q^{9}+O(q^{10})$$ q - b1 * q^2 + b2 * q^3 - q^4 - b3 * q^5 + b3 * q^6 + (-b2 + 2) * q^7 + b1 * q^8 - 3 * q^9 $$q - \beta_1 q^{2} + \beta_{2} q^{3} - q^{4} - \beta_{3} q^{5} + \beta_{3} q^{6} + ( - \beta_{2} + 2) q^{7} + \beta_1 q^{8} - 3 q^{9} + \beta_{2} q^{10} - \beta_{2} q^{12} + \beta_{2} q^{13} + ( - \beta_{3} - 2 \beta_1) q^{14} - 3 \beta_1 q^{15} + q^{16} - 2 \beta_{3} q^{17} + 3 \beta_1 q^{18} + 4 \beta_{2} q^{19} + \beta_{3} q^{20} + (2 \beta_{2} + 3) q^{21} + \beta_1 q^{23} - \beta_{3} q^{24} - 2 q^{25} + \beta_{3} q^{26} - 3 \beta_{2} q^{27} + (\beta_{2} - 2) q^{28} - 9 \beta_1 q^{29} - 3 q^{30} + 6 \beta_{2} q^{31} - \beta_1 q^{32} + 2 \beta_{2} q^{34} + ( - 2 \beta_{3} + 3 \beta_1) q^{35} + 3 q^{36} - 5 q^{37} + 4 \beta_{3} q^{38} - 3 q^{39} - \beta_{2} q^{40} - 7 \beta_{3} q^{41} + (2 \beta_{3} - 3 \beta_1) q^{42} + q^{43} + 3 \beta_{3} q^{45} + q^{46} - 5 \beta_{3} q^{47} + \beta_{2} q^{48} + ( - 4 \beta_{2} + 1) q^{49} + 2 \beta_1 q^{50} - 6 \beta_1 q^{51} - \beta_{2} q^{52} + 12 \beta_1 q^{53} - 3 \beta_{3} q^{54} + (\beta_{3} + 2 \beta_1) q^{56} - 12 q^{57} - 9 q^{58} + 2 \beta_{3} q^{59} + 3 \beta_1 q^{60} + 6 \beta_{3} q^{62} + (3 \beta_{2} - 6) q^{63} - q^{64} - 3 \beta_1 q^{65} - 4 q^{67} + 2 \beta_{3} q^{68} - \beta_{3} q^{69} + (2 \beta_{2} + 3) q^{70} + 12 \beta_1 q^{71} - 3 \beta_1 q^{72} + 4 \beta_{2} q^{73} + 5 \beta_1 q^{74} - 2 \beta_{2} q^{75} - 4 \beta_{2} q^{76} + 3 \beta_1 q^{78} - 10 q^{79} - \beta_{3} q^{80} + 9 q^{81} + 7 \beta_{2} q^{82} - 2 \beta_{3} q^{83} + ( - 2 \beta_{2} - 3) q^{84} + 6 q^{85} - \beta_1 q^{86} + 9 \beta_{3} q^{87} + 2 \beta_{3} q^{89} - 3 \beta_{2} q^{90} + (2 \beta_{2} + 3) q^{91} - \beta_1 q^{92} - 18 q^{93} + 5 \beta_{2} q^{94} - 12 \beta_1 q^{95} + \beta_{3} q^{96} - 7 \beta_{2} q^{97} + ( - 4 \beta_{3} - \beta_1) q^{98}+O(q^{100})$$ q - b1 * q^2 + b2 * q^3 - q^4 - b3 * q^5 + b3 * q^6 + (-b2 + 2) * q^7 + b1 * q^8 - 3 * q^9 + b2 * q^10 - b2 * q^12 + b2 * q^13 + (-b3 - 2*b1) * q^14 - 3*b1 * q^15 + q^16 - 2*b3 * q^17 + 3*b1 * q^18 + 4*b2 * q^19 + b3 * q^20 + (2*b2 + 3) * q^21 + b1 * q^23 - b3 * q^24 - 2 * q^25 + b3 * q^26 - 3*b2 * q^27 + (b2 - 2) * q^28 - 9*b1 * q^29 - 3 * q^30 + 6*b2 * q^31 - b1 * q^32 + 2*b2 * q^34 + (-2*b3 + 3*b1) * q^35 + 3 * q^36 - 5 * q^37 + 4*b3 * q^38 - 3 * q^39 - b2 * q^40 - 7*b3 * q^41 + (2*b3 - 3*b1) * q^42 + q^43 + 3*b3 * q^45 + q^46 - 5*b3 * q^47 + b2 * q^48 + (-4*b2 + 1) * q^49 + 2*b1 * q^50 - 6*b1 * q^51 - b2 * q^52 + 12*b1 * q^53 - 3*b3 * q^54 + (b3 + 2*b1) * q^56 - 12 * q^57 - 9 * q^58 + 2*b3 * q^59 + 3*b1 * q^60 + 6*b3 * q^62 + (3*b2 - 6) * q^63 - q^64 - 3*b1 * q^65 - 4 * q^67 + 2*b3 * q^68 - b3 * q^69 + (2*b2 + 3) * q^70 + 12*b1 * q^71 - 3*b1 * q^72 + 4*b2 * q^73 + 5*b1 * q^74 - 2*b2 * q^75 - 4*b2 * q^76 + 3*b1 * q^78 - 10 * q^79 - b3 * q^80 + 9 * q^81 + 7*b2 * q^82 - 2*b3 * q^83 + (-2*b2 - 3) * q^84 + 6 * q^85 - b1 * q^86 + 9*b3 * q^87 + 2*b3 * q^89 - 3*b2 * q^90 + (2*b2 + 3) * q^91 - b1 * q^92 - 18 * q^93 + 5*b2 * q^94 - 12*b1 * q^95 + b3 * q^96 - 7*b2 * q^97 + (-4*b3 - b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 8 q^{7} - 12 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 + 8 * q^7 - 12 * q^9 $$4 q - 4 q^{4} + 8 q^{7} - 12 q^{9} + 4 q^{16} + 12 q^{21} - 8 q^{25} - 8 q^{28} - 12 q^{30} + 12 q^{36} - 20 q^{37} - 12 q^{39} + 4 q^{43} + 4 q^{46} + 4 q^{49} - 48 q^{57} - 36 q^{58} - 24 q^{63} - 4 q^{64} - 16 q^{67} + 12 q^{70} - 40 q^{79} + 36 q^{81} - 12 q^{84} + 24 q^{85} + 12 q^{91} - 72 q^{93}+O(q^{100})$$ 4 * q - 4 * q^4 + 8 * q^7 - 12 * q^9 + 4 * q^16 + 12 * q^21 - 8 * q^25 - 8 * q^28 - 12 * q^30 + 12 * q^36 - 20 * q^37 - 12 * q^39 + 4 * q^43 + 4 * q^46 + 4 * q^49 - 48 * q^57 - 36 * q^58 - 24 * q^63 - 4 * q^64 - 16 * q^67 + 12 * q^70 - 40 * q^79 + 36 * q^81 - 12 * q^84 + 24 * q^85 + 12 * q^91 - 72 * q^93

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## Character values

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

 $$n$$ $$323$$ $$829$$ $$925$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
461.1
 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i
1.00000i 1.73205i −1.00000 1.73205 −1.73205 2.00000 + 1.73205i 1.00000i −3.00000 1.73205i
461.2 1.00000i 1.73205i −1.00000 −1.73205 1.73205 2.00000 1.73205i 1.00000i −3.00000 1.73205i
461.3 1.00000i 1.73205i −1.00000 −1.73205 1.73205 2.00000 + 1.73205i 1.00000i −3.00000 1.73205i
461.4 1.00000i 1.73205i −1.00000 1.73205 −1.73205 2.00000 1.73205i 1.00000i −3.00000 1.73205i
 $$n$$: 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
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.f.a 4
3.b odd 2 1 inner 966.2.f.a 4
7.b odd 2 1 inner 966.2.f.a 4
21.c even 2 1 inner 966.2.f.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.f.a 4 1.a even 1 1 trivial
966.2.f.a 4 3.b odd 2 1 inner
966.2.f.a 4 7.b odd 2 1 inner
966.2.f.a 4 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 3$$ acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T^{2} + 3)^{2}$$
$5$ $$(T^{2} - 3)^{2}$$
$7$ $$(T^{2} - 4 T + 7)^{2}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 3)^{2}$$
$17$ $$(T^{2} - 12)^{2}$$
$19$ $$(T^{2} + 48)^{2}$$
$23$ $$(T^{2} + 1)^{2}$$
$29$ $$(T^{2} + 81)^{2}$$
$31$ $$(T^{2} + 108)^{2}$$
$37$ $$(T + 5)^{4}$$
$41$ $$(T^{2} - 147)^{2}$$
$43$ $$(T - 1)^{4}$$
$47$ $$(T^{2} - 75)^{2}$$
$53$ $$(T^{2} + 144)^{2}$$
$59$ $$(T^{2} - 12)^{2}$$
$61$ $$T^{4}$$
$67$ $$(T + 4)^{4}$$
$71$ $$(T^{2} + 144)^{2}$$
$73$ $$(T^{2} + 48)^{2}$$
$79$ $$(T + 10)^{4}$$
$83$ $$(T^{2} - 12)^{2}$$
$89$ $$(T^{2} - 12)^{2}$$
$97$ $$(T^{2} + 147)^{2}$$