# Properties

 Label 966.2.i.i Level $966$ Weight $2$ Character orbit 966.i Analytic conductor $7.714$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ x^4 - x^3 + 5*x^2 + 4*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{2} - 1) q^{4} + \beta_1 q^{5} - q^{6} + (3 \beta_{2} - 1) q^{7} - q^{8} - \beta_{2} q^{9}+O(q^{10})$$ q + b2 * q^2 + (b2 - 1) * q^3 + (b2 - 1) * q^4 + b1 * q^5 - q^6 + (3*b2 - 1) * q^7 - q^8 - b2 * q^9 $$q + \beta_{2} q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{2} - 1) q^{4} + \beta_1 q^{5} - q^{6} + (3 \beta_{2} - 1) q^{7} - q^{8} - \beta_{2} q^{9} + (\beta_{3} + \beta_1) q^{10} + (3 \beta_{2} - 3) q^{11} - \beta_{2} q^{12} + (2 \beta_{3} + 2) q^{13} + (2 \beta_{2} - 3) q^{14} + \beta_{3} q^{15} - \beta_{2} q^{16} + ( - 3 \beta_{3} + 3 \beta_{2} + \cdots - 3) q^{17}+ \cdots + 3 q^{99}+O(q^{100})$$ q + b2 * q^2 + (b2 - 1) * q^3 + (b2 - 1) * q^4 + b1 * q^5 - q^6 + (3*b2 - 1) * q^7 - q^8 - b2 * q^9 + (b3 + b1) * q^10 + (3*b2 - 3) * q^11 - b2 * q^12 + (2*b3 + 2) * q^13 + (2*b2 - 3) * q^14 + b3 * q^15 - b2 * q^16 + (-3*b3 + 3*b2 - 3*b1 - 3) * q^17 + (-b2 + 1) * q^18 + (-2*b2 + 2*b1) * q^19 + b3 * q^20 + (-b2 - 2) * q^21 - 3 * q^22 + b2 * q^23 + (-b2 + 1) * q^24 + (b3 - b2 + b1 + 1) * q^25 + (2*b2 - 2*b1) * q^26 + q^27 + (-b2 - 2) * q^28 + (-2*b3 - 5) * q^29 - b1 * q^30 + (-3*b3 + 6*b2 - 3*b1 - 6) * q^31 + (-b2 + 1) * q^32 - 3*b2 * q^33 + (-3*b3 - 3) * q^34 + (3*b3 + 2*b1) * q^35 + q^36 + (-6*b2 + 2*b1) * q^37 + (2*b3 - 2*b2 + 2*b1 + 2) * q^38 + (-2*b3 + 2*b2 - 2*b1 - 2) * q^39 - b1 * q^40 + (2*b3 - 2) * q^41 + (-3*b2 + 1) * q^42 + (-4*b3 + 2) * q^43 - 3*b2 * q^44 + (-b3 - b1) * q^45 + (b2 - 1) * q^46 + (3*b2 - b1) * q^47 + q^48 + (3*b2 - 8) * q^49 + (b3 + 1) * q^50 + (-3*b2 + 3*b1) * q^51 + (-2*b3 + 2*b2 - 2*b1 - 2) * q^52 + (-b3 + 2*b2 - b1 - 2) * q^53 + b2 * q^54 + 3*b3 * q^55 + (-3*b2 + 1) * q^56 + (2*b3 + 2) * q^57 + (-5*b2 + 2*b1) * q^58 + (-b3 + 4*b2 - b1 - 4) * q^59 + (-b3 - b1) * q^60 + 2*b2 * q^61 + (-3*b3 - 6) * q^62 + (-2*b2 + 3) * q^63 + q^64 - 8*b2 * q^65 + (-3*b2 + 3) * q^66 + (-2*b2 + 2) * q^67 + (-3*b2 + 3*b1) * q^68 - q^69 + (2*b3 - b1) * q^70 + (b3 + 3) * q^71 + b2 * q^72 + (7*b3 - b2 + 7*b1 + 1) * q^73 + (2*b3 - 6*b2 + 2*b1 + 6) * q^74 + (b2 - b1) * q^75 + (2*b3 + 2) * q^76 + (-3*b2 - 6) * q^77 + (-2*b3 - 2) * q^78 + (3*b2 - 6*b1) * q^79 + (-b3 - b1) * q^80 + (b2 - 1) * q^81 + (-2*b2 - 2*b1) * q^82 + (b3 - 4) * q^83 + (-2*b2 + 3) * q^84 + 12 * q^85 + (2*b2 + 4*b1) * q^86 + (2*b3 - 5*b2 + 2*b1 + 5) * q^87 + (-3*b2 + 3) * q^88 + (2*b2 - 4*b1) * q^89 - b3 * q^90 + (-2*b3 + 6*b2 - 6*b1 - 2) * q^91 - q^92 + (-6*b2 + 3*b1) * q^93 + (-b3 + 3*b2 - b1 - 3) * q^94 + (8*b2 - 8) * q^95 + b2 * q^96 + (-3*b3 + 4) * q^97 + (-5*b2 - 3) * q^98 + 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + q^{5} - 4 q^{6} + 2 q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + q^5 - 4 * q^6 + 2 * q^7 - 4 * q^8 - 2 * q^9 $$4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + q^{5} - 4 q^{6} + 2 q^{7} - 4 q^{8} - 2 q^{9} - q^{10} - 6 q^{11} - 2 q^{12} + 4 q^{13} - 8 q^{14} - 2 q^{15} - 2 q^{16} - 3 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 10 q^{21} - 12 q^{22} + 2 q^{23} + 2 q^{24} + q^{25} + 2 q^{26} + 4 q^{27} - 10 q^{28} - 16 q^{29} - q^{30} - 9 q^{31} + 2 q^{32} - 6 q^{33} - 6 q^{34} - 4 q^{35} + 4 q^{36} - 10 q^{37} + 2 q^{38} - 2 q^{39} - q^{40} - 12 q^{41} - 2 q^{42} + 16 q^{43} - 6 q^{44} + q^{45} - 2 q^{46} + 5 q^{47} + 4 q^{48} - 26 q^{49} + 2 q^{50} - 3 q^{51} - 2 q^{52} - 3 q^{53} + 2 q^{54} - 6 q^{55} - 2 q^{56} + 4 q^{57} - 8 q^{58} - 7 q^{59} + q^{60} + 4 q^{61} - 18 q^{62} + 8 q^{63} + 4 q^{64} - 16 q^{65} + 6 q^{66} + 4 q^{67} - 3 q^{68} - 4 q^{69} - 5 q^{70} + 10 q^{71} + 2 q^{72} - 5 q^{73} + 10 q^{74} + q^{75} + 4 q^{76} - 30 q^{77} - 4 q^{78} + q^{80} - 2 q^{81} - 6 q^{82} - 18 q^{83} + 8 q^{84} + 48 q^{85} + 8 q^{86} + 8 q^{87} + 6 q^{88} + 2 q^{90} + 2 q^{91} - 4 q^{92} - 9 q^{93} - 5 q^{94} - 16 q^{95} + 2 q^{96} + 22 q^{97} - 22 q^{98} + 12 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + q^5 - 4 * q^6 + 2 * q^7 - 4 * q^8 - 2 * q^9 - q^10 - 6 * q^11 - 2 * q^12 + 4 * q^13 - 8 * q^14 - 2 * q^15 - 2 * q^16 - 3 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^20 - 10 * q^21 - 12 * q^22 + 2 * q^23 + 2 * q^24 + q^25 + 2 * q^26 + 4 * q^27 - 10 * q^28 - 16 * q^29 - q^30 - 9 * q^31 + 2 * q^32 - 6 * q^33 - 6 * q^34 - 4 * q^35 + 4 * q^36 - 10 * q^37 + 2 * q^38 - 2 * q^39 - q^40 - 12 * q^41 - 2 * q^42 + 16 * q^43 - 6 * q^44 + q^45 - 2 * q^46 + 5 * q^47 + 4 * q^48 - 26 * q^49 + 2 * q^50 - 3 * q^51 - 2 * q^52 - 3 * q^53 + 2 * q^54 - 6 * q^55 - 2 * q^56 + 4 * q^57 - 8 * q^58 - 7 * q^59 + q^60 + 4 * q^61 - 18 * q^62 + 8 * q^63 + 4 * q^64 - 16 * q^65 + 6 * q^66 + 4 * q^67 - 3 * q^68 - 4 * q^69 - 5 * q^70 + 10 * q^71 + 2 * q^72 - 5 * q^73 + 10 * q^74 + q^75 + 4 * q^76 - 30 * q^77 - 4 * q^78 + q^80 - 2 * q^81 - 6 * q^82 - 18 * q^83 + 8 * q^84 + 48 * q^85 + 8 * q^86 + 8 * q^87 + 6 * q^88 + 2 * q^90 + 2 * q^91 - 4 * q^92 - 9 * q^93 - 5 * q^94 - 16 * q^95 + 2 * q^96 + 22 * q^97 - 22 * q^98 + 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20$$ (-v^3 + 5*v^2 - 5*v + 16) / 20 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4 ) / 5$$ (v^3 + 4) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4\beta_{2} + \beta _1 - 4$$ b3 + 4*b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$5\beta_{3} - 4$$ 5*b3 - 4

## Character values

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

 $$n$$ $$323$$ $$829$$ $$925$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$ $$1$$

## 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
 −0.780776 − 1.35234i 1.28078 + 2.21837i −0.780776 + 1.35234i 1.28078 − 2.21837i
0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.780776 1.35234i −1.00000 0.500000 + 2.59808i −1.00000 −0.500000 0.866025i 0.780776 1.35234i
277.2 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.28078 + 2.21837i −1.00000 0.500000 + 2.59808i −1.00000 −0.500000 0.866025i −1.28078 + 2.21837i
415.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.780776 + 1.35234i −1.00000 0.500000 2.59808i −1.00000 −0.500000 + 0.866025i 0.780776 + 1.35234i
415.2 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.28078 2.21837i −1.00000 0.500000 2.59808i −1.00000 −0.500000 + 0.866025i −1.28078 2.21837i
 $$n$$: 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.i 4
7.c even 3 1 inner 966.2.i.i 4
7.c even 3 1 6762.2.a.bx 2
7.d odd 6 1 6762.2.a.br 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.i 4 1.a even 1 1 trivial
966.2.i.i 4 7.c even 3 1 inner
6762.2.a.br 2 7.d odd 6 1
6762.2.a.bx 2 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 $$S_{2}^{\mathrm{new}}(966, [\chi])$$:

 $$T_{5}^{4} - T_{5}^{3} + 5T_{5}^{2} + 4T_{5} + 16$$ T5^4 - T5^3 + 5*T5^2 + 4*T5 + 16 $$T_{11}^{2} + 3T_{11} + 9$$ T11^2 + 3*T11 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$T^{4} - T^{3} + \cdots + 16$$
$7$ $$(T^{2} - T + 7)^{2}$$
$11$ $$(T^{2} + 3 T + 9)^{2}$$
$13$ $$(T^{2} - 2 T - 16)^{2}$$
$17$ $$T^{4} + 3 T^{3} + \cdots + 1296$$
$19$ $$T^{4} + 2 T^{3} + \cdots + 256$$
$23$ $$(T^{2} - T + 1)^{2}$$
$29$ $$(T^{2} + 8 T - 1)^{2}$$
$31$ $$T^{4} + 9 T^{3} + \cdots + 324$$
$37$ $$T^{4} + 10 T^{3} + \cdots + 64$$
$41$ $$(T^{2} + 6 T - 8)^{2}$$
$43$ $$(T^{2} - 8 T - 52)^{2}$$
$47$ $$T^{4} - 5 T^{3} + \cdots + 4$$
$53$ $$T^{4} + 3 T^{3} + \cdots + 4$$
$59$ $$T^{4} + 7 T^{3} + \cdots + 64$$
$61$ $$(T^{2} - 2 T + 4)^{2}$$
$67$ $$(T^{2} - 2 T + 4)^{2}$$
$71$ $$(T^{2} - 5 T + 2)^{2}$$
$73$ $$T^{4} + 5 T^{3} + \cdots + 40804$$
$79$ $$T^{4} + 153 T^{2} + 23409$$
$83$ $$(T^{2} + 9 T + 16)^{2}$$
$89$ $$T^{4} + 68T^{2} + 4624$$
$97$ $$(T^{2} - 11 T - 8)^{2}$$