# Properties

 Label 966.2.i.k Level $966$ Weight $2$ Character orbit 966.i Analytic conductor $7.714$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.29428272.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 6x^{4} - 4x^{3} - 42x^{2} + 343$$ x^6 - 6*x^4 - 4*x^3 - 42*x^2 + 343 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 1) q^{2} - \beta_{3} q^{3} + \beta_{3} q^{4} + ( - \beta_{3} - 1) q^{5} + q^{6} + ( - \beta_{5} - 1) q^{7} - q^{8} + ( - \beta_{3} - 1) q^{9}+O(q^{10})$$ q + (b3 + 1) * q^2 - b3 * q^3 + b3 * q^4 + (-b3 - 1) * q^5 + q^6 + (-b5 - 1) * q^7 - q^8 + (-b3 - 1) * q^9 $$q + (\beta_{3} + 1) q^{2} - \beta_{3} q^{3} + \beta_{3} q^{4} + ( - \beta_{3} - 1) q^{5} + q^{6} + ( - \beta_{5} - 1) q^{7} - q^{8} + ( - \beta_{3} - 1) q^{9} - \beta_{3} q^{10} + (\beta_{5} - \beta_{2} + 1) q^{11} + (\beta_{3} + 1) q^{12} + ( - \beta_{5} - \beta_{3} - \beta_{2} + \cdots - 1) q^{13}+ \cdots + (\beta_{4} + \beta_1) q^{99}+O(q^{100})$$ q + (b3 + 1) * q^2 - b3 * q^3 + b3 * q^4 + (-b3 - 1) * q^5 + q^6 + (-b5 - 1) * q^7 - q^8 + (-b3 - 1) * q^9 - b3 * q^10 + (b5 - b2 + 1) * q^11 + (b3 + 1) * q^12 + (-b5 - b3 - b2 - b1 - 1) * q^13 + (b1 - 1) * q^14 - q^15 + (-b3 - 1) * q^16 + (2*b5 - b4 + 2*b3 - b2 + b1 + 1) * q^17 - b3 * q^18 + (-2*b3 - b2 + b1 - 2) * q^19 + q^20 + (-b5 - b1) * q^21 + (-b4 - b1) * q^22 + (b3 + 1) * q^23 + b3 * q^24 - 4*b3 * q^25 + (-b5 - b4 - b3 - 2) * q^26 - q^27 + (b5 + b1) * q^28 + (-b5 - b3 - b2 - b1 + 4) * q^29 + (-b3 - 1) * q^30 + (-b5 + 2*b4 - 2*b3 - b2 - 2*b1 + 1) * q^31 - b3 * q^32 + (b5 + b4 - b2 + b1 + 1) * q^33 + (b5 - 2*b4 + b3 + b2 - b1 - 1) * q^34 + (-b1 + 1) * q^35 + q^36 + (2*b5 + 2*b4 + 4*b3 - 3*b2 + 3*b1 + 6) * q^37 + (b5 - b4 - b3 + b1) * q^38 + (b4 - b2 - b1 + 1) * q^39 + (b3 + 1) * q^40 + (b5 - b4 + b3 + b2 - 6) * q^41 + (-b5 - 1) * q^42 + (-2*b4 - 2*b1) * q^43 + (-b5 - b4 + b2 - b1 - 1) * q^44 + b3 * q^45 + b3 * q^46 + (-b5 - b4 + b3 - b2 + b1) * q^47 - q^48 + (2*b5 - b4 + 3*b3 - 2*b2 + b1 + 2) * q^49 + 4 * q^50 + (b5 + b4 + b3 - 2*b2 + 2*b1 + 2) * q^51 + (-b4 + b2 + b1 - 1) * q^52 + (-2*b5 - b3 + 2*b2 - 2) * q^53 + (-b3 - 1) * q^54 + (b4 + b1) * q^55 + (b5 + 1) * q^56 + (-b5 + b4 - b3 - b2 - 2) * q^57 + (-b5 - b4 + 4*b3 + 3) * q^58 + (-2*b5 + b4 - 5*b3 + b2 - b1 - 1) * q^59 - b3 * q^60 + (-10*b3 - 10) * q^61 + (-2*b5 + b4 - 2*b3 - 2*b2 - b1) * q^62 + (-b1 + 1) * q^63 + q^64 + (b5 + b4 + b3 + 2) * q^65 + (b5 - b2 + 1) * q^66 + (b5 - 2*b4 - 5*b3 + b2 + 2*b1 - 1) * q^67 + (-b5 - b4 - b3 + 2*b2 - 2*b1 - 2) * q^68 + q^69 + (-b5 - b1) * q^70 + (b5 + 2*b4 + b3 + b2 + 3*b1 - 1) * q^71 + (b3 + 1) * q^72 + (b5 + 5*b3 - b2 + 1) * q^73 + (3*b5 - b4 + 5*b3 - 2*b2 + b1 + 2) * q^74 + (-4*b3 - 4) * q^75 + (b5 - b4 + b3 + b2 + 2) * q^76 + (-b5 + b4 + 4*b3 + 2*b2 + 5) * q^77 + (-b5 - b3 - b2 - b1 - 1) * q^78 + (-3*b5 - 3*b4 + 4*b3 + 2*b2 - 2*b1 + 1) * q^79 + b3 * q^80 + b3 * q^81 + (-6*b3 + b2 - b1 - 6) * q^82 + (b4 + b1 - 4) * q^83 + (b1 - 1) * q^84 + (-b5 + 2*b4 - b3 - b2 + b1 + 1) * q^85 + (-2*b5 - 2*b4 + 2*b2 - 2*b1 - 2) * q^86 + (b4 - 5*b3 - b2 - b1 + 1) * q^87 + (-b5 + b2 - 1) * q^88 + (-2*b5 - 2*b4 - 4*b3 + 3*b2 - 3*b1 - 6) * q^89 - q^90 + (2*b5 + 2*b4 + 8*b3 - 3*b2 + 2*b1 + 12) * q^91 - q^92 + (b5 + b4 + b2 - b1 + 1) * q^93 + (b5 - 2*b4 + 3*b3 + b2 + 2*b1 - 1) * q^94 + (-b5 + b4 + b3 - b1) * q^95 + (-b3 - 1) * q^96 + (-3*b5 - 3*b3 - 3*b2 - 3*b1) * q^97 + (b5 - 3*b4 + 2*b3 + b2 - b1 - 2) * q^98 + (b4 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{2} + 3 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} - 3 q^{7} - 6 q^{8} - 3 q^{9}+O(q^{10})$$ 6 * q + 3 * q^2 + 3 * q^3 - 3 * q^4 - 3 * q^5 + 6 * q^6 - 3 * q^7 - 6 * q^8 - 3 * q^9 $$6 q + 3 q^{2} + 3 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} - 3 q^{7} - 6 q^{8} - 3 q^{9} + 3 q^{10} + 3 q^{12} - 6 q^{13} - 3 q^{14} - 6 q^{15} - 3 q^{16} - 3 q^{17} + 3 q^{18} - 6 q^{19} + 6 q^{20} + 3 q^{23} - 3 q^{24} + 12 q^{25} - 3 q^{26} - 6 q^{27} + 24 q^{29} - 3 q^{30} + 3 q^{32} - 6 q^{34} + 3 q^{35} + 6 q^{36} + 12 q^{37} + 6 q^{38} - 3 q^{39} + 3 q^{40} - 36 q^{41} - 3 q^{42} - 3 q^{45} - 3 q^{46} + 3 q^{47} - 6 q^{48} - 3 q^{49} + 24 q^{50} + 3 q^{51} + 3 q^{52} + 3 q^{53} - 3 q^{54} + 3 q^{56} - 12 q^{57} + 12 q^{58} + 12 q^{59} + 3 q^{60} - 30 q^{61} + 3 q^{63} + 6 q^{64} + 3 q^{65} + 21 q^{67} - 3 q^{68} + 6 q^{69} - 6 q^{71} + 3 q^{72} - 15 q^{73} - 12 q^{74} - 12 q^{75} + 12 q^{76} + 24 q^{77} - 6 q^{78} + 12 q^{79} - 3 q^{80} - 3 q^{81} - 18 q^{82} - 24 q^{83} - 3 q^{84} + 6 q^{85} + 12 q^{87} - 12 q^{89} - 6 q^{90} + 33 q^{91} - 6 q^{92} - 3 q^{94} - 6 q^{95} - 3 q^{96} - 12 q^{98}+O(q^{100})$$ 6 * q + 3 * q^2 + 3 * q^3 - 3 * q^4 - 3 * q^5 + 6 * q^6 - 3 * q^7 - 6 * q^8 - 3 * q^9 + 3 * q^10 + 3 * q^12 - 6 * q^13 - 3 * q^14 - 6 * q^15 - 3 * q^16 - 3 * q^17 + 3 * q^18 - 6 * q^19 + 6 * q^20 + 3 * q^23 - 3 * q^24 + 12 * q^25 - 3 * q^26 - 6 * q^27 + 24 * q^29 - 3 * q^30 + 3 * q^32 - 6 * q^34 + 3 * q^35 + 6 * q^36 + 12 * q^37 + 6 * q^38 - 3 * q^39 + 3 * q^40 - 36 * q^41 - 3 * q^42 - 3 * q^45 - 3 * q^46 + 3 * q^47 - 6 * q^48 - 3 * q^49 + 24 * q^50 + 3 * q^51 + 3 * q^52 + 3 * q^53 - 3 * q^54 + 3 * q^56 - 12 * q^57 + 12 * q^58 + 12 * q^59 + 3 * q^60 - 30 * q^61 + 3 * q^63 + 6 * q^64 + 3 * q^65 + 21 * q^67 - 3 * q^68 + 6 * q^69 - 6 * q^71 + 3 * q^72 - 15 * q^73 - 12 * q^74 - 12 * q^75 + 12 * q^76 + 24 * q^77 - 6 * q^78 + 12 * q^79 - 3 * q^80 - 3 * q^81 - 18 * q^82 - 24 * q^83 - 3 * q^84 + 6 * q^85 + 12 * q^87 - 12 * q^89 - 6 * q^90 + 33 * q^91 - 6 * q^92 - 3 * q^94 - 6 * q^95 - 3 * q^96 - 12 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 6x^{4} - 4x^{3} - 42x^{2} + 343$$ :

 $$\beta_{1}$$ $$=$$ $$( 3\nu^{5} + 35\nu^{4} + 31\nu^{3} + 121\nu^{2} - 217\nu - 1519 ) / 490$$ (3*v^5 + 35*v^4 + 31*v^3 + 121*v^2 - 217*v - 1519) / 490 $$\beta_{2}$$ $$=$$ $$( 5\nu^{5} + 7\nu^{4} + 19\nu^{3} - 13\nu^{2} + 203\nu - 147 ) / 490$$ (5*v^5 + 7*v^4 + 19*v^3 - 13*v^2 + 203*v - 147) / 490 $$\beta_{3}$$ $$=$$ $$( -5\nu^{5} - 7\nu^{4} - 19\nu^{3} + 13\nu^{2} + 287\nu + 147 ) / 490$$ (-5*v^5 - 7*v^4 - 19*v^3 + 13*v^2 + 287*v + 147) / 490 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} - 7\nu^{4} - \nu^{3} + 11\nu^{2} + 91\nu + 245 ) / 70$$ (-v^5 - 7*v^4 - v^3 + 11*v^2 + 91*v + 245) / 70 $$\beta_{5}$$ $$=$$ $$( -13\nu^{5} - 35\nu^{4} + 29\nu^{3} - 81\nu^{2} + 637\nu + 1519 ) / 490$$ (-13*v^5 - 35*v^4 + 29*v^3 - 81*v^2 + 637*v + 1519) / 490
 $$\nu$$ $$=$$ $$\beta_{3} + \beta_{2}$$ b3 + b2 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + 2\beta_{4} - \beta_{2} + 2\beta _1 + 2$$ -b5 + 2*b4 - b2 + 2*b1 + 2 $$\nu^{3}$$ $$=$$ $$5\beta_{5} + \beta_{4} - 11\beta_{3} + \beta_{2} + 4\beta _1 - 3$$ 5*b5 + b4 - 11*b3 + b2 + 4*b1 - 3 $$\nu^{4}$$ $$=$$ $$\beta_{5} - 9\beta_{4} + 18\beta_{3} + 5\beta_{2} + 5\beta _1 + 40$$ b5 - 9*b4 + 18*b3 + 5*b2 + 5*b1 + 40 $$\nu^{5}$$ $$=$$ $$-23\beta_{5} + 14\beta_{4} - 24\beta_{3} + 44\beta_{2} - 17\beta _1 - 10$$ -23*b5 + 14*b4 - 24*b3 + 44*b2 - 17*b1 - 10

## Character values

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

 $$n$$ $$323$$ $$829$$ $$925$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{3}$$ $$1$$

## 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
 −0.0741344 + 2.64471i 2.63435 − 0.245357i −2.56022 − 0.667305i −0.0741344 − 2.64471i 2.63435 + 0.245357i −2.56022 + 0.667305i
0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000 −2.25332 + 1.38656i −1.00000 −0.500000 0.866025i 0.500000 0.866025i
277.2 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000 −1.10469 2.40409i −1.00000 −0.500000 0.866025i 0.500000 0.866025i
277.3 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000 1.85801 + 1.88356i −1.00000 −0.500000 0.866025i 0.500000 0.866025i
415.1 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000 −2.25332 1.38656i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
415.2 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000 −1.10469 + 2.40409i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
415.3 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000 1.85801 1.88356i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 277.3 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.k 6
7.c even 3 1 inner 966.2.i.k 6
7.c even 3 1 6762.2.a.ce 3
7.d odd 6 1 6762.2.a.cf 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.k 6 1.a even 1 1 trivial
966.2.i.k 6 7.c even 3 1 inner
6762.2.a.ce 3 7.c even 3 1
6762.2.a.cf 3 7.d odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} + T_{5} + 1$$ T5^2 + T5 + 1 $$T_{11}^{6} + 15T_{11}^{4} - 12T_{11}^{3} + 225T_{11}^{2} - 90T_{11} + 36$$ T11^6 + 15*T11^4 - 12*T11^3 + 225*T11^2 - 90*T11 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{3}$$
$3$ $$(T^{2} - T + 1)^{3}$$
$5$ $$(T^{2} + T + 1)^{3}$$
$7$ $$T^{6} + 3 T^{5} + \cdots + 343$$
$11$ $$T^{6} + 15 T^{4} + \cdots + 36$$
$13$ $$(T^{3} + 3 T^{2} - 24 T - 22)^{2}$$
$17$ $$T^{6} + 3 T^{5} + \cdots + 15876$$
$19$ $$T^{6} + 6 T^{5} + \cdots + 2304$$
$23$ $$(T^{2} - T + 1)^{3}$$
$29$ $$(T^{3} - 12 T^{2} + \cdots + 48)^{2}$$
$31$ $$T^{6} + 75 T^{4} + \cdots + 45796$$
$37$ $$T^{6} - 12 T^{5} + \cdots + 345744$$
$41$ $$(T^{3} + 18 T^{2} + \cdots + 128)^{2}$$
$43$ $$(T^{3} - 60 T + 48)^{2}$$
$47$ $$T^{6} - 3 T^{5} + \cdots + 19600$$
$53$ $$T^{6} - 3 T^{5} + \cdots + 121$$
$59$ $$T^{6} - 12 T^{5} + \cdots + 32400$$
$61$ $$(T^{2} + 10 T + 100)^{3}$$
$67$ $$T^{6} - 21 T^{5} + \cdots + 156816$$
$71$ $$(T^{3} + 3 T^{2} + \cdots - 726)^{2}$$
$73$ $$T^{6} + 15 T^{5} + \cdots + 3136$$
$79$ $$T^{6} - 12 T^{5} + \cdots + 1140624$$
$83$ $$(T^{3} + 12 T^{2} + 33 T - 2)^{2}$$
$89$ $$T^{6} + 12 T^{5} + \cdots + 345744$$
$97$ $$(T^{3} - 243 T + 108)^{2}$$