# Properties

 Label 966.2.g.b Level $966$ Weight $2$ Character orbit 966.g Analytic conductor $7.714$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(643,966)]

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, 1, 1]))

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

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

chi := DirichletCharacter("966.643");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.g (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: $$\Q(i, \sqrt{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 49$$ x^4 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
643.1
 1.87083 − 1.87083i −1.87083 + 1.87083i 1.87083 + 1.87083i −1.87083 − 1.87083i
−1.00000 1.00000i 1.00000 −3.74166 1.00000i 1.87083 1.87083i −1.00000 −1.00000 3.74166
643.2 −1.00000 1.00000i 1.00000 3.74166 1.00000i −1.87083 + 1.87083i −1.00000 −1.00000 −3.74166
643.3 −1.00000 1.00000i 1.00000 −3.74166 1.00000i 1.87083 + 1.87083i −1.00000 −1.00000 3.74166
643.4 −1.00000 1.00000i 1.00000 3.74166 1.00000i −1.87083 1.87083i −1.00000 −1.00000 −3.74166
 $$n$$: 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.b odd 2 1 inner
23.b odd 2 1 inner
161.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.g.b 4
3.b odd 2 1 2898.2.g.h 4
7.b odd 2 1 inner 966.2.g.b 4
21.c even 2 1 2898.2.g.h 4
23.b odd 2 1 inner 966.2.g.b 4
69.c even 2 1 2898.2.g.h 4
161.c even 2 1 inner 966.2.g.b 4
483.c odd 2 1 2898.2.g.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.g.b 4 1.a even 1 1 trivial
966.2.g.b 4 7.b odd 2 1 inner
966.2.g.b 4 23.b odd 2 1 inner
966.2.g.b 4 161.c even 2 1 inner
2898.2.g.h 4 3.b odd 2 1
2898.2.g.h 4 21.c even 2 1
2898.2.g.h 4 69.c even 2 1
2898.2.g.h 4 483.c odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 14$$ T5^2 - 14 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{4}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$(T^{2} - 14)^{2}$$
$7$ $$T^{4} + 49$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 4)^{2}$$
$17$ $$(T^{2} - 14)^{2}$$
$19$ $$(T^{2} - 14)^{2}$$
$23$ $$(T^{2} + 6 T + 23)^{2}$$
$29$ $$(T - 6)^{4}$$
$31$ $$(T^{2} + 16)^{2}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 4)^{2}$$
$43$ $$(T^{2} + 126)^{2}$$
$47$ $$(T^{2} + 36)^{2}$$
$53$ $$(T^{2} + 14)^{2}$$
$59$ $$(T^{2} + 196)^{2}$$
$61$ $$T^{4}$$
$67$ $$(T^{2} + 14)^{2}$$
$71$ $$(T - 6)^{4}$$
$73$ $$(T^{2} + 16)^{2}$$
$79$ $$(T^{2} + 14)^{2}$$
$83$ $$(T^{2} - 56)^{2}$$
$89$ $$(T^{2} - 14)^{2}$$
$97$ $$(T^{2} - 56)^{2}$$