# Properties

 Label 966.2.g.a Level $966$ Weight $2$ Character orbit 966.g 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(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.

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

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 0 1.00000i −1.87083 1.87083i −1.00000 −1.00000 0
643.2 −1.00000 1.00000i 1.00000 0 1.00000i 1.87083 + 1.87083i −1.00000 −1.00000 0
643.3 −1.00000 1.00000i 1.00000 0 1.00000i −1.87083 + 1.87083i −1.00000 −1.00000 0
643.4 −1.00000 1.00000i 1.00000 0 1.00000i 1.87083 1.87083i −1.00000 −1.00000 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.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.a 4
3.b odd 2 1 2898.2.g.g 4
7.b odd 2 1 inner 966.2.g.a 4
21.c even 2 1 2898.2.g.g 4
23.b odd 2 1 inner 966.2.g.a 4
69.c even 2 1 2898.2.g.g 4
161.c even 2 1 inner 966.2.g.a 4
483.c odd 2 1 2898.2.g.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.g.a 4 1.a even 1 1 trivial
966.2.g.a 4 7.b odd 2 1 inner
966.2.g.a 4 23.b odd 2 1 inner
966.2.g.a 4 161.c even 2 1 inner
2898.2.g.g 4 3.b odd 2 1
2898.2.g.g 4 21.c even 2 1
2898.2.g.g 4 69.c even 2 1
2898.2.g.g 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 $$S_{2}^{\mathrm{new}}(966, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{4}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 49$$
$11$ $$(T^{2} + 14)^{2}$$
$13$ $$(T^{2} + 4)^{2}$$
$17$ $$(T^{2} - 14)^{2}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 6 T + 23)^{2}$$
$29$ $$(T + 8)^{4}$$
$31$ $$(T^{2} + 16)^{2}$$
$37$ $$(T^{2} + 14)^{2}$$
$41$ $$(T^{2} + 4)^{2}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} + 64)^{2}$$
$53$ $$(T^{2} + 56)^{2}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - 126)^{2}$$
$67$ $$T^{4}$$
$71$ $$(T + 8)^{4}$$
$73$ $$(T^{2} + 100)^{2}$$
$79$ $$(T^{2} + 14)^{2}$$
$83$ $$(T^{2} - 14)^{2}$$
$89$ $$(T^{2} - 126)^{2}$$
$97$ $$T^{4}$$