# Properties

 Label 966.2.h.a Level $966$ Weight $2$ Character orbit 966.h Analytic conductor $7.714$ Analytic rank $0$ Dimension $24$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(827,966)]

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, 0, 1]))

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

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

chi := DirichletCharacter("966.827");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.h (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: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 4 q^{3} - 24 q^{4} - 4 q^{5} - 4 q^{9}+O(q^{10})$$ 24 * q + 4 * q^3 - 24 * q^4 - 4 * q^5 - 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 4 q^{3} - 24 q^{4} - 4 q^{5} - 4 q^{9} - 4 q^{12} + 8 q^{13} + 24 q^{14} - 4 q^{15} + 24 q^{16} + 32 q^{17} + 4 q^{18} + 4 q^{20} - 8 q^{23} - 12 q^{25} + 16 q^{27} - 4 q^{30} - 16 q^{31} + 20 q^{33} + 4 q^{36} - 8 q^{39} + 4 q^{42} + 24 q^{45} - 4 q^{46} + 4 q^{48} - 24 q^{49} + 24 q^{51} - 8 q^{52} + 24 q^{53} - 12 q^{54} + 16 q^{55} - 24 q^{56} + 4 q^{57} + 4 q^{58} + 4 q^{60} - 4 q^{63} - 24 q^{64} - 12 q^{66} - 32 q^{68} - 24 q^{69} - 4 q^{70} - 4 q^{72} - 32 q^{73} - 16 q^{74} + 48 q^{75} + 12 q^{78} - 4 q^{80} - 8 q^{81} - 8 q^{82} + 16 q^{83} - 16 q^{85} + 16 q^{86} + 20 q^{87} + 24 q^{89} - 28 q^{90} + 8 q^{92} + 16 q^{93} + 8 q^{94} - 48 q^{99}+O(q^{100})$$ 24 * q + 4 * q^3 - 24 * q^4 - 4 * q^5 - 4 * q^9 - 4 * q^12 + 8 * q^13 + 24 * q^14 - 4 * q^15 + 24 * q^16 + 32 * q^17 + 4 * q^18 + 4 * q^20 - 8 * q^23 - 12 * q^25 + 16 * q^27 - 4 * q^30 - 16 * q^31 + 20 * q^33 + 4 * q^36 - 8 * q^39 + 4 * q^42 + 24 * q^45 - 4 * q^46 + 4 * q^48 - 24 * q^49 + 24 * q^51 - 8 * q^52 + 24 * q^53 - 12 * q^54 + 16 * q^55 - 24 * q^56 + 4 * q^57 + 4 * q^58 + 4 * q^60 - 4 * q^63 - 24 * q^64 - 12 * q^66 - 32 * q^68 - 24 * q^69 - 4 * q^70 - 4 * q^72 - 32 * q^73 - 16 * q^74 + 48 * q^75 + 12 * q^78 - 4 * q^80 - 8 * q^81 - 8 * q^82 + 16 * q^83 - 16 * q^85 + 16 * q^86 + 20 * q^87 + 24 * q^89 - 28 * q^90 + 8 * q^92 + 16 * q^93 + 8 * q^94 - 48 * q^99

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
827.1 1.00000i −1.69648 0.349201i −1.00000 1.62492 −0.349201 + 1.69648i 1.00000i 1.00000i 2.75612 + 1.18483i 1.62492i
827.2 1.00000i −1.33692 1.10121i −1.00000 −0.136871 −1.10121 + 1.33692i 1.00000i 1.00000i 0.574695 + 2.94444i 0.136871i
827.3 1.00000i −1.01359 + 1.40450i −1.00000 −1.51059 1.40450 + 1.01359i 1.00000i 1.00000i −0.945264 2.84719i 1.51059i
827.4 1.00000i −0.798999 + 1.53675i −1.00000 2.11859 1.53675 + 0.798999i 1.00000i 1.00000i −1.72320 2.45572i 2.11859i
827.5 1.00000i −0.529264 1.64921i −1.00000 −2.93995 −1.64921 + 0.529264i 1.00000i 1.00000i −2.43976 + 1.74573i 2.93995i
827.6 1.00000i −0.0866531 1.72988i −1.00000 0.713996 −1.72988 + 0.0866531i 1.00000i 1.00000i −2.98498 + 0.299799i 0.713996i
827.7 1.00000i 0.656415 + 1.60285i −1.00000 −0.252332 1.60285 0.656415i 1.00000i 1.00000i −2.13824 + 2.10426i 0.252332i
827.8 1.00000i 0.696580 + 1.58580i −1.00000 −3.16093 1.58580 0.696580i 1.00000i 1.00000i −2.02955 + 2.20928i 3.16093i
827.9 1.00000i 1.23024 1.21923i −1.00000 0.666555 −1.21923 1.23024i 1.00000i 1.00000i 0.0269708 2.99988i 0.666555i
827.10 1.00000i 1.55095 0.771068i −1.00000 2.18892 −0.771068 1.55095i 1.00000i 1.00000i 1.81091 2.39178i 2.18892i
827.11 1.00000i 1.59575 + 0.673481i −1.00000 2.44752 0.673481 1.59575i 1.00000i 1.00000i 2.09285 + 2.14942i 2.44752i
827.12 1.00000i 1.73197 + 0.0164012i −1.00000 −3.75984 0.0164012 1.73197i 1.00000i 1.00000i 2.99946 + 0.0568128i 3.75984i
827.13 1.00000i −1.69648 + 0.349201i −1.00000 1.62492 −0.349201 1.69648i 1.00000i 1.00000i 2.75612 1.18483i 1.62492i
827.14 1.00000i −1.33692 + 1.10121i −1.00000 −0.136871 −1.10121 1.33692i 1.00000i 1.00000i 0.574695 2.94444i 0.136871i
827.15 1.00000i −1.01359 1.40450i −1.00000 −1.51059 1.40450 1.01359i 1.00000i 1.00000i −0.945264 + 2.84719i 1.51059i
827.16 1.00000i −0.798999 1.53675i −1.00000 2.11859 1.53675 0.798999i 1.00000i 1.00000i −1.72320 + 2.45572i 2.11859i
827.17 1.00000i −0.529264 + 1.64921i −1.00000 −2.93995 −1.64921 0.529264i 1.00000i 1.00000i −2.43976 1.74573i 2.93995i
827.18 1.00000i −0.0866531 + 1.72988i −1.00000 0.713996 −1.72988 0.0866531i 1.00000i 1.00000i −2.98498 0.299799i 0.713996i
827.19 1.00000i 0.656415 1.60285i −1.00000 −0.252332 1.60285 + 0.656415i 1.00000i 1.00000i −2.13824 2.10426i 0.252332i
827.20 1.00000i 0.696580 1.58580i −1.00000 −3.16093 1.58580 + 0.696580i 1.00000i 1.00000i −2.02955 2.20928i 3.16093i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 827.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
69.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.h.a 24
3.b odd 2 1 966.2.h.b yes 24
23.b odd 2 1 966.2.h.b yes 24
69.c even 2 1 inner 966.2.h.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.h.a 24 1.a even 1 1 trivial
966.2.h.a 24 69.c even 2 1 inner
966.2.h.b yes 24 3.b odd 2 1
966.2.h.b yes 24 23.b odd 2 1