# Properties

 Label 322.2.k.a Level $322$ Weight $2$ Character orbit 322.k Analytic conductor $2.571$ Analytic rank $0$ Dimension $80$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [322,2,Mod(83,322)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(322, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 21]))

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

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

chi := DirichletCharacter("322.83");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$322 = 2 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 322.k (of order $$22$$, degree $$10$$, 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: $$2.57118294509$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$8$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$80 q - 8 q^{2} - 8 q^{4} - 11 q^{7} - 8 q^{8} - 6 q^{9}+O(q^{10})$$ 80 * q - 8 * q^2 - 8 * q^4 - 11 * q^7 - 8 * q^8 - 6 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$80 q - 8 q^{2} - 8 q^{4} - 11 q^{7} - 8 q^{8} - 6 q^{9} + 11 q^{14} - 8 q^{16} - 6 q^{18} - 33 q^{21} + 18 q^{23} - 58 q^{25} - 11 q^{28} + 16 q^{29} + 22 q^{30} - 8 q^{32} + 47 q^{35} + 16 q^{36} - 22 q^{37} - 30 q^{39} - 22 q^{42} + 44 q^{43} - 4 q^{46} + 17 q^{49} + 8 q^{50} - 88 q^{51} + 44 q^{53} + 11 q^{56} - 22 q^{57} - 28 q^{58} + 121 q^{63} - 8 q^{64} - 154 q^{65} + 58 q^{70} + 42 q^{71} - 28 q^{72} + 22 q^{74} + 38 q^{77} + 58 q^{78} - 88 q^{79} - 26 q^{81} - 11 q^{84} + 26 q^{85} - 132 q^{86} + 22 q^{88} - 4 q^{92} + 152 q^{93} - 68 q^{95} - 27 q^{98} - 44 q^{99}+O(q^{100})$$ 80 * q - 8 * q^2 - 8 * q^4 - 11 * q^7 - 8 * q^8 - 6 * q^9 + 11 * q^14 - 8 * q^16 - 6 * q^18 - 33 * q^21 + 18 * q^23 - 58 * q^25 - 11 * q^28 + 16 * q^29 + 22 * q^30 - 8 * q^32 + 47 * q^35 + 16 * q^36 - 22 * q^37 - 30 * q^39 - 22 * q^42 + 44 * q^43 - 4 * q^46 + 17 * q^49 + 8 * q^50 - 88 * q^51 + 44 * q^53 + 11 * q^56 - 22 * q^57 - 28 * q^58 + 121 * q^63 - 8 * q^64 - 154 * q^65 + 58 * q^70 + 42 * q^71 - 28 * q^72 + 22 * q^74 + 38 * q^77 + 58 * q^78 - 88 * q^79 - 26 * q^81 - 11 * q^84 + 26 * q^85 - 132 * q^86 + 22 * q^88 - 4 * q^92 + 152 * q^93 - 68 * q^95 - 27 * q^98 - 44 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
83.1 −0.959493 + 0.281733i −1.73925 + 1.50707i 0.841254 0.540641i 0.406529 + 2.82747i 1.24421 1.93603i 0.697541 + 2.55214i −0.654861 + 0.755750i 0.326790 2.27288i −1.18665 2.59840i
83.2 −0.959493 + 0.281733i −1.70070 + 1.47367i 0.841254 0.540641i 0.0506866 + 0.352533i 1.21663 1.89312i −2.46109 0.971103i −0.654861 + 0.755750i 0.293752 2.04309i −0.147953 0.323973i
83.3 −0.959493 + 0.281733i −0.749737 + 0.649651i 0.841254 0.540641i −0.540672 3.76045i 0.536339 0.834560i −2.61856 0.378356i −0.654861 + 0.755750i −0.286885 + 1.99533i 1.57821 + 3.45580i
83.4 −0.959493 + 0.281733i −0.198408 + 0.171921i 0.841254 0.540641i −0.382697 2.66172i 0.141935 0.220855i 0.921868 + 2.47995i −0.654861 + 0.755750i −0.417136 + 2.90124i 1.11709 + 2.44608i
83.5 −0.959493 + 0.281733i 0.198408 0.171921i 0.841254 0.540641i 0.382697 + 2.66172i −0.141935 + 0.220855i 2.47792 0.927322i −0.654861 + 0.755750i −0.417136 + 2.90124i −1.11709 2.44608i
83.6 −0.959493 + 0.281733i 0.749737 0.649651i 0.841254 0.540641i 0.540672 + 3.76045i −0.536339 + 0.834560i −2.00073 1.73120i −0.654861 + 0.755750i −0.286885 + 1.99533i −1.57821 3.45580i
83.7 −0.959493 + 0.281733i 1.70070 1.47367i 0.841254 0.540641i −0.0506866 0.352533i −1.21663 + 1.89312i −2.34558 1.22403i −0.654861 + 0.755750i 0.293752 2.04309i 0.147953 + 0.323973i
83.8 −0.959493 + 0.281733i 1.73925 1.50707i 0.841254 0.540641i −0.406529 2.82747i −1.24421 + 1.93603i 2.38557 1.14413i −0.654861 + 0.755750i 0.326790 2.27288i 1.18665 + 2.59840i
97.1 −0.959493 0.281733i −1.73925 1.50707i 0.841254 + 0.540641i 0.406529 2.82747i 1.24421 + 1.93603i 0.697541 2.55214i −0.654861 0.755750i 0.326790 + 2.27288i −1.18665 + 2.59840i
97.2 −0.959493 0.281733i −1.70070 1.47367i 0.841254 + 0.540641i 0.0506866 0.352533i 1.21663 + 1.89312i −2.46109 + 0.971103i −0.654861 0.755750i 0.293752 + 2.04309i −0.147953 + 0.323973i
97.3 −0.959493 0.281733i −0.749737 0.649651i 0.841254 + 0.540641i −0.540672 + 3.76045i 0.536339 + 0.834560i −2.61856 + 0.378356i −0.654861 0.755750i −0.286885 1.99533i 1.57821 3.45580i
97.4 −0.959493 0.281733i −0.198408 0.171921i 0.841254 + 0.540641i −0.382697 + 2.66172i 0.141935 + 0.220855i 0.921868 2.47995i −0.654861 0.755750i −0.417136 2.90124i 1.11709 2.44608i
97.5 −0.959493 0.281733i 0.198408 + 0.171921i 0.841254 + 0.540641i 0.382697 2.66172i −0.141935 0.220855i 2.47792 + 0.927322i −0.654861 0.755750i −0.417136 2.90124i −1.11709 + 2.44608i
97.6 −0.959493 0.281733i 0.749737 + 0.649651i 0.841254 + 0.540641i 0.540672 3.76045i −0.536339 0.834560i −2.00073 + 1.73120i −0.654861 0.755750i −0.286885 1.99533i −1.57821 + 3.45580i
97.7 −0.959493 0.281733i 1.70070 + 1.47367i 0.841254 + 0.540641i −0.0506866 + 0.352533i −1.21663 1.89312i −2.34558 + 1.22403i −0.654861 0.755750i 0.293752 + 2.04309i 0.147953 0.323973i
97.8 −0.959493 0.281733i 1.73925 + 1.50707i 0.841254 + 0.540641i −0.406529 + 2.82747i −1.24421 1.93603i 2.38557 + 1.14413i −0.654861 0.755750i 0.326790 + 2.27288i 1.18665 2.59840i
111.1 0.415415 + 0.909632i −0.760851 2.59122i −0.654861 + 0.755750i 2.71881 + 1.74727i 2.04099 1.76853i 0.0953475 2.64403i −0.959493 0.281733i −3.61176 + 2.32114i −0.459941 + 3.19896i
111.2 0.415415 + 0.909632i −0.730873 2.48912i −0.654861 + 0.755750i −1.52995 0.983241i 1.96057 1.69884i −2.20702 + 1.45913i −0.959493 0.281733i −3.13780 + 2.01654i 0.258822 1.80015i
111.3 0.415415 + 0.909632i −0.229469 0.781499i −0.654861 + 0.755750i −1.74460 1.12119i 0.615551 0.533378i −0.251104 2.63381i −0.959493 0.281733i 1.96568 1.26326i 0.295134 2.05270i
111.4 0.415415 + 0.909632i −0.192661 0.656142i −0.654861 + 0.755750i −2.27924 1.46478i 0.516813 0.447821i 2.42953 + 1.04756i −0.959493 0.281733i 2.13036 1.36910i 0.385580 2.68177i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 293.8 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.d odd 22 1 inner
161.k even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.k.a 80
7.b odd 2 1 inner 322.2.k.a 80
23.d odd 22 1 inner 322.2.k.a 80
161.k even 22 1 inner 322.2.k.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.k.a 80 1.a even 1 1 trivial
322.2.k.a 80 7.b odd 2 1 inner
322.2.k.a 80 23.d odd 22 1 inner
322.2.k.a 80 161.k even 22 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{80} - 9 T_{3}^{78} + 119 T_{3}^{76} - 1342 T_{3}^{74} + 17316 T_{3}^{72} - 101532 T_{3}^{70} + 529201 T_{3}^{68} - 6314029 T_{3}^{66} + 59028322 T_{3}^{64} - 472420459 T_{3}^{62} + \cdots + 956720690641$$ acting on $$S_{2}^{\mathrm{new}}(322, [\chi])$$.