# Properties

 Label 322.2.i.d Level $322$ Weight $2$ Character orbit 322.i Analytic conductor $2.571$ Analytic rank $0$ Dimension $40$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [322,2,Mod(29,322)]

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([0, 18]))

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

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

chi := DirichletCharacter("322.29");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$322 = 2 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 322.i (of order $$11$$, 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: $$40$$ Relative dimension: $$4$$ over $$\Q(\zeta_{11})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 4 q^{2} - 4 q^{4} + 9 q^{5} + 4 q^{7} - 4 q^{8} - 8 q^{9}+O(q^{10})$$ 40 * q - 4 * q^2 - 4 * q^4 + 9 * q^5 + 4 * q^7 - 4 * q^8 - 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 4 q^{2} - 4 q^{4} + 9 q^{5} + 4 q^{7} - 4 q^{8} - 8 q^{9} - 2 q^{10} + 6 q^{11} + 2 q^{13} + 4 q^{14} - 3 q^{15} - 4 q^{16} - 13 q^{17} - 8 q^{18} - 22 q^{19} - 2 q^{20} - 16 q^{22} - 9 q^{23} + 22 q^{24} - 15 q^{25} - 9 q^{26} + 21 q^{27} + 4 q^{28} - 10 q^{29} - 14 q^{30} - 2 q^{31} - 4 q^{32} + 8 q^{33} - 2 q^{34} + 13 q^{35} - 8 q^{36} - 45 q^{37} + 11 q^{38} - 22 q^{39} + 9 q^{40} + 21 q^{41} + 31 q^{43} + 6 q^{44} + 2 q^{45} - 9 q^{46} + 64 q^{47} - 4 q^{49} + 7 q^{50} + 65 q^{51} + 2 q^{52} + 69 q^{53} + 21 q^{54} - 74 q^{55} + 4 q^{56} - 68 q^{57} + 12 q^{58} + 48 q^{59} - 3 q^{60} + 6 q^{61} - 13 q^{62} + 8 q^{63} - 4 q^{64} - 64 q^{65} - 69 q^{66} + 31 q^{67} - 2 q^{68} - 62 q^{69} + 2 q^{70} - 57 q^{71} - 19 q^{72} + 70 q^{73} - 45 q^{74} - 11 q^{75} + 22 q^{76} - 6 q^{77} + 33 q^{78} + 34 q^{79} - 13 q^{80} + 30 q^{81} - 12 q^{82} - 56 q^{83} - 17 q^{85} + 42 q^{86} - 3 q^{87} + 6 q^{88} + 16 q^{89} + 46 q^{90} - 46 q^{91} + 24 q^{92} + 48 q^{93} + 9 q^{94} - 42 q^{95} - 36 q^{97} - 4 q^{98} + 112 q^{99}+O(q^{100})$$ 40 * q - 4 * q^2 - 4 * q^4 + 9 * q^5 + 4 * q^7 - 4 * q^8 - 8 * q^9 - 2 * q^10 + 6 * q^11 + 2 * q^13 + 4 * q^14 - 3 * q^15 - 4 * q^16 - 13 * q^17 - 8 * q^18 - 22 * q^19 - 2 * q^20 - 16 * q^22 - 9 * q^23 + 22 * q^24 - 15 * q^25 - 9 * q^26 + 21 * q^27 + 4 * q^28 - 10 * q^29 - 14 * q^30 - 2 * q^31 - 4 * q^32 + 8 * q^33 - 2 * q^34 + 13 * q^35 - 8 * q^36 - 45 * q^37 + 11 * q^38 - 22 * q^39 + 9 * q^40 + 21 * q^41 + 31 * q^43 + 6 * q^44 + 2 * q^45 - 9 * q^46 + 64 * q^47 - 4 * q^49 + 7 * q^50 + 65 * q^51 + 2 * q^52 + 69 * q^53 + 21 * q^54 - 74 * q^55 + 4 * q^56 - 68 * q^57 + 12 * q^58 + 48 * q^59 - 3 * q^60 + 6 * q^61 - 13 * q^62 + 8 * q^63 - 4 * q^64 - 64 * q^65 - 69 * q^66 + 31 * q^67 - 2 * q^68 - 62 * q^69 + 2 * q^70 - 57 * q^71 - 19 * q^72 + 70 * q^73 - 45 * q^74 - 11 * q^75 + 22 * q^76 - 6 * q^77 + 33 * q^78 + 34 * q^79 - 13 * q^80 + 30 * q^81 - 12 * q^82 - 56 * q^83 - 17 * q^85 + 42 * q^86 - 3 * q^87 + 6 * q^88 + 16 * q^89 + 46 * q^90 - 46 * q^91 + 24 * q^92 + 48 * q^93 + 9 * q^94 - 42 * q^95 - 36 * q^97 - 4 * q^98 + 112 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 0.415415 0.909632i −2.49681 0.733130i −0.654861 0.755750i −1.90645 + 1.22520i −1.70409 + 1.96663i 0.142315 0.989821i −0.959493 + 0.281733i 3.17283 + 2.03906i 0.322515 + 2.24314i
29.2 0.415415 0.909632i −2.34488 0.688518i −0.654861 0.755750i 3.30897 2.12655i −1.60040 + 1.84696i 0.142315 0.989821i −0.959493 + 0.281733i 2.50063 + 1.60706i −0.559779 3.89335i
29.3 0.415415 0.909632i −0.195887 0.0575176i −0.654861 0.755750i −1.56656 + 1.00676i −0.133694 + 0.154291i 0.142315 0.989821i −0.959493 + 0.281733i −2.48870 1.59939i 0.265014 + 1.84322i
29.4 0.415415 0.909632i 2.84549 + 0.835512i −0.654861 0.755750i −0.221294 + 0.142217i 1.94207 2.24127i 0.142315 0.989821i −0.959493 + 0.281733i 4.87498 + 3.13296i 0.0374364 + 0.260376i
71.1 0.841254 + 0.540641i −0.473601 3.29397i 0.415415 + 0.909632i −2.68725 0.789049i 1.38243 3.02711i 0.654861 0.755750i −0.142315 + 0.989821i −7.74745 + 2.27486i −1.83407 2.11663i
71.2 0.841254 + 0.540641i −0.311523 2.16669i 0.415415 + 0.909632i 3.63410 + 1.06707i 0.909331 1.99116i 0.654861 0.755750i −0.142315 + 0.989821i −1.71902 + 0.504749i 2.48030 + 2.86241i
71.3 0.841254 + 0.540641i −0.0270781 0.188333i 0.415415 + 0.909632i −0.254438 0.0747098i 0.0790407 0.173075i 0.654861 0.755750i −0.142315 + 0.989821i 2.84374 0.834998i −0.173656 0.200410i
71.4 0.841254 + 0.540641i 0.341180 + 2.37296i 0.415415 + 0.909632i 1.96603 + 0.577277i −0.995899 + 2.18071i 0.654861 0.755750i −0.142315 + 0.989821i −2.63604 + 0.774012i 1.34183 + 1.54855i
85.1 −0.654861 + 0.755750i −0.863913 + 0.555203i −0.142315 0.989821i −0.415591 0.910017i 0.146148 1.01648i 0.959493 0.281733i 0.841254 + 0.540641i −0.808150 + 1.76960i 0.959899 + 0.281852i
85.2 −0.654861 + 0.755750i 0.124065 0.0797318i −0.142315 0.989821i 1.52064 + 3.32974i −0.0209881 + 0.145975i 0.959493 0.281733i 0.841254 + 0.540641i −1.23721 + 2.70911i −3.51226 1.03129i
85.3 −0.654861 + 0.755750i 1.39187 0.894499i −0.142315 0.989821i −1.27474 2.79128i −0.235462 + 1.63768i 0.959493 0.281733i 0.841254 + 0.540641i −0.109077 + 0.238845i 2.94429 + 0.864520i
85.4 −0.654861 + 0.755750i 2.64484 1.69974i −0.142315 0.989821i 0.940662 + 2.05976i −0.447428 + 3.11193i 0.959493 0.281733i 0.841254 + 0.540641i 2.85983 6.26216i −2.17267 0.637952i
127.1 0.841254 0.540641i −0.473601 + 3.29397i 0.415415 0.909632i −2.68725 + 0.789049i 1.38243 + 3.02711i 0.654861 + 0.755750i −0.142315 0.989821i −7.74745 2.27486i −1.83407 + 2.11663i
127.2 0.841254 0.540641i −0.311523 + 2.16669i 0.415415 0.909632i 3.63410 1.06707i 0.909331 + 1.99116i 0.654861 + 0.755750i −0.142315 0.989821i −1.71902 0.504749i 2.48030 2.86241i
127.3 0.841254 0.540641i −0.0270781 + 0.188333i 0.415415 0.909632i −0.254438 + 0.0747098i 0.0790407 + 0.173075i 0.654861 + 0.755750i −0.142315 0.989821i 2.84374 + 0.834998i −0.173656 + 0.200410i
127.4 0.841254 0.540641i 0.341180 2.37296i 0.415415 0.909632i 1.96603 0.577277i −0.995899 2.18071i 0.654861 + 0.755750i −0.142315 0.989821i −2.63604 0.774012i 1.34183 1.54855i
141.1 −0.142315 0.989821i −1.08968 + 2.38606i −0.959493 + 0.281733i 0.443664 0.512016i 2.51685 + 0.739013i −0.841254 + 0.540641i 0.415415 + 0.909632i −2.54129 2.93281i −0.569944 0.366281i
141.2 −0.142315 0.989821i −0.182677 + 0.400008i −0.959493 + 0.281733i 2.48992 2.87352i 0.421934 + 0.123891i −0.841254 + 0.540641i 0.415415 + 0.909632i 1.83795 + 2.12110i −3.19863 2.05563i
141.3 −0.142315 0.989821i 0.128569 0.281527i −0.959493 + 0.281733i −1.69136 + 1.95194i −0.296959 0.0871950i −0.841254 + 0.540641i 0.415415 + 0.909632i 1.90185 + 2.19486i 2.17278 + 1.39636i
141.4 −0.142315 0.989821i 1.27568 2.79334i −0.959493 + 0.281733i 0.685739 0.791385i −2.94645 0.865157i −0.841254 + 0.540641i 0.415415 + 0.909632i −4.21081 4.85954i −0.880921 0.566133i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.i.d 40
23.c even 11 1 inner 322.2.i.d 40
23.c even 11 1 7406.2.a.bu 20
23.d odd 22 1 7406.2.a.bv 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.i.d 40 1.a even 1 1 trivial
322.2.i.d 40 23.c even 11 1 inner
7406.2.a.bu 20 23.c even 11 1
7406.2.a.bv 20 23.d odd 22 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{40} + 10 T_{3}^{38} - 7 T_{3}^{37} + 29 T_{3}^{36} + 69 T_{3}^{35} + 300 T_{3}^{34} + 1336 T_{3}^{33} + 3538 T_{3}^{32} + 15964 T_{3}^{31} + 24792 T_{3}^{30} - 51392 T_{3}^{29} + 89706 T_{3}^{28} - 73250 T_{3}^{27} + \cdots + 11881$$ acting on $$S_{2}^{\mathrm{new}}(322, [\chi])$$.