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Properties

Label 322.2.i.d
Level $322$
Weight $2$
Character orbit 322.i
Analytic conductor $2.571$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{2} - 4 q^{4} + 9 q^{5} + 4 q^{7} - 4 q^{8} - 8 q^{9} - 2 q^{10} + 6 q^{11} + 2 q^{13} + 4 q^{14} - 3 q^{15} - 4 q^{16} - 13 q^{17} - 8 q^{18} - 22 q^{19} - 2 q^{20} - 16 q^{22} - 9 q^{23} + 22 q^{24}+ \cdots + 112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 29.4
Significant digits:
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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 \( T_{3}^{40} + 10 T_{3}^{38} - 7 T_{3}^{37} + 29 T_{3}^{36} + 69 T_{3}^{35} + 300 T_{3}^{34} + \cdots + 11881 \) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\). Copy content Toggle raw display