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Properties

Label 322.2.i.e
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{2} + 2 q^{3} - 4 q^{4} - 9 q^{5} - 2 q^{6} - 4 q^{7} + 4 q^{8} - 2 q^{10} + 2 q^{12} + 4 q^{13} + 4 q^{14} - 11 q^{15} - 4 q^{16} - 21 q^{17} + 4 q^{19} + 2 q^{20} + 2 q^{21} - 11 q^{23} - 2 q^{24}+ \cdots + 138 q^{99}+O(q^{100}) \) 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 −2.45665 − 0.721337i −0.654861 − 0.755750i −3.26944 + 2.10114i 1.67668 − 1.93499i −0.142315 + 0.989821i 0.959493 − 0.281733i 2.99103 + 1.92222i −0.553091 − 3.84683i
29.2 −0.415415 + 0.909632i −0.589310 − 0.173037i −0.654861 − 0.755750i 3.36368 − 2.16170i 0.402208 − 0.464173i −0.142315 + 0.989821i 0.959493 − 0.281733i −2.20642 − 1.41798i 0.569033 + 3.95771i
29.3 −0.415415 + 0.909632i −0.427601 − 0.125555i −0.654861 − 0.755750i −0.0300272 + 0.0192973i 0.291841 − 0.336802i −0.142315 + 0.989821i 0.959493 − 0.281733i −2.35668 − 1.51455i −0.00507970 − 0.0353301i
29.4 −0.415415 + 0.909632i 3.20046 + 0.939739i −0.654861 − 0.755750i −3.28037 + 2.10817i −2.18434 + 2.52086i −0.142315 + 0.989821i 0.959493 − 0.281733i 6.83607 + 4.39327i −0.554941 − 3.85970i
71.1 −0.841254 − 0.540641i −0.459025 − 3.19259i 0.415415 + 0.909632i −1.51548 − 0.444984i −1.33989 + 2.93395i −0.654861 + 0.755750i 0.142315 − 0.989821i −7.10346 + 2.08576i 1.03432 + 1.19367i
71.2 −0.841254 − 0.540641i −0.0909948 − 0.632882i 0.415415 + 0.909632i −0.867979 − 0.254862i −0.265612 + 0.581610i −0.654861 + 0.755750i 0.142315 − 0.989821i 2.48622 − 0.730020i 0.592402 + 0.683668i
71.3 −0.841254 − 0.540641i 0.0894077 + 0.621844i 0.415415 + 0.909632i −2.40216 − 0.705337i 0.260980 − 0.571466i −0.654861 + 0.755750i 0.142315 − 0.989821i 2.49978 − 0.734002i 1.63949 + 1.89207i
71.4 −0.841254 − 0.540641i 0.274220 + 1.90724i 0.415415 + 0.909632i 3.76154 + 1.10449i 0.800444 − 1.75273i −0.654861 + 0.755750i 0.142315 − 0.989821i −0.683890 + 0.200808i −2.56727 − 2.96279i
85.1 0.654861 − 0.755750i −1.71143 + 1.09987i −0.142315 − 0.989821i 0.475695 + 1.04163i −0.289523 + 2.01368i −0.959493 + 0.281733i −0.841254 − 0.540641i 0.473040 − 1.03581i 1.09872 + 0.322614i
85.2 0.654861 − 0.755750i −1.01640 + 0.653203i −0.142315 − 0.989821i −1.23775 − 2.71030i −0.171945 + 1.19590i −0.959493 + 0.281733i −0.841254 − 0.540641i −0.639842 + 1.40106i −2.85887 − 0.839440i
85.3 0.654861 − 0.755750i 2.09228 − 1.34463i −0.142315 − 0.989821i −0.762523 − 1.66969i 0.353951 − 2.46178i −0.959493 + 0.281733i −0.841254 − 0.540641i 1.32337 − 2.89778i −1.76122 − 0.517140i
85.4 0.654861 − 0.755750i 2.24991 − 1.44593i −0.142315 − 0.989821i 1.60528 + 3.51508i 0.380617 − 2.64725i −0.959493 + 0.281733i −0.841254 − 0.540641i 1.72514 − 3.77752i 3.70776 + 1.08870i
127.1 −0.841254 + 0.540641i −0.459025 + 3.19259i 0.415415 − 0.909632i −1.51548 + 0.444984i −1.33989 − 2.93395i −0.654861 − 0.755750i 0.142315 + 0.989821i −7.10346 − 2.08576i 1.03432 − 1.19367i
127.2 −0.841254 + 0.540641i −0.0909948 + 0.632882i 0.415415 − 0.909632i −0.867979 + 0.254862i −0.265612 − 0.581610i −0.654861 − 0.755750i 0.142315 + 0.989821i 2.48622 + 0.730020i 0.592402 − 0.683668i
127.3 −0.841254 + 0.540641i 0.0894077 − 0.621844i 0.415415 − 0.909632i −2.40216 + 0.705337i 0.260980 + 0.571466i −0.654861 − 0.755750i 0.142315 + 0.989821i 2.49978 + 0.734002i 1.63949 − 1.89207i
127.4 −0.841254 + 0.540641i 0.274220 − 1.90724i 0.415415 − 0.909632i 3.76154 − 1.10449i 0.800444 + 1.75273i −0.654861 − 0.755750i 0.142315 + 0.989821i −0.683890 − 0.200808i −2.56727 + 2.96279i
141.1 0.142315 + 0.989821i −1.36463 + 2.98813i −0.959493 + 0.281733i −1.28840 + 1.48690i −3.15192 − 0.925489i 0.841254 − 0.540641i −0.415415 − 0.909632i −5.10212 − 5.88817i −1.65512 − 1.06368i
141.2 0.142315 + 0.989821i −0.605793 + 1.32650i −0.959493 + 0.281733i 2.76217 − 3.18771i −1.39921 − 0.410846i 0.841254 − 0.540641i −0.415415 − 0.909632i 0.571960 + 0.660077i 3.54836 + 2.28039i
141.3 0.142315 + 0.989821i 0.527208 − 1.15442i −0.959493 + 0.281733i −2.43563 + 2.81087i 1.21770 + 0.357550i 0.841254 − 0.540641i −0.415415 − 0.909632i 0.909835 + 1.05001i −3.12889 − 2.01081i
141.4 0.142315 + 0.989821i 0.744281 − 1.62975i −0.959493 + 0.281733i 1.17446 − 1.35540i 1.71908 + 0.504768i 0.841254 − 0.540641i −0.415415 − 0.909632i −0.137541 − 0.158731i 1.50875 + 0.969614i
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.e ✓ 40
23.c even 11 1 inner 322.2.i.e ✓ 40
23.c even 11 1 7406.2.a.bt 20
23.d odd 22 1 7406.2.a.bs 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.i.e ✓ 40 1.a even 1 1 trivial
322.2.i.e ✓ 40 23.c even 11 1 inner
7406.2.a.bs 20 23.d odd 22 1
7406.2.a.bt 20 23.c even 11 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} - 2 T_{3}^{39} + 8 T_{3}^{38} - 49 T_{3}^{37} + 169 T_{3}^{36} - 217 T_{3}^{35} + \cdots + 33860761 \) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\). Copy content Toggle raw display