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Properties

Label 84.2.n.a
Level $84$
Weight $2$
Character orbit 84.n
Analytic conductor $0.671$
Analytic rank $0$
Dimension $24$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [84,2,Mod(11,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 84.n (of order \(6\), degree \(2\), 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: \(0.670743376979\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{4} - 2 q^{9} - 10 q^{10} - 12 q^{12} - 24 q^{13} - 10 q^{16} - 10 q^{18} - 6 q^{21} + 28 q^{22} - 14 q^{24} - 12 q^{25} + 10 q^{28} - 14 q^{30} + 10 q^{33} - 8 q^{34} + 44 q^{36} - 8 q^{37}+ \cdots + 72 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.32175 + 0.502962i −1.72058 − 0.199011i 1.49406 − 1.32958i 1.79791 − 1.03802i 2.37428 − 0.602344i 2.61347 + 0.412056i −1.30604 + 2.50883i 2.92079 + 0.684828i −1.85431 + 2.27629i
11.2 −1.29776 − 0.561968i 0.373317 − 1.69134i 1.36838 + 1.45860i 0.432549 − 0.249732i −1.43496 + 1.98517i −0.261928 − 2.63275i −0.956154 − 2.66191i −2.72127 − 1.26281i −0.701688 + 0.0810151i
11.3 −1.09645 + 0.893190i 1.72058 + 0.199011i 0.404424 − 1.95868i 1.79791 − 1.03802i −2.06429 + 1.31860i −2.61347 − 0.412056i 1.30604 + 2.50883i 2.92079 + 0.684828i −1.04417 + 2.74402i
11.4 −0.951145 − 1.04658i −1.66539 + 0.475894i −0.190646 + 1.99089i −2.36229 + 1.36387i 2.08209 + 1.29031i −1.89768 + 1.84359i 2.26495 − 1.69410i 2.54705 − 1.58510i 3.67427 + 1.17508i
11.5 −0.430790 − 1.34700i 0.420559 + 1.68022i −1.62884 + 1.16055i 2.36229 − 1.36387i 2.08209 − 1.29031i 1.89768 − 1.84359i 2.26495 + 1.69410i −2.64626 + 1.41326i −2.85479 − 2.59447i
11.6 −0.162204 + 1.40488i −0.373317 + 1.69134i −1.94738 − 0.455755i 0.432549 − 0.249732i −2.31558 − 0.798808i 0.261928 + 2.63275i 0.956154 − 2.66191i −2.72127 − 1.26281i 0.280683 + 0.648187i
11.7 0.162204 − 1.40488i 1.27809 − 1.16897i −1.94738 − 0.455755i −0.432549 + 0.249732i −1.43496 − 1.98517i 0.261928 + 2.63275i −0.956154 + 2.66191i 0.267006 − 2.98809i 0.280683 + 0.648187i
11.8 0.430790 + 1.34700i 1.66539 − 0.475894i −1.62884 + 1.16055i −2.36229 + 1.36387i 1.35846 + 2.03828i 1.89768 − 1.84359i −2.26495 − 1.69410i 2.54705 − 1.58510i −2.85479 − 2.59447i
11.9 0.951145 + 1.04658i −0.420559 − 1.68022i −0.190646 + 1.99089i 2.36229 − 1.36387i 1.35846 − 2.03828i −1.89768 + 1.84359i −2.26495 + 1.69410i −2.64626 + 1.41326i 3.67427 + 1.17508i
11.10 1.09645 − 0.893190i 1.03264 + 1.39056i 0.404424 − 1.95868i −1.79791 + 1.03802i 2.37428 + 0.602344i −2.61347 − 0.412056i −1.30604 − 2.50883i −0.867316 + 2.87189i −1.04417 + 2.74402i
11.11 1.29776 + 0.561968i −1.27809 + 1.16897i 1.36838 + 1.45860i −0.432549 + 0.249732i −2.31558 + 0.798808i −0.261928 − 2.63275i 0.956154 + 2.66191i 0.267006 − 2.98809i −0.701688 + 0.0810151i
11.12 1.32175 − 0.502962i −1.03264 − 1.39056i 1.49406 − 1.32958i −1.79791 + 1.03802i −2.06429 − 1.31860i 2.61347 + 0.412056i 1.30604 − 2.50883i −0.867316 + 2.87189i −1.85431 + 2.27629i
23.1 −1.32175 − 0.502962i −1.72058 + 0.199011i 1.49406 + 1.32958i 1.79791 + 1.03802i 2.37428 + 0.602344i 2.61347 − 0.412056i −1.30604 − 2.50883i 2.92079 − 0.684828i −1.85431 − 2.27629i
23.2 −1.29776 + 0.561968i 0.373317 + 1.69134i 1.36838 − 1.45860i 0.432549 + 0.249732i −1.43496 − 1.98517i −0.261928 + 2.63275i −0.956154 + 2.66191i −2.72127 + 1.26281i −0.701688 − 0.0810151i
23.3 −1.09645 − 0.893190i 1.72058 − 0.199011i 0.404424 + 1.95868i 1.79791 + 1.03802i −2.06429 − 1.31860i −2.61347 + 0.412056i 1.30604 − 2.50883i 2.92079 − 0.684828i −1.04417 − 2.74402i
23.4 −0.951145 + 1.04658i −1.66539 − 0.475894i −0.190646 − 1.99089i −2.36229 − 1.36387i 2.08209 − 1.29031i −1.89768 − 1.84359i 2.26495 + 1.69410i 2.54705 + 1.58510i 3.67427 − 1.17508i
23.5 −0.430790 + 1.34700i 0.420559 − 1.68022i −1.62884 − 1.16055i 2.36229 + 1.36387i 2.08209 + 1.29031i 1.89768 + 1.84359i 2.26495 − 1.69410i −2.64626 − 1.41326i −2.85479 + 2.59447i
23.6 −0.162204 − 1.40488i −0.373317 − 1.69134i −1.94738 + 0.455755i 0.432549 + 0.249732i −2.31558 + 0.798808i 0.261928 − 2.63275i 0.956154 + 2.66191i −2.72127 + 1.26281i 0.280683 − 0.648187i
23.7 0.162204 + 1.40488i 1.27809 + 1.16897i −1.94738 + 0.455755i −0.432549 − 0.249732i −1.43496 + 1.98517i 0.261928 − 2.63275i −0.956154 − 2.66191i 0.267006 + 2.98809i 0.280683 − 0.648187i
23.8 0.430790 − 1.34700i 1.66539 + 0.475894i −1.62884 − 1.16055i −2.36229 − 1.36387i 1.35846 − 2.03828i 1.89768 + 1.84359i −2.26495 + 1.69410i 2.54705 + 1.58510i −2.85479 + 2.59447i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.c even 3 1 inner
12.b even 2 1 inner
21.h odd 6 1 inner
28.g odd 6 1 inner
84.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.2.n.a ✓ 24
3.b odd 2 1 inner 84.2.n.a ✓ 24
4.b odd 2 1 inner 84.2.n.a ✓ 24
7.b odd 2 1 588.2.n.e 24
7.c even 3 1 inner 84.2.n.a ✓ 24
7.c even 3 1 588.2.e.e 12
7.d odd 6 1 588.2.e.d 12
7.d odd 6 1 588.2.n.e 24
12.b even 2 1 inner 84.2.n.a ✓ 24
21.c even 2 1 588.2.n.e 24
21.g even 6 1 588.2.e.d 12
21.g even 6 1 588.2.n.e 24
21.h odd 6 1 inner 84.2.n.a ✓ 24
21.h odd 6 1 588.2.e.e 12
28.d even 2 1 588.2.n.e 24
28.f even 6 1 588.2.e.d 12
28.f even 6 1 588.2.n.e 24
28.g odd 6 1 inner 84.2.n.a ✓ 24
28.g odd 6 1 588.2.e.e 12
84.h odd 2 1 588.2.n.e 24
84.j odd 6 1 588.2.e.d 12
84.j odd 6 1 588.2.n.e 24
84.n even 6 1 inner 84.2.n.a ✓ 24
84.n even 6 1 588.2.e.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.n.a ✓ 24 1.a even 1 1 trivial
84.2.n.a ✓ 24 3.b odd 2 1 inner
84.2.n.a ✓ 24 4.b odd 2 1 inner
84.2.n.a ✓ 24 7.c even 3 1 inner
84.2.n.a ✓ 24 12.b even 2 1 inner
84.2.n.a ✓ 24 21.h odd 6 1 inner
84.2.n.a ✓ 24 28.g odd 6 1 inner
84.2.n.a ✓ 24 84.n even 6 1 inner
588.2.e.d 12 7.d odd 6 1
588.2.e.d 12 21.g even 6 1
588.2.e.d 12 28.f even 6 1
588.2.e.d 12 84.j odd 6 1
588.2.e.e 12 7.c even 3 1
588.2.e.e 12 21.h odd 6 1
588.2.e.e 12 28.g odd 6 1
588.2.e.e 12 84.n even 6 1
588.2.n.e 24 7.b odd 2 1
588.2.n.e 24 7.d odd 6 1
588.2.n.e 24 21.c even 2 1
588.2.n.e 24 21.g even 6 1
588.2.n.e 24 28.d even 2 1
588.2.n.e 24 28.f even 6 1
588.2.n.e 24 84.h odd 2 1
588.2.n.e 24 84.j odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(84, [\chi])\).