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Properties

Label 168.2.t.a
Level $168$
Weight $2$
Character orbit 168.t
Analytic conductor $1.341$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,2,Mod(19,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.t (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: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} - 2 q^{4} + 16 q^{8} + 16 q^{9} - 18 q^{10} + 8 q^{11} - 10 q^{14} + 6 q^{16} + 2 q^{18} - 20 q^{22} - 18 q^{24} - 16 q^{25} - 30 q^{26} - 14 q^{28} - 8 q^{30} - 12 q^{32} - 24 q^{35} - 4 q^{36}+ \cdots + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.41257 + 0.0681843i 0.866025 − 0.500000i 1.99070 − 0.192630i 2.08776 − 3.61611i −1.18923 + 0.765334i −2.39694 + 1.12013i −2.79887 + 0.407838i 0.500000 − 0.866025i −2.70255 + 5.25036i
19.2 −1.40727 + 0.139971i −0.866025 + 0.500000i 1.96082 − 0.393955i 0.128707 − 0.222928i 1.14875 − 0.824854i 0.623918 − 2.57113i −2.70425 + 0.828860i 0.500000 − 0.866025i −0.149923 + 0.331735i
19.3 −1.04777 − 0.949827i 0.866025 − 0.500000i 0.195657 + 1.99041i 0.155280 − 0.268953i −1.38231 − 0.298688i 2.58581 + 0.560001i 1.68554 − 2.27133i 0.500000 − 0.866025i −0.418156 + 0.134312i
19.4 −1.03162 + 0.967346i 0.866025 − 0.500000i 0.128485 − 1.99587i −1.25150 + 2.16767i −0.409738 + 1.35356i −1.36321 + 2.26752i 1.79815 + 2.18327i 0.500000 − 0.866025i −0.805806 − 3.44685i
19.5 −0.765161 + 1.18934i −0.866025 + 0.500000i −0.829058 − 1.82007i 1.61398 − 2.79550i 0.0679787 − 1.41258i 1.82725 + 1.91341i 2.79905 + 0.406616i 0.500000 − 0.866025i 2.08984 + 4.05858i
19.6 −0.725478 − 1.21395i −0.866025 + 0.500000i −0.947364 + 1.76139i 1.14053 − 1.97545i 1.23526 + 0.688575i −1.95181 − 1.78618i 2.82554 − 0.127796i 0.500000 − 0.866025i −3.22553 + 0.0485996i
19.7 −0.647418 + 1.25732i −0.866025 + 0.500000i −1.16170 − 1.62802i −1.61398 + 2.79550i −0.0679787 − 1.41258i −1.82725 − 1.91341i 2.79905 − 0.406616i 0.500000 − 0.866025i −2.46991 − 3.83914i
19.8 −0.321935 + 1.37708i 0.866025 − 0.500000i −1.79272 − 0.886663i 1.25150 − 2.16767i 0.409738 + 1.35356i 1.36321 − 2.26752i 1.79815 − 2.18327i 0.500000 − 0.866025i 2.58215 + 2.42127i
19.9 −0.297109 − 1.38265i −0.866025 + 0.500000i −1.82345 + 0.821596i −1.44142 + 2.49662i 0.948630 + 1.04886i 2.63862 − 0.194181i 1.67775 + 2.27710i 0.500000 − 0.866025i 3.88021 + 1.25122i
19.10 0.221012 − 1.39684i 0.866025 − 0.500000i −1.90231 − 0.617436i 0.225540 − 0.390646i −0.507016 − 1.32020i −0.458196 − 2.60577i −1.28289 + 2.52075i 0.500000 − 0.866025i −0.495822 − 0.401380i
19.11 0.582416 + 1.28872i −0.866025 + 0.500000i −1.32158 + 1.50114i −0.128707 + 0.222928i −1.14875 − 0.824854i −0.623918 + 2.57113i −2.70425 − 0.828860i 0.500000 − 0.866025i −0.362252 − 0.0360307i
19.12 0.647235 + 1.25741i 0.866025 − 0.500000i −1.16217 + 1.62768i −2.08776 + 3.61611i 1.18923 + 0.765334i 2.39694 − 1.12013i −2.79887 − 0.407838i 0.500000 − 0.866025i −5.89822 − 0.284706i
19.13 1.09919 − 0.889821i 0.866025 − 0.500000i 0.416438 − 1.95616i −0.225540 + 0.390646i 0.507016 − 1.32020i 0.458196 + 2.60577i −1.28289 − 2.52075i 0.500000 − 0.866025i 0.0996941 + 0.630084i
19.14 1.34597 − 0.434022i −0.866025 + 0.500000i 1.62325 − 1.16836i 1.44142 − 2.49662i −0.948630 + 1.04886i −2.63862 + 0.194181i 1.67775 − 2.27710i 0.500000 − 0.866025i 0.856518 − 3.98597i
19.15 1.34646 + 0.432485i 0.866025 − 0.500000i 1.62591 + 1.16465i −0.155280 + 0.268953i 1.38231 − 0.298688i −2.58581 − 0.560001i 1.68554 + 2.27133i 0.500000 − 0.866025i −0.325396 + 0.294978i
19.16 1.41405 + 0.0213058i −0.866025 + 0.500000i 1.99909 + 0.0602550i −1.14053 + 1.97545i −1.23526 + 0.688575i 1.95181 + 1.78618i 2.82554 + 0.127796i 0.500000 − 0.866025i −1.65485 + 2.76909i
115.1 −1.41257 − 0.0681843i 0.866025 + 0.500000i 1.99070 + 0.192630i 2.08776 + 3.61611i −1.18923 − 0.765334i −2.39694 − 1.12013i −2.79887 − 0.407838i 0.500000 + 0.866025i −2.70255 − 5.25036i
115.2 −1.40727 − 0.139971i −0.866025 − 0.500000i 1.96082 + 0.393955i 0.128707 + 0.222928i 1.14875 + 0.824854i 0.623918 + 2.57113i −2.70425 − 0.828860i 0.500000 + 0.866025i −0.149923 − 0.331735i
115.3 −1.04777 + 0.949827i 0.866025 + 0.500000i 0.195657 − 1.99041i 0.155280 + 0.268953i −1.38231 + 0.298688i 2.58581 − 0.560001i 1.68554 + 2.27133i 0.500000 + 0.866025i −0.418156 − 0.134312i
115.4 −1.03162 − 0.967346i 0.866025 + 0.500000i 0.128485 + 1.99587i −1.25150 − 2.16767i −0.409738 − 1.35356i −1.36321 − 2.26752i 1.79815 − 2.18327i 0.500000 + 0.866025i −0.805806 + 3.44685i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
8.d odd 2 1 inner
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.2.t.a ✓ 32
3.b odd 2 1 504.2.bk.c 32
4.b odd 2 1 672.2.bb.a 32
7.c even 3 1 1176.2.p.a 32
7.d odd 6 1 inner 168.2.t.a ✓ 32
7.d odd 6 1 1176.2.p.a 32
8.b even 2 1 672.2.bb.a 32
8.d odd 2 1 inner 168.2.t.a ✓ 32
12.b even 2 1 2016.2.bs.c 32
21.g even 6 1 504.2.bk.c 32
24.f even 2 1 504.2.bk.c 32
24.h odd 2 1 2016.2.bs.c 32
28.f even 6 1 672.2.bb.a 32
28.f even 6 1 4704.2.p.a 32
28.g odd 6 1 4704.2.p.a 32
56.j odd 6 1 672.2.bb.a 32
56.j odd 6 1 4704.2.p.a 32
56.k odd 6 1 1176.2.p.a 32
56.m even 6 1 inner 168.2.t.a ✓ 32
56.m even 6 1 1176.2.p.a 32
56.p even 6 1 4704.2.p.a 32
84.j odd 6 1 2016.2.bs.c 32
168.ba even 6 1 2016.2.bs.c 32
168.be odd 6 1 504.2.bk.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.t.a ✓ 32 1.a even 1 1 trivial
168.2.t.a ✓ 32 7.d odd 6 1 inner
168.2.t.a ✓ 32 8.d odd 2 1 inner
168.2.t.a ✓ 32 56.m even 6 1 inner
504.2.bk.c 32 3.b odd 2 1
504.2.bk.c 32 21.g even 6 1
504.2.bk.c 32 24.f even 2 1
504.2.bk.c 32 168.be odd 6 1
672.2.bb.a 32 4.b odd 2 1
672.2.bb.a 32 8.b even 2 1
672.2.bb.a 32 28.f even 6 1
672.2.bb.a 32 56.j odd 6 1
1176.2.p.a 32 7.c even 3 1
1176.2.p.a 32 7.d odd 6 1
1176.2.p.a 32 56.k odd 6 1
1176.2.p.a 32 56.m even 6 1
2016.2.bs.c 32 12.b even 2 1
2016.2.bs.c 32 24.h odd 2 1
2016.2.bs.c 32 84.j odd 6 1
2016.2.bs.c 32 168.ba even 6 1
4704.2.p.a 32 28.f even 6 1
4704.2.p.a 32 28.g odd 6 1
4704.2.p.a 32 56.j odd 6 1
4704.2.p.a 32 56.p even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(168, [\chi])\).