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Properties

Label 168.3.g.a
Level $168$
Weight $3$
Character orbit 168.g
Analytic conductor $4.578$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,3,Mod(43,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.43");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 168.g (of order \(2\), degree \(1\), 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: \(4.57766844125\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 10 q^{4} + 12 q^{6} + 10 q^{8} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 10 q^{4} + 12 q^{6} + 10 q^{8} + 72 q^{9} + 12 q^{10} + 32 q^{11} + 24 q^{12} - 14 q^{14} + 66 q^{16} + 16 q^{17} - 6 q^{18} - 64 q^{19} + 20 q^{20} + 12 q^{22} - 36 q^{24} - 72 q^{25} + 100 q^{26} - 14 q^{28} + 72 q^{30} + 98 q^{32} - 108 q^{34} + 30 q^{36} - 72 q^{38} - 332 q^{40} - 80 q^{41} + 32 q^{43} - 292 q^{44} - 48 q^{48} - 168 q^{49} + 46 q^{50} + 192 q^{51} - 4 q^{52} + 36 q^{54} + 98 q^{56} - 96 q^{58} - 24 q^{60} - 16 q^{62} - 182 q^{64} - 192 q^{65} - 24 q^{66} - 32 q^{67} + 188 q^{68} - 84 q^{70} + 30 q^{72} - 240 q^{73} + 208 q^{74} - 384 q^{75} + 8 q^{76} + 168 q^{78} + 484 q^{80} + 216 q^{81} - 372 q^{82} - 320 q^{83} - 604 q^{86} + 468 q^{88} + 400 q^{89} + 36 q^{90} + 352 q^{92} - 72 q^{94} - 252 q^{96} + 144 q^{97} + 14 q^{98} + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.98610 − 0.235345i 1.73205 3.88923 + 0.934838i − 7.47825i −3.44003 − 0.407629i − 2.64575i −7.50440 − 2.77200i 3.00000 −1.75997 + 14.8526i
43.2 −1.98610 + 0.235345i 1.73205 3.88923 − 0.934838i 7.47825i −3.44003 + 0.407629i 2.64575i −7.50440 + 2.77200i 3.00000 −1.75997 − 14.8526i
43.3 −1.92787 − 0.532260i −1.73205 3.43340 + 2.05226i 0.560422i 3.33918 + 0.921901i − 2.64575i −5.52682 − 5.78396i 3.00000 0.298290 − 1.08042i
43.4 −1.92787 + 0.532260i −1.73205 3.43340 − 2.05226i − 0.560422i 3.33918 − 0.921901i 2.64575i −5.52682 + 5.78396i 3.00000 0.298290 + 1.08042i
43.5 −1.61092 − 1.18531i −1.73205 1.19010 + 3.81886i − 8.04930i 2.79019 + 2.05301i 2.64575i 2.60937 − 7.56248i 3.00000 −9.54089 + 12.9667i
43.6 −1.61092 + 1.18531i −1.73205 1.19010 − 3.81886i 8.04930i 2.79019 − 2.05301i − 2.64575i 2.60937 + 7.56248i 3.00000 −9.54089 − 12.9667i
43.7 −1.59973 − 1.20036i 1.73205 1.11825 + 3.84051i − 0.769463i −2.77081 − 2.07909i 2.64575i 2.82111 − 7.48608i 3.00000 −0.923636 + 1.23093i
43.8 −1.59973 + 1.20036i 1.73205 1.11825 − 3.84051i 0.769463i −2.77081 + 2.07909i − 2.64575i 2.82111 + 7.48608i 3.00000 −0.923636 − 1.23093i
43.9 −1.05960 − 1.69625i −1.73205 −1.75451 + 3.59468i 2.47223i 1.83527 + 2.93799i − 2.64575i 7.95653 − 0.832818i 3.00000 4.19351 − 2.61956i
43.10 −1.05960 + 1.69625i −1.73205 −1.75451 − 3.59468i − 2.47223i 1.83527 − 2.93799i 2.64575i 7.95653 + 0.832818i 3.00000 4.19351 + 2.61956i
43.11 −0.121960 − 1.99628i 1.73205 −3.97025 + 0.486933i 9.26683i −0.211241 − 3.45765i − 2.64575i 1.45627 + 7.86634i 3.00000 18.4992 − 1.13018i
43.12 −0.121960 + 1.99628i 1.73205 −3.97025 − 0.486933i − 9.26683i −0.211241 + 3.45765i 2.64575i 1.45627 − 7.86634i 3.00000 18.4992 + 1.13018i
43.13 −0.0411377 − 1.99958i −1.73205 −3.99662 + 0.164516i 3.19600i 0.0712526 + 3.46337i 2.64575i 0.493374 + 7.98477i 3.00000 6.39065 − 0.131476i
43.14 −0.0411377 + 1.99958i −1.73205 −3.99662 − 0.164516i − 3.19600i 0.0712526 − 3.46337i − 2.64575i 0.493374 − 7.98477i 3.00000 6.39065 + 0.131476i
43.15 0.434385 − 1.95226i −1.73205 −3.62262 − 1.69606i − 3.78720i −0.752377 + 3.38141i − 2.64575i −4.88476 + 6.33554i 3.00000 −7.39360 − 1.64510i
43.16 0.434385 + 1.95226i −1.73205 −3.62262 + 1.69606i 3.78720i −0.752377 − 3.38141i 2.64575i −4.88476 − 6.33554i 3.00000 −7.39360 + 1.64510i
43.17 1.18386 − 1.61198i 1.73205 −1.19696 − 3.81671i − 2.78060i 2.05050 − 2.79203i − 2.64575i −7.56949 − 2.58898i 3.00000 −4.48227 − 3.29183i
43.18 1.18386 + 1.61198i 1.73205 −1.19696 + 3.81671i 2.78060i 2.05050 + 2.79203i 2.64575i −7.56949 + 2.58898i 3.00000 −4.48227 + 3.29183i
43.19 1.81394 − 0.842398i 1.73205 2.58073 − 3.05611i 5.94177i 3.14183 − 1.45908i 2.64575i 2.10682 − 7.71760i 3.00000 5.00533 + 10.7780i
43.20 1.81394 + 0.842398i 1.73205 2.58073 + 3.05611i − 5.94177i 3.14183 + 1.45908i − 2.64575i 2.10682 + 7.71760i 3.00000 5.00533 − 10.7780i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.3.g.a ✓ 24
3.b odd 2 1 504.3.g.d 24
4.b odd 2 1 672.3.g.a 24
8.b even 2 1 672.3.g.a 24
8.d odd 2 1 inner 168.3.g.a ✓ 24
12.b even 2 1 2016.3.g.d 24
24.f even 2 1 504.3.g.d 24
24.h odd 2 1 2016.3.g.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.g.a ✓ 24 1.a even 1 1 trivial
168.3.g.a ✓ 24 8.d odd 2 1 inner
504.3.g.d 24 3.b odd 2 1
504.3.g.d 24 24.f even 2 1
672.3.g.a 24 4.b odd 2 1
672.3.g.a 24 8.b even 2 1
2016.3.g.d 24 12.b even 2 1
2016.3.g.d 24 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(168, [\chi])\).