↑

Properties

Label 168.3.e.f
Level $168$
Weight $3$
Character orbit 168.e
Analytic conductor $4.578$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,3,Mod(83,168)]
 
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, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.83");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 168.e (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: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 44 q^{4} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 44 q^{4} - 16 q^{9} - 76 q^{16} + 4 q^{18} + 32 q^{22} + 144 q^{25} + 108 q^{28} - 268 q^{30} - 176 q^{36} + 44 q^{42} + 128 q^{43} + 496 q^{46} - 464 q^{49} + 576 q^{51} + 128 q^{57} - 80 q^{58} - 244 q^{60} - 212 q^{64} - 640 q^{67} - 336 q^{70} + 220 q^{72} - 84 q^{78} + 208 q^{81} + 308 q^{84} + 56 q^{88} + 320 q^{91} - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1 −1.91650 − 0.571871i −0.759679 − 2.90222i 3.34593 + 2.19198i 4.48051i −0.203773 + 5.99654i −1.60136 − 6.81437i −5.15893 − 6.11436i −7.84577 + 4.40951i 2.56227 − 8.58688i
83.2 −1.91650 − 0.571871i 0.759679 + 2.90222i 3.34593 + 2.19198i − 4.48051i 0.203773 − 5.99654i 1.60136 − 6.81437i −5.15893 − 6.11436i −7.84577 + 4.40951i −2.56227 + 8.58688i
83.3 −1.91650 + 0.571871i −0.759679 + 2.90222i 3.34593 − 2.19198i − 4.48051i −0.203773 − 5.99654i −1.60136 + 6.81437i −5.15893 + 6.11436i −7.84577 − 4.40951i 2.56227 + 8.58688i
83.4 −1.91650 + 0.571871i 0.759679 − 2.90222i 3.34593 − 2.19198i 4.48051i 0.203773 + 5.99654i 1.60136 + 6.81437i −5.15893 + 6.11436i −7.84577 − 4.40951i −2.56227 − 8.58688i
83.5 −1.54451 − 1.27062i −2.22641 − 2.01074i 0.771038 + 3.92498i − 0.478551i 0.883824 + 5.93455i 3.37834 + 6.13081i 3.79629 − 7.04189i 0.913816 + 8.95349i −0.608057 + 0.739127i
83.6 −1.54451 − 1.27062i 2.22641 + 2.01074i 0.771038 + 3.92498i 0.478551i −0.883824 − 5.93455i −3.37834 + 6.13081i 3.79629 − 7.04189i 0.913816 + 8.95349i 0.608057 − 0.739127i
83.7 −1.54451 + 1.27062i −2.22641 + 2.01074i 0.771038 − 3.92498i 0.478551i 0.883824 − 5.93455i 3.37834 − 6.13081i 3.79629 + 7.04189i 0.913816 − 8.95349i −0.608057 − 0.739127i
83.8 −1.54451 + 1.27062i 2.22641 − 2.01074i 0.771038 − 3.92498i − 0.478551i −0.883824 + 5.93455i −3.37834 − 6.13081i 3.79629 + 7.04189i 0.913816 − 8.95349i 0.608057 + 0.739127i
83.9 −1.36669 − 1.46019i −2.87422 + 0.859555i −0.264297 + 3.99126i − 6.07480i 5.18330 + 3.02216i 1.74098 − 6.78004i 6.18920 − 5.06891i 7.52233 − 4.94111i −8.87034 + 8.30239i
83.10 −1.36669 − 1.46019i 2.87422 − 0.859555i −0.264297 + 3.99126i 6.07480i −5.18330 − 3.02216i −1.74098 − 6.78004i 6.18920 − 5.06891i 7.52233 − 4.94111i 8.87034 − 8.30239i
83.11 −1.36669 + 1.46019i −2.87422 − 0.859555i −0.264297 − 3.99126i 6.07480i 5.18330 − 3.02216i 1.74098 + 6.78004i 6.18920 + 5.06891i 7.52233 + 4.94111i −8.87034 − 8.30239i
83.12 −1.36669 + 1.46019i 2.87422 + 0.859555i −0.264297 − 3.99126i − 6.07480i −5.18330 + 3.02216i −1.74098 + 6.78004i 6.18920 + 5.06891i 7.52233 + 4.94111i 8.87034 + 8.30239i
83.13 −1.02282 − 1.71867i −1.99627 + 2.23940i −1.90766 + 3.51580i 2.25183i 5.89062 + 1.14043i −6.94082 + 0.908322i 7.99370 − 0.317389i −1.02980 − 8.94089i 3.87015 − 2.30322i
83.14 −1.02282 − 1.71867i 1.99627 − 2.23940i −1.90766 + 3.51580i − 2.25183i −5.89062 − 1.14043i 6.94082 + 0.908322i 7.99370 − 0.317389i −1.02980 − 8.94089i −3.87015 + 2.30322i
83.15 −1.02282 + 1.71867i −1.99627 − 2.23940i −1.90766 − 3.51580i − 2.25183i 5.89062 − 1.14043i −6.94082 − 0.908322i 7.99370 + 0.317389i −1.02980 + 8.94089i 3.87015 + 2.30322i
83.16 −1.02282 + 1.71867i 1.99627 + 2.23940i −1.90766 − 3.51580i 2.25183i −5.89062 + 1.14043i 6.94082 − 0.908322i 7.99370 + 0.317389i −1.02980 + 8.94089i −3.87015 − 2.30322i
83.17 −0.454905 − 1.94758i −0.0762703 + 2.99903i −3.58612 + 1.77193i − 4.74085i 5.87554 − 1.21573i 5.78723 + 3.93802i 5.08231 + 6.17820i −8.98837 − 0.457474i −9.23317 + 2.15664i
83.18 −0.454905 − 1.94758i 0.0762703 − 2.99903i −3.58612 + 1.77193i 4.74085i −5.87554 + 1.21573i −5.78723 + 3.93802i 5.08231 + 6.17820i −8.98837 − 0.457474i 9.23317 − 2.15664i
83.19 −0.454905 + 1.94758i −0.0762703 − 2.99903i −3.58612 − 1.77193i 4.74085i 5.87554 + 1.21573i 5.78723 − 3.93802i 5.08231 − 6.17820i −8.98837 + 0.457474i −9.23317 − 2.15664i
83.20 −0.454905 + 1.94758i 0.0762703 + 2.99903i −3.58612 − 1.77193i − 4.74085i −5.87554 − 1.21573i −5.78723 − 3.93802i 5.08231 − 6.17820i −8.98837 + 0.457474i 9.23317 + 2.15664i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
8.d odd 2 1 inner
21.c even 2 1 inner
24.f even 2 1 inner
56.e even 2 1 inner
168.e 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.e.f ✓ 48
3.b odd 2 1 inner 168.3.e.f ✓ 48
4.b odd 2 1 672.3.e.f 48
7.b odd 2 1 inner 168.3.e.f ✓ 48
8.b even 2 1 672.3.e.f 48
8.d odd 2 1 inner 168.3.e.f ✓ 48
12.b even 2 1 672.3.e.f 48
21.c even 2 1 inner 168.3.e.f ✓ 48
24.f even 2 1 inner 168.3.e.f ✓ 48
24.h odd 2 1 672.3.e.f 48
28.d even 2 1 672.3.e.f 48
56.e even 2 1 inner 168.3.e.f ✓ 48
56.h odd 2 1 672.3.e.f 48
84.h odd 2 1 672.3.e.f 48
168.e odd 2 1 inner 168.3.e.f ✓ 48
168.i even 2 1 672.3.e.f 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.e.f ✓ 48 1.a even 1 1 trivial
168.3.e.f ✓ 48 3.b odd 2 1 inner
168.3.e.f ✓ 48 7.b odd 2 1 inner
168.3.e.f ✓ 48 8.d odd 2 1 inner
168.3.e.f ✓ 48 21.c even 2 1 inner
168.3.e.f ✓ 48 24.f even 2 1 inner
168.3.e.f ✓ 48 56.e even 2 1 inner
168.3.e.f ✓ 48 168.e odd 2 1 inner
672.3.e.f 48 4.b odd 2 1
672.3.e.f 48 8.b even 2 1
672.3.e.f 48 12.b even 2 1
672.3.e.f 48 24.h odd 2 1
672.3.e.f 48 28.d even 2 1
672.3.e.f 48 56.h odd 2 1
672.3.e.f 48 84.h odd 2 1
672.3.e.f 48 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(168, [\chi])\):

\( T_{5}^{12} + 132T_{5}^{10} + 6448T_{5}^{8} + 142912T_{5}^{6} + 1387792T_{5}^{4} + 4299456T_{5}^{2} + 913536 \) Copy content Toggle raw display
\( T_{13}^{12} - 1160 T_{13}^{10} + 510220 T_{13}^{8} - 106236272 T_{13}^{6} + 10714276960 T_{13}^{4} + \cdots + 8005504946688 \) Copy content Toggle raw display
\( T_{17}^{12} - 1724 T_{17}^{10} + 1045940 T_{17}^{8} - 272542720 T_{17}^{6} + 32757624832 T_{17}^{4} + \cdots + 28369609555968 \) Copy content Toggle raw display