↑

Properties

Label 168.7.z.a
Level $168$
Weight $7$
Character orbit 168.z
Analytic conductor $38.649$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,7,Mod(73,168)]
 
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([0, 0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.73");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 168.z (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: \(38.6490860481\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 324 q^{3} - 126 q^{5} - 12 q^{7} + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 324 q^{3} - 126 q^{5} - 12 q^{7} + 2916 q^{9} - 1190 q^{11} + 2268 q^{15} - 1500 q^{17} - 13446 q^{19} - 2106 q^{21} + 21504 q^{23} + 22542 q^{25} - 85484 q^{29} + 6264 q^{31} + 32130 q^{33} + 32268 q^{35} - 46938 q^{37} - 17010 q^{39} - 19548 q^{43} - 30618 q^{45} - 167004 q^{47} + 250644 q^{49} + 13500 q^{51} - 258982 q^{53} + 242028 q^{57} + 744834 q^{59} - 390096 q^{61} + 59778 q^{63} - 19388 q^{65} + 62742 q^{67} - 1102984 q^{71} - 663534 q^{73} - 608634 q^{75} + 404298 q^{77} - 271032 q^{79} - 708588 q^{81} + 2540040 q^{85} + 1154034 q^{87} - 433740 q^{89} - 2142270 q^{91} - 56376 q^{93} + 2205360 q^{95} - 578340 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 0 −13.5000 − 7.79423i 0 −160.491 + 92.6595i 0 −338.827 + 53.3419i 0 121.500 + 210.444i 0
73.2 0 −13.5000 − 7.79423i 0 −151.084 + 87.2284i 0 180.229 + 291.833i 0 121.500 + 210.444i 0
73.3 0 −13.5000 − 7.79423i 0 −126.269 + 72.9015i 0 114.286 − 323.400i 0 121.500 + 210.444i 0
73.4 0 −13.5000 − 7.79423i 0 −118.587 + 68.4663i 0 −302.168 − 162.306i 0 121.500 + 210.444i 0
73.5 0 −13.5000 − 7.79423i 0 −56.9387 + 32.8735i 0 298.061 + 169.732i 0 121.500 + 210.444i 0
73.6 0 −13.5000 − 7.79423i 0 −51.6596 + 29.8257i 0 161.216 − 302.751i 0 121.500 + 210.444i 0
73.7 0 −13.5000 − 7.79423i 0 44.9687 − 25.9627i 0 −120.755 + 321.041i 0 121.500 + 210.444i 0
73.8 0 −13.5000 − 7.79423i 0 59.9565 − 34.6159i 0 9.46147 − 342.869i 0 121.500 + 210.444i 0
73.9 0 −13.5000 − 7.79423i 0 76.6020 − 44.2262i 0 −298.004 + 169.831i 0 121.500 + 210.444i 0
73.10 0 −13.5000 − 7.79423i 0 112.001 − 64.6636i 0 289.805 + 183.473i 0 121.500 + 210.444i 0
73.11 0 −13.5000 − 7.79423i 0 127.654 − 73.7013i 0 327.920 − 100.586i 0 121.500 + 210.444i 0
73.12 0 −13.5000 − 7.79423i 0 180.847 − 104.412i 0 −327.223 − 102.830i 0 121.500 + 210.444i 0
145.1 0 −13.5000 + 7.79423i 0 −160.491 − 92.6595i 0 −338.827 − 53.3419i 0 121.500 − 210.444i 0
145.2 0 −13.5000 + 7.79423i 0 −151.084 − 87.2284i 0 180.229 − 291.833i 0 121.500 − 210.444i 0
145.3 0 −13.5000 + 7.79423i 0 −126.269 − 72.9015i 0 114.286 + 323.400i 0 121.500 − 210.444i 0
145.4 0 −13.5000 + 7.79423i 0 −118.587 − 68.4663i 0 −302.168 + 162.306i 0 121.500 − 210.444i 0
145.5 0 −13.5000 + 7.79423i 0 −56.9387 − 32.8735i 0 298.061 − 169.732i 0 121.500 − 210.444i 0
145.6 0 −13.5000 + 7.79423i 0 −51.6596 − 29.8257i 0 161.216 + 302.751i 0 121.500 − 210.444i 0
145.7 0 −13.5000 + 7.79423i 0 44.9687 + 25.9627i 0 −120.755 − 321.041i 0 121.500 − 210.444i 0
145.8 0 −13.5000 + 7.79423i 0 59.9565 + 34.6159i 0 9.46147 + 342.869i 0 121.500 − 210.444i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.7.z.a ✓ 24
4.b odd 2 1 336.7.bh.h 24
7.d odd 6 1 inner 168.7.z.a ✓ 24
28.f even 6 1 336.7.bh.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.7.z.a ✓ 24 1.a even 1 1 trivial
168.7.z.a ✓ 24 7.d odd 6 1 inner
336.7.bh.h 24 4.b odd 2 1
336.7.bh.h 24 28.f even 6 1