↑

Properties

Label 688.2.bg.c
Level $688$
Weight $2$
Character orbit 688.bg
Analytic conductor $5.494$
Analytic rank $0$
Dimension $36$
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] = [688,2,Mod(17,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 0, 38]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("688.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.bg (of order \(21\), degree \(12\), not 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: \(5.49370765906\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(3\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 43)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9} + 4 q^{11} + 3 q^{15} - 10 q^{17} - 10 q^{19} - 21 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 9 q^{29} - 40 q^{31} - 11 q^{33} - 11 q^{35} - 19 q^{37} + q^{39} - 28 q^{41} + 8 q^{43} - 46 q^{45} + 30 q^{47} + 6 q^{49} - 57 q^{51} - 24 q^{53} - 14 q^{55} + 52 q^{57} + q^{59} - 14 q^{61} - 47 q^{63} + 38 q^{65} - 66 q^{67} - 7 q^{69} + 33 q^{71} + 29 q^{73} + 55 q^{75} - 27 q^{77} + 17 q^{79} + 38 q^{81} + 23 q^{83} - 56 q^{85} + 86 q^{87} - 19 q^{89} + 13 q^{91} - 30 q^{93} - q^{95} - 31 q^{97} + 31 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −1.28860 + 0.397480i 0 −1.48781 + 3.79089i 0 −1.38920 + 2.40617i 0 −0.976224 + 0.665578i 0
17.2 0 0.711521 − 0.219475i 0 0.511972 − 1.30448i 0 2.37928 − 4.12103i 0 −2.02062 + 1.37764i 0
17.3 0 1.63701 − 0.504949i 0 0.140805 − 0.358765i 0 −1.74586 + 3.02391i 0 −0.0539035 + 0.0367508i 0
81.1 0 −1.28860 − 0.397480i 0 −1.48781 − 3.79089i 0 −1.38920 − 2.40617i 0 −0.976224 − 0.665578i 0
81.2 0 0.711521 + 0.219475i 0 0.511972 + 1.30448i 0 2.37928 + 4.12103i 0 −2.02062 − 1.37764i 0
81.3 0 1.63701 + 0.504949i 0 0.140805 + 0.358765i 0 −1.74586 − 3.02391i 0 −0.0539035 − 0.0367508i 0
225.1 0 0.0129300 + 0.0119973i 0 −3.39819 + 0.512194i 0 0.134521 + 0.232998i 0 −0.224167 − 2.99130i 0
225.2 0 0.240184 + 0.222858i 0 0.0260188 − 0.00392170i 0 −1.56464 − 2.71003i 0 −0.216168 − 2.88456i 0
225.3 0 1.40948 + 1.30780i 0 2.95419 − 0.445272i 0 0.339884 + 0.588696i 0 0.0520854 + 0.695032i 0
273.1 0 −0.922165 + 2.34964i 0 −0.0373507 − 0.498411i 0 1.65334 − 2.86367i 0 −2.47126 − 2.29299i 0
273.2 0 0.528255 − 1.34597i 0 0.0684907 + 0.913945i 0 0.971539 − 1.68276i 0 0.666571 + 0.618487i 0
273.3 0 0.863608 − 2.20044i 0 −0.284956 − 3.80248i 0 −1.30981 + 2.26866i 0 −1.89695 − 1.76011i 0
289.1 0 −1.09131 + 0.744044i 0 −1.30537 − 1.21121i 0 0.00749281 + 0.0129779i 0 −0.458662 + 1.16865i 0
289.2 0 1.90223 − 1.29692i 0 2.37710 + 2.20563i 0 1.38418 + 2.39748i 0 0.840455 − 2.14144i 0
289.3 0 2.41097 − 1.64377i 0 −2.00127 − 1.85691i 0 −1.01083 − 1.75082i 0 2.01476 − 5.13352i 0
353.1 0 −2.66973 − 0.402397i 0 −2.09843 − 1.43068i 0 1.09365 − 1.89425i 0 4.09883 + 1.26432i 0
353.2 0 1.14296 + 0.172273i 0 −1.45700 − 0.993363i 0 0.297283 − 0.514909i 0 −1.59004 − 0.490464i 0
353.3 0 2.93359 + 0.442167i 0 −0.492700 − 0.335917i 0 −2.18154 + 3.77854i 0 5.54371 + 1.71001i 0
369.1 0 −1.09131 − 0.744044i 0 −1.30537 + 1.21121i 0 0.00749281 − 0.0129779i 0 −0.458662 − 1.16865i 0
369.2 0 1.90223 + 1.29692i 0 2.37710 − 2.20563i 0 1.38418 − 2.39748i 0 0.840455 + 2.14144i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.2.bg.c 36
4.b odd 2 1 43.2.g.a ✓ 36
12.b even 2 1 387.2.y.c 36
43.g even 21 1 inner 688.2.bg.c 36
172.o odd 42 1 43.2.g.a ✓ 36
172.o odd 42 1 1849.2.a.n 18
172.p even 42 1 1849.2.a.o 18
516.bb even 42 1 387.2.y.c 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.g.a ✓ 36 4.b odd 2 1
43.2.g.a ✓ 36 172.o odd 42 1
387.2.y.c 36 12.b even 2 1
387.2.y.c 36 516.bb even 42 1
688.2.bg.c 36 1.a even 1 1 trivial
688.2.bg.c 36 43.g even 21 1 inner
1849.2.a.n 18 172.o odd 42 1
1849.2.a.o 18 172.p even 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} - 16 T_{3}^{35} + 124 T_{3}^{34} - 612 T_{3}^{33} + 2052 T_{3}^{32} - 4171 T_{3}^{31} + \cdots + 1681 \) acting on \(S_{2}^{\mathrm{new}}(688, [\chi])\). Copy content Toggle raw display