↑

Properties

Label 968.2.k.f
Level $968$
Weight $2$
Character orbit 968.k
Analytic conductor $7.730$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $16$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [968,2,Mod(403,968)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(968, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("968.403");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 968.k (of order \(10\), degree \(4\), 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: \(7.72951891566\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{3} + 8 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{3} + 8 q^{4} - 8 q^{9} - 32 q^{12} + 16 q^{16} + 24 q^{20} + 16 q^{25} - 12 q^{26} - 32 q^{27} + 48 q^{34} + 8 q^{36} - 48 q^{38} - 24 q^{42} + 16 q^{48} + 8 q^{49} + 28 q^{58} + 72 q^{60} - 64 q^{64} - 288 q^{67} + 48 q^{70} + 112 q^{75} - 48 q^{80} - 8 q^{81} + 12 q^{82} - 24 q^{86} + 288 q^{89} + 24 q^{91} - 72 q^{92} - 48 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
403.1 −1.29195 + 0.575212i −0.226216 + 0.696222i 1.33826 − 1.48629i −0.157497 − 0.216775i −0.108216 − 1.02960i −0.756934 − 2.32960i −0.874032 + 2.68999i 1.99350 + 1.44836i 0.328169 + 0.189469i
403.2 −1.29195 + 0.575212i 0.844250 − 2.59833i 1.33826 − 1.48629i 2.19364 + 3.01929i 0.403867 + 3.84254i 0.756934 + 2.32960i −0.874032 + 2.68999i −3.61153 − 2.62393i −4.57081 − 2.63896i
403.3 −0.147826 + 1.40647i −0.226216 + 0.696222i −1.95630 − 0.415823i 0.157497 + 0.216775i −0.945772 − 0.421085i 0.756934 + 2.32960i 0.874032 − 2.68999i 1.99350 + 1.44836i −0.328169 + 0.189469i
403.4 −0.147826 + 1.40647i 0.844250 − 2.59833i −1.95630 − 0.415823i −2.19364 − 3.01929i 3.52967 + 1.57151i −0.756934 − 2.32960i 0.874032 − 2.68999i −3.61153 − 2.62393i 4.57081 − 2.63896i
403.5 0.147826 − 1.40647i −0.226216 + 0.696222i −1.95630 − 0.415823i 0.157497 + 0.216775i 0.945772 + 0.421085i −0.756934 − 2.32960i −0.874032 + 2.68999i 1.99350 + 1.44836i 0.328169 − 0.189469i
403.6 0.147826 − 1.40647i 0.844250 − 2.59833i −1.95630 − 0.415823i −2.19364 − 3.01929i −3.52967 − 1.57151i 0.756934 + 2.32960i −0.874032 + 2.68999i −3.61153 − 2.62393i −4.57081 + 2.63896i
403.7 1.29195 − 0.575212i −0.226216 + 0.696222i 1.33826 − 1.48629i −0.157497 − 0.216775i 0.108216 + 1.02960i 0.756934 + 2.32960i 0.874032 − 2.68999i 1.99350 + 1.44836i −0.328169 − 0.189469i
403.8 1.29195 − 0.575212i 0.844250 − 2.59833i 1.33826 − 1.48629i 2.19364 + 3.01929i −0.403867 − 3.84254i −0.756934 − 2.32960i 0.874032 − 2.68999i −3.61153 − 2.62393i 4.57081 + 2.63896i
475.1 −1.38331 + 0.294032i −2.21028 + 1.60586i 1.82709 − 0.813473i −3.54939 + 1.15327i 2.58532 − 2.87129i 1.98168 + 1.43977i −2.28825 + 1.66251i 1.37948 − 4.24561i 4.57081 − 2.63896i
475.2 −1.38331 + 0.294032i 0.592242 − 0.430289i 1.82709 − 0.813473i 0.254835 − 0.0828009i −0.692735 + 0.769360i −1.98168 − 1.43977i −2.28825 + 1.66251i −0.761449 + 2.34350i −0.328169 + 0.189469i
475.3 −0.946294 − 1.05097i −2.21028 + 1.60586i −0.209057 + 1.98904i 3.54939 − 1.15327i 3.77927 + 0.803309i −1.98168 − 1.43977i 2.28825 − 1.66251i 1.37948 − 4.24561i −4.57081 − 2.63896i
475.4 −0.946294 − 1.05097i 0.592242 − 0.430289i −0.209057 + 1.98904i −0.254835 + 0.0828009i −1.01265 − 0.215246i 1.98168 + 1.43977i 2.28825 − 1.66251i −0.761449 + 2.34350i 0.328169 + 0.189469i
475.5 0.946294 + 1.05097i −2.21028 + 1.60586i −0.209057 + 1.98904i 3.54939 − 1.15327i −3.77927 − 0.803309i 1.98168 + 1.43977i −2.28825 + 1.66251i 1.37948 − 4.24561i 4.57081 + 2.63896i
475.6 0.946294 + 1.05097i 0.592242 − 0.430289i −0.209057 + 1.98904i −0.254835 + 0.0828009i 1.01265 + 0.215246i −1.98168 − 1.43977i −2.28825 + 1.66251i −0.761449 + 2.34350i −0.328169 − 0.189469i
475.7 1.38331 − 0.294032i −2.21028 + 1.60586i 1.82709 − 0.813473i −3.54939 + 1.15327i −2.58532 + 2.87129i −1.98168 − 1.43977i 2.28825 − 1.66251i 1.37948 − 4.24561i −4.57081 + 2.63896i
475.8 1.38331 − 0.294032i 0.592242 − 0.430289i 1.82709 − 0.813473i 0.254835 − 0.0828009i 0.692735 − 0.769360i 1.98168 + 1.43977i 2.28825 − 1.66251i −0.761449 + 2.34350i 0.328169 − 0.189469i
699.1 −1.38331 − 0.294032i −2.21028 − 1.60586i 1.82709 + 0.813473i −3.54939 − 1.15327i 2.58532 + 2.87129i 1.98168 − 1.43977i −2.28825 − 1.66251i 1.37948 + 4.24561i 4.57081 + 2.63896i
699.2 −1.38331 − 0.294032i 0.592242 + 0.430289i 1.82709 + 0.813473i 0.254835 + 0.0828009i −0.692735 − 0.769360i −1.98168 + 1.43977i −2.28825 − 1.66251i −0.761449 − 2.34350i −0.328169 − 0.189469i
699.3 −0.946294 + 1.05097i −2.21028 − 1.60586i −0.209057 − 1.98904i 3.54939 + 1.15327i 3.77927 − 0.803309i −1.98168 + 1.43977i 2.28825 + 1.66251i 1.37948 + 4.24561i −4.57081 + 2.63896i
699.4 −0.946294 + 1.05097i 0.592242 + 0.430289i −0.209057 − 1.98904i −0.254835 − 0.0828009i −1.01265 + 0.215246i 1.98168 − 1.43977i 2.28825 + 1.66251i −0.761449 − 2.34350i 0.328169 − 0.189469i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 403.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
88.g even 2 1 inner
88.k even 10 3 inner
88.l odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.2.k.f 32
8.d odd 2 1 inner 968.2.k.f 32
11.b odd 2 1 inner 968.2.k.f 32
11.c even 5 1 968.2.g.b ✓ 8
11.c even 5 3 inner 968.2.k.f 32
11.d odd 10 1 968.2.g.b ✓ 8
11.d odd 10 3 inner 968.2.k.f 32
44.g even 10 1 3872.2.g.a 8
44.h odd 10 1 3872.2.g.a 8
88.g even 2 1 inner 968.2.k.f 32
88.k even 10 1 968.2.g.b ✓ 8
88.k even 10 3 inner 968.2.k.f 32
88.l odd 10 1 968.2.g.b ✓ 8
88.l odd 10 3 inner 968.2.k.f 32
88.o even 10 1 3872.2.g.a 8
88.p odd 10 1 3872.2.g.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
968.2.g.b ✓ 8 11.c even 5 1
968.2.g.b ✓ 8 11.d odd 10 1
968.2.g.b ✓ 8 88.k even 10 1
968.2.g.b ✓ 8 88.l odd 10 1
968.2.k.f 32 1.a even 1 1 trivial
968.2.k.f 32 8.d odd 2 1 inner
968.2.k.f 32 11.b odd 2 1 inner
968.2.k.f 32 11.c even 5 3 inner
968.2.k.f 32 11.d odd 10 3 inner
968.2.k.f 32 88.g even 2 1 inner
968.2.k.f 32 88.k even 10 3 inner
968.2.k.f 32 88.l odd 10 3 inner
3872.2.g.a 8 44.g even 10 1
3872.2.g.a 8 44.h odd 10 1
3872.2.g.a 8 88.o even 10 1
3872.2.g.a 8 88.p odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\):

\( T_{3}^{8} + 2T_{3}^{7} + 6T_{3}^{6} + 16T_{3}^{5} + 44T_{3}^{4} - 32T_{3}^{3} + 24T_{3}^{2} - 16T_{3} + 16 \) Copy content Toggle raw display
\( T_{17}^{16} - 12 T_{17}^{14} + 135 T_{17}^{12} - 1512 T_{17}^{10} + 16929 T_{17}^{8} - 13608 T_{17}^{6} + \cdots + 6561 \) Copy content Toggle raw display