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Properties

Label 968.2.o.c
Level $968$
Weight $2$
Character orbit 968.o
Analytic conductor $7.730$
Analytic rank $0$
Dimension $32$
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [968,2,Mod(245,968)]
 
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([0, 5, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("968.245");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 968.o (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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{4} + 16 q^{9} - 64 q^{12} - 8 q^{14} + 8 q^{15} + 8 q^{16} - 24 q^{20} + 32 q^{23} + 16 q^{25} - 8 q^{26} + 40 q^{31} - 160 q^{34} + 24 q^{38} - 40 q^{42} + 16 q^{47} - 8 q^{48} - 40 q^{49}+ \cdots + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
245.1 −1.40928 − 0.118079i −2.66099 + 0.864609i 1.97211 + 0.332813i −0.681209 + 0.937603i 3.85216 − 0.904264i −0.935846 + 2.88024i −2.73995 − 0.701891i 3.90628 − 2.83808i 1.07072 − 1.24090i
245.2 −1.32637 + 0.490644i 1.40150 − 0.455375i 1.51854 − 1.30156i −2.09113 + 2.87820i −1.63549 + 1.29164i 1.18995 − 3.66230i −1.37555 + 2.47141i −0.670212 + 0.486937i 1.36145 − 4.84357i
245.3 −0.876503 + 1.10984i −1.40150 + 0.455375i −0.463486 − 1.94555i 2.09113 − 2.87820i 0.723026 − 1.95458i 1.18995 − 3.66230i 2.56550 + 1.19089i −0.670212 + 0.486937i 1.36145 + 4.84357i
245.4 −0.323190 + 1.37679i 2.66099 − 0.864609i −1.79110 − 0.889929i 0.681209 − 0.937603i 0.330378 + 3.94306i −0.935846 + 2.88024i 1.80411 − 2.17835i 3.90628 − 2.83808i 1.07072 + 1.24090i
245.5 0.323190 − 1.37679i 2.66099 − 0.864609i −1.79110 − 0.889929i 0.681209 − 0.937603i −0.330378 − 3.94306i 0.935846 − 2.88024i −1.80411 + 2.17835i 3.90628 − 2.83808i −1.07072 − 1.24090i
245.6 0.876503 − 1.10984i −1.40150 + 0.455375i −0.463486 − 1.94555i 2.09113 − 2.87820i −0.723026 + 1.95458i −1.18995 + 3.66230i −2.56550 − 1.19089i −0.670212 + 0.486937i −1.36145 − 4.84357i
245.7 1.32637 − 0.490644i 1.40150 − 0.455375i 1.51854 − 1.30156i −2.09113 + 2.87820i 1.63549 − 1.29164i −1.18995 + 3.66230i 1.37555 − 2.47141i −0.670212 + 0.486937i −1.36145 + 4.84357i
245.8 1.40928 + 0.118079i −2.66099 + 0.864609i 1.97211 + 0.332813i −0.681209 + 0.937603i −3.85216 + 0.904264i 0.935846 − 2.88024i 2.73995 + 0.701891i 3.90628 − 2.83808i −1.07072 + 1.24090i
269.1 −1.20953 − 0.732823i −1.64458 + 2.26358i 0.925941 + 1.77275i 1.10222 + 0.358133i 3.64798 − 1.53268i −2.45008 + 1.78008i 0.179155 − 2.82275i −1.49207 − 4.59211i −1.07072 − 1.24090i
269.2 −0.784666 − 1.17656i 0.866175 − 1.19219i −0.768600 + 1.84642i 3.38352 + 1.09937i −2.08234 − 0.0836407i 3.11534 − 2.26343i 2.77552 − 0.544514i 0.255998 + 0.787881i −1.36145 − 4.84357i
269.3 −0.547790 + 1.30381i 1.64458 − 2.26358i −1.39985 − 1.42843i −1.10222 − 0.358133i 2.05039 + 3.38419i 2.45008 − 1.78008i 2.62923 − 1.04266i −1.49207 − 4.59211i 1.07072 − 1.24090i
269.4 −0.0567584 − 1.41307i −0.866175 + 1.19219i −1.99356 + 0.160408i −3.38352 − 1.09937i 1.73381 + 1.15630i 3.11534 − 2.26343i 0.339819 + 2.80794i 0.255998 + 0.787881i −1.36145 + 4.84357i
269.5 0.0567584 + 1.41307i −0.866175 + 1.19219i −1.99356 + 0.160408i −3.38352 − 1.09937i −1.73381 − 1.15630i −3.11534 + 2.26343i −0.339819 − 2.80794i 0.255998 + 0.787881i 1.36145 − 4.84357i
269.6 0.547790 − 1.30381i 1.64458 − 2.26358i −1.39985 − 1.42843i −1.10222 − 0.358133i −2.05039 − 3.38419i −2.45008 + 1.78008i −2.62923 + 1.04266i −1.49207 − 4.59211i −1.07072 + 1.24090i
269.7 0.784666 + 1.17656i 0.866175 − 1.19219i −0.768600 + 1.84642i 3.38352 + 1.09937i 2.08234 + 0.0836407i −3.11534 + 2.26343i −2.77552 + 0.544514i 0.255998 + 0.787881i 1.36145 + 4.84357i
269.8 1.20953 + 0.732823i −1.64458 + 2.26358i 0.925941 + 1.77275i 1.10222 + 0.358133i −3.64798 + 1.53268i 2.45008 − 1.78008i −0.179155 + 2.82275i −1.49207 − 4.59211i 1.07072 + 1.24090i
493.1 −1.20953 + 0.732823i −1.64458 − 2.26358i 0.925941 − 1.77275i 1.10222 − 0.358133i 3.64798 + 1.53268i −2.45008 − 1.78008i 0.179155 + 2.82275i −1.49207 + 4.59211i −1.07072 + 1.24090i
493.2 −0.784666 + 1.17656i 0.866175 + 1.19219i −0.768600 − 1.84642i 3.38352 − 1.09937i −2.08234 + 0.0836407i 3.11534 + 2.26343i 2.77552 + 0.544514i 0.255998 − 0.787881i −1.36145 + 4.84357i
493.3 −0.547790 − 1.30381i 1.64458 + 2.26358i −1.39985 + 1.42843i −1.10222 + 0.358133i 2.05039 − 3.38419i 2.45008 + 1.78008i 2.62923 + 1.04266i −1.49207 + 4.59211i 1.07072 + 1.24090i
493.4 −0.0567584 + 1.41307i −0.866175 − 1.19219i −1.99356 − 0.160408i −3.38352 + 1.09937i 1.73381 − 1.15630i 3.11534 + 2.26343i 0.339819 − 2.80794i 0.255998 − 0.787881i −1.36145 − 4.84357i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 245.8
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{3}^{16} - 10T_{3}^{14} + 83T_{3}^{12} - 660T_{3}^{10} + 5189T_{3}^{8} - 11220T_{3}^{6} + 23987T_{3}^{4} - 49130T_{3}^{2} + 83521 \) Copy content Toggle raw display
\( T_{5}^{16} - 14 T_{5}^{14} + 179 T_{5}^{12} - 2268 T_{5}^{10} + 28709 T_{5}^{8} - 38556 T_{5}^{6} + \cdots + 83521 \) Copy content Toggle raw display
\( T_{7}^{16} + 24 T_{7}^{14} + 440 T_{7}^{12} + 7296 T_{7}^{10} + 115264 T_{7}^{8} + 992256 T_{7}^{6} + \cdots + 342102016 \) Copy content Toggle raw display