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Properties

Label 1000.2.k.g
Level $1000$
Weight $2$
Character orbit 1000.k
Analytic conductor $7.985$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1000,2,Mod(307,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1000.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.k (of order \(4\), 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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{6} - 16 q^{11} - 16 q^{16} + 72 q^{26} + 16 q^{36} - 16 q^{41} + 112 q^{46} - 272 q^{51} - 48 q^{56} + 48 q^{66} - 176 q^{76} - 32 q^{81} - 104 q^{86} - 16 q^{91} - 48 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 −1.38341 − 0.293555i −2.23811 + 2.23811i 1.82765 + 0.812215i 0 3.75323 − 2.43921i 1.14088 − 1.14088i −2.28996 − 1.66014i − 7.01825i 0
307.2 −1.37255 − 0.340727i 1.22568 − 1.22568i 1.76781 + 0.935332i 0 −2.09993 + 1.26469i 2.24548 − 2.24548i −2.10772 − 1.88613i − 0.00456158i 0
307.3 −1.36369 + 0.374640i −0.370474 + 0.370474i 1.71929 − 1.02178i 0 0.366417 − 0.644006i −1.12342 + 1.12342i −1.96177 + 2.03751i 2.72550i 0
307.4 −1.36172 + 0.381735i 0.540520 − 0.540520i 1.70856 − 1.03963i 0 −0.529700 + 0.942371i 1.44802 − 1.44802i −1.92971 + 2.06790i 2.41568i 0
307.5 −1.27675 − 0.608205i 0.973971 − 0.973971i 1.26017 + 1.55305i 0 −1.83589 + 0.651141i 0.522974 − 0.522974i −0.664350 − 2.74930i 1.10276i 0
307.6 −1.18382 − 0.773670i −1.64664 + 1.64664i 0.802870 + 1.83178i 0 3.22328 − 0.675373i −2.95603 + 2.95603i 0.466734 − 2.78965i − 2.42284i 0
307.7 −1.13860 − 0.838801i −0.443206 + 0.443206i 0.592826 + 1.91012i 0 0.876397 − 0.132873i −2.04389 + 2.04389i 0.927218 − 2.67213i 2.60714i 0
307.8 −1.06641 + 0.928849i −1.78961 + 1.78961i 0.274480 − 1.98108i 0 0.246189 − 3.57075i −0.329013 + 0.329013i 1.54741 + 2.36760i − 3.40542i 0
307.9 −0.928849 + 1.06641i 1.78961 − 1.78961i −0.274480 − 1.98108i 0 0.246189 + 3.57075i −0.329013 + 0.329013i 2.36760 + 1.54741i − 3.40542i 0
307.10 −0.838801 − 1.13860i −0.443206 + 0.443206i −0.592826 + 1.91012i 0 0.876397 + 0.132873i 2.04389 − 2.04389i 2.67213 − 0.927218i 2.60714i 0
307.11 −0.773670 − 1.18382i −1.64664 + 1.64664i −0.802870 + 1.83178i 0 3.22328 + 0.675373i 2.95603 − 2.95603i 2.78965 − 0.466734i − 2.42284i 0
307.12 −0.608205 − 1.27675i 0.973971 − 0.973971i −1.26017 + 1.55305i 0 −1.83589 − 0.651141i −0.522974 + 0.522974i 2.74930 + 0.664350i 1.10276i 0
307.13 −0.381735 + 1.36172i −0.540520 + 0.540520i −1.70856 − 1.03963i 0 −0.529700 − 0.942371i 1.44802 − 1.44802i 2.06790 − 1.92971i 2.41568i 0
307.14 −0.374640 + 1.36369i 0.370474 − 0.370474i −1.71929 − 1.02178i 0 0.366417 + 0.644006i −1.12342 + 1.12342i 2.03751 − 1.96177i 2.72550i 0
307.15 −0.340727 − 1.37255i 1.22568 − 1.22568i −1.76781 + 0.935332i 0 −2.09993 − 1.26469i −2.24548 + 2.24548i 1.88613 + 2.10772i − 0.00456158i 0
307.16 −0.293555 − 1.38341i −2.23811 + 2.23811i −1.82765 + 0.812215i 0 3.75323 + 2.43921i −1.14088 + 1.14088i 1.66014 + 2.28996i − 7.01825i 0
307.17 0.293555 + 1.38341i 2.23811 − 2.23811i −1.82765 + 0.812215i 0 3.75323 + 2.43921i 1.14088 − 1.14088i −1.66014 − 2.28996i − 7.01825i 0
307.18 0.340727 + 1.37255i −1.22568 + 1.22568i −1.76781 + 0.935332i 0 −2.09993 − 1.26469i 2.24548 − 2.24548i −1.88613 − 2.10772i − 0.00456158i 0
307.19 0.374640 − 1.36369i −0.370474 + 0.370474i −1.71929 − 1.02178i 0 0.366417 + 0.644006i 1.12342 − 1.12342i −2.03751 + 1.96177i 2.72550i 0
307.20 0.381735 − 1.36172i 0.540520 − 0.540520i −1.70856 − 1.03963i 0 −0.529700 − 0.942371i −1.44802 + 1.44802i −2.06790 + 1.92971i 2.41568i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
8.d odd 2 1 inner
40.e odd 2 1 inner
40.k even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.k.g ✓ 64
5.b even 2 1 inner 1000.2.k.g ✓ 64
5.c odd 4 2 inner 1000.2.k.g ✓ 64
8.d odd 2 1 inner 1000.2.k.g ✓ 64
40.e odd 2 1 inner 1000.2.k.g ✓ 64
40.k even 4 2 inner 1000.2.k.g ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.2.k.g ✓ 64 1.a even 1 1 trivial
1000.2.k.g ✓ 64 5.b even 2 1 inner
1000.2.k.g ✓ 64 5.c odd 4 2 inner
1000.2.k.g ✓ 64 8.d odd 2 1 inner
1000.2.k.g ✓ 64 40.e odd 2 1 inner
1000.2.k.g ✓ 64 40.k even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\):

\( T_{3}^{32} + 184 T_{3}^{28} + 10570 T_{3}^{24} + 237140 T_{3}^{20} + 1930955 T_{3}^{16} + 4982684 T_{3}^{12} + \cdots + 15625 \) Copy content Toggle raw display
\( T_{7}^{32} + 508 T_{7}^{28} + 74586 T_{7}^{24} + 4153500 T_{7}^{20} + 84746675 T_{7}^{16} + \cdots + 23088025 \) Copy content Toggle raw display