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Properties

Label 1003.2.d.c
Level $1003$
Weight $2$
Character orbit 1003.d
Analytic conductor $8.009$
Analytic rank $0$
Dimension $38$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1003,2,Mod(237,1003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1003, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1003.237");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1003 = 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1003.d (of order \(2\), degree \(1\), 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: \(8.00899532273\)
Analytic rank: \(0\)
Dimension: \(38\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 2 q^{2} + 26 q^{4} + 6 q^{8} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q + 2 q^{2} + 26 q^{4} + 6 q^{8} - 32 q^{9} - 28 q^{13} + 10 q^{15} + 10 q^{16} + 6 q^{17} + 16 q^{18} + 26 q^{19} - 40 q^{21} - 12 q^{25} + 6 q^{26} - 30 q^{30} - 26 q^{32} - 14 q^{33} + 4 q^{34} + 26 q^{35} + 28 q^{36} - 50 q^{38} + 26 q^{42} + 22 q^{43} - 10 q^{47} - 12 q^{49} - 16 q^{50} + 22 q^{51} + 26 q^{52} + 44 q^{53} - 44 q^{55} + 38 q^{59} - 42 q^{64} - 72 q^{66} - 56 q^{67} + 36 q^{68} - 4 q^{69} + 98 q^{70} - 130 q^{72} + 54 q^{76} - 2 q^{77} + 14 q^{81} - 4 q^{83} - 64 q^{84} + 22 q^{85} + 64 q^{86} + 20 q^{87} - 32 q^{89} + 90 q^{93} - 40 q^{94} - 26 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
237.1 −2.63020 0.804884i 4.91796 − 1.64983i − 2.11701i − 3.03023i −7.67481 2.35216 4.33938i
237.2 −2.63020 − 0.804884i 4.91796 1.64983i 2.11701i 3.03023i −7.67481 2.35216 − 4.33938i
237.3 −2.26874 − 1.23744i 3.14717 − 3.13111i 2.80742i − 3.20219i −2.60263 1.46875 7.10366i
237.4 −2.26874 1.23744i 3.14717 3.13111i − 2.80742i 3.20219i −2.60263 1.46875 − 7.10366i
237.5 −2.16527 − 2.66229i 2.68839 1.47379i 5.76458i − 3.16761i −1.49055 −4.08780 − 3.19114i
237.6 −2.16527 2.66229i 2.68839 − 1.47379i − 5.76458i 3.16761i −1.49055 −4.08780 3.19114i
237.7 −1.57182 0.127694i 0.470607 1.80367i − 0.200711i 1.46477i 2.40392 2.98369 − 2.83504i
237.8 −1.57182 − 0.127694i 0.470607 − 1.80367i 0.200711i − 1.46477i 2.40392 2.98369 2.83504i
237.9 −1.36755 − 2.65891i −0.129813 − 0.695095i 3.63619i 2.91153i 2.91262 −4.06980 0.950575i
237.10 −1.36755 2.65891i −0.129813 0.695095i − 3.63619i − 2.91153i 2.91262 −4.06980 − 0.950575i
237.11 −1.35816 − 2.74744i −0.155390 − 0.918352i 3.73148i − 0.791623i 2.92737 −4.54843 1.24727i
237.12 −1.35816 2.74744i −0.155390 0.918352i − 3.73148i 0.791623i 2.92737 −4.54843 − 1.24727i
237.13 −0.758828 − 3.45415i −1.42418 3.39741i 2.62111i − 4.01989i 2.59836 −8.93117 − 2.57805i
237.14 −0.758828 3.45415i −1.42418 − 3.39741i − 2.62111i 4.01989i 2.59836 −8.93117 2.57805i
237.15 −0.681704 − 1.06799i −1.53528 3.83524i 0.728052i 2.27868i 2.41001 1.85940 − 2.61450i
237.16 −0.681704 1.06799i −1.53528 − 3.83524i − 0.728052i − 2.27868i 2.41001 1.85940 2.61450i
237.17 −0.278714 1.89805i −1.92232 − 2.24074i − 0.529012i 1.34650i 1.09320 −0.602594 0.624525i
237.18 −0.278714 − 1.89805i −1.92232 2.24074i 0.529012i − 1.34650i 1.09320 −0.602594 − 0.624525i
237.19 0.121058 − 0.364340i −1.98535 − 2.39380i − 0.0441061i 1.44583i −0.482456 2.86726 − 0.289788i
237.20 0.121058 0.364340i −1.98535 2.39380i 0.0441061i − 1.44583i −0.482456 2.86726 0.289788i
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 237.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1003.2.d.c ✓ 38
17.b even 2 1 inner 1003.2.d.c ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.2.d.c ✓ 38 1.a even 1 1 trivial
1003.2.d.c ✓ 38 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{19} - T_{2}^{18} - 25 T_{2}^{17} + 23 T_{2}^{16} + 256 T_{2}^{15} - 210 T_{2}^{14} - 1393 T_{2}^{13} + \cdots + 6 \) acting on \(S_{2}^{\mathrm{new}}(1003, [\chi])\). Copy content Toggle raw display