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Properties

Label 1716.2.z.g
Level $1716$
Weight $2$
Character orbit 1716.z
Analytic conductor $13.702$
Analytic rank $0$
Dimension $28$
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] = [1716,2,Mod(157,1716)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1716, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1716.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1716.z (of order \(5\), degree \(4\), 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: \(13.7023289869\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 7 q^{3} - 2 q^{5} - 5 q^{7} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 7 q^{3} - 2 q^{5} - 5 q^{7} - 7 q^{9} + 8 q^{11} - 7 q^{13} - 2 q^{15} + 4 q^{17} + 10 q^{21} + 26 q^{23} - 13 q^{25} - 7 q^{27} + 16 q^{29} - 16 q^{31} + 3 q^{33} - 24 q^{37} - 7 q^{39} - 16 q^{41} + 12 q^{43} + 8 q^{45} - 24 q^{47} - 52 q^{49} - q^{51} + 13 q^{53} + 6 q^{55} - 10 q^{57} - 19 q^{59} + 4 q^{61} - 5 q^{63} + 8 q^{65} + 40 q^{67} - 14 q^{69} - 68 q^{71} + 13 q^{73} - 13 q^{75} + 20 q^{77} - 11 q^{79} - 7 q^{81} + 4 q^{83} - 26 q^{85} - 4 q^{87} - 18 q^{89} - 5 q^{91} - 16 q^{93} + 27 q^{95} + 2 q^{97} - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1 0 0.309017 + 0.951057i 0 −2.66097 − 1.93331i 0 1.32690 − 4.08378i 0 −0.809017 + 0.587785i 0
157.2 0 0.309017 + 0.951057i 0 −2.42313 − 1.76050i 0 −1.22736 + 3.77743i 0 −0.809017 + 0.587785i 0
157.3 0 0.309017 + 0.951057i 0 −2.11469 − 1.53641i 0 0.0695547 − 0.214067i 0 −0.809017 + 0.587785i 0
157.4 0 0.309017 + 0.951057i 0 0.236907 + 0.172123i 0 −0.645989 + 1.98815i 0 −0.809017 + 0.587785i 0
157.5 0 0.309017 + 0.951057i 0 0.299270 + 0.217433i 0 1.43233 − 4.40826i 0 −0.809017 + 0.587785i 0
157.6 0 0.309017 + 0.951057i 0 1.77307 + 1.28821i 0 −1.20885 + 3.72047i 0 −0.809017 + 0.587785i 0
157.7 0 0.309017 + 0.951057i 0 3.27151 + 2.37689i 0 0.680469 − 2.09427i 0 −0.809017 + 0.587785i 0
313.1 0 −0.809017 − 0.587785i 0 −1.20740 + 3.71600i 0 0.900522 − 0.654268i 0 0.309017 + 0.951057i 0
313.2 0 −0.809017 − 0.587785i 0 −0.558385 + 1.71853i 0 −2.68818 + 1.95307i 0 0.309017 + 0.951057i 0
313.3 0 −0.809017 − 0.587785i 0 −0.373730 + 1.15022i 0 0.357194 − 0.259516i 0 0.309017 + 0.951057i 0
313.4 0 −0.809017 − 0.587785i 0 0.152253 − 0.468586i 0 2.52781 − 1.83656i 0 0.309017 + 0.951057i 0
313.5 0 −0.809017 − 0.587785i 0 0.360665 − 1.11001i 0 −3.79821 + 2.75956i 0 0.309017 + 0.951057i 0
313.6 0 −0.809017 − 0.587785i 0 1.09186 − 3.36039i 0 −1.00142 + 0.727576i 0 0.309017 + 0.951057i 0
313.7 0 −0.809017 − 0.587785i 0 1.15278 − 3.54788i 0 0.775227 − 0.563235i 0 0.309017 + 0.951057i 0
625.1 0 −0.809017 + 0.587785i 0 −1.20740 − 3.71600i 0 0.900522 + 0.654268i 0 0.309017 − 0.951057i 0
625.2 0 −0.809017 + 0.587785i 0 −0.558385 − 1.71853i 0 −2.68818 − 1.95307i 0 0.309017 − 0.951057i 0
625.3 0 −0.809017 + 0.587785i 0 −0.373730 − 1.15022i 0 0.357194 + 0.259516i 0 0.309017 − 0.951057i 0
625.4 0 −0.809017 + 0.587785i 0 0.152253 + 0.468586i 0 2.52781 + 1.83656i 0 0.309017 − 0.951057i 0
625.5 0 −0.809017 + 0.587785i 0 0.360665 + 1.11001i 0 −3.79821 − 2.75956i 0 0.309017 − 0.951057i 0
625.6 0 −0.809017 + 0.587785i 0 1.09186 + 3.36039i 0 −1.00142 − 0.727576i 0 0.309017 − 0.951057i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1716.2.z.g ✓ 28
11.c even 5 1 inner 1716.2.z.g ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.2.z.g ✓ 28 1.a even 1 1 trivial
1716.2.z.g ✓ 28 11.c even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{28} + 2 T_{5}^{27} + 26 T_{5}^{26} + 65 T_{5}^{25} + 492 T_{5}^{24} + 1280 T_{5}^{23} + \cdots + 2560000 \) acting on \(S_{2}^{\mathrm{new}}(1716, [\chi])\). Copy content Toggle raw display