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Properties

Label 1148.2.n.d
Level $1148$
Weight $2$
Character orbit 1148.n
Analytic conductor $9.167$
Analytic rank $0$
Dimension $24$
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] = [1148,2,Mod(57,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1148.57");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.n (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: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 10 q^{3} + 4 q^{5} - 6 q^{7} + 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 10 q^{3} + 4 q^{5} - 6 q^{7} + 38 q^{9} + 11 q^{11} - 4 q^{13} + 10 q^{15} + 9 q^{17} - 23 q^{19} + 5 q^{21} + 28 q^{23} - 10 q^{25} - 76 q^{27} + 28 q^{29} - 18 q^{31} - 27 q^{33} - q^{35} - 29 q^{37} - 6 q^{39} + 65 q^{41} - 15 q^{43} - 20 q^{45} - 11 q^{47} - 6 q^{49} - 18 q^{51} + 8 q^{53} - 50 q^{55} + 8 q^{57} + 55 q^{59} - 10 q^{61} - 2 q^{63} - 11 q^{65} + 65 q^{67} - 2 q^{69} - 14 q^{71} + 48 q^{73} - 77 q^{75} + 11 q^{77} + 22 q^{79} + 80 q^{81} - 22 q^{83} - 78 q^{85} - 4 q^{87} + 16 q^{89} - 4 q^{91} - 60 q^{93} + 56 q^{95} + 15 q^{97} + 80 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1 0 −3.15594 0 1.01403 + 0.736733i 0 0.309017 + 0.951057i 0 6.95994 0
57.2 0 −2.86083 0 −3.47370 − 2.52379i 0 0.309017 + 0.951057i 0 5.18432 0
57.3 0 −0.839141 0 3.00819 + 2.18558i 0 0.309017 + 0.951057i 0 −2.29584 0
57.4 0 1.14291 0 −1.88811 − 1.37180i 0 0.309017 + 0.951057i 0 −1.69375 0
57.5 0 1.35294 0 2.62263 + 1.90545i 0 0.309017 + 0.951057i 0 −1.16955 0
57.6 0 2.97808 0 0.835003 + 0.606666i 0 0.309017 + 0.951057i 0 5.86899 0
141.1 0 −3.15594 0 1.01403 − 0.736733i 0 0.309017 − 0.951057i 0 6.95994 0
141.2 0 −2.86083 0 −3.47370 + 2.52379i 0 0.309017 − 0.951057i 0 5.18432 0
141.3 0 −0.839141 0 3.00819 − 2.18558i 0 0.309017 − 0.951057i 0 −2.29584 0
141.4 0 1.14291 0 −1.88811 + 1.37180i 0 0.309017 − 0.951057i 0 −1.69375 0
141.5 0 1.35294 0 2.62263 − 1.90545i 0 0.309017 − 0.951057i 0 −1.16955 0
141.6 0 2.97808 0 0.835003 − 0.606666i 0 0.309017 − 0.951057i 0 5.86899 0
365.1 0 −3.33413 0 −0.141417 + 0.435238i 0 −0.809017 − 0.587785i 0 8.11640 0
365.2 0 −2.40864 0 1.08800 − 3.34852i 0 −0.809017 − 0.587785i 0 2.80153 0
365.3 0 −1.21617 0 0.00173847 − 0.00535046i 0 −0.809017 − 0.587785i 0 −1.52092 0
365.4 0 −0.0323142 0 −0.719662 + 2.21489i 0 −0.809017 − 0.587785i 0 −2.99896 0
365.5 0 1.51551 0 0.929786 − 2.86159i 0 −0.809017 − 0.587785i 0 −0.703221 0
365.6 0 1.85770 0 −1.27648 + 3.92860i 0 −0.809017 − 0.587785i 0 0.451065 0
953.1 0 −3.33413 0 −0.141417 − 0.435238i 0 −0.809017 + 0.587785i 0 8.11640 0
953.2 0 −2.40864 0 1.08800 + 3.34852i 0 −0.809017 + 0.587785i 0 2.80153 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.n.d ✓ 24
41.d even 5 1 inner 1148.2.n.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.n.d ✓ 24 1.a even 1 1 trivial
1148.2.n.d ✓ 24 41.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 5 T_{3}^{11} - 15 T_{3}^{10} - 94 T_{3}^{9} + 67 T_{3}^{8} + 616 T_{3}^{7} - 135 T_{3}^{6} + \cdots - 31 \) acting on \(S_{2}^{\mathrm{new}}(1148, [\chi])\). Copy content Toggle raw display