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Properties

Label 1848.2.cf.a
Level $1848$
Weight $2$
Character orbit 1848.cf
Analytic conductor $14.756$
Analytic rank $0$
Dimension $48$
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] = [1848,2,Mod(241,1848)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1848, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1848.241");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1848.cf (of order \(6\), 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: \(14.7563542935\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{7} + 24 q^{9} + 2 q^{11} + 4 q^{15} + 8 q^{17} - 10 q^{19} - 12 q^{21} - 12 q^{23} + 26 q^{25} - 6 q^{31} + 10 q^{33} - 16 q^{35} - 2 q^{37} + 12 q^{39} + 32 q^{41} + 38 q^{49} + 6 q^{51} - 12 q^{53} + 38 q^{55} + 4 q^{61} + 2 q^{63} + 48 q^{65} - 4 q^{67} + 44 q^{73} + 12 q^{75} - 4 q^{77} + 30 q^{79} - 24 q^{81} - 8 q^{83} - 4 q^{87} + 24 q^{89} + 4 q^{91} - 12 q^{93} - 72 q^{95} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1 0 −0.866025 − 0.500000i 0 −3.36627 + 1.94351i 0 −1.56862 − 2.13060i 0 0.500000 + 0.866025i 0
241.2 0 −0.866025 − 0.500000i 0 2.38183 − 1.37515i 0 2.63851 + 0.195584i 0 0.500000 + 0.866025i 0
241.3 0 −0.866025 − 0.500000i 0 −2.26399 + 1.30712i 0 0.955835 + 2.46706i 0 0.500000 + 0.866025i 0
241.4 0 −0.866025 − 0.500000i 0 −2.32516 + 1.34243i 0 2.60959 − 0.435946i 0 0.500000 + 0.866025i 0
241.5 0 −0.866025 − 0.500000i 0 −1.66549 + 0.961574i 0 2.15419 − 1.53606i 0 0.500000 + 0.866025i 0
241.6 0 −0.866025 − 0.500000i 0 1.78705 − 1.03175i 0 1.02905 − 2.43743i 0 0.500000 + 0.866025i 0
241.7 0 −0.866025 − 0.500000i 0 −0.848913 + 0.490120i 0 −1.39144 + 2.25031i 0 0.500000 + 0.866025i 0
241.8 0 −0.866025 − 0.500000i 0 0.500189 − 0.288784i 0 1.55651 + 2.13946i 0 0.500000 + 0.866025i 0
241.9 0 −0.866025 − 0.500000i 0 −1.05394 + 0.608492i 0 −2.62284 + 0.347403i 0 0.500000 + 0.866025i 0
241.10 0 −0.866025 − 0.500000i 0 1.04655 − 0.604228i 0 −2.54930 − 0.707869i 0 0.500000 + 0.866025i 0
241.11 0 −0.866025 − 0.500000i 0 1.19465 − 0.689729i 0 0.634068 − 2.56865i 0 0.500000 + 0.866025i 0
241.12 0 −0.866025 − 0.500000i 0 3.74747 − 2.16360i 0 −2.21350 − 1.44929i 0 0.500000 + 0.866025i 0
241.13 0 0.866025 + 0.500000i 0 −3.08452 + 1.78085i 0 0.0167662 + 2.64570i 0 0.500000 + 0.866025i 0
241.14 0 0.866025 + 0.500000i 0 2.36965 − 1.36812i 0 2.43649 + 1.03128i 0 0.500000 + 0.866025i 0
241.15 0 0.866025 + 0.500000i 0 0.0306583 − 0.0177006i 0 2.48876 − 0.897818i 0 0.500000 + 0.866025i 0
241.16 0 0.866025 + 0.500000i 0 0.543395 − 0.313729i 0 −2.04758 − 1.67554i 0 0.500000 + 0.866025i 0
241.17 0 0.866025 + 0.500000i 0 0.591612 − 0.341567i 0 −0.0197705 + 2.64568i 0 0.500000 + 0.866025i 0
241.18 0 0.866025 + 0.500000i 0 −1.37955 + 0.796484i 0 1.78299 − 1.95473i 0 0.500000 + 0.866025i 0
241.19 0 0.866025 + 0.500000i 0 −1.30334 + 0.752486i 0 1.59735 + 2.10914i 0 0.500000 + 0.866025i 0
241.20 0 0.866025 + 0.500000i 0 −1.63143 + 0.941905i 0 −2.20388 − 1.46387i 0 0.500000 + 0.866025i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 241.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1848.2.cf.a ✓ 48
7.d odd 6 1 1848.2.cf.b yes 48
11.b odd 2 1 1848.2.cf.b yes 48
77.i even 6 1 inner 1848.2.cf.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1848.2.cf.a ✓ 48 1.a even 1 1 trivial
1848.2.cf.a ✓ 48 77.i even 6 1 inner
1848.2.cf.b yes 48 7.d odd 6 1
1848.2.cf.b yes 48 11.b odd 2 1