Properties

Label 740.2.bf.a
Level $740$
Weight $2$
Character orbit 740.bf
Analytic conductor $5.909$
Analytic rank $0$
Dimension $76$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(97,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.bf (of order \(12\), 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: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(19\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 76 q - 2 q^{3} + 8 q^{13} + 2 q^{15} + 12 q^{19} - 4 q^{23} + 2 q^{25} + 28 q^{27} - 6 q^{29} + 16 q^{31} - 6 q^{33} + 20 q^{35} - 22 q^{37} + 8 q^{39} + 54 q^{41} - 16 q^{43} + 38 q^{45} + 8 q^{47} - 36 q^{49} - 24 q^{53} + 18 q^{55} - 16 q^{59} - 28 q^{61} - 6 q^{65} - 34 q^{67} + 64 q^{69} - 8 q^{71} + 4 q^{73} + 24 q^{75} - 2 q^{77} - 40 q^{79} + 2 q^{81} - 62 q^{83} + 108 q^{87} - 2 q^{89} - 20 q^{91} - 30 q^{93} - 70 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 0 −0.804236 + 3.00145i 0 0.433448 2.19366i 0 −0.0415533 + 0.155079i 0 −5.76383 3.32775i 0
97.2 0 −0.701381 + 2.61759i 0 1.90653 + 1.16840i 0 0.951054 3.54938i 0 −3.76176 2.17185i 0
97.3 0 −0.699874 + 2.61196i 0 −2.23588 + 0.0286679i 0 −1.17325 + 4.37863i 0 −3.73445 2.15609i 0
97.4 0 −0.603899 + 2.25378i 0 −1.48644 + 1.67048i 0 1.03426 3.85990i 0 −2.11677 1.22212i 0
97.5 0 −0.564174 + 2.10553i 0 1.58691 1.57535i 0 −0.320323 + 1.19546i 0 −1.51687 0.875765i 0
97.6 0 −0.364166 + 1.35909i 0 −1.96429 1.06845i 0 0.331427 1.23690i 0 0.883578 + 0.510134i 0
97.7 0 −0.216951 + 0.809671i 0 −0.720974 + 2.11665i 0 −0.208444 + 0.777924i 0 1.98958 + 1.14868i 0
97.8 0 −0.157156 + 0.586512i 0 1.83034 + 1.28447i 0 −0.887088 + 3.31066i 0 2.27878 + 1.31565i 0
97.9 0 −0.0867145 + 0.323623i 0 1.97109 1.05584i 0 0.189910 0.708753i 0 2.50086 + 1.44387i 0
97.10 0 0.0835144 0.311680i 0 0.0788587 2.23468i 0 −0.686223 + 2.56102i 0 2.50791 + 1.44794i 0
97.11 0 0.266873 0.995984i 0 0.861533 + 2.06343i 0 0.954075 3.56066i 0 1.67731 + 0.968397i 0
97.12 0 0.358039 1.33622i 0 −2.23587 + 0.0296847i 0 −0.151686 + 0.566098i 0 0.940782 + 0.543161i 0
97.13 0 0.368630 1.37574i 0 0.199835 2.22712i 0 1.19522 4.46063i 0 0.841290 + 0.485719i 0
97.14 0 0.377301 1.40811i 0 −0.531239 2.17205i 0 −1.28827 + 4.80788i 0 0.757669 + 0.437440i 0
97.15 0 0.405966 1.51509i 0 −1.67270 + 1.48394i 0 0.553968 2.06744i 0 0.467396 + 0.269851i 0
97.16 0 0.439069 1.63863i 0 1.82834 + 1.28731i 0 0.548032 2.04528i 0 0.105753 + 0.0610568i 0
97.17 0 0.670034 2.50060i 0 −0.737701 + 2.11088i 0 −1.21971 + 4.55201i 0 −3.20599 1.85098i 0
97.18 0 0.751881 2.80606i 0 2.02352 0.951503i 0 −0.118383 + 0.441812i 0 −4.71056 2.71964i 0
97.19 0 0.843267 3.14712i 0 −2.00134 0.997312i 0 0.336984 1.25764i 0 −6.59517 3.80772i 0
273.1 0 −2.89417 0.775491i 0 −0.787145 2.09294i 0 1.23641 + 0.331294i 0 5.17676 + 2.98880i 0
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.p even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.bf.a 76
5.c odd 4 1 740.2.bi.a yes 76
37.g odd 12 1 740.2.bi.a yes 76
185.p even 12 1 inner 740.2.bf.a 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.bf.a 76 1.a even 1 1 trivial
740.2.bf.a 76 185.p even 12 1 inner
740.2.bi.a yes 76 5.c odd 4 1
740.2.bi.a yes 76 37.g odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(740, [\chi])\).