Properties

Label 740.2.o.b
Level $740$
Weight $2$
Character orbit 740.o
Analytic conductor $5.909$
Analytic rank $0$
Dimension $32$
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(117,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([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("740.117");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.o (of order \(4\), 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: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{3} - 12 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{3} - 12 q^{5} + 4 q^{7} - 4 q^{13} + 4 q^{15} - 10 q^{19} + 20 q^{23} - 8 q^{25} - 26 q^{27} - 12 q^{29} + 40 q^{31} - 28 q^{33} + 22 q^{35} + 6 q^{37} - 38 q^{39} + 12 q^{43} + 18 q^{45} + 4 q^{47} - 26 q^{51} + 8 q^{53} - 22 q^{55} - 24 q^{57} + 16 q^{59} - 8 q^{61} + 18 q^{63} - 4 q^{65} - 18 q^{67} - 12 q^{69} - 4 q^{71} + 86 q^{75} - 4 q^{77} - 32 q^{79} - 76 q^{81} + 6 q^{83} - 4 q^{85} - 14 q^{91} - 8 q^{93} - 18 q^{95} + 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1 0 −2.20270 2.20270i 0 −0.620077 2.14837i 0 −3.05423 3.05423i 0 6.70378i 0
117.2 0 −1.87353 1.87353i 0 −2.22963 0.169498i 0 2.26756 + 2.26756i 0 4.02023i 0
117.3 0 −1.73414 1.73414i 0 0.452188 + 2.18987i 0 0.485073 + 0.485073i 0 3.01448i 0
117.4 0 −1.27465 1.27465i 0 −1.16447 + 1.90893i 0 −1.13749 1.13749i 0 0.249450i 0
117.5 0 −1.01791 1.01791i 0 1.45014 + 1.70209i 0 −0.533882 0.533882i 0 0.927718i 0
117.6 0 −0.872562 0.872562i 0 0.716816 2.11806i 0 0.815420 + 0.815420i 0 1.47727i 0
117.7 0 −0.808285 0.808285i 0 −2.04420 0.906222i 0 −3.10505 3.10505i 0 1.69335i 0
117.8 0 −0.204117 0.204117i 0 0.119792 2.23286i 0 −0.599893 0.599893i 0 2.91667i 0
117.9 0 −0.105893 0.105893i 0 1.97849 + 1.04190i 0 3.04554 + 3.04554i 0 2.97757i 0
117.10 0 0.387665 + 0.387665i 0 −1.63352 + 1.52697i 0 3.12801 + 3.12801i 0 2.69943i 0
117.11 0 0.527387 + 0.527387i 0 −1.30268 + 1.81742i 0 −2.46461 2.46461i 0 2.44373i 0
117.12 0 0.773163 + 0.773163i 0 −2.19545 0.424269i 0 0.328311 + 0.328311i 0 1.80444i 0
117.13 0 1.39300 + 1.39300i 0 1.28963 + 1.82670i 0 −0.266034 0.266034i 0 0.880892i 0
117.14 0 1.43758 + 1.43758i 0 −0.838464 2.07292i 0 2.28736 + 2.28736i 0 1.13328i 0
117.15 0 2.24402 + 2.24402i 0 2.10926 0.742314i 0 2.16004 + 2.16004i 0 7.07129i 0
117.16 0 2.33096 + 2.33096i 0 −2.08782 + 0.800627i 0 −1.35614 1.35614i 0 7.86678i 0
253.1 0 −2.20270 + 2.20270i 0 −0.620077 + 2.14837i 0 −3.05423 + 3.05423i 0 6.70378i 0
253.2 0 −1.87353 + 1.87353i 0 −2.22963 + 0.169498i 0 2.26756 2.26756i 0 4.02023i 0
253.3 0 −1.73414 + 1.73414i 0 0.452188 2.18987i 0 0.485073 0.485073i 0 3.01448i 0
253.4 0 −1.27465 + 1.27465i 0 −1.16447 1.90893i 0 −1.13749 + 1.13749i 0 0.249450i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.o.b yes 32
5.c odd 4 1 740.2.l.b 32
37.d odd 4 1 740.2.l.b 32
185.k even 4 1 inner 740.2.o.b yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.l.b 32 5.c odd 4 1
740.2.l.b 32 37.d odd 4 1
740.2.o.b yes 32 1.a even 1 1 trivial
740.2.o.b yes 32 185.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} + 2 T_{3}^{31} + 2 T_{3}^{30} + 6 T_{3}^{29} + 236 T_{3}^{28} + 528 T_{3}^{27} + 602 T_{3}^{26} + \cdots + 3600 \) acting on \(S_{2}^{\mathrm{new}}(740, [\chi])\). Copy content Toggle raw display