Properties

Label 740.2.bc.d
Level $740$
Weight $2$
Character orbit 740.bc
Analytic conductor $5.909$
Analytic rank $0$
Dimension $36$
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(81,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.bc (of order \(9\), degree \(6\), 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: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 12 q^{7} - 6 q^{11} - 12 q^{13} - 6 q^{17} + 15 q^{19} - 24 q^{23} + 12 q^{27} - 3 q^{29} + 30 q^{31} + 18 q^{33} - 3 q^{35} + 36 q^{37} - 6 q^{39} - 9 q^{41} + 12 q^{43} + 21 q^{45} - 21 q^{47} - 30 q^{49} - 24 q^{51} - 3 q^{53} + 3 q^{55} + 3 q^{57} - 27 q^{59} + 30 q^{61} - 27 q^{63} + 12 q^{65} - 21 q^{67} - 45 q^{69} + 24 q^{71} - 48 q^{73} - 6 q^{75} - 42 q^{77} + 108 q^{81} - 9 q^{83} + 3 q^{85} - 81 q^{87} - 60 q^{89} - 66 q^{91} + 75 q^{93} + 12 q^{95} - 36 q^{97} + 108 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 0 −2.51079 0.913852i 0 −0.173648 0.984808i 0 −0.774141 4.39037i 0 3.17079 + 2.66061i 0
81.2 0 −2.05044 0.746301i 0 −0.173648 0.984808i 0 0.299868 + 1.70063i 0 1.34922 + 1.13213i 0
81.3 0 −0.210487 0.0766109i 0 −0.173648 0.984808i 0 −0.202628 1.14916i 0 −2.25970 1.89611i 0
81.4 0 1.26990 + 0.462206i 0 −0.173648 0.984808i 0 −0.591039 3.35195i 0 −0.899118 0.754450i 0
81.5 0 2.06789 + 0.752652i 0 −0.173648 0.984808i 0 0.773432 + 4.38635i 0 1.41157 + 1.18444i 0
81.6 0 2.37361 + 0.863925i 0 −0.173648 0.984808i 0 −0.150921 0.855916i 0 2.58955 + 2.17289i 0
181.1 0 −0.593287 3.36470i 0 −0.766044 + 0.642788i 0 2.77336 2.32713i 0 −8.15013 + 2.96641i 0
181.2 0 −0.271716 1.54098i 0 −0.766044 + 0.642788i 0 0.504138 0.423022i 0 0.518293 0.188643i 0
181.3 0 −0.116920 0.663085i 0 −0.766044 + 0.642788i 0 −3.07084 + 2.57674i 0 2.39307 0.871005i 0
181.4 0 0.0765251 + 0.433995i 0 −0.766044 + 0.642788i 0 0.108070 0.0906818i 0 2.63658 0.959637i 0
181.5 0 0.223103 + 1.26528i 0 −0.766044 + 0.642788i 0 3.02839 2.54112i 0 1.26792 0.461486i 0
181.6 0 0.508647 + 2.88468i 0 −0.766044 + 0.642788i 0 −0.0561339 + 0.0471019i 0 −5.24358 + 1.90851i 0
201.1 0 −2.51079 + 0.913852i 0 −0.173648 + 0.984808i 0 −0.774141 + 4.39037i 0 3.17079 2.66061i 0
201.2 0 −2.05044 + 0.746301i 0 −0.173648 + 0.984808i 0 0.299868 1.70063i 0 1.34922 1.13213i 0
201.3 0 −0.210487 + 0.0766109i 0 −0.173648 + 0.984808i 0 −0.202628 + 1.14916i 0 −2.25970 + 1.89611i 0
201.4 0 1.26990 0.462206i 0 −0.173648 + 0.984808i 0 −0.591039 + 3.35195i 0 −0.899118 + 0.754450i 0
201.5 0 2.06789 0.752652i 0 −0.173648 + 0.984808i 0 0.773432 4.38635i 0 1.41157 1.18444i 0
201.6 0 2.37361 0.863925i 0 −0.173648 + 0.984808i 0 −0.150921 + 0.855916i 0 2.58955 2.17289i 0
441.1 0 −2.31219 + 1.94016i 0 0.939693 + 0.342020i 0 0.459860 + 0.167375i 0 1.06107 6.01760i 0
441.2 0 −1.32324 + 1.11033i 0 0.939693 + 0.342020i 0 −3.84213 1.39842i 0 −0.00281124 + 0.0159433i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.bc.d 36
37.f even 9 1 inner 740.2.bc.d 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.bc.d 36 1.a even 1 1 trivial
740.2.bc.d 36 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} - 10 T_{3}^{33} - 27 T_{3}^{32} + 15 T_{3}^{31} + 739 T_{3}^{30} - 672 T_{3}^{29} + \cdots + 3560769 \) acting on \(S_{2}^{\mathrm{new}}(740, [\chi])\). Copy content Toggle raw display