Properties

Label 740.2.ba.a
Level $740$
Weight $2$
Character orbit 740.ba
Analytic conductor $5.909$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(249,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.ba (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: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{5} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{5} + 22 q^{9} - 12 q^{11} - 3 q^{15} - 6 q^{19} + 12 q^{21} + 5 q^{25} + 21 q^{35} + 12 q^{39} + 32 q^{41} + 22 q^{49} + 30 q^{55} + 42 q^{59} + 42 q^{61} - 7 q^{65} - 54 q^{69} + 16 q^{71} - 50 q^{75} + 12 q^{79} - 56 q^{81} - 10 q^{85} + 30 q^{89} - 6 q^{91} - 7 q^{95} - 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
249.1 0 −2.92356 + 1.68792i 0 −1.87111 1.22432i 0 −3.35180 + 1.93516i 0 4.19812 7.27135i 0
249.2 0 −2.49411 + 1.43997i 0 2.16123 + 0.573672i 0 −0.134772 + 0.0778107i 0 2.64704 4.58481i 0
249.3 0 −2.10840 + 1.21728i 0 −0.272013 + 2.21946i 0 −0.549920 + 0.317496i 0 1.46355 2.53495i 0
249.4 0 −2.10722 + 1.21660i 0 −1.53139 + 1.62937i 0 4.29249 2.47827i 0 1.46025 2.52922i 0
249.5 0 −2.06209 + 1.19055i 0 2.05385 0.884133i 0 −0.155067 + 0.0895280i 0 1.33480 2.31194i 0
249.6 0 −1.27685 + 0.737190i 0 −2.02720 0.943639i 0 1.28215 0.740252i 0 −0.413101 + 0.715511i 0
249.7 0 −0.844980 + 0.487850i 0 −0.187442 2.22820i 0 −2.52014 + 1.45501i 0 −1.02401 + 1.77363i 0
249.8 0 −0.676419 + 0.390531i 0 −1.31592 + 1.80786i 0 −2.25770 + 1.30348i 0 −1.19497 + 2.06975i 0
249.9 0 −0.200277 + 0.115630i 0 1.15125 + 1.91693i 0 −0.567064 + 0.327394i 0 −1.47326 + 2.55176i 0
249.10 0 −0.0486079 + 0.0280638i 0 1.58113 1.58115i 0 4.16312 2.40358i 0 −1.49842 + 2.59535i 0
249.11 0 0.0486079 0.0280638i 0 2.15988 0.578723i 0 −4.16312 + 2.40358i 0 −1.49842 + 2.59535i 0
249.12 0 0.200277 0.115630i 0 −1.08448 1.95548i 0 0.567064 0.327394i 0 −1.47326 + 2.55176i 0
249.13 0 0.676419 0.390531i 0 −2.22361 + 0.235690i 0 2.25770 1.30348i 0 −1.19497 + 2.06975i 0
249.14 0 0.844980 0.487850i 0 1.83595 + 1.27643i 0 2.52014 1.45501i 0 −1.02401 + 1.77363i 0
249.15 0 1.27685 0.737190i 0 −0.196386 + 2.22743i 0 −1.28215 + 0.740252i 0 −0.413101 + 0.715511i 0
249.16 0 2.06209 1.19055i 0 1.79261 1.33662i 0 0.155067 0.0895280i 0 1.33480 2.31194i 0
249.17 0 2.10722 1.21660i 0 −2.17677 + 0.511535i 0 −4.29249 + 2.47827i 0 1.46025 2.52922i 0
249.18 0 2.10840 1.21728i 0 −2.05812 0.874161i 0 0.549920 0.317496i 0 1.46355 2.53495i 0
249.19 0 2.49411 1.43997i 0 0.583799 2.15851i 0 0.134772 0.0778107i 0 2.64704 4.58481i 0
249.20 0 2.92356 1.68792i 0 0.124735 + 2.23259i 0 3.35180 1.93516i 0 4.19812 7.27135i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 249.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.e even 6 1 inner
185.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.ba.a 40
5.b even 2 1 inner 740.2.ba.a 40
37.e even 6 1 inner 740.2.ba.a 40
185.l even 6 1 inner 740.2.ba.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.ba.a 40 1.a even 1 1 trivial
740.2.ba.a 40 5.b even 2 1 inner
740.2.ba.a 40 37.e even 6 1 inner
740.2.ba.a 40 185.l even 6 1 inner

Hecke kernels

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