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Properties

Label 740.4.i.b
Level $740$
Weight $4$
Character orbit 740.i
Analytic conductor $43.661$
Analytic rank $0$
Dimension $38$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,4,Mod(121,740)]
 
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, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.121");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 740.i (of order \(3\), 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: \(43.6614134042\)
Analytic rank: \(0\)
Dimension: \(38\)
Relative dimension: \(19\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 6 q^{3} + 95 q^{5} - 32 q^{7} - 187 q^{9} - 2 q^{11} + 93 q^{13} + 30 q^{15} + 38 q^{17} - 91 q^{19} - 230 q^{21} - 306 q^{23} - 475 q^{25} + 552 q^{27} + 156 q^{29} - 144 q^{31} - 484 q^{33} + 160 q^{35}+ \cdots + 721 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1 0 −4.92424 − 8.52903i 0 2.50000 + 4.33013i 0 −5.38226 − 9.32234i 0 −34.9962 + 60.6152i 0
121.2 0 −4.83172 − 8.36878i 0 2.50000 + 4.33013i 0 −1.66213 − 2.87889i 0 −33.1910 + 57.4886i 0
121.3 0 −3.84492 − 6.65960i 0 2.50000 + 4.33013i 0 −3.54678 − 6.14320i 0 −16.0668 + 27.8286i 0
121.4 0 −3.36414 − 5.82686i 0 2.50000 + 4.33013i 0 11.5801 + 20.0572i 0 −9.13484 + 15.8220i 0
121.5 0 −2.76970 − 4.79727i 0 2.50000 + 4.33013i 0 −14.1228 − 24.4613i 0 −1.84251 + 3.19132i 0
121.6 0 −2.52314 − 4.37020i 0 2.50000 + 4.33013i 0 −9.26690 − 16.0507i 0 0.767570 − 1.32947i 0
121.7 0 −1.62936 − 2.82214i 0 2.50000 + 4.33013i 0 −15.0273 − 26.0281i 0 8.19034 − 14.1861i 0
121.8 0 −1.50219 − 2.60187i 0 2.50000 + 4.33013i 0 4.84611 + 8.39370i 0 8.98684 − 15.5657i 0
121.9 0 −1.46436 − 2.53634i 0 2.50000 + 4.33013i 0 14.2772 + 24.7289i 0 9.21131 − 15.9545i 0
121.10 0 0.111910 + 0.193833i 0 2.50000 + 4.33013i 0 5.38871 + 9.33352i 0 13.4750 − 23.3393i 0
121.11 0 0.411849 + 0.713344i 0 2.50000 + 4.33013i 0 7.51095 + 13.0093i 0 13.1608 − 22.7951i 0
121.12 0 0.816236 + 1.41376i 0 2.50000 + 4.33013i 0 −6.46590 − 11.1993i 0 12.1675 − 21.0748i 0
121.13 0 1.52693 + 2.64471i 0 2.50000 + 4.33013i 0 −2.60491 − 4.51183i 0 8.83700 − 15.3061i 0
121.14 0 2.56266 + 4.43866i 0 2.50000 + 4.33013i 0 −9.93494 − 17.2078i 0 0.365526 − 0.633109i 0
121.15 0 2.83527 + 4.91084i 0 2.50000 + 4.33013i 0 −6.77514 − 11.7349i 0 −2.57754 + 4.46443i 0
121.16 0 3.13113 + 5.42328i 0 2.50000 + 4.33013i 0 13.8884 + 24.0555i 0 −6.10799 + 10.5794i 0
121.17 0 3.44241 + 5.96243i 0 2.50000 + 4.33013i 0 13.3083 + 23.0507i 0 −10.2003 + 17.6675i 0
121.18 0 4.24903 + 7.35954i 0 2.50000 + 4.33013i 0 −14.8184 − 25.6662i 0 −22.6086 + 39.1592i 0
121.19 0 4.76634 + 8.25554i 0 2.50000 + 4.33013i 0 2.80766 + 4.86301i 0 −31.9359 + 55.3147i 0
581.1 0 −4.92424 + 8.52903i 0 2.50000 − 4.33013i 0 −5.38226 + 9.32234i 0 −34.9962 − 60.6152i 0
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.4.i.b ✓ 38
37.c even 3 1 inner 740.4.i.b ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.4.i.b ✓ 38 1.a even 1 1 trivial
740.4.i.b ✓ 38 37.c even 3 1 inner