Properties

Label 354.5.d.a
Level $354$
Weight $5$
Character orbit 354.d
Analytic conductor $36.593$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [354,5,Mod(235,354)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(354, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("354.235");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 354.d (of order \(2\), degree \(1\), 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: \(36.5929669317\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 320 q^{4} - 80 q^{7} + 1080 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 320 q^{4} - 80 q^{7} + 1080 q^{9} - 144 q^{15} + 2560 q^{16} + 480 q^{17} - 792 q^{19} - 1024 q^{22} + 3400 q^{25} + 768 q^{26} + 640 q^{28} + 1608 q^{29} - 5760 q^{35} - 8640 q^{36} + 6264 q^{41} + 7040 q^{46} + 17912 q^{49} + 1296 q^{51} - 1104 q^{53} + 5040 q^{57} + 13584 q^{59} + 1152 q^{60} - 12288 q^{62} - 2160 q^{63} - 20480 q^{64} + 1152 q^{66} - 3840 q^{68} + 35352 q^{71} + 4608 q^{74} + 3168 q^{75} + 6336 q^{76} - 12672 q^{78} - 15720 q^{79} + 29160 q^{81} - 26872 q^{85} + 18432 q^{86} + 7776 q^{87} + 8192 q^{88} - 18432 q^{94} - 19128 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1 2.82843i 5.19615 −8.00000 −10.0054 14.6969i 10.1747 22.6274i 27.0000 28.2996i
235.2 2.82843i 5.19615 −8.00000 −10.0054 14.6969i 10.1747 22.6274i 27.0000 28.2996i
235.3 2.82843i −5.19615 −8.00000 21.2806 14.6969i −81.9829 22.6274i 27.0000 60.1907i
235.4 2.82843i −5.19615 −8.00000 21.2806 14.6969i −81.9829 22.6274i 27.0000 60.1907i
235.5 2.82843i −5.19615 −8.00000 0.141718 14.6969i −89.1589 22.6274i 27.0000 0.400839i
235.6 2.82843i −5.19615 −8.00000 0.141718 14.6969i −89.1589 22.6274i 27.0000 0.400839i
235.7 2.82843i −5.19615 −8.00000 −8.58312 14.6969i 95.1629 22.6274i 27.0000 24.2767i
235.8 2.82843i −5.19615 −8.00000 −8.58312 14.6969i 95.1629 22.6274i 27.0000 24.2767i
235.9 2.82843i −5.19615 −8.00000 −44.0140 14.6969i 46.8163 22.6274i 27.0000 124.490i
235.10 2.82843i −5.19615 −8.00000 −44.0140 14.6969i 46.8163 22.6274i 27.0000 124.490i
235.11 2.82843i −5.19615 −8.00000 0.345303 14.6969i 29.1133 22.6274i 27.0000 0.976665i
235.12 2.82843i −5.19615 −8.00000 0.345303 14.6969i 29.1133 22.6274i 27.0000 0.976665i
235.13 2.82843i 5.19615 −8.00000 −17.0264 14.6969i 47.1245 22.6274i 27.0000 48.1580i
235.14 2.82843i 5.19615 −8.00000 −17.0264 14.6969i 47.1245 22.6274i 27.0000 48.1580i
235.15 2.82843i −5.19615 −8.00000 −27.9908 14.6969i −34.1747 22.6274i 27.0000 79.1699i
235.16 2.82843i −5.19615 −8.00000 −27.9908 14.6969i −34.1747 22.6274i 27.0000 79.1699i
235.17 2.82843i 5.19615 −8.00000 −4.63644 14.6969i −33.3282 22.6274i 27.0000 13.1138i
235.18 2.82843i 5.19615 −8.00000 −4.63644 14.6969i −33.3282 22.6274i 27.0000 13.1138i
235.19 2.82843i 5.19615 −8.00000 26.1412 14.6969i −51.7193 22.6274i 27.0000 73.9384i
235.20 2.82843i 5.19615 −8.00000 26.1412 14.6969i −51.7193 22.6274i 27.0000 73.9384i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 354.5.d.a 40
3.b odd 2 1 1062.5.d.b 40
59.b odd 2 1 inner 354.5.d.a 40
177.d even 2 1 1062.5.d.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
354.5.d.a 40 1.a even 1 1 trivial
354.5.d.a 40 59.b odd 2 1 inner
1062.5.d.b 40 3.b odd 2 1
1062.5.d.b 40 177.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(354, [\chi])\).