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Properties

Label 560.3.bt.d
Level $560$
Weight $3$
Character orbit 560.bt
Analytic conductor $15.259$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,3,Mod(79,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.79");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 560.bt (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: \(15.2588948042\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - q^{5} - 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - q^{5} - 50 q^{9} + 12 q^{11} - 12 q^{19} - 34 q^{21} - 11 q^{25} + 40 q^{29} - 150 q^{31} + 69 q^{35} + 24 q^{39} - 20 q^{41} + 88 q^{45} + 52 q^{49} - 114 q^{51} - 66 q^{59} - 138 q^{61} + 38 q^{65} - 156 q^{69} - 465 q^{75} + 330 q^{79} + 120 q^{81} - 206 q^{85} + 66 q^{89} - 348 q^{91} - 321 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1 0 −2.56838 − 4.44857i 0 4.93730 − 0.789372i 0 −0.710759 − 6.96382i 0 −8.69319 + 15.0570i 0
79.2 0 −2.46461 − 4.26884i 0 −4.64528 + 1.84970i 0 6.97174 + 0.628402i 0 −7.64865 + 13.2479i 0
79.3 0 −2.04534 − 3.54264i 0 −3.62435 − 3.44443i 0 −6.85643 + 1.41043i 0 −3.86685 + 6.69759i 0
79.4 0 −2.02878 − 3.51395i 0 3.35750 − 3.70502i 0 4.05957 + 5.70262i 0 −3.73188 + 6.46380i 0
79.5 0 −1.30177 − 2.25474i 0 −1.72633 + 4.69252i 0 −4.95920 + 4.94028i 0 1.11078 − 1.92392i 0
79.6 0 −1.13568 − 1.96706i 0 2.03912 + 4.56530i 0 −2.83201 − 6.40154i 0 1.92044 − 3.32630i 0
79.7 0 −0.439688 − 0.761561i 0 3.06570 + 3.94987i 0 5.79142 + 3.93185i 0 4.11335 − 7.12453i 0
79.8 0 −0.319374 − 0.553171i 0 −0.658709 − 4.95642i 0 4.87282 − 5.02550i 0 4.29600 − 7.44089i 0
79.9 0 0.319374 + 0.553171i 0 4.62174 − 1.90775i 0 −4.87282 + 5.02550i 0 4.29600 − 7.44089i 0
79.10 0 0.439688 + 0.761561i 0 −4.95354 − 0.680044i 0 −5.79142 − 3.93185i 0 4.11335 − 7.12453i 0
79.11 0 1.13568 + 1.96706i 0 −4.97323 + 0.516717i 0 2.83201 + 6.40154i 0 1.92044 − 3.32630i 0
79.12 0 1.30177 + 2.25474i 0 −3.20068 + 3.84131i 0 4.95920 − 4.94028i 0 1.11078 − 1.92392i 0
79.13 0 2.02878 + 3.51395i 0 1.52990 − 4.76019i 0 −4.05957 − 5.70262i 0 −3.73188 + 6.46380i 0
79.14 0 2.04534 + 3.54264i 0 4.79514 + 1.41656i 0 6.85643 − 1.41043i 0 −3.86685 + 6.69759i 0
79.15 0 2.46461 + 4.26884i 0 0.720747 + 4.94778i 0 −6.97174 − 0.628402i 0 −7.64865 + 13.2479i 0
79.16 0 2.56838 + 4.44857i 0 −1.78503 − 4.67051i 0 0.710759 + 6.96382i 0 −8.69319 + 15.0570i 0
319.1 0 −2.56838 + 4.44857i 0 4.93730 + 0.789372i 0 −0.710759 + 6.96382i 0 −8.69319 − 15.0570i 0
319.2 0 −2.46461 + 4.26884i 0 −4.64528 − 1.84970i 0 6.97174 − 0.628402i 0 −7.64865 − 13.2479i 0
319.3 0 −2.04534 + 3.54264i 0 −3.62435 + 3.44443i 0 −6.85643 − 1.41043i 0 −3.86685 − 6.69759i 0
319.4 0 −2.02878 + 3.51395i 0 3.35750 + 3.70502i 0 4.05957 − 5.70262i 0 −3.73188 − 6.46380i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
28.g odd 6 1 inner
140.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.3.bt.d yes 32
4.b odd 2 1 560.3.bt.c ✓ 32
5.b even 2 1 inner 560.3.bt.d yes 32
7.c even 3 1 560.3.bt.c ✓ 32
20.d odd 2 1 560.3.bt.c ✓ 32
28.g odd 6 1 inner 560.3.bt.d yes 32
35.j even 6 1 560.3.bt.c ✓ 32
140.p odd 6 1 inner 560.3.bt.d yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.3.bt.c ✓ 32 4.b odd 2 1
560.3.bt.c ✓ 32 7.c even 3 1
560.3.bt.c ✓ 32 20.d odd 2 1
560.3.bt.c ✓ 32 35.j even 6 1
560.3.bt.d yes 32 1.a even 1 1 trivial
560.3.bt.d yes 32 5.b even 2 1 inner
560.3.bt.d yes 32 28.g odd 6 1 inner
560.3.bt.d yes 32 140.p odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(560, [\chi])\):

\( T_{3}^{32} + 97 T_{3}^{30} + 5660 T_{3}^{28} + 216587 T_{3}^{26} + 6151113 T_{3}^{24} + \cdots + 3797883801856 \) Copy content Toggle raw display
\( T_{11}^{16} - 6 T_{11}^{15} - 548 T_{11}^{14} + 3360 T_{11}^{13} + 225249 T_{11}^{12} + \cdots + 49338374160384 \) Copy content Toggle raw display