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Properties

Label 1716.3.s.a
Level $1716$
Weight $3$
Character orbit 1716.s
Analytic conductor $46.758$
Analytic rank $0$
Dimension $96$
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] = [1716,3,Mod(265,1716)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1716, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1716.265");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1716.s (of order \(4\), 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: \(46.7576133642\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 24 q^{5} + 16 q^{7} + 288 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 24 q^{5} + 16 q^{7} + 288 q^{9} - 16 q^{13} - 40 q^{19} + 80 q^{29} + 88 q^{31} + 112 q^{35} + 8 q^{37} - 144 q^{39} + 120 q^{41} - 72 q^{45} + 80 q^{47} + 64 q^{53} - 48 q^{57} - 16 q^{59} + 176 q^{61} + 48 q^{63} + 120 q^{65} + 24 q^{67} - 176 q^{73} - 208 q^{79} + 864 q^{81} - 64 q^{83} - 200 q^{85} + 240 q^{87} + 120 q^{89} + 168 q^{91} - 192 q^{93} - 120 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
265.1 0 −1.73205 0 6.46670 − 6.46670i 0 3.09717 + 3.09717i 0 3.00000 0
265.2 0 −1.73205 0 −5.88203 + 5.88203i 0 −8.06727 − 8.06727i 0 3.00000 0
265.3 0 −1.73205 0 5.68026 − 5.68026i 0 5.37559 + 5.37559i 0 3.00000 0
265.4 0 −1.73205 0 −5.61386 + 5.61386i 0 5.25643 + 5.25643i 0 3.00000 0
265.5 0 −1.73205 0 −4.83880 + 4.83880i 0 7.59021 + 7.59021i 0 3.00000 0
265.6 0 −1.73205 0 −3.89541 + 3.89541i 0 3.62314 + 3.62314i 0 3.00000 0
265.7 0 −1.73205 0 −3.58941 + 3.58941i 0 −6.11147 − 6.11147i 0 3.00000 0
265.8 0 −1.73205 0 −3.20295 + 3.20295i 0 −4.99272 − 4.99272i 0 3.00000 0
265.9 0 −1.73205 0 2.39752 − 2.39752i 0 7.69422 + 7.69422i 0 3.00000 0
265.10 0 −1.73205 0 −2.83535 + 2.83535i 0 0.860153 + 0.860153i 0 3.00000 0
265.11 0 −1.73205 0 2.13744 − 2.13744i 0 5.97512 + 5.97512i 0 3.00000 0
265.12 0 −1.73205 0 2.04087 − 2.04087i 0 2.75549 + 2.75549i 0 3.00000 0
265.13 0 −1.73205 0 −1.78379 + 1.78379i 0 −1.82078 − 1.82078i 0 3.00000 0
265.14 0 −1.73205 0 −0.614927 + 0.614927i 0 8.15993 + 8.15993i 0 3.00000 0
265.15 0 −1.73205 0 −0.421357 + 0.421357i 0 −0.916376 − 0.916376i 0 3.00000 0
265.16 0 −1.73205 0 −0.240294 + 0.240294i 0 3.09073 + 3.09073i 0 3.00000 0
265.17 0 −1.73205 0 −1.19533 + 1.19533i 0 −4.90121 − 4.90121i 0 3.00000 0
265.18 0 −1.73205 0 2.74539 − 2.74539i 0 −8.91154 − 8.91154i 0 3.00000 0
265.19 0 −1.73205 0 2.97197 − 2.97197i 0 −5.14635 − 5.14635i 0 3.00000 0
265.20 0 −1.73205 0 4.44458 − 4.44458i 0 −5.61255 − 5.61255i 0 3.00000 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 265.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1716.3.s.a ✓ 96
13.d odd 4 1 inner 1716.3.s.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.3.s.a ✓ 96 1.a even 1 1 trivial
1716.3.s.a ✓ 96 13.d odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1716, [\chi])\).