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Properties

Label 816.2.cj.d
Level $816$
Weight $2$
Character orbit 816.cj
Analytic conductor $6.516$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [816,2,Mod(65,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 0, 8, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.cj (of order \(16\), degree \(8\), not 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: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{16})\)
Twist minimal: no (minimal twist has level 204)
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 24 q^{15} + 24 q^{21} - 32 q^{25} + 32 q^{31} - 32 q^{37} + 16 q^{39} + 64 q^{43} - 32 q^{45} - 32 q^{49} + 32 q^{51} + 32 q^{55} - 48 q^{57} - 16 q^{61} + 72 q^{63} - 32 q^{69} + 64 q^{73} + 48 q^{75} - 32 q^{79} - 8 q^{81} + 80 q^{85} - 48 q^{87} - 160 q^{91} + 64 q^{93} + 64 q^{97} - 88 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 0 −1.60178 − 0.659014i 0 −0.354479 − 0.530515i 0 −1.29809 − 0.867357i 0 2.13140 + 2.11119i 0
65.2 0 −1.21054 + 1.23879i 0 1.29266 + 1.93461i 0 −1.07450 − 0.717960i 0 −0.0691980 − 2.99920i 0
65.3 0 −0.113941 − 1.72830i 0 2.32666 + 3.48210i 0 2.37259 + 1.58532i 0 −2.97403 + 0.393848i 0
65.4 0 0.644327 + 1.60774i 0 −1.29266 − 1.93461i 0 −1.07450 − 0.717960i 0 −2.16969 + 2.07183i 0
65.5 0 0.766659 − 1.55314i 0 −2.32666 − 3.48210i 0 2.37259 + 1.58532i 0 −1.82447 − 2.38145i 0
65.6 0 1.73205 + 0.00412537i 0 0.354479 + 0.530515i 0 −1.29809 − 0.867357i 0 2.99997 + 0.0142906i 0
113.1 0 −1.60178 + 0.659014i 0 −0.354479 + 0.530515i 0 −1.29809 + 0.867357i 0 2.13140 − 2.11119i 0
113.2 0 −1.21054 − 1.23879i 0 1.29266 − 1.93461i 0 −1.07450 + 0.717960i 0 −0.0691980 + 2.99920i 0
113.3 0 −0.113941 + 1.72830i 0 2.32666 − 3.48210i 0 2.37259 − 1.58532i 0 −2.97403 − 0.393848i 0
113.4 0 0.644327 − 1.60774i 0 −1.29266 + 1.93461i 0 −1.07450 + 0.717960i 0 −2.16969 − 2.07183i 0
113.5 0 0.766659 + 1.55314i 0 −2.32666 + 3.48210i 0 2.37259 − 1.58532i 0 −1.82447 + 2.38145i 0
113.6 0 1.73205 − 0.00412537i 0 0.354479 − 0.530515i 0 −1.29809 + 0.867357i 0 2.99997 − 0.0142906i 0
209.1 0 −1.72436 − 0.163016i 0 −0.523344 − 0.104100i 0 −0.587773 − 2.95493i 0 2.94685 + 0.562196i 0
209.2 0 −1.20610 + 1.24311i 0 1.05674 + 0.210198i 0 0.807215 + 4.05815i 0 −0.0906409 − 2.99863i 0
209.3 0 0.509278 − 1.65549i 0 0.523344 + 0.104100i 0 −0.587773 − 2.95493i 0 −2.48127 − 1.68621i 0
209.4 0 0.725559 + 1.57276i 0 3.25740 + 0.647936i 0 −0.219443 − 1.10321i 0 −1.94713 + 2.28225i 0
209.5 0 1.17538 + 1.27220i 0 −3.25740 − 0.647936i 0 −0.219443 − 1.10321i 0 −0.236970 + 2.99063i 0
209.6 0 1.61004 − 0.638575i 0 −1.05674 − 0.210198i 0 0.807215 + 4.05815i 0 2.18444 − 2.05626i 0
401.1 0 −1.72691 + 0.133346i 0 −0.330534 − 1.66171i 0 −0.223938 − 0.0445440i 0 2.96444 − 0.460552i 0
401.2 0 −0.589429 + 1.62867i 0 0.720265 + 3.62101i 0 −3.43219 − 0.682706i 0 −2.30515 − 1.91997i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
17.e odd 16 1 inner
51.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 816.2.cj.d 48
3.b odd 2 1 inner 816.2.cj.d 48
4.b odd 2 1 204.2.q.a ✓ 48
12.b even 2 1 204.2.q.a ✓ 48
17.e odd 16 1 inner 816.2.cj.d 48
51.i even 16 1 inner 816.2.cj.d 48
68.i even 16 1 204.2.q.a ✓ 48
204.t odd 16 1 204.2.q.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
204.2.q.a ✓ 48 4.b odd 2 1
204.2.q.a ✓ 48 12.b even 2 1
204.2.q.a ✓ 48 68.i even 16 1
204.2.q.a ✓ 48 204.t odd 16 1
816.2.cj.d 48 1.a even 1 1 trivial
816.2.cj.d 48 3.b odd 2 1 inner
816.2.cj.d 48 17.e odd 16 1 inner
816.2.cj.d 48 51.i even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} + 16 T_{5}^{46} + 232 T_{5}^{44} - 168 T_{5}^{42} - 18784 T_{5}^{40} + 133592 T_{5}^{38} + \cdots + 32255902407184 \) acting on \(S_{2}^{\mathrm{new}}(816, [\chi])\). Copy content Toggle raw display