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Properties

Label 768.2.s.a
Level $768$
Weight $2$
Character orbit 768.s
Analytic conductor $6.133$
Analytic rank $0$
Dimension $240$
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] = [768,2,Mod(47,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 11, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.s (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: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(30\) over \(\Q(\zeta_{16})\)
Twist minimal: no (minimal twist has level 192)
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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 8 q^{3} + 16 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 8 q^{3} + 16 q^{7} - 8 q^{9} - 16 q^{13} + 8 q^{15} + 16 q^{19} - 8 q^{21} - 16 q^{25} + 8 q^{27} + 32 q^{31} - 16 q^{37} + 8 q^{39} + 16 q^{43} - 8 q^{45} - 16 q^{49} + 8 q^{51} + 80 q^{55} - 8 q^{57} - 16 q^{61} + 144 q^{67} - 8 q^{69} - 16 q^{73} + 8 q^{75} + 48 q^{79} - 8 q^{81} - 16 q^{85} + 8 q^{87} + 16 q^{91} - 32 q^{93} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 −1.73124 − 0.0530351i 0 −1.55377 − 1.03819i 0 0.159818 + 0.385836i 0 2.99437 + 0.183633i 0
47.2 0 −1.69388 + 0.361602i 0 0.388565 + 0.259631i 0 −1.96536 − 4.74480i 0 2.73849 − 1.22502i 0
47.3 0 −1.67499 − 0.440918i 0 −1.07447 − 0.717940i 0 0.999337 + 2.41261i 0 2.61118 + 1.47707i 0
47.4 0 −1.62032 − 0.612015i 0 3.01614 + 2.01532i 0 0.924170 + 2.23114i 0 2.25087 + 1.98332i 0
47.5 0 −1.56005 + 0.752482i 0 1.30855 + 0.874343i 0 −0.168531 − 0.406870i 0 1.86754 − 2.34783i 0
47.6 0 −1.44627 + 0.953045i 0 −1.95251 − 1.30462i 0 0.960548 + 2.31897i 0 1.18341 − 2.75673i 0
47.7 0 −1.31348 − 1.12905i 0 −2.57009 − 1.71728i 0 −1.31022 − 3.16314i 0 0.450472 + 2.96599i 0
47.8 0 −1.04431 − 1.38182i 0 0.191752 + 0.128125i 0 −0.553671 − 1.33668i 0 −0.818831 + 2.88609i 0
47.9 0 −0.865146 + 1.50051i 0 2.09602 + 1.40051i 0 1.76691 + 4.26571i 0 −1.50305 − 2.59632i 0
47.10 0 −0.858398 − 1.50438i 0 2.38253 + 1.59195i 0 −0.537426 − 1.29746i 0 −1.52631 + 2.58271i 0
47.11 0 −0.816224 + 1.52767i 0 −2.90474 − 1.94088i 0 −0.941226 − 2.27232i 0 −1.66756 − 2.49384i 0
47.12 0 −0.585574 − 1.63006i 0 −3.57797 − 2.39072i 0 0.994439 + 2.40079i 0 −2.31421 + 1.90904i 0
47.13 0 −0.462320 − 1.66921i 0 0.780999 + 0.521847i 0 1.37838 + 3.32770i 0 −2.57252 + 1.54342i 0
47.14 0 −0.422356 + 1.67977i 0 1.22014 + 0.815271i 0 −0.731141 − 1.76513i 0 −2.64323 − 1.41892i 0
47.15 0 −0.252613 + 1.71353i 0 −1.22014 − 0.815271i 0 −0.731141 − 1.76513i 0 −2.87237 − 0.865720i 0
47.16 0 0.145818 − 1.72590i 0 0.731423 + 0.488721i 0 −0.683144 − 1.64925i 0 −2.95747 − 0.503337i 0
47.17 0 0.169478 + 1.72374i 0 2.90474 + 1.94088i 0 −0.941226 − 2.27232i 0 −2.94255 + 0.584272i 0
47.18 0 0.225071 + 1.71737i 0 −2.09602 − 1.40051i 0 1.76691 + 4.26571i 0 −2.89869 + 0.773059i 0
47.19 0 0.525755 − 1.65033i 0 −0.731423 − 0.488721i 0 −0.683144 − 1.64925i 0 −2.44716 − 1.73534i 0
47.20 0 0.971467 + 1.43396i 0 1.95251 + 1.30462i 0 0.960548 + 2.31897i 0 −1.11250 + 2.78610i 0
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
64.j odd 16 1 inner
192.s even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.s.a 240
3.b odd 2 1 inner 768.2.s.a 240
4.b odd 2 1 192.2.s.a ✓ 240
12.b even 2 1 192.2.s.a ✓ 240
64.i even 16 1 192.2.s.a ✓ 240
64.j odd 16 1 inner 768.2.s.a 240
192.q odd 16 1 192.2.s.a ✓ 240
192.s even 16 1 inner 768.2.s.a 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.2.s.a ✓ 240 4.b odd 2 1
192.2.s.a ✓ 240 12.b even 2 1
192.2.s.a ✓ 240 64.i even 16 1
192.2.s.a ✓ 240 192.q odd 16 1
768.2.s.a 240 1.a even 1 1 trivial
768.2.s.a 240 3.b odd 2 1 inner
768.2.s.a 240 64.j odd 16 1 inner
768.2.s.a 240 192.s even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(768, [\chi])\).