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Properties

Label 672.2.cf.a
Level $672$
Weight $2$
Character orbit 672.cf
Analytic conductor $5.366$
Analytic rank $0$
Dimension $512$
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] = [672,2,Mod(19,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([12, 21, 0, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.cf (of order \(24\), degree \(8\), 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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(512\)
Relative dimension: \(64\) over \(\Q(\zeta_{24})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 512 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 512 q + 64 q^{14} + 8 q^{16} - 8 q^{18} - 16 q^{22} - 16 q^{23} - 40 q^{28} + 48 q^{35} + 32 q^{43} + 8 q^{44} + 48 q^{50} - 72 q^{52} + 32 q^{53} + 32 q^{58} - 192 q^{59} - 48 q^{60} + 144 q^{64} - 144 q^{66} - 16 q^{67} - 72 q^{70} + 128 q^{71} + 32 q^{74} + 48 q^{78} - 72 q^{80} - 120 q^{82} + 88 q^{88} - 48 q^{91} + 128 q^{92} - 72 q^{94} - 120 q^{96} - 136 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.41296 − 0.0594356i 0.991445 − 0.130526i 1.99293 + 0.167961i 3.05491 + 0.402186i −1.40863 + 0.125502i −1.13609 + 2.38941i −2.80596 − 0.355773i 0.965926 − 0.258819i −4.29257 − 0.749845i
19.2 −1.41250 − 0.0696639i −0.991445 + 0.130526i 1.99029 + 0.196800i 2.22183 + 0.292510i 1.40951 − 0.115300i −2.31264 − 1.28518i −2.79757 − 0.416631i 0.965926 − 0.258819i −3.11796 − 0.567951i
19.3 −1.40924 − 0.118485i 0.991445 − 0.130526i 1.97192 + 0.333949i −1.89721 − 0.249772i −1.41265 + 0.0664712i 1.39993 − 2.24504i −2.73935 − 0.704258i 0.965926 − 0.258819i 2.64403 + 0.576780i
19.4 −1.40566 + 0.155314i −0.991445 + 0.130526i 1.95175 − 0.436638i −1.12456 − 0.148051i 1.37336 − 0.337461i 2.42588 + 1.05598i −2.67569 + 0.916900i 0.965926 − 0.258819i 1.60375 + 0.0334491i
19.5 −1.35963 − 0.389126i 0.991445 − 0.130526i 1.69716 + 1.05813i −0.123191 − 0.0162184i −1.39878 − 0.208330i −2.61557 − 0.398481i −1.89576 − 2.09907i 0.965926 − 0.258819i 0.161183 + 0.0699878i
19.6 −1.35116 − 0.417575i −0.991445 + 0.130526i 1.65126 + 1.12842i 2.43808 + 0.320979i 1.39410 + 0.237641i 1.56874 − 2.13051i −1.75992 − 2.21420i 0.965926 − 0.258819i −3.16020 − 1.45177i
19.7 −1.31827 + 0.512009i 0.991445 − 0.130526i 1.47569 − 1.34994i −2.69527 − 0.354839i −1.24017 + 0.679698i −2.62806 − 0.305445i −1.25419 + 2.53515i 0.965926 − 0.258819i 3.73478 − 0.912227i
19.8 −1.25444 + 0.652968i 0.991445 − 0.130526i 1.14726 − 1.63823i 1.53243 + 0.201748i −1.15848 + 0.811120i 0.822488 − 2.51466i −0.369470 + 2.80419i 0.965926 − 0.258819i −2.05409 + 0.747547i
19.9 −1.23967 − 0.680593i −0.991445 + 0.130526i 1.07359 + 1.68743i −0.611708 − 0.0805329i 1.31790 + 0.512961i 0.774535 + 2.52984i −0.182444 − 2.82254i 0.965926 − 0.258819i 0.703509 + 0.516159i
19.10 −1.23831 + 0.683067i −0.991445 + 0.130526i 1.06684 − 1.69170i −0.640996 − 0.0843887i 1.13856 − 0.838856i −0.302242 − 2.62843i −0.165535 + 2.82358i 0.965926 − 0.258819i 0.851396 − 0.333343i
19.11 −1.22263 + 0.710757i −0.991445 + 0.130526i 0.989649 − 1.73799i −4.34203 − 0.571639i 1.11940 − 0.864262i −1.90810 + 1.83280i 0.0253115 + 2.82831i 0.965926 − 0.258819i 5.71499 − 2.38722i
19.12 −1.20128 + 0.746270i −0.991445 + 0.130526i 0.886163 − 1.79296i 3.44217 + 0.453170i 1.09360 − 0.896684i 0.984703 + 2.45568i 0.273500 + 2.81517i 0.965926 − 0.258819i −4.47321 + 2.02440i
19.13 −1.15577 − 0.814978i 0.991445 − 0.130526i 0.671623 + 1.88386i −4.09233 − 0.538766i −1.25226 − 0.657147i 0.627676 + 2.57022i 0.759059 − 2.72467i 0.965926 − 0.258819i 4.29073 + 3.95785i
19.14 −1.14599 − 0.828678i 0.991445 − 0.130526i 0.626585 + 1.89931i 0.822731 + 0.108315i −1.24435 − 0.672007i 2.61930 − 0.373153i 0.855859 − 2.69583i 0.965926 − 0.258819i −0.853083 − 0.805906i
19.15 −1.09399 − 0.896209i 0.991445 − 0.130526i 0.393620 + 1.96088i 3.16702 + 0.416946i −1.20161 − 0.745747i 2.43117 + 1.04374i 1.32674 − 2.49795i 0.965926 − 0.258819i −3.09101 − 3.29445i
19.16 −1.04463 − 0.953282i −0.991445 + 0.130526i 0.182506 + 1.99166i 3.55006 + 0.467375i 1.16012 + 0.808775i −2.06807 + 1.65018i 1.70796 − 2.25452i 0.965926 − 0.258819i −3.26296 − 3.87245i
19.17 −1.01362 + 0.986194i 0.991445 − 0.130526i 0.0548434 − 1.99925i 1.47306 + 0.193933i −0.876222 + 1.11006i 2.63193 + 0.270060i 1.91606 + 2.08056i 0.965926 − 0.258819i −1.68438 + 1.25615i
19.18 −0.903219 − 1.08821i −0.991445 + 0.130526i −0.368389 + 1.96578i −1.62097 − 0.213404i 1.03753 + 0.961004i −2.59482 − 0.516646i 2.47191 − 1.37465i 0.965926 − 0.258819i 1.23186 + 1.95670i
19.19 −0.877083 − 1.10938i −0.991445 + 0.130526i −0.461451 + 1.94604i −3.02115 − 0.397742i 1.01438 + 0.985407i 2.56870 − 0.633878i 2.56363 − 1.19491i 0.965926 − 0.258819i 2.20855 + 3.70045i
19.20 −0.715695 − 1.21975i 0.991445 − 0.130526i −0.975562 + 1.74593i −0.187365 − 0.0246670i −0.868781 − 1.11589i −2.42163 + 1.06570i 2.82780 − 0.0596175i 0.965926 − 0.258819i 0.104008 + 0.246192i
See next 80 embeddings (of 512 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
32.h odd 8 1 inner
224.be even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.cf.a ✓ 512
7.d odd 6 1 inner 672.2.cf.a ✓ 512
32.h odd 8 1 inner 672.2.cf.a ✓ 512
224.be even 24 1 inner 672.2.cf.a ✓ 512
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.cf.a ✓ 512 1.a even 1 1 trivial
672.2.cf.a ✓ 512 7.d odd 6 1 inner
672.2.cf.a ✓ 512 32.h odd 8 1 inner
672.2.cf.a ✓ 512 224.be even 24 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(672, [\chi])\).