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Properties

Label 712.2.m.c
Level $712$
Weight $2$
Character orbit 712.m
Analytic conductor $5.685$
Analytic rank $0$
Dimension $344$
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] = [712,2,Mod(571,712)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(712, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 4, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("712.571");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 712 = 2^{3} \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 712.m (of order \(8\), degree \(4\), 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: \(5.68534862392\)
Analytic rank: \(0\)
Dimension: \(344\)
Relative dimension: \(86\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 344 q - 8 q^{2} - 24 q^{4} - 12 q^{6} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 344 q - 8 q^{2} - 24 q^{4} - 12 q^{6} - 8 q^{8} + 4 q^{10} - 4 q^{12} + 4 q^{14} - 40 q^{16} - 8 q^{17} + 20 q^{18} - 16 q^{19} + 56 q^{20} - 40 q^{24} + 8 q^{26} - 48 q^{27} + 44 q^{28} + 20 q^{30} - 8 q^{32} + 56 q^{33} - 20 q^{34} - 8 q^{35} - 4 q^{36} - 12 q^{38} - 20 q^{40} - 8 q^{41} + 72 q^{42} - 8 q^{43} + 36 q^{46} + 44 q^{48} - 8 q^{49} - 88 q^{51} - 24 q^{52} - 80 q^{54} + 8 q^{56} - 28 q^{58} + 80 q^{59} + 40 q^{60} + 12 q^{62} - 168 q^{64} + 32 q^{65} + 108 q^{66} - 16 q^{67} - 32 q^{68} - 80 q^{70} + 40 q^{72} + 12 q^{74} + 128 q^{75} + 12 q^{76} + 8 q^{78} + 16 q^{80} - 76 q^{82} - 88 q^{83} - 4 q^{84} - 16 q^{86} - 8 q^{89} + 288 q^{90} - 16 q^{91} - 24 q^{92} - 20 q^{94} - 16 q^{96} - 96 q^{97} - 64 q^{98} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
571.1 −1.40965 − 0.113581i 0.275005 + 0.113911i 1.97420 + 0.320216i 2.39648 + 2.39648i −0.374722 − 0.191809i −0.212443 + 0.512882i −2.74655 − 0.675622i −2.05867 − 2.05867i −3.10599 − 3.65038i
571.2 −1.40965 + 0.113581i 0.275005 + 0.113911i 1.97420 − 0.320216i −2.39648 − 2.39648i −0.400598 − 0.129339i 0.212443 − 0.512882i −2.74655 + 0.675622i −2.05867 − 2.05867i 3.65038 + 3.10599i
571.3 −1.40491 − 0.161937i 0.839855 + 0.347879i 1.94755 + 0.455014i −1.73744 − 1.73744i −1.12359 − 0.624743i −1.03002 + 2.48668i −2.66246 − 0.954635i −1.53698 − 1.53698i 2.15959 + 2.72230i
571.4 −1.40491 + 0.161937i 0.839855 + 0.347879i 1.94755 − 0.455014i 1.73744 + 1.73744i −1.23626 − 0.352736i 1.03002 − 2.48668i −2.66246 + 0.954635i −1.53698 − 1.53698i −2.72230 − 2.15959i
571.5 −1.37114 − 0.346387i 2.82387 + 1.16968i 1.76003 + 0.949887i 0.925026 + 0.925026i −3.46675 − 2.58195i 1.46178 − 3.52904i −2.08422 − 1.91208i 4.48474 + 4.48474i −0.947921 − 1.58875i
571.6 −1.37114 + 0.346387i 2.82387 + 1.16968i 1.76003 − 0.949887i −0.925026 − 0.925026i −4.27707 − 0.625647i −1.46178 + 3.52904i −2.08422 + 1.91208i 4.48474 + 4.48474i 1.58875 + 0.947921i
571.7 −1.34852 − 0.426028i −0.392897 − 0.162743i 1.63700 + 1.14901i −0.866280 − 0.866280i 0.460495 + 0.386847i 1.95711 − 4.72488i −1.71801 − 2.24687i −1.99344 − 1.99344i 0.799134 + 1.53725i
571.8 −1.34852 + 0.426028i −0.392897 − 0.162743i 1.63700 − 1.14901i 0.866280 + 0.866280i 0.599161 + 0.0520769i −1.95711 + 4.72488i −1.71801 + 2.24687i −1.99344 − 1.99344i −1.53725 − 0.799134i
571.9 −1.31978 − 0.508112i −1.68417 − 0.697607i 1.48365 + 1.34119i 0.165027 + 0.165027i 1.86828 + 1.77644i −0.420835 + 1.01599i −1.27661 − 2.52394i 0.228463 + 0.228463i −0.133947 − 0.301651i
571.10 −1.31978 + 0.508112i −1.68417 − 0.697607i 1.48365 − 1.34119i −0.165027 − 0.165027i 2.57720 + 0.0649412i 0.420835 − 1.01599i −1.27661 + 2.52394i 0.228463 + 0.228463i 0.301651 + 0.133947i
571.11 −1.30465 − 0.545797i 2.29437 + 0.950359i 1.40421 + 1.42415i −1.53551 − 1.53551i −2.47464 − 2.49214i −0.0718934 + 0.173566i −1.05471 − 2.62442i 2.23963 + 2.23963i 1.16522 + 2.84137i
571.12 −1.30465 + 0.545797i 2.29437 + 0.950359i 1.40421 − 1.42415i 1.53551 + 1.53551i −3.51205 + 0.0123767i 0.0718934 − 0.173566i −1.05471 + 2.62442i 2.23963 + 2.23963i −2.84137 − 1.16522i
571.13 −1.26320 − 0.635867i −1.75514 − 0.727004i 1.19135 + 1.60645i 2.54588 + 2.54588i 1.75482 + 2.03439i −0.598713 + 1.44542i −0.483416 − 2.78681i 0.430672 + 0.430672i −1.59711 − 4.83479i
571.14 −1.26320 + 0.635867i −1.75514 − 0.727004i 1.19135 − 1.60645i −2.54588 − 2.54588i 2.67937 − 0.197687i 0.598713 − 1.44542i −0.483416 + 2.78681i 0.430672 + 0.430672i 4.83479 + 1.59711i
571.15 −1.16202 − 0.806047i −3.10073 − 1.28437i 0.700578 + 1.87328i −2.93685 − 2.93685i 2.56785 + 3.99179i 0.839595 − 2.02696i 0.695868 − 2.74149i 5.84362 + 5.84362i 1.04544 + 5.77992i
571.16 −1.16202 + 0.806047i −3.10073 − 1.28437i 0.700578 − 1.87328i 2.93685 + 2.93685i 4.63837 − 1.00688i −0.839595 + 2.02696i 0.695868 + 2.74149i 5.84362 + 5.84362i −5.77992 − 1.04544i
571.17 −1.14883 − 0.824733i 1.11359 + 0.461262i 0.639630 + 1.89496i 0.493407 + 0.493407i −0.898905 − 1.44832i −1.77984 + 4.29692i 0.828009 − 2.70451i −1.09401 − 1.09401i −0.159913 − 0.973771i
571.18 −1.14883 + 0.824733i 1.11359 + 0.461262i 0.639630 − 1.89496i −0.493407 − 0.493407i −1.65974 + 0.388498i 1.77984 − 4.29692i 0.828009 + 2.70451i −1.09401 − 1.09401i 0.973771 + 0.159913i
571.19 −1.03480 − 0.963941i 0.523260 + 0.216742i 0.141637 + 1.99498i 1.45546 + 1.45546i −0.332546 − 0.728677i 0.678897 − 1.63900i 1.77647 − 2.20094i −1.89450 − 1.89450i −0.103138 − 2.90908i
571.20 −1.03480 + 0.963941i 0.523260 + 0.216742i 0.141637 − 1.99498i −1.45546 − 1.45546i −0.750398 + 0.280107i −0.678897 + 1.63900i 1.77647 + 2.20094i −1.89450 − 1.89450i 2.90908 + 0.103138i
See next 80 embeddings (of 344 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 571.86
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
89.d odd 8 1 inner
712.m even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 712.2.m.c ✓ 344
8.d odd 2 1 inner 712.2.m.c ✓ 344
89.d odd 8 1 inner 712.2.m.c ✓ 344
712.m even 8 1 inner 712.2.m.c ✓ 344
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
712.2.m.c ✓ 344 1.a even 1 1 trivial
712.2.m.c ✓ 344 8.d odd 2 1 inner
712.2.m.c ✓ 344 89.d odd 8 1 inner
712.2.m.c ✓ 344 712.m even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{172} + 8 T_{3}^{169} - 72 T_{3}^{167} + 72 T_{3}^{166} + 41056 T_{3}^{164} + \cdots + 96\!\cdots\!68 \) acting on \(S_{2}^{\mathrm{new}}(712, [\chi])\). Copy content Toggle raw display