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Properties

Label 712.2.ba.b
Level $712$
Weight $2$
Character orbit 712.ba
Analytic conductor $5.685$
Analytic rank $0$
Dimension $240$
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] = [712,2,Mod(9,712)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(712, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("712.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 712 = 2^{3} \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 712.ba (of order \(44\), degree \(20\), 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: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 2 q^{3} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 2 q^{3} + 4 q^{7} + 46 q^{13} + 4 q^{15} + 20 q^{19} - 38 q^{23} + 32 q^{25} + 10 q^{27} + 10 q^{29} + 4 q^{31} - 2 q^{33} + 54 q^{35} + 22 q^{37} - 24 q^{39} - 82 q^{41} - 4 q^{43} - 16 q^{45} - 88 q^{49} - 8 q^{51} + 44 q^{53} + 4 q^{57} - 60 q^{59} + 14 q^{61} + 228 q^{63} - 6 q^{65} - 132 q^{69} - 44 q^{71} - 4 q^{73} + 20 q^{75} - 176 q^{77} + 44 q^{81} + 20 q^{83} + 118 q^{85} + 40 q^{87} + 42 q^{89} - 4 q^{91} - 88 q^{93} - 8 q^{95} - 52 q^{97} + 88 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −3.17905 − 0.227370i 0 1.09967 − 3.74515i 0 1.93952 − 1.05906i 0 7.08518 + 1.01870i 0
9.2 0 −2.74867 − 0.196589i 0 −0.140771 + 0.479423i 0 −4.15284 + 2.26762i 0 4.54708 + 0.653771i 0
9.3 0 −2.58148 − 0.184631i 0 −0.801177 + 2.72856i 0 3.89883 − 2.12892i 0 3.66050 + 0.526300i 0
9.4 0 −1.15485 − 0.0825967i 0 −0.114048 + 0.388412i 0 0.210131 − 0.114740i 0 −1.64260 − 0.236170i 0
9.5 0 −0.851361 − 0.0608906i 0 0.195043 − 0.664254i 0 1.33684 − 0.729969i 0 −2.24836 − 0.323265i 0
9.6 0 0.456130 + 0.0326231i 0 −1.14462 + 3.89823i 0 −2.50868 + 1.36984i 0 −2.76247 − 0.397184i 0
9.7 0 1.00001 + 0.0715218i 0 0.971091 − 3.30723i 0 −4.45888 + 2.43473i 0 −1.97457 − 0.283900i 0
9.8 0 1.08208 + 0.0773917i 0 −0.475857 + 1.62062i 0 0.230286 − 0.125746i 0 −1.80456 − 0.259457i 0
9.9 0 1.46765 + 0.104969i 0 1.04507 − 3.55918i 0 3.98877 − 2.17803i 0 −0.826477 − 0.118829i 0
9.10 0 1.85536 + 0.132698i 0 0.0730555 − 0.248804i 0 −0.231138 + 0.126211i 0 0.455305 + 0.0654630i 0
9.11 0 3.09300 + 0.221216i 0 −1.03107 + 3.51150i 0 3.42514 − 1.87027i 0 6.54827 + 0.941500i 0
9.12 0 3.31206 + 0.236884i 0 0.409539 − 1.39476i 0 −1.66016 + 0.906517i 0 7.94420 + 1.14220i 0
17.1 0 −3.24327 − 0.705530i 0 −1.22855 + 1.06454i 0 −0.166651 + 2.33009i 0 7.29211 + 3.33020i 0
17.2 0 −2.11996 − 0.461170i 0 2.10029 − 1.81991i 0 0.300787 − 4.20555i 0 1.55267 + 0.709082i 0
17.3 0 −1.99108 − 0.433133i 0 0.626230 − 0.542631i 0 0.0198277 − 0.277228i 0 1.04791 + 0.478564i 0
17.4 0 −1.44311 − 0.313929i 0 −3.31488 + 2.87236i 0 0.264200 − 3.69399i 0 −0.744886 − 0.340178i 0
17.5 0 −0.928451 − 0.201972i 0 2.77013 − 2.40033i 0 −0.330614 + 4.62259i 0 −1.90767 − 0.871203i 0
17.6 0 −0.725500 − 0.157823i 0 0.203523 − 0.176354i 0 −0.242717 + 3.39362i 0 −2.22745 − 1.01724i 0
17.7 0 0.427907 + 0.0930855i 0 0.133063 − 0.115300i 0 0.155892 − 2.17965i 0 −2.55446 − 1.16658i 0
17.8 0 0.502136 + 0.109233i 0 −1.85558 + 1.60787i 0 −0.143867 + 2.01152i 0 −2.48869 − 1.13655i 0
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
89.g even 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 712.2.ba.b ✓ 240
89.g even 44 1 inner 712.2.ba.b ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
712.2.ba.b ✓ 240 1.a even 1 1 trivial
712.2.ba.b ✓ 240 89.g even 44 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{240} + 2 T_{3}^{239} + 2 T_{3}^{238} - 6 T_{3}^{237} - 178 T_{3}^{236} - 388 T_{3}^{235} + \cdots + 22\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(712, [\chi])\). Copy content Toggle raw display