Properties

Label 847.2.i.a
Level $847$
Weight $2$
Character orbit 847.i
Analytic conductor $6.763$
Analytic rank $0$
Dimension $24$
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] = [847,2,Mod(241,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.241");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.i (of order \(6\), degree \(2\), 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: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{4} + 8 q^{9} + 60 q^{12} - 44 q^{14} + 32 q^{15} - 8 q^{16} + 8 q^{23} + 4 q^{25} + 24 q^{31} + 24 q^{36} + 8 q^{37} + 12 q^{38} - 28 q^{42} - 72 q^{45} + 84 q^{47} + 36 q^{49} - 20 q^{53} - 68 q^{56} + 64 q^{58} - 48 q^{59} + 24 q^{60} + 48 q^{64} - 20 q^{67} - 60 q^{70} + 8 q^{71} - 96 q^{75} - 176 q^{78} - 84 q^{80} - 36 q^{81} + 12 q^{82} - 64 q^{86} + 60 q^{89} - 28 q^{91} - 160 q^{92} + 36 q^{93}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1 −2.13688 + 1.23373i 2.50488 + 1.44619i 2.04417 3.54061i 1.33074 0.768306i −7.13683 2.50945 + 0.838250i 5.15289i 2.68294 + 4.64699i −1.89576 + 3.28355i
241.2 −2.03242 + 1.17342i 0.219729 + 0.126861i 1.75383 3.03772i 0.452529 0.261268i −0.595443 2.16845 + 1.51586i 3.53824i −1.46781 2.54233i −0.613154 + 1.06201i
241.3 −1.38255 + 0.798214i −0.877650 0.506712i 0.274290 0.475085i −0.282779 + 0.163263i 1.61786 1.21424 2.35066i 2.31709i −0.986487 1.70864i 0.260637 0.451436i
241.4 −1.01511 + 0.586075i 1.45404 + 0.839493i −0.313032 + 0.542188i −1.91629 + 1.10637i −1.96802 −2.45295 0.991472i 3.07814i −0.0905026 0.156755i 1.29683 2.24618i
241.5 −0.589336 + 0.340253i −2.61084 1.50737i −0.768455 + 1.33100i −2.80754 + 1.62094i 2.05155 2.64379 0.101846i 2.40689i 3.04431 + 5.27290i 1.10306 1.91055i
241.6 −0.117445 + 0.0678066i −0.690164 0.398467i −0.990805 + 1.71612i 3.22334 1.86100i 0.108075 −0.139659 + 2.64206i 0.539959i −1.18245 2.04806i −0.252376 + 0.437127i
241.7 0.117445 0.0678066i −0.690164 0.398467i −0.990805 + 1.71612i 3.22334 1.86100i −0.108075 0.139659 2.64206i 0.539959i −1.18245 2.04806i 0.252376 0.437127i
241.8 0.589336 0.340253i −2.61084 1.50737i −0.768455 + 1.33100i −2.80754 + 1.62094i −2.05155 −2.64379 + 0.101846i 2.40689i 3.04431 + 5.27290i −1.10306 + 1.91055i
241.9 1.01511 0.586075i 1.45404 + 0.839493i −0.313032 + 0.542188i −1.91629 + 1.10637i 1.96802 2.45295 + 0.991472i 3.07814i −0.0905026 0.156755i −1.29683 + 2.24618i
241.10 1.38255 0.798214i −0.877650 0.506712i 0.274290 0.475085i −0.282779 + 0.163263i −1.61786 −1.21424 + 2.35066i 2.31709i −0.986487 1.70864i −0.260637 + 0.451436i
241.11 2.03242 1.17342i 0.219729 + 0.126861i 1.75383 3.03772i 0.452529 0.261268i 0.595443 −2.16845 1.51586i 3.53824i −1.46781 2.54233i 0.613154 1.06201i
241.12 2.13688 1.23373i 2.50488 + 1.44619i 2.04417 3.54061i 1.33074 0.768306i 7.13683 −2.50945 0.838250i 5.15289i 2.68294 + 4.64699i 1.89576 3.28355i
362.1 −2.13688 1.23373i 2.50488 1.44619i 2.04417 + 3.54061i 1.33074 + 0.768306i −7.13683 2.50945 0.838250i 5.15289i 2.68294 4.64699i −1.89576 3.28355i
362.2 −2.03242 1.17342i 0.219729 0.126861i 1.75383 + 3.03772i 0.452529 + 0.261268i −0.595443 2.16845 1.51586i 3.53824i −1.46781 + 2.54233i −0.613154 1.06201i
362.3 −1.38255 0.798214i −0.877650 + 0.506712i 0.274290 + 0.475085i −0.282779 0.163263i 1.61786 1.21424 + 2.35066i 2.31709i −0.986487 + 1.70864i 0.260637 + 0.451436i
362.4 −1.01511 0.586075i 1.45404 0.839493i −0.313032 0.542188i −1.91629 1.10637i −1.96802 −2.45295 + 0.991472i 3.07814i −0.0905026 + 0.156755i 1.29683 + 2.24618i
362.5 −0.589336 0.340253i −2.61084 + 1.50737i −0.768455 1.33100i −2.80754 1.62094i 2.05155 2.64379 + 0.101846i 2.40689i 3.04431 5.27290i 1.10306 + 1.91055i
362.6 −0.117445 0.0678066i −0.690164 + 0.398467i −0.990805 1.71612i 3.22334 + 1.86100i 0.108075 −0.139659 2.64206i 0.539959i −1.18245 + 2.04806i −0.252376 0.437127i
362.7 0.117445 + 0.0678066i −0.690164 + 0.398467i −0.990805 1.71612i 3.22334 + 1.86100i −0.108075 0.139659 + 2.64206i 0.539959i −1.18245 + 2.04806i 0.252376 + 0.437127i
362.8 0.589336 + 0.340253i −2.61084 + 1.50737i −0.768455 1.33100i −2.80754 1.62094i −2.05155 −2.64379 0.101846i 2.40689i 3.04431 5.27290i −1.10306 1.91055i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 241.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.b odd 2 1 inner
77.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.i.a 24
7.d odd 6 1 inner 847.2.i.a 24
11.b odd 2 1 inner 847.2.i.a 24
11.c even 5 4 847.2.r.e 96
11.d odd 10 4 847.2.r.e 96
77.i even 6 1 inner 847.2.i.a 24
77.n even 30 4 847.2.r.e 96
77.p odd 30 4 847.2.r.e 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.i.a 24 1.a even 1 1 trivial
847.2.i.a 24 7.d odd 6 1 inner
847.2.i.a 24 11.b odd 2 1 inner
847.2.i.a 24 77.i even 6 1 inner
847.2.r.e 96 11.c even 5 4
847.2.r.e 96 11.d odd 10 4
847.2.r.e 96 77.n even 30 4
847.2.r.e 96 77.p odd 30 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 16 T_{2}^{22} + 166 T_{2}^{20} - 1016 T_{2}^{18} + 4507 T_{2}^{16} - 12706 T_{2}^{14} + 25928 T_{2}^{12} - 32188 T_{2}^{10} + 28015 T_{2}^{8} - 11234 T_{2}^{6} + 3163 T_{2}^{4} - 58 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display