↑

Properties

Label 1098.2.x.c
Level $1098$
Weight $2$
Character orbit 1098.x
Analytic conductor $8.768$
Analytic rank $0$
Dimension $24$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1098,2,Mod(163,1098)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1098, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1098.163");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1098 = 2 \cdot 3^{2} \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1098.x (of order \(10\), 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: \(8.76757414194\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 122)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{4} - 2 q^{5} - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{4} - 2 q^{5} - 20 q^{7} - 10 q^{10} + 4 q^{13} - 16 q^{14} - 6 q^{16} + 10 q^{17} + 14 q^{19} - 8 q^{20} - 4 q^{22} - 22 q^{25} - 20 q^{28} + 10 q^{31} + 4 q^{34} - 10 q^{38} + 20 q^{41} - 20 q^{43} - 24 q^{46} - 48 q^{47} + 2 q^{49} - 4 q^{52} - 20 q^{53} + 30 q^{55} - 4 q^{56} + 28 q^{58} + 36 q^{61} + 40 q^{62} + 6 q^{64} + 46 q^{65} + 90 q^{67} - 10 q^{68} - 70 q^{71} + 8 q^{73} + 60 q^{74} + 26 q^{76} + 28 q^{77} - 20 q^{79} + 8 q^{80} + 42 q^{83} - 90 q^{85} - 24 q^{86} - 6 q^{88} - 70 q^{91} - 10 q^{92} - 20 q^{94} + 82 q^{95} - 72 q^{97} + 20 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1 −0.951057 + 0.309017i 0 0.809017 − 0.587785i −0.369267 + 0.268288i 0 −0.566575 − 0.184091i −0.587785 + 0.809017i 0 0.268288 − 0.369267i
163.2 −0.951057 + 0.309017i 0 0.809017 − 0.587785i 0.159756 − 0.116069i 0 −3.32270 − 1.07961i −0.587785 + 0.809017i 0 −0.116069 + 0.159756i
163.3 −0.951057 + 0.309017i 0 0.809017 − 0.587785i 1.46958 − 1.06772i 0 2.17336 + 0.706167i −0.587785 + 0.809017i 0 −1.06772 + 1.46958i
163.4 0.951057 − 0.309017i 0 0.809017 − 0.587785i 3.45228 − 2.50823i 0 −1.20496 − 0.391514i 0.587785 − 0.809017i 0 2.50823 − 3.45228i
163.5 0.951057 − 0.309017i 0 0.809017 − 0.587785i −1.06206 + 0.771633i 0 −4.15148 − 1.34890i 0.587785 − 0.809017i 0 −0.771633 + 1.06206i
163.6 0.951057 − 0.309017i 0 0.809017 − 0.587785i −3.03225 + 2.20306i 0 −0.163711 − 0.0531928i 0.587785 − 0.809017i 0 −2.20306 + 3.03225i
235.1 −0.587785 + 0.809017i 0 −0.309017 − 0.951057i 0.0330394 + 0.101685i 0 −1.46272 − 2.01327i 0.951057 + 0.309017i 0 −0.101685 − 0.0330394i
235.2 −0.587785 + 0.809017i 0 −0.309017 − 0.951057i 0.675165 + 2.07794i 0 0.778900 + 1.07206i 0.951057 + 0.309017i 0 −2.07794 − 0.675165i
235.3 −0.587785 + 0.809017i 0 −0.309017 − 0.951057i −0.929436 − 2.86051i 0 0.477429 + 0.657124i 0.951057 + 0.309017i 0 2.86051 + 0.929436i
235.4 0.587785 − 0.809017i 0 −0.309017 − 0.951057i −1.03010 − 3.17033i 0 −2.53391 − 3.48763i −0.951057 − 0.309017i 0 −3.17033 − 1.03010i
235.5 0.587785 − 0.809017i 0 −0.309017 − 0.951057i −1.18048 − 3.63316i 0 0.623192 + 0.857750i −0.951057 − 0.309017i 0 −3.63316 − 1.18048i
235.6 0.587785 − 0.809017i 0 −0.309017 − 0.951057i 0.813785 + 2.50457i 0 −0.646819 − 0.890270i −0.951057 − 0.309017i 0 2.50457 + 0.813785i
271.1 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i 0.0330394 − 0.101685i 0 −1.46272 + 2.01327i 0.951057 − 0.309017i 0 −0.101685 + 0.0330394i
271.2 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i 0.675165 − 2.07794i 0 0.778900 − 1.07206i 0.951057 − 0.309017i 0 −2.07794 + 0.675165i
271.3 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i −0.929436 + 2.86051i 0 0.477429 − 0.657124i 0.951057 − 0.309017i 0 2.86051 − 0.929436i
271.4 0.587785 + 0.809017i 0 −0.309017 + 0.951057i −1.03010 + 3.17033i 0 −2.53391 + 3.48763i −0.951057 + 0.309017i 0 −3.17033 + 1.03010i
271.5 0.587785 + 0.809017i 0 −0.309017 + 0.951057i −1.18048 + 3.63316i 0 0.623192 − 0.857750i −0.951057 + 0.309017i 0 −3.63316 + 1.18048i
271.6 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 0.813785 − 2.50457i 0 −0.646819 + 0.890270i −0.951057 + 0.309017i 0 2.50457 − 0.813785i
613.1 −0.951057 − 0.309017i 0 0.809017 + 0.587785i 0.159756 + 0.116069i 0 −3.32270 + 1.07961i −0.587785 − 0.809017i 0 −0.116069 − 0.159756i
613.2 −0.951057 − 0.309017i 0 0.809017 + 0.587785i −0.369267 − 0.268288i 0 −0.566575 + 0.184091i −0.587785 − 0.809017i 0 0.268288 + 0.369267i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.g even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1098.2.x.c 24
3.b odd 2 1 122.2.g.a ✓ 24
12.b even 2 1 976.2.bd.c 24
61.g even 10 1 inner 1098.2.x.c 24
183.l odd 10 1 122.2.g.a ✓ 24
183.r even 20 1 7442.2.a.t 12
183.r even 20 1 7442.2.a.v 12
732.y even 10 1 976.2.bd.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
122.2.g.a ✓ 24 3.b odd 2 1
122.2.g.a ✓ 24 183.l odd 10 1
976.2.bd.c 24 12.b even 2 1
976.2.bd.c 24 732.y even 10 1
1098.2.x.c 24 1.a even 1 1 trivial
1098.2.x.c 24 61.g even 10 1 inner
7442.2.a.t 12 183.r even 20 1
7442.2.a.v 12 183.r even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 2 T_{5}^{23} + 28 T_{5}^{22} + 10 T_{5}^{21} + 448 T_{5}^{20} + 580 T_{5}^{19} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(1098, [\chi])\). Copy content Toggle raw display