Properties

Label 1098.2.k.b
Level $1098$
Weight $2$
Character orbit 1098.k
Analytic conductor $8.768$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1098,2,Mod(217,1098)]
 
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, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1098.217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1098 = 2 \cdot 3^{2} \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1098.k (of order \(5\), 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: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \cdots - 1) q^{2}+ \cdots + \zeta_{10}^{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \cdots - 1) q^{2}+ \cdots + (3 \zeta_{10}^{2} + 2 \zeta_{10} + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - q^{4} - q^{5} + q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - q^{4} - q^{5} + q^{7} - q^{8} - q^{10} - 12 q^{13} + 6 q^{14} - q^{16} + 6 q^{17} + 3 q^{19} - q^{20} + 5 q^{22} - 7 q^{23} + 4 q^{25} + 3 q^{26} + q^{28} - 12 q^{29} + 5 q^{31} + 4 q^{32} + 6 q^{34} + q^{35} + 20 q^{37} + 8 q^{38} + 4 q^{40} + 7 q^{41} + 13 q^{43} - 5 q^{44} - 7 q^{46} + 38 q^{47} - 4 q^{49} - 16 q^{50} + 3 q^{52} + 17 q^{53} - 5 q^{55} - 4 q^{56} + 13 q^{58} - 11 q^{59} - 29 q^{61} + 20 q^{62} - q^{64} + 3 q^{65} - 12 q^{67} + 6 q^{68} - 4 q^{70} + 20 q^{71} - q^{73} - 10 q^{74} + 8 q^{76} + 5 q^{77} + 8 q^{79} - q^{80} - 8 q^{82} - 12 q^{83} + 6 q^{85} - 12 q^{86} - 5 q^{88} + 5 q^{89} - 3 q^{91} + 13 q^{92} - 7 q^{94} + 8 q^{95} - 12 q^{97} + 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1098\mathbb{Z}\right)^\times\).

\(n\) \(245\) \(307\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 + 0.951057i
−0.309017 0.951057i
−0.809017 + 0.587785i 0 0.309017 0.951057i 0.309017 0.951057i 0 −0.309017 0.224514i 0.309017 + 0.951057i 0 0.309017 + 0.951057i
253.1 −0.809017 0.587785i 0 0.309017 + 0.951057i 0.309017 + 0.951057i 0 −0.309017 + 0.224514i 0.309017 0.951057i 0 0.309017 0.951057i
325.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −0.809017 + 0.587785i 0 0.809017 2.48990i −0.809017 0.587785i 0 −0.809017 0.587785i
973.1 0.309017 0.951057i 0 −0.809017 0.587785i −0.809017 0.587785i 0 0.809017 + 2.48990i −0.809017 + 0.587785i 0 −0.809017 + 0.587785i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.e even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1098.2.k.b 4
3.b odd 2 1 1098.2.k.d yes 4
61.e even 5 1 inner 1098.2.k.b 4
183.n odd 10 1 1098.2.k.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1098.2.k.b 4 1.a even 1 1 trivial
1098.2.k.b 4 61.e even 5 1 inner
1098.2.k.d yes 4 3.b odd 2 1
1098.2.k.d yes 4 183.n odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1098, [\chi])\):

\( T_{5}^{4} + T_{5}^{3} + T_{5}^{2} + T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - T_{7}^{3} + 6T_{7}^{2} + 4T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} + 6 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$13$ \( (T + 3)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( T^{4} - 3 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$23$ \( T^{4} + 7 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T - 11)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$37$ \( T^{4} - 20 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$41$ \( T^{4} - 7 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{4} - 13 T^{3} + \cdots + 961 \) Copy content Toggle raw display
$47$ \( (T^{2} - 19 T + 89)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 17 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$59$ \( T^{4} + 11 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$61$ \( T^{4} + 29 T^{3} + \cdots + 3721 \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$71$ \( T^{4} - 20 T^{3} + \cdots + 9025 \) Copy content Toggle raw display
$73$ \( T^{4} + T^{3} + \cdots + 961 \) Copy content Toggle raw display
$79$ \( T^{4} - 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$83$ \( T^{4} + 12 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( T^{4} - 5 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$97$ \( T^{4} + 12 T^{3} + \cdots + 20736 \) Copy content Toggle raw display
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