Properties

Label 1098.2.d.a
Level $1098$
Weight $2$
Character orbit 1098.d
Analytic conductor $8.768$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1098,2,Mod(487,1098)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1098, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1098.487");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1098 = 2 \cdot 3^{2} \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1098.d (of order \(2\), degree \(1\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 122)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - i q^{2} - q^{4} + 3 q^{5} - 3 i q^{7} + i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - q^{4} + 3 q^{5} - 3 i q^{7} + i q^{8} - 3 i q^{10} - 3 i q^{11} - 5 q^{13} - 3 q^{14} + q^{16} - 6 i q^{17} - 2 q^{19} - 3 q^{20} - 3 q^{22} + 9 i q^{23} + 4 q^{25} + 5 i q^{26} + 3 i q^{28} - 6 i q^{29} - i q^{32} - 6 q^{34} - 9 i q^{35} - 6 i q^{37} + 2 i q^{38} + 3 i q^{40} + 9 q^{41} + 3 i q^{44} + 9 q^{46} - 2 q^{49} - 4 i q^{50} + 5 q^{52} - 9 i q^{55} + 3 q^{56} - 6 q^{58} + 3 i q^{59} + ( - 6 i - 5) q^{61} - q^{64} - 15 q^{65} - 3 i q^{67} + 6 i q^{68} - 9 q^{70} - 7 q^{73} - 6 q^{74} + 2 q^{76} - 9 q^{77} + 15 i q^{79} + 3 q^{80} - 9 i q^{82} + 12 q^{83} - 18 i q^{85} + 3 q^{88} + 15 i q^{91} - 9 i q^{92} - 6 q^{95} + 10 q^{97} + 2 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 6 q^{5} - 10 q^{13} - 6 q^{14} + 2 q^{16} - 4 q^{19} - 6 q^{20} - 6 q^{22} + 8 q^{25} - 12 q^{34} + 18 q^{41} + 18 q^{46} - 4 q^{49} + 10 q^{52} + 6 q^{56} - 12 q^{58} - 10 q^{61} - 2 q^{64} - 30 q^{65} - 18 q^{70} - 14 q^{73} - 12 q^{74} + 4 q^{76} - 18 q^{77} + 6 q^{80} + 24 q^{83} + 6 q^{88} - 12 q^{95} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(245\) \(307\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
487.1
1.00000i
1.00000i
1.00000i 0 −1.00000 3.00000 0 3.00000i 1.00000i 0 3.00000i
487.2 1.00000i 0 −1.00000 3.00000 0 3.00000i 1.00000i 0 3.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1098.2.d.a 2
3.b odd 2 1 122.2.b.a 2
12.b even 2 1 976.2.h.a 2
15.d odd 2 1 3050.2.d.c 2
15.e even 4 1 3050.2.c.b 2
15.e even 4 1 3050.2.c.c 2
61.b even 2 1 inner 1098.2.d.a 2
183.d odd 2 1 122.2.b.a 2
183.g even 4 1 7442.2.a.a 1
183.g even 4 1 7442.2.a.c 1
732.e even 2 1 976.2.h.a 2
915.e odd 2 1 3050.2.d.c 2
915.s even 4 1 3050.2.c.b 2
915.s even 4 1 3050.2.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
122.2.b.a 2 3.b odd 2 1
122.2.b.a 2 183.d odd 2 1
976.2.h.a 2 12.b even 2 1
976.2.h.a 2 732.e even 2 1
1098.2.d.a 2 1.a even 1 1 trivial
1098.2.d.a 2 61.b even 2 1 inner
3050.2.c.b 2 15.e even 4 1
3050.2.c.b 2 915.s even 4 1
3050.2.c.c 2 15.e even 4 1
3050.2.c.c 2 915.s even 4 1
3050.2.d.c 2 15.d odd 2 1
3050.2.d.c 2 915.e odd 2 1
7442.2.a.a 1 183.g even 4 1
7442.2.a.c 1 183.g even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 3 \) acting on \(S_{2}^{\mathrm{new}}(1098, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 9 \) Copy content Toggle raw display
$11$ \( T^{2} + 9 \) Copy content Toggle raw display
$13$ \( (T + 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 81 \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T - 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 9 \) Copy content Toggle raw display
$61$ \( T^{2} + 10T + 61 \) Copy content Toggle raw display
$67$ \( T^{2} + 9 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 7)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 225 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 10)^{2} \) Copy content Toggle raw display
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