Properties

Label 273.1.o.a
Level $273$
Weight $1$
Character orbit 273.o
Analytic conductor $0.136$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [273,1,Mod(83,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.83");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 273.o (of order \(4\), 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: \(0.136244748449\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.968877.1
Artin image: $C_4\wr C_2$
Artin field: Galois closure of 8.0.8719893.1

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + i q^{3} + i q^{4} - i q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} + i q^{4} - i q^{7} - q^{9} - q^{12} + i q^{13} - q^{16} + ( - i + 1) q^{19} + q^{21} - i q^{25} - i q^{27} + q^{28} + ( - i + 1) q^{31} - i q^{36} + (i - 1) q^{37} - q^{39} - i q^{48} - q^{49} - q^{52} + (i + 1) q^{57} + i q^{61} + i q^{63} - i q^{64} + ( - i - 1) q^{67} + ( - i - 1) q^{73} + q^{75} + (i + 1) q^{76} + q^{81} + i q^{84} + q^{91} + (i + 1) q^{93} + (i - 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} - 2 q^{12} - 2 q^{16} + 2 q^{19} + 2 q^{21} + 2 q^{28} + 2 q^{31} - 2 q^{37} - 2 q^{39} - 2 q^{49} - 2 q^{52} + 2 q^{57} - 2 q^{67} - 2 q^{73} + 2 q^{75} + 2 q^{76} + 2 q^{81} + 2 q^{91} + 2 q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(-1\) \(i\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1
1.00000i
1.00000i
0 1.00000i 1.00000i 0 0 1.00000i 0 −1.00000 0
125.1 0 1.00000i 1.00000i 0 0 1.00000i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
91.i even 4 1 inner
273.o odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.1.o.a 2
3.b odd 2 1 CM 273.1.o.a 2
7.b odd 2 1 273.1.o.b yes 2
7.c even 3 2 1911.1.cc.b 4
7.d odd 6 2 1911.1.cc.a 4
13.b even 2 1 3549.1.o.b 2
13.c even 3 2 3549.1.ca.b 4
13.d odd 4 1 273.1.o.b yes 2
13.d odd 4 1 3549.1.o.a 2
13.e even 6 2 3549.1.ca.c 4
13.f odd 12 2 3549.1.ca.a 4
13.f odd 12 2 3549.1.ca.d 4
21.c even 2 1 273.1.o.b yes 2
21.g even 6 2 1911.1.cc.a 4
21.h odd 6 2 1911.1.cc.b 4
39.d odd 2 1 3549.1.o.b 2
39.f even 4 1 273.1.o.b yes 2
39.f even 4 1 3549.1.o.a 2
39.h odd 6 2 3549.1.ca.c 4
39.i odd 6 2 3549.1.ca.b 4
39.k even 12 2 3549.1.ca.a 4
39.k even 12 2 3549.1.ca.d 4
91.b odd 2 1 3549.1.o.a 2
91.i even 4 1 inner 273.1.o.a 2
91.i even 4 1 3549.1.o.b 2
91.n odd 6 2 3549.1.ca.a 4
91.t odd 6 2 3549.1.ca.d 4
91.z odd 12 2 1911.1.cc.a 4
91.bb even 12 2 1911.1.cc.b 4
91.bc even 12 2 3549.1.ca.b 4
91.bc even 12 2 3549.1.ca.c 4
273.g even 2 1 3549.1.o.a 2
273.o odd 4 1 inner 273.1.o.a 2
273.o odd 4 1 3549.1.o.b 2
273.u even 6 2 3549.1.ca.d 4
273.bn even 6 2 3549.1.ca.a 4
273.ca odd 12 2 3549.1.ca.b 4
273.ca odd 12 2 3549.1.ca.c 4
273.cb odd 12 2 1911.1.cc.b 4
273.cd even 12 2 1911.1.cc.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.1.o.a 2 1.a even 1 1 trivial
273.1.o.a 2 3.b odd 2 1 CM
273.1.o.a 2 91.i even 4 1 inner
273.1.o.a 2 273.o odd 4 1 inner
273.1.o.b yes 2 7.b odd 2 1
273.1.o.b yes 2 13.d odd 4 1
273.1.o.b yes 2 21.c even 2 1
273.1.o.b yes 2 39.f even 4 1
1911.1.cc.a 4 7.d odd 6 2
1911.1.cc.a 4 21.g even 6 2
1911.1.cc.a 4 91.z odd 12 2
1911.1.cc.a 4 273.cd even 12 2
1911.1.cc.b 4 7.c even 3 2
1911.1.cc.b 4 21.h odd 6 2
1911.1.cc.b 4 91.bb even 12 2
1911.1.cc.b 4 273.cb odd 12 2
3549.1.o.a 2 13.d odd 4 1
3549.1.o.a 2 39.f even 4 1
3549.1.o.a 2 91.b odd 2 1
3549.1.o.a 2 273.g even 2 1
3549.1.o.b 2 13.b even 2 1
3549.1.o.b 2 39.d odd 2 1
3549.1.o.b 2 91.i even 4 1
3549.1.o.b 2 273.o odd 4 1
3549.1.ca.a 4 13.f odd 12 2
3549.1.ca.a 4 39.k even 12 2
3549.1.ca.a 4 91.n odd 6 2
3549.1.ca.a 4 273.bn even 6 2
3549.1.ca.b 4 13.c even 3 2
3549.1.ca.b 4 39.i odd 6 2
3549.1.ca.b 4 91.bc even 12 2
3549.1.ca.b 4 273.ca odd 12 2
3549.1.ca.c 4 13.e even 6 2
3549.1.ca.c 4 39.h odd 6 2
3549.1.ca.c 4 91.bc even 12 2
3549.1.ca.c 4 273.ca odd 12 2
3549.1.ca.d 4 13.f odd 12 2
3549.1.ca.d 4 39.k even 12 2
3549.1.ca.d 4 91.t odd 6 2
3549.1.ca.d 4 273.u even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{19}^{2} - 2T_{19} + 2 \) acting on \(S_{1}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$41$ \( T^{2} \) 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
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
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