Properties

Label 1155.1.e.b
Level $1155$
Weight $1$
Character orbit 1155.e
Analytic conductor $0.576$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -35, -231, 165
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1155,1,Mod(1154,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1155.1154");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1155.e (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: \(0.576420089591\)
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: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-35}, \sqrt{165})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.1634180625.2

$q$-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.

\(f(q)\) \(=\) \( q - i q^{3} + q^{4} - i q^{5} - i q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{3} + q^{4} - i q^{5} - i q^{7} - q^{9} + q^{11} - i q^{12} + i q^{13} - q^{15} + q^{16} - i q^{20} - q^{21} - q^{25} + i q^{27} - i q^{28} - q^{29} - i q^{33} - q^{35} - q^{36} + 2 q^{39} + q^{44} + i q^{45} + i q^{47} - i q^{48} - q^{49} + 2 i q^{52} - i q^{55} - q^{60} + i q^{63} + q^{64} + 2 q^{65} - i q^{73} + i q^{75} - i q^{77} - i q^{80} + q^{81} - q^{84} + 2 i q^{87} + 2 q^{91} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 2 q^{9} + 2 q^{11} - 2 q^{15} + 2 q^{16} - 2 q^{21} - 2 q^{25} - 4 q^{29} - 2 q^{35} - 2 q^{36} + 4 q^{39} + 2 q^{44} - 2 q^{49} - 2 q^{60} + 2 q^{64} + 4 q^{65} + 2 q^{81} - 2 q^{84} + 4 q^{91} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1154.1
1.00000i
1.00000i
0 1.00000i 1.00000 1.00000i 0 1.00000i 0 −1.00000 0
1154.2 0 1.00000i 1.00000 1.00000i 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
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
165.d even 2 1 RM by \(\Q(\sqrt{165}) \)
231.h odd 2 1 CM by \(\Q(\sqrt{-231}) \)
5.b even 2 1 inner
7.b odd 2 1 inner
33.d even 2 1 inner
1155.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.1.e.b yes 2
3.b odd 2 1 1155.1.e.a 2
5.b even 2 1 inner 1155.1.e.b yes 2
7.b odd 2 1 inner 1155.1.e.b yes 2
11.b odd 2 1 1155.1.e.a 2
15.d odd 2 1 1155.1.e.a 2
21.c even 2 1 1155.1.e.a 2
33.d even 2 1 inner 1155.1.e.b yes 2
35.c odd 2 1 CM 1155.1.e.b yes 2
55.d odd 2 1 1155.1.e.a 2
77.b even 2 1 1155.1.e.a 2
105.g even 2 1 1155.1.e.a 2
165.d even 2 1 RM 1155.1.e.b yes 2
231.h odd 2 1 CM 1155.1.e.b yes 2
385.h even 2 1 1155.1.e.a 2
1155.e odd 2 1 inner 1155.1.e.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.1.e.a 2 3.b odd 2 1
1155.1.e.a 2 11.b odd 2 1
1155.1.e.a 2 15.d odd 2 1
1155.1.e.a 2 21.c even 2 1
1155.1.e.a 2 55.d odd 2 1
1155.1.e.a 2 77.b even 2 1
1155.1.e.a 2 105.g even 2 1
1155.1.e.a 2 385.h even 2 1
1155.1.e.b yes 2 1.a even 1 1 trivial
1155.1.e.b yes 2 5.b even 2 1 inner
1155.1.e.b yes 2 7.b odd 2 1 inner
1155.1.e.b yes 2 33.d even 2 1 inner
1155.1.e.b yes 2 35.c odd 2 1 CM
1155.1.e.b yes 2 165.d even 2 1 RM
1155.1.e.b yes 2 231.h odd 2 1 CM
1155.1.e.b yes 2 1155.e odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1155, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{29} + 2 \) 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
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 4 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) 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
$97$ \( T^{2} \) Copy content Toggle raw display
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